Replenishment policy for a purchase-to-order seller: a tradeoff between ordering cost and delay cost

We consider replenishment decisions on when and how many goods to purchase for a seller under a purchase-to-order mode where there is no inventory and the seller purchases goods to fulfil orders already placed. For each purchase, there is a constant ordering cost. For each order, delay cost will be incurred if it is not fulfilled timely. Generally, the more frequent the replenishment, the higher the ordering cost but the lower the delay cost. Consequently, there is a tradeoff between the ordering cost and the delay cost for the seller to make replenishment decisions minimising the total cost. In this paper, we study three cases of the problem and investigate both offline versions and online versions according to the seller's knowledge about information of future orders. For offline versions with perfect information, we either develop an optimal policy, or prove it is NP-hard and develop an approximation policy. For online versions without any information about future orders, from the perspective of competitive analysis we prove the lower bound of competitive ratio for any possible online policy and present a 10-competitive online policy for the general case and a 2-competitive online policy for each of two special cases.