Effects of Probability Function on the Performance of Stochastic Programming

Document Type : Original Manuscript

Authors

1 Ph.D. Candidate in Structural Eengineering, Shiraz University, Shiraz, Iran.

2 Associate Professor, Department of Civil and Environment Engineering, Shiraz University, Shiraz, Iran.

Abstract

Stochastic programming is a valuable optimization tool where used when some or all of the design parameters of an optimization problem are defined by stochastic variables rather than by deterministic quantities. Depending on the nature of equations involved in the problem, a stochastic optimization problem is called a stochastic linear or nonlinear programming problem. In this paper,a stochastic optimization problem is transformed intoan equivalent deterministic problem,which can be solved byany known classical methods (interior penalty method is applied here).The paper mainly focuseson investigatingthe effect of applying various probability functions distributions(normal, gamma, and exponential) for design variables. The following basic required equations to solve nonlinear stochastic problems with various probability functionsfor random variables are derived and sensitivity analyses to studythe effects of distribution function typesand input parameterson the optimum solution are presented as graphs and in tables by studyingtwoconsidered test problems. It is concluded that thedifference between probabilistic and deterministic solutions toa problem, when the normal distribution ofrandom variables isused, is very different fromthe results when gamma and exponential distribution functions are used. Finally, it is shownthat the rate of solution convergence tothe normal distribution is faster than the other distributions.

Highlights

  • This article focuses on investigating the effect of applying various probability functions distributions (normal, gamma and exponential) for design variables on the stochastic programming solution.
  • Sensitivity analyses to study the effects of distribution function types and input parameters on the optimum solution are presented.
  • Difference between probabilistic and deterministic solutions (for various functions distributions) for some test problems is investigated.

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