Fuzzy forecasting based on fuzzy time series

This article presents an improved method of fuzzy time series to forecast university enrollments. The historical enrollment data of the University of Alabama were first adopted by Song and Chissom (Song, Q. and Chissom, B. S. (1993). Forecasting enrollment with fuzzy time series-part I, Fuzzy Sets and Systems, 54, 1–9; Song, Q. and Chissom, B. S. (1994). Forecasting enrollment with fuzzy time series-part II, Fuzzy Sets and Systems, 54, 267–277) to illustrate the forecasting process of the fuzzy time series. Later, Chen proposed a simpler method. In this article, we show that our method is as simple as Chen's method but more accurate. In forecasting the enrollment of the University of Alabama, the root mean square percentage error (RMSPE) of our method is 3.1113% while the RMSPE of Chen's method is 4.0516%, which shows that our method is doing much better.     

Solving parametric fuzzy systems of linear equations by a nonlinear programming method

Linear systems of equations, with uncertainty on the parameters, play a major role in various problems in economics and finance. In this paper parametric fuzzy linear systems of the general form A ₁ x + b ₁ = A ₂ x + b ₂, with A ₁, A ₂, b ₁ and b ₂ matrices with fuzzy elements, are solved by means of a nonlinear programming method. The relation between this methodology and the algorithm proposed in Muzzioli and Reynaerts [(2006) Fuzzy Sets and Systems, in press] is highlighted. The methodology is finally applied to an economic and a financial problem.     

Indistinguishability operators in measurement theory,Part I: Conversions of indistinguishability operators with respect to scales

For two given ordinal scales in a measurement process, the present paper investigates how an indistinguishability operator evaluated according to one of these ordinal scales can be converted to another indistinguishability operator w.r.t. the other ordinal scale, and establishes the mathematical base of these conversions under the framework of measurement theory [Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A. (1971) Foundations of Measurement, Vol. 1 (Academic Press, San Diego)]. Additionally, this work exposes the rudimentary facts behind the studies in [“Fuzzy Numbers and Equality Relations”, Proc. FUZZ'IEEE 93 (1993) 1298–1301; “Fuzzy Sets and Vague Environments”, Fuzzy Sets and Systems 66 (1994) 207–221; “Fuzzy Control on the Basis of Equality Relations-with an Example from Idle Speed Control”, IEEE Transactions on Fuzzy Systems 3 (1995) 336–350; and “T-partitions of the Real Line Generated by Idempotent Shapes”, Fuzzy Sets and Systems 91 (1997) 177–184], and points out the measurement theoretic derivations of the results in these studies.     

A goal programming approach for fuzzy multiobjective fractional programming problems

Fuzzy multiple objective fractional programming (FMOFP) is an important technique for solving many real-world problems involving the nature of vagueness, imprecision and/or random. Following the idea of binary behaviour of fuzzy programming (Chang 2007 Chang, C-T. 2007. Binary Behavior of Fuzzy Programming with Piecewise Membership Functions. IEEE Transactions on Fuzzy Systems, 15: 342–349.  [Google Scholar]), there may exist a situation where a decision-maker would like to make a decision on FMOFP involving the achievement of fuzzy goals, in which some of them may meet the behaviour of fuzzy programming (i.e. level achieved) or the behaviour of binary programming (i.e. completely not achieved). This is turned into a fuzzy multiple objective mixed binary fractional programming (FMOMBFP) problem. However, to the best of our knowledge, this problem is not well formulated by mathematical programming. Therefore, this article proposes a linearisation strategy to formulate the FMOMBFP problem in which extra binary variable is not required. In addition, achieving the highest membership value of each fuzzy goal defined for the fractional objective function, the proposed method can alleviate the computational difficulties when solving the FMOMBFP problem. To demonstrate the usefulness of the proposed method, a real-world case is also included.     

A new approach to generate weighted fuzzy rules using genetic algorithms for estimating null values

In this paper, we present a new method to generate weighted fuzzy rules using genetic algorithms for estimating null values in relational database systems, where there are negative functional dependency relationships between attributes. The proposed method can get higher average estimated accuracy rates than the method presented in [Chen, S. M., & Huang, C. M. (2003). Generating weighted fuzzy rules from relational database systems for estimating null values using genetic algorithms. IEEE Transactions on Fuzzy Systems, 11(4), 495–506].     

A hybrid fuzzy goal programming approach with different goal priorities to aggregate production planning

In this study a hybrid (including qualitative and quantitative objectives) fuzzy multi objective nonlinear programming (H-FMONLP) model with different goal priorities will be developed for aggregate production planning (APP) problem in a fuzzy environment. Using an interactive decision making process the proposed model tries to minimize total production costs, carrying and back ordering costs and costs of changes in workforce level (quantitative objectives) and maximize total customer satisfaction (qualitative objective) with regarding the inventory level, demand, labor level, machines capacity and warehouse space. A real-world industrial case study demonstrates applicability of proposed model to practical APP decision problems. GENOCOP III (Genetic Algorithm for Numerical Optimization of Constrained Problems) has been used to solve final crisp nonlinear programming problem.     

Definition and construction of fuzzy DI-subsethood measures

In this paper we study a method of construction of the fuzzy subsethood measures of V.R. Young [V.R. Young, Fuzzy subsethood, Fuzzy Sets and Systems 77 (1996) 371-384] and the fuzzy subsethood measures of J. Fan, X. Xie, and J. Pei [J. Fan, X. Xie, J. Pei, Subsethood measures: new definitions, Fuzzy Sets and Systems 106 (1999) 201-209]. We establish the conditions under which our constructions satisfy the axioms of Sinha and Dougherty’s inclusion measures and we present different methods for obtaining fuzzy entropies from said measures. Next we present a particular case of Young’s subsethood measures, DI-subsethood measures. For these we also analyze their construction and the conditions under which they satisfy different axioms.     

On set-valued fuzzy measures

So far, there have been many discussions on Sugeno’s fuzzy measures and fuzzy integrals, but most of them are concentrated on single-valued functions. Motivated by Aumann’s set-valued integral [J. Math. Anal. Appl. 12 (1965) 1], we have introduced the fuzzy integral of set-valued functions [Fuzzy Sets Syst. 56 (1993) 237; Fuzzy Sets Syst. 76 (1995) 365; Fuzzy Sets Syst. 78 (1996) 341; Ph.D. Dissertation, Harbin Institute of Technology, 1998]. It is a well-behaved extension of fuzzy integrals of single-valued functions. It is well known that Arstein’s set-valued measure [Trans. Am. Math. Soc. 165 (1972) 103] is an important branch in set-valued analysis [Set-valued Analysis, Birkhauser, Berlin, 1990] or theory of correspondences [Theory of Correspondences, Wiley, New York, 1984]. Compared to it, the present paper will try to establish the basic idea of set-valued fuzzy measures, which are monotone set-valued set-functions. It is also a natural generalization of (single-valued) fuzzy measures, as well as an extension of set-valued measures in the case of one-dimension. These works include the concept of set-valued fuzzy measures, set-valued fuzzy measures defined by set-valued fuzzy integrals and set-valued pseudo-additive measures. It can be viewed as a continuation of previous work [Fuzzy Sets Syst. 56 (1993) 237; Fuzzy Sets Syst. 76 (1995) 365; Fuzzy Sets Syst. 78 (1996) 341].     

Determinization of fuzzy automata with membership values in complete residuated lattices

In this paper we introduce a new method for determinization of fuzzy finite automata with membership values in complete residuated lattices. In comparison with the previous methods, developed by Bělohlávek [R. Bělohlávek, Determinism and fuzzy automata, Information Sciences 143 (2002), 205-209] and Li and Pedrycz [Y.M. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids, Fuzzy Sets and Systems 156 (2005), 68-92], our method always gives a smaller automaton, and in some cases, when the previous methods result in infinite automata, our method can result in a finite one. We also show that determinization of fuzzy automata is closely related to fuzzy right congruences on a free monoid and fuzzy automata associated with them, and in particular, to the concept of the Nerode’s fuzzy right congruence of a fuzzy automaton, which we introduce and study here.     

Piecewise regression for fuzzy input–output data with automatic change-point detection by quadratic programming

To handle the large variation issues in fuzzy input–output data, the proposed quadratic programming (QP) method uses a piecewise approach to simultaneously generate the possibility and necessity models, as well as the change-points. According to Tanaka and Lee [H. Tanaka, H. Lee, Interval regression analysis by quadratic programming approach, IEEE Transactions on Fuzzy Systems 6 (1998) 473–481], the QP approach gives more diversely spread coefficients than linear programming (LP) does. However, their approach only deals with crisp input and fuzzy output data. Moreover, their method is weak in handling fluctuating data. So far, no method has been developed to cope with the large variation problems in fuzzy input–output data. Hence, we propose a piecewise regression for fuzzy input–output data with a QP approach. There are three advantages in our method. First, the QP technique gives a more diversely spread coefficient than does a linear programming technique. Second, the piecewise approach is used to detect the change-points in the estimated model automatically, and handle the large variation data such as outliers well. Third, the possibility and necessity models with better fitness in data processing are obtained at the same time. Two examples are presented to demonstrate the merits of the proposed method.     

Implementation of piecewise linear fuzzy quantities

Fuzzy information in expert systems and in fuzzy database systems is often represented by means of piecewise linear fuzzy quantities. In this article we describe an implementation of piecewise linear fuzzy quantities and how to perform fuzzy operations on them. This article can be seen as a natural continuation of Baekeland and Kerre [“Operations on piecewise linear fuzzy quantities: A theoretical approach,” Fuzzy Sets Syst., submitted], in which the mathematical properties of piecewise linear fuzzy quantities are investigated. © 1995 John Wiley & Sons, Inc.     

Fuzzy measures for a correlation coefficient of fuzzy numbers under TW(the weakest t-norm)-based fuzzy arithmetic operations

Recently, the sup-min convolution based on Zadeh’s extension principle has been used by Liu and Kao [Fuzzy measures for correlation coefficient of fuzzy numbers, Fuzzy Sets and Systems 128 (2002) 267-275], to calculate a fuzzy correlation coefficient. They used a mathematical programming approach to derive fuzzy measures based on the classical definition of the correlation coefficient. It is well known that T_W (the weakest t-norm)-based addition and multiplication preserve the shape of L-R fuzzy numbers. In this paper, we consider the computational aspect of the T_W-based extension principle when the principle is applied to a correlation coefficient of L-R fuzzy numbers. We give the exact solution of a fuzzy correlation coefficient without programming or the aid of computer resources.     

A fuzzy vector valued KNN-algorithm for automatic outlier detection

The K nearest neighbors approach is a viable technique in time series analysis when dealing with ill-conditioned and possibly chaotic processes. Such problems are frequently encountered in, e.g., finance and production economics. More often than not, the observed processes are distorted by nonnormal disturbances, incomplete measurements, etc. that undermine the identification, estimation and performance of multivariate techniques. If outliers can be duly recognized, many crisp statistical techniques may perform adequately as such. Geno-mathematical programming provides a connection between statistical time series theory and fuzzy regression models that may be utilized e.g., in the detection of outliers. In this paper we propose a fuzzy distance measure for detecting outliers via geno-mathematical parametrization. Fuzzy KNN is connected as a linkable library to the genetic hybrid algorithm (GHA) of the author, in order to facilitate the determination of the LR-type fuzzy number for automatic outlier detection in time series data. We demonstrate that GHA[Fuzzy KNN] provides a platform for automatically detecting outliers in both simulated and real world data.     

Generalized Fuzzy Sub-hyperquasigroups of Hyperquasigroups

This paper concerns a relationship between fuzzy sets and algebraic hyperstructures. It is a continuation of ideas presented by Davvaz (Fuzzy Sets Syst 101: 191–195 1999) and Bhakat and Das (Fuzzy Sets Syst 80: 359-368 1996). In fact, the object of this paper is to study the notion of sub-hyperquasigroup in the ( q)-fuzzy setting.     

A two-phase approach for multi-objective programming problems with fuzzy coefficients

In this study, a two-phase procedure is introduced to solve multi-objective fuzzy linear programming problems. The procedure provides a practical solution approach, which is an integration of fuzzy parametric programming (FPP) and fuzzy linear programming (FLP), for solving real life multiple objective programming problems with all fuzzy coefficients. The interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision-maker (DM). The procedure can be also performed to obtain lexicographic optimal and/or additive solutions if it is needed. In the first phase of the procedure, a family of vector optimization models is constructed by using FPP. Then in the second phase, each model is solved by FLP. The solutions are optimal and each one is an alternative decision plan for the DM.     

The fuzzy approach to statistical analysis

For the last decades, research studies have been developed in which a coalition of Fuzzy Sets Theory and Statistics has been established with different purposes. These namely are: (i) to introduce new data analysis problems in which the objective involves either fuzzy relationships or fuzzy terms; (ii) to establish well-formalized models for elements combining randomness and fuzziness; (iii) to develop uni- and multivariate statistical methodologies to handle fuzzy-valued data; and (iv) to incorporate fuzzy sets to help in solving traditional statistical problems with non-fuzzy data. In spite of a growing literature concerning the development and application of fuzzy techniques in statistical analysis, the need is felt for a more systematic insight into the potentialities of cross fertilization between Statistics and Fuzzy Logic. In line with the synergistic spirit of Soft Computing, some instances of the existing research activities on the topic are recalled. Particular attention is paid to summarize the papers gathered in this Special Issue, ranging from the position paper on the theoretical management of uncertainty by the “father” of Fuzzy Logic to a wide diversity of topics concerning foundational/methodological/applied aspects of the integration of Fuzzy Sets and Statistics.     

Consistency test and weight generation for additive interval fuzzy preference relations

Some simple yet pragmatic methods of consistency test are developed to check whether an interval fuzzy preference relation is consistent. Based on the definition of additive consistent fuzzy preference relations proposed by Tanino (Fuzzy Sets Syst 12:117–131, 1984), a study is carried out to examine the correspondence between the element and weight vector of a fuzzy preference relation. Then, a revised approach is proposed to obtain priority weights from a fuzzy preference relation. A revised definition is put forward for additive consistent interval fuzzy preference relations. Subsequently, linear programming models are established to generate interval priority weights for additive interval fuzzy preference relations. A practical procedure is proposed to solve group decision problems with additive interval fuzzy preference relations. Theoretic analysis and numerical examples demonstrate that the proposed methods are more accurate than those in Xu and Chen (Eur J Oper Res 184:266–280, 2008b).     

Feature subset selection based on fuzzy entropy measures for handling classification problems

In this paper, we present a new method for dealing with feature subset selection based on fuzzy entropy measures for handling classification problems. First, we discretize numeric features to construct the membership function of each fuzzy set of a feature. Then, we select the feature subset based on the proposed fuzzy entropy measure focusing on boundary samples. The proposed method can select relevant features to get higher average classification accuracy rates than the ones selected by the MIFS method (Battiti, R. in IEEE Trans. Neural Netw. 5(4):537–550, 1994), the FQI method (De, R.K., et al. in Neural Netw. 12(10):1429–1455, 1999), the OFEI method, Dong-and-Kothari’s method (Dong, M., Kothari, R. in Pattern Recognit. Lett. 24(9):1215–1225, 2003) and the OFFSS method (Tsang, E.C.C., et al. in IEEE Trans. Fuzzy Syst. 11(2):202–213, 2003).

GENERALIZED FUZZY RELATIONAL INEQUALITIES AND RELATED PROBLEMS

Various authors investigated fuzzy relational equalities constructed on the basic of the classical max-min extension principle (Czogala, E.. Drewniak. J. and Pedrycz. W. (1982) Fuzzy Sets and Systems, 7,89-101; Sanchez, E. (1976) Information andComrol, 30, 38-48). In this contribution, generalized fuzzy relational inequalities constructed by making use of compensatory operators or t-norms are studied (for the introduction of these concepts (Schwcizer and Sklar, A. (I960) Pacific J. Math., 10,313-334; (Clement, E.P.. Mesiar. R. and Pap, A. (1994) Preprint). Fuzzy relations considered here have a finite support. Solvability of generalized fuzzy relational inequalities is investigated.     

A new approach based on artificial neural networks for high order multivariate fuzzy time series

Fuzzy time series methods have been recently becoming very popular in forecasting. These methods can be categorized into two subclasses that are univariate and multivariate approaches. It is a known fact that real time series data can actually be affected by many factors. In this case, the using multivariate fuzzy time series forecasting model can be more reasonable in order to get more accurate forecasts. To obtain fuzzy forecasts when multivariate fuzzy time series approach is adopted, the most applied method is using tables of fuzzy relations. However, employing this method is a computationally though task. In this study, we introduce a new method that does not require using fuzzy logic relation tables in order to determine fuzzy relationships. Instead, a feed forward artificial neural network is employed to determine fuzzy relationships. The proposed method is applied to the time series data of the total number of annual car road accidents casualties in Belgium from 1974 to 2004 and a comparison is made between our proposed method and the methods proposed by Jilani and Burney [Jilani, T. A., & Burney, S. M. A. (2008). Multivariate stochastic fuzzy forecasting models. Expert Systems with Applications, 35, 691–700] and Lee et al. [Lee, L.-W., Wang, L.-H., Chen, S.-M., & Leu, Y.-H. (2006). Handling forecasting problems based on two factors high order fuzzy time series. IEEE Transactions on Fuzzy Systems, 14, 468–477].