Ind. Eng. Chem. Res. 1993,32, 1383-1395
1383
Heuristic Synthesis in the Design of Noncontinuous Multiproduct Plants A n i l k u m a r N. Pate1 and Richard S . H.Mah’ Department of Chemical Engineering, Northwestern University, Evamton, Illinois 60208-3120 The preliminary design problem for multiproduct plants comprises of two subproblems: the synthesis subproblem and the sizing subproblem. While the sizing subproblem has been studied extensively, research work on the synthesis subproblem has been piecemeal. This work reports development of a procedure that addresses the two subproblems simultaneously. The procedure utilizes a number of newly developed heuristics which are based on physical understanding of noncontinuous processes. The implementation of these heuristics for multiproduct plants is carried out in three steps. In the first step, heuristics for each design option are analytically developed for single-product plants. The second step involves defining the multiproduct plant as a single hypothetical product plant. In the final step, we apply the single-productheuristics t o the hypothetical product plant. The performance of these heuristics in an integrated form has been evaluated on a number of test problems. Inclusion of the new heuristics enables the design procedure to solve larger, more general, and more realistic problems without requiring prohibitive computational effort. Introduction In recent years batch processing has received increased attention primarily due to its dominant use in the production of high-value-added,low-volumeproducts. The batch and semicontinuous plants (hereafter referred to as noncontinuous plants) are particularly suitable for industries producing multiple products with seasonal demands, periodic operation, and variable due dates. Another unique feature of noncontinuous plants is that often only one piece of equipment is used to accomplish several operations, resulting in lower capital investment. Due to their flexibility, versatility, and ease of scale up, noncontinuous plants are ideal for manufacturing pharmaceuticals, polymers, speciality chemicals, and food products. A noncontinuous plant producing multiple products can be classified as either a multiproduct plant or a multipurpose plant. Multiproduct plants are those in which all products follow essentially the same processing route, and typically at any given time only one product is produced. On the other hand, in a multipurpose plant each product may have one or more distinct processing routes. For a more extended discussion, please refer to Mah (1990). The design problem of a multiproduct plant can be decomposed into two subproblems: (i) the synthesis subproblem, which addresses the organization of processing steps into a task network and generates an appropriate equipment configuration for that network, and (ii) the sizing subproblem, which determines the sizes of batch units and the processing rates of the semicontinuous (SC) units in the configuration. Even though for computational expediency these two subproblems are frequently treated as forming two different levels of a hierarchical decision process, they are, in reality, quite interrelated and hence cannot be approached separately. The review of previous research in this field establishes that most of it has avoided the synthesis subproblem by assuming that the equipment configuration was given (see Appendix A). The few that dealt with the synthesis subproblem applied one or more of the following simplifying assumptions: (i) The presence and location of intermediate storage are given. (ii) Neither task splitting nor task merging is permitted. (iii) Only out-of-phase mode of operation is allowed. Use of intermediate storage, mixed mode of operation for parallel units, and task splitting and merging are very important design options and essential for solving real and complex problems.
However, the above assumptions restrict the designer from using these options. The reason for these restrictions lies in the methodology that most researchers use to solve the design problem. They formulate the design problem as a mixed integer nonlinear programming (MINLP) problem and employ mathematical programing methods to solve the MINLP formulation. There are several limitations of these methods. First of all,the mathematical programing methods require that all variables be numerical. Thus, any option that deals with the structural choice of the plant, e.g., assignment of a certain task to an equipment item, has to be represented as an integer variable. As a result, the number of integer variables in a formulation for a realistic problem involving several synthesis options will be very high. However, the current state of the art limits the MINLP formulation to about 100 integer variables. To satisfy this requirement, one is forced either to make simplifications in the problem definition or to make the problem less general (more specific). Second, the MINLP models are not comprehensive-different design options are covered by different models. Thus, one has to search for a suitable model every time the problem options change. Third, since these models do not incorporate all of the design options available, they give inferior design. Finally, even if a formulation incorporating all of the design options is developed and solved, the global optimum solution is not guaranteed. In this work we present a method for solving the synthesis subproblem without any of the above restrictions. The structure of the process network is not assumed in advance. The method utilizes a heuristic sizing procedure for solving the sizing subproblem and newly developed heuristics for solving the synthesis subproblem. Specifically, we have developed heuristics for storage insertion and location, parallel unit addition, and task merging and splitting. Sizing Subproblem Since the evaluation of a synthesis option requires the cost of the plant to be known, it is necessary to solve the sizing subproblem before a particular synthesis option is considered. We will, therefore, address the sizing subproblem first. Features of a Multiproduct Plant. A product recipe is a list of elementary tasks to be performed in the order specified. An elementary task is one that cannot be split
Q888-5885/93/2632-1383~Q4.QQlQ0 1993 American Chemical Society
1384 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 any further. However, a number of elementary tasks can be merged to form a composite task, A plant is comprised of a series of stages with or without intermediate storage. A stage consists of a set of groups, and a group consists of aset of equipment items (hereafter referred to as units) of the same type such that all units in a group operate in-phase and all groups in a stage operate out-of-phase with each other. The actiue production time of an equipment is defined as the total time for which the equipment is used during the horizon. Thus, the idle time of an equipment is the difference between the horizon and the active production time, The actiue capacity of a group is defined as the sum of the ratios of the actual group size utilized to the group capacity for all products. The set of stages between any two successive intermediate storages is referred to as a subprocess. If an intermediate storage separates two stages, then the two stages belong to two different subprocesses, each having its own limiting cycle time (LCT). A path is a set of groups, each taken from a stage, arranged in an order consistent with a product recipe. The batch size of a product is the amount of final product produced in one batch. The productiuity of a product in a subprocess is defined as the ratio of the batch size to the limiting cycle time. In a feasible design, each subprocess meets the horizon constraint and the productivities of all of the subprocesses must be equal for each product, since the intermediate storage only decouples the subprocesses and does not serve as a long-term storage. Assumptions. The following assumptions are made in this study: (1)The treatment is limited to serial multiproduct plants only. (2) Each equipment unit is used only once in a product recipe. (3) All groups within a stage and all units within a group are identical. (4) Intermediate storage is used only to decouple the upstream and downstream operations. It does not act as a long-term storage. (5) A batch is processed only once in each stage. (6) Only consecutive batch tasks can be merged to form a composite task. When two (or more) consecutive batch tasks are merged to form a composite task, the SC tasks in between are assumed to be performed by the batch stage which performs the composite task. (7) The parameters for the equipment which performs two tasks together (merged) are determined as the maximum values of the individual tasks. Thus, for example, if SIand SZare the size factors and a1 and a2 are the cost coefficients for tasks 1and 2, respectively, and if SI > SZand a2 > a l , then the equipment performing the merged tasks will have the size factor S1and cost efficient az.
Model Constraints. Let Cjg be the group capacity of group g of stage j . Then Cjg = max{BiSiJ i
g Ej , V i
(1)
where Bi is the batch size for product i and Sit is the size factor for a stage which performs task t of product i. Note that t can be either an elementary task or a composite task. Due to assumption 3, all groups in a stage have the same group capacity and thus we may refer to C j g as Cj for brevity hereafter. Let rjg indicate the total number of units in group g of stage j . Again due to assumption 3, all groups in a stage
have the same number of units and hence we may refer to r j g as rj. Then the size of a unit r, V,, can be determined as
Vr = C j h j r E g , g E j (2) The processing time on batch stage j for product i , Pij, is given by
qj
(3) Pij = + cipp The transfer time between stages j and j’ for product i, Bijj’, is given by
where l j f is the SC subtrain connecting batch stage j to j’ and k is a SC stage belonging to this subtrain. The limiting cycle time of product i,
e,is given by
where mj is the number of groups in stage j . The time required for meeting the specified demand, Qi, of product i is denoted Ti and can be determined as
Ti = Q i c l B i The design is feasible if it does not violate the horizon constraint represented below for any of the subprocesses
HLCT,
(7)
I
where H is the total available time (horizon). The size of the storage in such a plant is computed on the basis of the equation (Karimi and Reklaitis, 1983,1985)
c = max{SLIBi(l- x i ) + B’i(l - x’l)l)
Vs
(8)
i
where xi and x’i represent the fraction of the limiting cycle time used for filling the tank and for emptying the tank, respectively. Finally, the cost of the plant is computed as ri
8
The sizing procedure described below sets the initial and subsequent productivities of each product in such a way that feasibility is always maintained (eq 7 is always satisfied). An iterative procedure is used to compute the batch size of each product which satisfies eq 5 as well as the specified productivity. Next, group capacities and equipment sizes are computed using eqs 1and 2, respectively. Finally, the storage sizes and cost of the plant are computed using eqs 8 and 9, respectively. Sizing Procedure. Modi and Karimi (1989)presented the only heuristic sizing procedure that allows for intermediate storage. It gives fairly good results with low computational requirement. Since their procedure deals only with out-of-phase mode of operation and is applicable only for a single storage location, we cannot use that procedure directly. Therefore, the procedure was modified to overcome these difficulties. The modified procedure is shown in Figure 1. The sizing of the plant is carried out in two steps. First, we determine the best solution for an initial set of productivities. The initial productivities for
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1385
9 START
Set lnitlai productlvles for all products. Set all SC pmoasslng rates to maxlmum.
I
other words, no unit has either idle capacity or idle time. This, however, is almost impossible because of the differences in the size factors as well as the processing times of individual stages for the same product and of different products for the same stage. Thus, in the optimum design the idle capacity and idle time are reduced to the minimum possible limit. This is done by improving the size (capacity) and time utilization of units. To achieve these objectives,a designer has the followingdesign options available: (i) intermediate storage insertion, (ii) task merging or splitting, and (iii) parallel units with mixed mode of operation. In this work, the implementation of the heuristics for each of these options for multiproduct plants is carried out in three steps. In the first step, heuristics for each design option are analytically developed for single-product plants. The second step involvesdefining the multiproduct plant as a single hypothetical product plant. In the final step, we apply the single-product heuristics to the hypothetical product plant. The solution is then used for the synthesis subproblem of the multiproduct plants. The details for these steps are described in the following sections. Heuristics for Single-Product Plants. Since this section deals only with a single product, the index i for the product has been dropped in the following discussion. Thus,B indicates the batch size of the product, Sjindicates the size factor for stage j , and so on. (a) Storage Insertion. When intermediate storage is inserted in a subprocess, the original subprocess is divided into two subprocesses. One of these two subprocesses will have a smaller limiting cycle time than the other (which has the same limiting cycle time as the original subprocess). We will refer to the subprocess having the lower limiting cycle time as the new subprocess, NS, and the subprocess retaining the same limiting cycle time as the original subprocess as the old subprocess, OS. Because the productivities of the two subprocesses have to be the same, the new subprocess has lower batch size, which results in lower equipment sizes and savings. Let N~COfYI"I'ld and NSCOST,,, be the total cost of equipment in the new subprocess before and after storage, respectively. If we assume that the sizes of SC units do not have to change as aresult of storage insertion, then the savings in the new subprocess due to storage insertion can be estimated as
, 1
Select the parallel unit location that yields maximum savlngs. I -
II
Determinethe best values for SC processing rates using one dimensional
better set of productivities. !Resizethe plant.
Figure 1. Heuristic sizing procedure. all products are determined as the ratio of total production requirement of all products to the available production time (horizon). Next, we carry out a series of singlevariable line searches alternately on the productivity ratios and the processing rates, respectively. The line searches to obtain an improved set of productivities are carried out in two stages, starting with the product having largest production requirement. The first stage obtains bounds for the optimum productivity. The second stage uses a one-dimensional search between these bounds to achieve a productivity that yields lowest cost results. For SC units, the bounds for the processing rates are already known. Therefore, only the one-dimensional searches are needed for determining the best set of processing rates. These searches are carried out for all SC units starting from the most expensive SC unit. These steps are similar to the work of Modi and Karimi (1989). The major modifications introduced by us in the sizing procedure relate to the way parallel units are handled. In the original work, the decision to add a parallel unit to a stage is made whenever the required size of any batch equipment is higher than the upper bound. The location of the new parallel unit is decided by temporarily increasing the number of units at each of the time-limiting stages, followed by sizing and costing the plant, and selecting the stage that resulted in a lower cost. The sizing and costing of the whole plant for each time-limiting stage require a considerable computational effort. In addition, the original procedure is applicable only to parallel units with out-of-phase mode of operation. Our modified procedure makes use of the heuristics developed for the synthesis subproblem to determine when and where a parallel unit is to be added and also to find out which mode of operation it should have. With these modifications we need to size and cost the plant only once, which saves us considerable computational work.
Synthesis Subproblem The ideal design for a batch process is one in which all of the units are utilized fully over the whole horizon. In
NSCOST,,,
=
&
u~(B,,$~)"~
j€ S
savings in new subprocess cost = NSCOSTold- NSCOST,,
where
On the other hand, additional cost of storage can be estimated as cost of storage = c*(S*)y'(Bold+ B , ~ ~ ) Y * (13)
If estimated savings is less than the estimated cost of
1386 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
storage, then storage is not justified. Thus, the storage insertion is not justified when
5;saj(Sp,,)"'(l
- p"')
iF
Because p, aj, and y* are each less than 1 (1- p"j)/(l
+ P ) ~ *< 1
V j E NS
(15)
Hence, storage is not justified if
(16) The numerator in this expression is equal to NSCOST,ld and the denominator represents STGCOST, the cost of the storage of size V* = S*Bold. This enables us to propose the following heuristic. HEURISTIC 1: I f cost of the storage, STGCOST, is more than the total cost of the new subprocess, NSCOSTold, then storage insertion is not justified. This heuristic is obvious because even after the storage is inserted, the new subprocess cost cannot go to zero. Thus, storage that costa as high as the current cost of the new subprocess has to be rejected. Normally, however, the storage cost is lower than the total cost of the new subprocess. In that case, the ratio (1- p"j)/(l + p ) y < 1 becomes important. Let ajm== max{aj);then we establish the upper bound on possible sab%gs as maximum savings in cost of new subprocess = NSCOSTo1d(1- p"'"")
(17)
If the cost of the storage is higher than the maximum savings computed as above, then the storage should be rejected. Thus, we propose the following. HEURISTIC 2 The storage insertion is not justified when
Similar analysis can be performed to determine when the storage insertion is justified. The following heuristic is based on that analysis. HEURISTIC 3 The storage insertion is justified when
where ajmin = mini..). As can be 6% #rom the above heuristics, storage insertion can be determined without carrying out a detailed computation for the plant for a new configuration. The only parameter to be calculated for new configuration is p. If processing times are constant, then this ratio can be calculated very easily. On the other hand, if processing times are dependent on batch sizes, then an iterative procedure may need to be used. In our implementation we avoid the computation completely by selecting the largest stage cycle time of the new subprocess as the new cycle time. This, of course, will overestimate the cost of the storage and underestimate the savings. However, the approximation yields an acceptable level of accuracy. We can achieve better resulb by actually computing new batch sizes. Even in such cases, the computational effort is less
than traditional sizing procedures because one does not have to compute sizes and cost of all equipment in the plant. The savings in computational effort are compounded even further when more complex design options (e.g., task merging and splitting and mixed mode of operation) are present. (b) Storage Location. Storage should be located such that maximum savings are achieved in the non-timelimiting subprocess (new subprocess). Thus, the new subprocess should have as many stages as possible. The following heuristic tries to locate the storage such that the number of stages in the new subprocess is maximum. HEURISTIC 4: Storage should be located either immediately before or immediately afterthe time-limiting stage. It is possible to envisage a case where the stage cycle times of the time-limiting stage and its adjacent stage does not differ significantly. In such a case, the reduction achieved in the limiting cycle time is not large enough to justify an intermediate storage. As a result, location suggested by this heuristic is rendered ineffective. Therefore, the following analysis is carried out to explore other storage locations. In this analysis we assume that the choice is to be made between location 1and location 2, and location 2 results in a lower limiting cycle time than location 1. We will also assume that the storage cost does not change as a result of new location. The rationale behind this assumption is that the batch size for the old subprocess remains the same and the difference between the batch sizes of the new subprocesses will introduce a relatively small cost savings of storage compared to the cost savings of batch stages. Let TObe the old limiting cycle time. Let NS1 be the new subprocess after insertion of the storage at location 1, having the new limiting cycle time TI. Similarly, let NS2 be the new subprocess after insertion of the storage at location 2, having the new limiting cycle time T2. Also, let NS3 be the set of stages between location 1and location 2 and the cost of each of these subprocesses before storage insertion be NSCOST1, NSCOST2, and NSCOSTs respectively. By definition, NS1 = NS2 U NS3 and NSCOSTl = NSCOST2 + NSCOST3. Now, define the ratios p1= Tl/To, p2 = TdTo, and p3 = T1/T2. When location 2 is chosen instead of location 1, the minimum incremental savings can be computed as (NSCOST,)(p;m)(l- p i m ) (20) where a j m h = min(aj). Also, the maximum potential loss ENS, is given by (NSCOST3)(1- p:-) (21) where ajmax= max(aj). Thus, we arrive a t the following heuristic. HEURISTIC 5: A storage location other than immediately before or immediately after the time-limiting stage should be considered i f NSCOST, NSCOST,
(pi-
-
- pyj""9
(1- ,,;i-.)
(22)
(c)Parallelunits and Their Mode of Operation. As mentioned earlier, the optimum design strives to improve the time utilization and the capacity utilization of the units in the plant. The time utilization can be improved by reducing the limiting cycle time, whereas the capacity utilization can be improved by increasing the batch size. The design option of adding parallel units is very important as it can serve one or both of these purposes. The selection
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1387 of number of parallel units has been included as part of the basic sizing procedure. However, the synthesis subproblem must answer the location and the mode of operation of the new unit. We propose the following heuristic for this purpose. HE URISTIC6 The new parallel unit should be added either to the time-limiting stage with out-of-phasemode of operation or to the capacity-limiting stage with inphase mode of operation. The synthesis problem must also resolve the question of which of these two options is better. In our current implementation we select the option which yields more savings. However, the following heuristic may be applied to resolve this conflict. HEURISTIC 7 I f the ratio of the two highest size factors of the stages in the subprocess is higher than the ratio of the two highest stage cycle times, then the new parallel unit should be added to the capacity-limiting stage; otherwise, the new parallel unit should be added to the time-limiting stage. A high ratio of size factors implies that the reduction achievable in the batch size can be bigger by adding an in-phase unit than that obtained by adding an out-ofphase unit. Thus, savings are likely to be more in the former option, and hence that option is preferred. (d) Task Merging and Splitting. Choice of a particular synthesis option for a subprocess does not affect another subprocess in the plant. Therefore, the discussion in this section is focused on a single subprocess. The heuristics developed can be applied to each subprocess individually. Since a design with more units is likely to be more expensive than a design with fewer units, we should always start the synthesis process with the fewest units. This implies that all mergeable tasks should be fully merged. However, merging may not always be advantageous as the following cases demonstrate: First, merging of two tasks may result in a higher limiting cycle time, leading to a bigger batch size and ultimately a more expensive design. Second, merging of a task having a higher size factor with a task having a higher cost coefficient may result in a merged unit with higher cost than the total cost of the two items each performing these tasks separately. In such cases, splitting is justified. Splitting can be carried out either at the time-limiting stage or at the nontime-limiting stage. Yeh and Reklaitis (1987) suggested that splitting at the non-time-limiting stage did not affect the structure of the plant and hence could be considered during the final refinement phase. But this is not always true. For instance, splitting at a non-time-limiting stage with parallel units may result in the reduction of the number of parallel units operating out-of-phase. This changes the structure of the plant and, therefore, should be considered during the solution of the synthesis subproblem. Accordingly, in our synthesis phase, we will consider splitting at the time-limiting stage as well as at the non-time-limiting stage with parallel units. Yeh and Reklaitis also proposed that the most appropriate stage for splitting was the time-limiting stage. The logic behind their proposition is that reduction in the limiting cycle time results in cost savings. This proposition, however, excludes the possibility of cost savings through reduction in the number of parallel units at a stage. When a parallel unit is removed from a non-time-limiting stage, it does not reduce the limiting cycle time. On the contrary, it may increase the limiting cycle time and thus may require higher equipment sizes for the other units. Nonetheless, it may still result in more savings than the savings
achievable by splitting a t a time-limiting stage. We therefore propose the following heuristic, which is more general than the one proposed by Yeh and Reklaitis. HEURISTIC 8: The most appropriate stage for splitting is either ( i ) the time-limiting stage or (ii) the one for which the product of group cycle time (ctj) and the number of groups operating out-of-phase (mi) is maximum. Note that when no parallel units are present in the plant, both of these criteria give the same result. Therefore, we will consider (ii) in only those cases where mj > 1. Once the candidates for splitting are decided, we need to know if splitting is worthwhile or not. The following analysis answers that question. Although the analysis is carried out for only two elementary tasks, it can be proven that the results are true for any number of tasks. Consider two tasks tl and t2, which can be performed by two units E1 and Et, respectively. Let SIand St be the size factors and al and a2 be the cost coefficients for the two items, respectively. Also, let a be the cost exponent for these two items. Finally, we will denote the total cost of the rest of the units as COSTrwt, the merge option with subscript m and the split option with subscript s. CASE I: Splitting at the Time-Limiting Stage. Since splitting is at the time-limiting stage, the new limiting cycle time will be lower than the current limiting cycle time. This implies& < B,md (COST,,t), < (C0STrmt)m. Let A = (COSTre,t)m- (COST,,t),. Then splitting is not justified if COST, > COST, (23) But the cost of the plant when the tasks are merged is given by (assumption 7) COST, = [max(al,a2)l[max(Sl,S2)Ia(Bz)+ (COSTIwt), (24) and the cost of the plant when the tasks are split is given by COST, = (a,S: + a2S3(B3+ (COST,,,,), Therefore, splitting is not justified if
(25)
+ a2S3(pBm)*> [max(al,a2)l[max(S,,S,)I*(Bz) + &,= (26) where p = B$Bm and A- equals the maximum possible savingsin the rest of the equipment items. I t can be proven (Appendix B) that
4, = (COSTmt),(l - pa")
(27)
where amax= max{aj) j # splitting stage
(28)
j
Now consider the case when a1 = max(a1,az) and SI = max(S1,Sz). Then eq 26 becomes a2S;(pB,)*
> a1(SlB,)*(1 - pa) + (COST,,t),(1
- pa")
(29)
or a2S3pB,)*
> COST,(l - p-)
(30)
where CY-
= max{aj)
Vj
j
= max[a,a,,l (31) Next, consider the case when a1 = rnax(a1,az) and SZ=
1388 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
max(S1,Sz). Then eq 26 becomes
am
= min(aj)
Vj
j
azS;(pB,)" > a,[(S,Bm)* - (S1pBm)"I + (COSTr,Jm(l or azS;(pB,)*
> a1(SJ3,)*(l -
- pa"")
(33)
( m- l)az(pS.J3,)" + malB:[(Slp)* - 5';l + (C0STrat),(pa* - 1)> al(pS1Bm)"(44) Also, since p > pm
+
(COSTr,Sm(l - pa-) where pm is the modified ratio defined as pm
(32)
= min[a,amin] (43) Finally, consider the case when a1 = rnax(a1,az) and SZ = max(S1,Sz). Then eq 38 becomes
size factor for the unit with higher cost coeff size factor for the merged unit )p
=(
> pm, we can write eq 33 as azS;(pB,)* > COSTm(l- p y
(34)
and since pS1 = PmSz
Also, since p
)
(35)
But the left-hand side of the inequalities in eqs 30 and 35 can be viewed as the minimum additional cost of the new unit. Similarly, the right-hand side can be viewed as the maximum possible savings. Clearly, if the additional cost is higher than the possible savings, then splitting should not be carried out. This leads us to the followingheuristic. HEURISTIC 9 Splitting at the time-limiting stage is not justified if the cost of the new unit is higher than the maximum possible savings. The cost of the new unit is determined on the basis of the task with the minimum cost coefficient and the corresponding size factor. The possible savings are computed as a product of the cost of the merged subprocess and the factor (1 - p?). CASE 11: Splittingat the Non-Time-LimitingStage with Parallel Units. The purpose of splitting at the non-time-limiting stage with parallel units is to reduce the parallel items by one. Due to this, the new limiting cycle time may be higher than the current limiting cycle time. This impliesB, 2 B m and (COSTr& 1 (COSTrmt)m. Let A = (COSTr,J, - (COSTr,t)rnThe cost of the plant when tasks are merged is given by COST, = m[max(al,az)l[max(Sl,Sz)la(B~) + (COST,,,,), (36) and the cost of the plant when tasks are split is given by COST, = ( m- l)(alS; + a,S;)(B,") + (COST,,,), Therefore, from eq 23 splitting is not justified if
(37)
(m- l)(alSy + azS,*)(pB,)* + Amin> m[max(al,az)l[max(Sl,Sz)Ia(B~)(38) where (39) amin = min(aj)
j # splitting stage
(40)
j
Now consider the case when a1 = max(a1,az) and SI= max(S1,Sz). Then eq 38 becomes
( m- l)a,(pS.J3,)*
+ mal(SIB,)"(pa - 1)+
Once again, the left-hand side of the inequalities in eqs 42 and 46 can be viewed as the minimum additional cost.
The first term represents the cost of the new units, and the second term represents the minimum incremental cost for the existing equipment. Similarly, the right-hand side can be viewed as maximum possible savings. Note that for the first case pm is equal to p. Therefore, eqs 42 and 46 are equivalent and provide a new heuristic for the case when splitting is being considered to remove a parallel unit. This heuristic is as follows. HEURISTIC 1 0 Splitting at the non-time-limiting stage with parallel units is not justified if the sum of the total cost of the new ( m - 1) units and the minimum incremental cost is higher than the possible savings. The cost of the new units is computed on the basis of the task with the minimum cost coefficientand the corresponding size factor. The minimum incremental cost for the existing equipment is estimated as the product of the cost of the merged subprocess and the factor ( p p - 1). The possible savings are determined as the product of the cost of one merged unit and the factor p i . Heuristics 9 and 10 help us to decide when splitting the task chain is definitely not recommended. Alternatively, if we are interested in determining when task splitting is definitelyjustified, we can carry out an analysis analogous to the above analysis. We present below the results of such analyses in the form of two more heuristics. HEURISTIC 11: Splitting at the time-limiting stage is justified if the cost of the new unit is lower than the minimum possible savings. The cost of the new unit is determined on the basis of the task with the minimum cost coefficient and the corresponding size factor. The minimum possible savings are computed as a product of the cost of the merged subprocess and the factor (1 Pa").
HEURISTIC 1 2 Splitting at the non-time-limiting stage with parallel units is justified if the sum of the total cost of the new ( m - 1) units and the maximum incremental cost is lower than the possible savings. The cost of the new units is computed on the basis of the task with the minimum cost coefficient and the corresponding size factor. The maximum incremental cost for the existing equipment is estimated as the product of the cost of the merged subprocess and the factor ( p w - 1). The possible savings are determined as the product of the cost of one merged unit and the factor p i . Although these heuristics are applicable in most cases, there are some cases when none of these heuristics may yield a definite solution. For instance, consider the case of splitting a task at a non-time-limiting stage. Let the
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1389
Determine the bed locaclon for adding
betennine Iand where task splltllq h Justilled.
Heurlstka8,@,11,15.
I
J
J
I
I
e
F
lnserl the storage at this kartlar.
l
.( Fa ea* subprocesscafrj w( baln design
Figure 2. Overall design procedure.
new unit cost be $10000, minimum and maximum incremental cost in existing equipment be $5000 and $9 OOO respectively, and the cost of one merged unit be $16 500. We cannot say either that splitting is not justified (as per heuristic 10)or that splitting is justified (as per heuristic 12). This situation arises because of the high difference in the power coefficients of the cost function of different equipment. If the difference is small, this situation will not occur. Whenever higher difference is present and the above heuristics cannot yield a definite solution, we have to make a decision based on more detailed cost computations. Representation of a Multiproduct Plant as a Hypothetical Product Plant. In the previous section we developed heuristics for single-product plants. Most of the heuristics were derived on the basis of mathematical analysis. A similar approach can be used for multiproduct plants too, but the analysis is very complicated and the heuristics become complex. On the other hand, it would save us a great deal of work if we could use the same heuristics-in either the same form or a small variation of it-for multiproduct plants. This would be possible if the multiproduct plant data can be equivalently represented as single-product plant data. Heuristics for creating such a single hypothetical product must be developed. This development is presented below. The basic concept was proposed by Sparrow et al. (1975). Their method for the design of a multiproduct plant was based upon defining a new hypothetical product. The parameters for such a product were derived from those of the actual products. Once these parameters were specified, production requirements of each product could be converted into the equivalent data for the hypothetical product. Then the plant was sized for the hypothetical product. We propose to use a similar approach of defining a hypothetical product for our multiproduct plant. Next, we propose to use the characteristics of the hypothetical product with the single-product heuristics developed previously to solve the synthesis subproblem for multiproduct plant. Although our technique is similar to that
-
Repolt nrul cod Cost1 end tho correapondlng design.
Figure 3. Train design procedure.
of Sparrow et al. (19751,there are significant differences. The most notable differences are as follows: (i) We are solving the synthesis subproblem and not the sizing problem. (ii) We assume the processing times to be a function of batch size. SC units are also included in the plant design. (iii) In the earlier work weightage is given only to production requirements. We use both production requirements and the limiting cycle time. For determining the characteristics of the hypothetical products, a batch size conversion factor, Fi, is defined as the ratio of the batch size of the hypothetical product to the batch size of the ith product. Thus
(47) To compute Fi,we first determine the initial size factors for each stage in a heuristic manner from weighted sums of the S i j . The weighting scheme used in our work is as follows.
This weighting scheme is different from the one used by Sparrow et al. (1975).They used only the production requirements for estimating the size factors for the hypothetical product. By omitting the limiting cycle time in the weighting scheme, they made the computation of Fi independent of This is reasonable when processing times are constant. However, when processing time is dependent on batch size, the effect of should be considered when Fi is computed. Accordingly,we proposed the weighting scheme of eq 48. Both of these schemes have been tried in our implementation, and for most cases no significant differences were observed in the results obtained from the two schemes. However, we found that for problems involving large variations in with changes in batch sizes our scheme yielded better results. Therefore, we used eq 48 as the weighting scheme. Next, individual values of Fj are computed.
e.
q
e
Vi
(49)
The horizon constraint for the multiproduct plant can be
1390 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
Table I. Data for Problems 1 and 2 [Adapted from Yeh and Reklaitie (1987)l
recipe" S or D
sc1
C
2.74 -
d a or b a or j3
250 0.40
Po
-
B1 2.74 15.0 0.0172 0.865 592 0.65
B2 2.74 5.0 0.0 0.0 392 0.65
Problem 1 Q=120000 H = 7 2 0 0 Z = l 300 I Vj I 2 4 0 0 300 I R, I1800 B3 sc2 B4 2.74 2.74 1.44 5.0 10.0 0.0 0.612e-5 0.0 2.00 392 200 582 0.65 0.85 0.39
sc3 1.44 210 0.62
B5 1.20 6.0 0.0364 0.823 1200 0.52
B6 1.20 4.0 0.0 0.0 1200 0.52
sc4 1.20 -
B6 1.20 15.0 0.0364 0.823 1200 0.52
sc4 1.20
-
370 0.22
Problem 2 Same as Problem 1Except for the Following Parameters recipe
sc1
S or D
2.74
Po C
d a or b a or fi
-
250 0.40
B1 2.74 3.0 0.0172 0.865 592 0.65
sc2 2.74
-
0.0 200 0.85
B2 1.44 12.0 0.612e-5 2.0 582 0.39
B3 1.44 8.0 0.0 0.0 582 0.39
sc3 1.44 -
210 0.62
B4 1.20 10.0 0.0 0.0 1200 0.52
B5 1.20 10.0 0.0 0.0 1200 0.52
-
370 0.22
0 In all of the tables the following notation is used: (i) SC indicates semicontinuous task, B indicates batch task, and T indicates intermediate storage. Storage T and ita downstream SC unit are optional. Asterisks denote parameters for storage. (ii) Location of storage is unknown, but the parameters for its downstream SC units are considered the same as the upstream SC unit. If there is no upstream SC unit, then the duty factor for the downstream SC unit is assumed to be le-5 and the cost is assumed to be zero. (iii) Processing time is given as pij = p i cij(&)&. (iv) Cost of batch unit r = a,(V,)'+. (iv) Cost of SC unit k = bk(Rk)k (v) Cost of storage s =
+
C~(C)~*~.
expressed as N
Substitution of eq 47 into eq 50 gives N
or
For a single-product plant, the optimum design will use the horizon fully. Therefore, the above inequality can be replaced by an equality. The resulting equation gives us the estimate of the batch size of the hypothetical product. For the problems involving constant processing times, is already known, whereas for the problems having batch size dependent processing times one needs to recompute the limiting cycle times using an iterative procedure. Since our aim is to reduce computational effort, we propose to use the current values for estimating the batch size of the hypothetical product. Our approach does not require determination of individual equipment sizes. However, if desired, they can be computed as follows. If for any stage the size requirement of all products is less than that of the hypothetical product, then the size factor for the hypothetical product can be reduced until it constrains one of the real products. Therefore, we correct S H as ~ (53) Finally, the size of the batch equipment is calculated. (54)
Heuristics for Multiproduct Plants. We now describe the synthesis heuristics for the multiproduct plant for each of the design options discussed earlier. (a) Storage Insertion. To decide if the storage insertion is justified or not, we propose to use heuristics
1-3 developed earlier. It may be noted that these heuristics mainly require parameters related to the equipment items and not to the products. Hence, their use for multiproduct plant design is justified. However, we do need p, the ratio of new and old batch sizes. One alternative is to compute the ratios for each product and then use the extreme values for predicting the maximum or the minimum savings. This, of course, would involve considerable computational effort and also would give quite conservative results because of the extreme values of p. We believe that the development of the heuristics itself is sufficiently conservative, and hence no further cushion needs to be built. Therefore, the idea of using extreme values of p is rejected. Instead, we propose to determine p for the hypothetical product and use it in heuristics 1-3. With this, we avoid computing the new batch size and the ratio p for each product. This saves substantial computational effort and also gives very good results as shown in the later section. (b) Storage Location. For the single-product plant we proposed to insert the storage tank either immediately preceding or immediately succeeding the time-limiting stage. In the multiproduct plant, however, each product may have a different time-limiting stage. We also do not have a time-limiting stage for the hypothetical product because the processing times for each stage are not weighted separately, and hence we do not have equivalent hypothetical processing time. We, therefore have to develop another heuristic which is suitable for the multiproduct plant and does not involve much computation. An obvious idea is to determine a stage which is timelimiting for a maximum number of products. Although this is a good heuristic, there are two deficiencies. The first is that there may be a tie between a number of stages which would require another heuristic to break the tie. The second is that the proposed heuristic gives equal weight to products having vastly different production requirements. Thus, a more refined heuristic that takes the production requirement into consideration is desirable. The active production time of a stage combines both the demand as well as the limiting cycle time of all products. That leads us to the following heuristic, which determines the storage location on the basis of the time utilization of a stage.
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1391 Table 11. Data for Problems 3 and 4 [Adapted from Ravemark and Rippin (1991)p Problem 3 Q = [94366,100540,121538] H = 6000
B1
recipe
S
B2 7.022 10.545 0.0 0.60
6.263 3.099 0.0 0.60
Po C
a or fi
B3 1.457 4.043
B4 6.722 17.577
B5 6.552 6.037
B6 5.257 6.430
0.0
0.0
0.0
0.0
0.60
0.60
0.60
0.60
!
A = {atj)=
I=3 B7 17.647 12.285 0.0 0.60
B8 10.269 4.232 0.0
0.60
B9 12.275 12.937 0.0 0.60
B10 10.504 9.812 0.0 0.60
1
250 150 180 230 450 250 200
1
260 490 270 550 250
Problem 4 Same as Problem 3 Except Storage Parameters: c: = 278, 7: = 0.49, S: = 1.0 Vi a uti
is the coefficient for equipment j when used to perform task t .
Table 111. Results for Problems 1 and 2 problem no. reported obj fn heuristics obj fn
Vl VZ
1 147 397 147 406 tasks sizes performed 817 B1 B2,B3 817
v3
429
v4
358 1613 300 300 300
Ri Rz R3
R4
B4 B5,B6 SC1 sc2 sc3 sc4
2 155 510 155 514 tasks sizes performed 602 B1 316 B2,B3 316 300 B4,B5,B6 300 300
-
-
sc1
300 300 300 634
sc2 sc3 SC4
Table IV. Results for Problem 3 problem no. reported obj fn heuristics obj fn
3 488 155 488 334
tasks
VI v2 v3 v4 v6
ve VI
V8
vs VlO
sizes 6 076 7 502 11813 6 184
-
16 318 12 436 14 342 9 713
performed B1 B2, B3 B4 B5, B6
-
B7 B8 B9 B10
HEURISTIC 1 3 Storage should be located either immediately before or immediately after the stage for which the active production time is maximum. It can be observed that the additional computational effort required for this heuristic is minimal because individual stage cycle times have already been computed during the determination of the limiting cycle time. The success of this heuristic prompted us to utilize the same factor in the heuristics associated with parallel units. (c) Parallel Units and Their Mode of Operation. The difficulty in using the single-product plant heuristics
for parallel units is like the one we mentioned for the storage tank location, namely, there are multiple timelimiting and size-limiting stages. Since the two cases are analogous, it is not surprising that the following heuristic is quite similar to the earlier one. HEURISTIC 14: A new parallel unit should be added either operatingout-of-phase to the stage with maximum active production time or operating in-phase to the stage with groups having maximum active capacity. We pointed out previously that the conflict between these two options is resolved by selecting the option that gives better savings. The above heuristic will correctly identify the stage at which a parallel unit should be added for the following cases: (i) If one unit is capacity-limiting for most products, Le., it has the highest size factors for maximum number of products, then it is the most appropriate choice for addition of an in-phase unit. (ii) Similarly, if one unit is time-limiting for most products, then it is the most suitable choice for addition of an outof-phase unit. (d)Task Mergingand Splitting. The basic heuristic for the multiproduct plant is analogous to the singleproduct plant heuristic, namely, try to reduce the number of units in the plant. To achieve this, we always start the initial design with maximum level of merging for all tasks. As a result, we only need to develop heuristics for splitting. In the following discussion we first propose a heuristic for determining the task chain which is most suitable for splitting. In the next step we decide if splitting at that chain is favorable or not, by combining the hypothetical product parameters with the single-product heuristics. As we noted earlier, splitting can be carried out for the tasks associated with the time-limiting stage or with the non-time-limiting stage. In the first case, smaller limiting cycle time and smaller batch size after splitting require smaller equipment sizes for all of the existing units, resulting in cost savings. In the second case, although limiting cycle times does not reduce, savingscan be realized through elimination of a parallel unit from a non-timelimiting stage. In the first case, our problem is to determine a stage which is equivalent to the time-limiting stage of a single-product plant, since there may be more than one time-limiting stage for a multiproduct plant. This is
1392 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
Table V. Results for Problem 4 problem no. heuristics obj fn
4 457 550 sizes 5 972 6 695 10 118 5 445
v1
vz v3 v4 v6
VS
-
vs
12 384 10 960 12 629 7 969 1655
VI V8 VlO
T (after stage 3)
tasks perform ed
B1 B2,B3
-
B4
B5,B6
-
B7 B8 B9 B10
exactly the same problem we faced for the storage tank location heuristic. Therefore, we propose to use the same method here; we conclude that the stage having maximum active production time is equivalent to the time-limiting stage of a single product. This is also true for the second case as the active production time of a stage for any product is equal to the product of the group cycle time and the number of groups operating out-of-phase. Accordingly, we propose the following heuristic. HEURISTIC 15: The most appropriate stage for splittingis the stage for which the active production time is maximum. The next step is to decide if splitting at the stage determined by the above heuristic is justified or not. Once again, an analysis similar to the one for the single-product case can be carried out for each product individually, but the resulting heuristic would be very inefficient and hence is ruled out. To obtain a more efficient heuristic, we examine the corresponding heuristics developed for a single-product plant, i.e., heuristics 9-12. All parameters required in the analogous heuristics for multiproduct plant are known, except the ratio p. This ratio can easily be obtained for a hypothetical product if we were to represent our plant as a hypothetical single-product plant. Thus, our heuristics for multiproduct plants corresponding to heuristics 9-12 are the same with the ratio p defined for a hypothetical product. The representation of our multiproduct plant as a single hypothetical product plant has already been discussed. One should note that the representation scheme requires the knowledgeof limiting cycle time of individual product Therefore, determination of new values is needed whenever they are expected to change, for example, when storage is inserted or when a task is split. For constant processing times values are known. However, for processing times dependent on batch size, we must recalculate Instead of computing the new batch sizes and cycle times explicitly, we estimate the new as the maximum of the current stage cycle times. The inaccuracy introduced by this approximation is small enough to be justified against the savings in the computation time. Also, these values are used only for the purpose of choosing a synthesis option, and hence the final results obtained after sizing are more accurate. Use of these heuristics gives excellent results in a very reasonable computation time.
(e).
e
e.
e
Overall Design Procedure
A design procedure has been developed and implemented to test these heuristics. Figure 2 represents the
overall design procedure. The design process starts with an initial feasible design and a lower bound on the cost of the plant specified by the user. If an initial feasible design is not available, then it can be constructed by using storage at all possible locations and maximum number of parallel units with their maximum sizes. A lower bound, if unknown, can be obtained by inserting intermediate storage at each possible location. The cost of the storage in that case is assumed to be the minimum possible. If the design problem involves multiple products, then the characteristics for a hypothetical product are determined. Now, to improve the initial design, we invoke the storage option. Heuristic 4,5, or 13 is used first to determine the best possible location for the storage. Heuristics 1-3 are then used to determine if a storage insertion at such a location is justified or not. If storage insertion is favorable, then a storage is inserted. At this step, the number of subprocesses and the stages in each of the subprocesses are fixed. A train design procedure (described below) is used for each of the subprocesses, and the total cost of the plant is computed. The procedure is terminated when a satisfactory solution is obtained or when further cost reduction is not possible. When the design options of merging, splitting, and parallel unit addition are invoked for one subprocess, they do not affect the stages in the other subprocess. Thus, a design procedure involving all of these design options can be used for each of the subprocessesin asequential manner. Such a design procedure, referred to as the train design procedure, is represented in Figure 3. The procedure starts with a fully merged subprocess and without parallel units on any of the stages. Heuristics 8,9,11, and 15 are used to determine if and where task splitting is justified. The process is repeated until cost can no longer be reduced. Let this cost be denoted as COSTI. Heuristics 6,7, and 14 are used with a fully merged subprocess to determine if parallel units are to be added. Once again, heuristics for task splitting (8-12 and 15) are invoked and minimum possible cost is determined. Let this cost be COST2. If COST2 is less than COSTI, then addition of more parallel units is explored. The procedure is stopped when the cost cannot be reduced further by adding a parallel unit. COST1 is reported as the cost of the train.
Performance Evaluation No previous work exists for multiproduct plants which combine all of these design options into one formulation. Although the design options addressed in this study have been considered by Yeh and Reklaitis (1987) and Ravemark and Rippin (19911, there are significant differences between their works and ours. The work of Yeh and Reklaitis is for a single-product plant only and does not consider in-phase operation for parallel units. Similarly, the formulations developed by Ravemark and Rippin address either splitting/merging or parallel units and intermediate storage. They also assume constant processing times. Despite these contrasts, the problems solved in those papers are the only available test problems to evaluate the performance of our heuristics. Three problems are used from these papers. Data for problems 1and 2, given by Yeh and Reklaitis (1987), are shown in Table I. Data for problem 3 are obtained from Ravemark and Rippin (1991) and presented in Table 11. The results obtained are compared with the published results in Table I11 and IV, respectively. Excellent results are obtained for all of the teat problems. The plant structures for problems 1and 2, obtained using
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1393 Table VI. Data for Problem 5 [Adapted from Modi (1986)l H = 6OOO Z = 3 Q = [437000,324000,2580001 250 IVj I loo00 300 IR, Iloo00 c:
i=l
recipe
sc1
a or b a or 6
370 0.22
S or D
Po C
1.0
-
d
i=2
S or D
Po C
1.0
-
d
i=3
Sor D
Po C
d
1.0
-
= 278,
Stor*ageParameters: y, = 0.49, t$, = 1.0
Vi sc3 370 0.22
B1 250 0.60
B2 150 0.60
sc2 370 0.22
B3 250 0.60
8.28 0.35 0.0 0.0
5.0 0.80 0.20 0.40
-
1.0
9.70 9.86 0.24 0.33
-
5.58 2.95 0.0 0.0
3.0 3.00 0.15 0.40
1.0 -
8.09 7.01 0.35 0.33
-
1.28 1.76 0.0 0.0
2.34 2.20 0.34 0.40
1.0
10.3 6.01 0.50 0.33
-
-
our heuristics, turned out to be identical to those reported by Yeh and Reklaitis, and led to very similar final results. For problem 1,heuristics determined that splitting at the first component of the first batch stage is favorable, whereas merging is beneficial for the last two tasks. Also, the results show that parallel unit addition is no longer advantageous due to splitting. On the other hand, results for problem 2 indicate that splitting is not justified and parallel unit addition is the best option for cost reduction. Although Yeh and Reklaitis used storage option in their work, we do not consider storage for problems 1 and 2 because storage with no cost will always be justified, and thus the solution will have storage between each of the batch stages. This exercise is trivial and therefore need not be considered. The synthesized structure for problem 3is also the same as the optimum reported by Ravemark and Rippin (1991). It should be noted that this structure is better than the structure reported by Ravemark and Rippin when they used DICOPT++ to solve their formulation. These problems are good test cases for the heuristics, but do not explore the full potential of the heuristics. The first two problems are single-product problems, and the third problem, in spite of having multiple products, does not have optionsof intermediate storage and parallelunits. Therefore, problem 3 was modified. Data for problem 4 are the same as for problem 3 except we now allow the use of parallel units and also one intermediate storage. The results for problem 4, reported in Table V, show that although addition of a parallel unit is not useful, insertion of intermediate storage results in cost reduction from $488 334 to $457 550. Problem 4 emphasized two points: first, it showed the importance of allowing all design options in a design;second, it establishedthat the heuristics can successfully select the best option applicable. The number of binary variables in an MINLP formulation for such a problem is greater than 170 as estimated using the formula from Ravemark and Rippin. However, problem 4 is still not the best test problem because it does not involve SC units and involves only constant processing times. A more general problem that overcomesall of these restrictions is needed to test our heuristics. We modified the data of a problem reported by Modi (1986). The data for problem 5 are given in Table VI. The original problem had only four batch tasks. We have broken the first batch task of the original problem into
1.0
1.0
1.0
B4 250 0.60
sc4 370 0.22
B5 250 0.60
sc5 370 0.22
2.95 5.28 0.40 0.30
--
1.0
6.57 1.20 0.50 0.20
-
6.17 1.08 0.42 0.20
--
5.98 0.66 0.30 0.20
-
3.27 7.00 0.70 0.30 5.70 5.13 0.85 0.30
1.0
-
1.0
-
-
Table VII. Results for Problem 5 problem no. heuristics obi fn sizes 3 149 1693 1693 6 396 6 396 3 517 3 517 3 517 4 624 4 005 437 6 835 7 711 732 10 OOO
1.0
-
1.0
1.0
5 360 625
task8 performed B1 B2 B3 B4 B5 SCl sc2 sc3 sc4 sc5 SC6
1078
two subtasks such that if these two taskswere to be merged, we would have the original problem data. The results for problem 5 are presented in Table VII. The final solution involves splitting the first batch task, inserting an intermediate storage after stage 2, and adding parallel units at stages 2-4. Although both in-phase and out-of-phase modes of operation were allowed, the heuristics selected only out-of-phasemode of operation for the parallel units. The final cost is $360 625compared to $372 547achievable with merged formulation. The original work reported the optimum cost (NLPsolution)to be $369 728. The splitting of the first batch task results in additional savings. It should be noted that in the original work the location of storage was assumed in advance. This is equivalent to either avoiding a synthesis decision or repeating the procedure for all possible locations. In this work, we overcome this deficiency with the help of heuristics. The heuristics have also been tested on several additional problems adapted from the literature. In these cases the synthesized structure corresponded to either the same structure or a better structure than the one reported in the literature. The better structure is obtained due to the additional design options made available in this implementation.
1394 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
Conclusions In this paper we have solved the synthesis subproblem using a heuristic approach. New heuristics have been developed for (i) the location and insertion of intermediate storage, (ii) the addition of parallel units with their mode of operation, and (ii) task merging and splitting. None of the previously available procedures and formulations addressed these design options collectively. The premise of this work has been that, despite the advances in computer technology, mathematical programing techniques were not capable of solving general design problems. Therefore, we investigated the heuristic approach to the design problem. This study has shown that heuristics do solve the synthesis subproblem and obtain the optimal task network and equipment configuration. Heuristics may not always yield the global optimum solution. Fortunately for many engineering problems, including the preliminary design problem, it is not necessary to obtain the global optimum solution, due to two major reasons: first, the information used in the design may be inaccurate and uncertain, and second, the design problem is solved only for the purpose of early decision making and not for final implementation. Because of the heuristics, the search procedure becomes less iterative and requires significantly reduced computational effort. As a consequence, we are able to solve a more complex design problem. Heuristics may also contribute to the understanding of the knowledge by providing valuable insight into the design process. This advantage is in contrast to the mathematical formulations, where knowledge is very implicit and hence cannot be easily understood. Finally, if a global optimum solution is necessary and a corresponding (MINLP) formulation is available, then a procedure based on heuristics can be used to obtain an initial solution because heuristics give “near-optimal” solutions without requiring prohibitively large computation time. Though heuristics were proposed 20years ago, they were used to solve the sizing subproblem only. This work is the first attempt to collect and develop all heuristics that address both the sizing and synthesis subproblems simultaneously. We have successfullycombined these heuristics in one procedure and critically evaluated them. These heuristics enable the evolution of a method that solves quite general and large problems with moderate computer capabilities (such as low-end workstations). Acknowledgment This work is partially supported by the Steel Resource Center at Northwestern University and by the Shell Research Foundation.
Nomenclature aj = cost coefficient for batch stage j b k = cost coefficient for semicontinuous (SC)item k Bi = batch size of product i cr* = cost coefficient for an intermediate storage s cij = coefficient for processing time on stage j for product i, eq 3 Cj, = capacity of group g for stage j
Cj, = cj ct (j) = group cycle time for groupsof stagej for a given product
= power coefficientfor procssing time on stagej for product i, eq 3 Dih = duty factor for semicontinuous item k for product i g = index for group H = horizon; also used as an index for hypothetical product dij
i = index for product j = index for stage k = index for semicontinuous unit ljy = semicontinuous subtrain connecting stage j to j, LCT = limiting cycle time for a product mj = number of groups in stage j N = total number of products
NS = new subprocess = constant coefficient for processing time on stage j for product i, eq 3 Qi = demand for product i r = index for batch unit rj = total number of equipment units in a group g of stage p:.
i
r,# = r, Rk = processing rates for semicontinuous item k s = index for storage Sd = downstream subprocess s, = upstream subprocess SC = semicontinuous Sit = size factor for batch stage which performs task t for product i; characteristic size needed to produce unit mass of product i at batch stage performing task t Si, = size factor for storage s for product i Ti = time required to achieve the required production of product i = limiting cycle time for product i V, = size of batch unit r V,* = size of storage s xi = fraction of LCTi used for transferring to/from storage a, = cost exponent for batch stage j /??k = cost exponent for semicontinuous stage k ys* = cost exponent for storage s Bijj’ = transfer time from stage j to j’ for product i p = ratio of new batch size to old batch size pm = modified ratio of new batch size to old batch size, eq 34
Appendix A: Previous Works The common approach to the preliminary design problem of multiproduct plants has been to formulate it as an optimization problem with minimization of the capital cost as the objective. The original formulation was proposed by Sparrow et al. (1975). In their work a heuristic method was developed to obtain an estimate of the upper bound for the equipment cost. The formulation assumed processing times to be independent of the batch size and considered discrete equipment sizes. Grossmann and Sargent (1979)permitted the processing time to be a function of batch size. They formulated the problem as a nonlinear program and solved it using the combination of branch and bound technique with NLP algorithm. The optimal design of a single product batch process with intermediate storage tank was studied by Takamatsu et al. (1982). That work dealt only with single-product plants and did not consider semicontinuous units in the analysis. The problem of retrofit design of multiproduct batch plants was formulated as MINLP and solved with the outer approximation algorithm by Vaselenak et al. (1987). Fixed processing times and continuous range of equipment sizes were assumed, and semicontinuous equipment was not considered. The first work to address the synthesis issues in the design problem was that of Yeh and Reklaitis (1987). They partitioned the design problem into two parts: the network synthesis subproblem and the equipment sizing subproblem. Although their heuristic procedure performed well for sizing and parallel unit selection, it did not include intermediate storage. They also developed several heuristic synthesis rules for splitting and merging of tasks, reuse of equipment units, and installation of
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1395 intermediate storage. However, their work was not applicable to multiproduct plants. A heuristic procedure for the preliminary design of batch processes with and without intermediate storage was presented by Modi and Karimi (1989). Their procedure carried out a sequence of single-variableline searches and achieved excellent results with small computational effort. However, their work did not determine the location of storage. Recently, Patel et al. (1991)solved a more general design problem using the simulated annealing technique. Their work included design options such as mixed mode of operation, nonidentical parallel units, and location of intermediate storage. However, synthesis of the task network was not considered. Voudouris and Grossmann (1991) reformulated the earlier MINLP formulations as MILP formulations incorporating discrete sizes and obtained global minimum solutions. However, their work did not consider semicontinuous units and assumed constant processing times. None of the synthesis options were addressed. Ravemark and Rippin (1991) also presented various MINLP formulations incorporating features such as merging or splitting of subtasks, mixed mode of operation, and flexible production requirements. They found the computational requirements to be excessively large for problems involving storage and task network synthesis. No further work is reported that handles all of the design features including storage, task splitting and merging, and use of parallel items. Tricoire and Malone (1991) solved the design problem in which the objective is to maximize the net present value of the revenue minus raw material, processing, capital, setup, and inventory costs and penalty for early and late production, as well as shortage costs. They optimized the production planning, design, and flowsheet synthesis simultaneously using simulated annealing. Different products were allowed to have different due dates. The synthesis options not considered in their work include the intermediate storage and mixed mode of operation of parallel units. Their formulation also excluded semicontinuous units. The multiproduct plant design problem with uncertainties in the product demands and equipment failures was solved by Straub and Grossmann (1992). They proposed optimization models to establish the trade-offs between investment cost and stochastic flexibility. None of the synthesis options were considered in that work. The optimal retrofit design problem has also been considered by Fletcher et al. (1991). They proposed a MINLP formulation which eliminated the constraint imposed by previous researchers that new units must be used in the same manner for all product places. However, their work does not consider the design options of intermediate storgae and task merging and splitting. Besides, the formulation does not consider semicontinuous units. In summary, we can conclude that most researchers have addressed only the sizing subproblem. Those few who considered synthesis and sizing together did so in a very restrictive manner. Thus, either the work was applicable only to single-product plants (Yeh and Reklaitis, 1987)or all of the design options were not included in one formulation, for instance, task network synthesis was omitted (Patel et al., 1991) or parallel units and storage were excluded (Ravemark and Rippin, 1991). Moreover, no attempt has been reported so far to develop any heuristics for the synthesis subproblem of the multiproduct plant design.
Appendix B: Determination of A, Let the splitting stage be denoted SS, the difference in the total cost of the rest of the equipment due to splitting
be denoted A, and the ratio of the batch sizes with and without splitting be denoted p. Then
Since p
< 1 and a m -
= m-{aj), j # SS J
(1-pa-) 2 (1- p*j) V j , j # SS Therefore, the maximum difference in the total cost, Am,, can be estimated as
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Received for review August 13, 1992 Revised manuscript received March 10, 1993 Accepted March 18,1993
