Computers and Chemical Engineering 44 (2012) 45–57 Contents lists available at SciVerse ScienceDirect Computers and Chemical Engineering jo u rn al hom epa ge : www.elsevier.com/locate/compchemeng From time representation in scheduling to the solution of strip packing problems Pedro M. Castro a,b,∗ , Ignacio E. Grossmann b a Unidade de Modelac ¸ ão e Optimizac ¸ ão de Sistemas Energéticos, Laboratório Nacional de Energia e Geologia, 1649-038 Lisboa, Portugal b Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA a r t i c l e i n f o Article history: Received 14 November 2011 Received in revised form 30 April 2012 Accepted 3 May 2012 Available online 11 May 2012 Keywords: Optimization Integer programming 2D strip packing Scheduling Search algorithm Event points a b s t r a c t We propose two mixed-integer linear programming based approaches for the 2D orthogonal strip packing problem. Using knowledge from the alternative forms of time representation in scheduling formulations, we show how to efficiently combine three different concepts into the x- and y-dimensions. One model fea- tures a discrete representation on the x-axis (strip width) and a continuous representation with general precedence variables on the y-axis (strip height). The other features a full continuous-space representa- tion with the same approach for the y-axis and a single non-uniform grid made up of slots for the x-axis. Through the solution of a set of twenty nine instances from the literature, we show that the former is a better approach, even when compared to three alternative MILP models ranging from a pure discrete- space to a pure continuous-space with precedence variables in both dimensions. All models are available in www.minlp.org. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction Time representation is perhaps the most important classifica- tion criterion for scheduling approaches based on mathematical programming models. Mathematical formulations can either be classified as discrete- or continuous-time and several different alternatives have been proposed. Overall, there are essentially four main concepts being used by the process systems engineering com- munity, illustrated in Fig. 1, with no particular concept becoming more relevant with time, clearly reflecting that the best option for a problem is very much dependent on its features. While predicting the best performer is often difficult, some guidelines can be given. The discrete-time approach is perhaps the most powerful and has been shown capable of handling problems of industrial rele- vance (Wassick & Ferrio, 2011; Wassick, 2009). The time horizon of interest is divided into a fixed number of time slots of predeter- mined duration, with one knowing a priori the location of all time points. This makes it straightforward to handle holding and back- log costs (Sundaramoorthy & Maravelias, 2011), thus allowing for easy integration with the higher level planning model (Maravelias & Sung, 2009), intermediate events such as release/due dates, equipment maintenance as well as time-dependent utility pricing and availability (Castro, Harjunkoski, & Grossmann, 2009; Castro, ∗ Corresponding author at: Unidade de Modelac ¸ ão e Optimizac ¸ ão de Sistemas Energéticos, Laboratório Nacional de Energia e Geologia, 1649-038 Lisboa, Portugal. Tel.: +351 210924643. E-mail address: pedro.castro@lneg.pt (P.M. Castro). Harjunkoski, & Grossmann, 2011). On the downside, fixed process- ing times need to be assumed and approximated to a multiple of the interval length. Continuous-time models, on the other hand, are more accu- rate and sensitive to small changes in the duration of processing and changeover tasks, which can be of a different order of magni- tude. They are thus more appropriate for integration with the lower level control layer (Capón-Garcia, Moreno-Benito, & Espu ˜ na, 2011). Deciding for a continuous-time model needs to be followed by the choice of the concept used to keep track of events taking place. Precedence based models (Méndez, Henning, & Cerdá, 2000; Méndez, Henning, & Cerdá, 2001) were the first to appear and are known for their ability to provide high quality solutions with lim- ited computational resources, even though it may be difficult to prove optimality. The concept of general precedence is used more frequently when compared to immediate precedence since it gives rise to smaller models that typically perform better. Precedence based models tend to be less general than their time grid coun- terparts and are thus more commonly found for multistage plants, where they are more efficient. In facilities with a network structure involving resource constraints other than equipment and unit availability, i.e. mul- tipurpose plants, time grid based models become the only option. Due to process complexity they are linked to unified frameworks for process representation, the state-task (Kondili, Pantelides, & Sargent, 1993) and resource-task (Pantelides, 1994) networks. When compared to the discrete-time representation, the time hori- zon is also divided into a fixed number of slots but now the grid(s) is/are non-uniform with the duration of the slots being determined 0098-1354/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compchemeng.2012.05.002