math125-lecture 2.6 - An algebraic method for finding a...

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Unformatted text preview: An algebraic method for finding a maximum in a linear program with two or more variables Now we will learn a scaled down version. The full version is in Chapter 4 Sec 2.6 Friday, February 05, 2010 5:17 PM Section2dot6 Page 1 An algebraic method for finding a maximum in a linear program with two or more variables Now we will learn a scaled down version. The full version is in Chapter 4 The linear program must be a maximization The constraint inequalities must have the form: linear expression ≤ a positive # All variables must be non-negative 2.6.1 Friday, February 05, 2010 5:17 PM Section2dot6 Page 2 An algebraic method for finding a maximum in a linear program with two or more variables Now we will learn a scaled down version. The full version is in Chapter 4 The linear program must be a maximization The constraint inequalities must have the form: linear expression ≤ a positive # All variables must be non-negative A slack variable is an extra variable, which is added to an inequality to make the constraint an equality EX: 5 x 1 + x 2 ≤ 9 5 x 1 + x 2 + x 3 = 9 x 3 "picks up the slack" in the inequality 2.6.2 Friday, February 05, 2010 5:17 PM Section2dot6 Page 3 An algebraic method for finding a maximum in a linear program with two or more variables Now we will learn a scaled down version. The full version is in Chapter 4 The linear program must be a maximization The constraint inequalities must have the form: linear expression ≤ a positive # All variables must be non-negative A slack variable is an extra variable, which is added to an inequality to make the constraint an equality EX: 5 x 1 + x 2 ≤ 9 5 x 1 + x 2 + x 3 = 9 x 3 "picks up the slack" in the inequality This is the augmented matrix required for the simplex algorithm and formed from the linear system that arises when: 1. the objective function is written in standard form (as a linear equation) 2. all inequalities are converted to equations by introducing slack variables 2.6.3 Friday, February 05, 2010 5:17 PM Section2dot6 Page 4 z = 8 x 1 + 6 x 2 z- 8 x 1- 6 x 2 = 0 standard form 2.6.4 Friday, February 05, 2010 5:27 PM Section2dot6 Page 5 z = 8 x 1 + 6 x 2 z- 8 x 1- 6 x 2 = 0 standard form x 3- slack variable x 4- slack variable 2 x 1 + 4 x 2 ≤ 6 2 x 1 + 4 x 2 + x 3 = 6 x 1 + 7 x 2 ≤ 4 x 1 + 7 x 2 + x 4 = 4 2.6.5 Friday, February 05, 2010 5:27 PM Section2dot6 Page 6 z = 8 x 1 + 6 x 2 z- 8 x 1- 6 x 2 = 0 standard form x 3- slack variable x 4- slack variable 2 x 1 + 4 x 2 ≤ 6 2 x 1 + 4 x 2 + x 3 = 6 x 1 + 7 x 2 ≤ 4 x 1 + 7 x 2 + x 4 = 4 1 -8 -6 0 0 0 2 4 1 0 6 0 1 7 0 1 4 column headings : z x 1 x2 x3 x 4 solution objective function: constraint equalities: Initial Simplex Table 2.6.6 Friday, February 05, 2010 5:27 PM Section2dot6 Page 7 1 -8 -6 0 0 0 2 4 1 0 6 0 1 7 0 1 4 pivot column ratios 6/2 = 3 smallest 4/1 = 4 2.6.7 Friday, February 05, 2010 5:32 PM Section2dot6 Page 8 1 -8 -6 0 0 0 2 4 1 0 6 0 1 7 0 1 4 pivot column ratios 6/2 = 3 smallest 4/1 = 4 R' 2 = 1/2 R 2 1 -8 -6 0 0 0 1 2 1/2 0 3 0 1 7 0 1 4 2.6.8 Friday, February 05, 2010 5:32 PM Section2dot6 Page 9 1...
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This note was uploaded on 06/07/2011 for the course MATH 125 taught by Professor Tom during the Fall '07 term at University of Illinois at Urbana–Champaign.

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math125-lecture 2.6 - An algebraic method for finding a...

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