A local council owns an underground railway system and the railway"s managers have to make a decision on whether to lower fares in an attempt to increase passenger numbers. If they decide to reduce fares they will then have to decide whether to launch an advertising campaign on local television to increase awareness of the fare reduction. If fares remain the same then it is estimated that there is a 0.7 probability that the mean number of passengers carried per day over the next year will equal 20 000 and a 0.3 probability that it will decline to 15 000. The annual profits associated with these passenger numbers are estimated to be $3 million and $1 million, respectively. If fares are reduced, but television advertising is not used, then it is thought that there is a 0.6 probability that the mean number of passengers carried will increase to 25 000 and a 0.4 probability that it will increase to 22 000. The resulting profits generated by these passenger numbers are estimated to be $2 million and $1.7 million, respectively. Advertising the fare reduction on television would increase the probability of an increase to a mean of 25 000 passengers to 0.8 and reduce the probability that the mean will be 22 000 to 0.2. However, it would reduce the profits associated with these mean passenger numbers by $0.6 million. The railway"s objectives are to maximize profit and to maximize passenger numbers (since this

brings environmental benefits such as reduced traffic congestion).

(a) Utility functions for the mean numbers of passengers carried and profit have been elicited from the railway"s chief executive and these are given below.

Mean number of passengers	Utility
15 000	0
20 000	0.8
22 000	0.95
25 000	1
Profit	Utility
$1.0m	0
$1.1m	0.2
$1.4m	0.6
$1.7m	0.75
$2.0m	0.9
$3.0m	1.00

Plot these utility functions and interpret them.

(b) The elicitation session revealed that, for the chief executive, mean number of passengers and profit are mutually utility independent. You are reminded that, in this case, a two-attribute utility function can be obtained from:

u(x1, x2) = k1u(x1) + k2u(x2) + k3u(x1)u(x2)

where k3 = 1 - k1 - k2.

The elicitation session also revealed that k1 = 0.9 and k2 = 0.6, where attribute number 1 is the mean number of passengers. Determine the policy that the railway should pursue in the light of the above utilities and comment on your answer.

(c) Discuss briefly other multi-criteria decision making models or methods that could be used to assist managerial decision making in the context of above example.

A local council owns an underground railway system and the railway"s managers have to make a decision on whether to lower fares in an attempt to increase passenger numbers. If they decide to reduce fares they will then have to decide whether to launch an advertising campaign on local television to increase awareness of the fare reduction. If fares remain the same then it is estimated that there is a 0.7 probability that the mean number of passengers carried per day over the next year will equal 20 000 and a 0.3 probability that it will decline to 15 000. The annual profits associated with these passenger numbers are estimated to be $3 million and $1 million, respectively. If fares are reduced, but television advertising is not used, then it is thought that there is a 0.6 probability that the mean number of passengers carried will increase to 25 000 and a 0.4 probability that it will increase to 22 000. The resulting profits generated by these passenger numbers are estimated to be $2 million and $1.7 million, respectively. Advertising the fare reduction on television would increase the probability of an increase to a mean of 25 000 passengers to 0.8 and reduce the probability that the mean will be 22 000 to 0.2. However, it would reduce the profits associated with these mean passenger numbers by $0.6 million. The railway"s objectives are to maximize profit and to maximize passenger numbers (since this brings environmental benefits such as reduced traffic congestion). (a) Utility functions for the mean numbers of passengers carried and profit have been elicited from the railway"s chief executive and these are given below. Mean number of passengers Utilit y 15 000 0 20 000 0.8 22 000 0.95 25 000 1 Profit Utility $1.0m 0 $1.1m 0.2 $1.4m 0.6 $1.7m 0.75 $2.0m 0.9 $3.0m 1.00 Plot these utility functions and interpret them. (b) The elicitation session revealed that, for the chief executive, mean number of passengers and profit are mutually utility independent. You are reminded that, in this case, a two-attribute utility function can be obtained from: u(x1, x2) = k1u(x1) + k2u(x2) + k3u(x1)u(x2) where k3 = 1 - k1 - k2. The elicitation session also revealed that k1 = 0.9 and k2 = 0.6, where attribute number 1 is the mean number of passengers. Determine the policy that the railway should pursue in the light of the above utilities and comment on your answer. (c) Discuss briefly other multi-criteria decision making models or methods that could be used to assist managerial decision making in the context of above example