NC-Cross Entropy Based MADM under 1 Neutrosophic Cubic Set Environment 2

Surapati Pramanik 1*, Shyamal Dalapati 2, Shariful Alam 2, Florentin Smarandache 3 and 3 Tapan Kumar Roy 2 4 1 Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District–North 24 5 Parganas, Bhatpara 743126, West Bengal, India 6 2 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.7 Botanic Garden, Howrah 711103, West Bengal, India; shyamal.rs2015@math.iiests.ac.in (S.D.); 8 salam@math.iiests.ac.in (S.A.); tkroy@math.iiests.ac.in (T.K.R.) 9 3 Department of Mathematics & Science, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, 10 USA; smarand@unm.edu 11 * Correspondence: sura_pati@yahoo.co.in; Tel.: +91-9477035544 12 13 Abstract: Neutrosophic cubic set (NCS) is one of the important family members of neutrosophic 14 hybrid sets. Neutrosophic cubic set has more strength than other family members of neutrosophic 15 hybrid sets to express incomplete information due to the presence of interval valued neutrosophic 16 set (IVNS) and single valued neutrosophic set (SVNS) in its structure. Cross entropy measure is one 17 of the best way to calculate the divergence of any variable from the priori one variable. In this paper 18 we first define a new cross entropy measure under NCSs environment which we call NCcross 19 entropy measure. We investigate the basic properties of NC-cross entropy. We also propose weighted 20 NC-cross entropy and investigate its basic properties. We develop a novel multi attribute decision 21 making (MADM) strategy based on weighted NC-cross entropy. To show the feasibility and 22 applicability, we solve a MADM problem using the proposed strategy. 23

Ali et al. [29] proposed the concept of neutrosophic cubic set (NCS) by hybridizing NS and INS and defined external and internal neutrosophic cubic sets, and established some of their properties.In the same study, Ali et al. [29] proposed an adjustable strategy to NCS-based decision making.Jun et al. [30] also defined NCS by combining NS and INS.
Cross entropy measure is an important measure to calculate the divergence of any variable from prior one variable.In 1968, Zadeh [31] first proposed fuzzy entropy to calculate the divergence of two fuzzy The objectives of the paper are: i.
To introduce a cross entropy measure and prove its basic properties under NCS environment. ii.
To introduce a weighted cross measure and prove its basic properties under NCS environment. iii.
To develop a novel MADM strategy based on weighted NC-cross entropy measure under NCS environment.
To fill the research gap, we propose NC-cross entropy-based MADM.
The main contributions of this paper are summarized below: i.
We introduce a NC-cross entropy measure and prove its basic properties under NCS environment. ii.
We introduce a weighted NC-cross entropy measure and prove its basic properties under NCS environment. iii.
We develop a novel MADM strategy based on weighted NC-cross entropy to solve MADM problems. iv.
We solved a MADM problem using proposed strategy under NCS environment.
Rest of the paper is organized as follows.In section 2, we describe the basic definitions and operation of SVNS, IVNS, NCS.In section 3, we propose a NC-cross entropy measure and a weighted NC-cross entropy measure and establish their basic properties.Section 4 devotes to develop MADM strategy using NC-cross entropy.In section 5 an illustrative numerical example is provided to demonstrate the applicability and validity of proposed strategy under NCS environment.Section 6 offers conclusions and future direction of this research.

Preliminaries
In this section, some basic concepts and definitions of SVNS, INS and NCS are presented which will be utilized to develop the paper.

Definition 1. SVNS
Assume that U be a space of points (objects) with generic elements u  U. A SVNS [2] H in U is characterized by a truth-membership function TH(u), an indeterminacy-membership function IH(u), and a falsity-membership function FH(u), where TH(u), IH(u), FH(u) ∈ [0, 1] for each point u in U.
The order triplet < T, I, F > is called single valued neutrosophic number (SVNN).

Definition 2. Inclusion of SVNS
The inclusion of any two SVNSs [2] H1 and H2 in U is denoted by H1 ⊆ H2 and defined as follows: H1 ⊆ H2, iff for all u ∈ U.

Definition 3. Equality of two SVNS
The equality of any two SVNSs [2] H1 and H2 in U denoted by H1 = H2 and is defined as follows:

Definition 4. Complement of any SVNS
The complement of any SVNS [2] H in U denoted by c H and defined as follows: Let H be any SVNN in U presented as follows: H = < (0.7, 0.3, 0.5)>.Then compliment of H is obtained as c H = < (0.3, 0.7, 0.5) >.

Definition 8. INSs
Assume that U be a space of points (objects) with generic element u  U.An INS [22] J in U is characterized by a truth-membership function TJ(u), an indeterminacy-membership function IJ(u), and a falsity-membership function FJ(u), where for each point u in U. Therefore, a INSs J can be expressed as J = {u,

Definition 9. Inclusion of two INSs
| u∈ U} be any two INSs [22] in U, then

Definition 10. Complement of an INS
The complement J c of an INS [22] | u∈ U} is defined as follows:

Definition 11. Equality of two INSs
| u∈ U} be any two INSs [22] in U, then for all u∈ U.

Definition 12. Neutrosophic cubic set (NCS)
Assume that U be a space of points (objects) with generic elements ui  U. A NCS [29,30] Q in U is a hybrid structure of INS and SVNS that can be expressed as follows: ) and (TQ (ui), I Q (ui), FQ (ui) are INSs and SVNSs respectively in U. NCS can be simply presented as ), (TQ (u), I Q (u), FQ (u)) >, we call it as a neutrosophic cubic number (NCN).

MADM strategy using proposed NC-cross entropy measure in NCS environment
In MADM, decision maker evaluates the alternatives based on the attribute values.But it is not easy to rate the performance of alternatives with respect to predefined attributes in terms of crisp numbers due to the uncertain and inconsistent nature of decision making problem.NCS can express this type information.In this section, we develop a MADM strategy using the proposed NC-cross entropy measure.

Description of the MADM problem
The MADM problem can be consider as follows: be the discrete set of alternatives and attribute respectively.Let  Step: 2. Formulate priori/ ideal decision matrix In the MADM processes, the priori decision matrix is used to select the best alternatives among the set of collected feasible alternatives.In the decision making situation, we use the following decision matrix as priori decision matrix.Step: 3. Formulate the weighted NC-cross entropy matrix Using Equation (20), we calculate weighted cross entropy values between decision matrix and priori matrix.The cross entropy value can be presented in matrix form as follows: Step: 4. Rank the priority Smaller value of the cross entropy reflects that an alternative is closer to the ideal alternative.Therefore, the preference ranking order of all the alternatives can be determined according to the increasing order of the cross entropy values

Illustrative examples
In this section, we solve an illustrative example of MADM problem to reflect the feasibility and efficiency of our proposed strategy under NCSs environments.

Formulate the decision matrix
Step-1

Rank the priority
Step-4 Now, we use an example [59] for cultivation and analysis.A venture capital firm intends to make evaluation and selection to five enterprises with the investment potential:
The investment firm makes a panel of three decision makers .

The steps of decision making strategy to rank alternatives presented as follows:
Step: 1. Formulate the decision matrix The decision maker represents the rating values of alternative Ai (i = 1, 2, 3, 4, 5) with respect to the attribute Gj (j = 1, 2, 3, 4) in terms of NCNs and constructs the decision matrix M as follows: Step: 3. Formulate priori/ ideal decision matrix Priori/ ideal decision matrix Graphical representation of alternatives versus cross entropy is shown in Figure 2. From the Figure 2, we see that A2 is the best preference alternative and A4 is the least preference alternative.

Conclusion
In this paper we have introduced NC-cross entropy measure in NCS environment.We have proved the basic properties of the cross entropy measure.We have also introduced weighted NC-cross entropy measure and proved its basic properties.Based on the weighted NC-cross entropy measure, we proposed a novel MADM strategy.Finally, we solve a MADM problem to show the feasibility and efficiency of the proposed decision making strategy.The proposed NC-cross entropy based MADM strategy can be employed to solve a variety of problems such as fault diagnosis [15], logistics center selection [60], Weaver selection [61], teacher selection [62], brick selection [63] renewable energy selection [64], etc.The proposed NC-cross entropy based MADM strategy can also be extended to MAGDM strategy using suitable aggregation operators.
1, 2, 3, …, m).The smallest cross entropy value reflects the best alternative and the greatest cross entropy value reflects the worst alternative.A conceptual model of the proposed strategy is shown in Figure1.

Fig. 1 .
Fig.1.A flow chart of the NC-cross entropy based MADM strategy

Figure 3
Figure 3 presents relation between cross entropy value and preference ranking of the alternative.