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Real-world decision problems in decision analysis, system analysis, economics, ecology, and other fields are characterized by fuzziness and partial reliability of relevant information. In order to deal with such information, Prof. Zadeh suggested the concept of a Z-number as an ordered pair $Z=(A,B)$ of fuzzy numbers $A$ and $B$, the first of which is a linguistic value of a variable of interest, and the second one is a linguistic value of probability measure of the first one, playing a role of reliability of information. Decision making under Z-number based information requires ranking of Z-numbers. In this paper we suggest a human-like fundamental approach for ranking of Z-numbers which is based on two main ideas. One idea is to compute optimality degrees of Z-numbers and the other one is to adjust the obtained degrees by using a human being’s opinion formalized by a degree of pessimism. Two examples and a real-world application are provided to show validity of the suggested research. A comparison of the proposed approach with the existing methods is conducted.

This paper presents a method for ranking fuzzy numbers based on the comparison of expected intervals of these numbers. This relation is fuzzy and it verifies the distinguishability, rationality and robustness qualities. The term expected interval is extended to no normal fuzzy numbers, and this method then it allows to compare these type sets.

Zadeh introduced the concept of Z-numbers in 2011 to deal with imprecise information. In this regard, many research works have been published in an attempt to introduce some basic theoretical concepts of Z-numbers to model real-world problems. To understand the current challenges when dealing with Z-numbers and the feasibility of using Z-number in solving real-world problems, a comprehensive review of the existing work on Z-number is paramount. This paper consists of an overview of existing literature on Z-number and identifies some of the key areas that are required for further improvement.

The purpose of the paper is to study how to solve a type of matrix games with payoffs of triangular fuzzy numbers. In this paper, the value of a matrix game with payoffs of triangular fuzzy numbers has been considered as a variable of the triangular fuzzy number. First, based on two auxiliary linear programming models of a classical matrix game and the operations of triangular fuzzy numbers, fuzzy optimization problems are established for two players. Then, based on the order relation of triangular fuzzy numbers the fuzzy optimization problems for players are decomposed into three-objective linear programming models. Finally, using the lexicographic method maximin and minimax strategies for players and the fuzzy value of the matrix game with payoffs of triangular fuzzy numbers can be obtained through solving two corresponding auxiliary linear programming problems, which are easily computed using the existing Simplex method for the linear programming problem. It has been shown that the models proposed in this paper extend the classical matrix game models. A numerical example is provided to illustrate the methodology.

In this work, a fuzzy prey–predator system with time delay is proposed. The model consists of two preys and one predator. The biological coefficients/parameters are considered as imprecise in nature and quantified by triangular fuzzy numbers. We have studied the effect of gestation delay on the stability of the system in fuzzy environment. The signed distance method for the defuzzification of the proposed fuzzy prey–predator system is adopted. For the underlying fuzzy model, we have provided a solution procedure to find all possible equilibrium points and studied their stabilities in the fuzzy sense. It is observed that there are stability switches, and Hopf-bifurcation occurs when the delay crosses some critical value in fuzzy sense. Numerical illustrations are provided in crisp as well as fuzzy environment with the help of graphical presentations to support our proposed approach.

A fuzzy two person interval game problem is proposed and treated in this paper which is not easily tackled by the conventional methods. First, with respect to this pay-off values, a necessary and sufficient condition for the existence of a saddle point is proved. Based on interval value model, we are to find the value of interval game without saddle point. Finally, example is given to illustrate the procedure and to indicate the performance of the proposed method.

This paper proposes two new computational methods for solving linear programming problem under trapezoidal and triangular fuzzy uncertainties with equality constraints. The coefficients of the constraints and the objective functions are assumed to be crisp. However, the decision variables and the right-hand side vector of the constraints are considered as uncertain in nature. The concepts of fuzzy addition and subtraction have been used to develop the proposed methods. In the first method, the coefficients are considered as non-negative, whereas mixed coefficients (i.e. both negative and non-negative) are considered in the second method. The obtained results are compared with Behera et al. [D. Behera, K. Peters and S. A. Edalatpanah, Alternative methods for linear programming problem under triangular fuzzy uncertainty, Journal of Statistics and Management Systems, 25 (2022) 521–539; D. Behera, K. Peters, S. A. Edalatpanah and D. Qiu, New methods for solving imprecisely defined linear programming problem under trapezoidal fuzzy uncertainty, Journal of Information and Optimization Sciences, 42 (2021) 603–629], and Saati et al. [S. Saati, M. Tavana, A. Hatami-Marbini and E. Hajiakhondi, A fuzzy linear programming model with fuzzy parameters and decision variables, International Journal ofInformation and Decision Sciences, 7 (2015) 312–333.] for the validation.

This paper is devoted to investigating or computing the solution to one-dimensional partial fuzzy fractional order heat equation. In particular, one-dimensional fuzzy partial heat equation is hybridized into two equations by hybrid techniques along with the concept of parametric fuzzy number. For this investigation, a hybrid method of decomposition due to Adomian and Laplace transform is used. The considered techniques are presented for the computation of series of solutions of partial fractional order heat equation. The applied techniques have also provided the accuracy, simplicity and efficiency as compared to other existing methods. Finally, some illustrations are solved for the justification of our theoretical solution.

Let us consider a classical problem of decision-making with fuzzy rewards. In order to choose the alternative which corresponds to the best fuzzy reward, we need to use a total order for the fuzzy numbers involved in the problem. In this paper, we consider a definition of such a total order, which is based upon two subjective aspects: the degree of optimism/pessimism and the liking for risk/safety. We also use the graded numbers which are analogous to the fuzzy numbers. However, the operations with graded numbers are carried out as a simple extension of operations with real intervals.

Uncertainty is an unavoidable components of decision making process. The ranking of fuzzy numbers which deals with such uncertainties play again a significant role in the process. Fuzzy numbers must be ranked in order to take the appropriate action by a decision maker in any real life situation. A few numbers of ranking techniques have been encountered in last few decades. However, existing techniques are situation-dependent which have drawbacks/shortcomings. In this regard, this paper presents a sophisticated ranking method based on the concept of the exponential area of the fuzzy numbers. The outputs obtained from this approach are obtained to be more efficient in comparison to the other ranking methods and outperform in all situations. The novelty and validity have been established through comparison with existing works. Furthermore, the ranking approach has been applied in medical decision making problem and the results obtained by the approach absolutely conform with analytical results and human intuitions as well.

The aim of this note is to point out and correct some vital mistakes in the paper by P K Nayak and M Pal, "Linear programming technique to solve two person matrix (games with interval pay-offs). Asia-Pacific Journal of Operational Research, 26(2), 285–305". Lots of serious mistakes on the definitions, conclusions, models, methods, proofs and computing results have been corrected and modified in this note. We also indicate inappropriate formulations regarding their proposed linear programming models for solving generic matrix games with interval pay-offs and suggest a pair of linear programming models with any minimal acceptance degree of the interval inequality constraints which may be allowed to violate. The lexicographic method is suggested so that a rational and credible solution of the generic matrix game with interval pay-offs can be achieved.

In this paper, we propose a method to calculate the correlation coefficient of fuzzy numbers by means of "expected interval" (see [8], [9]). This value obtained from our formula tell us not only the strength of relationship between the fuzzy numbers, but also whether the fuzzy numbers are positively or negatively related. This approach looks better than previous methods which only evaluate the strength of the relation. Furthermore, we extend the "expected interval" method to interval-valued fuzzy numbers. The value of the correlation coefficient between interval-valued fuzzy numbers lies in the interval [-1,1], as computed from our formula.

The aim of this paper is to develop a fuzzy linear programming technique for multidimensional analysis of preference (FLINMAP) in multiattribute group decision making problems with linguistic variables and incomplete preference information. In this paper, linguistic variables are used to assess an alternative on qualitative attributes using fuzzy ratings corresponding to some triangular fuzzy numbers. Each alternative is assessed on the basis of its distance to a fuzzy positive ideal solution (FPIS) which is unknown a priori. The FPIS and the weights of attributes are calculated by constructing a new linear programming model based on the group consistency and inconsistency indices defined on the basis of preferences between alternatives given by the decision makers. The distance of each alternative to the FPIS can be calculated to determine the ranking order of all alternatives. The implementation process of this methodology is demonstrated with an example.

Fuzzy number approximation has been investigated by many researchers. It is still useful to develop new approximations in order to better fit real world problems. This paper proposes a method for symmetric trapezoidal fuzzy number approximation which preserves the expected interval. Some properties of approximation are proved and a fuzzy partition is generated by using the proposed method.

The relative magnitude of weights for defects has a substantial impact on the performance of attribute control charts. Apparently, the current demerit-chart approach is superior than the c-chart scheme, because it imposes different precise-weights on distinct types of nonconformities, enabling more severe defects to disclose the problems existing in the manufacturing or service processes. However, this crisp-weighting defect assignment, assuming defects are of equal degree of severity when classified into the same defect class, may be so subjective that it leads to the chart somewhat restricted in widespread applications. Since in many cases the severity of each defect is evaluated from practitioners' visual inspection on the key quality characteristics of products or services, when each defect is classified into one of several mutually-exclusive linguistic classes, a fuzzy-weighting defect assignment that represents a degree of seriousness of defects should be allotted in accordance. Therefore, in this paper a demerit-fuzzy rating mechanism and monitoring chart is proposed. We first incorporate a fuzzy-linguistic weight in response to the severe degree of defects. Then, we apply the resolution identity property in construction of fuzzy control limits, and further develop a new fuzzy ranking method in differentiation of the underlying process condition. Finally, the proposed fuzzy-demerit chart is elucidated by an application of TFT-LCD manufacturing processes for monitoring their LCD Mura-nonconformities conditions.

Recently, a new fuzzy fault tree analysis (FFTA) has been developed to propagate and quantify the epistemic uncertainties occurring in qualitative data such as expert opinions or judgments. It is well known that the weakest triangular norm (T_w) based fuzzy arithmetic operations preserve the shape of the fuzzy numbers, provide more exact fuzzy results and effectively reduce uncertainty range. The objective of this paper is to develop a novel T_w-based fuzzy importance measure to identify the critical basic events in FFTA. The proposed approach has been demonstrated by applying it to a case study to identify the critical components of the Group 1 of the U.S. Combustion Engineering Reactor Protection System (CERPS). The obtained results are then compared to the results computed by the existing well-known importance measures of conventional as well as FFTA. The computed results confirm that the proposed T_w -based importance measure is feasible to identify the critical basic events in FFTA in more exact way.

In this paper, we first investigate the relationship between various notions of fuzzy boundedness of linear operators in fuzzy normed linear spaces. We also discuss the fuzzy boundedness of fuzzy compact operators. Furthermore, the spaces of fuzzy compact operators have been studied.

In the real-world optimization problems, coefficients of the objective function are not known precisely and can be interpreted as fuzzy numbers. In this paper we define the concepts of optimality for linear programming problems with fuzzy parameters based on those for multiobjective linear programming problems. Then by using the concept of comparison of fuzzy numbers, we transform a linear programming problem with fuzzy parameters to a multiobjective linear programming problem. To this end, we propose several theorems which are used to obtain optimal solutions of linear programming problems with fuzzy parameters. Finally some examples are given for illustrating the proposed method of solving linear programming problem with fuzzy parameters.

In this paper we define a new set of spline functions called “Fuzzy Splines” to interpolate fuzzy data. Numerical examples will be presented to illustrate the differences between of using our spline and other interpolations that have been studied before.

In this article we introduce the concepts of λ-statistical convergence of order β and strong λp-summability of order β for sequences of fuzzy numbers. Also, we establish some relations between the λ-statistical convergence of order β and strong λp-summability of order β and present some interesting examples to show strictness of some inclusion relations.