Complexity of inventory routing problems when routing is easy

Abstract

In the inventory routing problem (IRP) inventory management and route optimization are combined. The traveling salesman problem (TSP) is a special case of the IRP, hence the IRP is NP-hard. We investigate how other aspects than routing influence the complexity of a variant of the IRP. We first study problem variants on a point and on the half-line. The problems differ in the number of vehicles, the number of days in the planning horizon and the service times of the customers. Our main result is a polynomial time dynamic programming algorithm for the variant on the half-line with uniform service times and a planning horizon of 2 days. Second, for nearly any problem in the class with nonfixed planning horizon, we show that the complexity is dictated by the complexity of the pinwheel scheduling problem, for which the complexity is a long-standing open research question. Third, NP-hardness is shown for problem variants with nonuniform servicing times. Finally, we prove strong NP-hardness of a Euclidean variant with uniform service times and an easily computable routing cost approximation, avoiding immediate NP-hardness via the TSP. KEYWORDSapproximation, computational complexity, dynamic programming, inventory routing, periodic replenishment, pinwheel schedulingThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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