# Multiattribute Decision Making Based on Entropy under Interval-Valued Intuitionistic Fuzzy Environment

Authors: Yingjun Zhang; Peihua Li; Yizhi Wang; Peijun Ma; Xiaohong Su
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

Multiattribute decision making (MADM) is one of the central problems in artificial intelligence, specifically in
management fields. In most cases, this problem arises from uncertainty both in the data derived from the decision
maker and the actions performed in the environment. Fuzzy set and high-order fuzzy sets were proven to be effective
approaches in solving decision-making problems with uncertainty. Therefore, in this paper, we investigate the
MADM problem with completely unknown attribute weights in the framework of interval-valued intuitionistic fuzzy
(IVIF) set (IVIFS). We first propose a new definition of IVIF entropy and some calculation methods for IVIF entropy.
Furthermore, we propose an entropy-based decision-making method to solve IVIF MADM problems with completely
unknown attribute weights. Particular emphasis is put on assessing the attribute weights based on IVIF entropy.
Instead of the traditional methods, which use divergence among attributes or the probabilistic discrimination of attributes
to obtain attribute weights, we utilize the IVIF entropy to assess the attribute weights based on the credibility
of the decision-making matrix for solving the problem. Finally, a supplier selection example is given to demonstrate
the feasibility and validity of the proposed MADM method.


---

## Body

## 1. Introduction

Atanassov and Gargov extended the intuitionistic fuzzy (IF) set (IFS) to the interval-valued intuitionistic fuzzy (IVIF) set (IVIFS) [1–3], characterized by membership and nonmembership functions whose values are intervals rather than real numbers. Since IFS and IVIFS were proposed, a great deal of literature abounds on both theoretical research of IFS and IVIFS as well as on application research in various fields, such as image processing [4–6], pattern recognition [7–9], clustering analysis [10–12], supplier selection, and decision-making analysis [11, 13, 14].In particular, IVIFS and IFS are effective in solving the fuzzy decision-making problems. In most fuzzy multi-attribute decision making (MADM) problems, the preference over alternatives provided by decision makers is usually not sufficient for the crisp membership and nonmembership degree values, because things are fuzzy, uncertain, and probably influenced by the subjectivity of the decision makers, or the knowledge and data about the problem domain are insufficient during the decision-making process [13–17]. Therefore, the preferences among alternatives with uncertainty may be denoted by IFS or IVIFS for decision-making problems [7, 11, 13–15]. Chen and Tan [18], Xu [11, 19], Ye [20], Lakshmana Gomathi Nayagam et al. [21], and Wang et al. [22] proposed some methods to rank alternatives expressed with IVIFSs/IFSs and discussed their application in the MADM field. Atanassov [23] introduced some aggregation operators on IFS and IVIFS. After the pioneering work of Atanassov, some aggregation operators were proposed and utilized to tackle the MADM problems with uncertainty [24–33]. Xu and Yager [34], Li et al. [35–38], and Dubey et al. [15] constructed IF and IVIF MADM models based on optimization theories, such as linear, nonlinear, and fractional programming. Meanwhile, decision-making methods based on some measures (such as distance, similarity degree, correlation coefficient, and entropy) were proposed in dealing with fuzzy IF and IVIF MADM problems [7, 19, 35, 39–41]. In [13, 17, 22, 34, 37], emphasis was given that not only the information of alternatives on attributes may be fuzzy but also the attribute weight information may be partially known or unknown in some situations. In fact, proper assessment of attribute weights plays an important and essential role in the decision-making process, because the variation of weight values may result in different final ranking order of alternatives [11, 14, 17].Generally speaking, the attribute weights in MADM can be classified as subjective and objective attribute weights based on the information acquisition approach [14]. The subjective attribute weights are determined by preference information on the attributes given by the decision maker, who provides subjective intuition or judgments on specific attributes. The objective attribute weights are determined by the decision-making matrix. Analytic hierarchy process (AHP) method [42] and Delphi method [16] are two classical methods for generating subjective attribute weights. In terms of determining objective attribute weights, one of the most famous approaches is the Shannon entropy method, which expresses the relative intensities of attribute importance to signify the average intrinsic information transmitted to the decision maker. So far, a lot of literature pertaining to MADM analysis under IF/IVIF environment has been published using subjective weights [11, 15–17]. In the course of determining objective attribute weights under IF environment, Chen and Li [14] utilized IF entropy to assess the objective attribute weights in dealing with the IF MADM problem with completely unknown attribute weights. Despite existing research effort, solving the fuzzy MADM problems with completely unknown attribute weights in the framework of IVIFS remains an open problem [11, 14, 17, 35, 37, 38].In an attempt to address such problems, we propose a novel entropy-based decision-making method under IVIF environment. Our focus is on the assessment of attribute weights. Meanwhile, we propose a definition of entropy on IVIFS, as well as a method to calculate it. Instead of using traditional methods, which use divergence among attributes or the probabilistic discrimination of attributes to obtain attribute weights, IVIF entropy is utilized to assess the objective weights based on the credibility of the input data. Furthermore, we construct a MADM model based on the IVIF weighted averaging (IIFWA) operator and the ranking functions on IVIFS.The rest of this paper is organized as follows: in Section2, we briefly review the concepts of IFS, IVIFS and some of their operations. In Section 3, we propose a definition of IVIF entropy and some calculation methods on entropy. An entropy-based MADM method within the framework of IVIFS is proposed in Section 4. A numerical example is utilized to illustrate the applicability of the proposed method in Section 5. Finally, conclusion is given in Section 6.


## 2. Preliminaries

In this section, we briefly review some basic concepts, for example, IFS, IVIFS, and their relevant operations.Notation.
X
=
{
x
1
,
x
2
,
…
,
x
N
} is the universal set (it is clear that N=Cardinal(X)). FSs(X), IFSs(X), and IVIFSs(X) denote all the fuzzy sets on X, all the IFSs on X, and all the IVIFSs on X, respectively.Definition 1 (see [1]).
LetX be a set. An IFS A in X is defined as
(1)A={(x,μA(x),νA(x))∣x∈X},
where
(2)μA:X⟶[0,1],νA:X⟶[0,1]
are two maps satisfying
(3)0≤μA(x)+νA(x)≤1,∀x∈X.μ
A
(
x
) and νA(x) denote the membership and nonmembership degrees of x to A, respectively. For each IFS A in X, we designate
(4)πA(x)=1-μA(x)-νA(x),
an intuitionistic index of x in A.The following expressions are defined in [1, 11] for all A and B belonging to IFSs(X):(1)
Ac={(x,νA(x),μA(x))∣x∈X},(2)
A≤B if and only if μA(x)≤μB(x) and νA(x)≥νB(x) for all x∈X,(3)
A⪯B if and only if μA(x)≤μB(x) and νA(x)≤νB(x) for all x∈X,(4)
A=B if and only if A≤B and B≤A.Definition 2 (see [2, 23]).
LetX be a set and Int[0,1] the set of all closed subintervals of [0,1]. An IVIFS A in X is defined as
(5)A={(x,μA(x),νA(x))∣x∈X},
where
(6)μA:X⟶Int[0,1],νA:X⟶Int[0,1],
for which
(7)0≤sup(μA(x))+sup(νA(x))≤1,∀x∈X.The intervalsμA(x) and νA(x) denote the membership and nonmembership degrees of x to A, respectively. Let
(8)A={(x,[μAL(x),μAU(x)],[νAL(x),νAU(x)])∣x∈X},
where μAL(x)=inf(μA(x)), νAL(x)=inf(νA(x)), μAU(x)=sup(μA(x)), and νAU(x)=sup(νA(x)). If μAL(x)=μAU(x) and νAL(x)=νAU(x), then IVIFS A reduces to an IFS.For each IVIFSA, we designate
(9)πA(x)=1-μA(x)-νA(x),
an intuitionistic interval of x in A, where
(10)πA(x)=[1-μAU(x)-νAU(x),1-μAL(x)-νAL(x)].The following expressions are defined for allA and B belonging to IVIFSs(X) [11]:(1)
Ac={(x,[νAL(x),νAU(x)],[μAL(x),μAU(x)])∣x∈X},(2)
A≤B if and only if μAL(x)≤μBL(x), μAU(x)≤μBU(x), νAL(x)≥νBL(x), and νAU(x)≥νBU(x) for all x∈X,(3)
A⪯B if and only if μAL(x)≤μBL(x), μAU(x)≤μBU(x), νAL(x)≤νBL(x), and νAU(x)≤νBU(x) for all x∈X,(4)
A=B if and only if A≤B and B≤A.Ranking the alternatives expressed with IVIFSs is necessary to deal with the MADM problem within the framework of IVIFS. We introduce the score function and accuracy function of IVIFS proposed by Xu [11].Definition 3 (see [11]).
Letα=([a,b],[c,d]) be an IVIF number. The score function s of α is defined as follows:
(11)s(α)=12(a+b-c-d),
where s(α)∈[0,1].Definition 4 (see [11]).
Letα=([a,b],[c,d]) be an IVIF number. The accuracy function h of α is defined as follows:
(12)h(α)=12(a+b+c+d),
where h(α)∈[0,1].Based on the score function and the accuracy function, Xu further introduced an algorithm of ranking alternatives expressed with IVIFSs.Definition 5 (see [11]).
Letα=([a,b],[c,d]) and β=([a1,b1],[c1,d1]) be two IVIF numbers; then the ranking order between α and β is defined as follows:  (1)
Ifs(α)>s(β), then α>β.(2)
Ifs(α)<s(β), then α<β.(3)
Ifs(α)=s(β), then
(a)
ifh(α)>h(β), then α>β;(b)
ifh(α)<h(β), then α<β;(c)
ifh(α)=h(β), then α~β.


## 3. Entropy on IVIFSs

The definition of entropy on IFSs/IVIFSs has a great importance in the theoretical research of IFSs/IVIFSs. In 1996, Burillo and Bustince [43] introduced the IF entropy for the first time, and, in 2001, Szmidt and Kacprzyk [44] proposed a non-probabilistic-type entropy measure for IFSs. Later, Hung [45] and Zhang et al. [46] constructed the IF entropy based on the distance measures among IFSs. Vlachos and Sergiadis [9] introduced the De Luca-Termini nonprobabilistic entropy for IFSs. Zeng and Su [32] introduced the IF entropy based on the similarity among IFSs. Ye [47] constructed two IF entropies based on the fuzzy entropy established by Zadeh [48]. Chen and Li [14] presented different entropies on IFSs. So far, a great deal of literature on entropy on IFSs is available, but, unfortunately, to our knowledge, only a few works on entropy on IVIFSs exist. Two published papers are found in [49] which extended De luca-Termini axioms and proposed an IVIF entropy, and in [41] which proposed an IVIF entropy in dealing with multiple-attribute decision-making problem.


### 3.1. Entropy on IFSs

Based on the definition of fuzzy entropy [48], Burillo and Bustince [43] defined the IF entropy as follows.Definition 66 (see [43]).
A real functionI:IFSs(X)→R+ is called an entropy if I has the following properties:(1)
I(A)=0 if and only if A∈FSs(X),(2)
I(A)=Cardinal(X)=N if and only if μA(x)=0 and νA(x)=0 for all x∈X,(3)
I(A)=I(Ac) for all A∈IVIFS(X),(4)
IfA⪯B, then I(A)≥I(B).To construct the entropy on IFSs, Burillo and Bustince introduced functionΦ:D→[0,1], where D={(x,y)∈[0,1]×[0,1]∣x+y≤1}. Meanwhile, function Φ satisfies the following conditions:(a)
Φ(x,y)=1, if and only if x+y=1,(b)
Φ(x,y)=0, if and only if x=0 and y=0,(c)
Φ(x,y)=Φ(y,x),(d)
ifx≤x′ and y≤y′, then Φ(x,y)≤Φ(x′,y′).According to Definition6 and function Φ, Bustince and Burillo proposed a theorem to construct the entropy on IFSs.Theorem 7 (see [43]).
LetX={x1,x2,…,xN}, I:IFSs(X)→R+, and A∈IFSs(X). If I(A)=∑i=1N‍(1-Φ(μA(xi),νA(xi))), where Φ satisfies conditions (a)–(d); then I is an entropy on A.


### 3.2. Entropy on IVIFSs

Based on Definition6 and some properties on IFS [50, 51], we define the entropy on IVIFSs as follows.Definition 8.
A real functionI:IVIFSs(X)→R+ is called an entropy, if I has the following properties:(1)
I(A)=0, if and only if A is a fuzzy set;(2)
I(A)=(1/N)Cardinal(X)=1, if and only if μA(x)=[0,0] and νA(x)=[0,0] for all x∈X;(3)
I(A)=I(Ac) for all A∈IVIFS(X);(4)
IfA⪯B, then I(A)≥I(B).For anyA∈IVIFS(X), we introduce the following operator:
(13)Fp(A)={(x,μAL(x)+pΔμA(x),νAL(x)+pΔνA(x))∣x∈X},
where ΔμA(x)=μAU(x)-μAL(x), ΔνA(x)=νAU(x)-νAL(x), and p∈(0,1). Obviously, Fp(A) is an IFS on X.The following theorem gives an expression that allows us to construct different entropies on IVIFSs.Theorem 9.
LetX={x1,x2,…,xN}, I:IVIFSs(X)→R+, and A∈IVIFSs(X). If
(14)I(A)=1N∑i=1N‍(1-Φ(μ-A(xi),ν-A(xi))),
then I is an entropy of A, where Φ satisfies conditions (a)–(d), μ-A(x)=μAL(x)+pΔμA(x), ν-A(x)=νAL(x)+pΔνA(x), and p∈(0,1).Proof.
(1)⇐ If A is a fuzzy set, then μAL(x)=μAU(x), νAL(x)=νAU(x), and μAU(x)+νAU(x)=1 for all x∈X. Clearly, I(A)=0 according to the condition (a) of Φ.
⇒
I
(
A
)
=
0 implies that Φ(μ-A(xi),ν-A(xi))=1(i=1,2,…,N) for any p∈(0,1). If A is not a fuzzy set, then μAL(xi*)+νAL(xi*)<1 for some xi*∈X  (i*∈{1,2,…,N}). As μAL(xi*)+νAL(xi*)<1 and μAU(xi*)+νAU(xi*)≤1, μ-A(xi*)+ν-A(xi*)<1 for any p∈(0,1). According to conditions (a) and (d) of Φ, we have Φ(μ-A(xi*),ν-A(xi*))<1. Thus, I(A)>0, which contradicts I(A)=0. Thus, A is a fuzzy set when I(A)=0.
(2)μA(x)=[0,0] and νA(x)=[0,0] for all x∈X which implies that μ-A(xi)=0, ν-A(xi)=0(i=1,2,…,N) for any p∈(0,1). Based on condition (b) of Φ, we have I(A)=1.
⇒
I
(
A
)
=
1 implies that Φ(μ-A(xi),ν-A(xi))=0(i=1,2,…,N) for any p∈(0,1). As 0≤μ-A(x)+ν-A(x)≤1, μ-A(x)+ν-A(x)=0 based on the condition (b) of Φ. If xi*∈X exists that satisfies μAU(xi*)+νAU(xi*)>0, it means that μ-A(xi*)+ν-A(xi*)>1 for any p∈(0,1). According to conditions (b) and (d) of Φ, we have Φ(μ-A(xi*),ν-A(xi*))>0. Obviously, I(A)<1, which contradicts I(A)=1. Thus, we have μA(x)=[0,0] and νA(x)=[0,0].
(3)   AsAc={(x,νA(x),μA(x))∣x∈X}, clearly, I(A)=I(Ac).
(4)   IfA⪯B, then μAL(x)≤μBL(x), μAU(x)≤μBU(x), νAL(x)≤νBL(x), and νAU(x)≤νBU(x);   then (1-p)(μBL(x)-μAL(x))+p(μBU(x)-μAU(x))+(1-p)(νBL(x)-νAL(x))+p(νBU(x)-νAU(x))≥0 for any p∈(0,1); then μAL(x)+p(μAU(x)-μAL(x))+νAL(x)+p(νAU(x)-νAL(x))≤μBL(x)+p(μBU(x)-μBL(x))+νBL(x)+p(νBU(x)-νBL(x)), that is, μ-A(x)+ν-A(x)≤μ-B(x)+ν-B(x). According to condition (d) of Φ, we have Φ(μ-A(xi),ν-A(xi))≤Φ(μ-B(xi),ν-B(xi))  (i=1,2,…,N), that is, 1-Φ(μ-A(xi),ν-A(xi))≥1-Φ(μ-B(xi),ν-B(xi)) implying I(A)≥I(B).
The proof is completed.According to (14), we can construct different entropies on IVIFSs based on various Φs. Some example functions that satisfy the previous conditions are presented as follows:
(15)Φn,cost(x,y)=(x+y)n,n=1,2,…,Φn,exp(x,y)=(x+y)n·e1-(x+y),n=1,2,…,Φn,sin(x,y)=(x+y)n·sin(π2(x+y)),n=1,2,….


### 3.3. Experimental Analysis of IVIF Entropies

#### 3.3.1. Examples

Example 10.
Let us assume six IVIFSs, namely,A1={([0,0],[0,0])}, A2={([1,1],[0,0])}, A3={([0.4,0.4],[0.6,0.6])}, A4={([0.1,0.2],[0.3,0.4])}, A5={([0.3,0.4],[0.1,0.2])}, and A6={([0.3,0.4],[0.5,0.6])} on X={x}.
In this example, we adoptIΦ1,cost, IΦ1,exp, and IΦ1,sin to calculate the entropies of Ai(i=1,…,6) by choosing different values of parameter p(p=0.1,0.2,…,0.9). The entropies of Ai(i=1,…,6) are shown in Table 1.Table 1
IVIF entropiesIΦ1,cost,  IΦ1,exp,  and  IΦ1,sin  of  Ai(i=1,…,6).
A
1
A
2
A
3
A
4
A
5
A
6
I
Φ
1
,
cost
p
=
0.1
1
0
0
0.5800
0.5800
0.1800
p
=
0.2
1
0
0
0.5600
0.5600
0.1600
p
=
0.3
1
0
0
0.5400
0.5400
0.1400
p
=
0.4
1
0
0
0.5200
0.5200
0.1200
p
=
0.5
1
0
0
0.5000
0.5000
0.1000
p
=
0.6
1
0
0
0.4800
0.4800
0.0800
p
=
0.7
1
0
0
0.4600
0.4600
0.0600
p
=
0.8
1
0
0
0.4400
0.4400
0.0400
p
=
0.9
1
0
0
0.4200
0.4200
0.0200
I
Φ
1
,
exp
p
=
0.1
1
0
0
0.2499
0.2499
0.0183
p
=
0.2
1
0
0
0.2297
0.2297
0.0143
p
=
0.3
1
0
0
0.2106
0.2106
0.0108
p
=
0.4
1
0
0
0.1926
0.1926
0.0078
p
=
0.5
1
0
0
0.1756
0.1756
0.0053
p
=
0.6
1
0
0
0.1596
0.1596
0.0034
p
=
0.7
1
0
0
0.1446
0.1446
0.0019
p
=
0.8
1
0
0
0.1305
0.1305
0.0008
p
=
0.9
1
0
0
0.1173
0.1173
0.0002
I
Φ
1
,
sin
p
=
0.1
1
0
0
0.7426
0.7426
0.2126
p
=
0.2
1
0
0
0.7195
0.7195
0.1864
p
=
0.3
1
0
0
0.6958
0.6958
0.1607
p
=
0.4
1
0
0
0.6714
0.6714
0.1356
p
=
0.5
1
0
0
0.6464
0.6464
0.1111
p
=
0.6
1
0
0
0.6209
0.6209
0.0873
p
=
0.7
1
0
0
0.5949
0.5949
0.0642
p
=
0.8
1
0
0
0.5685
0.5685
0.0419
p
=
0.9
1
0
0
0.5417
0.5417
0.0205Example 11.
The conditions of this example are the same as Example10. In this example, we adopt IΦn,cost, IΦn,exp, and IΦn,sin(n=1,2,3,4,5) with p=0.5 to calculate the entropies of Ai(i=1,…,6). The entropies of Ai(i=1,…,6) are shown in Table 2.Table 2
IVIF entropiesIΦn,cost,  IΦn,exp,  and  IΦn,sin  of Ai(i=1,…,6).
A
1
A
2
A
3
A
4
A
5
A
6
I
Φ
n
,
cost
n
=
1
1
0
0
0.5000
0.5000
0.1000
n
=
2
1
0
0
0.7500
0.7500
0.1900
n
=
3
1
0
0
0.8750
0.8750
0.2710
n
=
4
1
0
0
0.9375
0.9375
0.3439
n
=
5
1
0
0
0.9688
0.9688
0.4095
I
Φ
n
,
exp
n
=
1
1
0
0
0.1756
0.1756
0.0053
n
=
2
1
0
0
0.5878
0.5878
0.1048
n
=
3
1
0
0
0.7939
0.7939
0.1943
n
=
4
1
0
0
0.8970
0.8970
0.2749
n
=
5
1
0
0
0.9485
0.9485
0.3474
I
Φ
n
,
sin
n
=
1
1
0
0
0.6464
0.6464
0.1111
n
=
2
1
0
0
0.8232
0.8232
0.2000
n
=
3
1
0
0
0.9116
0.9116
0.2800
n
=
4
1
0
0
0.9558
0.9558
0.3520
n
=
5
1
0
0
0.9779
0.9779
0.4168


### 3.4. Analysis of the Experimental Results

The results of Example10 show that IΦn,cost, IΦn,exp, and IΦn,sin are decreasing functions of p(p∈(0,1)) and satisfy the four properties of Definition 8. The results of Example 11 show that IΦn,cost, IΦn,exp, and IΦn,sin can be regarded as increasing functions of n(n=1,2,…).


## 3.1. Entropy on IFSs

Based on the definition of fuzzy entropy [48], Burillo and Bustince [43] defined the IF entropy as follows.Definition 66 (see [43]).
A real functionI:IFSs(X)→R+ is called an entropy if I has the following properties:(1)
I(A)=0 if and only if A∈FSs(X),(2)
I(A)=Cardinal(X)=N if and only if μA(x)=0 and νA(x)=0 for all x∈X,(3)
I(A)=I(Ac) for all A∈IVIFS(X),(4)
IfA⪯B, then I(A)≥I(B).To construct the entropy on IFSs, Burillo and Bustince introduced functionΦ:D→[0,1], where D={(x,y)∈[0,1]×[0,1]∣x+y≤1}. Meanwhile, function Φ satisfies the following conditions:(a)
Φ(x,y)=1, if and only if x+y=1,(b)
Φ(x,y)=0, if and only if x=0 and y=0,(c)
Φ(x,y)=Φ(y,x),(d)
ifx≤x′ and y≤y′, then Φ(x,y)≤Φ(x′,y′).According to Definition6 and function Φ, Bustince and Burillo proposed a theorem to construct the entropy on IFSs.Theorem 7 (see [43]).
LetX={x1,x2,…,xN}, I:IFSs(X)→R+, and A∈IFSs(X). If I(A)=∑i=1N‍(1-Φ(μA(xi),νA(xi))), where Φ satisfies conditions (a)–(d); then I is an entropy on A.


## 3.2. Entropy on IVIFSs

Based on Definition6 and some properties on IFS [50, 51], we define the entropy on IVIFSs as follows.Definition 8.
A real functionI:IVIFSs(X)→R+ is called an entropy, if I has the following properties:(1)
I(A)=0, if and only if A is a fuzzy set;(2)
I(A)=(1/N)Cardinal(X)=1, if and only if μA(x)=[0,0] and νA(x)=[0,0] for all x∈X;(3)
I(A)=I(Ac) for all A∈IVIFS(X);(4)
IfA⪯B, then I(A)≥I(B).For anyA∈IVIFS(X), we introduce the following operator:
(13)Fp(A)={(x,μAL(x)+pΔμA(x),νAL(x)+pΔνA(x))∣x∈X},
where ΔμA(x)=μAU(x)-μAL(x), ΔνA(x)=νAU(x)-νAL(x), and p∈(0,1). Obviously, Fp(A) is an IFS on X.The following theorem gives an expression that allows us to construct different entropies on IVIFSs.Theorem 9.
LetX={x1,x2,…,xN}, I:IVIFSs(X)→R+, and A∈IVIFSs(X). If
(14)I(A)=1N∑i=1N‍(1-Φ(μ-A(xi),ν-A(xi))),
then I is an entropy of A, where Φ satisfies conditions (a)–(d), μ-A(x)=μAL(x)+pΔμA(x), ν-A(x)=νAL(x)+pΔνA(x), and p∈(0,1).Proof.
(1)⇐ If A is a fuzzy set, then μAL(x)=μAU(x), νAL(x)=νAU(x), and μAU(x)+νAU(x)=1 for all x∈X. Clearly, I(A)=0 according to the condition (a) of Φ.
⇒
I
(
A
)
=
0 implies that Φ(μ-A(xi),ν-A(xi))=1(i=1,2,…,N) for any p∈(0,1). If A is not a fuzzy set, then μAL(xi*)+νAL(xi*)<1 for some xi*∈X  (i*∈{1,2,…,N}). As μAL(xi*)+νAL(xi*)<1 and μAU(xi*)+νAU(xi*)≤1, μ-A(xi*)+ν-A(xi*)<1 for any p∈(0,1). According to conditions (a) and (d) of Φ, we have Φ(μ-A(xi*),ν-A(xi*))<1. Thus, I(A)>0, which contradicts I(A)=0. Thus, A is a fuzzy set when I(A)=0.
(2)μA(x)=[0,0] and νA(x)=[0,0] for all x∈X which implies that μ-A(xi)=0, ν-A(xi)=0(i=1,2,…,N) for any p∈(0,1). Based on condition (b) of Φ, we have I(A)=1.
⇒
I
(
A
)
=
1 implies that Φ(μ-A(xi),ν-A(xi))=0(i=1,2,…,N) for any p∈(0,1). As 0≤μ-A(x)+ν-A(x)≤1, μ-A(x)+ν-A(x)=0 based on the condition (b) of Φ. If xi*∈X exists that satisfies μAU(xi*)+νAU(xi*)>0, it means that μ-A(xi*)+ν-A(xi*)>1 for any p∈(0,1). According to conditions (b) and (d) of Φ, we have Φ(μ-A(xi*),ν-A(xi*))>0. Obviously, I(A)<1, which contradicts I(A)=1. Thus, we have μA(x)=[0,0] and νA(x)=[0,0].
(3)   AsAc={(x,νA(x),μA(x))∣x∈X}, clearly, I(A)=I(Ac).
(4)   IfA⪯B, then μAL(x)≤μBL(x), μAU(x)≤μBU(x), νAL(x)≤νBL(x), and νAU(x)≤νBU(x);   then (1-p)(μBL(x)-μAL(x))+p(μBU(x)-μAU(x))+(1-p)(νBL(x)-νAL(x))+p(νBU(x)-νAU(x))≥0 for any p∈(0,1); then μAL(x)+p(μAU(x)-μAL(x))+νAL(x)+p(νAU(x)-νAL(x))≤μBL(x)+p(μBU(x)-μBL(x))+νBL(x)+p(νBU(x)-νBL(x)), that is, μ-A(x)+ν-A(x)≤μ-B(x)+ν-B(x). According to condition (d) of Φ, we have Φ(μ-A(xi),ν-A(xi))≤Φ(μ-B(xi),ν-B(xi))  (i=1,2,…,N), that is, 1-Φ(μ-A(xi),ν-A(xi))≥1-Φ(μ-B(xi),ν-B(xi)) implying I(A)≥I(B).
The proof is completed.According to (14), we can construct different entropies on IVIFSs based on various Φs. Some example functions that satisfy the previous conditions are presented as follows:
(15)Φn,cost(x,y)=(x+y)n,n=1,2,…,Φn,exp(x,y)=(x+y)n·e1-(x+y),n=1,2,…,Φn,sin(x,y)=(x+y)n·sin(π2(x+y)),n=1,2,….


## 3.3. Experimental Analysis of IVIF Entropies

### 3.3.1. Examples

Example 10.
Let us assume six IVIFSs, namely,A1={([0,0],[0,0])}, A2={([1,1],[0,0])}, A3={([0.4,0.4],[0.6,0.6])}, A4={([0.1,0.2],[0.3,0.4])}, A5={([0.3,0.4],[0.1,0.2])}, and A6={([0.3,0.4],[0.5,0.6])} on X={x}.
In this example, we adoptIΦ1,cost, IΦ1,exp, and IΦ1,sin to calculate the entropies of Ai(i=1,…,6) by choosing different values of parameter p(p=0.1,0.2,…,0.9). The entropies of Ai(i=1,…,6) are shown in Table 1.Table 1
IVIF entropiesIΦ1,cost,  IΦ1,exp,  and  IΦ1,sin  of  Ai(i=1,…,6).
A
1
A
2
A
3
A
4
A
5
A
6
I
Φ
1
,
cost
p
=
0.1
1
0
0
0.5800
0.5800
0.1800
p
=
0.2
1
0
0
0.5600
0.5600
0.1600
p
=
0.3
1
0
0
0.5400
0.5400
0.1400
p
=
0.4
1
0
0
0.5200
0.5200
0.1200
p
=
0.5
1
0
0
0.5000
0.5000
0.1000
p
=
0.6
1
0
0
0.4800
0.4800
0.0800
p
=
0.7
1
0
0
0.4600
0.4600
0.0600
p
=
0.8
1
0
0
0.4400
0.4400
0.0400
p
=
0.9
1
0
0
0.4200
0.4200
0.0200
I
Φ
1
,
exp
p
=
0.1
1
0
0
0.2499
0.2499
0.0183
p
=
0.2
1
0
0
0.2297
0.2297
0.0143
p
=
0.3
1
0
0
0.2106
0.2106
0.0108
p
=
0.4
1
0
0
0.1926
0.1926
0.0078
p
=
0.5
1
0
0
0.1756
0.1756
0.0053
p
=
0.6
1
0
0
0.1596
0.1596
0.0034
p
=
0.7
1
0
0
0.1446
0.1446
0.0019
p
=
0.8
1
0
0
0.1305
0.1305
0.0008
p
=
0.9
1
0
0
0.1173
0.1173
0.0002
I
Φ
1
,
sin
p
=
0.1
1
0
0
0.7426
0.7426
0.2126
p
=
0.2
1
0
0
0.7195
0.7195
0.1864
p
=
0.3
1
0
0
0.6958
0.6958
0.1607
p
=
0.4
1
0
0
0.6714
0.6714
0.1356
p
=
0.5
1
0
0
0.6464
0.6464
0.1111
p
=
0.6
1
0
0
0.6209
0.6209
0.0873
p
=
0.7
1
0
0
0.5949
0.5949
0.0642
p
=
0.8
1
0
0
0.5685
0.5685
0.0419
p
=
0.9
1
0
0
0.5417
0.5417
0.0205Example 11.
The conditions of this example are the same as Example10. In this example, we adopt IΦn,cost, IΦn,exp, and IΦn,sin(n=1,2,3,4,5) with p=0.5 to calculate the entropies of Ai(i=1,…,6). The entropies of Ai(i=1,…,6) are shown in Table 2.Table 2
IVIF entropiesIΦn,cost,  IΦn,exp,  and  IΦn,sin  of Ai(i=1,…,6).
A
1
A
2
A
3
A
4
A
5
A
6
I
Φ
n
,
cost
n
=
1
1
0
0
0.5000
0.5000
0.1000
n
=
2
1
0
0
0.7500
0.7500
0.1900
n
=
3
1
0
0
0.8750
0.8750
0.2710
n
=
4
1
0
0
0.9375
0.9375
0.3439
n
=
5
1
0
0
0.9688
0.9688
0.4095
I
Φ
n
,
exp
n
=
1
1
0
0
0.1756
0.1756
0.0053
n
=
2
1
0
0
0.5878
0.5878
0.1048
n
=
3
1
0
0
0.7939
0.7939
0.1943
n
=
4
1
0
0
0.8970
0.8970
0.2749
n
=
5
1
0
0
0.9485
0.9485
0.3474
I
Φ
n
,
sin
n
=
1
1
0
0
0.6464
0.6464
0.1111
n
=
2
1
0
0
0.8232
0.8232
0.2000
n
=
3
1
0
0
0.9116
0.9116
0.2800
n
=
4
1
0
0
0.9558
0.9558
0.3520
n
=
5
1
0
0
0.9779
0.9779
0.4168


## 3.3.1. Examples

Example 10.
Let us assume six IVIFSs, namely,A1={([0,0],[0,0])}, A2={([1,1],[0,0])}, A3={([0.4,0.4],[0.6,0.6])}, A4={([0.1,0.2],[0.3,0.4])}, A5={([0.3,0.4],[0.1,0.2])}, and A6={([0.3,0.4],[0.5,0.6])} on X={x}.
In this example, we adoptIΦ1,cost, IΦ1,exp, and IΦ1,sin to calculate the entropies of Ai(i=1,…,6) by choosing different values of parameter p(p=0.1,0.2,…,0.9). The entropies of Ai(i=1,…,6) are shown in Table 1.Table 1
IVIF entropiesIΦ1,cost,  IΦ1,exp,  and  IΦ1,sin  of  Ai(i=1,…,6).
A
1
A
2
A
3
A
4
A
5
A
6
I
Φ
1
,
cost
p
=
0.1
1
0
0
0.5800
0.5800
0.1800
p
=
0.2
1
0
0
0.5600
0.5600
0.1600
p
=
0.3
1
0
0
0.5400
0.5400
0.1400
p
=
0.4
1
0
0
0.5200
0.5200
0.1200
p
=
0.5
1
0
0
0.5000
0.5000
0.1000
p
=
0.6
1
0
0
0.4800
0.4800
0.0800
p
=
0.7
1
0
0
0.4600
0.4600
0.0600
p
=
0.8
1
0
0
0.4400
0.4400
0.0400
p
=
0.9
1
0
0
0.4200
0.4200
0.0200
I
Φ
1
,
exp
p
=
0.1
1
0
0
0.2499
0.2499
0.0183
p
=
0.2
1
0
0
0.2297
0.2297
0.0143
p
=
0.3
1
0
0
0.2106
0.2106
0.0108
p
=
0.4
1
0
0
0.1926
0.1926
0.0078
p
=
0.5
1
0
0
0.1756
0.1756
0.0053
p
=
0.6
1
0
0
0.1596
0.1596
0.0034
p
=
0.7
1
0
0
0.1446
0.1446
0.0019
p
=
0.8
1
0
0
0.1305
0.1305
0.0008
p
=
0.9
1
0
0
0.1173
0.1173
0.0002
I
Φ
1
,
sin
p
=
0.1
1
0
0
0.7426
0.7426
0.2126
p
=
0.2
1
0
0
0.7195
0.7195
0.1864
p
=
0.3
1
0
0
0.6958
0.6958
0.1607
p
=
0.4
1
0
0
0.6714
0.6714
0.1356
p
=
0.5
1
0
0
0.6464
0.6464
0.1111
p
=
0.6
1
0
0
0.6209
0.6209
0.0873
p
=
0.7
1
0
0
0.5949
0.5949
0.0642
p
=
0.8
1
0
0
0.5685
0.5685
0.0419
p
=
0.9
1
0
0
0.5417
0.5417
0.0205Example 11.
The conditions of this example are the same as Example10. In this example, we adopt IΦn,cost, IΦn,exp, and IΦn,sin(n=1,2,3,4,5) with p=0.5 to calculate the entropies of Ai(i=1,…,6). The entropies of Ai(i=1,…,6) are shown in Table 2.Table 2
IVIF entropiesIΦn,cost,  IΦn,exp,  and  IΦn,sin  of Ai(i=1,…,6).
A
1
A
2
A
3
A
4
A
5
A
6
I
Φ
n
,
cost
n
=
1
1
0
0
0.5000
0.5000
0.1000
n
=
2
1
0
0
0.7500
0.7500
0.1900
n
=
3
1
0
0
0.8750
0.8750
0.2710
n
=
4
1
0
0
0.9375
0.9375
0.3439
n
=
5
1
0
0
0.9688
0.9688
0.4095
I
Φ
n
,
exp
n
=
1
1
0
0
0.1756
0.1756
0.0053
n
=
2
1
0
0
0.5878
0.5878
0.1048
n
=
3
1
0
0
0.7939
0.7939
0.1943
n
=
4
1
0
0
0.8970
0.8970
0.2749
n
=
5
1
0
0
0.9485
0.9485
0.3474
I
Φ
n
,
sin
n
=
1
1
0
0
0.6464
0.6464
0.1111
n
=
2
1
0
0
0.8232
0.8232
0.2000
n
=
3
1
0
0
0.9116
0.9116
0.2800
n
=
4
1
0
0
0.9558
0.9558
0.3520
n
=
5
1
0
0
0.9779
0.9779
0.4168


## 3.4. Analysis of the Experimental Results

The results of Example10 show that IΦn,cost, IΦn,exp, and IΦn,sin are decreasing functions of p(p∈(0,1)) and satisfy the four properties of Definition 8. The results of Example 11 show that IΦn,cost, IΦn,exp, and IΦn,sin can be regarded as increasing functions of n(n=1,2,…).


## 4. MADM Method within the Framework of IVIFS

In this section, we present a novel method to solve the decision-making problems with unknown attribute weights based on the ranking functions and IVIF entropy. LetA={A1,A2,…,An} be a set of alternatives and G={G1,G2,…,Gk} be a set of attributes. The IVIF decision matrix D of A on G is
(16)D=[α11α12⋯α1kα21α22⋯α2k⋮⋮⋱⋮αn1αn2⋯αnk],
where αij(i=1,2,…,n;j=1,2,…,k) denotes the IVIF numbers. In the following, we introduce our method.Definition 12.
LetD be the IVIF decision-making matrix. We call I=[Iij]n×k an IVIF entropy matrix of D, where Iij denotes the IVIF entropy of αij(i=1,2,…,n;l=1,2,…,k).Step 1 (determine the attribute weights).
Letαij=([aij,bij],[cij,dij])(i=1,2,…,n;j=1,2,…,k). We introduce an IVIF entropy matrix. From Definition 12, we obtain the entropy matrix I=[Iij]n×k of the decision matrix D. We normalize the entropy matrix I to get I-=[I-ij]n×k, where I-ij=Iij/max{I1j,I2j,…,Inj} for i=1,2,…,n and j=1,2,…,k.   Finally, the attribute weight wj(j=1,2,…,k) is calculated by
(17)wj=1-ρjk-Γ,
where ρj=∑t=1n‍I-tj(j=1,2,…,k) and Γ=∑j=1k‍ρj.Step 2.
Based on the attribute weights obtained in Step1, the weighted arithmetic average value expressed by αi=([ai,bi],[ci,di]) for Ai(i=1,2,…,n) is calculated using the IIFWA operator [29, 30]
(18)αi=IIFWAw(αi1,αi2,…,αik)=w1αi1⊕w2αi2⊕⋯⊕wkαik=([1-∏j=1k‍(1-aij)wj,1-∏j=1k‍(1-bij)wj],ffff[∏j=1k‍cijwj,∏j=1k‍dijwj]),
where ai=1-∏j=1k(1-aij)wj, bi=1-∏j=1k(1-bij)wj, ci=∏j=1kcijwj, and di=∏j=1kdijwj.Step 3.
Rank all the alternativesαi(i=1,2,…,n) according to Definition 5 and select the most desirable alternative.


## 5. Numerical Examples

### 5.1. Example

Example 13.
This example is adapted from [11]. In this supplier selection problem, five possible alternatives denoted by Ai(i=1,2,3,4,5) exist on the following six attributes denoted by Gj(j=1,2,3,4,5,6). The ratings of the alternatives are given in the IVIF decision matrix shown in Tables 3 and 4.
In this example, we adoptIΦn,cost(n=1,10,100;p=0.1,0.5,0.9) to solve the supplier selection problem. The decision-making process is as follows.Step 1.
Based onIΦn,cost(n=1,10,100;p=0.1,0.5,0.9), the attribute            weights are obtained, as shown in Table 5.Step 2.
The weighted arithmetic average valuesαi for Ai(i=1,2,3,4,5)   are calculated.Step 3.
The score valuess(αi)(i=1,2,3,4,5) are calculated, and the decision-making results are shown in Table 6.Table 3
IVIF decision matrixD.
X
1
X
2
X
3
A
1
([0.2, 0.5], [0.4, 0.5])
([0.3, 0.4], [0.3, 0.4])
([0.3, 0.4], [0.4, 0.6])
A
2
([0.3, 0.4], [0.3, 0.5])
([0.5, 0.8], [0.1, 0.2])
([0.5, 0.6], [0.2, 0.4])
A
3
([0.4, 0.5], [0.2, 0.3])
([0.7, 0.8], [0.0, 0.1])
([0.6, 0.7], [0.1, 0.3])
A
4
([0.4, 0.7], [0.0, 0.2])
([0.9, 1.0], [0.0, 0.0])
([0.7, 0.8], [0.0, 0.1])
A
5
([0.6, 0.7], [0.1, 0.3])
([0.4, 0.6], [0.1, 0.2])
([0.7, 0.8], [0.0, 0.1])Table 4
IVIF decision matrixD.
X
4
X
5
X
6
A
1
([0.3, 0.5], [0.4, 0.5])
([0.2, 0.4], [0.5, 0.6])
([0.3, 0.4], [0.4, 0.5])
A
2
([0.5, 0.6], [0.2, 0.3])
([0.4, 0.6], [0.3, 0.4])
([0.6, 0.7], [0.1, 0.3])
A
3
([0.7, 0.8], [0.1, 0.2])
([0.3, 0.4], [0.4, 0.5])
([0.5, 0.6], [0.1, 0.2])
A
4
([0.8, 0.9], [0.0, 0.1])
([0.5, 0.7], [0.2, 0.3])
([0.6, 0.8], [0.1, 0.2])
A
5
([0.3, 0.4], [0.2, 0.3])
([0.8, 0.9], [0.0, 0.1])
([0.4, 0.5], [0.3, 0.4])Table 5
Attributes weights based onIΦn,cost(n=1,10,100;p=0.1,0.5,0.9).
Entropy
w
1
w
2
w
3
w
4
w
5
w
6
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
1
0.1468
0.1382
0.2278
0.1109
0.2069
0.1694
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
1
0.1783
0.1486
0.2253
0.0953
0.1907
0.1618
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
1
0.2011
0.2144
0.2212
0.0751
0.1159
0.1724
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
10
0.1718
0.1556
0.1741
0.1596
0.1696
0.1693
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
10
0.1786
0.1562
0.1805
0.1472
0.1673
0.1703
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
10
0.2034
0.1934
0.1812
0.1195
0.1165
0.1860
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
100
0.1667
0.1667
0.1667
0.1667
0.1667
0.1667
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
100
0.1667
0.1665
0.1667
0.1667
0.1667
0.1667
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
100
0.1732
0.1592
0.1697
0.1622
0.1639
0.1718Table 6
Scores valuess(αi) (i=1,2,3,4,5).
Entropy
s
(
α
1
)
s
(
α
2
)
s
(
α
3
)
s
(
α
4
)
s
(
α
5
)
Ranking order ofAi(i=1,2,3,4,5)
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
1
−0.1168
0.2872
0.4647
0.8408
0.5785
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
1
−0.1139
0.2817
0.4643
0.8398
0.5754
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
1
−0.0968
0.3042
0.4965
0.8526
0.5412
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
10
−0.1044
0.2922
0.4855
0.8472
0.5399
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
10
−0.1050
0.2905
0.4829
0.8459
0.5429
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
10
−0.0951
0.3024
0.4977
0.8510
0.5213
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
100
−0.1021
0.2968
0.4917
0.8503
0.5349
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
100
−0.1021
0.2967
0.4916
0.8503
0.5349
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
100
−0.1029
0.2941
0.4883
0.8481
0.5351
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1


### 5.2. Experimental Analysis

Similarly, we adoptIΦn,exp and IΦn,sin(n=1,10,100;p=0.1,0.5,0.9) to implement the decision-making process for this supplier selection problem. All the ranking results are the same as the ranking order of IΦn,cost(n=1,10,100;p=0.1,0.5,0.9), that is, A4≻A5≻A3≻A2≻A1. By applying the methods in [11, 41] to Example 13, the ranking order is A4≻A5≻A3≻A2≻A1, and the most desirable one is A4 which is identical to ours. From the aforementioned results, the proposed decision-making method in this paper can be suitably utilized to solve the IVIF MADM problem with completely unknown attribute weights. Different IVIF entropies clearly offer various choices for assessing the attribute weights.


## 5.1. Example

Example 13.
This example is adapted from [11]. In this supplier selection problem, five possible alternatives denoted by Ai(i=1,2,3,4,5) exist on the following six attributes denoted by Gj(j=1,2,3,4,5,6). The ratings of the alternatives are given in the IVIF decision matrix shown in Tables 3 and 4.
In this example, we adoptIΦn,cost(n=1,10,100;p=0.1,0.5,0.9) to solve the supplier selection problem. The decision-making process is as follows.Step 1.
Based onIΦn,cost(n=1,10,100;p=0.1,0.5,0.9), the attribute            weights are obtained, as shown in Table 5.Step 2.
The weighted arithmetic average valuesαi for Ai(i=1,2,3,4,5)   are calculated.Step 3.
The score valuess(αi)(i=1,2,3,4,5) are calculated, and the decision-making results are shown in Table 6.Table 3
IVIF decision matrixD.
X
1
X
2
X
3
A
1
([0.2, 0.5], [0.4, 0.5])
([0.3, 0.4], [0.3, 0.4])
([0.3, 0.4], [0.4, 0.6])
A
2
([0.3, 0.4], [0.3, 0.5])
([0.5, 0.8], [0.1, 0.2])
([0.5, 0.6], [0.2, 0.4])
A
3
([0.4, 0.5], [0.2, 0.3])
([0.7, 0.8], [0.0, 0.1])
([0.6, 0.7], [0.1, 0.3])
A
4
([0.4, 0.7], [0.0, 0.2])
([0.9, 1.0], [0.0, 0.0])
([0.7, 0.8], [0.0, 0.1])
A
5
([0.6, 0.7], [0.1, 0.3])
([0.4, 0.6], [0.1, 0.2])
([0.7, 0.8], [0.0, 0.1])Table 4
IVIF decision matrixD.
X
4
X
5
X
6
A
1
([0.3, 0.5], [0.4, 0.5])
([0.2, 0.4], [0.5, 0.6])
([0.3, 0.4], [0.4, 0.5])
A
2
([0.5, 0.6], [0.2, 0.3])
([0.4, 0.6], [0.3, 0.4])
([0.6, 0.7], [0.1, 0.3])
A
3
([0.7, 0.8], [0.1, 0.2])
([0.3, 0.4], [0.4, 0.5])
([0.5, 0.6], [0.1, 0.2])
A
4
([0.8, 0.9], [0.0, 0.1])
([0.5, 0.7], [0.2, 0.3])
([0.6, 0.8], [0.1, 0.2])
A
5
([0.3, 0.4], [0.2, 0.3])
([0.8, 0.9], [0.0, 0.1])
([0.4, 0.5], [0.3, 0.4])Table 5
Attributes weights based onIΦn,cost(n=1,10,100;p=0.1,0.5,0.9).
Entropy
w
1
w
2
w
3
w
4
w
5
w
6
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
1
0.1468
0.1382
0.2278
0.1109
0.2069
0.1694
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
1
0.1783
0.1486
0.2253
0.0953
0.1907
0.1618
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
1
0.2011
0.2144
0.2212
0.0751
0.1159
0.1724
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
10
0.1718
0.1556
0.1741
0.1596
0.1696
0.1693
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
10
0.1786
0.1562
0.1805
0.1472
0.1673
0.1703
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
10
0.2034
0.1934
0.1812
0.1195
0.1165
0.1860
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
100
0.1667
0.1667
0.1667
0.1667
0.1667
0.1667
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
100
0.1667
0.1665
0.1667
0.1667
0.1667
0.1667
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
100
0.1732
0.1592
0.1697
0.1622
0.1639
0.1718Table 6
Scores valuess(αi) (i=1,2,3,4,5).
Entropy
s
(
α
1
)
s
(
α
2
)
s
(
α
3
)
s
(
α
4
)
s
(
α
5
)
Ranking order ofAi(i=1,2,3,4,5)
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
1
−0.1168
0.2872
0.4647
0.8408
0.5785
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
1
−0.1139
0.2817
0.4643
0.8398
0.5754
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
1
−0.0968
0.3042
0.4965
0.8526
0.5412
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
10
−0.1044
0.2922
0.4855
0.8472
0.5399
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
10
−0.1050
0.2905
0.4829
0.8459
0.5429
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
10
−0.0951
0.3024
0.4977
0.8510
0.5213
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.1
,
n
=
100
−0.1021
0.2968
0.4917
0.8503
0.5349
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.5
,
n
=
100
−0.1021
0.2967
0.4916
0.8503
0.5349
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1
I
Φ
n
,
cost
|
p
=
0.9
,
n
=
100
−0.1029
0.2941
0.4883
0.8481
0.5351
A
4
≻
A
5
≻
A
3
≻
A
2
≻
A
1


## 5.2. Experimental Analysis

Similarly, we adoptIΦn,exp and IΦn,sin(n=1,10,100;p=0.1,0.5,0.9) to implement the decision-making process for this supplier selection problem. All the ranking results are the same as the ranking order of IΦn,cost(n=1,10,100;p=0.1,0.5,0.9), that is, A4≻A5≻A3≻A2≻A1. By applying the methods in [11, 41] to Example 13, the ranking order is A4≻A5≻A3≻A2≻A1, and the most desirable one is A4 which is identical to ours. From the aforementioned results, the proposed decision-making method in this paper can be suitably utilized to solve the IVIF MADM problem with completely unknown attribute weights. Different IVIF entropies clearly offer various choices for assessing the attribute weights.


## 6. Conclusion

In this paper, we have focused on solving the IVIF MADM problem with completely unknown attribute weights. We first introduced a new definition of IVIF entropy and some calculation methods of different IVIF entropies. Subsequently, we proposed an entropy-based MADM method in the framework of IVIFS. The proposed method can be utilized to solve fuzzy and uncertain decision-making problems derived from supplier selection, public risk, medical diagnosis, and other problems in any aspects.



---

*Source: 526871-2013-09-01.xml*