Cross-Entropy and Prioritized Aggregation Operator with Simplified Neutrosophic Sets and Their Application in Multi-Criteria Decision-Making Problems

Simplified neutrosophic sets (SNSs) can effectively solve the uncertainty problems, especially those involving the indeterminate and inconsistent information. Considering the advantages of SNSs, a new approach for multi-criteria decision-making (MCDM) problems is developed under the simplified neutrosophic environment. First, the prioritized weighted average operator and prioritized weighted geometric operator for simplified neutrosophic numbers (SNNs) are defined, and the related theorems are also proved. Then two novel effective cross-entropy measures for SNSs are proposed, and their properties are proved as well. Furthermore, based on the proposed prioritized aggregation operators and cross-entropy measures, the ranking methods for SNSs are established in order to solve MCDM problems. Finally, a practical MCDM example for coping with supplier selection of an automotive company is used to demonstrate the effectiveness of the developed methods. Moreover, the same example-based comparison analysis of between the proposed methods and other existing methods is carried out.


Introduction
Fuzzy set (FS) theory was introduced by Zadeh [1] and used as a key method to solve multi-criteria decisionmaking (MCDM) problems [2], and pattern recognition [3].But, some issues, where the membership degree is difficult to be defined by one specific value, cannot be well dealt with by FSs.In order to overcome the shortcomings of Zadeh's FS theory, Atanassov [4] introduced intuitionistic fuzzy sets (IFSs) and Gau and Buehrer [5] defined vague sets, but in fact, IFSs and vague sets are mathematically equivalent collections.Because of the advantages that an IFS considers the membership-degree, non-membership degree and hesitation degree simultaneously, it is more flexible and useful to describe the uncertain information than a traditional FS.Thus, many methods based on IFSs have been put forward and widely applied to solve MCDM problems [6][7][8][9][10][11][12], medical diagnosis [13,14], pattern recognition [15,16], stock market prediction [17,18], and marketing strategy selection [19].However, in some real situations, the membership degree, non-membership degree and hesitation degree may be difficultly given by specific numbers; hence, interval-valued intuitionistic fuzzy sets (IVIFSs) [20] are developed and applied to solve such problems [21][22][23][24][25].In addition, Torra and Narukawa [26] proposed hesitant fuzzy sets (HFSs) to deal with the hesitant situation when people express their preferences for objects in a decision-making process.Since then, many researches on HFSs and their extensions have been carried out.Chen et al. [27] proposed interval-valued hesitant fuzzy sets (IVHFSs) and verified the effectiveness in solving MCDM problems.Wang et al. introduced several hesitant fuzzy linguistic aggregation operator-based methods [28,29], and Zhou et al. [30] proposed a linguistic hesitant fuzzy decision-making method based on evidential reasoning to solve MCDM problems.Moreover, Wang et al. [31] studied hesitant fuzzy linguistic term sets, Tian et al. [32] introduced gray linguistic sets based on gray sets and linguistic term sets, and Peng et al. [33] proposed intuitionistic hesitant fuzzy sets based on HFSs and IFSs.
Although FSs have been extended and generalized, they still cannot handle all types of problems with uncertainty in reality, especially those of the indeterminate and inconsistent information [34].For example, when an expert is asked for the opinion about a certain statement, he or she may say the possibility that the statement is true is 0.5, the possibility that the statement is false is 0.6, and the degree that he or she is not sure is 0.2.Such issues cannot be properly solved using HFSs and IFSs.Thus, a new theory is required.
Smarandache [35] proposed neutrosophic logic and neutrosophic sets, and then several researchers have made their efforts to enrich NSs [36][37][38][39][40][41][42].Recently, some methods on simplified neutrosophic sets (SNSs) and interval neutrosophic sets (INSs) have been put forward and used to solve MCDM problems [43][44][45][46][47][48][49][50][51].For example, Ye [43] defined the operational rules of SNSs and proposed a method with simplified neutrosophic information based on the weighted arithmetic average operator and the weighted geometric average operator.Ye [43,44] proposed different methods based on single valued neutrosophic measures: one is the cosine similarity-based measure method, and another is the logarithm-based cross-entropy measure method.The effectiveness of both methods for MCDM problems have been proved through the same illustrative example.However, Peng et al. [50,51] pointed out some limitations of previous research papers for SNSs [43][44][45], including the lacks of the SNS operation and cross-entropy measure, and brought forward an improved method of SNSs.In a word, it has been demonstrated that neutrosophic set-based methods are effective tools to handle indeterminate and inconsistent information, which cannot be achieved using HFSs and IFSs.
In this paper, in order to overcome the lacks of previous proposed methods [43][44][45], the prioritized weighted average operator (SNNPWA) and prioritized weighted geometric operator (SNNPWG) for SNS are defined and two novel cross-entropy measures are proposed.Moreover, based on the proposed operators and measures, the ranking methods are established.Then the assessment information of alternatives with respect to criteria is given by truthmembership degree, indeterminate-membership degree, and falsity-membership degree under simplified neutrosophic environment, and then the ranking of all alternatives is obtained using the developed approach.
The paper is organized as follows.Some concepts of NSs, SNSs, prioritized aggregation (PA) operator, and cross-entropy are introduced in Sect. 2. In Sect.3, the SNNPWA and SNNPWG operators are defined and proved, two novel cross-entropy measures are proposed and their effectiveness is verified.Section 4 provides the ranking method for MCDM problems with simplified neutrosophic information.Section 5 shows the illustration of our approaches and the comparison analysis between the proposed methods and other existing methods.Finally, conclusions are drawn in Sect.6.

Preliminaries
In this section, some basic concepts and definitions of NSs, SNSs, PA operator, cross-entropy, and cosine similarity measure are briefly reviewed.

NS and SNSs
In this subsection, the definitions and operations of NSs and SNSs are introduced.
Definition 1 [35] Let X be a space of points (objects), with a generic element in X denoted by x.A neutrosophic set A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x) and a falsity-membership function F A (x).The functions T A (x), I A (x) and F A (x) are real standard or nonstandard subsets of 0 ½, and F A ðxÞ : There is no restriction on the sum of T A (x), I A (x) and F A (x), so 0 Definition 2 [35] A neutrosophic set A is contained in the other neutrosophic set B, denoted by A B if and only if inf T A ðxÞ inf T B ðxÞ, sup T A ðxÞ !sup T B ðxÞ, inf I A ðxÞ !inf I B ðxÞ, sup I A ðxÞ !sup I B ðxÞ, inf F A ðxÞ !inf F B ðxÞ, and sup F A ðxÞ !sup F B ðxÞ for every x in X.
Since it is hard to use NSs to solve practical problems, so Ye [43] reduced NSs of nonstardard intervals into a kind of SNSs of standard intervals.
Definition 3 [43] Let X be a space of points (objects), with a generic element in X denoted by x.A neutrosophic set A in X is characterized by a truth-membership function T A (x), a indeterminacy-membership function I A (x) and a falsity-membership function F A (x).If the functions T A (x), I A (x) and F A (x) are singleton subintervals/subsets in the real standard 0; 1 ½ , that is, T A ðxÞ : XÀ !0; 1 ½ , I A ðxÞ : XÀ !0; 1 ½ and F A ðxÞ : XÀ !0; 1 ½ .Then, a simplification of the neutrosophic set A is denoted by A ¼ fhx; T A ðxÞ; I A ðxÞ; F A ðxÞijx 2 Xg which is called a SNS.It is a subclass of NSs.
Definition 4 [43].A SNS A is contained in the other SNS B, denoted by A B if and only if T A ðxÞ T B ðxÞ, I A ðxÞ !I B ðxÞ and F A ðxÞ !F B ðxÞ, for any x 2 X.Especially, A = B if A B and B A. The complement set of A denoted by A C is defined as A C ¼ f\x; F A ðxÞ; I A ðxÞ; T A ðxÞ [ jx 2 Xg: Definition 5 [43] Let A and B are two SNSs, the operations of SNSs are defined as follows. (1 However, Peng et al. [50,51] pointed out there are still some lacks in Definition 5.In some cases, the operations such as A þ B and A Á B might be impractical as presented in Example 1.
Example 1 Let A ¼ x; 0:5; 0:5; 0:5 h i f gand B ¼ x; 1; h f 0; 0ig be two SNSs.Obviously, B ¼ x; 1; 0; 0 h i f gis the largest SNSs.Theoretically, the sum of an arbitrary value and the maximum value should be equal to the maximum value.However, according to Definition 5, A þ B ¼ x; 1; h f 0:5; 0:5; 0:5ig 6 ¼ B. Thus, the operation ''?'' cannot be accepted.Similar contradictions exist in other operations of Definition 5, and thus the operations of SNSs need to be redefined.
Definition 6 [50,51] Let A and B be two SNSs, and the operations of SNSs can be defined as follows:

Prioritization Aggregation Operator
The prioritization aggregation (PA) operator was originally introduced by Yager [52], and is shown as follows.
Definition 7 [52] Let G ¼ G 1 ; G 2 ; Á Á Á ; G n f gbe a collection of criteria and there is a prioritization between the criteria expressed by the linear ordering which indicates the criteria G j has a higher priority than G k , if j\k.G j ðxÞ is an evaluation value denoting the performance of the alternative x under the criteria G j , and satisfies G j 2 0; 1 ½ , thus PAðG j ðxÞÞ ¼ X n j¼1 W j G j ðxÞ; where W j ¼ , T 1 ¼ 1 and T j ¼ Q jÀ1 k¼1 G j ðxÞ ðj ¼ 2; Á Á Á ; nÞ.Then PA is called the prioritized aggregation operator.

Cross-Entropy of FSs and SNSs
The cross-entropy measure was introduced by Kullback [53] and its definition is shown as follows.
Definition 9 [54].Assume that Þg are two FSs in the universe of discourse X ¼ fx 1 ; x 2 ; Á Á Á ; x n g, and the fuzzy cross-entropy of A from B is defined as follows: which indicates the degree of discrimination of A from B.
Similarly, considering the indeterminacy-membership and falsity-membership functions, Ye [44] proposed the cross-entropy measure of SNSs as follows: which also indicates the discrimination degree of the SNSs A from B. Moreover, it can be easily proved that EðA; BÞ !0 and EðA; BÞ ¼ 0 if and only if BÞ, where A C and B C are the complement of SNSs A and B, respectively.Then, EðA; BÞ is not symmetric, and similarly, it could be revised to a symmetric discrimination information measure for SNSs as DðA; BÞ ¼ EðA; BÞ þ EðB; AÞ.
The larger the difference between A and B is, the larger DðA; BÞ is.

Cosine Similarity Measure of SNSs
The cosine similarity measure of SNSs was introduced by Ye [43], which was induced from the correlation coefficient of Ye [45].To rank the alternatives in the decisionmaking process, Ye [43] defined the SNS value of ideal alternative as a Ã ¼ 1; 0; 0 h i, and the cosine similarity measure between SNSs a i i ¼ 1; 2; Á Á Á ; n ð Þ and a Ã is defined as follows: The bigger the measure value S i ða i ; a Ã Þ ði ¼ 1; 2; Á Á Á ; nÞ is, the better alternative A i is.However, the cosine measure above has the lacks when it is used in a real situation as demonstrated in Example 2.
Example 2 Let A 1 ¼ x; 0:8; 0; 0 h i f g and A 2 ¼ x; 0:2; 0; 0 h i f gbe two SNSs.Obviously, A 1 is superior than A 2 , that is, S 1 ða 1 ; a Ã Þ [ S 2 ða 2 ; a Ã Þ.However, according to the cosine measure of Ye [43], Therefore, the results in Example 2 cannot be accepted, and the measure given in [43] needs to be improved.

SNNPWA and SNNPWG Operators
and Cross-Entropy Measure for SNSs

SNNPWA and SNNPWG Operators
In this subsection, the score function of a simplified neutrosophic number (SNN) is first defined.Then, the SNNPWA and SNNPWG operators are defined, and their relative theorems are proved.
From the intuitive judgment, A SNN A, which is closer to the ideal SNN A þ ¼ 1; 0; 0 h i, should possess a higher score, thus, the score function S(A) can be defined as follows: Definition 10 Let A ¼ T A ; I A ; F A h i be a SNN, and the score function S(A) is represented as follows: Example 3 If A ¼ 0:8; 0:2; 0:2 h i , by applying Eq. ( 1), then SðAÞ ¼ 0:8þ1À0:2þ1À0:2 3 ¼ 0:8: In the following, the prioritized weighted average operator and prioritized weighted geometric operator under simplified neutrosophic environment are defined, and their related theorems are given.
Definition 11 Let A j ¼ T A j ; I A j ; F A j ðj ¼ 1; 2; Á Á Á ; nÞ be a collection of SNNs, and the SNNPWA operator can be defined as follows: where Theorem 1 For the collection of SNNs A ¼ fA j jj ¼ 1; 2; Á Á Á ; ng, the following aggregated results will be obtained by using the SNNPWA operator: Proof Clearly, according to Definition 11 and the operation of SNSs defined in Definition 6, Eq. ( 2) can be proven by utilizing mathematic induction.
(1) When n ¼ 2, we have (2) When n ¼ k, the following results can be obtained by applying Eq. ( 2) When n¼kþ1, by using Eqs.( 3) and (4), we can obtain The proof that Eq. ( 2) holds for any n is completed now.
According to Definition 4 and the induced result above, A À SNNPWA ðA 1 ; A 2 ; Á Á Á ; A n Þ holds.By the similar induction process, we obtain In accordance with Definition 4, SNNPWA ðA 1 ; A Thus, Property 2 is proved.

SNNPWG ðA
Proof According to Definition 12 and Definition 6, the proof of Eq. ( 6) can be proved in a similar proof manner.

Cross-Entropy Measure for SNSs
The cross-entropy measure for SNSs was proposed by Ye [44], but it cannot be accepted in some specific cases, as shown in the example given by Peng et al. [50].
gand A 2 ¼ x; 0:9; 0; h f 0ig be two SNSs, and B ¼ x; 1; 0; 0 h i f gbe the largest SNS.According to the cross-entropy measure for SNSs [44], S 1 ðA 1 ; BÞ ¼ S 2 ðA 2 ; BÞ ¼ 1 can be obtained, which indicates that A 1 is equal to A 2 .However, it is not possible to discern which one is the best.As T A 2 ðxÞ [ T A 1 ðxÞ, I A 2 ðxÞ ¼ I A 1 ðxÞ and F A 2 ðxÞ ¼ F A 1 ðxÞ for any x in X, it is clear that A 2 is superior to A 1 .
In order to overcome the shortcomings mentioned above, in this subsection, two new cross-entropy measures for SNSs are defined.Before defining the new cross-entropy measures, the following definition is required to be introduced to help us obtain the proof of the properties of the proposed cross-entropy measures later.

Definition 13 A SNS A is greater than or equal to the other SNS B, denoted by A ! B if and only if T A T B , I A ! I B and F A ! F B :
Next, the novel cross-entropy measures are defined.Definition 14 Let A and B be two SNSs, and then the cross-entropy between A and B can be defined as: which can indicate the degree of discrimination of A from B. However, I SNS 1 ðA; BÞ and I SNS 2 ðA; BÞ is not symmetric with respect to its argument.Therefore, a modified crossentropy measure based on I SNS 1 ðA; BÞ and I SNS 2 ðA; BÞ can be defined as follows: ( Proof Obviously, it can be easily verified that ( 1) and ( 2) hold.Next, the proofs of ( 3) and ( 4) are shown in the following.
Clearly, the problem pointed out by Peng et al. [50] can be solved by using the proposed cross-entropy measures.

The Ranking Method for MCDM Problems with Simplified Neutrosophic Information
The ranking method based on the SNNPWA and SNNPWG operators and the cross-entropy measure under simplified neutrosophic environment is presented to deal with MCDM problems.
and if i\j, then the priority of C i is higher than that of C j .The assessment value of the alternative A i on the criterion C j can be expressed in the following form: To rank the alternatives, we define a positive ideal solution and a negative ideal solution for SNNs denoted by A þ and A À , and they are A þ ¼ 1; 0; 0 h iand A À ¼ 0; 0; 1 h i: The decision process procedure of the proposed method is summarized as follows.
Step 1 Normalize the decision matrix.
First, the decision-making information b must be normalized.The criteria can be classified into the benefit and cost types.The evaluation information does not need to be changed for the benefittype criteria; however, for the cost-type criteria, it must be transformed with the complement set.The normalization of the decision matrices can be expressed as Step 2 Compute the aggregation results of each alternative A i Compute the aggregation values of each alternative A i by using Eq. ( 2) or ( 6), and then, the SNNPWA or SNNPWG operator aggregation values are obtained.
Step 3 Determine the cross-entropy and ranking value S b i of each alternative A i The cross-entropy values of each alternative A i from the positive ideal solution A ? and the negative solution A - are calculated by using Eqs.( 11) and (12).Then S b i can be obtained based on the following equation: Step 4 Select the best alternative by the ranking value S b i The smaller S b i is, the better the alternative is.According to S b i , the ranking of all alternatives is obtained and the best alternative is chosen.

Illustrated Example
The MCDM problem with simplified neutrosophic information example is used to demonstrate the application of the proposed approach and the relative comparison analysis.

An Illustrative Example with Simplified Neutrosophic Information
The illustrative example is the supplier-selection problem of an automotive company in reality.Actually, supplier selection of an automotive company is quite complex, and the number of the related criteria is up to fourteen [55], including quality, delivery, reputation, risk, security, service, and so forth.Furthermore, normally, considering real situations of a specific automotive company from different aspects of company strategies, product features, etc., different criteria sets should be constructed for different situations.In order to verify the effectiveness of the proposed method on the representative supplier-selection problems, a simplified supplier selection of an automotive company [56] with four essential criteria is adopted.Suppose that for an automotive company, which expects to select the most appropriate key components supplier, after first round assessment, five suppliers A i i ¼ 1; 2; ð Á Á Á ; 5Þ have been selected as alternatives for the final evaluation.During evaluation, four criteria are chosen, including product quality (c 1 ), relationship closeness (c 2 ), price (c 3 ), and delivery performance (c 4 ).The prioritization relationship of the criteria is The decision makers gave the evaluation values of all alternatives for each criterion with simplified netursophic information, and then, a simplified neutrosophic decision matrix D is provided as follows: : The following shows the decision-making procedure by means of the SNNPWA operator.
Step 4 Select the best alternative by the ranking value S b i .According to the S b i value in Tables 1 and 2, the ranking of five alternatives is In the following, we utilize the SNNPWG in the decision-making procedure.
Step 1 Normalize the decision matrix.This step is the same as that of the decision-making procedure by means of the SNNPWA operator and thus omitted here.
Step 2 Compute the aggregation results of each alternative A i .Compute the SNNPWG operator aggregation values b i for the alternative A i by applying Eq. ( 5 and obtain the value of S b i by applying Eq. ( 13).The results are shown in Tables 3 and  4.
Step 4 Select the best alternative by the ranking value S b i .According to the S b i value shown in Tables 3 and 4, the ranking of all alternatives is From the above results, we can see that the best alternative is A 5 , but the worst alternative is A 3 no matter which proposed aggregation operator is used.

A Comparison Analysis
In order to verify the effectiveness of proposed method, a comparison analysis is carried out with other four representative methods [43][44][45]57] by using the same illustrative example and the same weight.Among four representative methods, three methods are proposed by Ye [43][44][45], the other one is proposed by Liu and Wang [57].Meanwhile, the weight for each criterion in the illustrative example is still calculated using the PA operator.
Given the same decision information on the simplified supplier-selection problem under simplified neutrosophic environment, the final results of all compared methods is shown in Table 5.If the aggregation operators proposed by Ye [43] are used, for F w , the final ranking is Clearly, the best alternative is A 5 or A 2 , and the worst alternative is A 1 or A 4 .If the methods of Ye [44,45] are used, the final rankings are and the best alternative is A 5 while the worst alternative is A 1 or A 3 .However, if the proposed methods and Liu and Wang's methods [57] are utilized, the best alternative and worst alternative are the same, that is, A 5 and A 3 , but the final rankings are slightly different.
There are three reasons why different rankings exist in the proposed method and other previous methods: (1) The operations of SNSs [43] conflicts with the theory that the sum of an arbitrary value and the maximum value should be equal to the maximum one, as explained in Example 1.And the cross-entropy measure given in [44] has the lacks as discussed in   Example 4, that is, the cross-entropy of two different SNNs to the same SNN may be equal.(2) The cosine similarity measure [43] between SNSs the lacks as discussed in Example 2, that is, the similarity measure only considers the comparison with the positive ideal solution 1; 0; 0 h iand ignores the negative ideal solution 0; 0; 1 h i: (3) The previous methods were established by combining different operations of SNSs [43] with crossentropy measure [44], correlation coefficients [45], and aggregation operators [43,57].
Besides, the proposed method and the method of Liu and Wang [57] can obtain the same best and worst alternatives, but the final raking slightly varied.This is because the SNNPWA operator emphasizes the overall truthmembership of criteria, and the SNNPWG operator emphasizes the overall indeterminacy-membership and falsity-membership of criteria.
In summary, from the above analysis, it is concluded that the proposed method is more reasonable and reliable than the existing methods; meanwhile, the proposed method has several advantages: (1) The new proposed cross-entropy measures can overcome the shortcoming of the cross-entropy measure of Ye [44].(2) The SNNPWA and SNNPWG aggregation operators can compute the weighted vector of criteria, but they do not need to give the values by decision-maker in advance.(3) The improved operation of SNSs [50,51] is adopted to effectively make up the previous method's shortcomings [43].

Conclusions
SNSs can be utilized to solve the indeterminate and inconsistent information that exists in the real world but which FSs and IFSs cannot deal with.Considering the advantages of SNSs, several methods of SNSs were put forward and used to solve MCDM problems.However, there are some shortcomings in those methods [43,44].Therefore, two novel cross-entropy measures are put forward to overcome the shortcomings of the previously proposed cross-entropy measure [44].Based on the SNNPWA and SNNPWG aggregation operators, a MCDM method was established.Utilizing the proposed method, the best and the worst alternative can be identified easily.
In this paper, the main contributions are two novel cross-entropy measure that were put forward to overcome the shortcomings of the existing methods as discussed by Peng et al. [50,51], and the SNNPWA and SNNPWG operators that were inferred from the PA operator to solve the MCDM problems with incomplete weight information.Finally, the comparison results produced by different methods can show the effectiveness of the proposed method.
In the future, according to the different requirements in the real-world applications, how to optimize the score function could form the scope of discussion and further detailed study.
Table 5 The results of different methods for the illustrated example

Method
The final ranking Aggregation result F w of Ye [43] A 5 1 A 2 1 A 3 1 A 4 1 A 1 Aggregation result G w of Ye [43] A 2 1 A 5 1 A 1 1 A 3 1 A 4 Ye [44] A 5 1 A 2 1 A 1 1 A 4 1 A 3 Ye [45] A 5 1 A 4 1 A 3 1 A 2 1 A 1 Liu and Wang [57] A 5 1 A 2 1 A 1 1 A 4 1 A 3 The proposed approach based on the SNNPWA operator A 5 1 A 2 1 A 1 1 A 4 1 A 3 The proposed approach based on the SNNPWG operator and C ¼ T C ; I C ; F C h i be three SNSs.Assume A ! B ! C. then according to Definition 13, we have T A ! T B ! T C , I A I B I C , and F A F B F C .By using Eq. (

Table 3
The cross-entropy D SNS1 and ranking values S b i D SNS1 ðb i

Table 4
The cross-entropy D SNS2 and ranking values S b i

Table 2
The cross-entropy D SNS2 and the ranking valuesS b i D SNS2 ðb i ; A þ Þ D SNS2 ðb i ; A À Þ S b i

Table 1
The cross-entropy D SNS1 and the ranking valuesS b i D SNS1 ðb i ; A þ Þ D SNS1 ðb i ; A À Þ S b i