Multiple Objectives Satisficing Under Uncertainty

Abstract

We consider a multiple objectives problem in which the objectives are potentially uncertain when the decisions are made. The goal of the decision problem is to select a feasible solution so that all the objectives achieve their specified targets as well as possible. The targets may also be potentially uncertain. In order to deal with this multiple objectives problem, we propose a class of objective criteria, called multiple objectives satisficing (MOS) criteria. The MOS criteria provide efficacious evaluations of the level of compliance of the set of objectives in meeting their targets collectively under uncertainty. The MOS criteria include the joint targets achievement probability (or success probability) as a special case and also extend to situations when the probability distribution is not fully characterized (i.e. where only partial distributional information such as the support, mean and covariance are available). We specialize the MOS criteria to a sub-class of diversification favoring MOS (DMOS) criteria which has the potential to mitigate severe shortfalls in scenarios when an objective fails to achieve its target. Naturally, the DMOS criteria excludes success probability, which is inherently insensitive to the degree of shortfalls. A specific form of the DMOS criteria that inherits all the desirable characteristics of the MOS criteria and simultaneously incorporates diversification preferences is proposed. The shortfall-aware MOS criterion (S-MOS) is shown to be a lower bound to success probability and allows for the consideration of distributional ambiguity. As the consideration of distributional ambiguity may lead to intractable problems, we show how to construct tractable approximations for the S-MOS criterion. We also propose algorithms for evaluating the S-MOS criterion via solving sequences of convex optimization problems. We report encouraging results via an array of numerical experiments and case studies based on problems in the product development, finance and the oil and gas industry. The numerical studies clearly demonstrate the ability of the S-MOS criterion in overcoming the inherent deficiencies of success probability and expected value based criteria. Specifically, it is shown to mimic probability measures and yet possesses the qualities of sensitivity to the degree of shortfalls, ease of specifications and the ability to deal with decision problems where only partial distributional information are available.