Inventory Management

1. Inventory Management Operations Management Dr. Ron Tibben-Lembke

2. Purposes of Inventory • Meet anticipated demand • Demand variability • Supply variability • Decouple production & distribution • permits constant production quantities • Take advantage of quantity discounts • Hedge against price increases • Protect against shortages

3. 2006 13.81 1857 24.0% 446 801 58 1305 9.9 2007

5. Source: CSCMP, Bureau of Economic Analysis

6. Two Questions Two main Inventory Questions: • How much to buy? • When is it time to buy? Also: Which products to buy? From whom?

7. Types of Inventory • Raw Materials • Subcomponents • Work in progress (WIP) • Finished products • Defectives • Returns

8. Inventory Costs What costs do we experience because we carry inventory?

9. Inventory Costs Costs associated with inventory: • Cost of the products • Cost of ordering • Cost of hanging onto it • Cost of having too much / disposal • Cost of not having enough (shortage)

10. Shrinkage Costs • How much is stolen? • 2% for discount, dept. stores, hardware, convenience, sporting goods • 3% for toys & hobbies • 1.5% for all else • Where does the missing stuff go? • Employees: 44.5% • Shoplifters: 32.7% • Administrative / paperwork error: 17.5% • Vendor fraud: 5.1%

11. Inventory Holding Costs Category% of Value Housing (building) cost 4% Material handling 3% Labor cost 3% Opportunity/investment 9% Pilferage/scrap/obsolescence 2% Total Holding Cost 21%

12. Inventory Models • Fixed order quantity models • How much always same, when changes • Economic order quantity • Production order quantity • Quantity discount • Fixed order period models • How much changes, when always same

13. Economic Order Quantity Assumptions • Demand rate is known and constant • No order lead time • Shortages are not allowed • Costs: • S - setup cost per order • H - holding cost per unit time

14. EOQ Inventory Level Q* Optimal Order Quantity Decrease Due to Constant Demand Time

15. EOQ Inventory Level Instantaneous Receipt of Optimal Order Quantity Q* Optimal Order Quantity Time

16. EOQ Inventory Level Q* Reorder Point (ROP) Time Lead Time

17. EOQ Inventory Level Q* Average Inventory Q/2 Reorder Point (ROP) Time Lead Time

18. Total Costs • Average Inventory = Q/2 • Annual Holding costs = H * Q/2 • # Orders per year = D / Q • Annual Ordering Costs = S * D/Q • Cost of Goods = D * C • Annual Total Costs = Holding + Ordering + CoG

19. How Much to Order? Annual Cost Holding Cost = H * Q/2 Order Quantity

20. How Much to Order? Annual Cost Ordering Cost = S * D/Q Holding Cost = H * Q/2 Order Quantity

21. How Much to Order? Total Cost = Holding + Ordering Annual Cost Order Quantity

22. How Much to Order? Total Cost = Holding + Ordering Annual Cost Optimal Q Order Quantity

23. Optimal Quantity Total Costs = Take derivative with respect to Q = Set equal to zero Solve for Q:

24. d d Adding Lead Time • Use same order size • Order before inventory depleted • R = * L where: • = average demand rate (per day) • L = lead time (in days) • both in same time period (wks, months, etc.)

25. A Question: • If the EOQ is based on so many horrible assumptions that are never really true, why is it the most commonly used ordering policy? • Profit function is very shallow • Even if conditions don’t hold perfectly, profits are close to optimal • Estimated parameters will not throw you off very far

26. Quantity Discounts • How does this all change if price changes depending on order size? • Holding cost as function of cost: • H = I * C • Explicitly consider price:

27. Discount Example D = 10,000 S = $20 I = 20% Price Quantity EOQ c = 5.00 Q < 500 633 4.50 501-999 666 3.90 Q >= 1000 716

28. Discount Pricing Total Cost Price 1 Price 2 Price 3 X 633 X 666 X 716 Order Size 500 1,000

29. Discount Pricing Total Cost Price 1 Price 2 Price 3 X 633 X 666 X 716 Order Size 500 1,000

30. Discount Example Order 666 at a time: Hold 666/2 * 4.50 * 0.2= $299.70 Order 10,000/666 * 20 = $300.00 Mat’l 10,000*4.50 = $45,000.00 45,599.70 Order 1,000 at a time: Hold 1,000/2 * 3.90 * 0.2= $390.00 Order 10,000/1,000 * 20 = $200.00 Mat’l 10,000*3.90 = $39,000.00 39,590.00

31. Discount Model 1. Compute EOQ for next cheapest price 2. Is EOQ feasible? (is EOQ in range?) If EOQ is too small, use lowest possible Q to get price. 3. Compute total cost for this quantity • Repeat until EOQ is feasible or too big. • Select quantity/price with lowest total cost.

32. Inventory Management-- Random Demand

33. Random Demand • Don’t know how many we will sell • Sales will differ by period • Average always remains the same • Standard deviation remains constant

34. Impact of Random Demand How would our policies change? • How would our order quantity change? • How would our reorder point change?

35. Mac’s Decision • How many papers to buy? • Average = 90, st dev = 10 • Cost = 0.20, Sales Price = 0.50 • Salvage = 0.00 • Cost of overestimating Demand, CO • CO= 0.20 - 0.00 = 0.20 • Cost of Underestimating Demand, CU • CU = 0.50 - 0.20 = 0.30

36. Optimal Policy G(x) = Probability demand <= x Optimal quantity: Mac: G(x) = 0.3 / (0.2 + 0.3) = 0.6 From standard normal table, z = 0.253 =Normsinv(0.6) = 0.253 Q* = avg + zs = 90+ 2.53*10 = 90 +2.53 = 93

37. Optimal Policy • If units are discrete, when in doubt, round up • If u units are on hand, order Q - u units • Model is called “newsboy problem,” newspaper purchasing decision • By time realize sales are good, no time to order more • By time realize sales are bad, too late, you’re stuck • Similar to the problem of # of Earth Day shirts to make, lbs. of Valentine’s candy to buy, green beer, Christmas trees, toys for Christmas, etc., etc.

38. Random Demand – Fixed Order Quantity • If we want to satisfy all of the demand 95% of the time, how many standard deviations above the mean should the inventory level be?

39. Safety stock & Safety stock = zsL Therefore, z = sL Probabilistic Models Safety stock = x m From statistics, From normal table z.95 = 1.65 Safety stock = zsL= 1.65*10 = 16.5 R = m + Safety Stock =350+16.5 = 366.5 ≈ 367

40. Random Example • What should our reorder point be? • demand over the lead time is 50 units, • with standard deviation of 20 • want to satisfy all demand 90% of the time • (i.e., 90% chance we do not run out) • To satisfy 90% of the demand, z = 1.28 • Safety stock = zσL= 1.28 * 20 = 25.6 • R = 50 + 25.6 = 75.6

41. St Dev Over Lead Time • What if we only know the average daily demand, and the standard deviation of daily demand? • Lead time = 4 days, • daily demand = 10, • standard deviation = 5, • What should our reorder point be, if z = 3?

42. St Dev Over LT • If the average each day is 10, and the lead time is 4 days, then the average demand over the lead time must be 40. • What is the standard deviation of demand over the lead time? • Std. Dev. ≠ 5 * 4

43. St Dev Over Lead Time • Standard deviation of demand = • R = 40 + 3 * 10 = 70

44. Service Level Criteria • Type I: specify probability that you do not run out during the lead time • Probability that 100% of customers go home happy • Type II: proportion of demands met from stock • Percentage that go home happy, on average • Fill Rate: easier to observe, is commonly used • G(z)= expected value of shortage, given z. Not frequently listed in tables

45. Two Types of Service CycleDemand Stock-Outs 1 180 0 2 75 0 3 235 45 4 140 0 5 180 0 6 200 10 7 150 0 8 90 0 9 160 0 10 40 0 Sum 1,450 55 Type I: 8 of 10 periods 80% service Type II: 1,395 / 1,450 = 96%

46. Fixed-Time period models

47. Fixed-Time Period Model • Every T periods, we look at inventory on hand and place an order • Lead time still is L. • Order quantity will be different, depending on demand

48. Fixed-Time Period Model: When to Order? Inventory Level Target maximum Time Period

49. Fixed-Time Period Model: : When to Order? Inventory Level Target maximum Time Period Period

50. Fixed-Time Period Model:When to Order? Inventory Level Target maximum Time Period Period