Mathematical programming models for characterizing dominance and potential optimality when multicriteria alternative values and weights are simultaneously incomplete

This paper is concerned with the use of incomplete information on both multicriteria alternative value scores and importance weights in evaluating decision alternatives under certainty. Two decision concepts-dominance and potential optimality (PO)-are employed to determine outperforming alternatives or acts. Given incomplete weights and exact value scores, it is known that the set of nondominated acts implies the set of potentially optimal acts. We here rediscover this set inclusive relation in a different and computationally efficient manner that develops a mixed-integer programming approach to dominance. When both value scores and weights are incomplete, we show that the set inclusive relation between dominance and PO may or may not hold. This is a result of how we define PO. Two different definitions of PO-weak PO and strong PO-are made in the sense that an act is weak potentially optimal when there exists at least one exact vector of value scores in the given incomplete value scores which render this act potentially optimal. An act is strong potentially optimal when this act is potentially optimal for all possible value scores. Less formally, this means that under the given incomplete information, weak PO stands for sometimes good while strong PO stands for always good, so we can thus say that the latter outperforms than the former. As a result, we show that the set of nondominated acts implies the set of strong potentially optimal acts. No certain set inclusive relationship exists between nondominated acts and weak potentially optimal acts. We also present computational aspects of establishing dominance, weak and strong PO, which include transformations of nonlinear problems into linear programming problems.