Basic Results on Stochastic Programming

Abstract

Mathematical programming models have been applied to many problems in various fields. The data of real problems contain uncertainty and are thus represented as random variables. Stochastic programming deals with optimization under uncertainty. In Chap. 6, the basic model of the stochastic programming problem and solution algorithms are shown. Section 6.1 shows the basics of stochastic programming with recourse. This model is the most fundamental in the stochastic programming method, and its application to real problems is widely applied. In this model, when constraints are violated in accordance with probability fluctuations, new decisions are introduced to correct inconvenience. In this model, the total cost including the cost of additional decisions is minimised. The additional cost and its expected value are referred to as a recourse function, and it is shown that the recourse function becomes a convex function of the initial decision variable. In Sect. 6.2 we deal with the extension of stochastic programming with recourse to multistage planning. In multistage planning, problems generally include subproblems that correspond to a large number of scenarios. A basic method for decomposing such large scale problem is shown. Section 6.3 considers probabilistic programming including integer conditions. Especially the exact solution method for problem solving for probability planning problem including 0–1 variables. The 0–1 variable can be applied to actual problems such as various decision making, execution and interruption of specific decisions, and its application range is wide. Section 6.4 shows a stochastic programming model including stochastic constraints. This chance constraint gives the lower limit value of the probability of satisfying the constraint, and it has a deep relationship with the value at risk. We also consider the numerical integration method to calculate such probability. In Sect. 6.5, we consider stochastic programming considering risk. In particular, the problem considering variance of costs is dealt with. Since the objective function is a non-convex function, an exact solution by a branch cut method is presented which approximates objective function by a convex quadratic function.