Research ArticleEvaluation Model and Approximate Solution to InconsistentMax-Min Fuzzy Relation Inequalities in P2P File Sharing System

Xiao-Peng Yang

School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong 521041, China

Correspondence should be addressed to Xiao-Peng Yang; happyyangxp@163.com

Received 25 November 2018; Revised 2 February 2019; Accepted 25 February 2019; Published 14 March 2019

Academic Editor: Sing Kiong Nguang

Copyright © 2019 Xiao-Peng Yang. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

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 ina fuzzy relation system is the classical max-min one. Latersoon it was extended to the general max-𝑇 one, in which𝑇 represents a continuous triangular norm. The completesolution set of such max-𝑇 fuzzy relation equations orinequalities is completely determined by its unique greatestsolution (also named maximum solution) and all its minimalsolutions. A consistent system usually has a finite number ofminimal solutions. Solving a consistent max-𝑇 fuzzy relationsystem is equivalent to finding all itsminimal solutions.Thereexist many resolution methods for a consistent max-𝑇 fuzzyrelation system [3–23].

Inequality is a useful tool for describing various of quan-titative relations in the real world [24–32]. It plays importantrole in mathematics. Fuzzy relation inequalities, as a specialkind of inequalities, were also studied. It was recently foundthat a BitTorrent-like Peer-to-Peer (P2P) file sharing systemcould be reduced into a system of addition-min fuzzy relationinequalities [33–39]. Assume that there exist 𝑛 terminals,i.e., 𝐴1, 𝐴2, . . . , 𝐴𝑛, in the file sharing system. Based on theBitTorrent-like P2P transmission mechanism, each terminal

is able to receive (or download) file data from any otherterminal. Meanwhile, the 𝑗th terminal sends out file datawith quality level 𝑥𝑗 to any other terminal, 𝑗 ∈ {1, 2, . . . , 𝑛}.The bandwidth between terminals 𝐴 𝑖 and 𝐴𝑗 is 𝑎𝑖𝑗, 𝑖, 𝑗 ∈{1, 2, . . . , 𝑛}. Due to the bandwidth limitation, the networktraffic that 𝐴 𝑖 receives from 𝐴𝑗 is actually 𝑎𝑖𝑗 ∧ 𝑥𝑗, 𝑖, 𝑗 ∈{1, 2, . . . , 𝑛}.

In [33–36], the quality requirement of download trafficof 𝐴 𝑖 was considered as the total download speed that 𝐴 𝑖received file data from 𝐴1, 𝐴2, . . . , 𝐴𝑛, i.e.,

𝑎𝑖1 ∧ 𝑥1 + 𝑎𝑖2 ∧ 𝑥2 + ⋅ ⋅ ⋅ + 𝑎𝑖𝑛 ∧ 𝑥𝑛. (1)Suppose the quality requirement of download traffic of 𝐴 𝑖 isat least 𝑏𝑖. After normalization, the P2P file sharing systemcould be reduced into the following system of addition-minfuzzy relation inequalities:

𝑎11 ∧ 𝑥1 + 𝑎12 ∧ 𝑥2 + ⋅ ⋅ ⋅ + 𝑎1𝑛 ∧ 𝑥𝑛 ≥ 𝑏1,𝑎21 ∧ 𝑥1 + 𝑎22 ∧ 𝑥2 + ⋅ ⋅ ⋅ + 𝑎2𝑛 ∧ 𝑥𝑛 ≥ 𝑏2,

...𝑎𝑚1 ∧ 𝑥1 + 𝑎𝑚2 ∧ 𝑥2 + ⋅ ⋅ ⋅ + 𝑎𝑚𝑛 ∧ 𝑥𝑛 ≥ 𝑏𝑚.

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HindawiComplexityVolume 2019, Article ID 6901818, 11 pageshttps://doi.org/10.1155/2019/6901818

http://orcid.org/0000-0002-7516-5323https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/6901818

2 Complexity

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 terminalplans to obtain a file which cannot be separated, it wouldchoose only one other terminal to receive (download) thefile. In this situation, it is more reasonable to consider thequality requirement of download traffic of 𝐴 𝑖 as the biggest(highest) download speed that 𝐴 𝑖 receives file data from𝐴1, 𝐴2, . . . , 𝐴𝑛, i.e.,

𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛. (3)Correspondingly the P2P file sharing system is reduced intothe following max-min fuzzy relation inequalities:

𝑎11 ∧ 𝑥1 ∨ 𝑎12 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎1𝑛 ∧ 𝑥𝑛 ≥ 𝑏1,𝑎21 ∧ 𝑥1 ∨ 𝑎22 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎2𝑛 ∧ 𝑥𝑛 ≥ 𝑏2,

...𝑎𝑚1 ∧ 𝑥1 ∨ 𝑎𝑚2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑚𝑛 ∧ 𝑥𝑛 ≥ 𝑏𝑚.

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The matrix form of system (4) is

𝐴 ∘ 𝑥𝑇 ≥ 𝑏𝑇, (5)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 fuzzyrelation equation or inequality depended on the assumptionthat the system was consistent. However, as pointed outin [40], this assumption is often not the case in practicalapplications. Hence investigation on the inconsistent systemof fuzzy relation equations or inequalities is necessary andimportant. For a consistent system, the major objective isusually to obtain its solution(s), while for an inconsistentsystem, the major objective is often to find its approximatesolution(s). Approximate solution to inconsistent system offuzzy relation equations was studied by Pedrycz [41] for thefirst time. Modified Newton method was applied to findthe approximate solution. Another method based on thesolvability index was also proposed by Gottwald and Pedrcz[42] to deal with such problem. However, referring to thearguments in the work of Klir and Yuan [43], this method israther inefficient and may lead to a trivial solution. In [43],quality index was adopted to measure the goodness of anapproximate solution. In recent years other efficient methodswere proposed to solve the approximate solution, based onthe goodness measured by Euclidean distance [44], i.e.,

𝐷𝐸 = [[1𝑛∑𝑖∈𝐼

(⋁𝑗∈𝐽

(𝑎𝑖𝑗 ∧ 𝑥𝑗) − 𝑏𝑖)2]]1/2

, (6)or by Hamming distance [40, 45, 46], i.e.,

𝐷𝐻 = ∑𝑖∈𝐼

⋁𝑗∈𝐽 (𝑎𝑖𝑗 ∧ 𝑥𝑗) − 𝑏𝑖 . (7)

However, as we know there is no existing literatureinvestigating the approximate solution to inconsistent systemof fuzzy relation inequalities. In this paper we aim to establishan evaluation model to max-min fuzzy relation inequalitieswhich describe the P2P file sharing system. In such evaluationmodel, we will develop effective method for evaluating anypotential solution. Moreover, based on the evaluation modelwe may further define concept of approximate solution toinconsistent system of max-min fuzzy relation inequalities.

The rest of the paper is organized as follows. In Section 2we provide some necessary results onmax-min fuzzy relationinequality. In Section 3 an evaluation model is establishedto compare two arbitrary potential solutions, consideringthe application background in BitTorrent-like P2P file shar-ing system. Based on such evaluation model, approximatesolution of inconsistent system of max-min fuzzy relationinequalities is investigated in Section 4, with step-by-stepalgorithm and numerical illustrative example. Simple conclu-sion is set in Section 5.

2. Preliminaries

In this section we present some basic definitions and resultsrelated to max-min fuzzy relation inequality.

Denote𝑋 = [0, 1]𝑛.We call any 𝑥 ∈ 𝑋 a potential solutionof system (4). Correspondingly, 𝑋 is called the potentialsolution set.

Definition 1 (See [36]). Let 𝑥1 = (𝑥11, 𝑥12, . . . , 𝑥1𝑛), 𝑥2 = (𝑥21,𝑥22, . . . , 𝑥2𝑛) ∈ 𝑋; we define(i) 𝑥1 ≤ 𝑥2 if 𝑥1𝑗 ≤ 𝑥2𝑗 , ∀𝑗 ∈ 𝐽;(ii) 𝑥1 < 𝑥2 if 𝑥1 ≤ 𝑥2 and there are some 𝑗 ∈ 𝐽 such that𝑥1𝑗 < 𝑥2𝑗 .In what follows we shall denote the dual of order relation

‘’ 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 ofsystem (4).

Definition 2. System (4) is said to be consistent (or compati-ble) if 𝑋(𝐴, 𝑏) ̸= 0. Otherwise, it is said to be inconsistent.Definition 3. A solution 𝑥 ∈ 𝑋(𝐴, 𝑏) is said to be themaximum (or greatest) solution of system (4) if and only if𝑥 ≤ 𝑥 for all 𝑥 ∈ 𝑋(𝐴, 𝑏). A solution �̌� ∈ 𝑋(𝐴, 𝑏) is said tobe a minimal (or lower) solution of system (4) if and only if𝑥 ≤ �̌� implies 𝑥 = �̌� for any 𝑥 ∈ 𝑋(𝐴, 𝑏).

System (4) can be written as

𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛 ≥ 𝑏𝑖, 𝑖 ∈ 𝐼. (8)For given 𝑖 ∈ 𝐼, we define

𝐽𝑖 = {𝑗 | 𝑎𝑖𝑗 ≥ 𝑏𝑖} . (9)Theorem 4. For system (4), the following statements areequivalent:

Complexity 3

(i) system (4) is consistent;(ii) ⋁𝑗∈𝐽𝑎𝑖𝑗 ≥ 𝑏𝑖 holds for any 𝑖 ∈ 𝐼;(iii) 𝐽𝑖 ̸= 0 holds for any 𝑖 ∈ 𝐼;(iv) 1̂ = (1, 1, . . . , 1) ∈ 𝑋 is a solution of system (4).

Proof. The proof is trivial.

Remark 5 (see [47]). When system (4) is consistent, thevector 1̂ = (1, 1, . . . , 1) ∈ 𝑋 is always the unique maximumsolution of system (4).

Proposition 6. If 𝑥 ∈ 𝑋(𝐴, 𝑏), then 𝑥 ∈ 𝑋(𝐴, 𝑏) for any 𝑥satisfying 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 ofminimal solutions to system (4) when it is consistent. Wedenote the set of all minimal solutions of system (4) by�̌�(𝐴, 𝑏).Theorem 7 (see [47, 48]). If system (4) is consistent, then thesolution set of (4) is

X (𝐴, 𝑏) = ⋃�̌�∈�̌�(𝐴,𝑏)

{𝑥 | ̌𝑥 ≤ 𝑥 ≤ 𝑥} , (10)where �̌�(𝐴, 𝑏) is the minimal solution set and 𝑥 = 1̂ = (1, 1,. . . , 1) ∈ 𝑋 is the maximum solution.

It is expressed in Theorem 7 that the solution set ofsystem (4) is completely determined by a unique maximumsolution and a finite number of minimal solutions whenit is consistent. It follows from Theorem 7 that the uniquemaximum solution 𝑥 = 1̂ can be easily obtained. On the otherhand, theminimal solutions can be solved by the conservativepath method as presented in [48].

3. Evaluation Model to a System of Max-MinFuzzy Relation Inequalities

Obviously, each potential solution of system (4) representsa scheme of quality level on which the terminals send filedata. In this section, our purpose is to establish a rationalevaluation model for comparing the superiorities of any twopotential solutions.

As known to everybody, a solution of system (4) is avector satisfying all the inequalities in (4). However, an arbi-trary potential solution might not satisfy all the inequalities.The superiority of a potential solution depends on the degreethat satisfies system (4). We give the concept of satisfactiondegree below, based on which we can set up the evaluationmodel.

Definition 8 (satisfaction degree). Let 𝑥 = (𝑥1, 𝑥2, . . . , 𝑥𝑛) ∈𝑋 be a potential solution of system (4). Then 𝑑𝑖(𝑥) is said tobe the satisfaction degree function of 𝑥with respect to the 𝑖thinequality, where

𝑑𝑖 (𝑥) = {{{{{𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛𝑏𝑖 , if 𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛 < 𝑏𝑖,1, if 𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛 ≥ 𝑏𝑖,

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and 𝑑(𝑥) = (𝑑1(𝑥), 𝑑2(𝑥), . . . , 𝑑𝑚(𝑥)) = (𝑑𝑖(𝑥))𝑖∈𝐼 is said to bethe satisfaction degree (vector) of 𝑥.Remark 9. For any 𝑥 ∈ 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. Let 𝑥1, 𝑥2 ∈ 𝑋 be two arbitrary potentialsolutions of system (4), with satisfaction degree vectors 𝑑(𝑥1)and 𝑑(𝑥2), respectively. 𝑥1 is said to be (strictly) superior to𝑥2, denoted by 𝑥1 ⪰ 𝑥2 (𝑥1 ≻ 𝑥2), if and only if 𝑑(𝑥1) ≥ 𝑑(𝑥2)(𝑑(𝑥1) > 𝑑(𝑥2)). Besides, 𝑥1 is said to be approximativelyequal to 𝑥2, denoted by 𝑥1 ≈ 𝑥2, if and only if 𝑑(𝑥1) =𝑑(𝑥2).

Applying Definition 10, we are able to compare the supe-riorities of any two potential solutions of system (4). ThusDefinitions 8 and 10 form the evaluation model. Next weinvestigate some simple properties and provide a numericalexample to illustrate the above evaluating approach.

Theorem 11. System (4) is consistent if and only if there existssome 𝑥 ∈ 𝑋 such that 𝑑𝑖(𝑥) = 1 for all 𝑖 ∈ 𝐼. Furthermore, ifsystem (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 Theorem 4 andDefinition 8.

Theorem 12. Let 𝑥, 𝑦, 𝑧 ∈ 𝑋. �en(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 that 𝑑(𝑥) ≥ 𝑑(𝑥). Thus 𝑥 ⪰ 𝑥.(ii) 𝑥 ⪰ 𝑦 indicates 𝑑(𝑥) ≥ 𝑑(𝑦), while 𝑦 ⪰ 𝑥 indicates𝑑(𝑦) ≥ 𝑑(𝑥). According to Definition 1 it is easy to verify that𝑑(𝑥) = 𝑑(𝑦), which leads to 𝑥 ≈ 𝑦.(iii) It follows from 𝑥 ⪰ 𝑦 and 𝑦 ⪰ 𝑧 that 𝑑(𝑥) ≥ 𝑑(𝑦)

and 𝑑(𝑦) ≥ 𝑑(𝑧). So we get 𝑑(𝑥) ≥ 𝑑(𝑧). According toDefinition 10, it holds that 𝑥 ⪰ 𝑧.

4 Complexity

A2

A3 A4

A5

A1

Figure 1: Five-user BitTorrent-like P2P file sharing system.

It is shown in Theorem 12 that “⪰” is a partial order(relation) on the set𝑋.Example 13. Consider a BitTorrent-like P2P file sharingsystem including five users (terminals), denoted by 𝐴1, 𝐴2,. . . , 𝐴5 (see Figure 1).

Suppose the 𝑗th user 𝐴𝑗 sends the file data with qualitylevel 𝑥𝑗, 𝑗 = 1, 2, . . . , 5 and the system shown in Figure 1can be reduced into the following max-min fuzzy relationinequalities:

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.

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System (12) can be written as its matrix form, i.e.,

𝐴 ∘ 𝑥𝑇 ≥ 𝑏𝑇. (13)Three potential solutions are given as follows:

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) .

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We aim to evaluate the superiorities of 𝑥, 𝑦, 𝑧.Computing the satisfaction degree vectors of 𝑥, 𝑦, 𝑧 by

Definition 8, we get

𝑑 (𝑥) = 𝑑 (𝑦) = (1, 1, 1, 1, 1) ,𝑑 (𝑧) = (1, 1, 1, 0.83, 0.83) . (15)

Obviously 𝑑(𝑥) = 𝑑(𝑦) ≥ 𝑑(𝑧). It follows from Definition 10that 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 toTheorem 11, system (12) is consistent and both 𝑥 and 𝑦 aresolutions of (12).

Example 14. A six-user BitTorrent-like Peer-to-Peer file shar-ing system is reduced into the following max-min fuzzyrelation inequalities:

𝐴 ∘ 𝑥𝑇 ≥ 𝑏𝑇, (16)where

𝐴 = (𝑎𝑖𝑗) =[[[[[[[[[[[[

0 0.6 0.3 0.5 0.6 0.40.5 0 0.7 0.5 0.6 0.50.6 0.7 0 0.4 0.7 0.80.7 0.5 0.6 0 0.8 0.60.4 0.6 0.5 0.7 0 0.70.6 0.7 0.7 0.5 0.5 0

]]]]]]]]]]]]

,

𝑏 = (𝑏1, 𝑏2, . . . , 𝑏6) = (0.7, 0.8, 0.6, 0.6, 0.7, 0.8) ,𝑥 = (𝑥1, 𝑥2, . . . , 𝑥6) ∈ [0, 1]6 ,

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and ∘ is the max-min composition. Here, 𝑎𝑖𝑗 represents thebandwidth between 𝑖th user and 𝑗th user, 𝑥𝑗 is the qualitylevel on which the file data are sent from 𝑗th user, and 𝑏𝑖is the quality requirement of download traffic of 𝑖th user.Our purpose is to compare the superiorities of the followingpotential solutions:

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) .

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Solution. Since ⋁6𝑗=1𝑎1𝑗 = 0 ∨ 0.6 ∨ 0.3 ∨ 0.5 ∨ 0.6 ∨ 0.4 =0.6 < 0.7 = 𝑏1, it follows fromTheorem 4 that system (16) isinconsistent.

Nowwe compute the satisfaction degree vectors of 𝑥, 𝑦, 𝑧,respectively. According to Definition 8,

𝑑 (𝑥) = (0.8571, 0.75, 1, 1, 0.8571, 0.75) ,𝑑 (𝑦) = (0.8571, 0.625, 1, 1, 0.8571, 0.75) ,𝑑 (𝑧) = (0.8571, 0.75, 1, 1, 0.8571, 0.75) .

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It is obvious that 𝑑(𝑥) = 𝑑(𝑧) > 𝑑(𝑦). Hence 𝑥 is approx-imatively equal to 𝑧, while 𝑥 and 𝑧 are both strictly superiorto 𝑦, i.e.,

𝑥 ≈ 𝑧 ≻ 𝑦. (20)

Complexity 5

4. Approximate Solution toInconsistent System of Max-Min FuzzyRelation Inequalities

In this section, based on the evaluation model describedabove, we define concept of approximate solution in casesystem (4) is inconsistent. Moreover, solution method is con-structed to find all the approximate solutions to inconsistentsystem (4).

In this section we always assume that system (4) isinconsistent.

Denote 𝐼+ = {𝑖 ∈ 𝐼 | 𝑎𝑖1 ∨ 𝑎𝑖2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ≥ 𝑏𝑖} ,𝐼− = {𝑖 ∈ 𝐼 | 𝑎𝑖1 ∨ 𝑎𝑖2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 < 𝑏𝑖} . (21)

Since system (4) is inconsistent, it is clear that 𝐼− ̸= 0following Theorem 4.

Proposition 15. If 𝑖 ∈ 𝐼−, then 𝑑𝑖(𝑥) < 1 holds for any 𝑥 ∈ 𝑋.Proof. The proof is trivial.

Definition 16 (approximate solution). A vector 𝑥∗ ∈ 𝑋 is saidto be an approximate solution if and only if 𝑥∗ is superior to𝑥, i.e., 𝑥∗ ⪰ 𝑥, for any 𝑥 ∈ 𝑋.

In Definition 16, it means that an approximate solution𝑥∗ is a potential solution with the highest satisfaction degree(vector).

Proposition 17. Suppose both 𝑥∗ and 𝑦∗ are approximatesolutions to system (4). �en

(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 fromDefinitions 10 and 16.Now we check the second point. Since both 𝑥∗ and 𝑦∗ areapproximate solutions, it is obvious that 𝑥∗, 𝑦∗ ∈ 𝑋. It followsfrom 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 set𝑋, but not a totalorder. That is to say, not any two potential solutions can becompared under the order “⪰.” Hence, to make Definition 16reasonable, the existence of the approximate solution shouldbe checked first. Existence of the approximate solution to aninconsistent system of max-min fuzzy relation inequalities isshown in the following Theorem 18.

Theorem 18. �e vector 1̂ = (1, 1, . . . , 1) is always anapproximate solution to system (4).

Proof. For arbitrary 𝑥 ∈ 𝑋, 𝑥𝑗 ≤ 1 holds for any 𝑗 ∈ 𝐽. So wehave 𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛

≤ 𝑎𝑖1 ∧ 1 ∨ 𝑎𝑖2 ∧ 1 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 1 (22)Next we verify that 𝑑𝑖(𝑥) ≤ 𝑑𝑖(1̂) 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(1̂).Case 2. If 𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛 < 𝑏𝑖, then we have𝑑𝑖(𝑥) = (𝑎𝑖1 ∧𝑥1 ∨ 𝑎𝑖2 ∧𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧𝑥𝑛)/𝑏𝑖 ≤ (𝑎𝑖1 ∧ 1 ∨ 𝑎𝑖2 ∧1 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 1)/𝑏𝑖 and 𝑑𝑖(𝑥) ≤ 1. It follows fromDefinition 8that 𝑑𝑖(𝑥) ≤ 𝑑𝑖(1̂).Corollary 19. 1̂ = (1, 1, . . . , 1) is the maximum approximatesolution to system (4).

Until now, we are able to obtain the maximum approx-imate solution to system (4). However, is there anotherapproximate solution? If there exists, how to find out all theapproximate solutions? In the following we focus on thesetwo questions and propose effective method for solving allthe approximate solution.

Let 𝑏 = (𝑏1, 𝑏2, . . . , 𝑏𝑚), where

𝑏𝑖 = {{{{{𝑏𝑖, if 𝑖 ∈ 𝐼+,⋁𝑗∈𝐽

𝑎𝑖𝑗, if 𝑖 ∈ 𝐼−. (23)

Based on 𝑏, we construct the following system of max-minfuzzy relation inequalities.

𝑎11 ∧ 𝑥1 ∨ 𝑎12 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎1𝑛 ∧ 𝑥𝑛 ≥ 𝑏1,𝑎21 ∧ 𝑥1 ∨ 𝑎22 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎2𝑛 ∧ 𝑥𝑛 ≥ 𝑏2,

...𝑎𝑚1 ∧ 𝑥1 ∨ 𝑎𝑚2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑚𝑛 ∧ 𝑥𝑛 ≥ 𝑏𝑚.

(24)

Proposition 20. System (24) is consistent.

Proof. For 𝑖 ∈ 𝐼+, ⋁𝑗∈𝐽𝑎𝑖𝑗 ≥ 𝑏𝑖 = 𝑏𝑖 . For 𝑖 ∈ 𝐼+, ⋁𝑗∈𝐽𝑎𝑖𝑗 =𝑏𝑖 . That is, ⋁𝑗∈𝐽𝑎𝑖𝑗 ≥ 𝑏𝑖 holds for all 𝑖 ∈ 𝐼. According toTheorem 4, system (24) is consistent.

As a consistent system ofmax-min fuzzy relation inequal-ities, system (24) can be solved by the existing method(s),e.g., conservative path method [48]. Denote the solution setof system (24) by𝑋(𝐴, 𝑏).Theorem 21. Let 𝑥∗ ∈ 𝑋. �en 𝑥∗ is an approximate solutionto 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 1̂ ∈ 𝑋, we have

𝑑 (𝑥∗) ≥ 𝑑 (1̂) (25)Case 1. If 𝑖 ∈ 𝐼+, then ⋁𝑗∈𝐽𝑎𝑖𝑗 ≥ 𝑏𝑖, which indicates 𝑑𝑖(1̂) = 1.Combining Inequality (25), 𝑑𝑖(𝑥∗) ≥ 𝑑𝑖(1̂) = 1. ObservingRemark 9, it holds that 𝑑𝑖(𝑥∗) = 1 and

𝑎𝑖1 ∧ 𝑥∗1 ∨ 𝑎𝑖2 ∧ 𝑥∗2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥∗𝑛 ≥ 𝑏𝑖. (26)

6 Complexity

On the other hand, by equality (23), 𝑖 ∈ 𝐼+ indicates 𝑏𝑖 = 𝑏𝑖 .So we get

𝑎𝑖1 ∧ 𝑥∗1 ∨ 𝑎𝑖2 ∧ 𝑥∗2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥∗𝑛 ≥ 𝑏𝑖 . (27)Case 2. If 𝑖 ∈ 𝐼−, then ⋁𝑗∈𝐽𝑎𝑖𝑗 < 𝑏𝑖, which indicates𝑑𝑖 (1̂) = 𝑎𝑖1 ∧ 1 ∨ 𝑎𝑖2 ∧ 1 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 1𝑏𝑖 =

⋁𝑗∈𝐽𝑎𝑖𝑗𝑏𝑖 . (28)Moreover, since 𝑥∗𝑗 ≤ 1 for all 𝑗 ∈ 𝐽, it holds that

⋁𝑗∈𝐽

(𝑎𝑖𝑗 ∧ 𝑥∗𝑗 ) ≤ ⋁𝑗∈𝐽

(𝑎𝑖𝑗 ∧ 1) = ⋁𝑗∈𝐽

𝑎𝑖𝑗 < 𝑏𝑖. (29)According to Definition 8,

𝑑𝑖 (𝑥∗) = ⋁𝑗∈𝐽 (𝑎𝑖𝑗 ∧ 𝑥∗𝑗 )𝑏𝑖 . (30)

Combining inequalities (25), (28), and (30), we get

⋁𝑗∈𝐽 (𝑎𝑖𝑗 ∧ 𝑥∗𝑗 )𝑏𝑖 = 𝑑𝑖 (𝑥∗) ≥ 𝑑𝑖 (1̂) =⋁𝑗∈𝐽𝑎𝑖𝑗𝑏𝑖 , (31)

i.e.,⋁𝑗∈𝐽(𝑎𝑖𝑗 ∧ 𝑥∗𝑗 ) ≥ ⋁𝑗∈𝐽𝑎𝑖𝑗. Besides, again by equality (23),𝑖 ∈ 𝐼− indicates 𝑏𝑖 = ⋁𝑗∈𝐽𝑎𝑖𝑗. Hence⋁𝑗∈𝐽(𝑎𝑖𝑗 ∧ 𝑥∗𝑗 ) ≥ 𝑏𝑖 .Cases 1 and 2 show that ⋁𝑗∈𝐽(𝑎𝑖𝑗 ∧ 𝑥∗𝑗 ) ≥ 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 checkthat 𝑑𝑖(𝑥∗) ≥ 𝑑𝑖(𝑥) for all 𝑖 ∈ 𝐼.Case 1. If 𝑖 ∈ 𝐼+, then 𝑏𝑖 = 𝑏𝑖. Since 𝑥∗ is a solution to system(24),

⋁𝑗∈𝐽

(𝑎𝑖𝑗 ∧ 𝑥∗𝑗 ) ≥ 𝑏𝑖 = 𝑏𝑖. (32)According to Definition 8 and Remark 9, it is obvious that𝑑𝑖(𝑥∗) = 1 ≥ 𝑑𝑖(𝑥).Case 2. If 𝑖 ∈ 𝐼−, then ⋁𝑗∈𝐽𝑎𝑖𝑗 < 𝑏𝑖 and ⋁𝑗∈𝐽𝑎𝑖𝑗 = 𝑏𝑖 . It followsfrom 𝑥, 𝑥∗ ∈ 𝑋 = [0, 1]𝑛 that

⋁𝑗∈𝐽

(𝑎𝑖𝑗 ∧ 𝑥𝑗) ≤ ⋁𝑗∈𝐽

𝑎𝑖𝑗 < 𝑏𝑖,⋁𝑗∈𝐽

(𝑎𝑖𝑗 ∧ 𝑥∗𝑗 ) ≤ ⋁𝑗∈𝐽

𝑎𝑖𝑗 < 𝑏𝑖. (33)

According to Definition 8, we have

𝑑𝑖 (𝑥) = ⋁𝑗∈𝐽 (𝑎𝑖𝑗 ∧ 𝑥𝑗)𝑏𝑖 ,𝑑𝑖 (𝑥∗) = ⋁𝑗∈𝐽 (𝑎𝑖𝑗 ∧ 𝑥

∗𝑗 )𝑏𝑖 .

(34)

On the other hand, since 𝑥∗ is a solution to system (24), itfollows that

⋁𝑗∈𝐽

(𝑎𝑖𝑗 ∧ 𝑥∗𝑗 ) ≥ 𝑏𝑖 = ⋁𝑗∈𝐽

𝑎𝑖𝑗 = ⋁𝑗∈𝐽

(𝑎𝑖𝑗 ∧ 1)≥ ⋁𝑗∈𝐽

(𝑎𝑖𝑗 ∧ 𝑥𝑗) . (35)Combining (34) and (35), 𝑑𝑖(𝑥∗) = ⋁𝑗∈𝐽(𝑎𝑖𝑗 ∧ 𝑥∗𝑗 )/𝑏𝑖 ≥⋁𝑗∈𝐽(𝑎𝑖𝑗 ∧ 𝑥𝑗)/𝑏𝑖 = 𝑑𝑖(𝑥). The proof is complete.Corollary 22. �e set of all approximate solutions to system(4) is 𝑋(𝐴, 𝑏) = 𝑋(𝐴, 𝑏).Corollary 23. �e set of all approximate solutions to system(4) is determined by a unique maximum approximate solutionand 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 itsapproximate solutions as follows.

Step 1. Compute the index sets 𝐼+ and 𝐼− according to (21).Step 2. Compute the vector 𝑏 = (𝑏1, 𝑏2, . . . , 𝑏𝑚) by (23).Step 3. Based on 𝑏 and 𝑎𝑖𝑗, 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽, construct a consistentsystem of max-min fuzzy relation inequalities, i.e., system(24).

Step 4. Find out all the minimal solutions of system (24),denoted by �̌�1, �̌�2, . . . , �̌�𝑝, applying the conservative pathmethod.

Step 5. The set of all approximate solutions to system (4) is

𝑋 (𝐴, 𝑏) = 𝑝⋃𝑡=1

[�̌�𝑡, 1̂] , (36)where 1̂ = (1, 1, . . . , 1).

In the following a numerical example is provided toillustrate the above-proposed solution method.

Example 24. Find the set of all approximate solutions tosystem (16) in Example 14, which has been checked to beinconsistent.

Solution

Step 1. Compute the index sets 𝐼+ and 𝐼− according to (21).6⋁𝑗=1

𝑎1𝑗 = 0 ∨ 0.6 ∨ 0.3 ∨ 0.5 ∨ 0.6 ∨ 0.4 = 0.6 < 0.7= 𝑏1,

6⋁𝑗=1

𝑎2𝑗 = 0.5 ∨ 0 ∨ 0.7 ∨ 0.5 ∨ 0.6 ∨ 0.5 = 0.7 < 0.8= 𝑏2,

Complexity 7

6⋁𝑗=1

𝑎3𝑗 = 0.6 ∨ 0.7 ∨ 0 ∨ 0.4 ∨ 0.7 ∨ 0.8 = 0.8 ≥ 0.6= 𝑏3,

6⋁𝑗=1

𝑎4𝑗 = 0.7 ∨ 0.5 ∨ 0.6 ∨ 0 ∨ 0.8 ∨ 0.6 = 0.8 ≥ 0.6= 𝑏4,

6⋁𝑗=1

𝑎5𝑗 = 0.4 ∨ 0.6 ∨ 0.5 ∨ 0.7 ∨ 0 ∨ 0.7 = 0.7 = 𝑏5,6⋁𝑗=1

𝑎6𝑗 = 0.6 ∨ 0.7 ∨ 0.7 ∨ 0.5 ∨ 0.5 ∨ 0 = 0.7 < 0.8= 𝑏6.

(37)

Hence the index sets are

𝐼+ = {3, 4, 5} ,𝐼− = {1, 2, 6} . (38)

Step 2. Compute the vector 𝑏 = (𝑏1, 𝑏2, . . . , 𝑏6) by (23). For𝑖 ∈ 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 get𝑏1 = 6⋁𝑗=1

𝑎1𝑗 = 0.6,

𝑏2 = 6⋁𝑗=1

𝑎2𝑗 = 0.7,

𝑏6 = 6⋁𝑗=1

𝑎6𝑗 = 0.7.(39)

Consequently 𝑏 = (0.6, 0.7, 0.6, 0.6, 0.7, 0.7).Step 3. Based on 𝑎𝑖𝑗, 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 aconsistent system of max-min fuzzy relation inequalities asfollows:

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.

(40)

Step 4. Compute the minimal solutions of system (40) byapplying the conservative path method. The characteristicmatrix of system (40) is

𝐶 =[[[[[[[[[[[[

0 0.6 0 0 0.6 00 0 0.7 0 0 00.6 0.6 0 0 0.6 0.60.6 0 0.6 0 0.6 0.60 0 0 0.7 0 0.70 0.7 0.7 0 0 0

]]]]]]]]]]]]

(41)

There exist four conservative paths, i.e.,

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) .

(42)

Each conservative path corresponds to oneminimal solution.All the minimal solutions of system (40) are

̌𝑥1 = (0, 0.6, 0.7, 0.7, 0, 0) ,̌𝑥2 = (0, 0.6, 0.7, 0, 0, 0.7) ,̌𝑥3 = (0, 0, 0.7, 0.7, 0.6, 0) ,̌𝑥4 = (0, 0, 0.7, 0, 0.6, 0.7) .

(43)

Hence, the set of all minimal approximate solutions to system(16) is ̌̃𝑋(𝐴, 𝑏) = {�̌�1, �̌�2, �̌�3, �̌�4}.Step 5. The set of all approximate solutions to system (16) is

𝑋 (𝐴, 𝑏) = 4⋃𝑡=1

[�̌�𝑡, 1̂] , (44)where 1̂ = (1, 1, . . . , 1) is themaximum approximate solution,while �̌�𝑡 is theminimal approximate solution obtained in Step4, 𝑡 = 1, 2, 3, 4.Example 25. In this example we provide a simple real worldP2P file sharing system, including 6 users in the system. Theusers are represented by 𝐴1, 𝐴2, . . . , 𝐴6 (see Figure 2).

8 Complexity

A1

A2

A3 A4

A5

A6

Figure 2: P2P file sharing system with 6 users.

Table 1: Value of 𝑎𝑖𝑗 (measuring unit: Mbps).𝑖 𝑗

1 2 3 4 5 61 0 30 40 35 45 402 35 0 30 40 35 303 45 40 0 35 40 304 35 30 40 0 45 455 45 40 30 35 0 356 30 40 50 45 40 0

The bandwidth between 𝐴 𝑖 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 filedownloading are 𝑏 = (𝑏1, 𝑏2, . . . , 𝑏6), where

𝑏1 = 40,𝑏2 = 50,𝑏3 = 50,𝑏4 = 40,𝑏5 = 40,𝑏6 = 45.

(45)

Besides, the user 𝐴𝑗 sends out its local file data with qualitylevel 𝑥𝑗, 𝑗 = 1, 2, . . . , 6. As a consequence, if we denote 𝑥 =(𝑥1, 𝑥2, . . . , 𝑥6) and

𝐴 =[[[[[[[[[[[[

0 30 40 35 45 4035 0 30 40 35 3045 40 0 35 40 3035 30 40 0 45 4545 40 30 35 0 3530 40 50 45 40 0

]]]]]]]]]]]]

, (46)

then the above-given P2P file sharing system could bedescribed by

(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.

(47)

or

𝐴 ∘ 𝑥𝑇 ≥ 𝑏𝑇. (48)To convert it intomax-min fuzzy relation inequalities system,wenormalize all the parameters and variables.Wedivide eachquantity in system (47) by 50.Then system (47) (or (48)) turnsout to be

𝐴 ∘ 𝑥𝑇 ≥ 𝑏𝑇, (49)where 𝑥 = (𝑥1, 𝑥2, . . . , 𝑥6) = (𝑥1/50, 𝑥2/50, . . . , 𝑥6/50), 𝑏 =(𝑏1, 𝑏2, . . . , 𝑏6) = (0.8, 1, 1, 0.8, 0.8, 0.9), and

𝐴 =[[[[[[[[[[[[

0 0.6 0.8 0.7 0.9 0.80.7 0 0.6 0.8 0.7 0.60.9 0.8 0 0.7 0.8 0.60.7 0.6 0.8 0 0.9 0.90.9 0.8 0.6 0.7 0 0.70.6 0.8 1 0.9 0.8 0

]]]]]]]]]]]]

. (50)

Following Theorem 4, it is easy to check that system (49)is inconsistent. Moreover, applying the method proposed inSection 4, we are able to obtain the complete approximatesolution set of system (49) as [�̌�1, �̂�]∪[�̌�2, �̂�]∪[�̌�3, �̂�]∪[�̌�4, �̂�],where �̂� = [1, 1, . . . , 1] and

�̌�1 = (0.9, 0, 0.8, 0.9, 0, 0) ,�̌�2 = (0.9, 0, 0.9, 0.8, 0, 0) ,�̌�3 = (0.9, 0, 0, 0.9, 0.8, 0) ,�̌�4 = (0.9, 0, 0, 0.9, 0, 0.8) .

(51)

Complexity 9

Table 2: Value of the satisfaction degree 𝑑𝑖(𝑥𝑡).Satisfaction degree Potential solution𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6𝑑1(𝑥𝑡) 1 1 1 0.875 0.875 1𝑑2(𝑥𝑡) 0.8 0.8 0.8 0.8 0.6 0.7𝑑3(𝑥𝑡) 0.9 0.9 0.9 0.9 0.7 0.7𝑑4(𝑥𝑡) 1 1 1 0.875 0.875 1𝑑5(𝑥𝑡) 1 1 1 1 0.875 0.875𝑑6(𝑥𝑡) 1 1 1 1 0.778 1Correspondingly, the solution set of system (48) is [�̌�1, 𝑥] ∪[�̌�2, 𝑥] ∪ [�̌�3, 𝑥] ∪ [�̌�4, 𝑥], where 𝑥 = [50, 50, . . . , 50] and

�̌�1 = (45, 0, 40, 45, 0, 0) ,�̌�2 = (45, 0, 45, 40, 0, 0) ,�̌�3 = (45, 0, 0, 45, 40, 0) ,�̌�4 = (45, 0, 0, 45, 0, 40) .

(52)

Keep in mind that the measure unit isMbps.Now we consider the satisfaction degrees of 6 different

potential solutions as follows:

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) .

(53)

The satisfaction degree 𝑑𝑖(𝑥𝑡) represents the degree in whichthe potential solution 𝑥𝑡 satisfies the 𝑖th inequality in system(47), 𝑖 = 1, 2, . . . , 6, 𝑡 = 1, 2, . . . , 6. Detailed values of thesatisfaction 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 approxi-mate solutions, it owns the highest satisfaction degree, i.e.,

𝑑ℎ𝑖𝑔ℎ𝑒𝑠𝑡 = (1, 0.8, 0.9, 1, 1, 1) . (54)This indicates, when the system works with any approximatesolution, e.g., 𝑥1, 𝑥2 or 𝑥3, the quality requirements ofthe users 𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐴5, 𝐴6 are satisfied with degrees1,0.8,0.9,1,1,1, respectively. Obviously, the quality require-ments of the users 𝐴1, 𝐴4, 𝐴5, and 𝐴6 are completely satis-fied.

On the other hand, the vectors 𝑥4, 𝑥5, 𝑥6 are not approx-imate solutions of system (47). Their corresponding satisfac-tion degree is nomore than that of the approximate solutions.For example, when the system works with 𝑥5, the satisfactiondegree is

𝑑 (𝑥5) = (0.875, 0.6, 0.7, 0.875, 0.875, 0.778) . (55)

There is no user whose quality requirement is completelysatisfied. It is clear that

𝑑 (𝑥4) = (0.875, 0.8, 0.9, 0.875, 1, 1)< (1, 0.8, 0.9, 1, 1, 1) = 𝑑ℎ𝑖𝑔ℎ𝑒𝑠𝑡,

𝑑 (𝑥5) = (0.875, 0.6, 0.7, 0.875, 0.875, 0.778)< (1, 0.8, 0.9, 1, 1, 1) = 𝑑ℎ𝑖𝑔ℎ𝑒𝑠𝑡,

𝑑 (𝑥6) = (1, 0.7, 0.7, 1, 0.875, 1) < (1, 0.8, 0.9, 1, 1, 1)= 𝑑ℎ𝑖𝑔ℎ𝑒𝑠𝑡.

(56)

5. Conclusion

Considering the requirement of total download speed of theterminals, a BitTorrent-like P2P file sharing system could bereduced into a system of addition-min fuzzy relation inequal-ities. While considering the requirement of biggest downloadspeed, it can be reduced into a system of max-min fuzzyrelation inequalities. In order to investigate approximatesolution to the inconsistent system of max-min fuzzy relationinequalities, we proposed an evaluation model for comparingthe superiority of two arbitrary potential solutions, in senseof the satisfaction degree. Approximate solution was definedto be the vector with highest satisfaction degree basedon the evaluation model. According to our definition, wefurther proposed an effective algorithm to find out all theapproximate solutions of an inconsistent system of max-minfuzzy relation inequalities.

In the near future we are interested in the approximatesolution of max-min fuzzy relation equations. In fact, the def-initions of satisfaction degree and approximate solution formax-min fuzzy relation inequalities, proposed in this paper,could be generalized to those for the corresponding equalitysystem. A system of max-min fuzzy relation equations couldbe described as

𝑎11 ∧ 𝑥1 ∨ 𝑎12 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎1𝑛 ∧ 𝑥𝑛 = 𝑏1,𝑎21 ∧ 𝑥1 ∨ 𝑎22 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎2𝑛 ∧ 𝑥𝑛 = 𝑏2,

...𝑎𝑚1 ∧ 𝑥1 ∨ 𝑎𝑚2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑚𝑛 ∧ 𝑥𝑛 = 𝑏𝑚,

(57)

in which the limitations of the parameters and variables arethe same as in system (4). Next we suggest an alternativedefinition of approximate solution to system (57).

Definition 26. Let 𝑥 = (𝑥1, 𝑥2, . . . , 𝑥𝑛) ∈ 𝑋. The vector𝑑=(𝑥) = (𝑑=1 (𝑥), 𝑑=2 (𝑥), . . . , 𝑑=𝑚(𝑥)) is said to be the satisfac-tion degree (vector) of 𝑥 in system (57), if

10 Complexity

𝑑=𝑖 (𝑥) = {{{{{1 − 𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛 − 𝑏𝑖𝑏𝑖 ∧ 1, if 𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛 ̸= 𝑏𝑖,1, if 𝑎𝑖1 ∧ 𝑥1 ∨ 𝑎𝑖2 ∧ 𝑥2 ∨ ⋅ ⋅ ⋅ ∨ 𝑎𝑖𝑛 ∧ 𝑥𝑛 = 𝑏𝑖, (58)

where 𝑖 = 1, 2, . . . , 𝑚.Definition 27. In system (57), a vector 𝑥∗ ∈ 𝑋 is said to be anapproximate solution, if 𝑑=(𝑥∗) ≥ 𝑑=(𝑥) holds for all 𝑥 ∈ 𝑋.Here 𝑑=(𝑥∗) and 𝑑=(𝑥) represent the satisfaction degrees of𝑥∗ and 𝑥, respectively.

Although the definition of approximate solution may beextended to themax-min fuzzy relation equations, i.e., system(57), the resolution method proposed in this paper could notbe directly applied to system (57). For arbitrary 𝑥, 𝑦 ∈ 𝑋, itcould be easily found that the implication 𝑥 ≤ 𝑦 ⇒ 𝑑(𝑥) ≤𝑑(𝑦) holds for system (4). However, the relevant implicationthat 𝑥 ≤ 𝑦 ⇒ 𝑑=(𝑥) ≤ 𝑑=(𝑦) no longer holds for system(57). This is a key factor leading to the inapplicability of ourproposed method for system (57). In the future research,resolution of approximate solution(s) of system (57)might bean interesting topic.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Disclosure

This article does not contain any studies with human partici-pants or animals performed by the author.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (61877014), the Natural Sci-ence Foundation of Guangdong Province (2016A030307037,2017A030307020), and theNatural Science Foundations fromHanshan Normal University (2017KTSCX124, ZD201802,QD20171001).

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