Duality in nonlinear programming problems under fuzzy environment with exponential membership functions

In this paper, we have established appropriate duality relations for a general nonlinear optimization problem under fuzzy environment, taking exponential membership functions and using the aspiration level approach. A numerical example has also been shown to justify the results presented in the paper.


Introduction
Zadeh in 1965 introduced fuzzy set theory by publishing the first article in this area. He generalized the classical notion of a set and a proposition to accommodate fuzzyness. This has been applied in diverse fields such as machine learning, multi-attribute decision making, supply chain problems, management sciences, etc. Fuzzy control, which directly uses fuzzy rules, is the important application in fuzzy theory. Fuzzy set theory is also applicable in the real life case like controlling smart traffic light. The controller is designed in such a way that it changes the cycle time depending upon the densities of cars behind red and green lights.
The fuzzy set theory provides various logical operators that allow the aggregation of several criteria to just one criterion. These operators can be evaluated with respect to axiomatic requirements, numeric efficiency robustness, degree of compensation among the criteria, and ability to model expert behavior.
Bellman and Zadeh [1] proposed the idea of decision making in fuzzy environment. After the pioneering work on fuzzy linear programming problems (FLPP) in Tanaka et al. [2] and Zimmermann [3], several kinds of (FLPP) along with the different solution methodologies have been discussed in the literature. Many researchers, including Lai and Hwang [4], Shaochang [5], Buckley [6,7], and Negi [8], have considered the problems where all parameters are fuzzy. Lai and Hwang [4] assumed that the parameters have a triangular possibility distribution. Using multiobjective linear programming methods, they provided an auxiliary model related to it. Rodder and Zimmermann [9] were the first who studied the duality of (FLPP), considering the economic interpretation of the dual variables. After that, many interesting results regarding the duality of (FLPP) have been investigated by several researchers [10][11][12][13][14][15][16][17][18]. Zhang et al. [19] investigated the duality theory in fuzzy mathematical programming problems with fuzzy coefficients. Ovchinnikov [20] characterized Zadeh's extension principle in terms of the duality principle. Introducing the concept of convex fuzzy variables for fuzzy constrained programming, Yang [21] proved a convexity theorem with convex fuzzy parameters and a duality theorem for fuzzy linear constrained programming. Later on, Farhadinia and Kamyad [22] extended the duality theorems for the crisp conic optimization problems to the fuzzy conic programming problems based on the convexity-like concept of fuzzy mappings and the parameterized representation of fuzzy numbers.
The paper is organized as follows. In Sect. 2, we construct a general fuzzy nonlinear programming problem and formulate its Mangasarian type dual. Further, we prove duality theorems using exponential membership functions under convexity assumptions. In the next section, we illustrate a numerical example.

Definitions and preliminaries
In the crisp sense, a general nonlinear primal-dual pair can be expressed as follows: Let the aspiration levels corresponding to the objective function of primal (MP) and dual (MD) be denoted by z 0 and w 0 , respectively. Now, the above crisp pair (MP) and (MD) can be described in the fuzzy sense as the following pair ( MP) and ( DD): Primal Problem where " " and " " are the representations of inequalities "≥" and "≤" in the fuzzy sense, respectively, and have interpretation of "essentially greater than" and "essentially less than" in the sense of Zimmermann.
The exponential membership functions associated with the objective function and the ith constraint, i = 1, 2, . . . , m, are as follows: , and α, α i , 0 < α, α i < ∞ are fuzzy parameters, also called shape parameters as they measure the degree of vagueness. The constants p 0 , p i (i = 1, 2, . . . , m) are the allowed change or violations corresponding to the objective function and the constraints of (MP), respectively. The set of feasible solutions of the fuzzy nonlinear programming problem ( MP) is denoted and defined as follows: It represents the decision space with respect to the fuzzy constraints of ( MP). Its membership function μ D : R → [0, 1] can be determined from the membership functions of individual fuzzy sets as follows: For everyμ 0 ,μ i lying between 0 and 1, there exist uniquep 0 , 0 <p 0 < p 0 andp i , 0 <p i < p i such that Let X be the universe, whose generic elements are the sets FR(p 0 ,p). Then we define a membership function μ FR : X → [0, 1] by μ FR (FR(p 0 ,p)) = min{μ 0 ,μ 1 , . . . ,μ m }. Therefore, the set D can also be written as Here, D is a fuzzy set whose elements are the set of points in R n which are generated with the unique aspiration level z 0 +p 0 andp i . Now, following Bellman-Zadeh's maximization principle and using the fuzzy membership functions defined above, the crisp equivalent of ( MP) can be formulated as follows: The above problem can be equivalently expressed as follows: Similarly, if the constant q 0 denotes admissible violations of the objective function of the problem (DD), then the crisp equivalent of ( DD) can be obtained as follows: where H = β((w 0 -L(w, u))/q 0 ). This can be further re-written as ∇f (w) + u T ∇g(w) = 0, where β is a shape parameter that measures the degree of vagueness of the objective function of ( DD).
Since x 0 is a feasible solution of (MP), therefore we get Hence μ i (g i (x 0 )) = 1, ∀i.
If f (x 0 ) fully satisfies the goal z 0 , then f (x 0 ) ≤ z 0 . Therefore Also, the membership value of L(x 0 , u 0 ), Combining this with (8) yields Then That is, Hence This yields This completes the proof.

Theorem 2.3 (Modified weak duality)
Let (x, ξ ) and (w, u, ψ) be feasible solutions for (PP-1) and (DP-1), respectively. Further, assume that the functions f and g are convex at w. Then Proof Multiplying the constraint (2) of (PP-1) by u i ≥ 0 and further adding all the 'm inequalities, we obtain By the convexity of f and g at w, we have and Employing u ≥ 0 in (11) and then adding with inequality (10), we get Finally, using (5) in the addition of (9) and (12), we have Hence the result.