# Cosine Measures of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Some Basic Concepts of Cubic Sets and NCSs

- (i)
- $S=\{x,T(x),\mu (x)|x\in X\}$ an internal cubic set if ${T}^{-}(x)\le \mu (x)\le {T}^{+}(x)$ for x ∈ X;
- (ii)
- $S=\{x,T(x),\mu (x)|x\in X\}$ an external cubic set if $\mu (x)\notin \left({T}^{-}(x),{T}^{+}(x)\right)$ for x ∈ X.

- (i)
- An internal NCS $P=\{x,<T(x),U(x),F(x)>,<t(x),u(x),f(x)>|x\in X\}$ if ${T}^{-}(x)\le t(x)\le {T}^{+}(x)$, ${U}^{-}(x)\le u(x)\le {U}^{+}(x)$, and ${F}^{-}(x)\le f(x)\le {F}^{+}(x)$ for x ∈ X;
- (ii)
- An external NCS $P=\{x,<T(x),U(x),F(x)>,<t(x),u(x),f(x)>|x\in X\}$ if $t(x)\notin \left({T}^{-}(x),{T}^{+}(x)\right)$, $u(x)\notin \left({U}^{-}(x),{U}^{+}(x)\right)$, and $f(x)\notin \left({F}^{-}(x),{F}^{+}(x)\right)$ for x ∈ X.

_{1}= (<T

_{1}, U

_{1}, F

_{1}>, <t

_{1}, u

_{1}, f

_{1}>) and p

_{2}= (<T

_{2}, U

_{2}, F

_{2}>, <t

_{2}, u

_{2}, f

_{2}>) be two NCNs. Then, there are the following relations [38,39]:

- (1)
- ${p}_{1}^{c}=\left(\langle \left[{F}_{1}^{-},{F}_{1}^{+}\right],\left[1-{U}_{1}^{+},1-{U}_{1}^{-}\right],\left[{T}_{1}^{-},{T}_{1}^{+}\right]\rangle ,\langle {f}_{1},1-{u}_{1},{t}_{1}\rangle \right)$ (complement of p
_{1}); - (2)
- p
_{1}⊆ p_{2}if and only if ${T}_{1}\subseteq {T}_{2}$, ${U}_{1}\supseteq {U}_{2}$, ${F}_{1}\supseteq {F}_{2}$,${t}_{1}\le {t}_{2}$, ${u}_{1}\ge {u}_{2}$, and ${f}_{1}\ge {f}_{2}$ (P-order); - (3)
- p
_{1}= p_{2}if and only if p_{2}⊆ p_{1}and p_{1}⊆ p_{2,}i.e., <T_{1}, U_{1}, F_{1}> = <T_{2}, U_{2}, F_{2}> and <t_{1}, u_{1}, f_{1}> = <t_{2}, u_{2}, f_{2}>.

## 3. Cosine Measures of NCSs

**Definition**

**1.**

_{1}, x

_{2}, …, x

_{n}} be a finite set and two NCSs be P ={p

_{1}, p

_{2}, …, p

_{n}} and Q ={q

_{1}, q

_{2}, …, q

_{n}}, where p

_{j}= (<T

_{pj}, U

_{pj}, F

_{pj}>, <t

_{pj}, u

_{pj}, f

_{pj}>) and q

_{j}= (<T

_{qj}, U

_{qj}, F

_{qj}>, <t

_{qj}, u

_{qj}, f

_{qj}>) for j = 1, 2, …, n are two collections of NCNs. Then, three cosine measures of P and Q are proposed based on the included angle cosine of two vectors, distance, and cosine function, respectively, as follows:

- (1)
- Cosine measure based on the included angle cosine of two vectors$$\begin{array}{l}{S}_{1}(P,Q)\\ =\frac{1}{2n}\left\{{\displaystyle \sum _{j=1}^{n}\frac{{T}_{pj}^{-}{T}_{qj}^{-}+{T}_{pj}^{+}{T}_{qj}^{+}+{U}_{pj}^{-}{U}_{qj}^{-}+{U}_{pj}^{+}{U}_{qj}^{+}+{F}_{pj}^{-}{F}_{qj}^{-}+{F}_{pj}^{+}{F}_{qj}^{+}}{\left\{\begin{array}{l}\sqrt{{({T}_{pj}^{-})}^{2}+{({T}_{pj}^{+})}^{2}+{({U}_{pj}^{-})}^{2}+{({U}_{pj}^{+})}^{2}+{({F}_{pj}^{-})}^{2}+{({F}_{pj}^{+})}^{2}}\\ \times \sqrt{{({T}_{qj}^{-})}^{2}+{({T}_{qj}^{+})}^{2}+{({U}_{qj}^{-})}^{2}+{({U}_{qj}^{+})}^{2}+{({F}_{qj}^{-})}^{2}+{({F}_{qj}^{+})}^{2}}\end{array}\right\}}}+{\displaystyle \sum _{j=1}^{n}\frac{{t}_{pj}^{}{t}_{qj}^{}+{u}_{pj}^{}{u}_{qj}^{}+{f}_{pj}^{}{f}_{qj}^{}}{\left\{\sqrt{{t}_{pj}^{2}+{u}_{pj}^{2}+{f}_{pj}^{2}}\times \sqrt{{t}_{qj}^{2}+{u}_{qj}^{2}+{f}_{qj}^{2}}\right\}}}\right\}\end{array}$$
- (2)
- Cosine measure based on distance$${S}_{2}(P,Q)=\frac{1}{2n}{\displaystyle \sum _{j=1}^{n}\left\{\begin{array}{l}\mathrm{cos}\left(\frac{\left|{T}_{pj}^{-}-{T}_{qj}^{-}\right|+\left|{T}_{pj}^{+}-{T}_{qj}^{+}\right|+\left|{U}_{pj}^{-}-{U}_{qj}^{-}\right|+\left|{U}_{pj}^{+}-{U}_{qj}^{+}\right|+\left|{F}_{pj}^{-}-{F}_{qj}^{-}\right|+\left|{F}_{pj}^{+}-{F}_{qj}^{+}\right|}{12}\pi \right)\\ +\mathrm{cos}\left(\frac{\left|{t}_{pj}^{}-{t}_{qj}^{}\right|+\left|{u}_{pj}^{}-{u}_{qj}^{}\right|+\left|{f}_{pj}^{}-{f}_{qj}^{}\right|}{6}\pi \right)\end{array}\right\}}$$
- (3)
- Cosine measure based on cosine function$$\begin{array}{l}{S}_{3}(P,Q)\\ =\frac{1}{2n}\left\{\frac{1}{3(\sqrt{2}-1)}{\displaystyle \sum _{j=1}^{n}\left\{\begin{array}{l}\left[\sqrt{2}\mathrm{cos}\left(\frac{{T}_{pj}^{-}+{T}_{pj}^{+}-{T}_{qj}^{-}-{T}_{qj}^{+}}{8}\pi \right)-1\right]\\ +\left[\sqrt{2}\mathrm{cos}\left(\frac{{U}_{pj}^{-}+{U}_{pj}^{+}-{U}_{qj}^{-}-{U}_{qj}^{+}}{8}\pi \right)-1\right]\\ +\left[\sqrt{2}\mathrm{cos}\left(\frac{{F}_{pj}^{-}+{F}_{pj}^{+}-{F}_{qj}^{-}-{F}_{qj}^{+}}{8}\pi \right)-1\right]\end{array}\right\}}+\frac{1}{3(\sqrt{2}-1)}{\displaystyle \sum _{j=1}^{n}\left\{\begin{array}{l}\left[\sqrt{2}\mathrm{cos}\left(\frac{{t}_{pj}^{}-{t}_{qj}^{}}{4}\pi \right)-1\right]\\ +\left[\sqrt{2}\mathrm{cos}\left(\frac{{u}_{pj}^{}-{u}_{qj}^{}}{4}\pi \right)-1\right]\\ +\left[\sqrt{2}\mathrm{cos}\left(\frac{{f}_{pj}^{}-{f}_{qj}^{}}{4}\pi \right)-1\right]\end{array}\right\}}\right\}\end{array}$$

_{k}(P, Q) (k = 1, 2, 3) satisfy the following properties (S

_{1})–(S

_{3}):

- (S
_{1}) - 0 ≤ S
_{k}(P, Q) ≤ 1; - (S
_{2}) - S
_{k}(P, Q) = S_{k}(Q, P); - (S
_{3}) - S
_{k}(P, Q) = 1 if P = Q, i.e., <T_{pj}, U_{pj}, F_{pj}>, = <T_{qj}, U_{qj}, F_{qj}> and <t_{pj}, u_{pj}, f_{pj}> = <t_{qj}, u_{qj}, f_{qj}>.

**Proof.**

_{1})–(S

_{3}) of S

_{1}(P, Q).

- (S
_{1}) - The inequality S
_{1}(P, Q) ≥ 0 is obvious. Then, we only prove S_{1}(P, Q) ≤ 1.

_{1}, x

_{2}, …, x

_{n}) ∈ R

^{n}and (y

_{1}, y

_{2}, …, y

_{n}) ∈ R

^{n}, we can give the following inequality:

_{1}(P, Q) ≤ 1. Hence, 0 ≤ S

_{1}(P, Q) ≤ 1 holds.

- (S
_{2}) - It is straightforward.
- (S
_{3}) - If P = Q, there are <T
_{pj}, U_{pj}, F_{pj}> = <T_{qj}, U_{qj}, F_{qj}> and <t_{pj}, u_{pj}, f_{pj}> = <t_{qj}, u_{qj}, f_{qj}>. Thus T_{pj}= T_{qj}, U_{pj}= U_{qj}, F_{pj}= F_{qj}, t_{pj}= t_{qj}, u_{pj}= u_{qj}, and f_{pj}= f_{qj}for j = 1, 2, …, n. Hence S_{1}(P, Q) = 1 holds.

_{1})–(S

_{3}) of S

_{2}(P, Q).

- (S
_{1}) - Let ${x}_{1}=\left(\left|{T}_{pj}^{-}-{T}_{qj}^{-}\right|+\left|{T}_{pj}^{+}-{T}_{qj}^{+}\right|+\left|{U}_{pj}^{-}-{U}_{qj}^{-}\right|+\left|{U}_{pj}^{+}-{U}_{qj}^{+}\right|+\left|{F}_{pj}^{-}-{F}_{qj}^{-}\right|+\left|{F}_{pj}^{+}-{F}_{qj}^{+}\right|\right)/6$ and ${x}_{2}=\left(\left|{t}_{pj}^{}-{t}_{qj}^{}\right|+\left|{u}_{pj}^{}-{u}_{qj}^{}\right|+\left|{f}_{pj}^{}-{f}_{qj}^{}\right|\right)/3$. It is obvious that there exist 0 ≤ x
_{1}≤ 1 and 0 ≤ x_{2}≤ 1. Thus, there are 0 ≤ cos(x_{1}π/2) ≤ 1 and 0 ≤ cos(x_{2}π /2) ≤ 1. Hence, 0 ≤ S_{2}(P, Q) ≤ 1 holds. - (S
_{2}) - It is straightforward.
- (S
_{3}) - If P = Q, there are <T
_{pj}, U_{pj}, F_{pj}> = <T_{qj}, U_{qj}, F_{qj}> and <t_{pj}, u_{pj}, f_{pj}> = <t_{qj}, u_{qj}, f_{qj}>. Thus T_{pj}= T_{qj}, U_{pj}= U_{qj}, F_{pj}= F_{qj}, t_{pj}= t_{qj}, u_{pj}= u_{qj}, and f_{pj}= f_{qj}for j = 1, 2, …, n. Hence, S_{2}(P, Q) = 1 holds.

_{1})–(S

_{3}) of S

_{3}(P, Q).

- (S
_{1}) - Let ${y}_{1}=({T}_{pj}^{-}+{T}_{pj}^{+}-{T}_{qj}^{-}-{T}_{qj}^{+})/2$, ${y}_{2}=({U}_{pj}^{-}+{U}_{pj}^{+}-{U}_{qj}^{-}-{U}_{qj}^{+})/2$, ${y}_{3}=({F}_{pj}^{-}+{F}_{pj}^{+}-{F}_{qj}^{-}-{F}_{qj}^{+})/2$, ${y}_{4}={t}_{pj}^{}-{t}_{qj}^{}$, ${y}_{5}={u}_{pj}^{}-{u}_{qj}^{}$, and ${y}_{6}={f}_{pj}^{}-{f}_{qj}^{}$. Obviously, there exists −1 ≤ y
_{k}≤ +1 for k = 1, 2, ...., 6. Thus, $\sqrt{2}/2$ ≤ cos(y_{k}π/4) ≤ 1, and then there exists 0 ≤ S_{3}(P, Q) ≤ 1. - (S
_{2}) - It is straightforward.
- (S
_{3}) - If P = Q, there are <T
_{pj}, U_{pj}, F_{pj}> = <T_{qj}, U_{qj}, F_{qj}> and <t_{pj}, u_{pj}, f_{pj}> = <t_{qj}, u_{qj}, f_{qj}>. Thus T_{pj}= T_{qj}, U_{pj}= U_{qj}, F_{pj}= F_{qj}, t_{pj}= t_{qj}, u_{pj}= u_{qj}, and f_{pj}= f_{qj}for j = 1, 2, …, n. Hence, S_{3}(P, Q) = 1 holds. ☐

_{j}and q

_{j}(j = 1, 2, …, n) is taken into account,

**w**= {w

_{1}, w

_{2}, …, w

_{n}} is given as the weight vector of the elements p

_{j}and q

_{j}(j = 1, 2, …, n) with w

_{j}∈ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}=1$. Then, we have the following three weighted cosine measures between P and Q, respectively:

_{wk}(P, Q) (k=1, 2, 3) also satisfy the following properties (S

_{1})–(S

_{3}):

- (S
_{1}) - 0 ≤ S
_{wk}(P, Q) ≤ 1; - (S
_{2}) - S
_{wk}(P, Q) = S_{wk}(Q, P); - (S
_{3}) - S
_{w}_{k}(P, Q) = 1 if P = Q, i.e., <T_{pj}, U_{pj}, F_{pj}> = <T_{qj}, U_{qj}, F_{qj}> and <t_{pj}, u_{pj}, f_{pj}> = <t_{qj}, u_{qj}, f_{qj}>.

_{1})–(S

_{3}) for S

_{wk}(P, Q) (k = 1, 2, 3). Their proofs are omitted here.

## 4. Decision-Making Method Using Cosine Measures

_{1}, P

_{2}, …, P

_{m}} be a set of m alternatives and R = {R

_{1}, R

_{2}, …, R

_{n}} be a set of n attributes. The evaluation value of an attribute R

_{j}(j = 1, 2, …, n) with respect to an alternative P

_{i}(i = 1, 2, …, m) is expressed by a NCN p

_{ij}= (<T

_{ij}, U

_{ij}, F

_{ij}>, <t

_{ij}, u

_{j}, f

_{ij}>) (j = 1, 2, …, n; i = 1, 2, …, m), where ${T}_{ij},{U}_{ij},{F}_{ij}\subseteq [0,1]$ and ${t}_{ij},{u}_{ij},{f}_{ij}\in [0,1]$. Therefore, all the evaluation values expressed by NCNs can be constructed as the neutrosophic cubic decision matrix P = (p

_{ij})

_{m}

_{×n}. Then, the weight vector of the attributes R

_{j}(j = 1, 2, …, n) is considered as

**w**= (w

_{1}, w

_{2}, …, w

_{n}), satisfying w

_{j}∈ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}=1$. In this case, the proposed decision steps are described as follows:

**Step 1**:- Establish an ideal solution (ideal alternative) ${P}_{}^{*}=\{{p}_{1}^{*},{p}_{2}^{*},\dots ,{p}_{n}^{*}\}$ by the ideal NCN ${p}_{j}^{*}=\left(\langle \left[\underset{i}{\mathrm{max}}({T}_{ij}^{-}),\underset{i}{\mathrm{max}}({T}_{ij}^{+})\right],\left[\underset{i}{\mathrm{min}}({U}_{ij}^{-}),\underset{i}{\mathrm{min}}({U}_{ij}^{+})\right],\left[\underset{i}{\mathrm{min}}({F}_{ij}^{-}),\underset{i}{\mathrm{min}}({F}_{ij}^{+})\right]\rangle ,\langle \underset{i}{\mathrm{max}}({t}_{ij}),\underset{i}{\mathrm{min}}({u}_{ij}^{}),\underset{i}{\mathrm{min}}({f}_{ij}^{})\rangle \right)$ corresponding to the benefit type of attributes and ${p}_{j}^{*}=\left(\langle \left[\underset{i}{\mathrm{min}}({T}_{ij}^{-}),\underset{i}{\mathrm{min}}({T}_{ij}^{+})\right],\left[\underset{i}{\mathrm{max}}({U}_{ij}^{-}),\underset{i}{\mathrm{max}}({U}_{ij}^{+})\right],\left[\underset{i}{\mathrm{max}}({F}_{ij}^{-}),\underset{i}{\mathrm{max}}({F}_{ij}^{+})\right]\rangle ,\langle \underset{i}{\mathrm{min}}({t}_{ij}),\underset{i}{\mathrm{max}}({u}_{ij}^{}),\underset{i}{\mathrm{max}}({f}_{ij}^{})\rangle \right)$ corresponding to the cost type of attributes.
**Step****2:**- Calculate the weighted cosine measure values between an alternative P
_{i}(i = 1, 2, …, m) and the ideal solution P* by using Equation (4) or Equation (5) or Equation (6) and get the values of S_{w}_{1}(P_{i}, P*) or S_{w}_{2}(P_{i}, P*) or S_{w}_{3}(P_{i}, P*) (i = 1, 2, …, m). **Step****3:**- Rank the alternatives in descending order corresponding to the weighted cosine measure values and select the best one(s) according to the bigger value of S
_{w}_{1}(P_{i}, P*) or S_{w}_{2}(P_{i}, P*) or S_{w}_{3}(P_{i}, P*). **Step****4:**- End.

## 5. Illustrative Example and Comparison Analysis

#### 5.1. Illustrative Example

_{1}is a textile company; (b) P

_{2}is an automobile company; (c) P

_{3}is a computer company; (d) P

_{4}is a software company. The evaluation requirements of the four alternatives are on the basis of three attributes: (a) R

_{1}is the risk; (b) R

_{2}is the growth; (c) R

_{3}is the environmental impact; where the attributes R

_{1}and R

_{2}are benefit types, and the attribute R

_{3}is a cost type. The weight vector of the three attributes is

**w**= (0.32, 0.38, 0.3). When the expert or decision maker is requested to evaluate the four potential alternatives on the basis of the above three attributes using the form of NCNs. Thus, we can construct the following neutrosophic cubic decision matrix:

_{1}, R

_{2}, and the cost attribute R

_{3}, we establish an ideal solution (ideal alternative):

_{i}(i = 1, 2, 3, 4) and the ideal solution P* by using Equation (4) or Equation (5) or Equation (6), get the values of S

_{w}

_{1}(P

_{i}, P*) or S

_{w}

_{2}(P

_{i}, P*) or S

_{w}

_{3}(P

_{i}, P*) (i = 1, 2, 3, 4), and rank the four alternatives, which are shown in Table 1.

_{4}is the best one.

#### 5.2. Related Comparison

_{i}(i = 1, 2, …, m) and the ideal alternative (ideal solution) P* to rank all the alternatives; while the existing decision-making method with NCSs introduced in [40] firstly determines the Hamming distances of NCSs for weighted grey relational coefficients and standard (ideal) grey relational coefficients, and then derives the relative closeness coefficients in order to rank the alternatives. It is obvious that our decision-making method is simpler and easier than the existing decision-making method with NCSs introduced in [40]. But, our decision-making method can only deal with decision-making problems with exact/crisp weights, rather than NCS weights [40].

#### 5.3. Sensitive Analysis

_{4}into the external NCS and reconstruct the following neutrosophic cubic decision matrix:

_{w}

_{1}and the cosine measure based on cosine function S

_{w}

_{3}still hold the same ranking orders; while the cosine measure based on distance S

_{w}

_{2}shows another ranking form. In this case, S

_{w}

_{2}is sensitive to the change of the evaluation values, since its ranking order changes with the change of the evaluation values for the alternative P

_{4}.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Table 1.**All the cosine measure values between P

_{i}and P* and ranking orders of the four alternatives.

S_{wk}(P_{i}, P*) | Cosine Measure Value | Ranking Order | The Best Alternative |
---|---|---|---|

S_{w}_{1}(P_{i}, P*) | 0.9564, 0.9855, 0.9596, 0.9945 | P_{4} > P_{2} > P_{3} > P_{1} | P_{4} |

S_{w}_{2}(P_{i}, P*) | 0.9769, 0.9944, 0.9795, 0.9972 | P_{4} > P_{2} > P_{3} > P_{1} | P_{4} |

S_{w}_{3}(P_{i}, P*^{′}) | 0.9892, 0.9959, 0.9897, 0.9989 | P_{4} > P_{2} > P_{3} > P_{1} | P_{4} |

**Table 2.**All the cosine measure values between P

_{i}′ and P*′ and ranking orders of the four alternatives.

S_{wk}(P_{i}′, P*′) | Cosine Measure Value | Ranking Order | The Best Alternative |
---|---|---|---|

S_{w1}(P_{i}′, P*′) | 0.9451, 0.9794, 0.9524, 0.9846 | P_{4} > P_{2} > P_{3} > P_{1} | P_{4} |

S_{w2}(P_{i}′, P*′) | 0.9700, 0.9906, 0.9732, 0.9877 | P_{2} > P_{4} > P_{3} > P_{1} | P_{2} |

S_{w}_{3}(P_{i}′, P*′) | 0.9867, 0.9942, 0.9877, 0.9968 | P_{4} > P_{2} > P_{3} > P_{1} | P_{4} |

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**MDPI and ACS Style**

Lu, Z.; Ye, J.
Cosine Measures of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making. *Symmetry* **2017**, *9*, 121.
https://doi.org/10.3390/sym9070121

**AMA Style**

Lu Z, Ye J.
Cosine Measures of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making. *Symmetry*. 2017; 9(7):121.
https://doi.org/10.3390/sym9070121

**Chicago/Turabian Style**

Lu, Zhikang, and Jun Ye.
2017. "Cosine Measures of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making" *Symmetry* 9, no. 7: 121.
https://doi.org/10.3390/sym9070121