Similarity measure with indeterminate parameters regarding cubic hesitant neutrosophic numbers and its risk grade assessment approach for prostate cancer patients

PT-MARCOS multi-attribute decision-making method under neutrosophic cubic environment

Neutrosophic cubic set (NCS) can process complex information by combining interval neutrosophic set and single-valued neutrosophic set. It can simultaneously describe the uncertain and certain part of information. Prospect theory (PT) is based on bounded rationality and can reflect decision maker’s different risk attitudes to gains and losses. Measurement of Alternatives and Ranking according to COmpromise Solution (MARCOS) method can measure and rank the alternatives according to compromise solution. Considering the bounded rationality of decision makers and compromise solution of alternatives, this paper combines the PT with MARCOS method to neutrosophic cubic environment to solve multi-attribute decision-making problem. First, the theoretical basis of NCS is introduced. Second, the PT and MARCOS method are combined. To reflect subjective views of decision makers and the objectivity of decision-making information, this paper uses geometric average method to combine subjective weights (calculated by the best-worst method) and objective weights (calculated ed by the entropy method). Then, the PT-MARCOS method is applied to a decision-making problem. Further, a sensitivity analysis is conducted to study the influence of different attenuation factor values and different expectation coefficient on the ranking; and through comparative analysis to illustrate the superiority of the PT-MARCOS method. Finally is the conclusion.

Hybrid weighted aggregation operator of cubic fuzzy-consistency elements and their group decision-making model in cubic fuzzy multi-valued setting

The ambiguity and uncertainty of human cognition about actual engineering problems are very challenging and indispensable issues in the information expression and aggregation. However, existing various cubic (hesitant) concepts may not reasonably represent the hybrid information of both an interval-valued fuzzy value and a fuzzy sequence with identical and/or different fuzzy values, which commonly occurs in engineering fields. To express the hybrid information, this paper first proposes the notion of a cubic fuzzy multi-valued set as a new extension of existing cubic (hesitant) notions and defines operational relations of cubic fuzzy multi-valued elements. To obtain reasonable operations between different fuzzy sequence lengths in cubic fuzzy multi-valued elements, cubic fuzzy multi-valued elements are transformed into cubic fuzzy-consistency elements based on the average value and consistency degree/level (complement of standard deviation) of a fuzzy sequence in a cubic fuzzy multi-valued element. Next, we present operations of cubic fuzzy-consistency elements and an expected value of a cubic fuzzy-consistency element for ranking cubic fuzzy-consistency elements. Further, we propose a cubic fuzzy-consistency hybrid weighted arithmetic and geometric averaging operator, and then develop a multi-attribute group decision-making model using the cubic fuzzy-consistency hybrid weighted arithmetic and geometric averaging operator and expected value of cubic fuzzy-consistency elements to solve group decision-making problems under the cubic fuzzy multi-valued environment. To reflect the feasibility and effectiveness of the developed group decision-making model, the developed group decision-making model is utilized in an example on the selection problem of slope design schemes regarding an open pit mine in the cubic fuzzy multi-valued environment. Comparative analysis indicates the flexibility and rationality of the developed group decision-making model.

Particle swarm optimization for the shortest path problem

Solving the shortest path problem is very difficult in situations such as emergency rescue after a typhoon: road-damage caused by a typhoon causes the weight of the rescue path to be uncertain and impossible to represent using single, precise numbers. In such uncertain environments, neutrosophic numbers can express the edge distance more effectively: membership in a neutrosophic set has different degrees of truth, indeterminacy, and falsity. This paper proposes a shortest path solution method for interval-valued neutrosophic graphs using the particle swarm optimization algorithm. Furthermore, by comparing the proposed algorithm with the Dijkstra, Bellman, and ant colony algorithms, potential shortcomings and advantages of the proposed method are deeply explored, and its effectiveness is verified. Sensitivity analysis performed using a 2020 typhoon as a case study is presented, as well as an investigation on the efficiency of the algorithm under different parameter settings to determine the most reasonable settings. Particle swarm optimization is a promising method for dealing with neutrosophic graphs and thus with uncertain real-world situations.

Multiple attribute group decision-making based on cubic linguistic Pythagorean fuzzy sets and power Hamy mean

The linguistic Pythagorean fuzzy sets (LPFSs), which employ linguistic terms to express membership and non-membership degrees, can effectively deal with decision makers’ complicated evaluation values in the process of multiple attribute group decision-making (MAGDM). To improve the ability of LPFSs in depicting fuzzy information, this paper generalized LPFSs to cubic LPFSs (CLPFSs) and studied CLPFSs-based MAGDM method. First, the definition, operational rules, comparison method and distance measure of CLPFSs are investigated. The CLPFSs fully adsorb the advantages of LPFSs and cubic fuzzy sets and hence they are suitable and flexible to depict attribute values in fuzzy and complicated decision-making environments. Second, based on the extension of power Hamy mean operator in CLPFSs, the cubic linguistic Pythagorean fuzzy power average operator, the cubic linguistic Pythagorean fuzzy power Hamy mean operator as well as their weighted forms were introduced. These aggregation operators can effectively and comprehensively aggregate attribute values in MAGDM problems. Besides, some important properties of these operators were studied. Finally, we presented a new MAGDM method based on CLPFSs and their aggregation operators. Illustrative examples and comparative analysis are provided to show the effectiveness and advantages of our proposed decision-making method.

Multifuzzy Cubic Sets and Their Correlation Coefficients for Multicriteria Group Decision-Making

The notion of multifuzzy sets (MFSs) or multi-interval-valued fuzzy sets (MIVFSs) provides a new method to represent some problems with a sequence of the different and/or same fuzzy/interval-valued fuzzy membership values of an element to the set. Then, a fuzzy cubic set (FCS) consists of a certain part (a fuzzy value) and an uncertain part (an interval-valued fuzzy value) but cannot represent hybrid information of both MFS and MIVFS. To adequately depict the opinion of several experts/decision-makers by using a union/sequence of the different and/or same fuzzy cubic values for an object assessed in group decision-making (GDM) problems, this paper proposes a multifuzzy cubic set (MFCS) notion as the conceptual extension of FCS to express the hybrid information of both MFS and MIVFS in the fuzzy setting of both uncertainty and certainty. Then, we propose three correlation coefficients of MFCSs and then introduce correlation coefficients of MFSs and MIVFSs as special cases of the three correlation coefficients of MFCSs. Further, the multicriteria GDM methods using three weighted correlation coefficients of MFCSs are developed under the environment of MFCSs, which contains the MFS and MIVFS GDM methods. Lastly, these multicriteria GDM methods are applied in an illustrative example on the selection problem of equipment suppliers; then their decision results and comparative analysis indicate that the developed GDM methods are more practicable and effective and reflect that either different correlation coefficients or different information expressions can also impact on the ranking of alternatives. Therefore, this study indicates the main contribution of the multifuzzy cubic information expression, correlation coefficients, and GDM methods in the multifuzzy setting of both uncertainty and certainty.