Engineering Optimization: Theory and Practice, Fourth Edition

11 Stochastic Programming

11.1 Introduction

Stochasticorprobabilistic programmingdeals with situations where some or all of

the parameters of the optimization problem are described by stochastic (or random or

probabilistic) variables rather than by deterministic quantities. The sources of random

variables may be several, depending on the nature and the type of problem. For instance,

in the design of concrete structures, the strength of concrete is a random variable since

the compressive strength of concrete varies considerably from sample to sample. In

the design of mechanical systems, the actual dimension of any machined part is a

random variable since the dimension may lie anywhere within a specified (permissible)

tolerance band. Similarly, in the design of aircraft and rockets the actual loads acting

on the vehicle depend on the atmospheric conditions prevailing at the time of the flight,

which cannot be predicted precisely in advance. Hence the loads are to be treated as

random variables in the design of such flight vehicles.

Depending on the nature of equations involved (in terms of random variables) in

the problem, a stochastic optimization problem is called astochastic linear, geometric,

dynamic, ornonlinear programming problem. The basic idea used in stochastic pro-

gramming is to convert the stochastic problem into an equivalent deterministic problem.

The resulting deterministic problem is then solved by using familiar techniques such as

linear, geometric, dynamic, and nonlinear programming. A review of the basic concepts

of probability theory that are necessary for understanding the techniques of stochastic

programming is given in Section 11.2. The stochastic linear, nonlinear, and geometric

programming techniques are discussed in subsequent sections.

11.2 Basic Concepts of Probability Theory

The material of this section is by no means exhaustive of probability theory. Rather,

it provides the basic background necessary for the continuity of presentation of this

chapter. The reader interested in further details should consult Parzen [11.1], Ang and

Tang [11.2], or Rao [11.3].

11.2.1 Definition of Probability

Every phenomenon in real life has a certain element of uncertainty. For example,

the wind velocity at a particular locality, the number of vehicles crossing a bridge,

the strength of a beam, and the life of a machine cannot be predicted exactly. These

632 Engineering Optimization: Theory and Practice, Fourth Edition Singiresu S. Rao
Copyright © 2009 by John Wiley & Sons, Inc.