NN-Harmonic Mean Aggregation Operators-Based MCGDM Strategy in a Neutrosophic Number Environment

: A neutrosophic number ( a + bI ) is a signiﬁcant mathematical tool to deal with indeterminate and incomplete information which exists generally in real-world problems, where a and bI denote the determinate component and indeterminate component, respectively. We deﬁne score functions and accuracy functions for ranking neutrosophic numbers. We then deﬁne a cosine function to determine the unknown weight of the criteria. We deﬁne the neutrosophic number harmonic mean operators and prove their basic properties. Then, we develop two novel multi-criteria group decision-making (MCGDM) strategies using the proposed aggregation operators. We solve a numerical example to demonstrate the feasibility, applicability, and effectiveness of the two proposed strategies. Sensitivity analysis with the variation of “ I ” on neutrosophic numbers is performed to demonstrate how the preference ranking order of alternatives is sensitive to the change of “ I ”. The efﬁciency of the developed strategies is ascertained by comparing the results obtained from the proposed strategies with the results obtained from the existing strategies in the literature.


Introduction
Multi-criteria decision-making (MCDM), and multi-criteria group decision-making (MCGDM) are significant branches of decision theories which have been commonly applied in many scientific fields.They have been developed in many directions, such as crisp environments [1,2], and uncertain environments, namely fuzzy environments [3][4][5][6][7][8][9][10][11][12][13], intuitionistic fuzzy environments [14][15][16][17][18][19][20][21][22][23][24], and neutrosophic set environments .Smarandache [46,47] introduced another direction of uncertainty by defining neutrosophic numbers (NN), which represent indeterminate and incomplete information in a new way.A NN consists of a determinate component and an indeterminate component.Thus, the NNs are more applicable to deal with indeterminate and incomplete information in real world problems.The NN is expressed as the function N = p + qI in which p is the determinate component and qI is the indeterminate component.If N = qI, i.e., the indeterminate part reaches the maximum label, the worst situation occurs.If N = p, i.e., the indeterminate part does not appear, the best situation occurs.Thus, the application of NNs is more appropriate to deal with the indeterminate and incomplete information in real-world decision-making situations.
Information aggregation is an essential practice of accumulating relevant information from various sources.It is used to present aggregation between the min and max operators.The harmonic mean is usually used as a mathematical tool to accumulate the central tendency of information [48].
The harmonic mean (HM) is widely used in statistics to calculate the central tendency of a set of data.Park et al. [49] proposed multi-attribute group decision-making (MAGDM) strategy based on HM operators under uncertain linguistic environments.Wei [50] proposed a MAGDM strategy based on fuzzy-induced, ordered, weighted HM.In a fuzzy environment, Xu [48] studied a fuzzy-weighted HM operator, fuzzy ordered weighted HM operator, and a fuzzy hybrid HM operator, and employed them for MADM problems.Ye [51] proposed a multi-attribute decision-making (MADM) strategy based on harmonic averaging projection for a simplified neutrosophic sets (SNS) environment.
In a NN environment, Ye [52] proposed a MAGDM using de-neutrosophication strategy and a possibility degree ranking strategy for neutrosophic numbers.Liu and Liu [53] proposed a NN generalized weighted power averaging operator for MAGDM.Zheng et al. [54] proposed a MAGDM strategy based on a NN generalized hybrid weighted averaging operator.Pramanik et al. [55] studied a teacher selection strategy based on projection and bidirectional projection measures in a NN environment.
The strategies [52][53][54][55] cannot deal with the situation when larger values other than arithmetic mean, geometric mean, and harmonic mean are necessary for experimental purposes.To fill the research gap, we propose two MCGDM strategies.
In this paper, we develop two new MCGDM strategies based on a NN harmonic mean operator (NNHMO) and a NN weighted harmonic mean operator (NNWHMO) to solve MCGDM problems.We define a cosine function to determine unknown weights of the criteria.To develop the proposed strategies, we define score and accuracy functions for ranking NNs for the first time in the literature.
The rest of the paper is structured as follows: Section 2 presents some preliminaries of NNs and score and accuracy functions of NNs.Section 3 devotes NN harmonic mean operator (NNHMO) and NN weighted harmonic mean operator (NNWHMO).Section 4 defines the cosine function to determine unknown criteria weights.Section 5 presents two novel decision-making strategies based on NNHMO and NNWHMO.In Section 6, a numerical example is presented to illustrate the proposed MCGDM strategies and the results show the feasibility of the proposed MCGDM strategies.Section 7 compares the obtained results derived from the proposed strategies and the existing strategies in NN environment.Finally, Section 8 concludes the paper with some remarks and future scope of research.

Preliminaries
In this section, definition of harmonic and weighted harmonic mean of positive real numbers, concepts of NNs, operations on NNs, score and accuracy functions of NNs are outlined.

Harmonic Mean and Weighted Harmonic Mean
Harmonic mean is a traditional average, which is generally used to determine central tendency of data.The harmonic mean is commonly considered as a fusion method of numerical data.Definition 1. [48]: The harmonic mean H of the positive real numbers x 1 , x 2 , . . ., x n is defined as: Definition 2. [49]: The weighted harmonic mean H of the positive real numbers x 1 , x 2 , . . ., x n is defined as WH =

NNs
A NN [46,47] consists of a determinate component x and an indeterminate component yI, and is mathematically expressed as z = x + yI for x, y ∈ R, where I is indeterminacy interval and R is the set of real numbers.A NN z can be specified as a possible interval number, denoted by z = [x + yI L , x + yI U ] for z ∈ Z (Z is set of all NNs) and I ∈ [I L , I U ].The interval I ∈ [I L , I U ] is considered as an indeterminate interval.

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If yI = 0, then z is degenerated to the determinate component z = x Let two NNs be z 1 = x 1 + y 1 I and z 2 = x 2 + y 2 I for z 1 , z 2 ∈ Z, and I ∈ [I L , I U ]. Some basic operational rules for z 1 and z 2 are presented as follows: (1) Proof.Let z = x + yI.Then, Definition 3.For any NN z = x + yI = [x + yI L , x + yI U ], (x and y not both zeroes), its score and accuracy functions are defined, respectively, as follows: Theorem 2. Both score function Sc(z) and accuracy function Ac(z) are bounded.

Harmonic Mean Operators for NNs
In this section, we define harmonic mean operator and weighted harmonic mean operator for neutrosophic numbers.

NN-Harmonic Mean Operator (NNHMO)
Definition 5. Let z i = x i + y i I (i = 1, 2, . . ., n) be a collection of NNs.Then the NNHMO is defined as follows: Proof.
This shows that NNHMO is also a NN.
This shows that NNWHMO is also a NN.

P1. Idempotent law:
Hence, both the operators are bounded.

P3. Monotonicity:
This proves the monotonicity of the functions NNHMO(z 1 ,

Cosine Function for Determining Unknown Criteria Weights
When criteria weights are completely unknown to decision-makers, the entropy measure [56] can be used to calculate criteria weights.Biswas et al. [57] employed entropy measure for MADM problems to determine completely unknown attribute weights of single valued neutrosophic sets (SVNSs).Literature review reflects that, strategy to determine unknown weights in the NN environment is yet to appear.In this paper, we propose a cosine function to determine unknown criteria weights.

Definition 7. The cosine function of a NN
The weight structure is defined as follows: w j = COS j (P) ∑ n j=1 COS j (P) Axioms 2018, 7, 12 7 of 16 The cosine function COS j (P) satisfies the following properties: P1. COS j (P) = 1, if y ij = 0 and x ij = 0. P2.COS j (P) = 0, if x ij = 0 and y ij = 0. P3.COS j (P) ≥ COS j (Q), if x ij of P > x ij of Q or y ij of P < y ij of Q or both.

Multi-Criteria Group Decision-Making Strategies Based on NNHMO and NNWHMO
Two MCGDM strategies using the NNHMO and NNWHMO respectively are developed in this section.Suppose that A = {A 1 , A 2 , . . ., A m } is a set of alternatives, C = {C 1 , C 2 , . . ., C n } is a set of criteria and DM = {DM 1 , DM 2 , . . ., DM k } is a set of decision-makers.Decision-makers' assessment for each alternative A i will be based on each criterion C j .All the assessment values are expressed by NNs.
Steps of decision making strategies based on proposed NNHMO and NNWHMO to solve MCGDM problems are presented below.

MCGDM Strategy 1 (Based on NNHMO)
Strategy 1 is presented (see Figure 1) using the following six steps: Step 1. Determine the relation between alternatives and criteria.

DM k [A|C] =
Axioms 2018, 7, 12 8 of 16 Note 1: Here, x ij + y ij I k represents the NN rating value of the alternative A i with respect to the criterion C j for the decision-maker DM k .
Step 3. To fuse all the aggregation values (DM aggr k (A i )), corresponding to alternatives A i , we define the averaging function as follows: Here, w t (t = 1, 2, . . ., k) is the weight of the decision-maker DM t .
Step 4. Determine the preference ranking order.
Using Equation ( 1), determine the score values Sc(z i ) (accuracy degrees Ac(z i ), if necessary) (i = 1, 2, . . ., m) of all alternatives A i .All the score values are arranged in descending order.The alternative corresponding to the highest score value (accuracy values) reflects the best choice.
Step 5. Select the best alternative from the preference ranking order.

MCGDM Strategy 2 (Based on NNWHMO)
Strategy 2 is presented (see Figure 2) using the following seven steps: Step 1.This step is similar to the first step of Strategy 1.
Step 2. Determine the criteria weights.

Step 3. Determine the weighted aggregation values ( ) ( i waggr k A DM
).
Step 4. Determine the averaging values.
To fuse all the weighted aggregation values ( ), corresponding to alternatives Ai, we define the averaging function as follows: ( ) Here, wt (t = 1, 2, …, k) is the weight of the decision maker DMt.
Step 5. Determine the ranking order.

MCGDM Strategy 2 (Based on NNWHMO)
Strategy 2 is presented (see Figure 2) using the following seven steps: Step 1.This step is similar to the first step of Strategy 1.
Step 2. Determine the criteria weights.
Step 4. Determine the averaging values.
To fuse all the weighted aggregation values (DM waggr k (A i )), corresponding to alternatives A i , we define the averaging function as follows: Here, w t (t = 1, 2, . . ., k) is the weight of the decision maker DM t .
Step 5. Determine the ranking order.
Using Equation ( 1), determine the score values S(z i ) (accuracy degrees A(z i ), if necessary) (i = 1, 2, . . ., m) of all alternatives A i .All the score values are arranged in descending order.The alternative corresponding to the highest score value (accuracy values) reflects the best choice.
Step 6. Select the best alternative from the preference ranking order.

Simulation Results
We solve a numerical example studied by Zheng et al. [54].An investment company desires to invest a sum of money in the best investment fund.There are four possible selection options to invest the money.Feasible selection options are namely, A1: Car company (CARC); A2: Food company (FOODC); A3: Computer company (COMC); A4: Arms company (ARMC).Decision-making must be based on the three criteria namely, risk analysis (C1), growth analysis (C2), environmental impact analysis (C3).The four possible selection options/alternatives are to be selected under the criteria by the NN assessments provided by the three decision-makers DM1, DM2, and DM3.

Solution Using MCGDM Strategy 1
Step 1. Determine the relation between alternatives and criteria.
All assessment values are provided by the following three NN based decision matrices (shown in Equations ( 10)- (12).

Simulation Results
We solve a numerical example studied by Zheng et al. [54].An investment company desires to invest a sum of money in the best investment fund.There are four possible selection options to invest the money.Feasible selection options are namely, A 1 : Car company (CARC); A 2 : Food company (FOODC); A 3 : Computer company (COMC); A 4 : Arms company (ARMC).Decision-making must be based on the three criteria namely, risk analysis (C 1 ), growth analysis (C 2 ), environmental impact analysis (C 3 ).The four possible selection options/alternatives are to be selected under the criteria by the NN assessments provided by the three decision-makers DM 1 , DM 2 , and DM 3 .

Solution Using MCGDM Strategy 1
Step 1. Determine the relation between alternatives and criteria.
All assessment values are provided by the following three NN based decision matrices (shown in Equations ( 10)- (12).
Note 2: Here, DM 1 [L|C] , DM 2 [L|C] and DM 3 [L|C] are the decision matrices for the decision makers DM 1 , DM 2 and DM 3 respectively.
Step 5. Food company (FOODC) is the best alternative for investment.
Note 3: In Figure 3, we represent ranking order of alternatives with variation of "I" based on strategy 1. Figure 3 reflects that various values of I, ranking order of alternatives are different.However, the best choice is the same.
Step 6. End.Note 3: In Figure 3, we represent ranking order of alternatives with variation of "I" based on strategy 1. Figure 3 reflects that various values of I, ranking order of alternatives are different.However, the best choice is the same.

Solution Using MCGDM Strategy 2
Step 1. Determine the relation between alternatives and criteria.This step is similar to the first step of strategy 1.
Step 2. Determine the criteria weights.

Solution Using MCGDM Strategy 2
Step 1. Determine the relation between alternatives and criteria.
This step is similar to the first step of strategy 1.
Step 2. Determine the criteria weights.
Step 3. Determine the weighted aggregation values (DM waggr k Using Equation (1), we calculate the score values Sc(A i ) (i = 1, 2, 3, 4).Since scores values are different, accuracy values are not required.Sensitivity analysis and ranking order of alternatives are shown in Table 2 for different values of I.
Table 2. Sensitivity analysis and ranking order with variation of "I" on NNs for strategy 2.
Note 4: In Figure 4, we represent ranking order of alternatives with variation of "I" based on strategy 2. Figure 4 reflects that various values of I, ranking order of alternatives are different.However, the best choice is the same.
Step 4. Determine the averaging values.
Using Equation ( 9), we calculate the averaging (Considering equal importance of all the decision makers to fuse all the aggregation values corresponding to the alternative Ai. Step 6.Food company (FOODC) is the best alternative for investment.
Note 4: In Figure 4, we represent ranking order of alternatives with variation of "I" based on strategy 2. Figure 4 reflects that various values of I, ranking order of alternatives are different.However, the best choice is the same.

Comparison Analysis
In this subsection, a comparison analysis is conducted between the proposed MCGDM strategies and the other existing strategies in the literature in NN environment.Table 1 reflects that A 2 is the best alternative for I = 0 and I = 0 i.e., for all cases considered.Table 2 reflects that A 2 is the best alternative for any values of I. Ranking order differs for different values of I.
In strategy [52], deneutrosophication process is analyzed.It does not recognize the importance of the aggregation information.MCGDM due to Liu and Liu [53] is based on NN generalized weighted power averaging operator.This strategy cannot deal the situation when larger value other than arithmetic mean, geometric mean, and harmonic mean is necessary for experimental purpose.
The strategy proposed by Zheng et al. [54] cannot be used when few observations contribute disproportionate amount to the arithmetic mean.The proposed two MCGDM strategies are free from these shortcomings.

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NNHMO and NNWHMO in NN environment are firstly defined in the literature.We have also proved their basic properties.

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We have proposed score and accuracy functions of NN numbers for ranking.If two score values are same, then accuracy function can be used for ranking purpose.

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The proposed two strategies can also be used when observations/experiments contribute is disproportionate amount to the arithmetic mean.The harmonic mean is used when sample values contain fractions and/or extreme values (either too small or too big).

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To calculate unknown weights structure of criteria in NN environment, we have proposed cosine function.

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Steps and calculations of the proposed strategies are easy to use.

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We have solved a numerical example to show the feasibility, applicability, and effectiveness of the proposed two strategies.

Conclusions
In the study, we have proposed NNHMO and NNWHMO.We have developed two strategies of ranking NNs based on proposed score and accuracy functions.We have proposed a cosine function to determine unknown weights of the criteria in a NN environment.We have developed two novel MCGDM strategies based on the proposed aggregation operators.We have solved a hypothetical case study and compared the obtained results with other existing strategies to demonstrate the effectiveness of the proposed MCGDM strategies.Sensitivity analysis for different values of I is also conducted to show the influence of I in preference ranking of the alternatives.The proposed MCGDM strategies can be applied in supply selection, pattern recognition, cluster analysis, medical diagnosis, etc.

Figure 3 .
Figure 3. Ranking order with variation of 'I' based on strategy 1.

[Figure 3 .
Figure 3. Ranking order with variation of "I" based on strategy 1.

Figure 4 .
Figure 4. Ranking order with variation of 'I' on NNs for Strategy 2.

Figure 4 .
Figure 4. Ranking order with variation of "I" on NNs for Strategy 2.
accuracy function is bounded.Let two NNs be z1 = x 1 + y 1 I = [x 1 + y 1 I L , x 1 + y 1 I U ], and z 2 = x 2 + y 2 I = [x 2 + y 2 I L , x 2 + y 2 I U ],then the following comparative relations hold: 11 +y 11 I k x 12 +y 12 I k • • • x 1n +y 1n I k x 21 +y 21 I k x 22 +y 22 I k • • • x 2n +y 2n I k Determine the ranking order.Using Equation (1), we calculate the score values Sc(Ai) (i = 1, 2, 3, 4).Since scores values are different, accuracy values are not required.Sensitivity analysis and ranking order of alternatives are shown in Table2for different values of I.

Table 2 .
Sensitivity analysis and ranking order with variation of "I" on NNs for strategy 2.

Table 3 .
Comparison of ranking preference order with variation of "I" on NNs for different strategies.