The induced generalized interval-valued intuitionistic fuzzy Einstein hybrid geometric aggregation operator and their application to group decision-making

Abstract

For the multi-attribute group decision-making problems where attribute values are the interval-valued intuitionistic fuzzy numbers, the group decision-making method based on induced generalized Einstein geometric aggregation operators is developed. Firstly, induced generalized interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (I-GIVIFEOWG) aggregation operator and induced generalized interval-valued intuitionistic fuzzy Einstein hybrid weighted geometric (I-GIVIFEHWG) aggregation operator, were proposed. Some general properties such as, idempotency, commutativity, monotonicity and boundedness, were discussed and some special cases were analyzed. Furthermore, the method for multi-attribute group decision-making problems was developed, and the operational processes were illustrated in detail. The main advantage of using the proposed methods and operators is that these operators and methods give a more complete view of the problem to the decision makers. The proposed methods provide more general, more accurate and precise results. Therefore these methods play a vital role in real world problems. Finally the proposed operators have been applied to decision making problems to show the validity, practicality and effectiveness of the new approach.

Keywords

Group decision-making I-GIVIFEOWG aggregation operator I-GIVIFEHG aggregation operator some einstein operations

1 Introduction

Multi-criteria decision-making problems are of importance in most kinds of fields such as engineering, economics, and management. Traditionally, it has been assumed that the information which accesses the alternatives in term of criteria and weight are expressed in real numbers. But due to the complexity of the system day-by-day, it is difficult for the decision- makers to make a perfect decision, as most of the preferred value during the decision-making process imbued with uncertainty. In order to handle the uncertainties, Zadeh [1] introduced the notion of fuzzy set theory which has only one element called membership function. After the positive and successful growth of fuzzy set theory, Atanassov generalized the notion of fuzzy set and introduced the idea of intuitionistic fuzzy set theory [2, 3] which has two elements called membership function and non-membership respectively. Intuitionistic fuzzy set is more appropriate for dealing with fuzziness and uncertainty than the simple fuzzy set. The intuitionistic fuzzy set has gotten increasingly consideration and applications since its development. Xu [4] introduced the notion of some averaging operators, such as the intuitionistic fuzzy weighted averaging (IFWA) operator, the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid averaging (IFHA) operator and gave an application of the proposed operators to multiple attribute group decision making with intuitionistic fuzzy information. Xu and Yager [5] developed the idea of some geometric aggregation operators, namely the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator and gave an application of the intuitionistic fuzzy hybrid geometric operator to multiple attribute group decision making with intuitionistic fuzzy information. Li et al. [6] developed a linear programming method for multi-attribute group decision making problems using IFS. Wan and Li [7] extended the Linear Programming Technique for Multidimensional Analysis of Preference for solving heterogeneous multi-attribute decision making problems. Yu et al. [8, 9] introduced the idea of compromise-typed variable weight decision method for solving hybrid multiattribute decision making problems with multiple types of attribute values and a novel method for heterogeneous multi-attribute group decision making with preference deviation. Wang and Liu [10, 11] developed the concept of some new types aggregation operators based on Einstein sum and Einstein product, such as the intuitionstic fuzzy Einstein weighted averaging (IFEWA) operator, the intuitionstic fuzzy Einstein ordered weighted averaging (IFEOWA) operator, the intuitionstic fuzzy Einstein weighted geometric (IFEWG) operator, the intuitionstic fuzzy Einstein ordered weighted geometric (IFEOWG) operator and applied them on group decision making problem. Zhao and Wei [12] introduced the concept of some hybrid aggregation operators, such as the intuitionstic fuzzy Einstein hybrid averaging (IFEHA) operator and the intuitionstic fuzzy Einstein hybrid geometric (IFEHG) operator and applied these operators to group decision-making. Rahman et al. [13] developed the notion of some generalized intuitionistic fuzzy aggregation operators, such as the generalized intuitionistic fuzzy Einstein hybrid averaging (GIFEHA) operator and the generalized intuitionistic fuzzyEinstein hybrid geometric (GIFEHG) operator. Xu [14] introduced the idea of induced intuitionistic fuzzy Einstein ordered weighted averaging (I-IFEOWA) operator and gave an application of the proposed operators to multiple attribute group decision making with intuitionistic fuzzy information.

However, there are some cases in real life decision-making problems, due to insufficiency in available data, it is not easy for decision-makers to exactly compute their opinions with a crisp number, but they can be represented by an interval number within [0, 1]. Therefore it is more important to present the idea of interval-valued intuitionistic fuzzy set, which permit the membership degrees and non- membership degrees to a given set to have an interval-value [0, 1]. For this Atanassov and Gargov [15] introduced the notion of interval-valued intuitionistic fuzzy set. The aggregation operators based on interval-valued intuitionstic fuzzy numbers are the most important tools for decision-makers in the ranking of alternatives. Xu and Chen [16, 17] introduced the notion of several aggregation operators using interval-valued intuitionistic fuzzy numbers, including the interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator, the interval-valued intuitionistic fuzzy ordered weighted averaging (IVIFOWA) operator, the interval-valued intuitionistic fuzzy hybrid averaging (IVIFHA) operator, the interval-valued intuitionistic fuzzy weighted geometric (IVIFWG) operator, the interval-valued intuitionistic fuzzy ordered weighted geometric (IVIFOWG) operator, and the interval-valued intuitionistic fuzzy hybrid geometric (IVIFHG) operator along with their desirable properties, such as idempotency, boundedness and monotonicity. Zhao [18] introduced the notion of some generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging (GIFWA) operator, generalized intuitionistic fuzzy ordered weighted averaging (GIFOWA) operator, generalized intuitionistic fuzzy hybrid averaging (GIFHA) operator, generalized interval-valued intuitionistic fuzzy weighted averaging (GIVIFWA) operator, generalized interval-valued intuitionistic fuzzy ordered weighted averaging (GIVIFOWA) operator, generalized interval-valued intuitionistic fuzzy hybrid average (GIVIFHA) operator. Xu [19] developed the notion of interval-valued intuitionistic fuzzy weighted arithmetic aggregation (IVIFWAG) operator and interval-valued intuitionistic fuzzy weighted geometric aggregation (IVIEWGG) operator. Yu et al. [20] developed the concept of the interval-valued intuitionistic fuzzy prioritized weighted average (IVIFPWA) operator, the interval-valued intuitionistic fuzzy prioritized weighted geometric (IVIFPWG) operator along with their some basic properties, including idempotency, boundedness and monotonicity. Su et al. [21] introduced some induced aggregation operators, such as the induced intuitionistic fuzzy ordered weighted averaging (I-IFOWA) operator, the induced intuitionistic fuzzy hybrid averaging (I-IFHA) operator, the induced interval-valued intuitionistic fuzzy ordered weighted averaging (I-IVIFOWA) operator, and the induced interval-valued intuitionistic fuzzy hybrid averaging (I-IVIFHA) operator and gave an application of the proposed operators to multiple attribute group decision making with intuitionistic fuzzy information. Wei [22] introduced the notion of induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator, induced interval-valued intuitionistic fuzzy hybrid geometric (I-IVIFHG) operator along with their properties and also applied them on group decision making. Xuese and Liguo [23] introduced the notion of interval-valued intuitionistic fuzzy Einstein weighted averaging (IVIFEWA) operator, interval-valued intuitionistic fuzzy Einstein ordered weighted averaging (IVIFEOWA) operator, induced interval-valued intuitionistic fuzzy Einstein ordered weighted averaging (I-IVIFEOWA) operator and applied them to group decision making. Wang and Liu [24] developed some interval-valued Einstein aggregation operators, namely the interval-valued intuitionistic fuzzy Einstein weighted geometric (IVIFEWG) operator, the interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (IVIFEOWG) operator, the interval-valued intuitionistic fuzzy Einstein hybrid geometric (IVIFEHG) operator along with their desirable properties. Yang and Sheng [25] introduced the notion of induced interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (I-IVIFEOWG) operator, induced interval-valued intuitionistic fuzzy Einstein hybrid geometric (I-IVIFEHG) operator, and applied to group decision making.

Keeping the advantages of the above mentioned aggregation operators, in this paper, we introduce the notion of two new types operators based on based on interval-valued intuitionistic fuzzy numbers, such as the induced generalized interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (I-GIVIFEOWG) operator and the induced generalized interval-valued intuitionistic fuzzy Einstein hybrid geometric (I-GIVIFEHG) operator. This motivation comes from [25] in which the author introduced the induced interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (I-IVIFEOWG) operator and the induced interval-valued intuitionistic fuzzy Einstein hybrid geometric (I-IVIFEHWG) operator. But in this paper we generalized these operators and applied them on group decision-making. The proposed operators are more general and more flexible as compared to their existing methods, because the existing operators are the special cases of the new proposed operators. Of course, superficially, it is more complicated in calculation. However, in real applications, we need assign the specific parameter δ, firstly.

The remainder of this article is structured as follows. In Section 2, we give some basic definitions and results which will be used in our later sections. In Section 3, we introduce the notion of induced generalized interval-valued intuitionistic fuzzy Einstein ordered weighted geometric operator, and induced generalized interval-valued intuitionistic fuzzy Einstein hybrid weighted geometric operator along with their various properties including idempotency, boundedness, monotonicity and commutativity. In Section 4, we apply the operators to multiple attribute group decision making problems. In Section 5, we develop a numerical example. In Section 6, we have conclusion.

2 Preliminaries

Definition 1. [15] Let K be a universal set, then interval-valued intuitionistic fuzzy set can be defined as: $I = {〈 k, u_{I} (k), v_{I} (k) 〉 | k \in K},$ (1) where u_I (k) = [μ_I (k) , η_I (k)] ⊂ [0, 1] and ν_I (k)= [x_I (k) , y_I (k)] ⊂ [0, 1]. And also μ_I (k) = infμ_I (k), η_I (k) = sup η_I (k), x_I (k) = inf x_I (k), y_I (k)= sup y_I (k) with condition 0 ⩽ η_I (k) + y_I (k) ⩽ 1. If $π_{I} (k) = [π_{I}^{-} (k), π_{I}^{+} (k)]$ for all k ∈ K, then it is called the interval-valued intuitionistic fuzzy index of k to K, where $π_{I}^{-} (k) = 1 - η_{I} (k) - y_{I} (k)$ and $π_{I}^{+} (k) = 1 - μ_{I} (k) - x_{I} (k)$ .

Definition 2. [17] Let λ =〈 [μ_λ, η_λ] , [x_λ, y_λ] 〉 be the interval-valued intuitionistic fuzzy number, then the score function and accuracy degree of λ can be defined as: $S (λ) = \frac{1}{2} [μ_{λ} + η_{λ} - x_{λ} - y_{λ}]$ and $H (λ) = \frac{1}{2} [μ_{λ} + η_{λ} + x_{λ} + y_{λ}]$ respectively.

Note: If λ₁ and λ₂ are two interval-valued intuitionistic fuzzy numbers, then the following conditions always hold:

If S (λ₁) ≺ S (λ₂) , then we have λ₁ ≺ λ₂

If S (λ₁) = S (λ₂) , then we have the following three cases:

If H (λ₁) = H (λ₂) , then λ₁ = λ₂

If H (λ₁) ≺ H (λ₂) , then λ₁ ≺ λ₂

If H (λ₁) ≻ H (λ₂) , then λ₁ ≻ λ₂

Definition 3. [24] Let λ =〈 [μ, η] , [x, y] 〉 , λ₁ = 〈 [μ₁, η₁] , [x₁, y₁] 〉 , λ₂ = 〈 [μ₂, η₂] , [x₂, y₂] 〉 be three interval-valued intuitionistic fuzzy numbers and δ ≻ 0, then we have

$\begin{matrix} λ_{1} \oplus_{ɛ} λ_{2} = & 〈 [\frac{μ_{1} + μ_{2}}{1 + μ_{1} μ_{2}}, \frac{η_{1} + η_{2}}{1 + η_{1} η_{2}}], \\ [\frac{x_{1} x_{2}}{1 + (1 - x_{1}) (1 - x_{2})}, \frac{y_{1} y_{2}}{1 + (1 - y_{1}) (1 - y_{2})}] 〉 \end{matrix}$ (2)

\begin{matrix} λ_{1} \otimes_{ɛ} λ_{2} = & 〈 [\frac{μ_{1} μ_{2}}{1 + (1 - μ_{1}) (1 - μ_{2})}, \frac{η_{1} η_{2}}{1 + (1 - η_{1}) (1 - η_{2})}], \\ [\frac{x_{1} + x_{2}}{1 + x_{1} x_{2}}, \frac{y_{1} + y_{2}}{1 + y_{1} y_{2}}] 〉 \end{matrix}

(3)

\begin{matrix} δ λ = & 〈 [\frac{{(1 + μ)}^{δ} - {(1 - μ)}^{δ}}{{(1 + μ)}^{δ} + {(1 - μ)}^{δ}}, \frac{{(1 + η)}^{δ} - {(1 - η)}^{δ}}{{(1 + η)}^{δ} + {(1 - η)}^{δ}}], \\ [\frac{2 x^{δ}}{{(2 - x)}^{δ} + {(x)}^{δ}}, \frac{2 y^{δ}}{{(2 - y)}^{δ} + {(y)}^{δ}}] 〉 \end{matrix}

(4)

\begin{matrix} λ^{δ} = & 〈 [\frac{2 μ^{δ}}{{(2 - μ)}^{δ} + μ^{δ}}, \frac{2 η^{δ}}{{(2 - η)}^{δ} + η^{δ}}], \\ [\frac{{(1 + x)}^{δ} - {(1 - x)}^{δ}}{{(1 + x)}^{δ} + {(1 - x)}^{δ}}, \frac{{(1 + y)}^{δ} - {(1 - y)}^{δ}}{{(1 + y)}^{δ} + {(1 - y)}^{δ}}] 〉 \end{matrix}

(5)

Definition 4. [25] Let 〈u_j,λ_j 〉 (j = 1, 2, …, n) be a collection of 2-tuples, then the induced interval-valued intuitionistic fuzzy Einstein ordered weighted geometric aggregation operator can be definedas: $\begin{matrix} {I - IVIFEOWG}_{ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) \\ = 〈 \begin{matrix} [\frac{2 \prod_{j = 1}^{n} {(μ_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(2 - μ_{σ (j)})}^{ω_{j}} + \prod_{j = 1}^{n} {(μ_{σ (j)})}^{ω_{j}}}, \frac{2 \prod_{j = 1}^{n} {(η_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(2 - η_{σ (j)})}^{ω_{j}} + \prod_{j = 1}^{n} {(η_{σ (j)})}^{ω_{j}}}], \\ [\frac{\prod_{j = 1}^{n} {(1 + x_{σ (j)})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - x_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + x_{σ (j)})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - x_{σ (j)})}^{ω_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + y_{σ (j)})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - y_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + y_{σ (j)})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - y_{σ (j)})}^{ω_{j}}}] \end{matrix} 〉 \end{matrix}$ (6)

where (ω₁, ω₂, …, ω_n) ^T be the weighted vector of (λ₁, λ₂, …, λ_n) with some conditions ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ and also λ_σ(j) is the λ_j value of the IVIFEOWG pair 〈u_j, λ_j〉 having the jth largest of u_j and u_j in 〈u_j, λ_j〉 is referred to as the order inducing variable and λ_j as the intuitionistic fuzzy argument variable.

Definition 5. [25] Let 〈u_j,λ_j 〉 (j = 1, 2, …, n) be a collection of 2-tuples, then the induced interval-valued intuitionistic fuzzy Einstein hybrid geometric aggregation operator can be defined as: $\begin{matrix} {I - IVIFEHG}_{w, ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) \\ = 〈 \begin{matrix} [\frac{2 \prod_{j = 1}^{n} {({\dot{μ}}_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(2 - {\dot{μ}}_{σ (j)})}^{ω_{j}} + \prod_{j = 1}^{n} {({\dot{μ}}_{σ (j)})}^{ω_{j}}}, \frac{2 \prod_{j = 1}^{n} {({\dot{η}}_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(2 - {\dot{η}}_{σ (j)})}^{ω_{j}} + \prod_{j = 1}^{n} {({\dot{η}}_{σ (j)})}^{ω_{j}}}], \\ [\frac{\prod_{j = 1}^{n} {(1 + {\dot{x}}_{σ (j)})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{x}}_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{x}}_{σ (j)})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - {\dot{x}}_{σ (j)})}^{ω_{j}}}, \frac{\prod_{j = 1}^{n} {(1 + {\dot{y}}_{σ (j)})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{y}}_{σ (j)})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + {\dot{y}}_{σ (j)})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - {\dot{y}}_{σ (j)})}^{ω_{j}}}] \end{matrix} 〉 \end{matrix}$ (7) where ${\dot{λ}}_{σ (j)} 〈 {\dot{λ}}_{σ (j)} = {(λ_{j})}^{{nw}_{j}} 〉,$ ω = (ω₁, ω₂, …, ω_n) ^T be the associated vector of I-IVIFEHG operator such that ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1 .$ w =(w₁, w₂, …, w_n) ^T be the weighted vector of λ_j(j = 1, 2, …, n), such that w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ and n is the balancing coefficient.

3 Some induced generalized interval-valued intuitionistic fuzzy einstein geometric aggregation operators

In this section, we introduce the notion of two new types induced aggregation operators based on interval-valued intuitionistic fuzzy numbers, namely induced generalized interval-valued intuitionistic fuzzy Einstein ordered weighted geometric operator and induced generalized interval-valued intuitionistic fuzzy Einstein hybrid geometric aggregation operator along with their properties such as, idempotency, boundedness, monotonicity and commutativity.

Definition 6. Let 〈u_j,λ_j 〉 (j = 1, 2, …, n) be a collection of 2-tuples, then the induced generalized interval-valued intuitionistic fuzzy Einstein ordered weighted geometric aggregation operator can be define as:

$\begin{matrix} {I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) \\ = \frac{1}{δ} 〈 \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}} 〉 \end{matrix}$ (8) where (ω₁, ω₂, …, ω_n) ^T be the weighted vector of (λ₁, λ₂, …, λ_n) with some conditions ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ . And λ_σ(j) be the λ_j value of the IVIFEOWG pairs 〈u_j, λ_j〉 having the jth largest u_j and u_j in 〈u_j, λ_j〉 is referred to as the order inducing variable and λ_j as the intuitionistic fuzzy argument variable.

Theorem 1. Let 〈u_j,λ_j 〉 (j = 1, 2, 3) be a collection of 2-tuples, then the following conditions hold:

λ₁ ⊕ _ɛλ₂ = λ₂ ⊕ _ɛλ₁

λ ⊗ _ɛλ₂ = λ₂ ⊗ _ɛλ₁

(λ₁ ⊕ _ɛλ₂) ⊕ _ɛλ₃ = λ₁ ⊕ _ɛ (λ₂ ⊕ _ɛλ₃)

(λ₁ ⊗ _ɛλ₂) ⊗ _ɛλ₃ = λ₁ ⊗ _ɛ (λ₂ ⊗ _ɛλ₃)

$λ_{1}^{c} \oplus_{ɛ} λ_{2}^{c} = {(λ_{1} \otimes_{ɛ} λ_{2})}^{c}$

$λ_{1}^{c} \otimes_{ɛ} λ_{2}^{c} = {(λ_{1} \oplus_{ɛ} λ_{2})}^{c}$

Proof. Straightforward.

Theorem 2. Let 〈u_j,λ_j 〉 (j = 1, 2, 3) be a collection of 2-tuples, then the following conditions hold:

(λ₁ ∪ λ₂) ⊕ _ɛ (λ₁ ∩ λ₂) = λ₁ ⊕ _ɛλ₂

(λ₁ ∪ λ₂) ⊗ _ɛ (λ₁ ∩ λ₂) = λ₁ ⊗ _ɛλ₂

(λ₁ ∪ λ₂) ∩ λ₃ = (λ₁ ∩ λ₃) ∪ (λ₂ ∩ λ₃)

(λ₁ ∩ λ₂) ∪ λ₃ = (λ₁ ∪ λ₃) ∩ (λ₂ ∪ λ₃)

(λ₁ ∪ λ₂) ⊕ _ɛλ₃ = (λ₁ ⊕ _ɛλ₃) ∪ (λ₂ ⊕ _ɛλ₃)

(λ₁ ∩ λ₂) ⊕ _ɛλ₃ = (λ₁ ⊕ _ɛλ₃) ∩ (λ₂ ⊕ _ɛλ₃)

(λ₁ ∩ λ₂) ⊗ _ɛλ₃ = (λ₁ ⊗ _ɛλ₃) ∩ (λ₂ ⊗ _ɛλ₃)

(λ₁ ∪ λ₂) ⊕ _ɛλ₃ = (λ₁ ⊕ _ɛλ₃) ∪ (λ₂ ⊕ _ɛλ₃)

Theorem 3. Let 〈u_j, λ_j 〉 (j = 1, 2, …, n) be a collection of 2-tuples, then their aggregated valued by using the I-GIVIFEOWG operator is also interval-valued intuitionistic fuzzy number, and $\begin{matrix} {I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, 〈 u_{3}, λ_{3} 〉, \dots, 〈 u_{n}, λ_{n} 〉) \\ = 〈 [\begin{matrix} \frac{{(\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} - {(\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}, \\ \frac{{(\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} - {(\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {(a_{η_{j}})}^{ω_{j}} - \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}} \end{matrix}], \\ [\begin{matrix} \frac{2 {(\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}, \\ \frac{2 {(\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}} \end{matrix}] 〉 \end{matrix}$ (9) where a_{μ
_σ(j)} = (1 + μ_σ(j)) ^δ + 3 (1 - μ_σ(j)) ^δ, b_{μ
_σ(j)}= (1 + μ_σ(j)) ^δ - (1 - μ_σ(j)) ^δ, a_{η
_σ(j)} = (1 + η_σ(j)) ^δ +3 (1 - η_σ(j)) ^δ b_{η
_σ(j)} = (1 + η_σ(j)) ^δ - (1 - η_σ(j)) ^δ, c_{x
_σ(j)} = (2 - x_σ(j)) ^δ + 3 (x_σ(j)) ^δ, d_{x
_σ(j)} = (2 - x_σ(j)) ^δ - (x_σ(j)) ^δ c_{y
_σ(j)} = (2 - y_σ(j)) ^δ + 3 (y_σ(j)) ^δ, d_{y
_σ(j)} = (2 - y_σ(j)) ^δ - (y_σ(j)) ^δ

Proof. First we calculate δ . _ɛλ₁: $\begin{matrix} δ ._{ɛ} λ_{1} = & 〈 [\frac{{(1 + μ_{1})}^{δ} - {(1 - μ_{1})}^{δ}}{{(1 + μ_{1})}^{δ} + {(1 - μ_{1})}^{δ}}, \frac{{(1 + η_{1})}^{δ} - {(1 - η_{1})}^{δ}}{{(1 + η_{1})}^{δ} + {(1 - η_{1})}^{δ}}], \\ [\frac{2 {(x_{1})}^{δ}}{{(2 - x_{1})}^{δ} + {(x_{1})}^{δ}}, \frac{2 {(y_{1})}^{δ}}{{(2 - y_{1})}^{δ} + {(y_{1})}^{δ}}] 〉 \end{matrix}$

Now we calculate (δ . _ɛλ₁) ^{ω
₁}: $\begin{matrix} {(δ ._{ɛ} λ_{1})}^{ω_{1}} \\ = 〈 [\begin{matrix} \frac{2 {({(1 + μ_{1})}^{δ} - {(1 - μ_{1})}^{δ})}^{ω_{1}}}{{({(1 + μ_{1})}^{δ} + 3 {(1 - μ_{1})}^{δ})}^{ω_{1}} + {({(1 + μ_{1})}^{δ} - {(1 - μ_{1})}^{δ})}^{ω_{1}}}, \\ \frac{2 {({(1 + η_{1})}^{δ} - {(1 - η_{1})}^{δ})}^{ω_{1}}}{{({(1 + η_{1})}^{δ} + 3 {(1 - η_{1})}^{δ})}^{ω_{1}} + {({(1 + η_{1})}^{δ} - {(1 - η_{1})}^{δ})}^{ω_{1}}} \end{matrix}], \\ [\begin{matrix} \frac{{({(2 - x_{1})}^{δ} + 3 {(x_{1})}^{δ})}^{ω_{1}} - {({(2 - x_{1})}^{δ} - {(x_{1})}^{δ})}^{ω_{1}}}{{({(2 - x_{1})}^{δ} + 3 {(x_{1})}^{δ})}^{ω_{1}} + {({(2 - x_{1})}^{δ} - {(x_{1})}^{δ})}^{ω_{1}}}, \\ \frac{{({(2 - y_{1})}^{δ} + 3 {(y_{1})}^{δ})}^{ω_{1}} - {({(2 - y_{1})}^{δ} - {(y_{1})}^{δ})}^{ω_{1}}}{{({(2 - y_{1})}^{δ} + 3 {(y_{1})}^{δ})}^{ω_{1}} + {({(2 - y_{1})}^{δ} - {(y_{1})}^{δ})}^{ω_{1}}} \end{matrix}] 〉 \end{matrix}$

Again we calculate $\otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}}$ : $\begin{matrix} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}} \\ = 〈 [\frac{2 {(a_{μ_{σ (j)}})}^{ω_{j}}}{{(a_{μ_{σ (j)}})}^{ω_{j}} + {(b_{μ_{σ (j)}})}^{ω_{j}}}, \frac{2 {(b_{η_{σ (j)}})}^{ω_{j}}}{{(a_{η_{σ (j)}})}^{ω_{j}} + {(b_{η_{σ (j)}})}^{ω_{j}}}], \\ [\frac{{(c_{x_{σ (j)}})}^{ω_{j}} - {(d_{x_{σ (j)}})}^{ω_{j}}}{{(c_{x_{σ (j)}})}^{ω_{j}} + {(d_{x_{σ (j)}})}^{ω_{j}}}, \frac{{(c_{y_{σ (j)}})}^{ω_{j}} - {(d_{y_{σ (j)}})}^{ω_{j}}}{{(c_{y_{σ (j)}})}^{ω_{j}} + {(d_{y_{σ (j)}})}^{ω_{j}}}] 〉 \end{matrix}$

Then

$\begin{matrix} \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}} = & 〈 [\begin{matrix} \frac{2 \prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}}}, \\ \frac{2 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}} \end{matrix}], \\ [\begin{matrix} \frac{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}}} \end{matrix}] 〉 \end{matrix}$ (10)

Now we are going to prove (10) by using mathematical induction on n:

For n = 2 $\begin{matrix} δ ._{ɛ} λ_{1} = & 〈 [\frac{{(1 + μ_{1})}^{δ} - {(1 - μ_{1})}^{δ}}{{(1 + μ_{1})}^{δ} + {(1 - μ_{1})}^{δ}}, \frac{{(1 + η_{1})}^{δ} - {(1 - η_{1})}^{δ}}{{(1 + η_{1})}^{δ} + {(1 - η_{1})}^{δ}}], \\ [\frac{2 {(x_{1})}^{δ}}{{(2 - x_{1})}^{δ} + {(x_{1})}^{δ}}, \frac{2 {(y_{1})}^{δ}}{{(2 - y_{1})}^{δ} + {(y_{1})}^{δ}}] 〉 \end{matrix}$ and $\begin{matrix} δ ._{ɛ} λ_{2} = & 〈 [\frac{{(1 + μ_{2})}^{δ} - {(1 - μ_{2})}^{δ}}{{(1 + μ_{2})}^{δ} + {(1 - μ_{2})}^{δ}}, \frac{{(1 + η_{2})}^{δ} - {(1 - η_{2})}^{δ}}{{(1 + η_{2})}^{δ} + {(1 - η_{2})}^{δ}}], \\ [\frac{2 {(x_{2})}^{δ}}{{(2 - x_{2})}^{δ} + {(x_{2})}^{δ}}, \frac{2 {(y_{2})}^{δ}}{{(2 - y_{2})}^{δ} + {(y_{2})}^{δ}}] 〉 \end{matrix}$

Then $\otimes_{j = 1}^{2} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}} = 〈 \begin{matrix} [\frac{2 \prod_{j = 1}^{2} {(a_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{2} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{2} {(b_{μ_{σ (j)}})}^{ω_{j}}}, \frac{2 \prod_{i = 1}^{2} {(b_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{2} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{2} {(b_{η_{σ (j)}})}^{ω_{j}}}], \\ [\frac{\prod_{j = 1}^{2} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{2} {(d_{x_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{2} {(c_{x_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{2} {(d_{x_{σ (j)}})}^{ω_{j}}}, \frac{\prod_{j = 1}^{2} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{2} {(d_{y_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{2} {(c_{y_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{2} {(d_{y_{σ (j)}})}^{ω_{j}}}] \end{matrix} 〉$

Equation (10) is true for n = 2, now we assume that Equation (10) holds for n = k, then $\otimes_{j = 1}^{k} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}} = 〈 \begin{matrix} [\frac{2 \prod_{j = 1}^{k} {(a_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k} {(b_{μ_{σ (j)}})}^{ω_{j}}}, \frac{2 \prod_{i = 1}^{k} {(b_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k} {(b_{η_{σ (j)}})}^{ω_{j}}}], \\ [\frac{\prod_{j = 1}^{k} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{2} {(d_{x_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k} {(c_{x_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k} {(d_{x_{σ (j)}})}^{ω_{j}}}, \frac{\prod_{j = 1}^{k} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{k} {(d_{y_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k} {(c_{y_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k} {(d_{y_{σ (j)}})}^{ω_{j}}}] \end{matrix} 〉$

If Equation (10) is true for n = k, then we show that (10) is true for n = k + 1. Thus

\begin{array}{l} \otimes_{j = 1}^{k + 1} {(δ_{. ε} λ_{α (j)})}^{ω j} \\ = 〈 \begin{array}{l} [\frac{2 \prod_{j = 1}^{k} {(a_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k} {(b_{μ_{σ (j)}})}^{ω_{j}}}, \frac{2 \prod_{j = 1}^{k} {(a_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}], \\ [\frac{\prod_{j = 1}^{k} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{2} {(d_{x_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k} {(c_{x_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k} {(d_{x_{σ (j)}})}^{ω_{j}}}, \frac{\prod_{j = 1}^{k} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{k} {(d_{y_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k} {(c_{y_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k} {(d_{y_{σ (j)}})}^{ω_{j}}}] \end{array} 〉 \otimes \end{array} 〈 \begin{array}{l} [\frac{2 {(a_{μ_{k + 1}})}^{ω_{k + 1}}}{{(a_{μ_{k + 1}})}^{ω_{k + 1}} + {(b_{μ_{k + 1}})}^{ω_{k + 1}}}, \frac{2 {(b_{η_{k + 1}})}^{ω_{k + 1}}}{{(a_{η_{k + 1}})}^{ω_{k + 1}} + {(b_{η_{k + 1}})}^{ω_{k + 1}}}], \\ [\frac{{(c_{x_{k + 1}})}^{ω_{k + 1}} - {(d_{x_{k + 1}})}^{ω_{k + 1}}}{{(c_{x_{k + 1}})}^{ω_{k + 1}} + {(d_{x_{k + 1}})}^{ω_{k + 1}}}, \frac{{(c_{y_{k + 1}})}^{ω_{k + 1}} - {(d_{y_{k + 1}})}^{ω_{k + 1}}}{{(c_{y_{k + 1}})}^{ω_{i}} + {(d_{η_{y k + 1}})}^{ω_{k + 1}}}] \end{array} 〉 \begin{array}{l} = 〈 [\frac{\prod_{j = 1}^{k + 1} {(a_{μ_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{k + 1} {(b_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k + 1} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k + 1} {(b_{μ_{σ (j)}})}^{ω_{j}}}, \frac{\prod_{j = 1}^{k + 1} {(μ_{η_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{k + 1} {(b_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k + 1} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k + 1} {(b_{η_{σ (j)}})}^{ω_{j}}}], [\frac{2 \prod_{j = 1}^{k + 1} {(c_{x_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k + 1} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k + 1} {(b_{μ_{σ (j)}})}^{ω_{j}}}, \frac{2 \prod_{j = 1}^{k + 1} {(d_{y_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{k + 1} {(c_{y_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{k + 1} {(d_{y_{σ (j)}})}^{ω_{j}}}] 〉 \end{array}

Thus Equation (10) is true for n = k + 1, hence Equation (10) is true for all n.

$\otimes_{j = 1}^{n} {(δ_{. ε} λ_{α (j)})}^{ω j} = 〈 \begin{array}{l} [\frac{\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω j} - \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω j}}{\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω j} + \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω j}}, \frac{\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω j} - \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω j}}{\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω j} + \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω j}}] \\ [\frac{2 \prod_{j = 1}^{n} {(c_{c_{σ (j)}})}^{ω j}}{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω j} + \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω j}}, \frac{2 \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω j}}{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω j} + \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω j}}] \end{array} 〉$

Now we calculate $\frac{1}{δ} \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}}$ :

\frac{1}{δ} \otimes_{j = 1}^{n} {(δ_{. ε} λ_{σ (j)})}^{ω j} = {〈 \begin{array}{l} [\frac{2 \prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}}}, \frac{2 \prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}], \\ [\frac{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}, \frac{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}}}] \end{array} 〉}^{\frac{1}{δ}}

= 〈 [\frac{{(1 + (\frac{2 \prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}} - {(1 - (\frac{2 \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}}}{{(1 + (\frac{2 \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}} + {(1 - (\frac{2 \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{μ_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}}}, \frac{{(1 + (\frac{2 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}} - {(1 - (\frac{2 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}}}{{(1 + (\frac{2 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}} + {(1 - (\frac{2 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}}}] [\frac{[{(2 (\frac{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}}]}{{(1 - (\frac{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}} + {((\frac{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}}}, \frac{[{(2 (\frac{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}}]}{{(2 - (\frac{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{y_{j}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}} + {((\frac{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}}}{\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} + \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}}}))}^{\frac{1}{δ}}}] 〉

= 〈 [\begin{array}{l} \frac{{(\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} - {(\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {(a_{μ_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}, \frac{{(\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} - {(\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {(a_{η_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(b_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}} \end{array}], [\begin{array}{l} \frac{2 {(\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {(c_{x_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{x_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}, \\ \frac{2 {(\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {(c_{y_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {(d_{y_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}} \end{array}] 〉

The proof is completed.

Remark 1. In the following, let us look δλ and λ^δ some special cases of δ and λ.

If λ = ([μ, η] , [x, y]) = ([1, 1] , [0, 0]) i. e,. μ = η = 1 and x = y = 0, then

$\begin{matrix} λ^{δ} = & 〈 [\frac{2 μ^{δ}}{{(2 - μ)}^{δ} + μ^{δ}}, \frac{2 η^{δ}}{{(2 - η)}^{δ} + η^{δ}}], [\frac{{(1 + x)}^{δ} - {(1 - x)}^{δ}}{{(1 + x)}^{δ} + {(1 - x)}^{δ}}, \\ \frac{{(1 + y)}^{δ} - {(1 - y)}^{δ}}{{(1 + y)}^{δ} + {(1 - y)}^{δ}}] 〉 = 〈 [1, 1], [0, 0] 〉 \end{matrix}$

Thus λ^δ =〈 [1, 1] , [0, 0] 〉 and δλ = 〈 [0, 0] , [1, 1]〉.

If λ = ([μ, η] , [x, y]) = ([0, 0] , [1, 1]) i. e,. μ = η = 0 and x = y = 1, then

Thus λ^δ =〈 [0, 0] , [1, 1] 〉 and δλ = 〈 [1, 1] , [0, 0]〉.

If λ = ([μ, η] , [x, y]) = ([0, 0] , [0, 0]) i. e,. μ = η = 0 and x = y = 0, then

Thus λ^δ =〈 [0, 0] , [0, 0] 〉 and δλ = 〈 [0, 0] , [0, 0]〉.

If δ → 0 and 0 ⩽ μ, η, x, y ⩽ 1, then

Thus λ^δ =〈 [1, 1] , [0, 0] 〉 and δλ = 〈 [0, 0] , [1, 1]〉.

If δ→ + ∞ and 0 ⩽ μ, η, x, y ⩽ 1, then

Thus λ^δ =〈 [0, 0] , [1, 1] 〉 and δλ = 〈 [1, 1] , [0, 0]〉.

If δ = 1 and 0 ⩽ μ, η, x, y ⩽ 1, then

\begin{matrix} λ^{δ} = & 〈 [\frac{2 μ^{δ}}{{(2 - μ)}^{δ} + μ^{δ}}, \frac{2 η^{δ}}{{(2 - η)}^{δ} + η^{δ}}], [\frac{{(1 + x)}^{δ} - {(1 - x)}^{δ}}{{(1 + x)}^{δ} + {(1 - x)}^{δ}}, \\ \frac{{(1 + y)}^{δ} - {(1 - y)}^{δ}}{{(1 + y)}^{δ} + {(1 - y)}^{δ}}] 〉 = λ \end{matrix}

Thus λ^δ = λ and δλ = λ .

Theorem 4. Commutativity: Let (〈 u₁, λ₁ 〉 , 〈 u₂, λ₂ 〉 , … , 〈 u_n, λ_n 〉) and $(〈 u_{1}, λ_{1}^{*} 〉, 〈 u_{2}, λ_{2}^{*} 〉, \dots,$ $〈 u_{n}, λ_{n}^{*} 〉)$ be two set of 2-tuples, then

$\begin{matrix} {I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) \\ = {I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1}^{*} 〉, 〈 u_{2}, λ_{2}^{*} 〉, \dots, 〈 u_{n}, λ_{n}^{*} 〉) \end{matrix}$ (11)

where (〈 u₁, λ₁ 〉 , 〈 u₂, λ₂ 〉 , …, 〈 u_n, λ_n 〉) is any permutation of $(〈 u_{1}, λ_{1}^{*} 〉, 〈 u_{2}, λ_{2}^{*} 〉, \dots, 〈 u_{n}, λ_{n}^{*} 〉) .$

Proof. As

$\begin{matrix} {I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1}^{*} 〉, 〈 u_{2}, λ_{2}^{*} 〉, \dots, 〈 u_{n}, λ_{n}^{*} 〉) \\ = \frac{1}{δ} 〈 \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)}^{*})}^{ω_{j}} 〉 \end{matrix}$ (13)

Since (〈 u₁, λ₁ 〉 , 〈 u₂, λ₂ 〉 , …, 〈 u_n, λ_n 〉) is any permutation of $(〈 u_{1}, λ_{1}^{*} 〉, 〈 u_{2}, λ_{2}^{*} 〉, \dots, 〈 u_{n}, λ_{n}^{*} 〉),$ then $\frac{1}{δ} 〈 \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}} 〉 = \frac{1}{δ} 〈 \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)}^{*})}^{ω_{j}} 〉$

The proof is completed.

Theorem 5. Idempotency: Let 〈u_j, λ_j 〉 (j =1, 2, …, n) be a collection of 2-tuples, if λ_σ(j) = λ, then

$\begin{matrix} {I - GIVIFEOWG}_{ω} \\ (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) = λ \end{matrix}$ (14)

Proof. As

$\begin{matrix} {I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) \\ = \frac{1}{δ} 〈 \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}} 〉 \\ = \frac{1}{δ} 〈 {(δ ._{ɛ} λ)}^{\sum_{j = 1}^{n} ω_{j}} 〉 \\ = λ \end{matrix}$

Theorem 6. Boundedness: Let 〈u_j, λ_j 〉 (j = 1, 2, …, n) be a collection of 2-tuples, then

$\begin{matrix} λ_{min} ⩽ & {I - GIVIFEOWG}_{ω} \\ (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) ⩽ λ_{max} \end{matrix}$ (15) where $λ_{min} = min_{j} (λ_{σ (j)})$ and $λ_{max} = max_{j} (λ_{σ (j)})$ .

Proof. The proof is easy so it is omitted here.

Theorem 7. Monotonicity: Let 〈u_j, λ_j〉 and $〈 u_{j}, λ_{j}^{*} 〉$ be two set of 2-tuples, if $λ_{σ (j)} ⩽ λ_{σ (j)}^{*}$ , then

$\begin{matrix} {I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) \\ ⩽ {I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1}^{*} 〉, 〈 u_{2}, λ_{2}^{*} 〉, \dots, 〈 u_{n}, λ_{n}^{*} 〉) \end{matrix}$ (16)

Proof. As

$\begin{matrix} {I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1}^{*} 〉, 〈 u_{2}, λ_{2}^{*} 〉, \dots, \\ 〈 u_{n}, λ_{n}^{*} 〉) = \frac{1}{δ} 〈 \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)}^{*})}^{ω_{j}} 〉 \end{matrix}$ (18)

Since $λ_{σ (j)} ⩽ λ_{σ (j)}^{*},$ then $\frac{1}{δ} 〈 \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)})}^{ω_{j}} 〉 ⩽ \frac{1}{δ} 〈 \otimes_{j = 1}^{n} {(δ ._{ɛ} λ_{σ (j)}^{*})}^{ω_{j}} 〉$ .

Theorem 8. Shift Invariance: Let 〈u, ρ〉 be a 2-tuple, then

$\begin{matrix} {I - GIVIFEOWG}_{ω} (ρ ._{ɛ} 〈 u_{1}, λ_{1} 〉, ρ ._{ɛ} 〈 u_{2}, λ_{2} 〉, \dots, ρ ._{ɛ} 〈 u_{n}, λ_{n} 〉) \\ = ρ ._{ɛ} ({I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉)) \end{matrix}$ (19)

Theorem 9. Homogeneity: Let δ ≻ 0, is a positive real number, then

$\begin{matrix} {I - GIVIFEOWG}_{ω} ({〈 u_{1}, λ_{1} 〉}^{δ}, {〈 u_{2}, λ_{2} 〉}^{δ}, \dots, {〈 u_{n}, λ_{n} 〉}^{δ}) \\ = {({I - GIVIFEOWG}_{ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉))}^{δ} \end{matrix}$ (20)

Some special cases of I-GIVIFEOWG with respect to the given parameter δ .

When δ = 1, then I-GIVIFEOWG operator is reduced to I-IVIFEOWG operator.

When δ = 0, then I-GIVIFEOWG operator is reduced to I-IVIFEOWA operator.

Definition 7. Let 〈u_j,λ_j 〉 (j = 1, 2, …, n) be a collection of 2-tuples, then induced generalized interval-valued intuitionistic fuzzy Einstein hybrid geometric aggregation operator can be define as:

$\begin{matrix} {I - GIVIFEHG}_{w, ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) \\ = \frac{1}{δ} 〈 \otimes_{j = 1}^{n} {(δ ._{ɛ} {\dot{λ}}_{σ (j)})}^{ω_{j}} 〉 \end{matrix}$ (21) where ${\dot{λ}}_{σ (j)}$ is the jth largest of the weighted interval-valued Pythagorean fuzzy values ${\dot{λ}}_{σ (j)}$ $({\dot{λ}}_{σ (j)} = {(λ_{j})}^{{nw}_{j}}),$ ω = (ω₁, ω₂, …, ω_n) ^T be the associated vector of I-GIVIFEHWG operator such that ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ . w = (w₁, w₂, …, w_n) ^T be the weighted vector of λ_j (j = 1, 2, …, n) such that w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ , and n is the balancing coefficient, which plays a role of balancing. If the vector ω = (ω₁, ω₂, …, ω_n) ^T approaches ${(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T},$ then ((λ₁) ^{nw
₁}, (λ₂) ^{nw
₂}, …, (λ_n) ^{nw
_n}) ^T approaches (λ₁, λ₂, …, λ_n) ^T.

Theorem 10. Let 〈u_j,λ_j 〉 (j = 1, 2, …, n) be a collection of 2-tuples, then their aggregated valued by using I-GIVIFEHWG operator is also interval-valued intuitionistic fuzzy value, and $\begin{matrix} {l - GIVIFEHG}_{w, ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) = \\ 〈 [\begin{matrix} \frac{{(\prod_{j = 1}^{n} {({\dot{a}}_{μ_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {({\dot{b}}_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} - {(\prod_{j = 1}^{n} {({\dot{a}}_{μ_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {({\dot{b}}_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {({\dot{a}}_{μ_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {({\dot{b}}_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {({\dot{a}}_{μ_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {({\dot{b}}_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}, \\ \frac{{(\prod_{j = 1}^{n} {({\dot{a}}_{η_{σ (j)}})}^{ω_{j}} + 3 \prod_{i = 1}^{m} {({\dot{b}}_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} - {(\prod_{j = 1}^{n} {({\dot{a}}_{η_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {({\dot{b}}_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {({\dot{a}}_{η_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {({\dot{b}}_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {({\dot{a}}_{η_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {({\dot{b}}_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}} \end{matrix}], \\ [\begin{matrix} \frac{2 {(\prod_{j = 1}^{n} {({\dot{c}}_{μ_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {({\dot{d}}_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {({\dot{c}}_{μ_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {({\dot{d}}_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {({\dot{c}}_{μ_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {({\dot{d}}_{μ_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}, \\ \frac{2 {(\prod_{j = 1}^{n} {({\dot{c}}_{η_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {({\dot{d}}_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}}{{(\prod_{j = 1}^{n} {({\dot{c}}_{η_{σ (j)}})}^{ω_{j}} + 3 \prod_{j = 1}^{n} {({\dot{d}}_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}} + {(\prod_{j = 1}^{n} {({\dot{c}}_{η_{σ (j)}})}^{ω_{j}} - \prod_{j = 1}^{n} {({\dot{d}}_{η_{σ (j)}})}^{ω_{j}})}^{\frac{1}{δ}}} \end{matrix}] 〉 \end{matrix}$ (22) where ${\dot{a}}_{μ_{σ (j)}} = {(1 + {\dot{μ}}_{σ (j)})}^{δ} + 3 {(1 - {\dot{μ}}_{σ (j)})}^{δ}$ , ${\dot{b}}_{μ_{σ (j)}}$ $= {(1 + {\dot{μ}}_{σ (j)})}^{δ} - {(1 - {\dot{μ}}_{σ (j)})}^{δ}$ ${\dot{a}}_{η_{σ (j)}} = {(1 + {\dot{η}}_{σ (j)})}^{δ}$ $+ 3 {(1 - {\dot{η}}_{σ (j)})}^{δ}$ ${\dot{b}}_{η_{σ (j)}} = {(1 + {\dot{η}}_{σ (j)})}^{δ} - {(1 - {\dot{η}}_{σ (j)})}^{δ}$ , ${\dot{c}}_{x_{σ (j)}} = {(2 - {\dot{x}}_{σ (j)})}^{δ} + 3 {({\dot{x}}_{σ (j)})}^{δ}$ , ${\dot{d}}_{x_{σ (j)}} = {(2 - {\dot{x}}_{σ (j)})}^{δ}$ $- {({\dot{x}}_{σ (j)})}^{δ}$ ${\dot{c}}_{y_{σ (j)}} = {(2 - {\dot{y}}_{σ (j)})}^{δ} + 3 {({\dot{y}}_{σ (j)})}^{δ}, {\dot{d}}_{y_{σ (j)}}$ $= {(2 - {\dot{y}}_{σ (j)})}^{δ} - {({\dot{y}}_{σ (j)})}^{δ}$ and ${\dot{μ}}_{σ (j)} =$ $(\frac{2 {(μ_{j})}^{{nw}_{j}}}{{(2 - μ_{j})}^{{nw}_{j}} + {(μ_{j})}^{{nw}_{j}}})$ ${\dot{η}}_{σ (j)} = (\frac{2 {(η_{j})}^{{nw}_{j}}}{{(2 - (η_{j}))}^{{nw}_{j}} + {((η_{j}))}^{{nw}_{j}}})$ , ${\dot{x}}_{σ (j)} = (\frac{{(1 + x_{j})}^{{nw}_{j}} - {(1 - x_{j})}^{{nw}_{j}}}{{(1 + x_{j})}^{{nw}_{j}} + {(1 - x_{j})}^{{nw}_{j}}})$ , ${\dot{y}}_{σ (j)} = (\frac{{(1 + y_{j})}^{{nw}_{j}} - {(1 - y_{j})}^{{nw}_{j}}}{{(1 + y_{j})}^{{nw}_{j}} + {(1 - y_{j})}^{{nw}_{j}}})$

Proof. For proof follows Theorem 3.

Theorem 11. Idempotency: Let 〈u_j,λ_j 〉 (j = 1, 2, …, n) be a collection of 2-tuples, if ${\dot{λ}}_{σ (j)} = \dot{λ},$ then

$\begin{matrix} {I - GIVIFEHG}_{w, ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \\ \dots, 〈 u_{n}, λ_{n} 〉) = \dot{λ} \end{matrix}$ (23)

Proof. For proof follows Theorem 5.

Theorem 12. Boundedness: Let 〈u_j,λ_j 〉 (j = 1, 2, …, n) be a collection of 2-tuples, then

$\begin{matrix} {\dot{λ}}_{min} ⩽ & {I - GIVIFEHG}_{w, ω} \\ (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) ⩽ {\dot{λ}}_{max} \end{matrix}$ (24) where ${\dot{λ}}_{min} = min_{j} ({\dot{λ}}_{σ (j)})$ and ${\dot{λ}}_{max} = max_{j}$ $({\dot{λ}}_{σ (j)})$ .

Theorem 13. Monotonicity: Let 〈u_j, λ_j〉 and $〈 u_{j}, λ_{j}^{*} 〉$ be two set of 2-tuples, if $λ_{σ (j)} ⩽ λ_{σ (j)}^{*}$ , then

$\begin{matrix} {I - GIVIFEHG}_{w, ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉) \\ ⩽ {I - GIVIFEHG}_{w, ω} (〈 u_{1}, λ_{1}^{*} 〉, 〈 u_{2}, λ_{2}^{*} 〉, \dots, 〈 u_{n}, λ_{n}^{*} 〉) \end{matrix}$ (25)

Proof. For proof follows Theorem 7.

Theorem 14. Shift Invariance: Let 〈u, ρ〉 be an interval-valued intuitionistic fuzzy value, then

$\begin{matrix} {I - GIVIFEHG}_{w, ω} (ρ ._{ɛ} λ_{1}, ρ ._{ɛ} λ_{2}, \dots, ρ ._{ɛ} λ_{n}) \\ = ρ ._{ɛ} ({I - GIVIFEHG}_{w, ω} (λ_{1}, λ_{2}, \dots, λ_{n})) \end{matrix}$ (26)

Theorem 15. Homogeneity: Let δ ≻ 0, is a positive real number, then

$\begin{matrix} {I - GIVIFEHG}_{w, ω} ({〈 u_{1}, λ_{1} 〉}^{δ}, {〈 u_{2}, λ_{2} 〉}^{δ}, \dots, {〈 u_{n}, λ_{n} 〉}^{δ}) \\ = {({I - GIVIFEHG}_{w, ω} (〈 u_{1}, λ_{1} 〉, 〈 u_{2}, λ_{2} 〉, \dots, 〈 u_{n}, λ_{n} 〉))}^{δ} \end{matrix}$ (27)

Some special cases of I-GIVIFEHG with respect to the given parameter δ .

When δ = 1, then I-GIVIFEHG operator is reduced to I-IVIFEHG operator.

When δ = 0, then I-GIVIFEHG operator is reduced to I-IVIFEHA operator.

Theorem 16. The generalized interval-valued intuitionistic fuzzy Einstein weighted geometric aggregation operator is a special case of the induced generalized interval-valued intuitionistic fuzzy Einstein hybrid weighted geometric aggregation operator.

Theorem 17. The induced generalized interval-valued intuitionistic fuzzy Einstein ordered weighted geometric aggregation operator is a special case of the induced generalized interval-valued intuitionistic fuzzy Einstein hybrid weighted geometric aggregation operator.

Table 1

Decision matrix of D¹

	X ₁	X ₂	X ₃
A ₁	〈0.8, ([0.22, 0.31] , [0.23, 0.54])〉	〈0.7, ([0.20, 0.36] , [0.13, 0.53])〉	〈0.6, ([0.40, 0.56] , [0.12, 0.37])〉
A ₂	〈0.5, ([0.28, 0.41] , [0.33, 0.49])〉	〈0.9, ([0.20, 0.36] , [0.33, 0.53])〉	〈0.8, ([0.30, 0.46] , [0.12, 0.37])〉
A ₃	〈0.6, ([0.32, 0.41] , [0.23, 0.44])〉	〈0.5, ([0.16, 0.25] , [0.43, 0.53])〉	〈0.7, ([0.21, 0.37] , [0.23, 0.45])〉
A ₄	〈0.7, ([0.39, 0.47] , [0.18, 0.36])〉	〈0.8 ([0.27, 0.32] , [0.39, 0.53])〉	〈0.5, ([0.11, 0.23] , [0.28, 0.34])〉

4 Application of the proposed aggregation operators

This section, deals with multi-attribute group decision-making (MAGDM) problems based on the above mentioned operators, such as the induced generalized interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (I-GIVIFEOWG) operator and the induced generalized interval-valued intuitionistic fuzzy Einstein hybrid geometric (I-GIVIFEHG) operator using interval-valued intuitionstic fuzzy numbers. To show the viability and practicality of the proposed operators in daily life problems an example is also given.

Let A = {A₁, A₂, …, A_m} be a finite set of m alternatives, X = {X₁, X₂, …, X_n} be a finite set of n attributes, and D ={ D₁, D₂, …, D_k } be the set of k decision-makers. Let ω = (ω₁, ω₂, …, ω_n) ^T be the associated vector of the attributes X_j (j = 1, 2, …, n) , such that ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ . Let ϖ = (ϖ₁, ϖ₂, …, ϖ_k) ^T be the weighted vector of the decision makers D^s (s = 1, 2, …, k) , such that ϖ_s ∈ [0, 1] and $\sum_{s = 1}^{k} ϖ_{s} = 1$ . Let D = (a_ij) =〈 [μ_ij, η_ji] , [x_ij, y_ij] 〉, where [μ_ij, η_ij] indicates the interval degree that the alternative A_i (i = 1, 2, …, m) satisfies the attribute X_j (j = 1, 2, …, n) and [x_ij, y_ij] indicates the interval degree that the alternative A_i (i = 1, 2, …, m) does not satisfy the attribute X_j (j = 1, 2, …, n) with condition 0 ⩽ η_ij + y_ij ⩽ 1.

Step 1. Construct decision matrices, $D^{s} = {[a_{ij}^{(s)}]}_{m \times n} (s = 1, 2, \dots, k)$ for decision.

Step 2. If the criteria have two types such as, benefit criteria and cost criteria, then the interval-valued intuitionistic fuzzy decision-making matrices, $D^{s} = {[a_{ij}^{(s)}]}_{m \times n} (s = 1, 2, \dots, k)$ can be converted into the normalized interval-valued intuitionistic fuzzy decision-making matrices, $R^{s} = {[r_{i j}^{(s)}]}_{m \times n} (s = 1, 2, \dots, k)$ , where

$r_{ij}^{(s)} = {\begin{matrix} a_{ij}^{(s)}, {for benefit criteria X}_{j} \\ {\bar{a}}_{ij}^{(s)}, {for cost criteria X}_{j}, \end{matrix} (\begin{matrix} j = 1, 2, 3, \dots, n \\ i = 1, 2, 3, \dots, m \end{matrix})$

and ${\bar{a}}_{ij}^{(s)}$ is the complement of $α_{ij}^{s} .$ If all the criteria have the same type, then there is no need of normalization.

Step 3. Utilize the I-GIVIFEOWG operator to aggregate all the individual normalized interval-valued intuitionistic fuzzy decision-making matrices, $R^{s} = {[r_{ij}^{(s)}]}_{m \times n} (s = 1, 2, \dots, k)$ into a single interval-valued intuitionistic fuzzy decision-making matrix, R = [r_ij] _m×n.

Step 4. Utilize the I-GIVIFEOWG operator to get the collective overall preference values r_i (i = 1, 2, …, m) of the corresponding alternatives A_i (i = 1, 2, 3, …, m).

Step 5. Calculate the scores function of r_ij (i = 1, 2, …, m, j = 1, 2, …, n). If there is no difference between two or more than two scores, then we have to find out the accuracy degrees of the collective overall preference values.

Step 6. Arrange the scores of the all alternatives in the form of descending order and select that alternative which has the highest score function or accuracy degree.

5 Illustrative example

Table 2
Decision matrix of D²

X ₁ X ₂ X ₃

A ₁ 〈0.7, ([0.04, 0.21] , [0.35, 0.46])〉〈0.6, ([0.27, 0.45] , [0.10, 0.34])〉〈0.5, ([0.13, 0.20] , [0.32, 0.37])〉

A ₂ 〈0.4, ([0.32, 0.39] , [0.27, 0.39])〉〈0.7, ([0.30, 0.36] , [0.03, 0.57])〉〈0.6, ([0.14, 0.19] , [0.16, 0.25])〉

A ₃ 〈0.5, ([0.26, 0.37] , [0.21, 0.40])〉〈0.3, ([0.06, 0.15] , [0.23, 0.43])〉〈0.4, ([0.11, 0.29] [0.21, 0.35])〉

A ₄ 〈0.4, ([0.30, 0.43] , [0.19, 0.35])〉〈0.7, ([0.31, 0.34] , [0.28, 0.43])〉〈0.3, ([0.01, 0.17] , [0.39, 0.46])〉

	X ₁	X ₂	X ₃
A ₁	〈0.7, ([0.04, 0.21] , [0.35, 0.46])〉	〈0.6, ([0.27, 0.45] , [0.10, 0.34])〉	〈0.5, ([0.13, 0.20] , [0.32, 0.37])〉
A ₂	〈0.4, ([0.32, 0.39] , [0.27, 0.39])〉	〈0.7, ([0.30, 0.36] , [0.03, 0.57])〉	〈0.6, ([0.14, 0.19] , [0.16, 0.25])〉
A ₃	〈0.5, ([0.26, 0.37] , [0.21, 0.40])〉	〈0.3, ([0.06, 0.15] , [0.23, 0.43])〉	〈0.4, ([0.11, 0.29] [0.21, 0.35])〉
A ₄	〈0.4, ([0.30, 0.43] , [0.19, 0.35])〉	〈0.7, ([0.31, 0.34] , [0.28, 0.43])〉	〈0.3, ([0.01, 0.17] , [0.39, 0.46])〉

Table 3

Decision matrix of D³

	X ₁	X ₂	X ₃
A ₁	〈0.5, ([0.25, 0.27] , [0.23, 0.40])〉	〈0.5, ([0.26, 0.40] , [0.17, 0.27])〉	〈0.4, ([0.40, 0.56] , [0.21, 0.30])〉
A ₂	〈0.3, ([0.25, 0.29] , [0.33, 0.39])〉	〈0.4, ([0.43, 0.50] , [0.18, 0.46])〉	〈0.4, ([0.30, 0.46] , [0.06, 0.21])〉
A ₃	〈0.2, ([0.22, 0.27] , [0.27, 0.31])〉	〈0.2, ([0.16, 0.20] , [0.13, 0.37])〉	〈0.1, ([0.21, 0.37] , [0.11, 0.24])〉
A ₄	〈0.1, ([0.30, 0.48] , [0.09, 0.45])〉	〈0.6 ([0.20, 0.24] , [0.08, 0.53])〉	〈0.2, ([0.01, 0.09] , [0.32, 0.61])〉

Table 4

Normalized Decision Matrix R¹

	X ₁	X ₂	X ₃
A ₁	〈0.8, ([0.23, 0.54] , [0.22, 0.31])〉	〈0.7, ([0.20, 0.36] , [0.13, 0.53])〉	〈0.6, ([0.40, 0.56] , [0.12, 0.37])〉
A ₂	〈0.5, ([0.33, 0.49] , [0.28, 0.41])〉	〈0.9, ([0.20, 0.36] , [0.33, 0.53])〉	〈0.8, ([0.30, 0.46] , [0.12, 0.37])〉
A ₃	〈0.6, ([0.23, 0.44] , [0.32, 0.41])〉	〈0.5, ([0.16, 0.25] , [0.43, 0.53])〉	〈0.7, ([0.21, 0.37] , [0.23, 0.45])〉
A ₄	〈0.7, ([0.18, 0.36] , [0.39, 0.47])〉	〈0.8 ([0.27, 0.32] , [0.39, 0.53])〉	〈0.5, ([0.11, 0.23] , [0.28, 0.34])〉

Table 5

Normalized Decision Matrix R²

	X ₁	X ₂	X ₃
A ₁	〈0.7, ([0.35, 0.46] , [0.04, 0.21])〉	〈0.6, ([0.27, 0.45] , [0.10, 0.34])〉	〈0.5, ([0.13, 0.20] , [0.32, 0.37])〉
A ₂	〈0.4, ([0.27, 0.39] , [0.32, 0.39])〉	〈0.7, ([0.30, 0.36] , [0.03, 0.57])〉	〈0.6, ([0.14, 0.19] , [0.16, 0.25])〉
A ₃	〈0.5, ([0.21, 0.40] , [0.26, 0.37])〉	〈0.3, ([0.06, 0.15] , [0.23, 0.43])〉	〈0.4, ([0.11, 0.29] , [0.21, 0.35])〉
A ₄	〈0.4, ([0.19, 0.35] , [0.30, 0.43])〉	〈0.7, ([0.31, 0.34] , [0.28, 0.43])〉	〈0.3, ([0.01, 0.17] , [0.39, 0.46])〉

Table 6

DNormalized Decision Matrix R³

	X ₁	X ₂	X ₃
A ₁	〈0.5, ([0.23, 0.40] , [0.25, 0.27])〉	〈0.5, ([0.26, 0.40] , [0.17, 0.27])〉	〈0.4, ([0.40, 0.56] , [0.21, 0.30])〉
A ₂	〈0.3, ([0.33, 0.39] , [0.25, 0.29])〉	〈0.4, ([0.43, 0.50] , [0.18, 0.46])〉	〈0.4, ([0.30, 0.46] , [0.06, 0.21])〉
A ₃	〈0.2, ([0.27, 0.31] , [0.22, 0.27])〉	〈0.2, ([0.16, 0.20] , [0.13, 0.37])〉	〈0.1, ([0.21, 0.37] , [0.11, 0.24])〉
A ₄	〈0.1, ([0.09, 0.45] , [0.30, 0.48])〉	〈0.6, ([0.20, 0.24] , [0.08, 0.53])〉	〈0.2, ([0.01, 0.09] , [0.32, 0.61])〉

Table 7

Collective Decision Matrix R

	X ₁	X ₂	X ₃
A ₁	([0 . 264, 0 . 462] , [0 . 208, 0 . 275])	([0 . 241, 0 . 400] , [0 . 141, 0 . 410])	([0 . 207, 0 . 327] , [0 . 226, 0 . 350])
A ₂	([0 . 308, 0 . 420] , [0 . 285, 0 . 371])	([0 . 294, 0 . 400] , [0 . 241, 0 . 523])	([0 . 228, 0 . 297] , [0 . 119, 0 . 295])
A ₃	([0 . 235, 0 . 378] , [0 . 276, 0 . 360])	([0 . 116, 0 . 196] , [0 . 311, 0 . 454])	([0 . 148, 0 . 273] , [0 . 195, 0 . 366])
A ₄	([0 . 146, 0 . 383] , [0 . 339, 0 . 463])	([0 . 255, 0 . 296] , [0 . 295, 0 . 503])	([0 . 023, 0 . 153] , [0 . 332, 0 . 490])

Table 8

Comparative analysis at different values of δ

Different Methods	Score functions	Rank of Alternatives
	Existing Methods [25]
I-IVIFEOWG	S (r₁) ≻ S (r₂) ≻ S (r₃) ≻ S (r₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
I-IVIFEHG	S (r₁) ≻ S (r₂) ≻ S (r₃) ≻ S (r₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
	Proposed Methods
	Forδ → 0.01
I-GIVIFEOWG	S (r₁) ≻ S (r₂) ≻ S (r₃) ≻ S (r₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
I-GIVIFEHG	S (r₁) ≻ S (r₂) ≻ S (r₃) ≻ S (r₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
	Forδ → 0.05
I-GIVIFEOWG	S (r₁) ≻ S (r₃) ≻ S (r₂) ≻ S (r₄)	A₁ ≻ A₃ ≻ A₂ ≻ A₄
I-GIVIFEHG	S (r₁) ≻ S (r₃) ≻ S (r₂) ≻ S (r₄)	A₁ ≻ A₃ ≻ A₂ ≻ A₄
	Forδ → 1.0
I-GIVIFEOWG	S (r₁) ≻ S (r₂) ≻ S (r₃) ≻ S (r₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
I-GIVIFEHG	S (r₁) ≻ S (r₂) ≻ S (r₃) ≻ S (r₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
	Forδ → 1.5
I-GIVIFEOWG	S (r₁) ≻ S (r₂) ≻ S (r₃) ≻ S (r₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄
I-GIVIFEHG	S (r₁) ≻ S (r₂) ≻ S (r₃) ≻ S (r₄)	A₁ ≻ A₂ ≻ A₃ ≻ A₄

Suppose a Medicine company wants to appoint an expert manager for his company. For this purpose the company constructs a committee of three decision-makers, whose weighted vector is (0.314, 0.355, 0.331) ^T. There are many factors that must be considered while selecting the most expert manager, but here, the committee considers only the following three criteria, whose weighted vector is (0.40, 0.20, 0.40) ^T. After the first screening test four managers, A_j (j = 1, 2, 3, 4) are continue for further process. Committee must take a decision according to the following three attributes.

X₁: Salary and other facilities

X₂: Experience and dealing with public

X₃: Working skill

where X₁, is cost type criteria and X₂, X₃ are benefiting type criteria, the attributes have two types criteria, thus we have must to change the cost type criteria into benefit type criteria.

Step 1. Construct the decision matrices as shown in Tables 1–3.

Step 2. Normalized decision matrices as shown in Tables 4–6.

Step 3. Utilize the I-GIVIFEOWG operator to aggregate $R^{s} = {[r_{ij}^{(s)}]}_{m \times n}$ into R = [r_ij] _m×n.

Suppose $(δ = 2, ω = \frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ , the we have

Step 4. Again utilize the I-GIVIFEOWG aggregation operator to aggregate all preference values. Suppose $(δ = 2, ω = \frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ , then we have the result in Table 7.

$\begin{matrix} r_{1} = ([0.225, 0.368], [0.243, 0.426]), \\ r_{2} = ([0.261, 0.348], [0.313, 0.486]) \\ r_{3} = ([0.153, 0.258], [0.357, 0.461]), \\ r_{4} = ([0.092, 0.245], [0.375, 0.525]) \end{matrix}$

Step 5. Calculate the score functions: S (r₁) = -0.03, S (r₂) = -0.09, S (r₃) = -0.20, S (r₄) = -0.28

Step 6. Hence A₁ ≻ A₂ ≻ A₃ ≻ A₄ . Thus the best alternative is A₁ .

6 Conclusion

Induced aggregation operators are more suitable, more fitting and more appropriate for aggregating the individual preference relations into a collective fuzzy preference relation. In this paper, we have explored some induced generalized Einstein geometric aggregation operators based on interval-valued intuitionistic fuzzy numbers and applied them to the multi-attribute group decision-making problems where attribute values are the interval-valued intuitionistic fuzzy numbers. Firstly, induced generalized interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (I-GIVIFEOWG) aggregation operator and induced generalized interval-valued intuitionistic fuzzy Einstein hybrid geometric(I-GIVIFEHG) aggregation operator, were proposed. Some desirable properties, such as idempotency, boundedness and monotonicity corresponding to the proposed operators have been investigated. Furthermore, these operators are applied to decision-making problems in which experts provide their preferences in the intuitionistic fuzzy environment to show the validity, practicality, and effectiveness of the new approach, and some special cases of them were analyzed. Furthermore, a method to multi-criteria group decision-making was developed, and the operational processes were illustrated in detail. Finally, an illustrative example is given to show the decision steps of the proposed methods and to demonstrate their effectiveness.

In further research, it is necessary and meaningful to give the applications of these operators to the other domains such as pattern recognition, fuzzy cluster analysis, uncertain programming, logarithmic form, power, interval numbers and linguistic etc.

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