Guido Sand
7.1
Introduction
After more than two decades of academic research on mixed-integer programming in chemical batch scheduling, the relevant literature exhibits a variety of modeling frameworks, which claim to be “general” or “rather general”. A review and compar- ison of related modeling concepts can be found in the chapter “MILP Optimization Models for Short-Term Scheduling of Batch Processes”. However, the diversity of batch scheduling problems makes it impossible to include all potential problem characteristics in a unified model. Moreover, from a practical point of view this may even be undesirable as general, unspecific models typically suffer from their high computational effort. Nevertheless, the general modeling frameworks serve as an indispensable means to convey and to compare basic modeling concepts and techniques.
An alternative to mixed-integer programming based on general modeling frame- works is engineered mixed-integer programming based on tailored modeling and solution techniques [1]. In this chapter, a real-world case study is used to demon- strate how to develop and to solve a specific short-term scheduling problem. It will be shown that:
Ĺ The case study does not fit into the general modeling frameworks.
Ĺ The scheduling problem can be decomposed into a core problem and a subprob- lem.
Ĺ The specific problem characteristics are modeled most appropriately by a com- bination of concepts from various general modeling frameworks leading to a mixed-integernonlinearprogramming (MINLP) model.
Ĺ A mixed-integerlinearprogramming approximation can be derived following a problem specific approach.
* A list of symbols is given at the end of this chapter.
138 7 Engineered Mixed-Integer Programming in Chemical Batch Scheduling
This chapter is organized as follows. First, the case study, the short-term schedul- ing of the production of ten kinds of polymer in a multiproduct plant, is presented (Section 7.2). In Section 7.3, the engineered approach is first motivated, the core problem is then worked out, and the modeling approach is finally sketched. The engineered MINLP-model with its binary and continuous variables, its nonlinear and linear constraints and its objective is developed and discussed in Section 7.4.
In Section 7.5, a problem specific linearization approach is presented and applied, leading to a simplified mixed-integer linear programming (MILP) model. The so- lution of the MINLP-model and the MILP-model by various standard solvers is compared with respect to the solution quality and the computational effort (Section 7.6). In Section 7.7, some general conclusions on the application of engineered mixed-integer programming in chemical batch process scheduling are drawn.
7.2
The Case Study
The real-word case study considered here is the production of expandable polystyrene (EPS). Ten types of EPS are produced according to ten different recipes on a multiproduct plant which is essentially operated in batch mode. In this sec- tion, the multiproduct plant, the production process and the scheduling problem are presented.
7.2.1 Plant
The topology of the plant can be taken from Figure 7.1. It consists of a preparation stage for the production of two dispersion agents D1 and D2 and an organic phase OP, a polymerization stage and a finishing stage with two lines. The supply of the raw materials F1, F2 and F3 and the storage of the final products A1. . .A5, B1. . .B5 is assumed to be virtually unlimited. The preparation stage and the polymerization
Fig. 7.1 Flowchart of the EPS-plant.
7.2 The Case Study 139 stage are operated in batch mode, whereas the finishing stage is operated in con- tinuous mode.
The dispersion agents are produced in two stirred tank reactors with a capacity of two (D1) and four (D2) batches along with two (D1) and four (D2) storage tanks with a capacity of one batch each. The organic phase is produced in one out of two stirred tank reactors with a capacity of one organic phase batch each; no intermediate storage is provided for the organic phase.
The polymerization stage comprises four identical stirred tank reactors along with a common safety ventilation system designed for one runaway reaction (not shown in Figure 7.1). In the polymerization stage no intermediate storage is provided.
The preprocessing stage, the polymerization stage and the finishing stage are fully networked by dedicated piping such that several batches can be transferred simultaneously.
Each finishing line consists of a mixing vessel and a separation unit. The mixing process is assumed to be ideal such that the following relation holds:
mass of a product in the mixing vessel
total mass in the mixing vessel =feed rate of a product total feed rate
The mixing vessels serve as buffers between the polymerization stage operated in batch mode and the separation units operated continuously. The capacity of each mixing vessel is three polymerization batches and the minimal hold-up is 0.1 polymerization batches to ensure a sufficient mixing effect.
7.2.2
The Production Process
The plant is used to produce two chemically different EPS-types A and B in five grain size fractions each from raw materials F1, F2, F3. The polymerization reactions exhibit a selectivity of less than 100% with respect to the grain size fractions:
Besides one main fraction, they yield significant amounts of the other four fractions as by-products. The production processes are defined by recipes which specify the EPS-type (A or B) and the grain size distribution. For each EPS-type, five recipes are available with the grain size distributions shown in Figure 7.2 (bottom). The recipes exhibit the same structure as shown in Figure 7.2 (top) in state-task-network- representation (states in circles, tasks in squares). They differ in the parameters, e.g., the amounts of raw materials, and in the temperature profiles of the polymerization reactions.
The composition of the dispersion agents, which are produced in dedicated reactors, is the same for all recipes. The batch sizes may be one or two polymer- ization batch units for D1 and one, two, three or four polymerization batch units for D2. The processing times are ten hours for D1 and two hours for D2, and they do not depend on the batch sizes. The dispersion agents in their final states are unstable in the reactors and stable for limited periods of time in the storage
140 7 Engineered Mixed-Integer Programming in Chemical Batch Scheduling
Fig. 7.2 Recipes (top) and grain size distributions (bottom).
tanks (for 24 h in case of D1 and for 36 h in case of D2). The composition of the organic phase, which is produced in one out of two dedicated reactors, de- pends on the chosen recipe. The batch-size is not scalable, and the processing time is 1 h. The organic phase in its final state is stable for an unlimited period of time.
7.2 The Case Study 141 Each polymerization batch is produced from one unit of each dispersion agent and one batch of the organic phase. Each polymerization batch is processed in one out of the four dedicated reactors, and the product properties depend on the recipes (they differ in the EPS-type and the grain size distributions, see above). The batch-size is fixed, and the processing time is 17 h, where for the first phase of four hours there is a risk of a run-away reaction. Because of the limited capacity of the safety ventilation system only one reactor may perform the first stage of the polymerization process at a given point in time. A polymerization batch is unstable in its finite state and has to be transferred into the corresponding mixing vessel immediately.
Each of the two finishing lines is dedicated to one EPS-type A or B. The separation units have to be provided with a permanent feed with a rate between 0.10 and 0.25 polymerization batches per hour. The residence time in a separation unit is 24 h regardless of the feed rates. A start-up as well as a shut-down of a separation unit requires a set-up time of 24 h.
7.2.3
Scheduling Problem
The EPS-production is driven by customer demands. The scheduling problem exhibits the following degrees of freedom, which may be discrete or continuous in nature:
Ĺ number of polymerizations/organic phase batches (discrete), Ĺ number and size of dispersion agent batches (discrete),
Ĺ timings of batches in the preparation and the polymerization stages (continuous),
Ĺ assignment of recipes to polymerizations/organic phase batches (discrete),
Ĺ start-up and shut-down times of the finishing lines (continuous), Ĺ hold-up profiles (or feed flow rates) of the mixing vessels (continuous).
The decisions should be taken in an optimal fashion subject to the plant topol- ogy and the processing constraints with the objective to maximize the profit, given as the difference of revenues for products and costs for the production.
The demands are specified by their amounts and their due dates, where the rev- enues decrease with increasing lateness of the demand satisfaction. The pro- duction costs consist of fixed costs for each batch and for the start-up- and shut-down-procedures of the finishing lines, and variable costs for the product inventory.
The scheduling problem is complicated by the fact that the coupled production of grain size fractions and the mixing in the finishing lines prohibit a fixed assignment of recipes to products. Furthermore, there is neither a fixed assignment of storage tanks nor of polymerization reactors to batch processes.
142 7 Engineered Mixed-Integer Programming in Chemical Batch Scheduling 7.3
An Engineered Approach to Optimal Scheduling
7.3.1 Motivation
A specific feature of the EPS-production is the coupling of stages that are operated in batch mode with stages that are operated continuously. This hybrid character prohibits the complete classification of the EPS-scheduling problem according to general schemes for scheduling problems of batch plants as, e.g., the roadmap presented in the chapter “MILP Optimization Models for Short-Term Scheduling of Batch Processes”. As discussed below, this roadmap provides a suitable classifi- cation for the preparation stage and for the polymerization stage, but the finishing stage and the objective call for customized approaches.
1. Process topology: The EPS-process exhibits a sequential topology with multiple steps which are executed in a multiproduct plant. However, classical definitions as “flow-shop scheduling problem” or “job-shop scheduling problem” do not apply as the batches can no longer be identified in the finishing stage.
2. Options for assignment of the equipment: The assignment of the units to the processing steps is fixed with respect to the stages of the plant but variable with respect to particular units within the stages (e.g., the reactors of the polymer- ization stage).
3. Connectivity: The connectivity of the plant can be considered as full with respect to the given recipes as it does not constitute additional constraints.
4. Storage policies: In the preparation stage, finite intermediate storage (FIS) is provided by dedicated units; in the polymerization stage no intermediate storage (NIS) is allowed. The mixing vessels may be considered as special types of dedicated units which provide finite intermediate storage; the specialty here is the simultaneous execution of a mixing process.
5. Material transfer: Instantaneous material transfer can be assumed for the prepa- ration stage and for the polymerization stage. The organic phase is stable in its final state and the dispersion agents are stable for limited periods of time such that unlimited wait (UW) and finite wait (FW) material transfer strategies ap- ply in the preparation stage. The polymerization batches are unstable in their final states such that a zero wait (ZW) material transfer strategy applies in the polymerization stage. As the mixing vessels provide a permanent feed to the separation units, these types of material transfer strategies do not apply here.
6. Batch size: The batch sizes in the preparation stage and in the polymerization stage are variable as batches may be split in the preparation stage and mixed in the polymerization stage. However, the concept of batches does not apply in the finishing stage.
7. Batch processing times: The batch processing times in the preparation stage and in the polymerization stage are fixed and unit independent. Again, the concept of batches does not apply in the finishing stage.
7.3 An Engineered Approach to Optimal Scheduling 143 8. Demand patterns: Multiple product demands appear which are specified by
their amounts and their due dates.
9. Changeovers: No changeovers appear in the preparation stage and in the poly- merization stage. The start-ups and shut-downs of the finishing lines are changeovers with certain set-up times which cause costs (see below).
10. Resource constraints: In none of the stages resource constraints (other then equipment constraints) apply.
11. Time constraints: In none of the stages time constraints apply.
12. Costs: Fixed costs are caused by the use of equipment for reactions and by changeovers, and variable costs are incurred for inventory.
13. Degree of certainty: For the short-term scheduling problem (studied in detail in this chapter) the data is assumed to be deterministic. However, in the long term uncertainties in the demands and the capacity of the plant become relevant. The chapter “Stochastic Integer Programming in Uncertainty Conscious Schedul- ing” deals with a modeling and solution approach for scheduling problems with uncertain data.
7.3.2
Analysis of the Problem
After having motivated that the EPS-scheduling problem calls for a customized model, the interdependence of the scheduling decisions is analyzed with respect to feasibility and optimality. This analysis is performed for arbitrary demand profiles.
It exhibits that the scheduling decisions of the preparation stage are decoupled from the remainder such that they can be made based on rules once the decisions of the remaining core problem are fixed. Accordingly, the scheduling problem decomposes into a core problem and a subproblem.
For the dispersion agents finite intermediate storage is available, and the stored batches are stable for limited periods of time. Thus, a dispersion agent batch should be started when its storage runs empty such that it is finished just in time. With respect to the fixed production costs per batch, the batch size should be as large as possible; it is given by the number of polymerizations which are to be started within the period of stability. The processing times for the dispersion agents (10 h for D1, 2 h for D2) are smaller than the smallest interval between the starts of four polymerizations in case of D1 and two polymerizations in case of D2 which can be fed from the four (D1) and two (D2) storage tanks, namely 17 h in case of four polymerizations and 4 h in case of two polymerizations. Consequently, neither two batches of dispersion agents interact with each other nor are constraints on the polymerization stage imposed. The processing time of the organic phase is only one hour such that a just-in-time-strategy can be applied without constraining other decisions.
The decisions of the core problem may interact with each other over an infinite horizon. The number of polymerization reactors is the long-term bottleneck of the production process, whereas the capacity of the safety ventilation system imposes only short-term constraints. The run-away phase (four hours) of a polymerization
144 7 Engineered Mixed-Integer Programming in Chemical Batch Scheduling
(with a batch time of 17 h) constrains the next polymerization in each of the other reactors (4·4 h<17 h). The timing of a polymerization batch imposes lower bounds on the timings of all subsequent polymerizations in the same reactor.
Second, due to the coupled production, one polymerization batch corresponds to at least five demands with possibly very different due dates. That is why a very long horizon may have to be considered to assign the recipes optimally. Third, a finishing line which is out of operation or operated at its lower capacitive bound imposes constraints on the recipes and on the timings of the polymerizations. The duration of this effect is unbounded in principle such that the start-up and the shut-down times and the feed-rates may interact with infinitely many polymerization stage decisions.
In order to cover a long horizon on one hand and to provide feasible schedules on the other, the core problem is decomposed hierarchically into an aggregated scheduling problem and a detailed scheduling problem. Reasonable horizons are in the order of weeks and days, respectively. On the aggregated layer, the problem is modeled with a reduced temporal precision and solved to maximize the profit as was specified in Section 7.2.3. The long-term information is mapped onto the detailed scheduling horizon by means of demand profiles, the number of polymerization batches and start-up and shut-down times for the finishing lines. On the detailed scheduling layer, the timings of the polymerization batches, the assignments of recipes to polymerization batches and the holdup-profiles of the mixing vessels are optimized with respect to a simplified cost function. It is assumed that the profit is mainly determined on the aggregated scheduling layer such that the detailed scheduling problem aims at matching the demand profiles optimally. Its objective is to minimize a weighted sum of over- and underproduction of the demanded products at the respective due dates. In the following, the focus is on the detailed scheduling layer, for the aggregated layer the reader is referred to the chapter
“Stochastic Integer Programming in Uncertainty Conscious Scheduling” (see also [2–4]).
7.4
Nonlinear Short-Term Scheduling Model
7.4.1
Modeling Concept
The short-term scheduling model is based on a continuous representation of time.
Three distinct time axes are established corresponding to the polymerization stage, the finishing stage and the product storage units (see Figure 7.3). The events are represented following two different batch oriented concepts which both assume the number of batches to be given and their types (i.e., the recipes), sequencing and timings to be degrees of freedom. The assignment of recipes to batches is modeled on the polymerization stage by binary variables. For sequencing and timing, different concepts apply to the distinct stages.
7.4 Nonlinear Short-Term Scheduling Model 145
Fig. 7.3 Variables of engineered mixed-integer models.
The events on the polymerization stage time axis are modeled by an aggregated time slot approach: A single time axis is used to model the start times (but not the finish times) of the batch processes that are executed in the four polymerization reactors. The timings are degrees of freedom but the sequence is fixed. The events on the finishing stage time axis and the storage units time axis are modeled simi- larly: One time axis is used for the two finishing lines and one is used for the ten storage units. The events on the latter two axes are synchronized with the events on the polymerization stage time axis by fixed offsets such that they are no additional degrees of freedom. In contrast, a general precedence-based approach is applied on the storage units time axis to synchronize the variable event times with the fixed due dates of the demands: A binary variable indicates for each pair of event and due date if a certain event happens before or after a certain due date.
For the material balances, again two different concepts are applied to distinct stages of the plant. This mixed approach exploits that the number of batches produced is assumed to be given, but their types (i.e., the recipes) are degrees of freedom. As the capacity of the polymerization stage is not a function of the recipes, the material balance around the polymerization stage can be based on a batch oriented approach. In the finishing stage, batches are mixed and split into the grain size fractions which are stored and sold, such that the finishing stage and the product storage units require material balances based on network flow equations.
It is important to notice that the material balance around the finishing stage is accompanied by the nonlinear function characterizing the mixing process.
The presented concept leads to a mixed-integer nonlinear programming (MINLP) model. The binary variables representing assignment decisions and sequencing decisions are complemented by continuous variables representing timings, dura- tions, hold-ups, feed streams, supplied product, under- and overproduction. The