Article pubs.acs.org/IECR
Reformulations and Branching Methods for Mixed-Integer Programming Chemical Production Scheduling Models Sara Velez and Christos T. Maravelias* Department of Chemical and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, Wisconsin 53706, United States S Supporting Information *
ABSTRACT: Mixed-integer programs for chemical production scheduling are computationally challenging. One characteristic that makes them hard is that they typically have many symmetric solutions, that is, solutions that are different in terms of the values of the decision variables but have the same objective function value, which means that the algorithms used to solve these models must search through all such solutions before improving the bound on the objective. To address this challenge, we propose three reformulations of the widely used state−task network formulation. Specifically, we introduce additional constraints to define the number of batches of each task as an integer variable. Branching on this new integer variable quickly eliminates schedules that have the same number of batches, which, in turn, leads to the elimination of many symmetric solutions. We also study different branching strategies and variable selection rules and compare them. The proposed solution methods lead to orders-of-magnitude reductions in the computational requirements for the solution of scheduling problems.
1. INTRODUCTION Chemical production scheduling applications arise in a number of types of chemical manufacturing facilities, from batch manufacturing of low-volume high-value products (e.g., pharmaceuticals) to the continuous manufacturing of highvolume products (e.g., oil and gas in refineries).1−3 Early work in the area of process systems engineering (PSE) focused on the scheduling of sequential production environments, where it is assumed that the number of batches to be scheduled is fixed and each batch has to go through a series of processing stages without mixing with other batches or splitting toward multiple downstream batches.4−6 The models developed to address this class of problems rely on the modeling of batches and thus are termed batch-based. To address problems in which batches can be mixed or split and a task can produce or consume multiple materials, material-based models were developed in the early 1990s.7−9 The major characteristic of these approaches is that they rely on the modeling of materials through material balances enforced at different time points. Thus, all material-based models are timegrid-based: they employ one or more time grids, and variables are defined and constraints are expressed for every time point (or period) of the grid(s). The first material-based formulations were based on a uniform, common (across all units, tasks, and materials), and discrete time grid. Because the resulting mixedinteger programming (MIP) models were intractable for the solvers available at that time, researchers focused their efforts on developing smaller continuous-time models, starting with some early models in the mid-1990s,10,11 followed by approaches that focused on the development of smaller MIP models in the late 1990s and 2000s.12−18 Since then, the overwhelming majority of time-grid-based models proposed in the literature rely on continuous representations of time, with a few exceptions mainly from industrial practitioners.19,20 As mentioned in the preceding paragraph, the early batchbased approaches were limited to problems with fixed numbers © 2013 American Chemical Society
of batches (i.e., problems that did not consider batching decisions).21 In addition, these models considered only a subset of storage policies and assumed no resource constraints other than those on processing units. These modeling limitations have now been addressed. For problems with fixed numbers of batches, Ku and Karimi,22 Wu and He,23 and Mendez and Cerda24 developed models to account for some special cases of intermediate storage constraints, whereas Mendez et al.25 studied a special case of resource constraints in these problems. The problem of simultaneous batching and scheduling in sequential production environments was addressed recently by Maravelias and co-workers.26,27 A model for the treatment of all types of storage policies (including restrictions on the number and size of the vessels as well as timing restrictions) for simultaneous batching and scheduling in sequential environments was also proposed recently.28 Finally, the modeling of general storage and resource constraints in sequential environments was addressed by Sundaramoorthy and Maravelias.29 On the other hand, material-based models have been quite general from the beginning. In their seminal work, Kondili et al. showed how to model intermediate release and due times, semibatch and semicontinuous processes, utility constraints, variable resource consumption during the execution of a task, variable processing times, mixed intermediate storage policies, and changeovers between tasks.7 Furthermore, Kelly and Zyngier proposed models that account for variable processing times and flexible pools to store material.30 The modeling of pipelines has also been explored.30,31 In terms of material-based models employing continuous grid(s), Castro et al.32 proposed models to account for variable-recipe tasks, whereas Giménez et al.33,34 Received: Revised: Accepted: Published: 3832
December 12, 2012 January 30, 2013 February 19, 2013 February 19, 2013 dx.doi.org/10.1021/ie303421h | Ind. Eng. Chem. Res. 2013, 52, 3832−3841
Industrial & Engineering Chemistry Research
Article
through the introduction of new variables and constraints defining the number of batches of each task. These reformulations allow us to branch on constraints representing solution features, namely, the number of batches of each task, rather than just binary variables. This branching method results in dramatic computational enhancements. The article is structured as follows: In the next section, we present background material, including a motivating example for our reformulation. In section 3, we present three alternative reformulations and branching methods. In section 4, we present computational results, and in section 5, we discuss in detail the results for three examples. We use uppercase italic letters for variables, uppercase bold letters for sets, lowercase italic letters for indices, and lowercase Greek letters for parameters.
proposed models to account for noninstantaneous and resourceconstrained material transfers, storage of materials in processing units before and after the execution of a task, multiple material transfers to and from processing units, connectivity restrictions, and resource-constrained setup and changeover activities. Because batch-based models were designed to address problems solely in sequential environments and material-based models were designed to address problems in network environments, an outstanding modeling challenge appeared to be the lack of a model capable of representing problems in all production environments. This limitation was addressed by Sundaramoorthy and Maravelias, who proposed a framework for the representation of problems in all environments.35 Their approach relies on the modeling of materials but offers methods to enforce all material-handing restrictions present in a sequential environment (e.g., no batch or material mixing). Importantly, the proposed framework can be used to address scheduling problems in facilities that include sequential and network subsystems. Hence, it can be argued that problems in all facilities with all types of processing features can now be modeled effectively. Nevertheless, the solution of MIP scheduling models remains challenging. To address this challenge, PSE researchers have primarily focused on the development of smaller continuoustime models, although there have been a few theoretical,36−38 reformulation,39,40 and algorithmic20,41−45 efforts. However, a recent computational study showed that current MIP solvers can in fact effectively address large-scale MIP models based on discrete modeling of time.46 In particular, it was shown that the discrete-time state−task network (STN model) of Shah et al.8 is at least as effective as, if not more effective than, the model of Sundaramoorthy and Karimi,17 one of the most effective general continuous-time models. (Its generality is due to the fact that it relies on a single grid.) Most importantly, however, it was shown that the discrete-time STN can be easily extended, at no significant computational cost, to account for a number of common processing characteristics and constraints, such as intermediate release and due times, variable resource utilization during task execution, setup times, and modeling of inventory and utility costs. On the other hand, the incorporation of any of these characteristics makes all continuous-time models substantially harder; for example, the rigorous modeling of intermediate release and due times requires the introduction of a new set of binary variables and time-matching constraints, and the modeling of inventory and utility costs leads to nonlinear models. Thus, that study, which was the first extensive study of this type, strongly suggested that models developed using a discrete representation of time are currently likely to be better at addressing large-scale instances of industrial importance. This finding, coupled with the wide use of these models in industry, has motivated us to look into developing improved discrete-time models and methods. In a recent work, we proposed a demand propagation method for the calculation of parameters used for the generation of tightening constraints that result in dramatic computational enhancement.47 Also, we showed that the idea of specific grids can be employed in discrete time; we formulated models that employ not only unit-specific, but also task-specific and materialspecific grids, which, in addition, can also be discrete but nonuniform.48 Finally, we proposed tightening methods that exploit the restrictions posed by the production environment.49 In the present article, we continue to explore general solution methods for material-based MIP models relying on discrete time grids. Specifically, we propose reformulations of the STN model
2. BACKGROUND 2.1. Problem Statement. We consider the state−task network (STN) representation7 of a chemical manufacturing
Figure 1. Symmetric solutions of the linear programming (LP) relaxation of scheduling models. Four schedules with 3 batches of T1, 5.5 batches of T2, and 3 batches of T2. The shaded task represents a halfbatch, where the Xijt variable is 0.5. All four of these schedules and any other equivalent schedules would be eliminated by branching once on a variable for the number of batches of T1 (∑XT2,U2,t ≤ 5 and ∑XT2,U2,t ≥ 6).
Figure 2. Search tree showing the LP relaxation and the values of variables Ni at each node. Bounds on the number of batches used for branching are also included.
facility, but we use the term material instead of state. The STN was originally developed to address problems in network production environments, but it was recently shown that the STN can also be used to represent problems in sequential and hybrid environments, as well as in facilities that consist of 3833
dx.doi.org/10.1021/ie303421h | Ind. Eng. Chem. Res. 2013, 52, 3832−3841
Industrial & Engineering Chemistry Research
Article
different production environments.35 Thus, the methods we develop in this article are applicable to all classes of problems. The manufacturing facility is represented in terms of the following elements: i ∈ I tasks, which are unit operations that convert one or more materials into new materials; j ∈ J units, which are unary resources representing the machine or equipment where tasks take place; and k ∈ K materials, including feeds, intermediates, and final products. The production environment is defined in terms of the sets I+k / I−k of tasks producing/consuming material k and Ji of processing units that can process task i and the parameters max βmin i /βi , the minimum/maximum capacity of unit j; max γk , the maximum amount of material k that can be stored; and ρik, the fraction of material k produced (>0) or consumed (0) or final product k due (0) or consumed (
