Computer-Aided Civil and Infrastructure Engineering 22 (2007) 6–18

Robust Transportation Network Design UnderDemand Uncertainty

Satish V. Ukkusuri∗

Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute,4032 Jonsson Engineering Center, Troy, NY, USA

Tom V. Mathew

Department of Civil Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India

&

S. Travis Waller

Civil Engineering Department-TRAN, University of Texas at Austin, 1 University Station C1761, Austin, TX, USA

Abstract: This article addresses the problem of a trafficnetwork design problem (NDP) under demand uncer-tainty. The origin–destination trip matrices are taken asrandom variables with known probability distributions.Instead of finding optimal network design solutions for agiven future scenario, we are concerned with solutions thatare in some sense “good” for a variety of demand realiza-tions. We introduce a definition of robustness accountingfor the planner’s required degree of robustness. We pro-pose a formulation of the robust network design problem(RNDP) and develop a methodology based on genetic al-gorithm (GA) to solve the RNDP. The proposed modelgenerates globally near-optimal network design solutions,f, based on the planner’s input for robustness. The studymakes two important contributions to the network designliterature. First, robust network design solutions are sig-nificantly different from the deterministic NDPs and notaccounting for them could potentially underestimate thenetwork-wide impacts. Second, systematic evaluation ofthe performance of the model and solution algorithm is

∗To whom correspondence should be addressed. E-mail: ukkuss@rpi.edu.

conducted on different test networks and budget levelsto explore the efficacy of this approach. The results high-light the importance of accounting for robustness in trans-portation planning and the proposed approach is capableof producing high-quality solutions.

1 INTRODUCTION

Network design is pervasive in many application con-texts due to its ability to influence the full hierarchy ofstrategic, tactical, and operational decision-making in amultistage system. As is well known, transportation net-work design is defined as selecting arcs in the networkG(N, A), with a set of nodes N and links A, for addition(discrete) or improvement (continuous) to minimize theentire network cost subject to a budget constraint, to-gether with the requirement that the flows satisfy the userequilibrium conditions. The literature presents many for-mulations and solution algorithms to solve this nonlin-ear, nonconvex mathematical program, which is difficultto solve optimally (Abdulaal and LeBlanc, 1979; Chiou,2005; Davis, 1994; Friesz et al., 1993; LeBlanc, 1975).

C© 2007 Computer-Aided Civil and Infrastructure Engineering. Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA,

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Robust transportation network design 7

In this research, we study an important variation of thenetwork design problem (NDP). The proposed modelaccounts for the uncertainty in the O-D demand anddemonstrates that accounting for uncertainty in long-term demand results in significantly different networkdesigns as compared to the deterministic equilibriumnetwork design. Moreover, the model not only accountsfor the expected value of the network performance inthe objective function but also the standard deviation ofthe network performance. We suggest here that such amodel be referred to as a robust network design problem(RNDP). This model determines robust network designplans for a traffic network with many zones and inter-connected regions. In addition, the proposed model isflexible enough to account for the planner’s desired ro-bustness based on the sensitivity of the total networkperformance. The methodology used in this article yieldssolutions that are less sensitive to the future demand real-izations than the classical deterministic NDP by account-ing for moments of the distribution such as the expectedvalue and standard deviation of the total system traveltime (TSTT).

There is a vast amount of literature on develop-ing formulations and solution algorithms for the de-terministic NDP. LeBlanc (1975) presented a branchand bound algorithm for solving the upper-level prob-lem, but the bounding step was dependent on theassumption that additional link improvements wouldalways reduce total user cost. Dantzig et al. (1979)introduced a convex formulation assuming system op-timal (SO) flow patterns and allowing continuous anddiscrete improvements; the solution algorithm was anextension of the decomposition approach introduced bySteenbrink (1974). Abdulaal and LeBlanc (1979) intro-duced a Lagrange multiplier approach for the continu-ous SO NDP, while Hoang (1982) proposed a schemebased on Benders’ decomposition to solve an integerversion of the problem. LeBlanc and Abdulaal (1984)suggested using SO NDP as a lower bound to theuser equilibrium (UE) based formulation. Suwansirikulet al. (1987) proposed a heuristic solution approachbased on the decomposition of the original problem to aset of sub-problems and demonstrated that this performsbetter than Abdulaal and LeBlanc’s (1979) approach.Friesz et al. (1992) presented a promising heuristic ap-proach based on simulated annealing and a tabu search-based heuristic by Mouskos (1991). More recently, Davis(1994) presented a formulation and a heuristic algorithmbased on a stochastic UE model. Yang and Bell (1998)presented a formulation that accounts for the elastic-ity of the demand, for the mixed continuous and dis-crete NDP. Chiou (2005) presents four variants of thegradient-based method for solving the bi-level continu-ous NDP and tests the computational efficiency of these

approaches. Poorzahedy and Abulghasemi (2005) alsostudied the NDP and adapted the Ant system heuristicapproach to solve the problem.

Most of the work so far has primarily been concen-trated in developing methodologies for the deterministicNDP. To the best of our knowledge there is currently lim-ited work (Atamturk and Zhang, 2006; Atamturk, 2003;Santoso et al., 2005) on incorporating robustness intothe NDP. Previous work in this area deals with stochas-tic programming models, which deal with uncertainty intransportation network design (Ukkusuri et al., 2004;Waller and Ziliaskopoulos, 2001). The formulation is asingle level stochastic linear program (SLP) for the dy-namic NDP incorporating the cell transmission modelas the traffic flow model. The stochastic programmingmodels guarantee a solution that is best in the sense thatit minimizes the expected cost of the system for a givendistribution of the uncertain demand. However, the ex-pected cost minimizing solutions are not necessarily ro-bust, as they do not account for higher moments of thetotal system travel time.

Explicit consideration of uncertainty in the NDP isa critical aspect of investment decision-making. Not ac-counting for the uncertainty can lead to sub-optimal in-vestments, which may prove very expensive in terms ofthe level of service of the overall transportation networkif the anticipated solution is not realized. A clear distinc-tion between robust optimization (RO) and stochasticprogramming (SP) methods is needed as it defines theirapplicability. SP models account for uncertainty by theminimization of the expected value objective function;whereas RO, as defined here, considers higher momentsof the probability distribution in addition to the expectedvalue of the objective function. Although, both the ROand SP afford the NDP problem to account for uncer-tainty, the RO model has little sensitivity to demand.Given the resiliency of the RO solution, the expectedcost of this solution may be higher than that of the SPsolution. RO models allow determination of the trade-off between the expected performances of the networkagainst higher moments based on how the network be-haves in high-consequence scenarios.

The importance of controlling the variability (volatil-ity) of the solution (as opposed to just optimizing itsfirst moment) is well recognized in finance primarily dueto the seminal work of Markowitz (1959). The need forrobustness has been recognized previously in a num-ber of application areas: the capacity planning modelfor the plastics industry (Paraskevopoulos et al., 1991),stochastic programming models (Sengupta, 1991), theoutsourcing problem in manufacturing (Escudero et al.,1993), multinational production scheduling (Gutierrezand Kouvelis, 1995), the capacity expansion of thepower systems under uncertain load forecasts (Malcolm

8 Ukkusuri, Mathew & Waller

and Zenios, 1994), and reliability-based network design(Sumalee and Watling, 2004). A similar definition ofrobustness will be used to obtain capacity expansionpolicies for the transportation NDP under demand un-certainty. In some problems, the mean-variance tradeoffmay not be meaningful. Examples include minimizingthe variance of the user’s travel time in descriptive mod-els and mean-variance analysis of a system optimum ob-jective in fleet management problems (List et al., 2003).In the RNDP, however, the performance measure is asystem-wide cost and the uncertainty is long term, amean-variance analysis provides a good estimate of theeffect of underlying uncertainty.

The last 20 years has increasingly exploited the useof meta-heuristic evolutionary algorithms for multi-objective optimization problems. Recent surveys on thistopic include monographs by Coello et al. (2002), Deb(2002), and Zitzler (2005). Other meta-heuristics such assimulated annealing (Friesz et al., 1993) and genetic localsearch (Drezner and Salhi, 2002) have led to good resultsfor the NDP under deterministic conditions. The meta-heuristics reduce the convergence to local solutions andincrease the possibility of reaching a globally optimumsolution. Drezner and Salhi (2002) in a recent study com-pared the performance of heuristics, such as descent al-gorithm, tabu search, simulated annealing, and geneticalgorithm, for the one-way and two-way network prob-lem to find the best network configuration so as to mini-mize the total travel time of all users. The objective wasto minimize total vehicle miles traveled. For realistic net-work sizes it was found that GA outperformed the otheralgorithms in finding the best solution, while taking alonger computation time than other methods. In termsof the quality of the solutions, GA was found to be thebest for the test problems, followed by simulated anneal-ing, and tabu search. Meta-heuristics have been previ-ously used (Mouskos, 1991; Friesz et al., 1993; Xiong andSchneider, 1995; Poorzahedy and Abulghasemi, 2005) tosolve the deterministic equilibrium NDP.

In this article a RNDP, which optimizes capacity ex-pansion polices aimed at improving the performance ofa transport network, is attempted. A mathematical for-mulation of the problem and a solution methodologyusing genetic algorithm is presented. The primary con-tribution of this work is in extending the RO concept totransportation NDPs and in developing a methodologyto demonstrate the value of robustness that enables bet-ter strategic transportation planning. This article is orga-nized as follows: the next section describes the definitionof robustness considered in this article and the detailedmathematical formulation. Section 3 covers the solutionalgorithm employed for solving the RNDP and Section 4presents numerous computational results for differentexample networks. Conclusions and future work are dis-cussed in the last section.

2 ROBUST TRANSPORTATIONNETWORK DESIGN

2.1 Choice of robustness measure

There are a number of uncertainties in the NDP. Typi-cal uncertainties include the uncertainty in O-D demand,available link capacity, and link cost function parameters,which are primarily due to random effects (e.g., incidents,weather) on the network. Other types of uncertaintiescould be due to the lack of precision in the handled val-ues or due to some error in those values based on per-ception. Furthermore, the uncertainty can be classifiedbased on the time frame of planning. The modeling ap-proaches would be different for long-term (strategic) andshort-term (operational) uncertainties in the NDP. Un-certainty in long-term O-D demand is considered here.This is important because the investment made in thepresent time has a significant effect into the future, anddeveloping solutions that are resilient to future realiza-tions is desirable. The control variable for the problemis the demand, which is realized in the future. This is de-noted by ω ∈ �n to {ω1, ω2, . . . . . . , ωs} for each scenarios ∈ S. The compact planner objective function for theRO model is

MinimizeTSTT

ρσ (TSTT, ω1, ω2, . . . , ωs)

+ (1 − ρ)λ(TSTT, ω1, ω2, . . . , ωs),

where TSTT represents the TSTT for each realization ofthe demand. There are a number of choices for σ (·) andλ(·) depending on what the value of the various errorsis to the decision-maker and the engineering problem tobe modeled. However, it is desirable to choose measuresthat can lead to consistent preferences between alterna-tive solutions. One such measure of solution robustnesscould be to use σ (·) as the expected value of the TSTTthat would be experienced for all realizations of the de-mand and λ(·) to denote a measure of the variability ofthe TSTT for all the realized demands, that is, λ(·) de-notes the variance of the system over all future demands.It is this measure that we will use in our modeling withweights given to λ(·) and σ (·). The weight ρ can vary from[0, 1] ∈ �. Another possibility for measuring robustnesscould be to calculate the maximum regret that would beexperienced by not following the optimal scenario. Thisis equivalent to λ(·) = maxs{(ξs − ξ ∗

s )}, where ξ s is theuser optimal objective function value given the actionwe choose (for example, a particular capacity expansionpolicy) when scenario s becomes true and ξ ∗

s is the objec-tive function value for the optimal plan had we knownthat the scenario s was going to be true. This measure ofrobustness is, however, not considered in this article.

Robust transportation network design 9

2.2 The model formulation

The following notation will be used in the modelformulation:

tωa (xωa ) = the unit cost of transportation on arc a ∈ A,

assumed to be twice continuously differen-tiable in xω

a , given that the demand ω ∈ Ω isrealized;

Zω(x) = the user equilibrium objective for each de-mand realization ω ∈ Ω;

f rs,ωk = the flow on path k ∈ K between O-D pairs r ,

s ∈ R for demand realization ω ∈ Ω;

δrsa,k =

⎧⎪⎨⎪⎩1 if arc a ∈ A is on path k between

O-D pair r, s

0 otherwise

⎫⎪⎬⎪⎭xω

a = the flow on arc a ∈ A for demand realizationω ∈ Ω;

qωrs = the uncertain demand for transportation be-

tween O-D pair r , s ∈ R and demand real-ization ω ∈ Ω;

pω = probability that demand ω ∈ Ω is realized;γ a = the unit cost of improvement on arc a ∈ A;ya = the capacity of a single lane on arc a ∈ A;ρ = the weighting factor associated for the plan-

ner for expected cost Vs network robustness;B = the total budget available for improvements.

The discrete RNDP can be formulated as:P1:

U

miny

ρ∑∀ω

[pω

∑∀a

xωa tω

a

(xω

a , ya)]

+ (1 − ρ)

[∑∀ω

pω

{∑∀a

xωa tω

a

(xω

a , ya)

−∑∀ω

[pω

∑∀a

xωa tω

a

(xω

a , ya)]}2

⎤⎦ 12

subject to∑∀a

γa ya ≤ B ya = 0 or 1 ∀a ∈ A

L

min Zω(x)=∑∀a

∫ xa

0

tωa (ε) dε

subject to∑∀k

f rs,ωk = qω

rs k ∈ K; r, s ∈ R; ω ∈ Ω

xω

a=

∑r

∑s

∑k

δrs,ωa,k f rs,ω

k r, s ∈ R; a ∈ A; k ∈ K; ω ∈ Ω

f rs.ωk ≥ 0 r, s ∈ R; k ∈ K; ω ∈ Ω

xω

a≥ 0 a ∈ A; ω ∈ Ω

The formulation P1 comprises two levels. The upperlevel U refers to the system planner’s objective of min-imizing the weighted sum of the expected TSTT andthe standard deviation of the TSTT, while the flow tothe upper level is obtained from the user equilibriumin the lower level L for each demand realization ω ∈ Ω.The above model, P1, differs from the traditional NDPbecause of the stochastic nature of demand in L and theweighted stochastic objective function in U. Link costfunction in this formulation is the typical BPR function:ta(x) = FFTa + α( xa

Ca)β , where FFTa represents the free-

flow travel time on link a, Ca represents the capacity oflink a and α, and β represent the typical BPR parameters.

2.3 Motivating example

The effect of demand uncertainty is initially demon-strated on a small network to show the difference be-tween the results from optimal network expansions poli-cies for the expected value and the standard deviationof TSTT. The benefit of analysis on a small network isthat all the uncertain variables can be enumerated andthe solutions are not driven by sampling errors. Con-sider the small network shown in Figure 1. The net-work parameters used are shown in Table 1. y1 andy2 denote capacity improvements which will be deter-mined by NDP under different objectives. This net-work is assigned an origin–destination (OD) demandof 50 vehicle-trips between nodes C and D and 50vehicle-trips between A and D. The latter of thesedemands is taken as uncertain with possible valuesvarying uniformly between 0 and 100. A single uncer-tain variable is used here to enumerate all possibledemand outcomes and avoid any sampling errors forthis demonstration. As only discrete values are taken

Fig. 1. Sample test network.

10 Ukkusuri, Mathew & Waller

Table 1Link parameters for the simple test network

Free-flow Link Linktravel time parameter parameter Capacity

Link (FFT) (αk) (βk) (Ck)

1 1 0.15 4 20

2 1 0.15 4 20 + y1

3 1 0.15 4 30

4 1 0.15 4 30 + y2

5 1 0.15 4 40

for the possible demands, this results in 100 potentialfuture realizations, each with an equal probability of oc-curring. To see the effect of this demand variance onnetwork improvement decisions, 100 UE NDPs weresolved for this network, each with a different planningdemand level. The planning demand is an altered de-mand level used for the basis of a decision. Each prob-lem arrived at a network improvement policy (y1 andy2), which represents an increase in capacity and theirsums must not exceed a fixed budget of 10. For each ofthese proposed expansion policies (y1 and y2), all 100possible demand realizations were evaluated using de-terministic user equilibrium assignment. The average to-tal travel time represents the expected performance ofthe system, because each possibility was enumerated andeach scenario has an equal probability of being realized.These 100 scenarios were then averaged for each run.The results from the analysis shows that a policy expan-sion based on a demand of 95 yields a lower TSTT ascompared to the policies at the expected demand of 50.The difference in TSTT was modest in this network; itwas around 8.32 (y1 = 7.9, y2 = 2.1) for the former and8.98 (y1 = 0, y2 = 10) for the latter. However, it wasobserved that if the planning policy considers only thevariance of the TSTT in consideration for expansion, apolicy based on a demand of 8 gave the lowest varianceof travel time. The expected travel time was only slightlyhigher in this case (8.35) and the corresponding expan-sion policies were y1 = 9.8 and y2 = 0.2. This simpleexample provides two clear insights. Network expansionpolicies are significantly different in the presence of un-certainty and the network design decisions differ con-siderably based on the mean-variance considerations inthe objective function. Additionally, designing networksfor robustness yields solutions that are more resilient tofuture conditions; however, there is potentially a trade-off in terms of the increase in the expected value of totalsystem travel time.

The main difficulty in solving the robust network de-sign solution is in finding the network design solutionsfor each demand realization. With the increase in the

number of demand realizations the problem becomesextremely difficult to solve computationally, when it iswell known that even finding the capacity expansions inthe deterministic NDP is a difficult nonconvex problem.It is important to realize that the model presented herediffers from other equilibrium network design models,not only in accounting for uncertainty, but also in explic-itly capturing the solution robustness. As such, the modelpresented here is computationally more demanding thanother problems. In the next section, we tailor the geneticalgorithm to solve the RNDP efficiently. As we shall see,the set of network design solutions for RNDP on testnetworks demonstrate the value of accounting for ro-bustness in long-term transportation planning decisions.

3 SOLUTION METHODOLOGY

Traditional methods to solve the RNDP do not workwell due to the computational complexity of the nonlin-ear, nonconvex model in P1. The intricate nature of theproblem, primarily due to the existence of many localminima in typical transportation networks, and the sizeand complexity of the search space, cannot be handled bytraditional greedy search algorithms. Because the formu-lation is intractable with traditional mathematical pro-gramming methods, the RNDP is better suited for theapplication of metaheuristics. In this article, the geneticalgorithm (GA) is adopted for the RNDP and a briefoverview of the genetic algorithm and the solution algo-rithm is presented below.

3.1 Overview of GA implementation

GA finds the capacity improvements for RNDP by eval-uating the system objective to obtain better solutions ineach generation for each demand scenario. These are fi-nally used in a robust analysis (for mean and standarddeviation) returning the fitness function values of thetraffic network.

A genetic algorithm is a search algorithm, which worksstarting from an initial collection of strings represent-ing possible solutions of the problem. Each string of thepopulation is called a chromosome, and has an associ-ated value called a fitness function that contributes tothe generation of new populations by means of geneticoperators (denoted as reproduction, crossover, and mu-tation). The initial population is generated randomly,or it may consist of a number of known solutions, ora combination of both. The GA goes through a num-ber of steps in which the population at the beginningof each step is replaced with another population, whichit is hoped will include better solutions to the problem.The chromosomes at each new generation are produced

Robust transportation network design 11

by a process called reproduction, in which the chromo-somes of the old population are combined to create newones. A detailed explanation of the working of GA canbe found in Goldberg (1989) and Deb (2002). GA isapplicable to RNDP, which is a nonconvex and nonlin-ear problem because of its superiority over other searchtechniques, which are limited by the continuity, differ-entiability, and unimodality of the evaluated functions.GA handles these limitations by: (1) operating with thecodes of the parameter set and not with the parametersthemselves; (2) searching for a population of points andnot a single point; (3) using objective function informa-tion and not the derivative of the function; and (4) usingprobabilistic transition rules and not deterministic ones.All these features make GA an attractive choice in solv-ing the RNDP problem.

The working of GA involves coding of the solution, ini-tializing the coded solution, computing the fitness valueand the application of genetic operators to generate bet-ter offspring. In order for the GAs to solve the RNDP,the decision variables are first coded into a form whereGAs can operate. The decision variables are the capac-ity expansion ya for each link a. The decision is whetherthe link a will be expanded or not and is denoted bybinary states 1 or 0, respectively. Therefore, each vari-able is coded as 1 or 0 and the length of the string orchromosome will be the number of links in the network.Bit 1 indicates that the capacity of the link is expandedby 100% (i.e., the existing capacity is doubled) and a bit0 indicates that no improvement occurs. Once the cod-ing scheme is finalized; an initial population of solutioninstances is randomly created. It may be noted that thecoding scheme ensures that the randomly created popu-lation is feasible as well as drawn from diverse locationsof the search space.

GA works with coded variables and does not have anyinherent knowledge about the problem. The problem-specific information is provided by the fitness function,which is used for measuring the quality of individuals ineach generation. In our specific case this is the weightedsum of the TSTT represented in U. Conventionally, GAmaximizes fitness, but it is trivial to consider a cost func-tion minimization problem by assigning the fitness to bea negative of the cost function. The RNDP imposes con-straints on the acceptable total allowable capacity im-provements (budget constraint), hence it is possible thata chromosome maps to a solution would violate budgetconstraint, in some cases. The option of rejecting everyinfeasible solution that violates constraints may some-times lead to rejection of some good partial solutionsand may be computationally inefficient. The objectivefunction in the RNDP accounts for these constraints bypenalizing each solution that breaks the constraint by in-troducing a penalty term to the calculated fitness value,which depends on the constraint and the extent of vio-

lation. We adopt this “penalty method” that allows newconstraints to be added easily to the GA optimizationfor the RNDP.

Based on the fitness value, GA computes the nexttrial solution using genetic operators. Application ofthese genetic operators is expected to yield better off-spring and is repeated till convergence. The basic geneticoperations commonly used are reproduction, crossover,and mutation. Reproduction (or selection) is a proce-dure in which better fitness values are retained andinferior ones are eliminated from the current popula-tion (Michalewicz, 1992). A number of selection mech-anisms have been proposed in the literature and all ofthem attempt to achieve the correct balance betweenthe population diversity and selective criteria, which arefundamental in determining the convergence of the algo-rithm. In our solution approach, a popular and superiorselection mechanism called “remainder stochastic sam-pling without replacement” is used (Goldberg, 1989). Inthe crossover operation, a recombination process cre-ates individuals in the successive generation by com-bining information from two individuals of the previousgeneration. Crossover is done at the string level by ran-domly selecting two strings for crossover operation. Weuse a two-point crossover, a very effective solution forthe disruption of the schemata problem. Mutation addsnew information in a random way to the genetic searchprocess and ultimately helps to avoid getting trappedat local optima. It operates at the bit level, when thebits are being copied from the current string to the newstring. Mutation operates with a probability, usually avery small value called the mutation probability. A coin-toss mechanism is employed; if a random number gener-ated between 0 and 1 is less than the mutation probabilitythen the bits are flipped (i.e., zero become one and viceversa).

There are several strategies for stopping the evolutionprocess of the GA. Because it is difficult to define theoptimal solution, usually two procedures are adopted asconvergence criterion: (1) the GA procedure is stoppedwhen the variation in the fitness level among genera-tions is within a user-defined range; and (2) the iterationis stopped when the number of generations has accumu-lated to a predetermined level. In this research, the GAwas stopped when it reached a predefined number ofgenerations.

3.2 Proposed algorithm for RNDP

The algorithm of the RNDP is shown in Figure 2 and themain steps are discussed below. The input to the modelincludes GA parameters such as number of iterations forconvergence, population size, crossover probability, mu-tation probability, number of decision variables, lowerbound, upper bound, and the precision of each variable.

12 Ukkusuri, Mathew & Waller

Fig. 2. A GA-based solution algorithm for RNDP.

The GA parameters for the present study have beenarrived at by a limited set of experiments as describedlater. The network data include node and link data, costfunction parameters (length, alpha, beta, capacity, andfree-flow travel time) as described in the Appendix. Therobust data include the sample size, random seed, theweight ρ in the range [0, 1], percentage of demand vari-ation, and budget.

The next step in the algorithm is the GA coding. Here,the number of variables is equal to the number of links,and because it is a binary decision variable, the lowerbound of each variable is 0, upper bound is 1, and theprecision is 1. The string length is equal to the numberof links, where each bit corresponds to a link. A value of1 indicates that the link’s capacity will be doubled.

The next step is to initialize the population. This isnormally done by randomly initialing P strings with 1 or0. Note that, coding ensures every population is feasible.Then the GA iteration starts. First, for each string (cor-responding to an instance of the solution) it is decoded.The decoded values are the links whose capacities are tobe increased.

Then for every sample, a random sample of the ODmatrix is generated. This is done by multiplying everycell of the base OD matrix by a random factor. Sup-pose, if the demand variation is 50%, then this factoris a random value between 0.75 and 1.25. (i.e., MonteCarlo simulation for randomly selected demands in [l,u]of the expected value.) This sampled OD matrix is as-signed to the network using standard UE assignment,using Frank–Wolfe algorithm. This gives the TSTT foreach sample.

Next, the objective function value is computed byfirst calculating the expected value and the stan-dard deviation of the total system travel times for

each scenario and finding the expected performanceand robustness of the capacity expansion policiesusing the objective—miny ρE[

∑∀a xata(xa, ya)] + (1 −

ρ)Var[∑

∀a xata(xa, ya)]. The base problem is a con-strained one and this objective function is transformedinto an equivalent unconstrained problem by the follow-ing transformation:

�(x) = minya

ρE

[∑∀a

xata(xa, ya)

]

+ (1 − ρ)Var

[∑∀a

xata(xa, ya)

]+ ri (Max(Ci − B), 0)

where ri is the penalty term, Ci is the cost of capacityimprovement and B is the budget. The value of �(x) ispassed to GA to compute the fitness function value.

The fitness function value is compared with the cur-rent best solution and this solution is updated if thefitness function value is improved. This is followed bythe application of GA operators mentioned above. Thiscompletes one GA run and will be repeated till conver-gence. The primary output is the links that are chosenfor capacity expansion. In addition, other statistics suchas expected value of TSTT represented as E(TSTT) andstandard deviation of TSTT referred to as S(TSTT) canbe inferred.

4 COMPUTATIONAL RESULTS

To test the above model and solution approach forRNDP, we perform computational experiments on asmall-sized and a medium-sized example networks. Thefollowing parameters for the GA have been chosen forthese experiments: the number of generations requiredfor convergence gmax is 500, the population size or thenumber of parent solutions is 50, the mutation proba-bility is 0.001, and the crossover probability is 0.8. Theabove parameters for the GA were arrived at by con-ducting a series of systematic experiments based onvarying them over an interval in equidistant steps. Forexample, Appendix B in Ukkusuri (2005) shows theplot of number of generations required for convergencekeeping all the other things the same. Altogether, wehave conducted around 30 runs for finding good pa-rameters due to the limitations of computational time.Other approaches for determining the parameter val-ues for GA is based on the idea of applying a different(meta) evolutionary algorithm. This idea is outlined inHanne (2001). The parameters used in this analysis pro-duced reasonably robust results. The implementations

Robust transportation network design 13

have been conducted on Xenon 3.2 processor on a Linuxplatform using a gcc compiler and the model was codedin C/C++.

4.1 Experiment 1: Harker–Friesz (HF) network

Initially, we test our methodology for the determinis-tic case on the HF network. The motivation for thisanalysis is that for the HF network the deterministicNDP solution is available in the literature (Harker andFriesz, 1984; Chiou, 2005) and can be compared with theNDP solution using the GA. However, the solutions forthese test networks are in a deterministic and continu-ous network improvement setting and the comparisonwill be made accordingly. The data used for the HF net-work are presented in the Appendix. These data are thesame as those presented in Harker and Friesz (1984) forcomparison purposes.

The first experiment is conducted on the HF networks(Harker and Friesz, 1984) for which bounds on the deter-ministic continuous NDP solutions are available at dif-ferent demand levels. This comparison should prove thevalidity of the meta-heuristic approach for the proposedproblem. In reality, there are no rigorous techniques forvalidating the stated problem because of the lack of ex-act algorithms to solve the RNDP, however, the com-parison of the static NDP should provide reasonableconfidence on the proposed solution methodology. Fur-thermore, the proposed approach should facilitate thedevelopment of other efficient solution approaches forRNDP in the future.

The network is shown in Figure 3 with 6 nodes and16 arcs. There are two O-D pairs—(1,6) and (6,1). Thestatic continuous NDP is solved at different demand lev-els with q16 = 2.5, 3.75, and 5 and q61 = 5.0, 7.5, and 10.0,respectively. It was found in this study that at low lev-els of congestion the heuristic bounds derived are veryclose to optimal. A good lower bound of the continu-ous NDP can be found by solving the NDP based onthe system optimal network flows where in the lower

Table 2Comparison of results from Harker and Friesz (1984) and the GA approach for NDP

Total flow = 7.5 Total flow = 11.25 Total flow = 15.0 System cost

Improved HF GA HF GA HF GA HF GAarc result result result result result result result result

y3 (4.24, 4.24) 2.84 (4.17, 4.24) 3.70 (3.65, 4.24) 2.68 (63.28, 63.28) 63.38

(7.5)

y6 0 0.2 (0, 0.77) 0.1 (0, 6.07) 0.61 (99.14, 99.69) 101.03

(11.25)

y15 (13.14, 13.14) 13.10 (13.06, 13.14) 12.36 (12.51, 13.14) 13.1 (140.21, 147.59) 147.22

(15.0)

1

2

3

6

4

5

1

2

3

6

5

9

12

8

10 13

11

15

1614

7 4

Fig. 3. Harker–Friez (HF) network.

problem L is solved using the marginal travel time costfunction. Harker and Friesz (1984) and Suwansirikulet al. (1987) develop an inexact solution procedure basedon the iterative optimization assignment (IOA) methodto calculate the upper and lower bound for the continu-ous NDP as shown in Table 2. We compare the results forGA at low levels of demand; these results are presentedin Table 2. Note that the results from Harker and Friesz(1984) overestimate the NDP solution (Cournot Nashgame). However, as shown these values are tight andthe actual NDP solution lies in between the NDP solvedwith a system optimal objective and the Cournot–Nashgame. The results of the GA match very close to the NDPsolutions based on the bounds developed by Harker andFriesz (1984) as shown in Table 2. Furthermore, the sys-tem cost of the network at the three demand levels isalso close using the GA approach.

4.2 Experiment 2: Nguyen–Dupius network

The main results of the RNDP model are demonstratedon a middle-sized network shown in Figure 4, thenetwork of Nguyen and Dupius (1984), which has beenextensively used before by researchers for testing thetraffic equilibrium problems. With the validation of theGA approach on the test network in Figures 1 and 3,

14 Ukkusuri, Mathew & Waller

3

110

12

17

2

13

7

4

14

11

6

18

4 5

12

9

1

7

11

13

10

6 8

2

39

15

8

16

5

19

Fig. 4. Nguyen Dupius test network.

numerical experiments are conducted in this section tostudy the results obtained by the GA approach for theRNDP on the Nguyen–Dupius network shown in Fig-ure 4. Descriptions of the OD demand table and thenetwork parameters are given in the Appendix. Differ-ent sets of the values of ρ equal to 0, 0.25, 0.5, 0.75, and

Table 3RNDP solution using GA for different values of ρ

Base casecapacity NDP(veh/hr) ρ = 0 ρ = 0.25 ρ = 0.5 ρ = 0.75 ρ = 1 [E(OD)]

y1 2,200 0 1 0 0 1 0

y2 2,200 1 1 1 1 1 1

y3 2,200 0 1 1 0 0 0

y4 2,200 1 1 1 1 1 1

y5 2,200 0 0 1 0 0 0

y6 2,200 0 0 1 1 0 0

y7 2,200 1 1 0 1 1 1

y8 2,200 1 0 1 1 1 1

y9 2,200 0 0 0 0 0 0

y10 2,200 1 1 1 0 1 0

y11 2,200 1 0 1 1 1 1

y12 2,200 0 1 0 0 1 1

y13 2,200 0 1 1 1 1 1

y14 2,200 1 0 0 0 0 1

y15 2,200 1 0 0 0 1 0

y16 2,200 1 1 0 1 0 1

y17 2,200 0 0 0 1 0 0

y18 2,200 1 1 1 1 0 1

y19 2,200 1 1 1 1 1 1

E[TSTT] 857,046 571,022 561,710 563,749 556,102 549,881 539,536

S[TSTT] 0 59,191 61,596 61,809 60,237 59,834 0

# 1 11,016,028 1,611,749

Note: # denotes the number of user equilibrium assignments the inner optimization loop uses for RNDP. 0 implies that the link hasnot been improved and 1 implies that the link has been improved by a single lane.

1 are used to solve the discrete RNDP. When ρ equals0 it is the robust network design which accounts primar-ily for standard deviation and when ρ equals 1 it is thestochastic network design, where the planners objectiveonly accounts for the expected network-wide costs. Forthe OD demand matrix, a uniform distribution with theexpected values shown in the Appendix and a coefficientof variation (COV) of 0.5 is assumed.

Table 3 summarizes the main results obtained afterapplying the GA to the Nguyen–Dupius test network.It can be observed that accounting for different valuesof robustness yields different link capacity expansions.There is a significant tradeoff between accounting forjust the expected value of the network performance andthe risk (captured in terms of standard deviation) as-sociated with it. An important insight from this analy-sis is that the network design solutions are significantlydifferent based on the desired degree of robustness ac-counted for in the NDP. Figure 5 shows the convergenceof GA for a value of ρ equal to 1. The convergence ofGA is primarily visual with the number of generations.Other quantitative approaches for measuring the GAperformance are suggested by Veldhuizen and Lamont(2000).

Robust transportation network design 15

545000

550000

555000

560000

565000

570000

575000

580000

1 51 101 151 201 251 301 351 401 451

Generation number

Fit

ness f

un

cti

on

Rho=1.00

Fig. 5. Convergence of the RNDP fitness function for ρ = 1.

From Table 3 it can be observed that there is a sig-nificant improvement in the network performance inthe stochastic (ρ = 1) and the robust cases (ρ = 0)as compared to the base case network. The percent-age travel time savings for the stochastic and robustnetwork design solutions as compared to the base caseare 35.83% and 33.33%, respectively. The improvementin network resiliency by accounting for robustness inthis network is about 1.3%. The last column in Table 3represents the network design solution at the expectedvalue of demand. This quantifies the expected value so-lution of the problem. Although this network design so-lution performs well as compared to the base case, thereis a significant underestimation in the network perfor-mance. The comparison of this solution with the stochas-tic and robust solution shows that network performanceis underestimated by 1.89% and 5.60%, respectively.This underestimation can be greater based on the degreeof uncertainty and the network topology. Furthermore,there is no guarantee on the robustness of the obtainedsolution in the expected value solution.

From Table 3 it can be observed that by accountingfor standard deviation in the NDP there is a benefit ofimproving the resiliency of the network performance al-though there is a tradeoff in terms of the slight increase inthe expected network performance. In addition, it can benoticed that the design decisions on the network topol-ogy (link improvements) vary depending on the robust-ness accounted for in the RNDP. It is important to note,however, that there is no consistent relationship betweenthe expected network performance and the robustnesswith different values of ρ. For example, the stochasticcase (ρ = 1) has a lower standard deviation as comparedto the intermediate case when ρ = 0.75. This nondomi-nance effect should be explored further.

We also explored the network performance changesdue to the change in the budget. It is clear that in theextreme case of a very large budget the traffic condi-tions follow free flow with almost no variability in traveltimes. Table 4 shows the change in the expected networkperformance and the robustness for increasing value ofthe budget for the stochastic and the RNDPs. It can be

Table 4Change in network performance with budget

Robust (ρ = 0) Expected (ρ = 1)Budget(in $mil) E(TSTT) S(TSTT) E(TSTT) S(TSTT)

2 715,430.3 83,728.25 714,222 84,629.99

5 684,266.3 79,561.21 681,507.7 79,666.49

20 571,022.4 59,191.3 549,881.2 63,834.33

50 507,615.1 53,484.74 502,218.4 53,577.83

observed that with the increase in budget in both the net-work design solutions, the expected value and the stan-dard deviation of the network performance improves asexpected.

For the population size of 50, an evolution of 500 gen-erations, and a sample size of 5,000 it was noticed thatthe computational requirements for the C++ code arereasonable. A typical run takes 5.76 hours of CPU time.It was observed that about 95% of CPU time is requiredin evaluating the fitness function, that is, the weightedstochastic objective function in U considering the lowerlevel problem L. A sample of the number of user equilib-rium assignments performed to arrive at the final solu-tions is shown as # in Table 3. Further analysis of theC/C++ code has shown that major parts of the run-ning time can be minimized by using more efficient algo-rithms for evaluating the user equilibrium assignment.Furthermore, the GA can be sped by parallelizing thegenetic algorithm, improving the sampling techniques,and adopting efficient genetic programming. Further-more, it can be observed that the number of DUE (Sheffi,1985) iterations for convergence is very high. This is be-cause the convergence of the Frank–Wolfe is set equal to0.01, which is relatively high when many DUE iterationshave to be performed. We observe that by reducing theconverge criterion to 0.1 the whole process can be spedup considerably.

4.3 Evaluation of GA network design solutions

The GA solution procedure uses a smaller sample size(of 5,000 samples) because of the computational timeinvolved in arriving at the high-capacity network de-sign solutions within a pre-specified number of genera-tions. To determine the quality of the solutions obtainedfor the RNDP using the GA approach, we propose anevaluation approach. This section describes this evalua-tion approach.

For each RNDP solution in Table 3, we improve thenetwork with the “near optimal” capacity values. We testthis new capacity added network on a larger sample sizeof 25,000 samples to determine the expected value andstandard deviation of the TSTT (network performance).The intuition behind doing this evaluation is to test how

16 Ukkusuri, Mathew & Waller

Table 5Comparison of the network-wide travel time with GA and evaluation approach

Absolute percentEvaluation with difference

GA results 25,000 samples% Deviation % Deviation

E[TSTT] S[TSTT] E[TSTT] S[TSTT] of E[TSTT] of S[TSTT]

Base case (without 725,326 NA 736,268 86,322 1.5 NA

improvement)

ρ = 0 571,022 59,191 572,730 59,345 0.29 0.26

ρ = 0.25 561,710 61,596 563,589 61,839 0.33 0.39

ρ = 0.5 559,749 61,809 565,580 61,941 0.32 0.21

ρ = 0.75 556,102 62,237 557,924 62,436 0.33 0.33

ρ = 1 549,881 63,834 551,633 63,953 0.32 0.20

NDP [E(OD)] 626,354 NA 558,015 60,482 10.91 NA

well the RNDP solution obtained in Table 3 performs ata higher sample size and compare the differences. The“optimal” TSTTs for the RNDP using GA and the eval-uation approach are shown in Table 5. It is observedthat the RNDP capacity improvements from GA per-form very well on a higher sample size in terms of theexpected travel time and the standard deviation of thetravel time. However, this insight cannot be generalizedfor all transportation networks because of network spe-cific characteristics and the sample size used to performthe evaluation.

5 CONCLUSION

With the maturity of computational techniques and mod-eling approaches, new opportunities exist for studyingthe value of robustness in network design. Tractablemodel formulations, solution methodologies, and ad-vanced computational techniques are critical for devel-oping effective robust network solutions. The model for-mulation and solution approach presented in this articleprovide a means for accounting for uncertainty in net-work design decisions and supporting explicit decisionson the trade off between expected costs against risk,when the long-term O-D demand is a random variable.The problem is formulated as a nonlinear, nonconvex,multi-objective model where the planner’s objective is tominimize the weighted objective of the expected valueand the standard deviation of the network total traveltime and the user’s route choice is dependent on userequilibrium under finite scenarios of uncertain demand.

An efficient solution approach using a GA is proposedfor solving the RNDP. The key concept of the solutionmethod is in obtaining the solution for the RNDP by sys-tematically exploiting the randomness of the genetic evo-lution process. As demonstrated, the proposed approachcan produce optimal solutions with a reasonable compu-tational time and overhead cost. The main contribution

of this work is in formulating the RNDP and in propos-ing an evolutionary structure for solving the problem ina mathematical programming framework. Furthermore,the problem has been solved by adjusting suitable sys-tem parameters for the RNDP. Numerical comparisonshave been made on three test networks. First, for the4 node 5-link network the GA method produces simi-lar results to the deterministic continuous NDP resultsobtained by enumeration. Furthermore, in the 16-linkexample network of Harker and Friesz (1984) the GAapproach produced very close results as compared to theresults in the literature. Second, for the Nguyen–Dupiusnetwork the solution of the RNDP using a GA heuristicgives a very good framework for obtaining “good” net-work design solutions accounting for different degrees ofnetwork travel time variability. The computational timewas, however, high and it is difficult to solve large net-works with the proposed approach.

For the improvement of the RNDP models, this is-sue requires further research in incorporating a varietyof real-life networks in the future. In addition, efficientapproximation procedures such as sampling proceduresand the single point approximations (Ukkusuri andWaller, 2006) should be explored for solving large scaleRNDPs. Another issue deserves further remark. Asignificant bottleneck in implementing the problem com-putationally is in evaluating the fitness function for eachdemand realization and then solving the NDP. The fitnessevaluation procedure can be computed faster by using aparallelization approach to solve this problem and couldhelp in developing flexible RNDP solutions.

ACKNOWLEDGMENTS

This research is supported by National Science Founda-tion CAREER Award under Grant No. CMS 0347005.Any opinions, findings, and conclusions expressed in this

Robust transportation network design 17

article are those of the authors and do not necessarilyreflect the views of the National Science Foundation.

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APPENDIXTest network parameters for the HF network

FF PenaltyArc Length No. of speed associateda (mi) lanes α β Ca (mi/hr) with each arc

1 1 1 10 4 3 60 2

2 2 1 2.5 4 10 60 3

3 3 1 1 4 9 60 5

4 4 1 5 4 4 60 4

5 5 1 10 4 3 60 9

6 2 1 10 4 2 60 1

7 1 1 10 4 1 60 4

8 1 1 1 4 10 60 3

9 2 1 4 4 45 60 2

10 3 1 1 4 3 60 5

11 9 1 0.222 4 2 60 6

12 4 1 2.5 4 6 60 8

13 4 1 6.25 4 44 60 5

14 2 1 16.5 4 20 60 3

15 5 1 1 4 1 60 6

16 6 1 0.167 4 4.5 60 1

Test network parameters for the Nguyen–Dupius network

Length No. of FFFrom To (mi) lanes α β Ca speed

1 5 4.08 1 3.93 1 2,200 35

1 12 5.25 1 2.44 1 2,200 35

4 5 5.25 1 2.44 1 2,200 35

4 9 7 1 0.92 1 2,200 35

5 6 1.75 1 5.5 1 2,200 35

5 9 5.25 1 1.83 1 2,200 35

6 7 2.91 1 5.5 1 2,200 35

6 10 7.58 1 0.85 1 2,200 35

7 8 2.92 1 5.5 1 2,200 35

7 11 5.25 1 3.06 1 2,200 35

8 2 5.25 1 3.06 1 2,200 35

9 10 5.83 1 1.1 1 2,200 35

9 13 5.25 1 1.22 1 2,200 35

10 11 3.5 1 0.92 1 2,200 35

11 2 5.25 1 1.22 1 2,200 35

11 3 4.67 1 2.75 1 2,200 35

12 6 4.08 1 0.79 1 2,200 35

12 8 8.17 1 1.57 1 2,200 35

13 3 6.42 1 2 1 2,200 35

The expected OD demand table for the Nguyen–Dupius

network

From/To 1 4

2 1,900 2,300

3 2,100 1,700