04
Ind. Eng. Chem. Process Des. Dev. 1082, 21, 94-100
Lee. H. C.; Butt, J. 8. J . Catal. 1977,40, 320. Marquardt, D. W. Chem. Eng. Prog. 1959, 55(6),65. Marquardt, D. W. J . Soc.Ind. Appl. Math. 1963, 7 7 , 431. Nag, N. K.; Sapre, A. V.; Brcderlck, D. H.;Gates, 9. C. J . Catal. 1979,57,
509. Natlonal Academy of Sciences, “Assessment of Technology for Llquefactlon of Coal“; Washington, D.C., 1977. Owens, P. J.; Amberg, C. H. A&. Chem. Ser. 1961,No. 33, 182. Rlcardo, B. 0.; Pratt, K. C.; Trlmm. T. L. Fuel 1979,58, 309. Sapre, A. V.; Brodetick, D. H.;Fraenkel, D.; Gates, B. C.; Nag, N. K. AIChE J . 1980,26, 690. Sapre, A. V. Ph.D. Thesis, Universlty of Delaware, Newark, DE. 1980. Sapre, A. V.; Gates, B. C. Prepr., Dlv. Fuel Chem., Am. Chem. Soc. 1980, 25(I I), 66. Sapre, A. V.; Gates, B. C. Ind. Eng. Chem. Process Des. Dev. 1881, 20,
68.
Satterfleld, C. N.; Roberts, G. W. AIChE J . 1968, 74, 159. Unhrerslty of Delaware Computing Center, Technical MemorandumMT0108, 1977. Voorhoeve, R. J. H.; Stuiver,,:. C. M. J . Catal. 1971,23, 243. Welsser, 0.;Landa. S. Sulflde Catalyses. Their Properties and Appllcatlons”; Pergamon: London, 1973. Whkehurst, D. D.; Mkchell, T. 0. Prepr ., Div. fuel Chem ., Am. Chem. Soc. 1976,27(5), 127.
Received for review October 3, 1980 Revised manuscript received July 14, 1981 Accepted August 31, 1981
This work was supported by the Department of Energy.
Optimal Design of Multipurpose Batch Plants Iren Suhaml and Rlchard S. H. Mah’ Department of Chemical Engineering, Northwestern Universky, Evanston, Illinois 6020 1
The optimal design of a mMputpxe batch plant has been formulated as a mixed integer nonlinear program ( M I W ) . This problem differs from the multiproduct plant design in that two or more products may be produced at any one time. Consequentiy, the design is influenced by the sequencing of parallel productions as well as equipment size and number of parallel units. The strategy proposed consists of a method of generating feasible sequences and nonredundant horizon constraints, and a set of rules for selecting the optknal or near optimal configurations based on rigorous and heuristic considerations. The resulting MINLP is solved uslng a generalized reduced gradient code. The integrality of the integer variables is obtained without a tedious branch-and-bound search. If a configuration is associated only with mandatory constraints, its optimalky can be established unequivocally. Experience with problems involving up to 7 products and 10 unit types indicate that the procedure yields designs within 0.25% of the optimum.
Introduction Because continuously operated units dominate the large tonnage production of chemicals, there is a tendency to give batch plants much less than their fair share of attention in the technical literature. This certainly seems true with reference to the development and application of computer-aided design techniques for batch processes. On the other hand, in the manufacture of many chemicals of high value and low volume or products which require specially complex synthesis procedure, batch and semicontinuous processing are often the predominant modes of commercial production. Moreover, the design and scheduling of batch process facilities with the interaction of productive and non-productive (cleaning, preparation) time, overlapping cycles, parallel units, and multiple use of equipment pose technically interesting and challenging problems. These problems are only beginning to be studied systematically by chemical engineers in recent years (Loonkar and Robinson, 1970; Sparrow et al., 1974, 1975; Overturf et al., 1978; Grossmann and Sargent, 1979 Mauderli, 1979; Suhami and Mah, 1981). Batch chemical plants could be broadly classified according to the route which each product follows through the various processing units (Sparrow et al., 1975). In the multiproduct plant the processing sequence is the same for all products, the products are normally produced one at a time, and production runs or campaigns are carried out for each product in turn. Furthermore, the plant is usually designed for a fixed set of products. The multiproduct plant resembles a flowshop with no intermediate 0196-4305/82/ 1121-0094$01.25/0
storage (Suhami and Mah, 1981). The design of multiproduct plants has been formulated by Sparrow et al. (1975), who proposed a heuristic and a branch-and-bound procedure. Problems up to 4 products and 12 stages have been satisfactorily treated by these methods with a reported average computing time of 80 CP seconds on a CDC 6400 computer. The more general formulation of a multiproduct plant allows the processing time to be a function of the batch size. For this design problem Grossmann and Sargent (1979) gave a mixed integer nonlinear program (MINLP) formulation for which they were able to obtain either an optimal solution or a very good sub-optimal solution so that it is possible to circumvent a tedious branch-andbound search. They reported computational experience obtained from examples containing up to 3 products and 4 equipment types, but gave no indication of the performance of their method for higher problem dimensions for which the approximate integer solution is likely to be farther away from the continuous solution. In contrast to the multiproduct plant, the multipurpose plant resembles an engineering jobshop in which there is no common processing sequence for the products. In the most general case even successive batches of the same product may follow different sequences of processing steps. Although in this paper we shall confine our treatment to the case in which only one route is permitted for each product, it remains a characteristic of such a multipurpose plant that two or more products following different routes may be produced simultaneously. 0 1981 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982 95 BATCH
BATCH STAGE
R+T;+( p: R2 R3
I
I
. L _
CYCLE TIME
Figure 1. Gantt chart for successive batch production without parallel units.
In this paper we address ourselves to the optimal design of a multipurpose batch plant for which the objective is to minimize batch equipment cost. We shall begin with the formulation of this problem which will be followed by the presentation of the proposed scheme and our computational experience. Problem Description The plant in question consists of M types of batch equipment (Rj)to be used in producing N kinds of products. Processing of each product Pi requires ki I M types of batch equipment to be available, each corresponding to a stage j = 1, 2, ..., ki. In each stage j , defined by a prespecified type of batch equipment, nj units operate independently in parallel and out of phase, and all the units within a given stage j have the same size We shall further assume the ki equipment types to be distinct, and at any one instant, all parallel units of a given equipment type to be dedicated to the production of at most one product. In other words, each type of equipment is available only in one block of parallel units. This simplification precludes parallel processing of products which require the same equipment type at the same time. Define Ci = (RjIRjrequired for the production of Pi1 i = 1, 2, ...,N lCil = ki
where Sij is the characteristic size of equipment needed at that stage to produce unit mass of product Pi. The size Vj of each equipment type Rj must then be chosen to satisfy V j =max{Vij] j = 1, 2, ...,M i€ Uj
where Uj = (PilRjECiJ. Equipment sizes are available within given ranges. j = 1, 2, ..., M ViL IV j IVju
No intermediate storage is considered in this design. The problem is to choose V,, size of each equipment Rj, and number of equipment nj required per stage such that production requirement Qi per year for each Pi is met within the horizon H which is equal to the total time available for production in a year and such that the capital investment on R;s is minimized. Problem Formulation Figure 1depicts the processing of consecutive batches of a given product through 3 stages (units) with allowance for set-up times at each stage. Figure 2 depicts the same process with the addition of a parallel unit at the second stage. In either case the second stage is assumed to be rate
I
.
9
4
CYCLE TIME
Figure 2. Gantt chart for successive batch production with parallel units.
limiting. In general the time interval between successive batches, TLi, can be approximated by TLi
):{
= max
L*
= 1, 2,
RjECi
..., N
This approximation which was used by Sparrow et al. (1975) is a good one, since when the number of batches is large, the end effects of the schedule become negligible. Now consider the following MINLP: Choose Bi, TLi,Vi, nj to M
min
5.
For each product Pi,the sequence in which the ki types of batch equipment are going to be used is prespecified by technological considerations. Let T.. be the time required to process one batch of product Fi in a stage involving equipment type R,, and let Vij be the required unit size for processing one batch of size Bi for product Pi in stage j involving equipment type Rj. Then i = 1, 2, ...,N , RjECi Vij = BiSij
.
j=l
ajV,hj
(1)
subject to
V j =max (SijBi] j = 1, 2, ..., M
(3)
i€Uj
V j LIV j IVju
j = 1, 2,
..., M
(4)
n.L In . < nV I I I
j = 1, 2,
...,M
(5)
Ti = Qi TLi Bi
i = 1, 2, ...,N
N
CTiIH
(7)
i=l
C Ti IH
i€U,
j = 1, 2,
..., M
(8)
where aj, pi, Sij, Qi, H a r e given positive constants and nj is an integer. Constraints (7) and (8) shall be referred to as horizon constraints. Relations (1)-(7) constitute the formulation of the multiproduct plant design which Sparrow et al. (1975) and Grossmann and Sargent (1979) have dealt with and the objective is an upper bound to our multipurpose plant problem. We shall refer to this formulation as (P2). On the other hand, relations (1)through (6) and (8)represent the formulation if each product were to require only a single prespecified processor. Or to interpret it in another way, condition (8) implies that the solution is constrained independently by the time requirement for each type of equipment, determined only by the products using that equipment but taking no account of other interactions between products and equipment demands. We shall refer to this formulation as (P3). Problem (P3) yields a lower bound to our problem (Pl),because it assumes that each processor can be operated independently, whereas in reality further constraints may be imposed on (Pl) by the requirement of more than one processor generally for each product. Clearly the optimum of our problem (Pl) lies between the two. Thus, proper horizon constraints which reflect the "multipurpose" property are necessary. This can be
96
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982
Table I. Equipment Requirement for the Seven Products in Example 1
Figure 3. Configuration 1 to example 1.
$4. For example in Figure 3, PI which belongs to group 3
Table 11. Product Sets for Each Equipment type in Example 1
Ps, = P,>
u, = u 2
u, = u 4
us u6 u7
u9
p 7 >
P 2 ,
p,, p,>
= = = = =
6‘(1 ‘4,
‘6)
tPl, ‘3,
‘71
=
c p 2 > ‘3,
‘5)
Ul, =
P 2 , p 4 1
P4,
CPZ,
ps,p 7 1
{P3,
P‘1
achieved by grouping together products which can be produced simultaneously and by extracting the horizon constraints subsequently from the resulting configuration, The many different ways in which products can be grouped lead to different configurations. Our objective is to find a configuration which results in minimum equipment cost. Next, we shall explain the proposed strategy to find the best configuration. We shall also show the construction of the graph which leads to the horizon constraints and demonstrate their effectiveness with the examples below. Example 1. N = 7, M = 10 with Ci,U j sets given in Tables I and 11, respectively. The horizon constraints associated with the lower bound problem to example 1are T5 + TI IH (9)
T I IH T2 + T3 + T6 I H Tz + T4 I H
(10) (11)
T4 + T5 + T I I H
(13)
T6 IH Tz + T4 + T6 IH Ti + T3 + TI IH Tz + T3 + T5 IH T3 + T6 IH
(14)
(12)
(15) (16) (17)
(18)
where constraints (9), (lo), (12), (14), (18) are redundant. In Figure 3, compatible products (products which can be produced simultaneously) were gathered into three groups. Group 1contains P3,P4,group 2 contains P6, P5, P1, and group 3 contains PI, Pz. Given a particular sequence of the groups, a graph, in which nodes represent products and an arc ( il,i2) represents a conflict between products Pi, and Pi,, can be constructed. The arcs between nodes contained in different groups which are ordered, say, as 1,2, ...,are generated by comparing every node, say il, which belongs to groups u = 2, 3, ... against every other node i2 belonging to groups u-1, u-2, ..., 1 provided that there does not already exist a path between nodes il and iz. Nodes il and i2 are said to be in conflict if Ci,nCi, f
is first compared with P6, P5,and Pl in group 2. Since P7 is in conflict with Ps and PI,two arcs are generated. But PI is not compared with P3 and P4 in group 1, because there already exists a path between P3 and PI, and P4 and PI. The procedure continues with the next node in group 3. If source and sink nodes are added to Figure 3, together with arcs between the sink and the nodes in group 1 and between the nodes with no incoming arcs and the source, then there will be a unique horizon constraint associated with every path between the source and the sink of the form CiTi I H, summed over nodes (Pi’s)along that path. This procedure will not generate any redundant constraints, since nodes between which a path existed were not compared and as a result no possible extra arcs were generated. Horizon constraints resulting from Figure 3 are given below. Each constraint is comprised of products which need to be serially accommodated within the horizon. Note that these constraints are more restrictive than constraints (9)-(18).
T3 + T6 + T2 IH
(19)
T3 + T5 + Tz IH
(20)
T3 + T5 + TI IH
(21)
T3 + TI
(22)
+ TI IH T4 + T6 + Tz IH T4 + T5 + Tz I H T4 + T5 + T I 5 H
(23) (24) (25)
Constraints (2)-(6) together with (19)-(25) give the optimal answer to (Pl) for example 1. Let us now state the heuristic procedure to find the best configuration and clarify why Figure 3 proves to be the optimum configuration for example 1. Configuration Procedure The best configuration is chosen from a predetermined number of configurations which are generated randomly. Each configuration is generated by choosing a product type and assigning it to the first compatible group. To start with every product Pi has a probability of 1 / N of being selected. Once a product is selected, it is assigned to the first group with which it has no incompatibility. Thus if product Pi is selected to begin with, it becomes the first member of the first group. Each of the remaining ( N 1)products has a 1/(N - 1) chance of being selected next. The next selection Pi will be assigned to the first group, if it is compatible wit&Pi,. Otherwise, Pi, will become the first member of the second group. In general, if the selected product cannot be accommodated in any of the existing groups, a new group is started. Once a product is assigned to a group, its probability of being chosen is set to zero. All unassigned products have equal probabilities of being selected to be assigned next. A configu-
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982 97
Figure 7. Con9uration 5 to example 1.
a
Table 111. Horizon Constraints Associated with Figure 6 ZQj
constraints
Figure 4. Configuration 2 to example 1.
T,T, T, T, T,
PieH,
+ T , + T , + T, + T, + T , + T , + T, + T , g H + T, + T, Q H + T, + T, Q H
Q
586 000 596 000 606 000 512 000 522 000
H
Q H
Table IV. Horizon Constraints Associated with Figure 7
a
constraints
+ T, T, + T , T, + T, T, + T, T, + T, T, + T, T,
Figure 5. Configuration 3 to example 1,
+ T, + T, + T, Q H + T, Q H + T, g H + T, Q H + T, Q H
Q
746 000 436 000 490 000 480 000 522 000 512 000
H
Table V. Horizon Constraints Demonstrating Optimality Condition (ii) Figure 6. Configuration 4 to example 1.
ration will be completed when all products are assigned. Among the different configurations generated randomly, the inferior ones are eliminated after a less constrained configuration is identified. Our aim is to find the least constrained configuration. But how do we define "least constrained", since we do not know Ti beforehand? Let us define the set products associated with a given horizon constraint p by Hp. That is
Hp = (pi[(CTi)p 5 H1 I
Let us also indicate the set of Hp's associated with a configuration q by G,. We have G , = ( H IHp is a constraint resulting from configuration q}. Let f i e and Ge represent the respective sets for the lower bound problem. Note that Gt need contain the nonredundant constraints only. Configuration q1 is less constrained than configuration u, if either of the two following conditions holds true: (a) G,ICG 2; (b) for all h p l E G such that H p C H p 2 . Figures 4 and 5 stow two configurations which have been-eliminated from candidacy for the best configuration in example 1. They are labeled as configurations 2 and 3, respectively. The reason is condition (a). Each requires some horizon constraints in addition to the constraints (19)-(25). These are derived with reference to Figure 4. Among the remaining candidates, the best configuration, q*, is chosen according to the following heuristic rule a(q*) = min a(q) (30) -I
I
4
where
The application of this heuristic rule may be illustrated with reference to Figures 6 and 7 which also are associated with example 1. Note that these configurations would normally be eliminated, since configuration 1 (Figure 3) is superior to both due to condition (b). But, for con-
T, T, T, T, T, T,
+ T, + T , + T , + T , + T, Q H + T, + T, < H + T, + T , Q H + T, + T, Q H + T, Q H
Q
H
venience, we shall use these to illustrate the effectiveness of the heuristic rule. Tables I11 and IV give the horizon constraints and the quantity
C
Qi
P,€Hp
for each constraint associated with Figures 6 and 7, respectively. Note that neither of the conditions (a) nor (b) can be applied to differentiate between configurations 4 and 5. Although configuration 4 gives rise to two constraints with four elements in each, it yields a lower cost when compared to configuration 5, the reason being the large production requirement associated with the first constraint in Table IV. For a given configuration there is a horizon constraint which corresponds to maximum production requirement. By this heuristic rule, we pick the configuration which minimizes that quantity. If there is a tie in the first maximum, the second maximum is taken into consideration, and so forth. Given the best configuration q*, there are certain conditions in that if G,. satisfies any of them, then we can claim that q* is optimal. Before stating the optimality conditions, some preliminary explanation will be in order. It is clear that for any configuration q, G , must imply all lower bound constraints. Strictly speaking, for all HeEGe there exists H p E G p such that HeCH,. But sometimes one can do better in the sense that if certain conditions apply to lower bound constraints, one can derive mandatory horizon constraints which are more restrictive than some HeEG.p We define a mandatory constraint, H,, as a constraint which must be implied by any feasible configuration which is also more restrictive than some HI. If H,,, is mandatory, then, for any q , there exists H , E G , such that H,,,CHp. It should be noted that the derivation of mandatory constraints is itself a combinatorial problem.
98
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982
Table VI. Tii (hours/batch) for Products In Example 1 unit type
R,
product
R2
R3
p3 p4
P,
R,
R4
R6
7.143 4.36
2.595
P,
Table VII.
R9
R,o
5.719 7.318
2.297
R8
3.974 2.554
2.404
9.987 5.516
6.534
6.758 1.932 2.005
1.269
P6
R,
7.456
PI p2
6.855 Sij
5.469
5.062
7.326
7.725 6.65
(L/kg/batch) for Products in Example 1 unit type
product PI p* p, p,
PS P6
p,
R,
R2
R3
R,
R4
R6
R,
R9
R,,
3.304 4.529
8.163
R8
5.404 9.768 8,065
1.125
3.205 4.62
1.922
9.415 2.653
9.422
4.833 5.982
3.174 3.757
2.895
5.64
5.731
9.381
3.587 6.418
Table VIII. Yearly Production Requirements (kg) for Example 1 product yearly production
PI
P,
P,
Qi
300 000
150 000
200 000
p.
Let us illustrate mandatory constraints by an example. Suppose the constraints given by Table V constitute the set Gj for a given problem, Looking at the first three constraints in Table V, one can deduce that any configuration to this problem must have
T3 + T4 + TZ
+ TI f
T6 I H
(32)
as a horizon constraint for the following simple reason: P6 is in conflict with P3,P4.and P,, PI. Furthermore, P3,P4, P2,PI are in conflict with each other, thus making constraint (32) a mandatory one. Given constraint (32) and the fourth and fifth constraints in Table V, a similar argument can be put forward making also T3 T4 + TI + TG + TI I H (33)
+
mandatory. As a result, we can conclude that any configuration q with G, consisting of constraints (32)-(33) and the sixth constraint in Table V must be optimal. Now, we can state the sufficient conditions for optimality: confiiation q* is optimal if either of the following two conditions holds true: (i) G,. = G,; (ii) for all H E G ,, either there exits H e E G , such that H p = He, or = where H , is derived from Gj. These conditions follow directly from definitions of the lower bound problem (P3)and of mandatory constraints. Clearly, if the constraint set is the same as that of the lower bound problem (P3), the configuration q* must be optimal-condition (i). Alternatively, if it should turn out to consist of some constraints for (P3) and additional mandatory constraints derived from Gj, it will also represent a minimally constrained set, and hence the configuration q* will again be optimal. Notice that for a given problem the satisfaction of condition (i) automatically precludes the satisfaction of condition (ii). Moreover, there can only be at most one set of lower bound and mandatory constraints which satisfy condition (ii). We judge the different configurations by the objective value obtained when (Pl)is solved subject to the horizon constraints associated with that configuration. The con-
4
k,,
190 000
P, 140 000
P' 172 000
p, 106 000
tinuous and integer solutions for configurations 1-5 will be tabulated in the next section, which will also include the rest of the data for example 1. In addition, we shall present computational experience and state further details associated with the algorithm proposed. Computational Experience Numerical values of Tij(hours/batch), Sij(liters/kilogram/batch) and Qi (kilograms) in sample problems were generated from uniform distributions with [1.0, 10.01, [1.0, 10.01, and [lOOOOO, 5000001, respectively. The values for example 1are given in Tables VI-VIII. Number of stages required for each product was uniform on [1,5]. The equipment type for each of these stages is also generated randomly. Furthermore, N = 7, M = 10, H = 6200 h, aj = 250 SFr/L, = 0.6,l I nj I3, and 250 I Vj(L) 5 100oO for all j in all 10 problems on which computational experience was based. These parameters are similar to the ones used by Sparrow et al. (1975) and Grossmann and Sargent (1979). The dimensionality chosen was large enough to enable exploration of the complexity involved in multipurpose plants in terms of configurations and at the time to make it not prohibitively expensive to solve the MINLP. A few problems with N = 6, M = 8 were also tried, but they added nothing to conclusions derived from problems with N = 7, M = 10. Ten random configurations were generated for each problem and the best configuration was chosen by use of the procedure outlined in the previous section. The number of confiigurationswas determined empirically from the observation that the set usually contained a configuration which either satisfied the optimality condition or yielded an objective value very close to the lower bound. As a comparison, the total number of feasible configurations for example 1is 78. The quality of the best configuration was measured by comparing the objective values associated with the lower bound and continuous problems which are denoted by fi, fi, respectively. Generating 10 configurations proved quite adequate, since the resulting configuration either satisfied the optimality condition (ii) or was very close to the lower bound problem in each case.
a,
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982 99
Table IX. Objective Values, SFr, for Example 1
for all noninteger nj (34)
unique Kuhn-Tucker point, which makes a solution at a Kuhn-Tucker point a global optimum. The objective values for all five configurations, as well as for the upper and lower bound problems, are given in Table IX. Both the continuous and integer solutions, the number of constraints, the number of variables, d L , and d c are given in Table X. Configuration 5 can be observed to be inferior to configuration 4 as explained in the previous section. Configurations 1-3 seem to be equivalent, but configurations 2 and 3 involve a larger number of constraints. We shall now present the computational results of the other examples in a more abridged form. Table XI gives the mean and the range of d L and d c values for problems with both N = 7, M = 10, and N = 6, M = 8. Only two problems of the latter dimension were solved. Since the results based on two samples came out to be as expected, no further runs were made. Note that dc is slightly lower for thee following reason: 8 integer variables were involved instead of 10. d L will be about the same or slightly higher, since number of stages per product parameter was taken uniform on [1,5]. N being 6, the problem gets closer to the multiproduct situation. Both problems satisfied optimality condition (ii). The number of constraints varied from problem to problem and was 49 on the average. In Table XII, the objective values to the lower bound, continuous and integer solutions, d L and dc have been tabulated for each problem solved. As it can be observed, for N = 7, M = 10, 4 out of 10 problems satisfied the optimality condition (ii). It is these four problems that have raised the mean of d L to 9.13%. For the other six, dL averages to 0.25%. The problem for which d L = 50.16% was presented partially in Table V. It is clearly optimal. The GRG2 code used to solve the NLP was developed by Lasdon et al. (1978). It was found to be very efficient. The computing time on a CDC 6600 varied between 4 and 40 CP seconds depending on the starting point.
where 11, 11 represent smallest integer greater than or equal to and largest integer less than or equal to the operand, respectively. Constraint (34) forces the value of njto either the lower or upper bound. If the added constraints cause a previous integer variable to become noninteger, another run with a similar constraint is made until all n :s take on integer values. In most instances we were a61e to get integer solutions with one run in addition to the continuous solution. The quality of the sub-optimal integer solution can be measured by comparing it to the continuous solution. The percent deviation, d c , calculated as 100[f3/fi 11, where f 3 and f z are the objective values to the integer and continuous solutions, was within 7%. We found dc given in Tables X and XI quite satisfactory and felt a branch-and-bound procedure was unnecessary. Following a development used by Grossmann and Sargent (1979), problem (Pl) can be reformulated as a posynomial program. The objective function of this posynomial program has an exponent matrix with linearly independent columns. It can be shown (Suhami, 1980) that this is a sufficient condition for the program to have a
Closing Remarks The optimal design of multipurpose batch plants has been formulated as an MINLP. Multipurpose plants are similar to jobshops: more than one product may be produced simultaneously. Hence, the correct formulation requires the grouping of compatible products together. We propose a heuristic procedure in which randomly generated configurations are scanned as a result of which the best configuration is identified. Certain conditions, under which the optimality of the resulting best configuration can be determined, have been derived. Given a configuration in which products are grouped, a graph-theoretic approach is used to determine the horizon constraints associated with that configuration. Subsequently, the MINLP is solved. The integrality of integer variables is obtained without a tedious branch-and-bound search. The approximate integer solution, thus obtained, was within 7% of the continuous solution. In almost all problems solved the best configuration obtained was optimal or on the average yielded objective value within 0.25% of the optimal. This was measured by comparing it to the cor-
problem type
continuous
integer
upper bound lower bound configuration 1 configuration 2 configuration 3 configuration 4 configuration 5
616 378 353 126 354 770 354 770 354 780 379 524 389 177
355 505 355 505
Table X. Solution t o Example 1 Based on Configuration 1 continuous unit type Vi n; 1 3789.3 1.0 2 2976.6 1.0 7112.5 1.0 3 1.0 1330.6 4 6517.9 1.117 5 1690.1 1.0 6 3345.8 1.0 7 5316.5 1.0 8 9 2405.8 1.123 10 1.0 4336.2 34 no. of variables no. of constraints 54 d L = 0.47% % dev
integer
Vi ni 3991.6 1.0 1.0 3040.9 7270.7 1.0 1402.5 1.0 6870.0 1.0 1781.4 1.0 3526.6 1.0 5598.1 1.0 2549.0 1.0 4594.3 1.0 34 56 d c = 0.21%
The percent deviation, dL, calculated as 100[fz/fi - 11, is given in Tables X and XI. The MINLP was first solved without the integer restrictions. Then, the noninteger nis were forced to integer values by introducing the following constraints and making a subsequent run. Let nj*be the value of the variable nj in the most recent run. (Inj*l - nj)(nj- Inj*])= 0
Table XI. Deviations from Continuous and Lower Bound Pri3blems no. of constraints in continuous no. of no. of problem problem probvarimean dimension lems ables mean range N = 7 , M = 10 N=6,M=8 a
10 2
34 28
Optimality condition (ii) satisfied.
49 49
42-59 44-54
9.13 10.93
% dev 1OOX d c
1OOX dL
range
mean
range
0-50.16' 10.43a-11.43a
6.94 4.58
0.21-15.62 2.86-6.30
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982
100
Table XII. Objective Values and Deviations
for All Problems Solved prob-
lower
lem
bound
label
f , (SFr)
1a 2O 3 O
4a 5O 6O 7O 8O ga loa
contin-
353 126 354770 417073 417073 573301 860 850 647 550 647550 695890 695890 588430 628080 669610 669630 370220 407 330 769 400 945 340 361450 365070
l b 598860 Z b 509710 a N = 7,M = 10. tion (ii) satisfied.
integer
uous
f , (SFr) f , (SFr)
dc %
dL %
0.47 0.21 355505 2.17 426137 0 8.20 931 400 50.16 11.74 0 723590 15.62 0 804590 681790 6.74c 8.55 10.96 742990 0 2.74 418 490 10.02 4.65 989 260 22.87 4.52 381580 1.00
667 310 709 330 11.43 569 030 585 310 10.43
N = 6,M
= 8.
6.30 2.86
Optimality condi'
responding lower bound problem defined by assuming all processors can produce in parallel all the time. Computational experience was obtained on problems with 6, 7 products and 8, 10 stages, respectively. The number of constraints varied in each problem, depending on number of stages required per product. For problems solved, the number of constraints ranged from 42 to 69. Many problems of practical interest fall within the dimensional range covered in our investigation. In principle the method is extendable to problems of higher dimensions if a suitable nonlinear programming code is available. Acknowledgment This work was supported in part by the National Science Foundation, Grant No. 76-18852. Nomenclature Bi = batch size of Pi Ci= the set of Ris needed to produce Pi dc = percent deviation of the integer solution from the continuous solution dL = percent deviation of the continuous solution from the lower bound solution fi = objective value to the lower bound problem f z = objective value to the continuous problem f 3 = objective value to the integer problem G e = the set of H i s G, = the set of H 's resulting from configuration q G,**= the set of ffp's resulting from the best configuration 9
H = total production time in a year H , = the set of Pi's in a horizon constraint belonging to the lower bound problem H,,, = the set of Pi's in a mandatory horizon constraint m
H p = the set of Pis in a horizon constraint p il,iz = indices representing nodes ki = number of batch equipment needed to produce Pi M = number of batch equipment types N = number of product types nj = number of batch equipment Rj njL = lower bound to nj nju = upper bound to n . n.* = parameter defined by eq 34 = product type i p = index used to represent a horizon constraint Qi = yearly production requirement for Pi q = index used to represent a configuration q* = index used to represent the best configuration Rj = batch equipment type j Sij = characteristic size of equipment Rj needed to produce unit mass of Pi Tij = time required to process one batch of Pi in stage involving equipment type Rj T L = time interval between producing successive batches of
6
Pi
Ti = total production time required for Pi U . = the set of Pi's making use of equipment type Rj = size of batch equipment Rj v'. = lower bound to V , u'. = upper bound to $j L$. = unit size required to process one batch Biof Pi in stage involving equipment type Rj v = index used to represent a group Greek Letters aj = cost coefficient for V . p j = cost coefficient for ( (r = an upper bound defined by eq 31 Mathematical Symbols 1.1 = cardinality of a set Inj*] = largest integer less than or equal to nj* [nj*]= smallest integer greater or equal to nj* Literature Cited
v'
Grossmann, I . E.; Sargent, R. W. H. Ind. Eng. Chem. Process Des. Dev. 197% 18. 343-348. Lasdon, L. S.; Warren, A. D.; Ratner, M. W. "GRG2 User's Gulde", Technical Memorandum CIS-78-01, Computer and InformationSdence Department, College of Business Adminlstration, Cleveland State Unhrersity, 1978. Loonkar, Y. R.; Roblnson, J. D. Ind. Eng. Chem. Process Des. Dev. 1970,
9 , 625-629. Mauderll, A. M. Ph. D. dissertation, Chemical EngineeringDepartment, E@s noessischen Technlschen Hochschule, Zurich, 1979. Overturf, B. W.; Reklaltis, G. V.; Woods, J. M. Ind. Eng. Chem. Process Des. Dev. 1978. 17, 166-175. Sparrow, R. E.; Forder. G. F.; Rippin, D. W. T. Ind. Eng. Chem. Process D e s . D e v . 1975, 14, 197-203. Sparrow, R. E.; Forder, G. F.; Rippin, D. W. T. Chem. Eng. 1974, 289,
520-525. Suhaml, I. Ph.D. Thesis, Northwestern Unlverslty, Evanston, IL., 1980. Suhaml, I.; Mah, R. S. H. Chem. Eng. 1981, 5 , 83-91.
Received for review October 3, 1980 Accepted July 16,1981
