1. ISEN 315 Spring 2011 Dr. Gary Gaukler

2. Lot Size Reorder Point Systems Assumptions –Inventory levels are reviewed continuously (the level of on-hand inventory is known at all times) –Demand is random but the mean and variance of demand are constant. (stationary demand) –There is a positive leadtime, τ. This is the time that elapses from the time an order is placed until it arrives. –The costs are: Set-up each time an order is placed at $K per order Unit order cost at $c for each unit ordered Holding at $h per unit held per unit time ( i. e., per year) Penalty cost of $p per unit of unsatisfied demand

3. The Inventory Control Policy Keep track of inventory position (IP) IP = net inventory + on order When IP reaches R, place order of size Q

4. Inventory Levels

5. Solution Procedure The optimal solution procedure requires iterating between the two equations for Q and R until convergence occurs (which is generally quite fast). A cost effective approximation is to set Q=EOQ and find R from the second equation. In this class, we will use the approximation.

6. Example Selling mustard jars Jars cost $10, replenishment lead time 6 months Holding cost 20% per year Loss-of-goodwill cost $25 per jar Order setup $50 Lead time demand N(100, 25)

7. Example

9. Service Levels in (Q,R) Systems In many circumstances, the penalty cost, p, is difficult to estimate Common business practice is to set inventory levels to meet a specified service objective instead Service objectives: Type 1 and Type 2

10. Service Levels in (Q,R) Systems Type 1 service: Choose R so that the probability of not stocking out in the lead time is equal to a specified value. Type 2 service. Choose both Q and R so that the proportion of demands satisfied from stock equals a specified value.

11. Comparison Order Cycle Demand Stock-Outs 11800 2 750 3235 45 41400 51800 6200 10 71500 8 900 91600 10 400 For a type 1 service objective there are two cycles out of ten in which a stockout occurs, so the type 1 service level is 80%. For type 2 service, there are a total of 1,450 units demand and 55 stockouts (which means that 1,395 demand are satisfied). This translates to a 96% fill rate.

12. Type I Service Level Determine R from F(R) = a Q=EOQ E.g., if a = 0.95: Fill all demands in 95% of the order cycles

13. Type II Service Level a.k.a. Fill rate Fraction of all demands filled without backordering Fill rate = 1 – unfilled rate

14. Type II Service Level

15. Summary of Computations For type 1 service, if the desired service level is α, then one finds R from F(R)= α and Q=EOQ. For Type 2 service, set Q=EOQ and find R to satisfy n(R) = (1-β)Q.

16. Imputed (implied) Shortage Cost Why did we want to use service levels instead of shortage costs? Each choice of service level implies a shortage cost!

17. Imputed (implied) Shortage Cost Calculate Q, R using service level formulas Then, 1 - F(R) = Qh / (pλ)

18. Imputed (implied) Shortage Cost Imputed shortage cost vs. service level:

19. Exchange Curve Safety stock vs. stockouts: