On Optimal Allocation of Indivisibles Under Uncertainty

The optimal use of indivisible resources is often the central issue in the economy and management. One of the main difficulties is the discontinuous nature of the resulting resource allocation problems which may lead to the failure of competitive market allocation mechanisms (unless we agree to "divide" the indivisibles in some indirect way). The problem becomes even more acute when uncertainty of the outcomes of decisions is present. In this paper we formalize the problem as a stochastic optimization problem involving discrete decision variables and uncertainties. By using some concrete examples, we illustrate how some problems of "dividing indivisibles" under uncertainty can be formalized in such terms. Next, we develop a general methodology to solve such problems based on the concept of the branch and bound method. The main idea of the approach is to process large collections of possible solutions and to devote more attention to the most promising groups. By gathering more information to reduce the uncertainty and by specializing the solution the optimal decision can be found

Minorant methods for stochastic global optimization

We develop numerical methods for solution of stochastic global optimization problems: min$[F(x)=Ef(x,¦Ø)| xin X]$ and $min[F(x)=P{f(x, ¦Ø) ¡Ü0} | xin X]$, where x is a finite dimensional decision vector with possible values in the set X, ¦Ø is a random variable, $f(x,¦Ø)$ is a nonlinear function of variable x, E and P denote mathematical expectation and probability signs respectively. These methods are based on the concept of stochastic tangent minorant, which is a random function $¦Õ(x,y, ¦Ø)$ of two variables x and y with expected value $¦µ(x,y)=E ¦Õ(x,y, ¦Ø)$ satisfying conditions: (i) $¦µ(x,x)=F(x)$, (ii) $¦µ(x,y) ¡ÜF(x)$ for all x,y. Tangent minorant is a source of information on a function global behavior. We develop a calculus of (stochastic) tangent minorants. We develop a stochastic analogue of Pijavski¡¯s global optimization method and a branch and bound method with stochastic minorant bounds. Applications to optimal facility location and network reliability optimization are discussed

Adaptive Design in Discrete Stochastic Optimization

We present adaptive assignment rules for the design of the necessary simulations when solving discrete stochastic optimization problems. The rules are constructed in such a way, that the expected size of confidence sets for the optimizer is as small as possible

Modélisation, représentation et résolution de problèmes de partage équitable de biens indivisibles soumis au risque

Le développement et l utilisation de systèmes complexes multi-utilisateurs, ou encore la mise en réseau de systèmes d observation ou d information pose des problèmes complexes de partage de ressources entre les utilisateurs. La particularité de ces systèmes, impliquant plusieurs utilisateurs humains ou entités organisationnelles est que le partage des ressources doit satisfaire les préférences souvent antagonistes des utilisateurs et répondre à des exigences d équité. Ce travail de thèse a pour objet l étude des problèmes de partage de ressources indivisibles entre des agents ayant des préférences complexes sur ces ressources.Nous nous intéressons plus particulièrement à la modélisation de problèmes de partage en univers risqué.En effet, dans de nombreux problèmes d allocation de ressources réels, la part revenant réellement à chaque agent après le partage de la ressource dépend de facteurs exogènes. C est le cas par exemple dans les systèmes d observation (satellitaires, capteurs embarqués,...), dans lesquels la réalisation d une requête donnée dépend non seulement des conditions climatiques sur le secteur à observer, mais aussi du bon fonctionnement du capteur, de l absence de brouillage du signal, etc. L introduction de risque dans les problèmes de partage implique la redéfinition des notions classiques de choix social (utilité, absence d envie, ...), et l agrégation collective des préférences des agents s en trouve compliquée. Au cours de ce travail de thèse, nous nous sommes tout d abord intéressés à l étude de cette extension au risque du formalisme associé aux problèmes de partage classiques : nous proposons un modèle simple de problèmes de partages de biens indivisibles en présence de risque, toutefois assez général pour rester proche des applications réelles considérées, et nous introduisons une extension générale des méthodes d évaluation non risquées pour de tels partages. La seconde partie de ce travail de thèse porte sur l algorithmique associée à ces problèmes, dont la résolution est notablement complexifiée par la présence de ressources risquées. Pour plusieurs critères d évaluation (choisis car visant à garantir une certaine équité des solutions qu ils suggèrent), nous proposons des algorithmes de résolution exacte et approchée des problèmes de partage associés.The development and use of complex multi-user systems, or the networking of observation ou information systems raises complex resource allocation problems. The particularity of these systems, which involve several human users or organisational entities, rests in the fact that the share of resources must satisfy the often conflicting preferences of users and comply with equity exigences. This thesis deals with the problem of fairly allocating indivisible goods to a set of agents having complex preferences over these goods.We are more particularly interested in the modeling of fair allocation problems in a risky setting. In numerous real-world resource allocation problems, the actual share each agent receives after the allocation often depends on exogenous factors. This is for instance the case with the observation systems (satellites, embedded sensons, etc.) where the realisation of a request not only depends on weather conditions over the observation area, but also on the potential sensor malfunction, on the absence of jamming of the signal, etc. Introducing the risk in allocation problems implies the redefinition of classical social choice notions such as utility or envy-freeness for instance, and the collective aggregation of agents preferences becomes more complicated. We have studied in this thesis the extension of allocation problem formalism to a risky setting : we present a simple model for risky indivisible goods allocation problems, yet general enough to encompass most of the real-world applications, and we introduce a general extension of risk-free evalution methods for such allocations. The second part of the work concerns the algorithmical issues related to theses problems, whose resolution is significantly complexified because of the risky setting. Forseveral evaluation criteria (selected for the equity of the solutions they suggest) we present both exact and approached resolution algorithms for the related allocation problems.TOULOUSE-ISAE (315552318) / SudocSudocFranceF

Exact solutions to a class of stochastic generalized assignment problems

This paper deals with a stochastic Generalized Assignment Problem with recourse. Only a random subset of the given set of jobs will require to be actually processed. An assignment of each job to an agent is decided a priori, and once the demands are known, reassignments can be performed if there are overloaded agents. We construct a convex approximation of the objective function that is sharp at all feasible solutions. We then present three versions of an exact algorithm to solve this problem, based on branch and bound techniques, optimality cuts, and a special purpose lower bound. numerical results are reported.

Les taux de commission sur les transactions boursières au Canada : Acte II

This is the second part of a study about what could be called an economic play or drama: the deregulation of commission rates on stock exchange transactions. This article presents an economic analysis of the long-standing policy in Canada of charging minimum commission on stock transaction. The discussion draws heavily on the arguments put forward by the Montreal Exchange as a part of its recent ongoing defense of fixed commission. The arguments fall into three categories: (1) the economic approach to the analysis of the brokerage business (uncertainty in product quality) (2) the information produce by the brokerage industry are public goods because of externalities and (3) the structure of the brokerage industry. According to the Exchange's logic, the elimination of the practice of price fixing would lead to a less efficient capital market because of the reduction in the production of information and to an increase in the concentration in the brokerage business. The analysis presented in this article leads to the conclusion that the Exchange's case is faulty in terms of both its theory and its empirical proofs and that minimum commission rates on stock exchange transactions cannot be justified on economic grounds

Managing Water Quality under Uncertainty: Application of a New Stochastic Branch and Bound Method

The problem of water quality management under uncertain emission levels, reaction rates and pollutant transport is considered. Various performance measures: reliability, resiliency and vulnerability are taken into account. A general methodology for finding a cost-effective water quality management program is developed. The approach employs a new idea of the stochastic branch and bound method, which combines random estimates of the performance for subsets of decisions with iterative refinement of the most promising subsets

Environmental management problems, future generations and social decisions

The decisions of many individuals and social groups, taking according to well-defined objectives, are causing serious social and environmental problems, in spite of following the dictates of economic rationality. There are many examples of serious problems for which there are not yet appropriate solutions, such as management of scarce natural resources including aquifer water or the distribution of space among incompatible uses. In order to solve these problems, the paper first characterizes the resources and goods involved from an economic perspective. Then, for each case, the paper notes that there is a serious divergence between individual and collective interests and, where possible, it designs the procedure for solving the conflict of interests. With this procedure, the real opportunities for the application of economic theory are shown, and especially the theory on collective goods and externalities. The limitations of conventional economic analysis are shown and the opportunity to correct the shortfalls is examined. Many environmental problems, such as climate change, have an impact on different generations that do not participate in present decisions. The paper shows that for these cases, the solutions suggested by economic theory are not valid. Furthermore, conventional methods of economic valuation (which usually help decision-makers) are unable to account for the existence of different generations and tend to obviate long-term impacts. The paper analyzes how economic valuation methods could account for the costs and benefits enjoyed by present and future generations. The paper studies an appropriate consideration of preferences for future consumption and the incorporation of sustainability as a requirement in social decisions, which implies not only more efficiency but also a fairer distribution between generations than the one implied by conventional economic analysis.