Symmetry Measures of Simplified Neutrosophic Sets for Multiple Attribute Decision-Making Problems

A simplified neutrosophic set (containing interval and single-valued neutrosophic sets) can be used for the expression and application in indeterminate decision-making problems because three elements in the simplified neutrosophic set (including interval and single valued neutrosophic sets) are characterized by its truth, falsity, and indeterminacy degrees. Under a simplified neutrosophic environment, therefore, this paper firstly defines simplified neutrosophic asymmetry measures. Then we propose a normalized symmetry measure and a weighted symmetry measure of simplified neutrosophic sets and develop a simplified neutrosophic multiple attribute decision-making method based on the weighted symmetry measure. All alternatives can be ranked through the weighted symmetry measure between the ideal solution/alternative and each alternative, and then the best one can be determined. Finally, an illustrative example on the selection of manufacturing schemes (alternatives) in the flexible manufacturing system demonstrates the applicability of the proposed method in a simplified (interval and single valued) neutrosophic setting, and then the decision-making method based on the proposed symmetry measure is in accord with the ranking order and best choice of existing projection and bidirectional projection-based decision-making methods and strengthens the resolution/discrimination in the decision-making process corresponding to the comparative example.

As mentioned above, the measure method of SNSs is an important tool in decision-making.To develop new measures in simplified neutrosophic decision-making problems, this study proposes asymmetry measures of SNSs and their normalized symmetry measure of SNSs for the first time, and then develops a MADM method by using the weighted symmetry measure of SNSs.Therefore, this paper is presented as the following frame.Some definitions of asymmetry measures of SNSs are presented in Section 2. The normalized symmetry measure and weighted symmetry measure of SNSs are proposed in Section 3. A MADM method using the weighted symmetry measure is developed in Section 4. In Section 5, the practical example is provided in simplified neutrosophic environments to show the application, along with the sensitive analysis regarding to the attribute weight values, and then the feasibility and effectiveness are indicated by the comparative example.At last, Section 6 indicates conclusions and future work.

Asymmetry Measures of Simplified Neutrosophic Sets
In this section, asymmetry measures of SNSs are presented, including asymmetry measures of SVNSs and INSs.
Ye [2] presented an SNS as a subclass of a NS [1] and gave the following definition.
Definition 1 [2].A SNS is defined as , and h A (x) : U → [0, 1], which are described by the truth, indeterminacy and falsity-membership degrees, satisfying 0 ≤ sup u A (x) + sup v A (x) + sup h A (x) ≤ 3 for INS or 0 ≤ u A (x) + v A (x) + h A (x) ≤ 3 for SVNS and x ∈ U.
For convenience, an element in the SNS A is denoted by a = (u a , v a , h a ), which is called the simplified neutrosophic number (SNN), including a single valued neutrosophic number (SVNN) and an interval neutrosophic number (INN).
First, asymmetry measures of SVNSs are defined in the following.
If one considers the weight of each element b j or a j (j = 1, 2, . . ., n), the weighted asymmetry measure of SNSs can be introduced below.Definition 3. Let B = {b 1 , b 2 , . . ., b n } and A = {a 1 , a 2 , . . ., a n } be two SVNSs, where b j = (u bj , v bj , h bj ) and a j = (u aj , v aj , h aj ) are the j-th SVNNs (j = 1,2, . . ., n) of B and A respectively, and let the weight of an element b j or a j be w j , w j ∈ [0, 1], and ∑ n j = 1 w j = 1 .Then are called the weighted asymmetry measures of B and A.
By the similar way, the two asymmetry measures of SVNSs can be further extended to the asymmetry measures of INSs, which are given by the following definition.
) are the j-th INNs (j = 1, 2, . . ., n) of B and A respectively.Then, two asymmetry measures of B and A are defined as Similarly, if one considers the weight of each element b j or a j (j = 1, 2, . . ., n), the weighted asymmetry measures of INSs can be introduced below.
) are the j-th INNs (j = 1, 2, . . ., n) of B and A respectively, and let the weight of an element a j or b j be w j , w j ∈ [0, 1], and ∑ n j = 1 w j = 1.Thus, two weighted asymmetry measures of A on B are defined as

Normalized Symmetry Measures of Simplified Neutrosophic Sets
A normalized symmetry measure of SNSs is proposed in this section.Definition 6.Let B = {b 1 , b 2 , . . ., b n } and A = {a 1 , a 2 , . . ., a n } be two SNSs, where b j = (u bj , v bj , h bj ) and a j = (u aj , v aj , h aj ) are the j-th SNNs (j = 1, 2, . . ., n) of B and A respectively.Thus is called the normalized symmetry measure between B and A, where B = for INSs are the modules of B and A respectively, and is the inner product between B and A.
Therefore, the closer the value of M(B, A) is to 1, the closer B is to A, and then there are M(B, A) = M(A, B) = 1 if B = A, and it satisfies 0 ≤ M(B, A) ≤ 1 for any B and any A, which is a normalized symmetry measure.
The bellowing weighted symmetry measure between SNSs can be introduced if one considers the weight of each element b j or a j (j = 1, 2, . . ., n).Definition 7. Let B = {b 1 , b 2 , . . ., b n } and A = {a 1 , a 2 , . . ., a n } be two SNSs, where b j = (u bj , v bj , h bj ) and a j = (u aj , v aj , h aj ) are the j-th SNNs (j = 1, 2, . . ., n) of B and A respectively, and let the weight of an element b j or a j be w j , w j ∈ [0, 1], and is known as the weighted symmetry measure of B and A, where B w = for INSs are the weighted modules of B and A, and then (B for INNs is known as the weighted inner product of B and A.

Decision-Making Method Using the Weighted Symmetry Measure
In this section, the proposed weighted symmetry measure is utilized for simplified neutrosophic MADM problems.
Set a set of alternatives as S = {S 1 , S 2 , . . ., S m } and a set of attributes as A = {A 1 , A 2 , . . ., A n } in a MADM problem.Assume that the weight of the attribute A j is w j , w j ∈ [0, 1], and ∑ n j = 1 w j = 1.

Decision-Making Method Using the Weighted Symmetry Measure
In SNS setting, the satisfaction evaluations of an alternative S i (i = 1, 2, . . ., m) for an attribute A j (j = 1, 2, . . ., n) is expressed by an SNS S i = {s i1 , s i2 , . . . ,s in }, where Thus, the decision matrix of SNSs can be established as D = (s ij ) m×n .
In the MADM problem, the similarity measure between the ideal solution/alternative and an alternative can be used for determining the best one among all alternatives.By considering for INNs (j = 1, 2, . . ., n; i = 1, 2, . . ., m) as the ideal solution, an simplified neutrosophic ideal solution/alternative can be given as S * = s * 1 , s * 2 , . . ., s * n .Then by applying Equation (10), the weighted symmetry measure between S * and S i (i = 1, 2, . . ., m) is yielded by where ), for INNs.Thus, the greater the value of M w (S * , S i ) is, the closer S i is to S * , and then the better the alternative S i is.

Decision-Making Examples
A practical example about selecting the manufacturing schemes (alternatives) in the flexible manufacturing system is provided in SNS (SVNS and INS) environments to show the applications of the weighted symmetry measure-based MADM method in realistic scenarios, and then a comparative example with existing relative measures for SVNS is given to show the feasibility and effectiveness of the proposed method.

Practical Example
Assume that we consider a MADM problem in the flexible manufacturing system about the selection of manufacturing schemes (alternatives).Set a set of four alternatives for the flexible manufacturing system as S = {S 1 , S 2 , S 3 , S 4 }.They need to satisfy the three attributes: (i) A 1 is the improvement in quality; (ii) A 2 is the market response; (iii) A 3 is the manufacturing cost.In the decision-making problem, the decision maker/expert specifies the weight vector of the attributes as W = (0.36, 0.3, 0.34) corresponding to the importance of the three attributes.
Thus, the decision maker gives the satisfaction evaluation of an alternative S i (i = 1, 2, 3, 4) for an attribute A j (j = 1, 2, 3) by the evaluation information of SVNNs, and then single-valued neutrosophic decision matrix can be constructed as Thus, the developed approach is used for the MADM problem.
First, the single-valued neutrosophic ideal solution/alternative of s Then, according to Equation ( 11), the weighted symmetry measure values of S * and S i can be obtained as follows: M w (S * , S 1 ) = 0.8945, M w (S * , S 2 ) = 0.9964, M w (S * , S 3 ) = 0. 9717, and M w (S * , S 4 ) = 0.9730 Since the values of the weighted symmetry measure are M w (S * , S 2 ) > M w (S * , S 4 ) > M w (S * , S 3 ) > M w (S * , S 1 ), the four alternatives are ranked as S 2 > S 4 > S 3 > S 1 .Obviously, S 2 is the best one among the four alternatives.
In this practical example, there is little difference of ranking orders under SVNS and INS environments.
To indicate the sensitivity of the proposed MADM method, this work only considers that the attribute weights may affect the ranking of alternatives as the sensitive analysis because the attribute weights are given by the decision maker's subjective judgment/preference in this decision-making problem.If the weights of the three attributes are not considered in this decision-making problem, the three weight values in Equation ( 11) are reduced to w j = 1/n = 1/3 for j =1, 2, 3.
Clearly, there exists a little difference of the ranking orders with the given attribute weights and without the attribute weights under SVNS and INS environments, and then the best alternatives S 2 and S 4 in all ranking orders are still identical in these cases.
By the sensitive analysis regarding to the weights of the three attributes, it is obvious that the attribute weights can affect the ranking orders of four alternatives to some extent, which show some sensitivity to the attribute weights specified by the decision maker or expert.
All the measure values of both the proposed weighted symmetry measure M w (S * , S i ) and the various measures like the cosine measures of Cos w (S * , S i ) and C w (S * , S i ), the Dice measure of D w (S * , S i ), the Jaccard measure of J w (S * , S i ), the projection measure of Projw s* (S i ), and the bidirectional projection measure of BProj w (S * , S i ) in the literature [21] are shown in Table 1, where the average value (AV) and the standard deviation (SD) of S i for i = 1, 2, 3, 4 are also given.From Table 1, the ranking order based on the proposed symmetry measure is the same as the ones of the projection and bidirectional projection measures, but indicates a little difference of other ranking orders.However, the best one in all ranking orders is the same.It is obvious that the proposed symmetry measure is feasible and effective.According to their standard deviations, the SD value of the proposed symmetry measure is 0.0555, which is the biggest one among the SD values of various measures.In the MADM process, however, the cosine, Dice, and Jaccard measures indicate smaller SD values, which show lower resolution/discrimination; while the MADM methods using the projection and bidirectional projection measures also imply low resolution/discrimination due to their small SD values.Obviously, the new MADM method can strengthen the resolution/discrimination in the MADM process of the four alternatives so as to provide effective decision information for decision makers.

Conclusions
This paper firstly defined asymmetry measure of SNSs (SVNSs and INSs), and then developed the normalized symmetry measure and weighted symmetry measure of SNSs (SVNSs and INSs) and their MADM method with SNS information (interval and single valued neutrosophic information).Then the ranking of all alternatives and the best one can be given through the weighted symmetry measure between the ideal solution/alternative and each alternative.Finally, a practical example demonstrated the applications of the developed method for selecting the manufacturing schemes (alternatives) in the flexible manufacturing system under single-valued and interval neutrosophic environments, along with the sensitive analysis regarding to the attribute weights, and then the feasibility and effectiveness of the proposed method were indicated by the comparative example in single-valued neutrosophic setting.
Since the MADM method proposed in this study contains the biggest standard deviations among these existing related MADM methods, the higher resolution/discrimination given in the decision-making process is its main advantage.However, this study only proposes the simplified neutrosophic symmetry measure and its MADM method with the given (subjective) attribute weights for the first time, but it cannot handle simplified neutrosophic group decision-making problems.Therefore, in the future this study will be extended to simplified neutrosophic or simplified neutrosophic cubic group decision-making problems with given/unknown weights.

Table 1 .
Various measure results and ranking orders.AV: average value; SD: standard deviation.