Likelihood-based qualitative flexible approach to ranking of Z-numbers in decision problems

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.

Appendix I: Proof of the properties in Property 3

Proof

• 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.

• 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.

• 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, $L\left( {Z_{1} \succ Z_{2} } \right) = L\left( {Z_{2} \succ Z_{1} } \right) = 0.5$.