PLANNING, DESIGN AND SCHEDULING OF FLEX-ROUTE TRANSIT SERVICE

Transcription

1 PLANNING, DESIGN AND SCHEDULING OF FLEX-ROUTE TRANSIT SERVICE BY BAHA W. ALSHALALFAH A thess submtted n conformty wth the requrements for the degree of Doctor of Phlosophy Graduate Department of Cvl Engneerng Unversty of Toronto Copyrght by Baha W. Alshalalfah (2009)

2 Plannng, Desgn and Schedulng of Flex-Route Transt Servce Baha W. Alshalalfah Doctor of Phlosophy Department of Cvl Engneerng Unversty of Toronto 2009 ABSTRACT The rapd expanson of low-densty suburban areas n North Amerca has led to new travel patterns that requre transt servces to be more flexble. Flex-Route transt servce, whch combnes fxed-route transt servce wth elements of demand-responsve transt servce, has emerged as a vable transt opton to address the travel needs of the resdents of these areas. Exstng lterature n ths feld, however, s lmted and lacks any comprehensve analyss of Flex-Route plannng, desgn and schedulng. Ths research ams at explorng Flex-Route transt servce to provde detaled gudelnes for the plannng and desgn of the servce, as well as developng a new schedulng system for ths type of unque servce. Accordngly, the objectves of ths research are: assessng the practcalty of Flex-Route transt servce n servng low-densty suburban areas; dentfyng essental Flex-Route plannng steps and desgn parameters; determnng the feasblty and cost of replacng fxed-route transt wth Flex-Route servce; and developng a Flex-Route-specfc dynamc schedulng system that reles on recent developments n computer and communcaton technologes. In ths regard, we develop an analytcal model that addresses several desgn parameters and provde a detaled analyss that ncludes, among other parameters, fndng optmal values for Flex-Route servce area and slack tme. Furthermore, the analytcal model ncludes a feasblty and cost analyss that estmates the cost ncurred by several stakeholders f Flex-Route servce s chosen to replace fxed-route servce. The core of the schedulng system s a new developed algorthm the Constraned- Inserton Algorthm- that explots the powerful search technques of Constrant Programmng. The schedulng system can handle the daly operatons of Flex-Route transt

3 servces; t accepts daly (or dynamc) nputs and, n mnmal tme, produces very costeffectve and relable schedules. Moreover, the schedulng system has the ablty to be used as smulaton tool to allow transt operators to assess the feasblty and performance of proposed Flex-Route transt servces before mplementaton. The applcablty of the analytcal model as well as the performance of the schedulng system were subsequently evaluated and valdated through process that ncluded testng on a case study n the Cty of Oakvlle, Canada.

4 v ACKNOWLEDGEMENTS A journey that started n occuped Palestne and saw me get through so many ups and downs durng my Masters and PhD studes, now fnds me tryng, n few words, to thank every person who helped me through ths journey. It won t be easy, but I ll try my best. Frst, specal thanks to Prof. Amer Shalaby for hs advce, gudance, and genune support. Hs open-mndedness, dedcaton and humlty are examples to be followed. It was a great honour for me to get to know hm and learn from hm over the past sx years. Many thanks to the members of my examnaton commttee: Prof. Erc Mller, Prof. Baher Abdulha and Prof. Matthew Roorda, for ther valuable comments on my thess and for all ther gudance and support throughout my PhD studes. I am also thankful to Prof. Ngel Wlson for agreeng to be the external examner of my thess, and also for hs valuable feedback and comments. Specal thanks to Han Qaddum Scholarshp Foundaton for ther generous fundng of my Masters studes, whch paved the way for me to get ths PhD. To my brothers and ssters (Sana, Yousef, Mohammed, Alaa and Suhar) and ther famles (especally the lttle ones): I can t thank you enough for all your support and love. You always stood besde me, supported me and beleved n me. I love you all. To my father, who s no longer wth us on ths earth but lves n our hearts: I wsh you had the chance to be wth us n such joyous moments. Last but not least, to the one person whose devoton, perseverance, dedcaton and love has no lmts; the person who taught me that mracles, lke what she dd, could happen; the person who always beleved n me, and wthout her prayers I wouldn t be the person I am today: Mom, I cannot thank you enough for everythng you have done and had to go through to see me get ths degree. To the people of Palestne

5 v TABLE OF CONTENTS ACKNOWLEDGEMENTS IV TABLE OF CONTENTS V LIST OF FIGURES XI LIST OF TABLES XIII 1 INTRODUCTION POINT OF DEPARTURE - MOTIVATING SCENARIO OBJECTIVES THESIS SCOPE LIMITATIONS THESIS ORGANIZATION 11 2 LITERATURE REVIEW CHAPTER OVERVIEW DEMAND-RESPONSIVE TRANSIT FLEXIBLE TRANSIT SERVICES TYPES OF FLEXIBLE TRANSIT SERVICE FLEXIBLE TRANSIT SERVICE BENEFITS FLEX-ROUTE TRANSIT SERVICE FLEX-ROUTE TRANSIT SERVICE IN THE LITERATURE Optmal Slack Tme (Fu, 2002) A Mult-objectve Optmzaton Model Other Research Efforts FLEX-ROUTE TRANSIT SERVICE IN PRACTICE Madson, Wsconsn Potomac and Rappahannock Transportaton Commsson (PRTC), Vrgna System for Advanced Management of Publc Transport Operatons (SAMPO) 33

6 v The Moblty Allowance Shuttle Transt (MAST) ROUTING AND SCHEDULING EFFORTS THE DIAL-A-RIDE PROBLEM (DARP) SCHEDULING THE FLEX-ROUTE PROBLEM DYNAMIC VEHICLE ROUTING AND SCHEDULING Adaptaton of Statc Algorthms Stochastc Methods FLEX-ROUTE TRANSIT SCHEDULING LIMITATIONS OF EXISTING RESEARCH 47 3 FLEX-ROUTE PLANNING AND DESIGN PARAMETERS CHAPTER OVERVIEW SUBURBAN TRANSIT SERVICE: CHALLENGES AND OPPORTUNITIES FLEX-ROUTE SERVICE DESIGN AREA CONFIGURATION Spread-Out Areas Hard-to-Serve Areas SERVICE CHARACTERISTICS Vehcle route Boardng and alghtng locatons Schedule Advance-notce requrements FLEX-ROUTE SERVICE AREA Two-Sded Servce Areas Sngle-sded servce area SLACK TIME Allocaton of Slack Tme among Route Segments Slack Tme for A Sngle Route Segment wth Known Servce Area FIXED STOP LOCATION AND SPACING SERVICE FREQUENCY FLEX-ROUTE FEASIBILITY AND COST ANALYSIS OPERATOR COST Same Frequency (Same Headway) Lower Frequency (Increased Headway) 91

7 v OPERATOR BENEFIT Same Frequency (Same Headway) Lower Frequency (Increased Headway) FIXED-ROUTE USER COST Same Frequency (Same Headway) Lower Frequency (Increased Headway) TOTAL ADDITIONAL SYSTEM COST Same Frequency/ Same Number of Fxed Stops Lower Frequency/ Same Number of Fxed Stops Lower Frequency/ Reduced Number of Fxed Stops TOTAL ADDITIONAL SYSTEM BENEFIT STRATEGIES TO MAINTAIN THE LEVEL OF SERVICE FOR FIXED-ROUTE USERS FLEX-ROUTE PERFORMANCE PASSENGER BOARDINGS PER VEHICLE REVENUE HOUR (PVRH) PASSENGER BOARDINGS PER VEHICLE REVENUE KM (PVRKM) STRATEGIES TO IMPROVE THE PERFORMANCE OF FLEX-ROUTE SERVICE REAL-TIME DECISION-SUPPORT AND SCHEDULING SYSTEM FOR FLEX-ROUTE TRANSIT SERVICES CHAPTER OVERVIEW FLEX-ROUTE ROUTING AND SCHEDULING NEW APPROACH: CONSTRAINT PROGRAMMING ORIGINS OF CONSTRAINT PROGRAMMING FORMULATION CONNECTIONS TO INTEGER PROGRAMMING APPLICATIONS CONSTRAINT PROGRAMMING ALGORITHMS Doman Reducton and Arc Consstency Constrant Propagaton Constrant Programmng Algorthms for Optmzaton Programmng a Search Strategy PROPOSED DECISION-SUPPORT SYSTEM ARCHITECTURE TRIP RESERVATION ROUTING AND SCHEDULING ALGORITHM 140

8 v GEOGRAPHIC INFORMATION SYSTEM (GIS) PLATFORM MOBILE DATA TERMINALS (MDTS) DISPATCH CENTRE SCHEDULING METHODOLOGY FOR THE STATIC FLEX-ROUTE PROBLEM SYSTEM CHARACTERISTICS AND ASSUMPTIONS THE STATIC FLEX-ROUTE MODEL STATIC FLEX-ROUTE ROUTING AND SCHEDULING METHODOLOGY BUILD THE PROBLEM MODEL Modelng Objects INITIAL SOLUTION: THE CONSTRAINED-ASSIGNMENT ALGORITHM IMPROVE THE SOLUTION USING LOCAL SEARCH 157 NEIGHBORHOODS SCHEDULING METHODOLOGY FOR THE DYNAMIC FLEX-ROUTE PROBLEM DYNAMIC FLEX-ROUTE PROBLEM FEATURES DYNAMIC FLEX-ROUTE PROBLEM MODEL DYNAMIC FLEX-ROUTE MODEL FOR LATE ARRIVAL DYNAMIC FLEX-ROUTE SCHEDULING METHODOLOGY MODEL IMPLEMENTATION AND VALIDATION INTRODUCTION EVALUATION OF THE SCHEDULING MODEL CASE STUDY ASSUMPTIONS / SETTINGS COMPARATIVE ANALYSIS SENSITIVITY ANALYSIS Tme Wndow Duraton Excess Rde Tme VALIDATION OF THE ANALYTICAL MODEL AND SCHEDULING SYSTEM SENSITIVITY ANALYSIS EFFECT OF DESIGN PARAMETERS ON THE NUMBER OF FEASIBLE DEVIATIONS Effect of Servce Area Wdth on Feasble Devatons Effect of Slack Tme on Feasble Devatons Effect of On-Demand Request Rate on Feasble Devatons Effect of Fxed-Stop Spacng on Feasble Devatons DYNAMIC REQUESTS EFFECT AND LATE ARRIVAL 207

9 x Effects of Fxed-Stop Spacng on Dynamc Requests Acceptance CONCLUSIONS CASE STUDY - OAKVILLE TRANSIT CHAPTER OVERVIEW INTRODUCTION AND BACKGROUND THE GO TRANSIT PARKING PROBLEM EXPERIMENTAL STUDY SETTINGS EXISTING DESIGN OF THE STUDY S FIXED ROUTES PROPOSED FLEX-ROUTE SYSTEM: ASSUMPTIONS AND SPECIFICATIONS Fxed-Route On-demand Servce General Specfcatons ANALYSIS OF DESIGN PARAMETERS EFFECT OF SLACK TIME Route Number Route Number Route Number EFFECT OF ON-DEMAND REQUEST RATE FEASIBILITY ANALYSIS OF THE PROPOSED FLEX-ROUTE SERVICE SERVICE PROVIDER COST FIXED-ROUTE USER COST Unused Slack Tme Average Travel Tme FLEX-ROUTE PERFORMANCE VS. FIXED-ROUTE PERFORMANCE SUMMARY AND CONCLUSIONS CONCLUSIONS AND RECOMMENDATIONS SUMMARY AND CONCLUSIONS CONTRIBUTIONS DEVELOPING A FLEX-ROUTE ANALYTICAL MODEL BUILDING A SCHEDULING SYSTEM TO SUPPORT THE OPERATIONS OF FLEX-ROUTE TRANSIT SERVICE. 254

10 7.3 RECOMMENDATIONS FOR FUTURE RESEARCH 255 x

11 x LIST OF FIGURES Fgure 1-1 Scope of the research Fgure 2-1 Illustraton of a typcal fxed-route transt lne Fgure 2-2 Illustraton of a typcal DRT servce Fgure 2-3 Flexble transt servce categores (Source: Koffman, D. (2004). TCRP, Report Fgure 2-4 Fu s Flex-Route servce n an deal network Fgure 2-5 MAST System Fgure 3-1 A typcal Flex-Route servce area Fgure 3-2 T1wo-sded servce area of one route segment Fgure 3-3 One-sded servce area Fgure 3-4 Cases of (a) dfferent servce area wdth, and (b) fxed stop spacng among route segments Fgure 3-5 Calculaton of Average Access Dstance for Fxed Route Users Fgure 4-1: Constrant Programmng solvng procedure Fgure 4-2: Constrant propagaton and doman reducton are used to reduce the domans of the varables X and Y (source: Lustng and Puget, 2001) Fgure 4-3: A search tree Fgure 4-4: REFLEX structure Fgure 4-5: Schedulng Methodology Fgure 4-6: The Two-Opt neghbourhood Fgure 4-7: The Or-Opt neghbourhood Fgure 4-8: The Relocate neghbourhood Fgure 4-9: The Exchange neghbourhood Fgure 4-10 Bus route n dynamc envronment Fgure 4-11 Dynamc-request assgnment procedure Fgure 4-12 Dverson of a transt vehcle n real tme Fgure 5-1 Effect of tme wndow duraton and maxmum allowable rde tme on percentage reducton n cost: (a) based on 30km\hr vehcle-average-velocty, and (b) based on 20km\hr vehcle-average-velocty Fgure 5-2 Effect of tme wndow duraton and maxmum allowable rde tme on percentage reducton n vehcle used: (a) based on 30km\hr vehcle-average-velocty, and (b) based on 20km\hr vehcle-average-velocty Fgure 5-3 Map of the hypothetcal route for a servce area of 500m

12 x Fgure 5-4: Effect of slack tme and servce area wdth on the percentage of accepted ondemand requests for a demand rate of 10 requests /run Fgure 5-5: Effect of slack tme and servce area wdth on the percentage of accepted ondemand requests for a demand rate of 15 requests /run Fgure 5-6: Effect of on-demand request rate on the percentage of accepted requests for a slack tme of 3 mnutes Fgure 5-7: Effect of on-demand request rate on the percentage of accepted requests for a slack tme of 6 mnutes Fgure 5-8: Effect of fxed-stop spacng on the percentage of accepted requests for a slack tme of 6 mnutes Fgure 6-1 Map of route Fgure 6-2 Map of route Fgure 6-3 Map of route Fgure 6-4 Example of the produced map for route 21 (demand =6/run and slack tme = 5mn) Fgure 6-5: Relatonshp between slack tme and the number of accepted requests for Route Fgure 6-6: Relatonshp between slack tme and the number of accepted requests for Route Fgure 6-7: Relatonshp between slack tme and the number of accepted requests for Route Fgure 6-8: Relatonshp between slack tme and the unused slack tme for Route Fgure 6-9: Relatonshp between slack tme and the unused slack tme for Route Fgure 6-10: Relatonshp between slack tme and the unused slack tme for Route Fgure 6-11: Relatonshp between slack tme and the passenger per VRH for Route Fgure 6-12: Relatonshp between slack tme and the passenger per VRH for Route Fgure 6-13: Relatonshp between slack tme and the passenger per VRH for Route

13 x LIST OF TABLES Table 2-1: Number of transt systems usng each type of flexble transt servce Table 3-1 Number of transt agences usng flexble servces n dfferent suburban areas confguratons Table 3-2: Elements of servce desgn for fxed-route, demand-responsve and flexble transt servces Table 3-3: Maxmum extent of devatons reported for Flex-Route transt servce Table 3-4 Defntons of varables used n the cost-beneft analyss Table 5-1: Schedulng Results Obtaned for the Cases of 30km/hr Vehcle-Average-Velocty Table 5-2: Schedulng Results Obtaned for the Cases of 20km/hr Vehcle-Average-Velocty Table 5-3: Optmal slack tme values for each route segment (n mn) for dfferent combnatons of servce area wdth and on-demand request rate, usng equaton Table 5-4: Number of accepted requests for combnatons of slack tme and on-demand request rate for a servce area wdth of 250m (0.25 km) for each route segment, smulaton results Table 5-5: Number of accepted requests for combnatons of slack tme and on-demand request rate for a servce area wdth of 500m (0.5 km) for each route segment, smulaton results Table 5-6: Number of feasble devatons for 1 Km fxed-stop spacng (8-hour servce duraton) Table 5-7: Number of feasble devatons for 2 Km fxed-stop spacng (8-hour servce duraton) Table 5-8: Results obtaned for cases 1 to 3 (spacng =1 km) Table 5-9: Results obtaned for cases 4 to 6 (spacng =1 km) Table 5-10: Results obtaned for cases 7 to 9 (spacng = 2 km) Table 5-11: Results obtaned for cases 10 to 12 (spacng = 2 km) Table 6-1: Typcal parkng faclty fnancal costs (Source: Vctora Transport Insttute Table 6-2: GO Staton rdershp and parkng nformaton Table 6-3: Route Informaton Table 6-4 Number and percentage of accepted requests for dfferent slack tme and demand rates for route

14 xv Table 6-5 Number and percentage of accepted requests for dfferent slack tme and demand rates for route Table 6-6 Number and percentage of accepted requests for dfferent slack tme and demand rates for route Table 6-7 Average travel tme for fxed-stop passengers for dfferent slack tme and demand values for route Table 6-8 Average travel tme for fxed-stops passengers for dfferent slack tme and demand values for route Table 6-9 Average travel tme for fxed-stops passengers for dfferent slack tme and demand values for route

15 1 1 INTRODUCTION Publc transportaton has tradtonally been desgned to serve people n relatvely dense urban areas, where travel patterns and volumes enable servce along fxed routes to follow predetermned schedules. Recently, however, growth patterns and socal changes n Canada and the Unted States have led to more dspersed demand and low densty suburban developments. Suburban traffc congeston has also grown tremendously over the past two decades, causng much concern by varous governmental and professonal nsttutons. Accordngly, moblty plannng has shfted from emphaszng the automoble to enhancng transt servces and Transportaton Demand Management (TDM). These suburban land use and development patterns have several mplcatons for how transt servces are provded (Urbtran Assocates Inc, 1999). Some of these characterstcs nclude: 1. Demand s heavly peaked, and these peaks wll be at dfferent tmes of day. In a tradtonal central cty, the mx of employment, retal, and servce actvty means that demand exsts along a route throughout the day. In a suburban settng, an offce park wll have hgh employment-related peaks, whereas a shoppng center wll have mdday and evenng peaks. To mantan reasonable levels of servce effectveness, vehcles may need to serve dfferent routes and servce patterns at dfferent tmes. 2. Suburban regons cover far more land area than tradtonal ctes, whch result n development denstes that are lower than those of tradtonal urban centers. 3. The lower average denstes of suburban areas means not only that fewer orgns or destnatons are wthn walkng dstance of any transt route but also that the dstances traveled between ponts, on average, are longer. In addton, the lack of an nterconnected street system results n less drect routngs and more vehcle mles traveled to serve actvtes than n urban settngs. 4. The greater setbacks of buldngs from roadways means that more devaton off the prmary route may be requred The mpact of these suburban development patterns on North Amerca s transt ndustry has been sgnfcant. Where transt operators once had well-defned downtown cores and could provde networks that served them effectvely, the envronment wthn whch

16 2 transt exsts now ncludes multple centers, lower overall denstes, and multple orgndestnaton pars. Many urban transt provders are now faced wth the problem of declnng rdershp on tradtonal fxed route servces n low densty suburban areas. As a result, most fxed route servces n such areas are not economcally vable for the transt provder. As such, there s a need for the transt servce to adapt to these changes and fnd solutons to enhance the performance of transt servces n the suburban areas. Rdershp levels on under-performng fxed transt routes n the suburban areas could be mproved by addng more flexblty to the fxed route structure. Several flexble transt servces have been ntroduced and examned throughout North Amerca (Rosenbloom 1996, Koffman 2004). One of the most promsng flexble transt servces s Flex-Route transt servce (or route devaton servce). Flex-Route transt s a combnaton of fxed-route transt servce and demand-responsve transt servce. It s consdered fxed-route n that t has an establshed route wth a set of fxed stops and predetermned schedules. On the other hand, t s consdered demand responsve n that ts vehcles are allowed to devate from the man route to serve pont-to-pont requests. If there are no requests for devaton, the servce operates as a tradtonal fxed-route, fxed-schedule servce. Koffman s study (2004) provded some of the operatonal experences wth flexble transt servces n North Amerca. The study showed that transt provders have been usng flexble transt servces throughout North Amerca for decades. However, the current use of flexble transt servces, ncludng Flex-Route transt, s hndered by the fact that current experences of flexble servces are lackng n two aspects: 1. Servce plannng and desgn: Koffman s survey showed that most flexble transt servces are provded wthout servce plannng or usng desgn gudelnes. In fact, there s yet to be found a comprehensve gude of the desgn factors nvolved n the plannng and desgn of Flex-Route servces and ther mpact on the feasblty and operaton of the servce. 2. Specalzed schedulng systems: The survey also revealed that most flexble transt servces are scheduled and dspatched wthout the use of specalzed schedulng methods for Flex-Route transt or the use of advanced technology for real-tme operatons. The few transt

17 3 provders who have schedulng systems use schedulng systems that are desgned for demand-responsve servces. To address the needs of decson-makers n ths feld, there s a need for detaled gudelnes to help transt planners and operators n determnng the crtcal factors that should be consdered when Flex-Route transt s provded. These gudelnes should help answer such questons as: what mprovements over fxed-route does the servce brng? Whch routes are canddates for mplementng Flex-Route servce? What are the optmal values of slack tme, servce area and fare structure? Furthermore, for ths servce to be effectve there s a need for a schedulng system that s capable of handlng such complex servce both n statc and dynamc settngs. Ths schedulng system should be specfc to the Flex-Route problem and can capture the unque elements of the servce. In ths regard, ths thess s an attempt to advance the plannng, desgn and schedulng of Flex-Route transt servce ncludng an assessment of ts feasblty as a vable transt opton n low-densty suburban areas. It presents the results of a research project that amed at provdng a detaled gude of Flex-Route transt servce plannng and desgn, and developng an advanced schedulng and routng system that employs powerful algorthms to produce fast, cost-effectve and relable schedules. Throughout ths thess the followng terms are used: Flex-Route Transt Servce: Refers to a type of transt servce that ncorporates elements from both fxed-route and demand-responsve transt. It has a fxed-route component n that t has an establshed route wth a set of fxed stops and predetermned schedules. The demand responsve component ncludes on-demand requests that need to be served door-to-door. Door-to-Door Servce: The term door-to-door n ths research refers to the process of pckng up on-demand customers from ther homes (usually n the mornng) and transportng them to a fxed stop (usually the termnal). The same processes can be repeated n the afternoon, where on-demand customers can be pcked up from the termnal and then dropped off at ther homes. It should be noted that the term does not mean that an on-demand customer can be pcked up from home and dropped off at a locaton other than a fxed stop durng a sngle vehcle run.

18 4 Constrant Programmng: Is an emergent software technology orgnatng n the Artfcal Intellgence lterature n the Computer Scence communty. It uses powerful algorthms to provde solutons for large problems, especally n the area of schedulng problems. Slack Tme: The extra tme added to the runnng tme of a fxed transt route to ncorporate on-demand requests. 1.1 POINT OF DEPARTURE - MOTIVATING SCENARIO The prmary motvatng scenaro for ths research nvolves a hypothetcal transportaton engneer/planner who s responsble for montorng and evaluatng the transt operatons for a local transt servce provder n a typcal North Amercan cty. The engneer s faced wth the problem of declnng rdershp on one or more fxed transt routes n a suburban area n the cty. The engneer s lookng for solutons to ths problem, and wants to mprove the servce by ncreasng the rdershp, whle not compromsng the level of servce for the current transt users. Furthermore, the mprovement n the servce must take nto consderaton the effect of the changes n the servce on all the stakeholders nvolved such as exstng transt rders, potental rders and the servce provder. In order to study the avalable optons for mprovng the servce, the engneer must rely on a dverse body of knowledge. Ths knowledge s not only compled wthn some desgn gudelnes, but also n the form of best practces by other servce provders facng the same problem. Transt analysts face such scenaros qute frequently recently due to the deteroratng performance of fxed-route transt n suburban areas. The engneer decded that Flex-Route transt servce s an attractve opton to mprove the performance of the transt servce. Snce the evaluatng, plannng and schedulng of Flex- Route servce s very dfferent from fxed-route transt, the engneer wants a clearly-defned gude for the evaluaton and plannng process as well as a schedulng system that can be used to schedule ths type of servce. The schedulng system must be relable, fast, effectve, and easy-to-use user nterface.

19 5 1.2 OBJECTIVES The msson of ths research s to advance the applcaton of Flex-Route transt servce by nvestgatng the plannng, desgn and schedulng of the servce. Ths requres recognzng the necessary steps needed n the plannng of the servce whle also dentfyng the servce desgn parameters and ther effect on the servce. It also requres buldng a blueprnt for a decson support system (DSS) that can handle the operaton of ths servce n practce, and make use of the recent developments n advanced technology. One of the core components of the decson support system s the schedulng system, whch s used to produce cost-effectve schedules n both statc and real-tme operatonal envronments. The exploraton of ths scarcely researched feld, especally n recent years, s a major endeavour. Ths dssertaton focuses on developng a practcal understandng of Flex-Route servce; the reasons behnd the ntroducton of flexble transt servces; the best practces for the servce; the desgn parameters for the servce; the exstng schedulng algorthms; the use of advanced technology; and other topcs. In essence, the research has four man objectves whch are lsted below. 1. Identfyng the reasons behnd the ntroducton of Flex-Route transt servce as a vable transt opton n low-densty suburban areas. Ths objectve ams at revewng the hstory and the reasons behnd the ntroducton of flexble transt servces n general wth specal attenton gven to Flex-Route servce. Some of the most successful Flex-Route transt servces n North Amerca are revewed and analyzed to fnd out the reason behnd ther success. The desgn and performance of these servces s revewed to fnd out the crtcal desgn factors that could have a substantal effect on the operaton of the servce. 2. Identfyng the crtcal servce desgn parameters of Flex-Route servce and ther effect on the servce. Although conceptually smple and attractve, Flex-Route transt servce s much more dffcult to plan and desgn than regular fxed-route transt servce or demand-responsve transt. There s currently a lack of servce plannng and desgn gudelnes for Flex-Route transt servce. Experence from flexble transt provders (Rosenbloom 1996, Koffman 2004)

20 6 showed that most flexble servces provders do not follow a systematc manner of plannng and desgnng ther Flex-Route servces. Ths objectve ams at creatng a clearly-defned gude for the servce desgn parameters n the Flex-Route servce, the nteracton among these parameters, and the effect of the servce on the fxed-route component of the servce. The research also looks nto the feasblty and cost of the servce on several stakeholders nvolved n Flex-Route servce ncludng regular fxed-route customers, on-demand customers and the servce provder. 3. Buldng a blueprnt for a Decson Support System (DSS) to support the operatons of Flex-Route transt servce. Ths part of the research focuses on the development of a blueprnt for a new realtme decson support and schedulng system (REFLEX) for the operatons of Flex-Route transt servce. Key features of ths DSS system nclude: The system ncludes a ready-to-use schedulng mechansm for Flex-Route servce whch transt agences can use to produce relable, cost-effectve schedules both n statc and dynamc envronments. The proposed schedulng and routng methodology s developed usng a new technque from the feld of Artfcal Intellgence, namely Constrant Programmng. A model of the system s developed n a Geographc Informaton System (GIS) platform ncludng the bus routes and tmetables. Indvdual on-demand requests are nput to the system. The system blueprnt uses advanced communcaton technologes to boost the applcablty of Flex-Route servce n terms of producng the most cost-effectve, relable, real-tme schedules that wll enhance the productvty and performance of Flex-Route servce. Such technologes nclude Automatc Vehcle Locaton (AVL) and Moble Data Termnals (MDT). The system could also work as an expermental tool for transt agences to analyze the mpact of provdng Flex-Route servces and ther expected performance before mplementaton.

21 7 4. Buldng a routng and schedulng algorthm for Flex-Route servce usng advanced Constrant Programmng algorthms. Ths objectve ams at developng an advanced schedulng algorthm based on constrant Programmng, whch orgnates n the Artfcal Intellgence lterature n the Computer Scence communty. The schedulng methodology s developed by explotng the powerful search technques of Constrant Programmng. Fxed-route transt schedules and routes and ndvdual requests for travel are nput nto the schedulng algorthm to determne the optmal tnerares and schedules. The schedulng algorthm wll be able to handle cases of statc and dynamc Flex-Route operatons. In real-tme envronments, the schedulng algorthm wll be able to produce schedules n a very short tme when a real-tme request s receved. 1.3 THESIS SCOPE Fve major decsons were made early on to defne the scope of ths study. These decsons are based on the followng categores (see Fgure 1-1): 1. Type of flexble transt servce: Transt servces that combne elements from both conventonal fxed-route servces and demand-responsve servces have many dfferent structures and forms. It s not nfeasble to study and analyze all or most of the ssues of flexble transt servces found n practce. However, dong so requres an enormous amount of tme and effort due to the vast dspartes between the servces. Thus, a decson was made to lmt the scope of the thess to Flex-Route transt servce snce t s the most commonly-used flexble transt servce found n practce throughout North Amerca. 2. Servce for the general publc vs. servce for the elderly and dsabled: Flex-Route transt servce could be used as a servce for two groups of people: the general publc and the elderly/dsabled. In ths research, the focus of the study s not to replace the exstng demand-responsve servce of the elderly and dsabled; rather, t s ntended as an enhanced transt servce to attract the general publc to use transt. Specfcally, the objectve of the servce s to attract people who use

22 other modes of transportaton to use transt by offerng more flexble operatons of the servce, where people wll be able to use the servce door-to-door Statc vs. dynamc schedulng: A GIS-based decson-support system s developed n ths research for the schedulng and dspatchng of Flex-Route transt servce. The GIS-based support system ades n routng and schedulng the on-demand aspect of the servce wthn the bounds of the fxed-stop schedule constrants. Developng a system that s capable of handlng the operaton of Flex-Route servce n both statc and dynamc settngs s a tough task snce t requres dfferent algorthms for each settng. However, real-tme operatons could be a major porton of any Flex-Route transt servce gven the advancements n communcaton technologes. Thus, a decson s made that schedulng system should nclude a real-tme schedulng algorthm that can handle dynamc operatons of the servce. 4. New servce vs. replacement servce: Flex-Route transt servce could be ntroduced for dfferent reasons and under dfferent crcumstances. It could be ntroduced as a new servce n an area that dd not have any knd of transt servce before. It could also be ntroduced as a replacement of ether a fxed-route or demand-responsve transt servces for dfferent reasons. In ths research, we focus on the case where Flex-Route transt s replacng an exstng fxed-route transt servce n a suburban transt area to mprove the rdershp and performance of the servce. Ths s necessary to unfy the analyss and make t more concse and focused. 5. Smulaton tool: The decson support system can be used for real-world operatons of Flex- Route transt servces. It wll be able to provde daly and real-tme schedules and routes of the servce. However, the scope of the decson-support system ncludes also usng t as a smulaton tool to help make decsons on provdng Flex-Route transt servce. In several cases, the transt servce provder may need to study the

23 9 possble mpacts of ntroducng Flex-Route servce before mplementaton. For example, the servce provder may want to choose whch fxed routes to be replaced by Flex-Route transt servce out of several avalable optons. Conductng a smulaton on these routes wll help the decson-maker better understand of the possble mpacts and performance of the servce.

24 10 Fxed-Route Transt Flexble Transt Demand-Responsve Transt Flexble Route Segment Demand Responsve Connector Flex Route Request Stop Pont Devaton Zone Route New Servce Replacement Servce Dynamc Operatons General Publc Statc Operatons Elderly and Dsabled Legend Wthn Scope Beyond Scope Fgure 1-1 Scope of the research

25 LIMITATIONS The followng lsts the man lmtatons of ths research: 1. The research focuses on the plannng, desgn, operaton and schedulng of Flex-Route transt servce only. It does not nclude nvestgatng these ssues for other types of flexble transt servces. 2. The analytcal model developed for the servce desgn process consders prmarly an deal grd street network. Other types of street networks are beyond the scope of ths study. 3. The research does not nclude any formula for estmatng the potental demand for any Flex-Route servce. It s assumed that the servce provder has the ablty to estmate potental new demand usng methods such as statedpreference surveys. 4. The schedulng system does not guarantee that each on-demand request wll be accepted. Ths s due to the fact that the assgned slack tme mght not be enough to accommodate all requests. 5. The schedulng system does not guarantee optmal solutons when solvng the problem snce the Flex-Route problem s an NP-hard problem. 6. The schedulng system wll be able to provde schedules for Flex-Route transt operatons only. Other types of flexble transt servces are not addressed n ths research. 1.5 THESIS ORGANIZATION Ths thess s organzed nto seven chapters. Chapter two s a lterature revew that starts by revewng the hstory and development of demand-responsve transt servce. The chapter then ntroduces the reader to the concept of flexble transt servces and gves a succnct overvew of the dfferent types of flexble transt servces. Ths revew leads to a dscusson of Flex-Route transt servce and ts varous elements. The revew ncludes some of the most successful applcatons found n practce and n the lterature. The chapter concludes wth a revew of the vehcle routng and schedulng problem and the efforts to solve the problem n both statc and dynamc envronments.

26 12 Chapter three presents the frst component of ths research: an analytcal model for the plannng and desgn of Flex-Route transt servce. The chapter begns by provdng some gudelnes on the plannng of Flex-Route transt servce. Followng that, an analytcal model that covers the desgn parameters of Flex-Route transt s presented, ncludng the modfcatons that need to be made to the fxed-route structure to allow for the provson of Flex-Route operaton. The chapter proceeds to dscuss the costs ncurred by several stakeholders nvolved n the servce such as the transt operator and the fxedroute users, and what strateges can be mplemented to allevate ths cost. Followng that, two methods for montorng the performance of Flex-Route transt are presented. Fnally, the chapter concludes by conductng a senstvty analyss of the servce desgn parameters. Chapter four presents the Flex-Route schedulng methodology used n ths research. The chapter begns by ntroducng the concept of Constrant Programmng, whch s a core component of the schedulng system for Flex-Route servce bult n ths work. The orgns of Constrant Programmng and ts formulaton are ntroduced as well as the algorthms used n Constrant Programmng wth specal attenton gven to the dfference from mathematcal programmng. The chapter then gves an overvew of the blueprnt of the proposed decson support system for the operatons of Flex-Route transt servce wth ts varous sub-systems. The chapter then ntroduces the proposed schedulng methodology developed for Flex-Route transt servce. Ths ncludes separate algorthms for dealng wth the statc and dynamc verson of the Flex-Route problem. Chapter fve presents an evaluaton of both the analytcal model and the schedulng system. Two case studes were used to valdate the schedulng system performance as well as the valdty of the analytcal model developed n Chapter three. The chapter also ncludes a senstvty analyss of the dfferent desgn parameters and ther nfluence on the results. Chapter sx presents an analyss of a case study of Flex-Route transt servce n a suburban area. The case study evaluates the feasblty of Flex-Route servce f t were to replace an exstng fxed-route transt servce n the suburbs of the Greater Toronto Area. The analyss ncludes evaluaton of the performance of both systems and gves some

27 recommendatons on what factors contrbute to a successful Flex-Route transt system. Fnally Chapter seven presents a summary and conclusons of ths work. 13

28 14 2 LITERATURE REVIEW 2.1 CHAPTER OVERVIEW Ths chapter presents an analytcal revew of the man topcs that are addressed n the research; flexble transt servces, Flex-Route transt servce desgn, and Flex-Route servce routng and schedulng. The chapter begns by revewng some of the problems facng suburban transt n North Amerca and the proposed solutons to these problems throughout the years. The chapter then proceeds to revew the hstory and development of demand-responsve servce and the reasons behnd the recent growng demand for the servce. Followng ths revew, a concse ntroducton to flexble transt servce s gven, dentfyng the dfferent types of flexble transt servces found n practce. Ths revew leads to the dscusson of the concept of Flex-Route transt servce, whch ncludes a revew of the work done to tackle the desgn of the servce. The dscusson then leads to revewng the vehcle routng and schedulng problem and the efforts to solve the problem n both statc and dynamc envronments. A dscusson of the mportance of advanced technologes n the schedulng of Flex-Route servce follows. The chapter concludes by dentfyng the lmtatons of the exstng research n terms of Flex-Route servce desgn and schedulng, whch s addressed n ths thess. 2.2 DEMAND-RESPONSIVE TRANSIT Demand-responsve transt servce (DRT) s an alternatve travel method to personal vehcles, carpool/vanpool and regular transt servce. The DRT concept was ntroduced to mprove the transportaton accessblty for the elderly and persons wth dsabltes as well as to those who lve n small rural areas where transt demand s almost non-exstent. In larger urban areas n the Unted States, such servce s often provded as a complementary paratranst servce requred by the Amercans Wth Dsabltes Act (ADA). In smaller communtes, DRT may be provded as a replacement of fxed-route servce or to supplement other transt servce, and may be avalable to the general publc rather than a subset such as the elderly and persons wth dsabltes. In rural areas, demand-responsve servce may be the only servce avalable, and t may only be offered on certan days of the week or even month.

29 15 Demand-responsve transt, whch s often-called Dal-A-Rde servce, s comprsed of a number of customer requests that need to be served door-to-door or curbto-curb by a set of DRT vehcles. The servce operates n response to users requests, whch vary from day to day wth the routes and schedules of the vehcles varyng accordngly. Thus, a specfc vehcle route wll change daly based on the number of assgned customers and ther pck-up and drop-off locatons (see Fgures 2-1 and 2-2 for llustratons of fxed-route and demand-responsve transt servces). Although demand responsve transt systems have been n exstence for a long tme n several ctes around the US and Canada, serous research nto larger-scale demand-responsve transt dd not start untl the 1960s. The most ntensve academc research nto demand-responsve transt was at MIT startng n 1970, n the well-known project CARS. The work resulted n heurstc algorthms and a demonstraton project by Wlson and Colvn (1977). The growth of (DRT) began n the late 1970 s and early 1980 s wth large demonstraton projects developed n Rochester, NY and Santa Clara County, CA among others. These early systems faled to meet expectatons due to low demand requests and to defcences n communcaton and computer technology to effectvely manage such systems (Lave, Teal, and Pras, 1996). The popularty of DRT servce n the late 1970 s and early 1980 s brought much nterest among transt authortes and academcs to evaluate the concepts and develop nnovatve DRT systems. Numerous studes have been conducted on ths subject (Hall 1970; UMTA 1974; Louvere 1979; Demetsky et al. 1982; Jeffrey Parker and Assocates 1991; Mller 1989). These studes revealed that provdng DRT servce to the general publc s too expensve.

30 16 Legend Drecton of Travel W L Fxed Stop Termnal Route Servce Area Servce Area wdth Route Length W L Connectng Transt Lne Fgure 2-1 Illustraton of a typcal fxed-route transt lne

31 17 Legend Vehcle Depot Trp Orgn Trp Destnaton Drecton of Travel Fgure 2-2 A typcal vehcle route n DRT servce Fgure 2-2 Illustraton of a typcal DRT servce

32 18 In terms of system desgn, analytcal models for predctng fleet requrements, system capacty and qualty of servce measures are developed by Fu (2003) for specfc operatng condtons. Accordng to Fu, most paratranst servces have a fleet of varous types of vehcles because decsons on the types and number of vehcles are typcally made on ad-hoc bass. In hs study, Fu proposed a heurstc procedure for obtanng optmal fleet mx for specfc operatng condtons and the objectve s that of maxmzng the system s effcency. Karlafts (2005) developed an operatonal-level mathematcal model that jontly determnes paratranst servce frequences and vehcle types, both constraned by allowable frequences. An applcaton of the model to the Athens 2004 Olympcs scenaro and correspondng results are presented and dscussed. Aldahan and Dessouky (2003) proposed a heurstc algorthm that provdes an approxmate soluton to reduce the vehcle mles of the on-demand vehcles whle not sgnfcantly reducng the customer servce level. Sandln and Anderson (2004) presented a procedure for calculatng a servceablty ndex (SI) for DRT operators based on regonal socoeconomc condtons and nternal operaton data. The SI can be used to evaluate and compare DRT operatons. A survey by Palmer et al. (2004) nvolvng 62 transt agences showed that the use of paratranst computer aded dspatchng (CAD) system and a specalzed agency for servce delvery provde a productvty beneft. Dana et al. (2006) studed the problem of determnng the number of vehcles needed to provde a DRT servce wth a predetermned qualty for the user n terms of watng tme at the stops and maxmum allowed detour. Other studes nclude a study by Quadrfoglo et al. (2008a) who used smulaton methods to nvestgate the effect of usng a zonng vs. a no-zonng strategy and tme wndow settngs on performance measures such as total trp mles, deadhead mles and fleet sze. They dentfed quas lnear relatonshps between the performance measures and the ndependent varable, ether the tme-wndow sze or the zonng polcy. Other research tred to examne the sustanablty and envronmental effects of DRT servces. Dessouky et al. (2003) demonstrated through smulaton that t s possble to reduce envronmental mpacts substantally, whle ncreasng operatng costs and servce delays

33 19 only slghtly for the jont optmzaton of cost, servce, and lfe cycle envronmental consequences n vehcle routng and schedulng of a DRT system. 2.3 FLEXIBLE TRANSIT SERVICES In the urban transt context, most bus transt systems fall nto two categores: fxed-route transt (FRT) and demand-responsve transt (DRT) systems. Fxed-route bus systems are much more cost effcent (from the operator s perspectve), due to the relatvely large passenger loadng capacty of the buses and the consoldaton of many passenger trps onto a sngle vehcle (rdesharng). However, compared to the prvate automoble, they have major defcences as the general publc consders the servce to be nconvenent because of ts lack of flexblty. Moreover, the total trp tme s perceved as beng too long, and for longer trps there s often a need of transfers between vehcles. DRT systems, as dscussed n the prevous secton, provde the flexblty desred by the customers, but they tend to be much more costly; therefore, they are largely lmted to specalzed operatons such as the elderly and dsabled. Thus, both practtoners and researchers have been lookng for a transt servce that combnes the cost-effcency of fxed-route transt and the flexblty of DRT. Ther efforts started n the 1960s, when several types of servces that combne features of both conventonal fxed-route servce and demand-responsve servce were proposed. One of the earlest documented experments, reported by Flusberg (1976), s the Merrll-Go- Round n Merrll, Wsconsn, whch used a pont devaton mode of operaton. The servce, whch s stll n operaton, s comprsed of vehcles that serve some demandresponsve requests wthn a specfed zone, whle also servng a lmted number of fxed stops wthn the zone wthout any regular path between the stops. People who use the servce as demand-responsve customers pay hgher fares than those who use the fxed stops. Currently, demand-responsve customers pay $2 for one trp compared to $1 for customers who access the servce through the fxed stops. Another reason why transt provders were turnng to flexble transt servces n the Unted States was the passage of the Amercans wth Dsabltes Act (ADA) n 1990, whch has brought forth new and greater responsbltes for transt agences. Wth the law mandatng that certan dsabled persons must be provded complementary paratranst

34 20 servce at a nomnal cost, publc transportaton provders were suddenly faced wth the challenge of provdng tradtonal fxed-route transt servce whle also servng ndvduals wth dsabltes. Consequently, there has been a notable and steady ncrease n the demand for paratranst by dsabled people n the post-ada era n the Unted States. Snce the cost of provdng accessble paratranst s defntely hgher than the cost of accessble fxed route, the ncreased demand for paratranst was burdenng transt agences n the Unted States (Balog, 1997). Ths has led a number of transt provders to look for new optons to encourage paratranst rders to use fxed route servces. Most of these optons are centered on mprovng the level of servce of fxed route operatons and makng them more accessble to ndvduals wth dsabltes (Balog, 1997) Types of Flexble Transt Servce A survey conducted by Koffman (2004) for the Transt Cooperatve Research Program (TCRP) reported that flexble transt servces can be categorzed nto sx major servce types as llustrated n Fgure 2-3; these categores are: 1. Route devaton (Flex-Route) transt servce In ths type of servce, vehcles operate on a regular schedule along a well-defned path, wth or wthout marked bus stops, and devate to serve demand-responsve requests wthn a zone around the path. The wdth or extent of the zone may be precsely establshed or flexble. 2. Pont devaton In pont-devaton servce, vehcles serve demand-responsve requests wthn a zone and also serve a lmted number of stops wthn the zone wthout any regular path between the stops. 3. Demand-responsve connector servce (DRC) In demand-responsve connector servce, vehcles operate n demand-responsve mode wthn a zone, wth one or more scheduled transfer ponts that connect wth a fxed-route network. A hgh percentage of rdershp conssts of trps to or from the

35 21 transfer ponts. In most cases, the servce operates as an FRT servce durng daytme and swtches to a demand-responsve connector type of servce durng evenngs, nghts or early mornng, when the demand s lower. 4. Request stops In request stop flexble servce, vehcles operate n conventonal fxed-route, fxed-schedule mode and also serve a lmted number of defned stops near the route n response to passenger requests. (Request stops dffer from flag stops n not beng drectly on the route.) 5. Flexble-route segments In ths servce, vehcles operate n conventonal fxed-route, fxed-schedule mode, but swtch to demand-responsve operaton for a lmted porton of the route. 6. Zone route In zone route flexble servce, vehcles operate n demand-responsve mode along a corrdor wth establshed departure and arrval tmes at one or more end ponts. Other terms have been appled n the past to some of these servces. For example, demand-responsve connector servce has been called demand-responsve feeder servce. Indvdual transt systems call these servces by many dfferent names and do not follow any standard namng practce. These categores are useful n descrbng the flexble servces operated by most transt systems that use such servce. However, other desgns are possble, as are many varatons on the basc categores descrbed above. Table 2-1 provdes the number of each type of these servces as reported n the survey. As shown n the table, Flex-Route transt servce comprses the majorty of flexble servce n practce. In Koffman's survey (2003), 12 of the 24 transt agences that used flexble transt servce were usng Flex-Route servce as ther method of operaton. Demand-responsve connector occuped the second place wth 6 transt agences usng t as ther flexble method of operaton. Some experences on Flex-Route transt servce are dscussed n the next secton.

36 Fgure 2-3 Flexble transt servce categores (Source: Koffman, D. (2004). TCRP, Report 53 22

37 23 Table 2-1: Number of transt systems usng each type of flexble transt servce Type of Flexble Servce No. of Transt Systems Flex-Route (route devaton) 12 Pont devaton 3 Demand-responsve connector 6 Request stops 4 Flexble route segments 2 Zone route 1 Source: Koffman (2004) Specfc work on demand-responsve connector servce (DRC) has been conducted by Cayford and Ym (2004). The authors surveyed the customers demand for DRC n the cty of Mllbrae. They also desgned and mplemented an automated system used for the DRC servces. The servce uses an automated phone-n system for reservatons, computerzed dspatchng over a wreless communcaton channel to the bus drver and an automated call-back system for customer notfcatons. Some flexble transt servces nvolve checkponts. Daganzo (1984) descrbes a flexble system n whch the pck-up and drop-off ponts are concentrated at centralzed locatons called checkponts. Other deas of combnng DRT wth FRT nclude a hybrd flexble servce where demand-responsve vehcles are used to provde passengers wth access to a fxed-route stop to further contnue ther trps usng the fxed-route servce. Clearly, the DRT system as opposed to a hybrd system mnmzes the travel tme for the passenger. However, shftng some of the demand to fxed routes may allevate some of the demand pressure for the on-demand vehcles caused by ADA requrements. Some of the work n ths feld ncludes that of Aldahan et al. (10), Law, Whte, and Bander (1996), and Hckman and Blume (2000). Aldahan et al. (2003) developed an analytcal model that ads decson makers n desgnng a hybrd grd network that ntegrates a flexble demand responsve

38 24 servce wth a fxed-route servce. Ther model s to determne the optmal number of zones n an area, where each zone s served by a number of on-demand vehcles. Other efforts nclude that by Cortés and Jayakrshnan (2002) who proposed and smulated one type of flexble transt called Hgh-Coverage Pont-to-Pont Transt (HCPPT), whch requres the avalablty of a large number of transt vehcles. Khattak and Ym (2004) explored the demand for a consumer orented personalzed DRT (PDRT) servce n the San Francsco Bay Area. About 60% of those surveyed were wllng to consder PDRT as an opton, about 12% reported that they were very lkely to use PDRT. Many were wllng to pay for the servce and hghly valued the flexblty n schedulng the servce Flexble Transt Servce Benefts The growng nterest n flexble transt servce led transt operators to test the new servce and report ts effectveness. One beneft of ntroducng flexble transt servces s the ncrease n rdershp volumes. Several agences reported a steady ncrease n ther rdershp levels when fxed-route transt servce was modfed to flexble servce as n the case of Merrll-Go-Round transt n Merrll, Wsconsn (Smth, 1998), and Pennsula Transt n the ctes of Hampton, Newport News and York County (Durvasula, 1999). Other benefts of ntroducng flexble transt servce s a more cost-effectve and ntegrated servce for people wth dsabltes such as Pennsula Transt (Farwell, 1998), and as reported n the TCRP Report 53 (Koffman, 2004), evaluatng the transt operatons for people wth dsabltes. Another major beneft of flexble transt servce s to be used as part of the toolkt to help transt operators address suburbanzaton and dspersed travel patterns (Potomac and Rappahannock Transportaton Commsson 2003, Rosenbloom 1996). Other benefts of the flexble transt servce nclude: combnng the regularty of fxed-route servce wth the flexblty of demand-responsve servces (Loukakos, 2000), servng areas wth demand denstes too hgh for door-to-door servces but not hgh enough for fxed-route servce (Pratell, 2002) and makng transt more attractve to choce rders who have another mode of access (Potomac and Rappahannock Transportaton Commsson 2003). Rosenbloom (1996) ntervewed 40 transt systems wth flexble servce and found that most of them had adopted flexble servces as a way

39 25 to remove or reduce the need to provde complementary paratranst mandated by the Amercans wth Dsabltes Act (ADA). As mentoned n the objectves of ths research, the prmary focus of ths study s on Flex-Route transt servce (or route-devaton transt servce). Thereby, the next secton looks nto Flex-Route transt n more detal, defnng some of the elements of ths servce. 2.4 Flex-Route transt servce Flex-Route transt (also referred to as route-devaton) s an nnovatve publc transportaton approach n whch servce s provded at fxed stops on a predetermned schedule, whle also provdng on-demand servce to customers between the fxed stops. It s a hybrd of conventonal Fxed-Route transt and demand responsve transt servces. It s consdered conventonal transt n that t has an establshed route wth a set of fxed stops and predetermned schedules. On the other hand, t s consdered demand responsve n that ts vehcles are allowed to devate from the man route to serve doorto-door requests. If there are no requests for devaton, the servce operates as a tradtonal Fxed-Route, fxed-schedule servce. Although the concept of Flex-Route transt as an nnovatve transt servce has been around snce the late seventes, ts mplementaton has not been wdespread. Untl very recently, mplementaton of Flex-Route transt servce by transt agences was lmted to rural and small urban areas due to the complexty of schedulng such a servce n urban and suburban areas. However, wth the advent of new nformaton technologes the tools now exst to mplement route devaton n urban and suburban areas. Flex-Route transt has proven to be much more dffcult to plan, desgn, schedule and operate than regular transt and paratranst. Flex-Route servce has to mantan a balance between the needs of the two man groups of rders - the regular transt users who access the servce at the establshed fxed stops and the users who use the servce as on-demand customers. Under normal condtons, each transt vehcle should arrve at each fxed stop before the publshed departure tme at that stop for relablty ssues. On the other hand, demand-responsve users need to be served door-to-door durng the operaton perod of the fxed-route servce. These requests could be known n advance or requested n real-

40 26 tme and have a specfc tme wndow n whch to be served. The servce vehcles start operatng at a specfed departure tme from the frst stop and travel along the predefned route, wth a predetermned headway between the consecutve vehcles untl the end of the daly operaton perod. The created schedule should comply wth the Fxed-Route constrants that the servce vehcle must be at the fxed stops before the publshed departure tme, whle tryng to maxmze the number of demand-responsve requests that could be accommodated gven the avalable slack tme. Another mportant aspect of ths problem s tryng to mnmze the system cost, n terms of the total dstance (tme) traveled by all vehcles. One of the key servce desgn ssues n operatng flexble transt s determnng how much scheduled operatng tme needs to be reserved as slack tme to accommodate demand-responsve servce requests. Slack tme s the extra tme reserved for devated requests (statc and dynamc) n the schedule. The research n ths area s very scarce. The only notable work n ths area s that done by Fu (2002), who developed an advanced mathematcal model for fndng the optmal slack tme, whch s dscussed later n ths secton. Fu's analyss provded some nsghts nto varous ssues that may arse n desgnng a Flex-Route servce. A sgnfcant challenge to the mplementaton of Flex-Route transt has been the complexty nherent n the desgn process. Some of the key desgn parameters n Flex- Route transt servce are: servce area sze (the area between fxed stops where devatons are permtted), and slack tme dstrbuton (the method used to dstrbute among zones the total slack tme bult nto the schedule to allow for devatons), and the effect of the demand-responsve operatons on the fxed-route porton of the servce such as the effect on the headway, fxed stops spacng and vehcle operatons. The followng secton provdes some of the few efforts that studed the flex-servce desgn problem Flex-Route Transt Servce n the Lterature Optmal Slack Tme (Fu, 2002) One of the most ambtous efforts that deal wth the desgn of Flex-Route transt servces s the one conducted by Fu (2002). In hs work, Fu presented a theoretcal

41 27 nvestgaton of varous ssues nvolved n the plannng and desgn of Flex-Route transt servces. The author s analytcal model was set under an dealzed operatng envronment wth the objectve of determnng the optmal slack tme that should be allocated to a Flex-Route segment. The optmzaton objectve s to mnmze the total net cost that would be ncurred by the dfferent stakeholders ncludng the operator as well as the regular and DRT passengers. It should be noted that n Fu's model, the on-demand passengers are not the general publc; rather he assumed that the on-demand passengers conssts only of passengers who under normal crcumstances use DRT. To perform hs analyss, Fu reled on some assumptons, whch nclude: Both fxed-route and paratranst servces are provded by the same operator. All paratranst trps need to be accommodated ether by the Flex-Route servce or by a specalzed paratranst servce. The Flex-Route servce s to be operated wthn a rectangular servce area of length (l) and wdth (2w). The servce area s covered by a unformly dstrbuted grd road network wth a lnk travel speed of (v) (see Fgure 2-4). The man route of the Flex-Route servce s located at the mddle of the zone and ncludes one route segment (or one servce zone) between two fxed stops (A and B). Any servce vehcle departs from A and B at pre-scheduled departure tmes. The dfference between the departure tmes at B and A s the scheduled runnng tme, denoted as T, whch s also defned as the analyss perod n ths study. To accommodate possble devatons, the scheduled runnng tme (T) must be greater than the drect runnng tme between the fxed stops (T 0). The dfference between the scheduled runnng tme (T) and the drect runnng tme (T 0 ) s called slack tme (Δ) where (T = T 0 + Δ). There are N p paratranst stops that are expected durng the analyss perod (T). These devated stops are to be servced durng the servce headway. The stops are unformly dstrbuted over the servce zone. There are N t general publc transt rders travelng from A to B for each Flex- Route trp. Both transt and paratranst demands are perfectly nelastc, that s, they are not affected by servce qualty.

42 28 Fgure 2-4 Fu s Flex-Route servce n an deal network The model s objectve functon s defned as to mnmze the total operator and user costs. The problem of dentfyng the optmal slack tme was formulated as a lnear programmng problem (LP). Then, an equaton was derved for the relatonshp between the number of feasble devatons and varous system parameters such as slack tme, zone sze and dwell tme. Based on the analyss, the author concluded that the optmal slack tme should be determned wth a consderaton of the trade-off between the savngs that can be acheved from servng paratranst rders and the nconvenence that may result to the transt users. Also, Fu concluded that the crtcal factors that should be consdered nclude level of paratranst demand, zone sze and paratranst dwell tme. Moreover, n case the problem s just to dstrbute a gven amount of slack tme to ndvdual route segments, Fu concluded that the dstrbuton scheme should consder paratranst demand and zone sze. If the dstrbuton objectve s to maxmze the total number of feasble devatons, the optmal dstrbuton method should be based on the product of the expected paratranst demand and the average addtonal tme that s requred to vst a devated stop. Another mportant fndng s that dle tme at a fxed stop n the mddle of a Flex-Route has a negatve mpact on those rders who are already

43 on the bus and are headng to a destnaton beyond the fxed stop. Ths fndng suggests that the Flex-Route concept mght be vable only wth a mnmal number of fxed stops A Mult-objectve Optmzaton Model Few researchers tred to nvestgate the mpact some of the Flex-Route desgn parameters on the performance of the servce. Smth and Demetsky (2003) explored two of the key desgn parameters n the Flex-Route servce: servce zone sze (the area between fxed stops where devatons are permtted), and slack tme dstrbuton (the method used to dstrbute among zones the total slack tme bult nto the schedule to allow for devatons). The researchers used data from the Hampton Roads Transt (HRT), an agency that was consderng the ntroducton of Flex-Route servce n the Pennsula regon of Vrgna, USA. In addton to tradtonal fxed-route servce, HRT also provdes paratranst servces, HandRde, for the dsabled under the ADA mandate. Two of HRT s exstng fxed routes were chosen to serve as the routes for Flex-Route servce n the study. The two routes were chosen n consultaton wth the HRT management. Fxed stops wth low rdershp were elmnated from the two routes. Each route retaned up to a maxmum of fve major fxed stops. The schedules at these stops were changed to ncorporate some slack tme between the stops to allow for devaton. The researchers consdered three servce zones dstances n ther research: 400m, 800m, and 1,200m (1/4, 1/2, and 3/4 mle).the next step was to determne the best way to dstrbute the slack tme among a route s servce zones. In the study, two approaches to slack tme dstrbuton were consdered: Slack tme dstrbuton as a weghted average of the non-stop travel tme between the two fxed stops of a zone (SDWR), and Slack tme dstrbuton as a weghted average of the total number of orgns and destnatons of ADA certfed trps from HandRde logs (SDNP). The authors used two dfferent cost functons to study the effect of the desgn parameters on the cost ncurred by both the users and the operator. The maxmum feasble devatons per hour was chosen as the performance measure from the perspectve

44 30 of a Flex-Route provder, whle the cumulatve unused slack tme remanng at all fxed stops per route was chosen as the performance measure from the perspectve of a customer usng the Flex-Route servce. After performng the analyss, the authors concluded that for a constant slack tme, as the area of the servce zone s ncreased, the number of feasble devatons that can be accommodated decreases because of the lkely wder dsperson of the on-demand trp locatons. However, the excess slack tme left over ncreases wth the reducton n servce area. Thus, t was found that the objectves of maxmzng feasble devatons and mnmzng unused slack tme are conflctng and senstve to the desgn parameters and a trade-off between the two conflctng objectves needs to be made Other Research Efforts Other researchers tred to approach the Flex-Route desgn problem form a dfferent angle. One of the nterestng approaches to the problem s the study by Quadrfoglo et al. (2006). In the study, the authors developed a mathematcal formula to set bounds on the maxmum longtudnal velocty. The formula s used to evaluate the performance and help the desgn of Moblty Allowance Shuttle Transt (MAST) n Angeles County, Calforna. The lne operates as a regular fxed-route bus system durng the day but swtches to a MAST servce durng the nght. The authors argued that the man purpose of these servces should be to transport customers along a prmary drecton. The velocty along ths drecton should reman above a mnmum threshold value to mantan servce attractveness for customers. The authors use contnuous approxmatons to compute lower and upper bounds on the longtudnal velocty. The resultng narrow gap between them under realstc operatng condtons allows the authors to evaluate the servce n terms of velocty and capacty versus demand. The results show that a two-vehcle system, wth selected wdths of the servce area of 0.5 mles and 1 mle, s able to serve, respectvely, a demand of at least 10 and 7 customers per longtudnal mle of the servce area whle mantanng a reasonable forward progresson velocty of about 10 mles/hour. The relatonshps obtaned can be helpful n the desgn of MAST systems to set the man parameters of the servce, such as slack tme and headway.

45 31 All the aforementoned studes focused on the theoretcal sde of the problem suggestng that flexble transt servce could be an effectve transt soluton to many lowdemand urban areas. However, these studes dd not provde gudelnes on how to mplement such servces or the condtons under whch flexble transt servce could be provded. They also dd not provde a deep and comprehensve analyss of the factors nvolved n provdng flexble transt servce Flex-Route Transt Servce n Practce As mentoned earler, n the report by the TCRP that summarzed the operatng experences of several transt operators across the Unted States and Canada that mplemented flexble transt servces, 12 of the 24 surveyed agences used Flex-Route transt servce. The followng are some past and current experences of Flex-Route transt servce n North Amerca Madson, Wsconsn Madson Metro, the transt agency for the Cty of Madson, Wsconsn, currently operates a servce route system wth eght routes that are "fxed," wth publshed routes and schedules. The eght routes run on 60-mnute headways. The system provdes devaton servce to ADA-elgble customers that are wthn 3/4 mle of the fxed route. Currently, about 2.5 passengers per hour are beng served by devatons. A request for devaton must be made by 4:30 p.m. the day before the ntended trp. In restructurng ther system for the servce route concept, most "flag stops" were elmnated;.e., sgns for stops were removed. A few fxed stops were kept as tme checks. In performng a devaton, a bus must return to the fxed route at or before the next tme check. They ntend to provde "real-tme devaton servce (requests to be consdered up to one hour ahead of servce) wth future upgrades to ther current schedulng software, whch s developed by Trapeze, Inc Potomac and Rappahannock Transportaton Commsson (PRTC), Vrgna PRTC was created n 1986 to develop and operate transt servces n a rapdly growng suburban area approxmately 20 m southwest of Washngton, D.C. Untl 1995,

46 32 PRTC s servces conssted of express commuter bus and commuter ral servce, prmarly nto Washngton, and a rdeshare matchng program. OmnLnk servce was begun n 1995 n response to requests for local transt servce. PRTC determned that conventonal transt servce would not be attractve to rders n ts low-densty servce area. The area has grown rapdly n recent decades, wth pockets of development connected by an rregular and often crcutous road network. The area has no downtown and no major travel pattern focus other than Washngton, D.C. An affordable transt route network would not reach many resdental areas. Streets often lack sdewalks, so that walkng to bus stops would be dffcult. In addton to notng these dffcultes, PRTC realzed that provdng conventonal local transt servce would also brng wth t a requrement to provde ADAcomplementary paratranst servce. The Flex-Route concept was seen as resolvng these dffcultes. The devaton component made t possble to provde servce throughout the servce area, as well as to combne servce for the general publc and people wth dsabltes. The devaton component also addressed the dffculty of customers walkng to bus stops along streets wthout sdewalks. A further attracton of Flex-Route servce was that t responded to a desre by human servces agences n the area for addtonal capacty to serve ther transportaton-dsadvantaged clents PRTC operates a network of Flex-Route transt servces as the exclusve local mode for ts low-densty servce areas as a cost-effectve alternatve to provdng both fxed-route and paratranst servces. PRTC serves the countes of Stafford and Prnce Wllam n the outer Vrgna suburbs of Washngton, D.C. PRTC descrbes the servce as Flex-Route and uses the servce name OmnLnk. The system was formally deployed n October 1997 usng ntellgent transportaton systems (ITS) technologes such as automatc vehcle locaton (AVL) systems, computer aded dspatch (CAD) software, Moble Data Termnals and a propretary routng and schedulng software developed by Trapeze, Inc. PRTC clams that t has saved "roughly 50 percent from the cost of operatng both fxed route and paratranst servces, or an estmated $560,000 n annual operatng savngs" (ITS Amerca News, 1997). PRTC operates Flex-Route servce on fve routes usng 13 peak vehcles. On each route, buses stop at marked stops and can also devate up to three-quarters of a mle on

47 33 ether sde of the route n response to servce requests. Three- quarters of a mle s the same dstance as n the ADA requrement for complementary paratranst around fxed routes. PRTC does not operate separate ADA paratranst. There are also a lmted number of on-demand stops close to the man routes. The Omn-Lnk servce operates from 5:30 a.m. to 10:30 p.m., Monday through Frday. Passengers wantng an off-route stop are requred to call PRTC at least 2 hours n advance. However, PRTC advses that for best results, reservatons should be made 1 to 2 days n advance. PRTC lmts the number of off-route requests that wll be accepted on each vehcle trp and advses passengers that they may be asked to get on or off the bus at a locaton that s wthn a few blocks of ther orgn or destnaton, because some locatons are not accessble to OmnLnk buses. Passengers whose requests cannot be accommodated are advsed to ask for a dfferent tme or walk to a bus stop. Passengers can request servce to one of the on-demand bus stops when they board. The base fare for OmnLnk s $1.00 per trp. There s a devaton surcharge of $1.00 per trp, except for rders 60 years and older and those wth dsabltes System for Advanced Management of Publc Transport Operatons (SAMPO) In 1995, a demonstraton project called System for Advanced Management of Publc Transport Operatons (SAMPO) was ntated n Europe under the Transport Telematcs Program of the European Unon (Engels, 1997). The prmary objectve of SAMPO was to provde demand responsve transport servces (DRTS) usng telematc technologes. Specfcally, the goals of SAMPO were to: (a) Improve moblty for people n rural and urban areas; (b) Increase partcpaton of elderly and dsabled wthn ther communty; (c) Improve busness and vablty of publc transport operatons; and (d) Enhance rural and urban communty envronments, usng the "added potental and the effectveness of telematcs technology. The test stes for ths project are located n fve European countres: Fnland, Belgum, Ireland, Italy and Sweden. The objectve of DRTS under SAMPO s to provde "on demand" servces to passengers by ntegratng all the dfferent modes such as buses,

48 34 taxs, mnbuses, and ral servces. The DRTS system can be accessed by a passenger by callng a Travel Dspatch Center (TDC). A TDC wll have advanced bookng and reservaton systems that have the ablty to dynamcally assgn passengers to vehcles and optmze the routes and schedules. At the Fnnsh ste under the SAMPO project, there were essentally four generc concepts of DRTS servces wth ncreasng levels of flexblty (Engels, 1997). The four concepts are: 1. Predefned Tmetable and Route: ths s smlar to the tradtonal fxed route servce, 2. Partally Predefned Tmetable and Route wth Devatons to Predefned Stops: ths s smlar to Flex-Route servce, 3. Stops n a Regon: ths servce s smlar to a paratranst servce,.e., ondemand shared rde, 4. Ponts n a Regon: ths servce s smlar to a tax servce,.e., on-demand exclusve rde. All the above servces follow some common gudelnes. One such gudelne s that customers have to make a trp reservaton at least one hour n advance. Another type of DRTS servce mplemented was the concept of usng DRTS as a feeder servce to a major ral lne. Prelmnary results from a market survey of the SAMPO demonstraton ndcate that the publc s percepton of the fxed route servce has mproved remarkably. About 74% of the passengers ntervewed rated the new system as ether good or excellent The Moblty Allowance Shuttle Transt (MAST) The moblty allowance shuttle transt (MAST) system s a Flex-Route applcaton ntroduced n Los Angeles County, Calforna. The Metropoltan Transt Authorty (MTA) of Los Angeles County, Calforna, ntroduced MAST as part of ts feeder Lne 646. Durng the day, ths lne operates as a regular fxed-route bus system. At nght, the lne changes to a MAST servce wth three checkponts wth fxed scheduled departure tmes. The checkponts are convenently located at major connecton ponts or at hghdensty demand zones. The total servce area of the route s mles 2 (6 mles of

49 35 length between each par of checkponts). Buses are allowed to devate from the fxed path to pck up and drop off passengers at ther desred locatons. Customers make a reservaton to add ther desred pck-up and/or drop-off stops n the schedule of the servce. Regular customers do not need a bookng process to use the servce. 2.5 ROUTING AND SCHEDULING EFFORTS Vehcle routng and schedulng s a crtcal part of the desgn and operatons of any transport system. For conventonal transt systems where all transt vehcles stop at each servce stop/staton, schedules are easly produced and are adequate for operaton purposes. DRT transt systems, on the other hand, are more complcated than conventonal transt systems, and f proper care s not gven to the schedulng and routng of the DRT servce, very expensve and unrelable systems may result wth a hgh rsk that they wll ultmately fal to meet the requred level of servce expected by the passengers. DRT schedulng can be consdered an expanson of the travelng salesman problem, whch nvolves the calculaton of an optmum journey for vstng a number of predetermned nodes on a network. Determnng routes and schedules for DRT systems s known n the engneerng and operatons research communtes as the Dal-A-Rde problem- DARP The Dal-A-Rde Problem (DARP) The statc DARP s defned to construct a set of feasble and effcent routes and schedules to satsfy transportaton requests (trps) made by the system clents. The problem consders a case n whch there are N customers representng the demand for servce and M avalable DRT vehcles. A mxed fleet of vehcles that can accommodate the trps s avalable to operate the routes. Each trp has several attrbutes that should be known; these nclude: 1. The number of persons to be transported; 2. Seatng requrements; 3. Pck-up and drop-off locatons; and 4. The desred pck-up and/or drop-off tmes.

50 36 The DARP s commonly formulated to mnmze a general objectve functon (or cost functon) wth a set of servce qualty constrants (Jaw et al., 1986, Savelsbergh and Sol, 1995). The cost functon s usually defned as a weghted sum of the total clent nconvenence, as measured n terms of excess rde tme (the dfference between the scheduled rde tme and the rde tme wthout dversons to serve other passengers) and servce tme devaton (the dfference between the scheduled pck-up/drop-off tmes and ther most desred pck-up/drop-off tmes), and the cost to the servce provder, as often measured n terms of total vehcle travel tme and the number of vehcles needed. The servce qualty constrants specfy that the rde tme of each clent must be less than a maxmum allowable rde tme and that all clents must be pcked up (dropped off) after (before) ther most desred pck-up (drop-off) tmes wth servce tme devatons less than a maxmum allowable value. The latter defnes a tme nterval, or servce tme wndow, durng whch the servce must take place. Routng and schedulng are consdered central actvtes of DRT provders. For operatons n whch passenger requests are known n advance, proper optmzaton technques can be used to fnd routes and schedules that can perform well wth respect to specfc objectves. These objectves mght nclude smultaneously mnmzng customer nconvenence (travel tme and wat tme), tme wndow volatons, and vehcle mles traveled. In the earlest studes of DRT schedulng, manual technques were used to fnd the schedules and routes for paratranst servces. They nclude the followng steps: 1. Sortng trps chronologcally: Ths s a smple sort of all trp requests for a servce day; 2. Groupng trps geographcally: Trp groupng nvolves collectng parallel trps at smlar tmes of the day. Ths s done wth consderaton of geographcal obstacles and traffc condtons; 3. Assemblng groups nto manfests: Once trps are collected nto groups, the groups are sequenced chronologcally nto manfests; and

51 37 4. Resolvng excepton trps: Trps that do not ft nto groups, as well as ones that are handled outsde the rules are nserted nto the schedule as best as the scheduler can. The dsadvantages of manual schedulng nclude the ntensve labour requrements to schedule a large number of trps, when most are not standng requests. Furthermore, lke any manual process, the element of human error exsts, resultng n conflcts and nfeasbltes of schedules. Ths led to the development of a wde range of algorthms that can be effcently used to solve the schedulng of DRT servce. The problem of schedulng DRT systems s sometmes referred to n the lterature as the Dal-A-Rde Problem (DARP). It has been ntroduced n the research lterature snce the development of the DRT systems. It was well known snce the earlest studes that the DARP s an NP-hard problem (Savelsbergh and Sol, 1995), and only heurstc algorthms are feasble to solve ts real lfe nstances. Based on that, prevous research focused on developng effcent heurstc algorthms for solvng the statc DARP wth most requests known n advance. Several effcent heurstcs were developed, ncludng the popular nserton algorthm (Jaw et al., 1986). It s a heurstc algorthm that processes rde requests sequentally, nsertng one customer at a tme nto the work-schedule of some vehcle untl all rde requests have been processed. Central to the algorthm s a search for feasble nserton of customers n the work-schedules and an optmzaton step to fnd the best feasble nserton. An nserton of a customer nto the work-schedule of a vehcle s feasble only f t does not lead to volaton of any servce qualty constrants for the newly assgned customer and for all other customers already assgned to that vehcle. The optmzaton step deals wth mnmzng the addtonal cost due to nsertng the customer nto a vehcle s work schedule. The cost functon used s a weghted sum of dsutlty to the system s customers and system costs. One advantage of ths algorthm s that t has a lot of flexblty n the processng of requests. For example, customers can be processed ether accordng to ther earlest pck-up tme or latest delvery tme. Such processng method can be used to generate several alternatve solutons. The outcome of the

52 38 algorthm s a detaled work-schedule for each vehcle, lstng tmes and locatons for each of the pck-ups and drop-offs. Another wdely-used algorthm s the parallel nserton algorthm developed by Toth and Vgo (1997). The structure of the heurstc starts by ntalzng a small set of routes each wth a sngle pvot trp. The number of ntal routes (r) s gven by an estmate of the mnmum number of routes needed to serve a gven fracton of trps. Ths s done by consderng only the vehcle capacty constrants, and relaxng all other constrants. The vehcles are frst numbered n a decreasng order of an effcency score. These scores are used to dentfy the frst vehcles to be used for the ntal set of routes. After the determnaton of the set ntal routes, a number of pvot trps are chosen to be nserted nto each of the set of ntal routes. To do that, each trp s gven a score, whch s evaluated based on: 1. The attractveness of trp, defned as the number of trps whch have ther orgns or destnatons wthn a specfc amount of travel tme from the orgn or destnaton of trp (); 2. The average dstance between trp () and the pvot trps of the prevously consdered routes; and 3. The average tme between trp () and the depot of vehcle (v) servng the specfc route. The un-routed trps are then teratvely nserted nto the current routes, ntalzng a new route whenever a trp cannot be feasbly nserted nto any of the currently actve routes. The assgnment of the remanng un-routed trps (n) to the current routes s obtaned by solvng a mnmum-cost Rectangular Assgnment problem on (r*n) nserton cost matrx, where each row corresponds to a route and each column corresponds to an un-routed trp. Each element of the nserton cost matrx contans the extra cost correspondng to the best feasble nserton of a gven trp wthn a gven route. The soluton of the Rectangular Assgnment Problem can be computed by usng shortest path algorthm.

53 39 Other DRT schedulng heurstcs nclude the optmzaton-based method (Sexton and Bodn, 1985), and clusterng and mn-clusterng-based algorthm (Ioachm et al., 1995). More recent approxmaton soluton procedures nclude the column management scheme by Savelsbergh and Sol (1998) and the nserton heurstc developed by Madsen, Ravn, and Rygaard (1995). Some exact soluton procedures for solvng ths problem nclude the work of Psarafts (1980, 1983), Kalantar et al. (1985), Desrosers, Dumas, and Soums (1986), Fschett and Toth (1989), Dumas, Desrosers, and Soums (1991), and Ruland and Rodn (1997). Researchers also tred to tackle the stochastc nature of the operatons of DRT servces and ncluded that n the schedulng process. Such heurstcs nclude the Dal-A- Rde Problem wth Dynamc and Stochastc Travel Tmes (DARP_DS) developed by Fu (2002). The algorthm deals wth the DARP assumng tme-varyng, stochastc traffc congeston. It ncorporates a tme-dependent, stochastc travel tme model n the problem formulaton. The assumpton s based on the fact that travel tme between ndvdual locatons n urban traffc envronment are often subject to tme-varyng, stochastc varatons due to factors such as random fluctuatons n traffc volumes, frequent nterruptons of sgnal controls and unpredctable occurrences of traffc ncdents. As travel tmes are modeled as random varables, most of the performance measures and schedule varables become random varables and consequently the cost functon and servce constrants need to be redefned. The model assumes, for each O-D par, a stochastc process, representng the travel tme that a vehcle may experence when travelng from the orgn to the destnaton of that partcular OD par. For each tme nstance (t), at whch the vehcle may depart O to D, the mean and standard devaton of the travel tme between O and D are assumed to be gven as a set of varables for the schedulng process. Such a model s assumed to be suffcently accurate for representng the travel tme varatons under typcal traffc condtons wth tght trp servce tme wndows (less than 30mn) Schedulng the Flex-Route Problem Efforts for developng schedulng algorthms for the Flex-Route problem nclude the work done by Quadrfoglo et al. (2004), who proposed a schedulng heurstc for

54 40 schedulng the Moblty Allowance Shuttle Transt (MAST) system n Los Angeles County. The MAST system s a fxed-route transt servce that operates as a Flex-Route servce durng nght tme. The system conssts of a sngle vehcle, ntally assocated wth a predefned schedule along a fxed route, consstng of C checkponts, dentfed by c = 1, 2,, C; two of them are termnals located at the extremtes of the route (c = 1 and c = C) and the remanng C-2 ntermedate checkponts are dstrbuted along the route (see Fgure 2-5). The scheduled departure tmes at the fxed stops are assumed to be constrants of the system whch can not be volated. The model conssts of four dfferent types of requests: regular pck-up (P) and regular drop-off (D) representng customers pcked up/dropped off at checkponts; non-checkpont pck-up (NP) and non-checkpont drop-off (ND) representng customers pcked up/dropped off at any locaton n the servce area. For any request, the decson on whether to nsert the request nto the schedule s based on a cost functon that computes the cost for the possble nsertons, and chooses the one wth the mnmum value. The system s enttes consdered n ths algorthm are: 1. The customer requestng the nserton, n terms of how long the rde tme wll be. 2. The passengers already onboard and watng to be dropped off, n terms of how much longer they have to stay onboard, ncreasng ther rde tme. 3. The prevously nserted customers n the schedule watng to be pcked up at the NP stops, n terms of how much longer ther pck-up tme wll be delayed due to the re-routng procedure and also n terms of how much ther rde tme wll change. 4. The vehcle, n terms of how many extra mles t has to drve and therefore how much slack tme would need to be consumed.

55 41 Fgure 2-5 MAST System Based on these system enttes that wll be affected by an nserton of a new request, the algorthm computes the followng quanttes: PT: the sum over all passengers of the extra rde tme, ncludng the rde tme of the customer requestng the nserton. PW: the sum over all passengers of the extra watng tme at the already nserted NP stops. ΔT: the slack tme consumed by the nserton. The slack tme represents the resource needed by the system to serve more customers and T represents the consumpton of t. These cost quanttes, that represent the cost ncurred on the system enttes, are then combned together to provde the total cost functon as follows: Cost w PT w WT w T (Equaton 2-1) Where: w1, w2 and w3 are the weghts for each cost entty n the cost functon. In order to solve the problem the algorthm makes use of two control parameters that are a functon of the expected future demand and the relatve poston of the new

56 42 request wth respect to the already scheduled stops. The control parameter (π (0) 1) s multpled by the ntal slack tme and sets a cap on how much slack tme each nserton may requre. The (BACK) parameter (n mles) defnes the maxmum allowable backtrackng dstance avalable for each nserton. The dea behnd these control parameters s that a proper settng of these two parameters allows the system to control the consumpton of slack tme and mproves the overall performance Dynamc Vehcle Routng and Schedulng Dynamc vehcle routng and dspatchng problems have emerged as an area of ntense nvestgatons, due to recent advances n communcaton and nformaton technologes that now allow nformaton to be obtaned and processed n real-tme (Dror and Powell, 1993; Gendreau and Potvn, 1998; Powell et al., 1995). As compared to ther statc counterpart, these problems exhbt dstnctve features (Psarafts, 1995). In partcular, the data (e.g., customers to be servced) are not completely known before solvng the problem, but are dynamcally revealed as the current soluton, based on ncomplete and uncertan nformaton, s executed. Thus, t s not possble for the decson maker to solve the entre problem at once. Such problems are found n many dfferent applcaton domans, lke delvery of petroleum products and ndustral gases (Bausch et al., 1995; Bell et al., 1983), truckng (Powell et al., 1995; Powell, 1996), Dal- A-Rde systems (Wlson and Colvn, 1977) and emergency servces (Gendreau et al., 1997). In ths context, many dfferent factors must be consdered when a decson about the allocaton and schedulng of a new request s taken: the current locaton of the vehcle, the current planned route and schedule, characterstcs of the new request, travel tmes between the servce ponts, characterstcs of the underlyng road network, servce polcy of the transt agency and other related constrants. It s thus a complex decson problem where the decson must typcally be taken under consderable tme pressure. The earlest papers n the lterature on dynamc vehcle routng and dspatchng were presented n the seventes and were ether applcaton-orented (Wlson and Colvn, 1977) or analytcal (Daganzo, 1978; Sten, 1978). By the end of the eghtes, dynamc vehcle routng ganed an ncreasng attenton. Two major factors explan ths tendency:

57 43 new developments n nformaton technologes and the need for decson systems that could explot ths nformaton to better represent the real world. Interestng survey artcles on dynamc vehcle routng can be found n Psarafts (1995), Powell (1995) and Lund, Madsen, and Rygaard (1996). Because real-tme vehcle routng problems are NPhard and quck response tmes are requred, exact algorthms are not yet capable of handlng problems of realstc szes (Psarafts, 1980, 1983; et al Dal, 1995). Ths justfes the use of heurstcs n real-tme envronments. Neghbourhood search heurstcs, n partcular tabu search, were proposed as a means to effectvely and effcently tackle ths dynamc problem and optmze the planned routes between the occurrences of new events. Ths s acheved through neghbourhoods such as the ejecton chans, whch generate a sequence of nterrelated smple (component) moves to create a more complex compound move. Ejecton chans have seldom been mplemented n practce for solvng vehcle routng problems, wth notable exceptons for the Travelng Salesman Problem (Rego, 1998a), the Vehcle Routng Problem (Rego and Roucarol, 1996; Rego, 1998b) and the Vehcle Routng Problem wth Tme Wndows (Braysy, 2003; Caseau and Laburthe, 1999; Rousseau et al., 2002; Sontrop et al., 2005). Accordng to how they deal wth the dynamc aspects of the problem, the problem solvng approaches reported n the lterature can be classfed nto two major categores. They are reported n the followng subsectons Adaptaton of Statc Algorthms Ths approach s based on the noton of a rollng horzon. As tme unfolds, statc problems are solved repeatedly over events found wthn a horzon of length L that extends from the current tme t to t +L. Dfferent strateges emerge when the length L of the horzon s modfed. If L s very small, a myopc near-term strategy s observed. In some cases ths strategy can provde near-optmal solutons (Bell et al., 1983; Dal, 1995; Psarafts, 1985; Powell, 1988). In contrast, f L s very long, the problem consdered s rcher but, because t contans long-term events, the soluton obtaned s typcally weaker unless a fast and powerful soluton procedure s used (Trudeau et al., 1989; Gendreau et al., 1996a, 1996b). Adaptaton of statc procedures to dynamc vehcle routng problems can be dvded nto two classes:

58 44 1 A sophstcated statc problem-solvng procedure, whch typcally nvolves a reoptmzaton of the routes, s appled each tme an nput update occurs. Several researchers have used ths approach lke Bell et al. (1983), Hll et al. (1988), Brown et al. (1987), Psarafts (1980, 1983), Psarafts et al. (1985), Powell et al. (1988) and Dal (1995). The drawback of ths approach s the amount of computaton tme resultng from repeatedly executng the statc algorthm. Ths dsadvantage s more dramatc when new events occur frequently and when the executon of the statc algorthm requres more tme. 2 Fast local operatons (e.g., nserton) are used for reactng to any nput revson (Trudeau et al., 1989; Wlson and Colvn, 1977; Solonk, 1991; Madsen, Ravn, and Rygaard, 1995). Ths approach s easy to mplement and s approprate for a dynamc envronment where tme pressure s mportant (e.g., Lund, Madsen, and Rygaard, 1996). However, t s myopc because solutons are produced through consecutve nsertons (whereas a complete reorderng of the routes may lead to better solutons). To overcome ths weakness, some authors combne local operatons wth re-optmzaton procedures. Ths s often acheved by executng a set of successve nsertons followed by a local search (e.g., exchange procedures lke 2-opt, see Ln, 1965). For dfferent applcatons reported n the lterature, see Roy et al. (1985), Gendreau et al. (1996a, 1996b). All studes mentoned above gnore the potental benefts of consderng the stochastc aspects of the problems and tryng to forecast the future. Stochastc methods are amed at overcomng ths weakness Stochastc Methods Real-tme dspatchng problems have a stochastc nature (e.g., accdents, congeston, unexpected changes n meteorologcal condtons, etc.). Stochastc methods can be vewed as a natural way to judcously address these ssues. The goal s to react properly to an event to nsure a good qualty of servce to the customers dsturbed by these events, whle mnmzng ther undesrable mpact on the whole system. Two major classes of stochastc approaches are reported n the lterature: stochastc programmng and Markov decson processes.

59 45 Markov Decson Processes Formulatons based on ths modelng approach were proposed by Powell (1988), Bertsmas and Van Ryzn (1991, 1993). Unfortunately, Markov decson processes are confronted wth the followng lmtatons that often prevent them from beng appled to complex real-world problems: () the state space grows quckly wth problem sze; () smplfyng assumptons are often made to make the model more tractable. Stochastc Programmng The only work n ths category s the one done by Powell et al. (1988) n ther comparatve revew of dynamc vehcle allocaton problems. The authors proposed a hybrd model that combnes nsghts from Markov decson processes and classcal network formulatons. Other Methods A new generaton of approaches try to replcate a sklled dspatcher s decson makng process. Ths s acheved by automatng the decson procedure based on prevous decsons taken by a sklled dspatcher. Wthn ths framework, Shen and Potvn (1995) used a neural network to elaborate an expert consultng system for a dspatcher workng n a courer servce company. In the same context, Benyahya and Potvn (1995) proposed an approach based on genetc programmng Flex-Route Transt Schedulng Although the concept of Flex-Route transt servce as an nnovatve transt servce has been around snce the late seventes, ts mplementaton has not been wdespread. Untl very recently, the mplementaton of route devaton servce by transt agences was lmted to rural and small urban areas due to the complexty of schedulng such a servce n densely populated urban and suburban areas. Thus, the research on Flex-Route schedulng s scarce n the lterature and most of the proposed schedulng heurstcs are alteratons of the algorthms used for schedulng DRT servces. In a Flex-Route transt system, servce s provded at fxed tmes and locatons (fxed stops), whle also provdng an on-demand servce to customers off the fxed route. Thus, schedulng and dspatchng Flex-Route transt can pose a sgnfcant challenge due to demand at fxed-stops, the requests for devatons, and the dynamcs of bus schedule

60 46 adherence. Ths challenge has led to many proposed heurstcs for schedulng Flex-Route transt. Quadrfoglo, Dessouky, and Palmer (2007) developed a smlar nserton algorthm to schedule the MAST system dscussed earler, whle Zhao and Dessouky (2008) studed the optmal servce capacty through a stochastc approach. Cranc, Malucell, and Nonato (2001) descrbed the MAST concept and ncorporated t n a more general network settng whle also provdng a mathematcal formulaton. Other works can be found n Cortés and Jayakrshnan (2002), Horn (2002a, b), and Aldahan and Dessouky (2003), whch prmarly focus on the operatonal control and schedulng of such systems. Dessouky and Aldahan (2003) proposed a hybrd servce delvery method that ntegrates demand-responsve transt servce and fxed route transt to satsfy a set of demand-responsve trps. In the servce, DRT passengers use both DRT and FRT servces to make ther trps. The servce has a set of (M) paratranst vehcles wth known capacty that are used to pck up DRT passengers from ther orgns or from the fxed bus stops and drop them off at ther fnal destnatons or at the fxed bus stops. On the other hand, the fxed bus route system ncludes a set of (R) fxed bus routes. Each fxed bus route has a number of buses that travel through t, a set of bus stops and a tme schedule. Passengers that are strctly served by the DRT vehcles are referred to as door-to-door requests whle those passengers that transfer to a fxed route bus lne are referred to as hybrd requests. Whle n dal-a-rde servce the objectve s prmarly to determne the vehcles schedule, n a hybrd system the vehcles schedule as well as the delvery path for each request needs to be determned. The heurstc procedure determnes the ondemand vehcle schedule and the best canddate path for each request. The schedulng heurstc has four dstnct stages: the nserton procedure; the mprovement procedure; the re-sequencng step; and the re-assgnng step. In the nserton procedure, the canddate path wth the shortest on-demand vehcle travel dstance s selected for the hybrd requests. Ths may not be a good rule n terms of mnmzng the passenger trp tme snce the path wth the shortest on-demand vehcle travel dstance may connect to a fxed route bus lne that wll requre watng at the bus stop and have a long travel tme on the bus. So n the next step the mprovement

61 47 procedure searches for other canddate paths of each hybrd request that can reduce the passenger trp tme. In the re-sequencng step, a search technque s used n order to fnd an mproved requests sequence, whch leads to shorter vehcle dstance n every vehcle whle holdng the request to vehcle assgnment fxed. Ths leads to an mproved schedule by movng ndvdual predecessor (pckup pont) and/or successor (delvery pont) forward and/or backward n ther correspondng route. Three condtons need to be satsfed whle movng the pckup and delvery par n order to have a feasble schedule. The frst one s the precedence constrant where the pckup pont of any request must be vsted before the delvery pont of the same request. The second s the on-demand vehcle capacty constrant. The thrd one s the pckup tme wndow (and the delvery tme wndow for the frst leg of the hybrd requests). Fnally, the re-assgnng step tres to fnd whch request should be removed from ts current vehcle schedule, and where t should be nserted. It s worth mentonng that the above-mentoned studes focused prmarly on the statc type of the Flex-Route problem, where requests are known n advance and routng schedules to meet those requests were bult wthout allowng for addtonal real-tme requests. As such, there s a dearth of research of schedulng flexble servce n a realtme envronment, where last-mnute or real-tme requests for servce could orgnate after the statc schedule has been bult. Ths research focuses on several desgn and schedulng varables, and ther effects on the qualty and cost of Flex-Route servce. Specfcally, the effects of relaxng the departure tme constrants at the fxed stops of the servce, and those of the slack tme length were a major focus of ths study. In ths research, the real-tme part of the problem and ts mpact on the statc schedules were nvestgated through a senstvty analyss of the varables mentoned above. 2.6 LIMITATIONS OF EXISTING RESEARCH Most of the research on Flex-Route transt has focused ether on recognzng the potental of Flex-Route servce to play a vtal role n provdng transportaton servce for low-to-medum-demand urban areas, or on statng the mportance, or lack thereof, of

62 48 usng advanced technology n the feld of transportaton n the process of schedulng and routng of Flex-Route servce. However, exstng work lacks any comprehensve analyss of the mportant factors that affect Flex-Route servce and how they mght affect the operaton of such servces n terms of buldng schedules that capture the relatve mportance of all the nvolved factors. Currently, most agences that provde flexble servce do not have clear allocaton of scheduled tme for fxed and devated servce at all (Koffman, 2004). The allocaton of scheduled tme n these agences does not take nto consderaton any factors related to the cost and convenence of the rders or the agency tself. These agences would beneft from addtonal gudance n ths area. Furthermore, statc and dynamc operatons of Flex-Route servce requre a customzed decson-support system responsble for trp bookng, schedulng, routng and dspatchng. The real-tme schedulng and dspatchng task can become especally challengng when the system has to deal wth a large volume of requests for devaton and hghly vared demand at fxed stops. Currently, there are no robust and effcent algorthms avalable for schedulng trps for ths type of servces. Most of the exstng Flex-Route servce provders do not beneft from the recent advancements n communcaton technology snce they rely on ndvdual decsons made by the drvers n real-tme. There s, therefore, an urgent need to develop new schedulng algorthms that can be used to adaptvely adjust the exstng routes and schedules n response to specfc events such as traffc ncdents and congeston, trp cancellatons and new requests n real tme. Such schedulng mechansms should nclude the applcatons of advanced nformaton technologes such as automatc vehcle locaton and computer aded dspatch systems (AVL/CAD), geographc nformaton systems and dgtal telecommuncaton technologes. Wth the ablty to track vehcle locatons, communcate wth drvers and clents, and access traffc nformaton on a contnuous bass, Flex-Route transt systems could be expected to operate at a sgnfcantly mproved level of productvty, relablty and qualty of servce.

63 49 3 FLEX-ROUTE PLANNING AND DESIGN PARAMETERS 3.1 CHAPTER OVERVIEW Ths chapter presents the frst component of ths research: a detaled analytcal model for the plannng and desgn process of Flex-Route transt servce. The frst secton dscusses the need for Flex-Route transt servce n suburban areas and what factors should be consdered when plannng for transt servces n the suburbs. The chapter then presents Flex-Route-specfc characterstcs and those of suburban areas n whch Flex- Route transt can be applcable. Followng that, a detaled analytcal model of the desgn parameters of Flex-Route transt s presented, ncludng the modfcatons that need to be made to the fxed-route structure to allow for the provson of Flex-Route servces. The chapter proceeds to dscuss the costs ncurred by the stakeholders nvolved n the servce such as the transt operator and the fxed-route users, and what strateges can be mplemented to allevate ths cost. Followng that, we present two methods for montorng the performance of Flex-Route transt. 3.2 SUBURBAN TRANSIT SERVICE: CHALLENGES AND OPPORTUNITIES Tradtonally, Fxed-Route Transt (FRT) servces have been desgned to serve concentrated travel patterns that allow for large numbers of people to be conveyed along establshed routes followng set schedules. The man purpose of these systems s to move customers along a prmary drecton, whch may be around a loop or back and forth between two termnal checkponts. In densely bult-up ctes wth strongly focused travel patterns such as commutng to and from downtown areas, FRT systems usually have acceptable cost effcency rates due to the predetermned schedule, hgh demand served per vehcle and the large loadng capacty of the vehcles. However, urban sprawl has caused a steady declne n land use denstes of urban areas across North Amerca, gvng rse to suburban communtes wth low-densty and dspersed actvty patterns. In the US, populaton densty dropped 15% from 1960 to 2000 despte an average overall populaton growth of 86% ( Ths ncreasng dsperson of populaton causes conventonal fxed-route transt systems servng those areas to become progressvely more neffcent and relegated to a margnal role, snce they are usually

64 50 desgned to serve few establshed corrdors and they rely heavly on concentrated demand. As a result of the contnuous rse of low-densty urban development, the general publc s ncreasngly consderng transt to be nconvenent due to ts lack of flexblty, snce ether the locatons of pck-up and/or drop-off ponts or the servce s schedule do not match the ndvdual rder s desres. Therefore, an ncreasngly larger porton of the growng populaton reles almost exclusvely on prvate automobles for ther transportaton needs, causng modern urban areas to suffer from severe congeston and polluton problems. Therefore, the challenges facng transt agences of makng transt servce a vable opton n the suburbs are mmense. In suburban areas, transt servces are not only competng wth the automoble, but also have to deal wth low denstes and dspersed trp patterns. Therefore, the task of creatng effcent publc transt servces n suburban areas presents a sgnfcant challenge to servce provders. Nevertheless, effectve transt plannng and the provson of new types of transt servces have the potental to capture a greater share of the suburban travel market and present an alternatve to the prvate automoble. The vsble dfferences n trp patterns and n spatal arrangements between the suburbs and the tradtonal cty suggest that transt servces should adapt to these changes. In the suburbs, travel movement s usually characterzed by three dstnct patterns: 1. Trps from the suburbs to the urban core; 2. Reverse commute trps from the urban core to the suburbs, and 3. Suburb-to-suburb movements. Therefore, suburban transt servce plannng needs to take nto account the specfc travel patterns of people n these areas. Moreover, transt operators need also to recognze that rdershp volumes n suburban areas should not be compared to those of the tradtonal fxed-route rdershp found n a typcal large cty. Transt planners need also to recognze that the automoble domnates travel n suburban areas, and thus mprovng transt use n these areas requres recognzng the features that contrbute to ts domnance and how they can be addressed when new transt servces are beng ntroduced. Ths dfference mples that the crtera used to plan and evaluate servces n the suburban areas should be dfferent from those adopted n dense

65 51 urban areas. Furthermore, transt planners need to recognze that consumer appeal and acceptance are central to the success of new transt servces and thus must be central to the plannng effort. Some of the attrbutes that make the car more appealng for customers and need to be taken nto account when consderng a new transt servce nclude: Drectness and comparatve travel tme; Comfort and servce qualty; Convenence of the schedule (e.g., flexblty, mnmal transfer, connectvty); and Prcng, ncludng overall cost and smplfcaton of payments. Transt agences that want to employ new types of transt servce must account for these and other factors. A report by Urbtran Assocates Inc (1999) provded a lst of gudelnes that concentrate on servce modfcatons and nnovatons desgned to help transt servce provders create more effectve transt servce n the suburbs. The gudelnes were extracted through a survey and case studes n dfferent suburban transt markets to ntegrate transt nto overall moblty strateges. The report provded several key fndngs of successful transt strateges n suburban areas; some of these fndngs nclude: Operate along moderately dense suburban corrdors; Lnk suburban transt servces to the broader regonal lne-haul network; Adapt vehcle fleets to customer demand; Creatvely adapt transt servce practces to the landscape; Plan wth the communty; and Establsh realstc goals, objectves, and standards. In ths regard, one of the most notable nnovatons s testng new flexble transt concepts taken from experences wth paratranst servces. These servces are usually provded by modfyng fxed-route servces that work n urban settngs by addng some flexblty to the servce. Such flexble servces nclude Flex-Route transt servce, whch could be a vable transt opton n suburban areas. For Flex-Route transt to have a

66 52 comparatve and compettve servce wth the prvate automoble, t must be desgned to acheve several goals; for example: Mnmze overall travel tme by ensurng well-tmed connectons wth regonal ral servce and major transt networks; Provde these connectons as effortlessly as possble wth short walk dstances, tght schedulng, and approprate frequences; and Consder mechansms for sngle prcng of the entre trp. In ths context, the desgn of a Flex-Route transt servce follows from ts ntended role n a transt system s overall servce plan, the crcumstances that led to ts ntroducton, and the objectves t s ntended to serve. The remanng part of ths chapter dscusses the plannng and servce desgn of Flex-Route transt servce n suburban areas. The analyss s ntended for cases n whch Flex-Route servce s beng consdered to replace an exstng under-achevng fxed-route servce. 3.3 FLEX-ROUTE SERVICE DESIGN Surveyng the lterature for gudelnes on the desgn process of Flex-Route transt servce reveals that there s lttle lterature avalable that provdes such gudelnes n a comprehensve and systematc manner. Therefore, there s a need for a detaled lst of the desgn parameters of Flex-Route transt servce and what modfcatons need to be made to an exstng fxed-route servce to accommodate the new servce. In ths regard, several key ssues need to be addressed such as the optmal values of each desgn factor; the nteracton between these factors ncludng how the change n one desgn parameter wll affect the optmal value of another desgn factor; the effect on the operaton of the fxedroute part of the servce; and the effect of each desgn parameter on the operaton and performance of the Flex-Route system. In an artcle evaluatng the Potomac and Rappahannock Transportaton Commsson s (PRTC) Flex-Route servce, Farwell (1998) acknowledges ths fact. PRTC desgned ts Flex-Route system prmarly based on what t felt was rght. The same concluson was drawn by Koffman (2004) where the author found that the majorty of Flex-Route servce provders do not follow any specfc gudelnes when ntroducng the

67 53 servce. Ths reveals the crtcal need for some gudelnes on the desgn of Flex-Route transt servce. The study and analyss conducted n ths research wll help dentfy and shed some lght on the key desgn parameters of Flex-Route servce such as: servce area, slack tme, servce frequency, slack tme dstrbuton, and demand patterns. Furthermore, the research also ncludes a dscusson addressng the modfcatons that need to be made to the fxed-route part of the servce such as the servce headway, fxed stops locaton and spacng, and the effect on the level of servce and rdershp levels. Before we advance wth the servce desgn parameters of the Flex-Route servce, there are some assumptons and terms concernng Flex-Route transt consdered n ths research that should be defned frst (see Fgure 3-1). These are: A vehcle run (r) s defned as the contnuous travel of the transt vehcle from one end of the route (.e. termnal 1) to the other end (.e. termnal N) n one drecton. A complete cycle of the transt vehcle conssts of two consecutve vehcle runs startng and endng at the same termnal. The transt vehcle wll make mandatory stops at the fxed stops and adhere to the constrant of departng on schedule at these stops unless otherwse stated. Ths assumpton s a key constrant n the operaton of Flex-Route servce and should be respected when advsed to do so. However, there are cases n ths research n whch ths constrant wll be relaxed to examne the effects of the fxed-stop schedule adherence constrants on the servce. The total scheduled runnng tme of one vehcle run (T) s defned as the tme needed to travel from one end of the route to the other end n one drecton. Ths tme s fxed and ncludes the slack tme bult n the schedule (S) to pck up/drop off on-demand passengers n the servce area of the route. A route segment or servce area zone s defned as the area between a par of two consecutve fxed stops on a route where on-demand requests can be served. Two drectons of travel are usually used between the end ponts of the routes: nbound and outbound. The transt vehcle s sad to be travelng n the nbound drecton f t s travelng n a general drecton that s gong towards the end pont of the route (.e. termnal or fxed stop N). If the bus s travelng n the drecton out of the end fxed stop, t s sad to be n the outbound drecton.

68 54 Inbound Legend Outbound Fxed Stop On-Demand Request Frst Fxed Stop W Route Segment Route Segment W L Last Fxed Stop Servce Area Wdth Route Length W W L Fgure 3-1 A typcal Flex-Route servce area

69 Area Confguraton Flex-Route transt servce occupes a mddle ground between tradtonal fxedroute transt servce and demand-responsve transt servce. The suburban area confguratons, where Flex-Route transt servce can be applcable, and the servce characterstcs of Flex-Route transt servce are dscussed n ths secton. A survey by Koffman (2004) dentfed four prmary suburban area confguratons n whch flexble transt servces can be applcable. Table 3-1 provdes the number of agences that used flexble servces n each confguraton. For Flex-Route transt servce, the most commonly-used stuatons n whch the surveyed agences appled the servce were (1) large areas and (2) hard-to-serve areas. The followng s a dscusson of the two confguratons Spread-Out Areas Some transt systems mght choose to use Flex-Route servce as ther method of operaton for the entre transt system. Ths s usually the case n rural and small urban areas and n low-densty suburban areas that use Flex-Route as a way of ncreasng coverage. Gven a transt systems desre to serve as many ponts of nterest as possble n a spread-out area, Flex-Route servce s seen as more effectve than operatng a fxed route that attempts to connect all potental ponts of nterest regardless of actual demand. Another reason for usng Flex-Route servce n these areas s reducng or elmnatng the need for separate paratranst servce for people wth dsabltes. In some settngs, the cost savngs from provdng combned servce for people wth dsabltes and the general publc can be crucal n makng transt servce economcally vable Hard-to-Serve Areas In ths stuaton, Flex-Route servce s usually the only transt servce offered. Typcally, most of these servces have the same operatng hours as the transt agency s other local routes. In such cases, the servce s provded n hard-to-serve neghbourhoods and s connected to a regonal transt network. The motvatons for usng Flex-Route servce n these neghbourhoods could vary consderably, dependng on the goals of the transt servce. One of these reasons s that many transt operators have polcy mandates

70 56 and communty prortes to cover as much of ther servce area as possble. Flex-Route servce offers a way to provde such coverage n low-demand areas to establsh basc access and connectons to express routes or regonal ral lnes. Another motve for usng Flex-Route servce n these areas s layng the ground for future fxed-route transt. In such cases, Flex-Route servce can provde a transton between dal-a-rde, or no servce at all, and conventonal fxed-route transt servce. Resdents may be able to avod buyng second and thrd cars, and they may be more lkely to use conventonal transt when t s mplemented. As demand patterns become clearer through flexble operaton, effcent fxed routes can be desgned Servce Characterstcs Flex-Route transt servce dffers from fxed-route servce n many servce features. These dfferences n servce characterstcs can be characterzed by four elements as summarzed n Table 3-2. These elements are dscussed next Vehcle route In Flex-Route transt servce, vehcles operate along a defned route, as n fxedroute servce, but also respond to on-demand requests by dvergng from the route. The vehcles are not requred to follow a specfc route and could have a dfferent route from one day to another. The only constant n the servce s the fxed stops along the route. All vehcles are requred to be at the fxed stops before the publshed departure tme at these stops Boardng and alghtng locatons For fxed stops, whch may be along a defned path, passengers can board and alght wthout any knd of advance notce. On-demand customers on the other hand wll board the vehcles at the specfed address provded by customers when they request the servce Schedule The tmes when vehcles wll be at boardng and alghtng locatons are some mx of pre-scheduled tmes and tmes determned by demand. Fxed stops wll have daly fxed departure tmes that wll not be affected by the on-demand requests. Tmes at the

71 57 on-demand locatons may change daly based on the demand, although they are constraned by the fxed porton of the schedule Advance-notce requrements At fxed stops served on a schedule, there s no need for passengers to request a boardng or alghtng ahead of tme. At on-demand ponts, some type of advance notce s needed. Such notce may take the form of a phone call to a dspatch center or accessng a webste that provdes the requestng customer wth the decson to ether accept the request or reject t. A subscrpton that consttutes a standng order for the same trp every day or every week may be possble.

72 58 Table 3-1 Number of transt agences usng flexble servces n dfferent suburban areas confguratons Servce Confguraton Prmary servce n large area Demand- Responsve Connector Flexble-Route Segments Servce Type Pont Devaton Request Stop Flex- Route Zone Route Total 5 5 Prmary servce n lmted hard-to-serve area Servce at low-demand tmes n a large area Servce at low-demand tmes n a lmted area Total Source: Koffman (2004)

73 59 Table 3-2: Elements of servce desgn for fxed-route, demand-responsve and flexble transt servces Servce Type Element of Servce Fxed Route DRT Flexble Where vehcles On the defned route A geographc area A route plus off-route locatons or operate geographc areas Boardng and alghtng locatons Fxed stops Any locaton n the servce area Fxed stops plus other customers locatons Schedule Fxed Depends entrely on trps requested Fxed checkponts on the route, varable at other locatons Advance notce requrements Source: Koffman (2004) Not requred Always requred Requred for demand-responsve requests

74 Flex-Route Servce Area Once the decson to provde Flex-Route servce on a specfc route has been made, the next crtcal step s selectng the servce area of the Flex-Route servce. As mentoned earler, Flex-Route servce can be ntroduced as a replacement of an exstng fxed-route and paratranst servce area or a completely new servce n a new area. In our research, however, we focus on the case where Flex-Route transt servce replaces an exstng fxed-route transt servce. In such scenaro, several modfcatons have to be made to the fxed route structure to enable the provson of Flex-Route servce. Ths s requred because the servce characterstcs of a fxed-route and Flex-Route servce dffer n many ways. The servce area wdth of a Flex-Route servce defnes how far away from the standard route a vehcle may devate to pck up or drop off passengers. Under deal condtons, the servce area of Flex-Route transt servce (A) s represented by a rectangular regon as shown n Fgure 3-1, and s equal to (L x W), where A s the servce area of the route (n km 2 ); L s the total route length or the dstance between stops 1 and N (n km); and W s the wdth of the servce area or dstance that corresponds to the maxmum allowable devaton from the man route on ether sde (n km). Usually, the servce area of the route s dvded nto separate route segments. Each route segment can be gven a unque dentfcaton number to dfferentate t from the other segments; ths number can be based on the fxed stops t falls between. For example, route segment (1, 2) s the dentfcaton number of the segment between fxed stops 1 and 2. Ths defnton can be very useful n several scenaros; for example: a) In real tme operatons, ths dentfcaton number would be helpful to track the poston of the vehcle; b) The wdth of the servce area may dffer from one route segment to another due to factors such as topographcal features and street network. Thus, gvng a unque dentfcaton number to each zone wll be helpful to dstngush the dfferences among the zones along the route; and c) Ths defnton can be used to dentfy the locatons of trp requests wth respect to the fxed stops and wth respect to each other.

75 61 By vrtue of the defnton of the servce area, the total number of route segments for a gven route s always one less than the total number of fxed stops. In case there are N fxed stops as shown n Fgure 3-1, the number of route segments s N-1. In the Flex-Route transt servce lterature, the decson on how much a transt vehcle can devate from the man route to pck up on-demand customers has been commonly made an ad-hoc bass that does not follow systematc desgn gudelnes. Koffman (2004) reported that almost 75% of the reported Flex-Route transt servces have a formal polcy about how far the buses can devate from the route. However, there s great varaton n how the maxmum extent of devaton s defned. As shown n Table 3-3, the extent of devaton formally permtted ranges from 0.25 to 1.5 mle (400 to 2400m). The remanng systems have more flexble or nformal polces. Ths ad-hoc bass can be used f there s no lmt on the slack tme gven to a route or there s no nformaton avalable on the on-demand rate or dstrbuton. In ths case the transt agences usually set the servce area to what they judge as approprate. Table 3-3: Maxmum extent of devatons reported for Flex-Route transt servce Permtted Devaton Area 0.25 mle from route MTS (San Dego, CA) Transt System 0.50 mle from route Akron Transt (Akron, Oh), Mnnesota Transt (Burnsvlle, MN) 0.75 mle from route PRTC (Woodbrdge, VA), GRTC (Rchmond, VA) 1.50 mle from route Tllamook Transportaton (Tllamook, OR) Zones (unknown dstance) Cty lmts Informal Source: Koffman (2004) Rde Solutons (Palatka, FL) Napa Transt (Napa, CA), St. Joseph Transt (St. Joseph, MO) Madson County (Grante Cty, IL), Mason County (Shelton, WA), OTA (Ottumwa, IA)

76 62 However, decdng on the servce area wdth for each segment of the route depends on several factors that dffer from zone to zone, such as areas of potental ondemand rdershp, street network connectvty, topographcal lmtatons and assgned slack tme among other factors. Ths warrants the need to analyze the effect of these factors on the zone sze of a Flex-Route transt system. In ths research, two cases of servce area confguratons are studed: two-sded servce areas and one-sded servce areas. The followng sectons provde an analyss of each case Two-Sded Servce Areas The amount of slack tme assgned to a specfc route segment s a crtcal factor n determnng how far the transt vehcle can devate from the man route to pck up or drop off customers, and thus have a drect relatonshp wth the wdth of the servce area. In real-world applcatons of Flex-Route transt servce, the servce provder mght be bound by factors such as street network and topologcal features that requres the servce area to conform to the exstng area boundares. In such cases, the wdth of the servce area wll be determned based on these lmtatons. In other cases, however, the wdth of the servce area of a Flex-Route transt servce may be lmted by the amount of slack tme assgned to the route n general or to a specfc route segment. For a typcal route segment n two-sded servce areas, where the transt vehcle can devate on ether sde of the route to pck up or drop off customers, the servce area of ths segment conssts of two zones on each sde of the route, typcally wth the same wdth. In some areas, ths mght not be the case, and the wdth of the servce area on one sde could be dfferent from the other sde. In ths research, however, we lmt the analyss to cases n whch the wdth of the servce area for a specfc route segment s the same on both sdes of the route. In Flex-Route servces, one mportant assumpton that needs to be made s whether the transt vehcle wll be allowed to perform backward movements wth respect to the current locaton of a specfc on-demand request (.e. servng the customers n ther horzontal coordnates). In ths research, we lmt the analyss to the no-backtrackng polcy, whch mples that the transt vehcle can move only n the forward drecton to serve the on-demand requests (left to rght) as llustrated n Fgure 3-2.

77 63 Takng the above-mentoned assumptons and lmtatons nto consderaton, the followng parameters are defned: m = the average accepted customer demand of on-demand requests n route segment n vehcle run r (assumng unform demand dstrbuton); λ = the densty of the on-demand requests n route segment (n requests/km 2 ) h = the length of the rectlnear Hamltonan path of the transt vehcle n route segment (n km); W = the wdth of the servce area on ether sde of route segment (n km); the total wdth of the servce area s equal to 2W. L = the length of route segment (n km); V = the average operatng speed of the vehcle (n km/hr); DT = the drect runnng tme to travel from the frst fxed stop to the end fxed stop of route segment (n hours); T = the total tme avalable for the transt vehcle to travel through route segment (n hours); S = slack tme assgned to route segment (n hours); dy 1 = a random varable ndcatng the lateral dstance traveled between two random ponts n a rectangle (n km); and dy 2 = a random varable ndcatng the lateral dstance traveled between a fxed stop at ether end of the rectangle and any pont n the rectangle (n km). To fnd the dstance travelled by the transt vehcle n route segment, we assume that vehcles follow rectlnear paths wthn the rectangular servce area (see Dessouky et al., 2005). Based on ths assumpton, the vehcle travels a dstance of dy 2 n the lateral drecton from the frst fxed stop to the frst on-demand request and then travels a dstance of dy 1, (m -1) tmes between each par of on-demand requests. Fnally, the vehcle travels a dstance of dy 2 from the last on-demand request to the end fxed stop of the route segment. Also, the no-backtrackng polcy mples that n the horzontal drecton, the transt vehcle wll travel a dstance equal to the length of the route segment (L ). Therefore, the expected value of the rectlnear Hamltonan path (h ) of the transt vehcle n route segment can be gven by the followng relatonshp:

78 64 E h m 1Ed 2Ed L. (Equaton 3-1) y1 y2 Assumng the demand s unformly dstrbuted throughout the servce area of a route segment, the expected values of dy 1 and dy 2 for route segment are gven by: 2W E d y1.... (Equaton 3-2) 3 2W W E d y2... (Equaton 3-3) 4 2

79 65 Legend Drecton of Travel Fxed Stop Accepted On-Demand Request Rejected On-Demand Request W 2W /3 W W W /2 L Fgure 3-2 T1wo-sded servce area of one route segment

80 By substtutng equatons 3-2 and 3-3 nto Equaton 3-1, the expected value of (h ) s gven by the followng formula: 66 2W W L... (Equaton 3-4) 3 2 m 1 2 E h Equaton 3-4 provdes the expected length of the rectlnear Hamltonan path n route segment n terms of the servce area wdth (W ), the average demand-responsve customer demand (m) and the route segment length (L ). However, gven that DT - the tme needed to travel drectly from the frst fxed stop to the end fxed stop of route segment - s DT L, then the remanng part of the rectlnear Hamltonan path V has to be covered by the slack tme assgned to the route segment (S ) as follows: S m 2W W (Equaton 3-5) V It s worth mentonng that equaton 3-5 does not nclude an allowance for dwell tme at the on-demand stops. Although an average dwell tme can be approxmated for on-demand users, we felt that a better approach to ths ssue s to account for the dwell tme n the average operatng speed of the vehcle, gven that the average operatng speed wll account for all stoppage and turns n the devated part of the route. The estmaton of the average operatng speed could be performed through smulaton analyss that takes nto consderaton all external and stochastc factors that mght affect the operaton of the servce. Rearrangng the terms of Equaton 3-5 n terms of the wdth of the servce area (W ), the followng formula can be extracted: 3V S W.... (Equaton 3-6) 2m 1

81 67 Equaton 3-6 provdes a relatonshp that shows whch Flex-Route servce elements have a potental effect on the wdth of the servce area. The equaton reveals that the wdth of the servce area of a route segment depends on the transt vehcle operatng speed (V), the slack tme assgned to the route segment (S ) and the average customer demand of the on-demand requests (m ). The equaton shows that the wdth of the servce area of a route segment s postvely related to the amount of slack tme assgned to ths segment, and therefore s controlled by the value of the slack tme. Ths relatonshp can be expected as the more slack tme avalable for the transt vehcle wll allow the vehcle to reach farther dstances and thus expandng the wdth of the servce area and vce versa. The equaton also shows that the wdth of the servce area s postvely related to the vehcle operatng speed and nversely related to the average customer demand. Both relatonshps are expected and easly explaned: the hgher operatng speeds means the transt vehcle can travel farther dstances to serve customers gven the same amount of slack tme, whle hgher customer demand rate means there wll be enough demand wthn a smaller servce area and therefore lmt ts ablty to serve farther requests. Equaton 3-6 uses the average customer demand as a measure of the demand for the servce. In several cases, however, other measures of the demand could be used such as the demand densty of the on-demand customers, whch may depend on both W and L. In ths case, assumng that the average customer demand of the on-demand requests depends on a unform demand densty: m 2W L, then Equaton 3-6 can be rewrtten as follows: W 3V S 4 W L (Equaton 3-7) Rearrangng the terms of Equaton 3-7, the optmal maxmum value of W can be found usng the followng equaton:

82 68 W L V S 8 L... (Equaton 3-8) Equaton 3-8 gves the optmal maxmum wdth of a servce area for two-sded servce areas n terms of the demand densty (λ ), the route segment length (L ), the slack tme (S ) and the bus operatng speed (V). As wth Equaton 3-6, Equaton 3-8 shows that the maxmum wdth of the servce area (W ) ncreases as the values of S and V ncrease, whch s expected as explaned earler. Equaton 3-8 also shows that W s nversely related to λ and L, whch means that the hgher the demand rate the smaller W should be. Ths can be explaned by notng that hgher demand rates and longer route segments mean hgher number of demand requests that need to be served close to the route, and therefore t wll be benefcal to serve the requests close to the fxed-route than ncreasng the wdth of the servce area. We now can use Equaton 3-6 to provde a relatonshp for the total servce area of the whole route as follows: N 1 N 1 3V S L A L W 1 2m 1 1 N 1 3 S L A V... (Equaton 3-9) 2m 1 1 Equaton 3-9 can be used to gve the total servce area of the Flex-Route servce for any gven route gven the transt vehcle operatng speed, the length of each route segment, the slack tme and demand rate for each route segment. In cases where all the route segments are smlar n terms of the values of S, L and m, then Equaton 3-9 can be re-wrtten as follows: 3V S L A N 1... (Equaton 3-10) 2m 1

83 69 Where (N) s the total number of fxed stops along the route n each drecton. It should be noted that the above equatons are bult based on the assumpton of the rectlnear Hamltonan path assumpton made earler, and therefore t wll not be vald f other assumptons regardng the vehcle path are made. Ths relatonshp can be very helpful for transt agences as t gves a mathematcal relatonshp that demonstrates how the wdth of the servce area can be manpulated based on the values of several factors Sngle-sded servce area The prevous secton analyzed the wdth of the servce for two-sded servce areas, whch s usually the case n most stuatons. In ths secton, however, we analyze the case where the servce area of the route s only on one sde of the route (see Fgure 3-3. Ths case can be found n several stuatons n the suburbs where the street network and the area confguraton do not allow for the transt route to run along the center of the servce area. In ths case, the wdth of the servce area s gong to be dfferent from above. For sngle-sded servce areas, the same parameters used n the analyss of the two-sded servce area can be used except the wdth of the servce area, whch wll be changed from 2W to W snce the analyss wll be performed for one sde of the route. Ths mples that the expected values of dy1 and dy2 wll be dfferent from the prevous case as follows: W E d y1. (Equaton 3-11) 3 W E d y2. (Equaton 3-12) 2 Based on these formulas, the expected value of the rectlnear Hamltonan path of the transt vehcle s gven by: E h m 1Ed 2Ed L (Equaton 3-13) y1 y2

84 Usng the same procedure prevously used for the two-sded servce area, we are able to get the followng equaton for W : 70 3V S W.... (Equaton 3-14) m 2 Equaton 3-14 s smlar to equaton 3-6 n terms of the factors affectng the wdth of the servce area, whch expectedly dd not change gven the dfferent type of the servce area. In both cases the senstvty of the wdth of the servce area to these parameters s stll the same, whch mples that the dscusson made n the prevous secton apples also here. The only dfference, however, s n the denomnator whch ncludes the average demand of the on-demand requests. Ths mnor dfference s attrbuted to the dfference n the length of the Hamltonan path (h ) between the two types of servce area. Recallng that the average demand of the on-demand requests depends on the demand densty (λ ), and gven the new wdth of the servce area (W ) nstead of 2W, the average customer demand (m ) can be gven by the followng relatonshp (m = λ W L ). Substtutng ths relatonshp nto Equaton 3-14 and rearrangng the terms of the equaton, we get the followng formula for the optmal maxmum wdth of the servce area: W L V S 2 L... (Equaton 3-15) Equaton 3-15 gves the maxmum optmal value of the servce area wdth n a sngle-sded servce area. The equaton s smlar to Equaton 3-8 (the two-sded servce area) n terms of the senstvty of the wdth of the servce area to the dfferent parameters. However, the value of W wll be dfferent (hgher on a sngle sde) due to the fact the slack tme s used on only one sde of the route. In fact the only reason why W n a sngle-sded servce area s not twce that of the two-sded servce area s the dfference n the expected value of the rectlnear Hamltonan path.

85 71 As n the two-sded case, we now can use Equaton 3-14 to provde a relatonshp for the total servce area of the whole route as follows: N N m L S V W L A N m L S V A..... (Equaton 3-16) In cases where all the route segments are smlar n terms of the values of S, L and m, then Equaton 3-16 can be re-wrtten as follows: N m L S V A... (Equaton 3-17)

86 72 Legend Drecton of Travel Fxed Stop On-Demand Request W / 3 W W W / 2 L Fgure 3-3 One-sded servce area

87 Slack Tme Flex-Route operaton requres a fxed schedule that defnes when vehcles wll be at the fxed stops, but one that also leaves tme for respondng to demand-responsve servce requests. In the case where an exstng fxed route s beng converted to a Flex- Route servce, one key modfcaton that has to be made to the fxed route structure s addng a specfc amount of addtonal slack tme to the exstng scheduled runnng tme of the fxed-route servce. Ths added slack tme n the schedule s necessary to allow the transt vehcle to perform off-route devatons to pck up on-demand customers. Accordng to Koffman (2004), the degree of flexble and fxed-schedule operaton nherent n the desgn for Flex-Route transt operators vares from one system to another. The tme of flexble operatons n Flex-Route servce n the reported systems ranged from 20 mn out of every hour (33.3%), where demand-responsve operaton s a promnent part of the servce, to only 2.5 mn per hour (4.2%), where devatons play a much more lmted role. The total slack tme for a route can be computed from the relaton (S = T DT), where S s the slack tme assgned to a route (n hours), T s the scheduled runnng tme for one vehcle run n the Flex-Route servce (n hours) and DT s the drect runnng tme or the orgnal scheduled runnng tme for the fxed-route servce between fxed stops 1 and N (n hours). The decson on how much slack tme to add should be carefully examned. The slack tme bult nto each route (n each drecton,.e. nbound/outbound) should be enough to serve on-demand requests. However, addng too much slack tme wll result n a stuaton n whch vehcles have to stay dle at the fxed stops n the absence of ondemand requests. For slack tme, there are two cases that are studed: 1. Allocaton of slack tme among route segments gven the total slack tme assgned to the whole route; and 2. Fndng the optmal slack tme for a gven route segment based on a known servce area Allocaton of Slack Tme among Route Segments The frst stuaton analyzed s the case when the amount of slack tme for the whole route s determned beforehand. Such stuatons arse when the servce provder

88 74 decdes that there s a lmt on the amount of slack tme that can be used n the operatons of Flex-Route servce due to some cost or fleet constrants. In such a scenaro, the task here s to allocate ths slack tme among route segments based on some crtera rather than dstrbutng ths slack arbtrarly. There are several methods that could be used n these cases. These nclude: 1. Slack tme dstrbuton as a weghted average of the drect travel tme (or fxed stop spacng, L ) between any par of fxed stops (route segment). Consder the followng values: N-1 = the total number of route segments along the route; S = slack tme allocated for the whole route (n hours); DT = the drect runnng tme (under fxed route operatons) to travel from the frst fxed stop to the end fxed stop of route segment (n hours). Then, the slack tme assgned for route segment (S ) should be: S DT N 1 1 DT (Equaton 3-18) Equaton 3-18 guarantees that longer route segments wll have hgher amounts of slack tme allocated to them. Ths method can be used n cases where the wdth of the servce area s constant along all route segments, and all customer demand s dependent on the unform demand densty. Ths means that, gven the constant servce area wdth, longer route segments wll have a hgher demand rate, and therefore more slack tme s needed compared to shorter route segments. 2. Slack tme dstrbuton as a weghted average of the wdth of the servce area of a route segment (W ). The slack tme assgned for route segment (S ) wll be:

89 75 W S... (Equaton 3-19) N 1 1 W Equaton 3-19 guarantees that more slack tme s allocated to route segments wth wder servce areas. On the contrary to the prevous case, ths method can be used n cases where the length of the servce area s constant along all route segments, and n the same tme, customer demand s dependent on the unform demand densty. Ths mples that wder route segments wll have a hgher demand, and therefore more slack tme s needed compared to shorter route segments. In our case where the Flex-Route servce s used only by the general publc, the two methods can be applcable (see Fgure 3-4). However, both methods do not necessarly reflect the characterstcs of each route segment. Therefore, we propose another method that reles on both the servce area wdth and length. Ths could be done by ncludng the demand densty of each route segment. Thus, the thrd method of allocatng the slack tme s: 3. Slack tme dstrbuton as a weghted average of the total number of on-demand requests n a route segment. The slack tme assgned for route segment (S ) n ths case wll be: m S. (Equaton 3-20) m N 1 1 In terms of the demand rate dstrbuton, Equaton 3-20 can be expressed as follows: S N 1 1 W ( W L L ). (Equaton 3-21) Equaton 3-21 combnes the wdth of the servce area, the length of each route segment and the demand densty of the route segment nto one unversal equaton. Decdng on whch method to use for allocatng slack tme can vary from one case to

90 76 another dependng on the characterstcs of the servce area specfcatons and the demand patterns. It s clear, however, that Equaton 3-21 should be used as a method of allocatng slack tme among route segments n cases where the expected demand densty s known, snce t combnes all the major features of a route segment nto one equaton

91 77 L L L W 1 W 2 W 3 (a) W W L 1 L 2 L 3 L 4 (b) Fgure 3-4 Cases of (a) dfferent servce area wdth, and (b) fxed stop spacng among route segments

92 Slack Tme for A Sngle Route Segment wth Known Servce Area The prevous secton dscussed the several methods that can be used to dstrbute slack tme among route segments. Ths case could arse n real world applcatons where the servce provder wants to lmt the amount of ncrease n the total runnng tme of the transt vehcle and wants to dstrbute the avalable slack tme among the route segments. There mght be other cases, however, where there s no pre-specfed lmt on the slack tme avalable for the whole route. Instead, the only known s the servce area wdth (W), whch s determned beforehand. Ths s the exact opposte case to the optmal maxmum value of W dscussed n an earler secton. Thus, the optmal value of slack tme can be dfferentated as before nto two cases: two-sded servce areas and sngle-sded servce areas. Two-sded Servce Areas Recallng Equaton 3-6, the servce area wdth (W ) on ether sde of the route s gven n terms the slack tme (S ), the bus operatng speed (V) and the number of requests (m ). Rearrangng the equaton n terms of S : S 2m 1 W..... (Equaton 3-22) 3V Equaton 3-22 gves the optmal value of the slack tme as a functon of the number of requests (m ), the servce area wdth (W ) and the transt vehcle operatng speed (V). The equaton shows that the slack tme s postvely related to the average customer demand and the servce area wdth, and s nversely related to the transt vehcle operatng speed. Ths mples that the wder the servce area the more slack tme s needed to serve the requests n ths area, whch s expected as dscussed earler n the servce area analyss. The equaton also mples that the slack tme assgned to a specfc route segment ncreases as the customer demand ncreases and decreases as the vehcle operatng speed ncreases. These relatonshps are also expected snce a hgher demand rate means that more slack tme s needed to serve these requests, whle a hgher operatng speed means less slack tme s needed to travel.

93 To fnd the optmal slack tme n terms of the demand densty (λ ), Equaton 3-22 can be rearranged n terms of S as follows: 79 S 2 4 L W W (Equaton 3-23) 3V Equaton 3-23 gves the optmal slack tme for a gven route segment as a functon of the on-demand densty (λ ), the route segment length (L ), the wdth of the servce area (W ) and the bus operatng speed (V). The equaton shows that the slack tme ncreases as λ, W and L ncrease, and decreases as V ncreases. These relatonshps are expected as wder servce areas and longer route segments result n more on-demand requests scattered over a wder area, and therefore more slack tme s needed to serve all the requests Fxed Stop Locaton and Spacng Usually, the fxed routes chosen to be replaced by Flex-Route servce are those wth low rdershp at the majorty of the stops coupled wth low populaton densty along the routes. So once the decson to provde Flex-Route servce has been made, the next crtcal step s selectng what modfcatons, f any, to the fxed stops should be made to allow for the exstng fxed route to be converted to a Flex-Route servce. These modfcatons are requred because the routng and schedulng structures for a fxed-route and Flex-Route servce dffer n many ways. For nstance, a prmary standard n desgnng a fxed route servce s to locate the fxed stops wthn a 1/4-mle (250m) walkng dstance of the surroundng populaton (Gray and Hoel, 1992). For Flex-Route transt servce, t must be desgned wth the objectves of (1) servng hgh-actvty fxed stops and (2) provdng the necessary slack tme n the schedule to allow for door-to-door on-demand servce. Although there s no lterature drectly dealng wth how these modfcatons are to be made, there s one study that addresses removng some fxed stops from a fxed route (Welch et al., 1991) and used by the transt agency n San Dego, Calforna, to remove out-of-drecton (OOD) segments. An OOD segment can be defned as the

94 80 porton of the route that devates from the man lne of a fxed route servce. Removng these OOD segments can save some extra travel tme that can be used as a slack tme, whle those passengers who used to access the servce at the removed stops can ether use other fxed stops or swtch to the on-demand part of the servce. However, there s stll a chance that those passengers mght be lost and swtch to other modes of transportaton. Usng ths methodology, an OOD mpact ndex s calculated for each OOD segment. Based on the value of ths OOD ndex a decson on whether to retan or remove an OOD segment can be made. The OOD mpact ndex s defned as the weghted measure of travel tme as a functon of through rders, OOD rders (greater than 1/4 mle off manlne) and travel tme dfference between OOD and manlne. In short, t essentally measures the penalty to through rders mposed by OOD rders. Let s consder the followng varables: R D = through rders per day; R OOD = OOD rders per day; TT OOD = travel tme ncrease due to OOD segment (mnutes). The mpact ndex can be calculated as: R Impact Index D *TT R OOD OOD... (Equaton 3-24) Welch (1991) used the followng crtera to make a decson on retanng or removng an OOD segment: If (Impact Index < 5) retan the OOD segment; If (5 < Impact Index < 15) decde based on resource needs, operatng cost, effectveness, and qualtatve factors. If (Impact Index > 15) elmnate OOD segment. If the route has such OOD segments, then these segments can be evaluated based on the above crtera usng rdershp and travel tme nformaton from the servce provder. However, the above crtera mght not be applcable n all cases, snce some

95 routes may not have any OOD segments. In ths case, the followng factors could be used to evaluate f a specfc fxed stop can be removed: Rdershp data for each stop on the route; Populaton and populaton densty close to the stop; and Locaton of stops (.e. transfer stops, schools, hosptals, etc). Usually, the canddate fxed stops to be removed are those wth low rdershp numbers. Another crteron that could be used to retan a stop s ts proxmty to other fxed tme stops and whether t s an mportant landmark such as a hosptal, a school, a mall, a communty college, etc. Most of the tme, fxed stops that are landmarks wll also have a hgh rdershp, thus precludng the need for ther removal. However, there mght be cases n whch there s no need to remove fxed stops as the number of fxed stops on the route s already low Servce Frequency Once the decson to provde Flex-Route transt servce has been made, several features of the fxed-route structure of the route need to be modfed from ther orgnal settngs. One mportant feature that wll be affected s the servce headway at the fxed stops due to the addton of slack tme to the orgnal scheduled runnng tme of the route. In order to study ths effect, several defntons of the fxed-route servce should frst be ntroduced: K n K lne; T c = number of transt vehcles requred to operate the orgnal fxed-route lne; = new number of transt vehcles requred to operate the new Flex-Route = cycle tme of the orgnal fxed-route lne (n hours); 81 T c n = cycle tme of the new Flex-Route lne (n hours); H n H = orgnal servce headway of the fxed-route lne; (n hours); = new servce headway of the Flex-Route servce (n hours); S = the slack tme assgned to the entre route n one drecton (n hours);

96 82 One of the fundamental relatonshps of fxed route operatons s the relatonshp relatng the number of transt vehcles, cycle tme and headway: T K H c. In Flex- Route operatons, addng the slack tme wll result n ncrease of (2 x S) hours n the cycle tme of the route, assumng that there s an equal amount of slack tme added n each drecton. It s worth mentonng that each transt vehcle wll stll have a recovery (termnal) tme at each end of the route to account for all elements of stochastcty that mght affect the operaton of the servce. The only dfference (advantage) from fxedroute servce s that at each fxed stop, the vehcle mght use the slack tme assgned to each route segment to recover any delays that mght have happened along the route, whch means that the stochastcty element can be mnmzed at each stop. Therefore, the new cycle tme of the route, ncludng the added slack tme, wll be: T c n T 2 S.... (Equaton 3-25) c Ths ncrease n slack tme wll have an effect on the operaton of the servce as t has to be compensated by ether ncreasng the servce headway or, n some cases, ncreasng the number of transt vehcles requred to operate the servce. If the decson s to keep the servce headway at ts orgnal value, then the added slack tme wll result n an ncrease n the number of transt vehcles. Therefore, the new number of transt vehcles requred to operate the servce wll be: tme usng the formula found n Equaton 3-25: K n n T c. Substtutng for the cycle H K n 2 S K H (Equaton 3-26) Equaton 3-26 shows that f the headway of the servce t to be kept the same as the orgnal fxed-route settng, the new number of transt vehcles needed for the Flex-

97 83 Route servce wll ncrease by a fracton that depends on the added slack tme and orgnal headway. The relatonshp suggests that when the rato of the slack tme relatve to the headway s low, then the ncrease n the number of transt vehcles needed for the servce wll be also low, and vce versa. It s worth mentonng here that ths fractonal ncrease mght not warrant addng more vehcles to the transt fleet as transt agences may have extra vehcles or extra termnal tmes that can be used to allevate ths ncrease. Equaton 3-26 also reveals that f the servce headway s relatvely long, then addng more slack tme to the scheduled runnng tme wll not have the same substantal effect compared to cases where the servce headway has a lower value. For example, for headway values of 20 and 30 mnutes and a slack tme value of 10 mnutes, the ncrease n the number of transt vehcles wll be as follows: Case 1: Headway = 20 mnutes, slack tme = 10 mnutes. Usng Equaton 3-26: K n K n 2 S K H K K n K 1 The ncrease n the number of transt vehcles n ths case s (1). Case 2: Headway = 30 mnutes, slack tme = 10 mnutes. Usng Equaton 3-26: K n K K n K 0.67 The ncrease n the number of transt vehcles n ths case s (0.67), whch wll be rounded to 1 transt vehcle. The above two cases llustrate the senstvty of the ncrease n the number of transt vehcles to the values of the orgnal servce headway and slack tme. Therefore the longer the headway of the servce the less the effect of addng the slack tme on ncreasng the number of extra transt vehcles requred for the Flex-Route servce.

98 Another case that should be examned here s the case where the transt agency decdes aganst ncreasng the number of transt unts from the orgnal settng. The only opton here wll be to change the headway to account for the ncrease n the cycle tme. So for the transt agency to keep the same number of transt unts, then 84 n K must equal K. Tc T c 2S In other words, ths relatonshp should hold: n. Rearrangng the terms of H H ths relatonshp, we now can get the new value of the servce headway as follows: H n 2 S H H T c..... (Equaton 3-27) Equaton 3-27 shows that f the number of transt vehcles should reman the same as the orgnal fxed-route settng, then the headway has to be ncreased by a fracton that depends on the slack tme, the orgnal headway and the orgnal cycle tme. The relatonshp suggests that when the rato of the slack tme and the orgnal headway relatve to the orgnal cycle tme s low, the ncrease n the headway wll be also low, and vce versa. Therefore, the longer the cycle tme and the orgnal headway are, the lower the ncrease of the servce headway compared to ts orgnal value. Furthermore, the ncrease n the headway relatve to the orgnal headway can be found as follows: 2 S H T c Relatve Increase n Headway H Or: Relatve Increase n Headway 2 S (Equaton 3-28) T c Equaton 3-28 shows that the relatve ncrease n the headway depend only on the values of the slack tme and orgnal cycle tme. Ths means that the lower the rato of slack tme to the orgnal cycle tme, the lower the relatve ncrease n the servce headway. To llustrate ths concluson, let us consder two cases:

99 85 Case 1: Headway = 20 mnutes, cycle tme = 60 mnutes, slack tme = 5 mnutes. In ths case, the number of transt vehcles requred n the orgnal fxed-route settng s (K = 60/20 = 3). To keep the same number of transt vehcles n the new servce, then new headway usng Equaton 3-27 wll be: H H n n H n 23.33mn (24 mn) So n order to keep the same number of transt unts as the orgnal settng, then the headway has to be ncreased by 4 mnutes (24 mnutes compared to the orgnal headway of 20 mnutes), whch s a relatve ncrease of only (4/20 = 1/5 the orgnal headway), whch can be consdered a relatvely low ncrease relatve to the orgnal headway. Case 2: Headway = 30 mnutes, cycle tme = 60 mnutes, slack tme = 5 mnutes. In ths case, the number of transt vehcles requred n the orgnal fxed-route settng s (K = 60/30 = 2). To keep the same number of vehcles n the new servce, then new headway usng Equaton 3-27 wll be: H H n n H n 35mn 1 60

100 86 So n order to keep the number of transt unts wthout change for ths case, then the headway has to be ncreased by 5 mnutes, whch s a relatve ncrease of (5/30 = 1/6) - the same relatve ncrease as n case 1. Ths could be very mportant n the desgn of Flex-Route servce snce n real world applcatons, the relatve ncrease n the headway can be perceved to be more mportant for the transt users than the amount of the ncrease tself. Thus the ncrease n the headway wll be affected prmarly by the rato of slack tme to the orgnal cycle tme. 3.4 FLEX-ROUTE FEASIBILITY AND COST ANALYSIS The addton of slack tme to the orgnal scheduled runnng tme of the fxedroute servce nflcts addtonal costs on both the servce provder and the fxed-route customers, whle provdng some beneft to the on-demand passengers-through travel cost savngs- and the servce provder- through the addton of revenue from the on-demand passengers. Ths secton presents a cost-beneft analyss of replacng a fxed-route servce wth Flex-Route for several stakeholders nvolved n the process. The analyss ncludes examnaton of two case scenaros that can be adopted n practce, whch are based on the frequency of the servce. The decson s whether the orgnal servce headway of the fxed-route structure should be kept unchanged wth the mplementaton of the new Flex- Route servce (to mantan the same level of servce at the fxed stops) or to ncrease the servce headway (to mantan the same number of transt vehcles needed for the servce n case the transt agency s fleet s lmted). It should be emphaszed that the analyss wll be performed for a sngle bus run (n one-drecton) for a two-sded servce area. Table 3-4 lsts the varables that wll be used n ths analyss. Before we proceed wth the feasblty analyss, we need to pont out an mportant assumpton n advance: throughout the analyss we assume that fxed-stop users wll nether convert to on-demand servce nor they wll abandon the fxed-route servce. We acknowledge that ths assumpton mght not be fully realstc gven that the ntroducton of flex-route servce mght have negatve mpacts on servce relablty that could deter some fxed-route users from usng the servce. However, the addton of slack tme can counter ths problem by recoverng the schedule at each fxed stop, and therefore reducng the varablty and mprovng the relablty of the servce.

101 87 Some polces can be ntroduced by the servce provder to encourage the fxed-stop users to contnue usng the servce wthout swtchng to other travel modes or to the flexble servce along the same route. Such polces could nclude guaranteed specfc arrval tme at transfer ponts wth other major transt servces; reduced fare; and ncreased servce headway. Some of these polces are dscussed n further detal later n ths chapter. However, we acknowledge that ths fxed-route rdershp nelastcty assumpton s somewhat smplstc and requres more attenton, but gven the scope of the thess, whch focuses prmarly on the supply sde of the servce, we decded to leave ths part of the work for future research. Addressng the changes n fxed-stop user behavour can be accomplshed through studes that focus on the demand part of the servce usng methods such as rdershp elastcty analyss, stated-preference surveys and mode choce models that can estmate both fxedstop and on-demand rdershp. However, as dscussed earler, ths part of the analyss s beyond the scope of ths research and can be addressed n future research.

102 88 Table 3-4 Defntons of varables used n the cost-beneft analyss Unt of Symbol Defnton Measurement N-1 Number of route segments n one drecton L Length of the fxed-route n one drecton Km W Average wdth of the servce area along the route on one sde Km of the route (total average wdth = 2W) S Total slack tme added to one vehcle run n one drecton Hours (from n=1 to n=n), or total slack tme added to all route segments S Slack tme added to route segment Hours VR o The orgnal number of vehcle runs durng tme perod T VR n The new number of vehcle runs n tme perod T W The wdth of the servce area of route segment on one sde of the route Km D Extra dstance travelled (the rectlnear path length) due to the devaton of the servce n route segment Km m Expected number of on-demand passengers usng the servce n route segment C dr Cost coeffcent representng the margnal hourly cost of $$ / Hour vehcle operatons ( wage of bus drvers) C km Cost coeffcent representng the margnal cost of vehcle $$ / Km mleage (gasolne) O cost Operator s addtonal cost due to the change of the servce from fxed-route to Flex-Route $$ O beneft Operator s beneft due to the change of the servce from fxed-route to Flex-Route $$ m tot Total number of on-demand passengers usng the servce n one bus run B p Margnal beneft of servng a sngle on-demand passenger $$ FX cost Total fxed-stop user cost due to the change of the servce from fxed-route to Flex-Route $$ FXP cost Average cost ncurred by a sngle fxed-stop user due to the change of the servce from fxed-route to Flex-Route $$ A Number of fxed-stop passengers onboard the bus after leavng the frst fxed stop of route segment Y Total number of fxed-stop customers usng the servce n one bus run n one drecton. TT The ncrease n the average n-vehcle travel tme for a fxedstop Hours customer WT The ncrease n the average watng tme for a fxed-stop Hours customer AT The ncrease n the average access tme for a fxed-stop customer Hours

103 89 Table 3-4 Cont d C TT Cost coeffcent representng fxed-stop customer dsutlty $$ / Hour toward ncrease n n-vehcle travel tme C WT Cost coeffcent representng fxed-stop customer dsutlty toward ncrease n wat tme $$ / Hour C AT Cost coeffcent representng fxed-stop customer dsutlty toward ncrease n access tme; $$ / Hour V walk Average walkng speed of fxed-stop passengers Km/hr F o The orgnal fare value for fxed-stop customers before the ntroducton of Flex-Route servce $$ F n The new fare value for fxed-stop customers after the ntroducton of Flex-Route servce $$

104 Operator Cost The added slack tme to a typcal route segment ncurs addtonal costs to the servce operator. The ncrease n the operatng cost for the servce operator depends on the two cases mentoned above. Both cases wll be examned carefully Same Frequency (Same Headway) In case the transt agency decdes to mantan the same servce frequency as before ntroducng the Flex-Route servce, ths may result n an ncrease n the number of transt vehcles needed to operate the servce (fxed costs). Ths ncrease s equal to (2S/H) as shown n equaton In ths research, however, we are lmtng the analyss to the operatng costs of the servce, the reason beng that the transt agency mght be able to avod addng extra vehcles for several reasons. Frst, when determnng the number of transt vehcles needed for the exstng fxed-route servce, the servce provder usually rounds up the number of transt vehcles to the next nteger, whch wll result n extra unused termnal tme. For some low-demand routes, where headways are too long (e.g. 30 or 45 mn) and the termnal tme s exceedngly long, part of the termnal tme could be reallocated as slack tme for use n flex route servce, thus reducng the need for extra transt vehcles. Another reason s that the servce agency s transt fleet mght have extra vehcles that can be used to cover up the needed vehcles. Addtonally, the servce operator wll ncur operatng costs related to the extra mleage and extra operatng hours due to the devaton of the servce. The ncrease n mleage wll result n extra costs due to the extra gasolne requred to operate the servce, whle the ncrease n operatng hours wll result n extra wages pad to drvers. In all cases, the addtonal cost wll be proportonal to the added slack tme bult nto the schedule (S) as wll be shown next. It should be noted that the analyss always starts wth a sngle segment of the route, and then proceeds to the whole route. Usng the varables of Table 3-4, the extra operatng cost ncurred by the servce provder due to the devatons n a sngle route segment s gven by the followng equaton:

105 91 Operator cost (sngle route segment) = C dr S C D (Equaton 3-29) km Usng equaton 3-4 the expected extra dstance traveled due to (m ) on-demand requests n one route segment for two-sded servce areas (the rectlnear Hamlton path length) s gven by: D 2W W m.... (Equaton 3-30) Therefore, the operator s addtonal operatng cost due to devatons to serve ondemand requests for the whole route n one drecton s gven by: N 1 W 2m 1 O cos t C dr S Ckm (Equaton 3-31) 1 3 Gven that the total slack tme added to one bus run n one drecton s S, then equaton 3-31 can be re-wrtten as follows: N 1 Ckm O 2 cos 3 1 t Cdr S W m... (Equaton 3-32) 1 Equaton 3-32 gves the addtonal operatng costs ncurred by the servce provder due to the change of the servce to Flex-Route n case the servce frequency stays the same as the orgnal fxed-route system. The equaton does not nclude a representaton of the addtonal fxed costs for the servce provder for the reasons mentoned earler. However, the fxed costs can be calbrated and added to the equaton for a sngle bus run f requred Lower Frequency (Increased Headway) In case the servce provder can not afford addng new transt vehcles or the exstng transt fleet does not have extra vehcles to be used, the servce provder wll

106 92 have to lower the fxed-route servce frequency (ncrease the headway) to afford addng extra slack tme. In ths case, the assumpton s that there wll not be any ncrease n the fxed costs or total operatng hours snce the headway wll be ncreased to mantan the same total operatng hours for the operatng tme perod. In terms of the ncrease n mleage, the addtonal cost wll depend on the new number of bus runs, whch wll be fewer than the orgnal fxed-route servce, and thus reducng the total fxed-route vehcle mleage compared to the orgnal servce. Assumng that the orgnal number of vehcle runs durng a specfc tme perod (T) s VR o, and the new number of vehcle runs n tme perod T s (VR n ), then the total vehcle mleage for the orgnal and new servces wll be 2L(VR o ) and 2L(VR n ) respectvely. Therefore, the average relatve mleage for each vehcle run n each drecton for the new servce wll L VR be: VRo n, whch wll be lower than L. Usng the same procedure used for the prevous case (see equaton 3-32), the total operatng cost can be found as follows: O N 1 Ckm L VR W 3 1 o 1 cos t n 2m Ckm VR... (Equaton 3-33) Operator Beneft Same Frequency (Same Headway) The ntroducton of Flex-Route servce wll have some benefts to the servce provder n ths case. The amount and type of these benefts depend on the reason why Flex-Route was ntroduced and on what consttutes a beneft for the servce provder. For example, n some agences, the extra fare collected from the new passengers consttutes a beneft for the servce provder. The objectve of the servce n other cases mght be to encourage people to swtch from usng ther cars to usng transt when accessng a regonal ral network because of the lack of parkng spaces n the tran staton. In ths case, the beneft to the servce provder mght nclude the savngs n parkng costs ncludng the long-term cost of buldng new parkng spaces. Therefore, estmatng the benefts to the servce provder depends on the stuaton and the goals of the servce. In

107 general, the operator s beneft from a sngle bus run can be found usng the followng equaton: 93 O beneft = B m (Equaton 3-34) p tot In Chapter Sx, an example of evaluatng the beneft to the servce provder from servng on-demand requests wll be estmated for a real-world case scenaro Lower Frequency (Increased Headway) The same explanaton for the frst case apples also to ths case Fxed-route user cost Introducng Flex-Route servce as a replacement of an exstng fxed-route transt lne wll nflct some cost on the fxed-route passengers who use the servce regularly and access the servce at the establshed fxed stops. Each case mght have a dfferent amount of cost ncurred by the users, but usually ths cost s one or a combnaton of the followng three types of cost: 1. Increase n n-vehcle travel tme due to the added slack tme: The addtonal slack tme added to the orgnal scheduled runnng tme wll ncrease the average n-vehcle travel tme for fxed-route users. Ths ncrease n n-vehcle travel tme for all fxed-route customers wll vary dependng on the rdershp levels at each fxed stop. For example, f the rdershp levels are relatvely hgh near the startng pont of a transt route where most customers are destned for a major transfer pont at the end of the route, then the average ncrease n travel tme for all customers wll be hgher than the case wth more rdershp volumes close to the end of the route gven a relatvely constant amount of slack tme provded to each route segment. The reason for ths dfference s that f the majorty of fxed-route users access the servce early n the route, then they wll have to suffer the extra slack tme gven to each subsequent route segment, thus ncreasng ther average travel tme more than the other case.

108 94 2. Increase n wat tme due to the change n servce frequency: In suburban context, the servce headway usually has a hgher value compared to urban areas due to low levels of rdershp. Therefore ncreasng the headway mght not have a great effect on the behavour of fxed-route customers snce passengers usually tme ther arrval at the fxed stops n cases of low frequency (hgh servce headway values). However, ncreasng the headway wll ncrease the wat tme for a porton of the passengers who may stll arrve at random despte the low frequency. Moreover, ncreasng the headway wll reduce the avalablty of transt servces to the publc snce there wll be fewer bus runs n ths case. Therefore, n ths research we wll nclude the mpact of changng the headway on the wat tme of fxed-route customers to account for cases where the servce frequency mght play a role n the cost ncurred on fxed-route users. 3. Increase n access tme f the fxed-stop spacng changes: When ntroducng Flex-Route servce, there mght be cases n whch some fxed stops wll be removed due to ther low rdershp numbers and to mprove the performance of the servce. In such cases, the access tme (walkng) to the fxed stops wll ncrease snce passengers wll have to walk longer dstances to reach the retaned fxed stops assumng there s no loss n the orgnal fxed-route rdershp. Ths does not mply that the transt servce agency wll remove stops n all cases Same Frequency (Same Headway) In ths case, changng the fxed-route servce to Flex-Route operatons wll cause some nconvenence to the regular transt passengers because of the ncrease n the average n-vehcle rde tme, and n some cases, an ncrease n the average access tme f the fxed-stop spacng s changed (ncreased). It should be noted that there wll not be any ncrease n the average watng tme snce the servce frequency wll not change. The larger the amount of slack tme s, the hgher the nconvenence for fxed-stop passengers wll be. Such nconvenence could also lead to loss of regular transt rders. It s therefore necessary to consder ths factor when desgnng a Flex-Route servce and consder allevatng ths nconvenence by adoptng strateges such as ncreased frequency and

109 95 lower fares, whch wll be dscussed later. The analyss s further dvded here nto two cases: (1) no fxed stop s removed (same number of fxed stops) and (2) some fxed stops are removed (reduced number of fxed stops). 1. Same Number of Fxed Stops In ths scenaro, the cost ncurred by fxed-stop users wll be comprsed manly of the addtonal n-vehcle travel tme. Usng the varables n Table 3-4, the cost ncurred by fxed-stops users n a sngle route segment due to ncrease n n-vehcle travel tme s gven by the followng equaton: Fxed-user cost = A C TT (Equaton 3-35) TT In one route segment, TT wll amount to S snce the ncrease n n-vehcle rde tme n any route segment wll be equal to the slack tme gven to ths segment. Therefore, the total fxed-users cost due to the ncrease n n-vehcle travel tme s gven by the followng equaton: N 1 1 FX A C S (Equaton 3-36) cos t TT Gven that the total number of fxed-route users for a sngle bus run s Y, then the average cost ncurred on a sngle fxed-route user s gven by: N 1 CTT A S 1 FXPcos t (Equaton 3-37) Y 2. Reduced Number of Fxed Stops In case that the number of fxed stops has been reduced, there wll be an ncrease n average travel tme as well as an ncrease n the access tme for fxed-route s users assumng that there wll not be any loss n rdershp (.e. passengers who used to access a

110 96 removed fxed stop wll access the servce at the retaned fxed stops). However, there s also a small beneft ganed from the removal of fxed stops as the runnng tme of the vehcle wll be lower as a result of the elmnaton of dwell tme, deceleraton and acceleraton tmes at the removed fxed stops. For the sake of ths analyss, we wll not get nto the detals of ths amount of reducton, but we wll just assume that the orgnal runnng tme of the vehcle wll be reduced by T f for each fxed stop removed. Ths reducton can not be dealt wth on a route segment bass snce the only mpact t has on the servce s the runnng tme saved. Ths saved runnng tme can be ncorporated wth the slack tme assgned to the route by elmnatng the need for some of the needed slack tme, and therefore reducng the total runnng tme of the vehcle by an amount of T f tmes the number of removed fxed stops (N-N n ). There s also a small beneft ganed by the servce provder based on equatons 2, 3, and 4 because of the elmnaton of the fxed stop. When two route segments are combned together (.e. one fxed stop removed), the new combned route segment wll have two less (W /2) terms and one more (2W /3) term. Therefore the expected dstance traveled by the vehcle n route segment wll be reduced by an amount of (W /3). These benefts wll be ncorporated n the total cost functon of ths case (see Equaton 3-55). In ths case, the average ncrease n access tme wll depend on the number of fxed stops before and after the removal of some fxed stops. Gven the rectangular shape of the servce area of the route, and assumng that the fxed stop passengers are scattered unformly across the servce area of the route, then the orgnal average access dstance (AD o ) for any fxed-stop user can be found by assumng the fxed-stop user wll follow a rectlnear Hamltonan path to reach the fxed stop. As shown n Fgure 3, ths dstance (AD o ) s composed of two parts (AD 1 and AD 2 ). AD 1 can be found by assumng that the expected average horzontal access dstance s equal to ¼ of the average fxed-stop spacng (L ), whle AD 2 can be found by assumng that the expected average lateral access dstance s equal to ½ of the average servce area wdth on ether sde of the route (or maxmum lateral walkng dstance for fxed-route L customers). Assumng that the average fxed-stop spacng L s equal to and the N 1 average servce area wdth W (on ether sde of the route) s equal to W avg, AD o can be found usng the followng equaton:

111 AD 1 L 1 4 N 1 2 o W avg.... (Equaton 3-38) 97 Equaton 3-38 s bult based on the assumpton that the fxed-stop spacng s constant to lmt the scope of the analyss. Ths s because f the ndvdual route segment lengths were to be used, the analyss wll go nto the detals of how to dstrbute the users, who used to access the removed stops, among the retaned fxed stops. The equaton also uses an average wdth of the servce area for the whole route for the same reason. As mentoned earler, t s mportant to reterate that removng fxed stops s not recommended n all cases and careful attenton should be gven to ths decson. Fxed stops can be consdered for removal only f the stop demand and stop spacng are both very low that customers wll stll be wthn a reasonable walkng dstance of the remanng fxed stops. Therefore, the orgnal average access tme (AT o ) can then be found usng the followng formula: AT 1 L o W 4V walk N 1 2 avg..... (Equaton 3-39)

112 98 Legend Drecton of Travel Fxed Stop Fxed-Route Customer W AD 2= ½ W AD 1 = ¼ L W L Fgure 3-5 Calculaton of Average Access Dstance for Fxed Route Users

113 99 If the number of fxed stops s reduced from N to N n (.e. the new number of route segments = N n 1), then the new average access tme for the fxed-route users can be found as follows: AT 1 L n W 4V walk N n 1 2 avg.... (Equaton 3-40) AT n AT ) for fxed- Therefore the ncrease n the average access tme ( AT route users wll be gven by: o AT 1 4V walk L L.... (Equaton 3-41) N n 1 N 1 It should be noted that equatons 3-40 underestmate the new access dstance for fxed stop users snce t assumes that the remanng fxed stops wll be equally spaced as the orgnal settng, whch s not what actually happens snce some of the users wll have the same access dstance whle others wll suffer an ncrease n ther access dstance. Nevertheless, we wll use equaton 3-40 as an approxmate estmaton of the new access dstance. So n case that the number of fxed stops has been reduced to N n, then the total fxed-route user addtonal cost due to the ncrease n n-vehcle travel tme and access tme, s gven by the followng equaton: FX cos t 1 4V L L N n 1 N N n 1 A CTT S Y C AT 1 w 1.. (Equaton 3-42)

114 100 Rearrangng the terms of equaton 3-42, we get the followng expresson for the total cost ncurred on fxed-route users due to changng the servce from fxed-route to Flex-Route: FX cos t C TT Nn 1 C AT Y L L A S (Equaton 3-43) 1 4V walk N n 1 N 1 Gven that the total number of fxed-route users for a sngle bus run s Y, then the average cost ncurred on a sngle fxed-route user n ths case s gven by: Nn C AT Y L L CTT 1 A S V walk N n N FXP cos t.. (Equaton 3-44) Y Equaton 3-44 can be used as a bass for estmatng the mpact of dfferent Flex- Route desgn parameters on the cost ncurred by fxed-route users, and the senstvty of the cost to the change n these parameters. For example, the equaton can be used to examne what effect removng a fxed stop has on the average total cost ncurred by a fxed-route user. Another example s the case where the servce agency does not want the fxed-route user cost to exceed a specfc value, and want to see how many fxed stops t can remove wthout exceedng ths cost Lower Frequency (Increased Headway) When the headway of the servce s ncreased, the nconvenence to regular transt passengers wll be n terms of the ncrease n n-vehcle travel tme, the ncrease n watng tme, and n some cases the ncrease n access tme f the fxed-stop spacng s changed (ncreased) due to the removal of some fxed stops. As mentoned earler, low frequency fxed routes are the typcal canddates for converson nto flex route servces. Such routes have two types of passengers, namely aware passengers (those tmng ther arrval at stops accordng to the schedule) and unaware passengers. Arguably, aware passengers wll not see ther watng tme ncrease snce they usually tme ther arrval at

115 101 the fxed stops. On the other hand, unaware passengers are the ones whose wat tmes wll be affected by the headway ncrease assumng they arrve randomly at the fxed stops. Moreover, both groups passengers wll ncur addtonal cost because they wll have fewer bus runs n any gven tme perod (e.g. AM peak perod) whch lmts ther opportuntes to travel to ther desred actvtes. Therefore, the dfference from the prevous case s the ncrease n the average watng tme for the unaware passengers and the addtonal cost ncurred by all passengers due to the lower frequency. It should be noted, however, that the percentage of unaware passengers mght change gven a change n the headway of the servce as fxed-stop transt users mght start beng more aware of the schedule to avod long watng tmes. However, n ths research we assume that ths percentage wll reman the same even after the change n headway. 1. Same Number of Fxed Stops Frst, let s assume that the unaware passengers comprse a percentage of β from the total fxed route passengers (Y), and that the cost ncurred by all fxed-route passengers can be drectly related to the change n headway (or reducton n frequency). Usng, the varables n Table 3-4, the cost ncurred by fxed-stop users n one route segment (not ncludng access tme) s gven by the followng equaton: Fxed-route users cost = A C TT Y C WT Y C H TT (Equaton 3-45) WT F Where: C F = cost coeffcent representng fxed-stop customer dsutlty toward reducton n servce frequency; and H = the ncrease n the headway after changng the servce to Flex-Route transt servce. In route segment, TT wll amount to S as n the frst case. The average ncrease n wat tme wll depend on the value of the headway before and after the change of the

116 102 servce. Usng equaton 3-27 the ncrease n the headway after changng the servce to Flex-Route transt s gven by T c H S H 2. We can now use ths equaton to fnd the ncrease n the average wat tme (.e. half the headway) for a sngle fxed-route user as follows: WT= c T c H S T H S (Equaton 3-46) It s worth mentonng that the headway varance s not ncluded n the Equaton 3-46, whch mplctly assumes that the operaton s perfectly relable. Therefore, the total fxed-user cost due to the ncrease n both n-vehcle travel tme and wat tme for a sngle bus run can be expressed as follows: C F C WT N TT t T H S C Y T H S C Y S C A FX cos (Equaton 3-47) Rearrangng the terms of equaton 3-47, the total cost ncurred by fxed-route users n ths case wll be gven by the followng equaton: F WT C N TT t C C T H S Y S A C FX cos..... (Equaton 3-48) Gven that the total number of fxed-route users for a sngle bus run s Y, then the average cost ncurred on a sngle fxed-route user n ths case s gven by: Y C C T H S Y S A C FXP F WT C N TT t cos. (Equaton 3-49)

117 103 As wth the prevous case, Equaton 3-49 can be used to estmate the cost ncurred by fxed-route users and examne the senstvty of the cost to such factors as the slack tme, the headway and the fxed-route rdershp. 2. Reduced Number of Fxed Stops In case that, n addton to the ncreased headway, the number of fxed stops has been reduced, there wll be an ncrease n the access tme for fxed-route passengers assumng that there wll not be any loss n rdershp. As mentoned earler, there are also small tme and dstance savngs ganed from the removal of fxed stops. These benefts wll be ncorporated n the total cost functon of ths case (see Equaton 3-57). To fnd the ncrease n the average access tme for fxed-route passengers, we can use equaton 3-41 as n the prevous case. So n ths case the total fxed-users cost due to the ncrease n nvehcle rde tme, wat tme and access tme for one bus run, s gven by the followng equaton: Nn 1 Y S H 1 L L FXcost A 1. (Equaton 3-50) C TT S CWT 2CF Y CAT TC 4V w Nn 1 N 1 Rearrangng the terms of equaton 3-50, the total cost ncurred on fxed-route users due to the change of the servce types s gven by the followng expresson: Nn 1 S H CAT L L FX cost CTT A S Y CWT 2CF 1 TC 4V w Nn 1 N 1. (Equaton 3-51) Dvdng equaton 3-51 by the total number of fxed-route users (Y), we can fnd the average cost ncurred on a sngle fxed-stops user as follows:

118 104 Nn S H C L L C A S Y AT TT CWT CF 1 T 2 C V w Nn N 1 FXP cost Y. (Equaton 3-52) As can be seen n equaton 3-52, the cost ncurred on a sngle fxed-route user n ths case depends on several factors. The servce agency can manpulate the value of each factor and see the effect on the cost dependng on the stuaton that each transt agency fnds tself n Total Addtonal System Cost After fndng the operator and fxed-route user costs, we can now present the total addtonal system cost (C Total ) for a sngle bus run due to the change n the servce as follows: C Total = FX cos t Ocos t. (Equaton 3-53) Usng equaton 3-53, we can now estmate the total system cost for the dfferent cases analyzed above n a sngle expresson. Recall that the two man cases were based on whether to mantan the orgnal servce headway or change t after the ntroducton of the Flex-Route servce. These two cases are further dvded nto whether the transt servce agency decdes to remove some fxed stops or not after the ntroducton of the new servce. Therefore, we wll have four cases for whch the total system cost can be found as shown next Same Frequency/ Same Number of Fxed Stops Ths case assumes that the transt servce agency decdes to keep the same servce headway of the orgnal fxed-route servce and do not remove any exstng fxed stops. To fnd the total system cost n ths case, we substtute equatons 3-32 and 3-36 nto equaton 3-53 as follows:

119 105 C Total = N km dr N TT m W C S C S A C.. (Equaton 3-54) Same Frequency/ Reduced Number of Fxed Stops Ths case s smlar to the prevous case except that some fxed stops are removed from the orgnal fxed-route settng. To fnd the total system cost n ths case, we substtute equatons 3-32 and 3-42 nto equaton 3-54 as follows: C Total = N L N L V Y C S A C n w AT N TT n n N km n f dr N N W m W C N N T S C n (Equaton 3-55) Lower Frequency/ Same Number of Fxed Stops Ths case assumes that the transt servce agency decdes to reduce the servce frequency of the orgnal fxed-route servce and do not remove any exstng fxed stops. To fnd the total system cost n ths case, we substtute equatons 3-33 and 3-48 nto equaton 3-53 as follows: C Total = F WT C N TT C C T H S Y S A C o n km N km VR VR L C m W C (Equaton 3-56) Lower Frequency/ Reduced Number of Fxed Stops In ths case also there s no addtonal operatng cost and the total cost s composed only of the cost ncurred on fxed-route users gven by equaton Therefore the total cost can be calculated usng the followng equaton:

120 106 N 1 S H CAT L L WT F 1 TC 4V w Nn 1 N 1 n C Total = C A S Y C 2C TT..... (Equaton 3-57) Total Addtonal System Beneft As mentoned earler, the system beneft s composed manly of attractng the ondemand passengers as gven n equaton It s assumed that ths equaton apples to all cases dscussed above. Therefore, the total beneft s also gven by equaton 3-34 as follows: B Total = B m..... (Equaton 3-58) p tot The amount of ths beneft vares from one stuaton to another as mentoned earler. Therefore, equaton 3-58 does not specfy the money-value of the beneft and s left open to the decson-maker and future research efforts to decde on the amount of the beneft Strateges to Mantan the Level of Servce for Fxed-Route Users Under deal condtons, the transt servce agency would lke to keep the exstng fxed-route rdershp at the same level after ntroducng the Flex-Route servce. At the same tme, the transt agency would not want the fxed route customers to become ondemand customers. Ths requres fndng strateges to allevate the extra cost ncurred by fxed-route users due to the change of the servce type to Flex-Route. For each scenaro mentoned above, a dscusson of what type of strateges, f any, can be mplemented wll be dscussed next. Dependng on the dfferent cases dscussed n the prevous secton, the possble cost ncurred by the fxed-route users due to the ntroducton of Flex-Route servce can be one or a combnaton of the followng types of cost: Increase n n-vehcle travel tme due to the added slack tme:

121 107 Increase n wat tme due to the change n servce frequency: Increase n access tme f the fxed-stops spacng changes: Gven that the cost ncurred on fxed-route users mght be a combnaton of the above-mentoned cost types, we propose two types of strateges can be used (solely or combned) to lessen the effect of the addtonal cost on fxed-route users. These strateges nclude: 1. Increasng the frequency of the servce (decreasng the headway); and 2. Reducng the fare for fxed-route customers. The above two strateges can be adopted to serve two goals for the transt servce provder. Frst the two strateges can be used solely together to keep the level of servce for fxed-route customers at the same level as before ntroducng the new servce. Increasng the frequency can be a good measure to reduce the wat tmes of users so as to compensate for the longer n-vehcle and access tmes, whle lowerng the fare can be used as a measure to compensate them for the deteroratng servce. Lowerng the fare for fxed-route users could also serve the second objectve, whch s to keep the fxed-route customers from swtchng to the Flex-Route servce. Lowerng the fare for those users compared to the on-demand users, who wll have to pay a hgher fare because of the premum servce, can be a reasonable measure to encourage the fxed-route users to keep accessng the servce at the fxed stops. The followng s a dscusson of these strateges for the cases dscussed earler and how each strategy can be used to allevate the cost ncurred on fxed-route users Same Frequency/ Same Number of Fxed Stops In ths case, the cost ncurred by a sngle fxed-route user s comprsed manly of the extra n-vehcle travel tme due to the added slack tme and s gven by equaton The strateges to lessen ths cost are dscussed next. 1. Increasng the frequency of the servce Snce the only cost ncurred by the fxed-route users n ths case s due to the ncrease n n-vehcle travel tme, then ncreasng the servce frequency n the new servce could be a major factor n encouragng them to keep usng the servce and

122 108 allevate any travel tme ncrease. Accordngly, assumng the headway of the servce s decreased from H (n the orgnal fxed-route settng) to H n1 (for the new Flex-Route settng), then there wll be a reducton n the expected wat tme at the fxed stops for unaware customers, as well as a reducton n the servce headway (H-H n1 ) for all fxedroute customers. The reducton n wat tme amounts to: H H n 1 WT (Equaton 3-59) 2 Where WT 1 : s the average reducton n wat tme for an unaware fxed-route user; H n1 : s the new headway of the servce. Therefore, for the servce frequency ncrease to elmnate the effect of the ncrease n travel tme for fxed-route users, the followng nequalty should hold: C WT 2 H H n1 C F H H n1 C TT N 1 1 Y A S Or N 1 CTT A S H H 2 n1 1 CWT 2C F.... (Equaton 3-60) Y formula: Solvng the nequalty for the reducton n the headway, we get the followng H H n1 Y N 1 2 CTT C 2 WT C F 1 A S..... (Equaton 3-61)

123 109 Rearrangng equaton 3-61 and solvng for the new value of the headway, we get the followng formula: H n1 H Y N 2 C TT 1 A S CWT 2CF CWT (Equaton 3-62) Equaton 3-62 gves the new value of the headway, that f used wll elmnate the effect of the ncrease n n-vehcle travel tme for fxed-route users caused by the extra slack tme added to the servce. The amount of the reducton from the orgnal headway depends on the slack tme, the n-vehcle travel tme and wat tme cost coeffcents and the fxed-route rdershp. The equaton shows that the more slack tme added to the servce the more the reducton n the headway should be to lessen the effect of the cost on fxed-route users. 2. Reducng the fare If reducng the fare for fxed-route users was to be used as the only method to try to counter the effect of the ncrease n n-vehcle travel tme for fxed-stop users, then one way to do t s through usng the money-value estmate of the ncrease n n-vehcle travel tme found n equaton 3-37, and reducng the fare by an amount greater than that value as follows: F C TT N 1 1 Y A S.... (Equaton 3-63) Where F: s the reducton n fare for fxed-stop users.

124 Increasng the frequency of the servce and Reducng the fare If the two above strateges were to be used together to counter the cost effect of n-vehcle travel tme n order to mantan the same level of servce for fxed-route users, then the followng nequalty should hold: F N 1 WT F 1. (Equaton 3-64) CTT A S H H 2 n 1 C 2C Y Equaton 3-64 can be used to fnd a combnaton of fare reducton and ncreased frequency to counter the ncrease n travel tme for fxed-route users. For example, f the transt agency s lmted to decreasng the headway by a specfc amount, t can elmnate the remanng cost nflcted on fxed-route users by reducng the fare to satsfy the equaton Same Frequency/ Reduced Number of Fxed Stops In ths case, the cost ncurred by a sngle fxed-route user s comprsed of the extra n-vehcle travel tme due to the added slack tme plus the ncrease n average access tme due to the reduced number of fxed stops. The total cost to a sngle user n ths case s gven by equaton The strateges to counter ths cost effect are dscussed next. 1. Increasng the frequency of the servce In ths case, we also assume that the new headway of the servce after ntroducng the Flex-Route servce wll be H n1 compared to H n the orgnal settng. Thus, the reducton n the wat tme for fxed-route users s gven by equaton 3-59 as shown earler. For ths reducton n wat tme to elmnate the effect of both the ncrease n nvehcle travel tme and the ncrease n access tme, the cost functon of equaton 3-59 should be greater than that of equaton 3-53 as follows:

125 111 Y N L N L V Y C S A C C C H H n w AT N TT F WT n n (Equaton 3-65) Rearrangng the terms of equaton 3-65 and solvng for the new value of the headway, we get the followng equaton for (H n1 ): F WT n w AT N TT n C C Y N L N L V C Y S A C H H n (Equaton 3-66) Equaton 3-66 can be used to fnd the new value of the headway gven the values of the slack tme, fxed-route rdershp and route length among other factors. As mentoned n smlar cases, the equaton can be used to study the effect of each factor on the amount of headway reducton needed to counter the effect of ths factor. 2. Reducng the fare Usng the same procedure of the prevous case, the reducton n fare should be greater than the cost ncurred by a sngle fxed-stop user. The cost ncurred by a sngle fxed-route user n ths case s obtaned from equaton Therefore, to fnd the mnmum amount of reducton from the orgnal fare for fxed-route users, we can use the followng nequalty: Y N L N L V Y C S A C F n w AT N TT n (Equaton 3-67)

126 Increasng the frequency of the servce and Reducng the fare In ths strategy we combne the prevous two strateges nto a sngle strategy to offset the cost ncurred on fxed-route users. Therefore, we can combne the equatons obtaned for the prevous two strateges nto one equaton to fnd combnatons of the values of the new headway and fare that wll allevate the cost ncurred on fxed-route users as follows: F Nn CAT Y L L C 1 A S w Nn N TT H H V n CWT CF Y..... (Equaton 3-68) Reduced Frequency/ Same Number of Fxed Stops In the case where the transt agency has decded to lower the frequency (ncrease the headway) n order to mnmze the addtonal operatng costs, then the only strategy that the transt agency wll consder to mtgate the effects of the change of the servce on the fxed-route users wll be the fare reducton strategy. Usng the same procedure of the prevous cases, the reducton n fare should be greater than the cost ncurred by a sngle fxed-stop user. The cost ncurred by a sngle fxed-route user n ths case s obtaned from equaton Therefore, to fnd the mnmum amount of reducton from the orgnal fare for fxed-route users, we can use the followng nequalty: F C TT N 1 1 Y S H TC Y A S C 2 C WT F... (Equaton 3-69) Reduced Frequency/ Reduced Number of Fxed Stops In ths case, the cost ncurred by a sngle fxed-route user s comprsed of the extra n-vehcle travel tme due to the added slack tme, the cost ncurred by the change n servce frequency and the ncrease n average access tme due to the reduced number

127 113 of fxed stops. The total cost to a sngle user n ths case s gven by equaton The only strategy that the transt agency wll consder to mtgate the effects of the change of the servce on the fxed-route users wll be the fare reducton strategy. Usng the same procedure of the prevous cases, the reducton n fare should be greater than the cost ncurred by a sngle fxed-stop user. Therefore, to fnd the mnmum amount of reducton from the orgnal fare for fxed-route users, we can use the followng nequalty: F C TT Nn S H C L L A S Y AT 1 CWT CF T 2 C V w Nn N Y. (Equaton 3-70) 3.5 FLEX-ROUTE PERFORMANCE When desgnng and operatng Flex-Route systems, planners usually am at mprovng the performance of the Flex-Route servce compared to the orgnal fxedroute servce. The decson on what types of performance measures should be used depends on many factors such as the transt agency goals, the orgnal level of servce for users, and the orgnal rdershp. The most common measures n use are operatng cost per revenue vehcle hour, operatng cost per passenger boardng, cost recovery rato (the proporton of the operatng cost recovered by the fare-box revenue), passenger boardngs per revenue vehcle hour, and passenger boardngs per revenue vehcle mle. Wth the ultmate goal of mprovng the effcency and performance of transt servces, any of the aforementoned measures can be used. In ths work, however, we use two of these measures that are prmary related to the performance of the transt servce n terms of vehcle hours and vehcle klometres, namely passenger boardngs per revenue vehcle hour (PVRH), and passenger boardngs per revenue vehcle Klometre (PVRKm). These measures were chosen snce the cost and revenue of the orgnal fxed-route structure are not analyzed n ths research. Also, these two methods can be used n addton to the cost analyss provded n the prevous secton.

128 114 The analyss n ths secton can help decson-makers n ther assessment of how the servce performance of the flex-route servce compares to that of the orgnal fxedroute servce. However, the analyss does not nclude any measures of servce qualty to account for the customer perspectve. It s mportant to note that ths research s meant to be a frst step toward a more comprehensve understandng of flex-route servce and that t has some lmtatons, such as servce qualty measures, that can be addressed n future research whch should focus on the demand (customer) sde of the servce Passenger Boardngs per Vehcle Revenue Hour (PVRH) For the frst part of the analyss, the assumpton s that none of the exstng rdershp of the fxed-route servce s lost. Based on ths assumpton, consder the followng varables representng the average data for a sngle bus run n one drecton: R o : average exstng fxed-route rdershp; R add : addtonal rdershp from on-demand requests; R n : total rdershp from both fxed-route users and on-demand users; T o : exstng scheduled bus runnng tme of the fxed-route servce; S: slack tme added to serve the on-demand requests; and T n : new scheduled bus runnng tme of the Flex-Route servce. Usng these varables, the new rdershp of the Flex-Route servce (R n ) s defned by the followng equaton: R n = R o + R add..... (Equaton 3-71) Also, the new scheduled bus runnng tme (T n ), f Flex-Route servce replaces the fxed-route servce, s gven by: T n = T o + S (Equaton 3-72) Therefore, the PVRH value of the orgnal fxed-route servce (PVRH o ) and the new Flex-Route servce (PVRH n ) can be found usng the followng equatons:

129 115 R o PVRH o (Equaton 3-73) To R n PVRH n (Equaton 3-74) Tn For the Flex-Route servce to acheve a better performance than the orgnal fxed route servce n terms of PVRH then the followng nequalty should be satsfed: Rn R o (Equaton 3-75) T n T o Substtutng 3-71 and 3-72 nto equaton 3-75 and rearrangng the terms of the equaton, then the PVRH of the Flex-Route servce wll be greater than that of the orgnal fxed-route f the followng nequalty holds: R o T R o add S R T o o (Equaton 3-76) condtons: It can be easly shown that solvng the nequalty wll yeld the followng R add R o S (Equaton 3-77) T o Or n a dfferent format: R add S R o (Equaton 3-78) T o Equaton 3-77 states that the percent ncrease n rdershp f Flex-Route servce s mplemented should be more than the percent ncrease n the scheduled runnng tme, f

130 116 the PVRH of the Flex-Route servce s to exceed that of the orgnal fxed-route servce. In other words the slack tme allocated for the on-demand requests should brng enough on-demand passengers for the Flex-Route to have a better performance compared to the fxed-route servce. Equaton 3-78 states that the rate of accepted requests relatve to the added slack tme should be more than the rate of the exstng rdershp relatve to the exstng bus scheduled runnng tme n order for the Flex-Route servce to acheve a better performance. Both equatons can be used n Flex-Route applcatons to examne the performance of the Flex-Route servce and have a better understandng of what levels of addtonal rdershp are needed to justfy the added slack tme Passenger Boardngs per Vehcle Revenue Km (PVRKm) The process for analyzng the performance of the transt servce n terms of PVRKm s smlar to that performed for PVRH n the prevous secton. Let us use the same defntons for rdershp from the prevous secton, and consder the followng defntons for a sngle bus run: M o : exstng dstance traveled n a sngle bus run n one drecton (n vehcle-km); and M add : addtonal average dstance traveled due to devatons to serve on-demand requests (n vehcle-km). As n the case of PVRH, to acheve a hgher PVRM for the Flex-Route servce compared to that of the exstng fxed route servce, the followng nequalty should be satsfed: R M o o R M add add R M o o (Equaton 3-79) The same approach used for PVRH can be used n ths case also. Therefore, solvng the nequalty wll result n the followng two condtons:

131 117 R add R o M add (Equaton 3-80) M o Or n another format: R M add add R M o (Equaton 3-81) o Equaton 3-81 can be nterpreted n the same manner as equaton 3-77 n terms of how the percent ncrease n rdershp should be more than the percent ncrease n vehcleklometres n order to get a better performance from the Flex-Route servce n terms of PVRKm. Recallng that we are analyzng the performance for a sngle bus run, and consderng a two-sded servce area wth length L and wdth W on each sde of the route, the exstng dstance traveled by the transt vehcle n a sngle bus run (M o ) wll be equal to L. Also, the addtonal dstance traveled to serve the on-demand requests can be estmated usng equaton 3-4: D 2W W m.... (Equaton 3-4) Where D : s the extra dstance traveled n route segment ; M : s the number of on-demand requests served n route segment ; and W : s the wdth of the servce area for route segment. Therefore, the total extra dstance traveled for a sngle bus run (M add ) can be found usng the followng equaton: N -1 M.... (Equaton 3-82) add D 1

132 118 Substtutng equaton 3-4 nto equaton 3-82 and assumng that the number of route segments s N-1, we get the followng expresson for the extra dstance traveled to serve the on-demand requests for the whole route: M add N m 1 3 W (Equaton 3-83) Replacng M o wth L and substtutng equaton 3-83 nto equatons 3-80 and 3-81, we can re-wrte these two nequaltes as follows: R add R o N-1 1 2m 1 W 3 L (Equaton 3-84) N-1 1 R 2m 1 W L add 3 R o (Equaton 3-85) Equatons 3-84 and 3-85 can be used to estmate the performance of the Flex- Route servce compared to the orgnal route servce gven the rdershp numbers for both servces and the characterstcs of the route such as the route length and servce area wdth. It s worth mentonng that estmatng the addtonal on-demand rdershp s beyond the scope of ths thess, and that a dedcated effort s needed to develop mode choce models approprate for predctng on-demand customers as a functon of the servce characterstcs Strateges to Improve the Performance of Flex-Route Servce In real-world case scenaros, the number of on-demand requests served n each route segment may change from one day to another. Furthermore, the tme needed to serve these requests may also change dependng on the dstrbuton of these requests

133 119 wthn the servce area. The uncertanty found n such stuatons may lead to some problems to the servce provder n terms of the unused slack tme stemmng from stuatons where the transt vehcle arrves early at the fxed stops due to a low number of served on-demand requests, or a short tme needed to serve these requests. In such stuatons the transt vehcle wll have to spend some addtonal dwell tme at the fxed stops untl the publshed departure tme. Moreover, although the addtonal dwell tme does not affect the estmated tme of arrval at the customers destnatons, t may add to the perceved nconvenence of transt rders n such events of early arrval at the fxed stops. Early arrvals at fxed stops are always dscouraged n fxed-route servces, and are usually deterred n transt level of servce and transt network desgn (Gray and Hoel 1992). In fact, t could be argued that relatvely late arrval at the fxed stops s more favourable to customers. Therefore the transt servce provder should try to mnmze the stuatons of early arrval for each fxed stop. Ths leads to an mportant strategy that can be used to acheve several goals, namely fxed-stop constrant relaxaton (allowng late arrval at the fxed stops). In ths strategy, the transt vehcle s allowed to reach the fxed-stop later than ts assgned departure tme f that means the system wll perform better. Some of the advantages of late arrval nclude: 1. Mnmze the amount of unused slack tme due to the uncertanty n the amount of tme needed to serve requests n a route segment; 2. Mnmze the perceved nconvenence to the on-board transt rders by mnmzng the unnecessary dwell tme at the fxed stops; and 3. Maxmze the beneft of Flex-Route servce by consumng all the slack tme added, whch wll result n maxmzng the number of accepted ondemand requests. Reducng the unused slack tme s therefore desrable, even at the cost of arrvng a few mnutes late at the fxed stops. In today s transt operatons, late bus arrvals at fxed stops are expected occurrences by passengers who tolerate them to some extent, partcularly f real tme nformaton of expected arrval s provded. Thus relaxng the constrant of arrval tme at the fxed stops could be very useful n schedulng more

134 120 requests especally n dynamc settngs, where an extra arrval tme of less than a mnute or two at the fxed stop could result n answerng more requests. It should be also noted that the amount of late arrval should be controlled and not exceed a certan value. Puttng a hgher lmt on the amount of late arrval at the fxed stops should be studed carefully. That beng sad, t should be also noted that late arrval mght not be a favorable strategy n some cases snce t mght lead to hgher nconvenence to the fxed-route customers. Ths mght occur n stuatons where the rdershp at some fxed stops s hgh enough that late arrval at the fxed stops wll cause more nconvenence to the fxedroute customers overall. In such stuatons t would be preferred to keep the fxed-stop arrval constrant. Other cases for whch late arrval should not be mplemented are fxed stops that are located at mportant locatons such as schools, work places and hosptals. The most mportant case for whch late arrval should not be consdered at all s major transfer fxed stops that connect to other transt lnes, and n some cases connectons to ral lnes. Late arrval n these cases mght lead to mssed connectons for the passengers, whch wll severely affect the relablty of the servce and mght lead to passengers abandonng the servce. The valdty of the role that late arrval can play n mprovng the performance of the servce wll be examned n the senstvty analyss provded n the Chapter 5.

135 121 4 REAL-TIME DECISION-SUPPORT AND SCHEDULING SYSTEM FOR FLEX-ROUTE TRANSIT SERVICES 4.1 CHAPTER OVERVIEW Chapter three dealt wth the plannng, desgn and feasblty ssues of Flex-Route transt, whch s requred to make a decson whether or not Flex-Route transt s a vald opton for any specfc stuaton. In ths chapter, we deal wth the day-to-day schedulng problem (.e. tryng to produce daly schedules based on the daly demand). At ths stage, we assume that the servce elements (such as servce area, slack tme, servce frequency, etc.) are all fxed (.e. do not change from day to day), whch s requred to develop the route plans and schedules that are requred on a daly and real-tme bass to cater to ondemand requests. Secton 4.2 provdes the motvaton for developng a new schedulng algorthm for Flex-Route servce. Secton 4.3 provdes a comprehensve revew of Constrant Programmng and the role t plays n the schedulng system. Followng that, secton 4.4 presents a blueprnt for a decson-support system archtecture that can be used to assess and enhance the operatons of Flex-Route transt servces. The fnal two sectons present the proposed schedulng methodology for the Flex-Route problem for both ts statc and dynamc versons. Ths ncludes system models for each case wth approprate objectve functons. 4.2 FLEX-ROUTE ROUTING AND SCHEDULING Vehcle routng and schedulng s a crtcal part of the desgn and operatons of any transport system. For tradtonal fxed-route transt systems where all transt vehcles stop at every staton, schedules are easly developed for operatonal purposes. Other types of transt systems such as demand-responsve transt systems (DRT), however, are more complcated than tradtonal fxed-route transt systems, and f proper care s not gven to the routng and schedulng of the servce, very expensve and unrelable schedules may result wth a hgh rsk of falng to meet the requred level of servce expected by the passengers. For Flex-Route transt systems, the routng and schedulng task becomes

136 122 even more complex due to the exstence of fxed-route and demand-responsve components n the operaton of the servce. There are numerous vehcle routng and schedulng algorthms that are bult to produce solutons (schedules) for DRT systems. The problem of fndng optmal solutons for a DRT system, also known as the Dal-A-Rde Problem (DARP), can be consdered an expanson of the travelng salesman problem, whch nvolves the calculaton of an optmum journey for vstng a number of predetermned nodes on a network. Algorthms for schedulng the Flex-Route problem are very rare n the lterature. Most of the procedures adopted to solve the Flex-Route schedulng problem use the same algorthms bult for solvng the DARP, wth few varatons (see Chapter 2 for more ndepth revew). Therefore, there s a need for a schedulng methodology specfc to the Flex-Route problem that takes nto consderaton the unqueness of the servce. Ths s why one of the two major objectves of ths research s to develop such a methodology that can explot ths unqueness to fnd optmal schedules for the Flex-Route problem both n statc and dynamc settngs. Constrant Programmng, whch s used n the schedulng methodology proposed n ths research, s dscussed next. 4.3 NEW APPROACH: CONSTRAINT PROGRAMMING Constrant Programmng s often called constrant logc programmng, and t orgnates n the Artfcal Intellgence lterature n the Computer Scence communty. Unlke Mathematcal Programmng, the term programmng n Constrant Programmng refers to ts roots n the feld of programmng languages. Knuth (1968) defnes a computer program as an expresson of a computatonal method n a computer language. A computer program can be vewed as a plan of acton for the operatons of a computer, and hence the common concept of a plan s shared wth the plannng problems studed n the development of the smplex method. Constrant Programmng ganed ts name from the sprt of other programmng technques, such as object-orented programmng, functonal programmng, and structured programmng. Logc programmng s a declaratve, relatonal style of programmng based on frst-order logc, where smple resoluton algorthms are used to resolve the logcal statements of a problem. Constrant logc programmng extends ths

137 123 concept by usng more powerful algorthms to resolve these statements. Van Hentenryck (1999, p.4) wrtes: The essence of constrant programmng s a two-level archtecture ntegratng a constrant and a programmng component. The constrant component provdes the basc operatons of the archtecture and conssts of a system reasonng about fundamental propertes of constrant systems such as satsfablty and entalment. The constrant component s often called the constrant store, by analogy to the memory store of tradtonal programmng languages.... Operatng around the constrant store s a programmng-language component that specfes how to combne the basc operatons, often n non-determnstc ways. Hence, a constrant program s not a statement of a problem as n mathematcal programmng, but s rather a computer program that ndcates a method for solvng a partcular problem. It s mportant to emphasze the two-level archtecture of a constrant programmng system. Because t s frst and foremost a computer programmng system, the system ncludes representatons of programmng varables, whch are representatons of memory cells n a computer that can be manpulated wthn the system. The frst level of the constrant programmng archtecture allows users to state constrants over these programmng varables. The second level of ths archtecture allows users to wrte a computer program that ndcates how the varables should be modfed so as to fnd values of the varables that satsfy the constrants Orgns of Constrant Programmng The feld of Lnear Programmng (LP) has been n the lterature snce the 1940 s when Dantzg (1963) nvented the Smplex method for solvng a lnear program n Dantzg frst descrbed the Smplex method n a paper enttled Programmng n a lnear structure (Dantzg 1948, 1949). Throughout the years, LP has been a strategc technque used by thousands of busnesses, educaton nsttutes and researchers around the world tryng to optmze ther problems. The problems that Dantzg studed whle developng the Smplex algorthm were programmng problems, because the Unted States Defence Department n the post-world War II era was supportng research to devse programs of actvtes for future conflcts. Therefore, the term program became

138 124 assocated wth a specfc mathematcal problem n the Operatons Research lterature (Lustg and Puget, 2001). In the md-1980s, researchers developed Constrant Programmng (CP) as a Computer Scence technque by combnng developments n the Artfcal Intellgence communty wth the development of new computer programmng languages. Constrant Programmng can be consdered as the confluence of three major scences: Computer Scence, Artfcal Intellgence and Operatons Research. Ths makes t a powerful technology that makes use of software engneerng, logc, heurstcs and mathematcal algorthms. Constrant Programmng s now beng seen as an mportant technque that complements tradtonal mathematcal programmng technologes as researchers contnue to look for methods to tackle new types of optmzaton problems Formulaton In Lnear Program models (LP s) all the constrants are lnear and collectvely they defne a polyhedral set. There are powerful Smplex algorthms for determnng the consstency of such a collecton and for fndng solutons that are optmal for a lnear objectve functon. The key mathematcal tool s dualty theory whch s based on the fact that the constrants and the objectve functon are all lnear. Wth Integer Program models (IP s) for combnatoral problems, the set of feasble solutons s represented as the set of all nteger ponts n a polyhedral set and an optmal soluton s one among these wth best possble value for the objectve functon. The mathematcal bass for CP, however, s the Constrant Satsfacton Problem, whch s defned as follows: 1. Suppose a fnte set of varables (X 1,...,X n ) s gven. Wth each varable assocated a non-empty fnte doman (D 1,..., D n ). 2. A constrant on k varables (X 1,...,X k ) s a relaton: as follows R(X 1...,X k ) є (D 1,, D k ) 3. A constrant satsfacton problem (CSP) s gven by a fnte set of constrants on the varables of the problem.

139 A soluton to a CSP s an assgnment to all the varables wth values from ther respectve domans whch satsfy all the constrants. The extenson to the case where there s an objectve functon s straghtforward. In CP, each constrant R of a CSP s consdered as a sub-problem, and technques are developed for handlng frequently encountered constrants. Wth each constrant s assocated a doman reducton algorthm whch reduces the domans of the varables that occur n the constrant. The other key ssue s communcaton among the constrants or sub-problems. The basc method used s called constrant propagaton whch lnks the constrants through ther shared varables. The mportant thng about ths setup s that t s very modular and ndependent of the partcular structure of the ndvdual constrants. The next secton dscusses these algorthms n further detal Connectons to Integer Programmng It has been dscussed that constrant programmng can be appled to combnatoral optmzaton problems. The search strateges used n constrant programmng are related to those used when solvng mxed-nteger-programmng problems va branch-and-bound procedures. In fact, branch and bound, whch s an enumeratve search strategy, has been used to solve nteger programs snce the md 1960s. Lawler and Wood (1966) gave an early survey, whle Garfnkel and Nemhauser (1972) descrbed branch and bound n the context of an enumeratve procedure. Nemhauser and Wolsey (1988) provded a more recent dscusson. In systems developed for nteger programmng, users are often gven the opton of choosng a varable selecton strategy and a node selecton strategy. These are clearly equvalent to the search selectors and node evaluators descrbed earler. A constrant programmng framework extends the basc branch-and-bound procedures mplemented n typcal mxed nteger programmng solvers n two fundamental ways. Frst, n most mplementatons of branch-and-bound procedures for mxed-nteger programmng, the mplementaton creates two branches at each node after a varable x has been chosen to branch on. In the constrant programmng framework, the choces that are created can be any set of constrants that dvdes the search space. For

140 126 example, gven two nteger varables x 1 and x 2, one could create a choce pont consstng of the three choces (x 1 < x 2 ), (x 1 > x 2 ), (x 1 = x 2 ). The second way that a constrant programmng framework extends the basc branch-and-bound procedures s wth respect to the varable selecton strategy. In most branch-and-bound mplementatons, the varable selecton strategy uses no knowledge about the model of the problem to make the choce of varable to branch on. The nteger program s treated n ts matrx form, and dfferent heurstcs are used to choose the varable to branch on based on the soluton of the lnear programmng relaxaton that s solved at each node. In a constrant programmng approach, the user specfes the branchng strategy n terms of the formulaton of the problem. Because a constrant program s a computer program, the decson varables of the problem can be treated as computer programmng varables, and a strategy can be programmed usng the language of the problem formulaton. Hence, to effectvely apply constrant programmng technques, problem-specfc knowledge can be used to help gude the search strategy so as to effcently fnd a soluton. In ths way, a constrant programmng system, when combned wth a lnear programmng optmzer, can be vewed as a framework that allows users to program problem-specfc branch-and-bound search strateges for solvng mxed-nteger programmng problems by usng the same programmng objects for declarng the decson varables and for programmng the search strateges (for more detal, see Lustg and Puget, 2001) Applcatons Schedulng applcatons such as job shop, flow shop and varants have long been a subject of specal nterest n the OR communty and both IP technques and CP technques have been developed for these problems. In the classcal job schedulng problem, whch s embedded n some form or another n many current applcatons, the decson varables are typcally the start tmes of the tasks; the domans of these varables are the tmes when t s permssble to schedule them. The constrants usually nclude precedence constrants: lnear constrants that assert that one task must follow another. Another mportant type of constrant n schedulng problems s the resource constrant

141 127 whch asserts that a task requres a certan amount of a resource durng ts executon. For example, performng a task mght requre 2 workers wth certan sklls for 1 hour. Ths s a complex constrant. Another example that comes up n schedulng s the requrement that a task must be completed wthn one work shft or another tme frame. Snce IP s NP-Complete, these complex constrants can, of course, be coded n an IP model; however, n many stuatons the resultng model s not tght and the lnear relaxaton does not gude the search n the rght drecton. As wth IP, the soluton of a CP model wll requre a searchng mechansm. For ths, a wde range of strateges are used whch make use of the nformaton on the state of the decson varables, nformaton that s made avalable by the doman reducton process. For example, n job-shop schedulng applcatons, a strategy can make use of updates on earlest start and fnsh tmes of the tasks that reman to be scheduled. In general, CP s the method of choce for applcatons wth complex constrants where heurstcs can be enhanced by doman reducton and constrant propagaton. For a recent overvew of CP wth hstorcal perspectve, see Saraswat and vanhentenryck (1996). There are many schedulng problems whch are close n sprt to the job shop problem such as: The flow shop problem s the specal case of the job shop n whch all jobs requre the resources n the same order and the order of the jobs on the frst resource needed determnes the order on all other resources. The host schedulng problem s one where a host or robot shuffles back and forth movng materals onto the lne, from workstaton to workstaton and off the lne. The challenge s to fnd a cyclc schedule whch optmzes throughput Constrant Programmng Algorthms Constrant Programmng uses multple algorthms to fnd solutons to constrant satsfacton and optmzaton problems. In tradtonal constrant programmng systems, the user s requred to program a search strategy that ndcates how the values of the varables should change so as to fnd values that satsfy the constrants. Ths secton descrbes the fundamental algorthms underlyng a constrant programmng system and

142 then presents the methodologes used to program search strateges. Fgure 4-1 shows the solvng procedure used n Constrant Programmng to search for a soluton. 128 Fgure 4-1: Constrant Programmng solvng procedure Doman Reducton and Arc Consstency As explaned earler, a constrant s a mathematcal functon f (x 1, x 2,..., x n ) of the varables. Wthn ths envronment, an assumpton s made that there s an underlyng mechansm that allows the domans of the varables to be mantaned and updated. To mprove the search process, Constrant Programmng uses a technque known as doman reducton. Bascally, doman reducton means that at every node n the search tree, the algorthm wll remove from the doman of a varable any value whch can be confrmed not to be part of any feasble soluton to the constrants. For each constrant, a doman

143 129 reducton algorthm modfes the domans of all the varables n that constrant, gven the modfcaton of one of the varables n that constrant. The doman reducton algorthm for a partcular knd of constrant dscovers nconsstences among the domans of the varables n that constrant by removng values from the domans of the varables. If the algorthm dscovers that a partcular varable s doman becomes empty, then t can be determned that the constrant cannot be satsfed, and an earler choce can be undone. Dscardng values ths way can greatly reduce the number of choces to branch on for each varable. A smlar methodology s found n the bound-strengthenng algorthms used n modern mathematcal programmng solvers as dscussed by Brearley, Mtra, and Wllams (1975). A crucal dfference between the procedures used n mathematcal programmng pre-solve mplementatons and doman reducton algorthms s that n constrant programmng, the domans can have holes, whle n mathematcal programmng, domans are ntervals. There are several technques that are used for doman reducton n CP, and one of the most effcent algorthms s arc-consstency. As mentoned, a constrant on k varables (X 1,...,X k ) s a relaton as follows: R(X 1,,X k ) є (D 1,, D k ) The constrant R s arc-consstent f for every X and every d n D, there are values d j n D j for j such that R(d 1,...,d k ) holds. A constrant s then arc consstent f all of the values n the domans of all the varables nvolved n the constrant are consstent. A constrant system s arc consstent f all of the correspondng constrants are arc consstent. The term arc s used because the frst CSPs were problems wth constrants stated on pars of varables, and hence ths system can be vewed as a graph, wth nodes correspondng to the varables and arcs correspondng to the constrants. Arc consstency enables the domans of the varables to be reduced whle not removng potental solutons to the CSP. For a good exposton on arc-consstency, see (Mackworth 1993 and Freuder, 1977).

144 130 By way of example, let us consder a bnary constrant r (X, Y) on the two varables X and Y. The arc consstency of the constrant s mantaned as follows: For each element a n the doman of X, a s deleted from the doman unless there s an element b n the doman of Y such that r (a, b) holds; and conversely, b s removed from the doman of Y unless there s a n the doman of X such that r (a, b) holds. To see how ths works n practce, consder the varables X and Y wth doman [ ] and the bnary constrant: 7 X 5Y 27 Arc-consstency reduces the domans of X and Y respectvely to [1...1] and [4...4] n one smple step. Researchers have developed a number of algorthms to effcently propagate constrants and reduce domans so as to create systems that are arc-consstent. One algorthm, called AC-5, was developed by Van Hentenryck, Devlle, and Teng (1992). Ther artcle was mportant for constrant programmng because t unfed the research drectons of the constrant satsfacton communty and the logc programmng communty by ntroducng the concept of developng dfferent doman reducton algorthms for dfferent constrants as mplementatons of the basc constrant propagaton and doman reducton prncple. One of the most valuable constrants n Constrant Programmng s the all dfferent constrant (alldff constrant). Ths constrant asserts that the set of varables n that constrant must take dstnct values. The alldff constrant s especally nterestng from the pont of vew of arc-consstency. From an operaton research pont of vew, t can be easly seen that the satsfablty of the alldff (X 1,.,X k ) constrant taken n solaton reduces to the feasblty of an assgnment problem. Comparng the procedure to an assgnment problem, the varable X can be thought of as a supply node and the values of the X as demand nodes and to have an arc from X to each of the values n ts doman. The supply nodes produce 1 unt and the demand nodes consume at most 1 unt. If the

145 131 assgnment s feasble, then the alldff constrant tself s feasble and vce versa. The doman reducton needed to acheve arc-consstency can be descrbed as a sequence of these problems: to test a n the doman of X, reduce the capacty of all arcs from X j to a where j to 0, and test for feasblty of the resultng assgnment problem. For the alldff constrant, arc-consstency wll detect whenever a set of k values must be reserved for a set of k varables. As an example, suppose that the alldff constrant has been posted on the 10 varables (X 1,., X 10 ) all wth doman [1...10]. Then f the domans of X 1 and X 2 both become [1...2], arc-consstency of the alldff constrant wll remove 1 and 2 from the domans of the other X. Ths reflects the fact that the alldff constrant mples that the values 1 and 2 must be reserved for the varables X 1 and X Constrant Propagaton The prevous secton dscussed how ndvdual constrants perform doman reducton. The next thng s to consder how doman reducton nformaton can be propagated among the constrants to enable more doman reducton. Suppose two constrants R 1 (X, Y) and R 2 (X, Z) have the varable X n common. Let the doman of X be D X = {a, b, c, d} and let D Y = {u, v, w}. Suppose that after R 1 performs doman reducton, we have D X = {a, b, c} and D Y = {u, v, w}. Then, R 2 can perform doman reducton wth ths new doman for X; suppose now that ths reduces the domans to D X = {a, b} and D Z = {p, q, r}. The elmnaton of c from the doman of X can have mplcatons for the doman reducton machnery of R 1 ; re-applyng the doman reducton mechansm of R 1 mght reduce D Y to, say, {u, v}. If there s a constrant R 3 (Y, Z), ths reducton mght n turn reduce the doman of Z whch sends us back to R 2. Next, R 2 mght reduce the doman of X whch sends us back to R 1 ; and so on. Snce the domans of the varables are fnte, ths back-and-forth wll stablze reachng a pont where no further doman reducton s possble. Ths process where doman reducton s performed and constrants communcate teratvely through the changes n the domans of shared varables s known as constrant propagaton. When the process stablzes, t means a fxed pont has been reached. The process of doman reducton s often called flterng, whle the combned process of

146 132 doman reducton and constrant propagaton s often smply called constrant propagaton. Constrant propagaton can help solve a CSP n three ways. It can serve to detect nfeasbltes hgh n the search tree by reducng a doman to the empty set. Secondly, t can reduce the branchng factor at nodes n the search tree. Thrdly, t can help gude the search down the tree. For ths last pont, let us consder a classc heurstc. The frst-fal heurstc n a search chooses for the next branchng varable the one wth the smallest current doman (Haralck and Elot, 1980). Doman reducton n a CP system s dynamc and at each node the doman reducton process wll update nformaton on the domans of the varables and thus on the sze of these domans. The frst-fal heurstc wll reduce the branchng factor and also wll tend to work on a varable whch s nteractng sgnfcantly wth other varables of the problem. Also, the ntuton s that branchng on ths most restrcted varable s the most lkely way to dscern a contradcton snce t has the fewest degrees of freedom. The constrant propagaton methodology can be demonstrated wth a smple example (Fgure 4-2). Consder two varables x and y, where the domans of each varable are gven as follows: D x = {1,2,3, 4,,10} and D y = {1,2,3,4,...,10} Consder the sngle constrant: Y 2X If we frst consder the varable Y and ths constrant, we know that Y must be even, and hence the doman of Y can be changed to: D y = {2, 4, 6, 8, and 10}

147 133 Now, consderng the varable X, we see that snce y 10, t follows that X 5, and hence the doman of X can be changed to: D x = {1, 2, 3, 4, 5} Suppose that we add a constrant of the form (x modulo 2) = 1. Ths s equvalent to the statement that X s odd. We can then reduce the doman of X to be: D x = {1, 3, 5} Now, reconsderng the orgnal constrant Y = 2X, we can remove the values of 4 and 8 from the doman of Y and obtan: D y = {2, 6, 10} The above process that shows how doman reducton and constrant propagaton reduced the domans of the two varables n few smple steps s shown n Fgure 4-2. Ths approach wll be fully utlzed n the schedulng algorthm as t wll greatly reduce the search space of the problem, especally wth the exstence of a huge number of constrants. It s worth mentonng that some constrant programmng systems (for example, ILOG Solver) allow the programmer to take advantage of exstng propagators for bultn constrants that cause doman reductons and allow the programmer to buld hs or her own propagaton and doman reducton schemes for user-defned constrants. However, many constrant programmng systems (for example, OPL and ILOG Solver) are now powerful enough that they provde large lbrares of predefned constrants, wth assocated propagaton and doman reducton algorthms, and t s often not necessary to create new constrants wth specalzed propagaton and doman reducton algorthms. There are sgnfcant advantages to the constrant propagaton model of communcaton among constrants and the way t supports problem decomposton. Frstly, t s completely modular. New constrants can be added to the mx n an elegant

148 134 way. For example, ILOG Solver supports user defned constrants whch are automatcally lnked to the bult-n constrant propagaton machnery provded by the system. Another advantage s that t protects the sub-problem structure of the ndvdual constrants. For example, removng a value from the doman of a varable n the alldff constrant means reducng the flow along an arc to 0. In summary then, CP wll tend to perform well on applcatons where doman reducton and constrant propagaton are effectve and where ths up-to-date doman nformaton wll assst a search strategy. As one would expect, ths s the case for the classc schedulng applcatons where these condtons also apply.

149 135 D x Y = 2 X Y 10 X modulo 2 = 1 Y = 2 X D y Fgure 4-2: Constrant propagaton and doman reducton are used to reduce the domans of the varables X and Y (source: Lustng and Puget, 2001)

150 Constrant Programmng Algorthms for Optmzaton As compared to lnear and mxed-nteger programmng, a weakness of a constrant programmng approach when appled to a problem wth an objectve functon to mnmze s that a lower bound may not exst. A lower bound may be avalable f the expresson representng the objectve functon has a lower bound that can be derved from constrant propagaton and doman reducton. Ths s unlke nteger programmng, n whch a lower bound always exsts because of the lnear programmng relaxaton of the problem. Constrant programmng systems offer two methods for optmzng problems, called standard and dchotomc search. The standard search procedure s to frst fnd a feasble soluton to the CSP, whle gnorng the objectve functon, whch s a functon of the problem varables as follows: g (x 1, x 2,..., x n ). Let y 1, y 2,..., y n represent such a feasble pont. The search space can then be pruned by addng the followng constrant to the system and contnung the search: g (y 1, y 2,..., y n ) > g (x 1, x 2,..., x n ) The added constrant specfes that any new feasble pont must have a better objectve value than the current pont. Propagaton of ths constrant may cause the domans of the decson varables to be reduced, thus reducng the sze of the search space. As the search progresses, new ponts wll have progressvely better objectve values. The procedure concludes when no feasble pont s found. When ths happens, the last feasble pont can be taken as the optmal soluton. Dchotomc search depends on havng a good lower bound L on the objectve functon g (x 1, x 2,..., x n ). Before optmzng the objectve functon, a procedure must fnd an ntal feasble pont, whch determnes an upper bound U on the objectve functon. A dchotomc search procedure s essentally a bnary search on the objectve functon. The procedure computes the mdpont M = (U + L) / 2 of the two bounds and then solves a CSP by takng the orgnal constrants and addng the constrant:

151 137 g (x 1, x 2, x n ) < M If the search fnds a new feasble pont, then t updates the upper bound and contnues the search n the same way wth a new mdpont M. If t fnds the system to be nfeasble, then t updates the lower bound, and the search agan contnues wth a new mdpont M. Dchotomc search s effectve when the lower bound s strong, because the computaton tme to prove that a CSP s nfeasble can often be large. The use of dchotomc search n cooperaton wth a lnear programmng solver may be effectve f the lnear programmng representaton can provde a good lower bound. The dfference between ths procedure and a branch-and-bound procedure for mxed-nteger programmng s that the dchotomc search stresses the search for feasble solutons, whereas branch-and-bound procedures usually emphasze mprovng the lower bound Programmng a Search Strategy Gven a CSP, constrant propagaton and doman reducton algorthms to reduce the domans of the varables can be appled so as to arrve at an arc-consstent system. However, whle dong ths may determne whether the CSP s nfeasble, t does not necessarly fnd solutons to a CSP. To do ths, a search strategy must be used. Tradtonally, the search facltes that constrant programmng systems provde have been based on depth-frst search. The root node of the search tree (see Fgure 4-3) contans the ntal values of the varables. At each node, the user programs a goal, whch s a strategy that breaks the problem nto two (or more) parts and decdes whch part should be evaluated frst. A smple strategy mght be to pck a varable and to try to set that varable to the dfferent values n the varable s doman. Ths strategy creates a set of branches n the search tree and creates what s called a choce pont, wth each branch correspondng to a specfc choce. The goal also orders the branches amongst themselves wthn the choce pont. In the next level of the tree, the results of the choce made at the branch are propagated, and the domans are reduced locally n that part of the tree. Ths wll ether produce a smaller arc-consstent system or a proof that the choce made for ths leaf s not

152 138 possble. In ths case, the system automatcally backtracks to the parent and tres other leaves of that parent. The search thus proceeds n a depth-frst manner, untl t fnds a soluton at a node low n the tree or untl t explores the entre tree, n whch case t fnds the CSP to be nfeasble. The search strategy s enumeratve, wth constrant propagaton and doman reducton employed at each node to help prune the search space (for further nformaton, see Lustg and Puget, 2001). Fgure 4-3: A search tree 4.4 PROPOSED DECISION-SUPPORT SYSTEM ARCHITECTURE As mentoned n Chapter 1, ths research focuses on two complementary parts. The frst s concerned wth the plannng and desgn of Flex-Route transt servce n terms of defnng the key desgn factors of the servce and the modfcatons requred to allow a fxed-route structure to operate n a Flex-Route mode, as well as analyzng the mpacts on the dfferent stakeholders affected by the Flex-Route transt servce. Ths part wll be

153 139 addressed thoroughly n Chapter 5. The second part of the research s concerned wth developng a blueprnt for a decson-support system (DSS) responsble for trp bookng, schedulng and dspatchng of Flex-Route transt servce n both statc and dynamc settngs. The man component of the decson-support system s the schedulng system that can be used to produce optmal statc and dynamc schedules n mnmal computaton tme. In ths research, we develop a new schedulng methodology for schedulng the Flex- Route problem usng advanced algorthms from Artfcal Intellgence, namely Constrant Programmng. Ths schedulng methodology wll be presented and dscussed thoroughly later n ths chapter. The DSS framework could also be an excellent tool for expermentng wth dfferent types of Flex-Route servce and observng the changes n terms of benefts and costs to the system. The dynamc component s especally of great mportance, snce t allows for real tme control of the system wthout the need for ndvdual decsons by the drvers. The nput to the statc system conssts of: the fxed-route daly schedule (after determnng the approprate slack tme); the expected number of passengers on the fxed stops (wth ther orgn and destnaton stops); and the number of statc requests for devated stops (advance requests wth suffcently long lead tmes). Based on that, there exst two categores of data that need to be treated dfferently. The frst category of data, called statc data, conssts of those that are relatvely stable and need not be updated frequently. Examples nclude road network topology, customer addresses, and fleet and drver nformaton. The second category ncludes data such as vehcle locaton, traffc condtons, new requests and cancellaton, whch often change durng the tme of day and need to be updated on a contnuous bass. The archtecture for the proposed decson-support system (called REFLEX) wth ts varous sub-systems s shown n Fgure 4-4. The components of the system nclude: 1. Trp reservaton mechansm. 2. Routng and schedulng algorthm. 3. Geographc Informaton System (GIS) platform. 4. Moble Data Termnals (MDTs). 5. Dspatch center.

154 140 The followng s a bref descrpton of each component of the system Trp Reservaton In REFLEX, the reservaton system provdes the connecton between the operaton centre and the on-demand customers. It s responsble for recordng nformaton on each customer s request, ncludng pckup locaton, desred pck-up tme, number of people to be pcked up, and n the case of request cancellaton, the needed nformaton to update the system. Reservaton methods could nclude: automated telephone systems, text-messagng and web-based nteractve reservaton systems. All of these methods could also be used together to reach to a wde varety of the populaton Routng and Schedulng Algorthm The routng and schedulng algorthm n REFLEX, whch wll be presented later n ths chapter, wll be able to produce relable, cost-effectve schedules both n statc and dynamc settngs. The schedulng mechansm conssts of two major components: a statc component and a dynamc component (see Fgure 4-4). Both components have vrtually the same archtecture and functon (.e. producng a schedule that the transt vehcle has to follow). The excepton s that n the dynamc or real-tme settng, the schedule of the vehcle mght change several tmes durng the day based on real-tme events such as new real-tme requests, traffc condtons, trp cancellatons and so on. The nteracton between the statc model and the dynamc model can be seen n Fgure 4-4. If a real-tme event occurs, the schedule and route wll be updated n response to ths event Geographc Informaton System (GIS) Platform The GIS platform generates all nformaton regardng the street lnks (ncludng travel tme on each lnk), and how they relate to the map nodes (.e. ntersectons), and customers pck-up and drop-off locatons. Ths nformaton from the GIS platform wll be fed nto the schedulng and routng mechansm, where t can be used to fnd the statc or dynamc schedules. The schedulng algorthm s bult n ILOG Dspatcher TM to produce the schedules. The Constrant Programmng algorthms used to develop schedules for the Flex-Route transt servce receve data from the GIS platform and nformaton regardng the transt users to produce schedules and routes that optmze the

155 141 overall objectve functon, takng nto consderaton all varables and constrants representng the dfferent aspects of the servce. Geographc Informaton Systems allow users to store, manage, dsplay, manpulate and analyze spatal and attrbute data. The GIS software package selected for ths research s ArcGIS 9.2, developed by Envronmental Systems Research Insttute (ESRI). The necessary spatal and attrbute data can be stored n a database bult wthn ILOG Dspatcher TM. The spatal data dentfy locatons of features of nterest (such as bus stops and routes) whle the attrbute data nclude rdershp volumes, vehcle names, stop names, etc Moble Data Termnals (MDTs) Each servce vehcle would be equpped wth an n-vehcle computer or a Moble Data Termnal (MDT), to whch vehcle routes and schedules can be uploaded from the Dspatch centre through a wreless communcaton channel. The MDT s ntegrated wth an Automatc Vehcle Locatons (AVL) system that provdes the Dspatch centre wth the real-tme locaton of the vehcle, whch s used for real-tme montorng and vehcle dspatchng Dspatch Centre The dspatch centre wll contnuously montor any operatonal changes n the system such as request cancellatons and new requests. These changes may justfy modfcaton of exstng vehcle schedules such as dvertng en-route an on-road vehcle to servce a new request n the vcnty of the vehcle. Once a change s verfed, the modfed schedules are sent to the drvers and dsplayed on ther n-vehcle moble data termnals (MDT).

156 142 Trp Reservaton System Statc Requests Request Informaton Request Informaton Real Tme Requests Decson Data Inventory Decson Traffc Condtons Lnk Travel Tmes Fxed Stop Demand Updated Lnk Travel Tmes Real Tme Traffc Condtons Statc Settng Schedulng/Routng Framework [ILOG] Schedulng/Routng Algorthm Dynamc Settng Statc Daly Schedule AVL Dynamc Schedule MDT Vehcle MDT Fgure 4-4: REFLEX structure

157 SCHEDULING METHODOLOGY FOR THE STATIC FLEX-ROUTE PROBLEM The concept of Flex-Route schedulng s yet to be nvestgated n a rgorous way. To the authors knowledge, the research n ths area s very scarce. In fact, most of the currently used schedulng algorthms for solvng the Flex-Route problem are the same algorthms used for solvng the DARP wth some varatons. In ths work, a Flex-Routespecfc schedulng and routng algorthm s developed usng Constrant Programmng technques for both the statc and dynamc cases. The schedulng algorthm s bult n ILOG Dspatcher TM, a vehcle-routng C++ lbrary that uses the concepts and algorthms of Constrant Programmng to fnd solutons to optmzaton problems especally n areas of routng and schedulng. ILOG Dspatcher TM works smultaneously wth ILOG Solver, a Constrant Programmng lbrary, and as such t explots all the facltes of objectorentaton and Constrant Programmng. A two-stage method s used to fnd a soluton to a specfc problem, namely model and solve. The frst stage s to model the problem usng decson varables, constrants, and an objectve functon. ILOG Dspatcher TM offers features especally adapted to solvng vehcle routng problems. The decson varables n a Dspatcher TM model are the varables representng f a vst (request) s performed (performed varables) and varables representng the next vst (next varables). An ILOG Dspatcher TM model also ncludes constrants, such as capacty constrants, tme wndows, and precedence constrants. A soluton to a schedulng problem s defned by the values assgned to each of the decson varables. The second stage s to solve the problem usng ILOG Solver TM. Solvng the problem conssts of fndng a value for each decson varable whle smultaneously satsfyng the constrants and maxmzng or mnmzng an objectve functon. ILOG Solver TM uses two technques of Constrant Programmng to help solve optmzaton problems; they are search strateges and constrant propagaton. Addtonally, Constrant Programmng performs two types of constrant propagaton: ntal constrant propagaton and constrant propagaton durng search. Frst, Solver performs ntal constrant propagaton. The ntal constrant propagaton removes all values from domans that wll

158 144 not take part n any soluton. After ntal constrant propagaton, the search space s greatly reduced. For the remanng part of the search space, ILOG Solver TM uses a search strategy to fnd a soluton, whch s called the search tree. In the remanng part of ths secton, we present the statc Flex-Route schedulng problem, the varous components of the problem and the proposed schedulng methodology of the statc verson of the Flex-Route problem System Characterstcs and Assumptons The statc Flex-Route schedulng problem conssts of two components: a fxedroute component and a demand-responsve component. As mentoned earler, ths research focuses on the case where Flex-Route transt s replacng an exstng fxed-route servce as opposed to the schedulng of completely new routes. In our model, the fxedroute porton of the model conssts of a specfc transt route wth the followng characterstcs: The tme perod consdered s T, whch could be any tme perod durng the day, or the total operatng perod of the day. The route runs from a startng fxed stop (f = 1) to a fnal fxed stop, or a termnal (f = F). One drecton of travel s consdered n the model. Specfcally, we are nterested n the operaton of the route n the drecton leadng to a tran staton durng tme perod T. The other drecton s not consdered n ths case to lmt the scope of the problem. However, the problem can be easly extended to nclude both drectons. Several transt vehcles are used to provde the servce n the gven tme perod. The specfc number of transt vehcles used n ths tme perod s not mportant n the schedulng process. However, the number of vehcle runs (R) from the startng pont of the route to the end pont of the route s what s mportant at ths pont. In transt schedulng, the transt vehcles performng the runs can be found later after dentfyng all vehcle runs.

159 145 A vehcle run (r) s defned as a porton of the schedule begnnng at f = 1 and endng at f = F after vstng all the ntermedate fxed stops. Therefore, the total number of fxed stops to be vsted durng tme perod T s ( TF F R ). The ntal fxed-route schedule for each vehcle run s represented by an ordered sequence of stops ( = 1, 2,.., F) wth ther correspondng scheduled departure tmes (DT 1, DT 2,, DT F ). The departure tmes at the fxed stops are consdered as constrants of the system whch can not be volated. We treat the departure tmes at the fxed stops as hard constrants snce the fxed stops typcally represent major transfer ponts and late arrvals at these stops may result n passengers mssng ther connectons partcularly at the transfer staton. However, n Chapter 5 we wll nvestgate whether relaxng the departure tme constrants at some fxed stops wll mprove the performance of the servce or not. The servce vehcles start operaton at a specfed departure tme from the frst stop (DT 1 ) and travel along the route, wth a predetermned headway (h) between the consecutve vehcles, arrvng at the scheduled fxed stops at a specfc arrval tme AT untl the end of the operaton tme perod. The predetermned fxed-stop schedule s bult takng nto consderaton the amount of slack tme requred for the route to serve on-demand requests. The demand-responsve component of the model can be summarzed by the followng characterstcs: The model conssts of a set of on-demand requests (P) n tme perod (T), dentfed by O=O 1, O 2,.., O p. The on-demand requests orgnate n a defned servce area of the route and need to be served door-to-door. As explaned earler n Chapter one, door-to-door n ths research does not mean that customers wll be dropped off at locatons other than the fxed stops; rather t means that the on-demand customers wll be pcked up from home and then dropped off at a fxed stop (usually the termnal). Each request (j) has a specfc pck-up tme requested by the customer denoted by RT j. If the request s accepted, the customer wll be pcked up at a scheduled pck-

160 146 up tme denoted by PT j. Ths pck-up tme wll be constraned wthn a tme wndow (E j, L j ), where E j s the earlest servce tme and L j s the latest servce tme. Usually, RT j s at the md pont of the tme wndow (.e. RT j E j equals L j - RT j ). However, dependng on the customer request, other varatons of the tme wndow can be set. For example, a customer can request that E j be equal tort j, whch can be easly adjusted n the schedulng system. The beneft from servng the on-demand requests s gven by (B 1,.., B p ) The Statc Flex-Route Model In the statc verson of the Flex-Route problem, the transt servce provder usually wants to mnmze the total system cost n the form of a comprehensve cost functon. Ths cost functon can take many forms dependng on the transt agency s goals. For example, cost functons for the statc Flex-Route problem could am at maxmzng the number of accepted requests. The benefts and costs for the dfferent stakeholders consdered n the statc Flex-Route problem n our research are presented next. System Beneft In ths research, the beneft the transt servce provder gans from mplementng the Flex-Route servce n tme perod T s derved manly from servng the on-demand requests. Other types of benefts mght be consdered under dfferent crcumstances; however, n ths work we are nterested only n the beneft ganed from servng the ondemand requests. Gven that the number of on-demand requests n tme perod T s P, and the beneft from servng request j s B j, then the total operator beneft obtaned from servng the on-demand requests can be expressed as follows: B Total P j1 O j B j (Equaton 4-1) Where O O j j 1 0 If request (j) s accepted If request (j) s rejected... (Equaton 4-2)

161 147 System Cost There are three types of cost consdered n ths model representng the three stakeholders nvolved n the Flex-Route problem: the on-demand customers, the fxedroute customers and the transt servce operator. Each stakeholder has a dfferent cost functon as dscussed next: 1. The on-demand customers: Frst t s assumed here that f any on-demand request may be rejected wthout any cost mplcatons assumng any customer who requests the servce wll have other travel optons f hs request s rejected. Moreover, the schedulng system wll always try to maxmze the number of accepted requests gven the benefts ganed from acceptng the on-demand requests as shown n Equaton 4-1. If an on-demand customer request s accepted, then the cost to the customer s the dfference between the requested pck-up tme and the scheduled pck-up tme. Ths statement mples that the every mnute n the tme wndow has the same weght (.e. t does not matter whether the scheduled pck up tme s before or after the requested pck up tme). Also, t s assumed that the customers wll be gven an advance notce of ther exact scheduled pck up tme. Gven the requested pck-up tme for customer j s RT j and the scheduled pck-up tme s PT j, then the cost functon n ths case can be expressed as follows: P Cost (1) = O j1 j RT j PT j (Equaton 4-3) O j has the same defnton as n equaton The fxed-route passengers: The cost to those passengers s gven n terms of the extra dle tme at each fxed stop due to the unused slack tme after servng all requests. The basc assumpton here s that ths dle (unused) tme s consdered as nconvenence to fxed-route customers (as well as on-demand customers) and therefore t should be added to the cost functon. It s

162 worth mentonng that other assumptons regardng the cost functon of fxed stop customers could be made. However, n our research we used the followng cost functon: 148 TF Cost (2) = A DT AT (Equaton 4-4) Where: A = the number of passengers who arrve onboard the bus (both fxed stop and ondemand users) at stop () and whose ther destnaton sn t stop (). 3. The servce operator: The cost ncurred by the transt operator s consdered n terms of how much extra mleage s needed to serve the on-demand requests. It should be noted that the drvers extra hours are not ncluded n the cost snce the assumpton here s that the decson to provde Flex-Route servce has already been made, and extra drver operatng hours has been ncluded n the feasblty study of the servce as dscussed n Chapter 3. Gven that the extra dstance traveled to serve the accepted requests s D km, the cost ncurred by the operator wll be: Cost (3) = D km (Equaton 4-5) Based on these ndvdual cost functons found above, the net cost functon can be expressed as follows: C Total C C C C C C C B owt 1 dle 2 skm 3 on Total.... (Equaton 4-6) Where: c o-wt : s the cost coeffcent ndcatng the nconvenence to on-demand customers for the dfference between ther requested servce tme and the scheduled servce tme.

163 149 C dle : s the cost coeffcent ndcatng the nconvenence to fxed-route customers for dle tme of the vehcle at the fxed stops due to unused slack tme. c s-km : s the cost coeffcent ndcatng the cost ncurred by the servce provder for the extra dstance traveled due to the devaton of the servce. c on : s the cost coeffcent ndcatng the beneft of the servce from servng a sngle on-demand request. Substtutng equatons 4-1, and 4-3 to 4-5 nto equaton 4-6, we get the followng formula for the overall cost functon: C Total c owt P j1 O j RT j PT j c dle TF 1 A DT AT c skm Dkm con Oj B j P j (Equaton 4-7) These cost coeffcents can be modfed as needed to emphasze one factor over the others. It should be noted that the above cost functons gve a general ndcaton on what factors should be consdered n the cost and beneft estmaton. Therefore, these functons can be modfed to reflect the servce provders vew of the cost and benefts of provdng the servce and thus changng the terms of Equaton 4-7, whch mght nclude relaxng the fxed stops arrval constrants n case late arrval at the fxed stops s allowed. It s worth mentonng that the relatve value of these cost coeffcents s not addressed n ths research. Such evaluaton requres an ndependent and thorough travel demand study to estmate the dsutlty of each stakeholder towards these changes n servce frequency, watng tme among other changes. System Constrants There are three types of constrants that wll be consdered n our model based on the aforementoned assumptons:

164 150 For each fxed stop (), the arrval tme of the vehcle at that fxed stop (AT ) must be earler than the publshed departure tme (DT ;), whch can be expressed as follows: AT DT (1, TF) (Equaton 4-8) For each on-demand request (j), the arrval tme of the vehcle at that request locaton must be later than the lower bound of the tme wndow (E j ) and earler than the hgher bound of the tme wndow (L j ). E j AT j L j j (1, P) (Equaton 4-9) There s also the mplct constrant that at each on-demand or fxed stop, the total number of passengers on the bus should be less than or equal to the transt vehcle capacty. Overall System Model Gven all the ndvdual cost and beneft functons defned above, and gven all the constrants of the model, we can now represent the model of statc Flex-Route problem as follows: Mn C c O RT PT c A DT AT Total Subject to: owt P j1 j j j c skm Dkm con Oj B j P j1 dle TF (Equaton 4-10) AT DT (1, TF) (Equaton 4-11) E j AT j L j j (1, P) (Equaton 4-12)

165 151 Equaton 4-10 gves the objectve functon of the statc verson of the Flex-Route problem consderng all three stakeholders affected by the ntroducton of the servce. As mentoned earler, other formulatons of the objectve functon can be made dependng on the objectves of the servce provder. We thnk, however, that our model captures adequately the varous aspects of the servce and ncludes a representaton of the costs ncurred by all stakeholders affected by the servce. The objectve functon of equaton 4-10 can now be formulated wthn ILOG Dspatcher TM to be used n fndng the optmal statc schedules for any Flex-Route problem Statc Flex-Route Routng and Schedulng Methodology The Flex-Route schedulng problem could be consdered as a varaton of the vehcle routng problem. It s well known that the vehcle routng problem s an NP-hard problem, whch means that exact algorthms for fndng optmal solutons for a large scale problem can not be done n reasonable tme. Thus, only heurstc approaches can be used to fnd near-optmal solutons for the problem. Ths mples that heurstcs should be used to solve the Flex-Route problem. So, n our approach to solvng the statc Flex- Route schedulng problem, Constrant Programmng s used n the algorthm solvng process because of ts powerful problem-reducton technques as dscussed n ths chapter. Constrant Programmng wll be ncorporated n the algorthm as a tool of acceleratng and gudng the search process. Before we proceed further t s mportant to state some of the advantages of Constrant Programmng (mentoned earler) that led to the decson of ts use n ths research for schedulng the flex-route servce. Some of these advantages nclude: 1. Constrant Programmng s capable of usng any type of constrant to further lmt the search space of the problem, whether t s a small or large problem. As such, f the problem grows n sze, the ablty of the constrants to reduce the search space wll be of tremendous mportance. Other algorthms do not have ths capablty; 2. The order n whch the constrants are added to the problem s not mportant, and once a constrant s added, t wll start communcatng wth all exstng constrants to reduce the search space; and

166 The model and the soluton process are completely separate, so any changes to the model or the soluton process can be made wthout worryng about the nteracton wth the other part of the problem. Solvng the Flex-Route problem n our approach conssts of two stages. In the frst stage, a soluton to the problem s found usng an ntal soluton algorthm. In ths stage, we propose a new schedulng heurstc that uses elements of Constrant Programmng. Assumng that a frst soluton wth a specfc cost can be found, the second stage s to apply a local search procedure to mprove ths soluton. In ts basc form, the local search procedure s carred out by makng small changes (neghborhoods) to produce a new soluton. The process used n ths work for modelng and solvng the Flex-Route problem conssts of the followng steps: 1. Buld the problem model n ILOG Dspatcher TM 2. Generate an ntal soluton usng the Constraned-Assgnment Algorthm. a. Fnd a soluton usng the Constraned-Assgnment Algorthm that satsfes all the constrants whle explotng the Constrant Programmng algorthms. b. Set the decson varables to the values that satsfy the soluton. c. Store the frst soluton. 3. Generate a new, modfed soluton usng local search. a. Fnd a better soluton that satsfes all the constrants explotng the Constrant Programmng algorthms. b. Set the decson varables affected by local search to ther new values. c. Set the remanng decson varables to ther saved values. d. Repeat steps a through c untl no further mprovements can be made. e. Store the fnal soluton and save t as the optmal soluton. Fgure 4-5 llustrates the modellng and solvng process descrbed above. The output of ths process s a set of schedules and routes for every vehcle run n tme perod T. Each vehcle route has an ordered set of vsts to fxed stops and on-demand requests

167 153 wth the expected arrval and departure tmes at these vsts. Ths ncludes also the exact routes that must be followed to serve all these vsts (.e. the lnks from the startng pont to the end pont of each route). Usng the decson-support system descrbed earler n ths chapter, each route can be dsplayed on a GIS map and sent to each transt vehcle s Moble Data Termnal through advanced communcaton technologes. The followng secton ncludes a detaled presentaton of the above-mentoned stages used to model and solve the Flex-Route problem Buld the Problem Model As mentoned earler, ILOG Dspatcher TM s a C++ vehcle-routng software that explots all the powerful technques of object-orentaton and Constrant Programmng. ILOG Dspatcher TM offers features especally adapted to solvng problems n vehcle routng and mantenance-techncan dspatchng. There are, for example, classes of objects partcularly desgned to represent such aspects as vehcles, vsts and constrants such as capacty or tme-wndow constrants. The decson varables n a Dspatcher model are the varables representng f a vst s performed (performed varables) and the varables representng the vst mmedately after a gven vst (next varables). A soluton to a routng problem, a routng plan, s defned by the values assgned to each of these decson varables. A Dspatcher model also ncludes sde constrants, such as capacty constrants, tme wndows, and precedence constrants. The objectve s to lower the cost of a routng plan, whch depends on the partcular problem.

168 154 Fgure 4-5: Schedulng Methodology To fnd a soluton to a problem usng ILOG Dspatcher TM, two stages are used: modelng and solvng. The frst stage s to model the problem. In modelng the problem, the basc objects n ILOG Dspatcher TM such as nodes, vehcles and vsts are used to model the problem. For example, the locatons of the stops and customers wll be added as nodes n the model. The model s composed of the decson varables (such as varables representng f a vst s performed or not, or to whch vehcle a vst s

169 assgned), constrants (such as tme-wndow and capacty constrants) and the objectve functon Modelng Objects Dspatcher models nclude four basc objects: nodes, vsts, vehcles and dmensons. These objects help model the problem n a Constrant Programmng envronment. Nodes Flex-Route problems have a geometrc component: the stops must be performed at specfc physcal locatons. ILOG Dspatcher TM represents these physcal locatons as nodes. These nodes are then used to compute dstances and tmes (and subsequently cost) between stops. Vsts A vst represents an actvty that the vehcle has to perform. A vst occurs at a sngle node and s performed by only one vehcle. Many dfferent vsts can be created at the same node. For example, a specfc fxed stop along the route wll be vsted more than once gven ts schedule. Vsts have quanttes, whch can be weghts, volumes, numbers of objects, and so on. These quanttes n the Flex-Route problem are the number of people boardng and alghtng the vehcles at the stops along the route. Vehcles Vehcles are resources that perform the vsts. A vehcle has a start and an end vst and can have varable start and end tmes assocated wth those vsts. In the Flex- Route problem, those vsts correspond to the frst and last fxed stops of the route for any gven vehcle run. Vehcles can also be gven capactes. These capactes represent the total number of people the vehcle can carry at any pont along the route. Dmensons

170 156 Dmensons are objects closely assocated wth vsts and vehcles. The most common dmensons are volume, tme, and dstance. Dmensons are used to model sde constrants such as capacty, tme wndows, deadlnes, servce delays, and so on. For example, dmensons could be used to model number of passengers, dstance between nodes, travel tme between nodes, among other thngs. Dmensons are also used to model costs and the objectve functon Intal Soluton: The Constraned-Assgnment Algorthm In our model, each vehcle run was assgned a specfc number (r = 1, 2,...R). To smplfy the schedulng process, t s assumed that each run s performed by a dfferent vehcle (veh =1, 2,..., R). Ths of course does not mply that n real-world applcatons there wll be dfferent vehcles performng each run. Ths assumpton s just to ndcate that each run s dfferent from the other runs. In real-world applcatons, for example, any specfc vehcle wll perform several runs durng the tme perod under nvestgaton. Each vehcle has a start and end tmes correspondng to the departure tme from the frst fxed stop and the arrval tme at the last fxed stop of the routes for a specfc vehcle run. Ths constrant makes sure that each vehcle wll operate n ths specfc tme perod only. Each vehcle wll vst all the fxed stops along the route. Of course, each vehcle wll have a dfferent departure tme at a specfc fxed stop along the route. The dfference between the departure tmes of consecutve vehcles at a specfc fxed stop s the headway (h). Therefore, based on these assumptons, the algorthm for fndng the frst soluton s composed of 5 steps as follows: 1. Let Veh = (1, 2,, R) be the lst of all vehcles. 2. Let all vehcles have empty routes. 3. Let N = (1, 2,, N) be the lst of all unassgned fxed stop vsts along the route, where N = (F x R). The departure tmes at these fxed stops are (DT 1, DT 2,, DT N ) respectvely; 4. Prepare the fxed-route schedulng problem by assgnng every run to a specfc vehcle.

171 Let P = (1, 2,, P) be the lst of all on-demand requests for the tme perod T, and let the tme wndows for these requests be ([E 1, L 1 ],.., [E P, L P ]). For the on-demand requests 1 to P, do the followng: a. For on-demand request (j), f a feasble nserton s found, then nsert (j) n a poston where there wll be the least ncrease n cost, then remove (j) from P and go to step 5c. b. If no nserton of (j) s feasble, then remove request (j) from P and go back to step 5c. c. Repeat step 5a untl P becomes empty. The Constraned-Assgnment algorthm generates a frst soluton to the Flex- Route problem by fndng an ntal good feasble soluton. Ths soluton guarantees that all the constrants of the model are satsfed. These constrants nclude the fxed-stop constrant of the transt vehcle arrvng at these stops before the publshed departure tme of each run. The other constrant s that any accepted on-demand request must be vsted durng the tme wndow specfed earler n the model. Ths ntal soluton can be further mproved by usng local search. Local search uses neghbourhoods to reduce the costs of the routes they fnd. The next secton dscusses the local search procedures used to mprove the ntal soluton found by the Constraned-Assgnment algorthm Improve the Soluton Usng Local Search After a frst soluton s found usng the nserton algorthm, the local search procedure s performed usng the neghborhoods dscussed n the prevous secton to mprove the soluton. The local search procedure s performed by buldng neghborhoods that are composed of a combnaton of the neghborhoods Two-Opt, Or-Opt, Relocate, and Exchange, whch are defned later n ths secton. The search s teratve n that t tres a seres of moves n order to decrease the cost of the soluton found. In ths study, we used the frst accept search heurstc. Frst accept search takes the frst cost decreasng move encountered. The search contnues to take such moves untl the neghborhood contans no cost-reducng moves. Ths pont s usually termed a local mnmum. The

172 158 word local s used to sgnfy that ths pont s not guaranteed to be, and, n fact, s not usually, a global mnmum. Ths method accepts any mprovng move and so s not too expensve n terms of computatonal cost. The method wll frst see f t can make a costdecreasng move usng the neghborhood Two-Opt. If t cannot, t tres Or-Opt, and so on. The search heurstc mplements a greedy search heurstc. It accepts only new routng plans that strctly decrease the cost and, thus, allows only mprovements n the objectve. Throughout the local search procedure to fnd a better soluton, constrant propagaton s performed to reduce the search space by removng all values from the current domans that volate the constrants. Current domans are the domans of the varables at any pont durng the search process. When constrant propagaton removes values from the current domans durng search, values are only removed from these test domans and not from the orgnal doman. Each soluton s tested aganst the problem constrants for feasblty n every step of the search process to ensure that the soluton wll be feasble. If the new soluton s feasble and has reduced cost, t s accepted as the current one; otherwse, the current soluton remans unchanged. In ths way, only moves to mproved solutons are accepted. Solutons that do not mprove the cost or that volate problem constrants are rejected. Ths s known as a greedy mprovement algorthm, as t only makes changes to the solutons that mprove the cost. Ths process s repeated (usng dfferent soluton changes on subsequent attempts) untl a stoppng condton s met. The stoppng condton n our case s that no changes to the current soluton can be found and that no better solutons can be found wthout volatng the problem constrants. As mentoned earler, the product of ths process s a number of vehcle runs (Veh = 1, 2,, R) wth a set of ordered vsts to fxed stops and on-demand requests for each vehcle run. Neghborhoods The central dea of a neghborhood s to defne a set of soluton changes, or deltas, that represent alternatve moves that can be taken. In our research, a varety of

173 159 neghborhoods are used to mprove the soluton found n the frst step. The neghborhoods used here are classfed nto two groups: 1. Intra-route neghborhoods: They are neghborhoods that modfy only one route (one vehcle run) and nclude the Two-Opt neghborhood and the Or-Opt neghborhood, such as modfyng the order n whch on-demand requests are served. 2. Inter-route neghborhoods: These are the neghborhoods that make changes between dfferent routes (dfferent vehcle runs) and nclude the Relocate neghborhood and the Exchange neghborhood, such as relocatng an on-demand request from one vehcle run to another. In the ntal soluton, every par of stops/requests s connected by an arc representng the dstance traveled between these two stops. The neghborhoods wll try to mprove ths soluton by makng a number of changes to these vehcle runs that could nclude removng on-demand requests, addng on-demand requests, changng the locaton of the on-demand request n the vehcle schedule and other changes as dscussed next. Two-Opt Neghbourhood In a Two-Opt neghborhood two arcs n the route of a vehcle run are cut and reconnected to mprove the total cost of the vehcle run as follows: Gven the route of each vehcle run (r) 1. Remove two arcs from the route r, and try other possble reconnectons of the remanng stops of the vehcle run. 2. If the total system cost of the schedule has been reduced and f all constrants are satsfed, go back to step If the total system cost s ncreased, then reconnect the arcs at ther orgnal postons. 4. If all possble arcs are exhausted, then end the process. Wth ths neghbourhood, drectonal flows between stops may be reversed. Fgure 4-6 llustrates ths process. In ths fgure, the move destroys arcs 3 and 5 and

174 creates two new arcs 7 and 8. The move also reverses the drecton of arc 4. Therefore the new routes of the vehcle wll be shorter, and thus less costly. 160 Or-Opt Neghbourhood In an Or-Opt neghborhood, segments of stops n the same vehcle route are relocated as follows: Gven the route of each vehcle run (r) 1. Move the poston of one stop elsewhere n the route. 2. If the cost has been reduced and f all constrants are satsfed, go back to step If the total system cost s ncreased, then move the stop back to ts orgnal poston. 4. When all such moves have been tested, try movng parts of the route composed of two consecutve stops. 5. If the cost has been reduced and f all constrants are satsfed, go back to step If the total system cost s ncreased, then move the two stops back to ther orgnal postons. Fgure 4-7 llustrates the Or-Opt neghborhood. In ths case, the postons of requests D and E are changed from between requests C and F to between A and B, and therefore reducng the cost by reducng the dstance traveled to serve all requests. Relocate neghbourhood In a relocate neghborhood a request s removed from a route of a specfc vehcle run and nserted n the route of another vehcle run f the total cost s reduced. Ths relocaton process should satsfy all the constrants of the problem, such as tme constrants. Ths method can be generalzed f more than one request of a route s moved at the same tme. Fgure 4-8 llustrates the relocate neghborhood. In ths example, request number 3 s moved from run 1 to run 2 to reduce the total cost of the servce. Exchange neghbourhood In an exchange neghborhood, two requests from two dfferent runs swap places f all constrants are stll satsfed and the total cost s reduced. Ths method can also be generalzed f more than one request of a run s exchanged at the same tme. Fgure 4-9

175 llustrates the exchange neghborhood. In the fgure, requests number 3 and 8 exchange places between runs 1 and 2 respectvely Fgure 4-6: The Two-Opt neghbourhood

176 162 A B D E G F C A B E D G F C Fgure 4-7: The Or-Opt neghbourhood

177 Run Run Run Run 2 Fgure 4-8: The Relocate neghbourhood Run Run Run Run 2 Fgure 4-9: The Exchange neghbourhood

178 SCHEDULING METHODOLOGY FOR THE DYNAMIC FLEX-ROUTE PROBLEM Flex-Route routng problems may be classfed as statc or dynamc. In the statc case, all data are known before the routes are constructed and do not change afterward (e.g., locaton of customer requests, demand, etc.). In the dynamc case, however, all or a fracton of all requests are revealed as the servce s n progress and the servce dspatcher must be able to deal wth these new requests n a tmely manner. One crtcal use of realtme nformaton s to dvert a vehcle away from ts current destnaton to serve a request that just occurred n the vcnty of ts current poston or ahead n the route. Not only that, but dspatchers may also need to react to other events that occur n real tme such as unexpected delays, vehcle breakdowns, accdents, etc. In the past, technology has been a major mpedment n provdng Flex-Route transt servce n suburban areas. However, wth the recent mprovements n automatc vehcle locaton (AVL) technology and geographc nformaton systems (GIS), the tools now exst to effectvely support such a transt servce. These recent advances provde opportuntes for usng real-tme nformaton to enhance the performance of decson systems n the area of Flex-Route transt routng and schedulng. However, there stll s a lack of methodologes that can effcently solve Flex-Route routng problems through a judcous ntegraton of real-tme nformaton. In ths research, a new strategy for the dynamc assgnment of new requests n the Flex-Route problem s proposed and examned. An emprcal evaluaton s performed to test the effcency of the proposed strategy. The goal s to schedule a set of new real-tme on-demand requests n the already-bult daly schedule of the Flex-Route servce, whle mnmzng the cost nflcted on all customers Dynamc Flex-Route Problem Features In the dynamc verson of the Flex-Route problem, a number of servce requests are not known completely ahead of tme, but are rather dynamcally revealed as tme progresses. In a dynamc settng as llustrated n Fgure 4-20, the transt vehcle route at any tme nstant t, can be dvded nto three parts:

179 165 Completed route: ths part nclude all the fxed stops and on-demand requests that already have been served at ths pont n tme. Thereby, ths part of the route cannot be modfed anymore. Current movement to reach the current destnaton: ths part ncludes the current locaton of the vehcle between two consecutve vsts (fxed or on-demand stops) and ts planned route to the next vst. Remanng route: ths part consttutes the porton of the route that s yet to be executed by the transt vehcle followng the next vst. In other words, ths part ncludes the remanng route and schedule to vst all remanng stops accordng to the planned statc route. Legend Scheduled Stop Orgnal Scheduled Route Current Stop Completed Route Current Movement A Remanng Route D C B E G F Fgure 4-10 Bus route n dynamc envronment

180 166 Gven a new request at tme nstant t, the problem s to assgn ths request to the partcular transt vehcle n servce and nclude t n ts planned route at a mnmum cost. Apart from the studes of Regan et al. (1994, 1995) and Ichoua et al. (2000), most approaches to the dynamc vehcle routng problem fx the current destnaton of each vehcle and does not try to modfy t even f a new dynamc request occurred n the vcnty of the current poston of the vehcle whle the drver s on hs way to hs current destnaton. Dvertng a vehcle away from ts current destnaton to serve a nearby dynamc request may be benefcal to the servce as t may result n ncreasng the number of accepted requests and/or decreasng the travel tme and dstance to reach a new request Dynamc Flex-Route Problem Model The cost functon for the dynamc Flex-Route problem can take varous forms. Usually, the servce agency has an objectve functon that t wants to acheve based on some evaluaton crtera. The stakeholders affected by a dynamc request nserton n ths case are the same as n the statc case, and ther correspondng cost and beneft quanttes are dscussed next. System Beneft Smlar to the statc case, the beneft the transt servce provder gans from mplementng the Flex-Route servce n tme perod conssts manly from servng the ondemand requests. Therefore, gven a new request (k) n real tme, the total beneft to the system from servng ths request j s: B Total O B (Equaton 4-13) k k Where O O k k 1 0 If request (k) s accepted If request (k) s rejected.... (Equaton 4-14)

181 167 System Cost Three types of cost are consdered n ths model representng the three stakeholders nvolved n the Flex-Route problem as shown next. 1. The on-demand customers: The cost to the on-demand customers s caused by changng the scheduled pckup tme (PT j ) at the on-demand requests, f needed, to allow for the nserton of the dynamc request. It s also assumed here that any request can be rejected wthout any cost mplcatons. Gven the scheduled pck-up tme for customer j s PT j and the new scheduled pck-up tme s NPT j, then the cost functon n ths case can be expressed as follows: P Cost (1) = O j1 j NPT j PT j j (1, P) (Equaton 4-15) O j has the same defnton as n equaton The fxed-route passengers: The cost to the fxed-route passengers (as well as on-demand customers) s gven n terms of the extra dle tme at each fxed stop due to the unused slack tme after servng all the requests. The passengers n ths case actually beneft from servng the ondemand request snce servng the new request wll most probably reduce the total unused slack tme at the fxed stops. Gven the scheduled arrval tme at fxed stop from the statc soluton s AT, and gven the new scheduled arrval tme of the transt vehcle at s NAT, then the cost functon n ths case can be expressed as follows: TF Cost (2) = A AT NAT 1... (Equaton 4-16)

182 The servce operator: The cost ncurred by the transt operator s consdered n terms of how much extra mleage s needed to serve the on-demand requests. Gven that the extra dstance traveled to serve the new dynamc requests s D R, the cost ncurred by the operator wll be: Cost (3) = D R (Equaton 4-17) Based on these cost functons found above, the total cost functon can be expressed as follows: C Total C o wt C1 C dle C 2 C skm C (Equaton 4-18) Where: c o-wt : s the cost coeffcent ndcatng the nconvenence to an on-demand customer for the change of the scheduled pck-up tme. c f-wt : s the cost coeffcent ndcatng the nconvenence to fxed-route customers for dle tme of the vehcle at the fxed stops due to unused slack tme. c s-km : s the cost coeffcent ndcatng the cost ncurred on the servce provder for the extra dstance traveled due to the devaton of the servce. c on : s the cost coeffcent ndcatng the beneft of the servce from servng a sngle on-demand request. Substtutng equatons 4-15 to 4-17 nto equaton 4-18, we get the followng formula for the total system cost of the dynamc problem: C Total c owt P j1 O j NPT j PT j c dle TF 1 A NAT AT c D... (Equaton 4-19) skm R

183 169 And the total system beneft s gven by: B Total c O B... (Equaton 4-20) on k k Therefore, request k should be accepted, gven all the constrants are satsfed, f the total beneft from solvng the dynamc on-demand request s greater than the mnmum nserton cost obtaned from servng ths request. Therefore, the followng condton should hold: BTotal CTotal (Mn) (Equaton 4-21) Substtutng equatons 4-19 and 4-20 nto equaton 4-21, the followng formula can be used to determne f t s optmal to accept a dynamc request: P TF on k k owt j j j dle skm R j (Equaton 4-22) c O B Mn c O NPT PT c A NAT AT c D One mportant pont that should be mentoned here s that the value of each cost coeffcent may not be the same as n the statc problem. For example, the weght for the cost ncurred by on-demand customers for changng the scheduled pck-up tme n the dynamc settng should be more than ts statc counterpart. The reason behnd that s that n the statc settng, when customers are gven a dfferent pck-up tme than what they requested, they can accept that snce they now know ther exact pck-up tme the next day. In the dynamc settng, however, customers usually expect the transt vehcle to arrve at ts scheduled arrval tme, and any change n ths arrval tme mght cause

184 nconvenence to them. Therefore, future research should study the values of these cost coeffcents carefully n order to mantan an acceptable level of servce for all customers. 170 System Constrants The system constrants are the same constrants as n the statc model;.e. vehcle capacty constrant, fxed-stops constrants and on-demand tme wndow constrants as follows: AT DT (1, TF) (Equaton 4-23) E j AT j L j j (1, P) (Equaton 4-24) In addton to that, there s the tme wndow constrant for the dynamc request, whch s gven by: E k AT L (Equaton 4-25) k k Equaton 4-25 gves the condton for acceptng a dynamc on-demand request. Ths equaton and the constrants of equatons 4-23 and 4-24 are formulated n ILOG Dspatcher TM to be used n fndng optmal dynamc solutons n real tme Dynamc Flex-Route Model for Late Arrval A sgnfcant mprovement n the Flex-Route servce could be ganed by relaxng the fxed stop constrants (.e. allowng transt vehcles to arrve late at the fxed stops to maxmze the number of accepted on-demand requests). In real tme, ths stuaton could arse n the case where acceptng a dynamc request and nsertng t n the schedule would result n late arrval at the next fxed stop. When searchng for the best soluton n ths case, the algorthm would need to compute a cost functon for the stakeholders dfferent from that of the prevous case.

185 171 Assume that an nserton of dynamc request k nto the schedule of a transt vehcle wll result n the vehcle arrvng late at fxed stop by (LT ) tme unts. In ths case, the stakeholders affected by the dynamc request nserton wth ther correspondng cost quanttes are as shown next. System Beneft Ths s the same as the prevous case. Gven a new dynamc request (k), the total beneft to the system from servng ths request j s: B Total O B (Equaton 4-26) k k Where O O k k 1 0 If request (k) s accepted If request (k) s rejected.... (Equaton 4-27) System Cost Three types of cost consdered n ths model representng the three stakeholders nvolved n the Flex-Route problem as shown next. 1. The on-demand customers The cost to the on-demand customers s also the same as the prevous case. Gven the scheduled pck-up tme for customer j s PT j and the new scheduled pck-up tme s NPT j, then the cost functon n ths case can be expressed as follows: P Cost (1) = O j1 j NPT j PT j (Equaton 4-28) O j has the same defnton as n equaton 4-27.

186 The fxed-route passengers The cost to fxed stop passengers (as well as on-demand customers) n ths case s dfferent from the prevous case. The cost here s composed of the extra wat tme encountered by the fxed-route passengers (as well as on-demand customers) watng to board at the next fxed stop downstream. Assumng that the number of passengers watng at fxed stop s FP, then the cost functon for ths group can be expressed as follows: TF Cost (2) = FP LT (Equaton 4-29) 1 3. The servce operator: The cost ncurred by the transt operator s the same as the prevous case. Gven that the extra dstance traveled to serve the new dynamc requests s D R, the cost ncurred by the operator wll be: Cost (3) = D R (Equaton 4-30) 4. The passengers already on board Ths group of passengers (fxed stop and on-demand users) wll ncur some extra cost because of the extra travel tme they suffer due to the late arrval. Assumng that the number of passengers gettng off at fxed stop s Nb, then the cost functon for ths group can be expressed as follows: TF Cost (4) = Nb LT (Equaton 4-31) as follows: Based on these cost functons found above, the total system cost can be expressed

187 173 C Total C o wt C1 C f wt C 2 C skm C 3 C tt C (Equaton 4-32) Where c o-wt : s the cost coeffcent ndcatng the nconvenence to an on-demand customer for the change of the scheduled pck-up tme. c f-wt : s the cost coeffcent ndcatng the nconvenence to fxed-route customers (as well as on-demand customers) for extra watng tme at the fxed stops watng for a late transt vehcle. c s-km : s the cost coeffcent ndcatng the cost ncurred on the servce provder for the extra dstance traveled due to the devaton of the servce. c tt : s the cost coeffcent ndcatng the extra travel tme for the passengers on a transt vehcle. c on : s the cost coeffcent ndcatng the beneft of the servce from servng a sngle on-demand request. Substtutng equatons 4-28 to 4-31 nto equaton 4-32, we get the followng formula for the total system cost for ths type of Flex-Route dynamc problem: C Total c owt P j 1 O j NPT j PT TF c D c Nb LT skm R tt 1 j c f wt TF 1 FP LT (Equaton 4-33) And the total system beneft s gven by: B Total c O B... (Equaton 4-34) on k k Therefore, request k should be accepted, gven all the constrants are satsfed, f the total beneft from solvng the dynamc on-demand request s greater than the

188 mnmum nserton cost obtaned from servng ths request. Therefore, the followng condton should hold: 174 BTotal CTotal (Mn)..... (Equaton 4-35) Substtutng equatons 4-33 and 4-34 nto equaton 4-35, we get the followng decson formula for whether to accept a dynamc request or not: c on O k B k P Mn cowt Oj NPTj PTj f wt skm R tt b LT j1 c FPLT c D c N (Equaton 4-36) It should be mentoned here that the cost coeffcents can be modfed as needed to emphasze one factor over the others. For example, durng heavy demand perods at the fxed stops, hgher values of c f-wt and c tt can be used compared to c on. In contrast, durng low fxed-route demand perods, the opposte s true and the cost functon should emphasze the weght of acceptng a new request, rsng c on over c f-wt over c tt. System Constrants The constrants n ths case are the same as the statc problem except that the fxed-route departure tme constrant s dfferent snce the arrval tme of the transt vehcle at a fxed stop s allowed up to LT tme unts after the scheduled departure tme. Therefore, the constrants n ths case are as follows: For each fxed stop (), the arrval tme of the vehcle at that fxed stop (AT ) must be earler than the summaton of the publshed departure tme (DT ;) and the late arrval allowable tme perod (LT) and s expressed as follows:

189 175 AT DT LT (1, TF) (Equaton 4-37) For each statc on-demand request (j), the arrval tme of the vehcle at that request locaton must be later than the lower bound of the tme wndow (E j ) and earler than the hgher bound of the tme wndow (L j ). E j AT j L j j (1, P) (Equaton 4-38) In addton to the above two constrants, there s the tme wndow constrant for the dynamc request, whch s gven by: E k AT L (Equaton 4-39) k k Dynamc Flex-Route Schedulng Methodology In Regan et al. (1994, 1995), the authors proposed dfferent dverson strateges n the context of a truck-load carrer. Ther problem s a combned pck up and delvery problem wth no consoldaton, whch means that at any tme, a vehcle s ether empty or carryng a sngle load. Also, n the context of a truck-load carrer, requests are more dstant from each other and the actvtes take place over a longer tme horzon (.e. few days). In the Flex-Route problem, however, a transt vehcle may have a number of passengers on board whle en-route to the next stop. Also, on-demand real-tme requests for the servce take place n a very short tme perod, usually wthn mnutes or even seconds. These dscrepances reveal that there s a sgnfcant dfference n the applcaton of the two servces and a dfferent and fast algorthm should be used for the real-tme Flex-Route transt servce problem. The ablty of the schedulng and routng algorthm to dvert vehcles n real tme wll be advantageous to the servce as t enhances ts relablty n terms of ts ablty to accept and schedule more requests n real tme, whch may encourage people to use the servce n the future. So to avod a trade-off between computaton tme and soluton qualty, and snce vehcles are movng fast and dverson opportuntes may be quckly

190 176 lost f the evaluaton process takes too long, computaton tme s very crtcal n ths case. Ths s where Constrant Programmng wll be very helpful because of ts fast problem reducton and constrant propagaton technques as dscussed earler. An llustraton of the process of dynamc assgnment s shown n Fgure In the dynamc schedulng of the Flex-Route transt servce, the dynamc schedulng algorthm wll respond to three types of events: Event 1: A vehcle has fnshed servng ts current customer or fxed stop In ths case, the decson on the vehcle's next destnaton (.e., the next customer locaton or fxed stop) wll be made accordng to the saved optmal soluton (statc schedule) bult n the schedulng system snce there are no new requests. The schedule s then updated by removng the on-demand request or fxed stop that was just served from ts current poston n the remanng route and nsertng t n frst poston n the remanng planned route of the vehcle. Event 2: A new customer request has just occurred In ths case, the new request locaton s frst determned. The algorthm then examnes f the request can be nserted before the next destnaton n case t s n the vcnty of the last served stop or the vcnty of the planned destnaton (see Fgure 4-22). If the nserton does not succeed, then the algorthm tres to nsert the request n other places n the route. If the algorthm fnds a soluton that nserts the request whle satsfyng all the constrants, t s nserted n the last soluton saved n the memory of the schedulng system. If there s no feasble nserton poston n the soluton, the request s rejected. Event 3: A servce delay event just occurred Consderaton of addtonal stochastc elements or unexpected events such as changes n servce tmes, change n travel tme and street closures can also be handled n the dynamc problem. In ths case, the current locaton of the vehcle s determned and the schedule s modfed based on these events. For example, f a road s closed for any reason, ths event wll cause the schedulng algorthm to re-

191 177 optmze the remanng part of the route by fndng a new route to be followed and adjustng the arrval tmes at the remanng stops. Orgnal Scheduled Route Bus Locaton and Current Destnaton New Modfed Route wth New Request Optmzaton Procedure New Request Informaton Fgure 4-11 Dynamc-request assgnment procedure

192 178 Legend Scheduled Request New Request Current Locaton of Vehcle Orgnal Route Modfed Route A N D B L C Orgnal Route A N D B L C Modfed Route Fgure 4-12 Dverson of a transt vehcle n real tme

193 179 5 MODEL IMPLEMENTATION AND VALIDATION 5.1 INTRODUCTION Ths chapter presents the varous technques that were used to evaluate both the analytcal model and the developed schedulng system. Frst the performance of the schedulng system s tested on a hypothetcal statc dal-a-rde problem (DARP) to evaluate the schedulng system s ablty to produce better results than some of the exstng DARP algorthms. Followng that the valdty, applcablty and performance of both the analytcal model and the schedulng system, developed n Chapters 3 and 4 are evaluated usng a Flex-Route hypothetcal case. The frst part of the analyss focuses on the ablty of the developed schedulng algorthm to produce better solutons compared to the Inserton algorthm, whle the second part of the analyss focuses on the effect of dfferent nput and desgn factors on the schedulng results and the ablty of the schedulng system to reflect these effects. The chapter starts by examnng the analytcal model s capablty to estmate the optmal slack tme for a hypothetcal route. Followng that, secton 5.3 analyzes the results of the senstvty analyss conducted to examne the senstvty of the output to several nput and desgn factors. The analyss shows the valdty of the proposed model as well as the ablty of the schedulng system to produce the expected results. 5.2 EVALUATION OF THE SCHEDULING MODEL To the authors knowledge, no test nstances are avalable n the lterature for the statc verson of the flex-route or DARP problems (the schedulng problem for DRT servces). That means that the results obtaned usng the developed schedulng system can not be compared to results obtaned from other algorthms used to solve these two problems. Furthermore, algorthms that can specfcally solve the flex-route problem do not exst currently n the lterature, whch presents a serous challenge n evaluatng our schedulng system aganst competng approaches. As such, we decded to evaluate the performance of the schedulng system by testng t on the DARP and compare t to one of the most powerful and well-known algorthms used to solve the DARP, namely the Inserton Algorthm. Therefore, t s mportant, when analyzng and evaluatng the

194 180 results, to acknowledge that the results of the comparson are vald only for the DARP. However, t s safe to assume that, although the results of a comparson usng a flex-route problem mght vary, the performance of the schedulng system wll be at least comparable to the results of the DARP n terms of effcency. The nserton algorthm can be summarzed as follows: 1. Let all vehcles have empty routes. 2. Let L be the lst of unassgned vsts. 3. Take a vst v n L. 4. Insert v n a route at a feasble poston where there wll be the least ncrease n cost. If there s no feasble poston, then the goal fals. 5. Remove v from L. 6. If L s not empty, go to 3. Accordngly, based on some realstc assumptons, 24 DARP nstances were generated to analyze the behavor and effectveness of the schedulng methodology developed n Chapter 4. The DARP s a generalzaton of the TSP n that the DARP conssts of determnng m vehcle routes, subject to operatonal and schedulng constrants, where a route s a tour that begns at the depot, vsts a subset of the customers n a gven order and returns to the depot. The total customer demand of a route must not exceed the vehcle capacty at any pont durng the route. A number of sde constrants related to customers pck-up/drop-off tme wndows, and maxmum rde tme also exst n the model. Tme wndow s a perod of tme n whch the customer wll be pcked-up, and t s usually set by the servce provder. The maxmum excess rde tme s defned as the dfference between the scheduled rde tme of a specfc customer and the drect rde tme wthout dverson, f the customer were to use hs own vehcle. The DARP s formulated to mnmze an objectve functon (or cost functon) wth a set of servce constrants. The cost functon sometmes s formulated as a weghted sum of the total customer nconvenence, as measured n terms of excess rde tme and servce tme devaton and the cost to the servce provders, as often measured n terms of total vehcle traveled dstance and the number of vehcles used. In ths analyss, however, we

195 181 only consder the system cost, n terms of dstance traveled and number of vehcles used, as our prmary cost Case Study Assumptons / Settngs The nstances were generated by varyng the tme wndow duraton, the maxmum allowable rde tme, and the average vehcle velocty. These values are based on earler studes done n ths feld (Karlafts et al 2005; Fu 2002). The comparatve analyss focused on the dfference n vehcle productvty between ndvdual scenaros. All nstances contan 600 randomly-generated trps over a three-hour perod, wth unlmted number of vehcles, each wth a seatng capacty of 8. The vehcles start ther servce tme at 6:00 A.M. The pckup tme for all trps start at 8:00 A.M. and ends at 11:00 A.M. For each trp, orgn and destnaton locatons were generated randomly n a 20*20 km square area. To lmt the scope of our analyss, the servce area s assumed to be covered by a unform grd network wth a constant speed on all lnks, gnorng any stochastc varatons n network condtons (travel tmes). Each locaton of an orgn or a destnaton s dentfed by an (X, Y) coordnate. Each node (locaton) has a quantty of ether 1 or -1 dependng on whether the node s a pck-up node or a drop-off node (1 for pckup and -1 for drop-off). The locaton of the depot s assumed to be at the central pont of the servce area (.e. at X =10000 and Y =10000). For each par of nodes n the network, the cost s equal to the Manhattan dstance between these ponts, whch s used to fnd the travel tme between these nodes. The travel tme s found by dvdng the Manhattan dstance between the par of nodes by the constant speed assumed n the model. A tme wndow [E, L ] s also assocated wth each pck-up node. These nodes have tme wndows of [P, P + T], where P s the desred pck-up tme, and T s the tme wndow used n the model. The desred pck-up tme was generated unformly n the perod [120, 300] (.e. 8:00A.M. to 11:00 A.M.), whch resulted n 200 trps per hour (600 trps n three-hour perod). The objectve functon used n the schedulng procedure s to mnmze the total traveled dstance by all vehcles. A maxmum allowable rde tme rato of 0.5, 1.0, 1.5, and 2.0, and tme wndows of 20, 30, and 40 mnutes were used n schedulng the trps. The maxmum rde rato s defned by the followng equaton:

196 182 Scheduled Rde Tme - Drect Rde Tme Max Rde Rato Drect Rde Tme Comparatve Analyss The schedulng results obtaned by the developed schedulng system and the Inserton Algorthm for the dfferent case scenaros are summarzed n Tables 5-1 (for 30 km\he vehcle velocty) and Table 5-2 (for 20 km\he vehcle velocty). In general, t can be observed that schedulng system performs extremely well n terms of mnmzng the total cost of all vehcles. The dstance traveled by all vehcles, as well as the number of vehcles used to perform all trps, s sgnfcantly lower than the results obtaned by the Inserton Algorthm. For example, for 20mn tme wndow, 0.5 excess rde tme, and 30 km\hr average vehcle velocty (case 1), the dstance traveled found by the Inserton algorthm was ( km) whle after mprovng the soluton usng the schedulng system, the total dstance traveled was reduced to ( km). Ths reducton n cost s 30.5% from the total cost (or km), whch s a very sgnfcant reducton n operatonal cost consderng that ths reducton s only for a case of 600 trps n a perod of 3 hours. The same concluson could be drawn on all the 24 cases, where the reducton n cost ranges between 26.5 % for case 13 (20 mn tme wndow, 0.5 excess rde rato, and 20km\hr vehcle velocty), and 36.9 % for case 12 (40 mn tme wndow, 2.0 excess rde rato, and 30km\hr vehcle velocty). Ths leads to the obvous concluson that the schedulng methodology developed n ths research sgnfcantly mproves the soluton obtaned from the Inserton algorthm by a sgnfcant percentage, regardless of the dfferent values of the model parameters. Also, n terms of the number of vehcles used to schedule all trps, the number of vehcles found usng the Inserton Algorthm for case (1) was 176 vehcles, whle t s reduced to (88) vehcles usng our schedulng methodology. Ths means a reducton of 50 % (or 88 vehcles). Consderng the fxed cost assocated wth each vehcle, these savngs are even more sgnfcant than the savngs n the total dstance traveled. Ths s because that n our model, the objectve functon s mnmzng the total dstance traveled, not the number of vehcles used. Ths reducton n the number of vehcles occurs as a sde effect

197 183 of tryng to mnmze the dstance traveled. Ths concluson could be generalzed on all 24 cases, where all cases resulted n consderable reducton n the number of vehcles used. It should be noted that the amount of reducton n both the general cost and number of vehcles used, n the cases of 20km\hr vehcle-velocty, are lower than ther counterparts n the 30km\hr vehcle-velocty. Ths s due to the fact that when the velocty s lower, trps between par of nodes become longer, and the margn for mprovng the soluton due to the constrants become tghter.

198 184 Table 5-1: Schedulng Results Obtaned for the Cases of 30km/hr Vehcle-Average-Velocty Inserton Algorthm Schedulng Methodology % Reducton Excess Tme # Vehcles # Vehcles Vehcles Case # # Trps/hr Rde Tme Wndow Cost (km) Used Cost (km) Used Cost Used

199 185 Table 5-2: Schedulng Results Obtaned for the Cases of 20km/hr Vehcle-Average-Velocty Inserton Algorthm Schedulng Methodology % Reducton Excess Tme # Vehcles # Vehcles Vehcles Case # # Trps / hr Rde Tme Wndow Cost (km) Used Cost (km) Used Cost Used

200 Senstvty Analyss The objectve of ths secton s to analyze the schedulng methodology s senstvty to some the problem parameters. Two major factors are dentfed and dscussed, whch are: tme wndow duraton and excess rde tme. The tme wndow and excess rde tme values used n ths analyss were proposed as they were the most wdely used values n other studes Tme Wndow Duraton Ths secton examnes the possble mpact of the duraton of the tme wndow on the effcency of the developed schedulng methodology n reducng the cost found by the Inserton Algorthm. Fgure 5-1 shows the effect of ncreasng tme wndow duraton on the percentage reducton n total cost obtaned by the schedulng methodology compared to the results obtaned from the Inserton Algorthm. The observaton that can be made from the fgures s that there appears to be no crtcal mpact of tme wndow duraton on the percentage of reducton n total cost and number of vehcles used when the excess rde tme s fxed. For example, for a vehcle velocty of 30km\hr, f we fx the value of excess rde tme to 0.5, and have dfferent tme wndow duratons of 20, 30, and 40 mnutes, we can observe that the percentage reducton n total cost s almost the same n the three cases (30.5%, 33.4%, and 32% respectvely), whch means that the effect of dfferent tme wndow duratons on the schedulng methodology s not obvous. Ths can be generalzed on all cases where the dfference ranges between 1 to 5 %. In terms of the effect of tme wndow on the reducton n the number of vehcles used, Fgure 5-2 shows that effect. However, unlke cost, t s clear that the percentage reducton n number of vehcles used decreases as the tme wndow duraton ncreases. Ths trend s the same for both the 20 and 30km\hr vehcle velocty Excess Rde Tme Four dfferent values of the excess rde rato were nvestgated: 0.5, 1.0, 1.5, and 2.0. As can be seen n Fgure 5-1, the effect n terms of reducton n total cost s not qute

201 187 apparent. There appears to be a trend of more percentage reducton n total dstance traveled when ncreasng the maxmum allowable rde rato. Ths result was expected snce the more slack tme the vehcle has of droppng-off a passenger at hs destnaton, the more t can pck up other passengers on the way, and a result, less dstance s traveled. On the other hand, the percentage reducton n terms of the number of vehcles used s completely dfferent. As shown n Fgure 5-2, when ncreasng the maxmum allowable rde rato and controllng the tme wndow, the percentage reducton n number of vehcles used decreases notceably. For example, for 20 mnutes tme wndow and 30km\hr vehcle velocty, the percentage reducton n number of vehcles used fall from 50% for an 0.5 excess rde rato to 39.8% for an 2.0 excess rde rato. Whle ths mght look lke as a bad result, t s n fact an advantage of the schedulng methodology. The reason for ths mght be two-fold. Frst, n our model we are tryng to mnmze the total dstance traveled and not the number of vehcles used, as explaned earler. Second, the fall n reducton n number of vehcles shows that the effcency of the algorthm n tght constrants s much better than wth wde constrants. In summary, the results shows that the schedulng methodology developed n Chapter 4 sgnfcantly mprove the soluton obtaned by the Inserton Algorthm, whch valdates the schedulng methodology s ablty to produce relable and mproved results compared to exstng algorthms.

202 188 Percentage Reducton n Cost from Frst Soluton (%) Maxmum Allowable Rde Rato 20 mn Tme Wndow 30 mn Tme Wndow 40 mn Tme Wndow (a) Percentage Reducton n Cost from Frst Soluton (%) mn Tme Wndow 30 mn Tme Wndow 40 mn Tme Wndow Maxmum Allowable Rde Rato (b) Fgure 5-1 Effect of tme wndow duraton and maxmum allowable rde tme on percentage reducton n cost: (a) based on 30km\hr vehcle-average-velocty, and (b) based on 20km\hr vehcle-average-velocty.

203 189 Percentage Reducton n Vehcles Used from Frst Soluton (%) mn Tme Wndow 30 Mn Tme Wndow 40 mn Tme w ndow Maxmum Allowable Rde Rato (a) Percentage Reducton n Vehcles Used from Frst Soluton (%) Maxmum Allowable Rde Rato 20 mn Tme Wndow 30 mn Tme Wndow 40 mn Tme Wndow (b) Fgure 5-2 Effect of tme wndow duraton and maxmum allowable rde tme on percentage reducton n vehcle used: (a) based on 30km\hr vehcle-average-velocty, and (b) based on 20km\hr vehcle-average-velocty

204 VALIDATION OF THE ANALYTICAL MODEL AND SCHEDULING SYSTEM In Chapter 3, an analytcal model was developed to analyze the effect of several desgn and operatonal factors that are key to the desgn and operatons of Flex-Route transt servce. Such factors nclude optmal slack tme, allocaton of slack tme, servce frequency and servce feasblty and cost. Due to the fact that Flex-Route transt servce dffers from one place to another, and due to the lack of specfc gudelnes on the servce desgn n the lterature, the only avalable opton to analyze the senstvty of the results to each desgn factor s to perform a smulaton analyss usng the developed schedulng system. Therefore, n ths secton we try to examne the valdty of the model n estmatng optmal slack tmes and the ablty of the schedulng system to produce the expected results. We use the case of a two-sded servce area to demonstrate that the analytcal model proposed n ths research s a vald one and that the schedulng system correctly produces results that confrm wth the analytcal model estmates. The model for the case of a two-sded servce area (equaton 3-22 gves the optmal slack tme n terms of the servce area wdth and the ondemand rate (number of requests) n a typcal route segment. Accordngly, we test the model for several cases of servce area wdth and demand rate to show the valdty of the proposed model. Ths s done by usng the schedulng system developed n chapter 4 to perform a smulaton on a hypothetcal route. The smulaton s performed by runnng the schedulng system for dfferent case scenaros, where the results of each run are recorded automatcally n a Comma Separated Fle (CVS). The followng s a lst of the specfcatons assumed for ths hypothetcal route: 1. The route length s 10 km, wth a two-sded servce area. The wdth of the servce area on ether sde of the route ranges from 250m to 1000m for dfferent scenaros (see Fgure 5-3). 2. To lmt the scope of our analyss, the servce area s assumed to be covered by a unform grd network wth a constant speed of 20 km/hr on all lnks gnorng stochastc varatons n network condtons (travel tmes are constant).

205 Vehcles run from the frst stop to the end of the route wth a fxed headway of 1 hr for 8 consecutve hours. 4. All servce vehcles are dentcal and each vehcle s assumed to have an unlmted capacty so that no request s rejected due to a vehcle capacty constrant. It should be noted, however, that the vehcle capacty constrants can be added, f requred, wthout much effort as dscussed earler n Chapter The on-demand request rates tested range from 2 to 5 requests per route length per one bus run. 6. Each locaton of an orgn or a destnaton s dentfed by an (X, Y) coordnate at any lnk n the street network. The requests have been generated at random locatons wthn the specfed servce area. 7. For each par of nodes n the network, the travel dstance s equal to the rectlnear dstance between these ponts, whch s used to fnd the travel tme between these nodes. The rectlnear dstance s the dstance between two ponts measured along axes at rght angles. In a plane wth p 1 at (x 1, y 1 ) and p 2 at (x 2, y 2 ), t s x 2 x 1 + y 2 y The travel tme s found by dvdng the rectlnear dstance between the par of nodes by the constant speed assumed n the model. 9. For each smulaton experment settng, the servce system was smulated contnuously for 100 hours, whch should provde statstcally relable estmates of varous performance measures. As mentoned earler, the optmal slack tme for a two-sded servce area s gven by equaton 3-22 as follows: S 2m 1 W. (Equaton 3-22) 3V Where m: s the number of on-demand requests served; W: s the wdth of the servce area on one sde of the route (n km); and

206 192 V: s the operatng speed of the bus (n km/hr). Ths equaton s used to fnd the optmal slack tme for each combnaton of servce area wdth and demand rate examned n ths analyss. Table 5-3 provdes the optmal slack tme for these cases usng equaton 3-22.

207 Fgure 5-3 Map of the hypothetcal route for a servce area of 500m 193

208 194 The approach taken n ths smulaton analyss s to fnd the number of accepted requests gven a specfc servce area wdth, demand rate and slack tme, and then compare these results wth the correspondng values for the optmal slack tmes n Table 5-3. The results of the smulaton are shown n Tables 5-4 and 5-5 for servce area wdths of 250m (0.25 km) and 500m (0.5 km) respectvely. The results clearly show that the model correctly estmates the optmal slack tme gven the servce area wdth and the demand rate and the ablty of the schedulng system to produce the expected results based on equaton For example, for a servce area wdth of 250m and a demand rate of 2, the optmal slack tme for one route segment accordng to equaton 4-22 s 1.25 mnutes. Performng the smulaton usng ths value of slack tme resulted n an average of 1.88 requests beng accepted for each route segment compared to the demand rate of 2. The lttle varance of the result from the one obtaned from the analytcal model can be attrbuted to the varaton n the demand dstrbuton, whch may result n cases where not all the requests can be accepted. Table 5-4 also shows that for the same demand rate, ncreasng the slack tme wll result n ncreasng the number of accepted requests up to a specfc amount of slack tme, whch wll result n acceptng all the on-demand requests (2 mnutes n the case of 250m and demand rate of 2). Ths amount guarantees that every request n the servce area wll be accepted because there s enough slack tme to reach all requests regardless of the dstrbuton of the demand. Beyond ths value, ncreasng the slack tme wll be worthless snce t wll not add anythng to the servce. The analyss provded above s true for all the cases n Tables 5-4 and 5-5 as can be seen n both tables. Therefore, t can be concluded that the analytcal model correctly predcts the optmal amount of slack tme gven the servce area wdth and the demand rate, and that the schedulng system was able to produce the expected schedulng results.

209 195 Table 5-3: Optmal slack tme values for each route segment (n mn) for dfferent combnatons of servce area wdth and on-demand request rate, usng equaton 3-22 Demand Rate Servce Area Wdth Table 5-4: Number of accepted requests for combnatons of slack tme and on-demand request rate for a servce area wdth of 250m (0.25 km) for each route segment, smulaton results Slack Tme (mn) Demand Rate Value of S* At S* Table 5-5: Number of accepted requests for combnatons of slack tme and on-demand request rate for a servce area wdth of 500m (0.5 km) for each route segment, smulaton results Demand Rate Value of S* Slack Tme (mn) At S*

210 SENSITIVITY ANALYSIS The senstvty analyss s performed for the same hypothetcal route (see Fgure 5-3) used n the prevous secton wth the same assumptons. In addton, the followng two addtonal assumptons are made: The on-demand request rates tested are dfferent from the prevous secton. The rates tested here are 10, 15, and 20 passengers per route length per one bus run (or 1, 1.5 and 2 requests per route segment respectvely); and The fxed-stop spacng s tested for two cases: a constant spacng of 1 km and a constant spacng of 2 km. The computatonal performance of the algorthm was very encouragng for all cases. The runnng tme of the model takes less than 3 seconds n the worst case scenaro for one schedule to be completed. Ths could be a very mportant factor n real tme applcatons, where computatonal performance s a decdng factor. All runs were performed on an Intel Centrno Duo, 1.73 GHz wth RAM of The schedulng results obtaned by the schedulng system for some case scenaros are summarzed n Table 5-6 (for stop spacng of 1 Km) and 5-5 (for stop spacng of 2000m) n terms of the percentage of accepted on-demand requests. The schedulng algorthm guarantees that all vehcles wll be at the fxed stops at the publshed departure tmes at the respectve stops. In general, t can be observed that the schedulng algorthm performs extremely well n terms of the number of on-demand requests that can be scheduled durng the operaton perod. It should be noted that the low percentages of accepted requests n some cases are not related to the performance of the schedulng algorthm, but nstead to the amount of slack tme allocated for servng the on-demand requests, as wll be dscussed later. The effcency of the algorthm could be seen clearly n the smaller servce areas of 250m and 500m snce there s enough slack tme to use for servng the demand-responsve requests. As t can be seen, n almost all cases, the percentage of accepted requests s above 90%, whch s a very good result. The other 10% of unscheduled requests could be attrbuted to the varaton n demand durng the day. Regardless of that, the algorthm stll produces very encouragng results as t allows for schedulng a hgh percentage of the demand-responsve requests.

211 197 Table 5-6: Number of feasble devatons for 1 Km fxed-stop spacng (8-hour servce duraton) Slack Tme 2 Mnutes 3 Mnutes 6 Mnutes On-Demand Request Rate 10 Req/run # % # % Servce Accepted Accepted Accepted Accepted Area (m) Requests Requests Requests Requests # % Accepted Accepted Requests Requests Req/run Req/run

212 198 Table 5-7: Number of feasble devatons for 2 Km fxed-stop spacng (8-hour servce duraton) Slack Tme On-Demand 2 Mnutes 3 Mnutes Request Rate Servce Area (m) # Accepted Requests % Accepted Requests # Accepted Requests % Accepted Requests Req/run Req/run Req/run Effect of Desgn Parameters on the Number of Feasble Devatons The objectve of ths secton s to analyze the mpacts of the dfferent parameter values on the percentage of on-demand requests that are accommodated by the Flex- Route servce. The factors ncluded n ths dscusson are: servce area wdth, slack tme and stop spacng Effect of Servce Area Wdth on Feasble Devatons Ths secton examnes the possble effect of servce area wdth on the percentage of feasble devatons gven a specfc rate of on-demand requests whle fxng all other factors. The servce area wdth ranges from 250m to 2000 m. Fgures 5-4 and 5-5 show the expermental results for ths part of the analyss for dfferent slack tme values and

213 199 dfferent demand rates, whle fxng the fxed stop spacng at 1000m. The fgures are drawn for 10 and 15 requests/run demand rates respectvely. The effect of demand rate wll be dscussed later n ths secton. The fgure clearly shows that as the wdth of servce area ncreases, the percentage of feasble devatons decreases n a systematc way. For example, Fgure 5-4 shows that for a slack tme of 3 mnutes, the percentage of accepted requests decreases dramatcally from 93% for a servce area wdth of 250m, to 22% for a servce area wdth of 2000m. The same pattern s clear for the other values of slack tme. Ths s explaned by the fact that when the demand spreads out on a wder area (for the same slack tme), fewer trps wll be answered snce the amount of slack tme allocated to serve these trps s fxed regardless of the wdth of servce area Effect of Slack Tme on Feasble Devatons Fgures 5-4 and 5-5 show another mportant result. For the same servce area wdth, ncreasng the slack tme wll ncrease the percentage of feasble devatons for the same level of demand. Ths result can be seen for any wdth of servce area as shown n the fgures. However, the wder the servce area, the more ths pattern s apparent. Ths s because regardless of the amount of slack tme allocated to small servce areas, the vehcle can accommodate a specfc number of requests snce most the requests can be reached wthn the gven slack tme, even f t s low (.e. 2 and 3 mn). So ncreasng the slack tme wll not result n more answered requests, and t wll only result n more dle tme at the fxed stops. For example, for a servce area wdth of 250m n Fgure 5-4, the percentage of accepted requests are 92%, 93%, and 94% for slack tmes of 2, 3 and 6 mnutes. It should be noted that the percentage of accepted requests reaches 100% n some smulaton runs, but ths do not show n the results reported here because these results are the average results of multple smulaton runs not a sngle run. For the wder servce areas, ncreasng the slack tme wll ncrease the possblty for the vehcle to reach trps that can not be reached f the slack tme was lower. For example, n Fgure 5-4, for servce area of 750m, the percentages of answered trps are 41%, 64% and 92% for slack tmes of 2, 3 and 6mn, respectvely. These results show that slack tme should be carefully allocated dependng on the wdth of the servce area. Increasng the slack tme wll not necessarly ncrease the

214 number of answered requests. As the results show, slack tme s related to the servce area wdth, and they should be ntegrated n a proper way. 200

215 201 Percentage of Answered Requests (%) 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 2mn slack tme 3mn slack tme 4mn slack tme 5mn slack tme 6mn slack tme 0% Servce Area Wdth (m) Fgure 5-4: Effect of slack tme and servce area wdth on the percentage of accepted on-demand requests for a demand rate of 10 requests /run.

216 202 Percentage of Answered Requests (%) 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 2mn slack tme 3mn slack tme 6mn slack tme 0% Servce Area Wdth (m) Fgure 5-5: Effect of slack tme and servce area wdth on the percentage of accepted on-demand requests for a demand rate of 15 requests /run.

217 Effect of On-Demand Request Rate on Feasble Devatons Fgures 5-6 and 5-7 show the relatonshp between the on-demand request rate and percentage of accepted requests. The fgures show that the percentage of accepted requests remans approxmately the same for demand rates of 10 and 15 requests per hour. However, the number of answered requests obvously ncreases wth the hgher demand rate. For example, as shown n Fgure 5-6, for a 250m servce area, the percentage of accepted requests s 93% for both demand rates; however, 93% of 10 requests/run demand rate results n 75 accepted requests over 8 hours, whle 93% of 15 requests/run demand rate results n 112 accepted requests. The constant percentage of answered requests could be explaned by the fact that the lower request demand rate (.e. 10 requests per hour) was not enough to consume all the slack tme allocated for the ondemand requests, and that there was enough slack tme to cover more requests. The same can be sad about ths pattern for the demand rate of 20 requests /run, where the percentage of answered requests remans approxmately smlar to that of the other demand rates at the low servce area wdths. However, the percentage of accommodated requests becomes hgher than those of the 10 and 15 requests/run demand rates for wder servce area wdths. Ths s lkely due to the fact that for hgher demand rates and wder servce areas, the number of requested trps that could be reached gven the allocated slack tme becomes hgher (more trps could be reached wthn the specfed slack tme), and so the percentage of accepted requests becomes hgher. These results could be very nstrumental n decdng how much slack tme to allocate to a gven route based on the expected on-demand request rate as dscussed earler n ths chapter. It s mportant to allocate enough slack tme for requests that mght be requested at the last moment, or requests n real tme. Provdng enough slack tme wll allow for more requests to be answered and thus more satsfacton to the customers. On the other hand, provdng excessve slack tme wth very low demand could result n dssatsfacton and dscomfort for fxed stop passengers as the vehcle wll arrve earler at the stop, but t has to stay at the stop untl ts departure tme. So t s a trade-off between the two groups of customers, and a careful study should be made to determne the expected demand rate.

218 204 Percentage of Answered Requests (%) 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 10 requests/hr 15 requests/hr 20 requests/hr 0% Servce Area Wdth (m) Fgure 5-6: Effect of on-demand request rate on the percentage of accepted requests for a slack tme of 3 mnutes

219 205 Percentage of Answered Requests (%) 96% 95% 94% 93% 92% 91% 90% 89% 88% 10 requests/hr 15 requests/hr 20 requests/hr 87% Servce Area Wdth (m) Fgure 5-7: Effect of on-demand request rate on the percentage of accepted requests for a slack tme of 6 mnutes

220 Effect of Fxed-Stop Spacng on Feasble Devatons Fgure 5-8 shows the effect of fxed-stop spacng on the percentage of accepted on-demand requests. The fgure s plotted for a slack tme of 6 mnutes. The fgure shows that there s no apparent effect of fxed-stop spacng on the amount of accepted requests. However, an nherent sde effect should be noted, that could be very mportant n schedulng Flex-Route transt. The 6-mnute slack tme used n all cases could be related to the drect runnng tme between the fxed stops. For example, the 6-mnute slack tme n the case of 1000-m spacng s twce the drect runnng tme between the fxed-stops (.e. slack tme =150% drect runnng tme), whle for the 2000-m and 3000-m spacng, the values are 100% and 67% respectvely. So t s very mportant to notce ths result snce t plays an mportant part when t comes to fxed-stop customer convenence. Ths s because less slack tme allocated wth respect to drect runnng tme wll be more convenent for fxed-stop customers. However, on the other hand, ths means that ncreasng slack tme for the cases of hgher stop spacng (2000 and 3000m) could result n more on-demand requests to be accepted. Therefore, t s a trade-off between the two groups, whch hghly depends on the amount of demand by both groups of customers.

221 % Percentage of Answered Requests (%) 98% 96% 94% 92% 90% 88% 1000-m spacng 2000-m spacng 3000-m spacng 86% Servce Area Wdth (m) Fgure 5-8: Effect of fxed-stop spacng on the percentage of accepted requests for a slack tme of 6 mnutes Dynamc Requests Effect and Late Arrval The prevous secton analyzed the senstvty of the desgn parameters and ther effect on the number of accepted requests n statc settngs. In other words, all the requests are known n advance and no dynamc aspect s ncluded n the analyss. In ths secton, we analyze the effect of the presence of real-tme requests on the number of accepted requests, and how that mght mprove the performance of Flex-Route servce. The analyss n ths secton s based on a number of on-demand requests dstrbuted unformly along the route. These requests are dvded nto two types: statc requests wth a rate of 20 requests per route length per one bus run, whle the dynamc request rate s set at 5 requests per route length per one bus run. However, every smulaton run has ts own unformly random requests that could be located anywhere along the route. The number of customers at each fxed stop s assumed to be 3 to lmt the analyss. Ths assumpton s just to study the effect of unused slack tme on fxed-stop customers.

222 208 The settng of our man case scenaro s as follows: L=10km, W = 500m, statc demand = 20 requests/run, dynamc demand = 5 requests/run, headway = 60 mn, and fxed-stops spacng = 1 Km). One of the most mportant decsons n Flex-Route desgn s the amount of slack tme allocated to the route n general and to the ndvdual route segments separately. The servce provder should have an dea about the number of total expected on-demand requests (statc and dynamc) and ther locaton along the route to be able to make a decson regardng the length of the total slack tme and where to allocate t. However, spatal and temporal varatons of these requests could occur, but the slack tme amount wll not change from day-to-day. A slack tme of 30mn/route (3 mnutes per route segment) was consdered n ths case. The results obtaned for the man case scenaro are shown n Table 5-8, ncludng some performance measures used to check the valdty of the assgned slack tme and the effectveness of the schedulng system. The results show that the system was able to schedule an average of 17.2 statc requests (86%) and 2.3 dynamc requests (46%) n ths scenaro. These results are very encouragng consderng the spatal varaton of the demand. These results mean that the presence of dynamc requests allowed the Flex- Route servce to schedule an average 2.3 more requests per bus run. Ths wll result n an mprovement of the servce as more requests are accepted wth the same slack tme. Ths ncrease could also be attrbuted to the fact that the presence of dynamc requests add to the total on-demand request rate, and thus ncreasng the possblty of acceptng more requests. The amount of unused slack tme for the whole route s 8.83 mn representng 29.4% of the slack tme, whch s a farly hgh percentage. The effect of ths on fxedstop customers s an average of 26.5 passenger-mnutes lost due to the unused slack tme. Ths man case scenaro provdes an nsght of the valdty of the 30mn (3mn/stop) assumpton and the ablty of the schedulng system to schedule requests gven the hard constrant that the vehcle must be at the fxed stop on tme. That led to the second and thrd case scenaros, whch assumed that the vehcle can reach the fxed-stop later than ts assgned departure tme f that means the system wll be able to schedule more requests. The results for the second and thrd case scenaros are shown n Table 5-8. The only dfference from the frst case s that n the second case, the vehcle s allowed to be at the fxed stop after ts publshed departure tme up to 1 mn, whle n the thrd case

223 209 the vehcle s allowed to arrve at the fxed stop any tme after ts publshed departure tme up to 3mn. Ths s the case for every fxed stop except the fnal stop, whch s assumed to be a major transfer pont that the vehcle must be there on tme. Table 5-8: Results obtaned for cases 1 to 3 (spacng =1 km). Case 1 Case 2 Case 3 Statc Requests 20 req/run 20 req/run 20 req/run Slack Tme (per stop) 3mn/stop 3mn/stop 3mn/stop Fxed-Stop Spacng (m) Headway (mn) Fxed Stop Arrval Relaxed? No Yes (up to 1 mn) Yes (up to 3mn) Cycle Tme (mn) 120mn 120mn 120mn Accepted Requests % Statc Requests Made 86% 91.50% 98.50% Unperformed Requests Total Unused Slack Tme (mn) Total Passenger-mn Lost (Pass-mn) Total late Arrval Tme (mn) Maxmum Late Arrval Tme (mn) Average Travel Tme (mn) Dstance Traveled (km) After Dynamc Requests Accepted Dynamc Requests Unperformed Dynamc Requests Total Unused Slack Tme (mn) Total Passenger-mn Lost (Pass-mn) Total Late Arrval Tme (mn) Maxmum Late Arrval Tme (mn) Average Travel Tme (mn) Dstance Traveled (km)

224 210 The performance of these 2 cases s notceably better as can be seen n Table 5-8. For case 2 (up to 1mn), the relaxaton of the arrval tme at the fxed stops resulted n acceptng 18.3 statc requests and 2.3 dynamc requests, whle decreasng the amount of unused slack tme (7.52mn), and at the same tme arrvng late at the fxed stops wth a total of 1.4 mn and a maxmum arrval of 0.98 mn. However, for the thrd case (up to 3mn), the system was able to schedule 19.7 statc requests and 4.2 dynamc requests, whch means almost all the requests were answered n statc settng and more than 80% of the dynamc requests. Ths case also resulted n an unused slack tme of 6.7 mn (22% of the assgned slack tme), but also resulted n a total of extra 6.08 mn of late arrval at the fxed stop wth a maxmum of 2.04mn. These scenaros suggests that relaxng the constrant of arrval tme at the fxed stops could be very useful n schedulng more requests especally n dynamc settngs, where an extra arrval tme of less than a mnute or two at the fxed stop could result n acceptng more requests. One advantage of the proposed schedulng system s that t s very easy to relax any constrant or mpose new constrans n the Constrant Programmng envronment. To further test the effect of the assgned slack tme, relaxaton of fxed-stop arrval tme, and the ablty of the schedulng system to deal wth real tme requests, we used the same settng for cases 1-3 but wth dfferent slack tme. The total slack tme assgned was 20 mnutes (2 mnutes per route segment). Table 5-9 shows the results for these cases. One mportant thng to notce s that the cycle tme of the route s now (50 * 2 =100 mnutes), whch results n a 20 mnute decrease n cycle tme for the transt vehcle. In all the cases, the number of scheduled requests s lower than cases 1-3. Ths s expected because of the lower slack tme (20 mnutes compared to 30 mnutes). However, the advantage of relaxng the fxed-stop arrval tme by acceptng more requests and reducng unused slack tme s more apparent. In cases 5 and 6, the number of accepted statc requests ncreases from 11.1 n case 4 to 14.2 and 18.3 n cases 5 and 6 respectvely, whle the amount of unused slack tme dropped from 9.52 mnutes n case 4 to 4.45 mnutes and 1.65 mnutes n cases 5 and 6 respectvely. The only dsadvantage s that the amount of late arrval at the fxed-stops ncreases n cases 5 and 6. In case 6, the total amount of late arrval tmes at the fxed stops s mnutes wth a maxmum of

225 mnutes at one stop. Although ths amount could be consdered very large, t s mportant to consder that n the case of a major transfer pont at the end of the route whch most customers are destned to (lke a regonal tran servce wth a long headway), and n the presence of real-tme nformaton at the fxed stops, the man purpose of the customers s to reach the fnal stop on tme, whch s guaranteed n all cases. Also, reducng the assgned slack tme has resulted n a lower average travel tme for all passengers. In cases 4-6, the average travel tme s around 25 mnutes, whle n cases 1-3, the average travel tme s around 30 mnutes. As can be concluded from comparng cases1-3 wth cases 4-6, the amount of assgned slack tme effect the cycle tme of vehcles, the number of accepted statc and dynamc requests, the amount of unused slack tme, the effect of lost passenger mnutes due to unused slack tme, the amount of late arrval tme at fxed stops n case of constrant relaxaton, and the average travel tme of all passengers.

226 212 Table 5-9: Results obtaned for cases 4 to 6 (spacng =1 km) Case 4 Case 5 Case 6 Statc Requests 20 req/h 20 req/h 20 req/h Slack Tme (per stop) 2mn/stop 2mn/stop 2mn/stop Fxed-Stop Spacng (m) Headway (mn) Fxed Stop Arrval Relaxed? No Yes (up to 1 mn) Yes (up to 3mn) Cycle Tme (mn) Accepted Requests % Statc Requests Made 55.50% 71% 91.50% Unperformed Requests Total Unused Slack Tme (mn) Total Passenger-mn Lost (Pass-mn) Total late Arrval Tme (mn) Maxmum Late Arrval Tme (mn) Average Travel Tme (mn) Dstance Traveled (km) After Dynamc Requests Accepted Dynamc Requests Unperformed Dynamc Requests Total Unused Slack Tme (mn) Total Passenger-mn Lost (Pass-mn) Total Late Arrval Tme (mn) Maxmum Late Arrval Tme (mn) Average Travel Tme (mn) Dstance Traveled (km)

227 Effects of Fxed-Stop Spacng on Dynamc Requests Acceptance In order to study the effect of fxed-stop spacng on the number of accepted requests, both statc and dynamc, we used the same settng n cases 1-3 except that the fxed-stop spacng s now 2 km nstead of 1 km, and as such each new route segment has a slack tme of 6mn. The results are shown n Table 5-10 for cases 7-9. The results show that the number of accepted requests n case 7 has ncreased to a level where almost all the statc requests (18.9 out of 20) and dynamc requests (3.8 out of 5) are accepted. Ths s done wthout the need to relax the arrval tme constrant at the fxed stops. However, the unused slack tme s stll almost the same as n case 1. When relaxng the arrval tme at the fxed stops as shown n cases 8 and 9, the number of accepted statc requests s not apparent as wth the dynamc requests. Comparng cases 2 and 3 wth cases 8 and 9, the major dfference s that the number of accepted dynamc requests s much hgher n cases 8 and 9 (4.2 and 4.9 n cases 8 and 9 compared to 2.3 n cases 2 and 3). However, ths of course came at the expense of removng 5 fxed stops. These results suggest that f the number of expected dynamc requests s very hgh, t wll be better to have less fxed stops. Ths could be explaned by the fact that havng less fxed stops n real tme wll help the vehcle acceptng more dynamc requests as t has less fxed stops constrants to comply wth, and therefore more tme to accept nearby requests that otherwse would have been rejected f the number of fxed stops was hgher. In order to further test the effect of fxed-stops` spacng, we used the same settngs n cases 7-9 bus changed the slack tme from 30 mnutes to 15 mnutes for the whole route, whch means 3 mnute slack tme for each route segment nstead of 6 mnutes. The results for these cases (10-12) are shown n Table The results show that the number of accepted requests, both statc and dynamc, drops dramatcally snce the amount of avalable slack tme s lower. However, the cycle tme for a two-way trp s now 90 mnutes nstead of 120 mnutes. Ths could be crucal f the supply of transt vehcles s scarce. Also, ths settng reduces both the unused slack tme and the average travel tme for all passengers.

228 214 Table 5-10: Results obtaned for cases 7 to 9 (spacng = 2 km). Case 7 Case 8 Case 9 Statc Requests 20 req/h 20 req/h 20 req/h Slack Tme (per stop) 6mn/stop 6mn/stop 6mn/stop Fxed-Stop Spacng (m) Headway (mn) Fxed Stop Arrval Relaxed? No Yes (up to 1 mn) Yes (up to 3mn) Cycle Tme (mn) Accepted Requests % Statc Requests Made 94.5% 96.5% 99.5% Unperformed Requests Total Unused Slack Tme (mn) Total Passenger-mn Lost (Pass-mn) Total late Arrval Tme (mn) Maxmum Late Arrval Tme (mn) Average Travel Tme (mn) Dstance Traveled (km) After Dynamc Requests Accepted Dynamc Requests Unperformed Dynamc Requests Total Unused Slack Tme (mn) Total Passenger-mn Lost (Pass-mn) Total Late Arrval Tme (mn) Maxmum Late Arrval Tme (mn) Average Travel Tme (mn) Dstance Traveled (km)

229 215 Table 5-11: Results obtaned for cases 10 to 12 (spacng = 2 km). Case 10 Case 11 Case 12 Statc Requests 20 req/h 20 req/h 20 req/h Slack Tme (per stop) 3mn/stop 3mn/stop 3mn/stop Fxed-Stop Spacng (m) Headway (mn) Fxed Stop Arrval Relaxed? No Yes (up to 1 mn) Yes (up to 3mn) Cycle Tme (mn) Accepted Requests % Statc Requests Made 67.5% 72% 74% Unperformed Requests Total Unused Slack Tme (mn) Total Passenger-mn Lost (Pass-mn) Total late Arrval Tme (mn) Maxmum Late Arrval Tme (mn) Average Travel Tme (mn) Dstance Traveled (km) After Dynamc Requests Performed Dynamc Requests Unperformed Dynamc Requests Total Unused Slack Tme (mn) Total Passenger-mn Lost (Pass-mn) Total Late Arrval Tme (mn) Maxmum Late Arrval Tme (mn) Average Travel Tme (mn) Dstance Traveled (km)

230 CONCLUSIONS The results of the senstvty analyss performed n ths chapter support the results provded by the analytcal model developed n Chapter three. For example, the results show that ncreasng the slack tme would ncrease the number of accepted requests but wll result n longer travel tmes and addtonal unused slack tme. The results also show that ncreasng the fxed-stop spacng wll result n ncreasng the number of accepted requests snce the vehcle wll have more tme to navgate through the servce area to pck up more customers as t s not constraned by the fxed-stops constrants. Ths ncrease n stop spacng would be an opton n case the demand at a specfc fxed-stop s very low that ths stop could be removed. The results also show that the effect of relaxng the arrval tme of the vehcle at the fxed stops has a sgnfcant effect on the number of accepted on-demand requests and on the overall level of servce. Allowng late arrval at the fxed stops would result n lower cycle tme, less slack tme requred, less unused slack tme, less passenger-mnutes lost due to early arrval at the fxed-stops, and lower average travel tme. However, ths sgnfcant effect can lead to late arrval of the vehcle at some fxed stops. Ths late arrval could be constraned n the model to any amount of tme. It should be noted that n the case where the route leads to a major transfer pont (as n the case of ths ongong research), late arrval at the ntermedate fxed stops, where connecton protecton s not requred, would not consttute a major problem snce the man purpose of most passengers s to reach ths transfer pont on tme. Also, the presence of real-tme nformaton systems at the fxed-stops wll help allevate ths problem snce customers wll be able to know the expected arrval tme of the bus at all tmes. Therefore usng an approprate slack tme s essental n havng an acceptable level of servce for all customers. It s concluded that relaxng the fxed-stop constrants wll allow the servce provder to assgn less slack tme to the servce, and hence less operatng cost, whle at the same tme mantanng an acceptable level of servce for both fxed-route and demand-responsve customers.

231 217 6 CASE STUDY - OAKVILLE TRANSIT 6.1 CHAPTER OVERVIEW Ths chapter documents an attempt to examne the applcablty, feasblty and performance of the Flex-Route analytcal model developed n Chapter 3 as well as the ablty and performance of the schedulng system developed n Chapter 4 to handle real-world applcatons. The model s appled to a real-world context to evaluate the possblty of replacng a few fxedroute transt routes n the suburbs of the Greater Toronto Area wth Flex-Route servce. The results of ths applcaton demonstrate the valdty of both the proposed schedulng and analytcal models. Secton 6.2 gves a background on the applcaton context of the model. Secton 6.3 provdes some background on GO Transt commuter ral servce. It dscusses the potental of Flex-Route transt servce to produce better rdershp volumes than fxed-route transt and to be ntegrated wth GO Transt n a manner that could help allevate some of ts problems. Secton 6.4 ntroduces some of the assumptons and features specfc to the case study. Secton 6.5 presents the results of the expermental study that llustrate the applcablty of the servce and the senstvty of the desgn parameters to the nput parameters. Followng that, secton 6.6 analyzes the cost nflcted by the Flex-Route servce on the transt servce provder as well as the fxed-route customers. Fnally, secton 6.7summarzes the results and conclusons of ths case study 6.2 INTRODUCTION AND BACKGROUND As dscussed throughout ths thess, Flex-Route transt servce has the potental to meet the challenges faced by suburban transt provders n a more effcent manner than fxed-route transt. By ntegratng the regularty of tradtonal fxed-route transt servce and the flexblty of demand-responsve servce (DRT), Flex-Route transt servce can be a vtal transt opton for medum-to-low densty suburban areas where the demand for general publc transt s low to be effcently servced by conventonal fxed-route transt, and relatvely hgh and expensve to be served by DRT. As mentoned earler n the thess, Flex-Route transt has been used effectvely by a few transt agences n rural and small urban areas n the Unted States (see Rosenbloom 1996 and Koffman 2004).

232 218 To llustrate the feasblty of Flex-Route transt servce n suburban areas, the proposed Flex-Route servce desgn process dscussed n Chapter 3 s appled to a real-world case scenaro, whle the schedulng of the servce s performed usng the schedulng system developed n Chapter 4. The suburban area chosen for ths case study s the Cty of Oakvlle, located n the western part of the Greater Toronto Area (GTA), Canada. In the GTA, persstent urban sprawl over the past few decades has led to the present populaton dstrbuton pattern of more people lvng n the low-densty spread-out suburban areas around the Cty of Toronto than nsde the cty tself. In the Cty of Oakvlle, Oakvlle Transt s the publc transportaton provder snce Oakvlle Transt offers a conventonal fxed-route bus servce as well as door-to-door paratranst servce, called care-a-van, for the dsabled and elderly. Oakvlle Transt operates several bus routes that connect to the regonal commuter ral network (GO Transt), whch operates a comprehensve network of seven tran lnes and numerous bus routes lnkng towns and ctes across southern Ontaro s Greater Toronto Area (GTA) and the adjacent Cty of Hamlton. Currently, the rdershp volumes on many Oakvlle bus routes are very low even n the mornng and evenng peak perods due to the low densty areas and the hgh auto ownershp of most households. In Oakvlle, the percentage of people accessng GO Transt tran statons by car surpasses, by a great margn, the percentage of those who use publc transt to access the tran statons. Ths trend has led to a serous problem facng GO Transt n that the auto demand at parkng lots of GO tran statons are beyond capacty. Thus, there s a mutual problem facng both Oakvlle Transt and GO Transt that mght be solved by mplementng Flex-Route transt servce ( 2008) 6.3 THE GO TRANSIT PARKING PROBLEM The suburban populaton n the GTA reles heavly on GO Transt commuter ral for ther daly work trps to the downtown centre of Toronto. However, a sgnfcant problem facng GO Transt presently s that almost all ts parkng lots have reached capacty. Passengers who use ther cars to access GO statons have to be at the statons as early as possble n the mornng to fnd a parkng spot. Those who do not arrve suffcently early every mornng usually park ther cars at llegal spaces nsde the lot, or have to search for pad parkng at other places. The other

233 219 opton that those passengers have s usng local transt to access the statons. However, local transt n most suburban areas has relatvely low coverage so as to reach a majorty of the populaton, gven the low-densty demand areas and the street networks of the suburbs that make walkng to a transt stop rather burdensome, partcularly n cold wnter tmes. Therefore, there s a great opportunty that by enhancng the performance of suburban transt and makng t more attractve, some auto rders mght shft to transt f a relable and accessble transt system exsts. Ths would not only enhance the performance of suburban transt and make t more attractve to passengers, but also allevate some of the fnancal burden on GO Transt for expandng the capacty of ts parkng lots. As some estmates have suggested (Vctora Transport Insttute, 2008), addng one parkng space to a parkng faclty n a suburban area costs $1,818 to acqure the land, $3,000 n constructon costs and $300 n operatonal and management costs (See Table 6-1). Ths excludes any ndrect or envronmental costs. Therefore, t costs about $755 annually to buld one parkng space for the entre lfe span of the parkng space. It should be mentoned that these numbers are n 1999 dollars, so t s expected that the current costs are hgher than those numbers. Thus, every car removed from any parkng lot wll save GO Transt $755 (1999$) every year for the entre lfe span of the parkng space. Gettng these passengers to use transt and abandon usng ther cars to access GO statons also has a drect mpact on the envronment, snce emssons from cars are a major contrbutor to ar polluton and clmate change. The remander of ths chapter explores the mplementaton of Flex-Route transt servce n the Cty of Oakvlle. Three fxed-route bus lnes that connect dfferent areas of the cty to GO Transt tran statons wll be nvestgated to determne the applcablty and feasblty of Flex-Route servce n suburban areas.

234 220 Table 6-1: Typcal parkng faclty fnancal costs (Source: Vctora Transport Insttute EXPERIMENTAL STUDY SETTINGS The objectve of ths chapter s to nvestgate the feasblty and performance of Flex-Route servce n the suburban areas of the Greater Toronto Area. In order to do that, three fxed-route transt lnes, operated by Oakvlle Transt and connectng dfferent areas n the Cty of Oakvlle to GO Transt tran statons, have been selected as potental canddates for the mplementaton of Flex-Route servce. The routes were chosen due to ther low rdershp numbers especally n the mornng peak perod and also because they serve low-densty suburban areas. The routes chosen as case studes are routes 11, 21 and 25. The analyss examnes the feasblty of operatng the three routes n a Flex-Route servce mode n the mornng peak perod (6 8:30 AM), when commuters make ther mornng trps to the GO Transt statons. The performance of these routes n ther exstng fxed-route servce mode wll be compared to ther performance f they were to be replaced by Flex-Route servce.

235 Exstng Desgn of the Study s Fxed Routes Route 11 runs east-west through the Cty of Oakvlle and serves multple areas along ts route (see Fgure 6-1). At the end of ts run, the route connects to the Oakvlle GO staton located n the downtown area of the Cty of Oakvlle. Routes 21 and 25 (Fgures 6-2 and 6-3, respectvely) on the other hand, serve two separate zones on the east sde of the cty, where both routes connect to Clarkson GO staton located n the outskrts of the Cty of Mssssauga and close to the eastern boundares of the Cty of Oakvlle. Table 6-2 shows the number of parkng spaces avalable n each staton as of July 2008, as well as the number of passengers, from all regons, who access these statons usng dfferent modes, such as auto drver, auto passenger, and local transt among other types. Table 6-3 summarzes the desgn and rdershp nformaton of three routes. As shown n the table, the length of Route 11 (4975m) s almost half that of Routes 21 (9125m) and 25 (9705m), whle at the same tme, the route has more fxed stops (6) than Routes 21 (4) and 25 (4). Also, Route 11 has more frequent servce n the mornng peak perod (7 runs) compared to that of Route 21 (5) and Route 25 (4). The transt demand data for all three routes were obtaned from Oakvlle Transt (Oakvlle Transt Rdershp Counts, 2006) whle the general travel demand data wthn the servce area of each route were obtaned from the 2006 Transportaton Tomorrow Survey (TTS) database (Unversty of Toronto, 2006). Table 6-2: GO Staton rdershp and parkng nformaton Clarkson Staton Oakvlle Staton Number of Parkng Spaces Avalable Number of passengers Who Access the Staton as Auto Drvers Number of passengers Who Access the Staton as Auto Passengers Number of passengers Who Access the Staton as Transt Passengers Number of passengers Who Access the Staton by Other Modes

236 222 Table 6-3: Route Informaton Route Informaton Route 11 Route 21 Route 25 No. of Fxed Stops * Route Length (m) Runnng Tme (mn) Average Fxed-Stop Spacng (m) Mornng Peak Perod Rdershp (6-8:30AM) No. of bus runs (6-8:30AM) Number of passengers accessng GO staton by auto n the route s servce area Fxed stop rdershp for each bus run * The fxed stops exclude the frst stop (depot) and the last stop (GO staton)

237 Fgure 6-1 Map of route

238 Fgure 6-2 Map of route

239 Fgure 6-3 Map of route

240 Proposed Flex-Route System: Assumptons and Specfcatons In order to perform the analyss n a systematc way, several assumptons and specfcatons needed to be made; these are dscussed next Fxed-Route 1. An average operatng speed of 30 km/hr on all street lnks was assumed for Routes 21 and 25, whle for Route 11 the operatng speed was set at 20 km/hr. Ths assumpton s based on the scheduled runnng tme used by Oakvlle Transt. Ths speed was not obtaned from Oakvlle Transt, but rather obtaned by runnng a smulaton for each route. The smulaton used the publshed tmetables at each fxed stop along the route to determne what operatng speed would allow for the publshed tmetable to be attanable. After runnng the smulaton, t was found that the average operatng speed s approxmately 20km/hr for Route 11, and 30km/hr for Routes 21 and The departure tme at each fxed stop was determned on the bass of the drect runnng tmes between each par of fxed stops (.e. route segment) plus the added slack tme. The drect runnng tme was found usng the smulaton as dscussed earler. 3. The analyss was performed for a sngle bus run for each route. The assumpton s that the rdershp s dstrbuted unformly durng the study perod for the bus runs of each route, due to the lack of more detaled nformaton on the rdershp patterns. Ths s appled for both the fxed stops and the on-demand requests. 4. Based on the prevous assumpton, the fxed-route s mornng rdershp for each bus run of a specfc route s found by dvdng the mornng rdershp by the number of runs for that specfc route as shown n Table 6-3. For example, the mornng rdershp for a sngle bus run of Route 21 s equal to 60/5 = 12 passengers per bus run. 5. The fxed-route demand s assumed to be nsenstve to the change of the servce to Flex-Route transt servce. Ths assumpton presumes that fxed-route customers wll keep usng transt even though the servce wll be changed to Flex-Route servce and some extra cost wll be ncurred by them. Ths assumpton assumes also that none of the fxed-route passengers wll shft to on-demand servce. However, some strateges that

241 227 can be used to allevate the cost on fxed-route passengers wll be examned n the analyss. 6. The dstrbuton of fxed-route rdershp among the ndvdual fxed stops was performed n the smulaton gvng dfferent values of rdershp per stop. The values consdered ranged from 2 to 5 passengers per fxed stop due to the lack of more rdershp detals On-demand Servce 1. The servce area for Routes 21 and 25 was taken as the whole traffc zone for each route snce t s very dffcult to set dfferent servce area values gven the route desgn and the street network of these zones. For Route 21, the servce area dmensons are (1300m X 1750m), whle Route 25 servce area dmensons are (1500m X 1750m). For Route 11, the servce area was set to 500m on each sde of the route. 2. The on-demand requests were generated at random locatons wthn each zone at random tmes durng the specfed mornng peak perod (6-8:30AM). Ths assumpton means that some requests mght be located very close to the fxed stops, whch mght be unrealstc n real-world applcatons. It should be mentoned here that only statc requests were consdered n ths case study. 3. An mportant feature of the smulaton of on-demand requests s that the requests were generated as requests for specfc GO tran runs n the mornng (and n the evenng). In other words, the customer asks to be connected to a specfc GO tran n the mornng (for example the 7:30 tran gong to downtown Toronto) and also ncludes the nformaton about whch tran he wll use on hs way back n the evenng. The trp reservaton system then fnds the bus run that connects to the mornng tran and the bus run that connects to the evenng tran and adds that request to these specfc bus run. So a customer requests s accepted only f both ends of the trp are guaranteed. 4. Fndng random locatons for on-demand requests was a bt of a challenge because of the lack of geo-coded addresses n the area. The avalable data have only the number of houses on each street n the GIS map of the servce area of the route. Therefore, the method used for fndng the locatons of random on-demand requests s as follows:

242 228 The probablty of havng a request on each street lnk (every street mght have dfferent number of lnks n the GIS map) s found by dvdng the number of houses on that specfc street lnk by the total number of houses n the servce area. These probabltes were used n a roulette wheel model from 0 to 100, where each street lnk has a specfc range based on ts probablty. Then, a random number generator was used to produce random numbers that use the roulette wheel to produce a street lnk on whch a random on-demand request s generated. Followng that, a random number generator was used to fnd the exact address of the request on the street lnk. 5. The on-demand request rate for each route was chosen so as to represent a percentage of current auto drvers who mght be wllng to swtch to transt as on-demand customers. The percentages chosen for each route are 10%, 15% and 20 % (for routes 21 and 25) and 15%, 20% and 30% (for Route 11) of the total number of passengers who use ther cars to access GO statons. The hgher percentages used for Route 11 were because of the lower number of auto drvers who use ther cars to access GO statons n the areas served by Route 11. For Route 21 these percentages translate nto 20.7, 31.05, and 41.4 requests for the mornng peak perod. Snce there are 5 bus runs n the mornng peak perod, the number of demand requests for each run was approxmated to 4, 6 and 8 requests respectvely for each bus run. For Route 25 the percentages result n 24.3, 36.45, and 48.6 requests for the mornng peak perod. For the 4 bus runs, the number of requests for each run was approxmated to 6, 10 and 12 requests, respectvely. For Route 11 the percentages used resulted n 18.45, 24.6 and 36.9 requests for the mornng peak perod. That s dvded nto 4, 6, and 8 requests for each bus run. 6. For each value of the slack tme nvestgated n the analyss, the smulaton tested dfferent dstrbutons of the slack tme for each route segment, where a route segment spans the dstance between two consecutve fxed stops. For example, f the slack tme assgned to the whole route was 5 mnutes, then the amount assgned to each route segment of the route ranged from 0 to 2 to try to capture the varaton of the number of requests n each route segment along the route. For example, for Route 21 whch has 4 fxed stops (5 route segments), the dstrbuton of an assgned slack tme of 5 mnutes

243 among the route segments could have the followng slack tme dstrbutons: (1, 1, 1, 1, 1), (1, 0, 2, 2,0), (2, 1, 1, 0,1), etc General Specfcatons 1. All servce vehcles are dentcal and each has unlmted vehcle capacty so that no request s rejected due to a vehcle capacty constrant. Ths assumpton s vald snce, at the current rdershp levels, most transt vehcles carry very low number of passengers on each run. 2. For each scenaro, the schedulng system was run 100 tmes, whch should provde statstcally relable estmates of the varous varables. An example of the output of the schedulng system for a smulaton run for Route 21 s shown n Fgure 6-4 for a demand rate of 6/run and 5 mn slack tme. In the remanng sectons of the expermental case study, the senstvty of the new Flex- Route servce to desgn parameters such as the added slack tme and the on-demand request rate are examned. The analyss wll also look at the cost nflcted by the new system on the fxedroute users as well as the transt servce provder as a result of the change of the servce type. At the end, the performance of each proposed Flex-Route system wll be compared to that of the orgnal fxed-route system to assess whether the change n the servce type mproved the performance of the route or not.

244 Fgure 6-4 Example of the produced map for route 21 (demand =6/run and slack tme = 5mn) 230

245 ANALYSIS OF DESIGN PARAMETERS Ths part of the analyss wll look nto how the amount of slack tme gven to a route and the on-demand request rate affects the number of accepted on-demand request, and thus ncreasng the rdershp on that route. The followng s a dscusson of each desgn parameter Effect of Slack Tme The effect of the amount of slack tme gven to each route wll be examned n ths secton to evaluate how slack tme may determne the number of accepted on-demand requests. The analyss wll be performed for each route separately Route Number 11 Table 6-4 and Fgure 6-5 show the number of accepted on-demand requests under several cases of on-demand request rates and slack tme values for Route 11. It should be noted agan that the demand rates and slack tme values are for a sngle bus run to smplfy the analyss. As shown n the Table 6-4, the number of accepted requests ncreases steadly as the amount of slack tme ncreases n all cases. Ths of course s to be expected snce the extra slack tme wll gve the bus more tme to pck up more passengers. In fact, even at the hghest value of slack tme (12 mnutes), there stll was a porton of the demand that was not accepted. For a demand rate of 4, only 3.45 requests (86.3 % of the demand) were accepted gven the slack tme of 12 mnutes, whle for the demand rates of 6 and 8 the number of accepted requests were 4.38 and 5.13 request (72.9 % and 64.1% of the demand respectvely) were accepted. Ths shows that effect of addng slack tme to ths route s moderate n ts ablty to add a sgnfcant number of on-demand requests. Ths can be attrbuted to the wdth of the servce area (500m n ths case), and the low operatng speed (20 km/hr), whch makes t dffcult to reach some requests gven the avalable slack tme.

246 232 8 Number of Accepted Requests Demand =4 Demand =6 Demand = Assgned Slack Tme (mn) Fgure 6-5: Relatonshp between slack tme and the number of accepted requests for Route 11 Table 6-4 Number and percentage of accepted requests for dfferent slack tme and demand rates for route 11 No. of Accepted % Accepted Route # Requests Requests Slack Tme Demand Rate Per One Bus Demand Rate Per One Bus (mn) Run Run Route

247 Route Number 21 Table 6-5 and Fgure 6-6 show the number of accepted on-demand requests under several cases of on-demand request rates and slack tme for Route 21. As shown n Table 6-5, the number of accepted requests ncreases steadly as the amount of slack tme ncreases n all cases as n the prevous case. However, one pont that should be notced here s that addng more slack tme for the same demand rate wll reach a pont where more slack tme wll result n mnmal ncrease n the number of accepted requests, especally when the accepted requests come close to the potental demand. For example, for an on-demand request rate of 4, ncreasng the slack tme from 5 to 10 mnutes for the whole route wll result n a 0.56 ncrease n the amount of accepted requests (3.40 to 3.96). Ths s due to the fact that there wll be nstances where some requests wll be hard to accept gven ther locatons n the zone relatve to other requests. It s also mportant to notce that the 5-mnute slack tme n ths case was suffcent to schedule 3.40 out of 4 possble requests, whch means that there were cases n whch all the 4 requests were accepted. The same can be sad for the case of a demand rate of 6, where ncreasng the slack tme from 7 to 10 mnutes only ncreases the number of accepted requests by 0.60 (5.22 to 5.82). A slack tme of 7 mnutes, enough to serve an average of 5.22 requests, wll have nstances n whch all 6 requests wll be accepted. Table 6-5 Number and percentage of accepted requests for dfferent slack tme and demand rates for route 21 No. of Accepted % Accepted Route # Requests Requests Slack Tme Demand Rate Per One Demand Rate Per One (mn) Bus Run Bus Run Route

248 Number of Accepted Requests Demand =4 Demand =6 Demand = Assgned Slack Tme (mn) Fgure 6-6: Relatonshp between slack tme and the number of accepted requests for Route Route Number 25 The pattern of Route 21 s also apparent for Route 25, where there s a value of the slack tme beyond whch the effect of addng more slack tme s mnmal. Fgure 6-7 shows the number of accepted requests as a functon of the amount of added slack tme for Route 25. As mentoned earler, the number of accepted requests ncreases steadly untl the effect of addng more slack tme starts to dmnsh at a certan value. As shown n Fgure 6-7, a slack tme of 3 mnutes allow for 1.38, 1.63 and 1.72 requests to be accepted for the demand rates of 4, 6 and 8 respectvely. For Route 11, the above-mentoned pattern s not vsble n the range of slack tme nvestgated n ths analyss. In fact, the effect of addng more slack tme to the route s mnmal from the begnnng. If we compare Route 11 wth Route 21, we can see clearly that the performance of Route 21 s markedly better than that of Route 11. For example, for a slack tme

249 235 of 10 mnutes, the number of accepted requests for the demand rates of 4, 6, and 8 for Route 11 are 3.13, 3.63, and 4.50, whle those values for Route 21 are 3.96, 5.82, and 7.46 respectvely. Ths can be explaned by the fact that the operatng speed of Route 11 s lower than that of Route 21, whch means that the bus n Route 21 can reach further requests n less tme than Route 11. Another reason could be that the spacng between the fxed stops s shorter n the case of Route 11 than the case of Route 21 and 25 as mentoned n Table 6-3. Shorter fxed-stop spacng means that the bus has less chance of reachng further requests because t has to be at the fxed stop at ts publshed departure tme. Table 6-6 Number and percentage of accepted requests for dfferent slack tme and demand rates Route # Route 25 for route 25 No. of Accepted Requests % Accepted Requests Slack Tme (mn) Demand Rate Per One Bus Run Demand Rate Per One Bus Run

250 Number of Accepted Requests Demand =6 Demand =10 Demand = Assgned Slack Tme (mn) Fgure 6-7: Relatonshp between slack tme and the number of accepted requests for Route Effect of On-Demand Request Rate Tables 6-4 to 6-6 and Fgures 6-4 to 6-6 also show the relatonshp between the demand-responsve request rate and number of accepted requests. The fgures show that, for the same value of slack tme, the number of accepted requests ncreases as the demand rate ncreases. Ths s apparent for all values of the slack tme under all demand rates. However, the percentage of accepted requests from the total number of requests s lower when the demand rate ncreases. Ths result could be very useful n the desgn of Flex-Route transt. Dependng on the goal of the transt agency, ths can be looked at from two perspectves. Frst, f the transt agency s goal s to accept a certan percentage of the demand, then t has to look at the percentages of accepted requests and fnd out the amount of slack tme that satsfes that percentage. However, f the slack tme s constant, then the more there s demand the larger s the number of accepted requests. For example, f the transt agency has a polcy that the slack tme should be a maxmum of 5

251 237 mnutes for Route 21 (Table 6-5), then the expected number of accepted requests wll be 3.40, 4.13 and 4.90 for the demand rates of 4, 6 and 8 respectvely. 6.6 FEASIBILITY ANALYSIS OF THE PROPOSED FLEX-ROUTE SERVICE Ths secton analyzes the addtonal cost nflcted by the Flex-Route servce on the transt servce provder as well as the fxed-route customers. The analyss wll be based on the cost analyss provded n Chapter 4. Ths secton wll also look nto the performance of each proposed Flex-Route system and how t compares to the performance of the orgnal fxed-route system. Ths step wll help n assessng whether the change n the servce type mproved the performance of the route or not, and thus determnng the feasblty of the servce Servce Provder Cost In case the transt servce provder decdes to mantan the same servce frequency as before ntroducng the Flex-Route servce, ths may result n an ncrease n the number of transt vehcles needed to operate the servce. As shown n equaton 3-26, ths ncrease s equal to (2S/H) as follows: K n 2 S K H (Equaton 3-26) Where: K = Number of transt unts needed for the servce n the orgnal fxed-route settng; n K = New number of transt unts needed for the servce after the ntroducton of Flex-Route servce; H = Orgnal servce headway of the servce (n mnutes); S = the assgned slack tme to the entre route for one vehcle run n one drecton (n mnutes; Equaton 3-26 shows that the new number of transt vehcles needed for the new servce wll ncrease by a fracton that depends on the slack tme and orgnal headway.

252 238 Therefore, we can use equaton 3-26 to fnd the fractonal ncrease n the number of transt vehcle for each value of slack tme. For example, gven that the average servce headway of route 11 s 20 mnutes, the fractonal ncrease n the number of transt vehcles needed for the new servce ranges from 0.3 for a 3-mnute slack tme to 1.2 for a slack tme of 12 mnutes. It should me noted that ths ncrease can be absorbed f the servce provder has extra transt vehcles or extra termnal tme that can be used to cover ths ncrease. In terms of the extra operatng costs, the servce provder wll ncur an ncrease n the vehcle hours and vehcle mles due to the devatons to pck up and drop off ondemand customers. Ths cost can be found usng equaton 3-32 for a rectangular servce area as follows: N 1 Ckm O 2 cos 3 1 t Cdr S W m.. (Equaton 3-32) 1 However, ths equaton s not practcal snce the second term s for a rectangular servce area. Nevertheless, the second term of the equaton represents the ncrease n vehcle mles, whch was found n the smulaton process for each case. Therefore, we can replace the second term of the equaton as follows: O C S C D cos t dr km ext.. (Equaton 6-1) Where: O cost : s the cost ncurred by the servce provder due to the change of the servce from fxed-route to Flex-Route. S: s the slack tme added to the whole route C dr : s the cost coeffcent representng the margnal hourly cost of vehcle operatons -wage of bus drvers ($$ / Hour). C km : s the cost coeffcent representng the margnal cost of vehcle mleage gasolne ($$ / Km).

253 239 run. D ext : s the extra dstance traveled to serve the on-demand requests for a sngle bus Usng equaton 6-1, the operatng cost ncurred by the servce provder can be found usng approprate values for the cost coeffcents. Note that n ths study we are not ncludng the fxed-route cost ncurred by the servce provder snce t wll be dffcult to estmate ths effect on the servce provder n the long run, whch s beyond the scope of ths study Fxed-Route User Cost Unused Slack Tme Fgures 6-8 to 6-10 show the relatonshp between the assgned slack tme and the amount of slack tme that was not used after schedulng the accepted requests. Ths amount of unused tme wll go as extra watng tme for the bus at the fxed stops. Ths amount of extra watng tme should be carefully looked at, snce watng n the bus at the fxed stops wthout a reason could become a potental cause of dsutlty among fxedstops passengers. The three fgures show that the amount of unused slack tme ncreases as the value of added slack tme ncreases. Ths of course s expected because although ncreasng the amount of assgned slack tme to the route ncreases the number of accepted requests, ths ncrease s not enough to consume the ncreased amount of slack tme. As shown n the fgures, for low values of the assgned slack tme, the amount of unused slack tme ncreases at a low rate. However, when the amount of assgned slack tme ncreases to hgher values, the amount of unused slack tme starts ncreasng at hgher rates, especally n the cases of lower demand rate (.e. demand rate of 4). Ths of course s related to the number of accepted requests as dscussed earler, where at hgher values of slack tme the ncrease n the number of accepted requests s mnmal, thus resultng n hgher values of unused slack tme. Another observaton from the fgures s that for the same amount of assgned slack tme, the amount of unused slack tme decreases when the demand rate ncreases. Ths s also related to the number of accepted requests dscussed earler. For hgher rates of demand under the same value of slack tme, the number of accepted requests s hgher,

254 240 thus ncreasng the amount of slack tme consumed and lowerng the amount of unused slack tme. Ths could be very nstrumental n decdng how much slack tme to allocate to a gven route. It s mportant to allocate enough slack tme for requests that mght be placed n real tme. Provdng enough slack tme wll allow for more requests to be answered and thus more satsfacton to the customers. On the other hand, provdng excessve slack tme wth very low demand could result n dssatsfacton and dscomfort for fxed stop passengers as the vehcle wll arrve earler at the fxed stop, but t has to stay at the stop untl ts departure tme. So t s a trade-off between the two groups of people, and a careful study should be made to determne the expected demand rate Average Travel Tme The second performance measure used n ths study s the average travel tme for fxed-route customers under several cases of Flex-Route servce and the exstng fxedroute servce. Table 6-7 shows the results of the average travel tme for Route 11. The table shows that ncreasng the amount of slack tme ncreases the average travel tme for fxed-route customers as expected. However, the ncrease n average travel tme s not substantal; for example, for a demand rate of 4, the ncrease n average travel tme ranges from 1.01 mnutes for a slack tme of 3 mnutes to 4.23 mnutes for a slack tme of 12 mnutes. For a demand rate of 6, the ncrease n average travel tme ranges from 0.91 for a slack tme of 3 mnutes to 4.02 mnutes for a slack tme of 12 mnutes. However, t s mportant to menton that ths ncrease n average travel tme s not dstrbuted equally among fxed-stop passengers, snce some of them who board the bus at earler stops n the route wll face hgher average travel tme than those who board the bus at later stops, but ths of course s to be expected n such servce.

255 241 8 Unused Slack Tme (mn) Demand =4 Demand =6 Demand = Assgned Slack Tme (mn) Fgure 6-8: Relatonshp between slack tme and the unused slack tme for Route Unused Slack Tme (mn) Demand =4 Demand =6 Demand = Added Slack Tme (mn) Fgure 6-9: Relatonshp between slack tme and the unused slack tme for Route 21

256 242 Unused Slack Tme (mn) Demand =6 Demand =10 Demand = Assgned Slack Tme (mn) Fgure 6-10: Relatonshp between slack tme and the unused slack tme for Route 25 Table 6-7 Average travel tme for fxed-stop passengers for dfferent slack tme and demand values for route 11 Route # Slack Tme (mn) Average Travel Tme (mn) Demand Rate Per One Bus Run Route

257 243 Table 6-8 Average travel tme for fxed-stops passengers for dfferent slack tme and demand values for route 21 Route # Slack Tme (mn) Average Travel Tme (mn) Demand Rate Per One Bus Run Route Table 6-9 Average travel tme for fxed-stops passengers for dfferent slack tme and demand values for route 25 Route # Average Travel Tme Slack Tme (mn) (mn) Demand Rate Per One Bus Run Route

258 Flex-Route Performance vs. Fxed-Route Performance To test the performance of Flex-Route transt servce compared to the exstng fxed-route servce, the passenger per revenue vehcle hours (PVRH) performance measure was used to compare the rdershp performance of the proposed Flex-Route servce wth that of the exstng fxed-route system. Fgure 6-11 shows the PVRH results for Route 11 under several cases of on-demand request rate and assgned slack tme, ncludng the case of 0 slack tme (.e. the exstng fxed-route case). Fgure 6-11 shows that the performance of the exstng fxed-route servce, n terms of PVRH, s better than any case of Flex-Route desgn. In fact, the more slack tme added to the servce, the lower the value of PVRH becomes. As such, f the PVRH was used as the sole rdershp evaluaton crteron, then Flex-Route servce s not an effectve opton for Route 11. The reason behnd the low PVRH value for the Flex-Route cases s that, gven the current fxed-route rdershp, the operatng speed and the on-demand requests rate and dstrbuton, the relatve ncrease n rdershp from the on-demand rdershp s not large enough to allevate the slack tme that wll be added to the vehcle revenue hours. Ths concept was explaned n Chapter 3 where t was shown that for the PVRH of the new servce to exceed that of the exstng servce, the followng condton should be met: R add R o S (Equaton 3-75) T o Where: R o : s the exstng fxed-route rdershp; R add : s the addtonal rdershp from the on-demand requests; T o : s the exstng scheduled bus runnng tme of the fxed-route servce; and S: s the slack tme added to serve the on-demand requests. For example, f we look at Table 6-4, for a demand rate of 4 and a slack tme of 3 mnutes, the number of accepted requests s If we try to use equaton 3-75, then:

259 245 Whle R add R S T o o Usng these numbers, equaton 3-75 does not hold; therefore the PVRH of the Flex-Route servce s less than that of the exstng fxed-route servce. Ths result s true for all cases of Route 11, whch means that the Flex-Route servce rdershp numbers do not warrant usng Flex-Route transt for Route 11. For Route 21 (Fgure 6-12), the results show that, for the dfferent demand rates, the PVRH value under Flex-Route servce s hgher than that of the fxed-route servce for almost all the slack tme cases analyzed n ths study. However, the PVRH value starts ncreasng as the slack tme ncreases up to a maxmum performance pont, at whch the Flex-Route PVRH performance s at ts hghest value under a specfc demand rate. After ths pont the PVRH value, although stll above that of fxed-route, starts decreasng for hgher values of slack tme untl t crosses the fxed-route PVRH lne for very hgh values of slack tme. In the case of a demand rate of 4, the Flex-Route PVRH lne falls below that of the fxed-route lne at a slack tme value of 8 mnutes, whch means that the performance of Flex-Route n terms of PVRH becomes worse than that of the fxed-route for slack tme values above 8 mnutes for a demand rate of 4. For the demand rates of 4, 6 and 8, the maxmum PVRH for Route 21 occurs at a slack tme of 4, 6 and 8 mnutes, respectvely. Concdently, these values are the same values for the demand rate. Gong back to Table 6-5 the slack tme values of 4, 6, and 8 mnutes resulted n the followng numbers of accepted requests: 2.9, 4.79 and 5.9 respectvely, or 72.5%, 79.8% and 90% of the total demand for each case, respectvely. Ths ndcates that these values of slack tme result n at least 75% of the demand beng accepted whle at the same tme gettng the hghest PVRH performance for the Flex- Route servce. Route 25 has the same pattern as Route 21, where the PVRH of Flex-Route servce s generally hgher than that of the fxed-route servce, and also has a maxmum

260 pont of PVRH before t falls down. For the demand rates of 4, 6 and 8, the maxmum PVRH occurs at 5, 9 and 10 mnutes Passenger Per VRH Demand =4 Demand =6 Demand =8 Fxed-Route PVRH Assgned Slack Tme (mn) Fgure 6-11: Relatonshp between slack tme and the passenger per VRH for Route 11

261 Passenger Per VRH Demand =4 Demand =6 Demand =8 Fxed-Route PVRH Assgned Slack Tme (mn) Fgure 6-12: Relatonshp between slack tme and the passenger per VRH for Route Passenger Per VRH Demand =6 Demand =10 Demand =12 Fxed-Route PVRH Assgned Slack Tme (mn) Fgure 6-13: Relatonshp between slack tme and the passenger per VRH for Route 25.

262 SUMMARY AND CONCLUSIONS Ths chapter explored varous elements of mplementng Flex-Route transt servce n low-densty suburban areas n the Greater Toronto Area, Canada. The case study nvestgated the feasblty and performance of Flex-Route servce f t were to replace exstng low-performance fxed-routes n the Cty of Oakvlle, Canada. The results show that ncreasng the slack tme would ncrease the number of accepted requests but wll result n longer travel tmes and addtonal unused slack tme. A sgnfcant drop n the level of servce for fxed-route customers may lead to a drop n the number of fxed-route customers and deteroraton of the servce. The results also show that ncreasng the fxed-stop spacng wll result n ncreasng the number of accepted requests snce the vehcle wll have more tme to navgate through the servce area to pck up more customers. Ths ncrease n stop spacng would be an opton n case the demand at a specfc fxed-stop s very low that ths stop could be removed. The analyss shows that the performance of Flex-Route and fxed-route servces depends on several parameters such as slack tme, demand rate and the number of accepted requests. The most mportant concluson s that for the performance of Flex- Route servce to surpass that of the fxed-route servce n terms of PVRH, the ncrease n rdershp from the demand-responsve requests should be enough to allevate the effect of the assgned slack tme. The results show that the hgher the fxed-stop rdershp, the less lkely that Flex-Route servce performance wll be better than the exstng fxed-route servce. Ths result of course s based on the assumpton that the fxed-route demand s nsenstve to the change of the servce to Flex-Route servce, and that the fxed-stop customers have no opton but to use transt.

263 249 7 CONCLUSIONS AND RECOMMENDATIONS 7.1 SUMMARY AND CONCLUSIONS The objectve of ths research s to advance the applcaton of Flex-Route transt servce by provdng a thorough nvestgaton of the plannng, desgn and schedulng of the servce. Accordngly, the research was categorzed nto two parts: Identfyng the basc servce desgn parameters and servce-specfc characterstcs requred n the plannng and desgn process of Flex-Route servce, whle also nvestgatng the feasblty and cost of provdng the servce. Buldng an advanced schedulng system that can handle the daly operatons of Flex-Route servce, through the use of recent developments n advanced technology. In ths regard, the research focused on: dentfyng the ratonale and motvaton behnd Flex-Route attractveness and sutablty for suburban transt servce; recognzng the unqueness and servce-specfc characterstcs of the servce; dentfyng the desgn parameters and desgn gudelnes of the servce; examnng the feasblty and cost of provdng the servce; revewng the exstng schedulng algorthms used n schedulng Flex-Route servce; and developng a schedulng systems that explots recent advancements n computer and communcaton technologes, and other topcs. In Chapters one and two, the hstory and reasons behnd the ntroducton of flexble transt servces are revewed, wth specal attenton gven to Flex-Route servce. The plannng, desgn and performance of these servces as well as the feasblty of Flex- Route servce as a vable opton for transt servce n suburban areas were revewed to dentfy the crtcal servce-specfc and desgn parameters that mght have sgnfcant effect on the plannng and desgn processes. Ths revew gave a basc understandng of the necessary plannng process needed for Flex-route servce to succeed, and helped n formulatng a lst of some of the major servce desgn factors that should be addressed f the servce s to be mplemented. These desgn factors ncluded among other factors, slack tme, servce area, fxed-stop spacng.

264 250 Chapter 3 amed at creatng detaled plannng and desgn gudelnes for Flex- Route transt servce by developng an analytcal model. The chapter emphaszes and documents the dfferences between urban and suburban travel patterns as well as the dfferences between conventonal fxed-route transt and Flex-Route transt. Ths understandng s essental to develop adequate gudelnes for a successful Flex-Route transt servce. The chapter then addresses the desgn and operatons of Flex-Route transt by developng an analytcal model to estmate desgn factors such as the optmal slack tme and optmal servce area wdth. Ths analytcal model s a helpful tool n estmatng such values as optmal slack tme, optmal servce area wdth, and requred servce frequency changes among other factors. The analyss uses a cost functon that ncorporates the costs/benefts of the servce type change affectng the man stakeholders (.e. the transt operator, fxed-route users and on-demand customers). Chapter 4 focused on developng a new Flex-Route schedulng system. The developed schedulng system s ntended to be one component of a larger decsonsupport system (DSS) that can be used n real-world operatons of Flex-Route transt servce. In ths research, however, only a blueprnt of the system was dscussed, as buldng an actual DS system s beyond the scope of ths research. The schedulng methodology ncluded buldng a schedulng algorthm that ncludes a clear and concse representaton of all the varables and constrants of the servce. The transt vehcles, the road network, the fxed stops and the on-demand requests are all modelled n the schedulng system, ncludng all types of constrants. A new schedulng algorthm (the Constraned-Inserton Algorthm) s developed to schedule Flex-Route servce, and to produce solutons that can be mproved usng local search and Constrant Programmng algorthms. Two schedulng algorthms are developed to deal wth both the statc and dynamc verson of the Flex-Route problems. In each case, a representaton of all customers as well as the servce operator s ncluded n the cost functon to produce optmal schedules/solutons. Chapter 5 presented the practcal mplementaton of the analytcal model on a hypothetcal case study to examne the valdty, applcablty and performance of both the schedulng system and the analytcal model, developed n Chapters 3 and 4 respectvely. The analyss focused on the mpact of dfferent nput and desgn parameters on the

265 251 schedulng results and the ablty of the schedulng system to reflect these mpacts. The analyss provded practcal evdence on the valdty of the proposed analytcal model as well as the ablty of the schedulng system to produce the expected results. The results of the senstvty analyss show that ncreasng the slack tme wll ncrease the number of accepted on-demand requests but wll result n longer travel tmes and addtonal unused slack tme. Ths means a sgnfcant drop n the level of servce for fxed-route customers and may lead to a drop n the number of fxed-route customers and deteroraton of the servce. The results also show that ncreasng the fxed-stop spacng wll result n ncreasng the number of accepted requests snce the vehcle wll have more tme to navgate through the servce area to pck up more customers. Ths ncrease n stop spacng would be an opton n case the demand at a specfc fxed-stop s very low that ths stop can be removed. The results also show that the effect of relaxng the arrval tme of the vehcle at some fxed stops has a sgnfcant effect on the number of accepted on-demand requests and on the overall level of servce. However, late arrval s not recommended for all fxed stops especally at major transfer ponts where connectons protecton s requred. Allowng late arrval at the fxed stops would result n lower cycle tme, less slack tme requred, less unused slack tme, less passenger-mnutes lost due to early arrval at the fxed-stops, and lower average travel tme. The chapter concludes that usng an approprate slack tme s essental n havng an acceptable level of servce for all customers. Relaxng the fxed-stop constrants wll allow the servce provder to assgn less slack tme to the servce, and hence less operatng cost, whle at the same tme mantanng an acceptable level of servce for both fxed-route and demand-responsve customers. Chapter 6 documented an attempt to examne the applcablty, feasblty and performance of Flex-Route transt servce as a vable replacement of low-demand fxed transt routes n the suburban areas of the GTA. The results of ths applcaton demonstrated the valdty of both the proposed schedulng and analytcal models. The results of the analyss are consstent wth those produced n Chapter 5 n terms of the effect that each desgn and nput factor has on the servce.

266 252 The analyss ncluded a comparson between two operatonal scenaros for three exstng fxed bus routes to nvestgate the feasblty of Flex-Route transt servce. The two scenaros are: mantan the same fxed-route operatons on the three routes, or change the operatons of the routes to Flex-Route transt servce. The results show that the performance of Flex-Route and fxed-route servces depends on several parameters such as the amount of slack tme allocated to each route, on-demand request rate and the number of accepted requests. The most mportant concluson s that for the performance of Flex-Route servce to surpass that of the fxed-route servce, the ncrease n rdershp from the on-demand requests should be enough to allevate the effect of the ncrease n the bus operatng tme resultng from the assgned slack tme. The results show that the hgher the fxed-stop rdershp, the less lkely that Flex-Route servce performance wll be better than the exstng fxed-route servce. Ths result of course s based on the assumpton that the fxed-route demand s nsenstve to the change of the servce to Flex- Route servce, and that the fxed-stop customers have no opton but to use transt. 7.2 CONTRIBUTIONS The contrbuton of ths research can be broken nto two man categores Developng a Flex-Route analytcal model Experence from flexble transt provders (Rosenbloom 1996, Koffman 2004) showed that most flexble servce provders do not follow a systematc manner of plannng and desgnng ther Flex-Route servces. To tackle ths ssue, Chapter 3 creates a detaled analytcal model that addresses most of the plannng, desgn and operatonal ssues of Flex-Route transt servce. The model ncorporates several nput parameters and desgn factors to provde a detaled model that can be referred to n the plannng and desgn of Flex-Route transt servce. Based on the nput parameters used, the model can be used to produce optmal slack tme values and optmal servce area values. The model also looks nto the mpact of the servce on several stakeholders nvolved n Flex-Route servce ncludng regular fxed-route customers, on-demand customers and the servce provder. The model provdes clear and drect relatonshps that can be used to estmate the cost ncurred by these stakeholders. These relatonshps can be benefcal n assessng the feasblty and cost of replacng fxed-route transt wth

267 253 Flex-Route transt servce. Furthermore, the analyss ncludes nvestgatng several strateges that can be used to allevate the cost ncurred on fxed-route users by the change n servce type. The analyss looks nto two strateges: servce frequency mprovements and fare reducton strateges, and provde formulas that estmate the cost ncurred by the fxed-stop users and the amount of servce frequency mprovement and fare reducton needed to allevate ths cost. The model also presents methods for montorng the performance of Flex-Route transt servce snce transt planners usually am at mprovng the performance of the Flex-Route servce, especally when comparng Flex-Route transt to fxed-route servce. The decson on what types of performance measures should be used depends on many factors such as the transt agency goals, the orgnal level of servce for users, and the orgnal rdershp. In ths work, we use two measures that are prmarly related to the performance of the transt servce n terms vehcle hours and vehcle klometres, namely passenger boardngs per revenue vehcle hour (PVRH), and passenger boardngs per revenue vehcle Klometre (PVRKm). These two methods can be used n addton to the cost and feasblty analyss as mentoned earler. Usng such measures can help decsonmakers n ther assessment of how the performance of the Flex-Route servce compares to that of an exstng fxed-route servce. To examne the applcablty, feasblty and performance of the Flex-Route analytcal model developed n Chapter 4, a real-world scenaro was analyzed to evaluate the feasblty of Flex-Route transt servce to replace fxed-route transt servce n suburban areas. The case study nvestgates the possblty of replacng fxed-route transt routes n the suburbs of the Greater Toronto Area wth Flex-Route servce. The results of ths applcaton demonstrated the valdty of the analytcal model. One mportant concluson of ths case study s that for the performance of Flex-Route servce to surpass that of the fxed-route servce, the ncrease n rdershp from the demand-responsve requests should be enough to allevate the effect of the assgned slack tme.

268 Buldng a Schedulng System to support the operatons of Flex-Route transt servce. The lterature revew conducted n ths research revealed that most flexble transt servces are scheduled and dspatched wthout the use of specalzed schedulng algorthms for Flex-Route transt or the use of advanced technology for real-tme operatons. The few transt provders who have schedulng systems use schedulng systems that are desgned for demand-responsve servces, whch reveals the need for a specalzed schedulng system for Flex-Route transt servce. Chapter 4 tres to address ths problem by ntroducng a new real-tme decson support and schedulng system (REFLEX) for the operatons of Flex-Route transt servce. The decson-support system ncludes ready-to-use schedulng mechansm for Flex-Route servce whch enables transt agences to produce relable and cost-effectve schedules both n statc and dynamc envronments. The proposed schedulng and routng methodology s developed usng a new schedulng algorthm the Constraned Inserton Algorthm- that uses the powerful search technques of Constrant Programmng. Fxedroute transt schedules and routes and ndvdual requests for travel are nput nto the schedulng algorthm to determne the optmal tnerares and schedules. The schedulng algorthm s able to handle cases of statc and dynamc Flex-Route operatons. In realtme envronments, the schedulng algorthm s able to produce schedules n a very short tme when a real-tme request s receved. A model of the system s developed n a Geographc Informaton System (GIS) platform ncludng bus routes and tmetables. Indvdual on-demand requests are nput to the system. The decson support system blueprnt makes use of advanced communcaton technologes to boost the applcablty of Flex-Route servce n terms of producng the most cost-effectve, relable, real-tme schedules that can enhance the productvty and performance of Flex-Route servce. Such technologes nclude Automatc Vehcle Locaton (AVL) and Moble Data Termnals (MDT). The ablty of the schedulng algorthm to produce cost-effectve schedules was evaluated n Chapter 5. Usng a hypothetcal case study, a comparatve analyss of the results produced by the schedulng methodology developed n ths research and the results obtaned usng the Inserton Algorthm was conducted. The results showed that

269 255 the schedulng methodology developed n ths research produced much better results than the Inserton Algorthm, whch valdates the schedulng system s ablty to produce very cost-effectve schedules. Moreover, on the practcal sde, the schedulng system could also work as an expermental tool for transt agences to analyze the mpact of provdng Flex-Route servces and ther expected performance before mplementaton. In some cases, the transt servce provder may need to study the possble mpacts of ntroducng Flex-Route servce before mplementaton. For example, the servce provder may want to choose whch fxed routes are to be replaced by Flex-Route transt servce out of several avalable optons. Conductng a smulaton on these routes wll help the decson-maker have better understandng of the possble mpact and performance of the servce. 7.3 RECOMMENDATIONS FOR FUTURE RESEARCH Whle ths research has tred to nvestgate several aspects of Flex-Route transt servce, there are stll a number of ssues that need to be addressed n ths area that can potentally be a motvaton for future research. The followng are recommendaton for some areas that can be explored n future research: 1. In ths research, the focus of the study was to study the provson of an enhanced transt servce to attract the general publc to use transt, whle overlookng any requred demand-responsve servce of the elderly and dsabled. Future research can explore the feasblty of ncorporatng both servces n Flex-Route servce and examne whether such servce can be feasble wthout compromsng the level of servce requred for all customers. 2. In ths research, the feasblty study s focus was placed on the case where Flex- Route transt s replacng an exstng fxed-route transt servce n a suburban transt area. However, Flex-Route transt servce can also be ntroduced as a new servce n an area that dd not have any knd of transt servce before. Ths area can be addressed n future research so as to develop plannng and desgn gudelnes that can study areas such as the mnmum level of demand requred for such servce to succeed; at what level of demand the servce can be changed to

270 256 fxed-route; and the feasblty of multple routes of Flex-Route servce n a specfed area. 3. The decson support system bult n ths thess can be used for real-world operatons of Flex-Route transt servces. It has the ablty to provde daly and real-tme schedules and routes of the servce. However, n ths thess we only developed the schedulng system and the GIS platform needed to provde a vsual representaton of the servce. Future research can buld on ths research to try to develop the other components of the DSS system, namely buldng a web-based system that can be lnked to both the GIS platform and the schedulng system to llustrate the applcablty of the system to real world applcatons. 4. In ths work, we used two performance measures to study the feasblty of Flex- Route transt, namely passenger boardngs per revenue vehcle hour (PVRH), and passenger boardngs per revenue vehcle Klometre (PVRKm). The decson on what types of performance measures should be used depends on many factors such as the transt agency goals, the orgnal level of servce for users, and the orgnal rdershp. Thus future research can explore the feasblty of Flex-Route transt usng other performance measures, ndependently or by developng a more comprehensve performance ndcator usng several measures such as: operatng cost per revenue vehcle hour, operatng cost per passenger boardng, operatng cost recovery rato, passenger boardngs per revenue vehcle hour, and passenger boardngs per revenue vehcle mle. 5. The focus of ths research was on the feasblty of Flex-Route transt servce n suburban areas. Future research can explore the feasblty of usng Flex-Route servce as the entre transt servce for a small cty or rural area. In these cases, consoldaton wth paratranst servce can be a key feature of the servce. 6. Flex-Route transt s ntended to serve lmted portons of a large servce area and provde connectons wth a regonal network. As a result, schedulng needs to allow suffcent tme to provde relable transfers. Future research could study the benefts/costs of such coordnaton on the customers as well as the operator to determne what levels of synchronzaton and coordnaton could be benefcal.

271 One advantage of Flex-Route transt servce s that t has the ablty to attract new transt passengers, who otherwse mght use ther cars for ther trps. Future research can explore the envronmental benefts of such type of servce n terms of the GHG emssons, vehcle-mles travelled among other measures. 8. There are many dfferent structures and forms of flexble transt servces that combne elements from conventonal fxed-route servces and demand-responsve servces. Flex-Route transt servce s only one type of these servces. Future research can explore other flexble servces n the same manner as n ths research to study the plannng, desgn and schedulng aspects of these servces. Other research can focus on explorng whch types of flexble servce are approprate for varous land use and demand patterns. 9. As mentoned n Chapter 3, there s a need for a specal study to estmate approprate values for the cost coeffcents used n the analytcal model n Chapters 3. The estmaton of the values of these coeffcents s requred to conduct any Flex-Route feasblty study. 10. Throughout the analyss we assume that fxed-stop users wll nether convert to on-demand servce nor they wll abandon the fxed-route servce. Addressng the changes n fxed-stop user behavour can be accomplshed through rdershp elastcty analyss, stated-preference surveys and mode choce models n future research on the demand part of the servce.

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