# A New Approach to Artificial Intelligent Based Three-Way Decision Making and Analyzing S-Box Image Encryption Using TOPSIS Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. A Brief Review on the Development of Fuzzy Sets

#### 1.2. A Brief Review on the Aggregation Operators

#### 1.3. A Review on the New Version of Fuzzy Logic Systems

#### 1.4. A Review on the Three-Way Decisions

#### 1.5. A Review on the TOPSIS Method

#### 1.6. Motivation of the Study

- a.
- Extending the Einstein operations to aggregate double hierarchy hesitation fuzzy information.
- b.
- Artificial intelligence based three-way decision-making problem with double hierarchy hesitant linguistic terms estimate information.
- c.
- d.
- There are many methods in Artificial intelligence by which to find unknown experts and criteria weights to obtain unknown expert weights. When criterion are found, we must employ entropy methods.
- e.
- Using the popular technique for multi-criteria decision making is the TOPSIS technique. Based on a collection of parameters, the TOPSIS technique is used to select the best S-box for image encryption.

#### 1.7. Contribution of the Study

- a.
- Establish the basic concepts of DHHLTSs and introduce the Einstein operational rules of DHHLEs.
- b.
- Develop the score function, Einstein operational laws and Einstein aggregation operators for the double hierarchy hesitant linguistic term.
- c.
- Discuss the TOPSIS method for calculating conditional probability and a novel DHHLDTRS model using Einstein aggregation operations and their further expected losses and score functions.
- d.
- Develop the TWDs for the double hierarchy hesitant linguistic term.
- e.
- Use the process of TWDs for the s-box analysis for image encryption.

## 2. Preliminaries

#### Double Hierarchy Hesitant Linguistic Term Sets

## 3. The Einstein Operation Law of DHHLTs

- (i)
- ${P}_{1}{\oplus}_{\mathrm{E}}{P}_{2}=\underset{\begin{array}{c}\left({a}_{1},{a}_{2}\right)\in m\\ \left({b}_{1},{b}_{2}\right)\in n\end{array}}{{\displaystyle \cup}}\left[\frac{F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)+F\left({S}_{{a}_{2}}\u2329{O}_{{b}_{2}}\u232a\right)}{1+F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)F\left({S}_{{a}_{2}}\u2329{O}_{{b}_{2}}\u232a\right)}\right]$
- (ii)
- ${P}_{1}{\otimes}_{\mathrm{E}}{P}_{2}=\underset{\begin{array}{c}\left({a}_{1},{a}_{2}\right)\in m\\ \left({b}_{1},{b}_{2}\right)\in n\end{array}}{{\displaystyle \cup}}\left[\frac{F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)F\left({S}_{{a}_{2}}\u2329{O}_{{b}_{2}}\u232a\right)}{1+\left(1-F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)\right)\left(1-F\left({S}_{{a}_{2}}\u2329{O}_{{b}_{2}}\u232a\right)\right)}\right]$
- (iii)
- $\gimel {\odot}_{\mathrm{E}}{P}_{1}=\underset{\begin{array}{c}{a}_{1}\in m,\\ {b}_{1}\in n\end{array}}{{\displaystyle \cup}}\left[\frac{{\left(1+F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)\right)}^{\gimel}-{\left(1-F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)\right)}^{\gimel}}{{\left(1+F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)\right)}^{\gimel}+{\left(1-F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)\right)}^{\gimel}}\right]$
- (iv)
- ${P}_{1}{}^{\gimel}=\underset{\begin{array}{c}{a}_{1}\in m,\\ {b}_{1}\in n\end{array}}{{\displaystyle \cup}}\left[\frac{2{\left(F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)\right)}^{\gimel}}{{\left(2-F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)\right)}^{\gimel}+{\left(F\left({S}_{{a}_{1}}\u2329{O}_{{b}_{1}}\u232a\right)\right)}^{\gimel}}\right]$

- (1)
- If $S\left({P}_{1}\right)<S\left({P}_{2}\right)$, then ${P}_{1}<{P}_{2}$
- (2)
- If $S\left({P}_{1}\right)>S\left({P}_{2}\right)$, then ${P}_{1}>{P}_{2}$
- (3)
- If $S\left({P}_{1}\right)=S\left({P}_{2}\right)$, then ${P}_{1}\sim {P}_{2}$

**Theorem 1.**

**Proof.**

**Theorem 2**

**Proof.**

**Proof.**

- (i)
- ${P}_{1}{\oplus}_{E}{P}_{2}={P}_{2}{\oplus}_{E}{P}_{1;}$
- (ii)
- ${P}_{1}{\otimes}_{E}{P}_{2}={P}_{2}{\otimes}_{E}{P}_{1;}$
- (iii)
- $\gimel \left({P}_{1}{\oplus}_{E}{P}_{2}\right)=\gimel {P}_{1}{\oplus}_{E}\gimel {P}_{2};\gimel 0$
- (iv)
- ${\left({P}_{1}{\otimes}_{E}{P}_{2}\right)}^{\gimel}={P}_{1}{}^{\gimel}{\otimes}_{E}{P}_{2}{}^{\gimel};\gimel 0$

## 4. TOPSIS Method for Calculating Conditional Probability

## 5. A Novel DHHLDTRS Model Base on Einstein Aggregation Operators

- $\left(P\right)$ If $S\left(R\left({z}_{P}|{A}_{r}\right)\right)\le S\left(\left({z}_{B}|{A}_{r}\right)\right)$, and $S\left(R\left({z}_{P}|{A}_{r}\right)\right)\le S\left(R\left({z}_{N}|{A}_{r}\right)\right)$, decide ${A}_{r}\in POS\left(A\right)$;
- $\left(B\right)$ If $S\left(\left({z}_{B}|{A}_{r}\right)\right)\le S\left(R\left({z}_{P}|{A}_{r}\right)\right)$, and $S\left(\left({z}_{B}|{A}_{r}\right)\right)\le S\left(R\left({z}_{N}|{A}_{r}\right)\right)$, decide ${A}_{r}\in NEG\left(A\right)$; and
- $\left(N\right)$ If $S\left(\left({z}_{N}|{A}_{r}\right)\right)\le S\left(R\left({z}_{P}|{A}_{r}\right)\right)$, and $S\left(\left({z}_{N}|{A}_{r}\right)\right)\le S\left(R\left({z}_{B}|{A}_{r}\right)\right)$, decide ${A}_{r}\in BND\left(A\right)$.

#### The TWDs Procedure

## 6. Problem Statement

## 7. Discussion and Comparison

#### 7.1. Using the GRA Method for TWDs

#### 7.2. MADM Methods Are Used to Computed the Conditional Probability

#### 7.3. Discussion on the Advantages and Limitations

- (I)
- The FHHLT and SHHLT that make up the DHHLE provide for more flexible expressions of the estimation of DMs throughout the TWD procedure.
- (II)
- It is helpful to apply the TWD approach based on the DHHLEs when dealing with decision-making issues.
- (III)
- The conditional probability was calculated by using the TOPSIS method.

- (I)
- This work does not examine the scenario of group decisions or take into account varied expert weights in order to simplify the computation.
- (II)
- In the future, we will expand this concept to group choices and make it much more functional.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Yoon, K.P.; Hwang, C.L. Multiple Attribute Decision Making: An Introduction; Sage Publications: Thousand Oaks, CA, USA, 1995. [Google Scholar]
- Govindan, K.; Rajendran, S.; Sarkis, J.; Murugesan, P. Multi criteria decision making approaches for green supplier evaluation and selection: A literature review. J. Clean. Prod.
**2013**, 98, 66–83. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef][Green Version] - Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Atanassov, K.T. More on intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1989**, 33, 37–45. [Google Scholar] [CrossRef] - Zadeh, L.A. What is computing with words (CWW)? In Computing with Words; Springer: Berlin/Heidelberg, Germany, 2013; pp. 3–37. [Google Scholar]
- Xu, Z.; Wang, H. On the syntax and semantics of virtual linguistic terms for information fusion in decision making. Inf. Fusion
**2017**, 34, 43–48. [Google Scholar] [CrossRef] - Martinez, L.; Herrera, F. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst.
**2000**, 8, 746–752. [Google Scholar] [CrossRef][Green Version] - Xu, Z. A method based on linguistic aggregation operators for group decision making with linguistic preference relations. Inf. Sci.
**2004**, 166, 19–30. [Google Scholar] [CrossRef] - Gou, X.; Liao, H.; Xu, Z.; Herrera, F. Double hierarchy hesitant fuzzy linguistic term set and MULTIMOORA method: A case of study to evaluate the implementation status of haze controlling measures. Inf. Fusion
**2017**, 38, 22–34. [Google Scholar] [CrossRef] - Gou, X.; Xu, Z.; Liao, H.; Herrera, F. Multiple criteria decision making based on distance and similarity measures under double hierarchy hesitant fuzzy linguistic environment. Comput. Ind. Eng.
**2018**, 126, 516–530. [Google Scholar] [CrossRef] - Wang, H.; Xu, Z.; Zeng, X.-J. Modeling complex linguistic expressions in qualitative decision making: An overview. Knowl.-Based Syst.
**2018**, 144, 174–187. [Google Scholar] [CrossRef][Green Version] - Torra, V.; Narukawa, Y. On hesitant fuzzy sets and decision. In Proceedings of the 2009 IEEE International Conference on Fuzzy Systems, Jeju Island, Republic of Korea, 20–24 August 2009; pp. 1378–1382. [Google Scholar]
- Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 529–539. [Google Scholar] [CrossRef] - Ju, Y.; Liu, X.; Yang, S. Interval-valued dual hesitant fuzzy aggregation operators and their applications to multiple attribute decision making. J. Intell. Fuzzy Syst.
**2014**, 27, 1203–1218. [Google Scholar] [CrossRef] - Zhu, B.; Xu, Z.; Xia, M. Dual hesitant fuzzy sets. J. Appl. Math.
**2012**, 2012, 879629. [Google Scholar] [CrossRef][Green Version] - Wei, G.; Alsaadi, F.E.; Hayat, T.; Alsaedi, A. Hesitant Fuzzy Linguistic Arithmetic Aggregation Operators in Multiple Attribute Decision Making. Iran. J. Fuzzy Syst.
**2016**, 13, 1–16. [Google Scholar] - Zhao, X.; Wei, G. Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making. Knowl.-Based Syst.
**2013**, 37, 472–479. [Google Scholar] [CrossRef] - Wang, W.; Liu, X. Intuitionistic fuzzy geometric aggregation operators based on einstein operations. Int. J. Intell. Syst.
**2011**, 26, 1049–1075. [Google Scholar] [CrossRef] - Wang, W.; Liu, X. Intuitionistic Fuzzy Information Aggregation Using Einstein Operations. IEEE Trans. Fuzzy Syst.
**2012**, 20, 923–938. [Google Scholar] [CrossRef] - Wang, W.; Liu, X. Interval-valued intuitionistic fuzzy hybrid weighted averaging operator based on Einstein operation and its application to decision making. J. Intell. Fuzzy Syst.
**2013**, 25, 279–290. [Google Scholar] [CrossRef] - Wang, W.; Liu, X. The multi-attribute decision making method based on interval-valued intuitionistic fuzzy Einstein hybrid weighted geometric operator. Comput. Math. Appl.
**2013**, 66, 1845–1856. [Google Scholar] [CrossRef] - Zhang, S.; Yu, D. Some geometric Choquet aggregation operators using Einstein operations under intuitionistic fuzzy environment. J. Intell. Fuzzy Syst.
**2014**, 26, 491–500. [Google Scholar] [CrossRef] - Zhao, X.; Li, Q.; Wei, G. Model for multiple attribute decision making based on the Einstein correlated information fusion with hesitant fuzzy information. J. Intell. Fuzzy Syst.
**2014**, 26, 3057–3064. [Google Scholar] [CrossRef] - Zulqarnain, R.M.; Siddique, I.; Jarad, F.; Hamed, Y.S.; Abualnaja, K.M.; Iampan, A. Einstein Aggregation Operators for Pythagorean Fuzzy Soft Sets with Their Application in Multiattribute Group Decision-Making. J. Funct. Spaces
**2022**, 2022, 1358675. [Google Scholar] [CrossRef] - Huang, H.; Xu, H.; Chen, F.; Zhang, C.; Mohammadzadeh, A. An Applied Type-3 Fuzzy Logic System: Practical Matlab Simulink and M-Files for Robotic, Control, and Modeling Applications. Symmetry
**2023**, 15, 475. [Google Scholar] [CrossRef] - Wei, G.; Zhao, X. Induced hesitant interval-valued fuzzy Einstein aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst.
**2013**, 24, 789–803. [Google Scholar] [CrossRef] - Abdullah, S.; Al-Shomrani, M.M.; Liu, P.; Ahmad, S. A new approach to three-way decisions making based on fractional fuzzy decision-theoretical rough set. Int. J. Intell. Syst.
**2022**, 37, 2428–2457. [Google Scholar] [CrossRef] - Abosuliman, S.S.; Abdullah, S.; Qiyas, M. Three-Way Decisions Making Using Covering Based Fractional Orthotriple Fuzzy Rough Set Model. Mathematics
**2020**, 8, 1121. [Google Scholar] [CrossRef] - Liu, D.; Yao, Y.; Li, T. Three-way investment decisions with decision-theoretic rough sets. Int. J. Comput. Intell. Syst.
**2011**, 4, 66–74. [Google Scholar] - Yao, Y. Three-way decision: An interpretation of rules in rough set theory. In International Conference on Rough Sets and Knowledge Technology; Springer: Berlin/Heidelberg, Germany, 2009; pp. 642–649. [Google Scholar]
- Yao, Y. Three-way decisions with probabilistic rough sets. Inf. Sci.
**2010**, 180, 341–353. [Google Scholar] [CrossRef][Green Version] - Skowron, A.; Komorowski, J.; Pawlak, Z.; Polkowski, L. Rough sets perspective on data and knowledge. In Handbook of Data Mining and Knowledge Discovery; Springer: New York, NY, USA, 2002; pp. 134–149. [Google Scholar]
- Chien, Y. Pattern classification and scene analysis. IEEE Trans. Autom. Control
**1974**, 19, 462–463. [Google Scholar] [CrossRef] - Hu, J.; Yang, Y.; Chen, X. A Novel TODIM Method-Based Three-Way Decision Model for Medical Treatment Selection. Int. J. Fuzzy Syst.
**2017**, 20, 1240–1255. [Google Scholar] [CrossRef] - Wang, P.; Zhang, P.; Li, Z. A three-way decision method based on Gaussian kernel in a hybrid information system with images: An application in medical diagnosis. Appl. Soft Comput.
**2019**, 77, 734–749. [Google Scholar] [CrossRef] - Li, X.; Huang, X. A Novel Three-Way Investment Decisions Based on Decision-Theoretic Rough Sets with Hesitant Fuzzy Information. Int. J. Fuzzy Syst.
**2020**, 22, 2708–2719. [Google Scholar] [CrossRef] - Li, X.; Wang, H.; Xu, Z. Work Resumption After Epidemic Using Three-Way Decisions. Int. J. Fuzzy Syst.
**2021**, 23, 630–641. [Google Scholar] [CrossRef] - Ye, J.; Zhan, J.; Xu, Z. A novel decision-making approach based on three-way decisions in fuzzy information systems. Inf. Sci.
**2020**, 541, 362–390. [Google Scholar] [CrossRef] - Liang, D.; Liu, D.; Pedrycz, W.; Hu, P. Triangular fuzzy decision-theoretic rough sets. Int. J. Approx. Reason.
**2013**, 54, 1087–1106. [Google Scholar] [CrossRef] - Liang, D.; Liu, D. Deriving three-way decisions from intuitionistic fuzzy decision-theoretic rough sets. Inf. Sci.
**2015**, 300, 28–48. [Google Scholar] [CrossRef] - Liang, D.; Pedrycz, W.; Liu, D.; Hu, P. Three-way decisions based on decision-theoretic rough sets under linguistic assessment with the aid of group decision making. Appl. Soft Comput.
**2015**, 29, 256–269. [Google Scholar] [CrossRef] - Jia, F.; Liu, P. A novel three-way decision model under multiple-criteria environment. Inf. Sci.
**2018**, 471, 29–51. [Google Scholar] [CrossRef] - Liang, D.; Xu, Z.; Liu, D.; Wu, Y. Method for three-way decisions using ideal TOPSIS solutions at Pythagorean fuzzy information. Inf. Sci.
**2018**, 435, 282–295. [Google Scholar] [CrossRef] - Yang, B.; Yuan, J.; Ye, Z. Risk assessment of coal mining above confined aquifer based on maximizing deviation in a GIS environment. Arab. J. Geosci.
**2018**, 11, 299. [Google Scholar] [CrossRef] - Wei, G.-W. Gray relational analysis method for intuitionistic fuzzy multiple attribute decision making. Expert Syst. Appl.
**2011**, 38, 11671–11677. [Google Scholar] [CrossRef] - Wang, T.; Li, H.; Zhou, X.; Liu, D.; Huang, B. Three-way decision based on third-generation prospect theory with Z-numbers. Inf. Sci.
**2021**, 569, 13–38. [Google Scholar] [CrossRef] - Lin, F.; Xu, Y.; Yang, Y.; Ma, H. A Spatial-Temporal Hybrid Model for Short-Term Traffic Prediction. Math. Probl. Eng.
**2019**, 2019, 4858546. [Google Scholar] [CrossRef] - Susmaga, R.; Szczȩch, I.; Zielniewicz, P.; Brzezinski, D. MSD-space: Visualizing the inner-workings of TOPSIS aggregations. Eur. J. Oper. Res.
**2022**, 308, 229–242. [Google Scholar] [CrossRef] - Corrente, S.; Tasiou, M. A robust TOPSIS method for decision making problems with hierarchical and non-monotonic criteria. Expert Syst. Appl.
**2023**, 214, 119045. [Google Scholar] [CrossRef] - Erwansyah, K.; Anwar, B.; Kusnasari, S. Implementation decision support system in determining suppliers using wsm method. J. Mantik
**2023**, 6, 3793–3799. [Google Scholar] - Chourabi, Z.; Khedher, F.; Babay, A.; Cheikhrouhou, M. Multi-criteria decision making in workforce choice using AHP, WSM and WPM. J. Text. Inst.
**2018**, 110, 1092–1101. [Google Scholar] [CrossRef] - Hua, Z.; Jing, X. A generalized Shapley index-based interval-valued Pythagorean fuzzy PROMETHEE method for group decision-making. Soft Comput.
**2023**. [Google Scholar] [CrossRef] - Tariq, S.; Iqtadar, H.; Muhammad, A.G.; Hasan, M. Statistical analysis of S-box in image encryption applications based on majority logic criterion. Int. J. Phys. Sci.
**2011**, 6, 4110–4127. [Google Scholar] - Abdullah, S.; Ayub, S.; Hussain, I.; Bedregal, B.; Khan, M.Y. Analyses of S-boxes based on interval valued intuitionistic fuzzy sets and image encryption. Int. J. Comput. Intell. Syst.
**2017**, 10, 851. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**The suggested procedure. The double hierarchy hesitant linguistic term information system (DHHLTIS); the double hierarchy hesitant Einstein weights averaging (DHHLEWA); the technique for order of preference by similarity to ideal solution (TOPSIS) method.

**Figure 3.**The comparison between our proposed technique and GRA, BP, and weighted aggregation methods.

$\mathbf{Z}\left(\mathit{P}\right)$ | ${\mathbf{Z}}^{\mathit{c}}\left(\mathit{N}\right)$ | |
---|---|---|

${z}_{P}$ | ${p}_{\lambda PP}=F\left({S}_{mPP}^{k}\u2329{O}_{nPP}^{k}\u232a\right)$ | ${p}_{\lambda PN}=F\left({S}_{mPN}^{k}\u2329{O}_{nPN}^{k}\u232a\right)$ |

${z}_{B}$ | ${p}_{\lambda BP}=F\left({S}_{mBP}^{k}\u2329{O}_{nBP}^{k}\u232a\right)$ | ${p}_{\lambda BN}=F\left({S}_{mBN}^{k}\u2329{O}_{nBN}^{k}\u232a\right)$ |

${z}_{N}$ | ${p}_{\lambda NP}=F\left({S}_{mNP}^{k}\u2329{O}_{nNP}^{k}\u232a\right)$ | ${p}_{\lambda NN}=F\left({S}_{mNN}^{k}\u2329{O}_{nNN}^{k}\u232a\right)$ |

The Popular S-The Popular S-Boxes | ||
---|---|---|

${\mathit{A}}_{\mathbf{1}}$ | Advanced encryption standard | AES |

${\mathit{A}}_{\mathbf{2}}$ | Affine-power-affine | APA |

${\mathit{A}}_{\mathbf{3}}$ | Gray | G |

${\mathit{A}}_{\mathbf{4}}$ | Lui J | LJ |

${\mathit{A}}_{\mathbf{5}}$ | Residue Prime | RP |

${\mathit{A}}_{\mathbf{6}}$ | S8 | S8 |

Criteria | The Characteristics of the Popular S-Boxes | Calculated by |
---|---|---|

${\mathit{C}}_{\mathit{1}}$ | Correlation analysis | $Z=\frac{{{\displaystyle \sum}}_{\alpha}{{\displaystyle \sum}}_{\beta}\left({B}_{\alpha \beta}-\mathcal{B}\right)\left({A}_{\alpha \beta}-\mathcal{A}\right)}{\sqrt{{{\displaystyle \sum}}_{\alpha}{{\displaystyle \sum}}_{\beta}{\left({B}_{\alpha \beta}-\mathcal{B}\right)}^{2}{{\displaystyle \sum}}_{\alpha}{{\displaystyle \sum}}_{\beta}{\left({A}_{\alpha \beta}-\mathcal{A}\right)}^{2}}}$ |

${\mathit{C}}_{\mathit{2}}$ | Entropy analysis | $\mathrm{E}=-{\displaystyle {\displaystyle \sum}_{\mathrm{r}=1}^{\mathrm{k}}}\mathrm{g}\left({\mathrm{x}}_{\mathrm{r}}\right){\mathrm{log}}_{\mathrm{b}}\mathrm{g}\left({\mathrm{x}}_{\mathrm{r}}\right)$ |

${\mathit{C}}_{\mathit{3}}$ | Contrast analysis | $C={\displaystyle {\displaystyle \sum}_{r,j}}{\left|r-j\right|}^{2}g\left(r,j\right)$ |

${\mathit{C}}_{\mathit{4}}$ | Homogeneity analysis | $H={\displaystyle {\displaystyle \sum}_{r,j}}\frac{g\left(r,j\right)}{1+\left|i-j\right|}$ |

${\mathit{C}}_{\mathit{5}}$ | Energy analysis | $\mathrm{En}={\displaystyle {\displaystyle \sum}_{r,j}}g{\left(i,j\right)}^{2}$ |

${\mathbf{C}}_{1}$ | ${\mathbf{C}}_{2}$ | ${\mathbf{C}}_{3}$ | ${\mathbf{C}}_{4}$ | ${\mathbf{C}}_{5}$ | |
---|---|---|---|---|---|

${\mathbf{A}}_{1}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{1}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{0}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{0}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-1}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{-2}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{3}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{1}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-2}\u232a\right)\end{array}\right\}$ |

${\mathbf{A}}_{2}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{3}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{0}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{3}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{-2}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-1}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{2}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{0}\u2329{\mathrm{O}}_{1}\u232a\right)\right\}$ |

${\mathbf{A}}_{3}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{-2}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-1}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{0}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{-1}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{-2}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{2}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{0}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{-3}\u232a\right)\end{array}\right\}$ |

${\mathbf{A}}_{4}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{0}\u2329{\mathrm{O}}_{-2}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-2}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{0}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{-1}\u2329{\mathrm{O}}_{0}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{-2}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-1}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-3}\u232a\right)\right\}$ |

${\mathbf{A}}_{5}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{0}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-1}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{1}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{0}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-3}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-1}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{2}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{0}\u232a\right)\end{array}\right\}$ |

${\mathbf{A}}_{6}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{2}\u232a\right)\right\}$ | $\left\{\begin{array}{c}F\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{0}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-1}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{1}\u2329{\mathrm{O}}_{2}\u232a\right)\right\}$ | $\left\{\begin{array}{c}\mathrm{F}\left({\mathrm{S}}_{2}\u2329{\mathrm{O}}_{-3}\u232a\right),\\ \mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{2}\u232a\right)\end{array}\right\}$ | $\left\{\mathrm{F}\left({\mathrm{S}}_{3}\u2329{\mathrm{O}}_{0}\u232a\right)\right\}$ |

$\mathbf{Z}\left(\mathbf{P}\right)$ | ${\mathbf{Z}}^{c}\left(N\right)$ | |
---|---|---|

${\mathbf{z}}_{\mathbf{P}}$ | $\left\{\begin{array}{c}F\left({S}_{-3}\u2329{O}_{-2}\u232a\right),\\ F\left({S}_{-1}\u2329{O}_{3}\u232a\right)\end{array}\right\}$ | $\left\{\begin{array}{c}F\left({S}_{3}\u2329{O}_{2}\u232a\right),\\ F\left({S}_{1}\u2329{O}_{-3}\u232a\right)\end{array}\right\}$ |

${\mathbf{z}}_{\mathbf{B}}$ | $\left\{\begin{array}{c}F\left({S}_{2}\u2329{O}_{0}\u232a\right),\\ F\left({S}_{1}\u2329{O}_{2}\u232a\right)\end{array}\right\}$ | $\left\{\begin{array}{c}F\left({S}_{-2}\u2329{O}_{0}\u232a\right),\\ F\left({S}_{-1}\u2329{O}_{-2}\u232a\right)\end{array}\right\}$ |

${\mathbf{z}}_{\mathbf{N}}$ | $\left\{\begin{array}{c}F\left({S}_{3}\u2329{O}_{1}\u232a\right),\\ F\left({S}_{2}\u2329{O}_{-1}\u232a\right)\end{array}\right\}$ | $\left\{\begin{array}{c}F\left({S}_{-3}\u2329{O}_{-1}\u232a\right),\\ F\left({S}_{-2}\u2329{O}_{1}\u232a\right)\end{array}\right\}$ |

${\mathbf{A}}_{1}$ | ${\mathbf{A}}_{2}$ | ${\mathbf{A}}_{3}$ | ${\mathbf{A}}_{4}$ | ${\mathbf{A}}_{5}$ | ${\mathbf{A}}_{6}$ | |
---|---|---|---|---|---|---|

${d}_{r}^{+}$ | 0.0955 | 0.1765 | 0.1247 | 0.2492 | 0.1184 | 0.1357 |

${d}_{r}^{-}$ | 0.1488 | 0.2299 | 0.2060 | 0.0687 | 0.2491 | 0.2814 |

${D}_{r}$ | 0.6090 | 0.5657 | 0.6229 | 0.2162 | 0.6582 | 0.6747 |

$\mathrm{Pr}\left(Z|{A}_{r}\right)$ | 0.6090 | 0.5657 | 0.6229 | 0.2162 | 0.6582 | 0.6747 |

${\mathbf{A}}_{1}$ | ${\mathbf{A}}_{2}$ | ${\mathbf{A}}_{3}$ | ${\mathbf{A}}_{4}$ | ${\mathbf{A}}_{5}$ | ${\mathbf{A}}_{6}$ | |
---|---|---|---|---|---|---|

$\mathit{S}\left(\mathit{R}\left({\mathit{z}}_{\mathit{P}}|{\mathit{A}}_{\mathit{r}}\right)\right)$ | 0.0271 | 0.0220 | 0.0911 | 0.1729 | 0.1066 | 0.0686 |

$\mathit{S}\left(\mathit{R}\left({\mathit{z}}_{\mathit{B}}|{\mathit{A}}_{\mathit{r}}\right)\right)$ | 0.3271 | 0.2555 | 0.3496 | −0.3092 | 0.3121 | 0.2441 |

$\mathit{S}\left(\mathit{R}\left({\mathit{z}}_{\mathit{N}}|{\mathit{A}}_{\mathit{r}}\right)\right)$ | 0.3439 | 0.2494 | 0.2793 | −0.3629 | 0.3506 | 0.3827 |

V | ${\mathbf{A}}_{1}$ | ${\mathbf{A}}_{2}$ | ${\mathbf{A}}_{3}$ | ${\mathbf{A}}_{4}$ | ${\mathbf{A}}_{5}$ | ${\mathbf{A}}_{6}$ |
---|---|---|---|---|---|---|

${G}_{r}^{+}$ | 0.3116 | 0.3176 | 0.3236 | 0.1841 | 0.3192 | 0.3752 |

${G}_{r}^{-}$ | 0.2899 | 0.1571 | 0.1132 | 0.6683 | 0.246 | 0.1126 |

${H}_{r}$ | 0.5180 | 0.6690 | 0.7408 | 0.2160 | 0.5648 | 0.7692 |

$\mathrm{Pr}\left(Z|{A}_{r}\right)$ | 0.5180 | 0.6690 | 0.7408 | 0.2160 | 0.5648 | 0.7692 |

V | ${\mathbf{A}}_{1}$ | ${\mathbf{A}}_{2}$ | ${\mathbf{A}}_{3}$ | ${\mathbf{A}}_{4}$ | ${\mathbf{A}}_{5}$ | ${\mathbf{A}}_{6}$ |
---|---|---|---|---|---|---|

$S\left(R\left({z}_{P}|{A}_{r}\right)\right)$ | 0.1152 | 0.0818 | −0.0923 | 0.1732 | 0.0238 | −0.16571 |

$S\left(R\left({z}_{B}|{A}_{r}\right)\right)$ | 0.1719 | 0.2352 | 0.3440 | −0.3096 | 0.2540 | 0.3854 |

$S\left(R\left({z}_{N}|{A}_{r}\right)\right)$ | 0.2293 | 0.3717 | 0.4099 | −0.3636 | 0.2474 | 0.4583 |

V | $\mathbf{P}\mathbf{r}\left(\mathbf{Z}|{\mathbf{A}}_{1}\right)$ | $\mathbf{P}\mathbf{r}\left(\mathbf{Z}|{\mathbf{A}}_{2}\right)$ | $\mathbf{P}\mathbf{r}\left(\mathbf{Z}|{\mathbf{A}}_{3}\right)$ | $\mathbf{P}\mathbf{r}\left(\mathbf{Z}|{\mathbf{A}}_{4}\right)$ | $\mathbf{P}\mathbf{r}\left(\mathbf{Z}|{\mathbf{A}}_{5}\right)$ | $\mathbf{P}\mathbf{r}\left(\mathbf{Z}|{\mathbf{A}}_{6}\right)$ |
---|---|---|---|---|---|---|

GRA | 0.5180 | 0.6690 | 0.7408 | 0.2160 | 0.5648 | 0.7692 |

BP method | 0.2349 | 0.2704 | 0.2071 | 0.0448 | 0.2872 | 0.3565 |

Weighted aggregation method | 0.3718 | 0.3905 | 0.4084 | 0.3091 | 0.4179 | 0.4199 |

Our suggested method | 0.6090 | 0.5657 | 0.6229 | 0.2162 | 0.6582 | 0.6747 |

V | Ranking |
---|---|

GRA | ${A}_{6}>{A}_{3}>{A}_{2}>{A}_{5}>{A}_{1}>{A}_{4}$ |

BP method | ${A}_{6}>{A}_{5}>{A}_{2}>{A}_{1}>{A}_{3}>{A}_{4}$ |

Weighted aggregation method | ${A}_{6}>{A}_{5}>{A}_{3}>{A}_{1}>{A}_{2}>{A}_{4}$ |

Our proposed method | ${A}_{6}>{A}_{5}>{A}_{3}>{A}_{2}>{A}_{1}>{A}_{4}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abdullah, S.; Almagrabi, A.O.; Ullah, I.
A New Approach to Artificial Intelligent Based Three-Way Decision Making and Analyzing S-Box Image Encryption Using TOPSIS Method. *Mathematics* **2023**, *11*, 1559.
https://doi.org/10.3390/math11061559

**AMA Style**

Abdullah S, Almagrabi AO, Ullah I.
A New Approach to Artificial Intelligent Based Three-Way Decision Making and Analyzing S-Box Image Encryption Using TOPSIS Method. *Mathematics*. 2023; 11(6):1559.
https://doi.org/10.3390/math11061559

**Chicago/Turabian Style**

Abdullah, Saleem, Alaa O. Almagrabi, and Ihsan Ullah.
2023. "A New Approach to Artificial Intelligent Based Three-Way Decision Making and Analyzing S-Box Image Encryption Using TOPSIS Method" *Mathematics* 11, no. 6: 1559.
https://doi.org/10.3390/math11061559