# Multiattribute Decision-Making Problems in terms of the Weighted Mean Operation of Two Aggregation Operators of OrthopairZ-Numbers

Authors: Mailing Zhao; Jun Ye
Journal: Journal of Mathematics
 (2021)

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## Abstract

TheZ number defined by Zadeh can depict the fuzzy restriction/value and reliability measure by an ordered pair of fuzzy values to strengthen the reliability of the fuzzy restriction/value. However, there exist truth and falsehood Z-numbers in real life. Thus, the Z number cannot reflect both. To indicate both, this study presents an orthopair Z-number (OZN) set to depict truth and falsehood values (intuitionistic fuzzy values) and their reliability levels in uncertain and incomplete cases. Next, we define the operations, score and accuracy functions, and sorting rules of OZNs. Further, the OZN weighted arithmetic mean (OZNWAM) and OZN weighted geometric mean (OZNWGM) operators are proposed based on the operations of OZNs. According to the weighted mean operation of the OZNWAM and OZNWGM operators, a multiattribute decision-making (MADM) model is established in the case of OZNs. Lastly, a numerical example is presented to reflect the flexibility and rationality of the presented MADM model. Comparative analysis indicates that the presented MADM model can indicate its superiority in the reliability and flexibility of decision results. Meanwhile, the resulting advantage of this study is that the presented MADM model can strengthen the reliability level of orthopair fuzzy values and make the decision results more reliable and flexible.


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## Body

## 1. Introduction

In the case of fuzzy decision, fuzzy information expression and aggregation are two key issues. Then, a fuzzy set [1] only contains a truth-membership degree, but lacks a falsehood-membership degree. To make up for this issue, Atanassov [2] defined an intuitionistic fuzzy set (IFS) that contains truth- and falsehood-membership degrees. After that, some researchers proposed the weighted arithmetic and geometric mean operators of intuitionistic fuzzy values (IFVs) [3, 4] and utilized them in multiattribute decision-making (MADM), while other researchers further put forward different aggregation operators of IFVs [5–15] for MADM. Then, the truth- and falsehood-membership degrees in IFV are also named as an orthopair fuzzy value (OFV). However, the sum of the truth- and falsehood-membership degrees may be more than one in some cases. To represent such an OFV, some researchers defined Pythagorean fuzzy sets (PFSs) [16, 17] and Pythagorean fuzzy aggregation algorithms [18–23] for MADM. Then, other researchers also introduced a hybrid method of PFSs for MADM [24] and a digraph and matrix approach for risk evaluations [25]. As the generalization of PFSs, the notion of generalized OFVs [26–28] was introduced to flexibly present the hybrid information range of the truth- and falsehood-membership degrees regarding different values of the parameter q. Then, some aggregation operators of q-rung orthopair fuzzy sets/values (q-ROFSs/q-ROFVs) [29–31] were presented and applied in MADM. In the case of interval-valued q-rung orthopair dual-hesitant fuzzy sets, interval-valued q-rung orthopair dual-hesitant fuzzy Hamacher graphs were introduced for MADM [32]. Under the indeterminate case of the truth- and falsehood-membership degrees, Ye et al. [33] defined an orthopair indeterminate set/value (OIS/OIV) and some aggregation operators of OIVs and then established an orthopair indeterminate MADM model with indeterminate ranges/risks of decision makers (DMs) in the case of OISs.However, IFV, OFV,q-ROFV, and OIV cannot contain their reliability measures/levels. Then, both a fuzzy value (fuzzy constraint) and a reliability level contained in the Z-number (ZN) defined by Zadeh [34] are described by an ordered pair of fuzzy values in vague and uncertain cases. Clearly, ZN demonstrates its more reliable superiority over the classical fuzzy value. Therefore, ZNs have been used for some applications [35–37]. Although ZN implies the aforementioned superiority, it cannot reflect the truth and falsehood ZNs in real problems. Furthermore, the existing IFV, OFV, q-ROFV, and OIV cannot reflect the truth and falsehood ZNs due to the lack of their reliability measures. In this case, it is important we need to strengthen the reliability measure of IFV and to propose an orthopair Z-number (OZN) set for making up for the defect of the existing IFV/OFV. Motivated based on the idea of the truth and falsehood ZNs, this study proposes an orthopair Z-number (OZN) set to depict the truth and falsehood ZNs in uncertain and vague cases. Then, we define the operations and score and accuracy functions of OZNs and propose the OZN weighted arithmetic mean (OZNWAM) and OZN weighted geometric mean (OZNWGM) operators. Since the OZNWAM operator tends to group arguments and the OZNWGM operator tends to personal arguments, we can propose the weighted mean operation of the OZNWAM and OZNWGM operators to overcome both prejudices. According to the weighted mean operation of the two operators, a MADM model is established in the case of OZNs. A numerical example is presented to reflect the flexibility and rationality of the presented MADM model. Compared to existing related MADM models, the superiority of this study is that the presented MADM model can strengthen the reliability level of OFVs and make the decision results more reliable and flexible.The rest of this article contains the following parts: Section2 presents an OZN set and the operations, score and accuracy functions, and sorting rules of OZNs. Then, the OZNWAM and NZNWGM operators and their properties are presented in Section 3. In Section 4, a MADM model is established by using the weighted mean operation of the OZNWAM and OZNWGM operators. In Section 5, an illustrative example and comparative analysis are presented to demonstrate the flexibility and rationality of the presented MADM model in the case of OZNs. Lastly, this paper addresses some conclusions and further study in Section 6.


## 2. OrthopairZ-Number Sets

Inspired by the notion of ZNs [34], this section proposes the definition of an OZN set by a pair of truth and falsehood ZNs.Definition 1.
SetY as a universe set. Then, an OZN set in Y is defined below:(1)ZOZN=y,TP,Qy,FP,Qy|y∈Y,where T(P, Q)(y) = (TP(y), TQ(y)): Y ⟶ [0, 1]2 and F(P, Q)(y) = (FP(y), FQ(y)): Y ⟶ [0, 1]2 are the truth and falsehood ZNs. In T(P, Q)(y) and F(P, Q)(y), P is OFVs/IFVs for y ∈ Y, such that the condition 0≤TPy+FQy≤1; then, Q is the reliability measures of the truth and falsehood fuzzy values for P.
Then, the simplified form of the elementy,TP,Qy,FP,Qy in ZOZN is represented by zOZN=TP,Q,FP,Q=TP,TQ,FP,FQ, which is named OZN or an intuitionistic ZN (IZN).Definition 2.
LetzOZN1=T1P,Q,F1P,Q=TP1,TQ1,FP1,FQ1 and zOZN2=T2P,Q,F2P,Q=TP2,TQ2,FP2,FQ2 be two OZNs and α > 0. Thus, their operations are defined below:(1)
zOZN1 ⊇ zOZN2 ⟺ TP1 ≥ TP2, TQ1 ≥ TQ2, FP1 ≤ FP2, and FQ1 ≤ FQ2(2)
zOZN1 = zOZN2 ⟺ zOZN1 ⊇ zOZN2 and zOZN2 ⊇ zOZN1(3)
zOZN1∪zOZN2=TP1∨TP2,TQ1∨TQ2,FP1∧FP2,FQ1∧FQ2(4)
zOZN1∩zOZN2=TP1∧TP2,TQ1∧TQ2,FP1∨FP2,FQ1∨FQ2(5)
zOZN1c=FP1,FQ1,TP1,TQ1 (complement of zOZN1)(6)
zOZN1⊕zOZN2=TP1+TP2−TP1⋅TP2,TQ1+TQ2−TQ1⋅TQ2,FP1⋅FP2,FQ1⋅FQ2(7)
zOZN1⊗zOZN1=TP1⋅TP2,TQ1⋅TQ2,FP1+FP2−FP1⋅FP2,FQ1+FQ2−FQ1⋅FQ2(8)
α⋅zOZN1=1−1−TP1α,1−1−TQ1α,FP1α,FQ1α(9)
zOZN1α=TP1α,TQ1α,1−1−FP1α,1−1−FQ1α
To sort OZNszOZNi=TiP,Q,FiP,Q=TPi,TQi,FPi,FQi (i = 1, 2), the score and accuracy functions of OZNs are introduced, respectively, below:(2)CzOZNi=TPi⋅TQi−FPi⋅FQiforCzOZNi∈−1,1,(3)EzOZNi=TPi⋅TQi+FPi⋅FQiforEzOZNi∈0,1.
Thus, we can give the following sorting rules of the two OZNs:(1)
IfC(zOZN1) > C(zOZN2), then zOZN1 > zOZN2(2)
IfC(zOZN1) = C(zOZN2) and E(zOZN1) > E(zOZN2), then zOZN1 > zOZN2(3)
IfC(zOZN1) = C(zOZN2) and E(zOZN1) = E(zOZN2), then zOZN1 = zOZN2Example 1.
Set two OZNs aszOZN1 ≤ (0.8, 0.8), (0.1, 0.8)> and zOZN2 ≤ (0.7, 0.9), (0.2, 0.8)>. Thus, the sorting of both is yielded as follows:
Using equation (2), we have C(zOZN1) = 0.8 × 0.8–0.1 × 0.8 = 0.56 and C(zOZN2) = 0.7 × 0.9–0.2 × 0.8 = 0.47. Since C(zOZN1) > C(zOZN2), their sorting is zOZN1 > zOZN2.


## 3. Two Basic Aggregation Operators of OZNs

This section presents two basic aggregation operators of OZNs based on the operations (7)–(9) in Definition2 and their properties.


### 3.1. OZNWAM Operator

For a group of OZNs, the OZNWAM operator can be proposed from the operations (6) and (8) in Definition2.Definition 3.
SetzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs. Thus, the OZNWAM operator is defined below:(4)OZNWAMzOZN1,zOZN2,…,zOZNn=∑k=1nαkzOZNk,where αk (k = 1, 2, …, n) is the weight of zOZNk for 0 ≤ αk ≤ 1 and ∑k=1nαk=1.Theorem 1.
SetzOZNk=TPk,TQk,FRk,FMk (k = 1, 2, …, n) as a group of OZNs. Thus, the aggregated value of the OZNWAM operator is still OZN, which is calculated by the following formula:(5)OZNWAMzOZN1,zOZN2,…,zOZNn=∑k=1nαkzOZNk=1−∏k=1n1−TPkαk,1−∏k=1n1−TQkαk,∏k=1nFPkαk,∏k=1nFQkαk,where αk (k = 1, 2, …, n) is the weight of zOZNk for 0 ≤ αk ≤ 1 and ∑k=1nαk=1.Proof.
In terms of mathematical induction, equation (5) can be verified in the following way.(1)
Takingn = 2, based on the operations (6) and (8) in Definition 2, one obtains the following result:(6)OZNWAMzOZN1,zOZN2=α1zOZN1⊕α2zOZN2=1−1−TP1α1+1−1−TP2α2−1−1−TP1α11−1−TP2α2,1−1−TQ1α1+1−1−TQ2α2−1−1−TQ1α11−1−TQ2α2,FP1α1⋅FP2α2,FQ1α1⋅FQ2α2=1−∏k=121−TPkαk,1−∏k=121−TQkαk,∏k=12FPkαk,∏k=12FQkαk.(2)
Takingn = m, equation (4) can keep the result:(7)OZNWAMzOZN1,zOZN2,…,zOZNm=∑k=1mαkzOZNk=1−∏k=1m1−TPkαk,1−∏k=1m1−TQkαk,∏k=1mFPkαk,∏k=1mFQkαk.(3)
Takingn = m+1, based on the operations (6) and (8) in Definition 2 and equations (6) and (7), one obtains the following result:(8)OZNWAMzOZN1,zOZN2,…,zOZNm,zOZNm+1=∑k=1m+1αkzOZNk=1−∏k=1m1−TPkαk,1−∏k=1m1−TQkαk,∏k=1mFPkαk,∏k=1mFQkαk⊕αm+1zOZNm+1=1−∏k=1m+11−TPkαk,1−∏k=1m+11−TQkαk,∏k=1m+1FPkαk,∏k=1m+1FQkαk.
Regarding the above results, equation (5) can hold for any n.
Hence, the proof is completed.Theorem 2.
The OZNWAM operator of equation (5) implies the following properties:(a)
Idempotency: setzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs. If zOZNk = zOZN (k = 1, 2, …, n), thenOZNWAMzOZN1,zOZN2,…,zOZNn=zOZN(b)
Boundedness: letzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) be a group of OZNs, and let the minimum and maximum OZNs be in the following:(9)zOZNmin=minkTPk,minkTQk,maxkFPk,maxkFQk,zOZNmax=maxkTPk,maxkTQk,minkFPk,minkFQk.Thus, zOZNmin≤OZNWAMzOZN1,zOZN2,…,zOZNn≤zOZNmax can exist.(c)
Monotonicity: letzOZNk=TPk,TQk,FPk,FQk and zOZNk∗=TPk∗,TQk∗,FPk∗,FQk∗ (k = 1, 2, …, n) be two groups of OZNs. If zOZNj≤zOZNj∗, then OZNWAMzOZN1,zOZN2,…,zOZNn≤OZNWAMzOZN1∗,zOZN2∗,…,zOZNn∗.Proof.
(a)
WhenzOZNk = zOZN (k = 1, 2, …, n), one obtains the result of equation (5):(10)OZNWAMzOZN1,zOZN2,…,zOZNn=∑k=1nαkzOZNk=1−∏k=1n1−TPkαj,1−∏k=1n1−TQkαk,∏k=1nFPkαk,∏k=1nFQkαk=1−1−TP∑k=1nαk,1−1−TQ∑k=1nαk,FP∑k=1nαk,FQ∑k=1nαk=1−1−TP,1−1−TQ,FP,FQ=TP,TQ,FP,FQ=zOZN.(b)
SincezOZNmin and zOZNmax are the minimum and the maximum OZNs, there is zOZNmin ≤ zOZNk ≤ zOZNmax. Hence, ∑k=1nαkzOZNmin≤∑k=1nαkzOZNk≤∑k=1nαkzOZNmax can hold. Regarding the property (1), there exists zOZNmin≤∑k=1nαkzOZNk≤zOZNmax, i.e., zOZNmin≤OZNWAMzOZN1,zOZN2,…,zOZNn≤zOZNmax.(c)
IfzOZNk≤zOZNk∗, then ∑k=1nαkzOZNk≤∑k=1nαkzOZNk∗, i.e., OZNWAMzOZN1,zOZN2,…,zOZNn≤OZNWAMzOZN1∗,zOZN2∗,…,zOZNn∗.
Hence, the proof of Theorem2 is completed.


### 3.2. OZNWGM Operator

In this part, we can propose the OZNWGM operator according to the operations (7) and (9) in Definition2.Definition 4.
SetzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs. Thus, the OZNWGM operator is defined below:(11)OZNWGMzOZN1,zOZN2,…,zOZNn=∏k=1nzOZNkαk,where αk (k = 1, 2, …, n) is the weight of zOZNk for 0 ≤ αk ≤ 1 and ∑k=1nαk=1.Theorem 3.
SetzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs. Thus, the aggregated value of the OZNWGM operator is still OZN, which is calculated by the following formula:(12)OZNWGMzOZN1,zOZN2,…,zOZNn=∏k=1nzOZNkαk=∏k=1nTPkαk,∏k=1nTQkαk,1−∏k=1n1−FPkαk,1−∏k=1n1−FQkαk,where αk is the weight of zOZNk (k = 1, 2, …, n) for 0 ≤ αk ≤ 1 and ∑k=1nαk=1.Similar to the proof process of Theorem1, equation (12) can be also verified, which is omitted.Theorem 4.
The OZNWGM operator of equation (12) implies the following properties:(a)
Idempotency: setzOZNk=TPk,TQk,FPk,FQk (j = 1, 2, …, n) as a group of OZNs. If zOZNk = zOZN (k = 1, 2, …, n), thenOZNWGMzOZN1,zOZN2,⋯,zOZNn=zOZN.(b)
Boundedness: setzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs and take the following minimum and maximum OZNs:(13)zOZNmin=minkTPk,minkTQk,maxkFPj,maxkFQk,zOZNmax=maxkTPk,maxkTQk,minkFPk,minkFQk.Thus, there existszOZNmin≤OZNWGMzOZN1,zOZN2,…,zOZNn≤zOZNmax.(c)
Monotonicity: setzOZNk=TPk,TQk,FPk,FQk and zOZNk∗=TPk∗,TQk∗,FPk∗,FQk∗ (k = 1, 2, …, n) as two groups of OZNs. If zOZNk≤zOZNk∗, then OZNWGMzOZN1,zOZN2,…,zOZNn≤OZNWGMzOZN1∗,zOZN2∗,…,zOZNn∗.Clearly, the above properties of the OZNWGM operator can be also verified by the similar proof way of Theorem2, which is omitted.


## 3.1. OZNWAM Operator

For a group of OZNs, the OZNWAM operator can be proposed from the operations (6) and (8) in Definition2.Definition 3.
SetzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs. Thus, the OZNWAM operator is defined below:(4)OZNWAMzOZN1,zOZN2,…,zOZNn=∑k=1nαkzOZNk,where αk (k = 1, 2, …, n) is the weight of zOZNk for 0 ≤ αk ≤ 1 and ∑k=1nαk=1.Theorem 1.
SetzOZNk=TPk,TQk,FRk,FMk (k = 1, 2, …, n) as a group of OZNs. Thus, the aggregated value of the OZNWAM operator is still OZN, which is calculated by the following formula:(5)OZNWAMzOZN1,zOZN2,…,zOZNn=∑k=1nαkzOZNk=1−∏k=1n1−TPkαk,1−∏k=1n1−TQkαk,∏k=1nFPkαk,∏k=1nFQkαk,where αk (k = 1, 2, …, n) is the weight of zOZNk for 0 ≤ αk ≤ 1 and ∑k=1nαk=1.Proof.
In terms of mathematical induction, equation (5) can be verified in the following way.(1)
Takingn = 2, based on the operations (6) and (8) in Definition 2, one obtains the following result:(6)OZNWAMzOZN1,zOZN2=α1zOZN1⊕α2zOZN2=1−1−TP1α1+1−1−TP2α2−1−1−TP1α11−1−TP2α2,1−1−TQ1α1+1−1−TQ2α2−1−1−TQ1α11−1−TQ2α2,FP1α1⋅FP2α2,FQ1α1⋅FQ2α2=1−∏k=121−TPkαk,1−∏k=121−TQkαk,∏k=12FPkαk,∏k=12FQkαk.(2)
Takingn = m, equation (4) can keep the result:(7)OZNWAMzOZN1,zOZN2,…,zOZNm=∑k=1mαkzOZNk=1−∏k=1m1−TPkαk,1−∏k=1m1−TQkαk,∏k=1mFPkαk,∏k=1mFQkαk.(3)
Takingn = m+1, based on the operations (6) and (8) in Definition 2 and equations (6) and (7), one obtains the following result:(8)OZNWAMzOZN1,zOZN2,…,zOZNm,zOZNm+1=∑k=1m+1αkzOZNk=1−∏k=1m1−TPkαk,1−∏k=1m1−TQkαk,∏k=1mFPkαk,∏k=1mFQkαk⊕αm+1zOZNm+1=1−∏k=1m+11−TPkαk,1−∏k=1m+11−TQkαk,∏k=1m+1FPkαk,∏k=1m+1FQkαk.
Regarding the above results, equation (5) can hold for any n.
Hence, the proof is completed.Theorem 2.
The OZNWAM operator of equation (5) implies the following properties:(a)
Idempotency: setzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs. If zOZNk = zOZN (k = 1, 2, …, n), thenOZNWAMzOZN1,zOZN2,…,zOZNn=zOZN(b)
Boundedness: letzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) be a group of OZNs, and let the minimum and maximum OZNs be in the following:(9)zOZNmin=minkTPk,minkTQk,maxkFPk,maxkFQk,zOZNmax=maxkTPk,maxkTQk,minkFPk,minkFQk.Thus, zOZNmin≤OZNWAMzOZN1,zOZN2,…,zOZNn≤zOZNmax can exist.(c)
Monotonicity: letzOZNk=TPk,TQk,FPk,FQk and zOZNk∗=TPk∗,TQk∗,FPk∗,FQk∗ (k = 1, 2, …, n) be two groups of OZNs. If zOZNj≤zOZNj∗, then OZNWAMzOZN1,zOZN2,…,zOZNn≤OZNWAMzOZN1∗,zOZN2∗,…,zOZNn∗.Proof.
(a)
WhenzOZNk = zOZN (k = 1, 2, …, n), one obtains the result of equation (5):(10)OZNWAMzOZN1,zOZN2,…,zOZNn=∑k=1nαkzOZNk=1−∏k=1n1−TPkαj,1−∏k=1n1−TQkαk,∏k=1nFPkαk,∏k=1nFQkαk=1−1−TP∑k=1nαk,1−1−TQ∑k=1nαk,FP∑k=1nαk,FQ∑k=1nαk=1−1−TP,1−1−TQ,FP,FQ=TP,TQ,FP,FQ=zOZN.(b)
SincezOZNmin and zOZNmax are the minimum and the maximum OZNs, there is zOZNmin ≤ zOZNk ≤ zOZNmax. Hence, ∑k=1nαkzOZNmin≤∑k=1nαkzOZNk≤∑k=1nαkzOZNmax can hold. Regarding the property (1), there exists zOZNmin≤∑k=1nαkzOZNk≤zOZNmax, i.e., zOZNmin≤OZNWAMzOZN1,zOZN2,…,zOZNn≤zOZNmax.(c)
IfzOZNk≤zOZNk∗, then ∑k=1nαkzOZNk≤∑k=1nαkzOZNk∗, i.e., OZNWAMzOZN1,zOZN2,…,zOZNn≤OZNWAMzOZN1∗,zOZN2∗,…,zOZNn∗.
Hence, the proof of Theorem2 is completed.


## 3.2. OZNWGM Operator

In this part, we can propose the OZNWGM operator according to the operations (7) and (9) in Definition2.Definition 4.
SetzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs. Thus, the OZNWGM operator is defined below:(11)OZNWGMzOZN1,zOZN2,…,zOZNn=∏k=1nzOZNkαk,where αk (k = 1, 2, …, n) is the weight of zOZNk for 0 ≤ αk ≤ 1 and ∑k=1nαk=1.Theorem 3.
SetzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs. Thus, the aggregated value of the OZNWGM operator is still OZN, which is calculated by the following formula:(12)OZNWGMzOZN1,zOZN2,…,zOZNn=∏k=1nzOZNkαk=∏k=1nTPkαk,∏k=1nTQkαk,1−∏k=1n1−FPkαk,1−∏k=1n1−FQkαk,where αk is the weight of zOZNk (k = 1, 2, …, n) for 0 ≤ αk ≤ 1 and ∑k=1nαk=1.Similar to the proof process of Theorem1, equation (12) can be also verified, which is omitted.Theorem 4.
The OZNWGM operator of equation (12) implies the following properties:(a)
Idempotency: setzOZNk=TPk,TQk,FPk,FQk (j = 1, 2, …, n) as a group of OZNs. If zOZNk = zOZN (k = 1, 2, …, n), thenOZNWGMzOZN1,zOZN2,⋯,zOZNn=zOZN.(b)
Boundedness: setzOZNk=TPk,TQk,FPk,FQk (k = 1, 2, …, n) as a group of OZNs and take the following minimum and maximum OZNs:(13)zOZNmin=minkTPk,minkTQk,maxkFPj,maxkFQk,zOZNmax=maxkTPk,maxkTQk,minkFPk,minkFQk.Thus, there existszOZNmin≤OZNWGMzOZN1,zOZN2,…,zOZNn≤zOZNmax.(c)
Monotonicity: setzOZNk=TPk,TQk,FPk,FQk and zOZNk∗=TPk∗,TQk∗,FPk∗,FQk∗ (k = 1, 2, …, n) as two groups of OZNs. If zOZNk≤zOZNk∗, then OZNWGMzOZN1,zOZN2,…,zOZNn≤OZNWGMzOZN1∗,zOZN2∗,…,zOZNn∗.Clearly, the above properties of the OZNWGM operator can be also verified by the similar proof way of Theorem2, which is omitted.


## 4. MADM Model in terms of the Weighted Mean Operation of the OZNWAM and OZNWGM Operators

This section establishes a MADM model corresponding to the weighted mean operation of the OZNWAM and OZNWGM operators to perform MADM problems with the information of OZNs.In a MADM problem, a set of alternativesB = {B1, B2, …, Bm} is provided and evaluated by a set of attributes Y = {y1, y2, …, yn}. Meanwhile, the impotence of each attribute yk (k = 1, 2, …, n) is specified by the weight αk. DMs are invited to assess the suitability of each alternative Bi (i = 1, 2, …, m) over the attributes yk (k = 1, 2, …, n) by the truth and falsehood ZNs, which are constructed as the OZN zOZNik=TPik,TQik,FPik,FQik for TPik,TQik∈0,1, FPik,FQik∈0,1, and 0≤TPik+FPik≤1. Then, the decision matrix of OZNs can be denoted by ZOZN = (zOZNik)m×n. Thus, this MADM model is established and depicted by the following decision process:Step 1: using equations (5) and (12), the aggregated OZNs zOZN1i and zOZN2i are calculated by the following formulae:(14)zOZN1i=OZNWAMzOZNi1,zOZNi2,…,zOZNin=∑k=1nαkzOZNik=1−∏k=1n1−TPikαk,1−∏k=1n1−TQikαk,∏k=1nFPikαk,∏k=1nFQikαk,(15)zOZN2i=OZNWGMzOZNi1,zOZNi2,…,zOZNin=∏k=1nzOZNikαk∏k=1nTPikαk,∏k=1nTQikαk,1−∏k=1n1−FPikαk,1−∏k=1n1−FQikαk.Step 2: the weighted mean operation of the OZNWAM and OZNWGM operators with the weightsβ1 and β2 = 1 − β1 for β1 ∈ [0, 1] is calculated by the following equation:(16)HOZNi=β1zOZN1i⊕β2zOZN2i=β1zOZN1i⊕1−β1zOZN2i=1−1−TP1iβ11−TP2i1−β1,1−1−TQ1iβ11−TQ2i1−β1,FP1iβ1⋅FP2i1−β1,FQ1iβ1⋅FQ2i1−β1.Step 3: the values ofC(HOZNi) (E(HOZNi) if necessary) (i = 1, 2, …, m) are calculated by equation (2) (equation (3)).Step 4: the alternatives are sorted in terms of the sorting rules, and the best one is chosen.Step 5
: end.


## 5. Example and Comparative Analysis

### 5.1. Illustrative Example

Today, various robots have been used in industry, household services, and medical fields. They are changing our daily lives in various ways. Then, the performance evaluation of the robot is an important issue in the user’s selection process. To illustrate the applicability of the presented MADM model in the case of OZNs, this section applies the proposed MADM model to the selection of industrial robots to illustrate its application in the case of OZNs.An industrial company wants to purchase an industrial robot to improve its existing manufacturing system. The technical department initially considers four types of potential industrial robots (alternatives), which are denoted as a setB = {B1, B2, B3, B4}. To choose the most suitable one, they must satisfy four performance requirements (attributes): carrying capacity (y1), dexterity and work space (y2), operation accuracy (y3), and communication and control capabilities (y4). The weigh vector of the four attributes is specified by α = (0.28, 0.26, 0.24, 0.22). Thus, the experts/DMs are invited to evaluate the four alternatives over the four attributes by OFVs and corresponding reliability levels, which are expressed as OZNs zOZNik=TPik,TQik,FPik,FQik(i and k = 1, 2, 3, and 4) for TPik,TQik∈0,1, FPik,FQik∈0,1, and 0≤TPik+FPik≤1. Thus, all OZNs can be constructed as their decision matrix:(17)ZOZN=zOZNik4×4=0.7,0.8,0.2,0.80.7,0.7,0.2,0.80.7,0.8,0.2,0.90.7,0.8,0.2,0.70.8,0.7,0.1,0.90.7,0.8,0.1,0.80.8,0.7,0.1,0.80.7,0.7,0.1,0.80.7,0.7,0.3,0.80.6,0.7,0.2,0.70.7,0.9,0.2,0.70.7,0.9,0.2,0.70.8,0.8,0.2,0.70.6,0.9,0.1,0.80.6,0.7,0.3,0.80.7,0.7,0.2,0.7.In this case, the presented MADM model can be used for this MADM example, and then, its decision process is described in the following:Step 1: using equations (14) and (15), the aggregated OZNs zOZN1i and zOZN2i (i = 1, 2, 3, 4) are obtained below:zOZN11 = <(0.7000, 0.7778), (0.2000, 0.7991)>, zOZN12 = <(0.7570, 0.7300), (0.1000, 0.8268)>,zOZN13 = <(0.6767, 0.8190), (0.2240, 0.7267)>, and zOZN14 = <(0.6908, 0.7987), (0.1841, 0.7483)>,zOZN21 = <(0.7000, 0.7727), (0.2000, 0.8149)>, zOZN22 = <(0.7503, 0.7247), (0.1000, 0.8353)>,zOZN23 = <(0.6725, 0.7858), (0.2294, 0.7322)>, and zOZN24 = <(0.6728, 0.7757), (0.2011, 0.7551)>Step 2: using equation (16) with β1 = 0.5, the values of HOZNi are given below:HOZN1 = <(0.7000, 0.7752), (0.2000, 0.8069)>, HOZN2 = <(0.7537, 0.7274), (0.1000, 0.8310)>,HOZN3 = <(0.6746, 0.8031), (0.2267, 0.7294)>, and HOZN4 = <(0.6819, 0.7875), (0.1924, 0.7517)>Step 3: by equation (2), the values of C(HOZNi) for the alternatives Bi (i = 1, 2, 3, and 4) are obtained below:C(HOZN1) = 0.3813, C(HOZN2) = 0.4651, C(HOZN3) = 0.3764, and C(HOZN4) = 0.3924Step 4: sinceC(HOZN2) > C(HOZN4) > C(HOZN1) > C(HOZN3), the sorting order of the four applicants is B2 > B4 > B1 > B3. Hence, the best one is B2.To indicate the flexibility of the presented MADM model, we specify the weight values ofβ1 = 0, 0.5, and 1. Thus, the decision results of the presented MADM model are shown in Table 1.Table 1
The decision results of the presented MADM model with the weight values ofβ1 = 0, 0.5, and 1.
Weight valueC(zOZN1), C(zOZN2), C(zOZN3), C(zOZN4)SortingThe best oneβ1 = 00.3779, 0.4603, 0.3605, 0.3700B2 > B1 > B4 > B3B2β1 = 0.50.3813, 0.4651, 0.3764, 0.3924B2 > B4 > B1 > B3B2β1 = 10.3846, 0.4700, 0.3914, 0.4140B2 > B4 > B3 > B1B2Clearly, the sorting order of the alternatives in Table1 changes with the change of the weight value of β1, which verifies the decision-making flexibility. Then, the best selection is B2. However, the choice of the weight value of β1 depends on MD’s preference and/or requirements of actual applications. Under normal circumstances, the weight value is generally taken as β1 = 0.5 to reasonably make up for the prejudices of the two aggregation operators.


### 5.2. Comparative Analysis

For the convenience of comparative analysis with existing intuitionistic fuzzy MADM models [3, 4], if we ignore corresponding reliability levels and only consider IFVs in the decision matrix ZOZN = (zOZNik)4×4 as a special case of the above example, the decision matrix of OZNs degenerates into the decision matrix of IFVs:(18)ZIFN=zIFNik4×4=0.7,0.20.7,0.20.7,0.20.7,0.20.8,0.10.7,0.10.8,0.10.7,0.10.7,0.30.6,0.20.7,0.20.7,0.20.8,0.20.6,0.10.6,0.30.7,0.2,where zIFNik=TPik,FPik (i and k = 1, 2, 3, and 4) are IFVs for TPik,FPik∈0,1 and 0≤TPik+FPik≤1.Thus, equations (14) and (15) and equations (2) and (3) also degenerate into existing IFV arithmetic mean (IFVWAM) and IFV geometric mean (IFVWGM) operators and the score and accuracy functions of IFV [3, 4], respectively, below:(19)zIFVi=IFVWAMzIFVi1,zIFVi2,…,zIFVin=1−∏k=1n1−TPikαk,∏k=1nFPikαk,(20)zIFVi=IFVWGMzIFVi1,zIFVi2,…,zIFVin=∏k=1nTPikαk,1−∏k=1n1−FPikαk,(21)CzIFVi=TPi−FPiforCzIFVi∈−1,1,(22)EzIFVi=TPi+FPiforEzIFVi∈0,1.Using equations (19)–(22), the decision results obtained by the existing intuitionistic fuzzy MADM models [3, 4] are shown in Table 2. Then, the decision results obtained by the presented MADM model are also shown in Table 2.Table 2
Decision results regarding different MADM models.
MADM modelAggregated valueScore valueSortingPresented MADM model withβ1 = 1HOZN1 = <(0.7000, 0.7778), (0.2000, 0.7991)>HOZN2 = <(0.7570, 0.7300), (0.1000, 0.8268)>HOZN3 = <(0.6767, 0.8190), (0.2240, 0.7267)>HOZN4 = <(0.6908, 0.7987), (0.1841, 0.7483)>C(HOZN1) = 0.3846C(HOZN2) = 0.4700C(HOZN3) = 0.3914C(HOZN4) = 0.4140B2 > B4 > B3 > B1Presented MADM method withβ1 = 0.5HOZN1 = <(0.7000, 0.7752), (0.2000, 0.8069)>HOZN2 = <(0.7537, 0.7274), (0.1000, 0.8310)>HOZN3 = <(0.6746, 0.8031), (0.2267, 0.7294)>HOZN4 = <(0.6819, 0.7875), (0.1924, 0.7517)>C(HOZN1) = 0.3813C(HOZN2) = 0.4651C(HOZN3) = 0.3764C(HOZN4) = 0.3924B2 > B4 > B1 > B3Presented MADM method withβ1 = 0HOZN1 = <(0.7000, 0.7727), (0.2000, 0.8149)>HOZN2 = <(0.7503, 0.7247), (0.1000, 0.8353)>HOZN3 = <(0.6725, 0.7858), (0.2294, 0.7322)>HOZN4 = <(0.6728, 0.7757), (0.2011, 0.7551)>C(HOZN1) = 0.3779C(HOZN2) = 0.4603C(HOZN3) = 0.3605C(HOZN4) = 0.3700B2 > B1 > B4 > B3Existing MADM model using the IFNWAM operator [4]zIFV1 = <0.7000, 0.2000>zIFV2 = <0.7570, 0.1000>zIFV3 = <0.6767, 0.2240>zIFV4 = <0.6908, 0.1841>C(zIFV1) = 0.5000C(zIFV2) = 0.6570C(zIFV3) = 0.4527C(zIFV4) = 0.5067B2 > B4 > B1 > B3Existing MADM method using the IFNWGM operator [3]zIFV1 = <0.7000, 0.2000>zIFV2 = <0.7503, 0.1000>zIFV3 = <0.6725, 0.2294>zIFV4 = <0.6728, 0.2011>C(zIFV1) = 0.5000C(zIFV2) = 0.6503C(zIFV3) = 0.4431C(zIFV4) = 0.4716B2 > B1 > B4 > B3In Table2, there is the sorting difference between existing MADM models [3, 4] since there are different prejudices between the IFNWAM operator and the IFNWGM operator. Then, the presented MADM model can make up for the flaw. Meanwhile, the presented MADM model with different weight values of β1 can lead to different sorting orders, which demonstrates the flexible decision-making advantage, while the existing MADM models [3, 4] lack this flexibility. Especially, when β1 = 1, the sorting order of the presented MADM model using the unique OZNWAM operator, which is a special case of the presented MADM model, is different from the sorting orders of the existing MADM models [3, 4], which shows that the reliability measure information can affect the sorting order. However, the presented MADM model shows the importance and efficiency of the reliability measures and weight values of β1 in the MADM process. Then, the existing intuitionistic fuzzy MADM models [3, 4] only use the assessment information of the IFVs and the single aggregation operator, so they lack the reliability and flexibility of decision-making in the case of IFVs. Furthermore, the existing MADM models in the case of IFVs are only the special cases of the proposed MADM model and cannot perform MADM problems in the case of OZNs. Comparative analysis indicates that the presented MADM model can contain more advantages than the existing MADM models [3, 4] in the reliability and flexibility of decision results.


## 5.1. Illustrative Example

Today, various robots have been used in industry, household services, and medical fields. They are changing our daily lives in various ways. Then, the performance evaluation of the robot is an important issue in the user’s selection process. To illustrate the applicability of the presented MADM model in the case of OZNs, this section applies the proposed MADM model to the selection of industrial robots to illustrate its application in the case of OZNs.An industrial company wants to purchase an industrial robot to improve its existing manufacturing system. The technical department initially considers four types of potential industrial robots (alternatives), which are denoted as a setB = {B1, B2, B3, B4}. To choose the most suitable one, they must satisfy four performance requirements (attributes): carrying capacity (y1), dexterity and work space (y2), operation accuracy (y3), and communication and control capabilities (y4). The weigh vector of the four attributes is specified by α = (0.28, 0.26, 0.24, 0.22). Thus, the experts/DMs are invited to evaluate the four alternatives over the four attributes by OFVs and corresponding reliability levels, which are expressed as OZNs zOZNik=TPik,TQik,FPik,FQik(i and k = 1, 2, 3, and 4) for TPik,TQik∈0,1, FPik,FQik∈0,1, and 0≤TPik+FPik≤1. Thus, all OZNs can be constructed as their decision matrix:(17)ZOZN=zOZNik4×4=0.7,0.8,0.2,0.80.7,0.7,0.2,0.80.7,0.8,0.2,0.90.7,0.8,0.2,0.70.8,0.7,0.1,0.90.7,0.8,0.1,0.80.8,0.7,0.1,0.80.7,0.7,0.1,0.80.7,0.7,0.3,0.80.6,0.7,0.2,0.70.7,0.9,0.2,0.70.7,0.9,0.2,0.70.8,0.8,0.2,0.70.6,0.9,0.1,0.80.6,0.7,0.3,0.80.7,0.7,0.2,0.7.In this case, the presented MADM model can be used for this MADM example, and then, its decision process is described in the following:Step 1: using equations (14) and (15), the aggregated OZNs zOZN1i and zOZN2i (i = 1, 2, 3, 4) are obtained below:zOZN11 = <(0.7000, 0.7778), (0.2000, 0.7991)>, zOZN12 = <(0.7570, 0.7300), (0.1000, 0.8268)>,zOZN13 = <(0.6767, 0.8190), (0.2240, 0.7267)>, and zOZN14 = <(0.6908, 0.7987), (0.1841, 0.7483)>,zOZN21 = <(0.7000, 0.7727), (0.2000, 0.8149)>, zOZN22 = <(0.7503, 0.7247), (0.1000, 0.8353)>,zOZN23 = <(0.6725, 0.7858), (0.2294, 0.7322)>, and zOZN24 = <(0.6728, 0.7757), (0.2011, 0.7551)>Step 2: using equation (16) with β1 = 0.5, the values of HOZNi are given below:HOZN1 = <(0.7000, 0.7752), (0.2000, 0.8069)>, HOZN2 = <(0.7537, 0.7274), (0.1000, 0.8310)>,HOZN3 = <(0.6746, 0.8031), (0.2267, 0.7294)>, and HOZN4 = <(0.6819, 0.7875), (0.1924, 0.7517)>Step 3: by equation (2), the values of C(HOZNi) for the alternatives Bi (i = 1, 2, 3, and 4) are obtained below:C(HOZN1) = 0.3813, C(HOZN2) = 0.4651, C(HOZN3) = 0.3764, and C(HOZN4) = 0.3924Step 4: sinceC(HOZN2) > C(HOZN4) > C(HOZN1) > C(HOZN3), the sorting order of the four applicants is B2 > B4 > B1 > B3. Hence, the best one is B2.To indicate the flexibility of the presented MADM model, we specify the weight values ofβ1 = 0, 0.5, and 1. Thus, the decision results of the presented MADM model are shown in Table 1.Table 1
The decision results of the presented MADM model with the weight values ofβ1 = 0, 0.5, and 1.
Weight valueC(zOZN1), C(zOZN2), C(zOZN3), C(zOZN4)SortingThe best oneβ1 = 00.3779, 0.4603, 0.3605, 0.3700B2 > B1 > B4 > B3B2β1 = 0.50.3813, 0.4651, 0.3764, 0.3924B2 > B4 > B1 > B3B2β1 = 10.3846, 0.4700, 0.3914, 0.4140B2 > B4 > B3 > B1B2Clearly, the sorting order of the alternatives in Table1 changes with the change of the weight value of β1, which verifies the decision-making flexibility. Then, the best selection is B2. However, the choice of the weight value of β1 depends on MD’s preference and/or requirements of actual applications. Under normal circumstances, the weight value is generally taken as β1 = 0.5 to reasonably make up for the prejudices of the two aggregation operators.


## 5.2. Comparative Analysis

For the convenience of comparative analysis with existing intuitionistic fuzzy MADM models [3, 4], if we ignore corresponding reliability levels and only consider IFVs in the decision matrix ZOZN = (zOZNik)4×4 as a special case of the above example, the decision matrix of OZNs degenerates into the decision matrix of IFVs:(18)ZIFN=zIFNik4×4=0.7,0.20.7,0.20.7,0.20.7,0.20.8,0.10.7,0.10.8,0.10.7,0.10.7,0.30.6,0.20.7,0.20.7,0.20.8,0.20.6,0.10.6,0.30.7,0.2,where zIFNik=TPik,FPik (i and k = 1, 2, 3, and 4) are IFVs for TPik,FPik∈0,1 and 0≤TPik+FPik≤1.Thus, equations (14) and (15) and equations (2) and (3) also degenerate into existing IFV arithmetic mean (IFVWAM) and IFV geometric mean (IFVWGM) operators and the score and accuracy functions of IFV [3, 4], respectively, below:(19)zIFVi=IFVWAMzIFVi1,zIFVi2,…,zIFVin=1−∏k=1n1−TPikαk,∏k=1nFPikαk,(20)zIFVi=IFVWGMzIFVi1,zIFVi2,…,zIFVin=∏k=1nTPikαk,1−∏k=1n1−FPikαk,(21)CzIFVi=TPi−FPiforCzIFVi∈−1,1,(22)EzIFVi=TPi+FPiforEzIFVi∈0,1.Using equations (19)–(22), the decision results obtained by the existing intuitionistic fuzzy MADM models [3, 4] are shown in Table 2. Then, the decision results obtained by the presented MADM model are also shown in Table 2.Table 2
Decision results regarding different MADM models.
MADM modelAggregated valueScore valueSortingPresented MADM model withβ1 = 1HOZN1 = <(0.7000, 0.7778), (0.2000, 0.7991)>HOZN2 = <(0.7570, 0.7300), (0.1000, 0.8268)>HOZN3 = <(0.6767, 0.8190), (0.2240, 0.7267)>HOZN4 = <(0.6908, 0.7987), (0.1841, 0.7483)>C(HOZN1) = 0.3846C(HOZN2) = 0.4700C(HOZN3) = 0.3914C(HOZN4) = 0.4140B2 > B4 > B3 > B1Presented MADM method withβ1 = 0.5HOZN1 = <(0.7000, 0.7752), (0.2000, 0.8069)>HOZN2 = <(0.7537, 0.7274), (0.1000, 0.8310)>HOZN3 = <(0.6746, 0.8031), (0.2267, 0.7294)>HOZN4 = <(0.6819, 0.7875), (0.1924, 0.7517)>C(HOZN1) = 0.3813C(HOZN2) = 0.4651C(HOZN3) = 0.3764C(HOZN4) = 0.3924B2 > B4 > B1 > B3Presented MADM method withβ1 = 0HOZN1 = <(0.7000, 0.7727), (0.2000, 0.8149)>HOZN2 = <(0.7503, 0.7247), (0.1000, 0.8353)>HOZN3 = <(0.6725, 0.7858), (0.2294, 0.7322)>HOZN4 = <(0.6728, 0.7757), (0.2011, 0.7551)>C(HOZN1) = 0.3779C(HOZN2) = 0.4603C(HOZN3) = 0.3605C(HOZN4) = 0.3700B2 > B1 > B4 > B3Existing MADM model using the IFNWAM operator [4]zIFV1 = <0.7000, 0.2000>zIFV2 = <0.7570, 0.1000>zIFV3 = <0.6767, 0.2240>zIFV4 = <0.6908, 0.1841>C(zIFV1) = 0.5000C(zIFV2) = 0.6570C(zIFV3) = 0.4527C(zIFV4) = 0.5067B2 > B4 > B1 > B3Existing MADM method using the IFNWGM operator [3]zIFV1 = <0.7000, 0.2000>zIFV2 = <0.7503, 0.1000>zIFV3 = <0.6725, 0.2294>zIFV4 = <0.6728, 0.2011>C(zIFV1) = 0.5000C(zIFV2) = 0.6503C(zIFV3) = 0.4431C(zIFV4) = 0.4716B2 > B1 > B4 > B3In Table2, there is the sorting difference between existing MADM models [3, 4] since there are different prejudices between the IFNWAM operator and the IFNWGM operator. Then, the presented MADM model can make up for the flaw. Meanwhile, the presented MADM model with different weight values of β1 can lead to different sorting orders, which demonstrates the flexible decision-making advantage, while the existing MADM models [3, 4] lack this flexibility. Especially, when β1 = 1, the sorting order of the presented MADM model using the unique OZNWAM operator, which is a special case of the presented MADM model, is different from the sorting orders of the existing MADM models [3, 4], which shows that the reliability measure information can affect the sorting order. However, the presented MADM model shows the importance and efficiency of the reliability measures and weight values of β1 in the MADM process. Then, the existing intuitionistic fuzzy MADM models [3, 4] only use the assessment information of the IFVs and the single aggregation operator, so they lack the reliability and flexibility of decision-making in the case of IFVs. Furthermore, the existing MADM models in the case of IFVs are only the special cases of the proposed MADM model and cannot perform MADM problems in the case of OZNs. Comparative analysis indicates that the presented MADM model can contain more advantages than the existing MADM models [3, 4] in the reliability and flexibility of decision results.


## 6. Conclusion

In terms of the notion of truth and falsehood ZNs, the proposed OZN set can enhance the reliability measures of OFVs. Then, the presented score and accuracy functions of OZNs and weighted mean operation of the OZNWAM and OZNWGM operators provided reasonable modeling tools for MADM issues in the case of OZN sets. The established MADM model using the weighted mean operation of the OZNWAM and OZNWGM operators and score and accuracy functions solved such a flexible MADM problem with OZN information. Through an illustrative example and comparison with the existing MADM models, the presented MADM model indicated more advantages than the existing MADM models in the decision-making reliability and flexibility. In the application of the presented MADM model, different weight values ofβ1 can affect different sorting orders of alternatives and show some sensitivity to β1. Therefore, the presented MADM model with different weight values of β1 can satisfy the decision makers’ preference selection regarding some specified decision result and/or requirement of actual applications in uncertain and incomplete cases. Furthermore, the presented MADM model can overcome the flaw of using the single aggregation operator (either the OZNWAM operator or OZNWGM operator) and make up for the prejudices of both. Generally, the superiority of this study is that the presented MADM model not only strengthens the reliability level of OFVs but also makes the decision results more reliable and flexible than existing methods [3, 4, 24, 25, 28].However, the original research has contributed to the expression of OZN, the weighted average operation of the two OZN aggregation operators, and the MADM model with OZN information, but they cannot express and deal with decision-making problems with inconsistent and indeterminate information, which shows their limitations. In the future study, we shall further propose other aggregation operators and similarity measures of OZNs and apply them in pattern recognition, medical diagnosis/assessment, group decision modeling, etc.



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*Source: 6721297-2021-10-04.xml*