2003 Winter Simulation Conference

1
Simulation-based optimization for complex
stochastic systems
Yanto Prasetio, Joyce W. Yen, and Zelda B.
Zabinsky Industrial Engineering, University of
Washington
Objective Adapt existing search methodologies
for multimodal stochastic functions
Future application A future application of
interest to us is a weather forecast generator
for air traffic management. The goal is to
understand the impact of weather forecasting
accuracy on the ability of air traffic management
strategy to alleviate airspace weather-related
disturbances. We can stochastically simulate
dynamic changes in the airspace system via a
weather forecast generator. The simulation
represents the impact of a weather forecast on
the national airspace performance. As forecast
construction and its accuracy are governed by a
set of controllable parameters, we will apply our
simulation optimization search methodologies to
determine the best forecast generator parameters.
Simulation Average the output from R
replications of the simulation to estimate the
expected performance measure at a setting on the
kth iteration (?k) of the search methodology

• Problem
• Min EOf(?,?)
• subject to L ? U
• O is a random variable
• The expected performance measure of a stochastic
  system is minimized
• Eight multimodal test functions 1 with noise
  (error) are utilized for the objective function
• f(?,?) g(?) ?, where O uniform(-x,x),
• where f(?,?), which consists of a multimodal
  test function g(?) and the noise ?, represents
  the performance measure of the system at a
  setting, ?
• Each test function has continuous parameters,
  varied from two to ten parameters
• One hundred variations for each test function are
  generated
• Different lower and upper bounds, L and U
• Different parameter, x, associated with the noise
  in the test function
• Five replications are used to estimate the
  performance measure at a setting

Results effectiveness and efficiency of the
methodologies
Results performance of the explored search
methodologies

• Explored search methodologies
• Pure random search (PRS) exploration of the
  problem domain in a random manner to find a
  setting that optimizes the system performance
  measure
• Gradient-based search the use of a fundamental
  result from calculus, that the gradient points in
  the direction of the steepest descent, to
  determine how the simulation setting should be
  changed when optimizing the simulation model
• Variations finite-differences (G-FD) and
  simultaneous perturbations (G-SP)
• Response surface methodology the use of a
  sequential search utilizing a recursive set of
  localized experiments aimed at converging to
  global optimal setting
• Variations two-level and three-level
  full-factorial designs and balanced-star and non
  balanced-star designs (RSM-2 levels, RSM-3
  levels, RSM-Star-B, and RSM-Star-NB,
  respectively)
• Sequential grid search the use of a sequential
  partition of a feasible grid into sub-grids
• Variations two-level (SG-2 levels) and
  three-level (SG-3 levels) partitions for each
  parameter
• Sequential coordinate direction search the
  optimization of a multi-dimensional problem by
  optimizing along coordinate directions
  sequentially
• Variations uniform evaluations (SCD-U) and
  bisection (SCD-B)
• Fictitious-play the construction of a candidate
  setting from multiple settings that are obtained
  by optimizing multiple sub-problems via the
  decomposition of the original multi-dimensional
  problem at each iteration
• Variation uniform evaluations (FP-U) and
  bisection (FP-B)
• Improving hit-and-run the acceptance of a
  candidate setting only if such setting yields a
  better performance measure, where the candidate
  setting is generated from the random line segment
  defined by the current simulation setting
• Variation hyper-spherical direction (IHR-HD),
  coordinate direction (IHR-CD), and cyclic
  coordinate direction (IHR-cCD)
• Simulated annealing the use of acceptance
  probability for moving to a worse-performing
  candidate setting, where the probability is
  assigned based on a geometric cooling schedule
• Variations same as in improving hit-and-run
  (SA-HD, SA-CD, and SA-cCD)

• Observations
• From the test cases, we observed the following
• Sequential coordinate direction search with
  uniform evaluations (SCD-U) generally finds
  better solutions than the other methodologies,
  but with higher-than-average number of simulation
  settings evaluated
• Gradient-based search with simultaneous
  perturbation (G-SP) generally requires the least
  number of settings evaluated, but it does not
  typically yield good-performing solutions
• Sequential coordinate direction search with
  bisection (SCD-B) appears to perform well in
  terms of efficiency and effectiveness
• Fictitious-play (FP), improving hit-and-run
  (IHR), simulated annealing (SA), and sequential
  grid search (SG) yield some good solutions in our
  test cases as well

LEGEND PRS Pure random search G-FD
Gradient-based search, finite
differences G-SP Gradient-based search,
simultaneous perturbation RSM-2 levels Response
surface methodology, two-level full-factorial
design RSM-3 levels Response surface methodology,
three-level full factorial design RSM-Star-B Respo
nse surface methodology, balanced star
design RSM-Star-NB Response surface methodology,
non-balanced star design SG-2 levels Sequential
grid search, two-level partition SG-3
levels Sequential grid search, three-level
partition
SCD-U Sequential coordinate direction search,
uniform evaluations SCD-B Sequential coordinate
direction search, bisection FP-U Fictitious-play,
uniform evaluations FP-B Fictitious-play,
bisection IHR-HD Improving hit-and-run,
hyper-spherical direction (HD) IHR-CD Improving
hit-and-run, coordinate direction
(CD) IHR-cCD Improving hit-and-run, cyclical
coordinate direction (cCD) SA-HD Simulated
annealing, HD with geometric cooling
schedule SA-CD Simulated annealing, CD with
geometric cooling schedule SA-cCD Simulated
annealing, cCD with geometric cooling schedule
Acknowledgments This study was supported in part
by a grant from the Boeing Company for
collaborative research on air traffic flow
management under temporary capacity constraints.
We would like to thank the National Flow Model
team for their contribution and useful
discussions on this research.

• 1. M. Ali, C. Khompatraporn, and Z. Zabinsky. A
  Numerical Evaluation of Several Global
  Optimization Algorithms on Selected Test
  Problems. Journal of Global Optimization, to
  appear.