688
Ind. Eng. Chem. Res. 1991,30, 688-705
Wellons, M. C.; Reklaitis, G. V. Optimal Schedule Generation for a Single-Product Production Line, Part 1: Problem Formulation. Comput. Chem. Eng. 1989a,13 (l),201-212. Wellona, M.C.;Reklaitis, G. V. Optimal Schedule Generation for a Single-Product Production Line, Part 2: Identification of Dominant Unique Path Sequences. Comput. Chem. Eng. 1989b,13 (l),213-227. Wellons, M. C.; Reklaitis, G. V. Optimal Schedule Generation for a Single-Product Production Line, Part 3: MINLP Solution Stra-
tegies. Submitted to Comput. Chem. Eng. 198%. Wellons, M. C.; Reklaitis, G. V. Scheduling of Multipurpose Batch Chemical Plants. 2. Multiple-Product Campaign Formation and Production Planning. Ind. Eng. Chem. Res. 1991, following paper in this issue.
Received for review December 10,1989 Revised manuscript received July 26, 1990 Accepted August 10,1990
Scheduling of Multipurpose Batch Chemical Plants. 2. Multiple-ProductCampaign Formation and Production Planning Michael C. Wellons and Gintaras V. Reklaitis* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
An efficient procedure to identify the dominant multiple-product campaigns for multipurpose batch chemical plants is presented. On the basis of a linear dominance property, the dominant campaigns are identified as the noninferior extreme points of the associated multiobjective campaign formation problem. The noninferior set estimation method has been incorporated into a decomposition strategy which alternately solves an equipment group master problem to identify dominant equipment group profdes for the production lines of the campaign and a campaign formation subproblem with identifies the dominant campaigns for a particular equipment group profile. A multiperiod mixed integer linear programming production planning model for multipurpose batch plants that allots the available production time to a subset of the dominant campaigns and accounts for lost production time due to changeovers and startup times is also presented. The campaign formation and production planning procedures are illustrated with an example problem.
Introduction In the preceding paper (Wellons and Reklaitis, 1991), an efficient decomposition solution strategy was developed to solve the single-product campaign formation problem. In this paper, the decomposition strategy is extended to identify dominant multiple-product campaigns and a production planning model for multipurpose batch chemical plants is presented. A number of researchers have developed production planning models for batch chemical plants, including Mauderli and Rippin (1979),Lazar0 and Puigjaner (1985), Rich and Prokopakis (1986, 1987), Suhami and Mah (1984), Musier. and Evans (19881, and Vaselenak et al. (1987). Since a large number of alternative processing configurations are possible, a production planning model for multipurpose batch plants should have the capability to select from a set of alternative campaigns to meet the particular production requirements. Additionally, given the complexity of product recipes and the large changeover and startup times required for some products, the planning model should account for lost production time due to changeover and startup times for the production lines in each campaign. Unfortunately, most planning models only allow one processing configuration for each product and do not consider the effects of changeover and startup times. Rich and Prokapakis consider multiple production routes, but do not account for changeover and startup times. Mauderli and Rippin's planning model selects from a set of dominant campaigns, but allows only for a constant changeover time between all campaigns. In the following, the single-produd campaign formation decomposition algorithm is extended to identify dominant multiple-product campaigns based on the same linear dominance property used by Mauderli and Rippin. A production planning formulation for multipurpose batch planta is presented that selects from the set of previously
identified dominant campaigns and accounts for changeover and startup times of the production lines in each campaign. The dominant campaigns are then identified for the example multipurpose plant from Mauderli and Rippin, and several production planning problems are solved to illustrate the improvement in plant performance using the optimal dominant campaigns compared with the campaigns identified by Mauderli and Rippin.
Definition of Dominant Campaigns The goal of the campaign formation procedure is to identify a'small number of good campaigns to which the production planning procedure will assign the available production time. If the processing rates of the singleproduct production lines of each campaign are used as the selection criteria, then the set of good campaigns should consist of those campaigns that have the highest processing rates for a given product ratio. To quantify this idea, let Rc = (&, ...,Ri,, ...,Rpc) denote the vector of processing rates for each product in campaign c. Campaign 1dominates campaign 2 if R11 & or Ril 1 Ri2 for i = 1, ...,I. To satisfy a particular production requirement, any product ratio of I products can be achieved with Z (or fewer) suitable campaigns with the appropriate campaign lengths. Consequently, a campaign producing I products is inferior if a set of not more than I campaigns can be used to produce the same ratio of products but at a higher rate. Let CD denote the set of dominant campaigns. Neglecting the loss of production time due to changeovers and startup times, an inferior campaign q is dominated by a convex combination of the dominant campaigns.
The resulting linear dominance property is the following:
0888-588sf 91f 2630-0688$02.50/0 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 689
0 dominant campaign
maker as to the best campaign for the particular production requirements, the set of noninferior solutions (the dominant campaigns) for the above problem should be generated. A number of techniques to generate the noninferior solutions to multiobjective problems have been proposed, including the following (Cohon, 1978): i. the weighting method (Zadeh, 1963; Geoffrion, 1966), in which the individual objectives are combined into a weighted sum to form a single-objective optimization problem. The noninferior solutions are identified by solving the single-objective problem for a systematic variation of the weights. ii. the constraint method (Cohon and Marks, 19731, in which only one of the individual objectives is optimized subject to constraints on the remaining individual objectives. The noninferior solutions are identified by solving the single-objective problem for a systematic variation of the individual objective constraints. iii. the noninferior set estimation (NISE) method (Cohon et al., 1979), which also combines the individual objectives into a single weighted sum objective. However, the weights are generated on the basis of the current set of noninferior solutions so that each successive optimization problem attempts to generate a new noninferior solution. iv. the multiobjective simplex method (Yu and Zeleny, 1975; Zeleny, 1974), which is a specialized algorithm to identify all noninferior extreme points for problems that can be formulated as linear programs. The disadvantage of the first two techniques is that if the grid of weights or constraints is too coarse, some noninferior solutions may not be identified, while if the grid is too fine, many unnecessary problems will be solved. The NISE method does not suffer from these deficiencies, since successive optimization problems attempt to identify new noninferior solutions. Additionally, the NISE method can be easily modified to include the decomposition strategy to generate the dominant campaigns. In the following, the NISE method as developed by Cohon et al. (1979) is presented and modifications to the algorithm to incorporate the decomposition solution strategy are discussed.
The Noninferior Set Estimation Method The NISE method was originally developed for bicriteria problems but can be extended to problems with three or more objectives. The method for problems with two objectives is presented, and the modifications required for higher dimensional problems are discussed. It is assumed that the feasible region is convex and the objective functions are linear. The NISE method operates by identifying a number of noninferior extreme points w d examining the properties of the line segments between them. If two noninferior extreme points are known, then the line segment between these two points is feasible and either inferior or noninferior. If the line segment is noninferior, then moving in a direction out from the line segment is infeasible. If the line segment is inferior, then noninferior points can be found by moving in an outward direction. Noninferior points are found in the NISE method by selecting weighta so that the next noninferior point is the feasible solution farthest out in a perpendicular direction from the line segment. The method begins by optimizing the individual objectives yielding points A and B as shown in Figure 2. For generality, let Z1and 2, denote the individual objectives. The next solution is found by solving the weighted problem
690 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991
2. If \ki,i+l 5 9 " for i = 1, ..., n - 1, then STOP. Otherwise, go to step 3. 3. Search the \kij+l, i = 1,..., n - 1for the largest value. Solve the weighted problem for the line segment i,i+l that has the largest maximum possible error \ki,+p Let Z(i,i+l) denote the value of the objective at either Si or Si+l.If Z(i,i+l) > Z(i,i+l), denote the new inferior solution as Pn+l.Go to step 4. Otherwise Z(i,i+l) = Z(i,i+l). Set \ki,i+l = 0 and return to step 2. 4. Reorder the points P,, t = 1, ..., n + 1 according to the following scheme: s; = s, t = 1, ..., i
35 30
'\$#
0 0
I
I
I
I
I
5
10
15
20
25
"B
VA
30
I 35
R Figure 2. The noninferior set estimation method.
(objective Z = wlZl+ w2Z2)with weights w1and w2 that satisfy
--W1
-- slope of line AB
w2
For the problem in Figure 2, the next noninferior point identified would be point C. The algorithm would then continue by solving the weighted problem for line segments AC and CB to identify points D and E. For problems where the set of noninferior solutions is continuous and nonlinear, this process could go on indefintely. Consequently, Cohon (1978) presents an error estimation procedure to calculate the maximum error for a given set of noninferior extreme points. For linear problems or discrete problems (such as the campaign formation problem), the procedure could also be continued until all line segments have been shown to be noninferior. The NISE method is presented below. Notationally, let P, be the tth noninferior extreme point generated by the algorithm. As the algorithm progresses, only those line segments that are currently noninferior need to be examined. To aid the identification of those line segments, the current noninferior extreme points are ordered by decreasing value of 2,. Let Sibe the noninferior point with the ith highest value of 2,. After reordering, the noninferior line segments are those segments formed by adjacent extreme points. The weights are calculated on the basis of the slope of the line segment connecting points Si and Si+l,so that --W I - z2(Si) - z2(5'i+1) w2 Z l ( S i ) - Zl(Si+l)
-
or
w1 = &(Si) - z2(si+1) w2 = Zl(Si+l)- Zl(Si) Finally, let \kij+l be the maximum possible error in the line segment connecting Si and Si+l(see Cohon (1978) for details) and let be the maximum allowable error. The index n is used to indicate the number of noninferior solutions currently generated. The NISE algorithm is as follows: 1. Maximize Z1 and Z2 individually. Let P, be the optimal solution for objective Z1and let P2 be the optimal solution for objective Z2. Set S1= P2, S2 = Pl, and n = 2. Calculate \k12.
*-
S:+l = Pn+l t = i + 1, ..., n = s, A similar reordering of the terms \k,,,+l as well as calculation of \ki,i+l and \ki+l,i+2 is also required. Set n = n + 1 and return to step 2. To identify the entire noninferior set, simply set \kmsx = 0 and let \ki,i+l = 0 if line segment (i, i+l) is known to be noninferior and let \kij+l = 1otherwise. Step 3 of the algorithm is repeated until all \ki,i+l are zero. For higher dimensional problems, the NISE method proceeds in a similar fashion except that the weighted optimization problem must be solved for the noninferior faces of the polyhedron formed by the current set of noninferior extreme points. Thus in step 1of the algorithm, the single objective optimization problems are solved yielding n extreme points for a problem with n objectives. In step 2, the algorithm checks to see if the current set of noninferior faces provides a sufficient description of the efficient frontier. If not, then in step 3 a weighted objective problem is solved for one of the noninferior faces of the polyhedron of extreme points with the weights derived from the normal to the face. If a new extreme point is identified, then ;in step 4 the set of noninferior faces is updated. Yu and Zeleny (1975) present an efficient algorithm to identify the noninferior faces of a polyhedron formed from a set of noninferior extreme points.
NISE Decomposition Algorithm for Multiple-Product Campaign Formation The underlying assumption of the NISE method, that the noninferior set is convex, is met in our application involving the selection of dominant campaigns by virtue of the linear dominance construction. In the space of the rate variables, for single-product campaigns, there clearly can only be one dominant sbhtion and thus convexity holds trivially. For multiproduct campaigns, any linear combination of campaigns, single or multiproduct, with their associated rates can always be physically implemented and hence will be feasible. Thus, in the rate space the noninferior set is convex by construction. The key assumption, however, is that the campaigns are sufficiently long so that changeover times can be neglected. As shown in a later section, if the changeover time is a significant portion of the campaign length, then the convexity assumption may no longer hold and other campaign selection approches must be employed. Thus, under the above assumption, the NISE algorithm could be applied directly to the previously developed mixed integer nonlinear programming (MINLP) formulation for single-product campaign formation (suitably modified for multiple-product campaigns). However, it was shown in part 1that the original formulation required significant computation time. For single-productcampaign formation problems, at least an order of magnitude reduction in computation time was achieved by decomposing the
Ind. Eng. Chem. Res., Vol. 30, No. 4,1991 691 Dominant Single Product Campaigns
MJNLP
stop
NISE Decomposition Algorithm
r l Weighted NISE Equipment Group Master Problems
~
$
~
~
~
~
$
i I E/ Dominant EG Profile
'yes1' Weighted NISE
Update
Subproblems I
I
Identify Dominant Campaigns
1. Maximize Z1 and Z2 individually. Let P1 be the optimal solution for objective Z1and let Pzbe the optimal solution for objective Z2. Set S1= Pz and S2 = PI. Set PD = {1,2),n = nD = 2. Set = $12 = 1. s The equipment group master problem. 2. a. Solve the weighted EG master problem for any line segment (i,i+l) with \k$+l = 1. b. Let ZEG(l,l+l) denote the value of the objective at either Si or Si+l.If ZEG(i,i+l) > ZEG(i,i+l),then a dominant equipment group profile has been identified. Go to step 3. c. If ZEG(i,i+I) IZEG(i,i+l), set \kT,i+l= 0. If all = 0, then STOP. No EG profile can be found that dominates the current set of noninferior solutions. Otherwise, return to step 2a. 3. The campaign formation subproblem. a. Set \ki,i+l = \kT,i+l for i = 1, ..., nD - 1. b. Solve the weighted CF subproblem for any line segment (i,i+l) with \ki,i+l = 1. c. Let &F(1,1+1) denote the value of the objective at either Sior Si+l. If ZCF(i,i+l) > Z&,i+l), then denote the new dominant solution Pn+l.Go to step 4. d. If ZcF(i,i+l) IZcF(i,i+l), set \ki,i+l = 0. If all \ki,i+l = 0, then all new dominant points for the current EG profile have been identified. Add an integer cut to the EG master problem to eliminate the current EG profile and return to step 2. 4. Add the new dominant point to PD If the new point Pn+lhas the qth highest value of Z2,reorder the points, Pt,t = 1, ..., n + 1 according to the following scheme: s; = s, t = 1, ..., q - 1
Si = Pn+l = st t = q , ..., n Increment nD by one and go to step 5. 5. Check dominance of the current points in PD a. Set d- = 0. If line segment (q-d--2,q) dominates SH+ increment d- by one. Repeat until some point Sd-l dominates the line segment ( q 4 - 2 4 ) . b. Set d+ = 0. If line segment (q+d++2,q) dominates Sq+d++l, increment d+ by one. Repeat until some point Sq+d++l dominates the line segment (q+d++2,q). c. If d- > 0 or d+ > 0, then delete the points Sq+,t = 1, ..., d-, or S q + t , t = 1, ..., d+, respectively, from PD. Renumber the remaining points in PD according to t = 1, ..., q - d- - 1 S; = St Sh-dgt-d--d+
= st
= sq= Pn+l
t=
+ d+ + 1,
e..,
n
Reassign values to \kii+l and \kf'i+laccording to t =I, .-, 4 - d- - 2 *;,t+i = *t,t+i *?,\+I = *&+I \kb-d:l,q-d-
=1
\k&--l,q-d-
=1
\k&-,q-d-+l
=1
\k&j-,q-d-+l
=1
%-d--d+,t-d--d++l
=
@t,t+l
\k?Ld--d+,t-d--d++l
t =q
= *?,t+l
+ d+ + 1, ..., nD
Return to step 3b. The Multiple-Product Equipment Group Master Problem The equipment group master problem for single-product campaign formation developed earlier can be easily ex-
692 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991
v
vFwijsvijs
(po
Ti 5
1"
w-
C, v&"u~~,
vi,j
u= 1
+c..g+,
" N60.00 M&R campaigns 0.08 LP relaxation 852 737 -f 680/808 0.93 10.00 LPIMILP 1 767418 371249f 290 9.12 27.28 LPIMILP 2 774933 6413521416 8.77 full MILP 777 910 152/680/808 >60.00
h. The changeover cost is $5000. The production planning problem has been solved with both the optimal set of dominant campaigns and the Mauderli and Rippin campaigns. As for example 1, the LP/MILP strategy is compared with direct solution of the full MILP. Table XI11 shows the computational results for the planning problems. Since the CPU times for these problems are much larger than for example 1, several LP/MILP approaches are used. In the first LP/MILP strategy, campaign c is available in period p for the MILP problem only if campaign c had nonzero production time in period p for the relaxed LP. In the second LP/MILP strategy, campaign c is available for all periods in the MILP problem if campaign c was assigned nonzero production time in any period of the relaxed LP. As for example 2, the LP/MILP strategies required significantly less CPU time than the full MILP. The CPU time for the full planning problem was limited to 60 CPU min, and the profit from the LP/MILP 1strategy was used as a lower bound on the objective. The profit shown in Table XI11 for the full MILP is the objective value of the incumbent solution after 1 CPU h. All other problems were solved to optimality. The full MILP planning problem with the optimal campaigns was not able to improve on the objective obtained by the second LP/MILP strategy, while the planning problem with the M&R campaigns improved the objective by only 0.38%. The best production plan with the optimal dominant campaigns yields a 5.11% improvement over the best production plan with the Mauderli and Rippin campaigns. Since solution of the full MILP to optimality is likely to require significantly more CPU time, it is clear that if computation time is at a premium, the LP/MILP strategies are very attractive for larger production planning problems.
Table XIV compares the campaigns selected by the planning problems and the campaigns lengths and production times for each production line. Tables XV and XVI show the production, consumption, sales, and inventory of each product in each period for the optimal dominant campaigns and M&R campaigns, respectively. In the first period, the product combinations selected by the two planning problems are the same for all four campaigns. However, the optimal campaigns wR12 and wR35 dominate the corresponding Mauderli and Rippin campaigns MR14 and MR28, respectively. As a result, the optimal dominant campaigns produce an additional 2790 kg of D and 560 kg of G as well as additional amounts of the intermediates B, E, H, and J, while the M&R campaigns produce an extra 1720 kg of product F. A similar analyais of the other three periods shows that in most cases the two planning problems select similar product combinations and product ratios. Over the entire planning horizon, the optimal dominant campaigns produce an additional 5240 kg of D, 2510 kg of G, 5000 kg of C, 6070 kg of E, and 2620 kg of B, while the M&R campaigns produce an extra 8420 kg of F. Since the profit of product F is only $1.00 compared with $5.00 for product D and $3.50 for product G, the additional production of these highly profitable products far outweights the additional profit obtained with the M&R campaigns from the sale of product F.
Effect of Changeover and Startup Times on the Set of Dominant Campaigns As noted earlier, many of the previous production planning formulations assume that production occurs continuously at the nominal rate (batch size/cycle time). However, for large changeover and/or startup times, the effective production rate will be much less than the nominal rate. Let RIcpdenote the effective processing rate of product i in campaign c during period p . The effective rate is a function of the campaign length as well as the changover and startup times and is given by Ricp = Ri,-
T icp
CL,
=
T 1 icp
Ricmax t € I c COT^,
+ sui,+ T
~ ~ ~ )
Even more importantly, large changeover and startup times affect the set of noninferior processing rates, since a linear combination of q campaigns requires q changeovers and startups. To illustrate this observation, consider a simple example where products A and C (not the Mauderli and Rippin products) can be produced on four alternative campaigns: campaign 1, which produces only product A at a nominal rate of 40;campaign 2, which produces only product C with a nominal rate of 30; and campaigns 3 and 4, which produce both products at nominal rates of (15 C, 30 A) and (20 C, 15 A), respectively. Assuming a fixed campaign length of 200 h, Figure 10 illustrates the effect of a changeover time ranging from 0 to 30 h on the effective rates of the campaigns and the set of noninferior rates for this product combination. When the changeover time is 0, the effective rates of the campaigns are the same as the nominal rates, and campaign 4 is clearly dominated by a linear combination of campaigns 2 and 3. As the changeover time increases, more and more of the available production time is lost, so that the effective rate of the campaigns is reduced. Additionally, the effective rate of a combination of two campaigns is no longer the line segment joining the effective rates of these campaigns, since an additional changeover time is required. Consequently, for product ratios near one of the four campaigns, it is better to throttle back on the production of one of the
Ind. Eng. Chem. Res., Vol. 30,No.4, 1991 701 Table XIV. Comparison of the Campaigns Selected for Production, th;Campaign Example 3
WR12
425.76
WR13
626.30
WR20
317.75
WR35
630.19
WR4 WR8 WR13
494.28 393.85 577.63
WR16
604.45
WR30
229.79
WRlO
687.15
WR13
230.97
WR30
369.96
30.33 40.04 33.99 42.64 65.15 67.79 36.38 36.74
D A B E C E F G
97.09 55.96 33.99 42.64 77.94 42.64 46.69 71.45
Period 2 454.03 340.84 504.29 532.71 256.61 259.53 180.78 164.16
A
33.52 42.64 33.99 42.64 46.69 71.45 36.38 36.74
Period 3 596.66 643.23 157.63 186.05 320.96 265.92 656.95 653.27
55.96 33.52 42.64 33.99 42.64 77.94 42.64 65.15 67.79
Period 4 359.19 298.33 343.90 343.24 371.66 320.76 323.68 369.13 291.21
B E F H J
H
WRB WRlO
412.20 388.82
WR13
416.58
E B E
WR16
368.60
C
413.81
E D
WR20
campaign
A F B E D G H J
G 711.91
TiCP
Period 1 329.71 MR14 371.71 552.96 MR23 581.38 273.07 MR34 155.79 575.23 MR28 571.55
E
WR35
Mauderli and Rippin
Wellons and Reklaitis CL, Ric
campaign
A
G
Lengths, and the Production Times for
CLCP
Ticp
356.89 414.59 499.36 551.98 230.24 147.51 605.31 604.23
468.64
A F
598.89
B E D G H J
28.02 40.04 33.99 42.55 65.15 67.79 33.64 33.10
D H B E C E D G
91.98 55.96 33.99 42.55 74.24 41.89 65.15 67.79
304.90 350.92 614.66 641.09 269.40 278.48 194.72 177.02
A F
28.02 40.04 55.96
713.78 752.39 427.09
41.89 96.29 41.13 59.15
201.19 189.25 432.57 422.65
55.96 32.34 42.55 33.99 42.55 74.24 41.89 65.15 67.79
359.19 309.21 354.34 367.02 393.45 269.40 278.48 382.96 269.90
274.92 657.56
MR4 MR8 MR23
350.12 403.17 688.00
MR31
319.32
MR34
239.40
MR14
806.44
MR8
479.34
H
MR13
242.03
E G
MR18
Ric
472.19
E J
H
MR8 MRlO
411.44 401.25
A
MR23
440.36
E B E
MR31
319.32
C
MR34
427.64
E D G
Table XV. Production and Inventory Results for Example 3 Using the Optimal Dominant Campaigns product
init inventory
amt produced
amt consumed
amt sold
final inventory
amt produced
10 000 0 0 17 790 0 0 10 290 20 OOO 20 OOO
100 1oOOO 100 100 100 11815 370 1027 1099
0 17 141 20 OOO 44 082 33 781 8 441 11730 19073 0
0 0 22 041 5000 0 20 OOO 0 18131 28 781 5000 3 519 10 OOO 0 12OOO 0 20 OOO 0 0 Period 4
20 OOO 5 000 0 25 OOO 0 10 000 19OOO
100 458 100 1051 3 439 6 022 100 2000 100
10 000 11666 25 OOO 24 049 44312 0 19740
0 12024 0 0 42 652 5 922 0
10 OOO 0 25 OOO 25 OOO 5000 0 19741
100 100 100 100 100 100 100
20 100
0
22 OOO
100
0
0
0
100
Period 1 A
B C D E F G H J A B C D
E F G H J
100 100 100 100 100 100 100 100 100
10OOO 18795 0 17 790 24 789 14 883 10 560 20 927 20 999
0 8 895 0 0 24 789 3 168 0 0 0 Period
100 100 100 26051 100 6 737 100
20 OOO 5 358 0 0 35318 14 985 19OOO 23 900 24 001
0 0 0 0 31 979 5 700 0 0 0
100
lo99
two products than to incur an additional changeover time and use two campaigns in series. As a result, campaign 4, which was dominated for a zero changeover time, becomes a dominant campaign for changeover times larger than 15. Thus the set of dominant campaigns that are identified on the basis of the linear dominance property may not yield the best solution to the planning problem
amt sold
final inventory
Period 2
3
22 000 25 OOO
amt consumed
100 100 100 26051 100 6 737 100 100 1099
when changeover and startup times are large. Given the above observation, how does one identify these good but dominated campaigns that become dominant for large changeover times? Clearly the usefulness of these campaigns depends directly on the product ratios of the desired productg and the resulting campaign lengths selected by the planning problem. Rather than identify a
702 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 Table XVI. Production and Inventory Results for Edmple 3 Using the Campaigns from Mauderli and Rippin product
init inventory
A
100 100 100 100 100 100 100 100 100
10 OOO 16 973 0 15 OOO 23 486 16 600 10 OOO 20 363 20 OOO
100 5 100 100 25 150 3 197 100 100
20 000 0 0 0 26 219 30 126 18 223 23 900 25 OOO
B C D E F G
H J A
B
C D E F G H
amt produced
100 100
J
amt consumed
amt sold
Period 1 0 10 OOO 7 500 0 0 0 0 15 000 23 486 0 3000 0 0 10 OOO 0 20 000 0 20 OOO Period 3 0 20 OOO 0 5 000 0 0 0 25 000 28 755 0 5 467 15 000 0 18 223 0 22 000 0 25 000
final inventory
amt produced
100 9 573 100 100 100 13700 100 463 100
0 20 892 20 OOO 40 730 38 943 0 12 OOO 19 637 0
100 100 100 150 661 9 759 100
10 OOO 12 475 20 OOO 24 950 43 484 0 18 296 20 100 0
2000 100
amt consumed
amt sold
Period 2 0 0 20 365 5000 0 20 OOO 0 15680 30 846 5000 3 600 10 OOO 0 12 OOO 0 20 OOO 0 0 Period 4 0 10OOO 12 475 0 0 20 OOO 0 25 OOO 39 045 5000 5 489 4 170 0 18 296 0 22 OOO 0 0
final inventory 100 5 100 100 25150 3 197 100 100 100 100 100 100 100 100 100 100 100 100 100
C a m p a i g n length = 200
o COT = 0 0
COT = 16 COT = 30
Figure 11. Flow network for equipment assignment.
0
5
10
15
20
25
30
35
R Figure 10. Effect of changeover times on effective rate of product combination AC.
small set of dominant campaigns to which the planning model allots production time, it might be beneficial to examine an alternative decomposition of the production planning and campaign formation problems where a rough planning model including changeover and startup times is solved first to identify good product combinations and product ratios, and then solve the campaign formation problem only for the desired product combinations and product ratios. The conceptual advantages of this decomposition are good but dominated campaigns like campaign 4 in the above example could be identified by the rough planning model and the campaign formation problem would only have to be solved for those product combinations and product ratios that are needed for the particular production requirements. However, the conceptual disadvantages or difficulties with this approach include representation of the set of achievable noninferior rates in the rough planning model and the bilinearities that result when processing rate is treated as a variable (the product of rate and time) and their effect on the solution strategy. Limited computational studies (Wellons, 1989) have shown the potential for this approach to identify alternate campaigns, but rigorous algorithmic strategies remain to be developed.
Conclusions An efficient procedure to identify the dominant multiple-product campaigns for multipurpose batch chemical plants was presented. On the basis of a linear dominance property, the dominant campaigns are identified as the noninferior extreme of the associated multiobjective campaign formation problem. The noninferior set estimation method has been incorporated into a decomposition strategy that alternately solves an equipment group master problem to identify dominant equipment group profiles for the production lines of the campaign and a campaign formation subproblem that identifies the dominant campaigns for a particular equipment group profile. The optimal dominant campaigns were identified for the multipurpose batch plant from Mauderli and Rippin, and many of the Mauderli and Rippin campaigns were shown to be suboptimal. The decomposition strategy yields an order of magnitude reduction in computation time compared with direct solution of the multiple-product campaign formation with the NISE algorithm. A multiperiod MILP production planning model for multipurpose batch plants that allots the available production time to a subset of the dominant campaigns and accounts for lost production time due to changeover and startup times was presented. A comparison of the production plans for several production planning problems showed a 4-5'70 improvement in profit using the optimal dominant campaigns versus the Mauderli and Rippin campaigns. Finally, the set of dominant campaigns identified on the basis of the linear dominance property may not yield the optimal solution to the production planning
Ind. Eng. Chem. Res., Vol. 30,No. 4,1991 703 problem if the changeover and startup times are sufficient large. Acknowledgment This work was supported in part by a National Science Foundation Fellowship, National Science Foundation Grant No. CBT-8518175,and E. I. du Pont de Nemours and Co. We also thank J. Viswanathan and Prof. Ignacio Grossmann at Carnegie Mellon University for facilitating the implementation of DICOPT and DICPT++ at Purdue University and for their helpful advice during its use in this study. Nomenclature b index on units of a group B;: total batch size for product i for path sequence 8..*characteristic batch size for product i on stage j B P : maximum feasible batch size for product i on stage j Bijbk: the bth portion of the path k batch of product i processed on stage j 8,: characteristic batch size for product i, stage j , path k Bik: batch size for product i on path k c: index on campaigns CL : campaign length for campaign c, period p CO& changeover cost for campaign c COTi,: changeover time for product i, campaign c dCp: 0-1 assignment variable for campaign c, period p Aik: start time for the path k batch of product i e: index on equipment type Eij: set of feasible equipment types for product i, stage j E,: ending time for product i on stage j , path k f: index on equipment family fii: amount of product i required to product unit amount of product i* Pi: set of products that use product i as an intermediate g: index on groups of a stage Gi,-- binary variable denoting g groups on stage j for product
h,: holding costs for inventory of product i, period p i: index on products I . * inventory of product i at the end of period p minimum inventory of product i at the end of period P maximum inventory of product i at the end of period P j : index on stages JH: set of stages on which holding is allowed JEf set of stages that use the equipment types in family f k: index on paths LSi,,: level above which sales of product i, period p yield reduced profit Mi: number of stages in production line for product i NE,: number of equipment of type e NEf: number of equipment in equipment family f NP; number of paths in the sequence for product i NP-: maximum number of paths in the sequence for product i NG.,. number of equipment groups on stage j , product i N G P : maximum number of equipment groups on stage j , product i NU-: maximum number of units in an equipment group for stage j, product i Oijyk: overlapping time between stages j and j ’ on path k, product i qj., + eijyB$W relationship between overlapping time and batch size for product i P i l : processing time on stage j , path k, product i pd, + ci*: relationship between processing time and batch size for product i p: index on periods pi: profit for product i
e:
r:
rpc reduced profit for product i R; average processing rate for product i Ri,: processing rate of product i, campaign c S,: size factor for stage j , product i Si$ starting time on stage j , path k, product i S. * total sales of product i in period p lower bound on total sales of product i in period p upper bound on total sales of product i in period p Siijg: sequence starting time on group g, stage j , product i SE,: sequence ending time on group g, stage j , product i Sui,: startup time for product i, campaign c Ti: sequence cycle time for product i Ticp:production time for product i, campaign c, period p Vi,+ binary variable denoting u units on stage j for product
+. +: 1
vi: value of product i in inventory Vie: volume of equipment type e assigned to product i V... volume assigned to stage j , product i V% minimum volume to be assigned to stage j , product i maximum volume to be assigned to stage j with u units for product i V E * total volume of equipment items in equipment family
e?:
binary assignment variable for equipment type e, product i, stage j , group g, position b XSip: excess sales of product i in period p
&,&,:
Appendix A. Conditions for a Feasible Product Combination To show that the subset of products I, is a feasible product combination, it is sufficient to show that a feasible equipment assignment exists for a campaign producing all products i E I,. The conditions required for the existence of a feasible equipment assignment will be proved by showing that a feasible flow exists in the corresponding network flow problem. Consider the flow network (s,t, J,E, A) shown in Figure 11 with source s, terminal t, and nodes J corresponding to the stages of each production line in the campaign and nodes E corresponding to the feasible equipment types in the plant. The arcs a(j,e) E A exist if equipment type e is feasible for stage j of the production line. The arcs a(sJ have a maximum flow of 1,indicating that an equipment item can be assigned to each stage of the production line. The maximum flow of the arcs &‘,e) is 1, indicating that at most one unit is assigned to stage j , while the maximum flow of the arcs (e,t) is NE,, indicating that no more than NE,units of type e can be assigned to the production lines. A feasible equipment assignment for the campaign exists if a maximum flow in the network equal to the total number of stages can be found. The conditions required for a maximum flow in the network will be determined through application of the max-flow, min-cut theorem. An s-t cut of the flow network of the nodes into sets Wand W such is a partition ( W, that s E Wand t E W.The capacity C of the s-t cut is the capacity of the forward arcs: those arcs that go from W into W.Thus for the flow network shown in Figure 11, the capacity is
C
= la(s,j)l jea
+
+
la(j,e)l NEe j e W, e€ W eoW
(A.1)
The max-flow, min-cut theorem states that the maximum flow is equal to the minimum capacity cut. Let M,denote the total number of stages in the production lines for products i E I,. If there is to be maximum flow of M,, then C must be greater than or equal to M,. Let J, denote th? stages j E W and let JIBdenote those stages j E W. Furthermore, J , can be partitioned into J, and Ju, where Qw contains all those stages j E J, for which no arcs aG,e),
704 Ind. Eng. Chem. Res., Vol. 30,No. 4, 1991 e E W,exist and Jwecontains those stages j E J , with arcs a(j,e), e E W. Since
then
C IM, - lJwel+ C NE, e€
W
Consequently, a maximum flow of M, requires C 1 M, so that
C - M, 1 C NE, - lJwelI0 e€
W
(A.3) This condition states that for any subset W of equipment types, the number of stages whose set of feasible equipment types is completely contained in W must be less than the number of equipment items in W. Since the equipment types in the plant can be partitioned into equipment families E], with a corresponding set of stages JEfswhose set of feasible equipment types is completely contcuned in E the above condition can be expressed as follows: Let denote a subset of the equipment types in equipment family E, and let JWf denote the corresponding set of stages for products i E I, for which all feasible equipment types are contained in W,. The subset of products I, is a feasible product combination (there is a maximum flow of M,in the corresponding flow network) if
4
C NEe21JW,-j
e€
W,
(A.4)
ai" to satisfy the minimum sales requirement for period p . wp=
Let be the maximum amount of product i that can be produced in period p .
Let
- Z$=
+ i*€Fi C fii*(Spr + Z$r- Z$f)
Ginbe the minimum time required to produce Q p i " 6,
Bound 3: This bound calculates the cumulative production time available from the beginning of the planning horizon less the minimum time required to produce the minimum amount of all other products. Let C O T mand SUFinbe the minimum changeover and startup times of all the production lines that produce product i. Let NI$" be the largest number of products (including product i) that are produced simultaneously in any campaign that produces product i. Since the production of product i essentially uses (l/NIP")th of the available production time during the campaigns producing product i (One can think of the production horizon consisting of NI- parallel production slots, and the production line that produces product i occupies only one of those slots.), a lower bound estimate of the production time required to produce the minimum amount of product i in the first p periods, denoted T P T , is * D
Consequently, an upper bound on the available production time to produce product i during period p is T3$!
= P
Since the production requirements for all products produced in the same campaign c could be satisfied simultaneously with product i, only the production time required to produce the minimum amount of all products i E I, through the first p - 1 periods is subtracted.
Literature Cited
Appendix B. Bounds on the Production Time Let be the minimum amount of product i required
QV = S$= +
T 2 g = Qg=/Ric
P
= Qf"P'"/Rf""
where R Y is the single-product processing rate. Three upper bounds are generated for Tie,so that
QF = min ( T l z ! , T2gF, T 3 3 Bound 1: Regardless of the amount of the other products produced during period p , there must be at least CIFm - 1 other campaigns during period p in addition to the campaign that produces product i. Let CLminbe the smallest minimum campaign length for all campaigns. The first bound is the period length less the minimum production time allotted to the other campaigns in the period. TlEF = L, - CL""(C1,"" - 1)- COTi, - Sui, Bound 2: The second bound is the time required to produce the maximum amount of product i that cqn be produced in period p subject to sales and inventory constraints.
Cohon, J. L. Multiobjectiue Programing and Planning; Academic Press: New York, 1978. Cohon, J. L.; Marks, D. H. Multiobjective Screening Models and Water Resource Investment. Water Resour. Res. 1973, 9 (4), 826-836. Cohon, J. L.; Church, R. L.; Sheer, D. P. Generating Multiobjective Trade-offs: An Algorithm for Bicriterion Problems. Water Resour. Res. 1979, 15 (5), 1001-1010. Geoffrion, A. M. Solving Bicriterion Mathematical Program. Oper. Res. 1966, 15, 39-54. IBM. "IBM Mathematical Programming System Extended/370 (MPSX/370), Basic Reference Manual"; Technical Report: IBM White Plains, NY, 1979. Kocis, G. R.; Grossmann, I. E. Computational Experience with DICOPT Solving MINLP Problems in Process SystemsEngineering. Comput. Chem. Eng. 1989,13 (3), 307-316. Lazaro, M.; Puigjaner, L. Stimulation and Optimization of Multiproduct Plants for Batch and Semi-Batch Operation. I. Chem. E. Symp. Ser. 1985, 92, 209-222. Mauderli, A. M. Computer-Aided Process Scheduling and Production Planning for Multi-Purpose Batch Chemical Plants. Ph.D. Thesis, ETH Zurich, 1979, No. 6451. Mauderli, A. M.; Rippin, D. W. T. Production Planning and Scheduling for Multi-Purpose Batch Chemical Plants. Comput. Chem. Eng. 1979, 3, 199-206. Mwier, R. F. H.; Evans, L. B. Optimal Multi-period Production and Inventory Policies for Multi-Stage Chemical Process Plants. Presented at the AIChE Annual Meeting, Washington DC, November 1988. 1 Rich, S. M.; Prokopakis, G. J. Scheduling and Sequencing of Batch Operations in a Multipurpose Plant. Ind. Eng. Chem. Process Des. Deu. 1986,25,979-988. Rich, S. M.; Prokopakis, G. J. Multiple Routings and Reaction Paths in Project Scheduling. Ind. Eng. Chem. Res. 1987,26,1940-1943. Suhami, I.; Mah, R. H. S. Scheduling of Multipurpose Batch Plants with Product Precedence Constraints. Proceedings of the Second FOCAPD Conference; Westerberg, A. W., Chien, H. H., Ma.;
705
Ind. Eng. Chem. Res. 1991,30,705-713 American Institute of Chemical Engineers: New York, 1984. Vaselenak, J. A,; Groesmann,.I. E.; Westerberg, A. W. An Embedding Formulation for the Optimal Scheduling and Design of Multipurpose Batch Plants. Ind. Eng. Chem. Res. 1987,26,139-148. Viswanathan, J.; Grossmann, I. E. A Combined Penalty Function and Outer-Approximation Method for MINLP Optimization. Paper MD 18.5,presented at the CORS/TIMS/ORSA Meeting, Vancouver, May 1989. Wellons, M. C. Scheduling of Multipurpose Batch Plants. Ph.D. Dissertation, Purdue University, December 1989. Wellons, M. C.,Reklaitis, G.V. Scheduling of Multipurpose Batch Chemical Plants. 1. Formatiom of Single-Product Campaigns.
Ind. Eng. Chem. Res. 1991,preceding paper in this issue. Yu, P. L.;Zeleny, M. The Set of All Nondominatd Sohtions in Linear C a w and a Multicriteria Simplex Method. J. Math. Anal. Appl. 1976,49,430-468. Zadeh, L. A. Optimality and Non-Scalar-Valued Performance Criteria. IEEE Trans. Automatic Control 1963,AC-8, 5 M . Zeleny, M. Linear Multiobjective Programming; Springer-Verlag: Berlin and New York, 1974. Received for review December 10, 1989 Revised manuscript received July 26, 1990 Accepted August 10,1990
Control of a Binary Sidestream Distillation Column Heleni S. Papastathopoulou and William L. Luyben* Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, 111 Research Drive, Bethlehem, Pennsylvania 18015
This paper presents results of a comprehensive study of the dynamics and control of a large industrial distillation column that separates a binary mixture of propylene and propane into three products: polymer-grade propylene distillate, chemical-grade propylene liquid sidestream, and propane bottoms. The column is a superfractionator with 190 trays and a high reflux ratio. A two-stage vapor recompression system is used for energy conservation. Plant data, both steady state and dynamic, were used to validate steady-state and dynamic models of the column. Then steady-state rating programs were used to calculate steady-state gains for various alternative control structures, and several steady-state indices were used for initial screening of alternatives. The most promising structures were evaluated on the rigorous dynamic model. Use of sidestream location as a manipulative variable was included in the analysis. This strategy yielded an unusual dynamic response: a naturally occurring derivative action in the blending of liquid sidestreams. Several control objectivea were studied, ranging from control of only one composition to control of all three product compositions plus the sidestream flow rate.
Introduction An increasing number of sidestream columns are being used in industry. They offer an energy-efficient method for producing three products from a single column in some separations. Most of the applications are in multicomponent separations where the sidestream is used to remove an intermediate-boiling component. However, there are important industrial applications where a sidestream column is used in binary separations. The most important example is in the separation of propylene and propane when two propylene products are required at different purity levels. This is the system that was studied in this work. The literature on the control of sidestream distillation columns goes back many years: Luyben (1966),Buckley (19691,Mosler (19741,Tyreus and Luyben (19751,Doukas and Luyben (19801,Ogunnaike et al. (1983),and Alatiqi and Luyben (1986). The only paper dealing with binary sidestream columns is that of Tyreus and Luyben (1975), who studied the methanol-water separation with low-purity products. Both computer simulations and experimental tests on a small pilot-scale distillation column (24 trays) were conducted. Use of the location of the sidestream drawoff tray as an additional manipulated variable was recommended. The work reported in this paper is an extension of studies by Tyreus and Luyben (1975)to a very large industrial-scale column making fairly high-purity products *Towhom correspondence should be addressed.
Table I. Nominal Parameter Values for Sidestream Column F,(lb-mol)/h 1000 S, (lb-mol)/h 196 D,(lb-mol)/h 535 0.70 r,.mole fraction of propylene 0.995 xD, mole fraction of propylene 0.93 x g , mole fraction of propylene 0.0002 xB, mole fraction of propylene NT 200 NS 150 NF 60 110 P,psia 1060 MB,lb-mol 180 M D , lb-mol 29 M,,lb-mol 7450 R, (lb-mol)/h 13.3 RR 10 d, ft 1 wH,in. 6.41 WL, ft
Table 11. Stream Proaerties flow rate, stream no. (lb-mol)/h teme, OF 47.8 1 8913.6 ~
~
2 4 6 I 8 IO 11
~~~
~~
~
7800.3 1113.3 195.7 917.6 917.6 7800.3 7800.3
~~
~
89.2 105.8 100.1 100.1
47.8 87.2 47.8
0888-5885/91/2630-0705$02.50/00 1991 American Chemical Society
~~
pressure, Dsia 108.9 190.3 223.9 223.9 223.9 108.9 190.3 108.9
vapor fraction 1.0 1.0 1.0 0.0 0.0 0.248 0.0 0.179
