Luca Bertazzi

A Branch-and-Cut Algorithm for a Vendor-Managed Inventory-Routing Problem

We consider a distribution problem in which a product has to be shipped from a supplier to several retailers over a given time horizon. Each retailer defines a maximum inventory level. The supplier monitors the inventory of each retailer and determines its replenishment policy, guaranteeing that no stockout occurs at the retailer (vendor-managed inventory policy). Every time a retailer is visited, the quantity delivered by the supplier is such that the maximum inventory level is reached (deterministic order-up-to level policy). Shipments from the supplier to the retailers are performed by a vehicle of given capacity. The problem is to determine for each discrete time instant the quantity to ship to each retailer and the vehicle route. We present a mixed-integer linear programming model and derive new additional valid inequalities used to strengthen the linear relaxation of the model. We implement a branch-and-cut algorithm to solve the model optimally. We then compare the optimal solution of the problem with the optimal solution of two problems obtained by relaxing in different ways the deterministic order-up-to level policy. Computational results are presented on a set of randomly generated problem instances.

Deterministic Order-Up-To Level Policies in an Inventory Routing Problem

We consider a distribution problem in which a set of products has to be shipped from a supplier to several retailers in a given time horizon. Shipments from the supplier to the retailers are performed by a vehicle of given capacity and cost. Each retailer determines a minimum and a maximum level of the inventory of each product, and each must be visited before its inventory reaches the minimum level. Every time a retailer is visited, the quantity of each product delivered by the supplier is such that the maximum level of the inventory is reached at the retailer. The problem is to determine for each discrete time instant the retailers to be visited and the route of the vehicle. Various objective functions corresponding to different decision policies, and possibly to different decision makers, are considered. We present a heuristic algorithm and compare the solutions obtained with the different objective functions on a set of randomly generated problem instances.

A Hybrid Heuristic for an Inventory Routing Problem

We consider an inventory routing problem in discrete time where a supplier has to serve a set of customers over a multiperiod horizon. A capacity constraint for the inventory is given for each customer, and the service cannot cause any stockout situation. Two different replenishment policies are considered: the order-up-to-level and the maximum-level policies. A single vehicle with a given capacity is available. The transportation cost is proportional to the distance traveled, whereas the inventory holding cost is proportional to the level of the inventory at the customers and at the supplier. The objective is the minimization of the sum of the inventory and transportation costs. We present a heuristic that combines a tabu search scheme with ad hoc designed mixed-integer programming models. The effectiveness of the heuristic is proved over a set of benchmark instances for which the optimal solution is known.

Minimizing the Total Cost in an Integrated Vendor--Managed Inventory System

In this paper we consider a complex production-distribution system, where a facility produces (or orders from an external supplier) several items which are distributed to a set of retailers by a fleet of vehicles. We consider Vendor-Managed Inventory (VMI) policies, in which the facility knows the inventory levels of the retailers and takes care of their replenishment policies. The production (or ordering) policy, the retailers replenishment policies and the transportation policy have to be determined so as to minimize the total system cost. The cost includes the fixed and variable production costs at the facility, the inventory costs at the facility and at the retailers and the transportation costs, that is the fixed costs of the vehicles and the traveling costs. We study two different types of VMI policies: The order-up-to level policy, in which the order-up-to level quantity is shipped to each retailer whenever served (i.e. the quantity delivered to each retailer is such that the maximum level of the inventory at the retailer is reached) and the fill-fill-dump policy, in which the order-up-to level quantity is shipped to all but the last retailer on each delivery route, while the quantity delivered to the last retailer is the minimum between the order-up-to level quantity and the residual transportation capacity of the vehicle. We propose two different decompositions of the problem and optimal or heuristic procedures for the solution of the subproblems. We show that, for reasonable initial values of the variables, the order in which the subproblems are solved does not influence the final solution. We will first solve the distribution subproblem and then the production subproblem. The computational results show that the fill-fill-dump policy reduces the average cost with respect to the order-up-to level policy and that one of the decompositions is more effective. Moreover, we compare the VMI policies with the more traditional Retailer-Managed Inventory (RMI) policy and show that the VMI policies significantly reduce the average cost with respect to the RMI policy.

Reoptimizing the traveling salesman problem

In this paper, we study the reoptimization problems which arise when a new node is added to an optimal solution of a traveling salesman problem (TSP) instance or when a node is removed. We show that both reoptimization problems are NP-hard. Moreover, we show that, while the cheapest insertion heuristic has a tight worst-case ratio equal to 2 when applied to a TSP instance, it guarantees, in linear time, a tight worst-case ratio equal to 3/2 when used to add the new node and that also the simplest heuristic to remove a node from the optimal tour guarantees a tight ratio equal to 3/2 in constant time.

Analysis of the maximum level policy in a production-distribution system

We consider a production-distribution system, where a facility produces one commodity which is distributed to a set of retailers by a fleet of vehicles. Each retailer defines a maximum level of the inventory. The production policy, the retailers replenishment policies and the transportation policy have to be determined so as to minimize the total system cost. The overall cost is composed by fixed and variable production costs at the facility, inventory costs at both facility and retailers and routing costs. We study two different types of replenishment policies. The well-known order-up to level (OU) policy, where the quantity shipped to each retailer is such that the level of its inventory reaches the maximum level, and the maximum level (ML) policy, where the quantity shipped to each retailer is such that the inventory is not greater than the maximum level. We first show that when the transportation is outsourced, the problem with OU policy is NP-hard, whereas there exists a class of instances where the problem with ML policy can be solved in polynomial time. We also show the worst-case performance of the OU policy with respect to the more flexible ML policy. Then, we focus on the ML policy and the design of a hybrid heuristic. We also present an exact algorithm for the solution of the problem with one vehicle. Results of computational experiments carried out on small size instances show that the heuristic can produce high quality solutions in a very short amount of time. Results obtained on a large set of randomly generated problem instances are also shown, aimed at comparing the two policies.

Minimization of logistic costs with given frequencies

We study the problem of shipping products from one origin to several destinations, when a given set of possible shipping frequencies is available. The objective of the problem is the minimization of the transportation and inventory costs. We present different heuristic algorithms and test them on a set of randomly generated problem instances. The heuristics are based upon the idea of solving, in a first phase, single link problems, and of locally improving the solution in subsequent phases.

Inventory routing problems: an introduction

In this tutorial paper, we introduce the inventory routing problems (IRPs) with examples, we classify the characteristics of an IRP and present different models and policies for the class of problems where the crucial decision is when to serve customers. We call this class the problems with decisions over time only. The contributions are on the single link case, i.e., the problem where products are shipped from a supplier to a customer with capacitated vehicles, and on the IRPs with direct shipping. We overview the pioneering papers that appeared in the eighties, the literature on the single link and direct shipping problems, and cite the surveys and the tutorials available.

Reoptimizing the 0-1 knapsack problem

In this paper we study the problem where an optimal solution of a knapsack problem on n items is known and a very small number k of new items arrive. The objective is to find an optimal solution of the knapsack problem with n+k items, given an optimal solution on the n items (reoptimization of the knapsack problem). We show that this problem, even in the case k=1, is NP-hard and that, in order to have effective heuristics, it is necessary to consider not only the items included in the previously optimal solution and the new items, but also the discarded items. Then, we design a general algorithm that makes use, for the solution of a subproblem, of an @a-approximation algorithm known for the knapsack problem. We prove that this algorithm has a worst-case performance bound of 12-@a, which is always greater than @a, and therefore that this algorithm always outperforms the corresponding @a-approximation algorithm applied from scratch on the n+k items. We show that this bound is tight when the classical Ext-Greedy algorithm and the G^3^4 algorithm are used to solve the subproblem. We also show that there exist classes of instances on which the running time of the reoptimization algorithm is smaller than the running time of an equivalent PTAS and FPTAS.

Continuous and Discrete Shipping Strategies for the Single Link Problem

We consider the problem of shipping several products from a common origin to a common destination. Each product is offered at the origin and absorbed at the destination at a given constant rate. The objective of the problem is to determine a shipping strategy that minimizes the sum of the transportation and the inventory costs. We present a general framework of analysis from which we derive the known approaches with a continuous frequency and with a set of given frequencies as particular cases. Moreover, we derive from the general framework a new model for the case with discrete shipping times in which a shipment can take place at each discrete time instant. We prove that the optimal solutions of the three models can be ranked and that the distance between the optimal costs can be unlimited in the worst case. Finally, we evaluate, on the basis of a set of randomly generated problem instances, the ?average? distance between the optimal solutions of the models.