Abstract
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.
Highlights
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
The sum of the truth- and falsehood-membership degrees may be more than one in some cases. To represent such an orthopair fuzzy value (OFV), some researchers defined Pythagorean fuzzy sets (PFSs) [16, 17] and Pythagorean fuzzy aggregation algorithms [18,19,20,21,22,23] for multiattribute decision-making (MADM). en, other researchers introduced a hybrid method of PFSs for MADM [24] and a digraph and matrix approach for risk evaluations [25]
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
Summary
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. Under the indeterminate case of the truth- and falsehood-membership degrees, Ye et al [33] defined an orthopair indeterminate set/value (OIS/OIV) and Journal of Mathematics some aggregation operators of OIVs and established an orthopair indeterminate MADM model with indeterminate ranges/risks of decision makers (DMs) in the case of OISs. IFV, OFV, q-ROFV, and OIV cannot contain their reliability measures/levels. 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.
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