Assignment:
Problems
1. Even though independent gasoline stations have been having a difficult time, Susan Solomon has been thinking about starting her own independent gasoline station. Susan's problem is to decide how large her station should be. The annual returns will depend on both the size of her station and a number of marketing factors related to the oil industry and demand for gasoline. After a careful analysis, Susan developed the following table:
|
SIZE OF FIRST STATION
|
GOOD MARKET ($)
|
FAIR MARKET ($)
|
POOR MARKET ($)
|
|
Small
|
50,000
|
20,000
|
-10,000
|
|
Medium
|
80,000
|
30,000
|
-20,000
|
|
Large
|
100,000
|
30,000
|
-40,000
|
|
Very large
|
300,000
|
25,000
|
-160,000
|
For example, if Susan constructs a small station and the market is good, she will realize a profit of $50,000.
a. Develop a decision table for this decision.
b. What is the maximax decision?
c. What is the maximin decision?
d. What is the equally likely decision?
e. What is the criterion of realism decision? Use an α value of 0.8.
f. Develop an opportunity loss table.
g. What is the minimax regret decision?
2. Bill Holliday is not sure what he should do. He can build a quadplex (i.e., a building with four apartments), build a duplex, gather additional information, or simply do nothing. If he gathers additional information, the results could be either favorable or unfavorable, but it would cost him $3,000 to gather the information. Bill believes that there is a 50-50 chance that the information will be favorable. If the rental market is favorable, Bill will earn $15,000 with the quadplex or $5,000 with the duplex. Bill doesn't have the financial resources to do both. With an unfavorable rental market, however, Bill could lose $20,000 with the quadplex or $10,000 with the duplex. Without gathering additional information, Bill estimates that the probability of a favorable rental market is 0.7.
A favorable report from the study would increase the probability of a favorable rental market to 0.9. Furthermore, an unfavorable report from the additional information would decrease the probability of a favorable rental market to 0.4. Of course, Bill could forget all of these numbers and do nothing. What is your advice to Bill?
3. Peter Martin is going to help his brother who wants to open a food store. Peter initially believes that there is a 50-50 chance that his brother's food store would be a success. Peter is considering doing a market research study. Based on historical data, there is a 0.8 probability that the marketing research will be favorable given a successful food store. Moreover, there is a 0.7 probability that the marketing research will be unfavorable given an unsuccessful food store.
a. If the marketing research is favorable, what is Peter's revised probability of a successful food store for his brother?
b. If the marketing research is unfavorable, what is Peter's revised probability of a successful food store for his brother?
c. If the initial probability of a successful food store is 0.60 (instead of 0.50), find the probabilities in parts (a) and (b).
4. Mike Dreskin manages a large Los Angeles movie theater complex called Cinema I, II, III, and IV. Each of the four auditoriums plays a different film; the schedule is set so that starting times are staggered to avoid the large crowds that would occur if all four movies started at the same time. The theater has a single ticket booth and a cashier who can maintain an average service rate of 280 movie patrons per hour. Service times are assumed to follow an exponential distribution. Arrivals on a typically active day are Poisson distributed and average 210 per hour.
To determine the efficiency of the current ticket operation, Mike wishes to examine several queue operating characteristics.
a. Find the average number of moviegoers waiting in line to purchase a ticket.
b. What percentage of the time is the cashier busy?
c. What is the average time that a customer spends in the system?
d. What is the average time spent waiting in line to get to the ticket window?
e. What is the probability that there are more than two people in the system? More than three people? More than four?
5. The wheat harvesting season in the American Midwest is short, and most farmers deliver their truckloads of wheat to a giant central storage bin within a 2-week span. Because of this, wheat-filled trucks waiting to unload and return to the fields have been known to back up for a block at the receiving bin. The central bin is owned cooperatively, and it is to every farmer's benefit to make the unloading/storage process as efficient as possible. The cost of grain deterioration caused by unloading delays, the cost of truck rental, and idle driver time are significant concerns to the cooperative members.
Although farmers have difficulty quantifying crop damage, it is easy to assign a waiting and unloading cost for truck and driver of $18 per hour. The storage bin is open and operated 16 hours per day, 7 days per week, during the harvest season and is capable of unloading 35 trucks per hour according to an exponential distribution. Full trucks arrive all day long (during the hours the bin is open) at a rate of about 30 per hour, following a Poisson pattern.
To help the cooperative get a handle on the problem of lost time while trucks are waiting in line or unloading at the bin, find the
a. average number of trucks in the unloading system.
b. average time per truck in the system.
c. utilization rate for the bin area.
d. probability that there are more than three trucks in the system at any given time.
e. total daily cost to the farmers of having their trucks tied up in the unloading process.
The cooperative, as mentioned, uses the storage bin only two weeks per year. Farmers estimate that enlarging the bin would cut unloading costs by 50% next year. It will cost $9,000 to do so during the off-season. Would it be worth the cooperative's while to enlarge the storage area?
6. Billy's Bank is the only bank in a small town in Arkansas. On a typical Friday, an average of 10 customers per hour arrives at the bank to transact business. There is one single teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times can be described by the exponential distribution. Although this is the only bank in town, some people in the town have begun using the bank in a neighboring town about 20 miles away. If a single teller at Billy's is used, find
a. the average time in the line.
b. the average number in the line.
c. the average time in the system.
d. the average number in the system.
e. the probability that the bank is empty.
7. Refer to the Billy's Bank situation in Problem. Billy is considering adding a second teller (who would work at the same rate as the first) to reduce the waiting time for customers, and he assumes that this will cut the waiting time in half. A single line would be used, and the customer at the front of the line would go to the first available bank teller. If a second teller is added, find
a. the average time in the line.
b. the average number in the line.
c. the average time in the system.
d. the average number in the system.
e. the probability that the bank is empty.