Choosing realistic values of indifference,preference and veto thresholds for use with environmental criteria within ELECTRE

The ELECTRE III outranking model is particularly suited to aiding the choice between project alternatives on the basis of mainly environmental criteria. The model requires values of three criterion thresholds, the indifference threshold (q), the preference threshold (p) and the veto threshold (v). These allow the uncertainties inherent in the criteria valuations to be incorporated into the decision process. There is, at present, a high degree of subjectivity involved in determining these thresholds, which are expressed in terms of the error/uncertainty associated with the valuations of each of the criteria under scrutiny. If, however, the ELECTRE III outranking model is to be used within a formal environmental appraisal system, the thresholds which govern the outranking relationship of one project option over another must take account of the effect on human beings of the difference between any two criterion scores. The authors suggest a new method for applying the standard ELECTRE III model to decision-aid problems within the formal mechanism of environmental impact assessment. This involves a new, more comprehensive approach for specifying realistic limits for p, q and v, within the context of an environmental appraisal, where both criterion error/uncertainty and human sensitivity to differing levels of the criterion are taken into account. Threshold valuations for noise impacts from a highway project are used to illustrate the proposed method.     

The Proof of a Conjecture of Additive Number Theory

The aim of this paper is to show that for any n∈N, n>3, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having the same parity (see the text for the definition of the “length” of a natural number). Also we will show that for any n∈N, n>2, n≠5, 10, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having different parities. We will prove also that for any prime p≡7(mod 8) there exist a, b∈N* such that p=a²+b, the “length” of b being an even number.     

An axiomatic analysis of concordance–discordance relations

Outranking methods propose an original way to build a preference relation between alternatives evaluated on several attributes that has a definite ordinal flavor. Indeed, most of them appeal the concordance/non-discordance principle that leads to declaring that an alternative is “superior” to another, if the coalition of attributes supporting this proposition is “sufficiently important” (concordance condition) and if there is no attribute that “strongly rejects” it (non-discordance condition). Such a way of comparing alternatives is rather natural. However, it is well known that it may produce binary relations that do not possess any remarkable property of transitivity or completeness. This explains why the axiomatic foundations of outranking methods have not been much investigated, which is often seen as one of their important weaknesses. This paper uses conjoint measurement techniques to obtain an axiomatic characterization of preference relations that can be obtained on the basis of the concordance/non-discordance principle. It emphasizes their main distinctive feature, i.e. their very crude way to distinguish various levels of preference differences on each attribute. We focus on outranking methods, such as ELECTRE I, that produce a reflexive relation, interpreted as an “at least as good as” preference relation. The results in this paper may be seen as an attempt to give such outranking methods a sound axiomatic foundation based on conjoint measurement.     

Preference relations on actions and criteria in multicriteria decision making

Let A be a finite set of feasible actions which are judged following several criteria. An outranking relation is defined on A by considering preference of the decision maker as a weak order on each criterion and the relation among criteria as a semi-order on the given set of criteria.Several ways of constructing outranking relations have been proposed. One of the most popular, introduced by B. Roy, for instance ELECTRE(s), is based on the use of weights related to criteria. In our approach, the knowledge of weights is replaced by the existence of a semi-order.A case study is developed. It deals with a computer selection problem.     

Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions

We present a new method, called UTA^GMS, for multiple criteria ranking of alternatives from set A using a set of additive value functions which result from an ordinal regression. The preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives A^R ⊆ A, called reference alternatives. The preference model built via ordinal regression is the set of all additive value functions compatible with the preference information. Using this model, one can define two relations in the set A: the necessary weak preference relation which holds for any two alternatives a, b from set A if and only if for all compatible value functions a is preferred to b, and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b. These relations establish a necessary and a possible ranking of alternatives from A, being, respectively, a partial preorder and a strongly complete relation. The UTA^GMS method is intended to be used interactively, with an increasing subset A^R and a progressive statement of pairwise comparisons. When no preference information is provided, the necessary weak preference relation is a weak dominance relation, and the possible weak preference relation is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary relation and it is impoverishing the possible relation, so that they converge with the growth of the preference information. Distinguishing necessary and possible consequences of preference information on the complete set of actions, UTA^GMS answers questions of robustness analysis. Moreover, the method can support the decision maker when his/her preference statements cannot be represented in terms of an additive value function. The method is illustrated by an example solved using the UTA^GMS software. Some extensions of the method are also presented.     

As simple as possible but not simpler in Multiple Criteria Decision Aiding: the robust-stochastic level dependent Choquet integral approach

The level dependent Choquet integral has been proposed to handle decision making problems in which the importance and the interaction of criteria may depend on the level of the alternatives’ evaluations. This integral is based on a level dependent capacity, which is a family of single capacities associated to each level of evaluation for the considered criteria. We present two possible formulations of the level dependent capacity where importance and interaction of criteria are constant inside each one of the subintervals in which the interval of evaluations for considered criteria is split or vary with continuity inside the whole interval of evaluations. Since, in general, there is not only one but many level dependent capacities compatible with the preference information provided by the Decision Maker, we propose to take into account all of them by using the Robust Ordinal Regression (ROR) and the Stochastic Multicriteria Acceptability Analysis (SMAA). On one hand, ROR defines a necessary preference relation (if an alternative a is at least as good as an alternative b for all compatible level dependent capacities), and a possible preference relation (if a is at least as good as b for at least one compatible level dependent capacity). On the other hand, considering a random sampling of compatible level dependent capacities, SMAA gives the probability that each alternative reaches a certain ranking position as well as the probability that an alternative is preferred to another. A real-world decision problem on rankings of universities is provided to illustrate the proposed methodology.     

Robust ordinal regression for value functions handling interacting criteria

We present a new method called UTA^GMS–INT for ranking a finite set of alternatives evaluated on multiple criteria. It belongs to the family of Robust Ordinal Regression (ROR) methods which build a set of preference models compatible with preference information elicited by the Decision Maker (DM). The preference model used by UTA^GMS–INT is a general additive value function augmented by two types of components corresponding to “bonus” or “penalty” values for positively or negatively interacting pairs of criteria, respectively. When calculating value of a particular alternative, a bonus is added to the additive component of the value function if a given pair of criteria is in a positive synergy for performances of this alternative on the two criteria. Similarly, a penalty is subtracted from the additive component of the value function if a given pair of criteria is in a negative synergy for performances of the considered alternative on the two criteria. The preference information elicited by the DM is composed of pairwise comparisons of some reference alternatives, as well as of comparisons of some pairs of reference alternatives with respect to intensity of preference, either comprehensively or on a particular criterion. In UTA^GMS–INT, ROR starts with identification of pairs of interacting criteria for given preference information by solving a mixed-integer linear program. Once the interacting pairs are validated by the DM, ROR continues calculations with the whole set of compatible value functions handling the interacting criteria, to get necessary and possible preference relations in the considered set of alternatives. A single representative value function can be calculated to attribute specific scores to alternatives. It also gives values to bonuses and penalties. UTA^GMS–INT handles quite general interactions among criteria and provides an interesting alternative to the Choquet integral.     

Multiple criteria sorting with a set of additive value functions

We present a new multiple criteria sorting method that aims at assigning actions evaluated on multiple criteria to p pre-defined and ordered classes. The preference information supplied by the decision maker (DM) is a set of assignment examples on a subset of actions relatively well known to the DM. These actions are called reference actions. Each assignment example specifies a desired assignment of a corresponding reference action to one or several contiguous classes. The set of assignment examples is used to build a preference model of the DM represented by a set of general additive value functions compatible with the assignment examples. For each action a, the method computes two kinds of assignments to classes, concordant with the DM’s preference model: the necessary assignment and the possible assignment. The necessary assignment specifies the range of classes to which the action can be assigned considering all compatible value functions simultaneously. The possible assignment specifies, in turn, the range of classes to which the action can be assigned considering any compatible value function individually. The compatible value functions and the necessary and possible assignments are computed through the resolution of linear programs.     

Avoider-Enforcer: The rules of the game

An Avoider-Enforcer game is played by two players, called Avoider and Enforcer, on a hypergraph F⊆X². The players claim previously unoccupied elements of the board X in turns. Enforcer wins if Avoider claims all vertices of some element of F, otherwise Avoider wins. In a more general version of the game a bias b is introduced to level up the players' chances of winning; Avoider claims one element of the board in each of his moves, while Enforcer responds by claiming b elements. This traditional set of rules for Avoider-Enforcer games is known to have a shortcoming: it is not bias monotone.We relax the traditional rules in a rather natural way to obtain bias monotonicity. We analyze this new set of rules and compare it with the traditional ones to conclude some surprising results. In particular, we show that under the new rules the threshold bias for both the connectivity and Hamiltonicity games, played on the edge set of the complete graph Kn, is asymptotically equal to n/logn. This coincides with the asymptotic threshold bias of the same game played by two “random” players.     

Tangent circle graphs and ‘orders’

Consider a horizontal line in the plane and let γ(A) be a collection of n circles, possibly of different sizes all tangent to the line on the same side. We define the tangent circle graph associated to γ(A) as the intersection graph of the circles. We also define an irreflexive and asymmetric binary relation P on A; the pair (a,b) representing two circles of γ(A) is in P iff the circle associated to a lies to the right of the circle associated to b and does not intersect it. This defines a new nontransitive preference structure that generalizes the semi-order structure. We study its properties and relationships with other well-known order structures, provide a numerical representation and establish a sufficient condition implying that P is transitive. The tangent circle preference structure offers a geometric interpretation of a model of preference relations defined by means of a numerical representation with multiplicative threshold; this representation has appeared in several recently published papers.     

On L-functions of cyclotomic function fields

We study two criterions of cyclicity for divisor class groups of function fields, the first one involves Artin L-functions and the second one involves “affine” class groups. We show that, in general, these two criterions are not linked.     

On a combinatorial method for counting smooth numbers in sets of integers

In this paper we develop a method for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth” sums a+b, a∈A, b∈B. The proof of this result is completely combinatorial and elementary.     

A method for constructing t-norms

In this paper a new type of t-operators with double thresholda,b ∈ (0,1),a ≤b, is presented, each pair (t-norm,t-conorm) consisting of two dual elements with respect to a negation with double threshold.     

Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the n-dimensional hypercube so only two different in-degrees a and b occur. We show that this can be done, for two specified in-degrees, if and only if obvious necessary conditions hold. Namely, we need 0?a,b?n and also there exist non-negative integers s and t so that s+t=n² and as+bt=n²n−1. This is connected to a question arising from constructing a strategy for a “hat puzzle”.     

A Graded Quadrivalent Logic for Ordinal Preference Modelling: Loyola–Like Approach

We extend a quadrivalent logic of Belnap to graded truth values in order to handle graded relevance of positive and negative arguments provided in preferential information concerning ranking of a finite set of alternatives. This logic is used to design the preference modelling and exploitation phases of decision aiding with respect to the ranking problem. The graded arguments are presented on an ordinal scale and their aggregation leads to preference model in form of four graded outranking relations (true, false, unknown and contradictory). The exploitation procedure combines the min-scoring procedure with the leximin rule. Aggregation of positive and negative arguments as well as exploitation of the resulting outranking relations is concordant with an advice given by St. Ignatius of Loyola (1548) how to make a good choice.     

Symmetries in trees and parking functions

We investigate a particular symmetry in labeled trees first discovered by Gessel, which can be stated as follows: In the set of rooted labeled trees on n+1 vertices rooted at the smallest vertex, the number of trees with a descents and b+1 leaves equals the number of trees with b descents and a+1 leaves. We present two new ways to prove the symmetry resulting from decompositions of trees, which lead to three different bijections from trees to trees in which leaves and descents are swapped. We also interpret the symmetry in terms of parking functions: the number of parking functions on [n] with a descents and b unfavorable spaces (defined in this paper) equals the number of parking functions on [n] with b descents and a unfavorable spaces. We conclude with a generalization of these results to binary trees.RésuméNous étudions une symétrie particulière dans les arbres étiquetés, découverte par Gessel, qu'on peut énoncer comme suit: Dans l'ensemble des arbres étiquetés pointés avec n+1 sommets, pointés au sommet minimum, le nombre d'arbres avec a descentes et b+1 feuilles égale le nombre d'arbres avec b descentes et a+1 feuilles. Nous présentons deux nouvelles démonstrations de la symétrie, qui resultent des décompositions des arbres; à partir des décompositions, nous obtenons trois bijections des arbres sur les arbres qui échangent les feuilles et les descentes. De plus, nous interprétons la symétrie en termes des “fonctions de stationnement” (parking functions): le nombre des fonctions de stationnement avec a descentes et b positions défavorables (définies dans cette article) égale le nombre de fonctions de stationnement avec b positions défavorable et a descentes. Nous donnons aussi une généralisation de ces resultats aux arbres binaires.     

On nonlinear diffusion problems with strong degeneracy

In this paper, we study the “triply” degenerate problem: bt(v)−Δg(v)+divΦ(v)=f on Q:=(0,T)×Ω, b(v(0,⋅))=b(v₀) on Ω and “g(v)=g(a) on some part of the boundary (0,T)×∂Ω,” in the case of continuous nonhomogeneous and nonstationary boundary data a. The functions b,g are assumed to be continuous, locally Lipschitz, nondecreasing and to verify the normalization condition b(0)=g(0)=0 and the range condition R(b+g)=R. Using monotonicity and penalization methods, we prove existence of a weak renormalized entropy solution in the spirit of [K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations 228 (2006) 111-139].     

Interactive outranking approaches for multicriteria decision-making problems with imprecise information

PROMETHEE is a powerful method, which can solve many multiple criteria decision making (MCDM) problems. It involves sophisticated preference modelling techniques but requires too much a priori precise information about parameter values (such as criterion weights and thresholds). In this paper, we consider a MCDM problem where alternatives are evaluated on several conflicting criteria, and the criterion weights and/or thresholds are imprecise or unknown to the decision maker (DM). We build robust outranking relations among the alternatives in order to help the DM to rank the alternatives and select the best alternative. We propose interactive approaches based on PROMETHEE method. We develop a decision aid tool called INTOUR, which implements the developed approaches.