Fuzzy classifier identification using decision tree and multiobjective evolutionary algorithms

This paper presents a hybrid method for identification of Pareto-optimal fuzzy classifiers (FCs). In contrast to many existing methods, the initial population for multiobjective evolutionary algorithms (MOEAs) is neither created randomly nor a priori knowledge is required. Instead, it is created by the proposed two-step initialization method. First, a decision tree (DT) created by C4.5 algorithm is transformed into an FC. Therefore, relevant variables are selected and initial partition of input space is performed. Then, the rest of the population is created by randomly replacing some parameters of the initial FC, such that, the initial population is widely spread. That improves the convergence of MOEAs into the correct Pareto front. The initial population is optimized by NSGA-II algorithm and a set of Pareto-optimal FCs representing the trade-off between accuracy and interpretability is obtained. The method does not require any a priori knowledge of the number of fuzzy sets, distribution of fuzzy sets or the number of relevant variables. They are all determined by it. Performance of the obtained FCs is validated by six benchmark data sets from the literature. The obtained results are compared to a recently published paper [H. Ishibuchi, Y. Nojima, Analysis of interpretability-accuracy tradeoff of fuzzy systems by multiobjective fuzzy genetics-based machine learning, International Journal of Approximate Reasoning 44 (1) (2007) 4–31] and the benefits of our method are clearly shown.     

A problem-solving approach to product design using decision tree induction based on intuitionistic fuzzy

Customer complaint problem is a product design used to understand customer requirements. Furthermore, product design corresponding to customer requirement does not feel adequately solved for a cause of problem. The cause of the problem affecting product design is solved to prevent customer complaint from reoccurring. However, the problems by customer may have observation uncertainty and fuzzy. Fuzzy concept considers not only the degree of membership to an accept set, but also the degree of non-membership to a rejection set. Therefore, we present a new approach for problem solving using decision tree induction based on intuitionistic fuzzy sets in this paper. Under this approach, we first develop the problem formulation for the symptoms and causes of the problem based on intuitionistic fuzzy sets. Next, we identify the cause of the problem using intuitionistic fuzzy decision tree by the problem formulation. We then provide the approach to find the optimal cause of the problem for the consideration of product design. A numerical example is used to illustrate the approach applied for product design.     

A semi-automated design of instance-based fuzzy parameter tuning for metaheuristics based on decision tree induction

Two main concepts are established in the literature for the Parameter Setting Problem of metaheuristics: Parameter Tuning Strategies (PTS) and Parameter Control Strategies (PCS). While PTS result in a fixed parameter setting for a set of problem instances, PCS are incorporated into the metaheuristic and adapt parameter values according to instance-specific performance feedback. The idea of Instance-specific Parameter Tuning Strategies (IPTS) is aiming to combine advantages of both tuning and control strategies by enabling the adoption of parameter values tailored to instance-specific characteristics a priori to running the metaheuristic. This requires, however, a significant knowledge about the impact of instance characteristics on heuristic performance. This paper presents an approach that semi-automatically designs the fuzzy logic rule base to obtain instance-specific parameter values by means of decision trees. This enables the user to automate the process of converting insights about instance-specific information and its impact on heuristic performance into a fuzzy rule base IPTS system. The system incorporates the decision maker’s preference about the trade-off between computational time and solution quality.     

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 tree augmented classifier based on Extreme Imprecise Dirichlet Model

We present TANC, a TAN classifier (tree-augmented naive) based on imprecise probabilities. TANC models prior near-ignorance via the Extreme Imprecise Dirichlet Model (EDM). A first contribution of this paper is the experimental comparison between EDM and the global Imprecise Dirichlet Model using the naive credal classifier (NCC), with the aim of showing that EDM is a sensible approximation of the global IDM. TANC is able to deal with missing data in a conservative manner by considering all possible completions (without assuming them to be missing-at-random), but avoiding an exponential increase of the computational time. By experiments on real data sets, we show that TANC is more reliable than the Bayesian TAN and that it provides better performance compared to previous TANs based on imprecise probabilities. Yet, TANC is sometimes outperformed by NCC because the learned TAN structures are too complex; this calls for novel algorithms for learning the TAN structures, better suited for an imprecise probability classifier.     

Multicriteria decision analysis in a decision tree

The outcomes at the tips of a decision tree cannot always be represented by a single numerical value on a one-dimensional axis. In many decision problems they are multidimensional, and their performance is expressed in a mixture of verbal statements and physical or monetary values. We propose to use the scores of cardinal methods for multicriteria decision analysis in order to represent the relative performance of the outcomes. Thereafter, we evaluate the chance forks in the tree via the corresponding aggregation procedure: in the Multiplicative AHP via weighted geometric means of the scores and in SMART via weighted arithmetic means. The procedure is based on the idea that the numerical values of verbal quantifiers like somewhat more, more, … do not depend on what we compare, whether it is relative importance or relative likelihood.     

On fuzzy tree definition

The connectedness and acyclicity levels are presented for a fuzzy graph. Several fuzzy tree definitions are given too, and their relations with previous studies on the matter are analyzed.     

A fuzzy outranking relation in multicriteria decision making

It frequently happens that a decision maker must establish a ranking within a finite set of alternatives with respect to multiple criteria. The subjective evaluation of each alternative according to each criterion is expressed in the form of a distributive evaluation. To capture the preferences of one alternative over another, a concept of fuzzy outranking relation can be used. This fuzzy outranking relation is characterized by a degree of credibility which is computed from two indices: a confidence index and a doubt index. Each of these indices is calculated from the distributive evaluations over the various criteria. In this paper, such a fuzzy outranking relation (fuzzy binary relation) is constructed and an application is presented.     

A fuzzy relation space for group decision theory

In this paper reciprocal relations over n alternatives are defined, and their efficacy as a model of the group decision process is investigated. Several measures of individual preference and group consensus are defined and analyzed, and are used to generate associated measures of distance to consensus. Various decision oriented goals are identified, and we characterize these goals geometrically as subsets of a convex subset of a hyperplane in Rⁿ. An application of this model for assessing degree of consensus is given.     

Spatial fuzzy consumer's decision making: A multicriteria analysis

This paper is devoted to a multicriteria analysis of the consumer's behavior when the decision maker is acting in a fuzzy space and manifesting an imprecise attitude. At first, the process of decision making is described with the help of three relationships between the set of goods which are supplied in several locations, the set of their characteristics and the set of the consumer's a priori possible behaviors. All these relations are fuzzy. The model applies the theory of fuzzy relations equations. Then, the stages of the decision process are analyzed. Often fuzzy behavior relations are like ‘black boxes’. The mathematical solution of the model indicates in which conditions their valuations are possible. The main interest of this method is not to use additive operations on subjective items and to use operators which are coherent with the fuzzy nature of the variables.     

A fuzzy multiobjective programming algorithm in decision support systems

Multicriteria analysis is one of the analytical functions in the problem processing system of decision support systems (DSS). In this paper, an interactive and iterative fuzzy programming method for solving a quasi-optimization problem in complex decisions under constraints involving a multiple objective function is proposed. Comparing with an adapted gradient search method, a surrogate worth tradeoff method, and a Zionts—Wallenius method, an approximate preference structure is emphasized in the proposed method.     

A VIKOR-based method for hesitant fuzzy multi-criteria decision making

Since it was firstly introduced by Torra and Narukawa (The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378–1382), the hesitant fuzzy set has attracted more and more attention due to its powerfulness and efficiency in representing uncertainty and vagueness. This paper extends the classical VIKOR (vlsekriterijumska optimizacija i kompromisno resenje in serbian) method to accommodate hesitant fuzzy circumstances. Motivated by the hesitant normalized Manhattan distance, we develop the hesitant normalized Manhattan $L_p$ —metric, the hesitant fuzzy group utility measure, the hesitant fuzzy individual regret measure, and the hesitant fuzzy compromise measure. Based on these new measures, we propose a hesitant fuzzy VIKOR method, and a practical example is provided to show that our method is very effective in solving multi-criteria decision making problems with hesitant preference information.     

A survey of fuzzy implication algebras and their axiomatization

The theory of fuzzy implication algebras was proposed by Professor Wangming Wu in 1990. The present paper reviews the following two aspects of studies on FI-algebras: concepts, properties and some subclasses of FI-algebras; axiomatization of the class of FI-algebras and some of its important subclasses. The main results are summarized in the current paper, the relationships between FI-algebras and several classes of important fuzzy algebras are discussed, such as BL-algebras, MTL-algebras, and residuated lattices, and propositional calculus systems of several special classes of FI-algebras are shown.     

PAC-learning a decision tree with pruning

Empirical studies have shown that the performance of decision tree induction usually improves when the trees are pruned. Whether these results hold in general and to what extent pruning improves the accuracy of a concept have not been investigated theoretically. This paper provides a theoretical study of pruning. We focus on a particular type of pruning and determine a bound on the error due to pruning. This is combined with PAC (probably approximately correct) learning theory to determine a sample size sufficient to guarantee a probabilistic bound on the concept error. We also discuss additional pruning rules and give an analysis for the pruning error.     

A fuzzy soft set theoretic approach to decision making problems

The problem of decision making in an imprecise environment has found paramount importance in recent years. A novel method of object recognition from an imprecise multiobserver data has been presented here. The method involves construction of a Comparison Table from a fuzzy soft set in a parametric sense for decision making.     

Stability analysis of tree structured decision functions

In multicriteria decision problems many values must be assigned, such as the importance of the different criteria and the values of the alternatives with respect to subjective criteria. Since these assignments are approximate, it is very important to analyze the sensitivity of results when small modifications of the assignments are made. When solving a multicriteria decision problem, it is desirable to choose a decision function that leads to a solution as stable as possible. We propose here a method based on genetic programming that produces better decision functions than the commonly used ones. The theoretical expectations are validated by case studies.     

Application of fuzzy decision making to the fuzzy finite element method

In this paper we present a new approach to handle uncertainty in the Finite Element Method. As this technique is widely used to tackle real-life design problems, it is also very prone to parameter-uncertainty. It is hard to make a good decision regarding design optimization if no claim can be made with respect to the outcome of the simulation. We propose an approach that combines several techniques in order to offer a total order on the possible design choices, taking the inherent fuzziness into account. Additionally we propose a more efficient ordering procedure to build a total order on fuzzy numbers.     

A survey of quantum-like approaches to decision making and cognition

There has always been a steady interest in how humans make decisions amongst researchers from various fields. Based on this interest, many approaches such as rational choice theory or expected utility hypothesis have been proposed. Although these approaches provide a suitable ground for modeling the decision making process of humans, they are unable to explain the corresponding irrationalities and existing paradoxes and fallacies. Recently, a new formulation of decision theory that can correctly describe these paradoxes and possibly provide a unified and general theory of decision making has been proposed. This new formulation is founded based on the application of the mathematical structure of quantum theory to the fields of human decision making and cognition. It is shown that by applying these quantum-like models, one can better describe the uncertainty, ambiguity, emotions and risks involved in the human decision making process. Even in computational environments, an agent that follows the correct patterns of human decision making will have a better functionality in performing its role as a proxy for a real user. In this paper, we present a comprehensive survey of the researches and the corresponding recent developments. Finally, the benefits of leveraging the quantum-like modeling approaches in computational domains and the existing challenges and limitations currently facing the field are discussed.