Narrow Results

The problem of the Chinese postman is one of the types of the problem of the Chinese postman, which includes 4 models in this paper. In the first case, the postman moves from the origin, he must go through each edge at least once and then move to the destination, which in the first case, the destination is the same as the original origin. That is, the postman returns to the original origin. In the second case, the postman moves from the origin. We determine the destination so that the postman does not have to go back to the original source to incur a fee. In the third case, the postman moves from the origin to the destination that the computer program determines for him, in this way, he incurs a lower cost than in the second case because the GAMS programming determines a destination for us that costs less than all destinations. In the fourth case, both the origin and the destination are determined by programming so that all the edges are covered, which is less expensive than all the origins and destinations. The weight of edges is not constant due to different factors. Triangular fuzzy numbers are used in this study, which is included in the coefficients of the objective function of these fuzzy numbers. Two different ranking methods are used for fuzzification and the obtained results are compared with each other. The solution is done using GAMS software. Mathematical models cannot find solutions in large sizes and for this reason, the Lagrangian algorithm is used to solve these problems. After the problems are solved using the Lagrangian algorithm, the results of different fuzzy ranking methods are compared

A new way to study the prey–predator model with optimal harvesting is presented by analyzing the dynamics of prey–predator model in crisp and fuzzy environment. To incorporate the natural imprecision present in environmental factors, all parameters are considered first time as fuzzy numbers. We formulate the fuzzy model using the imprecise data as a form of fuzzy numbers since the biological characteristics such as prey population growth rate, predator population decay rate, and predation coefficients lack accurate numerical information. We discuss the positivity and boundedness of solutions of the proposed fuzzy model. Further, we investigate and discuss the local and global asymptotic fuzzy stability of the fuzzy equilibrium points to study the dynamical behavior of the proposed fuzzy model. Also, we study the fuzzy bionomic equilibrium; and investigate the fuzzy optimal harvest strategy and apply Pontryagin’s maximum principle to find the solution in the fuzzy equilibrium.

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.

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.

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.

By using the restricted and complementary relationship of the principle and secondary indexes, providing the description of the compound quantification of the fuzzy number, and analyzing the essential characteristic of fuzzy decision, we propose a kind of fuzzy genetic algorithm based on the principle index (PO-FGA for short) to deal with the fuzzy optimization and programming problems with fuzzy coefficients, fuzzy variables and fuzzy constraints. The concrete solution method is presented in accordance with the strategy of the unconditional penality transformation with conditional constrains. Then consider its convergence by using Markov chain theory and analyze its performance through two examples. All these indicate that this kind of algorithm is of faster speed of convergence, smaller number of iterations, has lower chances of trapping into the state of premature convergence and can be widely used in many problems of optimization.

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.

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.

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.

In this paper, we propose a method to calculate the correlation coefficient of fuzzy numbers by means of "α-cut". This value obtained from our formula tells 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 "α-cut" 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.

In this paper, we present the definition of integral of fuzzy mappings from the fuzzy number space E into E, and discuss its properties and calculation. And as an application of the integral, we provide a method of robustness estimates for the traffic volumes passing a road each day (24 hours). In addition, we also give another example of the integral characterizing the distance travelled in an imprecise or uncertain environment of time and speed.

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.

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.

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 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.

A new approach to solve Fuzzy Complex System of Linear Equations (FCSLE) based on fuzzy complex centre procedure is presented here. Few theorems related to the investigation are stated and proved. Finally the presented procedure is used to analyze an example problem of linear time invariant electric circuit with complex crisp coefficient and fuzzy complex sources. The results obtained are also compared with the known solutions and are found to be in good agreement.

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.

When Statistics deals with data which cannot be expressed in a numerical scale, the scale of fuzzy values (in particular, the scale of fuzzy numbers) often becomes a suitable tool to express such data. In this way, many ratings, opinions, judgements, etc. mostly coming from human valuations can be appropriately described in terms of fuzzy data. To summarize the central tendency of a fuzzy dataset, some measures have been suggested in the literature. This paper aims to review some of the main ones, and examine their properties in a comparative way. A real-life example illustrates their application. Furthermore, the paper shows the statistical robustness (both through the finite sample breakdown point and a simulation study) and the empirical “precision” of the fuzzy number-valued sample measures. Finally, some related developments and future directions are pointed out.

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.

Research in optimization under uncertainty is alive. It assumes different shapes and forms, all concurring to the general goal of designing effective and efficient tools for handling imprecision in an Optimization setting. In this paper we present a new approach for dealing with multiobjective programming problems with fuzzy objective functions. Similar to many approaches in the literature, our approach relies on the deffuzification of involved fuzzy quantities. Our improvement stem from the choice of a deffuzification operator that captures essential features of fuzzy parameters at hand rather than those that yield single values, leading to a loss of many useful information. Two oracles play a pivotal role in the proposed method. The first one returns a near interval approximation to a given fuzzy number. The other one delivers a Pareto Optimal solution of the resulting multiobjective program with interval coefficient. A numerical example is also provided for the sake of illustration.