Joint replenishment problem with truck cost structures

We consider inventory systems with multiple items in the presence of stochastic demand and jointly incurred order setup costs. The problem is to determine the replenishment policy that will minimize the total expected ordering, inventory holding and backorder costs; the so–called stochastic joint replenishment problem in the literature. In particular, we study the settings in which order setup costs reflect the transportation costs and have a step–wise cost structure, each step corresponding to an additional transportation vehicle. For this setting, we propose a new policy which we call the (s, Q) policy. Under this policy, a replenishment order of fixed size Q is triggered whenever the inventory position of one of the items drops to its reorder point s. The replenishment order is allocated to multiple items to equalize inventory positions of items to the extent possible. The policy is designed for settings in which the backorder and setup costs are high, as it allows the items to independently trigger replenishment orders and fully exploits the economies of scale by consistently ordering the same quantity. A numerical study is conducted to confirm that the policy works as designed and to compare its performance against the (Q, S) and (Q, s, S) policies that were suggested earlier in the literature. The study shows that the proposed (s, Q) policy outperforms the (Q, S) when the backorder and setup costs are high and when the vehicles are not capacitated. When the vehicles are capacitated, the new policy outperforms both other policies under the most settings considered.