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The Simplex Algorithm PDF

140 Pages·2017·0.41 MB·English
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Math 407: Linear Optimization Lecture 5: Simplex Algorithm I Lecture5:SimplexAlgorithmI Math407:LinearOptimization 1/35 1 Dictionaries, Augmented Matrices, the Simplex Tableau Dictionaries The Simplex Tableau 2 Basic Feasible Solutions (BFS) 3 The Grand Strategy: Pivoting 4 The Simplex Algorithm in Matrix Form 5 The Simplex Algorithm in Matrix Form Lecture5:SimplexAlgorithmI Math407:LinearOptimization 2/35 max5x +4x +3x 1 2 3 s.t. 2x +3x +x ≤ 5 1 2 3 4x +x +2x ≤ 11 1 2 3 3x +4x +2x ≤ 8 1 2 3 0 ≤ x ,x ,x 1 2 3 At this point we only have one tool for attacking linear systems. Gaussian elimination, a method for solving linear systems of equations. Let’s try to use it to solve LPs. The Simplex Algorithm We develop a method for solving standard form LPs. Lecture5:SimplexAlgorithmI Math407:LinearOptimization 3/35 At this point we only have one tool for attacking linear systems. Gaussian elimination, a method for solving linear systems of equations. Let’s try to use it to solve LPs. The Simplex Algorithm We develop a method for solving standard form LPs. max5x +4x +3x 1 2 3 s.t. 2x +3x +x ≤ 5 1 2 3 4x +x +2x ≤ 11 1 2 3 3x +4x +2x ≤ 8 1 2 3 0 ≤ x ,x ,x 1 2 3 Lecture5:SimplexAlgorithmI Math407:LinearOptimization 3/35 Gaussian elimination, a method for solving linear systems of equations. Let’s try to use it to solve LPs. The Simplex Algorithm We develop a method for solving standard form LPs. max5x +4x +3x 1 2 3 s.t. 2x +3x +x ≤ 5 1 2 3 4x +x +2x ≤ 11 1 2 3 3x +4x +2x ≤ 8 1 2 3 0 ≤ x ,x ,x 1 2 3 At this point we only have one tool for attacking linear systems. Lecture5:SimplexAlgorithmI Math407:LinearOptimization 3/35 Let’s try to use it to solve LPs. The Simplex Algorithm We develop a method for solving standard form LPs. max5x +4x +3x 1 2 3 s.t. 2x +3x +x ≤ 5 1 2 3 4x +x +2x ≤ 11 1 2 3 3x +4x +2x ≤ 8 1 2 3 0 ≤ x ,x ,x 1 2 3 At this point we only have one tool for attacking linear systems. Gaussian elimination, a method for solving linear systems of equations. Lecture5:SimplexAlgorithmI Math407:LinearOptimization 3/35 The Simplex Algorithm We develop a method for solving standard form LPs. max5x +4x +3x 1 2 3 s.t. 2x +3x +x ≤ 5 1 2 3 4x +x +2x ≤ 11 1 2 3 3x +4x +2x ≤ 8 1 2 3 0 ≤ x ,x ,x 1 2 3 At this point we only have one tool for attacking linear systems. Gaussian elimination, a method for solving linear systems of equations. Let’s try to use it to solve LPs. Lecture5:SimplexAlgorithmI Math407:LinearOptimization 3/35 max5x +4x +3x 1 2 3 s.t. 2x +3x +x ≤ 5 1 2 3 4x +x +2x ≤ 11 1 2 3 3x +4x +2x ≤ 8 1 2 3 0 ≤ x ,x ,x 1 2 3 Dictionaries We must first build a linear system of equations that encodes all of the information associated with the LP. Lecture5:SimplexAlgorithmI Math407:LinearOptimization 4/35 Dictionaries We must first build a linear system of equations that encodes all of the information associated with the LP. max5x +4x +3x 1 2 3 s.t. 2x +3x +x ≤ 5 1 2 3 4x +x +2x ≤ 11 1 2 3 3x +4x +2x ≤ 8 1 2 3 0 ≤ x ,x ,x 1 2 3 Lecture5:SimplexAlgorithmI Math407:LinearOptimization 4/35 x = 5 − [2x +3x +x ] ≥ 0, 4 1 2 3 x = 11 − [4x +x +2x ] ≥ 0, 5 1 2 3 x = 8 − [3x +4x +2x ] ≥ 0. 6 1 2 3 Slack Variables: x ,x ,x 4 5 6 Next we introduce a variable to represent the objective. z = 5x +4x +3x . 1 2 3 This system of equations is called a dictionary for the the LP. The slack variables and the objective are defined by the original decision variables. Slack Variables and Dictionaries For each linear inequality we introduce a new variable, called a slack variable, so that we can write each linear inequality as an equation. Lecture5:SimplexAlgorithmI Math407:LinearOptimization 5/35

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s.t. 2x1 + 3x2 + x3. ≤ 5. 4x1 + x2 + 2x3. ≤ 11. 3x1 + 4x2 + 2x3. ≤ 8. 0 ≤ x1,x2,x3. Lecture 5: Simplex Algorithm I. Math 407A: Linear Optimization.
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