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chap4_handout - Chapter 4: The Simplex Method Chapter 4:...

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Chapter 4: The Simplex Method Chapter 4: The Simplex Method I courtesty of xkcd.com Chapter 4: The Simplex Method 1
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Chapter 4: The Simplex Method Outline Outline Graphical Interpretation of the Simplex Method (Section 4.1) Setting up the Simplex Method (Section 4.2) The Algebra of the Simplex Method (Section 4.3) The Simplex Method in Tabular Form (Section 4.4) Tie Breaking in the Simplex Method (Section 4.5) Adapting to Other Model Forms (Section 4.6) Chapter 4: The Simplex Method 2
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Chapter 4: The Simplex Method Graphical Interpretation of the Simplex Method (Section 4.1) Outline Graphical Interpretation of the Simplex Method (Section 4.1) Introduction Important Properties Solving the Wyndor Glass Example Key Solution Concepts Setting up the Simplex Method (Section 4.2) The Algebra of the Simplex Method (Section 4.3) The Simplex Method in Tabular Form (Section 4.4) Tie Breaking in the Simplex Method (Section 4.5) Chapter 4: The Simplex Method 3
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Chapter 4: The Simplex Method Graphical Interpretation of the Simplex Method (Section 4.1) Introduction Recap from Chapter 3 I A linear program consists of an objective function and constraints I The constraints define the feasible region I In 2 dimensions, we can: I Draw the feasible region I Find the optimal solution graphically using parallel lines representing the objective function I In higher dimensions, we can’t draw the feasible region, but the intuition still holds I The simplex method is the most common algorithm for solving LPs I It is an algebraic method but it has a geometric interpretation I We’ll discuss the geometric interpretation first Chapter 4: The Simplex Method 4
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Chapter 4: The Simplex Method Graphical Interpretation of the Simplex Method (Section 4.1) Introduction Wyndor Case Study maximize 3 x 1 + 5 x 2 subject to x 1 4 2 x 2 12 3 x 1 + 2 x 2 18 x 1 0 x 2 0 The lines are the constraint boundaries . x 1 x 2 1 5 3 2 4 6 1 3 4 2 6 5 7 9 8 10 x 1 = 4 2 x 2 = 12 3 x 1 + 2 x 2 = 18 (0,0) (4,0) (6,0) (4,3) (2,6) (4,6) (0,6) (0,9) Chapter 4: The Simplex Method 5
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Chapter 4: The Simplex Method Graphical Interpretation of the Simplex Method (Section 4.1) Introduction Corner-Point Solutions 5 corner-point feasible (CPF) solutions : 1. (0,0) 2. (4,0) 3. (4,3) 4. (2,6) 5. (0,6) x 1 x 2 1 5 3 2 4 6 1 3 4 2 6 5 7 9 8 10 x 1 = 4 2 x 2 = 12 3 x 1 + 2 x 2 = 18 5 4 3 2 1 (0,0) (4,0) (6,0) (4,3) (2,6) (4,6) (0,6) (0,9) Chapter 4: The Simplex Method 6
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Chapter 4: The Simplex Method Graphical Interpretation of the Simplex Method (Section 4.1) Introduction Corner-Point Solutions 3 corner-point infeasible solutions : 6. (6,0) 7. (0,9) 8. (4,6) x 1 x 2 1 5 3 2 4 6 1 3 4 2 6 5 7 9 8 10 x 1 = 4 2 x 2 = 12 3 x 1 + 2 x 2 = 18 (0,0) (4,0) (6,0) (4,3) (2,6) (4,6) (0,6) (0,9) 6 7 8 Chapter 4: The Simplex Method 7
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Chapter 4: The Simplex Method Graphical Interpretation of the Simplex Method (Section 4.1) Introduction Corner-Point Solutions Each corner-point solution (feasible or infeasible) lies at the intersection of two constraint boundaries.
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This note was uploaded on 03/12/2009 for the course MSCI 331 taught by Professor Ostrowski during the Winter '09 term at Waterloo.

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chap4_handout - Chapter 4: The Simplex Method Chapter 4:...

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