We consider a single item Production-Inventory-Routing problem with asingle producer/supplier and multiple retailers. Inventory management constraints are considered both at the producer and at the retailers, following a vendor managed inventory approach, where the supplier monitors the inventory at retailers and decides on the replenishment policy for each retailer. We assume a constant production capacity.Based on the mathematical formulation we discuss a classical Lagrangian relaxationwhich allows to decompose the problem into four subproblems, and a new Lagrangiandecomposition which decomposes the problem into just a production-inventory subproblemand a routing subproblem. The new decomposition is enhanced with validinequalities. A computational study is reported to compare the bounds from the twoapproaches.
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