We consider a single item Production-Inventory-Routing problem with a
single 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 relaxation
which allows to decompose the problem into four subproblems, and a new Lagrangian
decomposition which decomposes the problem into just a production-inventory subproblem
and a routing subproblem. The new decomposition is enhanced with valid
inequalities. A computational study is reported to compare the bounds from the two
approaches