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381676 A Generalized Disjunctive Programming Model for Scheduling a Pulp Plant Under Energy Constraints

We address a real-life problem from the cooking section of an acid sulphite pulp plant featuring four parallel batch digesters and periods of insufficient steam availability, required to heat the digester contents up to the cooking temperature (Castro et al. 2002). The plant topology can be classified as a multistage multiproduct batch plant with a single unit per stage, where the parallel digesters are actually treated as production orders. Stage units involve the equipment resources that are shared amongst the digesters (e.g. conveyers, pumps, steam line, etc.). Such plant topology is best modeled with a multiple time grid continuous-time formulation (Harjunkoski et al., 2014), the challenge being the derivation of the timing constraints of the heating stage, which need to take into account the complex interaction of a particular digester with both its predecessor and successor in the processing sequence.

As highlighted in the recent review paper by Grossmann and Trespalacios (2013), the Generalized Disjunctive Programming (GDP) higher-level modeling framework (Raman and Grossmann, 1994) makes the formulating process more intuitive and systematic. Furthermore, it provides a structured way for deriving models that exhibit strong continuous relaxations, which often translates into shorter computational times. Another advantage of GDP is that it retains in the model the underlying logic structure of the problem, associating different constraints to alternative operating decisions. For this particular problem, the constraint that models the interacting heating tasks involving digesters in immediate positions in the sequence features embedded disjunctions (Vecchietti and Grossmann, 2000) so that heating tasks have variable processing times that are a function of the digester sequence.

The complete MILP was obtained following a compact convex hull reformulation (Balas 1985, Castro and Grossmann, 2012) of the disjunctions and conversion of the logic propositions into an equivalent set of integer inequalities. It can be classified as a multiple time grid continuous-time periodic scheduling formulation featuring the wrap-around operator (Shah et al., 1993) and timing constraints to maintain continuity of operation between cycles (Wu and Ierapetritou, 2004).

Compared to Resource-Task Network based discrete- and continuous-time formulations from the literature (Castro et al. 2003), the proposed model is tighter, does not require an iterative search procedure over the number of time slots in order to find the optimal cycle time and is at least two orders of magnitude faster. In fact, it was able to solve in fractions of a second two case studies corresponding to different values of constant steam availability. This made it possible to estimate the impact on productivity resulting from adding another digester to the plant, which was 10/20% (13/14 t/h steam availability) of what to expect from the ratio of batch sizes. The mathematical problem remains tractable up to as many as 60 digesters.

Acknowledgement

Financial support from FEDER (Programa Operacional Factores de Competitividade –COMPETE) and Fundação para a Ciência e Tecnologia through project FCOMP-01-0124-FEDER-020764.

References

E. Balas, 1985. Disjunctive Programming and a Hierarchy of Relaxations for Discrete Optimization Problems. SIAM Journal on Algebraic Discrete Methods, 6(3), 466–486.

P. M. Castro, Grossmann, I.E., 2012. Generalized Disjunctive Programming as a Systematic Modeling Framework to Derive Scheduling Formulations. Ind. Eng. Chem. Res., 51, 5781–5792.

P. Castro, H. Matos, A.P.F.D. Barbosa-Póvoa, 2002, Dynamic Modelling and Scheduling of an Industrial Batch System, Comput. Chem. Eng., 26, 671–686.

P.M. Castro, Barbosa-Póvoa, A.P. & Matos, H.A., 2003. Optimal Periodic Scheduling of Batch Plants Using RTN-Based Discrete and Continuous-Time Formulations: A Case Study Approach. Ind. Eng. Chem. Res., 42, 3346–3360.

I. E. Grossmann, F. Trespalacios, 2013, Systematic Modeling of Discrete-Continuous optimization Models through Generalized Disjunctive Programming. AIChE J., 59, 3276-3295.

I. Harjunkoski, C.T. Maravelias, P. Bongers, P.M. Castro, S. Engell, I.E. Grossmann, J. Hooker, C. Méndez, G. Sand, J. Wassick, 2014, Scope for Industrial Applications of Production Scheduling Models and Solution Methods, Comput. Chem. Eng, 62, 161-193.

R. Raman, Grossmann, I.E., 1994. Modelling and computational techniques for logic based integer programming. Comput. Chem. Eng. 18(7), 563–578.

N. Shah, C.C. Pantetides, R.W.H. Sargent, 1993. Optimal periodic scheduling of multipurpose batch plants. Annals of Operations Research, 42, 193–228.

A. Vecchietti, I.E. Grossmann, 2000, Modeling Issues and Implementation of Language for Disjunctive Programming. Comput. Chem. Eng., 24, 2143-2155.

D. Wu, M. Ierapetritou, 2004. Cyclic short-term scheduling of multiproduct batch plants using continuous-time representation. Comput. Chem. Eng., 28, 2271–2286.

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