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This paper considers a two-stage supply chain, consisting of a single warehouse and multiple retailers facing deterministic demands, under a vendor-managed inventory (VMI) policy.It presents a two-phase optimisation approach for coordinating the shipments in this VMI system.The first phase uses direct shipping from the supplier to all retailers to minimise the overall inventory costs.Then, in the second phase, the retailers are clustered using a construction heuristic in order to optimise the transportation costs while satisfying some additional restrictions.The improvement of the system's performance through coordinated VMI replenishments against the system with direct shipping only is shown and discussed in the comparative analysis section.

The inventory-routing problem (IRP) is a typical logistics optimisation problem that supply chains, implementing vendor managed inventory (VMI), are confronted with.It combines inventory control and vehicle routing.The main objective of the IRP is to jointly determine optimal quantities of the product to be delivered to the retailers, delivery periods and optimal vehicle routes for the shipment of these quantities.This paper considers a multiperiod inventory-routing problem with stochastic stationary demand rates (MP-SIRP).The problem is first formulated as a linear mixed-integer stochastic program for which we propose a deterministic equivalent approximation model (MP-DAIRP).This latter model can be decomposed into two well-know subproblems: an inventory allocation subproblem and a vehicle routing subproblem. The stochastic aspect of the demand is accounted for in the inventory allocation subproblem.The vehicle routing subproblem is solved as a deterministic mixed-integer problem.Lagrangian relaxation is used to determine close to optimal feasible solutions for the MP-DAIRP.Results of the proposed Lagrangian relaxation approach on some numerical examples are reported and thoroughly discussed.