Advances in Stochastic and Deterministic Global Optimization by Panos M. Pardalos Anatoly Zhigljavsky & Julius Žilinskas

Author:Panos M. Pardalos, Anatoly Zhigljavsky & Julius Žilinskas
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Solution Framework

A Branch and Price algorithm is used in this chapter. This consists of a master problem which is solved by Branch and Bound (B&B), with each node in the B&B tree being solved by Dantzig–Wolfe (DW) decomposition. Each column in the master problem corresponds to a tree of schedules for a ship. There is a huge number of these columns and if all were included explicitly the master problem would be impossible to solve. However the DW approach only generates the small subset of them that are needed, and is thus able to solve the full problem at each B&B node. In each iteration of DW a subproblem is solved for each ship to generate an attractive tree of schedules for that ship. In this chapter the subproblems are solved by stochastic dynamic programming.

At any stage in the solution of a master problem at a B&B node, a (finite) subset of the columns will have been generated. This problem, called a restricted master problem, is solved and the shadow prices of the constraints are then used to find the most negative reduced cost from among the columns that have not yet been generated. This can be done without explicitly generating any columns by solving a stochastic dynamic programming problem separately for each ship. The solution gives the tree of schedules for the ship. If this added as a column to the master problem, it would have the most negative reduced cost among all the possible columns for that ship. This procedure continues until no column with negative reduced cost can be generated, at which stage the master problem for that B&B node has been solved.

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