(471g) Simultaneous Scheduling and Dynamic Optimization of Discrete-Time Network Processes Using a Logic-Based Discrete Benders Decomposition Approach

The simultaneous scheduling and dynamic optimization (SSDO) of batch systems in chemical plants is a promising strategy to simultaneously optimize scheduling requirements and dynamic goals. Despite the advances in the field [1], the effective solution of SSDO problems in chemical batch systems still requires the development of reliable algorithms, specifically when considering variable processing times in discrete-time SSDO formulations. Discrete-time SSDO problems handle timing variables as discrete decisions after discretization of the time domain, which increases the combinatorial complexity of the nonlinear SSDO problem. This work studies the SSDO of discrete-time network production environments with variable processing times, which play an important role in chemical batch plants. For instance, short processing times may improve scheduling economics by increasing throughput, while long processing times may reduce the consumption of heating and cooling resources during the dynamic operation of a batch reactor.

Although continuous-time approaches for the SSDO of network systems have been widely studied in the literature, they may show a poor performance in terms of solution quality and computational time with respect to discrete-time formulations, e.g., the computational cost of continuous-time formulations may become prohibitive as the scheduling horizon increases, or when including side-constraints such as intermediate due dates [2]. In contrast, there are limited studies on the SSDO of discrete-time network problems that consider variable processing times. Previous works have proposed modified resource task network models to enable variable processing times in discrete-time SSDO [3]. Nevertheless, due to model inflation, only limited search spaces for processing times have been considered in practice, e.g., a binary variable that decides between a fast or slow operation mode of a task. This way of approximating variable processing times may result in a suboptimal or infeasible operation of the plant if fast or slow operation modes are not adequately selected, e.g., very short processing times may not be enough to meet the required conversion in a reaction, while very long processing times can lead to an economic loss. Therefore, research is needed to develop reliable modeling and solution frameworks for the discrete-time SSDO of network processes that provide more flexibility in the modeling of variable processing times. This would enable a more accurate representation of real-world batch chemical plants, which would return more accurate schedules.

This study brings novelty by presenting a Generalized Disjunctive Programming (GDP) modeling and solution framework for the SSDO of discrete-time State Task Network (STN) problems subject to variable processing times. The proposed GDP model has the flexibility to account for both discrete and continuous processing times, which is a feature that has not been considered in previous discrete-time SSDO network formulations. A refined extension of a Logic-based Discrete Benders Decomposition (LD-BD) method is considered in this work [4], which builds upon the general logic-based Benders decomposition (LBBD) framework proposed by Hooker [5]. LD-BD has only been previously used to solve nonlinear GDP superstructure design problems; hence, this work presents the first extension of this method into optimal process integration. The key idea of any LBBD algorithm is to design customized Benders cuts for each application, aiming to improve the efficiency of the solution process. Although many LBBD algorithms have been developed for Mixed-Integer Programming (MIP) scheduling applications, LD-BD is the first LBBD algorithm with Mixed-Integer Nonlinear Programming (MINLP) capabilities. Hence, this work presents the first application of an LBBD method in the context of a nonlinear scheduling application such as SSDO. Furthermore, decomposition solution strategies that rely on LBBD have shown potential in many applications involving sequential scheduling environments; however, applications to network scheduling environments such as that considered in this work are limited in the literature [5]. The proposed LD-BD strategy to address the SSDO of discrete-time STNs with variable processing times is explained next.

LD-BD performs a decomposition of the problem into a master Integer Programming problem that handles complicating variables and MINLP subproblems that solve for the remaining variables in an iterative fashion. A complicating variable is referred to as a variable that, when fixed, facilitates the solution of the subproblem. In brief, the master problem iteratively updates the values of complicating variables, based on a series of cuts that are inferred from the subproblems. The distinctive feature of LD-BD is that the cuts are specifically designed to handle ordered discrete decisions as complicating variables. In the context of discrete-time SSDO, these ordered discrete decisions are the discrete processing times (or the upper bound of processing times when processing times are treated as continuous variables) and the batching variables, i.e., the number of times each task is performed in each unit. The variables that are left to optimize by the subproblem are open loop control actions, as well as sequencing, and timing scheduling decisions. In contrast to other Benders-based strategies, LD-BD does not solve one but multiple subproblems at each iteration within a neighborhood of discrete decisions. This allows to generate cuts that consider the information of multiple subproblems at a time, which ultimately guarantees local optimality within a user-defined neighborhood of ordered discrete decisions. LD-BD has been substantially improved in this work to address discrete-time STN SSDO problems with variable processing times. The first improvement to LD-BD takes advantage of the fact that it solves multiple subproblems per iteration, which allows to generate multiple cuts per iteration to increase the convergence speed toward a locally optimal solution. The second improvement is the development of a pruning strategy that avoids solving every MINLP subproblem. An MINLP subproblem is pruned (avoided) based on a MIP approximation of the subproblem that incorporates scheduling information only and assumes that dynamic feasibility can be attained. For instance, if the solution from this approximated MIP subproblem is worse than the current best solution to the original scheduling and dynamic optimization problem, then the solution of the rigorous MINLP subproblem can be avoided.

The performance of the proposed framework was tested using a case study proposed by Chu and You [6] involving one mixing tank, two batch reactors that may perform three different reactions (tasks), a separation unit, and two packing units that generate two products. The objective function considered is the maximization of profits. The tasks performed by the reactors are batch reactions, whose dynamic models and variable processing times are included in the formulation. Both LD-BD and the traditional MINLP solver DICOPT were initialized from the solution reported by a sequential scheduling and dynamic optimization approach to showcase the advantages of LD-BD, and to illustrate the benefits of SSDO. The proposed LD-BD framework was able to improve the objective function by 13% with respect to the initialization; this solution was obtained in 18369 seconds. In contrast, DICOPT was unable to return a feasible solution within a CPU time limit of 50000 seconds. A similar performance was obtained from other MINLP solvers, e.g., SBB, BARON, and SCIP. This work shows that the SSDO problem with variable processing times within a discrete-time STN modeling framework can be addressed using a tailored LD-BD solution strategy.

References

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[4] D. A. Liñán and L. A. Ricardez-Sandoval, “A Benders decomposition framework for the optimization of disjunctive superstructures with ordered discrete decisions,” AIChE Journal, vol. n/a, no. n/a, p. e18008, 2023, doi: 10.1002/aic.18008.

[5] J. N. Hooker, “Logic-Based Benders Decomposition for Large-Scale Optimization,” in Large Scale Optimization in Supply Chains and Smart Manufacturing: Theory and Applications, J. M. Velásquez-Bermúdez, M. Khakifirooz, and M. Fathi, Eds., in Springer Optimization and Its Applications. Cham: Springer International Publishing, 2019, pp. 1–26.

[6] Y. Chu and F. You, “Integrated Scheduling and Dynamic Optimization of Complex Batch Processes with General Network Structure Using a Generalized Benders Decomposition Approach,” Ind. Eng. Chem. Res., vol. 52, no. 23, pp. 7867–7885, Jun. 2013, doi: 10.1021/ie400475s.