# A Novel MCDM Method Based on Plithogenic Hypersoft Sets under Fuzzy Neutrosophic Environment

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Soft Sets

#### 2.2. Hypersoft Sets

#### 2.3. Plithogenic Sets

**Remark**

**1.**

#### 2.4. Plithogenic Hypersoft Set (PHSS)

**Remark**

**2.**

#### 2.5. Illustrative Example

- ${a}_{1}$ = Processor power,
- ${a}_{2}$ = RAM,
- ${a}_{3}$ = Front camera resolution,
- ${a}_{4}$ = Screen size in inches.

- ${A}_{1}=\{$ dual-core, quad-core, octa-core},
- ${A}_{2}=\{$2GB, 4GB, 8GB, 16GB},
- ${A}_{3}=\{$2MP, 5MP, 8MP, 16MP},
- ${A}_{4}=\{$4, 4.5, 5, 5.5, 6}.

- 1.
- Soft setConsider $\mathcal{B}=\{{a}_{2},{a}_{3}\}\subseteq \mathcal{A}$. Afterwards, a soft set $(\mathcal{F},\mathcal{B})$, defined by the mapping $\mathcal{F}:\mathcal{B}\to P\left(\mathcal{U}\right)$, is given by$$(\mathcal{F},\mathcal{B})=\{({a}_{2},\{{m}_{2},{m}_{5}\}),({a}_{3},\{{m}_{2},{m}_{3},{m}_{8}\})\}$$$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathcal{B}}\left({a}_{2}\right)=\{{m}_{2},{m}_{5}\},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathcal{F}}_{\mathcal{B}}\left({a}_{3}\right)=\{{m}_{2},{m}_{3},{m}_{8}\}.\end{array}$$
- 2.
- Hypersoft setLet $C={A}_{1}\times {A}_{2}\times {A}_{3}\times {A}_{4}$. Then, a hypersoft set over $\mathcal{U}$ is a function $f:C\to P\left(\mathcal{U}\right)$. For an element $\left(\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right)\in C$, it is given by$$\begin{array}{c}\hfill f\left(\left\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right\}\right)=\{{m}_{5},{m}_{8}\}\end{array}$$
- 3.
- Plithogenic hypersoft setFor the same tuple $\left(\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right)\in C$, a plithogenic hypersoft set $F:C\to P\left(\mathcal{U}\right)$ is given by$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F\left(\right\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\left\}\right)=& \{{m}_{5}\left({d}_{{m}_{5}}\left(\mathrm{octa}\text{-}\mathrm{core}\right),{d}_{{m}_{5}}\left(8\mathrm{GB}\right),{d}_{{m}_{5}}\left(16\mathrm{MP}\right),{d}_{{m}_{5}}\left(5.5\right)\right),\hfill \\ & {m}_{8}\left({d}_{{m}_{8}}\left(\mathrm{octa}\text{-}\mathrm{core}\right),{d}_{{m}_{8}}\left(8\mathrm{GB}\right),{d}_{{m}_{8}}\left(16\mathrm{MP}\right),{d}_{{m}_{8}}\left(5.5\right)\right)\},\hfill \end{array}\end{array}$$

## 3. The Four Classifications of PHSS

#### 3.1. The First Classification

#### 3.1.1. Uni-Attribute Plithogenic Hypersoft Set

#### 3.1.2. Multi-Attribute Plithogenic Hypersoft Set

**Example**

**1.**

- 1.
- Uni-attribute plithogenic hypersoft setConsider the most demanding feature of a mobile phone given by the attribute ${a}_{3}$ that stands for front camera resolution. The set of attribute values of ${a}_{3}$ is ${A}_{3}=\left\{2\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5\mathrm{MP},\phantom{\rule{4.pt}{0ex}}8\mathrm{MP},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP}\right\}$. Then, the uni-attribute plithogenic hypersoft set $F:{A}_{3}\to P\left(\mathcal{U}\right)$ is given by$$\begin{array}{c}\hfill F\left(\gamma \right)=\{x\left({d}_{x}\left(\gamma \right)\right),\forall \phantom{\rule{4pt}{0ex}}\gamma \in {A}_{3},\phantom{\rule{4pt}{0ex}}x\in X\},\end{array}$$$$\begin{array}{c}\hfill F\left(16\mathrm{MP}\right)=\{{m}_{5}\left({d}_{{m}_{5}}\left(16\mathrm{MP}\right)\right),{m}_{8}\left({d}_{{m}_{8}}\left(16\mathrm{MP}\right)\right)\},\end{array}$$
- 2.
- Multi-attribute plithogenic hypersoft setLet $\mathcal{B}=\{{a}_{3},{a}_{4}\}$ be the set of attributes required by the customer. Therefore, we need ${A}_{3}$ and ${A}_{4}$ given by$$\begin{array}{cc}\hfill & {A}_{3}=\left\{2\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5\mathrm{MP},\phantom{\rule{4.pt}{0ex}}8\mathrm{MP},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP}\right\},\\ \hfill & {A}_{4}=\{4,4.5,5,5.5,6\}.\end{array}$$$$\begin{array}{c}\hfill \begin{array}{c}\hfill F\left(\{16\mathrm{MP},5.5\}\right)=\{{m}_{5}\left({d}_{{m}_{5}}\left(16\mathrm{MP}\right),{d}_{{m}_{5}}\left(5.5\right)\right),{m}_{8}\left({d}_{{m}_{8}}\left(16\mathrm{MP}\right),{d}_{{m}_{8}}\left(5.5\right)\right)\},\end{array}\end{array}$$

#### 3.2. The Second Classification

#### 3.2.1. Plithogenic Crisp Hypersoft Set

#### 3.2.2. Plithogenic Fuzzy Hypersoft Set

#### 3.2.3. Plithogenic Intuitionistic Fuzzy Hypersoft Set

#### 3.2.4. Plithogenic Neutrosophic Hypersoft Set

**Example**

**2.**

- 1.
- Plithogenic crisp hypersoft set$$\begin{array}{c}\hfill \begin{array}{c}\hfill F\left(\left\{\mathit{octa}\text{-}\mathit{core},\phantom{\rule{4.pt}{0ex}}\mathit{8}\mathit{GB},\phantom{\rule{4.pt}{0ex}}\mathit{16}\mathit{MP},\phantom{\rule{4.pt}{0ex}}\mathit{5.5}\right\}\right)=\{{m}_{5}(1,1,1,1),{m}_{8}(1,1,1,1)\}.\end{array}\end{array}$$
- 2.
- Plithogenic fuzzy hypersoft set$$\begin{array}{c}\hfill \begin{array}{c}\hfill F\left(\left\{\mathit{octa}\text{-}\mathit{core},\phantom{\rule{4.pt}{0ex}}\mathit{8}\mathit{GB},\phantom{\rule{4.pt}{0ex}}\mathit{16}\mathit{MP},\phantom{\rule{4.pt}{0ex}}\mathit{5.5}\right\}\right)=\{{m}_{5}(0.9,0.2,1,0.75),{m}_{8}(0.5,0.5,0.25,0.9)\}.\end{array}\end{array}$$
- 3.
- Plithogenic intuitionistic fuzzy hypersoft set$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F\left(\right\{\mathit{octa}\text{-}\mathit{core},\phantom{\rule{4.pt}{0ex}}\mathit{8}\mathit{GB},\phantom{\rule{4.pt}{0ex}}\mathit{16}\mathit{MP},\phantom{\rule{4.pt}{0ex}}\mathit{5.5}\left\}\right)=& \{{m}_{5}\left((0.9,0.1),(0.2,0.6),(1,0),(0.75,0.1)\right),\hfill \\ & {m}_{8}((0.5,0.25),(0.5,0.5),(0.25,0.1),(0.9,0))\}.\hfill \end{array}\end{array}$$
- 4.
- Plithogenic neutrosophic hypersoft set$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F\left(\right\{\mathit{octa}\text{-}\mathit{core},\phantom{\rule{4.pt}{0ex}}\mathit{8}\mathit{GB},\phantom{\rule{4.pt}{0ex}}\mathit{16}\mathit{MP},\phantom{\rule{4.pt}{0ex}}\mathit{5.5}\left\}\right)=& \{{m}_{5}((0.9,0.7,0.1),(0.2,0.3,0.6),(1,0.25,0),(0.75,0.3,0.1)),\hfill \\ & {m}_{8}((0.5,1,0.25),(0.5,0.9,0.5),(0.25,0.7,0.1),(0.9,0.8,0))\}.\hfill \end{array}\end{array}$$

#### 3.3. The Third Classification

#### 3.3.1. Plithogenic Refined Hypersoft Set

#### 3.3.2. Plithogenic Hypersoft Overset

#### 3.3.3. Plithogenic Hypersoft Underset

#### 3.3.4. Plithogenic Hypersoft Offset

#### 3.3.5. Plithogenic Hypersoft Multiset

#### 3.3.6. Plithogenic Bipolar Hypersoft Set

**Remark**

**3.**

#### 3.3.7. Plithogenic Complex Hypersoft Set

**Example**

**3.**

- 1.
- Plithogenic refined hypersoft setConsider an attribute ${a}_{4}$ = $\mathrm{screen}\phantom{\rule{4.pt}{0ex}}\mathrm{size}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{inches}$ whose attribute values belong to the set ${A}_{4}=$ $\{4,4.5,5,5.5,6\}$. A refinement of ${A}_{4}$ is given by$${A}_{4}=\{4,4.5,4.7,5,5.5,5.8,6\},$$$$d(x,\gamma )\in P\left({[0,1]}^{j}\right),\forall \phantom{\rule{4pt}{0ex}}\gamma \in {A}_{4}.$$$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {F}_{r}\left(\{4,4.5,4.7,5,5.5,5.8,6\}\right)=& \{{m}_{5}\left({d}_{{m}_{5}}\left(4\right),{d}_{{m}_{5}}\left(4.5\right),{d}_{{m}_{5}}\left(4.7\right),{d}_{{m}_{5}}\left(5\right),{d}_{{m}_{5}}\left(5.5\right),{d}_{{m}_{5}}\left(5.8\right),{d}_{{m}_{5}}\left(6\right)\right),\hfill \\ & {m}_{8}\left({d}_{{m}_{8}}\left(4\right),{d}_{{m}_{8}}\left(4.5\right),{d}_{{m}_{8}}\left(4.7\right),{d}_{{m}_{8}}\left(5\right),{d}_{{m}_{8}}\left(5.5\right),{d}_{{m}_{8}}\left(5.8\right),{d}_{{m}_{8}}\left(6\right)\right)\}.\hfill \end{array}\end{array}$$
- 2.
- Plithogenic hypersoft oversetLet each attribute value has a single-valued fuzzy degree of appurtenance to all the elements of X. Subsequently, for $\left(\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right)\in C$, a plithogenic hypersoft overset ${F}_{o}:C\to P\left(\mathcal{U}\right)$ is given by$$\begin{array}{c}\hfill \begin{array}{c}\hfill {F}_{o}\left(\left\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right\}\right)=\{{m}_{5}(0.9,0.2,1.3,0.75),{m}_{8}(0.5,0.5,0.25,0.9)\}.\end{array}\end{array}$$
- 3.
- Plithogenic hypersoft undersetA plithogenic hypersoft underset defined by the function ${F}_{u}:C\to P\left(\mathcal{U}\right)$ is given by$$\begin{array}{c}\hfill \begin{array}{c}\hfill {F}_{u}\left(\left\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right\}\right)=\{{m}_{5}(0.9,0.2,-0.3,0.75),{m}_{8}(0.5,0.5,0.25,0.9)\}.\end{array}\end{array}$$
- 4.
- Plithogenic hypersoft offsetA plithogenic hypersoft offset is a function ${F}_{\mathrm{off}}:C\to P\left(\mathcal{U}\right)$, as given by$$\begin{array}{c}\hfill \begin{array}{c}\hfill {F}_{\mathrm{off}}\left(\left\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right\}\right)=\{{m}_{5}(0.9,0.2,-0.3,0.75),{m}_{8}(0.5,1.5,0.25,0.9)\}.\end{array}\end{array}$$
- 5.
- Plithogenic hypersoft multisetA plithogenic hypersoft multiset ${F}_{m}:C\to P\left(\mathcal{U}\right)$ is given by$$\begin{array}{c}\hfill \begin{array}{c}\hfill {F}_{m}\left(\left\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right\}\right)=\{{m}_{5}(0.9,0.2,0.3,0.75),{m}_{5}(0.7,0.1,0.9,1),{m}_{8}(0.5,0.5,0.25,0.9)\}.\end{array}\end{array}$$
- 6.
- Plithogenic bipolar hypersoft setA plithogenic bipolar hypersoft set ${F}_{2}:C\to P\left(\mathcal{U}\right)$ is given by$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {F}_{2}\left(\left\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right\}\right)=& \{{m}_{5}(\{-0.1,0.9\},\{-1,0.2\},\{-0.9,0.3\},\{-0.5,1\}),\hfill \\ & {m}_{8}(\{-0.5,0\},\{-0.9,1\},\{-0.2,0.2\},\{-1,0.8\})\}.\hfill \end{array}\end{array}$$
- 7.
- Plithogenic complex hypersoft setA plithogenic complex hypersoft set ${F}_{\mathrm{com}}:C\to P\left(\mathcal{U}\right)$ is given by$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {F}_{\mathrm{com}}\left(\left\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right\}\right)=& \{{m}_{5}(0.9{e}^{0.5i},0.2{e}^{0.9i},0.3{e}^{0.25i},0.75{e}^{i}),\hfill \\ & {m}_{8}(0.5{e}^{0.5i},{e}^{0.3i},0.25{e}^{0.75i},0.9{e}^{0.1i})\}.\hfill \end{array}\end{array}$$

#### 3.4. The Fourth Classification

#### 3.4.1. Single-Valued Plithogenic Hypersoft Set

#### 3.4.2. Hesitant Plithogenic Hypersoft Set

#### 3.4.3. Interval-Valued Plithogenic Hypersoft Set

**Example**

**4.**

- 1.
- Single-valued plithogenic hypersoft set$$\begin{array}{c}\hfill \begin{array}{c}\hfill F\left(\left\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\right\}\right)=\{{m}_{5}(0.9,0.2,1,0.75),{m}_{8}(0.5,0.5,0.25,0.9)\}.\end{array}\end{array}$$
- 2.
- Hesitant plithogenic hypersoft set$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F\left(\right\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\left\}\right)=& \{{m}_{5}(\{0.9,0.75\},\{0.2,0.7\},\{1,0.9\},\{0.75,0.5\}),\hfill \\ & {m}_{8}(\{0.5,0.1\},\{0.5,0.9\},\{0.25,0\},\{0.9,1\})\}.\hfill \end{array}\end{array}$$
- 3.
- Interval-valued plithogenic hypersoft set$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F\left(\right\{\mathrm{octa}\text{-}\mathrm{core},\phantom{\rule{4.pt}{0ex}}8\mathrm{GB},\phantom{\rule{4.pt}{0ex}}16\mathrm{MP},\phantom{\rule{4.pt}{0ex}}5.5\left\}\right)=& \{{m}_{5}([0.25,0.75],[0.2,0.6],[0.1,0.9],[0.75,1]),\hfill \\ & {m}_{8}([0.5,0.6],[0.3,0.9],[0.25,0.8],[0.9,1])\}.\hfill \end{array}\end{array}$$

## 4. The Proposed PHSS-Based TOPSIS with Application to a Parking Problem

#### 4.1. Proposed PHSS-Based TOPSIS Algorithm

- S1: Choose an ordered tuple $({\gamma}_{1},{\gamma}_{2},\dots ,{\gamma}_{n})\in C$ and construct a matrix of order $n\times m$, whose entries are the neutrosophic degree of appurtenance of each attribute value $\gamma $, with respect to each alternative $x\in X$ under consideration.
- S2: Employ the newly developed plithogenic accuracy function ${A}_{p}$, to each element of the matrix obtained in S1, in order to convert each element into a single crisp value, as follows:$$\begin{array}{c}\hfill {A}_{p}({T}_{\gamma},{I}_{\gamma},{F}_{\gamma})=\frac{{T}_{\gamma}+{I}_{\gamma}+{F}_{\gamma}}{3}+\frac{{T}_{{\gamma}_{d}}+{I}_{{\gamma}_{d}}+{F}_{{\gamma}_{d}}}{3}\times {c}_{F}(\gamma ,{\gamma}_{d}),\end{array}$$
- S3: Apply the transpose on the plithogenic accuracy matrix to obtain the plithogenic decision matrix ${M}_{p}={\left[{m}_{ij}\right]}_{m\times n}$ of alternatives versus criteria.
- S4: A plithogenic normalized decision matrix ${N}_{p}={\left[{y}_{ij}\right]}_{m\times n}$ is constructed, which represents the relative performance of alternatives and whose elements are calculated as follows:$$\begin{array}{c}\hfill {y}_{ij}=\frac{{m}_{ij}}{\sqrt{{\displaystyle \sum _{i=1}^{m}}{m}_{ij}^{2}}},\phantom{\rule{1.em}{0ex}}j=1,2,3,\dots ,n.\end{array}$$
- S5: Construct a plithogenic weighted normalized decision matrix ${V}_{p}={\left[{v}_{ij}\right]}_{m\times n}={N}_{p}{W}_{n}$, where ${W}_{n}=[{w}_{1}\phantom{\rule{4pt}{0ex}}{w}_{2}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}{w}_{n}]$ is a row matrix of allocated weights ${w}_{k}$ assigned to the criteria ${a}_{k},\phantom{\rule{4pt}{0ex}}k=1,2,3,\dots ,n$ and $\sum {w}_{k}=1,k=1,2,\dots ,n$. Moreover, all of the selection criteria are assigned different weights by the decision maker, depending on their importance in the decision making process.
- S6: Determine the plithogenic positive ideal solution ${V}_{p}^{+}$ and plithogenic negative ideal solution ${V}_{p}^{-}$ by the following formula:$$\begin{array}{c}\hfill {V}_{p}^{+}=\left\{\underset{i=1}{\overset{m}{max}}\left({v}_{ij}\right)\phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}{a}_{j}\in \mathrm{benefit}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria},\phantom{\rule{4pt}{0ex}}\underset{i=1}{\overset{m}{min}}\left({v}_{ij}\right)\phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}{a}_{j}\in \mathrm{cos}\mathrm{t}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria},j=1,2,3,\dots ,n\right\},\\ \hfill {V}_{p}^{-}=\left\{\underset{i=1}{\overset{m}{min}}\left({v}_{ij}\right)\phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}{a}_{j}\in \mathrm{benefit}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria},\phantom{\rule{4pt}{0ex}}\underset{i=1}{\overset{m}{max}}\left({v}_{ij}\right)\phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}{a}_{j}\in \mathrm{cos}\mathrm{t}\phantom{\rule{4.pt}{0ex}}\mathrm{criteria},j=1,2,3,\dots ,n\right\}.\end{array}$$
- S7: Calculate plithogenic positive distance ${S}_{i}^{+}$ and plithogenic negative distance ${S}_{i}^{-}$ of each alternative from ${V}_{p}^{+}$ and ${V}_{p}^{-}$, respectively, while using the following formulas:$$\begin{array}{c}\hfill \begin{array}{cc}& {S}_{i}^{+}=\sqrt{\sum _{j=1}^{n}{({v}_{ij}-{v}_{i}^{+})}^{2}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}i=1,2,3,\dots ,m,\hfill \\ & {S}_{i}^{-}=\sqrt{\sum _{j=1}^{n}{({v}_{ij}-{v}_{i}^{-})}^{2}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}i=1,2,3,\dots ,m.\hfill \end{array}\end{array}$$
- S8: Calculate the relative closeness coefficient ${C}_{i}$ of each alternative by the following expression:$$\begin{array}{c}\hfill \begin{array}{c}\hfill {C}_{i}=\frac{{S}_{i}^{-}}{{S}_{i}^{+}+{S}_{i}^{-}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}i=1,2,3,\dots ,m.\end{array}\end{array}$$
- S9: The highest value from $\{{C}_{1},{C}_{2},\dots ,{C}_{m}\}$ belongs to the most suitable alternative. Similarly, the lowest value gives us the worst alternative.

#### 4.2. Parking Spot Choice Problem

#### 4.2.1. Case 1

**A. Application of PHSS-based TOPSIS for Case 1**

**B. Application of Fuzzy TOPSIS for Case 1**

#### 4.2.2. Case 2

**A. Application of PHSS-Based TOPSIS for Case 2**

**B. Application of Fuzzy TOPSIS for Case 2**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Sr. | Variables | ${\mathit{P}}_{1}$ | ${\mathit{P}}_{2}$ | ${\mathit{P}}_{3}$ | ${\mathit{P}}_{4}$ |
---|---|---|---|---|---|

1 | ${f}_{1}$ | (0.5, 0.1, 0.3) | (0.5, 0.0, 0.7) | (0.1, 0.4, 0.5) | (0.2, 0.1, 0.6) |

2 | ${f}_{2}$ | (0.7, 0.9, 0.1) | (0.6, 0.5, 0.2) | (0.2, 0.3, 0.6) | (0.7, 0.9, 0.3) |

3 | ${f}_{3}$ | (0.5, 0.5, 0.1) | (0.0, 0.1, 0.5) | (0.1, 0.1, 0.9) | (0.5, 0.7, 0.2) |

4 | ${r}_{1}$ | (0.8, 0.1, 0.7) | (0.9, 0.4, 0.5) | (0.9, 0.4, 0.0) | (0.8, 0.4, 0.2) |

5 | ${r}_{2}$ | (0.9, 0.3, 0.2) | (0.6, 0.1, 0.0) | (0.5, 0.2, 0.4) | (0.9, 0.1, 0.4) |

6 | ${r}_{3}$ | (0.9, 0.1, 0.3) | (0.8, 0.3, 0.1) | (0.6, 0.0, 0.6) | (0.2, 0.2, 0.5) |

7 | ${r}_{4}$ | (0.8, 0.3, 0.2) | (1.0, 0.1, 0.5) | (0.8, 0.5, 0.1) | (0.7, 0.3, 0.6) |

8 | ${r}_{5}$ | (1.0, 0.3, 0.2) | (1.0, 0.3, 0.2) | (0.8, 0.2, 0.8) | (0.6, 0.5, 0.6) |

9 | ${r}_{6}$ | (0.8, 0.1, 0.0) | (0.6, 0.8, 0.5) | (0.9, 0.7, 0.1) | (0.4, 0.8, 0.7) |

10 | ${s}_{1}$ | (0.0, 0.5, 0.5) | (0.4, 0.1, 0.6) | (0.2, 0.2, 0.7) | (0.8, 0.3, 0.4) |

11 | ${s}_{2}$ | (1.0, 0.8, 0.6) | (0.7, 0.5, 0.5) | (0.4, 0.4, 0.7) | (0.6, 0.5, 0.7) |

12 | ${d}_{1}$ | (0.1, 0.2, 1.0) | (0.3, 1.0, 0.6) | (0.7, 0.9, 0.2) | (0.9, 0.7, 0.5) |

13 | ${d}_{2}$ | (0.1, 0.4, 0.8) | (0.2, 0.2, 0.8) | (0.2, 0.6, 0.3) | (0.2, 0.8, 0.5) |

14 | ${d}_{3}$ | (0.5, 0.6, 0.9) | (0.9, 0.6, 0.3) | (0.9, 0.7, 0.5) | (0.6, 0.7, 0.6) |

$\mathit{Al}/\mathit{Cr}$ | ${\mathit{f}}_{2}$ | ${\mathit{r}}_{1}$ | ${\mathit{s}}_{2}$ | ${\mathit{d}}_{1}$ |
---|---|---|---|---|

${P}_{1}$ | 0.6667 | 0.5333 | 0.9667 | 0.4333 |

${P}_{2}$ | 0.5667 | 0.6000 | 0.7500 | 0.6333 |

${P}_{3}$ | 0.4778 | 0.4333 | 0.6833 | 0.6000 |

${P}_{4}$ | 0.7333 | 0.4667 | 0.8500 | 0.7000 |

${\mathit{S}}_{\mathit{i}}^{+}$ | ${\mathit{S}}_{\mathit{i}}^{-}$ | ${\mathit{C}}_{\mathit{i}}$ | Ranking | |
---|---|---|---|---|

${P}_{1}$ | 0.0697 | 0.0573 | 0.4511 | 3 |

${P}_{2}$ | 0.0601 | 0.0588 | 0.4944 | 2 |

${P}_{3}$ | 0.0320 | 0.0956 | 0.7494 | 1 |

${P}_{4}$ | 0.0986 | 0.0305 | 0.2366 | 4 |

${\mathit{S}}_{\mathit{i}}^{+}$ | ${\mathit{S}}_{\mathit{i}}^{-}$ | ${\mathit{C}}_{\mathit{i}}$ | Ranking | |
---|---|---|---|---|

${P}_{1}$ | 0.0888 | 0.0592 | 0.4000 | 3 |

${P}_{2}$ | 0.0591 | 0.0841 | 0.5872 | 2 |

${P}_{3}$ | 0.0320 | 0.1176 | 0.7863 | 1 |

${P}_{4}$ | 0.1170 | 0.0373 | 0.2417 | 4 |

Sr. | Parkings | PHSS-Based TOPSIS Ranking | Fuzzy TOPSIS Ranking |
---|---|---|---|

1 | ${P}_{1}$ | 3rd | 3rd |

2 | ${P}_{2}$ | 2nd | 2nd |

3 | ${P}_{3}$ | 1st | 1st |

4 | ${P}_{4}$ | 4th | 4th |

Sr. | Variables | ${\mathit{P}}_{1}$ | ${\mathit{P}}_{5}$ | ${\mathit{P}}_{6}$ | ${\mathit{P}}_{7}$ |
---|---|---|---|---|---|

1 | ${f}_{1}$ | (0.5, 0.1, 0.3) | (0.6, 0.6, 0.8) | (0.7, 0.2, 0.4) | (0.9, 0.5, 0.2) |

2 | ${f}_{2}$ | (0.7, 0.9, 0.1) | (0.8, 0.8, 0.5) | (0.4, 0.4, 0.7) | (0.7, 0.2, 0.1) |

3 | ${f}_{3}$ | (0.5, 0.5, 0.1) | (0.4, 0.2, 0.5) | (1.0, 0.5, 0.9) | (1.0, 0.7, 0.6) |

4 | ${r}_{1}$ | (0.8, 0.1, 0.7) | (0.9, 0.5, 0.2) | (0.5, 0, 0.9) | (0.8, 0.6, 0.1) |

5 | ${r}_{2}$ | (0.9, 0.3, 0.2) | (0.5, 0.4, 0.2) | (0.7, 0.5, 0.4) | (0.9, 0.6, 0.8) |

6 | ${r}_{3}$ | (0.9, 0.1, 0.3) | (0.5, 0.7, 0.3) | (0.9, 1.0, 0.6) | (0.2, 0, 1.0) |

7 | ${r}_{4}$ | (0.8, 0.3, 0.2) | (1.0, 0.2, 1.0) | (1.0, 0.5, 0.7) | (0.8, 0.8, 0.9) |

8 | ${r}_{5}$ | (0.2, 0.3, 0.9) | (1.0, 0.1, 0.8) | (0.4, 0.6, 0.8) | (0.8, 0.6, 0.6) |

9 | ${r}_{6}$ | (0.5, 0.7, 0.5) | (0.8, 0.2, 0.0) | (0.6, 0.3, 0.7) | (0.0, 0.9, 0.9) |

10 | ${s}_{1}$ | (0, 0.5, 0.5) | (0.8, 0.4, 0.6) | (0.9, 0.2, 0.2) | (0.8, 0.4, 0.7) |

11 | ${s}_{2}$ | (1.0, 0.8, 0.6) | (0.7, 1.0, 0.2) | (0.2, 0.4, 0.7) | (0.9, 0, 1.0) |

12 | ${d}_{1}$ | (0.1, 0.2, 1.0) | (1.0, 0.4, 0.3) | (0.7, 0.5, 0.6) | (0.8, 0.5, 0.7) |

13 | ${d}_{2}$ | (0.1, 0.4, 0.8) | (0.7, 1.0, 0.8) | (0.6, 0.6, 1.0) | (1.0, 0.8, 0.8) |

14 | ${d}_{3}$ | (0.5, 0.6, 0.9) | (1.0, 0.6, 0.5) | (1.0, 1.0, 0.5) | (1.0, 0.5, 0.8) |

${\mathit{S}}_{\mathit{i}}^{+}$ | ${\mathit{S}}_{\mathit{i}}^{-}$ | ${\mathit{C}}_{\mathit{i}}$ | Ranking | |
---|---|---|---|---|

${P}_{1}$ | 0.0561 | 0.0901 | 0.6163 | 3 |

${P}_{5}$ | 0.1306 | 0.0131 | 0.0912 | 4 |

${P}_{6}$ | 0.0496 | 0.0929 | 0.6518 | 2 |

${P}_{7}$ | 0.0576 | 0.1206 | 0.6769 | 1 |

${\mathit{S}}_{\mathit{i}}^{+}$ | ${\mathit{S}}_{\mathit{i}}^{-}$ | ${\mathit{C}}_{\mathit{i}}$ | Ranking | |
---|---|---|---|---|

${P}_{1}$ | 0.0908 | 0.0724 | 0.4439 | 3 |

${P}_{5}$ | 0.1453 | 0.0224 | 0.1337 | 4 |

${P}_{6}$ | 0.0694 | 0.0863 | 0.5543 | 2 |

${P}_{7}$ | 0.0516 | 0.1397 | 0.7301 | 1 |

Sr. | Parkings | PHSS-Based TOPSIS Ranking | Fuzzy TOPSIS Ranking |
---|---|---|---|

1 | ${P}_{1}$ | 3rd | 3rd |

2 | ${P}_{5}$ | 4th | 4th |

3 | ${P}_{6}$ | 2nd | 2nd |

4 | ${P}_{7}$ | 1st | 1st |

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## Share and Cite

**MDPI and ACS Style**

Ahmad, M.R.; Saeed, M.; Afzal, U.; Yang, M.-S.
A Novel MCDM Method Based on Plithogenic Hypersoft Sets under Fuzzy Neutrosophic Environment. *Symmetry* **2020**, *12*, 1855.
https://doi.org/10.3390/sym12111855

**AMA Style**

Ahmad MR, Saeed M, Afzal U, Yang M-S.
A Novel MCDM Method Based on Plithogenic Hypersoft Sets under Fuzzy Neutrosophic Environment. *Symmetry*. 2020; 12(11):1855.
https://doi.org/10.3390/sym12111855

**Chicago/Turabian Style**

Ahmad, Muhammad Rayees, Muhammad Saeed, Usman Afzal, and Miin-Shen Yang.
2020. "A Novel MCDM Method Based on Plithogenic Hypersoft Sets under Fuzzy Neutrosophic Environment" *Symmetry* 12, no. 11: 1855.
https://doi.org/10.3390/sym12111855