A linear programming formulation for an inventory management decision problem with a service constraint

Inventory systems with uncertainty go hand in hand with the determination of a safety stock level. The decision on the safety stock level is based on a performance measure, for example the expected shortage per replenishment period or the probability of a stock-out per replenishment period. The performance measure assumes complete knowledge of the probability distribution during lead time, which might not be available. In case of incomplete information regarding the lead-time distribution of demand, no single figure for the safety stock can de determined in order to satisfy a performance measure. However, an optimisation model may be formulated in order to determine a safety stock level which guarantees the performance measure under the worst case of lead-time demand, of which the distribution is known in an incomplete way. It is shown that this optimisation problem can be formulated as a linear programming problem.