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Inventory-routing problems (IRP) combine inventory control and vehicle routing, effectively optimizing inventory and replenishment decisions over several periods at a centralized level. In this paper we provide an exact formulation which includes several well-known valid inequalities for some classes of IRPs. We then propose three new valid inequalities based on the relation between demand and available capacities. Then, following an idea proposed for the binary clustering and for the job scheduling problems, we also show how the order of the input data can have a major effect on the linear relaxation of the proposed model for the IRP. Extensive computational experiments confirm the success of our algorithm. We have used two available datasets with new solutions identified as recently as 2013. On one set of benchmark instances with 249 open instances, we have improved 98 lower bounds, we have …