Some Duality Results on Linear Programming Problems with Symmetric Fuzzy Numbers

Abstract

Recently, linear programming problems with symmetric fuzzy numbers (LPSFN) have considered by some authors and have proposed a new method for solving these problems without converting to the classical linear programming problem, where the cost coefficients are symmetric fuzzy numbers (see in [4]). Here we extend their results and first prove the optimality theorem and then define the dual problem of LPSFN problem. Furthermore, we give some duality results as a natural extensions of duality results for linear programming problems with crisp data.

Duality in fuzzy linear programming: a survey

“…Unfortunately, there is some terminological confusion in the literature regarding the definition of fuzzy numbers as many authors, e.g., [7,15,18,34,46,50] allow in their definitions of fuzzy numbers a core of more than one element and refer to Dubois and Prade [19] fuzzy intervals as fuzzy numbers, which is not in accordance with Definition 3. In order to remain consistent with this widely used terminology and to avoid confusion, we adopt this (broader) understanding of fuzzy numbers in the remainder of this paper.…”

“…A key concept of ordering fuzzy numbers is the usage of a ranking function R : F(R) → R, with a b if and only if R( a) ≤ R( b) and a ≈ b if and only if R( a) = R( b) (see, for example, [50]). R is called linear if and only if R( a + λ b) = R( a) + λ R( b), λ ∈ R. A second option is to draw on a (lexicographic) ranking function R : F(R) → R × R [28] based on the concepts of possibilistic mean value and variance of a fuzzy number [8].…”

“…However, in most real-world applications decision makers require crisp solutions to make a decision. For example, Nasseri et al [51] consider a FLP profit maximization problem, where the produced units of products are represented as fuzzy sets. This implies that the optimal number of units to be produced is also a fuzzy number.…”

“…However, since most of the trapezoidal fuzzy numbers discussed in the presented works can be additionally restricted to have a zero left/right spread, we discuss the approaches further under the condition that they are only applicable with this additional restriction. An exception here is Nasseri et al [51] where the authors require the use of fuzzy numbers with positive left and right spread. As a result, we believe that the paper does not provide a useful contribution to duality in FLP and exclude it from further discussion.…”

“…The order of presentation is in the order of ascending numbers of classes with two exceptions. The first one is due to the fact that many papers draw on the early ideas of Hamacher et al [26] and Rödder and Zimmermann (2) impossible (3) impossible (4) impossible (5) impossible (6) impossible (7) impossible (8) [77] (9) (10) impossible (11) impossible (12) (13) (14) impossible (15) impossible (16) [44] (17) impossible (18) [77], [7] (4) (19) impossible (20) impossible (21) impossible (22) impossible (23) impossible (24 (2), (3),(4) (29) (30) [50] (1),(2), (4) , [51] (1),(2),(3),(4) (31) [87,31,61,60,62,25], [28] (1) , [46] (2),(4) (1) fuzzy decision variables (2) duality results are meaningful only when the left (right) spread of trapezoidal fuzzy numbers on right-hand side of ≤ (≥) constraints equals zero (3) excluded from further discussion due to theoretical weaknesses (4) the definition of fuzzy numbers differs from Definition 3 [63], which are used for deriving duality results in class 24. Thus, this class is presented first.…”