Ind. Eng. Chem. Res. 1989, 28,988-998
988
PROCESS ENGINEERING AND DESIGN A New Methodology for the Optimal Design and Production Schedule of Multipurpose Batch Plants Jaime CerdB* I N T E C , t Casilla de Correo 91, 3000 S a n t a Fe, Argentina
Manuel Vicente, Jose M. GutiGrrez, Santiago Esplugas, and Juan Mata Chemical Engineering Department, University of Barcelona, 08028 Barcelona, S p a i n
A nonlinear mathematical programming formulation for the multipurpose batch plant design problem is presented. It simultaneously provides both the optimal equipment sizes and the best series of multiproduct campaigns a t one step by taking into account the whole space of feasible production runs. By making minor changes to some problem constraints, the proposed modeling can consider the use of parallel units at certain stages. T h e units can be assigned to either the same or distinct products. Moreover, the proposed modeling can also handle practical situations where intermediate and salable products are to be manufactured. This algorithmic approach has been successfully applied to discover the best design and production policy in several examples. Optimal solutions to problems involving as many as 7 products and 10 stages have been determined by solving a single, small-size, nonlinear program. T h e method is computationally efficient even if the starting point is far from the optimum. Chemical batch plants are generally grouped into two classes, multiproduct and multipurpose batch facilities. In the former ones, a range of products are manufactured by running a sequence of single-product campaigns. Each of the N desired products undergoes a series of M processing tasks. These are accomplished in a set of M equipment modules or batch stages, each one carrying out a distinct physical/chemical task. A batch of product is transferred to the next stage after completion of the longest task. There is no intermediate storage between stages, and the plant is operated on the zero-wait mode. Important contributions to the mathematical problem description and the understanding of the optimal problem patterns have already been made by several authors (Sparrow et al., 1975; Grossmann and Sargent, 1979; Knopf et al., 1982; Vaselenak et al., 1987). When a single unit is available per stage, the batch cycle time for a product P, ( T L , , ) is equal to the time required for the limiting task, defined by T,, = max (T,),for any stage R,, where T,, is the processing time of a batch of product P, in stage R,. The value of T,, is usually a function of the product batch size (Grossmann and Sargent, 1979). By running two units in parallel working out-of-phase to perform the limiting task, the batch cycle time for a product can be diminished. It is generally assumed that the units in a given stage all have the same size and are dedicated to the production of only one product. Let B, be the amount of product P, produced per batch, and S,, represents the size of stage R, needed to produce a unit mass of product P, usually called the jth-stage size *To whom correspondence should be addressed. Also member of CONICET’s Research Staff and Professor at U.N.L. Instituto de Desarrollo Tecnoldgico para la Industria Quimica, Universidad Nacional del-Litoral (U.N.L.)and Consejo Nacional de Investigaciones Cientificas y TBcnicas (CONICET). 0888-5885/89/2628-0988$01.50/0
factor for product P,. Then, B,S, is the required size of stage R, for product P,. If all plant production requirements are to be satisfied, the size of stage R, (V,) should be large enough to process any of the desired products. Therefore, V, = max (B,S,,) for any product P,. The production time allocated to each product campaign is the kind of variable to be optimized. The best values are achieved by appropriately adjusting the product batch sizes so as to minimize the total investment cost. The series of processing tasks to be accomplished for the manufacture of the desired products are not unique in multipurpose batch plants. Depending upon the product, both the number and the arrangement of processing steps normally change. Because of that, processing a single product at a time is no longer required. Instead, several monoproduct campaigns can be run simultaneously. In other words, multiproduct campaigns are now allowed. During a campaign, however, each equipment item must still be assigned to a single product. Therefore, the simultaneous processing of various chemical species can be made only if their respective production lines do not share any unit, Le., nonconflicting production lines (Mauderli and Rippin, 1979). To make the prescribed range of products, a series of multiproduct campaigns must be carried out. As the number of species produced per campaign increases, fewer multiproduct production runs will be necessary. It may even happen that the required amount of a product be achieved through running two or more different campaigns. Generally speaking, an equipment module may be assigned to the completion of different types of tasks. The consideration of such a possibility adds new levels of complexity to the mathematical description of the multipurpose batch plant design problem. Alternative production lines for each product may be intended. In this paper, however, it is assumed a one-to-one relationship 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 7 , 1989 989 Table I. Data for E x a m d e s 1 and 2 product PI
p2 p3 p4
p5 p, p7
processing sequence RB Rn R3, R,, Rg Rg, Rim R3, RB R,, R5, R7 Rg, R5, Ri R39 KO,R,, Rs Ri, Rz, Rs, R5
Til 7.456 3.974, 7.143, 2.595, 5.719 7.318, 2.297, 4.36, 2.554 2.404, 9.987, 6.758 1.932, 5.516, 6.534 1.269, 7.725, 5.469, 2.005 6.855, 7.326, 6.65, 5.062
between processing tasks and batch stages. Furthermore, task merging and splitting are not considered. Mauderli and Rippin (1979, 1980) have developed the so-called BATCHMAN computer program, which searches for the best production planning of a multipurpose batch plant. Although they assumed that the equipment is already sized, the program accounts for the possibility of allocating each to a different type of task. After discarding inefficient campaigns, the program ends up with the optimal series of multiproduct runs and the assigned production times. Suhami and Mah (1982) presented a procedure to find both the optimal production strategy and equipment sizes for a multipurpose batch plant. It consists of solving a sequence of mixed-integer nonlinear programs (MINLP) where each MINLP is associated to a distinct feasible production scheduling. Similarly to this paper, each equipment can only accomplish a single type of task. Moreover, they assumed that parallel units in a given stage are always assigned to the production of the same product. Using some heuristics, the authors found a way to easily identifying the most promising series of campaigns and by so doing cutting down the computational effort. This work introduces an algorithmic procedure to simultaneously find the optimal production scheduling and the equipment sizes of a multipurpose batch plant. The proposed nonlinear mathematical programming (NLP) formulation accounts for all feasible multiproduct campaigns at once. In addition, minor modifications permit the handling of design options like the use of parallel units of equal size for the manufacture of distinct products in some stages or the production of intermediates.
Space of Feasible Campaigns Given the required not-yet-sized batch stages, the range of products to be manufactured, and their recipes, a wide variety of multiproduct campaigns can be run in a multipurpose batch plant. Each design problem has its own space of feasible campaigns from which one must choose a generally small set of time-efficient production runs that minimize the plant investment cost. The optimal sequence of multiproduct campaigns not only meets the production requirements but also significantly increases the average number of batches processed per unit time. In this manner, the average product batch size becomes much lower. Time-efficient production runs make a better utilization of the available equipment by reducing as much as possible the period of time during which each one stays idle. Usually, the incorporation of a nonconflicting production line to a campaign further diminishes the idle time of some units and consequently raises the productivity of the plant. By increasing the number of products being processed at the same time, the time efficiency of a campaign can be strongly improved. The total number of feasible campaigns is usually low when intermediate storage is not permitted, although it varies with the problem. Since a multiproduct campaign consists of several single-product production lines running simultaneously, then the nonoverlapping constraints
Sij
5.404 3.205, 9.768, 1.125, 3.304 4.529, 8.163, 8.065, 4.62 1.922, 9.415, 4.833 5.982, 2.653, 9.422 3.174, 3.587, 5.731, 2.895 3.757, 5.64, 6.418, 9.318
Table 11. Production Requirements for Examples 1 and 2" production needs Droduct examde 1 examde 2 800 000 1 300 000 2 150 000 150 000 700 000 3 200 000 590 000 4 190 000 5 140 000 140 000 6 172 000 172 000 406 000 7 106 000 aa,
= 250, p, = 0 . 6 , j = 1,2,...,10.
Table 111. Feasible Multiproduct Campaigns for Examples 1 and 2 (Single Unit per Stage) k Ch charact prod tb idle time,O h 1 1 1 7.456 9.0 2 7.143 2 2 7.28 7.742 3 7.318 3 3 4 4 4 8.082 9.987 5 5 5 7.86 6.534 6 7.728 6 6 7.869 7 7 7 6.465 7.326 1 7.456 8 1,2 6.28 9 1,4 1 7.456 7.082 1 10 1,5 6.86 7.456 1 11 1,6 7.456 6.869 2 12 2,7 7.143 3.745 3 13 3,4 5.824 7.318 5 14 5,6 6.534 5.729 6 15 6,7 4.334 7.725 1 16 1,5,6 4.729 7.456 a
Using expression 20.
forcing one to assign each unit to a single product significantly narrow the range of alternatives. Development of the whole set of feasible campaigns then becomes an affordable task. A computational procedure based on the notion of compatible products has already been proposed by Suhami and Mah (1982). A pair of products are said to be compatible with each other if their production lines do not overlap at all. Using this simple idea, one can sequentially generate compatible product sets comprising two, three, and a higher number of chemical species. Each one defines a distinct feasible multiproduct campaign. The lO-stage/7-product multipurpose batch plant problem studied by Suhami and Mah (1982) is presented in Table I. Data reported in Table I include the product recipes as well as the product size and time factors for every required task. Table I1 gives the production requirement for each product (see example 1). A total of 16 feasible campaigns including the monoproduct production runs are to be considered in the search for the best plant production policy (see Table 111). Only one three-product compatible set has been identified.
Mathematical Modeling (A) Problem Constraint Set. Although several products are manufactured during a multiproduct campaign h, for the sake of simplicity, each one will be characterized by a single batch cycle time ( t J . This can be made by simply picking up the limiting time, T L , i , of one of the
990 Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989
products yielded at campaign k. If Ck is a product set that includes all the products produced at campaign k , then any species 1 E Ck can play the role of kth-campaign characteristic product; Le., t k = TL,l. Let xk denote a continuous variable representing the number of batches of kth-campaign characteristic product processed during campaign k . Since t k x k stands for the time allotted to the production run k, it should then be satisfied that
v i E Ck
T L , i Z i ( k ) = tk X k
(1)
where z ~ ( is~ the ) number of batches of Pi produced a t campaign k. Equation 1 relates x k to the number of batches of the other species made a t campaign k. Furthermore, the overall number of batches of product Pi processed over the entire production period is given by z, =
Zi(k)
=
k€K,
tk --Xk k€K,tL,i
(2)
in which the campaign set Ki contains all the feasible problem campaigns producing Pi. For instead, the set K3 for product P3 comprises production runs 3 and 13 in example 1 (see Table 111). Available Production Time. If the available production time is equal to H , then tkXk
dH
(3)
k€K
where K includes all of the feasible problem campaigns. Then K i C K . By solving the proposed formulation, the optimal number of batches x k * for each active campaign is found. The production lines selected by the mathematical model are those featuring positive X k * at the optimum. Frequently, the optimal production strategy consists of running a small group of time-efficient multiproduct campaigns having nonoverlapping product sets. In other words, each product is made a t a single campaign. T o establish the optimal number of batches of product Pi E Ck yielded at campaign k,one should apply eq 1. Restriction 3 becomes nonlinear if the processing time is assumed to be a nonlinear function of the batch size. Production Requirements. Let Vj be a continuous variable standing for the size of the batch stage R,. In order to meet the production needs for every product, the size of stage Rj must satisfy the following condition,
QiSt,
v,2 -
i = 1,2....,N j
E Ji
(4)
zt
where the right-hand side of eq 4 gives the required jthstage size for product Pi. In turn, Qi is the total amount of product Pi to be manufactured and Ji is the set of processing tasks for Pi. Nonnegativity Constraints. zi 3 0 i = 1,2,...,A7
k E K
X k 2 0
VJ >, 0
j = 1,2,...,N
(5)
(B) Objective Function. The minimization of the annual total capital cost has been used as the problem design goal, even though operating costs could also be considered. Such a goal is achieved by an optimal choice of the sequence of multiproduct campaigns and the equipment sizes: M
Table IV. Additional Feasible Campaigns by Operating Two Parallel Equipments in Stages R, and R8 (Examples 1 and 2) k Ck charact prod tk idle time," h 17 1,3 1 7.456 8.742 18 1.7 1 7.456 7.465 19 3,7 3 7.318 6.207 9.987 7.942 20 4,5 4 4,7 4 21 9.987 6.547 1 22 1,2,7 7.456 4.745 23 1,3,4 1 7.456 6.824 24 1,6,7 1 7.456 5.334 25 1,4,5 4 9.987 6.942 9.987 5.547 26 1,4,7 4 9.987 4.289 27 3,4,7 4 Using expression 20.
In the first step, it is assumed that a single unit per stage is available. As a result, the multipurpose batch plant design problem is formulated through a low-size nonlinear mathematical program involving constraints 2-5 expressed in terms of the continuous variables v,,zi,and x k . The subsequent analysis of the optimal production strategy will indicate whether the operation of parallel units in certain stages may further reduce the annual investment cost. In case all batch stages exhibit a nonzero idle time at the optimum, the use of parallel units working out-of-phase to produce the same product hardly lessens the value of the problem objective. The next section presents the mathematical problem formulation when multiequipment stages dedicated to the manufacture of a single product or several products are available. By proceeding in two steps, much simpler formulations can be proposed to model the whole design problem. Moreover, the approach better resembles the way followed by a process designer who likes to know how much he would save by complicating more the plant operation.
Use of Parallel Units at Some Batch Stages The parallel operation of several equipment units a t a given stage can increase the number of compatible product sets if each one is assigned to the production of a different product. Consequently, the number of alternative multiproduct campaigns becomes larger and the value of the problem objective may be further lessened. Knopf et al. (1982) has already noted that the simultaneous processing of more than one product in a multiequipment stage can result in more efficient equipment utilization. In example 1, for instance, the introduction of a second unit in stage 8 causes the two-product sets [1,3], [1,7],and [3,7] to all be compatible. Since the production lines of such products all include stage 8, they cannot be run simultaneously unless a pair of units is available in stage 8. Therefore, two or more production lines sharing a batch stage are made compatible by operating two parallel equipment in that stage. However, the group [1,3,7] is still incompatible because three parallel units would be necessary to run the production lines simultaneously, yielding the products PI, P,,and P,. Table IV lists the additional feasible multiproduct campaigns if two pairs of units running in parallel in stages 5 and 8 are assigned to different production lines. For each new compatible product group, a member of the previously incompatible subset has been chosen as its characteristic product. For instance, product P4 play such a role for campaign [ 1,4,7] because [4,7] was incompatible before another unit in stage 5 was operated. Nonetheless, PI or P7 could have also been chosen as long as the selection of a characteristic campaign product is arbitrary and has no impact on the optimal solution.
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 991 There are a total of 11 feasible campaigns, more than before, some of which may be highly time-efficient. Despite that, the proposed modification does not always yield a lower investment cost because of the power-law opposite effect favoring the use of a fewer number of units. On the other hand, the parallel operation of two equipment units running out-of-phase in the limiting stage of a product yields a decrease of its batch cycle time and frequently a change of its controlling task if both are dedicated to the production of that product. For instance, the tasks being accomplished in stages 5 and 8 limit the batch cycle times for products P4and PI, respectively. By installing a second unit in such stages, the values of TL,4 and TL,1can be diminished to 6.758 and 3.728, respectively (see Table I). At the same time, stage 7 becomes the new limiting task for product Pq. Such reductions on the batch cycle times hold for any production run shown in Table I manufacturing products PI and/or P4. In the latter case, the compatible product sets are still those shown in Table 111. However, the operation of parallel units running out-of-phase in stage Rj could bring about a lower investment cost only if they are assigned to the product whose batch cycle time is limited by stage R,. Otherwise, the plant capital cost will surely grow. Therefore, campaigns producing P1and/or P4must only be considered if two-unit modules are opwated in stages 5 and 8 to process a single product. The other ones can be directly ignored. Analysis of the economical advantage of running two units in some stages can be made only if the enlarged set of feasible campaigns is taken into consideration. This requires us to make some changes in the design problem modeling presented in the previous section. New Set of Constraints. The available production time restriction is now given by (7)
u u
where I? = K K’ K” stands for the new campaign set instead of K and K’ is the set of additional feasible campaigns generated by simultaneously processing various products in the multiequipment stages j E J. K’usually depends upon the choice of J . A procedure for an optimal selection of d is presented in the next section. In turn, K” C K only includes those campaigns k E K , yielding at least a product, Pi, whose limiting stage j* belongs to J ( j * E 4. Similarly,
at the selected series of campaigns. When the parallel units are instead dedicated to the production of a single product, each one must process the same number of batches (zi/nj) and a similar amount of product Pi (Qi/nj).Therefore,
and eq 4 still holds. Problem Objective. M
2 n,a, V,pl J=l
min
where the coefficient n, is equal to two if a pair of units is running in stage R,. Otherwise, it is equal one. Optimal Selection of t h e Multiunit Batch Stages A mathematical formulation is introduced in this section to properly select the batch stages in which parallel units processing different products are operated. The problem goal is the generation of new feasible campaigns that are as time-efficient as possible. Single Equipment per Stage. Let’s define a set of binary variables y,, in such a way that y,, is equal to one whenever stage R, is assigned to the processing of P, at the current feasible campaign or is equal to zero otherwise. Such variables y,, will permit us to write the set of linear restrictions to be fulfilled by a feasible multiproduct campaign. Those constraints are the following: (1)Each stage is assigned to at most a single product
C y,, < 1
j = 1,2,...,A4
GI,
2 yv
= M,w,
i = 1,2,...,N
(11)
J€J,
where M , is the number of batch stages involved in the processing of P,. Besides, w,is a binary variable defined by C y,, 6 w,U 1 = 1,2,...,N (12) JEJ,
y,,=O,l
V j E J l i = 1 , 2 ,...,N i = 1,2,...,N
(13)
in which U is a large number. Therefore, w,is equal to one if at least a single batch stage has been assigned to PI. Otherwise, it is equal to zero. A measure of the overall equipment idle time at a multiproduct campaign is given by
where the first term provides the number of batch stages not being operated at all during the campaign. The other term accounts for the equipment remaining partially idle. The coefficient of each term is the fractional cycle idle time of the related unit. If the multiproduct campaign making the best utilization of the batch equipment is sought, then the overall idle time must be minimized. Therefore, the problem objective is M
where zi is the overall number of batches of Pi processed
(10)
(2) All the batch stages needed for the manufacture of a product P, yielded at the current campaign must be assigned to P,. In contrast, no equipment is allocated to a product not being produced during the campaign:
w,= 0, 1 where Ki= { k E Klcampaign k yields Pi). The coefficient ( t k / TLi)for a campaign k E K” should be properly adjusted with respect to eq 2 to account for the reduction of the batch cycle time of either Pi(TL,Jor the kth-campaign characteristic product ( t k ) . For instance, the coefficient for campaign 13 is equal to (7.318/6.758) = 1.083 since t13remains unchanged and TL,4decreases from 9.987 to 6.758 in example 1 (see Table 111). As long as the nj parallel units in a stage j E J all have the same size, nonlinear constraint 4,forcing the fulfillment of every production requirement, does not change at all. The subset of restrictions for the multiequipment stages j E J , if each unit manufactures a different chemical, is still given by
(9)
Til
max C C -yi;
j = l i € I , TL,i
(14)
992 Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989
subject to restrictions 10-13. In this way, a pure ILP formulation has been developed to search for the most efficient multiproduct campaign if a single equipment item is available per stage. The proposed mathematical model involves 2N + M linear constraints and N + C z , M i integer variables. This implies 30 integer variables, 17 linear inequalities, and 7 equality constraints when applied to example 1. T o find the best campaign for the manufacture of a particular product Pi, it should further be imposed that the related variable w i be equal to one. Operation of Parallel Units. Let’s now assume that parallel units are to be operated in certain batch stages to improve the plant efficiency. T o this end, different products will be simultaneously processed in the multiequipment stages. These should be appropriately chosen to get the largest efficiency improvement. Because of that, the previous mathematical model must be modified to account for the parallel units to be incorporated. Let y’ij be a binary variable set equal to one when the second unit in stage j is assigned to Pi and set equal to zero otherwise. For the new ILP problem, the set of constraints is given by C y i j d 1 j = 1,2,...,M (15) LEIJ
cy/, Q 1
j = 1,2,...,M
(16)
!€IJ
(yij
iEJ,
+ y’ij)
= Miwi
i = 1,2,...,A’
(17)
Since the parallel units process different products, then j i i + yti, < 1 j E Ji i = 1,2,...,N (18) Besides,
c (Y,j + ytij) 6 w i u
i
= 1,2,...,A’
jEA
(19)
Finally, the objective function is given by
If a t most p two-unit stages are going to be operated in the plant, then M
C Cy’ij < P
j=liEIJ
The proposed modeling will incorporate p additional equipment to operate in parallel only if by doing so the overall plant idle time is effectively decreased. On the other hand, the most time-efficient multiproduct campaign when p two-unit stages are run in the plant will be found by expressing (21) as an equality. Let’s suppose that the best campaign, k*, found by solving the proposed model, including (21) as an equality, yields a product set Ck.. To determine the production run ranked second in time efficiency, the following restriction should be added to the problem formulation:
EW
i€EC,.
l 3 l
(22)
In this manner, the most efficient production runs can be identified by repeatedly solving an ILP having an objective function expressed by eq 20 subject to constraints 15-19 and 21. An additional restriction, (22), is incorporated in the model before the next ILP is run. Table V shows the production runs ranked first to eighth in time efficiency, which becomes feasible by operating a couple of two-unit modules in example 1. Four of them
Table V. Time-Efficient Multiproduct Campaigns Generated by Running a Pair of Two-Unit Modules (Example 1 ) Ck
2,6,7 5,6J 3,4,7 4,6,7
1 ~ 7 L4,W 1,6,7 1,4,7
idle time, h 3.662 4.195 4.289 4.417 4.745 4.810 5.334 5.547
J
-
R3, R7 Ri, R j R5, R8 R63 R7
R,, Ra RE,, R7 Rj, Ra R,, R8
are brought about by installing parallel units in stages R5 and R8 (see Table V). Since the major goal is the generation of as many new time-efficient campaigns as possible, the allocation of the two parallel units to stages R5and R8 arises as the best choice. Indeed a significant number of promising campaigns (feasible integer solutions) are found a t once by solving a single ILP and similarly several constraints, (22), should be added to the formulation before the next run. Table IV also lists the total equipment idle time given by expression 20 for each campaign h E K’in example 1. As seen later, the two most efficient campaigns, [ 1,2,7] and [3,4,7], are run a t the optimal production strategy. Therefore, a small subset of efficient production runs, It* C K,is indeed really required to find the best production schedule and equipment sizes. Generation of t h e Campaign Set K * C K . An alternative procedure to that of Suhami and Mah (1982) consists of repeatedly running an ILP, trying to maximize expression 14 without violating the set of linear constraints 10-13. In each run, one or several restrictions similar t o eq 22 are being incorporated to find other efficient multiproduct campaigns. Table I11 provides the total equipment idle time for each campaign when a single unit is operated in each stage. Production run [1,5,6],ranked third in time efficiency, will be found after, a t most, three executions of the ILP solution method. At this point, one already knows that [1,5], [1,6], and [5,6] are also compatible product sets or campaigns. Inequality 22 prevents searching for them after finding [1,5,6]. In this way. the number of ILPs to be solved is reduced drastically. Intermediate P r o d u c t s Two types of compounds can be manufactured in a chemical batch plant, i.e., salable and intermediate products (Rich and Prokopakis, 1986). Intermediates are those chemical species used in the production of other intermediate or final products. Indeed, the consideration of intermediates does not introduce major modifications in the proposed mathematical formulation. However, it can have a great impact on the optimal problem solution. The handling of intermediates implies accounting for further problem constraints, which somewhat shrink the space of feasible multiproduct campaigns. If a single campaign per product is allowed, none of the feasible production runs can then carry out the manufacturing of a product and any of its intermediates simultaneously. They must be produced a t distinct campaigns. In addition, a product can be manufactured only if all of its intermediates are available. This brings about the notion of an optimal ordered sequence rather than a group of multiproduct campaigns. After finding the optimal solution, the campaigns producing the intermediates should be run first. If instead the required amount of a salable or intermediate product through running two or more multiproduct campaigns can be produced, the simultaneous production of both at some production run becomes feasible. How-
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 993 Table VI. Optimal Unit Sizes for Example 1 (Single Unit per Stage) Vjlsij uroduct
1
2
3
4
5
6
7
8
9
10
1036
PI
p2
744
p3
901
p, p5
1247
1101
772 1212
730
2289
p7
1063
539
Vj
3992
3041
426 615
615
732 7267
1403
6871
563
730
730
2590
424
p6
563
1281 872
1782
3527
5598
2549
4595
B; 1036 744 563 730 424 615 539
Z;
290 202 355 260 331 280 197
355521 obj value
ever, the maximum allowable production of each at those campaigns is set equal to l / r of the total production requirement, where r 3 2 is the minimum number of campaigns allowed per product. Otherwise, the production planning could eventually fail because of insufficient intermediate inventories. Assuming r = 2, campaigns producing salable and intermediate products can still be run but their productions are very bounded,
C ( t k / T L , m ) ~2,/2 k
V
m E [LlI
(23)
k€K,I
where Kil includes all the campaigns involving the simultaneous production of product Pi and its intermediate PI. A different inequality, (23), should be written for each product/intermediate pair. Of the r production runs allowed per product, only r - l can also yield some of its intermediates. But the amount of intermediates so produced is supposed to be consumed at subsequent campaigns. In order to illustrate the kind of changes caused by the presence of intermediates, it is temporarily assumed that a subset of the products listed in Table I is made up of intermediates. Only three are salable products, Le., P,, P4, and Pg. The manufacturing steps are given in Figure 1. Thus, P5 and P7 are intermediates of P,, while P1 is an intermediate of both P4 and Pg. In turn, Pz is the intermediate of P1. If a single production run per product is allowed, [1,2], [1,4], and [1,6] product pairs now become incompatible. Therefore, campaigns 8, 9, 11, and 16 are to be excluded from the feasible campaign set. If, instead, multiple production runs per product are permitted, the campaign subset Kil for the pair [Pl,Pg] contains the production runs 11 and 16. Overall, three linear constraints should be added to the problem formulation because no campaign produces, simultaneously, [3,5] or [3,7]. The generalization of eq 23 to account for the new feasible campaigns when parallel units are being run is straightforward.
Results The proposed nonlinear programming formulation has been applied to the solution of the multipurpose batch plant example problem first presented by Suhami and Mah (1982). The problem data are given in Tables I and 11. Processing sequences as well as time and size factors are provided in Table I, while Table I1 lists the production requirements (see example 1). When a single equipment unit per stage is available, a total of 16 different multiproduct campaigns can be run (see Table 111). Of these, only a few are really executed at the best production planning. The optimal series of campaigns, the proportion of time allocated to each production run, and the batch stage sizes are all simultaneously found by solving a nonlinear program involving 23 nonlinear and 8 linear constraints in 33 nonnegative real variables. The search for the optimum has been successfully made through the nonlinear optimization system MINOS 5.0 (Murtagh and
Figure 1. Manufacturing steps for the production of the salable products Pa, P,, and P6.
Saunders, 1983). No special effort to get a good starting point is really required as long as a fast convergence to the global optimum is achieved anyway. The optimal equipment sizes and number of batches for each product are shown in Table VI. There is full agreement with the previous results reported by Suhami and Mah (1982). Table VI also includes the maximum batch sizes of any product that can be processed in the different batch stages. There is always a t least one particular product determining the size of a given stage. Thus, product PI plays that role for stage R8(see Table VI). For such size-defining products (the “limiting” products), the equipment is used to full capacity, while it is partly empty for the others. Moreover, the corresponding production requirement constraint 4 is satisfied as a strict equality. The others are redundant, and their exclusion from the problem formulation would not affect the final outcome. In example 1, the jth-stage limiting products feature either the largest or the second largest (QiSij) among the products (Pi)processed in stage Rj. By use of this rule, a good initial (frequently feasible) solution can be found through solving a smaller nonlinear programming problem. The number of nonlinear constraints is reduced to 18 in example 1. The batch sizes for the limiting products are shown as italic in Table VI. When the number of stages exceeds the number of products to be manufactured, there are surely two or more stages having the same size-defining product. This is the case for stages R4 and R,. On the other hand, Figure 2 shows the optimal series of multiproduct campaigns for example 1 and the best allocation of the available production time among them. It is observed that the required amount of each chemical species has been achieved by running a single campaign. In other words, there is no pair of active campaigns featuring product sets with a common element (see Figure 2). For problems exhibiting this property, the max (QiS,) rule seems to work properly. Moreover, the optimal batch sizes and number of batches for products manufactured
994
Ind. Eng. Chem. Res., Vol. 28, No. 7 , 1989
Table VII. Optimal Solution for Example l b (Two Units Are Available in Stages R, and R,) VI IS,,
product PI p;
1
2
3
4
836 1012
p3
p4 p5
836
462
115;
,347
L’,
4348
1955
489 1736
499
1135 359
1941
3842
2304
2761
4070
426 836 499 489 462 670 347
2,
704 180 40 1 389 303 257 306
387868 obj value
Compo Qn
Compaign X 12
Compoigr
Compoign
CamDoign
4604
836 610
6 70
B,
10
462
6 70
940
9
795
491 8161
8 426
1199
2571
p:
7
499 489
P6
6
5
Compoign
X 13 Rll434
w44
5
P4
F7 i
0
(44C
4041
6200
Time ( h ) Figure 2. Optimal production schedule for example l a (a single unit
per stage).
at a given campaign k should satisfy the following relationships:
(24) where C k is the product set for the active campaign k and (QLl,Q12)are the required amounts of products Pll and PIP Usually, the optimal batch sizes are rather similar. But this optimal trend is counteracted by restrictions like eq 24 unless the production of some products a t campaign k can be fixed by the model. This becomes possible by distributing the required production of certain compounds among several campaigns as shown later. A simple reasoning can lead to the optimal set of multiproduct campaigns for example l. After the most efficient campaign, [2,7] (see Table 111),has been run, there remains five additional products to be manufactured. Since the required amount of P, is already available, campaign ranked second in time efficiency, [6,7],should be discarded. Therefore, campaigns [1,5,6] and [3,4] should be run. In this way, one ends up with the optimal series of campaigns. Frequently, however, the production needs of a particular compound are obtained through several production runs. If so, the identification of the best campaign set is no longer straightforward. This kind of situation is generated by simply introducing some perturbations on the production requirements of some products. As mentioned before, this especially happens when the values of Q , T L , I for the set of products processing at efficient campaigns are quite different. Figure 3 provides a picture of the time utilization of the different stages for example 1. None of the stages has a null idle time over the entire production period. Therefore, neither one can be regarded as the limiting stage of the batch facility. For this reason, the incorporation of parallel units running out-of-phase in some stages to produce the same product does not seem a t first sight economically attractive. From Figure 3, it is observed that stages R, and R8 present the lowest idle times. They stay idle a percentage of the available production period, as small as
1440
62CO
4041
Time ( h )
Figure 3. Stage idle times for example l a .
12.6% and 29.5’30, respectively. As discussed before, however, the use of parallel units in some stages could bring about new feasible multiproduct campaigns by simply assigning them to the manufacture of distinct products. By doing so, the time utilization of the plant may improve as long as the number of chemical species simultaneously processed increases. Then, there is still a chance to reach a better solution by operating two-unit modules in some carefully chosen stages. A good decision always favors the running of feasible campaigns featuring lower equipment idle times. One should operate parallel units where a significant number of such efficient campaigns is brought about. Accounting for that criterion and the time utilization values shown in Figure 3, stages R5 and R8were selected. Table IV lists the new feasible multiproduct campaigns that can be run if the parallel equipments are allotted to no longer incompatible products. In addition to them, the proposed model can still handle the possibility of assigning a two-unit module to the production of the same product. In order to tackle the new problem, some minor modifications to the nonlinear mathematical program solved before should be made. First, the number of real variables rises to 44 due to the new 11 production runs. On the contrary, there is no change in the number of constraints. Only additional terms related to the new xk’s are to be incorporated in restrictions 1-3. Moreover, an appropriate modification of the old x h coefficients should be made to consider the decrease on the limiting times, TL,:, for products PI and P4. These are just the products whose corresponding limiting stages, R, and R,, have been duplicated. When two-unit modules are running in stages R, and R8, the optimal stage sizes and number of batches are those given in Table VII. A rise of 9% is observed in the objective value, mostly caused by the type of cost function used by the model. The fact that the problem does not exhibit a limiting stage and that quite efficient campaigns are already available (see Table 111) has made the operation
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 995 Table VIII. Optimal Number of Batches and Production of Each Product per Campaign for Example l b campaign 13 16 22 product PI
2~,13
Qi.13
zi,16
QiJ6
zi,22
Qi,22
532
226 700
172 180
73 300 150 000
p2
270 293
p3
p4 p5
134 700 143 100 303 257
p6
21
zi,27
Q i p
131 96
65 300 46 900
131
95 300
140 000 172 000
P,
175
60 700
Table IX. ODtimal Unit Sizes for Examole 2a (Single Unit Der Stage)
V,/% product
1
2
4
5
6
7
8
9
B,
10
P2
1394 1689
p3
3070
2710
1797
1797
1689
1279
4291
p7
1804
1804
V,
6776
10173
61 7
1516
13620
3454 Campaign
3843 3309
1804
Campaign
1689
1797
6378
719
p6
Campaign
2315 4597
16920
1787
8686
21 239
7649
Campaign X 13
Campaign
13 786
z.
3930 1394 1689 1797 719 617 1804
3930
PI
p4 p5
3
204 108 415 328 195 279 225
616 362 obj value Campaipn # 22
Campaign # 16
Campagn # 27
IT(%)
P4
R l 13321
P2
R 2 163 91
P3
R314591
P4
A417741
P5
R5 (19 7 I
P6
R5 I65 81 R6 I91 7 )
P?
R7(2351
00
1977
5242
3960
6200
Time( h )
RB(3081
Figure 4. Optimal production schedule for example l b (two parallel units in stages R5 and RBI.
of parallel units less attractive. The results reported by Suhami and Mah (1982) already led to that conclusion. Table VI1 indicates that the assertion is still true even though the parallel units are assigned to the production of distinct products in example 1. Things may be different, however, if the increase of Tijwith Biis taken into account. The optimal series of campaigns is illustrated in Figure 4. Three out of four production runs involve the simultaneous processing of three products. Moreover, the required amounts of PI, p,, P4and P7 are obtained by running for each a pair of campaigns. Still, the limiting products are those ones featuring the two largest (QISij ) in each stage. Table VI11 gives both the number of batches and the production of each product per campaign at the optimum. The use of parallel units working out-of-phase in the limiting stages for PI and P4 significantly increases the number of batches of such chemical species processed during campaigns [3,4] and [1,5,6]. Indeed, an important fraction of the production time is dedicated to such production runs. Figure 5 depicts the time utilization of the different batch stages. The average idle time per stage has experienced little variation. There is a close dependency between the optimal sequence of campaigns and the production requirements due to restrictions like eq 24. To illustrate this point, a second example is presented where the production needs for some products have been increased. The new values are given in Table I1 (see example 2). Instead, there is no change in the time and size factors (see Table I). The dimension
R8 135 21 R 9 126 6 )
1977
00
p-4
,
5242
3960
R1015831
6200
Time (h)
Figure 5. Equipment idle times for example l b . Campaign #9
P1 P2
P3 P4
P5 P6 P7
0 246
4014
4047
4925
6200
Time(h)
Figure 6. Optimal production schedule for example 2a (a single unit per stage).
of the mathematical model remain the same. When a single unit is running per stage, the optimal solution is shown in Table IX and Figure 6. TWO additional campaigns have been incorporated to the optimal sequence found in example 1. Nevertheless, the allocated time and the level of production at those campaigns are rather low. Almost 82% of the production time is assigned to the triplet of campaigns, [12, 13, 161, favored in example 1.
996 Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 Table X. Optimal Unit Sizes for Example 2b (Two Unit Modules in Stages RSand R,)
ViISi, product
1
2
3
4
5
6
7
8
p* 1220 1478
p2 p3
2033
1795 4223
691
p7
1733
1070
Vj
6510
6038
1004
3363
1004
1194 11919
2287
1478
1119
3755
p6
1478
1190
1190
1070
11205
2906
5752
6870
6693
Bi
10
2026 1487
1190
p4
P6
9
1271
12064
ti
1271 1220 1478 1190 691 1004 1070
629 123 474 496 203 171 379
619953 obj value
Table XI. Optimal Equipment Sizes When Some of the Products Are Intermediates in Example 1
VIIS, product p, p2
1
2
4
5
6
7
8
9
10
1069
676
1286
1135
924
819
p3 p4 p5
3
1250 753
753
674
2671
510
2080
p6
p7
1279
654
v,
4807
3686
510
635
635
1534 900
755 6602
1447
7086
674
753
1838
3638
5775
3052
5501
Bl
21
1069 676 674 753 510 635 654
281 222 297 252 274 271 162
371549 obj value Campaign X15
2097
O
L
633
3421
4198
Time ( h 1
P1
P1
P2
P2
P3
P3
P4
P4
P5
P5
P6
P6
P7
P7
6200
Figure 7. Optimal production schedule for example 2b (two parallel units in stages R5and Rs).
The new campaigns intend to distribute the higher production requirements of p,, P4, and P7between two production runs. In this way, batch sizes of products manufactured during a particular campaign b,ecome less different. Still, B, is significantly larger. When two-unit modules are running in stages R5and R8,the optimal batch sizes look more similar. Moreover, the batch facility now involves equipment of much smaller size. The optimal solution is presented in Table X and Figure 7. There is a clear reduction of the overall size requirements. Despite that, the objective value for this option is slightly higher. The production scheduling has incoporated two additional production runs with respect to example 1. However, the four multiproduct campaigns favored in both cases account for 90% of the overall production time. Because of the relative large quantities of P3,P4,and P7needed, campaign 27 is the one taking the largest time period. During campaign 27, the pair of units operating in stage R5 are assigned products P3 and P,, respectively. Furthermore, the QSrule works perfectly. Finally, an example involving some intermediates is studied. It is based on example 1but assumes that only P3, P4,and P, are salable products. The manufacturing steps are depicted in Figure 1. Supplies of PI for the production of P4and P8 are proportional to the requirements of these salable products. As explained before, the handling of intermediates makes it necessary to include three additional linear constraints in the mathematical
0
886
1587
1937
298
3726 4027
Time ( h )
6200
Figure 8. Optimal series of campaigns (example 1 involving intermediates).
Production 0 1
I
250
Lo51
Stock P I the End 01
T
0
-
200
P
(z
z
c9
150
too
50
0
Figure 9. Evolution of the intermediate inventories along the production period (example 1 involving intermediates).
modeling. After the optimal solution given in Table XI is found, the sequence of active campaigns is appropriately ordered so that the intermediates are obtained first. The new optimal series of campaigns is shown in Figure 8. It includes seven different production runs instead of three. The variation of the. intermediate inventories over the production period is depicted in Figure 9. Production of any intermediate is assumed to be available for use just a t the end of each campaign. It is observed that the intermediate stock levels are never driven to zero after
,
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 997 running a campaign producing simultaneously a salable product and its intermediates. This proves the feasibility of the proposed production schedule. The intermediate constraints have increased the optimal objective value by a reasonable 4.5%. Although the results are not reported, the option of prohibiting every campaign that produces at the same time a salable product and its intermediates was also studied. It is important to mention, however, that the new values of the product batch sizes become much closer. A similar trend is observed in the number of batches of the different products. Moreover, the best sequence of campaigns includes production runs 12, 10, 14, 15, and 13. Not only does the optimal series of campaigns change, the objective value also rises to 12.8%.
Conclusions An efficient algorithmic technique for determining the optimal production schedule and equipment sizes for a multipurpose batch plant has been developed. It consists, in the solution, of a low-size nonlinear programming problem whose feasible region includes all possible multiproduct campaigns. Before deriving the mathematical modeling, therefore, the whole set of feasible campaigns is found through either the notion of compatible product sets introduced by Suhami and Mah (1982) or solving integer linear problems. This is an affordable task even for practical-size problems. The mathematical formulation can be easily generalized to consider either the operation of parallel units in some stages or the production of intermediates together with salable products. Application of the proposed method to a couple of 10stage/7-product examples yielded the optimal values a t a small computational cost. The efficiency of the technique is still good even if the starting point is far from the optimum. Analysis of the optimal results indicates that the limiting products defining the size of a certain batch stage generally feature the largest (sometimes the second largest) value of the parameter QS (production times the size factor) for that stage. Such a finding permits us to discard a significant number of nonlinear constraints and still frequently find the optimal solution. It is a matter of fact that the production runs featuring lower equipment idle times are favored a t the optimum. By performing them, the time utilization of the plant is greatly improved. A similar reason leads to the conclusion that efficient campaigns manufacturing products demanded in larger proportions are first selected. Application of these simple ideas generally yields a near-optimal set of production runs. The required amount of each product is often yielded by running a single campaign. However, this is not always true. When the parameters QTL(production times the limiting time) for a compatible group of products are sensibly different, they are produced through multiple campaigns. In this way, the optimal trend of using rather similar product batch sizes is better achieved. Generally, there is no limiting stage featuring zero idle time over the production period in a multipurpose batch plant. This is why the results systematically indicate that the operation of parallel units all dedicated to the same product is not recommended when the minimization of the capital cost is the problem goal. Even so, the allocation of parallel units to different products arises as an interesting mode of operation since the number of feasible multiproduct campaigns is significantly increased. The incorporation of the most efficient new ones in the selected strategy brings about a larger number of batches per product and smaller batch sizes. Nevertheless, the ob-
jective value may slightly increase because of the high number of units being run. T o handle the manufacturing of intermediate and salable products, additional linear constraints are to be added to the problem formulation. They preclude the possibility of failure in the manufacturing of the required amounts of salable products for the lack of intermediates. Even so, the use of multiproduct campaigns producing compatible salable and intermediate products is permitted within certain limits. Since the size of the problem remains rather low, the proposed technique still maintains a remarkable computational efficiency. An example involving four intermediates and three salable products is solved. From the results, it follows that the intermediate constraints may drastically change the production policy. It usually includes a higher number of shorter production runs in such a way that the rise on the optimal objective value is quite reasonable.
Acknowledgment This work has been carried out under support provided by Consejo Nacional de Investigaciones Cientificas y TBcnicas (CONICET) of Argentina, Universidad Nacional del Litoral (Santa Fe, Argentina), and Universidad de Barcelona (Spain).
Nomenclature B = batch size C = product set H = available production time I = product set J = stage set K = campaign set M = number of batch stages in the plant n = number of parallel units in a certain stage N = number of products to be manufactured P = product Q = production requirement R = batch stage S = size factor t = campaign cycle time TL = limiting time V = batch stage size w = binary variable x = number of batches per campaign y = binary variable z = total number of batches of a product over the production period Subscripts i = product i j = batch stage j h = campaign k Greek Letters a = cost coefficient B = cost coefficient
Literature Cited Grossmann, I. E.; Sargent, R. W. H. Optimum Design of Multipurpose Chemical Plants. Znd. Eng. Chem. Process Des. Deu. 1979, 18, 343-348. Knopf, F. C.; Okos, M. R.; Reklaitis, G. V. Optimal Design of Batch/Semicontinuous Processes. Znd. Eng. Chem. Process Des. Deu. 1982, 21, 79-86. Mauderli, A.; Rippin, D. W. T. Production Planning and Scheduling for Multipurpose Batch Chemical Plants. Comput. Chem. Eng. 1979, 3, 199-206.
Mauderli, A.; Rippin, D. W. T. Scheduling Production in Multipurpose Batch Plants: The Batchman Program. Chem. Eng. Prog. 1980, 76, 37-45.
I n d . E n g . Chem. Res. 1989, 28, 998-1003
998
Murtagh, B. A.; Saunders, M. A. Minos 5.0 User’s Guide. Technical Report Sol 83-20, Dec 1983; Stanford University. Rich, S. H.; Prokopakis, G. J. Scheduling and Sequencing of Batch Operations in a Multipurpose Plant. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 979-988. Sparrow, R. E.; Forder, G. J.; Rippin, D. W. T. The Choice of Equipment Sizes for Multiproduct Batch Plants. Heuristic vs Branch and Bound. Ind. Eng. Chem. Process Des. Dev. 1975,14, 197-203.
Suhami, I.; Mah, R. S. H. Optimal Design of Multipurpose Batch Plants. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 94-100. Vaselenak, J. A.; Grossmann, I. E.; Westerberg, A. W. Optimal Retrofit Design of Multiproduct Batch Plants. Ind. Eng. Chem. Res. 1987, 26, 718-726. Receiued for review September 1, 1988 Revised manuscript received February 21, 1989 Accepted March 9, 1989
Use of the Dimensional Relation Approach in a Computerized Model for the Simulation of a Batch Production Plant Beno Zaidman Casali I n s t i t u t e of Applied Chemistry, School of Applied Science a n d Technology, T h e Hebrew University of Jerusalem, 91904 Jerusalem, Israel
In order to facilitate an early evaluation of R & D projects, a computerized model which simulates a multistage batch process in the chemical industry has been produced. T h e costing of feedstocks and the sizing of stage reactors is done from input stoichiometric data. Dimensional relations between the stage reactors and the auxiliary equipment items are used in order to size the equipment. T h e costing of equipment items is based on established cost-dimension relations, and the investment estimate is made by the factorial method. Labor costs are calculated analytically. Other cost components are determined by built-in relations. T h e model uses a limited number of inputs that can be obtained even at the very beginning of a research operation, and i t generates as output the technical and economic information needed to evaluate the proposed process. Thus, with a n acceptable reliability (*25%), the model yields an assessment of the economics of the proposed process and can perform a n analysis of the respective influence of various parameters. The growing use of specialty chemicals with complex molecular structure, obtainable by multistage synthesis, has increased the interest in batch manufacturing processes. Literature on the planning and optimization of industrial batch processes is relatively abundant, especially in the area of multipurpose plants (Vaselenak 1987a,b; Kuriyan 1987), but no general simulation model has been proposed, at least as far as we know. A model for the simulation of an industrial multistage batch process has been built. The aim of this simulation is to provide technoeconomic data for the evaluation of R & D projects at their early stage of implementation. As such, the model uses the limited input information that can be obtained even at the very beginning of the research. (A complete list of the input information to be furnished by the user appears in the supplementary material.) This basic constraint of the model imposes the use of adequate, simple formulas in order to perform the calculation of the layouts of an industrial batch process, which are needed for the estimation of the potential profitability of the research projects. The use of such shortcut formulas is especially required when the estimation of the investment value-therefore sizing and costing of equipment items-is sought. The empirically determined shortcut formula used for the estimation of the investment in a continuous process (Zevnick and Buchanan, 1963; Allen, 1975) cannot be applied with good results to batch operations. It has been decided to tackle this problem by adopting an analytical approach that deals separately with each stage of operation. This approach has been named the dimensional relation approach, and it is based on the following assumptions: A batch process for the manufacture of a chemical product is composed of a number of consecutive production stages. In each stage, an intermediate product is obtained which is used as a raw material in the following stage.
In each production stage, the operation is performed in a batch stirred tank reactor with auxiliary standard equipment items. The dimensions of the reactors in each stage can be established according to stoichiometric and operational data obtained from input information or compiled in a previous stage. Quantitative relations can be established between one of the physical dimensions of the stage reactor and a functional dimension of the auxiliary equipment item. The delivered cost of the equipment items can be expressed as a relation connected to the functional dimensions. The application of the dimensional relation approach imposes the use of different algorithms and numerical factors, established in an empirical way. In the computerized simulation model, those relations that in our opinion, represent the state of the art, at the present time, have been introduced. Those relations can be updated, remaining in the frame of the general approach.
Determination of Quantities and Cost of Feedstocks A subroutine calculates the quantities and cost of the feedstocks needed for the manufacture of the desired product. The method is similar to one published (Silver and Bacher, 1969) and uses the following concepts: An intermediate is the product of a step of the manufacturing process and is related to the final product by a stoichiometric ratio SI and by a yield YT. A reactant is a feedstock related to an intermediate by a stoichiometric ratio SR and by a yield YI. An auxiliary is a feedstock related to an intermediate by a weight ratio AR. The quantity RM(1,NS) of a reactant (I) needed for the manufacture of the intermediate (NS) is evaluated with the relation
0888-5885/89/2628-0998$01.50/0 0 1989 American Chemical Society
