Hossein Arsham

Sensitivity analysis and the “what if” problem in simulation analysis

We discuss some known and some new results on the score function (SF) approach for simulation analysis. We show that while simulating a single sample path from the underlying system or from an associated system and applying the Radon-Nikodym measure one can: estimate the performance sensitivities (gradient, Hessian etc.) of the underlying system with respect to some parameter (vector of parameters); extrapolate the performance measure for different values of the parameters; evaluate the performance measures of queuing models working in heavy traffic by simulating an associated (auxiliary) queuing model working in light (lighter) traffic; evaluate the performance measures of stochastic models while simulating random vectors (say, by the inverse transform method) from an auxiliary probability density function rather than from the original one (say by the acceptance-rejection method). Applications of the SF approach to a broad variety of stochastic models are given.

Perturbation analysis of general LP models: A unified approach to sensitivity, parametric, tolerance, and more-for-less analysis

This paper develops an alternative approach to postoptimality analysis for general linear programming (LP) problems that provides a simple framework for the analysis of any single or simultaneous change of right-hand side (RHS) or cost coefficient terms for which the current basis remains optimal by solving the nominal LP problem with perturbed RHS terms. Postoptimality analysis of a row or column of the matrix coefficients is also discussed. The goal is a theoretical unification, as well as an advancement in the practical implementation of postoptimality analysis. Some common applications, such as ordinary sensitivity, the 100% rule, and parametric analysis, as well as extensions of recent developments such as tolerance analysis and the more-for-less paradox, are discussed in the context of numerical examples.

Kuiper's P-value as a measuring tool and decision procedure for the goodness-of-fit test

The need for computing P-values for the Kuiper statistic has been emphasised by Batschelet (1981). Some exact P-values, with useful interpretations for making inference about a probability model for circular data, are provided. Computation of the exact values are based on Durbin (1973) boundary crossing probabilities. A numerical example is used to demonstrate the usefulness of the results.

Managing project activity-duration uncertainties

The critical path method (CPM), project evaluation and review technique (PERT), and stochastic PERT, the most widely used tools for project management, each require different forms of activity duration information: fixed time, three-time estimates and an a priori distribution function, respectively. While PERT and stochastic PERT allow for uncertainty in activity durations unfortunately they require a number of strong statistical assumptions. In this paper, uncertainty in project activity duration is treated in a managerial context rather than as random factors. A linear programming (LP) formulation of the project activity network is constructed and a new simplex-type tabular solution algorithm is developed to find a critical path (CP). While the proposed approach requires a time estimate for each activity duration, it provides the manager with the allowable simultaneous, independent or dependent changes of the estimates that will preserve the current CP. The results of these analyses empower the project planner and manager to assess and monitor various types of activity-duration uncertainties encountered in real life situations. It is assumed the reader is familiar with LP terminology.

STOCHASTIC OPTIMIZATION OF DISCRETE EVENT SYSTEMS SIMULATION

Abstract The Score Function (SF) method has been proposed to estimate the gradient of a performance measure with respect to some continuous parameters in a stochastic system. In this paper we experiment with the use of this estimate in a stochastic approximation algorithm to perform a single-run optimization . The experiment is done on a simple M/M/1 queue. The performance measure involves the average system time per customer at steady-state, and the decision variable is the service rate. The optimal solution is easy to compute analytically, which facilitates the evaluation of the algorithm. Combined with appropriate variance reduction techniques, the method has been shown to be effective in the test problem.We study the algorithms properties and examine the validity of the estimates based on this single run procedure by performing some experimental studies. Implementation “details” necessary for packaging this method with existing simulation software are provided. Finally, there is a set of recommended directions for future research.

Performance extrapolation in discrete-event systems simulation

The use of simulation as an engineering tool to design complex computer stochastic systems is often inhibited by cost. Extensive computer processing is needed to recompute performance functions for changes in input parameters. Moreover, simulation models are often subject to errors caused by the input data in estimating the parameters of the input distributions. The ‘;what if analysis is needed to establish confidence in a models validity with respect to small changes in the system’s parameters. To solve the ‘what if problem for several scenarios requires a separate simulation run for each scenario. A method to estimate the performance for several scenarios using a single simulation run based on the efficient score function is developed. The approximate function is an exponential random function which is tangential to the performance function in expectation. We study the algorithms properties and examine the validity of the estimates based on this single run procedure by performing some experimental stud...

Algorithms for sensitivity information in discrete-event systems simulation

Abstract This paper considers the design, analysis, and operation of discrete event systems (DES) with performance J(θ) depending on the value of certain decision parameters θ ϵ Θ. Both engineers and managers are interested in information about the sensitivity of J(θ) with respect to certain continuous parameters θ. This sensitivity information is useful for what-if analysis, assessing the relative importance of each parameter θ, studying the local functional behavior, and construction of J(θ) over Θ. In addition to this valuable descriptive information, the sensitivity information is of prime importance in prescriptive analysis, namely optimization and goal-seeking problems where a ‘good enough’ solution is preferred. We present algorithms for obtaining sensitivity information on DES via simulation. The paper assumes an existing validated and verified simulation application in which the presented algorithms can be incorporated to provide powerful sensitivity information. Our focus is on enhancement, theoretical-unification, and some extensions of the existing algorithms. All algorithms are presented in English-like format and therefore can be implemented in a variety of operating systems and machines, providing unlimited portability.

Managing cost uncertainties in transportation and assignment problems

In a fast changing global market, a manager is concerned with cost uncertainties of the cost matrix in transportation problems (TP) and assignment problems (AP).A time lag between the development and application of the model could cause cost parameters to assume different values when an optimal assignment is implemented. The manager might wish to determine the responsiveness of the current optimal solution to such uncertainties. A desirable tool is to construct a perturbation set (PS) of cost coeffcients which ensures the stability of an optimal solution under such uncertainties.

Affine Geometric Method For Linear Programs

The algebraic (simplex) method is a complete enumerating algorithm to solve bounded linear programs (LP). It converts all the inequality constraints to equality to obtain a system of equations by introducing slack/surplus variables, converts all non-restricted (in sign) variables to restricted ones by substituting the difference of two new variables, and finally solves all of its square subsystems. This conversion of an LP into a pure algebraic version overlooks the original space of the decision variables and treats all variables alike throughout the process. We propose an effective three-phase algebraic method to overcome these deficiencies. Phase I concentrates on an early recognition of basic feasible solutions (BFS) which are located on the coordinate axis of each decision variable. Phase II finds the BFS which are on the faces of the coordinate system by solving some subsystems of equations which depend on the number of decision variables and constraints. Finally, in Phase III, it locates the BFS which are not on the faces of the coordinate system. The feasibility test is performed efficiently at the end of each phase. The algorithm works directly with the original variables even if some are unrestricted. The proposed method reduces considerably the computational complexity because it does not include any slack/surplus variables. The algorithmic strategy is to partition the set of BFS into three distinct sub-sets with common characteristics. Numerical examples illustrate the new solution algorithm.

Construction of the largest sensitivity region for general linear programs

This paper develops an alternative approach to post optimality analysis for general linear program models which provides the largest sensitivity region of any single or simultaneous change of right hand side of constraints and the coefficients of decision variables in the objective function. The goal is a theoretical unification of various types of sensitivity analyses, as well as advancement in the practical implementation of post optimality analysis. We extend our proposed approach in the construction of sensitivity region to maintain the degenerate vertex for models with degenerate optimal solution as well as maintaining the multiple solutions for the models with non-unique optimal solutions. As a by product, the paper resolves the paradoxical situation known as the more-for-less/less-for-more situations. The proposed method is based on the given optimal solution(s), therefore it is easy to understand, easy to implement, and provides useful information to the decision makers. The methodology and their computational algorithms are presented and discussed in the context of some illustrative numerical examples.