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Linear Programming independent research project Nick Yates 30 th April 2008

Linear Programming independent research project Nick Yates 30 th April 2008. Motivation. Linear programming = optimize a (linear) function given various (linear) constraints Applications business (maximize profit, with resources available)

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Linear Programming independent research project Nick Yates 30 th April 2008

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  1. Linear Programming independent research project Nick Yates 30th April 2008

  2. 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

  3. Linear Program Standard Form Maximize objective function Subject to constraints

  4. Duality Minimize related objective function Subject to constraints

  5. 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 z at every vertex!

  6. Example: Graphical Method • Our original system of constraints is graphed here, overlapping in a triangle. • z(0,0) = 2(0)+0 = 0 • z(1,0) = 2(1)+0 = 2 • z(0,.2) = 2(0)+.2 = .2

  7. 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)

  8. 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!

  9. 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

  10. Lesson Plan You set up a tutoring service in math and engineering for eleventh 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?

  11. Questions? Linear Programming independent research project Nick Yates 30th April 2008

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