Salvatore Greco

Jerzy Błaszczyński, Salvatore Greco, Benedetto Matarazzo et al.

Dominance-based Rough Approach (DRSA) has been proposed as a machine learning and knowledge discovery methodology to handle Multiple Criteria Decision Aiding (MCDA). Due to its capacity of asking the decision maker (DM) for simple preference information and supplying easily understandable and explainable recommendations, DRSA gained much interest during the years and it is now one of the most appreciated MCDA approaches. In fact, it has been applied also beyond MCDA domain, as a general knowledge discovery and data mining methodology for the analysis of monotonic (and also non-monotonic) data. In this contribution, we recall the basic principles and the main concepts of DRSA, with a general overview of its developments and software. We present also a historical reconstruction of the genesis of the methodology, with a specific focus on the contribution of Roman Słowiński.

Marko Palangetić, Chris Cornelis, Salvatore Greco et al.

In this article, a new Fuzzy Granular Approximation Classifier (FGAC) is introduced. The classifier is based on the previously introduced concept of the granular approximation and its multi-class classification case. The classifier is instance-based and its biggest advantage is its local transparency i.e., the ability to explain every individual prediction it makes. We first develop the FGAC for the binary classification case and the multi-class classification case and we discuss its variation that includes the Ordered Weighted Average (OWA) operators. Those variations of the FGAC are then empirically compared with other locally transparent ML methods. At the end, we discuss the transparency of the FGAC and its advantage over other locally transparent methods. We conclude that while the FGAC has similar predictive performance to other locally transparent ML models, its transparency can be superior in certain cases.

Henri Bollaert, Marko Palangetić, Chris Cornelis et al.

Interpretability is the next frontier in machine learning research. In the search for white box models - as opposed to black box models, like random forests or neural networks - rule induction algorithms are a logical and promising option, since the rules can easily be understood by humans. Fuzzy and rough set theory have been successfully applied to this archetype, almost always separately. As both approaches to rule induction involve granular computing based on the concept of equivalence classes, it is natural to combine them. The QuickRules\cite{JensenCornelis2009} algorithm was a first attempt at using fuzzy rough set theory for rule induction. It is based on QuickReduct, a greedy algorithm for building decision reducts. QuickRules already showed an improvement over other rule induction methods. However, to evaluate the full potential of a fuzzy rough rule induction algorithm, one needs to start from the foundations. In this paper, we introduce a novel rule induction algorithm called Fuzzy Rough Rule Induction (FRRI). We provide background and explain the workings of our algorithm. Furthermore, we perform a computational experiment to evaluate the performance of our algorithm and compare it to other state-of-the-art rule induction approaches. We find that our algorithm is more accurate while creating small rulesets consisting of relatively short rules. We end the paper by outlining some directions for future work.

Salvatore Corrente, Salvatore Greco, Roman Słowiński et al.

Composite indicators are widely used to score or classify units evaluated on multiple criteria. Their construction involves aggregating criteria evaluations, a common practice in Multiple Criteria Decision Aiding (MCDA). In MCDA, various methods have been proposed to address key aspects of multiple criteria evaluations, such as the measurement scales of the criteria, the degree of acceptable compensation between them, and their potential interactions. However, beyond producing a final score or classification, it is essential to ensure the explainability and interpretability of results as well as the procedure's transparency. This paper proposes a method for constructing explainable and interpretable composite indicators using "if..., then..." decision rules. We consider the explainability and interpretability of composite indicators in four scenarios: (i) decision rules explain numerical scores obtained from an aggregation of numerical codes corresponding to ordinal qualifiers; (ii) an obscure numerical composite indicator classifies units into quantiles; (iii) given preference information provided by a Decision Maker in the form of classifications of some reference units, a composite indicator is constructed using decision rules; (iv) the classification of a set of units results from the application of an MCDA method and is explained by decision rules. To induce the rules from scored or classified units, we apply the Dominance-based Rough Set Approach. The resulting decision rules relate the class assignment or unit's score to threshold conditions on values of selected indicators in an intelligible way, clarifying the underlying rationale. Moreover, they serve to recommend composite indicator assessment for new units of interest.

Henri Bollaert, Chris Cornelis, Marko Palangetić et al.

Interpretability is the next pivotal frontier in machine learning research. In the pursuit of glass box models - as opposed to black box models, like random forests or neural networks - rule induction algorithms are a logical and promising avenue, as the rules can easily be understood by humans. In our previous work, we introduced FRRI, a novel rule induction algorithm based on fuzzy rough set theory. We demonstrated experimentally that FRRI outperformed other rule induction methods with regards to accuracy and number of rules. FRRI leverages a fuzzy indiscernibility relation to partition the data space into fuzzy granules, which are then combined into a minimal covering set of rules. This indiscernibility relation is constructed by removing attributes from rules in a greedy way. This raises the question: does the order of the attributes matter? In this paper, we show that optimising only the order of attributes using known methods from fuzzy rough set theory and classical machine learning does not improve the performance of FRRI on multiple metrics. However, removing a small number of attributes using fuzzy rough feature selection during this step positively affects balanced accuracy and the average rule length.

Marko Palangetić, Chris Cornelis, Salvatore Greco et al.

In granular computing, fuzzy sets can be approximated by granularly representable sets that are as close as possible to the original fuzzy set w.r.t. a given closeness measure. Such sets are called granular approximations. In this article, we introduce the concepts of disjoint and adjacent granules and we examine how the new definitions affect the granular approximations. First, we show that the new concepts are important for binary classification problems since they help to keep decision regions separated (disjoint granules) and at the same time to cover as much as possible of the attribute space (adjacent granules). Later, we consider granular approximations for multi-class classification problems leading to the definition of a multi-class granular approximation. Finally, we show how to efficiently calculate multi-class granular approximations for Łukasiewicz fuzzy connectives. We also provide graphical illustrations for a better understanding of the introduced concepts.

Marko Palangetić, Chris Cornelis, Salvatore Greco et al.

Inconsistency in prediction problems occurs when instances that relate in a certain way on condition attributes, do not follow the same relation on the decision attribute. For example, in ordinal classification with monotonicity constraints, it occurs when an instance dominating another instance on condition attributes has been assigned to a worse decision class. It typically appears as a result of perturbation in data caused by incomplete knowledge (missing attributes) or by random effects that occur during data generation (instability in the assessment of decision attribute values). Inconsistencies with respect to a crisp preorder relation (expressing either dominance or indiscernibility between instances) can be handled using symbolic approaches like rough set theory and by using statistical/machine learning approaches that involve optimization methods. Fuzzy rough sets can also be seen as a symbolic approach to inconsistency handling with respect to a fuzzy relation. In this article, we introduce a new machine learning method for inconsistency handling with respect to a fuzzy preorder relation. The novel approach is motivated by the existing machine learning approach used for crisp relations. We provide statistical foundations for it and develop optimization procedures that can be used to eliminate inconsistencies. The article also proves important properties and contains didactic examples of those procedures.

Luca E. Schäfer, Tobias Dietz, Maria Barbati et al.

A variant of the classical knapsack problem is considered in which each item is associated with an integer weight and a qualitative level. We define a dominance relation over the feasible subsets of the given item set and show that this relation defines a preorder. We propose a dynamic programming algorithm to compute the entire set of non-dominated rank cardinality vectors and we state two greedy algorithms, which efficiently compute a single efficient solution.

Maria Barbati, Josè Rui Figueira, Salvatore Greco et al.

This paper presents a new methodology that combines a multiple criteria sorting or ranking method with a project portfolio selection procedure. The multicriteria method permits to compare projects in terms of their priority assessed on the basis of a set of both qualitative and quantitative criteria. Then, a feasible set of projects, i.e. a portfolio, is selected according to the priority defined by the multiple criteria method. In addition, the portfolio must satisfy a set of resources constraints, e.g. budget available, as well as some logical constraints, e.g. related to projects to be selected together or projects mutually exclusive. The proposed portfolio selection methodology can be applied in different contexts. We present an application in the urban planning domain where our approach allows to select a set of urban projects on the basis of their priority, budgetary constraints and urban policy requirements. Given the increasing interest of historical cities to reuse their cultural heritage, we applied and tested our methodology in this context. In particular, we show how the methodology can support the prioritization of the interventions on buildings with some historical value in the historic city center of Naples (Italy), taking into account several points of view.

Ana Sara Costa, Salvatore Corrente, Salvatore Greco et al.

Cat-SD is a multiple criteria decision aiding method for dealing with nominal classification problems. Actions are assessed according to multiple criteria and assigned to one or more categories. A set of reference actions is used for defining each category. The assignment of an action to a given category depends on the comparison of the action to each reference set according to likeness thresholds. Distinct sets of criteria weights, interaction coefficients, and likeness thresholds can be defined per category. We propose to apply Multiple Criteria Hierarchy Process (MCHP) to Cat-SD. An adapted MCHP is proposed to take into account possible interaction effects between criteria structured in a hierarchical way. On the basis of the known deck of cards method, we also consider an imprecise elicitation of parameters permitting to take into account interactions and antagonistic effects between criteria. The elicitation procedure we are proposing can be applied to any Electre method. With the purpose of exploring the assignments obtained by Cat-SD considering possible sets of parameters, we propose to apply the Stochastic Multicriteria Acceptability Analysis (SMAA). The SMAA methodology allows to draw statistical conclusions on the classification of the actions. The proposed method, SMAA-hCat-SD, helps the decision maker to check the effects of the variation of parameters on the classification at different levels of the hierarchy. We propose also a procedure, based on the concept of loss function, to obtain a final classification fulfilling some requirements given by the decision maker and taking into account the hierarchy of criteria and the probabilistic assignments obtained applying SMAA. Also this procedure can be applied to any classification Electre method.

Francesca Abastante, Salvatore Corrente, Salvatore Greco et al.

We propose a development of the Analytic Hierarchy Process (AHP) permitting to use the methodology also in cases of decision problems with a very large number of alternatives evaluated with respect to several criteria. While the application of the original AHP method involves many pairwise comparisons between alternatives and criteria, our proposal is composed of three steps: (i) direct evaluation of the alternatives at hand on the considered criteria, (ii) selection of some reference evaluations; (iii) application of the original AHP method to reference evaluations; (iv) revision of the direct evaluation on the basis of the prioritization supplied by AHP on reference evaluations. The new proposal has been tested and validated in an experiment conducted on a sample of university students. The new methodology has been therefore applied to a real world problem involving the evaluation of 21 Social Housing initiatives sited in the Piedmont region (Italy). To take into account interaction between criteria, the Choquet integral preference model has been considered within a Non Additive Robust Ordinal Regression approach.