# 11 Stochastic Programming

## 11.1 Introduction

*Stochastic* or *probabilistic programming* deals 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 a *stochastic linear*, *geometric*, *dynamic*, or *nonlinear programming problem*. The basic idea used in stochastic programming 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, ...

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