Development and optimization of unrestricted LR-type intuitionistic fuzzy mathematical programming problems

To the best of our knowledge, there is no method to find the product of unrestricted LR-type intuitionistic fuzzy numbers (IFNs) as well as the optimal solutions of LR-type intuitionistic fuzzy linear programming problems (IFLPPs) in which some or all the decision variables are unrestricted. Therefore it is necessary to pay attention to this issue and there is need to find the product of unrestricted LR-type IFNs as well as the optimal solutions of such programming problems. In this paper, the product of unrestricted LR-type IFNs based on the α-cut, β-cut and (α, β)-cut is proposed. Then with the help of proposed product, a new method is proposed to find the optimal solutions of unrestricted LR-type IFLPPs. Finally, an illustrative example is given to support the proposed method and investigated the applicability of existing approaches exist in the literature.     

On solutions of fuzzy random multiobjective quadratic programming with applications in portfolio problem

In this paper, a multiobjective quadratic programming problem fuzzy random coefficients matrix in the objectives and constraints and the decision vector are fuzzy variables is considered. First, we show that the efficient solutions fuzzy quadratic multiobjective programming problems series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. An application fuzzy portfolio optimization problem as a convex quadratic programming approach is discussed and an acceptable solution to such problem is given. At the end, numerical examples are illustrated in the support of the obtained results.     

Constrained fuzzy arithmetic approach to fuzzy transportation problems with fuzzy decision variables

Most of the existing methods for solving fully fuzzy mathematical programs are based on the standard fuzzy arithmetic operations and/or Zadeh's extension principle. These methods may produce questionable results for many real-life applications. Due to this fact, this paper presents a novel method based on the constrained fuzzy arithmetic concept to solve fully fuzzy balanced/unbalanced transportation problems in which all of the parameters (source capacities, demands of destinations, transportation costs etc.) as well as the decision variables (transportation quantities) are considered as fuzzy numbers. In the proposed method, the requisite crisp and/or fuzzy constraints between the base variables of the fuzzy components are provided from the decision maker according to his/her exact or vague judgments. Thereafter, fuzzy arithmetic operations are performed under these requisite constraints by taking into account the additional information while transforming the fuzzy transportation model into crisp equivalent form. Therefore, various fuzzy efficient solutions can be generated by making use of the proposed method according to the decision maker's risk attitude. In order to present the efficiency/applicability of the proposed method, different types of fully fuzzy transportation problems are generated and solved as illustrative examples. A detailed comparative study is also performed with other methods available in the literature. The computational analysis have shown that relatively more precise solutions are obtained from the proposed method for “risk-averse” and “partially risk-averse” decision makers. The proposed method also successfully provided fuzzy acceptable solutions for “risk seekers” with high degree of uncertainty similar to the other existing methods in the literature.     

大工业过程稳态优化的模糊双迭代法

在研究大工业过程稳态优化控制算法时, 针对模型–实际存在差异, 将子过程模型作为等式约束, 通过引入模糊系数使其转化为模糊等式约束, 同时对子过程的不等式约束进行模糊化处理, 提出具有模糊不等式约束的模糊双迭代法, 通过实际例子研究了模糊双迭代法. 仿真结果表明, 模糊双迭代法目标函数非常接近实际目标函数值、算法迭代次数较精确双迭代法有明显改善. 这对实际生产非常重要.     

A duality approach for solving bounded linear programming problems with fuzzy variables based on ranking functions and its application in bounded transportation problems

There are several methods, in the literature, for solving fuzzy variable linear programming problems (fuzzy linear programming in which the right-hand-side vectors and decision variables are represented by trapezoidal fuzzy numbers). In this paper, the shortcomings of some existing methods are pointed out and to overcome these shortcomings a new method based on the bounded dual simplex method is proposed to determine the fuzzy optimal solution of that kind of fuzzy variable linear programming problems in which some or all variables are restricted to lie within lower and upper bounds. To illustrate the proposed method, an application example is solved and the obtained results are given. The advantages of the proposed method over existing methods are discussed. Also, one application of this algorithm in solving bounded transportation problems with fuzzy supplies and demands is dealt with. The proposed method is easy to understand and to apply for determining the fuzzy optimal solution of bounded fuzzy variable linear programming problems occurring in real-life situations.     

Vector quantization in DCT domain using fuzzy possibilistic c-means based on penalized and compensated constraints

In this paper, fuzzy possibilistic c-means (FPCM) approach based on penalized and compensated constraints are proposed to vector quantization (VQ) in discrete cosine transform (DCT) for image compression. These approaches are named penalized fuzzy possibilistic c-means (PFPCM) and compensated fuzzy possibilistic c-means (CFPCM). The main purpose is to modify the FPCM strategy with penalized or compensated constraints so that the cluster centroids can be updated with penalized or compensated terms iteratively in order to find near-global solution in optimal problem. The information transformed by DCT was separated into DC and AC coefficients. Then, the AC coefficients are trained by using the proposed methods to generate better codebook based on VQ. The compression performances using the proposed approaches are compared with FPCM and conventional VQ method. From the experimental results, the promising performances can be obtained using the proposed approaches.     

An interactive fuzzy multi-objective optimization method for engineering design

The coupling of performance functions due to common design variables and uncertainties in an engineering design process will result in difficulties in optimization design problems, such as poor collaboration among design objectives and poor resolution of design conflicts. To handle these problems, a fuzzy interactive multi-objective optimization model is developed based on Pareto solutions, where the metric function and some additional constraints are added to ensure the collaboration among design objectives. The trade-off matrix at the Pareto solutions was developed, and the method for selecting weighting coefficients of optimization objectives is also provided. The proposed method can generate a Pareto optimal set with the maximum satisfaction degree and the minimum distance from ideal solution. The favorable optimal solution can be then selected from the Pareto optimal set by analyzing the trade-off matrix and collaborative sensitivity. Two examples are presented to illustrate the proposed method.     

Signed distance ranking based approach for solving bounded interval-valued fuzzy numbers linear programming problems

Fuzzy linear programming (FLP) problems with a wide varietyof applications in sciencesand engineering allow working with imprecise data and constraints, leading to more realistic models. The main contribution of this study is to deal with the formulation of a kind of FLP problems, known as bounded interval-valued fuzzy numbers linear programming (BIVFNLP) problems, with coefficients of decision variables in the objective function, resource vector, and coefficients of the technological matrix represented as interval-valued fuzzy numbers (IVFNs), and crisp decision variables limited to lower and upper bounds. Here, based on signed distance ranking to order IVFNs, the bounded simplex method is extended to obtain an interval-valued fuzzy optimal value for the BIVFNLP problem under consideration. Finally, one illustrative example is given to show the superiority of the proposed algorithm over the existing ones.     

Multi-objective decision making model under fuzzy random environment and its application to inventory problems

In this paper, we concentrate on developing a fuzzy random multi-objective model about inventory problems. By giving some definitions and discussing some properties of fuzzy random variable, we design a method of solving solution sets of fuzzy random multi-objective programming problems. These are applied to numerical inventory problems in which all inventory costs, purchasing and selling prices in the objectives and constraints are assumed to be fuzzy random variables in nature, and then the impreciseness of fuzzy random variables in the above objectives and constraints are transformed into fuzzy variables which are similar trapezoidal fuzzy numbers. The exact parameters of fuzzy membership function and probability density function can be obtained through fuzzy random simulating the past dates. By comparing the results with those from the fuzzy multi-objective models, we believe that the proposed fuzzy random multi-objective model and hybrid intelligent algorithm provide significant solutions to construct other inventory models with fuzzy random variables in real life.     

Solution of the problem concerning a conditional extremum under fuzzy constraints

The solution of the conditional extremum problem by Lagrange’s method of undetermined coefficients under fuzzy constraints on the expenditure function was proposed. The problem is solved by using a special example of the availability maximization of a double-link communication system. A disadvantage of the solution is pointed out, and problems for further study are stated.     

Interactive multiobjective fuzzy random programming through the level set-based probability model

This paper considers a multiobjective linear programming problem involving fuzzy random variable coefficients. A new fuzzy random programming model is proposed by extending the ideas of level set-based optimality and a stochastic programming model. The original problem involving fuzzy random variables is transformed into a deterministic equivalent problem through the proposed model. An interactive algorithm is provided to obtain a satisficing solution for a decision maker from among a set of newly defined Pareto optimal solutions. It is shown that an optimal solution of the problem to be solved iteratively in the interactive algorithm is analytically obtained by a combination of the bisection method and the simplex method.     

A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers

In a recent paper, Kaur and Kumar (2012) proposed a new method based on ranking function for solving fuzzy transportation problem (FTP) by assuming that the values of transportation costs are represented by generalized trapezoidal fuzzy numbers. Here it is shown that once the ranking function is chosen, the FTP is converted into crisp one, which is easily solved by the standard transportation algorithms. The main contribution here is the reduction of the computational complexity of the existing method. By solving two application examples, it is shown that it is possible to find a same optimal solution without solving any FTP. Since the proposed approach is based on classical approach it is very easy to understand and to apply on real life transportation problems for the decision makers.     

Random fuzzy multi-objective linear programming: Optimization of possibilistic value at risk (pVaR)

This paper considers multiobjective linear programming problems (MOLPP) where random fuzzy variables are contained in objective functions and constraints. A new decision making model optimizing possibilistic value at risk (pVaR) is proposed by incorporating the concept of value at risk (VaR) into possibility theory. It is shown that the original MOLPPs involving random fuzzy variables are transformed into deterministic problems. An interactive algorithm is presented to derive a satisficing solution for a decision maker (DM) from among a set of Pareto optimal solutions. Each Pareto optimal solution that is a candidate of the satisficing solution is exactly obtained by using convex programming techniques. A simple numerical example is provided to show the applicability of the proposed methodology to real-world problems with multiple objectives in uncertain environments.     

An exact penalty function algorithm for time-lag control problems with control and terminal equality constraints

A computational algorithm for a class of time-lag optimal control problems involving control and terminal equality constraints is presented. This algorithm is based on the exact penalty function method and the concept of control parametrization. The convergence properties of the algorithm are also investigated. Two examples are solved using the proposed algorithm.     

Linear programming with fuzzy coefficients in constraints

This paper presents a new method for solving linear programming problems with fuzzy coefficients in constraints. It is shown that such problems can be reduced to a linear semi-infinite programming problem. The relations between optimal solutions and extreme points of the linear semi-infinite program are established. A cutting plane algorithm is introduced with a convergence proof, and a numerical example is included to illustrate the solution procedure.     

A direct solution approach to fuzzy mathematical programs with fuzzy decision variables

In this paper, a direct solution approach is presented for solving fuzzy mathematical programming problems with fuzzy decision variables. In the proposed approach, a fuzzy ranking procedure for fuzzy numbers and a meta-heuristic algorithm is employed. A basic example is presented in the paper. It has been observed that fuzzy mathematical programs with fuzzy decision variables can be solved effectively by employing direct solution approaches which are based on fuzzy ranking procedures and meta-heuristics.     

Network flow problems with fuzzy arc lengths.

Network flow problems cover a wide range of engineering and management applications. Many streamlined solution methods have been devised for solving different types of the problems. This paper investigates the network flow problems in that the arc lengths of the network are fuzzy numbers. Based on the integer-solution property of the network flow problem, the Yager ranking indices can be calculated for the fuzzy arcs to change the fuzzy formulation of the problem to a crisp formulation. Consequently, the conventional streamlined solution methods can still be applied to find an optimal solution. This optimal solution is proved to be the same as that derived from an exhaustive comparison of all possible solutions. Two examples, one shortest path and one transshipment, discussed in some previous studies illustrate that the method proposed in this paper is able to find the optimal solution. To show that the proposed method is useful in solving real-world problems, the problem of multimedia transmission over the Internet is exemplified.     

An epsilon-constraint method for fully fuzzy multiobjective linear programming

Linear ranking functions are often used to transform fuzzy multiobjective linear programming (MOLP) problems into crisp ones. The crisp MOLP problems are then solved by using classical methods (eg, weighted sum, epsilon-constraint, etc), or fuzzy ones based on Bellman and Zadeh's decision-making model. In this paper, we show that this transformation does not guarantee Pareto optimal fuzzy solutions for the original fuzzy problems. By using lexicographic ranking criteria, we propose a fuzzy epsilon-constraint method that yields Pareto optimal fuzzy solutions of fuzzy variable and fully fuzzy MOLP problems, in which all parameters and decision variables take on LR fuzzy numbers. The proposed method is illustrated by means of three numerical examples, including a fully fuzzy multiobjective project crashing problem.     

A PSO-based adaptive fuzzy PID-controllers

In this paper, a novel design method for determining the optimal fuzzy PID-controller parameters of active automobile suspension system using the particle swarm optimization (PSO) reinforcement evolutionary algorithm is presented. This paper demonstrated in detail how to help the PSO with Q-learning cooperation method to search efficiently the optimal fuzzy-PID controller parameters of a suspension system. The design of a fuzzy system can be formulated as a search problem in high-dimensional space where each point represents a rule set, membership functions, and the corresponding system’s behavior. In order to avoid obtaining the local optimum solution, we adopted a pure PSO global exploration method to search fuzzy-PID parameter. Later this paper explored the improved the limitation between suspension and tire deflection in active automobile suspension system with nonlinearity, which needs to be solved ride comfort and road holding ability problems, and so on. These studies presented many ideas to solve these existing problems, but they need much evolution time to obtain the solution. Motivated by above discussions this paper propose a novel algorithm which can decrease the number of evolution generation, and can also evolve the fuzzy system for obtaining a better performance.     

A new fuzzy neural network model for solving fuzzy linear programming problems and its applications

This paper is concentrated on two types of fuzzy linear programming problems. First type with fuzzy coefficients in the objective function and the second type with fuzzy right-hand side values and fuzzy variables. Considering fuzzy derivative and fuzzy differential equations, these kinds of problems are solved using a fuzzy neural network model. To show the applicability of the method, it is applied to solve the fuzzy shortest path problem and the fuzzy maximum flow problem. Numerical results illustrate the method accuracy and it’s simple implementation.