NS-Cross Entropy-Based MAGDM under Single-Valued Neutrosophic Set Environment

A 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 produces an undefined phenomenon in some situations. In order to deal with these disadvantages, we propose a new cross entropy measure under a single-valued neutrosophic set (SVNS) environment, namely NS-cross entropy, and prove its basic properties. Also we define weighted NS-cross entropy measure and investigate its basic properties. We develop a novel multi-attribute group decision-making (MAGDM) strategy that is free from the drawback of asymmetrical behavior and undefined phenomena. It is capable of dealing with an unknown weight of attributes and an unknown weight of decision-makers. Finally, a numerical example of multi-attribute group decision-making problem of investment potential is solved to show the feasibility, validity and efficiency of the proposed decision-making strategy.


Introduction
To tackle the uncertainty and modeling of real and scientific problems, Zadeh [1] first introduced the fuzzy set by defining membership measure in 1965. Bellman and Zadeh [2] contributed important research on fuzzy decision-making using max and min operators. Atanassov [3] established the intuitionistic fuzzy set (IFS) in 1986 by adding non-membership measure as an independent component to the fuzzy set. Theoretical and practical applications of IFSs in multi-criteria decision-making (MCDM) have been reported in the literature [4][5][6][7][8][9][10][11][12]. Zadeh [13] introduced entropy measure in the fuzzy environment. Burillo and Bustince [14] proposed distance measure between IFSs and offered an axiomatic definition of entropy measure. In the IFS environment, Szmidt and Kacprzyk [15] proposed a new entropy measure based on geometric interpretation of IFS. Wei et al. [16] developed an entropy measure for interval-valued intuitionistic fuzzy set (IVIFS) and presented its applications in pattern recognition and MCDM. Li [17] presented a new multi-attribute decision-making (MADM) strategy combining entropy and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) in an IVIFS environment. Shang and Jiang [18] introduced the cross entropy in the fuzzy environment. Vlachos and Sergiadis [19] presented intuitionistic fuzzy cross entropy by extending fuzzy cross entropy [18]. Ye [20] defined a new cross entropy under an IVIFS environment and presented an optimal decision-making strategy. Xia and Xu [21] put forward a new entropy and a cross entropy and employed them for multi-attribute criteria group decision-making (MAGDM) strategy under an IFS environment. Tong and Yu [22] defined cross entropy under an IVIFS environment and applied it to MADM problems.
Majumdar and Samanta [143] defined an entropy measure and presented an MCDM strategy under SVNS environment. Ye [144] proposed cross entropy measure under the single-valued neutrosophic set environment, which is not symmetric straight forward and bears undefined phenomena. To overcome the asymmetrical behavior of the cross entropy measure, Ye [144] used a symmetric discrimination information measure for single-valued neutrosophic sets. Ye [145] defined cross entropy measures for SVNSs to overcome the drawback of undefined phenomena of the cross entropy measure [144] and proposed a MCDM strategy.
The aforementioned applications of cross entropy [144,145] can be effective in dealing with neutrosophic MADM problems. However, they also bear some limitations, which are outlined below: i.
The strategies [144,145] are capable of solving neutrosophic MADM problems that require the criterion weights to be completely known. However, it can be difficult and subjective to offer exact criterion weight information due to neutrosophic nature of decision-making situations. ii. The strategies [144,145] have a single decision-making structure, and not enough attention is paid to improving robustness when processing the assessment information. iii. The strategies [144,145] cannot deal with the unknown weight of the decision-makers.

Research gap:
MAGDM strategy based on cross entropy measure with unknown weight of attributes and unknown weight of decision-makers.
This study answers the following research questions: i. Is it possible to define a new cross entropy measure that is free from asymmetrical phenomena and undefined behavior? ii. Is it possible to define a new weighted cross entropy measure that is free from the asymmetrical phenomena and undefined behavior?
iii. Is it possible to develop a new MAGDM strategy based on the proposed cross entropy measure in single-valued neutrosophic set environment, which is free from the asymmetrical phenomena and undefined behavior? iv. Is it possible to develop a new MAGDM strategy based on the proposed weighted cross entropy measure in the single-valued neutrosophic set environment that is free from the asymmetrical phenomena and undefined behavior? v. How do we assign unknown weight of attributes? vi. How do we assign unknown weight of decision-makers?

Motivation:
The above-mentioned analysis describes the motivation behind proposing a comprehensive NS-cross entropy-based strategy for tackling MAGDM under the neutrosophic environment. This study develops a novel NS-cross entropy-based MAGDM strategy that can deal with multiple decision-makers and unknown weight of attributes and unknown weight of decision-makers and free from the drawbacks that exist in [144,145].
The objectives of the paper are: 1. To define a new cross entropy measure and prove its basic properties, which are free from asymmetrical phenomena and undefined behavior. 2. To define a new weighted cross measure and prove its basic properties, which are free from asymmetrical phenomena and undefined behavior. To fill the research gap, we propose NS-cross entropy-based MAGDM, which is capable of dealing with multiple decision-makers with unknown weight of the decision-makers and unknown weight of the attributes.
The main contributions of this paper are summarized below: 1. We define a new NS-cross entropy measure and prove its basic properties. It is straightforward symmetric and it has no undefined behavior. 2. We define a new weighted NS-cross entropy measure in the single-valued neutrosophic set environment and prove its basic properties. It is straightforward symmetric and it has no undefined behavior. 3. In this paper, we develop a new MAGDM strategy based on weighted NS cross entropy to solve MAGDM problems with unknown weight of the attributes and unknown weight of decision-makers. 4. Techniques to determine unknown weight of attributes and unknown weight of decisions makers are proposed in the study.
The rest of the paper is presented as follows: Section 2 describes some concepts of SVNS. In Section 3 we propose a new cross entropy measure between two SVNS and investigate its properties. In Section 4, we develop a novel MAGDM strategy based on the proposed NS-cross entropy with SVNS information. In Section 5 an illustrative example is solved to demonstrate the applicability and efficiency of the developed MAGDM strategy under SVNS environment. In Section 6 we present comparative study and discussion. Section 7 offers conclusions and the future scope of research.

Preliminaries
This section presents a short list of mostly known definitions pertaining to this paper.

Definition 3 [24] Inclusion of SVNSs. The inclusion of any two SVNS sets H1 and H2 in U is denoted by
H1 ⊆ H2 and defined as follows:

Definition 7 [24] Intersection. The intersection of two single-valued neutrosophic sets H1 and H2 denoted by H4 and defined as
Example 6. Let H1 and H2 be two SVNSs in U presented as follows: Then intersection of H1 and H2 is presented as follows:

NS-Cross Entropy Measure
In this section, we define a new single-valued neutrosophic cross-entropy measure for measuring the deviation of single-valued neutrosophic variables from an a priori one.   Hence complete the proof.

Definition 8 NS-cross entropy measure. Let H1 and H2 be any two SVNSs in
(iii) Using Definition 5, we obtain the following expression   (2)) satisfies the following properties:

Theorem 2. Single-valued neutrosophic weighted NS-cross-entropy (defined in Equation
i. Hence complete the proof. (iii) Using Definition 5, we obtain the following expression   T uu T  uu  TT  w  uu  T  T  T  T  u  u   I uu I  I   I  I    Hence complete the proof. ☐

MAGDM Strategy Using Proposed Ns-Cross Entropy Measure under SVNS Environment
In this section, we develop a new MAGDM strategy using the proposed NS-cross entropy measure.

Description of the MAGDM Problem
Assume that   (3)

Step 2. Formulate priori/ideal decision matrix
In the MAGDM, the a priori decision matrix has been 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 a priori decision matrix.

Step 3. Determinate the weights of decision-makers
To find the decision-makers' weights we introduce a model based on the NS-cross entropy measure. The collective NS-cross entropy measure between Thus, we can introduce the following weight model of the decision-makers: where, 01 K λ  and

Step 5. Determinate the weight of attributes
To find the attributes weight we introduce a model based on the NS-cross entropy measure. The collective NS-cross entropy measure between M (Weighted aggregated decision matrix) and P (Ideal matrix) for each attribute is defined by where, i = 1, 2, 3, …, m; j = 1, 2, 3, …, n. Thus, we defined a weight model for attributes as follows: where, 01 j w  and 1 1 n j j w    for j = 1, 2, 3, …, n.
Step 6. Calculate the weighted NS-cross entropy measure Using Equation (2), we calculate weighted cross entropy value between weighted aggregated matrix and priori matrix. The cross entropy values can be presented in matrix form as follows:  (11) Step 7. Rank the priority Smaller value of the cross entropy reflects that an alternative is closer to the ideal alternative. Therefore, the preference priority order of all the alternatives can be determined according to the increasing order of the cross entropy values w NS i CE (A ) (i = 1, 2, 3, …, m). Smallest cross entropy value indicates the best alternative and greatest cross entropy value indicates the worst alternative.

Step 8. Select the best alternative
From the preference rank order (from step 7), we select the best alternative.

Illustrative Example
In this section, we solve an illustrative example adapted from [12] of MAGDM problems to reflect the feasibility, applicability and efficiency of the proposed strategy under the SVNS environment. Now, we use the example [12] for cultivation and analysis. A venture capital firm intends to make evaluation and selection of five enterprises with the investment potential: (1) Social and political factor (G1) (2) The environmental factor (G2) (3) Investment risk factor (G3) (4) The enterprise growth factor (G4).
The investment firm makes a panel of three decision-makers. The steps of decision-making strategy (4.1.1.) to rank alternatives are presented as follows: Step: 1. Formulate the decision matrices Step: 2. Formulate priori/ideal decision matrix A priori/ideal decision matrix Please provide a sharper picture  (95) Step: 3. Determine the weight of decision-makers By using Equations (5) and (6), we determine the weights of the three decision-makers as follows: Step: 4. Formulate the weighted aggregated decision matrix Using Equation (7) the weighted aggregated decision matrix is presented as follows: Weighted Aggregated decision matrix Step: 5. Determinate the weight of the attributes By using Equations (9) and (10), we determine the weights of the four attribute as follows:  (11) Step: 7. Rank the priority The cross entropy values of alternatives are arranged in increasing order as follows: 0.151 < 0.168 < 0.184 < 0.195 < 0.198.
Step: 8. Select the best alternative From step 7, we identify A4 is the best alternative. Hence, Food enterprises (A4) is the best alternative for investment.
In Figure 2, we draw a bar diagram to represent the cross entropy values of alternatives which shows that A4 is the best alternative according our proposed strategy.
In Figure 3, we represent the relation between cross entropy values and acceptance values of alternatives. The range of acceptance level for five alternatives is taken by five points. The high acceptance level of alternatives indicates the best alternative for acceptance and low acceptance level of alternative indicates the poor acceptance alternative.
We see from Figure 3 that alternative A4 has the smallest cross entropy value and the highest acceptance level. Therefore A4 is the best alternative for acceptance. Figure 3 indicates that alternative A2 has highest cross entropy value and lowest acceptance value that means A2 is the worst alternative. Finally, we conclude that the relation between cross entropy values and acceptance value of alternatives is opposite in nature.

Comparative Study and Discussion
In literature only two MADM strategies [144,145] have been proposed. No MADGM strategy is available. So the proposed MAGDM is novel and non-comparable with the existing cross entropy under SVNS for numerical example. i.
The MADM strategies [144,145] are not applicable for MAGDM problems. The proposed MAGDM strategy is free from such drawbacks. ii.
Ye [144] proposed cross entropy that does not satisfy the symmetrical property straightforward and is undefined for some situations but the proposed strategy satisfies symmetric property and is free from undefined phenomenon. iii.
The strategies [144,145] cannot deal with the unknown weight of the attributes whereas the proposed MADGM strategy can deal with the unknown weight of the attributes iv.
The strategies [144,145] are not suitable for dealing with the unknown weight of decision-makers, whereas the essence of the proposed NS-cross entropy-based MAGDM is that it is capable of dealing with the unknown weight of the decision-makers.

Conclusions
In this paper, we have defined a novel cross entropy measure in SVNS environment. The proposed cross entropy measure in SVNS environment is free from the drawbacks of asymmetrical behavior and undefined phenomena. It is capable of dealing with the unknown weight of attributes and the unknown weight of decision-makers. We have proved the basic properties of the NS-cross entropy measure. We also defined weighted NS-cross entropy measure and proved its basic properties. Based on the weighted NS-cross entropy measure, we have developed a novel MAGDM strategy to solve neutrosophic multi-attribute group decision-making problems. We have at first proposed a novel MAGDM strategy based on NS-cross entropy measure with technique to determine the unknown weight of attributes and the unknown weight of decision-makers. Other existing cross entropy measures [144,145] can deal only with the MADM problem with single decision-maker and known weight of the attributes. So in general, our proposed NS-cross entropy-based MAGDM strategy is not comparable with the existing cross-entropy-based MADM strategies [144,145] under the single-valued neutrosophic environment. Finally, we solve a MAGDM problem to show the feasibility, applicability and efficiency of the proposed MAGDM strategy. The proposed NS-cross entropy-based MAGDM can be applied in teacher selection, pattern recognition, weaver selection, medical treatment selection options, and other practical problems. In future study, the proposed NS-cross entropy-based MAGDM strategy can be also extended to the interval neutrosophic set environment.