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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. [ABSTRACT FROM AUTHOR]

A new approach for vague risk analysis based on the ranking of trapezoidal vague sets is proposed. Firstly, a new method for ranking of vague sets is presented. Then, the proposed method is applied to developed a new method for dealing with vague risk analysis problems. This analysis helps us to find out the probability of failure of each components of combustion system, which could be used for managerial decision making and future system maintenance strategy. The proposed method provides a useful way for handling vague risk analysis problems. [ABSTRACT FROM AUTHOR]

Abstract: In the literature, several algorithms are proposed for solving the transportation problems in fuzzy environment but in all the proposed algorithms the parameters are represented by normal fuzzy numbers. Chen [Operations on fuzzy numbers with function principal, Tamkang Journal of Management Science 6 (1985) 13–25] pointed out that in many cases it is not to possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers. There are several papers in the literature in which generalized fuzzy numbers are used for solving real life problems but to the best of our knowledge, till now no one has used generalized fuzzy numbers for solving the transportation problems. In this paper, a new algorithm is proposed for solving a special type of fuzzy transportation problems by assuming that a decision maker is uncertain about the precise values of transportation cost only but there is no uncertainty about the supply and demand of the product. In the proposed algorithm transportation costs are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed algorithm a numerical example is solved and the obtained results are compared with the results of existing approaches. Since the proposed approach is a direct extension of classical approach so the proposed approach is very easy to understand and to apply on real life transportation problems for the decision makers. [Copyright &y& Elsevier]

Abstract: In the literature, several methods are proposed for solving transportation problems in fuzzy environment but in all the proposed methods the parameters are represented by normal fuzzy numbers. [S.H. Chen, Operations on fuzzy numbers with function principal, Tamkang Journal of Management Sciences 6 (1985) 13–25] pointed out that in many cases it is not to possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers. There are several papers in the literature in which generalized fuzzy numbers are used for solving real life problems but to the best of our knowledge, till now no one has used generalized fuzzy numbers for solving the transportation problems. In this paper, a new method is proposed for solving fuzzy transportation problems by assuming that a decision maker is uncertain about the precise values of the transportation cost, availability and demand of the product. In the proposed method transportation cost, availability and demand of the product are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method a numerical example is solved and the obtained results are compared with the results of existing methods. Since the proposed method is a direct extension of classical method so the proposed method is very easy to understand and to apply on real life transportation problems for the decision makers. [Copyright &y& Elsevier]

Abstract: Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. Cheng (Cheng, C. H. (1998). A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets and Systems, 95, 307–317) pointed out that the proof of the statement “Ranking of generalized fuzzy numbers does not depend upon the height of fuzzy numbers” stated by Liou and Wang (Liou, T. S., & Wang, M. J. (1992). Ranking fuzzy numbers with integral value. Fuzzy Sets and Systems, 50, 247–255) is incorrect. In this paper, by giving an alternative proof it is proved that the above statement is correct. Also with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Chen, S. M., & Chen, J. H. (2009). Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads. Expert Systems with Applications, 36, 6833–6842) is incorrect. The main aim of this paper is to modify the Liou and Wang approach for the ranking of L–R type generalized fuzzy numbers. The main advantage of the proposed approach is that the proposed approach provide the correct ordering of generalized and normal fuzzy numbers and also the proposed approach is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfy all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Wang, X., & Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets and Systems, 118, 375–385). [Copyright &y& Elsevier]

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Expert Syst Appl 36:6833-6842, ) is incorrect. The main aim of this paper is to propose a new approach for the ranking of L- R type generalized fuzzy numbers. The proposed ranking approach is based on rank and mode so it is named as RM approach. The main advantage of the proposed approach is that it provides the correct ordering of generalized and normal fuzzy numbers and it is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Fuzzy Sets Syst 118:375-385, ). [ABSTRACT FROM AUTHOR]

Abstract: Ranking of fuzzy numbers play an important role in decision-making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples it is proved that the results proposed by Chen and Tang [C.C. Chen, H.C. Tang, Ranking of nonnormal -norm trapezoidal fuzzy numbers with integral value, Computers and Mathematics with Applications 56 (2008) 2340–2346] are applicable only for the nonnormal -norm trapezoidal fuzzy numbers with equal heights and a new approach is proposed for the ranking of nonnormal -norm trapezoidal fuzzy numbers with different heights. The results proposed by Chen and Tang are modified and to illustrate the proposed approach the counter examples are solved using the proposed approach. It is also shown that the proposed approach and the results, obtained by using the proposed approach, are valid. [Copyright &y& Elsevier]