To celebrate the centenary of the local football club, a pottery manufacturer has decided to produce a commemorative plate. A choice has to be made on whether to have a large-scale production run or a small-scale run. For technical reasons only a large- or small-scale run is possible and it would not be economical, because of other commitments, to have more than one production run. The manufacturer has two objectives: (i) to maximize the profits earned from the plate and (ii) to minimize the number of customers who will be disappointed because insufficient plates were produced (a large number of disappointed customers would not be good for customer goodwill). For simplicity, the potential demand for the plates has 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 − 0.0375x2

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 disap pointed 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.

After further questioning he is also able to say that he is indifferent between options C and D below:

C: A production run which will certainly yield a profit of $0 but which is also certain to disappoint no customers.

D: A production run which will have a 0.6 probability of a profit of $40 000 with no customers disappointed and a 0.4 probability of a profit of $0 with 5000 customers disappointed.

(i) Determine whether the manufacturer should choose the large- or small-scale production run in order to maximize his expected utility.

(ii) Interpret the expected utilities which you obtained in part (i).

(d) Would the Simple Multi-attribute Rating Technique (SMART) have offered a better way of tackling the pottery manufacturer’s problem?