Abstract
FUZZY INFORMATION AND ENGINEERING 2019, VOL. 11, NO. 2, 175–185 https://doi.org/10.1080/16168658.2021.1886818 Duality in Bipolar Fuzzy Number Linear Programming Problem a b c Reza Ghanbari , Khatere Ghorbani-Moghadam and Nezam Mahdavi-Amiri Faculty of Mathematical Sciences, Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran; Member of Optimization Laboratory in Faculty of Mathematical Sciences, Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran; Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran ABSTRACT ARTICLE HISTORY Received 7 September 2017 We develop a linear ranking function for ordering bipolar fuzzy num- Revised 16 August 2017 bers and study its properties. Using this ranking function, we solve Accepted 22 August 2018 a bipolar fuzzy linear programing problem. Then, we present the dual of the problem and establish several duality results. Also, we KEYWORDS presented an application of bipolar fuzzy number in real life problem. Fuzzy linear programing problem; bipolar fuzzy number; duality; ranking function 1. Introduction In a human decision making, there is a bipolar judgmental thinking on a negative side and a positive side; for instance, see [1]. In bipolar information, two types of information (as positive and negative) must be distinguished [2,3]. Positive information is given by obser- vation or experimentation. But, negative information represents impossibility. This domain has recently invoked several interesting research areas such as psychology [4], image pro- cessing [5], human reasoning [6] and graph theory [7]. Zhang [8] initiated the concept of bipolar fuzzy set as a generalization of fuzzy set. He defined bipolar fuzzy set as an extension of fuzzy set whose range of membership degree is [−1, 1]. Akram [7,9] used the concept of bipolar fuzzy set in graph theory. Broumand [10] introduced the concept of bipolar-valued fuzzy sub-algebras of BCK/BCI-algebras and investigated some of their useful properties. Zhou and Li [1] presented the concepts of bipolar fuzzy h-ideals and normal bipolar fuzzy h-ideals. Then, they investigated characterizations of bipolar fuzzy h -ideals by means of positive t-cut, negative s-cut, homomorphism and equivalence relation. Some other works on bipolar fuzzy sets can be found in [11–15]. There are several methods for comparison of unipolar fuzzy numbers based on ranking functions [16,17] and most convenient methods for solving linear programing problems are based on the concept of ranking functions [16,18,19,20,21,22]. Inspired by ranking functions of unipolar fuzzy numbers, we develop a ranking function for bipolar fuzzy num- bers. Then, we solve a bipolar fuzzy linear programing problem by using a certain ranking function and we give duality results for bipolar fuzzy linear programing problems. CONTACT Reza Ghanbari rghanbari@um.ac.ir © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 176 R. GHANBARI ET AL. In Section 2, we review the fundamental notions of bipolar fuzzy sets. Then, we propose a linear ranking function to order bipolar fuzzy numbers. In Section 3, we investigate and characterize several properties for the bipolar fuzzy number linear programing problem by using a linear ranking function. In Section 4, we introduce the dual of a bipolar fuzzy number linear programing problem. In Section 5, we present an application of bipolar fuzzy number in a real life problem. We give our concluding remarks in Section 6. 2. Preliminaries 2.1. Definitions and Notations Here, some necessary definitions and relevant results of bipolar fuzzy sets are given. Definition 2.1: [8]Let X be a nonempty set. A bipolar fuzzy set B in X is an object with the following form P N B ={(x, µ (x), µ (x))|x ∈ X}, ˜ ˜ B B P N where, µ (x) : X → [0, 1] and µ (x) : X → [−1, 0]. ˜ ˜ B B P N Definition 2.2: [11]Given B = ((µ (x), µ (x)) a bipolar-valued fuzzy set and (s, t) ∈ ˜ ˜ B B P P N N [−1, 0] × [0, 1], the sets B = { x ∈ X|µ (x) ≥ t} and B = { x ∈ X|µ (x) ≤ s} are respec- t s ˜ ˜ B B ˜ ˜ tively called the positive t-cut of B and the negative s-cut of B, and for every k ∈ [0, 1], the set: P N B = B ∩ B k k is called the k-cut of B. We next define a bipolar triangular fuzzy number. L P N R A bipolar triangular fuzzy number is defined as a quadruple A = (a , a , a , a ) with P N positive and negative membership functions µ (x) and µ (x) as follows (see Figure 1): ˜ ˜ A A ⎪ x − a ⎪ L P , a ≤ x < a P L a − a P R µ (x) = x − a (1) ˜ P R ⎪ , a ≤ x < a P R a − a 0, Otherwise ⎪ −(x − a ) ⎪ L N , a ≤ x < a N L a − a N R µ (x) = −(x − a ) (2) ˜ N R ⎪ , a ≤ x < a N R a − a 0, Otherwise. P N Note 1: In a bipolar triangular fuzzy number, a and a can be equal (see Figure 2). Note 2: We denote the set of all bipolar triangular fuzzy numbers by F(R). FUZZY INFORMATION AND ENGINEERING 177 P N Figure 1. Triangular bipolar fuzzy number a = a . P N Figure 2. Bipolar triangular fuzzy number with a = a . Using similar argument for fuzzy arithmetic, we now give the following proposition. L P N R L P N R Proposition 2.1: Let a ˜ = (a , a , a , a ) and b = (b , b , b , b ) be two bipolar fuzzy num- bers. Define: L P N R x > 0, x ∈ R, xa ˜ = (xa , xa , xa , xa ), R N P L x < 0, x ∈ R, xa ˜ = (xa , xa , xa , xa ), L L P P N N R R a ˜ + b = (a + b , a + b , a + b , a + b ). 2.2. Bipolar Ranking Function There are several methods for solving unipolar fuzzy linear programing problems by using ranking function [16,18,19,20,21,22]. We can define ranking function on bipolar fuzzy num- bers and use it for solving bipolar fuzzy number linear programing problems. A bipolar ranking function R : F(R) → R maps bipolar fuzzy numbers into the real line, where a 178 R. GHANBARI ET AL. natural order exists. Orders on F(R) are defined as follows: b b ˜ ˜ a ˜ ≥ b, if and only if R (a ˜ ) ≥ R (b), b b ˜ ˜ a ˜ ≤ b, if and only if R (a ˜ ) ≤ R (b), b b ˜ ˜ a ˜ = b, if and only if R (a ˜ ) = R (b), where a ˜, b ∈ F(R). Inspired by a special version of the ranking function on unipolar fuzzy numbers proposed by Yager [17], we define ranking function on bipolar fuzzy numbers as follows: 0 1 (inf a ˜ + sup a ˜ ) (inf a ˜ + sup a ˜ ) s s t t R (a ˜ ) = ds + dt. 2 2 −1 0 By using Definition 2.2 we have: P L P L P R R a ˜ = [a + ta − ta , ta − ta + a ] and N N L L N R R a ˜ = [−sa + sa + a , −sa + sa + a ]. So, 0 N L L N R R (−sa + sa + a − sa + sa + a ) R (a) = ds −1 1 L P L P R R R N P L (a + ta − ta + ta − ta + a ) (a + a + a + a ) + dt = 2 2 L P N R L P N R Then, for bipolar triangular fuzzy numbers a ˜ = (a , a , a , a ) and b = (b , b , b , b ) we have: R N P L R N P L (a + a + a + a ) (b + b + b + b ) a ˜ ≥ b if and only if ≥ . 2 2 Lemma 2.1: Ranking function (3) is linear. b b b L P N ˜ ˜ Proof: We need to show R (ka ˜ + b) = kR (a ˜ ) + R (b), for any k ∈ R,and a ˜ = (a , a , a , R L P N R a ) and b = (b , b , b , b ) any two bipolar triangular fuzzy numbers. Assume k > 0(for k < 0 is similar). From Proposition 2.1, we have L L P P N N R R ka ˜ + b = (ka + b , ka + b , ka + b , ka + b ), andfrom(3),weget, L L P P N N R R ka + b + ka + b + ka + b + ka + b R (ka ˜ + b) = L P N R L P N R k(a + a + a + a ) (b + b + b + b ) = + 2 2 b b = kR (a ˜ ) + R (b). FUZZY INFORMATION AND ENGINEERING 179 Lemma 2.2: Let R be defined as (3). Then, ˜ ˜ ˜ ˜ (1) a≥ b if and only if a − b≥ 0 if and only if −b≥ − a. ˜ b b b R R R ˜ ˜ ˜ (2) If a≥ band c≥ d then a ˜ + c≥ b + d. b b b R R R Proof: It is straightforward, by using (3). 3. Linear Programming Problem with Bipolar Triangular Fuzzy Numbers A bipolar triangular fuzzy number linear programing problem (BTFNLPP) is defined to be min z ˜= c ˜ x, s.t. (BTFNLPP) (4) Ax= bb, ⎪ R x ≥ 0, m m×n n n b ˜ ˜ where, b ∈ F(R ), A ∈ F(R ), c ˜ ∈ F(R ) are given, x ∈ R is to determined, and R is a b n b m b m×n ˜ ˜ linear ranking function as defined by (3), where R (c ˜) ∈ R , R (b) ∈ R and R (A) ∈ R . The following result gives an alternative formulation of (4). Proposition 3.1: Problem (4) is equivalent to min z ˜ = c ˜ x, s.t. (5) b b R (A)x = R (b), x ≥ 0. Proof: Since R is a linear ranking function, we have: ⎛ ⎞ n n b T b b b T ⎝ ⎠ R (c ˜ x) = R c ˜ x = R (c ˜ )x = R (c ˜ )x, j j j j j=1 j=1 b b b T n where R (c ˜) = (R (c ˜ ), ... , R (c ˜ )) ∈ R . On the other hand 1 n ⎛ ⎞ n n n b b b ⎝ ⎠ R a ˜ x = R (a ˜ x ) = R (a ˜ )x . ij j ij j ij j j=1 j=1 j=1 So, if we denote the i-th row of A by a ˜ ,wehave ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ n n b b b T b b ˜ ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ R (Ax) = (R (a ˜ x), ... , R (a ˜ x)) = R a ˜ x , ... , R a ˜ x 1 m 1j j mj j j=1 j=1 ⎛ ⎞ n n b b b b T b ⎝ ⎠ ˜ = R (a ˜ )x , ... , R (a ˜ )x = (R (a ˜ )x, ... , R (a ˜ )x) = R (A)x. 1j j mj j 1 m j=1 j=1 180 R. GHANBARI ET AL. 4. Duality Similar to the duality theory in linear optimization (see, for example, Luenberger and Ye [23]), for every BTFNLPP there is an associated dual BTFNLPP (DBTFNLPP) which satisfies some important properties. Definition 4.1: Using the notation of (4) define max u ˜ = y b s.t. (DBTFNLPP) (6) T T ⎪ ˜ y A≤ bc ⎪ R Indeed, (DBTFNLPP) is the dual of (BTFNLPP) and appropriate duality results can be established. Next, we show the weak duality result. Lemma 4.1: Dual of DBTFNLPP is BTFNLPP. Proof: This can be shown by changing the inequalities in (6) into equalities and an appro- priate change of the y variables followed by an application of Proposition 3.1. Theorem 4.1: If x and y are respectively feasible solutions to BTFNLPP and DBTFNLPPs, then 0 0 T T c ˜ x ≥ y b. 0 0 T T T ˜ ˜ ˜ ˜ Proof: Multiplying Ax = b on the left by y ,wehave y Ax = y b and multiplying b b 0 0 0 0 0 R R T T T T T T ˜ ˜ ˜ y A≤ c ˜ on the right by x ≥ 0, we have y Ax ≤ c ˜ x . So, we get y b≤ c ˜ x . b b b 0 0 0 0 0 0 0 R R R Note 3: Similar to some duality results for unipolar fuzzy number linear programing prob- lem, the value of the ranking function for the bipolar fuzzy value of the objective function at any feasible solution to BTFNLPP is always bigger than or equal to the value of the rank- ing function for the bipolar fuzzy value of the objective function for any feasible solution to DBTFNLPP. Also, if x and y are feasible solutions to BTFNLP and DBTFNLP problems, 0 0 T T respectively, and y b= c ˜ x , then x and y are optimal solutions to their respective prob- 0 0 0 0 lems and if any one of the BTFNLPP or DBTFNLPP is unbounded, then the other problem has no feasible solution. Next, we define a basic solution and then establish the strong duality result. Definition 4.2: Let A = [a ] = R (A). Assume rank(A) = m, and partition A as [B, N] ij m×n ˜ ˜ where B, m × m, is nonsingular. It is obvious that rank(B) = m and B = R (B), where B is the fuzzy matrix in A corresponding to B.Let y be a solution of By = a . A basic solution is j j T −1 x = (x , ... , x ) = B b, x = 0, B B B N 1 m ¯ ¯ where B = (b , ... , b ) = (a , ... , a ) with B being the index corresponding to the 1 m B B i 1 m ¯ ˜ i-th column of B, that is, b = a and b = R (b).If x ≥ 0, then the basic solution is feasible i B B ˜ ˜ and the corresponding fuzzy objective value is z= bc x , where c ˜ = (c ˜ , ... , c ˜ ) . B B1 Bm Now, corresponding to every nonbasic variable x ,1 ≤ j ≤ n, j = B , i = 1, ... , m define j i T T −1 z ˜ = c ˜ y = c ˜ B a.(7) b b j j j R B R B FUZZY INFORMATION AND ENGINEERING 181 Theorem 4.2: Assume the BTFNLPP is non-degenerate (see [23]). A basic feasible solution x = −1 B b, x = 0 is optimal to (4) if and only if z ˜ ≤ c ˜ , for all j, 1 ≤ j ≤ n. N j j T T T −1 Proof: Let x = (x , x ) be an optimal solution to (4), where x = B b, x = 0, correspond- B N ing to an optimal basis B. Then, the corresponding optimal objective value is T T T −1 z ˜ = c ˜ x= c ˜ x = c ˜ B b.(8) b b b ∗ B B B R R R Now, since x is feasible, we have x ≥ 0, and based on Definition 4.2, b = Ax = Bx + Nx.(9) B N Hence, we can rewrite (9) as follows: −1 −1 x = B b − B Nx . (10) B N Substituting (10) in (8), we obtain T T T z ˜ = c ˜ x= c ˜ x + c ˜ x b b B N R R B N T −1 T −1 T = c ˜ B b − (c ˜ B N − c ˜ )x R B B N T −1 T −1 ˜ ˜ ˜ = bc B b − (c B a − c )x . j j j R B B j=1,j=B Thus, z ˜= bz ˜ − (z ˜ − c ˜ )x . (11) j j j j=1,j=B Now, from (11) it is obvious that if there is a nonbasic variable x with z ˜ > c ˜ , then we can j j b j enter x into the basis and obtain z ˜ > z ˜ (since the problem is non-degenerate and x > 0 j b j in the new basis). This is contrary to z ˜ being optimal and hence we must have z ˜ ≤ bc ˜ for j j all j,1 ≤ j ≤ n. The converse of the theorem is similar to Theorem 3.1 in [19]. Theorem 4.3: If any one of the BTFNLPP or DBTFNLPP has an optimal solution, then both problems have optimal solutions and the two optimal value of ranking functions for the fuzzy objective values are equal. b b ˜ ˜ Proof: Assume that BTFNLPP has an optimal solution, rank(R (A)) = m and B = R (B) is −1 B b ∗T ∗T T the basic matrix and (x , x ) = forms the basic optimal solution corresponding B N to BTFNLPP.Since x is an optimal solution, according to Theorem 4.1, we have T −1 ˜ ˜ c B a − c ≤ b0, j = 1, ... , n, j j and thus, b T −1 b T b T ˜ ˜ R (c B A) − R (c) ≤ bR (0) , B R 182 R. GHANBARI ET AL. ∗ b T −1 T −1 ∗T T ∗ where by letting y = R (c ˜ B ) = c B , we can write y A≤ c ˜ .Thus, y is a feasible B B solution to DBTFNLPP. Based on Definition 4.2, we have ∗T b ∗T T −1 T ∗ T ∗ T ∗ b T ∗ y R (b) = y b = c B b = c x = c x + c x = R (c ˜) x . B B B B B N N Hence, ∗T T ∗ y b= c ˜ x . Due to Lemma 3.1, the converse of the theorem follows similarly. Theorem 4.4: For any BTFNLPP and its corresponding DBTFNLPP, exactly one of the following statements is true. ∗ ∗ T ∗ ∗T (1) Both have optimal solutions x and y with c ˜ x = y b. (2) One problem is unbounded and the other is infeasible. (3) Both problems are infeasible. Proof: 1 and 2 are proven in Theorem 4.2 and 3.1 respectively and we give an example for 3. Example 4.1: Consider BTFNLPP and its corresponding DBTFNLPP as follows: ⎧ ⎫ min z= b (−6, −4, 4, 10)x + (−14, −4, 3, 5)x 1 2 ⎪ R ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎨ ⎬ (BTFNLPP) (−8, 0, 2, 4)x + (−3, 0, 1, 4)x = (1, 2, 3, 4), 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ (−5, 0, 1, 6)x + (−5, 0, 1, 2)x = (2, 4, 6, 8), ⎪ 1 2 ⎪ R ⎪ ⎩ ⎭ x , x ≥ 0, 1 2 ⎧ ⎫ max u ˜ = (1, 2, 3, 4)w + (2, 4, 6, 8)w ⎪ 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s.t. ⎨ ⎬ (DBTFNLPP) (−8, 0, 2, 4)w + (−5, 0, 1, 6)w ≤ (−6, −4, 4, 10), 1 2 b ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (−3, 0, 1, 4)w + (−5, 0, 1, 2)w ≤ (−14, −4, 3, 5), 1 2 b ⎪ ⎪ ⎩ ⎭ x , x ≥ 0, 1 2 We clearly see that both problems are infeasible. The following result gives the complementary slackness conditions for optimality. ∗ ∗ Theorem 4.5: The vectors x and y respectively feasible solutions to BTFNLPP and DBTFNLPP are optimal solutions of the corresponding problems if and only if T ∗T ∗ ˜ ˜ (c − y A)x = b0. Proof: For a primal optimal solution x ,wehave Ax = bb, or b ∗ b ˜ ˜ R (A)x = R (b), (12) FUZZY INFORMATION AND ENGINEERING 183 also, for a dual optimal solution y ,wehave ∗T T y A≤ c ˜ , or ∗T b b T y R (A) ≤ R (c ˜) , which can be put in equality form by adding nonnegative surplus variables as follows: ∗T b T b T y R (A) + v = R (c) (13) Using (12), we get, ∗T b ∗ ∗T b y R (A)x = y R (b), (14) ∗T b ∗ T ∗ b T ∗ y R (A)x + v x = R (c ˜) x . (15) Subtraction of (15) from (14) yields, T ∗ b T ∗ ∗T b v x = R (c ˜) x − y R (b) = 0, (16) where the latter equality follows from Theorem 3.1. Hence, using (14)-(16), we obtain ∗T b b T ∗ ∗T b ∗ b T ∗ T ∗T ∗ ˜ ˜ ˜ ˜ ˜ ˜ y R (b) = R (c )x → y R (A)x = R (c )x → (c − y A)x = b0, T ∗T ∗ ˜ ˜ which completes the proof for the ‘only if’ part. Conversely, (c − y A)x = b0, gives T ∗ ∗T ∗ T ∗ ∗T ˜ ˜ c x = by Ax → c x = by b, R R to complete the proof. 5. Application of Proposed Method in Real Life Problems Akram [9] studied an application of bipolar fuzzy set in graph theory. He used bipolar fuzzy set in a social group. Here, we demonstrate an application of bipolar fuzzy number in max- imum weighted matching problem; matching problem has some applications in different fields such as scheduling [24] and network [25] problems. We consider each vertex as a person and weight of each edge between two vertices shows the influence of each per- son (vertex) to another person. In general, influence can be positive or negative. Suppose G = (V, E) is an arbitrary weighted graph, where V ={1, ... , n} is the vertex set of G and E ⊆ V × V is the edge set of G. The maximum weighted matching problem is: min w ˜ (e)x(e) ⎪ e∈E s.t. x(e) ≤ 1, ∀u ∈ V, ⎪ e=(u,v)∈E x(e) ∈{0, 1}, ∀e ∈ E, where x(e) = 1 if two persons u and $v are matched to each other, and x(e) = 0, otherwise, and w ˜ (e) is the weight of edge e (giving the influence of one person to another person), considered as a bipolar fuzzy number, since influence of a person cannot always be positive. We want to match every person to another person so that they have a stable relation. 184 R. GHANBARI ET AL. 6. Conclusions and Future Work We presented a linear ranking function for ordering bipolar fuzzy numbers and studied some of its properties. We defined bipolar triangular fuzzy number linear programing prob- lems and established its dual. We then proved the common duality results for the primal and dual problems. Similar to fuzzy primal simplex algorithms (see [21,26]), simplex, dual sim- plex [21], VNS algorithm [27,28] and primal-dual algorithms can be developed for solving bipolar fuzzy linear programing problems. Acknowledgements The first and second authors thank the Research Council of Ferdowsi University of Mashhad and opti- mization laboratory of Ferdowsi University of Mashhad and the third author thanks the Research Council of Sharif University of Technology for supporting this work. Disclosure statement No potential conflict of interest was reported by the author(s). Notes on contributors Reza Ghanbari is an Associate Professor of Mathematical Sciences at Ferdowsi University of Mashhad, Iran. He received his B.S. degree in Applied Mathematics from Ferdowsi University of Mashhad, Iran, in 2002, and his M.S. and Ph.D. degrees in Applied Mathematics from Sharif University of Technology, Iran, in 2004 and 2009, respectively. He is president of Khorasan science and technology park. He also is the manager of the optimization laboratory of Ferdowsi University. His research interests include algorithmic operational research, optimization and soft computing. Khatere Ghorbani-Moghadam is a Ph.D. of Mathematical Sciences in Applied Mathematics at Sharif University of Technology, Iran. She received her B.S. and M.S. degrees in Applied Mathematics from Ferdowsi University of Mashhad, Iran, in 2009 and 2011, respectively. Nezam Mahdavi-Amiri is a distinguished professor of Mathematical Sciences at Sharif University of Technology. He received his Ph.D. degree in Mathematical Sciences from Johns Hopkins University in 1981. He has been on the editorial board of several mathematical and computing journals in Iran including Bulletin of the Iranian Mathematical Society (Style-Language Editor), the Iranian Journal of Operations Research (Editor-in-Chief) and the CSI Journal of Computer Science and Engineering. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Apr 3, 2019
Keywords: Fuzzy linear programing problem; bipolar fuzzy number; duality; ranking function