Simulation optimization (SO) refers to the optimization of an objective function subject to constraints, both of which can be evaluated through a stochastic simulation. To address specific features of a particular simulation—discrete or continuous decisions, expensive or cheap simulations, single or multiple outputs, homogeneous or heterogeneous noise—various algorithms have been proposed in the literature. As one can imagine, there exist several competing algorithms for each of these classes of problems. This document emphasizes the difficulties in SO as compared to algebraic model-based mathematical programming, makes reference to state-of-the-art algorithms in the field, examines and contrasts the different approaches used, reviews some of the diverse applications that have been tackled by these methods, and speculates on future directions in the field.

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Simulation optimization: a review of algorithms and applications

Simulation optimization refers to the optimization of an objective function subject to constraints, both of which can be evaluated through a stochastic simulation. To address specific features of a particular simulation—discrete or continuous decisions, expensive or cheap simulations, single or multiple outputs, homogeneous or heterogeneous noise—various algorithms have been proposed in the literature. As one can imagine, there exist several competing algorithms for each of these classes of problems. This document emphasizes the difficulties in simulation optimization as compared to algebraic model-based mathematical programming makes reference to state-of-the-art algorithms in the field, examines and contrasts the different approaches used, reviews some of the diverse applications that have been tackled by these methods, and speculates on future directions in the field.

Simulation Optimization (SO) refers to the optimization of an objective function subject to constraints, both of which can be evaluated through a stochastic simulation. To address specific features of a particular simulation---discrete or continuous decisions, expensive or cheap simulations, single or multiple outputs, homogeneous or heterogeneous noise---various algorithms have been proposed in the literature. As one can imagine, there exist several competing algorithms for each of these classes of problems. This document emphasizes the difficulties in simulation optimization as compared to mathematical programming, makes reference to state-of-the-art algorithms in the field, examines and contrasts the different approaches used, reviews some of the diverse applications that have been tackled by these methods, and speculates on future directions in the field.

/pdf/simulation-optimization-a-review-of-algorithms-and-4abmdikltr.pdf

Simulation optimization: A review of algorithms and applications

An entire airport terminal building is simulated to examine passenger flows, especially international departures. First, times needed for passengers to be processed in the terminal building are examined. It is found that the waiting time for check-in accounts for more than 80 percent of the total waiting time of passengers spent in the airport. A special-purpose data-generator is designed and developed to create experimental data for executing a simulation. It is found that the possible number of passengers missing their flights could be drastically reduced by adding supporting staff to and by making use of first- and business-class check-in counters for processing economy- and group-class passengers.

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Simulation analysis of international-departure passenger flows in an airport terminal

An entire airport terminal building is simulated to examine passenger flows, especially international departures. First, times needed for passengers to be processed in the terminal building are examined. It is found that the waiting time for check-in accounts for more than 80 percent of the total waiting time of passengers spent in the airport. A special-purpose data-generator is designed and developed to create experimental data for executing a simulation. It is found that the possible number of passengers missing their flights could be drastically reduced by adding supporting staff to and by making use of first-and business-class check-in counters for processing economy-and group-class passengers.

Modeling people flow: simulation analysis of international-departure passenger flows in an airport terminal

Air traffic delay is a growing and expensive problem. We investigated ways to reduce the cost and magnitude of such delays by trading gate delays against more expensive air delays. Air management and planning at this level can be facilitated by simulation, especially for strategies that alter controls on the system. We used the SIMMOD air traffic simulation to model the system. The objective was to determine a set of control measures that achieve the best system performance subject to restrictions on the decision parameters and selected system output measurements. Because observed system performance is "noisy," the problem is a constrained stochastic optimization problem with nonlinear objective function and nonlinear, stochastic constraints, which requires efficient stochastic optimization methods for its solution. Our approach used the simultaneous perturbation stochastic approximation (SPSA) algorithm with a penalty function to handle the difficult constraints.

Simulation optimization of airline delay with constraints

The practical application of stochastic approximation methods requires a reliable means to stop the iterative process when the estimate is close to the optimizer or when further improvement in the estimate is doubtful. Conventional ideas on stopping stochastic approximation algorithms employ criteria based on a proxy distribution — usually the asymptotic distribution. Yet difficulties may arise when applying such distributions to small (finite) samples. We propose an approach that uses the distribution of a statistically similar process called a surrogate for the proxy distribution rather than the asymptotic distribution. Under certain conditions, surrogate-based probability calculations are close to the actual probabilities. The question of how surrogate processes may be developed is also addressed. Two example applications are given.

/pdf/stopping-small-sample-stochastic-approximation-2l8tmjshg4.pdf

Stopping small-sample stochastic approximation

Airline and airport applications: simulation optimization of airline delay with constraints

: The practical application of stochastic approximation methods require a reliable means to stop the iterative process when the estimate is close to the optimal value or when further improvement of the estimate is doubtful. Conventional ideas on stopping stochastic algorithms employ probabilistic criteria based on the asymptotic distribution of the stochastic approximation process, often with the parameters of the distribution determined by sequential estimation. Difficulties may arise when this approach is applied to small (finite) samples. We propose a different approach that uses the notion of an idealized process as a companion to the stochastic approximation. A discussion of this approach to stopping stochastic approximation is offered in the context of a simple example, including some empirical results.

Stopping Stochastic Approximation

The practical application of stochastic approximation methods requires a reliable means to stop the iterative process when the estimate is close to the optimal value or when further improvement of the estimate is doubtful. Conventional ideas on stopping stochastic algorithms employ probabilistic criteria based on the asymptotic distribution of the stochastic approximation process, often with the parameters of that distribution estimated by sequential estimation. 
Difficulties may arise when this approach is applied to small (finite) samples or when the conditions for asymptotic convergence do not hold. Here is proposed a different approach that uses the notion of a surrogate process as a proxy for the stochastic approximation. 
A surrogate process is a parameterized stochastic process whose distribution is close to that of the original stochastic approximation process, but whose probabilities are easier to calculate. We show that, given certain conditions, the error in the probability that the estimate is close to the optimal solution is small when computed using a surrogate process proxy distribution. We include a discussion of this approach to stopping stochastic approximation in the context first of a simple example, and then a more realistic simulation optimization, including some empirical results. 
Surrogate processes contribute to the theory and practice of stopping for stochastic approximation is as follows by providing a systematic approach to obtaining a proxy for the distribution of the estimate qdk (given the appropriate conditions), and by providing a means to stop a process or compute statistics for the estimate qdk without relying on asymptotic performance of the process, including such cases as finite samples, non-converging processes, and constant step size processes.

David W. Hutchison

Papers

Simulation optimization of airline delay with constraints

Stopping small-sample stochastic approximation

Airline and airport applications: simulation optimization of airline delay with constraints

Stopping Stochastic Approximation

Stopping times and confidence bounds for small-sample stochastic approximation algorithms