We present a sparse HPC implementation for big outranking digraphs of huge orders, up to several millions of decision alternatives. Our sparse digraph model is based on a quantiles equivalence class decomposition of the underlying multicriteria performance tableau. When locally ranking each of these ordered components, we obtain an overall linear ranking of the complete given set of decision alternatives. Both, Copeland's as well as the Net-Flows ranking rules appear herefore to give the best compromise between, on the one side, the fitness of the overall ranking with respect to the given global outranking relation and, on the other side, computational tractability for very big outranking digraphs modelling up to several trillions of pairwise outranking situations.

We apply order statistics for sorting a set X of $ potential deicison actions, evaluated on p incommensurable performance criteria, into k quantile equivalence classes, based on pairwise outranking characteristics involving the quantile class limits observed on each criterion. Thus we may implement a weak ordering algorithm of linear complexity O(npk).

When modelling preferences following the outranking approach, the sign of the majority margins do sharply distribute validation and invalidation of pairwise outranking situations. How can we be confident in the resulting outranking digraph, when we acknowledge the usual imprecise knowledge of criteria significance weights and a small majority margin? To answer this question, we propose to model the significance weights as random variables following more less widespread distributions around an average weight value that corresponds to the given deterministic weight. As the bipolarly valued random credibility of an outranking statement results from a simple sum of positive or negative independent and similarly distributed random variables, we may apply the CLT for computing likelihoods that a given majority margin is indeed positive, respectively negative. To test the effective convergence of the CLT likelihoods, we apply Monte Carlo simulations of outranking digraph constructions. Our computational results confirm a satisfactory convergence even for a random performance tableau with only seven criteria.

These tutorials describe the usage of Python3 resources for implementing decision aid algorithms in the context of a bipolarly-valued outranking approach. These computing resources are useful in the field of algorithmic decision theory and more specifically in outranking based multiple criteria decision aid. Available topics are the following:

• Working with the digraphs module
• Manipulating Digraph objects
• Working with the graphs module
• Computing the winner of an election
• Working with the outrankingDigraphs module
• Computing a best choice recommendation

Polarised outrankings with considerable performance differences (weighted majority margins with vetoes and counter-vetoes) appear as weakly complete (and reflexive) relations (Bisdorff 2013). For any two potential decision actions x and y, if x does effectively not outrank x then it is not the case that y does effecively not outrank x. Constructing weak orderings (rankings with potential ties) from such outranking relations consists in computing, hence, a transitive closure of the given outranking relation. Determining optimal transitive closures of (weakly) complete relations is a computational difficult problem (Hudry 2013). However, global scoring methods based on average ranks (like Borda scores) or average netflows like in the PROMETHEE approach, may easily deliver such a heuristic closure. Now, such weak orderings may also result from the iterated application of a certain choice procedure, an approach called ranking-by-choosing (Bouyssou 2004). In this contribution we shall present such a new ranking-by-choosing approach based on the Rubis best choice method (Bisdorff 2008).

We generalize Kendall's rank correlation measure τ to bipolarly valued relations. Motivation for this work comes from the need to measure the level of ordinal approximation that is required when replacing a given bipolar outranking with a convenient weak ordering recommendation.

In Multicriteria Decision Aid (MCDA), when working with outranking methods, the conclusion that an alternative appears being 'at least as good as' another one, or not, depends on a clear setting of different parameters, especially the weights of the performance criteria. In this article, we present the concept of stability of the crisp median-cut outranking digraph with respect to chosen criteria weights. We show in particular that, when an outranking statement can be qualified as stable, it will be less important to precisely quantify these criteria weights. We give an intuitive formulation, as well as simple mathematical conditions, for computing the degree of stability of outranking situations. Moreover, we propose a protocol for eliciting criteria weights that will render the outranking modelling as much stable as possible.

Same subject as the note below. The model parameter section is more developped and a more extensive set of benchmark clusterings is shown here.

We propose a meta-heuristic for clustering objects that are described on multiple incommensurable attributes of nominal, ordinal and/or cardinal type. Our approach makes use of an innovative bipolar-valued dual similarity-dissimilarity relation characterized by pairwise weighted majority margins of similar minus dissimilar attribute evaluations. The clustering is computed in two steps. First, an evolutionary algorithm searches for a suitable subset of maximal similarity cliques that will best serve as cluster cores. In a second step, we construct with a greedy heuristic, around these initial cluster cores, a corresponding final partition which best fits the given bipolar-valued similarity relation.

The chapter concerns the attribution of the EURO Best Poster Award at the 20^th EURO Conference, held in Rhodes, July 2004. We present the historical decision making process leading to the selection of the winner, followed by a thorough discussion of the constructed outranking models and of the best choice recommendation.

In this research note, we explore the correspondance between the codual and the assymmetric part of a bipolar valued outranking relation.

In this research note, we introduce and discuss an algorithm using O(p) time and O(n(n+m)) space for enumerating the m >= 0 chordless circuits in a digraph of order n containing p chordless paths. Running it on random digraphs yields some statistical insight into its practical performance. Python resources are here.

This article presents the main features of XMCDA, a standardised XML proposal to represent objects and data issued from the Field of Multiple Criteria Decision Aid (MCDA). Its main objective is to allow different MCDA algorithms to interact and to be easily callable from a software like, e.g., the diviz platform of the Decision Deck project1 . We present the structure of XMCDA and detail the main underlying data structures via examples speaking for themselves. Note that this document does not replace a detailed documentation of the related XML schema.

A comment on Bernard Roy's seminal 1966 unpuplished paper concerning the ELECTRE methods. Here I shortly report both, on some confirming insights like Roy's fundamental methodological pragmatism and, on a new insight I have experienced and that concerns the operational difficulties inherent in the practical implementation of the discordance principle. To illustrate this new insight, I have re-solved Roy's original illustrative decision problem with our Rubis best choice method.

In this research note we introduce St-Nicolas graphs, i.e. circulant digraphs showing exactly n maximal independent sets, isomorph under the digraph’s automorphisms group. This class of digraphs represent a generalisation of Andrásfai graphs with interesting links to finite group theory.

This article presents the problem of the selection of k best decision alternatives in the context of multiple criteria decision aiding. We situate ourselves in the context of pairwise comparisons of alternatives and the underlying bipolar-valued outranking digraph. We present three formulations for the best k-choice problem and detail how to solve two of them directly on the outranking digraph