Abstract
A refined single-valued/interval neutrosophic set is very suitable for the expression and application of decision-making problems with both attributes and sub-attributes since it is described by its refined truth, indeterminacy, and falsity degrees. However, existing refined single-valued/interval neutrosophic similarity measures and their decision-making methods are scarcely studied in existing literature and cannot deal with this decision-making problem with the weights of both attributes and sub-attributes in a refined interval and/or single-valued neutrosophic setting. To solve the issue, this paper firstly introduces a refined simplified neutrosophic set (RSNS), which contains the refined single-valued neutrosophic set (RSVNS) and refined interval neutrosophic set (RINS), and then proposes vector similarity measures of RSNSs based on the Jaccard, Dice, and cosine measures of simplified neutrosophic sets in vector space, and the weighted Jaccard, Dice, and cosine measures of RSNSs by considering weights of both basic elements and sub-elements in RSNS. Further, a decision-making method with the weights of both attributes and sub-attributes is developed based on the weighted Jaccard, Dice, and cosine measures of RSNSs under RSNS (RINS and/or RSVNS) environments. The ranking order of all the alternatives and the best one can be determined by one of weighted vector similarity measures between each alternative and the ideal solution (ideal alternative). Finally, an actual example on the selecting problem of construction projects illustrates the application and effectiveness of the proposed method.
Highlights
Since fuzzy set theory was introduced by Zadeh [1] in 1965, it has been successfully applied to decision-making areas, and fuzzy decision-making has become a research focal point since .With the increasing complexity of decision-making problems in actual applications, the fuzzy set is not suitable for fuzzy expression, which involves the membership degree and non-membership degree
This paper firstly introduces a refined simplified neutrosophic set (RSNS), which includes the concepts of refined single-valued neutrosophic set (RSVNS) and refined interval neutrosophic set (RINS), and proposes the vector similarity measures of RSNSs based on the Jaccard, Dice, and cosine measures between SNSs in vector space [8]
This paper introduced RSNSs, including the concepts of RSVNSs and RINSs, and proposed the vector similarity measures of RSNSs, including the Jaccard, Dice, and cosine measures between
Summary
Since fuzzy set theory was introduced by Zadeh [1] in 1965, it has been successfully applied to decision-making areas, and fuzzy decision-making has become a research focal point since . In existing literature [22,23], the similarity measures of RSVNSs and RINSs and their decision-making methods only took into account the basic element (single-valued/interval neutrosophic number in RSVNS/RINS)/attribute weights rather than sub-element/sub-attribute weights (weights of refined elements/refined attributes) in the measures of RSVNSs and RINSs and their decision-making methods To overcome these drawbacks, this paper firstly introduces a refined simplified neutrosophic set (RSNS), which includes the concepts of RSVNS and RINS, and proposes the vector similarity measures of RSNSs based on the Jaccard, Dice, and cosine measures between SNSs in vector space [8]. A decision-making method is established based on the Jaccard/Dice/cosine measures between RSNSs to solve multiple attribute decision-making problems with both attribute weights and sub-attribute weights under refined simplified (interval and/or single-value) neutrosophic environments.
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