Φ1Φ1-concavity and fuzzy multiple objective decision making

Concave properties play a dominate role in solving both classic and fuzzy optimization problems. However, since fuzzy problems are generally represented by sets, not crisp numbers, various aggregation schemes are needed to manipulate and to combine the different elements in a fuzzy optimization problem. Based on these different aggregations, various concavity properties can be formulated and explored. In this paper, the intersection aggregation and the convex combination aggregation are explored based on the supp-Φ1Φ1-concave fuzzy sets. First, the concept of Φ1Φ1-convexity, which covers a wider class of sets and functions, is extended to fuzzy sets. Supp-Φ1Φ1-concave and supp-Φ1Φ1-quasiconcave fuzzy sets are then introduced; and some useful aggregation and composition rules are developed. Based on these aggregation and composition rules and the generalized concave properties, fuzzy multiple objective decision making problems are formulated and the conditions to ensure local–global maximum property are discussed.