Linguistic hesitant intuitionistic fuzzy cross-entropy measures

Wei Yang; Yongfeng Pang; Jiarong Shi

doi:10.2991/ijcis.2017.10.1.9

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Volume 10, Issue 1, 2017, Pages 120 - 139

Linguistic hesitant intuitionistic fuzzy cross-entropy measures

Authors

Wei Yang^*, Yongfeng Pang, Jiarong Shi

^*Corresponding author. E-mail: yangweipyf@163.com.

Corresponding Author

Wei Yang

Received 19 November 2015, Accepted 10 September 2016, Available Online 1 January 2017.

DOI: 10.2991/ijcis.2017.10.1.9 How to use a DOI?
Keywords: Hesitant fuzzy set; intuitionisic fuzzy set; linguistic argument; aggregation operator; linguistic hesitant intuitionistic fuzzy cross-entropy
Abstract: In this paper, several cross-entropy measures for linguistic hesitant intuitionistic fuzzy information have been developed which integrating cross-entropy measures of intuitionistic fuzzy sets and hesitant fuzzy linguistic sets. Some desirable properties of new cross-entropy measures have been studied. Two new multiple attribute decision making methods have been presented based on the new cross-entropy measures in which attribute values are given in the form of linguistic hesitant intuitionistic fuzzy values to reflect human hesitantation and fuzzy thinking comprehensively. We consider different attribute weight situations including attribute weights are completely known, partly known and completely unknown. An optimization model is established to determine attribute weights if they are partly known and a formula is given if attribute weights are completely unknown. Finally, a numerical example is presented to illustrate practical advantages and effectiveness of the proposed approaches.
Copyright: © 2017, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Fuzziness and uncertainty exists extensively in decision making process due to complicated decision problems, limited decision time and fuzzy nature of human thinking, etc. Many useful tools have been developed including fuzzy set, intuitionistic fuzzy set¹, hesitant fuzzy set^2–3, linguistic arguments^4–8, etc. In fuzzy set, the membership of each element is a real number between 0 and 1. Intuitionistic fuzzy set is the extension of fuzzy set, which is characterized by membership and non-membership summing less than 1. Hesitant fuzzy set is another extension of fuzzy set, in which several values are possible for the definition of a membership function of a fuzzy set. The envelope of hesitant fuzzy set is intuitionistic fuzzy set. Comparing with other tools to model fuzzy and uncertain information, hesitant fuzzy set is more powerful and accurate. The hesitant fuzzy set has been studied and applied extensively^9–14. Some hesitant aggregation operators have been proposed^15–17. Some classic multiple attribute decision making methods have been extended to hesitant fuzzy environment^18–20. Some distance measures, entropy measures and correlation coefficients have been generalized to accommodate hesitant fuzzy information^21–25. Hesitant fuzzy set has been extended to accommodate intuitionistic fuzzy values²⁶, interval-values²⁷, triangular fuzzy values²⁸, linguistic arguments^29–35, etc. Due to fuzzy nature of human thinking, complicated decision problems and limited decision time, decision makers would like to evaluate with linguistic terms rather than exact numerical values. Several types of linguistic models have been developed. Herrera and Martínez⁴ proposed 2-tuple linguistic model to avoid information distortion and loss. Dong et al.⁵ defined numerical scale and extended the 2-tuple fuzzy linguistic representation models under numerical scale. By using 2-tuple linguistic model, each evaluation value only has one linguistic evaluation value. Rodríguez et al.³² developed hesitant linguistic fuzzy set in which each element has several linguistic terms. Wang³⁵ generalized hesitant fuzzy linguistic term sets by enabling any non-consecutive linguistic terms in them. Pang et al.⁶ developed probabilistic linguistic term set in which possible linguistic values may have different importance degrees. The fuzzy linguistic modelling based on discrete fuzzy numbers is proposed by Riera et al.⁷ to manage hesitant fuzzy linguistic information. Comparing with fuzzy numbers, intuitionistic fuzzy values are more accurately to model hesitation. The concept of a possibility distribution for hesitant fuzzy linguistic information has been defined¹¹. But hesitation in decision making hasn’t been modeled properly by existing linguistic models. In decision making process, experts would express some hesitation in evaluating with linguistic terms. Intuitionistic fuzzy values can be used to model hesitation accurately. By using intuitionistic fuzzy values to model hesitation in linguistic evaluating process, fuzzy nature of human thinking can be reflected accurately and hesitation can be modeled properly. If an expert uses linguistic term s_α in evaluating some alternative with respect to some attribute and he/she thinks the membership of alternative satisfying the attribute is μ and non-membership is ν, then linguistic intuitionistic fuzzy element (LIFE) (s_α, (μ,ν)) can be got. If two experts use the same linguistic term and different intuitionistic fuzzy values, intuitionistic fuzzy values are merged together. For example, in evaluating performance of a candidate for a college dean, one expert thinks the degree of ‘slightly good (S₆)’ the candidate belonging to is (0.6,0.3) and degree of ‘good (S₇)’ is (0.7,0.2). Another expert thinks the degree of ‘fair (S₅)’ the candidate belonging to is (0.5,0.4), the degree of ‘slightly good’ is (0.7,0.1) and the degree of ‘good’ is (0.6,0.2). Then we can get linguistic hesitant intuitionistic fuzzy information as ȟ = {(S₅,(0.5,0.4)), (S₆,(0.6,0.3), (0.7,0.1)), (S₇,(0.6,0.2), (0.7, 0.2))}. The linguistic hesitant intuitionistic fuzzy set is developed by Yang et al.³⁶. Comparing with other tools, linguistic hesitant intuitionisic fuzzy values can model fuzzy and uncertain information existing in decision making process more accurate, which is the prerequisite to get scientific and reasonable decision-making results.

A growing number of studies focus on entropy measures and cross-entropy measures due to advantages of measuring fuzziness and discrimination information^37–49. Kullback and Leibler³⁹ defined a cross-entropy measure between two probability distribution. The fuzzy cross-entropy has been defined by Bhandari and Pal⁴⁰ by using its membership function. Zhang and Jiang⁴¹ developed vague cross-entropy by analogy with the cross-entropy of probability distributions. Chen et al.⁴² developed several cross-entropy measures for uncertain variables. Mao et al.⁴³ proposed a novel symmetric intuitionistic fuzzy cross-entropy formula taking into account intuitionistic fuzzy entropy and fuzzy entropy simultaneously. Xia and Xu⁴⁴ defined two cross-entropy measures for intuitionistic fuzzy values by normalizing the J-divergence intuitionistic fuzzy values introduced by Hung and Yang⁴⁵. Wang and Li⁴⁶ proposed a cross-entropy measure of the membership degree from the non-membership degree for intuitionistic fuzzy values. Qi et al.⁴⁷ defined a new generalized interval-valued intuitionistic fuzzy cross-entropy measure and gave a new method to determine unknown attribute weights and expert weights based on the new cross-entropy measure. Xu and Xia⁴⁸ defined two cross-entropy measures for hesitant fuzzy information. Peng et al.²⁶ have developed some fuzzy cross-entropy measures for intuitionistic hesitant fuzzy information. The cross-entropy methods have been used extensively, such as traffic signal optimization, portfolio selection, clustering, energy management, etc.

From the above analysis we can find that all the existing cross-entropy measures are based on exact numerical values, fuzzy values, intuitionistic fuzzy values, hesitant fuzzy values. Linguistic hesitant intuitionistic fuzzy values are more accurate and flexible in modeling fuzzy and uncertain information. However, the study on cross-entropy measures for linguistic hesitant intuitionistic fuzzy information hasn’t been found yet. Due to the fact hesitation is common existing in actual decision making process, it is necessary to develop some cross entropy measures for linguistic hesitant intuitionistic fuzzy information. The aim of this paper is to propose several linguistic hesitant intuitionistic fuzzy cross-entropy measures by extending intuitionistic fuzzy cross-entropy measures and hesitant fuzzy cross-entropy measures to linguistic hesitant intuitionistic fuzzy environment, which can comprehensively accommodate membership, nonmembership and hesitation degree in linguistic evaluation process. Based on the new cross-entropy measures, a programming model is set up to derive unknown attribute weights by considering deviation between attribute assessment values and a formula is given to derive attribute weights if they are completely unknown. Then we develop two algorithms based on the new cross-entropy measures integrating the afore presented methods considering different situations of attribute weights. A numerical example of supplier selection problem is presented to illustrate the new algorithms. Additionally, it is important to note that the decision making methods proposed in this paper can also be used to solve other decision making problems with high uncertainty and hesitation degrees.

In order to do so, the rest of the paper is organized as follows. In section 2, we first review some basic concepts on linguistic hesitant intuitionistic fuzzy set. Then we define several linguistic hesitant intuitionistic fuzzy aggregation operators. In section 3, several cross-entropy measures for linguistic hesitant intuitionistic fuzzy information have been developed and some desirable properties have been studied. In section 4, we propose two new multiple attribute decision making methods based on the new cross-entropy measures. In section 5, an example of supplier selection is given to illustrate feasibility and practical advantages of new methods. The conclusions are given in the last section.

2. Linguistic hesitant intuitionistic fuzzy term set

An HFS is defined in terms of a function that returns a set of membership values of each element in the domain.

Definition 2.1².

Let X be a reference set, an HFS A on X is a function h that returns a subset of values in [0,1] when it is applied to X:

A={<x,hA(x)>|x∈X},

where h_A(x) is a set of some different values in [0,1], representing the possible membership degrees of x ∈ X to A. h_A(x) is called a hesitant fuzzy element (HFE)¹⁴.

Suppose that S = {s_i| i = 1, …, g} is a finite and totally ordered discrete term set, where s_i represents a possible value for a linguistic variable. A set of nine terms S⁵⁰ can be expressed as follows S={s₁ = extremely poor, s₂ = very poor, s₃ = poor, s₄ = slightly poor, s₅ = fair, s₆ = slightly good, s₇ = good, s₈ = very good, s₉ = extremely good}. In order to preserve all information, the discrete linguistic term set S can be extended to a continuous one S¯={sα|s0≤sα≤sg,α∈[0,g]} .

By extending hesitant fuzzy set, Zhang andWu³¹ develop hesitant fuzzy linguistic term set (HFLS), in which a linguistic variable has several linguistic terms.

Definition 2.2.

Let X be a reference set and S¯={sα|s0≤sα≤sg} be a linguistic term set. A hesitant fuzzy linguistic term set A¯ on X is an ordered finite subset of the consecutive linguistic term set S¯

A¯={<xi,h¯A¯(xi)>|xi∈X,i=1,2,…,n},

where h¯A¯(xi):X→S¯ denotes all the possible linguistic evaluation values of element x_i ∈ X. For convenience, we call h¯A¯(xi) a hesitant fuzzy linguistic element (HFLE), which can be represented as

h¯A¯(xi)={si|si∈h¯A¯(xi)},

here s_i is a linguistic argument.

Since experts would express some hesitation in evaluation using linguistic terms, we use intuitionistic fuzzy value to model hesitation. If multiple experts evaluate alternatives with respect to attributes using different linguistic terms and different intuitionistic fuzzy values, we get linguistic hesitant intuitionistic fuzzy set, which can be defined as follows.

Definition 2.3³⁶.

Let X ={x₁, x₂, …, x_n} be a reference set and S¯={sα|s0≤sα≤sg} be a linguistic term set. A linguistic hesitant intuitionistic fuzzy set (LHIFS) Ǎ on X is defined as

Aˇ={<xi,hˇAˇ(xi)>|xi∈X,i=1,2,…,n},

where ȟ_Ǎ (x_i) : X → H denotes all possible linguistic intuitionistic fuzzy evaluation values of element x_i ∈ X. For convenience, we call ȟ_Ǎ (x_i) a linguistic hesitant intuitionistic fuzzy element (LHIFE), which can be represented as

hˇAˇ(xi)={(sθi,lh(sθi))|xi∈X},

s_{θ_i} is a linguistic argument and lh(sθi)={(μi(k),νi(k))} is the set of intuitionistic fuzzy membership values that s_{θ_i} satisfies x_i. (s_{θ_i}, lh(s_{θ_i})) is the linguistic intuitionistic fuzzy element (LIFE). Let H be the set of all LHIFEs.

Definition 2.4³⁶.

Let ȟ, ȟ₁ and ȟ₂ be LHIFEs, λ > 0. a_k = (s_{θ_k}, lh(s_{θ_k})) ∈ ȟ, a_i = (s_{θ_i}, lh(s_{θ_i})) ∈ ȟ₁, a_j = (s_{θ_j}, lh(s_{θ_j})) ∈ ȟ₂. Some operations on these LHIFEs can be defined as follows

(1)
hˇ1⊕hˇ2=∪ai∈hˇ1,aj∈hˇ2{(sθi+θj,∪(μi(l),νi(l))∈lh(sθi),(μj(m),νj(m))∈lh(sθj){(μi(l)+μj(m)−μi(l)μj(m),νi(l)νj(m))})} ,
(2)
hˇ1⊕hˇ2=∪ai∈hˇ1,aj∈hˇ2{(sθiθj,∪(μi(l),νi(l))∈lh(sθi),(μj(m),νj(m))∈lh(sθj){(μi(l)μj(m),νi(l)+νj(m)−νi(l)νj(m))})} ,
(3)
λhˇ=∪ak∈hˇ{(sλθk,∪(μk(n),νk(n))∈lh(sθk){(1−(1−μk(n))λ,(νk(n))λ)})} ,
(4)
(hˇ)λ=∪ak∈hˇ{(sθkλ,∪(μk(n),νk(n))∈lh(sθk){((μk(n))λ,1−(1−νk(n))λ)})} .

Theorem 1³⁶.

Let ȟ, ȟ₁ and ȟ₂ be LHIFEs and λ, λ₁, λ₂ > 0, then

(1)
ȟ₁ ⊕ ȟ₂ = ȟ₂ ⊕ ȟ₁,
(2)
ȟ₁ ⊗ ȟ₂ = ȟ₂ ⊗ ȟ₁,
(3)
λ (ȟ₁ ⊕ ȟ₂) = λȟ₁ ⊕ λȟ₂,
(4)
(ȟ₁ ⊗ ȟ₂)^λ = (ȟ₁)^λ ⊗ (ȟ₂)^λ,
(5)
(λ₁ + λ₂)ȟ = λ₁ȟ ⊕ λ₂ȟ,
(6)
hˇλ1+λ2=hˇλ1⊗hˇλ2 .

Definition 2.5³⁶.

Let a_i = (s_{θ_i}, lh(s_{θ_i})) be a LIFE, then the score function s(a_i) of a_i can be defined as

s(ai)=θig|lh(sθi)|∑(μi(k),νi(k))∈lh(sθi)(μi(k)−νi(k)),

and the accuracy function h(a_i) of a_i can be defined as

h(ai)=θig|lh(sθi)|∑(μi(k),νi(k))∈lh(sθi)(μi(k)+νi(k)),

where g is the number of linguistic arguments in linguistic term set S and | lh(s_{θ_i}) | is the number of intuitionistic fuzzy memberships in lh(s_{θ_i}).

Based on the score function s(a_i) and the accuracy function h(a_i), we can rank LIFEs as follows. Let a_i = (s_{θ_i}, lh(s_{θ_i})) and a_j = (s_{θ_j}, lh(s_{θ_j})) be two LIFEs, then

(1)
If s(a_i) < s(a_j), then a_i < a_j,
(2)
If s(a_i) = s(a_j) and h(a_i) < h(a_j), then a_i < a_j, else if s(a_i) = s(a_j) and h(a_i) = h(a_j), then a_i ∼ a_j.

Definition 2.6³⁶.

Let ȟ = {(s_{θ_i}, lh(s_{θ_i}))} be a LHIFE, the score function S(ȟ) can be defined as

S(hˇ)=1|hˇ|(∑θig|lh(sθi)|∑(μi(k),νi(k))∈lh(sθi)(μi(k)−νi(k))),

and the accuracy function A(ȟ) can be defined as

A(hˇ)=1|hˇ|(∑θig|lh(sθi)|∑(μi(k),νi(k))∈lh(sθi)(μi(k)+νi(k))),

where | ȟ | is the number of LIFEs in ȟ and g is the number of linguistic terms in linguistic term set S, | lh(s_{θ_i}) | is the number of intuitionistic fuzzy memberships in lh(s_{θ_i}).

Based on the score function and accuracy function, we present the following method to compare LHIFEs. Let ȟ₁ and ȟ₂ be two LHIFEs,

(1)
If S(ȟ₁) < S(ȟ₂), then ȟ₁ < ȟ₂;
(2)
If S(ȟ₁) = S(ȟ₂) and
1. (I)
  A(ȟ₁) < A(ȟ₂), then ȟ₁ < ȟ₂,
2. (II)
  A(ȟ₁) = A(ȟ₂), then ȟ₁ ∼ ȟ₂.

Definition 2.7.

Let ȟ_j (j = 1,2, …, n) be a collection of LHIFEs, w = (w₁, w₂, …, w_n) be the weight vector of ȟ_j (j = 1, 2, …, n) with w_j ≥ 0 (j = 1, 2, …, n) and ∑j=1nwj=1 . The linguistic hesitant intuitionistic fuzzy weighted averaging (LHIFWA) operator is a mapping LHIFWA: Hⁿ → H, which can be defined as follows:

(1)LHIFWAw(hˇ1,hˇ2,…,hˇn)=w1hˇ1⊕w2hˇ2⊕…⊕wnhˇn.

Theorem 2.

Let ȟ_j (j = 1,2, …, n) be a collection of LHIFEs, w = (w₁, w₂, …, w_n) be the weight vector of ȟ_j (j = 1,2, …, n) with w_j ≥ 0 (j = 1,2, …, n) and ∑j=1nwj=1 . The aggregated result of the LHIFWA operator is also a LHIFE, and

(2)LHIFWAw(hˇ1,hˇ2,…,hˇn)=∪ai∈hˇi{(s(∑j=1nwjθj),∪(μj(k),νj(k))∈lh(sθj){(1−∏j=1n(1−μj(k))wj,∏j=1n(νj(k))wj)})},

where a_i ={(s_{θ_i}, lh(s_{θ_i}))}, (μj(k),νj(k))∈lh(sθj) , k = 1, 2, …, l_j, l_j is the number of intuitionistic fuzzy memberships in lh(s_{θ_j}).

Definition 2.8.

Let ȟ_j (j = 1, 2, …, n) be a collection of LHIFEs, w = (w₁, w₂, …, w_n) be the weight vector of ȟ_j (j = 1,2, …, n) with w_j ≥ 0 (j = 1,2, …, n) and ∑j=1nwj=1 . The linguistic hesitant intuitionistic fuzzy weighted geometric (LHIFWG) operator is a mapping LHIFWG: Hⁿ → H, which can be defined as follows:

(3)LHIFWGw(hˇ1,hˇ2,…,hˇn)=hˇ1w1⊗hˇ2w2⊗…⊗hˇnwn.

Theorem 3.

Let ȟ_j (j = 1, 2, …, n) be a collection of LHIFEs, w = (w₁, w₂, …, w_n) be the weight vector of ȟ_j (j = 1, 2, …, n) with w_j ≥ 0 (j = 1,2, …, n) and ∑j=1nwj=1 . The aggregated result of the LHIFWG operator is also a LHIFE, and

(4)LHIFWGw(hˇ1,hˇ2,…,hˇn)=∪ai∈hˇi{(s(∏j=1n(θj)wj),∪(μj(k),νj(k))∈lh(sθj){(∏j=1n(μj(k))wj,1−∏j=1n(1−νj(k))wj)})},

where a_i ={(s_{θ_i}, lh(s_{θ_i}))}, (μj(k),νj(k))∈lh(sθj) , k = 1, 2, …, l_j, l_j is the number of intuitionistic fuzzy memberships in lh(s_{θ_j}).

Definition 2.9.

Let ȟ_j (j = 1,2, …, n) be a collection of LHIFEs, w = (w₁, w₂, …, w_n) be the weight vector of ȟ_j (j = 1,2, …, n) with w_j ≥ 0 (j = 1, 2, …, n) and ∑j=1nwj=1 , λ > 0. The generalized linguistic hesitant intuitionistic fuzzy weighted averaging (GLHIFWA) operator is a mapping GLHIFWA: Hⁿ → H, which can be defined as follows:

(5)GLHIFWAw,λ(hˇ1,hˇ2,…,hˇn)=(w1hˇ1λ⊕w2hˇ2λ⊕…⊕wnhˇnλ)1/λ.

Theorem 4.

Let ȟ_j (j = 1,2, …, n) be a collection of LHIFEs, w = (w₁, w₂, …, w_n) be the weight vector of ȟ_j (j = 1,2, …, n) with w_j ≥ 0 (j = 1,2, …, n) and ∑j=1nwj=1 . The aggregated result of the GLHIFWA operator is also a LHIFE, and

(6)GLHIFWAw,λ(hˇ1,hˇ2,…,hˇn)=∪ai∈hˇi{(s(∑j=1nwj(θj)λ)1/λ,∪(μj(k),νj(k))∈lh(sθj){((1−∏j=1n(1−(μj(k))λ)wj)1/λ,1−(1−∏j=1n(1−(1−νj(k))λ)wj)1/λ)})}.

where a_i = {(s_{θ_i}, lh(s_{θ_i})) ∈ ȟ_i}, (μj(k),νj(k))∈lh(sθj) , k = 1,2, …, l_j and l_j is the number of intuitionistic fuzzy memberships in lh(s_{θ_j}), λ > 0.

3. Cross-entropy measures for LHIFSs

Let P = {p₁, p₂, …, p_n} and Q = {q₁,q₂, …, q_n} be two probability distribution. In order to measure the divergence between P and Q, Kullback and Leibler³⁹ defined the cross-entropy measure as

(7)CE1(P,Q)=∑i=1npilnpiqi.

If n = 2, P = {p,1 − p},Q = {q,1 − q}, then CE1(P,Q)=plnpq+(1−p)ln1−p1−q .

Bhandari and Pal⁴⁰ generalized the cross-entropy measure based on probability distribution to accommodate fuzzy information. Let A and B be two fuzzy sets in the finite universe X = {x₁,x₂, …, x_n}, then the cross-entropy measure for fuzzy values can be defined as

(8)CE2(A,B)=∑i=1n(μA(xi)lnμA(xi)μB(xi)+ (1−μA(xi))ln1−μA(xi)1−μB(xi)).

Vlachos and Sergiadis⁴⁹ developed a cross-entropy measure for intuitionistic fuzzy information by extending fuzzy cross-entropy measure to intuitionistic fuzzy environment. Let A′ and B′ be two intuitionistic fuzzy sets in the finite universe X = {x₁, x₂, …, x_n}, then the cross-entropy measure for intuitionsitic fuzzy values can be defined as

(9)CE3(A′,B′)=∑i=1n(μA(xi)lnμA(xi)μB(xi)+νA(xi)lnνA(xi)νB(xi)).

If μ_B(x_i) = 0, μ_A(x_i) ≠ 0 or ν_B(x_i) = 0,ν_A(x_i) ≠ 0, CE₃(A′,B′) is undefined. Vlachos and Sergiadis further gave the following cross-entropy

(10)CE4(A′,B′)=∑i=1n(μA(xi)lnμA(xi)12(μA(xi)+μB(xi))+ νA(xi)lnνA(xi)12(νA(xi)+νB(xi))).

Uncertainty of intuitionistic fuzzy values is decomposed into intuitionism and fuzziness. Intuitionism is determined by hesitancy degree π_A(x_i) = 1 − μ_A(x_i) − ν_A(x_i) and fuzziness is determined by the closeness of membership μ_A(x_i) and non-membership ν_A(x_i) as Δ_A (x_i) = |μ_A(x_i) − ν_A(x_i)|. Mao et al.⁴³ presented the cross-entropy measure for intuitionistic fuzzy information by considering intuitionism and fuzziness simultaneously as follows

(11)CE5(A′,B′)=∑i=1n(πA(xi)lnπA(xi)12(πA(xi)+πB(xi))+ ΔA(xi)lnΔA(xi)12(ΔA(xi)+ΔB(xi))).

Xu and Xia⁴⁸ proposed some cross-entropy formulas for hesitant fuzzy information by using two concave-up functions f (x) = (1 + qx)ln(1 + qx) and g(x) = x^p. Let Ã = {ᾶ₁, ᾶ₂, …, ᾶ_n} and B˜={β˜1,β˜2,…,β˜n} be two hesitant fuzzy sets in the finite universe X = {x₁, x₂, …, x_n}. The number of elements in all ᾶ_i (i = 1, 2, …, n) and β˜i(i=1,2,…,n) is the same. |ᾶ_i| is the number of elements in ᾶ_i. ᾶ_σ(i) is the ith largest values in ᾶ_i. Then the cross-entropy measures can be defined as follows in Eq. (12) and Eq.(13). Here l = |ᾶ_i|, T = (1 + q) ln(1 + q) − (2 + q) (ln(2 + q) − ln 2), q > 0.

(12)CE5(A˜,B˜)=1lT∑i=1n∑j=1l((1+qα˜σ(j)(xi))ln(1+qα˜σ(j)(xi))+(1+qβ˜σ(j)(xi))ln(1+qβ˜σ(j)(xi))2−(2+qα˜σ(j)(xi)+qβ˜σ(j)(xi))2ln(2+qα˜σ(j)(xi)+qβ˜σ(j)(xi))2+(1+q(1−α˜σ(l−i+1)(xi))ln(1+q(1−α˜σ(l−j+1)(xi))+(1+q(1−β˜2σ(l−j+1)(xi))ln(1+q(1−β˜σ(l−j+1)(xi))2−2+q(1−α˜σ(l−i+1)(xi))+1−β˜σ(l−i+1)(xi))2ln2+q(1−α˜σ(l−i+1)(xi))+1−β˜σ(l−i+1)(xi))2).

(13)CE6(A˜,B˜)=1l(1−21−p)∑i=1n∑j=1l(α˜σ(j)p(xi)+β˜σ(j)p(xi)2+(1−α˜σ(l−j+1)(xi))p+(1−β˜σ(l−j+1)(xi))p2−(α˜σ(j)(xi)+β˜σ(j)(xi)2)p+(1−α˜σ(l−j+1)(xi)+1−β˜σ(l−j+1)(xi))p2),l=|α˜i|.

Peng et al.²⁶ developed several cross-entropy measures for intuitionistic hesitant fuzzy numbers (IHFNs) by extending cross-entropy measures for hesitant fuzzy elements introduced by Xu and Xia⁴⁸. Let ᾶ_j =< Γ_{ᾶ_j}, Ψ_{ᾶ_j} > (j = 1, 2) be IHFNs, Γ_{ᾶ_j} and Ψ_{ᾶ_j} denote the possible degrees of membership and non-membership, respectively. Π_{ᾶ_j} denotes the possible degrees of hesitation. Here T = (1 + q) ln(1 + q) − (2 + q) (ln(2 + q) − ln 2), q > 0.

(14)CE7(α˜1,α˜2)=1T(maxμα˜1∈Γα˜1{minμα˜2∈Γα˜2{(1+qμα˜1)ln(1+qμα˜1)+(1+qμα˜2)ln(1+qμα˜2)2−2+qμα˜1+qμα˜22ln(2+qμα˜1+qμα˜22)}}+maxνα˜1∈Ψα˜1{minνα˜2∈Ψα˜2{(1+qνα˜1)ln(1+qνα˜1)+(1+qνα˜2)ln(1+qνα˜2)2−2+qνα˜1+qνα˜22ln(2+qνα˜1+qνα˜22)}}+maxπα˜1∈Πα˜1{minπα˜2∈Πα˜2{(1+qπα˜1)ln(1+qπα˜1)+(1+qπα˜2)ln(1+qπα˜2)2−2+qπα˜1+qπα˜22ln(2+qπα˜1+qπα˜22)}}),p≥1.

(15)CE8(α˜1,α˜2)=(1|Γα˜1|∑μα˜1∈Γα˜11T({minμα˜2∈Γα˜2{(1+qμα˜1)ln(1+qμα˜1)+(1+qμα˜2)ln(1+qμα˜2)2−2+qμα˜1+qμα˜22ln(2+qμα˜1+qμα˜22)})p)1/p+(1|Ψα˜1|∑να˜1∈Ψα˜11T({minνα˜2∈Ψα˜2{(1+qνα˜1)ln(1+qνα˜1)+(1+qνα˜2)ln(1+qνα˜2)2−2+qνα˜1+qνα˜22ln(2+qνα˜1+qνα˜22)})p)1/p(1|Πα˜1|∑πα˜1∈Πα˜11T({minπα˜2∈Πα˜2{(1+qπα˜1)ln(1+qπα˜1)+(1+qπα˜2)ln(1+qπα˜2)2−2+qπα˜1+qπα˜22ln(2+qπα˜1+qπα˜22)})p)1/p,p≥1.

In the following, we propose the axiomatic definition of cross-entropy measure for linguistic hesitant intuitionistic fuzzy information as follows, motivated by Xu and Xia⁴⁸, Peng et al.²⁶, etc.

Definition 3.1.

Let ȟ₁, ȟ₂ ∈ H, CE′ : H × H → R⁺, then cross-entropy CE′(ȟ₁, ȟ₂) of ȟ₁ and ȟ₂ should satisfy the following conditions:

(1)
CE′(ȟ₁, ȟ₂) ≥ 0, ∀ ȟ₁, ȟ₂ ∈ H,
(2)
CE′(ȟ₁, ȟ₂) = 0, if ȟ₁ = ȟ₂,
(3)
CE′(hˇ1c,hˇ2c)=CE′(hˇ1,hˇ2) , ∀ ȟ₁, ȟ₂ ∈ H.

Here hˇic={(s(g−θi), lh(sθi)c)} , lh(sθi)c={(νi(k),μi(k))} , (μi(k),νi(k))∈lh(sθi) , i = 1, 2.

We develop several cross-entropy measures for linguistic hesitant intuitionistic fuzzy information, motivated by Vlachos and Sergiadis⁴⁹, Mao et al.⁴³, Xu and Xia⁴⁸, Peng et al.²⁶, etc.

Let ȟ₁ = {a_1i} = {(s_{θ_1i}, lh(s_{θ_1i}))}, (μ1i(k),ν1i(k))∈lh(sθ1i) , ȟ₂ = {a_2j} = {(s_{θ_2j}, lh(s_{θ_2j}))}, (μ2j(k),ν2j(k))∈lh(sθ2j) . Then two cross entropy measures can be defined as follows by considering linguistic variables, intuitionistic fuzzy memberships. Since each LHIFE has several linguistic terms and intuitionistic fuzzy memberships, we can define cross-entropy measures by considering the maximum value and the average value.

(16)CE1′(hˇ1,hˇ2)=maxaki∈hˇk{θ1σ(i)glog22θ1σ(i)θ1σ(i)+θ2σ(i)+(1−θ1σ(i)g)log22(g−θ1σ(i))2g−θ1σ(i)−θ2σ(i)}+max(μij(k),νij(k))∈lh(sθij){μ1σ(j)(k)log22μ1σ(j)(k)μ1σ(j)(k)+μ2σ(j)(k)+ν1σ(j)(k)log22ν1σ(j)(k)ν1σ(j)(k)+ν2σ(j)(k)}.

(17)CE2′(hˇ1,hˇ2)=1|a1i|∑aki∈hˇk((θ1σ(i)glog22θ1σ(i)θ1σ(i)+θ2σ(i)+(1−θ1σ(i)g)log22(g−θ1σ(i))2g−θ1σ(i)−θ2σ(i))1|lh(s(θki))|∑(μij(k),νij(k))∈lh(sθij)(μ1σ(j)(k)log22μ1σ(j)(k)μ1σ(j)(k)+μ2σ(j)(k)+ν1σ(j)(k)log22ν1σ(j)(k)ν1σ(j)(k)+ν2σ(j)(k))).

Similarly, other cross-entropy measures for linguistic hesitant intuitionistic fuzzy elements can be defined as follows. By using the generalized mean operator, we can get the cross-entropy measures CE3′(hˇ1,hˇ2) and CE4′(hˇ1,hˇ2) .

(18)CE3′(hˇ1,hˇ2)=(1|a1i|∑aki∈hˇk((θ1σ(i)glog22θ1σ(i)θ1σ(i)+θ2σ(i))p+((1−θ1σ(i)g)log22(g−θ1σ(i))2g−θ1σ(i)−θ2σ(i))p+1|lh(sθki)|∑(μij(k),νij(k))∈lh(sθij)((μ1σ(j)(k)log22μ1σ(j)(k)μ1σ(j)(k)+μ2σ(j)(k))p+(ν1σ(j)(k)log22ν1σ(j)(k)ν1σ(j)(k)+ν2σ(j)(k))p))1/p,p≥1.

(19)CE4′(hˇ1,hˇ2)=(1|a1i|∑aki∈hˇk(θ1σ(i)glog22θ1σ(i)θ1σ(i)+θ2σ(i))p)1/p+(1|a1i|∑aki∈hˇk((1−θ1σ(i)g)log22(g−θ1σ(i))2g−θ1σ(i)−θ2σ(i))p)1/p+(1|a1i||lh(s(θki))|∑(μij(k),νij(k))∈lh(sθij)(μiσ(j)(k)*log22μ1σ(j)(k)μ1σ(j)(k)+μ2σ(j)(k))p)1/p+(1|a1i||lh(sθki)|∑(μij(k),νij(k))∈lh(sθij)(νiσ(j)(k)log22ν1σ(j)(k)ν1σ(j)(k)+ν2σ(j)(k))p)1/p,p≥1.

Ye³⁸ developed the cross-entropy measure for intuitionistic fuzzy value by considering the complementary set of the intuitionistic fuzzy set to get

CE(A′,B′)=∑i=1n(μA(xi)+1−νA(xi)2*log2μA(xi)+1−νA(xi)12[(μA(xi)+1−νA(xi))+μB(xi)+1−νB(xi)]+1−μA(xi)+νA(xi)2*log21−μA(xi)+νA(xi)12[(1−μA(xi)+νA(xi))+1−μB(xi)+νB(xi)]).

We extend the cross-entropy measure CE(A′, B′) in Ye³⁸ to accommodate linguistic hesitant intuitionitic fuzzy values to get cross-entropy measures CE5′(hˇ1,hˇ2) and CE6′(hˇ1,hˇ2)

(20)CE5′(hˇ1,hˇ2)=maxaki∈hˇk{θ1σ(i)glog22θ1σ(i)θ1σ(i)+θ2σ(i)+(1−θ1σ(i)g)log22(g−θ1σ(i))2g−θ1σ(i)−θ2σ(i)}+max(μij(k),νij(k))∈lh(sθij){1+μ1σ(j)(k)−ν1σ(j)(k)2*log22(1+μ1σ(j)(k)−ν1σ(j)(k))2+μ1σ(j)(k)−ν1σ(j)(k)+μ2σ(j)(k)−ν2σ(j)(k)+1−μ1σ(j)(k)+ν1σ(j)(k)2log22(1−μ1σ(j)(k)+ν1σ(j)(k))2−μ1σ(j)(k)+ν1σ(j)(k)−μ2σ(j)(k)+ν2σ(j)(k)}.

(21)CE6′(hˇ1,hˇ2)=(1|a1i|∑aki∈hˇk(θ1σ(i)glog22θ1σ(i)θ1σ(i)+θ2σ(i))p)1/p+(1|a1i|∑a1i∈hˇ1((1−θ1σ(i)g)log22(g−θ1σ(i))2g−θ1σ(i)−θ2σ(i))p)1/p(1|a1i||lh(s(θki))|∑(μij(k),νij(k))∈lh(sθij)(μ1σ(j)(k)+1−ν1σ(j)(k)2log22(μ1σ(j)(k)+1−ν1σ(j)(k))2+μ1σ(j)(k)−ν1σ(j)(k)+μ2σ(j)(k)−ν2σ(j)(k))p)1/p+(1|a1i||lh(sθki)|∑(μij(k),νij(k))∈lh(sθij)(1−μ1σ(j)(k)+ν1σ(j)(k)2log22(1−μ1σ(j)(k)+ν1σ(j)(k))2−μ1σ(j)(k)+ν1σ(j)(k)−μ2σ(j)(k)−ν2σ(j)(k))p)1/p,p≥1

By using the concave-up function f (x) = x^p, we extend the cross-entropy CE6(A˜,B˜) developed by Xu and Xia⁴⁸ to linguistic hesitant intuitionistic fuzzy environment to get CE7′(hˇ1,hˇ2) . By using the concave-up function f (x) = x^p and the generalized mean operator, we can further get cross-entropy measure CE8′(hˇ1,hˇ2) .

(22)CE7′(hˇ1,hˇ2)=11−21−p(maxaki∈hˇk{(θ1σ(i)/g)p+(θ2σ(i)/g)p2−(θ1σ(i)/g+θ2σ(i)/g2)p}+maxaki∈hˇk{(1−θ1σ(i)/g)p+(1−θ2σ(i)/g)p2−(1−θ1σ(i)/g+θ2σ(i)/g2)p}+max(μij(k),νij(k))∈lh(sθij){(μ1σ(j)(k))p+(μ2σ(j)(k))p2−(μ1σ(j)(k)+μ2σ(j)(k)2)p}+max(μij(k),νij(k))∈lh(sθij){(ν1σ(j)(k))p+(ν2σ(j)(k))p2−(ν1σ(j)(k)+ν2σ(j)(k)2)p}),2≥p>1;

(23)CE8′(hˇ1,hˇ2)=(1|a1i|∑aki∈hˇk11−21−q((θ1σ(j)/g)q+(θ2σ(j)/g)q2−(θ1σ(j)/g+θ2σ(j)/g2)q)p)1/p+(1|a1i|∑aki∈hˇk11−21−q((1−θ1σ(j)/g)q+(1−θ2σ(j)/g)q2−(1−θ1σ(j)/g+θ2σ(j)/g2)q)p)1/p+(1|a1i||lh(s(θki))|∑(μij(k),νij(k))∈lh(sθij)11−21−q((μ1σ(j)(k))q+(μ2σ(j)(k))q2−(μ1σ(j)(k)+μ2σ(j)(k)2)q)p)1/p+(1|a1i||lh(s(θki))|∑(μij(k),νij(k))∈lh(sθij)11−21−q((ν1σ(j)(k))q+(ν2σ(j)(k))q2−(ν1σ(j)(k)+ν2σ(j)(k)2)q)p)1/p,2≥q>1,p≥1.

By using the concave-up function f (x) = (1 + qx)ln(1 + qx), we can extend the cross-entropy measures CE₇(ᾶ₁, ᾶ₂) and CE₈(ᾶ₁, ᾶ₂) for intuitionistic hesitant fuzzy values to linguistic hesitant intuitionistic fuzzy environment to get cross-entropy measures CE9′(hˇ1,hˇ2) and CE10′(hˇ1,hˇ2) .

(24)CE9′(hˇ1,hˇ2)=1T(maxaki∈hˇk{(1+q(θ1σ(i)/g)) ln(1+q(θ1σ(i)/g))+(1+q(θ2σ(i)/g)) ln(1+q(θ2σ(i)/g))2−2+q(θ1σ(i)/g)+q(θ2σ(i)/g)2 ln2+q(θ1σ(i)/g)+q(θ2σ(i)/g)2}+maxaki∈hˇk{(1+q(1−θ1σ(i)/g)) ln(1+q(1−θ1σ(i)/g))+(1+q(1−θ2σ(i)/g)) ln(1+q(1−θ2σ(i)/g))2−2+q(1−θ1σ(i)/g)+q(1−θ2σ(i)/g)2 ln2+q(1−θ1σ(i)/g)+q(1−θ2σ(i)/g)2}+max(μij(k),νij(k))∈lh(sθij){(1+qμ1σ(j)(k)) ln(1+qμ1σ(j)(k))+(1+qμ2σ(j)(k)) ln(1+qμ2σ(j)(k))2−2+qμ1σ(j)(k)+qμ2σ(j)(k)2 ln2+qμ1σ(j)(k)+qμ2σ(j)(k)2}+max(μij(k),νij(k))∈lh(sθij){(1+qν1σ(j)(k)) ln(1+qν1σ(j)(k))+(1+qν2σ(j)(k)) ln(1+qν2σ(j)(k))2−2+qν1σ(j)(k)+qν2σ(j)(k)2 ln2+qν1σ(j)(k)+qν2σ(j)(k)2}),q>0.

(25)CE10′(hˇ1,hˇ2)=(1|a1i|T∑aki∈hˇk((1+q(θ1σ(i)/g)) ln(1+q(θ1σ(i)/g))+(1+q(θ2σ(i)/g)) ln(1+q(θ2σ(i)/g))2−2+q(θ1σ(i)/g)+q(θ2σ(i)/g)2 ln2+q(θ1σ(i)/g)+q(θ2σ(i)/g)2)p)1/p+(1|a1i|T∑aki∈hˇk((1+q(1−θ1σ(i)/g)) ln(1+q(1−θ1σ(i)/g))+(1+q(1−θ2σ(i)/g)) ln(1+q(1−θ2σ(i)/g))2−2+q(θ1σ(i)/g)+q(1−θ2σ(i)/g)2 ln2+q(1−θ1σ(i)/g)+q(1−θ2σ(i)/g)2)p)1/p+(1|a1i||lh(sθki)|T∑(μij(k),νij(k))∈lh(sθij)((1+qμ1σ(j)(k)) ln(1+qμ1σ(j)(k))+(1+qμ2σ(j)(k)) ln(1+qμ2σ(j)(k))2−2+qμ1σ(j)(k)+qμ2σ(j)(k)2 ln2+qμ1σ(j)(k)+qμ2σ(j)(k)2)p)1/p+(1|a1i||lh(s(θki))|T∑(μij(k),νij(k))∈lh(sθij)((1+qν1σ(j)(k)) ln(1+qν1σ(j)(k))+(1+qν2σ(j)(k)) ln(1+qν2σ(j)(k))2−2+qν1σ(j)(k)+qν2σ(j)(k)2 ln2+qν1σ(j)(k)+qν2σ(j)(k)2)p)1/p.

Here q > 0, p ≥ 1 and T = (1 + q) ln (1 + q) − (2 + q)(ln(2 + q) − ln 2).

Theorem 5.

The measures defined in Eqs.(16)–(25) are linguistic hesitant intuitionisic fuzzy cross-entropy measures, which satisfy the conditions given in Definition 3.1.

Proof.

We prove the Eq.(16) and other equations can be proved similarly.

According to Shannon’s inequality, it is clear that CE1′(hˇ1,hˇ2)≥0 . If ȟ₁ = ȟ₂, then ∀ x ∈ X, ã_1σ(j) = ã_2σ(j), s_{θ_1σ(i)} = s_{θ_2σ(i)}, (μ1σ(j)(k),ν1σ(j)(k))=(μ2σ(j)(k),ν2σ(j)(k)) .

CE1′(hˇ1,hˇ2)=maxaki∈hˇk{θ1σ(i)glog22θ1σ(i)θ1σ(i)+θ2σ(i)+(1−θ1σ(i)g)log22(g−θ1σ(i))2g−θ1σ(i)−θ2σ(i)}+max(μij(k),νij(k))∈lh(sθij){μ1σ(j)(k)log22μ1σ(j)(k)μ1σ(j)(k)+μ2σ(j)(k)+ν1σ(j)(k)log22ν1σ(j)(k)ν1σ(j)(k)+ν2σ(j)(k)}=maxaki∈hˇk{θ1σ(i)glog22θ1σ(i)θ1σ(i)+θ1σ(i)+(1−θ1σ(i)g)log22(g−θ1σ(i))2g−θ1σ(i)−θ1σ(i)}+max(μij(k),νij(k))∈lh(sθij){μ1σ(j)(k)log22μ1σ(j)(k)μ1σ(j)(k)+μ1σ(j)(k)+ν1σ(j)(k)log22ν1σ(j)(k)ν1σ(j)(k)+ν1σ(j)(k)}=maxaki∈hˇk{0}+max(μij(k),νij(k))∈lh(sθij){0}=0.

CE1′(hˇ1c,hˇ2c)=maxakic∈hˇkc{g−θ1σ(i)glog22(g−θ1σ(i))(g−θ1σ(i))+(g−θ2σ(i))+(1−g−θ1σ(i)g)log22(g−(g−θ1σ(i)))2g−(g−θ1σ(i))−(g−θ2σ(i))}+max(νij(k),μij(k))∈lh(sθij)c{ν1σ(j)(k)log22ν1σ(j)(k)ν1σ(j)(k)+ν2σ(j)(k)+μ1σ(j)(k)log22μ1σ(j)(k)μ1σ(j)(k)+μ2σ(j)(k)},

CE1′(hˇ1c,hˇ2c)=maxaki∈hˇk{(1−θ1σ(i)g)log22(g−θ1σ(i))2g−θ1σ(i)−θ1σ(i)+θ1σ(i)glog22θ1σ(i)θ1σ(i)+θ1σ(i)}+max(μij(k),νij(k))∈lh(sθij){μ1σ(j)(k)log22μ1σ(j)(k)μ1σ(j)(k)+μ1σ(j)(k)+ν1σ(j)(k)log22ν1σ(j)(k)ν1σ(j)(k)+ν1σ(j)(k)}=CE1′(hˇ1,hˇ2).

The proof is thus complete.

Several linguistic hesitant intuitionisic fuzzy cross-entropy measures have been introduced in this paper, each cross-entropy measure has its own characteristics and emphasis. Decision makers can select the cross-entropy measure according to the real needs and his preference.

The proposed cross-entropy measures are the degrees of discrimination of ȟ₁ from ȟ₂. However, CEi′(hˇ1,hˇ2) (i = 1, 2, …, 10) are not symmetric with respect to their arguments. Symmetric form of cross-entropy measures for LHIFEs can be got by modifying the Eqs. (16)–(25) as follows:

(26)CEi*(hˇ1,hˇ2)=CEi′(hˇi,hˇ2)+CEi′(hˇ2,hˇ1).

Example:

Let ȟ₁ = {(s₅, (0.7,0.2), (0.6,0.3)), (s₆, (0.6,0.2), (0.5,0.4)), (s₇, (0.5,0.3), (0.6,0.4))}, ȟ₂ = {(s₆, (0.8,0.1), (0.7,0.3)), (s₇, (0.6,0.2), (0.5, 0.3)), (s₈, (0.5,0.2), (0.5,0.3))} be two patterns and ȟ = {(s₄, (0.7,0.1), (0.6,0.2)), (s₅, (0.7,0.2), (0.6,0.3)), (s₆, (0.6,0.2), (0.6, 0.3))} be a sample. p = q = 2, then the following results can be got:

CE1*(hˇ1,hˇ)=0.2636, CE1*(hˇ2,hˇ)=0.3359,CE2*(hˇ1,hˇ)=0.0480,CE2*(hˇ2,hˇ)=0.0978,CE3*(hˇ1,hˇ)=0.3152,CE3*(hˇ2,hˇ)=0.4811,CE4*(hˇ1,hˇ)=0.0571,CE4*(hˇ2,hˇ)=0.1227,CE5*(hˇ1,hˇ)=0.6137,CE5*(hˇ2,hˇ)=0.8592,CE6*(hˇ1,hˇ)=0.5824,CE6*(hˇ2,hˇ)=0.7987,CE7*(hˇ1,hˇ)=0.1867,CE7*(hˇ2,hˇ)=0.2969,CE8*(hˇ1,hˇ)=0.0394,CE8*(hˇ2,hˇ)=0.0810,CE9*(hˇ1,hˇ)=0.0840,CE9*(hˇ2,hˇ)=0.1322,CE10*(hˇ1,hˇ)=0.0432, CE10*(hˇ2,hˇ)=0.0830.

From the above results we can see that the sample ȟ belongs to the pattern ȟ₁ by using all the above cross-entropy measures.

4. New MADM methods based on the cross-entropy measures of LHIFEs

Considering a multiple attribute decision making problem, let A = {A₁,A₂, …, A_m} be a finite alternative set and C = {C₁,C₂, …, C_n} be a finite attribute set. The decision makers evaluate alternatives with respect to attributes with linguistic terms and intuitionisic fuzzy memberships. If two or more decision makers gave the same LIFE in evaluating the same alternative with respect to some attribute, it is counted only once. The linguistic hesitant intuitionistic fuzzy decision matrix can be got as D˜=(hˇij)m×n , where hˇij={(sθij,lh(sθij)|(μij(t),νij(t))∈lh(sθij),t=1,2,…,lij} is LHIFE. {s_{θ_ij}} is the linguistic terms given by experts to evaluate alternative A_i with respect to the attribute C_j, μij(t) indicates the degree of linguistic term s_{θ_ij} satisfying the attribute C_j and νij(t) indicates the degree of linguistic term s_{θ_ij} dissatisfying the attribute C_j, such that μij(t),νij(t)∈[0,1] and μij(t)+νij(t)≤1 .

Different LHIFEs may have different number of LIFEs and different LIFEs may have different number of intuitionistic fuzzy memberships. In order to define cross-entropy measures more accurately, we extend LHIFEs according to the risk attitudes of decision makers until all LHIFEs have the same number of LIFEs and all LIFEs have the same number of intuitionistic fuzzy memberships. If the decision maker is risk-seeking, the largest LIFE and largest intuitionistic fuzzy membership can be added; if the decision maker is risk-averse, the smallest LIFE and smallest intuitionistic fuzzy membership can be added; if the decision maker is risk-neutral, the average LIFE and average intuitionistic fuzzy membership can be added.

In some decision making process, information for attribute weight is partly known or unknown completely due to decision time pressure, decision makers’ lack of knowledge and expertise, complicated decision problems, etc. Generally, partly known attribute information can be expressed as a subset of the following relations: a weak ranking: {w_i ≥ w_j}, i ≠ j; a strict ranking: {w_i −w_j ≥ ε_i(> 0)}, i ≠ j; a ranking with multiples: {w_i ≥ α_iw_j},0 ≤ α_i ≤ 1, i ≠ j; an interval form: {β_j ≤ w_j ≤ β_j + ε_j},0 ≤ β_j < β_j + ε_j ≤ 1; a ranking of differences: {w_i − w_j ≥ w_k − w_l}, for i ≠ j ≠ k ≠ l. For a specific decision problem, attribute weight information can be described as a subset of the above relationships. The corresponding attribute weight information set can be denoted as H. According to information theory, an attribute should be assigned a larger weight if its evaluation values have obvious differences since it plays an important role in the priority procedure. Otherwise, it should be assigned a smaller weight⁵¹. Since several linguistic hesitant intuitionistic fuzzy cross-entropy measures have been developed, we only choose one cross-entropy measure for the convenience of calculation and analysis and we can calculate similarly if other cross-entropy measures are chosen. Then the deviation value d_j of attribute C_j can be calculated as

(27)dj=∑i=1m∑k=1mCE*(hˇij,hˇkj).

The weighted deviation value can be calculated as

(28)d=∑j=1nwjdj =∑j=1nwj(∑i=1m∑k=1mCE*(hˇij,hˇkj)) =∑j=1n∑i=1m∑k=1mwjCE*(hˇij,hˇkj).

A reasonable weight vector w = (w₁, w₂, …, w_n) should make the weighted deviation value as large as possible to differentiate the problem characteristics more effectively⁵¹. Thus we set up the following programming model to determine optimal attribute weights if attribute weights are partly known.

(M-1) ∑j=1n∑i=1m∑k=1mwjCE*(hˇij,hˇkj) s.t. w∈H, wj≥0,j=1,2,…,n, w1+w2+…+wn=1.

The model (M-1) is a linear programming model, which can be solved easily by using many existing methods.

If information for attribute weights is unknown completely, we set up the following model

(M-2) ∑j=1n∑i=1m∑k=1mwjCE*(hˇij,hˇkj) s.t. ∑j=1nwj2=1, wj≥0,j=1,2,…,n.

In order to solve model (M-2), we construct the following Lagrange function:

(29)L(W,λ)=∑j=1n∑i=1m∑k=1mwjCE*(hˇij,hˇkj)+λ2(∑j=1nwj2−1),

where λ is the Lagrange multiplier. Calculate the differentiation of Eq. (29) with respect to w_j (j = 1,2, …, n) and λ, and set these partial derivatives equal to zeros to get

(30){∂L∂wj=∑i=1m∑k=1mCE*(hˇij,hˇkj)+λwj=0,∂L∂λ=∑j=1nwj2−1=0.

By solving Eq.(30), we can get the formula for calculating attribute weights as

(31)wj=∑i=1m∑k=1mCE*(hˇij,hˇkj)∑j=1n(∑i=1m∑k=1mCE*(hˇij,hˇkj))2,j=1,2,…,n.

Normalize the attribute weights to get

(32)wj=∑i=1m∑k=1mCE*(hˇij,hˇkj)∑j=1n∑i=1m∑k=1mCE*(hˇij,hˇkj),j=1,2,…,n.

Based on the above analysis, we can present an effective approach to solve the multiple attribute decision making problem with linguistic hesitant intuitionistic fuzzy information.

Step 1. Construct the decision matrix D˜=(hˇij)m×n . Multiple decision makers evaluate the alternatives with respect to attributes with linguistic terms and intuitionistic fuzzy memberships. Then linguistic hesitant intuitionistic fuzzy elements are formed as ȟ_ij.
Step 2. If attribute weights are known completely, go to Step 3 directly. In order to calculate the cross-entropy measure more accurately, we extend the decision matrix according to the risk attitude of decision makers and the decision matrix D˜′=(hˇij′)m×n can be got. If attribute weights are known partly, we can solve model (M-1) to obtain them; if attribute weights are completely unknown, we can calculate them by using Eq.(32).
Step 3. Calculate the collective evaluation values of alternatives by using the decision matrix D˜=(hˇij)m×n . We can calculate by using the LHIFWA operator, the LHIFWG operator or the GLHIFWA operator.
Step 4. Calculate the scores S(ȟ_i) (i = 1,2, …, m) and the accuracy degrees A(ȟ_i) (i = 1, 2, …, m) of alternatives’ collective evaluation values ȟ_i (i = 1,2, …, m) by using the score function and accuracy function.
(33)S(hˇi)=1|hˇi|(∑θig|lh(sθi)|∑(αi(k),βi(k))∈lh(sθi)(μi(k)−νi(k))),

(34)A(hˇi)=1|hˇi|(∑θig|lh(sθi)|∑(αi(k),βi(k))∈lh(sθi)(μi(k)+νi(k))),
where | ȟ_i | and | lh(s_θ(i)) | are the cardinalities of ȟ_i and lh(s_{θ_i}), respectively. ȟ_i = {(s_{θ_i}, lh(s_{θ_i}))}, lh(sθi)={(μi(k),νi(k))} .
Step 5. Rank ȟ_i according the method given in Definition 2.6 and rank alternatives accordingly.

In LHIFEs, intuitionistic fuzzy memberships and nonmemberships have been considered besides linguistic evaluation values. Hence, aggregation of linguistic hesitant intuitionistic fuzzy information is more complex than linguistic hesitant fuzzy sets. By calculating the number of basic operations at each step, we can get the worst-case time complexity of our algorithm is O(m²nlt), where m is the number of alternatives, n is the number of attributes, l is the largest number of LIFEs in LHIFEs, t is the largest number of intuitionistic fuzzy values in LIFEs. Then the complexity of algorithm can tell us the new algorithm is an efficient and practical polynomial-time algorithm for solving multiple attribute decision making problems.

TOPSIS method was developed by Hwang and Yong⁵², which is based on the principle that the optimal alternative should have the shortest distance from the positive ideal solution and at the same time have the farthest distance from the negative ideal solution. From the risk viewpoint, decision makers are risk-averse since they choose the alternative which is not only making as much profit as possible, but also avoiding as much risk as possible. In the following, we present a new ranking method based on the cross-entropy measures and the idea of TOPSIS.

Step 1. As for Algorithm I.
Step 2. As for Algorithm I.
Step 3. Determine the linguistic hesitant intuitionistic fuzzy positive-ideal solution (LHIFPIS) as
hˇ+={(sg,{(1,0),…,(1,0)})},
and linguistic hesitant intuitionistic fuzzy negative-ideal solution (LHIFNIS) as
hˇ−={(s1,{(1,0),…,(1,0)})}.

Each ȟ⁺ and ȟ⁻ have the same number of LHIFEs and LIFEs as hˇij′ .
Step 4. Calculate the symmetric cross-entropy of hˇij′ from ȟ⁺ and ȟ⁻ as Gij+ , Gij− , respectively.
(35)Gij+=C2*(hˇij′,hˇ+)=C2(hˇij′,hˇ+)+C2(hˇ+,hˇij′),

(36)Gij−=C2*(hˇij′,hˇ−)=C2(hˇij′,hˇ−)+C2(hˇ−,hˇij′).
Step 4. Determine the relative closeness by using the following equation
(37)Gij=Gij−Gij++Gij−,i=1,2,…,m, j=1,2,…,n.

Then the weighted relative closeness coefficients of alternatives can be calculated as follows
(38)Gi=∑j=1nwjGij, i=1,2,…,m.
Step 5. Rank alternatives according to the ranking of G_i (i = 1, 2, …, m). The larger the G_i, the better the alternative A_i.

5. Numerical example

A numerical example adapted from Chen and Yang⁵³ is presented to illustrate efficiency and practical advantages of the proposed procedure.

Suppose that there is an architecture company wanting to select a company to supply an important material, such as cement. Experts from different departments have been invited and they mainly consider the following four attributes: C₁− the price of product, C₂−the quality of product, C₃−delivery time, C₄−risk. After pre-evaluation, there are still five alternatives A_i (i = 1,2, …, 5) left for further evaluation. We use the new algorithms to rank alternatives.

Step 1. The experts evaluate alternatives A_i (i = 1,2, …, 5) with respect to attributes C_j (j = 1, 2, …, 4) with linguistic terms and intuitionistic fuzzy memberships. The decision matrix is formed as D˜=(hˇij)4×5 in Table 1.
Step 2. Assume attribute weights are known completely as w₁ = (0.15,0.20,0.30,0.35).
Step 3. Calculate alternatives’ collective evaluation values. For example, we calculate ȟ₁ by using the LHIFWA operator as follows
hˇ1={(s5.15,{(0.6206,0.1803),(0.6418,0.1803)}),(s5.30,{(0.6331,0.1726),(0.6536,0.1726)}),(s5.45,{(0.5864,0.2219),(0.6095,0.2219)}),(s5.60,{(0.6000,0.2125),(0.6224,0.2125)})}.

Similarly, we can calculate other ȟ_i(i = 2,3, …, 5).
Step 4. Calculate scores of collective evaluation values ȟ_i (i = 1,2, …, 5) to get S(ȟ₁) = 0.2527,S(ȟ₂) = 0.2493, S(ȟ₃) = 0.2193, S(ȟ₄) = 0.2505, S(ȟ₅) = 0.2601. Rank S(ȟ_i) (i = 1,2, …, 5) to get
S(hˇ5)>S(hˇ1)>S(hˇ4)>S(hˇ2)>S(hˇ3).
Step 5. Rank ȟ_i according to the ranking of S(ȟ_i) to get
hˇ5>hˇ1>hˇ4>hˇ2>hˇ3.

Then alternatives can be ranked accordingly as
A5≻A1≻A4≻A2≻A3.

The optimal alternative is A₅.

	C₁	C₂

A₁	{(s₆,(0.5,0.4)),(s₇,(0.6,0.3))}	{(s₂,(0.6,0.2),(0.7,0.2))}
A₂	{(s₃,(0.5,0.2),(0.5,0.3)) }	{(s₇,(0.7,0.1)),(s₈,(0.8,0.2))}
A₃	{(s₄,(0.6,0.1))}	{(s₅,(0.7,0.3),(0.5,0.4))}
A₄	{(s₂,(0.6,0.2),(0.5,0.3))}	{(s₆,(0.8,0.2))}
A₅	{(s₅,(0.7,0.3)),(s₆,(0.6,0.2))}	{(s₃,(0.5,0.2),(0.6,0.3))}

	C₃	C₄

A₁	{(s₇,(0.7,0.1)),(s₈,(0.6,0.2))}	{(s₅,(0.6,0.2))}
A₂	{(s₆,(0.6,0.3),(0.7,0.2))}	{(s₄,(0.7,0.2)),(s₅,(0.5,0.3),(0.6,0.4))}
A₃	{(s₂,(0.5,0.4)),(s₃,(0.5,0.2))}	{(s₈,(0.6,0.1),(0.7,0.2))}
A₄	{(s₃,(0.7,0.2),(0.6,0.3))}	{(s₆,(0.6,0.2)),(s₇,(0.7,0.1))}
A₅	{(s₇,(0.8,0.2)),(s₈,(0.6,0.3))}	{(s₄,(0.5,0.2))}

Table 1:

Decision matrix D˜ .

We can use the LHIFWG operator or the GLHIFWA_λ operator in Step 3 and other steps are the same as above. Then results can be obtained as shown in Table 3. If attribute weights are partly known or completely unknown, we need to calculate them first. Assume decision makers are risk-averse, then the smallest intuitionistic fuzzy value and LIFE can be added to extend the decision matrix as D˜′ , which is shown in Table 2. Though several cross-entropy measures have been introduced in this paper, we only choose one to calculate attribute weights for space limit. Here we use cross-entropy measures CE2* and similar results can be got if other cross-entropy measures are used. If attribute weights are known partly, we can set up the following Model (M-3) to determine them as w₂ = (0.2782, 0.3062, 0.2279, 0.1877). If attribute weights are unknown completely, we determine them by using Eq.(33) to get w₃ = (0.30, 0.15, 0.35, 0.20). Other steps can be calculated similarly as that of completely known attribute weights method above and results are shown in Table 3. If attribute weight vector w₂ is used, A₂ becomes the optimal alternative and A₅ becomes the sub-optimal alternative in most case. For w₃, A₁ becomes the optimal alternative and A₅ becomes the sub-optimal alternative.

	C₁	C₂

A₁	{(s₆,(0.5,0.4),(0.5,0.4)),(s₇,(0.6,0.3),(0.5,0.4))}	{(s₂,(0.6,0.2),(0.7,0.2)),(s₂,(0.5,0.4),(0.5,0.4))}
A₂	{(s₃,(0.5,0.2),(0.5,0.3)),(s₂,(0.5,0.4),(0.5,0.4))}	{(s₇,(0.7,0.1),(0.5,0.4)),(s₈,(0.8,0.2),(0.5,0.4))}
A₃	{(s₄,(0.6,0.1),(0.5,0.4)),(s₂,(0.5,0.4),(0.5,0.4))}	{(s₅,(0.7,0.3),(0.5,0.4)),(s₂,(0.5,0.4),(0.5,0.4))}
A₄	{(s₂,(0.6,0.2),(0.5,0.3)),(s₂,(0.5,0.4),(0.5,0.4))}	{(s₆,(0.8,0.2),(0.5,0.4)),(s₂,(0.5,0.4),(0.5,0.4))}
A₅	{(s₅,(0.7,0.3),(0.5,0.4)),(s₆,(0.6,0.2),(0.5,0.4))}	{(s₃,(0.5,0.2),(0.6,0.3)),(s₂,(0.5,0.4),(0.5,0.4))}

	C₃	C₄

A₁	{(s₇,(0.7,0.1),(0.5,0.4)),(s₈,(0.6,0.2),(0.5,0.4))}	{(s₅,(0.6,0.2),(0.5,0.4)),(s₂,(0.5,0.4),(0.5,0.4))}
A₂	{(s₆,(0.6,0.3),(0.7,0.2)),(s₂,(0.5,0.4),(0.5,0.4))}	{(s₄,(0.7,0.2),(0.5,0.4)),(s₅,(0.5,0.3),(0.6,0.4))}
A₃	{(s₂,(0.5,0.4),(0.5,0.4)),(s₃,(0.5,0.2),(0.5,0.4))}	{(s₈,(0.6,0.1),(0.7,0.2)),(s₂,(0.5,0.4),(0.5,0.4))}
A₄	{(s₃,(0.7,0.2),(0.6,0.3)),(s₂,(0.5,0.4),(0.5,0.4))}	{(s₆,(0.6,0.2),(0.5,0.4)),(s₇,(0.7,0.1),(0.5,0.4))}
A₅	{(s₇,(0.8,0.2),(0.5,0.4)),(s₈,(0.6,0.3),(0.5,0.4))}	{(s₄,(0.5,0.2),(0.5,0.4)),(s₂,(0.5,0.4),(0.5,0.4))}

Table 2:

Extended decision matrix D˜′ .

		S(ȟ₁)	S(ȟ₂)	S(ȟ₃)	S(ȟ₄)	S(ȟ₅)	Rankings
w₁	LHIFWA	0.2527	0.2493	0.2193	0.2505	0.2126	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅
	LHIFWG	0.2269	0.2285	0.1901	0.2193	0.1884	A₂ ≻ A₁ ≻ A₄ ≻ A₃ ≻ A₅
	GLHIFWA₂	0.2709	0.2626	0.2445	0.2724	0.2283	A₄ ≻ A₁ ≻ A₂ ≻ A₃ ≻ A₅

w₂	LHIFWA	0.2258	0.2599	0.2018	0.2278	0.2057	A₂ ≻ A₄ ≻ A₁ ≻ A₅ ≻ A₃
	LHIFWG	0.1949	0.2297	0.1818	0.1922	0.1838	A₂ ≻ A₅ ≻ A₁ ≻ A₄ ≻ A₃
	GLHIFWA₂	0.2500	0.2791	0.2212	0.2525	0.2207	A₂ ≻ A₅ ≻ A₄ ≻ A₁ ≻ A₃

w₃	LHIFWA	0.2630	0.2215	0.1876	0.1887	0.2442	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃
	LHIFWG	0.2390	0.1987	0.1654	0.1614	0.2201	A₁ ≻ A₅ ≻ A₂ ≻ A₃ ≻ A₄
	GLHIFWA₂	0.2787	0.2363	0.2100	0.2118	0.2582	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃

Table 3:

The results of different aggregation operators with different attribute weights.

(M-3)1.9957w1+2.1970w2+1.6348w3+1.3467w4 s.t.0.15≤w1≤0.30,0.10≤w2≤0.25,0.20≤w3≤0.35,0.10≤w4≤020,2w2≤w3,w1+w2+…+w4=1.

If Algorithm II is used to rank alternatives, the first two steps are the same as that of Algorithm I. In Step 3, we determine the LHIFPIS ȟ⁺ and LHIFNIS ȟ⁻ as ȟ⁺ = {ǎ⁺}, ǎ⁺ = (s₉,{(1,0), (1,0)}), ȟ⁻ = {ǎ⁻}, ǎ⁻ =(s₁, {(0,1), (0,1)}). In Step 4, we calculate the symmetric cross-entropy of ȟ_ij from ȟ⁺ and ȟ⁻. For example, we calculate Gij+ and Gij− by using Eqs.(35)–(36) if CE2* is chosen. The relative closeness coefficients can be calculated by using Eq. (37). For completely known attribute weight vector w₁ = (0.15,0.20,0.30,0.35), the weighted relative closeness coefficients can be calculated by using Eq.(38) as G₁ = 0.5588,G₂ = 0.5414,G₃ = 0.4701,G₄ = 0.5273,G₅ = 0.5332. Then we can rank the relative closeness coefficients as G₁ > G₂ > G₅ > G₄ > G₃. Alternatives can be ranked accordingly as A₁ ≻A₂ ≻ A₅ ≻ A₄ ≻ A₃ and the optimal alternative is A₁. If other cross-entropy measures are used, we can calculate similarly and results are shown in Table 4, where p = q = 2. For attribute weights w₂ and w₃, we can get results as in Table 5 and Table 6, respectively.

	G₁	G₂	G₃	G₄	G₅	Rankings	Best alternative
CE1*	0.5588	0.5414	0.4701	0.5273	0.5332	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE2*	0.6048	0.5943	0.4865	0.5395	0.5698	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE3*	0.5410	0.5349	0.4696	0.5133	0.5241	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE4*	0.5808	0.5670	0.4740	0.5465	0.5439	A₁ ≻ A₂ ≻ A₄ ≻ A₅ ≻ A₃	A₁
CE5*	0.5321	0.5295	0.4633	0.4990	0.5162	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE6*	0.5264	0.5213	0.4515	0.4913	0.5114	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE7*	0.6046	0.5895	0.4876	0.5670	0.5751	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE8*	0.5512	0.5470	0.4763	0.5153	0.5341	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE9*	0.4120	0.4171	0.3913	0.4009	0.3940	A₂ ≻ A₁ ≻ A₄ ≻ A₅ ≻ A₃	A₂
CE10*	0.3854	0.3874	0.3713	0.3676	0.3644	A₂ ≻ A₁ ≻ A₃ ≻ A₄ ≻ A₅	A₂

Table 4:

The results with known attribute weight vector w₁.

	G₁	G₂	G₃	G₄	G₅	Rankings	Best alternative
CE1*	0.5572	0.5480	0.4558	0.4877	0.5339	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE2*	0.4756	0.4613	0.3137	0.3603	0.4364	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE3*	0.5392	0.5400	0.4575	0.4828	0.5239	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃	A₂
CE4*	0.5819	0.5689	0.4566	0.5012	0.5419	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE5*	0.5267	0.5339	0.4492	0.4656	0.5149	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃	A₂
CE6*	0.5442	0.5497	0.4629	0.4824	0.5321	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃	A₂
CE7*	0.6078	0.5929	0.4696	0.5169	0.5798	A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃	A₁
CE8*	0.5463	0.5520	0.4615	0.4796	0.5327	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃	A₂
CE9*	0.3955	0.4178	0.3797	0.3946	0.3889	A₂ ≻ A₁ ≻ A₄ ≻ A₅ ≻ A₃	A₂
CE10*	0.3727	0.3902	0.3531	0.3576	0.3586	A₂ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₃	A₂

Table 5:

The results with completely unknown attribute weight vector w₂.

	G₁	G₂	G₃	G₄	G₅	Rankings	Best alternative
CE1*	0.6049	0.5051	0.4501	0.4769	0.5807	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁
CE2*	0.5457	0.3975	0.3085	0.3498	0.5026	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁
CE3*	0.5764	0.5076	0.4501	0.4742	0.5601	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁
CE4*	0.6360	0.5213	0.4482	0.4876	0.6005	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁
CE5*	0.5691	0.5010	0.4418	0.4546	0.5533	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁
CE6*	0.5843	0.5139	0.4536	0.4703	0.5686	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁
CE7*	0.6660	0.5408	0.4617	0.5033	0.6380	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁
CE8*	0.5914	0.5160	0.4536	0.4684	0.5741	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁
CE9*	0.4158	0.4029	0.3623	0.3789	0.4099	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁
CE10*	0.3939	0.3774	0.3374	0.3425	0.3801	A₁ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₃	A₁

Table 6:

The results with partly known attribute weight vector w₃.

In Table 4, it can be seen that A₂ becomes the optimal alternative if cross-entropy measures CE9* and CE10* are used, which is quite different from other results. However, in most cases, the ranking is A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃ and A₁ becomes the best alternative and A₃ becomes the worst alternative. The subtle ranking differences are due to the different information fusion mechanisms. In Table 5, A₁ becomes the optimal alternative if CE1* , CE2* , CE4* , CE7* are used and A₂ becomes the best alternative for other cross-entropy measures. A₃ is still the worst alternative. In Table 6, we can get the same ranking A₁ ≻ A₂ ≻ A₅ ≻ A₄ ≻ A₃ for all the cross-entropy measures.

Since different cross-entropy measures may produce different ranking results and each cross-entropy has its own characteristics and emphasis, different cross-entropy measures can provide different views of the decision problem. Decision makers can choose the corresponding cross-entropy measure according to real needs, decision makers’ preference and interests.

In order to illustrate practical advantages of the new method, we compare it with the method of Peng et al.²⁶. In Peng et al.’s method, only hesitant intuitionistic fuzzy values are considered. If linguistic terms are omitted, CEi*(i=1,2,…,10) reduce to intuitionistic hesitant fuzzy cross-entropy measures CEi†(i=1,2,…,10) . Extend hesitant intuitionistic fuzzy evaluation elements according to the risk attitude of decision makers. Assume decision makers are risk-averse and the minimum intuitionistic fuzzy value is added until all the hesitant intuitionisic fuzzy elements have the same number of intuitionisic fuzzy values and the extended decision matrix D=(hij′)5×4 is formed. Determine the hesitant intuitionistic fuzzy positive ideal solution h⁺ and the hesitant intuitionistic fuzzy negative ideal solution h⁻ as h⁺ = {a⁺}, a⁺ = ((1,0), (1,0)), h⁻ = {a⁻}, a⁻ = ((0,1), (0,1)). Calculate the symmetric cross-entropy measures of hij′ from h⁺ and h⁻ as Gij+ , Gij− , respectively. Assume weight vector of attributes is also w₁ = (0.15,0.20,0.30,0.35) to facilitate comparison. We can calculate weighted relative closeness coefficients of alternatives by using Eq.(38) and results are shown in Table 7. From the results we can see different ranking results can be got in the proposed method and Peng et al.²⁶ method. In the proposed method, A₁ is the optimal alternative in most case and A₂ is optimal alternative in CE9* and CE10* . In Peng et al.²⁶ method, A₂, A₃ and A₄ become the optimal alternative for different cross-entropy measures. Different ranking results due to different decision information. Comparing evaluation information in Peng el al.’s method with that in the proposed algorithm, linguistic terms have been omitted in Peng et al.’s method. If all decision makers use the same linguistic term in evaluation in the proposed method, we can got the same ranking results in the two methods. Since different linguistic terms can be used in the proposed method, more information has been used and more accurate evaluation values can be got. Different evaluation information has been used in two methods and different results are reasonable. We further compare it with the method of Yang et al.⁵⁴. In Yang et al.’s method, each linguistic term only has one intuitionistic fuzzy membership. Hence we first aggregate the intuitionistic fuzzy memberships into a collective one by using intuitionistic fuzzy averaging operator (IFA) if a linguistic term has several intuitionistic fuzzy memberships. IFA(α1,α2,…,αn)=∑j=1n1nαj=(1−∏j=1n(1−μj(k))1n,∏j=1n(νj)1n) , α_j = (μ_j,ν_j). The decision matrix degenerates to Dˆ=(hˆij)5×4 . Then we aggregate the evaluation values by using the hesitant intuitionistic fuzzy linguistic weighted averaging (HIFLWA) operator, HIFLWAw(hˆ1,hˆ2,…,hˆn)=∑j=1nwjhˆj=∪ai∈hˆi{(s(∑j=1nwjθj),(1−∏j=1n(1−μj(k))wj,∏j=1n(νj(k))wj))} , the hesitant intuitionistic fuzzy linguistic weighted geometric (HIFLWG) operator HIFLWGw(hˆ1,hˆ2,…,hˆn)=∏j=1n(hˆj)wj=∪ai∈hˆi{(s(∏j=1n(θj)wj),(∏j=1n(μj(k))wj,1−∏j=1n(1−νj(k))wj))} , or the generalized hesitant intuitionistic fuzzy linguistic weighted averaging (GHIFLWA) operator GHIFLWAw(hˆ1,hˆ2,…,hˆn)=(∑j=1nwj(hˆj)λ)1/λ=∪ai∈hˆi{(s(∑j=1nwj(θj)λ)1/λ,((1−∏j=1n(1−(μj(k))λ)wj)1/λ,1−(1−∏j=1n(1−(1−νj(k))λ)wj)1/λ))} . The weight vector of attributes is also taken as w₁ = (0.15,0.20,0.30,0.35) to facilitate comparison. The results are shown in Table 8. From the results we can see that we can get similar ranking results in the proposed method and Yang et al.’s method⁵⁴. There is little difference in values to be aggregated since different intuitionistic fuzzy values in the proposed method are replaced by the average one. For space limit, we only present a simple example to illustrate the new algorithm. The results for other large-scale complex decision problems may have a greater difference. Since different intuitionistic fuzzy values can be used to model hesitation and uncertainty, the proposed method is more flexible and accurate.

	G₁	G₂	G₃	G₄	G₅	Rankings	Best alternative
CE1†	0.6797	0.6785	0.6763	0.6955	0.6685	A₄ ≻ A₁ ≻ A₂ ≻ A₃ ≻ A₄	A₄
CE2†	0.6654	0.6799	0.6579	0.6803	0.6496	A₄ ≻ A₂ ≻ A₁ ≻ A₃ ≻ A₅	A₄
CE3†	0.6079	0.6222	0.6008	0.6196	0.6009	A₂ ≻ A₄ ≻ A₁ ≻ A₅ ≻ A₃	A₂
CE4†	0.6636	0.6551	0.6660	0.6637	0.6111	A₃ ≻ A₄ ≻ A₁ ≻ A₂ ≻ A₅	A₃
CE5†	0.5600	0.5732	0.5541	0.5719	0.5538	A₂ ≻ A₄ ≻ A₁ ≻ A₃ ≻ A₅	A₂
CE6†	0.6068	0.6204	0.6007	0.6180	0.6000	A₂ ≻ A₄ ≻ A₁ ≻ A₃ ≻ A₅	A₂
CE7†	0.7206	0.7334	0.7517	0.7276	0.7376	A₃ ≻ A₅ ≻ A₂ ≻ A₄ ≻ A₁	A₃
CE8†	0.6043	0.6272	0.6196	0.6271	0.6128	A₂ ≻ A₄ ≻ A₃ ≻ A₅ ≻ A₁	A₂
CE9†	0.7280	0.7380	0.7284	0.7455	0.7205	A₄ ≻ A₂ ≻ A₃ ≻ A₁ ≻ A₅	A₄
CE10†	0.7046	0.7242	0.6949	0.7221	0.6922	A₂ ≻ A₄ ≻ A₁ ≻ A₃ ≻ A₅	A₂

Table 7:

The results of Peng et al.²⁹ method with known attribute weights w₁.

	S(ĥ₁)	S(ĥ₂)	S(ĥ₃)	S(ĥ₄)	S(ĥ₅)	Rankings
HIFLWA	0.2528	0.2499	0.2208	0.2420	0.2128	A₁ ≻ A₂ ≻ A₄ ≻ A₃ ≻ A₅
HIFLWG	0.2275	0.2302	0.1936	0.2021	0.1890	A₂ ≻ A₁ ≻ A₄ ≻ A₃ ≻ A₅
GHIFLWA₂	0.2709	0.2628	0.2454	0.2650	0.2282	A₁ ≻ A₄ ≻ A₂ ≻ A₃ ≻ A₅

Table 8:

The results of Yang et al.⁶⁸ with weights w₁.

From the above analysis we can see the proposed approaches have the following advantages. First, LHIFEs have been used to evaluate alternatives, which are more flexible since each LHIFE has several linguistic evaluation values and each linguistic evaluation value has several intuitionistic fuzzy memberships. The inherent fuzzy thought of the decision makers have been retained, which can guarantee accuracy of final results. Second, the cross-entropy measures are very important in decision making and we have found few study based on the linguistic hesitant intuitionistic fuzzy information. The new proposed cross-entropy measures can include the advantages of intuitionistic fuzzy cross-entropy measures and hesitant fuzzy cross-entropy measures. Finally, the proposed approaches can provide useful and flexible way to deal with multiple attribute decision making problem with different attribute weight situations including attribute weights partly known, completely known and completely unknown.

6. Conclusions

In this paper, some linguistic hesitant intuitionistic fuzzy cross-entropy measures have been proposed, which have the advantages of the intuitionistic fuzzy cross-entropy measures and hesitant fuzzy cross-entropy measures. We first introduce some aggregation operators including LHIFWA operator, LHIFWG operator and the GLHIFWA operator. Then we propose several linguistic hesitant intuitionistic fuzzy cross-entropy measures. The properties of new cross-entropy measures have been studied. Two new multiple attribute decision making methods have been proposed based on the proposed cross-entropy measures, in which attribute values are given as linguistic hesitant intuitionistic fuzzy elements. The supplier selection problem has been presented to illustrate feasibility and practical advantages of the new methods. The prominent feature of the new methods is that they can provide a flexible and useful way to deal with decision making problems within linguistic hesitant intuitionisic fuzzy environment.

Further improvements of our algorithm might include the application of our new method to more complex multiple attribute decision making problem in reality, such as the personnel selection, the product selection, and the environment evaluation, etc.

Acknowledgments

The authors deeply thank anonymous referees and the associate editor for their helpful remarks, which have helped us to improve the paper in both content and style. This work is partly supported by National Natural Science Foundation of China (No. 11401457, 61403298), Postdoctoral Science Foundation of China (2015M582624).

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 10 - 1
Pages: 120 - 139
Publication Date: 2017/01/01
ISSN (Online): 1875-6883
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TY  - JOUR
AU  - Wei Yang
AU  - Yongfeng Pang
AU  - Jiarong Shi
PY  - 2017
DA  - 2017/01/01
TI  - Linguistic hesitant intuitionistic fuzzy cross-entropy measures
JO  - International Journal of Computational Intelligence Systems
SP  - 120
EP  - 139
VL  - 10
IS  - 1
SN  - 1875-6883
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International Journal of Computational Intelligence Systems

Linguistic hesitant intuitionistic fuzzy cross-entropy measures

1. Introduction

2. Linguistic hesitant intuitionistic fuzzy term set

Definition 2.12.

Definition 2.2.

Definition 2.336.

Definition 2.436.

Theorem 136.

Definition 2.536.

Definition 2.636.

Definition 2.7.

Theorem 2.

Definition 2.8.

Theorem 3.

Definition 2.9.

Theorem 4.

3. Cross-entropy measures for LHIFSs

Definition 3.1.

Theorem 5.

Proof.

Example:

4. New MADM methods based on the cross-entropy measures of LHIFEs

5. Numerical example

6. Conclusions

Acknowledgments

References

Cite this article

Definition 2.1².

Definition 2.3³⁶.

Definition 2.4³⁶.

Theorem 1³⁶.

Definition 2.5³⁶.

Definition 2.6³⁶.