Relatively Robust Multicriteria Decisions

For a general multicriteria decision problem with linear scalarization and unknown weights, we propose relatively robust decisions, which are Pareto-efficient and at the same time maximize a performance index. The latter measures the worst-case ratio, attained by the weighted objective relative to its maximum value, with respect to all possible weights. The main results include a simple boundary representation of the performance index as the minimum of criterion-specific performance ratios, and a computationally simple method of determining a relatively robust decision up to any prespecified performance tolerance by maximizing an Formula: see text-augmented performance index. The proposed method relies merely on the continuity of all criterion functions and the compactness of the set of feasible decisions which may be nonconvex. This imposes no restrictions at all for any finite action set. A notable feature of our method is that it endogenously yields the tradeoffs between all criteria, including a performance guarantee relative to decisions justified by any other weighting. A number of structural results, examples, and applications are provided, as well as generalizations to allow for limited weight ambiguity, criterion ambiguity, and generalized aggregation of criteria based on an axiomatic foundation. This paper was accepted by Peng Sun, optimization and decision analytics.