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    Question one
    A local authority in the USA owns a tramway system; and the tram operators are
    under pressure to increase passenger numbers. They 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 a TV
    advertising campaign 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. In
    addition, a 0.3 probability that the number will decline to 15 000.
    The annual profits associated with these passenger numbers are estimated to be
    $3million and $1million, respectively.
    If the fares are reduced, but TV 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.
    In addition, a 0.4 probability that the number will increase to 22 000. The resulting
    profits generated by these passenger numbers are estimated to be $2million and
    $1.7 million, respectively.
    TV advertising of the fare reduction 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 tram operating company’s objectives are to
    [A] maximise profit and to [B] maximise passenger numbers. Note This second
    objective is in place as the local authority wish to bring forward social & environmental
    benefits {e.g. improved air quality, more people walking and reduced traffic congestion}.
    (a) Utility functions for the mean numbers of passengers carried and the profit
    have been obtained from the trams operato
    (b) The elicitation session revealed that, for the CEO, 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 the
    attribute number 1 is the mean number of passengers. Determine the policy
    that the tramway should undertake in the light of the above utilities; and
    comment on your answer.
    (30 marks)
    (c) Discuss briefly other multi-criteria decision making models or methods that
    could be used to assist managerial decision making in the context of the
    above example.
    (10 marks)

    Question Two
    A group of medical professionals (dentists, doctors, physiotherapists etc) is
    considering the construction of a small private hospital in a remote part of South
    America. If medical demand is high (i.e. .if there is a favourable market for the
    hospital’s services), the group could realise a net profit of £10m.
    If the market is not favourable, they could lose £4m.
    Of course, they do not have to proceed at all in which case there is no cost.
    In the absence of any market data, the best the group and their advisers can
    estimate is that there is a 50:50 chance that the hospital will be successful.

    (a) Draw a decision tree that describes the above decision problem and advise
    the group on the optimal decision.
    (10 marks)
    The same group of medical professionals, have been approached by a market
    research firm that offers to perform a study of the market (for a fee of £30,000). The
    market researchers claim their experience has enabled them to use Bayes’ theorem
    to make the following six statements of probability:
    1. Probability of a favourable research study = 0.55
    2. Probability of a favourable market given a favourable study = 0.82
    3. Probability of an unfavourable market given a favourable study = 0.18
    4. Probability of an unfavourable research study = 0.45
    5. Probability of a favourable market given an unfavourable study = 0.11
    6. Probability of an unfavourable market given an unfavourable study = 0.89
    (b) Develop a new decision tree for the medical professionals to reflect the
    options now open with the market study.
    (10 marks)
    (c) Use the Expected Monetary Value (EMV) approach and recommend an
    optimal strategy.
    (10 marks)
    (d) Advise the medical professionals on the maximum amount that they should
    be willing to pay for the market study. (5 marks)
    (e) List a number of potential risks that the group should consider given the
    proposed hospital is located abroad. (5 marks)

    (f) Compare and contrast decision tree analysis and use of influence diagrams in
    managerial decision making involving risk and uncertainty.
    (10 marks)
    Total Marks for Questions One and Two: 100

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