(411h) A Multi-Stage Stochastic Programming Approach for Scheduling of Batch Processes Under Decision-Dependent Uncertainty
The key challenge in developing scenario based stochastic frameworks for modelling type II endogenous uncertainty is to enforce non-anticipativity (ensuring that current decisions are made without anticipating information from the future) throughout the decision-making process. This often involves introduction of auxiliary binary variables and non-anticipativity constraints . Introducing such auxiliary variables and constraints result in large model size, which further increases with the number of scenarios , . In our previous work , we developed a novel two-stage stochastic approach to model type II endogenous uncertainty that ensures non-anticipativity implicitly via material balance constraints, without the introduction of any auxiliary variable or explicit non-anticipativity constraints. We proposed a large-scale scheduling model to allocate samples (materials) that has to be processed sequentially through a set of tasks and reported significant benefits in using the proposed two-stage approach in modelling actual industrial case studies. The proposed two-stage approach considered uncertainty in recycle rate, i.e. the fraction of samples returned back to a previous task, with a limiting assumption that the recycle rate remains constant throughout the time horizon, independent of how many times a task is being processed. Governed by various operating factors, uncertainties like recycle rate are unlikely to remain constant throughout the operating time horizon. To account for such possible fluctuations in the uncertainty realizations, in this study, we relax the above assumption and provides flexibility to the system by extending the framework to a multi-stage approach, where the time horizon is divided into multiple time periods and a different realization of uncertainty (recycle rate) can be considered in each time period. Enforcing non-anticipativity for a multi-stage framework becomes even more challenging when a type II endogenous uncertainty is considered. To address this challenge, the proposed model considers a node-based approach. Nodes represent the uncertainty realizations. For every time period, the number of nodes represent the number of unique uncertainty realizations of that time period. At every time period, the model solves for each node considering the current uncertainty realization and the realizations of the preceding nodes. Together, with the material balance constraints and the node-based approach, the proposed multi-stage framework enforces implicit non-anticipativity without the need to specify additional auxiliary binary variables and explicit non-anticipativity constraints.
The proposed model has been validated using two case studies - a motivating case study, one of the most widely studied process network from literature  and an actual large-scale industrial case study. Computational studies were conducted and value of stochastic solution (VSS) was calculated to estimate the benefits of the proposed approach. Further studies were conducted to analyze the model sensitivity to the number of stages, number of uncertainty realizations and the production capacity. The computational results obtained from both the case studies depict significant benefits in using the proposed multi-stage approach. The results from the motivating case study shows more than 11% increase in the production rate, whereas the industrial case study results shows an increase of 8% in the annual earnings.
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