Theories and an exact interactive paired-comparison approach for discrete multiple-criteria problems

Decisionmakers (DM’s) are usually faced with the selection of alternatives that are characterized by noncommensurate conflicting multiple criteria. An interactive approach is developed to help DM’s find the best alternatives with few questions without making stringent assumptions about their behavior. Theories and procedures are developed for ranking alternatives and eliminating suboptimal ones, assuming that the DM can respond to trade-off and paired comparison questions. It is assumed that the DM wishes to maximize an unknown quasi-concave utility function for discrete multiple criteria decisionmaking (MCDM) problems. Several tests are developed based on convex dominating cones. The paper discusses how trade-off questions can be generated for MCDM discrete problems. For the first itme, optimality conditions for discrete MCDM problems are given for extreme, nonextreme, and convex-dominated points without requiring the DM to enumerate the remaining set of discrete alternatives. This optimality condition is based on a branching technique which converts nonextreme points to extreme points. This substantially reduces the number of questions asked of the DM. The paper discusses approaches for identifying discrete points for the one-dimensional search. Finally, an exact discrete MCDM (EDMCDM) method is developed. Some computational experiments are provided which indicate that the method asks relatively few questions to obtain the most preferred alternative. Some examples are discussed.