Ind. Eng. Chem. Res. 1996,34, 3037-3051
3037
A Continuous Time Mixed Integer Linear Programming Model for Short Term Scheduling of Multistage Batch Plants Jose M. Pinto and Ignacio E. Grossmann* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
The problem of short term scheduling of batch plants consists of determining the optimal production policy for satisfying the production demands for different products at due dates and/ or a t the end of a given time horizon. The objective of this work is to propose a n optimization model and solution method to the short term scheduling of batch plants with multiple stages which may contain equipment in parallel. A large scale mixed integer linear programming (MILP) model with continuous time domain representation is proposed that relies on the use of parallel time axes for units and tasks. Although in principle a n LP-based branch and bound method can be used to solve the problem, there is a limitation when the instances become large. The first solution strategy that is proposed consists of the use of preordering constraints. Furthermore, a second strategy relies on a decomposition scheme for large systems which is based on the solution of a n MILP model that minimizes total in process time in which assignments are determined and the subsequent solution of a n LP to minimize earliness and to eliminate unnecessary setups. Several examples are presented, including a large real world problem, to illustrate the performance of the model and solution method.
Introduction The problem of short term scheduling of batch plants has as an objective to determine the optimal production plan for satisfying the production demands for different products at due dates and/or a t the end of a given time horizon. The short term scheduling is relevant for flexible plants, such as is the case of plants for manufacturing dyes and plastic compounds, the production of which are allocated in order to satisfy individual customer orders with no regular production pattern. It involves the allocation of equipment and resources to orders, the sequencing of these orders, and the determination of the material flows through the plant. Extensive reviews in batch processing have been recently reported in the literature (Reklaitis, 1991,1922; Zentner and Pekny, 1994). Many of these problems can be posed as mixed integer optimization problems, since the corresponding mathematical optimization models involve both discrete and continuous variables that must satisfy a set of equality and inequality constraints (Grossmann e t al., 1992). A major question in any scheduling algorithm deals with the time domain representation. Kondili et al. (1993a) present a general formulation to the modeling of scheduling problems that arise in batch plants, namely the state task network (STN). Two types of nodes are defined in the network: the state nodes that represent the feeds, intermediates, and product materials and the task nodes that represent the processing steps. Their representation is based on a uniform time discretization and on the assumption that events only happen a t the boundaries of these intervals. The problem is formulated as a mixed integer linear programming model (MILP). The MILP is able t o accommodate different storage modes, processing of orders in parallel, mixing and splitting of material between stages, limited availability of utilities, simultaneous production of different products, successive batches with different routes through the plant, and a variety of objective functions. However the main limitation of the
* Author to whom correspondence should be addressed.
model is the generation of a large number of integer variables and constraints in problems of industrial relevance. Following this work, several techniques were introduced with the objective of reducing the required computational effort. Shah et al. (1993) reformulated the allocation constraints (from operations to machines), greatly reducing the integrality gap, i.e., the difference between the optimal solutions of the relaxed linear program and the MILP. Reductions of computational times up to 3 orders of magnitude were obtained with the reformulation. A heuristic decomposition method based on the model by Kondili et al. (1993a) was proposed by Elkamel e t al. (1993) in which units of the plant are grouped according to the tasks they are able to perform and the overall time horizon is decomposed into smaller sub horizons whose corresponding scheduling problems are solved sequentially. The uniform discretization of time domain was also applied to parallel flowshop scheduling (Gooding et al., 1993). The mathematical model was transformed into a branch office traveling salesman problem which was solved exactly through implicit enumeration. The operations research literature provides a large number of mathematical programming formulations, the vast majority using continuous time representation (Blazewicz et al., 1991). A general review of the scheduling problem is given in French (1982). The simplest sequencing problem is the single machine scheduling. Despite its apparent simplicity, the problem can be regarded as a building block in the development and understanding of more complicated systems. For instance, a polyhedral description of the single machine scheduling problem was derived by Lasserre and Queyranne (1992) for the case of no release dates and infinite due dates. The authors indicate that their formulation, based on a continuous time formulation, can outperform similar models based on time discretization. Another important example in the scheduling literature is the general job shop problem (Carlier and Pinson, 1989;Adams et al., 1988). A specialized branch and bound method was developed by Carlier and Pinson
Q888-5885/95/2634-3037$09.00/0 0 1995 American Chemical Society
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3038 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995
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Figure 1. Production horizon and due dates.
(1989) based on one machine scheduling problems (Carlier, 1982). An extensive computational study of the problem was also done by Applegate and Cook (1991) in which the authors develop heuristics for finding feasible schedules, cutting planes for obtaining lower bounds and a specialized branch and bound solution method. In the context of chemical processing industries, continuous time representations were also developed for the short term scheduling of batch plants. Zentner and Reklaitis (1992) introduced the NUCM (nonuniform continuous time modeling), a formulation whose objective is to obtain optimal solutions from a more simplified model and use this as a preprocessing step for the discrete time model. Xueya and Sargent (1994) have considered a continuous time version of the STN model which relies on the use of variable time events and which leads t o a large scale MILP model. Algorithms for the scheduling of continuous plants were also presented with a continuous time representation. The simultaneous planning and scheduling for the case of plants with a single stage and equipment in parallel was studied by Sahinidis and Grossmann (1991). Pinto and Grossmann (1994)considered the case of multiple stages with intermediate storage considerations. The problems were modeled as mixed integer nonlinear programs (MINLP’s). The problem of scheduling continuous multiproduct plants under resource constraints was studied by Kondili et al. (1993b), who used a discrete time representation with uniform grid. This led to a mixed integer linear program that was applied to the short term production of a cement plant. The objective of this work is to propose a model and a solution method to the short term scheduling of multistage batch plants which may contain equipment in parallel, based on a continuous time representation. The problem is restricted to setups that are sequence independent, and no resource constraints are considered since the main goal of this work is to establish the extent to which a continuous time model can be solved for large scale industrial applications. An MILP model is proposed that relies on the use of time matching constraints given parallel time coordinates for the units and the tasks. It is shown first that the use of preordering constraints, which are heuristic in nature, can significantly reduce the computational time. Second, a decomposition method is developed for solving large systems which determines initially feasible assignments that minimize in-process time and further determines a schedule that minimizes earliness and eliminates unnecessary setups. Numerical results are presented, including the problem of a real world plant consisting of 5 stages and 25 pieces of equipment, and with up to 50 orders.
Problem Definition The problem of short term scheduling of batch plants considered in this work can be defined as follows. Specified is a production horizon composed by a certain number of time periods given by due dates to satisfy demands of several products (Figure 1). The plant has
a structure of multiple stages with equipment in parallel and intermediate storage with known topology. Figure 2 illustrates the case of a plant for manufacturing dyes with 5 stages and 25 machines. The schedulingproblem has the objective of determining a production scheme that minimizes earliness.
Mathematical Model In this section we first consider the general mathematical model. As described in the Introduction, a major question in any scheduling algorithm is concerned with the time domain representation. A continuous time representation is adopted here. The main assumptions of the model are: Al. Demands, due dates, processing times, and transition times are deterministic. A2. Transition times are only equipment dependent. A3. Batch sizes are fixed parameters. A4. Each order is to be processed only once by exactly one unit of every stage it must go through. A5. Each order represents one batch; therefore it is possible that there might exist several orders of the same product. A6. No resource constraints except equipment are considered. A7. There is unlimited intermediate storage between stages. Assumption A1 seems quite reasonable. Relaxing assumption A2 would considerablycomplicate the model. The next three assumptions can be relaxed a t the time of specifying the order requirements by breaking the demand for products, as described next. Assumption 6 cannot be easily relaxed and will be considered explicitly in a future publication. Assumption 7 can easily be relaxed for the cases of no storage and zero-wait transfer. The following are the basic ideas in the proposed model: (a) Continuous time domain representation. Two parallel time coordinates are used for handling the assignment of orders to machines. Orders are represented in one coordinate while the other coordinate handles units, as shown in Figure 3. Since we do not know a priori the assignment and sequencing of orders in each unit, we will consider a sequence of production slots, defined as time intervals for unit allocation with unknown durations. When stage 1 of order i is assigned to slot k of unit j , matching takes place by enforcing equality of the start and end times of these coordinates through mixed integer constraints (see eq 4). Binary variables Wykl are used to model the potential assignment of stage 1 of order i to time slot k of unit j . When stage 1 of order i is assigned to time slot k of u n i t j (Wljkl = l), the start times in both coordinates are enforced to be the same. Otherwise, the constraints are relaxed. Exactly one time slot of a machine must be assigned to one stage of an order (see assumption A4). However, in some situations no order is assigned to any time slot; this happens due to an overestimation of the number of time slots postulated for each unit. (b) Definition of time slots for the units. As described in Figure 3, a certain number of time slots KJ is postulated for each unit j . These time slots represent the sequence of the processing of the orders. Since in this problem there may be units in parallel, it is not known in principle which processing stage of
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Figure 2. Multistage plant with equipment in parallel.
K = time slot for equipment
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I, I' = processing stage for an order = stage at which order i is withdrawn The following are the sets: I = set of orders Ij = set of orders which can be processed in unit j (4 I) J = set of units J,= set of units which can process order i (J, J) Jl = set of units which belong to stage 1 (JIc J) K = set of time slots KJ = set of time slots postulated for u n i t j (4 K> L = set of stages L, = stages involved in the production of i (L, 2 L) L, = stage corresponding to unit j (LJ L) The following are the parameters: d, = due date of order i h, = weight for earliness of order i SU, = setup time in u n i t j Tv= processing time of order i in unit j U = upper bound on start times Ull = upper bound on start times for order i in stage I The following are the variables: TSJk= starting time in u n i t j during time slot k TeJk= finishing time in unit j during time slot 12 Tsi,l = starting time for order i in stage I Tei,l = finishing time for order i in stage I Wgkl= binary variable that assigns stage I of order i to time slot 12 of u n i t j Xu!= binary variable that assigns order i stage I to unit
