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Chapter 11: The Order Up-To Model Medtronic’s InSync pacemaker supply chain and objectives Look at problem from two persopectives … One distribution center (DC) in Mounds View, MN. About 500 sales territories throughout the country. Consider Susan Magnotto’s territory in Madison, Wisconsin. Objective: Because the gross margins are high, develop a system to minimize inventory investment while maintaining a very high service target, e.g., a 99.9% in-stock probability or a 99.9% fill rate. Timing in the order up-to model Time is divided into periods of equal length, e.g., one hour, one month. During a period the following sequence of events occurs: A replenishment order can be submitted. Inventory is received. Random demand occurs. Lead times: An order is received after a fixed number of periods, called the lead time. Let l represent the length of the lead time. Order Receive period 0 order Order Receive period 1 order Order Receive period 2 order An example with l = 1 Time Demand occurs Period 1 Demand occurs Period 2 Demand occurs Period 3 Order up-to model vs. newsvendor model Both models have uncertain future demand, but there are differences… Newsvendor Order up-to After one period Never Number of replenishments One (maybe two or three with some reactive capacity) Unlimited Demand occurs during replenishment No Yes Inventory obsolescence Newsvendor applies to short life cycle products with uncertain demand and the order up-to applies to long life cycle products with uncertain, but stable, demand. The Order Up-To Model: Choosing a demand distribution/forecast Can calculate the mean m and standard deviation s from empirical data Calculate from monthly numbers For DC: average weekly demand = 80.6 and weekly standard deviation = 58.81 For sales territory average daily demand = 0.29 units Distribution Center Sales territory 16 700 14 600 12 Units 10 400 8 6 300 4 200 2 100 Dec Oct Nov Sep Jul Month Aug Jun Apr May Mar Jan Dec Oct Nov Sep Jul Aug Apr May Mar Jan Jun Month Feb 0 0 Feb Units 500 The Order Up-To Model: Model design and implementation Order up-to model definitions On-order inventory [a.k.a. pipeline inventory] = the number of units that have been ordered but have not been received. On-hand inventory = the number of units physically in inventory ready to serve demand. Backorder = the total amount of demand that has has not been satisfied: All backordered demand is eventually filled, i.e., there are no lost sales. Inventory level = On-hand inventory - Backorder. Inventory position = On-order inventory + Inventory level. Order up-to level, S the maximum inventory position we allow. sometimes called the base stock level. Order up-to model implementation Each period’s order quantity = S – Inventory position Suppose S = 4. If a period begins with an inventory position = 1, then three units are ordered. (4 – 1 = 3 ) If a period begins with an inventory position = -3, then seven units are ordered (4 – (-3) = 7) A period’s order quantity = the previous period’s demand: Suppose S = 4. If demand were 10 in period 1, then the inventory position at the start of period 2 is 4 – 10 = -6, which means 10 units are ordered in period 2. The order up-to model is a pull system because inventory is ordered in response to demand. The order up-to model is sometimes referred to as a 1-for-1 ordering policy. The Order Up-To Model: Performance measures What determines the inventory level? Short answer: Inventory level at the end of a period = S minus demand over l +1 periods. Explanation via an example with S = 6, l = 3, and 2 units on-hand at the start of period 1 Period 1 Period 2 Period 3 Keep in mind: Period 4 At the start of a period the Inventory level + On-order equals S. Time D1 D2 D3 D4 ? All inventory on-order at the start of period 1 arrives before the end of period 4 Nothing ordered in periods 2-4 arrives by the end of period 4 All demand is satisfied so there Inventory level at the end are no lost sales. of period four = 6 - D1 – D2 – D3 – D4 Expected on-hand inventory and backorder … S Period 1 … Period 4 S – D > 0, so there is on-hand inventory D D = demand over l +1 periods Time S – D < 0, so there are backorders This is like a Newsvendor model in which the order quantity is S and the demand distribution is demand over l +1 periods. So … Expected on-hand inventory at the end of a period can be evaluated like Expected left over inventory in the Newsvendor model with Q = S. Expected backorder at the end of a period can be evaluated like Expected lost sales in the Newsvendor model with Q = S. Stockout and in-stock probabilities, on-order inventory and fill rate The stockout probability is the probability at least one unit is backordered in a period: Stockout probabilit y ProbDemand over l 1 periods S 1 ProbDemand over l 1 periods S The in-stock probability is the probability all demand is filled in a period: In - stock probabilit y 1 - Stockout probabilit y ProbDemand over l 1 periods S Expected on-order inventory = Expected demand over one period x lead time This comes from Little’s Law. Note that it equals the expected demand over l periods, not l +1 periods. The fill rate is the fraction of demand within a period that is NOT backordered: Fill rate 1- Expected backorder Expected demand in one period Demand over l +1 periods DC: The period length is one week, the replenishment lead time is three weeks, l =3 Assume demand is normally distributed: Mean weekly demand is 80.6 (from demand data) Standard deviation of weekly demand is 58.81 (from demand data) Expected demand over l +1 weeks is (3 + 1) x 80.6 = 322.4 Standard deviation of demand over l +1 weeks is 3 1 58.81 117.6 Susan’s territory: The period length is one day, the replenishment lead time is one day, l =1 Assume demand is Poisson distributed: Mean daily demand is 0.29 (from demand data) Expected demand over l +1 days is 2 x 0.29 = 0.58 Recall, the Poisson is completely defined by its mean (and the standard deviation is always the square root of the mean) DC’s Expected backorder assuming S = 625 Expected backorder is analogous to the Expected lost sales in the Newsvendor model: Suppose S = 625 at the DC Normalize the order up-to level: z S m s 625 322.4 2.57 117.6 Lookup L(z) in the Standard Normal Loss Function Table: L(2.57)=0.0016 Convert expected lost sales, L(z), for the standard normal into the expected backorder with the actual normal distribution that represents demand over l+1 periods: Expected backorder s L(z) 117.6 0.0016 0.19 Therefore, if S = 625, then on average there are 0.19 backorders at the end of any period at the DC. Other DC performance measures with S = 625 Fill rate 1- Expected backorder 0.19 1 99.76%. Expected demand in one period 80.6 So 99.76% of demand is filled immediately (i.e., without being backordered) (Note for below: inventory level + backorder = on hand) Expected on-hand inventory S-Expected demand over l 1 periods Expected backorder =625 - 322.4 0.19 302.8. So on average there are 302.8 units on-hand at the end of a period. Expected on-order inventory Expected demand in one period Lead time = 80.6 3 241.8. So there are 241.8 units on-order at any given time. Performance measures in Susan’s territory: ~~FYI but will not be covered on Exam Look up in the Poisson Loss Function Table expected backorders for a Poisson distribution with a mean equal to expected demand over l+1 periods: Mean demand = 0.29 Mean demand = 0.58 F(S) L (S ) F(S) L (S ) S S 0 0.74826 0.29000 0 0.55990 0.58000 1 0.96526 0.03826 1 0.88464 0.13990 2 0.99672 0.00352 2 0.97881 0.02454 3 0.99977 0.00025 3 0.99702 0.00335 4 0.99999 0.00001 4 0.99966 0.00037 5 1.00000 0.00000 5 0.99997 0.00004 F (S ) = Prob {Demand is less than or equal to S} L (S ) = loss function = expected backorder = expected amount demand exceeds S Suppose S = 3: Expected backorder = 0.00335 In-stock = 99.702% Fill rate = 1 – 0.00335 / 0.29 = 98.84% Expected on-hand = S – demand over l+1 periods + backorder = 3 – 0.58 + 0.00335 = 2.42 Expected on-order inventory = Demand over the lead time = 0.29 Problem 11.1 A furniture store sells study desks with normally distributed demand. The mean = 40 and the standard deviation = 20. The lead time is 2 weeks and inventory replenishments are ordered weekly. a) If S = 220 and your inventory = 100 with 85 desks on order, how many desks will you order? (35) b) Suppose S = 220 and you are about to place an order and your inventory level – 160 with 65 desks on order. How many desks will you order? (order 0) The Order Up-To Model: Choosing an order up-to level, S, to meet a service target Choose S to hit a target in-stock with normally distributed demand Suppose the target in-stock probability at the DC is 99.9%: From the Standard Normal Distribution Function Table, F(3.08)=0.9990 So we choose z = 3.08 To convert z into an order up-to level: S m z s 322.4 3.08 117.6 685 Note that m and s are the parameters of the normal distribution that describes demand over l + 1 periods. Problem 11.1 A furniture store sells study desks with normally distributed demand. The mean = 40 and the standard deviation = 20. The lead time is 2 weeks and inventory replenishments are ordered weekly. c) What is the optimal order-up-to level if you want to target a 98% in stock probability? (191.14) Choose S to hit a target fill rate with normally distributed demand Find the S that yields a 99.9% fill rate for the DC. Step 1: Evaluate the target lost sales note 1 period in the numerator and l + 1 periods in the denominator Expected demand in one period 1 Fill rate L( z ) Standard deviation of demand over l 1 periods 80.6 (1 0.999) 0.0007 117.6 Step 2: Find the z that generates that target lost sales in the Standard Normal Loss Function Table: L(2.81) = L(2.82) = L(2.83) = L(2.84) = 0.0007 Choose z = 2.84 to be conservative (higher z means higher fill rate) Step 3: Convert z into the order up-to level: S 322.4 2.84 117.62 656 Problem 11.1 A furniture store sells study desks with normally distributed demand. The mean = 40 and the standard deviation = 20. The lead time is 2 weeks and inventory replenishments are ordered weekly. d) What is the optimal order-up-to level if you want to target a 98% fill rate? (S = 175.77) The Order Up-To Model: Appropriate service levels Justifying a service level via cost minimization Let h equal the holding cost per unit per period e.g. if p is the retail price, the gross margin is 75%, the annual holding cost is 35% and there are 260 days per year, then h = p x (1 -0.75) x 0.35 / 260 = 0.000337 x p Let b equal the penalty per unit backordered e.g., let the penalty equal the 75% gross margin, then b = 0.75 x p “Too much-too little” challenge: If S is too high, then there are holding costs, Co = h If S is too low, then there are backorders, Cu = b Cost minimizing order up-to level satisfies Prob Demand over l 1 periods S Cu b (0.75 p) Co Cu h b (0.000337 p) (0.75 p) 0.9996 Optimal in-stock probability is 99.96% because In - stock probabilit y ProbDemand over l 1 periods S Critical ratio The optimal in-stock probability is usually quite high Suppose the annual holding cost is 35%, the backorder penalty cost equals the gross margin and inventory is reviewed daily. Optimal in-stock probability 100% 98% 96% 94% 92% 90% 88% 0% 20% 40% 60% Gross margin % 80% 100% Problem 11.1 A furniture store sells study desks with normally distributed demand. The mean = 40 and the standard deviation = 20. The lead time is 2 weeks and inventory replenishments are ordered weekly. e) Suppose S = 120. What is the expected on-hand inventory? (13.82) f) Suppose S = 200. What is the fill rate? (99.69%) g) Suppose you want to maintain a 95% in-stock probability. What is the expected fill-rate? (98.22%) The Order Up-To Model: Controlling ordering costs Impact of the period length Increasing the period length leads to larger and less frequent orders: The average order quantity = expected demand in a single period. The frequency of orders approximately equals 1/length of period. See “saw tooth” diagrams in the text Suppose there is a cost to hold inventory and a cost to submit each order (independent of the quantity ordered)… … then there is a tradeoff between carrying little inventory (short period lengths) and reducing ordering costs (long period lengths) Tradeoff between inventory holding costs and ordering costs 22000 20000 18000 Total costs 16000 14000 Cost Costs: Ordering costs = $275 per order Holding costs = 25% per year Unit cost = $50 Holding cost per unit per year = 25% x $50 = 12.5 12000 Inventory holding costs 10000 8000 6000 4000 Period length of 4 weeks minimizes costs: This implies the average order quantity is 4 x 100 = 400 units EOQ model offers an approximation for min Q: Q 2 K R h 2 275 5200 478 12.5 Ordering costs 2000 0 0 1 2 3 4 5 6 Period length (in weeks) 7 8 9 The Order Up-To Model: Managerial insights 140 120 100 Expected inventory Better service requires more inventory at an increasing rate 80 60 40 Inc re a s ing s ta nda rd de via tion 20 0 90% 91% 92% 93% 94% 95% 96% 97% 98% 99% 100% Fill rate 600 Expected inventory Shorten lead times and you will reduce inventory 500 400 300 200 100 0 0 5 10 Lead time 15 20 3000 Do not forget about pipeline inventory Inventory 2500 2000 1500 1000 500 0 0 5 10 Lead time 15 20 Order up-to model summary The order up-to model is appropriate for products with random demand but many replenishment opportunities. Expected inventory and service are controlled via the order up-to level: The higher the order up-to level the greater the expected inventory and the better the service (either in-stock probability or fill rate). The key factors that determine the amount of inventory needed are… The length of the replenishment lead time. The desired service level (fill rate or in-stock probability). Demand uncertainty. When inventory obsolescence is not an issue, the optimal service level is generally quite high. Problem 11.1 A furniture store sells study desks with normally distributed demand. The mean = 40 and the standard deviation = 20. The lead time is 2 weeks and inventory replenishments are ordered weekly. a) If S = 220 and your inventory = 100 with 85 desks on order, how many desks will you order? (35) b) Suppose S = 220 and you are about to place an order and your inventory level – 160 with 65 desks on order. How many desks will you order? (order 0) c) What is the optimal order-up-to level if you want to target a 98% in stock probability? (191.14) d) What is the optimal order-up-to level if you want to target a 98% fill rate? (S = 175.77) e) Suppose S = 120. What is the expected on-hand inventory? (13.82) f) Suppose S = 200. What is the fill rate? (99.69%) g) Suppose you want to maintain a 95% in-stock probability. What is the expected fill-rate? (98.22%)