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.     

Mehar’s method for solving fuzzy sensitivity analysis problems with LR flat fuzzy numbers

In published works on fuzzy linear programming there are only few papers dealing with stability or sensitivity analysis in fuzzy mathematical programming. To the best of our knowledge, till now there is no method in the literature to deal with the sensitivity analysis of such fuzzy linear programming problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method, named as Mehar’s method, is proposed for the same. To show the advantages of proposed method over existing methods, some fuzzy sensitivity analysis problems which may or may not be solved by the existing methods are solved by using the proposed method.     

Weighted portfolio selection models based on possibility theory

In this paper, we discuss portfolio selection problem in a fuzzy uncertain environment. Based on the Fullér’s and Zhang’s notations, we discuss some properties of weighted lower and upper possibilistic means and variances as in probability theory. We further present two weighted possibilistic portfolio selection models with bounded constraint, which can be transformed to linear programming problems under the assumption that the returns of assets are trapezoidal fuzzy numbers. At last, a numerical example is given to illustrate our proposed effective means and approaches.     

Editorial

Linear programming problems with fuzzy parameters are formulated by fuzzy functions. The ambiguity considered here is not randomness, but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership. Parameters on constraint and objective functions are given by fuzzy numbers. In this paper, our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters. This fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential.     

Relating and extending semantical approaches to possibilistic reasoning

Two main semantical approaches to possibilistic reasoning with classical propositions have been proposed in the literature. Namely, Dubois-Prade's approach known as possibilistic logic, whose semantics is based on a preference ordering in the set of possible worlds, and Ruspini's approach that we redefine and call similarity logic, which relies on the notion of similarity or resemblance between worlds. In this article we put into relation both approaches, and it is shown that the monotonic fragment of possibilistic logic can be semantically embedded into similarity logic. Furthermore, to extend possibilistic reasoning to deal with fuzzy propositions, a semantical reasoning framework, called fuzzy truth-valued logic, is also introduced and proved to capture the semantics of both possibilistic and similarity logics.     

Solving a multiobjective possibilistic problem through compromise programming

Real decision problems usually consider several objectives that have parameters which are often given by the decision maker in an imprecise way. It is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory.Here we propose a method for solving these kinds of models through a fuzzy compromise programming approach.To formulate a fuzzy compromise programming problem from a possibilistic multiobjective linear programming problem the fuzzy ideal solution concept is introduced. This concept is based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their α-cuts. The accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and a definition of discrepancy between intervals is introduced in our analysis.     

Fuzzy identification problem for continuous extremal fuzzy dynamic system

This work deals with the problems of the Continuous Extremal Fuzzy Dynamic System (CEFDS) optimization and briefly discusses the results developed by Sirbiladze (Int J Gen Syst 34(2):107–138, 2005a; 34(2):139–167, 2005b; 34(2):169–198, 2005c; 35(4):435–459, 2006a; 35(5):529–554, 2006b; 36(1): 19–58, 2007; New Math Nat Comput 4(1):41–60, 2008a; Mat Zametki, 83(3):439–460, 2008b). The basic properties of extended extremal fuzzy measures and Sugeno’s type integrals are considered and several variants of their representation are given. Values of extended extremal conditional fuzzy measures are defined as a levels of expert knowledge reflections of CEFDS states in the fuzzy time intervals. The notions of extremal fuzzy time moments and intervals are introduced and their monotone algebraic structures that form the most important part of the fuzzy instrument of modeling extremal fuzzy dynamic systems are discussed. A new approach in modeling of CEFDS is developed. Applying the results of Sirbiladze (Int J Gen Syst 34(2) 107–138, 2005a; 34(2):139–167, 2005b), fuzzy processes with possibilistic uncertainty, the source of which are expert knowledge reflections on the states on CEFDS in extremal fuzzy time intervals, are constructed (Sirbiladze in Int J Gen Syst 34(2):169–198, 2005c). The dynamics of CEFDS’s is described. Questions of the ergodicity of CEFDS are considered. A fuzzy-integral representation of a continuous extremal fuzzy process is given. Based on the fuzzy-integral model, a method and an algorithm are developed for identifying the transition operator of CEFDS. The CEFDS transition operator is restored by means of expert data with possibilistic uncertainty, the source of which is expert knowledge reflections on the states of CEFDS in the extremal fuzzy time intervals. The regularization condition for obtaining quasi-optimal estimator of the transition operator is represented by the theorems. The corresponding calculating algorithm is provided. The results obtained are illustrated by an example in the case of a finite set of CEFDS states.     

A cutting plane algorithm for MV portfolio selection model

This paper deals with a portfolio selection problem with fuzzy return rates. A possibilistic mean variance (FMVC) portfolio selection model was proposed. The possibilistic programming problem can be transformed into a linear optimal problem with an additional quadratic constraint by possibilistic theory. For such problems there are no special standard algorithms. We propose a cutting plane algorithm to solve (FMVC). The nonlinear programming problem can be solved by sequence linear programming problem. A numerical example is given to illustrate the behavior of the proposed model and algorithm.     

Robust Solution to Fuzzy Identification Problem with Uncertain Data by Regularization

This study considers the robust identification of the parameters describing a Sugeno type fuzzy inference system with uncertain data. The objective is to minimize the worst-case residual error using a numerically efficient algorithm. The Sugeno type fuzzy systems are linear in consequent parameters but nonlinear in antecedent parameters. The robust consequent parameters identification problem can be formulated as second-order cone programming problem. The optimal solution of this second-order cone problem can be interpreted as solution of a Tikhonov regularization problem with a special choice of regularization parameter which is optimal for robustness (Ghaoui and Lebret (1997). SAIM Journal of Matrix Analysis and Applications 18, 1035–1064). The final regularized nonlinear optimization problem allowing simultaneous identification of antecedent and consequent parameters is solved iteratively using a generalized Gauss–Newton like method. To illustrate the approach, several simulation studies on numerical examples including the modelling of a spectral data function (one-dimensional benchmark example) is provided. The proposed robust fuzzy identification scheme has been applied to approximate the physical fitness of patients with a fuzzy expert system. The identified fuzzy expert system is shown to be capable of capturing the decisions (experiences) of a medical expert.     

基于模糊收益率的组合投资模型

本文考虑了收益率为模糊数的投资组合选择问题,利用模型约束简化方差约束,建立了投资组合选择的模糊线性规划模型,然后引进模糊期望把模糊线性规划问题化为普通参数线性规划问题,最后给出了一个数值算例.     

A group of knowledge-incorporated multiple criteria linear programming classifiers

Classification is a main data mining task, which aims at predicting the class label of new input data on the basis of a set of pre-classified samples. Multiple criteria linear programming (MCLP) is used as a classification method in the data mining area, which can separate two or more classes by finding a discriminate hyperplane. Although MCLP shows good performance in dealing with linear separable data, it is no longer applicable when facing with nonlinear separable problems. A kernel-based multiple criteria linear programming (KMCLP) model is developed to solve nonlinear separable problems. In this method, a kernel function is introduced to project the data into a higher-dimensional space in which the data will have more chance to be linear separable. KMCLP performs well in some real applications. However, just as other prevalent data mining classifiers, MCLP and KMCLP learn only from training examples. In the traditional machine learning area, there are also classification tasks in which data sets are classified only by prior knowledge, i.e. expert systems. Some works combine the above two classification principles to overcome the faults of each approach. In this paper, we provide our recent works which combine the prior knowledge and the MCLP or KMCLP model to solve the problem when the input consists of not only training examples, but also prior knowledge. Specifically, how to deal with linear and nonlinear knowledge in MCLP and KMCLP models is the main concern of this paper. Numerical tests on the above models indicate that these models are effective in classifying data with prior knowledge.     

基于DEA博弈交叉效率和投资者心理的模糊投资组合研究

考虑到投资者并不是完全理性的,本文结合DEA博弈交叉效率方法研究了带有投资者心理因素的多目标模糊投资组合决策问题。首先,为了充分描绘投资者的心理因素和风险感知,本文基于可能性理论推导了带有风险态度的可能性均值和半绝对偏差。其次,将候选的风险资产视为互相竞争的博弈者,采用基于熵权法的DEA博弈交叉效率模型衡量它们的综合表现,从而得到每项资产的博弈交叉效率和奇异指数,并将其分别作为额外的收益和风险决策准则。基于此,提出了更加综合的可能性均值—半绝对偏差—博弈交叉效率—奇异指数模型。最后,通过一个应用实例验证了所提出的模型的合理性和有效性,从而为不同类型的投资者提供具有个性化的投资策略。     

Mixed fuzzy inter-cluster separation clustering algorithm

Based on inter-cluster separation clustering (ICSC) fuzzy inter-cluster separation clustering (FICSC) deals with all the distances between the cluster centers, maximizes these distances and obtains the better performances of clustering. However, FICSC is sensitive to noises the same as fuzzy c-means (FCM) clustering. Possibilistic type of FICSC is proposed to combine FICSC and possibilistic c-means (PCM) clustering. Mixed fuzzy inter-cluster separation clustering (MFICSC) is presented to extend possibilistic type of FICSC because possibilistic type of FICSC is sensitive to initial cluster centers and always generates coincident clusters. MFICSC can produce both fuzzy membership values and typicality values simultaneously. MFICSC shows good performances in dealing with noisy data and overcoming the problem of coincident clusters. The experimental results with data sets show that our proposed MFICSC holds better clustering accuracy, little clustering time and the exact cluster centers.     

On the possibilistic mean value and variance of multiplication of fuzzy numbers

In this paper, we introduce the definitions of the possibilistic mean, variance and covariance of multiplication of fuzzy numbers, and show some properties of these definitions. Then, we apply these definitions to build the possibilistic models of portfolio selection under the situations involving uncertainty over the time horizon, by considering the portfolio selection problem from the point of view of possibilistic analysis. Moreover, numerical experiments with real market data indicate that our approach results in better portfolio performance.     

An extension to possibilistic fuzzy cluster analysis

We explore an approach to possibilistic fuzzy clustering that avoids a severe drawback of the conventional approach, namely that the objective function is truly minimized only if all cluster centers are identical. Our approach is based on the idea that this undesired property can be avoided if we introduce a mutual repulsion of the clusters, so that they are forced away from each other. We develop this approach for the possibilistic fuzzy c-means algorithm and the Gustafson–Kessel algorithm. In our experiments we found that in this way we can combine the partitioning property of the probabilistic fuzzy c-means algorithm with the advantages of a possibilistic approach w.r.t. the interpretation of the membership degrees.     

模糊环境中带交易费用的投资组合策略问题

由于金融市场是波动的,风险资产的预期收益率由于很多不确定性是很难估计的,本文考虑预期收益率是可能性分布(模糊数),并且在此基础上用模糊数的可能性均值表示投资组合的收益,用模糊数的平均绝对偏差表示风险,考虑了交易费用后,得到投资组合模型,最后给出了数值计算的例子.     

Possibilistic linear-programming approach for supply chain networking decisions

Supply chain networking decisions are very important for the medium- and long-term planning success of manufacturing companies. The inputs to supply chain planning models are subject to environmental and system uncertainties. In this paper, a fuzzy set theory-based model is proposed to deal with those uncertainties. For this purpose, a possibilistic linear programming (PLP) model is used to make strategic resource-planning decisions using fuzzy demand forecasts and fuzzy yield rates as well as other inputs such as costs and capacities. The objective of the proposed PLP is to maximize the total profit of the enterprise. The model is applied to Mercedes–Benz Türk, one of the largest bus-manufacturing companies in the world, and conclusions and suggestions for further research are provided.     

A jump-diffusion model for option pricing under fuzzy environments

Owing to fluctuations in the financial markets from time to time, the rate λ of Poisson process and jump sequence {V_i} in the Merton’s normal jump-diffusion model cannot be expected in a precise sense. Therefore, the fuzzy set theory proposed by Zadeh [Zadeh, L.A., 1965. Fuzzy sets. Inform. Control 8, 338-353] and the fuzzy random variable introduced by Kwakernaak [Kwakernaak, H., 1978. Fuzzy random variables I: Definitions and theorems. Inform. Sci. 15, 1-29] and Puri and Ralescu [Puri, M.L., Ralescu, D.A., 1986. Fuzzy random variables. J. Math. Anal. Appl. 114, 409-422] may be useful for modeling this kind of imprecise problem. In this paper, probability is applied to characterize the uncertainty as to whether jumps occur or not, and what the amplitudes are, while fuzziness is applied to characterize the uncertainty related to the exact number of jump times and the jump amplitudes, due to a lack of knowledge regarding financial markets. This paper presents a fuzzy normal jump-diffusion model for European option pricing, with uncertainty of both randomness and fuzziness in the jumps, which is a reasonable and a natural extension of the Merton [Merton, R.C., 1976. Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125-144] normal jump-diffusion model. Based on the crisp weighted possibilistic mean values of the fuzzy variables in fuzzy normal jump-diffusion model, we also obtain the crisp weighted possibilistic mean normal jump-diffusion model. Numerical analysis shows that the fuzzy normal jump-diffusion model and the crisp weighted possibilistic mean normal jump-diffusion model proposed in this paper are reasonable, and can be taken as reference pricing tools for financial investors.     

Environmental risk modelling under probability-normal interval-valued fuzzy number

In almost all the realistic circumstances, such as health risk assessment and uncertainty analysis of atmospheric dispersion, it is very essential to include all the information into modelling. The parameters associated to a particular model may include different kind of variability, imprecision and uncertainty. More often, it is seen that available informations are interpreted in probabilistic sense. Probability theory is a well-established theory to measure such kind of variability. However, not all of available information, data or model parameters affected by variability, imprecision and uncertainty can be handled by traditional probability theory. Uncertainty or imprecision may occur due to incomplete information or data, measurement errors or data obtained from expert judgement or subjective interpretation of available data or information. Thus, model parameters, data may be affected by subjective uncertainty. Traditional probability theory is inappropriate to represent them. Possibility theory and fuzzy set theory is another branch of mathematics which is used as a tool to describe the parameters with insufficient or vague knowledge. In this paper, an attempt has been made to combine probability knowledge and possibility knowledge and draw the uncertainty. The paper describes an algorithm for combining probability distribution and interval-valued fuzzy number and applied to environmental risk modelling with a case study. The primary aim of this paper is to propagate the proposed method. Computer codes are prepared for the proposed method using MATLAB.