1. Inventory Management

12. Demand rate Time Lead time Lead time Order placed Order placed Order receipt Order receipt Inventory Level Reorder point, R Order quantity, Q 0

13. C o - cost of placing order D - annual demand C c - annual per-unit carrying cost Q - order quantity Annual ordering cost = C o D Q Annual carrying cost = C c Q 2 Total cost = + C o D Q C c Q 2

14. TC = + C o D Q C c Q 2 = + C o D Q 2 C c 2  TC  Q 0 = + C 0 D Q 2 C c 2 Q opt = 2 C o D C c Deriving Q opt Proving equality of costs at optimal point = C o D Q C c Q 2 Q 2 = 2 C o D C c Q opt = 2 C o D C c

15. Order Quantity, Q Annual cost ($) Total Cost Carrying Cost = C c Q 2 Slope = 0 Minimum total cost Optimal order Q opt Ordering Cost = C o D Q

17. Q (1- d/p ) Inventory level (1- d/p ) Q 2 Time 0 Order receipt period Begin order receipt End order receipt Maximum inventory level Average inventory level

18. p = production rate d = demand rate Maximum inventory level = Q - d = Q 1 - Q p d p Average inventory level = 1 - Q 2 d p TC = + 1 - d p C o D Q C c Q 2 Q opt = 2 C o D C c 1 - d p

19. TC = + + PD C o D Q C c Q 2 where P = per unit price of the item D = annual demand

20. Q opt Carrying cost Ordering cost Inventory cost ($) Q ( d 1 ) = 100 Q ( d 2 ) = 200 TC ( d 2 = $6 ) TC ( d 1 = $8 ) TC = ($10 ) ORDER SIZE PRICE 0 - 99 $10 100 – 199 8 ( d 1 ) 200+ 6 ( d 2 )

21. R = dL

22. Reorder point, R Q LT Time LT Inventory level 0

23. Reorder point, R Q LT Time LT Inventory level 0 Safety Stock

24. R = dL + z  d L where d = average daily demand L = lead time  d = the standard deviation of daily demand z = number of standard deviations corresponding to the service level probability z  d L = safety stock

25. Probability of meeting demand during lead time = service level Probability of a stockout R Safety stock d L Demand z  d L

26. The carpet store wants a reorder point with a 95% service level and a 5% stockout probability For a 95% service level, z = 1.65 d = 30 yards per day L = 10 days  d = 5 yards per day R = dL + z  d L = 30(10) + (1.65)(5)( 10) = 326.1 yards Safety stock = z  d L = (1.65)(5)( 10) = 26.1 yards

27. Q = d ( t b + L ) + z  d t b + L - I where d = average demand rate t b = the fixed time between orders L = lead time  d = standard deviation of demand z  d t b + L = safety stock I = inventory level