Ind. Eng. Chem. Process Des. Dev. 1984, 23, 845-847 Wang, J. S.; Van Daei, W.; Starling, K. E. Can. J. Chem. Eng. 1076, 54, 241. Warzinski, R. F.;Ruether, J. A. Paper presented at Direct Coal Liquefaction Contractors’ Meeting, U.S. Dept. of Energy, Pittsburgh, Nov 16, 1983 (and to appear in Fuel). Watanasiri, S.; Brul6, M. R.; Starling. K. E. AIChE J. 1982, 28(4), 626. Wleczorek, S. A.; Kobayashi, A. J. Chem. Eng. Data 1061, 26(1), 11. Wieczorek, S. A.; Kobayashl, R. J. Chem. Eng. Data 1081, 26(4), 8. Wilson. G. M.; Johnston. R. H.; Hwang, S. C.; Tsonopouios, C. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 94.
a45
Yesavage, V. F. Ph.D. Dissertation, University of Michigan, 1968.
Received for review October 11, 1982 Accepted October 24, 1983 Support for this work was provided by the U.S. Department of Energy Pittsburgh Energy Technology Center (DOE PETC), and the Oklahoma Mining and Mineral Resources Research Institute. This paper was presented in part at the AIChE 82nd Annual Meeting in Los Angeles, Nov 8-12, 1982.
COMMUNICATIONS New Procedure Generating Suboptimal Configurations to the Optimal Design of Multipurpose Batch Plants The optimal design of a multipurpose batch plant is considered. The problem has been formulated as a mixed integer nonlinear program (MINLP) by previous authors and they have proposed a strategy consisting of a method of generating the optimal or near-opthnel confgurations, from which proper horizon constraints are extracted. These constraints are then applied to MINLP. I n this paper, a new algorithmic solution procedure for generating the optimal or nearsptimal configurations is presented. The method is based on the rigorous solution procedure of a set partitioning problem of operations research. A simple example illustrates how the method proposed works. A more detailed analysis, especially in the application to large size problems, must be made before any generalization.
Introduction Suhami and Mah (1982) have formulated a problem of optimal design of the multipurpose batch plants as a mixed integer nonlinear program (MINLP). In order to design the multipurpose batch plant, the production scheduling must be made before determining the batch size, volume of the unit, and the number of batch equipments. The reason for this is that in the multipurpose batch plant, two or more products may be manufactured at any one time. To find the optimal or near-optimal production scheduling, Suhami and Mah have proposed a strategy consisting of a method of generating feasible sequences and nonredundant horizon constraints and a set of rules for selecting the opitmal or near-optimal configurations based on heuristic considerations. In this paper we propose a new method for determining the optimal or near-optimal configurations of the multipurpose batch plant design. The method is based on the solution procedure of the set partitioning problem of operations research. Problem Formulation The problem to be considered has been defined by Suhami and Mah (1982). Briefly, consider a plant consisting of M type of batch equipment (R,), which are used to produce N kinds of products. k, IM types of batch equipment are available for processing each product PI, each corresponding to a stage j = 1,2, ..., k,. In each stage j , the types of batch equipment are prespecified, nj units can be operated independently in parallel, and all units within the given stage j have the same size VI. I t is also assumed that ki types of batch equipment are all distinct, the intermediate tank is not considered, and the sequence in which the k , types of batch equipment are going to be used is specified beforehand for each product PI. With the above assumptions, the design problem is to determine VI, the size of each equipment R,, the number of equipment nl required per stage and the batch size B, for each product P, so as to minimize the capital investment, and satisfy Q,, yearly production requirement for 0196-4305/84/1123-0845$01.50/0
each Pi within the horizon H (total time available for production in a year). Suhami and Mah (1982) have formulated this problem as a mixed integer nonlinear program (MINLP) as follows. Determine Bi, TL,,V,, nj to minimize M
minCn,cr;V,B, ;=l
subject to
V, =max(S&) (j = I , 2, ..., M) i€UJ
(3)
vi” Ivj Ivju 0‘ = 1, 2, ..., M)
(4)
..., M )
(5)
nIL Inj 5 nju Qi
= 1, 2,
(i = 1, 2, ..., N)
Ti = -TLi Bi N
E T i IH
i=l
C Ti I H
0’ = 1, 2, ..., M )
(7) (8)
i€ UJ
where Tij = time required to process one batch of product Pi in a stage involving equipment type Rj; Si;= characteristic size of equipment needed at stage j to produce unit mass of product Pi;Ci= {RjlRjrequired for product of Pi); U, = (PiIR ECiJ;CY,, pi, Si,, Qi, H are given positive constants, and nj in integer. Constraints ( 7 ) and (8) are referred to as horizon constraints. As pointed out by Suhami and Mah (1982), the design problem of the multipurpose batch plants (referred to as (Pl))lies between two problems. Namely, one is constituted by relations (1)-(7), which corresponds to the formulation of the multiproduct plant design treated by 0 1984
American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
Table I. The Matrix A for the Example
[ R,
P P6
p5
,
p,
R, 0
R3
0
0
0 0 0
1 1 0
0 0 1
0 1 0
R'4 R, R6 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0
R7
0
1 0 1 0 1 0
R, 1 0 1 0 0 0 1
1
R , R,, 0 1 1 0 0 1 0 0 0
Sparrow et al. (1975) and Grossmann and Sargent (1979). This problem (referred to as (P2)) gives an upper bound to (Pl). The other problem consisted of relations (1) through (6) and (8), and it represents the formulation if each product were to require only a single prespecified processor. This problem is referred to as (P3),and it yields a lower bound to (Pl). To solve problem (Pl), proper horizon constraints have to be imposed. In order to find the proper horizon constraints, it is necessary to find a configuration which results in minimum investiment cost. The horizon constraints are drawn from the resulting configuration. The method of generating the best Configuration by Suhami and Mah is as follows. To start with, generate randomly groups of products which can be produced independently. A configuration is completed when all products are assigned to one of the groups. The best configuration is chosen from all configurations to have the least constraints and result in the small batch sizes to be produced within the given horizon. The resulting set of horizon constraints from the best configuration is feasible and nonredundant. Configuration Procedure In this section, let us now state a new procedure to find an optimal or near-optimal configuration and the corresponding horizon constraints. Consider a set P = (Pl,..., PN)and a set I = {I,, ..., Im), where I,(CP) is a set of compatible productions, I is a set of all Ij,j € J = (1,..., m)and m is the maximal number of compatible production sets. Let L be a partition of P. Step 1. Generate I, and Q, = maxp8El,Q,, j = 1, ..., m. Step 2. Solve the following problem. min 6(L) =
QJ
An Illustrative Example We shall now illustrate the application of the proposed procedure with the example treated by Suhami and Mah. Consider a matrix A = (uLJ), where aqi = 1 if R, is required for the production of P, a,, = 0 otherwise
Then let a, be the ith row vector of matrix A. The matrix A and yearly production requirement 9, for the example are given in-Tables I and Ii, respectiveiy. Step 1. If ajTak= 0
0' #
k)
(13)
then Pi and Pk can be produced independently. And if (13) and ( a j + akITai= 0 (i # j ; i # k )
6(L*)= 650000 (kg) where
subject to
LCJ UI,=P E L
0' z
k)
(12)
Let 6(L*) be the optimal solution of this problem, and define I* = {IjpEL*}. Step 3. Consider the set of constraints G resulting from configuration q. Let q* be a configuration which minimizes the number of constraints of G, in G,'s generated from I*. The resulting Q* is an optimal or near-optimal configuration. The problem formulated in step 2 is a set partitioning problem of operations research. The problem can be formulated as a linear integer programming problem. Several algorithms for a set partitioning problem have been presented. For further details, refer to Garfinkel et al. (1972).
I* = ({p&l, (p3941,(pi,P5,p6)1 t Step 3. With respect to the groups of compatible products obtained in step 2, we choose a sequence of the groups such that the number of the horizon constraints is minimal, since the number of them may be under the influence of the sequence. Let a sequence be I1*
- -I2*
...
I m**
where m* = IL*l, I,*, ...,I,**EI*, and the set of compatible productions for optimal partition L* is renumbered. In this example, m* = 3, and 11*,12*,and 13*are given by
I,* = {P,,p'7);I2* = {p3,p41;I3* = (p1,p59p61 If (P,,P,}GI,then P, is in conflict with P,; otherwise P, is not in conflict with P, where P,EIk*,k = 1, ..., m* - 1,and P,EI,*, 1 = k + 1, ..., m*. From this property, a set of horizon constraints G,* and the corresponding number of constraints can be determined. Applying this property to
Table 11. Yearly Production Requirement Q, (kg) for the Example -
Q,
PI 300 000
(14)
are true, then Pi, Pb, and Pi can be produced independently. Repeating the same procedure, a set I is generated. It is noted that I is hereditary, i.e., if X E I and Y C X , then YEZ. Using this relation, we may reduce the Zmputational requirement for generating I. Table I11 presents the resulting I ] and Q, for the example. Step 2. Applying an appropriate algorithm for a set partitioning problem to the example, an optimal solution is obtained as
(9)
JEL
I , n I k = 4; j , k E L ;
The objective function of the problem formulated in step 2 is defined in order that a set L* is selected such that a total of greatest production requirement Q, in each group I,EI is minimal. Although it is a heuristic consideration, we may choose a configuration which results in the smallest batch sizes to be produced within given horizon H. In other words, the set L* is selected so as to minimize the capital investment. The proposed method may be applied to generation of optimal or suboptimal configurations for the problem of large multipurpose batch plant design, since a feasible sequence and nonredundant horizon constraints can be obtained with systematic steps.
PZ
p3
p4
ps
p6
150000
200 000
190 000
140 000
172 000
PI 106 000
Ind. Eng. Chem. Process Des. Dev. 1984, 23, 847-849
Table 111. Set of Compatible Productions 4 and Greatest Production Requirement Qj for the Example Ij Qj 300000 150000 200000 190000 140000 172OOO 106000 300000 300 000 300OOO 300000 150OOO 200 000 172OOO 172OOO 300 000
Ci = the set of Rj’s needed to produce Pi G, = the set of constraints resulting from configuration q G,* = the set of constraints resulting from configuration q* H = total production time in a year I . = the set of compatible products r’=the set of all I,, j c J = (1,..., m ) I* = (IjljEL*) ki = number of batch equipment needed to produce Pi L = a partition of P L* = optimal partition of P M = number of batch equipment types m = maximum number of compatible production sets m* = IL*l N = number of product types nj = number of batch equipment Rj njL= lower bound to nj n g = upper bound to nj = product type i Qi = yearly production requirement for Pi Qi =. greatest production requirement in each group IjEI q = index used to represent a configuration q* = index used to represent an optimal or suboptimal configuration R, = batch equipment type j Si, = characteristic size of equipment Rj needed to produce unit mass of Pi Ti,= time required to process one batch of Piin stage involving equipment type Rj TL = time interval between producing successive batches of
8
all sequences, we can find a set of horizon constraints with a minimal number. The chosen horizon constraints for the example are
T3 + TG + TZ IH T3 + T5 + Tz I H T3 + T5 + T7 I H T3 + TI + T7 I H T4 + TS + Tz I H T4 + T5 + Tz I H T4 + T5 + T7 IH
847
Pi
(15)
These horizon constraints agree with the constraints generated by applying the Suhami and Mah’s method to the example. Conclusion Suhami and Mah have defined a problem of the optimal design of a multipurpose batch plant as a mixed integer nonlinear program (MINLP) and proposed a strategy of generating the optimal or near-optimal configurations for the problem. From the resulting configuration, proper horizon constraints for MINLP can be drawn. Although the method is rigorous aqd successful, it is rather complex to implement. By means of the procedure proposed here, the optimal or near-optimal configuration can be effectively obtained with systematic steps, since the method is based on the rigorous solution procedure of a set partitioning problem of operations research, for which several efficient algorithms are available. However, a more detailed analysis, especially in the application to large size problems, must be made before any generalization. Nomenclature Bi= batch size of Pi
Ti = total production time required for Pi U . = the set of Pi’s making use of equipment type Rj v‘ = size of batch equipment Rj = lower bound to Vi fu= upper bound to V j = unit size required to process one batch Biof Pi in stage involving equipment type Rj Greek Letters aj = cost coefficient for V . p, = cost coefficient for ( Mathematical Symbols 1.1 = cardinality of a set
fL d,,
Literature Cited Garfinkel, R. S.;Nemhauser, G. L. “Integer Programming”; Wiley: New York, 1972. Grossmann, I . E.; Sargent, R. W. H. Ind. Eng. Chem. Process Des. D e v . 1979, 18, 343. Sparrow, R. E.; Forder, G. F.; Rippin, D. W. T. Ind. Eng. Chem. Process Des. D e v . 1975. 14, 197. Suhami, I.; Mah, R. S.H. Ind. Eng. Chem. Process Des. D e v . 1982, 218 94.
Department of Management Science Science University of Tokyo 1-3 Kagurazaka, Shinjuku-ku Tokyo 162, Japan
Masaru Imai Naonori Nishida*
Received for review October 21, 1982 Accepted November 29, 1983
p-Xylene Crystallization in c8 Aromatlcs Mixtures p-Xylene equiilbrium concentration vs. temperature data are fit with a one-parameter model for mixed xylenes with compositions such as those found In feeds to p-xylene recovery plants. Addition of carbon tetrachloride results in the selective formation of a solid compound with p-xylene, improving the separation of this isomer. Values of the parameter of the model are given for different Initial concentrations of CCI,.
Introduction In a typical commercial xylene separation unit the feed, containing about 20% p-xylene, 45% m-xylene, 15% 00196-4305/84/ 1 123-0847$01.50/0
xylene, and 20% ethylbenzene is cooled to approximately 213 K to separate the most valuable isomer, p-xylene, as a solid. This solid is purified by washing, melting, and 0 1984 American Chemical Society
