Resolution of bipolar fuzzy relation equations with max-Łukasiewicz composition

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Resolution of bipolar fuzzy relation equations with max-Łukasiewicz composition ✩ Xiao-Peng Yang School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China Received 31 October 2018; received in revised form 6 August 2019; accepted 9 August 2019

Abstract The bipolar system of fuzzy relation equations with the max-Łukasiewicz composition is investigated in this work. A bipolar path approach is proposed for such a system. It is found that the complete solution set of the bipolar system is fully determined by its conservative bipolar paths, which are finite. Our proposed resolution approach is performed using the so-called path-based algorithm, step by step, and illustrated with numerical examples. Moreover, the global and local minimal (or maximal) solutions are discussed in this paper with a comparison to those of a classical unipolar system. © 2019 Elsevier B.V. All rights reserved. Keywords: Bipolar system; Fuzzy relation equation; Max-Łukasiewicz composition; Conservative bipolar path; Complete solution set

1. Introduction E. Sanchez [19] first opened the research direction on fuzzy relation equation (FRE). Because of its important application in various practical fields, many scholars have focused on the theoretical research of FRE, including its resolution approach and some specific optimization problems with FRE constraints. The solution set of a consistent system of FREs with the max-t-norm composition could be written as a finite number of closed intervals, whose begin point is one of the minimal solutions, and the end point is the unique maximum solution. Hence, for solving the FREs system, one should find out all its minimal solutions. However, solving all the minimal solutions has been proved to be an NP-hard problem [3], as it is equivalent to the set covering problem [13,18]. There is no polynomial-time algorithm. Resolution of an FREs system remains a hot research topic in the fuzzy set theory [17]. In addition to the resolution of the FREs, the optimization problem with an FREs constraint is another relevant research topic [9,21,22]. Most of the optimization models have been found and established in the practical application background. In these problems, the authors must provide an effective algorithm to compute the optimal solution [10]. ✩ Supported by the National Natural Science Foundation of China (61877014), the Natural Science Foundation of Guangdong Province (2016A030307037, 2017A030307020) and the Natural Science Foundation of Hanshan Normal University (2017KTSCX124, QD20171001). E-mail address: [email protected].

https://doi.org/10.1016/j.fss.2019.08.005 0165-0114/© 2019 Elsevier B.V. All rights reserved.

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In general, the solution algorithm is related to the feasible domain, i.e., the complete solution set of a group of FREs. The properties of the FRE greatly contribute to the solution algorithm. Bipolarity exists widely in human understanding of information and preference [6]. Bipolar representation seems to be useful in the development of intelligent technologies. In [6] the authors tried to show how the possibility theory framework is convenient for handling bipolar representations. There are three forms of bipolarity: symmetric univariate, dual bivariate (or symmetric bivariate unipolarity), and asymmetric (or heterogeneous) bipolarity [6]. D. Dubois and H. Prade shown an overview of the asymmetric bipolar representation of positive and negative information in possibility theory [7]. The bipolar representation was applied to distinguish between negative and positive information in preference modeling [4,7]. Recently, the concept of FRE was generalized to a so-called bipolar fuzzy relation equation. S. Freson and B. De Baets et al. [8] proposed the concept of bipolar fuzzy relation equation for the first time, with application in public awareness of the products for a supplier. The bipolar fuzzy relation equation with max-min composition could be expressed as [8] max

j =1,2,··· ,n

max{min{aij+ , xj }, min{aij− , x¯j }}, i = 1, 2, · · · , m,

where aij+ , aij− , xj , x¯j ∈ [0, 1] and x¯j = 1 − xj . To optimize (increase) the suppliers’ benefits, which are related to the public awareness, the authors introduced the corresponding optimization problem, maximizing a linear objective function with the bipolar fuzzy relation equations constraint. Until now, few studies have reported the relevant results on the bipolar fuzzy relation equation. Most of them focused on the optimization problems subject to bipolar fuzzy relation equations [1,2,16]. As noted in [11,14], there are three commonly used compositions in the bipolar fuzzy relation equation: maxmin [12,15], max-product [5] and max-Łukasiewicz [23]. The bipolar max-Łukasiewicz equation constrained linear optimization problem has been studied in [11,14]. After investigation of the structure and properties of the feasible domain (solution set of system of bipolar max-Łukasiewicz fuzzy relation equations), the authors searched the optimal solution form the potential maximal and minimal solutions of the constraints. Some basic properties of bipolar fuzzy relation equations [5,12,15], as well as its corresponding optimization problem [1,2,16] have been investigated. Similar to the unipolar fuzzy relation system, the resolution of a bipolar fuzzy relation system is also an important issue in the relevant research. However resolution method for obtaining the complete solution set of bipolar maxŁukasiewicz fuzzy relation equations (see system (6) in next section) has not been found in the existing works. It was shown in [11,14] that the optimal solution of the linear optimization problem with bipolar max-Łukasiewicz fuzzy relation equations constraint could be selected from the maximal solutions or the minimal solutions of system (6). Hence, resolution of system (6), especially all the maximal and minimal solutions, appears to be important. The purpose of this work is to propose effective method for obtaining all the solutions of system (6). In this paper, we aim to investigate the resolution of the bipolar fuzzy relation equation with the max-Łukasiewicz composition. The remainder of this paper is organized as follows. Sec. 2 shows the necessary definitions and basic properties of the bipolar fuzzy relation equations with the max-Łukasiewicz composition. The major result is presented in Sec. 3, where we develop the bipolar path approach in detail. Based on the bipolar path approach, the structure of the complete solution set is obtained. The path-based algorithm and illustrative examples are shown in Sec. 4. Finally, Secs. 5 and 6 show a simple discussion and the conclusion, respectively. 2. Bipolar fuzzy relation equations 2.1. Application background of the bipolar fuzzy relation equations In this subsection we first recall the application background of such bipolar fuzzy relation equations according to Ref. [8]. Consider a supplier of several products, denoted by p1 , p2 , · · · , pn . The supplier aims to optimize the public awareness and therefore attributes to all the products a degree of appreciation, x1 , x2 , · · · , xn . For the j th product, the degree of appreciation xj is reflected by a real number in the unit interval [0, 1]. Correspondingly, its degree of disappreciation is indeed x¯ j = 1 − xj . The degree of appreciation thus acquires a bipolar character on [0, 1], in the sense that value 0.5 means neutrality with respect to appreciation. According to [6], the bipolarity in system (6) belongs to Type I bipolarity, i.e. symmetric univariate.

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For the publicity effects of the products, there exists a basic threshold with respect to the degree of appreciation (or disappreciation). For example, assume that the products are promoted in m areas/markets, denoted by A1 , A2 , · · · , Am . The threshold degree of appreciation of pj at the ith market Ai is aij (aij ∈ [0, 1]). If the degree of appreciation xj is less than aij , then the people might not pay attention to the product. Hence we assume that the people do not notice the product and the corresponding public awareness could be neglected, when the degree of appreciation is below the threshold. If xj is no less than aij , then the effective public awareness is xj − aij . As a result, the effective public awareness could be represented by max{xj − aij , 0},

(1)

which is said to be relative degree of appreciation of pj at the market Ai . Analogously, if the threshold degree of disappreciation of pj at Ai is a¯ ij (a¯ ij ∈ [0, 1]), then the relative degree of disappreciation could be represented by max{x¯j − a¯ ij , 0}.

(2)

Denote aij+ = 1 − aij ∈ [0, 1] and aij− = 1 − a¯ ij ∈ [0, 1]. Then the relative degrees of appreciation (1) and disappreciation (2) could be written as max{aij+ + xj − 1, 0},

(3)

max{aij− + x¯j − 1, 0},

(4)

and

respectively. In general, the threshold with respect to the degree of appreciation (i.e. aij ) is different from that with respect to the degree of disappreciation (i.e. a¯ ij ). These two thresholds are independent. Hence the quantities aij+ and aij− are also independent. As pointed out in [8], “in practice, often bad publicity can be seen as a form of publicity too”. Hence, not only a high degree of appreciation is helpful for public awareness, but also a high degree of disappreciation can in some situations be worthwhile. Both these two aspects are effective for publicity of the products. The public awareness of product pj at market Ai could be represented by max{max{aij+ + xj − 1, 0}, max{aij− + x¯j − 1, 0}}.

(5)

Assume further that market research has revealed that public awareness at a level of bi is the best value for the products to be sold at market Ai . For better product sales, the company hopes the public awareness reaches such best value. For arbitrary market Ai , there should exist at least one product, of which the public awareness reaches the best value bi . Therefore, the above-mentioned conditions could be formulated as max{max{aij+ + xj − 1, 0}, max{aij− + x¯j − 1, 0}} = bi , j ∈J

i = 1, 2, · · · , m.

(6)

System (6) could be written as max{TL (aij+ , xj ), TL (aij− , x¯j )} = bi , j ∈J

i = 1, 2, · · · , m,

(7)

where TL is the max-Łukasiewicz composition. Hence (6) is system of bipolar fuzzy relation equations with the max-Łukasiewicz composition [11,14]. In system (6), aij+ , aij− , xj , bi ∈ [0, 1], x¯j = 1 − xj , i ∈ I = {1, 2, · · · , m}, j ∈ J = {1, 2, · · · , n}. Here, I and J are two index sets. The matrix form of system (6) is A+ ◦ x T ∨ A− ◦ x¯ T = bT ,

(8)

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where A+ = (aij+ )m×n , A− = (aij− )m×n , x = (x1 , x2 , · · · , xn ), x¯ = (x¯1 , x¯2 , · · · , x¯n ), b = (b1 , b2 , · · · , bm ), and ◦ represents the max-Łukasiewicz composition. Here, for arbitrary vectors X = (x1 , x2 , · · · , xn )T and Y = (y1 , y2 , · · · , yn )T , we have X ∨ Y = (max{x1 , y1 }, max{x2 , y2 }, · · · , max{xn , yn })T . To deal with the classical unipolar fuzzy relation equation, the conservative path approach is one of the effective resolution methods. The conservative path approach was first proposed by P.Z. Wang et al. [20] for the max-min fuzzy relation inequalities and widely applied to other kinds of fuzzy relation systems. Unfortunately, it is found that the conservative path approach is no longer effective for the bipolar fuzzy relation system. In this paper, motivated by the conservative path approach presented in [20], we propose the so-called conservative bipolar path approach for the bipolar fuzzy relation inequalities with the max-Łukasiewicz composition. In a bipolar path, there exist both “positive” index and “negative” index (see Section 3 below). Due to the positive and negative indices, the selection of the path is much different from that in a classical unipolar fuzzy relation system. 2.2. Basic results on the bipolar fuzzy relation equations Basic definitions and properties of the bipolar fuzzy relation equations, i.e. system (6), are presented in this subsection. In this paper, we always denote  fij+ (xj ) = max{aij+ + xj − 1, 0}, (9) fij− (xj ) = max{aij− + x¯j − 1, 0} = max{aij− − xj , 0}, and fij (xj ) = max{fij+ (xj ), fij− (xj )},

(10)

for any i ∈ I , j ∈ J . Based on notations (9) and (10), system (6) can be shortened as max{fij (xj )} = bi , j ∈J

i = 1, 2, · · · , m.

Definition 1. Let x, y ∈ [0, 1]n . We write x ≤ (or ≥, =)y if and only if xj ≤ (or ≥, =)yj holds for all j ∈ J . For convenience, we denote X = [0, 1]n and X(A+ , A− , b) = {x ∈ X|A+ ◦ x T ∨ A− ◦ x¯ T = bT }.

(11)

Obviously, X(A+ , A− , b) is the solution set of system (6). Definition 2 (Consistency). System (6) is said to be consistent (or inconsistent) if X(A+ , A− , b) = ∅ (or X(A+ , A− , b) = ∅). Definition 3 (Minimum solution and maximum solution). In system (6), a solution xˇ˙ ∈ X(A+ , A− , b) (or x˙ˆ ∈ X(A+ , A− , b)) is called minimum solution (or maximum solution) if x˙ˇ ≤ x (or x˙ˆ ≥ x) holds for all x ∈ X(A+ , A− , b). Theorem 1. [11] A vector x ∈ X is a solution of system (6) if and only if fij (xj ) ≤ bi for every i ∈ I and j ∈ J , and there exists an index ji ∈ J for each i ∈ I such that fiji (xji ) = bi . Proof. See Lemma 3 in [11].

2

Proposition 1. [14] If system (6) is consistent and x = (xj )j ∈J ∈ X(A+ , A− , b) is a solution, then max{aij− − bi , 0} ≤ xj ≤ min{1 − aij+ + bi , 1}, ∀j ∈ J. i∈I

i∈I

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Proof. See Lemma 2 in [14].

5

2

Definition 4 (Lower bound and upper bound). In system (6), vectors x˙ˇ = (x˙ˇ1 , x˙ˇ2 , · · · , x˙ˇn ) and x˙ˆ = (x˙ˆ1 , x˙ˆ2 , · · · , x˙ˆn )

(12)

are called lower bound (vector) and upper bound (vector), respectively, if x˙ˇj = max{aij− − bi , 0} and x˙ˆj = min{1 − aij+ + bi , 1},

(13)

i∈I

i∈I

j = 1, 2, · · · , n. Corollary 1. Let x be a solution of system (6). x˙ˇ and x˙ˆ are the lower bound and upper bound, respectively. Then, we have ˙ˆ x˙ˇ ≤ x ≤ x.

(14)

Proposition 2. Let x˙ˇ and xˆ˙ be the lower and upper bounds of system (6). If x˙ˇ ∈ X(A+ , A− , b), then x˙ˇ is the minimum solution, whereas if x˙ˆ ∈ X(A+ , A− , b), then x˙ˆ is the maximum solution. Proof. This result can be easily obtained by Proposition 1 and Definition 4.

2

Proposition 2 shows that the lower and upper bounds are the potential minimum and maximum solutions, respectively. Proposition 3. Let x˙ˇ and x˙ˆ be the lower and upper bounds of system (6). For any i ∈ I , j ∈ J and xj ∈ [x˙ˇj , x˙ˆj ], it holds that fij+ (xj ) ≤ bi , fij− (xj ) ≤ bi and fij (xj ) ≤ bi . Proof. We prove that fij+ (xj ) ≤ bi . The remainder of the proof is similar. If fij+ (xj ) = 0, it is obvious that fij+ (xj ) = 0 ≤ bi . Otherwise, according to (13), we have fij+ (xj ) = aij+ + xj − 1 ≤ aij+ + x˙ˆj − 1 ≤ aij+ + (1 − aij+ + bi ) − 1 = bi .

2

3. Bipolar path approach for solving system (6) In this section, we develop the bipolar path approach to solve the complete solution set of system (6). 3.1. Concept of the conservative bipolar path Let J −+ = {1− , 1+ , 2− , 2+ , · · · , n− , n+ }. For a given index i ∈ I , define a function gi : J −+ → {0, 1} as follows:  0, if fij− (x˙ˇj ) = bi , − gi (j ) = 1, if fij− (x˙ˇj ) = bi ,  0, if fij+ (x˙ˆj ) = bi , + gi (j ) = 1, if f + (x˙ˆj ) = bi ,

(15)

(16)

(17)

ij

where j = 1, 2, · · · , n. Based on the above functions {gi }i∈I , we subsequently define the following index sets Ji−+ = {j ∈ J −+ |gi (j ) = 1},

∀i ∈ I.

(18)

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For any j ∈ J , we denote the complement of the element in J −+ by (j − )c = j + ,

(j + )c = j − .

(19)

Definition 5 (Conservative bipolar path). A vector p = (p1 , p2 , · · · , pm ) ∈ J1−+ × J2−+ × · · · × Jm−+ is called a conservative bipolar path of system (6) if for any i ∈ I , ⎧ −+ ⎪ if i = 1, ⎪ ⎨∈ J1 , −+ c c c (20) pi ∈ Ji − {p1 , p1 , p2 , p2 , · · · , pi−1 , pi−1 }, if Ji−+ ∩ {p1 , p2 , · · · , pi−1 } = ∅, ⎪ ⎪ −+ ⎩= p , if J ∩ {p , p , · · · , p } = ∅, k

i

1

2

i−1

where p1 , p2 , · · · , pk−1 ∈ / Ji−+ and pk ∈ Ji−+ , 1 ≤ k ≤ i − 1. Furthermore, the set of all conservative bipolar paths is denoted by P . It is clear that pi ∈ Ji−+ ⊆ J −+ for any i ∈ I . Example 1. [11,14] Find all conservative bipolar paths of the following system of bipolar fuzzy relation equations with the max-Łukasiewicz composition T

− T T 1 A+ 1 ◦ x ∨ A1 ◦ x¯ = b ,

(21)

where ⎡

0.9 ⎢ 0.8 + A1 = ⎢ ⎣ 0.8 0.5

0.8 0.9 0.6 0.6

0.9 0.7 0.8 0.4

⎡ ⎤ 0.6 0.9 ⎢ 0.7 1.0 ⎥ − ⎥, A = ⎢ 1 ⎣ 0.6 0.4 ⎦ 0.4 0.6

0.7 0.8 0.9 0.7 0.8 0.9 0.9 0.8

⎡ ⎤ ⎤ 0.9 0.8 ⎢ 0.8 ⎥ T 0.8 ⎥ 1 ⎥, b = ⎢ ⎥. ⎣ 0.7 ⎦ 0.9 ⎦ 0.4 0.6

After the calculation, the lower bound of (21) is x˙ˇ = (0.1, 0.3, 0.2, 0.2), whereas its upper bound is x˙ˆ = (0.9, 0.9, 0.9, 0.8). − ˙ − (xˇ1 ) = max{a11 − x˙ˇ1 , 0} = 0.9 − 0.1 = 0.8 = b1 , we have g1 (1− ) = 1. Since f11 − ˙ − (xˇ2 ) = max{a12 − x˙ˇ2 , 0} = 0.7 − 0.3 = 0.6 = 0.8 = b1 , we have g1 (2− ) = 0. Since f12 − ˙ − Since f13 (xˇ3 ) = max{a13 − x˙ˇ3 , 0} = 0.8 − 0.2 = 0.6 = 0.8 = b1 , we have g1 (3− ) = 0. − ˙ − Since f14 (xˇ4 ) = max{a14 − x˙ˇ4 , 0} = 0.9 − 0.2 = 0.7 = 0.8 = b1 , we have g1 (4− ) = 0. + ˙ + (xˆ1 ) = max{a11 + x˙ˆ1 − 1, 0} = 0.9 + 0.9 − 1 = 0.8 = b1 , we have g1 (1+ ) = 1. Since f11 + ˙ + Since f12 (xˆ2 ) = max{a12 + x˙ˆ2 − 1, 0} = 0.8 + 0.9 − 1 = 0.7 = 0.8 = b1 , we have g1 (2+ ) = 0. + ˙ + Since f13 (xˆ3 ) = max{a13 + x˙ˆ3 − 1, 0} = 0.9 + 0.9 − 1 = 0.8 = b1 , we have g1 (3+ ) = 1. + ˙ + Since f14 (xˆ4 ) = max{a14 + x˙ˆ4 − 1, 0} = 0.6 + 0.8 − 1 = 0.4 = 0.8 = b1 , we have g1 (4+ ) = 0. According to (16)-(18), the above results indicate that J1−+ = {1− , 1+ , 3+ }. Analogously we are able to find the other index sets as J2−+ = {2+ , 4+ }, J3−+ = {1+ , 3− , 3+ , 4− }, J4−+ = {2− , 3− }.

(22)

Following Definition 5 (see (20)), all conservative bipolar paths could be obtained. Since J1−+ = {1− , 1+ , 3+ }, the first component in the conservative bipolar path p, i.e., p1 , has three choices: 1− , 1+ and 3+ . If p1 = 1− , then J2−+ ∩ {p1 } = {2+ , 4+ } ∩ {1− } = ∅. According to (20), p2 ∈ J2−+ − {p1 , p1c } = {2+ , 4+ } − − {1 , 1+ } = {2+ , 4+ }. This indicates p2 has two choices, i.e., 2+ and 4+ . The detailed selections of p2 , p3 and p4 are as shown in (23) below, when p1 = 1− .

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p1 = 1 − , p2 = 2+ ,

7

%p2 ∈ J2−+ − {p1 , p1c } = {2+ , 4+ }, since J2−+ ∩ {p1 } = ∅

p3 = 3− ,

%p3 ∈ J3−+ − {p1 , p1c , p2 , p2c } = {3− , 3+ , 4− }, since J3−+ ∩ {p1 , p2 } = ∅

p4 = 3− .

%p4 = p3 , since J4−+ ∩ {p1 , p2 , p3 } = {3− } = {p3 } = ∅

%p4 ∈ J4−+ − {p1 , p1c , p2 , p2c , p3 , p3c } = ∅, since J4−+ ∩ {p1 , p2 , p3 } = ∅

p3 = 3 + ,

p4 ∈ ∅ (this indicates the corresponding path is ineffective). p 3 = 4− , −

%p4 ∈ J4−+ − {p1 , p1c , p2 , p2c , p3 , p3c } = {3− }, since J4−+ ∩ {p1 , p2 , p3 } = ∅

(23)

p4 = 3 . p2 = 4 + , p3 = 3 − ,

%p3 ∈ J3−+ − {p1 , p1c , p2 , p2c } = {3− , 3+ }, since J3−+ ∩ {p1 , p2 } = ∅

p4 = 3− . p3 = 3 + , p4 = 2− .

%p4 = p3 , since J4−+ ∩ {p1 , p2 , p3 } = {3− } = {p3 } = ∅

%p4 ∈ J4−+ − {p1 , p1c , p2 , p2c , p3 , p3c } = {2− }, since J4−+ ∩ {p1 , p2 , p3 } = ∅

According to (23), if p1 = 1− , there are four conservative bipolar paths, p 1 = (1− , 2+ , 3− , 3− ), p 2 = p 3 = (1− , 4+ , 3− , 3− ), p 4 = (1− , 4+ , 3+ , 2− ). In a similar way, if p1 = 1+ , we are able to find three four conservative bipolar paths, i.e., p 5 = (1+ , 2+ , 1+ , 3− ), p 6 = (1+ , 4+ , 1+ , 2− ) and p 7 = (1+ , 4+ , 1+ , 3− ). While if p1 = 3+ , there exists a unique conservative bipolar path p 8 = (3+ , 4+ , 3+ , 2− ). As a consequence, there are 8 conservative bipolar paths totally. (1− , 2+ , 4− , 3− ),

p 1 = (1− , 2+ , 3− , 3− ), p 2 = (1− , 2+ , 4− , 3− ), p 3 = (1− , 4+ , 3− , 3− ), p 4 = (1− , 4+ , 3+ , 2− ), p 5 = (1+ , 2+ , 1+ , 3− ), p 6 = (1+ , 4+ , 1+ , 2− ),

(24)

p 7 = (1+ , 4+ , 1+ , 3− ), p 8 = (3+ , 4+ , 3+ , 2− ). 3.2. Induced solutions of the conservative bipolar path For a given conservative bipolar path p, define a pair of vectors, which are denoted by xˇ p and xˆ p , with respect to p: ⎧ − ˙ ⎪ ⎨xˇj , if there exists some i ∈ I such that pi = j , p (25) xˇj = x˙ˆj , if there exists some i ∈ I such that pi = j + , ⎪ ⎩˙ xˇj , otherwise, and

⎧ ˙ ⎪ ⎨xˇj , p xˆj = x˙ˆj , ⎪ ⎩˙ xˆj ,

if there exists some i ∈ I such that pi = j − , if there exists some i ∈ I such that pi = j + , otherwise.

(26)

According to Definition 5, for arbitrary p ∈ P , it is impossible that pi = j + and pi

= j − hold simultaneously, for some i , i

∈ I and j ∈ J . Hence the above definitions of xˇ p and xˆ p are reasonable. Lemma 1. For any i ∈ I , if pi = j − , then fij− (x˙ˇj ) = bi , while if pi = j + , then fij+ (x˙ˆj ) = bi . Proof. If pi = j − ∈ Ji−+ , it follows from (18) that gi (pi ) = gi (j − ) = 1. According to (16), we have fij− (x˙ˇj ) = bi . Similarly, it is easy to finish the proof for pi = j + . 2

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Let J ∗ = {j ∈ J | there exists i ∈ I such that pi = j − or j + }.

(27)

Then, we obtain Lemma 2. Lemma 2. For any j ∈ J ∗ , it holds that xˇj = xˆj . p

p

Proof. According to (27), j ∈ J ∗ indicates that there exists some i ∈ I such that pi = j − or pi = j + . When pi = j − p p p p holds, it follows from (25) and (26) that xˇj = xˆj = x˙ˇj , while pi = j + holds, we have xˇj = xˆj = x˙ˆj . Thus, the proof is complete. 2 Theorem 2. Suppose that xˇ p and xˆ p are as defined by (25) and (26) with respect to the path p. Then, for any x that satisfies xˇ p ≤ x ≤ xˆ p , x is a solution of system (6). Proof. Take an arbitrary i ∈ I . Next, we will check x ∈ X(A+ , A− , b) following Theorem 1. First, according to Proposition 3 and the definition of xˇ p and xˆ p , we have fij (xj ) ≤ bi for any j ∈ J . Meanwhile, it is obvious that there exists ji ∈ J , such that either pi = ji− or pi = ji+ holds. Thus, it holds that p p ji ∈ J ∗ by (27). Following Lemma 2, we obtain xji = xˇji = xˆji . Now, we verify that fiji (xji ) = bi in two cases. p Case 1. If pi = j − , then xˇ = xˇ˙j . Following Lemma 1 and Proposition 3, we have i

ji

i

p fiji (xji ) = fiji (xˇji ) = fiji (x˙ˇji ) = max{fij+i (x˙ˇji ), fij−i (x˙ˇji )} = max{fij+i (x˙ˇji ), bi } = bi .

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p Case 2. If pi = ji+ , then xˆji = x˙ˆji . Following Lemma 1 and Proposition 3, we have

fiji (xji ) = fiji (xˆji ) = fiji (x˙ˆji ) = max{fij+i (x˙ˆji ), fij−i (x˙ˆji )} = max{bi , fij−i (x˙ˆji )} = bi . p

Cases 1 and 2 contribute to fij (xj ) = bi . Thus, we obtain x ∈ X(A+ , A− , b), i.e., x is a solution of system (6).

(29) 2

Corollary 2. For a conservative bipolar path p, the pair of vectors induced by p, i.e., xˇ p and xˆ p , is the pair of solutions of system (6). In the following, solution xˇ p is called the minimal induced solution of p, whereas xˆ p is called the maximal induced solution. Moreover, [xˇ p , xˆ p ] is called the induced solution interval of p. Corollary 3. For each p ∈ P , there is an induced solution interval [xˇ p , xˆ p ] ⊆ X(A+ , A− , b) that corresponds to p. 3.3. Bipolar path approach to obtain the solution set Proposition 4. In system (6), if there exist some i ∈ I and ji ∈ J and x˙ˇji ≤ xji ≤ x˙ˆji , such that fiji (xji ) = bi > 0, then it holds that xji = x˙ˇji or xji = x˙ˆji . Proof. (By contradiction) Assume that neither xji = x˙ˇji nor xji = x˙ˆji holds. Then, it becomes x˙ˇji < xji < x˙ˆji .

(30)

Since 0 < bi = fiji (xji ) = max{max{aij+i + xji − 1, 0}, max{aij−i − xji , 0}},

(31)

we have either aij+i + xji − 1 = bi > 0,

(32)

aij−i − xji = bi > 0.

(33)

or

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If (32) holds, by combining Inequality (30) and Proposition 3, we have bi = aij+i + xji − 1 < aij+i + x˙ˆji − 1 ≤ fiji (x˙ˆji ) ≤ bi .

(34)

If (33) holds, again, by Inequality (30) and Proposition 3, we have bi = aij−i − xji < aij−i − x˙ˇji ≤ fiji (x˙ˇji ) ≤ bi .

(35)

Both (34) and (35) lead to a contradictory inequality that “bi < bi ”. Hence, we obtain either xji = x˙ˇji or xji = x˙ˆji , and the proof is complete. 2 Theorem 3. Let y ∈ X(A+ , A− , b) be a solution of system (6). If bi > 0 for all i ∈ I , then there is a conservative bipolar path p ∈ P , such that xˇ p ≤ y ≤ xˆ p . Proof. Take arbitrary i ∈ I . According to Theorem 1, there exists ji ∈ J such that fiji (yji ) = bi > 0.

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Following Corollary 1 and Proposition 4, we have yji = x˙ˇji or x˙ˆji . Construct j1 , j2 , · · · , jm as follows:  j − , if yji = x˙ˇji ,

ji = i+ ji , if yji = x˙ˆji . p

p

(37)

p

In addition, construct j1 , j2 , · · · , jm as follows: ⎧

⎪ ⎨j1 , if i = 1, p ji = ji , if Ji−1 = ∅, ⎪ ⎩ jt , if Ji−1 = ∅,

(38)

}|f (y ) = b } and j = min J where Ji−1 = {j ∈ {j1 , j2 , · · · , ji−1 ij j i i−1 . t p p p Let p = (j1 , j2 , · · · , jm ). Then, p is a conservative bipolar path. Next, we aim to verify that xˇ p ≤ y ≤ xˆ p . Denote

J ∗ = {j ∈ J | there exists i ∈ I such that ji = j − or j + }. p

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(i) For j ∈ J ∗ , it holds that yj = xˇj = xˆj . p p j ∈ J ∗ indicates that there exists some i ∈ I such that ji = j − or j + . However, it follows from (38) that ji = jk , p p p where k = i or t. If ji = jk = j − , then it holds that yj = x˙ˇj = xˇj = xˆj by (25), (26) and (37). Otherwise, if p p p p p j = j = j + , then we have yj = x˙ˆj = xˇ = xˆ . Both cases lead to the conclusion that yj = xˇ = xˆ , ∀j ∈ J ∗ . p

i

p

j

k

j

j

j

p p (ii) For j ∈ / J ∗ , it follows from (25), (26) and (39) that xˇj = x˙ˇj and xˆj = x˙ˆj . Moreover, following Corollary 1, we p p obtain xˇ = xˇ˙j ≤ yj ≤ x˙ˆj = xˆ . j

j

Cases (i) and (ii) contribute to the inequality that xˇ p ≤ y ≤ xˆ p . The proof is complete.

2

Based on the aforementioned Theorems 2 and 3, it is easy to obtain Theorems 4 and 5 Theorem 4. Suppose that bi > 0 for all i ∈ I . System (6) is consistent if and only if there exists at least one conservative bipolar path, i.e., P = ∅. Theorem 5 (Structure theorem). Let system (6) be consistent with all bi > 0, i ∈ I . Then, its complete solution set can be represented as

X(A+ , A− , b) = [xˇ p , xˆ p ], (40) p∈P

where P is the set of all conservative bipolar paths of (6), and [xˇ p , xˆ p ] is the induced solution interval of each p ∈ P .

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In fact, after obtaining the index sets J1−+ , J2−+ , · · · , Jm−+ , system (6) can be simplified if there exist i and i

in I such that Ji−+ ⊆ Ji−+

. Next, we introduce a rule to reduce the problem size before computing the set of all conservative bipolar paths, i.e., P .

⊆ Ji−+ Rule 1. If there exist i , i

in I such that Ji−+

, then the i th equation can be deleted from system (6) before applying the bipolar path approach to compute its solution set.

Proof. Suppose x ∈ X(A+ , A− , b) is a solution of (6). Then, it satisfies the i th equation. However, according to the ⊆ Ji−+ indicates that x also satisfies the i

th equation. Hence, the i

th equation can definition of the index set, Ji−+

be deleted in system (6) without changing its solution set. 2 In addition, some conservative bipolar paths in P may be unnecessary for computing the solution set. Rule 2 helps us further reduce the computation. Definition 6. For a conservative bipolar path p l ∈ P , if there exists p k ∈ P such that k = l and k l {p1k , p2k , · · · , pm } ⊆ {p1l , p2l , · · · , pm },

then path p l is said to be reducible in P . Otherwise, it is said to be irreducible. If we delete all reducible conservative bipolar paths in P , the remainder, which is denoted by P , is said to be irreducible. Rule 2. All reducible conservative bipolar paths can be deleted from P without changing the solution set of (6).







p









l } ⊆ {p l , p l , · · · , p l }. To comProof. Assume that there exist some p l , p l ∈ P , l = l

such that {p1l , p2l , · · · , pm m 1 2





p

p

p

plete the proof, we must only verify that [xˇ p , xˆ p ] ⊆ [xˇ p , xˆ p ], i.e., [xˇj , xˆj ] ⊆ [xˇj , xˆj ] for all j ∈ J . p p Case 1. If there exists i ∈ I , such that pi = j − (or j + ), then xˇj = xˆj = x˙ˇj (or x˙ˆj ) by (25) and (26). It follows





p

p

from {p l , p l , · · · , p l } ⊆ {p l , p l , · · · , p l } that p

= j − (or j + ). Again, by (25) and (26), we have xˇ = xˆ = x˙ˇj 1

m

2

1

(or x˙ˆj ). Hence, we have p

p

p

m

2

j

i

j

p

[xˇj , xˆj ] = [xˇj , xˆj ] = [x˙ˇj , x˙ˇj ] = x˙ˇj (or x˙ˆj ). Case 2. If there is no i ∈ I such that pi = j − or j + , then it holds that



p p xˇj = x˙ˇj and xˆj = x˙ˆj ,



because of (25) and (26). However, it follows from Theorem 2 that both xˇ p and xˆ p are solutions of system (6). According to Corollary 1, p

p

p

p

[xˇj , xˆj ] ⊆ [x˙ˇj , x˙ˆj ] = [xˇj , xˆj ]. Based on the above Cases 1 and 2, the proof is complete. 2 Theorem 6. Let P be an irreducible set of P , which is obtained by deleting all reducible conservative bipolar paths in P . Then, the solution set of (6) is

X(A+ , A− , b) = [xˇ p , xˆ p ]. (41) p ∈P

Proof. The proof lies in Rule 2.

2

Theorem 5 shows that the complete solution set of system (6) is exactly a union of a finite number of closed intervals, which are induced by the conservative bipolar paths. Hence, the solution set is completely determined by the conservative bipolar paths. In addition, in most cases, a system of bipolar fuzzy relation equations has no unique

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maximum solution or minimum solution, which is different from the unipolar fuzzy relation equation system, where the solution set is commonly determined by a unique maximum solution and a finite number of minimal solutions. The results in Theorems 3 and 5 are expressed with the assumption that bi > 0 for all i ∈ I . Here, we note that when bi > 0 does not hold for some i ∈ I , system (6) can also be solved by the aforementioned bipolar path approach, and the structure of its solution set is identical to that in Theorem 5. Next, we discuss the solution method for system (6) in two cases according to the right-side vector b. In the remainder of this section, x˙ˇ and x˙ˆ are always assumed to be the lower and upper bounds of system (6). Case 1. bi = 0 for all i ∈ I Before presenting the solution method, we first provide a relevant proposition. ˙ˇ x], ˙ˆ x is a solution of the ith Proposition 5. If there exists some i ∈ I such that bi = 0, then for arbitrary x ∈ [x, equation in system (6), i.e., max{max{aij+ + xj − 1, 0}, max{aij− + x¯j − 1, 0}} = bi .

(42)

j ∈J

Proof. It is obvious that max{max{aij+ + xj − 1, 0}, max{aij− + x¯j − 1, 0}} ≥ 0 = bi . Meanwhile, it follows from Proposition 3 that max{max{aij+ + xj − 1, 0}, max{aij− + x¯j − 1, 0} ≤ bi }. Hence, max{max{aij+ + xj − 1, 0}, max{aij− + x¯j − 1, 0}} = max{0} = 0 = bi . j ∈J

The proof is complete.

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j ∈J

2

˙ˇ x]. ˙ˆ Theorem 7. In system (6), if bi = 0 holds for all i ∈ I , then the solution set of (6) is [x, Proof. This result is a direct conclusion from Proposition 5 and Corollary 1.

2

Case 2. bi > 0 for some i ∈ I In this case, if all elements in {b1 , b2 , · · · , bm } are positive, then the solution set of (6) can be obtained following Theorem 5 using the bipolar path approach. Meanwhile, if some of the elements in {b1 , b2 , · · · , bm } are equal to zero, then we may transmit (6) into an equivalent system with all bi being positive in the right side. Denote I = {i ∈ I |bi = 0} = ∅. By deleting the i th equation in system (6), we obtain a generated system, which is denoted by (6)’. According to Propositions 3 and 5, the solution set of system (6)’ is identical to that of system (6). In other words, we may delete the i th equation in system (6) without changing its solution set, where i ∈ I = {i ∈ I |bi = 0}. As a result, system (6) in this case can be solved by the mentioned bipolar path approach. However, after computing the lower and upper bounds, we should first check the value of b. If some of the elements in {b1, b2 , · · · , bm } are equal to zero, the corresponding equations should be deleted from system (6) before applying the bipolar path approach. 4. Path-based algorithm and illustrative examples 4.1. Path-Based Algorithm To perform the presented bipolar path approach to solve system (6), we develop the following path-based algorithm step by step.

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Path-Based Algorithm (to obtain the solution set of system (6)) Step 1. Compute the lower bound x˙ˇ and upper bound x˙ˆ by (12) and (13). ˙ˆ If x˙ˇ ≤ x, ˙ˆ go to Step 3. Otherwise, if x˙ˇ > x, ˙ˆ according to Corollary 1, system (6) has no Step 2. Compare x˙ˇ and x. solution, and we stop. ˙ˇ x] ˙ˆ and stop. If Step 3. Check the value of components in b. If bi = 0 for all i ∈ I , then the solution set of (6) is [x, bi > 0 for all i ∈ I , continue to Step 4. Otherwise, delete the i th equation in system (6) for any i ∈ I = {i ∈ I |bi = 0}, and continue to Step 4 (for convenience, we reset I := I − I ). Step 4. Compute the index sets Ji−+ by (16)-(18) for each i ∈ I . ⊆ Ji−+ Step 5. Apply Rule 1 to reduce the computation. If there exist some i , i

∈ I such that Ji−+

, then omit the

i

th index set Ji−+

. Adjust the subscript, and reset I := I − {i }. Step 6. Find all conservative bipolar paths using (20). Denote the set of all conservative bipolar paths by P . If P = ∅, go to Step 7. Otherwise, if P = ∅, following Theorem 4, system (6) has no solution, and we stop. Step 7. Delete all reducible conservative bipolar paths in P according to Rule 2. Assume that P turns out to be P after deleting.

Step 8. For each irreducible conservative bipolar path p in P , compute the induced solution interval Xp =

[xˇ p , xˆ p ] by (25) and (26). Step 9. According to Theorems 5 and 6, generate the complete solution set of system (6) by

X(A+ , A− , b) = [xˇ p , xˆ p ], p ∈P

and we stop. • Computational complexity of the Path-Based Algorithm Assume that m represents the number of equations in system (6), n represents the number of variables, while k is the number of conservative bipolar paths. ˙ˆ Hence In Step 1, computation of the lower bound x˙ˇ costs 2mn operations. It is the same for the upper bound x. ˙ ˙ Step 1 costs 4mn operations. In Step 2, comparison of xˇ and xˆ costs n operations. Step 3 takes m operations for checking the value of components in b. Step 4 costs 9mn operations for computation of the index sets. Comparison of each pair of the index sets in Step 5 cost n2 m(m − 1) operations. In Step 6, in order to obtain a conservative bipolar path p = (p1 , p2 , · · · , pm ), we should find out all its components. Finding p1 costs n operations, while finding pi costs 6mn operations, where i = 2, 3, · · · , m. Hence in Step 6 it totally costs [n + 6mn(m − 1)]k operations. In Step 7, checking the inclusion relationship of each pair of conservative bipolar paths costs m2 operations. There are k(k−1) pairs of conservative bipolar paths we have to check. Hence the total computation is m2 k(k−1) in this step. 2 2

Computational operations of xˇ p and xˆ p are 2mn and 2mn respectively. Thus Step 8 costs 4mn operations. At last, Step 9 costs k − 1 operations to merge the solution intervals. As a result, all the 9 steps in our proposed Path-Based Algorithm costs k(k − 1) + 4mn + (k − 1) 2 1 1 =m + n + k − 1 + 17mn + nk + mn2 − 6mnk − m2 k + m2 n2 + m2 k 2 + 6m2 nk 2 2 operations. The computational complexity of the Path-Based Algorithm is O(m2 n2 + m2 k 2 + m2 nk). The flow chart of the Path-Based Algorithm is shown in Fig. 1. 4mn + n + m + 9mn + n2 m(m − 1) + [n + 6mn(m − 1)]k + m2

4.2. Illustrative examples Example 2. Considering the following system ⎧ ⎪ ⎨max{T (0.4, x1 ), T (0.3, x¯1 ), T (0.4, x2 ), T (0.2, x¯2 ), T (0.3, x3 ), T (0.3, x¯3 )} = 0, max{T (0.2, x1 ), T (0.4, x¯1 ), (T (0.3, x2 ), T (0.3, x¯2 ), T (0.3, x3 ), T (0.2, x¯3 )} = 0, ⎪ ⎩ max{T (0.4, x1 ), T (0.2, x¯1 ), (T (0.4, x2 ), T (0.4, x¯2 ), T (0.2, x3 ), T (0.5, x¯3 )} = 0,

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Fig. 1. Flow chart of the Path-Based Algorithm.

where T (a0 , x0 ) = max{a0 + x0 − 1} for any a0 , x0 ∈ [0, 1], and x¯j = 1 − xj , j = 1, 2, 3. Solution: Step 1. By computing the lower bound x˙ˇ and upper bound x˙ˆ by (12) and (13), we obtain x˙ˇ = (0.3, 0.4, 0.5), x˙ˆ = (0.6, 0.6, 0.7). ˙ˆ moreover, b1 = b2 = b3 = 0. Hence, the solution set of system (44) is [x, ˙ˇ x] ˙ˆ = Steps 2-3. It is obvious that x˙ˇ < x; ([0.3, 0.6], [0.4, 0.6], [0.5, 0.7]). Example 3. Compute the solution set of system (21) in Example 1.

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Solution: Steps 1-5. Following the result in Example 1, the lower bound and upper bounds are x˙ˇ = (0.1, 0.3, 0.2, 0.2) and ˙ˆ For any i ∈ I = {1, 2, 3, 4}, it holds that bi > 0. Moreover, x˙ˆ = (0.9, 0.9, 0.9, 0.8), respectively. Obviously, x˙ˇ < x. ⊆ Ji−+ according to (22), there does not exist any i , i

∈ I such that Ji−+

. Hence, we can continue to Step 6. 1 2 Step 6. The set of all conservative bipolar paths is P = {p , p , · · · , p 8 }, where p l is as shown in (24), l = 1, 2, · · · , 8. Step 7. Delete all reducible conservative bipolar paths in P according to Rule 2. Since {p11 , p21 , p31 , p41 } = {1− , 2+ , 3− } ⊆ {1− , 2+ , 3− , 4− } = {p12 , p22 , p32 , p42 }, and {p18 , p28 , p38 , p48 } = {2− , 3+ , 4+ } ⊆ {1− , 2− , 3+ , 4+ } = {p14 , p24 , p34 , p44 }, the paths p 2 and p 4 are reducible. By deleting p 2 and p 4 from P , we obtain the irreducible set P = {p 1 , p 3 , p 5 , p 6 , p 7 , p 8 }.

Step 8. For each irreducible conservative bipolar path p in P , compute the induced solution interval Xp =

p p [xˇ , xˆ ] by (25) and (26). 1

1

3

3

5

5

6

6

p7

p7

p8

p8

[xˇ p , xˆ p ] = (0.1, 0.9, 0.2, [0.2, 0.8]), [xˇ p , xˆ p ] = (0.1, [0.3, 0.9], 0.2, 0.8), [xˇ p , xˆ p ] = (0.9, 0.9, 0.2, [0.2, 0.8]), [xˇ p , xˆ p ] = (0.9, 0.3, [0.2, 0.9], 0.8),

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[xˇ , xˆ ] = (0.9, [0.3, 0.9], 0.2, 0.8), [xˇ , xˆ ] = ([0.1, 0.9], 0.3, 0.9, 0.8). Step 9. According to Theorem 6, the complete solution set of system (21) is X(A+ , A− , b) =(0.1, 0.9, 0.2, [0.2, 0.8]) ∪ (0.1, [0.3, 0.9], 0.2, 0.8) ∪ (0.9, 0.9, 0.2, [0.2, 0.8]) ∪ (0.9, 0.3, [0.2, 0.9], 0.8) ∪ (0.9, [0.3, 0.9], 0.2, 0.8) ∪ ([0.1, 0.9], 0.3, 0.9, 0.8). Example 4. [14] Find the solution set of the following system of bipolar fuzzy relation equations with the maxŁukasiewicz composition T

− T T 2 A+ 2 ◦ x ∨ A2 ◦ x¯ = b ,

where

⎡

0.18 ⎢ 0.23 ⎢ ⎢ 0.75 ⎢ ⎢ 0.43 ⎢ ⎢ 0.70 ⎢ A+ = 2 ⎢ 0.65 ⎢ ⎢ 0.42 ⎢ ⎢ 0.82 ⎢ ⎣ 0.35 0.45 ⎡ 0.23 ⎢ 0.13 ⎢ ⎢ 0.85 ⎢ ⎢ 0.28 ⎢ ⎢ 0.80 − A2 = ⎢ ⎢ 0.57 ⎢ ⎢ 0.54 ⎢ ⎢ 0.74 ⎢ ⎣ 0.41 0.58

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0.15 0.56 0.90 0.56 0.72 0.82 0.43 0.61 0.68 0.46

0.12 0.71 0.76 0.72 0.45 0.72 0.58 0.67 0.43 0.48

0.25 0.62 0.32 0.57 0.54 0.61 0.70 0.65 0.76 0.36

0.22 0.80 0.95 0.81 0.70 0.53 0.67 0.80 0.64 0.70

0.35 0.93 0.61 0.19 0.90 0.78 0.80 0.63 0.55 0.45

0.21 0.45 0.49 0.80 0.34 0.82 0.33 0.54 0.45 0.52

0.12 0.43 0.64 0.38 0.46 0.62 0.45 0.76 0.25 0.32

0.20 0.46 0.98 0.41 0.80 1.00 0.55 0.53 0.74 0.59

0.17 0.61 0.86 0.57 0.55 0.64 0.70 0.59 0.49 0.61

0.30 0.52 0.42 0.96 0.64 0.53 0.82 0.57 0.82 0.49

0.27 0.70 1.00 0.66 0.80 0.45 0.79 0.72 0.70 0.83

0.40 0.83 0.71 0.04 1.00 0.70 0.92 0.55 0.61 0.58

0.26 0.35 0.59 0.65 0.44 0.74 0.45 0.46 0.51 0.65

0.17 0.33 0.74 0.23 0.56 0.54 0.57 0.68 0.31 0.45

⎤ 0.31 0.38 ⎥ ⎥ 0.68 ⎥ ⎥ 0.47 ⎥ ⎥ 0.63 ⎥ ⎥, 0.72 ⎥ ⎥ 0.26 ⎥ ⎥ 0.42 ⎥ ⎥ 0.80 ⎦ 0.77 ⎤ 0.36 0.28 ⎥ ⎥ 0.78 ⎥ ⎥ 0.32 ⎥ ⎥ 0.73 ⎥ ⎥, 0.64 ⎥ ⎥ 0.38 ⎥ ⎥ 0.34 ⎥ ⎥ 0.86 ⎦ 0.90

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Table 1 All conservative bipolar paths. p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20

(3+ , 3+ , 2− , 1+ , 4+ , 5− ) (3+ , 3+ , 2− , 1+ , 4+ , 5+ ) (3+ , 3+ , 2− , 1+ , 4+ , 9− ) (3+ , 3+ , 2− , 1+ , 4+ , 9+ ) (3+ , 3+ , 2− , 1+ , 9+ , 9+ ) (3+ , 3+ , 2− , 8+ , 4+ , 5− ) (3+ , 3+ , 2− , 8+ , 4+ , 5+ ) (3+ , 3+ , 2− , 8+ , 4+ , 9− ) (3+ , 3+ , 2− , 8+ , 4+ , 9+ ) (3+ , 3+ , 2− , 8+ , 9+ , 9+ ) (5+ , 5+ , 2− , 1+ , 4+ , 5+ ) (5+ , 5+ , 2− , 1+ , 9+ , 5+ ) (5+ , 5+ , 2− , 8+ , 4+ , 5+ ) (5+ , 5+ , 2− , 8+ , 9+ , 5+ ) (6+ , 3+ , 2− , 1+ , 4+ , 5− ) (6+ , 3+ , 2− , 1+ , 4+ , 5+ ) (6+ , 3+ , 2− , 1+ , 4+ , 9− ) (6+ , 3+ , 2− , 1+ , 4+ , 9+ ) (6+ , 3+ , 2− , 1+ , 9+ , 9+ ) (6+ , 3+ , 2− , 8+ , 4+ , 5− )

p21 p22 p23 p24 p25 p26 p27 p28 p29 p30 p31 p32 p33 p34 p35 p36 p37 p38 p39 p40

(6+ , 3+ , 2− , 8+ , 4+ , 5+ ) (6+ , 3+ , 2− , 8+ , 4+ , 9− ) (6+ , 3+ , 2− , 8+ , 4+ , 9+ ) (6+ , 3+ , 2− , 8+ , 9+ , 9+ ) (6+ , 4− , 2− , 1+ , 9+ , 9+ ) (6+ , 4− , 2− , 8+ , 9+ , 9+ ) (6+ , 5+ , 2− , 1+ , 4+ , 5+ ) (6+ , 5+ , 2− , 1+ , 9+ , 5+ ) (6+ , 5+ , 2− , 8+ , 4+ , 5+ ) (6+ , 5+ , 2− , 8+ , 9+ , 5+ ) (6+ , 7+ , 2− , 1+ , 4+ , 5− ) (6+ , 7+ , 2− , 1+ , 4+ , 5+ ) (6+ , 7+ , 2− , 1+ , 4+ , 9− ) (6+ , 7+ , 2− , 1+ , 4+ , 9+ ) (6+ , 7+ , 2− , 1+ , 9+ , 9+ ) (6+ , 7+ , 2− , 8+ , 4+ , 5− ) (6+ , 7+ , 2− , 8+ , 4+ , 5+ ) (6+ , 7+ , 2− , 8+ , 4+ , 9− ) (6+ , 7+ , 2− , 8+ , 4+ , 9+ ) (6+ , 7+ , 2− , 8+ , 9+ , 9+ )

and b2 = (0.00, 0.55, 0.70, 0.56, 0.52, 0.72, 0.42, 0.64, 0.48, 0.45), x = (x1 , x2 , · · · , x9 ), x¯ = 1 − x. Solution: Step 1. Compute the lower bound x˙ˇ and upper bound x˙ˆ by (12) and (13) (see [14]). x˙ˇ = (0.28, 0.28, 0.28, 0.40, 0.38, 0.50, 0.26, 0.17, 0.45), x˙ˆ = (0.82, 0.80, 0.84, 0.72, 0.75, 0.62, 0.76, 0.88, 0.68). ˙ˆ Thus, proceed to Step 3. Step 2. Obviously, it holds that x˙ˇ ≤ x. Step 3. Check the value of components in b. Since b1 = 0, we delete the 1th equation in system (46). Reset I = {1, 2, · · · , 9}, and continue to Step 4. Step 4. Compute the index sets Ji−+ by (16)-(18) for each i ∈ I . J1−+ = {3+ , 5+ , 6+ },

J2−+ = {2− , 2+ , 5+ },

J3−+ = {3+ , 4− , 5+ , 7+ },

J4−+ = {1− , 1+ , 2− , 2+ , 6+ },

J5−+ = {2− },

J6−+ = {3− , 3+ , 4− , 4+ , 5+ , 6− , 6+ }, J7−+ = {1+ , 8+ },

J8−+ = {2+ , 4+ , 9+ },

J9−+ = {5− , 5+ , 9− , 9+ }. Step 5. Apply Rule 1 to reduce the computation. Since J5−+ ⊆ J2−+ , J5−+ ⊆ J4−+ and J1−+ ⊆ J6−+ , the index sets J2−+ , J4−+ and J6−+ can be omitted. Reset I = {1, 2, · · · , 6}, and denote J1−+ = {3+ , 5+ , 6+ }, J2−+ = {3+ , 4− , 5+ , 7+ }, J3−+ = {2− }, J4−+ = {1+ , 8+ }, J5−+ = {2+ , 4+ , 9+ }, J6−+ = {5− , 5+ , 9− , 9+ }. Step 6. Find out all conservative bipolar paths by (20). We obtain the conservative bipolar path set P = {p 1 , p 2 · · · , p 40 } as shown in Table 1.

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Table 2 All of the irreducible conservative bipolar paths. p1 p3 p5 p6 p7 p8 p10 p11 p12 p13

(3+ , 3+ , 2− , 1+ , 4+ , 5− ) (3+ , 3+ , 2− , 1+ , 4+ , 9− ) (3+ , 3+ , 2− , 1+ , 9+ , 9+ ) (3+ , 3+ , 2− , 8+ , 4+ , 5− ) (3+ , 3+ , 2− , 8+ , 4+ , 5+ ) (3+ , 3+ , 2− , 8+ , 4+ , 9− ) (3+ , 3+ , 2− , 8+ , 9+ , 9+ ) (5+ , 5+ , 2− , 1+ , 4+ , 5+ ) (5+ , 5+ , 2− , 1+ , 9+ , 5+ ) (5+ , 5+ , 2− , 8+ , 4+ , 5+ )

p14 p25 p26 p31 p33 p35 p36 p38 p40

(5+ , 5+ , 2− , 8+ , 9+ , 5+ ) (6+ , 4− , 2− , 1+ , 9+ , 9+ ) (6+ , 4− , 2− , 8+ , 9+ , 9+ ) (6+ , 7+ , 2− , 1+ , 4+ , 5− ) (6+ , 7+ , 2− , 1+ , 4+ , 9− ) (6+ , 7+ , 2− , 1+ , 9+ , 9+ ) (6+ , 7+ , 2− , 8+ , 4+ , 5− ) (6+ , 7+ , 2− , 8+ , 4+ , 9− ) (6+ , 7+ , 2− , 8+ , 9+ , 9+ )

Step 7. Delete all reducible conservative bipolar paths in P . Then, we obtain the irreducible set of P , i.e., P = {p 1 , p 3 , p 5 , p 6 , p 7 , p 8 , p 10 , p 11 , p 12 , p 13 , p 14 , p 25 , p 26 , p 31 , p 33 , p 35 , p 36 , p 38 , p 40 }, as shown in Table 2.

Step 8. By (25) and (26), compute the induced solution interval X p = [xˇ p , xˆ p ] for each p ∈ P . 1

X p = (0.82, 0.28, 0.84, 0.72, 0.38, [0.5, 0.62], [0.26, 0.76], [0.17, 0.88], [0.45, 0.68]), 3

X p = (0.82, 0.28, 0.84, 0.72, [0.38, 0.75], [0.5, 0.62], [0.26, 0.76], [0.17, 0.88], 0.45), 5

X p = (0.82, 0.28, 0.84, [0.4, 0.72], [0.38, 0.75], [0.5, 0.62], [0.26, 0.76], [0.17, 0.88], 0.68), 6

X p = ([0.28, 0.82], 0.28, 0.84, 0.72, 0.38, [0.5, 0.62], [0.26, 0.76], 0.88, [0.45, 0.68]), 7

X p = ([0.28, 0.82], 0.28, 0.84, 0.72, 0.75, [0.5, 0.62], [0.26, 0.76], 0.88, [0.45, 0.68]), 8

X p = ([0.28, 0.82], 0.28, 0.84, 0.72, [0.38, 0.75], [0.5, 0.62], [0.26, 0.76], 0.88, 0.45), 10

X p = ([0.28, 0.82], 0.28, 0.84, [0.4, 0.72], [0.38, 0.75], [0.5, 0.62], [0.26, 0.76], 0.88, 0.68), 11

X p = (0.82, 0.28, [0.28, 0.84], 0.72, 0.75, [0.5, 0.62], [0.26, 0.76], [0.17, 0.88], [0.45, 0.68]), 12

X p = (0.82, 0.28, [0.28, 0.84], [0.4, 0.72], 0.75, [0.5, 0.62], [0.26, 0.76], [0.17, 0.88], 0.68), 13

X p = ([0.28, 0.82], 0.28, [0.28, 0.84], 0.72, 0.75, [0.5, 0.62], [0.26, 0.76], 0.88, [0.45, 0.68]), 14

X p = ([0.28, 0.82], 0.28, [0.28, 0.84], [0.4, 0.72], 0.75, [0.5, 0.62], [0.26, 0.76], 0.88, 0.68), 25

X p = (0.82, 0.28, [0.28, 0.84], 0.4, [0.38, 0.75], 0.62, [0.26, 0.76], [0.17, 0.88], 0.68), 26

X p = ([0.28, 0.82], 0.28, [0.28, 0.84], 0.4, [0.38, 0.75], 0.62, [0.26, 0.76], 0.88, 0.68), 31

X p = (0.82, 0.28, [0.28, 0.84], 0.72, 0.38, 0.62, 0.76, [0.17, 0.88], [0.45, 0.68]), 33

X p = (0.82, 0.28, [0.28, 0.84], 0.72, [0.38, 0.75], 0.62, 0.76, [0.17, 0.88], 0.45), 35

X p = (0.82, 0.28, [0.28, 0.84], [0.4, 0.72], [0.38, 0.75], 0.62, 0.76, [0.17, 0.88], 0.68), 36

X p = ([0.28, 0.82], 0.28, [0.28, 0.84], 0.72, 0.38, 0.62, 0.76, 0.88, [0.45, 0.68]), 38

X p = ([0.28, 0.82], 0.28, [0.28, 0.84], 0.72, [0.38, 0.75], 0.62, 0.76, 0.88, 0.45), 40

X p = ([0.28, 0.82], 0.28, [0.28, 0.84], [0.4, 0.72], [0.38, 0.75], 0.62, 0.76, 0.88, 0.68). Step 9. According to Theorem 6, the complete solution set of system (46) is

X(A+ , A− , b) = Xp , p ∈P

where P = {p 1 , p 3 , p 5 , p 6 , p 7 , p 8 , p 10 , p 11 , p 12 , p 13 , p 14 , p 25 , p 26 , p 31 , p 33 , p 35 , p 36 , p 38 , p 40 }.

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Fig. 2. Publicity of 4 products supplying to 4 markets.

Example 5. Consider a supplier, who want to supply four products to four markets (see Fig. 2). The degrees of appreciation of these four products are x1 , x2 , x3 and x4 , while the degrees of disappreciation are x¯1 , x¯2 , x¯3 and x¯4 , respectively. The effective public awareness of pj at market Ai is max{max{xj − aij }, max{x¯j − a¯ ij }}, where i ∈ {1, 2, 3, 4}, j ∈ {1, 2, 3, 4}. Moreover, the values of aij and a¯ ij are as shown in the following matrices A = (aij ) and A¯ = (a¯ ij ). ⎡ ⎤ 0.1 0.2 0.1 0.4 ⎢ 0.2 0.1 0.3 0 ⎥ ⎥ (47) A = (aij )4×4 = ⎢ ⎣ 0.2 0.4 0.2 0.6 ⎦ , 0.5 0.4 0.6 0.6 ⎡ ⎤ 0.1 0.3 0.2 0.1 ⎢ 0.3 0.1 0.3 0.2 ⎥ ⎥ (48) A¯ = (a¯ ij )4×4 = ⎢ ⎣ 0.4 0.2 0.1 0.1 ⎦ . 0.4 0.1 0.2 0.6 On the other hand, the best public awareness for product sales at markets A1 , A2 , A3 , A4 are b1 = 0.8, b2 = 0.8, b3 = 0.7, b4 = 0.6. In order to achieve best publicity effects, there should be at least one product, whose effective public awareness reaches the best public awareness at each market. Now we aim to find out all feasible degrees of appreciation, on which the supplier have to publicize its products. Solution: Denote A+ = (1 − aij ), A− = (1 − a¯ ij ) and b = (b1 , b2 , b3 , b4 ), i.e. ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0.9 0.8 0.9 0.6 0.9 0.7 0.8 0.9 0.8 ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ 0.8 0.9 0.7 1.0 0.7 0.9 0.7 0.8 − ⎢ ⎥ ⎥ T ⎢ 0.8 ⎥ A+ = ⎢ ⎣ 0.8 0.6 0.8 0.4 ⎦ , A = ⎣ 0.6 0.8 0.9 0.9 ⎦ , b = ⎣ 0.7 ⎦ . 0.5 0.6 0.4 0.4 0.6 0.9 0.8 0.4 0.6 Then we get the bipolar max-Łukasiewicz fuzzy relation equations as system (21). According to Examples 1 and 3, the complete solution set is X(A+ , A− , b) =(0.1, 0.9, 0.2, [0.2, 0.8]) ∪ (0.1, [0.3, 0.9], 0.2, 0.8) ∪ (0.9, 0.9, 0.2, [0.2, 0.8]) ∪ (0.9, 0.3, [0.2, 0.9], 0.8) ∪ (0.9, [0.3, 0.9], 0.2, 0.8) ∪ ([0.1, 0.9], 0.3, 0.9, 0.8). Each solution in

X(A+ , A− , b)

represents a feasible degree of appreciation of the products.

5. Discussion 5.1. Global (or local) minimal (or maximal) solution • Phenomenon

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In the existing literature, the concept of “minimal solution” of a system of fuzzy relation equations or inequalities was defined as a special solution x, satisfying x ≤ xˇ ⇒ x = xˇ for arbitrary solution x. The concept of “maximal solution” could be defined in a similar way. However, the above classical defined concept of the “minimal (or maximal) solution” appears not suitable for a system of bipolar fuzzy relation equations. For example, in the previous Example 3, the solution of system (21) is X(A+ , A− , b) =[xˇ 1 , xˆ 1 ] ∪ [xˇ 2 , xˆ 2 ] ∪ [xˇ 3 , xˆ 3 ] ∪ [xˇ 4 , xˆ 4 ] ∪ [xˇ 5 , xˆ 5 ] ∪ [xˇ 6 , xˆ 6 ] =(0.1, 0.9, 0.2, [0.2, 0.8]) ∪ (0.1, [0.3, 0.9], 0.2, 0.8) ∪ (0.9, 0.9, 0.2, [0.2, 0.8]) ∪ (0.9, 0.3, [0.2, 0.9], 0.8) ∪ (0.9, [0.3, 0.9], 0.2, 0.8) ∪ ([0.1, 0.9], 0.3, 0.9, 0.8). The solutions xˇ 1 , xˇ 2 , · · · , xˇ 6 should be considered minimal solutions, whereas xˆ 1 , xˆ 2 , · · · , xˆ 6 are maximal solutions. However, xˇ 3 = (0.9, 0.9, 0.2, 0.2) does not satisfy the classical definition of a “minimal solution”, since there exists y = (0.1, 0.9, 0.2, 0.2) ∈ X(A+ , A− , b) such that y ≤ xˇ 3 and y = xˇ 3 . In addition, xˆ 1 = xˆ 2 = (0.1, 0.9, 0.2, 0.8) is not a “maximal solution” according to the classical definition, since there exists y

= (0.9, 0.9, 0.2, 0.8) ∈ X(A+ , A− , b) such that y

≥ xˇ 3 and y

= xˇ 3 . • Reason The described phenomenon is a result of the following property, which is valid for the classical unipolar system of fuzzy relation equations but invalid for the bipolar system. Property 1. Let both x and x

be solutions of a system of unipolar fuzzy relation equations. Then, any x ∈ [0, 1]n that satisfies x ≤ x ≤ x

is still a solution of the system. Considering Property 1, Example 3 can be demonstrated as a counterexample. Take xˇ 3 = (0.9, 0.9, 0.2, 0.2) and = (0.1, 0.9, 0.2, 0.2) in X(A+ , A− , b). It is easy to check that (0.5, 0.9, 0.2, 0.2) ∈ [y , xˇ 3 ] is not a solution of system (21).

y

• New definition To avoid the described phenomenon, we provide new definitions of minimal and maximal solutions in system (6), following the idea in analytics. Definition 7. In system (6), a solution xˇ ∈ X(A+ , A− , b) is a local minimal solution, if there exists a positive real number ε ∈ R + such that xˇ − (ε, ε, · · · , ε) ≤ x ≤ xˇ indicates x = xˇ for any x ∈ X(A+ , A− , b). A solution xˇ ∈ X(A+ , A− , b) is a global minimal solution, if x ≤ xˇ indicates x = xˇ for any x ∈ X(A+ , A− , b). The concept of the local (or global) maximal solution can be similarly defined. According to Definition 7, the following Remarks 1 and 2 can be easily obtained. Remark 1. The definition of the global minimal (or maximal) solution in the bipolar system (6) is equivalent to that of the “minimal (or maximal) solution” in the unipolar fuzzy relation equations. Remark 2. In system (6), a global minimal (or maximal) solution is also a local minimal (or maximal) solution. However, in reverse, a local minimal (or maximal) solution may not be a global one. Still, in Example 3, following Definition 7, it is not difficult to check that xˇ 1 = (0.1, 0.9, 0.2, 0.2) and xˇ 2 = (0.1, 0.3, 0.2, 0.8) are both global minimal solutions, and xˇ 3 = (0.9, 0.9, 0.2, 0.2), xˇ 4 = xˇ 5 = (0.9, 0.3, 0.2, 0.8) and xˇ 6 = (0.1, 0.3, 0.9, 0.8) are local minimal solutions. Meanwhile, xˆ 1 = xˆ 2 = (0.1, 0.9, 0.2, 0.8) is a local maximal solution, whereas xˆ 3 = xˆ 5 = (0.9, 0.9, 0.2, 0.8) and xˆ 4 = xˆ 6 = (0.9, 0.3, 0.9, 0.8) are global maximal solutions.

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5.2. Structure of the solution set of system (6) In this subsection, we compare the solution set of a bipolar system (6) to that of a unipolar one. The solution set of a unipolar system of fuzzy relation equations or inequalities with the max-t-norm composition is well-known to be commonly determined by a unique maximum (greatest) solution and a finite number of minimal solutions. Denote the solution set of a unipolar system by X(A, b). Then, X(A, b) may be represented by a union of a finite number of closed intervals when it is nonempty. For bipolar system (6), it is much more difficult to obtain its solution set X(A+ , A− , b), although X(A+ , A− , b) is also a union of a finite number of closed intervals. Unlike a unipolar system, its maximal solutions are not unique. In general, the consistent system (6) has a finite number of maximal solutions and minimal solutions. Moreover, these maximal or minimal solutions are different from the classical ones. As noted in Subsection 5.1, the minimal solutions are divided into global minimal solutions and local minimal solutions. The maximal solutions could be similarly classified. 6. Conclusion The structure and resolution method of the solution set for a system of bipolar fuzzy relation equations is important and fundamental for theoretical research in this field. It was one of the main research topics for system of FREs. It has been shown in [11,14] that the optimal solution of the linear optimization problem with bipolar max-Łukasiewicz fuzzy relation equations constraint could be selected from the maximal solutions or the minimal solutions of system (6). Hence, resolution of system (6), especially all the maximal and minimal solutions, appears to be important. Due to the practical application in publicity of products through public awareness as shown in Subsection 2.1, every solution of (6) is indeed a feasible degree of appreciation of the products. Resolution of system (6) provides the manager all the feasible strategies, in which he/she is able to select a best one according to specific objective. To obtain the complete solution set of the bipolar system, we define the concept of the conservative bipolar path and discuss the relevant properties. The corresponding relation between the conservative bipolar path and the solution interval is established. After obtaining all irreducible conservative bipolar paths, the solution set can be represented by a union of a finite number of induced solution intervals, i.e.,

X(A+ , A− , b) = [xˇ p , xˆ p ], p∈P

where P is the set of all (irreducible) conservative bipolar paths, and [xˇ p , xˆ p ] is the corresponding solution interval induced by p. Detailed resolution procedures are provided to compute the solution set and illustrated by several numerical examples. In addition, we define the concept of global and local minimal (or maximal) solutions of the bipolar fuzzy relation equations. It is shown that the concept of the global minimal (or maximal) solution is equivalent to the classical definition of “minimal (or maximal) solution”. Acknowledgements We would like to express our appreciation to the editor and the anonymous reviewers for their valuable comments, which have been very helpful in improving the paper. References [1] S. Aliannezhadi, S.S. Ardalan, A.A. Molai, Maximizing a monomial geometric objective function subject to bipolar max-product fuzzy relation constraints, J. Intell. Fuzzy Syst. 32 (2017) 337–350. [2] S. Aliannezhadi, A.A. Molai, B. Hedayatfar, Linear optimization with bipolar max-parametric Hamacher fuzzy relation equation constraints, Kybernetika 52 (4) (2016) 531–557. [3] E. Bartl, R. Belohlavek, Hardness of solving relational equations, IEEE Trans. Fuzzy Syst. 23 (6) (2015) 2435–2438. [4] S. Benferhat, D. Dubois, S. Kaci, H. Prade, Bipolar possibility theory in preference modeling: representation, fusion and optimal solutions, Inf. Fusion 7 (2006) 135–150. [5] M. Cornejo, D. Lobo, J. Medina, Bipolar fuzzy relation equations based on product t-norm, in: Proceedings of 2017 IEEE International Conference on Fuzzy Systems, 2017.

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