Ind. Eng. Chem. Res. 1994,33, 1188-1196
1188
Separation Sequence Synthesis with Linearly Dependent Products: A Problem Simplification Approach Yi-Ming Chen Department of Chemical Engineering, Prairie View A&M University, P.O. Box 397, Prairie View, Texas 77446-0397
L. T.Fan' Department of Chemical Engineering, Durland Hall, Kansas State University, Manhattan, Kansas 66506
The solution to a separation sequencing problem may be simplified or reduced if linear dependency exists among product streams. This linear dependency occurs whenever the number of product streams is greater than that of components in the feed stream. Nevertheless, if only sharp separators are involved, the cost of the optimal sequence for the simplified or reduced problem has been proved to be greater than that for the original problem. In the present work, the structures of all-sharp separation sequences for the original and reduced problems are examined. It has been found that the characteristic structure of the optimal separation sequence for the original problem is identical to that for the reduced problem. This has led to a simplification procdure for solving the separation sequencing problem in which the optimal sequence for the reduced problem is first synthesized and then transformed into the optimal sequence for the original problem with relatively little computational effort. The efficacy of the proposed approach is illustrated through an example.
Introduction Separation sequence synthesis is one of the most difficult and challenging problems in process synthesis. Its complexity is determined by, among other things, the size of a product set, i.e., the number of product streams involved (see, e.g., Muraki et al., 1986; Floudas, 1987). Thus, extensive research has been conducted to explore the possibilities of reducing the size of a given product set without additional separators (see, e.g., Bamopoulos, 1984; Cheng, 1987; Liu et al., 1990). So far, the focus has been on transforming the product set based on linear dependency among the products or product streams. Linear dependency of products refers to the situation where at least one product can be expressed as the linear combination of the other products (see, e.g., Liu et al., 1990;Chen and Fan, 1991). This linear dependencyoccurs whenever the number of product streams is greater than the number of components in the feed stream. Under such circumstances, a separation sequencing problem gives rise to a set of overdetermined algebraic equations of compositional relationahips, thereby enabling us to simplify the problem. When one or more products are linearly dependent, a set of base vectors containing the maximum number of linearly independent products of any choice can replace the larger, original product set. On the basis of this concept, Liu et al. (1990) proposed that if the base vectors are chosen so that the coefficients in the expressions of the nonbase vectors are all nonnegative, then only the products corresponding to the base vector will be produced through separators; the products corresponding to the nonbase vectors can be obtained by appropriately blending those corresponding to the base vectors (see Figure 1). This appears to be advantageous in simplifying a general separation sequencing problem involving n components and m products with n < m. Nevertheless, it has been rigorously proved (Chen and Fan, 1991)that the maximum ~
~
~~
* Author to whom correspondence should be addressed.
Figure 1. Production of alinearly dependent product stream through b1ending.
extent of stream bypassing attainable for the problem with the original number of products (the original problem) is greater than that for the problem with the reduced number of products (the reduced problem). In other words, the mass loads to separators in a separation sequence for the reduced problem proposed by Liu et al. (1990) are always greater than those in the corresponding sequence for the original problem. Hence,the optimal separation sequence for the original problem costs less than that for the reduced problem if only sharp separators are involved (Chen and Fan, 1991). A relationship must, therefore, be found between the optimal sequence for the reduced problem and that for the original problem so that the former can be converted to the latter. Otherwise the problem simplification approach based on the reduced problem as suggested by Liu et al. (1990) is meaningless since the solution space for the reduced problem does not contain the optimal sequence for the original problem. The structures of the sequence for the reduced problem and that for the original problem need to be examined to relate them to each other. As will be elaborated later, it
0888-5885/94/2633-1188$04.50/0 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1189 has been unequivocally proven that the characteristic structure of the optimal separation sequence for the reduced problem is identical with that for the original problem for a three-componentseparation process involving one feed and three products of arbitrary composition. Moreover, numerical evaluation and search of numerous separation sequencing problems involving a variety of components and products have indicated that this conclusion may be generally valid. This has naturally given rise to a novel problem simplification approach presented in this work for synthesizing an all-sharp separation sequence with linearly dependent products. The proposed approach consista of two steps. First, the original problem is simplified to the reduced problem producing only the linearly independent products, and the optimal separation sequence is synthesized. Since the reduced problem contains fewer product streams, it is simpler to solve. Second, the optimal sequence obtained for the reduced problem is converted to that for the original problem by retaining the characteristic sequence structure for the former and adding the additional streams needed for the latter. The proposed approach is useful in facilitating the solution to complex separation sequencing problems where the number of linearly dependent products is large. Representation for Original and Reduced Problems Consider a general separation sequencing problem in which P n-component product streams are produced from one n-component feed stream with no left-over products (hereafter, product, product stream, and product vector will be used interchangeably). We further assume that separators effecting the separation are simple and sharp ones, and all the components in the feed and product streams are ranked in the descending order of a selected physical or chemical property on which separation is based. Obviously, separation is not needed for P = 1 since f = p. For P = 2, the compositions of both products must be identical to that of the feed if they are linearly dependent; again, separation is unnecessary. For P > 2, assume that the maximum number of linearly independent products is r, i.e., r = rank(P)
= [fl
f2
= (1 +
P
cji)[pil pi2
... fnlT
while the remaining (P-r) products, PHI,Pr+2,...,P,, can be expressed in terms of these r products as (see, e.g., Amundson, 1966)
+ ci2p2+ ... + cirpr, i = r + 1, r + 2, ...,P
(5)
The summation in the parentheses indicates the excessive amounts of the product streams, PI, P2, ..., Pr, to be produced to form product streams, PHI, P H ~..., , P,, through blending. When any of the constants, Cil, ci2, ..., ci,, is negative, however, we cannot obtain (P - r) product streams without additional separators. Separation Sequence Structures Separation sequence structures are determined by the number and compositions of the feed and product streams; therefore, let us first consider a three-component separation problem involving one feed and three product streams. The purpose here is to derive certain relationships that can be generalized for an n-component system involving one feed and P product streams. Suppose that the maximum number of linearly independent products is two and that they are designated as p1 and p2. Then, product p3 can be expressed as a linear combination of products p1 and p2. In the light of eqs 3 and 4, the feed, f, and the products, p1, p2, and p3, in the original problem are expressed as
P131T
(74
P22 P231T
(7b)
P1 = [Pll Pl2 P2
= [‘31P11 + ‘ 3 9 2 1
= [P2l
(4)
where elements uj)and biji represent the molar flow rate of thejth component in the feed stream and the ith product stream, respectively. If the constants, Cil, ~ i 2 ..., , Cir, in the above equation are all nonnegative, the product streams, PHI,Pr+2, ...,P,, can be obtained by blending the r linearly independent product streams, PI,P2, ...,Pp In such cases, the reduced problem can be formed with the same feed as that represented by eq 2 and the following product
‘31P12
+ ‘3922
‘31p13
+ ‘39231T
(7c) If constants ‘31 and ~ 3 are 2 both nonnegative, the reduced problem can be found based on eq 5 with the same feed as that represented in eq 6 and the following products. PI* = b l l *
pi = cilpl
... pi,]T, 8* = 1,2, ...,r
j=Hl
(1)
and they are designated as PI,P2, ...,P,. For the original problem producing the specified P products directly from the feed stream, the feed, f, and the products, PI,P2, ..., P,, can be expressed as the vectors, f
streams.
Pl2* p13*lT=
(l + ‘31)[P11
PI2 P13IT
(8a) P2* = [&I*
P22* p23*lT =
(1 + c32)[P21
P22 &3lT
(8b) For the original problem, the fractions of the feed streams bypassing to the product streams, PI,p2 and pa, designated as 71,112, and 73, respectively, are determined as (Chen and Fan, 1991)
-) ’ ’ - -)
= min(- P11 fl
-
pl2 p13 f2
f3
q2 = min(- P2l p22 p23
fl ’ f 2 ’
f3
1190 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 Table 1. Compositions of the Product Streams for the Reduced and Orhiha1 Problems after Stream Byparring Droduct stream case 1 case 2 case 3 case 4 [O 0 Pl3’*lT [O P12’* OIT [O P12’* P13’*lT P1’* [O P12’* Pl3‘*lT [P211* P22’* OIT [PZl’* Pa’* OIT [P2l’* 0 PB’*lT P2’* [PI~ 3 or 2 with that for product pz‘ if ~ 3 I 1 ~ 3 2 ;in other words, if
and the two updated product streams are
Pli
11 = -
(15a)
fi
then, we have (Chen, 1992) for the reduced problem. Compositionsof the Product Streams after Stream Bypassing. In synthesizing an all-sharp separation sequence, the maximum stream bypassing is always desirable (Lu and Motard, 1985) since the increased amount of stream bypassing reduces the separation mass load while it has no effect on the sloppiness of the separation which is constant. For both the original and reduced problems, the maximum extent of stream bypassing is attained when the feed stream is bypassed to the product streams to the extent that at least one component in any product stream is completely matched by the feed stream (such a component is termed the piuot component); at this point, separation becomes unavoidable. In essence, therefore, separation sequence structures are determined by the composition of, and especially the pivot component(s) in, every updated
13
=
c31plj
+ c3&?j
for cal I c32
fj
The compositions of the updated product streams for the original problem, corresponding to those for the reduced problem, are also listed in Table 1. Note that there exist t w e cases for the original problem corresponding to one each in the table for the reduced problem since the pivot component for product p 3 ’ in the original problem is determined by the relative values of ~ 3 and 1 ~ 3 2 . Here, we will only examine the situation where C g l > c32; naturally, any conclusions drawn should be equally applicable to cases where ~ 3 5 1 C32 due to symmetry.
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1191
p* A 3 -B 5
C 22.5
bypassing, the composition of the updated feed stream is expressed in eq 13. The resultant separation sequences are sequences l.e, l.f, and 1.h. For illustration, sequence 1.e for the original problem and that for the reduced problem are shown in Figures 4 and 5, respectively. The readers are referred to Chen (1992) for schematics of sequences 1.f and 1.h as well as the discussions about the sequences characteristic of compositions for cases 2-4 in Table 1. In the following, we shall compare the mass load on each separator in the separation sequence for the original problem with that on the counterpart separator in the corresponding sequence for the reduced problem.
Mass Load Figure 2. Separation sequence and ita characteristic structure denoted by boldface lines.
Characteristic Structures. According to Chen et al. (1991), a separation sequence consists of a characteristic structure defining the interconnections among the separators and the relative locations of dividers and mixers preceding each separator (see Figure 2). This structure is independent of the product streams, which determine the remaining factors required to construct a complete separation sequence. Illustrated in Figure 3 are the eight possible characteristic sequence structures for a separation process with one three-component feed stream (Chen et al., 1991). However, not all of them are feasible for the product streams of given compositions. An n-component separation sequence is considered feasible if it requires (n - 1) or less distinct separators to produce the specified products. A close examination of the figure against product compositions listed in Table 1reveals that only structures e, f, and h are feasible for case 1,structures d, e, and f for case 2, structures e and f for case 3, and structures c, e, and f for case 4. The detailed procedure of obtaining feasible sequences for a three-component separation process is discussed in Chenet al. (1991). For convenience, a feasible sequence obtained for the original or reduced problem is labeled as n.x where n is the case number and x, the letter indicating the characteristic structure from which the sequence is derived. For example, let us consider the sequences characteristic of compositions in case 1 (see Table 1). For the original problem with ‘31 > ‘32, the fractions of feed stream bypassing to product streams p 1 , p 2 , and p 3 , based on eq 9 and the pivot components for product streams PI’, p2’, and p i , are determined by =P22
=-
(16b)
f2
93
=
‘31p11
= (l- 9 1 - 9 2 - 9 3 ) c f I
+ ‘3921 fi
The composition of the updated feed stream after stream bypassing is depicted in eq 10. The resultant separation sequences are sequences l.e, l.f, and 1.h. For the correspondingsequences characteristic of the compositions shown in case 1for the reduced problem (see Table 11, the fractions of feed stream bypassing to product streams p 1 * and p 2 * are evaluated through eq 14 with 7 1 and 9 2 determined in eqs 16aand 16b,respectively. After stream
+ f 2 + f3)
(17)
w 2
The corresponding terms for the reduced problem, WI* and W 2 * are obtained as (see Figure 5) W1*
= 11 - (1+ ~ 3 1 ) ~ 1(1 - + ~ 3 2 ) ~ 2 l ( f+i f2
+ f3)
(19)
wz*= ( l +c31)@12f3 -p13f2) + (1+ c32)@23- 9j3) (20) f3
It can be shown that (Chen, 1992)
By resorting to this and eqs 17 and 19, we obtain
wl*> w,
fl
92
w1
which implies that
P11
91
For sequence l.e, the mass loads on S1and S2for the original problem, designated as W 1 and WZ,respectively, are obtained as (see Figure 4)
From eqs 18 and 20, we have
1192 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
structure a
I
structure b
structure c
s:.
L
f
structured
1 dl
&
1
1
f
I
L st+cture
structure f
d!
I
g
l r -
structure h
Figure 3. Characteristic sequence structure for a three-component separation system.
Equations 22 and 25 can also be derived from sequences l.f, l.h, 2.d, 2.e, 2.f, 3.e, 3.f, 4.c, 4.e, and 4.f (see Chen,
Note that
= (l + '32)@23f2
- P22f3)
(24)
Wl* -=-= Wl
Insertion of this expression into eq 23 yields
w,*/w, = 1 + c32 thereby implying that
w,*> w-
in summary, mese two equations give
(25)
w,* wz
(26)
In other words, the ratio of the mass load on a separator for the original problem to that on the corresponding separator for the reduced problem remains constant regardless of sequence structures. Even though this observation has been derived only for the three-component
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1193 Rearrangement gives
From eq 26, we obtain
P,
where C is a positive constant. Since the composition of the feed stream to any separator in a sequence for the original problem is the same as that to ita counterpart separator in a corresponding sequence for the reduced problem (Chen and Fan, 1991), we also have
Figure 4. Sequence 1.e for the original problem.
p:
+bi'
(31a)
($ = b y
(31b)
Insertion of this and eq 30 into eq 29 yields SP P:
which reduces to Figure 5. Sequence 1.e for the reduced problem.
various problems. It appears that this observation is generally valid and thus can be extended heuristically to a general n-component separation problem involving one feed and P product streams. Consequently, the optimal separation sequence for the original problem in general will have the same characteristic structure as the reduced problem for a great variety of cost functions for the separation sequence, a typical example of which takes the form of SP
TAC =
[ai + bi(Wi)"l
1=1
Here, ai is the constant representing the fixed cost for separator Si; bi is the coefficient associated with the operating cost of separator Si, which can be a function of the operating parameters such as composition of the feed to the separator but not the mass load, Wi; and m is constant. The majority of the cost functions employed in the literature and industry fall into this category (e.g., Muraki and Hayakawa, 1984;Muraki et al., 1986;Floudas, 1987; Wehe and Westerburg, 1987; Floudas and Anastasiadis, 1988; Floudas and Plaules, 1988; Aggarwal and Floudas, 1990). For illustration, assume that s alternative separation sequencesexist for the original and reduced problem. We index these alternative separation sequences in such a way that the two sequences for both problems with the same index will have the same structure (e.g., parallel, in-series, or mixed). Assume that the optimal sequence for the original problem is sequence k. In other words, for any other sequence j , we have SP
SP +I
j = 1,2, ..., s ; j # k (28)
SP
X[bi'(Wi')"- b!'(q*)"]
10
(33)
r=l
This indicates that the optimal sequence for the reduced problem is also sequence k. The significance of the observation made in the preceding paragraph is that once the optimal separation sequence is obtained for the reduced problem, it can be converted into that for the original problem. The procedure is as follows: First, the streams constituting the characteristic structure of the optimal sequence for the reduced problem are identified and converted into those for the original problem by resorting to eq 26; this is due to the fact that the former has the same composition as the latter. Second, the additional flow streams needed to construct the complete sequence are determined in the light of the product streams of the original problem. The efficacy of the proposed simplification approach is demonstrated with the following example.
Example This example consists of one feed and five product streams, expressed as the following vectors f = [A B C DIfT = [15 20 10 15IT
(34)
p1 = [A B C DIDIT= [l 2 0.8 2IT
(35a)
= [2 2 1.2 1IT
(35b)
p3 = [ A B C DID: = 13 5 2.2 4.5IT
(35c)
p2 = [A
p4 = [ A B
B C
C DIP: = L3.5 4 2.2 2.5IT (36d)
1194 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
p5 = [A B C DIP: = 15.5 7 3.6 5IT (35e) Equation 1gives r = rank(P) =rank[pl p2 p3 p4 p51 2 3 3.5 2 2 5 4 = rank 0.8 1.2 2.2 2.2 73.6 1 L2 1 4.5 2.5 5 J =2 This indicates that the maximum number of linearly independent product vectors is two, and any pair of such vectors can be chosen to form a basis for the product vector space. If product vectors p1 and pz are selected to form a basis, the remaining product vectors can be expressed as a linear combination of these two, i.e.,
r1
5.4
I
P3 = c31P1 + c32PZ = 2P1 + 0.5PZ
(36a)
P4 = c41P1 + c42Pz = 0.5P1 + 1.5Pz
(36b)
~5
= c 5 1 ~ 1 +c52Pz = 1 . 5 +~ 2~ ~ 2
-Figure 6. Superstructure for a one-feed, two-product, four (lF2P,4)-component separation system.
(36~)
Since constants Cij are all positive, based on the formulation expressed in eq 5, the reduced problem can be formed with the same feed as that represented by eq 34 and the following product streams.
~f
= (1+ c31+ c41+ ~ 5 1 1 ~ 1
A
B
+
C
=(1 2 + 0 . 5 + 1.5)[1 2 0.8 2IT f-
= [5 10 4 103T
D
(37a)
= (1+ 0.5 + 1.5 + 2.0)[2 2 1.2 1IT = 110 10 6 5IT
(37b)
The cost function is assumed to take the form of (see, e.g., Floudas, 1987) TAC = C(fidi)".'
Figure 7. Superstructure for a one-feed, five-product, four (lF5P,4)-component separation system.
with d, = 1.5,
d, = 4.5,
d3 = 3.5
The solution space can be illustrated by a superstructure (see, e.g., Floudas, 1987)shown in Figure 6 for the reduced problem and in Figure 7 for the original problem. Comparison of these two figures indicates that the size of the solution space for the reduced problem is much smaller than that for the original problem. On the basis of the simple heuristic approach by Chen and Fan (19931, the optimal sequence for the reduced problem is obtained (see Figure 8). The characteristic structure of this sequence is employed to construct the optimal sequence for the original problem (see Figure 9). First, the maximum fraction of the feed stream that can be bypassed to the product streams, p1, p2, ...,p5 is determined based on Chen and Fan (1991). It is BPDrg= 0.833
--
(38)
r=1
Thus, the bypassing stream, di in Figure 9, can be expressed as
d: = 0.833f = 112.5 16.666 8.334 12.5IT
(39)
Next we have, from eq 26 di = Cdr
(40)
di = Cd:
(41)
The mass balance gives di + di + di = f
(42)
with di determined from eq 39; thus, we obtain from the above expression
Ind. Eng. Chem. Res,, Vol. 33, No. 5, 1994 1196 A 10
A 5 B 6.667 c 3.333
i * B 13.334 d l C 6.666 D 10
D5,
A 12.5 B 16.666
A 5 B 6.667 c 3.333 D 5
A
B
f-
C D
A 5
B IO
*
c 4 D 10
A IO B 10 C 6 D 5
3
TAC =
(f,d,)06
f-
A 15 B 20 c 10 D 15
p;
= ( 1 0 . 0 X 1 . 5 ) 0 6 + (4.5X4.5)06+ (12.5X3.5)06 = 20.88
t i l
Figure 8. Optimal separation sequence for the reduced problem.
3
TAC=
1(f,d,)06=
( 5 0 x 1 5 ) 0 6 + ( 2 2 5 x 4 5 ) 0 6 + ( 6 2 5 x 3 5 ) 0 6 = 1373
,=I
Figure 10. Optimal separation sequence for the original problem.
the remaining streams, di 0' = 3,4,..., 8;i = 1, 2, ..., 5). For illustration, let us determine streams d{ 0' = 3,4, ...)8)associated with product stream p1. First, the maximum extent of bypassing the feed stream to product p1 gives p6, determine
d; = [1.0 1.333 0.667 1.0IT
(48)
Apparently, component A is completely matched; hence,
Figure 9. Characteristic structure of the optimal sequence for the original problem.
di + di = C ( d r + d r ) = f - di = 60- 50 = 10 (43) Referring to Figure 8,we obtain d;
+ di'
= 10 + 10 = 20
(44)
d! = [O 0.267 0.133 0.2IT
(45)
Hence, we have
di = C d r = 0.512.5 3.333 1.667 2.5IT = 11.25 1.667 0.833 1.25IT (46)
di = C d r = 0.512.5 3.333 1.667 2.5IT = 11.25 1.667 0.833 1.25IT (47) Similarly, on the basis of eqs 26 and 45, streams d:, di, mi,si, si, s;, si, si, and si in Figure 9 can be obtained,
which in combination with the product streams PI, p2, ...,
(50)
The remainder of B in product p1 is to be matched by dt containing only pure component B; therefore, we have
dt = [O 0.401 0 0IT
Substitution of this equation into eq 43 gives
C = 0.5
Also, the remainder of component C in product p1 is to be matched by d: having the composition of BCD in feed ratio. Thus, we obtain
(51)
Finally, since the remainder of D in product p1 is to be matched by dt containing only pure component D,we obtain d; = [O 0 0 0.8IT
(52)
The resultant sequence is presented in Figure 10 with a cost index of 13.73 compared to 20.88 for the optimal sequence for the reduced problem. The optimal sequence shown in Figure 10 has been verified by directly solving the original problem by resorting to a heuristic-based approach (Chen and Fan, 1993). The result illustrates our early proof (Chen and Fan, 1991) that the cost of the
1196 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
optimal sequence for the original problem is less than that for the reduced problem.
TAC = total annualized cost of the separation sequence W{= molar flow rate of the feed to separator Si in sequence
Conclusions
vi = maximum fraction of the feed stream that can be bypassed
i
The solution to a separation sequencing problem can be simplified on the basis of the proposed approach if linear dependency exists among the product vectors and all the coefficients in the expressions for linear dependency are nonnegative. Linear dependency among the products can always be found as long as the number of products is larger than the number of components in the feed stream; this is very common in industrial separation sequencing problems. Thus, the proposed simplification procedure comprising two steps is broadly applicable. In the first step, the reduced problem producing only the linearly independent products is formed and solved; in the second step, the resultant separation sequence is converted into one for the original problem. The optimality is generally retained during this simplification procedure as long as the separatQrs involved are all sharp even though the products can be nonsharp. The proposed approachmay not be valid for a separation sequence involving any nonsharp or sloppy separator due to the trade-off between the extent of stream bypassing and the sharpness of separation. Examples of separation sequence synthesis with nonsharp separators can be found in, e.g., Bamopoulos et al. (1988), Muraki and Hayakawa (1988), Aggarwal and Floudas (19901, Liu et al. (19901, Wehe and Westerberg (1990),and Sheppard et al. (1991). Furthermore, unlike an all-sharp separation sequence,the number of separators required in a sloppy separation sequence often depends on the number of products involved. It would be of interest, therefore, to investigate the cost of the separation sequence for the originalproblem with that for the reduced problem in the presence of sloppy separators. Acknowledgment This is Contribution 92-237-5,Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506. Nomenclature BP- = maximum extent of bypassingfor the original problem ci = coefficient associated with the fixed cost for separator Si cij = coefficients in the expression to form product pi based on product p j di = degree of difficulty of the ith separation d( = ith stream from divider j di = molar flow rate of the ith stream from divider j Di = divider i f i = molar flow rate of the feed stream to separator i f = feed stream m[ = ith stream from mixer j mi.= molar flow rate of the ith stream from mixer j = mixer i n = number of components in the feed stream pi = ith product vector in the original problem pi* = ith product vector in the reduced problem P = number of products P = matrix with product vectors as its column vectors Si = separator that performs sharp Separation between components i and i + 1, ranked in descending order of a certain physical or chemical property
ti*
to product pi in the original problem = maximum fraction of the feed stream that can be bypassed to product pi* in the reduced problem
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Received for review January 19, 1993 Revised manuscript received January 31,1994 Accepted February 16,1994.
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Abstract published in Advance ACSAbstracts, April 1,1994.
