An Integrated Group Decision-Making Framework for the Evaluation of Artificial Intelligence Cloud Platforms Based on Fractional Fuzzy Sets
<p>Detailed stepwise algorithm of proposed FF-VIKOR Approach.</p> "> Figure 2
<p>Detailed Algorithm of the Proposed FF-Extended TOPSIS Approach.</p> "> Figure 3
<p>Outputs of the Proposed FF-VIKOR Approach.</p> "> Figure 4
<p>Outputs of the Proposed FF-Extended TOPSIS Approach.</p> ">
Abstract
:1. Introduction
1.1. Related Studies
1.2. Motivation of This Study
- To define new Einstein operational laws in an FFS environment.
- On the basis of these operational laws, define a new set of Einstein geometric aggregation operators (AoPs) in the same environment.
- To develop a new approach to the extended TOPSIS method and VIKOR method in an FFS environment.
- To verify and apply the proposed method to a real-life decision-making issue, as discussed below.
1.3. Contribution of This Study
- We defined new Einstein operational laws in an FFS environment.
- On the basis of these operational laws, we defined a new Einstein geometric aggregation operator (AoPs) in an FFS environment.
- The FF–VIKOR technique and FF-Extended TOPSIS approach were formulated for MCGDM issues in an FFS environment.
- We verified our proposed method with a real-life decision-making issue for the evaluation of artificial intelligence cloud platforms.
2. Fundamental Knowledge
- Case 1.
- If , then the FF set becomes the IF set.with subject for all .
- Case 2.
- If , then the FF set becomes the PyFS set.P with the condition that 01 for all .
- Case 3.
- If , then FFS reduces to Farmatean fuzzy set.for all .
- Case 4.
- If , then FFS reduces to 𝓺-ROFS.for all .
3. Fractional Fuzzy Einstein norms and Aggregation Operators (AoPs)
3.1. Fractional Fuzzy Einstein Operational Laws
3.2. Fractional Fuzzy Einstein Geometric Aggregation Operators
3.3. Fractional Fuzzy Einstein Weighted Geometric Aggregation Operator (AoP)
3.4. Fractional Fuzzy Einstein Order Weighted Geometric Aggregation Operator (AoP)
3.5. Fractional Fuzzy Einstein Hybrid Geometric Aggregation Operator (AoP)
- 1.1
- Idempotency: if = , then
- 1.2
- Boundedness:
- 1.3
- Monotonicity: let = be a collection of fractional fuzzy figures and , for all , then
4. Methodology-I Proposed VIKOR Method for FFS Information
5. Methodology-II Proposed Extended TOPSIS Method for FFS Information
6. Real Life Decision Making Problem
Case Study
7. Numerical Result of the Proposed VIKOR Method for FFS information
8. Numerical Results of Our Proposed Extended TOPSIS Method for FFS information
9. Comparative Analysis and Discussion
10. Conclusions and Future Research Directions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Talpur, N.; Abdulkadir, S.J.; Alhussian, H.; Hasan, M.H.; Aziz, N.; Bamhdi, A. Deep Neuro-Fuzzy System application trends, challenges, and future perspectives: A systematic survey. Artif. Intell. Rev. 2023, 56, 865–913. [Google Scholar] [CrossRef] [PubMed]
- George, N.; Kadan, A.B.; Vijayan, V.P. Multi-objective load balancing in cloud infrastructure through fuzzy based decision making and genetic algorithm based optimization. IAES Int. J. Artif. Intell. 2023, 12, 678–685. [Google Scholar] [CrossRef]
- Bansal, V.; Bhardwaj, A.; Singh, J.; Verma, D.; Tiwari, M.; Siddi, S. Using Artificial Intelligence to Integrate Machine Learning, Fuzzy Logic, and The IOT as A Cybersecurity System. In Proceedings of the 2023 3rd International Conference on Advance Computing and Innovative Technologies in Engineering (ICACITE), Greater Noida, India, 12–13 May 2023; IEEE: Toulouse, France, 2023; pp. 762–769. [Google Scholar]
- Aksjonov, A.; Nedoma, P.; Vodovozov, V.; Petlenkov, E.; Herrmann, M. Detection and evaluation of driver dis-traction using machine learning and fuzzy logic. IEEE Trans. Intell. Transp. Syst. 2018, 20, 2048–2059. [Google Scholar] [CrossRef]
- Peng, B.; Zhou, J.; Peng, D. Cloud model based approach to group decision making with uncertain pure linguistic information. J. Intell. Fuzzy Syst. 2017, 32, 1959–1968. [Google Scholar] [CrossRef]
- Shirazi, S.A.R.; Khan, A.H.; Rasool, S.; Anwar, A.; Ammar, M. Load Balancing of Cloud Computing Service Model Empowered with Fuzzy Logic. Sir Syed Res. J. Eng. Technol. 2023, 13, 10–16. [Google Scholar] [CrossRef]
- Yuan, X.; Shi, C.; Wang, Z. The Optimization of Hospital Financial Management Based on Cloud Technology and Wireless Network Technology in the Context of Artificial Intelligence. Wirel. Commun. Mob. Comput. 2022, 2022, 9998311. [Google Scholar] [CrossRef]
- Zadeh, L. Fuzzy sets. Inform. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
- Atanassov, K.T. On Intuitionistic Fuzzy Sets Theory; Springer: Berlin/Heidelberg, Germany, 2012; Volume 283. [Google Scholar]
- Atanassov, K. Intuitionistic fuzzy sets. Int. J. Bioautom. 2016, 20, 1. [Google Scholar]
- Yager, R.R. Pythagorean fuzzy subsets. In Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), Edmonton, AB, Canada, 24–28 June 2013; IEEE: Toulouse, France, 2013; pp. 57–61. [Google Scholar]
- Yager, R.R. Properties and applications of Pythagorean fuzzy sets. In Imprecision and Uncertainty in Information Rep-Resentation and Processing: New Tools Based on Intuitionistic Fuzzy Sets and Generalized Nets; Springer: Berlin/Heidelberg, Germany, 2016; pp. 119–136. [Google Scholar]
- Yager, R.R. Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst. 2016, 25, 1222–1230. [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]
- Zhao, X.; Wei, G. Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multi-ple attribute decision making. Knowl. Based Syst. 2013, 37, 472–479. [Google Scholar] [CrossRef]
- Rahman, K.; Abdullah, S.; Ahmed, R.; Ullah, M. Pythagorean fuzzy Einstein weighted geometric aggregation operator and their application to multiple attribute group decision making. J. Intell. Fuzzy Syst. 2017, 33, 635–647. [Google Scholar] [CrossRef]
- Ali, Z.; Mahmood, T.; Ullah, K.; Khan, Q. Einstein geometric aggregation operators using a novel complex intervalvalued pythagorean fuzzy setting with application in green supplier chain management. Rep. Mech. Eng. 2021, 2, 105–134. [Google Scholar] [CrossRef]
- Zulqarnain, R.M.; Ali, R.; Awrejcewicz, J.; Siddique, I.; Jarad, F.; Iampan, A. Some Einstein Geometric Aggregation Operators for q-Rung Orthopair Fuzzy Soft Set with Their Application in MCDM. IEEE Access 2022, 10, 88469–88494. [Google Scholar] [CrossRef]
- Wu, X.; Ali, Z.; Mahmood, T.; Liu, P. A Multi-attribute Decision-Making Method with Complex q-Rung Orthopair Fuzzy Soft Information Based on Einstein Geometric Aggregation Operators. Int. J. Fuzzy Syst. 2023, 25, 2218–2233. [Google Scholar] [CrossRef]
- Bali, V.; Bali, S.; Gaur, D.; Rani, S.; Kumar, R. Commercial-off-the Shelf Vendor Selection: A Multi-Criteria Deci-sion-Making Approach Using Intuitionistic Fuzzy Sets and TOPSIS. Oper. Res. Eng. Sci. Theory Appl. 2023. [Google Scholar] [CrossRef]
- Luo, X.; Guo, S.; Du, B.; Guo, J.; Jiang, P.; Tan, T. Multi-criteria decision-making of manufacturing resources alloca-tion for complex product system based on intuitionistic fuzzy information entropy and TOPSIS. Complex Intell. Syst. 2023, 9, 5013–5032. [Google Scholar] [CrossRef]
- Zhang, X.; Xu, Z. Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets. Int. J. Intell. Syst. 2014, 29, 1061–1078. [Google Scholar] [CrossRef]
- Ak, M.F.; Gul, M. AHP–TOPSIS integration extended with Pythagorean fuzzy sets for information security risk analysis. Complex Intell. Syst. 2019, 5, 113–126. [Google Scholar] [CrossRef]
- Kumar, R.; Gandotra, N. Suman A novel pythagorean fuzzy entropy measure using MCDM application in preference of the advertising company with TOPSIS approach. Mater. Today Proc. 2023, 80, 1742–1746. [Google Scholar] [CrossRef]
- Nver, M.; Olgun, M. Continuous Function Valued q-Rung Orthopair Fuzzy Sets and an Extended TOPSIS. Int. J. Fuzzy Syst. 2023, 25, 2203–2217. [Google Scholar]
- Więckowski, J.; Kizielewicz, B.; Sałabun, W. Handling decision-making in Intuitionistic Fuzzy environment: PyIFDM package. SoftwareX 2023, 22, 101344. [Google Scholar] [CrossRef]
- Rani, P.; Mishra, A.R.; Pardasani, K.R.; Mardani, A.; Liao, H.; Streimikiene, D. A novel VIKOR approach based on entropy and divergence measures of Pythagorean fuzzy sets to evaluate renewable energy technologies in India. J. Clean. Prod. 2019, 238, 117936. [Google Scholar]
- Mahmood, T.; Ali, Z.; Naeem, M. Aggregation operators and CRITIC-VIKOR method for confidence complex q-rung orthopair normal fuzzy information and their applications. CAAI Trans. Intell. Technol. 2023, 8, 40–63. [Google Scholar] [CrossRef]
- Salsabeela, V.; Athira, T.M.; John, S.J.; Baiju, T. Multiple criteria group decision making based on q-rung orthopair fuzzy soft sets. Granul. Comput. 2023, 8, 1067–1080. [Google Scholar]
- Akram, M.; Shumaiza Rodríguez Alcantud, J.C. VIKOR Method with Trapezoidal Bipolar Fuzzy Sets. In Multi-Criteria Decision Making Methods with Bipolar Fuzzy Sets 2023; Springer Nature: Singapore, 2023; pp. 67–91. [Google Scholar]
- Opricovic, S.; Tzeng, G.-H. Multicriteria Scheduling in Water Resources Engineering Using Genetic Algorithm. In Computing in Civil and Building Engineering; ASCE: New York, NY, USA, 2000; pp. 1434–1441. [Google Scholar]
- Khan, M.S.A.; Abdullah, S. Interval-valued Pythagorean fuzzy GRA method for multiple-attribute decision mak-ing with incomplete weight information. Int. J. Intell. Syst. 2018, 33, 1689–1716. [Google Scholar] [CrossRef]
DMs | Responsibilities | Major | Association | Years of Experience | Level of Education |
---|---|---|---|---|---|
1 | Enterprise owner | IT Supervision and Software Engineer | Member of Council | 20 | Postdoc |
2 | AI Developer | IT Supervision and Big Data Expert | Member of Council | 14 | PhD |
3 | Experienced Big Data Expert/Architect | IT Supervision and Big Data Scientist | Member of Council | 12 | PhD |
Alternatives | ||||||
---|---|---|---|---|---|---|
(0.8,0.5) | (0.9,0.3) | (0.7,0.5) | (0.2,0.6) | (0.2,0.4) | (0.3,0.2) | |
(0.3,0.7) | (0.7,0.4) | (0.1,0.4) | (0.3,0.4) | (0.4,0.3) | (0.6,0.4) | |
(0.4,0.1) | (0.2,0.6) | (0.3,0.8) | (0.6,0.2) | (0.3,0.4) | (0.4,0.1) | |
(0.1,0.6) | (0.5,0.8) | (0.6,0.4) | (0.2,0.8) | (0.4,0.1) | (0.1,0.9) | |
(0.3,0.9) | (0.1,0.3) | (0.2,0.6) | (0.3,0.5) | (0.4,0.5) | (0.5,0.7) |
Alternatives | ||||||
---|---|---|---|---|---|---|
(0.7,0.4) | (0.8,0.5) | (0.6,0.9) | (0.1,0.4) | (0.1,0.2) | (0.3,0.7) | |
(0.4,0.2) | (0.4,0.4) | (0.6,0.2) | (0.5,0.7) | (0.5,0.9) | (0.2,0.4) | |
(0.5,0.4) | (0.2,0.2) | (0.4,0.1) | (0.3,0.3) | (0.4,0.2) | (0.4,0.2) | |
(0.1,0.9) | (0.3,0.1) | (0.2,0.4) | (0.4,0.7) | (0.3,0.5) | (0.5,0.1) | |
(0.8,0.3) | (0.1,0.5) | (0.8,0.7) | (0.2,0.1) | (0.7,0.4) | (0.6,0.7) |
Alternatives | ||||||
---|---|---|---|---|---|---|
(0.6,0.3) | (0.3,0.4) | (0.9,0.3) | (0.1,0.8) | (0.3,0.4) | (0.7,0.1) | |
(0.2,0.) | (0.5,0.2) | (0.7,0.4) | (0.2,0.6) | (0.4,0.6) | (0.2,0.9) | |
(0.3,0.1) | (0.7,0.6) | (0.3,0.7) | (0.3,0.4) | (0.4,0.1) | (0.3,0.2) | |
(0.6,0.9) | (0.1,0.8) | (0.4,0.1) | (0.4,0.3) | (0.2,0.6) | (0.4,0.5) | |
(0.1,0.5) | (0.4,0.3) | (0.1,0.3) | (0.6,0.4) | (0.4,0.4) | (0.9,0.3) |
Alternatives | ||||||
---|---|---|---|---|---|---|
(0.533,0.394) | (0.501,0.382) | (0.546,0.680) | (0.124,0.519) | (0.178,0.309) | (0.369,0.462) | |
(0.276,0.502) | (0.445,0.342) | (0.341,0.309) | (0.296,0.539) | (0.387,0.673) | (0.276,0.472) | |
(0.359,0.258) | (0.296,0.446) | (0.311,0.575) | (0.348,0.269) | (0.338,0.269) | (0.337,0.157) | |
(0.184,0.745) | 0.239,0.590) | (0.338,0.337) | (0.303,0.653) | (0.275,0.394) | (0.268,0.641) | |
(0.286,0.654) | (0.158,0.368) | (0.254,0.564) | (0.314,0.347) | (0.425.0.407) | (0.517,0.603) |
Criteria | ||||||
---|---|---|---|---|---|---|
0.1527 | 0.2038 | 0.1022 | 0.1494 | 0.2482 | 0.1437 |
(0.532,0.258) | (0.501,0.342) | (0.546,0.309) | (0.348,0.269) | (0.425,0.269) | (0.517,0.157) | |
(0.183,0.745) | (0.158,0.590) | (0.254,0.680) | (0.124,0.653) | (0.178,0.673) | (0.286,0.641) |
0.378 | 0.655 | 0.359 | 0.763 | 0.524 |
0.101 | 0.225 | 0.114 | 0.193 | 0.135 |
0.004 | 0.951 | 0.090 | 0.765 | 0.283 |
Alternatives |
Alternatives | ||||||
---|---|---|---|---|---|---|
0.075 | 0.037 | 0.341 | 0.659 | 0.114 | 0.998 | |
0.206 | 0.053 | 0.129 | 0.193 | 0.351 | 0.203 | |
0.116 | 0.130 | 0.247 | 0.062 | 0.075 | 0.121 | |
0.486 | 0.241 | 0.130 | 0.324 | 0.106 | 0.362 | |
0.352 | 0.149 | 0.246 | 0.070 | 0.084 | 0.295 |
Alternatives | ||||||
---|---|---|---|---|---|---|
0.418 | 0.229 | 0.143 | 0.159 | 0.003 | 0.002 | |
0.307 | 0.235 | 0.348 | 0.141 | 0.065 | 0.185 | |
0.469 | 0.150 | 0.133 | 0.334 | 0.359 | 0.350 | |
0.007 | 0.030 | 0.333 | 0.040 | 0.290 | 0.017 | |
0.136 | 0.206 | 0.146 | 0.294 | 0.289 | 0.128 |
0.324 | 0.201 | 0.115 | 0.263 | 0.183 |
0.149 | 0.194 | 0.305 | 0.122 | 0.212 |
0.684 | 0.508 | 0.274 | 0.683 | 0.463 |
Alternatives |
Various Approaches | Score Values | ||||
---|---|---|---|---|---|
Extended GRA Method [32] | 0.252 | 0.185 | 0.108 | 0.128 | 0.084 |
Proposed FF-VIKOR Method | 0.004 | 0.951 | 0.090 | 0.765 | 0.283 |
Proposed FF-Extended TOPSIS Method | 0.684 | 0.508 | 0.274 | 0.683 | 0.463 |
Various Approaches | Ranking |
---|---|
Existing GRA Method [32] | |
Proposed FF-VIKOR Method | |
Proposed FF-Extended TOPSIS Method |
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Abdullah, S.; Saifullah; Almagrabi, A.O. An Integrated Group Decision-Making Framework for the Evaluation of Artificial Intelligence Cloud Platforms Based on Fractional Fuzzy Sets. Mathematics 2023, 11, 4428. https://doi.org/10.3390/math11214428
Abdullah S, Saifullah, Almagrabi AO. An Integrated Group Decision-Making Framework for the Evaluation of Artificial Intelligence Cloud Platforms Based on Fractional Fuzzy Sets. Mathematics. 2023; 11(21):4428. https://doi.org/10.3390/math11214428
Chicago/Turabian StyleAbdullah, Saleem, Saifullah, and Alaa O. Almagrabi. 2023. "An Integrated Group Decision-Making Framework for the Evaluation of Artificial Intelligence Cloud Platforms Based on Fractional Fuzzy Sets" Mathematics 11, no. 21: 4428. https://doi.org/10.3390/math11214428
APA StyleAbdullah, S., Saifullah, & Almagrabi, A. O. (2023). An Integrated Group Decision-Making Framework for the Evaluation of Artificial Intelligence Cloud Platforms Based on Fractional Fuzzy Sets. Mathematics, 11(21), 4428. https://doi.org/10.3390/math11214428