A novel cardinal-normalization method for Probabilistic Hesitant Fuzzy Elements with incomplete information

Abstract

Owing to the lack of information, it is more realistic that the sum of probabilities is less than or equal to one in the probabilistic hesitant fuzzy elements (P-HFEs). Probabilistic-normalization method and cardinal-normalization method are common processing methods for the P-HFEs with incomplete information. However, the existed probabilistic-normalization method of sharing the remaining probabilities will lose information and change the information integrity of the P-HFEs. The first existed cardinal-normalization method of adding maximum or minimum membership degree with probability zero are influenced by the subjectivity of the decision makers. And the second existed cardinal-normalization method named as reconciliation method only applicable to the P-HFEs with complete information. Aiming at solving those shortcomings, we propose a possibility degree method based on a novel cardinal-normalization method for the sake of comparing the P-HFEs in pairs. In the process of comparison, the information integrity remains unchanged. Then, we propose a multi-criteria decision making (MCDM) problem, where the attribute weight is determined by entropy measures of the integration results. Finally, an application case in green logistics area is given for the sake of illustrating the efficiency of the proposed method, where the evaluation values are given in the P-HFEs form with incomplete information. Numerical and theoretical results show that a MCDM problem based on the proposed cardinal-normalization method and possibility degree method have a wide range of application.