A Goal-Programming-Based Heuristic Approach to Deriving Fuzzy Weights in Analytic Form from Triangular Fuzzy Preference Relations
Triangular fuzzy preference relations (TFPRs) have long been viewed as an effective framework to express subjective preferences with ambiguity in fuzzy multicriteria decision-making systems. This paper focuses on solving two important and challenging issues: First, how to judge the quality of nondiagonal preferences in a TFPR; and second, how to find an analytic solution of optimal fuzzy weights for consistent TFPRs and an analytic solution of heuristic fuzzy weights for inconsistent TFPRs. This paper analyzes two existing normalization models of triangular fuzzy weights and illustrates their deficiency. Notions of normalized triangular fuzzy multiplicative weights (TFMWs) and basic TFMWs are then introduced and used to characterize a group of triangular fuzzy weight vectors with equivalency. This paper proposes transformation models among triangular fuzzy weights, normalized TFMWs and basic TFMWs, and employs basic TFMWs to define consistent TFPRs. Some important properties are presented for the fuzziness of a consistent TFPR and its corresponding basic TFMWs. These properties are subsequently used to develop a two-step approach consisting of three-goal programming (GP) models to find basic TFMWs of consistent TFPRs. By using the Lagrangian multiplier method, this paper finds analytic solutions of the three GP models and obtains optimized basic TFMWs denoted by three computation formulas for consistent TFPRs. By changing the constraints of the three GP models and using heuristics, three concise and visualized formulas are devised, respectively, to obtain the lower and upper support bounds and the modal values of heuristic-based TFMWs for any TFPR. A numerical example comprising of a consistent TFPR and two inconsistent TFPRs is supplied and a comparative study is conducted to show that the two challenging issues are reasonably solved, and a hierarchical fuzzy multicriteria decision-making example is offered to illustrate the applicability of the proposed fuzzy priority model.