Linear Programmingindependent research project

Nick Yates

30th April 2008

Motivation

• Linear programming = optimize a (linear) function given various (linear) constraints

• Applications– business (maximize profit, with resources

available)– engineering (design a prototype with best

possible performance, subject to constraints on size and material cost)

• Big Ideas• Graphical method• Simplex method

Linear Program Standard Form

Maximize objective function

Subject to constraints

212 xx 212 xx 212 xx

0,

15

54

332

42

2

21

21

21

21

21

21

xx

xx

xx

xx

xx

xx

Duality

Minimize related objective function

Subject to constraints

212 xx 212 xx 212 xx

0,

153

2422

534

21

4321

4321

4321

yy

yyyy

yyyy

yyyy

Example: Graphical Method

• Graph all constraints (inequalities)

• Creates polygon if 2D, polytope if more

• Find all vertices

• Maximal (optimal) value will be at a vertex

• So just evaluate your objective function at every vertex!

Example: Graphical Method

• Our original system of constraints is graphed here, overlapping in a triangle.

• (0,0) = 2(0)+0 = 0

• (1,0) = 2(1)+0 = 2

• (0,.2) = 2(0)+.2 = .2

Example: Simplex Method

• Add new “slack” variables to simplify constraints

• Now pick a starting point, identify a variable that can be increased, and use (matrix) row operations to move around variables (x’s and w’s)

0 somefor

42 42

1

12121

w

wxxxx

Benefits of Simplex Method

• Similar in theory—slides along edges of polytope from vertex to vertex

• Don’t have to draw the graph

• This is especially important for linear programs with more than 3 variables—since we can’t see in 4 dimensions!

Lesson Plan

• Use in an Algebra 2 class

• Focus on Graphical Method (Simplex Method too complicated and abstract)

• Review graphing inequalities

• Begin with real-world apps as motivation

• Provide several partly-done and highly-structured examples to ease into full-length project

Lesson PlanYou set up a tutoring service in math and engineering foreleventh grade students in CIM and Algebra 2. You charge $10 an hour for math help and $15 an hour for engineering help. You wish to take on no more than eight students total (math + engineering). Only three engineering students are In need of your services, while lots of math students are.

– Set up a table– Write the objective function P for profit (or pay).– Graph the constraints on graph paper.– Find the four vertices.– Evaluate the objective function at each vertex.– What is the most money you can make?– How many of each type of student have you taken on in that

situation?

Questions?

Linear Programmingindependent research project

Nick Yates30th April 2008