1. The executives of the General Products Company (GPC) have to decide which of three products to introduce, A, B, or C. Product C is essentially a risk-free proposition, from which the company will obtain a net profit of $1 million. Product B is considerably more risky. Sales may be high, with resulting net profit of $8 million, medium with net profit of $4 million, or low, in which case the company just breaks even. The probabilities for these outcomes are:
P(Sales High for B) = 0.38
P(Sales Medium for B) = 0.12
P(Sales Low for B) = 0.50
Product A poses something of a difficulty; a problem with the production system has not been solved. The engineering division has indicated its confidence in solving the problem, but there is a light (5%) chance that devising a workable solution may take a long time. In this event, there will be a delay in introducing the product, and that delay will result in lower sales and profits. Another issue is the price for Product A. The options are to introduce it at either high or low price; the price would not be set until just before the product is to be introduced. Both of these issues have an impact on the ultimate net profit.
Finally, once the product is introduced, sales can be either high or low. If the company decides to set a low price, then low sales are just as likely as high sales. If the company set a high price, the likelihood of low sales depends on whether the product was delayed by the production problem. If there was no delay and the company sets a high price, the probability is 0.4 that sales will be high. However, if there is a delay and the price is set high, the probability is only 0.3 that sales will be high. The following table shows the possible net profits (in millions) for Product A:
Price High Sales
($ Millions) Low Sales
Time delay High 5.0 (0.5)
Low 3.5 1.0
No delay High 8.0 0.0
Low 4.5 1.5
a. Draw a decision tree for GPC. What should GPC do?
b. One of the executives of GPC is considerably less optimistic about Product B and assesses the probability of medium sales as 0.3 and the probability of low sales as 0.4. What decision would this executives make?
2. Suppose that you are the manager for a manufacturing plant that produces CD drives for personal computers. One of your machines produces a part that is used in the final assembly. The width of this part is important to the CD drive’s operation; if it falls below 3.995 or above 4.005 millimeters (mm), the disk drive will not work properly and must be repaired at a cost of $10.40.
The machine can be set to produce parts with a width of 4 mm, but it is not perfectly accurate. In fact, the actual width of a part is normally distributed with mean 4 mm and a variance that depends on the speed of the machine. If the machine is run at a slower speed, the width of the produced parts has a standard deviation of 0.0019 mm. At the higher speed, however, the machine is less precise, producing parts with a standard deviation of 0.0026 mm.
Of course the higher speed means that more parts can be produced in less time, thus reducing the overall cost of the disk drive. In fact, it turns out that the cost of the CD drive when the machine is set at high speed is $20.45. At low speed, the cost is $20.75. Would it be better to run the machine at high or low speed?
3. Most airlines practice overbooking. That is, they are willing to make more reservations than they have seats on an airplane. Why would they do this? The basic reason is simple; on any given flight a few passengers are likely to be “no-shows.” If the airline overbooks slightly, then it still may be able to fill the airplane. Of course, this policy has its risks. If more passengers arrive to claim their reservations than there are seats available, the airline must “bump” some of its passengers. Often this is done by asking for volunteers. If a passenger with a reserved seat is willing to give up his or her seat, the airline typically will give a refund as well as provide a free ticket to the same or another destination. The fundamental trade-offs is whether the additional expected revenue gained by flying an airplane that is nearer to capacity on average is worth the additional expected cost of refunds and free tickets.
To study the overbooking policy, let us look at a hypothetical situation. AC Airlines has a small commuter airplane with places for 16 passengers. The airline uses this jet on a route for which it charges $225 for a one-way fare. Every flight has a fixed cost of $900 (for pilot’s salary, fuel, airport fees, and so on). Each passenger costs AC an additional $100. Finally, the no-shows rate is 4%. That is, on average approximately 4% of those passengers holding confirmed reservations do not show up. Refunds for unused tickets are made only if the reservation is canceled at least 24 hours before scheduled departure. Should AC overbook? By how much?
4. Weekly demand for Motorola cell phones at a Best Buy store has been captured for the past 100 weeks and are shown in the table below.
a. Find a continuous probability distribution that best fits demand.
b. Motorola takes two weeks to supply a Best Buy order, and Best Buy is targeting a service level of 95%. How much safety inventory of cell phones should Best Buy carry and what should its reorder point be if it monitors inventory continuously? Assume weekly demands are independent of each other.
5. The time it takes a mechanic at Golden Muffler Shop to install a muffler is exponentially distributed with an average rate of three per hour. Customers seeking this service arrive at the shop on the average of two per hour, following a Poisson distribution. They are served on a first-in, first-out basis and come from a very large (almost infinite) population of possible buyers.
Using Excel to generate at least 100 random arrivals and service times, estimate the average number of customers waiting in line for service. How long on average does a customer wait before getting service?