A single-variable method for solving min–max programming problem with addition-min fuzzy relational inequalities

• Published in: Fuzzy optimization and decision making Vol. 18; no. 4; pp. 433 - 449
• Main Authors: Chiu, Ya-Ling, Guu, Sy-Ming, Yu, Jiajun, Wu, Yan-Kuen
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2019; Springer Nature B.V
• Subjects: Artificial Intelligence; Calculus of Variations and Optimal Control; Optimization; Iterative methods; Lower bounds; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Nonlinear programming; Operations Research/Decision Theory; Optimization; Probability Theory and Stochastic Processes; Programming; Fuzzy relational inequalities; Min–max programming problem; Single-variable method; Addition-min composition
• ISSN: 1568-4539; 1573-2908
• DOI: 10.1007/s10700-019-09305-9

In this paper, we study the min–max programming problem with n addition-min fuzzy relational inequality constraints. We prove that when the problem is feasible, an optimal solution always exists with all variables being of the same value. Based on this result, the min–max programming problem can be simplified as a single-variable optimization problem with the same optimal objective value. To solve the corresponding single-variable optimization problem, we propose an analytical method and an iterative method by successively approximating the lower bound of the optimal value. Numerical examples are given to illustrate our methods.