Inventory Management

While most modules in an Advanced Planning System are based on the assumption that all planning data are deterministic, in reality usually random events occur. In order to buffer the production plan against these random events (such as random demands, machine breakdowns, late deliveries), often safety stock is used. The determination of the required level of safety stock in the dynamic planning environment of an Advanced Planning System is a non-trivial problem. For a single inventory location that serves a number of downstream nodes in the supply chain, several stochastic inventory policies can be applied.

The size of the safety directly depends on the type of the inventory policy that is in effect. The underlying conception for a single-stage inventory policy is as follows. An inventory node is supplied from a "source" which fulfills orders for the considered product after a certain replenishment lead time. If the source is a production segment or rather production stage of the same company, then the replenishment lead time is a function of the flow time of a production order and depends on numerous factors, the utilization of the production stage being one of them. If the source is another inventory node of the company, then the order is a demand observed by this inventory node and the replenishment lead time depends on the inventory available on hand as well as on the time required for material handling and transportation processes. If the source is an external supplier, then the replenishment lead time is equal to the customer order waiting time provided by the supplier, plus an additional time required for material handling and transportation. In all mentioned cases it is clear that the replenishment lead time may be subject to random variations.

For the correct calculation of the parameters of an inventory poliy it is crucial to model the time axis of the inventory process as precise as possible. Basically the time axis can be modeled as continous or as discrete. In practice, the time axis of logistical processes is discrete. The MRP (material requirements planning) calculations that are standard in all ERP/MRP/AP systems are based on a discrete times axis. By contrast, as far as inventory policies are supported, most software systems model the time axis as continuous. This may lead to significant planning errors with the result that the service levels are goals are missed.

Inventory policies differ in two aspects, namely the mechanism used to trigger replenishment orders and the decision rule that specifies the determination of the order size. The specific inventory policies are defined through the combination of the decision variables s (reorder point), r (review interval, order cycle), q (order quantity) and S (order level) as follows:

  • (s,q) policy,
  • (r,S) policy,
  • (s,S) policy.

(s,q) policy


Under the (s,q) policy, the point in time at which replenishment orders are triggered, depends on the size of the reorder point s, whereas the order quantity q is constant over time. In the ideal (textbook) form of the (s,q) policy, the inventory position is continuously monitored. The inventory position is the sum of the inventory on hand plus the inventory on order minus the outstanding backorders (backlog). The inventory management system (or the inventory manager) acts according to the following decision rule: If at a review instant the inventory position has reached the reorder point s (from above), then launch a replenishment order of size q.


In reality the inventory is not monitored continuously. In contrast, the replenishment decisions are made in discrete time intervals, usually at the end of a day. In addition, often demand sizes are greater than one unit. Under these conditions, the analysis of the (s,q) policy as presented in many textbooks in false, as the so-called undershoot is neglected. In the above figure, the undershoot is the difference between s and the inventory position at the moment immediately before a new replenishment order is released. Negleting the undershoot usually results in significant over-estimation of the service level (under-estimation of the required safety stock).

(r,S) policy


If an (r,S) inventory policy is in effect, the points in time at which replenishment orders are released are determined through the review interval r. The inventory management system proceeds according to the following decision rule: In constant intervals of r periods launch a replenishment order that raises the inventory position to the target order level S. Obviously, the (r,S) policy is an inventory policy with periodic review. The order size at a time of a review depends on the demands and the development of the inventory observed in the preceding periods. If r=1, then this poliy is called base-stock policy.

(s,S) policy


Under an (s,S) inventory policy, the points in time when an order is triggered are determined policy, i. e. through the reorder point s. However, the order quantity is now, similar to the (r,S) policy, a function of the inventory development over time. In the literature this policy is sometimes characterized with the help of a third parameter which specifies the length of the review interval r. In this notation the policy is called (r,s,S) policy. In the case of r=0, continuous review is in effect. If demands arrive unit-sized, then the (r=0,s,S) policy is identical to the (s,q) policy with continuous review.

For the determination of the optimum safety stock under conditions of uncertainty the demand during the risk period plays a central role.

The risk period is composed of

  • the review period and

  • the replenishment lead time.

Stochastic demand occurs within this time span that usually comprises several periods. In order to compute the parameters of an inventory policy, we must know the probability distribution of the demand during the risk period.