(751d) A Novel Two-Stage Stochastic Programming Approach for Batch Processes with Unknown Time of Uncertainty Realization

Scheduling is a decisive technique that enables companies to manage time and resources effectively to achieve desired targets at minimum cost or in short turnaround times. One of the challenging aspects of industrial scheduling is uncertainty modelling, which is key to obtain a realistically feasible solution. One of the prominent and most widely used uncertainty modelling approaches is stochastic programming, which allows a set of here and now and wait and see decisions to be made before and after the realization of uncertainty, respectively. A major factor that increases the complexity of uncertainty modelling is the type of uncertainty. Uncertainties are often classified as exogenous and endogenous. Exogenous uncertainties are those that are independent of the model decisions and only depend on the external factors (e.g. market demand), whereas endogenous uncertainties are those that are dependent on model decisions (e.g. product yield). Endogenous uncertainties can be further classified as type I (realizations are dependent on the model decisions) and type II (the time of realizations are dependent on the model decisions) [1]. Most of the available literature on stochastic programming focus on exogenous uncertainties. In this study, the focus is on endogenous uncertainty modelling (type II) using stochastic programming.

One of the key factors of a stochastic programming approach is to ensure non-anticipativity throughout the decision making process. All here and now decisions should be based on the information available at that time and without anticipating information from the future. This becomes challenging when the uncertainties are model dependent and the time of uncertainty realization is unknown. In most of the existing models with type II endogenous uncertainty, this is accomplished by introducing non-anticipativity constraints through auxiliary binary variables [1], [2], [3]. Introduction of auxiliary binary variables and constraints linking them to the model decisions results in a large model size, which further grows exponentially with the number of scenarios. Several studies have explored various relaxation and decomposition strategies to solve such models in reasonable computational times [4], [5].

In this study, we propose a novel stochastic programming approach to ensure non-anticipativity implicitly via material balance constraints for discrete-time batch process operations with type II endogenous uncertainty, without the introduction of any auxiliary variables or explicit non-anticipativity constraints. The proposed framework has been validated using actual data from a large-scale scientific services plant, where sets of samples has to go through a series of processes to complete the analysis [6]. In such industries, factors such as process parameters, manual and machine errors can affect the quality of the analysis and a fraction of samples are often recycled back to a previous process for reanalysis (retesting). Uncertainty is considered in the fraction of samples recycled to a previous process for retesting and the results obtained for a two-stage stochastic model are presented. The results from this implementation indicate benefits between 15-18% in throughput in comparison to the deterministic model, thus showing the potential of the proposed framework.

References:

1. Goel, V., & Grossmann, I. E. (2006). A class of stochastic programs with decision dependent uncertainty. Mathematical programming, 108(2-3), 355-394.
2. Jonsbråten, T. W., Wets, R. J., & Woodruff, D. L. (1998). A class of stochastic programs withdecision dependent random elements. Annals of Operations Research, 82, 83-106.
3. Tarhan, B., & Grossmann, I. E. (2008). A multistage stochastic programming approach with strategies for uncertainty reduction in the synthesis of process networks with uncertain yields. Computers & Chemical Engineering, 32(4-5), 766-788.
4. Colvin, M., & Maravelias, C. T. (2010). Modeling methods and a branch and cut algorithm for pharmaceutical clinical trial planning using stochastic programming. European Journal of Operational Research, 203(1), 205-215.
5. Gupta, V., & Grossmann, I. E. (2011). Solution strategies for multistage stochastic programming with endogenous uncertainties. Computers & Chemical Engineering, 35(11), 2235-2247.
6. Lagzi, S., Lee, D. Y., Fukasawa, R., & Ricardez-Sandoval, L. (2017). A computational study of continuous and discrete time formulations for a class of short-term scheduling problems for multipurpose plants. Industrial & Engineering Chemistry Research, 56(31), 8940-8953.