(658b) A Generic Unit-Specific Event-Based Mathematical Formulation for Short-Term Scheduling Multipurpose Batch Plants

Multipurpose batch plants widely exist in the chemical industry for producing low-volume, high-value products. To achieve better economic benefits, optimal scheduling solutions with high utilization of resources, high revenue and low makespan are desirable. A plethora of mathematical formulations¹ have been developed for short-term scheduling problems, which can be divided into discrete- and continuous-time models. They can be further classified into global event-based², slot-based including process-slot based³ and unit-slot based⁴, unit-specific event-based^5-6 and sequence-based⁷ models. These models can be also classified into single- and multiple- time grid mathematical models¹.

Among these models, the capabilities of the unit-specific event-based formulations are well established in literatures^5-6 as they are more flexible and efficient (e.g. smaller model size and less computational effort), compared to other continuous-time models. It is often more efficient¹ and accurate than the discrete-time variant in handling variable processing times⁸.

Although many unit-specific event-based models^1,5,6 have been developed for short-term scheduling of multipurpose batch plants, they still have several limitations. First, they still require a greater number of event points to generate the optimal solution, leading to computational inefficiency. They also led to sub-optimality in some cases, which attributes to the strict sequencing constraints imposed to avoid real-time storage violation. More importantly, almost all existing models did not allow a batch of materials to be temporarily stored in processing units or be partially transferred to downstream units after production, leading to sub-optimality and inefficient utilization of processing units.

In this work we develop a novel continuous-time formulation using the unit-specific event-based modelling approach for short-term scheduling multipurpose batch plants to address the above limitations. The concepts of indirect and direct material transfer are explicitly introduced to sequence different tasks in different units, which allow material transfer to be monitored between units rather than specific tasks to reduce the number of discrete decision variables and improve computational efficiency. These concepts also allow us to relax some sequencing constraints for different tasks in different units, leading to further reduction in the number of event points required. A batch of materials after production is allowed to be partially transferred to storage and downstream processing units or temporarily held in suitable processing units over multiple event points. A new continuous variable denoting the time when a state produced in a processing unit is available at an event point is introduced to assist the sequencing of different tasks related to the same states in different units, which overcomes the limitations in the existing unit-specific event-based models^5,9. To further reduce the computational expense, some new tightening constraints are proposed. A number of benchmark examples are solved to evaluate the performance of the proposed formulations. The computational results demonstrate the proposed models require a smaller number of event points in many cases to achieve optimality than existing unit-specific event-based models⁵. It is also interestingly found that no task is required to span over multiple event points to generate the optimal solution for all tested problem instances. As a result, computational efforts were significantly reduced by at least one order of magnitude. For some instances, literature models⁵ cannot prove optimality over one hour, whilst our model can converge using about 2000s. More importantly, our models can generate better optimal solutions compared to existing unit-specific event-based⁵ and slot-based³ models. Optimal solutions could be increased by a maximum improvement of 67%.

The industrial-scale Kallrath example with zero-wait policy and variable production fractions is addressed. The computational results show that our model leads to a reduction in the CPU time by over 45% for medium instances and reports smaller makespan by 5.2% for large instances, compared to the existing unit-specific event-based models⁵, Additionally, our model yields competitive solutions for very large instances within 40000 CPU seconds, whilst the discrete-time formulation¹⁰ fails to find a feasible solution or generate a worse solution within the same time resources. The developed models are incorporated into the rolling-horizon framework^11-12 to solve very large instances. The results demonstrated that with significant improvement in the short-term scheduling model, the rolling-horizon framework can find better solutions.

Reference

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