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

Authors: Xiaobin Yang; Haitao Lin; Gang Xiao; Huanbin Xue; Xiaopeng Yang
Journal: Complexity
 (2019)

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## Abstract

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.


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## Body

## 1. 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 self-evident [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 max-min 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 [5–23].The composition plays important role in fuzzy relation equations. It was extended from max-min to max-product [24–27] 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 two-subproblem approach was widely adopted to deal with linear-objective optimization problem subject to fuzzy relation equations or inequalities with various composition such as max-min, max-product and max-average [30, 31]. Nonlinear-objective 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 nonlinear-objective 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 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 [39–41]. Besides, bipolar fuzzy relation equations was another interesting research object. The scholars were interested in the relevant linear optimization problem [42–44].As a special form of fuzzy number, interval was a powerful tool in both theoretical and practical application aspects [45–47]. Considering the fuzziness in the parameters, interval-valued 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 containedmax-t-norm [50],min-s-norm [51], andmax-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.


## 2. 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[u_,u¯], where u_,u¯∈R and u_≤u¯, R represents the real number set.Definition 1 ([53] (fuzzy number)).
A fuzzy setA~ in R is called a fuzzy number, if it satisfies the following conditions:(i)
A~ is normal; i.e., the cut set A1 is nonempty,(ii)
theα-level cut set Aα is a closed interval for any α∈(0,1],(iii)
the support ofA~ is bounded.An interval-valued number could be viewed as a special type of fuzzy number [54]. For interval-valued number [u_,u¯], the membership function is (1)μx=1,ifu_≤x≤u¯,0,otherwise. It is easy to check that [u_,u¯] satisfies the conditions in Definition 1. For more detail, the readers could refer to [55–57].Denote the set of all interval-valued numbers byIV={[u_,u¯]∣u_≤u¯,u_,u¯∈R}. We first present two basic operations on the set IV, which are commonly used in the existing works [58]. Let u_,u¯,[v_,v¯]∈IV and k∈R. Then(a)
(addition)[u_,u¯]+[v_,v¯]=[u_+v_,u¯+v¯];(b)
(scalar-multiplication)ku_,u¯=u_,u¯k={[ku_,ku¯],ifk≥0;[ku¯,ku_],ifk<0}.Besides, supremum (∨) and infimum (∧) are basic operations for establishing the fuzzy relation equations. These two operations depend on the order relation on the set IV. Now we investigate the order relation on the set IV first.In general, an order relation is defined for comparing two elements in a set. In the setIV, the most common order relation is the classical “product” order [59], denoted by “≲” in this paper. Let [u_,u¯],[v_,v¯]∈IV. We have [u_,u¯]≲[v_,v¯] if and only if both u_≤v_ and u¯≤u¯ hold, while [u_,u¯]=[v_,v¯] if and only if both u_=v_ and u¯=u¯ hold. It is easy to check that “≲” is a partial order on IV, but not a total order. Apartial order on a set is a binary relation satisfyingreflexivity,antisymmetry, andtransitivity. Atotal order is a partial order which makes any two elements in the setcomparable. Under the above-mentioned order “≲”, the supremum (∨~) and infimum (∧~) operations on the set IV are as follows: (2)u_,u¯∨~v_,v¯=u_∨v_,u¯∨v¯,u_,u¯∧~v_,v¯=u_∧v_,u¯∧v¯.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 [0.2,0.3]. Both [0.1,0.6]≲[0.2,0.3] and [0.2,0.3]≲[0.1,0.6] are incorrect.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 [62–64]. We recall the definition of admissible order on IV below.Definition 2 ([62] (admissible order)).
Let(IV,⪯) 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 IV;
(ii) for any[u_,u¯],[v_,v¯]∈IV, [u_,u¯]≲[v_,v¯] implies that [u_,u¯]⪯[v_,v¯], where ≲ represents the classical product order.For more detailed examples on admissible order, the readers could refer to [59, 62–64].In this paper, motivated by the order defined in [61] for the intuitionistic fuzzy set and the order introduced in [65–69] for comparison of two fuzzy numbers in solving fuzzy linear programming, we define the following order relation “<” on the set IV.Definition 3.
For any[u_,u¯],[v_,v¯]∈IV, we say [u_,u¯] is less than [v_,v¯], denoted by [u_,u¯]<[v_,v¯], if it holds that(i)
1/2(u_+u¯)<1/2(v_+v¯); or(ii)
1/2(u_+u¯)=1/2(v_+v¯) and 1/2(u¯-u_)>1/2(v¯-v_);
Besides,[u_,u¯] is equal to [v_,v¯], i.e., [u_,u¯]=[v_,v¯], if both 1/2(u_+u¯)=1/2(v_+v¯) and 1/2(u¯-u_)=1/2(v¯-v_) hold. Inequality [u_,u¯]≤[v_,v¯] represents either [u_,u¯]<[v_,v¯] or [u_,u¯]=[v_,v¯] holds. The dual symbols of < and ≤ are > and ≥, respectively.Under the above-defined order relation, it is easy to check that[u_,u¯]=[v_,v¯] if and only if u_=v_ and u¯=v¯. This is coincident with the classical product order. In Definition 3, 1/2(u_+u¯) is the midpoint of [u_,u¯], while 1/2(u¯-u_) means the radius of [u_,u¯] [70].Proposition 4.
For arbitrary[u_,u¯],[v_,v¯],[w_,w¯]∈IV, the following statements are true:(i)
(reflexivity)[u_,u¯]≤[u_,u¯];(ii)
(antisymmetry)[u_,u¯]≤[v_,v¯] and [v_,v¯]≤[u_,u¯] indicate [u_,u¯]=[v_,v¯];(iii)
(transitivity)[u_,u¯]≤[v_,v¯] and [v_,v¯]≤[w_,w¯] indicate [u_,u¯]≤[w_,w¯];(iv)
(comparability) either[u_,u¯]≤[v_,v¯] or [v_,v¯]≤[u_,u¯] holds;(v)
(refinement of the classical product order) if[u_,u¯]≲[v_,v¯], then [u_,u¯]≤[v_,v¯].Proof.
Omitted.It is shown in Proposition4 that the order relation “≤” in Definition 3 is an admissible order. Moreover, (IV, ≤) forms a total order set. Under such order relation, any pair of interval-valued numbers could be compared. Next we provide the supremum (∨) and infimum (∧) operations on IV, based on the above-defined order “≤”. Let [u_,u¯],[v_,v¯]∈IV. Then we have(c)
(supremum)u_,u¯∨v_,v¯=v_,v¯,ifu_,u¯≤v_,v¯;[u_,u¯],otherwise;(d)
(infimum)u_,u¯∧v_,v¯=u_,u¯,ifu_,u¯≤v_,v¯;v_,v¯,otherwise.


## 3. 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.


### 3.1. Unit Interval-Valued Number

An interval-valued number[u_,u¯] is said to be unit interval-valued number, if 0≤u_≤u¯≤1. We denote the set of all unit interval-valued numbers by (3)IV0,1=u_,u¯∣0≤u_≤u¯≤1,u_,u¯∈R, and the set of all nonzero (or positive) unit interval-valued numbers by (4)IV0,1=u_,u¯∣u¯>0,u_,u¯∈IV0,1.Remark 5.
According to the order given in Definition3, the set IV((0,1]) can also be written as(5)IV0,1=u_,u¯∣u_,u¯≠0,0,u_,u¯∈IV0,1, or(6)IV0,1=u_,u¯∣u_,u¯>0,0,u_,u¯∈IV0,1.Obviously bothIV([0,1]) and IV((0,1]) are subsets of IV. Hence the operations and order relation defined on IV in Section 2 keep valid for IV([0,1]) and IV((0,1]).


### 3.2. Classical Max-Min and Max-Product Fuzzy Relation Equations

For the convenience of expression, we denote two index setsI and J as follows:(7)I=1,2,…,m,J=1,2,…,n.System of fuzzy relation equations, with the most commonly usedmax-min composition [4], could be written as [19–21, 27, 29, 31](8)a11∧x1∨a12∧x2∨⋯∨a1n∧xn=b1,a21∧x1∨a22∧x2∨⋯∨a2n∧xn=b2,⋮am1∧x1∨am2∧x2∨⋯∨amn∧xn=bm,where aij,xj,bi∈[0,1], i∈I, j∈J. But soon after, themax-min composition was extended to themax-product one. Analogously, system of max-product fuzzy relation equations was expressed by(9)a11x1∨a12x2∨⋯∨a1nxn=b1,a21x1∨a22x2∨⋯∨a2nxn=b2,⋮am1x1∨am2x2∨⋯∨amnxn=bm,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 (peer-to-peer) network system. Suppose there exist n terminals in such system, denoted by Tj, j∈J. Each pair of terminals was wireless-connected. They transformed their local resources through electromagnetic wave. The terminal Tj transmits the electromagnetic wave with intensity xj. Due to the decreasing of the intensity, associated with the distance, the intensity of the electromagnetic wave is no more than xj, when it is received by Ti, i∈J and i≠j. Hence the intensity of electromagnetic wave at Ti, which is send out by Tj, could be represented by(10)aijxj, where 0≤aij≤1. We call aij the decreasing coefficient. In general case, Ti 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 Ti is bi>0, then the intensities of electromagnetic wave should satisfy(11)ai1x1∨ai2x2∨⋯∨ainxn=bi. Here, without loss of generality, we assume the first m terminals require the download quality level. Then we have i∈I={1,2,…,m}. 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).


### 3.3. 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 aij is determined by the distance between the terminals Ti and Tj. The value of aij will become smaller, if the distance of Ti and Tj becomes farther. However, in real world application, the distance might not be the unique factor influencing the decreasing coefficient aij. 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 aij is not a precise number. It could take any possible value within limits. Here we assume aij as an interval-valued number, i.e., aij=[a_ij,a¯ij]. Correspondingly, the requirement of the terminal Ti is also assumed to be a value range and denoted by [b_i,b¯i], i∈I.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(12)a_11,a¯11x1∨a_12,a¯12x2∨⋯∨a_1n,a¯1nxn=b_1,b¯1,a_21,a¯21x1∨a_22,a¯22x2∨⋯∨a_2n,a¯2nxn=b_2,b¯2,⋮a_m1,a¯m1x1∨a_m2,a¯m2x2∨⋯∨a_mn,a¯mnxn=b_m,b¯m, where the parameters [a_ij,a¯ij],[b_i,b¯i]∈IV((0,1]) are unit interval-valued numbers, i∈I, j∈J. System (12) consists of m max-product fuzzy relation equations with interval-valued parameters. System (12) could also be written as the following matrix form, (13)A⊙xT=bT, where A=(aij)m×n=([a_ij,a¯ij])m×n, b=(bi)1×m=([b_i,b¯i])1×m, x=(xj)1×n. In system (12), the order relation and operations are as defined in Section 2.


## 3.1. Unit Interval-Valued Number

An interval-valued number[u_,u¯] is said to be unit interval-valued number, if 0≤u_≤u¯≤1. We denote the set of all unit interval-valued numbers by (3)IV0,1=u_,u¯∣0≤u_≤u¯≤1,u_,u¯∈R, and the set of all nonzero (or positive) unit interval-valued numbers by (4)IV0,1=u_,u¯∣u¯>0,u_,u¯∈IV0,1.Remark 5.
According to the order given in Definition3, the set IV((0,1]) can also be written as(5)IV0,1=u_,u¯∣u_,u¯≠0,0,u_,u¯∈IV0,1, or(6)IV0,1=u_,u¯∣u_,u¯>0,0,u_,u¯∈IV0,1.Obviously bothIV([0,1]) and IV((0,1]) are subsets of IV. Hence the operations and order relation defined on IV in Section 2 keep valid for IV([0,1]) and IV((0,1]).


## 3.2. Classical Max-Min and Max-Product Fuzzy Relation Equations

For the convenience of expression, we denote two index setsI and J as follows:(7)I=1,2,…,m,J=1,2,…,n.System of fuzzy relation equations, with the most commonly usedmax-min composition [4], could be written as [19–21, 27, 29, 31](8)a11∧x1∨a12∧x2∨⋯∨a1n∧xn=b1,a21∧x1∨a22∧x2∨⋯∨a2n∧xn=b2,⋮am1∧x1∨am2∧x2∨⋯∨amn∧xn=bm,where aij,xj,bi∈[0,1], i∈I, j∈J. But soon after, themax-min composition was extended to themax-product one. Analogously, system of max-product fuzzy relation equations was expressed by(9)a11x1∨a12x2∨⋯∨a1nxn=b1,a21x1∨a22x2∨⋯∨a2nxn=b2,⋮am1x1∨am2x2∨⋯∨amnxn=bm,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 (peer-to-peer) network system. Suppose there exist n terminals in such system, denoted by Tj, j∈J. Each pair of terminals was wireless-connected. They transformed their local resources through electromagnetic wave. The terminal Tj transmits the electromagnetic wave with intensity xj. Due to the decreasing of the intensity, associated with the distance, the intensity of the electromagnetic wave is no more than xj, when it is received by Ti, i∈J and i≠j. Hence the intensity of electromagnetic wave at Ti, which is send out by Tj, could be represented by(10)aijxj, where 0≤aij≤1. We call aij the decreasing coefficient. In general case, Ti 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 Ti is bi>0, then the intensities of electromagnetic wave should satisfy(11)ai1x1∨ai2x2∨⋯∨ainxn=bi. Here, without loss of generality, we assume the first m terminals require the download quality level. Then we have i∈I={1,2,…,m}. 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).


## 3.3. 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 aij is determined by the distance between the terminals Ti and Tj. The value of aij will become smaller, if the distance of Ti and Tj becomes farther. However, in real world application, the distance might not be the unique factor influencing the decreasing coefficient aij. 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 aij is not a precise number. It could take any possible value within limits. Here we assume aij as an interval-valued number, i.e., aij=[a_ij,a¯ij]. Correspondingly, the requirement of the terminal Ti is also assumed to be a value range and denoted by [b_i,b¯i], i∈I.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(12)a_11,a¯11x1∨a_12,a¯12x2∨⋯∨a_1n,a¯1nxn=b_1,b¯1,a_21,a¯21x1∨a_22,a¯22x2∨⋯∨a_2n,a¯2nxn=b_2,b¯2,⋮a_m1,a¯m1x1∨a_m2,a¯m2x2∨⋯∨a_mn,a¯mnxn=b_m,b¯m, where the parameters [a_ij,a¯ij],[b_i,b¯i]∈IV((0,1]) are unit interval-valued numbers, i∈I, j∈J. System (12) consists of m max-product fuzzy relation equations with interval-valued parameters. System (12) could also be written as the following matrix form, (13)A⊙xT=bT, where A=(aij)m×n=([a_ij,a¯ij])m×n, b=(bi)1×m=([b_i,b¯i])1×m, x=(xj)1×n. In system (12), the order relation and operations are as defined in Section 2.


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

### 4.1. Resolution of[a_0,a¯0]x0=[b_0,b¯0] and [a_0,a¯0]x0≤[b_0,b¯0]

Proposition 6.
Let[a_0,a¯0],[b_0,b¯0]∈IV((0,1]). The equation(14)a_0,a¯0x0=b_0,b¯0is solvable (has at least one solution) if and only if a_0b¯0=a¯0b_0.Proof.
(⇒) Notice that the equation [a_0,a¯0]x0=[b_0,b¯0] is equivalent to (15)a_0x0=b_0,a¯0x0=b¯0.
Ifa_0=0, then a_0x0=b_0 indicates b_0=0. So we get a_0b¯0=0=a¯0b_0.
Ifa_0≠0, then a_0x0=b_0 indicates x0=b_0/a_0, while a¯0x0=b¯0 indicates x0=b¯0/a¯0. Combinatorially, we have b_0/a_0=x0=b¯0/a¯0, i.e., a_0b¯0=a¯0b_0.
(⇐) Since [a_0,a¯0]∈IV((0,1]), it is clear that a_0+a¯0>0. Set (16)x0=b_0+b¯0a_0+a¯0. Then it follows from a_0b¯0=a¯0b_0 that(17)a_0,a¯0x0=a_0,a¯0b_0+b¯0a_0+a¯0=a_0b_0+b¯0a_0+a¯0,a¯0b_0+b¯0a_0+a¯0=a_0b_0+a_0b¯0a_0+a¯0,a¯0b_0+a¯0b¯0a_0+a¯0=a_0b_0+a¯0b_0a_0+a¯0,a_0b¯0+a¯0b¯0a_0+a¯0=a_0+a¯0b_0a_0+a¯0,a_0+a¯0b¯0a_0+a¯0=b_0,b¯0. Hence x0 is a solution of (14).Lemma 7.
Let[a_0,a¯0]∈IV((0,1]). We have(i)
ifx0≤y0, then [a_0,a¯0]x0≤[a_0,a¯0]y0;(ii)
ifx0<y0, then [a_0,a¯0]x0<[a_0,a¯0]y0.Proof.
The proof is trivial.According Lemma7, we get the following Corollary 8 immediately.Corollary 8.
Let[a_0,a¯0]∈IV((0,1]). If y1 is a solution of the inequality [a_0,a¯0]x0≤[b_0,b¯0], then any y2 satisfying y2≤y1 is also a solution of [a_0,a¯0]x0≤[b_0,b¯0].Furthermore, combining Proposition6 and Lemma 7, we get Corollary 9 below.Corollary 9.
Let[a_0,a¯0],[b_0,b¯0]∈IV((0,1]). If a_0b¯0=a¯0b_0, then the equation [a_0,a¯0]x0=[b_0,b¯0] has a unique solution x0=b_0+b¯0/a_0+a¯0.Proposition 10.
Let[a_0,a¯0],[b_0,b¯0]∈IV((0,1]) and x0∈[0,1]. The inequality(18)a_0,a¯0x0≤b_0,b¯0is solvable and its solution set is either in form of (19)0,y0, or in form of (20)0,y0, where y0=b_0+b¯0/a_0+a¯0∧1. In particular, the solution set of (18) is [0,y0) if and only if a_0b¯0>a¯0b_0, and a_0+a¯0≥b_0+b¯0.Proof.
We consider the solution set of (18) in three cases according to its parameters.
Case 1. If a_0b_0+b¯0/a_0+a¯0≤b_0, i.e., a_0b¯0≤a¯0b_0, then(21)a_0,a¯0y0≤a_0,a¯0b_0+b¯0a_0+a¯0=a_0b_0+b¯0a_0+a¯0,a¯0b_0+b¯0a_0+a¯0Denote [a_0(b_0+b¯0)/a_0+a¯0,a¯0(b_0+b¯0)/a_0+a¯0]=[u_,u¯]. Now we check that(22)u_,u¯≤b_0,b¯0by the following points (i) and (ii), according to Definition 3.(i)
1/2(u_+u¯)=1/2(a_0(b_0+b¯0)/a_0+a¯0+a¯0(b_0+b¯0)/a_0+a¯0)=1/2(b_0+b¯0).(ii)
1/2(u¯-u_)=1/2(a¯0(b_0+b¯0)/a_0+a¯0-a_0(b_0+b¯0)/a_0+a¯0)=1/2(a¯0b_0+a¯0b¯0-a_0b_0-a_0b¯0/a_0+a¯0). Since a_0b¯0≤a¯0b_0, we have(23)12u¯-u_=12a¯0b_0+a¯0b¯0-a_0b_0-a_0b¯0a_0+a¯0≥12a_0b¯0+a¯0b¯0-a_0b_0-a¯0b_0a_0+a¯0=12b¯0-b_0.
Combining (21) and (22) we get [a_0,a¯0]y0≤[u_,u¯]≤[b_0,b¯0]; i.e., y0 is a solution of (18). According to Corollary 8, [0,y0] is a subset of the solution set of (18). Next we verify that any y>y0 is not a solution of (18) in two situations.
Whenb_0+b¯0/a_0+a¯0≤1, we have y0=b_0+b¯0/a_0+a¯0. Since y>y0≥0 and a_0+a¯0>0, it is clear that (a_0+a¯0)y>(a_0+a¯0)y0. Hence the midpoint of [a_0,a¯0]y=[a_0y,a¯0y] is(24)a_0y+a¯0y2=a_0+a¯02y>a_0+a¯02y0=a_0+a¯02b_0+b¯0a_0+a¯0=b_0+b¯02. According to Definition 2, [a_0,a¯0]y>[b_0,b¯0]; i.e., y is not a solution of (18).
Whenb_0+b¯0/a_0+a¯0>1, we have y0=1, then it is obvious that any y>y0 is not a solution of (18), due to the assumption of the variable that x0∈[0,1].
Consequently, in this case, the solution set of (18) is [0,y0].
Case 2. If a_0b_0+b¯0/a_0+a¯0>b_0, i.e., a_0b¯0>a¯0b_0, and b_0+b¯0/a_0+a¯0>1, then y0=b_0+b¯0/a_0+a¯0∧1=1 and 1/2(a_0+a¯0)<1/2(b_0+b¯0). It follows from Definition 3 that [a_0,a¯0]<[b_0,b¯0], i.e., [a_0,a¯0]1<[b_0,b¯0]. Considering Lemma 7 and the assumption of the variable that x0∈[0,1], the solution set of (18) is [0,1]=[0,y0].
Case 3. If a_0b_0+b¯0/a_0+a¯0>b_0, i.e., a_0b¯0>a¯0b_0, and b_0+b¯0/a_0+a¯0≤1, then y0=b_0+b¯0/a_0+a¯0∧1=b_0+b¯0/a_0+a¯0.
Now we compare the interval-value number[a_0,a¯0]x0 with [b_0,b¯0] in three cases as below.
(i) Whenx0=b_0+b¯0/a_0+a¯0=y0, in the similar way of (i) and (ii) in case 1, the midpoint of [a_0,a¯0]x0 is exactly 1/2(u_+u¯)=1/2(b_0+b¯0), which is equal to that of [b_0,b¯0]. On the other hand, since a_0b¯0>a¯0b_0, the radius of [a_0,a¯0]x0 is (25)12u¯-u_=12a¯0b_0+a¯0b¯0-a_0b_0-a_0b¯0a_0+a¯0<12a_0b¯0+a¯0b¯0-a_0b_0-a¯0b_0a_0+a¯0=12b¯-b_, which is less than that of [b_0,b¯0]. Hence it holds that [a_0,a¯0]x0=[a_0,a¯0]y0>[b_0,b¯0].
(ii) Whenx0>b_0+b¯0/a_0+a¯0=y0, according to Lemma 7 and the result in the above point (i), it is clear that [a_0,a¯0]x0>[a_0,a¯0]y0>[b_0,b¯0].
(iii) Whenx0<b_0+b¯0/a_0+a¯0=y0, we have(26)12a_0+a¯0x0<12a_0+a¯0b_0+b¯0a_0+a¯0=12b_0+b¯0,i.e., the midpoint of [a_0,a¯0]x0 is less than that of [b_0,b¯0]. Hence [a_0,a¯0]x0<[b_0,b¯0].
Combining the above points (i)-(iii), it is obvious that the solution set of (18) is [0,y0).
The rest of the proof is evident based on cases 1, 2, and 3.Proposition 11.
For the inequality[a_0,a¯0]x0≤[b_0,b¯0], we have the following:(i)
if its solution set is[0,y0), then(27)a_0,a¯0x0<b_0,b¯0,if0≤x0<y0,>b_0,b¯0,ify0≤x0≤1;(ii)
if its solution set is[0,y0], then(28)a_0,a¯0x0≤b_0,b¯0,if0≤x0≤y0,>b_0,b¯0,ify0<x0≤1.
In particular, if[a_0,a¯0]x0=[b_0,b¯0], then the solution set should be [0,y0] and x0=y0.Proof.
The proof lies in the process of the proof of Proposition10.


### 4.2. Basic Concepts and Properties of Max-Product IPFRE

This subsection presents some basic concept and properties of the max-product fuzzy relation equations with interval-valued numbers, i.e., system (12).We adopt the following order relation for comparing two n-dimension vectors. For anyx=(x1,x2,…,xn),y=(y1,y2,…,yn)∈[0,1]n, x≤y (x<y, x=y) means xj≤yj (xj<yj, xj=yj) holds for all j∈J, while x⪇y means both x≤y and x≠y hold.Denote the complete solution set of system (12) by (29)XA,b=x∈0,1n∣A⊙xT=bT.Definition 12.
IfX(A,b)≠∅(X(A,b)=∅), system (12) is said to be consistent (inconsistent),.Definition 13.
A minimal solutionxˇ∈X(A,b) is a solution satisfying x∈X(A,b) and x≤xˇ indicate x=xˇ; a maximum solution x^∈X(A,b) is a solution satisfying x≤x^ for any x∈X(A,b).Theorem 14.
Letx∈[0,1]n be an n-dimension vector. Then x is a solution of system (12) if and only if for any i∈I, [a_ij,a¯ij]xj≤[b_i,b¯i] holds for all j∈J and there exists ji∈J such that [a_iji,a¯iji]xji=[b_i,b¯i].Proof.
(⇒) Since x is a solution of (12), it satisfies the equation s(30)a_i1,a¯i1x1∨a_i2,a¯i2x2∨⋯∨a_in,a¯inxn=b_i,b¯i,i∈I.According to the scalar-multiplication and supremum operations (see (b) and (c) in Section 2),(31)a_i1,a¯i1x1∨a_i2,a¯i2x2∨⋯∨a_in,a¯inxn=⋁j∈Ja_ij,a¯ijxj=⋁j∈Ja_ijxj,a¯ijxj, Hence (32)a_ijxj,a¯ijxj≤⋁j∈Ja_ijxj,a¯ijxj=b_i,b¯i for any j∈J and there exists some ji∈J such that (33)a_ijixji,a¯ijixji=⋁j∈Ja_ijxj,a¯ijxj=b_i,b¯i, i.e., [a_ij,a¯ij]xj≤[b_i,b¯i] and [a_iji,a¯iji]xji=[b_i,b¯i].
(⇐) Analogously, it is easy to check the feasibility of x as a solution of (12), according to the scalar-multiplication and supremum operations on IV.According to Proposition10, the single-variable inequality(34)a_ij,a¯ijxj≤b_i,b¯icould be solved, i∈I, j∈J. Assume that the solution set of (34) is(35)0,yijor0,yij,i∈I, j∈J. We define the vector x^ based on the above solution sets (35) as follows:(36)x^=x^1,x^2,…,x^n=⋀i∈Iyi1,⋀i∈Iyi2,…,⋀i∈Iyin.Theorem 15.
SupposeX(A,b)≠∅. Then for any z∈X(A,b), it holds that z≤x^.Proof.
Sincez=(z1,z2,…,zn,)∈X(A,b), according to Theorem 14 we have(37)a_ij,a¯ijzj≤b_i,b¯i,∀i∈I,j∈J. That is to say, zj is a solution of the equation [a_ij,a¯ij]xj≤[b_i,b¯i]. Notice that the solution set of [a_ij,a¯ij]xj≤[b_i,b¯i] is either [0,yij] or [0,yij). Hence zj≤yij for all i∈I,j∈J. So we further get(38)zj≤⋀i∈Iyij=x^j,j∈J, i.e., z≤x^.As shown in Theorem15, there does not exist any solution bigger that x^. We call x^ thepotential maximum solution of system (12). Moreover, x^ is calledmaximum solution if x^∈X(A,b), while calledpseudomaximum solution if x^∉X(A,b).Theorem 16.
Letz1,z2∈[0,1] and z1≤z≤z2. If z1,z2∈X(A,b), then z∈X(A,b).Proof.
According to Theorem14, z2∈X(A,b) indicates(39)a_ij,a¯ijzj2≤b_i,b¯i,∀i∈I. Besides, by Lemma 7, z≤z2 implies that(40)a_ij,a¯ijzj≤a_ij,a¯ijzj2≤b_i,b¯i,∀i∈I.On the other hand, again by Theorem 14, z1∈X(A,b) indicates, for any i∈I, there exists ji∈J such that [a_iji,a¯iji]zji1=[b_i,b¯i]. It follows from Lemma 7 and the inequality z1≤z that(41)a_iji,a¯ijizji≥a_iji,a¯ijizji1=b_i,b¯i.Considering inequality (40) we have [a_iji,a¯iji]zji≤[b_i,b¯i]. Thus it holds that [a_iji,a¯iji]zji=[b_i,b¯i]. Combining (40), it follows from Theorem 14 that z∈X(A,b).Next we define some new concepts and consider the consistency of system (12).Define theclose index set(42)Jclose=j∈J∣a_ij,a¯ijx^j≤b_i,b¯ifor  alli∈I,and theopen index set(43)Jopen=j∈J∣a_ij,a¯ijx^j>b_i,b¯ifor  somei∈I.It is obvious that (44)Jclose∩Jopen=∅,Jclose∪Jopen=J.Proposition 17.
In system (12), if Jopen≠∅, then x^∉X(A,b).Proof.
IfJopen≠∅, then there exists j∈J and i∈I such that [a_ij,a¯ij]x^j>[b_i,b¯i] by (43). According to Theorem 14, x^ is not a solution of system (12).The following Corollary18 could be easily obtained by Theorem 15 and Proposition 17.Corollary 18.
The vectorx^ is the unique maximum solution of system (12) if and only if Jopen=∅ and x^∈X(A,b).Based on the potential maximum solutionx^ and the open index set, we define the vector(45)x_^=x_^1,x_^2,…,x_^n,where(46)x_^j=0,ifj∈Jopen;x^j,ifj∉Jopen;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).Theorem 19.
System (12) is consistent, i.e., X(A,b)≠∅, if and only if x_^∈X(A,b).Proof.
(⇐) It is obvious.
(⇒) According to Theorem 14, we complete the proof from two aspects as follows:
(i) Take arbitraryi∈I, j∈J. If j∈Jopen, then [a_ij,a¯ij]x_^j=[a_ij,a¯ij]0=[0,0]≤[b_i,b¯i]. While if j∉Jopen, then j∈Jclose, and [a_ij,a¯ij]x_^j=[a_ij,a¯ij]x^j≤[b_i,b¯i].
(ii) Notice that system (12) is consistent. There exists a solution z∈X(A,b) to system (12). By Theorem 14, for arbitrary i∈I, there exists some ji∈J, such that [a_iji,a¯iji]zji=[b_i,b¯i].
Firstly, we verify thatzji=x^ji. Since [a_iji,a¯iji]zji=[b_i,b¯i], it follows from Proposition 11 that the solution of the inequality [a_iji,a¯iji]xji=[b_i,b¯i] is [0,yiji] and zji=yiji. So we get(47)zji=yiji≥⋀k=1mykji=x^ji. On the other hand, it follows from Theorem 15 that z≤x^, which indicates zji≤x^ji. Thus it holds that zji=x^ji.
Secondly, we proveji∉Jopen by contradiction. Assume that ji∈Jopen. Then there exists some k∈I such that [a_kji,a¯kji]x^ji>[b_k,b¯k]. Considering zji=x^ji, we have(48)⋁j=1na_kj,a¯kjzj≥a_kji,a¯kjizji=a_kji,a¯kjix^ji>b_k,b¯k.Inequality (48) shows that the vector z does not satisfy the kth equation in system (12). Hence z is not a solution of (12), which is conflict with the assumption that z∈X(A,b).
At last, sinceji∉Jopen, it follows from (46) that x_^ji=x^ji. So we get(49)a_iji,a¯ijix_^ji=a_iji,a¯ijix^ji=a_iji,a¯ijizji=b_i,b¯i
The above-proved points (i) and (ii) contribute tox_^∈X(A,b) 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.


### 4.3. Structure of the Solution Set of System (12)

LetP=J1×J2×⋯×Jm, where(50)Ji=j∈J∣j∈Jcloseanda_ij,a¯ijx^j=b_i,b¯i,i∈I.Each p∈P is called a path of the characteristic matrix C (or system (12)). Thus P is the set of all paths.Letp∈P be a path. Denote the index sets(51)Ijp=i∈I∣ji=j,j∈J.Based on the path P and its corresponding index sets I1p,I2p,…,Inp, we can define vector(52)xˇp=xˇ1p,xˇ2p,…,xˇnp,where(53)xˇjp=0,ifIjp=∅;x^j,otherwise.Theorem 20.
Letp∈P be a path and xˇp be the vector defined by (52) and (53). Then xˇp is a solution of system (12).Proof.
(i) Take arbitraryi∈I and j∈J.
IfIjp=∅, then xˇjp=0. Hence [a_ij,a¯ij]xˇjp=[a_ij,a¯ij]0=[0,0]≤[b_i,b¯i].
IfIjp≠∅, then xˇjp=x^j. By (51), there exists i∈I such that ji=j. Remind that p=(j1,j2,…,jm)∈J1×J2×⋯×Jm. It is clear ji∈Ji, which indicates ji∈Jclose,i.e.,j∈Jclose. According to the definition of Jclose (see (42)), we have [a_ij,a¯ij]xˇjp=[a_ij,a¯ij]x^j≤[b_i,b¯i].
(ii) Take arbitraryi∈I. Notice p=(j1,j2,…,jm). We denote ji=j′. It is clear that i∈Ij′p, which indicates Ij′p≠∅. Hence(54)xˇj′p=x^j′,i.e.,xˇjip=x^ji.On the other hand, ji∈Ji implies(55)a_iji,a¯ijix^ji=b_i,b¯i.Equations (54) and (55) contribute to [a_iji,a¯iji]xˇjip=[b_i,b¯i].
Based on (i) and (ii), it follows from Theorem14 that xˇp∈X(A,b).Theorem20 shows that, for any path p∈P of system (12), xˇp is a solution corresponding to p. Next we further construct the solution interval corresponding to p. Based on the path p∈P, define the vector Xp=(X1p,X2p,…,Xnp), where(56)Xjp=xˇjp,x^j,ifj∈Jclose;xˇjp,x^j,ifj∈Jopen.We callXp thesolution interval corresponding to the path p.Remark 21.
It could be easily found from (56) that(i)
Xp is a close set (or close interval) if and only if Jopen=∅;(ii)
Xp is an open set (or open interval) if and only if Jclose=∅.Theorem 22.
Letp∈P be a path and Xp be defined by (56) based on p. Then it holds that Xp⊆X(A,b).Proof.
Take arbitraryy=(y1,y2,…,yn)∈Xp. Then it follows from (56) that (57)j∈Jclose⇒xˇjp≤yj≤x^j,(58)j∈Jopen⇒xˇjp≤yj<x^j,
(i) We first prove it holds for alli∈I and j∈J that [a_ij,a¯ij]yj≤[b_i,b¯i] in tow cases.
Case 1. If j∈Jclose, then [a_ij,a¯ij]yj≤[a_ij,a¯ij]x^j≤[b_i,b¯i], ∀i∈I.
Case 2. If j∈Jopen, then the index set Ij>={i∈I|[a_ij,a¯ij]x^j>[b_i,b¯i]}≠∅. Obviously, when i≠Ij>, it holds that [a_ij,a¯ij]x^j≤[b_i,b¯i]. Combining (58) and Lemma 7, we have (59)a_ij,a¯ijyj<a_ij,a¯ijx^j≤b_i,b¯i.On the other hand, when i∈Ij>, we have(60)a_ij,a¯ijx^j>b_i,b¯i.Suppose the solution set of the inequality [a_ij,a¯ij]xj>[b_i,b¯i] is [0,yij) or [0,yij]. Then(61)x^j=⋀k∈Iykj≤yij.Thus(62)a_ij,a¯ijyij≥a_ij,a¯ijx^j>b_i,b¯i.According to Proposition 11, the solution set of [a_ij,a¯ij]xj>[b_i,b¯i] should be [0,yij). It follows from (58) and (61) that yj<yij. Again by Proposition 11 we get [a_ij,a¯ij]yj<[b_i,b¯i].
(ii) For anyi∈I, there exists ji∈Ji⊆J (due to the existence of p=(j1,j2,…,jm)∈J1×J2×⋯×Jm), such that [a_iji,a¯iji]xˇjip=[b_i,b¯i]. By (57) and (58),(63)a_iji,a¯ijiyji≥xˇjip=b_i,b¯i. On the other hand, as a result of (i), [a_iji,a¯iji]yji≤[b_i,b¯i] holds. These two aspects lead to [a_iji,a¯iji]yji=[b_i,b¯i].
Combining (i) and (ii), it follows from Theorem14 that y∈X(A,b). Due to the arbitrariness of y, the inclusion relation Xp⊆X(A,b) holds.Theorem 23.
For anyy∈X(A,b), there exists a path p∈P such that y∈Xp.Proof.
Sincey∈X(A,b), it follows from Theorem 14 that(64)a_ij,a¯ijyj≤b_i,b¯i,and for any i∈I, there exists ji∈J such that(65)a_iji,a¯ijiyji=b_i,b¯i.Let p=(j1,j2,…,jm).
(i) We first verify thatp∈P, i.e., ji∈Ji, ∀i∈I.
Notice that[a_iji,a¯iji]yji=[b_i,b¯i]. Considering the inequality [a_iji,a¯iji]xji=[b_i,b¯i], it follows from Proposition 11 that its solution set should be [0,yiji], and moreover,(66)yji=yiji.Thus (67)x^ji=⋀k∈Iykji≤yiji=yji.On the other hand, Theorem 15 indicates yji≤x^ji. So we get(68)x^ji=yji.As a result of (65) and (68),(69)a_iji,a¯ijix^ji=a_iji,a¯ijiyji=b_i,b¯i.Combining (64) and (68), we get(70)a_kji,a¯kjix^ji=a_kji,a¯kjiyji≤b_k,b¯k,∀k∈I,which indicates(71)j∈Jclose.(69) and (71) contribute to ji∈Ji.
(ii) We now checkxˇp≤y.
Take arbitraryj∈J. If Ijp≠∅, then xˇjp=x^j. Besides, since Ijp≠∅, there exists i∈Ijp⊆I such that ji=j. By (68) we have x^j=x^ji=yji=yj. Hence (72)xˇjp=x^j=yj.If Ijp=∅, then xˇjp=0≤yj. Consequently, it holds that xˇjp≤yj for all j∈J, i.e., xˇp≤y.
(iii) According to Theorem15,(73)yj≤x^j,∀j∈J.Obviously yj≤x^j holds for any j∈Jclose. While for j∈Jopen, we have to verify that yj≤x^j (by contradiction). Otherwise, due to (73), it turns out to be yj=x^j. By (43), j∈Jopen indicates there exists i∈I such that [a_ij,a¯ij]x^j>[b_i,b¯i]. So we get(74)a_i1,a¯i1y1∨a_i2,a¯i2y2∨⋯∨a_in,a¯inyn≥a_ij,a¯ijyj=a_ij,a¯ijx^j>b_i,b¯i. Thus y does not satisfy the jth equation in system (12). This is conflict with the assumption that y∈X(A,b).
The above points (i)-(iii) contribute toy∈Xp and the proof is complete.Theorem 24 (structure of the solution set).
Suppose system (12) is consistent. Then its complete solution set is(75)XA,b=⋃p∈PXp,where P=J1×J2×⋯×Jm is the set of all paths and Xp is the solution interval corresponding to the path p.Proof.
It is simple corollary of Theorems22 and 23.


### 4.4. Algorithm for Solving System (12) Based on the Index Sets

Based on the above-defined index setsJclose, Jopen, and Ji, i∈I, we propose the following algorithm for obtaining the solution set of system (12).Algorithm for Solving System (12)Step 1. Solve the inequality [a_ij,a¯ij]xj≤[b_i,b¯i], for any i∈I and j∈J, by Proposition 10. Suppose the solution set is Sij=[0,yij) or Sij=[0,yij], where i∈I and j∈J.Step 2. Compute the potential maximum solution x^ by (36).Step 3. Obtain the close index set Jclose and open index set Jopen by (42) and (43), respectively.Step 4. Compute the vector x_^ by (45) and (46).Step 5. Check the consistency of system (12) by the vector x_^ according to Theorems 14 and 19. If x_^∈X(A,b), then system (12) is consistent and continue to Step 6. Otherwise, system (12) is inconsistent and stop.Step 6. Compute the index sets J1,J2,…,Jm by (50). Let P=J1×J2×⋯×Jm.Step 7. For any p∈P, compute the index sets I1p,I2p,…,Inp by (51).Step 8. Compute the corresponding solution xˇp by (52) and (53).Step 9. Construct the solution interval Xp by (56).Step 10. According to Theorem 24, generate the complete solution set of system (12) by(76)XA,b=⋃p∈PXp.


## 4.1. Resolution of[a_0,a¯0]x0=[b_0,b¯0] and [a_0,a¯0]x0≤[b_0,b¯0]

Proposition 6.
Let[a_0,a¯0],[b_0,b¯0]∈IV((0,1]). The equation(14)a_0,a¯0x0=b_0,b¯0is solvable (has at least one solution) if and only if a_0b¯0=a¯0b_0.Proof.
(⇒) Notice that the equation [a_0,a¯0]x0=[b_0,b¯0] is equivalent to (15)a_0x0=b_0,a¯0x0=b¯0.
Ifa_0=0, then a_0x0=b_0 indicates b_0=0. So we get a_0b¯0=0=a¯0b_0.
Ifa_0≠0, then a_0x0=b_0 indicates x0=b_0/a_0, while a¯0x0=b¯0 indicates x0=b¯0/a¯0. Combinatorially, we have b_0/a_0=x0=b¯0/a¯0, i.e., a_0b¯0=a¯0b_0.
(⇐) Since [a_0,a¯0]∈IV((0,1]), it is clear that a_0+a¯0>0. Set (16)x0=b_0+b¯0a_0+a¯0. Then it follows from a_0b¯0=a¯0b_0 that(17)a_0,a¯0x0=a_0,a¯0b_0+b¯0a_0+a¯0=a_0b_0+b¯0a_0+a¯0,a¯0b_0+b¯0a_0+a¯0=a_0b_0+a_0b¯0a_0+a¯0,a¯0b_0+a¯0b¯0a_0+a¯0=a_0b_0+a¯0b_0a_0+a¯0,a_0b¯0+a¯0b¯0a_0+a¯0=a_0+a¯0b_0a_0+a¯0,a_0+a¯0b¯0a_0+a¯0=b_0,b¯0. Hence x0 is a solution of (14).Lemma 7.
Let[a_0,a¯0]∈IV((0,1]). We have(i)
ifx0≤y0, then [a_0,a¯0]x0≤[a_0,a¯0]y0;(ii)
ifx0<y0, then [a_0,a¯0]x0<[a_0,a¯0]y0.Proof.
The proof is trivial.According Lemma7, we get the following Corollary 8 immediately.Corollary 8.
Let[a_0,a¯0]∈IV((0,1]). If y1 is a solution of the inequality [a_0,a¯0]x0≤[b_0,b¯0], then any y2 satisfying y2≤y1 is also a solution of [a_0,a¯0]x0≤[b_0,b¯0].Furthermore, combining Proposition6 and Lemma 7, we get Corollary 9 below.Corollary 9.
Let[a_0,a¯0],[b_0,b¯0]∈IV((0,1]). If a_0b¯0=a¯0b_0, then the equation [a_0,a¯0]x0=[b_0,b¯0] has a unique solution x0=b_0+b¯0/a_0+a¯0.Proposition 10.
Let[a_0,a¯0],[b_0,b¯0]∈IV((0,1]) and x0∈[0,1]. The inequality(18)a_0,a¯0x0≤b_0,b¯0is solvable and its solution set is either in form of (19)0,y0, or in form of (20)0,y0, where y0=b_0+b¯0/a_0+a¯0∧1. In particular, the solution set of (18) is [0,y0) if and only if a_0b¯0>a¯0b_0, and a_0+a¯0≥b_0+b¯0.Proof.
We consider the solution set of (18) in three cases according to its parameters.
Case 1. If a_0b_0+b¯0/a_0+a¯0≤b_0, i.e., a_0b¯0≤a¯0b_0, then(21)a_0,a¯0y0≤a_0,a¯0b_0+b¯0a_0+a¯0=a_0b_0+b¯0a_0+a¯0,a¯0b_0+b¯0a_0+a¯0Denote [a_0(b_0+b¯0)/a_0+a¯0,a¯0(b_0+b¯0)/a_0+a¯0]=[u_,u¯]. Now we check that(22)u_,u¯≤b_0,b¯0by the following points (i) and (ii), according to Definition 3.(i)
1/2(u_+u¯)=1/2(a_0(b_0+b¯0)/a_0+a¯0+a¯0(b_0+b¯0)/a_0+a¯0)=1/2(b_0+b¯0).(ii)
1/2(u¯-u_)=1/2(a¯0(b_0+b¯0)/a_0+a¯0-a_0(b_0+b¯0)/a_0+a¯0)=1/2(a¯0b_0+a¯0b¯0-a_0b_0-a_0b¯0/a_0+a¯0). Since a_0b¯0≤a¯0b_0, we have(23)12u¯-u_=12a¯0b_0+a¯0b¯0-a_0b_0-a_0b¯0a_0+a¯0≥12a_0b¯0+a¯0b¯0-a_0b_0-a¯0b_0a_0+a¯0=12b¯0-b_0.
Combining (21) and (22) we get [a_0,a¯0]y0≤[u_,u¯]≤[b_0,b¯0]; i.e., y0 is a solution of (18). According to Corollary 8, [0,y0] is a subset of the solution set of (18). Next we verify that any y>y0 is not a solution of (18) in two situations.
Whenb_0+b¯0/a_0+a¯0≤1, we have y0=b_0+b¯0/a_0+a¯0. Since y>y0≥0 and a_0+a¯0>0, it is clear that (a_0+a¯0)y>(a_0+a¯0)y0. Hence the midpoint of [a_0,a¯0]y=[a_0y,a¯0y] is(24)a_0y+a¯0y2=a_0+a¯02y>a_0+a¯02y0=a_0+a¯02b_0+b¯0a_0+a¯0=b_0+b¯02. According to Definition 2, [a_0,a¯0]y>[b_0,b¯0]; i.e., y is not a solution of (18).
Whenb_0+b¯0/a_0+a¯0>1, we have y0=1, then it is obvious that any y>y0 is not a solution of (18), due to the assumption of the variable that x0∈[0,1].
Consequently, in this case, the solution set of (18) is [0,y0].
Case 2. If a_0b_0+b¯0/a_0+a¯0>b_0, i.e., a_0b¯0>a¯0b_0, and b_0+b¯0/a_0+a¯0>1, then y0=b_0+b¯0/a_0+a¯0∧1=1 and 1/2(a_0+a¯0)<1/2(b_0+b¯0). It follows from Definition 3 that [a_0,a¯0]<[b_0,b¯0], i.e., [a_0,a¯0]1<[b_0,b¯0]. Considering Lemma 7 and the assumption of the variable that x0∈[0,1], the solution set of (18) is [0,1]=[0,y0].
Case 3. If a_0b_0+b¯0/a_0+a¯0>b_0, i.e., a_0b¯0>a¯0b_0, and b_0+b¯0/a_0+a¯0≤1, then y0=b_0+b¯0/a_0+a¯0∧1=b_0+b¯0/a_0+a¯0.
Now we compare the interval-value number[a_0,a¯0]x0 with [b_0,b¯0] in three cases as below.
(i) Whenx0=b_0+b¯0/a_0+a¯0=y0, in the similar way of (i) and (ii) in case 1, the midpoint of [a_0,a¯0]x0 is exactly 1/2(u_+u¯)=1/2(b_0+b¯0), which is equal to that of [b_0,b¯0]. On the other hand, since a_0b¯0>a¯0b_0, the radius of [a_0,a¯0]x0 is (25)12u¯-u_=12a¯0b_0+a¯0b¯0-a_0b_0-a_0b¯0a_0+a¯0<12a_0b¯0+a¯0b¯0-a_0b_0-a¯0b_0a_0+a¯0=12b¯-b_, which is less than that of [b_0,b¯0]. Hence it holds that [a_0,a¯0]x0=[a_0,a¯0]y0>[b_0,b¯0].
(ii) Whenx0>b_0+b¯0/a_0+a¯0=y0, according to Lemma 7 and the result in the above point (i), it is clear that [a_0,a¯0]x0>[a_0,a¯0]y0>[b_0,b¯0].
(iii) Whenx0<b_0+b¯0/a_0+a¯0=y0, we have(26)12a_0+a¯0x0<12a_0+a¯0b_0+b¯0a_0+a¯0=12b_0+b¯0,i.e., the midpoint of [a_0,a¯0]x0 is less than that of [b_0,b¯0]. Hence [a_0,a¯0]x0<[b_0,b¯0].
Combining the above points (i)-(iii), it is obvious that the solution set of (18) is [0,y0).
The rest of the proof is evident based on cases 1, 2, and 3.Proposition 11.
For the inequality[a_0,a¯0]x0≤[b_0,b¯0], we have the following:(i)
if its solution set is[0,y0), then(27)a_0,a¯0x0<b_0,b¯0,if0≤x0<y0,>b_0,b¯0,ify0≤x0≤1;(ii)
if its solution set is[0,y0], then(28)a_0,a¯0x0≤b_0,b¯0,if0≤x0≤y0,>b_0,b¯0,ify0<x0≤1.
In particular, if[a_0,a¯0]x0=[b_0,b¯0], then the solution set should be [0,y0] and x0=y0.Proof.
The proof lies in the process of the proof of Proposition10.


## 4.2. Basic Concepts and Properties of Max-Product IPFRE

This subsection presents some basic concept and properties of the max-product fuzzy relation equations with interval-valued numbers, i.e., system (12).We adopt the following order relation for comparing two n-dimension vectors. For anyx=(x1,x2,…,xn),y=(y1,y2,…,yn)∈[0,1]n, x≤y (x<y, x=y) means xj≤yj (xj<yj, xj=yj) holds for all j∈J, while x⪇y means both x≤y and x≠y hold.Denote the complete solution set of system (12) by (29)XA,b=x∈0,1n∣A⊙xT=bT.Definition 12.
IfX(A,b)≠∅(X(A,b)=∅), system (12) is said to be consistent (inconsistent),.Definition 13.
A minimal solutionxˇ∈X(A,b) is a solution satisfying x∈X(A,b) and x≤xˇ indicate x=xˇ; a maximum solution x^∈X(A,b) is a solution satisfying x≤x^ for any x∈X(A,b).Theorem 14.
Letx∈[0,1]n be an n-dimension vector. Then x is a solution of system (12) if and only if for any i∈I, [a_ij,a¯ij]xj≤[b_i,b¯i] holds for all j∈J and there exists ji∈J such that [a_iji,a¯iji]xji=[b_i,b¯i].Proof.
(⇒) Since x is a solution of (12), it satisfies the equation s(30)a_i1,a¯i1x1∨a_i2,a¯i2x2∨⋯∨a_in,a¯inxn=b_i,b¯i,i∈I.According to the scalar-multiplication and supremum operations (see (b) and (c) in Section 2),(31)a_i1,a¯i1x1∨a_i2,a¯i2x2∨⋯∨a_in,a¯inxn=⋁j∈Ja_ij,a¯ijxj=⋁j∈Ja_ijxj,a¯ijxj, Hence (32)a_ijxj,a¯ijxj≤⋁j∈Ja_ijxj,a¯ijxj=b_i,b¯i for any j∈J and there exists some ji∈J such that (33)a_ijixji,a¯ijixji=⋁j∈Ja_ijxj,a¯ijxj=b_i,b¯i, i.e., [a_ij,a¯ij]xj≤[b_i,b¯i] and [a_iji,a¯iji]xji=[b_i,b¯i].
(⇐) Analogously, it is easy to check the feasibility of x as a solution of (12), according to the scalar-multiplication and supremum operations on IV.According to Proposition10, the single-variable inequality(34)a_ij,a¯ijxj≤b_i,b¯icould be solved, i∈I, j∈J. Assume that the solution set of (34) is(35)0,yijor0,yij,i∈I, j∈J. We define the vector x^ based on the above solution sets (35) as follows:(36)x^=x^1,x^2,…,x^n=⋀i∈Iyi1,⋀i∈Iyi2,…,⋀i∈Iyin.Theorem 15.
SupposeX(A,b)≠∅. Then for any z∈X(A,b), it holds that z≤x^.Proof.
Sincez=(z1,z2,…,zn,)∈X(A,b), according to Theorem 14 we have(37)a_ij,a¯ijzj≤b_i,b¯i,∀i∈I,j∈J. That is to say, zj is a solution of the equation [a_ij,a¯ij]xj≤[b_i,b¯i]. Notice that the solution set of [a_ij,a¯ij]xj≤[b_i,b¯i] is either [0,yij] or [0,yij). Hence zj≤yij for all i∈I,j∈J. So we further get(38)zj≤⋀i∈Iyij=x^j,j∈J, i.e., z≤x^.As shown in Theorem15, there does not exist any solution bigger that x^. We call x^ thepotential maximum solution of system (12). Moreover, x^ is calledmaximum solution if x^∈X(A,b), while calledpseudomaximum solution if x^∉X(A,b).Theorem 16.
Letz1,z2∈[0,1] and z1≤z≤z2. If z1,z2∈X(A,b), then z∈X(A,b).Proof.
According to Theorem14, z2∈X(A,b) indicates(39)a_ij,a¯ijzj2≤b_i,b¯i,∀i∈I. Besides, by Lemma 7, z≤z2 implies that(40)a_ij,a¯ijzj≤a_ij,a¯ijzj2≤b_i,b¯i,∀i∈I.On the other hand, again by Theorem 14, z1∈X(A,b) indicates, for any i∈I, there exists ji∈J such that [a_iji,a¯iji]zji1=[b_i,b¯i]. It follows from Lemma 7 and the inequality z1≤z that(41)a_iji,a¯ijizji≥a_iji,a¯ijizji1=b_i,b¯i.Considering inequality (40) we have [a_iji,a¯iji]zji≤[b_i,b¯i]. Thus it holds that [a_iji,a¯iji]zji=[b_i,b¯i]. Combining (40), it follows from Theorem 14 that z∈X(A,b).Next we define some new concepts and consider the consistency of system (12).Define theclose index set(42)Jclose=j∈J∣a_ij,a¯ijx^j≤b_i,b¯ifor  alli∈I,and theopen index set(43)Jopen=j∈J∣a_ij,a¯ijx^j>b_i,b¯ifor  somei∈I.It is obvious that (44)Jclose∩Jopen=∅,Jclose∪Jopen=J.Proposition 17.
In system (12), if Jopen≠∅, then x^∉X(A,b).Proof.
IfJopen≠∅, then there exists j∈J and i∈I such that [a_ij,a¯ij]x^j>[b_i,b¯i] by (43). According to Theorem 14, x^ is not a solution of system (12).The following Corollary18 could be easily obtained by Theorem 15 and Proposition 17.Corollary 18.
The vectorx^ is the unique maximum solution of system (12) if and only if Jopen=∅ and x^∈X(A,b).Based on the potential maximum solutionx^ and the open index set, we define the vector(45)x_^=x_^1,x_^2,…,x_^n,where(46)x_^j=0,ifj∈Jopen;x^j,ifj∉Jopen;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).Theorem 19.
System (12) is consistent, i.e., X(A,b)≠∅, if and only if x_^∈X(A,b).Proof.
(⇐) It is obvious.
(⇒) According to Theorem 14, we complete the proof from two aspects as follows:
(i) Take arbitraryi∈I, j∈J. If j∈Jopen, then [a_ij,a¯ij]x_^j=[a_ij,a¯ij]0=[0,0]≤[b_i,b¯i]. While if j∉Jopen, then j∈Jclose, and [a_ij,a¯ij]x_^j=[a_ij,a¯ij]x^j≤[b_i,b¯i].
(ii) Notice that system (12) is consistent. There exists a solution z∈X(A,b) to system (12). By Theorem 14, for arbitrary i∈I, there exists some ji∈J, such that [a_iji,a¯iji]zji=[b_i,b¯i].
Firstly, we verify thatzji=x^ji. Since [a_iji,a¯iji]zji=[b_i,b¯i], it follows from Proposition 11 that the solution of the inequality [a_iji,a¯iji]xji=[b_i,b¯i] is [0,yiji] and zji=yiji. So we get(47)zji=yiji≥⋀k=1mykji=x^ji. On the other hand, it follows from Theorem 15 that z≤x^, which indicates zji≤x^ji. Thus it holds that zji=x^ji.
Secondly, we proveji∉Jopen by contradiction. Assume that ji∈Jopen. Then there exists some k∈I such that [a_kji,a¯kji]x^ji>[b_k,b¯k]. Considering zji=x^ji, we have(48)⋁j=1na_kj,a¯kjzj≥a_kji,a¯kjizji=a_kji,a¯kjix^ji>b_k,b¯k.Inequality (48) shows that the vector z does not satisfy the kth equation in system (12). Hence z is not a solution of (12), which is conflict with the assumption that z∈X(A,b).
At last, sinceji∉Jopen, it follows from (46) that x_^ji=x^ji. So we get(49)a_iji,a¯ijix_^ji=a_iji,a¯ijix^ji=a_iji,a¯ijizji=b_i,b¯i
The above-proved points (i) and (ii) contribute tox_^∈X(A,b) 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.


## 4.3. Structure of the Solution Set of System (12)

LetP=J1×J2×⋯×Jm, where(50)Ji=j∈J∣j∈Jcloseanda_ij,a¯ijx^j=b_i,b¯i,i∈I.Each p∈P is called a path of the characteristic matrix C (or system (12)). Thus P is the set of all paths.Letp∈P be a path. Denote the index sets(51)Ijp=i∈I∣ji=j,j∈J.Based on the path P and its corresponding index sets I1p,I2p,…,Inp, we can define vector(52)xˇp=xˇ1p,xˇ2p,…,xˇnp,where(53)xˇjp=0,ifIjp=∅;x^j,otherwise.Theorem 20.
Letp∈P be a path and xˇp be the vector defined by (52) and (53). Then xˇp is a solution of system (12).Proof.
(i) Take arbitraryi∈I and j∈J.
IfIjp=∅, then xˇjp=0. Hence [a_ij,a¯ij]xˇjp=[a_ij,a¯ij]0=[0,0]≤[b_i,b¯i].
IfIjp≠∅, then xˇjp=x^j. By (51), there exists i∈I such that ji=j. Remind that p=(j1,j2,…,jm)∈J1×J2×⋯×Jm. It is clear ji∈Ji, which indicates ji∈Jclose,i.e.,j∈Jclose. According to the definition of Jclose (see (42)), we have [a_ij,a¯ij]xˇjp=[a_ij,a¯ij]x^j≤[b_i,b¯i].
(ii) Take arbitraryi∈I. Notice p=(j1,j2,…,jm). We denote ji=j′. It is clear that i∈Ij′p, which indicates Ij′p≠∅. Hence(54)xˇj′p=x^j′,i.e.,xˇjip=x^ji.On the other hand, ji∈Ji implies(55)a_iji,a¯ijix^ji=b_i,b¯i.Equations (54) and (55) contribute to [a_iji,a¯iji]xˇjip=[b_i,b¯i].
Based on (i) and (ii), it follows from Theorem14 that xˇp∈X(A,b).Theorem20 shows that, for any path p∈P of system (12), xˇp is a solution corresponding to p. Next we further construct the solution interval corresponding to p. Based on the path p∈P, define the vector Xp=(X1p,X2p,…,Xnp), where(56)Xjp=xˇjp,x^j,ifj∈Jclose;xˇjp,x^j,ifj∈Jopen.We callXp thesolution interval corresponding to the path p.Remark 21.
It could be easily found from (56) that(i)
Xp is a close set (or close interval) if and only if Jopen=∅;(ii)
Xp is an open set (or open interval) if and only if Jclose=∅.Theorem 22.
Letp∈P be a path and Xp be defined by (56) based on p. Then it holds that Xp⊆X(A,b).Proof.
Take arbitraryy=(y1,y2,…,yn)∈Xp. Then it follows from (56) that (57)j∈Jclose⇒xˇjp≤yj≤x^j,(58)j∈Jopen⇒xˇjp≤yj<x^j,
(i) We first prove it holds for alli∈I and j∈J that [a_ij,a¯ij]yj≤[b_i,b¯i] in tow cases.
Case 1. If j∈Jclose, then [a_ij,a¯ij]yj≤[a_ij,a¯ij]x^j≤[b_i,b¯i], ∀i∈I.
Case 2. If j∈Jopen, then the index set Ij>={i∈I|[a_ij,a¯ij]x^j>[b_i,b¯i]}≠∅. Obviously, when i≠Ij>, it holds that [a_ij,a¯ij]x^j≤[b_i,b¯i]. Combining (58) and Lemma 7, we have (59)a_ij,a¯ijyj<a_ij,a¯ijx^j≤b_i,b¯i.On the other hand, when i∈Ij>, we have(60)a_ij,a¯ijx^j>b_i,b¯i.Suppose the solution set of the inequality [a_ij,a¯ij]xj>[b_i,b¯i] is [0,yij) or [0,yij]. Then(61)x^j=⋀k∈Iykj≤yij.Thus(62)a_ij,a¯ijyij≥a_ij,a¯ijx^j>b_i,b¯i.According to Proposition 11, the solution set of [a_ij,a¯ij]xj>[b_i,b¯i] should be [0,yij). It follows from (58) and (61) that yj<yij. Again by Proposition 11 we get [a_ij,a¯ij]yj<[b_i,b¯i].
(ii) For anyi∈I, there exists ji∈Ji⊆J (due to the existence of p=(j1,j2,…,jm)∈J1×J2×⋯×Jm), such that [a_iji,a¯iji]xˇjip=[b_i,b¯i]. By (57) and (58),(63)a_iji,a¯ijiyji≥xˇjip=b_i,b¯i. On the other hand, as a result of (i), [a_iji,a¯iji]yji≤[b_i,b¯i] holds. These two aspects lead to [a_iji,a¯iji]yji=[b_i,b¯i].
Combining (i) and (ii), it follows from Theorem14 that y∈X(A,b). Due to the arbitrariness of y, the inclusion relation Xp⊆X(A,b) holds.Theorem 23.
For anyy∈X(A,b), there exists a path p∈P such that y∈Xp.Proof.
Sincey∈X(A,b), it follows from Theorem 14 that(64)a_ij,a¯ijyj≤b_i,b¯i,and for any i∈I, there exists ji∈J such that(65)a_iji,a¯ijiyji=b_i,b¯i.Let p=(j1,j2,…,jm).
(i) We first verify thatp∈P, i.e., ji∈Ji, ∀i∈I.
Notice that[a_iji,a¯iji]yji=[b_i,b¯i]. Considering the inequality [a_iji,a¯iji]xji=[b_i,b¯i], it follows from Proposition 11 that its solution set should be [0,yiji], and moreover,(66)yji=yiji.Thus (67)x^ji=⋀k∈Iykji≤yiji=yji.On the other hand, Theorem 15 indicates yji≤x^ji. So we get(68)x^ji=yji.As a result of (65) and (68),(69)a_iji,a¯ijix^ji=a_iji,a¯ijiyji=b_i,b¯i.Combining (64) and (68), we get(70)a_kji,a¯kjix^ji=a_kji,a¯kjiyji≤b_k,b¯k,∀k∈I,which indicates(71)j∈Jclose.(69) and (71) contribute to ji∈Ji.
(ii) We now checkxˇp≤y.
Take arbitraryj∈J. If Ijp≠∅, then xˇjp=x^j. Besides, since Ijp≠∅, there exists i∈Ijp⊆I such that ji=j. By (68) we have x^j=x^ji=yji=yj. Hence (72)xˇjp=x^j=yj.If Ijp=∅, then xˇjp=0≤yj. Consequently, it holds that xˇjp≤yj for all j∈J, i.e., xˇp≤y.
(iii) According to Theorem15,(73)yj≤x^j,∀j∈J.Obviously yj≤x^j holds for any j∈Jclose. While for j∈Jopen, we have to verify that yj≤x^j (by contradiction). Otherwise, due to (73), it turns out to be yj=x^j. By (43), j∈Jopen indicates there exists i∈I such that [a_ij,a¯ij]x^j>[b_i,b¯i]. So we get(74)a_i1,a¯i1y1∨a_i2,a¯i2y2∨⋯∨a_in,a¯inyn≥a_ij,a¯ijyj=a_ij,a¯ijx^j>b_i,b¯i. Thus y does not satisfy the jth equation in system (12). This is conflict with the assumption that y∈X(A,b).
The above points (i)-(iii) contribute toy∈Xp and the proof is complete.Theorem 24 (structure of the solution set).
Suppose system (12) is consistent. Then its complete solution set is(75)XA,b=⋃p∈PXp,where P=J1×J2×⋯×Jm is the set of all paths and Xp is the solution interval corresponding to the path p.Proof.
It is simple corollary of Theorems22 and 23.


## 4.4. Algorithm for Solving System (12) Based on the Index Sets

Based on the above-defined index setsJclose, Jopen, and Ji, i∈I, we propose the following algorithm for obtaining the solution set of system (12).Algorithm for Solving System (12)Step 1. Solve the inequality [a_ij,a¯ij]xj≤[b_i,b¯i], for any i∈I and j∈J, by Proposition 10. Suppose the solution set is Sij=[0,yij) or Sij=[0,yij], where i∈I and j∈J.Step 2. Compute the potential maximum solution x^ by (36).Step 3. Obtain the close index set Jclose and open index set Jopen by (42) and (43), respectively.Step 4. Compute the vector x_^ by (45) and (46).Step 5. Check the consistency of system (12) by the vector x_^ according to Theorems 14 and 19. If x_^∈X(A,b), then system (12) is consistent and continue to Step 6. Otherwise, system (12) is inconsistent and stop.Step 6. Compute the index sets J1,J2,…,Jm by (50). Let P=J1×J2×⋯×Jm.Step 7. For any p∈P, compute the index sets I1p,I2p,…,Inp by (51).Step 8. Compute the corresponding solution xˇp by (52) and (53).Step 9. Construct the solution interval Xp by (56).Step 10. According to Theorem 24, generate the complete solution set of system (12) by(76)XA,b=⋃p∈PXp.


## 5. Numerical Example

Example 1.
Consider the following interval-valued-parameter max-product fuzzy relation equations:(77)A⊙xT=bT,where (78)A=0.2,0.30.5,0.70,0.50.3,0.450.1,0.30.3,0.50.5,1.00.25,0.50.6,0.80.3,0.50.2,0.40.4,0.50.2,0.50.3,0.60.75,1.00.4,0.60.6,0.70.5,0.60.3,0.50.4,0.60.3,0.40.6,0.80.5,0.70.6,0.80.7,0.980.7,0.980.28,0.450.5,0.70.36,0.420.48,0.720.5,0.90.45,0.750.4,0.60.7,0.80.3,0.40.5,0.70.4,0.50.6,0.70.85,0.950.5,0.7,b=0.2,0.4,0.3,0.4,0.36,0.48,0.35,0.49,0.4,0.5,x=x1,x2,…,x8∈0,18.SolutionStep 1. Solve the inequality [a_ij,a¯ij]xj≤[b_i,b¯i] for i=1,2,…,5, j=1,2,…,8 by Proposition 10. Denote the corresponding solution set by Sij. After calculation we get all the solution sets as follows:(79)
S
i
j
=
0,1
0
,
1
2
0,1
0
,
4
5
0,1
0
,
3
4
0
,
2
5
0
,
4
5
0
,
1
2
0
,
7
8
0,1
0
,
7
9
0,1
0
,
7
9
0
,
2
5
0
,
7
10
0
,
42
65
0
,
42
55
0,1
0
,
21
25
0
,
1
0
,
3
5
0
,
7
10
0
,
3
5
0
,
1
2
0
,
1
2
0,1
0
,
7
10
0,1
0
,
7
10
0
,
3
5
0
,
7
10
0
,
9
10
0
,
3
5
0,1
0
,
3
4
0,1
0
,
9
13
0
,
1
2
0
,
3
4
.Step 2. After calculation by (36), the potential maximum solution of system (77) is(80)x^=0.5,0.5,1,0.7,1,0.6,0.4,0.6.Step 3. According to (42) and (43), we find the close index set and open index set as(81)Jclose=1,3,4,5,6,7,8,Jopen=2, respectively.Step 4. Computing the vector x_^ by (45) and (46), we get(82)x_^=0.5,0,1,0.7,1,0.6,0.4,0.6.Step 5. Check the consistency of system (77) by the vector x_^ according to Theorems 14 and 19. After detailed calculation, it is not difficult to check that A⊙x_^T=bT, i.e., x_^ is a solution of (77). Hence system (77) is consistent and we could continue to Step 6.Step 6. Computing the index sets J1,J2,…,J5 by (50), we get (83)J1=7;J2=1,7;J3=6,8;J4=1,4;J5=5. Let P=J1×J2×⋯×J5. Obviously the set P contains 8 elements, i.e., there exist 8 paths of system (77).Step 7. For any p∈P, compute the index sets I1p,I2p,…,I8p by (51).Forp1=(7,1,6,1,5), the corresponding index sets are (84)I1p1=2,4;I2p1=∅;I3p1=∅;I4p1=∅;I5p1=5;I6p1=3;I7p1=1;I8p1=∅.For p2=(7,1,6,4,5), the corresponding index sets are (85)I1p2=2;I2p2=∅;I3p2=∅;I4p2=4;I5p2=5;I6p2=3;I7p2=1;I8p2=∅.For p3=(7,1,8,1,5), the corresponding index sets are (86)I1p3=2,4;I2p3=∅;I3p3=∅;I4p3=∅;I5p3=5;I6p3=∅;I7p3=1;I8p3=3.For p4=(7,1,8,4,5), the corresponding index sets are (87)I1p4=2;I2p4=∅;I3p4=∅;I4p4=4;I5p4=5;I6p4=∅;I7p4=1;I8p4=3.For p5=(7,7,6,1,5), the corresponding index sets are (88)I1p5=4;I2p5=∅;I3p5=∅;I4p5=∅;I5p5=5;I6p5=3;I7p5=1,2;I8p5=∅.For p6=(7,7,6,4,5), the corresponding index sets are (89)I1p6=∅;I2p6=∅;I3p6=∅;I4p6=4;I5p6=5;I6p6=3;I7p6=1,2;I8p6=∅.For p7=(7,7,8,1,5), the corresponding index sets are (90)I1p7=4;I2p7=∅;I3p7=∅;I4p7=∅;I5p7=5;I6p7=∅;I7p7=1,2;I8p7=3.For p8=(7,7,8,4,5), the corresponding index sets are (91)I1p8=∅;I2p8=∅;I3p8=∅;I4p8=4;I5p8=5;I6p8=∅;I7p8=1,2;I8p8=3.Step 8. Compute the corresponding solution xˇp by (52) and (53). (92)xˇp1=0.5,0,0,0,1,0.6,0.4,0,xˇp2=0.5,0,0,0.7,1,0.6,0.4,0,xˇp3=0.5,0,0,0,1,0,0.4,0.6,xˇp4=0.5,0,0,0.7,1,0,0.4,0.6,xˇp5=0.5,0,0,0,1,0.6,0.4,0,xˇp6=0,0,0,0.7,1,0.6,0.4,0,xˇp7=0.5,0,0,0,1,0,0.4,0.6,xˇp8=0,0,0,0.7,1,0,0.4,0.6.Step 9. Construct the solution interval Xp by (56). (93)Xp1=0.5,0,0.5,0,1,0,0.7,1,0.6,0.4,0,0.6,Xp2=0.5,0,0.5,0,1,0.7,1,0.6,0.4,0,0.6,Xp3=0.5,0,0.5,0,1,0,0.7,1,0,0.6,0.4,0.6,Xp4=0.5,0,0.5,0,1,0.7,1,0,0.6,0.4,0.6,Xp5=0.5,0,0.5,0,1,0,0.7,1,0.6,0.4,0,0.6,Xp6=0,0.5,0,0.5,0,1,0.7,1,0.6,0.4,0,0.6,Xp7=0.5,0,0.5,0,1,0,0.7,1,0,0.6,0.4,0.6,Xp8=0,0.5,0,0.5,0,1,0.7,1,0,0.6,0.4,0.6.Step 10. Generate the complete solution set of system (77) according to Theorem 24. Since Xp1=Xp5, Xp3=Xp7, Xp2⊆Xp1, Xp4⊆Xp3, we have(94)XA,b=⋃p∈PXp=⋃t=18Xpt=Xp1∪Xp3∪Xp6∪Xp8,where Xp1,Xp3,Xp6,Xp8 are as shown in Step 9.


## 6. Discussion

### 6.1. Some Further Properties on System (12)

Proposition 1.
Ifxˇ=(xˇ1,xˇ2,…,xˇn) is a minimal solution of system (12), then it holds for any j∈J that xˇj∈{0,x^j}, where x^ is the potential maximum solution of (12) as defined by (36). In particular, when j∈Jopen, it holds that xˇj=0.Proof.
Based on Theorems23 and 24, any minimal solution could be selected from the set {xˇp∣p∈P}, i.e.,(95)xˇ∈xˇp∣p∈P. It follows from (53) that xˇj∈{0,x^j} for any j∈J. Moreover, when j∈Jopen, it should hold that xˇj=0. Otherwise, it turns out to be xˇj=x^j. According to (43), there exists some i∈I such that [a_ij,a¯ij]x^j>[b_i,b¯i]. So we get(96)a_i1,a¯i1xˇ1∨a_i2,a¯i2xˇ2∨⋯∨a_in,a¯inxˇn≥a_ij,a¯ijxˇj=a_ij,a¯ijx^j>b_i,b¯i. This is conflict with xˇ∈X(A,b).Proposition 2.
Let system (12) be consistent and Jopen is its open index set. There exists a unique maximum solution if and only if Jopen=∅. In particular, when the maximum solution exists, it should be x^.Proof.
(⇒) If system (12) has a maximum solution, according to the structure of its solution set, i.e., (97)XA,b=⋃p∈PXp,and the expression of Xp (see (56)), the maximum solution should be x^. Then it follows from Corollary 18 that Jopen=∅.
(⇐) If Jopen=∅, it holds that x_^=x^ by (46). Since system (12) is consistent, it follows from Theorem 19 that x^=x_^∈X(A,b). According to Corollary 18, x^ is the unique maximum solution of system (12).
Consequently, when the maximum solution exists, it should bex^.Theorem 3.
Let system (12) be consistent with solution set X(A,b). Then the following statements are equivalent:(i)
X(A,b) is a close set;(ii)
Jopen=∅;(iii)
x^∈X(A,b);(iv)
there exists a unique maximum solution of system (12).Proof.
The proof of (ii)⇔ (iii) ⇔ (iv) lies in Proposition 2.
(i)⇔ (ii) Considering X(A,b)=⋃p∈P‍Xp and the expression of Xp (see (56)), X(A,b) is a close set if and only if Xp is close set for any p∈P. This is equivalent to Jopen=∅.


### 6.2. Comparing the Solution Set of System (12) to That of the Classical Max-T Fuzzy Relation Equations

As well known to everyone, the solution set of a system of classical max-T 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)XA,b=⋃xˇ∈XˇA,bxˇ,x^.Differences between the solution set of the interval-valued-parameter max-product fuzzy relation equation (i.d. system (12)) and that of the classical max-T fuzzy relation equation are shown as below.(i) The classical max-T 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 Jopen is empty.(ii) As a union of finite close intervals, the solution set of classical max-T 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 Jopen=∅; i.e., the maximum solution exists, according to Theorem 3.


## 6.1. Some Further Properties on System (12)

Proposition 1.
Ifxˇ=(xˇ1,xˇ2,…,xˇn) is a minimal solution of system (12), then it holds for any j∈J that xˇj∈{0,x^j}, where x^ is the potential maximum solution of (12) as defined by (36). In particular, when j∈Jopen, it holds that xˇj=0.Proof.
Based on Theorems23 and 24, any minimal solution could be selected from the set {xˇp∣p∈P}, i.e.,(95)xˇ∈xˇp∣p∈P. It follows from (53) that xˇj∈{0,x^j} for any j∈J. Moreover, when j∈Jopen, it should hold that xˇj=0. Otherwise, it turns out to be xˇj=x^j. According to (43), there exists some i∈I such that [a_ij,a¯ij]x^j>[b_i,b¯i]. So we get(96)a_i1,a¯i1xˇ1∨a_i2,a¯i2xˇ2∨⋯∨a_in,a¯inxˇn≥a_ij,a¯ijxˇj=a_ij,a¯ijx^j>b_i,b¯i. This is conflict with xˇ∈X(A,b).Proposition 2.
Let system (12) be consistent and Jopen is its open index set. There exists a unique maximum solution if and only if Jopen=∅. In particular, when the maximum solution exists, it should be x^.Proof.
(⇒) If system (12) has a maximum solution, according to the structure of its solution set, i.e., (97)XA,b=⋃p∈PXp,and the expression of Xp (see (56)), the maximum solution should be x^. Then it follows from Corollary 18 that Jopen=∅.
(⇐) If Jopen=∅, it holds that x_^=x^ by (46). Since system (12) is consistent, it follows from Theorem 19 that x^=x_^∈X(A,b). According to Corollary 18, x^ is the unique maximum solution of system (12).
Consequently, when the maximum solution exists, it should bex^.Theorem 3.
Let system (12) be consistent with solution set X(A,b). Then the following statements are equivalent:(i)
X(A,b) is a close set;(ii)
Jopen=∅;(iii)
x^∈X(A,b);(iv)
there exists a unique maximum solution of system (12).Proof.
The proof of (ii)⇔ (iii) ⇔ (iv) lies in Proposition 2.
(i)⇔ (ii) Considering X(A,b)=⋃p∈P‍Xp and the expression of Xp (see (56)), X(A,b) is a close set if and only if Xp is close set for any p∈P. This is equivalent to Jopen=∅.


## 6.2. Comparing the Solution Set of System (12) to That of the Classical Max-T Fuzzy Relation Equations

As well known to everyone, the solution set of a system of classical max-T 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)XA,b=⋃xˇ∈XˇA,bxˇ,x^.Differences between the solution set of the interval-valued-parameter max-product fuzzy relation equation (i.d. system (12)) and that of the classical max-T fuzzy relation equation are shown as below.(i) The classical max-T 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 Jopen is empty.(ii) As a union of finite close intervals, the solution set of classical max-T 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 Jopen=∅; i.e., the maximum solution exists, according to Theorem 3.


## 7. 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 interval-valued fuzzy relation equation, which is different from that to the classical one. Detailed algorithm is developed for obtaining the complete solution set with illustrative example. In the future research we will focus on the optimization problem subject to interval-valued fuzzy relation equations.



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*Source: 8179763-2019-02-17.xml*