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

. 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 suchsystem. Besides, based on the evaluationmodel, concept ofapproximatesolutionisdefined.Itisindeedapotentialsolutionwithhighestsatisfactiondegree.Furthermore,effectivealgorithm isdevelopedforobtainingtheapproximatesolutionstoinconsistentsystem.Numericalexamplesareprovidedtoillustrateour proposedmodelandalgorithm.


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 classical max-min one.Later soon it was extended to the general max- one, in which  represents a continuous triangular norm.The complete solution set of such max- 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- fuzzy relation system is equivalent to finding all its minimal solutions.There exist many resolution methods for a consistent max- fuzzy relation system .
In [33][34][35][36], the quality requirement of download traffic of   was considered as the total download speed that   received file data from  1 ,  2 , . . .,   , i.e., Suppose the quality requirement of download traffic of   is at least   .After normalization, the P2P file sharing system could be reduced into the following system of addition-min fuzzy relation inequalities: where   ,   ∈ [0, 1],   ∈  + ,  ∈ , ∈ .Here  = {1, 2, . . ., } and  = {1, 2, . . ., } 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 of   as the biggest (highest) download speed that   receives file data from  1 ,  2 , . . .,   , i.e., Correspondingly the P2P file sharing system is reduced into the following max-min fuzzy relation inequalities: The matrix form of system ( 4) is where 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., or by Hamming distance [40,45,46], i.e., 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 Section 2 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.

Preliminaries
In this section we present some basic definitions and results related to max-min fuzzy relation inequality.
Definition .System (4) is said to be consistent (or compatible) if (, ) ̸ = 0. Otherwise, it is said to be inconsistent.
Definition .A solution x ∈ (,) is said to be the maximum (or greatest) solution of system (4) if and only if  ≤ x for all  ∈ (, ).A solution x ∈ (, ) is said to be a minimal (or lower) solution of system (4) if and only if  ≤ x implies  = x for any  ∈ (, ).
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 X(, ).
It is expressed in Theorem 7 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].

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.
In particular,   () = 1 holds if and only if Definition .Let  1 ,  2 ∈  be two arbitrary potential solutions of system (4), with satisfaction degree vectors ( 1 ) and ( 2 ), respectively. 1 is said to be (strictly) superior to  2 , denoted by Applying Definition 10, 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 ( ) is consistent if and only if there exists some  ∈  such that   () = 1 for all  ∈ .Furthermore, if system ( ) is consistent, then  ∈ (, ) if and only if   () = 1 for all  ∈ .(ii)  ⪰  indicates () ≥ (), while  ⪰  indicates () ≥ ().According to Definition 1 it is easy to verify that () = (), which leads to  ≈ .
Example .A six-user BitTorrent-like Peer-to-Peer file sharing system is reduced into the following max-min fuzzy relation inequalities: where and ∘ is the max-min composition.Here,   represents the bandwidth between th user and th user,   is the quality level on which the file data are sent from th user, and   is the quality requirement of download traffic of th user.

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 Since system ( 4) is inconsistent, it is clear that  − ̸ = 0 following Theorem 4.
Proof.The proof is trivial.
Definition (approximate solution).A vector  * ∈  is said to be an approximate solution if and only if  * is superior to , i.e.,  * ⪰ , for any  ∈ .
In Definition 16, it means that an approximate solution  * is a potential solution with the highest satisfaction degree (vector).Proposition 17. Suppose both  * and  * are approximate solutions to system ( ). en Notice that ⪰ is a partial order on the set , 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. e vector 1 = (1, 1, . . ., 1) is always an approximate solution to system ( ).
Proof.For arbitrary  ∈ ,   ≤ 1 holds for any  ∈ .So we have Next we verify that   () ≤   ( 1) in two cases.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.
Theorem 21.Let  * ∈ .en  * is an approximate solution to system ( ) if and only if  * is a solution to system ( ).
(⇐) Let  ∈  be an arbitrary potential solution to system (4).In order to complete the proof, we need to check that   ( * ) ≥   () for all  ∈ .

Corollary 23. e set of all approximate solutions to system ( ) 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 .Compute the index sets  + and  − according to (21).
Step .Based on   and   ,  ∈ ,  ∈ , construct a consistent system of max-min fuzzy relation inequalities, i.e., system Step .Find out all the minimal solutions of system (24), denoted by x 1 , x 2 , . . ., x  , applying the conservative path method.
In the following a numerical example is provided to illustrate the above-proposed solution method.
Example .Find the set of all approximate solutions to system (16) in Example 14, which has been checked to be inconsistent.

Solution
Step .Compute the index sets  + and  − according to (21).
Hence, the set of all minimal approximate solutions to system Step .The set of all approximate solutions to system (16) is where 1 = (1, 1, . . ., 1) is the maximum approximate solution, while x  is the minimal approximate solution obtained in Step 4,  = 1, 2, 3, 4.

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 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).

Table 2 :
Value of the satisfaction degree   (  ).