The Stochastic Inventory Routing Problem with Direct Deliveries

Vendor managed inventory replenishment is a business practice in which vendors monitor their customers’ inventories, and decide when and how much inventory should be replenished. The inventory routing problem addresses the coordination of inventory management and transportation. The ability to solve the inventory routing problem contributes to the realization of the potential savings in inventory and transportation costs brought about by vendor managed inventory replenishment. The inventory routing problem is hard, especially if a large number of customers is involved. We formulate the inventory routing problem as a Markov decision process, and we propose approximation methods to find good solutions with reasonable computational effort. Computational results are presented for the inventory routing problem with direct deliveries. ∗Supported by the National Science Foundation under grant DMI-9875400. The inventory routing problem (IRP) is one of the core problems that has to be solved when implementing the emerging business practice called vendor managed inventory replenishment (VMI). VMI refers to the situation where the replenishment of inventory at a number of locations is controlled by a central decision maker (vendor). The central decision maker can be the supplier, and the inventory can be kept at independent customers, or the central decision maker can be a manager responsible for inventory replenishment at a number of warehouses or retail outlets of the same company. Often the central decision maker manages a fleet of vehicles that make the deliveries. In this paper the central decision maker is called the supplier, and the inventory locations are referred to as the customers. VMI differs from conventional inventory management in the following way. In conventional inventory management, the customers monitor their own inventory levels, and when a customer thinks that it is time to reorder, an order for a quantity of the product is placed at the supplier. The supplier receives these orders from the customers, prepares the product for delivery, and makes deliveries using the fleet of vehicles. Conventional inventory management has several disadvantages. It is typical for orders not to arrive uniformly over time. For example, one of the suppliers we worked with used to be flooded with orders on Mondays. The conjecture was that many customers tend to check their inventory levels on Mondays, and then place orders. The result of this nonuniform order arrival pattern is that the supplier’s resources, such as the production and storage facilities, as well as transportation resources, cannot be utilized well over time. For example, the supplier’s resources would be stretched to the limit on Mondays and Tuesdays, after a large number of orders have arrived, and would be relatively idle during the rest of the week. Another related phenomenon causes a disadvantage for the customers. Some customers place apparently urgent orders when other customers place orders that are really urgent. Since the supplier does not know the inventory levels at the customers, the information needed to compare the real urgency of different orders is not available. Also, the supplier is only responsible for delivering product on order to the customer, and not for maintaining a desirable inventory level at the customer, and hence, even if the supplier were provided with the inventory level data, there would not be a strong incentive for the supplier to find the optimal trade-off between the inventory needs of the different customers. Consequently really urgent orders may be delayed because of a lack of information and incentive, and a high demand on the supplier’s resources. In VMI, the supplier monitors the inventory at the customers. This is made possible with modern equipment that can both measure the inventory at the customers and communicate with the supplier’s computer. The rapidly decreasing cost of this technology has probably made a significant contribution to the increasing popularity and success of VMI. The supplier is responsible for maintaining a desirable inventory level at each customer, and decides which customers should be replenished at which times, and with how much product. To make these decisions, the supplier has the benefit of access to a lot of relevant information, such as the current (and past) inventory levels at all the customers, the customers’ demand behavior, the customers’ locations relative to the supplier and relative to each other and the resulting transportation costs, and the capacity and availability of vehicles and drivers for delivery. It is thus not surprising that VMI has several advantages for the supplier over conventional inventory