This paper presents a novel approach for solving an integrated production planning and scheduling problem. In theory as well as in practice, because of their complexity, these two decision levels are most of the time treated sequentially. Scheduling largely depends on the production quantities (lot sizes) computed at the production planning level and ignoring scheduling constraints in planning leads to inconsistent decisions. Integrating production planning and scheduling is therefore important for efficiently managing operations. An integrated model and an iterative solution procedure were proposed in earlier research papers: The approach has limitations, in particular when solving the planning problem. In this paper, a new formulation is proposed to determine a feasible optimal production plan, i.e. lot sizes, for a fixed sequence of operations on the machines when setup costs and times are taken into account. Capacity constraints correspond to paths of the conjunctive graph associated to the sequence. An original Lagrangian relaxation approach is proposed to solve this NP-hard problem. A lower bound is derived and an upper bound is calculated using a novel constructive heuristic. The quality of the approach is tested on numerous problem instances. 
A Lagrangian heuristic for an integrated lot-sizing and fixed scheduling problem