Conducting fuzzy division by using linear programming MURAT ALPER BASARAN Department of Mathematics Nigde University Nigde TURKEY muratalper@yahoo.com CAGDAS HAKAN ALADAG Department of Statistics Hacettepe University Beytepe, Ankara TURKEY chaladag@gmail.com CEM KADILAR Department of Statistics Hacettepe University Beytepe, Ankara TURKEY kadilar@hacettepe.edu.tr Abstract: Some approximation methods have been proposed for fuzzy multiplication and division in the literature. Instead of doing arithmetic operations using fuzzy membership functions for fuzzy numbers, parameterized representation of fuzzy numbers have been used in arithmetic operations. The most applied parameterized fuzzy numbers used in many of the research papers are symmetric and asymmetric triangular and trapezoidal fuzzy numbers. In this study, we propose a new approximation method based on linear programming for fuzzy division. In order to show the applicability of the proposed method, some examples are solved using the proposed method and the results are compared with those generated by other methods in the literature. The proposed method has produced better results than those generated by the others. Key-words: Approximation method; Fuzzy arithmetic; Fuzzy division; Linear programming; Triangular fuzzy number 1 Introduction Instead of using membership functions in fuzzy arithmetic, parameterized form of fuzzy numbers have been used for arithmetic operations. Despite of the fact that the addition and subtraction of the parameterized fuzzy numbers result in closed form, the same does not hold for multiplication and division. However, fuzzy multiplication in closed form can be done under the weakest T norm as an exception [4]. Therefore, some approximation methods are proposed for multiplication and division of the parameterized fuzzy numbers [1,2,3,4]. The most applied parameterized fuzzy numbers are symmetric and asymmetric triangular and trapezoidal fuzzy numbers since they are simple and easy to implement for many purposes [8]. Dubois and Prade [1] first introduced arithmetic operations on parameterized fuzzy numbers. Then, Giachetti and Young [2,3] proposed a new method for fuzzy multiplication and division. Another method for fuzzy arithmetic operation is the weakest T norm [4]. In these studies, researchers have focused on decreasing the fuzziness of the resulting fuzzy number. It is assumed that two parameterized triangular fuzzy numbers are multiplied or divided. This computation can be done using one of the methods mentioned above [1,2,3,4]. However, the results obtained from these methods have different fuzzy end points or spread values except center value. Instead of using parameterized form of fuzzy numbers that consist of real numbers, the area and the proportions of center value to left and to right end points are used to do multiplication or division of two fuzzy numbers. We proposed a new method based on linear programming for parameterized fuzzy division. The parameterized fuzzy numbers are limited to symmetric and asymmetric and trapezoidal fuzzy numbers because of their simplicity and ease of use. In this paper, new fuzzy division method is applied to triangular fuzzy numbers. The essence of proposed method depends on utilizing geometric features and definition of triangular fuzzy numbers [5,7]. In the next section, some preliminary knowledge related to fuzzy set is given. Section 3 gives brief information about existing approaches available in the literature for fuzzy division. The new proposed method is introduced in Section 4. Section 5 contains the implementation of the proposed method and the comparisons with other available methods in the literature. Last section is discussion and conclusion. 2 Preliminaries A fuzzy set [6] A of the real line R with membership function [ ] 1 , 0 : → R A μ is called a fuzzy number if 1. A is normal, namely, there exist an element x such that 1 ) ( = x A μ 2. A is fuzzy convex, that is, ) ( ) ( ) ) 1 ( ( 2 1 2 1 x x x x A A A μ μ λ λ μ ∧ ≥ − + ] 1 , 0 [ , , 2 1 ∈ ∀ ∈ ∀ λ R x x 3. A μ is semi continuous 4. supp A is bounded where supp { } 0 ) ( : > ∈ = x R x A A μ WSEAS TRANSACTIONS on INFORMATION SCIENCE & APPLICATIONS Manuscript received Dec. 5, 2007; revised May 10, 2008 Murat Alper Basaran, Cagdas Hakan Aladag, Cem Kadilar ISSN: 1790-0832 Issue 6, Volume 5, June 2008 923