(626f) Efficient Solution of Enterprise-Wide Optimization Problems Using Nested Stochastic Blockmodeling

The integration of process operations has been considered as a promising avenue to improve process economics by considering simultaneously decisions at different scales [1], resulting in large optimization problems whose monolithic solution is challenging. Decomposition-based solution algorithms can be used to reduce the computational time [2-5], but a decomposition of the optimization problem itself is required. In general, the application of decomposition-based solution approaches is currently case specific and a general framework to determine the decomposition of optimization problems is lacking.

In this work, we propose the application of nested Stochastic Blockmodeling (nested SBM) for learning of the underlying multilevel hierarchical block structure of many integrated enterprise-wide optimization problems. Based on our previous work [6,7], the optimization problem (variables and constraints) is represented as a graph and its structure is learnt using statistical inference. Nested SBM is a random graph generative model which corresponds to a nested sequence of network ensembles (Stochastic Blockmodels) where the connections among the nodes at different levels depend only on their block affiliation [8]. Therefore, given the graph of an optimization problem the parameters of the nested SBM model, i.e. the node affiliation at different hierarchical levels, can be learnt through statistical inference and the learnt hierarchical structure of the graph reveals the structure of the problem at multiple scales.

We apply this approach first to the integration of scheduling and dynamic optimization for parallel lines. We assume that 6 products must be produced in two lines. The variable graph is decomposed into 8 blocks and the nested model is shown to have three levels. The first level has a core periphery structure which is used as the basis for the application of Generalized Benders Decomposition (GBD). The original graph has a hybrid multi-core community structure which is used as the basis for nested GBD, where the multi-core structure is used to decompose the scheduling problem. The nested GBD approach is shown to solve the problem faster compared to the GBD approach.

In a second case study, the integration of planning, scheduling and dynamic optimization is considered. For six products and two planning periods, the variable graph is decomposed into 14 blocks and the nested model is shown to have three levels. The variable graph has a multi-core community structure whereas levels 1 and 2 have a core periphery structure. We apply GBD based on this core periphery structure and a solution is obtained in reduced computational time compared to the monolithic solution.

Overall, we posit argue that application of nested SBM and statistical inference reveals the multiscale nature of a broad class of integrated optimization problems. Furthermore, the exploitation of the structure at different hierarchical levels is shown to reduce the computational time.

References:

[1]. Daoutidis, P., Lee, J. H., Harjunkoski, I., Skogestad, S., Baldea, M., & Georgakis, C. (2018). Integrating operations and control: A perspective and roadmap for future research. Computers & Chemical Engineering, 115, 179-184.

[2]. Terrazas‐Moreno, S., Flores‐Tlacuahuac, A., & Grossmann, I. E. (2008). Lagrangean heuristic for the scheduling and control of polymerization reactors. AIChE Journal, 54(1), 163-182.

[3]. Li, Z. and Ierapetritou, M.G. (2010). Production planning and scheduling integration through augmented Lagrangian optimization. Computers & Chemical Engineering, 34(6), pp.996-1006.

[4]. Chu, Y. and You, F., 2013. Integrated scheduling and dynamic optimization of complex batch processes with general network structure using a generalized benders decomposition approach. Industrial & Engineering Chemistry Research, 52(23), pp.7867-7885.

[5]. Nie, Y., Biegler, L. T., Villa, C. M., & Wassick, J. M. (2015). Discrete time formulation for the integration of scheduling and dynamic optimization. Industrial & Engineering Chemistry Research, 54(16), 4303-4315.

[6]. Allman, A., Tang, W., & Daoutidis, P. (2019). DeCODe: a community-based algorithm for generating high-quality decompositions of optimization problems. Optimization and Engineering, 20(4), 1067-1084.

[7]. Mitrai, I., Tang, W., & Daoutidis, P. (2021). Stochastic Blockmodeling for Learning the Structure of Optimization Problems, submitted to AIChE Journal

[8]. Peixoto, T.P. (2014). Hierarchical block structures and high-resolution model selection in large networks. Physical Review X, 4(1), p.011047.