# Some Improvedq-Rung Orthopair Fuzzy Aggregation Operators and Their Applications to Multiattribute Group Decision-Making

Authors: Lei Xu; Yi Liu; Haobin Liu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

As a generalization of intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS),q-rung orthopair fuzzy set (q-ROFS) is a new concept in describing complex fuzzy uncertainty information. The present work focuses on the multiattribute group decision-making (MAGDM) approach under the q-rung orthopair fuzzy information. To begin with, some drawbacks of the existing MAGDM methods based on aggregation operators (AOs) are firstly analyzed. In addition, some improved operational laws put forward to overcome the drawbacks along with some properties of the operational law are proved. Thirdly, we put forward the improved q-rung orthopair fuzzy-weighted averaging (q-IROFWA) aggregation operator and improved q-rung orthopair fuzzy-weighted power averaging (q-IROFWPA) aggregation operator and present some of their properties. Then, based on the q-IROFWA operator and q-IROFWPA operator, we proposed a new method to deal with MAGDM problems under the fuzzy environment. Finally, some numerical examples are provided to illustrate the feasibility and validity of the proposed method.


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## Body

## 1. Introduction

Decision-making is an inseparable science in human daily life, and its ideas and methods have been applied to many aspects of real life such as systems engineering, regional planning and design, and operational research.In the process of decision-making, because of the fuzziness of human thinking, the uncertainty of decision-making information and the complexity of objective things, it is difficult for decision makers to give accurate evaluation values for different schemes. Since Zadeh [1] put forward the concept of fuzzy set (FS), the discussions on MAGDM under the condition of fuzzy information have never stopped. However, the fuzzy set cannot reasonably express all the decision-making problems. For example, in the process of site selection, some evaluators agree with a scheme, while others oppose or even abstain from it. Therefore, how to use them to effectively deal with the fuzzy decision-making problem is directly related to people’s production, life, and social progress.For this, Bulgarian scholar Atanassov [2] proposed the intuitionistic fuzzy set (IFS) which is extended from FS. The IFS describes the fuzziness of things from three aspects, membership degree (MD), nonmembership degree (NMD), and hesitation degree; therefore, it has higher flexibility in fuzziness and uncertainty. After the introduction of IFS, scholars have conducted extensive and in-depth research [3–6]. The main results are as follows: (1) the related properties of IFS [7], operational rules (ORs) [8], similarity measurement [9, 10], distance measurement [11–13], information entropy [14], and so on. (2) Some extended decision-making method based on IFS, such as ELECTRE-III [15], Taguchi and the VIKOR methods [16, 17], and other decision-making approaches [18–22]. (3) The AOs, for example, IFWA operator and IFWG operator proposed by Xu [23, 24], Hamacher aggregation operators by Liu [25], and Frank aggregation operators by Zhang et al. [26].However, the application scope of the IFS is limited. MDu and NMD v satisfy the formula u+v≤1. Therefore, some special decision evaluation information cannot be expressed. For example, if MD and NMD are equal to 0.7 and 0.5, respectively, it is clear that the IFS cannot be used. After that, Yager and Abbasov [27] proposed the Pythagorean fuzzy set (PFS) to solve the above problem. Compared with the IFS, the PFS generalizes the MD and the NMD to the square sum less than or equal to 1. Therefore, it can solve more problems than the IFS [16, 28]. However, with the increase of the complexity of decision-making problems and the contradictory psychology of policy makers, it is still difficult for PFS to express corresponding information.Recently, Yager [29] again proposed the concept of the q-ROFS, in which MD u and NMD v satisfy uq+vq≤1 (q≥1). We can see that the IFS and PFS are special cases of q-ROFS. As q-rung increases, the range of processing fuzzy information increases. The acceptable space for different types of fuzzy numbers is shown in the Figure 1. IFN, PFN, and q-ROFN are abbreviations of intuitionistic fuzzy number, Pythagorean fuzzy number, and q-rung orthopair fuzzy number, respectively. Therefore, it can be seen that q-ROFS has stronger capability in dealing with uncertain information than the IFS and PFS.Figure 1
Comparison of grades space of IFN, PFN, andq-ROFN [30].In recent years, the topic of information aggregation has attracted a lot of attention and is one of the key research issues in the problems of MAGDM. As far asq-ROFS is concerned, different aggregation operators have been introduced and applied, such as q-ROFWA and q-ROFWG operator [31], q-ROFPA and q-ROFPMSM operator [32], q-ROFGBM and q-ROFWGBM operator [33], IFMM operator [34], q-ROFWPHM operator [30], and q-ROFIDHM operator [35].Although different aggregation operators have been introduced, these operators have some drawbacks. The above aggregation operators are carried out based on traditional arithmetic rules, i.e., Einstein ORs [36], Hamacher ORs [25], and Frank ORs [26]. However, these operational rules have some drawbacks. When NMD of any an IFN is 0, the aggregated result will be 0 no matter what value of another NMD is. In addition, when MD of any IFN is 1, the aggregated result will be 1 no matter what value of another MD is. Furthermore, aggregated results are obtained independently. The change of MD will not affect the aggregated result of NMD. Obviously, this is unreasonable.Motivated by the new operational laws developed by He et al. [37], we propose some improved operational laws for q-ROFS to surmount the drawbacks firstly, and then develop an improved q-rung orthopair fuzzy-weighted averaging (q-IROFWA) aggregation operator. Considering that the PA operator can reduce the impact of irrational data given by biased decision makers, we extend the power average (PA) operator [38] to deal with q-ROFS and based on the q-IROFWA and q-IROFWPA operator we proposed a new method to deal with MAGDM problems.The contributions of this paper are as follows: (1) some drawbacks of the existing aggregation operator are reviewed and analyzed; (2) proposed new aggregation operators containing theq-IROFWA operator and q-IROFWPA operator; (3) proposed a new method to deal with MAGDM problems based on the q-IROFWA and q-IROFWPA operator; (4) some numerical examples are provided to verify the feasibility and practicability of the new method.The remaining of this paper is organized as follows. In Section1, we briefly introduce the preliminary conceptions of q-ROFS, PA operator, and score function. In Section 2, we analyze some drawbacks of existing q-rung orthopair fuzzy aggregation operators. In Section 3, we propose a new q-IROFWA and q-IROFWPA operator based on new operational rules. In Section 4, we will give some examples to evaluate the feasibility and validity of the proposed method. In Section 5, the conclusions are given.


## 2. Preliminaries

In this section, some basic theories related toq-ROFS, PA operator, and score function are reviewed.


### 2.1.q-ROFS

Definition 1.
LetX=x1,x2,…,xn be a finite universe of discourse, a q-ROFS A in X characterized by a membership function μA:X⟶0,1, and a nonmembership function νA:X⟶0,1, which satisfies that 0≤μAxq+νAxq≤1, (q≥1). A q-ROFS is represented as(1)A=x,μAx,νAxx∈X,where πAx=μAxq+νAxq−μAxqνAxq1/q is the degree of indeterminacy, and we called μAx,νAx a q‐ROFN, denoted by μA,νA.


### 2.2. The Power Average Operator

Definition 2.
LetX=x1,x2,…,xn be a set of crisp numbers; the PA operator can be expressed as(2)PAx1,x2,…,xn=∑i=1n1+Txixi∑i=1n1+Txi,where(3)Txi=∑j=1,j≠inSupxi,xj.
Sup
x
i
,
x
j represents the support degree for xi from xj, which meets(1)
Supxi,xj∈0,1;(2)
Supxi,xj=Supxj,xi;(3)
Supa,b≥Supx,y, if a−b<x−y.


### 2.3. Score Function of Decision-Making Problem

Definition 3 (see [31])].
Letα=μ,ν be a q-ROFN (q≥1). The score function S of α is defined as follows:(4)Sα=μq−νq,where Sα∈−1,1.Definition 4 (see [31]).
Letα=μ,ν be a q-ROFN (q≥1). The accuracy function H of α is defined as follows:(5)Hα=μq+νq,where Hα∈0,1.Definition 5 (see [31]).
Letα1=μ1,ν1 and α2=μ2,ν2 be any two q-ROFN (q≥1). Sα1 and Sα2 be the score functions of α1 and α2, and Hα1 and Sα2 be the accuracy functions of α1 and α2.(1)
ifSα1>Sα2, then α1>α2(2)
ifSα1=Sα2 andifHα1>Hα2, then α1>α2ifHα1=Hα2, then α1=α2


## 2.1.q-ROFS

Definition 1.
LetX=x1,x2,…,xn be a finite universe of discourse, a q-ROFS A in X characterized by a membership function μA:X⟶0,1, and a nonmembership function νA:X⟶0,1, which satisfies that 0≤μAxq+νAxq≤1, (q≥1). A q-ROFS is represented as(1)A=x,μAx,νAxx∈X,where πAx=μAxq+νAxq−μAxqνAxq1/q is the degree of indeterminacy, and we called μAx,νAx a q‐ROFN, denoted by μA,νA.


## 2.2. The Power Average Operator

Definition 2.
LetX=x1,x2,…,xn be a set of crisp numbers; the PA operator can be expressed as(2)PAx1,x2,…,xn=∑i=1n1+Txixi∑i=1n1+Txi,where(3)Txi=∑j=1,j≠inSupxi,xj.
Sup
x
i
,
x
j represents the support degree for xi from xj, which meets(1)
Supxi,xj∈0,1;(2)
Supxi,xj=Supxj,xi;(3)
Supa,b≥Supx,y, if a−b<x−y.


## 2.3. Score Function of Decision-Making Problem

Definition 3 (see [31])].
Letα=μ,ν be a q-ROFN (q≥1). The score function S of α is defined as follows:(4)Sα=μq−νq,where Sα∈−1,1.Definition 4 (see [31]).
Letα=μ,ν be a q-ROFN (q≥1). The accuracy function H of α is defined as follows:(5)Hα=μq+νq,where Hα∈0,1.Definition 5 (see [31]).
Letα1=μ1,ν1 and α2=μ2,ν2 be any two q-ROFN (q≥1). Sα1 and Sα2 be the score functions of α1 and α2, and Hα1 and Sα2 be the accuracy functions of α1 and α2.(1)
ifSα1>Sα2, then α1>α2(2)
ifSα1=Sα2 andifHα1>Hα2, then α1>α2ifHα1=Hα2, then α1=α2


## 3. Analysis of the Existingq-ROFS Operators

In this section, we will review the formal representation of the typicalq-ROFS operators and analyze their drawbacks; then in Section 3, we will introduce methods in order to overcome those drawbacks.Definition 6 (see [31]).
Letαk=uk,vkk=1,2,…,n is a collection of q-ROFN (q≥1) and w=w1,w2,…,wnT be the weight vector of αkk=1,2,…,n, such that 0≤wk≤1 and ∑k=1nwk=1. Then, the aggregation result is still a q-ROFN and has(6)q‐ROFWAα1,α2,…,αn=1−∏k=1n1−ukqwk1/q,∏k=1nvkwk,(7)q‐ROFWGα1,α2,…,αn=∏k=1nukwk,1−∏k=1n1−vkqwk1/q.Example 1.
Letα1 = (0.6, 0), α2 = (0.5, 0.3), α3 = (0.7, 0.2), and α4 = (0.3, 0.5) be four q-ROFN and w=0.3,0.2,0.1,0.4 be the corresponding weight vector (WV), then suppose q = 3 and according to equation (6) we have(8)q‐ROFWAα1,α2,α3,α4=1−∏k=141−uk3wk1/3,∏k=14vkwk=1−1−0.630.3×1−0.530.2×1−0.730.1×1−0.330.41/3,00.3×0.30.2×0.20.1×0.50.4=0.521,0.
Drawback A: as can be seen from Example 1, although the degree of NMD of threeq-ROFNs α2, α3, and α4 are not equal to 0, the aggregated result is still 0. Obviously, this is an unreasonable result. Considering equation (2), if there only one NMD of q-ROFNs is 0, the aggregated result will be 0 even if the other n−1 q-ROFNs are not 0. Therefore, the q-ROFWA operator in [31] is not well defined. Accordingly using this ill-defined q-ROFWA operator will get an unreasonable ranking result of the alternatives in some situations.
In addition, from equation (11), we know that the MD and NMD of aggregated results are obtained independently from the MDs and NMDs of n q-ROFNs, respectively. It is shown as Example 2.Example 2.
Letα1 = (0.6, 0.3), α2 = (0.5, 0.3), α3 = (0.7, 0.2), and α4 = (0.3, 0.5) be four q-ROFNs and w=0.3,0.2,0.1,0.4 be the corresponding WV. Then, based on equation (6) and suppose q = 3, we have(9)q‐ROFWAα1,α2,α3,α4=1−∏k=141−uk3wk1/3,∏k=14vkwk=1−1−0.630.3×1−0.530.2×1−0.730.1×1−0.330.41/3,0.30.3×0.30.2×0.20.1×0.50.4=0.521,0.353.
When we change the value ofα3 and α4 to α3 = (0.8, 0.2) and α4 = (0.5, 0.5), then the aggregated result is(10)q‐ROFWAα1,α2,α3,α4=1−∏k=141−uk3wk1/3,∏k=14vkwk=1−1−0.630.3×1−0.530.2×1−0.830.1×1−0.530.41/3,0.30.3×0.30.2×0.20.1×0.50.4=0.586,0.353.
Drawback B: from Example 2 we can see that, the change of MD will not affect the aggregated result of NMD. In the calculation process, MD and NMD are independent. Therefore, the aggregated result cannot provide the right information for the decision makers.
In Section3 below, two new methods are proposed to surmount the abovementioned drawbacks of the existing methods.


## 4. Improvedq-ROFS Operators

In this section, two operatorsq-IROFWA and q-IROFWPA are proposed to overcome the drawbacks presented above.Because the aggregation operation is derived from the operational rules, the first step to improve the aggregation operator is to improve the operation rules. Some improved rules based on He et al. [37] are proposed as follows.Letα=u,v, α1=u1,v1, and α2=u2,v2 be any two q-ROFN (q≥1) and λ>0, then(11)aα1⊕α2=u1,v1⊕u2,v2=1−∏i=121−uiq1/q,∏i=121−uiq−∏i=121−uiq−viq1/q,(12)bα1⊗α2=u1,v1⊗u2,v2=∏i=121−viq−∏i=121−uiq−viq1/q,1−∏i=121−viq1/q,(13)cλα=1−1−uqλ1/q,1−uqλ−1−uq−vqλ1/q,(14)dαλ=1−vqλ−1−uq−vqλ1/q,1−1−vqλ1/q.Theorem 1.
Letα1=u1,v1, α2=u2,v2, and α3=u3,v3 be any three q-ROFN (q≥1) and λ,λ1,λ2>0, then(15)aα1⊕α2=α2⊕α1,(16)bα1⊗α2=α2⊗α1,(17)cα1⊕α2⊕α3=α1⊕α2⊕α3,(18)dα1⊗α2⊗α3=α1⊗α2⊗α3,(19)eλα1⊕α2=λα1⊕λα2,(20)fα1⊗α2λ=α1λ⊗α2λ,(21)gλ1+λ2α1=λ1α1⊕λ2α1,(22)hα1λ1+λ2=α1λ1⊗α1λ2.Proof of Theorem 1.
(1)
According to rules (11) and (12), it is obvious that equations (15)and (16) are kept.
(2)
Then, we prove equations (17), (19), and (21), and the proof of others are similar to equations (17) and (19), so omitted here.
(3)
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(4)
The proof of equation (18) is similar to that of equation (17), so it is omitted.
(5)
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1
2
1
−
u
i
q
λ
−
∏
i
=
1
2
1
−
u
i
q
−
v
i
q
λ
1
/
q
.
Thus, equation (19) is kept.
(6)
For the right-hand side of equation (21), we get
(27)
λ
1
α
1
⊕
λ
2
α
1
=
1
−
1
−
u
1
q
λ
1
1
/
q
,
1
−
u
1
q
λ
1
−
1
−
u
1
q
−
v
1
q
λ
1
1
/
q
⊕
1
−
1
−
u
2
q
λ
2
1
/
q
,
1
−
u
2
q
λ
2
−
1
−
u
2
q
−
v
2
q
λ
2
1
/
q
=
1
−
1
−
1
−
1
−
u
1
q
λ
1
1
/
q
q
1
−
1
−
1
−
u
1
q
λ
2
1
/
q
q
1
/
q
,
1
−
1
−
1
−
u
1
q
λ
1
1
/
q
q
1
−
1
−
1
−
u
1
q
λ
2
1
/
q
q
−
1
−
1
−
1
−
u
1
q
λ
1
1
/
q
q
−
1
−
u
1
q
λ
1
−
1
−
u
1
q
−
v
1
q
λ
1
/
q
1
−
1
−
1
−
u
1
q
λ
2
1
/
q
q
−
1
−
u
1
q
λ
2
−
1
−
u
1
q
−
v
1
q
λ
2
1
/
q
1
/
q
,
=
1
−
1
−
u
1
q
λ
1
+
λ
2
1
/
q
,
1
−
u
1
q
λ
1
+
λ
2
−
1
−
u
1
q
−
v
1
q
λ
1
+
λ
2
1
/
q
,
and for the left-hand side of equation (21), we have
(28)
λ
1
+
λ
2
α
1
=
1
−
1
−
u
1
q
λ
1
+
λ
2
1
/
q
,
1
−
u
1
q
λ
1
+
λ
2
−
1
−
u
1
q
−
v
1
q
λ
1
+
λ
2
1
/
q
.
Thus, equation (21) is kept.
(7)
The proof of equation (20) is similar to that of equation (21), so it is omitted.Example 3.
Letα1 = (0.6, 0) and α2 = (0.5, 0.3) be two q-ROFN, λ=3,andq=3, then we can get(29)α1⊕α2=1−1−0.63×1−0.531/3,1−0.63×1−0.53−1−0.63−03×1−0.53−0.331/3=0.6797,0.4707,α1⊗α2=1−03×1−0.33−1−0.63−03×1−0.53−0.331/3,1−1−03×1−0.331/3=0.5288,0.3000,λα1=1−1−0.6331/3,1−0.633−1−0.63−0331/3=0.8032,0,α1λ=1−033−1−0.63−0331/3,1−1−0331/3=0.8032,0.


## 5. Improvedq-ROFS Aggregation Operators for MADM Problems

### 5.1. Improvedq-Rung Orthopair Fuzzy-Weighted Averaging Operator

In the section, we will introduce and analyze theq-IROFWA operator based on the new ORs in Section 3.Theorem 2.
Letαk=uk,vkk=1,2,…,n be a collection of q-ROFN (q≥1) and w=w1,w2,…,wnT be the corresponding WV, and meet 0≤Wk≤1 and ∑k=1nwk=1. Then, the aggregation result is still a q-ROFN and has(30)q‐IROFWAα1,α2,…,αn=1−∏k=1n1−ukqwk1/q,∏k=1n1−ukqwk−∏k=1n1−ukq−vkqwk1/q.Proof of Theorem 2.
Using mathematical induction, it can be proved as follows:(1)
Whenn = 1, for the left-hand side of equation (30), we obtain(31)q‐IROFWAα1=w1α1=1−1−u1qw11/q,1−u1qw1−1−u1q−v1qw11/q.For the right-hand side of equation (30), we have(32)1−∏k=111−ukqwk1/q,∏k=111−ukqwk−∏k=111−ukq−vkqwk1/q=1−1−u1qw11/q,1−u1qw1−1−u1q−v1qw11/q.So, whenn = 1, equation (30) is kept.(2)
Whenn = 2,(33)q‐IROFWAα1,α2=w1α1⊕w2α2=1−1−u1qw11/q,1−u1qw1−1−u1q−v1qw11/q⊕1−1−u2qw21/q,1−u2qw2−1−u2q−v2qw21/q=1−1−u1qw11−u2qw2,1−u1qw11−u2qw2−1−u1q−v1qw11−u2q−v2qw2=1−∏k=121−ukqwk1/q,∏k=121−ukqwk−∏k=121−ukq−vkqwk1/q.So, whenn = 2, equation (30) is kept.(3)
Supposen=m, equation (30) is kept, that is,(34)q‐IROFWAα1,α2,…,αm=1−∏k=1m1−ukqwk1/q,∏k=1m1−ukqwk−∏k=1m1−ukq−vkqwk1/q.Then, whenn=m+1, we have(35)q‐IROFWAα1,α2,…,αm,αm+1=q‐IROFWAα1,α2,…,αm⊕wm+1αm+1=1−∏k=1m1−ukqwk1/q,∏k=1m1−ukqwk−∏k=1m1−ukq−vkqwk1/q⊕1−1−um+1qwm+11/q,1−um+1qwm+1−1−um+1q−vm+1qwm+11/q=1−1−1−∏k=1m1−ukqwk1/qq1−1−1−um+1qwm+11/qq1/q,1−1−∏k=1m1−ukqwk1/qq1−1−1−um+1qwm+11/qq−1−1−∏k=1m1−ukqwk1/qq−∏k=1m1−ukqwk−∏k=1m1−ukq−vkqwk1/q1−1−1−um+1qwm+11/qq−1−um+1qwm+1−1−um+1q−vm+1qwm+11/q1/q=1−∏k=1m1−ukqwk1−um+1qwm+11/q,∏k=1m1−ukqwk1−um+1qwm+1−∏k=1m1−ukq−vkqwk1−um+1q−vm+1qwm+11/q=1−∏k=1m+11−ukqwm+11/q,∏k=1m+11−ukqwm+1−∏k=1m+11−ukq−vkqwm+11/q.
So, whenn=m+1, equation (30) is kept.
Therefore, according to steps 1–3, equation (30) is kept for any k.
In the following, we will prove thatq‐IROFWAα1,α2,…,αm is still a q‐ROFN.
Letq‐IROFWAα1,α2,…,αm=a,b, then if q‐IROFWA is a q‐ROFN, it meets the following two conditions:(1)
0≤a,b≤1(2)
0≤aq+bq≤1(1)
Because0≤uk,vk≤1, we have(36)0≤1−ukq≤1⟹0≤1−ukqwk≤1⟹0≤∏k=1m1−ukqwk≤1⟹0≤1−∏k=1m1−ukqwk1/q≤1.So, we get0≤a≤1.In addition, since0≤ukq+vkq≤1, we have(37)0≤1−ukq+vkq≤1⟹0≤1−ukq−vkqwk≤1⟹0≤∏k=1m1−ukq−vkqwk≤1.Moreover, because1−ukq≥1−ukq−vkq, we have ∏k=1m1−ukqwk≥∏k=1m1−ukq−vkqwk, so 0≤∏k=1m1−ukqwk−∏k=1m1−ukq−vkqwk1/q≤1, and we get 0≤b≤1.Therefore, condition (1) is proved.(2)
Since0≤aq+bq=1−∏k=111−ukqwk+∏k=111−ukqwk−∏k=1m1−ukq−vkqwk⟹ 0≤aq+bq=1−∏k=1m1−ukq−vkqwk, we have 0≤∏k=1m1−ukq−vkqwk≤1⟹0≤1−∏k=1m1−ukq−vkqwk≤1 Therefore, condition (2) is proved.Thus,q‐IROFWAα1,α2,…,αm is still a q‐ROFN.Theorem 3 (idempotency).
Supposeαi=ui,vii=1,2,…,n is a collection of q-ROFNs, if αi=α=u,vi=1,2,…,n, then q‐IROFWAα1,α2,…,αm=αProof of Theorem 3.
(38)
q
‐
IROFWA
α
1
,
α
2
,
…
,
α
n
=
1
−
∏
k
=
1
n
1
−
u
k
q
w
k
1
/
q
,
∏
k
=
1
n
1
−
u
k
q
w
k
−
∏
k
=
1
n
1
−
u
k
q
−
v
k
q
w
k
1
/
q
=
1
−
∏
k
=
1
n
1
−
u
q
w
k
1
/
q
,
∏
k
=
1
n
1
−
u
q
w
k
−
∏
k
=
1
n
1
−
u
q
−
v
q
w
k
1
/
q
=
1
−
1
−
u
q
∑
k
=
1
n
w
k
1
/
q
,
1
−
u
q
∑
k
=
1
n
w
k
−
1
−
u
q
−
v
q
∑
k
=
1
n
w
k
1
/
q
=
u
,
v
.Theorem 4 (boundedness).
Supposeαi=ui,vii=1,2,…,n is a collection of q-ROFNs and α−=miniuiq,maxiuiq+viq−miniuiq1/q and α+=maxiuiq,G, then we have α−≤q‐IROFWAα1,α2,…,αn≤α+, where(39)G=0,ifminiuiq+viq≤maxiuiq,miniuiq+viq−maxiuiq1/q,ifminiuiq+viq>maxiuiq.Proof of Theorem 4.
(1)
For the MD ofq‐IROFWAα1,α2,…,αm=α, we have
(40)
1
−
∏
i
=
1
n
1
−
min
1
≤
i
≤
n
u
i
q
w
i
1
/
q
≤
1
−
∏
i
=
1
n
1
−
u
i
q
w
i
1
/
q
≤
1
−
∏
i
=
1
n
1
−
max
1
≤
i
≤
n
u
i
q
w
i
1
/
q
⟹
1
−
1
−
min
1
≤
i
≤
n
u
i
q
∑
i
=
1
n
w
i
1
/
q
≤
1
−
∏
i
=
1
n
1
−
u
i
q
w
i
1
/
q
≤
1
−
1
−
max
1
≤
i
≤
n
u
i
q
∑
i
=
1
n
w
i
1
/
q
⟹
min
1
≤
i
≤
n
u
i
q
≤
1
−
∏
i
=
1
n
1
−
u
i
q
w
i
1
/
q
max
1
≤
i
≤
n
u
i
q
.
(2)
For the NMD, we have
(41)
∏
i
=
1
n
1
−
max
1
≤
i
≤
n
u
i
q
w
i
−
∏
i
=
1
n
1
−
min
1
≤
i
≤
n
u
i
q
+
v
i
q
w
i
1
/
q
≤
∏
i
=
1
n
1
−
u
i
q
w
i
−
∏
i
=
1
n
1
−
u
i
q
−
v
i
q
w
i
1
/
q
≤
∏
i
=
1
n
1
−
min
1
≤
i
≤
n
u
i
q
w
k
−
∏
i
=
1
n
1
−
max
1
≤
i
≤
n
u
i
q
+
v
i
q
w
i
1
/
q
⟹
min
1
≤
i
≤
n
u
i
q
+
v
i
q
−
max
1
≤
i
≤
n
u
i
q
1
/
q
≤
∏
i
=
1
n
1
−
u
i
q
w
i
−
∏
i
=
1
n
1
−
u
i
q
−
v
i
q
w
i
1
/
q
≤
max
1
≤
i
≤
n
u
i
q
+
v
i
q
−
min
1
≤
i
≤
n
u
i
q
1
/
q
.
Because(42)∏i=1n1−uiqwi−∏i=1n1−uiq−viqwi1/q≥0,
then(43)G≤∏i=1n1−uiqwi−∏i=1n1−uiq−viqwi1/q≤max1≤i≤nuiq+viq−min1≤i≤nuiq1/q.Example 4.
For Example 2, we have(44)q‐IROFWAα1,α2,α3,α4=1−∏k=141−ukqwk1/q,∏k=141−ukqwk−∏k=141−ukq−vkqwk1/q=1−1−0.630.3×1−0.530.2×1−0.730.1×1−0.330.41/3,1−0.630.3×1−0.530.2×1−0.730.1×1−0.33−1−0.63−0.330.3×1−0.53−0.330.2×1−0.73−0.230.1×1−0.33−0.531/3=0.521,0.3795.
When we change the value ofα3 and α4 to α3=0.8,0.2 and α4=0.5,0.5, then the aggregated result is (0.5861, 0.4428).
Obviously, as the membership degrees of the aggregated parameters are changed, then the MD and NMD of the aggregated result are also changed simultaneously.


### 5.2. Improvedq-Rung Orthopair Fuzzy Power Averaging Operator

Definition 7.
Letαk=uk,vkk=1,2,…,n be a set of q-ROFN (q≥1), then q‐IROFPA is defined as follows:(45)q‐IROFPAα1,α2,…,αn=⊕i=1n1+Tαiαi∑i=1n1+Tαi,where(46)Tαj=∑i=1,i≠jnSupαj,αi.
T
α
i expresses the support degree of jth ROFN from all the other ROFNS, the closer the value, the more they support each other.Definition 8.
Letα1=μ1,ν1 and α2=μ2,ν2 be any two q-ROFNS (q≥1). Then, the generalized hesitance degree-preference distance of α1 and α2 is defined as follows:(47)dα1,α2=pμ1q−μ2q+ν1q−ν2q+1−pπ1q−π2q2,where p∈0,1,d∈0,1,πiq=1−μiq−νiqi=1,2.
For equation (47), we can get some different distance measures when setting different parameter values of p.Case 1.
Whenp=1, the generalized hesitance degree-preference distance will reduce to the Hamming degree-preference distance:(48)dα1,α2=μ1q−μ2q+ν1q−ν2q2.Case 2.
Whenp=0.5, the generalized hesitance degree-preference distance will reduce to the Hamming-indeterminacy degree-preference distance. Its normalization form is defined by(49)dα1,α2=μ1q−μ2q+ν1q−ν2q+π1q−π2q2.Theorem 5.
Letαk=uk,vkk=1,2,…,n be a set of q-ROFN (q≥1), then the aggregation result is still a q-ROFN and has(50)q‐IROFWAα1,α2,…,αn=1−∏k=1n1−ukq1+Tαiαi/∑i=1n1+Tαi1/q,∏k=1n1−ukq1+Tαiαi/∑i=1n1+Tαi−∏k=1n1−ukq−vkq1+Tαiαi/∑i=1n1+Tαi1/q.Proof of Theorem 5.
Letwi=1+Tαiαi/∑i=1n1+Tαi, then the proof is similar to the proof of Theorem 2, so it is omitted here.
Obviously, the weighted vector is not considered in Definition8. In many actual decision cases, the weights of attributes can affect the result. Therefore, we need to propose the q-IROFWPA operator as follows.Definition 9.
Letαk=uk,vkk=1,2,…,n be a set of q-ROFN (q≥1), then q‐IROFWPA is defined as follows:(51)q‐IROFWPAα1,α2,…,αn=⊕i=1nwk1+Tαkαk∑k=1nwk1+Tαk.
In order to simplify the proof, we letδk=⊕i=1nwk1+Tαkαk/∑k=1nwk1+Tαk,k=1,2,…,n and w=w1,w2,…,wnT be the weight vector of αkk=1,2,…,n, such that 0≤wk≤1 and ∑k=1nwk=1. Then, equation (51) has the following expression:(52)q‐IROFWPAα1,α2,…,αn=⊕i=1nwiαi.Theorem 6.
Supposeαk=uk,vkk=1,2,…,n be a set of q-ROFN (q≥1), then the aggregation result from Definition 8 is still a q-ROFN:(53)q‐IROFWPAα1,α2,…,αn=1−∏k=1n1−ukqδk1/q,∏k=1n1−ukqδk−∏k=1n1−ukq−vkqδk1/q.Proof of Theorem 6.
The proof is similar to Theorem2, and it can be omitted here.Theorem 7 (idempotency).
Letαi=ui,vii=1,2,…,n is a collection of q-ROFNs, if αi=α=u,vi=1,2,…,n, then q‐IROFWPAα1,α2,…,αm=α.Proof of Theorem 7.
The proof is similar to Theorem3, and it can be omitted here.Theorem 8 (boundedness).
Letα−=miniuiq,maxiuiq+viq−miniuiq1/q and α+=maxiuiq,G, then we have α−≤q‐IROFWPAα1,α2,…,αn≤α+, where G is consistent with the definition in Theorem 4.Proof of Theorem 8.
The proof is similar to Theorem4, and it can be omitted here.


## 5.1. Improvedq-Rung Orthopair Fuzzy-Weighted Averaging Operator

In the section, we will introduce and analyze theq-IROFWA operator based on the new ORs in Section 3.Theorem 2.
Letαk=uk,vkk=1,2,…,n be a collection of q-ROFN (q≥1) and w=w1,w2,…,wnT be the corresponding WV, and meet 0≤Wk≤1 and ∑k=1nwk=1. Then, the aggregation result is still a q-ROFN and has(30)q‐IROFWAα1,α2,…,αn=1−∏k=1n1−ukqwk1/q,∏k=1n1−ukqwk−∏k=1n1−ukq−vkqwk1/q.Proof of Theorem 2.
Using mathematical induction, it can be proved as follows:(1)
Whenn = 1, for the left-hand side of equation (30), we obtain(31)q‐IROFWAα1=w1α1=1−1−u1qw11/q,1−u1qw1−1−u1q−v1qw11/q.For the right-hand side of equation (30), we have(32)1−∏k=111−ukqwk1/q,∏k=111−ukqwk−∏k=111−ukq−vkqwk1/q=1−1−u1qw11/q,1−u1qw1−1−u1q−v1qw11/q.So, whenn = 1, equation (30) is kept.(2)
Whenn = 2,(33)q‐IROFWAα1,α2=w1α1⊕w2α2=1−1−u1qw11/q,1−u1qw1−1−u1q−v1qw11/q⊕1−1−u2qw21/q,1−u2qw2−1−u2q−v2qw21/q=1−1−u1qw11−u2qw2,1−u1qw11−u2qw2−1−u1q−v1qw11−u2q−v2qw2=1−∏k=121−ukqwk1/q,∏k=121−ukqwk−∏k=121−ukq−vkqwk1/q.So, whenn = 2, equation (30) is kept.(3)
Supposen=m, equation (30) is kept, that is,(34)q‐IROFWAα1,α2,…,αm=1−∏k=1m1−ukqwk1/q,∏k=1m1−ukqwk−∏k=1m1−ukq−vkqwk1/q.Then, whenn=m+1, we have(35)q‐IROFWAα1,α2,…,αm,αm+1=q‐IROFWAα1,α2,…,αm⊕wm+1αm+1=1−∏k=1m1−ukqwk1/q,∏k=1m1−ukqwk−∏k=1m1−ukq−vkqwk1/q⊕1−1−um+1qwm+11/q,1−um+1qwm+1−1−um+1q−vm+1qwm+11/q=1−1−1−∏k=1m1−ukqwk1/qq1−1−1−um+1qwm+11/qq1/q,1−1−∏k=1m1−ukqwk1/qq1−1−1−um+1qwm+11/qq−1−1−∏k=1m1−ukqwk1/qq−∏k=1m1−ukqwk−∏k=1m1−ukq−vkqwk1/q1−1−1−um+1qwm+11/qq−1−um+1qwm+1−1−um+1q−vm+1qwm+11/q1/q=1−∏k=1m1−ukqwk1−um+1qwm+11/q,∏k=1m1−ukqwk1−um+1qwm+1−∏k=1m1−ukq−vkqwk1−um+1q−vm+1qwm+11/q=1−∏k=1m+11−ukqwm+11/q,∏k=1m+11−ukqwm+1−∏k=1m+11−ukq−vkqwm+11/q.
So, whenn=m+1, equation (30) is kept.
Therefore, according to steps 1–3, equation (30) is kept for any k.
In the following, we will prove thatq‐IROFWAα1,α2,…,αm is still a q‐ROFN.
Letq‐IROFWAα1,α2,…,αm=a,b, then if q‐IROFWA is a q‐ROFN, it meets the following two conditions:(1)
0≤a,b≤1(2)
0≤aq+bq≤1(1)
Because0≤uk,vk≤1, we have(36)0≤1−ukq≤1⟹0≤1−ukqwk≤1⟹0≤∏k=1m1−ukqwk≤1⟹0≤1−∏k=1m1−ukqwk1/q≤1.So, we get0≤a≤1.In addition, since0≤ukq+vkq≤1, we have(37)0≤1−ukq+vkq≤1⟹0≤1−ukq−vkqwk≤1⟹0≤∏k=1m1−ukq−vkqwk≤1.Moreover, because1−ukq≥1−ukq−vkq, we have ∏k=1m1−ukqwk≥∏k=1m1−ukq−vkqwk, so 0≤∏k=1m1−ukqwk−∏k=1m1−ukq−vkqwk1/q≤1, and we get 0≤b≤1.Therefore, condition (1) is proved.(2)
Since0≤aq+bq=1−∏k=111−ukqwk+∏k=111−ukqwk−∏k=1m1−ukq−vkqwk⟹ 0≤aq+bq=1−∏k=1m1−ukq−vkqwk, we have 0≤∏k=1m1−ukq−vkqwk≤1⟹0≤1−∏k=1m1−ukq−vkqwk≤1 Therefore, condition (2) is proved.Thus,q‐IROFWAα1,α2,…,αm is still a q‐ROFN.Theorem 3 (idempotency).
Supposeαi=ui,vii=1,2,…,n is a collection of q-ROFNs, if αi=α=u,vi=1,2,…,n, then q‐IROFWAα1,α2,…,αm=αProof of Theorem 3.
(38)
q
‐
IROFWA
α
1
,
α
2
,
…
,
α
n
=
1
−
∏
k
=
1
n
1
−
u
k
q
w
k
1
/
q
,
∏
k
=
1
n
1
−
u
k
q
w
k
−
∏
k
=
1
n
1
−
u
k
q
−
v
k
q
w
k
1
/
q
=
1
−
∏
k
=
1
n
1
−
u
q
w
k
1
/
q
,
∏
k
=
1
n
1
−
u
q
w
k
−
∏
k
=
1
n
1
−
u
q
−
v
q
w
k
1
/
q
=
1
−
1
−
u
q
∑
k
=
1
n
w
k
1
/
q
,
1
−
u
q
∑
k
=
1
n
w
k
−
1
−
u
q
−
v
q
∑
k
=
1
n
w
k
1
/
q
=
u
,
v
.Theorem 4 (boundedness).
Supposeαi=ui,vii=1,2,…,n is a collection of q-ROFNs and α−=miniuiq,maxiuiq+viq−miniuiq1/q and α+=maxiuiq,G, then we have α−≤q‐IROFWAα1,α2,…,αn≤α+, where(39)G=0,ifminiuiq+viq≤maxiuiq,miniuiq+viq−maxiuiq1/q,ifminiuiq+viq>maxiuiq.Proof of Theorem 4.
(1)
For the MD ofq‐IROFWAα1,α2,…,αm=α, we have
(40)
1
−
∏
i
=
1
n
1
−
min
1
≤
i
≤
n
u
i
q
w
i
1
/
q
≤
1
−
∏
i
=
1
n
1
−
u
i
q
w
i
1
/
q
≤
1
−
∏
i
=
1
n
1
−
max
1
≤
i
≤
n
u
i
q
w
i
1
/
q
⟹
1
−
1
−
min
1
≤
i
≤
n
u
i
q
∑
i
=
1
n
w
i
1
/
q
≤
1
−
∏
i
=
1
n
1
−
u
i
q
w
i
1
/
q
≤
1
−
1
−
max
1
≤
i
≤
n
u
i
q
∑
i
=
1
n
w
i
1
/
q
⟹
min
1
≤
i
≤
n
u
i
q
≤
1
−
∏
i
=
1
n
1
−
u
i
q
w
i
1
/
q
max
1
≤
i
≤
n
u
i
q
.
(2)
For the NMD, we have
(41)
∏
i
=
1
n
1
−
max
1
≤
i
≤
n
u
i
q
w
i
−
∏
i
=
1
n
1
−
min
1
≤
i
≤
n
u
i
q
+
v
i
q
w
i
1
/
q
≤
∏
i
=
1
n
1
−
u
i
q
w
i
−
∏
i
=
1
n
1
−
u
i
q
−
v
i
q
w
i
1
/
q
≤
∏
i
=
1
n
1
−
min
1
≤
i
≤
n
u
i
q
w
k
−
∏
i
=
1
n
1
−
max
1
≤
i
≤
n
u
i
q
+
v
i
q
w
i
1
/
q
⟹
min
1
≤
i
≤
n
u
i
q
+
v
i
q
−
max
1
≤
i
≤
n
u
i
q
1
/
q
≤
∏
i
=
1
n
1
−
u
i
q
w
i
−
∏
i
=
1
n
1
−
u
i
q
−
v
i
q
w
i
1
/
q
≤
max
1
≤
i
≤
n
u
i
q
+
v
i
q
−
min
1
≤
i
≤
n
u
i
q
1
/
q
.
Because(42)∏i=1n1−uiqwi−∏i=1n1−uiq−viqwi1/q≥0,
then(43)G≤∏i=1n1−uiqwi−∏i=1n1−uiq−viqwi1/q≤max1≤i≤nuiq+viq−min1≤i≤nuiq1/q.Example 4.
For Example 2, we have(44)q‐IROFWAα1,α2,α3,α4=1−∏k=141−ukqwk1/q,∏k=141−ukqwk−∏k=141−ukq−vkqwk1/q=1−1−0.630.3×1−0.530.2×1−0.730.1×1−0.330.41/3,1−0.630.3×1−0.530.2×1−0.730.1×1−0.33−1−0.63−0.330.3×1−0.53−0.330.2×1−0.73−0.230.1×1−0.33−0.531/3=0.521,0.3795.
When we change the value ofα3 and α4 to α3=0.8,0.2 and α4=0.5,0.5, then the aggregated result is (0.5861, 0.4428).
Obviously, as the membership degrees of the aggregated parameters are changed, then the MD and NMD of the aggregated result are also changed simultaneously.


## 5.2. Improvedq-Rung Orthopair Fuzzy Power Averaging Operator

Definition 7.
Letαk=uk,vkk=1,2,…,n be a set of q-ROFN (q≥1), then q‐IROFPA is defined as follows:(45)q‐IROFPAα1,α2,…,αn=⊕i=1n1+Tαiαi∑i=1n1+Tαi,where(46)Tαj=∑i=1,i≠jnSupαj,αi.
T
α
i expresses the support degree of jth ROFN from all the other ROFNS, the closer the value, the more they support each other.Definition 8.
Letα1=μ1,ν1 and α2=μ2,ν2 be any two q-ROFNS (q≥1). Then, the generalized hesitance degree-preference distance of α1 and α2 is defined as follows:(47)dα1,α2=pμ1q−μ2q+ν1q−ν2q+1−pπ1q−π2q2,where p∈0,1,d∈0,1,πiq=1−μiq−νiqi=1,2.
For equation (47), we can get some different distance measures when setting different parameter values of p.Case 1.
Whenp=1, the generalized hesitance degree-preference distance will reduce to the Hamming degree-preference distance:(48)dα1,α2=μ1q−μ2q+ν1q−ν2q2.Case 2.
Whenp=0.5, the generalized hesitance degree-preference distance will reduce to the Hamming-indeterminacy degree-preference distance. Its normalization form is defined by(49)dα1,α2=μ1q−μ2q+ν1q−ν2q+π1q−π2q2.Theorem 5.
Letαk=uk,vkk=1,2,…,n be a set of q-ROFN (q≥1), then the aggregation result is still a q-ROFN and has(50)q‐IROFWAα1,α2,…,αn=1−∏k=1n1−ukq1+Tαiαi/∑i=1n1+Tαi1/q,∏k=1n1−ukq1+Tαiαi/∑i=1n1+Tαi−∏k=1n1−ukq−vkq1+Tαiαi/∑i=1n1+Tαi1/q.Proof of Theorem 5.
Letwi=1+Tαiαi/∑i=1n1+Tαi, then the proof is similar to the proof of Theorem 2, so it is omitted here.
Obviously, the weighted vector is not considered in Definition8. In many actual decision cases, the weights of attributes can affect the result. Therefore, we need to propose the q-IROFWPA operator as follows.Definition 9.
Letαk=uk,vkk=1,2,…,n be a set of q-ROFN (q≥1), then q‐IROFWPA is defined as follows:(51)q‐IROFWPAα1,α2,…,αn=⊕i=1nwk1+Tαkαk∑k=1nwk1+Tαk.
In order to simplify the proof, we letδk=⊕i=1nwk1+Tαkαk/∑k=1nwk1+Tαk,k=1,2,…,n and w=w1,w2,…,wnT be the weight vector of αkk=1,2,…,n, such that 0≤wk≤1 and ∑k=1nwk=1. Then, equation (51) has the following expression:(52)q‐IROFWPAα1,α2,…,αn=⊕i=1nwiαi.Theorem 6.
Supposeαk=uk,vkk=1,2,…,n be a set of q-ROFN (q≥1), then the aggregation result from Definition 8 is still a q-ROFN:(53)q‐IROFWPAα1,α2,…,αn=1−∏k=1n1−ukqδk1/q,∏k=1n1−ukqδk−∏k=1n1−ukq−vkqδk1/q.Proof of Theorem 6.
The proof is similar to Theorem2, and it can be omitted here.Theorem 7 (idempotency).
Letαi=ui,vii=1,2,…,n is a collection of q-ROFNs, if αi=α=u,vi=1,2,…,n, then q‐IROFWPAα1,α2,…,αm=α.Proof of Theorem 7.
The proof is similar to Theorem3, and it can be omitted here.Theorem 8 (boundedness).
Letα−=miniuiq,maxiuiq+viq−miniuiq1/q and α+=maxiuiq,G, then we have α−≤q‐IROFWPAα1,α2,…,αn≤α+, where G is consistent with the definition in Theorem 4.Proof of Theorem 8.
The proof is similar to Theorem4, and it can be omitted here.


## 6. Novel MAGDM Method Based on the Proposed Operators

In this section, based on theq-IROFWA operator and the q-IROFWPA operator, we proposed a novel method to deal with MAGDM problems. Let D=D1,D2,…,Dp be the set of experts (decision makers), A=A1,A2,…,Am be all alternatives, and C=C1,C2,…,Dn be all attributes. αijk=uijk,vijkk=1,2,…,p represents the value of attribute Cj from alternative Ai given by the expert Dk. Let Dk=Dijkm×n be the decision matrix from experts Dk. Let w=w1,w2,…,wp be the WV of experts, wk∈0,1, and ∑k=1pwk=1. Then, the details of the novel method are described as follows:Step 1: normalize the decision matrix.In MAGDM problem, the attribute values may have different types, such as benefit and cost. Since different types of attributes may be neutralized during the aggregating process, it needs to convert different attribute types into the same. Owing to cognitive habits, cost-based attributes are usually transformed into benefit ones by the following formula:(54)αijk=uijk,vijk=uijk,vijk,for benefit attributeAj,vijk,uijk,for cost  attributeAj.Step 2: aggregate all attribute valuesαijki=1,cdots,m;j=1,cdots,n;k=1,cdots,p; to a comprehensive value Zik by the q-IROFWA operator as follows:(55)Zik=q‐IROFWAαi1k,αi2k,…,αink=1−∏j=1n1−uijkqwk1/q,∏j=1n1−uijkqwk−∏j=1n1−uijkq−vijkqwk1/q.Step 3: calculate the supports:(56)SupZik,Zit=1−dZik,Zit,k,t=1,2,…,p;i=1,…,m.Step 4: calculate the supportsTZik and weights wik:(57)TZik=∑t=1,k≠tpSupZik,Zit,k=1,2,…,p;i=1,2,…,m,(58)wik=λk1+TZik∑k=1pλk1+TZik,wherewik≥0 and ∑k=1pwik=1Step 5: use theq-IROFWPA operator expressed by equation (53) to get the collective preference values Zi:(59)Zi=q‐IROFWPAαik,αik,…,αik=1−∏k=1p1−uikqwik1/q,∏k=1p1−uikqwik−∏k=1p1−uikq−vikqwik1/q.Step 6: use score function and accuracy function to rankZii=1,…,m.Step 7: select the best alternative based on the value ofZii=1,…,m.The flow chart of the novel MAGDM method is shown in Figure2.Figure 2
Flow chart of MAGDM.


## 7. Case Study

In this section, an example is used to illustrate the application of the method in detail, and then some examples are further adopted to verify the effectiveness and superiority of the proposed method with some existing MAGDM method.


### 7.1. The Application of the Proposed Method

Example 5.
This example is from [39]. There are five alternative companies Aii=1,2,…,5 which are evaluated by three experts Dkk=1,2,3 with respect to four attributes: the enterprise management level C1, the business growth ability C2, the economic benefit C3, and the corporate reputation C4, respectively. Di be the decision matrix obtained from expert Di by using q‐ROFN, as shown in Table 1.
Letλ=0.35,0.40,0.25 be WV of the three experts and w=0.2,0.1,0.3,0.4 be WV of the attributes. The detail of the proposed MAGDM method to select the best company is as below.
To further verify the effect of parameterq on the results, experiments were carried out using different q. The ranking results are shown in Table 3.
As can be seen from Table3 that although the aggregation result will change with the parameter q, the optimal result remains unchanged. Further, we can see that the value of the score function becomes smaller as q increases, so q is better in the interval 2 to 5.Table 1
Decision matrix expressed byq-ROFs.
C
1
C
2
C
3
C
4
D
1
A
1
(0.2, 0.5)
(0.4, 0.6)
(0.4, 0.3)
(0.5, 0.2)
A
2
(0.6, 0.3)
(0.4, 0.2)
(0.5, 0.1)
(0.5, 0.2)
A
3
(0.4, 0.2)
(0.5, 0.2)
(0.5, 0.3)
(0.4, 0.3)
A
4
(0.5, 0.3
(0.3, 0.5)
(0.5, 0.2)
(0.5, 0.2
A
5
(0.6, 0.1)
(0.4, 0.4)
(0.6, 0.2)
(0.6, 0.1)
D
2
A
1
(0.3, 0.6)
(0.2, 0.7)
(0.5, 0.4)
(0.5, 0.3
A
2
(0.5, 0.1)
(0.5, 0.3)
(0.6, 0.2)
(0.4, 0.3)
A
3
(0.5, 0.3)
(0.6, 0.2)
(0.6, 0.1)
(0.5, 0.4)
A
4
(0.5, 0.4)
(0.2, 0.6)
(0.5, 0.1)
(0.6, 0.3)
A
1
(0.2, 0.7)
(0.5, 0.4)
(0.4, 0.5)
(0.2, 0.6)
D
3
A
2
(0.5, 0.2)
(0.6, 0.4)
(0.5, 0.3)
(0.4, 0.1)
A
3
(0.6, 0.3)
(0.5, 0.2)
(0.5, 0.3)
(0.5, 0.4)
A
4
(0.4, 0.5)
(0.5, 0.4)
(0.6, 0.2)
(0.5, 0.2)
A
5
(0.6, 0.3)
(0.5, 0.1)
(0.6, 0.4)
(0.6, 0.1)Step 1: normalize the decision matricesD1, D2, and D3. In this example, the type of the four attributes is the same, so it can be omitted.
Step 2: according to theq‐IROFWA operator in equation (55), all the attributes matrices Dijk are aggregated into the integrated matrices Zik, as shown in Table 2.Table 2
Integrated decision matrixZik.
Z
1
Z
2
Z
3
Z
4
Z
5
Z
1
(0.4277, 0.3712)
(0.5170, 0.3630)
(0.4463, 0.3243)
(0.4872, 0.3661)
(0.5867, 0.4206)
Z
2
(0.4562, 0.3866)
(0.5069, 0.3194)
(0.5458, 0.4068)
(0.5337, 0.4471)
(0.6124, 0.4204)
Z
3
(0.3341, 0.4528)
(0.4805, 0.2868)
(0.5242, 0.4098)
(0.5215, 0.3625)
(0.5919, 0.4196)Step 3: according to equation (56), we can get the supports Sikt=SupZik,Zit as follows:(60)
S
1
12
=
S
1
21
=
0.9883
,
S
1
13
=
S
1
31
=
0.9791
,
S
1
23
=
S
1
32
=
0.9712
,
S
2
12
=
S
2
21
=
0.9884
,
S
2
13
=
S
2
31
=
0.9742
,
S
2
23
=
S
2
32
=
0.9859
,
S
3
12
=
S
3
21
=
0.9465
,
S
3
13
=
S
3
31
=
0.9551
,
S
3
23
=
S
3
32
=
0.9907
,
S
4
12
=
S
4
21
=
0.9617
,
S
4
13
=
S
4
31
=
0.9869
,
S
4
23
=
S
4
32
=
0.9740
,
S
5
12
=
S
5
21
=
0.9861
,
S
5
13
=
S
5
31
=
0.9973
,
S
5
23
=
S
5
32
=
0.9886.Step 4: according to equations (57) and (58), we can get the supports Tik=TZik and the weights wik as follows:(61)
T
1
k
=
∑
t
=
1
,
t
≠
1
3
Sup
Z
1
1
,
Z
1
t
=
Sup
Z
1
1
,
Z
1
2
+
Sup
Z
1
1
,
Z
1
3
=
0.9883
+
0.9791
=
1.9674
,
T
1
2
=
1.9595
,
T
1
3
=
1.9503
,
T
2
1
=
1.9626
,
T
2
2
=
1.9742
,
T
2
3
=
1.9601
,
T
3
1
=
1.9016
,
T
3
2
=
1.9373
,
T
3
3
=
1.9458
,
T
4
1
=
1.9486
,
T
4
2
=
1.9357
,
T
4
3
=
1.9609
,
T
5
1
=
1.9835
,
T
5
2
=
1.9748
,
T
5
3
=
1.9859
,
ω
1
1
=
λ
1
1
+
T
1
1
∑
k
=
1
3
λ
k
1
+
T
1
k
=
0.3509.Similarly, we have(62)
ω
1
2
=
0.3495
,
ω
1
3
=
0.3470
,
ω
1
4
=
0.3502
,
ω
1
5
=
0.3503
,
ω
2
1
=
0.3999
,
ω
2
2
=
0.4010
,
ω
2
3
=
0.4014
,
ω
2
4
=
0.3985
,
ω
2
5
=
0.3992
,
ω
3
1
=
0.2492
,
ω
3
2
=
0.2494
,
ω
3
3
=
0.2516
,
ω
3
4
=
0.2512
,
ω
3
5
=
0.2504.Step 5: according to theq‐IROFPWA operator in equation (59), we can get the collective preference values Zi as follows:(63)
Z
1
=
0.4217
,
0.3161
,
Z
2
=
0.5043
,
0.3136
,
Z
3
=
0.5104
,
0.3700
,
Z
4
=
0.5154
,
0.3540
,
Z
5
=
0.5986
,
0.3965
.Step 6: obtain the comprehensive evaluation valueZi by the score function of equation (4) as follows:(64)
L
s
Z
1
=
0.0434
,
L
s
Z
2
=
0.0974
,
L
s
Z
3
=
0.0823
,
L
s
Z
4
=
0.0925
,
L
s
Z
5
=
0.1521.Step 7: sort the calculation results of the score function, and we obtainA5≻A2≻A4≻A3≻A1, so A5 is the best choice.Table 3
Ranking results for differentq.
q
The score function
Preference order
2
S
Z
1 = 0.1017, SZ2 = 0.1876, SZ3 = 0.1576, SZ4 = 0.1757, SZ5 = 0.2486
A
5
≻
A
2
≻
A
4
≻
A
3
≻
A
1
3
S
Z
1 = 0.0434, SZ2 = 0.0974, SZ3 = 0.0823, SZ4 = 0.0925, SZ5 = 0.1521
A
5
≻
A
2
≻
A
4
≻
A
3
≻
A
1
5
S
Z
1 = 0.0082, SZ2 = 0.0260, SZ3 = 0.0230, SZ4 = 0.0256, SZ5 = 0.0565
A
5
≻
A
2
≻
A
4
≻
A
3
≻
A
1
8
S
Z
1 = 0.0007, SZ2 = 0.0035, SZ3 = 0.0035, SZ4 = 0.0037, SZ5 = 0.0126
A
5
≻
A
4
≻
A
3
≻
A
2
≻
A
1
10
S
Z
1 = 0.0001, SZ2 = 0.0009, SZ3 = 0.0010, SZ4 = 0.0010, SZ5 = 0.0046
A
5
≻
A
3
≻
A
4
≻
A
2
≻
A
1


### 7.2. The Verification of the Effectiveness

Example 6.
There are five companies as alternativesAii=1,2,3,4,5 and four attributes: the risk analysis (C1), the growth analysis (C2), the social political impact analysis (C3), and the environmental impact analysis (C4) to be evaluated by three decision makers. The decision matrices (see [40]) are listed in Table 3. The experimental results are listed in Table 4. It can be seen from Table 4 that all methods have the same result, thus demonstrating that the method proposed in this paper is effective.Table 4
A comparison of preference order for different methods for Example 6.
Methods
Preference order
Liu et al.’s IFWDBM operator [40]
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3
Liu et al.’s IFWDGBM operator [40]
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3
Xu’s IFPWA operator [38]
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3
Liu et al.’s ROFPWMSW operator [32]
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3
q-IROFWA and q-IROFWPA operator in this paper (when q = 4)
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3


### 7.3. Further Comparison with Other Methods

In the following, in order to explain the advantages of the proposed method compared with the existing method, such asq-ROFWA and q-ROFWG operator [31] and q-ROFPA and q-ROFPMSM operator [32], we will use a new example to illustrate it.Example 7.
This example is about a investment problem in renewable energy. There are three alternativesAii=1,2,3: (1) A1 is geothermal; (2) A2 is solar; and (3) A3 is biomass. The following three attributes C1,C2,andC3 are to be evaluated by three decision makers, where C1 denotes the risk factor, C2 denotes the growth rate in the sector, and C3 denotes the payback reliability. The three possible alternatives are to be evaluated using the q-ROFN by the DM, as shown in Table 5. Let λ=1/3,1/3,1/3 be WV of decision makers and w=1/3,1/3,1/3 be WV of the attributes Cjj=1,2,3.
The details of the calculation are the same as in Example 1, so skipped. In practical applications, it is often unavoidable that some NMD are 0 inq-ROFS. In this situation, the methods proposed by Liu and Wang [31] are not able to get an effective or even the wrong ranking results. The comparison results are shown in Table 6.
From the comparison results in Table6, we can see that when there is 0 in the NMD of q-ROFs, the existing method cannot be solved very well. The ranking results will change as the parameter q changes, and when q is equal to 3, there is a situation that cannot be distinguished. Obviously, this is unreasonable. However, for the method in this paper, the ranking result is the same. Therefore, the developed method in this paper can overcome Drawback A. In addition, regardless of how the q value changes, the proposed method can distinguish the priority of the alternative and the result will not change, and furthermore the ranking result is the same as the ones presented in [41].Table 5
Individualq-ROFS decision matrix Dkk=1,2,3.
C
1
C
2
C
3
D
1
A
1
(0.6, 0.1)
(0.5, 0.2)
(0.7, 0.2)
A
2
(0.4, 0)
(0.4, 0.5)
(0.4, 0.1)
A
3
(0.4, 0.3)
(0.5, 0.1)
(0.6, 0.3)
D
2
A
1
(0.6, 0.3)
(0.7, 0.1)
(0.6, 0.1)
A
2
(0.7, 0.2)
(0.1, 0.3)
(0.5, 0.3)
A
3
(0.5, 0.2)
(0.7, 0.2)
(0.1, 0)
D
3
A
1
(0.4, 0.5)
(0.6, 0.3)
(0.5, 0.4)
A
2
(0.5, 0.4)
(0.6, 0.2)
(0.7, 0.2)
A
3
(0.7, 0.1)
(0.5, 0.4)
(0.1, 0.5)Table 6
A comparison of preference order for different methods for Example 7.
Methods
q
Preference order
q-ROFWA [31]
q = 2
x
1
≻
x
3
≻
x
2
q = 3
x
1
≻
x
3
=
x
2
q = 4
x
1
≻
x
2
≻
x
3
q = 5
x
1
≻
x
2
≻
x
3
q-IROFWA and q-IROFWPA operator in this paper
q = 2
x
1
≻
x
3
≻
x
2
q = 3
x
1
≻
x
3
≻
x
2
q = 4
x
1
≻
x
3
≻
x
2
q = 5
x
1
≻
x
3
≻
x
2
Liu et al.’s MDIFWG operator [41]
x
1
≻
x
3
≻
x
2Example 8.
In this example, the decision matrix is shown in Table7. The other data are identical with Example 3. The comparison results of different methods are shown in Table 8. From Table 8, we can get conclusion that in some special cases the existing method cannot distinguish the preference order of alternatives. The root of this problem is the related aggregation operators. Compared with the existing methods, the MD and NMD in this paper are not independent during the aggregation process. It is also shown that our method can overcome effectively the Drawback B.
In the following, we analyze the relative shortcoming of Xu’s IFPWA [38], Liu’s q-ROFWA [31], and Liu et al.’s ROFPWMSW Operator [32] is shown as follows:(1)
The shortcoming of Xu’s IFPWA Operator [38]: first, information is described by IFN, which must meet the restriction that MD u and NMD v satisfy the formula u+v≤1. Second, it cannot consider the interrelationship between multiple attributes. Third, it cannot overcome the Drawbacks A and B mentioned in the paper.(2)
The shortcoming of Liu and Wang’sq-ROFWA [31]: first, it cannot overcome the Drawbacks A and B mentioned in the paper. Second, it cannot properly deal extreme data given by biased decision makers.(3)
The shortcoming of Liu et al.’s ROFPWMSW Operator [32]: although the method integrates the advantages of the MSM operator (can consider the interrelationship between multiple attributes), PA operator (can eliminate the effects of extreme values), and q-ROFS operator (can express a wider range of information), MD and NMD are independent. Therefore, in some special cases the ROFPWMSW operator in [32] cannot distinguish the preference order of alternatives.
More visual comparison results can be easily obtained from Table9.Table 7
Individualq-ROFs decision matrix Dkk=1,2,3.
C
1
C
2
C
3
D
1
A
1
(0.25, 0.7)
(0.5, 0.2)
(0.7, 0.2)
A
2
(0.5, 0.2)
(0.4, 0.5)
(0.4, 0.1)
A
3
(0.4, 0.3)
(0.5, 0.3)
(0.6, 0.3)
D
2
A
1
(0.6, 0.3)
(0.4, 0.1)
(0.6, 0.1)
A
2
(0.7, 0.1)
(0.25, 0.7)
(0.5, 0.3)
A
3
(0.5, 0.2)
(0.7, 0.2)
(0.4, 0.5)
D
3
A
1
(0.4, 0.5)
(0.7, 0.3)
(0.5, 0.3)
A
2
(0.6, 0.3)
(0.6, 0.3)
(0.7, 0.2)
A
3
(0.7, 0.1)
(0.6, 0.1)
(0.25, 0.7)Table 8
A comparison of preference order for different methods for Example 8.
Methods
Preference order
q-ROFWA [31]
x
1
=
x
2
=
x
3
q-ROFWA [31]
x
1
=
x
2
=
x
3
q-ROFPA [32]
x
1
=
x
2
=
x
3
q-ROFPMSM operator [32]
x
1
=
x
2
=
x
3
The proposed method in this paper
x
2
<
x
3
<
x
1Table 9
Comparison of the characteristics for different MAGDM operators.
IFPWA [38]
q-ROFWA [31]
ROFPWMSW [32]
The proposed method
Can consider the interrelationship between multiple attributes
No
No
Yes
No
Can eliminate the effects of extreme values
Yes
No
Yes
Yes
Can express a wider range of information
No
Yes
Yes
Yes
Can overcome the Drawbacks A and B
No
No
No
Yes


## 7.1. The Application of the Proposed Method

Example 5.
This example is from [39]. There are five alternative companies Aii=1,2,…,5 which are evaluated by three experts Dkk=1,2,3 with respect to four attributes: the enterprise management level C1, the business growth ability C2, the economic benefit C3, and the corporate reputation C4, respectively. Di be the decision matrix obtained from expert Di by using q‐ROFN, as shown in Table 1.
Letλ=0.35,0.40,0.25 be WV of the three experts and w=0.2,0.1,0.3,0.4 be WV of the attributes. The detail of the proposed MAGDM method to select the best company is as below.
To further verify the effect of parameterq on the results, experiments were carried out using different q. The ranking results are shown in Table 3.
As can be seen from Table3 that although the aggregation result will change with the parameter q, the optimal result remains unchanged. Further, we can see that the value of the score function becomes smaller as q increases, so q is better in the interval 2 to 5.Table 1
Decision matrix expressed byq-ROFs.
C
1
C
2
C
3
C
4
D
1
A
1
(0.2, 0.5)
(0.4, 0.6)
(0.4, 0.3)
(0.5, 0.2)
A
2
(0.6, 0.3)
(0.4, 0.2)
(0.5, 0.1)
(0.5, 0.2)
A
3
(0.4, 0.2)
(0.5, 0.2)
(0.5, 0.3)
(0.4, 0.3)
A
4
(0.5, 0.3
(0.3, 0.5)
(0.5, 0.2)
(0.5, 0.2
A
5
(0.6, 0.1)
(0.4, 0.4)
(0.6, 0.2)
(0.6, 0.1)
D
2
A
1
(0.3, 0.6)
(0.2, 0.7)
(0.5, 0.4)
(0.5, 0.3
A
2
(0.5, 0.1)
(0.5, 0.3)
(0.6, 0.2)
(0.4, 0.3)
A
3
(0.5, 0.3)
(0.6, 0.2)
(0.6, 0.1)
(0.5, 0.4)
A
4
(0.5, 0.4)
(0.2, 0.6)
(0.5, 0.1)
(0.6, 0.3)
A
1
(0.2, 0.7)
(0.5, 0.4)
(0.4, 0.5)
(0.2, 0.6)
D
3
A
2
(0.5, 0.2)
(0.6, 0.4)
(0.5, 0.3)
(0.4, 0.1)
A
3
(0.6, 0.3)
(0.5, 0.2)
(0.5, 0.3)
(0.5, 0.4)
A
4
(0.4, 0.5)
(0.5, 0.4)
(0.6, 0.2)
(0.5, 0.2)
A
5
(0.6, 0.3)
(0.5, 0.1)
(0.6, 0.4)
(0.6, 0.1)Step 1: normalize the decision matricesD1, D2, and D3. In this example, the type of the four attributes is the same, so it can be omitted.
Step 2: according to theq‐IROFWA operator in equation (55), all the attributes matrices Dijk are aggregated into the integrated matrices Zik, as shown in Table 2.Table 2
Integrated decision matrixZik.
Z
1
Z
2
Z
3
Z
4
Z
5
Z
1
(0.4277, 0.3712)
(0.5170, 0.3630)
(0.4463, 0.3243)
(0.4872, 0.3661)
(0.5867, 0.4206)
Z
2
(0.4562, 0.3866)
(0.5069, 0.3194)
(0.5458, 0.4068)
(0.5337, 0.4471)
(0.6124, 0.4204)
Z
3
(0.3341, 0.4528)
(0.4805, 0.2868)
(0.5242, 0.4098)
(0.5215, 0.3625)
(0.5919, 0.4196)Step 3: according to equation (56), we can get the supports Sikt=SupZik,Zit as follows:(60)
S
1
12
=
S
1
21
=
0.9883
,
S
1
13
=
S
1
31
=
0.9791
,
S
1
23
=
S
1
32
=
0.9712
,
S
2
12
=
S
2
21
=
0.9884
,
S
2
13
=
S
2
31
=
0.9742
,
S
2
23
=
S
2
32
=
0.9859
,
S
3
12
=
S
3
21
=
0.9465
,
S
3
13
=
S
3
31
=
0.9551
,
S
3
23
=
S
3
32
=
0.9907
,
S
4
12
=
S
4
21
=
0.9617
,
S
4
13
=
S
4
31
=
0.9869
,
S
4
23
=
S
4
32
=
0.9740
,
S
5
12
=
S
5
21
=
0.9861
,
S
5
13
=
S
5
31
=
0.9973
,
S
5
23
=
S
5
32
=
0.9886.Step 4: according to equations (57) and (58), we can get the supports Tik=TZik and the weights wik as follows:(61)
T
1
k
=
∑
t
=
1
,
t
≠
1
3
Sup
Z
1
1
,
Z
1
t
=
Sup
Z
1
1
,
Z
1
2
+
Sup
Z
1
1
,
Z
1
3
=
0.9883
+
0.9791
=
1.9674
,
T
1
2
=
1.9595
,
T
1
3
=
1.9503
,
T
2
1
=
1.9626
,
T
2
2
=
1.9742
,
T
2
3
=
1.9601
,
T
3
1
=
1.9016
,
T
3
2
=
1.9373
,
T
3
3
=
1.9458
,
T
4
1
=
1.9486
,
T
4
2
=
1.9357
,
T
4
3
=
1.9609
,
T
5
1
=
1.9835
,
T
5
2
=
1.9748
,
T
5
3
=
1.9859
,
ω
1
1
=
λ
1
1
+
T
1
1
∑
k
=
1
3
λ
k
1
+
T
1
k
=
0.3509.Similarly, we have(62)
ω
1
2
=
0.3495
,
ω
1
3
=
0.3470
,
ω
1
4
=
0.3502
,
ω
1
5
=
0.3503
,
ω
2
1
=
0.3999
,
ω
2
2
=
0.4010
,
ω
2
3
=
0.4014
,
ω
2
4
=
0.3985
,
ω
2
5
=
0.3992
,
ω
3
1
=
0.2492
,
ω
3
2
=
0.2494
,
ω
3
3
=
0.2516
,
ω
3
4
=
0.2512
,
ω
3
5
=
0.2504.Step 5: according to theq‐IROFPWA operator in equation (59), we can get the collective preference values Zi as follows:(63)
Z
1
=
0.4217
,
0.3161
,
Z
2
=
0.5043
,
0.3136
,
Z
3
=
0.5104
,
0.3700
,
Z
4
=
0.5154
,
0.3540
,
Z
5
=
0.5986
,
0.3965
.Step 6: obtain the comprehensive evaluation valueZi by the score function of equation (4) as follows:(64)
L
s
Z
1
=
0.0434
,
L
s
Z
2
=
0.0974
,
L
s
Z
3
=
0.0823
,
L
s
Z
4
=
0.0925
,
L
s
Z
5
=
0.1521.Step 7: sort the calculation results of the score function, and we obtainA5≻A2≻A4≻A3≻A1, so A5 is the best choice.Table 3
Ranking results for differentq.
q
The score function
Preference order
2
S
Z
1 = 0.1017, SZ2 = 0.1876, SZ3 = 0.1576, SZ4 = 0.1757, SZ5 = 0.2486
A
5
≻
A
2
≻
A
4
≻
A
3
≻
A
1
3
S
Z
1 = 0.0434, SZ2 = 0.0974, SZ3 = 0.0823, SZ4 = 0.0925, SZ5 = 0.1521
A
5
≻
A
2
≻
A
4
≻
A
3
≻
A
1
5
S
Z
1 = 0.0082, SZ2 = 0.0260, SZ3 = 0.0230, SZ4 = 0.0256, SZ5 = 0.0565
A
5
≻
A
2
≻
A
4
≻
A
3
≻
A
1
8
S
Z
1 = 0.0007, SZ2 = 0.0035, SZ3 = 0.0035, SZ4 = 0.0037, SZ5 = 0.0126
A
5
≻
A
4
≻
A
3
≻
A
2
≻
A
1
10
S
Z
1 = 0.0001, SZ2 = 0.0009, SZ3 = 0.0010, SZ4 = 0.0010, SZ5 = 0.0046
A
5
≻
A
3
≻
A
4
≻
A
2
≻
A
1


## 7.2. The Verification of the Effectiveness

Example 6.
There are five companies as alternativesAii=1,2,3,4,5 and four attributes: the risk analysis (C1), the growth analysis (C2), the social political impact analysis (C3), and the environmental impact analysis (C4) to be evaluated by three decision makers. The decision matrices (see [40]) are listed in Table 3. The experimental results are listed in Table 4. It can be seen from Table 4 that all methods have the same result, thus demonstrating that the method proposed in this paper is effective.Table 4
A comparison of preference order for different methods for Example 6.
Methods
Preference order
Liu et al.’s IFWDBM operator [40]
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3
Liu et al.’s IFWDGBM operator [40]
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3
Xu’s IFPWA operator [38]
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3
Liu et al.’s ROFPWMSW operator [32]
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3
q-IROFWA and q-IROFWPA operator in this paper (when q = 4)
A
2
≻
A
4
≻
A
5
≻
A
1
≻
A
3


## 7.3. Further Comparison with Other Methods

In the following, in order to explain the advantages of the proposed method compared with the existing method, such asq-ROFWA and q-ROFWG operator [31] and q-ROFPA and q-ROFPMSM operator [32], we will use a new example to illustrate it.Example 7.
This example is about a investment problem in renewable energy. There are three alternativesAii=1,2,3: (1) A1 is geothermal; (2) A2 is solar; and (3) A3 is biomass. The following three attributes C1,C2,andC3 are to be evaluated by three decision makers, where C1 denotes the risk factor, C2 denotes the growth rate in the sector, and C3 denotes the payback reliability. The three possible alternatives are to be evaluated using the q-ROFN by the DM, as shown in Table 5. Let λ=1/3,1/3,1/3 be WV of decision makers and w=1/3,1/3,1/3 be WV of the attributes Cjj=1,2,3.
The details of the calculation are the same as in Example 1, so skipped. In practical applications, it is often unavoidable that some NMD are 0 inq-ROFS. In this situation, the methods proposed by Liu and Wang [31] are not able to get an effective or even the wrong ranking results. The comparison results are shown in Table 6.
From the comparison results in Table6, we can see that when there is 0 in the NMD of q-ROFs, the existing method cannot be solved very well. The ranking results will change as the parameter q changes, and when q is equal to 3, there is a situation that cannot be distinguished. Obviously, this is unreasonable. However, for the method in this paper, the ranking result is the same. Therefore, the developed method in this paper can overcome Drawback A. In addition, regardless of how the q value changes, the proposed method can distinguish the priority of the alternative and the result will not change, and furthermore the ranking result is the same as the ones presented in [41].Table 5
Individualq-ROFS decision matrix Dkk=1,2,3.
C
1
C
2
C
3
D
1
A
1
(0.6, 0.1)
(0.5, 0.2)
(0.7, 0.2)
A
2
(0.4, 0)
(0.4, 0.5)
(0.4, 0.1)
A
3
(0.4, 0.3)
(0.5, 0.1)
(0.6, 0.3)
D
2
A
1
(0.6, 0.3)
(0.7, 0.1)
(0.6, 0.1)
A
2
(0.7, 0.2)
(0.1, 0.3)
(0.5, 0.3)
A
3
(0.5, 0.2)
(0.7, 0.2)
(0.1, 0)
D
3
A
1
(0.4, 0.5)
(0.6, 0.3)
(0.5, 0.4)
A
2
(0.5, 0.4)
(0.6, 0.2)
(0.7, 0.2)
A
3
(0.7, 0.1)
(0.5, 0.4)
(0.1, 0.5)Table 6
A comparison of preference order for different methods for Example 7.
Methods
q
Preference order
q-ROFWA [31]
q = 2
x
1
≻
x
3
≻
x
2
q = 3
x
1
≻
x
3
=
x
2
q = 4
x
1
≻
x
2
≻
x
3
q = 5
x
1
≻
x
2
≻
x
3
q-IROFWA and q-IROFWPA operator in this paper
q = 2
x
1
≻
x
3
≻
x
2
q = 3
x
1
≻
x
3
≻
x
2
q = 4
x
1
≻
x
3
≻
x
2
q = 5
x
1
≻
x
3
≻
x
2
Liu et al.’s MDIFWG operator [41]
x
1
≻
x
3
≻
x
2Example 8.
In this example, the decision matrix is shown in Table7. The other data are identical with Example 3. The comparison results of different methods are shown in Table 8. From Table 8, we can get conclusion that in some special cases the existing method cannot distinguish the preference order of alternatives. The root of this problem is the related aggregation operators. Compared with the existing methods, the MD and NMD in this paper are not independent during the aggregation process. It is also shown that our method can overcome effectively the Drawback B.
In the following, we analyze the relative shortcoming of Xu’s IFPWA [38], Liu’s q-ROFWA [31], and Liu et al.’s ROFPWMSW Operator [32] is shown as follows:(1)
The shortcoming of Xu’s IFPWA Operator [38]: first, information is described by IFN, which must meet the restriction that MD u and NMD v satisfy the formula u+v≤1. Second, it cannot consider the interrelationship between multiple attributes. Third, it cannot overcome the Drawbacks A and B mentioned in the paper.(2)
The shortcoming of Liu and Wang’sq-ROFWA [31]: first, it cannot overcome the Drawbacks A and B mentioned in the paper. Second, it cannot properly deal extreme data given by biased decision makers.(3)
The shortcoming of Liu et al.’s ROFPWMSW Operator [32]: although the method integrates the advantages of the MSM operator (can consider the interrelationship between multiple attributes), PA operator (can eliminate the effects of extreme values), and q-ROFS operator (can express a wider range of information), MD and NMD are independent. Therefore, in some special cases the ROFPWMSW operator in [32] cannot distinguish the preference order of alternatives.
More visual comparison results can be easily obtained from Table9.Table 7
Individualq-ROFs decision matrix Dkk=1,2,3.
C
1
C
2
C
3
D
1
A
1
(0.25, 0.7)
(0.5, 0.2)
(0.7, 0.2)
A
2
(0.5, 0.2)
(0.4, 0.5)
(0.4, 0.1)
A
3
(0.4, 0.3)
(0.5, 0.3)
(0.6, 0.3)
D
2
A
1
(0.6, 0.3)
(0.4, 0.1)
(0.6, 0.1)
A
2
(0.7, 0.1)
(0.25, 0.7)
(0.5, 0.3)
A
3
(0.5, 0.2)
(0.7, 0.2)
(0.4, 0.5)
D
3
A
1
(0.4, 0.5)
(0.7, 0.3)
(0.5, 0.3)
A
2
(0.6, 0.3)
(0.6, 0.3)
(0.7, 0.2)
A
3
(0.7, 0.1)
(0.6, 0.1)
(0.25, 0.7)Table 8
A comparison of preference order for different methods for Example 8.
Methods
Preference order
q-ROFWA [31]
x
1
=
x
2
=
x
3
q-ROFWA [31]
x
1
=
x
2
=
x
3
q-ROFPA [32]
x
1
=
x
2
=
x
3
q-ROFPMSM operator [32]
x
1
=
x
2
=
x
3
The proposed method in this paper
x
2
<
x
3
<
x
1Table 9
Comparison of the characteristics for different MAGDM operators.
IFPWA [38]
q-ROFWA [31]
ROFPWMSW [32]
The proposed method
Can consider the interrelationship between multiple attributes
No
No
Yes
No
Can eliminate the effects of extreme values
Yes
No
Yes
Yes
Can express a wider range of information
No
Yes
Yes
Yes
Can overcome the Drawbacks A and B
No
No
No
Yes


## 8. Conclusions

For express fuzzy information,q-ROFS is a good tool. It has a parameter q, so it holds a wider range of fuzzy information than IFS and PFS. In this paper, we reviewed and analyzed some drawbacks of the existing MAGDM methods. We improved the traditional operation rules to overcome these drawbacks, proposed a new method to deal with the MAGDM method based on the proposed q-IROFWA and q-IROFWPA operator. At the same time, some features of the newly proposed operators are proved. Moreover, some numerical examples are provided to show the application of the method in detail and further to verify the feasibility and the superiority of the proposed method. From the example results shown in Tables 6 and 8, we can come to the conclusion that the proposed methods can surmount the drawbacks of some existing MAGDM methods. For future research, we will apply the improved operation rules to more aggregation operators. In addition, we will introduce granular computing techniques [42] to MAGDM and develop the new MAGDM method.



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*Source: 2036728-2019-12-27.xml*