Robust decision making using a general utility set

Abstract

We address the problem of ambiguity and inconsistency in a decision maker’s (DM) assessed utility function by using a maxmin framework. In this framework DM’s utility function belongs to a set of functions. The member functions of the set are nondecreasing, and satisfy additional boundary and auxiliary conditions. The maxmin framework provides robustness in decision making. Alternatively, it allows us to perform sensitivity or parametric analysis for the optimal decision when a reference utility function is known. For this purpose we use a cost of ambiguity concept, and show that this cost is increasing and concave when extent of ambiguity is parametrically increased. Next we develop a Lagrangian based solution approach for the decision problem. We show that under suitable conditions a Sample Average Approximation (SAA) of the Lagrangian model can be solved using a mixed integer linear program (MILP). We also show that the set of optimal solutions of the SAA converges to that of its true counterpart, and the optimum objective value of the SAA converges to the true objective value at an exponential rate. We use this convergence property to develop a heuristic for identifying a solution of the SAA MILP with increasing sample size. We illustrate the properties of the maxmin model using two examples of portfolio investment and streaming bandwidth. We provide a discussion on the performance of a commercial solver used to solve SAA MILPs, and the quality of solutions generated by the proposed heuristic.