The elimination of distribution inefficiencies, occurring due to the timing of customers' orders is an important reason for companies to introduce vendor managed inventory programs. By managing their customers' inventories, suppliers may be able to reduce demand variability and therefore distribution costs. We develop technology to measure the effectiveness of distribution strategies. We develop a methodology that allows the computation of tight lower bounds on the total mileage required to satisfy customer demand over a period of time. As a result, companies will be able to gain insight into the effectiveness of their distribution strategy. This technology can also be used to suggest desirable delivery patterns and to analyze tactical and strategic decisions. Secondly, we study the inventory routing problem with continuous moves (IRP-CM). The typical inventory routing problem deals with the repeated distribution of a single product, from a single facility, with an unlimited supply, to a set of customers that can all be reached with out-and-back trips. Unfortunately, this is not always the reality. We introduce the IRP-CM to study two important real-life complexities: limited product availabilities at facilities and customers that cannot be served using out-and-back tours. We need to design delivery tours spanning several days, covering huge geographic areas, and involving product pickups at different facilities. We develop a heuristic and an optimization algorithm to construct distribution plans. The heuristic is an innovative randomized greedy algorithm, which includes linear programming based postprocessing technology. To solve the IRP-CM to optimality, we give a time-discretized integer programming model and develop a branch-and-cut algorithm. As instances of time-discretized models tend to be large we discuss several possibilities for reducing the problem size. We introduce a set of valid inequalities, called delivery cover inequalities, in order to tighten the bounds given by the LP relaxation of the time-discretized model. We also introduce branching schemes exploiting the underlying structure of the IRP-CM. An extensive computational study demonstrates the effectiveness of the optimization algorithm. Finally, we present an integrated approach using heuristics and optimization algorithms providing effective and efficient technology for solving inventory problems with continuous moves.

[pt] Esta pesquisa trata de duas variantes do conhecido Problema de Roteirização de Veículos com Estoque (do inglês Inventory Routing Problem – IRP). O problema nasce num contexto de um sistema de Vendor Managed Inventory (VMI) no qual o fornecedor é responsável pela gestão de estoques do cliente. Tal problema é a junção dos problemas de transporte e gestão de estoques, que correspondem aos maiores custos em uma operação logística. Destarte este trabalho apresenta um modelo matemático para uma variante do IRP que considera que o fornecedor tem clientes dentro e fora do sistema de VMI. Este caso surge quando para alguns clientes não é interessante a realização do controle de seus estoques dentro do sistema de VMI, somente o atendimento de suas demandas. Além disto, o modelo contempla três diferentes tipos de políticas de gestão de estoques e é capaz de lidar com casos contendo vários períodos e vários veículos. Após sua elaboração, o modelo foi validado em instâncias do IRP, do Problema de Roteamento de Veículos (do inglês Capacitated Vehicle Routing Problem - CVRP) e instâncias próprias para a variante. Foram realizados também estudos sobre os impactos das diferentes políticas de gestão de estoques. Além do modelo matemático, foi desenvolvida uma meta-heurística híbrida que resolve uma variante do IRP considerando vários períodos e vários veículos. Cada movimento considerado durante a meta-heurística é divido em duas etapas, a primeira sendo a modificação da posição de um ou mais clientes nos veículos e períodos e uma segunda etapa que resolve de forma exata um Problema de Fluxo Máximo a Custo Mínimo para a atribuição ótima do volume de carga transportada para cada cliente por cada veículo em cada período. Esta abordagem é então testada em instâncias clássicas para esta variante do IRP, obtendo resultados que comprovam a eficiência do algoritmo. / [en] This research deals with two variants of the Inventory Routing Problem (IRP). This problem comes from the context of a Vendor Managed Inventory (VMI) system in which the vendor is responsible for managing the customer s inventory. It is the combination of transportation and inventory management problems, which correspond to the higher costs in a logistics operation. Hence, this paper presents a mathematical model for an IRP variant, in which the vendor has customers inside and outside the VMI system. This situation is presented when it is not interesting to manage the inventories of some clients within the VMI system, resulting only in meeting their demands. In addition, the model considers three different types of stock management policies and it can comprehend multiple periods and multiple vehicles. After its modelling, the model was validated using IRP instaces, the Vehicle Routing Problem (CVRP) and specific instances for this variant. The impacts of different inventory management policies were also analyzed. In addition to the mathematical model, a hybrid meta-heuristic was developed, which solves an IRP variant considering several periods and several vehicles. Each iteration of the metaheuristic is divided into two stages: the first is modifying the position of one or more customers attended by the vehicles and periods, and a second step that solves a Maximum Flow at Minimum Cost problem, to optimally assign the load volumes transported to each customer in each vehicle in each period. Then, this approach is tested in classical instances for this IRP variant, obtaining results that prove the efficiency of the algorithm.

Le problème d’approvisionnement conjoint (JDP) proposé est un problème de planification des tournées de livraisons sur un horizon de temps décomposé en périodes élémentaires, l’horizon de temps étant la période commune de livraison de tous les produits,. La donnée de ces paramètres permet d’obtenir une formulation linéaire du problème, avec des variables de décision binaires. Le modèle intègre aussi des contraintes de satisfaction de la demande à partir des stocks et des quantités livrées, des contraintes sur les capacités de stockage et de transport.Afin de résoudre aussi le problème de choix des tournées de livraison, il est nécessaire d'introduire dans le modèle des contraintes et des variables liées aux sites visités au cours de chaque tour. Il est proposé de résoudre le problème en deux étapes. La première étape est le calcul hors ligne du coût minimal de la tournée associé à chaque sous-ensemble de sites. On peut observer que pour tout sous-ensemble donné de sites, le cycle hamiltonien optimal reliant ces sites à l'entrepôt peut être calculé à l'avance par un algorithme du problème du voyageur de commerce (TSP). Le but ici n'est pas d'analyser pleinement le TSP, mais plutôt d'intégrer sa solution dans la formulation de JRP. .Dans la deuxième étape, des variables binaires sont associées à chaque tour et à chaque période pour déterminer le sous-ensemble de sites choisi à chaque période et son coût fixe associé. / The proposed joint delivery problem (JDP) is a delivery tour planning problem on a time horizon decomposed into elementary periods or rounds, the time horizon being the common delivery period for all products. The data of these parameters provides a linear formulation of the problem, with binary decision variables. The model also incorporates the constraints of meeting demand from stock and the quantities supplied, storage and transport capacity constraints.In order to also solve the problem of choice of delivery rounds, it is necessary to introduce in the model several constraints and variables related to the sites visited during each round. It is proposed to solve the problem in two steps. The first step is the calculation of the minimum off-line cost of the tour associated with each subset of sites. One can observe that for any given subset of sites, the optimal Hamiltonian cycle linking those sites to the warehouse can be calculated in advance by a traveling salesman problem algorithm (TSP). The goal here is not to fully analyze the TSP, but rather to integrate its solution in the formulation of the JRP. In the second stage, binary variables are associated with each subset and each period to determine the selected subset of sites in each period and its associated fixed cost.