Downloadable - About University

Exercises 139
been classified as either high or low and it is estimated that there is
a 70% chance that demand will be high.
If the manufacturer opts for a large-scale production run and
demand is high then an estimated profit of $40 000 will be made,
but it is also estimated that 2000 customers who wished to buy the
plate would still be disappointed (production capacity constraints
mean that it would be impossible to meet all the potential demand).
If demand is low then the company would just break even, but no
customers would be disappointed.
If the manufacturer opts for a small-scale production run and
demand is high then an estimated profit of $30 000 will be made
but around 5000 customers would be disappointed. Low demand
would still yield profits of $10 000 and no customers would be
disappointed. It has been established that ‘profit’ and ‘number of
disappointed customers’ are mutually utility independent.
(a) Draw a decision tree to represent the manufacturer’s problem.
(b) The manufacturer’s utility function for profit can be approximated
by the function:
U(x) = 0.4x − 0.0375x 2
where: x = profit in tens of thousands of dollars (this function is
valid for profits from $0 to $40 000, i.e. x values from 0 to 4).
The manufacturer’s utility function for the number of disappointed
customers is given below:
Number of customers
disappointed Utility
0 1.0
2000 0.3
5000 0
Plot these two utility functions on separate graphs and explain
what they show.
(c) After much questioning the manufacturer is able to say that he
is indifferent between alternatives A and B below.
A: A production run which will yield a certain profit of $40 000,
but which will certainly disappoint 5000 customers.
B: A production run which will have a 0.8 probability of a
profit of $40 000 with no customers disappointed and a 0.2
probability of a profit of $0 with 5000 customers disappointed.