A Multi ‐ Criteria Decision ‐ Making Method Based on the Improved Single ‐ Valued Neutrosophic Weighted Geometric Operator

: The aggregation operator is one of the most common techniques to solve multi ‐ criteria decision ‐ making (MCDM) problems. The aim of this paper is to propose an MCDM method based on the improved single ‐ valued neutrosophic weighted geometric (ISVNWG) operator. First, the defects of several existing single ‐ valued neutrosophic weighted geometric aggregation operators in terms of producing uncertain results in some special cases are analyzed. Second, an ISVNWG operator is proposed to avoid the defects of existing operators. Further, the properties of the proposed ISVNWG operator, including idempotency, boundedness, monotonicity, and commutativity, are discussed. Finally, a single ‐ valued neutrosophic MCDM method based on the developed ISVNWG operator is proposed to overcome the defects of existing MCDM methods based on existing operators. Application examples demonstrate that our proposed operator and corresponding MCDM method are effective and rational for avoiding uncertain results in some special cases.

However, these existing single-valued geometric aggregation operators and their corresponding MCDM methods have some considerable defects as follows. (1) Existing aggregation operators may produce uncertain results in the aggregation process, that is, the aggregated terms are uncertain in some special cases. (2) Those MCDM methods based on existing aggregation operators can also produce unreasonable decision-making results, which are contradictory to the real decision-making. On account of this, an improved single-valued neutrosophic weighted geometric (ISVNWG) operator is developed to avoid the defects of existing aggregation operators presented above. Moreover, a novel MCDM method based on the proposed aggregation operator is proposed to overcome the defects of existing MCDM methods based on corresponding aggregation operators.
The structure of this article is organized as follows. Section 2 reviews the related concepts, including SVNSs, single-valued neutrosophic number (SVNNs), and the comparison method of SVNNs. Section 3 analyzes the defects of existing aggregation operators. Then, Section 4 proposes the ISVNWG operator, and discusses its related properties. Further, Section 5 puts forward the corresponding MCDM method based on the proposed aggregation operator. In Section 6, the rationality of the proposed MCDM method is investigated in combination with three different application examples. Finally, some conclusions are drawn in Section 7.

Preliminaries
In this section, some related concepts, including SVNSs, SVNNs, and the comparison method of SVNNs, are reviewed. Definition 1 [16,17]. Let X be a space of points (objects), with a generic element in X , denoted by x. An SVNS,  , in X is characterized by which is denoted by , , Definition 2 [18]. Let 1  and 2  be any two SVNNs . Then the comparison method of the two SVNNs is  1  1  1  1  1  1   2  1  1  1  1  , , ,  ,  ,  2  1  1  1  1   i  i  i  i  i   i  i  i  i  i  i   n  n  n  n  n   i  i  i  i  i  i  i  i  i  i  n  n  n  n  n  n  n   i  i  i However, the S N N W G  operator, i.e., Equation (3) Similarly, the SVNFWG operator, i.e., Equation (4) in Definition 6, is unreasonable. Apparently, the truth-membership value cannot be equal to 0, and the indeterminacy-membership and falsitymembership values cannot be equal to 1.

The Improved Single-Valued Neutrosophic Weighted Geometric Operator
In the following, the ISVNWG operator is defined, and the corresponding properties are discussed as well.
Specially, if 1 t  , then the ISVNWG operator, i.e., Equation (5), can be reduced to the SNNWG operator presented in Equation (2). The larger the parameter value of t , the higher the similarity between the ISVNWG operator and the SNNWG operator. Generally speaking, DMs can choose different parameter values of t based on their preferences.
Moreover, if the proposed ISVNWG operator is used to deal with the problems discussed in Definitions 3-5, and assuming = 0.95 t , then we can get the following results.
(1) If the same case in Definition 3 is considered, that is, 1 Apparently, the result by using the proposed ISVNWG operator is consistent with our intuition. Hence, the proposed ISVNWG operator can avoid the defects of the SVNWG operator.
(2) If the same case in Definitions 4 and 5 is considered, that is 1 The result by using the proposed ISVNWG operator is also consistent with our intuition. Hence, the proposed operator can avoid the defects of the SNNWG and the S N N W G  operators simultaneously.
(3) If the same case in Definition 6 is considered, that is, The result by using the proposed ISVNWG operator is also consistent with our intuition. Hence, the proposed ISVNWG operator can avoid the defects of the SVNFWG operator.
Thus, the proposed ISVNWG operator can overcome the defects of existing aggregation operators. Moreover, the proposed ISVNWG operator also satisfies some properties, including idempotency, boundedness, monotonicity, and commutativity, which are discussed in the following.

The Single-Valued Neutrosophic MCDM Method Based on the Improved Aggregation Operator
Assume   c . Then the steps to obtain the optimal alternative(s) are presented as follows: Step 1. Normalize the Decision Matrix.
Since the criteria may belong to the cost type or benefit type, the normalized method of the criteria can be determined as: Step 2. Determine the General Value.
Based on the ISVNWG operator, i.e., Equation (5) where 0.95 t  is assumed in the following calculation process.
Step 3. Calculate the Score, Accuracy, and Certainty Values. Apparently, the result by using the proposed MCDM method based on the developed ISVNWG operator is the same as that by using the MCDM method based on the SVNWG operator, i.e., 1 2    , which is more reasonable in the real decision-making environment. By contrast, the results by using the MCDM methods based on the SNNWG,


In order to further illustrate the advantages of the proposed ISVNWG operator and the corresponding MCDM method, comparison analysis is conducted in the following by using the MCDM methods based on different aggregation operators to deal with the same MCDM problem, respectively, and 2   , then the comparison results are shown in Table 2.