Chance-Constrained Models

Abstract

Chance-constrained programming was developed as a means of describing constraints in mathematical programming models in the form of probability levels of attainment. Consideration of chance constraints allows decision makers to consider mathematical programming objectives in terms of the probability of their attainment. If α is a predetermined confidence level desired by a decision maker, the implication is that a constraint will be violated at most (1 – α) of all possible cases.

A number of different types of models can be built using chance constraints. The first form is to maximize the linear expected return subject to attaining specified probabilities of reaching specified targets. The second is to minimize variance. This second form is not that useful, in that the lowest variance is actually to not invest. Here we forced investment of the 1000 capital assumed. The third form is to maximize probability of attaining some target, which in order to be useful, has to be infeasible.

Chance-constrained models have been used in many applications. Here we have focused on financial planning, but there have been applications whenever statistical data is available in an optimization problem.