Managing inventory systems with dual delivery modes and minimum order quantities

We consider an infinite‐horizon periodic‐review inventory system with dual delivery modes, each with a minimum order quantity (MOQ). The expedited mode provides a shorter lead time than the regular mode but has a higher unit ordering cost. As the optimal ordering policy for a system with dual delivery modes and MOQ requirements is unknown and expected to be very complicated, we propose a class of simple policies called single‐index (M,S)$$ \left(M,S\right) $$ policies, and we provide an exact procedure to compute the expected long‐run average total cost. Specifically, we first analyze the steady‐state distribution of the inventory position. Then we develop a recursive procedure to determine the steady‐state distribution of the inventory level. In addition, for a special case where ordering from the regular mode follows a base‐stock policy, we apply normal approximation to simplify the exact calculation of the cost. We also study a more complicated class of policies called dual‐index (M,S)$$ \left(M,S\right) $$ policies. In numerical studies, we first compare the average cost of the single‐index (M,S)$$ \left(M,S\right) $$ policy with that of a modified single‐index policy without MOQ consideration and the (M,S)$$ \left(M,S\right) $$ policy for the single‐mode system respectively to investigate the value of the single‐index (M,S)$$ \left(M,S\right) $$ policy. In addition, we find that the simpler single‐index (M,S)$$ \left(M,S\right) $$ policy performs close to the dual‐index (M,S)$$ \left(M,S\right) $$ policy and the optimal policy computed via dynamic programming. Finally, to assess the effectiveness of the normal approximation in the special case, we numerically compare the cost and policy parameters of the exact calculation with those of the normal approximation. The results illustrate that normal approximation not only improves the calculation speed, but also has near‐optimal solutions.