A note on “Price discount based on early order commitment in a single manufacturer-multiple retailer supply chain”

In a recent paper by Xie et al. Xie, J., Zhou, D., Wei, J.C., Zhao, X., 2010. Price discount based on early order commitment in a single manufacturer-multiple retailer supply chain. European Journal of Operational Research 200, 368-376], the authors have studied the early order commitment (EOC) strategy for a decentralized, two-level supply chain consisting of a single manufacturer and multiple retailers. They fail to provide an algorithm to determine the optimal EOC periods to minimize the total supply chain cost. This note proposes a polynomial-time algorithm to find the optimal solutions, and provides a new set of sufficient conditions under which the wholesale price discount scheme coordinates the whole supply chain.