International Journal of Computational Intelligence Systems

Volume 10, Issue 1, 2017, Pages 569 - 592

A theoretical development on the entropy of interval-valued intuitionistic fuzzy soft sets based on the distance measure

Authors
Yaya Liu1, yayaliu@my.swjtu.edu.cn, Junfang Luo1, junfangluo@163.com, Bing Wang2, bwang@my.swjtu.edu.cn, Keyun Qin1, *, keyunqin@263.net
College of Mathematics, Southwest Jiaotong University, Cheng Du, 610000, Sichuan, PR China
College of Information Science and Technology, Southwest Jiaotong University, Cheng Du, 610000, Sichuan, PR China
*Corresponding author.
Corresponding Author
Received 16 May 2016, Accepted 19 December 2016, Available Online 1 January 2017.
DOI
10.2991/ijcis.2017.10.1.39How to use a DOI?
Keywords
interval-valued intuitionistic fuzzy soft set; entropy; similarity measures; inclusion measures; fuzzy equivalences
Abstract

In this work, the axiomatical definition of similarity measure, distance measure and inclusion measure for interval-valued intuitionistic fuzzy soft set (IVIFSSs) are given. An axiomatical definition of entropy measure for IVIFSSs based on distance is firstly proposed, which is consistent with the axiomatical definition of fuzzy entropy of fuzzy sets introduced by De Luca and Termini. By different compositions of aggregation operators and a fuzzy negation operator, we obtain eight general formulae to calculate the distance measures of IVIFSSs based on fuzzy equivalences. Then we discuss the relationships among entropy measures, distance measures, similarity measures and inclusion measures of IVIFSSs. We prove that the presented entropy measures can be transformed into the similarity measures and the inclusion measures of IVIFSSs based on fuzzy equivalences.

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

Many new set theories treating imprecision and uncertainty have been proposed since fuzzy sets were introduced by Zadeh1. Atanassov’s intuitionistic fuzzy sets3 (IFSs), vague sets4 and interval-valued fuzzy sets20, 21 (IVFSs), as extensions of classic fuzzy set theory, are proved to be useful in dealing with imprecision and uncertainty. As a combining concept of IFSs and IVFSs, interval-valued intuitionistic fuzzy sets (IVIFSs) introduced by Atanassov5 greatly furnishes the additional capability to model non-statistical uncertainty by providing a membership interval and a non-membership interval. Therefore, IVIFSs play a significant role in the uncertain system and receives much attention. The concept of soft set theory, which can be used as a general mathematical tool for dealing with uncertainty, is initiated by Molodtsov6 in 1999. Since it has been pointed out that classical soft sets are not appropriate to deal with imprecise and fuzzy parameters, some fuzzy (or intuitionistic fuzzy, interval-valued fuzzy) extensions of soft set theory, yielding fuzzy (or intuitionistic fuzzy, interval-valued fuzzy) soft set theory6, 7, 8, 9, 10, 11 has been presented to deal with imprecise and fuzzy parameters. Recently, by combining the interval-valued intuitionistic fuzzy sets and soft sets, Jiang et al 12 propose a new soft set model: interval-valued intuitionistic fuzzy soft sets (IVIFSSs). Intuitively, interval-valued intuitionistic fuzzy soft set can be regarded as an interval-valued fuzzy extension of the intuitionistic fuzzy soft set 8,9,10 or an intuitionistic fuzzy extension of the interval-valued fuzzy soft set11.

Some scholars have already noticed and studied entropy measures based on distance for fuzzy sets and extensions of fuzzy sets. Mi13 extended De Lucas axioms2 to introduce an entropy of fuzzy set based on fuzzy distance. Later, Farhadinia16 propose a class of entropies of IVFSs based on the distance measure and investigate the relationship between the entropy measure and the similarity measure. Zhang et al17 propose an axiomatical definition of entropy measure for IVIFSs based on distances and discuss the relationship between entropy with similarity and inclusion measure. However, few scholars have paid attention to the entropy measures based on distance for fuzzy (or intuitionistic fuzzy, interval-valued fuzzy, interval-valued intuitionistic fuzzy) extensions of soft sets yet. In this work, we provide an axiomatic definition of entropy based on distance for IVIFSSs and discuss the relationship between entropy measure with similarity, distance and inclusion measures for IVIFSSs. There are several reasons that motivate us to do this research. Firstly, although there are a number of researches regarding entropy measures for hybrid fuzzy set theory, few literatures studied the entropy measure of IVIFSSs; Secondly, the uncertain measures of IVIFSSs have great application potential in many fields such as uncertain system control, decision-making and pattern recognition; Thirdly, the study of relationships between different measure benefits us in achieving as more information as possible through each measure. This new extension not only provides a significant addition to existing theories for handling uncertainties, but also leads to potential areas of further research field and pertinent applications. It is worth noticing that we give a method to construct the distance measures of IVIFSSs by aggregating fuzzy equivalencies and prove that the presented entropy measures can be transformed into the similarity measures and the inclusion measures of IVIFSSs based on fuzzy equivalences.

The structure of this paper is as follows. Section 2 reviews some concepts which are necessary for our paper. Section 3 provides the axiomatic definitions of similarity measure, distance measure and inclusion measure of IVIFSSs, an information entropy based on distance is also introduced to estimate uncertainty in IVIFSSs. Corresponding calculate formulae or construction methods of these measures are also given. In section 4, we investigate the relationship between the entropy measure and other uncertain measures of IVIFSSs, prove that both the similarity measures and the inclusion measures of IVIFSSs can be constructed by entropy measures of IVIFSSs. In section 5, an application of the entropy and the distance measure of IVIFSSs is given. This paper is concluded in Section 6.

2. Preliminaries

In this section, we shall recall several definitions which are necessary for our paper.

Let U be the universe of discourse and P be the set of all possible parameters related to the objects in U. In the following discussion, we assume that both U and P are nonempty finite sets.

Definition 1.

6 Let 𝒫(U) be the power set of U, a pair (F, A) is called a soft set in the universe U, where AP and F is a mapping given by

F:A𝒫(U)
In other words, the soft set is not a kind of set in ordinary sense, but a parameterized family of subsets of the set U. For any parameter eiA, F(ei) ⊆ U may be considered as the set of ei − approximate elements of the soft set (F, A).

Interval-valued intuitionistic fuzzy set was first introduced by Atanassov and Gargov18. It is characterized by an interval-valued membership degree and an interval-valued non-membership degree.

Definition 2.

5, 18 An interval-valued intuitionistic fuzzy set on a universe U is an object of the form A = {(x, uA(x), vA(x))/xU}, where uA : UInt([0, 1]) and vA : UInt([0, 1]) satisfy the following condition: ∀xU, sup(uA(x)) + sup(vA(x)) ⩽ 1. (Int([0, 1]) stands for the set of all closed subintervals of [0, 1]).

The class of all interval-valued intuitionistic fuzzy sets (IVIFSs) on U will be denoted by IVIFS(U).

For an arbitrary set A ⊆ [0, 1], define Ā = supA and A = infA. The interval-valued intuitionistic fuzzy set A can be written as

A={(x,[u_A(x),u¯A(x))],[v_A(x),v¯A(x))]/xU}
with the condition: 0u¯A(x)+v¯A(x)1 for all xU.

The union, intersection and complement of the interval-valued intuitionistic fuzzy sets are defined as follows: let A, BIVIFS(U), then

  1. 1)

    the union of A and B is denoted by AB where

    AB={x,[sup(u_A(x),u_B(x)),sup(u¯A(x),u¯B(x))],[inf(v_A(x),v_B(x)),inf(v¯A(x),v¯B(x))]|xU}.

  2. 2)

    the intersection of A and B is denoted by AB where

    AB={x,[inf(u_A(x),u_B(x)),inf(u¯A(x),u¯B(x)],[sup(v_A(x),v_B(x)),sup(v¯A(x),v¯B(x))]|xU}.

  3. 3)

    the complement of A is denoted by AC where

    AC={x,vA(x),uA(x)}.

Atanassov5 shows that AB, AB and AC are again interval-valued intuitionistic fuzzy sets.

Jiang et al.12 define interval-valued intuitionistic fuzzy soft sets (IVIFSSs) by combining interval-valued intuitionistic fuzzy sets and soft sets, and then give some operations on IVIFSSs.

Definition 3.

12 A pair (F, A) is an interval-valued intuitionistic fuzzy soft set over U, where AP and F is a mapping given by

F:AIVIFS(U)

The class of all interval-valued intuitionistic fuzzy soft sets over U will be denoted by IVIFSS(U).

An interval-valued intuitionistic fuzzy soft set is a parameterized family of interval-valued intuitionistic fuzzy subsets of U, thus, its universe is the set of all interval-valued intuitionistic fuzzy sets of U, i.e., IVIFS(U). For any parameter ei ∈ A , F(ei) is referred as the interval-valued intuitionistic fuzzy value set of parameter ei, it can be written as:

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU}
with the condition 0u¯F(ei)(xj)+v¯F(ei)(xj)1 . Here, uF(ei)(xj) is the interval-valued fuzzy membership degree that object xj holds on parameter ei, vF(ei)(xj) is the interval-valued fuzzy non-membership degree that object xj holds on parameter ei.

Definition 4.

19 Let [a1, b1], [a2, b2] ∈ Int([0, 1]), we define

[a1, b1] ⩽ [a2, b2]; iff a1a2; b1b2;

[a1, b1] ≼ [a2, b2]; iff a1a2; b1b2;

[a1, b1] = [a2, b2]; iff a1 = a2; b1 = b2.

Definition 5.

12 Let U be an initial universe and P be a set of parameters. Suppose that A, BP, (F, A) and (G, B) are two interval-valued intuitionistic fuzzy soft sets, we say that (F, A) is an interval-valued intuitionistic fuzzy soft subset of (G, B) if and only if

  1. (1)

    AB;

  2. (2)

    eiA, F(ei) is an interval-valued intuitionistic fuzzy subset of G(ei), that is, [uF(ei)(xj), ūF(ei)(xj)] ⩽ [uG(ei)(xj), ūG(ei)(xj)] and [v_F(ei)(xj),v¯F(ei)(xj)][v_G(ei)(xj),v¯G(ei)(xj)] for all xjU, eiA.

This relationship is denoted by (F, A) ⊆ (G, B). (F, A) and (G, B) are said to be intuitionistic equal if and only if (F, A) ⊇ (G, B) and (F, A) ⊆ (G, B) at the same time, we write (F, A) = (G, B).

The union and intersection of the interval-valued intuitionistic fuzzy soft sets are defined12 as follows: let (F, A), (G, B) ∈ IVIFSS(U), then

  1. 1)

    The union of (F, A) and (G, B) is an interval-valued intuitionistic fuzzy soft set (H, C), where

    • C = AB and eiC.

    • uH(ei)(xj) = uF(ei)(xj), vH(ei)(xj) = vF(ei)(xj), if eiA\B, xjU;

    • uH(ei)(xj) = uG(ei)(xj), vH(ei)(xj) = vG(ei)(xj), if eiB\A, xjU;

    • uH(ei)(xj) = [sup(uF(ei)(xj), uG(ei)(xj)), sup(ūF(ei)(xj), ūG(ei)(xj))],

    • vH(ei)(xj)=[inf(v_F(ei)(xj),v_G(ei)(xj)),inf(v¯F(ei)(xj),v¯G(ei)(xj))] if eiAB, xjU.

      We denote it by (F, A) ∪ (G, B) = (H, C).

  2. 2)

    The intersection of (F, A) and (G, B) is an interval-valued intuitionistic fuzzy soft set (H, C), where C = AB and eiC.

    • uH(ei)(xj) = uF(ei)(xj), vH(ei)(xj) = vF(ei)(xj), if eiA\B, xjU;

    • uH(ei)(xj) = uG(ei)(xj), vH(ei)(xj) = vG(ei)(xj), if eiB\A, xjU;

    • uH(ei)(xj) = [inf (uF(ei)(xj), uG(ei)(xj)), inf (ūF(ei)(xj), ūG(ei)(xj))],

    • vH(ei)(xj)=[sup(v_F(ei)(xj),v_G(ei)(xj)),sup(v¯F(ei)(xj),v¯G(ei)(xj))] ,

      if eiAB, xjU.

      We denote it by (F, A)∩(G, B) = (H, C).

Definition 6.

The relative complement of an interval-valued intuitionistic fuzzy soft set (F, A) is denoted by (F, A)C and is defined by (F, A)C = (FC, A), where FC : AIVIFS(U) is a mapping given by FC(ei) = {⟨xj, vF(ei)(xj), uF(ei)(xj)⟩|xjU} for all eiA.

Definition 7.

12 An interval-valued intuitionistic fuzzy soft set (F, A) over U is said to be a null interval-valued intuitionistic fuzzy soft set denoted by (∅, A), if uF(ei)(xj) = [0, 0], vF(ei)(xj) = [1, 1] for all eiA, xjU.

Definition 8.

12 An interval-valued intuitionistic fuzzy soft set (F, A) over U is said to be an absolute interval-valued intuitionistic fuzzy soft set denoted by (U, A), if uF(ei)(xj) = [1, 1], vF(ei)(xj) = [0, 0] for all eiA, xjU.

3. The distance, similarity, inclusion measure and entropy of IVIFSSs

3.1. Axiomatic definitions

In this subsection, we extend the axiomatic definitions of the distance, similarity, inclusion measure and entropy of IVIFSs in Ref. 17 to IVIFSSs.

Definition 9.

Let (F, P), (G, P) and (H, P) be interval-valued intuitionistic fuzzy soft sets over U, i.e., (F, P), (G, P), (H, P) ∈ IVIFSS(U). Let D be a mapping D : IVIFSS(U) × IVIFSS(U) → [0, 1]. If D((F, P), (G, P)) satisfies the following properties ((1)(4)):

  1. (1)

    D((F, P), (F, P)C) = 1, if (F, P) is a classical soft set;

  2. (2)

    D((F, P), (G, P)) = 0, iff (F, P) = (G, P);

  3. (3)

    D((F, P), (G, P)) = D((G, P), (F, P));

  4. (4)

    D((F, P), (H, P)) ⩾ D((F, P), (G, P)) and D((F, P), (H, P)) ⩾ D((G, P), (H, P)), if (F, P) ⊆ (G, P) ⊆ (H, P).

Then D((F, P), (G, P)) is a distance measure between interval-valued intuitionistic fuzzy soft sets (F, P) and (G, P).

Definition 10.

Let (F, P), (G, P) and (H, P) be interval-valued intuitionistic fuzzy soft sets over U, i.e., (F, P), (G, P), (H, P) ∈ IVIFSS(U). Let S be a mapping S : IVIFSS(U) × IVIFSS(U) → [0, 1]. If S((F, P), (G, P)) satisfies the following properties ((1)(4)):

  1. (1)

    S((F, P), (F, P)C) = 0, if (F, P) is a classical soft set;

  2. (2)

    S((F, P), (G, P)) = 1, iff (F, P) = (G, P);

  3. (3)

    S((F, P), (G, P)) = S((G, P), (F, P));

  4. (4)

    S((F, P), (H, P)) ⩽ S((F, P), (G, P)) and S((F, P), (H, P)) ⩽ S((G, P), (H, P)), if (F, P) ⊆ (G, P) ⊆ (H, P).

Then S((F, P), (G, P)) is a similarity measure between interval-valued intuitionistic fuzzy soft sets (F, P) and (G, P).

Definition 11.

A real function J : IVIFSS(U) × IVIFSS(U) → [0, 1] is named as the inclusion measure of interval-valued intuitionistic fuzzy soft sets, if J has the following properties:

  1. (1)

    If (F, P) = (U, P), (G, P) = (∅, P), then J((F, P), (G, P)) = 0;

  2. (2)

    J((F, P), (G, P)) = 1, iff (F, P) ⊆ (G, P);

  3. (3)

    If (F, P) ⊆ (G, P) ⊆ (H, P), then J((H, P), (F, P)) ⩽ J((G, P), (F, P)) and J((H, P), (F, P)) ⩽ J((H, P), (G, P)).

Then J((F, P), (G, P)) is called an inclusion measure of interval-valued intuitionistic fuzzy soft sets.

Definition 12.

Let (Q, P) be an interval-valued intuitionistic fuzzy soft set on U, s.t. for ∀eiP, Q(ei) = {⟨xj, [1/2, 1/2], [1/2, 1/2]⟩|xjU}. A real function I : IVIFSS(U)→[0, 1] is called an entropy for interval-valued intuitionistic fuzzy soft sets, if I has the following properties:

  1. (1)

    I((F, P)) = 0 if (F, P) is a classical soft set;

  2. (2)

    I((F, P)) = 1 iff uF(ei)(xj) = vF(ei)(xj) = [1/2, 1/2], ∀eiP, xjU;

  3. (3)

    I((F, P)) = I((F, P)C);

  4. (4)

    I((F, P)) ⩽ I((G, P)), if D((F, P), (Q, P)) ⩾ D((G, P), (Q, P)).

Here, the requirement (2) implies that entropy of (F, P) will be maximum if (F, P) is equal to (Q, P); the requirement (4) implies that the closer an interval-valued intuitiionistic fuzzy soft set (F, P) is to (Q, P), the more entropy of (F, P) should decrease.

3.2. Some general formulae to construct the distance measure of IVIFSSs

Before giving some general formulae to construct the distance measure of IVIFSSs, we review the notions of aggregation operators and equivalence operators.

Definition 13.

14 A function M : ∪nN[0, 1]n → [0, 1] is an aggregation operator if it satisfies the following properties: for each nN and xi, yi ∈ [0, 1],

  1. (1)

    M(xi) = xi.

  2. (2)

    M(0,0,,0ntimes)=0 .

  3. (3)

    M(1,1,,1ntimes)=1 .

  4. (4)

    M (x1, x2, …xn) ⩽ M (y1, y2, …yn) whenever xiyi, ∀i ∈ {1, 2, …n}.

    severe

    This definition allows us to introduce the following notions:

    An aggregation operator M : ∪nN[0, 1]n → [0, 1] is called a severe-aggregation operator if it satisfies properties: for each nN and xi ∈ [0, 1](i = {1, 2, …, n}),

  5. (5)

    M (x1, x2, …xn) < 1 if xi < 1, ∀i ∈ {1, 2, …n}.

  6. (6)

    M (x1, x2, …xn) > 0 if xi > 0, ∀i ∈ {1, 2, …n}.

    An aggregation operator M : ∪nN[0, 1]n → [0, 1] is called a top-aggregation operator if it satisfies property: for each nN and xi ∈ [0, 1](i = {1, 2, . . ., n}),

  7. (7)

    M (x1, x2, …xn) = 1 ⇔ xi = 1, ∀i ∈ {1, 2, …n}.

    An aggregation operator M : ∪nN[0, 1]n → [0, 1] is called a bottom-aggregation operator if it satisfies property: for each nN and xi ∈ [0, 1](i = {1, 2, …, n}),

  8. (8)

    M (x1, x2, …xn) = 0 ⇔ xi = 0, ∀i ∈ {1, 2, …n}.

    An aggregation operator M : ∪nN[0, 1]n → [0, 1] is called an idempotent-aggregation operator if it satisfies property: for each nN and x ∈ [0, 1],

  9. (9)

    M(x,x,,xntimes)=x for ∀x ∈ [0, 1].

Example 1.

As examples of the severe-aggregation operators, we take: for each nN and xi ∈ [0, 1](i = {1, 2, …, n}),

  1. (1)

    M(x1,x2,xn)=1ni=1nxi .

  2. (2)

    M (x1, x2, …xn) = λmin(x1, x2, …, xn) + (1 − λ)max(x1, x2, …, xn) with λ ∈ [0, 1].

  3. (3)

    M (x1, x2, …xn) = max(x1, x2, …, xn) / (max(x1, x2, …, xn) + max(1 − x1, 1 − x2, …, 1 − xn)).

As examples of the top-aggregation operators, we take: for each nN and xi ∈ [0, 1] (i = {1, 2, …, n}),

  1. (1)

    M(x1,x2,,xn)=(x1p+x2p++xnpn)1p,p1 , p ⩾ 1.

  2. (2)

    M(x1,x2,,xn)=x1px2pxnp,p1 , p ⩾ 1.

As examples of the bottom-aggregation operators, we take: for each nN and xi ∈ [0, 1](i = {1, 2, …, n}),

  1. (1)

    M(x1,x2,,xn)=(x1p+x2p++xnpn)1p,p1 , p ⩾ 1.

  2. (2)

    M(x1,x2,,xn)=x1px2pxnp,p1 , p ⩾ 1.

As examples of the idempotent-aggregation operators, we take: for each nN and xi ∈ [0, 1](i = {1, 2, . . ., n}),

  1. (1)

    M(x1,x2,,xn)=(x1p+x2p++xnpn)1p,p1 , p ⩾ 1.

  2. (2)

    M(x1, x2, …xn) = λmin(x1, x2, …, xn) + (1−λ)max(x1, x2, …, xn) with λ ∈ [0, 1].

  3. (3)

    M(x1, x2, …, xn) = x1x2 ∧…∧xn.

  4. (4)

    M(x1, x2, …, xn) = x1x2 ∨…∨xn.

Definition 14.

15 A function E : [0, 1]2 → [0, 1] is called a fuzzy equivalence if it satisfies the following properties:

  1. (1)

    E(x, y) = E(y, x) for all x, y ∈ [0, 1].

  2. (2)

    E(x, x) = 1 for all x ∈ [0, 1].

  3. (3)

    E(0, 1) = E(1, 0) = 0.

  4. (4)

    For all x, y, x′, y′ ∈ [0, 1], if xx′y′y, then E(x, y) ⩽ E(x′, y′).

In this article, we strength condition (2) to (2):

(2) For all x, y ∈ [0, 1], E(x, y) = 1 iff x = y.

Definition 15.

22 If a decreasing function n : [0, 1] → [0, 1] satisfies the boundary conditions n(0) = 1 and n(1) = 0, then n is called a fuzzy negation.

If a fuzzy negation n : [0, 1] → [0, 1] is a strictly decreasing function, it is called a strict fuzzy negation in this work.

By the compositions of three severe-aggregation operators and a strict fuzzy negation operator, we obtain eight general formulae to calculate the distance measures of IVIFSSs based on fuzzy equivalencies.

Definition 16.

Given U = {x1, x2, …, xn} and P = {e1, e2, …, em}. Let Mk (k = 1, 2, 3) be severe-aggregation operators. Let El (l =1, 2, 3, 4) be fuzzy equivalence operators and f be a strict fuzzy negation. Suppose Dq (q = 1, 2, …, 8) : IVIFSS(U) × IVIFSS(U) → [0, 1] are functions defined for all (F, P), (G, P) ∈ IVIFSS(U) as follows: for any eiP, xjU,

D1((F,P),(G,P))=M1j=1n(M2i=1m(M3(f(E1(u¯F(ei)(xj),u¯G(ei)(xj)),f(E2(u_F(ei)(xj),u_G(ei)(xj)),f(E3(v¯F(ei)(xj),v¯G(ei)(xj)),f(E4(v_F(ei)(xj),v_G(ei)(xj)))))).
D2((F,P),(G,P))=M2i=1m(M1j=1n(M3(f(E1(u¯F(ei)(xj),u¯G(ei)(xj)),f(E2(u_F(ei)(xj),u_G(ei)(xj)),f(E3(v¯F(ei)(xj),v¯G(ei)(xj)),f(E4(v_F(ei)(xj),v_G(ei)(xj)))))).
D3((F,P),(G,P))=M1j=1n(M2i=1m(f(M3(E1(u¯F(ei)(xj),u¯G(ei)(xj)),(E2(u_F(ei)(xj),u_G(ei)(xj)),E3(v¯F(ei)(xj),v¯G(ei)(xj)),(E4(v_F(ei)(xj),v_G(ei)(xj)))))).
D4((F,P),(G,P))=M2i=1m(M1j=1m(f(M3(E1(u¯F(ei)(xj),u¯G(ei)(xj)),E2(u_F(ei)(xj),u_G(ei)(xj)),E3(v¯F(ei)(xj),v¯G(ei)(xj)),E4(v_F(ei)(xj),v_G(ei)(xj)))))).
D5((F,P),(G,P))=M1j=1n(f(M2i=1m(M3(E1(u¯F(ei)(xj),u¯G(ei)(xj)),E2(u_F(ei)(xj),u_G(ei)(xj)),E3(v¯F(ei)(xj),v¯G(ei)(xj)),E4(v_F(ei)(xj),v_G(ei)(xj)))))).
D6((F,P),(G,P))=M2i=1m(f(M1j=1n(M3(E1(u¯F(ei)(xj),u¯G(ei)(xj)),E2(u_F(ei)(xj),u_G(ei)(xj)),E3(v¯F(ei)(xj),v¯G(ei)(xj)),E4(v_F(ei)(xj),v_G(ei)(xj)))))).
D7((F,P),(G,P))=f(M1j=1n(M2i=1m(M3(E1(u¯F(ei)(xj),u¯G(ei)(xj)),E2(u_F(ei)(xj),u_G(ei)(xj)),E3(v¯F(ei)(xj),v¯G(ei)(xj)),E4(v_F(ei)(xj),v_G(ei)(xj)))))).
D8((F,P),(G,P))=f(M2i=1m(M1j=1n(M3(E1(u¯F(ei)(xj),u¯G(ei)(xj)),E2(u_F(ei)(xj),u_G(ei)(xj)),E3(v¯F(ei)(xj),v¯G(ei)(xj)),f(E4(v_F(ei)(xj),v_G(ei)(xj)))))).

Theorem 1.

Dq((F, P), (G, P))(q ∈ {1, 2, …8}) in Definition 16 are distance measures between interval-valued intuitionistic fuzzy soft sets (F, P) and (G, P).

Proof.

  1. (1)

    If (F, P) is a classical soft set, we have

    • [uF(ei)(xj), ūF(ei)(xj)]=[1, 1], [v_F(ei)(xj),v¯F(ei)(xj)]=[0,0] or

    • [uF(ei)(xj), ūF(ei)(xj)]=[0, 0], [v_F(ei)(xj),v¯F(ei)(xj)]=[1,1] , ∀eiP, xjU.

    Then we get

    • [uFC(ei)(xj), ūFC(ei)(xj)]=[0, 0], [v_FC(ei)(xj),v¯FC(ei)(xj)]=[1,1] or

    • [uFC(ei)(xj), ūFC(ei)(xj)]=[1, 1], [v_FC(ei)(xj),v¯FC(ei)(xj)]=[0,0] , ∀eiP, xjU.

    By property (3) of fuzzy equivalence operators, we have

    E1(u¯F(ei)(xj),u¯FC(ei)(xj))=E2(u_F(ei)(xj),u_FC(ei)(xj))=E3(v¯F(ei)(xj),v¯FC(ei)(xj))=E4(v_F(ei)(xj),v_FC(ei)(xj))=0,eiP,xjU.

    Thus, we have

    • Dq ((F, P), (G, P)) = 1 (q ∈ {1, 2, …8}).

  2. (2)

    If (F, P) = (G, P), it is obviously that Dq ((F, P), (G, P)) = 0 (q ∈ {1, 2, …8}).

    For q ∈ {1, 2, …8}, assume that Dq ((F, P), (G, P)) = 0, if there exists a eiP, and a xjU, s.t.

    • E1(ūF(ei)(xj), ūG(ei)(xj))<1 or

    • E2(uF(ei)(xj), uG(ei)(xj))<1 or

    • E3(v¯F(ei)(xj),v¯G(ei)(xj))<1 or

    • E4(ūF(ei)(xj), vG(ei)(xj))<1,

    since f is a strict fuzzy negation, we get Dq ((F, P), (G, P)) > 0. It is a contradiction.

    So, we have

    E1(u¯F(ei)(xj),u¯G(ei)(xj))=E2(u_F(ei)(xj),u_G(ei)(xj))=E3(v¯F(ei)(xj),v¯G(ei)(xj))=E4(v_F(ei)(xj),v_G(ei)(xj))=1,eiP,xjU.

    Thus, we have for any eiP, xjU,

    • ūF(ei)(xj) = ūG(ei)(xj), uF(ei)(xj) = uG(ei)(xj),

    • v¯F(ei)(xj)=v¯G(ei)(xj) , ūF(ei)(xj), vG(ei)(xj), that is, (F, P) = (G, P).

  3. (3)

    By the commutative law of the fuzzy equivalence operators, we can easily get that

    • Dq((F, P), (G, P)) = Dq((G, P), (F, P))(q ∈ {1, 2, …8}).

  4. (4)

    Since (F, P) ⊆ (G, P) ⊆ (H, P), we have for any eiP, xjU,

    • [uF(ei)(xj), ūF(ei)(xj)] ⩽ [uG(ei)(xj), ūG(ei)(xj)] ⩽ [uH(ei)(xj), ūH(ei)(xj)],

    • [v_F(ei)(xj),v¯F(ei)(xj)][v_G(ei)(xj),v¯G(ei)(xj)][v_H(ei)(xj),v¯H(ei)(xj)] .

    By the property of fuzzy equivalence operators, we get for any eiP, xjU,

    • E1(ūF(ei)(xj), ūH(ei)(xj))⩽E1(ūF(ei)(xj), ūG(ei)(xj)),

    • E2(uF(ei)(xj), uH(ei)(xj))⩽E2(uF(ei)(xj), uG(ei)(xj)),

    • E3(v¯F(ei)(xj),v¯H(ei)(xj))E3(v¯F(ei)(xj),v¯G(ei)(xj)) ,

    • E4(ūF(ei)(xj), vH(ei)(xj)) ⩽ E4(ūF(ei)(xj), vG(ei)(xj)).

    Thus, Dq((F, P), (H, P)) ⩾ Dq((F, P), (G, P)) (q ∈ {1, 2, …8}).

Remark 1.

All of the distance measures for IVIFSSs are discussed on discrete universes here, the cases for continuous universes can be researched similarly.

Remark 2.

If the IVIFSSs degenerate to IVIFSs, the distance measures of IVIFSSs degenerate to the corresponding distance measures of IVIFSs.

Example 2.

Considering (F, P), (G, P) ∈ IVIFSS(U), let

  1. (1)

    M1(x1,x2,xn)=M2(x1,x2,xn)=1ni=1nxi , xi ∈ [0, 1], ∀nN;

  2. (2)

    E1(x1, x2) = E2(x1, x2) = E3(x1, x2) = E4(x1, x2) = 1 − |x1x2|, for any x1, x2 ∈ [0, 1];

  3. (3)

    f (x) = 1 − x, ∀x ∈ [0, 1],

then, we may construct the following distance measures for IVIFSSs by Eq. (2) in Definition 16.

  1. (1)

    Let M3(x1,x2,x3,x4)=[14(x12+x22+x32+x42)]12 , then we get the Normalized Euclidean distance

    d1((F,P),(G,P))={14mni=1mj=1n[(u¯F(ei)(xj)u¯G(ei)(xj))2+(u_F(ei)(xj)u_G(ei)(xj))2+v¯F(ei)(xj)v¯G(ei)(xj)2+(v_F(ei)(xj)v_G(ei)(xj))2]}12.

  2. (2)

    Let M3(x1,x2,x3,x4)=14(x1+x2+x3+x4) , then we get the Normalized hamming distance

    d2((F,P),(G,P))=14mnj=1ni=1m[|u¯F(ei)(xj)u¯G(ei)(xj)|+|(u_F(ei)(xj)u_G(ei)(xj)|+|v¯F(ei)(xj)v¯G(ei)(xj)|+|v_F(ei)(xj)v_G(ei)(xj))|].

  3. (3)

    Let M3(x1,x2,x3,x4)=12(x1x2+x3x4) , then we get the Normalized hamming distance measure induced by Hausdorff metric

    d3((F,P),(G,P))=12mnj=1ni=1m[(|u¯F(ei)(xj)u¯G(ei)(xj)||u_F(ei)(xj)u_G(ei)(xj)|)+(|v¯F(ei)(xj)v¯G(ei)(xj)||v_F(ei)(xj)v_G(ei)(xj))|)].

  4. (4)

    Let M3(x1,x2,x3,x4)=x12x22x32x42 , then we get the fourth distance

    d4((F,P),(G,P))=1mnj=1ni=1m[(u¯F(ei)(xj)u¯G(ei)(xj))2(u_F(ei)(xj)u_G(ei)(xj))2(v¯F(ei)(xj)v¯G(ei)(xj))2(v_F(ei)(xj)v_G(ei)(xj))2].

  5. (5)

    Let M3(x1,x2,x3,x4)=[14(x13+x23+x33+x43)]13 , then we get the fifth distance

    d5((F,P),(G,P))={14mni=1mj=1n[(u¯F(ei)(xj)u¯G(ei)(xj))3+(u_F(ei)(xj)u_G(ei)(xj))3+(v¯F(ei)(xj)v¯G(ei)(xj))3+v_F(ei)(xj)v_G(ei)(xj))3]}13.

If (F, P), (G, P) ∈ IVIFSS(U) are reduced to F, GIVIFS(U), we get the following distance measures of IVIFSs. Note that the similarity measures d′1d′3 of IVIFSs have been proposed in Ref. 17, whereas d′4d′5 are new for IVIFSs.

  1. (1)

    The Normalized Euclidean distance

    d1(F,G)={14nj=1n[(u¯F(xj)u¯G(xj))2+(u_F(xj)u_G(xj))2+(v¯F(xj)v¯G(xj))2+(v_F(xj)v_G(xj))2]}12.

  2. (2)

    The Normalized hamming distance

    d2(F,G)=14nj=1n[|u¯F(xj)u¯G(xj)|+|u_F(xj)u_G(xj)|+|v¯F(xj)v¯G(xj)|+|v_F(xj)v_G(xj))|].

  3. (3)

    The Normalized hamming distance measure induced by Hausdorff metric

    d3(F,G)=12nj=1n[(|u¯F(xj)u¯G(xj)||u_F(xj)u_G(xj)|)+(|v¯F(xj)v¯G(xj)||v_F(xj)v_G(xj))|)].

  4. (4)

    Let M3(x1,x2,x3,x4)=x12x22x32x42 , then we get the fourth distance

    d4(F,G)=1nj=1n[(u¯F(xj)u¯G(xj))2(u_F(xj)u_G(xj))2(v¯F(xj)v¯G(xj))2(v_F(xj)v_G(xj))2].

  5. (5)

    Let M3(x1,x2,x3,x4)=[14(x13+x23+x33+x43)]13 , then we get the fifth distance

    d5(F,G))={14nj=1n[(u¯F(xj)u¯G(xj))3+(u_F(xj)u_G(xj))3+(v¯F(xj)v¯G(xj))3+(v_F(xj)v_G(xj))3]}13.

Example 3.

Considering (F, P), (G, P) ∈ IVIFSS(U), let

  1. (1)

    M1(x1, x2, …, xn) = λmin(x1, x2, …, xn) + (1 − λ)max(x1, x2, …, xn) with λ ∈ [0, 1],

    M2(x1,x2,,xn)=1ni=1nxi ,

    M3(x1, x2, …, xn) = x1x2, …, ∨ xn,

    for each nN and xi ∈ [0, 1], i ∈ {1, 2, …, n}.

  2. (2)

    E1(x1,x2)=E2(x1,x2)=1|x12x22| ,

    E3(x1,x2)=E4(x1,x2)=2x1x2x12+x22 for any x1, x2 ∈ [0, 1].

  3. (3)

    f (x) = 1−x, for any x ∈ [0, 1].

We may construct the distance measure for IVIFSSs by Eq. (3) in Definition 16 as follows.

  • d6((F, P), (G, P)) = λmin(α1, α2, …, αn) + (1 − λ)max(α1, α2, …, αn),

    where λ ∈ [0, 1] and αj=1mi=1m{1[(1|u¯F(ei)(xj)2u¯G(ei)(xj)2|)(1|u_F(ei)(xj)2u_G(ei)(xj)2|)2v¯F(ei)(xj)v¯G(ei)(xj)v¯F(ei)(xj)2+v¯G(ei)(xj)22v_F(ei)(xj)v_G(ei)(xj)v_F(ei)(xj)2+v_G(ei)(xj)2]} , (j = 1, 2, …, n).

4. Relationships between distance, similarity, inclusion measures and entropy for IVIFSSs

4.1. Transformation of distance measures into similarity measures for IVIFSSs

Theorem 2.

Let f′ be a strict fuzzy negation and D be a distance measure of interval-valued intuitionistic fuzzy soft sets. Then a similarity measure S of interval-valued intuitionistic fuzzy soft sets can be deduced from the distance measure D as follows:

S((F,P)(G,P))=f(D((F,P),(G,P)))

Remark 3.

If we take the strict fuzzy negation f′(x) = 1 − x for all x ∈ [0, 1], by the distance measures Di((F, P), (G, P))(1 ⩽ i ⩽ 8) given in Definition 16, we can generate the corresponding similarity measures of interval-valued intuitionistic fuzzy soft sets as Si((F, P), (G, P)) = 1 − Di((F, P), (G, P)), (1 ⩽ i ⩽ 8).

Example 4.

Considering the distance measure given in Example 3, take f′(x) = 1 − x, one can get a similarity measure of IVIFSSs as follows.

  • S((F, P), (G, P)) = 1 − [λmin(α1, α2, …, αn) + (1 − λ)max(α1, α2, …, αn)],

    where λ ∈ [0, 1] and αj=1mi=1m{1[(1|u¯F(ei)(xj)2u¯G(ei)(xj)2|)(1|u_F(ei)(xj)2u_G(ei)(xj)2|)2v¯F(ei)(xj)v¯G(ei)(xj)v¯F(ei)(xj)2+v¯G(ei)(xj)22v_F(ei)(xj)v_G(ei)(xj)v_F(ei)(xj)2+v_G(ei)(xj)2]} , (j = 1, 2, …, n).

4.2. Transformation of distance measures into entropies for IVIFSSs

Now we present a transformation method for constructing entropy of IVIFSSs based on the distance measure of IVIFSSs as follows.

Theorem 3.

Let (Q, P) be an interval-valued intuitionistic fuzzy soft set on U, s.t. for any eiP, Q(ei) = {⟨xj, [1/2, 1/2]), [1/2, 1/2]⟩|xjU}. Suppose that

  1. (1)

    for each p ∈ {1, 2, 3}, Mp is both a bottom-aggregation operator and an idempotent-aggregation operator;

  2. (2)

    M3(x1, x2, x3, x4) = M3(x3, x4, x1, x2) for x1, x2, x3, x4 ∈ [0, 1];

  3. (3)

    E1(x1, x2) = E2(x1, x2) = E3(x1, x2) = E4(x1, x2) = 1− |x1 − x2| for any x1, x2 ∈ [0, 1];

  4. (4)

    f (x) = 1 − x, for any x ∈ [0, 1];

  5. (5)

    D1((F, P), (Q, P)) and D2((F, P), (Q, P)) are distance measures between (F, P) and (Q, P) constructed by Eq.(1) and Eq.(2) in Definition 16, respectively;

  6. (6)

    f′ is a strict fuzzy negation,

then for any (F, P) ∈ IVIFSS(U),

Iq((F,P))=f(2Dq((F,P),(Q,P)))(q=1,2)
are entropies for interval-valued intuitionistic fuzzy soft sets.

Proof.

It is sufficient to show that I((F, P)) satisfies the requirements (1)(4) listed in Definition 12.

  1. (1)

    If (F, P) is a classical soft set, we have

    • [uF(ei)(xj), ūF(ei)(xj)]=[1, 1], [v_F(ei)(xj),v¯F(ei)(xj)]=[0,0] or

    • [uF(ei)(xj), ūF(ei)(xj)]=[0, 0], [v_F(ei)(xj),v¯F(ei)(xj)]=[1,1], ∀eiP, xjU.

    Since E1(x1, x2) = E2(x1, x2) = E3(x1, x2) = E4(x1, x2) = 1−|x1x2| for any x1, x2 ∈ [0, 1], and f (x) = 1−x for any x ∈ [0, 1],

    we have f(E1(u¯F(ei)(xj),12))=f(E2(u_F(ei)(xj),12))=f(E3(v¯F(ei)(xj),12))=f(E4(v_F(ei)(xj),12))=12 , ∀eiP, xjU.

    Since Mp(p = 1, 2, 3) is an idempotent-aggregation operator, we have Dq((F,P),(Q,P))=12(q=1,2) , i.e., 2Dq((F, P), (Q, P)) = 1 (q = 1, 2).

    Thus, we get Iq((F, P)) = f′(1) = 0 (q = 1, 2).

  2. (2)

    Since Mp(p = 1, 2, 3) is a bottom-aggregation operator and f ′ is a strict fuzzy negation, we get

    • Iq((F, P)) = 1 (q = 1, 2)

    • ⇔ 2Dq((F, P), (Q, P)) = 0 (q = 1, 2)

    • Dq((F, P), (Q, P)) = 0 (q = 1, 2)

    • f(E1(u¯F(ei)(xj),12))=f(E2(u_F(ei)(xj),12))=

    • f(E3(v¯F(ei)(xj),12))=f(E4(v_F(ei)(xj),12))=0 , ∀eiP, xjU.

    • E1(u¯F(ei)(xj),12)=E2(u_F(ei)(xj),12)= E3(v¯F(ei)(xj),12)=E4(v_F(ei)(xj),12)=1 , ∀eiP, xjU.

    • uF(ei)(xj)=vF(ei)(xj)=[12,12] .

  3. (3)

    For any eiP,

    if F(ei) = ⟨xj, uF(ei)(xj), vF(ei)(xj)⟩, ∀xjU,

    then FC(ei) = ⟨xj, vF(ei)(xj), uF(ei)(xj)⟩, ∀xjU.

    Since E1(x1, x2) = E2(x1, x2) = E3(x1, x2) = E4(x1, x2) for any x1, x2 ∈ [0, 1], we have

    • E1(u¯F(ei)(xj),12)=E3(u¯F(ei)(xj),12) ,

    • E2(u_F(ei)(xj),12)=E4(u_F(ei)(xj),12) ,

    • E3(v¯F(ei)(xj),12)=E1(v¯F(ei)(xj),12) ,

    • E4(v_F(ei)(xj),12)=E2(v_F(ei)(xj),12) , ∀eiP, xjU.

    Since M3(x1, x2, x3, x4) = M3(x3, x4, x1, x2) for any x1, x2, x3, x4 ∈ [0, 1], we have

    • M3(f(E1(u¯F(ei)(xj),12)) , f(E2(u_F(ei)(xj),12)) , f(E3(v¯F(ei)(xj),12)) , f(E4(v_F(ei)(xj),12)))

    • =M3(f(E3(v¯F(ei)(xj),12)) , f(E4(v_F(ei)(xj),12)) , f(E1(u¯F(ei)(xj),12)) , f(E2(u_F(ei)(xj),12)))

    • =M3(f(E1(v¯F(ei)(xj),12)) , f(E2(v_F(ei)(xj),12)) , f(E3(u¯F(ei)(xj),12)) , f(E4(u_F(ei)(xj),12))) , ∀eiP, xjU.

    By Definition 16 we get

    Dq((F, P), (Q, P)) = Dq((FC, P), (Q, P)) (q = 1, 2),

    Thus, I((F, P)) = I((F, P)C).

  4. (4)

    Since f′ is a fuzzy negation,

    if Dq((F, P), (Q, P)) ⩾ Dq((G, P), (Q, P)) (q = 1, 2),

    then f ′(2Dq((F, P), (Q, P))) ⩽ f ′(2Dq((G, P), (Q, P))) (q = 1, 2), i.e., Iq((F, P)) ⩽ Iq((G, P)) (q = 1, 2).

Example 5.

Now we list some aggregation operators M3 which satisfy the conditions in Theorem 3: for any x1, x2, x3, x4 ∈ [0, 1],

  1. (1)

    M3(x1,x2,x3,x4)=((x1x2)p+(x3x4)p2)1p , p ⩾ 1.

  2. (2)

    M3(x1,x2,x3,x4)=((x1+x2)p(x3+x4)p2)1p , p ⩾ 1.

  3. (3)

    M3(x1,x2,x3,x4)=(x1p+x2p+x3p+x4p4)1p , p ⩾ 1.

Theorem 4.

Let (Q, P) be an interval-valued intuitionistic fuzzy soft set on U, s.t. for any eiP, Q(ei) = {⟨xj, [1/2, 1/2]), [1/2, 1/2]⟩|xjU}. Suppose that

  1. (1)

    for each p ∈ {1, 2}, Mp is both a bottom-aggregation operator and an idempotent-aggregation operator;

  2. (2)

    M3 is both a top-aggregation operator and an idempotent-aggregation operator;

  3. (3)

    M3(x1, x2, x3, x4) = M3(x3, x4, x1, x2) for any x1, x2, x3, x4 ∈ [0, 1];

  4. (4)

    E1(x1, x2) = E2(x1, x2) = E3(x1, x2) = E4(x1, x2) = 1−|x1x2| for any x1, x2 ∈ [0, 1];

  5. (5)

    f (x) = 1−x, for any x ∈ [0, 1];

  6. (6)

    D3((F, P), (Q, P)) and D4((F, P), (Q, P)) are distance measures between (F, P) and (Q, P) given by Eq.(3) and Eq.(4) in Definition 16, respectively;

  7. (7)

    f′ is a strict fuzzy negation,

then for any (F, P) ∈ IVIFSS(U),

Iq((F, P)) = f′(2Dq((F, P), (Q, P))) (q = 3, 4), is an entropy for interval-valued intuitionistic fuzzy soft sets based on the corresponding distance Dq (q = 3, 4).

Theorem 5.

Let (Q, P) be an interval-valued intuitionistic fuzzy soft set on U, s.t. for any eiP, Q(ei) = {⟨xj, [1/2, 1/2]), [1/2, 1/2]⟩|xjU}. Suppose that

  1. (1)

    M1 is both a bottom-aggregation operator and an idempotent-aggregation operator;

  2. (2)

    for each p ∈ {2, 3}, Mp is both a top-aggregation operator and an idempotent-aggregation operator;

  3. (3)

    M3(x1, x2, x3, x4) = M3(x3, x4, x1, x2) for any x1, x2, x3, x4 ∈ [0, 1];

  4. (4)

    E1(x1, x2) = E2(x1, x2) = E3(x1, x2) = E4(x1, x2) = 1−|x1x2| for any x1, x2 ∈ [0, 1];

  5. (5)

    f (x) = 1 − x, for any x ∈ [0, 1];

  6. (6)

    D5((F, P), (Q, P)) and D6((F, P), (Q, P)) are distance measures between (F, P) and (Q, P) given by Eq.(5) and Eq.(6) in Definition 16, respectively;

  7. (7)

    f′ is a strict fuzzy negation,

then for any (F, P) ∈ IVIFSS(U),

Iq((F, P)) = f′(2Dq((F, P), (Q, P))) (q = 5, 6), is an entropy for interval-valued intuitionistic fuzzy soft sets based on the corresponding distance Dq (q = 5, 6).

Theorem 6.

Let (Q, P) be an interval-valued intuitionistic fuzzy soft set on U, s.t. for any eiP, Q(ei) = {⟨xj, [1/2, 1/2]), [1/2, 1/2]⟩|xjU}. Suppose that

  1. (1)

    for each p ∈ {1, 2, 3}, Mp is both an idempotent-aggregation operator and a top-aggregation operator;

  2. (2)

    M3(x1, x2, x3, x4) = M3(x3, x4, x1, x2) for any x1, x2, x3, x4 ∈ [0, 1];

  3. (3)

    E1(x1, x2) = E2(x1, x2) = E3(x1, x2) = E4(x1, x2) = 1−|x1x2| for any x1, x2 ∈ [0, 1];

  4. (4)

    f (x) = 1 − x, for any x ∈ [0, 1];

  5. (5)

    Dq((F, P), (Q, P))(q = 7, 8) are distance measures between (F, P) and (Q, P) given by Eq.(7) and Eq.(8) in Definition 16, respectively;

  6. (6)

    f′ is a strict fuzzy negation,

then for any (F, P) ∈ IVIFS(U),

Iq((F, P)) = f′(2Dq((F, P), (Q, P))) (q = 7, 8), is an entropy for interval-valued intuitiionistic fuzzy soft sets based on the corresponding distance Dq (q = 7, 8).

Example 6.

Now we list some aggregation operators M3 which satisfies the conditions in Theorem 46: for any x1, x2, x3, x4 ∈ [0, 1],

  1. (1)

    M3(x1,x2,x3,x4)=((x1x2)p+(x3x4)p2)1p , p ⩾ 1.

  2. (2)

    M3(x1,x2,x3,x4)=((x1+x2)p(x3+x4)p2)1p , p ⩾ 1.

  3. (3)

    M3(x1,x2,x3,x4)=(x1p+x2p+x3p+x4p4)1p , p ⩾ 1.

Remark 4.

We can easily obtain a large number of distances by Definition 16, employing different aggregation operators. Furthermore, we can easily obtain a large number of entropies by Theorem 3–6, employing different distances.

4.3. Transformation of entropies into similarity measures for IVIFSSs

Next, we provide a transformational method of constructing similarity measure of IVIFSSs based on the entropy of IVIFSSs as below.

Definition 17.

Let (F, P), (G, P) ∈ IVIFSS(U), assume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[v_G(ei)(xj),v¯G(ei)(xj)]|xjU}.

Suppose that

  1. (1)

    M1 is a bottom-aggregation operator,

  2. (2)

    M1(x1, x2, x3, x4) ⩾ M2(x1, x2, x3, x4) for any x1, x2, x3, x4 ∈ [0, 1],

  3. (3)

    f is a strict fuzzy negation,

  4. (4)

    El (l = 1, 2, 3, 4) are fuzzy equivalence operators,

    then for any α ∈ [1, +∞), β ∈ [1, +∞), we can define a new interval-valued intuitionistic fuzzy set (ψ1(F, G), P) from (F, P) and (G, P) as follows: for any eiP, xjU,

    u_ψ1(F,G)(ei)(xj)=12{1[M1(f(E1(u_F(ei)(xj),u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯G(ei)(xj))))1/α};
    u¯ψ1(F,G)(ei)(xj)=12{1[M1(f(E1(u_F(ei)(xj),u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯G(ei)(xj))),f(E3(v_F(ei),v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯G(ei)(xj))))]};
    v_ψ1(F,G)(ei)(xj)=12{1+[M2(f(E1(u_F(ei)(xj),u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯G(ei)(xj))))]β};
    v¯ψ1(F,G)(ei)(xj)=12{1+[M2(f(E1(u_F(ei)(xj),u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯G(ei)(xj))),f(E3(v_F(ei),v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯G(ei)(xj))))]}.

Theorem 7.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft set. For (F, P), (G, P) ∈ IVIFSS(U), then I((ψ1(F, G), P)) is a similarity measure of (F, P) and (G, P).

Proof.

We only need to prove that all the properties in Definition 10 hold.

  1. (1)

    If (F, P) is a classical soft set, then for all eiP, xjU, we know

    • uF(ei)(xj) = [1, 1], vF(ei)(xj) = [0, 0], uFC(ei) (xj) = [0, 0], vFC(ei) (xj) = [1, 1], or

    • uF(ei)(xj) = [0, 0], vF(ei)(xj) = [1, 1], uFC(ei) (xj) = [1, 1], vFC(ei) (xj) = [0, 0],

    then we have

    E1(u_F(ei)(xj),u_FC(ei)(xj))=E2(u¯F(ei)(xj),u¯FC(ei)(xj))=E3(v_F(ei)(xj),v_FC(ei)(xj))=E4(v¯F(ei)(xj),v¯FC(ei)(xj))=0.

    Since M1, M2 are aggregation operators, we get

    • M1(f(E1(uF(ei)(xj), uFC(ei)(xj))), f (E2(ūF(ei)(xj), ūFC(ei)(xj))), f (E3(vF(ei)(xj), vFC(ei)(xj))), f(E4(v¯F(ei)(xj),v¯FC(ei)(xj)))=1 ,

    • M2(f (E1(uF(ei)(xj), uFC(ei)(xj))), f (E2(ūF(ei)(xj), ūFC(ei)(xj))), f (E3(vF(ei)(xj), vFC(ei)(xj))), f(E4(v¯F(ei)(xj),v¯FC(ei)(xj)))=1 , ∀eiP, xjU.

    hence we get

    • uψ1(F,FC)(ei)(xj) = ūψ1(F, FC)(ei)(xj) = 0,

    • v_ψ1(F,FC)(ei)(xj)=v¯ψ1(F,FC)(ei)(xj)=1 , ∀eiP, xjU.

    So, (ψ1(F, FC), P) is crisp soft set in U.

    By Definition 12 of entropy for IVIFSSs, we have

    S((F, P), (FC, P)) = I((ψ1(F, FC), P)) = 0.

  2. (2)

    S((F, P), (G, P)) = I((ψ1(F, G), P)) = 1.

    • uψ1(F,G)(ei)(xj)=vψ1(F,G)(ei)(xj)=[12,12] , ∀eiP, xjU.

    • M1(f (E1(uF(ei)(xj), uG(ei)(xj))), f (E2(ūF(ei)(xj), ūG(ei)(xj))), f (E3(vF(ei)(xj), vG(ei)(xj))), f(E4(v¯F(ei)(xj),v¯G(ei)(xj)))=0 and M2(f (E1(uF(ei)(xj), uG(ei)(xj))), f (E2(ūF(ei)(xj),ūG(ei)(xj))), f (E3(vF(ei)(xj), vG(ei)(xj))), f(E4(v¯F(ei)(xj),v¯G(ei)(xj)))=0 , ∀eiP, ∀xjU.

    • f (E1(uF(ei)(xj), uG(ei)(xj))) = 0, f (E2(ūF(ei)(xj), ūG(ei)(xj))) = 0, f (E3(vF(ei)(xj), vG(ei)(xj))) = 0, f(E4(v¯F(ei)(xj),v¯G(ei)(xj)))=0 , ∀eiP, ∀xjU.

    • E1(uF(ei)(xj), uG(ei)(xj)) = E2(ūF(ei)(xj), ūG(ei)(xj))

    • =E3(v_F(ei)(xj),v_G(ei)(xj))=E4(v¯F(ei)(xj),v¯G(ei)(xj))=1 , ∀eiP, ∀xjU.

    • uF(ei)(xj) = uG(ei)(xj), ūF(ei)(xj) = ūG(ei)(xj), vF(ei)(xj) = vG(ei)(xj), and v¯F(ei)(xj)=v¯G(ei)(xj) , ∀eiP, ∀xjU.

    • ⇔(F, P) = (G, P).

  3. (3)

    From the definition of (ψ1(F, G), E), we know for any eiP, xjU,

    • uψ1(F,G)(ei)(xj) = uψ1(G,F)(ei)(xj),

    • vψ1(F,G)(ei)(xj) = vψ1(G,F)(ei)(xj),

    that is, (ψ1(F, G), P) = (ψ1(G, F), P),

    then we get I((ψ1(F, G), P)) = I((ψ1(G, F), P))

    • S((F, P), (G, P)) = S((G, P), (F, P)).

  4. (4)

    If (F, P) ⊆ (G, P) ⊆ (H, P), we know for any eiP, xjU,

    • uF(ei)(xj)⩽ uG(ei)(xj)⩽ uH(ei)(xj),

    • ūF(ei)(xj)⩽ ūG(ei)(xj)⩽ ūH(ei)(xj),

    • vF(ei)(xj)⩾ vG(ei)(xj)⩾ vH(ei)(xj),

    • v¯F(ei)(xj)v¯G(ei)(xj)v¯H(ei)(xj) ,

    hence,

    • E1(uF(ei)(xj), uH(ei)(xj))⩽ E1(uF(ei)(xj), uG(ei)(xj)),

    • E2(ūF(ei)(xj), ūH(ei)(xj))⩽ E2(ūF(ei)(xj), ūG(ei)(xj)),

    • E3(vF(ei)(xj), vH(ei)(xj))⩽ E3(vF(ei)(xj), vG(ei)(xj)),

    • E4(v¯F(ei)(xj),v¯H(ei)(xj))E4(v¯F(ei)(xj),v¯G(ei)(xj)) ,

    from properties of aggregation operators and decreasing monotone property of f we have

    • M1(f (E1(uF(ei)(xj), uH(ei)(xj))), f (E2(ūF(ei)(xj),

    • ūH(ei)(xj))), f (E3(vF(ei)(xj), vH(ei)(xj))),

    • f(E4v¯F(ei)(xj),v¯H(ei)(xj)))M1(f(E1(u_F(ei)(xj),u_G(ei)(xj))) , f (E2(ūF(ei)(xj),

    • ūG(ei)(xj))), f (E3(vF(ei)(xj), vG(ei)(xj))),

    • f(E4(v¯F(ei)(xj),v¯G(ei)(xj)))) ,

    and

    • M2(f (E1(uF(ei)(xj), uH(ei)(xj))), f (E2(ūF(ei)(xj),

    • ūH(ei)(xj))), f (E3(vF(ei)(xj), vH(ei)(xj))),

    • f(E4(v¯F(ei)(xj),v¯H(ei)(xj))))M2(f(E1(u_F(ei)(xj),u_G(ei)(xj))) , f (E2(ūF(ei)(xj),

    • ūG(ei)(xj))), f(E3(ūF(ei)(xj), vG(ei)(xj))), f(E4(v¯F(ei)(xj),v¯G(ei)(xj)))) .

    Thus, we get

    • uψ1(F,H)(ei)(xj)uψ1(F,G)(ei)(xj)[12,12] , vψ1(F,H)(ei)(xj)vψ1(F,G)(ei)(xj)[12,12] , ∀eiP, ∀xjU.

    Let (Q, P) ∈ IVIFSS(U) and Q(ei) = {⟨xj, [1/2, 1/2]), [1/2, 1/2]⟩|xjU} for any eiP, then we get (ψ1(F, H), P) ⊆ (ψ1(F, G), P) ⊆ (Q, P).

    Similarly, we get (ψ1(F, H), P) ⊆ (ψ1(G, H), P) ⊆ (Q, P).

    By Definition 9 of distance measure for IVIFSSs, we know

    • D((ψ1(F, G), P), (Q, P))⩽ D((ψ1(F, H), P), (Q, P)),

    • D((ψ1(G, H), P), (Q, P))⩽ D((ψ1(F, H), P), (Q, P)).

    By Definition 12 of entropy for IVIFSSs, we conclude that

    • I((ψ1(F, H), P))⩽ I((ψ1(F, G), P)),

    • I((ψ1(F, H), P))⩽ I((ψ1(G, H), P)).

    Hence, I((ψ1(F, H), P))⩽ I((ψ1(F, G), P)) ∧ I((ψ1(G, H), P)), that is, S((F, P), (H, P))⩽ S((F, P), (G, P)) ∧ S((G, P), (H, P)).

Example 7.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft sets. For (F, P), (G, P) ∈ IVIFSS(U), assume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[v_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
let
  1. (1)

    M1(x1, x2, x3, x4) = (x1x2)∨(x3x4),

  2. (2)

    M2(x1, x2, x3, x4) = (x1x2)∧(x3x4),

  3. (3)

    E1(x1, x2) = E2(x1, x2) = E3(x1, x2) = E4(x1, x2) = 1 − |x1x2|,

  4. (4)

    α =β = 2,

  5. (5)

    f (x) = 1− x,

we get an interval-valued intuitionistic fuzzy soft set (ψ′1 (F, G), P) from (F, P) and (G, P) by Definition 17 as follows: for any eiP, xjU,

u_ψ1(F,G)(ei)(xj)=12{1[max((|u_F(ei)(xj)u_G(ei)(xj)||u¯F(ei)(xj)u¯G(ei)(xj)|),(|v_F(ei)(xj)v_G(ei)(xj)||v¯F(ei)(xj)v¯G(ei)(xj)|))1/2};
u¯ψ1(F,G)(ei)(xj)=12{1[max((|u_F(ei)(xj)u_G(ei)(xj)||u¯F(ei)(xj)u¯G(ei)(xj)|),(|v_F(ei)(xj)v_G(ei)(xj)||v¯F(ei)(xj)v¯G(ei)(xj)|))]};
v_ψ1(F,G)(ei)(xj)=12{1+[min((|u_F(ei)(xj)u_G(ei)(xj)||u¯F(ei)(xj)u¯G(ei)(xj)|),(|v_F(ei)(xj)v_G(ei)(xj)||v¯F(ei)(xj)v¯G(ei)(xj)|))]2};
v¯ψ1(F,G)(ei)(xj)=12{1+[min((|u_F(ei)(xj)u_G(ei)(xj)||u¯F(ei)(xj)u¯G(ei)(xj)|),(|u_F(ei)(xj)u_G(ei)(xj)||v¯F(ei)(xj)v¯G(ei)(xj)|))]},
then I((ψ′1(F, G), P)) is a similarity measure of (F, P) and (G, P).

Definition 18.

Let (F, P) and (G, P) be two IVIFSS(U) in universe U = {x1, x2, …xn}, assume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[u_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
Suppose that M1, M2 are aggregation operators which satisfy that
  1. (1)

    M1 is a top-aggregation operator,

  2. (2)

    M1(x1, x2, x3, x4)⩽ M2(x1, x2, x3, x4) for any x1, x2, x3, x4 ∈ [0, 1];

  3. (3)

    f is a strict fuzzy negation,

  4. (4)

    El (l = 1, 2, 3, 4) are fuzzy equivalence operators,

then for any α ∈ [1, +∞), β ∈ [1, +∞), we can define a new interval-valued intuitionistic fuzzy set (ψ2(F, G), P) from (F, P) and (G, P) as follows: for any eiP, xjU,

u_ψ2(F,G)(ei)(xj)=12{1[f(M1(E1(u_F(ei)(xj),u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯G(ei)(xj)),E3(v_F(ei)(xj),v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯G(ei)(xj))))]1/α};
u¯ψ2(F,G)(ei)(xj)=12{1[f(M1(E1(u_F(ei)(xj),u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯G(ei)(xj)),E3(v_F(ei)(xj),v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯G(ei)(xj))))]};
v_ψ2(F,G)(ei)(xj)=12{1+[f(M2(E1(u_F(ei)(xj),u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯G(ei)(xj)),E3(v_F(ei)(xj),v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯G(ei)(xj))))]β};
v¯ψ2(F,G)(ei)(xj)=12{1+[f(M2(E1(u_F(ei)(xj),u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯G(ei)(xj)),E3(v_F(ei)(xj),v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯G(ei)(xj))))]}.

Theorem 8.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft set. For (F, P), (G, P) ∈ IVIFSS(U), then I((ψ2(F, G), P)) is a similarity measure of (F, P) and (G, P).

Example 8.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft sets. For (F, P), (G, P) ∈ IVIFSS(U), assume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[v_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
let
  1. (1)

    M1(x1,x2,x3,x4)=(x1+x2)(x3+x4)2 for any x1, x2, x3, x4 ∈ [0, 1];

  2. (2)

    M2(x1,x2,x3,x4)=(x1+x2)(x3+x4)2 for any x1, x2, x3, x4 ∈ [0, 1];

  3. (3)

    E1(x1,x2)=E2(x1,x2)=E3(x1,x2)=E4(x1,x2)=2x1x2x12+x22 for any x1, x2 ∈ [0, 1].

  4. (4)

    α = 8, β = 4, f (x) = 1 − x,

we get an interval-valued intuitionistic fuzzy soft set (ψ′2 (F, G), P) from (F, P) and (G, P) by Definition 18 as follows: for any eiP, xjU,

u_ψ1(F,G)(ei)(xj)=12{1[112(2u_F(ei)(xj)u_G(ei)(xj)u_F(ei)(xj)2+u_G(ei)(xj)2+2u¯F(ei)(xj)u¯G(ei)(xj)u¯F(ei)(xj)2+u¯G(ei)(xj)2)(2v_F(ei)(xj)v_G(ei)(xj)v_F(ei)(xj)2+v_G(ei)(xj)2+2v¯F(ei)(xj)v¯G(ei)(xj)v¯F(ei)(xj)2+v¯G(ei)(xj)2)]1/8};
u¯ψ1(F,G)(ei)(xj)=12{1[112(2u_F(ei)(xj)u_G(ei)(xj)u_F(ei)(xj)2+u_G(ei)(xj)2+2u¯F(ei)(xj)u¯G(ei)(xj)u¯F(ei)(xj)2+u¯G(ei)(xj)2)(2v_F(ei)(xj)v_G(ei)(xj)v_F(ei)(xj)2+v_G(ei)(xj)2+2v¯F(ei)(xj)v¯G(ei)(xj)v¯F(ei)(xj)2+v¯G(ei)(xj)2)]};
v_ψ1(F,G)(ei)(xj)=12{1+[112(2u_F(ei)(xj)u_G(ei)(xj)u_F(ei)(xj)2+u_G(ei)(xj)2+2u¯F(ei)(xj)u¯G(ei)(xj)u¯F(ei)(xj)2+u¯G(ei)(xj)2)(2v_F(ei)(xj)v_G(ei)(xj)v_F(ei)(xj)2+v_G(ei)(xj)2+2v¯F(ei)(xj)v¯G(ei)(xj)v¯F(ei)(xj)2+v¯G(ei)(xj)2)]4};
v¯ψ1(F,G)(ei)(xj)=12{1+[112(2u_F(ei)(xj)u_G(ei)(xj)u_F(ei)(xj)2+u_G(ei)(xj)2+2u¯F(ei)(xj)u¯G(ei)(xj)u¯F(ei)(xj)2+u¯G(ei)(xj)2)(2v_F(ei)(xj)v_G(ei)(xj)v_F(ei)(xj)2+v_G(ei)(xj)2+2v¯F(ei)(xj)v¯G(ei)(xj)v¯F(ei)(xj)2+v¯G(ei)(xj)2)]},
then I((ψ′2 (F, G), P)) is a similarity measure of (F, P) and (G, P).

Definition 19.

Let (F, P), (G, P) ∈ IVIFSS(U), assume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[v_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
Suppose that,
  1. (1)

    M is a top-aggregation operator,

  2. (2)

    f is a strict fuzzy negation,

  3. (3)

    El (l = 1, 2, 3, 4) are fuzzy equivalence operators,

then for 0 < α1α2α4α3, we can define a new interval-valued intuitionistic fuzzy set (ψ3(F,G), P) from (F, P) and (G, P) as follows: for any eiP, xjU,

u_ψ3(F,G)(ei)(xj)=12{1[f(M(E1(u_F(ei)(xj),u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯G(ei)(xj)),E3(v_F(ei)(xj),v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯G(ei)(xj))))]α1};
u¯ψ3(F,G)(ei)(xj)=12{1[f(M(E1(u_F(ei)(xj),u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯G(ei)(xj)),E3(v_F(ei)(xj),v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯G(ei)(xj))))]α2};
v_ψ3(F,G)(ei)(xj)=12{1+[f(M(E1(u_F(ei)(xj),u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯G(ei)(xj)),E3(v_F(ei)(xj),v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯G(ei)(xj))))]α3};
v¯ψ3(F,G)(ei)(xj)=12{1+[f(M(E1(u_F(ei)(xj),u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯G(ei)(xj)),E3(v_F(ei)(xj),v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯G(ei)(xj))))]α4}.

Theorem 9.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft set. For (F, P), (G, P) ∈ IVIFSS(U), then I((ψ3(F,G), P)) is a similarity measure of (F, P) and (G, P).

Proof.

We only need to prove that all the properties in Definition 10 hold.

  1. (1)

    If (F, P) is a classical soft set, then for ∀eiP, xjU, we know

    • uF(ei)(xj) = [1, 1], vF(ei)(xj) = [0, 0], uFC(ei)(xj) = [0, 0], vFC(ei)(xj) = [1, 1] or

    • uF(ei)(xj) = [0, 0], vF(ei)(xj) = [1, 1], uFC(ei)(xj) = [1, 1], vFC(ei)(xj) = [0, 0],

    so we get

    E1(u_F(ei)(xj),u_FC(ei)(xj))=E2(u¯F(ei)(xj),u¯FC(ei)(xj))=E3(v_F(ei)(xj),v_FC(ei)(xj))=E4(v¯F(ei)(xj),v¯FC(ei)(xj))=0.

    For ∀αi ∈ {α1, α2, α3, α4} we have

    • [f (M(E1(uF(ei)(xj), uFC(ei)(xj)), E2(ūF(ei)(xj), ūFC(ei)(xj)), E3(vF(ei)(xj), vFC(ei)(xj)), E4(v¯F(ei)(xj),v¯FC(ei)(xj)))]αi=1 , ∀eiP, xjU.

    • Hence, uψ3(F, FC)(ei)(xj) = ūψ3(F, FC)(ei)(xj) = 0, v_ψ3(F,FC)(ei)(xj)=v¯ψ3(F,FC)(ei)(xj)=1 , ∀eiP, xjU.

    Thus, (ψ3(F, FC), P) is classical soft set in U.

    By Definition 12 of entropy for IVIFSSs, we have

    S((F, P), (FC, P)) = I((ψ3(F, FC), P)) = 0.

  2. (2)

    S((F, P), (G, P)) = I((ψ3(F,G), P)) = 1

    • uψ3(F,G)(ei)(xj)=vψ3(F,G)(ei)(xj)=[12,12] , ∀eiP, xjU,

    • ⇔ for ∀αi ∈ {α1, α2, α3, α4},

    • [ f (M(E1(uF(ei)(xj), uG(ei)(xj)), E2(ūF(ei)(xj), ūG(ei)(xj)), E3(vF(ei)(xj), vG(ei)(xj)), E4(v¯F(ei)(xj),v¯G(ei)(xj))))]αi=0 , ∀eiP, xjU,

    • M(E1(uF(ei)(xj), uG(ei)(xj)), E2(ūF(ei)(xj), ūG(ei)(xj)), E3(vF(ei)(xj), vG(ei)(xj)), E4(v¯F(ei)(xj),v¯G(ei)(xj))))=1 , ∀eiP, xjU,

    • E1(uF(ei)(xj), uG(ei)(xj)) = E2(ūF(ei)(xj), ūG(ei)(xj)) = E3(v_F(ei)(xj),v_G(ei)(xj))=E4(v¯F(ei)(xj),v¯G(ei)(xj))=1 , ∀eiP, xjU,

    • uF(ei)(xj) = uG(ei)(xj), ūF(ei)(xj) = ūG(ei)(xj), vF(ei)(xj) = vG(ei)(xj), v¯F(ei)(xj)=v¯G(ei)(xj) , ∀eiP, xjU,

    • ⇔(F, P) = (G, P).

  3. (3)

    From the definition of (ψ3(F,G), P) we easily know that

    • uψ3(F,G)(ei)(xj) = uψ3(G,F)(ei)(xj),

    • vψ3(F,G)(ei)(xj) = vψ3(G, F)(ei)(xj), i.e. (ψ3(F,G), P) = (ψ3(G, F), P).

    Thus, I((ψ3(F,G), P)) = I((ψ3(G, F), P))

    S((F, P), (G, P)) = S((G, P), (F, P)).

  4. (4)

    If (F, P) ⊆ (G, P) ⊆ (H, P), then we know

    • uF(ei)(xj)⩽ uG(ei)(xj)⩽ uH(ei)(xj),

    • ūF(ei)(xj)⩽ ūG(ei)(xj)⩽ ūH(ei)(xj),

    • vF(ei)(xj)⩾ vG(ei)(xj)⩾ vH(ei)(xj),

    • v¯F(ei)(xj)v¯G(ei)(xj)v¯H(ei)(xj) , ∀eiP, xjU,

    hence,

    • E1(uF(ei)(xj), uH(ei)(xj))⩽ E1(uF(ei)(xj), uG(ei)(xj)),

    • E2(ūF(ei)(xj), ūH(ei)(xj))⩽ E2(ūF(ei)(xj), ūG(ei)(xj)),

    • E3(vF(ei)(xj), vH(ei)(xj))⩽ E3(vF(ei)(xj), vG(ei)(xj)),

    • E4(v¯F(ei)(xj),v¯H(ei)(xj))E4(v¯F(ei)(xj),v¯G(ei)(xj)) , ∀eiP, xjU,

    then we have, for ∀αi ∈ {α1, α2, α3, α4},

    [f(M(E1(u_F(ei)(xj),u_H(ei)(xj)),E2(u¯F(ei)(xj),u¯H(ei)(xj)),E3(v_F(ei)(xj),v_H(ei)(xj)),E4(v¯F(ei)(xj),v¯H(ei)(xj))))]αi[f(M(E1(u_F(ei)(xj),u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯G(ei)(xj)),E3(v_F(ei)(xj),v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯G(ei)(xj))))]αi,eiP,xjU,

    so we get,

    • uψ3(F,H)(ei)(xj)uψ3(F,G)(ei)(xj)[12,12] ,

    • vψ3(F,H)(ei)(xj)vψ3(F,G)(ei)(xj)[12,12] , ∀eiP, ∀xjU.

    Let (Q, P) ∈ IVIFSS(U) and Q(ei) = {⟨xj, [1/2, 1/2]), [1/2, 1/2]⟩|xjU} for any eiP, then we get

    (ψ3(F, H), P) ⊆ (ψ3(F,G), P) ⊆ (Q, P).

    Similarly, we get

    (ψ3(F, H), P) ⊆ (ψ3(G, H), P) ⊆ (Q, P).

    By Definition 9 of distance measure for IVIFSSs, we know

    • D((ψ3(F,G), P), (Q, P)) ⩽ D((ψ3(F, H), P), (Q, P)),

    • D((ψ3(G, H), P), (Q, P)) ⩽ D((ψ3(F, H), P), (Q, P)).

    By Definition 12 of entropy for IVIFSSs, we conclude that

    • I((ψ3(F, H), P))⩽ I((ψ3(F,G), P)),

    • I((ψ3(F, H), P))⩽ I((ψ3(G, H), P)).

    Hence,

    I((ψ3(F, H), P)) ⩽ I((ψ3(F,G), P)) ∧ I((ψ3(G, H), P)),

    that is,

    S((F, P), (H, P))⩽ S((F, P), (G, P)) ∧ S((G, P), (H, P)).

Definition 20.

Let (F, P), (G, P) ∈ IVIFSS(U), assume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[v_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
Suppose that,
  1. (1)

    M is a bottom-aggregation operator,

  2. (2)

    f is a strict fuzzy negation,

  3. (3)

    El (l = 1, 2, 3, 4) are fuzzy equivalence operators,

then for 0 < α1α2α4α3, we can define a new interval-valued intuitionistic fuzzy set (ψ4(F, G), P) from (F, P) and (G, P) as follows: for any eiP, xjU,

u_ψ4(F,G)(ei)(xj)=12{1[M(f(E1(u_F(ei)(xj),u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯G(ei)(xj))))]α1};
u¯ψ4(F,G)(ei)(xj)=12{1[M(f(E1(u_F(ei)(xj),u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯G(ei)(xj))))α2]};
v_ψ4(F,G)(ei)(xj)=12{1[M(f(E1(u_F(ei)(xj),u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯G(ei)(xj))))]α3};
v¯ψ4(F,G)(ei)(xj)=12{1[M(f(E1(u_F(ei)(xj),u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯G(ei)(xj))))]α4}.

Theorem 10.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft set. For (F, P), (G, P) ∈ IVIFSS(U), then I((ψ4(F, G), P)) is a similarity measure of (F, P) and (G, P).

Example 9.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft sets. For (F, P), (G, P) ∈ IVIFSS(U), for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[v_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
Let
  1. (1)

    M(x1,x2,x3,x4)=x1+x2+x3+x44 for any x1, x2, x3, x4 ∈ [0, 1],

  2. (2)

    E1(x1, x2) = E2(x1, x2) = E3(x1, x2) = E4(x1, x2) = 1− |x1x2| for any x1, x2 ∈ [0, 1],

  3. (3)

    α1 = 2, α2 = 3, α3 = 5, α4 = 4,

  4. (4)

    f (x) = 1− x,

we get an interval-valued intuitionistic fuzzy soft set (ψ′4(F, G), P) from (F, P) and (G, P) by Definition 20 as follows: for any eiP, xjU,

u_ψ4(F,G)(ei)(xj)=12{1[14(|u_F(ei)(xj)u_G(ei)(xj)|+|u¯F(ei)(xj)u¯G(ei)(xj)|+|v_F(ei)(xj)v_G(ei)(xj)|+|v¯F(ei)(xj)v¯G(ei)(xj)|)]2};
u¯ψ4(F,G)(ei)(xj)=12{1[14(|u_F(ei)(xj)u_G(ei)(xj)|+|u¯F(ei)(xj)u¯G(ei)(xj)|+|v_F(ei)(xj)v_G(ei)(xj)|+|v¯F(ei)(xj)v¯G(ei)(xj)|)]3};
v_ψ4(F,G)(ei)(xj)=12{1+[14(|u_F(ei)(xj)u_G(ei)(xj)|+|u¯F(ei)(xj)u¯G(ei)(xj)|+|v_F(ei)(xj)v_G(ei)(xj)|+|v¯F(ei)(xj)v¯G(ei)(xj)|)]5};
v¯ψ4(F,G)(ei)(xj)=12{1+[14(|u_F(ei)(xj)u_G(ei)(xj)|+|u¯F(ei)(xj)u¯G(ei)(xj)|+|v_F(ei)(xj)v_G(ei)(xj)|+|v¯F(ei)(xj)v¯G(ei)(xj)|)]4},
then I((ψ′4(F, G), P)) is a similarity measure of (F, P) and (G, P).

Theorem 11.

If I is an entropy measure of IVIFSSs and (ψh(F, G), P)(h = 1, 2, 3, 4) is given by Definition 17–20, then I((ψh(F, G)C, P))(h = 1, 2, 3, 4) is also a similarity measure between (F, P) and (G, P).

Remark 5.

Based on Definition 1720, by selecting different aggregation operators and fuzzy equivalences, we can obtain a large number of IVIFSSs, which can be used to transform an entropy measure into a similarity measure for IVIFSSs.

Remark 6.

If (F, P), (G, P) ∈ IVIFSS(U) degenerate to F, GIVIFS(U), the specific interval-valued intuitionistic fuzzy soft set (ψ′1 (F, G), P) in Example 7 degenerates to ψ′1 (F, G) ∈ IVIFS(U). The entropy of ψ′1 (F, G) has been proven a similarity measure between F and G in Ref. 17. Our research in this subsection can be regarded as a generalization and extension of the research in Ref. 17 based on fuzzy equivalences and aggregation operators. However, even if it degenerates to the IVIFSs situation, all the formulae given by Definition 1820 in this work are new.

4.4. Transformation of entropies into inclusion measures for IVIFSSs

Definition 21.

Let (F, P), (G, P) ∈ IVIFSS(U), assume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[v_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
Suppose that,
  1. (1)

    M1 is a bottom-aggregation operator,

  2. (2)

    M1(x1, x2, x3, x4)⩾ M2(x1, x2, x3, x4) for any x1, x2, x3, x4 ∈ [0, 1],

  3. (3)

    f is a strict fuzzy negation,

  4. (4)

    El (l = 1, 2, 3, 4) are fuzzy equivalence operators,

then for any α ∈ [1, + ∞), β ∈ [1, + ∞), we can define a new interval-valued intuitionistic fuzzy set (ϕ1(F, G), P) from (F, P) and (G, P) as follows: for any eiP, xjU,

u_φ1(F,G)(ei)(xj)=12{1[M1(f(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]1/α};
u¯φ1(F,G)(ei)(xj)=12{1[M1(f(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]};
v_φ1(F,G)(ei)(xj)=12{1[M2(f(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]β};
v¯φ1(F,G)(ei)(xj)=12{1[M2(f(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]}.

Theorem 12.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft set. For (F, P), (G, P) ∈ IVIFSS(U), then I((ϕ1(F, G), P)) is an inclusion measure between (F, P) and (G, P).

Proof.

We only need to prove that all the properties in Definition 11 hold.

  1. (1)

    If (F, P) = (U, P), (G, P) = (∅, P),

    we get F(ei) = {⟨xj, [1, 1], [0, 0]⟩|xjU}, G(ei) = {⟨xj, [0, 0], [1, 1]⟩|xjU} for ∀eiP,

    then we have for any xjU, eiP

    • E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj)) = E1(1, 1∧0) = 0,

    • E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj)) = E2(1, 1∧0) = 0,

    • E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj)) = E3(0, 0∨1) = 0,

    • E4(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj)) = E4(0, 0∨1) = 0,

    so we get

    • M1(f (E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj))), f (E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj))), f (E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj))), f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))=M1(1,1,1,1)=1 and

    • M2(f (E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj))), f (E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj))), f (E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj))), f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))=M2(1,1,1,1)=1 .

    Thus, it is easy to get that

    • [uϕ1(F,G)(ei)(xj), ūϕ1(F,G)(ei)(xj)] = [0, 0],

    • [v_φ1(F,G)(ei)(xj),v¯φ1(F,G)(ei)(xj)]=[1,1] , ∀xjU, eiP.

    By Definition 12 of entropy for IVIFSSs, we know

    • I((ϕ1(F, G), P)) = 0⇔J((F, P), (G, P)) = 0.

  2. (2)

    I((ϕ1(F, G), P)) = J((F, P), (G, P)) = 1,

    • [u_φ1(F,G)(ei)(xj),u¯φ1(F,G)(ei)(xj)]=[12,12] ,

    • [v_φ1(F,G)(ei)(xj),v¯φ1(F,G)(ei)(xj)]=[12,12] , ∀xjU, eiP.

    • M1(f (E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj))),

    • f (E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj))),

    • f (E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj))),

    • f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))=0 ,

    and

    • M2(f (E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj))),

    • f (E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj))),

    • f (E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj))),

    • f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj)))=0 , ∀xjU, eiP.

    • f (E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj))) = 0,

    • f (E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj))) = 0,

    • f (E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj))) = 0,

    • f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj)))=0 ,

    • xjU, eiP.

    • E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj)) = 1,

    • E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj)) = 1,

    • E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj)) = 1,

    • E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))=1 ,

    • xjU, eiP.

    • uF(ei)(xj) = uF(ei)(xj) ∧ uG(ei)(xj),

    • ūF(ei)(xj) = ūF(ei)(xj) ∧ ūG(ei)(xj),

    • vF(ei)(xj) = vF(ei)(xj) ∨ vG(ei)(xj),

    • v¯F(ei)(xj)=v¯F(ei)(xj)v¯G(ei)(xj) ,

    • xjU, eiP.

    • ⇔[uF(ei)(xj), ūF(ei)(xj)]⩽ [uG(ei)(xj), ūG(ei)(xj)],

    • [v_F(ei)(xj),v¯F(ei)(xj)][v_G(ei)(xj),v¯G(ei)(xj)] ,

    • xjU, eiP.

    ⇔ (F, P) ⊆ (G, P)

  3. (3)

    If (F, P) ⊆ (G, P) ⊆ (H, P), then for any xjU, eiP,

    • [uF(ei)(xj), ūF(ei)(xj)]⩽ [uG(ei)(xj), ūG(ei)(xj)]⩽

    • [uH(ei)(xj), ūH(ei)(xj)] and [v_F(ei)(xj),v¯F(ei)(xj)][v_G(ei)(xj),v¯G(ei)(xj)][v_H(ei)(xj),v¯H(ei)(xj)] ,

    so we have

    • E1(uH(ei)(xj), uH(ei)(xj) ∧ uF(ei)(xj)) = E1(uH(ei)(xj), uF(ei)(xj))⩽ E1(uG(ei)(xj), uF(ei)(xj)) = E1(uG(ei)(xj), uG(ei)(xj) ∧ uF(ei)(xj)),

    • E2(ūH(ei)(xj), ūH(ei)(xj) ∧ ūF(ei)(xj)) = E2(ūH(ei)(xj), ūF(ei)(xj))⩽ E2(ūG(ei)(xj), ūF(ei)(xj)) = E2(ūG(ei)(xj), ūG(ei)(xj) ∧ ūF(ei)(xj)),

    • E3(vH(ei)(xj), vH(ei)(xj) ∨ vF(ei)(xj)) = E3(vH(ei)(xj), vF(ei)(xj))⩽ E3(vG(ei)(xj), vF(ei)(xj)) = E3(vG(ei)(xj), vG(ei)(xj) ∨ vF(ei)(xj)), E4(v¯H(ei)(xj),v¯H(ei)(xj)v¯F(ei)(xj))=E4(v¯H(ei)(xj) ,

    • v¯F(ei)(xj))E4(v¯G(ei)(xj),v¯F(ei)(xj))=E4(v¯G(ei)(xj),v¯G(ej)(xj)v¯F(ei)(xj)) ,

    then we get

    • f (E1(uH(ei)(xj), uH(ei)(xj) ∧ uF(ei)(xj))) ⩾ f (El(uG(ei)(xj), uG(ei)(xj) ∧ uF(ei)(xj))),

    • f (E2(ūH(ei)(xj), ūH(ei)(xj) ∧ ū (ei)(xj))) ⩾ f (E2(ūG(ei)(xj), ūG(ei)(xj) ∧ ūF(ei)(xj))),

    • f (E3(vH(ei)(xj), vH(ei)(xj) ∨ vF(ei)(xj))) ⩾ f (E3(vG(ei)(xj), vG(ei)(xj) ∨ vF(ei)(xj))),

    • f(E4(v¯H(ei)(xj),v¯H(ei)(xj)v¯F(ei)(xj)))f(E4(v¯G(ei)(xj),v¯G(ei)(xj)v¯F(ei)(xj))) .

    From the property of aggregation operators, we get

    [u_φ1(H,F)(ei)(xj),u¯φ1(H,F)(ei)(xj)][u_φ1(G,F)(ei)(xj),u¯φ1(G,F)(ei)(xj)][12,12],[v_φ1(H,F)(ei)(xj),v¯φ1(H,F)(ei)(xj)][v_φ1(G,F)(ei)(xj),v¯φ1(G,F)(ei)(xj)][12,12].

    Let (Q, P) ∈ IVIFSS(U) and Q(ei) = {⟨xj, [1/2, 1/2]), [1/2, 1/2]⟩|xjU} for any eiP,

    then we get

    (ϕ1(H, F), P) ⊆ (ϕ1(G, F), P) ⊆ (Q, P),

    thus,

    D((ϕ1(H, F), P), (Q, P)) ⩾ D((ϕ1(G, F), P), (Q, P)).

    By Definition 12 of entropy for IVIFSSs, we get

    • I((ϕ1(H, F), P))⩽ I((ϕ1(G, F), P))

    • J((H, P), (F, P))⩽ J((G, P), (F, P)).

    By the similar way, we get

    • I((ϕ1(H, F), P))⩽ I((ϕ1(H, G), P))

    • J((H, P), (F, P))⩽ J((H, P), (G, P)).

Definition 22.

Let (F, P), (G, P) ∈ IVIFSS(U), assume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[v_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
Suppose that,
  1. (1)

    M1 is a top-aggregation operator,

  2. (2)

    M1(x1, x2, x3, x4)⩽ M2(x1, x2, x3, x4) for any x1, x2, x3, x4 ∈ [0, 1],

  3. (3)

    f is a strict fuzzy negation,

  4. (4)

    El (l = 1, 2, 3, 4) are fuzzy equivalence operators,

then for any α ∈ [1, + ∞), β ∈ [1, +∞), we can define a new interval-valued intuitionistic fuzzy set (ϕ2(F, G), P) from (F, P) and (G, P) as follows: for any eiP, xjU,

u_φ2(F,G)(ei)(xj)=12{1[f(M1(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj)),E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]1/α};
u¯φ2(F,G)(ei)(xj)=12{1[f(M1(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj)),E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]};
v_φ2(F,G)(ei)(xj)=12{1+[f(M2(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj)),E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]β};
v¯φ2(F,G)(ei)(xj)=12{1+[f(M2(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj)),E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]}.

Theorem 13.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft set. For (F, P), (G, P) ∈ IVIFSS(U), then I((ϕ2 (F, G), P)) is an inclusion measure between (F, P) and (G, P).

Definition 23.

Let (F, P), (G, P) ∈ IVIFSS(U), asssume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[u_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
Suppose that,
  1. (1)

    M is a top-aggregation operator,

  2. (2)

    f is a strict fuzzy negation,

  3. (3)

    El (l = 1, 2, 3, 4) are fuzzy equivalence operators,

then for any 0 < α1α2α4α3, we can define a new interval-valued intuitionistic fuzzy set (ϕ3(F, G), P) from (F, P) and (G, P) as follows: for any eiP, xjU,

u_φ3(F,G)(ei)(xj)=12{1[f(M(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj)),E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]α1};
u¯φ3(F,G)(ei)(xj)=12{1[f(M(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj)),E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]α2};
v_φ3(F,G)(ei)(xj)=12{1+[f(M(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj)),E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]α3};
v¯φ3(F,G)(ei)(xj)=12{1+[f(M(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj)),E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj)),E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj)),E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]α4}.

Theorem 14.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft set. For (F, P), (G, P) ∈ IVIFSS(U), then I((ϕ3(F, G), P)) is an inclusion measure between (F, P) and (G, P).

Proof.

We only need to prove that all the properties in Definition 11 hold.

  1. (1)

    If (F, P) = (U, P), (G, A) = (∅, P),

    we get F(ei) = {⟨xj, [1, 1], [0, 0]⟩|xjU}, G(ei) = {⟨xj, [0, 0], [1, 1]⟩|xjU} for ∀eiP,

    then we have for any xjU, eiP,

    • E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj) = E1(1, 1 ∧ 0) = 0,

    • E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj)) = E2(1, 1 ∧ 0) = 0,

    • E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj)) = E3(0, 0 ∨ 1) = 0,

    • E4(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj)) = E4(0, 0 ∨ 1) = 0,

    so we get,

    • M(E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj)), E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj)), E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj)), E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj)))=M(0,0,0,0)=0 .

    Thus, it is easy to get that

    • [uϕ3(F,G)(ei)(xj), ūϕ3(F,G)(ei)(xj)] = [0, 0],

    • [v_φ3(F,G)(ei)(xj),v¯φ3(F,G)(ei)(xj)]=[1,1] , ∀xjU, eiP.

    From Definition 12 of entropy for IVIFSSs, we know

    • I((ϕ3(F, G), P)) = 0⇔J((F, P), (G, P)) = 0.

  2. (2)

    I((ϕ3(F, G), P)) = J((F, P), (G, P)) = 1,

    • [u_φ3(F,G)(ei)(xj),u¯φ3(F,G)(ei)(xj)]=[12,12] ,

    • [v_φ3(F,G)(ei)(xj),v¯φ3(F,G)(ei)(xj)]=[12,12] , ∀xjU, eiP.

    • f (M(E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj)), E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj)), E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj)), E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))=0 , ∀xjU, eiP.

    • M(E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj)), E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj)), E3(vF(ei)(xj), vF(ei)(xj) ∨ vG(ei)(xj)), E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj)))=1 , ∀xjU, eiP.

    • E1(uF(ei)(xj), uF(ei)(xj) ∧ uG(ei)(xj))=E2(ūF(ei)(xj), ūF(ei)(xj) ∧ ūG(ei)(xj)) = =E3(v_F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))=E4(v¯F(ei)(xj),v¯F(ej)(xj)v¯G(ei)(xj))=1 , ∀xjU, eiP.

    • uF(ei)(xj) = uF(ei)(xj) ∧ uG(ei)(xj), ūF(ei)(xj) = ūF(ei)(xj) ∧ ūG(ei)(xj), vF(ei)(xj) = vF(ei)(xj) ∨ vG(ei)(xj), v¯F(ei)(xj)=v¯F(ei)(xj)v¯G(ei)(xj) , ∀xjU, eiP.

    • ⇔[uF(ei)(xj), ūF(ei)(xj)]⩽ [uG(ei)(xj), ūG(ei)(xj)], and [v_F(ei)(xj),v¯F(ei)(xj)][v_G(ei)(xj),v¯G(ei)(xj)] , ∀xjU, eiP.

    • ⇔ (F, P) ⊆ (G, P).

  3. (3)

    If (F, E) ⊆ (G, P) ⊆ (H, P), then

    • [uF(ei)(xj), ūF(ei)(xj)]⩽ [uG(ei)(xj), uG(ei)(xj)]⩽ [uH(ei)(xj), ūH(ei)(xj)] and

    • [v_F(ei)(xj),v¯F(ei)(xj)][v_G(ei)(xj),v¯G(ei)(xj)][v_H(ei)(xj),v¯H(ei)(xj)] , ∀xjU, eiP.

    So we have for ∀xjU, eiP,

    • E1(uH(ei)(xj), uH(ei)(xj) ∧ uF(ei)(xj)) = E1(uH(ei)(xj), uF(ei)(xj))⩽ E1(uG(ei)(xj), uF(ei)(xj)) = E1(uG(ei)(xj), uG(ei)(xj) ∧ uF(ei)(xj)),

    • E2(ūH(ei)(xj), ūH(ei)(xj) ∧ ūF(ei)(xj)) = E2(ūH(ei)(xj), ūF(ei)(xj))⩽ E2(ūG(ei)(xj), ūF(ei)(xj)) = E2(ūG(ei)(xj), ūG(ei)(xj) ∧ ūF(ei)(xj)),

    • E3(vH(ei)(xj), vH(ei)(xj) ∨ vF(ei)(xj)) = E3(vH(ei)(xj), vF(ei)(xj))⩽ E3(vG(ei)(xj), vF(ei)(xj)) = E3(vG(ei)(xj), vG(ei)(xj) ∨ vF(ei)(xj)),

    • E4(v¯H(ei)(xj),v¯H(ei)(xj)v¯F(ei)(xj))=E4(v¯H(ei)(xj),v¯F(ei)(xj))E4(v¯G(ei)(xj),v¯F(ei)(xj))=E4(v¯G(ei)(xj),v¯G(ei)(xj)v¯F(ei)(xj))

    so we get

    • f (M(E1(uH(ei)(xj), uH(ei)(xj) ∧ uF(ei)(xj)), E2(ūH(ei) (xj), ūH(ei)(xj) ∧ ūF(ei)(xj)), E3(vH(ei)(xj), vH(ei)(xj) ∨ vF(ei)(xj)), E4(v¯H(ei)(xj),v¯H(ei)(xj)v¯F(ei)(xj)))f(M(E1(u_G(ei)(xj) ,

    • uG(ei)(xj) ∧ uF(ei)(xj)), E2(ūG(ei) (xj), ūG(ei)(xj) ∧ ūF(ei)(xj)), E3(vG(ei)(xj), vG(ei)(xj) ∨ vF(ei)(xj)), E4(v¯G(ei)(xj),v¯G(ei)(xj)v¯F(ei)(xj)))) .

    Hence,

    • [uϕ3(H, F)(ei)(xj), ūϕ3(H, F)(ei)(xj)]⩽ [uϕ3(G, F)(ei)(xj), u¯φ3(G,F)(ei)(xj)][12,12] and [v_φ3(H,F)(ei)(xj),v¯φ3(H,F)(ei)(xj)][v_φ3(G,F)(ei)(xj),v¯φ3(G,F)(ei)(xj)][12,12] , ∀xjU, eiP.

    Let (Q, P) ∈ IVIFSS(U) and Q(ei) = {⟨xj, [1/2, 1/2]), [1/2, 1/2]⟩|xjU} for any eiP,

    then we get

    (ϕ3(H, F), P) ⊆ (ϕ3(G, F), P) ⊆ (Q, P),

    by Definition 9 of distance measure for IVIFSSs, we get

    D((ϕ3(H, F), P), (Q, P)) ⩾ D((ϕ3(G, F), P), (Q, P)).

    Therefore, by Definition 12 of entropy for IVIFSSs,

    we get

    • I((ϕ3(H, F), P))⩽ I((ϕ3(G, F), P))

    • J((H, P), (F, P))⩽ J((G, P), (F, P)).

    By the similar way, we get

    • I((ϕ3(H, F), P))⩽ I((ϕ3(H, G), P))

    • J((H, P), (F, P))⩽ J((H, P), (G, P)).

Definition 24.

Let (F, P), (G, P) ∈ IVIFSS(U), assume that: for any eiP,

F(ei)={xj,uF(ei)(xj),vF(ei)(xj)|xjU}={xj,[u_F(ei)(xj),u¯F(ei)(xj)],[v_F(ei)(xj),v¯F(ei)(xj)]|xjU},
G(ei)={xj,uG(ei)(xj),vG(ei)(xj)|xjU}={xj,[u_G(ei)(xj),u¯G(ei)(xj)],[v_G(ei)(xj),v¯G(ei)(xj)]|xjU}.
Suppose that,
  1. (1)

    M is a bottom-aggregation operator,

  2. (2)

    f is a strict fuzzy negation,

  3. (3)

    El (l = 1, 2, 3, 4) are fuzzy equivalence operators,

then for any 0 < α1α2α4α3, we can define a new interval-valued intuitionistic fuzzy set (ϕ4(F, G), P) from (F, P) and (G, P) as follows: for any eiP, xjU,

u_φ4(F,G)(ei)(xj)=12{1[M(f(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]α1};
u¯φ4(F,G)(ei)(xj)=12{1[M(f(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]α2};
v_φ4(F,G)(ei)(xj)=12{1[M(f(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]α3};
v¯φ4(F,G)(ei)(xj)=12{1[M(f(E1(u_F(ei)(xj),u_F(ei)(xj)u_G(ei)(xj))),f(E2(u¯F(ei)(xj),u¯F(ei)(xj)u¯G(ei)(xj))),f(E3(v_F(ei)(xj),v_F(ei)(xj)v_G(ei)(xj))),f(E4(v¯F(ei)(xj),v¯F(ei)(xj)v¯G(ei)(xj))))]α4};

Theorem 15.

Let I be an entropy measure of interval-valued intuitionistic fuzzy soft set. For (F, P), (G, P) ∈ IVIFSS(U), then I((ϕ4(F, G), P)) is an inclusion measure between (F, P) and (G, P).

Theorem 16.

If I is an entropy measure of IVIFSSs and (ϕh(F, G), P)(h = 1, 2, 3, 4) is given by Definition 2124, then I((ϕh(F, G)C, P)) (h = 1, 2, 3, 4) is also an inclusion measure between (F, P) and (G, P).

Remark 7.

Based on Definition 2124, by selecting different aggregation operators and fuzzy equivalences, we can obtain a large number of IVIFSSs, which can be used to transform an entropy measure into an inclusion measure for IVIFSSs.

Remark 8.

In Ref. 17, the authors provided a specific interval-valued intuitionistic fuzzy set, the entropy of which have been proved the inclusion measure for IVIFSs. If we extend this intervalvalued intuitionistic fuzzy set into IVIFSSs, the corresponding interval-valued intuitionistic fuzzy soft set can be constructed by Definition 24, Theorem 15 and 16 in this work, by selecting a specific aggregation operator, a specific equivalence operator, a specific fuzzy negation operator and several specific power exponents. To a certain degree, our research is the extension of the research in Ref. 17 based on fuzzy equivalence and aggregation operators. However, even if it degenerates to the IVIFSs situation, all the formulae given by Definition 2123 in this work are new.

4.5. Transformation of similarity measures into inclusion measures for IVIFSSs

Theorem 17.

Let S be a similarity measure of interval-valued intuitionistic fuzzy soft sets and (F, P), (G, P) ∈ IVIFSS(U), then J((F, P), (G, P)) = S((G, P), (F, P) ∪ (G, P)) is an inclusion measure between (F, P) and (G, P).

Proof.

We only need to verify that the following three properties of inclusion measure hold.

  1. (1)

    J((U, P), (∅, P)) = S((∅, P), (U, P)) = 0;

  2. (2)

    J((F, P), (G, P)) = 1 ⇔ S((G, P), (F, P) ∪ (G, P)) = 1 ⇔ (G, P) = (F, P) ∪ (G, P) ⇔ (F, P) ⊆ (G, P).

  3. (3)

    If (F, P) ⊆ (G, P) ⊆ (H, P), we easily get that

    • J((H, P), (F, P)) = S((F, P), (H, P) ∪ (F, P)) = S((F, P), (H, P))⩽ S((F, P), (G, P))

    • = S((F, P), (G, P)∪(F, P)) = J((G, P), (F, P)),

    and

    • J((H, P), (F, P)) = S((F, P), (H, P) ∪ (F, P)) = S((F, P), (H, P))⩽ S((G, P), (H, P)) = S((G, P), (H, P)∪(G, P)) = J((H, P), (G, P)).

    So, we have J((H, P), (F, P))⩽J((G, P), (F, P)) and J((H, P), (F, P))⩽ J((H, P), (G, P)).

    Thus, J is an inclusion measure of IVIFSSs.

5. Disease diagnosis based on entropy and distance measure of IVIFSSs

An application of similarity measure of intuitionistic fuzzy soft set in disease diagnosis can be found in23. Benefiting from their idea, an application of the entropy and the distance measure of IVIFSSs in disease diagnosis is given. In oder to estimate if an ill person is suffering from a certain disease or not, with the help of experts, we will construct an intervalvalued intuitionistic fuzzy soft set for the disease and an interval-valued intuitionistic fuzzy soft set for the ill person, respectively. The algorithm is stated as follows:

Algorithm 1

  • Step 1. Select the threshold α ∈ [0, 1] for judging the sample set of a disease and the threshold β ∈ [0, 1] for assessing if a patient is suffering from a disease or not;

  • Step 2. Constructs an interval-valued intuitionistic fuzzy soft set (F, P) over U for the disease.

  • Step 3. Calculate the entropy of (F, P). If I((F, P)) < α, (F, P) can be regarded as a sample set for the disease; if else, collect more relevant information and reconstruct the interval-valued intuitionistic fuzzy soft set for the disease;

  • Step 4. Constructs an interval-valued intuitionistic fuzzy soft set (G, P) over U for the patient;

  • Step 5. Calculate the distance measure between (F, P) and (G, P), i.e., D((F, P), (G, P));

  • Step 6. We say the patient is suffering from the disease if D((F, P), (G, P)) < β ; if else, we say the patient is not suffering from the disease.

The thresholds α and β in Step 1 can be selected according to the actual situation with the help of experts. Step 2 is based on the consideration that if the uncertain degree of an interval-valued intuitionistic fuzzy soft set for the disease is too large, it maybe not suitable to be a reference sample.

Example 10.

Assume that our universal set contain three elements U = {x1, x2, x3}, where x1 = on the first day of illness, x2 = on the second day of illness, x3 = on the third day of illness. Here the set of parameters P is the set of certain visible symptoms, assume that P = {e1, e2, e3, e4, e5} where e1 = fever, e2 = cough, e3 = vomit, e4 = twitch, e5 = trouble breathing. We will try to estimate if a patient is suffering from a certain disease or not.

  • Step 1. Let α = 0.5 and β = 0.1.

  • Step 2. Constructs an interval-valued intuitionistic fuzzy soft set (F, P) over U for the disease which can be prepared with the help of experienced doctors:

    • F(e1) = {(x1, [0.7, 0.8], [0.15, 0.2]), (x2, [0.6, 0.7], [0.15, 0.21]), (x3, [0.55, 0.65], [0.15, 0.25])},

    • F(e2) = {(x1, [0.7, 0.8], [0.1, 0.2]), (x2, [0.55, 0.65], [0.2, 0.25]), (x3, [0.60, 0.70], [0.05, 0.1])},

    • F(e3) = {(x1, [0.7, 0.8], [0.1, 0.2]), (x2, [0.65, 0.75], [0.2, 0.25]), (x3, [0.77, 0.88], [0.1, 0.1])},

    • F(e4) = {(x1, [0.6, 0.7], [0.1, 0.2]), (x2, [0.55, 0.65], [0.2, 0.25]), (x3, [0.66, 0.7], [0.05, 0.1])},

    • F(e5) = {(x1, [0.6, 0.6], [0.2, 0.3]), (x2, [0.55, 0.60], [0.2, 0.25]), (x3, [0.7, 0.8], [0.05, 0.1])}.

  • Step 3. Calculate the entropy of (F, P). Here we use the entropy measure of IVIFSSs constructed by Theorem 3. Let D2((F, P), (Q, P)) be the Normalized hamming distance between (F, P) and (Q, P) and f′(x) = 1 − x for all x ∈ [0, 1]. Then we get I2((F, P)) = f′(2D2((F, P), (Q, P))) = 0.49 < 0.5, that is to say, (F, P) can be regarded as a sample set for the disease.

  • Step 4. Constructs an interval-valued intuitionistic fuzzy soft set (G, P) over U based on the data of a patient:

    • G(e1) = {(x1, [0.7, 0.8], [0.15, 0.2]), (x2, [0.6, 0.7], [0.15, 0.21]), (x3, [0.55, 0.75], [0.15, 0.25])},

    • G(e2) = {(x1, [0.6, 0.7], [0.2, 0.3]), (x2, [0.55, 0.65], [0.2, 0.25]), (x3, [0.7, 0.88], [0.05, 0.1])},

    • G(e3) = {(x1, [0.5, 0.6], [0.2, 0.3]), (x2, [0.45, 0.55], [0.2, 0.25]), (x3, [0.7, 0.78], [0.05, 0.1])},

    • G(e4) = {(x1, [0.3, 0.4], [0.3, 0.4]), (x2, [0.55, 0.65], [0.2, 0.25]), (x3, [0.7, 0.88], [0.05, 0.1])},

    • G(e5) = {(x1, [0.4, 0.5], [0.2, 0.3]), (x2, [0.35, 0.40], [0.2, 0.25]), (x3, [0.7, 0.88], [0.05, 0.1])}.

  • Step 5. Here we use the Normalized hamming distance between (F, P) and (G, P), which is denoted by D2((F, P), (G, P)). It is easy to get that

    D2((F, P), (G, P)) 0.067.

  • Step 6. We conclude that the patient is suffering from the disease since

    D2((F, P), (G, P)) < 0.1.

6. Conclusions and Discussion

In this paper, we give eight general formulae to calculate the distance measures of IVIFSSs by aggregating fuzzy equivalencies. Consistently with a new axiomatic definition of entropy for IVIFSSs, we prove some theorems which demonstrate that distance measures can be transformed into entropies for IVFSSs. Besides, we prove some theorems which demonstrate that entropies can be transformed into the inclusion measure and the similarity measure for IVIFSSs based on fuzzy equivalencies.

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (Grant Nos. 61473239, 61175044, 61175055, 61603307), the Fundamental Research Funds for the Central Universities of China (Grant Nos. 2682014ZT28, JBK160132) and the Open Research Fund of Key Laboratory of Xihua University(szjj2014-052).

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Journal
International Journal of Computational Intelligence Systems
Volume-Issue
10 - 1
Pages
569 - 592
Publication Date
2017/01/01
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.2991/ijcis.2017.10.1.39How to use a DOI?
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/).

Cite this article

TY  - JOUR
AU  - Yaya Liu
AU  - Junfang Luo
AU  - Bing Wang
AU  - Keyun Qin
PY  - 2017
DA  - 2017/01/01
TI  - A theoretical development on the entropy of interval-valued intuitionistic fuzzy soft sets based on the distance measure
JO  - International Journal of Computational Intelligence Systems
SP  - 569
EP  - 592
VL  - 10
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.2017.10.1.39
DO  - 10.2991/ijcis.2017.10.1.39
ID  - Liu2017
ER  -