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

Abstract

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.

1. 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 [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]. 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.

Only four [52,53,54,55] MCGDM strategies using NNs have been reported in the literature. Motivated from the works of Ye [52], Liu and Liu [53], Zheng et al. [54], and Pramanik et al. [55], we consider the proposed strategies to handle MCGDM problems 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.

2. 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.

2.1. 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 x₁, x₂, …, 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.

[49]: The weighted harmonic mean H of the positive real numbers x₁, x₂, …, 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.

Here, ${\displaystyle \sum _{i=1}^n{w}_i}=1.$

2.2. 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 $\in$ 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 $\in$ 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.

Let two NNs be z₁ = x₁ + y₁I and z₂ = x₂ + y₂I for z₁, z₂ $\in$ Z, and I $\in$ [I^L, I^U]. Some basic operational rules for z₁ and z₂ are presented as follows:

(1)

I² = I

(2)

I.0 = 0

(3)

I/I = Undefined

(4)

z₁ + z₂ = x₁ + x₂ + (y₁ + y₂)I = [x₁ + x₂ + (y₁ + y₂)I^L, x₁ + x₂ + (y₁ + y₂)I^U]

(5)

z₁ − z₂ = x₁ − x₂ + (y₁ − y₂)I = [x₁ − x₂ + (y₁ − y₂)I^L, x₁ − x₂ + (y₁ − y₂)I^U]

(6)

z₁ $\times$ z₂ = x₁x₂ + (x₁y₂ + x₂y₁)I + y₁y₂I² = x₁x₂ + (x₁y₂ + x₂y₁ + y₁y₂)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.

If z is a neutrosophic number then, $\frac{1}{(z){}^{-1}}=z, z\ne 0.$

Proof. 

Let z = x + yI. Then,

$$\frac{1}{z}=\left(z\right){}^{-1}=\frac{1}{x}+\frac{-y}{x(x+y)}I; x\ne 0, x\ne -y$$

$$\begin{gathered} \frac{1}{\left(z\right){}^{-1}}=\frac{1}{1/x}+\frac{\frac{y}{x(x+y)}}{1/x\left(1/x-\frac{y}{x(x+y)}\right)}I; x\ne 0 ,x\ne -y \\ =x+\mathrm{y}I=z. \end{gathered}$$

☐

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:

$$Sc(z)=\left|\frac{x+y(I^U-I^L)}{2\sqrt{x^2+y^2}}\right|$$

(1)

$$Ac(z)=1-\exp \langle -\left|x+y(I^U-I^L\right|\rangle$$

(2)

Theorem 2.

Both score function Sc(z) and accuracy function Ac(z) are bounded.

Proof. 

 

$x, y\in R \mathrm{and} I\in [0, 1]$

$\Rightarrow 0\le \frac{x}{\sqrt{x^2+y^2}}\le 1, 0\le \frac{y(I^U-I^L)}{\sqrt{x^2+y^2}}\le 1$

$\Rightarrow 0\le \left|\frac{x+y(I^U-I^L)}{\sqrt{x^2+y^2}}\right|\le 2\Rightarrow 0\le \left|\frac{x+y(I^U-I^L)}{2\sqrt{x^2+y^2}}\right|\le 1\Rightarrow 0\le S(z)\le 1.$

Since $0\le$ Sc(z) $\le$ 1, score function is bounded.

Again:

$\begin{array}{l}0\le \exp \langle -\left|x+y(I^U-I^L\right|\rangle \le 1\\ \Rightarrow -1\le -\exp \langle -\left|x+y(I^U-I^L\right|\rangle \le 0\\ \Rightarrow 0\le 1-\exp \langle -\left|x+y(I^U-I^L\right|\rangle \le 1\end{array}$

Since $0\le$ Ac(z) $\le$ 1, accuracy function is bounded.  ☐

Definition 4.

Let two NNs be z₁ = x₁ + y₁I = [x₁ + y₁I^L, x₁ + y₁I^U], and z₂ = x₂ + y₂I = [x₂ + y₂I^L, x₂ + y₂I^U], then the following comparative relations hold:

• If S(z₁) > S(z₂), then z₁ > z₂
• If S(z₁) = S(z₂) and A(z₁) < A(z₂), then z₁ < z₂
• If S(z₁) = S(z₂) and A(z₁) = A(z₂), then z₁ = z₂.

Example 1.

Let three NNs be z₁ = 10 + 2I, z₂ = 12 and z₃ = 12 + 5I and I $\in$ [0, 0.2]. Then,

S(z₁) = 0.5099, S(z₂) = 0.5, S(z₃) = 0.5577, A(z₁) = 0.999969, A(z₂) = 0.999994, A(z₃) = 0.999997.

We see that, $S(z{}_1)\succ S(z{}_2)=S(z{}_3)$, and $A(z{}_3)\succ S(z{}_2)$.

Using Definition 2, we conclude that, $z{}_1\succ z{}_3\succ z{}_2$.

3. Harmonic Mean Operators for NNs

In this section, we define harmonic mean operator and weighted harmonic mean operator for neutrosophic numbers.

3.1. NN-Harmonic Mean Operator (NNHMO)

Definition 5.

Let z_i = x_i + y_iI (i = 1, 2, …, n) be a collection of NNs. Then the NNHMO is defined as follows:

$$\mathrm{NNHMO}(z{}_1, z{}_2,\cdots ,z{}_n)=n\left({\displaystyle \sum _{i=1}^n\left(z{}_i\right){}^{-1}}\right){}^{-1}$$

(3)

Theorem 3.

Let z_i = x_i + y_iI (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. 

 

$\mathrm{NNHMO}(z{}_1, z{}_2,\cdots , z{}_n)=n\left({\displaystyle \sum _{i=1}^n\left(z{}_i\right){}^{-1}}\right){}^{-1}$

$=n\left({\displaystyle \sum _{i=1}^n\frac{1}{x{}_i}+{\displaystyle \sum _{i=1}^n\frac{-y{}_i}{x{}_i(x{}_i+y{}_i)}I}}\right){}^{-1}; x{}_i\ne 0, x{}_i\ne -y{}_i$

$=\frac{n}{{\displaystyle \sum _{i=1}^n\frac{1}{x{}_i}}}+\frac{-n{\displaystyle \sum _{i=1}^n\frac{-y{}_i}{x{}_i(x{}_i+y{}_i)}}}{\left({\displaystyle \sum _{i=1}^n\frac{1}{x{}_i}}\right)\left({\displaystyle \sum _{i=1}^n\frac{1}{x{}_i}}+{\displaystyle \sum _{i=1}^n\frac{-y{}_i}{x{}_i(x{}_i+y{}_i)}}\right)}I; {\displaystyle \sum _{i=1}^n\frac{1}{x{}_i}}\ne 0, {\displaystyle \sum _{i=1}^n\frac{1}{x{}_i}}\ne -{\displaystyle \sum _{i=1}^n\frac{-y{}_i}{x{}_i(x{}_i+y{}_i)}}$

This shows that NNHMO is also a NN. ☐

3.2. NN-Weighted Harmonic Mean Operator (NNWHMO)

Definition 6.

Let z_i = x_i + y_iI (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 ${\displaystyle \sum _{i=1}^n{w}_i}=1.$ Then the NN-weighted harmonic mean (NNWHMO) is defined as follows:

$$\mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)={\left({\displaystyle \sum _{i=1}^n\frac{w{}_i}{z{}_i}}\right)}^{{}^{-1}}, z{}_i\ne 0$$

(4)

Theorem 4.

Let z_i = x_i + y_iI (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. 

 

$\mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)={\left({\displaystyle \sum _{i=1}^n\frac{w{}_i}{z{}_i}}\right)}^{{}^{-1}}, z{}_i\ne 0$

$=\left({\displaystyle \sum _{i=1}^n{w}_i\left(\frac{1}{x{}_i}+\frac{-y{}_i}{x{}_i(x{}_i+y{}_i)}I\right)}\right){}^{-1}; x{}_i\ne 0, x{}_i\ne -y{}_i$

$= \left(w_i·{\displaystyle \sum _{i=1}^n\frac{1}{x{}_i}+w_i·{\displaystyle \sum _{i=1}^n\frac{-y{}_i}{x{}_i(x{}_i+y{}_i)}I}}\right){}^{-1}; x{}_i\ne 0, x{}_i\ne -y{}_i$

$=\frac{1}{w_i·{\displaystyle \sum _{i=1}^n}\frac{1}{x{}_i}}+\frac{-w_i·{\displaystyle \sum _{i=1}^n}\frac{-y{}_i}{x{}_i(x{}_i+y{}_i)}}{(w_i·{\displaystyle \sum _{i=1}^n}\frac{1}{x{}_i})\left(w_i·{\displaystyle \sum _{i=1}^n}\frac{1}{x{}_i}+w_i·{\displaystyle \sum _{i=1}^n}\frac{-y{}_i}{x{}_i(x{}_i+y{}_i)}\right)}I; w_i·{\displaystyle \sum _{i=1}^n}\frac{1}{x{}_i}\ne 0, w_i·{\displaystyle \sum _{i=1}^n}\frac{1}{x{}_i}\ne -w_i·{\displaystyle \sum _{i=1}^n}\frac{-y{}_i}{x{}_i(x{}_i+y{}_i)}; {\displaystyle \sum _{i=1}^n}{w}_i=1$

This shows that NNWHMO is also a NN. ☐

Example 2.

Let two NNs be z₁ = 3 + 2I and z₂ = 2 + I and I $\in$ [0, 0.2]. Then:

$$\mathrm{NNHMO}(z{}_1, z{}_2)=2\left(\frac{1}{z{}_1}+\frac{1}{z_2}\right){}^{-1}=2\left(\frac{1}{3+2I}+\frac{1}{2+I}\right){}^{-1}= 2.4 + 0.635I.$$

Example 3.

Let two NNs be z₁ = 3 + 2I and z₂ = 2 + I, I $\in$ [0, 0.2] and w₁ = 0.4, w₂ = 0.6, then:

$$\mathrm{NNWHMO}(z{}_1, z{}_2)=\left(w_1\frac{1}{z_1}+w_2\frac{1}{z_2}\right){}^{-1}=\left(0.4\frac{1}{3+2I}+0.6\frac{1}{2+I}\right){}^{-1}=2.308+1.370I.$$

The NNHMO operator and the NNWHMO operator satisfy the following properties.

P1. Idempotent law:

If z_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. 

For, z_i = z, ${\displaystyle \sum _{i=1}^n{w}_i}=1,$

$$\mathrm{NNHMO}(z{}_1, z{}_2,\cdots ,z{}_n)=n\left({\displaystyle \sum _{i=1}^n\left(z{}_i\right){}^{-1}}\right){}^{-1}=n\left({\displaystyle \sum _{i=1}^n\left(z\right){}^{-1}}\right){}^{-1}=\frac{n}{nz{}^{-1}}=z.$$

$$\mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)={\left({\displaystyle \sum _{i=1}^n\frac{w{}_i}{z{}_i}}\right)}^{{}^{-1}}, z{}_i\ne 0={\left({\displaystyle \sum _{i=1}^n\frac{w{}_i}{z}}\right)}^{{}^{-1}}={\left({\displaystyle \sum _{i=1}^n{w}_i}\right)}^{{}^{-1}}\left(z{}^{-1}\right){}^{-1}=z.$$

 ☐

P2. Boundedness:

Both the operators are bounded.

Proof. 

Let $z{}_{\min }=\min (z{}_1, z{}_2,\cdots ,z{}_n)$, $z{}_{\max }=\max (z{}_1, z{}_2,\cdots ,z{}_n)$ for i = 1, 2, …, n then,

$z{}_{\min }\le \mathrm{NNHMO}(z{}_1, z{}_2,\cdots ,z{}_n)\le z{}_{\max }$ and $z{}_{\min }\le \mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)\le z{}_{\max }$.

Hence, both the operators are bounded. ☐

P3. Monotonicity:

If $z_i\le z_i^{\ast }$ for i = 1, 2, …, n then, $\mathrm{NNHMO}(z{}_1, z{}_2,\cdots ,z{}_n)\le \mathrm{NNHMO}(z_1^{\ast }, z_2^{\ast }, \cdots , z_n^{\ast })$ and $\mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)\le \mathrm{NNWHMO}(z_1^{\ast }, z_2^{\ast },\cdots , z_n^{\ast })$.

Proof. 

$\mathrm{NNHMO}(z{}_1, z{}_2,\cdots ,z{}_n)-\mathrm{NNHMO}(z_1^{\ast }, z_2^{\ast },\cdots , z_n^{\ast })$ = $\frac{n}{\frac{1}{z{}_1}+\frac{1}{z{}_2}+\mathrm{\cdots }+\frac{1}{z{}_n}}-\frac{n}{\frac{1}{z{}_1^{\ast }}+\frac{1}{z{}_2^{\ast }}+\mathrm{\cdots }+\frac{1}{z{}_n^{\ast }}}\le 0$, since $z_i\le z_i^{\ast } \mathrm{or} \frac{1}{z_i}\ge \frac{1}{z_i^{\ast }}$, for i = 1, 2, …, n.

Again,

$\mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)-\mathrm{NNWHMO}(z_1^{\ast }, z_2^{\ast }, \cdots , z_n^{\ast })$ = $\frac{1}{\frac{w{}_1}{z{}_1}+\frac{w{}_2}{z{}_2}+\mathrm{\cdots }+\frac{w{}_n}{z{}_n}}-\frac{1}{\frac{w{}_1}{z{}_1^{\ast }}+\frac{w{}_2}{z{}_2^{\ast }}+\mathrm{\cdots }+\frac{w{}_n}{z{}_n^{\ast }}}\le 0$, since $z_i\le z_i^{\ast } \mathrm{or} \frac{1}{z_i}\ge \frac{1}{z_i^{\ast }}$, $\le 0$, for $z_i\le z_i^{\ast }$; ${\displaystyle \sum _{i=1}^n{w}_i}=1$; (i = 1, 2, …, n).

This proves the monotonicity of the functions $\mathrm{NNHMO}(z{}_1, z{}_2,\cdots ,z{}_n)$ and $\mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)$. ☐

P4. Commutativity:

If $(z_1^{\circ }, z_2^{\circ }, \cdots , z_n^{\circ })$ be any permutation of $(z{}_1, z{}_1,\cdots , z{}_n)$ then, $\mathrm{NNHMO}(z{}_1, z{}_2,\cdots ,z{}_n)=\mathrm{NNHMO}(z_1^{\circ }, z_2^{\circ },\cdots , z_n^{\circ })$ and $\mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)=\mathrm{NNWHMO}(z_1^{\circ }, z_2^{\circ }, \cdots , z_n^{\circ })$.

Proof. 

$\mathrm{NNHMO}(z{}_1, z{}_2,\cdots ,z{}_n)-\mathrm{NNHMO}(z_1^{\circ }, z_2^{\circ },\cdots , z_n^{\circ })$ = $n\left({\displaystyle \sum _{i=1}^n\left(z{}_i\right){}^{-1}}\right){}^{-1}-n\left({\displaystyle \sum _{i=1}^n\left(z{}_i^{\ast }\right){}^{-1}}\right){}^{-1}$ = 0, because, $(z_1^{\circ }, z_2^{\circ },\cdots , z_n^{\circ })$ is any permutation of $(z{}_1, z{}_2,\cdots ,z{}_n)$.

Hence, we have $\mathrm{NNHMO}(z{}_1, z{}_2,\cdots ,z{}_n)=\mathrm{NNHMO}(z_1^{\circ }, z_2^{\circ },\cdots , z_n^{\circ })$.

Again:

$\mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)-\mathrm{NNWHMO}(z_1^{\circ }, z_2^{\circ },\cdots , z_n^{\circ })$ = ${\left({\displaystyle \sum _{i=1}^n{w}_i\left(z{}_i\right){}^{-1}}\right)}^{{}^{-1}}-{\left({\displaystyle \sum _{i=1}^n{w}_i \left(z{}_i^{\ast }\right){}^{-1}}\right)}^{{}^{-1}}$ = 0, because, $(z_1^{\circ }, z_2^{\circ },\cdots , z_n^{\circ })$ is any permutation of $(z{}_1, z{}_2,\cdots ,z{}_n)$.

Hence, we have $\mathrm{NNWHMO}(z{}_1, z{}_2,\cdots ,z{}_n)=\mathrm{NNWHMO}(z_1^{\circ }, z_2^{\circ },\cdots , z_n^{\circ })$. ☐

4. 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 P = x_ij + y_ijI = [x_ij + y_ijI^L, x_ij + y_ijI^U], (i = 1, 2, …, m; j = 1, 2, …, n) is defined as follows:

$${COS}_j(P)=\frac{1}{n}{\displaystyle \sum _{i=1}^n\cos \frac{\pi }{2}}\left(\left|\frac{y{}_{ij}}{\sqrt{x_{ij}^2+y_{ij}^2}}\right|\right), \left(x_{ij} and y_{ij} are not both zeroes\right)$$

(5)

The weight structure is defined as follows:

$$w_j=\frac{{COS}_j(P)}{{\displaystyle {\sum }_{j=1}^n{COS}_j(P)}}; j=1, 2, \mathrm{\cdots }, n \& {\displaystyle \sum _{j=1}^n{w}_j}=1$$

(6)

The cosine function ${COS}_j(P)$ satisfies the following properties:

P1. 

${COS}_j(P)=1$, if $y_{ij}=0 \mathrm{and} x_{ij}\ne 0.$

P2. 

${COS}_j(P)=0$, if $x{}_{ij}=0 and y_{ij}\ne 0.$

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. 

$y_{ij}=0$ $\Rightarrow$ ${COS}_j(P)=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left[\cos 0\right]}=1$

P2. 

$x{}_{ij}=0$ $\Rightarrow$ ${COS}_j(P)=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left[\cos \frac{\pi }{2}\right]}=0$

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.

Let two NNs be z₁ = 3 + 2I, and z₂ = 3 + 5I, then, $COS(z{}_1)=0.9066$, $COS(z{}_2)=0.7817$.

Example 5.

Let two NNs be z₁ = 3 + I, and z₂ = 7 + I, then, $COS(z{}_1)=0.9693$, $COS(z{}_2)=0.9938$.

Example 6.

Let two NNs be z₁ = 10 + 2I, and z₂ = 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

Two MCGDM strategies using the NNHMO and NNWHMO respectively are developed in this section. Suppose that A = {A₁, A₂, …, A_m} is a set of alternatives, C = {C₁, C₂, …, C_n} is a set of criteria and DM = {DM₁, DM₂, …, 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)

Strategy 1 is presented (see Figure 1) using the following six steps:

Step 1. Determine the relation between alternatives and criteria.

Each decision-maker forms a NN decision matrix. The relation between the alternative A_i (i = 1, 2, …, m) and the criterion C_j (j = 1, 2, …, n) is presented in Equation (7).

$$DM{}_k[A|C]=\begin{array}{c}\\ A_1\\ A_2\\ ⋮\\ A_m\end{array}\left(\begin{array}{cccc}C{}_1& C{}_2& \cdots & C{}_n\\ {\langle x{}_{11}+y{}_{11}I\rangle }_k& {\langle x{}_{12}+y{}_{12}I\rangle }_k& \cdots & {\langle x{}_{1n}+y{}_{1n}I\rangle }_k\\ {\langle x{}_{21}+y{}_{21}I\rangle }_k& {\langle x{}_{22}+y{}_{22}I\rangle }_k& \cdots & {\langle x{}_{2n}+y{}_{2n}I\rangle }_k\\ ⋮& ⋮& \ddots & ⋮\\ {\langle x{}_{m1}+y{}_{m1}I\rangle }_k& {\langle x{}_{m2}+y{}_{m2}I\rangle }_k& & {\langle x{}_{mn}+y{}_{mn}I\rangle }_k\end{array}\right)$$

(7)

Note 1: Here, $\langle x{}_{ij}+y{}_{ij}I\rangle {}_k$ represents the NN rating value of the alternative A_i with respect to the criterion C_j for the decision-maker DM_k.

Step 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:

$$DM{}^{aggr}(A{}_i)={\displaystyle \sum _{t=1}^k{w}_t (DM{}_t{}^{aggr}(A{}_i))}; {\displaystyle \sum _{t=1}^k{w}_t}=1. \left(i=1, 2, \dots ,n; t=1, 2, \dots ,k\right)$$

(8)

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.

Step 6. End.

5.2. 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.

Using Equation (6), determine the criteria weights from decision matrices (${DM}_t[A|C]$), (t = 1, 2, …, k).

Step 3. Determine the weighted aggregation values ($DM{}_k{}^{waggr}(A{}_i)$).

Using Equation (4), determine the weighted aggregation values ($DM{}_k{}^{waggr}(A{}_i)$), (i = 1, 2, …, n) for all decision matrices.

Step 4. Determine the averaging values.

To fuse all the weighted aggregation values ($DM{}_k{}^{waggr}(A{}_i)$), corresponding to alternatives A_i, we define the averaging function as follows:

$$DM{}^{waggr}(A{}_i)={\displaystyle \sum _{t=1}^k{w}_t(DM{}_t{}^{waggr}(A{}_i))}\left(i=1, 2, \dots ,n; t=1, 2, \dots ,k\right)$$

(9)

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.

Step 7. End.

6. 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.

6.1. 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).

$${DM}_1[L|C]=\begin{array}{c}\\ A_1\\ A_2\\ A_3\\ A_4\end{array}\left(\begin{array}{ccc}C{}_1& C{}_2& C{}_3\\ 4+I& 5& 3+I\\ 6& 6& 5\\ 3& 5+I& 6\\ 7& 6& 4+I\end{array}\right)$$

(10)

$${DM}_2[L|C]=\begin{array}{c}\\ A_1\\ A_2\\ A_3\\ A_4\end{array}\left(\begin{array}{ccc}C{}_1& C{}_2& C{}_3\\ 5& 4& 4\\ 5+I& 6& 6\\ 4& 5& 5+I\\ 6+I& 6& 5\end{array}\right)$$

(11)

$${DM}_3[L|C]=\begin{array}{c}\\ A_1\\ A_2\\ A_3\\ A_4\end{array}\left(\begin{array}{ccc}C{}_1& C{}_2& C{}_3\\ 4& 5+I& 4\\ 6& 7& 5+I\\ 4+I& 5& 6\\ 8& 6& 4+I\end{array}\right)$$

(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 DM₁, DM₂ and DM₃ respectively.

Step 2. Determine the weighted aggregation values ($DM{}_k{}^{aggr}(A{}_i)$).

Using Equation (3), we calculate the aggregation values ($DM{}_k{}^{aggr}(A{}_i)$) as follows:

$$DM{}_1{}^{aggr}(A{}_1)=3.829+0.785I; DM{}_1{}^{aggr}(A{}_2)=5.625; DM{}_1{}^{aggr}(A{}_3)=4.285+0.214I; DM{}_1{}^{aggr}(A{}_4)=5.362+0.514I;$$

$$DM{}_2{}^{aggr}(A{}_1)=4.285; DM{}_2{}^{aggr}(A{}_2)=5.206+0.415I; DM{}_2{}^{aggr}(A{}_3)=4.196+0.532I; DM{}_2{}^{aggr}(A{}_4)=5.234+0.618I;$$

$$DM{}_3{}^{aggr}(A{}_1)=4.019+0.605I; DM{}_3{}^{aggr}(A{}_2)=5.817+0.433I; DM{}_3{}^{aggr}(A{}_3)=4.876+0.387I; DM{}_3{}^{aggr}(A{}_4)=6.023+0.257I.$$

Step 3. Determine the averaging values.

Using Equation (8), we calculate the averaging values (Considering equal importance of all the decision makers) to fuse all the aggregation values corresponding to the alternative A_i.

$$DM{}^{aggr}(A{}_1)=4.044+0.463I; DM{}^{aggr}(A{}_2)=5.549+0.282I; DM{}^{aggr}(A{}_3)=4.452+0.378I; DM{}^{aggr}(A{}_4)=5.539+0.463I.$$

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. Sensitivity analysis and ranking order with variation of “I” on NNs for strategy 1.

I	Sc(Ai)	Ranking Order
I = [0, 0]	S(A1) = 0.4988, S(A2) = 0.4993, S(A3) = 0.4982, S(A4) = 0.4983	A2 $\succ$ A1 $\succ$ A4 $\succ$ A3
I $\in$ [0, 0.2]	S(A1) = 0.5081, S(A2) = 0.5144, S(A3) = 0.5067, S(A4) = 0.5056	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4
I $\in$ [0, 0.4]	S(A1) = 0.5182, S(A2) = 0.5195, S(A3) = 0.5151, S(A4) = 0.5249	A2 $\succ$ A1 $\succ$ A4 $\succ$ A3
I $\in$ [0, 0.6]	S(A1) = 0.5289, S(A2) = 0.5346, S(A3) = 0.5236, S(A4) = 0.5233	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4
I $\in$ [0, 0.8]	S(A1) = 0.5396, S(A2) = 0.5497, S(A3) = 0.5320, S(A4) = 0.5316	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4
I $\in$ [0, 1]	S(A1) = 0.5503, S(A2) = 0.5547, S(A3) = 0.5405, S(A4) = 0.5399	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4

for different values of I.

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

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.

6.2. 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.

Using Equations (5) and (6), criteria weights are calculated as follows:

[w₁ = 0.3265, w₂ = 0.3430, w₃ = 0.3305] for DM₁,

[w₁ = 0.3332, w₂ = 0.3334, w₃ = 0.3334] for DM₂,

[w₁ = 0.3333, w₂ = 0.3335, w₃ = 0.3332] for DM₃.

Step 3. Determine the weighted aggregation values ($DM{}_k{}^{waggr}(A{}_i)$).

Using Equation (4), we calculate the aggregation values ($DM{}_k{}^{aggr}(A{}_i)$) as follows:

$$DM{}_1{}^{aggr}(A{}_1)=3.861+0.774I; DM{}_1{}^{aggr}(A{}_2)=6.006; DM{}_1{}^{aggr}(A{}_3)=4.307+0.234I; DM{}_1{}^{aggr}(A{}_4)=5.399+0.541I;$$

$$DM{}_2{}^{aggr}(A{}_1)=4.288; DM{}_2{}^{aggr}(A{}_2)=5.219+0.429I; DM{}_2{}^{aggr}(A{}_3)=4.206+0.541I; DM{}_2{}^{aggr}(A{}_4)=5.251+0.629I;$$

$$DM{}_3{}^{aggr}(A{}_1)=4.024+0.616I; DM{}_3{}^{aggr}(A{}_2)=5.824+0.445I; DM{}_3{}^{aggr}(A{}_3)=4.889+0.393I; DM{}_3{}^{aggr}(A{}_4)=6.029+0.265I.$$

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 A_i.

$$DM{}^{aggr}(A{}_1)=4.057+0.463I; DM{}^{aggr}(A{}_2)=5.568+0.291I; DM{}^{aggr}(A{}_3)=4.467+0.389I; DM{}^{aggr}(A{}_4)=5.559+0.478I.$$

Step 5. Determine the ranking order.

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. Sensitivity analysis and ranking order with variation of “I” on NNs for strategy 2.

I	Sc(Ai)	Ranking Order
I = 0	S(A1) = 0.4968, S(A2) = 0.4993, S(A3) = 0.4981, S(A4) = 0.4982	A2 $\succ$ A4 $\succ$ A3 $\succ$ A1
I $\in$ [0, 0.2]	S(A1) = 0.5081, S(A2) = 0.5095, S(A3) = 0.5068, S(A4) = 0.5067	A2 $\succ$ A1 $\succ$ A4 $\succ$ A3
I $\in$ [0, 0.4]	S(A1) = 0.5195, S(A2) = 0.5198, S(A3) = 0.5155, S(A4) = 0.5153	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4
I $\in$ [0, 0.6]	S(A1) = 0.5308, S(A2) = 0.5350, S(A3) = 0.5241, S(A4) = 0.5239	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4
I $\in$ [0, 0.8]	S(A1) = 0.5421, S(A2) = 0.5502, S(A3) = 0.5328, S(A4) = 0.5324	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4
I $\in$ [0, 1]	S(A1) = 0.5535, S(A2) = 0.5654, S(A3) = 0.5415, S(A4) = 0.5410	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4

for different values of I.

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

Step 7. End.

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.

7. Comparison Analysis and Contributions of the Proposed Approach

7.1. 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 A₂ is the best alternative for I = 0 and $I\ne 0$ i.e., for all cases considered. Table 2 reflects that A₂ is the best alternative for any values of I. Ranking order differs for different values of I.

The ranking results obtained from the existing strategies [52,53,54] are furnished in

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]	A2 $\succ$ A4 $\succ$ A3 $\succ$ A1	A2 $\succ$ A4 $\succ$ A3 $\succ$ A1	A2 $\succ$ A4 $\succ$ A1 $\succ$ A3	A2 $\succ$ A1 $\succ$ A4 $\succ$ A3	A2 $\succ$ A4 $\succ$ A3 $\succ$ A1
[0, 0.2]	A2 $\succ$ A4 $\succ$ A3 $\succ$ A1	A2 $\succ$ A4 $\succ$ A3 $\succ$ A1	A2 $\succ$ A3 $\succ$ A1 $\succ$ A4	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4	A2 $\succ$ A1 $\succ$ A4 $\succ$ A3
[0, 0.4]	A2 $\succ$ A4 $\succ$ A3 $\succ$ A1	A2 $\succ$ A4 $\succ$ A3 $\succ$ A1	A2 $\succ$ A3 $\succ$ A4 $\succ$ A1	A2 $\succ$ A1 $\succ$ A4 $\succ$ A3	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4
[0, 0.6]	A4 $\succ$ A2 $\succ$ A3 $\succ$ A1	A4 $\succ$ A2 $\succ$ A3 $\succ$ A1	A2 $\succ$ A3 $\succ$ A4 $\succ$ A1	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4
[0, 0.8]	A4 $\succ$ A2 $\succ$ A3 $\succ$ A1	A4 $\succ$ A2 $\succ$ A3 $\succ$ A1	A2 $\succ$ A3 $\succ$ A4 $\succ$ A1	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4
[0, 1]	A4 $\succ$ A2 $\succ$ A3 $\succ$ A1	A4 $\succ$ A2 $\succ$ A3 $\succ$ A1	A2 $\succ$ A4 $\succ$ A3 $\succ$ A1	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4	A2 $\succ$ A1 $\succ$ A3 $\succ$ A4

. The ranking orders of Ye [52] and Zheng et al. [54] are similar for all values of I considered. When I lies in [0, 0], [0, 0.2], [0, 0.4], A₂ is the best alternative for [52,53,54] and the proposed strategies. When I lies in [0, 0.6], [0, 0.8], [0, 1], A₄ is the best alternative for [52,54], whereas A₂ is the best alternative for [53], and the proposed strategies.

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.

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

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.