Equitable Efficiency and Decomposition of Preference Matrix in Multiobjective Optimization Problems

In equitable multiobjective optimization, all of the objectives are uniformly optimized, but in some cases, the decision maker believes that some of them should be uniformly optimized. In order to solve the proposed problem, we introduce the concept of equitable AP-efficiency, where P = {P1,P2,…,Pn} is a partition of the index set of objective functions and the preference matrix AP is the direct sum of the matrices A1,A2,…,An, in which Ak is a preference matrix for the objective functions in the class Pk for k = 1, 2,…,n. We examine some theoretical and practical aspects of equitably AP-efficient solutions and provide the some conditions that guarantee the relation of equitable AP-dominance is a P-equitable rational preference.Furthermore, we introduce the new problem with the preference matrix AP and we decompose it into a collection of smaller subproblems. In continuation, the subproblems are solved by the concept of equitable efficiency. Finally, two models are demonstrated to coordinate equitably efficient solutions of the proposed subproblems.