Review, extensions and computational comparison of MILP formulations for scheduling of batch processes

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Abstract

In this paper we investigate mixed-integer linear programs for minimizing the makespan of batch processes. Thereby it turns out that some additional valid constraints which were originally given for event driven model (EDM) also accelerate the running time for uniform discretization of time model (UDM) considerably. With help of an example for EDM we show that we can store some products in the batches if we prolong the processing times. For this reason EDM differs from the standard model. To achieve the same results as for UDM we introduce the ending times of the batches as additional variables. Moreover we extend the model by considering flexible proportions of output as well. However, computational tests indicate a rise of CPU time for the modified problem although the number of binary variables is smaller compared to UDM. Furthermore we point out that an upper bound on the total number of batchstarts is only possible for some production processes. Extensive computational tests on five benchmark problems proposed in the literature are reported. Thereby new optimal solutions could be found.

Introduction

In recent years algorithms for scheduling problems occurring in the chemical process industry have gained an increasing interest. In general, we distinguish between continuous and batch production systems, but due to the fact that plants with multi-purpose equipment, which are very common in practice, are often operated in batch mode, we will only consider this case. This means that we do not have a continuous stream of resources, but well-defined starting and ending times for the single steps in the production process. These points in time will be called events. This kind of problem was also considered in Blömer and Günther, 1998, Blömer and Günther, 1999, Blömer and Günther, 2000, Burkard, Hujter, Klinz, Rudolf, and Wennink (1998), Burkard, Kocher and Rudolf (1998), Burkard and Hatzl (2005), Hatzl (2004), Maravelias and Grossmann, 2003a, Maravelias and Grossmann, 2003b, Mockus and Reklaitis (1997), Pantelides (1994), Schilling and Pantelides, 1996, Schilling and Pantelides, 1999, Schwindt and Trautmann (2000), Shah et al., 1993a, Shah et al., 1993b and Vin and Ierapetritou (2000). For more extensive surveys on different techniques applied to tackle this question and further references see Kallrath (2002), Reklaitis, Sunol, Rippin, and Hortascu (1996) and Schilling (1997).

We suppose that the production process is described by the widely used state-task-network. This concept was originally introduced by Kondili, Pantelides and Sargent (1993) and is able to describe the major characteristics of real batch production processes involving variable batch sizes, non-preemptive processes, shared intermediates, flexible proportions of output products, cyclic material flows and non-storable products.

In this article we analyze different mixed-integer linear programs (MILP) to determine the minimal makespan for producing a given amount of final products. We distinguish two main classes of MILPs:

The models in the first class are based on a uniform discretization of time which involves restrictions on the times at which a batch could start. This idea goes back to Kondili et al. (1993) and was later slightly modified by Blömer and Günther, 1998, Blömer and Günther, 1999 and Blömer and Günther (2000). However, the large number of binary variables needed to state MILPs of that type and consequentially, the enormous computational time to solve only small problems encouraged to think about different formulations of the model and LP-based heuristics to obtain near-optimal solutions.

Shah et al. (1993a) reformulated some constraints and proposed a compact linear programming relaxation resulting in an acceleration of the branch and bound method. This method improves the computational time and still guarantees to find an optimal solution. On the other hand Blömer and Günther (1998) examined a heuristic based on the MILP-formulation to obtain sub-optimal solutions. Their two-phase method consists of initially restricting the time periods at which a batch may start. In the second phase the idle times of the machines are removed in order to reduce the overall makespan. Thus the obtained solution is improved. Burkard et al. (1998) investigated another LP-based solution method by using different rounding strategies based on optimal solutions of the corresponding linear relaxations.

The second class of models discussed in this paper avoids an external discretization of time. This idea has been suggested by Sahinidis and Grossmann, 1991a, Sahinidis and Grossmann, 1991b for the first time and has been revived again in different forms by Mockus and Reklaitis (1997) and Schilling and Pantelides (1999) leading to a nonlinear programming model. Another step forward was made in the papers by Ierapetritou and Floudas (1998a) and Ierapetritou and Floudas (1998b). They proposed an approach that leads to a smaller MILP which has fewer binary variables and constraints. Moreover, a small integrality gap guarantees a significant improvement of the solution process with respect to computational time. The drawback is that the exact number N of time intervals is required. Otherwise the model delivers a feasible solution, which may be not optimal with respect to the makespan.

Recently, Burkard, Fortuna and Hurkens (2002) proposed an alternative model called event-driven model (EDM) in which they added the starting times of the batches as continuous variables to the MILP. However, in their model it is allowed to prolong the batch processing times. Thus it is possible to store some products in the batches which saves storage capacities. We will give an example where this model leads to a different optimal solution. To avoid this disparity we add some additional constraints to EDM and obtain a new model, which we call continuous time model (CTM).

In the following we analyze these approaches. Special emphasis is laid on a small number of binary variables and on introducing valid constraints in order to speed up the solution process. The purpose of valid constraints is to give a closer approximation of the convex hull of the feasible integer points. Thus the integrality gap between the optimal solution of the relaxed linear programming representation and its mixed-integer counterpart can be reduced. A small gap normally accelerates the running times because fewer branch and bound iterations are required, because the bounds obtained during the process are more adequate. A detailed discussion on branch and bound methods can be found in Ibaraki (1987a) and Ibaraki (1987b). A theoretical background about valid constraints is given in Wolsey (1998).

During the tests it turned out that the modifications in the MILP proposed by Burkard et al. (2002) are responsible for a rise in computational time even if we add some valid constraints suggested in Burkard et al. (2002) to the modified MILP. However, these additional constraints also support the models which use a uniform discretization of time. Thereby we could significantly improve the results regarding the quality of the best known solution for bigger instances and the running times for smaller problems, which can be solved exactly within a reasonable amount of time.

Section snippets

The problem represented as state-task-network

The chemical batch process under investigation can be presented by a State-Task-Network (STN). An STN is a directed graph which has two different types of nodes. The state nodes represent the feeds, intermediate and final products whereas the task nodes represent the process operations which transform material from one or more input states into one or more output states. The arcs of the graph indicate the flow of material. Due to the fact that a particular state is either received from or used

A model using uniform discretization of time (UDM)

In this model we divide the entire time horizon [0,T] into a number of smaller periods, i.e., 0=t1<<tK=T, of equal length. Because of the fact that all processing times are assumed to be nonnegative integers, we can define tk+1tk for k=1,,K1 to be the greatest common divisor of all these times. As a consequence each batch starts and ends at a time tk and we only have to calculate the starting points of batch b. Thus we introduce the variable λb,k, which is equal to 1 if and only if a task

A continuous time model (CTM)

The stated model using an equidistant discretization of time is highly effected by the length of one interval. This length depends mainly on the processing times of the tasks. Thus the grid size can be small resulting in a high number of binary variables. As an alternative we state another model, whose number of binary variables is not so much influenced by a different choice of the processing times. Here we just deal with the real variables tn for n=1,,N representing the starting or ending

Modifications of the mixed-integer linear programs

The models above are stated in a compact form. However, this is sometimes not very useful if a standard optimization software solves the MILP.

Basically, there are two essential features for solving MILP within a reasonable amount of time. Because of the fact that most optimizers exploit a branch and bound algorithm the number of binary variables plays a key role. Thus we should try to keep it as small as possible. On the other hand, a small integrality gap also enhances the performance of the

Numerical results

At the end of this study we consider five different chemical batch processes, which differ significantly by their scope and complexity. Thus we are able to compare the performance of UDM and CTM in detail. The advantages of EDM compared to UDM are already reported in Burkard et al. (2002) and are consequently kept short in this paper.

The STNs outlined in Fig. A.2, Fig. A.3, Fig. A.4, Fig. A.5 are a collection of problems stated in the papers of Shah et al. (1993a), Kondili et al. (1993),

Concluding remarks

The main conclusion of this paper is that chemical batch processes given as STN of moderate size can be solved exactly with MILPs. Especially if the batch processing times do not diversify very much UDM seems to be a very powerful approach to find the minimum makespan. We added some strong valid constraints to UDM which were already known for EDM and modified the objective function in order to speed up the solution process. As a result of this modification we could find some optimal solutions

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