Modeling capacity flexibility of transportation networks

Transportation Research Part A 45 (2011) 105–117

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Transportation Research Part A

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Modeling capacity flexibility of transportation networks

Anthony Chen a,⇑, Panatda Kasikitwiwat b

a Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110, USAb Department of Civil Engineering, Kasetsart University, Kamphaeng Saen Campus 73140, Thailand

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 June 2009Received in revised form 10 September2010Accepted 10 November 2010

Keywords:System flexibilityNetwork capacityReserve capacityTraffic equilibriumBi-level program

0965-8564/$ - see front matter Published by Elseviedoi:10.1016/j.tra.2010.11.003

⇑ Corresponding author. Tel.: +1 435 797 7109; faE-mail address: [email protected] (A. Chen

Flexibility of the transportation system is one of the important performance measuresneeded to deal with demand changes. In this paper, we provide a quantitative assessmentof capacity flexibility for the passenger transportation network using bi-level networkcapacity models. Two approaches for assessing the value of capacity flexibility are pro-posed. One approach is based on the concept of reserve capacity, which reflects the flexi-bility with respect to changes in terms of demand volume only. The second approachallows for variations in the demand pattern in addition to changes in demand volume inorder to more fully capture demand changes. Two models are developed in the secondapproach to consider two types of capacity flexibility. The total capacity flexibility allowsall users to have both route choice and destination choice when estimating capacity flex-ibility. The limited capacity flexibility estimates how much more demand volume could beadded to a fixed demand pattern by allowing the additional demand to deviate from thefixed demand pattern. Numerical examples are provided to demonstrate the different con-cepts of capacity flexibility for a passenger transportation system under demand changes.

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1. Introduction

Flexibility, in the Webster’s Dictionary, is defined as ‘‘a ready capability to adapt to new, different, or changing requirements’’(Merriam-Webster, 2000). Flexibility is also the ability to accommodate, withstand or handle uncertainty. It describes thelevel of capability a system can handle or absorb uncertainties or changes. In systems engineering, flexibility is the charac-teristic of the interface between a system and its external environment (Correa, 1994). It has been widely researched in thefield of manufacturing. Many categories, such as machine flexibility, operation flexibility, and process flexibility, have beenadopted as key strategies for improving market responsiveness in uncertain demand. The typical reason that manufacturingindustries have adopted flexibility is to speed up the entire product cycles. In transportation, flexibility is one of the impor-tant performance measures needed to deal with demand changes due to several different reasons. One reason is the contin-uing increase in traffic as economic growth and technology evolve, while the infrastructures remain relatively stagnant.Another reason is changes in demand pattern because of external forces such as unusual events and land use developmentpolicies. Therefore, an important issue for a transportation system is to have adequate capacity to accommodate changes intraffic demand.

Different flexibility definitions and measures have been proposed in the literature. Feitelson and Salomon (2000) definednetwork flexibility as the ease with which a network can adjust to changing circumstances and demands, both in terms ofinfrastructures and operations. It consists of node flexibility, link flexibility, and temporal flexibility. Node flexibility is de-fined as the ease with which network nodes (points of access) can be located. Link flexibility depends on the ease and cost of

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Nomenclature

A set of links in the networkN set of nodes in the networkI set of all origin nodes, I # NJ set of all destination nodes, J # NR set of routes in the networkRij set of routes between origin i 2 I and destination j e Ji an origin node, i e Ij a destination node, j e Ja a link in the network, a e Ar a route, r e Rij

Z objective functionl O–D matrix multiplier for the whole networkCa capacity on link ava flow on link ata(va) travel time on link a�qij existing demand between O–D pair ij~qij additional demand between O–D pair ijqij total demand between O–D pair ij, qij ¼ �qij þ ~qij

q O–D demand matrix in vector formhij

r flow on route r between O–D pair ij associated with �qij

f ijr flow on route r between O–D pair ij associated with ~qij

dijar 1 if link a is on route r from origin i e I to destination j e J; 0 otherwise

�oi existing trip production at origin i~oi additional trip production at origin ioi total trip production at origin i, oi ¼ �oi þ ~oi

omaxi maximum trip production at origin i (a constant)

o trip production in vector form�dj existing trip attraction at destination j~dj additional trip attraction at destination jdj total trip attraction at destination j, dj ¼ �dj þ ~dj

dmaxj maximum trip attraction at destination j (a constant)

cj(dj) cost of destination jh impedance parameter for trip distribution

106 A. Chen, P. Kasikitwiwat / Transportation Research Part A 45 (2011) 105–117

locating an additional link between nodes, thus increasing network connectivity. Temporal flexibility refers to the ability tosequence infrastructure investments and the degree to which use of the infrastructure requires coordination among users.However, the approach is a qualitative assessment using a subjective rating scheme to measure flexibility. Values of flexi-bility might be varied depending on different perceptions.

Recently, capacity flexibility was proposed in a dissertation by Cho (2002) for the application in freight transportation. Hestated that transportation flexibility could be viewed as a multidimensional concept comprised of such parameters as per-formance, operating cost, and network design. Flexibility of a transportation system can be evaluated in typical performanceparameters such as system capacity, origin–destination (O–D) connectivity, and travel time. The overall performance flexi-bility of a transportation system should be determined by considering all of these dimensions. In his work, conceptual def-inition and operational means of measuring the system capacity flexibility of a freight transportation system weredeveloped. Morlok and Chang (2004) formalized the definition of capacity flexibility for a freight transportation system,which is ‘‘the ability of a transport system to accommodate changes in traffic demand while maintaining a satisfactory level ofperformance.’’ Capacity flexibility can be quantified by estimating the maximum system capacity or the amount of traffica system can handle. Two interpretations of flexibility are measured. The first is the range of changes in demand that thesystem can accommodate. In this case, the demand pattern is fixed. The second is the amount of traffic volume that canbe accommodated when deviation of the base traffic pattern is permitted. Sun et al. (2006) further extended the work ofMorlok and Chang (2004) by making three enhancements: treatment of uncertainty in future traffic pattern, incorporationof volume-delay functions to account for congestion effect and level of service constraint to model service quality, and adop-tion of a probit-based stochastic traffic assignment procedure to enhance its routing options. These enhancements are con-sidered useful for assessing a degradable transportation system with demand changes.

As noted by Yang et al. (2000), modeling characteristics of passenger and freight transportation are different in many as-pects: (1) the movement involves flow of people rather than physical commodities as in freight transportation, (2) traveldelay increases with increasing flow as a result of congestion, whereas freight treats travel time as a fixed cost, (3) routechoice behavior has to be considered in estimating network capacity as opposed to the system optimal routing option in

A. Chen, P. Kasikitwiwat / Transportation Research Part A 45 (2011) 105–117 107

freight transportation, and (4) multiple O–D pairs exist and the flows between different O–D pairs are not exchangeable orsubstitutable in a passenger transportation system. These differences make modeling capacity flexibility for passengers acomplex, yet intriguing problem to solve. To the best of our knowledge, no research has directly developed quantitative mea-sures of system capacity flexibility to a passenger transportation network. The aim of this paper is to apply the freight capac-ity flexibility definition to the passenger transportation system and to develop methods for assessing its network capacityflexibility quantitatively. An assessment methodology for measuring capacity flexibility of a passenger transportation systemwith demand changes in terms of demand volume and demand pattern is provided. Two approaches for assessing capacityflexibility are considered. The first approach follows the concept of reserve capacity (Wong and Yang, 1997), which assumesa fixed demand pattern. Because a common multiplier is used to scale all O–D pairs, this approach results in a conservativeestimate of capacity flexibility. The second approach allows for variations in the demand pattern in addition to changes indemand volume in order to more fully capture demand changes. This approach adopts a variant of the network capacity andlevel of service model proposed by Yang et al. (2000) to estimate capacity flexibility. Two models are developed in the secondapproach to consider two types of flexibility. The total flexibility allows all users to have both route choice and destinationchoice when estimating capacity flexibility. The limited flexibility estimates how much more demand volume could beadded to the first approach (fixed demand pattern) by allowing the additional demand to deviate from the fixed demandpattern.

This paper is organized as follows. After the introduction, Section 2 provides definitions and basic approaches to measur-ing capacity flexibility in a passenger transportation system. Section 3 describes the bi-level network capacity models basedon two approaches for assessing capacity flexibility of a passenger transportation system. Section 4 presents some numer-ical examples to illustrate the capacity flexibility assessment of the two approaches. Concluding remarks are provided inSection 5.

2. Definitions and basic approaches to measuring flexibility

2.1. Flexibility and reliability in transportation networks

In any transportation system, uncertainty, which is an uncontrolled condition, becomes an inseparable part of many prob-lems. It exists in both supply (infrastructure such as roadways) and demand (travelers’ demand). On the supply side, thecapacity of roadways in the transportation network can be degraded because of adverse weather, natural disasters andeveryday disturbances, such as traffic congestion arising from daily peak traffic or occasional traffic accidents. On the de-mand side, traffic demand is constantly changing over time due to technology, economic growth, and some usual/unusualevents. In order to handle these changes, both capacity reliability and capacity flexibility need to be considered. While capac-ity reliability reflects the performance of the system when the supply or the capacity of system components is uncertain(Chen et al., 1999, 2002), capacity flexibility of a transportation system should be considered to reflect the performanceof the system under demand changes in terms of traffic volume and pattern due to the imposed external changes.

In the literature, there has not been much research on transportation system flexibility as there has been for transporta-tion network reliability. For reliability, the widely accepted definition is: ‘‘the probability of a device performing its purposeadequately for the period of time intended under the operating conditions encountered’’ (Allan and Billington, 1992; Wakabayashiand Iida, 1992). It can be concluded that reliability studies are concerned with probabilities and reliability in the transpor-tation system describes this probability, under uncertainties, of successfully traveling from one place to another. From theliterature, three main aspects have been considered: (1) connectivity reliability, which is the probability that network nodesare connected (Iida and Wakabayashi, 1989), (2) travel time reliability, which is the probability that a trip between a givenO–D pair can be made successfully within a given time interval and a specified level of service (Asakura and Kashiwadanu,1991), and (3) capacity reliability, which is the probability that the network capacity can accommodate a certain volume oftraffic demand at a required service level (Chen et al., 1999, 2000, 2002). For other recently proposed transportation reliabil-ity measures, see Heydecker et al. (2007) for a review.

Flexibility refers to the ability of a system to accommodate, withstand, or handle changes. Therefore, the flexibility of thetransportation system describes the level of capability to handle or absorb changes. It determines how much change the sys-tem can accommodate. Similar to the transportation reliability studies, system flexibility in passenger transportation net-work should consider multidimensional performance measures as suggested by Cho (2002). In this study, we focus on thecapacity flexibility dimension.

2.2. Capacity flexibility definition for transportation networks

Capacity flexibility can be described as the characteristic of the interface between the network capacity and demandchanges or flexibility with respect to changes in demand. Changes in traffic demand of the transportation network includethe following:

� The overall quantity of demand volume.� The spatial demand pattern of traffic flow (e.g., shifting from one O–D pair to other O–D pairs in the network).


The definition of capacity flexibility for freight transportation network given by Morlok and Chang (2004) is adapted tothe problem of the passenger transportation network. It is defined as the ability of the passenger transportation network toaccommodate changes in traffic demand while maintaining a satisfactory level of performance. Level of performance can bethe network capacity and/or level of service. Note that the capacity flexibility of a passenger transportation system canbe varied due to the external changes in terms of traffic demand level and traffic demand patterns.

2.3. Basic approaches to measuring capacity flexibility

The measurement of flexibility borrows the concept of breakeven analysis from the engineering economic literature (alsomentioned in Morlok and Chang, 2004). The concept provides the range of sales volume needed in order to have a positiveprofit when facing an uncertain sales volume. This concept is illustrated in Fig. 1. The basic idea of the breakeven analysis isto determine the minimum level of performance necessary for a system to have benefits that equal the costs. In the sense ofthe flexibility measurement, this tool also provides the range of performance level that satisfies or benefits the system. Giventhat the capacity of sales is limited, flexibility is the range of the difference between the limited capacity line and the break-even point. In the transportation system, capacity flexibility can be the range of traffic demand changes that the current sys-tem can accommodate with an acceptable level of performance. Because changes in traffic demand can be in both volumeand pattern, two approaches of measuring flexibility are considered. One approach is based on the concept of reserve capac-ity, which reflects the flexibility to accommodate demand changes resulting from changes in demand volume only. The re-serve capacity approach is graphically shown in Fig. 2a. Flexibility is the range of the difference between the maximumcapacity and the current demand. The second approach, shown in Fig. 2b, allows for variations in the demand pattern in

Dollars

UnitsBreakevenin Units

BreakevenSales (in $)

Total Revenue

Total Cost

Variable Cost

Fixed CostLoss

Profit

Range of profitablesales

LimitedCapacity

Fig. 1. Concept of breakeven analysis as a basis for flexibility assessment (adapted from Morlok and Chang, 2004).

Demand Multiplier

Range of acceptable volume changes

Current demand

Maximum Capacity

Traffic Volume

Capacity

ParameterRange of acceptable

patterns

Capacities of different patterns

Desired Demand

Different ranges of acceptable volume changes

Current demand

Traffic demand

(a) (b)Fig. 2. Capacity flexibility measures: (a) reserve capacity approach and (b) performance level approach.


addition to changes in demand volume in order to more fully capture demand changes. This approach provides both range ofchanges in demand volume for different demand patterns and range of changes in demand pattern. Given the desired de-mand that the network should accommodate, range of changes in demand pattern provides the patterns that have capacitieshigher than the desired demand.

3. Capacity flexibility assessment of a passenger transportation system

In this section, we provide a quantitative assessment of capacity flexibility for the passenger transportation networkusing bi-level network capacity models. Two approaches of measuring network capacity flexibility are discussed. One isbased on the concept of reserve capacity, which reflects the flexibility with respect to demand changes in terms of de-mand volume only. The second approach allows for variations in the demand pattern in addition to changes in demandvolume.

3.1. Reserve capacity approach to measuring capacity flexibility

This approach follows the concept of reserve capacity, which requires preserving the fixed demand pattern when estimat-ing the network capacity. The measure provides information about the range of changes in demand volume that can beaccommodated (see Fig. 2a). The model used for this approach is the reserve capacity model originally proposed by Wongand Yang (1997) for a signal-controlled road network. The concept of reserve capacity is defined as the largest multiplierl applied to a given existing O–D demand matrix that can be allocated to a network without violating the link capacitiesCa or exceeding a pre-specified level of service. This network capacity model, with a uniform O–D growth, was used in Chenet al. (1999, 2000, 2002) as a core component in the reliability assessment procedure for estimating the capacity reliability ofa transportation network. A bi-level mathematical program for finding the reserve capacity l can be formulated as follows:

Max l ð1aÞs:t: vaðlqÞ 6 Ca; 8 a 2 A; ð1bÞ

where vaðlqÞ is obtained by solving the following user equilibrium problem:

MinXa2A

Z va

0taðxÞdx ð1cÞ

s:t:Xr2Rij

f ijr ¼ lqij; 8 i 2 I; j 2 J; ð1dÞ

va ¼Xi2I

Xj2J

Xr2Rij

f ijr dij

ar ; 8 a 2 A; ð1eÞ

f ijr P 0; 8 i 2 I; j 2 J; r 2 Rij: ð1fÞ

Route choice behavior and congestion effect are explicitly considered in the lower-level problem while the upper-levelproblem determines the maximum O–D matrix multiplier in Eq. (1a) subject to the roadway capacity constraints in Eq.(1b). The lower-level problem is a standard network equilibrium problem (Sheffi, 1985) for a given l value determined fromthe upper-level problem. Eq. (1c) is the user equilibrium objective function, which is a sum of the integrals of the link per-formance functions. Eq. (1d) is a set of flow conservation constraints scaled by the O–D matrix multiplier l. Eq. (1e) is theincidence relationship which expresses the link flows in terms of path flows. Eq. (1f) represents the non-negativity conditionto ensure a meaningful solution. The link flow vaðlqÞ represents the equilibrium link-flow pattern obtained from solving thelower-level problem for a given existing demand pattern q uniformly scaled by l. The largest value of l indicates whetherthe current network capacity has spare capacity or not. For example, if l > 1, then the network has a spare capacity amount-ing to 100(l � 1) percent of the existing O–D demand matrix q; otherwise, the network is overloaded by 100(1 � l) percentof the existing O–D demand matrix q. The above reserve capacity model has the following features: (1) the movements ofmulti-commodity flows are considered in the O–D demand matrix q, (2) congestion effect is captured by using an increasingtravel time function with respect to the flow for each link in the transportation network, (3) since path flows for each O–Dpair are explicitly determined in the route choice problem, the issue associated with exchangeable or substitutable flows ofdifferent O–D pairs does not exist, and (4) level of service (LOS) constraints such as volume to capacity ratio can be easilyadded to the bi-level mathematical program when determining the maximum throughput. This can be easily accomplishedby adding a LOS indicator to the roadway capacity constraint (1b). In the above model, LOS is set to 1, which is equivalent to aLOS E in the Highway Capacity Manual. For additional details on extending the reserve capacity model to consider differentLOS requirements, see Kasikitwiwat (2005).

3.2. Performance level approach to measuring capacity flexibility

While the first approach of capacity flexibility measures the range of demand volume that can be accommodated, the sec-ond approach allows for variations in the demand pattern (in addition to the demand volume changes) in order to better


capture demand changes. The measure provides the ranges of both pattern and volume changes that can be accommodatedwith an acceptable performance level in terms of network capacity. Fig. 2b illustrates the capacity flexibility measure usingthe second approach. In this approach, two models are considered for modeling two types of flexibility: total capacity flex-ibility and limited capacity flexibility.

The total capacity flexibility allows all network users to have both destination choice and route choice when estimatingnetwork capacity and hence capacity flexibility. The limited capacity flexibility estimates the amount of demand volume thatcould be added to the first approach (fixed demand pattern) by allowing only the additional demands to deviate from thefixed demand pattern. Both types of flexibility use a variant of the network capacity and level of service model developedby Yang et al. (2000) to measure capacity flexibility. The model for total flexibility is based on the ultimate capacity conceptand the model for limited flexibility is based on the practical capacity concept. These two models are described as follows.

3.2.1. Network capacity with total flexibilityFor total flexibility, the ultimate capacity concept, which is defined as the maximum throughput the system can handle

without violating the roadway and zonal capacity constraints, is used to estimate network capacity and hence to measurecapacity flexibility. This concept relaxes the common multiplier requirement in the reserve capacity concept by allowingthe maximum throughput to be scaled by individual O–D pairs. The network capacity model is a variant of the networkcapacity and level of service problem described in Yang et al. (2000), which integrates a combined distribution andassignment model to determine the maximum total zonal trip production in a bi-level optimization framework. This conceptallows all travelers in the network to choose both destination and route simultaneously to minimize their travel costs. Thebi-level mathematical program is formulated as follows:

MaxXi2I

Oi ð2aÞ

s:t: vaðoÞ 6 Ca; 8 a 2 A; ð2bÞoi ¼

Xj2J

qijðoÞ 6 omaxi ; 8 i 2 I; ð2cÞ

dj ¼Xi2I

qijðoÞ 6 dmaxj ; 8 j 2 J; ð2dÞ

oi P 0; 8 i 2 I; ð2eÞ

where qij(o) and va(o) are obtained by solving the combined trip distribution-assignment problem:

MinXa2A

Z va

0taðxÞdxþ 1

h

Xi2I

Xj2J

qijðln qij � 1Þ ð2fÞ

s:t:Xj2J

qij ¼ oi; 8 i 2 I; ð2gÞXr2Rij

f ijr ¼ qij; 8 i 2 I; j 2 J; ð2hÞ

va ¼Xi2I

Xj2J

Xr2Rij

f ijr dij

ar; 8 a 2 A; ð2iÞ

qij P 0; 8 i 2 I; 2 J; ð2jÞf ijr P 0; 8 i 2 I; j 2 J; r 2 Rij: ð2kÞ

Eq. (2a), in the upper-level problem, determines the maximum total trip productions from all origins subject to the road-way capacity constraints in Eq. (2b), maximum trip production and attraction constraints in Eqs. (2c) and (2d), and non-neg-ativity constraints on trip productions in Eq. (2e). The O–D demand matrix and equilibrium link-flow pattern are determinedby the lower-level problem. Both destination choice and route choice are simultaneously considered in the combined tripdistribution-assignment model of the lower-level problem in Eqs. (2f)–(2k). Eqs. (2g) and (2h) represent the flow conserva-tion constraints. Eq. (2i) is the incidence relationship that expresses the link flows in terms of path flows. Eqs. (2j) and (2k)are the non-negativity conditions for the O–D flows and path flows respectively. The impedance parameter h for trip distri-bution in Eq. (2f) reflects the sensitivity of network users to travel time from an origin to a destination.

3.2.2. Network capacity with limited flexibilityFor limited flexibility, the practical capacity concept, which is defined as the summation of the current O–D demand and

the additional demand that the network can accommodate, is used to estimate network capacity and hence to measurecapacity flexibility. This concept, using the network capacity and level of service problem described in Yang et al. (2000),allows only the additional demand to choose both route and destination based on the travel costs and the destination attrac-tiveness measures, while the current demand pattern is preserved. In this model, the upper-level problem maximizes theadditional total zonal trip production subject to the roadway and zonal capacity constraints, while the lower-level problem


is a combined trip distribution-assignment model with variable destination costs. The bi-level mathematical program is for-mulated as follows:

MaxXi2I

~oi ð3aÞ

s:t: vaðoÞ 6 Ca; 8 a 2 A; ð3bÞ~oi ¼

Xj2J

~qijðoÞ 6 omaxi � �oi; 8 i 2 I; ð3cÞ

~dj ¼Xi2I

~qijðoÞ 6 dmaxj � �dj; 8 j 2 J; ð3dÞ

~oi P 0; 8 i 2 I; ð3eÞ

where ~qijðoÞ and va(o) are obtained by solving the combined trip distribution-assignment problem with variable destinationcosts:

MinXa2A

Z va

0taðxÞdxþ 1

h

Xi2I

Xj2J

~qijðln ~qij � 1Þ þXj2J

Z Pi2I

ð~qijþ�qijÞ

0cjðyÞdy ð3fÞ

s:t:Xj2J

~qij ¼ ~oi; 8 i 2 I; ð3gÞ

Xr2Rij

hijr ¼ �qij; 8 i 2 I; j 2 J; ð3hÞ

Xr2Rij

f ijr ¼ ~qij; 8 i 2 I; j 2 J; ð3iÞ

va ¼Xi2I

Xj2J

Xr2Rij

ðf ijr þ hij

r Þdijar; 8 a 2 A; ð3jÞ

~qij P 0; 8 i 2 I; j 2 J; ð3kÞf ijr P 0; 8 i 2 I; j 2 J; r 2 Rij; ð3lÞ

hijr P 0; 8 i 2 I; j 2 J; r 2 Rij: ð3mÞ

Eq. (3a), in the upper-level problem, determines the maximum additional productions from all origins subject to the road-way and zonal capacity constraints given in Eqs. (3b)–(3d). Eq. (3b) is the roadway capacity constraint. Eq. (3c) states thatadditional trip production from each origin has to be less than or equal to the available capacity of the origin zone, and Eq.(3d) states that additional trip attraction to each destination has to be less than or equal to the available capacity of the des-tination zone. Eq. (3e) is the non-negativity constraints for the additional trip productions from each origin zone. The addi-tional O–D demands and link flows are determined in the lower-level problem together with the fixed demand pattern. Thetotal demands are assigned to the network according to the conventional deterministic user equilibrium (DUE) manner. Onthe other hand, the additional demand or the traffic growth at each origin zone is distributed among various destinationzones by a multinomial logit model based on the O–D travel times and the destination costs. The destination cost is anincreasing function of the total number of trips attracted to destination j (i.e., dj ¼

Pi2Ið~qij þ �qijÞ). Eq. (3f) is the objective

function of the equilibrium trip distribution-assignment with variable destination costs (ETDA–VDC) model given by Oppen-heim (1993). Eqs. (3g)–(3i) are the flow conservation constraints for the additional O–D demands, path flows associated withthe existing O–D demands, and path flows associated with the additional O–D demands. Eq. (3j) is the incidence relationshipthat expresses the link flows in terms of path flows. Eqs. (3k)–(3m) are the non-negativity conditions for the additional O–Ddemands, path flows associated with the existing O–D demands, and path flows associated with the additional O–Ddemands, respectively.

4. Numerical results

In this section, numerical results using two approaches to measuring capacity flexibility are presented. Solution algo-rithms used for solving the three network capacity models are described in Kasikitwiwat and Chen (2005). The reservecapacity model is solved by the Frank-Wolfe traffic assignment algorithm by incrementally increasing the maximum O–Dmatrix multiplier until at least one of the equilibrium link flows violates the capacity constraints. The network capacity mod-els for the ultimate capacity and practical capacity concepts are solved by a genetic algorithm combined with a partial lin-earization algorithm for solving the lower-level distribution-assignment problem. Readers can also refer to Kasikitwiwat(2005) for the numerical tests conducted to compare the solution quality obtained by the genetic algorithm and the succes-sive linear programming (SLP) approach developed by Yang et al. (2000) for the ultimate and practical network capacitymodels. In general, the genetic algorithm approach is capable of delivering near-optimal solutions (or at least not worse thanthe SLP approach).


For illustration purpose, a simple network given in Fig. 3 is adopted. The network consists of six nodes, seven links, twoorigins, two destinations, and four O–D pairs. Existing demand pattern of O–D 1–3, O–D 1–4, O–D 2–3, and O–D 2–4 are 20,30, 40, and 20, respectively. The link travel time function used is the standard Bureau of Public Road (BPR) function asfollows:

Table 2Effect o

O–D

1–31–42–32–4NetwO–D

ta ¼ tfa 1þ 0:15

va

Ca

� �4 !

;

where va, tfa, and Ca are the flow, free-flow travel time, and capacity on link a, respectively. Link characteristics are provided

in Table 1.

4.1. Capacity flexibility with the reserve capacity approach

In the network capacity flexibility evaluation with the reserve capacity approach, the difference between the reservecapacity (the maximum traffic demand that the network can accommodate) and the current demand volume given bythe existing demand pattern reflects the degree of capacity flexibility (see Fig. 2a). However, the reserve capacity valuecan vary with respect to various factors such as initial demand patterns, routing strategies, and levels of information pro-vided to road users. In this section, we conduct experiments to examine the effect of initial demand patterns on the reservecapacity and capacity flexibility. The results presented in Table 2 provide the maximum traffic (or flow) that the system canaccommodate and Fig. 4 presents the range that traffic demand volume can be increased for different initial demand pat-terns. These ranges reflect the capacity flexibility of the network. In the reserve capacity concept, traffic demand betweeneach O–D pair must increase at the same rate to preserve the initial demand pattern. Different initial patterns of O–D de-mands (with the same total demand) can be increased differently until the flow on some links reaches the capacity. TheO–D demand pattern that is more congruous with the network topology would achieve a higher network capacity and hence

4

5

31

2

6

Origin Destination

Origin Destination

1

2

3

7

6

4

5

Fig. 3. Test network.

Table 1Link characteristics.

Link # Free-flow travel time (FFTT, min) Link capacity (veh/min)

1 10.00 100.002 4.00 80.003 12.00 80.004 4.00 50.005 5.00 120.006 5.00 50.007 4.00 50.00

f initial demand patterns for the reserve capacity model.

Initial demandpattern 1

O–D flow result Initial demandpattern 2

O–D flow result Initial demandpattern 3

O–D flow result

40 82.88 30 61.20 25 41.6510 20.72 20 40.80 25 41.6510 20.72 20 40.80 30 49.9850 103.60 40 81.60 30 49.98

ork capacity 227.92 224.40 183.26matrix multiplier 2.072 2.040 1.666

0

50

100

150

200

250

Pattern 1 Pattern 2 Pattern 3

Net

wor

k F

low

Existing Demand Maximum Flow

Ranges of acceptable changes in traffic volume

Fig. 4. Effect of initial demand patterns on capacity flexibility.


a higher capacity flexibility. Among the three different initial demand patterns, pattern 1 gives the highest network capacityof 227.92 with an increased rate (or maximum O–D matrix multiplier) of 2.072. Therefore, this initial demand pattern hasthe largest flexibility range that can be used to accommodate the demand volume increase, which amounts to 117.92(227.92–110.00).

4.2. Capacity flexibility with the performance level approach

The second approach of measuring capacity flexibility allows variations of demand patterns. In addition to changes in de-mand volume, the measurement provides a range of pattern changes that can be accommodated with an acceptable level ofperformance in terms of network capacity. In order to assess capacity flexibility for this approach, different demand patterns(by using different impedance parameter values) of network capacities have to be estimated. Two types of capacity flexibilityare considered for this approach: total flexibility and limited flexibility.

4.2.1. Capacity flexibility measure for total flexibilityTotal flexibility based on the ultimate capacity concept allows all users in the network to have both destination choice

and route choice when assessing the network capacity and capacity flexibility. In the model, the impedance parameter inthe combined distribution-assignment model controls the distribution pattern of O–D flows, which affects the networkcapacity. This impedance parameter reflects the sensitivity of network users to travel time from an origin to a destination.As noted earlier, different O–D demand patterns give different values of network capacity. Fig. 5 provides the results of theultimate network capacity model and the O–D demand pattern for different impedance parameters. For this specific net-work, impedance parameter values from 0.01 to 6 are tested. With a low impedance parameter value, the demand patterntends to be more equally dispersed among the O–D pairs. With a high impedance parameter value, the O–D demand patterntends to concentrate on O–D (1–3) and O–D (2–4), which have lower O–D travel times. However, flow on O–D (1–3) stops

Fig. 5. Capacity flexibility measure for total flexibility.


increasing due to the roadway capacity constraint of link 1 when the impedance parameter value reaches 0.5. The demandpattern that fits best with the network topology gives the largest network capacity. For this network, the O–D demand pat-tern with an impedance parameter value of 0.25 gives the largest network capacity. From Fig. 5, values of capacity flexibilityin terms of demand volume change, which is the difference between the maximum capacity and the existing demand, areobtained. In the total flexibility, there is no existing demand since all travelers have both route choice and destination choice.For the purpose of comparison, we assume the existing demand is 110 (the same as the reserve capacity approach and thelimited capacity flexibility in the next section). The range of demand volume change varies with the impedance parameter.

To obtain the range for an acceptable demand pattern change, the performance level approach of capacity flexibility isexamined. With an acceptable level of performance (or acceptable network capacity) of 230, the range of impedance param-eter for this network is between 0.1 and 2. The value of capacity flexibility in terms of traffic demand pattern change wouldbe lower if the acceptable level of performance is higher.

4.2.2. Capacity flexibility measure for limited flexibilityWhile the total flexibility allows all users to have both route choice and destination choice when estimating capacity flex-

ibility, the limited flexibility based on the practical capacity concept estimates the amount of traffic demand that could beadded to a fixed demand pattern by allowing the additional demand to deviate from the fixed demand pattern. Thus, only theadditional demand has both destination choice and route choice. The destination cost function used to estimate the addi-tional demand is:

cjðdjÞ ¼ aj dbj

j � mj;

where aj, bj, and mj are parameters given in Table 3, and dj is the total number of trips attracted to destination j. For the fixeddemand pattern, we use demand pattern 2 from Table 2.

Because the flexibility in choosing both route and destination is limited to the additional demand, the difference amongthe various demand patterns for different impedance parameter values in the limited flexibility is less than the difference inthe total flexibility. Fig. 6 provides the results of network capacity and O–D demand patterns for different values of theimpedance parameter. For consistency, the same impedance parameter values from 0.01 to 6 are also tested. Similar tothe results of total flexibility, values of capacity flexibility in terms of demand volume change, which is the difference be-tween the maximum capacity and the existing traffic demand, are obtained. The range of flexibility is different for differentimpedance parameter values.

Because the effect of impedance parameter is limited to the additional network users, demand patterns and networkcapacities do not change or decrease significantly when the impedance parameter value is deviated from 0.5 (the parametervalue that gives the highest network capacity among the tested patterns). For a range of an acceptable pattern change withan acceptable network capacity of 230, the impedance parameter values of 0.1 or greater give a network capacity higher than

Table 3Destination cost data used in the practical network capacity model.

Destination mj aj bj

3 1.20 0.15 0.254 1.50 0.10 0.25

0

20

40

60

80

100

120

0.01 0.1 0.25 0.5 0.75 1 2 4 6Parameter

O-D

flo

w

0

50

100

150

200

250

300

Net

wor

k C

apac

ity

O-D (1-3) O-D (1-4) O-D (2-3) O-D (2-4) Capacity

Range of pattern change

Range of volume change Acceptable network capacity

Existing demand

Fig. 6. Capacity flexibility of limited flexibility.


230. The value of capacity flexibility in terms of pattern change would be lower if the acceptable level of performance werehigher. For example, with a higher value of acceptable network capacity of 245, the flexible range of impedance parameterfor this network is between 0.25 and 2.

4.2.3. Comparison between total flexibility and limited flexibilityIn the performance level approach of measuring the capacity flexibility, two types of flexibility (i.e., total flexibility and

limited flexibility) are implemented. From the results, three observations can be drawn. First, the differences in O–D distri-bution from the limited flexibility among different demand patterns are less than those from the total flexibility because theflexibility in choosing both route and destination is limited to the additional demand. Second, the largest network capacityamong different demand patterns from the total flexibility is higher than that from the limited flexibility. This is because allusers in the total flexibility are distributed to the destinations and assigned to the routes in such a way that the networkresources are more utilized while the existing demand pattern in the limited flexibility has to be preserved. Third, the de-mand pattern with the impedance parameter value of 2 or greater in the total flexibility may not be practical because flowson O–D (1–4) and O–D (2–3) are very low (almost nil).

4.3. Comparison of capacity flexibility between two approaches

The network capacity results of these two approaches are provided in Table 4 and the capacity flexibility in terms of therange of demand volume changes are graphically shown in Fig. 7. The capacity flexibility of the reserve capacity approach is117.92, while the capacity flexibility with the performance level approach for total flexibility gives the largest value, 152.54,an increase of 29.36% compared with the reserve capacity approach. For the limited flexibility, the capacity flexibility is147.58, an increase of 25.15% compared with the one estimated using the reserve capacity approach. Fig. 8 provides theO–D demand patterns and link volume to capacity (V/C) ratios from different approaches. The results show that the flows

Table 4Comparison of network capacity between two approaches.

System capacity Trip production Origin–destination flow

O1 O2 OD(1–3) OD(1–4) OD(2–3) OD(2–4)

Reserve capacity approachInputs (existing traffic pattern) – – – 40.00 10.00 10.00 50.00Flow results (multiplier = 2.072) 227.92 103.60 124.32 82.88 20.72 20.72 103.60

Performance level approach: total flexibility Omax1 Omax

2 OD(1–3) OD(1–4) OD(2–3) OD(2–4)

Inputs (Max. zonal capacity) – 150.00 150.00 – – – –Flow results 262.54 138.61 124.53 99.97 38.04 43.75 80.78

Performance level approach: limited flexibilityInputs (existing traffic pattern and Max. zonal capacity) – 150.00 150.00 40.00 10.00 10.00 50.00Flow results 257.58 137.83 119.75 100.80 37.03 31.36 88.39

0

50

100

150

200

250

300

Net

wor

k fl

ow

Capacity flexibility or range of acceptable change in demand volume

Existing demand

Reserve Capacity

Limited Flexibility Total Flexibility

Fig. 7. Comparison of three capacity flexibility measures.

0

20

40

60

80

100

O-D(1-3)

O-D(1-4)

O-D(2-3)

O-D(2-4)

Existing demandReserve capacity

Limited flexibilityTotal flexibility

Link Reserve Limit Flex Total Flex

1 0.83 1.00 1.00

2 0.26 0.45 0.47

3 1.00 0.99 1.00

4 0.89 0.84 0.91

5 0.54 0.65 0.69

6 0.41 0.65 0.88

7 0.89 0.91 0.78

Link V/C Ratio

Fig. 8. O–D demand patterns and link volume to capacity ratios of three capacity flexibility measures.


on O–D (1–3), O–D (1–4), and O–D (2–3) from the performance level approach (limited flexibility and total flexibility) arehigher than those in the reserve capacity approach, while flows on O–D (2–4) show the opposite. However, the net increasein network capacity is larger in the performance level approach because the models for limited flexibility and total flexibilitycan achieve more network capacity by having more demands in O–D (1–3) (link 1), O–D (1–4) (links 2–5–7), and O–D (2–3)(links 4–5–6). The routes’ capacities of these O–D pairs are underutilized in the reserve capacity model. Note that the volumeto capacity (V/C) ratio of link 4 for the practical capacity is less than the V/C ratio for the reserve capacity because link 4 isalso used by the demand in O–D (2–4) (alternative route) in the reserve capacity model.

5. Concluding remarks

In this paper, we provided an assessment of capacity flexibility for the passenger transportation network by using bi-levelnetwork capacity models to estimate network capacity for different situations. Two approaches of measuring capacity flex-ibility were developed. The first approach follows the concept of reserve capacity, which requires preserving a fixed demandpattern. This capacity flexibility measure provides information about the range of change in demand volume that can beaccommodated. The study also examined the effect of demand patterns on estimating the reserve capacity and on capacityflexibility. The results indicated that different demand patterns give different reserve capacities. The demand pattern that ismore congruous with the network topology can achieve a higher reserve capacity, and hence a higher capacity flexibility.

The second approach of the capacity flexibility measure allows variations of demand pattern. This capacity flexibilitymeasure provides a range of the demand pattern changes that can be accommodated with an acceptable level of perfor-mance in addition to changes in demand volume. Two types of capacity flexibility (i.e., total flexibility and limited flexibility)were considered. Comparisons were made between the two approaches. Capacity flexibility is underestimated with the re-serve capacity concept due to the requirement of preserving the fixed O–D demand pattern. However, the estimation ofcapacity flexibility using the performance level approach with total flexibility overestimates the network capacity flexibilitysince allowing all users in the network to choose both destination and route is not practical. The estimation of capacity flex-ibility that gives the most reasonable value is the limited flexibility in the performance level approach.

For future research, it is worthwhile to consider capacity flexibility as a potential performance measure for improving orupgrading the network. The problem can be posed as a network design problem to determine optimal strategies for enhanc-ing not only the efficiency, reliability, durability, resiliency, etc., but also the flexibility of roadway networks to accommodatedemand changes in demand volume and variations in demand pattern.

Acknowledgements

The authors would like to thank the editor and two reviewers for their constructive comments. The work described in thispaper was supported by a CAREER Grant from the National Science Foundation of the United States (CMS-0134161). The sec-ond author also would like to acknowledge the financial support from the Royal Thai Government Scholarship.

References

Allan, R.N., Billington, R., 1992. Probabilistic methods applied to electric power system, are they worth it? Power Engineering Journal 6 (3), 121–129.Asakura, Y., Kashiwadanu, M., 1991. Road network reliability caused by daily fluctuation of traffic flow. In: Proceeding of the 19th PTRC Summer Annual

Meeting, Brighton.


Chen, A., Yang, H., Lo, H.K., Tang, W., 1999. A capacity related reliability for transportation networks. Journal of Advanced Transportation 33 (2), 183–200.Chen, A., Tatineni, M., Lee, D.-H., Yang, H., 2000. Effect of route choice models on estimating network capacity reliability. Transportation Research Record

1733, 63–70.Chen, A., Yang, H., Lo, H.K., Tang, W., 2002. Capacity reliability of a road network: an assessment methodology and numerical result. Transportation Research

Part B 36 (3), 225–252.Cho, D.J., 2002. Three Papers on Measuring the Reliability and Flexibility of Transportation System Capacity. Ph.D. Dissertation. University of Pennsylvania,

Philadelphia, Pennsylvania.Correa, H.L., 1994. Linking Flexibility, Uncertainty and Variability in Manufacturing Systems. London, Avebury.Feitelson, E., Salomon, I., 2000. The implications of differential network flexibility for spatial structures. Transportation Research Part A 34 (6), 459–479.Heydecker, B., Lam, W.H.K., Zhang, N., 2007. Use of travel demand satisfaction to assess road network reliability. Transportmetrica 3 (2), 139–171.Iida, Y., Wakabayashi, H., 1989. An approximation method of terminal reliability of a road network using partial minimal path and cut set. In: Proceeding of

the Fifth WCTR, Yokohama, Japan.Kasikitwiwat, P. 2005. Capacity Reliability and Capacity Flexibility of a Transportation network. Ph.D. Dissertation. Utah State University, Logan, Utah.Kasikitwiwat, P., Chen, A., 2005. Analysis of transportation network capacity related to different system capacity concepts. Journal of Eastern Asia Society for

Transportation Studies 6, 1439–1454.Merriam-Webster, 2000. Webster’s Third New International Dictionary, Unabridged Merriam-Webster, Springfield, Massachusetts.Morlok, E.K., Chang, D.J., 2004. Measuring capacity flexibility of a transportation system. Transportation Research Part A 33 (6), 405–420.Oppenheim, N., 1993. Equilibrium trip distribution/assignment with variable destination costs. Transportation Research Part B 27 (3), 207–217.Sheffi, Y., 1985. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice Hall, England Cliffs, NJ.Sun, Y., Turnquist, M.A., Nozick, L.K., 2006. Estimating freight transportation system capacity, flexibility, and degraded-condition performance.

Transportation Research Record 1966, 80–87.Wakabayashi, H., Iida, Y., 1992. Upper and lower bounds of terminal reliability of road networks: an efficient method with Boolean algebra. Journal of

Natural Disaster Science 14, 29–44.Wong, S.C., Yang, H., 1997. Reserve capacity of a signal-controlled road network. Transportation Research Part B 31 (5), 397–402.Yang, H., Bell, M.G.H., Meng, Q., 2000. Modeling the capacity and level of service of urban transportation networks. Transportation Research Part B 34 (4),

255–275.

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Modeling capacity flexibility of transportation networks