Question 1- An entertainment company is organizing a pop concert in London. The company has to decide how much it should spend on publicizing the event, and three options have been identified:
Option 1: Advertise only in the music press;
Option 2: As option 1, but also advertise in the national press;
Option 3: As options 1 and 2, but also advertise on commercial radio.
For simplicity, the demand for tickets is categorized as low, medium or high. The payoff table below shows how the profit that the company will earn for each option depends on the level of demand:
|
Demand
|
|
Option
|
Low
|
Medium
|
High
|
Profits ($000s)
|
|
1
|
-20
|
-20
|
100
|
|
|
2
|
-60
|
-20
|
60
|
|
|
3
|
-100
|
-60
|
20
|
|
It is estimated that if option I is adopted the probabilities of low, medium and high demand are 0.4, 0.5 and 0.1 respectively. For option 2 the respective probabilities are 0.1, 03 and 0.6, while for option 3 they are 0.05, 0.15 and 0.8. Determine the option that will lead to the highest expected profit would you have any reservations about recommending this option to the company?
Question 2- A company has to decide whether to invest money in the development of a microbiological product. The company's research director has estimated that there is a 60% chance that a successful development could be achieved in 2 years. However, if the product had not been successfully developed at the end of this period, the company would abandon the project, which would lead to a loss in present value terms of $3 million. (Present value is designed to take the company's time preference for money into account. The concept is explained in Chapter 8.) In the event of a successful development, a decision would have to be made on the scale of production. The returns generated would depend on the level of sales that could be achieved over the period of the product's life. For simplicity, these have been categorized as either high or low. If the company opted for large-volume production and high sales were achieved, then net returns with a present value of $6 million would be obtained. However, large-scale production followed by low sales would lead to net returns with a present value of only $1 million. On the other hand, if the company decided to invest only in small-scale production facilities, then high sales would generate net returns with a present value of $4 million, and low sales would generate net returns with a present value of $2 million. The company's marketing manager estimates that there is a 75% chance that high sales could be achieved.
(a) Construct a decision tree to represent the company's decision problem.
(b) Assuming that the company's objective is to maximize its expected returns, determine the policy that it should adopt.
(c) There is some debate in the company about the probability that was estimated by the research director. Assuming that all other elements of the problem remain the same, determine how low this probability would have to be before the option of not developing the product should be chosen.
(d) Before the ?nal decision is made, the company is taken over by a new owner who has the utilities shown below for the sums of money involved in the decision. (The owner has no interest in other attributes that may be associated with the decision, such as developing a prestige product or maintaining employment.) What implications does this have for the policy that you identi?ed in (b) and why?
|
Present value of net returns
|
New owner's utility
|
|
-$3 m
|
0
|
|
$0 m
|
0.6
|
|
$1 m
|
0.75
|
|
$2 m
|
0.85
|
|
$4 m
|
0.95
|
|
$6 m
|
1
|
Question 3- A small retail store sells a particular brand of monochrome and color television sets. Each monochrome set that is sold earns a profit of $100, while each color set earns $200 profit. The manager of the store estimates that the weekly demands for each type of set follow the probability distributions shown below (it can be assumed that the demands for each type of set are independent, as is the week-to-week demand):
|
Probability
|
|
Demand per week
|
Monochrome sets
|
Color sets
|
|
0
|
0.2
|
0.4
|
|
1
|
0.5
|
0.6
|
|
2
|
0.3
|
|
(a) Determine the possible total profits that can be earned in any given week by selling television sets, and calculate the probability of each of these profits being earned.
(b) The following two-digit random numbers have been generated by a computer. Use these numbers to simulate the demand for the two types of set for a 10 week period and hence calculate the profit that will be earned in each week (the first set of numbers should be used for monochrome sets and the second for color):
|
Monochrome
|
71
|
82
|
19
|
50
|
67
|
29
|
95
|
48
|
84
|
32
|
|
Color
|
36
|
44
|
64
|
92
|
39
|
21
|
18
|
55
|
77
|
73
|
(c) Use your simulation results to estimate the probability of particular profits being earned in a given week. Now close are these probabilities to those that you calculated in (a)?