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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