OIDD615 practice questions - solutions

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1 OIDD615 Practice Questions Below is a sample of past exam questions. This is not meant to provide a comprehensive list of possible questions, but it does provide a decent representation of the type of questions that could be asked on the exam. Newsvendor (Q1-5) The National Football League (NFL) has granted Nike an exclusive license to sell NFL replica jerseys. Nike outsources the jersey cutting and sewing operations to an offshore contract manufacturer (CM). The jerseys are then delivered to Nike’s Distribution Center (DC). Because of the long production and shipment leadtimes, Nike must decide in advance how much inventory to hold at the DC in anticipation of retailers’ orders for the coming season. Nike developed the demand forecasts provided in the table for the Baltimore Ravens for the upcoming season, assuming independent normal demand distributions. Nike sells the NFL jerseys to retailers at a wholesale price of $30 per jersey. For popular players (like the first four listed above), the CM prints the player’s name and number on the jersey and ships the completely finished good, known as a dressed jersey, to Nike’s DC. The cost to Nike of a dressed jersey is $12. Nike does not have the opportunity to make a mid-season replenishment. At the end of the season, Nike sells its unsold jerseys at a discount price of $8 per jersey. Product Description Mean Standard Deviation Joe Flacco 45,000 30,000 Ray Rice 35,000 20,000 Haloti Ngata 25,000 15,000 Terrell Suggs 15,000 10,000 Other players 60,000 30,000 Q1. How many Joe Flacco jerseys should Nike order to maximize expected profit? Demand for his jersey is Normally distributed with a mean of 45,000 and a standard deviation of 30,000. Answer: 72,300. Underage cost is the marginal cost of making one less jersey than the true demand, Cu= 30 - 12 = 18; Overage cost is the marginal cost of making one more jersey than the true demand, Co= 12 - 8 = 4. Then the Critical Ratio = Cu / (Co+Cu) = 0.8182 Looking up the probability from the Standard Normal Distribution Function Table we obtain z = 0.91. Then the optimal production quantity, Q = µ + z*σ = 72,300.

2 Q2. Demand for Ray Rice jerseys in normally distributed with a mean of 35,000 and a standard deviation of 20,000. If Nike orders 45,000 Ray Rice jerseys, how many of these jerseys can Nike expect to sell at the full price ($30)? Q3. Demand for the Haloti Ngata jersey is Normally distributed with a mean of 25,000 and a standard deviation of 15,000. If Nike orders 28,000 Haloti Ngata jerseys, what is the probability that Nike will be able to satisfy all demand for this jersey? Express the probability as a number between 0 and 1 (and not as a %). Q4. Demand for the Terrell Suggs jersey is Normally distributed with a mean of 15,000 and a standard deviation of 10,000. If Nike orders 10,000 Terrell Suggs jerseys, how many of these jerseys will they have to sell (in expectation) at the discount price? Answer: 31,044. First find the z-score corresponding to the order quantity. z=(Q-µ)/σ = (45000-35000)/20000= 0.50 To find sales, first find expected leftover inventory. I(z) = I(0.5) = 0.6978. Expected left over inventory = s x I(z) = 20,000 x 0.6978 = 13956 Expected Sales = Q – Expected left over inventory = 45,000 – 13956 = 31,044. Answer: 0.5793. If Demand is less than 28,000, we satisfy all demand. Hence we need to find the probability F(28,000)= Pr(Demand<=28000). This is the in-stock probability. (If you satisfy all demand, you are “in-stock” at the end of the season.) Find the z-score corresponding to the order quantity. z=(Q-µ)/σ = 0.2 From the Standard Normal Distribution Function Table Φ(z) = 0.5793. Answer: 1,978. First we find the z-score corresponding to the order quantity. z=(Q-µ)/σ = -0.5. From the Normal Distribution Inventory Function Table we obtain I(z) = I(-0.5) = 0.1978 Expected leftover inventory = s x I(z) = 10,000 x 0.1978 = 1,978

3 Q5. For the less popular players (“Other Players”) the CM ships a blank jersey to Nike’s DC at a cost of $11. After receiving the orders from retailers, Nike prints the name of the player and number on the blank jersey, which costs Nike $2 per jersey. Assume the selling price of jerseys with the names is still the same. If Nike runs out of blank jerseys, they can order more blank jerseys from a local supplier with essentially an immediate response time. But that supplier charges $14 per blank jersey. If Nike has blank jerseys left over at the end of the season, they sell them blank for $7 each. Demand for “Other Players” is Normally distributed with a mean of 60,000 and a standard deviation of 30,000. How many blank jerseys should Nike purchase from the CM (for $11 each) to maximize their expected profit from selling “Other Players”? (Q6-7) (Simplified Hosting Problem) Vmail is a service provider of free email. It hosts all emails on servers on “the cloud”. The typical usage on Mondays is normally distributed with mean 300 million minutes and standard deviation 75 million minutes. For a particular Monday, Vmail can buy cloud capacity well in advance for $0.01 per minute. If it purchases more capacity than it needs, the capacity goes unused (and they cannot get a refund for the capacity they purchased). If demand on Monday exceeds the capacity they purchased in advance, they must purchase additional capacity as needed from a company called Mackspace. However, Mackspace charges $0.03 per minute. Q6. How much capacity should Vmail purchase in advance (at $0.01 per minute) to minimize its total expected capacity expense? Give your answer in units of million minutes. Answer: 54,600. Cu= 14-11=3. If they order one jersey fewer than demand, they order from the more expensive supplier which costs $14 (instead of $11 from the regular supplier). The $2 printing cost doesn’t matter because it is incurred whether $11 or $14 is paid for a blank jersey. Co = Cost – salvage = $11-$7 = $4. If they over order by one unit, it must be salvaged (but there is no printing on this blank jersey). The Critical Ratio = Cu / (Co + Cu) = 0.4286 From the Normal Distribution Function Table we obtain z = -0.18. Then Q = µ + z*σ = 54,600. Answer: 333. Cu= 0.03-0.01 = 0.02 because if you had known you would use the minute of capacity, you would have purchased it for $0.01 rather than having to buy it from Mackspace for $0.03. Co=0.01 because if you buy the minute of capacity but don’t use it, you would have just not purchased the capacity and saved yourself 0.01. Critical Ratio = 0.6667 From the Standard Normal distribution function table, z = 0.44. Then Q = µ + z*σ = 333M.

4 Q7. Suppose Vmail purchases 435 million minutes in advance. How much should Vmail expect to pay (in $s) to Mackspace (at $0.03 per minute) for the additional capacity required? Recall, their usage is Normally distributed with mean 300 million minutes and standard deviation 75 million minutes. Q8-10 The Penn Bookstore sells several magazines. A single stocking quantity is ordered for each issue. When a new issue arrives, any remaining copies of the old issue are returned to the publisher. If a magazine sells out, then it remains unavailable until the next issue arrives. Q8. Consider the following data on Bits and Bytes magazine at the Penn Bookstore: The forecast column lists their forecast of demand for that issue (when they ordered copies for that issue). It is generated by an internal computer system that accounts for seasonality and other Answer: 32,175. The charge to Mackspace equals the price per minute times the number of minutes needed. The number of minutes needed is the expected number of minutes used above 435M. z= (Q-µ)/σ = (435-300)/75 = 1.80. From the Standard Normal Inventory Function table we obtain, I(z) = I(1.80) = 1.81 Expected Leftover inventory = sigma x I(z) = 75 x 1.81 = 136.1 Expected Loss Sales = mu – Q + Exp Leftover = 300 – 435 + 136.1 = 1.1 Vmail expected to pay = Expected Loss Sales * 1,000,000 * (0.03) = 32,175.

5 events in the store (which is why the forecast varies from issue to issue). The quantity column is the actual number of issues stocked that week and the sales column provides the number of units actually sold. The estimated demand column provides an estimate of what sales could have been had there been no stockouts. (If sales is less than or equal to the order quantity, the estimated demand equals sales, otherwise it can be greater.) Q8. Suppose the week 37 forecast is for 20 copies. What would be the coefficient of variation of demand? Again, assume a normal distribution is chosen to model demand. Q9. Consider Modern Active MBA, a periodical dedicated to selecting an MBA program and getting the most out of the experience. The publisher charges the Penn Bookstore $1.25 for each copy of MAMBA sold. It costs the publisher $0.40 to print and deliver each copy of the magazine. The copies that are left over are discarded at a cost of $0.10 per copy to the publisher. The forecast of demand is normally distributed with a mean of 90 and a standard deviation of 22. What order quantity maximizes the publisher’s expected profit? (Leave your answer in decimal form, i.e., no need to round to an integer value.) Answer = 0.385 The average A/F ratio is 0.97 and the standard deviation of the A/F ratios is 0.37. So the coefficient of variation is the 0.37 / 0.97 = 0.385 Answer: 97.48. Cu =$1.25-$0.40 = $0.85 because if you under order by one unit, you could have purchased it for $0.40 and sold it for $1.25, earning a profit of $0.85 Co = Overage Cost = cost –salvage = 0.40 – (- 0.10) = 0.40+0.10 = $0.50. If an unit is purchased that is not needed, the publisher incurs the cost to print of $0.4 as well as the cost to dispose which is $0.10, making the publisher worse off by $0.50 The Critical Ratio=0.6296. Looking up Standard Normal distribution function table, this gives z=0.34. Then, the optimal quantity to order = µ + z*σ = 97.48.

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