Decision Maths Solving Linear Programming problems

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.

Final - ProblemLets consider the example we looked at in lesson 7.Maximise the profit function.P = 60x + 84ySubject to the constraints

2x + 3y 305x + 5y 60x 0, y 0

The Graph (drawing the lines)To draw 5x + 5y = 60 and 2x + 3y = 30 find the intersections with both axes. i.ex=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 (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.

The Graph (shading inequalities)We can now do a similar thing with the constraint inequalities.If 2x + 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. From this we know to shade in the region above the line.

The Feasible Region

We can do a similar thing with 5x + 5y 60, and again we shade in the region above the line.

Questions1 Indicate on a diagram the region for which5x + 3y 15x 0, y 02 - Indicate on a diagram the region for which4x + 3y 122x + 5y 10x 0, y 03 - Indicate on a diagram the region for which2x + y 8y 7, x 3x 0, y 0

Questions4 Indicate on a diagram the region for whichy + 2x 12x 2, y 45 - Indicate on a diagram the region for which3x + 2y 123x + y 6 x + y 46 - Indicate on a diagram the region for which3x + 2y 6y 2xy 0

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.

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.

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

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

Maximising the Profit Function Method iThe maximum profit will come from the values of x and y at one of the vertices.We can now find the value of the profit function at each of these points:

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

xyP =60x+84y00012072066864010840

Maximising the Profit Function Method iiThe ruler Method

You draw the object function 60x + 84y = P. (here pick P to be any value to help you, it makes no difference)

Maximising the Profit Function Method iiThe ruler Method

Once you have drawn the line move it to the last point available in the region keeping the line parallel.

Maximising the Profit Function Method iiThe 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