Bas (Eng. Appl. Artif. Intell. (2025) 144 (110168)) claimed that, to the best of their knowledge, no mathematical programming-based method has been proposed in the literature to determine optimal secret parameters of public key cryptosystems in an interval or fuzzy environment. To fill this gap, Bas proposed an interval mathematical programming problem (IMPP) and a fuzzy mathematical programming problem (FMPP) for the Rivest, Shamir, Adleman (RSA) public key cryptosystem. Bas also proposed a method to solve IMPP as well as a method to solve FMPP. In these methods, firstly, IMPP and FMPP are transformed into crisp mathematical programming problem (CMPP). Then, using an optimal solution of the transformed CMPP, an optimal solution of original IMPP and FMPP is obtained. In this note, it is pointed out that the subtraction operation of triangular fuzzy numbers (TFNs), used in the method to transform FMPP into CMPP, is not valid. Due to which the transformed CMPP is not equivalent to the original FMPP. Hence, Bas's method to solve FMPP is not valid in its present form. Keeping the same in mind, Bas's method is modified by replacing the invalid subtraction operation of TFNs with the valid subtraction operation. Furthermore, using the modified method, correct results for the RSA public key cryptosystem corresponding to existing data, considered by Bas to illustrate their proposed method, are obtained. It is pertinent to mention that the subtraction operation of intervals, used in Bas's method to transform IMPP into CMPP, is valid.

Karmakar et al. (Appl. Soft Comput. 106 (2021) 107357) proposed a distance measure to evaluate the distance between type-2 intuitionistic fuzzy (T2IF) sets. For proving the validity of proposed distance measure; Karmakar et al. proved that their proposed distance measure satisfies the four necessary properties (bounded, reflexive, symmetric and transitive). Karmakar et al. also proposed a similarity measure, based upon the proposed distance measure, to measure the degree of similarity between two T2IF sets. Furthermore, based upon the proposed similarity measure, Karmakar et al. proposed a method to solve matrix games with T2IF payoffs. Finally, Karmakar et al. solved a real-life Biogas plant implementation problem in West Bengal, India by their proposed method. In this paper, it is pointed out that Karmakar et al. have used a mathematically incorrect result to prove the bounded property. In actual case, the bounded property is not satisfying for Karmakar et al.’s distance measure. Therefore, the distance measure, proposed by Karmakar et al., is not valid. Moreover, it is pointed out that T2IF sets corresponding to linguistic terms (Very high, High, Medium, Low, Very low), considered to solve real-life Biogas plant implementation problem in West Bengal, India, are not correct. Hence, the result of the real-life Biogas plant implementation problem, obtained by Karmakar et al., is not correct. Finally, for resolving the invalidity of Karmakar et al.’s distance measure, the required modification is suggested.

Bera and Mondal (Eng Appl Artif Intell 123:106396, 2023) proposed an approach to transform a triangular intuitionistic fuzzy number (TIFN) into its equivalent real number. Bera and Mondal claimed that their proposed approach is better than the existing approach (Int J Intell Syst 34:3–23, 2019). To validate this claim, Bera and Mondal, firstly, transformed an intuitionistic fuzzy multi-objective transportation problem (TP) into its equivalent a crisp multi-objective TP by their proposed approach as well as by existing approach. Then, Bera and Mondal shown that the results of the crisp multi-objective TP, obtained on applying their proposed approach, are better than the results of the crisp multi-objective TP by existing approach. Keeping the same in mind, in future, other researchers may prefer to use Bera and Mondal’s approach instead of existing approach for transforming a real-life intuitionistic fuzzy mathematical model (IFMM) into its equivalent crisp mathematical model (CMM). In this note, it is pointed out that Bera and Mondal’s approach cannot be used to transform each TIFN into its equivalent real number. It can be used only to transform such a TIFN for which a specific condition will be satisfied. Hence, Bera and Mondal’s approach cannot be used to transform each IFMM into its equivalent CMM. It can be used only if for each TIFN, used in the IFMM, a specific condition will be satisfied.

Khatter (Soft Comput (2020) 24:16847–16867), firstly, proposed a ranking function to transform a single-valued triangular neutrosophic number into its equivalent real number. Then, using the proposed ranking function, Khatter (2020) proposed an approach to solve neutrosophic linear programming problems. Inspired by Khatter (2020)’s work, Tamilarasi and Paulraj (Soft Comput (2022) 26:8497–8507), proposed an alternative ranking function to transform a single-valued triangular neutrosophic number into its equivalent real number. Tamilarasi and Paulraj (2022) also shown that if the existing neutrosophic linear programming problems (Khatter 2020) are solved by considering their proposed ranking function in Khatter (2020)’s approach. Then, the obtained results are better than the existing results (Khatter 2020). In future, other researchers may use Tamilarasi and Paulraj (2022)’s ranking function in their research work. In this paper, it is shown that Tamilarasi and Paulraj (2022) have used incorrect definition of a single-valued triangular neutrosophic number to propose their ranking function. Therefore, Tamilarasi and Paulraj (2022)’s ranking function is not valid and hence, it is inappropriate to use Tamilarasi and Paulraj (2022)’s method for solving neutrosophic linear programming problems. Also, it is shown that if in the method, used by Tamilarasi and Paulraj (2022) to obtain their proposed ranking function, the correct definition (Soft Comput (2022) 26:921–936) of a single-valued triangular neutrosophic number is considered then Tamilarasi and Paulraj (2022)’s method fails to find a ranking function. Hence, it is impossible to resolve the inappropriateness of Tamilarasi and Paulraj (2022)’s method for solving neutrosophic linear programming problems. Furthermore, it is pointed out that one may use Bhatia et al.’s (Soft Comput (2022) 26:8479–8495) approach to solve real-life neutrosophic linear programming problems instead of Khatter (2020)’s approach as Bhatia et al. (2022) have pointed out that some mathematically incorrect results are considered in Khatter (2020)’s approach.

Chhibber et al. (2022) proposed a method to solve a special type of non-linear multi-objective intuitionisticfuzzy transportation problems (NLMOIFTPs). Based on the results of a numerical problem, Chhibber et al. (2022) claimed that their proposed method is better than several existing methods. In this paper, it is pointed out thatChhibber et al. (2022)’s method isnot valid as Chhibber et al. (2022) have considered several mathematicallyincorrectresults to propose their method. Also, it is pointed out that in theactualcase, the numerical problem, considered by Chhibber et al. (2022), is a combination of three independent non-linear single-objective intuitionisticfuzzy transportation problems (IFTPs). However, Chhibber et al. (2022) have solved the numerical problem as NLMOIFTP which is mathematically incorrect.

An approach (a generalization of the classical simplex algorithm) is proposed to solve hesitant fuzzy linear programming problems (HFLPPs). In this paper, we pointed out that much computational effort is required to solve HFLPPs by the existing approach. Moreover, to reduce the computational efforts, an alternative approach is proposed to solve HFLPPs. Furthermore, some other advantages of the proposed alternative approach (PrAlApp) over the existing approach are discussed. Finally, an existing HFLPP is solved by the PrAlApp.

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.

Seikh and Dutta (Soft Comput 26: 921–936, 2022) claimed that there does not exist any approach to solve single-valued trapezoidal neutrosophic (SVTrN) matrix games (matrix games in which each payoff is represented by a SVTrN number). To fill this gap, Seikh and Dutta, firstly, proposed SVTrN non-linear programming problems (NLPPs) corresponding to Player-I and Player-II. Then, Seikh and Dutta proposed two different approaches to transform the proposed SVTrN NLPPs into crisp linear programming problems (CLPPs). Finally, Seikh and Dutta claimed that an optimal solution of the transformed CLPPs also represents an optimal solution of SVTrN NLPPs. Brikaa (Soft Comput 26: 9137–9139, 2022) pointed out that a mathematically incorrect result is considered in Seikh and Dutta’s first approach to transform SVTrN NLPPs into CLPPs. Therefore, the transformed CLPPs are not equivalent to SVTrN NLPPs. Hence, it is mathematically incorrect to assume that an optimal solution of the transformed CLPPs also represents an optimal solution of SVTrN NLPPs. Brikaa also proposed an approach to transform the SVTrN NLPPs into CLPPs. In this paper, it is pointed out that on solving the CLPPs, obtained by Brikaa’s approach corresponding to SVTrN NLPPs of Player-I and Player-II, different optimal value is obtained. Also, it is pointed out that on solving the CLPPs, obtained by Seikh and Dutta’s second approach corresponding to SVTrN NLPPs of Player-I and Player-II, a different optimal value is obtained. However, in the actual case, the obtained optimal value should be the same as in the literature; it is proved that the CLPPs corresponding to Player-I and Player-II represent a primal–dual pair. This indicates that neither the CLPPs, obtained by Brikaa’s approach nor the CLPPs, obtained by Seikh and Dutta’s second approach, are equivalent to the SVTrN NLPPs of Player-I and Player-II. Hence, it is inappropriate to use the CLPPs, obtained by Brikaa’s approach as well as Seikh and Dutta’s second approach to find an optimal solution for the SVTrN NLPPs of Player-I and Player-II. Also, Brikaa’s approach as well as Seikh and Dutta’s second approach is modified to transform SVTrN NLPPs into their equivalent CLPPs. Furthermore, it is proved that the CLPPs corresponding to SVTrN NLPPs of Player-I and Player-II, obtained by the proposed modified approaches, represent a primal–dual pair. Finally, the correct result of a SVTrN matrix game, considered by Seikh and Dutta to illustrate their approaches, is obtained by the proposed modified approaches.

Abstract Muthuperumal et al. (Soft Comput (2020) 24: 18689–18698) proposed two different approaches to find an initial fuzzy basic feasible solution of unbalanced triangular fuzzy transportation problems (unbalanced transportation problems in which each parameter is represented by a triangular fuzzy number). Since, Muthuperumal et al. have claimed that less computational efforts are required to apply their proposed approaches as compared to other existing approaches. Therefore, in future, other researchers may use Muthuperumal et al.’s approaches to find an initial fuzzy basic feasible solution of real-life unbalanced triangular fuzzy transportation problems. However, it is observed that in actual case, the initial fuzzy basic solution, obtained by Muthuperumal et al.’s approaches, is not a fuzzy feasible solution. Hence, it is not appropriate to use Muthuperumal et al.’s approaches. The aim of this note is to point out the inconsistencies that exist in Muthuperumal et al.’s approaches.

Shakeel et al. (Soft Comput 24: 1585–1597, 2020) proposed the concept of a Pythagorean uncertain linguistic hesitant fuzzy set (PULHFS), some arithmetic operations of Pythagorean uncertain linguistic hesitant fuzzy sets (PULHFSs), an approach for comparing PULHFSs and a Pythagorean uncertain linguistic hesitant fuzzy weighted averaging (PULHFWA) operator as well as it extensions. Also, using the proposed comparing approach and the proposed aggregation operators, Shakeel et al. proposed a method for solving multi-attribute group decision making (MAGDM) problems. In future other researchers may use Shakeel et al.’s work in their research work. However, it is observed that the approach for comparing PULHFSs and aggregation operators, proposed by Shakeel et al. (Soft Comput 24: 1585–1597, 2020), are not appropriate. Hence, the method for solving MAGDM problems, proposed by Shakeel et al., is also not appropriate. The aim of this note is to make the researchers aware of the inappropriateness of Shakeel et al. work. Furthermore, point out that to resolve the inappropriateness of Shakeel et al. work (Soft Comput 24: 1585–1597, 2020) is a challenging open research problem.

The aim of each company/industry is to provide a final product to customers at the minimum possible cost, as well as to protect the environment from degradation. Ensuring the shortest travel distance between involved locations plays an important role in achieving the company's/industry's objective as (i) the cost of a final product can be minimized by minimizing the total distance travelled (ii) finding the shortest distance between involved locations will require less fuel than the longest distance between involved locations. This will eventually result in lesser degradation of the environment. Hence, in the last few years, various algorithms have been proposed to solve different types of shortest path problems. A recently proposed algorithm for solving interval-valued Pythagorean fuzzy shortest path problems requires excessive computational efforts. Hence, to reduce the computational efforts, in this paper, firstly, an alternative lexicographic method is proposed for comparing interval-valued Pythagorean fuzzy numbers. Then, using the proposed lexicographic comparing method, a new approach (named as Mehar approach) is proposed to solve interval-valued Pythagorean fuzzy shortest path problems. Furthermore, the superiority of the proposed lexicographic comparing method, as well as the proposed Mehar approach, is discussed.

The supply chain (SC) network is prone to disturbance due to various uncertainties associated with their subsystems. The COVID-19 outbreak has exposed the global vulnerability of the supply chain network. The current pandemic has severely affected almost every SC network because its members are situated at the international level. One of the reasons for SC network failure is the deterministic assumptions of different parameters. A realistic SC network model requires the use of the uncertain value of the parameters, which can be further captured by fuzzy numbers. This paper discusses the possibilistic moment of several nonlinear types of fuzzy numbers that are important for SC network modeling. We give closed-form possibilistic moments’ expression for various types of fuzzy numbers that are very similar to the moment’s properties in probability theory and stochastic process. We then illustrate the application of proposed fuzzy numbers by solving an inventory model. This paper also provides results related to the EPQ inventory model in a fuzzy possibilistic setup.

Yue (Computers & Industrial Engineering 102 (2016) 233-245) proposed a geometric approach for solving IVIFMAGDM (interval-valued intuitionistic fuzzy multiattribute group decision making) problems and used it to solve a real-life supplier selection problem. In future, the other researchers may use the same approach to solve other real-life IVIFMAGDM problems. However, after a deep study, it is observed that this approach fails to rank the alternatives. Therefore, the aim of this note is to make the researchers aware that it is scientifically incorrect to use this approach for solving real-life IVIFMAGDM problems.

Li etal. (2016) proposed a least absolute deviation method to find such fuzzy regression models whose all the parameters are trapezoidal fuzzy numbers. Since, the method, proposed by Li et al., is published in Engineering Applications of Artificial Intelligence (a very reputed International Journal) as well as it is easy to apply Li et al. method in real life problems. So, other researchers may be attracted to apply the method, proposed by Li et al., to find the solution of real life problems. However, after a deep study of the work done by Li et al., it is noticed that the multiplication of two trapezoidal fuzzy numbers, proposed by Li et al. as well as used in this method, is not valid. Hence, the aim of this note is to make the researchers aware about the valid multiplication of trapezoidal fuzzy numbers that should be used in Li et al. method.

Basaran [Calculating fuzzy inverse matrix using fuzzy linear equation system, Applied Soft Computing, 12 (2012), 1810–1813] proposed a method for finding the inverse of a fuzzy matrix by assuming all the elements of the fuzzy inverse matrix as non-negative fuzzy numbers, while some of the elements of fuzzy matrix inverse may also be negative fuzzy numbers. Keeping the same in mind, Mosleh and Otadi [A discussion on "Calculating fuzzy inverse matrix using fuzzy linear equation system", Applied Soft Computing, 28 (2015), 511–513] assumed (i, j) element x˜ij=(xij,αij,βij) of the fuzzy inverse matrix as a non-negative fuzzy number if the value of xij obtained by Basaran's approach, is a non-negative real number and a negative fuzzy number if the value of xij is negative real number. In this paper, it is shown that the fuzzy multiplicative inverse of a fuzzy matrix, obtained by considering this assumption, is also not an exact fuzzy multiplicative inverse. Furthermore, the required modifications, in Mosleh and Otadi's approach, to obtain the exact multiplicative inverse of a fuzzy matrix are suggested.