OPSM 301 Operations Management

1. Koç University OPSM 301 Operations Management Class 15: Inventory Management EOQ Model Zeynep Aksin zaksin@ku.edu.tr

2. Inventory • “The stock of any item or resource used in an organization” • “All the money that the system has invested in purchasing things it intends to sell”

3. The Material Flow Cycle

4. Types of Inventories • Inputs - Raw Materials • Processes - Work-in-Progress • Outputs - Finished Goods

5. Why do we need Inventory? • Variability (uncertainty) • Demand • Capacity availability • Materials and lead times • Processing times • Time • Delivery lead time, production lead time • Economies of Scale • Purchasing, production

6. Functions Provided by Inventories Purpose /Reason Type Cost Transportation Pipeline Transportation Costs Economies in Setups Cycle Stocks Setup/Order Costs Seasonality in Demand Seasonal Stock Smoothing Costs Uncertainty in Demand Safety Stock Shortage/Stock-out Costs Economies in Purchase Cycle Stocks Price Discounts Inflation and/or Price Fluctuations Speculative Stock Costs due to Price

7. Inventory Costs • Purchase Cost • Ordering Cost • Receiving and inspection • Transportation • Holding (Carrying) Cost • Cost of money • Insurance • Taxes • Shrinkage, spoilage, obsolescence • Stock-out (Shortage) Cost • Lost sales, customers etc. • Emergency shipment costs

8. warehouse retailer Economies of Scale:Inventory Management for a Retailer The South Face retail shop in the John Hancock Tower has observed a stable monthly demand for its line of Gore-Tex jackets on the order of 100 jackets per month. The retail shop incurs a fixed cost of $2,000 every time it places an order to the Berkeley warehouse for stock replenishment. The marginal cost of a jacket is $200, and South Face’s cost of capital is approximately 25%. What order size would you recommend for The South Face?

9. Parameters EOQ Model Ddemand rate (units per year) C unit production cost, not counting setup or inventory costs (dollars per unit) S fixed or setup cost to place an order (dollars) H holding cost (dollars per year); if the holding cost is consists entirely of interest on money tied up in inventory, then H = iC where i is an annual interest rate. Q the unknown size of the order or lot size

10. Order quantity = Q (maximum inventory level) Usage Rate AverageInventory (Q*/2) Inventory Level Minimum inventory 0 Time Inventory Usage Over Time

11. C O S T Total Cost (TC) Holding Cost Annual Cost of Items Ordering Cost QOPT Order Quantity (Q) Cost Minimization Goal

12. Annual Purchasing Cost Annual Ordering Cost Annual Holding Cost Total Annual Cost = + + Total Annual Cost • Using calculus, we can take the derivative of the total cost function and set the derivative (slope) equal to zero • We can also use economic intuition

13. Find most economical order quantity: Spreadsheet for The South Face

14. Deriving the EOQ

15. EOQ Model: if there is a lead time L Qopt # Units on hand ROP Time L L ROP = Reorder point L = Lead time (constant) Q = Economic order quantity

16. EOQ Example • Annual Demand = 1,000 units • Days per year considered in average daily demand = 250 • Cost to place an order = $10 • Holding cost per unit per year = $0.50 • Lead time = 7 days • Cost per unit = $15 Determine the economic order quantity and the reorder point

17. 2DS H Q* = 2(1,000)(10) 0.50 Q* = = 40,000 = 200 units An EOQ Example Determine optimal number of needles to order D = 1,000 units S = $10 per order H = $.50 per unit per year

18. Expected number of orders Demand Order quantity D Q* = N = = 1,000 200 N = = 5 orders per year An EOQ Example Determine optimal number of needles to order D = 1,000 units Q* = 200 units S = $10 per order H = $.50 per unit per year

19. Number of working days per year N Expected time between orders = T = 250 5 T = = 50 days between orders An EOQ Example Determine optimal number of needles to order D = 1,000 units Q* = 200 units S = $10 per order N = 5 orders per year H = $.50 per unit per year

20. Q 2 D Q 200 2 TC = S + H 1,000 200 TC = ($10) + ($.50) An EOQ Example Determine optimal number of needles to order D = 1,000 units Q* = 200 units S = $10 per order N = 5 orders per year H = $.50 per unit per year T = 50 days Total annual cost = Setup cost + Holding cost TC = (5)($10) + (100)($.50) = $50 + $50 = $100

21. Q* Slope = units/day = d Inventory level (units) ROP (units) Time (days) Lead time = L Reorder Point Curve Figure 12.5

22. D Number of working days in a year d = Reorder Point Example Demand = 8,000 iPods per year 250 working day year Lead time for orders is 3 working days = 8,000/250 = 32 units ROP = d x L = 32 units per day x 3 days = 96 units

23. Economic Order Quantity (EOQ) Model • Economic Order Quantity (EOQ) Model • Robust, widely used • Insensitive to errors in estimating parameters (40-20-2 Rule): • 40% error in one of the parameters • 20% error in Q • < 2% of total cost penalty

24. 1,500 units Q 2 D Q 200 2 TC = S + H 1,500 200 TC = ($10) + ($.50) = $75 + $50 = $125 An EOQ Example Management underestimated demand by 50% D = 1,000 units Q* = 200 units S = $10 per order N = 5 orders per year H = $.50 per unit per year T = 50 days Total annual cost increases by only 25%

25. 1,500 units 244.9 2 TC = ($10) + ($.50) Q 2 D Q TC = S + H 1,500 244.9 An EOQ Example Actual EOQ for new demand is 244.9 units D = 1,000 units Q* = 244.9 units S = $10 per order H = $.50 per unit per year Only 2% less than the total cost of $125 when the order quantity was 200 TC = $61.24 + $61.24 = $122.48