A Generalization of Some Distribution Aspects of Chance-Constrained Linear Programming

Abstract

AN ORDINARY LINEAR PROGRAMMING MODEL is said to be chance-constrained if its linear constraints are associated with a set of probability measures indicating the extent of violation of the constraints. When partial violation of the constraints is allowed for, the chance-constrained approach may be viewed as a method for providing appropriate safety margins. This approach has been generalized [9] in recent years in several directions. of which two are specially worth mentioning. First, although for reasons of simplicity solutions restricted to linear decision rules only have often been employed in chance-constrained programming, it is now possible to have solutions of a more general functional form, and this considerably enhances the scope of application of the chance-constrained approach in dynamic models with nonlinear objective functions. Second, it is not necessary in the chance-constrained approach to make the assumption that the decision maker's utility function is quadratic (or of a specific form), as it is required, for example, in the portfolio selection studies by Markowitz and others who base the analysis on the mean and variance of the probability distribution of net returns. Although the extent of violation of the constraints that would be tolerated is preassigned subjectively in this approach by the decision maker before the: actual solutions are computed, the tolerance measure may be parametrically varied, as in the revealed preference theory, and the resulting optimal solutions may help the decision maker move to the most preferred solution. This. formulation (see [8], [19]) also considerably helps the scope of applicability of the chance-constrained approach by developing suitable criteria for decisions under risk. Our objective here is to analyze the implications of a non-normal distribution of the random elements (A, b, c) of a linear program in the framework of probabilistic linear programming, where only the two approaches of chance-constrained programming (CCP) and stochastic linear programming (SLP) are considered. It is interesting to note that although the CCP and the SLP approaches are developed with different objectives in mind, they have frequently been applied under the assumption of normality of relevant