INSTITUTE OF TRANSPORTATION STUDIES UNIVERSITY OF CALIFORNIA, BERKELEY Public Tranportation Sytem: Baic Principle of Sytem Deign, Operation Planning and Real-Time Control Carlo F. Daganzo Coure Note UCB-ITS-CN-00- October 00
Intitute of Tranportation Studie Univerity of California at Berkeley Public Tranportation Sytem: Baic Principle of Sytem Deign, Operation Planning and Real-Time Control Carlo F. Daganzo COURSE NOTES UCB-ITS-CN-00- October 00 i
Preface Thi document i baed on a et of lecture note prepared in 007-00 for the U.C. Berkeley graduate coure CE59-Public Tranportation Sytem --a coure targeted to firt year graduate tudent with divere academic background. The document i different from other book on public tranportation ytem becaue it i informal, ha a narrower focu and look at thing in a different way. It focu i the planning, management and operation of public tranportation ytem. Important topic uch a financing, governance trategie and urban tranportation policy are not covered becaue they are not pecific to tranit ytem, and becaue other book and coure already treat them in depth. The document i alo different becaue it deemphaize fact in favor of idea. Fact that contantly change and can be found elewhere, uch a tranit uage tatitic and tranit ytem characteritic, are not covered. The document way of looking at thing, and it tructure, i imilar to the author previou book Logitic ytem analyi (Springer, 4 th edition, 005) from which many baic idea are borrowed. (Tranit ytem, after all, are logitic ytem for the movement of people.) Both document epoue a two-tep planning approach that ue idealized model to explore the larget poible olution pace of potential plan. The logical organization i alo imilar: in both document ytem are examined in order of increaed complexity o that generic inight evident in imple ytem can be put to ue a knowledge building block for the tudy of more complex ytem. The document i organized in 8 module: 5 on planning (general; huttle ytem; corridor; twodimenional ytem; and unconventional tranit); on management (vehicle; and employee); and on operation (how to keep bue on chedule). The planning module examine thoe apect of the ytem that are uually viible to the public, uch a routing and cheduling. The management and operation module analyze the more mundane apect required for the ytem to work a deigned. Two more module are in the work: management of pecial event (e.g., evacuation; Olympic); and operation in traffic. Although the document include new idea, which could be of ue to academic and profeional, it main aim i a a teaching aid. Thu, a companion document including 7 homework exercie and 3 mini-laboratory project directly related to the lecture i alo made available. It can be obtained by viiting the Intitute of Tranportation Studie web ite and ii
looking for a publication entitled: Public Tranportation Sytem: Mini-Project and Homework Exercie. Verion of thee exercie and mini-project were ued in the 009 and 00 intallment of CE59: a 4-week coure with two -hour lecture and one -hr dicuion eion per week. Sample olution to the mini-project and exercie can be obtained by univerity profeor by writing to the ITS publication office and requeting a third document entitled: Public Tranportation Sytem: Solution Set. The variou module were originally compiled by PhD tudent Eric Gonzale, Joh Pilachowki and Vikah Gayah, directly from the lecture. Subequently, my colleague Prof. Mike Caidy ued them in an intallment of CE59 and offered many comment. Thi publihed verion ha been edited and reflect the input of all thee individual. Their help i gratefully acknowledged. The error, of coure, are mine. The financial upport of the Volvo Reearch and Educational Foundation i alo gratefully acknowledged. Carlo F. Daganzo September, 00 Berkeley, California iii
CONTENTS Preface.. i Module : Planning General Idea..... - Coure ubtance and organization.... - Tranit Planning. - o Definition. - o How to account for politic... -3 o How to account for demand.. -6 o The hortightedne tragedy. -6 o Planning and deign approache -7 Appendix: Cla Syllabu... -0 Module : Planning Shuttle Sytem. - Overview.... - Shuttle Sytem.. - o Individual Tranportation... - Time-independent Demand.... - Time-Dependent Demand Evening (Queuing) -3 Time-Dependent Demand Morning (Vickrey) -4 o Collective Tranportation.. -7 Time-Independent Demand.. -7 Time-Dependent Demand. -8 o Comparion between Individual and Collective Tranportation. -0 Appendix A: Vickrey Model of the Morning Commute...- Module 3: Planning Corridor.. 3- Idealized Analyi 3- o Limit to The Door-to-Door Speed of Tranit 3- o The Effect of Acce Speed: Uefulne of Hierarchie.. 3-5 Realitic Analyi (patio-temporal) 3-8 o Aumption and Qualitative Iue 3-8 o Quantitative formulation 3- o Graphical Interpretation.. 3- o Dealing with Multiple Standard 3-3 o No tranfer 3-4 o Tranfer and Hierarchie.. 3-7 o Inight... 3- o Standard-Reviited... 3-4 o Space- and Time-Dependent Service... 3-6 Average Rate Analyi.. 3-6 Service Guarantee Analyi... 3-8 iv
Module 4: Planning Two Dimenional Sytem... 4- Idealized Cae (New -D Iue).... 4- o Sytem without Tranfer... 4- o The Role of Tranfer in -D Sytem.. 4-4 Realitic Cae (No Hierarchy)... 4-9 o Logitic Cot Function (LCF) Component.. 4-9 o Solution for Generic Inight.. 4-0 o Modification in Practical Application. 4- o General Idea for Deign 4-4 Realitic Cae (Hierarchie--Qualitative Dicuion). 4-6 Time Dependence and Adaptation 4-7 Capacity Contraint 4-9 Comparing Collective and Individual Tranportation 4-0 Module 5: Planning Flexible Tranit... 5- Way of delivering flexibility..... 5- o Individual Public Tranportation 5- o Collective Tranportation... 5- Taxi.. 5- Dial-a-Ride (DAR). 5-6 Public Car-Sharing... 5-0 Appendix: Determination of Expected Ditance to a Taxi.. 5-3 Module 6: Management Vehicle Fleet 6- Introduction.... 6- Schedule Covering Bu Route..... 6-3 o Fleet Size: Graphical Analyi... 6-4 o Fleet Size: Numerical Analyi.. 6-6 o Terminu Location.. 6-7 o Bu Run Determination.. 6-8 Schedule Covering N Bu Route... 6-9 o Single Terminu Cloe to a Depot.. 6-9 o Dipered Termini and Deadheading Heuritic.. 6-0 Dicuion: Effect of Deadheading.. 6- Appendix: The Vehicle Routing Problem and Meta-Heuritic Solution Method... 6-3 v
Module 7: Management Staffing.. 7- Recap.. 7- Staffing a Single Run.. 7- o Effect of Overtime.. 7-3 o Effect of Multiple Worker Type... 7-4 Staffing Multiple Run... 7-5 o Run-Cutting 7-5 o Covering. 7-6 o Simplified etimation of cot... 7-6 Chooing Worker-Type. 7-8 Dealing with Abenteeim.. 7-9 What i Still Left to be Done 7- Module 8: Reliable Tranit Operation.. 8- Reliability.. 8- Sytem of Sytem 8- o Example : a table ingle agent.... 8- o Example : an untable ingle agent... 8-4 o Example 3: two agent.... 8-5 Uncontrolled Bu Motion... 8-6 Conventional Schedule Control.. 8-8 o Optimizing the Slack.. 8-9 Dynamic (Adaptive) Control.... 8- o Forward looking Method.. 8- o Two Way Looking Method (Cooperative)... 8-4 vi
Public Tranportation Sytem: Planning General Idea Module : Planning General Idea (Originally compiled by Eric Gonzale and Joh Pilachowki, January 008) (Lat updated 9--00) Outline General coure info (admin) Coure ubtance and organization Tranit Planning o Definition o How to account for politic o How to account for demand o The hortightedne tragedy o Planning and deign approache Coure Subtance and Organization Goal of the Coure What tranit can and can t do realitically How to do it (large/mall cale) How to make it happen practically (focu on engineering) Brief Explanation of Syllabu (ee Appendix) The planning part of the coure explore what i poible and how to do it with building block of increaing realim and complexity; it how the limit of tranit ytem and give you the tool to develop ytematic plan. The management and operation part explore the plumbing of tranit ytem. Thi include management item that are hidden from the uer view uch a fleet izing/deployment and taffing plan, a well a more viible operational item uch a adaptive chedule control and traffic management. Planning idea will be reinforced with two lab project and five homework exercie. Management/operation idea will be reinforced with one lab project and two exercie. Imagine public tranit in a linear city. Many people travel between different origin and detination at different time (thin arrow in the time-pace diagram below). Note how people have to adapt their travel in pace to the location of top and in time to the cheduled ervice in order to ue tranit (thick arrow), and how thi adaptation could be reduced by providing more tranit ervice (more thick arrow). Unfortunately, the thick arrow cot money; and thi -
Public Tranportation Sytem: Planning General Idea competition between upply cot veru demand adaptation turn out alway to be at the heart of tranit planning. It will be a central theme in thi coure. x Uer deired top city top adaptation tranit veh trip t Tranit Planning Definition Guideway fixed part of a tranportation ytem, modeled a link and node (infratructure) Network et of link and node, uni- or multi-modal Path a equence of link and node Origin/Detination beginning and end of a path through a network Terminal node where uer can change mode Planning art of developing long term/large cale cheme for the future Mobility the ditance people can reach in a given time (e.g. VKT/VHT) Acceibility the opportunitie people can reach in a given time (depend on land ue) We can improve acceibility by improving mobility and/or by changing the ditribution of opportunitie. But if the opportunitie are fixed in pace, then a change in mobility i equivalent to a change in acceibility. A hown in the previou figure, there i a trade-off inherent in public tranportation becaue uer give up flexibility (uffering a level of ervice penalty) for economy. To trike thi balance between level of ervice (LOS) and upply cot in network for individual mode (e.g. highway, bike-lane, and idewalk), planner can only change the infratructure. But in collective tranportation, planner alo have control over the vehicle route and chedule. -
Public Tranportation Sytem: Planning General Idea The goal of planning i to achieve efficiency, meaured a a combination of LOS and upply cot. Cot come in different form, uch a time, T, comfort, afety, and money, $, and hould be reduced to ome common unit. The reult i called a generalized cot or diutility, which can be defined both for individual and group, and i uually expreed a a linear combination of component cot; e.g. for one individual experiencing time T and cot $ it could be: How to Take into Account Politic Generalized Cot = β T T + β $ $ Note that β T and β $ will vary between individual, o even though an individual may have a welldefined generalized cot, the choice of appropriate weight to repreent a divere group i alway a political deciion that cannot be reolved objectively. Note too that tranit ytem involve cot to non-uer energy, pollution, noie, etc. and that ince people alo diagree about how thee hould be valued, they further complicate the deciionmaking picture. Clearly, we need to implify thing! (but without ignoring the effect of politic). To thi end, we will aume in thi coure that there i a political proce that ha converged to the etablihment of ome tandard, which would apply to all the non-monetary output of the tranit ytem; e.g., T Door-to-door time (no more than a tandard, T 0 ) E Energy conumed (no more than E 0 ) M Mobility (at leat M 0 ) A Acceibility (at leat A 0 ) And our goal will be minimizing the cot, $, of meeting the tandard; i.e., Mathematical Program (MP): min{ $: T T 0 ; E E 0 ; M M 0 ; A A 0 } Note how each tandard i aociated with an inequality contraining the value of the performance output in quetion. Since thee output are uually directly connected to 4 key meaure of aggregate motion: VHT, VKT, PHT, PKT, we can often reformulate the tandard in term of paenger time (ditance) and vehicle time (ditance). Alternatively, ince all variable in thi MP (both monetary and non-monetary), which we collectively call y = ($, T, E, M, A), are function of the ytem deign, x, (i.e., the route and chedule ued for the whole ytem) and the demand, α (which we aume to be given), we can expre the MP in term of x and α. -3
Public Tranportation Sytem: Planning General Idea To make thi formulation more concrete, let u define thee relation by mean of a vector-valued function F m : where, y = F m (x, α) y performance output for the entire ytem (both monetary and non-monetary) m mode x deign variable for the entire ytem α demand We then look for the value of x that minimize the $-component of y while the other component atify the tandard contraint. The reult i a a bet deign, x * (α), which if implemented would yield y*(α) = F m (x * (α),α) = G m (α). Thi function repreent the bet performance poible from mode m with given demand α. We will, in thi coure, compare the G m (α) for different mode. To ee all thi more concretely, conider a imple tranit ytem where all uer are concentrated at two point. In thi cae we have: x frequency of ervice (a ingle deign variable: bue/hr) α demand (a ingle demand variable: pax/hr) Define now the component of F m. We aume that each vehicle dipatch cot c f monetary unit. Thu we have: $ = F m $ (x,α) = c f x/α [$/pax] Note: we have defined $ a an average cot per paenger. We could intead have defined $ a the total ytem cot per hour. Both definition lead to the ame reult ince they differ by a contant factor: the demand, α. If we now aume that headway are contant but the chedule i not advertied, we have: T = F m T (x,α) = /x [hr] (out of vehicle delay aume ½ headway at origin and ½ headway at the detination) And finally, if each vehicle trip conume c e joule of energy we alo have: E = c e x/α [joule/pax] -4
Public Tranportation Sytem: Planning General Idea If the political proce had ignored energy and imply yielded a tandard T 0 for T, and if we chooe the monetary unit o c f =, the MP would then be: min{ x/α: /x T 0 }. Note that the OF i minimized by the mallet x poible. Thu, the contraint mut be binding, and we have: x * = /T 0 Therefore the optimum monetary cot per paenger would be: $ * G m $ (α) = /(αt 0 ) We call the above the tandard approach to finding efficient plan. There i another approach, which we call the Lagrangian approach. It involve chooing ome hadow price, β, and minimizing a generalized cot with thee price without any contraint. Although the election of price cannot be made objectively, one can alway find price that will meet a et of tandard (ee your CE 5 note). So the Lagrangian approach i equivalent to the tandard approach. For example, we can formulate: The olution i: min x { $+βt x/α + β(/x) } x * = αβ You can verify that the tandard olution (x * = /T 0 and $ * = x * /α = /(αt 0 ) i achieved for β = (/ T 0 )(/ α). So no matter what tandard you chooe, there i a price that achieve it. In ummary, there are approache to obtain low cot deign that atify policy aim:. Standard: min { $.t. T T 0, E E 0 } Thi minimize the dollar cot ubject to policy contraint, e.g. for trip time, energy conumption and poibly other output. Uually, a hown in the example, contraint become binding when olved T = T 0, E = E 0. Lagrangian: min { $(x,α) + β T (T(x,α)) + β E (E(x,α)) } -5
Public Tranportation Sytem: Planning General Idea Thi minimize the generalized cot, and give the ame olution a the tandard method when uitable hadow price, β T and β E, are choen. The hadow price can be found by olving the Lagrangian problem for ome price, finding the optimum T and E and then adjuting the price until T and E meet the tandard. In imple cae, uch a the above example, thi can be done analytically in cloed form. How to Account for Demand: Some Comment about Demand Uncertainty and Endogeneity So far, we have aumed that the demand, α, i given, and critic could ay that thi i not realitic. However, if we are lucky and the deign one provide happen to be optimum for the demand that materialize, then the iue i moot. Suppoe we deign x for a choen level of demand, α, that i expected to materialize at ome point in the future. Normally, we expect realized demand to change with time, and for a well-deigned ytem that provide improved ervice thi demand hould be increaing. Then, the quetion of whether the ytem deign i optimal in reality (given that we aumed a demand α 0 ) i le a quetion of if, but of when, ince the demand α 0 will eventually be realized. Furthermore, we will learn later in the coure that the cot aociated with a deign, x*, that i optimal for α 0 i alo near-optimal for a broad range of value of α (within a factor of of α 0 ). Thu, if the realized demand doe not change quickly with time, the ytem deign i likely to produce near optimal cot for a long period of time. Furthermore, we hould remember that demand i difficult to predict in the long run. So, building complicated model that endogenize α in order to predict precie value i not a worthwhile activity in my opinion. Rough etimate of future demand are ufficient for deign purpoe. Thi i not to ay that a viion for the future i not important; only that it doe not need to be anticipated preciely. The following example illutrate what happen if one ignore the viion. The Shortightedne Tragedy Thi example how that when demand change with time, then incrementally chaing optimality with hort-term gain objective in mind can lead u to a much wore tate than if we deign from the tart with foreight and long term objective. Now, conider the invetment deciion for a ytem with potential for mode: automobile diviible capacity with cot per unit capacity, c g ubway indiviible and very large capacity with cot for a very large capacity, c 0 Politician, who make deciion about how much money to invet in tranportation infratructure, tend to focu on hort-run return becaue of the relatively hort political cycle. If election for city leader occur every couple of year, then politician have incentive to look at cot only in the near future. Thi can be tragic. Suppoe that demand for tranportation in a city i growing over time and i expected to continue growing long into the future (thi tend to be the cae in nearly all citie around the developing -6
Public Tranportation Sytem: Planning General Idea world). Suppoe too that the goal i upplying (at all time) enough capacity to meet demand. The politician mut decide whether to invet a large amount of money, c 0, in digging tunnel and laying track for a ubway that will have enormou capacity to handle demand for decade into the future or to incrementally expand road infratructure to handle the demand α i expected over the next political cycle, i. Thi would cot c i = c g α i monetary unit and will be the deciion made if c i < c 0 (auming cot i the main political iue.) The reult of thi periodic review deciion making i hown by thi figure: $ periodic review baed on political cycle $ auto (t) $ ubway (t) c 0 now c i = c g α i t t If the deciion rule for inveting in infratructure i to choe the lowet cot over the next political cycle and demand increae gradually, automobile will alway win becaue with gradual increae in demand: c i < c 0. In the long run, however, the cot of invetment in automobile infratructure i unbounded. Had deciion been made with a view to the long run (t > t ), the ubway (i.e. the le cotly invetment) would have been choen. Another point pertaining to the future demand viion i that ytem often create their own demand; and thi hould be recognized (even exploited) when developing deign target. Planning action that have long-term conequence hould be made with a long-term horizon and long term viion. Planning and Deign Approache Comparative Analye Thi i planning by looking at what imilar citie have done and trying to copy it. Although thi i ueful, afe and often done, it can exclude opportunitie to come up with innovative olution that may only be appropriate for the cae of concern. (We will not do thi in thi coure; we will intead create deign from cratch, ytematically.) -7
Public Tranportation Sytem: Planning General Idea Step-wie Approach Thi i how ytematic planning mut be done -- problem are too big to be explored in one hot. We firt plan generally for the big picture; then fill in the deign/engineering tep. In order to conduct broad planning for the large cale, it i ueful to implify the analye. Deciion variable, uch a number of bue, number of top, and number of bu route are integer value in reality, but we will treat them a diviible (continuou) variable. Thi greatly implifie matter, for example turning integer programming problem into linear program, o that complex problem can be olved much more eaily. Thi will work if the implification doe not introduce large error. Deciion Method. Planning Large/Long cale Simplified/Broad. Deign Detailed/Specific Example Conider a imple mathematical (integer) program, e.g. for maximizing peronal mobility ubject to a budget contraint: max { z = x + 8y }.t..x +.9y x, y Z (integer valued) Thi i o imple that the olution can be obtained graphically (try it); the olution i: x * = 0, y * =, z * = 8. Now, if we tart with the planning approach and implify the problem by treating x and y a continuou variable. We are now olving a linear program which ha the (optimitic) olution: x * = 0.95, y * = 0, z * = 0.95, (The olution i optimitic becaue it i the optimum for a problem with fewer contraint.) To obtain a feaible olution the LP olution can be rounded up or down. Becaue of the contraint, we mut round down and we obtain: x* = 0, y* = 0, z* = 0. -8
Public Tranportation Sytem: Planning General Idea Thi olution will be peimitic ince it i feaible, but not necearily optimal. In fact, thi i much wore than the optimum olution! So, the implifying aumption of the tep-wie approach do not work o well for thi mall cale problem. Now, if we do the ame problem on a much larger cale (e.g. for a budget that would cover a whole city) we would olve intead the mathematical program, max { z = x + 8y }.t..x +.9y 00 x, y Z (integer valued) Starting with a planning tep, auming the variable can take non-integer value (linear program), the (optimitic) olution i x * = 95., y * = 0, z * = 095. Rounding to the nearet integer value (the deign tep) give a peimitic final objective function value: x* = 95, y* = 0, z* = 090 Now the peimitic value aociated with the integer olution we obtained with the tep-wie approach i very cloe to the optimitic value, and therefore hould be even cloer to the real optimum that could have been obtained. So, implifying the problem for large-cale planning purpoe, a we will do in thi coure, i not detrimental to the reult of the analyi. -9
Public Tranportation Sytem: Planning General Idea Appendix: Cla Syllabu (pring 00) The chedule below lit the topic covered in -hr lecture period in the pring emeter (00) and how they were coordinated with the homework exercie and the mini-project activitie. Not lited, a -hr weekly dicuion eion wa alo cheduled to cover the homework exercie and the mini-project. Period Date Lecture ubject Problem Mini-project /9 Introduction: general idea, politic / Introduction: tandard, demand uncertainty 3 /6 Planning: huttle ytem, fixed demand (EOQ) 4 /8 Planning: huttle ytem, adaptive demand 5 /5 Planning: modal comparion, idealized corridor (Vickrey) 6 /4 Planning: idealized corridor hierarchie 7 /9 Planning: corridor (detailed analyi, tandard) 8 / Planning: corridor (tandard v. generalized cot) 9 /6 Planning: inhomogeneou corridor 3 (pacing only CA) 0 /8 Planning: idealized grid ytem (iue) 3 /3 Planning: realitic grid ytem (no hierarchy) /5 Planning: grid ytem (practical iue) 3 3/ Planning: hybrid ytem (modal comparion) 4 (modal competition) 4 3/4 Planning: hierarchical ytem, adaptation 4 5 3/9 Planning: paratranit (general concept; taxi) 5 (hierarchy deign) 6 3/ Planning: paratranit (dial-a-ride) 5 7 3/6 Planning: paratranit (car-haring) 8 3/8 Management: vehicle fleet ( route) SPRING BREAK -0
Public Tranportation Sytem: Planning General Idea Period Date Lecture ubject Problem Mini-project 9 3/30 Management: vehicle fleet (n route) 6 (feeder DAR) 0 4/ Management: methodology (meta-heuritic) 6 4/6 Management: taffing ( run) 3 4/8 Management: taffing (n run) 3 3 4/3 Operation: vehicle movement (theory, ytem of ytem) 3 4 4/5 Operation: vehicle movement (pairing) 3 5 4/0 Operation: vehicle movement (pairing avoidance) 7 (bu pairing) 6 4/ Operation: right-of-way (iue, node) 7 7 4/7 Operation: right-of-way (link, ytem) 8 4/9 Operation: pecial event (capacity management) -
Public Tranportation Sytem: Planning Shuttle Sytem Module : Planning--Shuttle Sytem (Originally compiled by Eric Gonzale and Joh Pilachowki, February, 008) (Lat updated 9--00) Outline Overview Shuttle Sytem o Individual Tranportation Time-independent Demand Time-Dependent Evening (Queuing), Morning (Vickrey) o Collective Tranportation Time-Independent Time-Dependent o Comparion and Competition Overview Recall from Module that public tranportation can be thought of a a ytem that conolidate individual trip in time and pace to exploit economie of cale that reult from collective travel. Since thi coure i about developing inight a well a recipe, we will analyze imple ytem tarting with point-to-point huttle, then expand to tranit in corridor, and finally build up to the more realitic cae of organizing public tranportation in dimenion.. Shuttle Sytem Aume the population i already conolidated at two point (an origin and detination) o that there i no patial conolidation of trip. Collective tranportation, in thi cae, will involve temporal conolidation a individual adjut their departure time to match the cheduled departure of tranit vehicle from the hared origin to the hared detination.. Corridor Aume now that the population i pread along a corridor o that all travel i made in dimenion along which tranit ervice i provided. Here, collective tranportation mut involve patio-temporal conolidation a individual mut travel to dicrete tation where they can board tranit vehicle which depart at dicrete time. -
Public Tranportation Sytem: Planning Shuttle Sytem 3. Citie Finally we conider the more realitic cae of a population pread acro dimenion. Now tranit ervice mut be aligned in a route tructure to cover the -D pace, and thi routing add circuity to travel a tranit ytem carry individual out of the way of their hortet path in order to conolidate trip patially. Shuttle Sytem We tart by analyzing point-to-point huttle ytem. For comparion purpoe we will do thi for both, individual and collective tranportation mode. In both cae we look firt at the timeindependent cae where we aume teady tate condition (upply and demand are contant over time). Thi i the way many economic model treat tranportation. We then look at the (more intereting) time-dependent cae. Individual mode, like private automobile, incur ignificant guideway cot in proportion to the capacity provided, which cannot be eaily adapted to a timedependent demand. Public tranit mode without extenive guideway will be hown to be more flexible, becaue a ignificant part of their cot come from vehicle operation. Individual Tranportation Mode Time-Independent Demand In order for individual to travel in private vehicle (uch a automobile) without much delay, ome amount of capacity, μ (pax/hr), mut be provided to erve the demand, λ (pax/hr). For private mode, there i a roughly contant infratructure cot, c g, per unit of capacity provided. There i alo a cot per vehicle trip, c f, that each driver perceive a a fixed cot of making a trip by private car. Auming a an approximation that there i no delay whatoever when the capacity exceed demand (μ λ), the cot per paenger of a private vehicle ytem i cg μ $ = + c f, for μ λ. λ -
Public Tranportation Sytem: Planning Shuttle Sytem In order to minimize thi cot, we would alway chooe to provide the leat poible capacity, which mean μ = λ. Therefore the minimum cot per paenger i given by $ = cg + c f which i independent of demand, o there are no economie of cale in our idealization of private tranportation; i.e., the total cot accrue at rate λ$. Doubling the number of driver on the road would double the total cot of tranportation when jut enough capacity i provided to erve demand. We now look at the time-dependent cae, both for the evening and morning ruh hour, which are different. Time-Dependent Demand The Evening Commute with Known Demand (Queuing Analyi) Until now, we have aumed that demand i time-independent o that a long a capacity matche demand there i no delay, but in reality travel demand rie and fall over the coure of a day. Below i a cumulative plot of demand howing the difference between the daily average demand, λ, and the maximum demand in the peak of ruh hour, λ m. We aume that the demand curve i given and (for implicity only) that the day ha a ingle ruh intead of two. Note that λ m λ, and that in a time-independent ytem where the demand rate doe not fluctuate over the coure of the day, λ m would equal λ. # V(t) λ m μ λ D(t) Figure. T D = 4 hour t The minimum monetary cot of providing ervice ubject to a travel delay tandard, T 0, can take a range of value depending on the tandard and the capacity it require. Thi range can be -3
Public Tranportation Sytem: Planning Shuttle Sytem eaily identified. A lower bound for the cot i obtained by relaxing the tandard and imply auming, T <. Thi relaxed tandard i achieved by providing jut enough capacity to meet the average daily demand (μ = λ ) uch that there are no unerved vehicle carrying over from day to day. Thi yield a lower bound equal to the monetary cot of the time-independent cae: c g + c f. An upper bound for the cot i obtained by tightening the tandard to T 0 = 0. Thi tandard i achieved by providing ufficient capacity o that there i never congetion: μ = λ m. The upper bound i therefore a hown below: m c g + c f min{$ : T T0 } cg + -4 λ λ Note that thee bound apply whether we interpret T a the average delay experienced by driver, or a the maximum delay experienced in the wort cae. The choice of which tandard to ue i a political deciion. But thee bound how that a ruh hour can only make cot wore than in the time-dependent cae becaue the cot of erving uniform demand i the lower bound of thi expreion. So, we till do not ee economie of cale. Aide (howing how to calculate the actual value T* and $*): If deired, one can alo etimate T* and $* (not jut the bound) by uing a cumulative plot diagram and/or a preadheet. For example, if T and T 0 are average acro driver, we would evaluate the total time delay, T T (μ), for a given capacity, μ, a the area between the arrival curve decribed by V(t) and the departure curve, D(t), determined by the capacity, μ. The average time delay per driver, T(μ), i thu given by TT ( μ) T ( μ) =. λ Note from the picture that the area between V(t) and D(t), and therefore T(μ) decline with μ; and ince the monetary cot of private tranportation alway increae with capacity, $(μ) c g μ/ λ, the contraint of our mathematical program mut be binding. Thu, which yield μ* (and $*). T ( μ *) = T Time-Dependent Demand The Morning Commute (Vickrey Model with Endogenou Demand) In our idealization of the morning commute the time at which people leave their home and would arrive at our mythical bottleneck are not given. Intead, the demand i driven by work appointment characterized by a cumulative curve of deired departure time through the bottleneck, which we call the wih curve, W(t). If the lope of the wih curve,, i le than the capacity of the bottleneck, μ, all driver can pa through the bottleneck exactly when they would 0 c f
Public Tranportation Sytem: Planning Shuttle Sytem like; then there would be no delay. Curve V(t), D(t) and W(t) would match. However, if the exceed capacity, ome driver would have to depart the bottleneck earlier or later than their wihed time and the three curve could not match. To ee what could happen a driver adjut their home departure time (over day) in repone to their delay, we uppoe that each driver value time in queue at a rate β ($/hr), time arriving early at rate eβ and time late at a rate Lβ. The contant e and L are dimenionle and uch that: e L According to Vickrey (969), if exceed μ and driver minimize their generalized cot including delay, earline, and latene, an equilibrium curve of arrival time to the bottleneck arie in which the order of arrival to the bottleneck i the ame a the order of wihed departure. The equilibrium principle i that no driver hould be able to decreae it generalized cot by changing their arrival time. In Vickrey equilibrium, hown in Fig., there i a critical driver, numbered N c in the equence of arrival and departure, who experience no earline or latene and whoe entire cot i time in queue. (Note how the departure curve D(t) croe W(t) for the ordinate of thi driver.) All driver who arrive before N c will depart the bottleneck before their wihed departure time. We will define N e a the count of uch driver. All driver who arrive after N c will depart the bottleneck after their deired departure time. We will define N L a the count of uch driver. If there are a total of N R driver then the following i true: N e + N L = N You can convince yourelve that the queuing diagram for the equilibrium i uniquely defined if you are given T, N e and N L. It can be hown (ee Appendix) that: R NRLe T = ; μ ( L + e) N e LN R = ; and L + e N L en R =. L + e It alo turn out that if >> μ, the generalized level of ervice cot (including both queuing delay and unpunctuality cot) i nearly the ame for all commuter, approximately βt. When L >> e, thi generalized cot i βn R /μ. -5
Public Tranportation Sytem: Planning Shuttle Sytem # N L /μ N L N R T N c V(t) D(t) N e μ W(t) N e /μ T D = 4 hour t Figure. The total cot of congetion in thi morning commute i the um of total queuing delay (the area between V(t) and D(t)), the total earline penalty (e time the area between D(t) and W(t) where D(t) > W(t)), and the total latene penalty (L time the area between W(t) and D(t) where D(t) < W(t)). Thi calculation can be mot eaily done baed on the geometry of the figure. A little reflection how that if we chooe a bottleneck capacity that minimize the out-of-pocket cot per peron $ required to cover the cot of aid capacity ubject to a time tandard (ay for the critical commuter), we obtain the ame bound a in the evening ruh: where λ = N / T. R D c g + c f min{$ : T T0 } cg + c f, λ So, in the morning ruh we continue to be wore-off than in the time-independent cae; and economie of cale till do not appear. Thi i true becaue the practical range of μ i [λ, ] and$ c + cg μ / λ =. f -6
Public Tranportation Sytem: Planning Shuttle Sytem Collective Tranportation We now repeat thi analyi for public tranit and find that the reult are quite different (and encouraging). Time-independent Demand Conider now a huttle ervice provided on an exiting guideway from a common origin to a common detination, where the frequency of ervice i the deciion variable that the tranit agency can determine. We aume that huttle vehicle (e.g., train) are large enough to carry any number of paenger that may how up and define: H headway between vehicle dipatche [hour] x frequency of vehicle dipatch [number of vehicle per hour] = H c f cot per vehicle dipatch of providing huttle ervice [dollar per vehicle] λ demand [number of paenger per hour] So, the monetary cot per paenger, $, of providing huttle ervice i given by the cot per hour of dipatching the tranit vehicle divided by the total number of paenger uing the ytem. c f x $ = λ The out-of-vehicle delay experienced by paenger in the ytem (ignoring the time in motion between the origin and detination, which i the ame for every traveler) i alway proportional to the headway of ervice. For example, if people know the headway but not the chedule and they have pecific appointment at the detination (a in the morning commute), they will leave home with at leat one headway of lack, which they will pend either at the origin or at the detination. Combined, their total delay would be H. If people do not have pecific appointment (a happen for many people in the evening commute) their delay would be ½H on average. Thu, for the wort-cae ituation (with appointment) the average delay T i: T = x So if we apply a tandard T 0 (a we did for individual mode) we have to olve: -7
Public Tranportation Sytem: Planning Shuttle Sytem min $ and ince the contraint i binding, we find: c f x : T0 λ x $* = c f λt Note: There are economie of cale in providing collective tranportation becaue the monetary cot, $*, decreae with the demand! Thi i the promie of public tranportation vi a vi individual tranportation. In reality the contrat i not o pronounced becaue a we hall ee there exit compenating complication, but the promie i real. The reaon i that with more demand more individual can conolidate their travel onto each vehicle without changing the number of vehicle run; and thi lower the cot of providing tranportation per peron. We now how that economie till arie if we allow the demand to vary with time. Time-Dependent Demand The analyi above aume that the demand i uniformly pread throughout the coure of the day, but in reality the demand for travel i concentrated into ruh hour. Let u now evaluate the cot of providing collective tranportation for thi cae, auming that the paenger arrival are given. Conider now a implified cae of a day with two demand period: a peak demand, λ p, for a period of T p hour of the day, and an off-peak demand, λ o, for the remaining T D T p hour. The cumulative plot of Fig. 3 how thi demand profile and that N p paenger travel in the peak, leaving N D N p paenger for the off-peak hour. 0 Thi aumption can now be ued for both the evening and morning commute (with and without appointment) becaue with our large-vehicle, paenger do not have to compete for limited ytem capacity. -8
Public Tranportation Sytem: Planning Shuttle Sytem # N D λ o λ p N p T p Figure 3. T D = 4 hour t To deign a tranit ytem for thi demand, we can break up the day into two regime and chooe a peak period headway, H p, and an off-peak headway, H o, to minimize the cot in providing tranit ervice over the coure of the whole day. Thi can be done by minimizing the total generalized cot by the Lagrangian approach with the two deciion variable, H p and H o : min { Z = β ( Total amount of waiting time) + c ( Number of bu dipatche)} f min Z = β H p N p + H o ( N D N p ) + c f T H p p TD T + H o p The headway that minimize the generalized cot are H p * = c f T βn D p p = p c f βλ c f ( TD Tp ) c f H o* = =. β ( N N ) βλ p o Uing thee optimal headway give a minimum total generalized cot of ( T N + ( T T )( N N ) ) Z* = β c f p p D p D p. -9
Public Tranportation Sytem: Planning Shuttle Sytem Note that for a given ratio N p /N D thi total generalized cot i proportional to generalized cot of collective tranportation per peron i proportional to / N D N D, o the ; i.e., it decreae with increaing riderhip, N D, and therefore with the average daily demand λ = N D /T D. So even with time-dependent demand, public tranit diplay economie of cale. Technical aide: Note that the optimum cot doe not change much if the demand i pread evenly acro the whole day. Suppoe, for example, that the coefficient β = and 30% of the trip are made in 4 of the 4 hour in a day (i.e., there i quite a bit of peaking). If we ue a dummy value N D = 0 in the formula, we find that the total generalized cot for thi time-dependent cae i ( 4 3 + (4 4)(0 3) ) 5. 30 =. Uing the ame logic we ee that if the N D = 0 trip had been pread uniformly acro the entire 4 hr, the generalized cot would have been: (4 0) ½ = 5.49. Note the very mall difference, and that peaking actually reduce the cot to ociety, which wa not the cae for individual mode! You can alo convince yourelf that the relative difference between thee two cot i independent of N D. The relative difference i o mall becaue we can adapt the proviion of tranit ervice to match demand. The mall and favorable relative error ugget that to plan collective tranportation ytem with dominant vehicle cot (a in our example) one can aume a time-independent demand a a implification. Infratructure cot, on the other hand, mut be provided in a time-invariant (non-adaptable) way, o the ame cannot be aid when guideway cot are important, a happen for tranportation by individual mode and ome collective kind (e.g., ubway). Comparion between Individual and Collective Tranportation Mode In many cae, individual mode are ued in parallel with public tranit line, and an equilibrium i reached in which ome trip are made by individual mode and the ret by tranit. If a traveler deciion of which mode to take i baed only on the level of ervice (LOS) cot (i.e. the delay time), the equilibrium will be reached when the level of ervice cot are the ame for both choice. We have een from Vickrey model that the generalized cot of delay for automobile commuter i approximately βn R /μ, when L >> e and >> μ. Note that thi cot increae proportionally with the number of individual uing the roadway, N R, and decreae a capacity, μ, i expanded. For collective tranportation, by contrat, the level of ervice cot i alway proportional to the ervice headway, H, and i independent of the number of individual uing the tranit ytem. It i βh if everyone ha appointment. Auming the vehicle are ufficiently large, thi make c f -0
Public Tranportation Sytem: Planning Shuttle Sytem ene becaue the time cot of riding a tranit huttle depend only on how long a rider mut wait for the vehicle, not on how many other people are haring the vehicle. So the following diagram plotting general cot v. number of uer help explain what happen when the two mode provide competing huttle ervice for a population of N R traveler and we have to decide where to allocate fund for increaed capacity. The increaing line correpond to automobile and the horizontal line to public tranit. Generalized Cot Initial Equilibrium c f (car) + βn/μ, low μ c f (car) + βn/μ, medium μ (initial value) c f (car) + βn/μ, high μ Improvement in generalized cot c f (tranit) + βh, high H c f (tranit) + βh, medium H (initial value) c f (tranit) + βh, low H N Car N Car N Tranit Figure 4. N R Aume now that the automobile and public tranit ytem are initially decribed by the two curve labeled medium in the figure. If people chooe huttle ervice baed on generalized cot, then the interection of thee two curve i the initial equilibrium. The total generalized cot i then the um of the total cot for all mode (which i the ame for all trip, regardle of mode), depicted by the haded area: N R (c f (tranit) + βh). Now, uppoe ome public fund become available and we can chooe whether to invet in public tranit or individual mode. We can chooe to improve the headway for tranit ervice, H, (option in the figure) or the roadway capacity, μ, (option ); o where hould we pend the money? An invetment in automobile infratructure lower the cot of driving which will caue a hift in mode hare to more driver (point ). The uer cot (haded area), however, remain unchanged becaue driver fill the new road capacity until the time delay i equivalent to the time cot of taking tranit. -
Public Tranportation Sytem: Planning Shuttle Sytem Inveting in public tranit, however, lower the uer cot for tranit rider by reducing the headway, and thi create a mode hare hift toward tranit (point ). In thi cae the improvement benefit both tranit rider and driver (by taking driver off the road). Therefore, in thi idealized example everyone benefit from inveting more fund in collective tranportation, even thoe people who never et foot on a tranit vehicle. Related Reading Vickrey, W.S. (969). Congetion theory and tranportation invetment. The American Economic Review, 59() 5 60. Appendix A: Vickrey Model of the Morning Commute We look for an equilibrium where the critical driver i indifferent to any arrival time, and the firt and lat driver to the bottleneck experience no delay. Thu, given a fixed lope, μ, of D(t), we can find thi equilibrium (ee Figure ) by etting the delay experienced by the critical driver, T, equal to the earline cot experienced by arriving firt or the latene cot experienced by arriving lat: T N ee = and μ T N L L =. μ With thee two equalitie and the relation N e + N L = N R we can olve for T, N e + N L, with the reult of the text: T = N R μ = + L e N R Le μ( L + e) ; N e LN R = and L + e N L = en R L + e So thi how that the critical driver would not have an incentive to change it arrival poition. But for the curve of Figure to be in equilibrium, other driver whether their wihed time are before or after the critical time would alo have to lack an incentive to change their arrival poition. A good way to verify thi i in two tep: (a) Draw an indifference curve for a generic non-critical driver (with a given wih time) howing for each poible arrival poition from 0 to N R the time at which the driver would have to join the virtual queue when arriving in thi poition to achieve the generalized cot currently experienced. (Note that each arrival poition ha a given earline or latene for thi driver.) -
Public Tranportation Sytem: Planning Shuttle Sytem (b) Noting that the latet time at which the queue can be joined for any poition i given by V(t); and that V(t) i never to the right of the indifference curve; i.e., the indifference time are not feaible and the driver cannot improve hi or her poition. Step (a) require ome care. The following reference can perhap help. They are not required reading, but they contain more detail and additional application. Related Reading Daganzo, C.F. (985). The uniquene of a time-dependent equilibrium ditribution of arrival at a ingle bottleneck. Tranportation Science. 9() 9 37. Daganzo C.F. and Garcia, R.C. (000). A Pareto improving trategy for the time-dependent morning commute problem. Tranportation Science. 34(3) 9. -3
Public Tranportation Sytem: Planning Corridor Module 3: Planning Corridor (Originally compiled by Eric Gonzale and Joh Pilachowki, February, 008) (Lat updated 9--00) Outline Idealized Analyi o Limit to The Door-to-Door Speed of Tranit o The Effect of Acce Speed: Uefulne of Hierarchie Realitic Analyi (patio-temporal) o Aumption and Qualitative Iue o Quantitative formulation o Graphical Interpretation o Dealing with Multiple Standard o No tranfer o Tranfer and Hierarchie o Inight o Standard-Reviited o Space- and Time-Dependent Service Average Rate Analyi Service Guarantee Analyi In the previou module we looked at the pecial cae where all trip originate at one point and end at another point. Now, we conider demand pread along a corridor, o trip mut be conolidated both in time and in pace. The deign of tranit ervice in a corridor require chooing a top pacing, S, and ervice headway, H. We will firt focu excluively on S in order to iolate the effect of patially ditributed demand from that of it temporal ditribution, which we aw in Module. Wherea temporal conolidation involved a trade-off between out-of-vehicle (waiting) time and vehicle operating cot, which had huge economie of cale a demand increaed, we will now ee that in the patial cae the trade-off i between out-of-vehicle (acce) time and in-vehicle time, and that thi tradeoff i le favorable to public tranit: it impoe a evere limit on door-to-door peed even if we make the mot favorable aumption poible for collective tranportation. 3-
Public Tranportation Sytem: Planning Corridor Idealized Analyi Limit to Door-to-Door Speed Conider a very long tranit corridor erving cutomer that travel from left to right. Cutomer origin are continuouly ditributed anywhere along the corridor and their trip can take any length up to a maximum l. The top are eparated by ditance, l. We are intereted in the tightet door-to-door travel time guarantee that can be extended to all cutomer. l Now we will make a number of optimitic (although unrealitic) aumption in order to identify thi guarantee while accounting for the fact that paenger mut acce the tranit top and then ride vehicle which make periodic top to pick up and drop of paenger. Thi bound will be independent of demand and many other parameter, o it i very general. Aume vehicle are dipatched o frequently that once a paenger arrive at a top, he or he doe not wait at all for the next vehicle; i.e., H = 0. Aume the door of the vehicle open and cloe intantly, and paenger take no time to get in or out of the vehicle. Finally aume that there i no upper bound to the peed that can be achieved by a tranit vehicle while traveling between top, o that v max =. Although we would agree that thee condition would favor operation extremely, the tranit ytem will till be limited by: A maximum acceleration above which paenger will feel phyical dicomfort from the force (a 0 m/ ). The average walking peed at which paenger travel to acce their nearet tranit top (v a m/). There are two component of travel time in thi cae: acce time, t a, and riding time, t r. In the wort cae, the acce time reult from a paenger walking half of a top pacing from the origin and another half top pacing to the detination. So: t a = v a 3-
Public Tranportation Sytem: Planning Corridor Riding time i the conequence of the commercial peed of tranit (the average peed of the vehicle v v ) which i affected by the top pacing. If there i no maximum peed, then the tranit vehicle will accelerate a it depart a top until it i half way between top. Then the vehicle will decelerate to make the next top (ee figure below). Under thee condition, the riding time t for a trip between top can be decompoed into two equal part of length: / = ½a 0 (t /). From thi we find: t =, a and the riding time t r for a trip of length l >> will be approximately l/ time longer; i.e.: 0 t r l. a 0 Note that the commercial peed i therefore: l t r a 0. x x(t) v v = l/ t r Figure 5. t We aume that people walk to the nearet tation. Then, you can verify that for any pacing you chooe, there alway i an unlucky paenger who would have to walk a ditance and then 3-3
Public Tranportation Sytem: Planning Corridor ride for a ditance l/. A a reult, the total door-to-door time for thi wort-cae paenger i: t = t a + t r = /v a + l/ /(a 0 )½. Thi function increae with except and decline only when i a ub-multiple of l. At thee point it take on the form: t = v a + l a 0. So we look for the minimum of thi expreion, and a (a very good) approximation we ignore the fact that hould be a ub-multiple of l. There i a trade-off here for chooing the top pacing. On the one hand, a longer top pacing increae the ditance paenger mut walk to acce the mode, o the acce time increae with. However, a greater pace between top allow vehicle to accelerate to higher peed o that riding time decreae with. Therefore, an optimal top pacing, *, can be choen to minimize the door-to-door travel time. The reult of thi optimization i: 3 v * = l a ; a0 t *( a v = 0, a, l ) 3 l vaa 0 3 Of coure, thi reult i valid only if * l, a we aumed; i.e., only if l v a /a 0. Fortunately, ince realitic value of v a /a 0 are comparable with m, thi requirement i comfortably atified for the trip length that interet u. Since the unluckiet paenger ha a trip length cloe to l we can approximate the peed of thi paenger by: l vˆ = ( lvaa0 )3, t * 3 Thi expreion can alo be interpreted a the door-to-door peed that can be guaranteed to all paenger with trip of length cloe to l. Let u plug in ome number to ee how thi upper bound of door-to-door peed change with the length of trip made. If paenger walk with peed v a = m/ and the maximum allowable To ee thi, draw a picture with an unlucky trip a follow: (i) an origin diplaced by an infiniteimal amount ε toward the left of a mid-point between tation, and (ii) a trip length, y = l if = l; or ele, y = l/ +ε if < l. (Thi i an admiible choice, ince for ufficiently mall ε the trip length i valid: y < l.) Now note that in both cae the trip length i a multiple of, o both the origin and the detination are near a mid-point and acce ditance i. Note too that both cae involve evere backtracking with total in-vehicle ditance l/ l. You can alo convince yourelve that l/ i alo an upper bound to the in-vehicle ditance traveled by any paenger; and that therefore, our unlucky paenger i actually the unluckiet. 3-4
Public Tranportation Sytem: Planning Corridor acceleration i a 0 = m/, the figure below how the fatet door-to-door peed that can be guaranteed. vˆ.8 m/ 7.5 mph 6.7 m/ 5 mph 4. m/ 7.5 mph km 8 km 50 km ~ mi ~ 5 mi ~ 30 mi l Thi reult i very low, even with all the favorable aumption we have made for tranit (including v max = ). Why? We are minimizing total travel time including the acce time (i.e. maximizing door-to-door travel peed) which relie on paenger walking to the top. Since people walk very lowly, the top mut be paced cloely enough to limit the time paenger pend acceing tranit. Thi pacing, along with the limit of acceleration, prevent the vehicle from achieving high peed. With individual tranport mode the reult are better. I there a way of improving collective tranportation o it can be more competitive? The anwer, a we hall ee next day, i ye. (Hint: the door-to-door peed of public tranit depend on the acce peed; and if we could increae thi peed by ome mean, the door-to-door peed would increae.) We will explore thi iue next, and how to exploit it. We will alo tudy how to plan real corridor ytem without the implifying aumption we have made fully recognizing patiotemporal effect. The Effect of Acce Speed: Uefulne of Hierarchie For the moment we continue with our idealized and favorable cenario for public tranit ervice. So far, our goal ha been to undertand how tranit door-to-door ervice peed depend on l. We If we made imilar favorable aumption for individual tranportation mode on uncongeted guideway, their commercial peed would be cloe to the mode maximum peed for all l; i.e., much better than for public tranit. The reaon i that by being individual thee mode do not require much of an acce diplacement: a great virtue. 3-5
Public Tranportation Sytem: Planning Corridor made a couple of aumption, hown below, in order to obtain an optimitic but very imple upper bound of door-to-door time. The demand, λ, doe not matter for thi bound. H 0 Recall that the door-to-door travel time for the unluckiet paenger wa hown to be: t v = 0 max = l t = +. v a a 0 By minimizing thi expreion with repect to we obtained the following approximate formulae for the door-to-door travel time and peed of the unluckiet paenger with trip length l: 3 l t( l ) = 3 and vˆ = ( lvaa0 )3. vaa0 3 Note how if we could increae the peed of acce the ituation would improve. We can do thi by uing another tranit ervice to provide acce! l 0 Let reexamine our logic auming thi i done. By providing a local tranit ervice with top pacing, 0, to acce an expre ervice with top pacing,, the acce peed would now be: v a ˆ = 3 = v vwa0 3 where v w i the peed of walking. The derivation of thi would actually be lightly different o we do not double-count acce time, o for implicity we will aume ome mall tranfer time 3-6
Public Tranportation Sytem: Planning Corridor 0 equal to. Thi will allow u to continue uing the ame equation. The improved door-todoor travel time i v w then: You can verify that: l 3 3 tl ( ) = + 3 ( vwa0 ) a0 < * * * 0 <. 3 t l ( ) will be the bet travel time for a fixed, auming that you have optimized 0 already. Note: you can notice that thi equation i in the form: n m z = Ax + Bx ; with n, m > 0 which we will be analyzing in more detail in Homework #. You will find that the optimum olution x * = Bm An m+ n i inenitive to different value of A and B. After we optimize t l with repect to we find the reult to be: t l 7 4 l 3 7 7 5.3 = 5.3 3 a0 vw l a0vw 4 7 4 7 Thi equation how that t l i of order l and l 3 vˆ 7 7 i of order l and of order v. By plotting vˆ with repect to l with and without a hierarchy we can ee for which trip length it i optimum to provide a local ervice. t l w 3-7
Public Tranportation Sytem: Planning Corridor vˆ hierarchy no hierarchy l * below thi you do not need any hierarchy l Realitic Analyi with Spatio-Temporal Effect We have o far made a number of favorable and unrealitic aumption about our tranit ytem in order to derive generic inight about the effect of the patial diperion of paenger along a corridor. So with thee inight in mind we now turn our attention to the development of pecific plan introducing more realim. The analyi will include both, the patial and temporal effect of dipered demand; combining the idea we have o far een with thoe of Module. We hall ee that in addition to l, two other important variable affect a corridor ytem tructure: the trip generation rate, λ, and the uer value of time β. Aumption and Qualitative Iue Here are the improvement to realim we now conider: ) Remove the aumption that v max = ; for example define v max = v auto (for bue) 3-8
Public Tranportation Sytem: Planning Corridor x x(t) where v max = x( t ) where v max = v auto v max t t a / /v t a / max ) Remove the aumption that t = 0. If we approximate the trajectory of the bu with piecewie linear egment of v max and top time then we can define t a the dwell time at a top plu the lo time due to acceleration and deceleration. The total travel time will then be: dit t = + (# top) v max t 3) Remove the aumption that H = 0 Before tarting quantitative analyi, let u compare the patio-temporal acceibility provided by different mode with a plot howing the area that a peron can reach in a given time depending on their mode of tranportation. 3-9
Public Tranportation Sytem: Planning Corridor x auto tranit v auto v tranit v walk pedetrian t H We can look at the area covered by a ingle top pacing and headway. Notice how a peron, depending on their origin in pace and time, will chooe a bu top baed on their acceibility: x v walk v tranit t H 3-0
Public Tranportation Sytem: Planning Corridor 3- Quantitative Formulation Let try to deign a realitic corridor without any hierarchy. We propoe chooing the H * and * that minimize the cot of ervice given ome door-to-door travel time tandard. For example: min {cot of ervice}.t. t(l) T 0 We aume for now that we focu on a ingle l ; e.g. the longet trip people make. To do thi, we need formulae for the cot of ervice and the contraint in term of our deciion variable: Cot of ervice = H c H c d λ λ + H v t v t a + + + = l l l max ) ( Note: λ i the average demand denity in the corridor (trip/time dit) and λh i the number of cutomer aociated with one top and one vehicle. The contant c and c d are unit cot for a bu top and a bu-mile. How would you derive thee? To olve the problem we can write the Lagrangian a below. Can you aociate the four term with pecific paenger activitie? max $ $ v v t H c H H c z LH AD IVD WD a d top moving β β λ β λ l l + + + + + = + + + + + which (ignoring the c term) ha the olution: ( ) * * ; l a d t v c H λβ giving u: ( ) + = upper bound lower bound t v c c c a d d * 0 $ l λ β λ β
Public Tranportation Sytem: Planning Corridor T * cd = λβ tl + v a l + v max Note: the UB olution i obtained by ticking H * and * into the neglected term and adding the reult to $ *. Graphical interpretation: Thi picture how how the olution depend on λ, l, and β. Where WD repreent waiting delay, AIVD repreent acce and in-vehicle-delay, and LH repreent line haul time. Note: β i a proxy for the wealth of a city and the diagram illutrate the kind of ytem that citie of wealth might ue to atify a demand characterized by λ and l. 3-
Public Tranportation Sytem: Planning Corridor * H * H * * β Dealing with Multiple Standard A more realitic ituation would require adherence to level of ervice for more than a ingle trip length. Let examine the ituation where we our contraint i: t l) T ( l); l ( 0 T 0 ) T 0( l given l 3-3
Public Tranportation Sytem: Planning Corridor We end up with a minimization problem that look like: min{ agency cot(, ) H, H (, H, l) T0 ( l); } ().t. T l () Note: There will alway be at leat one binding contraint when the problem i minimized. We will call thi (unknown) binding trip length l c. If we knew it and we knew thi length provided the only binding contraint (a reaonable aumption), we could formulate the problem a a ingle-contraint problem and olve it: min, H { $(, H )}.t. T ( l ) T (, H, l ) 0 0 c = c = Thi would be an eay tak becaue it can be done with the Lagrangian method we have jut een. Note that the remaining contraint would be atified a trict inequalitie. If we don t know the critical length, thi property of the optimal olution of the ingle-contraint problem can be ued to ee if a tet value for l i the correct one. So to olve the problem we can olve the ingle-contraint Lagrangian problem for different l until we find one that exhibit thi property. No Tranfer For our pecific corridor formulae and auming no tranfer, thi procedure can be implified even more and the reult i intuitive. Thi i now explained. Given our aumption, the mathematical program correponding to () and () i: l c min $ = d, H λ H l l.t. H + + t + T0 () l, l v v a max 3-4
Public Tranportation Sytem: Planning Corridor Notice that the cot function omit the component related to making a top ( c λ H ) becaue thi value i mall, and the objective function here give a lower bound. Notice too that the contraint eparate into a part that depend only on the choice of headway, H; we call thi the waiting delay (WD). The ret of the contraint depend only on the choice of top pacing, ; thi will be called the acce and in-vehicle time (AIVT T(l )). When we plot the expreion T(l ) for a fixed the reult i a traight line: the vertical intercept i the fixed maximum acce time (/v a ), and the lope the vehicle average pace (/v max + t /). The minimum vertical ditance between the travel time tandard, T 0 (l), and an AIVT line for a given, T(l ), repreent the fixed amount of waiting delay that can be added to every trip and till keep the travel time with the contraint. Note that thi minimum vertical ditance i the maximum vertical diplacement of our AIVT line until it become tangent from below to the T 0 (l) curve. Thi vertical diplacement i the maximum headway, H, that can be choen for a given and till meet the tandard, thu minimizing the cot of providing tranit ervice. Now, the AIVT line can be changed by our choice of, o let chooe the that give u the maximum diplacement o we can chooe the greatet poible H and therefore achieve the lowet poible operating cot. Thi i the ought reult. t T 0 ( l) convex hull T ( l 3) target LOS T( l ) H * ( ) T ( l ) LE of T ( l ) = mint ( l ) = T ( l) L l* l 3-5
Public Tranportation Sytem: Planning Corridor Thi optimization can be done in one hot by conidering the lower envelope (LE) of travel time acro all choice of. Lower Envelope of T ( l ) = min{ T ( l ) } = T ( l) To thi end, note that when an AIVT line i diplaced it cannot poibly touch T 0 (l) in an upward bulge; o we only need to look for point of tangency on the convex hull (CH) of T 0 (l). 3 So, we propoe the following: lide T L (l) up until it touche (and i tangent to) the convex hull of the time tandard T 0 (l). 4 Then, the diplacement i the optimum headway H*, and the tangent to the envelope at the point of contact (l = l*) i the optimum AIVT line (with = *). 5 Applying thi reult, L T L l l = + vmax * = l *t v a () lt v a To ummarize, we have plit the optimization into two part: (i) a patial tep to find a top pacing,, that minimize the acce and in-vehicle time and (ii) a temporal tep to find the headway, H, to minimize the cot of meeting the ervice contraint. Thi i approximate and work neatly becaue we left out the cot of the topping. So the analyi above give u a lower bound of cot. If the topping cot were left in the analyi, the mathematical program can till be olved with brute force in a preadheet, but thi give u very little inight. If we olve the implified formulation and then plug the reulting T L (l) and * into the cot function, we will get an upper bound for the cot. No further analyi i neceary when the lower bound and upper bound are cloe. What if bue run in both direction along a corridor? 3 The CH i the highet convex curve that can be drawn without exceeding T 0 (l).) 4 Note that thi point of tangency doe not have to be on T 0 (l), a occur on the figure. 5 Why i thi true? (i) You ee from the geometry of the picture that the diplacement of the optimum AIVT line (which i traight) to firt contact with T 0 (l), i.e. the optimum headway H(*) for the = *, i alway equal to the diplacement of the LE to firt contact with the CH; thu, the diplacement we propoe i the optimum headway for *. And (ii) * i the optimum pacing becaue no other AIVT line can be diplaced by a greater amount. 3-6
Public Tranportation Sytem: Planning Corridor l The top pacing will remain unchanged, becaue i choen only to minimize travel time, and the demand play no role in the travel time expreion. The cot of operating ervice will double, however, becaue twice a many bue are needed to erve the ame demand per unit length. Exercie: Conider tranit ervice in a loop demand uniformly ditributed between all point. Would we want to erve trip with bi-directional tranit route or i it better to reduce headway by putting all vehicle in ervice in the ame direction? You hould be able to convince yourelf that if the route ha 4 bue or more, it i alway better to operate bidirectional ervice. (Hint: If you had only one bu, it hould be obviou that it i mot time efficient to operate ervice in one direction. Likewie, if you had an infinite number of bue, it hould be obviou that bue hould be deployed in both direction to erve the demand. Where i the tipping point where it become more efficient to operate bue in the both direction?) Tranfer and Hierarchie Now, what if we introduce tranfer to an expre ervice operating in parallel to the local ervice with frequent top. There are couple way thi ervice could be tructured. So far, we have been looking at tranlationally ymmetric route pattern, but thi need not be the cae. We could run offet local-expre ervice a hown below. 3-7
Public Tranportation Sytem: Planning Corridor A B local expre The diadvantage of uch a network deign outweigh the benefit for cae where the demand i pread out becaue for trip between point uch a A and B we would require multiple tranfer. But if all the trip have a common detination (e.g., for feeder ytem that collect paenger from many detination and deliver them to a ingle hub) the trategy ha merit. For pread-out (many-to-many) ervice it make ene to conider a local bu ervice that i paralleled by an expre ervice where paenger can tranfer from one ervice to the other at deignated tranfer top. l 0 Aume that the headway are ynchronized with the ame H for local and expre ervice, but the local bue top with pacing, 0, and the expre bue make le frequent top with pacing. Even thi tructure of ervice can be operated in different way. Strategy : Expre bue are cheduled at conitent headway, and the local feeder are dipatched in to depart in both direction along the corridor every time an expre bu reache a tranfer tation. At ome point between tranfer tation, the local bue wait and then begin a return trip, bringing paenger to the tranfer tation jut in time for the arrival of the next expre bu. 3-8
Public Tranportation Sytem: Planning Corridor x e e ditribute collect Δ dead time 0 collect ditribute t H Strategy : Expre bue are again dipatched at a cheduled headway. Intead of running feeder bue in both direction, a bu i dipatched from the tranfer tation after the arrival of an expre bu, and a econd feeder i dipatched in the ame direction to collect paenger and drop them off at the downtream tranfer tation in time to catch the next arriving expre bu. 6 6 If ervice i not ynchronized there i no need for dead-time and bue can both collect and deliver paenger. The two bu ytem can even have different headway, H 0 and H. Could you draw a picture uch a thoe above? 3-9
Public Tranportation Sytem: Planning Corridor x e e dead time Δ collect 0 ditribute t H Both of thee operational trategie teellate acro time and pace and require two local bu dipatche for each expre bu dipatch. Therefore they require the ame number of vehicle kilometer of ervice, and a lower bound to the cot of providing ervice baed on vehicle-km i 3c $ = d λh for both timed-tranfer trategie. (Convince yourelve that the coefficient would be for unynchronized ervice with H 0 = H = H). To be complete we mut account for bu-hr while topping. Then, the cot in a ytem with timed tranfer i $ λh ( ) + +,, H 3c t c + ( dead time) 0 = d t ct 0 The unynchronized cae with H 0 = H = H would have a very imilar form except for ome of the coefficient: 3 would be, the next would be and the final would be 0. Tet yourelve and ee if you can derive the unynchronized expreion for H 0 H. The door-to-door travel time T i compoed of the following component: H = waiting delay 3-0
Public Tranportation Sytem: Planning Corridor 0 = acce time v w v 0 = average peed of local vehicle including top but not dead time = local in-vehicle travel time v0 v = average peed of expre vehicle including top but not dead time (v > v 0 ) l = expre in-vehicle travel time v = tranfer time where the vehicle pace v i = v max t + i So the door-to-door travel time i given by 7 T t t l ; l > > 0 0 ( ) = + Δ + + + + + 0,, H H vw 0 vmax vmax and we can optimize the ytem with a mathematical program of the familiar form: (,, H ) min $ 0,, H 0,, H T0 0 =.t. ( ) ( ) l T l, The lower bound of the cot i now 3c d /λh, and the door-to-door time, v T ( 0,, H ) = H + T ( l ). The maximum poible H can be determined by the ame method decribed for a ytem with only local ervice, although here we determine a lower envelope of travel time in parameter, 0 and. T L ( l) = min v { T ( l )} Example: Conidering for the time being a a contant, find the optimal 0 *. 0* = v t v w 7 The only change for the unynchronized cae involve the coefficient of H (or of H 0 and H, if H 0 H ). 3-
Public Tranportation Sytem: Planning Corridor T * ( ) l t l = Δ + + + + vmax vw vmax lt The /v max term i typically much le than approximate olution. t vw o we can ignore /v max and get an Inight (Comparion acro Countrie) T L l v ( t ) 3 l v * w 3 t l 3 w v w vmax () l = Δ + + + ( t l v ) 3 max Imagine combining the cot and time into a Lagrangian expreion of generalized cot when we value time at a rate of β dollar per unit time. If we neglect the (mall) effect of dead time and tranfer time, the reult i: 3cd z = λh + βh + β v + + v 0 t + lt w 0 max l + v max Three of the parameter that appear in thi expreion (λ, β and l) can vary by order of magnitude acro citie and countrie, and the other vary much le. Therefore, (λ, β and l) can be thought of a the main driver of ytem tructure or deign. Now, if we divide through the above expreion by β o that the generalized cot (GC) i alway expreed in unit of time, then λβ alway appear together o z*(λ, l, β)/β i really a function of only two driver of deign: (λβ and l). Thi generalized cot in unit of time i the total time required to make a trip including the time people mut pend working to afford ytem. We can think of the λβ driver a the wage generation rate per unit time and ditance becaue λ i the trip generation rate and β the value of time aociated with each trip generated, which hould be imilar to the wage rate. It i nice to ue intrinic unit that are independent of a currency or country. We can expre wage β in any equivalent unit we want. For example we could ue unit of c t (where c t i the operating cot per unit time of running a bu), uing β/c t a our wage metric. Note that thi ratio i the number of bue that can be continuouly operated with the wage of one peron. (In rich countrie the ratio can be cloe to and in poor countrie much, much le.) Thu, we can think of λβ/c t a the bu generation rate. Whether one ue intrinic unit or not, the fact that demand and wealth can be combined into a ingle driver mean that low-denity wealthy neighborhood in developed countrie and poor 3-
Public Tranportation Sytem: Planning Corridor dene neighborhood in developing countrie (with the ame bu-generation rate) hould have approximately the ame ytem tructure. And they hould alo hare the time-baed GC. (Thi happen becaue a we have een the time-baed GC depend only on the combined value of λβ.) In t it nice that we can ay thi even before optimizing the ytem? Example: Plugging ome number into thi model help illutrate the difference between tranit competitivene in wealthy veru poor countrie. Uing extrinic unit of hr, km, $: v w 3 km/hr v max 36 km/hr t 5 x 0-3 hr β ~ 0 $/hr c d $/km c 0 - $/top c t 0 $/hr l ~ 40 km λ ~,.5, 0, 0, 50, 00 trip/km The value with the greatet range of value (marked with ) are our driver of deign. The figure below how how the generalized cot (in unit of time) relate to the length of a trip for tranit erving neighborhood of different value of λβ and the cot of making the trip by car in a wealthy or poor country. More acceibility i aociated with greater trip length for a generalized cot. l 50 km car (wealthy) λβ = 00 λβ = 50 λβ = 0 5 km car (poor) Time, z*/β 00 min 40 min 3-3
Public Tranportation Sytem: Planning Corridor Standard Reviited (Two Additional Point) The firt point i that every length-baed tandard can be reduced to a imple tandard. Recall from the earlier dicuion how, for a defined political tandard T 0 (l) for door-to-door trip time, we were able to find the critical length of trip and critical headway to atify that tandard with the graphical contruction below. t imple tandard T ( l 0 ) convex hull of T ( l 0 ) * T 0 * H AIVD line * LE * l l Note that if we replace T 0 (l) with the imple tandard hown with it corner at point (l *, T 0 * ) we arrive at the ame olution! Thi imple tandard can be interpreted uch that all trip horter than a certain length (l * ) mut be completed within a certain time (T 0 * ) and longer trip can be ignored. The implification i ueful becaue it involve jut two parameter (l * and T 0 * ). Therefore, by exploring the tructure of optimum tranit ytem for all poible value of thee two parameter one would have explored all poible optimum olution. Note too from the figure that l * mut be the binding length and therefore we can treat it a the only (equality) contraint. A a reult, there i a : relationhip between (l *, T 0 * ) and (l *, β), and we ee that we can alternatively explore the pace of all olution by plotting the Lagrangian olution for all value of (l *, β), a we had uggeted earlier. 3-4
Public Tranportation Sytem: Planning Corridor The econd point i that there i a neater way of eliminating the ocioeconomic driver (λ and β) from the formulation of the problem, imply by working with the total ytem cot per day, rather than the unit cot per paenger carried. In the tandard formulation we wrote formula for $(, H) and T(, H) with unit per paenger. But if intead we had (equivalently) ued $ T (, H) λ$(, H), with unit of cot per unit time and length, then you can ee from the earlier note that the parameter λ would not appear in any of our formula for $ T (, H). In fact, the mathematical program: min $ T t. T T ( l); l. 0 would not include either of our ocioeconomic driver (λ or β) in it formulation! Thi allow you to find the optimum yearly cot and the ytem tructure by defining a tandard and nothing ele. The ocioeconomic variable enter the picture only when a city chooe the tandard it can afford. The average cot per paenger carried expreed in unit of local wage, which i $ / β $ T /( λβ ), hould be an important factor in any uch deciion. Example: (optional problem for tudent to olve to undertand thee two idea) Show that the equivalent imple tandard to the linear tandard T 0 ( l) = T 0 + P0l for the lower bound formulation of the cae with no tranfer i: * a l = if P 0 t v v max P 0 > v max and that: 3-5
Public Tranportation Sytem: Planning Corridor 3-6 = + = max 0 0 * 0 max 0 0 0 * 0 $ v P v t T c v P v t P T T a d a if max 0 0 v P v t T a > Note how the olution doe not involve λ or β. Then, ue the Lagrangian approach to how that the hadow price that would achieve the above i: max 0 0 * ) ( = v P v t T c a d λβ To repeat: The importance of thi i that tandard are connected to total cot, and you don t need anything ele to determine thi cot. Space- and Time-Dependent Service Auming we have a corridor, we want to ee how performance i affected by changing the deign variable in pace and time. Of our two deciion variable, and H, pacing i a phyical apect of the route, and o i only a function of pace: (x), while headway will remain contant a bue travel the route, and o i only a function of time: H(t). The (t, x) area of concern can be partitioned into pace (i) and time (j) lice a hown below and we can find the cot of delivering ervice for i and H j. We will do thi firt for average-cae analyi (which you hould know) and then for the ervice guarantee (tandard) approach. Average Cae Analyi For average cae analyi, demand play an important role, o we tart by defining an OD matrix of trip election rate. The OD matrix can be repreented a λ i i' j., where i i the origin, i the detination and j the time (the unit of λ would be pax/time dit ). We hall find that it i not neceary to ue the entire OD matrix, only the relevant part for which we want tandard.
Public Tranportation Sytem: Planning Corridor x i j t If we ignore the cot of top, the total cot of ervice i: $ T = j cd L T H j j Note: it doe not depend on the OD matrix. The generalized cot of waiting delay, where λ j i the total number of trip generated per unit time along the complete corridor during time lice j (unit of pax/time), i: β j H (λ T ) Similarly, the generalized cot of inbound acce i: j j j β i i 4 v a ( λ L ) i i where λ i i the total number of trip generated per unit ditance with detination for the whole corridor during the coure of a day (unit = pax/dit). Since the cot of egre hould be the 3-7
Public Tranportation Sytem: Planning Corridor ame, we can multiply thi equation by to account for the total acce cot. Finally, if we let Λ i be the number of people croing a creen-line in region i during the coure of a day (unit=pax/hr), we can expre the generalized cot of top a: β i Li i A you can ee, we don t need to know the whole OD matrix, only the ummary information embodied in {λ i, λ j, and Λ i }. Alo note that the optimization i very imple. The firt two equation are function of H j and not i and can be optimized alone and eparately for each time period. Likewie, the lat two equation are function of i and not H j and can be optimized alone and eparately for each location. Service Guarantee Analyi Intead of optimizing for the average cae with a choice of β, we can chooe a et of time tandard T 0 (i, i, j) for elected origin and detination pair and time of day. Then, there i no need to know the demand to etimate the optimum cot. It would be the job of policy-maker to decide on a reaonable tandard. The objective function i the ame a above, and the tandard would imply introduce contraint of the form: i t T ( i, i', j) AT + AT + IVTT + H 0 i' Λ i ii' j for relevant et of (i, i, j). Note that the four term of the RHS have imple ubcript. Thi MP can often be olved by introducing hadow price and decompoing the Lagrangian into part that can be optimized eparately. If thi doe not work we can reort to a numerical olution. Further Reading The following reading may be ueful to reinforce the concept you have learned in thi module. Claren, G. and Hurdle, V. (975) An operating trategy for a commuter bu ytem, Tranportation Science 9, -0. (Average-cae analyi of non-hierarchical many-to-one -D ytem with inhomogeneou demand.) Wirainghe, C.S., Hurdle, V.F. and Newell, G.F. (977) Optimal parameter for a coordinated rail and bu tranit ytem Tranportation Science, 359-74. (Average-cae analyi of a -mode hierarchy erving -D, many-to-one demand.) 3-8
Public Tranportation Sytem: Planning Two Dimenional Sytem Module 4: Planning Two-Dimenional Sytem (Originally compiled by Eric Gonzale and Joh Pilachowki, March, 008) (Lat updated 9--00) Outline Idealized Cae (New -D Iue) o Sytem without Tranfer o The Role of Tranfer in -D Sytem Realitic Cae (No Hierarchy) o Logitic Cot Function (LCF) Component o Solution for Generic Inight o Modification in Practical Application o General Idea for Deign Realitic Cae (Hierarchie--Qualitative Dicuion) Time Dependence and Adaptation Capacity Contraint Comparing Collective and Individual Tranportation Remember from previou module the type of ytem we have analyzed. Shuttle ytem had one deciion variable, H, and could only be optimized temporally. Corridor had two deciion variable, H and, and could be optimized temporally and patially. Thee deign deciion defined all the paenger travel choice; i.e., when and where to board a tranit vehicle. Think now about a two-dimenional ytem and the new travel choice available to paenger. Thi hould illuminate the extra iue that mut now enter into the analyi. They include conideration of total route length and layout, the role of tranfer and travel circuity. A before we tart with an idealized analyi that iolate the new iue and then proceed with a more realitic treatment that combine them all. Idealized Cae We will perform the idealized analyi in a imilar manner a the corridor analyi. We conider a ytem with a ingle line with no tranfer allowed and bi-directional ervice. We aume H=0 and t =0. For the two-dimenional ytem we will alo aume that a 0 =, which remove all penalty for topping meaning that v=v max at all time. We make thi aumption becaue if we had allowed a 0 = in the huttle and corridor analyi then the door-to-door peed would be v max. Yet, thi turn out not to be true in the two-dimenional cae. So, thi et of aumption allow u to iolate the new effect introduced by the econd patial dimenion. Let u ee 4-
Public Tranportation Sytem: Planning Two Dimenional Sytem Sytem without Tranfer Conider a quare city with ide φ, area R=φ and an infinitely dene grid of treet; ee figure below. No matter how long a tranit line i, it cannot cover all point. Therefore, we anticipate that coverage and acce become important iue in -D, and that our new deciion variable will be route length and placement. To minimize wort-cae acce time in -D we hould place top on a (quare) grid, with pacing to be determined. The wort-cae acce time would then be /v w ince there i an acce ditance of at both the origin and detination. What then about travel time? Note that ince top don t matter it will be the maximum ditance a peron pend in a vehicle, divided by v max. And ince ervice i bi-directional, the maximum ditance i ½ of the length of the line, which we denote L. Thu, IVTT=L/v max, and we minimize IVTT by chooing the hortet route to cover our lattice of top. The problem of hortet-path routing for pre-exiting point i a famou and complex problem known a the Traveling Saleman Problem. Fortunately for u, the olution for a two-dimenional lattice tructure with an even number of point, uch a the one hown above, i eay and efficient ince there alway i a path where the ditance between any two conecutive top along the route i (you can convince yourelf of thi.) φ R φ 4-
Public Tranportation Sytem: Planning Two Dimenional Sytem If you now imagine cutting the grid between parallel route line then the area can be imagined a a corridor with length L and width where L i the total length of the route. Thu, the area can be expreed a: L L = R, and the in-vehicle travel time for the wort cae peron would be: IVTT = L R =. v v max max The door-to-door travel time guarantee i then: t = + v w Note: Thi i an EOQ expreion with repect to the lattice pacing. When optimized the olution i: R v max t * R = vwv max = v φ w v max Thi give a door-to-door travel peed for the wort-cae peron: φ vˆ * = t v w v max If we aume value of v w =3kph and v max =36kph, then the reulting door-to-door peed i vˆ 5.5kph, which i not much fater than walking peed. The underperformance arie becaue to achieve low acce time the route need to be very winding. And bue in a windy route entrap paenger unfortunate to go a long ditance. So, how can we improve the ytem? If we allow for tranfer then paenger are no longer entrapped, and all we have to do i look for routing that give good coverage while providing good travel option to paenger that can tranfer. So what are thee routing? To get an undertanding of thi iue, we look at ome idealized ytem with one tranfer. 4-3
Public Tranportation Sytem: Planning Two Dimenional Sytem The Role of Tranfer in -D Sytem Two extreme poibilitie are conidered here. A hub and poke ytem (H) with only one tranfer point; and a grid ytem that allow for tranfer at every top. See the illutration below. Note that for the ame route pacing, the grid ytem require more route-kilometer; o it hould be more expenive to cover with vehicle. Another diadvantage of the grid ytem relative to the hub i that coordination i more difficult. An advantage i that uer can alway chooe a direct route without backtracking. We now compare the performance of thee two ytem (and of the no-tranfer, ingle-line ytem (O)) for different value of L. Thi i reaonable becaue if one hold H and the commercial peed of vehicle invariant acro cenario, then L i the mot important driver of cot. φ A Hub and Spoke Sytem (H) φ 4-4
Public Tranportation Sytem: Planning Two Dimenional Sytem φ φ A quare grid ytem (G) We now change notation and ue L to denote the kilometer of undirected ervice provided. A little bit of reflection how that the total length of ervice for the three cae are: L G H 4φ = L L 0 3 φ φ = = For the ame L, the three ervice provide different coverage, a repreented by : 4-5 L L L G H O 4 ; 3 ; φ φ φ = = = To get thee imple expreion, it i aumed that it take pacing to turn the bue at the end of each route.
Public Tranportation Sytem: Planning Two Dimenional Sytem Thee value repreent the ideway pacing between line achieved by the three ytem type. Thu, the wort-cae ideway acce time are: φ / Lv w, 3φ / Lvw, and 4φ / Lvw. If we ignore the longitudinal acce time (which hould be the ame for the three ytem) and focu on cro-town trip (of length l φ ), the wort-cae door-to-door travel time are then: T T T 0 H G φ = + Lv w 3φ = Lv w 4φ = Lv w L 4v max φ + v max φ + v vmax If we now chooe ~ 0 then we can compare the three cae baed on the dimenionle v variable φ L. The formulae become: w max T φ = v max φ 0 + L 4 φ 30 + L φ 40 + L L ( O ) φ ( H ) ( G ) Thee expreion can be expreed graphically a follow: 4-6
Public Tranportation Sytem: Planning Two Dimenional Sytem v T max φ 8 6 4 O H G 0 0 30 40 L φ Thi illutrate that hort ytem with few top, whoe total length i not much greater than the perimeter of their ervice region do not require tranfer. The figure alo illutrate that long ytem with many top do benefit, and that in thee cae the longer the ytem the greater the benefit. Jut o you get a feel for the meaning of φ L we look at four common routing example and their L value: φ 4-7
Public Tranportation Sytem: Planning Two Dimenional Sytem Campu periphery L 4 φ Small metro/mall town bu L φ Large ytem L 0 φ We find that the optimal routing layout depend on the value of φ L and, if we add a 5% acce penalty to the grid ytem to reflect the added cot of an uncoordinated tranfer, we find that the critical point are a follow: O H G 0 0 L/φ Thi explain why ytem in real-life often have the tructure hown in the above figure. Alo, note that when we allow one tranfer, then v ˆ = vmax a L for the grid ytem. So, tranfer really do help with performance in -D. 4-8
Public Tranportation Sytem: Planning Two Dimenional Sytem Realitic Cae No Hierarchie We could do a realitic analyi for each cae we introduced earlier, however in the interet of time we will be concentrating on the grid cae ince it i the mot ueful for larger network. Since we are dealing with wort cae analyi we will alo only concentrate on quare grid. A rectangular grid would introduce directionality and add unneeded complexity. Firt we will introduce top pacing,, within route pacing, S, uch that <S. We will alo ue S a a deciion variable intead of L. We need to make aumption about how people travel. In thi cae, we will aume that people only make one tranfer and they chooe their origin and detination top in order to minimize their acce ditance. We will then develop formula for agency cot and paenger time (acce + waiting + in-vehicle travel time) Logitic Cot Function (LCF) Component Recall that the tranit ervice in dimenion can be decribed by 3 deciion variable: top pacing, line pacing S, and ervice headway H. S The total cot for uch a ytem i cot of driving and topping a bu multiplied by the number of bue operating per unit area. 4 c cot $ T = cd + in unit of SH time dit 4 c cot $ = cd + in unit of λsh pax 4-9
Public Tranportation Sytem: Planning Two Dimenional Sytem Notice thi i very imilar to the cae for a corridor, the only difference being a factor 4/S, expreing the fact that cot depend on the number of line. The travel time i compoed of acce time (AT), waiting time (WT), and in-vehicle travel time (IVTT) jut a we aw for corridor. For the wort cae paenger whoe trip tart and end a far a poible from tranit ervice (the middle of the quare), AT S S + S = v + + + = w v w WT = H + Δ or H + Δ or 3 H + Δ where repreent time required to make a tranfer, uch a walking time from one top to another. The number of headway included in WT depend on the aumption we make about the ynchronization of chedule (H if ervice are perfectly ynchronized o that paenger only wait at the firt top where they board; H if ervice are not coordinated and paenger have to wait when they tranfer, or ele if ervice i coordinated but paenger have appointment at the detination; etc ). IVTT = l 0 v max t + where the longet poible trip length i l 0 φ. So, the wort cae time for a -D ytem i given by the um, T S + = H + Δ + + l v w 0 v max t + Notice again that thi i very imilar to the time aociated with tranit ervice in a corridor. The difference i the waiting time, H +, and an additional component of acce time, S/v w. Solution for Generic Inight If we conider the lower bound of cot, auming that the cot of topping i mall, the tandard approach i decribed by the following mathematical program: 4c min d SH (6.) S H + + T l vw T0 l) (6.).t. ( ) ( 4-0
Public Tranportation Sytem: Planning Two Dimenional Sytem 4- where ( ) + + = Δ + t T l l v v w max. The contraint will be an equality at optimality becaue for any T(l ) the cot i minimized by chooing the highet value of S and H. Therefore, the lower envelope method (explained in Module 3) can be ued to olve for *, and with T L (l) we can determine l*. The mathematical program can thu be obtained with pencil and paper. Alternatively, we can ue the Lagrangian approach, expreing the generalized cot in dollar per peron a + + + + Δ + + = t v v S H SH c z w d L l l max 4 β λ (6.3) which decompoe o that the top pacing,, i iolated. Solving for * and ubtituting, lt v w = * (6.4) + + + Δ + + = max 4 v v t v S H SH c z w w d L l l β λ The optimal headway, H*, and line pacing, S*, can be olved in cloed form. λsβ c H d * = (6.5) ( ) + Δ + + + = w w d L v t v v S S c S z l l 4 * max β β λ β 3 S 8 * = λβ d v w c (6.6) + Δ + + = w w d L v t v v c z l l 6 * max 3 β λ β We now compare thi cot to the generalized cot for corridor, auming the ame value a in module 3: v w 3 km/hr v max 36 km/hr
Public Tranportation Sytem: Planning Two Dimenional Sytem t 5 x 0-3 hr c d $/km In -D, the generalized cot i z L β * = 4. λ 3 + β 0.08 l l + 36 and in univeral unit of time: * 3 z L = 4. β λβ + 0.08 l l + 36 compared to a generalized cot in a corridor of z L β * = λ z * L = β λβ + β 0.08 + 0.08 l l + 36 l l + 36 Note that the univeral generalized cot per peron decline with demand, λ, and wealth, β, more lowly in the D cae than in the D cae. In other word, the econd dimenion omewhat dilute the economie of cale in collective tranportation. Note too that the effect of ditance i the ame in both cae. Remember, however, that λ i expreed in demand per area in the -D cae, and demand per ditance in the corridor cae, o thee expreion cannot be compared for the ame λ. For the hypothetical cae of long trip in a relatively poor city (l 0 = 40 km, β = $ / hour, and λ = 0 3 pax / hr km ), the generalized cot z L /β =. hour which decompoe to 0.7 hour of work, 0.9 hour of delay (acce, waiting, and in-vehicle topping), and. hour of travel time (like in a car). Modification for Practical Application ) Some line may require fixed infratructure (BRT, rail, etc.), o the cot of contruction, bond finance, etc. hould be amortized over the life of the infratructure. Convince yourelve that for an infratructure cot r $/hr top, thi contribute 4-
Public Tranportation Sytem: Planning Two Dimenional Sytem to the objective function. r λs ) Stop may be kipped if the demand i low. In thi cae we work with expectation. E(time topped per unit length) = E(# pax boarding/alighting move per ditance)t m + E(# top per ditance) t where t m i the marginal time for one paenger move and t i the marginal time for a vehicle top. The expectation are now given. Firt, note that E(# of pax move per top) = λhs Therefore, ince there are / top per km; E(# pax move per ditance) = E(# pax move per top) = λhs, and E(# top) = Pr{topping}, where Pr{topping} = ( e λhs ) if the demand for top follow a Poion proce with the given mean. 3) Citie have center, o we may want to orient our grid toward the center. Notice that if we zoom in on a part of a ring-radial network it look like a grid. Nothing prevent u from making a contant denity of ervice in a ring-radial network by adding radial line a we move out from the city center. 4-3
Public Tranportation Sytem: Planning Two Dimenional Sytem We can alo ue thi trategy if we want to have the flexibility to have different denitie of ervice and headway in different part of the city a hown in the figure below. To do thi ytematically, we can et different tandard for trip in different part of the city. For example, T 0 (A) (l) for all trip in A T 0 (B) (l) for all trip in B (or, even better, B A) T 0 (AB) (l) for all trip between A and B B S B, B, H B A S A, A, H A General Idea for Deign ) Think of a family of deign concept, qualitatively e.g. grid ytem, ring-radial network, etc. ) Identify member of the family by lit of deciion variable e.g. top pacing, line pacing S, and headway H. 4-4
Public Tranportation Sytem: Planning Two Dimenional Sytem 3) Etimate the cot and tranlate the pecific concept into a detailed plan e.g. OR Conidering all region (r = A, ) and time period of the day (j =,, ) olve the following mathematical program for the deciion variable: {, ( r, S r ), } and {, H rj, }: min$ T = ( r) $ T r= A,... j=,,....t. TA ( A, S A, H Aj ) T0 Aj T B ( A S A, B, S B, H Bj ) T0 Bj (, S A, B, S B H Aj, H Bj ) T ABj ( S, H ) r rj, j = ruh, off-peak, night,, j = ruh, off-peak, night TAB A 0,, j = ruh, off-peak, night You may want to include eparate contraint for acce time or waiting time, depending on what the city want, but you hould alway ue your judgement. Anything i poible, but the more complicated the problem, the more difficult it i to olve the problem exactly. Lagrangian decompoition can help u olve thi mathematical program. It may be poible to implify the problem and eliminate many of the deciion variable. So, we can ue hadow price to implify thee complicated mathematical problem by aigning a different β to each of the contraint. Increaing the value of β will reduce the left ide of the contraint when the Lagrangian i optimized, o we tart with an etimated value of β and then increae it until the contraint are met. If we have a cloed form for the optimal deciion variable value in term of β, it i eay to adjut the olution by changing the hadow price. Note: Thi approach can be ued to olve (6.,) by working intead with (6.3). All you hould have to do i plug in (6.4,5,6) into (6.) and find the β that olve (6.) a an equality. 4-5
Public Tranportation Sytem: Planning Two Dimenional Sytem -D Sytem: Realitic Cae (Hierarchie) Until now, we have looked only at local ytem in -D. However, we could introduce a hierarchy with the ame method a for corridor (ee module 3). There are now deciion variable for the top pacing and line pacing of both the local and expre ervice. Expre Line Local Line The introduction of hierarchie alo give u the flexibility to deign non-iotropic ytem. For example, a grid may erve a a bai for local bue guaranteeing a length-baed but uniform tandard for hort-medium trip. A radial expre ervice may be overlaid to provide better ervice for inter-zonal travel (e.g. Chicago). Such an expre network may be decribed in a few a 3 additional deciion variable: # of radial line, # of ring line, ervice headway. Expre Line Local Line Perhap one ytem can be deigned to act a radial network outide of the city center and look more like a grid in the city center (e.g. Wahington DC, London). The poibilitie are many, but in all cae the goal i to reduce thee concept to a few decriptor a poible which will decribe the hape and deign that the ytem hould have once the variable are choen. 4-6
Public Tranportation Sytem: Planning Two Dimenional Sytem -D Sytem: Time-Dependence and Adaptation Over time, demand for tranportation in a city change. Some of the deciion variable are eaier to change over time than other. The headway, H, can be varied very eaily even within the coure of a day. The top pacing,, can be changed with a little more effort, and line pacing, S, i relatively fixed. Suppoe we have a linear city of length one, and we place a tation to minimize acce ditance. A ingle tation divide the city into two halve and hould be placed in the center to minimize wore cae and average acce ditance; S* = /. S = / S = / A demand grow over time, we may want to add tation incrementally to the city, one at a time. If we can pull up old tation and alway re-optimize, the placement hould make the pacing follow the progreion, S* = /, /3, /4, /5, etc. However, if the tation are fixed once they are placed, ubequent placement of tation will not alway give u the minimum acce cot. For a wort cae analyi, imagine that we have the city above with n = pacing (S* = /) and we add one more tation. Only half of the city benefit, o the wort cae acce i unchanged. The wort cae acce cot i only improved when ymmetry i etablihed at n =, 4, 8, The incremental addition of tation following thi naïve approach i hown below: 4-7
Public Tranportation Sytem: Planning Two Dimenional Sytem S = / S = / S = /4 S = /4 S = / S = /4 S = /4 S = /4 S = /4 /8 /8 S = /4 S = /4 S = /4 /8 /8 /8 /8 S = /4 S = /4 /8 /8 /8 /8 /8 /8 S = /4 /8 /8 /8 /8 /8 /8 /8 /8 I there a way to place tation o that each incremental addition of a tation improve the wort cae acce? In fact there i! If tation are placed o that for n+ tation, the placement reult in pacing of length ( ) x n i ( i + n) log ( i + ) = log n for i =,, n n ( ) ( ( n+ ) ( n+ ) ( n) Note that thi alway atifie log i + n log i + n )=, and alo x + x = x. i= The latter equation how that the larget pacing become the um of the mallet two (after the plit). Thi i in fact how the equation i derived. The incremental addition of tation would now look like the progreion below. 0.585 0.45 0.63 0.3 0.45 0.63 0.3 0.93 0. n n If we plot the wort cae acce ditance againt number of top pacing, thi aymmetric approach i better than the ymmetric naïve approach mot of the time. 4-8
Public Tranportation Sytem: Planning Two Dimenional Sytem Acce Ditance 0.5 Naïve Approach Aymmetric Approach 0.5 0.065 4 8 n For average cae analyi, the average acce ditance when tation expanion follow thi pecific aymmetric recipe i.04/4n. If the tation could be picked up and re-placed optimally every time the ytem i expanded, the average acce ditance would be /4n. So the penalty for not being able to move tranit line once they are placed can be a low a 4%. Thi i good new, becaue it mean fixed infratructure that i extremely cotly to change (uch a ubway tunnel) are alway near optimal! Capacity Contraint One additional point that can be added to thi module i the idea of riderhip (i.e., the number of paenger in a bu). Thi can be ueful in planning becaue it allow for the addition of vehicle ize (capacity) contraint to the deign. For a grid ytem with pacing S, the average riderhip can be roughly approximated by: Riderhip λs Hl. 4S Thi expreion arie from conidering a ingle S S cell of our grid. The numerator i the number of pax-km generated in the cell in a ingle headway, and the denominator i the number A an exercie, derive the average cae analyi cot for the naïve cae where the tation cannot be moved. The reult may urprie you! 4-9
Public Tranportation Sytem: Planning Two Dimenional Sytem of bu-km traveled per cell in one headway. Auming optimal deign parameter developed in eq. 6.5 and 6.6 the reult i: Riderhip 3 c 3 d v w λ β 3 l. 6 If we adopt the typical number we have been uing, we ee that for a wealthy city (β ~ 0) with l ~ 0 km, a demand of λ ~ pax/km -hr would yield an average riderhip of about pax per bu. So, we would not expect collective tranportation (CT) to be a feaible option for delivering mobility if λ wa much maller than pax/km -hr. And what to do in thi cae will be the topic of the next module. Comparing Collective and Individual Tranportation Recall from the equation immediately following (6.6) that our average cot function for -D ytem with no hierarchie and allowing for tranfer could be expreed a: * z l = Δ + β v lt v w cd + 6 βλv w 3 (7.) The RHS i the extra cot of CT over and above the unavoidable (time) cot of overcoming ditance. We can ee that CT ha nice economie of cale. Demand and extra cot are inverely related, o a demand rie, the cot per paenger decreae. (In fact, if Δ and t are neglected, the extra cot tend to 0 a λ ; it can be virtually eliminated.) Thi i the beauty of CT. However, in cae of low demand (λ 0) the cot can become quite ubtantial. Thi i expected, becaue the ytem i there regardle of the riderhip it generate. In thee cae, collective tranportation in t very efficient. A a point of comparion, conider the cot function for individual tranit (IT), omething like a taxi, which look like thi: * z l = β v c I l β where c I i the cot per unit ditance of providing IT. Notice that the extra cot of IT no longer decline with increaing demand (a it did with CT); and that it doe not go to infinity in cae of low demand (a it did with CT). To further undertand the ource of thi difference we now how a more detailed comparion between the two ytem with one vehicle, auming λ 0. 4-0
Public Tranportation Sytem: Planning Two Dimenional Sytem MT We conider a quare of area R with an underlying grid of treet and aume that bue top on demand. With CT, the deciion variable i length of route and we ue the idealized analyi of D-ytem at the tart of thi module to evaluate the optimum trade-off between acce time and in-vehicle travel time. With an optimal length of route, L *, the expected door-to-door time i 3 : R E ( tm ) =. vv w IT With IT, the route can be directed where the people are, removing all acce time. Auming that the vehicle tart from a central location every time a requet i made, the expected door-todoor time can be hown to be: 7 R E( t I ) =. 6 v 3 The reult differ from our earlier reult by a factor of becaue (by topping on demand) the current ytem eliminate the acce ditance parallel to the tranit route. 4-
Public Tranportation Sytem: Planning Two Dimenional Sytem The ratio of expected time between the two ytem i approximately 3. Note that the cot of both ytem i imilar ince both ue one vehicle and one driver. So, in cae of low demand, IT will outperform CT by a large margin. Thi i due to it flexibility in routing. Therefore the next module will explore poible way of delivering thi flexibility for ytem with higher (albeit till low) demand. Further Reading The following reading may be ueful to reinforce the concept you have learned. Holroyd, E. M. (967) "The optimum bu ervice: a theoretical model for a large uniform urban area in Vehicular Traffic Science (L.C. Edie, R. Herman and R. Rothery, editor) Proc. 3 rd ISTTF pp. 308-38, Elevier. (Average-cae analyi of non-hierarchical many-tomany grid ytem with uniform demand. The paenger routing model in thi beautiful reference i complex; unfortunately thi complicate the formulae, obcuring poible inight.) Daganzo, C.F. (009) Structure of competitive tranit network Tranportation Reearch Part B (in pre). (Thi reading i of interet becaue it generalize the idea in thi module by exploring a general family of ytem that include the hub-and-poke and the grid concept a pecial cae. The imple formula it give, are ued to compare the performance of different technologie (Bu, BRT, LRT and Metro) in different urban context.) 4-
Public Tranportation Sytem: Planning Flexible Tranit Module 5: Planning Flexible Tranit (Originally compiled by Eric Gonzale and Joh Pilachowki, March, 008) (Lat updated 9--00) Outline Way of delivering flexibility Taxi Dial-a-Ride (DAR) Car-Share We aw in the lat module that conventional form of collective tranportation with fixed acce point and route deliver better ervice than individual tranportation when one or more of the following factor are high: the demand rate, the typical trip length and/or the time value of money. When thi happen traveler gain if they trade-off the flexibility in routing and timing of individual tranportation for the lower cot of collective tranportation. Since there i alo a grey area where the choice i not o clear, one may ak whether collective tranportation can be made more flexible o it can better compete with individual tranportation in ituation like thee. Thi i the quetion addreed in thi module. Way of Delivering Flexibility with Public Tranportation We divide public tranportation concept depending on whether or not people hare ride. Individual Public Tranportation (IPT) Taxi A driver take paenger directly from their origin to their detination Car-hare Uer pick up a vehicle at one of many predetermined location (pod), then return it to any pod when finihed Driverle Taxi (futuritic) Same a taxi, only without a driver required. The military i currently developing thee type of vehicle for urban warfare, but they could alo be ued during peace time. See http://www.darpa.mil/grandchallenge/overview.ap for more information. Peronal Rapid Tranit (PRT) (futuritic) Small occupancy vehicle would travel along exiting guideway. See http://en.wikipedia.org/wiki/ruf_%8dual_mode_tranit%9 for more information. 5-
Public Tranportation Sytem: Planning Flexible Tranit Collective Tranportation (CT) Dial-a-ride (DAR) Same a a taxi, but with multiple uer haring the vehicle. FIFO i not guaranteed. Long Ditance DAR with tranfer on-the-fly (futuritic) Train with paenger in car grouped by detination. Car would couple and decouple without lowing the train and then drop off and pick up individual a DAR. Paenger would walk in the train to the appropriate car. Thi could be ueful a a ubtitute to long ditance train. A t-x diagram can be ued to etimate feaible top pacing. Det. 3 Det. 4 Det. -3 (would become Det.4 after pax reditribute) The main advantage of flexible IPT over the automobile i that public ervice car are hared; by being ued mot of the time they require le parking infratructure. The added promie of flexible CT i that perhap it can reduce cot with only a mall degradation in LOS, and alo provide ome economie of cale with repect to demand. So, with thi in mind, we now examine what exiting flexible concept can do. Taxi Conider a region of area R in which we provide radio-taxi ervice to cutomer with demand denity λ and expected trip length l. We will aume that we provide enough taxi, m, to enure that every call immediately get aigned a taxi. 5-
Public Tranportation Sytem: Planning Flexible Tranit λ,l R - Random call for a taxi The fleet ize, m, can be divided into three type of taxi: n i idle taxi, n a aigned taxi, and n ervicing taxi. The cot of the ytem will be roughly proportional to m. We meaure LOS by the waiting time a uer experience between requeting a taxi and being picked up. The following diagram how the rate at which taxi witch from one tate to another. λr μ a μ n a n Uing Little Formula, and uing T a for the expected cutomer waiting time, we ee that: na μ a = ; with T a E ( d n T t i ) a ~ +, v where E(d ) i the expected ditance the cloet idle taxi to a requet will have to travel. An i n expreion for thi expectation i obtained by imagining a region of area R, with n i point and a circular dik with diameter x. Becaue the n i taxi are randomly ditributed we can write: Pr{ d n x} Pr{zero point in the dic} n i x π R. = From thi ditribution function we find (the proof for thi can be found in the appendix at the end of thi module): 5-3
Public Tranportation Sytem: Planning Flexible Tranit R (. n E d n ) i Alo uing Little Formula, and uing T for the expected cutomer ervice time, we ee that: i If the ytem i in equilibrium, then: n μ = T ; with l T ~ + t. v λ n a R = = T Thi give u two equation with three unknown: n i, n a and n. To get a third equation we aume a target ervice level with average waiting time T 0 : a l l T0 ( l ) = T0 + t + = t + + Ta Ta = T0. v v Uing thi a the third equation, and neglecting t a a reaonable firt approximation, we find that the equilibrium olution i: n n n i a R = (vt = n T ( ) R T λ 0 l = λr v which give the minimum fleet ize required to achieve the target level of ervice: 0 ) m = n i + n a + n R l = ( + λ R T0 +. vt v 0 ) Notice that m decline with T 0 up to a point, and then tart riing again. The riing branch i undeirable. 5-4
Public Tranportation Sytem: Planning Flexible Tranit m undeirable! T * 0 0. 8 λ 3 v 3 T 0 The hape of the curve mean that there i a minimum fleet ize required, m *.Rλ /3 v /3 + λr l /v, to enure that incoming call are aigned a taxi without delay, and that the wort LOS * thi kind of ytem hould provide,, i alo bounded. T 0 We ee that the leat poible extra generalized cot (in univeral unit) of delivering thi type of ervice i: * zt l = T β v + T + m γ 0.8 +. β λr * * 0 γ λ β 3 v 3 γ l + β v where γ i a taxi cot per unit time (which hould be greater than β). Note that the extra cot i at leat l/v even for λ 0. So taxi do not have ignificant economie of cale. Example: T t 0 = 0.hr = 0.0hr λ = pax / hr km R = 400km v = 0km / hr l ~ 0km na = 88 ni = 7 n = 80 m = 375 uch a low number of idle taxi i bad for new uer. 5-5
Public Tranportation Sytem: Planning Flexible Tranit Note, for thi ytem T * 0 0. change T = 0.08hr then the new reult i: 0 na = 40 ni = 30 n = 80 m = 350, o we hould be able to reduce both T 0 and m. So, if we intead * A better level of ervice and le taxi needed! Clearly, operating with T 0 > T 0 hould be avoided. If T 0 i too large the trip time to collect paenger will be long, requiring many taxi in collection mode, and the equilibrium become untable. Dial-a-Ride If taxi can have more than one paenger, then you can remove idle vehicle from the ytem and reduce cot. Service call would then go directly to vehicle currently in ervice. To analyze thi ytem, we again conider a region of area R with demand denity λ. We will aume that origin and detination are uniformly ditributed throughout the region. m vehicle with n r paenger each λ R n w waiting paenger requet A urge in demand would: further increae the number of taxi in collection mode; reduce the number of idle taxi, thereby increaing ditance to new uer and their collection time. With larger collection time, the number of taxi in collection mode would increae ome more, etc 5-6
Public Tranportation Sytem: Planning Flexible Tranit Paenger will be divided into thoe waiting at home, n w, and thoe inide vehicle, n r. The fleet ize i m. The following diagram how the flow of paenger from one tate to another: λr μ w μ r n w m x n r We ue the following aumption (favorable to DAR): After achieving a deired paenger load, n r, bue alternate between pickup and drop-off Bue pickup the paenger cloet to them Bue drop of the paenger with the cloet detination Uing Little Formula, equilibrium aumption, and the expected ditance equation from before we can aume that: m μ r = = λr, (7.) t + t p d where t p and t d are average time to pickup and drop-off repectively, and the ditance for each are: d d R R = ; d p =. (7.3) n n r w For any choice of m and n r (deciion variable) we can ue (7.) and (7.3) to find n w (n r, m). We then chooe the value of n r that would minimize the number of people in the ytem: A little bit of algebra how that the reult i: p = mn n ( n, m). r + w r c p * =, where m λ RR c =. (7.4) v Note: c ha the meaning of number of requet that the ytem would receive in the time it take a bu to travel acro the region (dimenionle demand). 5-7
Public Tranportation Sytem: Planning Flexible Tranit The generalized cot per unit time for the ytem would then be: * γ m + βp, where γ and β are the cot per unit time of one bu and one paenger. On a per paenger bai the cot i: Optimizing thi EOQ expreion give u: * * c = $ + T = γm + β λr m * z DAR / * βc m = ~ c γ $ * * γβ c γβr = T ~ =. (7.5) λr v Note: λ doe not appear in thi equation. So DAR ha no economie of cale. Recall however that the cot of taxi ervice would have been: $ * ( β + ) R + T ~. v * γ So, particularly when β <<γ (poor population), (7.5) i a ignificant improvement over taxi ervice. Could there be a form of DAR with economie of cale? We upect that introducing tranfer may produce a poitive anwer to thi quetion becaue, after all, fixed route CT doe not have economie of cale without tranfer but it doe with tranfer. So, perhap by introducing tranfer into the DAR concept economie could be achieved. One poibility i a follow. Remember our D network where we had to trade-off between number of top and commercial peed. We introduced ome flexibility by not topping at every top and howed how thi could be tudied. We can generalize thi by allowing bue to detour and erve paenger at their origin and detination a hown in the figure. (The figure how an N-S route; if a imilar pattern i ued for E-W route then the whole pace i covered.). Uer could activate a top near their origin and the bu would detour through the region to pick up each paenger. You can convince yourelve of thi by introducing demand into the idealized analyi of Module 4. 5-8
Public Tranportation Sytem: Planning Flexible Tranit S area erved by N-S bue A trade-off would arie between the detour to pick up paenger and the elimination of acce time. For any given detour the erved paenger (or cutomer ) would gain the benefit of zero acce time, while paenger on the bu would be penalized by the extra travel ditance. The average acce ditance aved by the cutomer (auming all hi/her location are equally likely) can be hown to be: 3 S/6. Thu, the average time benefit i: time benefit of a detour S. 6 v w Now let u examine the penalty. If the demand i o low that bue rarely make more than one top per interval, S, then the bu would return to it original route after each detour, and the ditance added by a detour would be twice a large a that aved by the cutomer; i.e., (S/6) on average. But thi ituation i peimitic. If bue make multiple top they do not have to return to the mainline after every detour (ee figure) and thi reduce the average detour ditance. In the mot optimitic cae, where bue make a very large number of detour, the average detour 3 By ymmetry, it uffice to conider cutomer in the bottom half of the haded quare. Let y (0, S/) be any uch cutomer vertical ditance from the bottom of the quare, and x hi/her acce ditance. Since all location are equally likely E(x y) = y/. Now, note that the p.d.f. of y i triangular becaue the number of poible location i proportional to y. A a reult E(y) = S/3. It then follow that E(x) = E(E(x y)) = E(y/) = S/6. 5-9
Public Tranportation Sytem: Planning Flexible Tranit ditance turn out to be: 4 S/9. So plitting the difference between the peimitic and optimitic cae, we etimate the ditance added by a detour a 5S/8, and the collective time penalty a: time penalty of a detour 5S (# of paenger in the bu). 8v So etting the penalty equal to the benefit we ee that the number of paenger in the bu hould be no greater than (3/5)v/v w ; e.g., 6 paenger if v/v w ~0. The econd equation at the outet of thi module how when thi condition may be met but, clearly, thi i a trategy for low demand ytem. There are many unreolved practical iue with thi type of ytem, but they would be worth ome tudy. Public Car-Sharing We now conider a region of area R in which to provide car-hare ervice, with demand denity λ and expected trip time τ. Pod where uer can pick up car will be ditributed through the region. The denity of pod i Δ. λ,τ R -Pod 4 The argument here parallel the one jut given for the acce ditance, after recognizing that when there are very many cutomer, conecutive cutomer have nearly the ame y. With thi in mind, we ee that the average ditance added by erving a cutomer near y i approximately equal to the average ditance between two random point in a egment of length y; i.e., E(x y) = y/3. Thu, again, ince the expected number of cutomer i proportional to y o that E(y) = S/3, we find: E(x) = E(E(x y)) = E(y/3) = S/9. 5-0
Public Tranportation Sytem: Planning Flexible Tranit The fleet will have m car, which are to be repoitioned evenly after time h; ay one day. We need to make ure that there are enough car at each pod o it i rare that one would run out. For implicity of explanation, we aume no ruh hour. There are three important cot for thi ytem: fleet cot; vehicle repoitioning cot; and uer acce cot. We propoe that the required fleet ize i: m λrτ + ( λrτ ) + λh ΔR Δ which can be broken down a: [the # of car in ue] + [95% CI (auming Poion arrival) of car in ue] + [95% CI for net change of car for a pod in a repoitioning interval, multiplied by the number of pod]. The firt two term are a probabilitic upper bound to the number of car in ue. A uch, thee term expre the required fleet ize for an ideal bet cae cenario with h = 0; i.e., where car could be contantly and intantaneouly reaigned acro pod. (Thi i a logical concluion becaue with h = 0 a pod would run out of car only if all pod did; i.e. if the demand for car in circulation exceeded the total available.) The third term expree the afety tock that pod mut collectively have to compenate for the fact that they are only rebalanced every h time unit. Thi third term i the product of the number of pod and twice the tandard deviation of the difference between car requeted and car returned at a pod in a rebalancing interval. (Thi i why the demand rate i multiplied by inide the quare root. 5 ) So if pod are given thi extra afety tock the expreion can be implified to: m λrτ + λrτ +.8 λτ Δ h τ The repoitioning cot in one h will be the product of the number of vehicle move per pod, which i a number cloe to the tandard deviation of the net change in a pod; i.e.: σ ~ λh Δ the number of pod ΔR, and the ditance to repoition the car (a problem famouly known a the tranportation problem of linear programming (TLP)). Reearch how (ee reference) that for thi type of problem: 5 Thi i quite accurate if we aume that cutomer can return car to any pod in the ytem, and omewhat conervative if we aume that car mut be returned to the pod from which they were checked out a exiting ytem operate. 5-
Public Tranportation Sytem: Planning Flexible Tranit repo dit. ( + 0.078log( ΔR )). Δ Δ The lat approximation i reaonable unle the number of pod i huge. Thu, the repoitioning cot per unit time will be: c d λh. ΔR.5c Δ Δ h d λh R. The acce (LOS) cot are proportional to: λ RΔ. Therefore, in ummary we ee that our cot component depend on h and Δ a follow: Fleet cot (hδ) / Vehicle repoitioning cot h / Uer acce cot Δ / The um of thee term (appropriately weighted) i a GEOQ expreion that ha a unique h * and Δ * and would yield the leat poible cot of a car-haring operation. Tet yourelve and ee if you can do it. Notice how car-haring beat taxi for large λ. Reference Daganzo, C.F. (977) An approximate analytic model of many-to-many demand reponive tranportation ytem, Tranportation Reearch (5), 35-333. (Average-cae analyi of many-to-many DAR ytem; trategy II of that reference i the one decribed in thi module.) Daganzo, C.F., Hendrickon, C.T. and Wilon, N.H.M. (977) An approximate analytic model of many-to-one demand reponive tranportation ytem, Proc. 7th Int. Symp. on the Theory of Traffic Flow and Tranportation, pp. 743-77, Kyoto, Japan. (Average-cae analyi of many-to-one DAR ytem; the trategie in thi reference are pecial cae of the one in the homework.) Daganzo, C.F. (984) "Check-Point Dial-a-Ride Sytem," Tranportation Reearch, 8B, 35-37. (The ytem analyzed in thi reference i a building block toward the -mode DAR decribed in thi module). Daganzo, C.F. and Smilowitz, K.S. (004) Bound and approximation for the tranportation problem of linear programming and other calable network problem Tranportation Science 38(3), 343-356 (Derive the TLP formula we ued in connection with the repoitioning cot for car-haring ytem). 5-
Public Tranportation Sytem: Planning Flexible Tranit Appendix: Determination of Expected Ditance to a Taxi We tart with the probability of zero taxi within a dic of radiu x: Pr{ d n x} Pr{zero point in the dic} n i x π R. = By integrating thi over the range of poible radii, we can calculate the expected ditance from a point to the cloet taxi a: R π πx E d n i dx R ( ) =. 0 π With a change of variable y = x, the integral can be changed to a impler form: R R ( ) dy y n π With another change of variable y = coθ, and uing the identity integral can be further implified to a known form: π 0 R = R Γ( n) ( inθ ) n dθ = π π π Γ( n + 0. n. ) ( co θ ) = in θ, the For value of n>>3, a reaonable aumption for a region that would employ a fleet of taxi, we can ue the approximation: which reult in an anwer of: Γ( n), Γ n + ) n ( R E( d n ). i n 5-3
Public Tranportation Sytem: Planning Flexible Tranit For a region with where vehicle move along a rectilinear grid, the ame method can be followed with a couple of ubtitution: R x E d n i R ( ) = 0 n dx y = x R With a final reult of: R E( d n ) π 0. 63 i n R n 5-4
Public Tranportation Sytem: Management Vehicle Fleet Module 6: Management Vehicle Fleet (Originally compiled by Eric Gonzale and Joh Pilachowki, April, 008) (Lat updated 9--00) Outline Introduction Schedule Covering Bu Route o Fleet Size: Graphical and Numerical Analye o Determination of Terminu Location and Bu Run Schedule Covering N Bu Route o Single Terminu Cloe to a Depot o Multiple Termini and Deadheading Heuritic Dicuion: Effect of Deadheading Appendix: The Vehicle Routing Problem and Meta-Heuritic Solution Method The block diagram below illutrate that a tranit agency i in eence a mechanim (the middle box) for tranforming money input from both, government and uer into tranportation ervice. In thi coure we focu on the working of thi mechanim, treating the arrow pointing into the middle box a contraint and thoe pointing out a the objective function. We have jut finihed a et of (planning) module that explore the ideal tructure of thi mechanim, focuing on the long term. We are now about to tart a et of module that will explore what need to be done in the hort and medium term to execute the long term plan. Thi involve medium-term invetment and deployment deciion of the tranit agency manageable reource, which mainly conit of vehicle and peronnel. Inviible to the public, we call thee action management deciion. Thi module deal with vehicle fleet management. Module 7 will deal with peronnel management. A tranit agency alo need to make other medium-term and hort-term operational deciion that are viible to the public. Module 8 will deal with thee. Although our attention will continue to be focued in the middle box we hould not loe ight that it i only part of the whole picture. A tranit agency i alo concerned with the arrow. The iue of finance and governance (inbound arrow) and public relation and information diemination (outbound arrow) are of much importance to the ucce of a tranit operation. Thee iue, however, are not tranit-pecific and will therefore not be addreed in thee note. So let u now return to the inide of the box, with a focu on management. 6-
Public Tranportation Sytem: Management Vehicle Fleet Government $ Tranit Agency: Expert Labor Infratructure Vehicle Service Uer Introduction To thi point, in the planning part of the coure, we have aumed that agency cot (including operating cot and amortized capital cot) are proportional to the vehicle hour and vehicle kilometer of ervice provided. Thi would be exact if vehicle could be rented for only the time that they are needed for ue in ervice, and driver could be hired and fired o that people only worked the hour that bue are in operation. In reality, vehicle are purchaed or leaed for more than a few hour at a time and labor union place retriction on the number of hour that driver can work. In thi and the next module we will develop vehicle operating plan and driver taffing plan recognizing thee limitation. Homework exercie will compare thee more realitic operating cot experienced by the agency and thoe aumed in the planning tage. We will find that the aumption made during the planning part of the coure were not bad approximation. Definition Thee definition will be ued in the two management Module. Schedule et of route and cheduled ervice advertied by the tranit agency. Depot location where bue are tored without driver. 6-
Public Tranportation Sytem: Management Vehicle Fleet Run time-pace path of one pecific tranit vehicle from and returning to a depot. The vehicle need a driver during the entire run. The run may include coverage of more than one tranit line. Terminu part of a tranit line (i.e., route) where bue can be changed. Loop part of the run between conecutive viit to a route terminu; it mut be covered by the ame bu. Driver Tak indiviible part of a run that mut be covered by the ame driver to be pecified later. Job et of tak covered by one pecific worker in a ingle day. Worker Type common work pattern characterized by pay rate and propertie of their hift. Allocating vehicle and driver to provide the chedule promied by the tranit agency i a two tep proce: ) Find a fleet operating plan to cover the chedule {# of vehicle, pecific run}. To do thi, ome vehicle may have to it unued for part of the time. ) Find a taffing plan to cover the run {# of worker by type, pecific job}. Thi involve cutting the given run into tak and then allocating worker to cover the tak. Thee tep are parallel in tructure. Both anwer the quetion: how many item are required to cover a et of requirement? Thi Module i concerned with tep. Schedule covering: bu route The data for thi problem (the chedule) can be repreented in a time pace diagram howing each of the bue traveling along a route from a terminu (at x = 0) and looping back to the terminu. Each bu require a cycle time T to make a full loop of length L and return to the terminal. Then, N(t, 0) i the cumulative number of dipatche from the origin over time (alo denoted D(t)), and N(t, L) i the cumulative number of return, which i D(t T) if the cycle time i fixed. 6-3
Public Tranportation Sytem: Management Vehicle Fleet x n= n= N(t,L) # bue returned D(t T), if cycle time i fixed L # bue returned D(t) H, headway N(t,0) t T, cycle time Fleet Size: Graphical Analyi We analyze thi ytem a a queuing ytem from the perpective of the terminu imagining for the time being that the depot i on top of the terminu and that the depot upplie bue when needed. What i the minimum number of bue needed to utain the chedule? Each bu can be claified a waiting in reerve at the terminal until dipatch or circulating in ervice. The tranition between reerve and circulation are the dipatche and return. return dipatche # bue in reerve Terminu # bue in circulation Route 6-4
Public Tranportation Sytem: Management Vehicle Fleet A cumulative plot of bue available A(t), bue dipatched D(t), and returned R(t) how graphically how the number of bue in reerve and ervice evolve over time. See below. The curve D(t) i given and the other two are derived. The number of available bue i equal to thoe initially available, M, and thoe returned: A(t) = M + D(t T). Note how for thi cloed queuing ytem the um of reerve and circulating bue i the total number of vehicle, M, which remain contant over the coure of the day. Note: curve A i obtained from curve R with a vector hift (T, M). Since the number of bue in reerve ha to be poitive, we require: A(t) D(t); o the minimum fleet ize i obtained when A(t) and D(t) are tangent, a hown. Note: The tangency point i where the cumulative dipatch and cumulative return curve are maximally eparated; i.e., where the number of circulating bue, U(t) i maximum. So, we have: { U ( t)} M = max. A an exercie, you can prove that thi formula reduce in the time-independent cae to the reult we know: t T D () t = M = H H. # A(t) = # available hift vector M # in reerve D(t) = N(t, 0) R(t) = D(t T) = N(t T, 0) T M # in circulation, U(t) M T t 6-5
Public Tranportation Sytem: Management Vehicle Fleet Fleet Size: Numerical Analyi Given a cumulative dipatch curve D(t) with (tentative) fleet ize j, we ee from the picture below that if T n+j T n T for all n, then A(t) i alway to the left of D(t) for all t. The condition i a precedence condition enuring that every bu i available before it i dipatched. # n+j n T j D(t) j T T n T n+j t The minimum fleet ize can be eaily determined with a preadheet that check the precedence condition for different tentative fleet ize, j. The lowet value of j correponding to column with all value greater than or equal to 0 i the minimum fleet ize which enure that the reerve of bue i never empty. j = j = n T n T T 0 T T 0 time data time data time data M M n + n T n + n T 0 6-6
Public Tranportation Sytem: Management Vehicle Fleet Terminu Location So far, we aumed that the terminu wa at x = 0. Could it be poible to reduce the number of bue by locating the terminu in a different place along the route? The anwer i no if bu trajectorie are the ame through the day. Here i why: Let n be the travel time of bu n from the old to the new terminu (ee figure) and note: T ' n = Tn + Δ n. Now, if Δ n = Δ, then T ' T ' = T T T becaue cancel out and we can put the terminu wherever we want n + j n n+ j n without a penalty. Thi i good becaue it give u the feaibility to put the termini at favorable location (e.g. where bue are nearly empty). # T n new terminu old terminu T n t Run Determination The bu cheduling problem i to determine which bu i aociated with each loop; i.e. figuring out what each bu will do: the bu run. We dicu two heuritic method for determining the pecific run for each vehicle, but there are many more. In method we ue the lat-in-firt-out (LIFO) trategy; new bue are only introduced when abolutely neceary. Thi method tep through time, o when a loop i cheduled to tart the mot recent bu that returned i dipatched. If there i no uch bu a bu i elected from the initial pool. Thi trategy i good becaue it keep ome individual bue running while other experience long period of idling. The latter can be returned to the depot for driver relief. 6-7
Public Tranportation Sytem: Management Vehicle Fleet An alternative trategy with the ame goal i a greedy trategy that tep through bue, aigning to each bu a many loop a poible. After each bu, the loop covered by the bu are removed and the next bu again cover a many loop a poible. To do thi, each bu i redeployed a early a poible after returning to the terminu. Thi can be performed graphically by hand by plotting each cheduled loop from the route terminu againt time a hown below. Could you organize thi in a preadheet? veh t veh veh 3 veh 4 veh 5 veh veh loop idle time veh It i perhap intuitive, and can be proven, that both the LIFO and greedy method are feaible with the minimum fleet ize we calculated earlier: M = max { U ( t)} Example: Qualitative difference between LIFO and Greedy Method The imple three-run, two-bu ytem below how why the two method differ: Bu Bu LIFO Bu Bu Greedy Bu Bu 6-8
Public Tranportation Sytem: Management Vehicle Fleet The LIFO method aign the third loop to the bu returning mot recently (Bu ). The Greedy method aign it to the lowet indexed available bu (Bu ). The method only differ in the bu that i elected from the pool of idle bue for the next run. Since we are auming that all bue are identical, the choice ha no future repercuion on the availability of bue. In fact, one could have elected the bu at random, or with any other rule, and the trategy would perform imilarly. Thu, the pecific bu choice can be made with other (non-bu) criteria in mind. Schedule covering N bu route Imagine a map of many route all paing converging at a centrally located terminu. We could imagine thi i a bu tation in the center of a city or at a buy rail tranfer tation. The terminu may be cloe or far from the depot. We imagine for now that it i cloe. hared terminu depot Single Terminu Cloe to the Depot We could treat each route independently a before, but on the other hand, it may be poible to reduce the fleet ize by haring bue between route. On the left below i the model for dedicating bu fleet to eparate route in iolation. On the right, thi model i modified o that rather than a reerve of bue for each route, the terminu hold a reerve of available vehicle for all route. Each route, i, i characterized by loop of different cycle time, T(i), o the time until a dipatched bu return i no longer uniform but depend on which route the bu ha been dipatched. 6-9
Public Tranportation Sytem: Management Vehicle Fleet D (t) D (t) # bue in reerve for # bue in circulation on # bue in circulation on D (t) # bue in hared reerve D (t) # bue in reerve for # bue in circulation on # bue in circulation on The aggregated cumulative count of dipatched and returned vehicle i now expreed a R ( ) = D ( t) D t i ( t) D ( t T ) = i And ince the fleet (M) i hared, the cumulative number of bue made available for collective ue i till: A ( t) = M + R( t) Curve D(t), R(t), and A(t) can be plotted a before to determine the minimum fleet ize, and the formula M = max{d(t) R(t)} continue to hold. The only difference i that R(t) i no longer related to D(t) by a hift. i i i Dipered Termini and Deadheading Heuritic Conider now the cae where the termini and depot are dipered. Perhap the termini are at end of the line, and there may be ome cot, c kk, of moving a bu from loop k to loop k. To include deadheading from the depot, we ue k = 0 for the depot and k =,, for the loop. 6-0
Public Tranportation Sytem: Management Vehicle Fleet k c kk c 0k depot k A imple heuritic method can be ued to olve thi problem approximately. Thi method i good if c kk << T (k). Otherwie, it produce olution that may need improvement. We imagine that all bue on route k are requeted a time Δ ( k ) max{ ckk '} ahead of their real dipatch time, recognizing that they could be coming from any other terminu. If we build thi lack into the chedule, i.e. we define: k ' T ' = + Δ ( k ) T( k ) ( k ) We can treat thi new problem (with T (k) ) a previouly (ignoring deadheading). Thi i a way to obtain a tentative fleet ize and et of bu run which can be improved uing a computer. Fortunately, the problem we are olving i analogou to the vehicle routing problem (VRP); a famou problem that ha been extenively tudied. So, we don t have to do thi from cratch. (The appendix give ome background on the VRP and a imple computer method that can be ued to improve tentative olution.) The VRP i analogou to the chedule covering problem that we want to olve becaue we are looking for the leat cotly way to cover a et of requirement. The analogy i preented in the table below. The penalty can be defined by any function that map idle time between loop to a penalty. Thi may be a function that increae a the idle time wate money until ome point when the bu can be returned to the depot. 6-
Public Tranportation Sytem: Management Vehicle Fleet Vehicle Routing Problem (VRP) Schedule Covering Problem point i,j loop k, k ditance, c ij if impoible penalty, p kk 0 if ckk' = 0 > 0 if feaible, but ckk ' > 0 vehicle Bu vehicle load loop covered by a bu; i.e., the bu run capacity Dicuion: Effect of Deadheading To illutrate the potential benefit of deadheading, uppoe we have two bu line with different peaking pattern, uch a a commuter route running heavily in the morning and evening, paired with a route that i run mot heavily during the middle of the day for omething like an athletic event. The figure below diplay thee pattern by mean of two olid curve. The dotted line (not drawn to cale) i the um of thee curve. U i (t) max{u (t)+u (t)} max{u (t)} max{u (t)} 6- U (t) U (t) Compare the fleet requirement if the route were conidered eparately (the um of the maximum route requirement conidering each route individually) to the fleet requirement if the two route t
Public Tranportation Sytem: Management Vehicle Fleet hare reource even if deadheading i required (the maximum of the um of route requirement given by the dotted curve). It alway happen that: i max t { U ( t)} max { U ( t)} i So, ome aving are poible with deadheading, but thee are offet againt the cot of deadheading itelf. The greatet benefit i from route that peak at different time. t i i Appendix: Introduction to the Vehicle Routing Problem and Meta-Heuritic Solution Method. Here we decribe ome combinatorial optimization concept that are ueful for cheduling public tranportation worker and vehicle. Local Search Method and Meta-Heuritic The baic idea behind local earch method i to gue olution that get increaingly better a the procedure develop. Solution are characterized by a tate which i a tring of number. Thi can be illutrated with the TSP. Given are N point (or citie), i =,. N, and a matrix of ditance {e ij } between every pair of point. In the TSP, we look for a tour that viit all the point with the leat total ditance (or cot ). Since the poition in which the citie appear in a tour are uniquely defined by an ordering of the firt N integer (a permutation), any uch ordering i a tate of the TSP problem. Example 3 below how poible tate for a 6 point TSP problem: (,, 3, 4, 5, 6) and (, 6, 3, 4, 5, ). It i aumed in thi example that cot are given by the Euclidean ditance of the link. Thu, the cot of each tate i the length of the tour one would meaure with a ruler. Any local earch i baed on perturbation that tranform a tate into a imilar tate, hopefully with leer cot. For the TSP, a perturbation could be chooing conecutive citie and wapping their order. For example, from (,, 3, 4, 5, 6) we could go to (, 3,, 4, 5, 6,) and from thi to (, 3, 4,, 5, 6). The et of tate that can be reached in one tep (one perturbation) i the tate local neighborhood. Perturbation hould be imple (o they are eay to make and evaluate), but alo comprehenive, in the ene that they hould allow the ytem to reach any tate from any other tate. Conecutive city wap have thee two propertie and are therefore acceptable perturbation for the TSP. 6-3
Public Tranportation Sytem: Management Vehicle Fleet Given a current tate, a greedy local earch would evaluate the cot of all the tate in it neighborhood and move to the one with the leat cot if uch a tate exit; otherwie the earch end. Thi procedure i then repeated uing thi new tate a the current tate, and then repeated iteratively until the earch end becaue no improvement can be found. The termination point i called a local optimum. Local optima are generally not unique for the TSP. For example, you can verify that the two tour of Example 3 are locally optimal, even though tour (, 6, 3, 4, 5, ) on the right i quite bad. In view of thi, people have created meta-heuritic method that in theory can avoid being trapped in local optima and converge to the global optimum. The implet meta-heuritic method i called imulated annealing (SA). It differ from the greedy method in that it randomly chooe a ingle perturbation from the current tate to identify a ingle new tate. A coin i then flipped to ee whether the new tate i accepted and become the new current tate, or one tay put. The probability of ucce p i choen to be the following function of the change in cot, Δe, and the iteration number, n: p =, if Δe 0; but if Δe > 0 then p = exp{-δe/(n+a)}, where a i a poitive contant. Note that at the tart of the earch (n = ) there can be a ignificant probability of accepting a more cotly tate (with Δe > 0) but thi probability decline a the imulation progree. Thi probabilitic feature of SA allow the algorithm to jump out of local optima and, given enough time, to converge to the global optimum. Unfortunately convergence i low for problem with more than (ay) 00 point. Even in thee cae, though, the method can be ued to fine tune olution obtained with other method. A large value of a i normally choen for thi type of application. The Vehicle Routing Problem (VRP) The VRP arie in practice more often than the TSP, and many variant of it exit (e.g. with route length retriction, time-window, etc.). In it mot baic form it eek vehicle route to erve a 6-4
Public Tranportation Sytem: Management Vehicle Fleet et of N cutomer ditributed in pace. Cutomer have item to be carried, which take up vehicle pace. Vehicle have finite capacity. Given are: N point, i =, N M vehicle, m =, M A deport at i = 0 A matrix of ditance, e ij A demand d i for every point (city) (in unit of quantity ) A vehicle capacity, V m for every vehicle (alo in unit of quantity) We look for: An allocation of point to vehicle and a et of vehicle route ending and beginning at the deport that minimize either vehicle ditance, number of vehicle or a combination of the two. The VRP can alo be attacked with meta-heuritic uch a imulation annealing (SA), and thee technique till give reaonable reult for problem with up to (about) 00 point. Intead of a ingle permutation, a tate now conit of an ordered allocation of citie to vehicle. Note, ome of thee tate may be infeaible--if the total demand for vehicle m exceed V m. The SA algorithm would work a before. One define perturbation, which can be wap of point (alo called cutomer ) within a tour, or wap of group of cutomer among tour. Example how the reult of wapping the lat cutomer of the tour on the left with the middle cutomer of the tour on the right. It hould be clear that any tate whatoever can be reached from any other tate if one ue a proper equence of wap. Therefore, the SA approach with random wap hould (theoretically) work. In practice, experience with the VRP ha been good with problem a large a ~00 point. For larger problem SA can be ued a a fine-tuning tool with a large value of it parameter a. A demontration of thi approach can be found in Robute, et al. (990), which applied the SA annealing algorithm to a problem with about 00 point). 6-5
Public Tranportation Sytem: Management Vehicle Fleet A i explained in the text, many tranit problem can be cat in the form of a VRP-like problem that can be olved or fine-tuned with SA. Thi technique can be quickly matered and applied. The cae tudy in Robute et al (990) took le than week from conception to completion. More Information: The following elementary reading could be of ue. Section 0.9 of Numerical Recipe: The Art of Scientific Computing by W. Pre et al., Cambridge 987, pp. 36-334, decribe imulated annealing in the context of the Traveling Saleman Problem (TSP), and how ome computer code. A hort decription can alo be found in Appendix B of Daganzo (005), Logitic Sytem Analyi, Springer. Section 4.5. of thi reference (Fine-tuning Poibilitie) ummarize the cae tudy in Robute et al, (990). 6-6
Public Tranportation Sytem: Management--Staffing Module 7: Management Staffing (Originally compiled by Eric Gonzale and Joh Pilachowki, May, 008) (Lat updated 9-8-00) Outline Recap Staffing a Single Run o Effect of Overtime o Effect of Multiple Worker Type Staffing Multiple Run o Run-Cutting o Covering Chooing Worker-Type Dealing with Abenteeim What i Still Left to be Done Recap Recall from lat lecture the -tep proce we ued to cover a chedule: Route chedule : Bu Loop : Bu Run Cutting problem Covering problem Bu Type We tart with a route chedule and cut it into loop that can be covered by the bue. Bue, categorized by peed, capacity, etc are then aigned o that each loop i covered. Thi can be olved a a Vehicle Routing Problem (VRP). The olution conit of all the run for each bu. If bue were automated vehicle thi would be the end of the problem. However, ince they are not, we mut figure how to cover the bu run with driver. To do thi we conider the driver and the contraint they add, uing the following equence: 7-
Public Tranportation Sytem: Management--Staffing Bu Run : Driver Tak : Driver Job Cutting problem Covering problem Worker Type Notice the imilaritie between the two formulation. The bu run obtained from the fleet cheduling tep are now cut into elementary driver tak and driver, categorized by hift length and wage rate, are then aigned to feaible et of tak (job) o that each tak i covered. Thi i what the focu of thi lecture will be. Becaue of the imilaritie between the chedule covering and taffing problem we can analyze them in a imilar way: firt by conidering each run eparately and then by pooling them, allowing driver to cover multiple run. Staffing a ingle run The reult of the chedule-covering analyi from lat lecture i a erie of run, r, each with a beginning time, B r, an ending time, E r, and a duration S r. We alo take a given the (continuou) work interval for worker of type i, w i. Normally, w i < S r for many run. B r E r S r w i We alo aume that we know the wage rate of driver of type i, a well a the premium added to their wage rate for working overtime. To implify the formula we ue monetary unit o that the wage i. In thoe unit, we define the overtime premium a π, and the overtime wage a (+π). Conider now the extra cot of wage over the lower bound (LB) obtained by auming that π = 0 (i.e. that driver can be hired and paid only when needed. We will examine how thi extra cot depend on the type of hift that are ued - auming that the tranit agency ha the flexibility 7-
Public Tranportation Sytem: Management--Staffing to ak worker to tart their hift at any time. Firt we look at a ingle run of length S r and et w = 8 hr and π = (no overtime offered). Then the extra cot over the LB, $ w, i decribed by the following curve, which i our bae cae; ee figure below. $ w extra cot over LB 8 - - 8 6 S r exact point with zero wate You can ee that the larget cot, w, i paid for run with length lightly longer than a multiple of w. Over multiple run of random duration, the average wated cot hould be about w/. We wate about one half of a hift per run. Effect of Overtime By allowing for overtime (and hiring hort-term driver at the overtime rate), a new extra cot curve with lope π become poible, ee figure below. The leat cot i then the minimum of the two curve. One would ue overtime only when the overtime curve i beneath the regular curve (i.e., where the latter i dotted.) Simple algebra reveal that the maximum cot i now: w π. Thu, the exce (wated) cot + π w π per run hould be about on average. Note how overtime reduce wate. + π 7-3
Public Tranportation Sytem: Management--Staffing $ w extra cot over LB overtime curve - - π w + π π π w S r Effect of Multiple Worker Type If we introduce more worker type to the ytem we can further reduce our cot. For example, by offering a horter hift, w = 3 hr, we can find exact point of zero extra cot for run with the following length: 3, 6 = (3 + 3), 8, 9 = (3 + 3 + 3), = (8 + 3), = (3 x 4), 4 = (8 + 3 + 3), 5 = (3 x 5) (continuing for all integer value greater than 5). You can ee from the figure below that the reulting extra hould be about 0.5 without overtime and about 0.5 if overtime with π i allowed. If we know the cumulative ditribution, F(x), of all run, the extra expected cot acro all run can be expreed more preciely a: E ($ ) = $ ( x) df( x). w 4 0 w 7-4
Public Tranportation Sytem: Management--Staffing $ w extra cot over LB π - w = 3 w = 8 S r Staffing Multiple Run A in the cae of the fleet cheduling problem, the ituation can till be improved by pooling: conidering all run together and allowing driver to erve more than one run. Run-Cutting To do thi, we firt cut the run into elementary tak, p, that can be covered by different driver - although each tak mut be done by the ame driver. Thee tak hould be a hort a poible, recognizing the practical contraint that apply to the agency. (Perhap, for example, driver can only be witched at certain point on the route.) The reult of the cutting proce can be expreed graphically a in the fleet cheduling problem: time, t tak, p t p 7-5
Public Tranportation Sytem: Management--Staffing Covering Data for the problem would include the tak duration, t p, the time, t pp, between the end of tak p and the beginning of p, and the real cot of moving a driver from tak p to tak p, c pp (thi cot i et to whenever the move i infeaible; e.g., for all move where t pp < 0). Thi covering problem can alo be formulated a a VRP, albeit a variant with different contraint. The analogy from the previou lecture i continued below: Vehicle Routing Schedule Covering Staffing Problem Problem (VRP) Problem depot depot worker home (or depot) point i,j loop k, k tak p, p vehicle bu worker vehicle load loop covered by a bu; i.e., the bu run tak covered by a worker; i.e., a job capacity time for a top, t i n/a t p time between top t i, j n/a t pp ditance for a leg, c i,j p kk c pp' time contraint, T n/a w i Depite the complication, the SA method can till be ued. We would ue the ame tate a in the previou lecture (I.e., ordered tring of number, p, for each worker), and would treat run independently. We could alo ue the ame et of perturbation. Only the cot evaluation tep would be lightly different ince violation of the time contraint would imply an infinite cot. The SA approach can be ued even if worker type are characterized by pecific beginning and ending time of their hift, and not only by their hift duration. Simplified etimation of cot We can ue a graphical method in order to obtain a quick etimate for a LB of total cot. To do thi we aume that worker can tart their hift at any time and can be reallocated acro run without a time penalty. Thi allow u to focu on the number of run, ignoring pecific run and where they take place in pace. By graphing the number of active run over time, the problem 7-6
Public Tranportation Sytem: Management--Staffing imply become one of covering the area under the graph with trip repreenting a work hift, with a height of one bu and a length, w. # active run w D D w w w cover by reallocating ome of the idle time t In the above figure, D i the number of active run during the morning peak, and D i the number of active run during the afternoon peak. The leat poible number of trip have been ued to cover the area under the graph. The darker part of the curve mark the beginning of the trip. The haded portion of the graph repreent the wated time uch that a driver i employed without having a bu to drive. The mall portion at the end of the day can be covered by idle worker from the afternoon peak. Note: The two peak are eparated by the length of a workday which of coure i cloe to w. Thi preent a problem, ince driver who tart their hift at the beginning of the peak mut be idle mot of their hift. From the picture you can ee that there will be (D + D )w driver-hour hired. By allowing ome horter hift of duration (w/ = 4) in addition to the regular hift (w = 8) we could ue the following olution: 7-7
Public Tranportation Sytem: Management--Staffing # active run D D w/ w/ w w cover by reallocating ome of the idle time t Thi reult in no driver being idle for more than 4 hour even though mot of the hift are of regular length. Thi i very intereting: a large cot reduction i achieved by hiring jut a few 4- hr worker. If we continue thi idea by allowing horter hift and overtime, the amount of wate can be reduced even more. The homework illutrate thi effect. Chooing Worker Type We have hown that it i beneficial to have horter hift; however there i the quetion of how to induce worker to chooe thee hift. There i the poibility of paying higher wage, but we can poibly provide an incentive other than money. You could offer a hift chedule uch that a driver can work 9 hour a day for four day, and then have a 4-hour hift on their fifth day. By partitioning 4-hour hift for each weekday over all the worker, each day would have a 4-hour hift for every four 9-hour hift. The normal and 9/4 rotation are hown in the table below: 7-8
Public Tranportation Sytem: Management--Staffing M T W Th F rot 8 8 8 8 8 rot. 9 9 9 9 4 rot. 9 9 9 4 9 rot.3 9 9 4 9 9 rot.4 9 4 9 9 9 rot.5 4 9 9 9 9 You could then allow driver to chooe their preferred rotation, in order of eniority, with the tandard 8-hour hift being the default. Thi trategy ha not been put into practice regarding tranit taffing, however it exit in other field. Reearch ugget that the idea could have merit for tranit ytem. Dealing with Abenteeim The above analyi aume that driver how up for work reliably. It doe not take into account ick leave, vacation time, or abenteeim. Given m, the number of job needed, we aume a probability, f, that people will how up to work. We will alo aume that abentee are paid. In the bet-cae cenario, with the ame number of abentee every day, we would need to hire n = m /f driver; e.g., if m = 60 and f = 0.9, we would need 60 / 0.9 = 67 driver. Coleman, R. M. (995) The 4 hr buine, AMACOM, N.Y. Muñoz, J. C. (00) Driver hift deign for ingle-hub tranit ytem under uncertainty PhD thei, Dept. of CEE, U. C. Berkeley, CA. 7-9
Public Tranportation Sytem: Management--Staffing If we leave n a a deciion variable but the number of people who how up to work, N, i random; and if the on-call worker who come in lat minute to cover a hift, are paid at a higher rate ($ 0 >$), then the expected cot would be: E(cot) = $n + $ 0 E(m-N) + Here, N i a binomial random variable B(n, f). For large n, N can be approximated a a normal random variable with mean (m-nf) and variance nf(-f). Note: The election of n preent a tradeoff imilar to the well-known newboy problem in which a newboy maximize hi expected profit by buying jut enough newpaper to balance the rik of either running out or having ome unold inventory at the end of the day. The formula for the expected cot i: E(cot) = $n + $ 0 Φ x m + nf nf ( f 0 ) dx where n i a deciion variable, m and f are data, and x i a dummy argument. Here we have ued the well known reult for the mean of a random variable in term of it cdf: E(x) = 0 F X ( x) dx + ( F ( x)) dx ; ee Figure. 0 X There exit a cloed form olution to expected cot integral above in term of Φ and φ, exploiting the fact that Φ ( x) dx = φ( x) + xφ( x). However ince the cot can be found numerically, the formula i omitted. c.d.f., F(x) + (x) f X E(x) = difference in haded area - 7-0
Public Tranportation Sytem: Management--Staffing Since for our application the area to the left of the axi in the above figure i mall, we will take the mean to be the area to the right of the axi. For the cae given at the beginning of thi ection with m = 60, f = 0.9, and a cot ratio of on-call driver $/$ 0 = /3, the optimal olution would be n = 64. Thi would give an expected cot of 67. driver, which i only marginally higher than our bet-cae cenario! Of coure, wore reult would be obtained with maller pool of driver. So, having a flexible workforce that can do many tak i better than having many mall pool of pecialized worker. The formula we have given quantifie thee effect. What i till Left to be Done Beide vehicle fleet and peronnel, a tranit agency alo need to manage other medium- and hort-term problem. Some are quite viible to the public and may have to be handled adaptively in real time. We call them operational deciion. They will be examined in the lat Module, albeit not exhautively. They include: Real-Time Control of Vehicle Schedule and Repone to Service Diruption (Module 8) Throughout thi coure we have dealt mainly with the aumption that bue run on chedule. In reality there are many poible diruption that can prevent thi from happening. Thee include vehicle breakdown, delay caued by ignal and congetion, and paenger with pecial need, all of which can caue bue to divert from chedule. We need to know how bet to recover from thee diruption. For example, uing GPS and communication technology it i poible to introduce real-time control which can minimize the effect of thee diruption. Interaction between Tranit and Other Mode (Module 9 in the work) Becaue many form of tranit exit within the traffic tream, there are many interaction between tranit and other mode. We need to learn how better to manage all mode together. We hould learn when and how to egregate different mode on the urface treet o the available treet pace i better ued. Special event (Module 9 in the work) Tranit can be ueful for pecial event uch a the Olympic or emergency evacuation and we need to know how to plan ahead for thee. 7-
Public Tranportation Sytem: Reliable Tranit Operation Module 8: Reliable Tranit Operation (Originally compiled by Vikah V. Gayah, April, 00) (Lat updated, 9--00) Outline Reliability Sytem of Sytem (book on claical dynamic and non-linear ocillation) Uncontrolled Bu Motion (reference, ) Conventional Schedule Control (reference 3, 7) Dynamic/Adaptive Control (reference 4-8) Lit of Reference Reliability Reliability in a tranit network refer to conitency in vehicle headway, arrival time, and chedule. When tranit uer are aked about the mot important iue relating to tranit, the number one repone i the reliability of the ytem. Therefore, it i important for agencie to deign ytem that have conitent headway and vehicle arrival time. A will be hown here, mot tranit ytem are inherently unreliable vehicle tend to bunch or pair, creating gap in ervice. For an animation that explain why, ee: http://www.ce.berkeley.edu/~daganzo/simulation/bu_bunching.html We will learn in thi module how to overcome thi difficulty. Sytem of Sytem Tranit ytem can be analyzed a a ytem of ytem (SoS). A ytem of ytem i a group of interconnected ytem (known a agent) that interact with decentralized agent-pecific rule. Our goal will be to undertand the macrocopic behavior of the SoS baed on the individual rule governing the agent. The rule governing the behavior of a particular agent ( ) depend on the current (and pat) tate,, of the particular agent, outide factor (repreenting the world) and the tate of other agent with which the agent interact. The figure below graphically diplay the generic tructure of a -agent ytem: Agent i on the left and agent on the right; arrow denote the input and output of each agent rule. A SoS i characterized by the mathematical function embodying thee rule (called the dynamic equation of the ytem). 8
Public Tranportation Sytem: Reliable Tranit Operation Although the number of poible interaction in thee ytem increae quadratically with the number of agent, in tranportation application we typically encounter ytem in which each agent only interact with a limited number of agent. In thi cae, the number of interaction i comparable to the number of agent. Can you think of example of SoS in the tranportation field and what the agent and rule would be for them? E.g., car in traffic, airplane nearing an airport, etc. Since SoS are decentralized, we need to undertand their behavior over time. An important quetion to ak about uch ytem i: if the world i fixed at a teady tate, doe the ytem have an equilibrium tate which i invariant in time? And if o, i thi equilibrium unique? And i it table? Stability mean that the ytem tend to the equilibrium tate when the overall tate,, i cloe to it. Thee quetion can uually be anwered in three tep: ) determine the dynamic equation for the ytem; ) determine if one or more equilibrium tate exit, and find them; and 3) determine which equilibria are table. To get ome inight into the meaning of tability and thi type of analyi, ome example of SOS are now preented. Example A table ingle-agent SoS: A parking lot with a fixed demand of vehicle entering the lot (λ = 000 vehicle per time period) where 0% of the vehicle in the lot at the beginning of each time period leave by the end of the period. Here, the agent i the parking lot, the world i the entity upplying the demand and the tate of the ytem i the number of vehicle in the parking lot at the beginning of any time period,. Step : the dynamic equation decribe how change. It can be written uing the given demand and upply rule. Note: the number of vehicle parked in the lot at the beginning of time period i imply equal to the number of vehicle in the lot at the beginning of time period, plu the number that enter during the time period, minu the number that leave. Therefore,,000 0.. () Thi i the dynamic equation. 8
Public Tranportation Sytem: Reliable Tranit Operation Now on to Step : at equilibrium, the number of vehicle in the lot will not change with time. Therefore, the equilibrium olution can be found by etting.,000 0. () Solving thi for, we find 0,000. Thi olution i unique. So we are now done with tep and now check for tability. Step 3: to determine if the equilibrium i table, we need to examine what happen when 0,000 but 0,000, i.e.,. The equilibrium will be table if 0,000. To graphically ee what happen, plot the tate of the ytem at time a a function of the tate of the ytem at time. Thi i the dark line below, given by (): Imagine now that the ytem i perturbed by a value from the equilibrium tate a in the figure. Uing the dynamic equation along with the dotted line, we can ee how the ytem evolve through time. A moment of thought reveal that it follow the grey arrow in the figure above. Clearly, the ytem move back to the equilibrium tate. Thi i alo true if 0. (Check it for yourelf.) Therefore, we ay that thi equilibrium i table ince the ytem return to after any minor initial perturbation,. Stability can alo be determined analytically by performing the ame tep algebraically. To do thi, define the reidual perturbation after tep:. The dynamic equation can then be rewritten in term of by ubtracting () from (). The reulting equation i: 0.9. (3) 8 3
Public Tranportation Sytem: Reliable Tranit Operation The reaon for doing thi i that we remove the independent contant from () and then the reulting equation become homogeneou and eaier to analyze. Note, 0.9 0. Thu, it i now clear that the perturbation decreae with time and tend to zero no matter the value of 0. Therefore, any perturbation will be reduced in ubequent time tep and the ytem will move back toward the equilibrium. Example an untable ingle-agent SoS: The econd example i a queuing ytem where the cutomer ervice time increae with queue length. (Thi could happen, for example, if the length of the queue reduced the erver efficiency.) In our example, cutomer arrive at a rate of,000 per time tep. The erver can proce,000 0. cutomer in each time tep, where i the number in queue at the beginning of time tep. For thi SoS, the erver i the unique agent, the world upplie the demand and the tate i. Step : noting that the number of cutomer in the ytem cannot be negative, we ee that the dynamic equation i imply:,000,000 0..,000. (4) Step : Now, replace and with and olve thi equation. The ytem i found to have two equilibria: 0and 0,000. Step 3: the tability of all the equilibria hould now be checked. Start with 0,000. A a firt ub-tep we rewrite the dynamic equation a a function of the reidual perturbation from equilibrium a we did for (3). To do thi (4) mut be linearized ; i.e. we mut remove the truncation and eliminate the independent contant. The truncation ha no effect cloe to 0,000 ince it argu ment i about 0,000. Thu, it i omitted. To eliminate the contant, we repeat the ame tep of example (thi i alwa y done); i.e., we rewrite (4) for the equilibrium olution of interet:. 000. 000 (5) and ubtract from (4). The reult, in term of i:.. (6) The econd ub-tep i analyzing (6). In thi cae,. 0. Thu, it i clear that any perturbation will continue to grow in ize with time, and the ytem will move further and further away from the equilibrium tate. Therefore, thi equilibrium i untable. Thi i alo confirmed with the graphical contruction of the tate of the ytem. A een in the graphical contruction below, the equilibrium tate 0,000 i untable. Minor perturbation move the ytem away from the equilibrium. 8 4
Public Tranportation Sytem: Reliable Tranit Operation Thi whole analyi would have to be repeated for 0, but we don t do it here. The figure clearly how that the ytem move toward the equilibrium 0 when cloe to it, o thi particular equilibrium i table. Example 3 agent: Thi i an example with multiple () agent. The ytem being tudied i a ytem of two queue in erie a hown below where the demand λ i a contant and the ervice rate depend on the queue length a follow: γ and γ where, 0. The term indicate that work proceing reource are contantly being moved from the mall pile to the large pile, preumably to balance them. In thi cae, the queue are the agent, the world upplie the demand and the tate are the queue length, and. Since we have agent, the graphical olution we have given cannot be ued. Therefore, we analyze the ytem algebraically. The method ued can be applied to any number of agent. For our example, we normalize the unit o that and aume that in thi ytem of unit γ λ. Step : the dynamic equation of the ytem become: λ λ λ λ, if 0, (7a) λ λ, if 0. (7b) 8 5
Public Tranportation Sytem: Reliable Tranit Operation Step : from thee equation, an equilibrium olution of the ytem with,,, 0 i:,, λ. Step 3, ub-tep : the dynamic equation are now rewritten a a function of the perturbation from equilibrium. The equilibrium verion of (7) i:, λ, λ, λ, (8a), λ, λ,. ( 8b) Now, letting and ubtracting (8) from (7) we find:, λ λ, (9a) λ λ. (9b) We are done with ub-tep and mut now check for tability. Note, we cannot draw a picture of (9) and it i not immediately obviou what happen to if the equation are iterated. Fortunately, linear algebra come to the recue! Equation (9) can be rewritten in matrix form a: λ λ where, (0) λ λ and in thi form, the equation i very imilar to (3) and (6). In thi cae 0. Thu, a in the previou cae, if 0 0 a, then the equilibrium i table. Thi 0 0 condition can be checked by analyzing the eigenvalue of the matrix. If the abolute value of all the eigenvalue are le than then, a you may recall from linear algebra, and perturbation will hrink with time. Thu, the equilibrium will be table. If any of the eigenvalue are greater than one, however, the perturbation will grow with time and the equilibrium will be untable. A an exercie, check what happen for λ. You will find that the ytem i untable. Alo, ee if you can determine for which value of λ the ytem i table. (If you can anwer thi econd quetion, you have reached a very good undertanding of thi method.) Uncontrolled Bu Motion We now apply thee idea to tudy an uncontrolled bu ytem, a hown below. In thi ytem, the bu travel between point (,-,,+, ) known a control point. Much of what follow i baed on [7] 8 6
Public Tranportation Sytem: Reliable Tranit Operation The ideal motion can be defined by the bu chedule in term of arrival time at ucceeding control point. Thi can be written in the following form:,,,, = 0,, () where i the bu number, i the control point, i the target headway, and i the travel time from control point to, including top. The firt term on the RHS of () i the time at which the firt bu arrive at the control point at the origin. The econd term i the time eparation between the firt and the bue at the origin. The lat term i the bu travel time from the origin to. We treat the bue,, a agent, the control point,, a time and the actual arrival time, which we denote, a the tate of the agent. We want to ee if the tay cloe to the, a the bue proceed forward (with ). We hall ue the notation, intead of for conitency with [7]. We are now ready to tart the analyi. Equation () define the equilibrium condition of the ytem. However, the dynamic equation till need to be derived. A a preliminary tep, note that the uncontrolled travel time for bu between top and,,, hould obey:,,,, () where,,, i the headway ahead of bu at control point, i the target travel time including top at equilibrium and i an experimentally determined contant (typically between 0.0-0. if the ditance between top i km). Thi contant capture the 8 7
Public Tranportation Sytem: Reliable Tranit Operation extra time that boarding and alighting paenger add to the bu trip. Since,,,,, where, i a random noie due to traffic or the type and number of paenger arriving at, it follow that the actual arrival time for bu at poin t i:,,,,,,, 0,,, (3) Thee are our dynamic equation. Note, (3) include our exogenou noie term contributed by the world. So now we proceed with the linearization tep. A before, we ubtract () from (3) to get the DE in term of perturbation away from the equilibrium,,,,. The reult after a little algebra i:,,,,,,,,,, 0,,, (4) Conider now the tability of bu, treating, and, a exogenou input. We are hoping that if thee input are turned off, then, 0 a. We ee from (4) that, and, atify:,,, ( 5) which i untable ince. Therefore, uncontrolled bu ytem are inherently untable! When one bu get behind, even jut a little bit, the bu will tend to get further and further behind until it become paired with the bu behind it. The oppoite happen if the bu run ahead of chedule. Let u now ee what can be done. Conventional Schedule Control Thi ection will examine a typically ued method to reduce the bu pairing phenomenon: conventional chedule control. In t hi type of control, lack i added to the bu chedule at predetermined control point along the bu route. Bue are held at thee control point if they arrive early to get them back on chedule. Nothing i done to bue that arrive late, but the ytem i deigned o that thi i a rare event. There i a tradeoff in applying chedule control, however. The lack added to the chedule reduce the commercial peed of the bue, increaing the in-vehicle travel time that paenger experience to their detination. So let u examine thi tradeoff. To recognize explicitly that the addition of lack change the travel time between control point, let u write the cheduled travel time a, where i the amount of lack. The value of hould be elected to be greater than typical diturbance that arie in the bu movement. For example, if, ~ 0, then can be et to 4 o that the bu will be able to arrive on chedule over 99% of the time. Since the lack ha been added, the equilibrium bu travel time are given by () with. They atify: 8 8
Public Tranportation Sytem: Reliable Tranit Operation,,,, 0,,, (6) We aume that bue are allowed to run free between control point but are held immediately before the control point o they will not pa through them ahead of chedule. With thi control trategy the dynamic equation are:, max,,,,,,, 0,,, (7) We now expre (7) in term of deviation from t he chedule a we did in the derivation of (4). To do thi, ubtract (6) from (7). The reult i:, max 0,,,,,, 0,,, (8) Now remember that, i related to the deviation by:,,,,,,, 0,,, (9) and ubtitute thi expreion into (8). The final reult i:,,,,., 0,,, (0) Thi expreion i very imilar to (4) if we treat not jut, but alo, a input from the world. Like (4), equation (0) have a table equilibrium at, 0. In fact, even if we allow for mall perturbation in, and, the term equal zero if, i mall. Thi mean that bu return to chedule immediately. Thi i good. However, like (4) equation (0) alo have an untable equilibrium,, / 0. Thi intability mean that if a bu i late arriving by more than /, then it cannot recover and it forever loe time. Thi reult how that conventional chedule control i not reilient to large perturbation, uch a thoe caued by bu breakdown. It explain why it i o difficult for tranit agencie to keep bue on chedule, depite the agencie bet effect, and why improved method are neceary. Before we look at thee method, however, let u now examine how and the length of the control interval hould be choen for chedule control. Optimizing the Slack In implementing conventional chedule control, a pertinent quetion become: how far apart (in number of top) hould the control point be placed? We now aume that the control point are paced top apart where i a deciion variable. The travel time between control point i now written a 4 to tre the dependence on. Recall i the variation in travel time between control point (not between top). If all the top are imilar we expect, where i the time per top. Now, let be the variance of the travel time noie between ucceive top. If the noie wa independent acro top and all the top were imilar, we would alo expect and If the control point i a top, the lack can be introduced at the top itelf while the bu door are open. 8 9
Public Tranportation Sytem: Reliable Tranit Operation. Thi independence aumption break down for large (a bue would begin to pair), but imulation in reference [7] how that it i good a long a 0.5/. We now ue thee formulae to evaluate the bu average inter-top travel time, /, a a function of. The reult i: / 4 / 4 /, if 0.5/. () If the contraint i violated, will be greater. Note, i the inter-top travel time paenger experience in the bu. Uer alo experience waiting time for the bu, and thi need to be accounted for. Let be the variance of the bu arrival at top after a control point. Since a headway involve two bue, the headway vary at that top with variance. Then, the average waiting time for random arrival at thi bu top,, can be written a: / /. () To get the average wait acro all top,, average the above formulae acro all integer contained in the interval [0, ]. The reult i: / / /, (3) where / /. If a paenger ride for top, the average travel time including riding and waiting i: 4 / /, if 0.5/ (4) Although thi can be optimized for, we alo wih to prevent the bue from catching up with each other jut by chance. To enure that thi i extremely rare, we ue the contraint: 4, which in term of become 6 ; i.e., / /6. To reduce one degree of freedom in the form of the mathematical program reulting from the combination of thi contraint and (4), let /. The reult i: min 8 /.. / 6, 0.5/ The olution of thi mathematical program give the optimal length of the control egment. You / can verify that the uncont rained olution i:.5 /. Thu, the actual olution recognizing the contraint i mid, min / 6, 0.5/,.5 / /. For mall and 0. (an intermediate value), the ratio / i: 0.5 for 3,. for 7 and 3. for 0. So it look like travel time ha to be increaed by or headway for reaonable value of if we want to achieve regularity. Thi indicate that conventional chedule control achieve regularity with a large travel time premium. And we aw earlier that it i not reilient to large diturbance. Therefore, a better control cheme would be appealing. The following are ome idea in thi repect. 8 0
Public Tranportation Sytem: Reliable Tranit Operation Dynamic (Adaptive) Control Schedule control i rigid. Bue are obliviou to what other bue are doing. But recent advance in GPS and communication technology have allowed the poibility of information-haring and adaptive control cheme. Perhap thi can improve performance. There are two approache that can be ued: Approach (e.g., reference [6]) optimize the holding time and total travel time baed on the current tate of the ytem and it expected evolution. Thi i like a DP with recoure approach. Lot of literature on thi topic but it i heuritic becaue nobody know the recoure function and a ction depend on aumption about demand. Proof have not been given that thi approach prevent bu bunching. Approach Control theory approach. Doe not optimize travel time; intead it focue on guaranteeing tandard of headway variance and commercial peed. Proof can be given howing that it prevent bu bunching and the tandard are met. Some example are: Forward looking headway control (reference [7]): it i adaptive o it will not ak a bu to low down unnecearily if the bu ahead i alo ahead of chedule. Reult in higher commercial peed but till uceptible to the ecape problem (low reiliency). Two-way looking pacing control (reference [8]): bue repond to their front and back pacing; bue can cooperate by lowing down to help following bue. Remedie ecape problem. More difficult analyi. Forward looking method Recall from the analyi of (4) that the motion of bu number i untable (perturbation grow), if. Clearly, if,0 thi problem would be eliminated. But in car- following theory, thi doe not guarantee that perturbation could not grow acro bue. To etablih thi reult we would have to expre (4) in matrix form and then look for eigenvalue, a we did with (0). Thi i difficult to do in thi cae becaue i unbounded (o the dimenion of our matrix i infinite). Intead of dealing with eigenvalue, we ue a more pecialized reult that applie to equation of the form (5) given below: The reult ay that if and (5) i iterated with increaing, then the maximum error acro all bue at any control point,,, i bounded and cannot increae with. Proof:,,,,, 0,,, (5),,,,,, (6) 8
Public Tranportation Sytem: Reliable Tranit Operation where i the wort bu at location and the lat inequality hold due to the triangle inequality. Clearly, the lat member of (6) cannot exceed,,, and ince it follow that,,,. In view of thi reult, we ee that (4) would be well behaved if there wa no noie and we could find a way of chooing,0 becaue then we would have 0, 0, for 0,,,3,. Therefore, the wort deviation in the noiele ytem would not grow! Reference [7] how that how thi analyi i extended if there i noie; i.e., if the equation i of the form:,,,,. (7) In thi cae, reference [7] how that the variance of the perturbation from the chedule grow but doe o very, very lowly. And more importantly, the variance of the headway i uniformly bounded by thi imple formula: (8) So armed with thi knowledge, we need to e e if can actually be changed to a value in,0. To do thi, we introduce lack into the travel time of the bue o bue can be accelerated when their headway (and, therefore, their workload) are high and vice vera. Propoe:,,, where:,, (9) i the extra delay added. Equation (9) i truncated becaue bue cannot take le time than,. Thi control law can be diplayed graphically a in the figure below. When (9) i truncated, bue run uncontrolled. We aume we chooe a large enough d to enure that thi rarely happen. Thu, we now write the dynamic equation auming no truncation. Note that,,,,,,,,. Now ue (9) and (9) to write:,,,,,, 0,,, (30),,,, Since,, at equilibrium, we ubtract thi from the above and get:,,,,,, 0,,, (3) Note thi matche (7). Thu, the bound (8) applie to control law (9). 8
Public Tranportation Sytem: Reliable Tranit Operation Recall from the figure above that the maximum feaible deviation one hould allow for the control to be in force i. Since the headway tandard deviation i /, we hould et 3 / to aure that the ytem remain in the controllable regime mot of the time. Therefore, et. Thi i the expected holding time; i.e., the in-vehicle delay rider experience for every control egment. Now the average waiting time will be / and the average riding time will be. With thee function, the average travel time can be minimized with repect to. It i hown in [7] that the added holding delay can be cut by a factor of 3 in a typical cae. So thi method i promiing. 8 3
Public Tranportation Sytem: Reliable Tranit Operation Two-way looking (cooperation) Reference [8] contain an analyi. It introduce a pacing-baed, two-way looking, linear control law (a if bue were attached to each other through pring) imilar to the model in the homework with pace a the tate, and time a the parameter. The following are ome key point: The coefficient of (5) are a Bernoulli pdf and the denominator of (8) i the variance of aid pdf. Thi reult i alo true for verion of (5) with more term and pdf coefficient. Phyic: a control law that look forward and backward lead to a dynamic equation like (7) but with 3 term. The coefficient form a pdf with larger variance better control under mall diturbance, It can alo introduce cooperation no critical gap and no ecape problem. It can reduce lightly relative to headway-baed law but provide continuou monitoring with GPS; therefore, it recognize large problem ooner. Lit of Reference [] Newell, G.F. and Pott, R. B. (964) Maintaining a bu chedule. Proc. nd Autralian Road Reearch Board, Vol., pp. 388-393. [] Newell, G.F. (977) Untable Brownian motion of a bu trip. Statitical Mechanic and Statitical Method in Theory and Application (ed. U. Landman). Plenum Pre, 645-667. [3] Daganzo, C.F. (997) Schedule intability and control. In: Daganzo, C.F. Fundamental of Tranportation and Tranportation Operation, Elevier, New York, N.Y. pp. 304-309. [4] Barnett, A. (974) On controlling randomne in tranit operation. Tranportation Science 8(), 0-6. [5] Newell, G.F. Control of pairing of vehicle on a public tranportation route, two vehicle, one control point. Tranportation Science 8(3), 48-64. [6] Eberlein, X.J., Wilon, N.H.M. and Berntein, D. (00) The holding problem with real-time information available. Tranportation Science 35(), -8. [7] Daganzo, C.F. (009) A headway-baed approach to eliminate bu bunching. Tranportation Reearch Part B 43(0), 93-9. [8] Pilachowki, J. and Daganzo, C. F. (00) Reducing bu bunching with bu-to-bu cooperation. Tranportation Reearch Part B (ubmitted) 8 4