CS149-Week1 - What is Combinatorial Optimization? Ch...

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Chapter 1 Linear Programming aragraph 1 Paragraph 1 First Insights What is b i t i l O t i i t i ? Combinatorial Optimization? Given a set of variables, each associated with a alue domain and given constraints over the value domain, and given constraints over the variables, find an assignment of values to variables uch that the constraints are satisfied and an such that the constraints are satisfied and an objective function over the variables is minimized (or maximized)! CS 149 - Intro to CO 2 Examples p napsack Problem Knapsack Problem Market Split Problem Network Problems • Maximum Flow • Minimum Spanning Tree Routing Problems • Shortest Path • Vehicle Routing • Travelling Salesman Problem atisfiability Problem CS 149 - Intro to CO 3 Satisfiability Problem Examples p ransportation Problem: A good produced at • Transportation Problem: A good produced at various factories needs to be distributed to ifferent retailers. Each factory provides a specific different retailers. Each factory provides a specific supply, and each retailer has a specific demand. Also, the transportation cost per unit of the good for each factory/retailer pair is known. How can the demand be met while minimizing ansportation costs? transportation costs? CS 149 - Intro to CO 4
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Examples – Transportation Problem 110 2 6 6 90 65 3 2 1 60 1 2 9 2 7 80 1 120 CS 149 - Intro to CO 5 4 3 95 Examples – Transportation Problem • Constants D r : demand of retailer r S f : supply of factory f cost of shipping one unit from f to r c fr : cost of shipping one unit from f to r • Variables X : How many units are sent from factory f to retailer r fr • Constraints Ê f X fr = D r for all retailers r Ê r X fr S f for all factories f • Objective inimize CS 149 - Intro to CO 6 Minimize Ê fr c fr X fr How do we solve such problems? p A 4 2 3 6 2 4 1 2 3 A234 1 5 8 7 B468 C527 B 2 5 4 3 6 7 C 2 2 CS 149 - Intro to CO 7 Heuristic: Matrix-Minimum-Method A 2 0 2 3 6 4 2 4 1 2 3 A 2 34 2 1 5 8 7 2 5 B 4 3 6 7 C 2 2 Cost: 0 CS 149 - Intro to CO 8
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Heuristic: Matrix-Minimum-Method A 2 0 3 6 2 4 1 2 3 A 2 34 2 1 5 8 7 B468 C5 2 7 3 2 5 B 4 3 6 0 7 C 2 2 2 Cost: 4 CS 149 - Intro to CO 9 Heuristic: Matrix-Minimum-Method A 0 2 0 3 6 2 4 1 2 3 A 2 3 4 2 1 5 8 7 2 7 1 3 2 B 4 3 6 2 0 7 C 2 2 Cost: 8 CS 149 - Intro to CO 10 Heuristic: Matrix-Minimum-Method A 0 0 3 6 2
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This note was uploaded on 11/03/2009 for the course CS CS 149 taught by Professor Meinolf during the Spring '09 term at Sanford-Brown College.

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CS149-Week1 - What is Combinatorial Optimization? Ch...

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