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Autograph _ LP - Linear Programming

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					Using Autograph in the Classroom                                                        Linear Programming


                                       Linear Programming
Linear Programming is a technique used to solve problems which involve maximising or
minimising an objective function subject to constraints in the form of inequalities. With two
variables (x and y) a graphical approach is a good way to solve the problem, and is easily achieved
using a graph plotter such as Autograph.

Example

The problem

A factory produces two types of drink, an ‘energy’ drink and a ‘refresher’ drink. The day’s output
is to be planned. Each drink requires syrup, vitamin supplement and concentrated flavouring, as
shown in the table.

                                                                Vitamin           Concentrated
                                                Syrup
                                                              supplement           flavouring

               1 litre of energy drink        0.25 litres       0.4 units             6 cc

              1 litre of refresher drink      0.25 litres       0.2 units             4 cc

                    Availabilities            250 litres        300 units           4.8 litres


 The last row in the table shows how much of each ingredient is available for the day’s production.

                Energy drink sells at £1 per litre : Refresher drink sells at 80 p per litre
                How can the factory manager decide how much of each drink to make?

The formulation

                              Let x represent number of litres of energy drink
                            Let y represent number of litres of refresher drink

Objective function:
                                             Maximise x + 0.8y

Subject to:

     Syrup constraint:
                                 0.25x + 0.25y  250  x + y  1000

     Vitamin supplement constraint:
                                     0.4x + 0.2y  300  2x + y  1500

     Concentrated flavouring constraint:
                                     6x + 4y  4800  3x + 2y  2400
Bob Francis                                           1                                          March 2006
Using Autograph in the Classroom                                                     Linear Programming

The solution

Draw the graphs representing the constraints; shade out the regions that do NOT satisfy the
inequalities.

            1600      y




            1400




            1200


                      A
            1000




             800


                                             B
             600




             400                                       C
             200


                                                               D              x

- 200   O                    200       400       600           800     1000   1200



            - 200




The unshaded polygon is called the feasible region.

Now work out the value of the objective function x + 0.8y at each of the vertices of the feasible
region:
                   At O:   x + 0.8y = 0 + 0.8  0             = 0
                   At A:   x + 0.8y = 0 + 0.8  1000          = 800
                   At B:   x + 0.8y = 400 + 0.8  600         = 880
                   At C:   x + 0.8y = 600 + 0.8  300         = 840
                   At D:   x + 0.8y = 750 + 0.8  0           = 750

The objective function, x + 0.8y, is a maximum at B(400, 600), giving an income of £880.

Alternatively, graph the objective function, x + 0.8y = k, and see how large you can make the
value of k without leaving the feasible region – this gives a value of k = 880 at point B.


Bob Francis                                                2                               March 2006
Using Autograph in the Classroom                                                    Linear Programming

Problem for you to solve

1.   A factory produces two types of toy: bicycles and trucks. In the manufacturing process of
     these toys, three machines are used. These are a moulder, a lathe and an assembler. The table
     shows the length of time (in hours) needed for each toy.

                                        Moulder          Lathe       Assembler
                          Bicycle           1              3              1
                           Truck            0              1              1

     The moulder can be operated for 3 hours per day, the lathe for 12 hours per day, and the
     assembler for 7 hours per day.

     Each bicycle made gives a profit of £15 and each truck made gives a profit of £11.

     Let x be the number of bicycles made per day and y be the number of trucks made per
     day.

     (i)      Write down the objective function to be maximised and three constraints.

     (ii)     Use Autograph to graph the inequalities, display the feasible region and find the
              combination of x and y that maximises the profit.




Bob Francis                                         3                                      March 2006
Using Autograph in the Classroom                                                    Linear Programming

Problem for you to solve

2.   An oil company has two refineries.
     The amount of oil, in barrels produced per day, is given in the table.

                                   High grade       Medium grade          Low grade
                 Refinery 1           100                 200                 300
                 Refinery 2           200                 100                 200

     An order is received for 2000 barrels of high grade oil, 2000 barrels of medium grade oil and
     3600 barrels of low grade oil.

     Refinery 1 costs £10000 per day to operate and Refinery 2 costs £9000 per day to operate.

     Let x be the number of days for which refinery 1 is operated and y be the number of
     days for which refinery 2 is operated.

     (i)      Write down the objective function to be minimised and three constraints.

     (ii)     Use Autograph to graph the inequalities, display the feasible region and find the
              combination of x and y that minimises cost.




Bob Francis                                         4                                      March 2006

				
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