Award Lecture . • •

76

COMPUTER-AIDED DESIGN AND OPERATION OF BATCH PROCESSES

G.V. Rex Reklaitis is Professor and Head of the School of Chemical Engineering at Purdue University. He received his BS from the Illinois Institute of Technology ('65) and his PhD from Stanford University

( '69). Following a year as NSF Postdoctoral Fellow at the Institut fur Operations Research and Elektronische Datenverarbeitung in Zurich, Switzerland, he joined the faculty at Purdue, where he was appointed full profes­sor in 1980 and served as Assistant Dean of Engineering for Graduate Education and Re­search from 1985 to 1988. He was named Head of the School in 1987.

Rex 's PhD thesis , with Douglass Wilde, addressed theoretical and algorithmic issues in nonlinear programming, in general, and geometric programming, in particular. He ini­tiated work on the computational component of this theme during his postdoctoral year. The nonlinear optimization thread continued at Purdue, where he developed an interdisci­plinary course that Jed to the book Engineer­ing Optimization. During his initial years of teaching the process design courses at Purdue, he noted the consistent difficulties that stu­dents had in specifying and solving basic process material and energy balances. This spurred the development of a suitable frame­work and led to teaching the associated course and the publication of the text Introduction to Material and Energy Balances. More re­cent educational interests include the codevelopment of video- and computer­graphics-based simulated industrial labora­tory modules and the initiation of a team­taught course on computer-integrated pro­cess operations.

© Copyright ChE Division of ASEE 1995

G. V. REKLAITIS

Purdue University • West Lafayette, IN 47907

T his article describes, at a conceptual level, the basic operational and design decisions that arise in batch chemical processing and will sum­marize the approaches that have been developed to employ computing

technology to facilitate these decision processes. It is not a comprehensive treatment of the available literature either from the perspective of problem formulations or solution methods; rather, the aim is simply to convey the richness of the domain, present the nature of some of the research issues that must be addressed, and sketch out a few of the successes that have been attained to date, all from an unapologetically personal perspective. The reader interested in more detailed technical reviews is invited to consult other refer­ences11 ·3J and the additional sources cited therein.

Batch chemical processing has been practiced by the chemical engineering profession for many decades; indeed, it precedes the birth of our discipline by centuries. It had long been neglected in process systems engineering research, perhaps, because it was viewed as but a temporary expedient in the transition to an automated "modern" continuous process. But the venerable batch process has received increased attention within the last decade or two because of the growing emphasis on high value-added products, notably in the food , pharma­ceutical, polymers, agricultural chemicals, and specialty chemicals domains.

Batch operations are typically employed when 1) the production volume of a product is too low to justify a dedicated plant (typically less than 1000 tons per year), 2) the complexity of the processing steps is too high and production scale too low to justify research and development expenditures sufficient to fully develop reaction engineering, physical properties, and engineering scaleup information, and 3) a high degree of flexibility is required to accommodate continual changes in product slate, grades, and demands. The batch plant, in fact, is now often viewed as the CPI version of the modern flexible manufac­turing facility of the future- a remarkable rehabilitation of an old workhorse! Unfortunately, that rehabilitation is not yet complete within chemical engi­neering professional training, as evidenced by the minimal coverage of batch operations in the typical undergraduate curriculum.

BATCH PROCESS FEATURES

What makes a batch process different? There is, of course, the obvious difference that batch operations are inherently non-steady-state and, thus, require the explicit consideration of time and, therefore, of the dynamics of the processing steps. Additional fundamental differences exist, however. The manu-

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facture of all chemical products involves three key elements: a process or recipe that describes the set of chemical and physical steps required to make product, a plant that consists of the set of equipment withjn wmch these steps are executed, and a market that defines the amounts, timing, and qualities of the product required.

A distinguisrung feature of continuous operations is the one-to-one correspondence between the recipe steps and the plant equipment items. In the continuous case, the flowsheet is the physical realization of the recipe and its structure remains fixed in time. In batch plants, the structure of the recipe and the plant equipment network structure are in general distinct. Moreover, the equipment configuration may change each time a different product is made. Thus, in the batch case there exists an additional engineering decision level: the assignment of recipe steps to equipment items over specific intervals of time. These assignment decisions are inherently djscrete in nature, introducing a combinatorial aspect not nor­mally present in the continuous process case.

To aid in our further exploration of the implications of the above djstinctions, we will first review some basic terminology. A recipe is a network of tasks that must be executed to produce a product. Each task consists of a sequence of chemical/physical steps wmch are executed in the same vessel (see Figure 1). Each step or subtask is described by a processing time, a size factor that defines the capacity required per unjt amount of task output; and input/output ratios that describe the propor­tions in wmch inputs must be supplied and outputs are generated. A production line is a set of equipment assigned to each task of a given recipe. Assuming that the identity of a batch is preserved in the production line, then the batch size will be the amount of final product made in one batch. If the production line is used to produce a series of identical batches, it is often convenient to operate the line in a cyclic fashjon. The cycle time is then the time between the completion of batches. A Gantt chart is an equipment occupation diagram in whjch time is the ordinate and the abscissa has an entry for each equipment item. A campaign is a time interval during which one or more production lines are dedicated to making a specific set of products.

Figure 2a shows a Gantt chart for a serial four-task recipe in which a distinct unjt is assigned to each task. Note that the transfer of a task output to the next task in the recipe is denoted by an arrow. The cycle time is 6, corresponding to the maximum of the processing times of the four tasks of the recipe. As is typical, several of the units are idle for a considerable portion of the time, but at least one is continuously engaged and becomes cycle time limiting. In trus illustration the campaign consists of three batches.

As noted earlier, a characteristic feature of batch production is the need to specify an assignment of units to tasks. In general, this assignment need not be one-to-one; rather, multiple tasks can be assigned to the same unit and multiple units can be assigned to execute the same task. For the recipe of Figure 2, task 4 can be executed in two different units (U l and U4). Since these two units are inefficiently used in the one-to-one assignment shown in Figure 2a, an improvement in equipment utilization can be achieved by assigning Ul to execute both the fust and fourth task, as shown in Figure 2b, thereby releasing U4 for other uses.

Trus multiple task assignment can be viewed as a form of recycle since the batch revisits a previously used unit. Of course, since the two tasks are

Spring 1995

temporarily displaced and there is no mixing of task 1 and task 4 materials, this does not consti­tute a recycle in the usual continuous sense. Im­provements can also be achieved by assigning multiple units to a task that is performance limit­ing. If a unit assigned to a task is batch size limiting, then assigning another unit which al­lows the batch at that task to be split and pro­cessed in parallel (parallel unit in-phase) will allow an increase in the batch size. (A set of in­phase units assigned to a task is called a

Figure 1. Recipe, tasks, and subtasks

Task 1

Time 2 Units Ul

U1

U2

U3

U4

Task 2

6 U2

Task 3

4 U3

Task 4

3

Ul , U4

a) One-to-one

- t ime - I-- s--l

b) Multiple Task Assignment

:: 17,..,,.=t:~ --- time -

Figure 2. One-to-one and many-to-one task to unit assignments

77

group.)Altematively, if the task is cycle time limiting, then adding another unit and alternating the processing of batches at that task (parallel unit out-of-phase) will effectively re­duce the task processing and thus the cycle time. As shown in Figure 3, the addition of a second U2 unit out-of-phase reduces the cycle time to 4.

Based on the nature of the product recipes and the allow­able task/unit assignmen_ts, batch operations can be roughly classified into three basic types: the multiproduct plant, the multipurpose plant under campaign mode, and the general multipurpose plant. The classical multiproduct plant is employed for a set of products whose recipe structure is the same (or nearly so), the production line employs fixed many­to-one unit/task assignments, the line is operated cyclically, and multiple products are accommodated through serial cam­paigns. It should be noted that the special case of the multiproduct plant, which occurs when campaigns are re­duced to single batches, is sometimes referred to as a flowshop. The multipurpose plant under campaign op­eration is appropriate for products with dissimilar recipe structures, allows many-to-many unit/task assignments, and employs multiple campaigns involving one or more produc­tion lines, each operated cyclically. Finally, the general multipurpose plant is a multipurpose plant operated with no defined production lines; rather, production occurs in an aperiodic fashion involving many-to-many unit/task assign­ments on an individual batch basis.

The distinction between these operational types is illus­trated in Figure 4. Two products, A and B, are to be pro­duced, each involving a two-step recipe. Three multipurpose units are available, each capable of accommodating all four tasks. Figure 4a shows a production line in which Ul is assigned to task Al, and U2 and U3 are assigned out-of­phase to task A2. If the same unit/task assignments were employed for product B, we would have a multiproduct

Task 1 Task 2 Task 3 Task 4

operation. In Figure 4b, a different assignment is selected for product B (U 1 and U2 are assigned to B 1, and U3 to B2). Both lines operate in campaign style with their own charac­teristic cycle times. For instance, a campaign of four batches of A might be followed by a campaign of three batches of B, followed by another campaign of six batches of A, etc., as required to meet specific product orders.

In Figure 4c, production is in the general multipurpose mode, with tasks assigned to units in a flexible fashion, no clearly defined production line, and certainly no cyclic patterns of batch completion. Note that as a result of the imposition of a cyclic production pattern, the equipment utilization in the campaign mode (as evident from the idle time gaps) is in general not as efficient as the utilization obtained when that constraint is relaxed. But if cross-con­tamination is a consideration, the flexible, acyclic operation would require more frequent equipment clean-out than in the regular campaign mode where clean-outs may only be re­quired between campaigns.

As noted by Lucet, et al., l4J the multi product mode typi­cally is employed for larger volume products (300 to 700 t/y) with similar recipes, such as might be the case with a plant that produces a family of grades of the same product. The multipurpose mode is prevalent in facilities which produce a large number of products of smaller volume (30 to 300 t/y). The campaign form of the multipurpose plant is used when product purity requirements are stringent (such as in phar­maceuticals production) for reasons of operational simplic­ity, or to facilitate batch consistency. The general form al­lows more effective use of capital equipment at the cost of operating complexity and additional change-over costs.

THE SCHEDULING PROBLEM

A key problem that arises in batch operations is schedul­ing of the plant to meet specified product requirements .

Al Al Al Al

~ Product A Campaign Ul

U2

U3

LSSSSSS1SSSSSS1S SSSSSiSSSSSS'I A2 AZ

Time

Units

Ul

UZA

UZB

U3

U4

2 6 U1 U2A, U2B

---time-

4 3 U3 U1, U4

Figure 3. (Above) One-to-many task to unit assign­ments

Figure 4 (right) Multipurpose plant operation 78

~2

A Recipe

Multipurpose Equipment Ul ,U2,U3

Product B Campaign

~B2

B Recipe

General Ul

Multipurpose U2 Operation

U3

Ul

U2

U3

Al

Bl

'~ ~ ~ si.ss~ss.sssss,~ss.sss~t

--rime

Bl

= Bl

Bl

= Bl

= B2 == B2 B2

• • • Al AZ A2 Bl

BZ Bl Bl BZ

AZ Al BZ Al

• • BA A B A

B2

t

B2

Al

A2

B

t

t

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Specifically, given the mode of operation, the product orders, the product recipes, the number and capacity of the various types of existing equipment, the list of equip­ment types allowed for assignment to each task, any limita­tions on shared resources (such as utilities or

to research) has been the detailed exploitation of problem structure. Indeed, as will be shown in the subsequent discus­sion, tailored approaches have been proposed for each of the types of operating modes, taking advantage of the occur-

rence of specific resource constraints types, manpower), and any operating or safety restrictions, the scheduling problem is to de­termine the order in which tasks use equip­ment and resources and the detailed timing of the execution of all tasks so as to optimize plant performance.

. .. the venerable batch inventory characteristics, and cost structures.

process has received increased attention

within the last decade

THE PRELIMINARY DESIGN PROBLEM

While the scheduling problem focuses on effective use of existing production resources to meet product requirements, the design problem involves determination of what the optimal level of those production resources should be. Thus, given the mode of opera­tion, the product orders, the product recipes, the list of equipment types allowed for as­signment to each task, any limitations on shared resources (such as utilities or man­power), and any operating or safety restric­tions, the preliminary design problem is to

The scheduling problem involves three closely lipked elements: assignment of units and resources to tasks, sequencing of the tasks assigned to specific units, and determination of the start and stop times for the execution of all tasks. For instance, given two reactors (Ul and U2) and six product batches (A through F) which need to be processed, the assignment step might involve allocating A through C to Ul, and D through F to U2. The sequencing

or two because of the growing emphasis on

high value-added products, notably in the food, pharmaceutical, polymers, agricultural

chemicals, and specialty chemicals

domains.

step would involve determining the processing order on each unit (e.g., first B, then C, and then A on Ul), while the timing step would assign specific start and stop times for each batch on each unit. The above problem elements are shared by scheduling problems arising in a wide range of applications, from machine shops to transportation systems to classroom assignments. Not surprisingly, a large literature (dating to the early 1950s) exists in the operations research domain on solution approaches to scheduling prob­lems. The batch processing related literature began its growth only in the mid- l 970s.

Note that in the above example, the assignment compo­nent at root involves binary decisions (assign Ul to task A, or not) as does the sequencing component (position A first in the sequence, or not) . The timing component can be a dis­crete decision problem, or not, depending on whether time is treated as a continuum or is divided into individual time quanta. It is the binary decisions that provide the challenge to scheduling problem solution. Indeed, theoretical worst­case (computational complexity) analysis has shown that even the conceptually simplest forms of scheduling prob­lems (those involving only sequencing considerations, such as the sequencing of jobs on a single machine with set-up costs that are dependent on the job order) can exhibit expo­nential growth in computational effort with increasing prob­lem size (e.g., number of jobs).

Fortunately, recent research experience has shown that through creative problem representation, clever exploitation of problem specific structure, and effective algorithm de­sign, practical problems can be solved before "hitting the wall" of exponential growth. The key to effective solution of scheduling problems (and thus the essence of the challenge

Spring 1995

determine the required number and capacity of the various types of equipment, the order in which tasks use equipment and resources, and the timing of the execu­tion of all tasks so as to optimize plant annualized cost.

Note that the principal difference between the earlier defi­nition of the scheduling problem and the above statement of the design problem lies in the relaxation of the equipment number and capacity from the status of problem parameters to optimization variables. Indeed, since how the plant is scheduled will determine its capacity, the design problem can be viewed as an upper-level decision problem which has imbedded in it the scheduling problem. Thus, to solve the former, we must necessarily also solve the latter. Of course, there are differences in the time scales that must be consid­ered; at the design stage product demands are not known at the level of individual orders , and instead might be aggregated at quarterly, seasonal, or annual requirements. Moreover, because of differences in the degree of certainty of the demand requirements (longer range forecasts in the design case versus concrete orders in the scheduling case) the scheduling subproblem solutions required in the design case may be less rigorous.

In principle, in defining the design problem one should also include the choice of mode of operation as one of the design optimization variables. After all, mode selection (e.g., cyclic vs. acyclic, multiproduct vs. multipurpose) is at root dictated by economic considerations such as cost of inven­tory, change-overs, complexity (measured in labor and auto­mation costs), and off-spec production. Indeed, since the general multipurpose operational mode can be viewed to encompass the other two limiting modes as special cases, the mode-specific design problems can in principle be subsumed by that of the general multipurpose plant. The direct optimi-

79

zation over operational mode proves impractical-first, be­cause all of the mode-dependent costs are difficult to quan­tify, and second, because more effective solution methods can and have been devised for mode-specific formulations.

In the next few sections, we will briefly visit some ap­proaches to the scheduling and design problems for each of the three types of operating modes. For simplicity, we will confine the discussion to recipe descriptions in which size factors and input/output ratios are known constants, and task processing times are constant or known functions of the batch size. Demands will be assumed to be deterministic.

THE MUL TIPRODUCT PLANT

This operating mode was the first to be addressed in the literature[5J and continues to receive the greatest attention. It has been investigated both in the campaign form and in the limiting flowshop form.

In the campaign form, if the equipment groups used out­of-phase for a given task are equivalent, the scheduling problem reduces to the straightforward determination of the maximum product batch size and minimum cycle time for each product. Determination of the campaign lengths is made by solving a linear programming planning model if the change-over times and costs between campaigns are inde­pendent of product order. If change-overs are sequence de­pendent, then the resulting sequencing problem can be trans­formed and solved as a traveling salesman problem (TSP).[6l

If unequal, out-of-phase groups are allowed and task times are dependent on the batch size, then cyclic operation is possible with different batch sizes and cycle times, depend­ing upon the path that a batch takes. 171 The problem can be posed as a mixed integer nonlinear programming problem (MINLP) and solved via decomposition methods.

In the flowshop form, the sarrie recipe structure is used for all products; thus the equipment network is fixed and, in addition, batches are scheduled individually rather than in campaigns. A variety of approximate and rigorous branch-and-bound approaches to this problem have been proposed for various types of network structures_l81 Approxi­mate approaches typically divided the problem into a se­quencing subproblem and a completion time computation problem_[9

·11 J Rigorous approaches to problems with serial

and with parallel network structures have used reformula­tion to TSP problem forms and specialized branch-and-bound solution methodsJ 12

•131 This work is notable not only because

of the efficient optimal solutions which are obtained, but also because of the bounds on attainable schedule perfor­mance that are provided if the solution process must be terminated before the optimum is reached.

The design problem has principally been attacked in its campaign form, beginning with the seminal paper by Spar­row, et al. r14l If for each task only out-of-phase parallel units of equal size are allowed, then assuming constant processing

80

times and no sequence dependent change-over losses, the capital cost minimization problem is simply stated as:

Minimize

subject to

Vi ~Bi Sij for all products i and tasks/units j (I)

Ti ~ t ii / mi for al I tasks j and each product i ( 2)

L,Q?/Bi ~H (3)

ymin ~v. ~ y _max J J J

where

Vi denotes size of the unit assigned to taskj

B; batch size of product i

T; cycle time

mi number of out-of-phase units assigned to task j

H available production time in hours per year

Q; annual demand for product i

t;i processing time

S;i size factor for task j of product i

The power law expression in the objective function is simply a correlated cost function for equipment assigned to task j.

Note that in this model the scheduling constraints consist only of (2) and (3). The former family of constraints, which define the cycle times for each product, derive their simple structure from the fact that each unit is assigned a unique task. Constraint (3) merely insures that the total plant utiliza­tion time assigned to each product does not exceed the total available production time. There are no explicit restrictions on the number of campaigns into which the production of any given product is divided, no cost of inventory of finished products, and no explicit consideration of the costs (in time or money) of transitioning from one product to another. These more detailed production planning considerations are all essentially lumped into the specification of the produc­tion horizon H.

The presence of the integer variables mi makes the above formulation an MINLP, whose solution requires use of some form of partial enumeration strategy. The additional restric­tion of the Vi variables to a discrete set of "standard" sizes increases the combinatorial dimension of the problem. One approximate approach to such combinatorial problems is to relax the discrete value restrictions on the variables, solve the resulting continuous nonlinear programming prob­lem (which in this case can be shown to have a unique optimal solution), and then round the solution up to the nearest discrete value. Given the structure of the above model, rounding up always leads to a feasible solution, but one which is usually not cost-optimal. Thus, some round-up/

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round-down trade-offs must be explored in either heuris­ticf 15J or rigorous (branch-and-bound) form. The above prob­lem has been extended to include semkontinuous equip­ment, batch-size-dependent processing times, and in-phase units while preserving the unique optimum property of the relaxed problem.c 16I

The above formulation implicitly assumes that a batch retains its identity in processing: the batch volume/mass simply expands or contracts from task to task, as determined by the Sii. But in practice, it may be advantageous to store a large batch from, say, a long duration fermentation task, and then to process it in several smaller batches in a successor task (centrifugation, filtration , etc.) . Furthermore, it is pos­sible that the intermediates produced as outputs of one or more tasks must be combined as ingredients to a successor task (e.g., tasks 1 and 2 in Figure 1). These batch splitting and mixing possibilities require introduction of suitably sized intermediate storage. Storage decouples the production line into trains, which have their own characteristic batch sizes and cycle times, but which are linked through material balances. The minimum required size of such storage facili­ties can be determined as a periodic function of the up­and down-stream train parametersY 71 But the joint determi­nation of the optimal locations in the recipe network for such intermediate storage, the sizing of such storage, and the sizing of the process units, requires solution of an augmented MINLP. This expanded problem is challenging even in the single-product caseC181 because of its dimension­ality and the presence of many local optima in the underly­ing relaxed problem.

An interesting review of alternative MINLP formula­tions of the multiproduct design problem in its various forms is given by Ravemark and Rippin.c191 Suffice it to note that much computational research remains to derive efficient solution methods to the large-scale MINLP prob-

product A I product A

product B

CAMPAIGN 1

I I I I I I I I I I I I I I I I I I I I

HORJZON

product C

CAMPAIGN 2

Figure 5. Multipurpose plant: operation modes

Spring 1995

lems that arise when batch mixing, splitting, intermediate storage, campaign change-over, and product inventory costs are considered.

THE MUL Tl PURPOSE PLANT: CAMPAIGN OPERATION

This mode of operation extends the multiproduct mode by allowing the reassignment of equipment to tasks as dictated by the specific recipe requirements of the individual prod­ucts. Since not all available equipment may be required by a given product, parallel production of compatible products can also be considered. But once configured, the resulting production lines are operated in a cyclic fashion . As illus­trated in Figure 5, decisions must be made on grouping of products for parallel production in the same campaign (e.g., products A and B in campaign 1), assignment of the avail­able equipment among the products in the campaign (seven of the units to product A and only three to product B), and detailed configuration and scheduling of the production lines. Thus, the overall scheduling problem for this form of multi­purpose plant inherently involves three decision levels: plan­ning of campaigns, formation of campaigns, and scheduling of the production lines.

Mauderli and Rippinc201 were the first to consider this problem, focusing particularly on the campaign formation problem. They used enumerative techniques to generate and evaluate alternative single-product production lines. The more efficient of these single-product lines were then combined in an enumerative fashion, aided by an LP screening proce­dure, to identify a set of dominant multiproduct campaigns. For instance, campaign 1 of Figure 5 would be considered dominant if the combined rate of production of A and B is higher than the average production rate obtained if A and B are produced sequentially, each using its own optimally configured single-product line.

Recognizing the limitations of the heuristic enumeration approach, Wellons and Reklaiti s1211 developed rigorous MINLP formulations for all three of the decision levels and solved them using decomposition-based mathematical pro­gramming techniques. A key feature of that work was the use of the Noninferior Set Estimation method to sequentially generate dominant campaigns starting with the set of opti­mized single-product campaigns. Using this approach, cam­paigns yielding production rates as much as 20% higher than those obtained in the earlier work could be generated. Given a set of dominant campaigns, the production planning prob­lem could then be posed and solved as a multi-time period MILP that selects the dominant campaigns and determines their optimal sequence and duration so as to meet production requirements while maximizing net profit.

The key limitation of both these approaches is the require­ment of first determining a set of dominant campaigns. As the number of products increases, the computational burden

81

associated with this step grows explosively, yet at the production planning level most of these campaigns will never be selected. Thus, a more effective strategy is to form campaigns as and when they are required for specific production needs. This strategy is exploited in Tsirukis, et al. , [221 to address a more general form of the problem, which also considers the assignment and use of constrained resources (such as utilities and operators) and product demands expressed in the form of orders with specified product amounts and due dates . The key decision variables of the Tsirukis, et al., formulation are the structural variables

X omegk which take on the value l if task m of order o is processed by unit type e of equipment group g in campaign k, and O otherwise ·

and the usual continuous variables describing the batch sizes, cycle times , campaign lengths, and production amounts. The number of batches produced in each campaign are integer, but for integer values sufficiently large, can be treated as continuous. Whether defined using a cost-based or a performance­based objective function (e.g., minimize total order tardiness), the formula­tion is a large-scale MINLP, whose solution requires some form of problem structure dependent decomposition. In the present instance this is accom­plished by a hierarchical decomposition involving two decision levels: an upper-level relaxation called the campaign formation subproblem (CFS) and a reduced dimensionality lower-level problem called the equipment and resources assignment problem (ERAS).

The role of the CFS subproblem is simply to assign orders to campaigns. In this problem, the equipment of a given type is considered to be a continu­ously divisible resource of constrained availability. A key feature of this subproblem is that it can be proven to be a proper relaxation of the original MINLP problem and thus will yield lower bound estimates of its solution. Since the number of campaigns required (K) is not known a priori, K is treated as an outer iteration variable that is adjusted, as shown in Figure 6. The role of the ERAS subproblem is to convert the campaign information to specific task and equipment assignments. Since it is a reduced dimensionality form of the original MINLP, it will yield an upper bound estimate of the solution of the original problem. Furthermore, it is interesting to note that by virtue of the underlying campaign structure, the ERAS subproblem really consists of a set of individual campaign assignment problems. Since the individual campaign problems decouple, they can be solved in parallel.

As is typical in decomposition approaches, the two levels must be solved recursively until the difference between the upper and lower bound estimates is sufficiently reduced, as also shown in Figure 6. It should be noted, how­ever, that due to the nonconvexity of the ERAS subproblems, convergence to the global optimum can not be guaranteed. An approach to obtaining the global .solution of the ERAs problem as been proposed and tested[23J using feature extraction methods.

Finally, the grass-roots and retrofit design forms of the campaigned multi­purpose plant can be treated using strategies similar to those for the underly­ing scheduling problem. The grass-roots design problem differs principally in that order information is typically not available and thus the design is tar­geted toward meeting annual production requirements, somewhat simplify­ing the campaign planning level. Also, at the grass-roots design level, resource constraints are normally not treated. The number and sizes of the equipment, however, become unknowns which must be determined. As shown in Papageorgaki,- et al., [241 the resulting MINLP problem can again be solved via a hierarchical decomposition strategy. The retrofit problem is

82

positioned somewhat between the grass-roots design and the scheduling case in that some of the equipment items exist and others may need to be added. In general, the problem must be posed with an annualized net profit objec­tive function , which accounts for the additional revenue produced by the retrofit and is nonconvex. Details of the retrofitting problem under the restriction that all groups assigned to a given task are identical can be found in Papageorgaki, et al. [25-261

GENERAL MULTIPURPOSE PLANT

If production requirements of individual prod­ucts are small and cross-contamination risks low, then it is advantageous to relax the stric­tures of the campaign production mode and to allow product tasks to be executed in an acyclic fashion as needed to meet specific or­der deadlines. The equipment utilization and resource utilization time profiles thus will appear as shown in Figure 7. Note that mul­tiple tasks of different products are assigned to a given unit and no periodic resource utilization structure is evident over time. The key challenge in formulating a scheduling model for this mode is to construct sets of constraints that insure that at each point in time in the production horizon each item of equip­ment is assigned to a single task and that the utilization level of each resource shared by the simultaneously active tasks does not exceed

UpdateK

FixK

K>K mar? Yes

1---------STOP

CFS (MJNLP)

Infeasible ?

Yes RL> Ru ?

Integer Cuts

Figure 6. Multipurpose plant schedul­ing problem decomposition .

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the available supply.

The classical approach to this problem was proposed in the early days of mathematical programming research127

•281

and was subsequently elaborated in the resource constrained scheduling context by Pritsker, et al., l291 and others. The modeling device employed is to di scretize time in some suitable fashion, to introduce assignment variables specific to each time interval, and then to write for each time period a constraint set that would insure resource re­strictions were not exceeded. This approach was first ap­plied in the context of the multipurpose batch plant by Sargent and coworkers. 130

·3 11

If time is subdivided into suitably small uniform time quanta, then a zero-one decision variable can be defined for each quantum:

W;j, which talces on the value 1 if task i is performed in unit j

in time quantum t, and O otherwise

Typical resource constraints might, for instance, take the form

" W.. :<; I, for each j and t £,_ i IJl

indicating that in time interval t, unit j can be assigned to at most one task. Similarly, one can write mass balance con­straints on the material resulting from a given task i, which expresses the fact that the material available at the start of an interval , plus that produced over the interval, minus that

U1

U2

U3

Resource

Level

A2 B1 B2 1sssssssss11222222221 ~

B1 B2 A1 t2777277A ~ aJIIIl

B2 A1 A2 ~ lIIIIlJ] 1ssssssssssi

time

Availability

1111111111111111111111111111111111111111.

Figure 7. General multipurpose plant schedule structure.

Dala-ralaled: I I I II II 1111 I

Model-related: I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 111111111111111 Figure 8. Uniform and nonuniform time discretization.

Spring 1995

consumed, must be equal to what is available to the next time interval. Assuming fixed task processing times, re­source utilization amounts, and size factors, it is possible to express all of the necessary constraints as linear functions of the 0-1 and continuous variables (batch sizes, material amounts, etc.). The resulting scheduling problem can thus be posed as a mixed integer linear program.

In general, the MILPs will be quite large. For instance, with 25 time intervals, 20 tasks, and only 4 unit choices allowed per task, the 0-1 variables will number 2000. With ten equipment items, the number of constraints of the above form alone would be 250. Because of this , solution using off-the-shelf MILP solvers is not efficient or reliable, be­yond problems of perhaps 100 to 200 0-1 variables. As noted in Pekny, et al. ,l321 it is possible to formulate the MILP constraints in alternative ways, some of which provide tighter relaxations and therefore lend themselves to more effective solution than others. Moreover, as shown in that work, it is critically important to develop solution methods that fully exploit the structure and data of these types of problems. A number of alternative uniform discretization (UDM) formu­lations have been recently proposed,133-361 with various means of representing key problem features such as sequence de­pendent change-over times and losses.

Collectively, these various UDM formulations offer the advantages of accommodating complex recipe structures, treating alternative intermediate storage policies and limita­tions as well as handling multiple task-unit assignments, partial equipment ~onnectivity, and batch/lot size selection. But all UDM forms share a common limitation-namely, approximation of the underlying problem that results from the use of time di scretization . In order to rigorously model the processing events that will take place, the size of the time quantum must be chosen to equal the shortest duration event. For instance, if task processing times range from 10 hours to 1/4 hour, the latter value must be chosen for the di scretization. If the scheduling horizon is 100 hours, a problem with 400 intervals is created. On the other hand, if a much coarser interval is selected, the schedule obtained may be quite slack, reducing considerably the value of the entire optimization exercise. To address this limitation, Zentner1371 proposed the notion of using nonuniform repre­sentations of time.

The motivation for nonuniform continuous time modeling (NUCM) is illustrated in Figure 8. The chart shows an avail­ability profile of a required resource and several shaded blocks representing tasks that require this resource. The width of each block represents the task duration and the height the level of the resource required. The UDM model uses the fine discretization in order to insure that all events are captured. Since the relevant events occur only at the beginning and end of tasks and at discontinuities in the resource profile, the problem data suggest that a much sparser,

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nonuniform time representation might suffice. Specifically, in Zentner' s formulation a set of 0-1 variables is only used to represent the sequence in which tasks are executed, and continuous time variables are used to represent the start times of these tasks. It could be shown that this approach allowed significant reduction in the number of 0-1 variables, especially for problems in which the processing time values ranged widely. But for problem instances in which all task durations are of unit length, the UDM formulation will still yield problems with fewer 0-1 variables. Nonetheless , the explicit sequencing variables of the NUCM formulation do offer important advantages in treating sequence dependent change-overs. The key limitation of Zentner' s formulation , namely that batch sizes must be specified, was removed in Mockus and Reklaitisr381 at the price of introducing some bilinear terms into the mixed integer formulation. An alter­native nonlinear nonuniform formulation was reported in Xueya and Sargentf39

l but no computational comparisons could yet be offered. While there is considerable scope for further work in exploring representation, formulation , and solution issues, it is clear that there is a role for both UDM and NUCM type formulations in process scheduling.

As noted in Shah and Pantelides, r4oi the design of the general multipurpose plant can be in principle accommo­dated within the scope of a UDM scheduling model by allowing the processing unit capacities to be variables that can take on any of a set of discrete values. An MILP formu­lation of the design problem can thus be obtained and solved to yield both the design and a suitable operating schedule. The key difficulty underlying this approach, however, is that the design specifications are normally defined for annual or seasonal capacity, while the scheduling model of necessity can only consider shorter time frames.

To address this difference between the capacity planning and plant scheduling time scales, Subrahmanyam, et a/.,f411

proposed a decomposition strategy in which plant capacity optimization is carried at the level of a Design Superproblem while the verification of the operational feasibility of the design is carried out at the detailed UDM scheduling level. The Design Superproblem is an MILP that accommodates demand changes over seasonal periods, but handles the sched­uling constraints in an aggregate form. The design solution is then used to create a series of scheduling problems that cover the seasonal periods in sufficient detail to allow effec­tive UDM solution. If one or more of these scheduling problems prove to be infeasible, then the parameters of the Design Superproblem must be modified and the design opti­mization repeated. The particular feedback strategy employed in this work focuses on identifying bottleneck resources and suitably reducing their effective availability at the Superproblem level. This hierarchical approach appears to be an effective means of extending the size of UDM formu­lations that can be treated in large-scale planning, design, and scheduling applications in general.c421

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CONCLUDING REMARKS

In this paper we have provided a highly personalized per­spective on modeling and optimization approaches to deter­ministic batch process scheduling and preliminary design prob­lems. We have sought to highlight the inherently discrete and combinatorial nature of these problems and the requirements for careful formulation and rigorous solution strategies, tai­lored to the specific features of the selected operating mode. While the optimization problems that are encountered are typi­cally large in dimensionality and their solution invariably very computationally intense, application in the field is now gener­ally feasible, although not yet with off-the-shelf technology. Indeed, commercial software suitable for these problems is limited to generic MILP solvers and rule-based systems. None­theless, over the past decade, a methodological foundation has been crafted that is rapidly leading to tools accessible to the practicing engineer. The field in its present state continues to offer excellent opportunities for academic research and indus­trial application, most especially in close university-industry research collaborations.

While the focus of this article has been on simplified, deter­ministic batch process scheduling and preliminary design problems, the range of research issues in batch process systems engineering extend much beyond these confines. It includes: treatment of uncertainty and variability at both the operational and the design levels; physical layout of plant equipment; dynamic simulation as well as control of plant operations; heat integration and waste minimization, integra­tion of monitoring, diagnosis, control, and scheduling levels; synthesis of operating procedures, batch process hazard and operability analysis ; coordination of multiple plant sites; and supply chain management.

Each of these issues itself constitutes an exciting area for research and development of a computational nature. Indeed, progress is being made today in each of these areas both at Purdue and elsewhere. But a discussion of these development must be deferred to other venues.

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