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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Harmonic Mean and Weighted Harmonic Mean

**Definition**

**1.**

_{1}, x

_{2}, …, x

_{n}is defined as: $\mathrm{H}=\frac{n}{\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}+\cdots +\frac{1}{{x}_{n}}}=\frac{n}{{\displaystyle \sum _{i=1}^{n}\frac{1}{{x}_{i}}}}$ ; i = 1, 2, …, n.

**Definition**

**2.**

_{1}, x

_{2}, …, x

_{n}is defined as $\mathrm{WH}=\frac{1}{\frac{{w}_{1}}{{x}_{1}}+\frac{{w}_{2}}{{x}_{2}}+\cdots +\frac{{w}_{n}}{{x}_{n}}}=\frac{1}{{\displaystyle \sum _{i=1}^{n}\frac{{w}_{i}}{{x}_{i}}}}$ ; i = 1, 2, …, n.

#### 2.2. NNs

^{L}, x + yI

^{U}] for z $\in $ Z (Z is set of all NNs) and I $\in $ [I

^{L}, I

^{U}]. The interval I $\in $ [I

^{L}, I

^{U}] is considered as an indeterminate interval.

- If yI = 0, then z is degenerated to the determinate component z = x
- If x = 0, then z is degenerated to the indeterminate component z = yI
- If I
^{L}= I^{U}, then z is degenerated to a real number.

_{1}= x

_{1}+ y

_{1}I and z

_{2}= x

_{2}+ y

_{2}I for z

_{1}, z

_{2}$\in $ Z, and I $\in $ [I

^{L}, I

^{U}]. Some basic operational rules for z

_{1}and z

_{2}are presented as follows:

- (1)
- I
^{2}= I - (2)
- I.0 = 0
- (3)
- I/I = Undefined
- (4)
- z
_{1}+ z_{2}= x_{1}+ x_{2}+ (y_{1}+ y_{2})I = [x_{1}+ x_{2}+ (y_{1}+ y_{2})I^{L}, x_{1}+ x_{2}+ (y_{1}+ y_{2})I^{U}] - (5)
- z
_{1}− z_{2}= x_{1}− x_{2}+ (y_{1}− y_{2})I = [x_{1}− x_{2}+ (y_{1}− y_{2})I^{L}, x_{1}− x_{2}+ (y_{1}− y_{2})I^{U}] - (6)
- z
_{1}$\times $ z_{2}= x_{1}x_{2}+ (x_{1}y_{2}+ x_{2}y_{1})I + y_{1}y_{2}I^{2}= x_{1}x_{2}+ (x_{1}y_{2}+ x_{2}y_{1}+ y_{1}y_{2})I - (7)
- $\frac{{z}_{1}}{{z}_{2}}=\frac{x{}_{1}+y{}_{1}I}{x{}_{2}+y{}_{2}I}=\frac{x{}_{1}}{x{}_{2}}+\frac{x{}_{2}y{}_{1}-x{}_{1}y{}_{2}}{x{}_{2}(x{}_{2}+y{}_{2})}I;x{}_{2}\ne 0,x{}_{2}\ne -y{}_{2}$
- (8)
- $\frac{1}{{z}_{1}}=\frac{1+0.I}{x{}_{1}+y{}_{1}I}=\frac{1}{x{}_{1}}+\frac{-y{}_{1}}{x{}_{1}(x{}_{1}+y{}_{1})}I;x{}_{1}\ne 0,x{}_{1}\ne -y{}_{1}$
- (9)
- ${z}_{1}^{2}={x}_{1}^{2}+(2x{}_{1}y{}_{1}+{y}_{1}^{2})I$
- (10)
- $\mathsf{\lambda}z{}_{1}=\mathsf{\lambda}x{}_{1}+\mathsf{\lambda}y{}_{1}I$

**Theorem**

**1.**

**Proof.**

**Definition**

**3.**

^{L}, x + yI

^{U}], (x and y not both zeroes), its score and accuracy functions are defined, respectively, as follows:

**Theorem**

**2.**

**Proof.**

**Definition**

**4.**

_{1}= 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:

- If S(z
_{1}) > S(z_{2}), then z_{1}> z_{2} - If S(z
_{1}) = S(z_{2}) and A(z_{1}) < A(z_{2}), then z_{1}< z_{2} - If S(z
_{1}) = S(z_{2}) and A(z_{1}) = A(z_{2}), then z_{1}= z_{2}.

**Example**

**1.**

_{1}= 10 + 2I, z

_{2}= 12 and z

_{3}= 12 + 5I and I $\in $ [0, 0.2]. Then,

_{1}) = 0.5099, S(z

_{2}) = 0.5, S(z

_{3}) = 0.5577, A(z

_{1}) = 0.999969, A(z

_{2}) = 0.999994, A(z

_{3}) = 0.999997.

## 3. Harmonic Mean Operators for NNs

#### 3.1. NN-Harmonic Mean Operator (NNHMO)

**Definition**

**5.**

_{i}= x

_{i}+ y

_{i}I (i = 1, 2, …, n) be a collection of NNs. Then the NNHMO is defined as follows:

**Theorem**

**3.**

_{i}= x

_{i}+ y

_{i}I (i = 1, 2, …, n) be a collection of NNs. The aggregated value of the $\mathrm{NNHMO}(z{}_{1},z{}_{2},\cdots ,z{}_{n})$ operator is also a NN.

**Proof.**

#### 3.2. NN-Weighted Harmonic Mean Operator (NNWHMO)

**Definition**

**6.**

_{i}= x

_{i}+ y

_{i}I (i = 1, 2, …, n) be a collection of NNs and w

_{i}(i = 1, 2, …, n) is the weight of z

_{i}(i = 1, 2, …, n) and $\sum _{i=1}^{n}{w}_{i}}=1.$ Then the NN-weighted harmonic mean (NNWHMO) is defined as follows:

**Theorem**

**4.**

_{i}= x

_{i}+ y

_{i}I (i = 1, 2, …, n) be a collection of NNs. The aggregated value of the $\mathrm{NNWHMO}(z{}_{1},z{}_{2},\cdots ,z{}_{n})$ operator is also a NN.

**Proof.**

**Example**

**2.**

_{1}= 3 + 2I and z

_{2}= 2 + I and I $\in $ [0, 0.2]. Then:

**Example**

**3.**

_{1}= 3 + 2I and z

_{2}= 2 + I, I $\in $ [0, 0.2] and w

_{1}= 0.4, w

_{2}= 0.6, then:

**Idempotent law**:

_{i}= z for i = 1, 2, …, n then, $\mathrm{NNHMO}(z{}_{1},z{}_{2},\cdots ,z{}_{n})=z$ and $\mathrm{NNWHMO}(z{}_{1},z{}_{2},\cdots ,z{}_{n})=z$.

**Proof.**

_{i}= z, $\sum _{i=1}^{n}{w}_{i}}=1,$

**Boundedness**:

**Proof.**

**Monotonicity**:

**Proof.**

**Commutativity**:

**Proof.**

## 4. Cosine Function for Determining Unknown Criteria Weights

**Definition**

**7.**

_{ij}+ y

_{ij}I = [x

_{ij}+ y

_{ij}I

^{L}, x

_{ij}+ y

_{ij}I

^{U}], (i = 1, 2, …, m; j = 1, 2, …, n) is defined as follows:

- P1.
- ${COS}_{j}(P)=1$, if ${y}_{ij}=0\mathrm{and}{x}_{ij}\ne 0.$
- P2.
- ${COS}_{j}(P)=0$, if $x{}_{ij}=0and{y}_{ij}\ne 0.$
- P3.
- ${COS}_{j}(P)\ge {COS}_{j}(Q)$, if x
_{ij}of P > x_{ij}of Q or y_{ij}of P < y_{ij}of Q or both.

**Proof.**

- P1.
- ${y}_{ij}=0$ $\Rightarrow $ ${COS}_{j}(P)=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left[\mathrm{cos}0\right]}=1$
- P2.
- $x{}_{ij}=0$ $\Rightarrow $ ${COS}_{j}(P)=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left[\mathrm{cos}\frac{\pi}{2}\right]}=0$
- P3.
- For, x
_{ij}of P > x_{ij}of Q

- ⇒
- Determinate part of P > Determinate part of Q
- ⇒
- ${COS}_{j}(Q)<{COS}_{j}(P)$.For, y
_{ij}of P < y_{ij}of Q - ⇒
- Indeterminacy part of P < Indeterminacy part of Q
- ⇒
- ${COS}_{j}(Q)>{COS}_{j}(P)$.For, x
_{ij}of P > x_{ij}of Q and y_{ij}of P < y_{ij}of Q - ⇒
- (Real part of P > Real part of Q) & (Indeterminacy part of P < Indeterminacy part of Q)
- ⇒
- ${COS}_{j}(Q)>{COS}_{j}(P)$. ☐

**Example**

**4.**

_{1}= 3 + 2I, and z

_{2}= 3 + 5I, then, $COS(z{}_{1})=0.9066$, $COS(z{}_{2})=0.7817$.

**Example**

**5.**

_{1}= 3 + I, and z

_{2}= 7 + I, then, $COS(z{}_{1})=0.9693$, $COS(z{}_{2})=0.9938$.

**Example**

**6.**

_{1}= 10 + 2I, and z

_{2}= 2 + 10I, then, $COS(z{}_{1})=0.9882$, $COS(z{}_{2})=0.7178$.

## 5. Multi-Criteria Group Decision-Making Strategies Based on NNHMO and NNWHMO

_{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.

#### 5.1. MCGDM Strategy 1 (Based on NNHMO)

**Step 1.**Determine the relation between alternatives and criteria.

_{i}(i = 1, 2, …, m) and the criterion C

_{j}(j = 1, 2, …, n) is presented in Equation (7).

_{i}with respect to the criterion C

_{j}for the decision-maker DM

_{k}.

**Step 2.**Using Equation (3), determine the aggregation values ($DM{}_{k}{}^{aggr}(A{}_{i})$), (i = 1, 2, …, n) for all decision matrices.

**Step 3.**To fuse all the aggregation values ($DM{}_{k}{}^{aggr}(A{}_{i})$), corresponding to alternatives A

_{i}, we define the averaging function as follows:

_{t}(t = 1, 2, …, k) is the weight of the decision-maker DM

_{t}.

**Step 4.**Determine the preference ranking order.

_{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.

**Step 6.**End.

#### 5.2. MCGDM Strategy 2 (Based on NNWHMO)

**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 ($DM{}_{k}{}^{waggr}(A{}_{i})$).

**Step 4.**Determine the averaging values.

_{i}, we define the averaging function as follows:

_{t}(t = 1, 2, …, k) is the weight of the decision maker DM

_{t}.

**Step 5.**Determine the ranking order.

_{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.

**Step 7.**End.

## 6. Simulation Results

_{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}.

#### 6.1. Solution Using MCGDM Strategy 1

**Step 1.**Determine the relation between alternatives and criteria.

_{1}, DM

_{2}and DM

_{3}respectively.

**Step 2.**Determine the weighted aggregation values ($DM{}_{k}{}^{aggr}(A{}_{i})$).

**Step 3.**Determine the averaging values.

_{i}.

**Step 4.**Using Equation (1), we calculate the score values Sc(A

_{i}) (i = 1, 2, 3, 4). Sensitivity analysis and ranking order of alternatives are shown in Table 1 for different values of I.

**Step 5.**Food company (FOODC) is the best alternative for investment.

**Step 6.**End.

#### 6.2. Solution Using MCGDM Strategy 2

**Step 1.**Determine the relation between alternatives and criteria.

**Step 2.**Determine the criteria weights.

_{1}= 0.3265, w

_{2}= 0.3430, w

_{3}= 0.3305] for DM

_{1},

_{1}= 0.3332, w

_{2}= 0.3334, w

_{3}= 0.3334] for DM

_{2},

_{1}= 0.3333, w

_{2}= 0.3335, w

_{3}= 0.3332] for DM

_{3}.

**Step 3.**Determine the weighted aggregation values ($DM{}_{k}{}^{waggr}(A{}_{i})$).

**Step 4.**Determine the averaging values.

_{i}.

**Step 5.**Determine the ranking order.

_{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.

**Step 6.**Food company (FOODC) is the best alternative for investment.

**Step 7.**End.

## 7. Comparison Analysis and Contributions of the Proposed Approach

#### 7.1. Comparison Analysis

_{2}is the best alternative for I = 0 and $I\ne 0$ 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.

_{2}is the best alternative for [52,53,54] and the proposed strategies. When I lies in [0, 0.6], [0, 0.8], [0, 1], A

_{4}is the best alternative for [52,54], whereas A

_{2}is the best alternative for [53], and the proposed strategies.

#### 7.2. Contributions of the Proposed Approach

- NNHMO and NNWHMO in NN environment are firstly defined in the literature. We have also proved their basic properties.
- 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.
- 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).
- To calculate unknown weights structure of criteria in NN environment, we have proposed cosine function.
- Steps and calculations of the proposed strategies are easy to use.
- We have solved a numerical example to show the feasibility, applicability, and effectiveness of the proposed two strategies.

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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I | Sc(A_{i}) | Ranking Order |
---|---|---|

I = [0, 0] | S(A_{1}) = 0.4988, S(A_{2}) = 0.4993, S(A_{3}) = 0.4982, S(A_{4}) = 0.4983 | A_{2} $\succ $ A_{1} $\succ $ A_{4} $\succ $ A_{3} |

I $\in $ [0, 0.2] | S(A_{1}) = 0.5081, S(A_{2}) = 0.5144, S(A_{3}) = 0.5067, S(A_{4}) = 0.5056 | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

I $\in $ [0, 0.4] | S(A_{1}) = 0.5182, S(A_{2}) = 0.5195, S(A_{3}) = 0.5151, S(A_{4}) = 0.5249 | A_{2} $\succ $ A_{1} $\succ $ A_{4} $\succ $ A_{3} |

I $\in $ [0, 0.6] | S(A_{1}) = 0.5289, S(A_{2}) = 0.5346, S(A_{3}) = 0.5236, S(A_{4}) = 0.5233 | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

I $\in $ [0, 0.8] | S(A_{1}) = 0.5396, S(A_{2}) = 0.5497, S(A_{3}) = 0.5320, S(A_{4}) = 0.5316 | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

I $\in $ [0, 1] | S(A_{1}) = 0.5503, S(A_{2}) = 0.5547, S(A_{3}) = 0.5405, S(A_{4}) = 0.5399 | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

I | Sc(A_{i}) | Ranking Order |
---|---|---|

I = 0 | S(A_{1}) = 0.4968, S(A_{2}) = 0.4993, S(A_{3}) = 0.4981, S(A_{4}) = 0.4982 | A_{2} $\succ $ A_{4} $\succ $ A_{3} $\succ $ A_{1} |

I $\in $ [0, 0.2] | S(A_{1}) = 0.5081, S(A_{2}) = 0.5095, S(A_{3}) = 0.5068, S(A_{4}) = 0.5067 | A_{2} $\succ $ A_{1} $\succ $ A_{4} $\succ $ A_{3} |

I $\in $ [0, 0.4] | S(A_{1}) = 0.5195, S(A_{2}) = 0.5198, S(A_{3}) = 0.5155, S(A_{4}) = 0.5153 | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

I $\in $ [0, 0.6] | S(A_{1}) = 0.5308, S(A_{2}) = 0.5350, S(A_{3}) = 0.5241, S(A_{4}) = 0.5239 | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

I $\in $ [0, 0.8] | S(A_{1}) = 0.5421, S(A_{2}) = 0.5502, S(A_{3}) = 0.5328, S(A_{4}) = 0.5324 | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

I $\in $ [0, 1] | S(A_{1}) = 0.5535, S(A_{2}) = 0.5654, S(A_{3}) = 0.5415, S(A_{4}) = 0.5410 | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

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

I | Ye [52] | Zheng et al. [54] | Liu and Liu [53] | Proposed Strategy 1 | Proposed Strategy 2 |
---|---|---|---|---|---|

[0, 0] | A_{2} $\succ $ A_{4} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{4} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{4} $\succ $ A_{1} $\succ $ A_{3} | A_{2} $\succ $ A_{1} $\succ $ A_{4} $\succ $ A_{3} | A_{2} $\succ $ A_{4} $\succ $ A_{3} $\succ $ A_{1} |

[0, 0.2] | A_{2} $\succ $ A_{4} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{4} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{3} $\succ $ A_{1} $\succ $ A_{4} | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} | A_{2} $\succ $ A_{1} $\succ $ A_{4} $\succ $ A_{3} |

[0, 0.4] | A_{2} $\succ $ A_{4} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{4} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{3} $\succ $ A_{4} $\succ $ A_{1} | A_{2} $\succ $ A_{1} $\succ $ A_{4} $\succ $ A_{3} | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

[0, 0.6] | A_{4} $\succ $ A_{2} $\succ $ A_{3} $\succ $ A_{1} | A_{4} $\succ $ A_{2} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{3} $\succ $ A_{4} $\succ $ A_{1} | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

[0, 0.8] | A_{4} $\succ $ A_{2} $\succ $ A_{3} $\succ $ A_{1} | A_{4} $\succ $ A_{2} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{3} $\succ $ A_{4} $\succ $ A_{1} | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

[0, 1] | A_{4} $\succ $ A_{2} $\succ $ A_{3} $\succ $ A_{1} | A_{4} $\succ $ A_{2} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{4} $\succ $ A_{3} $\succ $ A_{1} | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} | A_{2} $\succ $ A_{1} $\succ $ A_{3} $\succ $ A_{4} |

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## Share and Cite

**MDPI and ACS Style**

Mondal, K.; Pramanik, S.; Giri, B.C.; Smarandache, F.
NN-Harmonic Mean Aggregation Operators-Based MCGDM Strategy in a Neutrosophic Number Environment. *Axioms* **2018**, *7*, 12.
https://doi.org/10.3390/axioms7010012

**AMA Style**

Mondal K, Pramanik S, Giri BC, Smarandache F.
NN-Harmonic Mean Aggregation Operators-Based MCGDM Strategy in a Neutrosophic Number Environment. *Axioms*. 2018; 7(1):12.
https://doi.org/10.3390/axioms7010012

**Chicago/Turabian Style**

Mondal, Kalyan, Surapati Pramanik, Bibhas C. Giri, and Florentin Smarandache.
2018. "NN-Harmonic Mean Aggregation Operators-Based MCGDM Strategy in a Neutrosophic Number Environment" *Axioms* 7, no. 1: 12.
https://doi.org/10.3390/axioms7010012