A branch & bound algorithm for the 0–1 mixed integer knapsack problem with linear multiple choice constraints

This paper presents a branch and bound (B&B) algorithm for the 0–1 mixed integer knapsack problem with linear multiple choice constraints. The formulation arose in an application to transportation management for allocating funds to highway improvements. Several model properties are developed and utilized to design a B&B solution algorithm. The algorithm solves at each node of the B&B tree a linear relaxation using an adaptation of an existing algorithm for the linear multiple choice knapsack problem. The special relationship between the parent and children subproblems is exploited by the algorithm. This results in high efficiency and low storage space requirements. The worst case complexity of the algorithm is analyzed and computational results that demonstrate its efficiency in the average case are reported.