Abstract
Duality theorem is an attractive approach for solving fuzzy optimization problems. However, the duality gap is generally nonzero for nonconvex problems. So far, most of the studies focus on continuous variables in fuzzy optimization problems. And, in real problems and models, fuzzy optimization problems also involve discrete and mixed variables. To address the above problems, we improve the extended duality theory by adding fuzzy objective functions. In this paper, we first define continuous fuzzy nonlinear programming problems, discrete fuzzy nonlinear programming problems, and mixed fuzzy nonlinear programming problems and then provide the extended dual problems, respectively. Finally we prove the weak and strong extended duality theorems, and the results show no duality gap between the original problem and extended dual problem.
Highlights
Nonlinear programming problems (NLPs) play an important role in both manufacturing systems and industrial processes and have been widely used in the fields of operations research, planning and scheduling, optimal control, engineering designs, and production management [1,2,3,4]
We define extended duality theory of fuzzy nonlinear optimization with continuous, discrete, and mixed spaces and prove the weak and strong extended duality theorems, and the results show no duality gap between the original problem and extended dual fuzzy optimization problems
We proposed the fuzzy optimization problems in continuous, discrete, and mixed spaces and defined the extended duality problems, respectively
Summary
Nonlinear programming problems (NLPs) play an important role in both manufacturing systems and industrial processes and have been widely used in the fields of operations research, planning and scheduling, optimal control, engineering designs, and production management [1,2,3,4]. In 2008, Wu proposed continuous and differentiable fuzzyvalued objective function with real constraints and presented the sufficient optimality conditions for obtaining the nondominated solution of fuzzy optimization problem [15]. Later, he adopted the Karush-Kuhn-Tucker optimality conditions to solve the fuzzy optimization problems [16]. To deal with nonconvex fuzzy optimization problems with continuous, discrete, and mixed variable, we improve the extended duality theory by adding fuzzy objective functions. We define extended duality theory of fuzzy nonlinear optimization with continuous, discrete, and mixed spaces and prove the weak and strong extended duality theorems, and the results show no duality gap between the original problem and extended dual fuzzy optimization problems.
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