A note on capacitated lot sizing with setup carry-over

The "Capacitated lot sizing problem with setup carry-over" is based ort the well known "Capacitated lot sizing problem" and incorporates the possibility of preserving a setup-state from a period to the following. Sox and Gao [2] decompose their formulation of the first one (GCLP1) by Lagrangian relaxation. For obtaining a heuristic solution of the GCLP1 they use subgradient optimization and a dynamic programming algorithm to solve the corresponding subproblems optimally. The present paper elucidates that this algorithm does not necessarily provide the optimal solution of the subproblem. Additionally, two improved approaches are presented. The first is an extension of the one proposed by [2] while the second is supplementary based on a slight modification of the underlying model