International Journal of Computational Intelligence Systems

Volume 9, Issue 1, January 2016, Pages 133 - 152

Multi-attribute group decision making based on Choquet integral under interval-valued intuitionistic fuzzy environment

Authors
Jindong Qin1, qinjindongseu@126.com, Xinwang Liu1, *, xwliu@seu.edu.cn, Witold Pedrycz2, wpedrycz@ualberta.ca
1School of Economics and Management, Southeast University, Nanjing, 211189, China
2Department of Electrical and Computer Engineering, University of Alberta, Edmonton, T6G 2G7, Canada
*Corresponding author, Email: xwliu@seu.edu.cn (X.Liu).
Corresponding Author
Xinwang Liuxwliu@seu.edu.cn
Received 14 November 2014, Accepted 5 October 2015, Available Online 1 January 2016.
DOI
10.1080/18756891.2016.1146530How to use a DOI?
Keywords
Interval-valued intuitionistic fuzzy Choquet integral; interval-valued intuitionistic fuzzy sets; interval-valued intuitionistic fuzzy measure; multiple attribute group decision making
Abstract

In this paper, we propose new methods to represent interdependence among alternative attributes and experts’ opinions by constructing Choquet integral using interval-valued intuitionistic fuzzy numbers. In the sequel, we apply these methods to solve the multiple attribute group decision-making (MAGDM) problems under interval-valued intuitionistic fuzzy environment. First, the concept of interval-valued intuitionistic fuzzy Choquet integral is defined, and some elementary properties are studied in detail. Next, an axiomatic system of interval-valued intuitionistic fuzzy measure is established by delivering a series of mathematical proofs. Then, with fuzzy entropy and Shapely-values in game theory, we propose the interval-valued intuitionistic fuzzy measure development methods in order to form the importance measure of attributes and correlation measure of the experts, respectively. Based on the results of theoretical analysis, a new method is proposed to handle the interval-valued intuitionistic fuzzy group decision making problems. A numerical example illustrates the procedure of the proposed methods and verifies the validity and effectiveness of our new proposed methods.

Copyright
© 2016. the authors. Co-published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

With the increasing complexity and uncertainty of the social-economic environment and various limitations, such as time pressure, lack of knowledge of problem domain, difficulties in information extraction etc., there are numerous uncertain phenomena existing in our daily life. Therefore, in order to better understand the vagueness and uncertainty of the real world and being able to explain it, Zadeh 1 proposed fuzzy set theory in 1960s. Since its inception, the fuzzy set theory has shown convenience as a powerful tool for modeling vagueness and uncertainty in a variety domains, such as economics 2,3,4, management 5,6,7,8, artificial intelligence 16, processing control 10,11, pattern recognition 12, decision making 13,14,15,16 etc. In 1980s, Atanassov 17,18 generalized fuzzy set theory by bringing an idea of intuitionistic fuzzy set (IFS). IFS is characterized by three important notations: membership degree, non-membership degree and hesitancy degree. On the basis of these three indexes, IFS can cope with the vagueness and uncertainty characteristics of complex systems. Now, IFS has become an important area of research, and attracted more attention in various fields. For example, Xu 19,20 has completed some research in the field of IFS, especially in aggregation operators, developed a series of intuitionistic fuzzy information aggregation operators for aggregating intuitionistic fuzzy information and applied these operators to multiple attribute decision making (MADM) problems. Wang and Liu 22 proposed the intuitionistic fuzzy aggregation operators realized through Einstein operations and analyzed the relations between these operators and some existing intuitionistic fuzzy aggregation operators. Li 23 developed the concept of IFS, and further extended mathematical programming methodology of matrix games with payoffs represented by IFS. Furthermore, some desirable properties were discussed in detail. Farhadinia and Ban 24 developed some new similarity measures to deal with generalized intuitionistic fuzzy numbers.

However, in many practical situations, there are limitations given various factors, such as a lack of information, uncertainty of the decision making environment, difficulties in information extraction etc. Therefore, it is not easy for decision maker to determine exact values of membership degree, non-membership degree and hesitancy degree. To overcome these limitations, Atanassov and Gargov 18 proposed the concept of the interval-valued intuitionistic fuzzy set (IVIFS), in which the attributes are taken the form with interval-valued intuitionistic fuzzy number (IVIFN). Many studies were completed in recent years and various approaches to solve MADM problems under interval-valued intuitionistic fuzzy environment have been developed 24,25,26,27,28,29,30,31,32,33,34. For example, Farhadinia 31 generalized recent results for the entropy of interval-valued fuzzy set (IVFS) based on the intuitionistic distance measure and its relationship with the similarity measure, and then applied them to interval-valued intuitionistic fuzzy decision making problem. Liu 32 developed some geometric aggregation operators based on interval-valued intuitionistic uncertain linguistic variables and some desirable properties were discussed as well. Wang and Li 34 proposed a framework to handle multi-attribute group decision making (MAGDM) problems with incomplete pair-wise comparison preference over decision alternatives. Wu and Chiclana 33 defined interval-valued intuitionistic fuzzy COWA operator and a new score function, then proposed some non-dominance and attitudinal prioritization decision making methods for intuitionistic fuzzy preference relations.

As the most important branch of fuzzy mathematics, fuzzy integrals were originally investigated by Choquet 36 in 1950s, as a powerful tool for modeling interaction phenomena among various factors. The main difference between the fuzzy integral and the classic integral is that fuzzy integral focuses on non-additive cases while the classical integral only consider additive cases. On the basis of classic sets theory and measure theory, it can be easily proved the fact that the Choquet integral is a special case of classical integral when Choquet integral satisfies the condition of additivity, In this case Choquet integral reduces to the Lebesgue integral. Therefore, the Choquet integral is more general. Compared with the Sugeno integral 37, which is also known as a fuzzy integral for aggregating discrete sets, Choquet integral exhibits interesting mathematical properties, such as boundary conditions, monotonicity, continuity from below, continuity from above. Therefore, Choquet integral is more suitable to cope with fuzzy uncertainty problems in practical applications. Recently, Choquet integral has been widely used in decision making 35,38,45,46,48, especially in intuitionistic fuzzy MADM problems. Xu 40 investigated Choquet integral to propose some intuitionistic fuzzy aggregation operators, and then applied them to solve intuitionistic fuzzy multiple attribute decision making problems. Tan and Chen 42,43 developed an weight solution method based on Choquet integral and further proposed a new Choquet integral-TOPSIS approach to handle MADM problems. Meng and Tang 44 developed a interval-valued intuitionistic fuzzy multi-attribute group decision making approach based on cross entropy measure and Choquet integral. Compared with other existing methods, this algorithm can not only deal with the relevance among the indexes, but also can cope with dynamic fuzzy systems. Meng et al. 45 defined some new hybrid Choquet integral aggregation operators, and proposed a method to solve multiple criteria group decision making (MCGDM) based on intuitionistic fuzzy Choquet integral with respect to the generalized λ-Shapely index. Wu et al. 47 presented a detailed discussion on the integration properties of the interval intuitionistic fuzzy conjugate Choquet integral, and applied these theoretical analysis results to software development risks decision making. In addition, some authors have also applied Choquet integral to other fields, such as hesitant fuzzy set 49, game theory 50, neural networks 51,52, statistical learning 53, and combinatorial optimization 54,55,56.

According to the previous overview, intuitionistic fuzzy Choquet integral and its property has been widely used in decision making problems. However, there has been a limited research on Choquet integral under interval-valued intuitionistic fuzzy decision making environment in the literatures. When Choquet integral is used to handle MADM problems under interval-valued intuitionistic fuzzy environment, the most challenging difficulty is how to determine fuzzy measure with an effective and accurate method. For an MADM problem which contains n attributes, we should determine 2n − 2 elements of fuzzy measures in decision making process. When the problem dimensionality increases, the computational complexity increases rapidly. So far, in the current literature 42,43,44,45,57,58,59, the fuzzy measures are usually provided by decision makers (DMs) in advance, which maybe difficult to the DMs and is closely related the subjective preferences of the DMs, or can even lead to inconsistent results. If the fuzzy measures can be determined from the problem formulation or the decision maker’s known information directly, the decision making model solution can be more reliable and objective than the current ones. Therefore, how to determine the fuzzy measure based on the expert decision making information becomes a very important issue both in theory and practice. The motivation of this study is to develop a new method of constructing fuzzy measures based on interval-valued intuitionistic information and apply them to the group decision making problems.

The remainder of this paper is organized as follows. In Section 2, we briefly introduce some basic concepts and related operational laws of interval-valued intuitionistic fuzzy set (IVIFS) and the Choquet integral. In Section 3, we give the definition of interval-valued intuitionistic fuzzy measure with Choquet integral and propose some useful theoretical background. In Section 4, we develop new methods to determine fuzzy measure in real-world decision marking situations, by establishing interval-valued intuitionistic fuzzy measure to determine the measure of expert weights and attribute weights, and strict mathematical proof process is given. In Section 5, a new approach based on interval-valued intuitionistic fuzzy Choquet integral is proposed to solve MAGDM problems. In Section 6, an illustrative example is provided to illustrate the proposed method. Finally, we come up with some conclusions and point out the future research in Section 7.

2. Preliminaries

In this section, we briefly review some basic concepts of interval-valued intuitionistic fuzzy set (IVIFS) and fuzzy measure, which are extended to interval-valued intuitionistic fuzzy measure in next Section 3.

2.1. Interval-valued intuitionistic fuzzy set

Definition 1 18

Let X be a universe of discourse. Then

A={x,([μAL(x),μAR(x)],[νAL(x),νAR(x)])|xX}
is an IVIFS, where μA(x)=[μAL(x),μAR(x)] , νA(x)=[νAL(x),νAR(x)] is called as interval membership degree and non-membership degree of x. The following two conditions are satisfied:
  1. (1)

    For all xX, [μAL(x),μAR(x)][0,1] , and [νAL(x),νAR(x)][0,1] ;

  2. (2)

    xX, 0μAR(x)+νAR(x)1 .

In particular, if μAL(x)=μAR(x) , νAL(x)=νAR(x) , for each xX, then the IVIFS is reduced to IFS.

For convenience, Xu 20 called α = ([aL,aR],[bL,bR]) an interval-valued intuitionistic fuzzy number (IVIFN), where [aL,aR] ⊆ [0,1], [bL,bR] ⊆ [0,1],aR + bb ≤ 1, and let Θ be the set of all IVIFNs. Obviously, α+ = ([1,1],[0,0]) is the largest IVIFN, and α = ([0,0],[1,1]) is the smallest IVIFN.

Definition 2 20

Let α = ([aL, aR],[bL,bR]), β = ([cL,cR],[dL,dR]) be two IVIFNs. The operational laws are shown as follows:

  1. (1)

    αβ = ([aL + cLaLcL,aR + cRaRcR],[bLdL,bRdR]);

  2. (2)

    αβ = ([aLcL,aRcR],[bL + dLbLdL,bR + dRbRdR]);

  3. (3)

    αC = ([bL,bR],[aL,aR]);

  4. (4)

    aλ = ([(aL)λ,(aR)λ ],[1 − (1 − bL)λ,1 − (1 − bR)λ ]), λ > 0.

Definition 3 19

Let α = ([aL, aR],[bL,bR]) be an IVIFN. Then the score function of α is defined as follows:

s(α)=aL+aRbLbR2
where s(α) ∈ [−1,1]. Obviously, the greater s(α) is, the larger IVIFN α becomes.

Definition 4 21

Let A={xi,([aiL,aiR],[biL,biR])|xX} (i = 1,2,⋯,n) be a discrete IVIFS. The entropy of A is defined in the following form:

E(A)=1ni=1ncos|biLaiL|+|biRaiR|2(πiL+πiR+2)π

The interval-valued intuitionistic fuzzy entropy satisfies the following properties:

  1. (1)

    E(A) = 0 if and only if aiL=aiR=0 , biL=biR=1 or aiL=aiR=1 , biL=biR=0 ;

  2. (2)

    E(A) = 1 if and only if [aiL,aiR]=[biL,biR] , for any xiX;

  3. (3)

    E(A) = E(Ac);

  4. (4)

    E(A) ≤ E(B), ∀xiX, if aBLbBL , aBRbBR , then aALaBL , aARaBR , bALbBL , bARbBR or aBLbBL , aBRbBR , then aALaBL , aARaBR , bALbBL , bARbBR .

2.2. Fuzzy measure and Choquet integral

Definition 5 60

Let X be a universe of discourse, P(X) be the power set of X. A fuzzy measure on X is a mapping μ: P(X) ↦ [0,1] which satisfies the following properties:

  1. (1)

    μ(ϕ) = 0, μ(X) = 1 (boundary conditions);

  2. (2)

    if AB implies μ(A) ≤ μ(B), for all A,BX,(monotonicity);

  3. (3)

    μ(A + B) = μ(A) + μ(B) + λ μ(A)μ(B), where A,BP(X) and AB = ϕ,λ ≥ −1;

  4. (4)

    μ(n=1nAn)=limnμ(An) , whenever AnAn+1,AnX,nN (continuity from below);

  5. (5)

    μ(n=1An)=limnμ(An) , whenever AnAn+1,AnX,nN (continuity from above).

Definition 6 37

Let X = {x1, x2, ⋯,xn} be a universe of discourse, and X=i=1nxi , then λfuzzy measure μ satisfies following condition:

μ(X)={1λ(i=1n[1+λμ(xi)]1)ifλ0i=1nμ(xi)ifλ=0
where λ > −1. From Eq.(4), the value of λ can be uniquely determined by setting μ(X) = 1, which is equivalent by solving the following equation:
λ+1=i=1n(1+λμ(xi))

Definition 7 35

Let μ be a fuzzy measure on X, f is a nonnegative, real-valued, and measurable function. Then the Choquet integral is defined in the form:

(c)fdμ=0μ{Fα}dl
where Fα = {x|f (x) ≥ α} denotes the αcut of the function. When X = {x1,x2, ⋯,xn} be a discrete set, the discrete form of Choquet integral with respect to a fuzzy measure μ is defined as:
(c)fdμ=i=1n(f(xi*)f(xi1*))μ({xi*,xi+1,*,xn*})
Where A*={x1*,x2*,,xn*} is a monotonic non-decrease permutation of X = {x1,x2, ⋯,xn} such that f(x1*)f(x2*)f(xn*) and An+1*= .

3. Interval-valued intuitionistic fuzzy Choquet integral

In this section, we investigate the definition of the interval-valued intuitionistic fuzzy measure and the interval-valued intuitionistic fuzzy Choquet integral with such fuzzy measure. Some properties and characteristics of them are proposed and discussed in detail.

Definition 8

Let X be a universe of discourse, and (X,Σ, μ) be an interval intuitionistic fuzzy measure space, Σ is a σalgebra and μ = ([μ1, μ2],[μ3, μ4]), where fuzzy measure μ satisfies the following properties:

  1. (1)

    μ(ϕ) = ([μ1(ϕ), μ2(ϕ)],[μ3(ϕ), μ4(ϕ)]) = ([0,0],[1,1]);

  2. (2)

    μ(X) = ([μ1(X), μ2(X)],[μ3(X), μ4(X)]) = ([1,1],[0,0]);

  3. (3)

    If E,F ∈ Σ, and EF, then μ1(E) ≤ μ1(F), μ2(E) ≤ μ2(F) and μ3(E) ≥ μ3(F), μ4(E) ≥ μ4(F) ;

  4. (4)

    For any E ∈ Σ, μ2(E) + μ4(E) ≤ 1.

Especially, when μ1(E) = μ2(E), μ3(E) = μ4(E), for any E ∈ Σ, the interval-valued intuitionistic fuzzy measure reduces to the commonly encountered intuitionistic fuzzy measure.

Based on the decomposition theorem for the original Choquet integral and the extended intuitionistic fuzzy Choquet integral, the interval-valued intuitionistic fuzzy Choquet integral of f with respect to μ is defined as:

(c)fdμ=([aiLdμ1,[biL¯dμ3¯,aiRdμ2],biR¯dμ4¯])
where f=([aiL,aiR],[biL,biR]) , μ = ([μ1,μ2],[μ3,μ4]). For any set A ∈ Σ, which satisfies μ1(A) ≤ μ2(A), μ3(A) ≤ μ4(A), μ2(A) + μ4(A) ≤ 1, and “- “ denotes the dual operation, i.e.,
f¯(x)=1f(x)μ¯(A)=1μ(A)

Theorem 1

Let μ be an interval-valued intuitionistic fuzzy measure defined on X, and A is an interval-valued intuitionistic fuzzy set on P(X). Then the interval-valued intuitionistic fuzzy Choquet integral with respect to μ is also an IVIFN.

Proof.

Based on the definition of IVIFS, we have aLaR, bLbR. According to the monotonicity of fuzzy measure, we can prove that aiLdμ1aiRdμ1aiRdμ2 and biL¯dμ3¯¯=1biL¯dμ3¯=1(1biL)dμ3¯1(1biR)dμ3¯1(1biR)dμ4¯=1biR¯dμ4¯=biR¯dμ4¯¯ . Thus, Eq. (8) satisfies the partial order of the IVIFS.

For convenience, we denote f1=aiL , f2=aiR , f3=biL , f4=biR in the following proof process. Based on the definition of discrete Choquet integral described in Section 2, we have:

(c)f1μ1=μ1(f1(x)α1)dα1(c)f2μ2=μ2(f2(x)α2)dα2

So we only need to prove the following relationships:

(c)f3μ3=μ3(f3(x)α3)dα3(c)f4μ4=μ4(f4(x)α4)dα4

Based on Eq. (10), we have

f3¯μ3¯¯=101μ3¯(f3¯(x)α3)dα3=101[1μ3¯(f3¯(x)α3)]dα3=01μ3¯(f3¯(x)α3)dα3=01(1μ3)(f3(x)1α3)dα3=101μ3(f3(x)1α3)dα3=01μ3(f3(x)α3)dα3

In a similar way, it is easy to prove that (c) ∫ f4μ4 = ∫ μ4(f4(x) ≤ α4)4 holds.

Next, Based on the definition of interval intuitionistic fuzzy sets described in Section 2, we only need to prove that the following inequality holds:

(c)f2μ2+(c)f¯4μ4¯¯1

Since f2 + f4 ≤ 1, μ2 + μ4 ≤ 1, based on the monotonicity of Choquet integral, the following inequality ensues:

(c)f¯4μ4¯(c)f2μ4¯(c)f2μ2

Then

(c)f2μ2+(c)f¯4μ4¯¯=(c)f2μ2+1(c)f¯4μ4¯(c)f2μ2+1(c)f2μ2=1

The proof has been completed.

Theorem 2

Let X = {x1,x2, ⋯,xn} be the universe of discourse, then the interval-valued intuitionistic fuzzy Choquet integral of f with respect to μ can be expressed as the following form:

(c)fdμ=([i=1naiL(xi)(μ1(Ai)μ1(Ai+1)),i=1naiR(xi)(μ2(Ai)μ2(Ai+1))],[i=1nbiL(xi)(μ1(Bi+1)μ1(Bi)),i=1nbiR(xi)(μ4(Bi+1)μ4(Bi))])
where
Ai

= {xi, ⋯,xn | (aL + aR) (xi+1) ≥ (aL + aR) (xi)}

and
Bi

= {xi, ⋯,xn | (bL + bR) (xi+1) ≤ (bL + bR) (xi)} (i = 1,2, ⋯,n)

Proof.

Based on the definition of Choquet integral described in Section 2, it can be easily shown that the following two relationships are satisfied:

(c)f1dμ1=i=1nf1(xi)(μ1(Ai)μ1(Ai+1))=i=1naiL(xi)(μ1(Ai)μ1(Ai+1))(c)f2dμ2=i=1nf2(xi)(μ2(Ai)μ2(Ai+1))=i=1naiR(xi)(μ2(Ai)μ2(Ai+1))

Thus, we should only prove the following formulas:

(c)f¯3dμ¯3¯=i=1nf¯3(xi)(μ¯3(Bi+1)μ¯3(Bi))=i=1nbiL(xi)(μ3(Bi+1)μ3(Bi))(c)f¯4dμ¯4¯=i=1nf¯4(xi)(μ¯4(Bi+1)μ¯4(Bi))=i=1nbiR(xi)(μ4(Bi+1)μ4(Bi))

Based on Eqs.(9,10,12), we obtain

(c)bL¯dμ3¯¯=f3¯dμ3¯¯=1f3¯dμ3¯=1i=1nf3¯(xi)[μ3¯(Bi)μ3¯(Bi+1)]=1i=1nf3¯(xi)[(1μ3(Bi))(1μ3(Bi+1))]=i=1n(μ3(Bi+1)μ3(Bi))i=1nf3¯(xi)(μ3(Bi+1)μ3(Bi))=i=1n(μ3(Bi+1)μ3(Bi))(1f3¯(xi))=i=1nf3(xi)(μ3(Bi+1)μ3(Bi))

Similarly, it is easy to verify that the following formula holds:

(c)bR¯dμ4¯¯=i=1nbiR(xi)(μ4(Bi+1)μ4(Bi))

Thus, the proof has been completed.

In what follows, we investigate two useful theorems of the interval-valued intuitionistic fuzzy Choquet integral proposed above.

Theorem 3

Let X be a universe of discourse, f = ([ f1, f2],[ f3. f4]) be an interval intuitionistic fuzzy-valued function on X. 0¯=([0,0],[1,1]) and 1¯=([1,1],[0,0]) are the smallest and largest interval intuitionistic fuzzy-valued function, respectively. Then

  1. 1.

    (Idempotency). c(0¯,0¯,,0¯)=0¯,c(1¯,1¯,,1¯)=1¯,c(f¯,f¯,,f¯)=f¯ ;

  2. 2.

    (Boundary). min (f (1), f (2),⋯, f (n)) ≤ c (f (1), f (2) ⋯, f (n)) ≤ max (f (1), f (2),⋯, f (n));

  3. 3.

    (Monotonicity). If f (k)f (k*), then c (f (1), ⋯, f (k),⋯, f (n)) ≤ c (f (1),⋯, f (k*),⋯, f (n)).

Proof.

  1. (1)

    Based on the definition of Choquet integral, we have

    (c)(0¯,0¯,,0¯)=0¯×i=1n(μ(xi)μ(xi+1))=0¯×1=0¯(c)(1¯,1¯,,1¯)=1¯×i=1n(μ(xi)μ(xi+1))=1¯×1=1¯(c)(f,f,,f)=f×i=1n(μ(xi)μ(xi+1))=f×1=f

  2. (2)

    Based on Theorems 1 and 2, we obtain (c)(min1inf(i),,min1inf(i))(c)(f(1),,f(n))(c)(max1inf(i),,max1inf(i)) . Then according to the property of idempotency, we have (c)(min1inf(i),,min1inf(i))=min1inf(i) and (c)(max1inf(i),,max1inf(i))=max1inf(i) . Therefore, min(f(1),f(2),,f(n))(c)(f(1),f(2),,f(n))max(f(1),f(2),,f(n)) .

  3. (3)

    Based on the monotonicity of fuzzy integral and Theorem 2, the conclusion directly ensues.

which completes the proof of Theorem 3.

Theorem 4

Let X be a universe of discourse, P(X) is a power set of X, set function μ is an interval-valued intuitionistic fuzzy measure on X,let f = ([ f1, f2],[ f3, f4]) be an interval-valued intuitionistic fuzzy-valued function X. 0¯=([0,0],[1,1]) and 1¯=([1,1],[0,0]) are the smallest and largest interval-valued intuitionistic fuzzy measure, respectively. Then

  1. 1.

    (c)f1¯=maxxXf ;

  2. 2.

    (c)f0¯=minxXf ;

  3. 3.

    For any interval intuitionistic fuzzy measure μ on X, minxXf(c)fμmaxxXf .

Proof.

  1. (1)

    For any α = ([α1,α2],[α3,α4]) ∈ X, based on the definition of interval intuitionistic fuzzy Choquet integral described in Section 2, we have

    (c)fμ=([01μ1(f1α1)dα1,01μ2(f2α2)dα2],[01μ3(f3α3)dα3,01μ4(f4α4)dα4])
    where { fα} = {xX| f (x) ≥ α} = {xX|f1 (x) ≥ α1f2(x) ≥ α2f3(x) ≤ α3f4(x) ≤ α4}. When αβ, then {x|fα} ⊆ {x| fβ}, which implies μ{x| fα} ≤ μ {x| fβ}.

    If μ{x|fα}=1¯μ{x|fβ}=1¯ . Then,

    (c)fμ=([01μ1(f1α1)dα1,01μ2(f2α2)dα2],[01μ3(f3α3)dα3,01μ4(f4α4)dα4])
    (c)f1¯=supμ(fα)=1¯([01μ1(f1β1)dβ1,01μ2(f2β2)dβ2],[01μ3(f3β3)dβ3,01μ4(f4β4)dβ4])=supμ(fα)=1¯([α1,α2],[α3,α4])=supμ(fα)=1¯αsupμ(fα)=1¯infμ(fα)=1¯fsupμ(fα)=1¯supμ(fα)=1¯f=maxf

  2. (2)

    The proof of property 2 is similar to property 1, omitted.

  3. (3)

    Based on the monotonicity of fuzzy measure, we have 0¯μ1¯ , so we can easily prove the following inequality

    (c)f0¯(c)fμ(c)f1¯

    According to Theorem 3, we have minxXf(c)fμmaxxXf .

which completes the proof of Theorem 4.

4. A new method to determine fuzzy measure under interval-valued intuitionistic fuzzy environment

In most real-world decision making situations, where the measure of attributes usually does not satisfy the condition of non-additive property. However, there exist some cases where the attributes may interact. Therefore, how to derive the fuzzy measures to reflect the relationship of attributes becomes a vital problem in such practical problems. From the current literature 57,58,59, we can only find research on the determination of the fuzzy measures with the decision makers assignment directly, which not only raises difficulties for large dimensional problems and may also lead to information losses, distortion and inconsistencies. Here, we propose a new method to determine the fuzzy measures from the known information, which can avoid the shortcomings of the current direct assignment methods.

For convenience, we first introduce some new notations. Let D = {D1,D2, ⋯,Dp} be the set of experts, where Di indicates the i-th expert, and Ei indicates the average interval-valued intuitionistic fuzzy entropy provided by Di, which is calculated by Eq. (3).

4.1. Experts importance measure based on interval-valued intuitionistic fuzzy entropy

Definition 9

Let μ(Di) denote the importance measure of expert Di, and μ(Di1,Di2, ⋯,Din) denotes the joint absolute importance measure of experts Di1,Di2, ⋯,Din, the experts absolute importance measure can be defined as:

μ(Di)=1Ei
μ(Di,Dj)=1E(EiEj)
μ(Di1,Di2,,Din)=1E(Ei1Ei2Ein)
where {i1,i2, ⋯,in} is a subset of {1,2,⋯, p}, Ekij is the interval-valued intuitionistic fuzzy entropy provided by expert Dk for the alternative Ai with respect to the attribute Cj. It should be noted that E(Ei1Ei2 ∩ ⋯ ∩ Ein) indicate the average interval-valued intuitionistic fuzzy entropy provided by experts Di1,Di2, ⋯,Din, where Ei1Ei2 ∩ ⋯ ∩ Ein is a real-valued entropy matrix.

Remark 1

This definition is established based on interval-valued intuitionistic fuzzy entropy. As we know, the entropy can be regarded as an important measure that reflects the decision making information to some extent. The smaller the entropy is, the greater the information will be. Therefore, consider the characteristics of the group decision making problems under interval-valued intuitionistic fuzzy environment, we utilize the minimize entropy principle to fuse the decision making information among the experts and establish the model above to obtain the experts absolute importance measure.

Due to the fact that importance measure should satisfy normalization condition, we should normalize the absolute measure before calculation. Hence, we define a new concept as follows:

Definition 10

Let μ(Di) denote the relative importance measure of Di, and μ(Di1,Di2,,Din)¯ denotes the joint relative importance measure of Di1,Di2, ⋯,Din, the experts relative importance measure is defined as follows:

μ(Di1,Di2,,Din)¯=μ(Di1,Di2,,Din)μ(D1,D2,,Dn)

Theorem 5

μ(Di1,Di2,,Din)¯ satisfies the monotonicity of the fuzzy measure.

Proof.

Since {Di1,Di2, ⋯,Din} ⊆ {Di1,Di2, ⋯,Din+1}, then

μ(Di1,Di2,,Din,Din+1)¯μ(Di1,Di2,,Din)¯=μ(Di1,Di2,,Din,Din+1)μ(Di1,Di2,,Din)μ(D1,D2,,Dn)=1E(Ei1Ei2EinEin+1)(1E(Ei1Ei2Ein))μ(D1,D2,,Dn)=E(Ei1Ei2Ein)E(Ei1Ei2EinEin+1)μ(D1,D2,,Dn)0

So μ(Di1,Di2,,Din)¯ satisfies the monotonicity of the fuzzy measure. The proof has been completed.

Remark 2

Theorem 5 indicates the fact that with the increasing number of experts (or decision makers), the information proposed by experts (or decision makers) is also increasing. It also states that the uncertainty of information reduces. Therefore, the experts relative importance measure proposed here is reasonable in real life decision making situations.

4.2. Attributes importance measure based on weight information matrix

Attribute importance measure is a key issue of the decision making problems in practice. As a useful tool to solve the attribute importance measure, Shannon entropy was widely used to trade off the importance among the attributes in real life decision making. Let Ci denote the ith attribute, Hi denote the Shannon entropy with respect to Ci. We usually measure the importance of Ci based on the value of Hi. The smaller value of Hi, the great importance of Ci is. In what follows, we define some new joint attributes information entropy based on original Shannon entropy to measure the importance of the attributes.

Definition 11

Let Hi indicates the Shannon entropy with respect to the attribute Ci. Three types of joint attributes information entropy are defined as follows:

  1. (1)

    EΔ1(in)=i=1in(1Hi)π4 ;

  2. (2)

    EΔ2(in)=sin(i=i1in(1Hi)π4) ;

  3. (3)

    EΔ3(in)=tan(i=i1in(1Hi)π4) .

where In = {i1,i2, ⋯,in} is a index sequence such that in−1in, 1 ≤ inp.

It is clear that 0 ≤ EΔ1(in),EΔ2(in),EΔ3(in) < 1, and EΔ1(in),EΔ2(in),EΔ3(in) are all continuous, monotonic and decreasing functions of In. That is to say, if |In| < |In+1| (where |In| is the number of In), then EΔj(in+1) < EΔj(in), j = 1,2,3, which means, with an increasing of the number of attributes, the uncertainty of the information is reduced. Therefore, we can use these three formulas to measure the importance of the attributes. Let CI={{i1,i2, ⋯,in}|inin−1, 1 ≤ inp} be a set of index sequence, then we derive the following Theorem 6.

Theorem 6

For any InCI,EΔ2(In) < EΔ1(In) < EΔ3(In).

Proof.

Considering the sine function y = sinx, we can clearly see that when x(0,π2) ,the sine function is a continuous, monotonic increasing with respect to x. Based on the theory of trigonometric function, we can easily infer the fact that if x(0,π2) , then sinx < x < tanx. Let x=i=1in(1Hi)π4 , it is clear to see that 0<x=i=1in(1Hi)π4<π4<π2 . Thus, we obtain the following inequalities:

sin(i=1in(1Hi)π4)<i=1in(1Hi)π4<tan(i=1in(1Hi)π4)

Therefore, we have EΔ2(in) < EΔ1(in) < EΔ3(in). The proof has been completed.

To determine the importance of attributes, Xu 41 proposed a useful weight solving method with incomplete information. A weight information matrix W = (wi j)m×n is established based on a multiobjective optimization model, whose elements wi j is the optimal weight solution corresponding to the alternatives. In what follows, we consider a multiattribute group decision-making problem with attributes interaction information, and give a formulation based on weight information matrix and the three types of joint attributes information entropy we proposed to determine the attributes important measure under interval-valued intuitionistic fuzzy environment.

Definition 12

Let W = (wji)m×n be a weight information matrix, wji indicates the optimal weight for the alternative Aj with respect to the attribute Ci, μ(Ci) indicates the attribute importance measure with respect to the attribute Ci. μ(Ci1,Ci2, ⋯,Cin) indicates the attribute importance measure with respect to the attributes Ci1,Ci2, ⋯,Cin. Then the attributes importance measures are defined as:

μ(Ci)=([wi1λ,wi(1λ)EΔ3(i1)],[(1wi(1λ)EΔ2(i1))2,1wi(1λ)EΔ1(i1)])
μ(Ci1,Ci2,,Cin)=([(i=i1inwi)1λ,(i=i1inwi)(1λ)EΔ3(in)],[(1(i=i1inwi)(1λ)EΔ2(in))2,1(i=i1inwi)(1λ)EΔ1(in)])
where λ indicates an interaction coefficient.

Theorem 7

μ(Ci1,Ci2, ⋯,Cin) satisfies the monotonicity of the interval-valued intuitionistic fuzzy measure.

Proof.

Based on the theory of inequality, since −1 ≤ λ ≤ 1,0 ≤ H ≤ 1, and 0<i=i1inwi1 , so we have 0<i=1n(1Hi)<1 , 0<i=1n(1Hi)π4<π4 , and 0<tan(i=1n(1Hi)π4)<tanπ4=1 . Since i=i1inwi1 and (1λ)>(1λ)tan(i=i1i=in(1Hi)π4) , then

(i=i1inwi)1λ<(i=i1inwi)(1λ)tan(i=1n(1Hi)π4),(1(i=i1inwi)(1λ)sin(i=1n(1Hi)π4))2<1(i=i1inwi)(1λ)sin(i=1n(1Hi)π4)<1(i=i1inwi)(1λ)(i=1n(1Hi)π4)(i=i1inwi)(1λ)tan(i=1n(1Hi)π4)+1(i=i1inwi)(1λ)i=1n(1Hi)π4<(i=i1inwi)(1λ)i=1n(1Hi)π4+1(i=i1inwi)(1λ)i=1n(1Hi)π4=1

Then we prove the monotonicity of the interval-valued intuitionistic fuzzy measure as follows:

  1. (1)

    Let μ1(i)=(i=i1inwi)1λ , then we have μ1(i+1)μ1(i)=(i=i1in+1wi)1λ(i=i1inwi)1λ=eln(i=i1in+1wi)1λeln(i=i1inwi)1λ>0 .

    Thus, μ1(i) is an increasing function with respect to i.

  2. (2)

    Let μ2(i)=(i=i1inwi)(1λ)tan(i=i1in(1Hi)π4) . Then

    lnμ2(i+1)lnμ2(i)=(1λ)tan(i=i1in+1(1Hi)π4)ln(i=i1in+1wi)(1λ)tan(i=i1in(1Hi)π4)ln(i=i1inwi)=(1λ)(tan(i=i1in+1(1Hi)π4)ln(i=i1in+1wi)tan(i=i1in(1Hi)π4)ln(i=i1inwi))=(1λ)(ln(i=i1in+1wi)tan(i=i1in+1(1Hi)π4)ln(i=i1inwi)tan(i=i1in(1Hi)π4))=(1λ)ln(i=i1in+1wi)tan(i=i1in+1(1Hi)π4)(i=i1inwi)tan(i=i1in(1Hi)π4)>0

    From ln μ2(i + 1) − ln μ2(i) > 0 ⇒ μ2(i + 1) > μ2(i). So μ2(i) is an increasing function with respect to i.

  3. (3)

    Let μ3(i)=(1(i=i1inwi)(1λ)sin(i=1n(1Hi)π4))2

    μ3(i+1)μ3(i)=(1(i=i1in+1wi)(1λ)sin(i=i1in+1(1Hi)π4))2(1(i=i1inwi)(1λ)sin(i=i1in(1Hi)π4))2=(2(i=i1in+1wi)(1λ)sin(i=i1in+1(1Hi)π4)(i=i1inwi)(1λ)sin(i=i1in+1(1Hi)π4))×((i=i1inwi)(1λ)sin(i=i1in(1Hi)π4)(i=i1in+1wi)(1λ)sin(i=i1in+1(1Hi)π4))<0

    From μ3(i+1) − μ3(i) < 0, we can obtain μ3(i+1) < μ3(i). Hence, μ3(i) is a decreasing function with respect to i.

  4. (4)

    The proof of (4) is similar to the proofs of (1)-(3), omitted.

Hence, we verify the proposed fuzzy measure satisfies the properties of the interval-valued intuitionistic fuzzy sets and when i1 = i2 = ⋯ = in, Eq. (23) is reduced to Eq. (22). Thus, the proof is complete.

In this section, we give a new method to solve experts importance measure and attributes importance measure under interval-valued intuitionistic fuzzy environment. Since the given approach based on interval-valued intuitionistic fuzzy Choquet integral, which greatly reduces the complexity of determining the fuzzy measures, and improves the practicality of the method.

5. An approach to multi-attribute group decision-making based on interval-valued intuitionistic fuzzy Choquet integral

Interval-valued intuitionistic fuzzy multiple attribute group decision making task is a special case of fuzzy multiple attribute group decision making problems. In the sequel, based on the interval-valued intuitionistic fuzzy entropy and the interval-valued intuitionistic fuzzy measure we derived above, a new approach is proposed to solve multiple attribute group decision making problem under interval-valued intuitionistic fuzzy environment. We propose two methods to determine the fuzzy measure which are used to calculate the interactive importance measure of attributes and its correlation measure of experts (e.g. absolute measure, relative measure). Firstly, we describe the interval-valued intuitionistic fuzzy multiple attribute group decision making (MAGDM) problems in this paper.

5.1. The description of MAGDM problems

For an interval-valued intuitionistic fuzzy multiple attribute group decision making problem. Let A = {A1,A2, ⋯,Am} be a discreet set of alternatives, C = {C1,C2, ⋯,Cn} be a set of attributes, and D = {D1,D2, ⋯,Dp} be a set of experts. The performance of the alternative Ai with respect to the attribute Cj which is provided by expert Dk is expressed by an IVIFN Aij(k)=([aijkL,aijkR],[bijkL,bijkR]) , where [aijkL,aijkR] indicates the interval-valued degrees that the alternative Ai satisfies the attribute Cj, and [bijkL,bijkR] indicates the interval-valued degrees that the alternative Ai does not satisfies the attribute Cj, such that [aijkL,aijkR][0,1] , [bijkL,bijkR][0,1] and 0aijkR+bijkR1 . For all elements Aij(k) are contained in the decision matrix DM(k) which is provided by expert Dk. Based on the analysis presented before, we develop a new method for MAGDM problems under interval-valued intuitionistic fuzzy environment. Based on these necessary conditions, the ranking of alternatives is required.

5.2. Decision making steps

The interval-valued intuitionistic fuzzy Choquet integral method is illustrated in what follows. The algorithm involves the following steps:

  • Step 1

    Construct the individual decision matrix DM(k)(k = 1,2,⋯, p).

  • Step 2

    Calculate the entropy of each decision matrix Ek(k = 1,2,⋯, p).

  • Step 3

    Calculate the importance measure based on the entropy of experts in accordance with decision matrix.

    1. (1)

      Calculate the absolute measure

      With the use of Definition 9 and Eqs. (1719), we obtain the experts absolute importance measure.

    2. (2)

      Calculate the relative measure

      Using Definition 10 and Eq. (20), we determine the experts relative importance measure.

  • Step 4

    Calculate the weights of experts.

    Based on game theory, we get the expert weights with Shapely value 60.

    σk(ν)=(p|k|)!(|k|1)!p!(μ(DS)¯μ(DS{i})¯)
    where σk(v) indicates the expert weight of Dk.

  • Step 5

    Aggregate the decision information of each expert to collect into group decision matrix G by using the IVIFWA operator.

    Utilize the IVIFWA operator to arrange the individual decision matrices into a collective (aggregate) decision matrix as follows:

    G=([gμijL,gμijL],[gνijL,gνijL])m×n

  • Step 6

    Calculate the group decision score matrix and the entropy matrix, respectively.

    With Eqs. (2) and (3), we calculate the score matrix SG and entropy matrix EG respectively, which are defined as follows:

    SG=[s11s12s1ns21s22s2nsm1sm2smn]EG=[E11E12E1nE21E22E2nEm1Em2Emn]

  • Step 7

    Construct the programming model of attribute weights based on the relative entropy measure.

    Based on the definition of relative entropy, and with the purpose to determine the optimal weight vector, we have the basic idea to maximize the score function of each alternative, and minimize the entropy function of each alternative. On the basis of the above analysis, we establish the following linear programming model corresponding to the i-th alternative:

    minWi=j=1nwijln|Eijsij|,i=1,2,ms.t.{j=1nwij=1wij0

    By solving the above sated linear programming model, we produce the solution of the attribute weight vectors W(i)=(w1(i),w2(i),,wn(i)) , where i = 1,2, ⋯,m.

  • Step 8

    Solve the optimal attribute weights vector.

    Based on the definition of Shannon entropy, we can determine the attribute weight vector as follows:

    Hj=1mln2i=1mwij(i)lnwij(i)(j=1,2,,n)
    wj=1Hjnk=1nHk(j=1,2,,n)

  • Step 9

    Calculate the interaction coefficient λ.

    In virtue of Eq. (5), calculate the value of parameter λ by solving the following non-linear multiobjective programming problem:

    According to the principle of optimization, we can construct a single-objective programming model to solve this multiple-objective programming:

    min12[(i=1n(1+λHi)(λ+1))2i=1nHi+i=1n1λ(i=1n[1+λHi]1)ln(1λ(i=1n[1+λHi]1))]s.t.{1λ1λ0

  • Step 10

    Calculate the importance measure of attribute weights.

    Based on Eqs. (2223), form the fuzzy measure of attribute weights.

  • Step 11

    Utilize the Choquet integral to aggregate the interval-valued intuitionistic fuzzy information of decision-making matrix.

    Based on Definition 8 and Eq. (16), the aggregate value of the alternatives is formed.

  • Step 12

    Use the score function of IVIFS to obtain the score of overall alternatives.

    Based on Eq. (2), we form the score value of the alternatives.

  • Step 13

    Utilize the score value to rank the alternatives in the descending order.

    Rank all the alternatives by Eq. (2), the larger the score value is, the better the alternative is.

  • Step 14

    End.

6. Numerical example

In this section, we provide an example to show the application of the proposed method for supplier selection problem.

6.1. The supplier selection problem description

With the continuous development of economic globalization, the supply chain management has played an important role in marketing economic and become the most hot research topic in modern management science, which directly impact on the manufactures performance. Consider a problem in a shipbuilding company, which aims to search for the best supplier for purchasing ship equipments. Three potential ship equipments suppliers have been identified. The attributes to be considered in the evaluation process are: C1: business management capacity; C2: quality; C3: transportation and delivery ability; C4: service ability. The three shipbuilding companies are to be evaluated using the interval-valued intuitionistic fuzzy information by three experts Dk(k = 1,2,3).

6.2. Illustration of the proposed method

  • Step 1

    Construct the individual decision matrix DM(k)(k = 1,2,3), respectively.

    DM(1)=(([0.3,0.4],[0.4,0.6])([0.5,0.6],[0.1,0.2])([0.6,0.8],[0.1,0.2])([0.6,0.7],[0.2,0.3])([0.5,0.8],[0.1,0.2])([0.7,0.8],[0.0,0.2])([0.6,0.7],[0.2,0.3])([0.7,0.8],[0.0,0.1])([0.2,0.3],[0.4,0.6])([0.5,0.6],[0.1,0.3])([0.5,0.5],[0.4,0.5])([0.2,0.3],[0.2,0.4]))DM(2)=(([0.4,0.5],[0.3,0.4])([0.5,0.6],[0.1,0.2])([0.6,0.8],[0.1,0.2])([0.5,0.6],[0.3,0.4])([0.5,0.6],[0.3,0.4])([0.5,0.7],[0.1,0.2])([0.6,0.7],[0.2,0.3])([0.7,0.8],[0.1,0.2])([0.4,0.5],[0.3,0.4])([0.4,0.6],[0.3,0.4])([0.5,0.6],[0.3,0.4])([0.3,0.4],[0.2,0.5]))DM(3)=(([0.4,0.6],[0.3,0.4])([0.5,0.7],[0.0,0.2])([0.5,0.8],[0.1,0.2])([0.3,0.5],[0.2,0.3])([0.5,0.6],[0.0,0.1])([0.5,0.8],[0.1,0.2])([0.5,0.6],[0.2,0.4])([0.6,0.8],[0.1,0.2])([0.3,0.6],[0.2,0.4])([0.4,0.5],[0.2,0.4])([0.4,0.7],[0.2,0.3])([0.2,0.4],[0.2,0.3]))

  • Step 2

    Calculate the entropy of each individual decision matrix Ek(k = 1,2,3), respectively.

    E1=(0.97910.88550.84120.60870.73080.84130.95100.90480.77050.63100.99720.9985)E2=(0.99140.88540.84120.65480.73080.95940.99140.97900.95940.84430.95940.9927)E3=(0.97910.82290.94220.73080.79330.98480.98230.98220.84670.79330.92380.9985)

  • Step 3

    Calculate the importance measure based on the entropy of expert in accordance with the obtained decision matrix.

    Based on Defintion 10 and Eqs.(1113), we calculate the experts importance measure. The obtained results are shown as follows:

    μ(D1)=0.1551,μ(D2)=0.1009,μ(D3)=0.1017
    μ(D1,D2)=1E[0.97910.88540.84120.60870.73080.84130.95100.90480.77050.63100.95940.9927]=10.8413=0.1587

    Similarly, we form other fuzzy measures based on the entropy of interval-valued intuitionistic fuzzy information. The results are calculated as follows:

    μ(D1,D3)=0.1664,μ(D2,D3)=0.1245,μ(D1,D2,D3)=0.1669

    Then we use Eq. (20) to normalize the absolute measure, and obtain the experts importance measures:

    μ(D1)¯=0.9292,μ(D2)¯=0.6045,μ(D3)¯=0.6093,μ(D1,D2)¯=0.9508,μ(D1,D3)¯=0.9970,μ(D2,D3)¯=0.7459,μ(D1,D2,D3)¯=1

  • Step 4

    Calculate the weight of experts.

    According to the results of game theory, we get the expert weight based on Shapely-value. The results read shown as follows:

    σ1(v)=(31)!(11)!3!μ(D1)¯+(32)!(21)!3!(μ(D1,D2)¯μ(D2))¯+(32)!(21)!3!(μ(D1,D3)¯μ(D3))¯+(33)!(31)!3!(μ(D1,D2,D3)¯μ(D2,D3))¯=13×0.9292+16×(0.95080.6045)+16×(0.99700.6093)+13×(10.7459)=0.4451σ2(v)=13×0.6045+16×(0.95080.9292)+16×(0.74590.6093)+13×(10.9970)=0.2288σ3(v)=13×0.6093+16×(0.99700.9292)+16×(0.74590.6045)+13×(10.9508)=0.2543

    Normalize the results of σi(v)(i = 1,2,3), which leads to the weight information corresponding to the three experts:

    wD1=σ1(v)σ1(v)+σ2(v)+σ3(v)=0.4795wD2=σ2(v)σ1(v)+σ2(v)+σ3(v)=0.2465wD3=σ3(v)σ1(v)+σ2(v)+σ3(v)=0.2740

  • Step 5

    Aggregate the decision information of each expert to collect a group decision matrix G by using the IVIFWA operator.

    Utilize the IVIFWA operator, and then structure the decision information in the form of the group decision matrix.

    G=[([0.35,0.48],[0.34,0.48])([0.50,0.63],[0.00,0.20])([0.57,0.80],[0.10,0.20])([0.51,0.63],[0.22,0.32])([0.50,0.71],[0.00,0.19])([0.61,0.77],[0.00,0.20])([0.57,0.68],[0.20,0.32])([0.67,0.80],[0.00,0.14])([0.36,0.44],[0.31,0.48])([0.45,0.57],[0.16,0.35])([0.47,0.59],[0.30,0.41])([0.25,0.35],[0.20,0.39])]

  • Step 6

    Calculate the group decision score matrix and the entropy matrix, respectively.

    Based on the Definitions 3 and 4, the score matrix SG and entropy matrix EG come in the form:

    SG=[0.0050.4650.3650.6650.5350.3000.0050.2550.5100.5900.1750.005]EG=[0.9990.8540.8710.6420.7510.9180.9980.9470.8160.7210.9690.998]

  • Step 7

    Construct the programming model of attribute weights based on relative entropy.

    The information about attribute weights given by the decision makers can be described as follows:

    • ΩD1: 0.1 ≤ w1 ≤ 0.3, w2w1 ≥ 0.15

    • ΩD2: 0.1 ≤ w2 ≤ 0.4, 0.2 ≤ w3 ≤ 0.3

    • ΩD3: w3w4, w4w2w3w1

    Then we establish a linear programming problem:

    minWi=5.29w1+6.07w2+0.87w30.035w4s.t.{w1+w2+w3+w4=1wi0,ΩDk,k=1,2,3

    By solving this problem, we can get the optimal weight vector is W(1) = (0.10,0.25,0.25,0.40). Similarly, we get W(2) = (0.23,0.37,0.20,0.20),W(3) = (0.20,0.40,0.20,0.20). Hence, the optimal weight information matrix becomes:

    W=[0.100.250.250.400.230.370.200.200.200.400.200.20]

  • Step 8

    Solve the optimal attribute weights vector.

    Based on Eq. (28), we derive the value of Hi(i = 1,2,3,4) as follows: H1 = 0.428, H2 = 0.265, H3 = 0.399, H4 = 0.376. Utilize Eq. (29) to calculate the weights of attributes:

    w1=1H14k=13Hk=10.42841.468=0.2259,w2=1H24k=13Hk=10.26541.468=0.2902,w3=1H34k=13Hk=10.39941.468=0.2373,w4=41H43k=13Hk=10.37641.468=0.2466.

  • Step 9

    Calculate the interaction coefficient λ.

    Based on Eq. (5), we calculate the interaction coefficient to be λ = −0.6925.

  • Step 10

    Establish the importance measure of attribute weights.

    Based on Eqs. (2223), we obtain the fuzzy measures directly. The results are shown as follows:

    μ(C1)=([0.0806,0.2800],[0.4318,0.6773]),μ(C2)=([0.1232,0.2261],[0.4463,0.7014]),μ(C3)=([0.0876,0.2696],[0.4367,0.6831]),μ(C4)=([0.0935,0.2619],[0.4394,0.6869]),μ(C1,C2)=([0.3264,0.6813],[0.0927,0.3090]),μ(C1,C3)=([0.2718,0.6973],[0.0861,0.2965]),μ(C1,C4)=([0.2811,0.6940],[0.0877,0.2993]),μ(C2,C3)=([0.3387,0.6761],[0.0948,0.3131]),μ(C2,C4)=([0.3489,0.6726],[0.0961,0.3157]),μ(C3,C4)=([0.2927,0.6889],[0.0900,0.3036]),μ(C1,C2,C3)=([0.6192,0.9081],[0.0127,0.0907]),μ(C1,C2,C4)=([0.6322,0.9086],[0.0126,0.0901]),μ(C1,C3,C4)=([0.7098,0.8143],[0.0714,0.1855]),μ(C2,C3,C4)=([0.6483,0.9091],[0.0124,0.0896]),μ(C1,C2,C3,C4)=([1,1],[0,0]).

  • Step 11

    Utilize Choquet integral to aggregate the interval-valued intuitionistic fuzzy information of decision matrix.

    Take A1 as an example. From Definition 8 and Eq. (16), we have:

    μA1L=0.35×(μ(C)μ(C2,C3,C4))+0.50×(μ(C2,C3,C4)μ(C3,C4))+0.57×(μ(C3,C4)μ(C4))+0.67×(μ(C4)μ(ϕ))=0.35×(10.6483)+0.50×(0.64830.2927)+0.57×(0.29270.0935)+0.67×(0.09350)=0.4771μA1R=0.48×(μ(C)μ(C2,C3,C4))+0.63×(μ(C2,C3,C4)μ(C3,C4))+0.68×(μ(C3,C4)μ(C4))+0.80×(μ(C4)μ(ϕ))=0.48×(10.9091)+0.63×(0.90910.6889)+0.68×(0.68890.2619)+0.80×(0.26190)=0.6843νA1L=0.34×(μ(C1,C2,C4)μ(C))+0.2×(μ(C2,C4)μ(C1,C2,C4))+0×(μ(C4)μ(C2,C4))+0×(μ(ϕ)μ(C4))=0.34×(0.01260)+0.2×(0.09610.0126)+0×(0.43940.0961)+0×(10.4394)=0.0210νA1R=0.48×(μ(C2,C3,C4)μ(C))+0.32×(μ(C2,C4)μ(C2,C3,C4))+0.2×(μ(C4)μ(C2,C4))+0.14×(μ(ϕ)μ(C4))=0.48×(0.08960)+0.32×(0.31570.0896)+0.2×(0.68690.3157)+0.14×(10.6869)=0.2334

    So the aggregation value of A1 based on interval-valued intuitionistic fuzzy Choquet integral is:

    (c)A1=([0.4771,0.6843],[0.0210,0.2334])

    Similarly, we can calculate the aggregation values of A2 and A3, respectively:

    (c)A2=([0.4738,0.6607],[0.2652,0.3055])(c)A3=([0.4631,0.6524],[0.2266,0.3287])

  • Step 12

    Utilize the score function to get the score value of the overall alternatives.

    Based on Eq. (2), we can calculate the score value of overall alternatives, respectively:

    S(A1)=0.4518,S(A2)=0.2819,S(A3)=0.2801

  • Step 13

    Utilize the score value to rank the alternatives.

    Since S(A1) > S(A2) > S(A3), we have A1A2A3. Thus, we choose A1 as the best alternative.

6.3. Comparisons and further discussion

In what follows, we compare our method with other existing methods including interval-valued intuitionistic fuzzy group decision making with Choquet integral-based TOPSIS 42, and the interval-valued intuitionistic fuzzy Choquet integral with respect to the generalized lambda-Shapley index method 50. The results are shown in Table 1.

Methods Order of alternatives
The proposed method A1A2A3
Choquet integral-based TOPSIS A1A2A3
lambda-Shapley index method A1A2A3
Table 1.

Comparasions with Choquet integral- TOPSIS and generalized lambda-Shapley index methods.

From Table 1, it is clear to see that three methods have the same ranking results. This verifies the method we proposed is reasonable and validity in this study.

  1. (1)

    Compared with the Choquet integral-based TOPSIS which was proposed by Tan 42. The main advantage of our method is that we develop a new method to determine the fuzzy measures directly based on decision information, while in Tan’s method, the fuzzy measures need provided by the decision maker which maybe difficult to the DMs and is closely related the subjective preferences of the DMs, or even lead to inconsistent results. In addition, our method makes full use of weights information to characterize the interrelationships among the attributes, which ensures the ranking results are more objective and accurate.

  2. (2)

    Compared with the generalized lambda-Shapley index method which was proposed by Meng et al. 50. The computational complexity of our method is much lower than Mengs method. The reason is that our method is a constructive method, which only needs to determine a single parameter by solving a simple linear programming model, and then the fuzzy measures can be easily obtained, whereas Meng’s method needs to solve complex programming model, which leads to high computational complexity and information loss. Moreover, our method can solve the MAGDM with incomplete known information, while generalized lambda-Shapley index method does not deal with this problem. Therefore, our method is more general.

According to the comparisons and analysis above, the method we proposed is better than the other two methods. Therefore, our method is more suitable to solve interval-valued intuitionistic fuzzy multiple attribute group decision making.

7. Conclusions

In this study, we have presented a new approach to determine the fuzzy measure of Choquet integral on the basis of interval-valued intuitionistic fuzzy decision matrix. Considering the interval-valued intuitionistic fuzzy entropy theory, we give a simple solution method to determine the interval-valued intuitionistic fuzzy measures and present a method to obtain the experts weights and attributes weights with fuzzy measures directly. The strict mathematical reasoning shows that our new proposed method satisfies the axioms of interval-valued intuitionistic fuzzy measures. Then, the experts weights and attributes weights are aggregated by using the Shapely values. The prominent advantage of such fuzzy measure method is that it can ease the burden of the decision maker, avoid the information losing and distortion and improve the accuracy of the decision making results. Based on our theoretical analysis results, we develop a new Choquet integral-based approach to solve the interval-valued intuitionistic fuzzy multiple attribute decision making problems. The fuzzy measures to determine the attribute weights and experts weights are generated by relative entropy models. A solution process is proposed to solve the intervalCvalued intuitionistic fuzzy multiple attribute group decision making problems. Our new method is simple prerequisite conditions and can be easily integrated into other multi-attribute group decision-making techniques, which are effective expansions on the current interval-valued intuitionistic fuzzy decision making models and solution methods. Furthermore, our new methods can also be applied to the similar problems in the forms of linguistic variables hesitate fuzzy sets and type-2 fuzzy sets. These will be considered in the future research.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (NSFC) under Projects 71171048 and 71371049, Ph.D. Program Foundation of Chinese Ministry of Education 20120092110038, the Scientific Research and Innovation Project for College Graduates of Jiangsu Province CXZZ13_0138, and the Scientific Research Foundation of Graduate School of Southeast University YBJJ1454.

References

37.M Sugeno, Theory of fuzzy integrals and its applications, Tokyo Institute of Technology, 1974.
60.Z Wang and G Klir, Fuzzy measure theory, Springer Science and Business Media, 2013.
Journal
International Journal of Computational Intelligence Systems
Volume-Issue
9 - 1
Pages
133 - 152
Publication Date
2016/01/01
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.1080/18756891.2016.1146530How to use a DOI?
Copyright
© 2016. the authors. Co-published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Jindong Qin
AU  - Xinwang Liu
AU  - Witold Pedrycz
PY  - 2016
DA  - 2016/01/01
TI  - Multi-attribute group decision making based on Choquet integral under interval-valued intuitionistic fuzzy environment
JO  - International Journal of Computational Intelligence Systems
SP  - 133
EP  - 152
VL  - 9
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.1080/18756891.2016.1146530
DO  - 10.1080/18756891.2016.1146530
ID  - Qin2016
ER  -