On Bounds Available from Branch-and-Price Decomposition of the Multi-dimensional Knapsack Problem with Generalized Upper Bound Constraints

This paper describes and evaluates alternative branch-and-price decompositions (B&P-Ds) of the multi-dimensional knapsack problem with generalized upper bound constraints both analytically and computationally. As part of our theoretical analysis we compare the bounds available from B&P-Ds with three alternative relaxations (Lagrangean relaxation, Lagrangean decomposition, and Surrogate relaxation) and whether incorporating a surrogate constraint can make an improvement or not. Our computational tests compare alternative ways of implementing B&P-D to assess the trade off between the tightness of resulting bounds and the run times required to obtain them.