Posynomial Geometric Programming with Fuzzy Coefficients
In practice, there are many problems in which all decision parameters are fuzzy numbers, and such problems are usually solved by either possibilistic programming or multiobjective programming methods. Unfortunately, all these methods have shortcomings. In
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Abstract In practice, there are many problems in which all decision parameters are fuzzy numbers, and such problems are usually solved by either possibilistic programming or multiobjective programming methods. Unfortunately, all these methods have shortcomings. In this note, using the concept of comparison of fuzzy numbers, we introduce a very effective method for solving these problems. Then we propose a new method for solving posynomial geometric programming problems with fuzzy coefficients. Keywords Fuzzy number · Posynomial geometric programming · Fuzzy posynomial geometric programming.
1 Introduction Fuzzy geometric programming was first proposed by Cao Bingyuan in 1987 in Tokyo, in the second session of the International Fuzzy Systems Association (IFSA) conference held in Japan. The direct algorithm and dual algorithm of fuzzy geometric programming were studied in references [1, 2]. By mainly utilizing original algorithm and duality algorithm to the multi-objective fuzzy geometric programming, the multi-objective fuzzy geometric programming has been solved in [3–5]. The referR. Hu · G. Zhang (B) School of Management, Guangdong University of Technology, 510520 Guangdong, China e-mail: [email protected] G. Zhang e-mail: [email protected] B. Cao School of Mathematics and Information Science,Guangzhou University, 510006 Guangdong, China e-mail: [email protected]
B.-Y. Cao and H. Nasseri (eds.), Fuzzy Information & Engineering and Operations Research & Management, Advances in Intelligent Systems and Computing 211, DOI: 10.1007/978-3-642-38667-1_5, © Springer-Verlag Berlin Heidelberg 2014
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ences [6] studied the method for solving fuzzy posynomial geometric programming based on comparison method of fuzzy numbers developed by Roubens. The references [7] studied the method for solving fuzzy posynomial geometric programming based on comparison method of fuzzy numbers developed by Renjie Hu. In this note, we discuss the problem on how to solve the fuzzy posynomial geometric programming with fuzzy coefficients. For solving this problem, we proposed a new method on comparison of fuzzy numbers. By utilizing the concept of comparison of fuzzy numbers; posynomial geometric programming with fuzzy coefficients is reduced to a posynomial geometric programming with crisp coefficient. In other words, we get a new method different from the method mentioned in [1, 2, 6, 7]. And the method is testified to be effective by numerical examples.
2 Preliminaries ∈ F(X ), ∀α ∈ [0, 1] , written down as Aα = x ∈ X μ A Definition 2.1 Let A (x) ≥ α } , Aα is said to be α − cut set of a fuzzy set A. be a fuzzy number, i.e. a convex normalized fuzzy subset of the Definition 2.2 Let A real line in the sense that: (a) ∃x0 ∈ R and μ A(x0 ) = 1 , where μ A(x) is the membership function specifying to what degree x belongs to A. (b) μ A is a piecewise continuous function. AαL
L R According 2.1 and 2.2, definition definition the α − cut of A is Aα = [Aα , Aα ], R = inf x ∈ X μ A˜ (x) ≥ α ,
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