A column generation approach to delivery planning over time with inhomogeneous service providers and service interval constraints

We consider a problem of delivery planning over multiple time periods. Deliveries must be made to customers having nominated demand in each time period. Demand must be met in each time period by use of some combination of inhomogeneous service providers. Each service provider has a different delivery capacity, different cost of delivery to each customer, a different utilisation requirement, and different rules governing the spread of deliveries in time. The problem is to plan deliveries so as to minimise overall costs, subject to demand being met and service rules obeyed. A natural integer programming model was found to be intractable, except on problems with loose demand constraints, with gaps between best lower bound and best feasible solution of up to 35.1%, with an average of 15.4% over the test data set. In all but the problem with loosest demand constraints, Cplex 6.5 applied to this formulation failed to find the optimal solution before running out of memory. However a column generation approach improved the lower bound by between 0.6% and 21.9%, with an average of 9.9%, and in all cases found the optimal solution at the root node, without requiring branching.