Abstract
Information reliability has a remarkable effect on decision-making outcome. Zadeh’s Z-number considers information fuzziness and reliability, and can, therefore, help the decision-maker to manage complicated problems. However, due to its complex construct, several issues concerning the computation of Z-numbers require further research. This study develops a novel likelihood-based method of comparing two Z-numbers to solve multi-criteria decision-making (MCDM) problems. Four main parts can be outlined. First, the likelihood of fuzzy restriction of Z-numbers is defined based on the conversion method of Z-number. Second, the likelihood of underlying probability distributions of Z-numbers is also proposed to compare the difference of randomness of Z-information. Third, by adding to a risk preference parameter, this study constructs a comprehensive weighted likelihood of Z-numbers. Finally, a likelihood-based qualitative flexible approach is extended to address the MCDM problems under Z-evaluation. In addition, a numerical example of the selection of ERP systems for ABC enterprise is placed to illustrate the applicability, validity, and effectiveness of the proposed method.
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Acknowledgements
The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions to help improve the overall quality of this paper. This work was supported by the National Natural Science Foundation of China (No. 71871228).
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Appendix I: Proof of the properties in Property 3
Appendix I: Proof of the properties in Property 3
Proof
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1.
According to Properties 1 and 2, \(0 \le L^{f} \left( {Z_{1} \succ Z_{2} } \right) \le 1\) and \(0 \le L^{p} \left( {Z_{1} \succ Z_{2} } \right) \le 1\) exists. In addition, \(L\left( {Z_{1} \succ Z_{2} } \right) = \omega L^{f} \left( {Z_{1} \succ Z_{2} } \right) + \left( {1 - \omega } \right)L^{p} \left( {Z_{1} \succ Z_{2} } \right)\). Therefore, \(0 \le L\left( {Z_{1} \succ Z_{2} } \right) \le 1\) can be proven.
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2.
From Properties 1 and 2, \(L^{f} \left( {Z_{1} \succ Z_{2} } \right) + L^{f} \left( {Z_{2} \succ Z_{1} } \right) = 1\) and \(L^{p} \left( {Z_{1} \succ Z_{2} } \right) + L^{p} \left( {Z_{2} \succ Z_{1} } \right) = 1\) exist. Moreover, \(L\left( {Z_{1} \succ Z_{2} } \right) = \omega L^{f} \left( {Z_{1} \succ Z_{2} } \right) + \left( {1 - \omega } \right)L^{p} \left( {Z_{1} \succ Z_{2} } \right)\). Thus, \(L\left( {Z_{1} \succ Z_{2} } \right) + L\left( {Z_{2} \succ Z_{1} } \right) = 1\) can be obtained.
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3.
Based on Definitions 7, 8, and 9, if \(Z_{1}\) and \(Z_{2}\) are the same, then both the likelihoods of fuzzy restriction and underlying probability distributions are the same between two Z-numbers. Therefore, \(L\left( {Z_{1} \succ Z_{2} } \right) = L\left( {Z_{2} \succ Z_{1} } \right) = 0.5\).
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Qiao, D., Wang, Yt., Wang, Jq. et al. Likelihood-based qualitative flexible approach to ranking of Z-numbers in decision problems. Comp. Appl. Math. 39, 134 (2020). https://doi.org/10.1007/s40314-020-01167-x
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DOI: https://doi.org/10.1007/s40314-020-01167-x