A relaxed cutting plane algorithm for solving fuzzy inequality systems

This paper studies a system of infinitely many fuzzy inequalities with concavemembership functions. By using the tolerance approach, we show that solving such system can be reduced to a semi-infinite programming problem. A relaxed cutting plane algorithm is proposed. In each iteration, we solve a finite convex optimization problem and add one or two more constraints. The proposed algorithm chooses a point at which the infinite constraints are violated to a degree rather than at which the violation is maximized. The iterative process ends when an optimal solution is identified. A convergence proof, under some mild conditions, is given. An efficient implementation based on the "method of centres" with "entropic regularization" techniques is also included. Some computational results confirm the efficiency of the proposed method and show its potential for solving large scale problems.     

Scalarization of the fuzzy optimization problems

Scalarization of the fuzzy optimization problems using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Two solution concepts are proposed by considering two convex cones. The set of all fuzzy numbers can be embedded into a normed space. This motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to solve the fuzzy optimization problems. By applying scalarization to the optimization problem with fuzzy coefficients, we obtain its corresponding scalar optimization problem. Finally, we show that the optimal solution of its corresponding scalar optimization problem is the optimal solution of the original fuzzy optimization problem.     

Bilevel Decision with Generalized Semi-infinite Optimization for Fuzzy Mappings as Lower Level Problems

In this paper, we introduce the bilevel decision problems with parametric generalized semi-infinite optimization for fuzzy mappings as the lower-level problem, and its corresponding MPEC problems. For these problems, we establish two models which are different in the feasible region setting of lower-level problems. Some new existence results are obtained in rather weak conditions.     

Examples of fuzzy metrics and applications

In this paper we present new examples of fuzzy metrics in the sense of George and Veeramani. The examples have been classified attending to their construction and most of the well-known fuzzy metrics are particular cases of those given here. In particular, novel fuzzy metrics, by means of fuzzy and classical metrics and certain special types of functions, are introduced. We also give an extension theorem for two fuzzy metrics that agree in its nonempty intersection. Finally, we give an application of this type of fuzzy metrics to color image processing. We propose a fuzzy metric that simultaneously takes into account two different distance criteria between color image pixels and we use this fuzzy metric to filter noisy images, obtaining promising results. This application is also illustrative of how fuzzy metrics can be used in other engineering problems.     

Duality Theory in Fuzzy Optimization Problems

A solution concept of fuzzy optimization problems, which is essentially similar to the notion of Pareto optimal solution (nondominated solution) in multiobjective programming problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. We also introduce a concept of fuzzy scalar (inner) product based on the positive and negative parts of fuzzy numbers. Then the fuzzy-valued Lagrangian function and the fuzzy-valued Lagrangian dual function for the fuzzy optimization problem are proposed via the concept of fuzzy scalar product. Under these settings, the weak and strong duality theorems for fuzzy optimization problems can be elicited. We show that there is no duality gap between the primal and dual fuzzy optimization problems under suitable assumptions for fuzzy-valued functions.     

Discrete optimization algorithms and problems of decision making in a fuzzy environment

An approach to solving optimization problems with fuzzy coefficients is described. It consists in formulating and analyzing one and the same problem within the framework of mutually related models by constructing equivalent analogs with fuzzy coefficients in objective functions alone. Since the approach is applied within the context of fuzzy discrete optimization problems, modified algorithms of discrete optimization are discussed. These algorithms are based on a combination of formal and heuristic procedures and allow one to obtain quasi-optimal solutions after a small number of steps, thus overcoming the computational complexity posed by the NP-completeness of discrete optimization problems. The subsequent contraction of the decision uncertainty regions is associated with reduction of the problem to multiobjective decision making in a fuzzy environment using techniques based on fuzzy preference relations. The results of the paper are of a universal character and are already being used to solve practical problems in several fields.     

Solving Large-Scale Fuzzy and Possibilistic Optimization Problems

Fuzzy and possibilistic optimization methods are demonstrated to be effective tools in solving large-scale problems. In particular, an optimization problem in radiation therapy with various orders of complexity from 1000 to 62,250 constraints for fuzzy and possibilistic linear and nonlinear programming implementations possessing (1) fuzzy or soft inequalities, (2) fuzzy right-hand side values, and (3) possibilistic right-hand side is used to demonstrate that fuzzy and possibilistic optimization methods are tractable and useful. We focus on the uncertainty in the right side of constraints which arises, in the context of the radiation therapy problem, from the fact that minimal and maximal radiation tolerances are ranges of values, with preferences within the range whose values are based on research results, empirical findings, and expert knowledge, rather than fixed real numbers. The results indicate that fuzzy/possibilistic optimization is a natural and effective way to model various types of optimization under uncertainty problems and that large fuzzy and possibilistic optimization problems can be solved efficiently.     

On the Use of Augmented Lagrangians in the Solution of Generalized Semi-Infinite Min-Max Problems

We present an approach for the solution of a class of generalized semi-infinite optimization problems. Our approach uses augmented Lagrangians to transform generalized semi-infinite min-max problems into ordinary semi-infinite min-max problems, with the same set of local and global solutions as well as the same stationary points. Once the transformation is effected, the generalized semi-infinite min-max problems can be solved using any available semi-infinite optimization algorithm. We illustrate our approach with two numerical examples, one of which deals with structural design subject to reliability constraints.     

Fuzzy expectation values in multistage optimization problems

The study demonstrates the use of fuzzy expectation values in problems of multistage optimization under uncertainty. A practicable procedure is presented for the case where the optimization objective can be decomposed into a series of single-stage decision goals. Instead of probability theory, the uncertainty resolution is accomplished by fuzzy expectation values. In essence, then, the risk aversion is emboided in the selection of the fuzzy integration measure. If for example, the primary goal of the optimization is to achieve a strict cost minimum, then in the lack of information, a weaker goal can be introduced that corresponds to balancing the anticipated costs to the risk of exceeding these in reality. In a number of common optimization problems the method proposed facilitates a rapid solution with clear information on the risk involved.     

Models for a fuzzy inventory of two replaceable merchandises without backorder based on the signed distance of fuzzy sets

This paper investigates the inventory problems for two mutually complementary merchandises. We first consider the merchandises in a monopoly market, and then in a perfect competitive market. With the fuzzy sets concept, we discuss how to determine the optimal ordering policy for the aforementioned inventory problem such that the total related cost is minimum. Three results are obtained and the numerical examples are provided to illustrate these results. In contrast with the previous studies that employed the extension principle and centroid method to derive the estimate of the total cost in the fuzzy sense, we show that using the decomposition principle and the signed distance can attain it easier.     

Identification of fuzzy relation models using hierarchical fair competition-based parallel genetic algorithms and information granulation

The paper is concerned with a hybrid optimization of fuzzy inference systems based on hierarchical fair competition-based parallel genetic algorithms (HFCGA) and information granulation. The process of information granulation is realized with the aid of the C-Means clustering. HFCGA being a multi-population based parallel genetic algorithms (PGA) is exploited here to realize structure optimization and carry out parameter estimation of the fuzzy models. The HFCGA becomes helpful in the context of fuzzy models as it restricts a premature convergence encountered quite often in optimization problems. It concerns a set of parameters of the model including among others the number of input variables to be used, a specific subset of input variables, and the number of membership functions. In the hybrid optimization process, two general optimization mechanisms are explored. The structural development of the fuzzy model is realized via the HFCGA optimization and C-Means, whereas to deal with the parametric optimization we proceed with a standard least square method and the use of the HFCGA technique. A suite of comparative studies demonstrates that the proposed algorithm leads to the models whose performance is superior in comparison with some other constructs commonly used in fuzzy modeling.     

An elaborative unit cost structure-based fuzzy economic production quantity model

The purpose of this paper is to investigate and propose a fuzzy extended economic production quantity model based on an elaboratively modeled unit cost structure. This unit cost structure consists of the various lot-size correlative components such as on-line setups, off-line setups, initial production defectives, direct material, labor, and depreciation in addition to lot-size non-correlative items. Thus, the unit cost is correlatively modeled to the production quantity. Therefore, the modeling or the annual total cost function developed consists of not only annual inventory and setup costs but also production cost. Moreover, via the concept of fuzzy blurred optimal argument and the vertex method of the α-cut fuzzy arithmetic (or fuzzy interval analysis), two solution approaches are proposed: (1) a fuzzy EPQ and (2) a compromised crisp EPQ in the fuzzy sense. An optimization procedure, which can simultaneously determine the α-cut-vertex combination of fuzzy parameters and the optimizing decision variable value, is also proposed. The sensitivity model for the fuzzy total cost and thus EPQ to the various cost factors is provided. Finally, a numerical example with the original data collected from a firm demonstrates the usefulness of the new model.     

A hybrid intelligent algorithm for portfolio selection problem with fuzzy returns

Portfolio selection theory with fuzzy returns has been well developed and widely applied. Within the framework of credibility theory, several fuzzy portfolio selection models have been proposed such as mean–variance model, entropy optimization model, chance constrained programming model and so on. In order to solve these nonlinear optimization models, a hybrid intelligent algorithm is designed by integrating simulated annealing algorithm, neural network and fuzzy simulation techniques, where the neural network is used to approximate the expected value and variance for fuzzy returns and the fuzzy simulation is used to generate the training data for neural network. Since these models are used to be solved by genetic algorithm, some comparisons between the hybrid intelligent algorithm and genetic algorithm are given in terms of numerical examples, which imply that the hybrid intelligent algorithm is robust and more effective. In particular, it reduces the running time significantly for large size problems.     

Fuzzy linear programming problems as bi-criteria optimization problems

In this paper, we consider fuzzy linear programming (FLP) problems which involve fuzzy numbers only in coefficients of objective function. First, we shall give concepts of optimal solutions to (FLP) problems and investigate their properties. Next, in order to find all optimal solutions, we define three types of bi-criteria optimization problems.     

Duality theory in fuzzy optimization problems formulated by the Wolfe’s primal and dual pair

The weak and strong duality theorems in fuzzy optimization problem based on the formulation of Wolfe’s primal and dual pair problems are derived in this paper. The solution concepts of primal and dual problems are inspired by the nondominated solution concept employed in multiobjective programming problems, since the ordering among the fuzzy numbers introduced in this paper is a partial ordering. In order to consider the differentiation of a fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the Wolfe’s dual problem can be formulated by considering the gradients of differentiable fuzzy- valued functions. The concept of having no duality gap in weak and strong sense are also introduced, and the strong duality theorems in weak and strong sense are then derived naturally.     

Higher order duality for a new class of nonconvex semi-infinite multiobjective fractional programming with support functions

In the paper, a new class of semi-infinite multiobjective fractional programming problems with support functions in the objective and constraint functions is considered. For such vector optimization problems, higher order dual problems in the sense of Mond-Weir and Schaible are defined. Then, various duality results between the considered multiobjective fractional semi-infinite programming problem and its higher order dual problems mentioned above are established under assumptions that the involved functions are higher order $\left(\Phi,\rho,\sigma^{\alpha}\right)$-type I functions. The results established in the paper generalize several similar results previously established in the literature.     

直觉模糊子环及其同构

利用既约集合套讨论了直觉模糊子环和直觉模糊理想的一些性质,并在两个经典环同态与同构意义下,定义了直觉模糊商环,商直觉模糊子环,以及直觉模糊子环的直和,并讨论了其相关性质.     

Application of fuzzy decision making to the fuzzy finite element method

In this paper we present a new approach to handle uncertainty in the Finite Element Method. As this technique is widely used to tackle real-life design problems, it is also very prone to parameter-uncertainty. It is hard to make a good decision regarding design optimization if no claim can be made with respect to the outcome of the simulation. We propose an approach that combines several techniques in order to offer a total order on the possible design choices, taking the inherent fuzziness into account. Additionally we propose a more efficient ordering procedure to build a total order on fuzzy numbers.     

Bounding Contingent Claim Prices via Hedging Strategy with Coherent Risk Measures

We generalize the notion of arbitrage based on the coherent risk measure, and investigate a mathematical optimization approach for tightening the lower and upper bounds of the price of contingent claims in incomplete markets. Due to the dual representation of coherent risk measures, the lower and upper bounds of price are located by solving a pair of semi-infinite linear optimization problems, which further reduce to linear optimization when conditional value-at-risk (CVaR) is used as risk measure. We also show that the hedging portfolio problem is viewed as a robust optimization problem. Tuning the parameter of the risk measure, we demonstrate by numerical examples that the two bounds approach to each other and converge to a price that is fair in the sense that seller and buyer face the same amount of risk.     

Fuzzy modelling environment for designing fuzzy controllers

Fuzzy control algorithms are developed based on fuzzy models of systems. The control issues are posed as multiobjective optimization problems involving goals and constraints imposed on system's variables. Two basic design modes embrace on- and off-line modes of control development. The first type of design deals with the time and state-dependent objectives and pertains to control determination based upon the current state of the fuzzy model. The second design mode gives rise to an explicit form of a fuzzy controller that is learned based on a given list of state-control associations. Both the fuzzy models as well as fuzzy controllers are realized as logic processors.