Ranking Alternatives Using a Fuzzy Preference Relation-Based Fuzzy VIKOR Method
Abstract
:1. Introduction
2. Literature Review
2.1. Fuzzy VIKOR
2.2. Fuzzy Preference Relation
3. Fuzzy Set Theory
3.1. Fuzzy Set and Fuzzy Numbers
3.2. Operations on Fuzzy Numbers
3.3. Linguistic Values
4. Model Establishment
- Step 1. Establish the alternatives and criteria.
- Step 2. Determine the average ratings of alternatives versus qualitative criteria.
- Step 3. Normalize the values under quantitative criteria.
- Step 4. Determine the fuzzy best values using the extended fuzzy preference relation.
- Step 5. Determine the dominance of over .
- Step 6. Determine the importance weights of criteria.
- Step 7. Compute the separation measures of and .
- Step 8. Compute the separation measures of .
- Step 9. Propose a compromise solution by ranking the values of in the ascending order. The conditions that should be satisfied are as follows:
- -
- Condition 1: acceptable advantage. The following conditions should be satisfied: , where .
- -
- Condition 2: acceptance stability in decision-making. The best alternative must also be the best ranked by or/and . This result should be stable within a decision-making process.
5. Numerical Example and Comparison
5.1. A Numerical Example
- Step 1. Determine the alternatives and criteria.
- Step 2. Determine the average ratings of alternatives versus qualitative criteria.
- Step 3. Normalize the values under quantitative criteria.
- Step 4. Determine the fuzzy best value using the extended fuzzy preference relation.
- Step 5. Determine the dominance of over .
- Step 6. Determine the importance weights of criteria.
- Step 7. Compute the separation measures of and .
- Step 8. Compute the separation measures of .
5.2. A Comparison with Fuzzy VIKOR Method [4]
- Step a. Determine the alternatives and criteria—same as Step 1 above.
- Step b. Aggregate the ratings of the alternatives against criteria.
- Step c. Aggregate the weighted weightings.
- Step d. Determine the fuzzy best value and the fuzzy worst value.
- Step e. Compute the fuzzy difference .
- Step f. Compute the separation measures.
- Step g. Compute the value of .
- Step h. Defuzzify the values of , , and .
- Step i. Propose a compromise solution by ranking the defuzzified values of , , and in the ascending order. The following conditions need to be satisfied:
- -
- Condition 1: acceptable advantage. The following conditions should be satisfied: , where
- -
- Condition 2: acceptance stability in decision-making. The alternative Ai must also be the best ranked by S or/and R. This result should be stable within a decision-making process.
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Opricovic, S.; Tzeng, G.-H. Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. Eur. J. Oper. Res. 2004, 156, 445–455. [Google Scholar] [CrossRef]
- Opricovic, S.; Tzeng, G.-H. Extended VIKOR method in comparison with outranking methods. Eur. J. Oper. Res. 2007, 178, 514–529. [Google Scholar] [CrossRef]
- Opricovic, S. A fuzzy compromise solution for multicriteria problems. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 2007, 15, 363–380. [Google Scholar] [CrossRef]
- Opricovic, S. Fuzzy VIKOR with an application to water resources planning. Expert Syst. Appl. 2011, 38, 12983–12990. [Google Scholar] [CrossRef]
- Shemshadi, A.; Shirazi, H.; Toreihi, M.; Tarokh, M. A fuzzy VIKOR method for supplier selection based on entropy measure for objective weighting. Expert Syst. Appl. 2011, 38, 12160–12167. [Google Scholar] [CrossRef]
- Ploskas, N.; Papathanasiou, J. A decision support system for multiple criteria alternative ranking using TOPSIS and VIKOR in fuzzy and nonfuzzy environments. Fuzzy Sets Syst. 2019, 377, 01–30. [Google Scholar] [CrossRef]
- Sanayei, A.; Mousavi, S.F.; Yazdankhah, A. Group decision making process for supplier selection with VIKOR under fuzzy environment. Expert Syst. Appl. 2010, 37, 24–30. [Google Scholar] [CrossRef]
- Taghavifard, M.T.; Majidian, S. Identifying Cloud Computing Risks based on Firm’s Ambidexterity Performance using Fuzzy VIKOR Technique. Glob. J. Flex. Syst. Manag. 2022, 23, 113–133. [Google Scholar] [CrossRef]
- Kaufmann, A.; Gupta, M. Introduction to Fuzzy Arithmetic Theory and Applications; Van Nostrand Reinhold: New York, NY, USA, 1991. [Google Scholar]
- Yuan, Y.-F. Criteria for evaluating fuzzy ranking methods. Fuzzy Sets Syst. 1991, 44, 139–157. [Google Scholar] [CrossRef]
- Lee, H.-S. On fuzzy preference relation in group decision making. Int. J. Comput. Math. 2005, 82, 133–140. [Google Scholar] [CrossRef]
- Wang, Y.-J. Ranking triangle and trapezoidal fuzzy numbers based on the relative preference relation. Appl. Math. Model. 2014, 39, 586–599. [Google Scholar] [CrossRef]
- Li, R.-J. Fuzzy Method in Group Decision Making. Comput. Math. Appl. 1999, 38, 91–101. [Google Scholar] [CrossRef]
- Liou, T.S.; Wang, M.J. Ranking Fuzzy Numbers with Integral Value. Fuzzy Sets Syst. 1992, 50, 247–255. [Google Scholar] [CrossRef]
- Gul, M.; Celik, E.; Aydin, N.; Gumus, A.T.; Guneri, A.F. A state of the art literature review of VIKOR and its fuzzy extensions on applications. Appl. Soft Comput. J. 2016, 46, 60–89. [Google Scholar] [CrossRef]
- Opricovic, S. Multicriteria Optimization of Civil Engineering Systems. Ph.D. Thesis, University of Belgrade, Faculty of Civil Engineering, Belgrade, Serbia, 1998. [Google Scholar]
- Duckstein, L.; Opricovic, S. Multi-objective Optimization in River Basin Development. Water Resour. Res. 1980, 16, 14–20. [Google Scholar] [CrossRef]
- Yu, P.L. A class of solutions for group decision problems. Manag. Sci. 1973, 19, 936–946. [Google Scholar] [CrossRef]
- Zeleny, M. Compromise Programming. In Multiple Criteria Decision Making; Cochrane, J.L., Zeleny, M., Eds.; University of South Calorina Press: Columbia, SC, USA, 1973; pp. 262–301. [Google Scholar]
- Opricovic, S.; Tzeng, G.-H. Multicriteria Planning of Post-Earthquake Sustainable Reconstruction. Comput.-Aided Civ. Infrastruct. Eng. 2002, 17, 211–220. [Google Scholar] [CrossRef]
- Chang, T.-H. Fuzzy VIKOR method: A case study of the hospital service evaluation in Taiwan. Inf. Sci. 2014, 271, 196–212. [Google Scholar] [CrossRef]
- Zadeh, L. Fuzzy Logic and Approximate Reasoning. Synthese 1975, 30, 407–428. [Google Scholar] [CrossRef]
- Jeya Girubha, R.; Vinodh, S. Application of fuzzy VIKOR and environmental impact analysis for material selection of an automotive component. Mater. Des. 2012, 37, 478–486. [Google Scholar] [CrossRef]
- Bahadori, M.; Hosseini, S.M.; Teymourzadeh, E.; Ravangard, R.; Raadabadi, M.; Alimohammadzadeh, K. A supplier selection model for hospitals using a combination of artificial neural network and fuzzy VIKOR. Int. J. Healthc. Manag. 2017, 13, 286–294. [Google Scholar] [CrossRef]
- Jing, S.; Tang, Y.; Yan, J. The Application of Fuzzy VIKOR for the Design Scheme Selection in Lean Management. Math. Probl. Eng. 2018, 2018, 9253643. [Google Scholar] [CrossRef]
- Rathore, R.; Thakkar, J.J.; Jha, J.K. Evaluation of risks in foodgrains supply chain using failure mode effect analysis and fuzzy VIKOR. Int. J. Qual. Reliab. Manag. 2019, 38, 551–580. [Google Scholar] [CrossRef]
- Ikram, M.; Zhang, Q.; Sroufe, R. Developing integrated management systems using an AHP-Fuzzy VIKOR approach. Bus. Strategy Environ. 2020, 29, 2265–2283. [Google Scholar] [CrossRef]
- Arslankaya, S. Catering Company Selection with Fuzzy AHP, ELECTRE and VIKOR Method for a Company Producing Trailer. Eur. J. Sci. Technol. 2020, 18, 413–423. [Google Scholar] [CrossRef]
- Li, H.; Wang, W.; Fan, L.; Li, Q.; Chen, X. A novel hybrid MCDM model for machine tool selection using fuzzy DEMATEL, entropy weighting and later defuzzification VIKOR. Appl. Soft Comput. J. 2020, 91, 106207. [Google Scholar] [CrossRef]
- Kotb, K.M.; Elkadeem, M.; Khalil, A.; Imam, S.M.; Hamada, M.A.; Sharshir, S.W.; Dan, A. A fuzzy decision-making model for optimal design of solar, wind, diesel-based RO desalination integrating flow-battery and pumped-hydro storage: Case study in Baltim, Egypt. Energy Convers. Manag. 2021, 235, 113962. [Google Scholar] [CrossRef]
- Yang, H.; Luo, Q.; Sun, X.; Wang, Z. Comprehensive evaluation of urban waterlogging prevention resilience based on the fuzzy VIKOR method: A case study of the Beijing-Tianjin-Hebei urban agglomeration. Environ. Sci. Pollut. Res. 2023, 30, 112773–112787. [Google Scholar] [CrossRef] [PubMed]
- Orlovsky, S.A. Decision-making with a fuzzy preference relation. Fuzzy Sets Syst. 1978, 1, 155–167. [Google Scholar] [CrossRef]
- Kołodziejczyk, W. Orlovsky’s concept of decision-making with fuzzy preference relation-further results. Fuzzy Sets Syst. 1986, 19, 11–20. [Google Scholar] [CrossRef]
- Nakamura, K. Preference Relations On A Set Of Fuzzy Utilities As A Basis For Decision Making. Fuzzy Sets Syst. 1986, 20, 147–162. [Google Scholar] [CrossRef]
- Tanino, T. Fuzzy Preference Relations in Group Decision Making. In Non-Conventional Preference Relations in Decision Making. Lecture Notes in Economics and Mathematical Systems; Kacprzyk, J.R., Ed.; Springer: Berlin/Heidelberg, Germany, 1988; Volume 301. [Google Scholar] [CrossRef]
- Hipel, K.W.; Kilgour, D.M.; Bashar, M.A. Fuzzy preferences in multiple participant decision making. Sci. Iran. 2011, 18, 627–638. [Google Scholar] [CrossRef]
- Liu, W.; Dong, Y.; Chiclana, F.; Cabrerizo, F.J.; Herrera-Viedma, E. Group decision-making based on heterogeneous preference relations with self-confidence. Fuzzy Optim. Decis. Mak. 2017, 16, 429–447. [Google Scholar] [CrossRef]
- Sadiq, M.; Jain, S.K. Applying fuzzy preference relation for requirements prioritization in goal oriented requirements elicitation process. Int. J. Syst. Assur. Eng. Manag. 2014, 5, 711–723. [Google Scholar] [CrossRef]
- Roldán López de Hierro, A.F.; Sánchez, M.; Roldán, C. Multi-criteria decision making involving uncertain information via fuzzy ranking and fuzzy aggregation functions. J. Comput. Appl. Math. 2020, 404, 113–138. [Google Scholar] [CrossRef]
- Chen, S.H. Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets Syst. 1985, 17, 113–129. [Google Scholar] [CrossRef]
- Chu, T.C.; Yeh, W.C. Fuzzy multiple criteria decision-making via an inverse function-based total utility approach. Soft Comput. 2018, 22, 7423–7433. [Google Scholar] [CrossRef]
- Dubois, D.; Prade, H. Operations on fuzzy numbers. Int. J. Syst. Sci. 1978, 9, 613–626. [Google Scholar] [CrossRef]
- Chu, T.-C.; Charnsethikul, P. Ordering Alternatives under Fuzzy Multiple Criteria Decision Making via a Fuzzy Number Dominance Based Ranking Approach. Int. J. Fuzzy Syst. 2013, 15, 263–273. [Google Scholar]
Criteria | Alternatives | Decision-Makers | ||
---|---|---|---|---|
D1 | D2 | D3 | ||
M | P | M | ||
G | G | G | ||
VG | VG | G | ||
M | G | M | ||
P | M | VP | ||
G | M | G | ||
G | G | G | ||
P | M | M | ||
VP | P | M | ||
G | VG | VG | ||
VG | VG | VG | ||
G | G | G | ||
M | P | VP | ||
G | M | M | ||
VG | G | G | ||
G | P | M | ||
M | M | G | ||
G | VG | VG | ||
G | G | G | ||
VG | G | G | ||
M | M | P | ||
G | VG | G | ||
VG | G | G | ||
VP | P | VP | ||
M | P | P | ||
VG | G | G | ||
VG | VG | VG | ||
M | G | M | ||
P | P | VP | ||
G | M | G | ||
G | VG | VG | ||
M | G | M | ||
M | P | P | ||
M | VG | G | ||
G | VG | M | ||
G | P | M | ||
P | P | M | ||
G | G | VG | ||
G | VG | VG | ||
M | M | M |
Criteria | Alternatives | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
A1 | A2 | A3 | A4 | |||||||||
0.300 | 0.483 | 0.667 | 0.600 | 0.750 | 0.900 | 0.733 | 0.850 | 0.967 | 0.433 | 0.617 | 0.800 | |
0.217 | 0.367 | 0.517 | 0.517 | 0.683 | 0.850 | 0.600 | 0.750 | 0.900 | 0.300 | 0.483 | 0.667 | |
0.217 | 0.367 | 0.517 | 0.800 | 0.800 | 0.917 | 0.800 | 0.867 | 0.967 | 0.600 | 0.700 | 0.850 | |
0.217 | 0.367 | 0.517 | 0.433 | 0.617 | 0.800 | 0.667 | 0.800 | 0.933 | 0.383 | 0.550 | 0.717 | |
0.433 | 0.617 | 0.800 | 0.733 | 0.850 | 0.967 | 0.600 | 0.750 | 0.900 | 0.667 | 0.800 | 0.933 | |
0.300 | 0.483 | 0.667 | 0.667 | 0.800 | 0.933 | 0.667 | 0.800 | 0.933 | 0.133 | 0.250 | 0.367 | |
0.250 | 0.417 | 0.583 | 0.667 | 0.800 | 0.933 | 0.800 | 0.900 | 1.000 | 0.433 | 0.617 | 0.800 | |
0.167 | 0.300 | 0.433 | 0.517 | 0.683 | 0.850 | 0.733 | 0.850 | 0.967 | 0.433 | 0.617 | 0.800 | |
0.250 | 0.417 | 0.583 | 0.583 | 0.733 | 0.883 | 0.583 | 0.733 | 0.883 | 0.383 | 0.550 | 0.717 | |
0.217 | 0.367 | 0.517 | 0.517 | 0.683 | 0.850 | 0.600 | 0.750 | 0.900 | 0.300 | 0.483 | 0.667 |
Criteria | Alternatives | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
A1 | A2 | A3 | A4 | |||||||||
12 | 14 | 16 | 20 | 21 | 22 | 17 | 19 | 21 | 17 | 19 | 21 | |
5 | 7 | 10 | 11 | 12 | 13 | 6 | 7 | 9 | 8 | 10 | 12 |
Criteria | Alternatives | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
A1 | A2 | A3 | A4 | |||||||||
0.000 | 0.200 | 0.400 | 0.800 | 0.900 | 1.000 | 0.500 | 0.700 | 0.900 | 0.500 | 0.700 | 0.900 | |
0.375 | 0.750 | 1.000 | 0.000 | 0.125 | 0.250 | 0.500 | 0.750 | 0.875 | 0.125 | 0.375 | 0.625 |
j | j | j | j |
---|---|---|---|
1 | 0.450 | 0.625 | 0.800 |
2 | 0.250 | 0.500 | 0.688 |
3 | 0.517 | 0.675 | 0.833 |
4 | 0.408 | 0.571 | 0.733 |
5 | 0.604 | 0.683 | 0.813 |
6 | 0.425 | 0.583 | 0.742 |
7 | 0.608 | 0.754 | 0.900 |
8 | 0.442 | 0.583 | 0.725 |
9 | 0.538 | 0.683 | 0.829 |
10 | 0.463 | 0.613 | 0.763 |
11 | 0.450 | 0.608 | 0.767 |
12 | 0.500 | 0.654 | 0.808 |
0.000 | 1.000 | 0.686 | 0.686 | |
0.852 | 0.000 | 0.916 | 0.290 | |
0.083 | 0.722 | 0.945 | 0.339 | |
0.050 | 0.795 | 0.922 | 0.270 | |
0.000 | 0.997 | 0.999 | 0.571 | |
0.035 | 0.595 | 0.974 | 0.401 | |
0.156 | 0.811 | 0.486 | 0.655 | |
0.229 | 0.983 | 0.983 | 0.000 | |
0.008 | 0.844 | 0.995 | 0.312 | |
0.000 | 0.706 | 0.996 | 0.512 | |
0.072 | 0.837 | 0.837 | 0.331 | |
0.026 | 0.892 | 0.970 | 0.239 |
0.800 | 0.900 | 1.000 | |
0.500 | 0.750 | 0.875 | |
0.733 | 0.850 | 0.967 | |
0.600 | 0.750 | 0.900 | |
0.800 | 0.867 | 0.967 | |
0.667 | 0.800 | 0.933 | |
0.733 | 0.850 | 0.967 | |
0.667 | 0.800 | 0.933 | |
0.800 | 0.900 | 1.000 | |
0.733 | 0.850 | 0.967 | |
0.583 | 0.733 | 0.883 | |
0.733 | 0.850 | 0.967 |
1.000 | 0.500 | 0.955 | 0.955 | |
0.500 | 1.000 | 0.500 | 0.973 | |
1.000 | 0.818 | 0.500 | 0.981 | |
1.000 | 0.695 | 0.500 | 0.985 | |
1.000 | 0.759 | 0.500 | 0.979 | |
1.000 | 0.924 | 0.500 | 0.990 | |
0.981 | 0.500 | 0.818 | 0.686 | |
1.000 | 0.500 | 0.500 | 1.000 | |
1.000 | 0.850 | 0.500 | 1.000 | |
1.000 | 0.928 | 0.500 | 0.981 | |
1.000 | 0.500 | 0.500 | 0.924 | |
1.000 | 0.686 | 0.500 | 0.999 |
Importance Weights of Criteria | |||
---|---|---|---|
D1 | D2 | D3 | |
VI | VI | VI | |
MI | IM | MI | |
MI | VI | IM | |
VI | VI | IM | |
MI | VI | MI | |
MI | IM | IM | |
UI | MI | IM | |
MI | IM | MI | |
VI | VI | IM | |
UI | MI | IM | |
VI | VI | VI | |
MI | IM | MI |
Average Important Weight | |||
---|---|---|---|
0.775 | 0.875 | 1.000 | |
0.517 | 0.675 | 0.825 | |
0.583 | 0.725 | 0.867 | |
0.650 | 0.775 | 0.908 | |
0.642 | 0.775 | 0.917 | |
0.458 | 0.625 | 0.775 | |
0.367 | 0.517 | 0.650 | |
0.517 | 0.675 | 0.825 | |
0.650 | 0.775 | 0.908 | |
0.367 | 0.517 | 0.650 | |
0.775 | 0.875 | 1.000 | |
0.517 | 0.675 | 0.825 |
6.551 | 8.136 | 9.725 | |
4.820 | 6.044 | 7.256 | |
3.877 | 4.803 | 5.736 | |
6.547 | 8.138 | 9.732 |
0.775 | 0.875 | 1.000 | |
0.553 | 0.659 | 0.773 | |
0.740 | 0.835 | 0.955 | |
0.740 | 0.835 | 0.955 |
1.000 | |
0.729 | |
0.725 | |
0.996 |
Criteria | Alternatives | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
A1 | A2 | A3 | A4 | |||||||||
0.000 | 0.200 | 0.400 | 0.800 | 0.900 | 1.000 | 0.500 | 0.700 | 0.900 | 0.500 | 0.700 | 0.900 | |
0.375 | 0.750 | 1.000 | 0.000 | 0.125 | 0.250 | 0.500 | 0.750 | 0.875 | 0.125 | 0.375 | 0.625 | |
0.000 | 0.483 | 0.750 | 0.600 | 0.750 | 0.900 | 0.600 | 0.850 | 1.000 | 0.350 | 0.617 | 0.900 | |
0.000 | 0.367 | 0.750 | 0.350 | 0.683 | 0.900 | 0.350 | 0.683 | 0.900 | 0.200 | 0.483 | 0.750 | |
0.000 | 0.367 | 0.750 | 0.600 | 0.850 | 1.000 | 0.800 | 0.900 | 1.000 | 0.600 | 0.750 | 0.900 | |
0.000 | 0.367 | 0.750 | 0.350 | 0.617 | 0.900 | 0.600 | 0.800 | 1.000 | 0.200 | 0.550 | 0.900 | |
0.000 | 0.617 | 0.900 | 0.600 | 0.850 | 1.000 | 0.600 | 0.750 | 0.900 | 0.600 | 0.800 | 1.000 | |
0.000 | 0.483 | 0.750 | 0.600 | 0.800 | 1.000 | 0.600 | 0.800 | 1.000 | 0.200 | 0.350 | 0.500 | |
0.000 | 0.417 | 0.750 | 0.600 | 0.800 | 1.000 | 0.800 | 0.900 | 1.000 | 0.350 | 0.617 | 0.900 | |
0.000 | 0.300 | 0.500 | 0.350 | 0.683 | 0.900 | 0.600 | 0.850 | 1.000 | 0.350 | 0.617 | 0.900 | |
0.000 | 0.417 | 0.750 | 0.350 | 0.733 | 1.000 | 0.350 | 0.733 | 1.000 | 0.200 | 0.550 | 0.900 | |
0.000 | 0.417 | 0.750 | 0.600 | 0.800 | 1.000 | 0.600 | 0.850 | 1.000 | 0.350 | 0.550 | 0.750 |
Average Important Weight | |||
---|---|---|---|
0.775 | 0.875 | 1.000 | |
0.400 | 0.675 | 0.875 | |
0.400 | 0.725 | 0.725 | |
0.400 | 0.775 | 0.725 | |
0.575 | 0.775 | 0.875 | |
0.400 | 0.625 | 0.725 | |
0.125 | 0.517 | 0.725 | |
0.400 | 0.675 | 0.875 | |
0.400 | 0.775 | 0.725 | |
0.125 | 0.517 | 0.725 | |
0.775 | 0.875 | 1.000 | |
0.400 | 0.675 | 0.875 |
0.800 | 0.900 | 1.000 | |
0.500 | 0.750 | 1.000 | |
0.600 | 0.850 | 1.000 | |
0.350 | 0.683 | 0.900 | |
0.800 | 0.900 | 1.000 | |
0.600 | 0.800 | 1.000 | |
0.600 | 0.850 | 1.000 | |
0.600 | 0.800 | 1.000 | |
0.800 | 0.900 | 1.000 | |
0.600 | 0.850 | 1.000 | |
0.350 | 0.733 | 1.000 | |
0.600 | 0.850 | 1.000 |
0.000 | 0.200 | 0.400 | |
0.000 | 0.125 | 0.250 | |
0.000 | 0.483 | 0.750 | |
0.000 | 0.367 | 0.750 | |
0.000 | 0.367 | 0.750 | |
0.000 | 0.367 | 0.750 | |
0.000 | 0.617 | 0.900 | |
0.000 | 0.350 | 0.500 | |
0.000 | 0.417 | 0.750 | |
0.000 | 0.300 | 0.500 | |
0.000 | 0.417 | 0.750 | |
0.000 | 0.417 | 0.750 |
Alternatives | A1 | A2 | A3 | A4 | ||||||||
Criteria | ||||||||||||
0.400 | 0.700 | 1.000 | −0.200 | 0.000 | 0.000 | −0.100 | 0.200 | 0.500 | −0.100 | −0.556 | 0.100 | |
−0.500 | 0.000 | 0.625 | 0.250 | 0.625 | 0.750 | −0.375 | 0.000 | 0.500 | −0.125 | −0.333 | 0.375 | |
−0.150 | 0.367 | 1.000 | −0.300 | 0.100 | 0.100 | −0.400 | 0.000 | 0.400 | −0.300 | −0.157 | 0.100 | |
−0.444 | 0.352 | 1.000 | −0.611 | 0.000 | 0.000 | −0.611 | 0.000 | 0.611 | −0.444 | −0.171 | 0.167 | |
0.050 | 0.533 | 1.000 | −0.200 | 0.050 | 0.000 | −0.200 | 0.000 | 0.200 | −0.100 | −0.426 | 0.100 | |
−0.150 | 0.433 | 1.000 | −0.300 | 0.183 | 0.100 | −0.400 | 0.000 | 0.400 | −0.300 | −0.229 | 0.100 | |
−0.300 | 0.233 | 1.000 | −0.400 | 0.000 | 0.000 | −0.300 | 0.100 | 0.400 | −0.400 | −0.216 | 0.000 | |
−0.150 | 0.317 | 1.000 | −0.400 | 0.000 | 0.000 | −0.400 | 0.000 | 0.400 | 0.100 | 0.000 | 0.500 | |
0.050 | 0.483 | 1.000 | −0.200 | 0.100 | 0.000 | −0.200 | 0.000 | 0.200 | −0.100 | −0.222 | 0.100 | |
0.100 | 0.550 | 1.000 | −0.300 | 0.167 | 0.100 | −0.400 | 0.000 | 0.400 | −0.300 | −0.373 | 0.100 | |
−0.400 | 0.317 | 1.000 | −0.650 | 0.000 | 0.000 | −0.650 | 0.000 | 0.650 | −0.550 | −0.182 | 0.100 | |
−0.150 | 0.433 | 1.000 | −0.400 | 0.050 | 0.000 | −0.400 | 0.000 | 0.400 | −0.150 | −0.157 | 0.250 |
−0.594 | 3.398 | 9.522 | |
−1.646 | 0.845 | 0.874 | |
−1.898 | 0.227 | 4.211 | |
−1.177 | −2.172 | 1.683 |
0.310 | 0.613 | 1.000 | |
0.100 | 0.422 | 0.656 | |
−0.038 | 0.175 | 0.650 | |
0.040 | 0.000 | 0.438 |
−0.126 | 0.539 | 1.000 | |
−0.273 | 0.335 | 0.456 | |
−0.350 | 0.189 | 0.498 | |
−0.281 | 0.000 | 0.386 |
i | 1 | 2 | 3 | 4 |
---|---|---|---|---|
3.931 | 0.230 | 0.691 | −0.959 | |
0.634 | 0.400 | 0.241 | 0.119 | |
0.488 | 0.213 | 0.132 | 0.026 |
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
Le, H.-T.; Chu, T.-C. Ranking Alternatives Using a Fuzzy Preference Relation-Based Fuzzy VIKOR Method. Axioms 2023, 12, 1079. https://doi.org/10.3390/axioms12121079
Le H-T, Chu T-C. Ranking Alternatives Using a Fuzzy Preference Relation-Based Fuzzy VIKOR Method. Axioms. 2023; 12(12):1079. https://doi.org/10.3390/axioms12121079
Chicago/Turabian StyleLe, Hanh-Thao, and Ta-Chung Chu. 2023. "Ranking Alternatives Using a Fuzzy Preference Relation-Based Fuzzy VIKOR Method" Axioms 12, no. 12: 1079. https://doi.org/10.3390/axioms12121079