Interval Solution to Fuzzy Relation Inequality With Application in P2P Educational Information Resource Sharing Systems

Max-min fuzzy relation inequalities have recently been introduced to describe the peer-to-peer (P2P) educational information resource sharing systems. It is well known that the complete solution set of the max-min fuzzy relation system is fully determined by its minimal solutions. However, solving all the minimal solutions has been proven to be equivalent to the set-covering problem, which is NP-hard. Without solving the complete solution set, some specific solutions can be obtained through the corresponding fuzzy relation optimization problems. However, these solutions are usually unstable and fragile. Any minor fluctuations to the components of these specific solutions will cause them to be no longer feasible. In this work, we define and study the widest interval solution of a max-min fuzzy relation inequality system for the first time. The interval solution allows the solution to fluctuate within some range. The fluctuation range is determined by the width of the interval solution. We propose a novel resolution method for searching for the widest interval solution. The resolution method is carried out by detailed procedures and illustrated by a numerical example.


I. INTRODUCTION
In classical algebra, the composition operation is typical addition-multiplication. However, in various application fields, it has been found that the addition-multiplication composition is defective and unsuitable for modeling the quantitative relation. The max-min composition became an effective substitute for the addition-multiplication relation. Correspondingly, the relevant max-min algebra has attracted some scholars' attention [14]- [17], [35]. The max-min composition and mechanism have been widely applied in engineering management and control [8]- [13].
The max-min composition was introduced to the linear equation system by Sanchez [19], [20] for the first time. It was named max-min fuzzy relation equations. The structure of the solution set for a max-min fuzzy relation equation system was much different from that for a classical linear equation The associate editor coordinating the review of this manuscript and approving it for publication was Yiming Tang . system. It has been formally proven that the complete solution set of a consistent max-min system is a nonconvex set in most cases (when its minimal solutions are not unique). The solution set is fully determined by its unique maximum solution and a finite number of minimal solutions.
When first investigated by E. Sanchez, fuzzy relation equations were applied to medical diagnoses [20]. Fuzzy relation systems, including both equation systems and inequality systems, have been successfully applied for dealing with various kinds of practical problems, such as fuzzy infer-(see Fig. 1.). Assume the educational information resources stored in terminals, denoted by A 1 , A 2 , . . . , A n , are unbalanced. Each terminal is connected to any other terminal and free to download its required educational information resources. The bandwidth between the terminals A i and A j is assumed to be a ij . That is, when the ith terminal A i downloads its required resources from the jth terminal A j , the actual quality level is where x j (measure: Mbps) represents the quality level on which A j shares (sends out) its local resources. a ij ∧x j denotes the receiving quality level at A i from A j . In general, A i will select the terminal with the highest receiving quality level to download the resources. Additionally, we assume that the download traffic requirement is no less than b i and no more than d i . Then, the A i requirement can be written as Without loss of generality, we also assume that some of the terminals, i.e., {A 1 , A 2 , . . . , A m }, request their required resources for downloading. As a consequence, their requirements can be represented by the following max-min fuzzy relation inequalities after normalization: · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b m ≤ (a m1 ∧ x 1 where a ij , x j ∈ [0, 1], 0 < b i ≤ d i ≤ 1, i ∈ I , j ∈ J , and I = {1, 2, . . . , m}, J = {1, 2, . . . , n}. In [18], the authors investigated the approximate solution(s) of the max-min fuzzy relation equations system in cases where the system is inconsistent. In some existing works, the total deviations, i.e., the sum of all deviations with respect to each equation, were applied to characterize the approximate solution. Instead of the total deviations, the authors defined the approximate solution based on the largest deviation [18]. They also proposed a linear searching algorithm for obtaining the approximate solution of the inconsistent system. Different from the inconsistent system studied in [18], the above system (1) considered in this paper is assumed to be consistent. We investigate the interval solution of system (1) in this work.
The novelty and contribution of this work can be summarized as follows.
(i) In this paper, the fluctuation of a given solution to the max-min fuzzy relation inequality system, i.e., system (1), is considered for the first time.
(ii) We define the interval solution concept for system (1). The width of an interval solution is designed to characterize the corresponding fluctuation range. (iii) To maximize the solution fluctuation range, we define and investigate the widest interval solution of system (1). An effective and detailed resolution algorithm is proposed for searching the widest interval solution of system (1).
The remainder of this manuscript is organized as follows. Section 2 presents some existing concepts and results on max-min fuzzy relation inequalities. In Section 3, we define the widest interval solution and provide a sufficient and necessary condition for its existence. In Section 4, we propose a novel resolution method for obtaining the widest interval solution of system (1). We also give detailed resolution procedures and a numerical illustrative example. Section 5 concludes.

II. PRELIMINARY
System (1) can be written in its matrix form as . . , d m ) and ''•'' represents the max-min composition. Here, the order relation ≤ is defined as follows. For any x = (x 1 , x 2 , . . . , x n ), y = (y 1 , y 2 , . . . , y n ) ∈ [0, 1] n , x ≤ y means that x j ≤ y j for all j ∈ J . In what follows, the set of all solutions of system (1) is denoted by X (A, b, d), i.e., To check the consistency of system (1), we introduce the vectorx as follows. For arbitrary j ∈ J , let Furthermore, denotex = (x 1 ,x 2 , . . . ,x n ), wherê Then, we obtain the following Theorem 1. Theorem 1 ([5]- [7]): System (1) is consistent if and only if the above-defined vectorx is a solution of system (1). Moreover, if system (1) is consistent,x is exactly its unique maximum solution.
In fact, when system (1) is consistent, it always has a unique maximum solution and a finite number of minimal solutions. Moreover, the complete solution set of system (1) can be represented as indicated in the following Theorem 2 Theorem 2 ([5]- [7]): If system (1) is consistent, then its solution set can be represented by Here,x is its unique maximum solution, andX (A, b, d) denotes the set of all its minimal solutions.
As shown in Theorem 2, solving system (1) is equivalent to searching all its minimal solutions. However, it is difficult to obtain all the minimal solutions since they have been proven to be highly related to the set-covering problem [1]- [4]. Moreover, the number of minimal solutions to a consistent max-min fuzzy relation inequality (or equation) system is exponentially associated with the size of the system. It is hard and unnecessary to compute and represent the complete solution set of system (1).

III. WIDEST INTERVAL SOLUTION DEFINITION
As noted in the previous section, obtaining all the (minimal) solutions is difficult and unnecessary. Some researchers have paid attention to optimization problems subject to the fuzzy relation system [5], [6], [27], [35], [42]- [45]. By solving the corresponding fuzzy relation optimization problems, some specific solutions to the fuzzy relation system can be found. However, these specific solutions are usually unstable and fragile. When some minor fluctuation or perturbation occurs in any component of the specific solution, it might no longer be a max-min fuzzy relation system solution. To study the fluctuation in the solution of system (1), we define and investigate the widest interval solution in this work as follows.

Definition 3 (Width of an Interval
Proof: The proof is straightforward following Definition 3.
According to Theorem 2, x ≤ x .
(6) and (7) indicates Sincex (8), (9) and (10) It is indicated in Theorem 3 that when system (1) is consistent, the widest interval solution can be obtained by selecting it from the minimal solution set by pairwise comparison. However, as noted in the last section, obtaining the minimal solution set is hard to achieve. To overcome such hardness, we propose a novel resolution method to find the widest interval solution of system (1) in the following.

IV. RESOLUTION METHOD BASED ON THE MAXIMUM SOLUTION AND INDEX SETS
In this section, to obtain the widest interval solution of system (1), we propose a resolution method based on the maximum solution and some index sets.

A. THEORETICAL ANALYSIS AND PROOF
Based on the maximum solutionx, we define the following index sets: for i = 1, 2, . . . , m. Moreover, denote Proposition 2: If system (1) is consistent, then it holds that J i = ∅ for any i ∈ I , i.e., P = ∅. Proof: If system (1) is consistent, then it follows from Hence, for any i ∈ I , there exists some j i ∈ J such that a ij i ∧x j i ≥ b i . By (11), we have i.e., P = ∅.
Proof: Since [x , x ] is an interval solution andx is the maximum solution of system (1), it is obvious that Hence, for any i ∈ I , it holds that and there exists p i ∈ J such that Since x ≤x, we have It follows from (11) that p i ∈ J i , ∀i ∈ I . Thus, p = (p 1 , p 2 , . . . , p m ) ∈ P.
Next, we check that x p ≤ x . Take arbitrary j ∈ J . Case 1.
it is clear that p i = j. It follows from (19) that This implies that x j ≥ i∈I p j b i = x p j . Following Cases 1 and 2, it holds that x p j ≤ x j , ∀j ∈ J . So we obtain Inequalities (16) and (22) contribute to Furthermore, we define the following vector x p * based on p * . Denote Next, we verify that the above-defined vector x p * is exactly a solution of system (1). As a consequence, we obtain an interval solution of system (1) as [x p * ,x].
Lemma 1: For arbitrary i ∈ I and j ∈ J , it holds that a ij ∧ x j ≤ d i .
Proof: Take arbitrary j ∈ J . Case 1. If I j = ∅, then by (3) and (4),x j = 1 and a ij ≤ d i for all i ∈ I . So we have is an interval solution of system (1).
Proof: In fact, we only have to verify that x p * is a solution of system (1). Take arbitrary k ∈ I . Denote It follows from j k ∈ J k and (11) that a kj k ∧x j k ≥ b k . This indicates that Inequalities (27) and (28) contribute to For any j ∈ J , we further check the inequality that a kj ∧ x p * j ≤ d k in two cases. Case 1. If I Furthermore, according to (11), we havê This indicates that Following Lemma 1, Due to the arbitrariness of j, we obtain Considering inequalities (29) and (34), it is obvious that x p * is a solution of system (1). Hence, [x p * ,x] is an interval solution.
Theorem 6: The width of the interval solution [ where J p * = {p * 1 , p * 2 , . . . , p * m } ⊆ J . Proof: According to Definition 3, the width of the interval If j ∈ J − J p * , then j / ∈ J p * . This indicates that there does not exist any i ∈ I such that p * i = j. Note that I If j ∈ J p * , then there exists i ∈ I such that p * i = j. Thus, For any j ∈ J p * , since I p * j = ∅, it follows from (25) that x According to I p * j = {i ∈ I |p * i = j}, it holds for arbitrary j ∈ J p * that Hence, for any j ∈ J p * , we have Considering (37), (38) and (40), we further obtain Equalities (35), (36) and (41) contribute to The proof is complete.
Then, it holds that w[ Take arbitrary k ∈ J . Case 1. If k / ∈ J p * , i.e., k ∈ J − J p * , then according to the proof of Theorem 6, we have I p * k = ∅ and x p * k = 0. Hencê Case 2. If k ∈ J p * , then according to the proof of Theorem 6, we have I Denote It is clear that k, k ∈ J i . It follows from (24) that Considering (46)- (49), we havê Combining Cases 1 and 2, we havê . Theorem 8: Let x p * be defined by (23) and (25), andx be the maximum solution of system (1). Then, [x p * ,x] is the widest interval solution of (1).
Proof: According to Theorem 5, [x p * ,x] is an interval solution of system (1). Moreover, its width is Due to the arbitrariness of [x , x ], it follows from Definition 5 that [x p * ,x] is the widest interval solution of system (1).

B. RESOLUTION PROCEDURES
Based on the theoretical results presented in the previous subsection, we summarize the resolution procedures for the widest interval solution of system (1) as follows.
Step 3. Following Theorem 1, check the consistency of system (1) by the above-obtained vectorx. Ifx ∈ X (A, b, d), then system (1) is consistent and goes to the next step. Otherwise, system (1) is inconsistent and has no widest interval solution.
Step 8. Combining x p * andx, we find the widest interval solution of system (1) as [x p * ,x] according to Theorem 8.
Our proposed resolution procedures are represented in Fig. 2.

C. NUMERICAL EXAMPLE
Example 1: Assume a P2P educational information resource sharing system with six terminals is reduced into the following max-min fuzzy relation inequalities: system (53).
We aim to find the widest interval solution of the system (53).

Solution:
The matrix form of the above system (53) is Next, we attempt to find the widest interval solution of system (53) following our proposed resolution procedures.
Step  Step 2. Based on the index sets obtained in Step 1, we can compute the potential maximum solutionx by (4) as follows.
Step 3. It is easy to determine thatx satisfies all inequalities in (53). Thus,x ∈ X (A, b, d) is a solution of system (53). It follows from Theorem 1 that system (53) is consistent, and we continue to the next step.
Step  is the widest interval solution (see Fig. 3.) of system (53) with width 0.1.

V. COMPARISON WITH THE EXISTING WORKS
In this section, we compare our studied problems with those in the relevant existing works.
(i) In the existing works [18], [46], [47], the system of max-min fuzzy relation inequalities or equations was assumed to be inconsistent. Under this assumption, there was no solution for such a system. As a consequence, the so-called approximate solution was defined and studied for the inconsistent fuzzy relation system with max-min composition [18], [47].
(ii) Solving the complete solution set of max-min fuzzy relation inequalities or equations is always an important research topic [1], [19], [26], [35]. The solution set of a consistent max-min system can be represented by a union of a finite number of closed intervals. However, it has been verified that solving the complete solution set is hard in most cases [1], [4]. Instead of solving the complete solution set, we focus only on the widest interval solution of the max-min fuzzy relation system in this paper.
(iii) Some of the existing works focused on the minimal solution(s) of a fuzzy relation system [48]- [51]. It is well known that the set of all minimal solutions is exactly a finite set. However, the number of minimal solutions might increase exponentially associated with the increase in the problem size. Hence, it is not easy and it is unnecessary to obtain all the minimal solutions. In this paper, our resolution method successfully avoids obtaining all the minimal solutions.
(iv) Other works have attempted to search for the optimal solution of the optimization problem subject to fuzzy relation inequalities or equations [5]- [7], [27], [28], [35], [45]. In fact, these optimal solutions can be viewed as specific solutions to the fuzzy relation constraints. However, these optimal solutions are usually unstable and fragile. Any minor fluctuations in the components of these optimal solutions will make them no longer feasible. Any minor fluctuation is not permitted for the optimal solution. To overcome the frangibility of the optimal solution, this paper further studies the so-called interval solution, which allows the solution to fluctuate in a certain range. VOLUME 9, 2021

VI. CONCLUSION
In the existing works [18], [46], [47], the max-min fuzzy relation inequalities, i.e., system (1), were introduced to model a P2P sharing system. The authors only considered the inconsistent case and studied the approximate solution of the max-min fuzzy relation system. In theory, the complete solution of system (1) can be obtained; however, this is difficult, since it is equivalent to the set-covering problem, a famous NP-hard problem. Instead of solving system (1) completely, optimal solutions to some kinds of optimization problems with fuzzy relation constraints were investigated [5], [6], [27], [35], [42]- [45]. However, no perturbation was permitted to these optimal solutions. Minor fluctuations in their components will make them no longer optimal or even no longer feasible in the max-min fuzzy relation system. In this paper, we study the interval solution, which allows the solution to fluctuate to some degree. The fluctuation range is determined by the width of the interval solution. To maximize the allowable fluctuation range, we define and investigate the widest interval solution. A detailed resolution method is proposed to find the widest interval solution of system (1). The resolution procedures are illustrated by a numerical example. In the future, we will further extend the concept of the widest interval solution to other types of fuzzy relation systems. LEI ZHANG received the M.S. degree from Northeast Normal University, Changchun, China, in 2006. He is currently a Mathematical Teacher with Hanshan Normal University, Chaozhou, China. He has published more than 30 research articles. His research interests include mathematical education, mathematical modeling, and data-driven mathematical resources management. VOLUME 9, 2021