The document discusses (Q,R) inventory systems which generalize the economic order quantity (EOQ) model to allow for stochastic demand and a reorder point R. Key aspects include: 1) The system uses continuous review with fixed lead times, stochastic demand, ordering costs, holding costs, and shortage costs. 2) Decision variables are the order quantity Q and reorder point R. The total cost considers holding, ordering, and shortage costs. 3) Service level can be measured in two ways - type 1 is the probability of no shortage in the lead time, type 2 is the proportion of demands met from stock. 4) The optimal solution balances minimizing total costs while meeting a specified service level

1. Lot size reorder point systems (Q, R) system

3. Changes in Inventory Over Time for Continuous-Review (Q, R) System Fig. 5-5

6. Holding cost λ τ R- λτ Q + R- λτ Q + R - λτ R- λτ Q/2 + R - λτ

7. Penalty cost x

8. Expected number of shortages

9. Total cost function Holding cost Ordering cost Shortage cost

10. Necessary conditions for optimality

11. Optimal solution