# Interval Entropy of Fuzzy Sets and the Application to Fuzzy Multiple Attribute Decision Making

Authors: Yiying Shi; Xuehai Yuan
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A series of new concepts including interval entropy, interval similarity measure, interval distance measure, and interval inclusion measure of fuzzy sets are introduced. Meanwhile, some theorems and corollaries are proposed to show how these definitions can be deduced from each other. And then, based on interval entropy, a fuzzy multiple attribute decision making (FMADM) model is set up. In this model, interval entropy is used as the weight, by which the evaluation values of all alternatives can be obtained. Then all alternatives with respect to each criterion can be ranked as the order of the evaluation values. At last, a practical example is given to illustrate an application of the developed model and a comparative analysis is made.


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## Body

## 1. Introduction

Multiple attribute decision making (MADM) problems existed in the economic, management, and various social fields. And it refers to making preference decisions over the available alternatives that are characterized by multiple attributes. But for practical needs and peoples’ more profound understanding, there is much uncertain information included in the decision making process. How to handle the uncertain information is the issue that decision makers must be concerned about. In 1965, the fuzzy set theory was proposed by Zadeh, which provided effective methods to solve the issue. Several years later, Bellman and Zadeh [1] put forward a fuzzy model based on MADM method by combining fuzzy set and decision making. In the model, the attributes which cannot be defined exactly will be expressed as some proper fuzzy sets and converted into classical decision making problems by use of level sets. Because of its great flexibility and adaptability, the model has been widely viewed as the basis of fuzzy decision making. The FMADM method, proposed by Bass and Kwakernaak in 1977 [2], is regarded as the classical method in fuzzy decision making. In recent years, many new methods have been applied to FMADM problems (such as hesitant fuzzy theory [3–6], TOPSIS method [7, 8], some operators [9–11], preference relations [12], and intuitionistic fuzzy decision making [13]). Hadi-Vencheh and Mirjaberi [14] developed an approach to solve MADM problems considering distances both to the positive ideal solution and to the negative ideal solution. Zhang and Xu [15] proposed an extended technique for order preference based on Pythagorean fuzzy set. In this approach, a score function based comparison method is proposed to identify the Pythagorean fuzzy positive ideal solution and the Pythagorean fuzzy negative ideal solution. And a distance measure is defined to calculate the distances between each alternative and the Pythagorean fuzzy positive ideal solution as well as the Pythagorean fuzzy negative ideal solution, respectively. And the comparative analysis had been made among these methods [16].As Zimmermann [17] pointed out, the existing methods can be divided into two steps: the first is to determine the weights of all alternatives and then combine them into the evaluation values with fuzzy operator; the second is to rank the order of all alternatives according to the evaluation values. For MADM problems, when the weight of alternative is defined, results of decision making depended on the values of alternatives. And the alternative weights in MADM can be classified as subjective and objective alternative weights based on the information acquisition approach. The subjective alternative weights are determined by preference information on the alternatives given by the decision maker. The objective alternative weights are determined by the decision making matrix. In terms of determining objective alternative weights, one of the most famous approaches is the entropy method, which expresses the relative intensities of alternative importance to signify the average intrinsic information transmitted to the decision maker. Entropy concept was used in various scientific fields, especially in information theory, and used to refer to a general measure of uncertainty [18]. In MADM, the greater the value of the entropy corresponding to a special attribute is, which means the smaller attribute’s weight, the less the discriminate power of that attribute in decision making process is. So far, a lot of literature pertaining to MADM analysis has been published using entropy weights, for instance, the cross entropy [19, 20], fuzzy entropy [21], Shannon entropy [22], maximizing fuzzy entropy [23], and sine entropy [24]. Jin et al. [25] proposed the interval value intuitionistic fuzzy continuous weighted entropy which generalizes intuitionistic fuzzy entropy measures defined by Szmidt and Kacprzyk on the basis of the continuous ordered weighted averaging operator. Zhang et al. [26] investigated the MADM problem with completely unknown attribute weights in the framework of interval value intuitionistic fuzzy sets. Using a new definition of interval value intuitionistic fuzzy entropy and some calculation methods for interval value intuitionistic fuzzy entropy, an entropy-based decision making method to solve interval value intuitionistic FMADM problems with completely unknown attribute weights is proposed. Besides the above results, there are also some papers focused on the relationships between entropy and other concepts (such as distance measure [27], similarity measure [28, 29], and inclusion measure [30–32]).For the above methods, the common point is to use different entropies as weights. That is to say, the weight value is one specific number. But in practice, because of uncertainty of people’s cognition, the data of the decision making processes cannot be measured precisely and there may be some other types of data, for instance, interval value data. In other words, the decision maker would prefer to express the point of view in this form rather than a real number because of the uncertainty and the lack of certain data, especially when data are known to lie within bounded variables, or when facing missing data, judgment data, and so forth. In MADM it is most probable that we confront such a case. So when the weight is an interval value, how to use entropy to represent it is worthy of discussion.In this paper, according to the definition of fuzzy set entropy, the concept of interval entropy on fuzzy sets is proposed, and its application in MADM is introduced. There are four sections in the paper. Firstly, Section2 is preliminary, and interval entropy of fuzzy set is proposed in Section 3. Secondly, the relationships among interval entropy, interval similarity measure, interval distance measure, and interval inclusion measure are discussed in Section 4. Finally, the FMADM analysis has been conducted in risk assessment of Taiwan railway reconstruction project in Section 5.


## 2. Preliminary

In this section, some definitions are introduced. Here, letX be a set. The mapping μA:X→[0,1] is called a fuzzy subset of X. Let F(X) and P(X) denote the class of all fuzzy sets and crisp sets over X, respectively.Definition 1 (see [33]).
A fuzzy number is defined as a fuzzy subset with the membership functionμA(x), for any x∈R satisfies the following properties:(1)
A is a normal fuzzy set; that is, ∃x∈R, μA(x)=1.(2)
A is a convex fuzzy set; that is to say, for any x1, x2 and λ∈[0,1],(1)μAλx1+1-λx2≥min⁡μAx1,μAx2.(3)
The support ofA is bounded; that is to say, the set Supp(A)={x∈R∣μA(x)>0} is bounded.Definition 2 (see [7]).
The fuzzy numberA is called the trapezoidal fuzzy number and denoted by [a,b,c,d], where the membership function can be expressed as the following:(2)μAx=x-ab-aa≤x≤b1b≤x≤cd-xd-cc≤x≤d0else.If b=c, then A is called the triangular fuzzy number and denoted by [a,m,d]. Therefore, a triangular fuzzy number is a special case of the trapezoidal fuzzy number.Definition 3 (see [7, 34, 35]).
Forα∈(0,1], the α-cut set Aα of A is a classic set defined as (3)Aα=x∈R∣μAx≥α.If A is the trapezoidal fuzzy number, then Aα can be denoted by(4)Aα=bα+a1-α,cα+d1-αα∈0,1b,cα=1.Let D([0,1])={[a-,a+]∣0≤a-≤a+≤1} be the interval value set; we set [25] (1)
[a-,a+]≤[b-,b+]⇔a-≤b-,a+≤b+;(2)
[a-,a+]=[b-,b+]⇔a-=b-,a+=b+;(3)
[a-,a+]c=[1-a+,1-a-].


## 3. Interval Entropy of Fuzzy Sets

In fact, entropy of fuzzy sets can be used to describe the general measure of fuzziness through the mapping between fuzzy numbers and real numbers on[0,1], just like the process of the defuzzification, in which only one point is used to represent the fuzzy number. But, in many real life problems, the data of the decision making processes cannot be measured precisely. For instance, when the fuzziness of the fuzzy set can be expressed as a maximum of 0.8 and a minimum of 0.2, how about it? It is necessary to extend the value of entropy from the number to interval value, and then the definition of interval entropy is proposed as follows.Definition 4.
A real functionIE:F(X)→D([0,1])A↦IE(A) is called interval entropy on F(X), if IE has the following properties:(EP1)
IfA∈P(X), then IE(A)=[0,0].(EP2)
If∀x∈X, μA(x)≡1/2, then IE(A)=[1,1].(EP3)
If∀x∈X, μA2(x)≤μA1(x)≤1/2 or μA2(x)≥μA1(x)≥1/2, then(5)IEA2≤IEA1.(EP4)
IE(A)=IE(Ac).Remark 5.
It can be concluded that ifIE(A1)=[a1-,a1+], IE(A2)=[a2-,a2+]∈D([0,1]), then we have(6)IEA1≤IEA2⟺a1-≤a2-,a1+≤a2+.We can construct some interval entropy formulas based on Definition4 as follows.LetX={x1,x2,…,xn}; (7)IE1A=1n∑i=1n2μAxi∧μAcxi,1n∑i=1n4μAxi·μAcxi;(8)IE2A=1n∑i=1n4μAxi2∧μAcxi2,1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi;(9)IE3A=1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi,1n∑i=1n2μAxi∧μAcxi;(10)IE4A=1n∑i=1n2μAxi∧μAcxi,1n∑i=1n2μAxi·μAcxiμAxi2+μAcxi2;(11)IE5A=1n∑i=1n2μAxi·μAcxiμAxi2+μAcxi2,1n∑i=1n4μAxi·μAcxi;(12)IE6A=1n∑i=1nμAxi∧μAcxiμAxi∨μAcxip,1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi1/pp≥1.If μA(x) can be integrated over the considered interval [a,b], then(13)IE7A=1b-a∫ab2μAx∧μAcxdx,1b-a∫ab4μAx·μAcxdx,IE8A=1b-a∫ab4μAx2∧μAcx2dx,1b-a∫abμAx∧μAcxμAx∨μAcxdx;IE9A=1b-a∫abμAx∧μAcxμAx∨μAcxdx,1b-a∫ab2μAx∧μAcxdx;IE10A=1b-a∫ab2μAx∧μAcxdx,1b-a∫ab2μAx·μAcxμAx2+μAcx2dx;IE11A=1b-a∫ab2μAx·μAcxμAx2+μAcx2dx,1b-a∫ab4μAx·μAcxdx;IE12A=1b-a∫abμAx∧μAcxμAx∨μAcxpdx,1b-a∫abμAx∧μAcxμAx∨μAcx1/pdxp≥1.The following theorem shows that the above formulas are all interval entropies.Theorem 6.
I
E
i
(
A
,
B
)  (i=1,2,…,12) are interval entropies.Proof.
Wheni=7, by (7), we have(14)IE7A=1b-a∫ab2μAx∧μAcxdx,1b-a∫ab4μAx·μAcxdx.Firstly, if 0≤μA(x)≤1/2, then 2(μA(x)∧μAc(x))=2μA(x) and 1/2≤μAc(x)≤1, so (15)2μAx≤4μAx·μAcx≤1.Namely,(16)0≤1b-a∫ab2μAx∧μAcxdx≤1b-a∫ab4μAx·μAcxdx≤1.So(17)IE7A=1b-a∫ab2μAx∧μAcxdx,1b-a∫ab4μAx·μAcxdx∈D0,1.To (EP1): if A∈P(X), then μA(x)=0 or μA(x)=1. So(18)2μAx∧μAcx=0,4μAx·μAcx=0.Namely, IE7(A)=[0,0].
(EP2) If∀x∈X, μA(x)≡1/2, then 2(μA(x)∧μAc(x))=1, 4μA(x)·μAc(x)=1.
Namely,IE7(A)=[1,1].
(EP3) If∀x∈X, μA2(x)≤μA1(x)≤1/2, then(19)IE7A1=1b-a∫ab2μA1x∧μA1cxdx,1b-a∫ab4μA1x·μA1cxdx=1b-a∫ab2μA1xdx,1b-a∫ab4μA1x·μA1cxdx,IE7A2=1b-a∫ab2μA2x∧μA2cxdx,1b-a∫ab4μA2x·μA2cxdx=1b-a∫ab2μA2xdx,1b-a∫ab4μA2x·μA2cxdx.For μA2(x)≤μA1(x)≤1/2, then we have (20)2μA2x≤2μA1x,4μA2x·μA2cx≤4μA1x·μA1cx.Namely,(21)IE7A2≤IE7A1.When μA2(x)≥μA1(x)≥1/2, the proof is similar.
(EP4) It can be easily concluded thatIE7(A)=IE7(Ac).
The proofs for other cases are similar.
In particular, ifA is a trapezoidal fuzzy number and denoted bys A=[a,b,c,d], we have(22)IE7A=b-a-c+d2d-a,2b-a-c+d3d-a;IE8A=b-a-c+d3d-a,2ln⁡2-1b-a-c+dd-a;IE9A=2ln⁡2-1b-a-c+dd-a,b-a-c+d2d-a;IE10A=b-a-c+d2d-a,π-2b-a-c+d2d-a;IE11A=π-2b-a-c+d2d-a,2b-a-c+d3d-a.


## 4. Property

### 4.1. Other Definitions to Describe the Uncertainty

In fact, the conceptions to describe the general measure of uncertainty such as similarity measure, distance measure, and inclusion measure have also built the mapping between two fuzzy numbers and real numbers on[0,1], in which only one point is used to represent two fuzzy numbers (see appendix). But when the values are interval, it is necessary to extend these conceptions to interval values, and, in this section, the definitions of interval similarity measure, interval distance measure, and interval inclusion measure are proposed as follows.Definition 7.
A real functionIS:F(X)×F(X)→D([0,1]) is called interval similarity measure on F(X) if IS satisfies the following properties:(SP1)
IS(A,Ac)=[0,0] if A is a crisp set.(SP2)
IS(A,A)=[1,1].(SP3)
IS(A,B)=IS(B,A).(SP4)
For allA, B, C∈F(X), if A⊆B⊆C, then(23)ISA,C≤ISA,B,ISA,C≤ISB,C.We can construct some interval similarity measure formulas based on Definition7 as follows.For instance, letX={x1,x2,…,xn}; we set(24)IS1A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1-maxxi∈X⁡μAxi-μBxi;IS2A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2;IS3A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1-1n∑i=1nμAxi-μBxiμAxi+μBxi;IS4A,B=1-1n∑i=1nμAxi-μBxiμAxi+μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2;IS5A,B=1np∑i=1nμAxi∧μBxiμAxi∨μBxip,1n1/p∑i=1nμAxi∧μBxiμAxi∨μBxi1/pp≥1.If μA(x) and μB(x) can be integrated over the considered interval [a,b], then (25)IS6A,B=1b-a∫abμAx∧μBxμAx∨μBxdx,1-maxx∈a,b⁡μAx-μBx;IS7A,B=1b-a∫abμAx∧μBxμAx∨μBxdx,1b-a∫ab2μAx·μBxμAx2+μBx2dx;IS8A,B=1b-a∫abμAx∧μBxμAx∨μBxdx,1-1b-a∫abμAx-μBxμAx+μBxdx;IS9A,B=1-1b-a∫abμAx-μBxμAx+μBxdx,1b-a∫ab2μAx·μBxμAx2+μBx2dx;IS10A,B=1b-ap∫abμAx∧μBxμAx∨μBxpdx,1b-a1/p∫abμAx∧μBxμAx∨μBx1/pdxp≥1.The following theorem shows that the above formulas are all interval similarity measures.Theorem 8.
I
S
i
(
A
,
B
)  (i=1,2,…,10) are interval similarity measures.Proof.
Wheni=2, we have(26)IS2A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2.If μA(xi)≤μB(xi), then μA(xi)∧μB(xi)/μA(xi)∨μB(xi)=μA(xi)/μB(xi),(27)2μAxi·μBxiμAxi2+μBxi2-μAxiμBxi=μAxiμBxi2-μAxi2μAxi2+μBxi2μBxi≥0.Namely,(28)0≤1n∑i=1nμAxi∧μBxiμAxi∨μBxi≤1n∑i=1n2μAxi·μBxiμAxi2+μBxi2≤1.That is to say,(29)IS2A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2∈D0,1.The proof for μA(xi)≥μB(xi) is similar.
(SP1) IfA∈P(X), then μA(x)=0 or μA(x)=1. (30)1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi=0,1n∑i=1n2μAxi·μAcxiμAxi2+μAcxi2=0.Namely, IS2(A,Ac)=[0,0].
(SP2) Consider(31)IS2A,A=1n∑i=1nμAxi∧μAxiμAxi∨μAxi,1n∑i=1n2μAxi·μAxiμAxi2+μAxi2=1,1.
(SP3) It is obvious thatIS2(A,B)=IS2(B,A).
(SP4) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x). So(32)IS2A,C=1n∑i=1nμAxi∧μCxiμAxi∨μCxi,1n∑i=1n2μAxi·μCxiμAxi2+μCxi2=1n∑i=1nμAxiμCxi,1n∑i=1n2μAxi·μCxiμAxi2+μCxi2,IS2A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2=1n∑i=1nμAxiμBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2,for the reason that(33)2μAxi·μCxiμAxi2+μCxi2-2μAxi·μBxiμAxi2+μBxi2=2μAxiμCxi-μBxiμAxi2-μBxiμCxiμAxi2+μCxi2μAxi2+μBxi2<0.Namely,(34)μAxμCx≤μAxμBx,2μAxi·μCxiμAxi2+μCxi2≤2μAxi·μBxiμAxi2+μBxi2.That is to say, IS2(A,C)≤IS2(A,B) and IS2(A,C)≤IS2(B,C). And the proofs for other cases are similar.Definition 9.
A real functionID:F(X)×F(X)→D([0,1]) is called interval distance measure on F(X), if ID satisfies the following properties: (DP1)
ID(A,Ac)=[1,1] ifA is a crisp set.(DP2)
ID(A,A)=[0,0].(DP3)
ID(A,B)=ID(B,A).(DP4)
For allA,B,C∈F(X), if A⊆B⊆C, then (35)IDA,B≤IDA,C,IDB,C≤IDA,C.We can construct some interval distance measure formulas based on Definition9 as follows.LetX={x1,x2,…,xn}; we set(36)ID1A,B=maxi=1n⁡μAxi-μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi;ID2A,B=1-1n∑i=1n2μAxi·μBxiμAxi2+μBxi2,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi;ID3A,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi;ID4A,B=1-1n∑i=1n2μAxi·μBxiμAxi2+μBxi2,1n∑i=1nμAxi-μBxiμAxi+μBxi;ID5A,B=1-1n1/p∑i=1nμAxi∧μBxiμAxi∨μBxi1/p,1-1np∑i=1nμAxi∧μBxiμAxi∨μBxipp≥1.If μA(x) and μB(x) can be integrated over the considered interval [a,b], then(37)ID6A,B=maxx∈a,b⁡μAx-μBx,1-1b-a∫abμAx∧μBxμAx∨μBxdx;ID7A,B=1-1b-a∫ab2μAx·μBxμAx2+μBx2dx,1-1b-a∫abμAx∧μBxμAx∨μBxdx;ID8A,B=1b-a∫abμAx-μBxμAx+μBxdx,1-1b-a∫abμAx∧μBxμAx∨μBxdx;ID9A,B=1-1b-a∫ab2μAx·μBxμAx2+μBx2dx,1b-a∫abμAx-μBxμAx+μBxdx;ID10A,B=1-1b-a1/p∫abμAx∧μBxμAx∨μBx1/pdx,1-1b-ap∫abμAx∧μBxμAx∨μBxpdxp≥1.The following theorem shows that the above formulas are all interval distance measures.Theorem 10.
I
D
i
(
A
,
B
)  (i=1,2,…,10) are interval distance measures.Proof.
Wheni=3, we have (38)ID3A,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi.If μA(xi)≤μB(xi), then(39)μAxi-μBxiμAxi+μBxi=μBxi-μAxiμAxi+μBxi,1-μAxi∧μBxiμAxi∨μBxi=1-μAxiμBxi=μBxi-μAxiμBxi.Namely,(40)0≤1n∑i=1nμAxi-μBxiμAxi+μBxi≤1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi≤1.That is to say, (41)ID3A,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi∈D0,1.(DP1) If A∈P(X), then μA(x)=0 or μA(x)=1,(42)1n∑i=1nμAxi-μAcxiμAxi+μAcxi=1,1-1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi=1.Namely, ID3(A,Ac)=[1,1].
(DP2) Consider(43)ID3A,A=1n∑i=1nμAxi-μAxiμAxi+μAxi,1-1n∑i=1nμAxi∧μAxiμAxi∨μAxi=0,0.
(DP3) It is obvious thatID3(A,B)=ID3(B,A).
(DP4) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x) and (44)ID3A,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi=1n∑i=1nμBxi-μAxiμAxi+μBxi,1n∑i=1n1-μAxiμBxi,ID3A,C=1n∑i=1nμAxi-μCxiμAxi+μCxi,1-1n∑i=1nμAxi∧μCxiμAxi∨μCxi=1n∑i=1nμCxi-μAxiμAxi+μCxi,1n∑i=1n1-μAxiμCxi,for the reason that (45)μBxi-μAxiμAxi+μBxi=1-2μAxiμAxi+μBxi,μCxi-μAxiμAxi+μCxi=1-2μAxiμAxi+μCxi.We have(46)μBxi-μAxiμAxi+μBxi≤μCxi-μAxiμAxi+μCxi,1-μAxi∧μBxiμAxi∨μBxi≤1-μAxi∧μCxiμAxi∨μCxi.Namely, ID3(A,B)≤ID3(A,C), ID3(B,C)≤ID3(A,C).
The proofs for other cases are similar.Definition 11.
A real functionII:F(X)×F(X)→D([0,1]) is called interval inclusion measure on F(X), if II satisfies the following properties: (IP1)
II(X,Φ)=[0,0].(IP2)
II(A,B)=[1,1]⇔A⊆B.(IP3)
For allA, B, C∈F(X), if A⊆B⊆C, then (47)IIC,A≤IIB,A,IIC,A≤IIC,B.We can construct some interval inclusion measure formulas based on Definition11 as follows.LetX={x1,x2,…,xn}, (48)II1A,B=1n∑i=1nμAxi∧μBxiμAxip,1n∑i=1nμAxi∧μBxiμAxi1/pp≥1;II2A,B=1n∑i=1nμBxiμAxi∨μBxip,1n∑i=1nμBxiμAxi∨μBxi1/pp≥1.If μA(x) and μB(x) can be integrated over the considered interval [a,b], we set (49)II3A,B=1b-a∫abμAx∧μBxμAxpdx,1b-a∫abμAx∧μBxμAx1/pdxp≥1;II4A,B=1b-a∫abμBxμAx∨μBxpdx,1b-a∫abμBxμAx∨μBx1/pdxp≥1.The following theorem shows that the above formulas are all interval inclusion measures.Theorem 12.
I
I
i
(
A
,
B
)  (i=1,2,3,4) are interval inclusion measures.Proof.
Wheni=1, we have(50)II1A,B=1n∑i=1nμAxi∧μBxiμAxip,1n∑i=1nμAxi∧μBxiμAxi1/pp≥1.It is obvious that(51)0≤1n∑i=1nμAxi∧μBxiμAxip≤1n∑i=1nμAxi∧μBxiμAxi1/p≤1.(IP1) Consider(52)II1X,ϕ=1n∑i=1nμXxi∧μϕxiμXxip,1n∑i=1nμXxi∧μϕxiμXxi1/p=0,0.(IP2) Consider(53)II1A,B=1,1⟺1n∑i=1nμAxi∧μBxiμAxip,1n∑i=1nμAxi∧μBxiμAxi1/p=1,1⟺μAxi∧μBxiμAxi=1⟺A⊆B.(IP3) If A⊆B⊆C, then μA(x)≤μB(x)≤μC(x),(54)II1C,A=1n∑i=1nμCxi∧μAxiμCxip,1n∑i=1nμCxi∧μAxiμCxi1/p=1n∑i=1nμAxiμCxip,1n∑i=1nμAxiμCxi1/p,II1B,A=1n∑i=1nμBxi∧μAxiμBxip,1n∑i=1nμBxi∧μAxiμBxi1/p=1n∑i=1nμAxiμBxip,1n∑i=1nμAxiμBxi1/p.So(55)1n∑i=1nμAxiμCxip≤1n∑i=1nμAxiμBxip,1n∑i=1nμAxiμCxi1/p≤1n∑i=1nμAxiμBxi1/p.Namely, II1(C,A)≤II1(B,A) and II1(C,A)≤II1(C,B). The proofs for other cases are similar.


### 4.2. The Relationships with the above Definitions

In this section, some conclusions about the relationships with those definitions are presented. From the follow theorems and corollaries, any two can be transformed into each other.Theorem 13.
IfIE is interval entropy of fuzzy sets, ∀A,B∈F(X), ∀x∈X, (56)f1A,Bx=μAx∧μBx2μAx∨μBx,f2A,Bx=1-μAx-μBx2,f3A,Bx=1+μAx-μBx2,then(1)
IE(fi(A,B))  (i=1,2,3) are interval similarity measures of fuzzy sets,(2)
IEc(fi(A,B))  (i=1,2,3) are interval distance measures of fuzzy sets,(3)
IE(fi(B,A∪B))  (i=1,2,3) are interval inclusion measures of the fuzzy sets.Proof.
(1) Wheni=3, we have the following:(SP1)
IfA∈P(X), then ∀x∈X,(57)f3A,Acx=1+μAx-μAcx2=1.Namely, f3(A,Ac)∈P(X), so IE(f3(A,Ac))=[0,0].(SP2)
∀x∈X, we have (58)f3A,Ax=1+μAx-μAx2=12.
SoIE(f3(A,Ac))=[1,1].(SP3)
It is obvious thatIE(f3(A,B))=IE(f3(B,A)).(SP4)
IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x), so(59)12≤1+μAx-μBx2≤1+μAx-μCx2,12≤1+μBx-μCx2≤1+μAx-μCx2.Namely, (60)IEf3A,B≥IEf3A,C,IEf3B,C≥IEf3A,C.The proofs for other cases are similar.
(2) Wheni=1, let ID(A,B)=IEc(f1(A,B)).
(DP1)∀A∈P(X), we have f1(A,A)(x)=0, so f1(A,A)∈P(X) and(61)IEcf1A,A=1,1.
(DP2) LetID(A,A)=IEc(f1(A,A)), for the reason that f1(A,A)(x)=1/2, so(62)IEcf1A,A=0,0.
(DP3) It is obvious thatID(A,B)=ID(B,A).
(DP4) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x). For the reason that(63)μAx∧μBx2μAx∨μBx=μAx2μBx,μAx∧μCx2μAx∨μCx=μAx2μCx,μBx∧μCx2μBx∨μCx=μBx2μCx,we have(64)μAx2μCx≤μAx2μBx≤12,μAx2μCx≤μBx2μCx≤12.Namely,(65)IEcf1A,B≤IEcf1A,C,IEcf1B,C≤IEcf1A,C.The proofs for other cases are similar.
(3) Wheni=1, we set II(A,B)=IE(f1(B,A∪B)).
(IP1)II(X,Φ)=IE(f1(Φ,X∪Φ))=IE(f1(Φ,X)), ∀x∈X, f1(Φ,X)=0, so f1(Φ,X)∈P(X) and II(X,Φ)=IE(f1(Φ,X))=[0,0].
(IP2) Consider(66)IIA,B=1,1⟺IEf1B,A∪B=1,1⟺f1B,A∪Bx≡12,∀x∈X⟺μBx∧μAx∨μBx2μAx∨μBx≡12,∀x∈X⟺μAx∨μBx=μBx,∀x∈X⟺μAx≤μBx,∀x∈X⟺A⊆B.
(IP3) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x) and(67)IIC,A=IEf1A,C∪A=IEf1A,C,IIB,A=IEf1A,B∪A=IEf1A,B,IIC,B=IEf1B,C∪B=IEf1B,C.
Owing to the fact that(68)μAx∧μBx2μAx∨μBx=μAx2μBx,μAx∧μCx2μAx∨μCx=μAx2μCx,μBx∧μCx2μBx∨μCx=μBx2μCx,then ∀x∈X, μA(x)/2μC(x)≤μA(x)/2μB(x)≤1/2, μA(x)/2μC(x)≤μB(x)/2μC(x)≤1/2.
Namely,II(C,A)≤II(B,A), II(C,A)≤II(C,B).
The proofs for other cases are similar.Corollary 14.
I
E
(
f
i
(
A
∩
B
,
A
)
)  (i=1,2,3) are interval inclusion measures of the fuzzy sets.Proof.
Wheni=1, we set II(A,B)=IE(f1(A∩B,A)).
(IP1)II(X,Φ)=IE(f1(X∩Φ,X))=IE(f1(Φ,X)), ∀x∈X, f1(Φ,X)=0, so f1(Φ,X)∈P(X) and II(X,Φ)=IE(f1(Φ,X))=[0,0].
(IP2) Consider(69)IIA,B=1,1⟺IEf1A∩B,A=1,1⟺f1A∩B,Ax≡12,∀x∈X⟺μAx∧μBx2μAx≡12,∀x∈X⟺μAx∧μBx=μAx,∀x∈X⟺μAx≤μBx,∀x∈X⟺A⊆B.
(IP3) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x) and(70)IIC,A=IEf1C∩A,C=IEf1A,C;IIB,A=IEf1B∩A,B=IEf1A,B;IIC,B=IEf1C∩B,C=IEf1B,C.In view of the fact that(71)μAx∧μBx2μAx∨μBx=μAx2μBx,μAx∧μCx2μAx∨μCx=μAx2μCx,μBx∧μCx2μBx∨μCx=μBx2μCx,then we have ∀x∈X, μA(x)/2μC(x)≤μA(x)/2μB(x)≤1/2, μA(x)/2μC(x)≤μB(x)/2μC(x)≤1/2.
Namely,II(C,A)≤II(B,A), II(C,A)≤II(C,B).
The proofs for other cases are similar.
For example, if(72)IEA=1n∑i=1n4μAxi2∧μAcxi2,1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi,then(73)IS11A,B=IEf3A,B=1n∑i=1n1-μAxi-μBxi2,1n∑i=1n1-μAxi-μBxi1+μAxi-μBxiis an interval similarity measure. And(74)ID11A,B=IEcf3A,B=1n∑i=1n2μAxi-μBxi1+μAxi-μBxi,1-1n∑i=1n1-μAxi-μBxi2is an interval distance measure. And(75)II5A,B=IEf3B,A∪B=1n∑i=1n1-μBxi-μAxi∨μBxi2,1n∑i=1n1-μBxi-μAxi∨μBxi1+μBxi-μAxi∨μBxi,II6A,B=IEf3A∩B,A=1n∑i=1n1-μAxi∧μBxi-μAxi2,1n∑i=1n1-μAxi∧μBxi-μAxi1+μAxi∧μBxi-μAxiare interval inclusion measures.Theorem 15.
ForA,B∈F(X), IS is interval similarity measure of fuzzy sets; then(1)
IS(A,Ac) is interval entropy of the fuzzy sets;(2)
ISc(A,B) is interval distance measure;(3)
IS(B,A∪B) is interval inclusion measure.Proof.
The proof is as follows:(1)(EP1)
IfA∈P(X), then IS(A,Ac)=[0,0].(EP2)
If∀x∈X, μA(x)≡1/2, then A(X)=Ac(X) and IS(A,Ac)=IS(A,A)=[1,1].(EP3)
When∀x∈X, μA2(x)≤μA1(x)≤1/2, we have μA2(x)≤μA1(x)≤1/2≤μA1c(x)≤μA2c(x), so (76)A2⊆A1⊆A1c⊆A2c.(EP4)
It is obvious thatIS(A,Ac)=IS(Ac,A).Namely,(77)ISA2,A2c≤ISA1,A2c≤ISA1,A1c.
When∀x∈X, μA2(x)≥μA1(x)≥1/2, the proof is similar.(2)
LetID(A,B)=ISc(A,B).
(DP1)
IfA∈P(X), then ISc(A,Ac)=0,0c=[1,1].(DP2)
ISc(A,A)=1,1c=[0,0].(DP3)
It is obvious thatISc(A,B)=ISc(B,A).(DP4)
IfA⊆B⊆C, then IS(A,C)≤IS(A,B), IS(A,C)≤IS(B,C). Namely,(78)IScA,B≤IScA,C,IScB,C≤IScA,C.(3)
LetII(A,B)=IS(B,A∪B).
(IP1)
II(X,Φ)=IS(Φ,X∪Φ)=IS(Φ,X), for Φ,X∈P(X), so II(X,Φ)=IS(Φ,X)=[0,0].(IP2)
II(A,B)=[1,1]⇔IS(B,A∪B)=[1,1]⇔A∪B=B⇔A⊆B.(IP3)
IfA⊆B⊆C, then IS(A,C)≤IS(A,B), IS(A,C)≤IS(B,C) and (79)IIC,A=ISA,A∪C=ISA,C;IIB,A=ISA,A∪B=ISA,B;IIC,B=ISB,B∪C=ISB,C.Namely, II(C,A)≤II(B,A), II(C,A)≤II(C,B).Corollary 16.
I
S
(
A
∩
B
,
B
) is interval inclusion measure.Proof.
LetII(A,B)=IS(A∩B,B).
(IP1)II(X,Φ)=IS(X∩Φ,X)=IS(Φ,X). For Φ, X∈P(X), we have II(X,Φ)=IS(Φ,X)=[0,0].
(IP2)II(A,B)=[1,1]⇔IS(A∩B,A)=[1,1]⇔A∩B=A⇔A⊆B.
(IP3) IfA⊆B⊆C, then IS(A,C)≤IS(A,B), IS(A,C)≤IS(B,C), and(80)IIC,A=ISA∩C,C=ISA,C;IIB,A=ISA∩B,B=ISA,B;IIC,B=ISB∩C,C=ISB,C.Namely, II(C,A)≤II(B,A), II(C,A)≤II(C,B).
For example, if(81)ISA,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2,then(82)IE13A=ISA,Ac=1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi,1n∑i=1n2μAxi·μAcxiμAxi2+μAcxi2is an interval entropy. And(83)ID12A,B=IScA,B=1n∑i=1nμAxi-μBxi2μAxi2+μBxi2,1n∑i=1nμAxi-μBxiμAxi∨μBxiis an interval distance measure. And (84)II7A,B=ISB,A∪B=1n∑i=1nμBxiμAxi∨μBxi,1n∑i=1n2μBxi·μAxi∨μBxiμBxi2+μAxi∨μBxi2,II8A,B=ISA∩B,A=1n∑i=1nμAxi∧μBxiμAxi,1n∑i=1n2μAxi·μAxi∧μBxiμAxi2+μAxi∧μBxi2are interval inclusion measures.Theorem 17.
ForA,B∈F(X), ID is interval distance measure of fuzzy sets; then(1)
IDc(A,B) is interval similarity measure;(2)
IDc(A,Ac) is interval entropy;(3)
IDc(B,A∪B), IDc(A∩B,A) are interval inclusion measures.Proof.
(1) It can be easily concluded by Theorem15.
(2) It can be easily concluded by Theorem13.
(3) It can be easily concluded by Theorems13 and 15.
For example, if(85)IDA,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi,then(86)IS12A,B=IDcA,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1-1n∑i=1nμAxi-μBxiμAxi+μBxiis an interval similarity measure. And(87)II14A=IDcA,Ac=1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi,1-1n∑i=1nμAxi-μAcxiis an interval entropy. And(88)II9A,B=IDcB,A∪B=1n∑i=1nμBxiμAxi∨μBxi,1-1n∑i=1nμBxi-μAxi∨μBxiμBxi+μAxi∨μBxiII10A,B=IDcA∩B,A=1n∑i=1nμAxi∧μBxiμAxi,1-1n∑i=1nμAxi∧μBxi-μAxiμAxi∧μBxi+μAxiare interval inclusion measures.Theorem 18.
ForA,B∈F(X), II is interval inclusion measure of fuzzy sets; then(1)
II(A∪B,A∩B) is interval similarity measure;(2)
II(A∪Ac,A∩Ac) is interval entropy;(3)
IIc(A∪B,A∩B) is interval distance measure.Proof.
(1) LetIS(A,B)=II(A∪B,A∩B).(SP1)
IfA∈P(X), then IS(A,Ac)=II(A∪Ac,A∩Ac)=II(X,ϕ)=[0,0].(SP2)
IS(A,A)=II(A∪A,A∩A)=II(A,A)=[1,1].(SP3)
It is obvious thatIS(A,B)=IS(B,A).(SP4)
IfA⊆B⊆C, then II(C,A)≤II(B,A), II(C,A)≤II(C,B) and(89)ISA,C=ISA∪C,A∩C=IIC,A;ISA,B=ISA∪B,A∩B=IIB,A;ISB,C=ISB∪C,B∩C=IIC,B.Namely,(90)ISA,C≤ISA,B,ISA,C≤ISB,C.
(2) LetIE(A)=II(A∪Ac,A∩Ac).(EP1)
IfA∈P(X), then IE(A)=II(A∪Ac,A∩Ac)=II(X,ϕ)=[0,0].(EP2)
If∀x∈X, μA(x)≡1/2, then A(x)=Ac(x) and(91)IEA=IIA∪Ac,A∩Ac=IIA,A=1,1.(EP3)
When∀x∈X, μA2(x)≤μA1(x)≤1/2, we have μA2(x)≤μA1(x)≤1/2≤μA1c(x)≤μA2c(x); then(92)A2⊆A1⊆A1c⊆A2c.Namely,(93)IEA1=IIA1∪A1c,A1∩A1c=IIA1c,A1,IEA2=IIA2∪A2c,A2∩A2c=IIA2c,A2.
ForII(A2c,A2)≤II(A2c,A1)≤II(A1c,A1), we have IE(A2)≤IE(A1).
When∀x∈X, μA2(x)≥μA1(x)≥1/2, the proof is similar.(EP4)
It is obvious thatIE(A)=IE(Ac).
(3) It can be easily concluded by Theorems15 and 17.
For example, if(94)IIA,B=1n∑i=1nμAxi∧μBxiμAxip,1n∑i=1nμAxi∧μBxiμAxi1/pp≥1,then(95)IS13A,B=IIA∪B,A∩B=1n∑i=1nμAxi∧μBxiμAxi∨μBxip,1n∑i=1nμAxi∧μBxiμAxi∨μBxi1/pp≥1is an interval similarity measure. And(96)IE15A,B=IIA∪Ac,A∩Ac=1n∑i=1nμAxi∧μAcxiμAxi∨μAcxip,1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi1/pp≥1is an interval entropy. And(97)ID13A,B=IIcA∪B,A∩B=1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi1/p,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxipp≥1is an interval distance measure.
Above all, it can be concluded that, among interval entropy, interval similarity measure, interval distance measure and interval inclusion measure, any two can be transformed into each other.


## 4.1. Other Definitions to Describe the Uncertainty

In fact, the conceptions to describe the general measure of uncertainty such as similarity measure, distance measure, and inclusion measure have also built the mapping between two fuzzy numbers and real numbers on[0,1], in which only one point is used to represent two fuzzy numbers (see appendix). But when the values are interval, it is necessary to extend these conceptions to interval values, and, in this section, the definitions of interval similarity measure, interval distance measure, and interval inclusion measure are proposed as follows.Definition 7.
A real functionIS:F(X)×F(X)→D([0,1]) is called interval similarity measure on F(X) if IS satisfies the following properties:(SP1)
IS(A,Ac)=[0,0] if A is a crisp set.(SP2)
IS(A,A)=[1,1].(SP3)
IS(A,B)=IS(B,A).(SP4)
For allA, B, C∈F(X), if A⊆B⊆C, then(23)ISA,C≤ISA,B,ISA,C≤ISB,C.We can construct some interval similarity measure formulas based on Definition7 as follows.For instance, letX={x1,x2,…,xn}; we set(24)IS1A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1-maxxi∈X⁡μAxi-μBxi;IS2A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2;IS3A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1-1n∑i=1nμAxi-μBxiμAxi+μBxi;IS4A,B=1-1n∑i=1nμAxi-μBxiμAxi+μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2;IS5A,B=1np∑i=1nμAxi∧μBxiμAxi∨μBxip,1n1/p∑i=1nμAxi∧μBxiμAxi∨μBxi1/pp≥1.If μA(x) and μB(x) can be integrated over the considered interval [a,b], then (25)IS6A,B=1b-a∫abμAx∧μBxμAx∨μBxdx,1-maxx∈a,b⁡μAx-μBx;IS7A,B=1b-a∫abμAx∧μBxμAx∨μBxdx,1b-a∫ab2μAx·μBxμAx2+μBx2dx;IS8A,B=1b-a∫abμAx∧μBxμAx∨μBxdx,1-1b-a∫abμAx-μBxμAx+μBxdx;IS9A,B=1-1b-a∫abμAx-μBxμAx+μBxdx,1b-a∫ab2μAx·μBxμAx2+μBx2dx;IS10A,B=1b-ap∫abμAx∧μBxμAx∨μBxpdx,1b-a1/p∫abμAx∧μBxμAx∨μBx1/pdxp≥1.The following theorem shows that the above formulas are all interval similarity measures.Theorem 8.
I
S
i
(
A
,
B
)  (i=1,2,…,10) are interval similarity measures.Proof.
Wheni=2, we have(26)IS2A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2.If μA(xi)≤μB(xi), then μA(xi)∧μB(xi)/μA(xi)∨μB(xi)=μA(xi)/μB(xi),(27)2μAxi·μBxiμAxi2+μBxi2-μAxiμBxi=μAxiμBxi2-μAxi2μAxi2+μBxi2μBxi≥0.Namely,(28)0≤1n∑i=1nμAxi∧μBxiμAxi∨μBxi≤1n∑i=1n2μAxi·μBxiμAxi2+μBxi2≤1.That is to say,(29)IS2A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2∈D0,1.The proof for μA(xi)≥μB(xi) is similar.
(SP1) IfA∈P(X), then μA(x)=0 or μA(x)=1. (30)1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi=0,1n∑i=1n2μAxi·μAcxiμAxi2+μAcxi2=0.Namely, IS2(A,Ac)=[0,0].
(SP2) Consider(31)IS2A,A=1n∑i=1nμAxi∧μAxiμAxi∨μAxi,1n∑i=1n2μAxi·μAxiμAxi2+μAxi2=1,1.
(SP3) It is obvious thatIS2(A,B)=IS2(B,A).
(SP4) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x). So(32)IS2A,C=1n∑i=1nμAxi∧μCxiμAxi∨μCxi,1n∑i=1n2μAxi·μCxiμAxi2+μCxi2=1n∑i=1nμAxiμCxi,1n∑i=1n2μAxi·μCxiμAxi2+μCxi2,IS2A,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2=1n∑i=1nμAxiμBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2,for the reason that(33)2μAxi·μCxiμAxi2+μCxi2-2μAxi·μBxiμAxi2+μBxi2=2μAxiμCxi-μBxiμAxi2-μBxiμCxiμAxi2+μCxi2μAxi2+μBxi2<0.Namely,(34)μAxμCx≤μAxμBx,2μAxi·μCxiμAxi2+μCxi2≤2μAxi·μBxiμAxi2+μBxi2.That is to say, IS2(A,C)≤IS2(A,B) and IS2(A,C)≤IS2(B,C). And the proofs for other cases are similar.Definition 9.
A real functionID:F(X)×F(X)→D([0,1]) is called interval distance measure on F(X), if ID satisfies the following properties: (DP1)
ID(A,Ac)=[1,1] ifA is a crisp set.(DP2)
ID(A,A)=[0,0].(DP3)
ID(A,B)=ID(B,A).(DP4)
For allA,B,C∈F(X), if A⊆B⊆C, then (35)IDA,B≤IDA,C,IDB,C≤IDA,C.We can construct some interval distance measure formulas based on Definition9 as follows.LetX={x1,x2,…,xn}; we set(36)ID1A,B=maxi=1n⁡μAxi-μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi;ID2A,B=1-1n∑i=1n2μAxi·μBxiμAxi2+μBxi2,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi;ID3A,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi;ID4A,B=1-1n∑i=1n2μAxi·μBxiμAxi2+μBxi2,1n∑i=1nμAxi-μBxiμAxi+μBxi;ID5A,B=1-1n1/p∑i=1nμAxi∧μBxiμAxi∨μBxi1/p,1-1np∑i=1nμAxi∧μBxiμAxi∨μBxipp≥1.If μA(x) and μB(x) can be integrated over the considered interval [a,b], then(37)ID6A,B=maxx∈a,b⁡μAx-μBx,1-1b-a∫abμAx∧μBxμAx∨μBxdx;ID7A,B=1-1b-a∫ab2μAx·μBxμAx2+μBx2dx,1-1b-a∫abμAx∧μBxμAx∨μBxdx;ID8A,B=1b-a∫abμAx-μBxμAx+μBxdx,1-1b-a∫abμAx∧μBxμAx∨μBxdx;ID9A,B=1-1b-a∫ab2μAx·μBxμAx2+μBx2dx,1b-a∫abμAx-μBxμAx+μBxdx;ID10A,B=1-1b-a1/p∫abμAx∧μBxμAx∨μBx1/pdx,1-1b-ap∫abμAx∧μBxμAx∨μBxpdxp≥1.The following theorem shows that the above formulas are all interval distance measures.Theorem 10.
I
D
i
(
A
,
B
)  (i=1,2,…,10) are interval distance measures.Proof.
Wheni=3, we have (38)ID3A,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi.If μA(xi)≤μB(xi), then(39)μAxi-μBxiμAxi+μBxi=μBxi-μAxiμAxi+μBxi,1-μAxi∧μBxiμAxi∨μBxi=1-μAxiμBxi=μBxi-μAxiμBxi.Namely,(40)0≤1n∑i=1nμAxi-μBxiμAxi+μBxi≤1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi≤1.That is to say, (41)ID3A,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi∈D0,1.(DP1) If A∈P(X), then μA(x)=0 or μA(x)=1,(42)1n∑i=1nμAxi-μAcxiμAxi+μAcxi=1,1-1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi=1.Namely, ID3(A,Ac)=[1,1].
(DP2) Consider(43)ID3A,A=1n∑i=1nμAxi-μAxiμAxi+μAxi,1-1n∑i=1nμAxi∧μAxiμAxi∨μAxi=0,0.
(DP3) It is obvious thatID3(A,B)=ID3(B,A).
(DP4) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x) and (44)ID3A,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi=1n∑i=1nμBxi-μAxiμAxi+μBxi,1n∑i=1n1-μAxiμBxi,ID3A,C=1n∑i=1nμAxi-μCxiμAxi+μCxi,1-1n∑i=1nμAxi∧μCxiμAxi∨μCxi=1n∑i=1nμCxi-μAxiμAxi+μCxi,1n∑i=1n1-μAxiμCxi,for the reason that (45)μBxi-μAxiμAxi+μBxi=1-2μAxiμAxi+μBxi,μCxi-μAxiμAxi+μCxi=1-2μAxiμAxi+μCxi.We have(46)μBxi-μAxiμAxi+μBxi≤μCxi-μAxiμAxi+μCxi,1-μAxi∧μBxiμAxi∨μBxi≤1-μAxi∧μCxiμAxi∨μCxi.Namely, ID3(A,B)≤ID3(A,C), ID3(B,C)≤ID3(A,C).
The proofs for other cases are similar.Definition 11.
A real functionII:F(X)×F(X)→D([0,1]) is called interval inclusion measure on F(X), if II satisfies the following properties: (IP1)
II(X,Φ)=[0,0].(IP2)
II(A,B)=[1,1]⇔A⊆B.(IP3)
For allA, B, C∈F(X), if A⊆B⊆C, then (47)IIC,A≤IIB,A,IIC,A≤IIC,B.We can construct some interval inclusion measure formulas based on Definition11 as follows.LetX={x1,x2,…,xn}, (48)II1A,B=1n∑i=1nμAxi∧μBxiμAxip,1n∑i=1nμAxi∧μBxiμAxi1/pp≥1;II2A,B=1n∑i=1nμBxiμAxi∨μBxip,1n∑i=1nμBxiμAxi∨μBxi1/pp≥1.If μA(x) and μB(x) can be integrated over the considered interval [a,b], we set (49)II3A,B=1b-a∫abμAx∧μBxμAxpdx,1b-a∫abμAx∧μBxμAx1/pdxp≥1;II4A,B=1b-a∫abμBxμAx∨μBxpdx,1b-a∫abμBxμAx∨μBx1/pdxp≥1.The following theorem shows that the above formulas are all interval inclusion measures.Theorem 12.
I
I
i
(
A
,
B
)  (i=1,2,3,4) are interval inclusion measures.Proof.
Wheni=1, we have(50)II1A,B=1n∑i=1nμAxi∧μBxiμAxip,1n∑i=1nμAxi∧μBxiμAxi1/pp≥1.It is obvious that(51)0≤1n∑i=1nμAxi∧μBxiμAxip≤1n∑i=1nμAxi∧μBxiμAxi1/p≤1.(IP1) Consider(52)II1X,ϕ=1n∑i=1nμXxi∧μϕxiμXxip,1n∑i=1nμXxi∧μϕxiμXxi1/p=0,0.(IP2) Consider(53)II1A,B=1,1⟺1n∑i=1nμAxi∧μBxiμAxip,1n∑i=1nμAxi∧μBxiμAxi1/p=1,1⟺μAxi∧μBxiμAxi=1⟺A⊆B.(IP3) If A⊆B⊆C, then μA(x)≤μB(x)≤μC(x),(54)II1C,A=1n∑i=1nμCxi∧μAxiμCxip,1n∑i=1nμCxi∧μAxiμCxi1/p=1n∑i=1nμAxiμCxip,1n∑i=1nμAxiμCxi1/p,II1B,A=1n∑i=1nμBxi∧μAxiμBxip,1n∑i=1nμBxi∧μAxiμBxi1/p=1n∑i=1nμAxiμBxip,1n∑i=1nμAxiμBxi1/p.So(55)1n∑i=1nμAxiμCxip≤1n∑i=1nμAxiμBxip,1n∑i=1nμAxiμCxi1/p≤1n∑i=1nμAxiμBxi1/p.Namely, II1(C,A)≤II1(B,A) and II1(C,A)≤II1(C,B). The proofs for other cases are similar.


## 4.2. The Relationships with the above Definitions

In this section, some conclusions about the relationships with those definitions are presented. From the follow theorems and corollaries, any two can be transformed into each other.Theorem 13.
IfIE is interval entropy of fuzzy sets, ∀A,B∈F(X), ∀x∈X, (56)f1A,Bx=μAx∧μBx2μAx∨μBx,f2A,Bx=1-μAx-μBx2,f3A,Bx=1+μAx-μBx2,then(1)
IE(fi(A,B))  (i=1,2,3) are interval similarity measures of fuzzy sets,(2)
IEc(fi(A,B))  (i=1,2,3) are interval distance measures of fuzzy sets,(3)
IE(fi(B,A∪B))  (i=1,2,3) are interval inclusion measures of the fuzzy sets.Proof.
(1) Wheni=3, we have the following:(SP1)
IfA∈P(X), then ∀x∈X,(57)f3A,Acx=1+μAx-μAcx2=1.Namely, f3(A,Ac)∈P(X), so IE(f3(A,Ac))=[0,0].(SP2)
∀x∈X, we have (58)f3A,Ax=1+μAx-μAx2=12.
SoIE(f3(A,Ac))=[1,1].(SP3)
It is obvious thatIE(f3(A,B))=IE(f3(B,A)).(SP4)
IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x), so(59)12≤1+μAx-μBx2≤1+μAx-μCx2,12≤1+μBx-μCx2≤1+μAx-μCx2.Namely, (60)IEf3A,B≥IEf3A,C,IEf3B,C≥IEf3A,C.The proofs for other cases are similar.
(2) Wheni=1, let ID(A,B)=IEc(f1(A,B)).
(DP1)∀A∈P(X), we have f1(A,A)(x)=0, so f1(A,A)∈P(X) and(61)IEcf1A,A=1,1.
(DP2) LetID(A,A)=IEc(f1(A,A)), for the reason that f1(A,A)(x)=1/2, so(62)IEcf1A,A=0,0.
(DP3) It is obvious thatID(A,B)=ID(B,A).
(DP4) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x). For the reason that(63)μAx∧μBx2μAx∨μBx=μAx2μBx,μAx∧μCx2μAx∨μCx=μAx2μCx,μBx∧μCx2μBx∨μCx=μBx2μCx,we have(64)μAx2μCx≤μAx2μBx≤12,μAx2μCx≤μBx2μCx≤12.Namely,(65)IEcf1A,B≤IEcf1A,C,IEcf1B,C≤IEcf1A,C.The proofs for other cases are similar.
(3) Wheni=1, we set II(A,B)=IE(f1(B,A∪B)).
(IP1)II(X,Φ)=IE(f1(Φ,X∪Φ))=IE(f1(Φ,X)), ∀x∈X, f1(Φ,X)=0, so f1(Φ,X)∈P(X) and II(X,Φ)=IE(f1(Φ,X))=[0,0].
(IP2) Consider(66)IIA,B=1,1⟺IEf1B,A∪B=1,1⟺f1B,A∪Bx≡12,∀x∈X⟺μBx∧μAx∨μBx2μAx∨μBx≡12,∀x∈X⟺μAx∨μBx=μBx,∀x∈X⟺μAx≤μBx,∀x∈X⟺A⊆B.
(IP3) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x) and(67)IIC,A=IEf1A,C∪A=IEf1A,C,IIB,A=IEf1A,B∪A=IEf1A,B,IIC,B=IEf1B,C∪B=IEf1B,C.
Owing to the fact that(68)μAx∧μBx2μAx∨μBx=μAx2μBx,μAx∧μCx2μAx∨μCx=μAx2μCx,μBx∧μCx2μBx∨μCx=μBx2μCx,then ∀x∈X, μA(x)/2μC(x)≤μA(x)/2μB(x)≤1/2, μA(x)/2μC(x)≤μB(x)/2μC(x)≤1/2.
Namely,II(C,A)≤II(B,A), II(C,A)≤II(C,B).
The proofs for other cases are similar.Corollary 14.
I
E
(
f
i
(
A
∩
B
,
A
)
)  (i=1,2,3) are interval inclusion measures of the fuzzy sets.Proof.
Wheni=1, we set II(A,B)=IE(f1(A∩B,A)).
(IP1)II(X,Φ)=IE(f1(X∩Φ,X))=IE(f1(Φ,X)), ∀x∈X, f1(Φ,X)=0, so f1(Φ,X)∈P(X) and II(X,Φ)=IE(f1(Φ,X))=[0,0].
(IP2) Consider(69)IIA,B=1,1⟺IEf1A∩B,A=1,1⟺f1A∩B,Ax≡12,∀x∈X⟺μAx∧μBx2μAx≡12,∀x∈X⟺μAx∧μBx=μAx,∀x∈X⟺μAx≤μBx,∀x∈X⟺A⊆B.
(IP3) IfA⊆B⊆C, then μA(x)≤μB(x)≤μC(x) and(70)IIC,A=IEf1C∩A,C=IEf1A,C;IIB,A=IEf1B∩A,B=IEf1A,B;IIC,B=IEf1C∩B,C=IEf1B,C.In view of the fact that(71)μAx∧μBx2μAx∨μBx=μAx2μBx,μAx∧μCx2μAx∨μCx=μAx2μCx,μBx∧μCx2μBx∨μCx=μBx2μCx,then we have ∀x∈X, μA(x)/2μC(x)≤μA(x)/2μB(x)≤1/2, μA(x)/2μC(x)≤μB(x)/2μC(x)≤1/2.
Namely,II(C,A)≤II(B,A), II(C,A)≤II(C,B).
The proofs for other cases are similar.
For example, if(72)IEA=1n∑i=1n4μAxi2∧μAcxi2,1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi,then(73)IS11A,B=IEf3A,B=1n∑i=1n1-μAxi-μBxi2,1n∑i=1n1-μAxi-μBxi1+μAxi-μBxiis an interval similarity measure. And(74)ID11A,B=IEcf3A,B=1n∑i=1n2μAxi-μBxi1+μAxi-μBxi,1-1n∑i=1n1-μAxi-μBxi2is an interval distance measure. And(75)II5A,B=IEf3B,A∪B=1n∑i=1n1-μBxi-μAxi∨μBxi2,1n∑i=1n1-μBxi-μAxi∨μBxi1+μBxi-μAxi∨μBxi,II6A,B=IEf3A∩B,A=1n∑i=1n1-μAxi∧μBxi-μAxi2,1n∑i=1n1-μAxi∧μBxi-μAxi1+μAxi∧μBxi-μAxiare interval inclusion measures.Theorem 15.
ForA,B∈F(X), IS is interval similarity measure of fuzzy sets; then(1)
IS(A,Ac) is interval entropy of the fuzzy sets;(2)
ISc(A,B) is interval distance measure;(3)
IS(B,A∪B) is interval inclusion measure.Proof.
The proof is as follows:(1)(EP1)
IfA∈P(X), then IS(A,Ac)=[0,0].(EP2)
If∀x∈X, μA(x)≡1/2, then A(X)=Ac(X) and IS(A,Ac)=IS(A,A)=[1,1].(EP3)
When∀x∈X, μA2(x)≤μA1(x)≤1/2, we have μA2(x)≤μA1(x)≤1/2≤μA1c(x)≤μA2c(x), so (76)A2⊆A1⊆A1c⊆A2c.(EP4)
It is obvious thatIS(A,Ac)=IS(Ac,A).Namely,(77)ISA2,A2c≤ISA1,A2c≤ISA1,A1c.
When∀x∈X, μA2(x)≥μA1(x)≥1/2, the proof is similar.(2)
LetID(A,B)=ISc(A,B).
(DP1)
IfA∈P(X), then ISc(A,Ac)=0,0c=[1,1].(DP2)
ISc(A,A)=1,1c=[0,0].(DP3)
It is obvious thatISc(A,B)=ISc(B,A).(DP4)
IfA⊆B⊆C, then IS(A,C)≤IS(A,B), IS(A,C)≤IS(B,C). Namely,(78)IScA,B≤IScA,C,IScB,C≤IScA,C.(3)
LetII(A,B)=IS(B,A∪B).
(IP1)
II(X,Φ)=IS(Φ,X∪Φ)=IS(Φ,X), for Φ,X∈P(X), so II(X,Φ)=IS(Φ,X)=[0,0].(IP2)
II(A,B)=[1,1]⇔IS(B,A∪B)=[1,1]⇔A∪B=B⇔A⊆B.(IP3)
IfA⊆B⊆C, then IS(A,C)≤IS(A,B), IS(A,C)≤IS(B,C) and (79)IIC,A=ISA,A∪C=ISA,C;IIB,A=ISA,A∪B=ISA,B;IIC,B=ISB,B∪C=ISB,C.Namely, II(C,A)≤II(B,A), II(C,A)≤II(C,B).Corollary 16.
I
S
(
A
∩
B
,
B
) is interval inclusion measure.Proof.
LetII(A,B)=IS(A∩B,B).
(IP1)II(X,Φ)=IS(X∩Φ,X)=IS(Φ,X). For Φ, X∈P(X), we have II(X,Φ)=IS(Φ,X)=[0,0].
(IP2)II(A,B)=[1,1]⇔IS(A∩B,A)=[1,1]⇔A∩B=A⇔A⊆B.
(IP3) IfA⊆B⊆C, then IS(A,C)≤IS(A,B), IS(A,C)≤IS(B,C), and(80)IIC,A=ISA∩C,C=ISA,C;IIB,A=ISA∩B,B=ISA,B;IIC,B=ISB∩C,C=ISB,C.Namely, II(C,A)≤II(B,A), II(C,A)≤II(C,B).
For example, if(81)ISA,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1n∑i=1n2μAxi·μBxiμAxi2+μBxi2,then(82)IE13A=ISA,Ac=1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi,1n∑i=1n2μAxi·μAcxiμAxi2+μAcxi2is an interval entropy. And(83)ID12A,B=IScA,B=1n∑i=1nμAxi-μBxi2μAxi2+μBxi2,1n∑i=1nμAxi-μBxiμAxi∨μBxiis an interval distance measure. And (84)II7A,B=ISB,A∪B=1n∑i=1nμBxiμAxi∨μBxi,1n∑i=1n2μBxi·μAxi∨μBxiμBxi2+μAxi∨μBxi2,II8A,B=ISA∩B,A=1n∑i=1nμAxi∧μBxiμAxi,1n∑i=1n2μAxi·μAxi∧μBxiμAxi2+μAxi∧μBxi2are interval inclusion measures.Theorem 17.
ForA,B∈F(X), ID is interval distance measure of fuzzy sets; then(1)
IDc(A,B) is interval similarity measure;(2)
IDc(A,Ac) is interval entropy;(3)
IDc(B,A∪B), IDc(A∩B,A) are interval inclusion measures.Proof.
(1) It can be easily concluded by Theorem15.
(2) It can be easily concluded by Theorem13.
(3) It can be easily concluded by Theorems13 and 15.
For example, if(85)IDA,B=1n∑i=1nμAxi-μBxiμAxi+μBxi,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi,then(86)IS12A,B=IDcA,B=1n∑i=1nμAxi∧μBxiμAxi∨μBxi,1-1n∑i=1nμAxi-μBxiμAxi+μBxiis an interval similarity measure. And(87)II14A=IDcA,Ac=1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi,1-1n∑i=1nμAxi-μAcxiis an interval entropy. And(88)II9A,B=IDcB,A∪B=1n∑i=1nμBxiμAxi∨μBxi,1-1n∑i=1nμBxi-μAxi∨μBxiμBxi+μAxi∨μBxiII10A,B=IDcA∩B,A=1n∑i=1nμAxi∧μBxiμAxi,1-1n∑i=1nμAxi∧μBxi-μAxiμAxi∧μBxi+μAxiare interval inclusion measures.Theorem 18.
ForA,B∈F(X), II is interval inclusion measure of fuzzy sets; then(1)
II(A∪B,A∩B) is interval similarity measure;(2)
II(A∪Ac,A∩Ac) is interval entropy;(3)
IIc(A∪B,A∩B) is interval distance measure.Proof.
(1) LetIS(A,B)=II(A∪B,A∩B).(SP1)
IfA∈P(X), then IS(A,Ac)=II(A∪Ac,A∩Ac)=II(X,ϕ)=[0,0].(SP2)
IS(A,A)=II(A∪A,A∩A)=II(A,A)=[1,1].(SP3)
It is obvious thatIS(A,B)=IS(B,A).(SP4)
IfA⊆B⊆C, then II(C,A)≤II(B,A), II(C,A)≤II(C,B) and(89)ISA,C=ISA∪C,A∩C=IIC,A;ISA,B=ISA∪B,A∩B=IIB,A;ISB,C=ISB∪C,B∩C=IIC,B.Namely,(90)ISA,C≤ISA,B,ISA,C≤ISB,C.
(2) LetIE(A)=II(A∪Ac,A∩Ac).(EP1)
IfA∈P(X), then IE(A)=II(A∪Ac,A∩Ac)=II(X,ϕ)=[0,0].(EP2)
If∀x∈X, μA(x)≡1/2, then A(x)=Ac(x) and(91)IEA=IIA∪Ac,A∩Ac=IIA,A=1,1.(EP3)
When∀x∈X, μA2(x)≤μA1(x)≤1/2, we have μA2(x)≤μA1(x)≤1/2≤μA1c(x)≤μA2c(x); then(92)A2⊆A1⊆A1c⊆A2c.Namely,(93)IEA1=IIA1∪A1c,A1∩A1c=IIA1c,A1,IEA2=IIA2∪A2c,A2∩A2c=IIA2c,A2.
ForII(A2c,A2)≤II(A2c,A1)≤II(A1c,A1), we have IE(A2)≤IE(A1).
When∀x∈X, μA2(x)≥μA1(x)≥1/2, the proof is similar.(EP4)
It is obvious thatIE(A)=IE(Ac).
(3) It can be easily concluded by Theorems15 and 17.
For example, if(94)IIA,B=1n∑i=1nμAxi∧μBxiμAxip,1n∑i=1nμAxi∧μBxiμAxi1/pp≥1,then(95)IS13A,B=IIA∪B,A∩B=1n∑i=1nμAxi∧μBxiμAxi∨μBxip,1n∑i=1nμAxi∧μBxiμAxi∨μBxi1/pp≥1is an interval similarity measure. And(96)IE15A,B=IIA∪Ac,A∩Ac=1n∑i=1nμAxi∧μAcxiμAxi∨μAcxip,1n∑i=1nμAxi∧μAcxiμAxi∨μAcxi1/pp≥1is an interval entropy. And(97)ID13A,B=IIcA∪B,A∩B=1-1n∑i=1nμAxi∧μBxiμAxi∨μBxi1/p,1-1n∑i=1nμAxi∧μBxiμAxi∨μBxipp≥1is an interval distance measure.
Above all, it can be concluded that, among interval entropy, interval similarity measure, interval distance measure and interval inclusion measure, any two can be transformed into each other.


## 5. The Application to Fuzzy Multiple Attribute Decision Making

Givenm alternatives and n criteria, a typical FMADM problem can be expressed in matrix format as(98)X~=C1⋯CnA1⋮Amx~11⋯x~1n⋮⋮⋮x~m1⋯x~mn,where A1,A2,…,Am are the alternatives to be chosen, C1,C2,…,Cn denote the evaluation criteria, x~ij represents the value of alternative Ai with respect to criterion Cj evaluated, and the set W={ω1,ω2,…,ωn} is the weight of criterion [4–6, 8, 9, 11, 12]. Then the alternatives can be ranked based on the given information [36–38]. In this section, the steps of the proposed method are described with an example. How to determine the weights of all alternatives is described in Section 5.1. And the method to compute the evaluation values of all alternatives with respect to criteria is listed in Section 5.2. The method to rank the order of all alternatives named as possibility-based comparison relation is expressed in Section 5.3. Finally, a FMADM problem with thirty-one alternatives and twenty-three criteria is used to explain the process of the decision making.


### 5.1. Internal Entropy Weight

It is well known that entropy is a method in obtaining the weights for a MADM problem especially when obtaining suitable weights based on the preferences and decision making experiments are not possible [22]. The original procedure of entropy weight can be expressed in a series of steps [39]:(1)
Compute entropyEj.(2)
Setωj=1-Ej/E, E=∑s=1n(1-Es), j=1,2,…,n, as the weight of attribute j.Now suppose that the weight of the elements of decision matrix is difficult and uncertain, as a result, their values are considered as intervals. Entropy can be extended to interval entropy. The steps are as follows:(I1)
Compute interval entropyIEij=[IEijL,IEijR], i=1,2,…,m, j=1,2,…,n.(I2)
Setωij=[ωijL,ωijR]=[1-IEijR,1-IEijL], i=1,2,…,m, j=1,2,…,n, as the weight of attribute j.For instance, if A=[0.5,0.6,0.7,0.8], by (7), then interval entropy and interval entropy weight can be shown as [0.5,0.6667] and [0.3333,0.5], respectively.


### 5.2. The Evaluation Values Obtained by the Weight Average

To the givenα value, the α-cut sets of the weight ωj and the alternatives x~ij can be denoted by (ωj)α=[(ωj)αL,(ωj)αR] and (x~ij)α=[(x~ij)αL,(x~ij)αR], respectively. So the evaluation values of the given α value can be denoted by the following:(99)Fiα=FiαL,FiαR=∑j=1nωjαLxijαL,∑j=1nωjαRxijαR.Remark 19 (see [40]).
Here, two reasons are responded to the unnormalization. Firstly, the evaluation values after normalization are not always in[0,1]. Secondly, the normalization can only effect the proportion changes of the evaluation values, without any changes to the actual ranking results.
For instance, ifA=[0.5,0.6,0.7,0.8], then interval entropy weight can be shown as [0.3333,0.5] and the α-cut sets can be denoted by(100)Aα=0.6λ+0.51-λ,0.7λ+0.81-λλ∈0,10.6,0.7λ=1.So(101)Fα=0.05556α+0.2778,0.53336-0.06667α.


### 5.3. The Possibility-Based Method for Ranking Fuzzy Numbers [41]

In this section, possibility-based comparison relation is presented. Since theα-cut sets of fuzzy numbers can be expressed as the interval value sets, possibility-based comparison relation on interval values is proposed in Section 5.3.1, and then the comparison relation on fuzzy numbers is presented in Section 5.3.2.


#### 5.3.1. Interval Values

AssumingX=[x-,x+], Y=[y-,y+] are the interval values, Xλ, Yλ (λ∈R) can be constructed as follows:(102)Xλ=1x-≥λ12x-<λ≤x+0x+<λ,Yα=1y-≥λ12y-<λ≤y+0y+<λ.Set(103)Gλ=1Xλ>Yλ0Xλ≤Yλ.P
(
X
≻
Y
) is used to represent the degree of interval values “X≻Y” which is named as possibility-based comparison relation on interval values:(104)PX≻Y=1x+∨y+-x-∧y-∫x-∧y-x+∨y+Gλdλ.


#### 5.3.2. Fuzzy Numbers

IfA and B are two fuzzy numbers, then the possibility-based comparison relation between fuzzy numbers can be denoted by P(A≻B) as follows:(105)PA≻B=∫01PAα≻Bαdα.LetS={S1,S2,…,Sn} be a set which contains finite fuzzy numbers to be ranked; we can select function T as the evaluation of comparisons [42]. Then the fuzzy numbers can be ranked as the values of P(Si≻T). Here, T can be seen as an attitude which comes from the decision maker. Usually, the attitude of the decision maker can be classified into three groups: neutral, optimistic, and pessimistic. The optimistic attitude reflects the preferences of decision makers to the maximal payoff, and it is often used as follows: (106)μAoptx=x-xminxmax-xminxmin≤x≤xmax0else.The neutral attitude reflects the preferences of decision makers for not only the payoff which could be obtained but also the risk which could be faced, and it is often used as follows:(107)μAneux=1xmin≤x≤xmax0else.And the last attitude is the pessimistic one reflecting the preferences of decision makers for not only the payoff which could be gained but also the risk which could be faced. It is often used as follows:(108)μApessx=xmax-xxmax-xminxmin≤x≤xmax0else.For example, the ranking results with respect to three attitudes are shown in Table1.Table 1
The ranking results.
A
=
[
0,0.1,0.5,1
]
B
=
[
0.3,0.5,0.8,0.9
]
Ranking order
Optimistic
0
0.065
A
≺
B
Neutral
0.5
0.4
A
≺
B
Pessimistic
0.4547
0.7723
A
≺
B


### 5.4. A Numerical Example

In this section, the steps of the proposed method are described with an example. Suppose that there is a FMADM problem with thirty-one alternatives and twenty-three criteria. Data are presented in Tables2-3 [43] which can be represented as seven linguistic variables to evaluate degree of impact for railway reconstruction projects and their corresponding trapezoid fuzzy numbers are listed in Table 4. The decision making can be conducted as follows.Table 2
The evaluation values ofC1–C12.
C
1
C
2
C
3
C
4
C
5
C
6
C
7
C
8
C
9
C
10
C
11
C
12
A
1
A
11
S
S
VS
VS
A
A
A
U
S
A
U
A
A
12
A
AS
AS
AS
VS
A
VS
A
VS
S
S
VS
A
13
VS
VS
VS
VS
S
VS
S
S
AS
S
VS
VS
A
14
VS
VS
VS
A
A
S
A
A
S
U
A
VS
A
2
A
21
A
S
S
S
S
A
A
VS
A
U
S
S
A
22
A
A
S
VS
S
U
A
S
S
U
S
S
A
23
A
A
S
S
A
U
A
VS
A
U
S
S
A
24
S
A
S
AS
A
A
S
S
A
A
VS
S
A
3
A
31
VS
U
AS
AS
VS
S
A
AS
A
S
S
S
A
32
A
U
VS
S
S
S
A
VS
A
A
S
VS
A
33
A
A
S
VS
VS
A
U
VS
A
A
A
VS
A
34
S
A
VS
A
VS
A
A
AS
S
A
S
S
A
4
A
41
U
VS
VS
AS
VS
S
A
AS
A
VS
VS
AS
A
42
A
VS
A
S
S
S
S
VS
VS
S
A
S
A
43
A
VS
S
VS
S
A
A
S
VS
S
S
S
A
44
A
S
VS
S
VS
S
A
S
A
A
S
S
A
45
A
VS
AS
AS
VS
S
S
AS
A
A
A
AS
A
46
A
S
VS
S
VS
S
VS
AS
A
A
S
VS
A
5
A
51
S
U
A
VS
S
VS
S
AS
VS
S
U
AS
A
52
VS
A
S
S
A
A
A
S
S
A
S
AS
A
53
VS
A
S
U
A
A
A
A
S
A
U
S
A
54
S
S
S
A
S
S
A
VS
AS
S
S
VS
A
6
A
61
A
S
AS
VS
VS
S
S
S
S
A
A
S
A
62
A
S
VS
S
S
S
S
S
S
A
A
S
A
63
A
VS
VS
S
S
A
A
A
A
A
A
S
A
64
A
VS
VS
A
S
S
S
VS
A
A
A
S
A
65
A
VS
AS
VS
S
S
A
AS
A
S
S
VS
A
7
A
71
A
S
AS
VS
S
A
S
U
A
A
A
A
A
72
A
VS
AS
AS
S
A
VS
AS
A
S
A
S
A
73
U
A
AS
S
VS
U
S
S
A
A
A
VS
A
74
U
U
AS
S
S
A
A
S
S
S
S
VSTable 3
The evaluation values ofC12–C23.
C
13
C
14
C
15
C
16
C
17
C
18
C
19
C
20
C
21
C
22
C
23
A
1
A
11
A
S
A
S
A
S
VS
A
A
S
S
A
12
S
AS
A
S
U
S
VS
AS
S
U
VS
A
13
VS
VS
S
VS
VS
S
VS
S
VS
S
S
A
14
U
A
S
A
A
A
S
A
U
A
S
A
2
A
21
VS
S
U
S
VS
VS
A
S
A
A
S
A
22
VS
VS
A
S
S
S
S
AS
U
A
VS
A
23
S
A
U
A
S
VS
S
S
VU
A
S
A
24
VS
S
U
S
S
S
S
VS
A
A
VS
A
3
A
31
VS
VS
A
S
A
S
A
S
U
A
VS
A
32
VS
S
S
S
A
S
S
S
S
A
S
A
33
S
S
A
S
A
S
S
VS
A
U
S
A
34
VS
VS
A
S
A
S
A
S
VS
A
S
A
4
A
41
AS
VS
A
A
A
S
U
VS
VS
U
VS
A
42
S
VS
S
S
A
S
A
VS
S
S
S
A
43
VS
S
S
S
A
S
AS
S
AS
A
S
A
44
S
VS
U
A
A
S
U
S
A
U
S
A
45
AS
AS
U
A
A
VS
U
VS
S
VU
S
A
46
VS
VS
S
A
A
VS
A
S
A
VU
VS
A
5
A
51
AS
VS
U
S
S
S
VU
VS
VU
S
S
A
52
VS
VS
U
S
S
S
VU
S
AU
S
S
A
53
A
S
U
A
S
A
U
S
VU
S
S
A
54
VS
VS
A
A
S
S
VU
VS
VU
S
S
A
6
A
61
AS
VS
A
S
S
VS
A
AS
U
A
S
A
62
VS
S
U
A
S
S
VU
A
S
A
S
A
63
VS
VS
A
S
S
S
VU
S
AS
A
VS
A
64
AS
VS
U
A
S
S
VU
VS
VU
A
VS
A
65
AS
AS
U
A
S
S
S
AS
S
S
VS
A
7
A
71
AS
S
A
A
VS
A
S
S
VU
VS
S
A
72
A
VS
U
S
VS
A
VU
VS
AU
VS
S
A
73
A
S
A
A
VS
A
VU
VS
AU
VS
S
A
74
VS
S
A
A
VS
S
U
S
U
A
STable 4
Semantic scale and corresponding trapezoidal fuzzy numbers.
Linguistic variables
Trapezoidal fuzzy numbers
AS (absolutely serious)
[
0.8,0.9,1
,
1
]
VS (very serious)
[
0.7,0.8,0.8,0.9
]
S (serious)
[
0.5,0.6,0.7,0.8
]
A (average)
[
0.4,0.5,0.5,0.6
]
U (unserious)
[
0.2,0.3,0.4,0.5
]
VU (very unserious)
[
0.1,0.2,0.2,0.3
]
AU (absolutely unserious)
[
0.0,0.0,0.1,0.2
]Step 1.
Compute interval entropy weight of all alternatives according to the corresponding theα-cut sets. Here, to the trapezoid fuzzy number [a,b,c,d], the interval entropy IE7(A) can be expressed as(109)IE7A=b-a-c+d2d-a,2b-a-c+d3d-a.Then the correspondingα-cut sets and interval entropy weights are listed in Table 5.Table 5
Theα-cut sets of the linguistic variables and the corresponding interval entropy weights.
Linguistic variables
α-cut sets
Weights
AS
[0.9α+0.8(1-α), 1]
[
0.6667,0.75
]
VS
[0.8α+0.7(1-α),  0.8α+0.9(1-α)]
[
0.3333,0.5
]
S
[0.6α+0.5(1-α),  0.7α+0.8(1-α)]
[
0.5556,0.6667
]
A
[0.5α+0.4(1-α),  0.5α+0.6(1-α)]
[
0.3333,0.5
]
U
[0.3α+0.2(1-α),  0.4α+0.5(1-α)]
[
0.5556,0.6667
]
VU
[0.2α+0.1(1-α),  0.2α+0.3(1-α)]
[
0.3333,0.5
]
AU
[0,0.1α+0.2(1-α)]
[
0.6667,0.75
]Step 2.
Compute the evaluation values.
The alternative valuesFiα can be drawn by (99) and the results are listed in Table 6.Table 6
The evaluation values of all alternatives.
F
11
α
[
0.98889
α
+
4.4777,29.28358
-
1.3167
α
]
F
12
α
[
1.11113
α
+
6.48898,11.51686
-
1.03336
α
]
F
13
α
[
α
+
6.06659,11.40024
-
1.25003
α
]
F
41
α
[
0.94443
α
+
4.12212,8.76685
-
1.28336
α
]
F
21
α
[
1.03335
α
+
4.86672,9.9003
-
1.35004
α
]
F
22
α
[
1.08892
α
+
5.24456,10.38365
-
1.31671
α
]
F
23
α
[
1.03335
α
+
4.40007,9.25029
-
1.35004
α
]
F
24
α
[
1.04446
α
+
5.28896,10.31695
-
1.28337
α
]
F
31
α
[
1.06668
α
+
5.73339,10.70022
-
1.15003
α
]
F
32
α
[
1.05558
α
+
5.17788,10.33367
-
1.36671
α
]
F
33
α
[
0.96666
α
+
4.53327,9.35022
-
1.30003
α
]
F
34
α
[
0.97777
α
+
5.12219,9.96688
-
1.23336
α
]
F
41
α
[
1.0111
α
+
5.78879,10.61677
-
1.03335
α
]
F
42
α
[
1.05558
α
+
5.44455,10.68368
-
1.36671
α
]
F
43
α
[
1.10003
α
+
6.0016,11.20032
-
1.25004
α
]
F
44
α
[
1.05558
α
+
4.74453,9.78365
-
1.36671
α
]
F
45
α
[
1.10001
α
+
6.30009,11.05014
-
0.95002
α
]
F
46
α
[
0.93331
α
+
5.03321
,
9.80016
-
1.20002
α
]
F
51
α
[
1.13337
α
+
5.80017,10.90029
-
1.20004
α
]
F
52
α
[
1.03336
α
+
5.10014,10.10031
-
1.32504
α
]
F
53
α
[
1.03335
α
+
4.1334,8.90028
-
1.35004
α
]
F
54
α
[
1.04446
α
+
5.35565,10.36696
-
1.28337
α
]
F
61
α
[
1.08891
α
+
5.94456,10.98359
-
1.1667
α
]
F
62
α
[
1.07781
α
+
5.02239,10.11703
-
1.38338
α
]
F
63
α
[
0.95554
α
+
4.87772,9.58352
-
1.21669
α
]
F
64
α
[
0.95554
α
+
4.71104,9.38351
-
1.21669
α
]
F
65
α
[
1.15559
α
+
6.74464,11.88359
-
1.0667
α
]
F
71
α
[
1.01111
α
+
5.05558,9.76687
-
1.18336
α
]
F
72
α
[
0.96666
α
+
5.33331,10.05015
-
1.12502
α
]
F
73
α
[
0.92221
α
+
4.41104,9.0335
-
1.24169
α
]
F
74
α
[
1.11115
α
+
5.12237,10.267
-
1.33338
α
]Step 3.
Ranking of all alternatives: the alternatives can be ranked by (105).Remark 20 (Lu’s method).
Lu’s method is as follows:(1)
The linguistic variables are transferred into corresponding trapezoid fuzzy numbers:(110)x~j=1m∑i=1mx~ij.(2)
Defuzzify each aggregated trapezoid fuzzy number into a crisp value:(111)xj=∫x∈Uxμx~jxdx∫x∈Uμx~jxdx.(3)
Normalize the degree of impact of risk factors for each risk aspect level:(112)Rj=xj∑j=1nxj.The results of local ranking and global ranking under different attitudes are listed in Tables7 and 8, respectively. And there are some different results with Lu’s method. The values of all linguistic variables are determined by interval entropy weight. Here, the values are the most in AS and AU, the following are S and U, and then the least are VS, A, and VU. Among all these linguistic variables, U, VU, and AU are inclined to the optimistic attitude, and S, VS, and AS are inclined to the opposite. And the numbers of all linguistic variables are listed in Table 9. For example, to the second category, the numbers of U and A in F21 and F23 are 2 and 3, respectively. So, for the optimistic attitude, the decision maker would prefer F23 rather than F21. The explanations to other cases are similar.Table 7
The local ranking order.
Opt.
Local ranking order
Pess.
Local ranking order
Neu.
Local ranking order
Lu local ranking order
1
F
11
0.001224
3
1.2839
3
0.1095
3
3
F
12
0.06217
1
1.41998
2
0.3765
1
2
F
13
0.04008
2
1.4397
1
0.3149
2
1
F
41
0
4
1.25
4
0.0608
4
4
2
F
21
0.0055
4
1.3316
3
0.1625
3
3
F
22
0.01291
2
1.3638
1
0.2148
2
2
F
23
0.0075
3
1.2790
4
0.1024
4
4
F
24
0.01389
1
1.3574
2
0.2176
1
1
3
F
31
0.0273
1
1.3759
1
0.2763
1
1
F
32
0.0113
2
1.3644
2
0.2040
2
3
F
33
0.001636
4
1.2914
4
0.1152
4
4
F
34
0.0100
3
1.3312
3
0.1918
3
2
4
F
41
0.0290
3
1.3673
4
0.2799
3
1
F
42
0.0180
4
1.3909
2
0.2384
4
4
F
43
0.03792
2
1.4155
1
0.3130
2
2
F
44
0.0038
6
1.322
5
0.1482
6
6
F
45
0.051896
1
1.38098
3
0.3515
1
3
F
46
0.00821
5
1.3208
6
0.1775
5
5
5
F
51
0.0300
1
1.3902
1
0.2892
1
1
F
52
0.0096
3
1.3429
3
0.1926
3
3
F
53
0.000001
4
1.2526
4
0.0680
4
4
F
54
0.01558
2
1.3598
2
0.2262
2
2
6
F
61
0.03546
2
1.3948
2
0.3050
2
2
F
62
0.0082
3
1.344988
3
0.18542
3
5
F
63
0.005616
4
1.3012
4
0.1589
4
3
F
64
0.0034
5
1.2867
5
0.1374
5
4
F
65
0.07828
1
1.4417
1
0.4123
1
1
7
F
71
0.008728
3
1.3081
3
0.18540
3
2
F
72
0.01482
1
1.3308
2
0.2183
2
1
F
73
0.0008
4
1.2661
4
0.09663
4
4
F
74
0.0102
2
1.3546
1
0.2005
1
3Table 8
The global ranking order of all alternatives.
Alternative
Opt.
Pess.
Neu.
Lu ranking order
F
11
27
27
27
27
F
12
2
3
2
3
F
13
4
2
4
1
F
41
31
31
31
29
F
21
23
18
22
19
F
22
14
12
14
17
F
23
29
28
28
30
F
24
13
14
13
13
F
31
9
9
9
8
F
32
15
11
15
16
F
33
26
25
26
22
F
34
17
19
18
11
F
41
8
10
8
4
F
42
10
6
10
9
F
43
5
4
5
5
F
44
24
21
24
26
F
45
3
8
3
6
F
46
20
22
21
10
F
51
7
7
7
12
F
52
30
17
17
24
F
53
18
30
30
31
F
54
11
13
11
14
F
61
6
5
6
7
F
62
21
16
19
25
F
63
22
24
23
18
F
64
25
26
25
20
F
65
1
1
1
2
F
71
19
23
20
21
F
72
12
20
12
15
F
73
28
29
29
28
F
74
16
15
16
23Table 9
The statistical numbers of linguistic variables.
Alternative
AS
AU
S
U
VS
A
VU
1
F
11
8
2
3
10
F
12
5
6
2
6
4
F
13
1
9
13
F
41
5
3
4
11
2
F
21
10
2
4
7
F
22
1
10
3
4
5
F
23
9
3
2
8
1
F
24
1
10
1
4
7
3
F
31
3
7
2
5
6
F
32
12
1
4
6
F
33
7
2
5
9
F
34
1
8
5
9
4
F
41
4
2
3
9
5
F
42
13
5
5
F
43
2
12
4
5
F
44
10
3
3
7
1
F
45
6
4
2
4
6
1
F
46
1
6
8
7
1
5
F
51
9
3
5
1
2
F
52
1
1
11
1
3
5
1
F
53
8
4
1
9
1
F
54
1
11
5
4
2
6
F
61
3
9
1
4
6
F
62
13
1
2
6
1
F
63
1
7
5
9
1
F
64
1
6
1
6
7
2
F
65
5
9
1
4
4
7
F
71
2
7
1
3
9
1
F
72
3
1
5
1
6
6
1
F
73
1
1
5
2
5
8
1
F
74
1
10
4
3
5


## 5.1. Internal Entropy Weight

It is well known that entropy is a method in obtaining the weights for a MADM problem especially when obtaining suitable weights based on the preferences and decision making experiments are not possible [22]. The original procedure of entropy weight can be expressed in a series of steps [39]:(1)
Compute entropyEj.(2)
Setωj=1-Ej/E, E=∑s=1n(1-Es), j=1,2,…,n, as the weight of attribute j.Now suppose that the weight of the elements of decision matrix is difficult and uncertain, as a result, their values are considered as intervals. Entropy can be extended to interval entropy. The steps are as follows:(I1)
Compute interval entropyIEij=[IEijL,IEijR], i=1,2,…,m, j=1,2,…,n.(I2)
Setωij=[ωijL,ωijR]=[1-IEijR,1-IEijL], i=1,2,…,m, j=1,2,…,n, as the weight of attribute j.For instance, if A=[0.5,0.6,0.7,0.8], by (7), then interval entropy and interval entropy weight can be shown as [0.5,0.6667] and [0.3333,0.5], respectively.


## 5.2. The Evaluation Values Obtained by the Weight Average

To the givenα value, the α-cut sets of the weight ωj and the alternatives x~ij can be denoted by (ωj)α=[(ωj)αL,(ωj)αR] and (x~ij)α=[(x~ij)αL,(x~ij)αR], respectively. So the evaluation values of the given α value can be denoted by the following:(99)Fiα=FiαL,FiαR=∑j=1nωjαLxijαL,∑j=1nωjαRxijαR.Remark 19 (see [40]).
Here, two reasons are responded to the unnormalization. Firstly, the evaluation values after normalization are not always in[0,1]. Secondly, the normalization can only effect the proportion changes of the evaluation values, without any changes to the actual ranking results.
For instance, ifA=[0.5,0.6,0.7,0.8], then interval entropy weight can be shown as [0.3333,0.5] and the α-cut sets can be denoted by(100)Aα=0.6λ+0.51-λ,0.7λ+0.81-λλ∈0,10.6,0.7λ=1.So(101)Fα=0.05556α+0.2778,0.53336-0.06667α.


## 5.3. The Possibility-Based Method for Ranking Fuzzy Numbers [41]

In this section, possibility-based comparison relation is presented. Since theα-cut sets of fuzzy numbers can be expressed as the interval value sets, possibility-based comparison relation on interval values is proposed in Section 5.3.1, and then the comparison relation on fuzzy numbers is presented in Section 5.3.2.


### 5.3.1. Interval Values

AssumingX=[x-,x+], Y=[y-,y+] are the interval values, Xλ, Yλ (λ∈R) can be constructed as follows:(102)Xλ=1x-≥λ12x-<λ≤x+0x+<λ,Yα=1y-≥λ12y-<λ≤y+0y+<λ.Set(103)Gλ=1Xλ>Yλ0Xλ≤Yλ.P
(
X
≻
Y
) is used to represent the degree of interval values “X≻Y” which is named as possibility-based comparison relation on interval values:(104)PX≻Y=1x+∨y+-x-∧y-∫x-∧y-x+∨y+Gλdλ.


### 5.3.2. Fuzzy Numbers

IfA and B are two fuzzy numbers, then the possibility-based comparison relation between fuzzy numbers can be denoted by P(A≻B) as follows:(105)PA≻B=∫01PAα≻Bαdα.LetS={S1,S2,…,Sn} be a set which contains finite fuzzy numbers to be ranked; we can select function T as the evaluation of comparisons [42]. Then the fuzzy numbers can be ranked as the values of P(Si≻T). Here, T can be seen as an attitude which comes from the decision maker. Usually, the attitude of the decision maker can be classified into three groups: neutral, optimistic, and pessimistic. The optimistic attitude reflects the preferences of decision makers to the maximal payoff, and it is often used as follows: (106)μAoptx=x-xminxmax-xminxmin≤x≤xmax0else.The neutral attitude reflects the preferences of decision makers for not only the payoff which could be obtained but also the risk which could be faced, and it is often used as follows:(107)μAneux=1xmin≤x≤xmax0else.And the last attitude is the pessimistic one reflecting the preferences of decision makers for not only the payoff which could be gained but also the risk which could be faced. It is often used as follows:(108)μApessx=xmax-xxmax-xminxmin≤x≤xmax0else.For example, the ranking results with respect to three attitudes are shown in Table1.Table 1
The ranking results.
A
=
[
0,0.1,0.5,1
]
B
=
[
0.3,0.5,0.8,0.9
]
Ranking order
Optimistic
0
0.065
A
≺
B
Neutral
0.5
0.4
A
≺
B
Pessimistic
0.4547
0.7723
A
≺
B


## 5.3.1. Interval Values

AssumingX=[x-,x+], Y=[y-,y+] are the interval values, Xλ, Yλ (λ∈R) can be constructed as follows:(102)Xλ=1x-≥λ12x-<λ≤x+0x+<λ,Yα=1y-≥λ12y-<λ≤y+0y+<λ.Set(103)Gλ=1Xλ>Yλ0Xλ≤Yλ.P
(
X
≻
Y
) is used to represent the degree of interval values “X≻Y” which is named as possibility-based comparison relation on interval values:(104)PX≻Y=1x+∨y+-x-∧y-∫x-∧y-x+∨y+Gλdλ.


## 5.3.2. Fuzzy Numbers

IfA and B are two fuzzy numbers, then the possibility-based comparison relation between fuzzy numbers can be denoted by P(A≻B) as follows:(105)PA≻B=∫01PAα≻Bαdα.LetS={S1,S2,…,Sn} be a set which contains finite fuzzy numbers to be ranked; we can select function T as the evaluation of comparisons [42]. Then the fuzzy numbers can be ranked as the values of P(Si≻T). Here, T can be seen as an attitude which comes from the decision maker. Usually, the attitude of the decision maker can be classified into three groups: neutral, optimistic, and pessimistic. The optimistic attitude reflects the preferences of decision makers to the maximal payoff, and it is often used as follows: (106)μAoptx=x-xminxmax-xminxmin≤x≤xmax0else.The neutral attitude reflects the preferences of decision makers for not only the payoff which could be obtained but also the risk which could be faced, and it is often used as follows:(107)μAneux=1xmin≤x≤xmax0else.And the last attitude is the pessimistic one reflecting the preferences of decision makers for not only the payoff which could be gained but also the risk which could be faced. It is often used as follows:(108)μApessx=xmax-xxmax-xminxmin≤x≤xmax0else.For example, the ranking results with respect to three attitudes are shown in Table1.Table 1
The ranking results.
A
=
[
0,0.1,0.5,1
]
B
=
[
0.3,0.5,0.8,0.9
]
Ranking order
Optimistic
0
0.065
A
≺
B
Neutral
0.5
0.4
A
≺
B
Pessimistic
0.4547
0.7723
A
≺
B


## 5.4. A Numerical Example

In this section, the steps of the proposed method are described with an example. Suppose that there is a FMADM problem with thirty-one alternatives and twenty-three criteria. Data are presented in Tables2-3 [43] which can be represented as seven linguistic variables to evaluate degree of impact for railway reconstruction projects and their corresponding trapezoid fuzzy numbers are listed in Table 4. The decision making can be conducted as follows.Table 2
The evaluation values ofC1–C12.
C
1
C
2
C
3
C
4
C
5
C
6
C
7
C
8
C
9
C
10
C
11
C
12
A
1
A
11
S
S
VS
VS
A
A
A
U
S
A
U
A
A
12
A
AS
AS
AS
VS
A
VS
A
VS
S
S
VS
A
13
VS
VS
VS
VS
S
VS
S
S
AS
S
VS
VS
A
14
VS
VS
VS
A
A
S
A
A
S
U
A
VS
A
2
A
21
A
S
S
S
S
A
A
VS
A
U
S
S
A
22
A
A
S
VS
S
U
A
S
S
U
S
S
A
23
A
A
S
S
A
U
A
VS
A
U
S
S
A
24
S
A
S
AS
A
A
S
S
A
A
VS
S
A
3
A
31
VS
U
AS
AS
VS
S
A
AS
A
S
S
S
A
32
A
U
VS
S
S
S
A
VS
A
A
S
VS
A
33
A
A
S
VS
VS
A
U
VS
A
A
A
VS
A
34
S
A
VS
A
VS
A
A
AS
S
A
S
S
A
4
A
41
U
VS
VS
AS
VS
S
A
AS
A
VS
VS
AS
A
42
A
VS
A
S
S
S
S
VS
VS
S
A
S
A
43
A
VS
S
VS
S
A
A
S
VS
S
S
S
A
44
A
S
VS
S
VS
S
A
S
A
A
S
S
A
45
A
VS
AS
AS
VS
S
S
AS
A
A
A
AS
A
46
A
S
VS
S
VS
S
VS
AS
A
A
S
VS
A
5
A
51
S
U
A
VS
S
VS
S
AS
VS
S
U
AS
A
52
VS
A
S
S
A
A
A
S
S
A
S
AS
A
53
VS
A
S
U
A
A
A
A
S
A
U
S
A
54
S
S
S
A
S
S
A
VS
AS
S
S
VS
A
6
A
61
A
S
AS
VS
VS
S
S
S
S
A
A
S
A
62
A
S
VS
S
S
S
S
S
S
A
A
S
A
63
A
VS
VS
S
S
A
A
A
A
A
A
S
A
64
A
VS
VS
A
S
S
S
VS
A
A
A
S
A
65
A
VS
AS
VS
S
S
A
AS
A
S
S
VS
A
7
A
71
A
S
AS
VS
S
A
S
U
A
A
A
A
A
72
A
VS
AS
AS
S
A
VS
AS
A
S
A
S
A
73
U
A
AS
S
VS
U
S
S
A
A
A
VS
A
74
U
U
AS
S
S
A
A
S
S
S
S
VSTable 3
The evaluation values ofC12–C23.
C
13
C
14
C
15
C
16
C
17
C
18
C
19
C
20
C
21
C
22
C
23
A
1
A
11
A
S
A
S
A
S
VS
A
A
S
S
A
12
S
AS
A
S
U
S
VS
AS
S
U
VS
A
13
VS
VS
S
VS
VS
S
VS
S
VS
S
S
A
14
U
A
S
A
A
A
S
A
U
A
S
A
2
A
21
VS
S
U
S
VS
VS
A
S
A
A
S
A
22
VS
VS
A
S
S
S
S
AS
U
A
VS
A
23
S
A
U
A
S
VS
S
S
VU
A
S
A
24
VS
S
U
S
S
S
S
VS
A
A
VS
A
3
A
31
VS
VS
A
S
A
S
A
S
U
A
VS
A
32
VS
S
S
S
A
S
S
S
S
A
S
A
33
S
S
A
S
A
S
S
VS
A
U
S
A
34
VS
VS
A
S
A
S
A
S
VS
A
S
A
4
A
41
AS
VS
A
A
A
S
U
VS
VS
U
VS
A
42
S
VS
S
S
A
S
A
VS
S
S
S
A
43
VS
S
S
S
A
S
AS
S
AS
A
S
A
44
S
VS
U
A
A
S
U
S
A
U
S
A
45
AS
AS
U
A
A
VS
U
VS
S
VU
S
A
46
VS
VS
S
A
A
VS
A
S
A
VU
VS
A
5
A
51
AS
VS
U
S
S
S
VU
VS
VU
S
S
A
52
VS
VS
U
S
S
S
VU
S
AU
S
S
A
53
A
S
U
A
S
A
U
S
VU
S
S
A
54
VS
VS
A
A
S
S
VU
VS
VU
S
S
A
6
A
61
AS
VS
A
S
S
VS
A
AS
U
A
S
A
62
VS
S
U
A
S
S
VU
A
S
A
S
A
63
VS
VS
A
S
S
S
VU
S
AS
A
VS
A
64
AS
VS
U
A
S
S
VU
VS
VU
A
VS
A
65
AS
AS
U
A
S
S
S
AS
S
S
VS
A
7
A
71
AS
S
A
A
VS
A
S
S
VU
VS
S
A
72
A
VS
U
S
VS
A
VU
VS
AU
VS
S
A
73
A
S
A
A
VS
A
VU
VS
AU
VS
S
A
74
VS
S
A
A
VS
S
U
S
U
A
STable 4
Semantic scale and corresponding trapezoidal fuzzy numbers.
Linguistic variables
Trapezoidal fuzzy numbers
AS (absolutely serious)
[
0.8,0.9,1
,
1
]
VS (very serious)
[
0.7,0.8,0.8,0.9
]
S (serious)
[
0.5,0.6,0.7,0.8
]
A (average)
[
0.4,0.5,0.5,0.6
]
U (unserious)
[
0.2,0.3,0.4,0.5
]
VU (very unserious)
[
0.1,0.2,0.2,0.3
]
AU (absolutely unserious)
[
0.0,0.0,0.1,0.2
]Step 1.
Compute interval entropy weight of all alternatives according to the corresponding theα-cut sets. Here, to the trapezoid fuzzy number [a,b,c,d], the interval entropy IE7(A) can be expressed as(109)IE7A=b-a-c+d2d-a,2b-a-c+d3d-a.Then the correspondingα-cut sets and interval entropy weights are listed in Table 5.Table 5
Theα-cut sets of the linguistic variables and the corresponding interval entropy weights.
Linguistic variables
α-cut sets
Weights
AS
[0.9α+0.8(1-α), 1]
[
0.6667,0.75
]
VS
[0.8α+0.7(1-α),  0.8α+0.9(1-α)]
[
0.3333,0.5
]
S
[0.6α+0.5(1-α),  0.7α+0.8(1-α)]
[
0.5556,0.6667
]
A
[0.5α+0.4(1-α),  0.5α+0.6(1-α)]
[
0.3333,0.5
]
U
[0.3α+0.2(1-α),  0.4α+0.5(1-α)]
[
0.5556,0.6667
]
VU
[0.2α+0.1(1-α),  0.2α+0.3(1-α)]
[
0.3333,0.5
]
AU
[0,0.1α+0.2(1-α)]
[
0.6667,0.75
]Step 2.
Compute the evaluation values.
The alternative valuesFiα can be drawn by (99) and the results are listed in Table 6.Table 6
The evaluation values of all alternatives.
F
11
α
[
0.98889
α
+
4.4777,29.28358
-
1.3167
α
]
F
12
α
[
1.11113
α
+
6.48898,11.51686
-
1.03336
α
]
F
13
α
[
α
+
6.06659,11.40024
-
1.25003
α
]
F
41
α
[
0.94443
α
+
4.12212,8.76685
-
1.28336
α
]
F
21
α
[
1.03335
α
+
4.86672,9.9003
-
1.35004
α
]
F
22
α
[
1.08892
α
+
5.24456,10.38365
-
1.31671
α
]
F
23
α
[
1.03335
α
+
4.40007,9.25029
-
1.35004
α
]
F
24
α
[
1.04446
α
+
5.28896,10.31695
-
1.28337
α
]
F
31
α
[
1.06668
α
+
5.73339,10.70022
-
1.15003
α
]
F
32
α
[
1.05558
α
+
5.17788,10.33367
-
1.36671
α
]
F
33
α
[
0.96666
α
+
4.53327,9.35022
-
1.30003
α
]
F
34
α
[
0.97777
α
+
5.12219,9.96688
-
1.23336
α
]
F
41
α
[
1.0111
α
+
5.78879,10.61677
-
1.03335
α
]
F
42
α
[
1.05558
α
+
5.44455,10.68368
-
1.36671
α
]
F
43
α
[
1.10003
α
+
6.0016,11.20032
-
1.25004
α
]
F
44
α
[
1.05558
α
+
4.74453,9.78365
-
1.36671
α
]
F
45
α
[
1.10001
α
+
6.30009,11.05014
-
0.95002
α
]
F
46
α
[
0.93331
α
+
5.03321
,
9.80016
-
1.20002
α
]
F
51
α
[
1.13337
α
+
5.80017,10.90029
-
1.20004
α
]
F
52
α
[
1.03336
α
+
5.10014,10.10031
-
1.32504
α
]
F
53
α
[
1.03335
α
+
4.1334,8.90028
-
1.35004
α
]
F
54
α
[
1.04446
α
+
5.35565,10.36696
-
1.28337
α
]
F
61
α
[
1.08891
α
+
5.94456,10.98359
-
1.1667
α
]
F
62
α
[
1.07781
α
+
5.02239,10.11703
-
1.38338
α
]
F
63
α
[
0.95554
α
+
4.87772,9.58352
-
1.21669
α
]
F
64
α
[
0.95554
α
+
4.71104,9.38351
-
1.21669
α
]
F
65
α
[
1.15559
α
+
6.74464,11.88359
-
1.0667
α
]
F
71
α
[
1.01111
α
+
5.05558,9.76687
-
1.18336
α
]
F
72
α
[
0.96666
α
+
5.33331,10.05015
-
1.12502
α
]
F
73
α
[
0.92221
α
+
4.41104,9.0335
-
1.24169
α
]
F
74
α
[
1.11115
α
+
5.12237,10.267
-
1.33338
α
]Step 3.
Ranking of all alternatives: the alternatives can be ranked by (105).Remark 20 (Lu’s method).
Lu’s method is as follows:(1)
The linguistic variables are transferred into corresponding trapezoid fuzzy numbers:(110)x~j=1m∑i=1mx~ij.(2)
Defuzzify each aggregated trapezoid fuzzy number into a crisp value:(111)xj=∫x∈Uxμx~jxdx∫x∈Uμx~jxdx.(3)
Normalize the degree of impact of risk factors for each risk aspect level:(112)Rj=xj∑j=1nxj.The results of local ranking and global ranking under different attitudes are listed in Tables7 and 8, respectively. And there are some different results with Lu’s method. The values of all linguistic variables are determined by interval entropy weight. Here, the values are the most in AS and AU, the following are S and U, and then the least are VS, A, and VU. Among all these linguistic variables, U, VU, and AU are inclined to the optimistic attitude, and S, VS, and AS are inclined to the opposite. And the numbers of all linguistic variables are listed in Table 9. For example, to the second category, the numbers of U and A in F21 and F23 are 2 and 3, respectively. So, for the optimistic attitude, the decision maker would prefer F23 rather than F21. The explanations to other cases are similar.Table 7
The local ranking order.
Opt.
Local ranking order
Pess.
Local ranking order
Neu.
Local ranking order
Lu local ranking order
1
F
11
0.001224
3
1.2839
3
0.1095
3
3
F
12
0.06217
1
1.41998
2
0.3765
1
2
F
13
0.04008
2
1.4397
1
0.3149
2
1
F
41
0
4
1.25
4
0.0608
4
4
2
F
21
0.0055
4
1.3316
3
0.1625
3
3
F
22
0.01291
2
1.3638
1
0.2148
2
2
F
23
0.0075
3
1.2790
4
0.1024
4
4
F
24
0.01389
1
1.3574
2
0.2176
1
1
3
F
31
0.0273
1
1.3759
1
0.2763
1
1
F
32
0.0113
2
1.3644
2
0.2040
2
3
F
33
0.001636
4
1.2914
4
0.1152
4
4
F
34
0.0100
3
1.3312
3
0.1918
3
2
4
F
41
0.0290
3
1.3673
4
0.2799
3
1
F
42
0.0180
4
1.3909
2
0.2384
4
4
F
43
0.03792
2
1.4155
1
0.3130
2
2
F
44
0.0038
6
1.322
5
0.1482
6
6
F
45
0.051896
1
1.38098
3
0.3515
1
3
F
46
0.00821
5
1.3208
6
0.1775
5
5
5
F
51
0.0300
1
1.3902
1
0.2892
1
1
F
52
0.0096
3
1.3429
3
0.1926
3
3
F
53
0.000001
4
1.2526
4
0.0680
4
4
F
54
0.01558
2
1.3598
2
0.2262
2
2
6
F
61
0.03546
2
1.3948
2
0.3050
2
2
F
62
0.0082
3
1.344988
3
0.18542
3
5
F
63
0.005616
4
1.3012
4
0.1589
4
3
F
64
0.0034
5
1.2867
5
0.1374
5
4
F
65
0.07828
1
1.4417
1
0.4123
1
1
7
F
71
0.008728
3
1.3081
3
0.18540
3
2
F
72
0.01482
1
1.3308
2
0.2183
2
1
F
73
0.0008
4
1.2661
4
0.09663
4
4
F
74
0.0102
2
1.3546
1
0.2005
1
3Table 8
The global ranking order of all alternatives.
Alternative
Opt.
Pess.
Neu.
Lu ranking order
F
11
27
27
27
27
F
12
2
3
2
3
F
13
4
2
4
1
F
41
31
31
31
29
F
21
23
18
22
19
F
22
14
12
14
17
F
23
29
28
28
30
F
24
13
14
13
13
F
31
9
9
9
8
F
32
15
11
15
16
F
33
26
25
26
22
F
34
17
19
18
11
F
41
8
10
8
4
F
42
10
6
10
9
F
43
5
4
5
5
F
44
24
21
24
26
F
45
3
8
3
6
F
46
20
22
21
10
F
51
7
7
7
12
F
52
30
17
17
24
F
53
18
30
30
31
F
54
11
13
11
14
F
61
6
5
6
7
F
62
21
16
19
25
F
63
22
24
23
18
F
64
25
26
25
20
F
65
1
1
1
2
F
71
19
23
20
21
F
72
12
20
12
15
F
73
28
29
29
28
F
74
16
15
16
23Table 9
The statistical numbers of linguistic variables.
Alternative
AS
AU
S
U
VS
A
VU
1
F
11
8
2
3
10
F
12
5
6
2
6
4
F
13
1
9
13
F
41
5
3
4
11
2
F
21
10
2
4
7
F
22
1
10
3
4
5
F
23
9
3
2
8
1
F
24
1
10
1
4
7
3
F
31
3
7
2
5
6
F
32
12
1
4
6
F
33
7
2
5
9
F
34
1
8
5
9
4
F
41
4
2
3
9
5
F
42
13
5
5
F
43
2
12
4
5
F
44
10
3
3
7
1
F
45
6
4
2
4
6
1
F
46
1
6
8
7
1
5
F
51
9
3
5
1
2
F
52
1
1
11
1
3
5
1
F
53
8
4
1
9
1
F
54
1
11
5
4
2
6
F
61
3
9
1
4
6
F
62
13
1
2
6
1
F
63
1
7
5
9
1
F
64
1
6
1
6
7
2
F
65
5
9
1
4
4
7
F
71
2
7
1
3
9
1
F
72
3
1
5
1
6
6
1
F
73
1
1
5
2
5
8
1
F
74
1
10
4
3
5


## 6. Conclusions

It is well known that entropy is a famous method in obtaining the weights for a MADM problem especially when obtaining suitable weights based on the preferences and decision making experiments are not possible. And most of the literatures pertaining to MADM analysis have been published using entropy weights. Among all these literatures, the result of entropy weight can be expressed as the certain number. But, in practice, because of uncertainty of people’s cognition, when the weight of the alternatives cannot be measured precisely and be expressed as some uncertain types of data, how to use entropy to represent it is worthy of discussion. In this paper, the concept of interval entropy is proposed and the relationships among other conceptions such as interval entropy, interval similarity measure, interval distance measure, and interval inclusion measure are investigated in detail. And then, a FMADM model based on interval entropy is set up. In this model, interval entropy is used as the weight, by which the evaluation values can be obtained. And all alternatives with respect to each criterion can be ranked using the method based on the possibility degree. The achievements in the presented work will provide a new model for FMADM based on interval entropy weight. Because of the relationship between the interval entropy, interval similarity measure, interval distance measure, and inclusion measure, in the future, other types of uncertain weights will be researched in detail.



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*Source: 451987-2015-10-04.xml*