Multi-criteria classification – A new scheme for application of dominance-based decision rules

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Abstract

We are considering the problem of multi-criteria classification. In this problem, a set of “if  then …” decision rules is used as a preference model to classify objects evaluated by a set of criteria and regular attributes. Given a sample of classification examples, called learning data set, the rules are induced from dominance-based rough approximations of preference-ordered decision classes, according to the Variable Consistency Dominance-based Rough Set Approach (VC-DRSA). The main question to be answered in this paper is how to classify an object using decision rules in situation where it is covered by (i) no rule, (ii) exactly one rule, (iii) several rules. The proposed classification scheme can be applied to both, learning data set (to restore the classification known from examples) and testing data set (to predict classification of new objects). A hypothetical example from the area of telecommunications is used for illustration of the proposed classification method and for a comparison with some previous proposals.

Introduction

We are considering the Dominance-based Rough Set Approach (DRSA), and its extension called Variable Consistency Dominance-based Rough Set Approach (VC-DRSA), to multi-criteria classification problems (sometimes called also sorting problems). Those problems concern an assignment of objects evaluated by a set of criteria and regular attributes to pre-defined preference-ordered decision classes [1], [5]. The difference between criteria and regular attributes is such that domains (scales) of criteria are preference ordered, while domains of regular attributes are not (e.g. customer’s dues = criterion, while customer’s sex = regular attribute). In DRSA and VC-DRSA, the process of classification involves a use of decision rules. In this paper, we concentrate on this process and we propose a new classification method based on decision rules.

The principle of rough approximation of a set [12] is to include in its lower approximation only those objects which are consistent from the viewpoint of available information; the inconsistent objects enter the upper approximation, in addition to consistent ones. Analysis of large real-life data sets shows, however, that for some multi-criteria classification problems, the application of DRSA identifies large differences between lower and upper approximations of decision classes. In consequence, decision rules induced from lower approximations are weak, that is supported by few objects only. For this reason, a variant of DRSA, called VC-DRSA, has been proposed in [3]. This variant enables relaxation of the conditions for assignment of objects to lower approximations. In this way, one allows for some inconsistency also in lower approximations of sets, however, in a controlled way. In VC-DRSA, the allowed level of inconsistency is controlled by a parameter called consistency level.

In all rough set approaches to analysis of classification data, computation of approximations is followed by induction of a set of decision rules. The decision rules are logic statements of the type “if conditions, then decision”. The set of decision rules represents a preference model of the decision maker who made the classification decisions described by a data set. It is generally acknowledged that people prefer to give examples of decisions rather than explain those decisions in terms of specific parameters required by a supposed model of their preferences. For this reason, the idea of considering the examples of classification as a learning data set, and construction of the preference model based on it, is very attractive. The exploration of this preference model has two aspects. It explains the past decisions in terms of the circumstances in which they were made and gives recommendation how to make a new decision under specific circumstances.

The preference model in terms of decision rules has several advantages over the classical models, which are the Multi-Attribute Utility Theory (MAUT) [10] and the outranking approach [16]:

  • the decision rules do not convert ordinal information into numeric one but keep the ordinal character of input data due to the syntax proposed; in this sense, DRSA or VC-DRSA is concordant with the paradigm of computing with words which are hardly convertible to numerical scales,

  • heterogeneous information (qualitative and quantitative, ordered and non-ordered) and scales of preference (ordinal, cardinal) can be processed within DRSA or VC-DRSA, while classical methods consider only quantitative ordered evaluations, with rare exceptions,

  • the decision rule preference model resulting from DRSA or VC-DRSA can represent even inconsistent preferences, i.e., preferences not satisfying the Pareto-dominance principle (the principle says: if object x is at least as good as object y on all the considered criteria, then x should be assigned to a class at least as good as y).

Furthermore, the equivalence of preference representation by a general non-additive and non-transitive utility function, by an outranking relation, and by decision rules, was proved in [2], [6], [9], [17]. Most of the well known multi-criteria aggregation procedures can be represented in terms of the decision rule model; in these cases, the decision rules decompose the synthetic aggregation formula used by these procedures into a set of simple “if …, then …” statements representing preferences in particular areas of evaluation space. Therefore, the decision rule approach is not discordant with MAUT and outranking classical models while it presents the decision model in a natural and understandable language.

An example of a decision rule belonging to a preference model of a car buyer, obtained from DRSA, is as follows:

  • if

    • maximum speed of car x is at least 175 km/hour and

    • its price is at most $12,000,

  • then car x is comprehensively at least medium”.

Decision rules of the form above involve partial profiles defined for subsets of criteria plus a dominance relation on these profiles. In the example above, the partial profile is composed of two conditions: 175 km/hour for maximum speed and $12,000 for price. This profile is partial because it is based on a subset of two criteria only, from among many criteria which are considered while taking a decision about buying a car (other criteria that may enter a model of a car buyer are: fuel consumption, space, acceleration, comfort and so on). The dominance relation says that car x satisfies the “if” part of the decision rule if its maximum speed and its price are at least as good as the values specified in the profile.

The main question to be answered in this paper is how to classify an object using decision rules in one of three possible situations: it is covered by (i) no rule, (ii) exactly one rule, (iii) several rules. We propose a classification method which can be applied to both, learning data set (to restore the classification known from examples) and testing data set (to predict classification of new objects). A hypothetical example from the area of telecommunications is used for illustration of the proposed classification method and for a comparison with a rule classifier previously used in DRSA [7], and with the C4.5 tree classifier [14]. We show that the new method outperforms the two previous proposals.

The paper is organized as follows. In Section 2, principles of DRSA and VC-DRSA are reminded. We concentrate on notions of rough approximations, approximation measures and decision rules. Section 3 contains a description of the classification method. An example of application of the new classification method, as well as its comparison with two other methods, is presented in Section 4. We conclude with a discussion in Section 5.

Section snippets

Dominance-based Rough Set Approaches

For algorithmic reasons, the learning data set is represented in the form of an information table. The rows of the table are labelled by objects, whereas columns are labelled by attributes and entries of the table are attribute-values. Formally, by an information table we understand the 4-tuple S = U, A, V, f〉, where U is a finite set of objects, A is a finite set of attributes, V = aAVa and Va is a domain of the attribute a and f : U × A  V is a total function such that f(x, a)  Va for every a  A, x  U,

The classification method

In this section a new method of classification using decision rules is proposed. We also compare this method to the one presented in previous works [7], [8]. The previously presented method was proposed for DRSA and will be referenced as standard classification method. The new method is particularly suitable for VC-DRSA but, obviously, it can be applied within DRSA as well.

To start presentation of the new method we distinguish three basic situations that occur while classifying new objects

Illustrative example

In this section, the VC-DRSA methodology and the standard and new classification methods presented in this paper are illustrated by a simple example. All results presented in this section can be reproduced using a software system called JAMM [13]. The example refers to a real problem solved by marketing departments of telecommunication operators. The operators are trying to maintain long-term relationship with their clients, so they are looking to convince clients to continue using their

Conclusions

The main result presented in this paper is the new classification method based on the use of a set of dominance-based decision rules. The rules are induced from rough approximations of preference-ordered decision classes, according to Variable Consistency Dominance-based Rough Set Approach. The set of rules induced in this way constitutes a preference model of the decision maker who gave classification examples. The proposed classification method, which is using this model, can be applied to

Acknowledgements

The first and the third author wish to acknowledge financial support from the State Committee for Scientific Research (KBN 3T11F 02127). The research of the second author has been supported by Italian Ministry of Education, University and Scientific Research (MIUR).

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