Just For You Ltd
Recently, your town has built a new venue to be used by small bands, comedians and
speakers to provide students with a new activity to entertain them. The first event being
held is going to be by a local band.
The designers of the venue have been asked to arrange the audience area of the venue. This
involves deciding the correct amount of floor-level seating and raked seating to incorporate
into their design, in order for the band to make as much profit from the event as possible.
They do not have a budget, therefore they can buy as many materials as they like for this
project, however they need to consider the actual cost of the floor-level seating and raked
seating, the ticket price of an audience member in either
one of these seated positions and what effect this will all
have on the overall profit.
Can you help the designers come to a conclusion as to the
correct amount of floor-level seating and raked seating to
incorporate into their plan, in order to reach their main aim
of maximising the profit?
Modelling
First of all, there are a few bits of information that we must consider before we start.
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The venue size is 16m x 11m therefore it is important to ensure that the standing
space and seating space fits into this size fully (i.e. you cannot have half a chair, nor
half a standing position).
There must be a 1m gap around the edge of the venue, and a 1m aisle down the
middle.
The company specifies that there must be floor-level seating and raked seating. The
floor-level seating must be positioned together at the front, and the raked seating
must be at the back of the venue.
The band specifies that the seating should be organised in rows of 16 minimum.
There must be a 0.5m gap between each row of floor-level seating.
The size of a floor-level chair is 0.5m x 0.5m.
The space which a raked seat takes up is 1mx1m.
1. What are the different combinations of floor-level seating and raked seating
possible? (i.e. ensuring that the whole room is filled as much as possible). Use the
modelling packs to help you investigate.
(Note: After you have found a few combinations, it may be easier to think of a
formula to use to calculate the rest of the combinations.)
We are now given the following costs of the materials:
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A floor-level chair costs £15.
A raked seat costs £8.
The prices of tickets are also given as follows:
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A ticket to sit in the floor-level seating area is £25.
A ticket to sit in the raked seating area is £23.
2. Using this information about prices, what combination of floor-level seating and
raked seating give the greatest profit? (Use your results from question 1.)
We want to make the most amount of profit as possible, therefore it is important to find the
best design for the audience area as possible, with these costs and prices in mind. At the
moment, the only thing stopping us from doing this, is the size of the room (16m x 11m, as
given), and the fact that the band would like the seating to be organised into rows of 16
minimum.
The council have also told us now however, that the maximum capacity of the venue is 220
people.
3. Looking at your table, what combination of floor-level seating and raked seating
gives the greatest profit now?
These are called constraints on the resources.
We can model constraints using algebra.
The constraints for this model can be modelled using two variables:
R – the number of people in raked seating
F – the number of people in floor-level seating
4. Recalling that seating must be organised into rows of 16 minimum, and that there
must be floor-level seating and raked seating, can you write an equation which
shows the greatest amount rows of raked seating which could be used?
5. Similarly, can you write an equation which shows the greatest amount of rows of
floor-level seating possible?
6. What equation can you write to show the minimum amount of raked seating area
and floor-level seating area possible, within the total venue capacity of 220 people?
Graph Work
7. Draw the three constraints on a graph. Choose appropriate scales to use.
8. Find the points on the graph which represent the different combinations of raked
seating and floor-level seating which you worked out earlier. (Remember, your
values do not take into consideration the fact that the total venue capacity is 220
people.)
This area (i.e. where all of the points you have found are positioned) is called the feasible
region. This region contains all of the possible solutions to the problem, and one of them
will be the best. The best solution is called the optimum solution.
The objective is the thing you are aiming for in the problem. In this case, your objective is to
find the combination that will make the greatest profit.
9. Which point on the graph represents the combination that has the greatest profit?
(Recall the constraints we found earlier and remember that we can only have integer
values.)
Modelling the Problem in Excel
This problem can be modelled in Excel and we can use this to help answer several “what-if”
questions, if we wanted to change the problem slightly.
Excel has a what-if analysis tool called Solver that will find the optimal value by changing
values in different cells.
What is the best solution for the designers if:
10. The venue has a maximum capacity of 200 people.
(Use the sheet in your Excel workbook entitled “copy of model 1” to show this.)
11. For every row of raked seating, there has to be two rows of floor-level seating.
(Use the sheet in your Excel workbook entitled “copy of model 2” to show this.)
Relation to Operational Research (OR)
Operational Research involves applying appropriate, and often advanced, analytical
methods to real life problems in order to help make better decisions. This is a standard OR
problem called linear programming. It is called this because all the constraints are straight
lines.
Linear Programming Case Study
Here is an article from “Chron” (a small business website) which highlights one use of linear
programming in business.
How Do Restaurants Use Linear Programming for Menu
Planning?
by Diane Perez, Demand Media
Restaurants use linear programming for menu planning. It uses basic algebra to optimize
meal production and thereby increase restaurant profits. Linear algebra reflects a direct
relationship between an increase or decrease in food resources, and an increase or decrease in
meal production. For example, if the kitchen has only half its needed supply of cream base,
then it can only prepare half its normal amount of cream soups. Additionally, management
can determine the cost of preparing different menu items to decide how many of each menu
item to prepare for optimal profit.
http://smallbusiness.chron.com/restaurants-use-linear-programming-menu-planning-37132.html
Can you think of any more examples where linear programming is used in real life?
Evaluation
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What did you feel confident doing during this lesson?
What do you think you need to improve on?
How do you think you could use linear programming in the future?
What is the benefit of linear programming?