THE SAMPLE AVERAGE APPROXIMATION …transctr/pdf/trb_paper_gopal.pdfTHE SAMPLE AVERAGE APPROXIMATION METHOD FOR FLEXIBLE NETWORK DESIGN PROBLEM Gopal R. Patil PostDoctoral Researcher,

THE SAMPLE AVERAGE APPROXIMATION METHOD FOR FLEXIBLE NETWORKDESIGN PROBLEM

Gopal R. PatilPostDoctoral Researcher, UVM Transportation Center,

University of Vermont, Burlington, VT, USAPhone: 1-802-656-8567, Fax: 1-802-656-9892

Email: [email protected]

Satish V. UkkusuriAssistant Professor, Department of Civil and Environmental Engineering,

Rensselaer Polytechnic Institute, Troy, NY, USAPhone: 1-518-276-6033, Fax: 1-518-276-4833

Email: [email protected]

Transportation Research Board 87th Annual MeetingJanuary 13-17, 2008 Washington, D. C.

Word Count: 4150 + 12*250 = 7150Date of submission: November 15, 2007

TRB 2008 Annual Meeting CD-ROM Paper revised from original submittal.

Patil and Ukkusuri 2

ABSTRACT

Finding an optimal investment strategy to efficiently use the scarce resources is challenging, sincethe transportation network parameters such as demand, capacity and travel cost are uncertain. Se-quencing investment over time can give flexibility to planner to change, delay, or even abandon thefuture investment based on system realization. In this paper, we present a stochastic mathematicalprogram with equilibrium constraints (STOCH-MPEC) formulation for a multi-stage network de-sign problem, called flexible network design problem (FNDP), accounting for demand stochasticityand demand elasticity. STOCH-MPEC problems can be computationally intractable, if number ofscenarios are large and/or study network is bigger. In order to reduce the associated complexityof FNDP, we use the sample average approximate method (SAA), to solve the proposed FNDP.We implement the SAA to a test network and compare the performance of SAA with differentsample sizes. We show that SAA can produce solutions which are close to the true solutions withconsiderably less number of scenarios.



1 Introduction

Sequencing infrastructure investment over time gives flexibility to transportation planners whennetwork parameters such as demand, travel cost, and link costs are uncertain, and are difficult toforecast accurately. Such an investment strategy gives planner options to change, delay, or evencancel the future investment under unfavorable network realizations, there by making an efficientuse of the available resources (1; 2). These options are of great value mainly because, (a) trans-portation networks parameters are inherently uncertain , (b) network improvement investments arehuge, and (c) the investments are irreversible. Adopting point estimates of the uncertain parame-ters, based on past trends, results in sub-optimal decisions (3; 4). Making investments based on thepoint estimates of the uncertain parameters is risk-prone, and often lead to significant economiclosses (e.g. see 5; 6).

Finding quantities of link improvements in a transportation network under budget con-straint is called the network design problem (NDP) in transportation engineering literature. InNDPs planner’s objective is to improve network considering network wide benefits, and travelersare selfishly behaving to reduce their own travel cost. The network design problem in which plan-ners flexibility is introduced under stochasticity by sequencing the capacity improvements overtime is defined as Flexible network design problem (FNDP) (1; 2). FNDP in which demand isassumed to be stochastic and travel cost and capacities are taken deterministic is studied in thispaper. The resulting of the NDP studied is a stochastic program because origin-destination (OD)demand is assumed to be stochastic. This problem can either be formulated as a bi-level mathemat-ical program or a single level mathematical program with equilibrium constraints (MPEC); boththe formulations are non-convex and are very difficult to solve. Our formulation comes under thelatter category. Since the demand is stochastic, the formulation is called stochastic mathematicalprogram with equilibrium constraints (STOCH-MPEC) (see 7–9). Stochastic NDP formulationscan be thought as here-and-now type two-stage stochastic programming problems. In the first stagethe improvement decisions are made without knowing the demand realization. When the demandis realized in the second stage, UE is solved to get the actual demand and link flows. The evaluationof the above objective functions require numerous solutions of the second stage problem. There-fore, when the scenarios are large, it is not easy to evaluate the objective functions. One of thetools that has successfully used to reduce problem size without sacrificing much on accuracy is theSample average approximation (SAA) method. It has been shown theoretically and with numeri-cal examples, SAA can be very efficient in solving large scale stochastic programming problems(10–12).

In SAA, a random sample of size P is generated from the total scenarios, and the expectedvalue function is approximated by the corresponding sample average function. The resulting sam-ple average problem, which will be less computationally intensive, is solved. This procedure isrepeated many times to meet certain level of accuracy. Verweij et al. (12) show SAA can be usedto solve Stochastic routing problems with up to 21694 scenarios within an estimated optimality of1%. The application of SAA to STOCH-MPEC problem is studied in Shapiro and Xu (9) andBirbil et al. (13). Shapiro and Xu (9) study the convergence of optimal values, optimal solutions,and generalized Karush-Kuhn-Tucker points of the SAA to their true values.

In addition to incorporating demand stochasticity the FNDP in this paper is developed



incorporating demand elasticity, that is, the actual demand will be the function of capacity im-provements. The FNDP in this paper can be thought as a game in which planner decides, (a) thetime of improvement, and (b) the quantity of improvements; and travelers decide (a) ‘to travel’ or‘not to travel’, and ii) path to travel to destination. Essentially, the problem studied is a multi-stagestochastic-elastic demand NDP.

Different variation of deterministic demand single stage NDPs are widely studied in thepast (see for example 14–18). A thorough review of NDP models is found in Magnanti and Wong(19) and Yang and Bell (20). Recently, a few studies incorporate demand stochasticity in NDPs(e.g. 4; 21–23). The variations of FNDP formulations are presented in (1; 2), in which the authorsshow the benefits of FNDP over the single stage NDP. The study on FNDP in this paper differs intwo ways: (1) the past FNDP formulations are developed assuming demand as a stochastic process(Geometric Brownian Motion (GBM)) and scenario tree; in both the cases demand at time τ +1 isa function of demand at time τ . The scenario tree approach is a better representation of the demandstochasticity, but the downside of that approach is that the number of scenarios grows quickly. Inthis paper potential demand (discussed later) is assumed to follow an independent distribution ateach time period, (2) SAA, a sampling method is used to address the computational complexity ofthe FNDP formulations. By assuming an independent distribution, it is possible to consider largernumber demand realizations at any given time.

In the next section, the approach used to model demand stochasticity and demand elasticityis presented. The problem is defined with notations in section 3. The objective function of FNDPis derived in section 4, which is followed by a section on developing FNDP formulation. Section 6gives the background information on SAA. SAA algorithm is discussed in section 7. SAA imple-mentation and results are presented in section 8. Conclusions and future directions are discussedin section 9.

2 Demand modeling

In this paper, we incorporate both demand stochasticity as well as demand elasticity. The potentialdemand constitutes, (a) present travelers, (b) users of other networks (for example, dedicated busroutes, and railroads) that are willing to shift, and (c) the persons not traveling presently becauseof the high travel costs on all networks. The improvement is expected to reduce the travel cost,thus, a part of users from (b) and (c) will start using the improved network. The potential demandis assumed to be stochastic and the actual demand is obtained based on demand function.

2.1 Demand stochasticity

Demand stochasticity can be classified into three groups: (i) Longitudinal demand stochasticity,(ii) Latitudinal demand stochasticity, and (iii) Two-way stochasticity (Scenario tree) (see 2). Inthe first case, the demand is modeled as a stochastic process (e.g., Geometric Brownian Motion(GBM) or a mean reverting process (MPR)) over the planning horizon. The demand at a discretetime τ +1 is the function of the demand at τ only. This approach yields one demand value at anytime, τ . FNDP formulation with Longitudinal demand stochasticity is given in Patil and Ukkusuri(1). In the third approach, demand at (τ + 1) depends on demand at τ only, but many demand



values are possible at a given time. The scenario tree approach is computationally more intensivethan the others. The number of scenario grows quickly with the increase in implementation andevaluation stages and the possible demand level. FNDP with two-way demand stochasticity isstudied in Patil and Ukkusuri (2). In the second approach, demand is independently distributed ateach stage; thus there are many possible demand values at each stage, and the values at any twostages are independent. However, the mean of the distribution at τ can depend on the means of theother time periods. The formulation in this paper model latitudinal demand stochasticity.

We assume that for an OD rs the potential demand at time τ follows Gamma distributiondistribution with mean E[qτ

rs] and standard deviation σ τrs. The estimates of these values can be

obtained from past demand trend. A typical longitudinal demand stochasticity structure is shownin Figure 1.

T T +L1 t

0rsq

Time

Figure 1: Latitudinal demand stochasticity

2.2 Demand elasticity

The following exponential demand function is used:

qτrs(πrs) = qτ

rs exp(−θπτrs), ∀rs,τ (1)

where, qτrs is the potential level of demand between rs; πτ

rs is the minimum travel cost between rs;and θ is a positive constant.

Equation 1 can also be written as:



lnqrs = ln qrs− θπrs (2)

Now the inverse demand function becomes,

πrs = q−1rs(qrs) =

1θ

[ln qrs− lnqrs] (3a)

=1θ

[ln

(qrs

qrs

)](3b)

3 Problem definition

Take a network G (N ,A ) for improvements, where N is a set of nodes, and A is set of arcs.The maximum budget available for improvements in base year is B. This amount can be spent atany implementation stage t over an implementation period T . The improvement is assumed to beinstantaneous and can last efficiently for L years after improvement. The latest period by whicha link can be improved is T , and the improvement will last for another L years. Thus, the totalplanning horizon is T +L years. Let Ξτ is a set of demand realizations at time τ; all OD pairs areassumed to have equal number of demand realization at all evaluation stages. In order to evaluatethe improvements, the network is evaluated at regular interval of ∆τ over the planning horizon. Thefuture benefits are discounted at the discount rate of γ . The mean values of the demand at futuretimes can be estimated from the past trends, and are assumed to be available for the present study.The potential demand is modeled as latitudinal stochasticity, and the actual demand is elastic tothe improvements. The aim of FNDP is to obtain a matrix of capacity improvements y, so as tomaximize the benefits without exceeding the available budget.

Following are the factors that add complexity to FNDP, thus requiring to use approximatemethods to solve it.

• large number of scenarios as a result of demand stochasticity,

• non-convexity of MPEC problems because of the complementarity constraints

• multi-stage evaluation over planning horizon

• many links, nodes, and OD pairs in the network

• demand elasticity

Following notations are used:

n: number of implementation stagesΘ: a set of implementation stages, 1,2, · · · , t, t +1, · · · ,nT : a set of network evaluation stages 1,2, · · · ,τ,τ +1, · · ·L: useful life after improvementdi j: marginal cost of adding capacity on link (i, j)



γ: risk neutral discount rate∆τ: length of network evaluation time interval∆t: length of implementation time intervalqξ ,τ

rs : demand between r and s at time τ for realization ξ

τ: absolute time at τ starting from base periodpξ : is the probability of realization ξ

B : total budget availablef ξ ,τi j,s : flow on link (i, j) going to destination s at time τ

xξ ,τi j : after improvement flow on link (i, j) at time τ

yti j: capacity addition on link (i, j) at time t

Y τi j: cumulative capacity added for link (i, j) at time τ

ymin: lower bound on one time capacity addition for any link at any stageymax : upper bound on one time capacity addition for any link at any stageYmax : upper bound on cumulative capacity addition during time period T for any linkcξ ,τ

i j (x,Y ): travel cost function for link (i, j)

4 Performance evaluation of FNDP

Many possible performance measures, such as reduction in system travel time, increase in travelersutility, environmental justice, and increase in the consumer surplus, can be used to evaluate thebenefits of capacity improvements . We use consumer surplus (CS) as the performance measure ofFNDP. CS for an OD pair rs is given as:

CSrs =∫ qrs

0q−1

rs(ϖ)dϖ −qrsπrs (4)

where q−1rs(ϖ) is the inverse demand function ( equations 3a–3b). The first term of the consumer

surplus equation 4 is the willingness to pay (WP) to travel from r to s, and the second term is theminimum travel cost from r to s. The CS for the entire network, that is, system CS is obtained by,

SCS = ∑rs

∫ qrs

0q−1

rs(ϖ)dϖ −∑rs

qrsπrs (5)

The first term of equation 5 is the system willingness to pay (SWP), and the second term isthe system travel cost (STC). In order to obtain STC in equation 5 it is necessary to calculate theminimum travel cost for each OD pair. Note, STC can also be calculated from link flows and linkcosts. It is usually easy to find link costs than path costs. At equilibrium ∑rs qrsπrs = ∑i j ci j(x)xi j.The SCS using link costs is calculated as,

SCS = ∑rs

∫ qrs

0q−1

rs(ϖ)dϖ −∑i j

ci j(x)xi j (6)

The solution of FNDP formulation in this gives path costs in addition to other parameters,thus we use SCS derived from equation 5. Using equation 3a for inverse demand, SWP can be



written as,

SWP =1θ

∑rs

∫ qrs

0

1θ

[ln qrs− lnϖ ]dϖ (7a)

=1θ

∑rs

[ϖ ln(qrs)−ϖ lnϖ +ϖ ]qrs0 (7b)

=1θ

∑rs

[qrs ln(q)−qrs ln(qrs)+qrs] (7c)

= ∑rs

qrs

θ[ln(qrs)− ln(q)+1] (7d)

= ∑rs

qrs

θ

[ln

(qrs

qrs

)+1

](7e)

Now equation 5 for SCS becomes,

SCS = ∑rs

qrs

θ

[ln

(qrs

qrs

)+1

]−∑

rsqrsπrs (8)

Putting the value of πrs (equation 3a) in equation 8,

SCS = ∑rs

qrs

θ

[ln

(qrs

qrs

)+1

]−∑

rsqrs

1θ

[ln

(qrs

qrs

)](9a)

= ∑rs

qrs

θ

[ln

(qrs

qrs

)− ln

(qrs

qrs

)+1

](9b)

= ∑rs

qrs

θ(9c)

The demand distribution in the network depends on the network realization ξ . The SCS forany network realization ξ is,

SCSξ = ∑rs

qξrs

θ= ∑

rs

qξrs

θexp(−θπ

ξrs) (10)

FNDP aims to maximize the benefits of improvements. The benefits are measured by takingthe difference in SCS with improvements and without improvements. The change in CS for an ODpair rs at time τ is shown in the figure 2. In order to find the benefits over planning period, theCS is calculated for all OD pairs, all time intervals, and all scenarios. The SCS of future times arediscounted to the present value. The resulting value is called present expected system consumersurplus (PESCS). The change in PESCS due to improvements is given by,

∆PESCS = PESCSFNDP−PESCSNI (11)

Note that the second term, that is, PESCS under ‘no improvement’, is not dependent on theimprovements, but is a constant. Thus, the objective function of FNDP is to maximize the PESCS



after improvements, that is, to maximize PESCSFNDP or to minimize the negative of PESCSFNDP.With latitudinal demand stochasticity, the objective function of FNDP is given as,

Z = min− 1θ

T+L

∑τ

∑ξ

∑rs

pξ qξ ,τrs

∆τ

(1+ γ)τ(12)

Figure 2: Change in Consumer SurplusArea ABCD: CS with FNDP, Area ABEF: CS with no improvement,

Area EFCD: increase in CS due to improvements

5 FNDP formulation with latitudinal demand stochasticity

The FNDP formulation using the objective function derived in section 4 is presented below.



Z = min − 1θ

T+L

∑τ

∑ξ

∑rs

pξ ,τqξ ,τrs

∆τ

(1+ γ)τ(13a)

subject to ∑i j

∑t

di jyti j ≤ B (13b)

qξ ,τrs = qξ ,τ

rs exp(−θπξ ,τks ), ∀s,τ,ξ ∈ Ξ

τ (13c){cξ ,τ

i j (x,Y )+πξ ,τks −π

ξ ,τis

}f ξ ,τi j,s = 0, ∀(i, j),s, i 6= s,ξ (13d)

cξ ,τi j (x,Y )+π

ξ ,τks ≥ π

ξ ,τis , ∀(i, j),s, i 6= s,τ,ξ (13e)

∑i j∈A+

i

f ξ ,τi j,s − ∑

ji∈A−i

f ξ ,τji,s = qξ ,τ

is , ∀h,s, i 6= s,τ,ξ (13f)

∑s

f ξ ,τi j,s = xξ ,τ

i j , ∀(i, j),τ,ξ (13g)

f ξ ,τi j,s ≥ 0, ∀(i, j),s,τ,h, (13h)

Y τi j = ∑

t≤τ,t∈Θ

yti j, (i, j) ∈A ,τ ≤ T,τ ∈T (13i)

Y τi j = Y T

i j , (i, j) ∈A ,T < τ ≤ (L−T ) (13j)

Y τi j = Y τ−1

i j −Y τ−L, (i, j) ∈A ,τ > T if τ ∈T (13k)

ymin ≤ yti j ≤ ymax, ∀(i, j), t (13l)

Y τi j ≤ Ymax, ∀τ (13m)

The objective function of the FNDP is to maximize the PESCS during the planning hori-zon. Equations 13d through 13h is the nonlinear complementarity formulation for user equilibrium(1; 2; 24; 25). These complementarity conditions, instead of Beckman’s UE formulation, enableconverting the NDP problem into a single stage formulation. Equations 13i-13k calculate cumu-lative capacity added to a link. The limits on capacity additions are set with constraints 13l and13m.

The cost function used in conditions 13d and 13e is a BPR type continuous monotonicallyincreasing function given by:

cξ ,τi j (x,Y ) = T 0

i j +βi j

xξ ,τi j

(Ki j +Y τi j)

α

(14)

6 SAA background

SAA is a Monte Carlo simulation based sampling method, in which the expected value of objectivefunction of stochastic program is approximated by solving the problem for a sample of scenarios.The sample size is much smaller than the actual number of scenarios. This smaller problem is



then solved using using any suitable standard tool such as KNITRO 5.1 or metaheuristics such assimulated annealing and tabu search. This process of solving for a sample is repeated many timesand a best solution is selected. SAA is also known as “stochastic counterpart method”, “sample-path method”, and “simulated likelihood method”. SAA converges exponentially fast to the trueproblem with the increase of sample size (26; 27). SAA is used in Santoso et al. (11) to solverealistic size supply chain network design with a huge number of scenarios. Norkin et al. (28) andMak et al. (10) give procedure to obtain candidate solutions. Shapiro and Xu (9) show convergenceof SAA for stochastic MPEC problems which are typically nonconvex, and can have many localoptima.

In SAA method a random sample of P demand realizations are selected. The expectedPESCS function 15 is approximated with

ZP = min− 1Pθ

T+L

∑τ

P

∑h=1

∑rs

qh,τrs

∆τ

(1+ γ)τ(15)

Solving SAA problem of sample, P results the objective value, Zp capacity improvements,yp, and link flows xp. The process of taking sample of P realizations and solving to obtain Zp, yp,and, xp is repeated M times.

The selection of sample size P for SAA is a trade-off between computational complexityand accuracy. A larger value of P can give a more accurate estimate of the true value of objectivefunction and a better solution for capacity improvements. However, the computational complexityincreases with the sample size. Therefore, the choice of sample size should be such that it ispossible to solve the FNDP with reasonably difficulty and accuracy. The choice can be madedynamically. The SAA problem is solved for a smaller size and then increased subsequently basedon the achieved accuracy and underlying computational difficulty.

SAA algorithm needs evaluation of objective function for a given solution, that is, PESCSfor an estimated capacity improvements needs to be evaluated. The evaluation of objective functionis done by taking sample size of P′ > P. Note that variable q is scenario dependent. When the prob-lem is solved for P samples, we have demand values only for P scenarios. Since P′ > P, we needto obtain demand, q for remaining scenarios also. A close look at the above FNDP formulationwill reveal that if capacity, y is fixed the resulting problem is the elastic demand user equilibriumproblem. User equilibrium problems for P′ is computationally less complex than FNDP for thesame number of scenarios. Additionally, note that y is the only scenario independent variable inthe above FNDP formulations. Demand, link flow, link cost are specific to each scenario. Thus, ify is fixed, the problem can be solved independently for each scenario. Thus evaluating objectivefunction for P′ is not very difficult. The problem for each scenarios is shown below.



min − 1θ

T+L

∑τ

∑rs

qτrs

∆τ

(1+ γ)τ(16a)

qτrs = qτ

rs exp(−θπτks), ∀s,τ (16b){

cτi j(x,Y )+π

τks−π

τis}

f τi j,s = 0, ∀(i, j),s, i 6= s (16c)

cτi j(x,Y )+π

τks ≥ π

τis, ∀(i, j),s, i 6= s,τ (16d)

∑i j

f τi j,s−∑

i jf τ

ji,s = qτis, ∀s, i 6= s,τ (16e)

∑s

f τi j,s = xτ

i j, ∀(i, j),τ (16f)

f τi j,s ≥ 0, ∀(i, j),s,τ (16g)

(16h)

Once the demand is know for each scenario, the estimate of true objective function isobtained as,

Z∗p′ = − 1

p′θ

T+L

∑τ

∑rs

qτrs

∆τ

(1+ γ)τ(17)

The capacity improvements obtained with sample size P is used here. ZP′ is obtained forall samples 1, · · · ,M. Since the value P′ can usually be much larger than P, the objective valueobtained, ZP′ is considered as an estimator of the true objective function 13a. Among all trials,the solution, y∗ with minimum objective value, Z∗

p′ , or maximum PESCS is selected as an optimalcapacity improvement.

7 SAA algorithm

Based on the above discussion, the SAA algorithm is described below:

1. Choose initial sample sizes P and P′. Also choose M and rules to increase the sample sizesof P and P′.

2. For m = 1,2, · · ·M, generate samples of size P and solve the SAA problems to get the objec-tive values Z1

P,Z2P, · · ·ZM

P , and solutions y1, y2, · · · yM.

3. Calculate the average of the optimal objective function values of the M, SAA problems asZP = 1

M ∑M1 Zm

P and the variance of estimator ZP as σ2ZP

= 1(M−1)M ∑

Mm=1(Z

mP − ZP)2.

4. For m = 1,2, · · · ,M, solve UE problem 16a- 16g for P′ and obtain ZP′ = 1P′ ∑

P′p=1 Zp

P′(ym).

5. Calculate variance σ2ZP′

= 1(P′−1)M ∑

Pp=1(Z

pP′− ZP′)2.

6. Select capacity improvement y∗ ∈ argmin ZP′(y) : y ∈ x1, x2, · · · , xM and calculate ZP′ .



Table 1: Mean Demand and Standard Deviationstime, τ E[qτ

1,6] σ τ1,6 E[qτ

6,1] σ τ6,1

0 10.000 0.000 20.000 0.0001 10.200 7.000 20.400 12.0002 10.404 7.000 20.808 12.0003 10.612 7.000 21.224 12.0004 10.824 7.000 21.649 12.0005 11.041 7.000 22.082 12.0006 11.262 7.000 22.523 12.0007 11.487 7.000 22.974 12.000

7. Evaluate the quality of the solution by calculating the optimality gap ZP′(y∗)− Z and thevariance of optimality gap σ2

ZP′+ZP= σ2

ZP′+ σ2

ZZP.

8. If the convergence criteria is met then stop, otherwise change P and P′ and repeat steps fromstep 2.

8 Computational results

This section presents the analysis of FNDP results and implementation of SAA on two test net-works. The values of T and L are assumed to be 2 and 5 years respectively. Thus, there are threestages for improvements: ST I (base year), ST II (year 1), and ST III (year 2). Demand in thenetwork is assumed to follow Gamma distribution with mean mean E[qτ

rs] and variance σ2τ

rs.Test network-1, as shown in Figure 3, has 6 nodes, 16 links and 2 OD pairs: 1 to 6 (OD1)

and 6 to 1 (OD2). Network parameters (except demand) for the test network are reproduced fromSuwansirikul et al. (29) in Table 2. Parameter di j in the table is the marginal cost of capacityaddition for link (i, j). The base year demand for OD 1-6 and 6-1 is taken 10 and 20 respectively.Results are obtained for two budget levels of 100 and 200. The value of θ is assumed to be 0.01.

Figure 3: Test network-1

For network-1, 100 values of demand are generated with the mean and standard deviationgiven in Table 1. For the implementation of SAA, the value of M is taken 10, and is kept constant



Table 2: Data for Test Network-1Arc (i, j) T 0

i j βi j Ki j di j

(1,2) 1.0 10.0 3.0 2.0(1,3) 2.0 5.0 10.0 3.0(2,1) 3.0 3.0 9.0 5.0(2,3) 4.0 20.0 4.0 4.0(2,4) 5.0 50.0 3.0 9.0(3,1) 2.0 20.0 2.0 1.0(3,2) 1.0 10.0 1.0 4.0(3,5) 1.0 1.0 10.0 3.0(4,2) 2.0 8.0 45.0 2.0(4-5) 3.0 3.0 3.0 5.0(4,6) 9.0 2.0 2.0 6.0(5,3) 4.0 10.0 6.0 8.0(5,4) 4.0 25.0 44.0 5.0(5,6) 2.0 33.0 20.0 3.0(6,4) 5.0 5.0 1.0 6.0(6,5) 6.0 1.0 4.5 1.0

Source: Suwansirikul et al.(1987)

for all sample sizes. The sample size is varied from 20 to 50 with an interval of 10. The value of P′

is taken 50 for sample sizes 20, 30, and 40. A higher value of P′ equal to 75 is taken for sample size50. The results for budget 100 are presented in Table 3 and for budget 200 are presented in Table4. The upper section of tables give approximate objective function values for each samples. Thesevalues for sample sizes 20, 30, 40, and 50 are plotted for budget 100 in Fig. 4 and for budget 200in Fig. 5. The minimum, maximum, and average values of the objective function and the variancesof these trials are also given in the table. It is observed that the variance is decreasing consistentlywith the increase in the sample sizes. The variance of the true estimate for sample size 50 is higherthan that for 20 and 30 because P′ in the former case is 75 against 50 for the latter. The estimatedoptimality gap is reducing considerable with the increase in the sample size. The optimality gapfor sample size 50 is only 0.23% for budget 100 and is 0.05% when the budget is 200. We takeresults of SAA with P = 50 as the solution of FNDP using SAA for the test network-1.

Since the network is reasonably small, and all the number of scenarios are not very large, wewere able to solve the FNDP for all scenarios and get the true solutions. We also solved the FNDPwith average demand values. Capacity improvements with the average demand are evaluated forall scenarios to check the quality of average demand solutions. The performance of network designsolutions obtained with average demand, and SAA approach is evaluated by taking the capacityimprovements with the corresponding approach and assigning all demand scenarios to calculatePESCS. These results are presented in table 5. It is clearly seen that the average demand solutionoverestimate the benefits of improvements. However, the actual benefits of average demand so-lution, obtained by the equilibrium assignment for all demand scenarios and taking average, areconsiderably low. It is encouraging to note that the actual benefits evaluated for all scenarios using



Table 3: SAA Results for Test Network-1 (Budget=100)Negative of objective function value (PESCS)

Sample P = 20 P = 30 P = 40 P = 501 15,760.828 16,095.755 16,103.693 16,113.1812 16,505.837 16,147.478 16,233.444 16,117.2853 15,813.132 15,859.801 16,173.994 16,155.8324 16,044.213 15,946.979 16,250.932 15,664.6275 16,254.757 16,572.287 16,105.303 15,821.1036 16,035.829 16,191.555 15,986.539 15,802.8727 15,863.336 16,275.082 16,204.067 16,221.1258 15,787.712 16,079.058 15,972.691 16,217.8899 15,985.693 16,262.425 15,754.853 15,736.048

10 15,421.104 15,499.644 15,687.862 15,852.028−Zmin 15,421.104 15,499.644 15,687.862 15,664.627−Zmax 16,505.837 16,572.287 16,250.932 16,221.125−ZP 15,947.244 16,093.007 16,047.338 15,970.199σZP

87,485.5674 81,359.1811 38,585.1917 45,916.1420−ZP′ 16,237.589 15,980.340 15,981.973 15,933.266σ2

ZP′3,688,717.71 3,438,058.02 3,442,405.49 4,499,666.18

Opt. gap -290.344 112.667 65.365 36.933%gap 1.786 0.705 0.409 0.231

σ2ZP′+ZP

3,776,203.28 3,519,417.20 3,480,990.69 4,545,582.32

Figure 4: Variation in the objective function values for test network-1 (Budget=100)



Table 4: SAA Results for Test Network-1 (Budget=200)Negative of objective function values (PESCS)

Sample P = 20 P = 30 P = 40 P = 501 16,223.46 16,591.59 16,589.46 16,603.602 17,057.97 16,622.66 16,705.40 16,468.033 16,166.45 16,345.34 16,708.66 16,664.404 16,536.16 16,293.74 16,725.70 16,000.255 16,778.99 17,057.30 16,613.61 16,292.096 16,506.61 16,674.85 16,349.28 16,143.857 16,317.25 16,772.96 16,689.09 16,704.638 16,244.27 16,573.23 16,456.70 16,737.979 16,333.58 16,755.81 16,083.48 16,078.45

10 15,756.06 15,938.03 16,148.31 16,305.47−Zmin 15,756.06 15,938.03 16,083.48 16,000.25−Zmax 17,057.97 17,057.30 16,725.70 16,737.97−ZP 16,345.34 16,540.66 16,458.40 16,366.53σZP

126,856.90 94,782.94 57,180.00 74,711.33−ZP′ 16,742.05 16,410.63 16,428.64 16,358.98σ2

ZP′4,320,390.31 3,961,173.11 3,968,814.29 5,338,919.79

Opt. gap -396.71 130.02 29.76 7.55% gap 2.20 0.79 0.181 0.05σ2

ZP′+ZP4,447,247.21 4,055,956.05 4,025,994.29 5,413,631.12

Figure 5: Variation in the objective function values for test network-1 (Budget=200)



SAA approach capacity improvements are very close to the true benefits.

Table 5: Comparison of ResultsNetwork Improvements

Stage Links Avg. demand SAA method True problemI 1 3 1.243 1.713 2.107I 2 1 0.000 5.217 4.614I 3 1 9.456 5.893 5.923I 3 2 0.000 0.000 0.000I 3 5 0.000 0.000 0.000I 5 6 0.000 0.000 0.000I 6 4 10.894 0.000 0.000I 6 5 0.000 13.499 12.931II 1 3 0.000 1.233 0.000II 2 1 0.000 0.055 0.599II 3 1 0.000 1.535 1.445II 3 2 0.000 0.000 0.000II 3 5 0.000 0.227 0.000II 5 6 0.000 0.064 0.000II 6 4 0.000 0.000 0.000II 6 5 0.000 0.852 1.337III 1 3 0.000 1.310 1.135III 2 1 0.797 3.524 4.328III 3 1 7.985 7.780 7.475III 3 2 0.000 0.000 0.000III 3 5 0.000 0.000 0.000III 5 6 0.000 0.000 0.000III 6 4 0.000 0.000 0.000III 6 5 9.48 12.824 13.459−Z 16,171.39 15,933.27 15,765.30

actual PESCS 15,242.18 15,758.60

The proposed SAA algorithm is run for network-2, shown in figure 6. The budget is as-sumed to be 1000. The demand is assumed to follow Gamma distribution. Total 200 demandscenarios are considered. The SAA algorithm is applied for sample sizes 20, 30, 40, and 50. Theresults are presented in Table 6. It can be observed that the estimated optimality gap is less than1% for all sample sizes. This cleary shows a good solution can be obtained with a considerablysmaller sample size using SAA method.

9 Conclusions

The main focus of this paper is to develop and solve flexible network design problem (FNDP)with Sample Average Approximation (SAA) method. The demand is assumed to be stochastic,



Figure 6: Test network-2Source: Nguyen and Dupuis (30)

Table 6: SAA Results for Test Network-2 (Budget=1000)Negative of objective function values (PESCS)

Sample P = 20 P = 30 P = 40 P = 501 791,189.0 791,065.2 797,740.9 790,549.32 783,769.1 784,898.7 789,849.7 789,147.53 794,364.5 798,049.2 787,828.9 780,232.44 786,112.5 802,384.9 786,197.6 789,366.95 777,546.3 786,990.1 794,892.7 794,800.36 793,058.9 791,538.4 792,396.3 782,499.27 788,163.4 782,906.9 782,364.1 779,986.08 784,843.0 787,391.2 781,600.6 789,062.59 794,690.0 776,021.6 779,228.2 781,625.6

10 783,608.1 779,199.1 786,910.5 790,507.3−Zmin 777,546.3 776,021.6 779,228.2 779,986.0−Zmax 794,690.0 802,384.9 797,740.9 794,800.3−ZP 787,734.5 788,044.5 787,900.9 786,777.7σZP

3.1149E+07 6.5299E+07 3.5464E+07 2.7095E+07−ZP′ 788,375.9 789,303.4 788,021.0 787,723.6σ2

ZP′1.5023E+09 1.0633E+09 1.3763E+09 1.2231E+09

Optimality gap -641.4 -1,258.9 -120.1 -945.9% gap 0.081 0.159 0.015 0.120σ2

ZP′−ZP1.5334E+09 1.1286E+09 1.4117E+09 1.2502E+09



and is also assumed to be elastic to the improvements. The FNDP problem is formulated as astochastic mathematical programs with equilibrium constraints (STOCH-MPEC). The FNDP isvery computationally intensive, and is impractical to solve directly when the number of scenariosare large and/or when the network is big. Thus, the sample average approximation (SAA) methodis used to reduce the computational burden and get good approximate solutions.

We apply the SAA algorithm to 2 test networks with different budget levels. The results ofSAA approach is obtained with the sample sizes 20, 30, 40 and 50. For network-1, the estimatedoptimality gap with SAA solution for sample size of 30, 40, and 50 is is less than 1.0% for bud-gets 100 and 200. For network-2, the estimated optimality gap for samples sizes 20, 30, 40, and50 is also less than 1.0%. This clearly shows that SAA can give good approximate solutions atthe same time considerably reducing problem complexity. We observed that the finding capacityimprovements with average demand over estimates the benefits of improvements considerably.

Apart from its contribution transportation network modeling research, this study can haveimportant implications on the infrastructure investment decision making in the practice. The net-works in practice are usually large, thus need efficient approaches to solve larger problem instancesfor which SAA can successfully be used.

Although, the present study clearly shows the benefits of SAA approach in FNDP, the realneed SAA comes for a large size problems. SAA is perhaps not the best approach to solve the casestudy presented. The limitation on the size is mainly because of the constraints of NEOS server.We used solver KNITRO 5.1 available on NEOS server, which can not handle larger problem.Nevertheless the same approach adopted in this paper can be used for larger networks.

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THE SAMPLE AVERAGE APPROXIMATION …transctr/pdf/trb_paper_gopal.pdfTHE SAMPLE AVERAGE APPROXIMATION METHOD FOR FLEXIBLE NETWORK DESIGN PROBLEM Gopal R. Patil PostDoctoral Researcher,