Novel distance,similarity and entropy measures for interval valued intuitionistic fuzzy soft set

Abstract

Soft and fuzzy sets are generic tools to deal with uncertainty. Both contemporary sets are not suitable to deal with all type of uncertain parameters. In this paper the hybridization of soft with extended fuzzy set information measures are derived. Interval-valued intuitionistic fuzzy soft set theory is a powerful tool for dealing with uncertainty of knowledge in information systems. In this paper, firstly some distance and similarity measures for interval-valued intuitionistic fuzzy soft sets were proposed. Further, some new entropy measures are also introduced by using the similarity measures. The validity of these measures is also proved. Applications of the distance measures is also used in the field of multi attribute decision making and medical diagnosis. The proposed measures are also compared with an existing measure to prove its significance.

Keywords

Interval valued intuitionistic fuzzy soft set (IVIFSS)interval number hesitant factor Fuzzy soft set (FSS)

1 Introduction

Soft set theory (SST) is very important concept to deal with the problem of dimension reduction and decision-making where uncertainty is present. Molodtsov [4] introduced soft set theory which is very useful to solve the unruly system identification which involves characteristics that is tedious and non- probabilistic in nature. SST is simple tool to deal with uncertainty or vagueness in the data in comparison with probabilistic information as introduced by Shannon [5], fuzzy set theory (FST) Zadeh [20], the intuitionistic fuzzy set theory (IFS) by Atanassov [12], another point of view rough set (RS) theory by Pawlak [40]. To deal with incompleteness of the hybridization for the model fuzzy rough set (FRS) by Nakamura [1] introduced fuzzy rough set to deal with incompleteness of hybridization related to fuzzy rough set theory [2]. Further Dubois and Prade [3] worked on hybridization extended the work given by Nakamura [1], which is widely used by Nanda and Majumdar [30]. The intuitionistic fuzzy set (IFS) was proposed by Atanassov [12] which is the conventional extension of fuzzy sets. Interval-valued fuzzy set (IVFS) was apparently discussed independently in the mid- seventies by Grattan-Guinness [9], Jahn [17] and Zadeh [21]. Yang et al. [34] introduced the concept interval valued fuzzy soft set as the combination of interval-valued fuzzy set and the soft set. Atanassov and Gargov [14] further proposed interval-valued intuitionistic fuzzy sets (IVIFSs) based on IFSs. After their pioneering work, both IFSs and IVIFSs got more attention and have been extensively used in number of fields, such as industrial control by Cheng et al. [31], pattern classification by Lin [16], pattern recognition, medical diagnosis, and cluster analysis by Huang et al. [18] and Milošević et al. [26], system modeling by Papageorgiou & Iakovidis [6], and decision-making analysis by Atanassov et al. [13], Zhou & Wang[22], Jiang et al. [29], Xu & Yager [37] and Yang et al. [38]. Generalized entropy, distance, and similarity measures for IVIFS with application in decision-making was given by Tiwari and Gupta [27].

Aygunoglu [11] worked on hybridization of fuzzy with soft sets termed as fuzzy soft group and discussed some properties and structural characteristics of fuzzy soft groups. Yang et al. [33] defined the operations on fuzzy soft sets, which are based on three fuzzy logic operations: negation, triangular norm and triangular co-norm. Zou and Xiao [36] introduced the soft sets and fuzzy soft sets into the incomplete environment respectively. Xiao et al. [39] proposed a combined forecasting approach based on fuzzy soft sets. Feng et al. [7] present an adjustable approach to fuzzy soft sets-based decision making and give some illustrative examples. Xu et al. [32] introduced the extension of soft set as vague soft sets and their notation. Majumdar and Samanta [25] further generalized the concept of fuzzy soft sets as introduced by Maji et al. [23, 24], in other words, a degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. Jiang et al. [35] used the concepts of Description Logics (DLs) by Baader et al. [8] to play as the parameters of soft sets.

Hybridization of the interval-valued intuitionistic fuzzy sets Deschrijver et al. [10], Atanassov and Gargov [14], Atanassov [15] and soft sets, from which another soft set model: interval-valued intuitionistic fuzzy soft set theory has been derived.

The purpose of the paper is to develop new information measures for IVIFSS. This paper is organized as follows. The following section briefly reviews some background on soft sets, fuzzy soft sets, intuitionistic fuzzy soft sets and interval-valued intuitionistic fuzzy soft sets. In section 2, preliminaries are given those are related to the work. In Section 3, proposes the distance measures and based on these measures’ similarity measures of interval-valued intuitionistic fuzzy soft sets is proposed with their properties. In section 4, entropy measures are derived and then their proof is detailed. Finally, Section 5, applies these distance measures in medical area and entropy measures are used in decision-making. At the last the conclusion of the work done is given.

2 Preliminary

Preliminary definitions and perceptions related to soft, fuzzy soft, interval- valued intuitionistic fuzzy and interval valued intuitionistic fuzzy sets are presented in this section. Let initial universal set Φ ={ φ₁, φ₂, φ₃ … . φ_n } and a parameter set Θ ={ θ₁, θ₂, θ₃ … . θ_m } under deliberation with respect to Φ . Parameter set includes characteristics by attributes or properties of objects in Φ . Thespace (Φ, Θ) is called a soft space.

Definition 2.1. (Molodtsov [4]): A duos of (Γ, Π) is called a soft set(SS) over Φ, where Π ⊆ Θ and Γ is a mapping given by $Γ \sim Π \to P (Φ)$

Otherwise, a SS under Φ is a parameterize family of subset of power set P (Φ). Π is called the parameter set of the SS (Γ, Π). Where θ_i ∈ Π, Γ (θ_i) may be considered as the set of θ-approximate elements of (Γ, Π).

Definition 2.2. (The choice value Algorithm) Let (Γ, Π) is a SS, (Γ, Π) can be disclosed as a binary table, and if φ_i ∈ Γ_{θ
_i} then φ_ij = 1 otherwise φ_ij = 0. The choice value c_i = ∑_jφ_ij, the object with the maximum choice value is selected as the optimal decision. The algorithm is as follows:

The choice value algorithm

Input the soft values

Compute the choice values c_i for each object φ_i, where c_i = ∑_jφ_ij

The decision is φ_i if $c_{i} = \max_{j} (c_{j})$

If i has more than one value any one of φ_i may be chosen.

Example 2.1. Let us consider Φ is set of houses under consideration. Now Θ is the set of parameters set. $Θ = {\begin{matrix} expensive, beautiful, wooden, cheap, \\ in the green surroundings, modern, \\ in good repair, in bad repair \end{matrix}}$

The soft set (Γ, Π) describes “attractiveness of the houses” which someone is going to buy.

Now let us consider are six houses in the universe Φ given by Φ ={ φ₁, φ₂, φ₃, φ₄, φ₅, φ₆ } and five parameters Θ ={ θ₁, θ₂, θ₃, θ₄, θ₅ }

θ₁ = stands for the parameter expensive,

θ₂ = stands for the parameter beautiful

θ₃ = stands for the parameter wooden

θ₄ = stands for the parameter cheaps

θ₅ = stands for the prameter in the green sroundings

Suppose that Γ_{θ
₁} ={ φ₁, φ₂ } $\begin{matrix} Γ_{θ_{2}} = {φ_{1}, φ_{3}, φ_{4}}, Γ_{θ_{3}} = {φ_{4}, φ_{5}, φ_{6}}, \\ Γ_{θ_{4}} = {φ_{1}, φ_{2}}, Γ_{θ_{5}} = {φ_{1}, φ_{5}, φ_{6}} \end{matrix}$

The soft set (Γ, Π) pparametrized family Γ_{θ
_i}; i = 1, 2, 3 … 5 of subsets of the set Φ denotes the approximate descriptions of an object. Consider the mapping Γ which is “house(.)” where dot(.) is filled up by a parameter θ ∈ Θ therefore Γ_{θ
_i} means “house, (expensive)” and its functional value is {φ₁, φ₂}. Tabular form of representation of a soft set.

According to the above Table 1 first house φ₁ is chosen.

Table 1
Representation of soft set

Θ Expensive θ₁ Beautiful θ₂ Wooden θ₃ Cheap θ₄ In green surrounding θ₅ Choice value

φ ₁ 1 1 0 1 1 4

φ ₂ 1 0 0 1 0 2

φ ₃ 0 1 0 0 0 1

φ ₄ 0 1 1 0 0 2

φ ₅ 0 0 1 0 1 2

Θ	Expensive θ₁	Beautiful θ₂	Wooden θ₃	Cheap θ₄	In green surrounding θ₅	Choice value
φ ₁	1	1	0	1	1	4
φ ₂	1	0	0	1	0	2
φ ₃	0	1	0	0	0	1
φ ₄	0	1	1	0	0	2
φ ₅	0	0	1	0	1	2

Table 1 Soft set (Γ, Π) represented as a real valued information system.

Definition 2.2. (Maji et al. [24]): Let (Φ, Θ) is a soft space. A pair (Γ, Π) is called a FSS over Φ, where Π ⊆ Θ and Γ is a mapping given by $Γ \sim Π \to Γ (Φ)$

and Γ (Φ) is the set of all fuzzy subsets (FSs) on Φ.

Example 2.2. Let universe Φ ={ φ₁, φ₂, φ₃, φ₄ } and a parameter set Θ ={ θ₁, θ₂, θ₃, θ₄, θ₅ } under consideration with respect to Φ then the fuzzy soft set is defined as follows:

Table 2

Fuzzy soft set (Γ, Π) represented as a real valued information system

Θ	θ ₁	θ ₂	θ ₃	θ ₄
φ ₁	0.21	0.31	0.45	0.51
φ ₂	0.35	.260	0.460	0.8
φ ₃	0.35	0.11	0.7	0.68
φ ₄	0.68	0.67	0.87	0.346
φ ₅	0.38	0.76	.43	0.186

Definition 2.3. (Maji et al. [24]): (Γ, Π), (I, Λ) be two FSS over a common universe Φ. Then

The fuzzy soft space (Γ, Π) is said to be a fuzzy soft subset FSS of the fuzzy soft space (I, Λ). That is given (Γ, Π) ⊆ (I, Λ), if Π ⊆ Λ and ∀θ ∈ Π then Γ (θ) ⊆ Λ (θ).

If Γ (θ) =∅ for any θ ∈ Π then the fuzzy soft space (Γ, Π) is called null fuzzy soft set and denoted by ∅.

If Γ (θ) = Φ for any θ ∈ Π the space (Γ, Π) is equal to absolute fuzzy soft set, denoted by Φ.

The complement of fuzzy soft set (Γ, Π) is denoted by (Γ, Π) ^c = 1 - (Γ, Π) where Π ⊆ Θ.

Definition 2.4. Atanassov and Gargov [14]: An interval valued intuitionistic fuzzy set, Π over the universe Φ can be defined as follows: $Π = {φ, ((Γ^{l} (φ)_{j}, Γ^{u} (φ_{j})), (Γ^{l *} (φ_{j}), Γ^{u *} (φ_{j}))) : φ ɛ Φ}$ where Γ^l (φ_j) : Φ → [0, 1] , Γ^u (φ_j) : Φ → [0, 1], Γ^l* (φ_j) : Φ → [0, 1], Γ^u* (φ_j) : Φ → [0, 1], Γ^l (φ_j): is the lower bound of the membership value, Γ^u (φ_j) is the upper bound of the membership value, Γ^l* (φ_j) s the lower bound of the non-membership value, Γ^u* (φ_j) is the upper bound of the non-membership value with the property 0 ⩽ Γ^l (φ_j) , Γ^u (φ_j) , Γ^l* (φ_j) , Γ^u* (φ_j) ⩽ 1, ∀ φɛΦ here Γ (φ_j) = (Γ^l (φ_j) , Γ^u (φ_j)) and Γ^* (φ_j) = (Γ^l* (φ_j) , Γ^u* (φ_j)) represents the degree of membership and non-membership functions respectively. Γ^l** (φ_j) = 1 - (Γ^u (φ_j) + Γ^u* (φ_j)) and Γ^u** (φ_j) = 1 - (Γ^l (φ_j) + Γ^l* (φ_j)) are called the intuitionistic fuzzy index.

Definition 2.5. Yang et al. [34]: A pair (Γ, Π) is called Interval valued intuitionistic fuzzy soft set, ΠɛΓ is set parameters over the universe Φ, where Γ is mapping given by can be defined as follows: $Π = {φ, ((Γ_{θ_{i}}^{l} φ_{j}, Γ_{θ_{i}}^{u} φ_{j}), (Γ_{θ_{i}}^{l *} φ_{j}, Γ_{θ_{i}}^{u *} φ_{j})) : φ ɛ Φ}$ where $Γ_{θ_{i}}^{l} φ_{j} : Φ \to [0, 1], Γ_{θ_{i}}^{u} φ_{j} : Φ \to [0, 1]$ , $Γ_{θ_{i}}^{l *} φ_{j} : Φ \to [0, 1]$ , $Γ_{θ_{i}}^{u *} φ_{j} : Φ \to [0, 1]$ , $Γ_{θ_{i}}^{l} φ_{j} : Φ \to [0, 1]$ is the lower bound of the membership value, $Γ_{θ_{i}}^{u} φ_{j}$ is the upper bound of the membership value, $Γ_{θ_{i}}^{l *} φ_{j}$ s the lower bound of the non-membership value, $Γ_{θ_{i}}^{u *} φ_{j}$ is the upper bound of the non-membership value with the property $0 ⩽ Γ_{θ_{i}}^{l} φ_{j}, Γ_{θ_{i}}^{u} φ_{j}, Γ_{θ_{i}}^{l *} φ_{j}, Γ_{θ_{i}}^{u *} φ_{j} ⩽ 1, \forall φ ɛ Φ$ here $Γ_{θ_{i}} φ_{j} = (Γ_{θ_{i}}^{l} φ_{j}, Γ_{θ_{i}}^{u} φ_{j})$ and $Γ_{θ_{i}}^{*} φ_{j} = (Γ_{θ_{i}}^{l *} φ_{j}, Γ_{θ_{i}}^{u *} φ_{j})$ represents the degree of membership and non-membership functions respectively. $Γ_{θ_{i}}^{l * *} φ_{j} = 1 - (Γ_{θ_{i}}^{u} φ_{j} + Γ_{θ_{i}}^{u *} φ_{j})$ and $Γ_{θ_{i}}^{u * *} φ_{j} = 1 - (Γ_{θ_{i}}^{l} φ_{j} + Γ_{θ_{i}}^{l *} φ_{j})$ are called the intuitionistic fuzzy index.

Definition 2.6. Union of two IVIFSS (Γ, Π), (I, Λ) over a common universe Φ is Γ ∪ I $Γ \cup I = {〈 \begin{matrix} φ, ((sup (Γ_{θ_{i}}^{l} φ_{j}, I_{θ_{i}}^{l} φ_{j}), sup (Γ_{θ_{i}}^{u} φ_{j}, Γ_{θ_{i}}^{u} φ_{j}))), \\ ((inf (Γ_{θ_{i}}^{l *} φ_{j}, I_{θ_{i}}^{l *} φ_{j}), \inf (Γ_{θ_{i}}^{u *} φ_{j}, I_{θ_{i}}^{u *} φ_{j}))) \end{matrix} | φ ɛ Φ 〉}$

Definition 2.7. Intersection of two IVIFSS (Γ, Π), (I, Λ) over a common universe Φ is Γ ∩ I $Γ \cap I = {〈 \begin{matrix} φ, ((inf (Γ_{θ_{i}}^{l} φ_{j}, I_{θ_{i}}^{l} φ_{j}), inf (Γ_{θ_{i}}^{u} φ_{j}, Γ_{θ_{i}}^{u} φ_{j}))), \\ ((sup (Γ_{θ_{i}}^{l *} φ_{j}, I_{θ_{i}}^{l *} φ_{j}), \sup (Γ_{θ_{i}}^{u *} φ_{j}, I_{θ_{i}}^{u *} φ_{j}))) \end{matrix} | φ ɛ Φ 〉}$

Definition 2.8. Compliment of IVIFSS (Γ, Π) over a common universe Φ is Γ^c $Γ^{c} = {φ, ((Γ_{θ_{i}}^{l *} φ_{j}, Γ_{θ_{i}}^{u *} φ_{j}), (Γ_{θ_{i}}^{l} φ_{j}, Γ_{θ_{i}}^{u} φ_{j})) : φ ɛ Φ}$

Definition 2.9. Yaya et al. [19]: Consider (Γ, Π) and (I, Π) be two interval valued intuitionistic fuzzy soft set over the same finite universe Φ. Then the following conditions are hold for distance measure:

0 ⩽ D ((Γ, Π)) ⩽ 1

D ((Γ, I)) = 0iffΓ = I

D ((Γ, Π)) = D ((I, Π))

If Γ ⩽ I ⩽ H then D (Γ, I) ⩽ D (Γ, H) then D (I, H) ⩽ D (Γ, H)

Definition 2.10. Yaya et al. [19]: Consider (Γ, Π) and (I, Π) be two interval-valued intuitionistic FSS over the same finite universal set Φ. Then the following conditions are hold for similarity measure:

0 ⩽ S ((Γ, Π)) ⩽ 1

S ((Γ, I)) = 1iffΓ = I

S ((Γ, Π)) = S ((I, Π))

If Γ ⩽ I ⩽ H then S (Γ, I) ⩾ S (Γ, H) then S (I, H) ⩾ S (Γ, H)

Definition 2.11. Yaya et al. [19]: Consider (H, Π) be an interval valued intuitionistic fuzzy soft set on the universe Φ such that $\forall θ_{i} \in Π, H (θ_{i}) = {(φ_{i}, [\frac{1}{2}, \frac{1}{2}], [\frac{1}{2}, \frac{1}{2}]), φ_{i} \in Φ}$ . A real function E: IVIFSS Φ → [0, 1] is called an entropy for interval-valued intuitionistic fuzzy soft sets, if E has the following properties:

E (Γ, Π) = 0 if (Γ, Π) is classical soft set

E (Γ, Π) = 1 iff $Γ_{θ_{i}} φ_{j} = Γ_{θ_{i}}^{*} φ_{j} = [\frac{1}{2}, \frac{1}{2}] \forall$ θ_i ∈ Π, φ_i ∈ Φ

E (Γ, Π) = E (Γ, Π) ^c

E (Γ, Π) ⩽ E (I, Π), If D ((Γ, Π) , (H, Π)) ⩾ D ((I, Π) , (H, Π))

Here (2) implies that entropy of (Γ, Π) will be maximum if (Γ, Π) is equal to (H, Π); and (4) implies that the closer an interval valued intuitionistic fuzzy soft set (Γ, Π) is to (H, Π), the more entropy of (Γ, Π) should decrease.

3 Novel distance and similarity for IVIFSS

Distance and similarity are dual of each other. Larger the distance is smaller the similarity between any two sets. This concept is used to define below some similarity measures based on defined distance measures. These measures are used extensively in FSS to compare two sets. Suppose that the IVIFSS (Γ, Π) and (I, Λ) have the same parameter set, namely Π = Λ. Three distance measures are derived. The average distance measure is the combination of Hamming distance measure and normalized distance induced by Hausdoff metric measure those are suggested by Feng et al. [28]. Maximin and convex distance measures are the generalizations of normalized distance induced by Hausdorff metric measure Feng et al. [28].

Based on these distance measures three similarity and entropy measures are also derived. Three information measures were defined as follows:

3.1 Distance measures for IVIFSS

“Maximin Distance Measure” $\begin{matrix} D_{1} ((Γ, Π), (I, Λ)) = \frac{1}{6} {Max}_{i} {{Min}_{j} (| Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | \\ + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} |)} \dots \end{matrix}$ (3.1)

“Average Distance Measure” $\begin{matrix} D_{2} ((Γ, Π), (I, Λ)) \\ = \frac{1}{6 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} \\ + Max (\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}} \dots \end{matrix}$ (3.2)

“Convex Distance Measure” $\begin{matrix} D_{3} ((Γ, Π), (I, Λ)) \\ = {Max}_{i} {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) \end{matrix}} \\ \dots \end{matrix}$ (3.3)

In the following theorems it will be proved that all the above measures satisfy all the axioms stated in definition (2.9) which are essential for the validity of the distance measures for the IVIFSS.

Theorem 3.1. D₁ is the valid maximin distance measure between two IVIFSSs (Γ, Π) and (I, Λ).

Proof. Let us consider (Γ, Π), (I, Λ) be two interval valued intuitionistic fuzzy soft sets over a common universal set Φ, where θ_i ∈ Π and φ_j ∈ Λ.

d₁: Clearly $0 ⩽ Γ_{θ_{i}}^{l} φ_{j} ⩽ 1, 0 ⩽ Γ_{θ_{i}}^{u} φ_{j} ⩽ 1, 0 ⩽ I_{θ_{i}}^{l} φ_{j} ⩽ 1, 0 ⩽ I_{θ_{i}}^{u} φ_{j} ⩽ 1$ , $\begin{matrix} 0 ⩽ Γ_{θ_{i}}^{l *} φ_{j} ⩽ 1, 0 ⩽ Γ_{θ_{i}}^{u *} φ_{j} ⩽ 1 and 0 ⩽ I_{θ_{i}}^{l *} φ_{j} \\ ⩽ 1, 0 ⩽ I_{θ_{i}}^{u *} φ_{j} ⩽ 1 \end{matrix}$ $\begin{matrix} 0 ⩽ Γ_{θ_{i}}^{l * *} φ_{j} ⩽ 1, 0 ⩽ I_{θ_{i}}^{l * *} φ_{j} ⩽ 1, \\ 0 ⩽ Γ_{θ_{i}}^{u * *} φ_{j} ⩽ 1, and \\ 0 ⩽ I_{θ_{i}}^{u * *} φ_{j} ⩽ 1 \end{matrix}$

Thus $0 ⩽ | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | ⩽ 1, 0 ⩽ | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | ⩽ 1$ , $\begin{matrix} 0 ⩽ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | ⩽ 1, 0 \\ ⩽ | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | ⩽ 1 \end{matrix}$ $0 ⩽ {Min}_{j} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{6} ⩽ 1$ $0 ⩽ {Max}_{i} {{Min}_{j} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{6}} ⩽ 1$ $0 ⩽ D_{1} ((Γ, Π), (I, Λ)) ⩽ 1$

Thus d₁ is fulfilled.

d₂: Assume that $\begin{matrix} Γ = I then D_{1} ((Γ, Π), (Γ, Π)) \\ = {Max}_{i} {{Min}_{j} (\frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{6})} = 0 \end{matrix}$

Thus, d₂ is satisfied. $\begin{matrix} d_{3} : Consider D_{1} ((Γ, Π), (I, Λ)) = \\ {Max}_{i} {{Min}_{j} (\frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{6})} \end{matrix}$ $= {Max}_{i} {{Min}_{j} (| \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{6} |)}$ $D_{1} ((Γ, Π), (I, Λ)) = D_{1} ((I, Λ), (Γ, Π))$

Thus, d₃ is proved.

d₄: If Γ ⩽ I ⩽ H then D (Γ, I) ⩽ D (Γ, H) then D (I, H) ⩽ D (Γ, H) $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} ⩽ I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} ⩽ I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j}, \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l *} φ_{j} ⩽ I_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j}, \\ Γ_{θ_{i}}^{u *} φ_{j} ⩽ I_{θ_{i}}^{u *} φ_{j} ⩽ H_{θ_{i}}^{u *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l * *} φ_{j} ⩽ I_{θ_{i}}^{l * *} φ_{j} ⩽ H_{θ_{i}}^{l * *} φ_{j}, \\ Γ_{θ_{i}}^{u * *} φ_{j} ⩽ I_{θ_{i}}^{u * *} φ_{j} ⩽ H_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} I_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j}, \\ I_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} \end{matrix}$ $\begin{matrix} I_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j}, I_{θ_{i}}^{u *} φ_{j} \\ - Γ_{θ_{i}}^{u *} φ_{j} ⩽ H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} \end{matrix}$ $\begin{matrix} I_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} ⩽ H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j}, I_{θ_{i}}^{u * *} φ_{j} \\ - Γ_{θ_{i}}^{u * *} φ_{j} ⩽ H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} |, \\ | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | ⩽ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} |, \\ | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | ⩽ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} |, \\ | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | \\ + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | \\ + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | \\ + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | \end{matrix}$ ${Max}_{i} {{Min}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6})}$ $⩽ {Max}_{i} {{Min}_{j} (\frac{(\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6})}$ $D_{1} (Γ, I) ⩽ D_{1} (Γ, H)$ $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} ⩽ I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} ⩽ I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j}, \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l *} φ_{j} ⩽ I_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j}, \\ Γ_{θ_{i}}^{u * *} φ_{j} ⩽ I_{θ_{i}}^{u * *} φ_{j} ⩽ H_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l * *} φ_{j} ⩽ I_{θ_{i}}^{l * *} φ_{j} ⩽ H_{θ_{i}}^{l * *} φ_{j}, \\ Γ_{θ_{i}}^{u * *} φ_{j} ⩽ I_{θ_{i}}^{u * *} φ_{j} ⩽ H_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} ⩽ H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j}, \\ Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} \\ ⩽ H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} |, \\ | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | ⩽ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} |, \\ | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | ⩽ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} |, \\ | H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | \\ + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | \\ + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ ${Max}_{i} {{Min}_{j} (\frac{\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{6})}$ $⩽ {Max}_{i} {{Min}_{j} (\frac{(\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6})}$ $D_{1} (I, H) ⩽ D_{1} (Γ, H)$

Hence d₄ is fulfilled.

Since all the axioms d_k, k = 1, 2, 3, 4 are satisfied stated in the definition (2.9). Hence D₁ ((Γ, Π) , (I, Λ)) is the valid maximin distance measure between two IVIFSSs (Γ, Π), (I, Λ).

Theorem 3.2. D₂ is the valid average distance measure between two IVIFSSs (Γ, Π), (I, Λ).

Proof. Let (Γ, Π), (I, Λ) be two interval valued intuitionistic fuzzy soft sets over a common universe Φ, where θ_i ∈ Π and φ_j ∈ Φ $\begin{matrix} d_{1} : D_{2} ((Γ, Π), (I, Λ)) = \\ \frac{1}{6 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} + \\ Max (\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}} \end{matrix}$ $\begin{matrix} 0 ⩽ Γ_{θ_{i}}^{l} φ_{j} ⩽ 1, 0 ⩽ Γ_{θ_{i}}^{u} φ_{j} ⩽ 1 and 0 ⩽ I_{θ_{i}}^{l} φ_{j} \\ ⩽ 1, 0 ⩽ I_{θ_{i}}^{u} φ_{j} ⩽ 1 \end{matrix}$ $\begin{matrix} 0 ⩽ Γ_{θ_{i}}^{l *} φ_{j} ⩽ 1, 0 ⩽ Γ_{θ_{i}}^{u *} φ_{j} ⩽ 1 and 0 ⩽ I_{θ_{i}}^{l *} φ_{j} \\ ⩽ 1, 0 ⩽ I_{θ_{i}}^{u *} φ_{j} ⩽ 1 \end{matrix}$ $\begin{matrix} 0 ⩽ Γ_{θ_{i}}^{* * l} φ_{j} ⩽ 1, 0 ⩽ I_{θ_{i}}^{* * l} φ_{j} ⩽ 1, 0 ⩽ Γ_{θ_{i}}^{* * u} φ_{j} \\ ⩽ 1, and 0 ⩽ I_{θ_{i}}^{* * u} φ_{j} ⩽ 1 \end{matrix}$

Thus $0 ⩽ | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | ⩽ 1, 0 ⩽ | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | ⩽ 1$ , $\begin{matrix} 0 ⩽ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | ⩽ 1, 0 \\ ⩽ | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | ⩽ 1 \end{matrix}$ $\begin{matrix} 0 ⩽ | Γ_{θ_{i}}^{* * l} φ_{j} - I_{θ_{i}}^{* * l} φ_{j} | ⩽ 1, 0 \\ ⩽ | Γ_{θ_{i}}^{* * u} φ_{j} - I_{θ_{i}}^{* * u} φ_{j} | ⩽ 1 \end{matrix}$ $0 ⩽ \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} ⩽ \frac{6}{n} < 6$

and $\begin{matrix} 0 ⩽ Max (| Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} |) ⩽ 6 \end{matrix}$ $0 ⩽ \sum_{i = 1}^{m} (\sum_{j = 1}^{n} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n}) < 6$ and $\begin{matrix} 0 ⩽ (\sum_{i = 1}^{m} Max (| Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} |) < 6 \end{matrix}$ $\begin{matrix} 0 ⩽ \frac{1}{6 m} {\sum_{i = 1}^{m} (\sum_{j = 1}^{n} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n})} \\ < 1 . . . \end{matrix}$ (3.4) $\begin{matrix} 0 ⩽ \frac{1}{6 m} {\sum_{i = 1}^{m} Max (| Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | \\ + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} |)} \\ < 1 \dots \end{matrix}$ (3.5)

Now adding (3.4) and (3.5), $\begin{matrix} 0 ⩽ \frac{1}{12 m} \sum_{i = 1}^{m} (\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} + \\ Max (\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}) \\ ⩽ \frac{1}{2} < 1 \end{matrix}$ $0 ⩽ D_{2} ((Γ, Π) (I, Λ)) ⩽ 1$

d₁ is fulfilled.

d₂: Assume that Γ = I $\begin{matrix} D_{2} ((Γ, Π), (I, Λ)) \\ = \frac{1}{12 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} + \\ Max (\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}} \end{matrix}$ $D_{2} ((Γ, Π) (I, Λ)) = 0$ d₂ is fulfilled.

$\begin{matrix} d_{3} : Consider D_{2} ((Γ, Π), (I, Λ)) = \\ \frac{1}{12 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} + \\ Max (\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}} \end{matrix}$ $\begin{matrix} = \frac{1}{12 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | - (Γ_{θ_{i}}^{l} φ_{j} + I_{θ_{i}}^{l} φ_{j}) | + (| - Γ_{θ_{i}}^{u} φ_{j} + I_{θ_{i}}^{u} φ_{j} |) + \\ | - (Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j}) | + | - (Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j}) | + \\ | - (Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j}) | + | - (Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j}) | \end{matrix}}{n} + \\ Max (\begin{matrix} | - (Γ_{θ_{i}}^{l} φ_{j} + I_{θ_{i}}^{l} φ_{j}) | + (| - Γ_{θ_{i}}^{u} φ_{j} + I_{θ_{i}}^{u} φ_{j} |) + \\ | - (Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j}) | + | - (Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j}) | + \\ | - (Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j}) | + | - (Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j}) | \end{matrix}) \end{matrix}} \\ = D_{2} ((I, Λ) (Γ, Π)) \end{matrix}$

Thus, D₂ ((Γ, Π) (I, Λ)) = D₂ ((I, Λ) (Γ, Π))

d₃ is fulfilled.

d₄: If Γ ⩽ I ⩽ H then D (Γ, I) ⩽ D (Γ, H) then D (I, H) ⩽ D (Γ, H) $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} ⩽ I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} ⩽ I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j}, \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l *} φ_{j} ⩽ I_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j}, \\ Γ_{θ_{i}}^{u *} φ_{j} ⩽ I_{θ_{i}}^{u *} φ_{j} ⩽ H_{θ_{i}}^{u *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l * *} φ_{j} ⩽ I_{θ_{i}}^{l * *} φ_{j} ⩽ H_{θ_{i}}^{l * *} φ_{j}, \\ Γ_{θ_{i}}^{u * *} φ_{j} ⩽ I_{θ_{i}}^{u * *} φ_{j} ⩽ H_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} |, \\ | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | ⩽ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} |, \\ | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | ⩽ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} |, \\ | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | \\ + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | \\ + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\frac{1}{12 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} + \\ Max (\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}}$ $⩽ \frac{1}{12 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | + \\ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} + \\ Max (\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | + \\ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}}$ $D_{2} (Γ, I) ⩽ D_{2} (Γ, H)$ $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} ⩽ I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} ⩽ I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j}, \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l *} φ_{j} ⩽ I_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j}, \\ Γ_{θ_{i}}^{u *} φ_{j} ⩽ I_{θ_{i}}^{u *} φ_{j} ⩽ H_{θ_{i}}^{u *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l * *} φ_{j} ⩽ I_{θ_{i}}^{l * *} φ_{j} ⩽ H_{θ_{i}}^{l * *} φ_{j}, \\ Γ_{θ_{i}}^{u * *} φ_{j} ⩽ I_{θ_{i}}^{u * *} φ_{j} ⩽ H_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j}, \\ Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} \\ ⩽ H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} |, \\ | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | ⩽ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} |, \\ | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | ⩽ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} |, \\ | H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | \\ + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | \\ + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\frac{1}{12 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} + \\ Max (\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}}$ $⩽ \frac{1}{12 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | + \\ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} + \\ Max (\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | + \\ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}}$

$D_{2} (I, H) ⩽ D_{2} (Γ, H)$

d₄ is fulfilled.

Since all the axioms d_k, k = 1, 2, 3, 4 are fulfilled. Hence D₂ is the valid average distance measure between two IVIFSS (Γ, Π), (I, Λ).

Theorem 3.3. D₃ is the valid convex distance measure between two IVIFSSs (Γ, Π), (I, Λ).

Proof. Let us consider (Γ, Π), (I, Λ) be two IVIFSSs over a common universe Φ, where θ_i ∈ Π and φ_j ∈ Λ. It is simple and calm d₁ property (similar to theorem 3.2). Now in order to check that d₂ is satisfied. Assume that (Γ, Π) = (I, Λ)

$\begin{matrix} D_{3} ((Γ, Π), (I, Λ)) \\ = {Max}_{i} {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) \end{matrix}} = 0 \end{matrix}$

Hence d₂ is satisfied.

d₃: Consider $\begin{matrix} D_{3} ((Γ, Π), (I, Λ)) \\ = {Max}_{i} {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) \end{matrix}} \end{matrix}$ $\begin{matrix} D_{3} ((Γ, Π), (I, Λ)) \\ = {Max}_{i} {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | I_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | I_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | I_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | I_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | + \\ | I_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | I_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | I_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | I_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | I_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | I_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | + \\ | I_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | I_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) \end{matrix}} \end{matrix}$ $= D_{3} ((I, Λ), (Γ, Π))$

Hence d₃ is fulfilled.

d₄: If Γ ⩽ I ⩽ H then D (Γ, I) ⩽ D (Γ, H) then D (I, H) ⩽ D (Γ, H) $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} ⩽ I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} ⩽ I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j}, \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l *} φ_{j} ⩽ I_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j}, \\ Γ_{θ_{i}}^{u *} φ_{j} ⩽ I_{θ_{i}}^{u *} φ_{j} ⩽ H_{θ_{i}}^{u *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l * *} φ_{j} ⩽ I_{θ_{i}}^{l * *} φ_{j} ⩽ H_{θ_{i}}^{l * *} φ_{j}, \\ Γ_{θ_{i}}^{u * *} φ_{j} ⩽ I_{θ_{i}}^{u * *} φ_{j} ⩽ H_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j}, \\ Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} \\ ⩽ H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | ⩽ | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} |, \\ | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | ⩽ | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} |, \\ | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | ⩽ | H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} |, \\ | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\begin{matrix} {Max}_{i} {α {Max}_{j} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | + (1 \\ - α) {Min}_{j} (| Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} |)} \\ ⩽ {Max}_{i} {α {Max}_{j} (| H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | \\ + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | \\ + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} |) + (1 \\ - α {Min}_{j} (| H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} |)} \end{matrix}$ $D_{3} (Γ, I) ⩽ D_{3} (Γ, H)$ $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} ⩽ I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} ⩽ I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j}, \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l *} φ_{j} ⩽ I_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j}, \\ Γ_{θ_{i}}^{u *} φ_{j} ⩽ I_{θ_{i}}^{u *} φ_{j} ⩽ H_{θ_{i}}^{u *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l * *} φ_{j} ⩽ I_{θ_{i}}^{l * *} φ_{j} ⩽ H_{θ_{i}}^{l * *} φ_{j}, \\ Γ_{θ_{i}}^{u * *} φ_{j} ⩽ I_{θ_{i}}^{u * *} φ_{j} ⩽ H_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} ⩽ H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j}, \\ Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} ⩽ H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} ⩽ H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j}, \\ Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} \\ ⩽ H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} \end{matrix}$ $\begin{matrix} Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} ⩽ H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j}, \\ Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} \\ ⩽ H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} |, \\ | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | ⩽ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} |, \\ | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | ⩽ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} |, \\ | H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + | H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \\ ⩽ | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | \\ + | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | \\ + | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | \\ + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}$ $\frac{1}{6} {Max}_{i} {\begin{matrix} α {Max}_{j} (\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) + \\ (1 - α) {Min}_{j} (\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | H_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}}$ $\begin{matrix} ⩽ \frac{1}{6} {Max}_{i} {\begin{matrix} α {Max}_{j} (\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | + \\ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) + \\ (1 - α) {Min}_{j} (\begin{matrix} | H_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{l} φ_{j} | + | H_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{u} φ_{j} | + \\ | H_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l *} φ_{j} | + | H_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u *} φ_{j} | + \\ | H_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * *} φ_{j} | + | H_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}} \end{matrix}$ $D_{3} (I, H) ⩽ D_{3} (Γ, H)$

d₄ is fulfilled.

Since all the axioms d_k, k = 1, 2, 3, 4 are fulfilled. Hence D₃ is the convex Distance Measure between (Γ, Π), (I, Λ).

3.2 Similarity measures for IVIFSS

“Maximin similarity measure” $\begin{matrix} S_{1} ((Γ, Π), (I, Λ)) = 1 \\ - \frac{1}{6} {Max}_{i} {{Min}_{j} (| Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | \\ + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | \\ + | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | \\ + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | \\ + | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | \\ + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} |)} \dots \end{matrix}$ (3.6)

“Average similarity measure” $\begin{matrix} S_{2} ((Γ, Π), (I, Λ)) = 1 \\ - \frac{1}{12 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}}{n} \\ + Max (\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix}) \end{matrix}} \\ \dots \end{matrix}$ (3.7)

“Convex similarity measure” $\begin{matrix} S_{3} ((Γ, Π), (I, Λ)) = 1 \\ - {Max}_{i} {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - I_{θ_{i}}^{l} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - I_{θ_{i}}^{u} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - I_{θ_{i}}^{l *} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - I_{θ_{i}}^{u *} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - I_{θ_{i}}^{l * *} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - I_{θ_{i}}^{u * *} φ_{j} | \end{matrix})}{6}) \end{matrix}} \\ \dots \end{matrix}$ (3.8)

All the axioms stated in the definition (2.10) which are essential for the validity of the similarity measures for the IVIFSS.

3.3 Entropy measures for IVIFSS

Maximin entropy is presented as: $\begin{matrix} E_{1} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - (\frac{1}{6}) {Max}_{i} {{Min}_{j} (| 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 |)} \dots \end{matrix}$ (3.9)

Average entropy is presented as: $\begin{matrix} E_{2} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - \frac{1}{12 m} \sum_{i = 1}^{m} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix}}{n} \\ + Max (\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix}) \end{matrix}} \end{matrix}$ (3.10)

Convex entropy is presented as: $\begin{matrix} E_{3} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - \frac{1}{6} {Max}_{i} {\begin{matrix} α {Max}_{j} (\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{* l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{* u} φ_{j} - 1 | \end{matrix}) + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | + \\ (1 - α) {Min}_{j} (\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{* l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{* u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix}) \end{matrix}} \end{matrix}$ (3.11)

Theorem 3.4. E₂ ((Γ, Π) , (Γ^c, Π)) is the valid maximin entropy measure for IVIFSSs. $\begin{matrix} E_{1} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - (\frac{1}{6}) {Max}_{i} {{Min}_{j} (| 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | f + | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 |)} \\ - D_{1} ((Γ, Π) (H, A)) \\ ⩽ - D_{1} ((H, A) (I, Λ)) \end{matrix}$ $1 - D_{1} ((Γ, Π) (H, A)) ⩽ 1 - D_{1} ((H, A) (I, Λ))$

For entropy E₁ ((Γ, Π) , (Γ^c, Π)) ⩽ E₁ ((I, Λ) , (I^c, Λ))

e₄ is fulfilled.

Subsequently all the axioms e_k, k = 1, 2, 3, 4 are fulfilled. Hence E₁ ((Γ, Π) , (Γ^c, Π)) is the valid maximin entropy measure (Γ, Π).

Theorem 3.5. E₂ ((Γ, Π) , (Γ^c, Π)) is the valid entropy measure for IVIFSS.

Proof. Let (Γ, Π), be an interval valued intuitionistic FSS over a common universe Φ, where θ_i ∈ Π and φ_j ∈ Φ. It is simple and easy to prove e₁ property assume that (Γ, Π) is a classical soft set that is Γ^l = Γ^u = Γ^l* = Γ^u* = Γ^l** = Γ^u** = 0 then E₁ ((Γ, Π) , (Γ^c, Π)) = 0.

Now in order to square that e₂ is fulfilled. $Γ^{l} = Γ^{u} = Γ^{l *} = Γ^{u *} = Γ^{l * *} = Γ^{u * *} = \frac{1}{2}$ $\begin{matrix} E_{1} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - (\frac{1}{6}) {Max}_{i} {{Min}_{j} (| 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 |)} = 1 \end{matrix}$

e₂ is satisfied.

e₃: Consider $\begin{matrix} E_{1} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - - (\frac{1}{6}) {Max}_{i} {{Min}_{j} (| 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | \\ + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 |)} = 1 \end{matrix}$ $\begin{matrix} E_{1} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - - (\frac{1}{6}) {Max}_{i} {{Min}_{j} ((| - Γ_{θ_{i}}^{l} φ_{j} + Γ_{θ_{i}}^{lc} φ_{j} | \\ + | (- Γ_{θ_{i}}^{u} φ_{j} + Γ_{θ_{i}}^{uc} φ_{j}) | + | (- Γ_{θ_{i}}^{l *} φ_{j} + Γ_{θ_{i}}^{l * c} φ_{j}) | \\ + | (- Γ_{θ_{i}}^{u *} φ_{j} + Γ_{θ_{i}}^{u * c} φ_{j}) | \\ + | (- Γ_{θ_{i}}^{l * *} φ_{j} + Γ_{θ_{i}}^{l * * c} φ_{j}) | \\ + | (- Γ_{θ_{i}}^{u * *}) φ_{j} + Γ_{θ_{i}}^{u * * c} φ_{j} |))} = E_{1} ((Γ, Π), (Γ^{c}, Λ)) \end{matrix}$

e₃ is satisfied.

e₄: E₁ ((Γ, Π) , (Γ^c, Π)) ⩽ E₁ ((I, Λ) , (I^c, Λ)) if D₁ ((Γ, Π) (H, A)) ⩾ D₁ ((H, A) (I, Λ)) is the average entropy measure for IVIFSS. $\begin{matrix} E_{3} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - \frac{1}{12 m} \sum_{j = 1}^{n} {\begin{matrix} \sum_{j = 1}^{n} \frac{\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix}}{n} \\ + Max (\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix}) \end{matrix}} \end{matrix}$

Proof. Let (Γ, Π), be an IVIFSS over a common universal Φ, where θ_i ∈ Π and φ_j ∈ Φ. It is simple and easy to prove e₁ property assume that (Γ, Π) is a classical soft set that is Γ^l = Γ^u = Γ^l* = Γ^u* = Γ^l** = Γ^u** = 0 then E₂ ((Γ, Π) , (Γ^c, Π)) = 0.

Now in order to square that e₂ is satisfied. $Γ^{l} = Γ^{u} = Γ^{l *} = Γ^{u *} = Γ^{l * *} = Γ^{u * *} = \frac{1}{2}$ $\begin{matrix} E_{2} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - {Max}_{i} {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix})}{6}) \end{matrix}} = 1 \end{matrix}$

Hence e₂ is satisfied.

e₃: Consider $\begin{matrix} E_{2} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - {Max}_{i} {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{lc} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{uc} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l * c} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u * c} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * * c} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * * c} φ_{j} | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{lc} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{uc} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l * c} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u * c} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * * c} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * * c} φ_{j} | \end{matrix})}{6}) \end{matrix}} \end{matrix}$ $\begin{matrix} E_{2} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - {Max}_{i} \\ {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | - Γ_{θ_{i}}^{l} φ_{j} + Γ_{θ_{i}}^{lc} φ_{j} | + | (- Γ_{θ_{i}}^{u} φ_{j} + Γ_{θ_{i}}^{uc} φ_{j}) | + \\ | (- Γ_{θ_{i}}^{l *} φ_{j} + Γ_{θ_{i}}^{l * c} φ_{j}) | + | (- Γ_{θ_{i}}^{u *} φ_{j} + Γ_{θ_{i}}^{u * c} φ_{j}) | + \\ | (- Γ_{θ_{i}}^{l * *} φ_{j} + Γ_{θ_{i}}^{l * * c} φ_{j}) | + | (- Γ_{θ_{i}}^{u * *}) φ_{j} + Γ_{θ_{i}}^{u * * c} φ_{j} | \end{matrix})}{6}) \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | - Γ_{θ_{i}}^{l} φ_{j} + Γ_{θ_{i}}^{lc} φ_{j} | + | (- Γ_{θ_{i}}^{u} φ_{j} + Γ_{θ_{i}}^{uc} φ_{j}) | + \\ | (- Γ_{θ_{i}}^{l *} φ_{j} + Γ_{θ_{i}}^{l * c} φ_{j}) | + | (- Γ_{θ_{i}}^{u *} φ_{j} + Γ_{θ_{i}}^{u * c} φ_{j}) | + \\ | (- Γ_{θ_{i}}^{l * *} φ_{j} + Γ_{θ_{i}}^{l * * c} φ_{j}) | + | (- Γ_{θ_{i}}^{u * *}) φ_{j} + Γ_{θ_{i}}^{u * * c} φ_{j} | \end{matrix})}{6}) \end{matrix}} \end{matrix}$

$= E_{2} ((Γ, Π), (Γ^{c}, Π))$

Hence e₃ is satisfied.

e₄: E₂ ((Γ, Π) , (Γ^c, Π)) ⩽ E₂ ((I, Λ) , (I^c, Λ)) if D₂ ((Γ, Π) (H, A)) ⩾ D₂ ((H, A) (I, Λ)) $- D_{2} ((Γ, Π) (H, A)) ⩽ - D_{2} ((H, A) (I, Λ))$ $1 - D_{2} ((Γ, Π) (H, A)) ⩽ 1 - D_{2} ((H, A) (I, Λ))$

For entropy E₂ ((Γ, Π) , (Γ^c, Π)) ⩽ E₂ ((I, Λ) , (I^c, Λ))

Hence e₄ is fulfilled.

Subsequently, all the axioms e_k, k = 1, 2, 3, 4 are fulfilled stated in the definition (2.11). Hence E₂ ((Γ, Π) , (Γ^c, Π)) is the valid average entropy measure (Γ, Π).

Theorem 3.6. E₃ ((Γ, Π) , (Γ^c, Π)) is the valid convex entropy measure for IVIFSS. $\begin{matrix} E_{3} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - {Max}_{i} {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix})}{6}) \end{matrix}} \end{matrix}$

Proof. Let (Γ, Π), be an interval valued intuitionistic fuzzy soft sets over a common universe Φ, where θ_i ∈ Π and φ_j ∈ Φ. It is simple and easy to prove e₁ property assume that (Γ, Π) is a classical soft set that is Γ^l = Γ^u = Γ^l* = Γ^u* = Γ^l** = Γ^u** = 0 then E₃ ((Γ, Π) , (Γ^c, Π)) = 0.

Now in order to square that e₂ is fulfilled. $Γ^{l} = Γ^{u} = Γ^{l *} = Γ^{u *} = Γ^{l * *} = Γ^{u * *} = \frac{1}{2}$ $\begin{matrix} E_{3} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - {Max}_{i} {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | 2 Γ_{θ_{i}}^{l} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u *} φ_{j} - 1 | + \\ | 2 Γ_{θ_{i}}^{l * *} φ_{j} - 1 | + | 2 Γ_{θ_{i}}^{u * *} φ_{j} - 1 | \end{matrix})}{6}) \end{matrix}} = 1 \end{matrix}$

Hance e₂ is satisfied.

e₃: Consider $\begin{matrix} E_{3} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - {Max}_{i} \\ {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{lc} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{uc} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l * c} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u * c} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * * c} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * * c} φ_{j} | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | Γ_{θ_{i}}^{l} φ_{j} - Γ_{θ_{i}}^{lc} φ_{j} | + | Γ_{θ_{i}}^{u} φ_{j} - Γ_{θ_{i}}^{uc} φ_{j} | + \\ | Γ_{θ_{i}}^{l *} φ_{j} - Γ_{θ_{i}}^{l * c} φ_{j} | + | Γ_{θ_{i}}^{u *} φ_{j} - Γ_{θ_{i}}^{u * c} φ_{j} | + \\ | Γ_{θ_{i}}^{l * *} φ_{j} - Γ_{θ_{i}}^{l * * c} φ_{j} | + | Γ_{θ_{i}}^{u * *} φ_{j} - Γ_{θ_{i}}^{u * * c} φ_{j} | \end{matrix})}{6}) \end{matrix}} \end{matrix}$ $\begin{matrix} E_{3} ((Γ, Π), (Γ^{c}, Π)) = 1 \\ - {Max}_{i} \\ {\begin{matrix} α {Max}_{j} (\frac{(\begin{matrix} | - Γ_{θ_{i}}^{l} φ_{j} + Γ_{θ_{i}}^{lc} φ_{j} | + | (- Γ_{θ_{i}}^{u} φ_{j} + Γ_{θ_{i}}^{uc} φ_{j}) | + \\ | (- Γ_{θ_{i}}^{l *} φ_{j} + Γ_{θ_{i}}^{l * c} φ_{j}) | + | (- Γ_{θ_{i}}^{u *} φ_{j} + Γ_{θ_{i}}^{u * c} φ_{j}) | \\ + | (- Γ_{θ_{i}}^{l * *} φ_{j} + Γ_{θ_{i}}^{l * * c} φ_{j}) | + | (- Γ_{θ_{i}}^{u * *}) φ_{j} + Γ_{θ_{i}}^{u * * c} φ_{j} | \end{matrix})}{6}) + \\ (1 - α) {Min}_{j} (\frac{(\begin{matrix} | - Γ_{θ_{i}}^{l} φ_{j} + Γ_{θ_{i}}^{lc} φ_{j} | + | (- Γ_{θ_{i}}^{u} φ_{j} + Γ_{θ_{i}}^{uc} φ_{j}) | + \\ | (- Γ_{θ_{i}}^{l *} φ_{j} + Γ_{θ_{i}}^{l * c} φ_{j}) | + | (- Γ_{θ_{i}}^{u *} φ_{j} + Γ_{θ_{i}}^{u * c} φ_{j}) | + \\ | (- Γ_{θ_{i}}^{l * *} φ_{j} + Γ_{θ_{i}}^{l * * c} φ_{j}) | + | (- Γ_{θ_{i}}^{u * *}) φ_{j} + Γ_{θ_{i}}^{u * * c} φ_{j} | \end{matrix})}{6}) \end{matrix}} \end{matrix}$ $= E_{3} ((Γ, Π), (Γ^{c}, Π))$

e₃ is satisfied.

s₄: E₃ ((Γ, Π) , (Γ^c, Π)) ⩽ E₃ ((I, Λ) , (I^c, Λ)) if D₃ ((Γ, Π) (H, A)) ⩾ D₃ ((H, A) (I, Λ)) $- D_{3} ((Γ, Π) (H, A)) ⩽ - D_{3} ((H, A) (I, Λ))$ $1 - D_{3} ((Γ, Π) (H, A)) ⩽ 1 - D_{3} ((H, A) (I, Λ))$

For entropy E₃ ((Γ, Π) , (Γ^c, Π)) ⩽ E₃ ((I, Λ) , (I^c, Λ))

Hence e₄ is satisfied.

Subsequently, all the maxims e_k, k = 1, 2, 3, 4 are fulfilled. Hence E₃ ((Γ, Π) , (Γ^c, Π)) is the valid convex entropy measure (Γ, Π).

4 Applications

4.1 Interval-valued intuitionistic fuzzy soft entropy in decision making

Study an IVIFSS (Γ, Π), where U is a set of six houses under the consideration of a decision make to purchase, which is denoted by Φ ={ φ₁, φ₂, φ_3,φ₄, φ₅, φ₆ } and a parameter set Θ = { θ₁, θ₂, θ₃, θ₄, θ₅ } = {expensive, beautiful, wooden, in good repair, in the green surroundings}. The interval-valued intuitionistic fuzzy soft set (Γ, Π) describes the “attractiveness of the houses” to the decision makers.

Table 3
Information from the customers

φ ₁ φ ₂ [0.6,0.7] φ ₄ φ ₅ φ ₆

θ ₁ [0.6,0.8] [0.8,0.9] [0.6,0.7], [0.65,0.65] [0.56,0.6], [0.68,0.82]

[0.1,0.2] ,[0.05,0.1] [0.2,0.025] [0.15,0.21] [0.2,0.3], [0.11,0.18]

[0,0.3] [0,0.15] [0.09,0.2] [0.14,0.2] [0.1,0.24] [0.02,0.21]

θ ₂ [0.5,0.6] [0.25,0.3], [0.35,0.4], [0.5,0.6] [0.6,0.7], [0.4,0.7],

[0.2,0.35] [0.5,0.7], [0.4,0.6] ,[0.1,0.3] [0.216,0.27] [0.15,0.18]

[0.05,0.3] [0,0.25] [0,0.25] [0.1,0.4] [0.03,0.18] [0.12,0.45]

θ ₃ [0.16,0.17] [0.31,0.46] [0.43,0.49] [0.346,0.45] [0.11,0.16], [0.16,0.48]

[0.11,0.16] [0.315,0.43] [0.16,0.5] [0.05,0.16] [0.56,0.58] [0.16,0.35]

[0.67,0.73] [0.11,0.375] [0.01,0.41] [0.39,0.604] [0.26,0.33] [0.17,0.68]

θ ₄ [0.19,0.2], [0.15,0.23] [0.56,0.7] [0.18,0.19], [0.19,0.45], [0.19,0.48]

[0.25,0.36] [0.3,0.46] [0.18,0.2] [0.36,0.4] [0.25,0.35] [0.3,0.4]

[0.44,0.56] [0.31,0.55] [0.1,0.26] [0.41,0.46] [0.2,0.56] [0.12,0.51]

θ ₅ [0.45,0.51] [0.45,0.49] [0.39,0.45], [0.23,45] [0.2,0.35] [0.3,0.34]

[0.35,0.36] [0.4,0.51] [0.46,0.51] [0.43,0.49] [0.3,0.6] [0.16,0.35]

[0.13,0.2] [0.04,0.15] [0.04,0.15] [0.06,0.34] [0.05,0.5] [0.31,0.54]

	φ ₁	φ ₂	[0.6,0.7]	φ ₄	φ ₅	φ ₆
θ ₁	[0.6,0.8]	[0.8,0.9]	[0.6,0.7],	[0.65,0.65]	[0.56,0.6],	[0.68,0.82]
	[0.1,0.2]	,[0.05,0.1]	[0.2,0.025]	[0.15,0.21]	[0.2,0.3],	[0.11,0.18]
	[0,0.3]	[0,0.15]	[0.09,0.2]	[0.14,0.2]	[0.1,0.24]	[0.02,0.21]
θ ₂	[0.5,0.6]	[0.25,0.3],	[0.35,0.4],	[0.5,0.6]	[0.6,0.7],	[0.4,0.7],
	[0.2,0.35]	[0.5,0.7],	[0.4,0.6]	,[0.1,0.3]	[0.216,0.27]	[0.15,0.18]
	[0.05,0.3]	[0,0.25]	[0,0.25]	[0.1,0.4]	[0.03,0.18]	[0.12,0.45]
θ ₃	[0.16,0.17]	[0.31,0.46]	[0.43,0.49]	[0.346,0.45]	[0.11,0.16],	[0.16,0.48]
	[0.11,0.16]	[0.315,0.43]	[0.16,0.5]	[0.05,0.16]	[0.56,0.58]	[0.16,0.35]
	[0.67,0.73]	[0.11,0.375]	[0.01,0.41]	[0.39,0.604]	[0.26,0.33]	[0.17,0.68]
θ ₄	[0.19,0.2],	[0.15,0.23]	[0.56,0.7]	[0.18,0.19],	[0.19,0.45],	[0.19,0.48]
	[0.25,0.36]	[0.3,0.46]	[0.18,0.2]	[0.36,0.4]	[0.25,0.35]	[0.3,0.4]
	[0.44,0.56]	[0.31,0.55]	[0.1,0.26]	[0.41,0.46]	[0.2,0.56]	[0.12,0.51]
θ ₅	[0.45,0.51]	[0.45,0.49]	[0.39,0.45],	[0.23,45]	[0.2,0.35]	[0.3,0.34]
	[0.35,0.36]	[0.4,0.51]	[0.46,0.51]	[0.43,0.49]	[0.3,0.6]	[0.16,0.35]
	[0.13,0.2]	[0.04,0.15]	[0.04,0.15]	[0.06,0.34]	[0.05,0.5]	[0.31,0.54]

The interval-valued intuitionistic fuzzy soft set (Γ, Π) is a parameterized family {Γ (θ_i) , i = 1, 2, 3, 4, 5} of interval-valued intuitionistic fuzzy sets on Φ, and (Γ, Π) = Obviously, the precise evaluation for each object on each parameter is unknown while the lower and upper limits of such an evaluation are given. Now entropy corresponding to each house is determined using the proposed entropy measures the corresponding values are given in Table 4.

Table 4

Entropy for the houses

	E ₁	E ₂	E ₃	E ₃	E ₃	E ₃	E ₃	E ₃	E ₃	E ₃	E ₃	E ₃
α	–	–	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
Value	0.63	0.54	0.62	0.6	0.58	0.56	0.53	0.51	0.48	0.45	0.43	0.4

All entropy values verify the data. Here first entropy gives the maximum information about houses.

4.2 Interval-valued intuitionistic fuzzy soft distance measures in medical diagnosis

Let Γ and I are the set that represent the set of diagnosis and symptoms respectively given as Γ ={ 〈Γ₁viral fever〉, 〈Γ₂, Malaria〉, 〈Γ₃, Typhoid, 〉 } and I ={ 〈I₁, Temprature〉, 〈I₂, Headache〉, 〈I₃, cough〉 } Assume the patient is represented by $P = {\begin{matrix} 〈 I_{1}, [0.4, 0.5], [0.4, 0.3], [0.2, 0.2] 〉, 〈 I_{2}, [0.4, 0.6], [0.2, 0.4], [0.4, 0], 〉 \\ 〈 I_{3}, [0.4, 0.8], [0.1, 0.2], [0.5, 0] 〉 \end{matrix}}$

And weight agreeing to each attribute is equal and each diagnosis is given by the following $Γ_{1} = {\begin{matrix} 〈 I_{1}, [0.3, 0.6], [0.3, 0.4], [0.4, 0] 〉, 〈 I_{2}, [0.5, 0.6], [0.3, 0.4], [0.2, 0], 〉 \\ 〈 I_{3}, [0.4, 0.5], [0.1, 0.3], [0.5, 0.2] 〉 \end{matrix}}$ $Γ_{2} = {\begin{matrix} 〈 I_{1}, [0.3, 0.5], [0.3, 0.4], [0.4, 0] 〉, 〈 I_{2}, [0.5, 0.6], [0.3, 0.4], [0.2, 0], 〉 \\ 〈 I_{3}, [0.4, 0.5], [0.1, 0.3], [0.5, 0.2] 〉 \end{matrix}}$ $Γ_{3} = {\begin{matrix} 〈 I_{1}, [0.7, 0.8], [0.1, 0.2], [0.2, 0] 〉, 〈 I_{2}, [0.6, 0.7], [0.1, 0.3], [0.3, 0], 〉 \\ 〈 I_{3,} [0.3, 0.4], [0.1, 0.2], [0.6, 0.3] 〉 \end{matrix}}$

Using the proposed similarity measures, we classify the patients P in one of the diagnoses Γ₁, Γ₂, Γ₃. The results are as follows:

The proposed distance measures, classify the patient P in one of the diagnoses Γ₁, Γ₂, Γ₃, results are shown in the Table 5. Patients P can be diagnosed with typhoid using first distance measure and according to second and third distance measures patient P can be diagnosed with viral fever.

Table 5
Application to medical diagnosis

(Γ₁, P) (Γ₂, P) (Γ₃, P)

D ₁ 0.1333 0.55835 0.06

D ₂ 0.1 0.55 0.3278

D₃, α = 0 0.1333 0.55835 0.06

α = 0.1 0.0733 0.54335 0.16

α = 0.2 0.08 0.545 0.13

α = 0.3 0.0767 0.54665 0.13

α = 0.4 0.0933 0.54835 0.14

α = 0.5 0.1 0.55 0.15

α = 0.6 0.0533 0.55165 0.16

α = 0.7 0.0733 0.5475 0.17

α = 0.8 0.12 0.555 0.18

α = 0.9 0.1267 0.55665 0.19

α = 1 0.1333 0.5575 0.2

Normalized 0.05 0.05 0.091667

Hamming

Distance

(Feng et al.

[28])

	(Γ₁, P)	(Γ₂, P)	(Γ₃, P)
D ₁	0.1333	0.55835	0.06
D ₂	0.1	0.55	0.3278
D₃, α = 0	0.1333	0.55835	0.06
α = 0.1	0.0733	0.54335	0.16
α = 0.2	0.08	0.545	0.13
α = 0.3	0.0767	0.54665	0.13
α = 0.4	0.0933	0.54835	0.14
α = 0.5	0.1	0.55	0.15
α = 0.6	0.0533	0.55165	0.16
α = 0.7	0.0733	0.5475	0.17
α = 0.8	0.12	0.555	0.18
α = 0.9	0.1267	0.55665	0.19
α = 1	0.1333	0.5575	0.2
Normalized	0.05	0.05	0.091667
Hamming
Distance
(Feng et al.
[28])

According to normalized hamming distance measure (Feng et al. [28]) patient P suffering from viral fever and malaria simultaneous. The proposed measures were able to distinguish and identify the medical diagnosis whereas the existing measures is not able to do so.

5 Conclusion

The interval-valued intuitionistic fuzzy soft set theory is an amalgamation of an interval-valued intuitionistic fuzzy set theory and a soft set theory. Interval-valued intuitionistic fuzzy soft set theory is an interval-valued fuzzy extension of the intuitionistic fuzzy soft set theory or an intuitionistic fuzzy extension of the interval-valued fuzzy soft set theory. In this paper three different distance measures are defined along with their properties. Further based on these measures three similarity and entropy measures are also derived. Thereafter a hypothetical data is used to demonstrate applicability of proposed entropy measure in decision making. Finally, to check the validity of proposed distance measures, these measures are applied to the field of multi attribute decision making and medical diagnosis. The proposed measures are also compared with normalized hamming distance measure proposed by Feng et al. [28].

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Novel distance,similarity and entropy measures for interval valued intuitionistic fuzzy soft set

Abstract

Keywords

1 Introduction

2 Preliminary

Table 1 Representation of soft set Θ Expensive θ1 Beautiful θ2 Wooden θ3 Cheap θ4 In green surrounding θ5 Choice value φ 1 1 1 0 1 1 4 φ 2 1 0 0 1 0 2 φ 3 0 1 0 0 0 1 φ 4 0 1 1 0 0 2 φ 5 0 0 1 0 1 2

3.1 Distance measures for IVIFSS

4.1 Interval-valued intuitionistic fuzzy soft entropy in decision making

References

Table 1
Representation of soft set

Θ Expensive θ₁ Beautiful θ₂ Wooden θ₃ Cheap θ₄ In green surrounding θ₅ Choice value

φ ₁ 1 1 0 1 1 4

φ ₂ 1 0 0 1 0 2

φ ₃ 0 1 0 0 0 1

φ ₄ 0 1 1 0 0 2

φ ₅ 0 0 1 0 1 2