Comment on “Duality theory in fuzzy linear programming problems with fuzzy coefficients”

This note provides a counterexample to illustrate the incorrectness of the proof of Proposition 3.3 that was presented by Wu (Fuzzy Optim Decis Mak 2:61–73, 2003). The original proof of Proposition 3.3 by Wu can only be correct when the extra assumption \(\mu _{\widetilde{y}_i}(0)= 1\) is added. The correct proof of Proposition 3.3 is also presented in this note.     

Application of Atanassov’s I-fuzzy set theory to matrix games with fuzzy goals and fuzzy payoffs

We aim to extend some results in [6, 7, 8, 2] on two person zero sum matrix games (TPZSMG) with fuzzy goals and fuzzy payoffs to I-fuzzy scenario. Because the payoffs of the matrix game are fuzzy numbers, the aspiration levels of the players are fuzzy as well. It is reasonable to believe that there is some indeterminacy in estimating the aspiration levels of both players from their respective expected pay offs. This situation is modeled in the game using Atanassov??s I-fuzzy set theory. A new solution concept is proposed for such games and a procedure is outlined to obtain the degrees of suitability of the aspiration levels for each of the two players.     

A multi-objective wholesaler–retailers inventory-distribution model with controllable lead-time based on probabilistic fuzzy set and triangular fuzzy number

This paper develops a single wholesaler and multi retailers mixture inventory distribution model for a single item involving controllable lead-time with backorder and lost sales. The retailers purchase their items from the wholesaler in lots at some intervals throughout the year to meet the customers’ demand. Not to loose the demands, the retailers offer a price discount to the customers on the stock-out items. Here, it is assumed that the lead-time demands of retailers are uncertain in both stochastic and fuzzy sense, i.e., these are simultaneously random and imprecise. To implement this behavior of the lead-time demands, at first, these demands are assumed to be random, say following a normal distribution. With these random demands, the expected total cost for each retailer is obtained. Now, the mean lead-time demands (which are crisp ones) of the retailers are fuzzified. This fuzzy nature of the lead-time demands implies that the annual average demands of the retailers must be fuzzy numbers, suppose these are triangular fuzzy numbers. Using signed distance technique for defuzzification, the estimate of total costs for each retailer is derived. Therefore, the problem is reduced to optimize the crisp annual costs of wholesaler and retailers separately. The multi-objective model is solved using Global Criteria method. Numerical illustrations have been made with the help of an example taking two retailers into consideration. Mathematical analyses have been made for global pareto-optimal solutions of the multi-objective optimization problem. Sensitivity analyses have been made on backorder ratio and pareto-optimal solutions for wholesaler and different retailers are compared graphically.     

Food chain model with optimal harvesting in fuzzy environment

A multispecies harvesting model with mutual interactions is formulated based on Lotka–Voltera model with three competing species which are affected not only by harvesting but also by the presence of prey, predator and the third species, which is super predator. In order to understand the dynamics of the system, it is assumed that the super predator follows the logistic growth. Further, there is demand for all the above three species in the market and hence harvesting of all species is performed. We derive the condition for global stability of the system using a suitable Lyapunov function. The possibility of existence of bioeconomic equilibrium is discussed. The optimal harvest policy is studied and the solution is derived under imprecise inflation in fuzzy environment using Pontryagin’s maximal principle. Finally some numerical examples are discussed to illustrate the model.     

L—fuzzy Ring and L—fuzzy Module

Ｌ－ｆｕｚｚｙＲｉｎｇａｎｄＬ－ｆｕｚｚｙＭｏｄｕｌｅＬ－ｆｕｚｚｙＲｉｎｇａｎｄＬ－ｆｕｚｚｙＭｏｄｕｌｅ￥ＳｕｎＲｏｎｇｇｕａｎｇＪｉａｎｇＢａｏｑｉｎｇ（ＨｕｍａｎＵｎｉｖｅｒｓｉｔｙ，Ｋａｉｆｅｎｇ，４７５００１）ＲｅｎＸｕｅｊｕｎ（Ｍｉｎｑ...     

Synchronization of Takagi–Sugeno fuzzy stochastic discrete-time complex networks with mixed time-varying delays

In this paper, we propose and investigate a new general model of fuzzy stochastic discrete-time complex networks (SDCNs) described by Takagi–Sugeno (T–S) fuzzy model with discrete and distributed time-varying delays. The proposed model takes some well-studied models as special cases. By employing a new Lyapunov functional candidate, we utilize some stochastic analysis techniques and Kronecker product to deduce delay-dependent synchronization criteria that ensure the mean-square synchronization of the proposed T–S fuzzy SDCNs with mixed time-varying delays. These sufficient conditions are computationally efficient as it can be solved numerically by the LMI toolbox in Matlab. A numerical simulation example is provided to verify the effectiveness and the applicability of the proposed approach.     

分配格的L－fuzzy拓扑表示与L－fuzzy理想

对任一具有逆序对合对应的完备Ｈｅｙｔｉｎｚ代数Ｌ，文中通过构造分配格的Ｌ－ｆｕｚｚｙ谱空间建立了分配格的Ｌ－ｆｕｚｚｙ拓扑表示定理．作为Ｌ－ｆｕｚｚｙ谱理论的代数准备，文中引入了分配格的Ｌ－ｆｕｚｚｙ理想的概念，并较为系统地讨论了它的运算性质，这些结果主要涉及保有限并映射的结构性质．     

Exponential stability of fuzzy Cohen–Grossberg neural networks with time delays and impulsive effects

In this paper, we investigate a class of fuzzy Cohen-Grossberg neural networks with time delays and impulsive effects. By employing an inequality technique, we find sufficient conditions for the existence, uniqueness, global exponential stability of the equilibrium without using the M-matrix theory. An example is given to illustrate the effectiveness of the obtained results.     

F—fuzzy演绎系统

本文建立了一种演绎系统FFCS,在该演义系统中可以处理具有模糊性的推理过程,区别与其他对模糊推理进行形式化的逻辑系统,FFCS对模糊假言推理FMP做了完全形式化的处理。     

Synchronization analysis for complex networks with coupling delay based on T–S fuzzy theory

In this paper, synchronization problems for a general complex networks are investigated by Takagi–Sugeno (T–S) fuzzy theory. A novel concept named linear approximation method is firstly proposed to solve the synchronization problems for T–S fuzzy complex networks. This novel method can linearize the system into some time-delay subsystems, which can effectively simplify the complicated system and then be easy to acquire the synchronization results. Since the expression based on linear matrix inequality (LMI) is used, the synchronization criteria can be easily checked in practice. Numerical simulation examples are provided to verify the effectiveness and the applicability of the proposed method.     

Some new results in linear programs with trapezoidal fuzzy numbers: Finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method

In a recent paper, Ganesan and Veermani [K. Ganesan, P. Veeramani, Fuzzy linear programs with trapezoidal fuzzy numbers, Ann. Oper. Res. 143 (2006) 305–315] considered a kind of linear programming involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems and then proved fuzzy analogues of some important theorems of linear programming that lead to a new method for solving fuzzy linear programming (FLP) problems. In this paper, we obtain some another new results for FLP problems. In fact, we show that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it has an optimal fuzzy basic solution too. We also prove that in the absence of degeneracy, the method proposed by Ganesan and Veermani stops in a finite number of iterations. Then, we propose a revised kind of their method that is more efficient and robust in practice. Finally, we give a new method to obtain an initial fuzzy basic feasible solution for solving FLP problems.     

Dynamical behaviors of fuzzy reaction–diffusion periodic cellular neural networks with variable coefficients and delays

When modeling neural networks in a real world, not only diffusion effect and fuzziness cannot be avoided, but also self-inhibitions, interconnection weights, and inputs should vary as time varies. In this paper, we discuss the dynamical behaviors of delayed reaction–diffusion fuzzy cellular neural networks with varying periodic self-inhibitions, interconnection weights as well as inputs. By using Halanay’s delay differential inequality, M$M$-matrix theory and analytic methods, some new sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of the periodic solution, and the exponentially convergent rate index is also estimated. In particular, the traditional assumption on the differentiability of the time-varying delays is no longer needed. The methodology developed in this paper is shown to be simple and effective for the exponential periodicity and stability analysis of neural networks with time-varying delays. Two examples are given to show the usefulness of the obtained results that are less restrictive than recently known criteria.     

Linear programming with triangular fuzzy numbers—A case study in a finance and credit institute

The objective of this paper is to deal with a kind of fuzzy linear programming problem involving triangular fuzzy numbers. Then some interesting and fundamental results are achieved which in turn lead to a solution of fuzzy linear programming models without converting the problems to the crisp linear programming models. Finally, the theoretical results are also supported by a real case study in a banking system. The same idea is emphasized to be also useful when a general LR fuzzy numbers is given.     

Stability analysis of fuzzy Markovian jumping Cohen–Grossberg BAM neural networks with mixed time-varying delays

In this paper, we investigate the robust stability of uncertain fuzzy Markovian jumping Cohen–Grossberg BAM neural networks with discrete and distributed time-varying delays. A new delay-dependent stability condition is derived under uncertain switching probabilities by Takagi–Sugeno fuzzy model. Based on the linear matrix inequality (LMI) technique, upper bounds for the discrete and distributed delays are calculated using the LMI toolbox in MATLAB. Numerical examples are provided to illustrate the effectiveness of the proposed method.     

H∞ control for uncertain Takagi–Sugeno fuzzy systems with time-varying delays and nonlinear perturbations

This paper investigates the robust H_∞ control for Takagi–Sugeno (T–S) fuzzy systems with time-varying delays and nonlinear perturbations. Delay-dependent criteria for uncertain fuzzy systems have been proposed. The LMI optimization approach is applied to solve the H_∞ control problem and find the minimization of H_∞ norm bound. Computer simulations show that significant improvement over the previous results can be obtained.