MIT1_204S10_lec17

MIT1_204S10_lec17 - 1.204 Lecture 17 Branch and bound...

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1.204 Lecture 17 Branch and bound: Method Warehouse location problem Breadth first search Breadth first search manages E-nodes in the branch and bound tree An E node is the node currently being explored In breadth first search, E-node stays live until all its children have been generated The children are placed on a queue, stack or heap Typical strategies to select E-nodes Choose node with largest upper bound (in a maximization problem), using a heap Choose node likely to be optimal, even if we can’t prove it’s optimal immediately Use problem specific, heuristic rule It can be a previous optimal result to similar problem Choose ‘quick improvement’ node based on a gradient estimate from the upper and lower bounds on a node 1
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Branching on nodes Several strategies are used to decide which branch (0 or 1) to take: User specified rules. Again, heuristics are used Set a group of 0-1 variables to given values, not just one This seems to perform better in many problems Our code in this lecture does not do this Other strategies to improve performance: If dual can be computed If dual can be computed, it provides a lower bound it provides a lower bound Bound tightening, such as truncation (like we used in the last lecture on the knapsack problem) Adding linear constraints (0<=x<=1) in subproblems in hopes that integer answers are obtained Greedy heuristics, including dual descent and others… Facility location problem Warehouse 1 Customer 4 Customer 1 Customer 3 Customer 2 8 18 14 10 Warehouse 0 Warehouse 2 Warehouse 3 Customer 0 3 e.g., Amazon Intel Tropicana 2
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3 Facility location example Warehouse Fixed cost kf [k] 0123 4 Cost to ship to customer j Set of 4 possible warehouses (0-3) to serve 5 possible customers (0-4) 04 3 1 0 8 1 8 1 4 16 9 4 6 5 5 26 1 2 6 1 0 4 8 38 8 6 5 1 2 9 Table gives annual capital (fixed) cost of warehouse if it is built, and the annual cost of shipping to each customer via that warehouse Decision is whether to build (x i = 1) or not build (x i =0) each warehouse Objective is to minimize fixed plus shipping costs Computing bounds Lower (optimistic) bound at each node is sum of: (p ) Minimum transport cost over all built or unknown warehouses Fixed cost of built warehouses Upper (pessimistic) bound at each node is sum of Minimum transport cost over all built warehouses Fixed cost of built warehouses Pruning rules If minimum (pessimistic) savings from building a warehouse are greater than its fixed cost we build it are greater than its fixed cost, we build it If maximum (optimistic) savings from building a warehouse are less than its fixed cost, we don’t build it All combinations are feasible in this problem, so there is no reduction in the size of the tree from feasibility constraints We can introduce capital budget constraints in some cases Pruning rules from Akinc, Khumwala
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Computational strategy Start at root node
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MIT1_204S10_lec17 - 1.204 Lecture 17 Branch and bound...

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