optimal production strategy and design of multiproduct batch plants

590 I n d . E n g . Chem. R e s

the minimum value of LB*2 can be determined by opti- mally scheduling the products in S ' on unit 2 alone with due dates dl2* = d , -- Ck2 - ak2, i E S ' .

Li tera ture Cited

Bernardo, J . J.; Lin, K.S. An Optimal Tardiness Algorithm for Scheduling N Jobs on M Processors. Adu. Manage. Stud. 1982, 1 (3) , 291.

Bernardo, J. d.; Lin, K. S. Scheduling Independent Jobs on Nonu- niform, Unequal Processors. J . Oper. Manage. 1984, 4 (41, 305.

Chaudhary, J . Batch Plants Adapt to CPI's Flexible Game Plans. Chem. Eng. 1988, 95 (21, 31.

Elmaghraby, S. E. The One Machine Sequencing Problem with Delay Costs. J . Ind . Eng. 1968, 19 (2), 105.

Garey, M. R.; Johnson, D. S.; Sethi, R. Complexity of Flowshop and Jobshop Scheduling. Math. Oper. Res. 1976, 1 , 117.

Held, M.; Karp, R. M. A Dynamic Programming Approach to Se- quencing Problems. J . Soc. Ind. A p p l . Math. 1962, 10 (l), 196.

Johnson, S. M. Optimal Two and Three Stage Production Schedules with Set-up Times Included. Nau. Res. Logist. Q. 1954, 1, 61.

Kan, R.; Lageweg, B. J.; Lenstra, J. K. Minimizing Total Costs in One-Machine Scheduling. Oper. Res. 1975, 23 (51, 908.

Karp, R. M. Reducibility among Combinatorial Problems. Com- pienit) of Computer Computations; Plenum Press: New York. 1972.

Ku, H. M.; Rajagopalan, D.; Karimi, I. A. Scheduling in Batch Pro- cesses. Chem. Eng. Prog. 1987, 83 (8), 35.

1990. 29, 590-600

Lawler, E. On Scheduling Problems with Deferral Costs. Manage. Sci. 1964, 11 (2) , 280.

McNaughton, R. Scheduling with Deadlines and Loss Functions. Manage. Sci. 1959, 6 (11, 1.

Musier, R. F. H.; Evans, L. B. Approximate Methods for the Pro- duction Scheduling of Industrial Batch Processes with Parallel Units. Comp. Chem. Eng. 1989, 13 (1 /2 ) , 229.

Parakrama, R. Improving Batch Chemical Processes. Chem. Eng. 1985, Sept. 24.

Rajagopalan, D.; Karimi, I. A. Completion Times in Serial Multi- product Batch Processes with Transfer and Set-Up Times. Comp. Chem. Eng. 1989, 13 (1/2), 175.

Sen, T.; Gupta, S. K. A State-of-Art Survey of Static Scheduling Research Involving Due Dates. Manage. Sei. 1984, 12 (11, 63.

Shwimer, J . On the N-Job, One-Machine, Sequence-Independent Scheduling Problem with Tardiness Penalites: A Branch-Bound Solution. Manage. Sci. 1972, 18 (6), B-301.

Smith, W. Various Optimizers for Single Stage Production. Nav. Res. Logist. Q. 1956, 3 (2), 59.

Townsend, W. Sequencing N Jobs on M Machines to Minimize Maximum Tardiness: A Branch-and-Bound Solution. Manage. Sci. 1977. 23 (9), 1016.

Wellons, M. C.; Reklaitis, G. V. Problems in the Scheduling of Batch Processes. Presented at the TIMS/ORSA Joint National Meet- ing, Washington, D.C., 1988.

Receioed for reuiew July 11, 1989 Revised manuscript received November 27, 1989

Accepted December 12, 1989

Optimal Production Strategy and Design of Multiproduct Batch Plants

J. C e r d i * INTEC,i Casilla de Correo No. 91. 3000 Santa Fe, Argentina

M. Vicente , J. Gut ie r r ez , S. Esplugas, a n d J. Mata Chemical Engineering Department , University o f Barcelona, 08028 Barcelona, S p a i n

Methods of sizing batch equipment for multiproduct batch plants usually assume production of a single product at a time. In some cases, however, lower unit sizes are achieved instead by running multiproduct campaigns. This paper proposes a nonlinear mathematical program to search for both the best production strategy and the minimum equipment sizes simultaneously. Since the number of multiproduct campaigns becomes extremely high for large problems, a preliminary step eliminating nonefficient campaigns is performed. Besides that , one usually can still find the problem optimal solution. On the basis of the equal-batch-size assumption, simple analytical expressions are derived to find an initial feasible solution. Several examples have been successfully solved in a short number of iterations. Analysis of the results shows that the use of multiproduct campaigns often brings about a significant savings in capital cost.

1 . Introduct ion Scheduling and design of batch chemical plants used for

the manufacture of fine chemicals in low quantities have gathered much attention in recent years. A set of N chemicals is produced in a batch plant by processing a discrete number of batches. Each batch should undergo a series of M processing tasks in a specified order to attain the desired product. In the so-called multiproduct batch plants, both the processing scheme (type and sequence of tasks) and the physical equipment where each task is carried out are the same for any product to be manufactured. Therefore, there is a one-to-one relationship between the equipment unit and the processing task it ac- complishes. As a result, the notion of batch stage to refer

*Member of CONICETs Research Staff and Professor at JJNL. t Instituto de Desarrollo Tecnoltigico para la Industria Quimica.

Universidad Nacional de! Litoral (UNL) and Consejo Nacional de Investigaciones Cientificas y TBcnicas (CONICET).

0888-58~5/90/2629-0590$02.50/0

to each pair (equipment/unit task) is introduced (Sparrow et al., 1975). A batch stage is identified by an index j that indicates its location at the processing scheme where j = 1 denotes the earliest task and j = M the last one. Any batch follows the same path through a series of batch stages independently of the product being attained. The operating time required to process a batch of product P, in stage j is denoted by T,,. It is the time necessary to complete all the subtasks that a processing task normally includes, like filling, processing, emptying, and cleaning. The characteristic processing time for a batch of P, in the j th stage (T,, ) generally varies with the size of the batch (BJ.

At any time, a set of M batches are simultaneously being processed in a multiproduct batch plant comprising M stages. Assuming that intermediate storages are not available, the processing of the entire sequence of M batches must be completed before transferring each one to the next stage and simultaneously introducing a new batch to the first; i.e., the plant is operated on the zero-wait

C 1990 American Chemical Society

Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 591

the current production cycle. Also, X has as many com- ponents as the number of stages in the plant (MI, with the j t h component denoting the product (Pi) currently processed in stage j . Each component of X can take any value from 1 to N.

In multiproduct batch plants manufacturing one product at a time, the state of the plant remains unchanged at each campaign. Then the status of a multiproduct batch plant being so operated and comprising four stages to produce three different products {I, 2, 3) is successively described by the vectors XT = (l,l,l,l), XT = (2,2,2,2), and XT = (3,3,3,3) during the manufacture of the products 1, 2, and 3, respectively. Therefore, a set of three state vectors becomes enough to represent at any time the status of the plant as long as the end effects are neglected by assuming a large number of batches per campaign.

On the other hand, the production cycle time, Tc(X), defined by stage j * accomplishing the longest processing task is given by

(2)

where x j is the product being processed in stage j a t the present production cycle, X. It should be emphasized that the notion of the time-limiting stage is primarily associated to a particular production cycle rather than a campaign involving a finite number of production cycles. Generally, T, is a function of X.

If a multiproduct batch plant operating by campaigns is manufacturing product Pi, then a single-state vector XT = (i,i, ..., i) portrays the status of the batch plant during the entire production run. Consequently,

Tc(X) = maxj=l,2 ,..., M TxJ, j = Tx,*,,*

- Tc,i = Tc(Xi) = maxj=l,z ,..., M T(xJ),, j -

maxj=l.z ,..., M Tij = Ti, j , *

since (xjIi = i, for j = 1, 2, ..., M

Both the operating cycle time (TC,J and the time-limiting stage Gi*) do not undergo any change along a single- product campaign. A different situation arises when multiproduct instead of single-product campaigns are ex- ecuted. To illustrate the point, a multiproduct campaign producing simultaneously products 1, 2, and 3 is considered. I t is assumed that a particular production strategy has been selected by which a batch of product 1 is successively followed by batches of products 2 and 3 before manufacturing another batch of product 1. In this case, a set of four state vectors, (1,2,3,llT, (1,l,2,3lT, (3,1,1,2)*, and (2,3,1,1)T, is needed to portray the status of the plant a t any campaign production cycle. Such four plant states are cyclically repeated along the production run. There are 24 alternative campaigns and a total of 81 different state vectors to consider in a batch plant processing 3 products in 4 stages. Since the sequence of batches changes from one state to the next, modifications in both the time-limiting stage and the cycle time can also be expected. Thus, the production cycle time at state Xk is given by

Tc(xk) = maxj=l,2 ,.._, M T ( x , ) k , j = T.zJk*, jk*

where jk* is the time-limiting stage a t state xk. Table I presents the data for a four-stage multiproduct

batch plant manufacturing three different compounds, introduced by Sparrow et al. (1975). It has been decided to fulfill the production needs through monoproduct campaigns. An analysis of the Ti, values in Table I indicates that the cycling time, although different in each case, is determined by stage 2 in the three campaigns (see

mode. The period of time between two successive batches that is equal for any stage becomes fixed by the longest processing task. I t is named in this paper the production cycle time (TJ. In fact, a batch plant is cyclically operated, and the processing of a new batch indicates the start of another production cycle. Stages carrying out short tasks stay idle a large fraction of the production cycle time while waiting for the completion of the time-limiting task (“time-limiting stage”).

A distinct set of M processing tasks is generally carried out in each production cycle, the longest one setting the production cycle time. I t is then expected that the production cycle time varies during plant operation unless the sequence of M batches being processed is steadily repeated.

Usually, a multiproduct batch plant is operated by single-product campaigns. In this case, a number of batches of a certain product high enough to reach production requirements is processed before starting the manufacture of another chemical. At any time, only one product is produced in the plant. Consequently, a sequence of M batches of Pi is repeatedly processed at every production cycle of the campaign for Pi. Therefore, each single- product campaign features an unique production cycle time (TJ defined by the longest processing task for that product. Then, it can also be said that each campaign has a time-limiting stage featuring zero idle time. However, it may be possible that each product has a distinct time- limiting stage. If so, every stage stays idle sometimes during the production period.

After the batch size for Pi has been selected, the number of production cycles or batches to be processed in the campaign becomes fixed by the production needs. Moreover, the time allotted to the i th product campaign can be computed by multiplying the number of batches by the production cycle time (TC,J.

The equipment size required to process a unit amount of final product Pi in stage J , commonly called the size factor Sij, is another problem parameter. By assuming the equipment sizes Vi, j = 1,2, ..., M a s known data, the batch size of Pi is then determined by the stage featuring the smallest capacity for P, (the capacity-limiting stage), i.e.,

At the design stage, the values of Bi, i = 1, 2, ..., N , are generally chosen to meet production requirements and simultaneously minimize the plant capital cost.

The residence time of a batch in the plant (TR) is the elapsed time from the start of the first task to the end of the last one. But M production cycles are necessary for a batch to pass completely through the process. By adding such M cycle times, one can evaluate T R .

During a multiproduct campaign, the batch plant can be producing two or more chemicals simultaneously. As a result, the campaign no longer exhibits a unique cycle time but a sequence of cycle times steadily repeated during the production run. A similar statement holds for the time-limiting stage. However, nonequal cycle times do not necessarily mean different time-limiting stages and vice versa.

This work presents a mathematical formulation for the multiproduct batch plant design problem which permits the consideration of multiproduct campaigns to further reduce equipment sizes. The proposed approach can also be extended to multipurpose batch plants.

2. State of a Multiproduct Batch Plant A batch plant state vector, X , is defined to provide a

picture of the present status of a multiproduct batch plant. I t reports the sequence of M batches being processed a t

Bi = minj=l,2,,,,,M (Vj/Sij) (1)

592 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990

Table I. Set of Data for Examole 1 (Soarrow et al. . 1975) stage 1 stage 2 stage 3 stage 4

products S,1 T#l , h s,, Ti*, h st3 T23, h si4 Ti,, h 1 0 - 3 ~ ~ PI 8.28 1.15 3.70 9.86 2.95 5.27 6.56 5.30 436 rj, 5.57 5.95 4.09 7.01 3.27 6.99 6.16 1.08 324 P3 2.34 3.95 0.80 6.00 ,5.69 ,5.13 5.98 0.65 258

a1 400 4 0.60

400 0.60

400 0.60

Table 11. Set of Data for Examoles 2 and 3 (Other Soecifications As Given in Table I)

400 0.60

processing time ( T L , ) , h example 2 example 3

products stage 1 stage 2 stage 3 stage 4 stage 1 stage 2 p, 1.15 9.86 5.27 5.30 5.0 8.0 p 2 7.01 5.95 6.99 1 0 8 6.0 4.0 pi 3.95 5.11 6.00 0.65 5.0 4.0

Table 111. Values of T, and TR for Monoproduct and Some Multiproduct Campaigns

campaign production cycle time batch residence time

state example 1 example 2 example 1 example 2 9.86 7.01 6.00 7.01 9.86 9.86 9.86 7.01 7.01

9.86 39.44 39.44 7.01 28.04 28.04 6.00 24.00 24.00 5.95 33.74 31.62 9.86 9.86 9.86 33.74 33.72 7.01 6.99

PRODUCT PRODUCT PRO DUCT

1'1 - - - - - - - - -

1'2

j=3

1'4

Tc,2 --- TC,3 - T Z , ~ - --- -.._.

Figure 1 . Operating scheme of a multiproduct batch plant performing single-product campaigns (an equal time-limiting stage at each campaign).

Figure 1). It is the time-limiting stage featuring no idle time during the entire operation of the plant. A set of monoproduct campaigns is then the best option. No multiproduct campaign is able to produce an improvement on the time utilization of the plant and, consequently, a decline on the required capital cost. Moreover, the incorporation of an additional equipment unit running out-of-phase in the time-limiting stage yields a simulta- neous decrease of the batch size in the three single-product campaigns.

Table I1 shows the data for a slightly different problem. Values of T,, have been changed in such a way that each single-product campaign features a distinct time-limiting stage. Now every stage stays idle during at least two of the three campaigns (see Figure 2). Better equipment utilization can be achieved by running multiproduct campaigns featuring lower batch residence time. This production strategy also provides the largest improvement when another unit is incorporated to run out-of-phase in the limiting stage. By reducing the average batch residence time, the use of multiproduct campaigns permits an in-

TC,2 ~ f- TC,3 1 -- TC,I - Figure 2. Operating scheme of a multiproduct batch plant performing single-product campaigns (a distinct time-limiting stage per campaign).

MULTIPRODUCT CAMPAIGN p1 ,P2 ( 1 1 )

05 T, ~- -- r- ~- - -

Figure 3. Operating scheme of a multiproduct batch plant producing products P , and P2 simultaneously a t a ratio of 1:l (example 2 )

crease in the number of batches produced during the available production time (H).

In example 2, an alternative multiproduct campaign producing PI and P2 simultaneously at a product ratio of


Table IV. Set of Dominant Multiproduct Campaigns for Example 2 (n2 = 1) ~~

campaign product ratio sequence of batches TR,I TR,? TR,l - TR,i no. of states 2:l:l 30.94 32.73 -1.79 4

a Residence time if monoproduct campaigns were carried out.

1:1 (see Table 111) is represented cyclically by the pair of state vectors, (1,31,2)T and (2,1,2,1)T. I t requires less time to produce a batch of both products P1 and P2 than monoproduct campaigns (see Table 111). The time-limiting stage for each state has been underlined. I t is stage 2 in both cases (see Figure 3). Thus, the processing of product 2 in the second stage limits the production cycle time at the first of the two states. Then T, is equal to 5.95 (see Table 111). In the other one, the cycle time increases up to 9.86 as long as the treatment of product 1 again in the second stage is the longest state processing task. From the start of the first processing task to the end of the last one for a batch of either PI or P2, the state of the plant is represented by a sequence of M (=4) vectors. In this case, the subsequence [(1,2,1,2), (2,1,2,1)] is repeated twice during TR. Therefore, the batch residence time for a campaign manufacturing (PI, P2) simultaneously at a product ratio of 1:l is equal to

against residence times for the monoproduct campaigns given by

(TR)plp2 = 2(5.95 + 9.86) = 31.62

(TR)1 = 4(9.86) = 39.44 (TR)2 = 4(7.01) = 28.04

This implies that the overall time required for manufacturing two batches of both products P, and P2 decreases from TR = 2(9.86 + 7.01) = 33.74 to T R = 31.62 when the proposed multiproduct campaign instead of monoproduct production runs is carried out. This is because the idle time a t stage 2 has been eliminated.

3. Problem State Vector Space Since there is a strong relationship between them, the

search for both the optimal production strategy and equipment sizes, Vi, j = 1,2, ..., M , should be made at the same time. A new mathematical formulation for the multiproduct batch plant design problem which accounts for any feasible multiproduct campaign to further reduce the stage sizes, V,, j = 1, 2, ..., M , is to be developed. A representation of a campaign by the model is attained through the corresponding set of state vectors. To avoid the exclusion of any campaign candidate, the entire space of state vectors for the problem must be generated. This is easily made by changing each component of vector X one a t a time from 1 to N. Therefore, a total number of N M state vectors is required to represent the complete set of campaigns for a multiproduct batch plant comprising M stages to manufacture N different products. At example 2, the number of state vectors climbs up to 34 = 81.

3.1. Elimination of Noneconomic Multiproduct Campaigns. There is a simple way to cut the number of

28.82 28.14 31.62 35.53 30.87 30.26 34.88 27.13 26.02 28.02 26.00

29.88 28.87 33.74 36.59 30.89 31.72 35.58 27.86 26.02 27.03 25.01

-1.06 -0.73 -2.12 -1.06 -0.02 -1.46 -0.70 -0.73 0.00

+0.99 +0.99 total:

4 4 2 4 4 2 4 4 2 4 4

42

state vectors down. Two alternative multiproduct campaigns producing (Pl, P2) simultaneously at a product ratio of 1:l are included in Table 111. The first one featuring a batch residence time TR = 31.62 has already been analized in the previous section. Since the other presents a larger T R = 33.72, then the latter campaign and the four states that describe it can be discarded. From the set of campaigns producing a similar range of products a t the same product ratios, the one featuring the lowest TR, called the “dominant” campaign, should only be considered (Mauderli and Rippin, 1979).

Table IV lists the dominant multiproduct campaigns for example 2. By ignoring nondominant production strategies, the number of state vectors required to represent monoproduct and multiproduct campaigns is decreased from 81 to 45, Le., a 44% reduction. Moreover, some dominant campaigns need more time than monoproduct ones to manufacture a range of products at certain product ratios. This is the case for the campaigns producing (P2, P3) simultaneously (see Table IV). No time improvement with regard to monoproduct campaigns (TR - TR) is achieved, so one can also discard them and their corresponding state vectors. In this way, the number of relevant states for the problem falls down to 35. A systematic procedure for generating time-efficient multiproduct campaigns when the problem is of large size is presented in section 5 .

On the other hand, it can be anticipated that production strategies making better use of the available production time are going to be selected by the proposed mathematical model (Mauderli and Rippin, 1980). A good example is the campaign producing (PI, P2) at a ratio of 1:l (see Table IV). At each production cycle, stage 2 is steadily the time-limiting stage. I t is therefore the campaign time- limiting stage. Other efficient campaigns are those producing PI, P3 at a ratio of 1:l and (Pl, Pz, PJ a t a ratio of 2:1:1, respectively (see Figure 4). In both cases, however, no stage features zero idle time a t every production cycle, but stage 2 presents the least one, so it can still be regarded as the time-limiting stage at those campaigns.

By properly combining multiproduct campaigns producing various chemical species a t different ratios and monoproduct campaigns, one can always find a way to meet all production requirements. Such a feasible production strategy is automatically developed by solving the proposed mathematical formulation.

3.2. Handling of Cases Featuring M < N . To show that the approach can also handle batch plant design problems featuring fewer stages than products, it has included example 3 involving two stages and three products (see Table 11). The set of multiproduct campaigns for example 3, of which only the sequences ( P I , Pz) (1:l) and


MULTIPRODUCT CAMPAIGN

P, , Pz ,P3 ( 2 1 1 )

Figure 4. Operating scheme of a multiproduct batch plant producing P I , P,, and P3 simultaneously a t a ratio of 2:l:l (example 2) .

Table V. Set of Multiproduct Campaigns for Example 3 sequence

campaign product ratio of batches TR,, PR,r TR,I - PR,! 1:l 13 14 -1

p29 p3 1:l (2,3) 11 11 0 ( 3 2 )

PI, P,, P3 1:l:l (12) 19 19 0 (3,1) (2,3)

P,, P,, P, 1:l:l (13) 18 19 -1 (2,1) (32)

a Batch residence time if monoproduct campaigns were carried

(PI , PA, Pz) (1:l:l) are really time-efficient, is listed in Table V. Other product retids are obtained by combining single-product campaigns and those shown in Table V, so they have not been explicitly considered. If the product requirement ratio is different from l:l:l, the optimal solution to example 3 will surely include monoproduct campaigns to meet the problem constraints.

4. Mathematical Model 4.1. Problem Constraints. 4.1.1. Fulfillment of

Production Requirements. Let t, be a problem variable denoting the total number of batches of PI being produced through a series of campaigns over the production period H . Then,

out.

i = l , 2 ,..., N, j = 1 , 2 ,..., M (3)

where Q, is the production requirement and V,/S,, is the j th stage capacity for product P,. Only the capacity-limiting stage restriction for each PI becomes a strict equality at the optimum. Unlike constraints (31, the other ones are all linear.

4.1.2. Available Production Time ( H ) . Let u,' be the number of times the multiproduct batch plant is a t each state Xk during a campaign 1 E C. Then,

Q,s,, 2, 2 -

VI

E ( c T c , k ) U l f H (4) 1€C K L ,

where H is the available production time and Tc,k is the cycling time of state Xk. Besides, the set of distinct states describing campaign 1 is denoted by Ll, and C includes all feasible campaigns. At the optimal solution, restriction (4) always turns into an equation. When campaign 2 exhibits a set ( L J involving M different state vectors, it can be written as

where TR,, is the batch residence time for campaign 1. Some campaigns, however, are described by a set (L,) comprising Mlm, state vectors, where both M/ml and ml are always integers. However, monoproduct campaigns can be represented by a single state vector that repeats itself M times (ml = M). In example 2, the dominant campaign producing (Pl, P2) simultaneously at a ratio of 1:l exhibits a set Ll = {(1,2,1,2)T, (2,1,2,1)T} comprising ml = 2 different state vectors. In such cases,

Then,

By defining U , = ulf/ml, constraint (4) becomes

where ul denotes the number of lth campaign production steps. Values of TR,, for any multiproduct campaign 1 E C is computed a priori.

4.1.3. Relationships between the Sets of Variables lzi) and { v,). Let ail be the number of batches of Pi produced by performing a single production step of campaign 1. Therefore,

( 7 ) aipl 2 t i , i = 1, 2, ..., N l € C

4.1.4. Nonnegativity Condition. z , b O , i = l , 2 ,..., N

u / b 0, 1 E c (8) V j > O , j = l , 2 ,..., M

As long as the optimal values of {ti) and lull are either relatively large or equal to zero, the integer conditions on such variables can be avoided. Usually, a near-optimal integer solution is attained by rounding off the global optimum provided by the model.

4.2. The Objective Function. A production strategy is commonly defined in terms of a series of campaigns and a number of production steps carried out per campaign. In the present design problem, a production strategy is sought that minimizes the plant capital cost subject to the inequality constraints (3) and (6)-(8). Then, the objective function is given by

where the cost coefficients aj and Dl are known.

5. Handling of Large-Size Problems In this section, an algorithmic procedure is given for a

fast identification of the most promising multiproduct campaigns. They are then considered for the determination of the best production strategy and equipment sizes. Although one can no longer guarantee the optimality of the solution found, this is surely a near-optimal one never worse and frequently much better than those provided by prior approaches.

A multiproduct campaign manufacturing a pair of chemical species (il, iz) is generally a promising campaign candidate if the following occur.


Table VI. Data for Example 4 Drocessine time ( T J S h

products stage 1 stage 2 stage 3 stage 4 stage 5 stage 6 10-3Qi, kg PI 2.3 5.6 1.8 8.4 6.9 1.3 820

4.2 3.7 3.2 5.8 7.5 0.7 675 p3 6.7 1.4 0.9 4.4 5.7 4.9 750 p4 1.5 8.2 7.1 3.3 4.2 5.3 460

p2

p5 3.0 4.6 1.7 6.5 2.8 9.1 1020

Table VII. Set of Time-Efficient Campaigns for Example 4

campaign A.3 h sequence of batches TR,l TR,I - pR,l

0.60 45.9 -1.80 0.90 1.50 0.90 0.40 1.00 0.20 0.90 0.40 0.60 0.90 0.20 0.40 0.40

(Al) The related couple of monoproduct campaigns exhibits distinct limiting stages GIl f j l $ .

(Bl) The two limiting processing tasks ill and ir2 are the longest ones among all those required for the manufacture of il and i2 , e.g.,

aTL1,Iz = Tll,,,*l - Imax,#,l*, Tt2,,1 > 0 AT12,d = T12,,L** - b X , # , , * , Tll,,J ' 0

-12 = min PT,142, A7-,2,,1J > 0 Then,

(9)

must be positive. (Cl) The sequence of batches of L~ and i 2 has been

properly chosen. Table VI gives the stage processing times for a problem

involving five products and six stages (example 4). By use of the above criteria, the most efficient two-product campaigns have been found and listed in Table VII. Those ones producing (Pl, P3) and (P3, P4) that feature AE < 0 are the less efficient two-product campaigns as shown later.

On the other hand, multiproduct campaigns manufacturing simultaneously n chemical species, where 2 < n S N , rank generally among the best ones if (A2) all campaigns producing different subsets of n - 1 out of such n products are also promising campaigns since

*1,2, ,n = min ( M I 2 9 m13, ...! mn-l,n)

and if (B2) an adequate sequence of batches has been selected.

For example, a campaign producing (Pl, P2, P3) is generally time-efficient if the two-product campaigns (Pl, P2), (Pl, P3), and (P2 , P3) are also. Using criterion (A2), one can sequentially generate the promising campaign candidates involving three, four, or more products. For example 4, they are all listed in Table VII. In particular, campaign (PI, P2, P3) has been rejected because -123 0.

The best sequence of batches to use for each efficient multiproduct campaign still remains undefined. To this end, a mixed-integer linear programming problem is for- mulated. From the optimal solution, one can derive the set of state vectors defining the campaign as well the batch residence time (TR,J that it exhibits. The maximization

46.3 48.7 41.4 41.4 44.7 42.0 47.3 41.4 43.6 41.6 42.4 42.4 41.6

-3.70 -3.80 -2.00 -5.00 -5.10 -5.40 -4.60 -6.80 -6.40 -9.80 -5.80 -7.20 -9.80

of the time savjng (TR,l- pR,J has been adopted as the problem goal. TR,/ is the average amount of time required to produce a batch of product if monoproduct campaigns were used. Therefore, the proposed mathematical mod- eling to find the best batch sequencing for a multiproduct campaign (1) involving n d N products is given by

M n M

k = l r = l ,=1 min (TR,I - pR,l) = T c , k - T I , , ~ * { ~ ~ I , )

subject to

. - k = 1, 2, ..., M (11)

y l j = O , l , i = l , 2 ,..., M, j = l , 2 ,..., M (12)

where Tc,k is the batch cycling time for the kth plant state a t campaign 1. As mentioned earlier, campaign 1 can be represented by a set of M vector states, each one featuring a characteristic cycle time. The subscript j+k-1 in eq 11 should be replaced by j+k-1-M whenever j+k-1 > M. Furthermore, the overall number of constraints does not depend on the range of products manufactured during the campaign.

The optimal batch sequencing and residence time for each promising multiproduct campaign in example 4 are included in Table VII. In each case, the best solution has been found in a short-computer time averaging 40 s. Determination of T R j - TR,l for the rejected campaign (PI, P,, P3) yields a value of -3.10, the lowest one among the three-product campaigns (see Table VII). Similarly, the two-product campaigns (Pl, P3) and (P3, P4) that fea tye 1E < 0 are less efficient as long as the value of 7'R.L - TR,l amounts to -1.80 and -1.20, respectively.

As the number of products involved in_ a campaign grows, the cycle time improvement ( T R , l - T R , J rises, but a t a slower rate. Above a certain n*, no further time- savings is attained. In example 4, n* is equal to 3 (see Table VII). In turn, criterion (Bl ) would reject all three-product campaigns in example 2 (see Table IV). For the lO-product/ 10-stage multiproduct batch problem, the


number of MILPs to be solved amounts to 45 if the set of criteria for rejecting inefficient campaigns never works. In the general case, it will be much smaller. But if it still is too high, one can stop generating efficient campaigns when the following condition holds:

miniE.(-n-l (TRJ - f ' R , J - minlEcn (TRJ - p R , J 6 t (13)

where C " stands for the set of campaigns involving n products and t is a small positive number arbitrarily chosen.

6. A Good, Feasible Initial Solution 6.1. Running Monoproduct Campaigns. Product

batch sizes are generally nearly equal a t the optimum. Then one can make the following approximation:

(14)

where B* is the common batch size. If monoproducts campaigns are carried out,

B, = Q , / Z , = B*, i = 1, 2, ...) N

1=1

Tc,l being the production cycle time for product PI. By expressing 2, in terms of B * , one obtains

1 N

At the optimal solution, the constraint is satisfied as an strict equality. Therefore,

and

2 , = L__ Q1 H N (16)

The ratio Qi/C(T&) is the proportion of the total available time allocated to the production of Pi.

Initial values for Vj, j = 1,2, ..., M , can be readily found through

At the same time, the capacity-limiting stages for each product are already established.

The procedure can also be applied even if multiple units are running in parallel a t some stages. I t is only required to have the new production cycle time for each product available. The expressions for B * and 2, do not change a t all.

6.2. Running Multiproduct Campaigns. A similar initialization procedure can be developed to find a starting feasible solution involving multiproduct campaigns. Since the optimal solution always includes a number of active campaigns equal to N , one can still assign a different product to each active campaign and vice versa. Let zL,,[ and Q,,,[ stand for the number of batches and the production of PI[, respectively, during campaign 1 where p,, is the product assigned to campaign 1. Since the equal- batch-size assumption works better when multiproduct production runs are allowed, one can write

where C * is the set of active campaigns selected in the

proposed initial solution. The choice of C * is explained later through an example. But

ai,,["[ = Zi1,i (18)

At the optimum, the available production time constraint becomes a strict equality. Then,

and

(21)

Values for T R , i - f'R,l in Table IV indicate that multiproduct campaigns producing ( P I , Pz) or (PI, P3) simultaneously at a ratio of 1:1 are highly time-efficient. By use of them, a good production strategy can be generated. Such a pair of campaigns is going to be identified through the subscripts 1 = 1 and 1 = 2, respectively. The most efficient campaign is used first until the production requirements of one of the products is reached; then

QI1 = Qzl = min (Q, , Q z ) = min (436 000, 324 000) = 324 000

Since P, is only produced during campaign 1 = 1, it is therefore assigned to it (i, = 2). Next, campaign 1 = 2 is running until reaching the production requirement of P, or p3, Qi2 = Q32 = min (Qi - Qii, Qd =

min (112000, 258000) = 112000

Then iz = 1. Up to this point, production needs for both P, and P, have been entirely satisfied, but it still remains to produce an amount of P3 as large as 146 000. The only way is to run a single-product campaign for P3, which becomes campaign 1 = 3. Then Q33 = 146000 and i3 = 3.

Now the above approximate expressions (19), (20), and (21) for B *, ul, and z,, can be used to get a good starting solution. After the values of (z i l ) are set, the initial ap- proximations for {VI) are computed as shown for monoproduct campaigns.

7 . Processing Time as a Function of the Batch Size

If the time required to process one batch of product PI in stage j varies with the batch size, as suggested by Grossmann and Sargent (1979),

T , , = T ~ + C , , B ~ z = l , 2 ,..., N , j = 1 , 2 ,..., M (22)

then the production cycle time Tc(X) and the batch residence time for monoproduct and multiproduct campaigns both become functions of B,. Now

TC(W = max1=1,2, &f (T:,,] + C*,,,&Jk

or


Table VIII. Optimal Solution for Example 1 with n2 = 1 (Capital Cost = $355676) stage capacity (V,/S,,)

product 2 , ZINIT 0: B, 1 2 3 4 PI 317 322 3126 1376 1376 1620 2252 1376 p2 221 239 1549 1466 2 046 1466 2031 1466 p3 221 191 1325 1167 4 869 7493 1167 1510 vINIT u:

Time allocated to each monoproduct campaign.

However, the batch residence time for campaign 1, TR,l, is given by

By including the set of inequalities (23) that set the value of TR,l in the model, changes in Tij produced by variations on the batch size can also be taken into account.

Optimal batch sizes usually show among them slight differences. Therefore, variation of { Tij} with the batch size is expected not to have a major impact on the optimal series of campaigns provided by the model presented in section 4. The only requirement is that the different T, values be given for a similar batch size. However, optimd {zi} and {Bi} will surely experience some changes.

8. Operation of Two Units in Parallel at the Problem Time-Limiting Stage

By solving the proposed mathematical formulation, the problem time-limiting stage featuring the least idle time over the production horizon can be identified. I t usually presents a zero idle time a t some active campaigns and relatively low waiting times a t the others. No better solution can be found if a single unit is to be operated at each stage. The production bottleneck, however, is overcome by running two units in parallel at the time-limiting stage. They should be working out-of-phase to alternatively re- ceive a batch of material from the preceding stage. In this way, the “effective” time required to treat a batch of product in the problem limiting stage is reduced by half. As a result, the total number of batches (2,) can be raised further, producing a decrease in both equipment sizes and plant capital cost. The installation of an additional equipment a t any other stage generally brings about no improvement on capital investment. This type of approach permits handling parallel units without using integer variables and determining a t each step the capital cost savings achieved through an additional equipment. One can stop introducing further units when the economical gain becomes small.

A minor modification makes the mathematical model given in section 4 still works. The production cycle time for each relevant state is to be computed again after reducing the times required for the processing tasks in the limiting stage by two. The new objective function is

M u~

1=lp=l min C aj(Vj)@J

11 211 5544 7686 8882 11 394 5994 6642 9027

I I I p2 I p3 I

I O 0 0

0 , 0 T i m e ( h ) 62?o

L 8 1 +ee2-e3- Figure 5. Optimal batch sizes for example 1 (n2 = 1).

where u, is the number of units running in parallel in stage j. The set of inequality constraints is still given by (3) and (6)--(8).

The incorporation of a second unit running in parallel a t the problem limiting stage does not generally modify the capacity-limiting stages for each product as long as the size factors (Si,) remain unchanged. If the design problem involving a single equipment per stage has already been solved, then the capacity-limiting stages are known. This is still true even if monoproduct campaigns are only run. Let Gil, J i 2 ) be a pair of capacity-limiting stages for Pi. Then,

Therefore,

By substitution, the variable is excluded from the problem formulation. In examples 1 and 2, this kind of relationship permits eliminating two unit size variables. Reduction in the number of variables is of great help for large problems.

9. Results and Discussion The proposed NLP formulation has first been applied

to the solution of example 1 (see Table I). The resulting NLP is solved with the computer code MINOS/AUGMENTED (Murtagh and Saunders, 1981). The evaluation of the batch residence time for each campaign reveals that no multiproduct production run can improve the time efficiency of single-product campaigns. Then, the number of campaign candidates is sharply reduced to 3. Moreover, the model includes 16 equality and inequality constraints and 10 variables. The initialization procedure provides an initial feasible solution {zINIT} quite close to the optimum (see Table VIII). The common batch size B * = 1353 is also a good approximation to the optimal values which look quite similar to each other (see Figure 5). Initial values


Table IX. Optimal Solution for Example 1 with n 2 = 2 (Capital Cost = $352324) stage capacity (V , /S , , )

I ., vroduct 2. 0." B. 1 2 3 4 Z!NIT

p , 464 445 2456 941 g4J 1107 1990 941 p2 323 31 7 2261 1002 1398 g&2 1795 g&2 p3 250 253 1283 1032 3328 5121 1032

"1

Time allocated to each monoproduct campaign.

CAMPAIGN FOR PRODUCT PI

} ,=2

,=3

PRODUCT PRODUCT PRODUCT

b -

I

I ' 3

1'4

- TcJ,, -- T&2 -- T;,3 - Figure 6. Operating scheme of a multiproduct batch plant performing monoproduct campaigns when n2 = 2 (example 1)

for the stage sizes look impressive. Table VI11 shows the optimal solution for example 1. The same result has been obtained when all efficient and nonefficient campaigns were included in the problem formulation. Stage 2 is the problem time-limiting stage featuring no idle time over the production period (see Figure 1). The capacity-limiting stage of each product is underlined in Table VIII. Thus, stages 1 and 4 play that role for P,, stages 2 and 4 for Pp, and so on. The third column reports the proportion of the available production time H = 6000 allocated to each product.

To further lessen the plant capital cost, a second unit is installed in stage 2. The time required for each processing task in that stage is then decreased by two, and consequently stage 3 becomes practically the new time- limiting stage for the three products (see Figure 6). In- deed, stage 4 limits the production cycle time for P,, but the idle time of stage 3 per cycle is extremely low. As a result, no multiproduct campaigns practically get an effective improvement on the batch residence time, and again single-product campaigns are only considered by the model. The number of batches for P, and P2 being pro-

7788 2 X 4097 5871 6171

I PRODUCT PRODUCT PRODUCT

p2 I p3

I Time (h) 6200 0

0 , -- i - - e2 - - e 3 - c

Figure 7 Optimal batch sizes for example 1 (n, = 2).

cessed during the production period have increased significantly (see Table IX). This yields a considerable reduction on the larger batch sizes for n2 = 1 but only a slight savings on capital cost. Furthermore, optimal batch sizes become much closer, making more realistic the equal-batch-size assumption (see Table IX and Figure 7). The common batch size, B *, provided by the initialization procedure is in this case equal to 1021, while the initial stage size values are equal to 8113, 4180, 5802, and 6427, respectively. Another interesting point is that the capacity-limiting stage(s) for each product does(do) not change a t all. On the contrary, an additional one for P, arises.

Things are rather different in example 2 (see Table 11). There is no longer an equal time-limiting stage for the three products, and multiproduct campaigns also become better options. A total of 12 campaigns are handled by the model including the monoproduct production runs (see Table IV). The size of the problem grows since it now comprises 16 constraints in 19 real variables. Nonetheless, it has been solved in 12 s using a VAX 11/780. As expected, the optimal solution includes highly time-efficient multiproduct campaigns (see Table X). They simultaneously produce (Pl, P2) and ( P I , P3), respectively, both at a product ratio of 1:l. The total demand of P2 and a large proportion of the required P1 production is satisfied by the campaign (Pl, Pz) featuring the highest time efficiency (see Table IV). Since the production of P2 is no longer needed, the multiproduct campaign ranking second that manufactures (P,, P2, P3) at a ratio of 2:l:l cannot be performed. The remaining amount of P1 is then produced through the third best campaign, yielding (PI , P3) at a ratio of 1:l. Finally, the production requirement for P3 is completed by running a single-product campaign. This series of campaigns yields a sensible reduction on the batch sizes (see Figure 8).

Therefore, a simple way to establish a near-optimal set of active campaigns and the amounts of products each one is producing has been found. I t is the basis of a heuristic rule stating that the use of the most efficient campaign available to meet not-yet satisfied production requirements should always be favored. Such a campaign must be run as much as possible. This heuristic rule combined with the equal-batch-size assumption has permitted developing the initialization procedure for batch plant design problems


Table X. ODtimal Solution for Examole 2 with n, = 1 (Capital Cost = $344 668) stage capacity (V,/S,,)

Droduct 2 , ZINIT B, 1 2 3 4 p1 337 340 1294 1294 1523 2208 1294

p3 225 200 1145 4 578 7044 1145 1419

Vi 10712 5635 6513 8487

p2 235 252 1378 1923 1378 1992 1378

active campaigns

(1,2,1,2) (1,3,1,3) (3,3,3,3) 118 51 31 126 44 28

81 3719 1543 738

Table XI. Set of Dominant Multioroduct Camoainns for Examole 2 (n, = 2)

campaign product ratios sequence of batches TR,l ik,l TR,l - Pl, P2, P3 2:l:l (1,2,1,3) 24.54 23.61 +0.93

1:2:1 (1,2,3,2)

Z , B: 1 2 3 4 P, 446 976 976 1149 2065 976 P2 312 1039 1450 1039 1863 1039 P3 241 1070 3453 5312 " v, 8079 2 X 4250 6091 6401

active campaigns

" I 156 67 27 4 3833 1515 652

Table XIII. Optimal Solution for Example 3 ( S , = 10, for Any i , j ) "

Pl p2 p3 z, 422 314 250

active camDaiens

"1 54 125 314

O B i = 10320, i = 1-3; V j = 10320, j = 1, 2.

involving multiproduct campaigns presented in section 6. Table X shows a remarkably close to the optimum initial solution for example 2 provided by the procedure. The value of the common size, B*, is equal to 1282.

Since the cycle time for each single-product campaign is the same a t examples 1 and 2, then both have the same optimal solution if monoproduct production runs are only

25.29 25.27 24.56 24.57 28.02 22.54 21.87 23.27 26.02 28.02 26.00

25.32 24.31 24.62 22.91 26.33 22.60 21.90 23.30 26.02 27.03 25.01

-0.03

-0.06 +0.96

+1.66 +1.69 -0.06 -0.03 -0.03 0.00

+0.99 +0.99

allowed (see Table VIII). A comparison of the results included in Tables VI11 and X shows that the use of multiproduct campaigns a t example 2 brings about a no- ticeable reduction on the required unit sizes.

As long as both examples share the set of size factors (Sij) , it is not surprising that their optimal solutions include the same capacity-limiting stages for each product (see Tables VIII, IX, and X). They are underlined in those tables. On the basis of the knowledge of the capacity- limiting stages, the number of problem variables can be lessened significantly as explained in section 8.

Table XI lists the dominant multiproduct campaigns when two units are running in parallel at stage 2. It is clear that the time savings being achieved by running multiproduct campaigns instead of monoproduct campaigns is by far much smaller (see Tables IV and XI). Moreover, a few campaigns are now able to improve the time efficiency of single-product productions runs. I t seems to be that the economical advantage of performing multiproduct campaigns diminishes drastically when an additional unit is operated a t the problem limiting stage. Nonetheless, those campaigns producing (PI , Pz) and (PI , P3) a t a product ratio of 1:l remain among the most efficient ones.

The optimal solution shown in Table XI1 confirms the good performance of the proposed heuristic procedure. The additional unit does not introduce any change in either the production strategy or the capacity-limiting stages for each product. Merely, a new capacity-limiting stage for P3 arises as observed at example 1. Furthermore, the required capital investment increases. Therefore, the use of multiproduct campaigns has an economical edge over the installation of a parallel unit in the time-limiting stage.

Table XIV. Optimal Solution for Example 4 (S,, = 1, for Any i , j ) "

PI P, p , p* ps 2 ; 195 160 178 109 242

active campaigns

VI 15.5 18.7 15.8 42.7 54.6

"Bi = 4210.6, i = 1-5; V, = 4210.6, j = 1-6.


8 , 2000

1000

0 Time ( h l 6200 0

- 0, .__ +- e2 -+---e,- - Figure 8. Optimal batch sizes for example 2 ( n 2 = 1)

Optimal solutions for examples 3 and 4 are shown in Tables XI11 and XIV. They have been found in 8.7 and 20.8 s, respectively, on a VAX 11/780. The size factor (S , ) has been taken to be equal to 10.0 and 1.0 in Examples 3 and 4, respectively, for any 1 = 1, ..., N, j = 1, 2, ..., M . It can be observed that the most efficient campaigns are run to get a much better utilization of the batch equipments. The use of multiproduct campaigns has reduced the common size of the batch equipments by 4.8% and 13.1% in examples 3 and 4, respectively.

10. Conclusions It has been shown that the use of campaigns producing

simultaneously two or more products in multiproduct batch plants can frequently be cost-effective. Considera- tion of multiproduct campaigns requires the solution of a new mathematical formulation for the problem. It is a nonlinear mathematical program that is going to be extended to the design of multipurpose batch plants. Since there are alternative ways of reaching production requirements, both the optimal series of campaigns and the equipment sizes are sought simultaneously. Previously, a MILP formulation helps determine the time-efficient multiproduct campaigns to be considered and their associated batch residence times. Efficiency of the search method is greatly improved by making use of three im- portant findings to establish a good initial solution. First optimal product batch sizes are quite similar to each other (“the equal-batch-size assumption”). Second, a heuristic rule favoring the use of the most efficient campaign available to meet not-yet satisfied production requirements performs quite well in all the examples. Last, capacity- limiting stages for each product experience no change if either the series of campaigns or the number of units being operated a t the problem limiting stage is modified. In- dentification of the capacity-limiting stages permits eliminating many equipment size variables (V,) and a significant number of constraints from the problem formulation. The first two findings are the basis for both a preliminary

campaign screening procedure and a close-to-optimum initialization method.

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 = campaign set H = available production time i = product index j = stage index 1 = campaign index m = number of state vectors describing a campaign Q = production requirement S = size factor T, = production cycle time T, = batch residence time u = number of units in a batch stage u = number of production steps per campaign V = batch stage size z = number of batches over the production period

Subscripts i = denotes product i j = denotes batch stage j 1 = denotes campaign 1

Greek Le t t e r s a = cost coefficient @ = cost coefficient

Li terature Cited Grossmann, I. E.; Sargent, R. W. H. Optimum Design of Multipur-

pose Chemical Plants. Ind. Eng. Chem. Process Des . Dec. 1979, 18, 343.

Mauderli, A . ; Rippin, D. W. T. Production Planning and Scheduling for Multipurpose Batch Chemical Plants. Comput. Chem. Eng. 1979, 3, 199.

Maucierli, A.; Rippin, D. W. T. Scheduling Production in Multi- purpose Batch Plants: The Batchman Program. Chem. Eng. Progress 1980, 76, 37.

Murtagh, B. A.; Saunders, M. A. A Projected Lagrangian Algorithm and its Implementation for Sparse Nonlinear Constraints. MI- N O S ~ A U G M E N T E D User’s Manual; System Optimization Labora- tory, Department of Operations Research, Stanford University: Stanford, CA, 1981.

Sparrow, R. E.; Forder, G. J.; Rippin, D. W. T. The Choice of Equipment Sizes for Multiproduct Batch Plants. Heuristic V.S. Branch and Bound. Ind. Eng. Chem. Process Des. Dec. 1975, 14, 197.

Received for review December 19, 1988 Accepted November 8, 1989

optimal production strategy and design of multiproduct batch plants

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