A Multi-Criteria Decision-Making Method Based on Single-Valued Neutrosophic Partitioned Heronian Mean Operator

: A multi-criteria decision-making (MCDM) method with single-valued neutrosophic information is developed based on the Partitioned Heronian Mean (PHM) operator and the Shapley fuzzy measure, which recognizes correlation among the selection criteria. Motivated by the PHM operator and Shapley fuzzy measure, two new aggregation operators, namely the single-valued neutrosophic PHM operator and the weighted single-valued neutrosophic Shapley PHM operator, are deﬁned, and their corresponding properties and some special cases are investigated. An MCDM model is applied to solve the single-valued neutrosophic problem where weight information is not completely known. An example is provided to validate the proposed method.


Introduction
Zadeh first put forward the notion of fuzzy sets (FSs) [1]. Since then, multi-criteria decision-making (MCDM) methods based on FSs have been well developed and applied to hotel selection [2], investment project selection [3], supplier selection [4], solar power station site selection [5], recycling waste resource evaluation [6], and others [7][8][9][10][11][12][13]. However, due to the inherent subjectivity in the preferences of the decision makers (DMs), a single membership degree of FSs cannot adequately capture the subjectivity and uncertainty in the decision-making process. In view of this, Atanassov [14] introduced intuitionistic fuzzy sets (IFSs), including membership and non-membership degrees and a hesitation index, as an extension of FSs. However, both FSs and IFSs are not adept at tackling problems involving information uncertainty. For example, when we ask an expert about a certain statement, the expert may say the probability that the statement is true, false, and unsure is 0.6, 0.5, and 0.1 respectively [15]. Clearly, the solution to this problem is beyond the scope of FSs and IFSs. Smarandache et al. [16] constructed neutrosophic sets (NSs) that involve three membership functions: truth, indeterminacy, and falsity. It is noted that NSs lie on a non-standard unit interval ]0 − , 1 + [ [17], which is an extension of the standard interval [0.1] of IFSs. The uncertainty presented here, i.e., the indeterminacy factor, depends on the truth and falsity values while the incorporated uncertainty depends on the membership and non-membership degrees of the IFSs [18]. Thus, the earlier example of NSs can be expressed as x(0.6, 0.1, 0.5). While some MCDM methods with neutrosophic information have been investigated [19][20][21], their applicability is restricted because of the non-standard unit interval. As such, single-valued neutrosophic sets (SVNSs) were proposed, as a special case of NSs [22]. Definition 2 ([54]). Suppose µ is a fuzzy measure on X. The corresponding Möbius transformation can be expressed as β ⊂ X, m(β) = α⊂β (−1) |β\α| µ(α) (1) If β = k, m(β) = 0 and there exists at least one subset γ γ = k satisfying m(γ) 0, then µ is called a k-order additive fuzzy measure. Definition 3 ([50]). Suppose µ is a fuzzy measure on X; the Shapley value to measure the average importance degree of S is: where n, m, and s denote the cardinalities of X, M, and S, respectively. As noted in [54], τ S (µ, X) ≥ 0 and S⊆X τ S (µ, X) = 1. τ S (µ, X) is called Shapley fuzzy measures [53].
Definition 5 ([49]). Let χ i (i = 1, 2, . . . , n) be a set of inputs that can be partitioned into t categories P l (l = 1, 2, . . . , t). The PHM operator is defined as: PHM p,q (χ 1 , χ 2 , . . . , χ n ) = 1 t where p, q ≥ 0, p + q > 0, t l=1 |P l | = n, and P i ∩ P j = ∅, and |P l | denotes the cardinality of P l . Example 2. If C = {c 1 , c 2 , c 3 , c 4 , c 5 } is a set of criteria that can be partitioned into two categories P 1 = {c 1 , c 2 , c 3 } and P 2 = {c 4 , c 5 }, and the assessment values provided by the DMs are χ = {0.7, 0.5, 0.4, 0.6, 0.8} (for convenience, let p = q = 1), then, the aggregated results using the PHM operator are written as: Moreover, The reason for the difference in the results obtained by the PHM operator and those obtained by the HM operator is that the PHM operator partitions the input values into categories based on the relationship of the values, whereas the HM operator presupposes the condition that each input value is correlated with the other values. Therefore, the PHM operator is more reasonable than the HM operator.

NSs and SVNSs
Definition 6 [16]. An NSS in X = {x 1 , x 2 , . . . , x n } can be characterized asS = x,TS(x),ĨS(x),FS(x) x ∈ X , whereTS(x),ĨS(x), andFS(x) denote the truth, indeterminacy, and falsity memberships respectively. Furthermore,TS(x),ĨS(x), andFS(x) are subsets of ]0 − , 1 + [, that is, Since it is impractical for NSs to tackle real-life problems because of their nonstandard intervals, Majumdar and Samant [18] defined SVNSs based on standard intervals, and Ye [19] developed the corresponding properties for SVNSs. 1] . If X has only one element, then S is a single-valued neutrosophic number (SVNN). For convenience, we denote the SVNN by S = T S , I S , F S . Definition 8 ([22]). Let S = T S , I S , F S , S 1 = T S 1 , I S 1 , F S 1 , and S 2 = T S 2 , I S 2 , F S 2 be three SVNNs. With λ > 0, the following properties hold: However, as stated in [19], the above operations are unreasonable. In view of this, Peng et al. [20] improved the properties of SVNNs as well as the corresponding comparison method. Definition 9 ([23]). Let S = T S , I S , F S , S 1 = T S 1 , I S 1 , F S 1 , and S 2 = T S 2 , I S 2 , F S 2 be three SVNNs. With λ > 0, the properties of the SVNNs are defined as follows:

Single-Valued Neutrosophic PHM Operators
Through the PHM operator and Shapley fuzzy measure, the SVNPHM and WSVNSPHM operators are, respectively, defined, and their corresponding properties are discussed in this section.

SVNPHM Operator
Definition 12. Let S i = (T i , I i , F i )(i = 1, 2, . . . , n) be a set of SVNNs that can be partitioned into categories P l (l = 1, 2, . . . , t). The SVNPHM operator is defined as where p, q ≥ 0, p + q > 0, t l=1 |P l | = n, and P i ∩ P j = ∅. |P l | represents the cardinality of P l .
. . , n) be a set of SVNNs. Then, the results under the SVNPHM operator also produce an SVNN, i.e., Proof.
Based on Definition 9, we have S Next, we present some special cases with regard to the parameters.

WSVNSPHM Operator
Since the importance of each input value varies according to the decision-making situation, we propose a WSVNSPHM operator in this subsection. 1, 2, . . . , n) is a set of SVNNs that can be divided into categories P l (l = 1, 2, . . . , t), and τ i (µ, P l ) is the Shapley fuzzy measure on P l for S i = (T i , I i , F i )(i = 1, 2, . . . , n) in the l-th partition. The WSVNSPHM operator is defined as: where p, q ≥ 0, p + q > 0, t l=1 |P l | = n, and P i ∩ P j = ∅. |P l | represents the cardinality of P l .
If S − = min j T j , max j I j , max j F j and S + = max j T j , min j I j , min j F j , then S − ≤ WSVNSPHM p,q (S 1 , S 2 , . . . , S n ) ≤ S + .

Single-Valued Neutrosophic MCDM Method with Incomplete Weight Information
Suppose S = {S 1 , S 2 , . . . , S n } is a group of candidates and C = {c 1 , c 2 , . . . , c m } is the set of the corresponding selection criteria. Then R = S ij n×m is the single-valued neutrosophic decision matrix, whereby S ij = T ij , I ij , F ij (i = 1, 2, . . . , n; j = 1, 2, . . . , m) can be provided by DMs with respect to S i for the criterion c j in the form of SVNNs. Based on the relationships among the criteria, S ij can be partitioned into t categories P l (l = 1, 2, . . . , t) where P i ∩ P j = ∅. If the criteria are correlated with each other, then the Shapley fuzzy measure is the weight of the criteria and t = 1. Further, if the Shapley fuzzy measure of the criteria is known, the corresponding aggregation operators can be used directly to obtain the aggregated values. If it is partly or fully unknown, then the Shapley fuzzy measure of the criteria should be found first.
The flowchart of the proposed method is shown in Figure 1 and the steps to finding the optimal candidate(s) are as follows.

Step 1. Construct and normalize decision matrix
The DMs evaluate the criteria for each candidate and construct the decision-matrix. As the selection criteria will always involve the benefit type and cost type in MCDM problems, if the criteria belong to the benefit type, then it is not necessary to normalize the decision matrix. The cost type criteria should be transformed into the associated benefit type criteria as: for benefit criterion , for cost criterion

Step 1. Construct and normalize decision matrix
The DMs evaluate the criteria for each candidate and construct the decision-matrix. As the selection criteria will always involve the benefit type and cost type in MCDM problems, if the criteria belong to the benefit type, then it is not necessary to normalize the decision matrix. The cost type criteria should be transformed into the associated benefit type criteria as: for benefit criterion c j S ij c , for cost criterion c j , (i = 1, 2, . . . , n; j = 1, 2, . . . , m), where S ij Then, the normalized decision matrixR = S ij n×m can be obtained.
Step 2. Determine closeness coefficients . . . ,S − n be the positive and negative ideal solutions respectively, 1, 2, . . . , n; j = 1, 2, . . . , m). The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [55] is one of the key techniques in dealing with MCDM problems and it is very intuitive and simple. It can provide a ranking method by the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). Then the closeness coefficient of the candidate from the PIS can be found as follows: where d ij S ij ,S + can be obtained by using Equation (5).

Step 3. Determine Shapley fuzzy measures
According to TOPSIS [55], the smaller the value of D ij + S ij ,S + , the betterS ij is. If the weight of the criteria is partly known, then a model based on the fuzzy measure can be constructed as: where τ c j (µ, C) denotes the weight of criterion c j , and G j represents the weight information. Next, the fuzzy measure and the corresponding Shapley fuzzy measure are obtained by solving linear programming model (18).
Step 5. Find values of score, accuracy, and certainty Based on Definition 10, the values of score s(ς i ), accuracy a(ς i ), and certainty c(ς i ) of S i (i = 1, 2, . . . , n) can be achieved.
Step 6. Rank candidates According to Step 5, all candidates S i (i = 1, 2, . . . , n) are ranked, and the best selected.

Example
Hww is a large telecommunication technology player based in China. Hww produces and sells telecommunication equipment. To enhance the competitiveness of its products, the company intends to replace an existing electronic components supplier to improve the product quality. Thus, the decision-making department has to choose a suitable supplier from several candidates. Following preliminary surveys, five suppliers are considered, denoted by S i (i = 1, 2, . . . , 5). The assessment values are provided in the form of SVNNs with respect to five factors, namely: c 1 : cost, c 2 : quality, c 3 : service performance, c 4 : supplier's profile, and c 5 : risk. From the relationship amongst the five criteria, these criteria can be partitioned into two categories: P 1 = {c 1 , c 2 , c 5 } and P 2 = {c 3 , c 4 }. Only the range of the weights of these criteria are known, with is constructed as presented in Table 1.

Decision-Making Process
The decision-making process, using the proposed method, is as follows.

Step 1. Construct and normalize decision matrix
The DMs assess the values as SVNNs, and criteria c 1 , c 2 , and c 5 belong to the cost type. The normalized decision matrixR = S ij n×m is obtained as shown in Table 2.

Step 2. Compute closeness coefficients
Using Equation (17), the closeness coefficients of the candidates from the positive ideal solution are determined as given in Table 3.

Sensitivity Analysis
Next, a sensitivity analysis can be conducted to investigate the influence of the values of p and q on the final rankings. Table 4 shows the score values of the five candidates using the WSVNPHM operator. As can be seen, if p = q = 1, the final rank order is S 5 S 4 S 3 S 1 S 2 . However, when p and q are equal to the other values, the final rank order is S 5 S 4 S 1 S 3 S 2 . Although the rank positions of S 1 and S 3 will change with p and q, the best candidate is always S 5 while the worst is S 2 . Table 4 shows that the gap between the first and second rank positions increases with p and q, demonstrating the choice of candidate S 5 as an optimal scheme. Figures 2-6 show how the score values of the five candidates change with p and q in the interval [0, 1] under the WSVNPHM operator.

Comparison Analysis
To further validate the proposed MCDM method, we compared it against some of the existing methods based on aggregation operators. Since most methods cannot handle cases when there is only partial information on the weights of the criteria, the weights were first set as w = (0.5083, 0.4083, 0.3250, 0.6750, 0.0833) T using the optimal Shapley fuzzy measure found in Section 5.1.
For the proposed MCDM method, the weights found can be used to aggregate the preference information in Step 4 with p = q = 1. For the MCDM methods based on the Frank aggregation [24], Hamacher [25], and Bonferroni mean [27,28] operators, the corresponding parameters are determined as λ = 2 and p = q = 1, respectively. Table 5 shows the comparison results of the different methods used. Clearly, the final results found through the proposed method are the same as those by the methods employed in [24,27,28], and the best candidate is S 5 . However, for the methods employed in [22,23,25], the best candidate is S 4 . Notably, while the methods in [24,27,28] yield reasonable results, they do not factor in the correlation or the categories of the selection criteria. Furthermore, as discussed in [23], the rules of the corresponding operations in [22] are unreasonable, which leads to unreasonable algebraic operators. In actual decision-making instances, not all selection criteria correlate with each other. Our method can partition the criteria into distinct categories, considering not only the interrelationship of the criteria but also the independence of the criteria.

Conclusions
A single-valued neutrosophic MCDM problem with interdependent characteristics was investigated in this paper. Through the PHM operator and Shapley fuzzy measure, the SVNPHM and WSVNSPHM aggregation operators were defined, and their corresponding properties were discussed. An integrated MCDM method was then developed to solve single-valued neutrosophic problems where the weights of the selection criteria may not be completely known a priori. A mathematical programming model based on fuzzy measures was formed to obtain the optimal Shapley fuzzy measure. Next, the aggregation operators were used to aggregate DMs' preference information. Finally, an example was presented to validate the proposed method, yielding reasonable outcomes. Thus, our proposed aggregation operators recognize the correlation of the selection criteria, unlike previous techniques. In future, other aggregation operators of SVNNs based on the Shapley fuzzy measure can be studied.
Author Contributions: C.T. and J.J.P. proposed methodology and provided the original draft preparation. Z.Q.Z. analyzed the data. M.G. and J.Q.W. designed the study. All authors approved the publication work. All authors have read and agreed to the published version of the manuscript.