Shipping Multiple Items by Capacitated Vehicles: An Optimal Dynamic Programming Approach

We consider a system in which multiple-items are transferred from a warehouse or a plant to a retailer through identical capacitated vehicles, or freight wagons. Any mixture of the items may be loaded on a vehicle. The retailer is facing dynamic deterministic demand for several items, over a finite planning horizon. A vehicle incurs a fixed cost for each trip made from the warehouse to the retailer. In addition, there exist item dependent variable shipping costs and inventory holding costs at the retailer, which are both constant over time. The objective is to find a shipment schedule that minimizes the total cost, while satisfying demand on time. We address and partially resolve the question regarding the problem’s complexity by introducing a dynamic programming algorithm whose complexity is polynomial for a fixed number of items, but exponential otherwise. Our dynamic programming formulation is based on properties satisfied by the optimal solution, and is using an innovative way for partitioning the problem into sub-problems. (Inventory/Production; Multi-Item; Dynamic Programming) Introduction and Literature Review Consider a system in which items of several types are transferred from a warehouse or a plant to a retailer through identical vehicles, or by identical freight wagons, each with a finite capacity. We assume that the transportation activity is outsourced to exogenous freight carriers so that the number of vehicles available in a period is unlimited. Any mixture of items may be loaded on a vehicle as long as the capacity restriction is not violated. The retailer is facing dynamic deterministic demands for several items, over a finite planning horizon. The carrier charges a fixed cost for each vehicle that is dispatched from the warehouse to the retailer, in addition to item-dependent variable shipping costs. Items that are stored at the retailer at the end of a period incur item-dependent inventory holding costs. Both the item-specific shipping costs and inventory holding costs are constant over time. The objective is to find a shipment schedule that minimizes the total cost, while satisfying demand on time. The same problem may arise in a production environment, in which production decisions of multiple items have to be made, using a certain resource. A fixed cost is associated with the production of a batch (or part of it) of items, where each batch is of limited quantity. A specific variant of either the transportation or the production setting that we refer to in the sequel occurs when there is a capacity restriction in each of the periods, i.e., the number of available vehicles or batches in each period is limited to one. In the case where multiple vehicles may be used in each period (each of limited capacity, and each incurs a fixed cost), we refer to the problem as the Multi-Item with Multiple Vehicles (MIMV) problem, which is the focus of this paper. The related problem, in which only one vehicle may be used, is referred to as the Multi-Item with a Single Vehicle (MISV) problem. Capacity restrictions exist in most realistic transportation or production systems, but are often ignored. One apparent reason for this phenomenon is the difficulty associated with the consideration of capacitated resources, together with non-stationary demand, even in relatively simple systems, see the literature review below. Another issue, which is not well addressed in the literature, is the consideration of transportation in multiple batches, and in particular when multiple items are involved. In this paper, we analyze the problem scenario described above, which includes all three issues, namely, transportation of several different items by multiple capacitated vehicles.