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

The Z 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.


Introduction
In the case of fuzzy decision, fuzzy information expression and aggregation are two key issues. en, a fuzzy set [1] only contains a truth-membership degree, but lacks a falsehoodmembership degree. To make up for this issue, Atanassov [2] defined an intuitionistic fuzzy set (IFS) that contains truthand 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][6][7][8][9][10][11][12][13][14][15] for MADM. en, 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][19][20][21][22][23] for MADM. en, 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][27][28] was introduced to flexibly present the hybrid information range of the truth-and falsehood-membership degrees regarding different values of the parameter q. en, some aggregation operators of q-rung orthopair fuzzy sets/ values (q-ROFSs/q-ROFVs) [29][30][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. en, 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. erefore, ZNs have been used for some applications [35][36][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. en, 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. e rest of this article contains the following parts: Section 2 presents an OZN set and the operations, score and accuracy functions, and sorting rules of OZNs. en, 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.

Orthopair Z-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.

Two Basic Aggregation Operators of OZNs
is section presents two basic aggregation operators of OZNs based on the operations (7)- (9) in Definition 2 and their properties.

OZNWAM Operator.
For a group of OZNs, the OZNWAM operator can be proposed from the operations (6) and (8) in Definition 2.
. ., n) as a group of OZNs. us, the aggregated value of the OZNWAM operator is still OZN, which is calculated by the following formula: where α k (k � 1, 2, . . ., n) is the weight of z OZNk for 0 ≤ α k ≤ 1 and n k�1 α k � 1.
Proof. In terms of mathematical induction, equation (5) can be verified in the following way.
Hence, the proof is completed.
e OZNWAM operator of equation (5) implies the following properties: . ., n) be a group of OZNs, and let the minimum and maximum OZNs be in the following:
Similar to the proof process of eorem 1, equation (12) can be also verified, which is omitted.

Theorem 4.
e OZNWGM operator of equation (12)  Clearly, the above properties of the OZNWGM operator can be also verified by the similar proof way of eorem 2, which is omitted.

MADM Model in terms of the Weighted Mean Operation of the OZNWAM and OZNWGM Operators
is 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.
Step 4: the alternatives are sorted in terms of the sorting rules, and the best one is chosen.

Illustrative Example.
Today, various robots have been used in industry, household services, and medical fields. ey are changing our daily lives in various ways. en, 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. e technical department initially considers four types of potential industrial robots (alternatives), which are denoted as a set B � {B 1 , B 2 , B 3 , B 4 }. To choose the most suitable one, they must satisfy four performance requirements (attributes): carrying capacity (y 1 ), dexterity and work space (y 2 ), operation accuracy (y 3 ), and communication and control capabilities (y 4 ). e weigh vector of the four attributes is specified by α � (0.28, 0.26, 0.24, 0.22). us, 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 z OZNik � 〈 (TP ik , TQ ik ), (FP ik , FQ ik )〉(i and k � 1, 2, 3, and 4) for TP ik , TQ ik ∈ [0, 1], FP ik , FQ ik ∈ [0, 1], and 0 ≤ TP ik + FP ik ≤ 1. us, all OZNs can be constructed as their decision matrix: 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 3: by equation (2), the values of C(H OZNi ) for the alternatives B i (i � 1, 2, 3, and 4) are obtained below: Step 4: since C(H OZN2 ) > C(H OZN4 ) > C(H OZN1 ) > C(H OZN3 ), the sorting order of the four applicants is Hence, the best one is B 2 .
To indicate the flexibility of the presented MADM model, we specify the weight values of β 1 � 0, 0.5, and 1.
us, the decision results of the presented MADM model are shown in Table 1.
Clearly, the sorting order of the alternatives in Table 1 changes with the change of the weight value of β 1 , which verifies the decision-making flexibility. en, the best selection is B 2 . However, the choice of the weight value of β 1 depends on MD's preference and/or requirements of actual applications. Under normal circumstances, the weight value is generally taken as β 1 � 0.5 to reasonably make up for the prejudices of the two aggregation operators.

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 Z OZN � (z OZNik ) 4×4 as a special case of the above example, the decision matrix of OZNs degenerates into the decision matrix of IFVs:    (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: Using equations (19)- (22), the decision results obtained by the existing intuitionistic fuzzy MADM models [3,4] are shown in Table 2.
en, the decision results obtained by the presented MADM model are also shown in Table 2.
In Table 2, there is the sorting difference between existing MADM models [3,4] since there are different prejudices between the IFNWAM operator and the IFNWGM operator. en, the presented MADM model can make up for the flaw. Meanwhile, the presented MADM model with different weight values of β 1 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 β 1 � 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 β 1 in the MADM process. en, 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.

Conclusion
In terms of the notion of truth and falsehood ZNs, the proposed OZN set can enhance the reliability measures of OFVs. en, 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. e 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.
rough 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 of β 1 can affect different sorting orders of alternatives and show some sensitivity to β 1 . erefore, the presented MADM model with different weight values of β 1 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.

Data Availability
e data used to support the findings of this study are included within this article. 8 Journal of Mathematics