An exact solution for vehicle routing problems with semi-hard resource constraints

We develop an exact approach to solve the problem. We use the branch cut and price procedure to solve the problem modeling the pricing problem as an elementary shortest path problem with semi hard resource constraints. The modeling of the subproblem provides a tight lower bound to reduce the computation time. We solve this subproblem using a label setting algorithm, in which we form the labels in a compact way to facilitate incorporation of the resources requirement relaxation information into it, develop extension rules that generate labels with possible relaxations, and develop dominance criteria that reduce the computation time. The lower bound is improved by applying the subset-row inequalities.