# Evaluation Model and Approximate Solution to Inconsistent Max-Min Fuzzy Relation Inequalities in P2P File Sharing System

Authors: Xiao-Peng Yang
Journal: Complexity
 (2019)

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## Abstract

Considering the requirement of biggest download speed of the terminals, a BitTorrent-like Peer-to-Peer (P2P) file sharing system can be reduced into a system of max-min fuzzy relation inequalities. In this paper we establish an evaluation model by the satisfaction degree, for comparing two arbitrary potential solutions of such system. Besides, based on the evaluation model, concept of approximate solution is defined. It is indeed a potential solution with highest satisfaction degree. Furthermore, effective algorithm is developed for obtaining the approximate solutions to inconsistent system. Numerical examples are provided to illustrate our proposed model and algorithm.


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## Body

## 1. Introduction

Fuzzy relation equation was first introduced by Sanchez [1] with application in medical diagnosis in Brouwerian logic [2]. The first proposed and commonly used composition in a fuzzy relation system is the classicalmax-min one. Later soon it was extended to the general max-T one, in which T represents a continuous triangular norm. The complete solution set of such max-T fuzzy relation equations or inequalities is completely determined by its unique greatest solution (also named maximum solution) and all its minimal solutions. A consistent system usually has a finite number of minimal solutions. Solving a consistent max-T fuzzy relation system is equivalent to finding all its minimal solutions. There exist many resolution methods for a consistent max-T fuzzy relation system [3–23].Inequality is a useful tool for describing various of quantitative relations in the real world [24–32]. It plays important role in mathematics. Fuzzy relation inequalities, as a special kind of inequalities, were also studied. It was recently found that a BitTorrent-like Peer-to-Peer (P2P) file sharing system could be reduced into a system of addition-min fuzzy relation inequalities [33–39]. Assume that there exist n terminals, i.e., A1,A2,…,An, in the file sharing system. Based on the BitTorrent-like P2P transmission mechanism, each terminal is able to receive (or download) file data from any other terminal. Meanwhile, the jth terminal sends out file data with quality level xj to any other terminal, j∈{1,2,…,n}. The bandwidth between terminals Ai and Aj is aij, i,j∈{1,2,…,n}. Due to the bandwidth limitation, the network traffic that Ai receives from Aj is actually aij∧xj, i,j∈{1,2,…,n}.In [33–36], the quality requirement of download traffic of Ai was considered as the total download speed that Ai received file data from A1,A2,…,An, i.e., (1)ai1∧x1+ai2∧x2+⋯+ain∧xn.Suppose the quality requirement of download traffic of Ai is at least bi. After normalization, the P2P file sharing system could be reduced into the following system of addition-min fuzzy relation inequalities:(2)a11∧x1+a12∧x2+⋯+a1n∧xn≥b1,a21∧x1+a22∧x2+⋯+a2n∧xn≥b2,⋮am1∧x1+am2∧x2+⋯+amn∧xn≥bm.where aij,xj∈[0,1], bi∈R+, i∈I,j∈J. Here I={1,2,…,m} and J={1,2,…,n} are two index sets.However in some cases, a file should be downloaded from another terminal in whole. That is to say, when a terminal plans to obtain a file which cannot be separated, it would choose only one other terminal to receive (download) the file. In this situation, it is more reasonable to consider the quality requirement of download traffic ofAi as the biggest (highest) download speed that Ai receives file data from A1,A2,…,An, i.e.,(3)ai1∧x1∨ai2∧x2∨⋯∨ain∧xn. Correspondingly the P2P file sharing system is reduced into the following max-min fuzzy relation inequalities:(4)a11∧x1∨a12∧x2∨⋯∨a1n∧xn≥b1,a21∧x1∨a22∧x2∨⋯∨a2n∧xn≥b2,⋮am1∧x1∨am2∧x2∨⋯∨amn∧xn≥bm.The matrix form of system (4) is(5)A∘xT≥bT, where A=(aij)m×n∈[0,1]m×n,x=(x1,x2,…,xn)∈[0,1]n,b=(b1,b2,…,bm)∈(0,1]m.In most of the existing works, theoretical results of fuzzy relation equation or inequality depended on the assumption that the system was consistent. However, as pointed out in [40], this assumption is often not the case in practical applications. Hence investigation on the inconsistent system of fuzzy relation equations or inequalities is necessary and important. For a consistent system, the major objective is usually to obtain its solution(s), while for an inconsistent system, the major objective is often to find its approximate solution(s). Approximate solution to inconsistent system of fuzzy relation equations was studied by Pedrycz [41] for the first time. Modified Newton method was applied to find the approximate solution. Another method based on the solvability index was also proposed by Gottwald and Pedrcz [42] to deal with such problem. However, referring to the arguments in the work of Klir and Yuan [43], this method is rather inefficient and may lead to a trivial solution. In [43], quality index was adopted to measure the goodness of an approximate solution. In recent years other efficient methods were proposed to solve the approximate solution, based on the goodness measured by Euclidean distance [44], i.e.,(6)DE=1n∑i∈I⋁j∈Jaij∧xj-bi21/2,or by Hamming distance [40, 45, 46], i.e.,(7)DH=∑i∈I⋁j∈Jaij∧xj-bi.However, as we know there is no existing literature investigating the approximate solution to inconsistent system of fuzzy relation inequalities. In this paper we aim to establish an evaluation model to max-min fuzzy relation inequalities which describe the P2P file sharing system. In such evaluation model, we will develop effective method for evaluating any potential solution. Moreover, based on the evaluation model we may further define concept of approximate solution to inconsistent system of max-min fuzzy relation inequalities.The rest of the paper is organized as follows. In Section2 we provide some necessary results on max-min fuzzy relation inequality. In Section 3 an evaluation model is established to compare two arbitrary potential solutions, considering the application background in BitTorrent-like P2P file sharing system. Based on such evaluation model, approximate solution of inconsistent system of max-min fuzzy relation inequalities is investigated in Section 4, with step-by-step algorithm and numerical illustrative example. Simple conclusion is set in Section 5.


## 2. Preliminaries

In this section we present some basic definitions and results related to max-min fuzzy relation inequality.DenoteX=[0,1]n. We call any x∈X a potential solution of system (4). Correspondingly, X is called the potential solution set.Definition 1 (See [36]).
Letx1=(x11,x21,…,xn1), x2=(x12,x22,…,xn2)∈X; we define
(i)x1≤x2 if xj1≤xj2, ∀j∈J;
(ii)x1<x2 if x1≤x2 and there are some j∈J such that xj1<xj2.In what follows we shall denote the dual of order relation ‘<’ and ‘≤’ by the symbol ‘>’ and ‘≥’, respectively. Obviously, the operator ‘≤’ forms a partial order relation onX and (X,≤) becomes a lattice [34, 36].LetX(A,b)={x∈X∣A∘xT≥bT} be the solution set of system (4).Definition 2.
System (4) is said to be consistent (or compatible) if X(A,b)≠∅. Otherwise, it is said to be inconsistent.Definition 3.
A solutionx^∈X(A,b) is said to be the maximum (or greatest) solution of system (4) if and only if x≤x^ for all x∈X(A,b). A solution xˇ∈X(A,b) is said to be a minimal (or lower) solution of system (4) if and only if x≤xˇ implies x=xˇ for any x∈X(A,b).System (4) can be written as(8)ai1∧x1∨ai2∧x2∨⋯∨ain∧xn≥bi,i∈I.For given i∈I, we define(9)Ji=j∣aij≥bi.Theorem 4.
For system (4), the following statements are equivalent:(i)
system (4) is consistent;(ii)
⋁j∈Jaij≥bi holds for any i∈I;(iii)
Ji≠∅ holds for any i∈I;(iv)
1^=(1,1,…,1)∈X is a solution of system (4).Proof.
The proof is trivial.Remark 5 (see [47]).
When system (4) is consistent, the vector 1^=(1,1,…,1)∈X is always the unique maximum solution of system (4).Proposition 6.
Ifx∈X(A,b), then x′∈X(A,b) for any x′ satisfying x≤x′≤x^, i.e., [x,x^]⊆X(A,b).Proof.
The proof is trivial.It is shown in [48] that there exist a finite number of minimal solutions to system (4) when it is consistent. We denote the set of all minimal solutions of system (4) by Xˇ(A,b).Theorem 7 (see [47, 48]).
If system (4) is consistent, then the solution set of (4) is(10)XA,b=⋃xˇ∈XˇA,bx∣xˇ≤x≤x^,where Xˇ(A,b) is the minimal solution set and x^=1^=(1,1,…,1)∈X is the maximum solution.It is expressed in Theorem7 that the solution set of system (4) is completely determined by a unique maximum solution and a finite number of minimal solutions when it is consistent. It follows from Theorem 7 that the unique maximum solution x^=1^ can be easily obtained. On the other hand, the minimal solutions can be solved by the conservative path method as presented in [48].


## 3. Evaluation Model to a System of Max-Min Fuzzy Relation Inequalities

Obviously, each potential solution of system (4) represents a scheme of quality level on which the terminals send file data. In this section, our purpose is to establish a rational evaluation model for comparing the superiorities of any two potential solutions.As known to everybody, a solution of system (4) is a vector satisfying all the inequalities in (4). However, an arbitrary potential solution might not satisfy all the inequalities. The superiority of a potential solution depends on the degree that satisfies system (4). We give the concept of satisfaction degree below, based on which we can set up the evaluation model.Definition 8 (satisfaction degree).
Letx=(x1,x2,…,xn)∈X be a potential solution of system (4). Then di(x) is said to be the satisfaction degree function of x with respect to the ith inequality, where(11)dix=ai1∧x1∨ai2∧x2∨⋯∨ain∧xnbi,ifai1∧x1∨ai2∧x2∨⋯∨ain∧xn<bi,1,ifai1∧x1∨ai2∧x2∨⋯∨ain∧xn≥bi,and d(x)=(d1(x),d2(x),…,dm(x))=(di(x))i∈I is said to be the satisfaction degree (vector) of x.Remark 9.
For anyx∈X, it holds that 0≤di(x)≤1, ∀i∈I. In particular, di(x)=1 holds if and only if ai1∧x1∨ai2∧x2∨⋯∨ain∧xn≥bi.Definition 10.
Letx1,x2∈X be two arbitrary potential solutions of system (4), with satisfaction degree vectors d(x1) and d(x2), respectively. x1 is said to be (strictly) superior to x2, denoted by x1⪰x2 (x1≻x2), if and only if d(x1)≥d(x2) (d(x1)>d(x2)). Besides, x1 is said to be approximatively equal to x2, denoted by x1≈x2, if and only if d(x1)=d(x2).Applying Definition10, we are able to compare the superiorities of any two potential solutions of system (4). Thus Definitions 8 and 10 form the evaluation model. Next we investigate some simple properties and provide a numerical example to illustrate the above evaluating approach.Theorem 11.
System (4) is consistent if and only if there exists some x∈X such that di(x)=1 for all i∈I. Furthermore, if system (4) is consistent, then x∈X(A,b) if and only if di(x)=1 for all i∈I.Proof.
It can be easily obtained from Theorem4 and Definition 8.Theorem 12.
Letx,y,z∈X. Then(i)
x⪰x;(ii)
x⪰y and y⪰x imply x≈y;(iii)
x⪰y and y⪰z imply x⪰z.Proof.
(i) It is obvious thatd(x)≥d(x). Thus x⪰x.
(ii)x⪰y indicates d(x)≥d(y), while y⪰x indicates d(y)≥d(x). According to Definition 1 it is easy to verify that d(x)=d(y), which leads to x≈y.
(iii) It follows fromx⪰y and y⪰z that d(x)≥d(y) and d(y)≥d(z). So we get d(x)≥d(z). According to Definition 10, it holds that x⪰z.It is shown in Theorem12 that “⪰” is a partial order (relation) on the set X.Example 13.
Consider a BitTorrent-like P2P file sharing system including five users (terminals), denoted byA1,A2,…,A5 (see Figure 1).
Suppose thejth user Aj sends the file data with quality level xj, j=1,2,…,5 and the system shown in Figure 1 can be reduced into the following max-min fuzzy relation inequalities:(12)0.5∧x1∨0.8∧x2∨0.7∧x3∨0.6∧x4∨0.4∧x5≥0.6,0.3∧x1∨0.5∧x2∨0.4∧x3∨0.7∧x4∨0.6∧x5≥0.5,0.6∧x1∨0.5∧x2∨0.6∧x3∨0.8∧x4∨0.3∧x5≥0.7,0.4∧x1∨0.6∧x2∨0.5∧x3∨0.5∧x4∨0.8∧x5≥0.6,0.8∧x1∨0.7∧x2∨0.5∧x3∨0.5∧x4∨0.6∧x5≥0.6.System (12) can be written as its matrix form, i.e.,(13)A∘xT≥bT.Three potential solutions are given as follows:(14)x=0.4,0.6,0.3,0.9,0.6,y=0.3,0.7,0.5,0.8,0.4,z=0.5,0.5,0.8,0.7,0.4.We aim to evaluate the superiorities of x,y,z.
Computing the satisfaction degree vectors ofx,y,z by Definition 8, we get (15)dx=dy=1,1,1,1,1,dz=1,1,1,0.83,0.83.Obviously d(x)=d(y)≥d(z). It follows from Definition 10 that x is approximatively equal to y, i.e., x≈y. Besides, both x and y are superior to z, i.e., x,y⪰z. According to Theorem 11, system (12) is consistent and both x and y are solutions of (12).Figure 1
Five-user BitTorrent-like P2P file sharing system.Example 14.
A six-user BitTorrent-like Peer-to-Peer file sharing system is reduced into the following max-min fuzzy relation inequalities:(16)A∘xT≥bT,where (17)A=aij=00.60.30.50.60.40.500.70.50.60.50.60.700.40.70.80.70.50.600.80.60.40.60.50.700.70.60.70.70.50.50,b=b1,b2,…,b6=0.7,0.8,0.6,0.6,0.7,0.8,x=x1,x2,…,x6∈0,16,and ∘ is the max-min composition. Here, aij represents the bandwidth between ith user and jth user, xj is the quality level on which the file data are sent from jth user, and bi is the quality requirement of download traffic of ith user. Our purpose is to compare the superiorities of the following potential solutions: (18)x=0.7,0.6,0.5,0.6,0.6,0.5,y=0.8,0.6,0.5,0.6,0.5,0.6,z=0.6,0.5,0.5,0.6,0.7,0.4.
Solution. Since ⋁j=16a1j=0∨0.6∨0.3∨0.5∨0.6∨0.4=0.6<0.7=b1, it follows from Theorem 4 that system (16) is inconsistent.
Now we compute the satisfaction degree vectors ofx,y,z, respectively. According to Definition 8,(19)dx=0.8571,0.75,1,1,0.8571,0.75,dy=0.8571,0.625,1,1,0.8571,0.75,dz=0.8571,0.75,1,1,0.8571,0.75.It is obvious that d(x)=d(z)>d(y). Hence x is approximatively equal to z, while x and z are both strictly superior to y, i.e., (20)x≈z≻y.


## 4. Approximate Solution to Inconsistent System of Max-Min Fuzzy Relation Inequalities

In this section, based on the evaluation model described above, we define concept of approximate solution in case system (4) is inconsistent. Moreover, solution method is constructed to find all the approximate solutions to inconsistent system (4).In this section we always assume that system (4) is inconsistent.Denote(21)I+=i∈I∣ai1∨ai2∨⋯∨ain≥bi,I-=i∈I∣ai1∨ai2∨⋯∨ain<bi.Since system (4) is inconsistent, it is clear that I-≠∅ following Theorem 4.Proposition 15.
Ifi∈I-, then di(x)<1 holds for any x∈X.Proof.
The proof is trivial.Definition 16 (approximate solution).
A vectorx∗∈X is said to be an approximate solution if and only if x∗ is superior to x, i.e., x∗⪰x, for any x∈X.In Definition16, it means that an approximate solution x∗ is a potential solution with the highest satisfaction degree (vector).Proposition 17.
Suppose bothx∗ and y∗ are approximate solutions to system (4). Then(i)
d(x∗)≥d(x) for all x∈X, i.e. d(x∗)=maxx∈X{d(x)}.(ii)
d(x∗)=d(y∗).Proof.
(i) can be obtained directly from Definitions10 and 16. Now we check the second point. Since both x∗ and y∗ are approximate solutions, it is obvious that x∗,y∗∈X. It follows from Definition 16 that d(x∗)≥d(y∗) and d(y∗)≥d(x∗), which indicate d(x∗)=d(y∗).Notice that⪰ is a partial order on the set X, but not a total order. That is to say, not any two potential solutions can be compared under the order “⪰.” Hence, to make Definition 16 reasonable, the existence of the approximate solution should be checked first. Existence of the approximate solution to an inconsistent system of max-min fuzzy relation inequalities is shown in the following Theorem 18.Theorem 18.
The vector1^=(1,1,…,1) is always an approximate solution to system (4).Proof.
For arbitraryx∈X, xj≤1 holds for any j∈J. So we have (22)ai1∧x1∨ai2∧x2∨⋯∨ain∧xn≤ai1∧1∨ai2∧1∨⋯∨ain∧1Next we verify that di(x)≤di(1^) in two cases.
Case 1. If ai1∧x1∨ai2∧x2∨⋯∨ain∧xn≥bi, then ai1∧1∨ai2∧1∨⋯∨ain∧1≥bi and di(x)=1=di(1^).
Case 2. If ai1∧x1∨ai2∧x2∨⋯∨ain∧xn<bi, then we have di(x)=ai1∧x1∨ai2∧x2∨⋯∨ain∧xn/bi≤ai1∧1∨ai2∧1∨⋯∨ain∧1/bi and di(x)≤1. It follows from Definition 8 that di(x)≤di(1^).Corollary 19.
1 ^ = ( 1,1 , … , 1 ) is the maximum approximate solution to system (4).Until now, we are able to obtain the maximum approximate solution to system (4). However, is there another approximate solution? If there exists, how to find out all the approximate solutions? In the following we focus on these two questions and propose effective method for solving all the approximate solution.Letb′=(b1′,b2′,…,bm′), where(23)bi′=bi,ifi∈I+,⋁j∈Jaij,ifi∈I-.Based on b′, we construct the following system of max-min fuzzy relation inequalities.(24)a11∧x1∨a12∧x2∨⋯∨a1n∧xn≥b1′,a21∧x1∨a22∧x2∨⋯∨a2n∧xn≥b2′,⋮am1∧x1∨am2∧x2∨⋯∨amn∧xn≥bm′.Proposition 20.
System (24) is consistent.Proof.
Fori∈I+, ⋁j∈Jaij≥bi=bi′. For i∈I+, ⋁j∈Jaij=bi′. That is, ⋁j∈Jaij≥bi′ holds for all i∈I. According to Theorem 4, system (24) is consistent.As a consistent system of max-min fuzzy relation inequalities, system (24) can be solved by the existing method(s), e.g., conservative path method [48]. Denote the solution set of system (24) by X(A,b′).Theorem 21.
Letx∗∈X. Then x∗ is an approximate solution to system (4) if and only if x∗ is a solution to system (24).Proof.
(⇒) If x∗ is an approximate solution to system (4), then d(x∗)≥d(x) holds for any x∈X. Since 1^∈X, we have(25)dx∗≥d1^
Case 1. If i∈I+, then ⋁j∈Jaij≥bi, which indicates di(1^)=1. Combining Inequality (25), di(x∗)≥di(1^)=1. Observing Remark 9, it holds that di(x∗)=1 and(26)ai1∧x1∗∨ai2∧x2∗∨⋯∨ain∧xn∗≥bi.On the other hand, by equality (23), i∈I+ indicates bi=bi′. So we get(27)ai1∧x1∗∨ai2∧x2∗∨⋯∨ain∧xn∗≥bi′.
Case 2. If i∈I-, then ⋁j∈Jaij<bi, which indicates(28)di1^=ai1∧1∨ai2∧1∨⋯∨ain∧1bi=⋁j∈Jaijbi.Moreover, since xj∗≤1 for all j∈J, it holds that(29)⋁j∈Jaij∧xj∗≤⋁j∈Jaij∧1=⋁j∈Jaij<bi.According to Definition 8,(30)dix∗=⋁j∈Jaij∧xj∗bi.Combining inequalities (25), (28), and (30), we get(31)⋁j∈Jaij∧xj∗bi=dix∗≥di1^=⋁j∈Jaijbi,i.e., ⋁j∈J(aij∧xj∗)≥⋁j∈Jaij. Besides, again by equality (23), i∈I- indicates bi′=⋁j∈Jaij. Hence ⋁j∈J(aij∧xj∗)≥bi′.
Cases 1 and 2 show that⋁j∈J(aij∧xj∗)≥bi′ for any i∈I. Consequently x∗ is a solution to system (24).
(⇐) Let x∈X be an arbitrary potential solution to system (4). In order to complete the proof, we need to check that di(x∗)≥di(x) for all i∈I.
Case 1. If i∈I+, then bi′=bi. Since x∗ is a solution to system (24), (32)⋁j∈Jaij∧xj∗≥bi′=bi.According to Definition 8 and Remark 9, it is obvious that di(x∗)=1≥di(x).
Case 2. If i∈I-, then ⋁j∈Jaij<bi and ⋁j∈Jaij=bi′. It follows from x,x∗∈X=[0,1]n that (33)⋁j∈Jaij∧xj≤⋁j∈Jaij<bi,⋁j∈Jaij∧xj∗≤⋁j∈Jaij<bi.According to Definition 8, we have(34)dix=⋁j∈Jaij∧xjbi,dix∗=⋁j∈Jaij∧xj∗bi.On the other hand, since x∗ is a solution to system (24), it follows that(35)⋁j∈Jaij∧xj∗≥bi′=⋁j∈Jaij=⋁j∈Jaij∧1≥⋁j∈Jaij∧xj.Combining (34) and (35), di(x∗)=⋁j∈J(aij∧xj∗)/bi≥⋁j∈J(aij∧xj)/bi=di(x). The proof is complete.Corollary 22.
The set of all approximate solutions to system (4) is X~(A,b)=X(A,b′).Corollary 23.
The set of all approximate solutions to system (4) is determined by a unique maximum approximate solution and a finite number of minimal approximate solutions.Based on the above results related to inconsistent system (4), we propose some solution procedures for solving all its approximate solutions as follows.Step 1.
Compute the index setsI+ and I- according to (21).Step 2.
Compute the vectorb′=(b1′,b2′,…,bm′) by (23).Step 3.
Based onb′ and aij, i∈I,j∈J, construct a consistent system of max-min fuzzy relation inequalities, i.e., system (24).Step 4.
Find out all the minimal solutions of system (24), denoted by xˇ1,xˇ2,…,xˇp, applying the conservative path method.Step 5.
The set of all approximate solutions to system (4) is(36)X~A,b=⋃t=1pxˇt,1^,where 1^=(1,1,…,1).In the following a numerical example is provided to illustrate the above-proposed solution method.Example 24.
Find the set of all approximate solutions to system (16) in Example 14, which has been checked to be inconsistent.SolutionStep 1.
Compute the index setsI+ and I- according to (21).(37)⋁j=16a1j=0∨0.6∨0.3∨0.5∨0.6∨0.4=0.6<0.7=b1,⋁j=16a2j=0.5∨0∨0.7∨0.5∨0.6∨0.5=0.7<0.8=b2,⋁j=16a3j=0.6∨0.7∨0∨0.4∨0.7∨0.8=0.8≥0.6=b3,⋁j=16a4j=0.7∨0.5∨0.6∨0∨0.8∨0.6=0.8≥0.6=b4,⋁j=16a5j=0.4∨0.6∨0.5∨0.7∨0∨0.7=0.7=b5,⋁j=16a6j=0.6∨0.7∨0.7∨0.5∨0.5∨0=0.7<0.8=b6.Hence the index sets are(38)I+=3,4,5,I-=1,2,6.Step 2.
Compute the vectorb′=(b1′,b2′,…,b6′) by (23). For i∈I+={3,4,5}, we have bi′=bi. Thus b3′=0.6,b4′=0.6,b5′=0.7. On the other hand, for i∈I-={1,2,6}, we get(39)b1′=⋁j=16a1j=0.6,b2′=⋁j=16a2j=0.7,b6′=⋁j=16a6j=0.7.Consequently b′=(0.6,0.7,0.6,0.6,0.7,0.7).Step 3.
Based onaij, i,j∈{1,2,…,6} and b′=(0.6,0.7,0.6,0.6,0.7,0.7) obtained in Step 2, we construct a consistent system of max-min fuzzy relation inequalities as follows:(40)0∧x1∨0.6∧x2∨0.3∧x3∨0.5∧x4∨0.6∧x5∨0.4∧x6≥0.6,0.5∧x1∨0∧x2∨0.7∧x3∨0.5∧x4∨0.6∧x5∨0.5∧x6≥0.7,0.6∧x1∨0.7∧x2∨0∧x3∨0.4∧x4∨0.7∧x5∨0.8∧x6≥0.6,0.7∧x1∨0.5∧x2∨0.6∧x3∨0∧x4∨0.8∧x5∨0.6∧x6≥0.6,0.4∧x1∨0.6∧x2∨0.5∧x3∨0.7∧x4∨0∧x5∨0.7∧x6≥0.7,0.6∧x1∨0.7∧x2∨0.7∧x3∨0.5∧x4∨0.5∧x5∨0∧x6≥0.7.Step 4.
Compute the minimal solutions of system (40) by applying the conservative path method. The characteristic matrix of system (40) is (41)C=00.6000.60000.70000.60.6000.60.60.600.600.60.60000.700.700.70.7000There exist four conservative paths, i.e., (42)p1=2,3,2,3,4,3,p2=2,3,2,3,6,3,p3=5,3,5,3,4,3,p4=5,3,5,3,6,3.Each conservative path corresponds to one minimal solution. All the minimal solutions of system (40) are (43)xˇ1=0,0.6,0.7,0.7,0,0,xˇ2=0,0.6,0.7,0,0,0.7,xˇ3=0,0,0.7,0.7,0.6,0,xˇ4=0,0,0.7,0,0.6,0.7.Hence, the set of all minimal approximate solutions to system (16) is X~ˇ(A,b)={xˇ1,xˇ2,xˇ3,xˇ4}.Step 5.
The set of all approximate solutions to system (16) is(44)X~A,b=⋃t=14xˇt,1^,where 1^=(1,1,…,1) is the maximum approximate solution, while xˇt is the minimal approximate solution obtained in Step 4, t=1,2,3,4.Example 25.
In this example we provide a simple real world P2P file sharing system, including 6 users in the system. The users are represented byA1,A2,…,A6 (see Figure 2).
The bandwidth betweenAi and Aj is denoted by aij, i=1,2,…,6, j=1,2,…,6. Values of aij are shown in Table 1, with measure unitMbps.
The expected requirements of the users for their file downloading areb=(b1,b2,…,b6), where(45)b1=40,b2=50,b3=50,b4=40,b5=40,b6=45.Besides, the user Aj sends out its local file data with quality level xj, j=1,2,…,6. As a consequence, if we denote x=(x1,x2,…,x6) and (46)A=030403545403503040353045400354030353040045454540303503530405045400,then the above-given P2P file sharing system could be described by(47)0∧x1∨30∧x2∨40∧x3∨35∧x4∨45∧x5∨40∧x6≥40,35∧x1∨0∧x2∨30∧x3∨40∧x4∨35∧x5∨30∧x6≥50,45∧x1∨40∧x2∨0∧x3∨35∧x4∨40∧x5∨30∧x6≥50,35∧x1∨30∧x2∨40∧x3∨0∧x4∨45∧x5∨45∧x6≥40,45∧x1∨40∧x2∨30∧x3∨35∧x4∨0∧x5∨35∧x6≥40,30∧x1∨40∧x2∨50∧x3∨45∧x4∨40∧x5∨0∧x6≥45.or(48)A∘xT≥bT.To convert it into max-min fuzzy relation inequalities system, we normalize all the parameters and variables. We divide each quantity in system (47) by 50. Then system (47) (or (48)) turns out to be(49)A_∘x_T≥b_T,where x_=(x_1,x_2,…,x_6)=(x1/50,x2/50,…,x6/50), b_=(b_1,b_2,…,b_6)=(0.8,1,1,0.8,0.8,0.9), and (50)A_=00.60.80.70.90.80.700.60.80.70.60.90.800.70.80.60.70.60.800.90.90.90.80.60.700.70.60.810.90.80.
Following Theorem4, it is easy to check that system (49) is inconsistent. Moreover, applying the method proposed in Section 4, we are able to obtain the complete approximate solution set of system (49) as [x_ˇ1,x_^]∪[x_ˇ2,x_^]∪[x_ˇ3,x_^]∪[x_ˇ4,x_^], where x_^=[1,1,…,1] and (51)x_ˇ1=0.9,0,0.8,0.9,0,0,x_ˇ2=0.9,0,0.9,0.8,0,0,x_ˇ3=0.9,0,0,0.9,0.8,0,x_ˇ4=0.9,0,0,0.9,0,0.8.Correspondingly, the solution set of system (48) is [xˇ1,x^]∪[xˇ2,x^]∪[xˇ3,x^]∪[xˇ4,x^], where x^=[50,50,…,50] and (52)xˇ1=45,0,40,45,0,0,xˇ2=45,0,45,40,0,0,xˇ3=45,0,0,45,40,0,xˇ4=45,0,0,45,0,40.Keep in mind that the measure unit isMbps.
Now we consider the satisfaction degrees of 6 different potential solutions as follows:(53)x1=50,50,50,50,50,50,x2=45,0,0,45,0,40,x3=48,36,46,42,27,33,x4=50,35,30,50,35,30,x5=30,35,35,30,25,35,x6=35,0,45,30,0,0.The satisfaction degree di(xt) represents the degree in which the potential solution xt satisfies the ith inequality in system (47), i=1,2,…,6, t=1,2,…,6. Detailed values of the satisfaction degrees are presented in Table 2.
In fact,x1,x2,x3 are all approximate solutions of system (47) (i.e., (48)). When the system works with these approximate solutions, it owns the highest satisfaction degree, i.e.,(54)dhighest=1,0.8,0.9,1,1,1.This indicates, when the system works with any approximate solution, e.g., x1,x2 or x3, the quality requirements of the users A1,A2,A3,A4,A5,A6 are satisfied with degrees 1,0.8,0.9,1,1,1, respectively. Obviously, the quality requirements of the users A1,A4,A5, and A6 are completely satisfied.
On the other hand, the vectorsx4,x5,x6 are not approximate solutions of system (47). Their corresponding satisfaction degree is no more than that of the approximate solutions. For example, when the system works with x5, the satisfaction degree is(55)dx5=0.875,0.6,0.7,0.875,0.875,0.778.There is no user whose quality requirement is completely satisfied. It is clear that (56)dx4=0.875,0.8,0.9,0.875,1,1<1,0.8,0.9,1,1,1=dhighest,dx5=0.875,0.6,0.7,0.875,0.875,0.778<1,0.8,0.9,1,1,1=dhighest,dx6=1,0.7,0.7,1,0.875,1<1,0.8,0.9,1,1,1=dhighest.Table 1
Value ofaij (measuring unit: Mbps).
i j 1 2 3 4 5 6 1 0 30 40 35 45 40  2 35 0 30 40 35 30  3 45 40 0 35 40 30  4 35 30 40 0 45 45  5 45 40 30 35 0 35  6 30 40 50 45 40 0Table 2
Value of the satisfaction degreedi(xt).
Satisfaction degree Potential solution x 1 x 2 x 3 x 4 x 5 x 6 d1(xt) 1 1 1 0.875 0.875 1  d2(xt) 0.8 0.8 0.8 0.8 0.6 0.7  d3(xt) 0.9 0.9 0.9 0.9 0.7 0.7  d4(xt) 1 1 1 0.875 0.875 1  d5(xt) 1 1 1 1 0.875 0.875  d6(xt) 1 1 1 1 0.778 1Figure 2
P2P file sharing system with 6 users.


## 5. Conclusion

Considering the requirement of total download speed of the terminals, a BitTorrent-like P2P file sharing system could be reduced into a system of addition-min fuzzy relation inequalities. While considering the requirement of biggest download speed, it can be reduced into a system of max-min fuzzy relation inequalities. In order to investigate approximate solution to the inconsistent system of max-min fuzzy relation inequalities, we proposed an evaluation model for comparing the superiority of two arbitrary potential solutions, in sense of the satisfaction degree. Approximate solution was defined to be the vector with highest satisfaction degree based on the evaluation model. According to our definition, we further proposed an effective algorithm to find out all the approximate solutions of an inconsistent system of max-min fuzzy relation inequalities.In the near future we are interested in the approximate solution of max-min fuzzy relation equations. In fact, the definitions of satisfaction degree and approximate solution for max-min fuzzy relation inequalities, proposed in this paper, could be generalized to those for the corresponding equality system. A system of max-min fuzzy relation equations could be described as(57)a11∧x1∨a12∧x2∨⋯∨a1n∧xn=b1,a21∧x1∨a22∧x2∨⋯∨a2n∧xn=b2,⋮am1∧x1∨am2∧x2∨⋯∨amn∧xn=bm,in which the limitations of the parameters and variables are the same as in system (4). Next we suggest an alternative definition of approximate solution to system (57).Definition 26.
Letx=(x1,x2,…,xn)∈X. The vector d=(x)=(d1=(x),d2=(x),…,dm=(x)) is said to be the satisfaction degree (vector) of x in system (57), if (58)di=x=1-ai1∧x1∨ai2∧x2∨⋯∨ain∧xn-bibi∧1,ifai1∧x1∨ai2∧x2∨⋯∨ain∧xn≠bi,1,ifai1∧x1∨ai2∧x2∨⋯∨ain∧xn=bi,where i=1,2,…,m.Definition 27.
In system (57), a vector x∗∈X is said to be an approximate solution, if d=(x∗)≥d=(x) holds for all x∈X. Here d=(x∗) and d=(x) represent the satisfaction degrees of x∗ and x, respectively.Although the definition of approximate solution may be extended to the max-min fuzzy relation equations, i.e., system (57), the resolution method proposed in this paper could not be directly applied to system (57). For arbitrary x,y∈X, it could be easily found that the implication x≤y⇒d(x)≤d(y) holds for system (4). However, the relevant implication that x≤y⇒d=(x)≤d=(y) no longer holds for system (57). This is a key factor leading to the inapplicability of our proposed method for system (57). In the future research, resolution of approximate solution(s) of system (57) might be an interesting topic.



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