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4120_lecture19-F10

# 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
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
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.
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 16 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
Representation as Integer program 1 2 3 4 5 6 7 8 9 11 10 12 14 15 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 1 2 3 4 5 6 7 8 9 11 10 12 14 15 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
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 (DEN) Chicago ORD)
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 11. SEA–LAX 2 2 4 4 2

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

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