S olving LP relaxations of airline crew scheduling models is computationally challenging due to the presence of a large number of variables, complex feasibility rules to generate columns, and nonlinear cost. We perform computational experiments with a nonlinear pricing strategy. We develop a column generation scheme that uses several pruning rules to fathom column enumeration. The pruning rules are categorized into approximate and exact, where the approximate rules might prune columns that would yield an improved objective value. The pruning rules use the fact that columns are paths in a network, and we use shortest path algorithms and their extensions to obtain bounds.
The source-to-all maximum cost-to-time ratio problem is the problem of finding the maximum cost-to-time ratio path from a source node to every other node. The motivation comes from an application in large-scale linear programming. We present three algorithms for solving the problem. We give proofs of correctness and we analyze the running times. One of the algorithms is polynomial and the remaining two are pseudopolynomial. We present extensive computational results on several networks.
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