Exact algorithms for the order picking problem

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixed-integer programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde

Algorithms for elementary path problem : application to kidney exchange

Cette thèse traite de problèmes de chemins élémentaires et leur application au problème d’échange de reins. Nous nous concentrons sur des programmes d’échange de reins qui incluent des donneurs altruistes, qui sont essentiels pour les patients avec une maladie rénale, mais représentent un défi pour les méthodes de recherche opérationnelle. Notre objectif est de développer un algorithme efficace qui pourra être utilisé pour résoudre des instances futures, qui sont susceptibles d’impliquer un grand nombre de participants. Nous rencontrons des problèmes étroitement lié au notre : problèmes de packing, de tournée de véhicules, de stable. Pour ce dernier, nous présentons une nouvelle formulation étendue et prouvons qu’elle est idéale et compacte pour les graphes parfaits sans griffe. Nous nous focalisons ensuite sur la conception d’une génération de colonnes dédiée au problème d’échange de reins et nous attaquons à son problème de pricing, NP-difficile. Nous abordons le problème du chemin élémentaire minimum avec contrainte de taille, qui modélise la recherche de chaînes de dons intéressantes à ajouter dans la phase du pricing. Nous étudions des approches dynamiques, en particulier la relaxation NG-route et l’heuristique de color coding, et les améliorons en exploitant la contrainte de taille et la faible densité des graphes considérés. Nous nous intéressons ensuite au color coding dans un contexte plus général, proposant de nouvelles stratégies randomisées qui apportent une garantie d’amélioration. Ces stratégies s’appuient sur un ordonnancement du graphe et introduisent un biais dans la loi de probabilité pour augmenter les chances de trouver une solution optimale.This thesis deals with elementary path problems and their application to the kidney exchange problem. We focus on kidney exchange programs including altruistic donors, which are crucial for patients with renal disease and challenging for operations research methods. The goal of this work is to develop an efficient algorithm that can be used to solve future instances, which are likely to involve a large number of donors and patients. While we progress on this topic, we encounter closely related problems on packing, vehicle routing and stable set. For this last problem, we introduce a new extended formulation and prove it is ideal and compact for claw-free perfect graphs by characterizing its polytope. We then concentrate on the design of a column generation dedicated to the kidney exchange problem and confront its NP-hard pricing problem. The specific problem that we address is the elementary path problem with length constraint, which models the search for interesting chains of donation to add during the pricing step. We investigate dynamic approaches, in particular the NG-route relaxation and the color coding heuristic, and improve them by exploiting the length constraint and sparsity of graphs. We study the color coding in a more general context, providing a guaranteed improvement by proposing new randomized strategies. They are based on ordering the graph before coloring it and introduce a bias in the probability distribution to increase the probability of finding an optimal solution

Algorithmes de chemin élémentaire : application aux échanges de reins

This thesis deals with elementary path problems and their application to the kidney exchange problem. We focus on kidney exchange programs including altruistic donors, which are crucial for patients with renal disease and challenging for operations research methods. The goal of this work is to develop an efficient algorithm that can be used to solve future instances, which are likely to involve a large number of donors and patients. While we progress on this topic, we encounter closely related problems on packing, vehicle routing and stable set. For this last problem, we introduce a new extended formulation and prove it is ideal and compact for claw-free perfect graphs by characterizing its polytope. We then concentrate on the design of a column generation dedicated to the kidney exchange problem and confront its NP-hard pricing problem. The specific problem that we address is the elementary path problem with length constraint, which models the search for interesting chains of donation to add during the pricing step. We investigate dynamic approaches, in particular the NG-route relaxation and the color coding heuristic, and improve them by exploiting the length constraint and sparsity of graphs. We study the color coding in a more general context, providing a guaranteed improvement by proposing new randomized strategies. They are based on ordering the graph before coloring it and introduce a bias in the probability distribution to increase the probability of finding an optimal solution.Cette thèse traite de problèmes de chemins élémentaires et leur application au problème d’échange de reins. Nous nous concentrons sur des programmes d’échange de reins qui incluent des donneurs altruistes, qui sont essentiels pour les patients avec une maladie rénale, mais représentent un défi pour les méthodes de recherche opérationnelle. Notre objectif est de développer un algorithme efficace qui pourra être utilisé pour résoudre des instances futures, qui sont susceptibles d’impliquer un grand nombre de participants. Nous rencontrons des problèmes étroitement lié au notre : problèmes de packing, de tournée de véhicules, de stable. Pour ce dernier, nous présentons une nouvelle formulation étendue et prouvons qu’elle est idéale et compacte pour les graphes parfaits sans griffe. Nous nous focalisons ensuite sur la conception d’une génération de colonnes dédiée au problème d’échange de reins et nous attaquons à son problème de pricing, NP-difficile. Nous abordons le problème du chemin élémentaire minimum avec contrainte de taille, qui modélise la recherche de chaînes de dons intéressantes à ajouter dans la phase du pricing. Nous étudions des approches dynamiques, en particulier la relaxation NG-route et l’heuristique de color coding, et les améliorons en exploitant la contrainte de taille et la faible densité des graphes considérés. Nous nous intéressons ensuite au color coding dans un contexte plus général, proposant de nouvelles stratégies randomisées qui apportent une garantie d’amélioration. Ces stratégies s’appuient sur un ordonnancement du graphe et introduisent un biais dans la loi de probabilité pour augmenter les chances de trouver une solution optimale

Exact algorithms for the picking problem

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixed-integer programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde

Exact algorithms for the picking problem

International audienc

Exact algorithms for the picking problem

International audienc

New randomized strategies for the color coding algorithm

International audienceThe color coding technique is used to solve subgraph isomorphism problems, in particular path problems. One color among C is randomly assigned to each vertex of the graph and if distinct colors are given to the vertices of the desired subgraph, it can be found efficiently by dynamic programming. These two phases are repeated until the subgraph is found with a high probability, which can require a large number of iterations. We propose new coloring strategies that take advantage of the graph structure to increase this probability and thus reduce the number of iterations. They provide a guaranteed improvement over the original color coding technique based on a particular structural parameter related to the bandwidth. When this parameter is smaller than the number C of colors, we prove that only C calls to the dynamic program are needed to find the subgraph

Column Generation for the Kidney Exchange Problem

International audienceThe Kidney Exchange Problem (KEP) aims at finding the best exchanges in a barter market where agents are patients with a willing but incompatible donor. (Abraham et al. 2007) introduced a natural (exponential) integer programming formulation called the cycle formulation that they could solve efficiently by a branch-and-price approach. Recently, several countries allowed for the participation of altruistic donors in the exchanges. The corresponding variant of KEP is harder to solve as the pricing problem becomes N P-complete. In this work, we study and experiment a column generation approach that takes into account altruistic donors. We use advanced techniques to circumvent the N P-hardness of the pricing problem and show that the corresponding method can provide excellent guaranteed feasible solutions in a small amount of time

Solving the Maximum Weighted Length-Constrained Simple Path

National audienc

Kidney Exchange Problem: models and algorithms

International audienc