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Lecture10 - MS&E111 Introduction to Optimization Prof Amin...

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MS&E111 Lecture 10 Introduction to Optimization May 15, 2006 Prof. Amin Saberi From Prof. Van Roy’s Notes 1 Network Flows A wide variety of engineering and management problems involve optimization of network flows that is, how objects move through a network. Examples include coordination of trucks in a transportation system, routing of packets in a communication network, and sequencing of legs for air travel. Such problems often involve few indivisible objects, and this leads to a finite set of feasible solutions. For example, consider the problem of finding a minimal cost sequence of legs for air travel from Nukualofa to Reykjavik. Though there are many routes that will get a traveller from one place to the other, the number is finite. This may appear as a striking difference that distinguishes network flows problems from linear programs the latter always involves a polyhedral set of feasible solutions. Surprisingly, as we will see in this chapter, network flows problems can often be formulated and solved as linear programs. 1.1 Networks A network is characterized by a collection of nodes and directed edges, called a directed graph. Each edge points from one node to another. The figure below offers a visual rep- resentation of a directed graph with nodes labelled 1 through 8. We will denote an edge pointing from a node i to a node j by ( i,j ). In this notation, the graph of the below figure can be characterized in terms of a set of nodes V = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } and a set of edges E = { (1 , 2) , (1 , 3) , (1 , 6) , (2 , 5) , (3 , 4) , (4 , 6) , (5 , 8) , (6 , 5) , (6 , 7) , (7 , 8) } . Graphs can be used to
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2 MS&E 111: Introduction to Optimization - Spring 06 model many real networked systems. For example, in modelling air travel, each node might represent an airport, and each edge a route taken by some flight. Note that, to solve a specific problem, one often requires more information than the topology captured by a graph. For example, to minimize cost of air travel, one would need to know costs of tickets for various routes. 1.2 Min-Cost-Flow Problems Consider a directed graph with a set V of nodes and a set E of edges. In a min-cost-flow problem, each edge ( i,j ) E is associated with a cost c ij and a capacity constraint u ij .
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