Effects of Probability Function on the Performance of Stochastic Programming

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 into an equivalent deterministic problem, which can be solved by any known classical methods (interior penalty method is applied here).The paper mainly focuses on investigating the 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 functions for random variables are derived and sensitivity analyses to study the effects of distribution function types and input parameters on the optimum solution are presented as graphs and in tables by studying two considered test problems. It is concluded that the difference between probabilistic and deterministic solutions to a problem, when the normal distribution of random variables issued, is very different from the results when gamma and exponential distribution functions are used. Finally, it is shown that the rate of solution convergence tithe normal distribution is faster than the other distributions