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?
μ = 100 σ = 34 LT = 1 week T = 1 week μL+T = 200 σL+T = 48.08 Expected Demand in One Period = μ
s = μL+T + z x σL+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 = μL+T + z x σL+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?
μ = 100 σ = 34 LT = 1 week T = 1 week μL+T = 200 σL+T = 48.08 s = μL+T + z x σL+T SL = 0.99
z = 2.33
Mean Demand Over Lead Time Plus Periods = μL+T
= 100 x 1 +1
= 100 x 2
= 200
Standard Deviation of Demand Over Lead Time Plus Periods = σL+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 = μL+T + z x σL+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 = μ / 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?
μ = 415 σ = 154 LT = 4 T = 1 μL+T = 159.62 σL+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 = μ / quarterly weeks Demand Over Five Weeks = μL+T
= 415 / 13 = 31.92 x (4+1)
= 31.92 = 31.92 x 5
= 159.6
Weekly Standard Deviation = σ / √quarterly weeks Standard Deviation Over Five Weeks = σL+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 = μL+T + z x σL+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?
μ = 415 σ = 154 LT = 4 T = 1 μL+T = 159.62 σL+T = 95.51 z = 2.43 SL = 0.9925 s = 392
Weekly Demand = μ / quarterly weeks Demand Over Five Weeks = μL+T
= 415 / 13 = 31.92 x (4+1)
= 31.92 = 31.92 x 5
= 159.6
Weekly Standard Deviation = σ / √quarterly weeks Standard Deviation Over Five Weeks = σL+T
= 154 / √13 = 42.71 x √(4+1)
= 42.71 = 42.71 x √5
= 95.51
Locate corresponding zscore from theNormal Distribution Function Table for 0.9925:
0.9925 = 2.43
z = 2.43
Convert z to an order UpTo level:
s = μL+T + z x σL+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?
μ = 415 σ = 154 LT = 4 T = 1 μL+T = 159.62 σL+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 = μL+T + z x σL+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?
μ = 415 σ = 154 LT = 4 T = 1 μL+T = 159.62 σL+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 μL+T / σL+T L(z) in Standard Normal Loss Function Table:
= (250 – 159.62) / 95.51 L(0.9463) = 0.0933
= 0.9463
Expected Backorder = σL+T x L(z)
= 95.51 x 0.0933
= 8.911 or 9
Expected OnHand Inventory = s μL+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?
μ = 5200 σ = 3800 LT = 3 T = 1 μL+T = 20800 σL+T = 7600 FR = 0.999 L(z) = 2.84 s = 42384
Demand Over LeadTime + Periods = μL+T Standard Deviation Over LeadTime + Periods = σL+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 = σL+T x L(z) Convert z to order upto level:
= 7600 x 0.00070 s = μL+T + z x σL+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?
