The assignment and loading transportation problem

We consider a generalization of the multiple knapsack problem that combines assignment and loading. The problem can arise in military and emergency situations in which one is required to refurnish a unit with a number of different goods available at different locations. We present a mathematical model and study Lagrangian and surrogate relaxations. We propose heuristic and metaheuristic approaches which we use to develop two overall approximation algorithms: a self-contained polynomial-time heuristic and a more time consuming matheuristic approach that makes use of a MILP solver. Solution times and accuracy of lower and upper bounds are computationally evaluated on a real military data set and on sets of both realistic and randomly generated instances.