Linear Programming

Week 9 Lecture 1

Linear programming (LP)

• LP is a method used to model a problem where maximum or minimum values of some variable need to be identified

• LP is used to model real-world problems

Linear Programming

• A linear program consists of:– An object function (a maximisation or

minimisation function)– A number of constraints

• A number of issues related to linear algebra need to be recalled to help solve linear programs

Linear algebra

• The equation of a line can be written as:ax+by=c

• The values of ‘a’ and ‘b’ determine the slope of this line and the value of ‘c’ determines its actual position.

• Consider some examples:– Draw the following lines on the x-y axis

• 5x+10y=50• 5x+10y=200

Linear algebra

• The slope of a line:– Rewrite the equation for a line in the form of y=mx+c.

Now m=slope.– Take the previous examples:

• 5x+10y = 50• 10y = -5x+50• y = -0.5x+5• Slope= -0.5• Changing the value of c (50) will not change the slope of the

line• Draw some more lines on the x-y axis that have a slope of

-0.5

Linear algebra

• Draw 2 lines that each have the following slope:m=2

m=-2

m=0.5

m=-5

• What can be said about the following line:ax+by=0 ( can also be written as: y=(-a/b)x )

Linear Programming

• A objective function (a minimisation function or a minimisation function) normally looks like the following:– Maximise: 2x+5y

• Can you tell the slope of all lines that look like this?

• Can you draw some of them?• From the lines you have drawn, can you tell

where 2x+5y is at a maximum?

Linear Programming

• Draw the following lines:x+y=6

-x-2y=-18

x=0

y=0

Linear Programming

• Where do each of the following inequalities hold? (explain):(1) x+y >= 6

(2) -x-2y >= -18

(3) x >= 0

(4) y >= 0

• Where do (1) AND (2) hold?

• Where do all four inequalities hold?

Linear Programming

• The inequalities on the previous slide are typical of the constraints in a linear program.

• Now draw these four inequalities and a line with a slope corresponding to the objective function

• Can you now tell where 2x+5y has a maximum value that satisfies the constraints?

• Where will 2x+5y have a minimum value and still satisfy the constraints?

Linear Programming

• The point(s) at which the objective function reaches a maximum of minimum value is either:– The point of intersection of two of the lines

representing the constraints (inequalities).– A set of points along one of the constraints

that is still part of the feasible region• When can this occur?

Linear algebra

• Find the point of intersection of the following two lines:x=0y=0

• Find the point of intersection of the following two lines:x=0x+y=6

• Find the point of intersection of the following two lines:x+y=6-x-2y=-18

Practical example

• Use linear programming to model the following problem. Use both a mathematical model and a diagram to illustrate the problem

• A television set manufacturing company has to decide on the mix of LCD and Plasma screen TVs to be produced. Market Research indicates that at most 1000 and 4000 units of LCD and Plasma screens respectively can be sold per month. The maximum man-hours available per month is 50,000. A plasma screen requires 15 man-hours and an LCD requires 20 man hours. The unit profit is 180 and 90 euros for LCD and Plasma respectively. How many units of each TV should be produced to maximise the profit?

Some Links

• http://www.youtube.com/watch?v=M4K6HYLHREQ

• http://www.purplemath.com/modules/linprog3.htm

• http://people.brunel.ac.uk/~mastjjb/jeb/or/morelp.html

Summary

• Linear programming provides a way of finding the best solution to certain problems

• A linear program is a mathematical model• It can also be represented visually using a graphical

model (a diagram)• Linear algebra can help solve the linear program. The

graphical model can be used to solve the linear program also but it needs to be drawn very accurately. It is normally best to use linear algebra to find the best solution and to check this against the graphical model.

• Note: If this type of question is asked in the exam, graph paper will NOT be available to draw an accurate graphical model and so the graphical model should be used to help visualise the problem and to verify your answer.