Surapati Pramanik, Ph. D.

The purpose of this study is to propose new similarity measures namely rough variational coefficient similarity measure under the rough neutrosophic environment. The weighted rough variational coefficient similarity measure has been also defined. The weighted rough variational coefficient similarity measures between the rough ideal alternative and each alternative are calculated to find the best alternative. The ranking order of all the alternatives can be determined by using the numerical values of similarity measures. Finally, an illustrative example has been provided to show the effectiveness and validity of the proposed approach. Comparisons of decision results of existing rough similarity measures have been provided.

This paper is devoted to propose triangular fuzzy number neutrosophic sets by combining triangular fuzzy numbers with single valued neutrosophic set and define some of its operational rules. Then, triangular fuzzy number neutrosophic weighted arithmetic averaging operator and triangular fuzzy number neutrosophic weighted geometric averaging operator are defined to aggregate triangular fuzzy number neutrosophic sets. We have also established some of their properties of the proposed operators. The operators have been employed to multi attribute decision making problem to aggregate the triangular fuzzy neutrosophic numbers based rating values of each alternative over the attributes. The collective rating values of each alternative have been ordered with the help of score and accuracy values to find out the best alternative. Finally, an illustrative example has been provided to validate the proposed approach for multi attribute decision making problem.

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a popular strategy for Multi-Attribute Decision Making (MADM). In this paper, we extend the TOPSIS strategy of MADM problems in trapezoidal neutrosophic number environment.

In this paper, the sine, cosine and cotangent similarity<br>measures of interval rough neutrosophic sets is proposed. Some properties of the proposed measures are discussed. We have proposed multi attribute decision making approaches based on proposed similarity measures. To demonstrate the applicability, a numerical example is solved.

In this paper, we define correlation coefficient measure between any two rough neutrosophic sets. We also prove some of its basic properties. We develop a new multiple attribute group decision making method based on the proposed correlation coefficient measure. An illustrative example of medical diagnosis is solved to demonstrate the applicability and effecriveness of the proposed method.

The notion of neutrosophic cubic set is originated from the hybridization of the concept of neutrosophic set and interval valued neutrosophic set. We define similarity measure for neutrosophic cubic sets and prove some of its basic properties. We present a new multi criteria group decision making method with linguistic variables in neutrosophic cubic set environment. Finally, we present a numerical example to demonstrate the<br> usefulness and applicability of the proposed method.

Single-valued neutrosophic hesitant fuzzy set is a merged form of single-valued neutrosophic sets and hesitant fuzzy sets. This set is a useful tool to handle imprecise, incomplete and inconsistent information existing in multi-attribute decision making problems. In multi-attribute decision making, distance measures play an important role to take a decision regarding alternatives. In this paper we propose a variety of distance measures for single valued neutrosophic sets. Furthermore, we apply these measures to multi-attribute decision making problem with singlevalued neutrosophic hesitant fuzzy set environment to find out the best alternative. We provide an illustrative example to validate and to show fruitfulness of the proposed approach. Finally, we compare the proposed approach with other existing methods for solving multi-attribute decision making problems.

In this study, we give some concepts concerning the neutrosophic sets, single valued neutrosophic sets, interval-valued neutrosophic sets, bipolar neutrosophic sets, neutrosophic hesitant fuzzy sets, inter-valued neutrosophic hesitant fuzzy sets, refined neutrosophic sets, bipolar neutrosophic refined sets, multi-valued neutrosophic sets, simplified neutrosophic linguistic sets, neutrosophic over/off/under sets, rough neutrosophic sets, rough bipolar neutrosophic sets, rough neutrosophic hyper-complex set, and their basic operations. Then we introduce triangular neutrosophic numbers, trapezoidal neutrosophic fuzzy number and their basic operations. Also some comparative studies between the existing neutrosophic sets and neutrosophic number are provided.

The paper proposes TOPSIS method for solving multi-objective multi-level programming problem (MO-MLPP) with fuzzy parameters via fuzzy goal programming (FGP). At first, - cut method is used to transform the fuzzily described MO-MLPP into deterministic MO-MLPP. Then, for specific , we construct the membership functions of distance functions from positive ideal solution (PIS) and negative ideal solution (NIS) of all level decision makers (DMs). Thereafter, FGP based multi-objective decision model is established for each level DM for obtaining individual optimal solution. A possible relaxation on decisions for all DMs is taken into account for satisfactory solution. Subsequently, two FGP models are developed and compromise optimal solutions are found by minimizing the sum of negative deviational variables. To recognize the better compromise optimal solution, the concept of distance functions is utilized. Finally, a novel algorithm for MO-MLPP involving fuzzy parameters is provided and ...

The objective of the paper is to introduce new similarity measure for single valued neutrosophic sets based on logarithm function. We define logarithm similarity measure and their weighted similarity measure for single valued neutrosophic sets. Then we define hybrid logarithm similarity measure and weighted hybrid logarithm similarity measure for single valued neutrosophic sets. We prove the basic properties of the proposed measures. We then define an entropy function using logarithm function to determine unknown attribute weights. We develop a novel multi attribute group decision making strategy for single valued neutrosophic sets based on the weighted hybrid logarithm similarity measure. We address an illustrative example to demonstrate the effectiveness and aptness of the proposed strategies. We conduct a sensitivity analysis of the developed strategy. We also make a comparison between the obtained results from proposed strategies and different existing strategies in the literature.

Bipolar neutrosophic set is an important extension of bipolar fuzzy set. This set is a hybridization of bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic set consists of three independent positive membership functions and three independent negative membership functions. In this paper, we develop cross entropy measures of bipolar neutrosophic sets and prove its properties. We also define cross entropy measures of interval bipolar neutrosophic sets and prove its properties. Thereafter, we develop two novel multi-attribute decision making methods based on the proposed cross entropy measures. In the decision making framework, we calculate the weighted cross entropy measures between each alternative and the ideal alternative to rank the alternatives and choose the best one. We solve two illustrative examples of multi-attribute decision making problems and compare the obtained result with the results of other existing methods to show the applicability and eff...

Single valued neutrosophic set has king power to express uncertainty characterized by indeterminacy, inconsistency and incompleteness. Most of the existing single valued neutrosophic cross entropy bears an asymmetrical behavior and produce an undefined phenomenon in some situations. In order to deal with these disadvantages, we propose a new cross entropy measure under single valued neutrosophic set (SVNS) environment namely SN- cross entropy and prove its basic properties. Also we define weighted SN-cross entropy measure and investigate its basic properties. We develop a new multi attribute group decision making (MAGDM) strategy for ranking of the alternatives based on the proposed weighted SN-cross entropy measure between each alternative and the ideal alternative. Finally, a numerical example of MAGDM problem of investment potential is solved to show the validity and efficiency of proposed decision making strategy. We also present comparative anslysis of the obtained result with ...

The concept of neutrosophic number is a significant mathematical tool to deal with real scientific problems because it can tackle indeterminate and incomplete information which exists generally in real problems. In this article, we use neutrosophic numbers (a + bI), where a and bI denote determinate component and indeterminate component respectively. We explore the situations in which the input information is needed to express in terms of neutrosophic numbers. We define score functions and accuracy functions for ranking neutrosophic numbers. We then define a cosine function to determine unknown criteria weights. We define neutrosophic number harmonic mean operators and proved their basic properties. Then, we develop two novel MCGDM strategies using the proposed aggregation operators. We solve a numerical example to demonstrate the feasibility and effectiveness of the proposed two strategies. Sensitivity analysis with variation of “I” on neutrosophic numbers is performed ...

The paper proposes two fuzzy goal programming (FGP) models to multi-level linear fractional programming problem (MLFPP) with single decision maker (DM) at each level. The linear fractional membership functions for the fuzzily described linear fractional objective functions of level decision makers are developed first. First order Taylor polynomial series is employed in order to transform the linear fractional membership functions into equivalent linear membership functions. Possible relaxation on decision of each level DM is considered for overall benefit of the hierarchical organization. Then the FGP technique is utilized to achieve compromise optimal solution of the multi-level system by minimizing negative deviational variables. The proposed FGP approach is then extended to solve multi-objective MLFPP. Multi-objective decision making model for the each level DM is formulated in order to yield his or her own satisfactory solution in the decision making process. Then, two new FGP m...

In this paper, multi attribute decision making problem based on grey relational analysis in neutrosophic cubic set environment is investigated. In the decision making situation, the attribute weights are considered as single valued neutrosophic sets. The neutrosophic weights are converted into crisp weights. Both positve and negative GRA coefficients, and weighted GRA coefficients are determined.

Many multi-attribute group decision-making (MAGDM) strategies have been introduced in the literature to deal with decision-
making problems in uncertain environment. Many of them are based on fuzzy numbers and they are not able to cope with
indeterminacy and inconsistency involving in decision making. In recent years, some neutrosophic multi-attribute group
decision-making strategies have been successfully developed to deal with uncertainty, indeterminacy, and inconsistency in
decision making. Among them, TODIM (an acronym in Portuguese of interactive and multiple attribute decision making)
strategy based on prospect theory has received more attention due to its great performance in considering the bounded
rationality of decision makers. In this paper, we develop a TODIM strategy to deal with multi-attribute group decision-
making problem in trapezoidal neutrosophic numbers environment. To establish the TODIM strategy, we employ score
function, accuracy function, and Hamming distance function for trapezoidal neutrosophic numbers. Lastly, we solve an
illustrative numerical example to show the applicability and usefulness of the proposed strategy. A comparison analysis is
also provided.

Abstract
Purpose: The purpose of the paper is to propose goal programming strategy to multi-objective linear programming problem with neutrosophic numbers which we call NN-GP. The coefficients of objective functions and the constraints are considered as
neutrosophic numbers of the form (m+nI), where m, n are real numbers and I denotes indeterminacy.
Design: For this study, the neutrosophic numbers are converted into interval numbers. Then, the problem reduces to multi-objective linear interval programming problem. Employing interval programming technique, the target interval of the objective function is determined. For the sake of achieving the target goals, the goal achievement
functions are constructed. Three new neutrosophic goal programming models are developed using deviational variables to solve the reduced problem.
Findings: Realistic optimization problem involves multiple objectives. Crisp multi-objective optimization problems involve deterministic objective functions and/or constrained functions. However, uncertainty involves in real problems. Hence, several strategies dealing with uncertain multi-objective programming problems have been
proposed in the literature. Multi-objective linear programming has evolved along with different paradigms and in different environment. Goal programming and fuzzy goal programming have been widely used to solve the multi-objective linear programming problems. In this paper goal programming in neutrosophic number environment has
been developed. It deals with effectively multi-objective linear programming problem with neutrosophic numbers. We solve a numerical example to illustrate the proposed NN-GP strategy.
Originality: There are different Schools in optimization field and each has their own distinct strategy. In neutrosophic number environment goal programming for multi-objective programming problem is proposed here at first.

The objective of the paper is to introduce a new cross entropy measure in a neutrosophic cubic set (NCS) environment, which we call NC-cross entropy measure. We prove its basic properties. We also propose weighted NC-cross entropy and investigate its basic properties. We develop a novel multi attribute decision-making (MADM) strategy based on a weighted NC-cross entropy measure. To show the feasibility and applicability of the proposed multi attribute decision-making strategy, we solve an illustrative example of the multi attribute decision-making problem. Keywords: single valued neutrosophic set (SVNS); interval neutrosophic set (INS); neutrosophic cubic set (NCS); multi attribute decision-making (MADM); NC-cross entropy measure

This paper is devoted to propose triangular fuzzy number neutrosophic sets by combining triangular fuzzy numbers with single valued neutrosophic set and define some of its operational rules. Then, triangular fuzzy number neutrosophic weighted arithmetic averaging operator and triangular fuzzy number neutrosophic weighted geometric averaging operator are defined to aggregate triangular fuzzy number neutrosophic sets. We have also established some of their properties of the proposed operators. The operators have been employed to multi attribute decision making problem to aggregate the triangular fuzzy neutrosophic numbers based rating values of each alternative over the attributes. The collective rating values of each alternative have been ordered with the help of score and accuracy values to find out the best alternative. Finally, an illustrative example has been provided to validate the proposed approach for multi attribute decision making problem. Keywords: Triangular fuzzy number neutrosophic set, Score and accuracy function, Triangular fuzzy number neutrosophic weighted arithmetic averaging operator, Triangular fuzzy number neutrosophic weighted geometric averaging operator, Multi-attribute decision making problem.

The objective of the paper ARE to introduce single-valued trapezoidal neutrosophic numbers(SVTrNNs), which is a special case of single-valued neutrosophic numbers and to develop a ranking method for ranking SVTrNNs. Some operational rules as well as cut sets of SVTrNNs have been introduced. The value and ambiguity indices of truth, indeterminacy, and falsity membership functions of SVTrNNs have been defined. A new ranking method has been proposed by using these two indices and applied the ranking method to multi attribute decision making problem in which the ratings of the alternatives over the attributes are expressed in terms of TrNFNs. Finally , an illustrative example has been provided to demonstrate the validity and applicability of the proposed approach.

For modeling imprecise and indeterminate data for multi-objective decision making, two different methods: neutrosophic multi-objective linear/non-linear programming, neutrosophic goal programming, which have been very recently proposed in the literatuire. In many economic problems, the well-known probabilities or fuzzy solutions procedures are not suitable because they cannot deal the situation when indeterminacy inherently involves in the problem. In this case we propose a new concept in optimization problem under uncertainty and indeterminacy. It is an extension of fuzzy and intuitionistic fuzzy optimization in which the degrees of indeterminacy and falsity (rejection) of objectives and constraints are simultaneously considered together with the degrees of truth membership (satisfaction/acceptance). The drawbacks of the existing neutrosophic optimization models have been presented and new framework of multi-objective optimization in neutrosophic environment has been proposed. The essence of the proposed approach is that it is capable of dealing with indeterminacy and falsity simultaneously.

This paper presents solution methodology for assignment problem with fuzzy cost. The fuzzy costs are considered as trapezoidal fuzzy numbers. Ranking method (introduced by S. Abbasbandy and T. Hajjary, 2009) has been used for ranking the trapezoidal fuzzy ...

Many multi-attribute group decision-making (MAGDM) strategies have been introduced in the literature to deal with decision-making problems in uncertain environment. Many of them are based on fuzzy numbers and they are not able to cope with indeterminacy and inconsistency involving in decision making. In recent years, some neutrosophic multi-attribute group decision-making strategies have been successfully developed to deal with uncertainty, indeterminacy, and inconsistency in decision making. Among them, TODIM (an acronym in Portuguese of interactive and multiple attribute decision making) strategy based on prospect theory has received more attention due to its great performance in considering the bounded rationality of decision makers. In this paper, we develop a TODIM strategy to deal with multi-attribute group decision-making problem in trapezoidal neutrosophic numbers environment. To establish the TODIM strategy, we employ score function, accuracy function, and Hamming distance function for trapezoidal neutrosophic numbers. Lastly, we solve an illustrative numerical example to show the applicability and usefulness of the proposed strategy. A comparison analysis is also provided.