Scheduling continuous time representation

Mockus, L. and Reklaitis, G.V. (1999a) Continuous-time representation approach to batch and continuous process scheduling. 1. MINLP formulation. Ind. Eng. Chem. Res., 38, 197-203. 

R. Rejowski Jr. and J.M. Pinto. A novel continuous time representation for the scheduling of pipeline systems with pumping yield rate constraints. Comp, and Chem. Eng. (2007), doi 10.1016/j.compchemeng.2007.06.021. 

The continuous-time representation has the same ambition events are allowed to take place at any point in the continuous domain of time. This is carried out by introducing the concept of variable event time that can be defined either globally or for each unit. For this purpose, additional variables are necessary to determine the timings of events. Since a major fraetion of the inactive event-time interval assignments is eliminated with the continuoustime approach, the resulting mathematical models require less computational effort for their solution of the associated scheduling task involved in design/retrofit problems. However, the mathematical models involved in the continuous-time approach may embed more complicated structures compared with their discrete-time counterparts. 

The main challenge in short-term scheduling emanates from time domain representation, which eventually influences the number of binary variables and accuracy of the model. Contrary to continuous-time formulations, discrete-time formulations tend to be inaccurate and result in an explosive binary dimension. This justifies recent efforts in developing continuous-time models that are amenable to industrial size problems. 

In this chapter, state sequence network (SSN) representation has been presented. Based on this representation, a continuous-time formulation for scheduling of multipurpose batch processes is developed. This representation involves states only, which are characteristic of the units and tasks present in the process. Due to the elimination of tasks and units which are encountered in formulations based on the state task network (STN), the SSN based formulation leads to a much smaller number of binary variables and fewer constraints. This eventually leads to much shorter CPU times as substantiated by both the examples presented in this chapter. This advantage becomes more apparent as the problem size increases. In the second literature example, which involved a multipurpose plant producing two products, this formulation required 40 binary variables and gave a performance index of 1513.35, whilst other continuous-time formulations required between 48 (Ierapetritou and Floudas, 1998) and 147 binary variables (Zhang, 1995). 

A new continuous approach to the scheduling of a single multiproduct pipeline has been presented. By adopting a continuous representation in both time and volume, a more rigorous problem representation and a severe reduction in binary variables and CPU time have simultaneously been achieved. 

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