The Order Up-To Inventory Model

1 1 The Order Up-To Inventory Model Inventory Control Best Order Up To Level Best Service Level Impact of Lead Time11. Medtronic s InSync Pacemaker Model 7272. Implanted in a patient after a cardiac surgery. One distribution center (DC) in Mounds View, Minnesota. About 500 sales territories. Majority of FGI is held by sales representatives. Consider Susan Magnotto s territory in Madison, and Inventory at the DCAvg. monthly demand = 349 Std. Dev. of monthly demand = weekly demand = 349 = deviation of weekly demand = (Assume weeks per month and independent weekly demands.) 0100200300400500600700 JanFebMarAprMayJunJulAugSepOctNovDecMont hUnitsMonthly implants (columns) and end of month Inventory (line)3 Demand and Inventory in Susan s TerritoryTotal annual demand = 75 Average daily demand = units (75/260), assuming 5 days per demand distribution works better for slow moving items0246810121416 JanFebMarAprMayJunJulAugSepOctNovDecMont hUnitsMonthly implants (columns) and end of month Inventory (line)43 Medtronic s Inventory Problem Patients and surgeons do not tolerate backorders.

2 The pacemaker is small and has a long shelf life. Sales incentive system. Each representative is given a par level which is set quarterly based on previous sales and anticipated : Because the gross margins are high, Medtronic wants an Inventory control policy to minimize Inventory investment while maintaining a very high fill rate. 5 Review: Reasons to Hold Inventory Pipeline Inventory Seasonal Inventory Cycle Inventory Decoupling Inventory /Buffers Safety Inventory64 Review: Reasons to Hold Less Inventory7 Inventory might become obsolete. Inventory might perish. Inventory might disappear. Inventory requires storage space and other overhead cost.

3 Opportunity : Inventory Costs Holding or Carrying costOverestimate the demandstorage cost: facility, handlingrisk cost: depreciation, pilferage, insuranceopportunity cost Ordering or Setup costcost placing an Order or changing machine setups Shortage costs or Lost SalesUnderestimate the demandcosts of canceling an Order or penaltyAnnual cost 20% to 40% of the Inventory s worth5 Review: Inventory Performance9 Throughput rate = average daily sales Throughput time = days of supply avg. Inventory value= avg. daily sales avg. throughput time Days of supply =average daily salesaverage Inventory value_____OrderReceiptOn ShelfSalesReview: Inventory Performance10 Monthly Inventory turn = Service level = in stock probability before the replenishment Order arrives Fill rate =Cost of Goods Sold in one month_____average Inventory valuenumber of demandsnumber of sales_____611 Review: Multi-Period Inventory ModelsQ Model : fixed Order quantityRis the reorder point and is based on lead time Land the forecast.

4 Place a new Order whenever the Inventory level drops to : P Model : fixed time period Tis the review period. Sis the target Inventory level determined by the place an Order to bring the Inventory level up to S. 7 amount of Inventory carried in addition to the expected demand, in Order to avoid shortages when demand increases depends on service level, demand variability, Order lead time service level depends on Holding cost Shortage cost13 Review: Safety Stocksafety stockService level=probability of no shortage=P (demand Inventory )=P(demand E(D)+safety stock)14 Review: Q Models with Safety Stockd =daily demand d=std dev.

5 Of daily demand dLzLd Service level or probability of no shortage =95% (99%) z= ( )Timespan=Lead time L(in days)R=expected demand during L + safety stock815 Review: P Models with Safety StockEx = length of review period + lead time = T+ L Order Quantity = target Inventory Inventory positiondLTzLTd )(Target Inventory = expected demand + safety stock2. The Order Up-To Model (P models) Time is divided into periods of equal length, , one week. During a period the following sequence of events occurs: A replenishment Order can be submitted. A previous Order is received. (lead times = l) Random demand 1 OrderReceiveperiod 0orderDemandoccursOrderReceiveperiod 1orderDemandoccursOrderReceiveperiod 2orderDemandoccursTimePeriod 1 Period 2 Period 3169 Order Up-To Model Definitions On Order Inventory (pipeline Inventory ) = the number of units that have been ordered but have not been received.

6 On hand Inventory = number of units physically in stock Backorder=total amount of demand yet to be satisfied. Inventory level = On hand Inventory Backorder. Inventory position = On Order Inventory + Inventory level. Order up to level, Sis the maximum Inventory position or target Inventory level or base stock Up-To Model ImplementationEach period s Order quantity= S Inventory position Suppose S = 4. If begins with an Inventory position = 1, Order 4 1 = 3 If begins with an Inventory position = 3, Order 4 ( 3) = 7 S = 4. We begin with an Inventory position = 1 and Order 3. If demand were 10in period 1, then the Inventory position at the start of period 2 is 1 10 + 3 = 6.

7 Order 4 ( 6)=10unitsPull system: Order quantity = the previous period s demand1810 Solving the Order - Up-To Model Given an Order up to level S What is the average Inventory ? What is the expected lost sale? What is the best Order up to level?19 Order Up-To Level and Inventory Level On Order Inventory + Inventory level at the beginning of Period 1 = S Inventory level at the end of Period l +1 = S demand over recent l +1 periods. Ex: S= 6, l= 3, and 2 units on hand at the start of period 1D1D2D3D4?Period 1 TimeInventory level at the end of period 4 = 6 - D1 D2 D3 D4 Period 2 Period 3 Period 42011 Similarity with a Newsvendor ModelThis is like a Newsvendor Model in which the Order quantity is Sand the demand distribution is demand over l+1 4DS D> 0, so there is on hand inventoryS D< 0, so there are backordersD = demand over l+1 periods21 Expected On-Hand Inventory and Backorder Expected on hand inventoryat the end of a period can be evaluated like Expected left over inventoryin the Newsvendor Model with Q= S.

8 Expected backorderat the end of a period can be evaluated like Expected lost salesin the Newsvendor Model with Q= S. Expected on Order Inventory = Expected demand in a periodx lead timeThis comes from Little s Law. Note that it equals the expected demand over lperiods, not l +1 Performance Measures The stockout probability is the probability at least one unit is backordered in a period: The in stock probability is the probability all demand is filled in a period: The fill rate is the fraction of demand within a period that is NOT backordered:Expected backorderFill rate 1-Expected demand in one period S periods 1l over DemandProb S periods 1l over DemandProb y probabilit Stockout 1 S periods 1l over DemandProbyprobabilit Stockout-1 y probabilit stock-In 23 Medtronic.

9 Demand over l+1 Periods at DCDC The period length is one week, the replenishment lead time is three weeks, l= 3 Assume demand is normally distributed: Mean weekly demand is (from demand data) Standard deviation of weekly demand is Expected demand over l +1 weeks is (3 + 1) x = Standard deviation of demand over l +1 weeks is 2413DC s Expected Backorder Assuming S= 625 Expected backorder Expected lost salesin a Newsvendor Model : Suppose S = 625 at the DC Normalize the Order up to level: Lookup L(z) in the Standard Normal Loss Function Table: L( )= Convert expected lost sales, L(z), into the expected backorder with the actual normal distribution that represents demand over l+1 (z) backorder Expected 25 Other DC Performance Measures % of demand is filled immediately (not backorders) average number of units on hand at the end of a period.

10 There are units on Order at any given backorderFill rate -Expected demand in one period 1625 - on-hand Inventory S -Expected demand over l periods Expected backorder= 3 on- Order Inventory Expected demand in one period Lead time= 2614 Choose Swith Normally Distributed DemandSuppose the target in stock probability at the DC is From the Standard Normal Distribution Function Table, ( )= So we choose z= To convert zinto an Order up to level: Note that and are the parameters of the normal distribution that describes demand over l + 1 27 Medtronic: Demand at Susan s RegionSusan s territory: The period length is one day, the replenishment lead time is one day, l=1 Assume demand is Poissondistributed.