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Unformatted text preview: Chapter 5 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 Nuku’alofa 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. 5.1 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. Fig ure 5.1 offers a visual representation of a directed graph with nodes la belled 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 Figure 5.1 can be character ized 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 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 107 108 1 2 5 8 7 4 3 6 Figure 5.1: A directed graph. example, to minimize cost of air travel, one would need to know costs of tickets for various routes. 5.2 MinCostFlow Problems Consider a directed graph with a set V of nodes and a set E of edges. In a mincostflow problem, each edge ( i, j ) ∈ E is associated with a cost c ij and a capacity constraint u ij . There is one decision variable f ij per edge ( i, j ) ∈ E . Each f ij is represents a flow of objects from i to j . The cost of a flow f ij is c ij f ij . Each node j ∈ V \ { s, d } satisfies a flow constraint: X { k  ( j,k ) ∈ E } f jk X { i  ( i,j ) ∈ E } f ij = b j , where b j denotes an amount of flow generated by node j . Note that if b j is negative, the node consumes flow. The mincostflow problem is to find flows that minimize total cost subject to capacity and flow conservation constraints. It can be written as a linear program: minimize ∑ ( i,j ) ∈ E c ij f ij subject to ∑ { k  ( j,k ) ∈ E } f jk ∑ { i  ( i,j ) ∈ E } f ij = b j , ∀ j ∈ V ≤ f ij ≤ u ij , ∀ ( i, j ) ∈ E....
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 Spring '06
 UNKNOWN
 Operations Research, Linear Programming, Optimization, Flow network, Benjamin Van Roy, Kahn Mason

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