The effect of discreteness in vendor-buyer relationships

To facilitate integrated vendor-buyer modeling, it has been assumed that the ratio between the lengths of their inventory cycles is integral. This integer-ratio policy has been proven to be optimal for a vendor who is instantly replenished. Unfortunately, the proof is not extendable to a manufacturer who produces at a finite rate. Thus, this case is considered here. First, a proper analytical methodology is developed, which enables us to relax this assumption and characterize the effect of the cycles' ratio on inventory levels. This characteristic fully reveals the effect of discreteness; while the average levels of both the joint inventory and the vendor's inventory follow the general pattern and grow as a function of the production lot-size, both grow in leaps. The average level of both inventories jumps up when the production lot-size reaches an integral multiple of the delivery quantity and decreases between these points. It then follows that once the lower bound of a leap is passed, it is better to round the vendor lot-size up to the next multiple of the delivery quantity. Alternatively, a fractional ratio can be applied as an average of integer ratios by using variable lot sizes. Evidently, this solution reduces inventories but not enough; the associated cost is shown to be higher than the cost associated with either the next or the previous integer ratio.