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

This paper presents an exact solution procedure for a vehicle routing problem with semi-hard resource constraints where each resource requirement can be relaxed to a pre-fixed extent at a predefined cost. This model is particularly useful for a supply chain coordination when a given number of vehicles cannot feasibly serve all the customers without relaxing some constraints.It is different from VRP with soft time windows in that the violation is restricted to a certain upper bound, the penalty cost is flat, and the number of relaxations allowed has an upper bound.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.