Management Science
1. There is a fixed cost of $50,000 to start a production process. Once the process has begun, the variable cost per unit is $25. The revenue per unit is projected to be $45. Find the break-even point.
2. Administrators at a university are planning to offer a summer seminar. It costs $3000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $25 per student for the administrators to provide the course materials. If we know that 20 people will attend, what price should be charged per person to break even?
Probability and Statistics
3. An automotive center keeps track of customer complaints received each week. The probability distribution for complaints can be represented as a table, shown below. The random variable xi represents the number of complaints, and p(xi) is the probability of receiving xi complaints.
|
xi
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
|
p(xi)
|
.10
|
.15
|
.18
|
.20
|
.20
|
.10
|
.07
|
a. What is the probability that they receive less than 3 complaints in a week?
b. What is the average number of complaints received per week?
4. A loaf of bread is normally distributed with a mean of 22 ounces and a standard deviation of 0.5 ounces. What is the probability that a loaf is larger than 21 ounces?
Decision Analysis
5. A payoff table (profits) is shown below.
|
|
States of Nature
|
|
Decisions
|
S1
|
S2
|
S3
|
|
D1
|
10
|
8
|
6
|
|
D2
|
14
|
15
|
2
|
|
D3
|
7
|
8
|
9
|
a. Using the maximax criterion, what decision should be made by the decision maker?
b. Using the maximin criterion, what decision should be made by the decision maker?
c. Using an equal likelihood criterion, what decision should be made by the decision maker?
d. Using minimax regret criterion, what decision should be made by the decision maker?
e. If the probabilities of s1, s2, and s3 are 0.2, 0.4, and 0.4, respectively, what decision should be made by the decision maker?
6. The local operations manager for the IRS must decide whether to hire 1, 2, or 3 temporary workers. He estimates that net revenues will vary with how well taxpayers comply with the new tax code. The probabilities of low, medium and high compliance are 0.3, 0.4, and 0.3, respectively, and the payoff table is shown below. Using expected values, determine how many workers the company should hire.
|
# of workers
|
Low compliance
|
Medium compliance
|
High compliance
|
|
1
|
50
|
50
|
50
|
|
2
|
20
|
60
|
100
|
|
3
|
-10
|
70
|
150
|
Simulation
7. The number of cars arriving at Joe Kelly's oil change and tune-up place during the last 200 hours of operation is observed to be the following.
|
Number of cars arriving
|
Frequency
|
|
4
|
10
|
|
5
|
30
|
|
6
|
70
|
|
7
|
50
|
|
8
|
40
|
a. Determine the probability distribution, and the cumulative probability distribution of car arrivals.
b. Simulate 20 hours of car arrivals at Joe Kelly's oil change and tune-up place.
c. For the simulation in (b), what is the average number of cars arriving per hour?
Forecasting
8. Given the following data on hotel check-ins for a 6-month period:
|
Month
|
Number of rooms
|
|
July
|
70
|
|
August
|
105
|
|
September
|
90
|
|
October
|
120
|
|
November
|
110
|
|
December
|
115
|
a. What is the 3-month moving average for January?
b. What is the 5-month moving average for January?
9. Recent actual and forecasted data for product XYZ is given in the following table.
|
Month
|
Actual Demand
|
3-month Forecasted Demand
|
|
February
|
20
|
-
|
|
March
|
22
|
-
|
|
April
|
33
|
-
|
|
May
|
35
|
25
|
|
June
|
31
|
30
|
|
July
|
48
|
33
|
|
August
|
41
|
38
|
|
September
|
-
|
40
|
Determine the MSE, MAD, cumulative error and average error.
10. The following table summarizes data between money spent on gambling and winnings for Robert.
|
Money Spent
|
Money Won
|
|
x
|
y
|
|
12
|
62
|
|
10
|
54
|
|
16
|
86
|
|
18
|
100
|
|
15
|
80
|
|
10
|
57
|
|
5
|
26
|
|
12
|
60
|
|
22
|
105
|
|
25
|
140
|
Develop a linear regression equation for these data and forecast how much money Robert will win if he spends $30.