Similarity measures based on the generalized distance of neutrosophic Z-number sets and their multi-attribute decision making method

Zadeh firstly proposed a Z-number with help of both a fuzzy value and a reliability measure in uncertain environment. Its highlight is that the fuzzy value is closely related to the reliability measure. However, the Z-number lacks the indeterminacy and falsity fuzzy values and reliability measures. Although a single-valued neutrosophic set is described by the truth, indeterminacy and falsity membership degrees, it lacks the reliability measures of the truth, indeterminacy and falsity membership degrees. By the hybrid information of the truth, indeterminacy and falsity Z-numbers, a neutrosophic Z-number (NZN) can be introduced to express the inconsistent, incomplete and indeterminate knowledge and judgments of human cognitions in the real world. Then, the similarity measure is an important mathematical tool for decision making (DM) and pattern recognition, but there is no similarity measure of any NZN set in the existing literature. Motivated by this idea, this study firstly proposes NZN sets and generalized distance and similarity measures between NZN sets and then further introduces cosine and cotangent similarity measures based on the weighted generalized distance of NZN sets. Next, a multi-attribute DM method using the proposed similarity measures is established in the setting of NZN sets. An illustrative example is provided to demonstrate the feasibility and rationality of the proposed method, and then the comparative analysis indicates the two highlighting advantages: (a) NZN enriches the neutrosophic information expression by means of the truth, indeterminacy and falsity Z-numbers and (b) the established DM method based on the proposed similarity measures not only enhances the reliability of DM problems, but also shows its superiority over existing neutrosophic DM methods.