Stochastic Linear Programming with Chance Constraints

Abstract

An ordinary linear programming model is said to be chance-constrained if its linear constraints are associated with a set of measures indicating the extent of violation of the constraints. The CCP approach usually assumes the resource vector to be normally and mutually independently distributed and then derives a deterministic concave programming problem. In the SLP approach the tolerance measure for the linear constraints is not preassigned by the decision maker and the approach seeks to derive the statistical distribution of the optimal solution vector and also of the optimal objective function under the assumption that the set (A, b, c) of parameters contains elements with known probability distributions. Some basic differences of the CCP and the SLP approaches may be noted at the outset. First, the CCP approach utilizes the distribution properties of relevant random variables statisfying preassigned tolerance limits to specify a deterministic nonlinear program, whereas the SLP approach starts from a deterministic linear program (e.g., a program where all random elements are replaced by their expected values) and admits the random variations around its optimal basis to derive the probability distribution of the optimal solution satisfying (if necessary at a later stage) some tolerance measures if and when feasible. Second, nonlinearities are introduced in both approaches, although the initial problem in both cases is a linear programming problem. Third, the CCP approach restricts decision rules within a certain class (e.g.,