A differential harmony search based hybrid interval type2 fuzzy EGARCH model for stock market volatility prediction

In this paper a new hybrid model integrating an interval type2 fuzzy logic system (IT2FLS) with a computationally efficient functional link artificial neural network (CEFLANN) and an Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH) model has been proposed for accurate forecasting and modeling of financial data with changing variance over time. The proposed model denoted as IT2F-CE-EGARCH helps to enhance the ability of EGARCH model through a joint estimation of the important features of EGARCH like leverage effect, asymmetric shock by leverage effect with the secondary membership functions of interval type2 TSK FLS and the functional expansion and learning component of a CEFLANN. The secondary membership functions with upper and lower limits of IT2FLS provide a forecasting interval for handling more complicated uncertainties involved in volatility forecasting compared to type1 FLS. The performance of the proposed model has been observed with two membership functions i.e. Gaussian with fixed mean, uncertain variance and Gaussian with fixed variance and uncertain mean. The proposed model has also been compared with a few other fuzzy time series models and GARCH family models based on four performance metrics: MSFE, RMSFE, MAFE and Rel MAE. Again a differential harmony search (DHS) algorithm has been suggested for optimizing the parameters of all the fuzzy time series models. The results indicate that the proposed IT2F-CE-EGARCH model offers significant improvements in volatility forecasting performance in comparison with all other specified models over BSE Sensex and CNX Nifty dataset.     

Some robust objectives of FCM for data analyzing

There are many data clustering techniques available to extract meaningful information from real world data, but the obtained clustering results of the available techniques, running time for the performance of clustering techniques in clustering real world data are highly important. This work is strongly felt that fuzzy clustering technique is suitable one to find meaningful information and appropriate groups into real world datasets. In fuzzy clustering the objective function controls the groups or clusters and computation parts of clustering. Hence researchers in fuzzy clustering algorithm aim is to minimize the objective function that usually has number of computation parts, like calculation of cluster prototypes, degree of membership for objects, computation part for updating and stopping algorithms. This paper introduces some new effective fuzzy objective functions with effective fuzzy parameters that can help to minimize the running time and to obtain strong meaningful information or clusters into the real world datasets. Further this paper tries to introduce new way for predicting membership, centres by minimizing the proposed new fuzzy objective functions. And experimental results of proposed algorithms are given to illustrate the effectiveness of proposed methods.     

Differential Equations With Fuzzy Parameters

This paper is concerned with systems of ordinary differential equations with fuzzy parameters. Applying the Zadeh extension principle to the equations, we introduce the notions of fuzzy solutions and of componentwise fuzzy solutions. The fuzzy extension of the solution operator is shown to provide the unique fuzzy solution as well as the maximal componentwise fuzzy solution. A numerical algorithm based on monotonicity properties of membership functions is presented, together with a proof of convergence. In an interplay of interval analysis and possibility theory, these methods allow to process subjective information on the possible fluctuations of parameters in models involving ordinary differential equations. This is demonstrated in two engineering applications: a queueing model for earthwork and a model of oscillations of bell-towers.     

Modelling and long-term simulation of a heat recovery steam generator

Developing accurate non-linear dynamical models for heat recovery steam generator (HRSG) units is presented in this article. The common non-linear autoregressive with exogenous input (NARX) system topology was employed to develop the neuro-fuzzy models based on the experimental data taken during field experiments. In this structure, the non-linear behaviours of the HRSG unit can be characterized through interpolation of local linear models associated with different operating regions via fuzzy inference mechanism. The operating regimes were recognized by applying a genetic algorithm-based fuzzy clustering technique to the prepared data sets. The structures of the fuzzy models are defined with respect to the obtained optimal cluster centres and the corresponding membership functions. The parameters of fuzzy rules were adjusted by recursive least-squares estimation method to fit the model responses to real data. The performances of developed models were evaluated by performing a comparison between the model responses and the responses of the real plant. In addition, the stability of the developed models was assessed by perturbing the model inputs from the nominal values. This guarantees the long-term simulation capabilities of the developed models. A comparison between the responses of the corresponding models and the models obtained from some recent modelling approaches was performed to show the advantages of the developed models. The results show the accuracy and reliability of the developed models at transient and steady-state conditions.     

Type-2 T–S fuzzy impulsive control of nonlinear systems

A novel impulsive control approach based on interval Type-2 T–S fuzzy model has been presented for nonlinear systems in this paper. This approach makes up for the drawback of Type-1 fuzzy impulsive control, which cannot fully handle the uncertainties in describing the complex nonlinear systems by Type-1 fuzzy membership functions and cannot give rigorous fuzzy rules. Further more, this approach uses the “broad band” effect of the Type-2 membership functions to solve the noise of training data and exterior disturbance of the Type-1 fuzzy impulsive control. By using Lyapunov theory and Lipschitz condition, which is combined with integrated approaches such as comparison methods and linear matrix inequalities, the Type-2 fuzzy impulsive controller is designed and the general asymptotical stability analysis of the systems is given. Finally, the simulation of the inverted pendulum model demonstrates the validity and superiority of the proposed method by easily determining the membership functions and choosing minimum number of fuzzy rules and the method can handle random disturbance and data uncertainties very well.     

An interactive method for inventory control with fuzzy lead-time and dynamic demand

Normally, the real-world inventory control problems are imprecisely defined and human interventions are often required to solve these decision-making problems. In this paper, a realistic inventory model with imprecise demand, lead-time and inventory costs have been formulated and an inventory policy is proposed to minimize the cost using man–machine interaction. Here, demand increases with time at a decreasing rate. The imprecise parameters of lead-time, inventory costs and demand are expressed through linear/non-linear membership functions. These are represented by different types of membership functions, linear or quadratic, depending upon the prevailing supply condition and marketing environment. The imprecise parameters are first transformed into corresponding interval numbers and then following the interval mathematics, the objective function for average cost is changed into respective multi-objective functions. These functions are minimized and solved for a Pareto-optimum solution by interactive fuzzy decision-making procedure. This process leads to man–machine interaction for optimum and appropriate decision acceptable to the decision maker’s firm. The model is illustrated numerically and the results are presented in tabular forms.     

A displayed inventory model with L–R fuzzy number

In this study, we formulate a multi-item displayed inventory model under shelf-space constraint in fuzzy environment. Here demand rate of an item is considered as a function of the displayed inventory level. The problem is formulated to maximize average profit. In real life situation, the goals and inventory parameters are may not precise. Such type of uncertainty may be characterized by fuzzy numbers. Here, the constraint goal and the inventory cost parameters are assumed to be triangular shaped fuzzy numbers with different types of left and right membership functions. The fuzzy numbers are then approximated to a nearest interval number. Using arithmetic of interval numbers, the problem is described as a multi-objective inventory problem. The problem is then solved by fuzzy geometric programming approach. Finally a numerical example is given to illustrate the problem.     

Upper and lower approximation models in interval regression using regression quantile techniques

In this paper, we propose new interval regression analysis based on the regression quantile techniques. To analyze a phenomenon in a fuzzy environment, we propose two interval approximation models. Without using all data, we first identify the main trend from the designated proportion of the given data. To select the main part of data to be analyzed, we introduce the regression quantile techniques. The obtained model is not influenced by extreme points since it is formulated from the center-located main proportion of the given data. After that, the interval regression model including all data can be identified based on the acquired main trend. The obtained interval regression model by the main proportion of the given data is called the lower approximation model, while interval regression model by all data is called the upper approximation model for the given phenomenon. Also it is shown that, from the lower approximation model (main trend) and the upper approximation model, we can construct a trapezoidal fuzzy model. The membership function of this fuzzy model is useful to obtain the locational information for each observation. The characteristic of our approach can be described as obtaining the upper and lower approximation models and combining them to be a fuzzy model for representing the given phenomenon in a fuzzy environment.     

Extracting interpretable fuzzy models for nonlinear systems using gradient-based continuous ant colony optimization

This paper exploits the ability of a novel ant colony optimization algorithm called gradient-based continuous ant colony optimization, an evolutionary methodology, to extract interpretable first-order fuzzy Sugeno models for nonlinear system identification. The proposed method considers all objectives of system identification task, namely accuracy, interpretability, compactness and validity conditions. First, an initial structure of model is obtained by means of subtractive clustering. Then, an iterative two-step algorithm is employed to produce a simplified fuzzy model in terms of number of fuzzy sets and rules. In the first step, the parameters of the model are adjusted by utilizing the gradient-based continuous ant colony optimization. In the second step, the similar membership functions of an obtained model merge. The results obtained on three case studies illustrate the applicability of the proposed method to extract accurate and interpretable fuzzy models for nonlinear system identification.     

Fuzzy confidence interval for pH titration curve

In this paper we present a new method of confidence interval identification for Takagi–Sugeno fuzzy models in the case of the data with regionally changeable variance. The method combines a fuzzy identification methodology with some ideas from applied statistics. The idea is to find, on a finite set of measured data, the confidence interval defined by the lower and upper bounds. The confidence interval which defines the band that contains the measurement values with certain confidence. The method can be used when describing a family of uncertain nonlinear functions or when the systems with uncertain physical parameters are observed. In our example the proposed method is applied to model the pH-titration curve.     

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.     

Fuzzy pricing,marketing and service planning in a fuzzy inventory model: A geometric programming approach

In the real world markets, demand is influenced by different parameters. Recently, many researchers have been interested in integrated production and marketing planning strategies in inventory models where demand depends on different parameters such as price and/or marketing expenditure. The quality of services that are offered to customers of a product is one of the most important parameters that affects demand in the real markets and has not been considered in development of inventory models. On the other hand, the cost parameters in real inventory systems and other parameters such as price, marketing and service elasticity to demand are imprecise and uncertain in nature. So, the notion of fuzziness can be applied to cope with this uncertainty. In this paper, a new fuzzy profit maximization inventory model with shortages is proposed. The demand is considered as a power function of price, marketing expenditure and service expenditure. Furthermore, unit cost is determined as a power function of order quantity. Since the proposed model is in a fuzzy environment, a fuzzy decision should be made to meet the decision criteria, and the results should be fuzzy. Therefore, the proposed model is formulated and solved using geometric programming and fuzzy optimization techniques to derive an approximation of the results’ membership functions. The model is illustrated with a numerical example and finally a case study is provided for evaluation and validation of the results of model.     

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.     

Multi-item inventory models with price dependent demand under flexibility and reliability consideration and imprecise space constraint: A geometric programming approach

In this paper, multi-item economic production quantity (EPQ) models with selling price dependent demand, infinite production rate, stock dependent unit production and holding costs are considered. Flexibility and reliability consideration are introduced in the production process. The models are developed under two fuzzy environments–one with fuzzy goal and fuzzy restrictions on storage area and the other with unit cost as fuzzy and possibility–necessity restrictions on storage space. The objective goal and constraint goal are defined by membership functions and the presence of fuzzy parameters in the objective function is dealt with fuzzy possibility/necessity measures. The models are formed as maximization problems. The first one—the fuzzy goal programming problem is solved using Fuzzy Additive Goal Programming (FAGP) and Modified Geometric Programming (MGP) methods. The second model with fuzzy possibility/necessity measures is solved by Geometric Programming (GP) method. The models are illustrated through numerical examples. The sensitivity analyses of the profit function due to different measures of possibility and necessity are performed and presented graphically.     

模糊运算和模糊有限元静力控制方程的求解

根据模糊数的区间形式表达和区间运算的性质,给出了模糊数和模糊变量的运算规则.据此并依据区间有限元理论,提出了结构模糊有限元静力控制方程的几种求解方法.方法可根据输入模糊数的隶属函数,给出结构响应量的可能性分布.且计算量小,易于实施.算例分析说明了方法是实用和可行的.     

Intelligent data analysis and model interpretation with spectral analysis fuzzy symbolic modeling

This paper proposes fuzzy symbolic modeling as a framework for intelligent data analysis and model interpretation in classification and regression problems. The fuzzy symbolic modeling approach is based on the eigenstructure analysis of the data similarity matrix to define the number of fuzzy rules in the model. Each fuzzy rule is associated with a symbol and is defined by a Gaussian membership function. The prototypes for the rules are computed by a clustering algorithm, and the model output parameters are computed as the solutions of a bounded quadratic optimization problem. In classification problems, the rules’ parameters are interpreted as the rules’ confidence. In regression problems, the rules’ parameters are used to derive rules’ confidences for classes that represent ranges of output variable values. The resulting model is evaluated based on a set of benchmark datasets for classification and regression problems. Nonparametric statistical tests were performed on the benchmark results, showing that the proposed approach produces compact fuzzy models with accuracy comparable to models produced by the standard modeling approaches. The resulting model is also exploited from the interpretability point of view, showing how the rule weights provide additional information to help in data and model understanding, such that it can be used as a decision support tool for the prediction of new data.     

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.     

Process planning in a fuzzy environment

Mixed-integer optimization models for chemical process planning typically assume that model parameters can be accurately predicted. As precise forecasts are difficult to obtain, process planning usually involves uncertainty and ambiguity in the data. This paper presents an application of fuzzy programming to process planning. The forecast parameters are assumed to be fuzzy with a linear or triangular membership function. The process planning problem is then formulated in terms of decision making in a fuzzy environment with fuzzy constraints and fuzzy net present value goals. The model is transformed to a deterministic mixed-integer linear program or mixed-integer nonlinear program depending on the type of uncertainty involved in the problem. For the nonlinear case, a global optimization algorithm is developed for its solution. This algorithm is applicable to general possibilistic programs and can be used as an alternative to the commonly used bisection method. Illustrative examples and computational results for a petrochemical complex with 38 processes and 24 products illustrate the applicability of the developed models and algorithms.     

Fuzinsumo - Fuzzy investigation with surrogate models

In this contribution, we investigate the accuracy of surrogate models for fuzzy systems. Within a virtual tolerance analysis, unknown uncertainties are considered by fuzzy numbers. These fuzzy numbers consist of higher-dimensional intervals, where each interval is assigned a membership value. To reproduce fuzzy numbers correctly, a large number of system evaluations is needed. Instead of evaluating numerically expensive full systems, we employ surrogate models with radial basis functions. In order to find sufficiently accurate surrogate models with as few as possible sampling points, we investigate the influence of the sampling points on the respective fuzzy number intervals. As an example, we investigate the tilting of an X-ray aperture, which shows different accuracy for different sampling strategies. The reference results for the nonlinear multi-body system are calculated by a Matlab optimizer and are compared to the results of the surrogate models. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

具有模糊目标的多目标双矩阵对策集结与求解

针对不确定性多冲突环境,建立了多个具有模糊目标的多目标双矩阵对策的综合集结模型.在假定局中人各模糊目标的隶属函数为线性函数的情形下,基于总体模糊目标的可达度,给出了纳什均衡解的定义,并应用粒子群优化算法对集结模型求解.最后,给出一个军事例子说明了模型的实用有效性和粒子群优化算法求解的高效性.