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.
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
 x c (x
SO : Min
Rutgers, The State University of New Jersey
a a
•Nguyen–Dupuis Network
a
a
a
) or UE : Min  ca ( )d
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.