Inventory control in multi‐channel retail

In recent years multi‐channel retail systems have received increasing interest. Partly due to growing online business that serves as a second sales channel for many firms, offering channel specific prices has become a common form of revenue management. We analyze conditions for known inventory control policies to be optimal in presence of two different sales channels. We propose a single item lost sales model with a lead time of zero, periodic review and nonlinear non‐stationary cost components without rationing to realistically represent a typical web‐based retail scenario. We analyze three variants of the model with different arrival processes: demand not following any particular distribution, Poisson distributed demand and a batch arrival process where demand follows a Pòlya frequency type distribution. We show that without further assumptions on the arrival process, relatively strict conditions must be imposed on the penalty cost in order to achieve optimality of the base stock policy. We also show that for a Poisson arrival process with fixed ordering costs the model with two sales channels can be transformed into the well known model with a single channel where mild conditions yield optimality of an (s, S) policy. Conditions for optimality of the base stock and (s, S) policy for the batch arrival process with and without fixed ordering costs, respectively, are presented together with a proof that the batch arrival process provides valid upper and lower bounds for the optimal value function.