Some Duality Results for Fuzzy Nonlinear Programming Problem

Available online at www.ispacs.com/jfsva Volume 2012, Year 2012 Article ID jfsva-00112, 9 pages doi:10.5899/2012/jfsva-00112 Research Article

Some Duality Results for Fuzzy Nonlinear Programming Problem G. Panda 1∗, S. Jaiswal

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(1) Department of Mathematics, Indian Institute of Technology, Kharagpur, India-721302. (2) Department of Mathematics, BITS, Goa Campus, India.

c G. Panda and S. Jaiswal. This is an open access article distributed under the CreCopyright 2012 ⃝ ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract The concept of duality plays an important role in optimization theory. This paper discusses some relations between primal and dual nonlinear programming problems in fuzzy environment. Here, fuzzy feasible region for a general fuzzy nonlinear programming is formed and the concept of fuzzy feasible solution is defined. First order dual relation for fuzzy nonlinear programming problem is studied. Keywords : Fuzzy mathematical programming; Membership function; Fuzzy inequality; First order dual.

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Introduction

In a general linear/nonlinear programming problem, if uncertainty appears in the form of fuzzy inequality or fuzzy numbers in the coefficients or fuzzy valued functions, either in the objective function or in the constraints or in both then that problem is treated as a fuzzy linear/nonlinear programming problem. Several duals for general crisp linear and nonlinear programming problems exist in literature. Duality for the fuzzy linear programming was first introduced by Rodder and Zimmerman [2]. Wu [3] studied weak and strong duality results for fuzzy linear programming problem where the coefficients are fuzzy numbers. Zhang, Yuan and Lee [5] used the concept of derivatives for convex fuzzy mapping and developed the necessary and sufficient optimality condition for the existence of solution for fuzzy mathematical programming using Lagrange dual function which are parallel to KKT condition in crisp case. Wu [4] defined fuzzy valued lagrange function and proved that the primal and dual fuzzy mathematical programming problems have ∗

Corresponding author. Email address: [email protected], Tel:+913222283680

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Journal of Fuzzy Set Valued Analysis

no duality gap under suitable convexity assumption. Panda and Jaiswal [1] established duality result between the fuzzy nonlinear programming and its Lagrange dual. However very few developments are made in the context of duality for fuzzy nonlinear programming problems to study the primal dual relationship. In this paper, Wolfe’s first order dual for a general fuzzy nonlinear programming problem is constructed, which is different from the approaches of Wu and Zhang, Yuan and Lee. They have considered fuzzy mappings and fuzzy coefficients in the optimization problems, but here we have considered fuzzy inequalities, by accepting some goals of the objective function and the constraints of the fuzzy nonlinear programming problem. This paper is divided into 4 sections. First section is introductory by nature. In Section 2, feasible region for a fuzzy nonlinear programming problem is discussed. In Section 3, we formulate the first order fuzzy dual and establish some modified duality results for fuzzy primal and dual pairs. Section 4 discusses some future research scope. Throughout the paper we consider the following concepts of fuzzy set theory. Let X be a universal set. Then a fuzzy set A˜ in X is the set of ordered pairs A˜ = {(x, µ Ã˜ (x))|x ∈ X}, where µ Ã˜ (x) is called the membership function or grade of member˜ ship of x in A which maps X to [0, 1]. In this paper we have used the fuzzy inequality due to Zimmermann [6]. The fuzzy inequality a ≼ b, a, b ∈ R, where ≼ is the fuzzified version of ≤, is a fuzzy set whose membership function µ ˜(a ≼ b) is defined by   1 a≤b    ∈ [0, 1] b ≤ a ≤ b + α µ ˜(a ≼ b) =     0 a≥b+α where α is the tolerance limit for the inequality a ≼ b at b. ˜ Let ⊙ be any binary operations ⊕ or ⊗ between two fuzzy sets A˜ and B,corresponding to ◦ as + and ×. Then from extension principle, µA˜ ⊙ B˜ (z) = sup min{µA˜ (x), µB˜ (y)}. x◦y=z

Also µA∩ ˜ (x)}. ˜ B ˜ (x) = min{µA ˜ (x), µB

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Fuzzy nonlinear programming and its feasible region

Consider a general nonlinear programming problem, (N P ) min

f (x)

subject to g(x) ≤ 0 where f : Rn → R, g = (g1 , g2 , ..., gm ), gi : Rn → R. If f and gi are differentiable functions then its first order Wolfe’s dual (ND) is, (N D)

max

L(x, u),

subject to

∇L(x, u) = 0, u ≥ 0,

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where L(x, u) = f (x) + uT g(x), u ∈ Rm , s ∈ Rn . In a fuzzy nonlinear programming problem, vagueness is present in the objective function or constraints or in both. Here we consider the presence of vagueness in the objective function as well as constraints of (NP ) in the form of fuzzy inequalities. If it is required to solve this problem where minimum value of f (x) is essentially less than (≼) some goal Z0 which is prefixed by the decision maker and the constraints are not exactly less than equal to zero rather than essentially less than equal to (≼) zero, in that case, (NP ) takes the following form in fuzzy environment and called as fuzzy nonlinear programming(FNP ). (F N P ) F ind

x ∈ Rn

such that f (x) ≼ Z0 gi (x) ≼ 0, where Z0 is the aspiration level of the primal objective function. The fuzzy inequalities, f (x) ≼ Z0 and gi (x) ≼ 0 hold with certain degree of satisfaction, which can be determined from their corresponding membership functions, µi : R → [0, 1], i = 0, 1, ..., m.

2.1

Feasible region for (FNP)

Feasible region for crisp nonlinear programming problem is a crisp set. But, due to the presence of flexible fuzzy constraints or fuzzy objective function or both in the optimization problem, feasible region, which is the set of feasible solutions of (FNP ) , is a fuzzy set, ˜ = {x ∈ Rn | f (x) ≼ Z0 , gi (x) ≼ 0, i = 1, 2, . . . , m}, D which is the decision space corresponding to the fuzzy constraints of (F N P ). Its membership function, µD˜ : R → [0, 1] can be determined from the membership functions of individual fuzzy sets as, µD˜ (x) = min{µ0 (f (x)), µ1 (f1 (x)), . . . , µm (fi (x))}. So the crisp equivalent of (FNP ) becomes, (CF N P )

max µD˜ (x), µD˜ (x) ≤ µ0 (f (x)), µD˜ (x) ≤ µ1 (fi (x)), f or each i.

x∈Rn

˜ of (FNP ) may also be interpreted as follows. The fuzzy feasible region, D, Let pi (i = 0, 1, . . . , m) are subjectively chosen nonzero positive constants of admissible violations associated with the fuzzy inequalities (FNP ). For some x ∈ Rn , if f (x) fully satisfies the goal Z0 , then µ0 (f (x)) = 1, if f (x) partially satisfies the goal within its flexibility limit, then 0 ≤ µ0 (f (x)) ≤ 1 and if f (x) goes beyond the flexibility limit of the goal, then µ0 (f (x)) = 0. Hence µ0 (f (x)) ∈ [0, 1], Z0 ≤ f (x) ≤ Z0 + p0 . Similar argument can be made for the fuzzy sets gi (x) ≼ 0 and we conclude, µi (gi (x)) ∈ [0, 1], 0 ≤ gi (x) ≤ pi for i = 1, 2, . . . , m. If f (x) ≼ Z0 is α% satisfied and each gi (x) ≼ 0 is βi % satisfied, then the region satisfying both fuzzy events has satisfaction level γ = min{α%, βi %, i = 1, 2, . . . , m}. This is a particular feasible region with degree of membership value γ. Corresponding to every γ ∈ [0, 1], there exists a feasible region.

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For every µ∗0 , µi ∗ ∈ [0, 1], there exists unique p∗0 ∈ (0, p0 ) and p∗i ∈ (0, pi ) such that µ0 (Z0 + p∗0 ) = µ∗0 and µi (p∗i ) = µ∗i . Corresponding to p∗ = (p1 ∗ , p2 ∗ , . . . , pm ∗ )T , denote F R(p∗0 , p∗ ) = {x ∈ Rn |f (x) ≤ Z0 + p∗0 , g(x) ≤ p∗ }. Suppose X be the universe, whose generic elements are the sets F R(p∗0 , p∗ ). We define a function µFg : X → [0, 1] by µFg (F R(p∗0 , p∗ )) = min{µ∗0 , µ∗1 , µ∗2 , . . . , µ∗m }. Then the R R ˜ of (FNP ) is equivalent to feasible region, D ˜ = D

∪

{(F R(p∗0 , p∗ ), µFg (F R(p∗0 , p∗ )))|F R(p∗0 , p∗ ) ∈ X}, R

(p∗0 ,p∗ )

˜ is a set of points in Rn (with same membership which is a fuzzy set in X. A member of D value) which is generated with unique aspiration level Z0 +p∗0 and p∗i that is f (x) ≤ Z0 +p∗0 and gi (x) ≤ p∗i . Conversely any feasible point x of (FNP ) corresponds to unique aspiration levels p∗0 and p∗i for which, f (x) ≤ Z0 +p∗0 and gi (x) ≤ p∗i at x for each i = 1, . . . , m. That is, x ∈ F R(p∗0 , p∗ ) for some p∗0 and p∗i . This concept may be explained through the following example. Example 2.1. Consider the following fuzzy nonlinear programming problem F ind

(x, y) ∈ R2

such that 2x + 3y ≼ 5 x2 + y 2 ≼ 1 with membership function, µ0 and µ1 defined by ,   1 if 2x + 3y ≤ 5    6 − 2x − 3y if 5 ≤ 2x + 3y ≤ 6 µ0 (2x + 3y) =     0 if 2x + 3y ≥ 6 and

  1 if x2 + y 2 ≤ 1    2 − (x2 + y 2 ) if 1 ≤ x2 + y 2 ≤ 2 µ1 (x2 + y 2 − 1) =     0 if x2 + y 2 ≥ 2.

Consider the admissible violations for 2x + 3y ≼ 5 and x2 + y 2 ≼ 1 as p0 = 1 and p1 = 1 respectively. In particular for p∗0 = 0.8, p∗1 = 0.25, we have µ0 (5.8) = 0.2 and µ1 (.25) = 0.75. Then F R(0.8, 0.25) = {(x, y) ∈ R2 , 2x + 3y ≤ 5.8, x2 + y 2 − 1 ≤ 0.25} with membership value µFg (F R(0.8, 0.25)) = min{0.2, 0.75} = 0.2. R


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The fuzzy feasible region is, ˜ = {(x, y) ∈ R2 , 2x + 3y ≼ 5, x2 + y 2 ≼ 1} D = {F R(p∗0 , p∗ )| 2x + 3y ≤ 5 + p∗0 , x2 + y 2 − 1 ≤ p∗1 } and µFg (F R(p∗0 , p∗ )) = min{µ0 , µ1 }. R We prove the following relation between (FNP ) and (CFNP ), which is used to prove the duality results in next section. Theorem 2.1. For every feasible solution of (FNP) there exists λ ∈ [0, 1] such that (x, λ) is a feasible solution of (CFNP). ˜ Then following the above discussion for fuzzy feasible region, we Proof. Let x ∈ D. conclude that there exist some p∗0 ∈ [0, p0 ], p∗i ∈ [0, pi ], i = 1, 2, . . . , m, such that x ∈ F R(p∗0 , p∗ ) whose membership value is µFg (F R(p∗0 , p∗ )) = min{µ∗0 , µ∗1 , µ∗2 , . . . , µ∗m }. R ∗ ∗ x ∈ F R(p0 , p ) implies f (x) ≤ Z0 + p∗0 , g(x) ≤ p∗ . For some λ0 , λi ∈ [0, 1], let p∗0 = (1 − λ0 )p0 and p∗i = (1 − λi )pi , i = 1, 2, . . . , m. Hence the above inequalities become (λ0 − 1)p0 ≤ Z0 − f (x), (λi − 1)pi ≤ −gi (x), i = 1, 2, . . . , m. For λ = min(λ0 , λi ) we have, (λ − 1)p0 ≤ Z0 − f (x), (λ − 1)pi ≤ −gi (x). Hence (x, λ) is a feasible solution of (CFNP ).

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First order dual of (FNP )

Since the functions f and gi have vagueness due to the presence of the fuzzy inequality “essentially less than”, so the Lagrange function L(x, u) = f (x) + uT g(x) have certain degree of vagueness. It is required to maximize L(x, u) while formulating its dual. In case, the decision maker agrees to accept the maximum of L(x, u) essentially greater than equal to some goal W0 , with subjectively chosen nonzero positive constant, q0 , of the admissible violation of the fuzzy constraint L(x, u) ≽ W0 , then the objective function can be treated as L(x, u) ≽ W0 , where “ ≽ ” means “ essentially greater than equal to”, which is a linguistic relation with degree of satisfaction η0 (L(x, u)), whose value is zero of if this fuzzy inequality is not satisfied, 1 if it is fully satisfied, and lies in [0,1] if it is partially satisfied. Hence   1 L(x, u) ≥ W0    ∈ [0, 1] W0 − q0 ≤ L(x, u) ≤ W0 η0 (L(x, u)) =     0 L(x, u) ≤ W0 − q0 .

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In the light of (ND), the corresponding Wolfe’s dual in fuzzy sense and its crisp equivalent takes the following form. (F N D) F ind (x, u) ∈ Rn × Rm (CF N D)

subject to L(x, u) ≽ W0 , ∇L(x, u) = 0, u ≥ 0.

max(x,u)∈Rn ×Rm η0 (L(x, u)) subject to ∇L(x, u) = 0, u ≥ 0.

From well known duality relation between (NP ) and (ND), it is true that if x0 is a feasible solution of (NP ) and u0 is a feasible solution of (ND), then f (x0 ) ≥ L(x0 , u0 ) and (ND) is a strong dual of (NP ) if it is a convex programming problem. Here we see the relation between (FNP ) and (FND) in terms of the goals Z0 and W0 . Theorem 3.1. Suppose x0 is a feasible solution of (NP) and u0 is a feasible solution of N D1 . If the corresponding objective value of (NP) fully satisfies the goal Z0 then the weak duality between (FNP) and (FND) holds. That is Z0 ≥ W0 . If the corresponding objective value of ( NP) partially satisfies the goal Z0 then the weak duality between (FNP) and (FND) partially holds. That is Z0 ≥ W0 − (p0 + q0 ). Proof. By weak duality result between (NP ) and (ND), f (x0 ) ≥ L(x0 , u0 ). Since x0 is the feasible solution of (NP ), so gi (x0 ) ≤ 0. Hence µi (gi (x0 )) = 1,

i = 1, 2, . . . , m.

If f (x0 ) fully satisfies the goal Z0 , then f (x0 ) ≤ Z0 . Hence µ0 (f (x0 )) = 1. η(L(x0 , u0 ))) = η(f (x0 ) + uT0 g(x0 )) = min{µ0 (f (x0 )), µ1 (g( x0 )), . . . , µm (gm (x0 ))} = 1. So L(x0 , u0 ) ≥ W0 . Combining all results we have Z0 ≥ f (x0 ) ≥ L(x0 , u0 ) ≥ W0 . If f (x0 ) partially satisfies the goal Z0 , then Z0 ≤ f (x0 ) ≤ Z0 +p0 . Hence µ0 (f (x0 )) ∈ [0, 1]. η(L(x0 , u0 ))) = η(f (x0 ) + u0 T g(x0 )) = min{µ0 (f (x0 )), µ1 (g( x0 )), . . . , µm (gm (x0 ))} ∈ [0, 1]. So W0 − q0 ≤ L(x0 , u0 ) ≤ W0 . Combining above results we have Z0 + p0 ≥ f (x0 ) ≥ L(x0 , u0 ) ≥ W0 − q0 . Hence Z0 ≥ W0 − (p0 + q0 ). Theorem 3.2. Suppose x0 is a feasible solution of (FNP) and u0 is a feasible solution of (FND). If f and g are differentiable convex functions at x0 and the degree of satisfaction of the fuzzy inequalities in both (FNP) and (F N D1 ) are linear functions then there exists a positive number K such that W0 ≤ Z0 + K. Proof. The membership functions µ0 (f (x)), µi (gi (x)) and η0 (L(x, u)) are 1 if the respective fuzzy inequalities f (x) ≼ Z0 , gi (x) ≼ 0 and L(x, u) ≽ W0 are fully satisfied, and are 0 if

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they violate their flexibility bound. In addition to this, the degree of satisfaction of the inequalities are linear functions. Hence the corresponding membership functions become,  1 f (x) ≤ Z0     0 µ0 (f (x)) = 1 − f (x)−Z Z0 ≤ f (x) ≤ Z0 + p0 p0     0 f (x) ≥ Z0 + p0  1     µi (gi (x)) = 1−     0

gi (x) ≤ 0 gi (x) pi

0 ≤ gi (x) ≤ pi gi (x) ≥ pi

  1    1 − (W0 − L(x, u))/q0 η0 (L(x, u)) =     0

L(x, u) ≥ W0 W0 − q0 ≤ L(x, u) ≤ W0 L(x, u) ≤ W0 − q0 .

Using the above membership functions, (CFNP ) and (CF N D) becomes, (CF N P ) max

λ

subject to (λ − 1)p0 + f (x) ≤ Z0 (λ − 1)pi + gi (x) ≤ 0, (CF N D) max

i = 1, 2, . . . , m,

0≤λ≤1

η

subject to (η − 1)q0 ≤ L(x, u) − W0 ∇L(x, u) = 0,

u ≥ 0,

0 ≤ η ≤ 1.

Since x0 is a feasible solution of (FNP ) , there exists 0 ≤ λ0 ≤ 1 such that (x0 , λ0 ) is the feasible solution of (CFNP ). Also since u0 is feasible solution of (FND), there exists 0 ≤ η0 ≤ 1 such that (x0 , u0 , η0 ) is the feasible solution of (CF N D). Hence (λ0 − 1)p0 ≤ Z0 − f (x0 )

(3.1)

(λ0 − 1)p ≤ −g(x0 )

(3.2)

(η0 − 1)q0 ≤ L(x0 , u0 ) − W0

(3.3)

∇f (x0 ) + u0 T ∇g(x0 ) = 0

(3.4)

Adding Inequalities (3.1), (3.2) and (3.3), (λ0 − 1)p0 + (λ0 − 1)u0 T p + (η0 − 1)q0 ≤ Z0 − (f (x0 ) + u0 T g(x0 )) + L(x0 , u0 ) − W0 = Z0 − W0 + f (x0 ) − f (x0 ) +u0 T (g(x0 ) − g(x0 )) (3.5)

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Since f and g are convex and differentiable at x0 , so f (x0 ) − f (x0 ) ≥ (∇f (x0 ))T (x0 − x0 ) and g(x0 ) − g(x0 ) ≥ (∇g(x0 ))T (x0 − x0 ). Hence right hand side of Inequality (3.5) becomes, Z0 − W0 − (∇f (x0 ) + u0 T ∇g(x0 ))T (x0 − x0 ). Using Eq. (3.4), Inequality (3.5) reduces to (λ0 − 1)p0 + (λ0 − 1)u0 T p + (η0 − 1)q0 ≤ Z0 − W0 . Let K = (1 − λ0 )p0 u0 T p + (1 − η0 )q0 . Then K ≥ 0. Hence Z0 ≥ W0 − K.

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Conclusion

In this paper, duality relation between the fuzzy primal nonlinear programming problem and the fuzzy version of Wolfe’s first order dual are studied. We may observe that if the fuzzy inequalities of (FNP ) are fully satisfied then weak duality between (FNP ) and (FND) holds(Z0 ≥ W0 ), which is parallel to the weak duality concept between (NP ) and (ND). But this is not true in case the fuzzy inequalities are partially satisfied. In that case the duality gap is expressed in terms of the tolerance levels, p0 , pi of the fuzzy inequalities. Also if the fuzzy nonlinear programming problem is defined in other fuzzy environments, in the presence of fuzzy valued functions or in the presence of fuzzy coefficients, then the modified weak duality results may not follow directly. Several type of duals like higher order dual, symmetric dual etc. for a general nonlinear programming problem exist. We have studied the weak duality result between fuzzy primal dual pair. In the light of the concept of this paper it is possible to construct other type duals in fuzzy environment develop strong duality result between these pairs.

References [1] G. Panda, S. Jaiswal, Lagrangian dual of fuzzy nonlinear programming problem and some duality results, Journal Fuzzy Mathematics, 18(2) (2010) 263-273. [2] W. Rodder, H.J. Zimmermann, Duality in fuzzy linear programming, in: AV, Fiacco, K.O.Kortanek(Eds), External Methods and System Analysis, Berlin, New York, 415429 (1980). [3] H.C. Wu, Duality theory in fuzzy linear programming problems with fuzzy coefficients, Fuzzy Optimzation and Decision Making, 2 (2003a) 61-73. http://dx.doi.org/10.1023/A:1022852314914. [4] H.C. Wu, Duality theorems in fuzzy mathematical programming problems based on the concept of necessity, Fuzzy Sets and Systems, 139 (2003b) 363-377. http://dx.doi.org/10.1016/S0165-0114(02)00575-4.


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[5] C. Zhang, X.H. Yuan and E.S. Lee, Duality theory in fuuzy mathematical programming problems with fuzzy coeffcients, Computer and Mathematics with Applications, 49 (2005) 1709-1730. http://dx.doi.org/10.1016/j.camwa.2004.12.012. [6] H.J. Zimmermann, Fuzzy Set Theory and its Applications, Kluwer academic publishers, 1991.