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The reliability of an industrial system plays a significant role in new technological era where everyone is concerned about the performance of associated industry. With the frequent demands of the customers, the job of the production industries becomes more tedious to produce the required products at a high rate. To fulfill the customer's demand, the initial focus of the industries is to work well without any interruption/failure in the entire production process. In this paper, Reliability, Availability, MTTF, MTTR, MTBF, ENOF is proposed for an industrial system namely "Injection moulding machine". For this, the membership function for right triangular generalized fuzzy numbers (RTrGFN) is proposed with the certain level of confidence. The real data of the Injection moulding machine is taken to validate the proposed methodology. The input data is extracted from the records/maintenance sheets of several years and found uncertainty due to the loss of any information or human mistakes. The parameters of the system like failure rate and repair time is retrieved. Based on data, AND-OR combination for the system is constructed. The lambda-tau methodology along with RTrGFN and its corresponding arithmetic operations is used for the performance analysis of the considered "Injection moulding machine". For better understanding the results are discussed with the aid of graphs. Also after seeing the result one can conclude that this methodology is one of the best methodology for the performance analysis of a conventional system. Also authors have added the highlights and future scope of the research work at the end of the manuscript. [ABSTRACT FROM AUTHOR]

To the best of author's knowledge, only three approaches are proposed in the literature to solve fully fuzzy linear fractional transportation problems. (Linear fractional transportation problems in which each known parameter is represented by a fuzzy number.) In this paper, it is pointed out that it is inappropriate to use any of these existing approaches as some mathematical incorrect assumptions are considered in these existing approaches. Also, a new approach (named as Mehar approach) is proposed to solve fully fuzzy linear fractional transportation problems. Furthermore, an exact fuzzy optimal solution of an existing fully fuzzy linear fractional transportation problem is obtained by the proposed Mehar approach. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to investigate the performance of a turbine structure of the oil and gas Egyptian company in terms of reliability, mean time to failure (MTTF), mean time to repair (MTTR) and mean time between failures (MTBF) under fuzzy environment and working criteria. This paper examines the impact of the failure of various components on the complete turbine structure of the oil and gas system. Design/methodology/approach: To overcome the problem of uncertain behavior of available data for various components, the right triangular generalized fuzzy number (RTrGFN) is proposed to be taken into the account to express the uncertainty which attains some tolerance in data. Furthermore, reliability indices are calculated with the help of the Lambda Tau method and the arithmetic operations on right generalized triangular fuzzy numbers (RTrGFN). Findings: This paper explores the reliability of a repairable 3 out of 4 structure of turbines and along with the other parameters namely MTTF, MTTR and MTBF; under a fuzzy environment. Failure rates and repair times are expected to be exponential. The ranking of components of the structure is being found to decide the priority for maintenance. Originality/value: This paper investigates the performance of the system with different spread/tolerance like 15%, 25% and 50% of crisp data. It helps to predict realistic results in the range value. To enhance the system's performance, the most important item of the system requires greater attention. For this, the authors find the sensitive part by ranking. For ranking, an extended approach has been developed to find the sensitive unit of the system by using the right triangular generalized fuzzy number. This paper explores the most and least sensitive component of the system, which helps the maintenance department to plan the maintenance action. [ABSTRACT FROM AUTHOR]

To the best of author's knowledge, only one approach is proposed in the literature to solve fuzzy linear fractional minimal cost flow problems (minimal cost flow problems in which each known arc cost is represented either by a triangular fuzzy number or a trapezoidal fuzzy number). In this paper, the mathematical incorrect assumptions, considered in the existing approach to solve fuzzy linear fractional minimal cost flow problems, are pointed out. Also, by generalizing an existing approach for solving fuzzy linear fractional programming problems, an approach (named as Mehar approach) is proposed to solve fuzzy linear fractional minimal cost flow problems. Furthermore, two numerical examples are solved to illustrate the proposed Mehar approach. [ABSTRACT FROM AUTHOR]

Enayattabr et al. (Journal of Intelligent and Fuzzy Systems 37 (2019) 6865– 6877) claimed that till now no one has proposed an approach to solve interval-valued trapezoidal fuzzy all-pairs shortest path problems (all-pairs shortest path problems in which distance between every two nodes is represented by an interval-valued trapezoidal fuzzy number). Also, to fill this gap, Enayattabr et al. proposed an approach to solve interval-valued trapezoidal fuzzy all-pairs shortest path problems. In this paper, an interval-valued trapezoidal fuzzy shortest path problem is considered to point out that Enayattabr et al.'s approach fails to find correct shortest distance between two fixed nodes. Hence, it is inappropriate to use Enayattabr et al.'s approach in its present from. Also, the required modifications are suggested to resolve this inappropriateness of Enayattabr et al.'s approach. [ABSTRACT FROM AUTHOR]

Classical sets are used commonly to consider reliability. Because of the uncertainty in the data (which considered in the present paper) classical sets fail to describe the reliability accurately. Uncertainty leads to fluctuation in the actual situation of the structure. Fuzzy logic method attempts to test system reliability with the benefit of membership function. Within this context, specific problems of reasoning-based approaches are studied, explored and correlated with standard reliability approaches. In this paper Generalized Trapezoidal Fuzzy numbers (GTrFN) are used to assess the structure's fuzzy reliability. The reliability of each event is assigned with different level of satisfaction and some improved operations on the generalized trapezoidal fuzzy numbers (GTrFN) are used to calculate the fuzzy boundaries for the resultant reliability of the final event along with the degree of satisfaction. Also the results are compared to demonstrate the application of the improved operations on Generalized Trapezoidal Fuzzy Numbers (GTrFN). The obtained results converge to more precise interval values as compare to the vague fuzzy number. [ABSTRACT FROM AUTHOR]

Mahmoodirad et al. (Soft Comput, 2018. 10.1007/s00500-018-3115-z) proposed an approach for solving fully intuitionistic fuzzy transportation problems (FIFTPs) (transportation problems in which each parameter is represented as a triangular intuitionistic fuzzy number). In this approach, firstly, an unbalanced fully intuitionistic fuzzy transportation problem (FIFTP) is transformed into a balanced FIFTP, and then the intuitionistic fuzzy (IF) optimal solution of the transformed balanced FIFTP is obtained. In this paper, it is shown that Mahmoodirad et al.'s approach fails to transform an unbalanced FIFTP and hence, Mahmoodirad et al.'s approach fails to find the IF optimal solution of unbalanced FIFTPs. It is obvious that to overcome this limitation of Mahmoodirad et al.'s approach, there is need to propose a method to transform an unbalanced FIFTP into a balanced FIFTP. Therefore, in this paper, a new method (named as JMD method) is proposed to transform an unbalanced FIFTP into a balanced FIFTP. [ABSTRACT FROM AUTHOR]

To the best of our knowledge, only the existing approach (Rabiei et al. in Soft Comput 18(10), 2043–2059, 2014) has been proposed to construct a full interval-valued fuzzy linear regression model [a regression model when the observation of the response, independent variables as well as the regression coefficients are triangular interval-valued fuzzy numbers (TIVFNs)]. However, after a deep study, it is observed that a mathematical incorrect assumption has been considered in this approach. Furthermore, it is observed that to resolve this mathematical incorrect assumption, there is a need to propose the multiplication of an unrestricted TIVFN (regression coefficient) with a restricted TIVFN [observed values of independent variable(s)]. Keeping the same in mind, in this paper, the same type of multiplication is proposed, and with the help of proposed multiplication, a modified approach is proposed to construct a full interval-valued fuzzy regression model. Also, the modified results of some existing real-life problems are obtained with the help of the modified approach. [ABSTRACT FROM AUTHOR]

Bharti and Singh (International Journal of Fuzzy Systems20 (2018), 1511-1522) proposed a method for solving a special type of interval-valued intuitionistic fuzzy transportation problems (IVIF-TPs) (transportation problems (TPs) in which the quantity of the product to be supplied is represented as a real number, whereas, all the other parameters are represented as interval-valued triangular intuitionistic fuzzy numbers (IVTIFNs)). In this note, an interval-valued intuitionistic fuzzy transportation problem (IVIF-TP) is solved by Bharti and Singh's method and shown that more than one IVTIFNs, representing the optimal interval-valued intuitionistic fuzzy (IVIF) transportation cost is obtained, which is mathematically incorrect as the obtained distinct IVTIFNs has different physical meanings. Also, it is pointed out that to resolve this flaw of Bharti and Singh's method may be considered as a challenging open research problem. [ABSTRACT FROM AUTHOR]

Abdel-Basset et al. (Neural Computing and Applications, 2018, https://doi.org/10.1007/s00521-018-3404-6) proposed methods for solving different types of neutrosophic linear programming problems (NLPPs) (NLPPs in which some/all the parameters are represented as trapezoidal neutrosophic numbers (TrNNs)). Abdel-Basset et al. also pointed out that as a trapezoidal fuzzy number is a special case of trapezoidal neutrosophic number. Therefore, the fuzzy linear programming problems which can be solved by the existing methods (Ganesan and Veermani, Ann Oper Res, 2006, 143 : 305-315; Ebrahimnejad and Tavana, Appl Math Model, 2014, 38 : 4388-4395; Kumar et al., 2011, Appl Math Model, 35 : 817-823; Satti et al., Int J Decis Sci, 7 : 312-33) can also be solved by thier proposed method. In addition to that, to show the advantages of their proposed method over the existing methods (Ganesan and Veermani, Ann Oper Res, 2006, 143 : 305-315; Ebrahimnejad and Tavana, Appl Math Model, 2014, 38 : 4388-4395; Kumar et al., 2011, Appl Math Model, 35 : 817-823; Satti et al., Int J Decis Sci, 7 : 312-33), Abdel-Basset et al. solved the same fuzzy linear programming problems by their proposed method as well as the existing methods (Ganesan and Veermani, Ann Oper Res, 2006, 143 : 305-315; Ebrahimnejad and Tavana, Appl Math Model, 2014, 38 : 4388-4395; Kumar et al., 2011, Appl Math Model, 35 : 817-823; Satti et al., Int J Decis Sci, 7 : 312-33) and shown that the results, obtained on applying by their proposed method are better than the results, obtained on applying the existing methods (Ganesan and Veermani, Ann Oper Res, 2006, 143 : 305-315; Ebrahimnejad and Tavana, Appl Math Model, 2014, 38 : 4388-4395; Kumar et al., 2011, Appl Math Model, 35 : 817-823; Satti et al., Int J Decis Sci, 7 : 312-33). After a deep study of Abdel-Basset et al. 's method, it is observed that Abdel-Basset et al. have considered several mathematical incorrect assumptions in their proposed method and hence, it is scientifically incorrect to use Abdel-Basset et al. 's method in its present form. The aim of this paper is to make the researchers aware about the mathematical incorrect assumptions, considered by Abdel-Basset et al. in their proposed method, as well as to suggest the required modifications in Abdel-Basset et al. 's method. [ABSTRACT FROM AUTHOR]

Bhardwaj and Kumar (J. Optim. Theory Appl. 163(2), 685-696 (2014)) pointed out that Khan et al. (J. Optim. Theory Appl. 159(2), 536-546 (2013)) have assumed some mathematical incorrect assumptions in their proposed method for solving fully fuzzy linear programming problems (FFLPPs). However, Khan et al. (J. Optim. Theory Appl. 173(1), 353-356 (2017)) replied that their proposed method is valid and no mathematical incorrect assumptions have been considered. The aim of this commentary is to make the researchers aware that Khan et al. (J. Optim. Theory Appl. 173(1), 353-356 (2017)) are misleading the others. In actual case, Khan et al. (J. Optim. Theory Appl. 159(2), 536-546 (2013)) have considered mathematical incorrect assumptions in their proposed method due to which some of the elements of the optimal simplex table, obtained by Khan et al. (J. Optim. Theory Appl. 159(2), 536-546 (2013)), are not triangular fuzzy numbers. [ABSTRACT FROM AUTHOR]

Dhanasekar et al. (International Journal of Fuzzy Systems 19 (2017) 1479-1491) proposed a fuzzy Hungarian MODI algorithm to solve fully fuzzy transportation problems. Dhanasekar et al. have used the standard multiplication of trapezoidal fuzzy numbers in their proposed method. In this paper, it is pointed out that the method, proposed by Dhanasekar et al., is not valid for standard multiplication of trapezoidal fuzzy numbers and is valid only if a special type of multiplication of fuzzy numbers is used. [ABSTRACT FROM AUTHOR]

There are very few methods in literature to deal with fully fuzzy linear fractional programming (FFLFP) problems. In this paper, it is pointed out that in the existing methods (Journal of Applied Mathematics and Computing, 27 (2008), 227-242; Yugoslav Journal of Operations Research, 22 (2012), 41-50) for solving FFLFP problems, an inappropriate ranking property is used which may lead to the erroneous results. To resolve the flaw of the existing methods, a new method is proposed for solving FFLFP problems, and is illustrated with the help of a numerical problem. [ABSTRACT FROM AUTHOR]

In this paper, it is pointed out that in the existing methods [Sarangam Majumdar, Numerical solutions of fuzzy complex system of linear equations, German Journal of Advanced Mathematical Sciences 1 (2013), 20-26] for solving fuzzy complex system of linear equations in which all the parameters except variables are represented by triangular/trapezoidal fuzzy complex numbers as well as the fuzzy complex system of linear equations in which all the parameters except coefficients are represented by triangular/trapezoidal fuzzy complex numbers, Majumdar has assumed that on subtracting a fuzzy number from itself, the obtained value is a zero fuzzy number as well as on dividing a fuzzy number by itself, the obtained value is a unit fuzzy number. To resolve these flaws of the existing methods, two new methods (named as Mehar method and Keerat method) are proposed for solving the same systems of fuzzy complex linear equations. [ABSTRACT FROM AUTHOR]

Suresh et al. (Solving intuitionistic fuzzy linear programming problems by ranking function, Journal of Intelligent and Fuzzy Systems, 27, 3081-3087, 2014) proposed the ranking function for comparing triangular intuitionistic fuzzy numbers (TIFNs) and applied this ranking function to solve different types of intuitionistic fuzzy linear programming problems (IFLPPs). In this note, it is pointed out that the ranking function proposed by authors is not valid. Hence, the results of intuitionistic fuzzy linear programming problems, obtained by using this ranking function, are also not valid. Further, the exact ranking function is obtained by modifying existing ranking function and using the exact ranking function, the exact results of intuitionistic fuzzy linear programming problems considered by Suresh et al., are obtained. [ABSTRACT FROM AUTHOR]

Li and Yang [D.F. Li and J. Yang, A difference-index based ranking bilinear programming approach to solving bimatrix games with payoffs of trapezoidal intuitionistic fuzzy numbers, Journal of Applied Mathematics 2013 (2013), 1-10] pointed out that there is no method in the literature for solving such bimatrix games in which payoffs are represented by intuitionistic fuzzy numbers and proposed a method for the same. In this paper, it is pointed out that Li and Yang have considered some mathematical incorrect assumptions in their proposed method. For resolving the shortcomings of Li and Yang's method, a new method (named as Mehar method) is proposed. Also, the exact optimal solution of the numerical problem, solved by Li and Yang by their proposed method, is obtained by the proposed Mehar method. [ABSTRACT FROM AUTHOR]

Ezzati et al. (2013) proposed a method for comparing triangular fuzzy numbers and using it, propose a new algorithm to find the fuzzy optimal solution of fully fuzzy linear programming problems with equality constraints. Ezzati et al. also claimed that the fuzzy optimal solution of fully fuzzy linear programming problems with inequality constraints can also be obtained by the same algorithm by transforming it into fully fuzzy linear programming problems with equality constraints. In this note, it is proved that the fully fuzzy linear programming problems with inequality constraints cannot be transformed into fully fuzzy linear programming problems with equality constraints and hence, the algorithm, proposed by Ezzati et al. to find the fuzzy optimal solution of fully fuzzy linear programming problems with equality constraints, cannot be used for finding the fuzzy optimal solution of fully fuzzy linear programming problems with inequality constraints. [ABSTRACT FROM AUTHOR]

We propose a novel approach to the solution of fuzzy matrix games with payoffs given as trapezoidal intuitionistic fuzzy numbers. We extend Li’s [36, Chapter 9] work based on a cut-set based method for finding an optimal solution to overcome the fact that the assumptions and properties assumed therein do not guarantee in general, first, the very existence of an optimal solution, and second, its attainment via a mathematical programming formulation proposed. We first briefly mention those problems in Li’s [36] approach, and then propose a new, corrected and general method, called the Mehar mehod, based on a modified mathematical programming formulation of a matrix game with payoffs represented by trapezoid intuitionistic fuzzy numbers. For illustration, we solve Li’s [36] example, and compare his and our results. [ABSTRACT FROM AUTHOR]

Li et al. (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. [ABSTRACT FROM AUTHOR]

It is well known that the ranking of generalized fuzzy numbers depend upon the height of fuzzy numbers. In this note, it is shown that the method, proposed by Rezvani [Applied Mathematics and Computation 262 (2015) 191-198] for ranking of generalized exponential trapezoidal fuzzy numbers, is independent from height of fuzzy numbers. Hence, it is not genuine to use this method for ranking of generalized exponential trapezoidal fuzzy numbers. [ABSTRACT FROM AUTHOR]

Several authors have proposed different methods for solving fuzzy minimum cost flow (MCF) problems. In this paper, some single and multi-objective fuzzy MCF problems are chosen which cannot be solved by using any of the existing methods and a new method is proposed for solving such type of problems. The main advantage of the proposed method over existing methods is that the fuzzy MCF problems which can be solved by using the existing methods can also be solved by the proposed method. But, there exist several fuzzy MCF problems which can be solved only by using the proposed method i.e., it is not possible to solve these problems by using the existing methods. To illustrate the proposed method and also to show the advantages of the proposed method over existing methods some single and multi-objective fuzzy MCF problems which cannot be solved by using the existing methods are solved by using the proposed method and the obtained results are discussed. [ABSTRACT FROM AUTHOR]

In this paper, the limitations and shortcomings of the existing methods for solving fuzzy solid transportation problem are pointed out and to overcome these shortcomings, a new method is proposed for solving fuzzy solid transportation problem. The advantages of the proposed method over the existing methods are discussed. To illustrate the proposed method, an existing fuzzy solid transportation problem is solved. Also, to show the application of the proposed method in real life problems an existing real life fuzzy solid transportation problem is solved by the proposed method. [ABSTRACT FROM AUTHOR]

The drawbacks of the existing methods to obtain the fuzzy optimal solution of such linear programming problems, in which coefficients of the constraints are represented by real numbers and all the other parameters as well as variables are represented by symmetric trapezoidal fuzzy numbers, are pointed out, and to resolve these drawbacks, a new method (named as Mehar method) is proposed for the same linear programming problems. Also, with the help of proposed Mehar method, a new method, much easy as compared to the existing methods, is proposed to deal with the sensitivity analysis of the same type of linear programming problems. [ABSTRACT FROM AUTHOR]

In this paper, a new method is proposed to find the fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems with mixed constraints. By using the proposed method the fuzzy optimal solution of FFLP problems with mixed constraints occurring in real life situations can be easily obtained. To illustrate the proposed method a numerical example is solved. The proposed method can be easily applied to find the fuzzy optimal solution of FFLP problems occurring in real life situations. [ABSTRACT FROM AUTHOR]

In this paper limitations of an existing method (M. Otadi and M. Mosleh, Solving fully fuzzy matrix equations, Applied Mathematical Modelling 2012) are pointed out and to overcome these limitations a new method is proposed. To illustrate the proposed method numerical examples are solved and the obtained results are discussed. [ABSTRACT FROM AUTHOR]

Abstract: To the best of our knowledge, there is no method in literature for solving such fully fuzzy linear programming (FLP) problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers. Also, to propose such a method, there is need to find the product of unrestricted L-R flat fuzzy numbers. However, there is no method in the literature to find the product of unrestricted L-R flat fuzzy numbers. In this paper, firstly the product of unrestricted L-R flat fuzzy numbers is proposed and then with the help of proposed product, a new method (named as Mehar’s method) is proposed for solving fully FLP problems. It is also shown that the fully FLP problems which can be solved by the existing methods can also be solved by the Mehar’s method. However, such fully FLP problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers can be solved by Mehar’s method but can not be solved by any of the existing methods. [Copyright &y& Elsevier]

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]

In this paper, we discuss some new numerical methods to solve a fully fuzzy linear system (FFLS) with triangular fuzzy numbers of the form $$ ( {m,\alpha ,\beta }) $$. Almost every existing method that intends to solve a FFLS confines the coefficient matrix and the solutions to be non-negative fuzzy numbers. The main intent of the proposed methods is to remove these restrictions and widen the scope of fuzzy linear systems in scientific applications. The methods are illustrated with the help of numerical examples and are conceptually easy to understand and apply in real life situations. [ABSTRACT FROM AUTHOR]

Abstract: In the literature, it is pointed out that it is better to use vague sets instead of fuzzy sets. Several authors have proposed different methods for solving such differential equations in which all the parameters are represented by fuzzy numbers but to the best of our knowledge till now no one have represented the same as vague sets. In this paper, a new representation of (α, β)-cut, named as JMD (α, β)-cut, is proposed and with the help of JMD (α, β)-cut a new method is proposed to find the analytical solution of vague differential equations. To show the application of proposed method in real life problems the vague Kolmogorov’s differential equations, obtained by using vague Markov model of piston manufacturing system, are solved by proposed method. Also, to show the advantage of JMD (α, β)-cut over existing (α, β)-cut the same vague Kolmogorov’s differential equations are solved by using the proposed method with the help of existing (α, β)-cut and it is shown that the obtained results are not necessarily vague sets while the results, obtained by using JMD (α, β)-cut, are always vague sets. [Copyright &y& Elsevier]

In this paper, we discuss two new computational techniques for solving a generalized fully fuzzy linear system (FFLS) with arbitrary triangular fuzzy numbers (m, ɑ, β). The methods eliminate the non-negative restriction on the fuzzy coefficient matrix that has been considered by almost every method in the literature and relies on the decomposition of the dual FFLS into a crisp linear system that can be further solved by a variety of classical methods. To illustrate the proposed methods, numerical examples are solved and the obtained results are discussed. The methods pose several advantages over the existing methods to solve a simple or dual FFLS. [ABSTRACT FROM AUTHOR]

Abstract: In published works on fuzzy linear programming there are only few papers dealing with stability or sensitivity analysis in fuzzy mathematical programming. To the best of our knowledge, till now there is no method in the literature to deal with the sensitivity analysis of such fuzzy linear programming problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method, named as Mehar’s method, is proposed for the same. To show the advantages of proposed method over existing methods, some fuzzy sensitivity analysis problems which may or may not be solved by the existing methods are solved by using the proposed method. [Copyright &y& Elsevier]

To the best of our knowledge, there is no method in the literature for solving such fuzzy transportation problems in which the different cost parameters are represented by different type of LR flat fuzzy numbers. In this paper, a new method is proposed for solving similar type of fuzzy transportation problems. To illustrate the proposed method and also to show the advantages of the proposed method some fuzzy transportation problems which can not be solved by using any of the existing method are solved. Also, it is shown that the fuzzy transportation problems which can be solved by the existing methods can also be solved by using the proposed method and the same results are obtained. [ABSTRACT FROM AUTHOR]

Abstract: There are several methods in the literature for solving transportation problems by representing the parameters as normal fuzzy numbers. Chiang [J. Chiang, The optimal solution of the transportation problem with fuzzy demand and fuzzy product, J. Inform. Sci. Eng. 21 (2005) 439–451] pointed out that it is better to represent the parameters as (λ, ρ) interval-valued fuzzy numbers instead of normal fuzzy numbers and proposed a method to find the optimal solution of single objective transportation problems by representing the availability and demand as (λ, ρ) interval-valued fuzzy numbers. In this paper, the shortcomings of the existing method are pointed out and to overcome these shortcomings, a new method is proposed to find solution of a linear multi-objective transportation problem by representing all the parameters as (λ, ρ) interval-valued fuzzy numbers. To illustrate the proposed method a numerical example is solved. The advantages of the proposed method over existing method are also discussed. [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]

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 (A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets and Systems 95 (1998) 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 (Ranking fuzzy numbers with integral value. Fuzzy Sets and Systems 50 (1992) 247-255) is not ture. 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 the results proposed by Chen and Chen (Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads. Expert Systems with Applications 36 (2009) 6833-6842) are not accurate. 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. [ABSTRACT FROM AUTHOR]

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 maximal flow problems. In this paper, the existing algorithm is modified to find fuzzy maximal flow between source and sink by representing all the parameters as generalized triangular fuzzy numbers. To illustrate the modified algorithm a numerical example is solved and the obtained results are compared with the existing results. If there is no uncertainty about the flow between source and sink then the proposed algorithm gives the same result as in crisp maximal flow problems. [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]

The aim of this note is to point out the error in the existing method [J.-X. Nan, M.-J. Zhang and D.-F. Li, A methodology for matrix games with payoffs of triangular intuitionistic fuzzy number, Journal of Intelligent and Fuzzy Systems, DOI: 10.3233/IFS-130956] and to suggest the required modifications to resolve this error. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]