Question

An amateur theatre company wishes to mount a play. A three night run is planned,and a particular play has been chosen. They have already spent or have committed to spend $2500 for such things as costumes, makeup, royalties to the copyright owners, and so on. They are definitely going ahead with the play; the only decision they must make is where to hold it. Small, medium, and large theatres are available for rent which hold 100, 400, and 1200 people respectively. Three nights rent at each theatre would cost $600, $1800, and $4700 respectively. They must make a commitment to one of these theatres several weeks before the run begins.

The theatre company has already decided to price all the tickets at $10.00 each. Because everyone in the theatre company is a volunteer, they can price the tickets at an affordable price. All they care about from a financial point of view is to at least cover their expenses over the long term.

The demand for the play is uncertain until the run begins. Demand is heavily influenced by the critics’ reviews. The critics will attend a dress rehearsal the night before the first performance, and their opinions will be printed and broadcast in the media the next morning.

The directors of the company know from experience that demand for plays falls into four broad categories of interest: fringe; average; great; and heavy. The director has decided to keep the run short and has chosen 3 nights to run the play. We will assume that the demand is spread equally across the three nights. The number of people who wish to see a play each night over a short run is typically 85 for fringe, 270 for average, 775 for great, and 1500 for heavy. These are demand levels, not necessarily the number of tickets sold. For example, if a play sells every seat in a 250 seat theatre for three nights, and if another 50 people were wait-listed for tickets but could not obtain them, then 750 tickets were sold, but the demand was for 800 tickets.

The demand is an event in which one of four outcomes will occur. To estimate the probabilities of these four outcomes, the theatre company could look at the historical data for plays of this type with tickets sold in this price range. Suppose that of one hundred plays in the past, the interest attracted was twenty for fringe, seventy for average, nine for great, and one for heavy. We would then estimate the chance of the next play attracting fringe interest as ??????? ????????=20100=0.20.P(fringe interest)=20/100=0.20.   Continuing in this manner we would estimate the probabilities for average, great, and heavy as 0.70, 0.09, and 0.01 respectively.

Using historical data to estimate probabilities ignores such factors as changing consumer tastes and economic conditions, but we have to start somewhere. Using these numbers we will obtain one conclusion after solving the model, but another set of numbers will often lead to a different conclusion.

This model has been kept simple in that everything has been decided except one thing – which theatre to rent. This is the problem which we shall now solve.

Build a Payoff or Payback Matrix including the salvage value of the discounted (last-minute) tickets.

Build a Decision Matrix to include the Pecimistic (MaxiMin), Optimistic (MaxiMax), Hurwicz (Coefficient of Optimism) and the Laplace (balanced) criteria.

Answer 1

In the given problem the cost of choosing the alternatives are

$600 for 100 seats, $1800 for 400 seats and $4700 for 1200 seats apart from this fee they have already spent or are committed to spend $2500. Thus the total cost for each alternative is $3100 for 100 seats, $4300 for 400 seats and $7200 for 1200 seats.

The revenue earned under all criteria is as follows

The Profit is as follows

Using Pecimistic criteria

The minimum loss is incurred with an option to choose 100 seaters small theatre.

Using Optimist criteria

The Maximum profit of 28800 is incurred using 1200 seaters large theatre.

Using Laplace (balanced) criteria

The maximum profit using Laplace criteria is obtained by using a large theatre of 1200 capacity

Using Hurwicz criteria

The maximum profit incurred using Hurwicz criteria is by renting 400 seaters medium-size theatre.