Ind. Eng. Chem. Res. 1989, 28, 1191-1202
1191
Optimal Design of Batch Plants with Single Production Routes? Naim M. Faqir and Iftekhar A. Karimi* Department of Chemical Engineering, Northwestern University, Evanston, Illinois 60208
A rigorous analysis for incorporating production planning considerations into the design of multipurpose batch plants with a single production route for each product is presented. A new set of horizon constraints is derived, which can be used as such in the MINLP design formulation to inscribe the superstructure of all production plans involving long campaigns. The resulting formulation guarantees a unique minimizer and gives the optimal campaign structure directly. The constraint set can also be transformed into an equivalent set with a lesser number of variables. The equivalence conditions are derived, and two procedures for obtaining the complete reduced set are developed using the theory of linear inequalities. A set of dominant production plans is also identified, and the procedures for deriving the corresponding constraints are given. Several examples are solved to illustrate the procedures.
Batch plants are economically very attractive for producing multiple products using the same set of equipment. These plants have generally been divided (Rippin, 1983) into two categories; multiproduct and multipurpose. In a multiproduct plant, only one product is produced at a time, and all products follow essentially the same path (the sequence of units) through the plant. In a multipurpose plant, however, multiple products may be produced at a time and the products usually follow different paths through the plant. Due to the unsteady-state nature of the batch plants and the need for allocating the available production time among various products, it is important to plan, at least roughly, how the production will be carried out, while designing the batch plants. These production planning considerations can lead to lower cost designs both in the case of the multiproduct (Birewar and Grossmann, 1987) and the multipurpose (Suhami and Mah, 1982; Vaselenak et al., 1987) plants. In this paper, we address the design of a special type of multipurpose plant. The plant consists of several processing stages with each stage having multiple identical batch units operating in parallel, out of phase. Every product passes through a unique and fixed subset of these stages, i.e., has only one production route and one batch size in which it is produced. Because all products do not use all the stages, products with no common stages can be produced simultaneously. We call such a plant a plant with single production routes, because there is a single pre-fixed path that a product follows through the plant and it is always produced in the form of identical batches of a fixed size. Another reason for using this terminology is to distinguish this special type of plant from the more general multipurpose plant in which a product can be produced via multiple routes with different batch sizes. In the batch plant design research, the multiproduct plant has received the most attention (Modi and Karimi, 1988; Birewar and Grossmann, 1987; Yeh and Reklaitis, 1985, 1987). On the other hand, the multipurpose plant has been studied (Suhami and Mah, 1982; Vaselenak et al., 1987; Klossner and Rippin, 1984) only in the special form (the multipurpose plant with single production routes) described above. The common approach has been to formulate its design problem as a mixed-integer non-
* Author to whom correspondence should be addressed.
Presented at the AIChE Annual Meeting, Nov 1988; paper
19a.
linear program (MINLP). Since the planning considerations mentioned earlier are important, these works have used a rough production plan for the design formulation. This plan involves long production campaigns (Mauderly and Rippin, 1979). A campaign is a set of production runs of one or more products which can be produced simultaneously by using the given plant equipment, while a production run is a series of identical batches of a single product produced by using a fixed set of units. This production plan is inscribed in the MINLP formulation by means of several constraints called the horizon constraints. For any given design problem, a number of such production plans are possible. Recently, Vaselenak et al. (1987) proposed the idea of a superstructure (i.e., a super production plan) which can accommodate all such possible production plans. However, since the number of horizon constraints and the number of optimization variables describing this superstructure were huge, they proposed a procedure to derive a much reduced set of horizon constraints involving total production times allotted to various products as variables. However, it was not clear how the set of constraints describing the superstructure and the reduced set were equivalent. Moreover their procedure did not work uniformly for all problems. In fact, in some problems, it failed to derive a reduced set of constraints involving the total allotted production time variables only. This led to the undesirable possibility of multiple local minimizers in the MINLP formulation for the design. In this paper, we do use the superstructure proposed by Vaselenak et al. (1987) but propose a much better (much less variables and constraints) set of horizon constraints to describe it. This original set can be used in the design formulation as such or can be reduced to an equivalent set involving only the total allotted production time variables. We clearly define the equivalence between the original and reduced sets and present two procedures for deriving the latter, which work uniformly for all problems. These procedures are based on the theory of linear inequalities and involve the identification of the vertices of a parametric linear program. We also identify a set of dominant (i.e., those that are candidates for the optimal design) production plans and procedures for deriving their constraint sets. We begin with a description of the design problem with its underlying assumptions and notation and then proceed to the development of our formulation and procedures. We will illustrate the procedures via several examples during the course of this paper.
0888-5885/89/2628-1l91$01.50/0 0 1989 American Chemical Society
1192 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 STAGE 2
STAGE 1
Units
m,
"z
STRGE J
m.
J
STAGE
M
m M
Figure 1. Multipurpose batch plant with single production routes.
Two or more products are compatible, if they can be produced simultaneously; i.e., no stage is needed by more than one product. From these sets, identify the maximal sets, i.e., the sets that are not subsets of any other set. For instance, let us say that we have three products A, B, and C, and the compatible product sets are (A],(B],(C],(A,B), and (B,C]. Then the maximal sets are (A,B]and (B,C]only. These maximal sets are not difficult to derive, and a systematic procedure for the same is given by Vaselenak et al. (1987). Let n denote the number of maximal sets for the given N products and let Gi, i = 1, n, denote the ith maximal set. Let us divide the total available production time H into n campaigns with normalized (with respect to H ) lengths ti, i = 1, n. Clearly we must have tl + t z + t , + t , + ... t , I1 or et 5 1 (2)
+
Problem Description A schematic diagram of the M-stage multipurpose plant with single production routes is shown in Figure 1. Only the batch stages are considered in the design, as we assume that the cost of the semicontinuous equipment is negligible as compared to that of the batch equipment. There are mj identical pieces of batch items with size V, in stage j , which are operated in parallel and out of phase. The plant is to be designed to produce N products. As shown in Figure 1,the products may follow different paths through the plant, but the sequence of stages through which a product passes is known a priori for every product. Let Ri(i = 1,N ) be the set of stages required by product i and U j ( j = 1,M ) be the set of products that use stage j for their production. Further define an equipment utilization matrix E with elements eij as follows: e . .= '1
1
1 if product i uses stage j for its production 0 otherwise i=l,N; j = l , M
(1)
Since every product is produced always using the same stages, and all items in a stage are used by only one product at a time, every product i has a unique batch size B , which is the amount of fiial product i produced in a single batch. Let pi, be the time required to process a batch of product i in an item of stage j . This includes the times required to fill the batch unit, to process the batch, and to transfer it to the next stage. We assume that pi, depends on Bivia the following known relationship: p I.] . = p 110 . . + -y..B.&j I] I where, pijotai, and yij are known constants specific to each product i and stage j . With the above notation and assumptions, the design problem is stated as follows. Given the N products with production requirements Qi, i = 1, N , and the total available production time H , determine the number mj of items in each stage j , the size Vi of each item in stage j , the batch size Bi for each product i, and the details of the rough production plan. MINLP Formulation Let us first derive a new set of constraints for inscribing the production plans in the design formulation. As mentioned earlier, we assume that the production occurs in the form of long campaigns. To include all such possible production plans in our formulation, we use the superstructure (Vaselenak et al., 1987). An intuitive procedure to derive the superstructure for a given set of N products is to first identify all possible sets of compatible products.
where e is a row vector (1 X n ) with elements 1, and t ( n X 1) is the vector of normalized campaign lengths t,, i = 1, n. Since the products in Gi can be produced simultaneously, assign their production to the campaign i; i.e., the campaign i will produce only the products in Gi. However, it is not necessary that it must produce all the products in Gi or that the products must be produced for the full length tiH of the campaign. Thus, campaign i may produce one or more products from Gi for differing amounts of time not exceeding tiH. Note that a product may be produced in more than one campaign. Let us define a campaign matrix C ( N X n) with elements cij as follows: 1 if product i can be produced in campaign j
Let T,H denote the total time for which product i will be produced, and T ( N X 1)be the vector of these normalized production times T,, i = 1, N. For every product i, we know the campaigns in which it can be produced and hence the total available production time for it in terms of the appropriate campaign lengths. For instance, if product i can be produced in campaigns 2 , 3 , and 6, and we need to allocate a total production time of TIto satisfy its production requirement and then we must choose t2, t,, and t, such that t z+ t3 + t6 I TI. The production plan defined in terms oft, will be feasible, only if such a condition is satisfied for each product. Thus, the campaign lengths t, must satisfy the constraints cIltl + cI2t2+ ... + c,,t, I T I i = 1, N or Ct 2 T (4 1
Equations 2 and 4 involving ( N + n) variables and ( N + 1) linear inequalities are the new horizon constraints which completely describe the superstructure. Given the campaign lengths t,, i = 1, n, it is easy to distribute the production times T , allotted to various products to get the desired production plan. Vaselenak et al. (1987) used a different set of variables to describe the superstructure. It involved t, and the times for which products are produced in individual campaigns as variables. This amounted to n + gl + g2 + g3 + ... + g, variables and 1 + g, + g2 + g3 + ... + g, constraints, where g, is the number of products in G,. They did not use this set of constraints in their MINLP formulation, because it involved too many variables and constraints and it also introduced the possibility of local minimizers in the optimization problem. Clearly, our horizon constaint set is much smaller, uses much less variables, and, as indicated later in a lemma, has a unique minimizer; thus, it can be used as such in the design formulation.
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1193 The horizon constaints specify the possible production plans in the formulation. To satisfy the production requirements of products, we must allocate the total production times to products by using the constraints i = 1, N BiTi/Pi IQ i / H
uniformly, guarantee the completeness of the reduced set, and actually give a set of constraints different from that of Vaselenak et al. (1987) for some problems.
where Pi, the limiting cycle time of product i, is given by Pi = max [ p i j / m j ] j E Ri, i = 1, N
If the horizon constraints are to be in terms of T only, then T would be the only variable inscribing the production plans in the formulation, in contrast to both t and T as earlier. Clearly then, we should only allow those T’s that are feasible. Let S denote the set of feasible T’s. But what does the feasibility of T mean? Since we allow production plans involving campaigns only, a given vector T would be feasible, if it is possible for us to devise a valid production plan involving the campaigns of compatible products. Mathematically stated, the feasibility criterion is then as follows. Feasibility Criterion. A T represents a feasible production plan if there exists a nonnegative t satisfying eq 11 and 12. With the above criterion, our task is now to find out those T’s for which eq 11 and 12 have a guaranteed nonnegative solution t . To this end, consider the T variables as parameters satisfying Ti I1,so eq 11 and 1 2 become linear constraints in t only. To examine if a nonnegative t satisfies them, we use the following theorem (Strang, 1980) about the existence of a nonnegative solution to a system of linear inequalities. Theorem 1. Either A x 2 b has a nonnegative solution or else there is a nonpositive y such that y A I0 and y b < 0, where A is a matrix ( m X n) of coefficients, b is a vector ( m X 1)of constants, and x ( n X 1) and y (1X m) are vectors of variables. By applying theorem 1 to eq 11 and 12, we get the following lemma for the feasibility of a given T (for proof, see Appendix B). Proposition 1. A T represents a feasible production plan if the following inequalities have no nonnegative solution y (1 X N ) :
The batch unit sizes must accommodate the batch sizes of all products; thus, we must have V j 1 SijBi i E Uj, j = 1, M where Si.,called the size factor, is the volume of materials required to be processed in stage j to produce a unit amount of final product i. Assembling all the constraints developed above and imposing other constraints on the available batch unit sizes, number of parallel units, and the nonnegativity of variables, we obtain the following MINLP formulation for the design of the multipurpose plant with single production routes for minimizing the total capital cost: M
min CaJmJ(VJ)*J J=1
subject to
VI IS,B, ’t
PL]/mJ
V
i E U, j = 1, M
V jER, i=l,N
BLTl/PlIQ,/H
i = 1, N
VIL 5 VI IVIu
j = 1,M
mJL ImJ ImJL
j = 1,M
Ct IT
Equivalence of Constraint Sets
yCI e
(134
yT> 1
(13b)
et5 1
In the above formulation, all the variables except for mi are continuous, and the mi variables are integers. As indicated earlier and shown in the following lemma (for proof, see Appendix A), our above formulation eliminates the possibility of multiple minimizers. Lemma 1. The continuous relaxation of the MINLP formulation (eq 5-12) for the design problem has a unique minimizer. As mentioned earlier, Vaselenak et al. (1987) proposed a procedure to derive a reduced set of constraints for the superstructure. This set involved linear constraints in terms of T only. They did state a condition under which the reduced set would be equivalent to the original set of constraints describing the superstructure; however, we do not known of any proof for the equivalence, and hence there seems to be no guarantee that the reduced constraint set derived by their procedure is complete, i.e., no constraint is left out. In such circumstances, use of their reduced constraint set in the design formulation may eliminate a subset of the feasible production plans embedded in the superstructure from consideration or may even generate an infeasible production plan. We now develop a rigorous treatment of the problem of deriving a set of constraints equivalent to eq 11 and 12. We first define exactly what equivalence means and then develop two procedures for deriving the reduce set. As we show by several examples, the procedures work on all problems
The above proposition can be easily used to determine the feasibility of a given T by solving the following L P for it: maxyT subject to y I 0 and y C I e (LP1) Let y * denote the optimal solution to LP1. If y * T 5 1,then the given T i s feasible, otherwise not. But this gives us a way to check the feasibility of a given T. In order to derive the feasibility conditions defining the set S , we must force yT I1 for all T E S. Clearly y * will change as we change T. But then can we identify all possible y*? For a general optimization problem, this would not be possible, but fortunately, for the present problem, it is made possible by the following theorem from the L P theory (Reklaitis et al., 1983). Theorem 2. If an L P has an optimal solution, then at least one of the vertices of the polyhedron defining its feasible region qualifies to be an optimal solution. Evidently then if we identify all the vertices of the feasible region of LP1, we would get all the possible y*. By forcing y * T a t all the vertices to be less than 1, we will get a set of constraints in terms of T . For the T’s that satisfy these constraints, a nonnegative y satisfying eq 13a and 13b clearly cannot exist, and hence those T’s must be feasible from proposition 1. Alternatively, the constraints thus obtained define the set S and are the desired
1194 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 Table I. Data for Example 1 product campaign matrix C 1
A
1 1
B C
0 0 0 0 0
D E F G 1
A B C D E F G
equipment utilization matrix 5 6
0 0 0 0 1 0
1
horizon constraints. Let us now see how we can identify the vertices of LP1.
Horizon Constraint Procedures The feasible region of LP1 is a convex polyhedron with many vertices. Unless a convex polyhedron consists of one or two variables, it is almost impossible to identify all its vertices manually, and the use of an algorithm is warranted. Several such algorithms exist in the literature (see Matheiss and Rubin (1980)). Although it is much more convenient to use such an alogrithm, it is not a must. This is because for most problems of interest to us, the vertices can also be enumerated by a manual procedure, which we call the tearing parameter procedure. Thus, we present two procedures for deriving the horizon constaints: computer-aided and manual. We use two examples from the literature to explain and illustrate them. Example 1 is the 7-product ( N = 7), 10-stage ( M = 10) design problem considered by Suhami and Mah (1982). The equipment utilization matrix E for this problem is shown in Table I, from which the maximal compatible sets are L V I , (A,DI, (A,E,FJ, (B,GJ, (C,DI, and (F,GJ; and the corresponding campaign matrix C is also shown in Table I. Example 2 is the “cycle” problem considered by Vaselenak et al. (1987). The data for this example are shown in Table 11, from which the maximal compatible sets are ( A B ) , (B,CI, IC,DI, P,EI, and (E,AJ. Vertex Enumeration Algorithm. Two types of vertex enumeration algorithms are available (Matheiss and Rubin, 1980): pivoting and nonpivoting. The pivoting algorithms make use of the simplex method for linear programming and operate directly on the polyhedron of interest. Most of them assume that the constraint set is nondegnerate, and hence special treatment is required for problems with degeneracy. In contrast, the nonpivoting algorithms do not work with the polyhedron directly but transform it to a convex polyhedral cone, and they are not affected by the presence of degeneracy. Our experience with limited but typical problems of interest suggests the presence of alot of degeneracy; hence, we recommend the use of a nonpivoting algorithm for the present application. We very briefly mention the idea behind one such algorithm (Chernikova, 1965). The theory behind the algorithm and its implementational details are beyond the scope of this paper and are available in Matheiss and Rubin (1980). Chernikova’s algorithm (1965) is a special case of the Double Description method (Motzkin et al., 1953) for enumerating the vertices of convex polyhedra. The basic idea behind the algorithm is to convert a given convex
0 0 0 1
1 0 1
0 0 0 0 0 1 0
E 7
8
9
10
0 1 0 1 0 1 0
1 0 1 0 0
0 1 1 0 1 0 0
0 0 1 0 0 1 0
0 1
Table 11. Data for Examde 2 product campaign matrix C 1
A B C D E
A B C D
E
2
3
4
0 0 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 eauipment utilization matrix E 1 2 3 4
1
1
0 0 1 0
0 1 0 0
0 1 0 1 0
0 1 0 0 1
5 1
0 0 0
1 5 0 0
1 0 1
polyhedron into a cone and enumerate its extreme rays. Let P = Ix IAx I b , x 1 0) be the given polyhedron, where A ( m X n ) and b ( m X 1)are constants and x ( n X 1) is the variable. The corresponding convex polyhedral cone (Hadley, 1962) is defined as C = ( ( x , { ) l b {- Ax 1 0, x 1 0, { 2 01, where { is a scalar variable. The algorithm finds all the extreme rays (Hadley, 1962) of the cone C. The extreme rays that have { > 0 correspond to the vertices of P. Specifically if (x,{)is an extreme ray with { > 0, then x f { is the corresponding vertex of P. In this work, we used an implementation (De la Llata, 1988) of Chernikova’s algorithm. All the vertices that are candidates for the maxima and result in nontrivial horizon constraints are listed in Tables I11 and IV for the two examples, respectively. By forcing y * T a t these vertices to be, at the most, 1, we obtain the horizon constaints listed in Table V. Note that the horizon constraints developed by our procedure are identical with those of Vaselenak et al. (1987) for example 1but not so for example 2. Their procedure cannot derive the last constraint of example 2. To remedy the problem, they defined two classes of problems, which they called “noncycle” problems and “cycle” problems (example 2 being one of them) with special consideration required for the latter. Our procedure does not make any such differentiation and suggests that the required constraints can be derived for any problem. It also suggests that such constraints could be many, as there could be many nontrivial vertices of a polyhedron. Tearing Parameter Procedure. Since it is possible to identify the vertices of a polyhedron of one or two dimensions, is it possible to simplify the polyhedron of LPl? In most problems of interest, this is possible. The basic
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1195 Table 111. Nonredundant Vertices for Example 1 variable ~
no.
YA
1 2 3 4 5 6 7 8 9 1 1 12 13 14 15 16 17 18 19 20
0 0 0 0 0 0 0 1 0 0 1
~
YB 1 1
YE
YF
YG
0 0 0 0 0 0 0 0 0
1 0 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1
0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0
0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1
0 0
0 0 1 1 1 1 1 1 1 1 1
0 1 0
1 0 0 0 0 0 0
1
0 1 0 0 0
1 0
Table IV. Nonredundant Vertices for Example 2 variable no.
YA
YB
Yc
YD
YE
1
1 1 0 0 0
0 0
1
0 1 1 0 0
0 0 0 1 1
2 3 4 5 6
0 0 0 1
1 1 0
If2
2
‘I2
+ Tc + TG 5 1 + TF 5 1 TB + T D + TE I 1 TD + TE + TG 5 1 TB + TC + TE I 1 TB+ Tc + TFI 1 Tc + TE + TG I1
TA + Tc I TB + TE I TB + T D I
TB + TD
TA
+
1 1 1 TD I 1 TE I 1
TC + TA + TB + 7 ’ + ~ TO
+ TE I
2
idea behind our manual procedure is then to select intellignetly a few variables of y and treat them as arbitrary given t parameters. This is to decompose the given polyhedron into several one- and two-dimensional polyhedra described in terms of the 6 parameters. Since we can identify the vertices of these polyhedra, we can evaluate y T a t all these vertices in terms of the t parameters. Then we can choose particular values for these parameters so as to maximize y T . This would give us all the possible maxima for yT in terms of T and thus the constraints. Let us illustrate the procedure by solving example 1. From theorem 1,the parametric LP for this problem is as follows. maxy TAYA + TBYB+ CCYC+ TDYD + TEYE+ TFYF+ TGYG subject to YA
+ YB 5 1
..., G
yAoccurs in the most number of constraints and offers the maximum possibility of decomposition, so let YA = cl. With this, the above constraints reduce to the following: YE
+ Y F I 1 - €1
YB
+ YG 5 1
YC
+ YD 5 1
YF
+ YG
YB
I 1 - €1
yD
I 1 - 61
From the above constraints, variables YG and Y F occur in the most constraints. Let us assign YF = t2. Then we have the following reduced and disjoint sets of constraints: YE 5 1 - 6 1 - €2
y D 5 1-61
Table V. Horizon Constraints for Examples 1 and 2 example 1 example 2 TA
yi L 0 i = A, B,
~~
YD
0 1 1 0 1 0 0 0 0
~
YC
y C + y D I
1
Since yT is separable in terms of y and the cost coefficients T of y are all nonnegative, the maximum of yT will be obtained when the y’s assume their maximum values subject to the above disjoint constraint sets. From the first set, for yT to be at its maximum, yE must be 1 - q - t2. From the second set, three possibilities exist: (1) YB = 1 - €1, YG = €1; (2) YB = €2, YG = 1- t2; and (3) YB = 1 - el, YG = 1 - e2. Two possibilities also exist from the third set: (1)yc = 1,YD = 0; and (2) yc = el, YD = 1 - el. Thus, we have six possible parametric solution sets for maximizing yT as listed in Table VI. Let us consider the first set of solutions from the above six. Clearly the t parameters cannot be arbitrary and must satisfy some constraints. These constraints can be derived simply by substituting the values of y back into constraints 13a and 13b. Then t l and t2 must satisfy 0 Itl I 1 , 0 I c2 I 1, and t1 t2 I1. For the solution set 1, the objective function yT = T B + T D + T E + t l ( T A - TB + Tc + T G T D - T E ) + t g ( T F - T E ) . Now we should select c1 and c2 to maximize yT subject to the constraints 0 I el 5 1,0 5 t2 I1,and tl + c2 I 1. Clearly there are three corner points of this feasible region, namely, (1)el = 1, c2 = 0; (2) t1 = 0, t2 = 1; and (3) t! = 0, t2 = 0. Evaluating yT at these points and forcing it to be, at the most, 1,we get the first three constraints of Table V. Following the same procedure for all other sets, we get the complete set of constraints given in Table V. The derivation of the constraints for example 2 is given in Appendix C. It should be evident from the above example that one needs to be clever in picking the right variables as parameters. The idea is to be able to decompose the constraint set 13a into one- or two-dimensional disjoint sets by selecting as few parameters as possible. The philosophy behind picking these variables is essentially the same as in solving a set of sparse equations by tearing (Westerberg et al., 1979); hence, we call this procedure a tearing parameter procedure. It is clear that the manual procedure can involve the examination of a number of combinations and could be tedious for complex problems; however, we have solved several problems easily. Of course, a vertex enumeration
+
1196 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 Table VI. Parametric Solutions for Example 1 variable no. 1
YA
YB
e1
YC e,
YD
2
fI
1 - €1 1-q
3
el
62
€1
0 1 - tl
4 5
€1
€2
1
0
el
l - f l
€1
1 - 6 1
6
el
1-€1
1
0
1
1-
61
YF
YE
1 - €1 1 - e1 -
YC
€2
€2
€1
€2
€2 e2
€1
1 - e1 - e2 1 - e l - €2 l-tl-cz 1 - €1 - €2
e2
11-
e2
1-tp
e*
1-
€2 c2
€2
algorithm, if accessible, would be clearly desirable. Let us outline the main steps of the tearing parameter procedure. (1) Select the y variable that appears in the most number of constraints, assign an 6 parameter to it, and generate the reduced set of constraints in terms of the parameter. In case of a tie, select that which results in most one- to two-variable constraints. Repeat this procedure, until the set of constraints is divided into disjoint sets of one or two variables. (2) Identify all the vertices of the disjoint sets, which are candidates for the maxima, in terms of the t parameters. List all the combinations of these vertices, Le., the possible parametric y * . ( 3 ) Consider a parametric y * . Identify the constraints that the 6 parameters should satisfy for the given y * by substituting the values of the y variables in the constraint 13a. Express y * T in terms of the 6 parameters, and evaluate it at all possible vertices of the constraints on the t parameters. Obtain a horizon constraint y * T 5 1 by using specific values of the E’S a t each vertex. (4) Repeat step 3 for all the parametric y * ,and generate all the constraints. Eliminate the redundant ones.
Horizon Constraints for Dominant Plans The horizon contraints that we developed thus far embed all possible production plans involving campaigns. For the design problem as formulated earlier, it is not required to consider all of these plans, only a subset of them. We call such production plans dominant. By dominant we mean that an optimal design with a production plan from the dominant subset always exists. Thus, we need to consider only the dominant plans for locating the optimal design. The following proposition characterizes the dominant production plans. Proposition 2. For every feasible solution to the MINLP formulation, in which one of the superstructure constraints (Ct 1 T , et I 1)is inactive, there exists another feasible solution with Ct = T and et = 1, which has the same or better objective function. Clearly from proposition 2, it is sufficient to use the following superstructure constraints in the MINLP formulation to get the optimal solution: Ct = T (14a) et = 1
(14b)
These constraints can either be used as such in the MINLP formulation or reduced to an equivalent set of constraints involving only the T variables. To derive the latter, let us first state a feasibility criterion for the T’s defining the dominant plans, as we did for the general plans. Feasibility Criterion. A T represents a feasible dominant plan if a nonnegative t satisfies eq 14a and 14b. We found it better to divide the problems into two categories because it is easier to solve one than the other. To this end, let us define an augmented matrix D = [CT
eTITand T = [TT 1IT. The problems in which rank D = n belong to the first category, while those in which rank D C n belong to the second. The former are simpler because the desired constraints can be derived just by solving a set of linear equations. In contrast, the latter require the use of a vertex enumeration algorithm in addition to solving linear equations and hence are more involved than the former. We begin with the simpler problems. Case 1: rank D = 11. Since our goal is to identify the set of T’s representing the feasible dominant plans, let us consider T as some given parameters. Then eq 14a and 14b are just (N 1)equations in n unknowns, which define the relation between T and t . When rank D = n, they have a unique solution (Rust and Burrus, 1972): t = (DTD)-lDTT.Since t L 0, the first subset of constraints is simply
+
(DTD)-lDTL 0 When n = N + 1, D is a square matrix and hence the above constraints reduce to D-lT 1 0. But when n < N + 1,we have more equations and less unknowns, and hence additional ( N + 1 - n ) constraints must be satisfied for the solution t = (DTD)-lDT’i’to be consistent. These additional constraints can be easily derived by substituting for t into any ( N + 1 - n) equations from the system Dt =
T.
An alternate approach for deriving the horizon constraints for this case is to select n equations from the ( N + 1) equations and solve for t in terms of T . Then the first subset of the constraints is simply t L 0, when t is substituted in terms of T , and the additional consistency constraints are obtained by substituting for t in the remaining ( N + 1 - n) equations. In many cases, this procedure can be carried out manually because of the sparsity of D, but of course, an algorithmic procedure such as the LU factorization (Strang, 1980) can also be used. Let us consider example 1 in which N = 7, n = 6, and the superstructure constraints are ti + tz t 3 = TA (1 5 4
+
ti
t2
+ t , = TB
(15b)
t , = Tc
(15~)
+ t j = TD
(1 5 4
t, =
t,
TE
(15e)
t3
+ t e = TF
(15f)
t4
+ t,j = TG
(15g)
+ t 2 + t 3 + t, + t 5 + t , = 1
(15h)
Since we have two extra equations, we solve eq 15b-g to obtain t = (TB-TG+TB-TE TD-Tc TE TG-TF+TE Tc TF-TE)T By substituting the above in eq 15a and 15h, we get TB
+ TD+ TF = T A+ Tc + TG = 1. Forcing t 2 0 along with
these two constraints and eliminating the trivial and nonredundant constraints, we obtain the final constraints for the dominant plans as given in Table VII. Following a similar procedure for example 2, we get the constraints for the dominant plans as given in Table VII. Case 2: rank D C II. In this category of problems, there is an infinite number of t which satisfy eq 14a and
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1197 Table VII. Horizon Constraints for Dominant Plans for Examples 1 and 2
14b for any given T . Therefore, for a given T to be feasible, at least one of these t must be nonnegative. In contrast to the first case, it is not possible to solve for t in terms of T uniquely in this case, and hence the strategy needs to be different. As we did for the general production plans, we employ the following theorem (Strang, 1980), which is similar to theorem 1. Theorem 3. Either A x = b has a nonnegative solution or else there is a y such that y A 2 0 and y b < 0, where A is a matrix ( m X n) of coefficients, b is a vector ( m X 1) of constants, and x ( n X 1)and y (1 X m) are vectors of variables. By applying the above theorem to eq 14a and 14b, we obtain the following proposition (proof in Appendix E) for the feasibility of a T representing a dominant production plan. Proposition 3. A T represents a feasible dominant production plan if none of the following sets of inequalities has a solution y : (1)y T > 1, yC 5 e ; (2) y T > -1, yC I -e; (3) y T = 0, yC I0. Then from the above proposition, we need to solve the following two LP’s to determine if a given T represents a feasible dominant plan: max y T
subject to yC Ie
(LP2)
max y T
subject to yC I-e
(LP3)
If u * and v* denote the optimal solutions to the above LP’s, respectively, then a Tis feasible, if u*T I 1and v*T I-1. In order to derive the constraints defining the feasible T , we again make use of theorem 2 for LP2 and LP3. Thus, we must use the procedures described earlier to identify all the vertices of LP2 and force u * T I1 at all of them, and then we should repeat for LP3 and force v*T I-1 at all the vertices. The constraints thus derived will then be the desired horizon constraints. To avoid enumerating the vertices of both LP2 and LP3, we relax eq 14b to be an inequality. By doing this, we need to solve only LP2 to determine if a given Tis feasible for the constraints eq 14a and et I1. By using the vertex enumeration procedure, we can derive the horizon constraints in T which are equivalent to the constraints eq 14a and et I1. Then to obtain the complete set of constraints equivalent to eq 14a and 14b, we need only to add constraints to force et = 1. This can be done simply by representing et in terms of T and forcing it to be 1. Let us postulate that there exists at least one vector a (1X N) of coefficients such that aT = 1. By using eq 14a and 14b, we must have aC = e , which are n equations in N unknowns. By employing Gaussian elimination or manual elimination of a variables, we can reduce the system to a system of? ( r. Let us illustrate the procedure using example 3, for which C is shown in Table VIII. For this system, N = 6, n = 9, and aC = e are aA aB = 1
+
+
+ aA + aD = 1
+ aD = 1 ac + UF = 1
ac
aE
+ aF = 1
From the first five equations, we get aB = aD = aF,and = ac = aE. Then the system reduces to aA + aB = 1, for which the linearly independent solutions are (a, = 0, aB = 1) and (aA = 1, aB = 0), from which the linearly independent constraints aT = 1are TA + Tc + TE = 1and TB + TD + TF = 1. UA
Examples We solved two design problems (examples 1and 2, numerical data in Tables IX and x) using the four different
1198 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 Table IX. Numerical Data for Example 1 processing times, p i i , h/batch 4 5 6
3
2
1
A B C D
7.143 4.36
2.595
6.534
3.974 9.987 5.516
A B C D E F G
7.326
1
2
A 300 000
9.415 2.653
A 300000
8 5.404
5.64
9
10
3.304 4.529
8.163
4.833 5.982 2.895
5.731
9.381 yearly production requirements,Qi,kg
3.587 6.418
B
C
D
E
F
G
150 000
200 000
190 000
140000
172000
106000
Table X. Numerical Data for Example 2 processing times, pij, h/ batch 1 2 3 4 A 9.0 6.0 B 3.9 6.2 C 5.5 D 7.5 4.5 E 7.1 size factors, Si;, L/kg/batch 1 2 3 4 A 3.0 2.5 B 1.0 1.5 C 2.7 1.1 D 3.1 E
7 3.205
3.174 3.757
7.725
4.62
9.422
2.297
6.65
1.125 1.922
5.719 7.318
5.469
5.062 size factors, Sij, L/kg/batch 4 5 6
9.768 8.065
10
1.932 2.005
3
9
6.758
1.269 6.855
8 7.456
2.554 2.404
E F G
7
1.7
5
3.5 4.0 5
2.3 2.8
vearlv production requirements, Q;,kg B C D E 195000 220000 190000 170000 ~
sets of horizon constraints that we have developed in this paper. We index these sets as follows: set 1, constraints Ct 2 T a n d et I1; set 2, constraints in Tequivalent to set 1; set 3, constraints Ct = T a n d et = 1; and set 4, constraints in T equivalent to set 3. Thus sets 1 and 2 represent all the possible production plans, while sets 3 and 4 represent the dominant ones. Sets 2 and 4 represent the reduced sets of constraints that can be derived by using the different procedures developed earlier. For both design problems, we used the same data as used by Veselanak et al. (1987), i.e., H = 6200 h, aJ = 250 Swiss Franc, PI = 0.6, 1 Im, 5 3, and 250 IVI I 10 000. The integer values for the mJ variables were obtained by first solving the MINLP without the integrality restrictions and then forcing them to be integers by using the constraints ((m,*)" - mJ)(mJ - (mJ*)L) = O for all noninteger mJ
where (m,*)L = trunc (mi*),(mJ*)"= trunc (m,*)+ 1,and mJ* is the optimal noninteger solution for m, from the relaxed MINLP. We used the GAMS 2.05/hnNOS 5 . l / z O o ~ 2.1 package on a VAXIVMS 111785 (Murtagh and Saunders, 1983). As expected, all four sets give the same optimal solutions (Tables XI and XII) for the two problems. The production
Table XI. Solution to Example 1 integer continuous stage Vj, L Vj, L mj mj 1 1.000 3991.6 1.000 3789.0 3040.8 1.000 1.000 2 2976.2 7266.4 1.000 1.000 3 7112.0 1402.5 1.000 1.000 4 1331.3 1.000 1.116 6870.1 5 6521.3 1.000 1.000 1781.5 6 1691.0 1.000 3526.6 7 3347.6 1.000 1.ooo 5313.9 1.000 5598.1 8 1.000 1.123 2549.0 9 2405.6 1.000 4594.4 10 1.000 4335.9 integer solution product Be kg Ti,h A 1036.0 2159.2 B 743.9 1440.3 C 562.8 2600.4 D 729.7 2600.4 E 423.6 2159.2 F 615.4 2159.2 G 539.2 1440.3 campaign length t h ,h campaign 1 0.0 2 3 4 5 6
obj function value
0.0 2159.2 1440.3 2600.4 0.0 continuous integer 354 778 355 516
plans corresponding to the optimal designs are shown in Figure 2 and 3. Note that sets 1 and 3 produce the production plans directly as they generate the t values also, while for sets 2 and 4 one needs to derive a production plan corresponding to the T values. This process of deriving the appropriate plan from the T values is not necessarily a trivial task, and in that sense sets 1 and 3 are easier to use. Tables XI11 and XIV show the CPU times required to solve the design problems using the different constraints sets. At least for these two problems, the CPU times are essentially the same for all four sets. This is not surprising because the number of constraints in all four sets is approximately the same. However, the situation is different
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1199 Product
Table XII. Solution to Example 2 continuous ~~
v,,L
stage 1 2
1665.6 1388.0 591.0 717.9 1182.3
3 4 5
integer
Campaign
Vj,L
mi
1.264 1.000 1.000 1.313 1.000
mi
t3
1.000 1.000 1.000 1.000 1.000
2035.2 1696.0 722.2 914.0 1444.8
= 2159.2
t4 =
1440.3
integer solution product A
B C
D
E campaign
Ti,h 3979.9 1984.0 1926.3 2170.5 2339.2
Bi, kg 678.4 609.4 628.2 656.5 516.0
campaign length
tk,
Product
1934.5 49.5 1876.7 293.8 2045.4 obj function value
integer
92207
93 409
Table XIII. CPU Times for Example 1 CPU time, s MINLP continuous integer 11.95 set 1 15.47 set 2 15.14 14.01 18.86 11.43 set 3 set 4 16.61 13.87
27.42 29.15 30.29 30.48
Table XIV. CPU Times for Examde 2 CPU time, s MINLP continuous integer set 1 6.59 4.63 set 2 6.33 4.48 set 3 7.19 4.82 6.61 5.08 set 4
11.22 10.81 12.01 11.69
R
campaign
h
continuous
tl
-
t2
-
1934.5
49.5
t 3 = 1876.7
t4
t5
-
293.8
I
B
C
D
E
I t
I
-1
4-k
"2"
u
2045.4
total
Figure 3. Optimal production plan for example 2.
ready made, Le., there is no need for using any other procedure such as the tearing or the vertex enumeration, so there is no preprocessing needed. This was not possible to do with the superstructure representation of Vaselenak et al. (1987). Secondly, there is no postprocessing needed after the problem solution, because the production pian does not need to be derived; it is obvious from the optimal t values. In contrast, using sets 2 and 4, we require both post- as well as preprocessing. In addition to the two advantages, we do not have the problem of multiple minimizers when using sets 1or 3. Thus, sets 1and 3 clearly seem the desirable choices at least for small n and N.
total
when n or N is large, as shown by example 4. Example 4 involves 12 products whose utilization matrix E is shown in Table XV. A total of 24 campaigns (Table XVI) is possible. If we use the original sets (1and 3), we need 36 optimization variables and 13 constraints, while if we use the reduced sets (2 and 4), we need only 12 variables and 9 constraints (Table XVII). Clearly there is a significant reduction in the number of variables and constraints, so the reduced sets will be preferred in this case. Thus, the reduced sets are very useful especially for large values of n or N. However, it seems clear that if n and N are small, it is better to use the original constraint sets, i.e., either set 1or 3, as these have two advantages. Firstly, they are Table XV. Equipment Utilization Matrix E for Example 4 1 2 3 4 A 1 1 0 0 B 1 0 1 0 0 0 1 C 1 0 D 0 0 0 0 E 0 0 0 0 0 0 F 0 0 G 0 0 0 0 H 0 0 0 0 0 0 I 0 1 0 0 J 0 0 K 0 0 1 0 0 1 L 0
-
Figure 2. Optimal production plan for example 1.
Conclusions We have presented a rigorous analysis for incorporating production planning into the design of batch plants with single production routes. Our contributions in this respect are as follows. Firstly, we have developed a new set of horizon constraints for describing the superstructure of all possible production plans involving campaigns. In contrast to an existing representation of the supersturcture, it involves considerably less numbers of constraints and variables, guarantees a unique global minimizer in the opti-
5 0 0 0 1 1 1 0 0 0 1 1 1
6
7
8
9
10
0
0 0
0
0 0
0 0 1 0 0 1 0 0 0 0
0
0 0 0 1
0 0 1 0 0 1 0 0 0 0 0
0
0 0 1 0 0 1 0 0 0
1 1 0 0 0 0 0 0
0
0 0 0 1 1 1 1 1 1
1200 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 Table XVI. Campaigns for Example 4 no. no. camDaien
camDaien
transformations and taking the log on both sides of several constraints, we get the following MINLP: M
1
2 3 4 5 6 7 8 9 10 11 12
min
aj exp[fij
+ @jV,]
j=l
subject to
V j IBj + In Sij
j = 1, M
fij+pi21nPij
i=l,N jERi
i E Uj
pj- Bi In Tj + In ( Q j / H )I0 rfij
IIn miu
i = 1, N
j = 1, M
In VjLIV IIn Viu
j = 1, M
Ct 2 T et 5 1
Table XVIII. Parametric Solutions for Example 2 variable
In the above formulation, the nonlinearities appear only in the objective function and the production requirement constraints, both of which are convex. Thus, the formulation is a convex program with a unique minimizer.
Appendix B: Proof of Proposition 1 By use of the feasibility criterion for T and application of theorem 1 to the inequalities 11 and 12, a given T i s feasible, if there is no w = [ z 61 such that
mization formulation, and, being expressible in matrix form, needs no derivation. We have found that in small size problems, it is best to use it as such in the design formulation. Secondly, we have presented several proven procedures for reducing it to an alternate set involving only the variables representing the production times allotted to various products. In the process, we defined exactly what the equivalence between the original and the reduced sets means. In contrast to the existing work, our procedure guarantees the completeness of the reduced set and works uniformly on all problems. Thirdly, we identified the dominant production plans and their horizon constraints. Lastly, the results of this paper can be extended to incorporate planning in the design of general multipurpose batch plant with multiple routes, which we will report in the near future.
ZC + 6eT I0
(Bla)
zT+6 0. Then dividing the inequalities by 6 and defining y = -216, we obtain proposition 1.
Appendix C: Horizon Constraints for Example 2 by the Tearing Parameter Procedure The parametric LP for this problem is as follows: maxy TAYA + TELYB + TCYC+ TDYD + TEYE subject to
Acknowledgment Acknowledgement is made to the University of Jordan and the National Science Foundation (Grant CBT8810174) for partial support of this research. Acknowledgment is also made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research.
Appendix A: Proof of Lemma 1 The continuous relaxation of the MINLP formulated in the text can in principle have multiple minimizers because of the nonconvex objective function (eq 5). In order to convert it into a convex program, we define the exponential transformations for all variables except t and T . Thus, let Bj = exp(Bi),Pi = exp(pi),for i = 1,N and mi = exp(fij), Vj = exp(Vj), for j = 1, M . Using these
YA
+ YB
YD
+ YE 5 1
YE
+ YA
yi 2 0
i = A, B, ..., E
Since all variables appear equally in all constraints and each of them results in one-variable constraints, select yA = el; then the above constraints reduce to the following. YB
5 1- e13
YB
+ YC
5 1,
YC
+ YD
5 1
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1201 Clearly no decomposition has taken place. Let us assign yD = t2. Then we have the following disjoint sets of constraints: yE Imin (1- el, 1 - t2) YB I1 - €1,
yc I1 - €2,
YB
+ yc I1
[N(A)] = n - 12. Let xobe a particular solution to A x = b obtained by assigning zeros to the last n - k variables. Let zi, i = 1, n - k , be the basis of N(A). From theorem 4, xi = xo+ zi, i = 1, n - k , are also solutions to A x = b . To show that xi,i = 0, n - k , are linearly independent, let ai, i = 0, n - k , be such that + alxl + + (Yn-kxn-k = 0 (F1) .a.
From the first set, two possibilities for the maxima exist: (1)yE = 1 - t l and (2) yE = 1 - t2. From the second set, three possibilities exist: (1)YB = 1 - el, yc = el; (2) YB = t2, yc = 1 - t2; and (3) YB = 1 - el, yc = 1 - €2. Thus, we have six possible parametric solution sets for maximizing the objective function as listed in Table XVIII. For solution set l , the constraints on the € parameters are tl + t2 I1,t l Ic2, and 0 Iel, c2 I1, for which the three corner points are (1) tl = 1, t2 = 0; (2) el = 0, e2 = 0; and (3) el = l I 2 , c2 = lI2. Substituting for the y in terms of the t's and forcing yT to be less than 1, we obtain TB TE t 1 ( T ~+ Tc - T B- TE) + € ~ T I D1
+
+
Writing the above constraint for the three specific corner points for set 1, we obtain the first two and the last constraints of Table V. Repeating the same procedure for the remaining five solution sets, we obtain all the constraints of Table V.
Appendix D: Proof of Proposition 2 Let us consider an arbitrary feasible design with fixed mj and V,, j = 1,M , and objective function (capital cost) F. Let I be a feasible production plan such that Ct' ITI, et' < 1. Let us define another production plan I1 such that = ti' for i = 1, n - 1, tnI1= t,' + (1 - et'), and TI1= TI + (Ct" - TI). Clearly, in production plan 11, Ct" = TI1and et" = 1. From the above definition of the two production plans, it should be obvious that T?IITi' for all i = 1, N . If B: denotes the batch sizes of products in plan I, then plan I1 is also feasible if we choose BiI1= B:, because it also satisfies the production requirements Qi.However, since we may be overproducing the products, we can choose some suitable BiII IBi' such that we do not overproduce. Note that while doing all this, we have not changed the mj and V j (thus F ) and generated a feasible production plan such that inequalities 11 and 12 are all strict equalities and still BiI15 Bil. Clearly, based on plan 11, we may be able to generate another design with lower V,, i.e., lower objective function; hence, proposition 2 is proved.
Appendix E: Proof of Proposition 3 By use of the feasibility criterion and application of theorem 3 to equalities 14 and 14b, a given vecotr T is feasible, if there is no vector w = [ z 61 such that
zC + 6eT IO
(EM
zT+ 6 < 0
(Elb)
where w (1 x N + 1) and z (1 X N ) are unrestricted vectors and 6 is a scalar variable. Let us define a vector y = -216. By considering the three possible values for 6, we obtain proposition 3, Le., (1)if 6 > 0, t h e n y T > 1 a n d y C I e; (2) if 6 < 0, t h e n y T > -1 and yC 5 -e; (3) if 6 = 0, then yT = 0 and yC 5 0.
Appendix F: Proof of Proposition 4 Let us prove the proposition for a general system Ax = b in which A is ( k X n) and rank A = k . Then dim
Premultiplying eq F1 by A, substituting xi = xo + zi, and using the fact that Azi = 0, we obtain a1 + ... + CYn-k ao)Axo= 0. But since A x o = b and b = 0, it follows that (Y1 (Yn-k + (YO = 0
+
+ + e..
and eq F1 reduces to alzl
+ + an-kzn-k ..a
=0
But the linear independence of zi, i = 1, n - k , implies that a1 = a2 = ... = CYn+ = 0. Then from eq F2, we get cyo = 0, implying that the xi,i = 0, n - k , are linearly independent.
Literature Cited Birewar, D. B.; Grossmann, I. E. Incorporating Scheduling in the Optimal Design of Multiproduct Batch Plants. Presented at the AIChE Annual Meeting, New York, Nov 1987; paper 92e. Chernikova, N. V.Algorithm for Finding a General Formula for the Nonnegative Solutions of a System of Linear Inequalities. USSR Comput. Math. Math. Phys. (Engl. Transl.) 1965, 5, 228-233. De la Llata, R. Multiobjective Linear Programming under Implicit Concave Utility and Partial Preference Information. Ph.D. Dissertation, Department of Industrial Engineering and Management Science, Northwestern University, Evanston, IL, 1988. Hadley, G. Linear Programming; Addison-Wesley: Reading, MA, 1962. Kendrick, D.; Meeraus, A. GAMS An Introduction. Development and Research Department of the World Bank, 1987. Klossner, J.; Rippin, D. W. T. Combinatorial Problems in the Design of Multiproduct Batch Plants - Extension to Multiplant and Partly Parallel Operations. Presented at the AIChE Annual Meeting, San Francisco, Nov, 1984; paper 104b. Matheiss, T. H.; Rubin, D. A Survey and Comparison of Methods for Finding all Vertices of Convex Polyhedral Sets. Math. Oper. Res. 1980,5, 167-185. Mauderly, A.;Rippin, D. W. T. Production Planning and Scheduling for Multipurpose Batch Chemical Plants. Comput. Chem. Eng. 1979, 3, 199-206. Modi, A. K.; Karimi, I. A. Design of Multiproduct Batch Processes with Finite Intermediate Storage. Submitted for publication in Comput. Chem. Eng. 1988. Motzkin, T. S.; Raiffa, H.; Thompson, G. L.; Thrall, R. M. The Double Decomposition Method, in Contributions to the Theory of Games. In Annals of Math. Study; Kuhn, H. W., Tucker, A. W., Eds.; Princeton University Press: Princeton, NJ, 1953; No. 28. Murtagh, B. A.; Saunders, M. A. MINOS 5.0 User's Guide. Technical Report Sol 83-20, Dec 1983; Systems Optimization Laboratory, Department of Operations Research, Stanford University. Noble, B. Applied Linear Algebra; Prentice-Hall: Englewood Cliffs, NI, 1969. Reklaitis, G. V.; Ravindran, A.; Ragsdell, K. M. Engineering Optimization: Methods and Applications; Wiley: New York, 1983. Rippin, D. W. T. Design and Operation of Multiproduct and Multipurpose Batch Chemical Plants-An Analysis of Problem Structure. Comput. Chem. Eng. 1983, 7, 463-481. Rust, B. W.; Burrus, W. R. Mathematical Programming and the Numerical Soltuion of Linear Equations; American Elsevier Publishing Co.: New York, 1972. Strang, G. Linear Algebra and its Applications; Academic Press: New York, 1980. Suhami, I.; Mah, R. S. H. Optimal Design of Multipurpose Batch Plants. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 94-100. Vaselenak, J. A.; Grossmann, 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.
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1202
Westerberg, A. W.; Hutchison, H. P.; Motard, R. L.; Winter, P. Process Flowsheeting; Cambridge University Press: Cambridge, MA, 1979. Yeh, N. C. C.; Reklaitis, G . V. Synthesis and Sizing of Batch/Semicontinuous Processes. Presented a t the AIChE Annual Meeting, Chicago, Nov 1985; paper 35a. Yeh, N. C. C.; Reklaitis, G. V. Synthesis and Sizing of Batchisem-
icontinuous Processes: Single Product Plants. Conzput. Chem. Eng. 1987, 11, 639-654.
Received for reuiew October 17, 1988 Reoised manuscript receiued April 13, 1989 Accepted May 15, 1989
Dynamics of the Flash Fermentor System with Recycle Shuji Ohbayashi and Kazuyuki Shimizu* Department of Chemical Engineering, Nagoya University, Chikusa, Nagoya 464, Japan
Masakazu Matsubara Department of Electrical Engineering, Daido Institute of Technology, Minami-ku, Nagoya 457, J a p a n
The dynamic characteristics of the immobilized-cell flash fermentor system with recycle were investigated in terms of the frequency responses, the pole/zero locations, and the step responses. All eigenvalues are real in the left half of the complex plane for the case of a single-stage flash column, while N - 1 pairs of complex conjugates appear as the recycle ratio is increased for the case where the number of stages of the flash column, N , is greater than one. This causes the appearance of the resonant peak(s) in the frequency responses. It is also shown that the amplitude ratio becomes greater than the steady-state gain and that the phase lead occurs a t certain frequency ranges for the case where the recycle flow was considered to be the manipulated variable. Some physical interpretation on the dynamic behavior was made based on the step responses. 1. Introduction
In our previous study (Shimizu et al., 19881, we found the promising feature of the immobilized-cell type of flash fermentor system with recycle as compared to the conventional systems without a flash column. The performance evaluation was made in terms of a vector-valued objective function whose components are the productivity, the product concentration or the energy required for the product separation, and the substrate conversion. The next problem is the understanding of the dynamic characteristics of the flash fermentor system with recycle, which is essential to the control system design. Although very few papers have appeared so far on the dynamics of the system with recycle, it has been known that the system having recycle streams may show significantly different dynamic characteristics as compared with the system without any closed loops. Gilliland et al. (1964) have studied the effect of recycle structure on the process dynamics. Attire and Denn (1978) observed a significant change in the response time of an activated sludge plant with recycle. Kapoor et al. (1986) studied the effect of the recycle structure on the distillation tower time constants. Denn and Lavie (1982) showed that the dynamics of a plant with recycle are equivalent to those of the system with a feedback controller and that the general effect of recycle is to increase the steady-state gain and the dominant plant time constant. Papadourakis et al. (1987) have shown that the presence of recycle loops in a process can have a significant effect on the relative gain array (RGA) which is often used to assess the static interaction between the single-input/single-output(SISO) control loops for the control system design of multiinput/multioutput (MIMO) systems.
* Corresponding author. 0888-5885/89/2628-1202$01.50/0
In the present paper, we investigate the dynamic characteristics of the flash fermentor process with recycle from various points of view such as the frequency response, the pole/zero location, and the step responses in order to make clear the dynamic characteristics inherent in this very promising process.
2. Mathematical Model Equations The following assumptions were made in deriving the mathematical equations that describe the dynamic behavior of the flash fermentor system as shown in Figure 1:
(i) The dynamic behavior of the immobilized-cell concentration is slow enough to be negligible. (ii) The fermentor and the liquid holdups in the flash column are perfectly mixed. (iii) The vapor holdups are negligible. (iv) The hydraulic delay occurring in the liquid flows is negligible. (v) The liquid holups are constant with respect to time. (vi) The feed stream of the flash column is introduced at the top of the column with saturated liquid. (vii) The feed stream of the flash column is a ternary mixture of glucose, ethanol, and water. (viii) The vapor is in equilibrium with the liquid leaving the stage. (ix) No glucose is present in the vapor flow streams. (x) The immobilized-cell system is assumed to respond instantaneously to any perturbation in the substrate or ethanol concentration. Assumption iv will be dropped later, and the effect of the time delay on the dynamics will be discussed. Several other assumptions made are mentioned after the description of the mathematical equations. Numbering the stages of the flash column from the top as stage 1and the reboiler 0 1989 American Chemical Society
