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Rutgers Intelligent Transportation Systems (RITS) Laboratory Department of Civil & Environmental Engineering An Investigation of the Convergence and Accuracy Properties of Latin Hypercube Sampling Technique for Traffic Equilibrium Problem under Capacity Uncertainty Jian Li, M.Sc. and Kaan Ozbay, Ph.D. Paper: 10-3504 Abstract Methodology The traffic equilibrium problem under capacity uncertainty (TEPCU) has drawn significant interest in recent years, mainly because of the need to incorporate the stochastic nature of link capacities in the transportation planning process. A common approach for solving TEPCU is to use a sampling technique that randomly selects subsets of the uncertainty set to obtain approximate solutions. Latin hypercube sampling (LHS) is one of the most frequently used sampling methods, and it can provide accurate approximations. However, the main concern when using LHS is the large required sample size, which is important in the application of LHS for TEPUC because of the high computational time required for large networks. The main objective of this paper is to conduct an in-depth analysis of the convergence and approximation accuracy properties of LHS. Several computational tests are conducted using two different networks, with the goal of determining an efficient sample size that can be used to obtain an accurate approximate solution of TEPUC at a given level of confidence. The results provide us with a better understanding of the requirements of an appropriate experimental design for applying LHS to TEPUC on large transportation networks. •Formulation Introduction For modeling purposes, roadway capacity uncertainty is usually treated as a random variable under certain distribution instead of as a deterministic value. A common approach for solving the TEPCU problem is to use sampling techniques that randomly select subsets of the uncertainty set to obtain approximate solutions. Latin hypercube sampling (LHS) is a stratified sampling method that can reduce the variance in the MC estimate of the integrand significantly. An example of LHS with two input variables and five bin fractions is shown in Figure 1. Rutgers, The State University of New Jersey NUMERICAL EXPERIMENTS The traffic assignment formulated by Wardrop (1952) has been widely used under the standard assumption of deterministic origin–destination (OD) demand and link capacity conditions. x •Nguyen–Dupuis Network a SO : Min x a ca ( x a ) or UE: Min ca ( )d a 0 Approximation Accuracy of LHS For test networks with predetermined numbers of bin fractions, the value of approximation accuracy consistently decreased when the sample size is increased. Moreover, the predetermined number of bin fractions had a significant effect on the approximation accuracy. For the same sample size, when the number of bin fractions increased, the value of approximation accuracy improved. However, the approximation accuracy did not change when the number of bin fractions was large. s.t. rs h k qrs r, s k hkrs 0 r, s x a h rsk δ a,rsk a r s a Incorporating above link capacity distributions, and supposing that there are m links in the network, the uncertain capacity parameter vector (1 , 2 , , m ) can be viewed as a variable vector, with each item having a probability distribution. The following stochastic programming problem is then formulated: •Performance Measures Error of the Estimate Min { f (x) E[F (x, ())]} Where F(x , ξ) is the objective function that will derive the traffic assignment toward either SO or UE, and ξ(ω) is link capacity, a random vector calculated on the basis of certain probability distributions. For solving approach, sample-average approximation (SAA), was employed and then minimized using a deterministic optimization algorithm. Where S Error 1.96 * K S = the sample standard deviation K = the sample size. Accuracy of Approximation fK f * Accuracy f* Where, f K = the expected value of K sample size f * = the value of benchmark The true expected value is approximated by the benchmark obtained by running the MC algorithm for 100,000 capacity realizations. Parameter Initialization p=0 Link Capacity Realization i=0 FIGURE 4 Change in the Approximation Accuracy for Different Bin Fractions and Sample Sizes Comparison of Performance of LHS and MC The output results by LHS quickly converged and became stable when an acceptably small sample size was used and when the number of bin fractions was large. This represents a significant advantage over MC. Although the approximation accuracy of LHS is slightly worse than that of MC, this can be considered as a compromise between the approximation accuracy and computational time requirements. •Results p=p+1 Random Selected Capacity Realization Required Sample Size for the Convergence of LHS For both networks, the convergence rate was related only to the sample size and that there was no significant difference among the different number of bin fractions. When the sample size was increased, the error of the estimate decreased and the convergence rate gradually slowed down. The inflection point of the curve shown in Figure 3 (i.e., the point where the slope of the curve changed from sharp to flat) is between 500 and 2,000 sample sizes. i=i+1 Random Selected Capacity Realization Standard Traffic Assignment FIGURE 5 Comparison of the Approximation Accuracy of LHS and MC for Different Bin Fractions and Sample Sizes •Conclusion FIGURE 1 Example of LHS with 2 Variables and 5 Intervals The main shortcoming of the LHS stratification scheme is: • “it is one-dimensional and does not provide good uniform properties on a multi-dimensional unit hypercube” (Diwekar, 2003). • The outcome of LHS can be considered only as a probabilistic output. NO LHS Done? p=m? YES Sample Size Satisfied? Thus, the first and most important questions related to the application of repeated LHS procedure for TEPUC are these: • How should one determine the number of bin fractions? • How many corresponding sample sizes or additions of LHS are required to produce the desired output accuracy at the minimum level of confidence acceptable to the transportation planner? NO p*m=K? p=m? YES FIGURE 2 Flowchart of repeated LHS based SAA Solution Steps FIGURE 3 Change in Error of the Estimate for Different Bin Fractions and Sample Sizes This study investigated the convergence and approximation accuracy properties of LHS for TEPCU. The results indicate that: • Convergence rate was related only to the sample size and that no significant difference was observed as a result of the differences in the size of the predetermined bin fractions. • When the sample size was increased, the error of the estimate decreased and the convergence rate gradually slowed down. • The predetermined number of bin fractions had a significant effect on approximation accuracy. • For the same sample size, when the number of bin fractions increased, the value of approximation accuracy improved. However, the approximation accuracy exhibited a stable trend when the number of bin fractions was large.