OPSM 301 Operations Management Class 15: Inventory Management EOQ Model Koç University Zeynep Aksin

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”

The Material Flow Cycle

Types of Inventories  Inputs - Raw Materials  Processes - Work-in-Progress  Outputs - Finished Goods

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

Functions Provided by Inventories Purpose /ReasonType 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

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

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? retailer warehouse

Parameters EOQ Model D demand 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) Hholding 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. Qthe unknown size of the order or lot size

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

Cost Minimization Goal Ordering Cost Holding Cost Q OPT Order Quantity (Q) COSTCOST Annual Cost of Items Total Cost (TC)

Total Annual Cost Total Annual Cost = Annual Purchasing Cost Annual Ordering Cost Annual Holding 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

Find most economical order quantity: Spreadsheet for The South Face

Deriving the EOQ

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

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

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

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 = N = = Expected number of orders Demand Order quantity DQ* N = = 5 orders per year 1,000200

An EOQ Example Determine optimal number of needles to order D = 1,000 unitsQ*= 200 units S = $10 per orderN= 5 orders per year H = $.50 per unit per year = T = Expected time between orders Number of working days per year N T = = 50 days between orders 2505

An EOQ Example Determine optimal number of needles to order D = 1,000 unitsQ*= 200 units S = $10 per orderN= 5 orders per year H = $.50 per unit per yearT= 50 days Total annual cost = Setup cost + Holding cost TC = S + H DQQ2 TC = ($10) + ($.50) 1, TC = (5)($10) + (100)($.50) = $50 + $50 = $100

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

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

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

An EOQ Example Management underestimated demand by 50% D = 1,000 units Q*= 200 units S = $10 per orderN= 5 orders per year H = $.50 per unit per yearT= 50 days TC = S + H DQQ2 TC = ($10) + ($.50) = $75 + $50 = $125 1, ,500 units Total annual cost increases by only 25%

An EOQ Example Actual EOQ for new demand is units D = 1,000 units Q*= units S = $10 per order H = $.50 per unit per year TC = S + H DQQ2 TC = ($10) + ($.50) 1, ,500 units TC = $ $61.24 = $122.48