Application of genetic algorithm for fuzzy rules optimization on semi expert judgment automation using Pittsburg approach

Genetic algorithm is well-known of its best heuristic search method. Fuzzy logic unveils the advantage of interpretability. Genetic fuzzy system exploits potential of optimization with ease of understanding that facilitates rules optimization. This paper presents the optimization of fourteen fuzzy rules for semi expert judgment automation of early activity based duration estimation in software project management. The goal of the optimization is to reduce linguistic terms complexity and improve estimation accuracy of the fuzzy rule set while at the same time maintaining a similar degree of interpretability. The optimized numbers of linguistic terms in fuzzy rules by 27.76% using simplistic binary encoding mechanism managed to improve accuracy by 14.29% and reduce optimization execution time by 6.95% without compromising on interpretability in addition to promote improvement of knowledge base in fuzzy rule based systems.     

A Proposal for the Genetic Lateral Tuning of Linguistic Fuzzy Systems and Its Interaction With Rule Selection

Linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings. However, since the accuracy and the interpretability of the obtained model are contradictory properties, the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled. To solve this problem, one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level. In this paper, a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions, with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability. To do so, we consider a new rule representation scheme base on the linguistic 2-tuples representation model which allows the lateral variation of the involved labels. Furthermore, the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper, presenting results on different real applications. The obtained results show the good performance of the proposed approach in high-dimensional problems and its ability to cooperate with methods to remove unnecessary rules.     

Fuzzy rule weight modification with particle swarm optimisation

The most challenging problem in developing fuzzy rule-based classification systems is the construction of a fuzzy rule base for the target problem. In many practical applications, fuzzy sets that are of particular linguistic meanings, are often predefined by domain experts and required to be maintained in order to ensure interpretability of any subsequent inference results. However, learning fuzzy rules using fixed fuzzy quantity space without any qualification will restrict the accuracy of the resulting rules. Fortunately, adjusting the weights of fuzzy rules can help improve classification accuracy without degrading the interpretability. There have been different proposals for fuzzy rule weight tuning through the use of various heuristics with limited success. This paper proposes an alternative approach using Particle Swarm Optimisation in the search of a set of optimal rule weights, entailing high classification accuracy. Systematic experimental studies are carried out using common benchmark data sets, in comparison to popular rule based learning classifiers. The results demonstrate that the proposed approach can boost classification performance, especially when the size of the initially built rule base is relatively small, and is competitive to popular rule-based learning classifiers.     

Extracting Takagi-Sugeno Fuzzy Rules with Interpretable Submodels via Regularization of Linguistic Modifiers

In this paper, a method for constructing Takagi-Sugeno (TS) fuzzy system from data is proposed with the objective of preserving TS submodel comprehensibility, in which linguistic modifiers are suggested to characterize the fuzzy sets. A good property held by the proposed linguistic modifiers is that they can broaden the cores of fuzzy sets while contracting the overlaps of adjoining membership functions (MFs) during identification of fuzzy systems from data. As a result, the TS submodels identified tend to dominate the system behaviors by automatically matching the global model (GM) in corresponding subareas, which leads to good TS model interpretability while producing distinguishable input space partitioning. However, the GM accuracy and model interpretability are two conflicting modeling objectives, improving interpretability of fuzzy models generally degrades the GM performance of fuzzy models, and vice versa. Hence, one challenging problem is how to construct a TS fuzzy model with not only good global performance but also good submodel interpretability. In order to achieve a good tradeoff between GM performance and submodel interpretability, a regularization learning algorithm is presented in which the GM objective function is combined with a local model objective function defined in terms of an extended index of fuzziness of identified MFs. Moreover, a parsimonious rule base is obtained by adopting a QR decomposition method to select the important fuzzy rules and reduce the redundant ones. Experimental studies have shown that the TS models identified by the suggested method possess good submodel interpretability and satisfactory GM performance with parsimonious rule bases.     

Context adaptation of mamdani fuzzy rule based systems

Context adaptation is certainly a promising approach in the development of fuzzy rule based systems (FRBSs). First, an initial rule base is extracted from heuristic knowledge of the application domain. Meanings of linguistic terms are defined so as to guarantee high interpretability of the FRBSs. Then, meanings are adapted to a specific context through the use of operators that, using a set of known input–output patterns, appropriately modify the corresponding fuzzy sets. The choice of the specific operators and their parameters is context based and optimized so as to obtain a good interpretability–accuracy trade‐off. In this paper, we propose a set of operators that, starting from a given FRBS, adapt the FRBS to the specific context by adjusting the universes of the input and output variables, and modifying the core, the support and the shape of the fuzzy sets which compose the partitions of these universes. The operators are defined so as to preserve ordering of the linguistic terms, universality of rules, and interpretability of partitions. The choice of the parameters used in the operators is performed by a genetic optimization process aimed at maximizing the accuracy and preserving the interpretability of the FRBS. We finally describe the application of our context adaptation approach to two Mamdani fuzzy systems developed, respectively, for two different domains, namely, regression and data modeling. © 2008 Wiley Periodicals, Inc.     

不确定性数据中基于GSO优化MF的模糊关联规则挖掘方法

针对不确定性数据中模糊关联规则的挖掘问题,提出一种基于群搜索优化(GSO)算法优化隶属度函数(MF)的模糊关联规则挖掘方法。首先,将不确定性数据通过三元语言表示模型进行表示;然后,给定一个初始MF,并以最大化模糊项集支持度和语义可解释性作为适应度函数,通过GSO算法的优化学习获得最佳MF;最后,根据获得的最佳MF,利用改进型的FFP-growth算法来从不确定数据中挖掘模糊关联规则。实验结果表明,该方法能够根据数据集自适应优化MF,以此实现从不确定数据中有效地挖掘关联规则。     

Increasing fuzzy rules cooperation based on evolutionary adaptive inference systems

This article presents a study on the use of parametrized operators in the Inference System of linguistic fuzzy systems adapted by evolutionary algorithms, for achieving better cooperation among fuzzy rules. This approach produces a kind of rule cooperation by means of the inference system, increasing the accuracy of the fuzzy system without losing its interpretability. We study the different alternatives for introducing parameters in the Inference System and analyze their interpretation and how they affect the rest of the components of the fuzzy system. We take into account three applications in order to analyze their accuracy in practice. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1035–1064, 2007.     

多维背包问题的一个蚁群优化算法

蚁群优化(ACO)是一种通用的启发式方法,已被用来求解很多离散优化问题.近年来,已提出几个ACO算法求解多维背包问题(MKP).这些算法虽然能获得较好的解但也耗用太多的CPU时间.为了降低用ACO求解MKP的复杂性,文章基于一种已提出但未实现过的MKP的信息素表示定义了新的选择概率的规则和相应的基于背包项的一种序的启发式信息,从而提出了一种计算复杂性较低、求解性能较好的改进型蚁群算法.实验结果表明,无论串行执行还是虚拟并行执行,在计算相同任务时,新算法耗用时间少且解的价值更高.不仅如此,在实验中,文中的新算法获得了ORLIB中测试算例5.250-22的两个"新"解.     

Interpretability of linguistic fuzzy rule-based systems: An overview of interpretability measures

Linguistic fuzzy modelling, developed by linguistic fuzzy rule-based systems, allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings. Linguistic fuzzy modelling comes with two contradictory requirements: interpretability and accuracy. In recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown.Whereas the measures of accuracy are straightforward and well-known, interpretability measures are difficult to define since interpretability depends on several factors; mainly the model structure, the number of rules, the number of features, the number of linguistic terms, the shape of the fuzzy sets, etc. Moreover, due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem.In this paper, we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule-based systems. To this end, we will propose a taxonomy based on a double axis: “Complexity versus semantic interpretability” considering the two main kinds of measures; and “rule base versus fuzzy partitions” considering the different components of the knowledge base to which both kinds of measures can be applied. The main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works.     

基于AQ覆盖框架的蚁群规则集学习算法

针对规则集学习问题,提出一种遵循典型AQ覆盖算法框架（AQ Covering Algorithm）的蚁群规则集学习算法（Ant-AQ）。在Ant-AQ算法中,AQ覆盖框架中的柱状搜索特化过程被蚁群搜索特化过程替代,从某种程度上减少了陷入局优的情况。在对照测试中,Ant-AQ算法分别和已有的经典规则集学习算法（CN2、AQ-15）以及R.S.Parpinelli等提出的另一种基于蚁群优化的规则学习算法 Ant-Miner在若干典型规则学习问题数据集上进行了比较。实验结果表明：首先,Ant-AQ算法在总体性能比较上要优于经典规则学习算法,其次,Ant-AQ算法在预测准确度这样关键的评价指标上优于Ant-Miner算法。     

Learning concurrently data and rule bases of Mamdani fuzzy rule-based systems by exploiting a novel interpretability index

Interpretability of Mamdani fuzzy rule-based systems (MFRBSs) has been widely discussed in the last years, especially in the framework of multi-objective evolutionary fuzzy systems (MOEFSs). Here, multi-objective evolutionary algorithms (MOEAs) are applied to generate a set of MFRBSs with different trade-offs between interpretability and accuracy. In MOEFSs interpretability has often been measured in terms of complexity of the rule base and only recently partition integrity has also been considered. In this paper, we introduce a novel index for evaluating the interpretability of MFRBSs, which takes both the rule base complexity and the data base integrity into account. We discuss the use of this index in MOEFSs, which generate MFRBSs by concurrently learning the rule base, the linguistic partition granularities and the membership function parameters during the evolutionary process. The proposed approach has been experimented on six real world regression problems and the results have been compared with those obtained by applying the same MOEA, with only accuracy and complexity of the rule base as objectives. We show that our approach achieves the best trade-offs between interpretability and accuracy.     

Genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction

Tuning fuzzy rule-based systems for linguistic fuzzy modeling is an interesting and widely developed task. It involves adjusting some of the components of the knowledge base without completely redefining it. This contribution introduces a genetic tuning process for jointly fitting the fuzzy rule symbolic representations and the meaning of the involved membership functions. To adjust the former component, we propose the use of linguistic hedges to perform slight modifications keeping a good interpretability. To alter the latter component, two different approaches changing their basic parameters and using nonlinear scaling factors are proposed. As the accomplished experimental study shows, the good performance of our proposal mainly lies in the consideration of this tuning approach performed at two different levels of significance. The paper also analyzes the interaction of the proposed tuning method with a fuzzy rule set reduction process. A good interpretability-accuracy tradeoff is obtained combining both processes with a sequential scheme: first reducing the rule set and subsequently tuning the model.     

A PSO-aided neuro-fuzzy classifier employing linguistic hedge concepts

The present paper proposes the development of an adaptive neuro-fuzzy classifier which employs two relatively less explored and comparatively new problem solving domains in fuzzy systems. The relatively less explored field is the domain of the fuzzy linguistic hedges which has been employed here to define the flexible shapes of the fuzzy membership functions (MFs). To achieve finer and finer adaptation, and hence control, over the fuzzy MFs, each MF is composed of several piecewise MF sections and the shape of each such MF section is varied by applying a fuzzy linguistic operator on it. The system employs a Takagi–Sugeno based neuro-fuzzy system where the rule consequences are described by zero order elements. This proposed linguistic hedge based neuro-fuzzy classifier (LHBNFC) employs a relatively new field in the area of combinatorial metaheuristics, called particle swarm optimization (PSO), for its efficient learning. PSO has been employed in this scheme to simultaneously tune the shape of the fuzzy MFs as well as the rule consequences for the entire fuzzy rule base. The performance of the proposed system is demonstrated by implementing it for two classical benchmark data sets: (i) the iris data and (ii) the thyroid data. Performance comparison vis-à-vis other available algorithms shows the effectiveness of our proposed algorithm.     

A multi-objective genetic optimization of interpretability-oriented fuzzy rule-based classifiers

The paper presents a multi-objective genetic approach to design interpretability-oriented fuzzy rule-based classifiers from data. The proposed approach allows us to obtain systems with various levels of compromise between their accuracy and interpretability. During the learning process, parameters of the membership functions, as well as the structure of the classifier's fuzzy rule base (i.e., the number of rules, the number of rule antecedents, etc.) evolve simultaneously using a Pittsburgh-type genetic approach. Since there is no particular coding of fuzzy rule structures in a chromosome (it reduces computational complexity of the algorithm), original crossover and mutation operators, as well as chromosome-repairing technique to directly transform the rules are also proposed. To evaluate both the accuracy and interpretability of the system, two measures are used. The first one – an accuracy measure – is based on the root mean square error of the system's response. The second one – an interpretability measure – is based on the arithmetic mean of three components: (a) the average length of rules (the average number of antecedents used in the rules), (b) the number of active fuzzy sets and (c) the number of active inputs of the system (an active fuzzy set or input means a set or input used by at least one fuzzy rule). Both measures are used as objectives in multi-objective (2-objective in our case) genetic optimization approaches such as well-known SPEA2 and NSGA-II algorithms. Moreover, for the purpose of comparison with several alternative approaches, the experiments are carried out both considering the so-called strong fuzzy partitions (SFPs) of attribute domains and without them. SFPs provide more semantically meaningful solutions, usually at the expense of their accuracy. The operation of the proposed technique in various classification problems is tested with the use of 20 benchmark data sets and compared to 11 alternative classification techniques. The experiments show that the proposed approach generates classifiers of significantly improved interpretability, while still characterized by competitive accuracy.     

Boosting fuzzy rules in classification problems under single‐winner inference

In previous studies, we have shown that an Adaboost‐based fitness can be successfully combined with a Genetic Algorithm to iteratively learn fuzzy rules from examples in classification problems. Unfortunately, some restrictive constraints in the implementation of the logical connectives and the inference method were assumed. Alas, the knowledge bases Adaboost produces are only compatible with an inference based on the maximum sum of votes scheme, and they can only use the t‐norm product to model the “and” operator. This design is not optimal in terms of linguistic interpretability. Using the sum to aggregate votes allows many rules to be combined, when the class of an example is being decided. Because it can be difficult to isolate the contribution of individual rules to the knowledge base, fuzzy rules produced by Adaboost may be difficult to understand linguistically. In this point of view, single‐winner inference would be a better choice, but it implies dropping some nontrivial hypotheses. In this work we introduce our first results in the search for a boosting‐based genetic method able to learn weighted fuzzy rules that are compatible with this last inference method. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 1021–1034, 2007.     

Complexity reduction and interpretability improvement for fuzzy rule systems based on simple interpretability measures and indices by bi-objective evolutionary rule selection

The aim of this paper is to develop a general post-processing methodology to reduce the complexity of data-driven linguistic fuzzy models, in order to reach simpler fuzzy models preserving enough accuracy and better fuzzy linguistic performance with respect to their initial values. This post-processing approach is based on rule selection via the formulation of a bi-objective problem with one objective focusing on accuracy and the other on interpretability. The latter is defined via the aggregation of several interpretability measures, based on the concepts of similarity and complexity of fuzzy systems and rules. In this way, a measure of the fuzzy model interpretability is given. Two neuro-fuzzy systems for providing initial fuzzy models, Fuzzy Adaptive System ART based and Neuro-Fuzzy Function Approximation and several case studies, data sets from KEEL Project Repository, are used to check this approach. Both fuzzy and neuro-fuzzy systems generate Mamdani-type fuzzy rule-based systems, each with its own particularities and complexities from the point of view of the fuzzy sets and the rule generation. Based on these systems and data sets, several fuzzy models are generated to check the performance of the proposal under different restrictions of complexity and fuzziness.     

深度神经模糊系统算法及其回归应用

深度神经网络是人工智能的热点, 可以很好处理高维大数据, 却有可解释性差的不足. 通过IF-THEN规则构建的模糊系统, 具有可解释性强的优点, 但在处理高维大数据时会遇到“维数灾难”问题. 本文提出一种基于ANFIS (Adaptive network based fuzzy inference system)的深度神经模糊系统(Deep neural fuzzy system, DNFS)及两种基于分块和分层的启发式实现算法: DNFS1和DNFS2. 通过四个面向回归应用的数据集的测试, 我们发现: 1)采用分块、分层学习的DNFS在准确度与可解释性上优于BP、RBF、GRNN等传统浅层神经网络算法, 也优于LSTM和DBN等深度神经网络算法; 2)在低维问题中, DNFS1具有一定优势; 3)在面对高维问题时, DNFS2表现更为突出. 本文的研究结果表明DNFS是一种新型深度学习方法, 不仅可解释性好, 而且能有效解决处理高维数据时模糊规则数目爆炸的问题, 具有很好的发展前景.     

Fuzzy multi‐attribute cost–benefit analysis of e‐services

E‐service evaluation is a complex problem in which many qualitative attributes must be considered. These kinds of attributes make the evaluation process hard and vague. Cost–benefit analyses applied to various areas are usually based on the data under certainty or risk. In case of uncertain, vague, and/or linguistic data, the fuzzy set theory can be used to handle the analysis. In this article, after the evaluation attributes of e‐services and the fuzzy multi‐attribute decision‐making methods are introduced, a fuzzy hierarchical TOPSIS model is developed and applied to an e‐service provider selection problem with some sensitivity analyses. The developed model is a useful tool for the companies that prefer outsourcing for e‐activities. It is shown that service systems can be effectively evaluated by the proposed method. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 547–565, 2007.     

Multiobjective evolutionary feature selection and fuzzy classification of contact centre data

In this work, a data set describing phone interactions arising in a multichannel and multiskill contact centre is considered with the aim of classifying inbound sessions into those that will be eventually managed by an agent and those that, instead, will be abandoned before. More precisely, the goal of the work is to extract interpretable pieces of information that allow us to predict whether a user will or will not abandon a call, which may turn out to be very useful for the purpose of contact centre managing. To this end, the performance of two well‐known, state‐of‐the‐art evolutionary algorithms for feature selection (evolutionary nondominated radial slots based algorithm and nondominated sorted genetic algorithm) is compared for the task of feature selection, under the criteria of accuracy and cardinality of the selection, as well as for the task of fuzzy rule extraction, under the criteria of interpretability, accuracy, and hypervolume test. The best obtained fuzzy classifier, chosen after a decision making process, is validated and interpreted by domain experts.