Interval-valued intuitionistic fuzzy continuous ...

Knowledge-Based Systems 59 (2014) 132–141

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Interval-valued intuitionistic fuzzy continuous weighted entropy and its application to multi-criteria fuzzy group decision making Feifei Jin, Lidan Pei, Huayou Chen ⇑, Ligang Zhou School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601, China

a r t i c l e

i n f o

Article history: Received 18 March 2013 Received in revised form 15 January 2014 Accepted 16 January 2014 Available online 24 January 2014 Keywords: Multi-criteria decision making Group decision making Interval-valued intuitionistic fuzzy set Continuous weighted entropy COWA operator

a b s t r a c t In this paper, we propose the interval-valued 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 (COWA) operator. It is shown that the continuous entropy of interval-valued intuitionistic fuzzy set is the average of the entropies of its interval-valued intuitionistic fuzzy values (IVIFVs). We also establish the programming model to determine optimal weight of criteria with the principle of minimum entropy. Furthermore, we investigate the multi-criteria group decision making (MCGDM) problems in which criteria values take the form of interval-valued intuitionistic fuzzy information. An approach to interval-valued intuitionistic fuzzy multi-criteria group decision making is given, which is based on the weighted relative closeness and the IVIFV attitudinal expected score function. Finally, emergency risk management (ERM) evaluation is provided to illustrate the application of the developed approach. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction The theory of fuzzy sets (FSs) put forward by Zadeh [1] has achieved a great success in various fields. Atanassov [2] introduced the concept of intuitionistic fuzzy sets (IFSs), which is the generalization of the FSs. The introduction of IFSs proved to be very meaningful and practical, and has been found to be highly useful to deal with vagueness. In IFSs, the data information is expressed by means of 2-tuples, and each 2-tuples is characterized by the degree of membership and non-membership. Furthermore, Atanassov and Gargov [3] introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs), whose components are intervals rather than exact numbers. In order to calculate the aggregation values of the alternatives, a lot of works [4–7,25,28–30,32,33] have been done about the aggregation operators of the IVIFSs. Atanassov [4] proposed some operational laws for IVIFSs. Xu and Chen [5] developed some intervalvalued intuitionistic fuzzy aggregation operators, such as the interval-valued intuitionistic fuzzy weighted averaging (IIFWA) operator, the interval-valued intuitionistic fuzzy ordered weighted averaging (IIFOWA) operator and the interval-valued intuitionistic fuzzy hybrid aggregation (IIFHA) operator, and then, they gave an application of the IIFHA operator to multiple attribute group decision making with interval-valued intuitionistic fuzzy information. ⇑ Corresponding author. Tel.: +86 136 156 90958. E-mail addresses: [email protected] (F. Jin), [email protected] (H. Chen), [email protected] (L. Zhou). http://dx.doi.org/10.1016/j.knosys.2014.01.014 0950-7051/Ó 2014 Elsevier B.V. All rights reserved.

Park et al. [6] presented a method for multi-person multi-attribute decision making based on the interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator and the interval-valued intuitionistic fuzzy weighted geometric (IIFWG) operator. Wei and Wang [7] investigated the interval-valued intuitionistic fuzzy ordered weighted geometric aggregation (IIFOWGA) operator and the interval-valued intuitionistic fuzzy hybrid geometric aggregation (IIFHGA) operator for multiple attribute group decision making under interval-valued intuitionistic fuzzy environment. Wang et al. [8] proposed an approach to multi-attribute decision making with incomplete attribute weighted information where individual assessments are provided as IVIFSs. Park et al. [9] presented an improved correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multi-attribute group decision-making problems with partially known attribute weight information. As an important topic in the theory of fuzzy sets, entropy measures have been investigated widely from different points of view, such as decision making [10–12] and pattern recognition [13–15]. The fuzzy entropy was first introduced by Zadeh [16]. Moreover, Luca and Termini [17] presented the axioms with which the fuzzy entropy should comply, and defined the entropy of a fuzzy set based on Shannon’s function. Szmidt and Kacprzyk [18] constructed the axiomatic requirements of intuitionistic fuzzy entropy measure and proposed a non-probabilistic-type entropy measure for IFSs based on the ratio of intuitionistic fuzzy cardinalities. Hung and Yang [19] gave their axiomatic definitions of entropy of IFS by exploiting the concept of probability.

F. Jin et al. / Knowledge-Based Systems 59 (2014) 132–141

However, due to the increasing complexity of the social-economic environment and the lack of knowledge or data about the problem domain, the decision information may be provided with IVIFSs. For example, emergency risk management (ERM) evaluation aims evaluating and improving social preparedness and organizational ability of an emergency operating center (EOC) in identifying, analyzing and treating emergency risks to the community arising from emergency events [35]. Suppose that there are five pre-determined alternatives (emergency operating centers), the criteria associated with alternatives include energy, food, health and medical services, etc., the evaluation values may be given with IVIFSs because of fuzzy environment in the process of multi-criteria decision making. Therefore, it is necessary and important to extend the entropy measures to accommodate the situation with interval-valued intuitionistic fuzzy information. Ye [20] proposed two entropy measures for IVIFSs and established an entropy weighted model to determine the entropy weights with respect to a decision matrix provided as IVIFSs. Wei et al. [21] developed an entropy measure for IVIFSs, which generalized three entropy measures for IFSs. Wu [22] introduced the concept of intuitionistic fuzzy weighted entropy, and gave a new method for intuitionistic fuzzy multi-criteria decision making problems. Motivated by the concept of intuitionistic fuzzy weighted entropy proposed by Wu [22], we develop a new formula to calculate the entropy of an interval-valued intuitionistic fuzzy set based on the continuous ordered weighted averaging (COWA) operator [23], called interval-valued intuitionistic fuzzy continuous entropy, which generalizes intuitionistic fuzzy entropy measures defined by Szmidt and Kacprzyk [18]. Then, we introduce the concept of the interval-valued intuitionistic fuzzy continuous weighted entropy. According to the principle of minimum entropy, we establish the programming model to determine optimal weight of criteria. Moreover, we give an approach to multi-criteria group decision making based on the proposed continuous entropy measures under interval-valued intuitionistic fuzzy environment and the TOPSIS method, in which we utilize the weighted relative closeness for each alternative with respect to ideal alternative. The novelty of this weighted relative closeness is that it can take into account the decision makers’ attitudinal character. The rest of the paper is organized as follows. In Section 2, we briefly review some concepts of IFSs and IVIFSs. Section 3 presents the concepts of the interval-valued intuitionistic fuzzy continuous entropy and the interval-valued intuitionistic fuzzy continuous weighted entropy. In Section 4, an approach to interval-valued intuitionistic fuzzy multi-criteria group decision making is proposed, which is based on the interval-valued intuitionistic fuzzy continuous weighted entropy. Section 5 provides a numerical example of emergency operating center evaluation to illustrate the application of the developed method. In Section 6, we end the paper by summarizing the main conclusions. 2. Preliminaries

with the condition

0 6 lA ðxi Þ þ mA ðxi Þ 6 1; 8xi 2 X:

Let C(X) be the set of all the IFSs on X. The operations of IFSs [2] are defined as follows. If A 2 C(X), B 2 C(X), then

Ac ¼ fhxi ; mA ðxi Þ; lA ðxi Þijxi 2 Xg;

ð4Þ

A \ B ¼ fhxi ; minflA ðxi Þ; lB ðxi Þg; maxfmA ðxi Þ; mB ðxi Þgijxi 2 Xg;

ð5Þ

A [ B ¼ fhxi ; maxflA ðxi Þ; lB ðxi Þg; minfmA ðxi Þ; mB ðxi Þgijxi 2 Xg:

ð6Þ

For an IFS A = {hxi, lA(xi), mA(xi)ijxi 2 X}, Szmidt and Kacprzyk [18] first axiomatized intuitionistic fuzzy entropy measure. Definition 2.2. Suppose that I is a real-valued function I:C(X) ? [0, 1]. I is an entropy measure of IFSs if it satisfies the following axiomatic requirements: (1) (2) (3) (4)

I(A) = 0 if and only if A is a crisp set; I(A) = 1 if and only if lA(xi) = mA(xi) for all xi 2 X; I(A) = I(Ac); I(A) 6 I(B) if A is less fuzzy than B, i.e., lA(xi) 6 lB(xi) and mA(xi) P mB(xi) for lB(xi) 6 mB(xi) or lA(xi) P lB(xi) and mA(xi) 6 mB(xi) for lB(xi) P mB(xi) for all xi 2 X.

Using the biggest cardinality (max-sigma-count) of IFSs, Szmidt and Kacprzyk also introduced an entropy measure for IFSs as follows:

Definition 2.3 [18]. The Szmidt and Kacprzyk entropy of an IFS A 2 C(X) is defined as

n max count Ai \ Aci 1X ISK ðAÞ ¼ ; n i¼1 max countðAi [ Aci Þ

A ¼ fhxi ; lA ðxi Þ; mA ðxi Þijxi 2 Xg;

ð1Þ

ð7Þ

where P Ai = {hxi, lA(xi), mA(xi)i} and max countðAÞ ¼ ni¼1 ðlA ðxi Þ þ pA ðxi ÞÞ. For an IFS A, Wang and Lei [25] gave a different entropy formula by

IWL ðAÞ ¼

n 1X minflA ðxi Þ; mA ðxi Þg þ pA ðxi Þ : n i¼1 maxflA ðxi Þ; mA ðxi Þg þ pA ðxi Þ

ð8Þ

Huang and Liu [26] introduced the concept of vague fuzzy entropy. Based on the equivalence of two theories of vague sets and intuitionistic fuzzy sets [27], we can transform the vague fuzzy entropy into an intuitionistic fuzzy entropy formula [26] for an IFS A by the following equation:

IHL ðAÞ ¼

Definition 2.1 [2]. Let X = {x1, x2, . . . , xn} be a fixed set. An intuitionistic fuzzy set (IFS) A over X is an object having the form:

ð3Þ

For each xi 2 X, the numbers lA(xi) and mA(xi) represent the membership and non-membership degrees of xi to A, respectively. For each IFS A in X, let pA(xi) = 1 lA(xi) mA (xi),which is called the hesitancy degree of xi to A. It is obvious that 0 6 pA(xi) 6 1, xi 2 X.

2.1. IFSs and the entropy of IFSs In the following, we introduce some basic concepts related to IFSs and several methods for calculating entropy measures for IFSs.

133

n 1X 1 jlA ðxi Þ mA ðxi Þj þ pA ðxi Þ : n i¼1 1 þ jlA ðxi Þ mA ðxi Þj þ pA ðxi Þ

ð9Þ

The entropy formulas (7)–(9) are introduced from different points of view. It is interesting to study their relations. Note that in [21], Wei et al. proved that ISK(A) = IWL(A) = IHL(A) for an IFS A. 2.2. IVIFSs and the operations between IVIFSs

where

lA : X ! ½0; 1; mA : X ! ½0; 1

ð2Þ

In some cases, it is not appropriate to assume that the membership degrees for certain elements of IFSs are exactly defined.

134


Therefore, Atanassov and Gaegov introduced the concept of interval-valued intuitionistic fuzzy sets. Definition 2.4 [3]. Let D[0, 1] be the set of all closed subintervals of the interval [0, 1], and X be a universe of discourse. An intervale over X is given by: valued intuitionistic fuzzy set A

e ¼ fhx; l ðxÞ; m ðxÞijx 2 Xg; A e e A

ð10Þ

A

where

leA : X ! D½0; 1; meA : X ! D½0; 1

ð11Þ

Denoting k ¼

A

8x 2 X:

A

ð12Þ

The intervals le ðxÞ and me ðxÞ denote the degree of membership and A A e respectively. the degree of non-membership of element x to A, Suppose that le ðxÞ ¼ l ðxÞ; lþ ðxÞ ; me ðxÞ ¼ m ðxÞ; mþ ðxÞ , e e e e A A A A A A then

e¼ ðxÞ; lþ ðxÞ ; me ðxÞ; mþ ðxÞ jx 2 X ; A x; le eA eA A A

ð13Þ

with the condition that

0 6 lþ ðxÞ þ mþ ðxÞ 6 1; eA eA

le ðxÞ P 0; me ðxÞ P 0: A

ð14Þ

A

Clearly, if l ðxÞ ¼ l ðxÞ and m ðxÞ ¼ mþ ðxÞ for each x 2 X, then eAe eA eA eA the given IVIFS A is reduced to an IFS. ~ convenience, For

an IVIFV will be denoted by a ¼ hla~ ; ma~ i ¼ la~ ; lþa~ ; ma~ ; mþa~ . þ

Definition 2.5 [3]. For two IVIFSs x; l ðxÞ; lþ ðxÞ ; eA eA þ þ þ me ðxÞ; me ðxÞ jx 2 X and Be ¼ x; le ðxÞ; le ðxÞ ; me ðxÞ; me ðxÞ B B A B B A e¼ A

e#B e (1) A

if

and

only

if

le ðxÞ 6 le ðxÞ; leþ ðxÞ 6 leþ ðxÞ; A

B

A

B

me ðxÞ P me ðxÞ; meþ ðxÞ P meþ ðxÞ for each x 2 X; A

B

e¼B e#B e e if and only if A e and B e # A; (2) A ec ¼ x; m ðxÞ; mþ ðxÞ ; l ðxÞ; lþ ðxÞ jx 2 X . (3) A eA eA eA eA

where k is the attitudinal character of Q. Thus, FQ ([a ,a ]) is the weighted average of the end points of the closed interval with attitudinal character parameter, and it is called the attitudinal expected value of [a, a+]. Based on the COWA operator, we give a measure formula for IVIFSs as follows:

e ¼ Eð AÞ

n minfF ðl Þ; F ðm ~ Þg þ p ~j Þ Q Q aj F Q ða 1X a~ j ; n j¼1 maxfF Q ðla~ j Þ; F Q ðma~j Þg þ pF Q ða~ j Þ

where 2; . . . ; n.

pF Q ða~j Þ ¼ 1 F Q ðla~j Þ F Q ðma~j Þ

In this section, we propose the continuous entropy and the continuous weighted entropy of IVIFSs. 3.1. Interval-valued intuitionistic fuzzy continuous entropy The continuous ordered weighted averaging (COWA) operator was developed by Yager [23], which extends the OWA operator [34]. Definition 3.1 [23]. A COWA operator is a mapping F:M ? R+ associated with a basic unit interval monotonic (BUM) function Q, such that

0

1

dQðyÞ þ ða yðaþ a ÞÞdy; dy

pF Q ða~j Þ 2 ½0; 1; j ¼ 1;

and

e is an IVIFS, then Eð AÞ e also has the folTheorem 3.1. Suppose that A lowing expression

e ¼ Eð AÞ

n 1 jF ðl Þ F ðm ~ Þj þ p ~j Þ Q Q aj F Q ða 1X a~ j : n j¼1 1 þ jF Q ðla~j Þ F Q ðma~ j Þj þ pF Q ða~j Þ

ð18Þ

Proof. Suppose that F Q ðla~ j Þ P F Q ðma~j Þ, then

e ¼ Eð AÞ

n minfF ðl Þ; F ðm ~ Þg þ p ~j Þ Q Q aj F Q ða 1X a~ j n j¼1 maxfF Q ðla~ j Þ; F Q ðma~j Þg þ pF Q ða~ j Þ

¼

n n F Q ðma~j Þ þ pF Q ða~j Þ 1 X 2ðF Q ðma~j Þ þ pF Q ða~j Þ Þ 1X ¼ n j¼1 F Q ðla~j Þ þ pF Q ða~j Þ n j¼1 2ðF Q ðla~j Þ þ pF Q ða~j Þ Þ

¼

n ðF Q ðma~ j Þ þ pF Q ða~ j Þ Þ þ ðF Q ðma~j Þ þ pF Q ða~j Þ Þ 1X n j¼1 ðF Q ðla~ j Þ þ pF Q ða~ j Þ Þ þ ðF Q ðla~ j Þ þ pF Q ða~ j Þ Þ

¼

n ð1 F ðl ÞÞ þ ðF ðm ~ Þ þ p ~j Þ Þ Q Q aj F Q ða 1X a~ j n j¼1 ð1 F Q ðma~j ÞÞ þ ðF Q ðla~ j Þ þ pF Q ða~ j Þ Þ

¼

n 1 jF ðl Þ F ðm ~ Þj þ p ~j Þ Q Q aj F Q ða 1X a~ j n j¼1 1 þ jF Q ðla~j Þ F Q ðma~ j Þj þ pF Q ða~j Þ

Similarly, when conclusion. h

F Q ðla~j Þ 6 F Q ðma~ j Þ,

j

Z

ð17Þ

we

can

get

the

same

e ¼ fa e ¼ fb ~ ;b ~ ;...; ~1 ; a ~2 ; . . . ; a ~ g # X, B Theorem 3.2. Suppose that A Dhn h1 2 i þ ~ n g # X, b where a~ j ¼ hla~j ; ma~ j i ¼ la~ j ; la~ j ; ma~j ; mþa~j i; e ~j ¼ hl~ ; m~ i ¼ b lb~ ; lþb~ ; mb~ ; mþb~ ; j ¼ 1; 2; . . . ; n. Then Eð AÞ bj bj

3. Interval-valued intuitionistic fuzzy continuous weighted entropy

F Q ðaÞ ¼ F Q ð½a ; aþ Þ ¼

ð16Þ +

In practice, the set of all IVIFVs can be denoted as X, then we have the following theorem.

B

A

Q ðyÞdy, then we have:

þ

) jx 2 X , then

0

F Q ðaÞ ¼ F Q ð½a ; a Þ ¼ kaþ þ ð1 kÞa :

with the condition

0 6 sup le ðxÞ þ sup meðxÞ 6 1;

R1

j

j

j

satisfies the following axiomatic requirements: (i) (ii) (iii) (iv)

e ¼ 0 if and only if A e Eð AÞ h is a crisp i set; h i þ þ e e ~ Eð AÞ ¼ 1 if and only if l a~ j ; la~ j ¼ ma~ j ; ma~ j for each aj 2 A; e ¼ Eð A e c Þ; Eð AÞ e 6 Eð BÞ e if Eð AÞ þ þ e e e ~ A # B when l ~ 6 mb ~ and lb ~ 6 mb ~ for each bj 2 B, b j j j j or e when l P m and lþ P mþ for each b e#A e ~j 2 B. B ~ ~ ~ ~ b b b b j

j

j

j

Proof.

ð15Þ

where a = [a,a+] 2 M, and M is the set of all nonnegative interval numbers.

e ¼ 0. Since every term in the summation of (i) Assume that Eð AÞ e is nonnegative, we deduce that every term should be Eð AÞ zero, i.e.,

135


minfF Q ðla~ j Þ; F Q ðma~ j Þg ¼ 0;

e pF Q ða~j Þ ¼ 0 for each a~ j 2 A:

If F Q ðla~ j Þ 6 F Q ðma~ j Þ, then. minfF Q ðla~j Þ; F Q ðma~ j Þg ¼ F Q ðla~ j Þ, that is F Q ðla~ j Þ ¼ 0. Notice that pF Q ða~ j Þ ¼ 1 F Q ðla~ j Þ F Q ðma~ j Þ, and pF Q ða~j Þ ¼ 0. Therefore,

e ~ j 2 A: F Q ðma~ j Þ ¼ 1 for each a i

h

So we have.

Hence,

i

e la~j ; lþa~j ¼ ½0; 0 and ma~j ; mþa~j ¼ ½1; 1 for each a~ j 2 A;

e is a crisp set. which implies that A e is a crisp set. Similarly, if F Q ðla~ j Þ P F Q ðma~ j Þ, we also get that A h i e is a crisp set, and l ; lþ ¼ ½1; 1 , On the other hand, if A h i a~ j a~ j e then. ma~j ; mþa~j ¼ ½0; 0 for each a~ j 2 A,

and we have

i

h

i

e la~j ; lþa~j ¼ ma~j ; mþa~j for each a~ j 2 A.

e c ¼ a ~ c1 ; a ~ c2 ; . . . ; a ~ cn , (iii) From Definition 2.5, we know that A Dh i h iE þ þ ~ cj ¼ m . where a a~ j ; ma~ j ; la~ j ; la~ j Therefore,

F Q ðla~c Þ ¼ F Q ðma~j Þ;

F Q ðma~cj Þ ¼ F Q ðla~ j Þ;

j

j ¼ 1; 2; . . . ; n:

la~j 6 lb~j ; lþa~j 6 lþb~j ; ma~j P mb~j ; mþa~j P mþb~j ; j ¼ 1; 2; . . . ; n:

pF Q ða~j Þ ¼ 1 F Q ðla~j Þ F Q ðma~j Þ ¼ 0; j ¼ 1; 2; . . . ; n: e ¼ 0. From Eq. (17), we obtain that Eð AÞ h i e Similarly, if A is a crisp set, and la~j ; lþa~j ¼ ½0; 0 , h i e ¼ 0. ma~j ; mþa~j ¼ ½1; 1; j ¼ 1; 2; . . . ; n, we also have that Eð AÞ h i h i þ þ e ~ (ii) If l a~ j ; la~ j ¼ ma~ j ; ma~ j , for each aj 2 A, then F Q ðla~ j Þ ¼ F Q ðma~ j Þ. e ¼ 1. According to Eq. (18), we have Eð AÞ On the other hand, by Eq. (18), it is obvious that e 6 1. If Eð AÞ e ¼ 1, then we obtain that 0 6 Eð AÞ

1 jF Q ðla~ j Þ F Q ðma~j Þj þ pF Q ða~ j Þ 1 þ jF Q ðla~ j Þ F Q ðma~j Þj þ pF Q ða~ j Þ

Then it follows that

F Q ðla~ j Þ 6 F Q ðlb~j Þ 6 F Q ðmb~j Þ 6 F Q ðma~j Þ; Thus, from Eq. (17) and

e ¼ Eð AÞ ¼

¼ 1; j ¼ 1; 2; . . . ; n;

8k

j ¼ 1; 2; . . . ; n:

pF Q ða~j Þ ¼ 1 F Q ðla~j Þ F Q ðma~j Þ, we have

n minfF ðl Þ; F ðm ~ Þg þ p ~j Þ Q Q aj F Q ða 1X a~ j n j¼1 maxfF Q ðla~ j Þ; F Q ðma~ j Þg þ pF Q ða~ j Þ n F ðl Þ þ ð1 F ðl Þ F ðm ~ ÞÞ n 1 F Q ðma~ j Þ Q Q Q aj 1X 1X a~ j a~ j ¼ n j¼1 F Q ðma~j Þ þ ð1 F Q ðla~ j Þ F Q ðma~ j ÞÞ n j¼1 1 F Q ðla~ j Þ

Similarly, we have

2 ½0; 1

e ¼ Eð BÞ

i.e.,

1 jF Q ðla~ j Þ F Q ðma~j Þj þ pF Q ða~ j Þ ¼ 1 þ jF Q ðla~j Þ F Q ðma~j Þj þ pF Q ða~ j Þ ;

j ¼ 1; 2; . . . ; n:

n 1 F ðm~ Þ Q 1X bj : n j¼1 1 F Q ðlb~j Þ

Since F Q ðma~j Þ P F Q ðmb~j Þ and F Q ðla~j Þ 6 F Q ðlb~j Þ; j ¼ 1; 2; . . . ; n, we have

e 6 Eð BÞ: e Eð AÞ

It follows that

jF Q ðla~ j Þ F Q ðma~ j Þj ¼ 0;

h

e ¼ Eð A e c Þ. According to Eq. (17), we have Eð AÞ þ þ e # B, e and A e ~j 2 B b (iv) Suppose that l 6 m ; l 6 m for each ~ ~ ~ ~ bj bj bj bj i.e.,

F Q ðla~ j Þ ¼ 1 and F Q ðma~ j Þ ¼ 0;

j ¼ 1; 2; . . . ; n; for 8k 2 ½0; 1;

i.e.,

þ þ e ~ Similarly, when l ~ P mb ~ and lb ~ P mb ~ for each bj 2 B and b j j e one can also prove e 6 jEð BÞ. e # A, e j h B that Eð AÞ

Definition 3.2. Suppose that F is a COWA operator associated with

F Q ðla~ j Þ F Q ðma~ j Þ ¼ 0;

j ¼ 1; 2; . . . ; n; for 8k 2 ½0; 1:

Let fj ðkÞ ¼ F Q ðla~ j Þ F Q ðma~ j Þ;

j ¼ 1; 2; . . . ; n, then.

fj ðkÞ ¼ F Q ðla~ j Þ F Q ðma~ j Þ ¼ 0;

e ¼ fa ~1; a ~2; . . . ; a ~ n g is an IVIFS, where BUM function Q, and A D E Dh i h iE þ e is a~ j ¼ la~j ; ma~j ¼ la~j ; la~ j ; ma~ j ; mþa~j ; j ¼ 1; 2; . . . ; n, then Eð AÞ e called the continuous entropy of IVIFS A.

j ¼ 1; 2; . . . ; n; for 8k 2 ½0; 1:

According to Eq. (16), we have

fj ðkÞ ¼ F Q ðla~ j Þ F Q ðma~ j Þ ¼ klþa~j þ ð1 kÞla~j kmþa~ j þ ð1 kÞma~j ¼ lþa~ j mþa~j la~ j þ ma~j k þ la~ j ma~j : Therefore,

ðlþa~j

lþa~j mþa~j la~j þ ma~j ¼ 0 and la~j ma~j ¼ 0; j ¼ 1; 2; . . . ; n;

la~j ¼ ma~j and lþa~j ¼ mþa~j ; j ¼ 1; 2; . . . ; n:

i.e.,

h

then

mþa~j la~j þ ma~j Þk þ la~j ma~j ¼ 0; for 8k 2 ½0; 1;

Remark 3.1. If an IVIFS reduces to be an IFS, then the continuous entropy of IVIFS defined by Eq. (17) or Eq. (18) reduces to the IF entropy measure defined by Eq. (8) or Eq. (9), respectively. Hence, in some sense, the continuous entropy formula for IVIFSs is a generalization of the entropy for IFSs. e ¼ fa e ¼ fb ~1 ; b ~2 ; . . . ; ~1; a ~2; . . . ; a ~ n g and B Example 3.1. Suppose that A ~n g are two IVIFSs, where a ~j ¼ ~ j ¼ hla~ ; ma~ j i ¼ h½0:3; 0:4; ½0:2; 0:4i; b b j hlb~j ; mb~j i ¼ h½0; 0:1; ½0:9; 0:9i; j ¼ 1; 2; . . . ; n. e is more fuzzy than IVIFS B e Intuitively, we can see that IVIFS A R1 from the point of hesitancy. Let k ¼ 0 Q ðyÞdy, then we can get the e and Eð BÞ e as follows: continuous entropy Eð AÞ

136


e ¼1 Eð AÞ n

n minfF ðl Þ; F ðm ~ Þg þ p X ~j Þ Q Q aj F Q ða a~ j j¼1

maxfF Q ðla~ j Þ; F Q ðma~ j Þg þ pF Q ða~j Þ

n o n o þ þ þ n min l ; m X a~ j a~ j þ min la~ j ; ma~ j þ pa~ j þ pa~ j 1 e ¼ n o n o EWCP ð AÞ n j¼1 max l ; m þ max lþ ; mþ þ p þ pþ a~ j a~ j a~ j a~ j a~ j a~ j n 2 l m þ lþ mþ þ p þ pþ a~ a~ j a~ j a~ a~ j a~ j 1X j j ¼ n j¼1 2 þ l m þ lþ mþ þ p þ pþ a~ j a~ j a~ j a~ j a~ j a~ j

7k ¼ 8 2k

and

e ¼ Eð BÞ

n minfF ðl~ Þ; F ðm~ Þg þ p ~Þ Q Q 1X 1 bj F Q ðb bj j ¼ ; n j¼1 maxfF Q ðlb~j Þ; F Q ðmb~j Þg þ pF Q ðb~j Þ 10 k

If k = 0.5, then 1 k = 0.5. From Eq. (18), we have

then we have

n 1 jF ðl Þ F ðm ~ Þj þ p ~j Þ Q Q aj F Q ða 1X a~ j n j¼1 1 þ jF Q ðla~j Þ F Q ðma~j Þj þ pF Q ða~j Þ h i n 1 klþ þ ð1 kÞl kmþ þ ð1 kÞm þ 1 klþ þ ð1 kÞl kmþ þ ð1 kÞm a~ j a~ j a~ j a~ j a~ j a~ j a~ j a~ j 1X h i ¼ n j¼1 1 þ klþ þ ð1 kÞl kmþ þ ð1 kÞm þ 1 klþ þ ð1 kÞl kmþ þ ð1 kÞm a~ j a~ j a~ j a~ j a~ j a~ j a~ j a~ j n 1 0:5lþ þ 0:5l 0:5mþ þ 0:5m þ 0:5pþ þ 0:5p a~ j a~ j a~ j a~ j a~ j a~ j 1X ¼ n j¼1 1 þ 0:5lþ þ 0:5l 0:5mþ þ 0:5m þ 0:5pþ þ 0:5p a~ j a~ j a~ j a~ j a~ j a~ j þ þ þ n 1 X 1 0:5 la~j ma~ j þ 0:5 la~ j ma~j þ 0:5pa~j þ 0:5pa~ j ¼ n j¼1 1 þ 0:5 l m þ 0:5 lþ mþ þ 0:5pþ þ 0:5p a~ j a~ j a~ j a~ j a~ j a~ j þ þ þ n 1 X 2 la~ j ma~ j þ la~j ma~ j þ pa~ j þ pa~ j ¼ n j¼1 2 þ l m þ lþ mþ þ pþ þ p a~ j a~ j a~ j a~ j a~ j a~ j

e ¼ Eð AÞ

2 e Eð BÞ e ¼ 7 k 1 ¼ ðk 7:5Þ þ 5:6 : Eð AÞ 8 2k 10 k ð8 2kÞð10 kÞ

Since

e Eð BÞ e > Eð BÞ, e > 0. i.e., Eð AÞ e Since k 2 [0, 1], then we obtain that Eð AÞ which is consistent with our intuition. Motivated by intuitionistic fuzzy entropy measures which was introduced by Wang and Lei [25], Wei et al. [21] proposed an e ¼ fa ~1; a ~2 ; . . . ; a ~ n g as follows: entropy measure for IVIFS A

n o n o þ þ þ n min l ; m X a~ j a~ j þ min la~ j ; ma~ j þ pa~ j þ pa~ j 1 e ¼ n o n o EWCP ð AÞ : n j¼1 max l ; m þ max lþ ; mþ þ p þ pþ a~ j

a~ j

a~ j

a~ j

a~ j

ð19Þ

a~ j

In the following, we will prove that the continuous entropy is the generalization of the entropy measure defined by Wei et al.[21] under some conditions.

ð22Þ

la~ j ma~ j lþa~ j mþa~ j P 0; j ¼ 1; 2; . . . ; n, then

la~j ma~ j þ lþa~ j mþa~j ¼ la~ j ma~ j þ lþa~ j mþa~ j ; j ¼ 1; 2; . . . ; n:

ð23Þ

Therefore,

n 2 l X a~ e ¼1 j Eð AÞ n j¼1 2 þ l a~ j n 2 l a~ j 1X ¼ n j¼1 2 þ l a~ j

ma~ j þ lþa~ j mþa~j þ pþa~ j þ pa~j ma~ j þ lþa~ j mþa~j þ pþa~ j þ pa~j ma~ j þ lþa~j mþa~ j þ pa~j þ pþa~ j ma~ j þ lþa~j mþa~ j þ pa~j þ pþa~ j

e ¼ EWCP ð AÞ:

ð24Þ

which completes the proof of Theorem 3.3. h e is an IVIFS, then EWCP ð AÞ e also has Lemma 3.1 [21]. Suppose that A the following expression

n 2 l m þ lþ mþ þ p þ pþ X a~ j a~ j a~ j a~ j a~ j a~ j 1 e ¼ EWCP ð AÞ : n j¼1 2 þ l m þ lþ mþ þ p þ pþ a~ j a~ j a~ j a~ j a~ j a~ j

ð20Þ

Theorem 3.3. Suppose that F is a COWA operator associated with e ¼ fa ~1; a ~2; . . . ; a ~ n g is an IVIFS, where BUM function Q. A Dh i h iE a~ j ¼ hla~j ; ma~ j i ¼ la~j ; lþa~ j ; ma~ j ; mþa~j j ¼ 1; 2; . . . ; n. If k = 0.5 and la~j ma~j lþa~j mþa~j P 0; j ¼ 1; 2; . . . ; n, then

e ¼ EWCP ð AÞ: e Eð AÞ Proof. According to Lemma 3.1, we have

ð21Þ

In general, the parameter k can be viewed as the measure of the decision maker’s attitudinal character. If k < 0.5, it means that the decision maker is risk averse. k = 0.5 indicates that the decision maker is risk neutral. If k > 0.5, it shows that the decision maker is with risk appetite. Therefore, each decision maker can choose the different parameter k according to his/her own risk attitude. e ¼ fa e ¼ fb ~1 ; b ~2 ; . . . ; ~1 ; a ~2 ; . . . ; a ~ n g and B Example 3.2. Suppose that A ~n g are two IVIFSs, and b ~j ¼ hl~ ; m~ i ¼ h½0:3; 0:5; ½0:1; 0:5i; b bj bj j ¼ 1; 2; . . . ; n. e and B, e we If we use Eq. (19) to calculate the entropy of IVIFSs A can get that

e ¼ EWCP ð AÞ

5 ; 8

e ¼ EWCP ð BÞ

6 : 7

e and B, e Now, using Eq. (17) to calculate the entropy of IVIFSs A we have

137


e ¼ Eð AÞ

6 2k ; 9 2k

e ¼ Eð BÞ

7 2k : 9 4k

e ¼ 5 ¼ EWCP ð AÞ; e Eð BÞ e ¼ 5 ¼ EWCP ð BÞ. e If k = 0.5, then Eð AÞ 8 8 62k 3 72k 1 5 e e Since Eð AÞ ¼ 92k ¼ 1 92k and Eð BÞ ¼ 94k ¼ 2 þ 188k , then it is e e is a obvious that Eð AÞ is a strictly decreasing function, and Eð BÞ strictly increasing function. e In fact, the relationship between the continuous entropy Eð AÞ e and the entropy measure EWCP ð AÞ depends on the decision maker’s attitudinal character as follows: e > EWCP ð AÞ; e (1) If k < 0.5, then Eð AÞ e ¼ EWCP ð AÞ; e (2) If k = 0.5, then Eð AÞ e e (4) If k > 0.5, then Eð AÞ < EWCP ð AÞ. e and the The relationship between the continuous entropy Eð BÞ e entropy measure EWCP ð BÞ depends on the decision maker’s attitudinal character as follows: e < EWCP ð BÞ; e (1) If k < 0.5, then Eð BÞ e ¼ EWCP ð BÞ; e (2) If k = 0.5, then Eð BÞ e e (3) If k > 0.5, then Eð BÞ > EWCP ð BÞ.

j

e in Theorem 3.3, then we are also unable to get satisfied for IVIFS A e and Eð AÞ. e the relationship between EWCP ð AÞ 3.2. Interval-valued intuitionistic fuzzy continuous weighted entropy In this section, we first introduce the concept of continuous entropy measure for IVIFV, which is in the analogous manner of the continuous entropy measure for IVIFS, and then we propose the interval-valued intuitionistic fuzzy continuous weighted entropy. ~ ¼ hla~ ; ma~ i ¼ Let e be a real-valued function e: X ? [0, 1]. For a Dh i h iE

þ þ þ þ ~ 2 X, if e h l ; l ; m ; m i; b ¼ hl~ ; m~ i ¼ l ; l ; m ; m a~

a~

a~

a~

~ b

b

b

~ b

~ b

~ b

satisfies the four axiomatic requirements: (1) (2) (3) (4)

eða~ Þ ¼ 0 if and only if la~ ; lþa~ ¼ ½1; ; mþ ¼ ½1; 1; 1 or m

a~ a~ eða~ Þ ¼ 1 if and only if la~ ; lþa~ ¼ ma~ ; mþa~ ; eða~ Þ ¼ eða~ c Þ; ~ if eða~ Þ 6 eðbÞ la~ 6 lb~ 6 mb~ 6 ma~ and lþa~ 6 lþb~ 6 mþb~ 6 mþa~ . or

la~ P lb~ P mb~ P ma~ and lþa~ P lþb~ P mþb~ P mþa~ . Then e is called an entropy measure of IVIFVs.

þ þ ~ ¼ hla~ ; ma~ i ¼ l Suppose that a is an IVIFV, we a~ ; la~ ; ma~ ; ma~ ~ as follows: develop an entropy measure for a

minfF Q ðla~ Þ; F Q ðma~ Þg þ pF Q ða~ Þ eða~ Þ ¼ maxfF Q ðla~ Þ; F Q ðma~ Þg þ pF Q ða~ Þ

ð25Þ

where pF Q ða~ Þ ¼ 1 F Q ðla~ Þ F Q ðma~ Þ. Using the similar proof of Theorem 3.1, we can get that

1 jF Q ðla~ Þ F Q ðma~ Þj þ pF Q ða~Þ eða~ Þ ¼ : 1 þ jF Q ðla~ Þ F Q ðma~ Þj þ pF Q ða~Þ

Definition 3.3. Suppose that F is a COWA operator associated with

þ þ ~ ¼ hla~ ; ma~ i ¼ l 2 X, then BUM function Q and a a~ ; la~ ; ma~ ; ma~ eða~ Þ, defined by Eq. (25), is called the continuous entropy of IVIFV a~ . The continuous entropy measures of IVIFS mentioned above are the average of the continuous entropy measures for IVIFVs in the IVIFS, in which each element has equal weight. While different elements may play different roles, they have different importance in real-life problems. Therefore, each IVIFV produces different effect on the results. Thus, it is necessary to introduce the concept of the interval-valued intuitionistic fuzzy continuous weighted entropy which is based on the continuous entropy of IVIFVs.

e ¼ fa ~1; a ~2; . . . ; a ~ n g # X, Definition 3.4. Suppose that A e:X ? [0, 1] is continuous entropy measures for IVIFV, then

From the above analysis, we can know that the monotonicity with respect to the attitudinal character k of the continuous e is changing with the different IVIFS A, e and we are entropy Eð AÞ e unable to determine the ordering relationship between EWCP ð AÞ e If k – 0.5 in Theorem 3.3, then Eq. (21) does not hold. If and Eð AÞ. þ þ the conditions of l a~ j ma~ j la~ ma~ P 0; j ¼ 1; 2; . . . ; n are not j

Proof. The proof of Theorem 3.4 is similar to Theorem 3.2, it is ~ Þ satisfies the four axiomatic easy to know that the mapping eða requirements (1)–(4) of IVIFV entropy measure, respectively. h

ð26Þ

e ¼ EW ð AÞ

n X ~jÞ wj eða

ð27Þ

j¼1

is called the continuous weighted entropy of interval-valued intuie where 0 < wj 6 1; j ¼ 1; 2; . . . ; n; Pn wj ¼ 1. tionistic fuzzy set A, j¼1 Example 3.3. Let X = {x1, x2, x3} be a universe of discourse. Suppose e ¼ fa ~1 ; a ~2; a ~ 3 g is an IVIFS in X, where a ~ 1 ¼ hx1 ; ½0:2; 0:3; that A ~ 2 ¼ hx2 ; ½0:3; 0:4; ½0:3; 0:5i and a ~ 3 ¼ hx3 ; ½0:3; 0:4; ½0:4; 0:6i; a e ½0:4; 0:6i. W = (0.3,0.4,0.3)T is a weight vector associated with A and k = 0.5. Now, using the continuous weighted entropy of IVIFS e given by Eq. (27), we have A

e ¼ EW ð AÞ

3 X 104 ~jÞ ¼ wj eða : 130 j¼1

e ¼ fa e ~1 ; a ~2 ; . . . ; a ~ n g # X. Then EW ð AÞ Theorem 3.5. Suppose that A satisfies four axiomatic requirements in Theorem 3.2. Proof. The proof of Theorem 3.5 is similar to that of Theorem 3.2. So it is omitted here. h Remark 3.2. In Definition 3.4, if there exists that wj = 0, j 2 {1, 2, . . . , n}, then we cannot ensure that the continuous weighted entropy satisfies the axiomatic requirements (i) and (ii) for IVIF entropy measure in Theoerm 3.2, which can be shown in the following Example 3.4. e ¼ fa ~1; a ~2 ; . . . ; a ~ n g # X, Example 3.4. Suppose that A e ¼ fb ~1 ; b ~2 ; . . . ; b ~n g # X, and a ~ 1 ¼ h½0:1; 0:2; ½0:3; 0:4i; a ~2 ¼ a ~3 ¼ . . . ¼ B a~ n ¼h½1;1;½0;0i; b~1 ¼h½0:1;0:2;½0:1;0:2i; b~2 ¼ b~3 ¼...¼ b~n ¼h½0:2;0:3; ½0:4;0:5i. If W1 = (0, 1/(n 1), ... , 1/(n 1))T and W2 = (1, 0, 0, ... , 0)T, then we have

e ¼ EW 1 ð AÞ

n n X X e ¼ ~j Þ ¼ 1: ~ j Þ ¼ 0; EW 2 ð BÞ wj eða wj eðb j¼1

e is not a crisp set and But A ~ Þ, defined by Eq. (25), is the entropy Theorem 3.4. The mapping eða measure for IVIFVs.

and

h

j¼1

i

h

i

l lþb~j – mb~j ; mþb~j ; j ¼ 2; 3; . . . ; n for ~ ; b j

e which means that if w1 = 0, the continuous weighted entropy B, Eq. (27) does not satisfy the IVIF entropy measure axiomatic requirements (i) and (ii) in Theoerm 3.2.

138


4. An approach to multi-criteria group decision making based on the IVIF continuous weighted entropy With economy development, the problems facing decision makers are becoming more complicated, uncertain and fuzzy than ever. It is obvious that an increasing amount of information provided by decision maker will be given in interval-valued intuitionistic fuzzy arguments. In this section, we present a handling method for multi-criteria fuzzy group decision making problems. Let X = {x1, x2, . . . , xm} be a set of alternatives, C = {C1, C2, . . . , Cn} be a set of criteria, E = {e1, e2, . . . , et} be a set of decision makers, and L = (l1, l2, . . . , lt)T be a weight vector of decision makers, where P ~ ijðkÞ lk P 0, k = 1, 2, . . . , t, and tk¼1 lk ¼ 1. Suppose that DðkÞ ¼ a mn

is an interval-valued intuitionistic fuzzy decision matrix, given by Dh i ðkÞ ðkÞ ðkÞþ ~ ðkÞ ; the decision maker ek, where a lðkÞ ; lij ij ¼ hlij ; mij i ¼ ij h iE ðkÞþ is an IVIFV for the alternative x mðkÞ ; m with respect to the i ij ij criterion Cj by the decision maker ek. Assume that the weight vector of criteria is W = (w1, w2, . . . , wn)T, where 0 < wj 6 1, P j = 1, 2, . . . , n, and nj¼1 wj ¼ 1. 4.1. The method of determing the criteria weight vector

k¼1

ð28Þ

i¼1

According to the entropy theory, if the entropy value for a criterion is smaller across alternatives, the criterion should be assigned a bigger weight. i.e., the smaller E(Cj) is, the bigger weight we should assign to the criterion Cj. If the information about weight wj of the criterion Cj is completely unknown, we can establish the following equations for determining criterion weights:

Pm ~ ðkÞ i¼1 1 e aij ; wj ¼ P P P n t m ~ ðkÞ j¼1 k¼1 lk i¼1 1 e aij Pt

k¼1 lk

j ¼ 1; 2; . . . ; n:

ð29Þ

Sometimes, the decision makers may only possess partial knowledge about criteria weights. In such a case, let H be the set of incomplete information about criteria weights, in order to get the optimal weight vector, the following model can be constructed as follows:

min EW

! n t m t m X n X X X X X ðkÞ ~ ij ~ ijðkÞ ¼ wj lk e a lk wj e a ¼ j¼1

k¼1

i¼1

8 W 2 H; > > > n > j¼1 > > > : wj > 0; j ¼ 1; 2; . . . ; n:

k¼1

~ i1 ; a ~ i2 ; . . . ; a ~ in g, we transform xi into a For an alternative xi ¼ fa ~ i1 Þ; Sða ~ i2 Þ; . . . ; Sða ~ in ÞÞ and Sða ~ ij Þ score vector S(xi), where Sðxi Þ ¼ ðSða ~ ij , i.e., is the attitudinal expected score function of a

~ ij Þ ¼ Sða

F Q ðlij Þ þ 1 F Q ðmij Þ : 2

Based on the TOPSIS method, a weighted relative closeness for each alternative xi with respect to ideal alternative xþ 0 is defined as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Pn ~ w Sð a Þ S a~ 0j ij j¼1 j ffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T xi ; xþ0 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ffi ; Pn Pn þ ~ ~ ~ ~ þ w Sð a Þ S a w Sð a Þ Sð a Þ j ij j ij j¼1 j¼1 0j 0j

~ ij Þ ¼ where Sða i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n. According to Eq. (31), we can easy get that

~ þ0j Þ ¼ 1; Sða

~ 0j Þ ¼ 0; Sða

j ¼ 1; 2; . . . ; n:

Thus, from Eq. (33), we have

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Pn ~ ij ÞÞ2 wj ðSða j¼1 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; T i ¼ T xi ; xþ0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pn 2 ~ 2 ~ j¼1 wj ðSðaij ÞÞ þ j¼1 wj ðSðaij Þ 1Þ

i ¼ 1; 2; . . . ; m:

ð34Þ

The larger the value of the weighted relative closeness Tðxi ; xþ 0 Þ, the better of the alternative xi. Therefore, all the alternatives can be ranked according to the value of T xi ; xþ 0 , and the best alternative can be selected. It is easy to check that T xi ; xþ has the following properties: 0 (a) 0 6 T xi ; xþ 0 6 1; þ (b) T xi ; xþ 0 ¼ T x0 ; xi ; (c) T xi ; xþ ¼ 1 if xi ¼ xþ 0 0. Then, we utilize the IVIF continuous weighted entropy to develop an approach to multi-criteria group decision making under interval-valued intuitionistic fuzzy environment, the main steps are as follows:

ð30Þ

Step 1: Utilize Eq. (29) or model (30) to determine the weight vector of criteria, W = (w1, w2, . . . , wn). Step 2: Utilize the IIFWA operator [5,31] to aggregate the whole individual interval-valued intuitionistic fuzzy decision ~ ðkÞ matrices DðkÞ ¼ a ðk ¼ 1; 2; . . . ; tÞ into a collective ij mn interval-valued intuitionistic fuzzy decision matrix ~ ij Þmn , where D ¼ ða

~ ð2Þ ~ ðtÞ a~ ij ¼ IIFWAL a~ ð1Þ ij ; aij ; . . . ; aij *"

First, we introduce the following score function of an IVIFV:

þ ~ ¼ hla~ ; ma~ i ¼ l Definition 4.1 [24]. Suppose that a a~ ; la~ ;

þ ma~ ; ma~ 2 X, the attitudinal expected score degree of a~ is

F Q ðla~ Þ þ 1 F Q ðma~ Þ : 2

ð33Þ

F Q ðlij Þþ1F Q ðmij Þ ; 2

i¼1 j¼1

4.2. An approach to multi-criteria group decision making based on the IVIF continuous weighted entropy and TOPSIS method

~Þ ¼ Sða

ð32Þ

i ¼ 1; 2; . . . ; m:

Considered the continuous weighted entropy of criteria, the overall continuous weighted entropy E(Cj) of the criterion Cj is given as: t m X X ~ ijðkÞ : EðC j Þ ¼ lk e a

In multi-criteria fuzzy decision making problems, the concept of ideal and anti-ideal points has been used to identify the best alternative. As we know, the ideal alternative may not exist in our real-life, but it does provide a useful tool for evaluating alternatives. Now, wedefine the ideal alternative anti-ideal and alterþ ~þ þ ~ ~ ~ ~ ~ natives as xþ ¼ a ; a ; . . . ; a ¼ a ; a ; . . . ; a ; x 0 01 02 0n , where 0 01 02 0n a~ þ0j ¼ hlþ0j ; mþ0j i ¼ h½1; 1; ½0; 0i; a~ 0j ¼ hl0j ; m0j i ¼ h½0; 0; ½1; 1i; j ¼ 1; 2; . . . ; n.

ð31Þ

1

¼

t Y

1l

ðkÞ ij

lk

# t l Y ðkÞþ k ; ;1 1 lij

k¼1

"

t Y k¼1

k¼1 t lk Y

mijðkÞ ;

mijðkÞþ

k¼1

lk

#+ ð35Þ

139


ðkÞ

~ ij (3) The decision matrix DðkÞ ¼ a is provided by kth eval85 uator, and the evaluated values are expressed by interval~ ijðkÞ denotes valued intuitionistic fuzzy information, where a the evaluated value of the emergency operating center x j under criteria C i provided by the kth evaluator ek(i = 1, 2, . . . , 8, j = 1, 2, . . . , 5, k = 1, 2, 3). The decision matrices D(1), D(2), D(3) are shown in Tables 1–3, respectively.

Step 3: Based on the interval-valued intuitionistic fuzzy deci~ ij Þmn , we can calculate the attitudision matrix D ¼ ða ~ ij ði ¼ 1; 2; . . . ; m; nal expected score function of a j ¼ 1; 2; . . . ; nÞ by using Eq. (32), and get the attitudinal ~ ij ÞÞmn . expected score matrix S ¼ ðSða Step 4: Calculate Ti, i = 1, 2, . . . , m by using Eq. (34) from attitu~ ij ÞÞmn . dinal expected score matrix S ¼ ðSða Step 5: Rank all the weighted relative closeness Ti (i = 1, 2, . . . , m) in descending order. Step 6: Select the best alternative (s) in accordance with the weighted relative closeness Ti(i = 1, 2, . . . , m). The best alternative (s) is the one with maxiTi. Step 7: End.

With this information, we can use the proposed decision making method to get the ranking of the emergency operating centers. The following steps are involved: Step 1: Take the BUM function Q(y) = y, then the attitudinal character k = 0.5 (medium value). Due to each criteria weight is completely unknown, then we use the following equations to determine the weight vector of criteria:

5. Illustrative example Emergency risk management (ERM) is a process which involves dealing with risks to the community arising from emergency events. Emergency management evaluation as one of the important parts of ERM, aims evaluating and improving social preparedness and organizational ability of an emergency operating center (EOC) in identifying, analyzing and treating emergency risks to the community arising from emergency events [35]. Fuzzy multicriteria decision making method is widely used for emergency management evaluation, and there are some pre-determined alternatives, associated with the criteria, the evaluation results are to be made [36,37]. According to Zhang et al. [35], There are five emergency operating centers (EOCs) X = {x1, x2, . . . , x5} to be evaluated by three evaluators E = {e1, e2, e3}. we suppose that the emergency management evaluation task has the following four features:

~ ðkÞ 1e a ij ; wi ¼ P P P5 8 3 ~ ðkÞ i¼1 k¼1 lk j¼1 1 e aij P3

k¼1 lk

P5 j¼1

i ¼ 1; 2; . . . ; 8

and we can obtain the criteria weights as

w1 ¼ 0:1537; w2 ¼ 0:1052; w3 ¼ 0:0933; w4 ¼ 0:1646; w5 ¼ 0:1170; w6 ¼ 0:1458; w7 ¼ 0:1337; w8 ¼ 0:0867 Step 2: Use the IIFWA operator to aggregate all individual interval-valued intuitionistic fuzzy decision matrices into a collective interval-valued intuitionistic fuzzy decision ~ ij Þ85 , which is shown in Table 4. matrix D ¼ ða Step 3: Calculate the attitudinal expected score function of a~ ij ði ¼ 1; 2; . . . ; 8; j ¼ 1; 2; . . . ; 5Þ, and we can get the ~ ij ÞÞ85 , which attitudinal expected score matrix S ¼ ðSða is shown in Table 5. Step 4: Based on the attitudinal expected score matrix S, we can get the weighted relative closeness Ti(i = 1, 2, . . . , 5) as follows:

(1) There are 8 criteria to evaluate five emergency operating centers, including C1: energy, C2: food, C3: health and medical services, C4: communication equipment, C5: emergency medical personnel, C6: human resource coordinator, C7: cars, C8: generators. Each criteria weight vector W = (w1, w2, . . . , w8) is completely unknown; (2) There are three evaluators E = {e1, e2, e3} associated with weighting vector L = (l1, l2, l3)T = (0.35, 0.40, 0.25)T;

T 1 ¼ 0:6316; T 2 ¼ 0:5757; T 3 ¼ 0:6249; T 4 ¼ 0:5521; and T 5 ¼ 0:6951

Table 1 e ð1Þ provided by e1. Interval-valued intuitionistic fuzzy decision matrix D

C1 C2 C3 C4 C5 C6 C7 C8

x1

x2

x3

x4

x5

([0.6, 0.8], [0.1, 0.2]) ([0.4, 0.7], [0, 0.1]) ([0.3, 0.7], [0.2, 0.3]) ([0.7, 0.8], [0.1, 0.2]) ([0.5, 0.6], [0.3, 0.4]) ([0.7, 0.8], [0.1, 0.2]) ([0.2, 0.4], [0.6, 0.6]) ([0.1, 0.4], [0.1, 0.5])

([0.2, 0.4], [0.4, 0.5]) ([0.5, 0.7], [0.1, 0.2]) ([0.2, 0.4], [0.4, 0.5]) ([0.2, 0.3], [0.4, 0.6]) ([0.7, 0.8], [0, 0.1]) ([0.3, 0.4], [0.4, 0.5]) ([0.2, 0.3], [0.3, 0.5]) ([0.4, 0.5], [0.4, 0.5])

([0.6, 0.7], [0.2, 0.3]) ([0.6, 0.6], [0.3, 0.4]) ([0.1, 0.4], [0.4, 0.5]) ([0.6, 0.8], [0, 0.2]) ([0.2, 0.4], [0.4, 0.5]) ([0.4, 0.5], [0.2, 0.3]) ([0.7, 0.9], [0.1, 0.1]) ([0.4, 0.4], [0.3, 0.4])

([0.4, 0.5], [0.2, 0.4]) ([0.7, 0.8], [0.1, 0.2]) ([0.3, 0.4], [0.5, 0.6]) ([0.6, 0.8], [0, 0.2]) ([0.1, 0.3], [0.4, 0.6]) ([0.2, 0.4], [0.4, 0.6]) ([0.6, 0.7], [0, 0.1]) ([0.5, 0.5], [0.5, 0.5])

([0.7, 0.8], [0.1, 0.1]) ([0.6, 0.8], [0, 0.2]) ([0.2, 0.4], [0.3, 0.4]) ([0.3, 0.4], [0.3, 0.5]) ([0.5, 0.5], [0, 0.3]) ([0.3, 0.3], [0.4, 0.5]) ([0.6, 0.7], [0.2, 0.3]) ([0.3, 0.4], [0.3, 0.4])


C1 C2 C3 C4 C5 C6 C7 C8

x1

x2

x3

x4

x5

([0.3, 0.4], [0.4, 0.6]) ([0.3, 0.5], [0.2, 0.3]) ([0.3, 0.5], [0.4, 0.4]) ([0.60.8], [0, 0.1]) ([0.1, 0.3], [0.5, 0.6]) ([0.8, 0.9], [0.1, 0.1]) ([0.1, 0.4], [0.3, 0.5]) ([0.3, 0.4], [0.3, 0.3])

([0.3, 0.3], [0.4, 0.4]) ([0.3, 0.5], [0.1, 0.2]) ([0.3, 0.4], [0.4, 0.6]) ([0.6, 0.7], [0.1, 0.1]) ([0.4, 0.5], [0.2, 0.3]) ([0.2, 0.3], [0.3, 0.5]) ([0.4, 0.7], [0, 0.1]) ([0.4, 0.5], [0.4, 0.5])

([0.4, 0.6], [0.2, 0.3]) ([0.5, 0.5], [0.2, 0.2]) ([0.3, 0.4], [0.1, 0.2]) ([0.7, 0.9], [0.1, 0.1]) ([0.5, 0.5], [0.5, 0.5]) ([0.2, 0.5], [0.3, 0.5]) ([0.1, 0.3], [0.4, 0.6]) ([0.5, 0.7], [0.2, 0.3])

([0.4, 0.5], [0.4, 0.5]) ([0.1, 0.2], [0.4, 0.7]) ([0.3, 0.3], [0.3, 0.5]) ([0.5, 0.6], [0.2, 0.3]) ([0.2, 0.4], [0, 0.3]) ([0.1, 0.3], [0.5, 0.6]) ([0.5, 0.7], [0.2, 0.2]) ([0.4, 0.5], [0.3, 0.5])

([0.1, 0.2], [0.5, 0.7]) ([0.4, 0.6], [0.1, 0.2]) ([0.3, 0.4], [0.3, 0.5]) ([0.1, 0.1], [0.6, 0.7]) ([0.2, 0.4], [0.4, 0.5]) ([0.3, 0.4], [0.5, 0.6]) ([0.1, 0.3], [0.4, 0.6]) ([0.3, 0.5], [0.3, 0.5])

140



C1 C2 C3 C4 C5 C6 C7 C8

x1

x2

x3

x4

x5

([0.2, 0.4], [0.3, 0.5]) ([0.2, 0.3], [0.6, 0.7]) ([0.6, 0.7], [0.2, 0.2]) ([0.8, 0.8], [0.1, 0.2]) ([0.3, 0.5], [0.4, 0.5]) ([0.7, 0.8], [0.1, 0.2]) ([0.5, 0.7], [0.2, 0.3]) ([0.1, 0.3], [0.5, 0.6])

([0.4, 0.5], [0.5, 0.5]) ([0.3, 0.5], [0.1, 0.2]) ([0.4, 0.5], [0.3, 0.3]) ([0.7, 0.8], [0.1, 0.2]) ([0.4, 0.6], [0.2, 0.3]) ([0.3, 0.5], [0.3, 0.5]) ([0.6, 0.7], [0.2, 0.2]) ([0.3, 0.4], [0.4, 0.5])

([0.2, 0.3], [0.5, 0.6]) ([0.3, 0.4], [0.5, 0.6]) ([0.8, 0.9], [0.1, 0.1]) ([0.9, 0.9], [0, 0.1]) ([0.5, 0.6], [0.2, 0.3]) ([0.7, 0.8], [0.1, 0.2]) ([0.7, 0.7], [0.2, 0.3]) ([0.1, 0.4], [0.2, 0.4])

([0.1, 0.4], [0.4, 0.5]) ([0.6, 0.8], [0, 0.2]) ([0.2, 0.4], [0.5, 0.6]) ([0.5, 0.7], [0.1, 0.2]) ([0.7, 0.8], [0.1, 0.2]) ([0.1, 0.4], [0.2, 0.5]) ([0.2, 0.3], [0.4, 0.6]) ([0.3, 0.5], [0.4, 0.5])

([0.8, 0.9], [0, 0.1]) ([0.3, 0.8], [0, 0.1]) ([0.6, 0.7], [0.2, 0.3]) ([0.5, 0.7], [0.1, 0.2]) ([0.2, 0.4], [0.4, 0.5]) ([0.4, 0.7], [0.2, 0.3]) ([0.6, 0.7], [0.2, 0.3]) ([0.3, 0.4], [0.3, 0.5])

Table 4 e Collective interval-valued intuitionistic fuzzy decision matrix D.

C1 C2 C3 C4 C5 C6 C7 C8

x1

x2

x3

x4

x5

([0.41, 0.59], [0.23, 0.39]) ([0.31, 0.54], [0, 0.25]) ([0.52, 0.61], [0.12, 0.27]) ([0.47, 0.64], [0, 0.20]) ([0.42, 0.57], [0.21, 0.38]) ([0.82, 0.84], [0.03, 0.12]) ([0.34, 0.46], [0.43, 0.53]) ([0.21, 0.42], [0.36, 0.49])

([0.29, 0.39], [0.42, 0.46]) ([0.38, 0.58], [0.10, 0.20]) ([0.34, 0.47], [0.42, 0.46]) ([0.57, 0.76], [0.14, 0.21]) ([0.31, 0.45], [0, 0.43]) ([0.23, 0.47], [0.47, 0.69]) ([0.44, 0.56], [0, 0.27]) ([0.59, 0.66], [0.23, 0.32])

([0.44, 0.55], [0.25, 0.36]) ([0.50, 0.56], [0.29, 0.34]) ([0.65, 0.76], [0.09, 0.22]) ([0.76, 0.84], [0, 0.11]) ([0.43, 0.53], [0.34, 0.41]) ([0.66, 0.72], [0.15, 0.23]) ([0.23, 0.43], [0.41, 0.52]) ([0.24, 0.35], [0.41, 0.57])

([0.34, 0.48], [0.31, 0.46]) ([0.50, 0.65], [0, 0.33]) ([0.13, 0.24], [0.46, 0.67]) ([0.44, 0.65], [0.22, 0.34]) ([0.45, 0.67], [0.21, 0.31]) ([0.23, 0.36], [0.34, 0.54]) ([0.51, 0.73], [0, 0.21]) ([0.14, 0.28], [0.47, 0.52])

([0.58, 0.71], [0, 0.22]) ([0.46, 0.74], [0, 0.17]) ([0.57, 0.71], [0.11, 0.16]) ([0.43, 0.54], [0.22, 0.31]) ([0.49, 0.62], [0, 0.13]) ([0.54, 0.63], [0.23, 0.31]) ([0.54, 0.67], [0.12, 0.20]) ([0.28, 0.47], [0, 0.39])

Acknowledgments

Table 5 The attitudinal expected score matrix S.

C1 C2 C3 C4 C5 C6 C7 C8

x1

x2

x3

x4

x5

0.5950 0.6500 0.6850 0.7275 0.6000 0.8775 0.4600 0.4450

0.4500 0.6650 0.4825 0.7450 0.5825 0.3850 0.6825 0.6750

0.5950 0.6075 0.7750 0.8725 0.5525 0.7500 0.4325 0.4025

0.5125 0.7050 0.3100 0.6325 0.6500 0.4275 0.7575 0.3575

0.7675 0.7575 0.7525 0.6100 0.7450 0.6575 0.7225 0.5900

Step 5: Rank the weighted relative closeness Ti (i = 1, 2, . . . , 5) in descending order:

T5 P T 1 P T3 P T2 P T4: Step 6:

Rank all the emergency operating centers xi (i = 1, 2, . . . , 5) and select the best one (s) in accordance with the weighted relative closeness Ti(i = 1, 2, . . . , 5). Therefore, we have

x5 x1 x3 x2 x4 ; and thus the most desirable emergency operating center is x5. 6. Conclusions At present, many entropy measures are applied to group decision making problems with intuitionistic fuzzy information, but they could not be used to deal with the group decision making problems with interval-valued intuitionistic fuzzy information. We develop a new entropy measure of IVIFSs, called interval-valued intuittionistic fuzzy continuous weighted entropy. We give the axiomatic requirements for continuous entropty of IVIFVs and show that the continuous entropy of interval-valued intuitionistic fuzzy set is the average of the entropies of its IVIFVs. Furthermore, a fuzzy multi-criteria group decision making method is presented, which is based on interval-valued intuitionistic fuzzy continuous weighted entropy. The proposed method provides a useful way to deal with multi-criteria fuzzy group decision making under interval-valued intuitionistic fuzzy environment.

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