Integration of planning, scheduling and control problems using data-driven feasibility analysis and surrogate models
Introduction
US Industrial manufacturers are facing numerous challenges such as increasing complexity of production processes, fluctuating customer demands, and expansion of supply chains. Foreign trade is at historically low levels, and nationalist governments around the world are threatening to further undermine the free flow of goods, creating more uncertainty and constraints upon manufacturing growth. In such a slow-growth environment, increases in production efficiency and cost reductions are essential. In the field of operations research and process systems engineering, the main strategy to combat the emerging challenges is the pursue of optimal operating conditions through an enterprise-wide optimization (EWO).
Enterprise-wide optimization proposes to optimize decision-making processes related to supply, manufacturing and distribution within a company. The major operational decisions include planning, scheduling and process control (Grossmann, 2005; Dias and Ierapetritou, 2017), usually represented in a hierarchical way. On one end, tactical decisions determine allocation of resources on a time scale of months and years; on the other end, operational decisions address disturbances on a time scale of seconds.
Traditionally, planning, scheduling and control problems are considered individually and solved in a sequential way. Planning problems usually set production targets for different manufacturing facilities which, in their turn, try to achieve the targets by defining an appropriate schedule. Scheduling decisions can be translated to setpoints and production sequences, which are transmitted to the control layer. The control attempts to implement the scheduling decisions, while handing disturbances over time. Such sequential strategies result in upper-level decisions made with little knowledge of the production constraints at the lower levels of the decision-making hierarchy. The decisions are usually assumed to be inflexible, and lower-levels decision makers either achieve the targets at higher costs, or fail to achieve those targets (Engell and Harjunkoski, 2012).
Driven by the possibility of determining an overall optimal and feasible solution, many researchers have explored the problems of integrating two or more decision making process, and techniques to solve the complex resulting problems have been developed. One of the major challenges in this integration is dealing with the different time scales related to each individual level. Usually, the simplest alternative for solving integrated problems is to adopt the time scale of the lower-level problem and formulate a single simultaneous model by incorporating the detailed lower-level problem as constraints in the upper-level model. However, this approach becomes computationally intractable when applied to large time horizons and high-dimensional problems (Grossmann, 2005). To address these challenges, several approaches have been proposed in the literature.
At the scheduling and control level, initial efforts towards integration followed the intuitive route of including the dynamic model of the process as an additional set of constraints in the scheduling problem. The result is a mixed-integer dynamic optimization problem (MIDO), and its solution provides the optimal production sequence and optimal control moves required to implement the schedule. The MIDO problem is then discretized into a Mixed Integer Nonlinear Programming (MINLP) (Flores-Tlacuahuac and Grossmann, 2006; Zhuge and Ierapetritou, 2012). Alternatively, decoupled modeling approaches consisting of formulating the scheduling problem (master problem) as a Mixed Integer Linear Programming (MILP) and the control problem (primal problem) as Dynamic Optimization have also been proposed (Nyström et al., 2005; Nyström et al., 2006). This problem is solved through iterations between the master and primal problems. These approaches, however, face considerable challenges associated with the use of high-fidelity representations of the process dynamics and the complexity, nonlinearities and discontinuities that this brings to the scheduling problem. While decomposition techniques (Terrazas-Moreno et al., 2008; Chu and You, 2013; Chu and You, 2013) and time-scale bridging models (Du et al., 2015; Pattison et al., 2016) have been proposed to address these challenges, the computational cost of performing the integrated scheduling/control calculations in real-time remains an issue. Additionally, there is a need for the consideration of stability issues related to the dynamic behavior of the system and incorporation of advanced control techniques that have been well established in the industry.
Further up in the decision-making hierarchy, we face the problem of integrating production planning and scheduling. As identified by Maravelias and Sung (2009), a general formulation for the planning problem is given by resource constraints, feasibility of production targets, production and holding costs constraints and material balances. The integration of planning and scheduling translates to incorporating scheduling information in order to determine the resource and production cost constraints. The formulations can be grouped into three categories: (a) detailed scheduling models, (b) relaxations/aggregations of scheduling models, and (c) surrogate models. The first group includes intuitive ideas such as replacing the resource and production cost constraints by a monolithic scheduling model over the entire planning horizon (full-space models). Clearly, such methods result in complex mathematical formulations that can become intractable for large time horizons and high-dimensional problems. The second group tries to handle the problem's complexity by removing some of the scheduling constraints (Harjunkoski and Grossmann, 2002; Jain and Grossmann, 2001; Maravelias, 2006; Roe et al., 2005), or by aggregating some of the decisions of the original scheduling formulation (Wilkinson et al., 1995; Birewar and Grossmann, 1990; Wellons and Reklaitis, 1991; Erdirik-Dogan and Grossmann, 2006). Alternatively, the use of surrogate models have been proposed to generate constraints that define the feasible region of the scheduling model and the production cost as function of production targets (Sung and Maravelias, 2007; Sung and Maravelias, 2009; Li and Ierapetritou, 2010; Li and Ierapetritou, 2009). Such methods can generate more accurate and computationally tractable descriptions of resource and production costs constraints.
While previous research brought significant advances to the area of enterprise wide optimization, few works attempted to achieve the overall integration of planning, scheduling and control (Gutiérrez-Limón et al., 2014; Shi et al., 2015; Charitopoulos et al., 2018). Furthermore, the similarities between the integration of the different decision-layers have not been recognized. Planning, scheduling and control problems are modeled using different mathematical optimization concepts and address problems with different time scales. However, the underlying problems of integrating planning-scheduling and scheduling-control have several common aspects that can be addressed with a standard framework. In practice, the efficient integration of any two decision-making processes translates to the aggregation and inter-exchange of the proper amount of information between the different processes, in order to ensure that feasibility is maintained while preventing unnecessary rise of the problem complexity.
Therefore, the objective of this work is to present a systematic framework to achieve the overall integration of decision-making processes. The integrated problem is first formulated as a grey-box optimization problem. Then, classification and regression methods are employed to approximate the unknown black-box constraints. Specifically, classification methods are employed following a data-driven feasibility analysis framework to approximate inequality constraints, while regression models are used to approximate equality constraints. We follow a systematic procedure to achieve the overall integration, consisting of two building blocks (Fig. 1): first, we address the integration of scheduling and control. Then, the integrated scheduling and control problem is treated as a black box on the integration of planning and scheduling. To handle dimensionality issues, we introduce the concept of feature selection when building the surrogate models. The methodology is applied to the optimization of an enterprise of air separation plants.
This paper is organized as follows. In section two, a background in grey-box optimization, data-driven feasibility analysis and feature selection methodologies is provided, building the theoretical basis of the proposed framework. In section three, the air separation problem is described, and section four presents the methodology proposed to integrate planning, scheduling and control problems. The performance of the proposed framework is demonstrated through case studies in section five. Final conclusions and future work are discussed in section six.
Section snippets
Grey-box optimization
Grey-box problems are characterized by partial or total lack of closed-form equations describing the constraints and the objective of the problem. Such problems arise in a variety of fields including chemical engineering, geosciences, financial management, molecular engineering and aerospace engineering (Boukouvala et al., 2017). Grey-box problems primarily rely on expensive simulations, input/output data, or phenomena which have not yet been defined by physics-based mathematical equations.
A
Problem definition
In this work, the ideas of decision-making integration using grey-box optimization, surrogate models and data-driven feasibility analysis are applied to an air separation unit that produces gas nitrogen (GN2), gas oxygen (GO2), and liquid nitrogen (LN2). The unit operates by first compressing ambient air in a large multistage compressor, followed by removal of water, carbon dioxide and hydrocarbons, and by cooling in a multi-stream heat exchanger. This air feed mixture of oxygen, nitrogen and
Framework for the integration of planning, scheduling and control
The systematic methodology for the integration of planning, scheduling and control includes two building blocks. First, the integration of scheduling and control is performed by treating the control problem as a black box. Input-output data is obtained by simulating the control problem. Algebraic expressions for unknown constraints are obtained using feasibility analysis and surrogate models. Therefore, an algebraic model for the scheduling-control problem is obtained, and can be solved to
Case study
The framework for the integration of planning, scheduling and control was implemented in the enterprise of air separation plants described in Section 3. In this section, we give a brief overview of the dynamic control simulation, followed by the results of integrating scheduling and control. Finally, we discuss the results of the overall planning, scheduling and control optimization.
Conclusions
This work presented a systematic framework for the integration of planning, scheduling and control problems. The framework consists of two-building blocks, which are addressed following principles of gray-box optimization, feasibility analysis, feature selection, and surrogate modeling. The proposed framework was implemented in the problem of optimization of an enterprise of air separation plants, considering deterministic electricity prices that vary in an hourly basis and flexible operation
CRediT authorship contribution statement
Lisia S. Dias: Conceptualization, Methodology, Software, Validation, Writing - original draft. Marianthi G. Ierapetritou: Conceptualization, Methodology, Supervision, Project administration, Funding acquisition, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
L.S.D. gratefully acknowledges financial support from CNPQ - Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brazil. M.G.I. acknowledges financial support from NSF under grant CBET 1159244, grant CBET 1839007 and grant CBET 1547171.
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