(554a) Optimal Scheduling of Multistage Batch Plants. A Comparative Study

Scheduling in the process industry has received considerable attention during the last 15 years and extensive reviews can be found in the literature. Because of the great variety of aspects that need to be taken into consideration, many approaches have been proposed to handle specific problem types. However, the main developments have resulted from more general approaches, those that can effectively handle a large number of problem features. Important landmarks are the unified frameworks for process representation: the state-task (STN) and the resource task networks (RTN).

Scheduling models are first characterized by the representation of time. Discrete-time approaches can sometimes be a good option but, most of the times, the characteristics of the problem compels us to employ a continuous-time model, which can be either time grid or sequence based. Nowadays, it is clear that single time grid, continuous-time models based on unified frameworks are the most general since they can consider resource constraints other than equipment, together with various storage policies. However, it is also true that they are usually not the most efficient for some classes of problems. Recent detailed computational studies have shown that unit-specific formulations can be orders of magnitude faster in problems arising from multipurpose batch plants (Shaik and Floudas, Ind. Eng. Chem. Res. 2006, 45, 6190 and Comput. Chem. Eng. 2008, 32, 260) and also from multistage multiproduct batch plants (Castro and Grossmann, Ind. Eng. Chem. Res. 2005, 44, 9175 and Liu and Karimi, Chem. Eng. Sci. 2007, 62, 1549), under an unlimited intermediate storage policy.

Traditional approaches for multistage multiproduct batch plants consider that the number and size of batches is known a priori. Different batches of the same product are treated as distinct entities that maintain their identity throughout the processing stages, similarly to batches belonging to different products. Hence, they can be viewed as single batch approaches. Decoupling batching from scheduling decisions may not be an easy task in processes with parallel units of dissimilar capacities. Furthermore, it often leads to suboptimal solutions since one cannot take advantage of batch mixing/splitting to capitalize on the different capacities from one stage to the next, as noted by Sundaramoorthy and Maravelias (Ind. Eng. Chem. Res. 2008, 47, 1546).

In this paper, we propose a new multiple time grid (a.k.a. unit-specific) formulation that can simultaneously determine the optimal set of batches (number and size) together with unit assignment and sequencing of batches. It can be viewed as an extension of our previous work (Castro et al. Ind. Eng. Chem Res. 2006, 45, 6210), which dealt with problems involving a single product batch per stage. The main novelty is the definition of a single intermediate storage unit per stage to which an explicit storage unit is associated. The events belonging to the time grid of a particular storage unit are then related to the corresponding event points of those processing units feeding to and receiving material from it. Such timing constraints, together with the material balances, guarantee that a product can only start to be produced in stage k+1 after enough intermediate has been produced in stage k. For problems with sequence dependent changeovers, the model uses 4-index binary variables to identify the execution of combined processing and changeover tasks. Wherever changeovers are not an issue, it can be simplified to a 3-index model.

The new formulation is thoroughly compared to the unit-specific approaches of our previous work (Castro and co-workers, 2005 and 2006) and that of Shaik and Floudas (2008). A variety of scenarios are considered involving single and multiple batch cases, alternative objective functions and problems with and without sequence-dependent changeovers. The results show that its higher generality, when compared to that of Castro et al. (2005, 2006), comes at the expense of a small increase in the computational cost for single batch problems. However, when compared to the one of Shaik and Floudas (2008) it can be up to 3 or 4 orders of magnitude faster due to lower integrality gap, to the use of a smaller number of event points to find the global optimal solution, and to fewer constraints. On the other hand, the latter has the advantage of being suitable for multipurpose plants and other storage policies.