Developments in fuzzy multicriteria analysis

Selecting or ranking available alternatives (observations/objects) with respect to multiple, often conflicting criteria in a fuzzy environment usually referred to as fuzzy multicriteria analysis is a problem of a major interest in information and engineering. Methodologies for addressing this problem have been developed from a variety of research disciplines, including statistics, econometrics, artificial intelligent, and operations research. This paper presents an overview of the developments in fuzzy multicriteria analysis. It discusses the complexity of fuzzy multicriteria analysis and analyses the existing approaches from four different perspectives for facilitating a better understanding of the recent development in this domain. Finally, the paper elaborates on the future research areas in fuzzy multicriteria analysis.     

European option pricing model in a stochastic and fuzzy environment

The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.     

Interval optimization based line sampling method for fuzzy and random reliability analysis

For structural system with fuzzy variables as well as random variables, a novel algorithm for obtaining membership function of fuzzy reliability is presented on interval optimization based Line Sampling (LS) method. In the presented algorithm, the value domain of the fuzzy variables under the given membership level is firstly obtained according to their membership functions. Then, in the value domain of the fuzzy variables, bounds of reliability of the structure are obtained by the nesting analysis of the interval optimization, which is performed by modern heuristic methods, and reliability analysis, which is achieved by the LS method in the reduced space of the random variables. In this way the uncertainties of the input variables are propagated to the safety measurement of the structure, and the membership function of the fuzzy reliability is obtained. The presented algorithm not only inherits the advantage of the direct Monte Carlo method in propagating and distinguishing the fuzzy and random uncertainties, but also can improve the computational efficiency tremendously in case of acceptable precision. Several examples are used to illustrate the advantages of the presented algorithm.     

A stopping game in a stochastic and fuzzy environment

A zero-sum stopping game for a sequence of fuzzy-valued random variables is discussed. The fuzzy random variables are estimated by probabilistic expectation and fuzzy expectation. A saddle point is given for the stopping game.     

Risk model with fuzzy random individual claim amount

In this paper, we consider a risk model in which individual claim amount is assumed to be a fuzzy random variable and the claim number process is characterized as a Poisson process. The mean chance of the ultimate ruin is researched. Particularly, the expressions of the mean chance of the ultimate ruin are obtained for zero initial surplus and arbitrary initial surplus if individual claim amount is an exponentially distributed fuzzy random variable. The results obtained in this paper coincide with those in stochastic case when the fuzzy random variables degenerate to random variables. Finally, two numerical examples are presented.     

Stability analysis of sampled-data fuzzy controller for nonlinear systems based on switching T–S fuzzy model

This paper investigates the system stability of a sampled-data fuzzy-model-based control system, formed by a nonlinear plant and a sampled-data fuzzy controller connected in a closed loop. The sampled-data fuzzy controller has an advantage that it can be implemented using a microcontroller or a digital computer to lower the implementation cost and time. However, discontinuity introduced by the sampling activity complicates the system dynamics and makes the stability analysis difficult compared with the pure continuous-time fuzzy control systems. Moreover, the favourable property of the continuous-time fuzzy control systems which is able to relax the stability analysis result vanishes in the sampled-data fuzzy control systems. A Lyapunov-based approach is employed to derive the LMI-based stability conditions to guarantee the system stability. To facilitate the stability analysis, a switching fuzzy model consisting of some local fuzzy models is employed to represent the nonlinear plant to be controlled. The comparatively less strong nonlinearity of each local fuzzy model eases the satisfaction of the stability conditions. Furthermore, membership functions of both fuzzy model and sampled-data fuzzy controller are considered to alleviate the conservativeness of the stability analysis result. A simulation example is given to illustrate the merits of the proposed approach.     

A new solution method for fuzzy chance constrained programming problem

This paper deals with an optimization model, where both fuzziness and randomness occur under one roof. The concept of fuzzy random variable (FRV), mean and variance of FRV is used in the model. In particular, the methodology is developed in the presence of FRV in the constraint. The methodology is verified through numerical examples.     

Fuzzy periodic review system with fuzzy random variable demand

In this paper, a periodic review inventory system has been analyzed in a mixed imprecise and uncertain environment where fuzziness and randomness appear simultaneously. A model has been developed with customer demand assumed to be a fuzzy random variable. The lead-time has been assumed to be a constant. The lead-time demand and the lead-time plus one period’s demand have also been assumed to be fuzzy random variables. A methodology has been developed to determine the optimal inventory level and the optimal period of review such that the total expected annual cost in the fuzzy sense is minimized. A numerical example has been presented to illustrate the model.     

A fuzzy closeness approach to fuzzy multi-attribute decision making

The aim of this paper is to develop a new fuzzy closeness (FC) methodology for multi-attribute decision making (MADM) in fuzzy environments, which is an important research field in decision science and operations research. The TOPSIS method based on an aggregating function representing “closeness to the ideal solution” is one of the well-known MADM methods. However, while the highest ranked alternative by the TOPSIS method is the best in terms of its ranking index, this does not mean that it is always the closest to the ideal solution. Furthermore, the TOPSIS method presumes crisp data while fuzziness is inherent in decision data and decision making processes, so that fuzzy ratings using linguistic variables are better suited for assessing decision alternatives. In this paper, a new FC method for MADM under fuzzy environments is developed by introducing a multi-attribute ranking index based on the particular measure of closeness to the ideal solution, which is developed from the fuzzy weighted Minkowski distance used as an aggregating function in a compromise programming method. The FC method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum individual regret for the “opponent”. A real example of a personnel selection problem is examined to demonstrate the implementation process of the method proposed in this paper.     

A class of multiobjective linear programming model with fuzzy random coefficients

The aim of this paper is to deal with a multiobjective linear programming problem with fuzzy random coefficients. Some crisp equivalent models are presented and a traditional algorithm based on an interactive fuzzy satisfying method is proposed to obtain the decision maker’s satisfying solution. In addition, the technique of fuzzy random simulation is adopted to handle general fuzzy random objective functions and fuzzy random constraints which are usually hard to be converted into their crisp equivalents. Furthermore, combined with the techniques of fuzzy random simulation, a genetic algorithm using the compromise approach is designed for solving a fuzzy random multiobjective programming problem. Finally, illustrative examples are given in order to show the application of the proposed models and algorithms.     

Numerical solution of fuzzy Camassa-Holm equation by using Homotopy analysis methods

In this paper, a fuzzy Camassa-Holm equation is solved by using the  homotopy analysis method (HAM). The approximation solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The existence and uniqueness of the solution and the convergence of the proposed method are proved.     

Self-tuning of fuzzy belief rule bases for engineering system safety analysis

A framework for modelling the safety of an engineering system using a fuzzy rule-based evidential reasoning (FURBER) approach has been recently proposed, where a fuzzy rule-base designed on the basis of a belief structure (called a belief rule base) forms a basis in the inference mechanism of FURBER. However, it is difficult to accurately determine the parameters of a fuzzy belief rule base (FBRB) entirely subjectively, in particular for complex systems. As such, there is a need to develop a supporting mechanism that can be used to train in a locally optimal way a FBRB initially built using expert knowledge. In this paper, the methods for self-tuning a FBRB for engineering system safety analysis are investigated on the basis of a previous study. The method consists of a number of single and multiple objective nonlinear optimization models. The above framework is applied to model the system safety of a marine engineering system and the case study is used to demonstrate how the methods can be implemented.     

The bi-objective fuzzy c-means cluster analysis for TSK fuzzy system identification

For conventional fuzzy clustering-based approaches to fuzzy system identification, a fuzzy function is used for cluster formation and another fuzzy function is used for cluster validation to determine the number and location of the clusters which define IF parts of the rule base. However, the different fuzzy functions used for cluster formation and validation may not indicate the same best number and location of the clusters. This potential disparity motivates us to propose a new fuzzy clustering-based approach to fuzzy system identification based on the bi-objective fuzzy c-means (BOFCM) cluster analysis. In this approach, we use the BOFCM function for both cluster formation and validation to simultaneously determine the number and location of the clusters which we hope can efficiently and effectively define IF parts of the rule base. The proposed approach is validated by applying it to the truck backer-upper problem with an obstacle in the center of the field.     

On fuzzy random multiobjective quadratic programming

In this paper, a multiobjective quadratic programming problem having fuzzy random coefficients matrix in the objective and constraints and the decision vector are fuzzy pseudorandom variables is considered. First, we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. At the end, numerical examples are illustrated in the support of the obtained results.     

Random fuzzy differential equations under generalized Lipschitz condition

We present the studies on two kinds of solutions to random fuzzy differential equations (RFDEs). The different types of solutions to RFDEs are generated by the usage of two different concepts of fuzzy derivative in the formulation of a differential problem. Under generalized Lipschitz condition, the existence and uniqueness of both kinds of solutions to RFDEs are obtained. We show that solutions (of the same kind) are close to each other in the case when the data of the equation did not differ much. By an example, we present an application of each type of solutions in a population growth model which is subjected to two kinds of uncertainties: fuzziness and randomness.     

A new methodology for crisp equivalent of fuzzy chance constrained programming problem

This paper deals with a chance constrained programming model, where both fuzziness and randomness are present in the objective function and constraints. The concept of fuzzy random variable, mean and variance of fuzzy random variable, minimum of fuzzy numbers are used in the model. The methodology is verified through a numerical example.     

Some comments on fuzzy variables

Nahmias introduced the concept of a fuzzy variable as a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. In this paper we attempt to shed more light on fuzzy variables in analogy with random variables. In particular, we study the problem: if X₁, X₂,…,X_n are mutually unrelated fuzzy variables with common membership function μ and α₁, α₂,…,α_n are real numbers satisfying α_i ? o for every i and Σ_i=1ⁿα_i=1, when does does Z = Σ_{i = 1}ⁿα_iX_i have the same membership function μ?     

Supply chain coordination for fuzzy random newsboy problem with imperfect quality

Facing to imperfect quality and fuzzy random market demand in the real-life inventory management, a two-echelon supply chain system with one retailer and one manufacturer for perishable products is considered. Two fuzzy random models for the newsboy problem with imperfect quality in the decentralized and centralized systems are presented. The expectation theory and signed distance are employed to transform the fuzzy random model into crisp model. The optimal policies in the two decision-making systems are derived and analyzed contrastively. The theoretical analysis shows that manufacturer’s repurchase strategy can achieve the increase in the whole supply chain profit. The influence of the fuzzy randomness of the demand and the defective rate on the optimal order quantity, the whole supply chain profit and the repurchasing price is analyzed via numerical examples.     

Modelling redundancy allocation for a fuzzy random parallel-series system

Due to subjective judgment, imprecise human knowledge and perception in capturing statistical data, the real data of lifetimes in many systems are both random and fuzzy in nature. Based on the fuzzy random variables that are used to characterize the lifetimes, this paper studies the redundancy allocation problems to a fuzzy random parallel-series system.Two fuzzy random redundancy allocation models (FR-RAM) are developed through reliability maximization and cost minimization, respectively. Some properties of the FR-RAM are obtained, in which an analytical formula of reliability with convex lifetimes is derived and the sensitivity of the reliability is discussed. To solve the FR-RAMs, we first address the computation of reliability. A random simulation method based on the derived analytical formula is proposed to compute the reliability with convex lifetimes. As for the reliability with nonconvex lifetimes, the technique of fuzzy random simulation together with the discretization method of fuzzy random variable is employed to compute the reliability, and a convergence theorem of the fuzzy random simulation is proved. Subsequently, we integrate the computation approaches of the reliability and genetic algorithm (GA) to search for the approximately optimal redundancy allocation of the models. Finally, some numerical examples are provided to illustrate the feasibility of the solution algorithm and quantify its effectiveness.     

Measurability criteria for fuzzy random vectors

Fuzzy random vector is a measurable map from a probability space to a collection of fuzzy vectors. Our aim in this paper is to discuss the measurability criteria for fuzzy random vectors, and show that under mild assumption, the measurability criteria for upper semicontinuous fuzzy random vectors can be expressed in several different but equivalent formulations. Finally, applying the obtained results, we resolve an open problem about the relationship between fuzzy random vectors and fuzzy random variables.