A Study and Application Analysis Exploring Pythagorean Fuzzy Set Distance Metrics in Decision Making
<p>Framework of microgeneration technology selection.</p> "> Figure 2
<p>Rankings visualization for the problem of microgeneration technology selection.</p> "> Figure 3
<p>Framework of renewable energy source selection.</p> "> Figure 4
<p>Rankings visualization for the problem of renewable energy source selection.</p> ">
Abstract
:1. Introduction
- Developing a new distance measure with a simple mathematical expression to address the challenges of solving complex real-world problems accurately and efficiently.
- Demonstrating the proposed measure’s applicability, stability, and reliability in solving multi-criteria decision-making problems.
- Comparing the proposed measure with the existing distance measure and other multi-criteria decision-making methods, emphasizing its advantages and superiority.
2. Preliminaries
2.1. Distance for Pythagorean Fuzzy Set
- i
- Boundedness: .
- ii
- Reflexivity: .
- iii
- Symmetry: .
- iv
- Separability: If then, .
- i
- Boundedness: From the definition of the Pythagorean fuzzy set, we know that:Therefore, we have:
- ii
- Reflexivity: .
- iii
- Symmetry: .
- iv
- Separability: , if .
2.2. PF-TOPSIS
2.3. PF-VIKOR
2.4. Similarity Coefficients
3. Methodology
- Step 1:
- The first step to consider a problem in multi-criteria decision making in the Pythagorean fuzzy environment is to establish a decision matrix. Considering that, establish a Pythagorean fuzzy decision matrix where denotes alternatives () and criteria () where each element contains the value of membership and non-membership:
- Step 2:
- The next step requires the calculation of a positive ideal PFS () and a negative ideal PFS (). This determination can be made using the following equations.
- Step 3:
- Next, determine the distances from the positive and negative ideal solution. The distance measure is a weighted distance calculated with a given weight vector using the following equation.
- Step 4:
- The similarity measure S is then calculated using the distance determined in the previous step. In this context, different functional types of similarity measures can be explored. However, the following equation is used in this study to calculate the similarity.
- Step 5:
- In the fifth step, the relative similarity is calculated using the following formula.
- Step 6:
- Lastly, rank the alternatives in descending order based on their relative similarity measure value.
4. Numerical Examples
4.1. Microgeneration Technology Selection
Comparison
4.2. Renewable Energy Source Selection
Comparison
5. Conclusions and Future Directions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
MCDA | Multi-criteria Decision Analysis |
MCDM | Multi-criteria Decision Making |
FS | Fuzzy Set |
IFS | Intuitionistic Fuzzy Set |
PFS | Pythagorean Fuzzy Set |
TOPSIS | Technique for Order of Preference by Similarity to Ideal Solution |
SWARA | Stepwise Weight Assessment Ratio Analysis |
PF-TOPSIS | Pythagorean Fuzzy Technique for Order of Preference by Similarity to Ideal Solution |
PF-VIKOR | Pythagorean Fuzzy VIseKriterijumska Optimizacija I Kompromisno Resenje |
References
- Aruldoss, M.; Lakshmi, T.M.; Venkatesan, V.P. A survey on multi criteria decision making methods and its applications. Am. J. Inf. Syst. 2013, 1, 31–43. [Google Scholar]
- Jato-Espino, D.; Castillo-Lopez, E.; Rodriguez-Hernandez, J.; Canteras-Jordana, J.C. A review of application of multi-criteria decision making methods in construction. Autom. Constr. 2014, 45, 151–162. [Google Scholar] [CrossRef]
- Kabir, G.; Sadiq, R.; Tesfamariam, S. A review of multi-criteria decision-making methods for infrastructure management. Struct. Infrastruct. Eng. 2014, 10, 1176–1210. [Google Scholar] [CrossRef]
- Marsh, K.; Lanitis, T.; Neasham, D.; Orfanos, P.; Caro, J. Assessing the value of healthcare interventions using multi-criteria decision analysis: A review of the literature. Pharmacoeconomics 2014, 32, 345–365. [Google Scholar] [CrossRef] [PubMed]
- Jamwal, A.; Agrawal, R.; Sharma, M.; Kumar, V. Review on multi-criteria decision analysis in sustainable manufacturing decision making. Int. J. Sustain. Eng. 2021, 14, 202–225. [Google Scholar] [CrossRef]
- Shekhovtsov, A.; Więckowski, J.; Wątróbski, J. Toward reliability in the MCDA rankings: Comparison of distance-based methods. In Intelligent Decision Technologies, Proceedings of the 13th KES-IDT 2021 Conference; Springer: Berlin/Heidelberg, Germany, 2021; pp. 321–329. [Google Scholar]
- Kizielewicz, B.; Więckowski, J.; Shekhovtsov, A.; Wątróbski, J.; Depczyński, R.; Sałabun, W. Study towards the time-based mcda ranking analysis—A supplier selection case study. Facta Univ. Ser. Mech. Eng. 2021, 19, 381–399. [Google Scholar] [CrossRef]
- Thakur, P.; Kizielewicz, B.; Gandotra, N.; Shekhovtsov, A.; Saini, N.; Saeid, A.B.; Sałabun, W. A new entropy measurement for the analysis of uncertain data in mcda problems using intuitionistic fuzzy sets and copras method. Axioms 2021, 10, 335. [Google Scholar] [CrossRef]
- Sałabun, W.; Wątróbski, J.; Shekhovtsov, A. Are mcda methods benchmarkable? A comparative study of topsis, vikor, copras, and promethee ii methods. Symmetry 2020, 12, 1549. [Google Scholar] [CrossRef]
- Zhou, P.; Ang, B.W. Comparing MCDA aggregation methods in constructing composite indicators using the Shannon-Spearman measure. Soc. Indic. Res. 2009, 94, 83–96. [Google Scholar] [CrossRef]
- Antucheviciene, J.; Zakarevicius, A.; Zavadskas, E.K. Measuring congruence of ranking results applying particular MCDM methods. Informatica 2011, 22, 319–338. [Google Scholar] [CrossRef]
- Zadeh, L.A. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1978, 1, 3–28. [Google Scholar] [CrossRef]
- Kaya, I.; Colak, M.; Terzi, F. A comprehensive review of fuzzy multi criteria decision making methodologies for energy policy making. Energy Strategy Rev. 2019, 24, 207–228. [Google Scholar] [CrossRef]
- Pelissari, R.; Oliveira, M.C.; Abackerli, A.J.; Ben-Amor, S.; Assumpção, M.R.P. Techniques to model uncertain input data of multi-criteria decision-making problems: A literature review. Int. Trans. Oper. Res. 2021, 28, 523–559. [Google Scholar] [CrossRef]
- Mardani, A.; Jusoh, A.; Zavadskas, E.K. Fuzzy multiple criteria decision-making techniques and applications—Two decades review from 1994 to 2014. Expert Syst. Appl. 2015, 42, 4126–4148. [Google Scholar] [CrossRef]
- Shekhovtsov, A.; Paradowski, B.; Więckowski, J.; Kizielewicz, B.; Sałabun, W. Extension of the SPOTIS method for the rank reversal free decision-making under fuzzy environment. In Proceedings of the 2022 IEEE 61st Conference on Decision and Control (CDC), Cancun, Mexico, 6–9 December 2022; pp. 5595–5600. [Google Scholar]
- Yatsalo, B.; Korobov, A. Different approaches to fuzzy extension of an MCDA method and their comparison. In Intelligent and Fuzzy Techniques: Smart and Innovative Solutions, Proceedings of the INFUS 2020 Conference, Istanbul, Turkey, 21–23 July 2020; Springer: Berlin/Heidelberg, Germany, 2021; pp. 709–717. [Google Scholar]
- Yatsalo, B.; Korobov, A.; Martínez, L. Fuzzy multi-criteria acceptability analysis: A new approach to multi-criteria decision analysis under fuzzy environment. Expert Syst. Appl. 2017, 84, 262–271. [Google Scholar] [CrossRef]
- Atanassov, K.T.; Stoeva, S. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
- Yager, R.R.; Abbasov, A.M. Pythagorean membership grades, complex numbers, and decision making. Int. J. Intell. Syst. 2013, 28, 436–452. [Google Scholar] [CrossRef]
- Peng, X.; Selvachandran, G. Pythagorean fuzzy set: State of the art and future directions. Artif. Intell. Rev. 2019, 52, 1873–1927. [Google Scholar] [CrossRef]
- Lin, M.; Chen, Y.; Chen, R. Bibliometric analysis on Pythagorean fuzzy sets during 2013–2020. Int. J. Intell. Comput. Cybern. 2021, 14, 104–121. [Google Scholar] [CrossRef]
- Jafar, M.N.; Muniba, K.; Saqlain, M. Enhancing Diabetes Diagnosis through an Intuitionistic Fuzzy Soft Matrices-Based Algorithm. Spectr. Eng. Manag. Sci. 2023, 1, 73–82. [Google Scholar]
- El-Morsy, S. Stock portfolio optimization using pythagorean fuzzy numbers. J. Oper. Strateg. Anal. 2023, 1, 8–13. [Google Scholar] [CrossRef]
- Rani, P.; Mishra, A.R.; Saha, A.; Pamucar, D. Pythagorean fuzzy weighted discrimination-based approximation approach to the assessment of sustainable bioenergy technologies for agricultural residues. Int. J. Intell. Syst. 2021, 36, 2964–2990. [Google Scholar] [CrossRef]
- Paul, T.K.; Jana, C.; Pal, M. Enhancing Multi-Attribute Decision Making with Pythagorean Fuzzy Hamacher Aggregation Operators. J. Ind. Intell. 2023, 8, 372–383. [Google Scholar] [CrossRef]
- Hussain, Z.; Alam, S.; Hussain, R.; ur Rahman, S. New similarity measure of Pythagorean fuzzy sets based on the Jaccard index with its application to clustering. Ain Shams Eng. J. 2023, 15, 102294. [Google Scholar] [CrossRef]
- Garg, H. Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process. Comput. Math. Organ. Theory 2017, 23, 546–571. [Google Scholar] [CrossRef]
- Karsak, E.E. Distance-based fuzzy MCDM approach for evaluating flexible manufacturing system alternatives. Int. J. Prod. Res. 2002, 40, 3167–3181. [Google Scholar] [CrossRef]
- Sałabun, W. The characteristic objects method: A new distance-based approach to multicriteria decision-making problems. J. Multi-Criteria Decis. Anal. 2015, 22, 37–50. [Google Scholar] [CrossRef]
- Ejegwa, P.A. Distance and similarity measures for Pythagorean fuzzy sets. Granul. Comput. 2020, 5, 225–238. [Google Scholar] [CrossRef]
- Soltani, M.; Aouag, H.; Anass, C.; Mouss, M.D. Development of an advanced application process of lean manufacturing approach based on a new integrated mcdm method under pythagorean fuzzy environment. J. Clean. Prod. 2023, 386, 135731. [Google Scholar] [CrossRef]
- Chen, T.-Y. Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis. Inf. Fusion 2018, 41, 129–150. [Google Scholar] [CrossRef]
- Hussian, Z.; Yang, M.-S. Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS. Int. J. Intell. Syst. 2019, 34, 2633–2654. [Google Scholar] [CrossRef]
- Agheli, B.; Adabitabar Firozja, M.; Garg, H. Similarity measure for Pythagorean fuzzy sets and application on multiple criteria decision making. J. Stat. Manag. Syst. 2022, 25, 749–769. [Google Scholar] [CrossRef]
- Zeng, W.; Li, D.; Yin, Q. Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making. Int. J. Intell. Syst. 2018, 33, 2236–2254. [Google Scholar] [CrossRef]
- Ohlan, A. Multiple attribute decision-making based on distance measure under pythagorean fuzzy environment. Int. J. Inf. Technol. 2022, 14, 2205–2217. [Google Scholar] [CrossRef]
- Yin, L.; Zhang, Q.; Zhao, F.; Mou, Q.; Xian, S. A new distance measure for Pythagorean fuzzy sets based on earth mover’s distance and its applications. J. Intell. Fuzzy Syst. 2022, 42, 3079–3092. [Google Scholar] [CrossRef]
- Zhou, F.; Chen, T.-Y. A novel distance measure for pythagorean fuzzy sets and its applications to the technique for order preference by similarity to ideal solutions. Int. J. Comput. Intell. Syst. 2019, 12, 955–969. [Google Scholar] [CrossRef]
- Wang, F.; Zhao, X. Prospect-theory and geometric distance measure-based Pythagorean cubic fuzzy multicriteria decision-making. Int. J. Intell. Syst. 2021, 36, 4117–4142. [Google Scholar] [CrossRef]
- Sun, G.; Wang, M. Pythagorean fuzzy information processing based on centroid distance measure and its applications. Expert Syst. Appl. 2024, 236, 121295. [Google Scholar] [CrossRef]
- Lai, Y.-J.; Liu, T.-Y.; Hwang, C.-L. Topsis for MODM. Eur. J. Oper. Res. 1994, 76, 486–500. [Google Scholar] [CrossRef]
- Krohling, R.A.; Pacheco, A.G. A-TOPSIS–an approach based on TOPSIS for ranking evolutionary algorithms. Procedia Comput. Sci. 2015, 55, 308–317. [Google Scholar] [CrossRef]
- Yang, S.; Pan, Y.; Zeng, S. Decision making framework based Fermatean fuzzy integrated weighted distance and TOPSIS for green low-carbon port evaluation. Eng. Appl. Artif. Intell. 2022, 114, 105048. [Google Scholar] [CrossRef]
- Salih, M.M.; Zaidan, B.; Zaidan, A.; Ahmed, M.A. Survey on fuzzy TOPSIS state-of-the-art between 2007 and 2017. Comput. Oper. Res. 2019, 104, 207–227. [Google Scholar] [CrossRef]
- Akram, M.; Kahraman, C.; Zahid, K. Extension of TOPSIS model to the decision-making under complex spherical fuzzy information. Soft Comput. 2021, 25, 10771–10795. [Google Scholar] [CrossRef]
- Ilieva, G.; Yankova, T. Extension of interval-valued Fermatean fuzzy TOPSIS for evaluating and benchmarking COVID-19 vaccines. Mathematics 2022, 10, 3514. [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]
- Opricovic, S. Fuzzy VIKOR with an application to water resources planning. Expert Syst. Appl. 2011, 38, 12983–12990. [Google Scholar] [CrossRef]
- Hezer, S.; Gelmez, E.; Özceylan, E. Comparative analysis of TOPSIS, VIKOR and COPRAS methods for the COVID-19 Regional Safety Assessment. J. Infect. Public Health 2021, 14, 775–786. [Google Scholar] [CrossRef]
- Gou, X.; Xu, Z.; Liao, H.; Herrera, F. Probabilistic double hierarchy linguistic term set and its use in designing an improved VIKOR method: The application in smart healthcare. J. Oper. Res. Soc. 2021, 72, 2611–2630. [Google Scholar] [CrossRef]
- Saqlain, M. Sustainable hydrogen production: A decision-making approach using VIKOR and intuitionistic hypersoft sets. J. Intell. Manag. Decis. 2023, 2, 130–138. [Google Scholar] [CrossRef]
- Ju, Y.; Wang, A. Extension of VIKOR method for multi-criteria group decision making problem with linguistic information. Appl. Math. Model. 2013, 37, 3112–3125. [Google Scholar] [CrossRef]
- Riaz, M.; Tehrim, S.T. A robust extension of VIKOR method for bipolar fuzzy sets using connection numbers of SPA theory based metric spaces. Artif. Intell. Rev. 2021, 54, 561–591. [Google Scholar] [CrossRef]
- Bakioglu, G.; Atahan, A.O. AHP integrated TOPSIS and VIKOR methods with Pythagorean fuzzy sets to prioritize risks in self-driving vehicles. Appl. Soft Comput. 2021, 99, 106948. [Google Scholar] [CrossRef]
- Pinto da Costa, J.; Soares, C. A weighted rank measure of correlation. Aust. N. Z. J. Stat. 2005, 47, 515–529. [Google Scholar] [CrossRef]
- Sałabun, W.; Urbaniak, K. A new coefficient of rankings similarity in decision-making problems. In Proceedings of the Computational Science–ICCS 2020: 20th International Conference, Amsterdam, The Netherlands, 3–5 June 2020; Proceedings, Part II 20. Springer: Berlin/Heidelberg, Germany, 2020; pp. 632–645. [Google Scholar]
- Kamali Saraji, M.; Streimikiene, D.; Ciegis, R. A novel Pythagorean fuzzy-SWARA-TOPSIS framework for evaluating the EU progress towards sustainable energy development. Environ. Monit. Assess. 2022, 194, 42. [Google Scholar] [CrossRef] [PubMed]
- Zhang, C.; Wang, Q.; Zeng, S.; Baležentis, T.; Štreimikienė, D.; Ališauskaitė-Šeškienė, I.; Chen, X. Probabilistic multi-criteria assessment of renewable micro-generation technologies in households. J. Clean. Prod. 2019, 212, 582–592. [Google Scholar] [CrossRef]
- Mahanta, J.; Panda, S. Distance measure for Pythagorean fuzzy sets with varied applications. Neural Comput. Appl. 2021, 33, 17161–17171. [Google Scholar] [CrossRef]
- Lee, H.C.; Chang, C.T. Comparative analysis of MCDM methods for ranking renewable energy sources in Taiwan. Renew. Sustain. Energy Rev. 2018, 92, 883–896. [Google Scholar] [CrossRef]
- Xu, L.; Shah, S.A.A.; Zameer, H.; Solangi, Y.A. Evaluating renewable energy sources for implementing the hydrogen economy in Pakistan: A two-stage fuzzy MCDM approach. Environ. Sci. Pollut. Res. 2019, 26, 33202–33215. [Google Scholar] [CrossRef]
- Ezbakhe, F.; Pérez-Foguet, A. Decision analysis for sustainable development: The case of renewable energy planning under uncertainty. Eur. J. Oper. Res. 2021, 291, 601–613. [Google Scholar] [CrossRef]
C1 | C2 | C3 | C4 | |
---|---|---|---|---|
A1 | (0.54, 0.63) | (0.29, 0.61) | (0.32, 0.27) | (0.73, 0.44) |
A2 | (0.23, 0.61) | (0.49, 0.54) | (0.39, 0.64) | (0.60, 0.63) |
A3 | (0.10, 0.25) | (0.44, 0.56) | (0.48, 0.31) | (0.34, 0.43) |
A4 | (0.61, 0.54) | (0.73, 0.42) | (0.28, 0.44) | (0.53, 0.29) |
A5 | (0.80, 0.12) | (0.27, 0.68) | (0.45, 0.66) | (0.30, 0.64) |
Ci | C1 | C2 | C3 | C4 |
---|---|---|---|---|
M+ | (0.80, 0.12) | (0.73, 0.42) | (0.48, 0.27) | (0.73, 0.29) |
M− | (0.10, 0.63) | (0.27, 0.68) | (0.28, 0.66) | (0.30, 0.64) |
Ai | A1 | A2 | A3 | A4 | A5 |
---|---|---|---|---|---|
D+ | 0.2287 | 0.3636 | 0.2296 | 0.1919 | 0.2736 |
D− | 0.2414 | 0.1064 | 0.2405 | 0.2782 | 0.1964 |
S(R) | 0.5042 | 0.4159 | 0.5036 | 0.5282 | 0.4748 |
Ai | Proposed Distance | Mahanta Distance | PF-TOPSIS | PF-VIKOR |
---|---|---|---|---|
A1 | 0.5042 | 0.5278 | −0.1258 | 0.3196 |
A2 | 0.4159 | 0.3557 | −1.5212 | 0.8506 |
A3 | 0.5036 | 0.5420 | −0.5991 | 0.0000 |
A4 | 0.5282 | 0.5401 | 0.0000 | 0.2784 |
A5 | 0.4748 | 0.4578 | −0.9118 | 1.0000 |
Statistical Measure | Proposed Distance | Mahanta Distance | PF-TOPSIS | PF-VIKOR |
---|---|---|---|---|
Min | 0.4159 | 0.3557 | −1.5212 | 0.0000 |
Mean | 0.4853 | 0.4847 | −0.6316 | 0.4897 |
Max | 0.5282 | 0.5420 | 0.0000 | 1.0000 |
Std | 0.0386 | 0.0715 | 0.5524 | 0.3752 |
Coefficient | Mahanta Distance | PF-TOPSIS | PF-VIKOR |
---|---|---|---|
Weighted Spearman | 0.6 | 1.0 | 0.5500 |
WS coefficient | 0.6667 | 1.0 | 0.6380 |
C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
---|---|---|---|---|---|---|---|---|---|---|
A1 | (0.42, 0.60) | (0.34, 0.23) | (0.60, 0.16) | (0.70, 0.23) | (0.32, 0.42) | (0.36, 0.54) | (0.53, 0.70) | (0.45, 0.25) | (0.24, 0.35) | (0.45, 0.45) |
A2 | (0.72, 0.52) | (0.12, 0.35) | (0.33, 0.22) | (0.55, 0.51) | (0.36, 0.54) | (0.14, 0.51) | (0.21, 0.65) | (0.23, 0.50) | (0.72, 0.45) | (0.26, 0.41) |
A3 | (0.56, 0.54) | (0.54, 0.12) | (0.33, 0.59) | (0.36, 0.45) | (0.70, 0.18) | (0.56, 0.45) | (0.60, 0.45) | (0.39, 0.36) | (0.60, 0.12) | (0.20, 0.10) |
A4 | (0.34, 0.45) | (0.30, 0.70) | (0.36, 0.21) | (0.45, 0.54) | (0.35, 0.60) | (0.26, 0.34) | (0.25, 0.36) | (0.35, 0.26) | (0.27, 0.34) | (0.14, 0.39) |
A5 | (0.23, 0.45) | (0.64, 0.45) | (0.55, 0.35) | (0.25, 0.32) | (0.12, 0.35) | (0.47, 0.52) | (0.21, 0.51) | (0.43, 0.54) | (0.24, 0.36) | (0.32, 0.60) |
A6 | (0.52, 0.10) | (0.56, 0.30) | (0.45, 0.60) | (0.40, 0.25) | (0.50, 0.70) | (0.58, 0.34) | (0.36, 0.74) | (0.46, 0.50) | (0.32, 0.54) | (0.60, 0.36) |
A7 | (0.70, 0.33) | (0.26, 0.36) | (0.16, 0.36) | (0.60, 0.11) | (0.22, 0.70) | (0.50, 0.35) | (0.54, 0.70) | (0.50, 0.15) | (0.56, 0.60) | (0.31, 0.54) |
A8 | (0.66, 0.21) | (0.75, 0.24) | (0.41, 0.55) | (0.51, 0.36) | (0.70, 0.60) | (0.51, 0.41) | (0.66, 0.16) | (0.52, 0.36) | (0.52, 0.23) | (0.71, 0.44) |
A9 | (0.22, 0.69) | (0.60, 0.61) | (0.33, 0.60) | (0.45, 0.31) | (0.55, 0.29) | (0.35, 0.21) | (0.53, 0.25) | (0.24, 0.29) | (0.31, 0.54) | (0.16, 0.36) |
A10 | (0.16, 0.47) | (0.13, 0.33) | (0.26, 0.70) | (0.23, 0.31) | (0.30, 0.39) | (0.25, 0.50) | (0.31, 0.60) | (0.13, 0.50) | (0.36, 0.60) | (0.27, 0.35) |
Ci | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 |
---|---|---|---|---|---|---|---|---|---|---|
M+ | (0.72, 0.10) | (0.75, 0.12) | (0.60, 0.16) | (0.70, 0.11) | (0.70, 0.18) | (0.58, 0.21) | (0.66, 0.16) | (0.52, 0.15) | (0.72, 0.12) | (0.71, 0.10) |
M− | (0.16, 0.69) | (0.12, 0.70) | (0.16, 0.70) | (0.23, 0.54) | (0.12, 0.70) | (0.14, 0.54) | (0.21, 0.74) | (0.13, 0.54) | (0.24, 0.60) | (0.14, 0.60) |
Ai | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 |
---|---|---|---|---|---|---|---|---|---|---|
D+ | 0.2591 | 0.3110 | 0.2052 | 0.3398 | 0.3170 | 0.2780 | 0.2750 | 0.1350 | 0.3030 | 0.3927 |
D− | 0.2646 | 0.2127 | 0.3185 | 0.1849 | 0.2067 | 0.2457 | 0.2487 | 0.3887 | 0.2207 | 0.1309 |
S(R) | 0.5019 | 0.4667 | 0.5384 | 0.4472 | 0.4626 | 0.4891 | 0.4911 | 0.5859 | 0.4722 | 0.4113 |
Ai | Proposed Distance | Mahanta Distance | PF-TOPSIS | PF-VIKOR |
---|---|---|---|---|
A1 | 0.5019 | 0.5261 | −1.4433 | 0.3742 |
A2 | 0.4667 | 0.3892 | −2.3021 | 0.4269 |
A3 | 0.5384 | 0.6010 | −1.1194 | 0.6948 |
A4 | 0.4472 | 0.3675 | −2.2322 | 0.5495 |
A5 | 0.4626 | 0.3863 | −1.8938 | 0.4737 |
A6 | 0.4891 | 0.5057 | −1.8033 | 0.4977 |
A7 | 0.4911 | 0.5074 | −1.8559 | 0.4759 |
A8 | 0.5859 | 0.6758 | 0.0000 | 0.0000 |
A9 | 0.4722 | 0.4447 | −2.0014 | 0.4722 |
A10 | 0.4113 | 0.2321 | −2.9347 | 0.9960 |
Statistical Measure | Proposed Distance | Mahanta Distance | PF-TOPSIS | PF-VIKOR |
---|---|---|---|---|
Min | 0.411 | 0.232 | −2.935 | 0.000 |
Mean | 0.487 | 0.464 | −1.759 | 0.496 |
Max | 0.586 | 0.676 | 0.000 | 0.996 |
Std | 0.046 | 0.121 | 0.748 | 0.237 |
Coefficient | Mahanta Distance | PF-TOPSIS | PF-VIKOR |
---|---|---|---|
Weighted Spearman | 1.0 | 0.95 | 0.47 |
WS coefficient | 1.0 | 0.98 | 0.72 |
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. |
© 2024 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
Thakur, P.; Paradowski, B.; Gandotra, N.; Thakur, P.; Saini, N.; Sałabun, W. A Study and Application Analysis Exploring Pythagorean Fuzzy Set Distance Metrics in Decision Making. Information 2024, 15, 28. https://doi.org/10.3390/info15010028
Thakur P, Paradowski B, Gandotra N, Thakur P, Saini N, Sałabun W. A Study and Application Analysis Exploring Pythagorean Fuzzy Set Distance Metrics in Decision Making. Information. 2024; 15(1):28. https://doi.org/10.3390/info15010028
Chicago/Turabian StyleThakur, Palvinder, Bartosz Paradowski, Neeraj Gandotra, Parul Thakur, Namita Saini, and Wojciech Sałabun. 2024. "A Study and Application Analysis Exploring Pythagorean Fuzzy Set Distance Metrics in Decision Making" Information 15, no. 1: 28. https://doi.org/10.3390/info15010028
APA StyleThakur, P., Paradowski, B., Gandotra, N., Thakur, P., Saini, N., & Sałabun, W. (2024). A Study and Application Analysis Exploring Pythagorean Fuzzy Set Distance Metrics in Decision Making. Information, 15(1), 28. https://doi.org/10.3390/info15010028