Lecture7 - Advanced Operations Research Techniques IE316...

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Advanced Operations Research Techniques IE316 Lecture 7 Dr. Ted Ralphs
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IE316 Lecture 7 1 Reading for This Lecture Bertsimas 3.2-3.4
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IE316 Lecture 7 2 The Simplex Method A typical iteration of the simplex method: 1. Start with a specified basis matrix B and a corresponding BFS x 0 . 2. Compute the reduced cost vector ¯ c . If ¯ c 0 , then x 0 is optimal . 3. Otherwise, choose j for which ¯ c j < 0 . 4. Compute u = B - 1 A j . If u 0 , then θ * = and the LP is unbounded . 5. Otherwise, θ * = min { i | u i > 0 } x 0 B ( i ) u i . 6. Choose l such that θ * = x 0 B ( l ) u l and form a new basis matrix, replacing A B ( l ) with A j . 7. The values of the new basic variables are x 1 j = θ * and x 1 B ( i ) = x 0 B ( i ) - θ * u i if i 6 = l .
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IE316 Lecture 7 3 Some Notes on the Simplex Method We will see later how to construct an initial basic feasible solution. We saw last time that each iteration of the simplex methods ends with a new basic feasible solution. This is all we need to prove the following result: Theorem 1. Consider a linear program over a nonempty polyhedron P and assume every basic feasible solution is nondegenerate . Then the simplex method terminates after a finite number of iterations in one of the following two conditions: We obtain an optimal basis and a corresponding optimal basic feasible solution. We obtain a vector d R n such that Ad = 0 , d 0 , and c T d < 0 , and the LP is unbounded .
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IE316 Lecture 7 4 Pivot Selection The process of removing one variable and replacing from the basis and replacing it with another is called pivoting . We have the freedom to choose the entering variable from among a list of candidates. How do we make this choice? The reduced cost tells us the cost in the objective function for each unit of change in the given variable. Intuitively, c j is the cost for the change in the variable itself and - c T B B - 1 A j is the cost of the compensating change in the other variables. This leads to the following possible rules: Choose the column with the most negative reduced cost . Choose the column for which θ * | ¯ c j | is largest .
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IE316 Lecture 7 5 Other Pivoting Rules In practice, sophisticated pivoting rules are used.
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