Some comments on the economic lot size of a three-stage supply chain with backordering derived without derivatives

Chung and Wee [Chung, C.J., Wee, H.M., 2007. Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives. European Journal of Operational Research 183, 933-943] investigate the economic lot size of a three-stage supply chain with backordering derived without derivatives. Their inventory model is correct and interesting. Basically, they adopt an algebraic approach by using the method of complete squares to locate the optimal solution of the integrated system's total cost TC(q,B,n,M) and ignore the role of the functional behavior of TC(q,B,n,M) in locating the optimal solution of it. However, as argued in this paper [Chung, C.J., Wee H.M., 2007. Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives. European Journal of Operational Research 183, 933-943] need to explore the functional behavior of TC(q,B,n,M) to justify their solution. So, from the viewpoint of logic, the derivations about the optimal solution have some shortcomings such that the validity of the solution procedure in Chung and Wee [Chung, C.J., Wee H.M., 2007. Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives. European Journal of Operational Research 183, 933-943] is questionable. The main purpose of this paper is to overcome those shortcomings and present complete proofs for Chung and Wee [Chung, C.J., Wee H.M., 2007. Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives. European Journal of Operational Research 183, 933-943].