Resolution of Max-Product Fuzzy Relation Equation with Interval-Valued Parameter

Considering the application background on P2P network system, we investigate the max-product fuzzy relation equation with interval-valued parameter in this paper. Order relation on the set of all interval-valued numbers plays key role in the construction and resolution of the interval-valued-parameter fuzzy relation equation (IPFRE). The basic operations supremum ( 𝑎 ∨ 𝑏 ) and infimum ( 𝑎 ∧ 𝑏 ) in the IPFRE should be defined depending on the order relation. A novel total order is introduced for establishing the IPFRE. We also discuss some properties of the IPFRE system, including the consistency and structure of the complete solution set. Concepts of close index set and open index set are defined, helping us to construct the resolution method of the IPFRE system. We further provide a detailed algorithm for obtaining the complete solution set. Besides, the solution set is compared to that of the classical max-𝑇 fuzzy relation equations system.


Introduction
A crisp relation represents the presence or absence of association, interaction, or interconnectedness between the elements of two or more sets [1].Its importance is almost selfevident [2].Relation exists everywhere widely and trying to discovery the relation is one of the most important targets in science research.In classical two-valued logic relation, one object is either relevant or irrelevant to another one.The degree or strength of the relation is not able to be described.To overcome such shortage, the typical crisp relation was naturally extended to fuzzy relation [3].As inverse problem of fuzzy linear system, fuzzy relation equations were first proposed by E. Sanchez [4], motivated by its application in medical diagnosis.The potential maximal solution could be easily obtained, used for checking the consistency of a maxmin fuzzy relation equations system.However, searching all the minimal solutions is much more difficult and important for the complete solution set.Since the traditional linear algebra methods are no longer effective for the fuzzy relation equations, various resolution methods were proposed and illustrated [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].
The composition plays important role in fuzzy relation equations.It was extended from max-min to max-product [24][25][26][27] and other ones [21].It has been demonstrated the max-product composition is superior to the max-min one in some cases [25].J. Loetamonphong and S.-C.Fang first investigated the optimization problem with linear-objective function and max-product fuzzy relation equations constraint [28].The resolution idea was picked from [29], in which the main problem was divided into two subproblems and solved by the branch-and-bound method.In fact the twosubproblem approach was widely adopted to deal with linearobjective optimization problem subject to fuzzy relation equations or inequalities with various composition such as max-min, max-product and max-average [30,31].Nonlinearobjective optimization problem subject to fuzzy relation system was also studied.There exist three kinds of solution methods for such nonlinear optimization problem: (i) genetic algorithm for optimization problem with general nonlinearobjective function and fuzzy relation constraint.By this kind of method, only approximate solution could be found [32,33]; (ii) method to separate the main problem into a finite number of subproblems according to the minimal solutions of the 2 Complexity constraint [34,35]; (iii) specific method for the fuzzy relation problem with special objective functions [36,37].
In recent years some new types of fuzzy relation inequalities or equations appeared.Fuzzy relation inequalities with addition-min composition were employed to describe the P2P file sharing system.Some properties of its solution set [38] and corresponding optimization models were studied [39][40][41].Besides, bipolar fuzzy relation equations was another interesting research object.The scholars were interested in the relevant linear optimization problem [42][43][44].
As a special form of fuzzy number, interval was a powerful tool in both theoretical and practical application aspects [45][46][47].Considering the fuzziness in the parameters, intervalvalued fuzzy relation equations attracted some researchers' attentions [48,49].The authors investigated three types of solutions, i.e., tolerable solution set, united solution set, and controllable solution set, respectively.Before definition of these solutions, a group of fuzzy relations was converted into two groups of fuzzy relation inequalities.In the existing works on interval-valued fuzzy relation equations, the compositions in the system contained max-t-norm [50], min-s-norm [51], and max-plus [52].In these existing works, an interval-valued number was considered as a set of real numbers, but not a fuzzy number.Analogously, the interval-valued vector and matrix were both considered as some sets.However as we know, an interval-valued number is indeed a fuzzy number.There exists its own operations, such as addition and multiplication.Based on such consideration, we introduce the max-product fuzzy relation equations with interval-valued parameters in this paper.The rest part is organized as follows.
In Section 2 we define some basic operations and an order relation on the set of all interval-valued numbers.Section 3 is the formulation of max-product fuzzy relation equation with interval-valued parameter.In Section 4 we provides its solution method based on concepts of close index set and open index set.Moreover, structure and resolution algorithm of the complete solution set is investigated.A numerical example is given in Section 5 to illustrated the algorithm.Simple discussion and conclusion are set in Sections 6 and 7 respectively.

Basic Operation and Order Relation on the Set of All Interval-Valued Numbers
Interval-valued number is an important extension of the precise real number for describing the quantities in the real world.An interval-valued number is usually with the form of [, ], where ,  ∈  and  ≤ ,  represents the real number set.
Definition ( [53] (fuzzy number)).A fuzzy set Ã in  is called a fuzzy number, if it satisfies the following conditions: (i) Ã is normal; i.e., the cut set  1 is nonempty, (ii) the -level cut set   is a closed interval for any  ∈ (0, 1], (iii) the support of Ã is bounded.
An interval-valued number could be viewed as a special type of fuzzy number [54].For interval-valued number [, ], the membership function is It is easy to check that [, ] satisfies the conditions in Definition 1.For more detail, the readers could refer to [55][56][57].
Denote the set of all interval-valued numbers by  = {[, ] |  ≤ , ,  ∈ }.We first present two basic operations on the set , which are commonly used in the existing works [58].
Besides, supremum (∨) and infimum (∧) are basic operations for establishing the fuzzy relation equations.These two operations depend on the order relation on the set .Now we investigate the order relation on the set  first.
In general, an order relation is defined for comparing two elements in a set.In the set , the most common order relation is the classical "product" order [59], denoted by "≲" in this paper.Let and only if both  = V and  =  hold.It is easy to check that "≲" is a partial order on , but not a total order.A partial order on a set is a binary relation satisfying reflexivity, antisymmetry, and transitivity.A total order is a partial order which makes any two elements in the set comparable.Under the abovementioned order "≲", the supremum (∨ ∼ ) and infimum (∧ ∼ ) operations on the set  are as follows: The advantage of the product order "≲" lies in its convenience in computation.However, it is not a total order.Some pairs of interval-valued numbers could not be compared by the product order.For example, under such order relation, we are not able to compare [0.1, 0.6] and However, as pointed out in [60], a total order is indispensable in some situations.It enables us to compare any pair of interval-valued numbers.Specific total order was introduced for comparing any two intuitionistic fuzzy sets [61].Moreover, admissible order, which refined both the total order and the classical product order, was defined on the interval-valued number set [62][63][64].We recall the definition of admissible order on  below.
Definition ( [62] (admissible order)).Let (, ⪯) be a poset, with partial order "⪯".The order ⪯ is said to be admissible order, if it satisfies the following conditions: (i) ⪯ is a total (linear) order on ; (ii) for any where ≲ represents the classical product order.
In this paper, motivated by the order defined in [61] for the intuitionistic fuzzy set and the order introduced in [65][66][67][68][69] for comparison of two fuzzy numbers in solving fuzzy linear programming, we define the following order relation "<" on the set .
The dual symbols of < and ≤ are > and ≥, respectively.
Under the above-defined order relation, it is easy to check that [, ] = [V, V] if and only if  = V and  = V.This is coincident with the classical product order.In Definition 3, It is shown in Proposition 4 that the order relation "≤" in Definition 3 is an admissible order.Moreover, (, ≤) forms a total order set.Under such order relation, any pair of intervalvalued numbers could be compared.Next we provide the supremum (∨) and infimum (∧) operations on , based on the above-defined order "≤".Let [, ], [V, V] ∈ .Then we have

Form of Max-Product Fuzzy Relation Equation with Interval-Valued Parameter
Motivated by the application background of P2P network system, we introduce the form of max-product fuzzy relation equations with interval-valued parameters in this section.
. .Unit Interval-Valued Number.An interval-valued number [, ] is said to be unit interval-valued number, if 0 ≤  ≤  ≤ 1.We denote the set of all unit interval-valued numbers by and the set of all nonzero (or positive) unit interval-valued numbers by Remark .According to the order given in Definition 3, the set ((0, 1]) can also be written as or Obviously both ([0, 1]) and ((0, 1]) are subsets of .Hence the operations and order relation defined on  in Section 2 keep valid for ([0, 1]) and ((0, 1]).

. . Classical Max-Min and Max-Product Fuzzy Relation
Equations.For the convenience of expression, we denote two index sets  and  as follows: System of fuzzy relation equations, with the most commonly used max-min composition [4], could be written as [19-21, 27, 29, 31] where   ,   ,   ∈ [0, 1],  ∈ ,  ∈ .But soon after, the max-min composition was extended to the max-product one.

Complexity
Analogously, system of max-product fuzzy relation equations was expressed by with the same parameters and variables appear in system (8).
In [71], the max-product fuzzy relation equation was applied to describe the quantitative relation in a P2P (peerto-peer) network system.Suppose there exist  terminals in such system, denoted by   ,  ∈ .Each pair of terminals was wireless-connected.They transformed their local resources through electromagnetic wave.The terminal   transmits the electromagnetic wave with intensity   .Due to the decreasing of the intensity, associated with the distance, the intensity of the electromagnetic wave is no more than   , when it is received by   ,  ∈  and  ̸ = .Hence the intensity of electromagnetic wave at   , which is send out by   , could be represented by where 0 ≤   ≤ 1.We call   the decreasing coefficient.In general case,   selects the terminal with highest intensity, to download the target resource.The intensity of electromagnetic wave reflects the download quality level.Hence if the requirement of download quality level of   is   > 0, then the intensities of electromagnetic wave should satisfy Here, without loss of generality, we assume the first  terminals require the download quality level.Then we have  ∈  = {1, 2, . . ., }.After normalization of the parameters and variables, the requirements of download quality level of the terminals could be described the max-product fuzzy relation equations as system (9).

. . Max-Product Fuzzy Relation Equation with
Interval-Valued Parameter.When applying system (8) to describe the P2P network system as shown in last subsection, the decreasing coefficient   is determined by the distance between the terminals   and   .The value of   will become smaller, if the distance of   and   becomes farther.However, in real world application, the distance might not be the unique factor influencing the decreasing coefficient   .For example, weather condition and artificially disturbing of the electromagnetic wave are also potential influence factors.Notice that these influence factors are usually uncertain or random.Hence the decreasing coefficient   is not a precise number.It could take any possible value within limits.Here we assume   as an interval-valued number, i.e.,   = [  ,   ].Correspondingly, the requirement of the terminal   is also assumed to be a value range and denoted by [  ,   ],  ∈ .
As a consequence, considering the uncertain potential influence factors of the decreasing coefficient, the corresponding system for describing the P2P network system could be written as where the parameters [  ,   ], [  ,   ] ∈ ((0, 1]) are unit interval-valued numbers,  ∈ ,  ∈ .System (12) consists of  max-product fuzzy relation equations with interval-valued parameters.System (12) could also be written as the following matrix form, where In system (12), the order relation and operations are as defined in Section 2.

Resolution of Max-Product Fuzzy Relation
Equation with Interval-Valued Parameter ). e equation is solvable (has at least one solution) if and only if Then it follows from  0  0 =  0  0 that Hence  0 is a solution of ( 14).
According Lemma 7, we get the following Corollary 8 immediately.
Proof.We consider the solution set of (18) in three cases according to its parameters.
The rest of the proof is evident based on cases 1, 2, and 3.
Proof.The proof lies in the process of the proof of Proposition 10.
. .Basic Concepts and Properties of Max-Product IPFRE.This subsection presents some basic concept and properties of the max-product fuzzy relation equations with intervalvalued numbers, i.e., system (12).
Definition .A minimal solution x ∈ (, ) is a solution satisfying  ∈ (, ) and  ≤ x indicate  = x ; a maximum solution x ∈ (, ) is a solution satisfying  ≤ x for any  ∈ (, ).Theorem 14.Let  ∈ [0, 1]  be an n-dimension vector.en  is a solution of system ( ) if and only if for any  ∈ , [  ,   ]  ≤ [  ,   ] holds for all  ∈  and there exists Proof.(⇒) Since  is a solution of (12), it satisfies the equation s According to the scalar-multiplication and supremum operations (see (b) and (c) in Section 2), Hence for any  ∈  and there exists some   ∈  such that (⇐) Analogously, it is easy to check the feasibility of  as a solution of (12), according to the scalar-multiplication and supremum operations on .

Complexity 7
According to Proposition 10, the single-variable inequality could be solved,  ∈ ,  ∈ .Assume that the solution set of ( 34) is ∈ ,  ∈ .We define the vector x based on the above solution sets (35) as follows: i.e.,  ≤ x.
As shown in Theorem 15, there does not exist any solution bigger that x.We call x the potential maximum solution of system (12).Moreover, x is called maximum solution if x ∈ (, ), while called pseudomaximum solution if x ∉ (, ).
Next we define some new concepts and consider the consistency of system (12).
Define the close index set and the open index set It is obvious that Proposition 17.In system ( ), if   ̸ = 0, then x ∉ (, ).
The following Corollary 18 could be easily obtained by Theorem 15 and Proposition 17.

Corollary 18. e vector x is the unique maximum solution of system ( ) if and only if 𝐽 𝑜𝑝𝑒𝑛 = 0 and x ∈ 𝑋(𝐴, 𝑏).
Based on the potential maximum solution x and the open index set, we define the vector where The vector x could be used to check the consistency of system (12) by the following Theorem 19.We call x the maximum close solution, if it is a solution of system (12).
(⇒) According to Theorem 14, we complete the proof from two aspects as follows: (i) Take arbitrary  ∈ ,  ∈ .
(ii) Notice that system (12) is consistent.There exists a solution  ∈ (,) to system (12).By Theorem 14, for arbitrary  ∈ , there exists some   ∈ , such that Firstly, we verify that On the other hand, it follows from Theorem 15 that  ≤ x, which indicates   푖 ≤ x 푖 .Thus it holds that   푖 = x 푖 .Secondly, we prove   ∉   by contradiction.Assume that   ∈   .Then there exists some  ∈  such that Inequality (48) shows that the vector  does not satisfy the th equation in system (12).Hence  is not a solution of ( 12), which is conflict with the assumption that  ∈ (, ).
At last, since   ∉   , it follows from ( 46) that x 푖 = x 푖 .So we get The above-proved points (i) and (ii) contribute to x ∈ (, ) by Theorem 14.
Note.In general, x is not the maximum solution even if system (12) is consistent.Moreover, it is possible that system (12) has no maximum solution.

. . Structure of the Solution Set of System ( ). Let
Each  ∈  is called a path of the characteristic matrix  (or system (12)).Thus  is the set of all paths.
Let  ∈  be a path.Denote the index sets Based on the path  and its corresponding index sets   1 ,   2 , . . .,    , we can define vector where Theorem 20.Let  ∈  be a path and x  be the vector defined by ( ) and ( ). en x  is a solution of system ( ).
Theorem 20 shows that, for any path  ∈  of system ( 12), x  is a solution corresponding to .Next we further construct the solution interval corresponding to .Based on the path  ∈ , define the vector where We call   the solution interval corresponding to the path .
Remark .It could be easily found from ( 56) that (i)   is a close set (or close interval) if and only if   = 0; (ii)   is an open set (or open interval) if and only if   = 0.
Theorem 22.Let  ∈  be a path and   be defined by ( ) based on .en it holds that   ⊆ (, ).
Proof.Take arbitrary  = ( 1 ,  2 , . . .,   ) ∈   .Then it follows from (56) that (i) We first prove it holds for all  ∈  and  ∈  that (58) and Lemma 7, we have On the other hand, when  ∈  >  , we have Suppose the solution set of the inequality Thus and for any  ∈ , there exists   ∈  such that Let  = ( Thus On the other hand, Theorem 15 indicates   푖 ≤ x 푖 .So we get As a result of ( 65) and ( 68), Combining ( 64) and ( 68), we get Thus  does not satisfy the th equation in system (12).This is conflict with the assumption that  ∈ (, ).
The above points (i)-(iii) contribute to  ∈   and the proof is complete.
Theorem 24 (structure of the solution set).Suppose system ( ) is consistent.en its complete solution set is where  =  1 ×  2 × ⋅ ⋅ ⋅ ×   is the set of all paths and   is the solution interval corresponding to the path .
Proof.It is simple corollary of Theorems 22 and 23.
. .Algorithm for Solving System ( ) Based on the Index Sets.Based on the above-defined index sets   ,   , and   ,  ∈ , we propose the following algorithm for obtaining the solution set of system (12).
Step .Compute the potential maximum solution x by (36).
Step .Obtain the close index set   and open index set   by ( 42) and (43), respectively.
Step .Check the consistency of system (12) by the vector x according to Theorems 14 and 19.If x ∈ (, ), then system ( 12) is consistent and continue to Step 6.Otherwise, system (12) is inconsistent and stop.
Step .For any  ∈ , compute the index sets  Step .Compute the corresponding solution x  by ( 52) and (53).
Step .According to Theorem 24, generate the complete solution set of system (12) by (76)
Proposition 1.If x = ( x 1 , x 2 , . . ., x  ) is a minimal solution of system ( ), then it holds for any  ∈  that x  ∈ {0, x }, where x is the potential maximum solution of ( ) as defined by ( ).In particular, when  ∈   , it holds that x  = 0.
Proof.Based on Theorems 23 and 24, any minimal solution could be selected from the set { x  |  ∈ }, i.e., and the expression of   (see (56)), the maximum solution should be x.Then it follows from Corollary 18 that   = 0.
Consequently, when the maximum solution exists, it should be x.Theorem 3. Let system ( ) be consistent with solution set (, ).en the following statements are equivalent: (i) (, ) is a close set; (ii)   = 0; (iii) x ∈ (, ); (iv) there exists a unique maximum solution of system ( ).
(i) ⇐⇒ (ii) Considering (, ) = ⋃ ∈   and the expression of   (see (56)), (, ) is a close set if and only if   is close set for any  ∈ .This is equivalent to   = 0.
. .Comparing the Solution Set of System ( ) to at of the Classical Max- Fuzzy Relation Equations.As well known to everyone, the solution set of a system of classical max- fuzzy relation equations, when nonempty, is determined by a unique maximum solution and a finite number of minimal solutions.The solution set could also be considered as a union of finite close intervals in form of  (, ) = ⋃ (98) Differences between the solution set of the intervalvalued-parameter max-product fuzzy relation equation (i.d.system (12)) and that of the classical max- fuzzy relation equation are shown as below.
(i) The classical max- fuzzy relation equation always has a maximum solution, when it is consistent.But in the consistent system (12), this property no longer holds.According to Proposition 2, the maximum solution exists if and only if its open index set   is empty.
(ii) As a union of finite close intervals, the solution set of classical max- fuzzy relation equation is always a close set.However, the solution set of system (12) might not be a close set.It turns out to be close set unless   = 0; i.e., the maximum solution exists, according to Theorem 3.

Conclusion
This paper pay attention to interval-valued fuzzy relation equations with max-product composition.Basic operations and order relation of the interval-valued numbers are introduced before description of the fuzzy relation equations system.In order to deal with such kind of system, we define concepts of close index set and open index set.We characterize the structure of the complete solution set to the intervalvalued fuzzy relation equation, which is different from that