Copyrght by Satsh V S K Ukkusur 2005
The Dssertaton Commttee for Satsh V S K Ukkusur certfes that ths s the approved verson of the followng dssertaton: Accountng for Uncertanty, Robustness and Onlne Informaton n Transportaton Networks Commttee: S. Travs Waller, Supervsor Chandra R. Bhat Mchael C. Walton Davd P. Morton Srnvas Peeta
Accountng for Uncertanty, Robustness and Onlne Informaton n Transportaton Networks by Satsh V S K Ukkusur, B.Tech.; M.S. Dssertaton Presented to the Faculty of the Graduate School of The Unversty of Texas at Austn n Partal Fulfllment of the Requrements for the Degree of Doctor of Phlosophy The Unversty of Texas at Austn August, 2005
Dedcaton To my brother Ramesh U V Chandra who always told me, Learn more, be deal
Acknowledgements An endeavor near complete My debts are manfold and varous: Frst, Travs who took me under hs strde Nurtured my speedy PhD wth copous prase Oh, what a rde! Second, my commttee members Srn, Chandra, Mke and Dave Sharpened my thnkng wth ther astute nsghts And all the advce, grats they gave Thrd, to my research group, frends and UT colleagues who Helped me n remanng sane When the gong was aganst the gran Fourth, to my brothers Madhu and Ramesh For refuge, frendshp, ears and love And, fnally to my parents and grandmother Who backed my brashness, verve and vm Wth ther frm support, love, compasson and belef n hm v
Accountng for Uncertanty, Robustness and Onlne Informaton n Transportaton Networks Publcaton No. Satsh V S K Ukkusur, Ph.D. The Unversty of Texas at Austn, 2005. Supervsor: S Travs Waller Transportaton equlbrum problems wth determnstc forecasts of O-D demand yeld unsatsfactory results. Accurate estmaton of transportaton network performance helps n mprovng network reslency and reducng network-wde congeston. Accountng for uncertanty and rsk n transportaton networks facltate effcent evaluaton and desgn of transportaton networks and ths has emerged as a recent topc of nterest. Central problems n ths area are quantfyng network performance, desgnng robust networks, and modelng nformaton recourse to optmze the performance of transportaton systems. In ths dssertaton, dfferent approaches for evaluatng network performance under stochastc orgn-destnaton (OD) demand condtons are presented. Specfcally, two fundamentally dfferent approaches - analytcal expressons and sngle pont approxmatons - for evaluatng transportaton network performance under uncertan demand are dscussed. Computatonal results on multple v
transportaton test networks demonstrate the beneft of ncorporatng demand uncertanty n the model. A natural extenson of the stochastc network evaluaton model s the robust network desgn model. Ths model determnes lnk mprovement polces for the network consderng not only the expected network performance but also ts volatlty under a budget constrant. A soluton procedure based on a multobjectve evolutonary algorthm that computes the hgh performance network desgns for a stochastc objectve functon s dscussed. Computatonal results for the robust network desgn problem demonstrate the value of ncorporatng robustness. Accountng for dynamcs and stochastcty based on user equlbrum condtons are studed by developng a lnear programmng based network model where the cell transmsson model s used as the embedded traffc flow model. Computatonal results from ths model are demonstrated. Fnally, nformaton recourse s proposed as one potental strategy for mtgatng transportaton network uncertanty. An onlne equlbrum model where travelers have the ablty to take recourse enroute s developed as a fxed pont formulaton. A heurstc soluton approach based on the method of successve averages (MSA) s proposed to solve ths problem. Key fndngs from ths problem relate to studyng the beneft of onlne nformaton provson as compared to off lne network equlbrum problems. Further, opportuntes for usng these methodologes n other areas, and open problems of nterest n ths area are dscussed. In the overall, ths research s envsoned as an mportant frst step n the development of fundamentally new network assgnment models that account for uncertanty, robustness and nformaton recourse n stochastc transportaton networks. v
Table of Contents Table of Contents... v Lst of Tables... x Lst of Fgures...xv Part I Prelmnares...1 Chapter 1 Introducton...2 1.1 Introducton...2 1.2 Motvatng Examples...4 1.2.1 Long Range Network Performance...4 1.2.2 Onlne Informaton Provson as Recourse (UER)...6 1.3 Contrbutons...9 1.4 Dssertaton Organzaton...10 1.4.1 Presentaton Overvew...10 1.4.2 Dependences...11 1.4.3 Bblographc Informaton...11 Chapter 2 Crtcal Evaluaton and Comparson to Prevous Work...12 2.1 Sources of Uncertanty n Transportaton Decson Makng...12 2.2 Uncertanty Modelng n Transportaton...15 2.3 Revew of Traffc Equlbrum Models...18 2.3.1 Statc Traffc Equlbrum...18 2.3.2 Dynamc Traffc Assgnment...23 2.3.3 Network Desgn...25 2.3.4 Heurstc Soluton Approaches for Dscrete NDP (DNDP)...28 2.4 Revew of Soluton approaches for dealng wth uncertanty...33 2.4.1 Stochastc Programmng...33 2.4.2 Robust Optmzaton...36 2.5 Revew of Onlne Equlbrum Problems...37 2.6 Summary...40 Part II Evaluaton and Desgn under Uncertan Demand...41 v
Chapter 3 Two Methodologes for Evaluatng Network Performance under Stochastc O-D Demand...42 3.1 Introducton and Problem Defnton...42 3.2 Methodology 1: Sngle Pont Approxmatons...44 3.2.1 Approxmaton of Uncertan Demand by a Sngle Pont Estmate...46 3.2.2 Expermental Setup...50 3.3 Results...51 3.4 Methodology 2: Analytcal Expressons for Network Performance Experencng Uncertan Demand...54 3.4.1 Our Contrbutons...54 3.4.2 Proposed Model...55 3.4.3 Plannng Demand and Performance Measure...56 3.5 Mathematcal Calculatons of Network Performance...57 3.5.1 Dervaton of Path and Lnk travel tme Uncertantes...58 3.5.2 Computng Network Robustness...63 3.6 Numercal Experments...66 3.6.1 Results...69 3.6.2 Key Insghts...72 3.7 Concludng Remarks...73 Chapter 4 Desgnng Robust Networks for Improved Performance...76 4.1 Introducton...76 4.2 Robust Transportaton Network Desgn Model...78 4.2.1 Robust Optmzaton (RO)...78 4.2.2 Choce of Robustness Measure...80 4.3 RNDP Model formulaton...81 4.3.1 Notaton...81 4.3.2 The Lower-level User Equlbrum Assgnment under Uncertan Demand...82 x
4.3.3 Motvatng Example...84 4.4 Evolutonary Algorthm for the Robust Network Desgn Problem...86 4.4.1 Overvew of GA...88 4.4.2 Codng...89 4.4.3 The Objectve Functon and Constrant Satsfacton...90 4.4.4 The Reproducton Mechansm...90 4.4.5 The Crossover Mechansm...91 4.4.6 The Mutaton Mechansm...91 4.4.7 The Convergence Crteron...92 4.4.8 The ntal populaton...92 4.4.9 Proposed Algorthm for RNDP usng GA...92 4.5 Computatonal Results...95 4.5.1 Test Networks and Parameters...96 4.5.1.1 Experment 1...96 4.5.1.2 Experment 2...96 4.5.1.3 Experment 3...98 4.6 Evaluaton of GA Network Desgn solutons...104 4.7 Concludng Remarks...105 Part III Accountng for Dynamcs...107 Chapter 5 Dynamc Stochastc Network Desgn Model...108 5.1 Introducton...108 5.2 Embedded Traffc Flow Model The Cell Transmsson Model...110 5.2.1 Basc Prncples of CTM...110 5.2.2 Network Representaton...111 5.2.3 Modelng Transportaton Networks usng CTM...116 5.3 The Model: Problem Formulaton...118 5.3.1 Dynamc System Optmal Network Desgn Problem (DSO NDP) Formulaton...118 5.3.2 Dervaton and Interpretaton of the cost vector M t...119 x
5.3.3 Dynamc User Optmal Network Desgn Problem (DUO NDP) Formulaton...120 5.3.4 Dual Formulaton of the DUO NDP...122 5.4 Computatonal Results for the Dynamc Network Desgn problem...126 5.4.1 Results from DUO NDP...126 5.4.2 Comparson of the DUO and DSO NDP...129 5.4.3 Implementng the DUO NDP on the Nguyen Dupus network...130 5.5 Comparson and dscusson of results...133 5.6 Extenson of the DUO NDP Models: Accountng for Stochastcs...140 5.6.1 Chance constraned DUO NDP model...141 5.6.2 Two Stage DUO NDP Recourse model...143 5.7 Concludng Remarks...148 Part IV Copng wth Uncertanty: Onlne Informaton as Recourse...150 Chapter 6 A stochastc traffc equlbrum model wth nformaton recourse...151 6.1 Introducton...151 6.2 Problem Formulaton...152 6.2.1 Problem Defnton...152 6.2.2 UER Formulaton...155 6.2.3 UER Demonstraton...157 6.3 UER Paradox...159 6.4 Soluton Algorthm and Network Loadng Process...161 6.5 Computatonal Tests...163 6.6 Concludng Remarks...167 Chapter 7 Conclusons and Future work...168 7.1 Research Summary...169 7.1.1 Methodologes for network evaluaton under uncertan demand...169 7.1.2 Robust network desgn...169 x
7.1.3 Accountng for Informaton Recourse...170 7.2 Recent work n ths area...171 7.3 Open questons and future drectons...174 Appendx A Computatonal Results for sngle pont approxmatons for dfferent sample szes...179 Appendx B Dervaton of Central Moments of Normal Dstrbuton...181 Appendx C Sample plot to determne the sample sze n GA evaluaton for RNDP...183 Appendx D Data for the HF test network n Chapter 4...184 Appendx E Notaton and the TD-OSP Algorthm...185 References...188 Vta...196 x
Lst of Tables Table 1.1 Fve scenaros for the year 2025 for the example network (Source: DVRPC)...6 Table 3.1 Demand approxmaton functons...47 Table 3.2 O-D dstrbutons n the expermental setup...50 Table 3.3 Computatonal results for Cauchy dstrbuton, Sample sze = 100...52 Table 3.4 Computatonal results for Exponental dstrbuton, Sample sze = 100...52 Table 3.5 Computatonal results for Logstc dstrbuton, Sample sze = 100...53 Table 3.6 Computaton Tme of varous test networks...69 Table 3.7 Comparatve solutons of network performance from experment and analytcal expressons for the Test Network 1...70 Table 3.8 Network Performance Comparsons from experment and analytcal expressons for the Nguyen Dupus Test Network...71 Table 3.9 Comparatve results for the Soux Falls Network...71 Table 4.1 Lnk parameters for the smple test network...86 Table 4.2 Overvew of the GA lterature wth problem applcatons...87 Table 4.3 Comparson of the enumeraton result wth the GA on the small test network...96 Table 4.4 Comparson of results from Harker and Fresz (1983) and the GA approach for NDP...98 Table 4.5 RNDP soluton usng GA for dfferent values of...101 Table 4.6 Change n network performance wth budget...104 Table 4.7 Comparson of the network wde travel tme wth MOEA and evaluaton approach...105 Table 5.1 Defnton of varables and notaton...113 Table 5.2 Tme Invarant parameters of the test network...127 Table 5.3 Value of M t wth tme...127 Table 5.4 Optmal values of b for B = 120 n DUO NDP...128 x
Table 5.5 Baselne Case wth no mprovements...134 Table 5.6 Effect of Budget Levels for Peak dstrbuton, Demand = 308 vehcle trps...134 Table 5.7 Effect of Dstrbuton Types and Congeston Levels...135 Table 5.8 Cell Expanson polces for chance constrant Model n $ Mllons for dfferent values of alpha...147 xv
Lst of Fgures Fgure 1.1 Motvatng example for evaluatng network performance...5 Fgure 1.2 Motvatng example for studyng onlne network equlbrum...7 Fgure 2.1 Key classfcatons of Transportaton Network uncertantes...16 Fgure 3.1 Conceptual fgure to determne the range of approxmaton estmates...45 Fgure 3.2 Approxmaton of demand procedure...47 Fgure 3.4 Framework for evaluatng the robust network performance...68 Fgure 3.5 Test Network 1...70 Fgure 3.6 Soux FallsTest Network...72 Fgure 4.1 Smple test network...86 Fgure 4.2 The proposed algorthm...93 Fgure 4.3 HF Test Network...97 Fgure 4.4 Nguyen Dupus test network for the RNDP...99 Fgure 4.5 Convergence of the RNDP ftness functon for = 0...102 Fgure 4.6 Convergence of the RNDP ftness functon for = 0.5...102 Fgure 4.7 Convergence of the RNDP ftness functon for = 1...103 Fgure 5.2 Illustraton of the buldng block of CTM...114 Fgure 5.3 Network representaton of ordnary, source, and snk cells...115 Fgure 5.4 Network representaton of merge and dverge cells...116 Fgure 5.5 Conceptual lnk segments...116 Fgure 5.6 Illustraton of ordnary cells and cell connectors...117 Fgure 5.7 Illustraton of source cells and cell connectors...117 Fgure 5.8 Illustraton of snk cell and cell connectors...118 Fgure 5.9 Cell Representaton and Tme Invarant parameters of the test network127 Fgure 5.10 DSO and DUO TSTT wth the ncrease n total budget B...129 Fgure 5.11 Tractable M t vector for the DUO NDP...132 Fgure 5.12 Nguyen and Dupus s Test Network: (a) Lnk-Node Verson and (b) Cell Verson...133 xv
Fgure 5.13 TSTT as a functon of budget for DUO and DSO NDP...137 Fgure 5.14 Spendng polces for the unform and peak loadng demand patterns for UO and DSO NDP, 308 trps and 2000 Thousand Dollars...139 Fgure 5.15 Expected System Performance for the DUONDP...148 Fgure 6.1 Example Network...153 Fgure 6.2 Example network for demonstratng user equlbrum wth recourse paradox...160 xv
Part I Prelmnares 1
Chapter 1 Introducton Any study whch throws lght upon the nature of order or pattern n the unverse s surely nontrval - Gregory Bateson 1.1 Introducton Optmal evaluaton and desgn s central to the effcent plannng, ncreased effcency, operaton, mantenance and retrement of transportaton networks. Gven the ncreasng capabltes n characterzng uncertantes and the maturty n nformaton technologes and onlne control opportuntes, there s an ncreasng need to account for and compensate for the nherent stochastcty by developng new methodologes to enable the effcency and longevty of transportaton systems. Ths dssertaton s prmarly motvated by the need to mtgate ths nherent uncertanty through new traffc assgnment methodologes that explctly account for demand uncertanty. Specfcally, new models are developed for evaluaton consderng long-term demand uncertanty, robust optmzaton for network desgn, and user equlbrum wth recourse. Wth these new models t s antcpated that the effcency, reslency, and compettveness of stochastc transportaton networks wll mprove. Often, the tradtonal transportaton plannng approach nvolves developng a 20- year forecast of demand based on key socoeconomc varables, such as populaton, ncome and employment for dfferent zones. The tradtonal four step process s appled to deduce the forecasted traffc and lnk volumes on the metropoltan transportaton network. Typcally, planners base ther master plan to meet the condtons forecasted to occur. However, t s clear that there are uncertantes and bases nvolved n the decson makng process. Almost certanly, gnorng these uncertantes leads to systematcally erroneous results whch does not justfy the decson makng process. 2
In ths research we study modelng technques to deal wth uncertanty and n specfc propose robustness as a goal n the evaluaton and desgn of transportaton networks. The models developed n ths work are at the boundary of two dfferent branches of network scence (.e. transportaton network analyss and stochastc modelng). On the one hand the traffc assgnment models are an mportant component of the transportaton plannng whch places them n the feld of transportaton systems. On the other hand there s nherent uncertanty n many of the parameters of the transportaton network. Ths requres solutons n the present stage whch to be nherently reslent (.e. robust) or n some sense good for many realzatons of the system state nto the future, whch places these models n the broad area of stochastc programmng. Due to the numerous uncertantes that travelers routnely face, a contnung effort s placed nto strateges for managng onlne network traffc condtons. Wth the ncreasng avalablty of nformaton-based technologes and communcaton medums a wde opportunty exsts for mprovng and desgnng these systems. As these approaches have evolved, ncreased nterest needs to be focused on the ablty to account for recourse n network behavor and traveler route choce. In part, ths results from our ncreasng ablty to gather and dssemnate nformaton n a sophstcated manner on a real tme bass. However, to derve the true benefts from such onlne systems for mprovng network performance, we must enhance related equlbrum models and fundamentally devse new strateges to account for recourse. Ths queston of developng novel modelng approaches s crtcal n stochastc transportaton networks to evaluate the value of nformaton provson. The nformaton can be obtaned by varous means: rado broadcasts, varable message sgns (VMS), on board vehcular unts etc. Two types of nformaton strateges are usually studed: a pror nformaton and onlne nformaton. It s easy to magne future transportaton systems havng onlne nformaton devces whch can provde nformaton about traffc condtons on a specfc road segment durng all perods of tme. Travelers routng decsons n a stochastc transportaton network wth onlne 3
nformaton are concevably dfferent from those n a determnstc network. It has been demonstrated that onlne routng wll save travel tme as compared to a pror routng n transportaton networks (Waller and Zlaskopoulos, 2002). The onlne equlbrum research explored n ths work wll help us n understandng the network wde mpacts f all the travelers n a network are allowed to make onlne routng decsons. Our specfc focus s on tryng to quantfy the equlbrum process acheved by ths onlne routng gven uncertan network condtons n congested transportaton networks. Formulatng such a model and understandng ts propertes provde a useful applcaton n the evaluaton of nformaton systems and judcous nvestment of scarce resources n Advanced Traveler Informaton Systems (ATIS). 1.2 Motvatng Examples Two motvatng examples lead us to nvestgate the mportance of robustness and value of nformaton n transportaton networks. The frst s essentally the traffc equlbrum problem under uncertan demand solved on an example network. The second example s network equlbrum wth onlne nformaton provson n a transportaton network. 1.2.1 Long Range Network Performance Consder an example network connected by two non-nterferng hghways and drvers commutng between the node s and t as shown n Fgure 1.1. Suppose the frst hghway s narrow wth the cost on t represented by c 1 ( ) whch represents the delay experenced whle drvng from s to t. Smlarly, suppose that the second hghway s 1.618 tmes wder than the frst hghway and ts cost functon s represented by c 2 ( ). The network s shown n Fgure 1.1 and c( ) represents the cost ncurred by drvers due to the presence of other drvers on the hghway. Typcal 4
cost functons smlar to c (, ) = a b k k k k + k k k smplcty we wll assume that a k = 0 and b k = 1. ( ) 4 c ( ) 1 1 1 = 100 4 are consdered here. For s t ( ) 4 c ( ) 2 2 2 = 685 Fgure 1.1 Motvatng example for evaluatng network performance Tradtonally long term transportaton plannng s performed for a sngle forecasted value based on soco-economc characterstcs and other parameters for 20 years nto the future. However, suppose that by some exogenous analyss, the long term plannng organzaton of the example network has arrved at fve dfferent What If Scenaros for the long term s-t demand. Many possble combnatons of scenaros could be possble dependng on the analyss. We wll assume that all the scenaros are all equally lkely; these are shown n Table 1.1. Ths type of scenaro plannng process s ganng sgnfcance n recent tmes, but models to study the network performance are stll lackng. If the transportaton planner performs a tradtonal network analyss, he/she would plan for scenaro 1. The Delaware Long term regonal analyss group s one such Metropoltan Plannng Organzaton (MPO) where smlar studes are ganng promnence. 5
Table 1.1 Fve scenaros for the year 2025 for the example network (Source: DVRPC) No. Scenaro Demand 1 2025 forecast prevals 1000 2 Regonal economy grows (Boom) 1500 3 Energy cost rses causng out-mgraton (Bust) 700 4 Informaton technology amentes grow 1300 5 Homeland securty tghtened 500 In all the analyses that ensue, we wll assume that drvers am to mnmze the tme taken to travel from s to t. Naturally, ths would push more vehcles onto lnk 2 where there s greater capacty untl both the lnks ncur the same travel tme. For example for scenaro 1, we arrve at two smultaneous equatons at equlbrum - 1 + 2 = 1000 and 2 = 1.618* 1. Solvng these equatons gves us, 1 = 381.97 and 2 = 618.03, wth a total network cost of 4.25 10 8 unts. Now suppose that gven the What If Scenaros nto the future, we evaluate the total network travel tme for all the fve scenaros and compute ther average. Ths yelds a total network cost of 7.85 10 8 unts. We have thus underestmated the network performance by 3.50 10 8 unts when we consder only one scenaro (note here that scenaro 1 also represents the average demand case), a sgnfcant underestmaton. 1.2.2 Onlne Informaton Provson as Recourse (UER) The prevous example demonstrates the need for mtgatng uncertanty n transportaton plannng by accountng for uncertanty n the network assgnment stage. However, n addton to accountng for and assessng uncertanty we must also be able to manage ths uncertanty. Provdng nformaton on a real tme bass s one way to manage stochastc transportaton networks. Modelng such onlne nformaton provson forms an ntal and crtcal step before dfferent nformaton strateges can be developed and deployed. The example presented here shows the beneft n terms of travel tme savngs n onlne equlbrum problems. Consder the example network shown n Fgure 1.2 wth fve nodes and three drected paths: 1-2-3-5, 1-2-4-5 and 1-2-4-3-5. Assume that 6
there are 100 vehcles whch should be assgned from node 1 (orgn) to node 5 (destnaton). The travel tmes on lnks a, b, c, d, e and f are gven as follows: C a ( x ) = 2 C b ( x 2 ) = 2.5x 2 Cc ( x 3 ) = 2x 3 Ce ( x 4 ) = 3x 4 C f ( x 5 ) = 2.5x 5 1 + x 6 w. p. 0.8 C d ( x 6 ) = 1 + 100 x 6 w. p. 0.2 3 1 C a =2 a 2 b c d e f 5 4 Fgure 1.2 Motvatng example for studyng onlne network equlbrum Lnks a, b, c, e and f have determnstc travel tme functons whle lnk d has a random travel tme functon as shown above. The capacty on lnk d takes two dfferent values based on whether t s a sunny or a rany day. Ths leads to dfferent cost functons as shown n C d. Assume that the rany day occurs wth a probablty of 0.2 and ths reduces the capacty of the road by 100 tmes causng greater delays. The probablty of a sunny day s 0.8 wth a correspondng travel tme functon of 1+ x 6. Gven ths network topology, cost structure and a total demand of 100 unts from 1-5, f we were to perform an a pror equlbrum based on the expected path cost (n other words, all used paths have the same expected cost), we would arrve at the followng set of equatons for paths 1-2-3-5, 1-2-4-3-5 and 1-2-4-5: 7
2 + 2.5x 2 + 3x 4 = 2 + 2.5x 5 + 2x 3 = 2 + 2x 3 + (1+20.8x 6 ) + 3x 4 ; and x 2 + x 3 = 100 (the flow conservaton equaton). Solvng these equatons we get a path cost of 249.498 on all the used paths. Now can we mprove ths travel tme for each vehcle when we have addtonal nformaton at node 4? We assume that when a vehcle reaches node 4 t has nformaton about the condtons on lnk d (.e., whether t s rany or sunny day on lnk d). Based on ths nformaton each user wll decde on the next lnk to choose, for example f the user learns a cost of 1 + 100x 6 he/she wll take lnk f, otherwse he/she would take lnk d. Ths assgnment s a condtonal assgnment based on the nformaton learned at node 4. In ths example the vehcles route themselves based on strateges nstead of paths. These strateges can be represented usng hyperpaths. The dfferent strateges n ths network are: If lnk d s sunny, then avalable paths and the correspondng realzed costs are: 2.5x 2 3 3x 4 1 2 2 2x 3 1+x 6 2.5x 5 5 4 and f lnk d s rany, the avalable paths and the realzed costs are : 2.5x 2 3 3x 4 1 2 2 5 2x 3 2.5x 5 4 8
Analyzng the equlbrum soluton for all the strateges n ths network, the result s a path cost of 242.796. We have thus ganed a travel tme of nearly 7 unts for each vehcle, thereby ganng a travel tme savngs of nearly 680 unts for all the vehcles n the entre network n ths small example. Ths example demonstrates the beneft of usng onlne nformaton n equlbrum traffc networks. However, one should be cautous n generalzng the demonstrated mpact as a beneft when the network s characterzed by decentralzed traffc behavor. In Part IV of ths dssertaton, we wll revst ths example and aspre to develop algorthms to solve ths problem and quantfy the mpact of onlne equlbrum assgnment as compared to tradtonal equlbrum assgnment. 1.3 Contrbutons To acheve better modelng approaches for the above mentoned problems, the goal of ths research s two fold: contrbute to the area of transportaton network modelng by () modelng uncertanty and devsng strateges to account for robustness and () establshng a modelng approach to account for the beneft of onlne nformaton. Ths dssertaton substantally adds to the state of knowledge and to the methodologes n transportaton network modelng by addressng the followng objectves: 1. Develop a framework for studyng uncertanty n statc and dynamc network equlbrum models. 2. Develop approxmate analytcal expressons for the network performance when the nput demand s a stochastc random varable. 3. Propose a robust network desgn model under uncertan demand employng evolutonary algorthm as the soluton technque. 4. Develop a dynamc network desgn model accountng for user equlbrum condtons and extend ths framework to account for demand uncertanty. 5. Propose a novel equlbrum model and soluton approach whch accounts for nformaton recourse n stochastc networks. 9
The potental treatment of each of these topcs s comprehensve and prescrptve. It s fully antcpated that the results from ths research wll motvate new research drectons n transportaton network modelng and ther applcaton n the overall plannng process. 1.4 Dssertaton Organzaton 1.4.1 Presentaton Overvew Chapter 2 descrbes the prelmnares of the problem defnton and provdes a crtcal revew of the prevous work n transportaton equlbrum modelng wth a partcular emphass on stochastcty. Chapter 3 presents two methodologes the frst s an approxmaton procedure for solvng the traffc equlbrum problem under uncertan demand and the second s based on analytcal dervatons to account for robustness n transportaton networks. The long term demand s assumed to be uncertan and uncorrelated; closed form expressons are derved for the overall network performance. Chapter 4 develops a robust network desgn model when the long term demand s a stochastc random varable. Specfcally, the soluton method s based on a multobjectve evolutonary algorthm for solvng the non-convex, non-dfferentable optmzaton problem. The formulaton of the problem and computatonal analyss are presented. Chapter 5 formulates a network desgn model under tme varyng condtons (DTA) and extends ths model to a stochastc programmng formulaton to account for demand uncertantes n the network desgn problem. Chapter 6 deals wth the user equlbrum wth recourse as a modelng framework for ncorporatng onlne nformaton avalablty n transportaton networks. The fundamental nvestgaton n ths lne of work s n fndng onlne equlbrum polces where the network users have the ablty to gan nformaton as they traverse the network. Fnally, n Chapter 7 we conclude ths dssertaton wth a summary of contrbutons and the future extensons of ths lne of research. 10
1.4.2 Dependences Chapter 2 provdes a quck revew of the mportant contrbuton of ths dssertaton; though some of the sectons are specfc to partcular chapters. Ths s further elaborated n Chapter 2. Chapter 4 should follow chapter 3 for a logcal understandng of the problem. However, chapters 3, 4, 5 and 6 can be read ndependently of each other although each of them complements the other. Chapters 3 and 4 are hghly related to each other. 1.4.3 Bblographc Informaton Some of the work presented n ths dssertaton has appeared prevously n research papers (Ukkusur and Waller (2004a), Waller et al (2004a)). Some of the other work s n revew (Ukkusur and Waller (2004b), Ukkusur et al. (2004); Waller et al. (2004b); Unnkrshnan et al. (2004); Ukkusur et al. (2004); Ukkusur and Waller (2004c)). Most of the work s n jont work wth Prof. S. Travs Waller. The results of Chapter 3 appeared n TRISTAN V proceedngs. An ntal verson of ths work s presented at the INFORMS Annual (2003, 2004) conferences and the Transportaton Research Board (TRB) doctoral semnar. 11
Chapter 2 Crtcal Evaluaton and Comparson to Prevous Work If I have seen further t s by standng on the shoulders of Gants n 'Letter to Robert Hooke, February 5, 1675/1676 - Newton, Sr Isaac Modelng uncertanty and n specfc robustness has been a sought after goal n many domans. A good non-techncal readng for a comprehensve understandng of the hstorcal underpnnngs of uncertanty and rsk can be found n Bernsten (1998). Ths dssertaton bulds upon four prmary areas n optmzaton: traffc equlbrum (statc and dynamc), stochastc programmng, robust optmzaton and onlne equlbrum problems. Ths chapter revews these topcs and provdes a crtcal evaluaton of related lterature to provde a bass for understandng the models proposed n ths work. Ths chapter s organzed as follows. Secton 2.1 begns the revew wth dentfyng dfferent sources of uncertanty n transportaton decson makng. Secton 2.2 revews the dfferent modelng approaches studed n the lterature for modelng uncertanty. A revew of traffc equlbrum models (statc and dynamc) s conducted n secton 2.3. Stochastc programmng approaches and the defnton of robust optmzaton are dscussed n secton 2.4. Secton 2.5 revews the lterature on onlne equlbrum models n transportaton networks. Secton 2.6 concludes ths chapter wth a summary and a drect connecton between the past work and ths research. The purpose of the revew s prmarly expostory rather than comprehensveness. The nterested reader can broaden hs/her knowledge from the works mentoned n ths revew. 2.1 Sources of Uncertanty n Transportaton Decson Makng In ths dssertaton we prmarly deal wth demand uncertanty, however, a comprehensve lstng of dfferent sources of uncertanty s n order for the sake of completeness. Further, some of the proposed methodologes n ths work could 12
potentally be drectly transferable to account for some of the uncertantes dscussed below. Uncertanty n transportaton systems plannng s one of the major factors that planners and decson makers face both n the short term and ncreasngly so n the long term. In fact, the tact assumpton n urban transportaton plannng process s that we can forecast the future wth certanty. The process typcally develops 20-year forecasts of the key ndependent varables, such as populaton, employment and demand for each traffc analyss zone of a metropoltan area. Based on the 20 year forecasts dsaggregated to a zonal level, the planners am to produce forecasts of traffc volumes on each lnk of the transportaton network. Some reports from the Florda s The Tampa Trbune gve us ample evdence of the massve uncertanty prevalent n forecastng land use, transportaton demand and economc actvty. A few snppets demonstrate the uncertantes: 1. A December 18, 1994 report says, Hllsborough growth slower than expected, noted that almost two years ago, the plannng commsson receved a great deal of crtcsm for recommendng the county adopt lower growth estmates. As t has turned out, even the low projectons were optmstc. 2. A May 21, 1993 artcle quotes Florda s fnances lookng better, whch quoted the drector of Florda s legslature s economc research unt as sayng, The problem wth our forecastng technque s that we are always wrong. 3. An October 10, 1992 artcles chroncles County planners to recommend reducng populaton estmate, whch noted that Hllsborough county populaton forecasts for the year 2015 are expected to be 166,000 fewer than had earler been offcally forecasted for the year 2010. These artcles demonstrate the gross underestmaton of forecasted demand n crtcal plannng decsons. In the lterature dfferent researchers have classfed uncertanty based on the parameter/decson of nterest n transportaton plannng/operatons. Neumann (1976) focused prmarly on accountng for uncertantes n resource constrants, mplementaton tmes, expected mpacts and 13
poltcal acceptablty of alternatve transportaton plans. Pecknold (1970) dealt manly wth uncertanty n future demand usng transportaton plannng. Mahmassan (1984) presented an overvew of evaluaton approaches for uncertanty n transportaton systems. He categorzed fve dfferent types of uncertantes n the evaluaton of transportaton systems. 1. Unexpected events and unforeseen stuatons; these nclude major poltcal upheavals or unantcpated technologcal breakdowns. Ths knd of uncertanty s nearly mpossble to nclude n transportaton decson makng because the modeler or forecaster s not cognzant of the possblty of such occurrences. 2. The exogenous states affectng the transportaton systems, such as new admnstraton at the federal level, economc boom or bust etc. These can have causal relatonshps on transportaton systems, e.g. a hgh economc growth may mply hgh automoble ownershp levels and thereby ncreased nter-zonal demand. Ths type of uncertanty can be represented ether through dscrete states of nature or a scenaro approach as proposed n chapter 3 of ths dssertaton. 3. Uncertanty n the values of measured or predcted mpacts usually as a result of the modelng actvty. Ths s prmarly referred to as descrptve uncertanty n the knowledge of the phenomena modeled. Good examples of ths nclude the estmates of demands, flows on varous network lnks, beneft measures, costs and others. Ths type of uncertanty can be represented as nterval estmates (or bounded uncertanty), probablty mass functons n dscrete or contnuous forms. Agan, ths uncertanty s modeled n chapter 3 both under dscrete and contnuous forms for O-D demand. 4. Fuzzness or vagueness characterzed wth the descrpton of a performance measure n transportaton systems. Ths s prmarly because the system optons are ll-defned by the planner. Fuzzy set theory and assocated algebra can be used to study such problems (see Ukkusur et al. (2005) for an 14
applcaton of such a technque to the congeston prcng technology evaluaton under fuzzness of the crtera). Ths methodology wll not be consdered n ths dssertaton. 5. Uncertanty as to the preferental or normatve bass of the evaluaton. Ths ncludes the rsk atttudes of the decson makers n the decson process, the bases of the actors n the plannng process. A very good example of such an uncertanty s the outrght bas and poltcs nvolved n the choce of lght ral transt n Sacramento (Johnson et al., 1988). The representaton and quantfcaton of ths type of uncertanty s a hghly complex task. However, ts exstence s a fundamental characterstc of transportaton decson makng and cannot be gnored completely. Lowe and Rchards (1983) dentfed three sources of uncertanty n transportaton plannng models; specfcaton error, calbraton error and the error n forecasted exogenous nputs. Specfcaton uncertantes recognze that wth the complexty of human behavor, t would be unlkely that the models would nclude all relevant factors. Calbraton errors on the other hand recognze that the estmated coeffcents are based on sample values, ntroducng addtonal error. Fnally, the thrd source of uncertanty deals wth the dffculty n forecastng future exogenous varables (such as O-D demand) at the network level. Gven these uncertantes, how can we develop models to evaluate, desgn for and mtgate uncertan future condtons? In ths dssertaton, we prncpally address the thrd source of uncertanty by developng new network models for evaluaton, desgn and mtgaton of uncertanty. 2.2 Uncertanty Modelng n Transportaton Transportaton systems are nherently characterzed by stochastcty due to the human-system nteracton assocated wth the overall problem. These systems nvolve uncertanty n the network topology, nput parameters, actvty patterns of 15
users, drver behavor and land use changes whch cannot be captured adequately wth the exstng methodologes. In ths study we classfy network uncertanty nto four broad categores; based on tme and network parameters as shown n Fgure 2.1. Sgnfcant other uncertantes exst n measurng network performance such as the cost parameters and drver behavor; however, these wll not be modeled explctly n ths dssertaton. We assume that all the other uncertantes manfest themselves n the stochastcty of the O-D demand. Network nputs Bell et al., 1999 Du and Ncholson, 1997 Demand Waller and Zlaskopoulos, 2000 Focus of ths work Short Term Long Term Tme Bell and Ida, 1997 Chen et al.,2002 Lo and Tung, 2003 Capacty Fgure 2.1 Key classfcatons of Transportaton Network uncertantes The second quadrant n Fgure 2.1 s concerned wth short term demand uncertantes whch are usually solved as stochastc user equlbrum problems or travel tme relablty methods (Bell et al., 1999; Du and Ncholson, 1997; Yang et al., 2000). A probablstc lnk travel tme functon s drectly assumed and an equlbrum senstvty analyss s performed to overcome some of the computatonal burden as n Bell et al. (1999). Du and Ncholson (1997) used a conventonal 16
ntegrated equlbrum model wth varable demand to descrbe flows n a network wth degradable lnk capactes. Ther focus was however on a multmodal context and employed dfferental senstvty analyss for the equlbrum problem. The thrd quadrant encompasses the models whch deal wth short term capacty uncertantes. In the lterature these models are usually referred to as connectvty relablty (Ida and Wakabayash, 1989; Bell and Ida, 1997) or capacty relablty (Chen et al., 1999, 2002; Lo and Tung, 2003) models. Connectvty relablty as defned by Ida and Wakabayash (1989) deals wth the probablty that the network nodes reman connected, whch s used as a measure of the performance of a lnk to reflect the flow to capacty rato beng less than a specfc value. Chen et al. (1999, 2002) defned capacty relablty as the probablty that the network can accommodate a certan demand at a gven level of servce n short tme duratons. They studed small network examples and extendng ther work to large scale networks s a computatonally burdensome task. Lo and Tung (2003) developed a model to study day-to-day capacty degradatons whch reduce to travel tme varablty. They characterze the uncertan travel tme problem as probablstc user equlbrum (PUE) and use Mellns Transform to derve expressons for the capacty uncertanty. They solve a numercal example. However, the use of the Mellns transforms cannot be generalzed to all of the probablstc dstrbutons. Before proceedng further t s mportant to note that all the prevous studes assume the normalty of the lnk flows gven any dstrbuton of demand or capacty. Ths study renforces and brdges an mportant gap n Chapter 3 by showng the bass of the normalty assumpton when the demand s a stochastc varable. To the best of our knowledge there has been no work n the lterature whch deals wth long term capacty uncertanty as shown n the fourth quadrant n Fgure 2.1. The methodologes proposed n chapter 3 and 4 fall wthn the frst quadrant of Fgure 1 and deal wth the network performance and desgn when the long term OD demand s a stochastc varable. In a transportaton network, the demand between any gven OD pars s not constant over tme. OD demand patterns show sgnfcant 17
varatons wthn a day, and exhbt long-term, yearly varatons. Earler work n evaluatng the traffc equlbrum problem under uncertan demand was conducted by Waller and Zlaskopoulos (2000). An mportant nsght from ths work was that solvng the traffc equlbrum problem at the expected demand sgnfcantly underestmates the true expected performance of the network. A stochastc programmng formulaton was proposed for the dynamc network desgn problem under demand uncertanty for system optmal condtons (Waller, 2000). Ths was formulated as a lnear program and the cell transmsson model was embedded n the formulaton. For the User Equlbrum condtons, a smlar analyss was done by Ukkusur et al. (2004). 2.3 Revew of Traffc Equlbrum Models The stochastc models proposed n ths dssertaton are based on three prmary areas n traffc equlbrum () statc traffc assgnment whch s the startng pont for ths study because of the tremendous body of work n place and ts current acceptablty by practtoners; () dynamc traffc assgnment and the () Network desgn problem. 2.3.1 Statc Traffc Equlbrum Traffc n transportaton networks can be represented from two perspectves, traffc maxmzaton and travel tme mnmzaton. Traffc maxmzaton nvolves the determnaton of maxmal traffc demand that the network can support between two gven nodes, whereas the cost mnmzaton mnmzes transportaton costs gven a demand between two nodes. When congested travel tme s chosen as the objectve varable, the problem becomes more complex, as congested travel tme s a functon of the volume of travelers usng a lnk. The Bureau of Publc Records (BPR) formulaton (see U.S. Department of Commerce, 1964) s commonly used n transportaton plannng applcatons. Ths formula computes the congested tmes as: 18
Tc = T o 1 + ( V / C) (2.1) where Tc s the congested travel tme on a lnk, To s the free-flow travel tme, V s the hourly volume, C s the hourly practcal capacty, and and are parameters. Typcally, =0.15 and =4, although n some nstances other values may be used (for e.g. =5.5 for freeways). Other formulatons, such as those based on fundamental dagrams of traffc flow may also be used (Sheff, 1985). Horowtz (1997) descrbes some of the challenges n usng delay relatons descrbed by 2.1 n the 1994 update to the Hghway Capacty Manual. Horowtz also advocates usng delay functons that take as arguments the volume on the lnk of nterest and volumes on nearby lnks, to more accurately model the performance at sgnalzed ntersectons, at two-way and fourway stops, and on rural two-lane roads where passng n the oncomng travel lane s permtted. Methods to solve the user-equlbrum problem when lnk travel tmes are functons of volumes on other lnks are dscussed below. For a thorough treatment on these methods, the nterested reader s referred to Sheff, 1985 and Bell and Ida (1997). To solve the flow-dependent shortest path problems, an teratve approach s often used, alternatng between fndng shortest paths for fxed travel tme, and recalculatng travel tmes based on new lnk volumes. An equlbrum results between travelers choces of routes (whch depend on travel tme), and lnk traversal tmes (whch depends on the volumes of drvers usng them). When travelers, who seek to mnmze travel tme, have no ncentve to change routes, the teraton calculatons have reached a state correspondng to Wardrop s (1952) frst prncple. 19
An alternatve way of statng ths prncple s that for a gven O-D par, all used routes have the same travel tme, and unused routes have travel tmes greater than or equal to that of the used routes. Ths stuaton descrbed by Wardrop s frst prncple s often called user equlbrum, as t depends on ndvduals mnmzng ther own travel tmes. In comparson to the above, Wardrop s second prncple, whch descrbes the stuaton where total travel tme on the network s mnmzed, s often referred to as a system optmal (SO) assgnment. To acheve a SO assgnment, travelers must be assgned to lnks consderng the margnal cost an addtonal traveler nduces to lnk travel tmes, rather than the average cost descrbed by the congested tme of the BPR functon. Snce the BPR functon (and other congeston relatonshps, such as those descrbed by queung models) s concave upwards, the margnal cost of travel wll be greater than the average cost. The dfference between margnal and average travel cost s a negatve externalty to other travelers, and thus, the network-wde travel tme under user equlbrum wll be greater than under a SO assgnment. Some researchers have advocated that ATIS could be used to persuade travelers to use SO routes. In contrast, Hall (1996) argues that democratc socetes value honest travel nformaton, and therefore travelers wll resst SO paths that are not personally optmal. Hall asserts that travelers who perceve nformaton to be naccurate wll ether gnore or act contrary to that nformaton. Any user equlbrum assgnment routne can be used to produce a SO assgnment by replacng average lnk costs wth margnal costs. The margnal cost, MC, accordng to the BPR formulaton can be shown to be: MC = T o 1 + ( + 1)( V / C) (2.2) Lnk delay relatons can be descrbed as symmetrc or asymmetrc, dependng on the nterrelatonshps of volumes on varous lnks to correspondng travel tmes on those lnks. In the symmetrc case, the mpact of lnk A s volume, V A on lnk B s 20
travel tme, T B s dentcal (or symmetrc) to the mpact of lnk B s volume, V B on lnk A s travel tme, T A. Mathematcally, ths can be wrtten as: T V B A T = V A B (2.3) The famlar case assumed by the BPR functon, where a lnk s travel tme depends only on ts volume, s a specal case of symmetrc delay relatons. (In ths case the partal dervatve terms T V B A T and V A B are zero for all pars of lnks). The symmetrc desgnaton therefore descrbes the property of the Jacoban matrx of travel tme functons, that s, the square matrx of (partal) dervatves of the vector of lnk travel tmes wth respect to the vector of lnk volumes. If a mnmzaton formulaton s desred, the objectve functon must be such that ts second dervatve (the Hessan) matrx s the same as the Jacoban of travel tme wth respect to lnk volumes. Alternatvely stated, the objectve functon may be any lne ntegral of the lnk delay functon. (For more detals, see Secton 8.1 n Sheff, 1985). Two approaches for solvng the flow-dependent traffc assgnment problem are the method of convex combnatons popularly called the Frank-Wolfe Algorthm and the Method of Successve Averages (MSA). The method of convex combnatons was frst proposed by Frank and Wolfe (1956) as a general procedure for solvng a nonlnear optmzaton problem by decomposton. The problem s transformed nto a lnear program and a onedmensonal non-lnear problem or lne search. The lnear program step s called drecton fndng, as t represents a search for a new feasble soluton by whch to mprove the objectve functon. The lne search step determnes weghts to average the current drecton-fndng soluton wth prevous results to obtan a new mnmum of the objectve functon. Bruynoughe (1968) was the frst to propose applyng ths method to the traffc assgnment problems. In ths problem, the drecton-fndng step 21
s solved by a shortest path calculaton (usng for example, Djkstra s algorthm) assumng fxed travel tmes. The MSA (see Almond, 1967) s a more general and robust approach than the Frank-Wolfe algorthm. It mantans the drecton-fndng step, but nstead of calculatng a weght n a lne-search step, the MSA uses predetermned, fxed weghts. Therefore, the MSA, unlke the Frank-Wolfe Algorthm, does not requre a mnmzaton formulaton, and may therefore be easer to program. Sheff (1985) dscusses the regularty condtons under whch the MSA s guaranteed to converge to a soluton. However, for problems where the Frank-Wolfe Algorthm can be used, the MSA s often slower, because t cannot take advantage of optmzed step szes from the lne search routne. The mathematcal programmng formulaton of the statc UO and SO formulatons from Sheff (1985) are gven as: 22
UE : mn z( x) = t ( ) d SO : mn z( x) = x t ( x ) a s. t. s. t. k x a 0 a a a a a rs rs f = q, r, s f = q, k rs k rs k r, s rs rs f 0, k, r, s f 0, k, r, s k k x = f a x = f a rs rs rs rs a k a, k a k a, k r s k r s k where destnaton s); qrs s the orgn-destnaton matrx (trp rate between the orgn r and xa and ta represent the flow and travel tme respectvely on lnk a. Furthermore, t = t ( x ) represents the relatonshp between the flow and travel tme a a a on lnk a. Smlarly, rs fk represents the flow on path k connectng orgn r and destnaton s and ak rs s a bnary value ndcatng that lnk a exsts on path k between r and s. 2.3.2 Dynamc Traffc Assgnment Recently, Dynamc Traffc Assgnment (DTA) models have receved ncreasng attenton from transportaton planners and researchers because of the lmtaton n the exstng statc transportaton smulaton models n the real world. Transportaton agences and practtoners have realzed that DTA can gve better solutons to real world problems, whle the exstng statc plannng applcatons have unrealstc assumptons that traffc nformaton and drver s behavor are perfectly known and not changed durng assgnment (Ran and Boyce, 1996a). To overcome the lmtatons of the exstng statc models, there have been many efforts n developng theores of DTA and ts applcatons, whch can be deployable n a largescale network n real-tme. Most researchers n transportaton plannng area recognze that the development of a stable theory of DTA s stll the most 23
challengng problem, consderng problem tractablty and the realsm of the underlyng traffc flow model. The key feature of DTA s that vehcles on a traffc network can be tracked n tme varyng network condtons by capturng traffc flow and route choce. Yagar (1970, 1971) s one of the earlest contrbutors n the development of DTA models, and he used the smulaton technque n order to optmze the ndvdual path choce based on Wardrop s frst prncple. Yagar also contrbuted a heurstc soluton algorthm to the problem. Merchant (1974) and Merchant and Nemhauser (1978a, 1978b) proposed the frst mathematcal programmng problem for the statc dynamc assgnment wth a sngle orgndestnaton and fxed demand. They formulated a macroscopc model to mnmze total travel cost, n the form of a dscrete-tme, nonlnear, and non-convex mathematcal programmng model. The non-convexty of the model (Merchant and Nemhauser, 1978a, 1978b) causes analytcal and computatonal dffcultes. Ho (1980) and Carey (1986) show that the Merchant and Nemhauser formulaton satsfed a certan constrant qualfcaton needed for Kuhn-Tucker optmalty condtons to ensure the exstence of optmum. Carey (1987) reformulates the dynamc least-cost network flow problem as convex nonlnear programmng, to provde an alternatve formulaton for the Merchant and Nemhauser problem. Snce DTA models requre dfferent sets of decson varables based on the traffc behavoral and system assumptons, and tme-varyng data to represent the traffc dynamcs or controls that are a dstngushng feature dfferent from exstng statc assgnment adopted by the statc traffc nformaton, DTA models are consdered as an emergng applcaton for Intellgent Transportaton Systems (ITS). To run DTA models, tme-dependent trp tables and outputs as well as tme dependent lnk flows are needed. DTA models can capture valuable traffc nformaton such as traffc dynamcs, supply dynamcs and varyng user classes, by consderng traffc control schemes such as traffc sgnals, ramp meterng, varable message sgns and other nformaton provson devces. The lterature on DTA and ts related applcatons s vast and we wll not delve further nto the dfferent models. 24
A very good revew of the dfferent DTA models, the solutons approaches and areas of future research are presented n Peeta and Zlaskopoulos (2001). In chapter 5 of ths dssertaton, we aspre to develop a network desgn model whch satsfes user equlbrum DTA propertes. The proposed model dffers from the prevous NDP models n two prmary ways: () t accounts for a dynamc evoluton of traffc and () t captures lnk spllover and shockwave propagaton more realstcally because of the embedded traffc flow model. A stochastc programmng model wll be developed as an extenson of ths NDP model to account for demand uncertanty n the transportaton decson makng. 2.3.3 Network Desgn The NDP offers an abstracton of the problem often faced by agences responsble for managng hghway systems: namely how best to expand those systems n response to growng demand for travel. The NDP can be posed n two dfferent forms: a dscrete form dealng wth the addton of new roadway segments and a contnuous form dealng wth the optmal capacty ncreases of exstng roadway segments. Numerous formulatons of both types have been proposed n the lterature over the last thrty years. The dscrete verson of NDP s dealt wth n ths research n Chapter 4 and the contnuous verson n Chapter 5. A bref revew of the current state of the art s revewed n the followng paragraphs for the sake of completeness. Bascally, the NDP assumes that a subset of the network s nodes serves as ponts of trp orgn, whle another subset of nodes serves as trp destnatons. It s further assumed that the total demand for travel between each O-D par over some plannng nterval s known apror and quantfed n the form of an O-D matrx. Now, suppose the road network has a total of n lnks, and let the lnks be ndexed by the varable k = 1,2,.n. Each of these lnks has three varables, k,! k and " k where k, denotes the volume of traffc on lnk k,! k denotes the proposed expanson of lnk k s capacty, so that the capacty after expanson wll be 25
! k + " k and " k denotes the current capacty on lnk k. and # can be used to denote n-dmensonal vectors contanng the lnk volumes, capacty expansons and orgnal capactes, respectvely. Each lnk s also assumed to have a travel cost functon c (, + " ), whch gves the cost of traversng lnk k. A common choce k k k k for the travel cost functons s the Bureau of Publc Roads (BPR) relatonshp shown n (2.1). Fnally, the objectve of the NDP s to select a set of expanson values! k that mnmzes the total system cost. It s usually assumed that each lnk has a known constructon cost functon g ( ) whch gves the cost of ncreasng lnk k s capacty by! k. The objectve of the NDP s n [ kck k k + " k + $ g k k ] mn (, ) ( ) k=1 k k (2.4) where $ denotes a scalng coeffcent whch converts unts of constructon nto unts of daly travel cost. A common assumpton n network desgn (and n much of transportaton plannng and socal scences) s that, when left to ther own decsons, travelers ndependently search for the lowest-cost route from orgn to destnaton, wth the search termnatng when no traveler can decrease hs/her travel cost by unlaterally swtchng to another route. Ths results n the Determnstc User Equlbrum (DUE) assgnment as dscussed n secton 2.2. Soluton of the NDP requres selectng a set of capacty ncreases, whch, when the lnk volumes are the outcome of a DUE assgnment, mnmzes the cost gven n (2.4) above. Leblanc et al. (1975) showed that the Frank-Wolfe algorthm can be used to solve ths problem. Ths gves the DUE vector of lnk volumes as a well-defned functon and the capacty ncreases. Although a closed form relaton between the capacty ncreases and the DUE lnk volumes s not known, the functon can at least be evaluated for a gven va the Frank-Wolfe algorthm. But, unfortunately, the functon relatng the DUE to the capacty ncreases s non-dfferentable (Adbulaal and Leblanc, 1979), whch 26
means that numercal search methods whch do not requre dervatves of the cost functon must be employed to solve the NDP. Ths, n turn, means that numerous DUE assgnments must be performed at each teraton of the search routng, resultng n algorthms whch are computatonally demandng. In ths respect, Abdulaal and Leblanc (1979) provde the poneerng work on ths approach and report that of the methods they tested, one known as the Hooke-Jeeves algorthm appeared to perform best. Gradent based search methods were reported by Tan et al. (1979) where a set of constrants were constructed, but, the number of constrant functons needed to characterze the DUE equaled the number of paths n the network of nterest. Snce the number of paths n networks of realstc sze s very large, ths approach can only be employed for small, hypothetcal networks. Alternatvely, the fact that the DUE assgnment can be characterzed by varatonal nequalty (Dafermos, 1980; Smth, 1979) can be used to construct a set of nequalty constrants (Marcotte, 1983). Agan, however, the potental number of these constrants s very large so that, lke the Tan et al. approach, ths technque was used on small networks. All the soluton methods descrbed so far guarantee fndng a soluton whch s at least locally optmal, and when algorthms whch provde such guarantees prove to be awkward, one has to resort to heurstcs whch can produce acceptable solutons. One source of heurstcs for the NDP has been va b-level programmng, an approach whch recognzes that the NDP s n essence a constraned optmzaton problem, the feasble solutons beng determned by the soluton of the lower level problem. In Marcotte (1986), a detaled analyss of the DUE-constraned NDP s gven, producng several heurstcs based on the b-level programmng dea; whle n Leblanc and Boyce (1986) lnear approxmatons to the travel and constructon cost functons are used to produce a b-level lnear programmng heurstc. Suwansrkul et al. (1987) developed a heurstc method, called equlbrum decomposed optmzaton (EDO), whch does not clam to fnd locally optmal solutons, but does appear to fnd solutons whch are near optmal, wth markedly less computatonal 27
effort. A promsng heurstc approach based on smulated annealng was presented by Fresz et al. (1992) and a tabu-search based heurstc was presented by Mouskos (1992). More recently, Davs (1994) presented the formulaton and a heurstc algorthm based on a stochastc UE model, whch adds to the realsm of the drver behavor. A mult-objectve desgn was formulated by Fresz et al. (1993) and further extended by Tseng and Tsuar (1997). Yang and Bell (1998) presented a formulaton that accounts for the elastcty of the demand, for the mxed contnuous and dscrete NDP. The above approaches demonstrate that substantal work has been performed n the area of NDP, but most of these formulatons focus entrely on the statc traffc condtons and have some sort of lnk performance functons lke the BPR functons, whch requre that the problem has some defnte set of propertes lke dfferentablty. The only research whch captures traffc dynamc n network desgn s by Janson (1995). Although, ths approach s lmted to usng DTA for evaluatng alternatves rather than computng them, the paper provdes varous nsghts on the effect of traffc dynamcs on network desgn decsons. In the next secton we wll concentrate prmarly on meta-heurstc approaches for solvng the dscrete NDP. 2.3.4 Heurstc Soluton Approaches for Dscrete NDP (DNDP) DNDP allows for the addton of a new lnk to the exstng congested networks, or for buldng addtonal lanes to the exstng lnks. The followng examples can be consdered as DNDPs: an addton of a lane, a road closure scheme, the provson of a new publc transport servce such as a new set of lnks, and the constructon of new road and ral lnks (Bell and Ida, 1997). DNDPs are well known as dffcult and complex problems to solve by usng tradtonal local search methods such as branch-and-bound, etc. These local search methods were appled to partcular problem classes, however, and dd not explore beyond the frst local optmum. Moreover, these generally requre sgnfcant computatonal tmes. Due to these 28
dffcultes many researchers have appled metaheurstc and other approxmaton technques, such as tabu search, genetc algorthms, and smulated annealng, to solve the statc DNDPs. These are consdered as combnatorally hard problems (Magnant and Wong, 1984; Xong and Schneder, 1992; Solank et al., 1998). Although t s possble to fnd near-optmal solutons to these NP-hard problems, there s no guarantee that the solutons found are optmal. As for sequental branchand-bound, t cannot solve larger dmenson problems. The frst known soluton approach for the DNDP s by LeBlanc (1975) who developed a nonlnear mxed nteger programmng model and soluton strateges usng a branch-and-bound approach to determnng whch lnks should be mproved n an urban network. The mathematcal formulaton for DNDPs wth a budget constran and fxed demands under user equlbrum s as follows: x a Mn Z( x ) c ( w) dw (2.5.1) = a' A 0 a a a ( (2.5.2) a' P Subject to e z B x ( Mz a ' P (2.5.3) a a za = 0 or 1 a ' P (2.5.4) where x a s an user equlbrum flow pattern on lnk a ; z a s a decson varable and 1 f lnk a s added to the network, and 0 otherwse; e a s the fxed constructon cost on lnk a ; B s total budget; M s a very large constant; A s the set of all lnks; P s the set of proposed lnks for addton to the exstng network; P ) A. Abdulaal and LeBlanc (1979) & Dantzg and Maer (1979) have modeled wth contnuous budget functons. Boyce and Janson (1980), however, ponted out that snce contnuous NDP (CNDP) models gve the soluton n a way of fractons of hghway lanes or fractons of ralway lnes, DNDP model s more approprate for transportaton networks. However, the approprateness of modelng CNDP s or 29
DNDP s for transportaton networks s stll an open queston for debate and s dependent on the research phlosophy and problem applcaton. Unlke the exstng DNDPs models, Chan et al. (1989) have developed the network desgn model n a way of mnmzng budget expendture objectve functon, under the condton that the total travel tme cannot exceed a certan level of congeston. Chen and Alfa (1991) proposed the two-level nonlnear nteger programmng model for solvng a logt-based network desgn algorthm ncorporatng a stochastc ncremental traffc assgnment approach, by ntroducng a converson factor between constructon cost and travel tme (Dantzg and Maer, 1979). Some studes were made usng metaheurstc approaches to escape the bane of local optmalty. Mouskos (1991) studes the dscrete transportaton equlbrum network desgn problem by usng the tabu-based metaheurstc search strategy and by accountng for a lnk average travel speed and the rato of lnk volume to capacty under statc traffc assgnment. Xong and Schneder (1992) dscuss the dscrete network desgn problem by usng a genetc algorthm, based on statc traffc assgnment. Cranc and Gendreau (2000) have proposed that a smplex-based tabu search procedure for capactated mult-commodty network desgn problems provdes a near optmal soluton, and path-based mathematcal formulaton as follows: p p Mn z( h, y) = ca ya + k h a' A p' P p l l l' L (2.6.1) (2.6.2) p Subject to hl = d p ' P p l' L p p hl al ( ua ya a ' A (2.6.3) p' P p l' L p p h 0 p ' P, l ' L (2.6.4) l y = {0, 1} a ' A (2.6.5) a 30
where N s the set of nodes; A { a (, j) N, j N} = = ' ' s the set of arcs; P s the set of commodtes to be dstrbuted; r ' N s the orgn node; s ' N s the destnaton node; p d s the demand of commodty p ' P ; p e a s the unt cost of movng commodty p ' P through the lnk a ' A ; c a s the fxed-cost of ncludng the lnk n the desgn of the network(or to ntroduce addtonal capacty on the lnk); k p = e ; = 1 f lnk a ' A s the varable cost of p p p l a al a' A al th l path for commodty p ' P, and s 0 otherwse h s the flow of commodty p ' P on path p l l y = p ' L ; a 1 f lnk a ' A s ncluded (opened), and 0 otherwse. In ths study, feasble solutons are obtaned by usng a tabu search metaheurstc that explores the soluton space of the contnuous flow varables by combnng smplex pvot-type moves wth column generaton, whle evaluatng the actual mxed nteger objectve of the fxed-charge, capactated, mult-commodty network desgn problem wth lnear costs. Very few evolutonary algorthms for transportaton network desgn problems have been proposed, but GAs n transportaton optmzaton problems are gettng more attenton than ever before. The frst attempt to solve transportaton network desgn problems by usng genetc algorthms was done by Xong and Schneder (1992). Ths study uses a traned neural network for traffc assgnment to calculate smplfed total travel tmes. They use the cumulatve genetc algorthm that mantans the hstorcal non-domnated soluton set, nstead of the sngle-objectve genetc algorthm that can handle only one objectve (ftness) value to represent the performance of the problem. The cumulatve genetc algorthm uses the domnance comparson methods to account for two ftness values,.e., constructon cost and total travel tme. Ths study uses a ftness value F = (M - # of tmes domnated), where M s the largest number of tmes domnated among all the solutons n the concerned generaton. Furthermore, lmtatons n ths approach are the use of a statc traffc assgnment model that does not account for traffc realtes such as traffc nteracton on transportaton networks and the total system travel tme does 31
not account for evaluaton of the ftness value. In chapter 4 of ths dssertaton, an mportant varant of the dscrete network desgn problem, by ncorporatng uncertanty drectly nto the formulaton wll be formulated and solved. A soluton technque based on genetc algorthm s explored. For bus network desgn problems, Pattnak et al. (1998) proposed the network desgn model for urban bus transt paths usng a genetc algorthm. In the frst phase of ths approach, a set of canddate paths s generated, then the optmum paths set s selected usng GA n the second phase. Two codng (representaton) schemes are proposed, namely, the fxed strng length codng whch s smple and gves a better soluton, but requres more computatonal tme, and the varable strng length codng whch can handle smultaneously selecton of the path set sze and the set of paths, but requres complex codng. A hybrd scheme ntegratng the fxed strng length codng and varable strng length codng s necessary n analyzng whether a hybrd scheme gves better solutons than the proposed two codng schemes. Bell et al. (2002) proposed the genetc algorthms for optmzaton of transt networks, especally bus networks. In case of transt network desgn problems, t s relatvely smple to search for the network desgn solutons, because the capacty of a transt network s generally fxed, and traffc nteractons on the bus network are much smpler than general transportaton networks. Furthermore, snce the proposed genetc algorthm s not senstve to the ntal traffc assgnment, the proposed model suggests an almost globally optmal soluton. Drezner and Salh (2002) have compared the performance of metaheurstcs, such as descent algorthm, tabu search, smulated annealng and genetc algorthm, for the one-way and two-way network problem to fnd the best confguraton of the network, so as to mnmze the total travel tme of all users, based on the followng smple objectve functon: Z( x)= f - d ( x ), where (, j ) are nodes;, j j j f j s the number of vehcles travelng from orgn to destnaton j ; x s the vector of varables x j ; x j s a varable defnng a partcular one-way confguraton for the 32
network for (, j) ' L ; L s the set of all lnks; d ( x ) s the shortest dstance from orgn to destnaton j for the network defned by x. The shortest paths between all par of nodes are found by usng dynamc programmng. For more detaled algorthms of metaheurstcs, the reader s referred to the paper (Drezner and Salh, 2002). For small problems, all algorthms are able to fnd the best soluton. In case of computaton tme, a descent algorthm was the best and a GA takes longer than others. For large problems, however, GA was found to outperform other algorthms n fndng the best soluton, whle takng a longer computaton tme (a few seconds) than others. As a result by assessng the qualty of the solutons, the GA was the best for the test problems, followed by smulated annealng, and then tabu search. Even f the tests have been conducted by usng a smple mathematcal functon, ths study shows the superorty of the GA when comparng other metaheurstcs under the same test condton. To the best of our knowledge, none of the works n lterature account for uncertanty n the network desgn problem. Ths problem wll be explore n Chapters 4 and 5, ntally wth statc equlbrum problems (as a startng pont) and then an NDP model n a tme dependent network under user equlbrum condtons wll be explored. j 2.4 Revew of Soluton approaches for dealng wth uncertanty 2.4.1 Stochastc Programmng Stochastc programmng models drectly ncorporate the uncertanty n the formulaton. Two types of models are usually studed: () Chance constraned problems and () Mult-stage recourse problems. The chance constraned formulaton for the dynamc network desgn problem wll be explored n chapter 5. The two stage recourse problem s dscussed here. A smple recourse model can be formulated as an extenson of the general LP problem of the form: 33
Mnmze cx Subject to Ax = b and L x U Suppose that the decson maker specfes a sub vector of x, say x 1, as the frst-stage decson varables. These varables cannot be postponed untl better nformaton s avalable, whereas the remanng varables, say x 2 can be postponed. Wth ths temporal dvson of the problem, the LP constrants are dvded nto two types: one nvolvng only the frst stage varables (x 1 ), and constrants that may nvolve both the set of varables. Thus, the LP can be wrtten as: Mnmze c 1 x 1 + c 2 x 2 Subject to: A 1 x 1 = b 1 Bx 1 + A 2 x 2 = b 2 and L 1 x 1 U 1, L 2 x 2 U 2. It s convenent to thnk of ths determnstc LP as the core problem from whch the stochastc LP wll be derved. A general recourse problem s sad to have complete recourse f for any choce of x 1, a feasble recourse decson s possble for all outcomes '#. The smple recourse formulaton possesses complete recourse. A slghtly less restrctve property s that of relatvely complete recourse whereby one requres that a feasble recourse decson be possble for all outcomes provded the frst-stage decson satsfes the frst-stage constrants. Of course, by usng penalty for devatons from constrant satsfacton, one can ensure complete recourse n any problem. One mportant noton ncorporated wthn any SLP s that of nonantcpatvty, whch reflects the requrement that under uncertanty, the plannng decsons(x 1 ) must be mplemented before an outcome of the random varable s observed. Ths means that the plannng decson s made whle the random varable s stll unknown, and therefore t cannot be based on any partcular outcome of the random varable. An alternate statement of ths requrement gves us the scenaro formulaton as: 34
Mnmze p [ c1 x1 + c2 x2 ] '# Subject to: A 1 x 1 = b 1 B x 1 + A 2 x 2 = b 2 and x 1 x 1 = 0 L 1 x 1 U 1, L 2 x 2 U 2 Note, however that the general recourse problem s fnte-dmensonal lnear program whenever # s a fnte set. However, whenever the random varable s contnuous these formulatons lead to nfnte dmensonal problems. Under, these crcumstances t s more convenent to state the model n the followng decomposed form: Mnmze cx 1 + E [h(x 1 )] Subject to: A 1 x 1 b 1 and L 1 x 1 U 1, where each outcome h ( x) of the random varable h(x) s a functon of the LP defned by the outcome ( c2, A2, B, b2) of the random varables (c 2, a 2, B, b 2 ). Ths mples that h ( x) = Mnmze c 2 x 2 ; Subject to: A 2 x 2 = b 2 - B x 1 and L 2 x 2 U 2. Ths decomposed formulaton s convenent when the sample space # contans ether a large number of sample ponts or a contnuum. Specfcally, the functon E [h(x 1 )] s referred to as the recourse functon. Ths formulaton emphaszes the tme-staged nature of the decson problem. Most of the dffculty assocated wth recourse models may be traced to dffcultes assocated wth evaluatng and approxmatng the recourse functon. In essence, the dffculty n solvng the recourse problem may be attrbuted to the evaluaton of the expectaton of the random LP value functon whch nvolves multdmensonal ntegraton. However, when the scenaros are lmted n nature t may be easy to solve these problems. Notwthstandng the mpractcalty of ths ntegraton of the recourse functon, t possesses one of the most sought-after propertes n mathematcal programmng, leadng to ts wde applcablty, namely convexty. The recourse functon E [h(x 1 )], s convex over ts effectve doman D = 35
{ x X E[ h( x )] } ' 1 < / (for a detaled proof of ths refer to (Wets, 1974)). The general two-stage SLP wth recourse s shown below n detal. th Let SP denote the problem solved at the stage (=1, 2, 3 ). Then the mathematcal statement of the two-stage stochastc LP (SLP) problem wth recourse s: mn T x c x + E [ Q( x, )] s. t : Ax = b ( 1 SP ) x 0 where 2 Q( x, ) = mn q( ) T y y SP s. t : T ( ) x + W ( ) y = h( ) where '#represents the number of scenaros (system realzaton), x and y are varables, T( ) represents the technology matrx and W represents the recourse matrx. Further, W, A and b are parameters; T and h are realzaton dependent parameters. 2.4.2 Robust Optmzaton Robust Optmzaton (RO) models are more constructve technques than senstvty analyss, for nstance, and deal wth problems that evolve over tme and where system varablty s a concern. In these approaches, the decson maker s afforded the flexblty of recourse varables. They provde the mechansm for adjustng model recommendatons to account for nformaton realzatons. SLP, however, typcally optmzes only the frst moment of the dstrbuton of the objectve value and gnores the hgher moments and the decson maker s atttude towards rsk. The mportance of controllng varablty (or volatlty) of the soluton (as opposed to just optmzng ts frst moment) s well recognzed n fnance, for nstance, prmarly due to the semnal work of Markowtz (1959). The need for 36
robustness has been recognzed prevously n number of applcaton areas. Paraskevopoulos, Karaktsos and Rustem (1991) propose a capacty plannng model for the plastcs ndustry. They show that the robust approach s superor to the equvalent determnstc demand method. Usng numercal results from a nonlnear programmng capacty plannng model, t was shown that as cauton aganst demand uncertanty ncreases, the varance of the total objectve functon decreases. Sengupta (1991) dscusses the noton of robustness for stochastc programmng models. They develop non-parametrc methods whch are relevant n stuatons of ncomplete nformaton and partal uncertanty. Mulvey et al. (1995) develop robust optmzaton models for large-scale systems, Escudero et al. (1993) presents an RO formulaton for the problem of outsourcng n manufacturng, and Escudero (2000) uses RO technques for studyng water resources system plannng under water exogenous flow and demand uncertanty. The uncertanty n the last work s treated va scenaro analyss and by consderng a full recourse scheme. Due to the specal structure of the problem, the soluton methodology nvolves usng a decomposton framework based on the augmented Lagrangan approach. Kouvels (1995) develop RO models for multnatonal producton schedulng. 2.5 Revew of Onlne Equlbrum Problems Traffc equlbrum models form the core of transportaton plannng n both the evaluaton and desgn of transportaton systems. To mprove these systems, t s desrable to have onlne nformaton capabltes about network condtons whch potentally wll help n reducng each user s travel tme. A precse defnton of the onlne equlbrum problem s postponed to chapter 6. A lterature revew on onlne equlbrum shows very lttle work on such problems. We begn from the early developments of percepton n equlbrum models and revew the latest work on modelng of nformaton provson n transportaton networks. The early approaches ncorporate the concept of perceved travel tme to ncorporate the element of uncertanty nvolved n drver s error n the knowledge of 37
the true travel tme of a lnk/path. Ths led to the development of Stochastc User Equlbrum (SUE) models (Daganzo and Sheff, 1977; Cascetta (1989)). Of the SUE models, the most relevant models to the onlne equlbrum models are the condtonal stochastc user equlbrum models developed by Hazelton (1998). The man proposton of the model s that, a user selects the routes he/she perceves to have the mnmum cost condtonal on all the travelers choces. A numercal example s used to show that ths model gves dfferent results form the equlbrum model developed by Daganzo and Sheff (1977). Both the models acheve the same results when there s low demand. In the transt assgnment lterature, Nguyen and Pallottno (1988), presented a model and algorthm for the transt equlbrum problem employng shortest hyperpaths (the precse defnton of a hyperpath vares somewhat by the specfc mplementaton, but t s essentally a collecton of possble paths between a gven orgn-destnaton par). Another promsng attempt was contrbuted n a book chapter by Marcotte and Nguyen (1998) who developed an approach for traffc equlbrum employng hyperpaths agan for transt networks. Whle ths work focused on transt and capactated networks where an arc may or may not exst upon arrval at a node (as opposed to general nformaton learned en route), valuable nsghts were gven for network hyperpaths and ther applcaton. Cantarella (1997) presented a general fxed-pont approach for statc traffc assgnment models n a general network where the user s choce set nclude hyperpaths. However, by hs defnton, lnk access probabltes are not functons of lnk flows, whch mght not be a realstc assumpton. Furthermore, lnks flows and costs are not treated as random varables, although t s easly magnable that flows on the hyperpaths are stochastc. Instead, some approxmatons are taken n relatng expected lnk costs wth expected lnk flows. Spess and Floran (1989) proposed a transt assgnment model where travelers choose strateges that allow them to reach the destnaton at mnmum expected cost. A label-settng algorthm s developed that solves the 38
transt assgnment problem n polynomal tme. Clearly, ths problem s dfferent from the one studed n ths dssertaton. Most of the hyperpath based models for the transt networks are statc formulatons (e.g. Marcotte and Nguyen (1998)) whch wll also be the focus n Chapter 6. Further, although these models have a drect relaton wth the noton of strategy n transt models, ther source of randomness s drven by bus lne frequences whle the noton of uncertanty n our model s due to costs or capactes whch are clearly dstnct. However, a fnal goal of ths work wll also be to provde nsghts nto developng a modelng framework consderng traffc dynamcs. There are only two works known to ths author whch model traffc dynamcs as part of the onlne equlbrum process. Both these works are stll prelmnary. Gao (2005) proposed a flexble framework for dynamc traffc assgnment where user behavor s dctated by travel polces. At each node, a polcy specfes the next outgong lnk dependng on the current network state; path choce, whch s performed en route, s stochastc. Equlbrum n ths model s acheved when expected delays of actve polces are equal and less than those of nactve polces. Ther concept of polcy s related to that of a strategy (whch wll be dscussed later n Chapter 6) wth the followng key dfferences (1) probablty densty functons descrbng the occurrence of lnk ncdents are assumed to be known; (2) the state of the network s flow-ndependent; and (3) a sngle destnaton s consdered. The second work by Hamdouch et al. (2004) propose a strategc dynamc traffc assgnment model where strategc choces are ncorporated drectly nto the user behavor. They developed a wthn-day model where strategc volumes are loaded onto the network n accordance wth the FIFO dscplne and user preferences. An equlbrum assgnment defnton s acheved when the expected delays of actve strateges s mnmal for every O-D par (smlar to the defnton of our equlbrum model). They prove the exstence of such an assgnment based on a fxed pont formulaton. 39
2.6 Summary A summary of the related lterature shows that substantal work has been done n traffc equlbrum modelng n both the statc and dynamc contexts. However, there s lmted work on the network evaluaton and desgn under uncertan condtons. Further, to the best of our knowledge there s no publshed work on user equlbrum wth recourse where the traveler learns the costs or capactes enroute n makng hs/her routng decsons and the nteracton between demand and supply s studed. The problems studed n Chapters 3, 4, 5 and 6 seek to contrbute to ths mportant area by proposng modelng approaches and related nsghts nto these problems. 40
Part II Evaluaton and Desgn under Uncertan Demand 41
Chapter 3 Two Methodologes for Evaluatng Network Performance under Stochastc O-D Demand Greed s good. Greed s rght. Greed works. Mchael Douglas as Gordon Gekko n Wall Street (1987) Chapter 2 characterzed the dfferent uncertantes n transportaton plannng. Ths chapter proposes two methodologes wth fundamentally dfferent underlyng phlosophes for treatng O-D demand uncertanty n the traffc equlbrum problem. The frst methodology dentfes dfferent technques to approxmate the random OD demand matrx wth a sngle pont estmate that yelds solutons that perform well under uncertanty. The prmary motvaton for ths technque s to solve large nstances of the problem and to provde an alternatve n the absence of a stochastc solver. The second methodology s developed to drectly ascertan the mpact of demand uncertanty on network performance. Three key deas are presented n the second model. The uncertanty n the objectve functon s quantfed for a gven underlyng uncertanty n the nput demand. The relatonshp between O-D uncertanty and lnk/path uncertanty s examned for dfferent dstrbutons. Closed form analytcal expressons for the frst (expected value) and the second moment (varance) of the network performance are derved. These expressons are then verfed on real test networks. Ths technque s descrbed n detal and the results from the model are demonstrated to show ts potental applcablty. Towards the end of ths chapter a bref overvew of the results wll be presented. In addton, other potental methods for solvng ths problem wll be dentfed for future research. 3.1 Introducton and Problem Defnton Transportaton plannng n the presence of uncertan and ncomplete network characterstcs has been a central problem of nterest to the transportaton communty. It s well known that both the long-term and short-term demands n a 42
transportaton network are uncertan and can at best be forecasted based on past trends. Here, short-term demand s defned n the context of studyng day to day/hour-to-hour (real tme) fluctuatons n demand whereas long-term demand uncertanty occurs over a perod of perhaps months/years. The short term demand varatons arse due to daly fluctuatons n actvty patterns, whch are manfested n the orgn-destnaton (O-D) matrx. For long-term demand uncertanty n the traffc equlbrum problem, the demand level s uncertan when some decson must be assessed n the present stage but the demand s realzed n the future. Capturng ths uncertanty and network robustness as defned n ths chapter wll have sgnfcant beneft for transportaton plannng n quantfyng the network performance wth a possblty of drectng future research efforts n the effcent and flexble desgn of such networks. By measurng the network robustness, the planner can desgn the network for a specfc demand realzaton whch accounts for mnmum network performance fluctuatons nto the long term. Further, ths could beneft n the gudance of transportaton polcy measures towards an mproved understandng of long term uncertanty. Prevous efforts have predomnantly concentrated on short-term uncertantes n percepton such as stochastc user equlbrum (SUE) models (Bell, 1995, Hazelton, 1998). More recent work n transportaton modelng concentrated on the development of models that account for relablty n terms of the uncertanty n lnk capactes (Lo and Tung (2003), Chen et al. (2002), Bell (2000), Du and Ncholson (1997)). Lo and Tung (2003) propose a model to study network dsruptons, wthn day traffc operatons under the assumpton that the capacty s unformly dstrbuted usng Melln s transforms. An equally mportant problem s to estmate the mpact of the long term demand uncertanty n the traffc equlbrum problem (TEP) proposed by Beckmann (1952). Ths problem s especally ntractable because there exsts no drect relaton between the plannng demand and the total system travel tme n the convex formulaton proposed by Beckmann (1952). 43
The fundamental queston that ths chapter addresses s: what would be the mpact on the network performance by accountng for the long-term uncertanty n O-D demand? A frst step to answer ths queston would be to address the followng the queston: Can we derve closed form analytcal solutons for the mean and varance of the travel tme when the long term demand s uncertan n the statc traffc assgnment problem? We assume that the set of demand realzatons s a mult-dmensonal random varable, q that follows an unknown probablty dstrbuton. In the frst approach, ~ rs { =,,...,, '# 0 1} we assume ndependent samples, q 1 2 k rs { qrs qrs qrs}, can be taken from ths dstrbuton. Dfferent approxmaton schemes that use the nformaton of the samples to obtan solutons to the TEP under uncertan demand are presented. The approxmaton schemes are used to fnally solve the determnstc TEP. These approaches are useful when a two stage stochastc programmng problem for the TEP s prohbtvely expensve or mpossble. In the second approach, a closed form expressons for the system s expected value and varance are obtaned. By developng such expressons, the stochastc system can be analyzed more drectly. Further, the expressons for system varance facltate robust optmzaton approaches for transportaton networks. 3.2 Methodology 1: Sngle Pont Approxmatons A smple and straghtforward approach to address uncertanty n the TEP s to plug n the stochastc nput wth an approprate estmate and solve the determnstc verson of the problem. However, as shown n Secton 1.2, evaluatng the network performance at the expected value wll not yeld effcent solutons for the dfferent realzatons of the demand. As shown n Fgure 1.1 solvng the TEP for all realzatons of demand wll be underestmated when ths s replaced wth the expected value. In ths methodology, we seek to fnd approxmate determnstc solutons whch wll use the determnstc TEP to produce solutons whch are close 44
to the expected performance (E[f(q 1, q 2,,q k )] n fgure 3.1). We propose seven dfferent approxmaton approaches. These approaches are computatonally effcent although the results may be far from optmal. The key challenge n devsng these approxmaton methods s n fndng estmates that perform better than the sample average. We explore seven approaches essentally producng a sngle estmate of the random parameter. Towards the end of ths secton, we provde a bref dscusson on dscrete approxmatons (usually referred to as scenaro generaton n stochastc programmng lterature) methods whch are applcable to ths problem. In ths approach, the degree of conservatsm depends upon the sze of the problem and the accuracy needed n approxmatng the true dstrbuton. These approaches are very effcent when the dmenson of the random O-D matrx s small. For large networks, the dscrete approxmaton approach becomes computatonally nfeasble to solve. E [f (q 1, q 2,,q k )] Dstrbuton of objectve functon: f (.) f (E[q]) Dstrbuton of q: p (q) q* E[q] q Fgure 3.1 Conceptual fgure to determne the range of approxmaton estmates 45
It s well know that the TEP can be solved effcently under determnstc condtons. We prmarly utlze ths determnstc TEP solver to arrve at an estmate of the network performance under demand uncertanty. Hence, rather than mplementng a computatonally expensve stochastc solver (mplemented n Methodology 3), we seek to dentfy approxmaton estmates around the determnstc solver to produce good solutons. Ths methodology s applcable n stuatons where the stochastc program s undesrable to solve for all demand realzatons: e.g. n solvng the Chcago regonal network for 10,000 realzatons from a normally dstrbuted demand. 3.2.1 Approxmaton of Uncertan Demand by a Sngle Pont Estmate In ths demand approxmaton approach, we generate a sngle estmate of the stochastc demand to be solved by the determnstc TEP solver. The key dea of the approxmaton here s to generate pont estmates n the range (q*-e[q], q*+e[q]), where q* s the expected network performance due to all the demand realzatons as shown n Fgure 3.1. As the TEP formulaton s convex, the closer the estmates are to q*, the better the qualty of the fnal performance measure. We ncorporate a demand approxmaton functon,, such that ^ 1 2 q = ( q, q,..., q,.., q ), where ^ q s an approxmated demand estmate. procedure to estmate the expected network performance s shown n Fgure 3.2. The problem of senstvty of an estmator to the presence of outlers (Hampel et al., 1986) has lead to the development of robust locaton measures. We develop seven approaches for the approxmaton functon: the medan demand, most lkely demand, rsk-neutral trmmed mean, rsk-averse trmmed mean, rsk-prone trmmed mean. Trmmed L-mean and the Tanh mean of the nput demand realzatons. Smlar approxmaton estmators have been used for capacty approxmaton of the resourceconstraned assgnment problem n Toktas (2004) and n obtanng event-related potentals n neuroscence (Leonowcz, 2005). 46 The
Input Demand realzatons (q1,q2,...,qk) Demand Approxmaton Procedure Approxmate Demand Estmate (q^) Solve determnstc TEP Arrve at estmate of Stochastc network performance Fgure 3.2 Demand approxmaton procedure for evaluatng network performance The seven approxmaton schemes are lsted n Table 3.1. Approxmaton Scheme Medan demand 1 Most lkely demand Rsk-neutral trmmed mean Rsk-averse trmmed mean Rsk-prone trmmed mean Trmmed L-mean (TL Mean) Table 3.1 Demand approxmaton functons Functon Functon Descrpton 0 1 02 1 q rs 2 (arg max { I }) 3 4 5 Tanh mean 7 md 2'2 l qrs l '3 '2 (1 5 4) (1 4) 1 5 + (( 5 2 2 0 1 0 1 (041 q 0p 1 rs 0(1 54) 1 0p 1 0p 1 q rs 1 6 5 N5 ( p )( p ) N ( 2 p+ 1) p+ 1(( N 5p q rs q rs (tanh[ k( )] 5 s) q + 5tanh 0 k( 5 ) + sq 2 1 rs < 2 2 rs 47
The frst and second approxmaton functons (medan and the mode) are useful when we have nformaton that the O-D demand dstrbuton s skewed. The medan s the md-value of the set of demand realzatons whereas the mode s defned as the value wth most number of observatons. When the range of the demand varable s large, we dvde the nterval nto sub-ntervals and take the mdpont of the nterval wth the most number of observatons as the mode of the O-D demand realzatons. A recent study by Ozdamar and Kalayc (1999) demonstrated the advantages of medan averagng over conventonal averagng for audtory bran stem responses where hgh numbers of epochs have to be averaged. One of the dsadvantages of medan averagng s that t does not only remove the outlers but uses the rest of the data n the order of the arranged values. It s clear that some useful nformaton mght be lost by ths procedure as compared to other technques, whch employ the data values nstead of ther order. Further, t should be noted that the value of the medan wll not change even wth the addton of a large value to one of the data values above or below the medan. To crcumvent ths shortcomng, we would lke to combne the advantages of the mean and medan averagng. Ths estmator whch les between these two extremes already exsts and s called the trmmed mean. We used dfferent varatons of the trmmed mean such as the Trmmed L mean and the Tanh mean to evaluate the stochastc network performance. It has been observed that the trmmed mean and ts modfed verson, Wnsorzed mean are effcent robust estmators (Stuart and Ord, 1994). To the best of our knowledge, these sngle pont approxmatons have never been reported to estmate the transportaton network performance under demand uncertanty. In the next three approxmaton technques, we select a proporton, p 48 = 4 of the samples for each O-D. The approxmated O-D demand s set equal to the average of the selected samples. The rsk-neutral trmmed mean, 3 selects samples that are concentrated prmarly around the medan and does not consder outlers away from the medan. The rsk-averse trmmed mean functon, 5 takes the p lowest samples away from the medan and the rsk-prone trmmed mean functon
takes the p hghest samples (Toktas, 2004). Applcaton of trmmng lowers the nfluence of extreme data values on the result of averagng. However, unlke the medan, substantal part of the data can be ncluded nto the average. 6 represents the trmmed L (TL)-mean recently proposed by Elamr and Seheult (2003) as a generalzaton of the L-moments (Hoskng, 1990). The equaton 6 represents the connecton between the TL-mean and the trmmed mean. In both the estmators, the extreme observatons are gnored. The man dfference however s that the trmmed mean apples equal weght to the remanng observatons whereas the TL-mean uses hgher weght for the observatons near the medan. Elamr and Seheult (2003) derve analytcal expressons for the varance of the TL-mean and demonstrate that the TL-mean s more robust than other estmators n the presence of outlers. It was found to perform reasonably well especally for normal and heavy-taled dstrbutons. Ths nformaton however s not accurate enough for the purpose of dervng nsghts nto ts applcablty for evaluatng transportaton network performance and further nvestgaton s necessary. The tanh mean, 7 s a new estmator proposed to allevate the problem of nose n trmmed means and medan. The averagng s done usng a hyperbolc tangent functon as shown n Table 3.1, where k s the factor controllng the slope of the weghts for extreme values n data and s determnes the vertcal shft. The robustness of ths estmator les n the fact that all the negatve values are gven a zero weght so that the extreme observatons wll be neglected smlar to other estmators. Such an analytcal structure provdes the beneft of controllng the nfluence of extreme values on the fnal estmate of the averaged tral (Leonowcz et al, 2005). The shape of the weghts assocated wth the values depends on k and s. Ths method can be hghly benefcal by optmzng the parameters of k and s for the traffc equlbrum problem. In order to remove the bas from ths estmator we need to optmze ts performance by choosng values of k and s that provde a good estmate of the network performance by extensve expermental analyss. As wll be dscussed n the next secton, our analyss wth dfferent values of k and s dd not yeld satsfactory results for the traffc equlbrum problem. 49
3.2.2 Expermental Setup Our expermental test s performed on the Soux Falls test network whch has been extensvely used n the lterature (e.g. Chou, 2005). The Soux Falls network has 24 nodes, 76 lnks and 16 OD pars. For ths network, we consder three dfferent values of = {100, 400, 1000} and for each value of, we select 4 = {0.1, 0.25, 0.4, 0.6}. We consder three dfferent probablty dstrbutons for the O-D demand: logstc, exponental and cauchy. For a gven O-D demand, these dstrbutons are shown n table 3.2. Table 3.2 O-D dstrbutons n the expermental setup EXPONENTIAL x 1 5 µ e f x 0 f ( x) = µ 0 o.w. LOGISTIC ( x5 ) 1 5 ( ) e f ( x) where - < <, >0, - < x< = / / / / ( x5 ) 5 2 (1 + e ) CAUCHY -1 2 x 5 7 01+ 1 where -/ < < /, >0, -/ < x< / f ( x) = 0 1 Ths gves us a total of 36 cases for analyss from the correspondng probablty dstrbutons. For each dstrbuton, the proposed approxmaton schemes wll be compared wth the stochastc value soluton for all the sample values. Ths wll be used n comparng the best approxmaton procedure based on the gap for dfferent values of4. 50
3.3 Results For all the dstrbutons shown n Table 3.2 we perform the approxmaton procedures shown n Table 3.1 usng the determnstc verson of the traffc equlbrum problem. Ths was coded and compled usng a gcc compler and the tests were performed on a Pentum Xeon 2.2 GHz machne. The total travel tme from each approxmaton method ( 8 ^a ) s compared wth the expected value of the total system travel tme, whch s represented as 8a = x t ( x ). The dfference between these two measures ( 8 ^a 58a ) wll be used to measure the accuracy of the approxmaton procedures. Ths s represented as the percent devaton n Tables 3.3, 3.4 and 3.5 of the computatonal results. The approxmaton method wth a consstently (n most cases) lower ^ 8a 58a wll be the best approxmaton method. We summarze the results obtaned from each of the dstrbutons n Tables 3.3, 3.4 and 3.5 for problems wth exponental, logstc, and cauchy dstrbuted O-D demands. In these tables we compare the performance measure for dfferent values of 4 for each approxmaton procedure. We test the dfferent dstrbutons at dfferent sample szes (100, 400 and 1000). The results are consstent wth dfferent sample szes and the results wth sample sze 1000 are shown n Appendx A. We do not compare the performance of the approxmaton method 7 because the effcency of ths method greatly depends on the parameters of k and s selected for ths problem. We have performed sgnfcant expermental analyss for dfferent values of k and s and no sngle set of parameters worked well for all values of 4 and a gven dstrbuton. Fndng effcent sets of k and s should be further explored and s left as a future pursut of ths research. 51
Table 3.3 Computatonal results for Cauchy dstrbuton, Sample sze = 100 Functon 4 = 0.1 % devaton 4 = 0.25 % devaton 4 = 0.4 % devaton 4 = 0.6 % devaton 220.94 40.85 220.94 40.85 220.94 40.85 220.94 40.85 1 211.47 43.38 211.47 43.38 211.47 43.38 211.47 43.38 2 207.62 44.41 221.16 40.79 236.36 36.72 469.73 25.76 3 239.24 35.95 287.56 23.01 287.39 23.05 299.73 19.75 4 1057.06 183.01 698.67 87.06 569.24 52.41 472.98 26.63 5 210.36 43.68 216.78 41.96 215.67 42.26 273.47 26.78 6 Table 3.4 Computatonal results for Exponental dstrbuton, Sample sze = 100 Functon 4 = 0.1 % devaton 4 = 0.25 % devaton 4 = 0.4 % devaton 4 = 0.6 % devaton 273.55 29.8 273.55 29.8 273.55 29.8 273.55 29.8 1 234.47 39.83 241.67 37.98 239.47 38.55 267.32 31.4 2 24.29 93.77 59.59 84.71 96.47 75.24 155.1 60.2 3 224.98 42.26 241.98 37.9 289.71 25.65 297.74 23.59 4 1370.58 251.73 940.92 141.47 750.67 92.64 584.63 50.03 5 204.66 47.48 178.89 54.09 256.76 34.11 251.39 35.49 6 52
Table 3.5 Computatonal results for Logstc dstrbuton, Sample sze = 100 Functon 4 = 0.1 % devaton 4 = 0.25 % devaton 4 = 0.4 % devaton 1 4 = 0.6 % devaton 393.45 14.58 393.45 14.58 393.45 14.58 393.45 14.58 274.43 40.42 296.77 35.57 324.12 29.64 298.98 35.09 2 294.23 36.12 314.17 31.8 331.31 28.07 347.2 24.62 3 319.9 30.55 384.55 16.52 382.82 16.89 393.93 14.48 4 1091.58 136.98 760.64 65.13 643.48 39.7 554.5 20.38 5 308.66 32.99 324.17 29.62 348.83 24.27 388.99 15.55 6 The man nsght from the above analyss s that, among all the demand approxmaton procedures studed above, the rsk-averse trmmed mean approxmaton procedure ( 4 ) consstently performs better than all the other approxmaton procedures regardless of the value of 4 53 and the probablty dstrbuton. The ntuton behnd ths result s that although ths method underestmates the true expected network performance, ts assgns traffc n such a way that t compensates for the nherent stochastcty by consderng only those samples whch contrbutes to the true network performance. It s also further observed that 5, the rsk-prone trmmed mean whch s expected to provde a good estmate of the true stochastc performance actually overestmates the network performance by a greater degree as compared to the underestmaton by 4. We also note that for values of 4 greater than 0.4, the rsk-averse trmmed mean performs well as compared to the expected value problem. Hence, for ths method to be useful a value of 4 greater than 0.4 should be chosen. Another measure of performance to test the approxmaton would be the computatonal tme nvolved n applyng each of these procedures. It s clear that the frst two approaches, the medan and the mode are the quckest n computatonal tmes but the accuracy of these estmates s poor. All the other approxmaton
technques requre approxmately the same amount of computatonal tme based on the value of4. In summary, t s concluded the rsk-averse trmmed mean approxmaton procedure performs reasonably well n approxmatng the stochastc traffc equlbrum problem under uncertan demand. The beneft of ths approach s that by choosng a fracton of the O-D demand scenaros and applyng the rsk-averse trmmed mean procedure on the transportaton network, we can get a reasonable estmate of the network performance. The next methodology dscussed n ths chapter focuses on dervng closed form analytcal expressons for the mean and varance of the network performance when the long term O-D demand s dstrbuted normally. In ths method, we solve the traffc equlbrum problem for all realzatons of demand and compare the performance of the derved analytcal expressons wth the true network performance. 3.4 Methodology 2: Analytcal Expressons for Network Performance Experencng Uncertan Demand The second methodology focuses on dervng analytcal expressons for the dfferent moments of network performance for the user equlbrum problem under uncertan demand. In ths technque, we account for all the realzatons of the O-D demand n the formulatons wthout resortng to approxmatng the nput dstrbuton as n Methodology 1. We assume that the demand dstrbuton s..d and normally dstrbuted. We present approxmate expressons for expected value and varance of network performance and ts senstvty. 3.4.1 Our Contrbutons The exstng studes do not account for long term demand uncertanty explctly n the plannng process. As dscussed earler, not accountng for the future realzatons of demand n traffc equlbrum modelng underestmates the network 54
performance. There s sgnfcant need to develop modelng technques to study ths problem systematcally usng analytcal dervable expressons, approxmaton technques (Methodology 1 and samplng technques dscussed later) and smulaton technques. Ths study contrbutes to the lterature by proposng a methodologcal approach n whch closed form expressons can be derved for the traffc equlbrum problem under realstc assumptons for the long term demand. The expressons capture the expected system cost and ts varablty due to the uncertanty n the O-D matrx. An ntutve nterpretaton of these expressons s descrbed under long term demand uncertanty. Extensve computatonal analyss s performed ntally on small networks to gan nsghts nto the problem. Further, computatonal analyss was performed on very large scale networks wth realstc data from transportaton test network bed provded by Hllel Bera (2002). Ths analyss demonstrates the need for capturng uncertanty and n specfc robustness explctly n the plannng stage whch we beleve to be a stronger component to model uncertan demand. 3.4.2 Proposed Model The basc User Equlbrum formulaton wth stochastc long term demand s presented as P1 before proceedng wth further analyss of the problem. Notaton and formulaton Defne t ( ) a x = the unt cost of transportaton on arc a ' A, assumed to be twce contnuously dfferentable n x Z ( x) = the user equlbrum objectve for each demand realzaton ' rs f k = the flow on path k 'K between O-D pars r, s ' R rs a, k = 1 f arc a ' A s on path k between O - D par r,s 0 otherwse 55
x a = the flow on arc a ' A q rs = the uncertan demand for transportaton between O-D par r, s ' R and demand realzaton ' P1: mn Z ( x) = t ( s) ds x a a 0 a s.t. rs fk = qrs k ' K; r, s ' R; ' k rs rs xa = a, k fk r, s ' R; a ' A; k ' K r s k f 0 r, s ' R; k ' K rs k xa 0 a 'A Note that the demand s represented as ~ qrs to dstngush t as a random varable nto the long-term decson makng process. A smplstc way of approachng the problem (whch s common n practce) s to substtute the long term forecasted demand wth ts expected value. However, as explaned before, t s known that when a functon 9 ( q ) s convex, and then ~ rs ~ 9 ( ( )) underestmates E q rs ~ the true system performance whch s E( 9 ( q rs )). Thus, we need a new methodology ~ whch can explctly calculate E( 9 ( q rs )). 3.4.3 Plannng Demand and Performance Measure The proposed method dvdes the uncertanty nto two elements; one concernng the uncertanty n the demand to the path flow, the uncertanty n the path flow to lnk flow and fnally the relatonshp of the uncertanty n lnk flow to 56
network performance. As descrbed later, the proposed technque specfcally derves closed form analytcal expressons for the expected value and the varance of the network performance when the O-D demand s a stochastc varable. Ths approach dffers from prevous studes of uncertanty (relablty) by showng emprcally the relatonshp between demand and lnk flows nstead of drectly assumng the normalty of the lnk flows. Further, the approach derves the network performance robustness and nether uses senstvty analyss (Chen et al, 2002) nor Monte Carlo smulatons as done prevously n accountng for uncertanty. 3.5 Mathematcal Calculatons of Network Performance From secton 3.3 t s clear that the crux of estmatng robustness of the network performance les n computng the expected value and the hgher moments of the quantty of nterest. We use a two pronged strategy for arrvng at the desred result; frstly, we emprcally derve the relatonshp between the stochastc demand and the path flow vector and secondly we use the fundamental probablty and convoluton propertes to derve the relatonshp between lnk flows and network performance. Ths approach gves ether the probablty dstrbuton of the network performance or a pont estmate n terms of mean and varance of the parameter n queston. Further, the beneft of ths approach s ts applcablty n solvng large scale networks n the framework of tradtonal statc assgnment (wth uncertan demand) and avods the use of senstvty analyss and samplng technques to assess network robustness. Before proceedng wth the dervaton of network robustness, the key assumptons n the modelng are dscussed. 1. The demand between all O-D pars r, s ' R s assumed to be dstrbuted wth 2 a normal dstrbuton wth a known mean and varance; N (m, d ). 2. The O-D demand s assumed to be ndependent among all realzatons and O- D pars r, s ' R. 3. The traffc assgnment s solved as an uncapactated network equlbrum problem. 57
Assumpton 1 presupposes that the demand s normally dstrbuted. Ths has an mportant mplcaton n the dervaton of the path and lnk travel tmes. Ths assumpton s relaxed towards the end of the chapter and nsghts nto other dstrbutons wth specfc propertes are dscussed. Other dstrbutons whch were hypotheszed by researchers based on earler drafts of ths work (and conference presentatons) nclude lognormal, Posson or a mxed dstrbuton. The second assumpton mples that there s no correlaton between the dfferent realzatons of a specfc O-D demand and the many O-D pars n the network. Ths assumpton of dstrbutonal ndependence across nter-zonal movements s typcal n network equlbrum; however ntal nsghts and modelng approaches for addressng correlated demand are mentoned n the fnal secton (for more nformaton on ths see Waller, Ukkusur and Duthe; 2004 and Duthe, 2004). Assumpton 3 s the typcal traffc equlbrum problem wthout sde constrants on the capacty. The mpact of capacty constrants on network robustness s outsde the scope of ths chapter. 3.5.1 Dervaton of Path and Lnk travel tme Uncertantes In ths research, we also assume that the lnk cost functon s monotonc and contnuous and s gven as a polynomal functon of the form t ( ) = t[1 + $ ( ) ], C where t s the free flow travel tme on lnk, C s the capacty of lnk whch s assumed to be determnstc, and and are determnstc lnk specfc parameters. Therefore, for the User Equlbrum under uncertan demand formulaton of P1 the objectve functon on ntegraton reduces to x a ta ( a ) d = t0[1 + $ ( ) ] da C a 0 a x a (3.1) 0 Ths ntegrated expresson above s referred as: a 58
1 Z ( x) = $ + t0 xa + [ xa ] a ( + 1) C (3.2) a The above equaton represents the objectve of the UE wth the costs gven by the specfed lnk performance functons. To fnd network robustness, the queston remans as to how to fnd the relaton between the random OD demand and Z(x). As the capacty of each lnk s determnstc, the only varable affected by the change n long term demand would be long term lnk flows on each of the lnk. It s qute ntutve that f the long term projected demand s more then the correspondng lnk flows on all the used paths would correspondngly ncrease or reman the same as compared to the base case n an uncapactated network. From the convoluton theorem, t s well known that the sum of ndependent normal random varables s also a normal random varable. Ths key property can be used n obtanng the relatonshp between O-D demand and lnk flows. In P1, f the demand s assumed to be a random normal varable, then one soluton (although clearly not unque) s that the path flows are also random normal varables wth a specfc mean and varance. Whle clearly not conclusve, ths wll be confrmed n fgure 3.3 wth numercal expermentaton on multple networks. The key nsght n ths reasonng s that an ncrease n demand between two ponts ncreases the congeston on the used lnks (and possbly pushes flow nto the unused lnks) n the network. Further, applyng the same property t s easy to show that the lnk flows, x a are also random normal varables. Ths s compellng however because the lnk flows are the sum of path flows, whch are ndependent normal random varables from the above observaton. Based on the above dscusson, the uncertanty n O-D demand reduces to the uncertanty n the lnk flows. The network performance s shown n (3.2) where the lnk flows are random varables. As explaned, ths work assumes that lnk travel tmes are ndependent of one another as done n most of the prevous studes. Our key nterest from (3.2) s n deducng ts non-central moments. In partcular, we are nterested n the expected value and the varance of the expresson (3.2). It s 59
mportant to clarfy a potental msunderstandng before proceedng. Equaton (3.2) captures the varaton of the user equlbrum objectve functon. It s well known that the user equlbrum objectve functon does not have an ntutve meanng; however, we show n appendx B the relatonshp between the user equlbrum objectve functon and the total system travel tme. Ths s used here as a measure of the network performance as mentoned before. In short, we show that the user equlbrum objectve functon underestmates the network performance for every demand scenaro. To fnd the nose (volatlty) and the expected value of the objectve functon due to the stochastcty n the long term demand, each of the lnk flows are derved to be normal random varables dstrbuted wth mean m and standard devaton d for all n A. The expectaton operator s appled on (3.2) to obtan the mean of the objectve functon: 1 E[ Z ( x) ] = E[ $ + t0 x + [ x ] ( + 1) C ] : : t $ t E x + E x 0 + 1 0 [ ] [ ( )] ( + 1) C t $ t m + E x 0 + 1 0 [ ( )] ( + 1) C (3.3) 1 Now, the key ssue s to fnd E[ ( x + )]. Ths s calculated by transformng the central moments of the normal dstrbuton to non-central moments. In partcular, we deduce the expectaton by usng the expectatons of the form E x 5 m + 1 [( ) ] E x + = 1 [ a ] / + 1 ( xa ) ( xa ) dxa 5/ (3.4) where ( x a ) represents the normal dstrbuton functon for lnk a. Ths s calculated by the moment about the mean (the central moments of the normal dstrbuton). The 60
dervaton of the central moments of the normal dstrbuton s shown n Appendx B. The general result for moments of a multvarate Gaussan dstrbuton has been a problem of nterest and has only recently been solved wth success. A recent result by Trantafyllopoulous (2002) gves a more effcent procedure for calculatng the moments of the multvarate Gaussan dstrbuton, when correlatons among the lnk flows are ncorporated. We propose an alternatve method for the unvarate Gaussan dstrbuton whch s smpler yet provdes an elegant closed form soluton for ths problem. We convert the non-central moments n terms of the moments of the standard normal varable usng Bnomal Theorem. Now, by Bnomal expanson t follows that: E x + = 1 [ ( )] E x 5 m + m + = 1 [ ( ) ] + 1 k + 15k E 0 ( x 5 m ) m k 1 (3.5) k + 1 + 1 + 15k k = m E ( x m ) k 5 k = 0 As shown n Appendx A, the central moments of the lnk flows are equal to k 2 k 2 d k + 1 k E[( x 5 m ) ] = 9( ) 7 2 (3.6) where k s even, 9(.) represents the Gamma functon. For odd k, the moments are zero. Lnk 60 µ: 1706.48 : 306.6286 Lnk 54 µ: 1795.02 : 306.6286 6 0 5 4 61
Lnk 52 µ: 3124.44 : 273.16 Lnk 37 µ: 2474.49 : 334.87 5 2 3 7 Lnk 30 µ: 3124.40 : 273.15 Lnk 20 µ: 3841.69 : 320.29 3 0 2 0 Lnk 7 µ: 2474.49 : 334.87 Lnk 6 µ: 6838.27 : 478.93 7 6 Lnk 2 µ: 4247.44 : 410.68 Lnk 1 µ: 2148.98 : 292.22 2 1 62
Lnk 9 µ: 3613.87 : 284.98 Lnk 16 µ: 3841.69 : 320.30 9 1 6 Fgure 3.3 Testng the hypothess: Representatve sample of the lnk flows on the Soux Falls test network when the O-D s normally dstrbuted 3.5.2 Computng Network Robustness Based on (3.5), closed form analytcal expressons for the mean and varance of the network performance are derved. These are smplfed further usng the propertes of Gamma functons. For values of = 2 and 4, the expressons are explaned below. Performng the bnomal expanson as n (3.5) and usng (3.6) yelds k + 1 + 1 2 + 15k m k k = 0 k 2 d k + 1 9( ) 7 2 (3.7) a 9 ( a + 1) Usng the property =, (3.7) s smplfed as k k! 9( a 5 k + 1) E x + = 1 [ ( )] k + 1 2 + 15k k 2 m d 9 ( + 2) k + 1 k! 7 k = 0 9( ) ; for even k 9( 5 k + 2) 2 k + 1 k! 7 Ths s further smplfed by usng the property9 ( ) = 2 k k 2 ( )! 2 whch yelds 5k + 1 1 E[ ( x + 2 + 15k k 2 m d 9 ( + 2) )] = ; k = 0, 2, 4,... (3.8) k k = 0 ( )! 9( 5 k + 2) 2 63
Hence, fnally the system s expected cost gven fnte realzatons of demand s 5k 1 2 + 15k k 0 + t0$ 2 m d 9 ( + 2) 1 [ Z( )] = 0 o + ; 0,2,4... ( 1) C k 1 = + k = 0 9( 5 k + 2) E x t m k 0 ( )! 1 0 2 1 The above result s demonstrated for two typcal cases, one for lnear cost functons and the other for typcal BPR type functons wth =4. (3.9) When =1, for all arcs the above expresson reduces to, t0$ 2 2 E[ Z( x)] = 0tom + { m + d } 1 (2) C and for =4, (3.10) t0$ 5 3 2 4 E[ Z ( x) ]= t0 m + [ m + 10m d + 15 md ] (3.11) ( 1) C + The above expresson (3.9) shows that as the mean and varance of the lnk flows ncrease there s a correspondng ncrease n the expected value of the network performance, whch would descrbe a trend towards hgher total system travel tme. Ths result s consstent wth the expectaton that as there s greater expected travel tme on the ndvdual lnks (congeston), there s a correspondng ncrease n the average total system travel tme. Now, the next task s n computng network robustness n terms of the varance of the network performance. Defne the varance of the system as: Var x E x E x 1 2( 1) 2 1 [ + ] [ + ] [ + = 5 ] 2( 1) E[ x ] The only expresson that needs dervaton s +. Ths can be computed usng the same trck used to derve (3.8). 64
Calculatng ths yelds, 5k 2 + 2 2 2 + 25k k 2 + 2 2 m d 9 (2 + 3) [ ] = ; k = 0, 2, 4,... k k = 0 9 (2 + 35 k) E x ( )! 2 Combnng both the terms gves the varance as Var x 2 5k 5k 2 + 2 2 2 + 25 k k + 1 2 + 15 k k 1 2 (2 3) 2 ( 2) + m d 9 + m d 9 + [ ] = 5 ; k 0,2,4,... k k 0 (2 3 k) k = = 9 + 5 k= 0 9( 5 k+ 2) (3.12) ( )! ( )! 2 2 Smlarly, the varance of the system s gven as: 5k 2 + 2 2 2 + 25k k 0 2 m d 9 (2 + 3) 1 0 k k = 0 ( )! 9 (2 + 3 5 k ) 1 0 2 2 2 1 2 t0$ Var[ Z( x)] = 0t0m 1 + 2 2 2 ; k = 0,2, 4,... 0 ( + 1) C 5k 1 0 + 1 2 + 15k k 2 m d 9 ( + 2) 1 0 5 k 1 0 k = 0 ( )! 9( 5 k + 2) 1 0 2 1 (3.13) Equaton (3.13) shows the varablty of the objectve functon wth the mean and varance of the lnk travel tmes. It s consstent wth our expectaton that the varablty of the whole system would ncrease f there s a larger varablty on each of the lnks. For example, f the demand s unknown wth a larger varance the system has to be desgned takng nto consderaton the ncreased varance n travel tme apart from the ncrease n expected total system travel tme. The expresson (3.13) s demonstrated on two typcal values of. When =1, for all arcs the above expresson reduces to, 65
and for =4, 2 2 2 2 t0$ 4 2 2 4 2 2 2 Var[ Z( x)] = 0t0 d + 2 {( m + 6m d + 3 d ) 5 ( m + d ) } 1 4C 2 2 10 8 2 6 4 4 6 2 8 10 2 2 t0$ ( m + 45m d + 630m d + 3150m d + 4725m d + 945 d ) Var[ Z( x)] = 0t0d + 8 5 3 2 4 2 1 (3.14) 0 25 C 5 ( m + 5m d + 15 m d ) 1 Another measure to assess the varablty of the network s the coeffcent of varaton of the performance measure. Ths s expressed as the quotent between the standard devaton and the expected value of the objectve functon as: COV [ Z( x)] = 2 0 2 9 (2 + 3) 2 9 ( + 2) 1 0 1 0 ( + 1) C ( )! 9 (2 + 3 5 k ) ( )! 9( 5 k + 2) 1 0 2 2 1 5k 5k 2 2 2 2 2 2 2 k k 1 2 1 k k 2 t0$ + + 5 m d + + 5 m d t0 m + 2 2 5 k k k = 0 k = 0 5 k + 1 2 + 15 k k 0 t0$ 2 m d 9 ( + 2) 1 0tom + ( + 1) C k 1 0 k = 0 9( 5 k + 2) 1 ( )! 2 (3.15) The mean, varance and the coeffcent of varaton provde mportant nformaton regardng the mpact on the network equlbrum soluton due to random O-D demand. In other words, f the O-D uncertanty can be captured, then the varaton of the equlbrum soluton can be acheved approxmately by the methodology descrbed here. 3.6 Numercal Experments The analytcal expressons derved n the prevous sectons are verfed and nsghts are derved nto the behavor of network equlbrum under stochastc demand. The performance of the networks s examned through numercal experments on multple networks. Intally, small networks are tested to compare the results obtaned from numercal expermentaton and the analytcal expressons derved n the prevous secton. The advantage of usng small networks s t enables us to enumerate all potental mpacts on all the paths n the network. Later, the above methodology wll be used to evaluate demand uncertanty on well known test networks. 66
The normal random demand s generated for each O-D par by the Box-Muller transformaton. The nput varance matrx s generated by defnng a scalng factor, whch factors the expected demand to generate a varance matrx. For example a scalng factor of 0.01 produces random normal demand matrces wth a greater range of uncertanty than those wth a scalng factor of 1. The scalng factor represents the varablty n O-D demand. The results for each network are compared wth the derved expressons n Secton 3.6.1. In addton t wll be demonstrated that solvng the problem at the expected value of demand sgnfcantly underestmates the network wde mpact and thus would lead to sub-optmal plannng. Specfcally, we are nterested n testng the followng hypothess: (a) How well does (3.9) represent the expected system cost for the traffc equlbrum as the network sze grows (b) How well does the varance expresson (3.13) approxmate the robustness of the traffc network as the network sze grows (c) What s the varaton n the system expected and robustness as the scalng factor s vared. (d) What s the closeness of the approxmate expressons for mean and varance of the system cost as the scalng factor s vared (e) What s the ncrease n computatonal tme as the network sze vares for the same sample sze The methodology for evaluatng robustness s shown n Fgure 3.4 67
Select a Sample Sze n, and Scalng Factor based on possble demand realzatons of future socoeconomc, demographc and land use growth Generate Random Demand Matrces p = 1 Perform User Equlbrum Assgnment For scenaro p If p = n If p < n; p++ Compute the Mean and the varance of the Lnk flows Compute the Mean and varance of the Traffc Assgnment objectve and compare the dfference between the results from Equatons and experments Report the Dfference Between the results Fgure 3.4 Framework for evaluatng the robust network performance The essental characterstcs of the networks are shown n Table 3.6. It also shows the computaton tme on a Pentum Xeon 2.2 GHz machne wth 4GB RAM for dfferent network szes and ten thousand demand realzatons to a relatve gap of 0.001. Ths was solved usng the generalzed nonlnear optmzaton method of Frank and Wolfe (FW) (1956). These results show that for practcal problems of nterest, t would be almost mpossble to evaluate the robustness of the traffc 68
equlbrum unless ths subroutne can be solved extremely quckly. Effcent samplng technques to solve large networks so that they represent system uncertanty wthn a good confdence level can be developed to overcome ths, however ths s beyond the scope of the proposed work. Intal drectons for applyng samplng technques can be found n Unnkrshnan et al. (2004). Table 3.6 Computaton Tme of varous test networks Network Zones Nodes Lnks OD pars CPU Tme Relatve Gap (hours) Nguyen Dupus 13 13 19 16 3.04 0.001 Soux Falls 24 24 76 528 3.28 0.001 Barcelona 110 1020 2522 7922 82.74 0.001 Wnnpeg 147 1052 2836 4345 59.44 0.001 Chcago Sketch 387 933 2950 93513 1124.35* 0.001 *Ths value was arrved at by aggregatng the tme taken on 12 machnes n a dstrbuted envronment 3.6.1 Results The frst demonstraton wll be performed on a small network wth 4 nodes and 5 lnks, shown n Fgure 3.5. The network has nne OD pars and the sample sze used for testng the results s 10,000. In ths study we do not conduct a systematc study of how to choose the optmal sample sze, the sample sze chosen s ad-hoc. However, t was observed that ncreasng the sample sze for demand realzatons has margnal mprovement on the soluton qualty beyond 10,000 samples. For large networks, we choose a smaller sample sze because of the computatonal tme requred to run these networks (see Table 3.6). We test ths small network wth dfferent scalng factors: 1, 0.5, 0.1, 0.05, 0.01 and dfferent levels of expected demand. Ths analyss allows us to characterze two thngs; the behavor of the network uncertanty wth the ncrease n mean and varance of the O-D demand (wthout correlatons) and the tghtness of the analytcal results wth the decrease n the varance of the O-D demand. It was observed that n dfferent scalng factors the soluton at the expected value of 69
demand s sgnfcantly lesser than explctly capturng the robustness of the network performance. It was also observed that all the lnk flows have a p-value greater than 0.5, ndcatng that they are normal when the nput demand s normal. Two of the lnks have zero flow n all the cases. C 1 = 1+0.15(x 1 /20) 4 B C 4 = 1+0.15(x 4 /30+z 2 ) 4 A C 3 = 1+0.15(x 3 /30) 4 D C 2 = 1+0.15(x 2 /20+z 1 ) 4 C C 5 = 1+0.15(x 5 /40) 4 Fgure 3.5 Test Network 1 Table 3.7 Comparatve solutons of network performance from experment and analytcal expressons for the Test Network 1 Sample Sze = Expermental Expresson Result % Dff % Dff Expm Expresson 10000 Result Scalng Factor Mean St. Dev Mean St. Dev Mean Stdev COV COV 0.01 6396.51 635.54 6396.20 611.13 0.0048 3.8402 0.099 0.096 0.05 6397.38 885.22 6397.37 848.25-0.0005 4.1765 0.138 0.133 0.1 6405.38 630.06 6405.42 608.95-0.0007 3.3503 0.098 0.095 0.5 6394.42 284.13 6394.43 276.22-0.0001 2.7868 0.044 0.043 1 6392.01 199.39 6392.03 199.72-0.0003-0.1689 0.031 0.031 2 6395.74 141.35 6395.73 147.22 0-4.1600 0.022 0.023 The results from the Nguyen Dupus test network and the Soux Falls test network are shown n Tables 3.8 and 3.9 respectvely. In the Nguyen Dupus test network t can be observed that the mean expresson gves tght results where as the varance expresson underestmates the true network varance. It was further observed that three of the nneteen lnk flows are not normal n all the scenaros. Smlar trends are 70
observed wth the Soux Falls test network. The Barcelona test network was tested wth a scalng factor of 1 and the expected value of objectve functon s 3.5 10 7 unts and the expected value from the expresson s around 4.06 10 7 unts. The analytcal expresson underestmates the varance by about 17.91%. One key mplementaton dffculty wth the Barcelona and Wnnpeg networks s n performng the analytcal expresson calculatons because of the varable betas n the cost functon for all the lnks. The test on the small Chcago network reveals that about 40% of the lnk flows are not normal. We solve about 4725 samples on the Chcago network n a dstrbuted envronment. The experment was conducted wth a scalng factor of 0.1. The mean from the expresson s 6.99 10 7 unts whereas the expected network performance from experment s around 8.43 10 7 unts. The robustness s sgnfcantly underestmated by around 75% by the expresson. Table 3.8 Network Performance Comparsons from experment and analytcal expressons for the Nguyen Dupus Test Network Sample Sze = Expermental Result Expresson Result % Dff % Dff Expm Expre 10000 Scalng Factor Mean St. Dev Mean St. Dev Mean Stdev COV COV 0.01 211523.8 58805.99 212989.3 17878.44-0.693 69.598 0.278 0.084 0.05 204603.9 26504.09 205279.7 7758.751-0.33 70.726 0.13 0.038 0.1 204146.8 18841.26 204786.2 5459.374-0.313 71.024 0.092 0.027 0.5 203825.5 8334.434 204420.6 2429.324-0.292 70.852 0.041 0.012 1 203625.1 5954.456 204198 1726.577-0.281 71.004 0.029 0.008 2 203626 4156.079 204197.4 1217.577-0.281 70.704 0.02 0.006 Table 3.9 Comparatve results for the Soux Falls Network Sample Sze = Expermental Result Expresson Result % Dff % Dff Expm Expre 10000 Scalng Factor Mean St. Dev Mean St. Dev Mean Stdev COV COV 0.01 125643.4 16088.87 129734 12000.6-3.256 25.411 0.128 0.093 0.05 124759.8 26504.09 127226.6 9002.056-1.977 66.035 0.212 0.071 0.1 124799.2 8156.194 128460 5043.089-2.933 38.169 0.065 0.039 0.5 124785 7402.472 128495.4 2240.781-2.973 69.729 0.059 0.017 1 124682.2 7357.296 128424.4 1587.547-3.001 78.422 0.059 0.012 2 124711.7 7348.311 128485.5 1114.715-3.026 84.83 0.059 0.009 71
2 1 5 3 1 2 4 14 3 8 4 11 5 15 6 6 9 13 23 12 16 19 7 35 10 31 9 21 8 25 26 24 22 47 33 27 48 12 36 11 32 10 29 16 51 49 52 30 34 40 28 43 17 17 20 55 50 7 18 54 18 37 38 53 58 41 57 14 44 15 45 19 42 71 72 46 67 23 22 59 61 70 63 73 76 69 65 68 56 60 13 74 24 66 21 62 20 39 75 64 Fgure 3.6 Soux Falls Test Network 3.6.2 Key Insghts The man nsght from the above analyss s that the assumpton of lnk flow normalty s an approxmaton wth the ncrease n network sze. Hence, the resultng expressons derved n 3.9 and 3.13 approxmate the true stochastc network performance. The mean expresson was found to be tght n all the networks whereas the robustness expresson was not very tght. However, the robustness expresson 72
gves us correct ordnal rankngs n terms of makng nvestment decsons based on robustness. It was also observed that the network uncertanty s postvely correlated wth the OD uncertanty. An ncrease n OD uncertanty ncreases the network wde uncertanty. The derved analytcal expressons can be used to estmate the mean network performance under stochastc OD demand condtons. As the COV of the OD ncreases t s observed that the COV of the network ncreases. In other words the COV of the OD and the network are postvely correlated. 3.7 Concludng Remarks Ths research developed two methodologes for quantfyng network performance when the long term O-D demand s stochastc. The presented methodology dffers from earler work on uncertanty by developng sngle pont approxmatons and closed form approxmate expressons for evaluatng network robustness rather than usng senstvty analyss or crude Monte Carlo smulatons. The relatonshp between the demand and path flows n the network s obtaned by numercal expermentaton and the expected value and the varance of the system s obtaned n a closed form from uncertan lnk flows. A clear advantage of the proposed approach s that such drect expressons greatly reduce the effort requred to analyze stochastc network condtons for plannng applcatons and should facltate work on numerous problems such as robust network desgn. Numercal results show that the expresson for expected network performance s tght, but the network robustness (varance) s an approxmaton. The results demonstrate that evaluatng network robustness usng expresson (3.13) becomes an approxmaton wth the ncrease n network sze. Two reasons contrbute to ths approxmaton; as the network sze grows the assumpton of lnk normalty was observed not to hold on all the lnks when the demand s dstrbuted normally. Further, sgnfcant correlatons could exst among dfferent O-D demands and neglectng ths could lead to approxmate results from the expresson as observed. 73
The derved analytcal expressons are tractable, n the sense that t can be talored for other demand dstrbutons. The only requrement s the addtve property of random varables to the same random varable. Prelmnary tests show that these results can easly be extended when the demand s Posson or Cauchy dstrbuted. Addtonally, the expressons can easly be appled to large networks to get an estmate of the mpact of plannng demand uncertanty n transportaton plannng applcatons. The numercal tests have demonstrated the ablty of the derved expressons n computng the mpacts of demand uncertanty on well known test networks. The shape of the network performance under uncertanty s dervable from the dervaton (Secton 3.5.1) whch wll gve an overall understandng beyond the mean and varance. The coeffcent of varaton s observed to be a good ft from the expressons to the actual condtons whch gves a good confdence on the overall network performance varaton. There are sgnfcant opportuntes for future research along the lnes of ths work. An mmedate drecton of ths work to gan better nsghts nto modelng uncertanty would be to capture O-D demand correlatons due to the underlyng external factors. Smlar closed form analytcal expressons ncorporatng correlatons could be derved and statstcal analyss of such results may provde nsghts nto the effect of OD correlatons on network uncertanty. Intal results show that ncorporaton of O-D correlatons gves better results (Waller et al, 2004, Duthe, 2004). To gan practcal nsghts nto network uncertanty for transportaton plannng, efforts must be expended n ascertanng O-D correlatons by ether data collecton or statstcal testng of network flow data. As observed n Table 3.6, the proposed methodology s prohbtvely expensve when the number of demand realzatons s huge. An alternate method s to develop samplng methods or bounded solutons for measurng network performance. The am of such methods should be to reduce computatonal tme at the same tme gvng a near optmal soluton wth good confdence nterval (Unnkrshnan et al, 2004). There are further opportuntes to extend the proposed model to ncorporate capacty 74
uncertanty n addton to stochastc demand. These types of models are applcable n evaluatng network performance under sesmc condtons when the demand n addton to the capacty cannot be estmated wth certanty (Km et al., 2005). Another specfc methodology can be explored to account for demand uncertanty n the traffc equlbrum problem. The Response Surface Analyss (RSA) can be another methodology that can be appled for the purpose of provdng the senstvty nformaton about the network performance wth the change n demand. In other words t gves us nformaton that estmates the objectve functons Z(x) behaves n the neghborhood of x*. By usng the traffc equlbrum problem as a black box valuable modelng nsghts can be obtaned from the response surface. Ths, n combnaton wth an effcent samplng technque lke the Latn Hypercube method should provde effcent and valuable nsghts nto the traffc equlbrum under stochastc nputs. A natural extenson of ths work on evaluaton s n developng models n the strategc stage: network desgn solutons that account for network robustness whch wll be the focus of the next chapter. We beleve that these solutons wll be dfferent from the solutons obtaned by determnstc demand between O-D pars. It s hoped that ths research wll nstgate nterest n developng soluton methodologes to solve these hard problems n capturng network uncertanty whch wll be the focus of the next chapter. 75
Chapter 4 Desgnng Robust Networks for Improved Performance Varance s everythng. Queung Theorst 4.1 Introducton Ths chapter of the dssertaton complements the prevous chapter on evaluatng network performance by proposng a model to assst the network manager (planner) n decdng the optmal nvestments to ensure a certan level of performance when faced wth uncertanty. Ths s a strategc decson makng stage n the transportaton plannng process. We term ths problem the robust network desgn model (RNDP). The mathematcal formulaton for RNDP s ntroduced and a soluton approach based on evolutonary algorthms s proposed. Computatonal results on three test networks are presented to demonstrate the feasblty of ths approach. Network Desgn s pervasve n many applcaton contexts due to ts ablty to nfluence the full herarchy of strategc, tactcal and operatonal decson-makng n any mult-stage system. As s well known, transportaton network desgn s defned as selectng arcs n the network G(N, A) for addton (dscrete) or mprovement (contnuous) to mnmze the total systems cost subject to a budget constrant, together wth the requrement that the flows satsfy the user equlbrum condtons. The lterature presents many formulatons and soluton algorthms to solve ths nonlnear, non-convex mathematcal program whch s dffcult to solve optmally (Chou, 1999; Davs, 1993; Dantzg et al., 1976; Fresz et al., 1993; Solank et al., 1998). In ths research, we study an mportant varaton of the network desgn problem. We account for the uncertanty n the O-D demand and demonstrate that accountng for uncertanty n long term demand results n sgnfcantly dfferent network desgns as compared to the determnstc equlbrum network desgn. We deal wth the dscrete verson of the network desgn problem under demand uncertanty. The developed model determnes robust network desgn plans for a 76
large traffc network wth many zones and nterconnected regons. In addton, the proposed model s flexble to account for the planner s desred robustness based on the senstvty of the total network performance. The methodology used n ths chapter yelds solutons that are less senstve to the future demand realzatons than the classcal determnstc NDP, by accountng not only for the expected value but also for the varance of the total system travel tme (TSTT). It s shown that ths model s effectve and solvable for large sze network desgn problems. As revewed n chapter 2, there s a vast amount of lterature on developng formulatons and solutons algorthms for the determnstc NDP. However, there s lmted lterature n the transportaton area dealng wth long-term demand uncertanty and robust capacty plannng. Prevous work n ths area deals wth stochastc programmng models, whch deal wth uncertanty n transportaton network desgn (Waller and Zlaskopoulos, 2000). The formulaton s a sngle level stochastc lnear program (SLP) for the dynamc network problem ncorporatng the cell transmsson model as the traffc flow model. The stochastc programmng models guarantee a soluton that s best n the sense that t mnmzes the expected cost of the system for a gven dstrbuton of the uncertan demand. However, the expected cost mnmzng solutons are not necessarly robust as they do not account for hgher moments of the total system travel tme. The prmary contrbuton of ths work s to provde a mult-objectve Genetc Algorthm (GA) methodology for transportaton NDP under demand uncertanty. In ths chapter we defne a methodology, based on genetc operators, that allows the generaton of robust solutons from an ntal set of solutons for the RNDP n order to mprove the performance of the traffc network. In specfc, we develop a methodology whch ncorporates uncertanty nto the statc transportaton network desgn problem, wth a dstnct emphass on robust optmzaton. The formulaton presented n ths research accounts for future O-D demand uncertanty whch are realzed nto the future when the network desgn decsons are made n the present tme perod. A robust soluton to ths problem s defned as one that 77
mnmzes network volatlty for dfferent realzatons of the demand. It s expected that these methodologes wll gve tremendous capabltes n desgnng transportaton networks to account explctly for the mean-varance and other rsk measures (Szego, 2005) of the objectve functon. 4.2 Robust Transportaton Network Desgn Model 4.2.1 Robust Optmzaton (RO) Explct consderaton of uncertanty n the NDP s a crtcal aspect of nvestment decson makng. Not accountng for the uncertanty can lead to suboptmal nvestments whch may prove very expensve n terms of the level of servce of the overall transportaton network f the antcpated soluton s not realzed. A clear dstncton between Robust Optmzaton (RO) and Stochastc Programmng (SP) methods s needed as t defnes ther applcablty. Although, both the RO and SP afford the NDP problem to account for uncertanty, the RO model has lttle senstvty wth demand. Gven the reslency of the RO soluton the expected cost of ths soluton may be hgher than that of the SP soluton. RO models allow n determnng the tradeoff between the expected performance of the network aganst hgher moments based on how the network behaves n hgh consequence scenaros. An alternate defnton of robustness recently ntroduced by Bertsmas and Sm (2003, 2004) proposes an approach that nvestgates the flexblty of the robust solutons n terms of probablstc bounds on the constrant volatons gven the bounded uncertanty n the nput data. Ths defnton s however not used n ths chapter. The mportance of controllng varablty (volatlty) of the soluton (as opposed to just optmzng ts frst moment) s well recognzed n fnance prmarly due to the semnal work of Markowtz (1959). The need for robustness has been recognzed prevously n number of applcaton areas. Paraskevopoulos et al. (1991) proposed a capacty plannng model for the plastcs ndustry. They showed that the robust approach s better than the equvalent determnstc demand method. Usng 78
numercal results from a nonlnear programmng capacty plannng model, t was shown that as cauton aganst demand uncertanty ncreases, the varance of the total objectve functon decreases. Sengupta (1991) dscusses the noton of robustness for stochastc programmng models. They developed non-parametrc methods, whch are relevant n stuatons of ncomplete nformaton and partal uncertanty. Escudero et al. (1993) presented an RO formulaton for the problem of outsourcng n manufacturng. Escudero (2000) uses RO technques for studyng water resources system plannng under water exogenous flow and demand uncertanty. The uncertanty was treated va scenaro analyss and by consderng a full recourse scheme. Because of the specal structure of the problem, the soluton methodology nvolves usng a decomposton framework based on the augmented Lagrangan approach. Kouvels (1995) develop RO models for multnatonal producton schedulng. Malcolm and Zenos (1994) develop an RO model for the capacty expanson of the power systems under uncertan load forecasts. They use pecewse lnear load curves to arrve at the trade-offs between the nablty to recover fully costs for excess capacty versus the need to purchase outsde power. Most of the applcatons descrbed above use varance as a measure of soluton robustness and establsh mean-varance tradeoff. A smlar defnton of robustness wll be used to obtan capacty expanson polces for the transportaton network desgn problem under demand uncertanty. In some problems, the mean-varance tradeoff may not be meanngful. For example n mnmzng the varance of user s travel tme n descrptve models or n the mean-varance analyss of a system optmum objectve (Lst et al., 2003). In the RNDP, however, the performance measure s a system wde cost and the uncertanty s long term, a mean-varance analyss provdes a good estmate of the effect of underlyng uncertanty. The formulaton proposed here accomplshes ths goal usng an evolutonary algorthm to evaluate NDP solutons. The prmary contrbuton of ths work s n extendng the RO concept to transportaton network desgn problems and n developng a methodology to 79
demonstrate the value of robustness that enables better strategc transportaton plannng. Prevous models for the NDP whch account for uncertanty deal wth stochastc programmng technques (Waller, 2000, Waller et al., 2000). Waller et al. (2000) evaluates the traffc assgnment problem under demand uncertanty showng that the expected demand cannot be used wth dsregard to varance n demand forecasts. Nakayama and Takayama (2003) propose an analytcal approach to deal wth demand uncertanty n traffc networks. They assume that the demand follows a bnomal dstrbuton and derve a stochastc network equlbrum model to estmate varances of the lnk travel tmes and evaluate the network uncertanty. Ukkusur and Waller (2004) develop closed form analytcal expressons for the expected and varance of the system costs for the traffc equlbrum problem when the long term demand s uncertan. However, the expressons for the varance are approxmate as the network sze grows. Clark and Watlng (2004) estmate the probablty dstrbuton of total system travel tme consderng the day-to-day varatons of traveler O-D matrx. The travel tme dstrbutons moments are obtaned usng the results from Isserls for a multvarate dstrbuton. All these papers deal wth network evaluaton and not wth RNDP as addressed n ths research. 4.2.2 Choce of Robustness Measure There are number of uncertantes n the NDP. Typcal uncertantes nclude the uncertanty n O-D demand, network capacty and lnk cost functon parameters. Further, the uncertanty can also be classfed based on the tme-frame of plannng. The modelng approaches would be dfferent for long term (strategc) and short term (operatonal) uncertantes n the network desgn problem. Uncertanty n long term demand s consdered here. Ths s mportant because the nvestment made n the present tme has a sgnfcant effect nto the future and developng solutons whch are reslent to future realzatons s desrable. The control varable for the problem s the demand, whch s realzed n the future. Ths s denoted by n q '; to 80
{ q1, q2,..., qs} for each scenaro s ' S. The compact planner objectve functon for the robust optmzaton model s: Mnmze 4 ( TSTT, q, q,..., q ) + (1 5 ) < ( TSTT, q, q,..., q ) TSTT 1 2 s 1 2 s There are number of choces for choosng 4 (.) and < (.) dependng on what the value of the varous errors s to the decson maker. However, t s desrable to choose measures, whch can lead to consstent preferences between alternatve solutons. One such measure of soluton robustness could be to use 4 (.) as the expected value of the Total System Travel Tme (TSTT) that would be experenced for all realzatons of the demand and < (.) denotes a measure of the varablty of the TSTT for all the realzed demands, e.g., < (.) denotes the varance of the system over all future demands. It s ths measure that we wll use n our modelng wth weghts gven to < (.) and 4 (.). The weght can vary from [0, 1]';. Another possblty for measurng robustness could be to calculate the maxmum regret that would be experenced by not followng the optmal scenaro. Ths s equvalent to < (.) = * max{( = s = s )} s mnmze and 5, where = s the user optmal objectve functon that we are tryng to = s s the objectve functon value gven the acton we choose (for example, a partcular capacty expanson polcy) and then scenaro s becomes true * and = s s the objectve functon value for the optmal plan had we known that the scenaro s was gong to be true. Ths measure of robustness s however not consdered n ths partcular research. 4.3 RNDP Model formulaton 4.3.1 Notaton The followng notaton wll be used n the model formulaton: t ( ) a = the unt cost of transportaton on arc a ' A, assumed to be twce contnuously dfferentable n 81
Z ( x) = the user equlbrum objectve for each demand realzaton ' rs f k = the flow on path k 'K between O-D pars r, s ' R rs a, k x a 1 f arc a ' A s on path k between O - D par r,s = 0 otherwse = the flow on arc a ' A q rs = the uncertan demand for transportaton between O-D par r, s ' R and demand realzaton ' a = the unt cost of mprovement on arc a ' A = the weghtng factor assocated for the planner for expected cost Vs network robustness B = the total budget avalable for mprovements r k functon = penalty factor to convert constraned to an unconstraned objectve The dscrete equlbrum network desgn under demand uncertanty (RNDP) can be formulated as shown n P1. 4.3.2 The Lower-level User Equlbrum Assgnment under Uncertan Demand The network desgn problem under determnstc demand can be represented as a two-player game where the transportaton planner (network manager) s a leader and the users freely choose the paths are the followers. However, n the RNDP the analogy stll holds except wth the addtonal complexty that the network manager has to plan wthout complete nformaton about the demand. In other words, the payoffs assocated wth the leader s actons are stochastc. The lower level problem represented by L n the RNDP represents the network flow pattern characterzed by user equlbrum under uncertan demand. The addton of new network lnks to a congested network wthout the consderng the response of network users may lead to paradoxcal results lke the well known 82
Braess s paradox. The predcton of network flows must follow a behavoral model that capture s user s selfshness. In the RNDP, the users at the lower-level are assumed to follow the user-equlbrum prncple of Wardrop wth the well known assumptons of traffc equlbrum (Sheff, 1985). The upper level of the RNDP shown as U represents the network manager s objectve of mprovng the system performance by reducng the congeston n the entre network. However, n the RNDP we are concerned wth a weghed sum of the expected value and the varance of the total system travel tme n the network as aganst a sngle estmate of the total system travel tme (as consdered n determnstc NDP (Magnant and Wong, 1984, Yang and Bell, 1998)). The network manager at the upper level s assumed to make decsons of the capacty mprovements n order to mnmze the weghted objectve functon. The budget constrant ensures that the total constructon does not exceed the total budget. The fnal constrant s a bnary restrcton on the decson varables. P1: U mn y E 0x t ( x, ) (1 ) (, ) a a a ya 1+ 5 Var 0x t x a a a ya 1 a a subject to y ( B a a y a a = 0 or 1 a ' A L x a mn t ( s, y ) ds a 0 a subject to a rs, fk = qrs k ' K; r, s ' R; ' k rs, rs, x a a k fk r s a k r s k rs, fk 0 r, s ' ; k ' ; ' x 0 a ' A; ' a R A K =,, ' ; ' ; ' ; ' R K 83
The above formulaton descrbes the RNDP model proposed n ths research. In summary, P1 comprses of two levels. The upper level U refers to the system planner s objectve of mnmzng the weghted sum of the total system cost, whle the flow to the upper level s obtaned from the user equlbrum n the lower level L for each demand realzaton '. The above model dffers from the tradtonal NDP because of the stochastc nature of demand n L and the weghted stochastc objectve functon n U. 4.3.3 Motvatng Example The effect of demand uncertanty on a small network s demonstrated to show the dfference between the results from optmal network expansons polces for the expected value and the varance of TSTT. The beneft of analyss on a small network s that all the uncertan varables can be enumerated and the solutons are not drven by samplng errors. Consder the small network shown n Fgure 4.1 wth the network characterstcs as shown n Table 4.1. Z 1 and Z 2 denote capacty mprovements whch wll be determned by NDP under dfferent objectves. Ths network s assgned an Orgn-Destnaton (OD) demand of 50 vehcle-trps between nodes C and D and 50 vehcle-trps between A and D. The latter of these demands s taken as uncertan wth possble values varyng unformly between 0 and 100 wth a mean of 50. A sngle uncertan varable s used here to enumerate all possble demand outcomes and avod any samplng errors for ths demonstraton. Snce only dscrete values are taken for the possble demands, ths results n 100 potental future realzatons, each wth an equal probablty of occurrng. To see the effect of ths demand varance on network mprovement decsons, 100 UE network desgn problems were solved for ths network, each wth a dfferent plannng demand level. The plannng demand s an altered demand level used for the bass of a decson. Each problem arrved at a network mprovement polcy (Z 1 and Z 2 ), whch represent an ncrease n capacty and ther sums must not exceed a fxed budget of 10. For each of these proposed polces, all 100 possble demand realzatons were evaluated 84
usng determnstc user equlbrum assgnment. The average total travel tme represents the expected performance of the system, snce each possblty was enumerated and each scenaro has an equal probablty of beng realzed. These 100 scenaros were then averaged for each run. The results from the analyss shows that a polcy expanson based on a demand of 95 yelds a lower total system travel tme as compared to the polces at the expected demand of 50. Ths dfference n TSTT was modest n ths network; t was around 8.32 (Z 1 = 7.9, Z 2 = 2.1) for the former and 8.98(Z 1 = 0, Z 2 = 10) for the latter. However, t was observed that f the plannng polcy consders only varance of the total system travel tme nto consderaton for expanson, a polcy based on a demand of 8 gave the lowest varance of travel tme. The expected travel tme was only slghtly hgher n ths case (8.35) and the correspondng expanson polces were Z 1 = 9.8 and Z 2 = 0.2. Ths smple example provdes two clear nsghts. Network expanson polces are sgnfcantly dfferent n the presence of uncertanty and the network desgn decsons dffer consderably based on the mean-varance consderatons n the objectve functon. In other words, desgnng networks for robustness yelds solutons whch are more reslent to future condtons, however there s a tradeoff n terms of the ncrease n the expected value of total system travel tme. The man dffculty n solvng the robust network desgn soluton s n fndng the network desgn solutons for each demand realzaton. Wth the ncrease n number of demand realzatons the problems becomes extremely dffcult to solve computatonally, when t s well known that even fndng the capacty expansons n the determnstc NDP s a dffcult non-convex problem. It s mportant to realze that the model presented here dffers from other equlbrum network desgn models, not only n accountng for uncertanty, but also n explctly capturng the soluton robustness. As such, the model presented here s computatonally more demandng than other problems. In the next secton, we talor the multobjectve evolutonary algorthm to solve the robust network desgn problem effcently. As we shall see, 85
the set of network desgn solutons for RNDP on test networks demonstrate the value of accountng for robustness n long term transportaton plannng decsons. C 1 = 1+0.15(x 1 /20) 4 B C 4 = 1+0.15(x 4 /30+z 2 ) 4 A C 3 = 1+0.15(x 3 /30) 4 D C 2 = 1+0.15(x 2 /20+z 1 ) 4 C C 5 = 1+0.15(x 5 /40) 4 Fgure 4.1 Smple test network Table 4.1 Lnk parameters for the smple test network Lnk Free-Flow Travel Lnk Parameter (a k ) Lnk Parameter (b k ) Capacty (y k ) Tme (c ok ) 1 1.15 4 20 2 1.15 4 20 + Z1 3 1.15 4 30 4 1.15 4 30 + Z2 5 1.15 4 40 4.4 Evolutonary Algorthm for the Robust Network Desgn Problem Tradtonal methods to solve the RNDP do not work well due to the dauntng computatonal complexty of the non-lnear, non-convex model n P1. The ntrcate nature of the problem, prmarly due to the exstence of many local mnma n typcal transportaton networks, and the sze and complexty of the search space, cannot be handled by tradtonal greedy search algorthms. Although, the formulaton s ntractable wth tradtonal mathematcal programmng methods, the RNDP s better 86
suted for the applcaton of meta-heurstcs. The meta-heurstcs reduce the convergence to local solutons and ncrease the possblty of reachng a globally optmum soluton. Drezner and Salh (2002) n a recent study compared the performance of heurstcs, such as descent algorthm, tabu search, smulated annealng and genetc algorthm, for the one-way and two-way network problem to fnd the best network confguraton so as to mnmze total travel tme of all users. The objectve was to mnmze total vehcle mles traveled. For realstc network szes t was found that GA outperformed the other algorthms n fndng the best soluton, whle takng a longer computaton tme (a few seconds) than other methods. In terms of the qualty of the solutons, GA was found to be the best for the test problems, followed by smulated annealng, and tabu search. Optmzaton wth GA has been dentfed for a number of applcaton areas ncludng capacty expanson, vehcle routng, schedulng and phase equlbra problems as shown n Table 4.2. Meta-heurstcs have been prevously used (Mouskos, 1991; Fresz et al., 1993; Xong and Schneder, 1992; Poorzahedy and Abulghasem, 2005) to solve the determnstc equlbrum network desgn problem. In ths research, the GA evolutonary algorthm s adapted for the RNDP and the soluton methodology s presented. Table 4.2 Overvew of the GA lterature wth problem applcatons Author(s) - (year of Problem Area Genetc Algorthm Study publcaton) Goldberg (1985) General Descrbes the overall GA methodology Xong and Schneder (1993) Network Desgn Transportaton Network Desgn - Statc condtons Cree et al. (1996) Network Desgn Generc Network Desgn problem -statc Gen et al. (2001) Network Desgn Communcaton network desgn statc condtons Cantarella and Vtetta (1994) Network desgn Urban network desgn problem statc condtons Kwan and Wren (1994) Schedulng problem Bus drver schedulng problem Tom and Pattnak Network Desgn Transt network desgn - statc Hseh and Lu (2004) Investment decsons Infrastructure nvestment under tme resource constrants Montastruc et al. (2004) Phase Equlbrum Optmzaton of calcum phosphate precptaton Berger and Barkaou (2004) Vehcle routng Hybrd GA for vehcle routng wth tme wndows Capr and Ignaccolo (2004) Ar Transport Arcraft sequencng problem dynamc model Norman and Bean (1999) Schedulng Machne schedulng to mnmze tardness Hartmann (2002) Schedulng Project schedulng under resource constrants 87
4.4.1 Overvew of GA In ths secton a bref descrpton of the algorthm s gven. GA fnds the capacty mprovements for NDP by evaluatng the system objectve to obtan better solutons n each generaton for each demand scenaro. These are fnally used n a robust analyss (for mean and varance) returnng the ftness functon values for each of the traffc network. A genetc algorthm s a local search algorthm whch works startng from an ntal collecton of strngs representng possble solutons of the problem. Each strng of the populaton s called a chromosome, and has assocated a value called ftness functon that contrbutes n the generaton of new populatons by means of genetc operators (denoted as reproducton, crossover, and mutaton). The ntal populaton can be generated randomly, or t may consst of a number of known solutons, or a combnaton of both. The GA goes through a number of steps n whch the populaton at the begnnng of each step s replaced wth another populaton, whch t s hoped wll nclude better solutons to the problem. The populaton produced at each step s called a generaton and t s numbered accordngly. The chromosomes at each new generaton are produced by a process called reproducton, n whch the chromosomes of the old populaton are combned to create new ones. A detaled explanaton of the workng of GA can be found n Goldberg (1989) and Deb (2002). GA s applcable to solve RNDP whch s nonconvex and non-lnear problem because of ts superorty over other local search technques whch are lmted by the contnuty, dfferentablty and the unmodalty of the evaluated functons. GA handles these lmtatons by; () operatng wth codes of parameter set and not wth the parameter themselves; () searchng for a populaton of ponts and not a sngle pont; () usng objectve functon nformaton and not the dervatve of the functon; (v) usng probablstc transton rules and not determnstc ones. All these features make GA an attractve choce n solvng the RNDP problem. 88
The workng of GA nvolves codng of the soluton, ntalzng the coded soluton, computng the ftness value, the applcaton of genetc operators to generate off-sprngs. Applcaton of these genetc operators s expected to yeld better offsprngs and s repeated tll convergence. The basc genetc operatons commonly used are reproducton, crossover and mutaton whch are dscussed below brefly. 4.4.2 Codng In order to use GAs to solve the RNDP, the decson varables (capacty mprovements) are frst coded as strng structures. Bnary-coded strngs havng 1's and 0's are used. Each varable s coded and the length of the sub-strng s usually determned accordng to the desred soluton accuracy of that varable the longer the strng length, the more the accuracy. Usually, a lnear mappng rule s used to convert coded varables nto ther real values. Once the codng of the varables s complete, the soluton strng s formed by concatenatng all the sub-strngs. For any combnaton of 1 s and 0 s n the strng, there exst correspondng values of each varable. They can be estmated usng the nverse mappng rule. There after, the functon value correspondng to pont can be calculated by substtutng n the varables n the gven objectve functon. For the present study, the decson varable s the hgh capacty network desgn solutons on each lnk. There are two dfferent versons of the problem; dscrete and contnuous. In the dscrete case, the decson s whether an addtonal lnk has to be added or not to a partcular lnk n the network. Ths decson s therefore bnary and can be represented by a sngle bt. Bt 1 ndcates that the capacty of the lnk s expanded by 100 % and a bt 0 ndcates that no mprovement occurs. The sub-strng length s 1 and the strng length s same as the number of lnks. 89
4.4.3 The Objectve Functon and Constrant Satsfacton GA works wth only bnary strngs and does not have any nherent knowledge about the problem. The problem specfc nformaton s provded by the ftness functon, whch s used for measurng the qualty of ndvduals n each generaton. In our specfc case ths s the weghted sum of the total system travel tme represented n U. Conventonally, GA maxmzes ftness, but t s trval to consder a cost functon mnmzaton problem by assgnng the ftness to be a negatve of the cost functon. The RNDP mposes constrants on the acceptable total allowable capacty mprovements (budget constrant), hence t s possble that the soluton that a chromosome maps to would not be feasble, n some cases. The opton of rejectng every nfeasble soluton that volates constrants may sometmes lead to rejecton of some good partal solutons and may be computatonally neffcent. The objectve functon n the RNDP accounts for these constrants by penalzng each soluton that breaks a constrant by ntroducng a penalty term to the calculated ftness a value that depends on the constrant and the extent of volaton. We adopt ths penalty method that allows new constrants to be added easly to the GA optmzaton for the RNDP. 4.4.4 The Reproducton Mechansm Reproducton (or selecton) s a procedure n whch better ftness values are retaned and nferor ones are elmnated from the current populaton (Mchalewcz, 1992). A strng wth a hgh ftness value has a greater probablty of contrbutng one or more soluton whle strngs wth a low ftness value have a low probablty of contrbutng to the next populaton. The mechansm starts wth the creaton of a matng pool by selectng ndvduals from the current populaton wth the hghest ftness value. A number of selecton mechansms have been proposed n the lterature and all of them attempt to acheve the correct balance between the populaton dversty and selectve crtera whch are fundamental n determnng the convergence of the algorthm. In our soluton approach, a popular and superor 90
selecton mechansm called remander stochastc samplng wthout replacement s used (Booker, 1982). In ths technque, an ndvdual s guaranteed representaton n the matng pool to the extent of the nteger value of ts expected occurrence. However, after each ndvdual has been allocated space accordng to the nteger porton of ts expected value, remanng space n the matng pool s flled usng a weghted con toss based on the fractonal component of the ndvdual s expected value n the matng pool. Ths selecton contnues wth addtonal Bernoull trals untl the matng pool s full. 4.4.5 The Crossover Mechansm In the crossover operaton, a recombnaton process creates ndvduals n the successve generaton by combnng nformaton from two ndvduals of the prevous generaton. A crossover operator s used to recombne two strngs to get a better strng. Crossover s usually performed wth a probablty called crossover probablty to preserve some of the good strngs found prevously. Crossover s done at the strng level by randomly selectng two strngs for crossover operaton. There are many types of cross over operatons avalable: one pont, two pont, unform, and mxed crossovers. We use a two pont crossover, a very effectve soluton for the dsrupton of the schemata problem. 4.4.6 The Mutaton Mechansm Mutaton adds new nformaton n a random way to the genetc search process and ultmately helps to avod gettng trapped at local optma. It operates at the bt level, when the bts are beng coped from the current strng to the new strng. Mutaton operates wth a probablty, usually a very small value called the mutaton probablty. A con-toss mechansm s employed; f a random number generated between 0 and 1 s less then the mutaton probablty then the bt are flpped (.e. zero become ones and vce versa). 91
4.4.7 The Convergence Crteron There are several strateges for stoppng the evoluton process of the GA. Because t s dffcult to defne the optmal soluton, usually two procedures are adopted as convergence crteron: (1) when the varaton n the ftness level among generatons s wthn a user defned range, the GA procedure s stopped; and (2) when the number of generatons has accumulated to a predetermned level, the teraton s stopped. In ths research, the GA was stopped when t reached a predefned number of 500 generatons. However, the number of generatons s arrved at after sgnfcant expermental runs on the qualty of the soluton on dfferent test networks and sample szes of demand. A sample plot to determne the sample sze for a good soluton qualty s shown n the Appendx C. 4.4.8 The ntal populaton The ntal populaton s generated randomly from a sngle head ndvdual calculated by the soluton of the NDP usng contnuous varables rather than nteger varables. 4.4.9 Proposed Algorthm for RNDP usng GA Fgure 4.2 shows the algorthm of the RNDP and the man steps are dscussed below. 92
( 1) Input ( 2) GA parameters ( 3) Network data ( 4) Robust data ( 5) Codng of varables ( 6) Intalze populaton ( 7) for every generaton ( 8) for every populaton j ( 9) Decode populaton (10) Modfy network capacty (11) for every sample k (12) Sample random demand (13) Perform UE assgnment (14) end (15) Compute objectve functon (16) end (17) Computed ftness functon (17) Update soluton (18) Apply GA operators (19) Reproducton (20) Crossover (21) Mutaton (22) end (23) Output Fgure 4.2 The proposed algorthm The nput to the model ncludes GA parameters such as number of teratons for convergence, populaton sze, crossover probablty, mutaton probablty, number of decson varables, lower bound, upper bound, and the precson of each varable. The GA parameters for the present study are adopted from the experence from networks of smlar sze from an earler work. The network data ncludes nodes, lnk length, alpha, beta, capacty, and speed lmt. The robust data ncludes the sample sze, random seed, value n [0,1], percentage demand varaton, budget, etc. The next step s the algorthm s the GA codng. Here, the number of varables s equal to the number of lnks, and snce t s a bnary decson varable, the lower bound of each varable s 0, upper bound s 1, and the precson s 1. The 93
strng length s equal to the number of lnks, where each bt corresponds to a lnk. A value of 1 ndcates that lnk s capacty wll be doubled. The next step s to ntalze the populaton. Ths s normally done by randomly ntalzng the P strngs wth 1 or 0. Note that codng ensures every populaton s feasble. Then the GA teraton starts. Frst, for each strng (correspondng to an nstance of the soluton) s decoded. The decoded values are the lnks whose capactes are to be ncreased. Then for every sample, a random sample of the OD matrx s generated. Ths s done by multplyng every cell of the base OD matrx by a random factor. Suppose, f the demand varaton s 50 percent, then ths factor s a random value between 0.75 and 1.25. (.e. Monte-Carlo smulaton for randomly selected demands n [l,u] of the expected value). Ths sampled OD matrx s assgned to the network usng standard UE assgnment by employng the Frank-Wolfe algorthm. Ths wll gve the total system travel tme for each sample. Next, the objectve functon value s computed by frst calculatng the expected value and the varance of the total system travel tmes for each scenaro and fndng the expected performance and robustness of the capacty expanson polces usng the objectve - mn y E 0 xata ( xa, ya ) 1+ (1 5 ) Var 0 xata ( xa, ya ) 1 a a. The base problem s a constraned one and ths objectve functon s transformed nto an equvalent unconstraned problem by the followng transformaton: Ph(x) = mn y E 0 xata ( xa, ya ) 1+ (1 5 ) Var 0 xata ( xa, ya ) 1 a a +r k (Max((C -B),0) Where r k s the penalty term, C s the cost of capacty mprovement and B s the budget. The Ph(x) s returned to GA to compute ftness functon value. The soluton s checked wth the earler values and the best values and the correspondng chromosome. Ths s followed by the applcaton of GA operators mentoned above. Ths completes one GA run and wll be repeated tll convergence. 94
The prmary output s the lnks that are chosen for capacty expanson. In addton, other statstcs such as E (TSST), S (TSTT), etc can be nferred. 4.5 Computatonal Results To test the above model and soluton approach proposed n ths paper, we perform computatonal experments on three dfferent test networks: smple test network (Fgure 4.1), Harker and Fresz (HF) network, and the Nguyen Dupus test network. Intally we test our methodology for the determnstc case on the smple test network and HF network. The motvaton for ths analyss s that for these networks the NDP soluton s avalable n the lterature (Harker and Fresz, 1984; Chou, 2005) and can be used to compare our NDP soluton usng the GA. However, the solutons for these test networks, are n a determnstc and contnuous network mprovement settng and the comparson wll be made accordngly. The man results of the RNDP model are demonstrated on a mddle szed network shown n fgure 4.4, the network of Nguyen Dupus, whch has been extensvely used before by researchers for testng the traffc equlbrum problems (Nguyen and Dupus, 1984). The detaled data for the HF network and the Nguyen Dupus network s presented n Appendx D. A set of desgned experments were performed to determne the optmal values of the model parameters of GA. It was expermentally determned that the best populaton sze wth the lmted computatonal resources was 50. The populaton sze and the number of generatons determne the fnal amount of search, the optmal soluton and the executon tme. However, n problems of nterest, the computatonal tme s lmted. In ths work, therefore, we select the approprate GA parameters when the total number of chromosomes generated s fxed. For a gven executon tme (and correspondng number of trals), the GA parameters that acheve the best capacty expansons are dentfed for dfferent values of. The model was coded n C++ and the mplementatons have been conducted on Pentum PC wth 686 processor on a GNU/Lnux platform usng a gcc compler. 95
4.5.1 Test Networks and Parameters 4.5.1.1 Experment 1 The frst experment s conducted on the example network n Fgure 4.1 under determnstc condtons wth the same parameters as n Table 4.1. The results from the GA and complete enumeraton wth 100 demand realzatons are shown n Table 4.3. In ths experment and the experments that follow, the number of GA generatons was set equal to 500 and the stoppng crteron n performng the nner loop of the equlbrum traffc assgnment s set equal to 0.005, whch s the absolute dfference between the equlbrum soluton n step and +1. Table 4.3 Comparson of the enumeraton result wth the GA on the small test network GA Enumeraton TSTT(UE) 336.16 335.54 Z 1 6.21 6.30 Z 2 3.79 3.70 4.5.1.2 Experment 2 The second experment s conducted on the HF networks (Harker and Fresz, 1984) for whch bounds on the determnstc contnuous NDP solutons are avalable at dfferent demand levels. Ths comparson should prove the valdty of the meta-heurstc approach for the proposed problem. In realty, there s no way of truly valdatng the stated problem because of the lack of exact algorthms to solve the RNDP, however, the comparson of the statc NDP should provde reasonable confdence on the proposed soluton methodology. Further, the proposed approach should facltate n the development of other effcent soluton approaches n the future. The network s shown n Fgure 4.3 wth 6 nodes and 16 arcs. There are two O-D pars (1,6) and (6,1). The statc contnuous NDP n solved at dfferent 96
demand levels wth q 16 = 2.5, 3.75 and 5 and q 61 = 5.0, 7.5 and 10.0 respectvely. It was found n the paper that at low levels of congeston the heurstc bounds derved are very close to optmal. A good lower bound of the contnuous NDP can be found by solvng the NDP based on the system optmal network flows where n the lower problem L s solved usng the margnal travel tme cost functon. Harker and Fresz (1984) and Suwansrkul et al., develop an nexact soluton procedure based on the teratve optmzaton assgnment (IOA) method to calculate the upper and lower bound for the contnuous NDP as shown n Table 4.3. We compare the results for GA at low levels of demand; these results are presented n Table 4.4. Note that the results from Harker and Fresz (1984) over estmate the NDP soluton (Cournot Nash game), however as proved these values are tght and the actual NDP soluton les n between the NDP solved wth a system optmal objectve and the Cournot Nash game. The results of the GA match very close to the NDP solutons based on the bounds developed by Harker and Fresz as shown n Table 4.4. Further, the system cost of the network at the three demand levels s also close usng the GA approach. 1 2 5 9 4 11 1 2 3 6 7 3 4 12 10 13 5 15 16 14 6 8 Fgure 4.3 HF Test Network 97
Table 4.4 Comparson of results from Harker and Fresz (1983) and the GA approach for NDP Improved arc Total Flow =7.5 Total Flow =11.25 Total Flow =15.0 System Cost HF Result GA Result HF Result GA Result HF Result GA Result HF Result y 3 (4.24,4.24) 2.84 (4.17,4.24) 3.70 (3.65,4.24) 2.68 (63.28,63.28) (7.5) y 6 0 0.2 (0,0.77) 0.1 (0,6.07) 0.61 (99.14,99.69) (11.25) y 15 (13.14,13.14) 13.10 (13.06,13.14) 12.36 (12.51,13.14) 13.1 (140.21,147.59) (15.0) GA Result 63.38 101.03 147.22 4.5.1.3 Experment 3 Wth the valdaton of the GA approach on the two test networks above, numercal experments are conducted n ths secton to study the results yelded by the GA approach for the RNDP on the Nguyen Dupus network shown n Fgure 4.4. Descrptons of the OD demand table and the network parameters are gven n Appendx D. Dfferent sets of the values of equal to 0, 0.25, 0.5, 0.75 and 1 are used to solve the dscrete RNDP. When equals to 0 t s the robust network desgn whch accounts prmarly for varance and when equals to 1 t s the stochastc network desgn, where the planners objectve only accounts for the expected network wde costs. For the OD demand matrx a unform dstrbuton wth the expected value of 1900 and 2300 are used for the O-D pars 1-2 and 1-3 & 2100 and 1700 for the O-D pars 4-2 and 4-3. A coeffcent of varaton (COV) of 0.5 s assumed. The followng parameters for the GA have been chosen for ths problem: No. of generatons: g max = 500 Populaton sze (No. of parent solutons) = 50 No. of offsprng solutons = 50 Mutaton probablty = 0.001 Crossover probablty = 0.8 The above parameters for the GA were arrved at by conductng a seres of systematc experments based on varyng them over an nterval n equdstant steps. 98
For example, the appendx shows the plot of arrvng at the requred number of generatons requred for convergence keepng all the other thngs the same. Altogether, we have conducted around 30 runs for fndng good parameters due to the computatonal tme ssues. Other approaches for determnng the parameter values for GA s based on the dea of applyng another (meta) evolutonary algorthm. Ths s outlned n Hanne (2001). The parameters used n ths analyss produced reasonably robust results and further analyss was not necessary. 1 1 12 19 10 12 2 4 5 3 4 5 6 7 8 17 11 13 14 9 6 10 7 11 8 2 18 15 16 13 9 3 Fgure 4.4 Nguyen Dupus test network for the RNDP Table 4.5 summarzes the man results obtaned after applyng the MOEA to the Nugyen Dupus test network. It can be observed that the accountng for dfferent values of robustness yelds dfferent lnk capacty expansons. There s a sgnfcant tradeoff between accountng for just the expected value of the network performance and the rsk (captured n terms of varance) assocated wth t. An mportant nsght from ths analyss s that the network desgn solutons are 99
sgnfcantly dfferent based on the desred degree of robustness accounted n the NDP. Fgures 4.6, 4.7 and 4.8 shows the convergence of GA for dfferent value of. The convergence of GA s prmarly vsual wth the number of generatons. Other quanttatve approaches for measurng the MOEA performance are suggested by Veldhuzen and Lamont (2005). From Table 4.5 t can be observed that there s a sgnfcant mprovement n the network performance n the stochastc ( = 1) and the robust cases ( = 0) as compared to the base case network. The percentage travel tme savngs for the stochastc and robust network desgn solutons as compared to base case are 35.83% and 33.33% respectvely. The mprovement n network reslency by accountng for robustness n ths network s about 1.3%. The last column n Table 4.5 represents the network desgn soluton at the expected value of demand. Ths quantfes the expected value soluton of the problem. We observe that ths network desgn soluton performs very poorly as compared to the NDP solutons accountng for uncertanty n the network performance. The comparson of ths soluton wth the stochastc and robust soluton shows that network performance s degraded by 8.83% and 12.21 % respectvely. Further, there s no guarantee on the robustness of the obtaned soluton n the expected value soluton. For the populaton sze of 50 and an evoluton of 500 generatons t was notced that the computatonal requrements are qute hgh. After performng some mprovements n the data structures for the lower level problem n P1, the runnng tmes are stll around 20 hours for each value of for the MOEA wth the above parameters. It was observed that about 95% of CPU tme s requred n evaluatng the ftness functon,.e., the weghted stochastc objectve functon n U consderng the lower level problem L. A sample of the number of user equlbrum assgnments performed to arrve at the fnal solutons s shown as # n Table 4.5. Further analyss of the C++ code has shown that major parts of the runnng tme can be mnmzed by usng more effcent algorthms for evaluatng the user equlbrum assgnment. For nstance, the user equlbrum can be effcently solved usng ts dual. Further, the 100
MOEA can be sped by parallelzng the algorthm and replacng the neffcent data structures: ths wll be an ssue of our future work. Table 4.5 RNDP soluton usng GA for dfferent values of Base Case Capacty(veh/hr) = 0 = 0.25 = 0.5 = 0.75 = 1 NDP[E(OD)] y 1 2200 0 1 0 0 1 0 y 2 2200 1 1 1 1 1 1 y 3 2200 0 1 1 0 0 0 y 4 2200 1 1 1 1 1 1 y 5 2200 0 0 1 0 0 0 y 6 2200 0 0 1 1 0 0 y 7 2200 1 1 0 1 1 1 y 8 2200 1 0 1 1 1 1 y 9 2200 0 0 0 0 0 0 y 10 2200 1 1 1 0 1 0 y 11 2200 1 0 1 1 1 1 y 12 2200 0 1 0 0 1 1 y 13 2200 0 1 1 1 1 1 y 14 2200 1 0 0 0 0 1 y 15 2200 1 0 0 0 1 0 y 16 2200 1 1 0 1 0 1 y 17 2200 0 0 0 1 0 0 y 18 2200 1 1 1 1 0 1 y 19 2200 1 1 1 1 1 1 E[TSTT] 857,046 571,022 561,710 563,749 556,102 549,881 626,354 S[TSTT] 0 59,191 61,596 61,809 60,237 59,834 0 # 1 11,016,028 1,611,749 Note: # denotes the number of user equlbrum assgnments the nner optmzaton loop uses for RNDP 0 mples that the lnk has not been mproved and 1 mples that the lnk has been mproved by a sngle lane 101
Ph(x) 63500 63000 62500 62000 61500 61000 60500 60000 59500 59000 1 51 101 151 201 251 301 351 401 451 Generaton number Rho=0.0 Fgure 4.5 Convergence of the RNDP ftness functon for = 0 Ph(x) 320000 319000 318000 317000 316000 315000 314000 313000 312000 Rho=0.5 1 51 101 151 201 251 301 351 401 451 Generaton number Fgure 4.6 Convergence of the RNDP ftness functon for = 0.5 102
Ftness functon 580000 575000 570000 565000 560000 555000 550000 545000 Rho=1.00 1 51 101 151 201 251 301 351 401 451 Generaton number Fgure 4.7 Convergence of the RNDP ftness functon for = 1 From Table 4.5 t can be observed that by accountng for varance n the NDP there s a beneft of mprovng the reslency of the network performance although there s a tradeoff n terms of the slght ncrease n the expected network performance. In addton, t can be notced that the desgn decsons on the network topology (lnk mprovements) vary dependng on the robustness accounted for n the RNDP. It s mportant to note however that there s no consstent relatonshp between the expected network performance and the robustness wth dfferent values of. For example, the stochastc case ( =1) has a lower varance as compared to the ntermedate case when = 0.75. Ths non-domnance effect should be explored further. We also explored the network performance changes due to the change n the budget. It s clear that n the extreme case of a very large budget the traffc condtons follow free flow wth almost zero varance n network performance. Table 4.6 shows the change n the expected network performance and the robustness for ncreasng value of the budget for the stochastc and the robust NDPs. It can be observed that wth the ncrease n budget n both the network desgn solutons, the expected value and the varance of the network performance mproves as expected. 103
Table 4.6 Change n network performance wth budget Budget Robust ( = 0 ) Expected ( = 1) (n $ml) E(TSTT) S(TSTT) E(TSTT) S(TSTT) 2 715430.3 83728.25 714222 84629.99 5 684266.3 79561.21 681507.7 79666.49 20 558183.4 61278.3 554920.2 61388.33 50 507615.1 53484.74 502218.4 53577.83 4.6 Evaluaton of GA Network Desgn solutons The GA soluton procedure uses a smaller sample sze because of the computatonal tme nvolved n arrvng at the hgh capacty network desgn solutons wthn a pre-specfed number of generatons. To determne the qualty of the soluton obtaned for the RNDP usng the GA approach, we propose an evaluaton approach. For each RNDP soluton n Table 4.5, we mprove the network wth the optmal capacty values. We test ths capacty added network on a larger sample sze of 25,000 samples to determne the expected value and standard devaton of the total system travel tme (network performance). The optmal TSTT s for the RNDP usng GA and the evaluaton approach are shown n Table 4.7. It s observed that the RNDP capacty mprovements from MOEA perform very well on a hgher sample sze n terms of the expected travel tme and the varance of the travel tme. 104
Table 4.7 Comparson of the network wde travel tme wth MOEA and evaluaton approach GA Results Evaluaton wth 25,000 samples E [TSTT] S[TSTT] E[TSTT] S[TSTT] % Devaton of E[TSTT] Absolute Percent Dfference %Devaton of S[TSTT] Base case 725,326 NA 736,268 86,322 1.5 NA = 0 571,022 59,191 572,730 59,345 0.29 0.26 = 0.25 561,710 61,596 563,589 61,839 0.33 0.39 = 0.5 559,749 61,809 565,580 61,941 0.32 0.21 = 0.75 556,102 60,237 557,924 60,436 0.33 0.33 = 1 549,881 59,834 551,633 59,953 0.32 0.20 NDP [E(OD)] 626,354 NA 558,015 60,482 10.91 NA 4.7 Concludng Remarks Wth the maturty of computatonal technques and modelng approaches, new opportuntes exst for studyng the value of robustness n network desgn. Tractable model formulatons, soluton methodologes and advanced computatonal technques are crtcal for developng effectve robust network solutons. The model formulaton and soluton approach presented n ths paper provde a means for accountng for uncertanty n network desgn decsons and supportng explct decsons on the trade-off between expected costs aganst rsk, when the long term O-D demand s a random varable. The problem s formulated as a non-lnear, nonconvex, multobjectve model where the planner s objectve s to mnmze the weghted objectve of the expected value and the varance of the network total travel 105
tme and the user s route choce s dependent on user equlbrum under fnte scenaros of uncertan demand. An effcent soluton approach usng a MOEA s proposed for solvng the RNDP. The key concept of the soluton method s n obtanng the soluton for the RNDP by systematcally explotng the randomness of the genetc evoluton process. As demonstrated, the proposed approach can produce optmal solutons wth a reasonable computatonal tme and overhead cost. The man contrbuton of ths work s n formulatng the RNDP and n proposng an evolutonary structure for solvng the problem n a mathematcal programmng framework. Further, the problem has been solved by adjustng sutable system parameters for the robust network desgn problem. Numercal comparsons have been made on three test networks. Frst, for the 4-node 5-lnk network the MOEA method produces smlar results to the determnstc contnuous NDP results obtaned by enumeraton. Furthermore, n the 16-lnk example network of Harker and Fresz, the MOEA approach produced very close results as compared to the results and n lterature. Second, for the Nguyen Dupus network the soluton of the RNDP usng a MOEA heurstc gves a very good framework for obtanng good network desgn solutons accountng for dfferent degree of network travel tme varablty. The computatonal tme, however, was hgh and t s dffcult to solve large network wth the proposed approach. For the mprovement of the RNDP models, ths ssue requres further research n ncorporatng a varety of real-lfe networks n the future. In addton, effcent approxmaton procedures such as samplng procedures and the sngle pont approxmatons for solvng the RNDP should be explored for solvng large scale RNDPs. Another ssue deserves further remark: a sgnfcant computatonal bottleneck n mplementng the problem s n the evaluaton of the ftness functon for each demand realzaton and then solvng the NDP. The ftness evaluaton procedure can be computed faster by usng a parallelzaton approach to solve ths problem could help n developng flexble and quck RNDP solutons. 106
Part III Accountng for Dynamcs 107
Chapter 5 Dynamc Stochastc Network Desgn Model A man wth genus s unendurable f he does not also possess at least two thngs: grattude and cleanlness Netzsche n Beyond good and Evl (1886) 5.1 Introducton In ths chapter, we pursue a mathematcal programmng approach to deal wth uncertanty whle accountng for dynamc traffc condtons. The prevous chapters prmarly dealt wth accountng for stochastc demand condtons n statc transportaton networks. These methodologes are mportant before ncludng traffc dynamcs to motvate ths area of research and to place them on a strong footng. Further, the statc models are readly accepted n practce and demonstratng the sgnfcance of uncertanty n them wll help us n pushng ths feld nto practce. Ths research s concerned wth formulatng a network desgn model n whch the traffc flows satsfy dynamc user optmal condtons for a sngle destnaton. The model presented here ncorporates the Cell Transmsson Model (CTM); a traffc flow model capable of capturng shockwaves and lnk spllovers. Comparsons wll be made between the propertes of the Dynamc User Optmal Network Desgn Problem (DUO NDP) and an exstng Dynamc System Optmal (DSO) NDP formulaton. Both network desgn models have dfferent objectve functons wth smlar constrant sets whch are lnear and convex. We propose a formulaton and show computatonal results on two test networks. Further, we extend the dynamc formulaton to a two stage stochastc lnear programmng formulaton wth the random varable as the long term demand. We demonstrate the beneft of accountng for uncertanty n the DNDP. The opportuntes of new technologes and mproved computatonal resources have encouraged consderable research n the development of methodologes and algorthms n dynamc transportaton networks. Transportaton nvestments form the core of decson makng n current and future projects. Decsons made n the current stage of capacty mprovements must take nto account the dynamc nature of demand. Ths chapter formulates the Dynamc User Optmal Network Desgn 108
Problem (DUO NDP) for a sngle destnaton usng a Lnear Programmng (LP) approach. Results are compared wth the known Dynamc System Optmal Network Desgn Problem (DSO NDP) formulaton (Waller and Zlaskopoulos, 2001). The dfferences between capacty mprovements under DUO and DSO behavor are analyzed usng tests on a small network. An approxmaton technque s descrbed to solve the model for larger networks. An expermental test s performed on a network smlar to the Nguyen and Dupus (1984) test network for varous levels of traffc demand, pattern and budget. The results for the two models are compared. Fnally, the proposed DUO NDP formulaton s extended to account for long term demand uncertanty to demonstrate flexblty of the proposed lnear programmng formulaton. Further, ths extenson demonstrates the mportance of ncorporatng uncertanty n network desgn problems. Whle sngle-destnaton DTA problems are nfrequently found n actual applcatons, the approach can be used to evaluate and calbrate other models and may potentally be used as a subroutne n large-scale mult-destnaton soluton approaches (as dscussed n Waller and Zlaskopoulos, 2001 and Golan and Waller, 2004). Furthermore, analytcal approaches, despte ther lmtatons, contrbute to the better understandng of the problem and the development of operatonal analytcal approaches. Moreover, snce ths model can propagate traffc accordng to the Cell Transmsson Model (CTM) (Daganzo, 1994, 1995), t can capture the queue evoluton along an arc, whch s not generally possble wth models relyng on lnk ext functons. Although the proposed analytcal model s lmted to a sngle destnaton, t can capture realtes of actual networks better than many other analytcal formulatons, because ts underlyng structure s based on traffc flow theoretcal models. Note that wth ths formulaton, the dynamc user optmal network desgn problem s a lnear program, when even the smplest statc traffc network desgn problem s an extremely dffcult nonlnear (commonly non-convex) mathematcal program. 109
5.2 Embedded Traffc Flow Model The Cell Transmsson Model CTM (Daganzo, 1993 and 1994) provdes a convergent numercal approxmaton to Lghthll and Whtham (1955) and Rchards (1956) (LWR) hydrodynamc model to smple dfference constrants by assumng a pecewse lnear relatonshp between traffc flow and densty for each cell (or segment). Although CTM assumes pecewse lnearty at the cell level, the model well llustrates the traffc flow propagaton on networks and captures other traffc phenomena such as the spatal as well as the temporal formaton and dsperson of queues, n other words, the propagaton of traffc flow dsturbance and formaton of shockwaves because the model dvdes the road nto short and dscrete cells. Moreover, CTM reasonably captures the non-lnearty between speed-densty and travel tme-densty at the lnk level reasonably well. Zlaskopoulos (2000), Ukkusur (2002), and Lo and Szeto (2002) use the CTM concepts n modelng DTA formulatons. Ths secton addresses the basc prncples of CTM and llustrates a modelng network for a general transportaton network, and then presents a modelng ntersecton for easy understandng of traffc movements on cell-based transportaton networks. 5.2.1 Basc Prncples of CTM The hydrodynamc model (Lghthll and Whtham, 1955; Rchards, 1956) of traffc flow s known to be powerful n capturng the formaton, propagaton, and dsspaton of queues on lnks. The key components of the hydrodynamc model of traffc flow are the followng constrants, q k + x t = 0 and q = Q( k,, tx ) (5.2.1.1) where q s the traffc flow; k s the densty; x and t are the space and tme varable, respectvely, and Q s a functon relatng q and k. The frst partal dfferental constrant states the flow conservaton that defnes the relatonshp between flow ( q ) and densty ( k ) over locaton ( x ) and tme (t ). Ths constrant s 110
usually supplemented by the assumpton that traffc flow ( q ) at locaton x s a functon of traffc densty ( k ), q = Q( k,, tx ). Daganzo (1993, 1994) smplfed the soluton scheme by adoptng the followng relatonshp between traffc flow ( q ) and densty ( k ): where k j, q max { max j } q = mn vk, q, w( k 5 k) (5.2.1.2), v, w denote the maxmum (or jam) densty, nflow capacty (or maxmum allowable nflow), free-flow speed, and the speed wth whch dsturbances propagate backward when traffc congested (the backward wave speed), respectvely. The LWR constrants for a sngle hghway lnk can be approxmated by a set of dfference constrants. Constrant (5.2.1.2) approxmates the fundamental dagram of flow-densty shown n Fgure 5.1.1 by a pecewse lnear model shown n Fgure 5.1.2. Flow q Flow q q max k j 1 + 1 v w q max q max w( k 5 k) j vk v -w Densty k k j Densty k k j Fgure 5.1.1 q-k dagram n LWR Fgure 5.1.2 q-k dagram n CTM 5.2.2 Network Representaton To facltate cross-reference, ths Secton adopts smlar notaton as n Daganzo (1993, 1994) and Zlaskopoulos (2000). The model assumes that the road s dvded nto homogeneous cells. The length of cells s the same as the dstance traveled at free-flow speed n one tme nterval. The results of LWR model (1956) can be approxmated by the followng set of recursve relatonshps n CTM, and the 111
basc relatonshps of the cell transmsson model that descrbe the evoluton of traffc flow are extensvely dscussed n Daganzo (1993, 1994), Zlaskopoulos (2000), Lo et al. (2001) and Lo and Szeto (2002) as follows: x t+ 1 = x t + y t k 5 y t j k ' 9 51 ( ), j ' 9( ), ' C O, t ' T (5.2.2.1) y t k t t t t t t { x,mn[ Q, Q ], [ N 5 x ]} ( k, ) ' E t ' T = mn, (5.2.2.2) k k O where y s the nflow to cell from cell k at tme nterval t (from t to t + 1); t k t x s number of vehcles n cell at tme t ; t y j s the flow from cell to cell j at tme t ; t Q s the maxmum number of vehcles that can flow nto or out of cell when the tme advances from tme t to tme t + 1; t N s the maxmum number of vehcles t that can be present n cell at tme t ; s the rato (v/w) of forward to backward propagaton speed for each cell and tme t ; 9 51 )( s the set of predecessor cells to cell ; 9 () s the set of successor cells to ; C s the set of cells: ordnary cells C, 9 51 ( ) = 1, 9( ) = 1), dvergng cells ( C D, 9 51 ( ) = 1, 9( ) > 1), mergng cells ( O C, 9 51 ( ) > 1, 9( ) = 1), source cells ( C R, 9 51 ( ) = 0, 9( ) = 1) and snk cells ( M C, 9 51 ( ) = 1, 9( ) = 0 ); E s the set of cell connectors: ordnary cell connectors ( S E ), mergng cell connectors ( E M ), dvergng cell connectors ( E D ), source cell ( O connectors ( E ), and snk cell connectors ( E ). Defnton of varables and R notatons used here are summarzed n Table 5.1 as follow: S 112
q : Table 5.1 Defnton of varables and notaton Lnk flow q max : Maxmum flow k : k m : k j : Densty Optmum densty when flow s at a maxmum rate Jam densty u m : Optmum speed when flow s at a maxmum rate u f : Free-flow speed t M : Cost per tme nterval that wll yeld user equlbrum flows C : Set of cells C O : C D : Ordnary cells Dvergng cells C M : Mergng cells C R : C S : T : Source cells Snk cells Set of dscrete tme ntervals r : Source cell, r ' CR s : Snk cell, s ' CS t x : Number of vehcles n cell at current tme nterval t 0 A : Intal cell occupances ( x = A t N : t y j : E : E O : ), ' C Maxmum number of vehcles n cell at current tme nterval t Number of vehcles movng from cell to cell j from current tme nterval t to t +1 Set of cell connectors Ordnary cell connectors 113
E D : Dvergng cell connectors E M : Mergng cell connectors E R : E S : t Q : Source cell connectors Snk cell connectors Maxmum number of vehcles that can flow nto or out of cell at current tme nterval t t : Rato of forward to backward propagaton speed for cell at current tme nterval t ; t t t t = 1 f x ( mn{ Q, Q } and = v / w f x > mn{ Q, Q } k k 9 () : Set of successor cells to cell 9 51 ( ) : Set of predecessor cells to cell k k t d : Demand (nflow) at cell at current tme nterval t, ' CR z : Bnary varable; f a cell s selected for mprovement, z =1, otherwse 0 B j : Unt ncrease parameter n the jam densty of cell j when z j =1 C j : Unt ncrease parameter n the saturaton flow of cell j when z j =1 t : D : b : B : Current tme nterval, t =1,, T Dscretzed tme nterval Postve constant and represents a known constructon cost to expand one lane of cell Total budget t N t y 51, t x t N + 1 t y, + 1 t x + 1 t y+1, + 2 Cell Cell + 1 Fgure 5.2 Illustraton of the buldng block of CTM 114
Fgure 5.2 shows the buldng block representng constrants (5.2.2.3) and (5.2.2.4). A road segment conssts of a seres of homogenous cells wth the physcal length. A lnk between cells has no physcal length and s used just to depct connectvty between cells. Furthermore, cells can hold traffc flows, but traffc flows does not exst physcally. Two basc relatonshps (.e., constrants (5.2.2.3) and (5.2.2.4)) mantan the traffc flow of CTM. Constrant (5.2.2.3) constrans the cell mass conservaton and represents the traffc movements on the ordnary cell type. Also, the flow between two cells s constraned and determned by the number t t t of vehcles ( x ) occupyng the begnnng cell, the remanng capacty ( [ N 5 x ] ) t k determned by the rato ( ) of forward to backward propagaton speed for each cell t and tme t at the endng cell, and the mnmum of the maxmum flow t t ( mn[ Q, Q ] ) that can get out of the begnnng cell and nto the endng cell. Ths k relatonshp s shown n constrant (5.2.2.4) for the ordnary type cell. Fgures 5.3 and 5.4 show the network representaton n CTM for general transportaton networks. The traffc movement on lnks s done by CTM (Daganzo, 1994; Zlaskopoulos, 2000; Waller, 2000) durng each smulaton tme nterval. A general transportaton network that s represented by a set of cells wll be consdered. Traffc starts from the source cell ( C ) and ends at the snk cell ( C ). R S E O C O E O C O E O C R ER ES C S (a) Ordnary cell (b) Source cell (b) Snk cell Fgure 5.3 Network representaton of ordnary, source, and snk cells 115
EM CM CD ED (a) Mergng cell (b) Dvergng cell Fgure 5.4 Network representaton of merge and dverge cells 5.2.3 Modelng Transportaton Networks usng CTM Unlke exstng traffc assgnment models that are based on a lnk ext functon, the model presented here uses CTM concepts. Road segments n networks can be dvded nto homogenous cells shown n Fgure 5.5 (b), such that the length of each cell s set equal to the dstance of free-flow travel. As presented n Secton 5.2, traffc flow characterstcs n networks based on CTM shown n Fgures 5.1.1(a) and 5.5(a) are much dfferent from those used n typcal smplfed models shown n Fgures 5.1.1(b) and 5.5(b). Snce traffc flows based on CTM can descrbe more detals than typcal lnk based models, the traffc propagaton n DTA model proposed n ths study uses concepts of Daganzo s CTM. Ths secton llustrates the cell network representaton for modelng networks. Upstream of Lnk Downstream of Lnk Movng Vehcles Vehcles n Queue (a) Conceptual segments on a lnk n typcal statc model Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 (b) Conceptual segments on a lnk n CTM Fgure 5.5 Conceptual lnk segments 116
(1) Ordnary Cells and Cell Connectors Followng constrants (5.2.2.1) and (5.2.2.2) and Fgure 5.6 represent key components of the cell traffc flow theory, the flow relatonshp and the conceptual dagram for these relatonshps at the ordnary cells and on ts cell connectors. Constrant (5.2.2.1) constrans the cell mass conservaton. As presented n constrant (5.2.2.2), the flow between two cells (.e. cells k and ) depends on t t mn[ Q, Q ] and t [ N t 5 x t ]. k t x k, t Q k, t N k t Q, t N t Q j, t N j t y k 51, k t x k t yk t x t yj t x j t y j, j+ 1 Cell k Cell Cell j Fgure 5.6 Illustraton of ordnary cells and cell connectors (2) Source Cells and Cell Connectors Followng constrants (5.2.2.3) and (5.2.2.4) represent the flow constrants and the conceptual dagram for these relatonshps at the source cells and on ts cell connectors. The source cells have unlmted capacty, but the maxmum output flows are fnte. The ntal value ( x, 0 ' CR ) can be set to ntal traffc condtons at the begnnng of the tme perod of nterest (the traffc peak perod). y t j t t t t t t t t t 5 x ( 0, y ( Q, y ( Q, y ( ( N 5 x ) (, j) ' E, ' C, t ' T (5.2.2.3) j j j j j j j R R x t = x t51 + d t51 5 y t51 j (, j) ' E, ' C, t ' T, R R x 0 = A, ' C (5.2.2.4) t Q, = / t N t Q j, t N j t x t y j t x j t y, j + 1 Source cell Cell connector Ordnary cell j Cell connector from from source cell ordnary cell j Fgure 5.7 Illustraton of source cells and cell connectors 117
(3) Snk Cell and Cell Connectors Followng constrants (5.2.2.5) represent the flow constrants and the conceptual dagram for these relatonshps at the snk cells and on ts cell connectors. The t capacty of snk cells s nfnte (.e. N j = /, j ' CS, t ' T ), and the maxmum t nput flows are unlmted (.e. Q = /, j ' CS, t ' T ). Therefore, ther occupances on these cells are not constraned. j y t j t t t ( x, y ( Q (, j) ' E, j ' C, t ' T (5.2.2.5) j j S S t Q, t t N Q = /, = / t j N j t y t 51, x t yj t x j Cell connector to Ordnary cell Cell connector to Snk cell j ordnary cell snk cell j Fgure 5.8 Illustraton of snk cell and cell connectors 5.3 The Model: Problem Formulaton Ths secton descrbes the notaton used for the DUO NDP, and brefly defnes the DUO Network Desgn Problems (NDPs) and SUO NDP models. It then dscusses ssues relevant to the formulaton of these problems. Table 5.1 summarzes the notaton used n the network desgn formulatons. 5.3.1 Dynamc System Optmal Network Desgn Problem (DSO NDP) Formulaton The objectve functon of the DSO NDP as formulated by Waller and Zlaskopoulos (2001) s mentoned here for the sake of completeness. The formulaton s based on the SO DTA LP of Zlaskopoulos (2000) whch uses a mnor relaxaton of the CTM (Daganzo 1994, 1995). The objectve of the DSO NDP s to mnmze the total system travel tme and therefore produce DSO cell denstes and flows, as well as an optmal set of cell capacty mprovements. The objectve t functon can be stated mathematcally as Mnmze t - yj whch s equvalent to 118 t' T (, j) ' ES
the prevously employed Mnmze t' T ' C / Cs x t ( Zlaskopoulos, 2000) as proven n Ukkusur (2002). The constrant set for the DSO NDP s the same as for DUO NDP and s shown n Equatons 5.3.3.2-5.3.3.12. Ths DSO NDP formulaton wll be used for comparson wth the DUO NDP n Sectons 4 and 5. 5.3.2 Dervaton and Interpretaton of the cost vector M t The formulaton of the DUO NDP s based on the User Optmal Dynamc Traffc Assgnment Lnear Program (UODTA LP) formulaton, gven n Waller and Ukkusur (2003) and Ukkusur (2002). A bref descrpton of the User Optmal DTA objectve s provded before developng the DUO NDP formulaton. For DUO condtons, costs must be assgned so that the objectve wll value an ndvdual s earler arrvals to such an extent that t wll sacrfce system-wde costs. Alternatvely, f a vehcle has the opportunty to arrve at some mnmum tme t, t wll do so even at the cost of all flow whch arrves after tme t. Under the stated strategy, the beneft for an arrval flow at tme 1 versus tme 2 should be equal to the maxmum possble value n the formulaton (.e., f any flow can leave the network at tme 1 t wll do so regardless of any other condtons). After the relatve cost dfference between tmes 1 and 2 have taken a value, the same argument then apples to tme ntervals 3 and 4 and so on. The vector of costs whch yeld ths behavor s denoted by M t. The vector M t must reward vehcles for takng shortest tme paths by overcomng the natural system costs whch exst due to the sngle objectve requred for an LP. For example, take a system optmal soluton whch has a vehcle arrvng at tme 6. Assume the vehcle could arrve at tme 5 n a UO soluton but would cause a delay for many vehcles arrvng after t. To capture UO condtons the LP objectve should vew a transfer from arrvng at tme 6 to tme 5 as beng so great that t outweghs all else wthn the network occurrng at a later tme. 119
Defnton 1 A vector of lnk flows t yj s at user equlbrum, f the objectve functon (5.3.3.1) M t vector satsfes: M t M t-1 > (M S M t ) D (1) where, D s the total demand wthn the network and S s the maxmum tme nterval n the modelng perod. From complementary slackness condtons, ths bound was proven to be suffcent for dynamc user optmal flows n Waller and Ukkusur (2003) and Ukkusur (2002) and wll be used n the frst example demonstrated n ths paper. An approxmaton scheme for ths M t vector wll be used later to solve a more reasonable szed test network. These results are shown n Secton 6. 5.3.3 Dynamc User Optmal Network Desgn Problem (DUO NDP) Formulaton The objectve of the DUO NDP s to determne the optmal decson for capacty mprovements of lnks (street segments) contnuously, so as to mnmze the total travel cost of each vehcle (the UO objectve), whle accountng for the User Optmum (UO) DTA behavor. The total amount spent on a network s constraned by B, whch s a user-specfed budget. The varable b j s defned as the amount of budget spent on each cell j. Parameter B specfes the ncrease n the jam densty of j cell j for one unt of ncrease n budget b j, whlec j represents the cost parameter specfyng the ncrease n saturaton flow of cell j for one unt of ncrease n b j In t t the NDP model, N whch appears n constrants 5.3.3.3, 5.3.3.5 and 5.3.3.8 s j j t t replaced wth ( N + b B ), j j j j t Q j whch appears n constrants 5.3.3.3 to 5.3.3.8 s replaced wthq t j + bjc j. A new sngle constrant s added requrng that the sum of b j must be less than the total budget B. The complete formulaton of the DUO NDP for a sngle destnaton s shown n 5.3.3.1 to 5.3.3.12 120
Objectve Functon : Mnmze ' t T (, j) ' Es M y t t j (5.3.3.1) subject to: \{ R, S}, x 5x 5 y + y = 0, t t51 t51 t51 k j 51 k'9 ( ) j'9( ) ' C C C ' t T (5.3.3.2) t t t t t t t t t t t y 5 x ( 0, y ( Q + b C, y ( Q + bc, y + x ( ( N + b B ), j j j j j j j j j j j j j (5.3.3.3) (, j) ' E EE, t ' T y t 5 x t ( 0, y t ( Q t + bc, (, j) ' E, t ' T (5.3.3.4) j j s o R y t ( Q t + b C, y t + t x t ( t ( N t + b B ), (, j) ' E, t ' T (5.3.3.5) j j j j j j j j j j j D y 5 x ( 0, y ( Q + bc, ' C, ' t T (5.3.3.6) t t t t j j D '9 j ( ) '9 j ( ) y t 5 x t ( 0, y t ( Q t + bc, (, j) ' E, t ' T (5.3.3.7) j j M y ( Q + b C, y + x ( ( N + b B ), ' j C, ' t T (5.3.3.8) t t t t t t t j j j j j j j j j j j M 51 '9 ( j) 51 '9 ( j) x 5 x + y = d, j '9( ), ' C, t ' T, x = F (5.3.3.9) t t51 t51 t51 0 j R T y = 0, (, j) ' E, x = 0, ' C / C, (5.3.3.10) 0 j S t x 0, ' C, ' t T, y 0, (, j) ' E, t ' T (5.3.3.11) t j b ( B; b 0 (5.3.3.12) 121
5.3.4 Dual Formulaton of the DUO NDP The dual formulaton of the DUO NDP has an ntutve economc nterpretaton n terms of the margnal contrbuton to the system by spendng unt cost on the network. Ths nterpretaton s more readly seen from another problem whch s smultaneously solved along wth the orgnal problem and ths s the dual problem of the LP. The dual formulaton for the program (5.3.3.1 to 5.3.3.12) s shown n 5.3.4.1 to 5.3.4.14 and ths s followed by an economc nterpretaton. The followng are the assocatons of the dual varables: the dual varables t <, (, j) ' C / C, t ' T wth constrants 5.3.3.2 and 5.3.3.9; 1t 2t 3t 4t j j j j o s µ, µ, µ and µ, (, j) ' E, t ' T, wth the set of constrants 5.3.3.3, respectvely; respectvely; respectvely; µ andµ (, j) ' E, t ' T wth the set of constrants 5.3.3.4, 1t 3t j j s µ, µ (,j)' E, t ' T, wth the set of constrants 5.3.3.5, 2t 4t j j D 1t 3 t 7 and 7, ' C, ' t T, wth the set of constrants D 5.3.3.6,respectvely; 5.3.3.7, respectvely; 1t 3 t µ andµ, (, j) ' E, t ' T wth the set of constrants j j M 2t 4 t 7 and7, ' j C, ' t T, wth the set of constrants 5.3.3.8, M respectvely; Gj wth the set of constrants 5.3.3.10, corresponds to constrant 5.3.3.12. Max t 2t t 3t t t 4t t 2t t t 4t t 3t { Qjµ j + Q µ j + jnjµ j } + { Qjµ j + jnjµ j } + Q 7 (, j) ' EO (, j) ' ED ' C D + B (5.3.4.1) t' T { } t 3t t 2t t t 4t t 3t t51 t + Q µ j + Q 7 + N 7 + Q µ j + d < (, j) ' EM ' EM (, j) ' ES ' CR < 5< 5 µ + µ ( 0 t t+ 1 1t t 4t j k < 5< 57 + µ ( 0 t t+ 1 1t t 4t k j '9 k '9 ' C t ' T 51 ( ), ( ), o, 5 k '9 1 ( ), ' C, t ' T D (5.3.4.2) (5.3.4.3) 122
< 5< 5 µ + 7 ( 0 j '9( ), ' C, t ' T t t+ 1 1t t 4t j < 5< 5 µ ( 0 j '9( ), ' C, t ' T t t+ 1 1t j < 5 < + µ + µ + µ + µ ( 0 (, j) ' E, t ' T t+ 1 t+ 1 1t 2t 3t 4t j j j j j < 5 < + 7 + µ + 7 + µ ( 0 (, j) ' E, t ' T t+ 1 t+ 1 1t 2t 3t 4t j j j < 5 < + µ + 7 + µ + 7 ( 0 (, j) ' E, t ' T t+ 1 t+ 1 1t 2t 3t 4t j j j j j < + + µ + µ ( M (, j) ' E, t ' T t 1 1t 3t j j t S < < C 1 0 5 + ( 0 ' o D M M R (5.3.4.4) (5.3.4.5) (5.3.4.6) (5.3.4.7) (5.3.4.8) (5.3.4.9) (5.3.4.10) t t <, < j, ' t T, ' C unrestrcted n sgn Gj, (, j) ' C unrestrcted n sgn (5.3.4.11) 7, 7 ( 0, ' C, t ' T 1t 3t D 7, 7 ( 0, ' C, t ' T (5.3.4.12) 2t 4t M µ, µ ( 0, (, j) ' E EE EE E E, t ' T 1t 3t j j o M R S µ, µ ( 0, (, j) ' E EE EE E E, t ' T 2t 4t j j o D R S (5.3.4.13) 2t 4t 3t 2t 3t 4t 5 1 ( C µ k + µ k ) + C µ j + C + C + ( 0, 5 ' t T k' () j' () ' C \{C,C } R S (5.3.4.14) The dual formulaton presented above dffers from the orgnal dual UO DTA n the addton of B to the objectve functon, and a new constrant for each of the cells 123
except the source and snk cells, and the non-postve constrant for. The dual varable < t can be nterpreted as the margnal contrbuton of an addtonal unt of demand at cell and tme nterval t to the total cost of the network. However, ths contrbuton s not measured n terms of margnal travel tme, as n the case of the DSO NDP (Waller, 2000), but n terms of the UO objectve functon. The physcal nterpretaton of ths objectve functon s dffcult, n a manner smlar to the statc case. Defnton 2 The margnal contrbuton of < t corresponds to the tme-dependent least tme path for an addtonal unt of flow. Ths follows drectly from the defnton of the dual varable. Smlarly, the duals µ and 7 can be seen as the change n the objectve functon due to the ncrease n the value of the correspondng rght-hand sde constrant by one 2t unt. For example, µ j s an estmate of the change n the objectve due to unt ncrease n the capacty of nflow nto cell j. Snce duals µ and 7 are non-postve, they never ncrease the value of the objectve functon. The new dual varable can be nterpreted as a bound on the change n the objectve functon value, f the budget B s ncreased by one unt. A smple analyss of the extreme cases of the allowable budget B, gves a few nsghts. When B s zero, there s no contrbuton of ths to the objectve functon, and the LP can freely set wthout penalty. Therefore the constrant wll always be non-bndng (ths constrant can be vewed as redundant constrant), and the soluton of the resultng LP wll be dentcal to the UO DTA problem. Lkewse, when the budget B s very large, the potental mpact of B wll domnate all the other costs n the objectve functon (5.3.3.1). Ths forces the non-postve dual varable to approxmately equal zero, and hence bnds the constrant (5.3.3.12), whch forces the 124
non-postve 2t 3t µ j, µ j, 4t µ j and 2t 7, 7, 7 values to zero. These zero valued 3t varables can be nterpreted as addtonal budget, jam densty, and saturaton flow, whch have no mpact on the soluton whatsoever. Thus, the objectve functon reduces to ' C R d < t51 t + B, where constrants. The dual constrants nvolvng the dual to: 4t t < corresponds to the prmal flow conservaton t < (5.3.3.2 5.3.3.10) reduce < 5< 5 µ ( 0 t t+ 1 1t j j '9 k '9 ' C t ' T (5.3.5.1) 51 ( ), ( ), o, < 5< 57 ( 0 t t+ 1 1t 5 k '9 1 ( ), ' C, t ' T (5.3.5.2) D < 5< 5 µ ( 0 j '9( ), ' C, t ' T (5.3.5.3) t t+ 1 1t j M < 5< 5 µ ( 0 j '9( ), ' C, t ' T (5.3.5.4) t t+ 1 1t j R < 5 < + µ ( 0 (, j) ' E, t ' T (5.3.5.5) t+ 1 t+ 1 1t j j o < 5 < + 7 ( 0 (, j) ' E, t ' T (5.3.5.6) t+ 1 t+ 1 1t j D < 5 < + µ ( 0 (, j) ' E, t ' T (5.3.5.7) t+ 1 t+ 1 1t j j M < + + µ ( M (, j) ' E, t ' T (5.3.5.8) t 1 1t j t S < < C 1 0 5 + ( 0 ' (5.3.5.9) where, the other remanng non-zero duals ( µ, 7 ) correspond to the constrants: 1t j 1t 125
t t yj 5 x ( 0, ' CD, ' t T (5.4) '9 j ( ) y t j t 5 x ( 0, (, j) ' E, t ' T (5.5) M The only remanng prmal constrants (5.3.3.4, 5.3.3.2, 5.3.3.9) restrct the flow between cells smply to the upstream densty and ensure flow conservaton at each cell. As a result the dual varable t < corresponds to the free-flow least tme path from each cell to the destnaton. Ths agrees wth ntuton, wth what would happen from havng a large budget, because gven enough capacty any network would reach free-flow condtons. 5.4 Computatonal Results for the Dynamc Network Desgn problem Ths secton dscusses the computatonal experence to solve the NDP s mentoned n the prevous sectons. As a startng pont, the fundamental dfference between the DUO NDP and DSO NDP s shown by computng the optmal lnk nvestments for dfferent levels of budget. The spendng dfferences under DUO and DSO condtons are also demonstrated. 5.4.1 Results from DUO NDP The DUO NDP model s tested for dfferent levels of budget. The cell verson of the network s shown n Fgure 5.9 and the tme nvarant cell saturaton flows and jam denstes are shown n Table 5.3. The value of the model saturaton flow rate Q depends on the lnk saturaton flow rate and the sze of the tme nterval. The tmedependent demand from source cells 1 and 14 s (2, 2, 1) vehcles when t = 1, 2 and 3 respectvely. The optmal M t vector used for ths network s from the optmalty condtons derved n Ukkusur (2002) and s shown n Table 5.4. In the latter subsecton, an approxmaton M t vector wll be presented to solve a more reasonable network. 126
2 4 6 8 10 12 1 3 5 7 9 11 13 14 15 16 Fgure 5.9 Cell Representaton and Tme Invarant parameters of the test network Table 5.2 Tme Invarant parameters of the test network Cell 2 3 4 5 6 7 8 9 10 11 12 15 16 t N 4 4 4 4 4 2 4 4 4 4 4 4 4 j t Q 1 2 2 2 1 2 1 2 1 2 1 1 1 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 Table 5.3 Value of Mt wth tme Tme M value 1 1 2 1 3 1 4 1 5 1 6 9375001.063 7 9960938.505 8 9997559.595 9 9999848.413 10 9999991.464 11 9999999.405 12 9999999.964 13 9999999.999 14 10000000.000 127
The proposed cell expendtures translate nto ncreases n jam densty and saturaton flow rates for the specfed cells. Ths could be nterpreted as a lane addton, f the expendture warrants such an ncrease or other smaller capacty ncreasng measures such as lane clearance changes, addton of medan etc. The model s mplemented for dfferent budget levels, ncreasng at the level of 0.1 unts. The budget s ncreased from 0 to 200 unts, thus gvng 20,000 data ponts for analyss. Ths solves 20,000 lnear programs wth budget ncrement of 0.1 each tme. The TSTT s calculated for each budget, and ths s plotted wth the budget ncrease. A sample calculaton for a budget value of 120 unts s shown n Table 5.4, where all the other cells not shown have no mprovements. The results show that at a total budget value of 120 unts, t should be spent completely on the merge ramp, comprsng of cells 14-15-16-7-9-11-13 and the optmal values of b are as shown n Fgure 5.10. Table 5.4 Optmal values of b for B = 120 n DUO NDP Cell No Optmal Budget spent (b ) under DUO condtons Optmal Budget spent (b ) under DSO condtons 7 40 48.8889 9 20 15.5556 11 20 15.5556 13 10 10 14 10 10 15 10 10 16 10 10 128
Fgure 5.10 DSO and DUO TSTT wth the ncrease n total budget B Further, Fgure 5.10 shows the change n total system travel tme as a functon of the total budget B. The free-flow condtons are acheved when B s approxmately 134 unts. The 20,000 budget ponts gve a plot of the TSTT vs B. The margnal beneft of addng addtonal budget s greater after 45 unts, untl about 120 unts. But addng an extra unt after 120 unts has a lesser margnal beneft than before. Ths beneft s n fact less than that n the 0-45 unts range. Further, t s mportant to note that at a budget of around 134 unts, the network experences free-flow condtons and any further addton of budget wll have a dmnshng beneft. Ths can be observed n the dual varable of the formulaton as well. 5.4.2 Comparson of the DUO and DSO NDP A sample result of the DSO NDP s gven n Table 5.4, where the total budget B s the same as before (120 unts) for comparson purposes. Ths table shows the budget s spent entrely on the mergng ramp, but t s mportant to note that the values of capacty mprovements n each cell are sgnfcantly dfferent. It s also nterestng to note that the capacty mprovement on the bottleneck cell 7 s hgher n the DSO case as compared to the UO case and vce versa on the freeway cells 9 and 129
11. The plot from the 20,000 data ponts for the DSO NDP s shown n Fgure 5.10. It can be observed that the mprovement n TSTT n the ntal regon s greater for the same ncrease n budget under DSO condtons as compared to under DUO condtons Further, as would be expected t can be observed that at large budgets both the DUO and DSO NDP soluton solutons converge to the same soluton snce they reach the free flow condtons. 5.4.3 Implementng the DUO NDP on the Nguyen Dupus network As dscussed n Waller and Ukkusur (2003), one of the lmtatons of the DUO NDP formulaton s the nature of M t cost vector to solve large-scale networks. The rate of growth of the values of the M t vector decreases exponentally and t becomes dffcult to capture ths for larger networks. In ths secton, an approxmaton procedure for calculatng M t s s presented and computatonal experence on a non-trval network s presented. By the defnton of the M t vector, all the flows arrvng at a later tme nterval are delayed and prorty of route choce s gven to the vehcles n the tme nterval n consderaton. However, n real traffc networks, not all the flows may be delayed. For example, a vehcle whle choosng ts route n the CBD area may have lttle/no effect on the travel tme of a vehcle/s n the later tme ntervals far away from the CBD. Ths allows the relaxaton of the defnton of M t and allows the development of better approxmaton technques, whch can stll capture UO condtons. The method employed here uses approxmaton technques to get a 130
closed form expresson for a tractable M t vector. Ths functonal form of M t s used on larger networks and equlbraton of flows s analyzed. The behavor of the M t s a strctly monotonc and nearly concave n nature. Dfferent functonal forms to capture M t exst. The ntal values of M t are derved from the orgnal defnton of M t (Waller and Ukkusur, 2003), but an approxmaton (n terms of the M t vector, the UO flows are found to be optmal for ths example, not approxmate) s used to capture the values of M t vector n the hgher tme ntervals. The hgher values of M t are chosen such that the dfference between the successve M t values follows an arthmetc progresson and the dfference s such that t s decreasng wth ncreasng M t values. For nstance, for an assgnment tme of 70 tme ntervals n the followng example, the value of M 1 to M 8 s 1 because no vehcle reaches the destnaton cell untl t = 13. M 9 to M 12 are derved from the optmal M t vector whereas M 13 to M 70 are approxmated so that they capture UO condtons. The magntude of M 13 to M 70 s ncreasng monotoncally such that the dfference between them s decreasng 23 23 rapdly. For nstance, M 21 s 1.91 10 and M 22 s1.92 10, the dfference s10 whereas the dfference between M 70 and M 69 tapers off to 12 10. The plot of the M t s shown n the Fgure 5.11. 21 131
Fgure 5.11 Tractable M t vector for the DUO NDP Gven ths tractable vector, the DUO NDP s solved. Whle the employed M t vector does not follow the bound proved n Waller and Ukkusur (2003), t follows a smlar pattern. The basc process adopted here for the larger network s that the NDP LP s solved wth the tractable vector, then the flows are verfed to be user optmal by solvng the combnatoral approach for dynamc UO DTA presented n Waller and Zlaskopoulos (2003). It should be noted that the combnatoral approach restrcts flows to take ntegral values, though, so the behavor could only be closely but approxmately verfed as user optmal (as the flows generated from the LP take contnuous values). In general, for any gven example, an approxmate M t vector s derved, the NDP model solved, and then the flows verfed wth the combnatoral algorthm. If the flows are not found to be DUO n nature, a new approxmate M t vector could be calculated. 132
Fnally, the DUO and DSO NDP models are tested on network smlar to the Nguyen and Dupus (1984) network (as shown n Fgure 5.12). The network has 13 nodes, 20 lnks and 2 O-D. The cell representaton of ths network consstng of 63 cells s shown n Fgure 5.12 (b). The length of each cell n the network s 440 ft. Each cell has two lanes n the network wth a speed lmt of 30 m/hr, a saturaton flow of 1800 vphpl, and jam densty of 200 veh/m. The network has two O-D pars and the demand s 8.5 and 12.3 respectvely durng the peak congeston for each tme nterval for O-Ds 1-13 and 14-13. 1 1 12 1 2 3 49 4 5 53 6 7 10 12 20 50 54 8 2 4 5 3 4 5 6 7 8 14 15 27 28 29 16 30 31 32 33 34 35 36 37 38 39 51 55 57 9 10 18 11 13 15 17 17 52 56 58 11 9 6 10 7 11 8 2 18 19 40 41 42 19 16 43 44 45 46 47 48 12 13 20 59 63 21 13 9 21 62 60 61 3 22 23 24 25 26 (a) (b) Fgure 5.12 Nguyen and Dupus s Test Network: (a) Lnk-Node Verson and (b) Cell Verson 5.5 Comparson and dscusson of results The models are tested wth three demand scenaros; a lght congeston scenaro, where a total of 116 vehcle-trps are generated over all the orgns, a moderate congeston case wth 186 vehcle trps, and a heavy congeston case wth 133
308 vehcle-trps. For each congeston level, two demand-loadng patterns are tested: a unform and a peak pattern. In the unform pattern, the demand s unformly dstrbuted over the assgnment perod, whle n the peak pattern 50% of the demand s generated n the mddle assgnment perod and the rest s unformly dstrbuted wthn the remanng assgnment perod. The TSTT s of the varous scenaros are presented n Tables 5.5, 5.6, and 5.7 wth the amount of budget used. Demand Level (Vehcle Trps) Table 5.5 Baselne Case wth no mprovements Dstrbuton Type Budget n $ Thousand (B) Total System Travel Tme UO(TSTT) DSO TSTT 308 Unform 0 11912.5 11908.4 186 Unform 0 3853.02 3853.02 116 Unform 0 1858.46 1858.46 308 Peak 0 11998.2 11762.90 186 Peak 0 3938.6 3938.60 116 Peak 0 1946.64 1936.64 Table 5.6 Effect of Budget Levels for Peak dstrbuton, Demand = 308 vehcle trps Max Budget n $ Thousand (B) Budget spent n $ Thousand ( b ) 134 DUO TSTT (seconds) % of Improvement 0 0 11998.2 0 100 100 7634.48 36.4 200 200 7198.45 40.0 300 300 6345.8 47.1 500 500 5774.76 51.9 800 800 5568.8 53.6 1200 1200 5036.31 58.0 1600 1600 4826.53 59.8 2000 2000 4703.34 60.8 2500 2500 4489.25 62.6
3000 3000 4397.74 63.3 4000 4000 4302.87 64.1 4500 4500 4194.91 65.0 5500 5500 4140.24 65.5 5700 5700 4134.7 65.5 5800 5800 4131.94 65.6 5900 5900 4130.4 65.6 6000 5990 4130.4 65.6 Demand Type Table 5.7 Effect of Dstrbuton Types and Congeston Levels Dstrbuton Type Max Budget n $ Thousand (B) Budget Spent n$ Thousand ( b ) DUO TSTT % of Improvement 308 Peak 5800 5800 4131.94 65.6 308 Unform 5800 5523 4078.40 66.0 308 Peak 200 200 7198.45 40.0 308 Unform 200 200 7024.40 41.6 308 Peak 100 100 7634.48 36.4 186 Peak 100 100 3250.87 17.5 116 Peak 100 100 1659.06 14.8 308 Peak 200 200 7198.45 40.0 186 Peak 200 200 2888.28 26.7 116 Peak 200 200 1591.32 18.3 The above analyss shows that the budget level, demand-loadng pattern, and congeston level sgnfcantly nfluence the DUO and DSO NDP recommended spendng polces. The dfferences and effects of each of these factors on the spendng polcy are dscussed next. Budget Level The network s tested at dfferent levels of budget values. Fgure 5.13 shows the change n total system travel tme as a functon of the budget spent. The 135
analyss tested thrteen dfferent budget szes for the same demand level and pattern (308 vehcles, peak hour dstrbuton) for the DUO and DSO NDP. The maxmum achevable mprovement for the DUO NDP s 65.6% and for the DSO NDP s 64.8 %. It s worth notng, however, that the margnal beneft of an addtonal thousand dollars dmnshes rapdly after the frst twelve hundred s spent. It was observed that after a suffcently hgh budget both the DUO and DSO TSTT s converge to a value of 4130.4 seconds because the network experences free flow condtons. Congeston Level Table 7 shows that the heaver the congeston, the hgher the benefts are n the total travel tme savngs for the same amount spent. At a very low congeston level (116 vehcles), there s only a 14.8% mprovement n the TSTT observed, compared to 36.4% for 308 vehcles at the same budget level. These results are ntutve, but stll useful n performng a beneft-cost analyss for varous scenaros of demand realzaton. Demand Pattern It was observed that the unform demand shows a hgher TSTT percentage mprovement (41.6%) at sgnfcantly lower budget than the peak demand for the DUO NDP. The dfference between the unform and peak demand decreases at hgher budgets. A smlar trend s observed for the DSO NDP. Ths s ntutve snce congeston s lower under unformly generated demand patterns as aganst the 136
peak demand scenaro. Note further that the spendng polces under the two scenaros are substantally dfferent as shown n Table 5.7. 11600 11100 Total System Travel Tme (sec) 10600 10100 9600 9100 8600 8100 7600 7100 6600 6100 5600 5100 4600 UONDP SONDP 4100 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Budget ($ Thousands) Fgure 5.13 TSTT as a functon of budget for DUO and DSO NDP Spendng Polcy The spendng polces of the DUO NDP and DSO NDP for the unform and peak scenaros at a budget of 2 mllon dollars and a demand level of 308 vehcles are shown n Fgure 5.14. The nsghts obtaned from ths are dscussed n ths secton. Frstly, we note that all the freeway cells are mproved n all the scenaros. Secondly, n the peak demand case both the DUO and DSO cases mprove capactes on dfferent routes. It s observed that the DUO NDP mproves along the least travel tme paths whch are the freeway segments (cells 1 through 26) and the path 4-9-10-137
11-2. Note that cell 18 s the bottleneck cell and DUO spends a small porton on path 4-9-10-11-2 so that t can equlbrate the flow onto the freeway and ths path thereby mnmzng the travel tme of each vehcle. On the other hand, the DSO NDP spends the entre budget on the freeway lnks as t benefts the overall system by mprovng the margnal least costs. Thrdly, t can be observed that n the unform case both the DUO and DSO NDP spend the budget on the same routes but at dfferent levels. Ths s because of the dfference n flows n both the DUO and DSO assgnments. Further, t can be observed that the mprovement appled on all the cells s consstent for a partcular lnk. For example, cells 26, 61, 62, 63 that correspond to lnk 21 (Fgure 5.12) are expanded equally n all scenaros. Cells 20, 21, 22, 23, 24 correspondng to lnks 19 and 9 receve a smaller amount, most lkely because they carry less traffc under all the scenaros. Furthermore, all the cells n both the DUO and DSO cases that receve resources for a gven budget level, receve at least the same amount of resources at hgher budget levels. 138
Budget Spent ($ Thousand) 180 160 140 120 100 80 60 40 UO Peak SO Peak UO Unform SO Unform 20 0 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 61 62 63 Cell Number Fgure 5.14 Spendng polces for the unform and peak loadng demand patterns for DUO and DSO NDP, 308 trps and 2000 Thousand Dollars Evaluaton of DSO capacty mprovements wth DUO flows Both the proposed models (DUO and DSO NDP) make smplfyng assumptons for modelng tractablty. It would be benefcal to assess whch would perform better for proposng capacty mprovements. Ideally, a b-level approach would be desrable where the planner takes the system s perspectve and follows a DSO approach whle the flows n the network follow DUO behavor. However, ths s beyond the scope of ths paper. An alternate approach to study an approxmate beneft of the NDP models would be to observe how the UO DTA behaves under DUO and DSO NDP capacty mprovements. The former s smply the DUO NDP and the latter s the evaluaton of the UO DTA under DSO NDP capacty 139
mprovements. Ths analyss shows that the evaluaton of the UO DTA wth DSO capacty mprovements yelds better total system travel tme as compared to the DUO NDP. For example at a budget level of 800,000 Dollars, the DUO NDP and DSO NDP gves TSTTs of 5568.80 and 5113.58 seconds respectvely. However, the evaluaton of the UO DTA wth DSO NDP capacty mprovements gves a TSTT of 5456.55 seconds. Smlar results were observed at dfferent budget levels. It was observed that the TSTT under UO DTA evaluaton moves farther away from the DUO NDP TSTT to a value n between the DUO and DSO NDP TSTTs. Ths result shows that n plannng applcatons, f no b-level model s avalable, t s better to use DSO NDP rather than DUO NDP because ths would gve a better TSTT even under the DUO flows. But t s mportant to note that ths result was observed for the Nguyen Dupus network studed here and the results may not be generalzed for all traffc networks. 5.6 Extenson of the DUO NDP Models: Accountng for Stochastcs One of the advantages of the lnear formulaton of the DUO NDP problem s the potental extenson of the model to ncorporate stochasttes at dfferent levels of decson makng. Many models n transportaton over the past several decades have been solved under determnstc condtons. Notwthstandng the success of both the statc and dynamc models, however, the assumpton that all model parameters are known wth certanty lmts ts usefulness n plannng under uncertanty. In ths secton, we demonstrate the mpact of ths uncertanty n a dynamc traffc network 140
envronment. We propose models that nclude the long term demand as a random varable wthn the DUO NDP optmzaton problem dscussed n the prevous sectons. The emphass here s on formulaton and potental soluton methodologes rather than n depth analyss of the value of capturng demand uncertanty. Ths wll reman an actve area of future research. We propose two methods to handle demand uncertanty n dynamc networks: Chance constraned model and the two stage recourse model. Each of them s dscussed brefly n the followng secton. 5.6.1 Chance constraned DUO NDP model Most of the Stochastc Lnear Programmng (SLP) methods elmnate the possblty of second-stage nfeasblty completely. However, n some stuatons t may be more approprate to accept the possblty of nfeasblty under some crcumstances, provded the probablty of ths event s restrcted below a gven threshold. These lead to mathematcal programs wth probablstc constrants or Chance constrants where the nfeasblty s accepted but only wth a fnte probablty. The two major advantages of Chance constrant formulatons are (1) the ablty to ntroduce relablty constrants explctly and (2) the ablty to derve robust solutons wth exogenously specfed functonal forms. A good revew of the SLP methods can be found n Hgle and Sen (1999). The general method of solvng Chance constraned formulatons s to formulate the determnstc equvalents to the chance constrants and the objectve functon, and to solve the resultng 141
mathematcal program wth approprate soluton technques. Ths s demonstrated for the DUO NDP model. In the determnstc DUO NDP, there s a sngle set of constrants that generate the demand at each orgn for each tme perod. The demand constrant can be wrtten as an nequalty wthout the loss of generalty as: t t 51 t 51 t 5 x 5 x + y d 1, j '9( ), ' C, t ' T (5.6) j R If t 1 d 5 s a random varable wth the probablty dstrbuton t 1 d F 5 (whch can be dscrete or contnuous), the equvalent chance constrant of (7) can be wrtten as: x 5 x + y d $ j '9 ' C t ' T (5.7) t t51 t51 t51 Pr[ j ], ( ), R, Further, t can be seen that by the defnton of the dstrbuton functon: t 51 d t t51 t51 t51 t t51 t51 Pr[ x x yj d ] F ( x x yj ), 5 + = 5 + (5.8) whch n combnaton wth (8) results n: t 1 5 F x 5 x + y $ (5.9) d 1 1 ( t t 5 t 5 j ) or t51 d t t51 t51 51 ( j ) ( ) ( ) x 5 x + y F $ (5.10) Relaton (5.10) s the determnstc equvalent of the chance constrant (5.7). Includng t n the DUO NDP nstead of the constrant (5.6) produces the equvalent determnstc LP for the Chance constraned UO DTA. Constrant (5.10) contans parameter$, whch corresponds to the network relablty factor. The hgher the 142
value of $ chosen, the greater the demand value t51 d 1 ( 5 ) ( ) F $ becomes and the more relable the soluton becomes, perhaps, at the expense of optmalty. By usng the chance constrant and ncreasng the confdence level$, the lnear program solves the NDP wth demands greater than the expected value. The desgn strateges nferred from the soluton should then be acceptable for any demand up to the value used and some confdence level exsts for these control strateges. Further, by solvng for hgher demand levels, potental bottlenecks can be found. These bottlenecks mght manfest themselves wth any varaton n demand ether spatally or temporally, but are not present at the expected value demand case. Whle a sngle program executon may be suffcent for gettng a soluton when confdence level s exogenously provded, addtonal computatons are necessary f probablstc system performance data s requred. Ths s demonstrated on the network n Fgure 1 to hghlght the mportance of capturng demand uncertanty. 5.6.2 Two Stage DUO NDP Recourse model Recourse models are dynamc models that capture the decson maker s ablty to take correctve acton nto the future. The nfeasblty of the soluton s corrected n the second stage at a certan cost once the uncertanty s realzed. A revew of the recourse models and the assocated soluton methodologes can be found n Brge and Louveaux (1997). The general mult-stage SLP wth recourse can be formulated as shown below. 143
Let SP denote the problem solved at the th stage (=1, 2, 3 ). Then the mathematcal statement of the two-stage stochastc LP (SLP) problem wth recourse s: T mn x c x + E [ Q( x, )] s. t : Ax b = where T Q( x, ) mn ( ) 2 = y q y SP s. t : T ( ) x + W ( ) y = h( ) 1 ( SP ) x 0 where '#represents the number of scenaros (system realzaton), x and y are varables, T ( ) represents the technology matrx and W represents the recourse matrx. Further, W, A and b are parameters; T and h are realzaton dependent parameters. For the DUO NDP, the second-stage classcal T parameter s not realzaton dependent and corresponds to the NDP parameters Q and N. The classcal H parameter corresponds to the realzed NDP demand d, the classcal W parameter corresponds to both and, and b corresponds to the NDP parameter B. The frststage classcal varables x, correspond to the desgn budgets b, and the second-stage classcal varables y correspond to the basc DUO NDP densty and flow varables. Fnally, there are no frst-stage costs n the objectve functon and the second-stage costs correspond to the expected value of the DUO NDP objectve over all 144
realzatons. If dscrete values are taken for potental demand realzatons, the twostage stochastc formulaton for the dynamc DUO NDP s as shown: Mnmze subject to: E 0 M ( ) t t y j 1 (5.6.2.1) '# t' T ' C x( ) 5x( ) 5 y( ) + y( ) = 0, t t51 t51 t51 k j 51 k'9 ( ) j'9( ) ' C \{ C, C }, ' t T, '# (5.6.2.2) R S t t t t t t y( ) 5 x( ) ( 0, y( ) ( Q + b C, y( ) ( Q + bc, j j j j j j t t t t t y( ) + x( ) ( ( N + b B ), j j j j j j j (, j) ' E EE, t ' T, '# (5.6.2.3) t t t t y( ) 5 x( ) ( 0, y( ) ( Q + bc, (, j) ' E, t ' T, '# (5.6.2.4) j j s t t t t t t t y( ) ( Q + b C, y( ) + x( ) ( ( N + b B ), (, j) ' E, t ' T, '# (5.6.2.5) j j j j j j j j j j j D o R y( ) 5 x( ) ( 0, y( ) ( Q + bc, ' C, ' t T, '# (5.6.2.6) t t t t j j D '9 j ( ) '9 j ( ) t t t t y( ) 5 x( ) ( 0, y( ) ( Q + bc, (, j) ' E, t ' T, '# (5.6.2.7) j j M y( ) ( Q + bc, y( ) + x( ) ( ( N + b B ), ' j C, ' t T, '# (5.6.2.8) t t t t t t t j j j j j j j j j j j M 51 '9 ( j) 51 '9 ( j) x 5 x + y = d j'9 ' C t ' T x = F ' C '# (5.6.2.9) t t51 t51 t51 0 ( ) ( ) ( ) j ( ), ( ), R,,,, y = j ' E '# (5.6.2.10) 0 ( ) j 0, (, ), T x( ) = 0, ' C / C, '# (5.6.2.11) S t x( ) 0, ' C, ' t T, '# (5.6.2.12) t y( ) 0, (, j) ' E, t ' T, '# (5.6.2.13) ' C j b ( B (5.6.2.14) 145
Based on the formulaton (5.6.2.1-5.6.2.14), general SLP soluton methods can be appled (see Brge and Louveaux, 1997) such as L-shaped Methods, Nested decomposton, Monte Carlo samplng or stochastc decomposton. The effect of onlne nformaton can be formulated as two stage recourse problem usng the formulaton presented before. The ntuton for the formulaton s based on realzaton that the nformaton s used as a recourse varable,.e., nformaton used n the frst stage s used n the decson makng (or route choce by UO DTA pattern) n the second stage. 5.6.3 Example Demonstraton In ths subsecton, the chance constraned model s mplemented on the small four node- four lnk network n Fgure 5.1. The value of the M t vector used for the test network s the same as n Table 5.2. All the other parameters are the same as used n the demonstraton n 4.1. The model s run at dfferent values of alpha. Table 5.8 gves the expanson polces for the mproved cells. There are only a few cells whch consstently have mprovements for all demand scenaros. It s mportant to note that the capacty mprovements presented here are for the base case wthout samplng for dfferent values of alpha. An nterestng trend observed n the capacty mprovements s that the freeway mergng ramp s mproved n almost all the scenaros. All the cells on the mergng ramp have dentcal expanson polces for dfferent levels of relablty consdered here. Ths trend ndcates that these cells would be the canddates for expanson under all scenaros even under the realzaton of hgher demands and dfferent values of alpha. 146
Table 5.8 Cell Expanson polces for chance constrant Model n $ Mllons for dfferent values of alpha Cell No = 0.5 = 0.75 = 1 7 40.52 53.23 36.52 9 20 19.89 21.91 11 20 19.89 21.91 13 10 5 11.91 14 10 7.5 9.41 15 10 7.5 9.41 16 10 7.5 9.41 Fgure 5.15 represents the approxmate expected system performance as a functon of the -level used to generate the respectve solutons. From the graph we can observe that the use of the expected value of the demand does not perform optmally under all scenaros. By usng an -value of 0.75, a gan of 12.4% s realzed. However, t s mportant to realze that the results presented here do not sample the entre sample space and only a few alpha values are consdered. Hence, there mght be some other value of alpha whch could perform better under all demand scenaros, but we can conclusvely say that t s not the expected value of demand. The key result from the Chance constrant analyss s that under demand uncertanty plannng for the expected value of demand wll not yeld optmal solutons. The results are smlar to the System Optmal case as demonstrated by Waller and Zlaskopoulos (2001) but ganng addtonal nsght nto the DUO varaton s crtcal. 147
Alpha 90 Total Travel tme (Mn) 80 70 60 50 40 30 20 10 0 0.5 0.6 0.7 0.8 0.9 1 Fgure 5.15 Expected System Performance for the DUONDP 5.7 Concludng Remarks The models presented n ths research are a step towards buldng NDP models that adequately account for dynamc traffc propagaton. Some benefts of the models nclude the ncorporaton of the Cell Transmsson Model as the traffc flow model rather than usng lnk performance functons, capturng the tme dependent traffc characterstcs and n proposng a model for dynamc plannng applcatons. Addtonally, we extend the dynamc network desgn model to account for demand uncertanty. The major dfference between the DUO and DSO NDP model s that n the former t s observed that the capacty mprovements are made so that vehcles take the least cost path where as n the latter t was observed vehcles prefer the least margnal cost path. Further, for the examples consdered n ths research, t was observed that the margnal beneft from capacty mprovements s greater durng the ntal spendng polces and ths beneft decreases wth ncrease n budget. It s also observed that for the networks examned under a non b-level model, t would be better to make plannng decsons based on DSO NDP snce ths gves a better overall system mprovement than the DUO NDP. As mentoned earler the formulatons presented n ths paper are lmted to a sngle destnaton. 148
Further, due to the relaxaton of the flow conservaton relatonshp, t could result n the holdng back of traffc. However, t s mportant to note that n the sngledestnaton DUO verson of the model no unt of flow wll be held f t causes any delay n that unt s arrval tme. The holdng, therefore, only occurs n the case of a non-unque soluton and correct flows can be found n post-processng. Although, these are clear lmtatons of ths formulaton, the models presented here are an ntal step n modelng NDPs that account for dynamc and stochastc traffc condtons. Numerous future research efforts reman. Snce the proposed formulaton s lnear, extensons to the model can be developed to study fxed-arrval tme demand and numerous other realtes made possble wth the benefcal mathematcal structure. Furthermore, specalzed decomposton approaches (L et al., 2001) may be developed to explot the specal structure of the problem to solve the dynamc network desgn problem more effcently. The proposed DUO NDP models and the nsghts obtaned from them can be used n developng approaches that are more effcent by explotng related optmzaton technques. Chapter 7 dscusses some of the recent computatonal analyss of the stochastc formulatons proposed n ths chapter. 149
Part IV Copng wth Uncertanty: Onlne Informaton as Recourse 150
Chapter 6 A stochastc traffc equlbrum model wth nformaton recourse If error s corrected, whenever t s recognzed as such, the path of error s the path of truth Hans Rechanbach 6.1 Introducton Recent advances n ATIS technology n provdng onlne network nformaton has motvated researchers to develop technques to evaluate and desgn such strateges. For nstance, on the hardware front, CoPlot, a technology used n the ATIS project n the Northeast corrdor of the U.S. (Lst et al., 2005) provdes onlne route gudance (trp plannng) to travelers usng a handheld devce. CoPlot s an embedded trp plannng software that uses the centralzed computng power avalable from a remote locaton n Prnceton, New Jersey to update onlne network condtons. It s antcpated that these technologes wll contnue to grow n the future to optmze the exstng network nfrastructure and n further reducng network wde travel tmes (congeston). To evaluate, crtcally understand and desgn these systems, advanced onlne network assgnment models are necessary. Ths chapter deals wth the mpact of uncertanty alone on the equlbrum decsons of a motorst travelng through a network when the probablty densty functon of the arc capactes are known only en-route and can be updated based on ths nformaton. As shown n example 1.2, n a stochastc network t s better to take onlne equlbrum strateges than equlbrum paths as n tradtonal determnstc equlbrum problems. We refer to ths problem as the onlne traffc equlbrum problem or the User equlbrum wth recourse (UER). The problem explored here s mportant because UER s a potental strategy to cope wth uncertanty and n devsng robust assgnment polces n the face of uncertanty. Ths problem complements the work on evaluatng and desgnng transportaton networks for uncertanty by gvng a model to cope wth uncertanty and take necessary recourse 151
n transportaton networks. Further, t readly helps n developng a model that helps n evaluaton of nformaton provson for network users thereby helpng network planners n devsng approprate ATIS strateges for reducng network wde congeston. In the context of UER, recourse s vewed as the ablty for the drver to reevaluate hs/her remanng path at each node based on the knowledge obtaned durng hs trp. Dfferent spatal and topologcal dependences can be captured as varatons of the UER. For example, a drver s knowledge of the state of a partcular lnk may reveal nformaton about other lnks n the vcnty. Ths llustrates a trval example of spatal dependence, where nformaton s obtaned about successor arcs whle travelng a network. In temporal dependency, there s an arc correlaton wthn successve tme ntervals and ths s accounted by vehcles on the network n choosng ther routes. In ths chapter rather than routng we are concerned about the nteracton between the supply and demand n a transportaton network when onlne nformaton s provded. It wll be shown that an equlbrum wth recourse affords better savngs n travel tme as compared to offlne equlbrum studed n determnstc networks (or varatons there of n stochastc networks). Ths chapter s organzed as follows. In secton 6.2, we explore the problem defnton and propose a formulaton for the problem. In secton 6.3, we demonstrate a paradox n onlne networks wth user equlbrum where provdng nformaton may not be useful always. The network loadng model and a heurstc soluton approach are proposed n Secton 6.4. Concludng remarks are summarzed n secton 6.5. 6.2 Problem Formulaton 6.2.1 Problem Defnton Ths work wll begn wth the tradtonal determnstc equlbrum model. Whle employng a dynamc (rather than statc) model for evaluatng the mpact of nformaton would seem deal, t wll be shown that certan problem varatons usng the statc approach lead to tractable mathematcal models for analyss and should, 152
therefore, also facltate potental future developments. The smple Determnstc User Equlbrum (DUE) problem can be stated as (Bell and Ida, 1997): Mnmze c ( x) dx v j j (6.1) j x= 0 v = Ah t = Bh h 0 where v s the vector of lnk flows, h path flows, A the lnk-path adjacency matrx, t orgn-destnaton trp vector, B trp-path adjacency matrx, and c the vector of arc cost functons. An ndvdual element of a matrx or vector s represented as v j, c j, etc. C A B E D Fgure 6.1 Example Network User Equlbrum wth Recourse (UER) can be defned n numerous ways. The varatons explored here treat the functonal form of the lnk cost functons as unknown a-pror and only learned when the upstream node of a gven lnk s reached. Ths general case encompasses stuatons where the functonal forms are known but lnk capactes are uncertan. For nstance take the trval network from above; when a traveler departs node A, they do not know wth certanty the specfc capactes for arcs (B,C) and (B,D). They, nstead, have some probablty 153
dstrbuton for the capactes. When the traveler reaches node B, the capactes would become known and they could then choose the next arc to follow along ther route to E. If nstead of capactes, the traveler learned the cost they would ncur, ths basc varaton would reduce to the shortest path problem wth recourse. In ths example, though, the cost wll depend on the number of other network users who also choose to use a partcular arc (resultng n an equlbrum flow). Such a problem can be formulated by expandng the sngle arc flows n the UE formulaton wth condtonal flows. A formal defnton of the problem follows. The notaton used n ths chapter s chosen to be consstent wth the work n Waller and Zlaskopoulos (2002) for easy reference as the temporal dependent onlne shortest path (TD-OSP) wll be used as a sub problem to the UER. Gven a probablstc drected network G = (N,A,S,P,C,U) where N ( N =n) descrbes the set of nodes, A the set of arcs ( A = m), S the set of possble states for each arc, P to be the set of assocated probabltes, C the set of arc-state cost functons, and U to be the set of possble states for each node. For the ntal analytcal development t wll be assumed that G s acyclc, numercal methods wthn the sub-problem soluton methodology wll be presented though for later crcumventng ths requrement. Denote wth 9 51 )( the set of predecessor nodes of arc 'N, and 9() the set of all successor nodes. On G, a destnaton node d'n s defned. The expected cost from node j to the destnaton s denoted by E[j]. E[j,s] denotes the expected cost from j to the destnaton node, gven that (,j) s at state s. In addton, let S,j denote the set of all possble states for the arc (,j); arc cost refers to the cost of arc (,j) at state k. Fnally, an arc cost does not become determnstc after t has been vsted, but t remans probablstc wth the same dstrbuton, makng successve vsts result n ndependent stochastc trals. The notaton and the onlne TD-OSP are summarzed n Appendx E. For each arc (,j)'v, denote as S j the set of possble states (each state representng a dfferent cost functonal form) for arc (,j) and P js as the assocated j c, k 154
probabltes for each possble arc-state s's j. For each node 'N, denote U as the set of all combnatons of dscrete states for arcs emanatng from node whch wll be referred to as the node state. Note, that U = (,j) S j and gven a node state u'u, all arc states emanatng from node can be determned. To fnd a specfc arc state gven the emanatng node state, denote uj to be the state on arc (,j) gven node s n state u. Fnally, denote C js ( ) to be the cost functon on arc (,j) gven the arc s n state s. Note that t wll ntally be assumed C js ( ) represents a lnear cost functon. Methods for numercally approxmatng non-lnear costs n the soluton method wll be provded after the ntal analytcal formulaton. Upon arrval at a node each unt of flow would learn the node state u'u and then select the succeedng node to vst. Therefore, the flows n the network are condtoned on the node-states. Note, t s assumed that the flows wll be condtoned on the node state of the mmedate upstream node only. Denote the condtonal expected flow as v ju (the expected flow on arc (,j) gven node was found to be n state u'u ) whch experences a state specfc cost c js (v ju ) where s= u. Defne H as the hyperpath-arc-state adjacency matrx whch wll characterze the probablty a partcular hyperpath ncludes a specfc condtonal lnk. As mentoned prevously, t s ntally assumed that H s taken from an acyclc network and numercal methods for crcumventng ths requrement are subsequently presented n the OSP soluton method. Defne the elements of H as: H ju =Probablty arc (,j) s vsted gven U =u Note that an equvalent H exsts defned by: H ju = Probablty node s vsted or 0 6.2.2 UER Formulaton Take the followng mathematcal program based on condtonal flows 155
Mnmze v ju ju x= 0 v = Hh t = Bh h 0 Pj! J cj! ( x) dx (6.2) uj uj The User Equlbrum wth Recourse (UER) problem can be extended from the DUE formulaton as shown n (6.2). The dffculty n ths formulaton s n calculatng the optmal condtonal path flows and then fndng the correspondng hyperpath-arcstate adjacency matrx (H) rather than the tradtonal lnk-path adjacency matrx A. Some analoges could be drawn to certan Stochastc User Equlbrum (SUE) methods n replacng A wth a probablstc value. However, the cause of the assocated probabltes s the onlne routng of users and not dscrete choce. As prevously dscussed, H n ths formulaton descrbes the condtonal probablty a unt of flow wll travel on a certan arc and when substtuted for A creates a formulaton whose sub-problem must ensures that the expected costs of all used condtonal paths are equal; ths sub-problem corresponds to the Onlne Shortest Path Problem (OSP). The UER Sub-problem Gven the prevous formulaton (6.2), a lnearzaton problem at a soluton uj! u ju Mnmze vju * Pj * cj ( vk! ju ) (6.3) v = Hh t = Bh h 0 vk ju yelds 156
The formulaton (6.3) corresponds to the problem of fndng the optmal hyperpath gven a set of probablstc lnk costs. Ths can also be solved va an Onlne Shortest Path algorthm such as n Waller and Zlaskopoulos (2002). Gven the set of probablstc costs, the algorthm typcally provdes the expected cost labels for each node n the network gven a user follows the optmal onlne shortest path. Here, we requre the probablty a traveler wll vst each lnk condtoned on the upstream node state. Therefore, addtonal nformaton must be stored from the algorthm. Gven the condtonal probabltes from the OSP algorthm, a straghtforward soluton method can be devsed based on the Method of Successve Averages (Powell and Sheff, 1982). Other more complex methods are feasble as well f the objectve from formulaton (2) s explctly mantaned. However, the number of condtonal lnks may grow excessvely large thereby decreasng the computatonal gans from non-msa methods. Ths, however, remans an open queston for future research efforts. No theoretcal results wll be further presented n ths chapter. The followng property s conjectured from the lnearzaton of (6.2). Property 6.1: A traffc network s n UER f each user follows a route that guarantees the mnmum hyperpath (strategy) avalable and no user can unlaterally change hs/her route to mprove ther travel tme 6.2.3 UER Demonstraton Gven the smple network n Fgure 6.2, take a basc cost functon as: C js (x) = 1 + x / k js Wth arc states and probabltes specfed by S AB = S CE = S BC = S BD = 3 S DC = 10 and S DE = {1,2}, P DC1 = P DC2 =.5 Wth capactes: k j1 = 1 k j2 = 4 157
Note that, U A = 1, U B = 1, U C = 1, U D = 2 There are fve hyperpaths wthn ths network. Here, a gven hyerpath can be vewed as a set of elementary paths defned by a sequence of node states /u where 'N and u'u (when more than 1 possble node state exsts, otherwse smply the node wll be lsted): Hyperpath Node/State of Elementary Path Probablty of followng an Elementary Path wthn a Hyperpath 1: A-B-C-E 1 2: A-B-D/1-E.5 A-B-D/2-E.5 3: A-B-D/1-C-E.5 A-B-D/2-C-E.5 4: A-B-D/1-E.5 A-B-D/2-C-E.5 5: A-B-D/1-C-E.5 A-B-D/2-E.5 If a partcular traveler were followng hyerpath four, for nstance, they would depart node A employng arc (A,B), then (B,D). Upon arrvng at node D they would learn f ths node were n state 1 or 2 (mplyng arc (D,C) has a capacty of ether 1 or 2). If node D s found to be n state 1, the traveler would follow arc (D,E) to arrve at the destnaton. If D s found to be n state 2, then the traveler would follow arc (D,C) followed by (C,E). Ths s very smlar to the onlne shortest path problem whch s n fact a sub-problem to UER as wll be dscussed later. The matrx H s shown below 158
H n ths example s the matrx: H1 H1 H3 H4 H5 A-B 1 1 1 1 1 B-C 1 0 0 0 0 B-D 0 1 1 1 1 C-E 1 0 1.5.5 D/1-C 0 0 1 0 1 D/2-C 0 0 1 1 0 D/1-E 0 1 0 1 0 D/2-E 0 1 0 0 1 6.3 UER Paradox Ths secton wll provde a counter-ntutve example when all the users n the network adopt onlne strateges mnmzng ther own expected travel tme. A possble explanaton for ths paradox s explaned towards the end of ths secton. In the motvatng example, we demonstrated that there s a beneft for each motorst n takng an onlne user equlbrum route as compared to the statc traffc assgnment route. However, we have left an open queston n the example whether the beneft of nformaton can be generalzed n any gven graph G. In ths secton we revst that example for a slghtly modfed cost structure on the arcs. The same network s shown n Fgure 6 except that we show lnks c and e n bold to ndcate a decrease n capacty. We decrease the capacty on both the lnks by a factor of ten as compared to the motvatng example. The new cost functons for the example n Fgure 6 are: C a ( x ) = 2 C b ( x 2 ) = 2.5x 2 Cc ( x 3 ) = 20x 3 Ce ( x 4 ) = 30x 4 C f ( x 5 ) = 2.5x 5 1 + x 6 w. p. 0.8 C d ( x 6 ) = 1 + 100 x 6 w. p. 0.2 159
1 C a =2 a 2 b c d 3 f e 5 4 Fgure 6.2 Example network for demonstratng user equlbrum wth recourse paradox Solvng the a pror assgnment based on expected costs, we arrve at a expected user cost of 234.923 unts. On the other hand a condtonal assgnment based on the equal hyperpath concept, yelds an expected user cost of 239.895 unts. The mportant pont to note here s that by provdng nformaton may not be benefcal always. A decrease n the capacty or change n the cost functons can worsen the delays to each user even under the provson of nformaton a seemngly counter-ntutve result, smlar to the Braess s paradox. Ths paradox, can however be readly explaned prmarly because of the selfsh routng of users where each motorst tres to mnmze hs or her own travel tme. Thereby wth nformaton provson each user wll route n a fashon wthout consderaton of the effect of hs or her acton on other users, thereby worsenng the system for everyone. Ths s the classcal example of the Tragedy of Commons. An nterestng queston to explore would be to dentfy, what nformaton strateges and network topologes would cause ths paradox and remedal measures to mtgate such occurrences. However, ths queston wll note be addressed n ths dssertaton but wll be left for future study. 160
Gven ths selfsh behavor of users n the network under nformaton provson, n ths research, we develop a formulaton and a soluton approach to arrve at dervng the optmal lnk flows and network wde performance. We compare the network performance wth apror equlbrum flows and study the propertes of the UER problem. 6.4 Soluton Algorthm and Network Loadng Process After studyng the paradox wth an llustratve example, we proceed to present a soluton algorthm for UER to solve reasonably large scale networks. The nput to the UER model s the O-D demand D, dfferent functonal costs whch represent the lnk states C a s, each state occurrng wth probablty p s. The demand s assumed to be nelastc,.e., the demand s fxed. The random supply s captured n the cost functon. Ths represents a random occurrence, severty of an accdent, reducton n capacty due to weather condtons, closure of a lane due to work zone condtons. The nteracton between the supply and demand gves us the output whch s the equlbrum dstrbuton of the lnk travel tme and the correspondng optmal strateges. There are two man components of the proposed algorthm to solve UER. They are: (1) the onlne routng algorthm that can gve optmal flows for cost ndependent costs whch are represented as TD-OSP and (2) the optmal routng polcy algorthm. The second part of the algorthm uses the method of successve averages (MSA) to solve the fxed pont problem. There s no proof of convergence; nevertheless, the propertes can be studed wth computatonal tests whch are performed n the next secton. The man features of the proposed algorthm are presented below: 161
ALGORITHM STATUER Step 0: (Intalzaton) 0.1: Input O-D and C a s, p s, and 0.2 : Set n = 0 Perform all or nothng assgnment usng the Temporal dependence Onlne shortest path algorthm (TD-OSP) based on C a s = C a s (0), a. Ths gves{x a, s 1 }. Set counter n =1 Step 1 : (Update) Set C a s = C a s (X a, s 1 ),, a Step 2 : (Drecton Fndng) Perform TD-OSP[C a s (X a, s 1 )], a to get a set of auxlary flows { a, n s Y } Step 3 : (Averagng usng MSA) Fnd new flow pattern by settng 1 X X n Y X a,n+1 = a,n + a, n 5 a,n s s s s Defne a, n an an a an Z(X s ) = ps xs cs ( xs ) as the total expected travel tme ncurred by all a users n the network n teraton n a, n+ 1 a, n Step 4 : (Convergence crteron) IF ( Z(X ) 5 Z(X ) <, STOP Otherwse, set n = n + 1 and go to step 1. The core part of the above soluton algorthm s n solvng the network loadng process. In the standard statc models, the lnk flow nformaton can be readly recovered from the path flow nformaton. But n the UER the condtonal path flow has to be derved for all the lnks. The teratve process descrbed above wll stop after the convergence crteron s reached. The convergence s a measure of the change n the expected total network travel tme n two successve teratons. 162 s s
Typcal values of used n ths study are 0.001 and 0.01. The network loadng process (TD-OSP) s explaned further n ths secton. The lmted Temporal Dependence (TD-OSP) algorthm can be seen as a frst towards a more general temporal dependence f ths model s combned wth the spatal dependence. Prevous approaches to study temporal dependence model (Andreatta and Romeo (1988); Polychronopoulos and Tstskls (1996)) have produced computatonally ntractable algorthms. The TD-OSP s an all-to-one label correctng algorthm n whch each node has a sngle cost value whch descrbes the shortest path from that node to the destnaton. However, n the UER we also need the path ponter nformaton to load the flow onto the OSP. For dong ths we make a mnor modfcaton to the algorthm by storng the lnk flows for each state usng a path ponter. Ths storage of the path ponter nformaton ncreases the computatonal requrements of ths algorthm. The fnal soluton from the TD-OSP algorthm wll consst of a set of the expected costs for each node to reach the destnaton and the correspondng lnk flows on each arc of the network and the path ponters correspondng to the next arc that the traveler wll traverse. The TD-OSP algorthm pseudo code s presented n Appendx E. The next secton demonstrates ntal computatonal results of the proposed algorthm. 6.5 Computatonal Tests The computatonal results n ths secton ntend to support the prevously stated fndngs as well as evaluate numercally the performance of the proposed algorthm n secton 6.4. We demonstrate that the computatonal tme s reasonably fast for UER on medum szed networks. An mportant nsght ganed from the computatonal analyss s the quantfcaton of the beneft of the UER assgnment versus the off-lne (a pure apror assgnment). The effcency of the network s calculated n terms of the expected total system travel tme. Dervng tght bounds on the dfference between the UER and the off-lne assgnment wll be a nterestng study for future research. To shed lght on the potental advantages of the UER 163
approach computatonal results on two test networks are performed. The STATUER algorthm s mplemented keepng track of the path ponters n each teraton of the algorthm. For each network, the STATUER s compared wth the tradtonal a pror assgnment (equvalent to the expected value of the UER problem). The prmary consderaton n the analyss was not that of computatonal tme because t s obvous that the a pror algorthm converges much qucker (n the order of 8-10 teratons for small networks) as compared to the STATUER. However, t should be noted that the computatonal tme of the STATUER was not prohbtve; t was n the order of 3-4 mnutes on the networks consdered n ths research. The tests were conducted on a Pentum PC wth 686 processor on a GNU/Lnux platform usng a gcc compler. The model was coded n C++ and tested on the network shown n fgures 6.3 and 6.4. 2 1 3 Fgure 6.3 Small test network to mplement STATUER 2 4 1 5 3 6 164
Fgure 6.3 Test Network for mplementng STATUER a a The cost functon used n both the networks s,0 a c = c (1 + x ). The frst k k k experment s conducted on the test network n Fgure 6.2 wth the network parameters smlar to the frst three rows n Table 6.1. The beneft of analyzng the small network s that we can enumerate all the possbltes and check the results wth hand calculatons. The expected network performance from the STATUER was about 55,732.5 and t took about 5547 teratons for the soluton to converge. The offlne algorthm gave a total network performance of about 207,093 and requred just 95 teratons for convergence. The second experment s performed on the test network shown n Fgure 6.3. Each of the lnks s assumed to be n two states and the cost functons and the probablty matrx are shown n table 6.1. Table 6.1 Network parameters for test network n Fgure 6.3 Arc Cost Probablty 1--2 (5,10) (0.2,0.8) 1--3 (8,12) (0.6,0.4) 2--4 (1,3) (0.5,0.5) 2--5 (6,8) (0.2,0.8) 3--2 (10,20) (0.4,0.6) 3--5 (10,12) (0.3,0.7) 3--6 (2,4) (0.8,0.2) 4--5 (4,6) (0.25,0.75) 6--5 (1,3) (0.9,0.1) The experment compares the dfference between the average total network performance usng the STATUER and the offlne equlbrum assgnment. It was observed that the STATUER at convergence ( = 0.01) gves an expected total travel tme of 242, 892 unts whle the offlne equlbrum at the expected costs gves a total travel tme of 636, 311. Ths s total savngs of about 61.83% as compared to the offlne equlbrum assgnment, a sgnfcant savngs. 165
Fgure 6.5 MSA convergence of the STATUER and OFFLINE algorthms We also compared the convergence of the STATUER and the OFFLINE algorthms on dfferent test networks, cost functons and stoppng crteron. It was observed that n almost all the cases the STATUER convergence was much slower than the OFFLINE algorthm. Fgure 6.5 shows the convergence of these algorthms for the test network n Fgure 6.4. The X-axs of the plot represents the number of teratons of the algorthm and the Y-axs corresponds to the expected network performance (objectve functon Z(X a, n s ) ). In the above plot, t can be observed that STATUER takes lesser teratons for convergence as compared to the OFFLINE, but these cases were qute rare n other experments. Both the experments demonstrate the beneft of accountng for onlne strateges and the beneft of nformaton n transportaton networks. The raton of the STATUER and the OFFLINE total network travel tmes can be used as a measure of the beneft of nformaton recourse n transportaton networks. Addtonal computatonal results to study the dfference n the STATUER and 166
OFFLINE algorthms wth ncrease n network sze, connectvty, O-D demand, probablty states are requred for better nsghts nto the problem. Further, the mpact of cost varance due to onlne algorthm should be studed to realze the robustness of the network performance n STATUER algorthms. 6.6 Concludng Remarks Ths chapter was concerned wth the statc user equlbrum wth recourse problem consderng temporal dependency. In ths problem the user assgnment s studed under onlne nformaton provson. General propertes of the STATUER algorthm are studed and an ntal fxed pont formulaton s proposed. The mpact of nformaton studed here, modeled as the dfferent probablty states n a network was shown to yeld sgnfcant travel tme savngs n the expected sense. The benefts could depend on the network topology and the quantfcaton of beneft cannot be generalzed for all networks. A certan paradox for the problem s dentfed where nformaton provson could ncrease the total network travel tme. A soluton approach based on the MSA s proposed for the problem. Apparently, the proposed algorthm provdes nsghts and potental tools to study the beneft of onlne nformaton n transportaton networks. However, future studes should concentrate on general network dependences, addtonal forms of nformaton recourse and account for traffc dynamcs. 167
Chapter 7 Conclusons and Future work Mere facts are for chldren only. As they begn to pont towards conclusons they become food for men. Edmund Selous Modelng stochastcty n transportaton networks s ganng promnence for obtanng mproved nsghts nto network wde traffc mpacts due to the nherent uncertanty n the O-D demand. Further, wth the accelerated growth n real-tme nformaton dssemnaton technologes, accountng for avalable nformaton n an onlne fashon could potentally lead to better traffc management n congested areas. In ths dssertaton, we have studed new modelng approaches for dealng wth uncertanty, robustness and nformaton recourse n transportaton networks. We have developed technques for evaluatng network uncertanty n statc networks usng sngle pont approxmatons and dervng analytcal expressons for the system performance. Further, we have devsed network desgn formulatons that account for uncertanty and robustness, user equlbrum wth recourse n transportaton networks to optmze system performance. Whle we have explored dfferent methodologes n answerng a few basc questons n modelng uncertanty and robustness, we realze that ths work s only an mportant frst step n understandng the mpact of uncertanty and a desre to model robustness n transportaton networks. To hghlght ths vew, n ths concludng chapter of the dssertaton, we suggest some open questons and future problems of research n ths enrchng area of transportaton networks. Our lst s not exhaustve; t only attempts to ndcate the turf left unexplored for future research. We also summarze some of the recent results related to ths work. The rest of ths chapter presents an overvew of the methodologcal contrbutons of ths study, summarzes the recent work, and fnally, dentfes ten future research questons whch wll be crtcal n ths lne of research. 168
7.1 Research Summary Ths secton dscusses the methodologcal contrbutons of ths dssertaton. The dssertaton provdes a framework for dealng wth uncertanty n network evaluaton, strategc network desgn and the onlne equlbrum stage of transportaton plannng. These are dscussed under three man headngs: (1) Methodologes for network evaluaton under uncertan demand, (2) Robust network desgn, and (3) Accountng for nformaton recourse. 7.1.1 Methodologes for network evaluaton under uncertan demand The frst part of the dssertaton focused on developng technques for modelng network evaluaton under O-D demand uncertanty. Specfcally, we developed two methodologes for dealng wth long term demand uncertanty. The frst approach develops dfferent sngle pont approxmatons for the random O-D demand matrx. The determnstc traffc equlbrum problem s used to solve the sngle pont approxmatons. It was found that the qualty of the soluton s hghly senstve to the sample sze ncluded n the approxmaton procedure. The rsk averse trmmed mean approxmaton procedure was found to perform reasonably well as compared to all the other approxmaton procedures. Ths methodology s useful n solvng large scale equlbrum problems under uncertan demand. The second approach derves analytcal expressons of the expected value and the varance of the network performance when the O-D demand s an ndependent normal dstrbuton. Expermental analyss on multple networks reveals that the expressons are approxmate wth ncrease n network sze. The man underlyng reason for ths approxmaton s that O-D and lnk level correlatons are gnored n the approxmatons. 7.1.2 Robust network desgn The second part of the dssertaton concentrated on mprovng transportaton network performance by accountng for the uncertanty n the strategc desgn stage. 169
An mportant varaton of the NDP accountng for uncertanty and robustness s formulated as an extenson of the determnstc NDP. The formulaton s multobjectve n nature, where the tradeoff between the expected value and the varance are studed wth dfferent weghts assocated to the two objectves. Ths problem s dffcult to solve because of the non-smooth nature of the optmzaton problem. We propose an evolutonary algorthm to solve ths problem and explore the dentfcaton of effcent GA parameters to solve the NDP. 7.1.3 Accountng for Informaton Recourse The thrd part of ths dssertaton focused on onlne nformaton based strateges to cope wth uncertanty n transportaton network. The fundamental queston addressed n ths lne of work s to study the traffc condtons when travelers are provded real-tme nformaton and are allowed to take adaptve routng strateges n the network. A fxed pont formulaton of the problem and a heurstc soluton algorthm based on MSA are proposed. The man sub-routne n the proposed algorthm for studyng nformaton recourse s the network loadng. Ths procedure s performed usng the Lmted Temporal Dependence (TD-OSP) algorthm whch assumes that the cost of an arc (a,b) s known once node a s reached. Ths type of nformaton s referred to here as temporal dependency, snce t can be nterpreted as yeldng nformaton on shortterm future arc costs based on current observatons from the same arc. To the best of our knowledge, the equlbrum problem studed here n stochastc transportaton networks s one of the frst of ts knd and can be solved n polynomal tme n acyclc networks. Computatonal results support the ntuton that an onlne strategy based assgnment s better than the offlne apror equlbrum n stochastc transportaton networks. For each of the computatonal test, our man goal was to compare the UER soluton wth the tradtonal apror equlbrum wth the expected costs n the network. It was found that the travel tme savngs were hgh when drvers routed accordng to onlne strateges. 170
The developed model was also explored to study the presence of any paradoxes due to the selfsh routng of drvers. It was found that under certan cost functons, the onlne equlbrum soluton may not be the best soluton as compared to an apror equlbrum soluton. However, these cost functons and network structures could be pathologcal cases and further analyss s requred to fundamentally understand the nature of these paradoxes. 7.2 Recent work n ths area The methodologes developed n ths dssertaton have facltated the development of many related extensons to ths work by researchers n our group and elsewhere. Ths secton s ntended to provde a synopss of the recent developments of ths work. The O-D demand n chapter 3 was assumed to be ndependent and dentcally dstrbuted. In the conclusons t was conjectured that the approxmatons are due to gnorng the nherent correlatons n O-D demand. Duthe (2004) explores a methodology usng the Hypersphere Decomposton approach for treatng correlated long term O-D demand n the statc traffc equlbrum problem. The effects of dfferent demand varances, types of correlatons and congeston levels are performed on two test networks. It was noted that, the more postve the correlatons, the more the underestmaton n the varance of the TSTT assumng O- D demand ndependence. It was also observed that as congeston ncreases the percent ncrease n standard devaton decreases. On gong work related to ths ncludes developng samplng based methods for effcent evaluaton of network performance wth correlated demand (Duthe et al., 2005) Another approxmaton procedure whch complements the work dscussed n chapter 3 s developng effcent samplng based technques to evaluate the network performance under demand uncertanty. Unnkrshnan et al. (2005) proposed the use of fve samplng technques - Monte Carlo, Antthetc, Latn Hypercube, randomzed Quas Monte Carlo and Control Varates as approxmaton 171
methods for evaluatng the traffc equlbrum problem to obtan an approxmate estmate of the expected future performance. The numercal results appear nonntutve n the sense that the methods often perform dfferently for the stated problem versus typcal stochastc problems n other domans. Latn Hypercube was observed to perform well for low sample szes. Antthetc samplng technque was found to outperform all other examned methods for larger sample szes on the traffc equlbrum problem. Chapter 4 n ths dssertaton developed a robust network desgn problem. To the best of our knowledge the RNDP model s the frst of ts knd n the network modelng lterature. Sumalee and Watlng (2005) develop a mathematcal formulaton to help the network planner to decde on the most approprate mprovements to ensure a certan level of servce. However, ther man objectve s to maxmze the total probablty of TSTT s under a specfed threshold. The problem defnton was presented, but no specfc formulatons, theoretcal developments or soluton approaches were presented n ths work. In chapter 5, we proposed the two stage stochastc programmng formulaton of the transportaton network desgn problem satsfyng user optmal condtons. We presented ntal results on a test network. Karoonsoontawong and Waller (2005) performed further systematc computatonal analyss on the same formulatons and compared the system optmal and user optmal stochastc NDPs. The common random numbers and ndependent random numbers are used to generate the stochastc demand. They perform numercal analyss on the test network shown n Fgure 5.9 and demonstrate that the common random numbers strategy outperforms the ndependent random numbers strategy. Other man nsghts from the analyss are smlar to the ones obtaned n chapter 5: (1) the SO and UO models allocate nvestment dfferently for certan budgets, and the stochastc but determnstc solutons may lead to a bottleneck, (2) for the SO models, t should be more valuable to solve the stochastc than the determnstc models, but t s not always the case for the UO models; (3) the SO models appear more desrable than 172
the UO models; and (4) t should be more valuable to solve the stochastc model accountng for more randomness. Kyunghw et al. (2005) studed a blevel formulaton of the dscrete verson of the UO and SO models proposed n chapter 5. A heurstc approach based on GA s used to solve the problem. The ntal populaton s generated from the cell transmsson based dynamc traffc assgnment model wthout mprovement. The heurstc procedure s also used to create feasble desgn solutons by accountng for a budget constrant. Expermental desgn and tests were conducted to choose approprate operators and parameters used n GA. Fnally, ntal test results were presented by comparng the network desgn solutons of NDP based on statc traffcs wth dynamc traffc condtons on the Soux Falls test network. The models developed n chapter 3 of ths research were also used n developng ntegrated state of art models for evaluatng transportaton network performance under sesmc condtons. Km et al. (2005) developed modelng technques to assess network wde mpacts under dfferent earthquake scenaros whch are a result of the stochastc capacty reductons. They proposed dfferent samplng technques for evaluatng the network mpacts and the best methodology s dentfed. The statc traffc equlbrum problem s used and the resultng probablstc lnk flows are used to detect the canddate lnks for mprovement n the transportaton network. The methodology s demonstrated on Charlotte, South Carolna test network and the results were analyzed. Ths nformaton was used not only n mprovng servce, plannng and repar schedulng of nfrastructure/brdges but also n determnng where an nvestment n mprovng the overall network performance s warranted. In chapter 6, we developed an equlbrum model wth onlne nformaton to drvers based where the routng problem s prmarly an onlne shortest path algorthm wth lmted temporal dependence. The prmary assumpton n ths algorthm s that travelers consder only expected costs n choosng the optmal routng strateges n the network. However onlne travel tme varablty s another mportant crteron besdes expected travel tme that travelers wll consder n routng 173
decsons n a stochastc network. Boyles (2005) s explorng the varaton of the user equlbrum wth recourse consderng robustness of the routng strategy. It would be a thought provokng study to nclude relablty n routng strateges wth onlne nformaton. 7.3 Open questons and future drectons As mentoned n the ntroducton, the research of studyng uncertanty, robustness and nformaton recourse s a burgeonng feld wth several avenues for future research. These further studes can sgnfcantly contrbute to the growng body of work n understandng the mportance of uncertanty and robustness n transportaton networks. A lst of ten most mportant future research questons (FRQ) s dscussed below: FRQ 1Boundng Technques In chapter 3 we studed the evaluaton of network performance under O-D demand uncertanty mostly usng two dfferent approxmaton schemes. Snce we were more nterested n evaluatng network performance rather than studyng the worst-possble degradaton of network performance we were naturally led to follow ths approach. A growng area n nternet routng s to quantfy the worst-possble ncrease n network performance due to selfsh routng (Roughgarden, 2002) as compared to coordnated routng (system optmal assgnment). Roughgarden (2002) derved bounds for the same model consdered n chapter 3 but wthout uncertanty. For example, for the lnear cost functons he proved that the worst possble degradaton due to selfsh routng s 4. However, t s clear that we cannot gnore the long term uncertantes present n 3 nfrastructure networks. Naturally t would be worthwhle to develop worst case bounds on the degradaton of network performance by selfsh routng under uncertanty. It s apparent that the same bound holds (although ts stll an open queston f a tghter bound can be obtaned) for short term uncertantes,.e. 174
00 xc ( x ) 1 1 0 ' A UE 1 4 E 0 ( 1 3 00 xc ( x ) 1 1 0 ' A 1 SO. However, t would be nterestng to obtan the stochastc selfsh routng bounds consderng long term uncertanty. In other words, gven a network, G (N, A) wth cost functons c (x ), what can be sad about the rato - E 0 xc ( x ) 1 ' A E 0 xc ( x ) 1 ' A UE SO FRQ 2 Other O-D dstrbutons In chapter 3, we hypotheszed the dstrbuton of network performance when the O-D demand s an ndependent - normal dstrbuton, posson and cauchy dstrbuton. A further queston of nterest s what can be sad about the dstrbuton of network performance when the O-D demand has some reasonable dstrbuton. Some headway n ths drecton has been made by Clark and Watlng (2004) who have used regresson analyss to ft probablty dstrbuton consderng the random O-D demand n a stochastc user equlbrum problem under demand uncertanty. FRQ 3 Computatonal effcency of RNDP In chapter 4, we have presented a robust network desgn model based on an evolutonary algorthm. The computatonal tests show that ths algorthm can be further speeded by usng effcent data structures, tranng the GA nto a dstrbuted and parallelzable algorthm and by a usng an effcent DUE soluton method (Hllel-Bera and Boyce, 2003) whch gves lnk flows on the fly. FRQ 4 Approxmaton Robust Network Desgn Methods In chapter 4, we developed a multobjectve stochastc formulaton of the network desgn problem and have been successful n obtanng network desgn solutons usng a traned GA 175
procedure. However, t s well known that the GA procedure has no theoretcal foundatons on the exstence, stablty and convergence of the obtaned solutons. Most of the work n ths area s based on extensve expermental analyss. Future work should explore other smpler approxmaton methods smlar to the ones consdered n Methodology 1 of chapter 3, samplng based technques to extract smaller number of realzatons for DUE evaluaton and other constructve approaches (Toktas, 2004). FRQ 5 Incorporatng other measure of rsk for RNDP For the robust network desgn, we have prmarly consdered varance as the measure of rsk. However, n transportaton networks t s clear that n some nstances we would want to penalze postve devatons from the expected value of total system travel tme (congeston) rather than earler arrvals. Ths would requre us to explore other measures of rsk such as condtonal varance (CVar), trmmed varances etc. whch have been recently found to be more robust measures of rsk (Szego, 2005). FRQ 6 Large scale optmzaton methods for dynamc NDPs In the dynamc network desgn model formulated n chapter 5, we have developed a sngle level lnear programmng formulaton accountng for the user optmal condtons. However, to get better nsghts nto the model we need to formulate the b-level NDP. Further, effcent soluton approaches such as decomposton technques, L shaped methods whch explot the structure of the formulaton are needed to solve large scale transportaton networks. FRQ 7 Theoretcal ssues n UER The User equlbrum wth recourse model proposed n chapter 6 s formulated as a fxed pont formulaton and the soluton approaches employs an MSA heurstc. However, no theoretcal study was performed to analyze the exstence, unqueness and stablty of the recourse problem. Instead, ths work focused on the development of the soluton approach that works 176
reasonably well to provde nsghts nto the proposed model. The theoretcal study of the problem would place t on a stronger footng. Some of the earler work on the hyperpath based user equlbrum models n transt can be used as an ntal step n dervng the optmalty condtons. It s speculated that dervng the optmalty condtons would requre relaxng the assumptons (on lnk dstrbutons, capactes) to make the model more tractable. FRQ 8 Dynamc UER The model developed n chapter 6 s a statc verson of the onlne equlbrum problem. However, a more realstc based strategc equlbrum model that accounts for traffc dynamcs (DTA) and drver behavor s needed. As mentoned n the lterature revew n Chapter 2 some of the work n ths area develops smlar soluton approaches accountng for dynamc traffc assgnment condtons (Marcotte and Hamdouch (2004)). FRQ 9 Extent of lnk dependence n UER The UER formulaton n chapter 6 consders only lmted temporal dependences (TD-OSP) where t s assumed that the cost of an arc s realzed only after arrvng at the upstream of the node. Ths lmted temporal dependence can be seen as a frst step toward a more general temporal dependence. Other types of spatal dependence (one lnk on the other neghborng lnks) and combned temporal-spatal dependence models can be explored wth the approach proposed n chapter 6. Further, a related queston n modelng s to understand the degree of dependence separaton. To what extent does a lnk n the transportaton network have nfluence on the neghborng lnks under nformaton provson? Insghts nto ths dependence can be obtaned by performng expermental observatons on real transportaton networks or by analyzng the data from smulaton models. FRQ 10 ATIS technology placement/network desgn In chapter 6, the current nformaton on each node s assumed to be provded by ATIS technologes such as 177
VMS, on board devces etc. However to make the model more competent and complete, dfferent nformaton strateges and ther market penetraton should be explored further. For example, addtonal study s needed n understandng how we can gan real-tme nformaton and how do travelers perceve ths nformaton. Wth the development n sensor technologes, n the future, low cost sensor dust can be mplanted on the roadsde to gather onlne condtons and then be used to communcate wth the drvers. Nevertheless, the models presented n chapter 6 can be used to evaluate such systems. A related queston s the effcent placement of these nformaton devces on the roadsde/vehcle. Ths type of network desgn problem wth onlne nformaton wll help us n optmal onlne routng n addton to the effcent placement of the nformaton centrc devces n the transportaton network. 178
Appendx A Computatonal Results for sngle pont approxmatons for dfferent sample szes Table 3.3 Computatonal results for Cauchy dstrbuton, Sample sze = 1000 Functon 4 = 0.1 % devaton 4 = 0.25 % devaton 4 = 0.4 % devaton 4 = 0.6 % devaton 241.98 40.85 241.98 40.85 241.98 40.85 241.98 40.85 1 211.47 43.38 211.47 43.38 211.47 43.38 211.47 43.38 2 207.62 44.41 221.16 40.79 236.36 36.72 469.73 25.76 3 239.24 35.95 287.56 23.01 287.39 23.05 299.73 19.75 4 1057.06 183.01 698.67 87.06 569.24 52.41 472.98 26.63 5 210.36 43.68 216.78 41.96 215.67 42.26 273.47 26.78 6 Table 3.4 Computatonal results for Exponental dstrbuton, Sample sze = 1000 Functon 4 = 0.1 % devaton 4 = 0.25 % devaton 4 = 0.4 % devaton 4 = 0.6 % devaton 273.55 29.8 273.55 29.8 273.55 29.8 273.55 29.8 1 234.47 39.83 241.67 37.98 239.47 38.55 267.32 31.4 2 24.29 93.77 59.59 84.71 96.47 75.24 155.1 60.2 3 224.98 42.26 241.98 37.9 289.71 25.65 297.74 23.59 4 1370.58 251.73 940.92 141.47 750.67 92.64 584.63 50.03 5 204.66 47.48 178.89 54.09 256.76 34.11 251.39 35.49 6 179
Table 3.5 Computatonal results for Logstc dstrbuton, Sample sze = 1000 Functon 4 = 0.1 % devaton 4 = 0.25 % devaton 4 = 0.4 % devaton 4 = 0.6 % devaton 393.45 14.58 393.45 14.58 393.45 14.58 393.45 14.58 1 274.43 40.42 296.77 35.57 324.12 29.64 298.98 35.09 2 294.23 36.12 314.17 31.8 331.31 28.07 347.2 24.62 3 319.9 30.55 384.55 16.52 382.82 16.89 393.93 14.48 4 1091.58 136.98 760.64 65.13 643.48 39.7 554.5 20.38 5 308.66 32.99 324.17 29.62 348.83 24.27 388.99 15.55 6 180
Appendx B Dervaton of Central Moments of Normal Dstrbuton The dervaton below of the central moments s a standard exercse problem n probablty textbooks. 1 ( ) E x m x m dx L 2 + 1 + 1 x [( ) ] ( ) exp{ a 5 m a 5 = a 5 5 2} 2 a 27 d 2d 5L Substtutng ( xa 5 m) = z d : d + 1 L 2 + 1 ( z) 27 5L ( z) exp{ 5 } dz 2 When + 1 = odd number, then the ntegral s a product of an odd functon and an even functon. So, the ntegraton s symmetrc and s zero. When + 1 = even number, for example 2n, then, E x m + 1 [( 5 ) ] = d + 1 L 2 + 1 ( z) 27 5L ( z) exp{ 5 } dz 2 = + 1 d 27 1.3... 1 7 { } + 1 2 2 2 ( + 2) 5 2 : [1.3... ] 1 d + Dervaton of the Moments of the Postve Normal Dstrbuton: 5( x 5m ) E[( x m ) ] ( x m ) exp dx / 2 k k 1 5 = 5 2 0 2 1 27 d 2d 5/ Normalzng the above normal dstrbuton gves k / 2 k d k 5z E[( x 5 m ) ] = ( z) exp dz 27 0 2 1 5/ 181
For k = odd, the above ntegral s zero, for k = even t s a symmetrc ntegral. Substtutng z 2 /2 wth p, we get k / k 51 k 2* d 2 E[( x 5 m ) ] = (2 p) exp[ 5 p] dp 27 5/ k 2 k / k 51 2 k 2 d E[( x 5 m ) ] = ( p) exp[ 5 p] dp 7 k 2 0 k / k 51 2 k 2 d E[( x 5 m ) ] = ( p) exp[ 5 p] dp 7 The ntegral s smply the Gamma Functon and can be evaluated as a closed form for all k 0 k 2 k 2 d k + 1 k E[( x 5 m ) ] = 9( ) 7 2 182
Appendx C Sample plot to determne the sample sze n GA evaluaton for RNDP 183
Appendx D Data for the HF test network n Chapter 4 Arc a length (m) No of lanes alpha Beta Capacty FF speed (m/hr) 1 1 1 10 4 3 60 2 2 2 1 2.5 4 10 60 3 3 3 1 1 4 9 60 5 4 4 1 5 4 4 60 4 5 5 1 10 4 3 60 9 6 2 1 10 4 2 60 1 7 1 1 10 4 1 60 4 8 1 1 1 4 10 60 3 9 2 1 4 4 45 60 2 10 3 1 1 4 3 60 5 11 9 1 0.222 4 2 60 6 12 4 1 2.5 4 6 60 8 13 4 1 6.25 4 44 60 5 14 2 1 16.5 4 20 60 3 15 5 1 1 4 1 60 6 16 6 1 0.167 4 4.5 60 1 Penalty assocated wth each arc 184
Appendx E Notaton and the TD-OSP Algorthm Ths Appendx shows the notaton of the Onlne Shortest Path (OSP) algorthm used n Chapter 6 and shows the pseudo code of the temporal dependent Onlne Shortest Path (TD-OSP) algorthm. OSP Notaton: o : Orgn node t : Destnaton node E[b] : Expected cost to the destnaton node from node b E[b a,s] : Expected cost to the destnaton node from node b, gven that predecessor arc (a,b) s traversed n state s a b p, k : Probablty that arc (a,b) s n state k a b c p,, s k S a,b :, : Probablty that arc (b,c) s n state k, gven that predecessor arc (a,b) was n state s Set of all possble states for the arc (a,b) a b c, k : Cost of arc (a,b) n state k SEL : A Scan elgble lst 9 51 ( a) : Set of all predecessor (upstream) nodes of a; '9-1 (a) : arc (,a) exsts 9 (a) : Set of all successor (downstream) nodes of a; j'9(a) :arc (a,j) exsts M : Vector of probabltes for possble states at a node 7 : Vector of Mnmum Costs for possble states at a node 185
TD-OSP Algorthm Pseudo-Code Set E[t]=0 for all 'N/t, let E[]= / SEL:= 5 9 1 ( t) whle SEL, N Remove an element, q, from the SEL M := [1] 7 := [/] for all j'9(q) tempm := [N] temp7 := [N] r:=0 for all k's(q,j) for all l' 1.. M tempm r := M l * q, j P k f c qj k + E[j] < 7 l else temp7 r := c qj k + E[j] temp7 r := 7 l r:=r+1 M, 7 := call Reduce(tempM, temp7) tempe[q]:= M k7 k k' M f tempe[q] < E[q] E[q] = tempe[q] End SEL:=SEL E 9 51 (q) 186
Functon Reduce(tempM, temp7) M := [N] 7 := [N] for k' 1.. temp7 j := Call Search(7,temp7 k ) 7 j := temp7 k M j := tempm k + M j return(m, 7) End Where Search() performs a search on vector 7 for value temp7 k and returns the locaton f found. If not found, t returns the frst unused locaton wthn vector 7. 187
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Vta Satsh Ukkusur was born n Chodavaram, Inda as a Scorpo n 78. After completng hs schoolng n Rajahmundry, he entered the Indan Insttute of Technology, Madras, Inda. In May 2001, he graduated wth the degree of Bachelor of Technology n Cvl Engneerng. He enjoyed the Mdwestern charm of the Unted States when he entered the top cvl engneerng program at the Unversty of Illnos at Urbana Champagn. After fnshng hs Masters (n December 2002), he moved down south to the Unversty of Texas at Austn wth hs advsor to pursue a PhD n transportaton. He began hs doctoral study n Transportaton Engneerng at The Unversty of Texas at Austn n August2003. After hs PhD, Dr. Ukkusur wll be headed to the pcture perfect snow clmes of the North east Unted States and wll be jonng the faculty at Rensselaer Polytechnc Insttute n Troy, NY. He can be reached va emal at satsh.ukkusur@gmal.com Permanent address: 100 McChesney Avenue, Apt E1 Troy, NY 12180 USA. Ths dssertaton was typed by the author. 196