Multi-objective stochastic linear programming with recourse and flexible decision making

Abstract

Optimal inventory leads to stochastic optimization problems where deterministic delivery decisions have to be made in advance of stochastic demand realizations. Similarly, risk deposits have to be given before the random outcomes of investments are known. In this paper, multi-criteria versions of such stochastic recourse problems are studied. In addition to traditional concepts like Pareto-optimality, a decision maker for the multi-criteria problem may have a preference for greater flexibility in the second stage decision. This idea leads to a first stage optimization problem with a set-valued objective instead of a mere multi-criteria one. Under linearity assumptions, this problem becomes a polyhedral convex set optimization problem instead of a multi-objective linear program. Solution concepts for multi-objective/set-valued recourse problems are given as well as deterministic surrogates of the stochastic problem such as its deterministic equivalent, the so-called wait-and-see problem and the expected-value problem for the multi-objective case. Managerial decision making guidelines are obtained based on set optimization methods and the preference-for-flexibility approach: choose the deterministic first stage variable such that a maximum of flexibility is combined with a guarantee for Pareto minimal objective values. Two major examples illustrate the findings, a multi-objective newsvendor problem with an additional health/sustainability objective and a risk compensation problem where the availability of more than one asset for risk compensation, e.g., several currencies, leads to multiple objectives.