optimization of transit route network, vehicle headways

7/29/2019 Optimization of Transit Route Network, Vehicle Headways

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O.R. Applications

Optimization of transit route network, vehicle headwaysand timetables for large-scale transit networks

Fang Zhao a,*, Xiaogang Zeng b

a Department of Civil and Environmental Engineering, Florida International University, Miami, FL 33199, USAb EMS Consultants, P.O. Box 1898, Miami, FL 33156, USA

Received 13 September 2005; accepted 15 February 2007Available online 4 March 2007

Abstract

This paper presents a metaheuristic method for optimizing transit networks, including route network design, vehicleheadway, and timetable assignment. Given information on transit demand, the street network of the transit service area,and total fleet size, the goal is to identify a transit network that minimizes a passenger cost function. Transit network opti-mization is a complex combinatorial problem due to huge search spaces of route network, vehicle headways, and timeta-bles. The methodology described in this paper includes a representation of transit network variable search spaces (routenetwork, headway, and timetable); a user cost function based on passenger random arrival times, route network, vehicleheadways, and timetables; and a metaheuristic search scheme that combines simulated annealing, tabu, and greedy search

methods. This methodology has been tested with problems reported in the existing literature, and applied to a large-scalerealistic network optimization problem. The results show that the methodology is capable of producing improved solutionsto large-scale transit network design problems in reasonable amounts of time and computing resources. 2007 Elsevier B.V. All rights reserved.

Keywords: Combinatorial optimization; Transit network; Metaheuristics; Routing/timetable

1. Introduction

In large urban areas, there has been a growing recognition that public transportation is an important partof solutions to traffic congestion. For a public transit system to be a viable alternative to travelers, it must beable to provide its users with reasonable travel time and convenience. Travel time and convenience are largelydetermined by the types of services provided and available budget. However, given a budget, the configurationof a transit network (TN), which includes the network layout, service frequencies, and schedules, may alsohave significant effects on transit service level and quality. The proper design of these network componentscontributes to both reducing user cost and providing high level of transit services, with a positive impacton the ridership.

0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2007.02.005

* Corresponding author. Tel.: +1 305 348 3821; fax: +1 305 348 2802.E-mail address: [email protected] (F. Zhao).

Available online at www.sciencedirect.com

European Journal of Operational Research 186 (2008) 841855

www.elsevier.com/locate/ejor
mailto:[email protected]:[email protected]


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To achieve an optimal design of a transit system, researchers have in the last several decades proposedmany TN design and optimization approaches. Ceder and Wilson (1986) outlined a conceptual five-leveldesign model for a complete TN planning process that includes, for instance, route network design, fre-quency/headway setting, and timetable development. Wu and Murray (2005) proposed an exact approachto solve multiple-route TN optimization problems. It was applied to a dense two-route TN system, and good

improvements over the existing system were reported. Recently, Matisziw et al. (2006) described an exactmathematical approach to find the optimal route extensions of an existing TN. A practical TN problem oflarge-scale was solved using the approach and significant improvements were obtained over the existing sys-tem. The success of mathematical optimization methods, however, has been limited in dealing with large-scale,realistic TN optimization problems. Baaj and Mahmassani (1991) observed that mathematical approaches toTN optimization have a number of difficulties, such as non-linearity, multi-objectives, and combinatorialintractability due to the discrete nature of TN systems. Similar observations were also made by Gao et al.(2002) and Fan and Machemehl (2004).

The majority of the previous solution methods to TN optimization problems has either relied on problem-specific heuristics or design guidelines, or has been limited to idealized or coarse networks for small or med-ium-sized urban areas. Mandl (1979) presented a heuristic algorithm to compare and select better route setsbased on reductions in average costs. An artificial intelligence based heuristic approach was developed by Baaj

and Mahmassani (1991), which involved the evaluation of objective functions and feasibility constraints atvarious solution search stages. Their approach was extended by Shih and Mahmassani (1994) with the conceptof transfer center. GIS was employed in TN design to improve transit coverage for heavy traffic locations(Ramirez and Seneviratne, 1996). Gao et al. (2002) presented a bi-level programming model for TN designproblems that included the interactions between the demand and supply, which was illustrated through a smallexample. Zhao (2006) described a metaheuristic search method for optimizing both transit network layout andheadways based on a user cost function. This method is, however, a local search scheme, since the algorithmused in the headway search is a greedy-type scheme that cannot escape from a local optimum. Zhao and Zeng(2006) proposed a global search method that combined GA and simulated annealing (SA) to improve themethod described in Zhao (2006). The GA was used to identify promising search locations whenever a localsearch process slows down or stalls. In both Zhao and Zeng (2006) and Zhao (2006), a static, daily passenger

trip demand is assumed, and schedule design is not considered. Reviews of various earlier works in the TNoptimization field may be found in Fan and Machemehl (2004).

This paper describes a metaheuristic algorithm for optimizing TN, including the design of route network,vehicle headway, and timetable assignment based on a daily, time-period based transit demand. This researchis motivated by the following facts. First, with the rapid advancement of computing technologies, the prospectof solving TN optimization problems of realistic sizes with mathematical approaches that have relative com-plete solution search spaces has becoming increasingly promising. Second, although various solution methodshave been reported in the existing TN literature, there seems to be a lack of effective and systematic optimi-zation procedures for the simultaneous design of route network, vehicle frequencies, and timetables for large-scale TN problems of realistic sizes. The proposed method is developed based on the following considerations:(a) The method should be generally applicable to the design and optimization of a wide range of practical TNproblems, and should not favor particular TN configurations. (b) Solutions obtained from this method shouldgive reasonably good results in a reasonable amount of time, as permitted by the current computing poweraffordable to most transit agencies. (c) The results should improve and approach a global optimum as com-puter resources (mainly CPU processing speed) increase.

The proposed methodology is based on an iteratively defined local solution search space combined with anintegrated simulated annealing, tabu, and greedy search algorithm (ISTG). The local spaces are designed to betractable for existing computational resources to perform local searches, while the algorithm is aimed at avoid-ing solution search processes becoming trapped at a poor local optimum. Simulated annealing (SA) is a MonteCarlo simulation based stochastic search algorithm (Kirkpatrick et al., 1983). Under fairly general conditions,it has been shown that a global optimal will be obtained with probability 1 (Hajek, 1988). However, the weak-ness of SA search methods has been recognized in various application fields, which is the exponential compu-tational intractability with respect to the problem size. This makes SA in the original form of little practical

use for large TN problems. In this research, interventions or enhancements including tabu and greedy inter-

842 F. Zhao, X. Zeng / European Journal of Operational Research 186 (2008) 841855


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vention are introduced to optimize or to accelerate the solution selecting process in SA search method.Detailed description of SA, tabu, and greedy search methods and local spaces used in this study may be foundin Zhao and Gan (2003).

2. Transit network optimization problem statement

A TN optimization problem can be stated as the determination of a set of transit routes, the correspondingroute headways and timetables, given the street network and the transit demand of a transit service area andsubject to the operator budgetary constraint and a set of route, headway, and timetables feasibility con-straints, to minimize the total user cost of the TN service area. Mathematically, the optimization processcan be stated as

Minimize: ucostT; h;S;O 8T2 T; h 2 H and S2 S 1

Subject to:

Route headway; route length; and total TN vehicle fleet size constraints :

hq

min6 h

q6 hq

max; l

q

min6 l

q6 lq

maxq 1; 2; . . . ; T and XT

q1

vq6 v

max;

Route load factor constraints: Lq 6 Lqmax q 1; 2; . . . ; T; 2

Route directness constraints: dq P dqmin q 1; 2; . . . ; T; 3

Route geometry or topology constraints: gqT q 1; 2; . . . ; T; 4

where T, h, and Sare, respectively, the unknown route network, vehicle headway vector, and vehicle timetablematrix; Orepresents a given transit demand origindestination (OD) matrix; T, H, and Sare the search spacesfor unknown variables T , h, and S, respectively; hq, h

qmin, and h

qmax represent, respectively, the route headway,

the minimum and the maximum route headway constraints on route q; Tis the total number of routes in T; lq,l

qmin, and l

qmax are, respectively, the route length, and the minimum and maximum route length constraints; vq

and vmax are the route vehicle fleet size and the maximum allowed total vehicle fleet size of TN; Lq and L

q

max arethe vehicle load factor and the maximum load factor constraint; dq and dqmin are the route directness and min-

imum route directness constraints, respectively. In the following sections, the user cost objective function,decision variables, and their search spaces, as well as various constraints in the optimization statement, arediscussed. For simplicity, the following conventions are adopted: (a) one-dimensional variables (vectors,routes, paths, etc.) are represented with lower case bold-faced letters, while matrices and spaces are repre-sented with upper case bold-faced letters; (b) plain letters represent the population sizes of variables withthe same names.

3. Constraints and assumptions

The following assumptions are made in this study: (a) vehicle headways and timetables for individual routes

remain unchanged in the time period under analysis; (b) passengers route choices are based on the shortesttravel time; (c) buses have the same seating capacity and can be assigned to any routes; (d) any OD transitdemand trips that require more than two transfers to complete are considered as not served by the transit sys-tem; (e) passengers arrive at transit stops randomly for initial boardings. For subsequent boardings requiredfor transfer(s), the user cost includes walking time (from a stop on the first route to one on the second route),transfer penalty time, and waiting time to board a vehicle on the second route; (f) the following vehicle loadfactor and route headway relationship is used:

Lq hq cmaxVSeat

and hq 2lqvq

; 5

where VSeat is the vehicle seating capacity, cmax is the critical link passenger flow of a given route; (g) the num-

ber of transit routes, T, remains fixed during an optimization process.

F. Zhao, X. Zeng / European Journal of Operational Research 186 (2008) 841855 843


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The route geometry/topology constrains gq(T) given in (4) can be used to represent the following con-straints: (a) routes with fixed paths; (b) route that must start from, and/or pass through, and/or end at certainpre-defined street node sets; (c) route with fixed segments; (d) route with or without repeating street nodes. Ifsome of the routes are fixed, the optimization process will expand the existing TN consisting of these fixedroutes by adding new routes. When some routes have fixed segments, the search process will try to find the

optimal or sub-optimal extensions of these existing routes. Finally, if some of the routes must past throughnodes on a pre-defined route (lightrail, subway, etc.), the optimization process corresponds to the design ofa feeder-trunk system.

In the route directness constraint (3), route directness is measured using the following formula:

d dr Xr1i1

Xrji1

uSij =u

Rij

r2 r=2; 6

where r is the number of nodes on route r, uRij is the user travel cost between nodes ni and nj along the route,

uSij is the user travel cost following the shortest path in the street network between these two nodes. Route

directness d(r) is the average ratio between the travel cost along a route segment and that along the shortestpath connecting the two end nodes of the route segment. The route directness d(r) has the property ofdr 6 1, with dr 1 being the optimum when all route segments coincide with the corresponding shortestpaths between the ending points of these segments.

4. Representations of TN variables and their local search spaces

The method of defining local search spaces developed in this study is similar to the branch-and-boundmethods in the network optimization field and the rollout algorithms described in Bertsekas (1998). Inbranch-and-bound methods, some integer sets and their descendant subsets are discarded if they have nochance of containing optimal results under given constraints. In rollout algorithms, instead of searchingthrough the entire branch-and-bound subsets, the search is performed along a sequence of subsets that are

most likely to contain better solutions.

4.1. Representation of TN service area and route network

A transit street network is defined by a set of street nodes N that are connected to each other by a set ofstreet segments. Street nodes in Nare street corners/points that are suitable to serve as transit stops and satisfycertain criteria (spacing, accessibility, etc.). Street segments must be suitable for transit vehicle operation. Adiscussion of transit stop spacing and algorithms for optimal route stop arrangement may be found in Wira-singhe and Ghoneim (1981). For brevity, n denotes a street node in N. A street segment is a vector defined byits two end street nodes (ni1, ni2). The length of a street segment is measured by the average in-vehicle travel

time between its two end nodes. A path or a route between any two street nodes is defined as a sequence ofstreet nodes, p pn1; n2; . . . ; np, where there is one street segment connecting two neighboring nodes on thepath. The set of all possible paths of the street network defines a global or complete path set, denoted as Pc. Itis assumed that the street network is connected, thus any two nodes in the street network are connected by atleast one path. A route network is represented by a set of route vectors:

T r1; r2; . . . ; rT; rq rnq1; nq2; . . . ; nqQ q 1; 2; . . . ;T; 7

where nqg is the street node number of the gth stop on route rq. To avoid double subscripts, the size (i.e., thenumber of nodes) of route rq is written as Q rq. Subscript g in nqg will be referred to as the gth stop numberof route rq (or simply, the stop number of route q), and nqg the street node number of the gth stop on route rq(or simply, the node number of route q). It is easy to see that a route network T is a subset of the global path

set Pc, i.e., T& Pc, and a transit route is a member of the combinatorial set rq 2 Pc.

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4.2. Representation of TN route vehicle timetable

A round trip departure timetable of a transit vehicle traveling along route q defined in (7) can be expressedas a vector:

tdeptq

1

; tdeptq

2

; . . . ; tdeptqQ ; tdept

qQ

1

; . . . ; tdeptq

2Q

1 ; 8

where tdeptqg is the actual vehicle departure time at the gth stop of route q. Vector (8) includes two one-way

trips as part of a round trip, i.e., an up-trend (from stop 1 to stop Q) trip and a down-trend (from stop Q tostop 1) trip. A stop in (8) with stop number g> Q corresponds to a down-trend trip, and the correspondingup-trend route stop number defined in (7) is (2Q g). Vector (8) represents the departure timetable of a typ-ical (or reference) round trip along route q. Departure times of other trips along the same route can be ob-tained by adding to (or subtracting from) the departure times defined in (8) with various multiples ofvehicle headway hq. The actual vehicle departure and arrival times at stop g of route q are

tdeptqg t

sdept

qg trand

qg and tarvl

qg tdept

qg thold

qg ; 9

where tsdept

qg is the scheduled vehicle departure time at stop gof route q, and trand

qg is a small random var-

iable to reflect the uncertainty in maintaining vehicle schedule, while tholdqg is the vehicle holding time. With(8) and (9), a typical round trip (departure or arrival) timetable of a transit vehicle traveling along route q canbe defined through the following vector sequence (route timetable matrix):

Sq sq1; sq2; . . .; where sqg tsdept

qg ; trand

qg ; thold

qg

g 1; 2; . . . ; 2Q 1: 10

The vehicle schedule timetable of the entire TN can be represented as

S SS1;S2; . . . ;ST; 11

where Sq is the route timetable matrix defined in (10). A vehicle timetable matrix for a route (or a network) willbe referred to as a route (or a TN) timetable.

4.3. Representation of TN vehicle headway and transit demands in a TN service area

Vehicle headways of a TN system are expressed as a vector, h h1; h2; . . . ; hT. For a given T, using con-straint relationship (5), a headway vector can be derived from a vehicle assignment vector vv1; v2; . . . ; vT withthe relationship hq 2lq=vq, where 2lq is the vehicle round trip travel time of route q.

The transit demand of a given time period can be represented by an OD matrix:

O Ooij; oij o1ij ; o

2ij ; . . . ; o

oijij

; 12

where oij is the OD trip distribution vector (or simply, an OD vector) that defines the arrival times of all thetrips originating from node ni and destined for node nj, oij oij is the number of trips in vector oij, and o

sij

represents the arrival time at the originating node ni of the sth trip (or passenger) in vector oij. Trip arrivaltimes in vector oijcan be obtained either directly from transit survey data or through a random trip assignmentprocess with a user defined probability distribution function (uniform distribution assumed in this study).

4.4. Representation of local search spaces for route network variable T

A local path space is generated from a master path and a sequence of local node spaces. A master path is apath from which a local path space is generated. A local node space consists of two components: a masternode and the nodes adjacent to the master node (i.e., nodes connected to the master node by one street seg-ment). Fig. 1 illustrates a master path (solid line) pn1; n2; . . . ; nj; . . . ; np and three local node spacesfns; ns1; ns2; ns3; ns4g s 1;j;p associated with the two ending nodes n1, np and the intermediate node nj onthe master path. The procedure to generate a local path space from the master path p shown in Fig. 1 has

two steps:



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(1) Connect a node in fn1; n11; n12; n13; n14g to a node in fnj; nj1; nj2; nj3; nj4g with a shortest path segment, andextend the resulted path segment to a node in fnp; np1; np2; np3; np4g to obtain a piecewise shortest pathsegment.

(2) Repeat step (1) for all possible node connections to obtain a path space. This path space, denoted asP1p, is the local path space of the master path p based on one intermediate node. For this exampleshown in Fig. 1, P1p has a population size of P1 5 5 5 5

3.

The procedure to generate the local path space in this example is easily extended to cases where zero ormultiple intermediate nodes are used, and the number of intermediate nodes may be flexibly defined basedon available computing resources. In this study, intermediate nodes are selected such that they are distributedevenly along a master path.

A drawback of the local path space P1p is that unless a large number of intermediate nodes are used,paths in P1p may not be flexible enough to reach all nodes in the master paths adjacent neighborhood(defined as nodes that can be connected to nodes of the master path with one street segment). This drawbackmay cause convergence difficulty when the master path is near an optimal or a local optimal path. To remedythis deficiency, an auxiliary local path space P2p is generated to ensure that all nodes in the master pathsadjacent neighborhood are reachable. The procedure to generate P2p from p is illustrated with the simpleexample shown in Fig. 1 and following the steps below:

(a) Select a node, for example node nj, from the master path p shown in Fig. 1.(b) Generate the local node space fnj; nj1; nj2; nj3; nj4g from the selected node nj.(c) Find the two adjacent nodes of node nj on the master path p (nodes nj1 and nj3 in this example).(d) Replace the existing path segment (nj3, nj, nj1) in p with a new piecewise shortest path segment that starts

from node nj3, ends at node nj1, and passes through a node in set {nj1, nj2, nj3, nj4} to obtain a new path. Ifnj is an ending node, for example nj np (refer to Fig. 1), then replace the existing ending path segment(np3, np) in p with a new shortest path segment that starts from node np3 and ends at a node in setfnp1; np2; np3; np4g.

(e) Repeat the procedure in (d) for all possible nodes in node (nj)s local node space and denote the resultedpath space as Pp; nj.

(f) Repeat the procedures of (d) and (e) for all nodes in p to obtain the auxiliary local path spaceP2p fPp; njj8nj 2 fn1; n2; . . . ; npgg.

One feature of the local path space P2(p) defined above is that P2(p) includes paths generated by removing anode from the master path (p)s end, or by extending the master paths end to a node in that ending nodeslocal node space. The resulted local path space of the master path p is defined as Pp P1p [ P2p,and the local route network space of a master route network T r1; r2; . . . ; rT is defined as the combinatorialspace:

TT fTp1;p2; . . . ;pTj8pq 2 Prqq 1; 2; . . . ; Tg; 13

where P(rq) is a local path space of the master path rq. In a solution search process, Prq will be the local route

search space of route rq, and T(T) be the local network search space of network T. It can be seen that T(T) still

n1 n11

n12

n13

n14

np np1

np2

np3

np4

nj nj1

nj2

nj3

nj4

Fig. 1. A path and three local node spaces.



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have an intractable population size for large-scale TNs. To illustrate, consider the local path space P1(p) of thepath p shown in Fig. 1. P1(p) has a population size of 5

3. For a route network ofT routes, the correspondinglocal network search space will have a population size in the order of 53T, which will be a number of astro-nomical order if, for example, TP 20. In this study, a flexible defined m-subspace TmT TT is used,which is defined as

TmT fTp1;p2; . . . ;pTj8pq 2 Prq; q q1; q2; . . . ; qm; 1 6 q1 < q2 < < qm 6 Tg: 14

Depending on available computing resources, m could be any number between 1 and T. An important featureof the local path (and network) space defined above is that it has completeness in a rollout sense. A local pathspace is considered as rollout complete if for any two paths p1;p2 2 Pc, there are a path sequence(u1; u2; . . . ; us in Pc and an associated local path space sequence Pu1;Pu2; . . . ; Pus, where u1 p1and us p2, such that path uj+1 is a member of path ujs local space, i.e., uj1 2 Puj j 1; 2; . . . ;s 1.The rollout completeness of a local space is a necessary condition for any locally defined search scheme tobe a global scheme, since without this condition, some solution candidates will be inaccessible during a search.The physical meaning of the rollout completeness is that starting from any path in Pc, one can always reachany other path in Pc by repeating the process of replacing a path with another from the former paths local

path search space (i.e., local search space rollout). To illustrate, assume that p1n11; n12; . . . ; n1p1 andp2n21; n22; . . . ; n2p2 are two paths in Pc. Since the street network is connected, there is a path p in Pc thatconnects node n1(p1) in path p1 and node n21 in path p2, i.e., p can be written as n1p1; n1; n2; . . . ; np; n21.Because any node in a path is a node of its neighboring nodes local node space, the path sequence consistingof u1n11; n12; . . . ; n1p1; u2n12; n13; . . . ; n1p1; n1; u3n13; n14; . . . ; n1p1; n1; n2; . . . ; us1np; n21; n22; . . . ; n2p21,and usn21; n22; . . . ; n2p2 satisfies the rollout conditions, i.e., u1 p1; us p2, and path uj+1 is a member ofpath ujs local space, i.e., uj1 2 Puj j 1; 2; . . . ;s 1. In fact, path uj+1 is generated by removing a nodefrom one end ofuj and/or extending the path to a node of the other ending nodes local node space of uj.

4.5. Representation of search spaces for TN headway

This section deals only with vehicle assignment vector v, since for a given T, any TN headway vector h canbe represented by vehicle assignment vector through relationship (5). Assume v vv1; v2; . . . ; vT is a vehicleassignment vector obtained from a previous search process, where vq represents the number of vehicles onroute q. The local space ofv is defined as

Vmv Vma v [ Vmr v [ V

mar v; 15

where

Vma v v1; v2; . . . ; vq s; . . . ; vTj8XT

j1;j6q

vj vq s 6 vmax;s 6 m; 1 6 q 6 T

!( );

Vm

r v fv1; v2; . . . ; vq s; . . . ; vTj8vq > s;s 6 m; 1 6 q 6 Tg;Vmar v fv1; v2; . . . ; vq s; . . . ; vj s; . . . ; vTj8vj > s;s 6 m; 1 6 q;j 6 T; q 6 jg:

In the above relationships, m and s are positive integers. The vector v will be referred to as the master vector ofthe local vector space Vmv. Vma v consists of all the possible cases of adding up to m vehicles to the indi-vidual components of the master vehicle assignment vector v. Vmr v includes all the possible cases of remov-ing up to m vehicles from vector vs individual components. Vmar v is made up by all the possible cases ofremoving up to m vehicles from one of the vector vs components, and adding them to another component.It is easy to see that Vmv is a large combinatorial integer space. In a TN solution search process, a smallerlocal space ofVmv, for example m = 1, may be used in a search iteration. The local vehicle assignment vec-tor space Vmv also has rollout completeness. The proof of the rollout completeness ofVmv is straight-

forward and can be carried out in a manner similar to that for local path search spaces.



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4.6. Representation of search spaces for TN timetables

The following assumptions are made: (a) timetables are rounded to the whole minute; (b) the search processinvolves only the shifting of various route vehicle timetables to obtain a TN timetable (or a combination ofroute timetables) that has minimum total user cost. Based on the above assumption, for a reference vehicle

departure timetable given in Eq. (9), all the possible departure and arrival times for routeq

at stopg

canbe written as

tjdept

qg t

sdept

qg j trand

qg and t

jarvl

qg t

jdept

qg thold

qg j 0; 1; 2; . . . ; hq 1:

In matrix form, the route timetable search space of route q becomes

Sq s0q1 ; s

0q2 ; . . .

; s

1q1 ; s

1q2 ; . . .

; s

2q1 ; s

2q2 ; . . .

; . . .

n o; 16

where

sjqg tsdept

qg j; trand

qg ; thold

qg

g 1; 2; . . . ; 2Q 1 j 0; 1; 2; . . . ; hq 1:

ts

dept

q

g

j in the above expression represents the shifted scheduled vehicle departure time (by j minutes) atstop g of route q. From (16) and in view of (11), the vehicle timetable search space of the entire TN can bewritten as

S fSS1;S2; . . . ;STj8Sq 2 Sq q 1; 2; . . . ; Tg: 17

It can be seen from (16) that the route timetable search space Sq contains hq possible vehicle schedule time-tables for route q, and the TN timetable search space Sis a combinatorial space that contains

QTq1hq TN time-

table solution candidates, which is a number of astronomic order for a large T. In a local timetable solutionsearch process, a flexibly defined m-subspace Sm S can be used in a search iteration, where

Sm fSS1;S2; . . . ;STj8Sq 2 Sq; q q1; q2; . . . ; qm; 1 6 q1 < q2 < < qm 6 Tg: 18

5. Formulation of total user cost objective function

The total user cost function used in this study is the summation of individual transit users travel timesbetween all the OD pairs in the transit service area:

ucostT;O; h;S XN

i;j1;i6j

Xoijs1

uij;sT; h;S

" #; 19

where uij;sT; h;S s 1; 2; . . . ; oij is the travel time of the sth trip (or passenger) between nodes ni and nj.Note that, even for the same OD node pair ni and nj, the travel time uij,s may have different values for differents (i.e., different passengers), since these passengers may arrive at the originating node ni at different times. De-note t

kij;s (k= 0,1,2) as the travel time of a k-transfer trip (i.e., a trip that may be accomplished with k vehicle

transfers) between a demand node pair ni and nj. For zero-, one- and two-transfer trips, the travel time for ansth trip can be expressed as follows:

t0ij;s twait

qi;s tinvh

qi!j; 20

t1ij;s twait

q1i;s tinvh

q1i!1 ttran

q1!q21;s tinvh

q21!j; 21

t2ij;s twait

q1i;s tinvh

q1i!1 ttran

q1!q21;s tinvh

q21!2 ttran

q2!q32;s tinvh

q32!j; 22

where twaitqi;s is the waiting time to board a vehicle at node ni of route q; tinvh

qi!j is the in-vehicle travel time

from nodes ni to nj on route q; and ttranq1!q21;s is the transfer time from route q1 to q2 at the common street

node n1. All the three travel time variables rely on individual passengers arrival time at the originating node ni.The following is a brief description of the relationship between user travel time and timetable and headway

variables described in Sections 4.2 and 4.3. To avoid confusion due to too many indices, the subscript s is



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omitted, and it is assumed by default that all the time values described here are based on individual passengertrips. The transfer time at street node n1 can be further expressed as

ttranq1!q21 twalk

q1!q21 twait

q21 tpenl

q1!q21 ; 23

where twalkq1!q21 is the walking time from route (q1)s stop at street node n1 to route (q2)s stop of the same

street node; twaitq21 is the waiting time to board a vehicle at route (q2)s stop of street node n1; and tpenl

q1!q21

represents the transfer penalty between route q1 and route q2, expressed in equivalent travel time. Dependingon the particular context, a time value (vehicle departure time, passenger waiting time, etc.) may either refer toa stop of a route or to a street node of a route. Fig. 2 shows the difference between a route stop and a routenode, where n1 is a common node of both routes r1 and r2; and g11,g12 and g21,g22 are, respectively, the up-trend and down-trend stop numbers of routes r1 and r2 at the street node n1. Although route r1 and route r2intersect at the same location represented by street node n1, their corresponding stops may not be at the samestreet corner, side, or point. Therefore, for transfer passengers, user cost must include the corresponding walk-ing time. In this study, the walking time between two transfer stops and the corresponding transfer penaltytime are considered as input data provided by transit planners based on survey data and statistical analysis.

The waiting time for a transit user to board a vehicle at stop g on route q, twaitqg , is evaluated with the

following relationship:

twaitqg tdept

qg mhq uarvl

qg ; 24

where uarvlqg represents a transit users arrival time at stop gof route q, and m is an integer selected such that

the passenger boards the first departing vehicle after arriving at a route stop, i.e., tdeptqg m 1hq

optimization of transit route network, vehicle headways

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