# Single-Valued Neutrosophic Linguistic Logarithmic Weighted Distance Measures and Their Application to Supplier Selection of Fresh Aquatic Products

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Single-Valued Neutrosophic Set (SVNS)

**Definition**

**1**

**.**A single-valued neutrosophic set (SVNS) $\eta $ in a finite set $X$ denoted by a mathematical form:

#### 2.2. The Linguistic Set

**Definition**

**2**

- (1)
- ${s}_{\alpha}\oplus {s}_{\beta}={s}_{\alpha +\beta}$;
- (2)
- $\mu {s}_{\alpha}={s}_{\mu \alpha}$,$\mu \ge 0$;

#### 2.3. The Single-Valued Neutrosophic Linguistic Set (SVNLS)

**Definition**

**3**

**.**A single-valued neutrosophic linguistic set (SVNLS) $\varphi $ in $X$ is defined as:

- (1)
- ${x}_{1}\oplus {x}_{2}=\langle {s}_{\theta ({x}_{1})+\theta ({x}_{2})},({T}_{{x}_{1}}+{T}_{{x}_{2}}-{T}_{{x}_{1}}\ast {T}_{{x}_{2}},{I}_{{x}_{1}}\ast {I}_{{x}_{2}},{F}_{{x}_{1}}\ast {F}_{{x}_{2}})\rangle ;$
- (2)
- $\lambda {x}_{1}=\langle {s}_{\lambda \theta ({x}_{1})},(1-{(1-{T}_{{x}_{1}})}^{\lambda},{({I}_{{x}_{1}})}^{\lambda},{({F}_{{x}_{1}})}^{\lambda})\rangle ;$
- (3)
- ${x}_{1}^{\lambda}=\langle {s}_{{\theta}^{\lambda}({x}_{1})},({({T}_{{x}_{1}})}^{\lambda},1-{(1-{I}_{{x}_{1}})}^{\lambda},1-{(1-{F}_{{x}_{1}})}^{\lambda})\rangle $.

**Definition**

**4**

**.**Let ${x}_{i}=\langle {s}_{\theta ({x}_{i})},({T}_{{x}_{i}},{I}_{{x}_{i}},{F}_{{x}_{i}})\rangle (i=1,2)$ be SVNLNs and $p>0$, then the distance measure between ${x}_{1}$ and ${x}_{2}$ is given by the mathematical form:

#### 2.4. The Ordered Weighted Logarithmic Averaging Distance (OWLAD) Measure

**Definition**

**5**

**.**Let $U=\left\{{u}_{1},{u}_{2},\dots ,{u}_{n}\right\}$ and $V=\left\{{v}_{1},{v}_{2},\dots ,{v}_{n}\right\}$ be two crisp sets, ${d}_{i}=\left|{u}_{i}-{v}_{i}\right|$ be the distance between ${u}_{i}$ and ${v}_{i}$, then the OWAD measure is defined as:

**Definition**

**6**

**.**Let $U=\left\{{u}_{1},{u}_{2},\dots ,{u}_{n}\right\}$ and $V=\left\{{v}_{1},{v}_{2},\dots ,{v}_{n}\right\}$ be two crisp sets, ${d}_{i}=\left|{u}_{i}-{v}_{i}\right|$ be the distance between ${u}_{i}$ and ${v}_{i}$, then the OWLAD measure is defined as:

## 3. SVNL Weighted Logarithmic Distance Measures

#### 3.1. SVL Weighted Logarithmic Averaging Distance (SVNLWLAD) Measure

**Definition**

**7.**

**Example**

**1.**

- (1)
- Calculate the individual distances ${d}_{SVNL}({x}_{i},{y}_{i})$ $(i=1,2,\dots ,5)$ according to Equation (5) (let $p=1$):$$\begin{array}{c}{d}_{SVNL}({x}_{1},{y}_{1})=\left|2\times 0.6-4\times 0.2\right|+\left|2\times 0.5-4\times 0.7\right|+\left|2\times 0.1-4\times 0\right|=2.4,\\ {d}_{SVNL}({x}_{2},{y}_{2})=\left|5\times 0.6-6\times 0.3\right|+\left|5\times 0.3-6\times 0.7\right|+\left|5\times 0.5-6\times 0.1\right|=5.8,\\ {d}_{SVNL}({x}_{3},{y}_{3})=\left|4\times 0.7-7\times 0.6\right|+\left|4\times 0.2-7\times 0.4\right|+\left|4\times 0.1-7\times 0.5\right|=6.5,\\ {d}_{SVNL}({x}_{4},{y}_{4})=\left|3\times 0.9-1\times 0.1\right|+\left|3\times 0.1-1\times 0.7\right|+\left|3\times 0.6-1\times 0.2\right|=4.2,\\ {d}_{SVNL}({x}_{5},{y}_{5})=\left|4\times 0.3-3\times 0.1\right|+\left|4\times 0.1-3\times 0.5\right|+\left|4\times 0.3-3\times 0.6\right|=2.6.\end{array}$$
- (2)
- Utilize the SVNLWLAD defined in Equation (9) to aggregate the individual distances:$$\begin{array}{l}SVNLWLAD\left(({x}_{1},{y}_{1}),\dots ,({x}_{5},{y}_{5})\right)=\mathrm{exp}\left\{{\displaystyle \sum _{j=1}^{n}{w}_{j}\mathrm{ln}\left({d}_{SVNL}({x}_{j},{y}_{j})\right)}\right\}\\ =\mathrm{exp}\left\{{\displaystyle \sum _{j=1}^{n}(0.15\times \mathrm{ln}(2.4)+0.25\times \mathrm{ln}(5.8)+0.25\times \mathrm{ln}(6.5)+0.15\times \mathrm{ln}(4.2)+0.2\times \mathrm{ln}(2.6)}\right\}\\ =4.2423\end{array}$$

#### 3.2. SVL Ordered Weighted Logarithmic Averaging Distance (SVNLOWLAD) Measure

**Definition**

**8.**

**Example**

**2.**

- (1)
- Compute the individual distances ${d}_{SVNL}({x}_{i},{y}_{i})$ $(i=1,2,\dots ,5)$ according to Equation (5) (obtained from example 1):$$\begin{array}{c}{d}_{SVNL}({x}_{1},{y}_{1})=2.4,{d}_{SVNL}({x}_{2},{y}_{2})=5.8,{d}_{SVNL}({x}_{3},{y}_{3})=6.5,\\ {d}_{SVNL}({x}_{4},{y}_{4})=4.2,\text{}{d}_{SVNL}({x}_{5},{y}_{5})=2.6\end{array}$$
- (2)
- Rank the ${d}_{SVNL}({x}_{i},{y}_{i})$ $(i=1,2,\dots ,5)$ in decreasing order:$$\begin{array}{c}{d}_{SVNL}({x}_{\sigma (1)},{y}_{\sigma (1)})={d}_{SVNL}({x}_{3},{y}_{3})=6.5,\text{}{d}_{SVNL}({x}_{\sigma (2)},{y}_{\sigma (2)})={d}_{SVNL}({x}_{2},{y}_{2})=5.8,\\ {d}_{SVNL}({x}_{\sigma (3)},{y}_{\sigma (3)})={d}_{SVNL}({x}_{4},{y}_{4})=4.2,\text{}{d}_{SVNL}({x}_{\sigma (4)},{y}_{\sigma (4)})={d}_{SVNL}({x}_{5},{y}_{5})=2.6,\\ {d}_{SVNL}({x}_{\sigma (5)},{y}_{\sigma (5)})={d}_{SVNL}({x}_{1},{y}_{1})=2.4.\end{array}$$
- (3)
- Utilize the SVNLOWLAD to aggregate the ordered distances:$$\begin{array}{l}SVNLOWLAD\left(({x}_{1},{y}_{1}),\dots ,({x}_{5},{y}_{5})\right)=\mathrm{exp}\left\{{\displaystyle \sum _{j=1}^{5}{w}_{j}\mathrm{ln}\left({d}_{SVNL}({x}_{\sigma (j)},{y}_{\sigma (j)})\right)}\right\}\\ =\mathrm{exp}\left\{0.1\times \mathrm{ln}(6.5)+0.2\times \mathrm{ln}(5.8)+0.25\times \mathrm{ln}(4.2)+0.3\times \mathrm{ln}(2.6)+0.15\times \mathrm{ln}(2.4)\right\}\\ =3.7266\end{array}$$

#### 3.3. SVL Combined Weighted Logarithmic Averaging Distance (SVNLCWLAD) Measure

**Definition**

**9.**

**Example**

**3.**

- The SVNLOWLAD and SVNLWLAD measures are obtained when $\gamma =1$ and $\lambda =0$, respectively. Moreover, the more lager $\gamma $, the more importance focused on the SVNLOWLAD.
- If $w={(1,0,0,\dots ,0)}^{T}$, then max-SVNLCWLAD measure is formed.
- If $w={(0,\dots ,0,1)}^{T}$, then the min-SVNLCWLAD is rendered.
- The step-SVNLCWLAD measure is obtained by designing ${w}_{1}=\cdots ={w}_{k-1}=0$, ${w}_{k}=1$ and ${w}_{k+1}=\cdots ={w}_{n}=0$.

- (1)
- Monotonicity: If ${d}_{SVNL}({x}_{i},{y}_{i})\ge {d}_{SVNL}({{x}^{\prime}}_{i},{{y}^{\prime}}_{i})$ for $i=1,2,\dots ,n,$ then$$SVNLCWLAD\left(({x}_{1},{y}_{1}),\dots ,({x}_{n},{y}_{n})\right)\ge SVNLCWLAD\left(({{x}^{\prime}}_{1},{{y}^{\prime}}_{1}),\dots ,({{x}^{\prime}}_{n},{{y}^{\prime}}_{n})\right)$$
- (2)
- Idempotency: If ${d}_{SVNL}({x}_{i},{y}_{i})=d$ for $i=1,2,\dots ,n,$ then$$SVNLCWLAD\left(({x}_{1},{y}_{1}),\dots ,({x}_{n},{y}_{n})\right)=d$$
- (3)
- Commutativity: If $\left(({x}_{1},{{x}^{\prime}}_{1}),\dots ,({x}_{n},{{x}^{\prime}}_{n})\right)$ is any permutation of $\left(({y}_{1},{{y}^{\prime}}_{1}),\dots ,({y}_{n},{{y}^{\prime}}_{n})\right)$, then$$SVNLCWLAD\left(({x}_{1},{{x}^{\prime}}_{1}),\dots ,({x}_{n},{{x}^{\prime}}_{n})\right)=SVNLCWLAD\left(({y}_{1},{{y}^{\prime}}_{1}),\dots ,({y}_{n},{{y}^{\prime}}_{n})\right)$$
- (4)
- Boundedness: Let ${d}_{\mathrm{min}}=\underset{i}{\mathrm{min}}\left(d({y}_{i},{{y}^{\prime}}_{i})\right)$ and ${d}_{\mathrm{max}}=\underset{i}{\mathrm{max}}\left(d({y}_{i},{{y}^{\prime}}_{i})\right)$, then$${d}_{\mathrm{min}}\le SVNLCWLAD\left(({y}_{1},{{y}^{\prime}}_{1}),\dots ,({y}_{n},{{y}^{\prime}}_{n})\right)\le {d}_{\mathrm{max}}$$

## 4. Application in MAGDM

**Step 1:**Let each expert ${e}_{q}$ $(q=1,2,\dots ,t)$(whose weight is ${\tau}_{q}$, with ${\tau}_{q}\ge 0$ and $\sum _{q=1}^{t}{\tau}_{q}=1$) expresses his or her assessment for different alternatives under given attributes by means of SVNLNs, thus formulate SVNL individual decision matrix ${R}^{q}={\left({r}_{ij}^{(q)}\right)}_{m\times n}$.

**Step 2:**The collective decision matrix $R={\left({r}_{ij}\right)}_{m\times n}$ is calculated by using the SVNL weighted average (SVNLWA) operator [8] to aggregate individual assessment, where ${r}_{ij}={\displaystyle \sum _{q=1}^{t}{\tau}_{q}}{r}_{ij}^{(q)}$.

**Step 3:**Set the ideal performances for each attribute to construct the ideal scheme (Table 1).

**Step 4:**Apply the SVNLCWLAD measure to compute the distances between the alternative ${B}_{i}(i=1,2,\dots ,m)$ and the ideal scheme $I$:

**Step 5:**Sort the alternatives according to the lowest value of distance obtained in the previous step and hence, select the best one(s).

**Step 6:**End.

## 5. Numerical Example for Supplier Selection of Fresh Aquatic Products

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | $\cdots $ | ${\mathit{A}}_{\mathit{n}}$ | |
---|---|---|---|---|

$I$ | ${I}_{1}$ | ${I}_{2}$ | $\dots $ | ${I}_{n}$ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${B}_{1}$ | $\langle {s}_{4}^{(1)},(0.6,0.1,0.2)\rangle $ | $\langle {s}_{6}^{(1)},(0.6,0.1,0.2)\rangle $ | $\langle {s}_{5}^{(1)},(0.7,0.0,0.1)\rangle $ | $\langle {s}_{3}^{(1)},(0.3,0.1,0.2)\rangle $ |

${B}_{2}$ | $\langle {s}_{5}^{(1)},(0.6,0.1,0.2)\rangle $ | $\langle {s}_{3}^{(1)},(0.6,0.2,0.4)\rangle $ | $\langle {s}_{6}^{(1)},(0.6,0.1,0.2)\rangle $ | $\langle {s}_{4}^{(1)},(0.5,0.2,0.2)\rangle $ |

${B}_{3}$ | $\langle {s}_{4}^{(1)},(0.5,0.2,0.3)\rangle $ | $\langle {s}_{5}^{(1)},(0.3,0.5,0.2)\rangle $ | $\langle {s}_{4}^{(1)},(0.3,0.2,0.3)\rangle $ | $\langle {s}_{3}^{(1)},(0.5,0.3,0.1)\rangle $ |

${B}_{4}$ | $\langle {s}_{5}^{(1)},(0.4,0.2,0.3)\rangle $ | $\langle {s}_{4}^{(1)},(0.5,0.3,0.3)\rangle $ | $\langle {s}_{5}^{(1)},(0.4,0.2,0.3)\rangle $ | $\langle {s}_{3}^{(1)},(0.3,0.2,0.5)\rangle $ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${B}_{1}$ | $\langle {s}_{4}^{(3)},(0.5,0.2,0.2)\rangle $ | $\langle {s}_{5}^{(3)},(0.7,0.2,0.1)\rangle $ | $\langle {s}_{4}^{(3)},(0.6,0.1,0.2)\rangle $ | $\langle {s}_{3}^{(3)},(0.4,0.1,0.1)\rangle $ |

${B}_{2}$ | $\langle {s}_{4}^{(3)},(0.7,0.2,0.2)\rangle $ | $\langle {s}_{6}^{(3)},(0.4,0.6,0.2)\rangle $ | $\langle {s}_{5}^{(3)},(0.5,0.2,0.3)\rangle $ | $\langle {s}_{5}^{(3)},(0.7,0.2,0.1)\rangle $ |

${B}_{3}$ | $\langle {s}_{5}^{(3)},(0.6,0.1,0.3)\rangle $ | $\langle {s}_{4}^{(3)},(0.3,0.6,0.2)\rangle $ | $\langle {s}_{6}^{(3)},(0.5,0.1,0.3)\rangle $ | $\langle {s}_{4}^{(3)},(0.6,0.2,0.1)\rangle $ |

${B}_{4}$ | $\langle {s}_{6}^{(3)},(0.6,0.2,0.4)\rangle $ | $\langle {s}_{4}^{(3)},(0.5,0.2,0.3)\rangle $ | $\langle {s}_{6}^{(3)},(0.5,0.2,0.3)\rangle $ | $\langle {s}_{5}^{(3)},(0.2,0.1,0.6)\rangle $ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${B}_{1}$ | $\langle {s}_{5}^{(2)},(0.7,0.2,0.3)\rangle $ | $\langle {s}_{6}^{(2)},(0.6,0.3,0.3)\rangle $ | $\langle {s}_{4}^{(2)},(0.8,0.1,0.2)\rangle $ | $\langle {s}_{4}^{(2)},(0.4,0.2,0.2)\rangle $ |

${B}_{2}$ | $\langle {s}_{6}^{(2)},(0.7,0.2,0.3)\rangle $ | $\langle {s}_{4}^{(2)},(0.5,0.4,0.2)\rangle $ | $\langle {s}_{6}^{(2)},(0.7,0.2,0.3)\rangle $ | $\langle {s}_{5}^{(2)},(0.6,0.2,0.2)\rangle $ |

${B}_{3}$ | $\langle {s}_{6}^{(2)},(0.6,0.3,0.4)\rangle $ | $\langle {s}_{5}^{(2)},(0.4,0.4,0.1)\rangle $ | $\langle {s}_{6}^{(2)},(0.4,0.2,0.4)\rangle $ | $\langle {s}_{4}^{(2)},(0.6,0.1,0.3)\rangle $ |

${B}_{4}$ | $\langle {s}_{6}^{(2)},(0.5,0.1,0.2)\rangle $ | $\langle {s}_{3}^{(2)},(0.7,0.1,0.1)\rangle $ | $\langle {s}_{5}^{(2)},(0.4,0.3,0.4)\rangle $ | $\langle {s}_{5}^{(2)},(0.3,0.1,0.6)\rangle $ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${B}_{1}$ | $\langle {s}_{4.33},(0.611,0.155,0.229)\rangle $ | $\langle {s}_{5.70},(0.633,0.180,0.186)\rangle $ | $\langle {s}_{4.37},(0.714,0.000,0.155)\rangle $ | $\langle {s}_{3.67},(0.365,0.128,0.163)\rangle $ |

${B}_{2}$ | $\langle {s}_{4.70},(0.666,0.155,0.229)\rangle $ | $\langle {s}_{4.23},(0.514,0.350,0.258)\rangle $ | $\langle {s}_{5.70},(0.611,0.155,0.258)\rangle $ | $\langle {s}_{2.37},(0.602,0.200,0.162)\rangle $ |

${B}_{3}$ | $\langle {s}_{4.96},(0.566,0.186,0.330)\rangle $ | $\langle {s}_{4.70},(0.335,0.491,0.159)\rangle $ | $\langle {s}_{5.26},(0.399,0.163,0.330)\rangle $ | $\langle {s}_{3.37},(0.566,0.185,0.144)\rangle $ |

${B}_{4}$ | $\langle {s}_{5.63},(0.450,0.159,0.286)\rangle $ | $\langle {s}_{3.67},(0.578,0.185,0.209)\rangle $ | $\langle {s}_{5.30},(0.432,0.229,0.330)\rangle $ | $\langle {s}_{2.37},(0.271,0.129,0.561)\rangle $ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

$I$ | $\langle {s}_{7},(1,0,0.1)\rangle $ | $\langle {s}_{7},(0.9,0.1,0)\rangle $ | $\langle {s}_{6},(0.9,0,0)\rangle $ | $\langle {s}_{7},(0.9,0,0.1)\rangle $ |

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## Share and Cite

**MDPI and ACS Style**

Wang, J.; Zeng, S.; Zhang, C.
Single-Valued Neutrosophic Linguistic Logarithmic Weighted Distance Measures and Their Application to Supplier Selection of Fresh Aquatic Products. *Mathematics* **2020**, *8*, 439.
https://doi.org/10.3390/math8030439

**AMA Style**

Wang J, Zeng S, Zhang C.
Single-Valued Neutrosophic Linguistic Logarithmic Weighted Distance Measures and Their Application to Supplier Selection of Fresh Aquatic Products. *Mathematics*. 2020; 8(3):439.
https://doi.org/10.3390/math8030439

**Chicago/Turabian Style**

Wang, Jiefeng, Shouzhen Zeng, and Chonghui Zhang.
2020. "Single-Valued Neutrosophic Linguistic Logarithmic Weighted Distance Measures and Their Application to Supplier Selection of Fresh Aquatic Products" *Mathematics* 8, no. 3: 439.
https://doi.org/10.3390/math8030439