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Operations Research: Network Problems 2

                              Brady Hunsaker


                            Novemb...
Min cost network flows

    Each node has a supply or demand or neither
    Each arc has a unit cost
    Each arc may have ...
Formulating Transportation Problems




    Brady Hunsaker ()   Operations Research: Network Problems 2   November 15, 200...
Shortest path

    Given in Section 8.2
    Want the shortest length path from origin node to destination node
    The sho...
Max flow

   Usually a directed graph
   One source node, one sink node
   Each arc has an upper bound on flow
   Find the fl...
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Operations Research: Network Problems 2

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Operations Research: Network Problems 2

  1. 1. Operations Research: Network Problems 2 Brady Hunsaker November 15, 2006 Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 1 / 10 Main concepts for network problems Identify real-world problems that can be modeled as network problems Formulate network problems as LPs or IPs Understand the hierarchy of problems—which are special cases of other problems? Understand that faster specialized algorithms are available for each type of problem; explain the tradeoff between using a specialized algorithm versus a more general one Know the basic ideas of greedy algorithms for min spanning tree and Dijkstra’s algorithm for shortest path Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 2 / 10
  2. 2. Min cost network flows Each node has a supply or demand or neither Each arc has a unit cost Each arc may have an upper bound (optional) Determine the flow along each arc to meet supply/demand at minimum cost Can be formulated as an LP: Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 3 / 10 Specialized algorithms exist for network problems For min cost network flows, sets of basic arcs have a special structure: they are always spanning trees This structure makes simplex pivots easier A special version of the simplex algorithm, often called the network simplex algorithm, takes advantage of this structure and runs about 10 times faster than the general simplex algorithm Why might we prefer to use a more general algorithm (like a simplex algorithm) instead of a network simplex algorithm? The network simplex algorithm uses only additions and subtractions (no divisions), which leads to an important result: if the supplies, demands, and upper bounds are all integers, then the network simplex algorithm will find an optimal flow for which all flow values are integer. Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 4 / 10
  3. 3. Formulating Transportation Problems Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 5 / 10 Formulating Assignment Problems Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 6 / 10
  4. 4. Shortest path Given in Section 8.2 Want the shortest length path from origin node to destination node The shortest path problem can also be modeled as a network flow (and therefore as an LP). This is only true because of the integrality property of network flows. Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 7 / 10 Shortest path: Dijkstra’s algorithm Dijkstra’s algorithm is much more efficient than even the network simplex algorithm for large instances. At the ith iteration, find and “label” the ith closest node to the origin with the distance from the origin (and the path to it) This is done by considering all nodes that are one arc away from one of the i − 1 nodes we have already labeled Which of them is the next closest? Label it and repeat. This algorithm assumes there are no negative length arcs. With that assumption, we can prove that it will give the correct answer quickly (though we won’t prove it for this class). Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 8 / 10
  5. 5. Max flow Usually a directed graph One source node, one sink node Each arc has an upper bound on flow Find the flows through each arc to maximize the amount of flow from source to sink LP formulation: Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 9 / 10 Max flow, Min Cut There are special algorithms for the max flow problem, which are faster than the simplex algorithm. The text describes the Ford-Fulkerson Method, which is based on the idea of augmenting flows. We may also consider the dual LP, which may be interpreted as a minimum cost cut of the network. Strong duality tells us that the maximum flow value will be equal to the minimum cut value. Brady Hunsaker () Operations Research: Network Problems 2 November 15, 2006 10 / 10

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