lecture9-4(matrix)

# lecture9-4(matrix) - ISyE Lecture#9 Outline Outline ISyE...

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ISyE 323 Lecture #9 Prof. Jeﬀ Linderoth October 1, 2009 ISyE Lecture #9 Outline Outline 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) Postoptimality Analysis (Section 4.7) ISyE Lecture #9 2 ISyE Lecture #9 Outline Quiz #1! ± In-class, A week from today. 10/8. ± Closed book, closed notes, no laptops. Calculator allowed. ± Responsible for material in. .. ± Lecture notes ± Chapters 1,2,3 (All). ± 4.1, 4.2, 4.3, 4.4. (All) ± 4.5, 4.6 (to a lesser degree). ± 5.1, 5.2 ± Practice quiz posted. (I indicated which problems you may not need to know how to do). ± Review session (in class) next Tuesday. Will post some extra practice modeling problems. ISyE Lecture #9 3 ISyE Lecture #9 Outline More Information Homeworks ± Homework #4 Due Tuesday ± All homework solutions will be posted soon as well. ± Bond discussion? Oﬃce Hours? ± If possible, I’d like to cut oﬃce hours short today to go to a meeting with the Dean. ISyE Lecture #9 4

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ISyE Lecture #9 Foundations of the Simplex Method (Section 5.1) Outline The Algebra of the Simplex Method (Section 4.3) Introduction Initialization Step 1 Step 2 Step 3 Optimality Test Simplex Iteration: Summary Iteration 2 The Simplex Method in Tabular Form (Section 4.4) Tie Breaking in the Simplex Method (Section 4.5) ISyE Lecture #9 5 ISyE Lecture #9 Foundations of the Simplex Method (Section 5.1) Properties of CPF Solutions Property 1 Property 1 (a) If there is exactly one optimal solution, then it is a CPF solution. (b) If there are multiple optimal solutions, then at least two must be adjacent CPF solutions. ± The geometric intuition should be clear. ± It can also be proved algebraically (see book). ISyE Lecture #9 6 ISyE Lecture #9 Foundations of the Simplex Method (Section 5.1) Properties of CPF Solutions Property 2 Property 2 There are a ﬁnite number of CPF solutions. ± This narrows down the number of solutions we need to examine. ± But we still can’t examine all of the CPF solutions. ISyE Lecture #9 7 ISyE Lecture #9 Foundations of the Simplex Method (Section 5.1) Properties of CPF Solutions Property 3 Property 3 If a CPF solution has no adjacent CPF solutions that are better, then there are no better solutions anywhere. ± This allows the simplex method to stop when it ﬁnds a solution with no better neighbors. ± It follows from the fact that the feasible region is convex for any LP. ISyE Lecture #9 8
ISyE Lecture #9 Foundations of the Simplex Method (Section 5.1) Augmented Form of the LP Augmented Form ± Augmented form is: a 11 x 1 + a 12 x 2 + . . . + a 1 n x n + x n +1 = b 1 (1) a 11 x 1 + a 12 x 2 + . . . + a 1 n x n + x n +2 = b 2 (2) . . . . . . a m 1 x 1 + a m 2 x 2 + . . . + a mn x n + x n + m = b m ( m ) plus nonnegativity constraints. ± x n +1 , . . . , x n + m are slack/artiﬁcial variables.

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lecture9-4(matrix) - ISyE Lecture#9 Outline Outline ISyE...

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