I n d . Eng. Chem. Res. 1992,31, 1325-1337
Grossmann, I. E.; Sargent, R. W. H. Optimum Design of Multipurpose Chemical Plants. Znd. Eng. Chem. Process Des. Dev. 1979, 18, 343-348. Grossmann, I. E.; Voudouris, V. T.; Ghattas, 0. Mixed-Integer Linear Programming Reformulation for Some Nonlinear Discrete Design Optimization Problems. In Recent Advances in Global Optimization; Floudas, C. A., Pardalos, P. M., Eds.;Princeton University Press: Princeton, NJ, 1992;pp 478-512. Knopf, F. C., Okos, M. R.; Reklaitis, G. V. Optimal Design of Batch/Semicontinuous Processes. Znd. Eng. Chem. Process Des. Dev. 1982,21, 79-86. Kocis, G. R.; Grossmann, 1. E. Global Optimization of Nonconvex MINLP Problems in Process Synthesis. Znd. Eng. Chem. Res. 1988,27, 1407-1421. Modi, A. K.; Karimi, I. A. Design of Multiproduct Batch Processes with Finite Intermediate Storage. Comput. Chem. Eng. 1989,13 (1/2),127-139. Nemhauser, G. L.; Wolsey, L. A. Integer and Combinatorial Optimization; Wiley: New York, 1988. Papadimitriou, C. H.; Steiglitz, K. Combinatorial Optimization: Algorithms and Complexity; Prentice Hall Englewood Cliffs, NJ, 1982. Papageorgaki, S., Reklaitis, G. V. Optimal Design of Multipurpose Batch Plants. 1. Problem Formulation. Znd. Eng. Chem. Res. 1990,29, 2054-2062. Patel, A. N.; Mah, R. S. H.; Karimi, I. A. Preliminary Design of Multiproduct Noncontinuous Plants Using Simulated Annealing. Comput. Chem. Eng. 1991,15 (7),451-469. Reklaitis, G. V. Progress and Issues in Computer-Aided Batch Process Design. FOCAPD Proceedings; Elsevier: New York, 1990;
1325
pp 241-275. Robinson, J. Do;Loonkar, Y. R. Minimizing Capital Investment for Multi-Product Batch Plants. Process Technol. Znt. 1972,17 (11), 861. Shah, N.; Pantelides, C. C. Optimal Long-Term Campaign Planning and Design of Batch Operations. Znd. Eng. Chem. Res. 1991,30, 2308-2321. Sparrow, R. E.; Forder, G. J.; Rippin, D. W. T. The Choice of Equipment Sizes for Multiproduct Batch Plant. Heuristic vs Branch and Bound. Znd. Eng. Chem. Process Des. Dev. 1975,14, 197. Suhami, I.; Mah, R. S. H. Optimal Design of Multipurpose Batch Plants. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 94-100. Tomlin, J. A. Special Ordered Sets and an Application to Gas Supply Operations Planning. Math. Program. 1988,42, 69-84. User’s Guide, SCICONIC/VM 2.11; Scicon Ltd, 1991. Vaselenak, J. A.; Grossmann, I. E.; Weaterberg, A. W. An Embedding Formulation for the Optimal Scheduling and Design of Multipurpose Batch Plants. Znd. Eng. Chem. Res. 1987,26,139-148. Wiede, W., Jr.; Yeh, N. C.; Reklaitis, G. V. Discrete Variable Optimization Strategies for the Design of Multiproduct processes. Paper presented at AIChE National Meeting, New Orleans, LA. Williams, H. P. Model Building in Mathematical Programming, 2nd ed.; Wiley: New York, 1985. Yeh, N. C.; Reklaitis, G. V. Synthesis and Sizing of Batch/Semicontinuous Processes. Comput. Chem. Eng. 1987,ll (6),639-654. Received for review August 21, 1991 Revised manuscript received February 10, 1992 Accepted February 25, 1992
Design of Multipurpose Batch Plants with Uncertain Production Requirements Nilay Shah and Constantinos C. Pantelides* Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BY, U.K.
The uncertainty involved in predicting product market trends may cause problems during batch plant design, with traditional approaches to this problem usually relying on simple risk analysis and the introduction of overdesign factors. We present a design procedure for batch plants where such uncertainty exists. It requires the specification of a variety of possible operating scenaria (such as different annual production requirements for all products). The plant design ensuing from the solution is able to cope with any one of the operating scenaria, without any arbitrary overdesign. The optimal solution t o this problem is obtained by solving a multiperiod design problem, based on a detailed deterministic multipurpose plant design formulation. For large designs, and/or problems involving many different scenaria, the solution of this multiperiod problem may not be feasible. Consequently, efficient techniques for obtaining good upper and lower bounds on the solution are also presented. 1. Introduction Batch processes are particularly suitable where a large number of products must be manufactured in relatively low amounts in the same facility. If the plant is flexible enough, different products may be manufactured either successively or simultaneously, and production lead times will tend to be low, with rapid adaptation to market trends. A key element in the operation of batch plants is the efficient use of multipurpose equipment and intermediate storage. In the design of such plants, it is therefore necessary to determine how the equipment may best be utilized; i.e. plant scheduling and production must form an integral part of the design problem. The most common form of batch plant design formulation considered in the literature is a deterministic one,
* To whom correspondence should be addressed.
in which fixed production requirements of each product must be fulfilled. However, it is often the case that no precise product demand predictions are available at the design stage. Indeed, the ability of batch plants to deal with irregular product demand patterns reflecting market uncertainties or seasonal variations is one of the main reasons for the recently renewed interest in batch operations. In this paper, we consider the design of plants which can fulfil any one of a set of production requirements for the various products that they can, in principle, manufacture. In the rest of this section, we review briefly the literature on both deterministic and stochastic design approaches. We then examine some characteristics of the feasible operating regions of batch plants and finally present a detailed statement of the problem of interest to this work. 1.1. Deterministic Design Problem. A comprehensive review of this area has recently been given by Reklaitis
0888-5885/92/2631-1325$03.Q0~0 0 1992 American Chemical Society
1326 Ind. Eng. Chem. Res., Vol. 31, No. 5 , 1992
(1989). Most of the early work on multipurpose batch plant design (see, e.g., Suhami and Mah (1982), Imai and Nishida (1984),and Vaselenak et al. (1987))was restricted to unique unit-to-task assignments, thus not taking full advantage of the multipurpose nature of the equipment. Parallel production of products was only allowed if the products did not share any equipment. Units assigned to perform the same task in parallel were usually assumed to be identical. Recently, Papageorgaki and Reklaitis (1990) presented a more general problem formulation which allows flexible unit-to-task allocations and nonidentical parallel units. Equipment may be allocated to a task in groups of nonidentical units operating in-phase, within a number of campaigns of varying length. Furthermore, identical groups operating in an out-of-phase mode may be allocated to the same task. The approaches above assume that the manufacture of each product involves a linear sequence of processing tasks with no intermediate storage between them, a consequence of which is that the minimum number of processing units is determined by the product with the maximum number of tasks. Shah and Pantelides (1991) take into account the effects of intermediate storage in decoupling the manufacture of each product into stages, each consisting of several tasks, so that each stage can be run independently in campaign mode. Since the same equipment can be used in two or more nonoverlapping stages, this additional flexibility may potentially lead to more economical designs. This formulation forms the basis for the work presented in this paper. 1.2. Uncertain Parameter Design Problem. A number of papers on plant design under conditions of uncertainty have also been published. Although they mainly consider continuous plants, their basic ideas are also applicable to batch plant design. An extensive review of work in this area may be found in Grossmann et al. (1983). Most of the work published considers problems expressed in the following nonlinear programming (NLP) form: min f(d,z,O) (1) subject to h(d,z,6) = 0 (2) g(d,z,O) I0 (3) where f is the objective function (usually a combination of capital and operating costs), d are the design variables, z are the state variables of the plant, and 6 are the uncertain parameters (e.g. process disturbances or varying product specifications). The latter are usually assumed to be bounded within given lower and upper bounds: 6L I6 I6”
(4)
Constraints 2 and 3 must be satisfied for all realizations of 6. This gives rise to a set of infinite dimensional constraints, which severely complicates the solution of the mathematical optimization problem. A number of solution techniques have been applied to the above problem. Wen and Chang (1968)assume known probability distributions for the uncertain parameters and attempt to minimize the expected cost. Nishida et al. (1974) choose the design variables so that the plant cost is minimized for the most adverse set of parameter values; i.e. they solve the minimax problem: min max f(d,z,O) (5) d
e
subject to the constraints above. The design produced,
although probably conservative, will not necessarily be feasible for all realizations of 6. Grossmann and Sargent (1978) presented a formulation whereby the design variable vector, d, is partitioned into the fixed design variables, u (e.g. those corresponding to equipment sizes), and the operating or control variables, u,the values of which may be varied during operation to take account of changes in 6. They then solved the problem of minimizing the expected cost over the entire range of operating conditions (the “here and now” approach). This formulation was modified by Halemane and Grossmann (1981) who discretized the space in 6 and solved the problem as a deterministic multiperiod design problem, similar to that presented by Grossmann and Sargent (1979). Swaney and Grossmann (1985a,b)presented a two-stage approach based on a scalar index of flexibility that attempts to provide a measure of the proportion of the uncertain parameter space within which feasible operation may be guaranteed. A subset of the possible realizations of 6 is used to arrive at a first-stage design. The index of flexibility of this design is calculated. If it is found to be too low, the limiting realization is added to the initial subset and the process is repeated. Reinhart and Rippin (1986, 1987) and Fichtner et al. (1990) also presented a here and now formulation specifically for batch plant design, in which the expected performance (i.e. profit less cost (including operating cost)) of the plant is maximized, with market sizes expressed in terms of nominal values and variances. Wellons and Reklaitis (1989) considered the problem of batch plant design under uncertainty where the uncertain parameters (size factors, processing times, demands, and production times) are assumed to be normally distributed. They distinguished the uncertainties into short and long term. The long-term uncertainties are accommodated by staged (i.e. future) expansion. 1.3. Feasible Operating Regions of Batch Plants. One measure of the flexibility of a batch plant is provided by its ability to guarantee feasible operation for different production requirements. In this section, we examine the feasible production region of a given plant. We also consider the variation in the feasible region of a plant being designed for different levels of capital expenditure. We determine the feasible region by using the deterministic production maximization algorithm of Shah and Pantelides (1991) to identify a number of operating points that are extreme with respect to production amounts. Consider the two-product plant 1. The recipes for each of the two products are defined in Table I, while the available equipment is shown in Table 11. To facilitate the definition of unit-to-task suitabilities, we introduce a notion of “task types”, whereby each item of equipment is suitable for all tasks of one or more types. If the plant is dedicated to either product 1or product 2 over a period of operation, for example, 1 year, the amounts produced represent the maximum possible production amounts for either product, denoted by Rl- and R2-, respectively. For the problem considered here, R1” = 640 tons and Rzmm= 233 tons. For every point R1 in the range [0, ...,Illmar],there exists a maximum amount R2of product 2 that may be produced. The feasible region may be delineated by systematically fixing R1 to different values and solving a campaign planning problem to maximize the amount of product 2 produced. Any point lying between this locus of the extreme points and the axes represents a point of feasible operation. This is due to the fact that, in contrast to most continuous plants, batch plants can decrease their pro-
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1327 a
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600
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700
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503 300
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Figure 1. Feasible production reagions for batch plants. Table I. Reciw Details for Plant 1 prerequisite product stage stage 1 1
2
2
1
3
2
1
2
1
3
2
processing time, h 11 10 11 13 10 9 8 8 7 9 11 9 11 14
task reaction 1 reaction 2 reaction 1 reaction 2 crystallization reaction 1 reaction 2 crystallization reaction 1 filtration 1 reaction 1 crystallization reaction 1 crystallization
1 1
1 1
material balance factof 0.233 0.388 0.949 0.699 0.386 0.927 0.997 0.371 0.386 0.944 0.483 0.156 0.277 0.279
task type 1
3 3 2 7 2 1 7 4 8 3 6 5 6
In tons of final product per ma of processing equipment for the task. Table 11. Processing Eauisment Details for Plant 1 equipment volume, m3 suitable task types 4 1, 4 reactor l a 6 2,3 reactor lb 4 2, 5 reactor 3a 6 2, 3 reactor 4b reactor 4d 10 1, 4 crystallizer la 8 6 crystallizer 2a 8 7 filter l a 8 8
duction rate by effectively any arbitrary factor simply by reducing either the batch sizes or the number of batches produced. The feasible production region for a two-product plant can be represented as a two-dimensional plot of Rz against
R1.If the two products are completely decoupled in the sense that their processing steps use completely disjoint subsets of the available equipment, then the feasible region is rectangular, as shown in Figure la. On the other hand, if the two produde are in fact identical in terms of recipes and equipment utilization, the feasible region is simply triangular (Figure lb). Although these two limiting cases might seem to suggest that the feasible region of batch plant operation is convex, this is not always so. For instance, the region for plant 1 is as shown in Figure IC. In general, as the demand R1on product 1is increased, production time is taken away from the limiting stage of product 2 to be allocated to that of product 1. This results in a smooth linear segment of the feasible region boundary. However, beyond a certain limit, the demand on product
1328 Ind. Eng. Chem. Res., Vol. 31, No. 5,1992 Table 111. Potentially Available EauiDment for Plant 1 suitable eauiDment volume. m3 fixed cost task twes 4 reactor l a 0.75 1, 4 6 1.00 reactor lb 2, 3 1.25 reactor IC 8 2, 3 0.75 4 reactor 3a 2, 5 6 1.00 reactor 3b 2, 5 6 reactor 4b 1.00 2, 3 10 1.50 reactor 4d 1, 4 8 6 1.00 crystallizer l a 10 1.50 6 crystallizer l b 8 7 crystallizer 2a 1.00 7 crystallizer 2b 10 1.50 8 8 filter l a 1.25 10 filter l b 8 1.50
P-od0
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7
_________ _______
Cost=B Coot=7 Cost=8 Capitd Cost=9 Capitd Cost=x) Capitd Coot=ll
_____. ..,,
600
......__ ...............
ProdA
0
1 can only be satisfied by reallocating equipment from product 2 to product 1, which results in a kink in the boundary. This is then followed by another smooth linear segment, and so on, until Illrnaxis reached. For a general n-product plant, a systematic procedure that fixes the production amounts of n - 1products while maximizing the production of the nth product can, in principle, be utilized, ultimately providing the n-dimensional feasible operating space. Figure I d shows the feasible region for a three-product plant with different lines corresponding to different levels of production of the third product. Again the irregular form of the feasible region is apparent. The feasible operating region for a plant which is still at the design stage obviously depends on the acceptable level of capital expenditure. For an n-product plant, the effects of the latter may be studied by fixing a ceiling on the available capital as well as the production amounts of n - 1 products, while maximizing the production of product n. As an illustration, we apply the above procedure to the design of a two-product plant corresponding to the processes of Table I. The plant equipment is to be selected from a list of potentially available items given in Table 111. We consider capital ceilings varying from 6 to 11cost units in steps of 1unit. The results are illustrated in Figure 2. The information obtained by this procedure would be most useful at an early stage in the design process, providing an idea of the relationship between investment and feasible operating regions. It must be stressed that, along any line of constant capital ceiling, the actual equipment makeup of the plant may vary. 1.4. Flexible Design Problem. The formulations reviewed in section 1.2 are similar in that a number of uncertain parameters are identified, and their probability distribution patterns are assumed to be known. The latter may not always be a realistic assumption, particularly with respect to product markets which tend to be heavily influenced by many external factors. However, although the plant designers may be uncertain as to the exact probability distributions of the production requirements that w i l l have to be satisfied, we assume that they will, at least, be able to specify a number of likely production scenaria, any of which may be realized. For example, consider a plant that must produce products A, B, and C. The deterministic design specification typically includes a statement of the type, “The plant must be able to produce 27 tons and A, 30 tons of B, and 45 tons of C per annum”. Considering now the uncertainties involved in the production requirements, the following alternative design specification may be made: “The plant must be able to produce either (i) 27 tons of A, 30 tons of B, and 45 tons of C per annum; or (ii) 22 tons of A, 32 tons of B, and 47
. . . . Capitd . . . Capitd . .. .. Capitd
100
200
300
400
500
600
700
800
900
Figure 2. Feasible production region for plant 1 for different capital ceilings.
tons of C per annum; or (iii) 19 tons of A, 35 tons of B, and 55 tons of C per annum; or (iv) 35 tons of A, 25 tons of B, and 42 tons of C per annum.” That is, the plant should be designed to cope with any of the above demand patterns, without the need for quantitative knowledge of their probability distribution pattern. One disadvantage of the lack of such detailed information is the inability to take proper account of operating costs in the objective function used for determining the optimal design, since this normally involves the expected value of such costs over the range of the uncertain parameters 6. However, if the operating costa are assumed to be approximately proportional to the throughput, then they are not immediately affected by decisions taken at the design stage. Using the terminology of the detailed deterministic design and campaign planning formulation of Shah and Pantelides (1991), the problem to be solved can now be stated as follows. Given a number of possible scenaria reflecting product demands, the characteristics of the potentially available equipment, the product recipes in terms of stages and their constituent tasks, and a time horizon of interest, determine the necessary equipment and, for each scenario, the allocation of the equipment to the various processing tasks in each stage, and the starting and finishing times of each stage, so as to minimize the capital cost. We note that, as seen in section 1.3, the feasible operating regions for batch plants of the kind considered here are neither convex nor concave in general. For this reason, simplistic approaches (such as designing a plant to accommodate production requirements averaged over all scenaria) are by no means guaranteed to lead to satisfactory, or even feasible, solutions. The rest of this paper is organized as follows. The next section reviews the important elements of the deterministic design formulation that forms the basis of the current work. Section 3 starts by considering the stochastic design problem in its rigorous form and then establishes tight lower and upper solution bounds which may be used if obtaining the rigorous solution of this problem is computationally impractical. Finally, section 4 presents the application of the techniques of section 3 to some representative design problems. 2. Key Features of the Deterministic Design
Formulation As already described, the deterministic design formulation is based on the decomposition of the production process into stages separated by finite or unlimited in-
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1329 termediate storage. Each stage is assumed to consist of a linear sequence of tasks operating in a no-wait mode. In this section, we summarize those features of the formulation of Shah and Pantelides (1991) which are necessary for the purposes of this paper. The interested reader is referred to the original publication. The optimal design problem involves the following three types of decisions: (i) the selection of the plant equipment items out of a set of potentially available equipment of given types, capacities, and costa; (ii) the allocation of equipment to the fundamental processing tasks; and (iii) the allocation of processing time to the different stages. Interaction occurs between these decisions as the operation of stages that share common equipment cannot overlap. The following key variables are defined: Ej
1 = 0
ujhi
1 = 0
(I
if unit j is included in the design otherwise
Finally, the objective function is the capital cost of the equipment: min C C j E j (11)
if unit j is to be used by the ith task of stage k otherwise
where Cj is the fixed cost of unit j .
I
I
q*i = length of time for which u n i t j is allocated to the ith task of stage k
tk’
= starting time of stage k
t t = finishing time of stage k
I
is the given capacity of item j . Aik is the set of units that are suitable for task i of stage k. K p is the set of stages k involved in the manufacture of product p and Pk is the set of tasks in stage k. Tkis the cycle time of stage k and is equal to the maximum of the processing times of the tasks in stage k. Task processing times are assumed constant to maintain the linearity of the formulation. This assumption is valid where processing times are weakly dependent on the batch sizes or approximate batch sizes are known. The Ujki are defined by
1
3’m = 0
if stage K starts after stage k’has finished otherwise
The main constraints are summarized below. Allocation constraints state that each item of equipment j can be allocated to at most one task i of each stage k, as different tasks of the same stage are operating in overlapping mode. This allocation can only happen if the item is actually included in the final design. Therefore the constraints are written as
C
i€njh
Ujki I Ej
V i, k
(6)
where Qjk is the set of tasks of stage k for which item j is suitable. If the same item of equipment is allocated to tasks in two different stages, then the latter cannot overlap in time:
where Ah is the set of units that are suitable for at least one task in stage k. This implies that Ykk’ and Ykrk cannot both be zero if item j is allocated to tasks in both k and k’. Note that the above constraint is only written for k’ < k to avoid duplication, and for j E Ab flAhJsince only these units can possibly be used for both stages. The Ykk’ variables affect the stage timings via the following constraints:
where Tkkt is the interstage set-up time required if stage k is to follow stage k ’in the same vessel. The above is only constraining if Ykk‘ = 1, in which case is ensures that t k s 2 tk’f Tkk’. The amount R of product p produced depends on the allocation of bot[ processing capacity and time to stage tasks:
+
where 4ikis an appropriate material balance fador and Vj
3. Dealing with Uncertain Production Requirements We now consider how the formulation presented in the previous section can be extended to design plants which can handle any one of a set of production requirementa. For notational simplicity, we summarize the constraints of the deterministic design formulation in the following form: AE+BzIa(R) (12) where A and B are given matrices, and E is the vector of equipment existence variables, while zcomprises all other variables in the formulation. The right-hand side vector a is a function of the vector of product demands R. 3.1. Optimal Formulation. The design problem subject to a set of different production requirement scenaria can be written concisely as problem P*
$*
min CC,Ej I
(13)
subject to
AE + B Z [ ~I] a(RI’1) V s = 1, NS (14) where NS is the number of different scenaria to be considered. In principle, the above is a straightforward optimization problem of the same type as the deterministicproblem-in this case, a mixed integer linear program (MILP). In some cases, the size of the problem may be reduced by removing one or more scenaria from consideration. In particular, a certain scenario 1may be omitted if it is dominated by another scenario 2, i.e. if, for every product p under consideration, the following holds: Rp[ll 5 Rp(21 (15) The above is simply a consequence of the ability of batch plants to reduce their production rates by essentially any required factor, as already mentioned in section 1.3hence, any plant which can cope with scenario 2 can certainly accommodate scenario 1 as well. However, for large designs (involving many stages and/or potentially available equipment items), and also when many different nondominated scenaria are to be taken into account, the size of the above problem may be such as to render its practical solution difficult or even impossible. This well-known complication invmiably arises with all multiperiod formulations of this type. In their
1330 Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992
work on flexible design, Grossmann and Sargent (1979) reduced the problem size by using the equality constraints to eliminate many of the variables in their formulation, thus leaving a relatively small number of variables to be considered at the outer optimization level. Groasmann and Halemane (1980) later extended this idea to eliminate both the equality and the active inequality constraints. Unfortunately these techniques are not directly applicable to the problem considered here, as most of the constraints are inequalities which are inactive at the solution. Instead, in the rest of this section, we address the problem of deriving good lower and upper bounds of the objective function, as well as obtaining a feasible equipment selection corresponding to the upper bound. 3.2. Lower Solution Bounds. A lower bound on the cost of the required plant may be obtained by solving the deterministic design problem separately for each scenario S:
problem PSI $[si
r min
CCFj[s1 J
(16)
subject to
to the proviso that this unit actually exists. This may be expressed as
f i j , the inner summation in (21) must also Since f i j , K I be bounded from above by HE? Constraint 21 may thus be modified to vpj I-1 fikp (23)
c
H
j€Am
c
(k,i)EKI
Constraints 23 relate Ej to the production requirements R[sl. For them to be effective in excluding as many equipment selections as possible, the right-hand sides must be as large as possible while the number of terms in the sum of the left-hand side must be kept to a minimum. One way of ensuring this is to select the set KI so that Aik AKI V (k,i) E KI, i.e. with all the tasks in KI using the same set of equipment items. This is likely to be the case if all tasks in KI are of the same type (e.g. filtration). All-integer "cover" constraints of the form EjIm (24) jE%
AErsl + Bds] ICY(R[~I) (17) Since the cost of the flexible plant cannot be less than that of a plant that can deal with only one of the given scenaria, a lower bound, $L, to the solution $* of problem PC may be obtained from = max $La] (18) S
However, for problems with scenaria involving widely different production requirements which, in turn, result the in very different individual equipment selections above bound may be too low. This is not unexpected, as the solution of the problem defined by eqs 16 and 17 for one scenario does not take any account of the requirements of any other. This deficiency may partly be remedied by augmenting problem PSIwith constraints which reflect the production requirements for all other scenaria s' (s' # s). This is achieved by analyzing the production requirements of each scenario separately in order to derive a relatively small number of all-integer constraints on the unit existence variables E? The union of these constraints over all scenaria under consideration is then appended to each of the individual optimization problems defined by (16) and (17). We consider the material balance eq 9 for a particular scenario s and rearrange it to the form 0.Jki.rslVj1 fiki"' v p , k E K p , i E Pk (19)
c
j € b
where = Rp["]Tk/r$ik.Let the pair (k,i) denote task i of stage k. Now a subset KI of the tasks ( k , i ) in the problem is considered. Summing both sides of (19) over all the elements of KI, we obtain Ojki[slvj1
(k,i)€KI j € A d
fiki'"
( k i )€ K I
(20)
We define fij,KI as the subset of tasks in KI that can be performed by unit j , i.e. QjW= KI n 0; also Am U(kj)EKI Aik. The order of the summations in (20) can then be reversed to
Clearly, the total time that any unit j spends performing various tasks cannot exceed the time horizon, H, subject
where m is an integer and OKI hKImay be created from (23) (see, for example, Nemhauser and Wolsey (1988) and Williams (1990)). These essentially provide lower bounds on the processing capacity that must exist in the plant in order to enable it to achieve the required production. A simple search procedure has been derived to generate all nonredundant constraints 24 from constraint 23. The cover constraints 24 may be used in a number of different ways. First, although they are redundant with respect to the scenario from which they were generated, they may be appended to its constrainta in order to reduce the integrality gap of the MILP formulation. This results in faster solution of the deterministic design problem described in section 2 by branch-and-bound type algorithms. Second, a similar acceleration effect may be achieved by including all the constraints derived from all scenaria within the optimal flexible design problem P" (eqs 13 and 14) considered in section 3.1. A third alternative is to append the constraints derived from all the different scenaria to the optimization problem Prsl for each individual scenario s (eqs 16 and 17). Thus, we ensure that the solution of the latter takes some account of the requirements of the other scenaria, thus resulting in a better lower bound derived from eq 18, Lastly, the cover constraints may be used to tighten the mathematical programming formulation used to generate an upper solution bound. This is discussed in the next section. 3.3. Upper Solution Bounds. A very simple upper solution bound may be obtained by applying the deterministic algorithm to design a plant which can cope with a maximal production requirement Rpmaxfor each and every product p. Rpmmis defined by
Rpma 3 max Rpbl S
(25)
The ability of this design to cope with the requirements of every individual scenario is a direct consequence of the dominance property discussed earlier. Unfortunately, this bound may be quite crude if the production requirements for one or more products vary greatly over the set of given scenaria. An alternative bound may be obtained by solving the individual design problems PIs] and then setting the equipment selection vedor E to the union of the individual
I I
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I
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I I
I
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I 9 I Stage3 I
* Slot2
Ind. Eng. Chem. Res., Vol. 31, NO. 5, 1992 1331
slot3
slot4 I
Slot5
I 1 I
Figure 3. Example of production slots.
selections constraints
This is equivalent to the mathematical
The upper solution bound is the corresponding value of the objective function (13). Once again, this may be too high two different units with equivalent functionality may be selected in two different scenaria, thus resulting in both of them being included in the design, whereas either of them would suffice for either scenario. In other cases, it may be advantageous to replace some units that were used by one or more scenaria by other units of similar functionality, even if the latter have not been selected for inclusion in the optimal solution of any individual scenario. It is obvious from the above discussion that a systematic yet efficient approach is required to yield an effective upper bound estimate. Two formulations that achieve this aim are described below. Both retain certain elements from the solutions of the individual problems PC81, thus limiting the complexity of the remaining optimization problem in comparison to the truly optimal formulation described in section 3.1. 3.3.1. Formulation UBl. This formulation is based on retaining the relative timing of the stages as determined by the solution to the individual problems, while allowing the stage durations and the selection and allocation of equipment to vary so as to minimize the overall capital cost. From the solution for an individual scenario, we determine a number of production "slots" such that the start and finish of each stage coincides with a slot boundary. Figure 3 illustrates an example with three stages and five slots. We now consider all scenaria simultaneously and seek to determine a set of required equipment and its task allocation so as to fulfil all the production requirements for all scenaria The number of slots and the stages active over each slot for each scenario will be the same as in the solution of P I . However, the duration of each slot is allowed to vary, provided the total production time does not exceed the available horizon. The following data are extracted from the solution of the individual problems PSI:lrSl = number of slots in the = set of stages that optimal solution for scenario s. are active in slot 1 of scenario s. KIli[Bl = set of tasks (k,i) which are active in slot 1 of scenario s for which unit j is suitable. = set of slots of scenario s over which stage k is active. In order to solve the problem, the following variables are defined: ujkpl = 1if Unit j is allocated to task i of stage k in scenario s; 0 otherwise. E j I 1if unit j is included in the final design; 0 otherwise. U'ki[']= amount of time that unit j is allocated to task i odstage k in scenario s. DlL81 = duration of slot E in scenario s. The following constraints are then formulated. Existence/AllocationConstraints, These state that (i) a unit must exist if it is to be allocated to any task for which is suitable in any scenario and (ii) a unit cannot be allocated to more than one active task in any production slot.
The constraints are of the form
Timing Constraints. The total time that a unit is allocated to a task is determined by (i) whether or not the unit is allocated to the task and (ii) the total duration of all the slots within which the task is active. Two sets of constraints are formulated: 0. 1k1.[SI 5 HUjki['1 (284
Time Horizon Constraints, The total length of all slots in each scenario cannot exceed the duration of the time horizon: p1
vs
C D p IH
1=1
(29)
Material Balance Constraints. These ensure that the minimum production requirements for each product in each scenario are fulfilled:
Objective Function. The objective is again the minimization of fixed cost in the final design; i.e. $J-'
1
min CC,Ej I
(31)
This formulation differs from the multiperiod one in that the binary Ykk' variables and the corresponding timing constaints 8 have been dropped. While this simplication often results in a much smaller MILP problem, the solution of the latter may still prove too expensive for large problems. A considerably smaller, albeit more restrictive, upper bounding formulation that may be used under such conditions is described below. 3.3.2. Formulation UB2. This formulation is based on retaining the absolute stage timings determined by the solutions to the individual scenaria, while allowing flexibility with respect to the allocation of equipment to the different tasks so as to minimize the total capital cost. The data extracted from the solution of the individual problems PSIare the same as those for formulation UB1 plus the duration 7k['] of each stage k in each scenario s. Similarly, the variables to be determined by the optimization are the same as for UB1 with the exception of the slot durations DP1, which are now fixed and related to the stage durations through 7 k [ 8 1 1 tkfM- tkS'[S1= E D,['I (32) 1ELp
The allocation constraints and the objective function for formulation UB2 are the same as for UB1, as given by eq 27 and 31. However, by virtue of (32), constraints 28 and 30 can now be merged to form the simpler constraint (33) while the horizon constraints 29 are now redundant. Even for large design problems with many different scenaria to be considered, the above is a reasonably small purely integer programming problem which can be solved relatively easily by a straightforward branch-and-bound procedure. Both upper bounding formulations may be made more efficient by including the cover constraints 24 already
1332 Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 Table IV. Recipe Details for Plant 2 prerequisite product stage stage 1
2
3
1
2
1
3
2
1 2
1
3
2
1
2
1
3
2
task reaction 1 reaction 1 crystallization 1 reaction 1 crystallization 1 reaction 1 reaction 1 crystallization 1 drying reaction 1 crystallization 1 reaction 1 reaction 1 crystallization 1 drying reaction 1 Crystallization 1
processing time, h
material balance facto9
task type
9 10
0.0365 0.1240 0.0980 0.0781 0.0675 0.0435 0.1980 0.1270 0.9300 0.0923 0.0763 0.0567 0.2010 0.1190 0.8760 0.1040 0.1230
4 6 7 4 7 2 1 7 8 2 7 3 2 7 8 3 7
11
9 11
10 10 9 9 11 10 10 10 9 8 10 11
In tons of final product per m3 of processing equipment for the task.
derived for the unit existence variables Ej; further cover Constraints for the allocation variables Ujki[81 may be derived from (30) or (33). In both cases, the solution obtained not only provides an upper bound on the optimal cost, I,P, but is also a feasible solution to the flexible design problem considered in section 3.1. 3.4. Procedure for Flexible Batch Plant Design. A procedure for obtaining good, if not always optimal, solutions to the flexible design problem can now be formalized, integrating the techniques discussed earlier in this section. (1)Check the given production scenaria to identify and remove from further consideration any that are dominated by others (eq 15). (2) Using the techniques of section 3.2, derive a set of cover constraints for each scenario. (3) For each individual scenario s, (a) form the deterministic design problem PI,(b) append to it the cover constraints derived from all scenaria in the problem, and (c) solve the resulting MILP to determine the optimal individual equipment selection and production plan. (4) Obtain a lower bound, $L, on the capital cost from eq 18. (5) Using the optimal solution for the individual scenaria (step 3c above), (a) set up either of the upper bounding formulations, UB1 or UB2 and (b) solve the above to obtain the upper bound qU on the capital cost, a revised equipment selection, and modified unit-to-task allocations. The solution obtained in step 5b of the above procedure constitutes a feasible solution to the flexible design problem. An upper bound on its fractional departure from true optimality is given by f
*u - +L -
$u
(34)
4. Results In this section, we consider the design of two different plants in order to demonstrate the applicability of the design procedure presented in section 3.4. 4.1. Plant 2. A plant that must be able to produce at least 38 tons per annum of a t least any two of three producta is to be designed. The recipe details for the three products are shown in Table IV. Table V shows the details of the potentially available equipment while the three productions requirement scenaria are shown in Table VI. If the design problem for each of the three scenaria is solved separately without using the cover constraints de-
Table V. Processing Equipment Details for Plant 2 suitable equipment volume, m3 fixed cost task types reactor 1 1 0.50 1, 5 reactor l a 5 1.00 1, 5 reactor 2 1 0.50 2, 3 reactor 2a 5 1.00 2, 3 reactor 3 2 1.25 4, 6 reactor 3a 10 1.50 4, 6 reactor 4 5 1.00 5, 6 reactor 4a 7 1.25 5, 6 crystallizer 1 5 1.25 7 crystallizer 2 8 1.50 7 drier 1 10 1.00 8 drier 2 5 0.75 8 Table VI. Production Requirement Scenaria for Plant 2 production requirements, tons (over 5000 h) product scenario 1 scenario 2 scenario 3 1 2 3
38 38 0
38 0 38
0 38 38
Table VII. Lower Bound Solution for Plant 2 without Cover Constraints scenario 1 scenario 2 scenario 3 total total total cost equipment cost equipment cost equipment 4.75 reactor 1 4.25 reactor 2a 3.50 reactor 1 reactor 2a reactor 3 reactor 2a reactor 3 crystallizer 1 crystallizer 1 crystallizer 1 drier 2 drier 2 drier 2
scribed in section 3.2 (i.e. without considering the implications of the production requirements for the other two scenaria), the solution obtained is as presented in Table VII. The solution of these three problems using a branch-and-bound approach required the solution of a total of 245 linear programs (LPs) taking 67 CPU s on a SUN SparcStation2 workstation (all problems in this section are solved within an optimality margin of 0.025). The Gantt charts that illustrate the corresponding schedules are presented in Figure 4. The numbers against each bar in these charta denote the product, stage, and task to which the relevant item of equipment is allocated (e.g. 3.2.1. denotes the first task of the second stage of the third product, as given in Table IV). If the three problems are solved separately while all production requirements are taken into account by using
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1333 UIYT
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Figure 4. Gantt charta for solution of plant 2 without cover constraints. Table VIII. Lower Bound Solution for Plant 2 with Cover Constraints scenario 2 scenario 3 scenario 1 total total total cost equipment cost equipment cost equipment 4.75 reactor 1 4.75 reactor 1 4.75 reactor 1 reactor 2a reactor 2a reactor 2a reactor 3 reactor 3 reactor 3 crystallizer 1 crystallizer 1 crystallizer 1 drier 2 drier 2 drier 2
the cover constraints, the augmented solution obtained is as presented in Table VIII. In this case, a total of 7 LPs and 4 CPU s are required-one effect of the cover constraints is to restrict the feasible search space, thus accelerating the branch-and-bound solution process. The
Figure 5. Gantt charta for solution of plant 2 with cover constraints.
Gantt charta that illustrate the corresponding schedules are presented in Figure 5. It is seen that now the same pieces of equipment are required for each of the three scenaria; this is because the cover constraints ensure that, in each case, sufficient equipment for the product not being produced in the scenario under consideration is nonetheless included in the design. Furthermore, in this case, the equipment selection obviously corresponds to the optimal solution, as it can satisfy all three scenaria without any redundancy. Therefore, there is no need to continue the design procedure beyond this stage. Comparing the corresponding Gantt charts in Figures 4 and 5 , it is interestng to note that the actual allocation of equipment items to tasks is the same. This is not surprising since the equipment selected for the flexible design (Table VIII) is a superset of the equipment required for carrying out individual scenaria (Table VII). On the
1334 Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992
a
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Figure 6. Gantt charts for sequential solution of plant 3. Table IX. Production Requirement Scenaria for Plant 3 production requirements, tons (over 5000 h) product scenario 1 scenario 2 scenario 3 scenario 4 110 190 150 1 280 170 210 130 2 60 170 210 130 3 60
other hand, we note that the durations of the various stages are different in Figures 4 and 5. Of course, the products of the durations and the corresponding batch sizes are exactly the same as they are determined by the production requirements imposed on the plant. In practice, it would probably be desirable to operate the plant so as to minimize the durations and maximize the batch sizes. This can be achieved by simply postprocessing the solutions presented in Figures 4 and 5. 4.2. Plant 3. We consider the problem of designing a three-product plant subject to the four possible scenaria defined in Table IX. The details of the product recipes and the potentially available equipment are outlined in Tables X and XI, respectively. First, we attempt to derive a lower bound on the solution of the multiperiod problem P.When the design problems for the different scenaria are solved individually (but with the cover constraints for all scenaria included), the results obtained are as shown in Table XII. The problem sizes and computational requirements for the four cases are shown in the first four rows of Table XIII. The Gantt charts of the production schedules corresponding to these different cases are shown in Figure 6. The lower bound on the cost of the flexible plant is given by the highest of the individual capital costs for each scenario, i.e. 6 cost units. The union of the equipment set
selected for the individual designs comprises eight items of equipment with a capital cost of 7.75 units-this provides an initial upper bound on the cost of the optimal design. The upper bounding formulation UB1 described in section 3.3.1 provides a more economical solution, comprising six items at a cost of 6.5 units. More specifically, the items chosen are reactor IC, reactor 2b, reactor 3a, crystallizer lb, crystallizer 2a, and filter la. The solution of this problem, which involves 341 binary variables, required 9861 CPU s on a SUN SparcStation2 workstation. The corresponding Gantt charts are shown in Figure 7. The solution of the upper bounding formulation UB2 described in section 3.3.2 is much more economical, only requiring 40 CPU s while obtaining the same value of the objective function as UB2, i.e. 6.50 cost units. The rest of the solution statistics for UB2 are shown in the penultimate row of Table XIII. The solution given by either of the two upper bounding procedures is feasible with respect to the flexible design requirements. From eq 34, we can calculate that its departure from optimality is less than 8%. For comparison purposes, the optimal formulation of section 3.1 was also solved. This was found to yield a solution with an objective function value also of 6.5 cost units, albeit with a different equipment selection, namely, reactor la, reactor 2b, reador 3b, y t a l l i z e r lb, crystalher 2b, and filter la. This serves to illustrate the potential degeneracy of these types of problems. The Gantt charts of the production schedules are shown in Figure 8. The corresponding CPU requirement was 43 272 s. It is seen from the Gantt charta that the relative order of the stages is the same for both upper bounding formu-
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1335 Table X. Recipe Detailee for Plant 3 prerequisite product stage stage 1 1
2
1
3
2
1
3
a
2
task reaction 1 reaction 2 reaction 1 reaction 2 crystallization reaction 1 reaction 2 crystallization reaction 1 filtration 1 reaction 1 crystallization reaction 1 crystallization reaction 1 filtration 1 reaction 1 crystallization reaction 1 crystallization
2
1
3
2
1
2
1
3
2
processing time, h 10 11 13 8 10 7 8 9 9 8 9 11 12 14 5 6 5 5 3 4
1 1
1
1
1 1
material balance facto? 0.233 0.398 0.949 0.699 0.386 0.927 0.997 0.371 0.386 0.944 0.483 0.156 0.217 0.279 0.997 0.886 0.423 0.958 0.254 0.281
task type 1 3 3 2 7 2 1
7 4 8 3 6 5 6 3 8 4 7 5 6
In tons of final product per m3 of processing equipment for the task.
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lations and the true optimal one. It is this feature that resulta in the former arriving at the optimal solution. This need not always be the case in general. Nevertheless, the lower and upper bounding techniques may be used in cases where the optimal multiperiod problem is too large to be tractable within reasonable computational effort. For instance, the total cost of the lower bounding procedure coupled with the upper bounding procedure UB2 for the example considered here is less than 30 CPU min (see last
row of Table XIII),compared to the more than 12 CPU h required for the solution of the rigorous multiperiod problem.
5. Conclusions and Significance The problem of designing flexible multipurpose batch plants where the production requirements are subject to some uncertainties has been considered. The deterministic design formulation of Shah and Pantelides (1991) forming
1336 Ind. Eng. Chem. Res., Vol. 31, No. 5 , 1992
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Figure 8. Gantt charts for optimal solution of plant 3. Table XI. Processing EauiDment Details for Plant 3 suitable eauioment volume. m3 fixed cost task twes 4 0.75 reactor l a 1, 4 6 1.00 reactor l b 2, 3 1.25 8 reactor IC 1, 4 0.75 4 reactor 2a 1, 2 1.00 6 reactor 2b 3, 5 1.25 reactor 2c 8 2, 4 reactor 3a 4 0.75 2, 5 1.00 reactor 3b 6 1, 2 4 0.75 reactor 4a 1, 5 1.00 reactor 4b 6 2, 3 6 1.00 8 crystallizer l a 6 1.25 11 crystallizer l b 6 1.50 13 crystallizer IC 7 1.00 crystallizer 2a 8 7 1.25 crystallizer 2b 11 7 crystallizer 2c 13 1.50 filter l a 8 1.25 8 filter l b 10 1.50 8
the basis of this work has been modified to take account of multiple potential operating scenaria, each with its own vector of production requirements. The final plant design Table XII. Sequential Solution Details for Plant 3 scenario 1 scenario 2 total cost eauipment total cost eauipment 5.75 reactor l a 5.75 reactor l a reactor 2a reactor 2a reactor 2b reactor 2b crystallizer l a crystallizer l a crystallizer 2a Crystallizer 2a filter l a filter l a
must be able to fulfillany of the possible operating scenaria should the need arise. In order to obtain the optimal solution to this design problem, a multiperiod MILP, each period of which corresponds to a different scenario with ita corresponding production requirements, must be solved. Upper and lower bounds on this optimal solution can be obtained by solving a deterministic design problem for each of the scenaria in sequence. The lower bound on the optimal capital cost is then the highest of the capital costs of the different designs. The upper bound is obtained by the solution of a smaller multiperiod MILP or IP which retains as fixed certain elements of the solutions produced by the individual single case designs. In general, the upper bounding formulation UB2 which retains the stage durations as determined by the single scenario solutions has been found to be much more efficient than the alternative formulation UB1 which attempts to reoptimize those durations. On the other hand, the upper bounds obtained by the two seldom differ significantly. Therefore, the use of UB2 is recommended. In this case, the total cost of the flexible design procedure is likely to be dominated by the cost of obtaining the single scenario 3 total cost eauipment 6.00 reactor l a reactor 2a reactor 2b crystallizer l b crystallizer 2a filter l a
scenario 4 total cost eauipment 6.00 reactor l a reactor 2b reactor 3a crystallizer l b crystallizer 2a filter l a
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1337 Table XIII. Problem Sizes and Computational Reauirements for Plant 3 Sequential Solution" scenario NIV NV NC NCOV LP 234 425 6 73 1 139 234 425 6 324 2 139 234 425 6 343 3 139 234 425 6 430 4 139 IP 290 290 377 6 221 total 1391
CPU 192 418 496 639 40 1785
" Key to abbreviations: NIV = number of integer variables; NV = total number of variables; NC = number of constraints; NCOV = number of nonredundant cover constraints; LP = number of linear programs required; CPU = CPU time (s) on SUN SparcStation2; IP = upper bounding integer program. scenario solutions. Fortunately, this step may be parallelized for solution on MIMD type computers as the separate deterministic design problems are independent of each other. Thus,large numbers of different scenaria m a y be considered provided a sufficient number of processors are available.
Nomenclature Cj = cost of unit j d = design variables D/sl = duration of slot 1 of scenario s Ej = decision variable denoting whether unit j is purchased E,!] = decision variable denoting whether unit j is purchased in scenario s
H = time horizon i = standard subscript for processing tasks j = standard subscript for equipment units k = standard subscript for processing stages = number of slots in optimal solution for scenario s LkLS1= set of slots of scenario s in which stage k is active K/sl = set of stages active in slot 1 of scenario s KI$l = set of tasks active in slot I of scenario s for which unit J is suitable
KI = any subset of tasks (k,i) in the problem pk = set of tasks of stage k R, = amount of final product p produced Rpma = maximum amount of final product p produced over all scenaria Rpi81 = amount of final product p produced in scenario s s = standard superscript for production requirement scenaria tks = starting time of stage k tkf = finishing time of stage k Tk = cycle time for stage k Ujki = decision variable denoting whether unit j is allocated to task i of stage k Ujkr[8]= decision variable denoting whether unit j is allocated to task i of stage k in scenario s o j k i = length of time during which unit j is allocated to task i of stage k = length of time during which unit j is allocated to task i of stage k in scenario s V j = volume of unit j Ykk' = decision variable denoting whether stage k starts after stage k' has finished Greek Letters
ojkl[sl
t* = fractional departure of upper bound from optimality Aik = set of equipment units suitable for task i of stage k Ak = set of equipment units which may be used for stage k AKI = set of equipment units which may be used for tasks in KI 13 = uncertain parameters BL = lower bounds of uncertain parameters Bu = upper bounds of uncertain parameters
8 = subset of AKI in cover constraints $1 = duration of stage k in scenario s f$ik = material balance factor for task i of stage k +* = optimal multiperiod design objective function = optimal design objective function for scenario s J" = lower bound on optimal multiperiod design objective function
+" = upper bound on optimal multiperiod design objective function = set of all tasks for which unit j is suitable Qj,KI = set of all tasks in KI for which unit j is suitable Qj
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