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. Optimal resource allocation is one of the most widely studied areas in mathematical programming. We introduce a single resource allocation model that considers both discrete and continuous activities. The model is a natural extension of the knapsack problem with both binary and continuous variables. It has application in transportation management for allocating funds to highway improvements. We explore in depth the special structure of the problem and we present important theory that arises from its study. After identifying the fundamental properties of the problem, we present an efficient solution procedure that outperforms existing commercial software packages.