4120_lecture19-F10 - Computational Methods for Management...

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Computational Methods for Management and Economics Carla Gomes Lecture 19 Reading: Section 11.1-11.3 of text book.
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Special kinds of IP programs
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Applications of Binary Variables Making “yes-or-no” type decisions Build a factory? Manufacture a product? Do a project? Assign a person to a task? Fixed costs If a product is produced, must incur a fixed setup cost. If a warehouse is operated, must incur a fixed cost. Either-or constraints Production must either be 0 or ≥ 100. Subset of constraints meet 3 out of 4 constraints.
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Special Kinds of Integer Programming Models Knapsack Problem Set Covering Problem Set Partitioning Problem Set Packing Problem The Traveling Salesman Problem The Quadratic Assignment Problem
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Set Covering Problem We are given a set of objects S = {1, 2, 3, …, n}. We are also given a set of subsets of S, S . Each subset has a cost associated with it. Problem: to “cover” all the members of S at the minimum cost using members of S. Properties: The problem is a minimization and all constraints are >=; All RHS coefficients are 1; All other matrix coefficients are 0 or 1.
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Fire Station Problem Set Covering Problem 1 2 3 4 5 6 7 8 9 11 10 12 14 15 13 16 Locate fire stations so that each district has a fire station in it, or next to it. Minimize the number of fire stations needed.
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Representation as Set Covering Problem 1 2 3 4 5 6 7 8 9 11 10 12 14 15 13 16 Set Covers 1 1, 2, 4, 5 2 1, 2, 3, 5, 6 3 2, 3, 6, 7 16 13, 15, 16
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Representation as Graph Cover Problem A node covers itself and its neighbors. Thus, node 16 covers nodes 13, 15, 16. 1 2 3 4 5 6 7 8 9 11 10 12 14 15 13 What is the minimum size of a subset of nodes that covers all of the nodes? Replace each district with a node. Two nodes are adjacent if their districts are adjacent 16
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Representation as Integer program 2 3 4 5 6 8 9 10 12 14 13 16 x j = 1 if node j is selected x j = 0 otherwise Minimize x 1 + x 2 + … + x 16 s.t. x 1 + x 2 + x 4 + x 5 1 x 1 + x 2 + x 3 + x 5 + x 6 1 x 13 + x 15 + x 16 1 x j {0, 1} for each j. 1 7 11 15
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Representation as Integer program 2 3 4 5 6 8 9 10 12 14 13 16 x j = 1 if node j is selected x j = 0 otherwise Minimize x 1 + x 2 + … + x 16 s.t. x 1 + x 2 + x 4 + x 5 1 x 1 + x 2 + x 3 + x 5 + x 6 1 x 13 + x 15 + x 16 1 x j {0, 1} for each j. 1 7 11 15
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Southwestern Airways Crew Scheduling Southwestern Airways needs to assign crews to cover all its upcoming flights. We will focus on assigning 3 crews based in San Francisco (SFO) to 11 flights. Question: How should the 3 crews be assigned 3 sequences of flights so that every one of the 11 flights is covered?
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Southwestern Airways Flights Seattle (SEA) San Francisco (SFO) Los Angeles (LAX) Denver Chicago ORD)
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Data for the Southwestern Airways Problem Feasible Sequence of Flights (pairings) Flights 1 2 3 4 5 6 7 8 9 10 11 12 1. SFO–LAX 1 1 1 1 2. SFO–DEN 1 1 1 1 3. SFO–SEA 1 1 1 1 4. LAX–ORD 2 2 3 2 3 5. LAX–SFO 2 3 5 5 6. ORD–DEN 3 3 4 7. ORD–SEA 3 3 3 3 4 8. DEN–SFO 2 4 4 5 9. DEN–ORD 2 2 2 10. SEA–SFO 2 4 4 5
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Algebraic Formulation Let x j =
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This note was uploaded on 02/25/2011 for the course AEM 4120 taught by Professor Gomes,c. during the Fall '08 term at Cornell University (Engineering School).

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4120_lecture19-F10 - Computational Methods for Management...

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