Short-Term Scheduling

During the last two decades, the problem of short-term scheduling of multiproduct and multipurpose batch plants has gained increasing attention in the academic, research, and manufacturing communities, predominantly because of the challenges and the high economic incentives. In the last 10 years, numerous formulations have been proposed in the literature based on continuous representations of time. The continuous-time formulations have proliferated because of their established advantages over discrete-time representations and in the quest to reduce the integrality gap and the resulting computational complexities. The various continuous-time models can be broadly classified into three distinct categories: slot-based, global event-based, and unit-specific event-based formulations. In this paper, we compare and evaluate the performance of six such models, based on our implementations using several benchmark example problems from the literature. Two different objective functions, maximization of profit and minimization of makespan, are considered, and the models are assessed with respect to different metrics such as the problem size (in terms of the number of binary variables, continuous variables, and constraints), computational times (on the same computer), and number of nodes needed to reach zero integrality gap. Two additional computational studies with resource constraints such as utility requirements are also considered.

Modeling and Optimization for Short-term Scheduling of Multipurpose Batch Plants

A novel mathematical formulation for short-term scheduling of multipurpose batch plants in chemical manufacturing

While the part 1 focuses on the application of the proposed formulation to scheduling, this paper focuses mainly on the integration of planning and scheduling in multipurpose batch plants. In dealing with this problem, the method presented in part 2 exploits the mathematical structure of the overall plant model. It is discovered that the overall model exhibits a block angular structure that is decomposed by raw material allocation. If raw materials can be allocated optimally to individual plants, solving individual models for each plant can produce the same results as solving an overall model for the site. This discovery leads to a decomposition strategy that consists of two levels. In the first level, only planning decisions are made, and the objective function is the maximization of the overall profit. The results from solving the planning model give optimal raw material allocation to different plants. In the second level, the raw material targets from the first (planning) level are incorporated into the scheduling submodels for each plant, which are solved independently without compromising global optimality. The objective function for each scheduling submodel is the maximization of product throughput. The scheduling level uses the concept of the state sequence network presented in part 1. Solving scheduling submodels for individual plants rather than the overall site model leads to problems with much a smaller number of binary variables and, hence, shorter CPU times. If conflicts arise, i.e., the planning targets are too optimistic to be realized at the scheduling level, the planning model is revisited with more realistic targets. This eventually becomes an iterative procedure that terminates once the planning and scheduling solutions converge within a specified tolerance. In this way, the planning model acts as coordination for scheduling models for individual plants. An industrial case study with three chemical processes is presented to demonstrate the effectiveness of this approach.

Design, synthesis and scheduling of multipurpose batch plants via an effective continuous-time formulation

A novel MILP formulation for short-term scheduling of multi-stage multi-product batch plants with sequence-dependent constraints

Effective scheduling of a large-scale paint production system

The increasing interest in multipurpose batch plants is evident, because of their inherent flexibility to cope with an ever changing market environment. These plants are easily reconfigured for product modifications, to cover a wide range of operating conditions. They are also suitable for producing different products within the same facility. In spite of this advantage, the design, synthesis, and scheduling of multipurpose batch plants can be a challenging task. This paper addresses design, synthesis, and scheduling simultaneously. It is known that the scheduling platform employed has a significant impact on the computational performance of the overall model in the design and synthesis of batch plants. The recent scheduling model that results in better computational efficiencies is extended to incorporate design and synthesis. Computational studies are presented to illustrate the effectiveness of the proposed model. A comparison with earlier formulations showed that better computational times and objective functions were obtained by this work.

Quadratization of symmetric pseudo-Boolean functions

Multi-valued decision diagrams (MDDs) are a generalization of binary decision diagrams (BDDs). They often allow efficient representation of functions with multi-valued input variables similar to BDDs in the binary case. Therefore they are suitable for several applications in synthesis and verification of integrated circuits. MDD sizes counted in number of nodes vary from linear to exponential dependent on the variable ordering used. In all these applications, minimization of MDDs is crucial. In many cases, multi-valued variables are composed by a certain number of binary variables, and so the multi-valued inputs arise by grouping binary variables. The selection of these groups, that is, the decision which variables to merge, has enormous impact on MDD sizes. Techniques for finding variable groupings before starting MDD minimization have been proposed recently. In this paper we present a new method that uses re-encoding, i.e. dynamic variable grouping. We don't choose one fixed variable grouping before minimizing MDDs, but allow to change the binary variables to be considered together during the minimization process. This is possible since MDDs are simulated on top of BDDs. By this, the underlying binary variables remain accessible throughout the minimization process. This technique is described in detail and we also show experimental results that demonstrate the efficiency of our approach.

ADVERTISEMENT RETURN TO ISSUEPREVCorrespondenceComments on “New General Continuous-Time State−Task Network Formulation for Short-Term Scheduling of Multipurpose Batch Plants” by Christos T. Maravelias and Ignacio E. Grossmann and on “Enhanced Continuous-Time Unit-Specific Event-Based Formulation for Short-Term Scheduling of Multipurpose Batch Processes: Resource Constraints and Mixed Storage Policies” by Stacy L. Janak, Xiaoxia Lin, and Christodoulos A. FloudasChristodoulos A. Floudas and Stacy L. JanakView Author Information Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544-5263 Cite this: Ind. Eng. Chem. Res. 2005, 44, 6, 1985–1986Publication Date (Web):February 19, 2005Publication History Published online19 February 2005Published inissue 1 March 2005https://doi.org/10.1021/ie050135jCopyright © 2005 American Chemical SocietyRequest reuse permissions This publication is free to access through this site. Learn MoreArticle Views290Altmetric-Citations1LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (37 KB) Get e-AlertscloseSUBJECTS:Mathematical methods,Theoretical models Get e-Alerts

An effective linear approximation method for separable programming problems

There exists a wide range of constraint programming (CP) problems defined on Boolean functions depending on binary variables. One of the approaches to solving CP problems is using specific appropriate solvers, e.g., SAT solvers. An alternative is using the generic solvers for mixed-integer linear programming problems (MILP), but they require transforming expressions with Boolean functions to linear equations or inequalities. Here, we present two methods of such a transformation which applies to any Boolean function defined by explicit rules giving values of the Boolean function for all combinations of its Boolean variables. The first method represents the Boolean function as a linear equation in the original binary variables and, possibly, binary ancillaries, which become additional variables of the MILP problem being composed. The second method represents the Boolean function as a set of linear inequalities in the original binary variables and one additional continuous variable (representing the value of the function). The choice between the first or second method is a trade-off between the number of binary variables and number of linear constraints in the emerging MP problem. The advantage of the proposed approach is that both methods reduce important cryptanalysis problems, such as the preimaging of hash functions or breaking symmetric ciphers as the MILP problems, which are solved by the generic MILP solvers. Furthermore, the first method enables to reduce the binary linear equations to quadratic unconstrained binary optimization (QUBO), by the quantum annealer, e.g., D-Wave.

Optimal scheduling of batch plants satisfying multiple product orders with different due-dates

A Continuous-time MILP Model for Direct Heat Integration in Batch Plants