Compact versus noncompact LP formulations for minimizing convex Choquet integrals

AbstractWe address here the problem of minimizing Choquet Integrals (also known as “Lovász Extensions”) over solution sets which can be either polyhedra or (mixed) integer sets. Typical applications of such problems concern the search of compromise solutions in multicriteria optimization. We focus here on the case where the Choquet Integrals to be minimized are convex, implying that the set functions (or “capacities”) underlying the Choquet Integrals considered are submodular. We first describe an approach based on a large scale LP formulation, and show how it can be handled via the so-called column-generation technique. We next investigate alternatives based on compact LP formulations, i.e. featuring a polynomial number of variables and constraints. Various potentially useful special cases corresponding to well-identified subclasses of underlying set functions are considered: quadratic and cubic submodular functions, and a more general class including set functions which, up to a sign, correspond to capacities which are both (k+1)−additive and k-monotone for k≥3. Computational experiments carried out on series of test instances, including transportation problems and knapsack problems, clearly confirm the superiority of compact formulations. As far as we know, these results represent the first systematic way of practically solving Choquet minimization problems on solution sets of significantly large dimensions

Sorted-pareto dominance and qualitative notions of optimality

Pareto dominance is often used in decision making to compare decisions that have multiple preference values – however it can produce an unmanageably large number of Pareto optimal decisions. When preference value scales can be made commensurate, then the Sorted-Pareto relation produces a smaller, more manageable set of decisions that are still Pareto optimal. Sorted-Pareto relies only on qualitative or ordinal preference information, which can be easier to obtain than quantitative information. This leads to a partial order on the decisions, and in such partially-ordered settings, there can be many different natural notions of optimality. In this paper, we look at these natural notions of optimality, applied to the Sorted-Pareto and min-sum of weights case; the Sorted-Pareto ordering has a semantics in decision making under uncertainty, being consistent with any possible order-preserving function that maps an ordinal scale to a numerical one. We show that these optimality classes and the relationships between them provide a meaningful way to categorise optimal decisions for presenting to a decision maker

Optimising a nonlinear utility function in multi-objective integer programming

In this paper we develop an algorithm to optimise a nonlinear utility function of multiple objectives over the integer efficient set. Our approach is based on identifying and updating bounds on the individual objectives as well as the optimal utility value. This is done using already known solutions, linear programming relaxations, utility function inversion, and integer programming. We develop a general optimisation algorithm for use with k objectives, and we illustrate our approach using a tri-objective integer programming problem.Comment: 11 pages, 2 tables; v3: minor revisions, to appear in Journal of Global Optimizatio

Descriptive Profiles for Sets of Alternatives in Multiple Criteria Decision Aid

International audienceIn the context of Multiple Criteria Decision Aid, a decision-maker may be faced at any time with the task of analyzing one or several sets of alternatives, irrespective of the decision he is about to make. As in this case the alternatives may express contrasting gains and losses on the criteria on which they are evaluated, and while the sets that are presented to the decision-maker may potentially be large, the task of analysing them becomes a difficult one. Therefore the need to reduce these sets to a more concise representation is very important. Classically, profiles that describe sets of alternatives may be found in the context of the sorting problem, however they are either given beforehand by the decision-maker or determined from a set of assignment examples. We would therefore like to extend such profiles, as well as propose new ones, in order to characterize any set of alternatives. For each of them, we present several approaches for extracting them, which we then compare with respect to their performance

Aggregation of measures to produce an overall assessment of animal welfare. Part 1: a review of existing methods

Several systems have been proposed for the overall assessment of animal welfare at the farm level for the purpose of advising farmers or assisting public decision-making. They are generally based on several measures compounded into a single evaluation, using different rules to assemble the information. Here we discuss the different methods used to aggregate welfare measures and their applicability to certification schemes involving welfare. Data obtained on a farm can be (i) analysed by an expert who draws an overall conclusion; (ii) compared with minimal requirements set for each measure; (iii) converted into ranks, which are then summed; or (iv) converted into values or scores compounded in a weighted sum (e.g. TGI35L) or using ad hoc rules. Existing methods used at present (at least when used exclusively) may be insufficiently sensitive or not routinely applicable, or may not reflect the multidimensional nature of welfare and the relative importance of various welfare measures. It is concluded that different methods may be used at different stages of the construction of an overall assessment of animal welfare, depending on the constraints imposed on the aggregation proces

Combining machine learning and metaheuristics algorithms for classification method PROAFTN

© Crown 2019. The supervised learning classification algorithms are one of the most well known successful techniques for ambient assisted living environments. However the usual supervised learning classification approaches face issues that limit their application especially in dealing with the knowledge interpretation and with very large unbalanced labeled data set. To address these issues fuzzy classification method PROAFTN was proposed. PROAFTN is part of learning algorithms and enables to determine the fuzzy resemblance measures by generalizing the concordance and discordance indexes used in outranking methods. The main goal of this chapter is to show how the combined meta-heuristics with inductive learning techniques can improve performances of the PROAFTN classifier. The improved PROAFTN classifier is described and compared to well known classifiers, in terms of their learning methodology and classification accuracy. Through this chapter we have shown the ability of the metaheuristics when embedded to PROAFTN method to solve efficiency the classification problems

Electre Methods: Main Features and Recent Developments

We present main characteristics of Electre family methods, designed for multiple criteria decision aiding. These methods use as a preference model an outranking relation in the set of actions - it is constructed in result of concordance and non-discordance tests involving a specific input preference information. After a brief description of the constructivist conception in which the Electre methods are inserted, we present the main features of these methods. We discuss such characteristic features as: the possibility of taking into account positive and negative reasons in the modeling of preferences, without any need for recoding the data; using of thresholds for taking into account the imperfect knowledge of data; the absence of systematic compensation between "gains" and "losses". The main weaknesses are also presented. Then, some aspects related to new developments are outlined. These are related to some new methodological developments, new procedures, axiomatic analysis, software tools, and several other aspects. The paper ends with conclusions