Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Duality in Fuzzy Quadratic Programming with Exponential Membership Functions

Duality in Fuzzy Quadratic Programming with Exponential Membership Functions Fuzzy Inf. Eng. (2010) 4: 337-346 DOI 10.1007/s12543-010-0054-5 ORIGINAL ARTICLE Duality in Fuzzy Quadratic Programming with Exponential Membership Functions S. K. Gupta· Debasis Dangar Received: 16 April 2010/ Revised: 4 November 2010/ Accepted: 21 November 2010/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2010 Abstract In this paper, we have presented fuzzy primal-dual quadratic program- ming problems and proved appropriate duality results taking exponential membership function. Keywords Fuzzy quadratic programming· Primal-dual problems· Duality results 1. Introduction Quadratic programming is one of the important optimization problems in operation research. The inventory management [1], portfolio selection [2,3], engineering design [4,5], molecular study [6] and economics [7] are some of its interesting applications. The classical quadratic programming problem is to find the minimum or maximum values of quadratic objective function subject to linear constraints. However, in many real world applications, the decision maker may not be in a position to specify the objective and/or constraint functions precisely but rather can specify them in a “fuzzy sense”. Therefore it is desirable to use fuzzy optimization type modelling so that a decision maker can have some flexibility. Fuzzy set theory has been extensively employed in linear and non-linear optimiza- tion problems [8-17]. Bellman and Zadeh [18] inspired the development of fuzzy op- timization by providing the aggregation operators, which combined the fuzzy goals and fuzzy decision space. Duality in fuzzy linear programming was first studied by Rodder and Zimmermann [19] considering the economic interpretation of the dual variables. Hamacher et al [20] also have given some results on duality in fuzzy linear programming and mainly devoted to sensitivity analysis. Bector and Chandra [21] introduced a linear pair of fuzzy primal-dual problem and obtained duality results under linear membership S. K. Gupta () · Debasis Dangar Department of Mathematics, Indian Institute of Technology Patna, Patna-800 013, India email: skgiitr@gmail.com debasis@iitp.ac.in 338 S. K. Gupta· Debasis Dangar (2010) function. The concept of duality for a fuzzy environment used in the present study is well supported by a significant amount of prior research, e.g. Hamacher et al [20], Rodder and Zimmermann [19], Liu et al [22], Bector and Chandra [21,23], Bector et al [24,25], Vijay et al [26]. Recently Gupta and Mehlawat [27] established the duality results for fuzzy linear programming problem using exponential membership functions. The duality results for a fuzzy quadratic programming problem has been studied in Bector and Chandra [23] using a linear membership function. However, the linear membership function is not properly represent problems in many practical situations [28]. Moreover, unlike this membership function, for non linear membership func- tions the marginal rate of increase (or decrease) of membership values as a function of model parameters is not constant. Therefore the nonlinear membership functions reflect reality better than the linear case. In this paper, we have formulated a pair of fuzzy primal-dual quadratic program- ming problems in which vague aspiration levels are represented by an exponential membership function. Further, appropriate duality results are established under fuzzy environment. 2. Problem Formulation and Duality Relations Consider the following crisp primal-dual quadratic programming problems: T T (QP) Max c x− x Hx s.t. Ax ≤ b, x ≥ 0 and T T (QD) Min b u+ w Hw s.t. A u+ Hw ≥ c, u ≥ 0, n m m n n n where c, x, w ∈ R , b, u∈ R , A is a matrix in R × R and H∈ R ×R is a positive semi definite matrix. All vectors are considered as column vectors. The dual (QD) of (QP) can be obtained using Mangasarian [29]. Remark 1 It may be noted that unlike Bector and Chandra [23], the constraint w ≥ 0 in (QD) will not be obtained. For the proof of this one can see Mangasarian [29]. Let z and w be the aspiration levels for the objective function of (QP) and (QD) 0 0 respectively. Now consider the fuzzy version of (QP) and (QD) as defined in Bector and Chandra [23]. Fuzzy Inf. Eng. (2010) 4: 337-346 339 (QP) Find x ∈ R , such that T T g(x) = c x− x Hx  z , Ax  b, x ≥ 0 and m n (QD) Find (u, w) ∈ R ×R , such that T T G(u, w) = b u+ w Hw  w , A u+ Hw  c, u ≥ 0, where “” and “” are fuzzy version of symbols “≥” and “≤”, respectively with in- terpretation of “essentially greater than” and “essentially less than”, in the sense of Zimmermann [30]. We take the following form of the exponential membership function for the objec- tive function and the system’s constraints: 1, if g(x) ≥ z , ⎪ 0 −α((g(x)−z )/−p ) −α ⎪ 0 0 ⎨ e − e μ (x) = 0 ⎪ , if z − p < g(x) < z , 0 0 0 ⎪ −α 1− e 0, if g(x) ≤ z − p 0 0 and 1, if A x ≤ b, ⎪ i i −α ((b −A x)/−p ) −α ⎪ i i i i i ⎨ e − e μ (x) = i ⎪ , if b < A x < b + p, i i i i ⎪ −α 1− e 0, if A x ≥ b + p, i i i whereα, α, 0<α, α < ∞ are fuzzy parameters measuring the degree of vagueness i i and are called shape parameters. Also p , p (i = 1, 2,..., m) are subjectively chosen 0 i constants of admissible violations associated with the objective function and the con- straints of (QP), respectively. Now by following Bellman-Zadeh’s maximization principle [18] and using the fuzzy membership functions defined above, the crisp equivalent of (QP) is as follows: 340 S. K. Gupta· Debasis Dangar (2010) (MP) Max λ −α((g(x)−z )/−p ) −α 0 0 e − e s.t. λ ≤ , −α 1− e −α ((b −A x)/−p ) −α i i i i i e − e λ ≤ (i = 1, 2,..., m), −α 1− e λ ∈ [0, 1], x ≥ 0 or (MP-1) Max λ −α −α s.t. p log(λ(1− e )+ e ) ≤ α(g(x)− z ), (1) 0 0 −α −α i i p log(λ(1− e )+ e ) ≤ α (b − A x)(i = 1, 2,..., m), (2) i i i i λ ∈ [0, 1], x ≥ 0. (3) Similarly, let q , q ( j = 1, 2,..., n) be subjectively chosen constants of admissible 0 j violations of objective function and system constraints of (QD). Then the crisp equiv- alent of (QD) can be obtained as follows (MD) Min −η −β((w −G(u,w))/−q ) −β 0 0 e − e s.t. η ≤ , −β 1− e T T −β ((A u+H w−c )/−q ) j j j −β j j j e − e η ≤ ( j = 1, 2,..., n), −β 1− e η ∈ [0, 1], u ≥ 0. This can be written as (MD-1) Min −η −β −β s.t. q log(η(1− e )+ e ) ≤ β(w − G(u, w)), (4) 0 0 −β −β T T j j q log(η(1−e )+e ) ≤ β (A u+H w−c )( j = 1, 2,..., n), (5) j j j j j η ∈ [0, 1], u ≥ 0, (6) where β, β ,0 <β, β < ∞ ( j = 1, 2,..., n) are shape parameters that measure the j j degree of vagueness of objective function and system’s constraints of (QD), respec- tively. The exponential membership function may change shape according to the param- eters α, α , β and β . By giving values for these parameters, the aspiration levels of i j the objective function and the system constraints may be described more accurately. Fuzzy Inf. Eng. (2010) 4: 337-346 341 Exponential membership functions can explore the different fuzzy utilities of the de- cision maker. Remark 2 Taking H = 0, the dual pair (QP) and (QD) reduce to (LP) and (LD) studied in Gupta and Mehlawat [27]. Theorem 1 (Modified weak duality) Let (x,λ) be feasible for (MP-1) and (u, w,η) be feasible for (MD-1). Then, m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x i i j j α β i j i=1 j=1 ≤ G(u, w)− g(x). (7) Proof Multiplying (2) by u ≥ 0 and summing all the ‘m’ inequalities, we obtain −α −α i i log(λ(1− e )+ e ) T T T p u ≤ b u− x A u. (8) i i i=1 Similarly, from (3) and (5), we get −β −β j j log(η(1− e )+ e ) T T T q x ≤ u Ax− c x+ w Hx. (9) j j j=1 Adding (8) and (9), we have m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x i i j j α β i j i=1 j=1 T T T ≤ b u− c x+ w Hx. T 1 T T n Since, w Hx ≤ (w Hw+ x Hx) for any w, x ∈ R , therefore we obtain m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x i i j j α β i j i=1 j=1 ≤ G(u, w)− g(x). Hence the result holds. T T Remark 3 (i) For λ = 1, η = 1 and H = 0, the inequality (7) reduces to c x ≤ b w, which is the weak duality result for a crisp linear programming problem. (ii) If λ = 1 and η = 1, then the inequality (7) becomes g(x) ≤ G(u, w), which is the weak duality theorem for crisp quadratic programming problem [29]. Theorem 2 Let (¯ x,λ) be feasible for (MP-1) and let (¯ u, w ¯,η¯) be feasible for (MD-1). Then 342 S. K. Gupta· Debasis Dangar (2010) m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η¯(1− e )+ e ) (i) p u ¯ + q x¯ i i j j α β i j i=1 j=1 = G(¯ u, w ¯ )− g(¯ x). −α −α −β −β log(λ(1− e )+ e ) log(η¯(1− e )+ e ) (ii) p + q 0 0 α β = {g(¯ x)− G(¯ u, w ¯ )}+{w − z }. 0 0 (iii) The aspiration levels z and w satisfy z − w ≤ 0, then (¯ x,λ) is optimal to 0 0 0 0 (MP-1) and (¯ u, w ¯,η¯) is optimal to (MD-1). Proof Let (x,λ) be feasible for (MP-1) and let (u, w,η) be feasible for (MD-1). Then we get m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x − G(u, w) i i j j α β i j i=1 j=1 +g(x) ≤ 0. The above inequality together with (1) yield m n −α −α −β −β i i  j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x − G(u, w)+ g(x) i i j j α β i j i=1 j=1 m n −α −α  −β −β ¯ i i j j log(λ(1− e )+ e ) log(η¯(1− e )+ e ) ≤ p u ¯ + q x¯ i i j j α β i j i=1 j=1 −G(¯ u, w ¯ )+ g(¯ x) = 0, (using hypothesis (i)) which implies (x¯,λ, u ¯, w ¯,η¯) is optimal to the following problem whose maximum value is zero: m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) (MP ) Max p u + q x i i j j α β i j i=1 j=1 −G(u, w)+ g(x) −α −α log(λ(1− e )+ e ) s.t. p ≤ g(x)− z , 0 0 −β −β log(η(1− e )+ e ) q ≤ w − G(u, w), 0 0 −α −α i i log(λ(1− e )+ e ) p ≤ b − A x (i = 1, 2,..., m), i i i −β −β j j log(η(1− e )+ e ) T T q ≤ A u+ H w− c ( j = 1, 2,..., n), j j j j j Fuzzy Inf. Eng. (2010) 4: 337-346 343 λ ∈ [0, 1],η ∈ [0, 1], x, u ≥ 0. Now adding the hypothesis (i) and (ii), we have m n −α −α  −β −β ¯ i i j j log(λ(1− e )+ e ) log(η¯(1− e )+ e ) p u ¯ + q x¯ + i i j j α β i j i=1 j=1 −α −α −β −β log(λ(1− e )+ e ) log(η¯(1− e )+ e ) p + q +{z − w } = 0. 0 0 0 0 α β But each term above is non positive (sinceλ, η¯ ≤ 1). Therefore each term is equal to zero and hence −α −α i i log(λ(1− e )+ e ) p u ¯ = 0, i i i=1 −β −β j j log(η¯(1− e )+ e ) q x¯ = 0, j j j=1 −α −α log(λ(1− e )+ e ) p = 0, −β −β log(η¯(1− e )+ e ) q = 0. Since −α −α −α −α log(λ(1− e )+ e ) log(λ(1− e )+ e ) p ≤ 0 = p , 0 0 α α therefore −α −α −α −α log(λ(1− e )+ e ) ≤ log(λ(1− e )+ e ), which implies λ ≤ λ. Similarly, η ≤ η¯ implies −η≥−η. ¯ This proves the result. 3. Numerical Example ∗ ∗ Consider the problem (QP ) and its dual (QD ) as follows. ∗ 2 (QP ) Max x− x s.t. 2x ≤ 1, x ≥ 0 and ∗ 2 (QD ) Min u+ w s.t. 2u+ 2w ≥ 1, u ≥ 0. 344 S. K. Gupta· Debasis Dangar (2010) Now taking p = 2, p = 2, z = 1,α = 2 and α = 1 for (QP ), the correspond- 0 1 0 1 ing problem (MP-1) becomes (MP ) Max λ s.t. log(0.864665λ+ 0.1353353) ≤ (x− x − 1), 2log(0.632121λ+ 0.367879) ≤ (1− 2x), λ ∈ [0, 1], x ≥ 0. Similarly, considering q = 1, q = 3, w = 1,β = 2 and β = 1 for (QD ), the 0 1 0 1 corresponding problem (MD-1) becomes (MD ) Min −η s.t. log(0.864665η+ 0.1353353) ≤ 2(1− u− w ), 3log(0.632121η+ 0.367879) ≤ (2u+ 2w− 1), η ∈ [0, 1], u ≥ 0. Using MAPLE 12, we solve these nonlinear optimization problems and the optimal ∗ ∗ ∗ ∗ ∗ ∗ solution of (MP ) and (MD ) obtained are x = 0.5,λ = 0.3898 and η = 1, u = 0.6980, w = 0.5495, respectively. 4. Conclusion In this paper, we studied quadratic programming duality under fuzzy environment taking an exponential membership function. One of the advantages of using the func- tion is the flexibility in changing the shape parameters, by which we can explore the different utilities of a decision maker. Also the function can be applied to an op- timal design problem. Gupta and Mehlawat [27] have proved duality results for a linear programming problem using the function. We have extended these results for a quadratic programming problem and illustrate it by an example. Acknowledgments The authors wish to thank the referees for several valuable suggetions which have considerably improved the presentation of this paper. The second author is also thank- ful to Ministry of Human Resource development, New Delhi (India) for financial support. References 1. Abdel-Malek L L, Areeractch N (2007) A quadratic programming approach to the multi-product newsvendor problem with side constraints. European Journal of Operational Research 176: 855-861 Fuzzy Inf. Eng. (2010) 4: 337-346 345 2. Ammar E, Khalifa H A (2003) Fuzzy portfolio optimization a quadratic programming approach. Chaos Solitons and Fractals 18: 10-54 3. Zhang W G, Nie Z K (2005) On admissible efficient portfolio selection policy. Applied Mathematics and Computation 169: 608-623 4. Dwyer T, Koren Y, Marriott K (2006) Drawing directed graphs using quadratic programming. IEEE Transaction Visualization and Computer Graphics 12: 536-548 5. PetersenJAM, Bodson M (2006) Constrained quadratic programming techniques for control allo- cation. IEEE Transaction on Control System Technology 14: 91-98 6. Pavlovi L, Divni T (2007) A quadratic programming approach to the randi index. European Journal of Operational Research 176: 435-444 7. Schwarz H G (2006) Economic material-product chain models: current status, further development and an illustrative example. Ecological Economics 58: 373-392 8. Ammar E (2000) Interactive stability of multiobjective NLP problems with fuzzy parameters in the objective functions and constraints. Fuzzy Sets and Systems 109: 83-90 9. Chen S P (2004) Parametric nonlinear programming for analyzing fuzzy queues with finite capacity. European Journal of Operational Research 157: 429-438 10. Liu S T (2004) Fuzzy geometric programming approach to a fuzzy machining economics model. International Journal Production Research 42: 3253-3269 11. Nieto J J, Rodaiguez-Lopez R (2006) Bounded solutions for fuzzy differential and integral equations. Chaos, Solitons and Fractals 27: 1376-1386 12. Roman-Florse H, Chalco-Cano Y (2006) Some chaotic properties of Zaheh’s extensions. Chaos, Solitons and Fractals 35: 452-459 13. Sakawa M (1993) Fuzzy sets and interactive multiobjective optimization. New York: Plenum Press 14. Soleimani-damaneh M (2006) Fuzzy upper bounds and their applications. Chaos, Solitons and Frac- tals 36: 217-225 15. Stefanini L, Sorini L, Guerra M L (2006) Solution of fuzzy dynamical systems using the LU-representation of fuzzy numbers. Chaos, Solitons and Fractals 29: 638-652 16. Bellman R E, Zadeh L A (1970) Decision Making in a Fuzzy Environment. Management Science 17: 141-164 17. Slowinski R (ed.) (1998) Fuzzy sets in decision analysis, Operations Research and Statistics. Boston: Kluwer Academic Publishers 18. Delgado M, Kacprzyk J, Verdegay J-L, Vila M A (ed.) (1994) Fuzzy optimization: Recent advances. New York, Physica-Verlag 19. Rodder W, Zimmermann H-J (1980) Duality in fuzzy linear programming. In: A.V. Fiacco, K.O. Kortanek (Eds.), Extremal Methods and System Analysis, Berlin, New York, , pp. 415-429 20. Hamacher H, Leberling H, Zimmermann H J (1978) Sensitivity analysis in fuzzy linear programming. Fuzzy Sets and Systems 1: 269-281 21. Bector C R, Chandra S (2002) On duality in linear programming under fuzzy environment. Fuzzy Sets and Systems 125(3): 317-325 22. Liu Y J, Shi Y, Liu Y H (1995) Duality of fuzzy MC2 linear programming: a constructive approach, Journal of Mathematical Analysis and Applications 194: 389-413 23. Bector C R, Chandra S (2005) Fuzzy mathematical programming and fuzzy matrix games. Springer, Berlin, Heidelberg 24. Bector C R, Chandra S, Vijay V (2004) Matrix games with fuzzy goals and fuzzy linear programming duality. Fuzzy Optimization and Decision Making 3(3): 255-269 25. Bector C R, Chandra S, Vijay V (2004) Duality in linear programming with fuzzy parameters and matrix games with fuzzy pay-offs. Fuzzy Sets and Systems 146(2): 253-269 26. Vijay V, Chandra S, Bector C R (2005) Matrix games with fuzzy goals and fuzzy pay offs. Omega 33(5): 425-429 27. Gupta Pankaj, Mehlawat Mukesh Kumar (2009) Bector-Chandra type duality in fuzzy linear pro- gramming with exponential membership functions. Fuzzy Sets and Systems 160: 3290-3308 28. Hersh H M, Caramazza A (1976) A fuzzy set approach to modifiers and vagueness in natural lan- guage. Journal of Experimental Psychology 105(3): 254-276 346 S. K. Gupta· Debasis Dangar (2010) 29. Mangasarian O L (1994) Nonlinear Programming, SIAM, Philadelphia, PA 30. Zimmermann H J (1978) Fuzzy programming and linear programming with several objective func- tions. Fuzzy Sets and Systems 1: 45-55 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Duality in Fuzzy Quadratic Programming with Exponential Membership Functions

Gupta, S. K.; Dangar, Debasis
Fuzzy Information and Engineering , Volume 2 (4): 10 – Dec 1, 2010
Download PDF
10 pages

Loading next page...
 
/lp/taylor-francis/duality-in-fuzzy-quadratic-programming-with-exponential-membership-0Ppqlcc79Y

References (33)

Publisher
Taylor & Francis
Copyright
© 2010 Taylor and Francis Group, LLC
ISSN
1616-8666
eISSN
1616-8658
DOI
10.1007/s12543-010-0054-5
Publisher site
See Article on Publisher Site

Abstract

Fuzzy Inf. Eng. (2010) 4: 337-346 DOI 10.1007/s12543-010-0054-5 ORIGINAL ARTICLE Duality in Fuzzy Quadratic Programming with Exponential Membership Functions S. K. Gupta· Debasis Dangar Received: 16 April 2010/ Revised: 4 November 2010/ Accepted: 21 November 2010/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2010 Abstract In this paper, we have presented fuzzy primal-dual quadratic program- ming problems and proved appropriate duality results taking exponential membership function. Keywords Fuzzy quadratic programming· Primal-dual problems· Duality results 1. Introduction Quadratic programming is one of the important optimization problems in operation research. The inventory management [1], portfolio selection [2,3], engineering design [4,5], molecular study [6] and economics [7] are some of its interesting applications. The classical quadratic programming problem is to find the minimum or maximum values of quadratic objective function subject to linear constraints. However, in many real world applications, the decision maker may not be in a position to specify the objective and/or constraint functions precisely but rather can specify them in a “fuzzy sense”. Therefore it is desirable to use fuzzy optimization type modelling so that a decision maker can have some flexibility. Fuzzy set theory has been extensively employed in linear and non-linear optimiza- tion problems [8-17]. Bellman and Zadeh [18] inspired the development of fuzzy op- timization by providing the aggregation operators, which combined the fuzzy goals and fuzzy decision space. Duality in fuzzy linear programming was first studied by Rodder and Zimmermann [19] considering the economic interpretation of the dual variables. Hamacher et al [20] also have given some results on duality in fuzzy linear programming and mainly devoted to sensitivity analysis. Bector and Chandra [21] introduced a linear pair of fuzzy primal-dual problem and obtained duality results under linear membership S. K. Gupta () · Debasis Dangar Department of Mathematics, Indian Institute of Technology Patna, Patna-800 013, India email: skgiitr@gmail.com debasis@iitp.ac.in 338 S. K. Gupta· Debasis Dangar (2010) function. The concept of duality for a fuzzy environment used in the present study is well supported by a significant amount of prior research, e.g. Hamacher et al [20], Rodder and Zimmermann [19], Liu et al [22], Bector and Chandra [21,23], Bector et al [24,25], Vijay et al [26]. Recently Gupta and Mehlawat [27] established the duality results for fuzzy linear programming problem using exponential membership functions. The duality results for a fuzzy quadratic programming problem has been studied in Bector and Chandra [23] using a linear membership function. However, the linear membership function is not properly represent problems in many practical situations [28]. Moreover, unlike this membership function, for non linear membership func- tions the marginal rate of increase (or decrease) of membership values as a function of model parameters is not constant. Therefore the nonlinear membership functions reflect reality better than the linear case. In this paper, we have formulated a pair of fuzzy primal-dual quadratic program- ming problems in which vague aspiration levels are represented by an exponential membership function. Further, appropriate duality results are established under fuzzy environment. 2. Problem Formulation and Duality Relations Consider the following crisp primal-dual quadratic programming problems: T T (QP) Max c x− x Hx s.t. Ax ≤ b, x ≥ 0 and T T (QD) Min b u+ w Hw s.t. A u+ Hw ≥ c, u ≥ 0, n m m n n n where c, x, w ∈ R , b, u∈ R , A is a matrix in R × R and H∈ R ×R is a positive semi definite matrix. All vectors are considered as column vectors. The dual (QD) of (QP) can be obtained using Mangasarian [29]. Remark 1 It may be noted that unlike Bector and Chandra [23], the constraint w ≥ 0 in (QD) will not be obtained. For the proof of this one can see Mangasarian [29]. Let z and w be the aspiration levels for the objective function of (QP) and (QD) 0 0 respectively. Now consider the fuzzy version of (QP) and (QD) as defined in Bector and Chandra [23]. Fuzzy Inf. Eng. (2010) 4: 337-346 339 (QP) Find x ∈ R , such that T T g(x) = c x− x Hx  z , Ax  b, x ≥ 0 and m n (QD) Find (u, w) ∈ R ×R , such that T T G(u, w) = b u+ w Hw  w , A u+ Hw  c, u ≥ 0, where “” and “” are fuzzy version of symbols “≥” and “≤”, respectively with in- terpretation of “essentially greater than” and “essentially less than”, in the sense of Zimmermann [30]. We take the following form of the exponential membership function for the objec- tive function and the system’s constraints: 1, if g(x) ≥ z , ⎪ 0 −α((g(x)−z )/−p ) −α ⎪ 0 0 ⎨ e − e μ (x) = 0 ⎪ , if z − p < g(x) < z , 0 0 0 ⎪ −α 1− e 0, if g(x) ≤ z − p 0 0 and 1, if A x ≤ b, ⎪ i i −α ((b −A x)/−p ) −α ⎪ i i i i i ⎨ e − e μ (x) = i ⎪ , if b < A x < b + p, i i i i ⎪ −α 1− e 0, if A x ≥ b + p, i i i whereα, α, 0<α, α < ∞ are fuzzy parameters measuring the degree of vagueness i i and are called shape parameters. Also p , p (i = 1, 2,..., m) are subjectively chosen 0 i constants of admissible violations associated with the objective function and the con- straints of (QP), respectively. Now by following Bellman-Zadeh’s maximization principle [18] and using the fuzzy membership functions defined above, the crisp equivalent of (QP) is as follows: 340 S. K. Gupta· Debasis Dangar (2010) (MP) Max λ −α((g(x)−z )/−p ) −α 0 0 e − e s.t. λ ≤ , −α 1− e −α ((b −A x)/−p ) −α i i i i i e − e λ ≤ (i = 1, 2,..., m), −α 1− e λ ∈ [0, 1], x ≥ 0 or (MP-1) Max λ −α −α s.t. p log(λ(1− e )+ e ) ≤ α(g(x)− z ), (1) 0 0 −α −α i i p log(λ(1− e )+ e ) ≤ α (b − A x)(i = 1, 2,..., m), (2) i i i i λ ∈ [0, 1], x ≥ 0. (3) Similarly, let q , q ( j = 1, 2,..., n) be subjectively chosen constants of admissible 0 j violations of objective function and system constraints of (QD). Then the crisp equiv- alent of (QD) can be obtained as follows (MD) Min −η −β((w −G(u,w))/−q ) −β 0 0 e − e s.t. η ≤ , −β 1− e T T −β ((A u+H w−c )/−q ) j j j −β j j j e − e η ≤ ( j = 1, 2,..., n), −β 1− e η ∈ [0, 1], u ≥ 0. This can be written as (MD-1) Min −η −β −β s.t. q log(η(1− e )+ e ) ≤ β(w − G(u, w)), (4) 0 0 −β −β T T j j q log(η(1−e )+e ) ≤ β (A u+H w−c )( j = 1, 2,..., n), (5) j j j j j η ∈ [0, 1], u ≥ 0, (6) where β, β ,0 <β, β < ∞ ( j = 1, 2,..., n) are shape parameters that measure the j j degree of vagueness of objective function and system’s constraints of (QD), respec- tively. The exponential membership function may change shape according to the param- eters α, α , β and β . By giving values for these parameters, the aspiration levels of i j the objective function and the system constraints may be described more accurately. Fuzzy Inf. Eng. (2010) 4: 337-346 341 Exponential membership functions can explore the different fuzzy utilities of the de- cision maker. Remark 2 Taking H = 0, the dual pair (QP) and (QD) reduce to (LP) and (LD) studied in Gupta and Mehlawat [27]. Theorem 1 (Modified weak duality) Let (x,λ) be feasible for (MP-1) and (u, w,η) be feasible for (MD-1). Then, m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x i i j j α β i j i=1 j=1 ≤ G(u, w)− g(x). (7) Proof Multiplying (2) by u ≥ 0 and summing all the ‘m’ inequalities, we obtain −α −α i i log(λ(1− e )+ e ) T T T p u ≤ b u− x A u. (8) i i i=1 Similarly, from (3) and (5), we get −β −β j j log(η(1− e )+ e ) T T T q x ≤ u Ax− c x+ w Hx. (9) j j j=1 Adding (8) and (9), we have m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x i i j j α β i j i=1 j=1 T T T ≤ b u− c x+ w Hx. T 1 T T n Since, w Hx ≤ (w Hw+ x Hx) for any w, x ∈ R , therefore we obtain m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x i i j j α β i j i=1 j=1 ≤ G(u, w)− g(x). Hence the result holds. T T Remark 3 (i) For λ = 1, η = 1 and H = 0, the inequality (7) reduces to c x ≤ b w, which is the weak duality result for a crisp linear programming problem. (ii) If λ = 1 and η = 1, then the inequality (7) becomes g(x) ≤ G(u, w), which is the weak duality theorem for crisp quadratic programming problem [29]. Theorem 2 Let (¯ x,λ) be feasible for (MP-1) and let (¯ u, w ¯,η¯) be feasible for (MD-1). Then 342 S. K. Gupta· Debasis Dangar (2010) m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η¯(1− e )+ e ) (i) p u ¯ + q x¯ i i j j α β i j i=1 j=1 = G(¯ u, w ¯ )− g(¯ x). −α −α −β −β log(λ(1− e )+ e ) log(η¯(1− e )+ e ) (ii) p + q 0 0 α β = {g(¯ x)− G(¯ u, w ¯ )}+{w − z }. 0 0 (iii) The aspiration levels z and w satisfy z − w ≤ 0, then (¯ x,λ) is optimal to 0 0 0 0 (MP-1) and (¯ u, w ¯,η¯) is optimal to (MD-1). Proof Let (x,λ) be feasible for (MP-1) and let (u, w,η) be feasible for (MD-1). Then we get m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x − G(u, w) i i j j α β i j i=1 j=1 +g(x) ≤ 0. The above inequality together with (1) yield m n −α −α −β −β i i  j j log(λ(1− e )+ e ) log(η(1− e )+ e ) p u + q x − G(u, w)+ g(x) i i j j α β i j i=1 j=1 m n −α −α  −β −β ¯ i i j j log(λ(1− e )+ e ) log(η¯(1− e )+ e ) ≤ p u ¯ + q x¯ i i j j α β i j i=1 j=1 −G(¯ u, w ¯ )+ g(¯ x) = 0, (using hypothesis (i)) which implies (x¯,λ, u ¯, w ¯,η¯) is optimal to the following problem whose maximum value is zero: m n −α −α  −β −β i i j j log(λ(1− e )+ e ) log(η(1− e )+ e ) (MP ) Max p u + q x i i j j α β i j i=1 j=1 −G(u, w)+ g(x) −α −α log(λ(1− e )+ e ) s.t. p ≤ g(x)− z , 0 0 −β −β log(η(1− e )+ e ) q ≤ w − G(u, w), 0 0 −α −α i i log(λ(1− e )+ e ) p ≤ b − A x (i = 1, 2,..., m), i i i −β −β j j log(η(1− e )+ e ) T T q ≤ A u+ H w− c ( j = 1, 2,..., n), j j j j j Fuzzy Inf. Eng. (2010) 4: 337-346 343 λ ∈ [0, 1],η ∈ [0, 1], x, u ≥ 0. Now adding the hypothesis (i) and (ii), we have m n −α −α  −β −β ¯ i i j j log(λ(1− e )+ e ) log(η¯(1− e )+ e ) p u ¯ + q x¯ + i i j j α β i j i=1 j=1 −α −α −β −β log(λ(1− e )+ e ) log(η¯(1− e )+ e ) p + q +{z − w } = 0. 0 0 0 0 α β But each term above is non positive (sinceλ, η¯ ≤ 1). Therefore each term is equal to zero and hence −α −α i i log(λ(1− e )+ e ) p u ¯ = 0, i i i=1 −β −β j j log(η¯(1− e )+ e ) q x¯ = 0, j j j=1 −α −α log(λ(1− e )+ e ) p = 0, −β −β log(η¯(1− e )+ e ) q = 0. Since −α −α −α −α log(λ(1− e )+ e ) log(λ(1− e )+ e ) p ≤ 0 = p , 0 0 α α therefore −α −α −α −α log(λ(1− e )+ e ) ≤ log(λ(1− e )+ e ), which implies λ ≤ λ. Similarly, η ≤ η¯ implies −η≥−η. ¯ This proves the result. 3. Numerical Example ∗ ∗ Consider the problem (QP ) and its dual (QD ) as follows. ∗ 2 (QP ) Max x− x s.t. 2x ≤ 1, x ≥ 0 and ∗ 2 (QD ) Min u+ w s.t. 2u+ 2w ≥ 1, u ≥ 0. 344 S. K. Gupta· Debasis Dangar (2010) Now taking p = 2, p = 2, z = 1,α = 2 and α = 1 for (QP ), the correspond- 0 1 0 1 ing problem (MP-1) becomes (MP ) Max λ s.t. log(0.864665λ+ 0.1353353) ≤ (x− x − 1), 2log(0.632121λ+ 0.367879) ≤ (1− 2x), λ ∈ [0, 1], x ≥ 0. Similarly, considering q = 1, q = 3, w = 1,β = 2 and β = 1 for (QD ), the 0 1 0 1 corresponding problem (MD-1) becomes (MD ) Min −η s.t. log(0.864665η+ 0.1353353) ≤ 2(1− u− w ), 3log(0.632121η+ 0.367879) ≤ (2u+ 2w− 1), η ∈ [0, 1], u ≥ 0. Using MAPLE 12, we solve these nonlinear optimization problems and the optimal ∗ ∗ ∗ ∗ ∗ ∗ solution of (MP ) and (MD ) obtained are x = 0.5,λ = 0.3898 and η = 1, u = 0.6980, w = 0.5495, respectively. 4. Conclusion In this paper, we studied quadratic programming duality under fuzzy environment taking an exponential membership function. One of the advantages of using the func- tion is the flexibility in changing the shape parameters, by which we can explore the different utilities of a decision maker. Also the function can be applied to an op- timal design problem. Gupta and Mehlawat [27] have proved duality results for a linear programming problem using the function. We have extended these results for a quadratic programming problem and illustrate it by an example. Acknowledgments The authors wish to thank the referees for several valuable suggetions which have considerably improved the presentation of this paper. The second author is also thank- ful to Ministry of Human Resource development, New Delhi (India) for financial support. References 1. Abdel-Malek L L, Areeractch N (2007) A quadratic programming approach to the multi-product newsvendor problem with side constraints. European Journal of Operational Research 176: 855-861 Fuzzy Inf. Eng. (2010) 4: 337-346 345 2. Ammar E, Khalifa H A (2003) Fuzzy portfolio optimization a quadratic programming approach. Chaos Solitons and Fractals 18: 10-54 3. Zhang W G, Nie Z K (2005) On admissible efficient portfolio selection policy. Applied Mathematics and Computation 169: 608-623 4. Dwyer T, Koren Y, Marriott K (2006) Drawing directed graphs using quadratic programming. IEEE Transaction Visualization and Computer Graphics 12: 536-548 5. PetersenJAM, Bodson M (2006) Constrained quadratic programming techniques for control allo- cation. IEEE Transaction on Control System Technology 14: 91-98 6. Pavlovi L, Divni T (2007) A quadratic programming approach to the randi index. European Journal of Operational Research 176: 435-444 7. Schwarz H G (2006) Economic material-product chain models: current status, further development and an illustrative example. Ecological Economics 58: 373-392 8. Ammar E (2000) Interactive stability of multiobjective NLP problems with fuzzy parameters in the objective functions and constraints. Fuzzy Sets and Systems 109: 83-90 9. Chen S P (2004) Parametric nonlinear programming for analyzing fuzzy queues with finite capacity. European Journal of Operational Research 157: 429-438 10. Liu S T (2004) Fuzzy geometric programming approach to a fuzzy machining economics model. International Journal Production Research 42: 3253-3269 11. Nieto J J, Rodaiguez-Lopez R (2006) Bounded solutions for fuzzy differential and integral equations. Chaos, Solitons and Fractals 27: 1376-1386 12. Roman-Florse H, Chalco-Cano Y (2006) Some chaotic properties of Zaheh’s extensions. Chaos, Solitons and Fractals 35: 452-459 13. Sakawa M (1993) Fuzzy sets and interactive multiobjective optimization. New York: Plenum Press 14. Soleimani-damaneh M (2006) Fuzzy upper bounds and their applications. Chaos, Solitons and Frac- tals 36: 217-225 15. Stefanini L, Sorini L, Guerra M L (2006) Solution of fuzzy dynamical systems using the LU-representation of fuzzy numbers. Chaos, Solitons and Fractals 29: 638-652 16. Bellman R E, Zadeh L A (1970) Decision Making in a Fuzzy Environment. Management Science 17: 141-164 17. Slowinski R (ed.) (1998) Fuzzy sets in decision analysis, Operations Research and Statistics. Boston: Kluwer Academic Publishers 18. Delgado M, Kacprzyk J, Verdegay J-L, Vila M A (ed.) (1994) Fuzzy optimization: Recent advances. New York, Physica-Verlag 19. Rodder W, Zimmermann H-J (1980) Duality in fuzzy linear programming. In: A.V. Fiacco, K.O. Kortanek (Eds.), Extremal Methods and System Analysis, Berlin, New York, , pp. 415-429 20. Hamacher H, Leberling H, Zimmermann H J (1978) Sensitivity analysis in fuzzy linear programming. Fuzzy Sets and Systems 1: 269-281 21. Bector C R, Chandra S (2002) On duality in linear programming under fuzzy environment. Fuzzy Sets and Systems 125(3): 317-325 22. Liu Y J, Shi Y, Liu Y H (1995) Duality of fuzzy MC2 linear programming: a constructive approach, Journal of Mathematical Analysis and Applications 194: 389-413 23. Bector C R, Chandra S (2005) Fuzzy mathematical programming and fuzzy matrix games. Springer, Berlin, Heidelberg 24. Bector C R, Chandra S, Vijay V (2004) Matrix games with fuzzy goals and fuzzy linear programming duality. Fuzzy Optimization and Decision Making 3(3): 255-269 25. Bector C R, Chandra S, Vijay V (2004) Duality in linear programming with fuzzy parameters and matrix games with fuzzy pay-offs. Fuzzy Sets and Systems 146(2): 253-269 26. Vijay V, Chandra S, Bector C R (2005) Matrix games with fuzzy goals and fuzzy pay offs. Omega 33(5): 425-429 27. Gupta Pankaj, Mehlawat Mukesh Kumar (2009) Bector-Chandra type duality in fuzzy linear pro- gramming with exponential membership functions. Fuzzy Sets and Systems 160: 3290-3308 28. Hersh H M, Caramazza A (1976) A fuzzy set approach to modifiers and vagueness in natural lan- guage. Journal of Experimental Psychology 105(3): 254-276 346 S. K. Gupta· Debasis Dangar (2010) 29. Mangasarian O L (1994) Nonlinear Programming, SIAM, Philadelphia, PA 30. Zimmermann H J (1978) Fuzzy programming and linear programming with several objective func- tions. Fuzzy Sets and Systems 1: 45-55

Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Dec 1, 2010

Keywords: Fuzzy quadratic programming; Primal-dual problems; Duality results

There are no references for this article.