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

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Ch t 1 Chapter 1 Linear Programming Paragraph 1 First Insights What is C bi t i l O ti i ti ? Combinatorial Optimization? Given a set of variables, each associated with a value domain and given constraints over the value domain, and given constraints over the variables, find an assignment of values to variables such 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 Knapsack Problem Market Split Problem Network Problems Maximum Flow Minimum Spanning Tree Routing Problems Shortest Path V hi l R ti Vehicle Routing • Travelling Salesman Problem Satisfiability Problem CS 149 - Intro to CO 3 Examples • Transportation Problem: A good produced at Transportation Problem: A good produced at various factories needs to be distributed to different 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 transportation costs? CS 149 - Intro to CO 4
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Examples – Transportation Problem 110 2 6 6 90 65 3 2 1 9 60 1 2 2 7 80 3 1 120 CS 149 - Intro to CO 5 4 95 Examples – Transportation Problem Constants D r : demand of retailer r S f : supply of factory f c : cost of shipping one unit from f to r c fr : cost of shipping one unit from f to r Variables X fr : How many units are sent from factory f to retailer r Constraints Ê f X fr = D r for all retailers r Ê X S f ll f t i f r fr f for all factories f Objective Minimize Ê f c f X f CS 149 - Intro to CO 6 Minimize fr fr fr How do we solve such problems? A 4 6 1 2 3 1 2 3 2 4 8 A 2 3 4 5 4 5 3 7 B 4 6 8 C 5 2 7 B 7 2 6 C 2 2 CS 149 - Intro to CO 7 Heuristic: Matrix-Minimum-Method A 2 6 4 1 2 3 0 1 2 3 2 4 8 A 2 3 4 2 5 4 5 3 7 B 4 6 8 C 5 2 7 2 B 7 6 C 2 2 Cost: 0 CS 149 - Intro to CO 8
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Heuristic: Matrix-Minimum-Method A 2 6 1 2 3 0 1 3 2 4 8 A 2 3 4 2 3 5 4 5 3 7 B 4 6 8 C 5 2 7 2 B 7 6 2 0 C 2 2 Cost: 4 CS 149 - Intro to CO 9 Heuristic: Matrix-Minimum-Method A 0 2 6 1 2 3 0 1 3 2 4 8 A 2 3 4 2 1 3 4 5 3 7 B 4 6 8 C 5 2 7 2 B 7 6 2 2 0 C 2 Cost: 8 CS 149 - Intro to CO 10 Heuristic: Matrix-Minimum-Method A 0 6 1 2 3 0 1 3 2 4 8 A 2 3 4 2 1 4 5 3 7 B 4 6 8 C 5 2 7 2 B 7 6 2 2 0 C 2 Cost: 14 CS 149 - Intro to CO 11 Heuristic: Matrix-Minimum-Method A 0 6 1 2 3 0 1 3 2 4 8 A 2 3 4 2 0 1 4 5 3 7 B 4 6
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