D1,L9 Solving Linear Programming Problems

7/29/2019 D1,L9 Solving Linear Programming Problems

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Decision Maths

Solving Linear

Programming problems


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Solving the problem.

There are a number of ways in which to solve the

linear programming problem that we looked at in the

last lesson.

1 You can draw the equations graphically and theni Solve by evaluating the vertices.

ii Solve using the ruler method.

2 You can apply the Simplex Algorithm (this is not

in the D1 syllabus and we will not be studying this) Taking last weeks example, we are going to look at

representing the algebra graphically and then we will

consider both methods to solve this problem.


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Final - Problem

Lets consider the example we looked at in lesson 7.

Maximise the profit function.

P = 60x + 84y

Subject to the constraints

2x + 3y 30

5x + 5y 60

x 0, y 0


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To draw 5x + 5y = 60 and 2x + 3y = 30 find the intersections with both axes.

i.e x=0 y=12 and y=0 x=12 the line joins (0,12) and (12,0).

And x=0 y=10 and y=0 x=15 the line joins (0,10) and (15,0)

The Graph

(drawing the lines)

0

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4

68

10

12

14

0 5 10 15 20

CraftsmanMachine


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The Graph

(shading inequalities)

As we know that x and y are both 0, we can shade in the

region of co-ordinates that we do not require.

0

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4

68

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0 5 10 15 20

CraftsmanMachine


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The Graph

(shading inequalities)

We can now do a similar thing with the constraint inequalities.

If2x + 3y 30, then choose any point and plug in to the inequality.

Using the point (2,2), 2 x 2 + 3 x 2 = 10, which is less than 30. Fromthis we know to shade in the region above the line.

0

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0 5 10 15 20

CraftsmanMachine


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The Feasible Region

We can do a similar thing with 5x + 5y 60, and

again we shade in the region above the line.

0

2

4

6

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0 5 10 15 20

CraftsmanMachine


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Questions

1 Indicate on a diagram the region for which

5x + 3y 15

x 0, y 0

2 - Indicate on a diagram the region for which4x + 3y 12

2x + 5y 10

x 0, y 0

3 - Indicate on a diagram the region for which2x + y 8

y 7, x 3

x 0, y 0


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Questions

4 Indicate on a diagram the region for which

y + 2x 12

x 2, y 4

5 - Indicate on a diagram the region for which

3x + 2y 12

3x + y 6

x + y 4

6 - Indicate on a diagram the region for which

3x + 2y 6

y 2x

y 0


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The Feasible Region

The feasible region is all of the co-ordinates that lie in the un-

shaded area. Remember that each co-ordinate (x,y) represents

the number of each shed made.

0

2

4

68

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0 5 10 15 20

CraftsmanMachine


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The Feasible Region

The vertices of the feasible region are: (0,0), (12,0), (6,6) and (0,10).

The co-ordinate (6,6) can be found by solving the simultaneous

equations 2x + 3y = 30 and 5x + 5y = 60.

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0 5 10 15 20

CraftsmanMachine


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Simultaneous Equations

(6,6) is found by solving 2x+3y=30 and 5x+5y=60simultaneously.

2x + 3y = 30

5x + 5y = 60

2x + 3y = 30

3x + 3y = 36

x = 6

2 x 6 + 3y = 30

12 + 3y = 30

3y = 18

y = 6


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Maximising the Profit

FunctionMethod i

The maximum profit will come from the values of x and y at oneof the vertices.

We can now find the value of the profit function at each of thesepoints:

the maximum profit of 864 can be obtained by making 6sheds of each type each day.

x y P =60x+84y0 0 0

12 0 720

6 6 8640 10 840


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FunctionMethod ii

0

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0 5 10 15 20

Craftsman

Machine

The ruler Method

You draw the object function 60x + 84y = P. (here pick P to be

any value to help you, it makes no difference)


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FunctionMethod ii

0

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Craftsman

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The ruler Method

Once you have drawn the line move it to the last point available

in the region keeping the line parallel.


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FunctionMethod ii

The co-ordinate that it lands on will give you

the values of x and y that will optimise the

function.

Here, as we know already x = 6, y = 6. So P = 60x + 84y

= 60 x 6 + 84 x 6

= 864

D1,L9 Solving Linear Programming Problems

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