Abstract
Marimuthu and Mahapatra (Soft Comput 25:9859–9871, 2021) claimed that several methods are proposed in the literature to solve such multi-criteria decision-making problems in which the rating value of each alternative over each attribute is represented by a generalized trapezoidal fuzzy number. However, the ranking methods, used in existing fuzzy multi-criteria decision-making methods, fail to distinguish two distinct generalized trapezoidal fuzzy numbers. Therefore, it is inappropriate to use existing fuzzy multi-criteria decision-making methods. To resolve the inappropriateness of fuzzy multi-criteria decision-making methods, Marimuthu and Mahapatra, first, defined some score functions to transform a generalized trapezoidal fuzzy number into its equivalent real number. Then, Marimuthu and Mahapatra stated and proved some results regarding their proposed score functions. Thereafter, using the proposed results, Marimuthu and Mahapatra proposed a ranking method for comparing generalized trapezoidal fuzzy numbers. Marimuthu and Mahapatra also proved that their proposed ranking method will never fail to distinguish two distinct generalized trapezoidal fuzzy numbers. Finally, Marimuthu and Mahapatra proposed a method, based on their proposed ranking method, to solve fuzzy multi-criteria decision-making problems. Jeevaraj (Soft Comput 26:11225–11230, 2022) considered some counterexamples to show that Marimuthu and Mahapatra’s results are not correct. Jeevaraj also considered some counterexamples to show that Marimuthu and Mahapatra’s ranking method also fails to distinguish two distinct generalized trapezoidal fuzzy numbers. Furthermore, Jeevaraj proposed the correct results corresponding to Marimuthu and Mahapatra’s results as well as Jeevaraj suggested the required modification in Marimuthu and Mahapatra’s ranking method. Finally, Jeevaraj proved that the modified ranking method will never fail to distinguish two distinct generalized trapezoidal fuzzy numbers. In the future, researchers may use the results and the ranking method, proposed by Jeevaraj, to solve real-life fuzzy multi-criteria decision-making problems. However, in this paper, some counterexamples are considered to show that Jeevaraj’s modified results are also not correct. In addition, some counterexamples are considered to show that the modified ranking method, proposed by Jeevaraj, also fails to distinguish two distinct generalized trapezoidal fuzzy numbers. Hence, it is inappropriate to use the results and the ranking method, proposed by Jeevaraj, for solving real-life fuzzy multi-criteria decision-making problems. Furthermore, the correct results, corresponding to Marimuthu and Mahapatra’s results, are stated and proved. Finally, it is proved that Marimuthu and Mahapatra’s ranking method as well as Jeevaraj’s ranking method will never fail to distinguish two distinct generalized trapezoidal fuzzy numbers having the same heights. However, both methods may fail to distinguish two distinct generalized trapezoidal fuzzy numbers having different heights. It is pertinent to mention that as there exist several ranking methods which will never fail to distinguish two distinct generalized trapezoidal fuzzy numbers. So, one may use any such ranking method to compare generalized trapezoidal fuzzy numbers.
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References
Abbasbandy S, Hajjari T (2009) A new approach for ranking of trapezoidal fuzzy numbers. Comput Math Appl 57(3):413–419
Abbasi S, Rahmani AM (2023) Artificial intelligence and software modeling approaches in autonomous vehicles for safety management: a systematic review. Information 14(10):555
Abbasi S, Daneshmand-Mehr M, Ghane Kanafi A (2022) Designing sustainable recovery network of end-of-life product during the COVID-19 pandemic: a real and applied case study. Discrete Dyn Nat Soc. https://doi.org/10.1155/2022/6967088
Abu Arqub O (2017) Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations. Neural Comput Appl 28:1591–1610
Abu Arqub O, Singh J, Alhodaly M (2023a) Adaptation of kernel functions-based approach with Atangana–Baleanu–Caputo distributed order derivative for solutions of fuzzy fractional Volterra and Fredholm integrodifferential equations. Math Methods Appl Sci 46(7):7807–7834
Abu Arqub O, Singh J, Maayah B, Alhodaly M (2023b) Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag–Leffler kernel differential operator. Math Methods Appl Sci 46(7):7965–7986
Alshammari M, Al-Smadi M, Arqub OA, Hashim I, Alias MA (2020) Residual series representation algorithm for solving fuzzy duffing oscillator equations. Symmetry 12(4):572
Chen L-H, Lu H-W (2001) An approximate approach for ranking fuzzy numbers based on left and right dominance. Comput Math Appl 41(12):1589–1602
Cheng C-H (1998) A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets Syst 95(3):307–317
Chu T-C, Tsao C-T (2002) Ranking fuzzy numbers with an area between the centroid point and original point. Comput Math Appl 43(1):111–117
Deng Y, Zhenfu Z, Qi L (2006) Ranking fuzzy numbers with an area method using radius of gyration. Comput Math Appl 51(6):1127–1136
Jeevaraj S (2022) A note on multi-criteria decision-making using a complete ranking of generalized trapezoidal fuzzy numbers. Soft Comput 26:11225–11230
Kumar A, Singh P, Kaur P, Kaur A (2010) Ranking of generalized trapezoidal fuzzy numbers based on rank, mode, divergence and spread. Turk J Fuzzy Syst 1(2):141–152
Kumar A, Singh P, Kaur P, Kaur A (2011) A new approach for ranking of L–R type generalized fuzzy numbers. Expert Syst Appl 38(9):10906–10910
Marimuthu D, Mahapatra GS (2021) Multi-criteria decision-making using a complete ranking of generalized trapezoidal fuzzy numbers. Soft Comput 25:9859–9871
Rezvani S (2013) A new method for ranking in areas of two generalized trapezoidal fuzzy numbers. Int J Fuzzy Log Syst 3:17–24
Yi P, Wang L, Li W (2019) Density-clusters ordered weighted averaging operator based on generalized trapezoidal fuzzy numbers. Int J Intell Syst 34(11):2970–2987
Yu VF, Chi HTX, Dat LQ, Phuc PNK, Wen Shen C (2013) Ranking generalized fuzzy numbers in fuzzy decision making based on the left and right transfer coefficients and areas. Appl Math Model 37(16):8106–8117
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Authors would like to thank to Area Editor “Prof. Yichuan Yang” and the anonymous reviewers for their valuable and constructive suggestions to improve the quality of the paper.
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Ahuja, R., Kumar, A. & Appadoo, S.S. Multi-criteria decision-making using a complete ranking of generalized trapezoidal fuzzy numbers: modified results. Soft Comput (2024). https://doi.org/10.1007/s00500-023-09607-6
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DOI: https://doi.org/10.1007/s00500-023-09607-6