(696g) A Mixed-Integer Programming Model and Tightening Methods for Scheduling in General Chemical Production Environments
A Mixed-integer Programming Model and Tightening Methods for Scheduling in General Chemical Production Environments
Sara Velez and Christos T. Maravelias
Department of Chemical and Biological Engineering, University of Wisconsin – Madison
1415 Engineering Dr., Madison, WI 53706, USA
Most existing models for chemical production scheduling are applicable to either purely sequential or purely network processes. Batch-based models are used for sequential processes where batches of material must remain intact. For network processes, where batches of material can be mixed and split, material-based approaches are used. Many production environments combine sequential and network subsystems; e.g., upstream fermentation and purification steps (sequential) followed by mixing steps (network). Hybrid production environments may have tasks that can produce and consume multiple materials with a variety of batching restrictions. With the exception of the approach of Sundaramoorthy and Maravelias (2011), which considers processes with sequential and network subsystems but not more general hybrid systems, no methods are available for these problems.
Here, we propose a general model to address these limitations by using material handling constraints to define the type of production environment. Each material in the process is classified as having no batching restrictions, just no-mixing restrictions, just no-splitting restrictions, or both no-mixing and no-splitting restrictions. We add constraints for each material to enforce the restrictions. A sequential process can be modeled by classifying all intermediates as having both no-mixing and no-splitting restrictions. Similarly, a purely network process can be modeled as having no material handling restrictions. More generally, any material in the process can have any type of restriction, enabling us to model any type of production environment through material handling constraints. The model is extended to consider storage in processing units and variable processing times that do not depend on the batch size. Other processing constraints and characteristics can be incorporated into the model using known modeling techniques. We use discrete-time representation to develop the model, but the methods can be applied to continuous-time formulations as well.
To address large-scale instances we extend the propagation algorithm and tightening methods of Velez et al. (2013) to facilities with special material handling restrictions. The extensions include (1) attainable batch-sizes, (2) updated customer demands, and (3) updated minimum numbers of batches. We first calculate attainable batch-sizes based on the material handling restrictions. For example, for materials with both no-mixing and no-splitting restrictions, the batch-size must remain constant; therefore the attainable batches-sizes for all units that produce or consume the material are the same regardless of the unit capacities. Next, when batches of final products cannot be split among customers, we can find tighter bounds on the customer demand based on the attainable batch-sizes. Finally, the total minimum number of batches depends on the material handling restrictions. For example, the tasks producing and consuming a material with both no-mixing and no-splitting restrictions must process the same number of batches regardless of their capacities. Using this information we can write constraints bounding the total production and total number of batches for each task and material. Adding these constraints reduces the computational time by several orders of magnitude. Many problems that cannot be solved to optimality within 1 day without the constraints are solved in just a few seconds once the constraints are added.
Sundaramoorthy, A. and Maravelias, C.T. A General Framework for Process Scheduling. AIChE J., 57(3), 695-710, 2011.
Velez, S., Sundaramoorthy, A., and Maravelias, C. T. Valid Inequalities Based on Demand Propagation for Chemical Production Scheduling MIP Models. AIChE J., 59 (3), 872-887, 2013