Novel entropy and distance measures of linguistic interval-valued q-Rung orthopair fuzzy sets

Gong, Zhiwei

doi:10.3233/JIFS-223729

Novel entropy and distance measures of linguistic interval-valued q-Rung orthopair fuzzy sets

Article type: Research Article

Affiliations: College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou, Fujian, P.R. China

Correspondence: [*] Corresponding author. Zhiwei Gong. E-mail: [email protected].

Keywords: Entropy measure, distance measure, linguistic interval-valued q-Rung orthopair fuzzy set, group decision making

DOI: 10.3233/JIFS-223729

Journal: Journal of Intelligent & Fuzzy Systems, vol. 44, no. 5, pp. 7865-7876, 2023

Published: 04 May 2023

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Abstract

Entropy is an important tool to describe the degree of uncertainty of fuzzy sets. In this study, we first define a new entropy and distance measure in the linguistic q-Rung orthopair fuzzy (LIVqROF) environment, and verify its correctness and rationality. Secondly, in the LIVqROF environment, the new entropy formula is effectively applied to the multi-attribute decision making (MADM) with unknown attribute weights, which provides a new idea for solving the MADM problems. Finally, the feasibility and effectiveness of the proposed method are verified by a numerical example.

1Introduction

Zadeh [1] proposed the fuzzy set (FS), which only has the membership degree, and it is difficult to describe the negative degree of the decision maker to the evaluation information. Atanassov [2, 3] proposed intuitionistic fuzzy set (IFS) on the basis of fuzzy set, and creatively added non-membership degree to the fuzzy sets. However, intuitionistic fuzzy numbers can only be used to describe the evaluation information in the form of real numbers. Due to the complexity of the decision-making environment, the decision making information is often uncertain. Sometimes it is very difficult to express the membership and non-membership degree with real numbers. Therefore, Atanassov [4] also proposed interval intuitionistic fuzzy set (IVIFS), which membership and non-membership are interval numbers rather than real numbers. In real life, the form of linguistic terms is often used to qualitatively evaluate information. Zadeh [5] proposed the fuzzy linguistic method, and Zhang [6] defined the concept of linguistic intuitionistic fuzzy set (LIF). Similar to intuitionistic fuzzy sets, linguistic intuitionistic fuzzy sets are also limited by the representation range. Inspired by the concepts of Pythagorean fuzzy set (PFS) and q-Rung orthopair fuzzy set (qROFS) introduced by Yager in [7] and [8], Garg [9] proposed linguistic Pythagorean fuzzy set (LPFS) and Lin [10] developed the concept of linguistic q-Rung orthopair fuzzy set (LqROFS). Linguistic interval intuitionistic fuzzy numbers can better deal with the uncertainty and fuzziness of decision information, and can truly reflect the objective world. Garg proposed linguistic interval intuitionistic fuzzy set (LIVIFS) and linguistic interval Pythagorean fuzzy set (LIVPFS) in [11] and [12], respectively. Khan [13] established linguistic interval q-Rung orthopair fuzzy set (LIVqROFS).

Information measure plays an important role in fuzzy set theory, and many researchers have explored it from different perspectives. In order to deal with the fuzziness measurement between FSs, De Luca and termini [14] gave an axiomatic definition of fuzzy entropy based on Shannon [15]. Since then, Hooda et al. [16, 17], Mishra et al. [18–21], Pal [22] have studied various entropy measures of FSs. Burillo and Bustance [23] defined the intuitionistic fuzzy entropy to measure the hesitation degree of intuitionistic fuzzy sets. Liu [24] presented the interval intuitionistic fuzzy entropy.Li [25, 26] gave the study of slope entropy and fractional slope entropy. Zhang and Jiang [27], Zhang [28], Wei [29], and Wei and Zhang [30] introduced the entropy measure of IVIFSs and applied it in solving MADM problems. Rani [31] gave a new entropy measure of IVIFS, Kumar et al. [32] gave a new Pythagorean fuzzy entropy, Sonia et al. [33] gave a entropy measures for interval valued intuitionistic fuzzy soft set and Ohlan [34] presented a distance measure on IVIFS. Inspired by their research, this paper defines an entropy measure and distance measure in LIVqROF environment. The relationship between the proposed entropy measure and distance measure are studied.

The rest of this paper is arranged as follows. In Section 2, we introduce some definitions and operation rules on IVIFSs and LIVqROFSs. In Section 3, the axiomatic definitions of entropy and distance in LIVqROF environment are given, a new entropy and distance are proposed, and their rationality and relevance are proved. In Section 4, a multi-attribute decision making method is proposed in LIVqROF environment by using the proposed entropy. A numerical example is used to verify the effectiveness and rationality of the method.

2Preliminary

The linguistic assessment scale is the basis of linguistic decision making. In [35], Bordogna et al. defined an additive linguistic scale.

Definition 2.1.Let

S={sα|α=0,1,⋯,t}

be an additive linguistic assessment scale with a linguistic term subscript non-negative integer, and let S satisfy the following conditions:

(1) The set is ordered: s_α < s_β if α < β.

(2) There is a negative operator: neg (s_α) = s_β such that α + β = t.

where t is an even number and s_α refers to linguistic terms.

After that, Xu [36] extended the discrete term set S to a continuous term set S¯={sα|α∈[0,τ]} , where τ (τ > t) is a sufficiently large natural number. If s_α ∈ S, then we call s_α the original linguistic term, otherwise, we call s_α the virtual linguistic term.

Combining the concept of intuitionistic fuzzy sets [2] with the definition of linguistic terms [35], Zhang [6] first proposed the concept of linguistic intuitionistic fuzzy sets(LIFSs), which is defined as follows.

Definition 2.2.Let X = {x₁, x₂, ⋯ , x_n} be a finite universe discourse and S¯={sα|α∈[0,τ]} be a continuous linguistic term set with a positive integer τ. A LIFS on the set X is defined as

(1)

I={(xi,sθ(xi),sσ(xi))|xi∈X},

where sθ:X→S¯ denotes the degree of linguistic membership and sσ:X→S¯ denotes the degree of linguistic nonmembership of the element x_i ∈ X to the set I, respectively, with the condition that 0 ≤ θ + σ ≤ τ. The degree of linguistic indeterminacy is given as s_π (x_i) = s_τ-θ-σ.

For convenience, the pairs of (s_θ (x_i) , s_σ (x_i)) are called as linguistic intuitionistic fuzzy value(LIFV) or linguistic intuitionistic fuzzy number(LIFN).

On the basis of LIFSs and the concept of Pythagorean fuzzy sets [7], Garg [9] proposed the concept of linguistic Pythagorean fuzzy sets(LPFSs), which is defined as follows.

Definition 2.3.Let X = {x₁, x₂, ⋯ , x_n} be a finite universe discourse and S¯={sα|α∈[0,τ]} be a continuous linguistic term set with a positive integer τ. A LPFS on the set X is defined with the form

(2)

P={(xi,sθ(xi),sσ(xi))|xi∈X},

where sθ(xi),sσ(xi)∈S¯ stand for the linguistic membership degree and linguistic nonmembership degree of the element x_i to P, respectively, with the condition that 0 ≤ θ² + σ² ≤ τ². The degree of linguistic indeterminacy is given as: sπ(xi)=sτ2-θ2-σ2 .

Similarly, the pairs of (s_θ (x_i) , s_σ (x_i)) are called as linguistic Pythagorean fuzzy value(LPFV) or linguistic Pythagorean fuzzy number(LPFN).

On the basis of LIFSs, LPFSs and q-rung orthopair fuzzy sets [8], Lin [10] proposed linguistic q-rung orthopair fuzzy sets(LqROFSs), which is defined as follows.

Definition 2.4.Let X = {x₁, x₂, ⋯ , x_n} be a finite universe discourse and S¯={sα|α∈[0,τ]} be a continuous linguistic term set with a positive integer τ. Then the form of a LqROFS on X is defined as

(3)

L={(xi,sθ(xi),sσ(xi))|xi∈X},

where sθ(xi),sσ(xi)∈S¯ stand for the linguistic membership degree and linguistic nonmembership degree of the element x_i to L, respectively, with the condition that 0 ≤ θ^q + σ^q ≤ τ^q. The degree of linguistic indeterminacy is given as: sπ(xi)=sτq-θq-σqq .

Similarly, the orthopair of (s_θ (x_i) , s_σ (x_i)) is called as linguistic q-rung orthopair fuzzy value(LqROFV) or linguistic q-rung orthopair fuzzy number(LqROFN).

Obviously, when q = 1 and q = 2, the two special cases of LqROFSs are LIFSs and LPFSs, respectively.

In [4], Atanassov and Gargov proposed the concept of interval-valued intuitionistic fuzzy sets(IVIFSs), which is defined as follows.

Definition 2.5.Let X = {x₁, x₂, ⋯ , x_n} be a finite universe discourse. Then the form of a IVIFS on Y is defined as

(4)

Iˆ={(xi,θˆ(xi),σˆ(xi))|xi∈X},

where θˆ(xi)=[θˆL(xi),θˆU(xi)],σˆ(xi)=[σˆL(xi),σˆU(xi)] are all subsets of [0, 1] and are said to be the membership degree and nonmembership degree of the element x_i to Iˆ , respectively, with the condition that 0≤θˆU+σˆU≤1 . The degree of indeterminacy is given as: πˆIˆ(xi)=[πˆL(xi),πˆU(xi)]=[1-θˆU(xi)-σˆU(xi),1-θˆL(xi)-σˆL(xi)] .

For convenience, the pairs of (θˆ(xi),σˆ(xi)) are called as interval-valued intuitionistic fuzzy value(IVIFV) or interval-valued intuitionistic fuzzy number(IVIFN).

If θˆL(xi)=θˆU(xi) and σˆL(xi)=σˆU(xi) , then the given IVIFSs can be reduced to intuitionistic fuzzy sets(IFSs).

In [11, 37], Garg, Liu et al. defined the concept of linguistic interval-valued intuitionistic fuzzy sets(LIVIFSs). In [12], Garg proposed the concept of linguistic interval-valued Pythagorean fuzzy sets(LIVPFSs). On this basis, Khan et al. [13] further proposed linguistic interval-valued q-rung orthopair fuzzy sets(LIVqROFSs), which is defined as follows.

Definition 2.6.Let X = {x₁, x₂, ⋯ , x_n} be a finite universe discourse and S¯={sα|α∈[0,τ]} be a continuous linguistic term set with a positive integer τ. Then the form of a LIVqROFS on X is defined as

(5)

Lˆ={(xi,sθˆ(xi),sσˆ(xi))|xi∈X},

where sθˆ(xi)=[sθˆL(xi),sθˆU(xi)],sσˆ(xi)=[sσˆL(xi), sσˆU(xi)] are all subsets of [s₀, s_τ] and are said to be the membership degree and nonmembership degree of the element x_i to Lˆ , respectively, with the condition that 0≤θˆUq+σˆUq≤τq . The degree of indeterminacy is given as:

sπˆ(xi)=[sπˆL(xi),sπˆU(xi)]=[sτq-θˆUq-σˆUqq,sτq-θˆLq-σˆLqq].

Obviously, when q = 1 and q = 2, the two special cases of LIVqROFSs are LIVIFSs and LIVPFSs, respectively. If sθˆL(xi)=sθˆU(xi) and sσˆL(xi)=sσˆU(xi) , then the given LIVqROFSs can be reduced to an ordinary LqROFSs.

For convenience, the pairs of (sθˆ(xi),sσˆ(xi)) are called as linguistic interval-valued q-rung orthopair fuzzy value (LIVqROFV) or linguistic interval-valued q-rung orthopair fuzzy number (LIVqROFN).

Definition 2.7.[38] Let Aˆi=(sθˆi,sσˆi)=([sθˆiL, sθˆiU],[sσˆiL,sσˆiU]),i=1,2 be two LIVqROFNs. Then

(1) If sθˆ1L=sθˆ2L,sθˆ1U=sθˆ2U,sσˆ1L=sσˆ2L,sσˆ1U&sσˆ2U , then Aˆ1=Aˆ2 ;
(2) If sθˆ1L≤sθˆ2L,sθˆ1U≤sθˆ2U,sσˆ1L≥sσˆ2L,sσˆ1Usσˆ2U , then Aˆ1≤Aˆ2 ;
(3) Negation of Aˆ1 is defined as Aˆ1C=([sσˆ1L,sσˆ1U],[sθˆ1L,sθˆ1U]) .

In [39], Bustince and Burillo proposed the axiomatic definition of entropy measure for IFSs. Analogously, in [24], Liu, Zheng and Xiong proposed the entropy measure of IVIFSs, which is defined as follows.

Definition 2.8. An entropy measure of IVIFS (X) is a real-valued function F : IVIFS (X) → [0, 1], if it satisfies the following conditions:

(F1) F(Aˆ)=0 if and only if Aˆ=([1,1],[0,0]) or Aˆ=([0,0],[1,1]) for each x_i ∈ X;
(F2) F(Aˆ)=1 if and only if [θˆL(xi),θˆU(xi)]=[σˆL(xi),σˆU(xi)] for each x_i ∈ X;
(F3) F(Aˆ)=F(AˆC) , where AˆC={(xi,σˆ(xi),^θ(xi))|xi∈X} ;
(F4) F(Aˆ1)≤F(Aˆ2) if Aˆ1≤Aˆ2 (i.e. θˆ1L(xi)≤θˆ2L(xi),θˆ1U(xi)≤θˆ2U(xi),σˆ1L(xi)≥σˆ2L(xi),σˆ1U(xi)≥σˆ2U(xi) ) when θˆ2L(xi)leqσˆ2L(xi) and θˆ2U(xi)≤σˆ2U(xi) for each x_i ∈ X, or Aˆ2≤Aˆ1 when θˆ2L(xi)≥σˆ2L(xi) and θˆ2U(xi)≥σˆ2U(xi) .

In [40], D üg˘ enci proposed the distance measure on IVIFS(X), defined as follows.

Definition 2.9. Let Aˆ1,Aˆ2∈IVIFS(X) , a mapping D : IVIFS (X) × IVIFS (X) → [0, 1] is called a distance measure between Aˆ1 and Aˆ2 if D(Aˆ1,Aˆ2) satisfies the following properties:

(D1) 0≤D(Aˆ1,Aˆ2)≤1 ;
(D2) D(Aˆ1,Aˆ2)=0 if Aˆ1=Aˆ2 ;
(D3) D(Aˆ1,Aˆ2)=D(Aˆ2,Aˆ1) ;
(D4) If Aˆ1≤Aˆ2≤Aˆ3,Aˆ1,Aˆ2,Aˆ3∈IVIFS(X) , then D(Aˆ1,Aˆ2)≤D(Aˆ1,Aˆ3) and D(Aˆ2,Aˆ3)≤D(Aˆ1,Aˆ3) .

3Entropy and distance measures for LIVqROFSs

Pal [41] proposed the exponential entropy measure of a fuzzy set A (A ∈ F (X) = {(x_i, μ (x_i)) |x_i ∈ X}) as

(6)

E(A)=1n(e-1)∑i=1n(μ(xi)e1-μ(xi)+(1-μ(xi))eμ(xi)-1).

Rani et al. [31] proposed the exponential entropy and distance measure for IVIFSs. Inspired by this, this study will discuss the problem of LIVqROFSs on entropy and distance measure. In the following, we give the axiomatic definition of entropy and distance measure for LIVqROFSs.

Definition 3.1.An entropy measure of LIVqROFS (X) is a real-valued function E : LIVqROFS (X) → [0, 1], if it satisfies the following conditions:

(E1) E(Aˆ)=0 if and only if Aˆ=([sτ,sτ],[s0,s0]) or Aˆ=([s0,s0],[sτ,sτ]) for each x_i ∈ X(i.e. Aˆ is a crisp set);
(E2) E(Aˆ)=1 if and only if [sθˆL(xi),sθˆU(xi)]=[sσˆL(xi),sσˆU(xi)] for each x_i ∈ X;
(E3) E(Aˆ)=E(AˆC) ;
(E4) E(Aˆ1)≤E(Aˆ2) if Aˆ1≤Aˆ2 when sθˆ2L(xi)≤sσˆ2L(xi) and sθˆ2U(xi)≤sσˆ2U(xi) for each x_i ∈ X, or Aˆ2≤Aˆ1 when sθˆ2L(xi)≥sσˆ2L(xi) and sθˆ2U(xi)≥sσˆ2U(xi) .

Definition 3.2. Let Aˆ1,Aˆ2,Aˆ3∈LIVqROFS(X) , a mapping D : LIVqROFS (X) × LIVqROFS (X) → [0, 1] is called a distance measure on LIVqROFS (X), if it satisfies the following properties:

(D1) 0≤D(Aˆ1,Aˆ2)≤1 ;
(D2) D(Aˆ1,Aˆ2)=0 if Aˆ1=Aˆ2 ;
(D3) D(Aˆ1,Aˆ2)=D(Aˆ2,Aˆ1) ;
(D4) If Aˆ1≤Aˆ2≤Aˆ3 , then D(Aˆ1,Aˆ2)≤D(Aˆ1,Aˆ3) and D(Aˆ2,Aˆ3)≤D(Aˆ1,Aˆ3) .

Definition 3.3. Let sα∈S¯ , φ:S¯→[0,τ] is a mapping, such that φ (s_α) = α.

Definition 3.4. Let X={x1,x2,⋯,xn},Aˆ∈LIVqROFS(X) , the entropy measure is defined by

(7)

E(Aˆ)=1nq(e-1){∑i=1n(ΨAˆ(xi)e(1-ΨAˆ(xi))+(1-ΨAˆ(xi))eΨAˆ(xi)-1)q}1q,

where

ΨAˆ(xi)=φ(sθˆL(xi))+φ(sθˆU(xi))+2τ-φ(sσˆL(xi))-φ(sσˆU(xi))4τ.

In the following, we prove the validity of our proposed entropy measure (7).

Theorem 3.5. The mapping E(Aˆ) defined in Equation (7) is a linguistic interval-valued q-Rung orthopair fuzzy entropy measure.

Proof. Obviously to prove that E(Aˆ) is an entropy measure of LIVqROFS(X), we just need to prove that it satisfies (E1)-(E4) in Definition 3.

(E1)Let Aˆ be a crisp set then we have Aˆ=([sτ,sτ],[s0,s0]) or Aˆ=([s0,s0],[sτ,sτ]) , that is [sθˆL(xi),sθˆU(xi)]=[sτ,sτ] and [sσˆL(xi),sσˆU(xi)]=[s0,s0] or [sθˆL(xi),sθˆU(xi)]=[s0,s0] and [sσˆL(xi),sσˆU(xi)]=[sτ,sτ] for each x_i ∈ X, so we get ΨAˆ(xi)=1 or ΨAˆ(xi)=0 . From equation (7), we obtain that E(Aˆ)=0 .

On the other hand, if E(Aˆ)=0 , from the definition of exponential entropy given by Pal in [41], we know that ΨAˆ(xi)=0 or 1 for each x_i ∈ X. i.e.,

(8)

φ(sθˆL(xi))+φ(sθˆU(xi))+2τ-φ(sσˆL(xi))-φ(sσˆU(xi))4τ=0

(9)

φ(sθˆL(xi))+φ(sθˆU(xi))+2τ-φ(sσˆL(xi))-φ(sσˆU(xi))4τ=1

for each x_i ∈ X. Now Equations (8) and (9) hold if either Aˆ=([sτ,sτ],[s0,s0]) or Aˆ=([s0,s0],[sτ,sτ]) i.e. Aˆ is a crisp set.

(E2)Let [sθˆL(xi),sθˆU(xi)]=[sσˆL(xi),sσˆU(xi)] , i.e. sθˆL(xi)=sσˆL(xi),sθˆU(xi)=sσˆU(xi) for each x_i ∈ X. Apply the condition to Equation (7), we can get E(Aˆ)=1 .

On the other hand, let

(10)

f(ΨAˆ(xi))=1(e-1)(ΨAˆ(xi)e(1-ΨAˆ(xi))+(1-ΨAˆ(xi))eΨAˆ(xi)-1),

if E(Aˆ)=1 , that is 1n∑i=1nf(ΨAˆ(xi))=1 , so we can get f(ΨAˆ(xi))=1 for each x_i ∈ X. Take the partial derivative of equation (10) with respect to ΨAˆ(xi) and set it equal to zero, we have

∂f(ΨAˆ(xi))∂(ΨAˆ(xi)))=1(e-1)((1-ΨAˆ(xi))e1-ΨAˆ(xi)-ΨAˆ(xi)eΨAˆ(xi))=0,

it implies (1-ΨAˆ(xi))e1-ΨAˆ(xi)=ΨAˆ(xi)eΨAˆ(xi) for each x_i ∈ X. Since g (x) = xe^x is a bijection function, we get 1-ΨAˆ(xi)=ΨAˆ(xi) , that is ΨAˆ(xi)=0.5 for each x_i ∈ X. Since

∂2f(ΨAˆ(xi))∂(ΨAˆ(xi)))2=1(e-1)((ΨAˆ(xi)-2)e(1-ΨAˆ(xi))-(1+ΨAˆ(xi))eΨAˆ(xi)),

we get

[∂2f(ΨAˆ(xi))∂(ΨAˆ(xi)))2]ΨAˆ(xi)=0.5<0,

for each x_i ∈ X. So f(ΨAˆ(xi)) is a concave function and has a maximum at ΨAˆ(xi)=0.5 . By Equation (7), E(Aˆ) achieves the maximum at ΨAˆ(xi)=0.5 which implies that [sθˆL(xi),sθˆU(xi)]=[sσˆL(xi),sσˆU(xi)] .

(E3) Form AˆC={(xi,[sσˆL(xi),sσˆU(xi)], [sθˆL(xi),sθˆU(xi)])|xi∈X} and Equation (7), we can easily get that E(Aˆ)=E(AˆC) .

(E4)If we take x=φ(sθˆL(xi))+φ(sθˆU(xi)) and y=φ(sσˆL(xi))+φ(sσˆU(xi)) , let

(11)

g(x,y)=(x+2τ-y4τ)e(1-x+2τ-y4τ)+(1-x+2τ-y4τ)e(x+2τ-y4τ)-1,

where x, y ∈ [0, τ].

Taking the partial derivatives of g with respect to x and y, respectively, we get

(12)

∂g∂x=14τ[(y+2τ-x4τ)e(y+2τ-x4τ)-x+2τ-y4τe(x+2τ-y4τ)],

(13)

∂g∂y=14τ[(x+2τ-y4τ)e(x+2τ-y4τ)-y+2τ-x4τe(y+2τ-x4τ)],

Set ∂g∂x=0 and ∂g∂y=0 to find the critical points, we get x = y. From Equation (12) and x = y, we obtain

∂g∂x≥0,when x≤y

and

∂g∂x≤0,when x≥y,

for any x, y ∈ [0, τ].

Thus, g (x, y) is increasing with respect to x when x ≤ y and is decreasing when x ≥ y. Similarly, we obtain that

∂g∂y≤0,when x≤y

and

∂g∂y≥0,when x≥y.

Now if Aˆ1≤Aˆ2 , with sθˆ2L(xi)≤sσˆ2L(xi) and sθˆ2U(xi)≤sσˆ2U(xi) for each x_i ∈ X. Then we have

sθˆ1L(xi)≤sθˆ2L(xi)≤sσˆ2L(xi)≤sσˆ1L(xi),

sθˆ1U(xi)≤sθˆ2U(xi)≤sσˆ2U(xi)≤sσˆ1U(xi).

It implies that sθˆ1L(xi)≤sσˆ1L(xi), and sθˆ1U(xi)≤sσˆ1U(xi) . Thus, from the monotonic of g (x, y) and Equation (7), we obtain E(Aˆ1)≤E(Aˆ2) .

Similarly, when Aˆ2≤Aˆ1 with sθˆ2L(xi)≥sσˆ2L(xi) and sθˆ2U(xi)≥sσˆ2U(xi) for each x_i ∈ X and thus, it can be prove that E(Aˆ1)≤E(Aˆ2) .

By calculating the weight of each element x_i ∈ X, a weighted exponential entropy measure of LIVqROFS is proposed as follows:

(14)

EW(Aˆ)=1e-1{∑i=1nwi(ΨAˆ(xi)e(1-ΨAˆ(xi))+(1-ΨAˆ(xi))eΨAˆ(xi)-1)q}1q,

where w_i ≥ 0, ∑i=1nwi=1 , and

ΨAˆ(xi)=φ(sθˆL(xi))+φ(sθˆU(xi))+2τ-φ(sσˆL(xi))-φ(sσˆU(xi))4τ.

It is clear, if wi=1n , then EW(Aˆ)=E(Aˆ) . It can be easily checked that the mapping EW(Aˆ) , defined by Equation (14), is an entropy measure for LIVqROFS.

For A₁, A₂ ∈ FS (X) , X = {x₁, x₂, ⋯ , x_n}, Fan and Xie [42] proposed the fuzzy information for discrimination of Aˆ1 against Aˆ2 is defined by

(15)

I1(A1,A2)=∑i=1n(1-(1-μA1(xi))e(μA1(xi)-μA2(xi))-μA1(xi)e(μA2(xi)-μA1(xi))).

The fuzzy distance between A₁ and A₂ is defined by

(16)

D1(A1,A2)=I1(A1,A2)+I1(A2,A1).

For Aˆ1,Aˆ2∈IVIFS(X), X={x1,x2,⋯,xn} , Ohlan [34] proposed the interval-valued intuitionistic fuzzy information for discrimination of Aˆ1 against Aˆ2 is defined by

(17)

I2(Aˆ1,Aˆ2)=∑i=1n(1-(1-HAˆ1(xi))e(HAˆ1(xi)-HAˆ2(xi))-HAˆ1(xi)e(HAˆ2(xi)-HAˆ1(xi))),

where

HAˆ1(xi)=θˆ1L(xi)+θˆ1U(xi)+2-σˆ1L(xi)-σˆ1U(xi)4,

HAˆ2(xi)=θˆ2L(xi)+θˆ2U(xi)+2-σˆ2L(xi)-σˆ2U(xi)4.

The distance measure for IVIFSs is defined as

(18)

D2(Aˆ1,Aˆ2)=I2(Aˆ1,Aˆ2)+I2(Aˆ2,Aˆ1).

Inspired by the above distance measures of FSs and IVIFSs, now the distance measures of LIVqROFSs can be defined as the following.

Definition 3.6. Let X={x1,x2,⋯,xn},Aˆ1,Aˆ2∈LIVqROFS(X) , the linguistic interval-valued q-Rung orthopair fuzzy information for discrimination of Aˆ1 against Aˆ2 is defined by

(19)

I(Aˆ1,Aˆ2)={∑i=1n(1-(1-ΨAˆ1(xi))e(ΨAˆ1(xi)-ΨAˆ2(xi))-ΨAˆ1(xi)e(ΨAˆ2(xi)-ΨAˆ1(xi)))q}1q,

where

ΨAˆ1(xi)=φ(sθˆ1L(xi))+φ(sθˆ1U(xi))+2τ-φ(sσˆ1L(xi))-φ(sσˆ1U(xi))4τ,

ΨAˆ2(xi)=φ(sθˆ2L(xi))+φ(sθˆ2U(xi))+2τ-φ(sσˆ2L(xi))-φ(sσˆ2U(xi))4τ.

Theorem 3.7. If q = 1, the relation between E(Aˆ) and I(Aˆ,Bˆ) is given by the following formula

E(Aˆ)=1-en(e-1)I(Aˆ,Bˆ),

where ΨBˆ(xi)=12. .

Proof.

eI(Aˆ,Bˆ)=e∑i=1n(1-(1-ΨAˆ(xi))e(ΨAˆ(xi)-ΨBˆ(xi))-ΨAˆ(xi)e(ΨBˆ(xi)-ΨAˆ(xi)))=e∑i=1n(1-(1-ΨAˆ(xi))e(ΨAˆ(xi)-12)-ΨAˆ(xi)e(12-ΨAˆ(xi)))=∑i=1n(e-(1-ΨAˆ(xi))eΨAˆ(xi)-ΨAˆ(xi)e(1-ΨAˆ(xi)))=n(e-1)-∑i=1n((1-ΨAˆ(xi))eΨAˆ(xi)+ΨAˆ(xi)e(1-ΨAˆ(xi))-1)=n(e-1)-∑i=1n((1-ΨAˆ(xi))eΨAˆ(xi)+ΨAˆ(xi)e(1-ΨAˆ(xi))-1)=-∑i=1n((1-ΨAˆ(xi))eΨAˆ(xi)+ΨAˆ(xi)e(1-ΨAˆ(xi))-1)=n(e-1)-n(e-1)E(Aˆ)=n(e-1)(1-E(Aˆ)).

Thus, we get E(Aˆ)=1-en(e-1)I(Aˆ,Bˆ).

Definition 3.8. Let X={x1,x2,⋯,xn},Aˆi=(sθˆi,sσˆi),i=1,2 be two LIVqROFSs, the distance measure D(Aˆ1,Aˆ2) between the LIVqROFS (X) A₁ and A₂ is defined as follows:

(20)

D(Aˆ1,Aˆ2)=12nq(1-e-1){∑i=1n(2-(1-ΨAˆ1(xi)+ΨAˆ2(xi))e(ΨAˆ1(xi)-ΨAˆ2(xi))-(1-ΨAˆ2(xi)+ΨAˆ1(xi))e(ΨAˆ2(xi)-ΨAˆ1(xi)))q}1q.

In particular, if q = 1, then

(21)

D(Aˆ1,Aˆ2)=I(Aˆ1,Aˆ2)+I(Aˆ2,Aˆ1).

Theorem 3.9. The mapping D(Aˆ1,Aˆ2) defined in Equation (20) is a distance measure on LIVqROFS (X).

Proof. In order for Equation (20) to be qualified as a sensible measure of LIVqROFSs, it must satisfy the conditions (D1)-(D4) in Definition 3.2.

(D1)It is clear that the function h (t) =2 - (1 - t) e^t - (1 + t) e^-t in interval [-1,1] has maximum value f_max (t) =2 - 2e^-1 and minimum value f_min (t) =0. h (t) is decreasing in [-1,0] and is increasing in [1]. Since -1≤ΨAˆ1(xi)-ΨAˆ2(xi)≤1 , we get 0≤D(Aˆ1,Aˆ2)≤1 .

(D2)Let Aˆ1 and Aˆ2 be two LIVqROFSs, if Aˆ1=Aˆ2 then φ(sθˆ1L(xi))=φ(sθˆ2L(xi)) , φ(sθˆ1U(xi))=φ(sθˆ2U(xi)) , φ(sσˆ1L(xi))=φ(sσˆ2L(xi)) , φ(sσˆ1U(xi))=φ(sσˆ2U(xi)) . It is clear ΨAˆ1(xi)-ΨAˆ2(xi)=0 , therefore, D(Aˆ1,Aˆ2)=0 .

(D3) D(Aˆ1,Aˆ2)=D(Aˆ2,Aˆ1) is easily known from Equation (20).

(D4)If Aˆ1≤Aˆ2≤Aˆ3 , we have 0≤ΨAˆ2(xi)-ΨAˆ1(xi)≤ΨAˆ3(xi)-ΨAˆ1(xi) and 0≤ΨAˆ3(xi)-ΨAˆ2(xi)≤ΨAˆ3(xi)-ΨAˆ1(xi) . Then it is easy to see that D(Aˆ1,Aˆ2)≤D(Aˆ1,Aˆ3) and D(Aˆ2,Aˆ3)≤D(Aˆ1,Aˆ3) .

It can be said that D(Aˆ1,Aˆ2) is a distance measure between LIVqROFSs Aˆ1 and Aˆ2 since D(Aˆ1,Aˆ2) satisfies (D1)-(D4).□

4Application of the proposed entropy measure of LIVqROFSs

Now we apply the proposed LIVqROFSs based weighted entropy measure to the Multi-criteria group decision making (MCGDM) problem. Therefore, we introduce a MCGDM model based on LIVqROFS weighted entropy measure.

4.1A method of group decision making of LIVqROFS setting based on the proposed weighted entropy measure

We provide a group decision making method based on LIVqROFSs with the known experts and unknown criteria weights. Let Xˆ={xˆ1,xˆ2,⋯,xˆmitsc} be a set of m feasible alternatives, and Cˆ={cˆ1,cˆ2,⋯,cˆnitsc} be a set of attributes. w = (w₁, w₂, ⋯ , w_n) ^T is the weighting vector of attributes satisfying w_j ≥ 0 and ∑j=1nwj=1 . Let D = {d₁, d₂, ⋯ , d_l} be a set of decision makers with the weighting vector V = (v₁, v₂, ⋯ , v_l) ^T satisfying v_k ≥ 0 and ∑k=1lvk=1 . Assume that each decision maker gives their own decision matrix Rk=(rˆij(k))m×n , where

rˆij(k)=([sθˆijL(k),sθˆijU(k)],[sσˆijL(k),sσˆijU(k)])

is the evaluation result given by decision maker d_k ∈ D under the attribute cˆj∈Cˆ for the alternative xˆi∈Xˆ .

Step 1. Construct the linguistic interval-valued q-rung orthopair fuzzy decision matrices Rk=(rˆij(k))m×n (k = 1, ⋯ , l).

Step 2. Based on the given expert weights, apply the proposed LIVqROF operators(Khan et al. [13]) to aggregate the LIVqROFVs. We choose the LIVqROFWA operator to aggregate all the decision matrices R_k into a collective decision matrix R=(rˆij)m×n , where

rˆij=([sθˆijL,sθˆijU],[sσˆijL,sσˆijU])=LIVqROFWA(rˆij(1),rˆij(2),⋯,rˆij(l))=v1rˆij(1)⊕v2rˆij(2)⊕⋯⊕vlrˆij(l)=([sτ1-∏k=1l(1-(θˆijL(k)τ)q)vkq,sτ1-∏k=1l(1-(θˆijU(k)τ)q)vkq],[sτ∏k=1l(σˆijL(k)τ)vk,sτ∏k=1l(σˆijU(k)τ)vk]).

Step 3. Calculate the weight vector of the criterion. The weight formula of the criterion is determined as follows.

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wj=1-E(cˆj)n-∑j=1nE(cˆj),j=1,2,⋯,n;

where E(cˆj) is the entropy value of the jth attribute, which is calculated by Equation (7).

Step 4. For each alternative xˆi , Equation (14) is used to calculate the weighted LIVqROF information measure.

Step 5. Rank all the alternative xˆi according to the EW(xˆi) , the smaller the value of EW(xˆi) , the better the alternative xˆi .

4.2Illustrative example

In the following, we apply the proposed group decision method to the evaluation of college teachers. Five college teachers {xˆ1,xˆ2,xˆ3,xˆ4,xˆ5} who participated in the evaluation from four main evaluation indexes: Teaching ability ( cˆ1 ), student training and service ( cˆ2 ),cientific research ability ( cˆ3 ), discipline construction ( cˆ4 ). Assume that, three decision makers {d₁, d₂, d₃} evaluated five teachers with four attributes using the following linguistic scale

S={s0=extrmely poor,s1=very poor,s2=poor,s3=slightly poor,s4=fair,s5=slightly good,s6=good,s7=very good,s8=extremly good}.

Step 1. The decision matrices given by decision makers are shown in Tables 1, 2 and 3.

Table 1

Decision matrix R₁ provided by decision maker d₁

	cˆ1	cˆ2	cˆ3	cˆ4
xˆ1	([s₂, s₃] , [s₃, s₄])	([s₃, s₄] , [s₁, s₄])	([s₂, s₆] , [s₁, s₃])	([s₃, s₅] , [s₂, s₄])
xˆ2	([s₂, s₄] , [s₂, s₆])	([s₂, s₆] , [s₁, s₃])	([s₃, s₅] , [s₂, s₅])	([s₃, s₆] , [s₃, s₅])
xˆ3	([s₃, s₅] , [s₂, s₆])	([s₄, s₆] , [s₃, s₅])	([s₃, s₄] , [s₁, s₂])	([s₃, s₅] , [s₂, s₆])
xˆ4	([s₁, s₃] , [s₃, s₄])	([s₃, s₅] , [s₂, s₄])	([s₁, s₃] , [s₃, s₆])	([s₃, s₄] , [s₂, s₆])
xˆ5	([s₂, s₅] , [s₃, s₆])	([s₂, s₄] , [s₂, s₆])	([s₂, s₆] , [s₃, s₅])	([s₂, s₄] , [s₃, s₆])

Table 2

Decision matrix R₂ provided by decision maker d₂

	cˆ1	cˆ2	cˆ3	cˆ4
xˆ1	([s₂, s₄] , [s₃, s₅])	([s₂, s₄] , [s₂, s₆])	([s₂, s₃] , [s₃, s₄])	([s₃, s₅] , [s₁, s₄])
xˆ2	([s₃, s₆] , [s₂, s₅])	([s₃, s₅] , [s₃, s₅])	([s₃, s₆] , [s₃, s₅])	([s₄, s₆] , [s₃, s₄])
xˆ3	([s₃, s₅] , [s₂, s₆])	([s₂, s₅] , [s₃, s₄])	([s₃, s₄] , [s₁, s₃])	([s₃, s₆] , [s₂, s₅])
xˆ4	([s₂, s₃] , [s₃, s₅])	([s₃, s₅] , [s₄, s₅])	([s₁, s₄] , [s₂, s₆])	([s₂, s₆] , [s₂, s₅])
xˆ5	([s₃, s₄] , [s₂, s₅])	([s₃, s₄] , [s₃, s₆])	([s₃, s₆] , [s₃, s₅])	([s₂, s₅] , [s₄, s₅])

Table 3

Decision matrix R₃ provided by decision maker d₃

	cˆ1	cˆ2	cˆ3	cˆ4
xˆ1	([s₁, s₄] , [s₃, s₅])	([s₂, s₄] , [s₃, s₆])	([s₂, s₃] , [s₄, s₅])	([s₃, s₄] , [s₁, s₅])
xˆ2	([s₃, s₆] , [s₂, s₅])	([s₂, s₆] , [s₃, s₅])	([s₂, s₅] , [s₃, s₆])	([s₂, s₄] , [s₃, s₅])
xˆ3	([s₃, s₆] , [s₁, s₄])	([s₃, s₅] , [s₂, s₆])	([s₃, s₅] , [s₂, s₃])	([s₂, s₆] , [s₄, s₅])
xˆ4	([s₂, s₃] , [s₃, s₅])	([s₃, s₄] , [s₃, s₅])	([s₄, s₆] , [s₄, s₅])	([s₁, s₅] , [s₃, s₄])
xˆ5	([s₂, s₄] , [s₃, s₆])	([s₁, s₅] , [s₃, s₆])	([s₃, s₄] , [s₃, s₅])	([s₂, s₅] , [s₁, s₃])

Step 2. The weighting vector v = (0.3, 0.5, 0.2) of decision maker is given. Let q = 3, the LIVqROFWA operator is utilized to aggregate all the decision matrices R_k (k = 1, 2, 3) into a collective decision matrix R, as shown in Table 4.

Table 4

Collective LIVqROF assessment information

	cˆ1	cˆ2	cˆ3	cˆ4
xˆ1	([s_1.8765, s_3.7612] ,	([s_2.3973, s_4.0000] ,	([s_2.0000, s_4.5433] ,	([s_3.0000, s_4.8418] ,
	[s_3.0000, s_4.6762])	[s_1.7617, s_5.3128])	[s_2.2855, s_3.8367])	[s_1.2311, s_4.1826])
xˆ2	([s_2.7753, s_5.6127] ,	([s_2.6008, s_5.5779] ,	([s_2.8545, s_5.5779] ,	([s_3.4835, s_5.7531] ,
	[s_2.0000, s_5.2811])	[s_2.1577, s_4.2896])	[s_2.6564, s_5.1857])	[s_3.0000, s_4.4721])
xˆ3	([s_3.0000, s_5.2560] ,	([s_3.0802, s_5.3706] ,	([s_3.0000, s_4.2529] ,	([s_2.8545, s_5.7611] ,
	[s_1.7411, s_5.5326])	[s_2.7663, s_4.6382])	[s_1.1487, s_2.6564])	[s_2.2974, s_5.2811])
xˆ4	([s_1.8080, s_3.0000] ,	([s_3.0000, s_4.8418] ,	([s_2.4256, s_4.4764] ,	([s_2.3146, s_5.4101] ,
	[s_3.0000, s_4.6762])	[s_3.0673, s_4.6762])	[s_2.5946, s_5.7852])	[s_2.1689, s_5.0506])
xˆ5	([s_2.6008, s_4.3664] ,	([s_2.5316, s_4.2529] ,	([s_4.7083, s_6.0774] ,	([s_2.0000, s_4.7567] ,
	[s_2.4495, s_5.4772])	[s_2.6564, s_6.000])	[s_3.0000, s_5.0000])	[s_2.7808, s_4.7682])

Step 3. By using Equation (22), take q = 3, we get the criteria weight as:

w=(0.2729,0.1427,0.3908,0.1936).

Step 4. Take q = 3, calculate Ew(xˆi) for each the alternative using Equation (14).

Ew(xˆ1)=0.9911,Ew(xˆ2)=0.9944,Ew(xˆ3)=0.9807,Ew(xˆ4)=0.9884,Ew(xˆ5)=0.9859.

Step 5. Since Ew(xˆ3)<Ew(xˆ5)<Ew(xˆ4)<Ew(xˆ1)<Ew(xˆ2). We get a descending order of xˆi .

xˆ3≻xˆ5≻xˆ4≻xˆ1≻xˆ2.

Therefore, we get xˆ3 is the best alternative.

4.3Comparative analyses

In this subsection, we compare the proposed weighted entropy measure model for LIVqROFSs with other decision tools VIKOR model for LIVqROFSs proposed by Khan et al. in [13] and TOPSIS model for LIVqROFSs proposed by Gurmani et al. in [38] to illustrate the effectiveness of our proposed method.

Steps of Khan’s approach are as under:

Step 1. Construct the decision matrices. For better comparison, we select the data in Tables 1, 2 and 3.

Step 2. The collective matrix is presented in Table 4 using the given expert weights.

Step 3. According to Table 4, we get the LIVqROFS positive ideal solution and LIVqROFS negative ideal solution respectively as:

r+={([s3.0000,s5.6127],[s1.7411,s4.6762]),([s3.0802,s5.5779],[s1.7617,s4.2896]),

([s4.7083,s6.0774],[s1.1487,s2.6564]),([s3.4835,s5.7611],[s1.2311,s4.1826])}

r-={([s1.8080,s3.0000],[s3.0000,s5.5326]),([s2.3973,s4.0000],[s3.0673,s6.0000]),

([s2.0000,s4.2529],[s3.0000,s5.7852]),([s2.0000,s4.7567],[s3.0000,s5.2811])}

Step 4. For better comparison, we give the criteria weight as:

w=(0.2729,0.1427,0.3908,0.1936).

Step 5. Calculate the distance between the LIVqROFS positive ideal solution r⁺ and the alternative xˆi , and the distance between the LIVqROFS negative ideal solution r^- and the alternative xˆi respectively as:

d1+(xˆ1,r+)=0.1570,d1-(xˆ1,r-)=0.1235;

d2+(xˆ2,r+)=0.1010,d2-(xˆ2,r-)=0.1065;

d3+(xˆ3,r+)=0.1537,d3-(xˆ3,r-)=0.1852;

d4+(xˆ4,r+)=0.1437,d4-(xˆ4,r-)=0.0650;

d5+(xˆ5,r+)=0.1072,d5-(xˆ5,r-)=0.1172.

Step 6. Calculate the relative closeness coefficient of each alternatives as follows:

Λ(xˆ1)=0.4403,Λ(xˆ2)=0.5133,Λ(xˆ3)=0.5465,Λ(xˆ4)=0.3115,Λ(xˆ5)=0.5222.

Step 7.Rank the relative closeness coefficients in descending order to select the best alternative.

Λ(xˆ3)>Λ(xˆ5)>Λ(xˆ2)>Λ(xˆ1)>Λ(xˆ4).

Therefore, we get xˆ3 is the best alternative.

Steps of Gurmani’s approach are as under:

In order to make the comparison result more reasonable, we chose the same data, so steps 1, 2, 3 and 4 are the same as Khan’s approach.

Step 5. Calculate the group utility measure Φ_i and individual regret measure y_i of alternatives xˆi respectively as:

Φ1=0.5971,Φ2=0.3393,Φ3=0.4214,Φ4=0.7355,Φ5=0.5832.

y1=0.2421,y2=0.2375,y3=0.1978,y4=0.3657,y5=0.1833.

Step 6. Calculate the compromise measure Qiitsc of alternatives xˆi .

Q1=0.4865,Q2=0.1485,Q3=0.1434,Q4=1,Q5=0.3077.

Step 7. Rank the compromise measure in ascending order to select the best alternative.

Q3<Q2<Q5<Q1<Q4.

Therefore, we get xˆ3 is the best alternative.

The comparison results of the proposed approach and the existing approachs are shown in Fig. 1.

Fig. 1

The ranking of alternatives obtained by the three approaches.

According to the optimal results of the three methods, there are some similarities among the three methods, but there are also differences among the three methods from the final ranking of the alternatives. The group decision making method based on weighted exponential entropy measure in LIVqROFS environment is more efficient, simple and consistent than the existing measures and methods to solve the decision problem.

5Conclusion

In this study, a new multi-attribute group decision making method is proposed under the linguistic interval-valued q-rung orthopair fuzzy environment. We propose novel entropy and distance measure for linguistic interval-valued q-rung orthopair fuzzy sets. The main innovations and advantages of this study are shown as follows:

(1) The axiomatic definitions of entropy and distance are proposed under the LIVqROF environment. It provides some important and reliable reference results for the subsequent research of complex information measures and distance measures in the LIVqROF environment.

(2) The proposed information measure is applied to the problem of multi-attribute decision making. An effective linguistic multi-attribute decision making model is established, which enriches the theory and method of qualitative decision making in the LIVqROF environment.

Theoretical analysis and numerical results show that the method is simple and intuitive without information loss. The model can be applied to medical diagnosis, personnel assessment, pollution treatment, quality evaluation and other fields. We will continue to extend the proposed method to complex decision information environment.

Conflicts of interest

I declare that I have no financial and personal relationships with other people or organizations that can inappropriately influence my work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Funding

This work was supported by the Natural Science Foundation of Fujian Province (No. 2020J01576) and the Science and Technology Innovation Special Fund Project of Fujian agriculture and Forestry University (No. CXZX2020110A).

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Abstract

1Introduction

2Preliminary

(1)

(2)

(3)

(4)

(5)

3Entropy and distance measures for LIVqROFSs

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

4Application of the proposed entropy measure of LIVqROFSs

4.1A method of group decision making of LIVqROFS setting based on the proposed weighted entropy measure

(22)

4.2Illustrative example

Table 1

Table 2

Table 3

Table 4

4.3Comparative analyses

Fig. 1

5Conclusion

Conflicts of interest

Funding

References

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