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Chapter 1

Chapter 1. Introduction Basic formulations Applications. Notation for Defining a Network. G = ( N , A ) is a directed network N = {1, 2, …, n } is the set of nodes A = {( i , j )} is the set of m arcs ; ( i , j ) is directed from node i (the tail ) to node j (the head )

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Chapter 1

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  1. Chapter 1 Introduction Basic formulations Applications

  2. Notation for Defining a Network • G = (N, A) is a directed network • N = {1, 2, …, n} is the set of nodes • A = {(i, j)} is the set of marcs; (i, j) is directed from node i (the tail) to node j (the head) • Arc data • cij is the cost per unit flow along (i, j) • lij is the lower bound for the flow along (i, j) • uij is its upper bound, or the arc’s capacity • Node data • b(i) is the supply if b(i) > 0 or the opposite of demand if b(i) < 0; if b(i) = 0 then i is called a transshipment node.

  3. Minimum Cost Flow Problem Linear programming formulation: flow along (i,j) is Two sets of constraints: conservation of flow, and bounds on the flows.

  4. Shortest Path Problem Define a source node s and a sink node t; redefine cij to be the length of arc (i, j). Set b(s) = 1, b(t) = -1, b(i) = 0 for all the other nodes i; lij = 0, uij = . In an optimal solution,

  5. Maximum Flow Problem Define a source node s and a sink node t; cij = 0 for all (i, j). Set b(i) = 0 for all nodes i. Add arc (t, s) with cts = -1, uts = . By conservation of flow, maximizing the flow from t to s has the effect of maximizing the flow across the original arcs from s to t. Costs are irrelevant.

  6. Transportation Problem Partition the nodes into two sets N1 and N2, where nodes in N1 are supply nodes and nodes in N2 are demand nodes, and for each arc (i, j), i is in N1 and j is in N2.

  7. Assignment Problem Partition the nodes into two sets N1 and N2 of equal size, where nodes in N1 have a supply of 1 and nodes in N2 have a demand of 1. For each arc (i, j), i is in N1 and j is in N2, and uij = 1. The variable xij = 1 if node i is assigned to node j.

  8. Undirected Network Problems • Minimum spanning tree problem • find a set of arcs that “connect” all the nodes in the network at minimum cost • Matching problems • cardinality: pair up as many nodes as possible • weighted: pair up nodes to minimize or maximize the weights on the arcs used (minimum weight matching on a bipartite graph = assignment)

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