SCM 403 Homework 4 Problem 1 Dr. Jack is in charge of the Blood Bank at the local hospital. Blood is collected in the regional blood center 200 miles away and is delivered to the hospital by airplane. Dr. Jack reviews the inventory and places order every Monday morning for delivery the following Monday morning. If demand begins to exceed supply, surgeons postpone nonurgent procedures, in which case blood is back ordered. The demand for blood in every given week is normal with mean 100 pints and standard deviation 34 pints. The demands are independent across weeks. 1a) On Monday morning Dr. Jack reviews his reserves and observes 200 pints in onhand inventory, no back orders, and 73 pints in pipeline inventory. Suppose the order upto level is 285. How many pints will he order? Inventory Level = OnHand Inventory - Back Order Inventory Level (IL) = 200 Inventory Position = Inventory Level + Pipeline Inventory Inventory Position = 273 Back Orders = 0 Pipeline Inventory (PI) = 73 Order UpToLevel or Base Stock (s) = 285 Quantity to Order = Order UpToLevel or Base Stock (s) - Inventory Position Inventory Position = IL + PI = 200 + 73 = 273 Quantity to Order = s IP = 285 - 273 = 12 Thus, accounting for the 273 pints of blood on hand or en route to the hospital, Dr. Jack must order 12 pints of blood to bring the stock back up to the predetermined base stock level of 285. 1b) Dr. Jack targets a 99% expected fill rate. What order upto level should he choose? = 100 = 34 LT = 1 week T = 1 week L+T = 200 L+T = 48.08 Expected Demand in One Period = s = L+T + z x L+T FR = 0.99 L(z) = 0.0208 z = 1.65 Fill Rate (FR) = 1 - Expected backorder / Expected Demand in One Period .99 = 1 Expected backorder / 100 Expected Backorder = 1 .99 x 100 Expected Backorder = .01 x 100 Expected Backorder = 1 Targeted Backorder Level with Standard Normal Distribution or Choose s for a Target Fill Rate of 99%: L(z) = (Expected Demand in One Period / Standard Deviation of Demand over L+T Periods) x (1 - Fill Rate) = (100 / 48.08) x (1 - 0.99) = 2.08 x 0.01 = .0208 Locate corresponding zscore from Normal Loss Function Table for L(z) 0.208: L(0.0208) = 1.65 z = 1.65 Convert z to Order UpTo Level: s = L+T + z x L+T = 200 + 1.65 x 48.08 = 200 + 79.33 = 279.33 or 280 Thus, in order for Dr. Jack to potentially reach a 99% fill rate, he should choose an order upto level of 280 pints (units) of blood. 1c) Dr. Jack targets a 99% service level. What order upto level should he choose? = 100 = 34 LT = 1 week T = 1 week L+T = 200 L+T = 48.08 s = L+T + z x L+T SL = 0.99 z = 2.33 Mean Demand Over Lead Time Plus Periods = L+T = 100 x 1 +1 = 100 x 2 = 200 Standard Deviation of Demand Over Lead Time Plus Periods = L+T = 34 x 1+1 = 34 x 2 = 48.08 Locate corresponding zscore from the Normal Distribution Function Table for 0.99: 0.9901 = 2.33 z = 2.33 Convert z to an order UpTo level: s = L+T + z x L+T = 200 + 2.33 x 48.08 = 200 + 112.03 = 312.03 or 313 Thus, in order for Dr. Jack to potentially satisfy a 99% service level, he should choose an order upto level of 313 pints (units) of blood. 1d) Dr. Jack is planning to implement a computer system that will allow daily ordering, seven days per week, and that the lead time will also be reduced to one day. What will be the average order quantity? The average order quantity is simply average demand (the mean) during a single order period. Therefore, convert average weekly demand to average daily demand during the order period: Average Order Quantity = / days per week = 100 / 7 = 14.28 or 15 units of blood Problem 2 You are the owner of Hotspices.com, an online retailer of hip, exotic, and hardtofind spices. Consider your inventory of saffron, generally worth more by weight than gold. You order saffron from an overseas supplier with a shipping leadtime of four weeks and that you order weekly. Your average quarterly demand is normally distributed with a mean of 415 ounces and a standard deviation of 154 ounces. The holding cost per ounce per week is $0.75. You estimate that your backorder penalty cost is $50 per ounce. Assume that there are 4.33 weeks per month. 2a) If you wish to minimize inventory holding costs while maintaining 99.25% fill rate, then what should your order upto level be? = 415 = 154 LT = 4 T = 1 L+T = 159.62 L+T = 95.51 SL = 0.9925 L(z) = 0.0025 z = 2.43 52 weeks in a year & quarterly is the same as every four months. So, 52 / 4 = 13 weeks Must convert from quarterly into weekly demand & demand over leadtime plus periods: Weekly Demand = / quarterly weeks Demand Over Five Weeks = L+T = 415 / 13 = 31.92 x (4+1) = 31.92 = 31.92 x 5 = 159.6 Weekly Standard Deviation = / quarterly weeks Standard Deviation Over Five Weeks = L+T = 154 / 13 = 42.71 x (4+1) = 42.71 = 42.71 x 5 = 95.51 Expected Back Order (a.k.a. target lost sales) for targeted 99.25% fill rate: L(z) = (Expected Demand in One Period / Standard Deviation of Demand over L+T Periods) x (1 - Fill Rate) = (31.92 / 95.51) x (1 - 0.9925) = .3342 x .0075 = 0.0025 & corresponding zscore 2.43 from Standard Normal Loss Function Table Convert z to order upto level: s = L+T + z x L+T = 159.62 + 2.43 x 95.51 = 391.71 or 392 Thus, in order to minimize inventory holding while maintaining a 99.25% fill rate, then Hotspices.com should utilize an order up to level of 392 units. 2b) If you wish to minimize inventory holding costs while maintaining 99.25% instock probability, then what should your order upto level be? = 415 = 154 LT = 4 T = 1 L+T = 159.62 L+T = 95.51 z = 2.43 SL = 0.9925 s = 392 Weekly Demand = / quarterly weeks Demand Over Five Weeks = L+T = 415 / 13 = 31.92 x (4+1) = 31.92 = 31.92 x 5 = 159.6 Weekly Standard Deviation = / quarterly weeks Standard Deviation Over Five Weeks = L+T = 154 / 13 = 42.71 x (4+1) = 42.71 = 42.71 x 5 = 95.51 Locate corresponding zscore from the Normal Distribution Function Table for 0.9925: 0.9925 = 2.43 z = 2.43 Convert z to an order UpTo level: s = L+T + z x L+T = 159.62 + 2.43 x 95.51 = 159.62 + 232.09 = 391.71 or 392 Thus, in order to minimize inventoryholding costs while maintaining a 99.25% instock probability, then Hotspices.com should utilize an order upto level of 392 units. 2c) If you wish to minimize inventory holding and backorder penalty costs, then what should your order upto level be? = 415 = 154 LT = 4 T = 1 L+T = 159.62 L+T = 95.51 z = 2.18 SL = 0.9854 CR =0.9852 Holding = 0.75 Penalty Cost = 50 s = 368 Calculate critical ratio to get zscore (similar to Newsvendor): CR = CU / CU + CO = 50 / 50 + .75 = 50 / 50.75 = 0.9852 Locate corresponding zscore from the Normal Distribution Function Table: 0.9854 = 2.18 z = 2.18 Convert z to an order UpTo level: s = L+T + z x L+T = 159.62 + 2.18 x 95.51 = 159.62 + 208.212 = 367.832 or 368 Thus, in order to minimize inventoryholding costs and backorder penalty costs, then Hotspices.com should utilize an order up to level of 368 units. 2d) If you arbitrarily decide an order upto level of 250, what fraction of the demand will not be met immediately? What is the expected onhand inventory at the beginning of a period? = 415 = 154 LT = 4 T = 1 L+T = 159.62 L+T = 95.51 z = 2.18 s = 250 Onhand inventory = Inventory position Onorder Inventory + Backorder OnHand Inventory = Inventory physically on premise to serve demand immediately with no delay! Expected OnHand Inventory = s - Expected Demand over (l+1) periods + Expected Backorder or Expected OnHand Inventory can be evaluated from: Inventory position = Onorder inventory + OnHand Inventory - Backorder z = s L+T / L+T L(z) in Standard Normal Loss Function Table: = (250 - 159.62) / 95.51 L(0.9463) = 0.0933 = 0.9463 Expected Backorder = L+T x L(z) = 95.51 x 0.0933 = 8.911 or 9 Expected OnHand Inventory = s L+T + expected backorder = 250 - 159.62 + 9 = 99.38 or 100 Fraction or Percent of Demand Not Met at Beginning of Period with S = 250: x = 1 159.62 - 9 / 159.62 = 1 0.9436 = .0564 or .06 Thus, if the order upto level is 250 units, then at least 6% of the demand will not be met immediately with an expected onhand inventory of 100 units at the beginning of a period. Problem 3 Livingston Tools, a manufacturer of batteryoperated, handheld power tools for consumer markets, has a problem. Its two biggest customers are \"big box\" discounters. Because the customers are fiercely price competitive, each wants exclusive products, thereby preventing consumers from making price comparisons. For example, Livingston will sell the exact same power drill to each retailer, but Livingston will use packing customized to each retailer (including two different product identification numbers). Suppose weekly demand for each product to each retailer is normally distributed with mean 5200 and standard deviation 3800. Livingston makes stocking decisions on a weekly basis and has a replenishment leadtime of three weeks. Because these two retailers are quite important to Livingston, it has set a target fill rate of 99.9 percent. 3a) Based on the order upto model, what is Livingston's average inventory of each of the two versions of the power drill? = 5200 = 3800 LT = 3 T = 1 L+T = 20800 L+T = 7600 FR = 0.999 L(z) = 2.84 s = 42384 Demand Over LeadTime + Periods = L+T Standard Deviation Over LeadTime + Periods = L+T = 5200 x (3+1) = 3800 x (3+1) = 20800 = 7600 Expected Back Order (a.k.a. target lost sales) for targeted 99.9% fill rate: L(z) = (Expected Demand in One Period / Standard Deviation of Demand over L+T Periods) x (1 - Fill Rate) = (5200 / 7600) x (1 - 0.999) = 0.6842 x 0.001 = 0.00068 or 0.00070 & corresponding zscore 2.84 from Standard Normal Loss Function Table Expected Backorder = L+T x L(z) Convert z to order upto level: = 7600 x 0.00070 s = L+T + z x L+T = 5.32 = 20800 + 2.84 x 7600 = 42384 Expected OnHand Inventory = s - Expected Demand over (l+1) periods + Expected Backorder = 42384 - 20800 + 5.32 = 21578.68 or 21579 Thus, Livingston Tools average inventory for each version of their power tool will be 21,579 units 3b) A supply chain analyst at Livingston suggests that it stock drills without putting them into their specialized packing. As the orders are received from the two retailers, Livingston will fulfill those orders from the same stockpile of inventory, since it doesn't take much time to actually package the tools. For simplicity, assume that the two demands are independent. By how much would this new system reduce inventory? Would you expect a higher or a lower savings if the two demands were negatively correlated? Problem 4 Dave Jones manages the warehouse inventory for Athletics, a distributor for sport watches. From his experience, Dave knows that PR5 jogging watch has an annual demand of 40,000 units. The fixed cost of placing an order with the manufacturer (Casio) is $50, while the holding cost per watch is $90/year. The leadtime for replenishment is 8 days and that Dave uses realtime monitoring of inventory. 4a) What are the optimal reorder point and optimal order quantity for Dave assuming that the demand is fixed? What is the safety stock? 4b) Dave's boss is concerned that Dave is treating the demand as fixed. She suggests looking into the demand data more closely. On further investigation, Dave found that the demand is actually random with an annual average of 40,000 units and an annual standard deviation of 8,000 units. How would this new information change Dave's optimal policy calculated in part (a) when Athletics' policy is to provide 99% service level? How much is the safety stock now? 4c) Athletics plans to install a new information system (ERP, SCM, and eprocurement software) that will achieve better information flow throughout its supply chain. The immediate benefit of this will be that the fixed cost of placing an order will be reduced to zero. Further, better information sharing will allow the less expensive periodic review policy to work perfectly for Athletics. Assume that the leadtime for replenishment is still 8 days but Dave now reviews inventory once every 3 days. The demand is random as described in part (b) above. Calculate and characterize Dave's optimal inventory policy. How much is the safety stock now