Stochastic Programming

With the requirement of considering randomness, different types of stochastic programming have been developed to suit the different purposes of management. The first type of stochastic programming is the

expected value model

, which optimizes the expected objective functions subject to some expected constraints. The second,

chance-constrained programming

, was pioneered by Charnes and Cooper [37] as a means of handling uncertainty by specifying a confidence level at which it is desired that the stochastic constraint holds. After that, Liu [174] generalized chance-constrained programming to the case with not only stochastic constraints but also stochastic objectives. In practice, there usually exist multiple events in a complex stochastic decision system. Sometimes the decision-maker wishes to maximize the chance functions of satisfying these events. In order to model this type of problem, Liu [166] provided a theoretical framework of the third type of stochastic programming, called

dependent-chance programming

.

This chapter will give some basic concepts of probability theory and introduce a spectrum of stochastic programming. A hybrid intelligent algorithm is also documented.