Minimizing the logistic ratio in the inventory routing problem

Inventory routing problems (IRPs) aim at minimizing the cost of the total distance traveled over a time horizon discretized in periods, while guaranteeing that the customers do not incur a stock-out event. In an optimal solution of an IRP, the customers in general have no inventory at the end of the horizon. Some inventory may remain only if this does not increase the cost of the distance traveled. To avoid this ending drawback, we consider in this paper as objective function the so-called logistic ratio, which is the ratio of the total routing cost to the total quantity distributed. The logistic ratio gives rise to a new optimization problem whose mathematical programming formulation is non-linear. Using a classical method, we can solve exactly instances with up to 5 vehicles and 15 customers over 3 periods. The solutions are compared with those of a classical IRP, both from the worst-case point of view and computationally. The results show that on average the logistic ratio increases by 20.4 % in the classical IRP on instances with 3 periods and that the percentage decreases when the horizon length increases.