Miłosz Kadziński

ELECTREGKMS: Robust ordinal regression for outranking methods

We present a new method, called ELECTREGKMS, which employs robust ordinal regression to construct a set of outranking models compatible with preference information. The preference information supplied by the decision maker (DM) is composed of pairwise comparisons stating the truth or falsity of the outranking relation for some real or fictitious reference alternatives. Moreover, the DM specifies some ranges of variation of comparison thresholds on considered pseudo-criteria. Using robust ordinal regression, the method builds a set of values of concordance indices, concordance thresholds, indifference, preference, and veto thresholds, for which all specified pairwise comparisons can be restored. Such sets are called compatible outranking models. Using these models, two outranking relations are defined, necessary and possible. Whether for an ordered pair of alternatives there is necessary or possible outranking depends on the truth of outranking relation for all or at least one compatible model, respectively. Distinguishing the most certain recommendation worked out by the necessary outranking, and a possible recommendation worked out by the possible outranking, ELECTREGKMS answers questions of robustness concern. The method is intended to be used interactively with incremental specification of pairwise comparisons, possibly with decreasing confidence levels. In this way, the necessary and possible outranking relations can be, respectively, enriched or impoverished with the growth of the number of pairwise comparisons. Furthermore, the method is able to identify troublesome pieces of preference information which are responsible for incompatibility. The necessary and possible outranking relations are to be exploited as usual outranking relations to work out recommendation in choice or ranking problems. The introduced approach is illustrated by a didactic example showing how ELECTREGKMS can support real-world decision problems.

Robust ordinal regression in preference learning and ranking

Multiple Criteria Decision Aiding (MCDA) offers a diversity of approaches designed for providing the decision maker (DM) with a recommendation concerning a set of alternatives (items, actions) evaluated from multiple points of view, called criteria. This paper aims at drawing attention of the Machine Learning (ML) community upon recent advances in a representative MCDA methodology, called Robust Ordinal Regression (ROR). ROR learns by examples in order to rank a set of alternatives, thus considering a similar problem as Preference Learning (ML-PL) does. However, ROR implements the interactive preference construction paradigm, which should be perceived as a mutual learning of the model and the DM. The paper clarifies the specific interpretation of the concept of preference learning adopted in ROR and MCDA, comparing it to the usual concept of preference learning considered within ML. This comparison concerns a structure of the considered problem, types of admitted preference information, a character of the employed preference models, ways of exploiting them, and techniques to arrive at a final ranking.

Robust ordinal regression for multiple criteria group decision: UTAGMS-GROUP and UTADISGMS-GROUP

We introduce the principle of robust ordinal regression to multiple criteria group decision, and we present two new methods using a set of additive value functions as a preference model, called UTA^G^M^S-GROUP and UTADIS^G^M^S-GROUP. With respect to the set of decision makers (DMs), we consider two levels of certainty for the results. The first level is related to the necessary or possible consequences of indirect preference information provided by each DM, whereas the other refers to the subset of DMs agreeing for a specific outcome. In this way, we investigate spaces of consensus and disagreement between the DMs. The proposed methods are illustrated by examples showing how they can support real-world group decision.

Robust multi-criteria ranking with additive value models and holistic pair-wise preference statements

We consider a problem of ranking alternatives based on their deterministic performance evaluations on multiple criteria. We apply additive value theory and assume the Decision Maker’s (DM) preferences to be representable with general additive monotone value functions. The DM provides indirect preference information in form of pair-wise comparisons of reference alternatives, and we use this to derive the set of compatible value functions. Then, this set is analyzed to describe (1) the possible and necessary preference relations, (2) probabilities of the possible relations, (3) ranges of ranks the alternatives may obtain, and (4) the distributions of these ranks. Our work combines previous results from Robust Ordinal Regression, Extreme Ranking Analysis and Stochastic Multicriteria Acceptability Analysis under a unified decision support framework. We show how the four different results complement each other, discuss extensions of the main proposal, and demonstrate practical use of the approach by considering a problem of ranking 20 European countries in terms of 4 criteria reflecting the quality of their universities.

Stochastic ordinal regression for multiple criteria sorting problems

We present a new approach for multiple criteria sorting problems. We consider sorting procedures applying general additive value functions compatible with the given assignment examples. For the decision alternatives, we provide four types of results: (1) necessary and possible assignments from Robust Ordinal Regression (ROR), (2) class acceptability indices from a suitably adapted Stochastic Multicriteria Acceptability Analysis (SMAA) model, (3) necessary and possible assignment-based preference relations, and (4) assignment-based pair-wise outranking indices. We show how the results provided by ROR and SMAA complement each other and combine them under a unified decision aiding framework. Application of the approach is demonstrated by classifying 27 countries in 4 democracy regimes.

Robust ordinal regression for Dominance-based Rough Set Approach to multiple criteria sorting

We present a new multiple criteria sorting method deriving from Dominance-based Rough Set Approach (DRSA). The preference information supplied by the Decision Maker (DM) is a set of possibly imprecise and inconsistent assignment examples on a subset of reference alternatives relatively well-known to the DM. To structure the data we use DRSA, and subsequently, represent the assignment examples by all minimal sets of rules covering all alternatives from the lower approximations of class unions. Such a set of rules is called minimal-cover set – it is one of the instances of the preference model compatible with DM’s preference information. In this way, we implement the principle of Robust Ordinal Regression (ROR) to decision rule preference model. For each alternative, we derive the necessary and possible assignments specifying the range of classes to which the alternative is assigned by all or at least one compatible set of rules, respectively, as well as class acceptability indices. We also introduce the notion of a representative compatible minimal-cover set of rules whose selection builds on the results of ROR, addressing the robustness concern. Application of the approach is demonstrated by classifying 69 land zones in 4 classes representing different risk levels.

Modeling assignment-based pairwise comparisons within integrated framework for value-driven multiple criteria sorting

We introduce a new preference disaggregation modeling formulations for multiple criteria sorting with a set of additive value functions. The preference information supplied by the Decision Maker (DM) is composed of: (1) possibly imprecise assignment examples, (2) desired class cardinalities, and (3) assignment-based pairwise comparisons. The latter have the form of imprecise statements referring to the desired assignments for pairs of alternatives, but without specifying any concrete class. Additionally, we account for preferences concerning the shape of the marginal value functions and desired comprehensive values of alternatives assigned to a given class or class range. The exploitation of all value functions compatible with these preferences results in three types of results: (1) necessary and possible assignments, (2) extreme class cardinalities, and (3) necessary and possible assignment-based preference relations. These outputs correspond to different types of admitted preference information. By exhibiting different outcomes, we encourage the DM in various ways to enrich her/his preference information interactively. The applicability of the framework is demonstrated on data involving the classification of cities into liveability classes.

DIS-CARD: a new method of multiple criteria sorting to classes with desired cardinality

In this paper, we present a new preference disaggregation method for multiple criteria sorting problems, called DIS-CARD. Real-life experience indicates the need of considering decision making situations in which a decision maker (DM) specifies a desired number of alternatives to be assigned to single classes or to unions of some classes. These situations require special methods for multiple criteria sorting subject to desired cardinalities of classes. DIS-CARD deals with such a problem, using the ordinal regression approach to construct a model of DM’s preferences from preference information provided in terms of exemplary assignments of some reference alternatives, together with the above desired cardinalities. We develop a mathematical model for incorporating such preference information via mixed integer linear programming (MILP). Then, we adapt the MILP model to two types of preference models: an additive value function and an outranking relation. Illustrative example is solved to illustrate the methodology.

Preferential reducts and constructs in robust multiple criteria ranking and sorting

We revisit the multiple criteria ranking and sorting methods based on ordinal regression, which accept preference information in the form of, respectively, pairwise comparisons or assignment examples for some reference alternatives. Robust ordinal regression methods consider the whole set of value functions reproducing these holistic statements provided at the input. Its impact on the recommendation is expressed in terms of the necessary and possible preference relations or assignments. We propose methods for generating explanations of this impact, showing pieces of preference information provided by the decision maker (DM), which led to the observed outcomes. In particular, the minimal set of preference information pieces, called preferential reduct, is identified to justify some result observable for the whole set of compatible value functions (e.g., the truth of the necessary relation for some pair of alternatives). Further, the maximal set of preference information pieces, called preferential construct, is discovered to reveal the conditions under which some result non-observable for the whole set of compatible value functions (e.g., the falsity of the possible relation for some pair of alternatives) is possible. Knowing such explanations, the DM can better understand the impact of each piece of preference information on the result and, in consequence, get conviction about the obtained recommendation.

Multiple criteria ranking and choice with all compatible minimal cover sets of decision rules

We consider all minimal-cover sets of rules compatible with the pairwise comparisons.We illustrate an undesired arbitrariness of the existing rule-based algorithms.Robustness of the delivered ranking/choice recommendation is examined in several ways.We introduce a constructivist paradigm to Dominance-based Rough Set Approach.The proposed method is applied for evaluating the innovativeness of Polish cities. We introduce a new multiple criteria ranking/choice method that applies Dominance-based Rough Set Approach (DRSA) and represents the Decision Makers (DMs) preferences with decision rules. The DM provides a set of pairwise comparisons indicating whether an outranking (weak preference) relation should hold for some pairs of reference alternatives. This preference information is structured using the lower and upper approximations of outranking (S) and non-outranking ( S c ) relations. Then, all minimal-cover (MC) sets of decision rules being compatible with this preference information are induced. Each of these sets is supported by some positive examples (pairs of reference alternatives from the lower approximation of a preference relation) and it does not cover any negative example (pair of alternatives from the upper approximation of an opposite preference relation). The recommendations obtained by all MC sets of rules are analyzed to describe pairwise outranking and non-outranking relations, using probabilistic indices (estimates of probabilities that one alternative outranks or does not outrank the other). Furthermore, given the preference relations obtained in result of application of each MC set of rules on a considered set of alternatives, we exploit them using some scoring procedures. From this, we derive the distribution of ranks attained by the alternatives. We also extend the basic approach in several ways. The practical usefulness of the method is demonstrated on a problem of ranking Polish cities according to their innovativeness.