A vendor managed inventory model using continuous approximations for route length estimates and Markov chain modeling for cost estimates

- This paper suggests a modification to the joint replenishment problem (JRP).
- Including routing decisions compounds the complexity of JRP models.
- Our Markov chain model avoids this complexity by using route lengths estimates.
- We develop a distance estimate for retailers located on a linear structure.
- Our model is effective if demand is infrequent or the costs per km are low.

We consider a vendor who supplies goods to a set of geographically dispersed retailers and can monitor and control the inventory levels at the retailers. Such an arrangement is often called vendor managed inventory (VMI). The decisions in this set-up are the inventory levels at the warehouse and at the retailers and the routing along the retailers. Normally, the inventory levels at the vendor's warehouse and at the retailers are established by modeling the problem as a joint replenishment problem (JRP). Such a model ignores the differences in distances, number of retailers visited, and vehicle loads that may occur, in particular when these retailers are served on joint delivery trips. Some approaches that take routing and inventory decisions into account jointly, but these are so complex that only relatively small instances can be solved.This paper develops a more detailed specification of the transport costs than other JRP models. In order to ensure that the complexity of the problem does not become overwhelming, we assume that the retailers are identical and uniformly distributed across an area, which can either be a two-dimensional area or a one-dimensional line structure. First of all, we decide to construct zones of retailers to be replenished jointly. The expected travel distances across a given number of retailers are estimated analytically using results from the field of continuous approximation for two-dimensional areas and using our own approximation for one-dimensional ones. Using a Markov chain approach we develop a model to minimize transport and inventory costs simultaneously. Our experiments show that our detailed specification gives more accurate set-up costs (and thereby different control policies) than regular JRP approaches when demand is infrequent and the transportation costs are low.