1. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.1 Table of Contents CD Chapter 18 (Inventory Management with Known Demand) A Case Study—The Atlantic Coast Tire Corp. (ACT) Problem (Section 18.1)18.2–18.4 Cost Components of Inventory Models (Sec.18.2)18.5–18.6 The Basic Economic Order Quantity (EOQ) Model (Section 18.3)18.7–18.11 The Optimal Inventory Policy for the Basic EOQ Model (Section 18.4)18.12–18.16 The EOQ Model with Planned Shortages (Section 18.5)18.17–18.22 The EOQ Model with Quantity Discounts (Section 18.6)18.23–18.29 The EOQ Model with Gradual Replenishment (Section 18.7)18.30–18.35

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.2 The Atlantic Coast Tire Corp. (ACT) Problem The Atlantic Coast Tire Corporation (ACT) is the east coast distributor of Eversafe tires, supplying 1500 retail stores and service stations When the inventory level of a particular size tire gets low, ACT places a large order with Eversafe to replenish the inventory. Shipments arrive by truck 9 working days after the placement of the order. Data for the 185/70 R13 size of Eversafe tires: –These tires sell at a regular rate of about 500 per month. –Policy has been to order 1,000 tires as needed every couple months. –The order is placed just in time to have the delivery arrive as inventory runs out.

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.3 Pattern of Inventory Levels for the 185/70 R13 Tire

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.4 Cost Components of Inventory for 185/70 R13 Tires The purchase price from Eversafe is $20 per tire. The administrative cost for placing an order is $115, due to the following: –A purchase order is initiated and processed. –The shipment must be received and placed into storage. –The computerized information processing system must be updated. The annual cost of holding tires in inventory is $4.20 per tire. This includes: –The cost of capital tied up in inventory (estimated at 15% of cost per annum) –The cost of leasing warehouse space. –The cost of insurance against loss by fire, theft, vandalism, etc. –The cost of personnel who oversee and protect the inventory. –Taxes that are based on the value of inventory. The annual cost of being out of stock is $7.50 per tire short, based on these consequences: –Customer dissatisfaction –Potential price drop to placate a customer due to late deliveries. –Delayed revenue –The cost of additional record keeping required for out-of-stock tires.

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.5 Cost Components of Inventory Models Acquisition Cost. The direct cost (c) of replenishing inventory, whether through purchasing or manufacturing of the product. –For the ACT Example: c = $20 per tire. Setup Cost. When purchasing the product, this cost consists of various administrative costs associated with initiating and processing the purchase order, receiving the shipment, and processing the payment. When a manufacturer is replenishing its inventory by manufacturing more of the product, the setup cost consists of the cost of setting up the manufacturing process for another production run. –For the ACT Example: K = $115 per order. Holding Cost. The cost of holding units in inventory, including the cost of capital tied up in invenoty as well as the cost of space, insurance, protection, and taxes attributed to storage. –For the ACT Example, h = $4.20 per unit per year. Shortage Cost. The cost incurred when there is a need for product, but none available in inventory. Possible consequences include lost sales, lost future sales, etc. –For the ACT Example, p = $7.50 per unit short

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.6 Combining the Cost Components Annual acquisition cost = c times number of units added to inventory per year. Annual Setup cost = K times number of setups per year. Annual holding cost = h times average number of units in inventory. Annual shortage cost = p times average number of units short throughout year. TC = total inventory cost per year = sum of above four annual costs. TVC = total variable inventory cost per year = sum of the variable annual costs.

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.7 The Basic Economic Order Quantity (EOQ) Model A constant demand rate. The order quantity to replenish inventory arrives all at once just when desired. Planned shortages are not allowed. D = annual demand rate. Q = order quantity (the decision to be made). Assumptions:

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.8 Reorder Point for 185/70 R13 Tires Reorder point = (24 tires/day) (9 days) = 216 tires.

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.9 Pattern of Inventory Levels for the EOQ Model

10. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.10 Total Variable Cost for the 185/70 R13 Tire Current policy: Q = 1000 tires. Number of setups (order placements) per year is D/Q = 6000/1000 = 6. Average inventory = Q/2 = 500. TVC= annual setup cost + annual holding cost = 6K + 500h = 6 ($115) + 500 ($4.20) = $2,790

11. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.11 Optimal Solution with Different Unit Profits Unit Profit DoorsWindowsObjective FunctionOptimal Solution $400 Profit = 400D + 400W(2, 6) $500$300Profit = 500D + 300W(4, 3) $300–$100Profit = 300D – 100W(4, 0) –$100$500Profit = –100D + 500W(0, 6) –$100 Profit = –100D – 100W(0, 0)

12. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.12 Spreadsheet Analysis with Q = 1000

13. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.13 Data Table (Cost vs. Order Quantity)

14. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.14 Using Solver to Minimize Cost

15. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.15 The Square Root Formula Number of setups per year = D/Q. Average inventory level = Q/2. Total variable cost (TVC) = The value of Q that minimizes the TVC is For ACT’s problem, Which gives TVC = $115 (6,000 / 573) + $4.20 (573 / 2) = $2407, a 14% savings over current policy.

16. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.16 Sensitivity Analysis for ACT

17. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.17 The EOQ Model with Planned Shortages A constant demand rate. The order quantity to replenish inventory arrives all at once just when desired. Planned shortages are allowed. When a shortage occurs, the affected customers will wait for the product to become available again. Their backorders are filled immediately when the order quantity arrives. Q = order quantity. S = maximum shortage (units backordered). Assumptions: Decision Variables:

18. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.18 Pattern of Inventory Level with Planned Shortages

19. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.19 The Objective of the Model: Minimize TVC Annual Setup cost = Annual holding cost = h times (average inventory level when positive) times (fraction of time inventory level is positive) Annual shortage cost = p times (average shortage level when a shortage occurs) times (fraction of time shortage is occurring) Combining these gives

20. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.20 The Optimal Inventory Policy where D = annual demand rate, K = setup cost, h = unit holding cost, p = unit shortage cost Maximum inventory level = Q* – S*

21. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.21 Application to the ACT Case Study

22. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.22 Comparison of Basic EOQ and EOQ with Planned Shortages QuantityBasic EOQ Model EOQ Model with Planned Shortages Order quantity573716 Maximum shortage0257 Maximum inventory level 573459 Reorder point216–41 Annual setup cost$1,204$964 Annual holding cost$1,204$618 Annual shortage cost0$346 Total variable cost$2,407$1,928

23. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.23 The Quantity Discounts Being Offered to ACT Discount Quantity Order QuantityDiscountUnit Cost 10 to 7490$20.00 2750 to 1,9991%$19.80 32,000 or more2%$19.60

24. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.24 Cost Analysis TVC = where c = unit acquisition cost (as given in Table 18.2) D = annual demand rate = 6,000 K = setup cost = $115 Q = order quantity (the decision variable) h = unit holding cost I = inventory holding cost rate = 0.21. h = Ic = 0.21c

25. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.25 The Unit Holding Cost for ACT Discount Category Price (c) Unit Holding Cost h = Ic = 0.21c 1$20.000.21($20) = $4.20 2$19.800.21($19.80) = $4.158 3$19.600.21($19.60) = $4.116

26. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.26 Total Variable Cost vs. Order Quantity

27. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.27 Cost Comparison for Discount Categories Annual Costs Discount Category Best Order Quantity Acquisition 6,000c Setups $115(6000/Q) Holding h(Q/2) Total (TVC) = Sum 1Q = 573$120,000$1,204 $122,407 2Q = 750$118,800$920$1,559$121,279 3Q = 2,000$117,600$345$4,116$122,061

28. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.28 Spreadsheet Analysis with Quantity Discounts

29. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.29 Conclusion of the ACT Cast Study A substantial reduction in the order quantity (from the current 1,000 down to 750) would provide a significant reduction in average inventory level and holding cost. The threat to reduce the order quantity even further (as suggested by the basic EOQ model) has prodded Eversafe to provide quantity discounts to ACT. The resulting reduction in the total annual inventory cost from that for the current policy would exceed $1,5000 for just this one size of tire. Extending this approach to the other sizes should greatly multiply this savings.

30. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.30 The EOQ Model with Gradual Replenishment A constant demand rate. A production run is scheduled to begin each time the inventory level drops to 0, and this production replenishes inventory at a constant rate throughout the duration of the run. Planned shortages are not allowed. Q = production lot size. Assumptions: Decision Variable:

31. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.31 Pattern of Inventory Level with Gradual Replenishment

32. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.32 The SOCA Problem SOCA, a television manufacturing company, produces its own speakers for assembly into television sets. Current policy for managing SOCA’s inventory of speakers: –Daily demand rate = 1,000 speakers per day. –Daily production rate = 3,000 speakers per day (when producing). –The production facilities get set up to start a production run each time the inventory level is scheduled to drop to 0. –Each production run produces 30,000 speakers over a period of 10 working days, so another 20 working days elapse before the next production run is needed.

33. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.33 TVC for the Current Policy Maximum inventory level= production lot size minus demand during production run = 30,000 – (10 days)(1,000 speakers / day) = 20,000 speakers Average inventory level = (1/2) (maximum inventory level) = 10,000 speakers Annual setup cost Annual holding cost TVC = annual setup cost + annual holding cost = $136,000.

34. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.34 The Optimal Inventory Policy where D = annual demand rate, R = annual production rate (if produce continuously) K = setup cost, h = unit holding cost, For SOCA,

35. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.35 Spreadsheet Analysis with Gradual Replenishment