An Investigation of the Convergence and Accuracy Properties of

1
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
Paper 10-3504 Jian Li, M.Sc. and
Kaan Ozbay, Ph.D. Rutgers, The State
University of New Jersey

• NUMERICAL EXPERIMENTS
• NguyenDupuis Network
• Performance Measures

• 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.
• FIGURE 4 Change in the Approximation Accuracy for
  Different Bin Fractions and Sample Sizes

Abstract 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.

• Methodology
• Formulation
• The traffic assignment formulated by Wardrop
  (1952) has been widely used under the standard
  assumption of deterministic origindestination
  (OD) demand and link capacity conditions.
• Incorporating above link capacity distributions,
  and supposing that there are m links in the
  network, the uncertain capacity parameter vector
  can be viewed as a variable vector,
  with each item having a probability distribution.
  The following stochastic programming problem is
  then formulated
• Where F(x , ?) is the objective function that
  will derive the traffic assignment toward either
  SO or UE, and ?(?) 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.

• 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.
• FIGURE 1 Example of LHS with 2 Variables and 5
  Intervals
• The main shortcoming of the LHS stratification
  scheme is