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
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.
Preliminaries
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
where (ω1, ω2, …, ω
n
)
T
be the weighted vector of (λ1, λ2, …, λ
n
) with some conditions ω
j
∈ [0, 1],
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.
λ1 ⊕
ɛ
λ2 = λ2 ⊕
ɛ
λ1
λ ⊗
ɛ
λ2 = λ2 ⊗
ɛ
λ1
(λ1 ⊕
ɛ
λ2) ⊕
ɛ
λ3 = λ1 ⊕
ɛ
(λ2 ⊕
ɛ
λ3) (λ1 ⊗
ɛ
λ2) ⊗
ɛ
λ3 = λ1 ⊗
ɛ
(λ2 ⊗
ɛ
λ3)
(λ1 ∪ λ2) ⊕
ɛ
(λ1 ∩ λ2) = λ1 ⊕
ɛ
λ2
(λ1 ∪ λ2) ⊗
ɛ
(λ1 ∩ λ2) = λ1 ⊗
ɛ
λ2
(λ1 ∪ λ2) ∩ λ3 = (λ1 ∩ λ3) ∪ (λ2 ∩ λ3) (λ1 ∩ λ2) ∪ λ3 = (λ1 ∪ λ3) ∩ (λ2 ∪ λ3) (λ1 ∪ λ2) ⊕
ɛ
λ3 = (λ1 ⊕
ɛ
λ3) ∪ (λ2 ⊕
ɛ
λ3) (λ1 ∩ λ2) ⊕
ɛ
λ3 = (λ1 ⊕
ɛ
λ3) ∩ (λ2 ⊕
ɛ
λ3) (λ1 ∩ λ2) ⊗
ɛ
λ3 = (λ1 ⊗
ɛ
λ3) ∩ (λ2 ⊗
ɛ
λ3) (λ1 ∪ λ2) ⊕
ɛ
λ3 = (λ1 ⊕
ɛ
λ3) ∪ (λ2 ⊕
ɛ
λ3)
Now we calculate (δ .
ɛ
λ1)
ω
1
:
Again we calculate
Then
Now we are going to prove (10) by using mathematical induction on n:
For n = 2
Then
Equation (10) is true for n = 2, now we assume that Equation (10) holds for n = k, then
If Equation (10) is true for n = k, then we show that (10) is true for n = k + 1. Thus
Thus Equation (10) is true for n = k + 1, hence Equation (10) is true for all n.
Now we calculate
The proof is completed.
If λ = ([μ, η] , [x, y]) = ([1, 1] , [0, 0]) i. e,. μ = η = 1 and x = y = 0, then
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
Thus λ δ = λ and δλ = λ .
where (〈 u1, λ1 〉 , 〈 u2, λ2 〉 , …, 〈 u
n
, λ
n
〉) is any permutation of
Since (〈 u1, λ1 〉 , 〈 u2, λ2 〉 , …, 〈 u
n
, λ
n
〉) is any permutation of
The proof is completed.
Since
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.
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.
Decision matrix of D1
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
and
Illustrative example
Decision matrix of D2
Decision matrix of D2
Decision matrix of D3
Normalized Decision Matrix R1
Normalized Decision Matrix R2
DNormalized Decision Matrix R3
Collective Decision Matrix R
Comparative analysis at different values of δ
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.
Suppose
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.
