Abstract
As a useful mathematical tool to solve the multi-attribute decision-making (MADM) problems, Pythagorean hesitant fuzzy set (PHFS) permits decision makers (DMs) to give several evaluation values in membership and non-membership degrees. Given that much cognitive preferences of DMs may be hidden in the original evaluation information and cannot be fully expressed, we propose the normal wiggly Pythagorean hesitant fuzzy set (NWPHFS) to comprehensively mine the uncertain preferences from original Pythagorean hesitant fuzzy information. NWPHFS can help DMs more accurately express their potential valuable preferences for objects while giving the original evaluation information. We put forward some operational laws and comparison rules of NWPHFS, then present the formulas of projection and bidirectional projection measures between normal wiggly Pythagorean hesitant fuzzy elements (NWPHFEs). Furthermore, we propose the normal wiggly Pythagorean hesitant fuzzy bidirectional projection (NWPHFBP) method and its specific application steps to solve MADM problems. The proposed approach can accurately reflect the projection relationship between the alternatives evaluated and ideal solutions by the NWPHF information. Through constructing a reasonable criteria system, we apply the NWPHFBP method to the issue of electric vehicle (EV) power battery recycling mode selection. Finally, we conduct the sensitivity analysis and verify the effectiveness of the proposed method by comparisons with other approaches.
Highlights
Multi-attribute decision-making (MADM) is a complex and common problem in daily life
We present the formula of projection measure between normal wiggly Pythagorean hesitant fuzzy elements (NWPHFEs), and based on which, we derive the bidirectional projection measure to reflect the projection relationship between alternatives and ideal solutions
We should not expect that the PHFEs include all of uncertain information of decision makers (DMs)
Summary
Multi-attribute decision-making (MADM) is a complex and common problem in daily life. NWPHFSs can dig the deeper uncertain information from the original Pythagorean hesitant fuzzy information automatically In this way, NWPHFS allows DMs to flexibly assign evaluation values like PHFS, and reflects the potential preferences of DMs, which facilitates them to obtain a more accurate result. Given that the above methods only consider the original evaluation information, but ignore the impact of DMs’ concealed preferences on evaluation results, we propose the normal wiggly Pythagorean hesitant fuzzy bidirectional projection (NWPHFBP) method, and based on which, construct the optimization weighting model. We put forward the normal wiggly Pythagorean hesitant fuzzy bidirectional projection (NWPHFBP) method and its specific steps to solve the MADM problems.
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