Operational Research

1
Operational Research

• Marketing and Business Systems
• Linear programming

2
Linear Programming

• Linear programming is, like most OR techniques,
  about optimising systems to ensure that profit is
  maximised
• The power of computers has really brought about
  the widespread use of OR and Linear Programming
  has become one of its main features.

3
A Simple Example

• Suppose a factory produces two different types of
  ipods, Ant and Dec. Each of the ipods require 3
  different factory operations for their
  production Moulding, Electronics and Assembling
• To produce one Ant requires 1 hour in Moulding, 3
  hours in Electronics and 1 hour in Assembling
• To produce one Dec requires 0 hours in Moulding,
  1 hour in Electronics and 1 hour in Assembling
• Clearly it looks as if it is cheaper to produce a
  Dec ipod

4
Some Constraints

• But as with all businesses there are constraints
  on some operations
• The Moulding Department can only operate for 3
  hours per day
• The Electronics Department can operate for a
  maximum of 12 hours per day
• The Assembling Department can operate for up to 7
  hours per day

5
Further Assumptions

• We are going to assume that everything in the
  factory is sold and that each Ant sold yields a
  profit of 8, while each Dec sold gives a profit
  of 5.
• The factory managers problem is determining how
  best to allocate the three separate operations to
  maximise profit.

6
Pulling together the information
7
The Profit Function

• The manufacturer will make a profit of (8x) on
  Ant per day and (5y) on Dec per day where
• x is the number of Ants sold per day
• y is the number of Decs sold per day
• So the total profit is
• P 8x 5y

8
The Profit Function

• So, for example if 4 Ants and 10 Decs were sold
  in one day then the total profit for the day
  would be
• P 8(4) 5(10) 82
• Or, if 9 Ants and 3 Decs were sold in one day
  then the total profit for the day would be
• P 8(9) 5(3) 87
• Clearly knowing how many of each product to make,
  given the constraints, is vital

9
More on the constraints

• If we could now make P as big as we like we would
  have no restriction on the profit we could make.
• But we know that Moulding is only operating for 3
  hours each day, so
• x lt 3
• (this symbol means that x must be less than or
  equal to 3)
• Also Electronics needs 3x hours to produce x Ants
  and y hours to produce y Decs and we are
  restricted to only 12 hours per day
• Thus we have another inequality 3xy lt 12

10
Nearly There!

• A similar argument for assembling, which needs
  one hour to produce an Ant and one for a Dec, and
  only operates for 7 hours per day gives
• x y lt 7
• And of course x and y cannot be negative so
• 0 lt x and 0 lt y
• This can now be solved graphically although, in
  general, we would use a computer

11
The solution graph
y
A constraint line is added one at a time
7
Here the number of x (Ant) and y (Dec) cannot be
more than 7
5
x y 7
3
0
x
7
5
3
2
1
12
The solution graph
y
12
Another constraint line is added
3x y 12
Here the number of 3x (Ant) and y (Dec) cannot
be more than 12
7
5
3
x y 7
0
x
7
5
3
2
1
4
13
One more constraint line is added
x 3 is added
12
The Feasible Region e.g any solution must be in
here
7
x
0
3
4
7
14
Computers can now pinpoint the optimium solution
exactly

• Remember that we are trying to maximise the
  profit which is P 8x 5y
• And we only want a solution of this in the
  feasible region. This occurs generally at a
  vertex which, in this case, gives the optimal
  solution as x 2.5 and y 4.5
• In other words the maximum profit of 42.50 a day
  is obtained by making 2.5 Ants and 4.5 Decs per
  day (or 5 and 9 in two days)

15
Optimum Solutions

• Note that this is an exact solution in the sense
  that there is no way, given the constraints, that
  the business can make more profit
• Other types of optimisation problems are also
  suited to this procedure
• Pension Funds
• Transport solutions
• Cattle Food

16
Further Reading

• Any book on OR will give much more detail and
  highlight the additional features of operational
  research
• Many of the recent books on OR also come with
  computer programmes that solve more complex
  problems instantly

17
Seminar Problem

• A Bordeaux winemaker grows three grape varieties
  Cabernet Sauvignon, Malbec and Merlot
• He uses combinations of these for wines of
  differing quality
• The blends are, in respect to Cabernet Sauvignon,
  Malbec, Merlot
• Premium Quality, ratio of (421)
• Standard Quality, ratio of (122)

18
Seminar Problem

• He can sell the wine at a profit of 15 euro per
  bottle of premium wine and 10 euro per bottle of
  standard wine
• The current 2004 vintage is expected to produce
  24000 litres of juice from the Cabernet Sauvignon
  grapes, 30000 litres from the Malbec and 28000
  from the Merlot
• How much of each wine should be produced to
  maximise profit?

19
Seminar Problem

• Perform all the calculations in bottles, using 1
  bottle 0.75 litre (So 24000 litres of Cabernet
  Sauvignon is the same as 32000 bottles)
• Let x be the number of bottles of premium wine
  and y be the number of bottles of standard wine
• Make a table