Integration of Production Planning and Scheduling

We start our presentation with an overview of supply chain management, the associated planning problems, and a motivation for the integration of medium-term production planning and short-term scheduling decisions (Maravelias and Sung, 2008). We briefly review existing modeling and solution methods and then present four new approaches to this integrated problem. First, we discuss how lot-sizing problems can be extended to address features often found in process networks (Sung and Maravelias, 2008). Specifically, we discuss the problem where a) multiple items can be produced in each planning period, b) sequence-independent setups can carry over from previous periods, c) setups can cross over planning period boundaries, and d) setups can be longer than one period. We develop a mixed-integer programming (MIP) formulation for this generalized problem and derive a number of optimality properties that can be used to tighten the formulation. We also discuss a network representation and the resulting MIP formulation. We present examples that illustrate how the proposed model can be used to address medium -size single- and multi-unit problems. Second, we develop polyhedral results for a discrete-time state-task network (STN) formulation for the production planning of multi-stage continuous processes (Maravelias and Papalamprou, 2009). Specifically, we express the feasible region of the LP-relaxation as the intersection of two sets. We show that the constraints describing the first set yield the convex hull of its integer points. For the second set, we show that for integral data the constraint matrix is kappa-regular, and that the corresponding polyhedron is integral if the length of the planning period is selected appropriately. We use this result to prove that for rational data, integer variables assume integral values at the vertices of the polyhedron. We also discuss special cases of this problem, for which we derive stronger results. We illustrate how these results can be used to effectively address large-scale production planning problems. Third, we present a projection-based method where a detailed MIP scheduling model is used off-line to obtain a convex approximation of feasible production levels and a convex underestimation of total production cost as a function of production levels (Sung and Maravelias, 2007). These approximations involve only planning variables yet provide all the relevant scheduling information necessary to solve the integrated problem with high quality. We then develop two MIP formulations for the representation of nonconvex regions: the first expresses non-convex regions as unions of polytopes, while the second as difference of polytopes. We use these formulations to generate a non-convex projection of the scheduling feasible region onto the space of production level variables (Sung and Maravelias, 2009). We discuss how this method can be used in a rolling-horizon manner to address hard planning-scheduling problems, as well as a process design tool. Finally, we discuss methods for the decomposition of planning-scheduling problems in multi-stage batch processes (Ferris et al., 2009). We generate subproblems based on fixing: a) batching decisions, b) batch-unit assignment variables in a bottlenecking stage, c) the last batch in the sequence on each unit of the bottlenecking stage, and d) strong branching. We discuss how the above methods can be combined and carried out in a dynamic fashion exploiting the natural hierarchy of decisions (batching -> unit-task assignment -> sequencing). We also discuss the implementation of this dynamic decomposition algorithm within GAMS using the master-worker paradigm: the master processor generates and spawns all the tasks to worker processors, collects the results, and communicates information among workers; worker processors are part of a computational grid. We present computational results on a collection of six instances of varying difficulty.