2062
I n d . Eng. Chem. Res. 1990,29, 2062-2073
Klossner, J.; Rippin, D. W. T. Combinatorial Problems in the Design of Multiproduct Batch Plants. Presented a t the AICHE Annual Meeting, San Francisco, CA, 1984; Paper 104e. Kocis, G. R.; Grwmann, I. E. Relaxation Strategy for the Structural Optimization of Process Flow Sheets. Ind. Eng. Chem. Res. 1987, 26, 1869-1880. 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. Wellons, H.C.;Reklaitis, G . V. The Design of Multiproduct Batch
Plants Under Uncertainty with Staged Expansion, I. Comput. Chem. Eng. 1989a, 13, 115-126. Wellons, M. C.; Reklaitis, G. V. Optimal Schedule Generation for a Single-Product Production Line-I. Problem Formulation. Comp u t . Chem. Eng. 1989b, 13 (1/2), 201-212. Yeh, N. C.;Reklaitis, G. V. Synthesis and Sizing of Batch/Semicontinuous Processes: Single Product Plants. Comput. Chem. Eng. 1987, I 1 (61, 639-654. Received for review February 15, 1990 Revised manuscript received May 23, 1990 Accepted June 4, 1990
Optimal Design of Multipurpose Batch Plants. 2. A Decomposition Solution Strategy Savoula Papageorgaki and Gintaras V. Reklaitis* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
A mixed integer nonlinear programming (MINLP) formulation for the optimal design of a multipurpose plant has been developed in part 1 of this series. The complexity of the model makes the problem computationally intractable for direct solution using existing MINLP solution techniques. Consequently, a decomposition strategy is presented that alternately solves a MILP master problem, which determines the values of the binary assignment variables for fixed compaign lengths, and a NLP subproblem, which performs equipment sizing and determines the values of the campaign lengths. The effectiveness of the decomposition procedure is demonstrated with a number of test problems that were solved in reasonable computation times. Introduction In part 1 of this series (Papageorgaki and Reklaitis, 1990), we presented a mixed integer nonlinear programming (MINLP) formulation for the design of a multipurpose batch plant. The complexity of the model and the combinatorially large feasible solution space suggest that a more specialized solution procedure should be developed to allow this problem to be solved routinely in practice. The proposed model will serve as a basis for such a procedure, since it adequately represents the most important structural options that must be selected at the design stage, as well as the interactions between the different decision levels. In this paper, we develop a solution methodology based on decomposition of the original MINLP problem into two subproblems: a MILP master problem that determines the values of the binary assignment variables and a NLP subproblem that optimizes the values of the continuous variables. An iteration procedure is devised that solves an alternating sequence of these subproblems to identify the design configuration with the least capital cost. Three test examples are solved to illustrate the effectiveness of the proposed solution strategy. Alternative Solution Techniques Mixed integer nonlinear problems can be solved via a variety of methods that include branch and bound procedures (Beale, 1977; Gupta, 1980), Generalized Benders Decomposition (Geoffrion, 1972; Floudas et al., 1989),and the Outer-Approximation/EqualityRelaxation algorithm (Duran and Grossmann, 1986; Kocis and Grossmann, 1987a,b). The OA/ER algorithm can be classified as a decomposition scheme that solves an alternating sequence of nonlinear subproblems and mixed integer linear master problems. In the NLP phase, the binary variables are fixed
* To whom correspondence should be addressed. 0888-5885/90/2629-2062$02.50/0
and the continuous variables are optimized to yield an upper bound on the solution of the MINLP problem. The MILP master problem contains the linear constraints of the original MINLP and the linear approximations of the nonlinear constraints derived at the solution of the NLP subproblem. The MILP phase is used to optimize the discrete (typically binary) variables and to provide an increasing sequence of lower bounds on the solution of the MINLP problem. The Generalized Benders Decomposition is a similar decomposition scheme that is based on a dual representation of the MILP master problem. In general, branch and bound techniques are required to obtain the values of the integer variables. The branch and bound strategy is based on the solution of continuous relaxations of the original MINLP problem. However, this method can guarantee attainment of the global optimum only when the continuous relaxations at each step are convex or can be convexified through suitable variable transformations (for instance exponential transformations). In this case, the necessary and sufficient Kuhn-Tucker conditions are satisfied and the global optimum solution is obtained. Alternatively, when the nonlinear relaxations exhibit nonconvexities that cannot be transformed to a convex form, the branch and bound procedure can only guarantee local optimal solutions. The OA/ER algorithm attempts to overcome this disadvantage by applying the branch and bound strategy only for the solution of the MILP master problems, for which the continuous relaxations are linear and, hence, convex. However, the NLP subproblem, which is first solved to provide the linearization points for the MILP problem, may still be nonconvex and, hence, exhibit multiple local optima. Recently, Kocis and Grossmann (1987a,b) incorporated the OA/ER algorithm into a two-phase strategy that applies local and global tests to identify nonconvexities in the constraints and the objective function and attempts to eliminate the impact of these nonconvexities by suitably modifying the 0 1990 American Chemical Society
Ind. Eng. Chem. Res,, Vol. 29, No. 10, 1990 2063 MILP master problem. Although several numerical examples show that in many cases the global optimum can be identified, the proposed method cannot guarantee the global optimal solution. The Generalized Benders Decomposition exhibits the same characteristics, and thus, attainment to the global optimal solution cannot be guaranteed with this strategy either. The MINLP formulation proposed in part 1 was first solved via the OA/ER algorithm implemented in the computer code DICOPT (Kosis and Grossmann, 1987a,b), which employs the modeling language GAMS (Generalized Algebraic Modeling Systems (Brooke et al., 1988)). DICOPT uses MINOS 5.0 (Murtagh and Saunders, 1983) to solve the NLP subproblems and MPSX to solve the MILP master problem. Several test examples have been used to illustrate the performance of the OA/ER algorithm. In all cases, since DICOPT can only handle binary variables, the nonbinary integer variables were handled by using the usual two-stage approach involving relaxed solution followed by solution using rounding of the continuous values of the relaxed variables to the nearest integer. In almost all cases, DICOPT did not proceed further than the initial guess with respect to the number of active campaigns and the allocation of products to campaigns. The product task-equipment assignment was slightly modified to accommodate the production requirements. Apparently, the nonconvexities of the constraint set force the algorithm to terminate prematurely. In addition, the relatively poor continuous relaxation of the proposed formulation due to the logical conditions and the bounds in constraints (1.9), (I.lO), (1.12), (1.13), (1.14), and (1.16) (Appendix I) cause the branch and bound procedure to enumerate a significant fraction of the integer solutions of the MILP. Consequently, a large part of the solution time is spent in solving the MILP master problems. As an alternative, a problem-specific decomposition scheme can be developed. In general, an efficient decomposition strategy is based on the identification of the appropriate set of complicating variables for a particular problem and the formation of two or more subproblems, which are relaxations of the original formulation but contain all the necessary information about the problem and can be solved more efficiently. The complicating variables are defined as the variables that, when fixed, render the remaining optimization problem considerably more tractable. Clearly, the choice of complicating variables depends on the problem special features, and in general, different problem classes require different sets of complicating variables. The natural 0-1 vs continuous variable decomposition suggested by the Generalized Benders Decomposition and the OA/ER algorithm can create a combinatorially large number of alternatives for the present problem. As an alternative, either heuristic rules should be applied to reduce the number of solution candidates or alternative decomposition strategies should be developed. A solution strategy based on decomposition of the original problem into two smaller subproblems and the selection of an appropriate set of complicating variables is presented in the following section.
solution of the original MINLP. In the following sections, the master problem and NLP subproblem will be presented, and their properties will be discussed.
Solution Approach The algorithm constitutes a decomposition scheme that solves an alternating sequence of two subproblems: (1) a master problem that is a broad relaxation of the original MINLP formulation and yields a lower bound on the optimal solution to the original MINLP (a minimization problem); (2) a NLP subproblem that corresponds to the original MINLP formulation with the values of the binary variables fixed and yields an upper bound on the optimal
CAP, = max CAP,(t)
Master Problem The master problem determines the best product-campaign and product-equipment assignment (values of the binary variables) during each iteration of the algorithm. This problem is a broad relaxation of the original MINLP and contains the linear assignment and connectivity constraints of the original MINLP as well as constraints resulting from relaxation of nonlinear constraints in the original formulation. It may also contain integer cuts corresponding to infeasible assignments identified by the NLP subproblem or cuts corresponding to previously considered assignments. Let CAP, denote the total capacity of equipment type e , which is equal to the product of the size of each unit of type e , V,, and the number of units of that type, Ne; namely, CAP, = VJV, (11.1) In this problem, the e equipment types are considered as continuously divisible, renewable, but limited resources having at every point in time a maximum availability value C A P F . According to the task recipe information, certain amounts of recipe-specified resources should be allocated to each product task for its processing. Assuming that equipment type e is feasible for the performance of task m of product i, the distribution profile of the level of resource e for this particular task can be expressed as a function of time CAP,(t;i,m) t E [O,H] e = 1, E (i,m)E Ue The various distributions CAP, interact in three different manners: (i) the sum of the integrals (weighted by certain mass balance factors) of the distributions that are allocated to task m of product i over the entire production horizon H, representing the amount of product i that is produced within the prespecified time limits, must satisfy the production requirements, Qi, dt I Qi
i
= 1, N
m
E TAi
(11.2)
(ii) There is a maximum availability of resource e , CAPE" for which the product tasks that require e must compete. The distribution of resource e at time t is expressed as a sum of the component distributions CAP:*"
2 CAP,(t) I
C CAP,(t;i,m)
t
E [OJI]
(i.m)EUe
(11.3) (iii) The maximum of the function CAP,(t) over t enters the cost function for resource e ; namely, (11.4)
Then, the master problem will attempt to determine the product-campaign and product-equipment assignments that balance the following opposing trends: (i) smoothing out of the resource (equipment availability) distribution profiles and consequently reduction of resource underutilization and (ii) satisfaction of the production requirements and resource availability constraints.
2064 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
Let VCimek denote the capacity of equipment type e assigned to task m of product i in campaign k; namely, VCimek = NUimekNGimk v e (11.5) Since the only resources considered in this formulation are the equipment types, this variable corresponds to the function CAP,(t;i,m)as described above. Then, constraints (11.2)-(11.4) can be restated as follows
5 E
TkVCimek
Simetime
k = l eEP,,
1 Qi
i = 1,..., N
Bound
m f TAi (11.6)
N
E
CAP, L
i-1 ,ETA,
VCimek
e = 1, ..., E
k = 1, ..., K
MINLP
(11.7) 0 5 CAP, IV F B X N F
e = 1, ..., E
(11.8)
Additional logical constraints are needed to impose logical conditions on the partial capacity VCimek; namely, VCimek 2
i = 1, N e f Pi, k = 1, ..., K (11.9) m f TA;
VFinXimek
e..,
The model also contains the assignment and connectivity constraints (1.2)-(1.6) of the original formulation, which is summarized in Appendix I. Finally, the following variables bounds are used: O 5 Vimek 5 VFaxNFax
(11.11)
0 5 NC, 5 NFax
(11.12)
The objective function for this problem is a linear underestimation of the nonlinear objective of the original formulation. The form of the objective function is as follows: E
min C(c,CAP,
+dye)
(11.13)
e=l
where ( V y ) b , - (v?in)b,
c, = a,
v me a x - V me i n
In the above objective function, the second term accounts for the economies of scale for units in the same equipment family. Therefore, the master problem will be able to differentiate designs with the same total capacity CAP, for unit e but different number of units Ne. To illustrate this point, consider the unit types R1 and R2 which belong to the same equipment family. Assume that the lower and upper bounds on unit capacity are lo00 and 3000, respectively, and the upper bound on the number of units is 1 for both unit types. Since both units belong to the same equipment family, the associated cost coefficients a, and be are the same for both types. In addition, the maximum and minimum unit capacities and are the same for both R1 and R2. Consequently, the cost coefficients cR1 and cR2 will be equal. Now assume that a total capacity of 2000 is needed for the processing of some product task. Economies of scale imply that one large unit, say unit R1, with size 2000 has lower cost than two units, each having size 1000. This first part of the objective
Terminate ? No
Yes
1
Figure 1. Basic decomposition algorithm.
function will not make the distinction between the two cases, because it will assume equal values; namely, cR1 (1000) + CR2 (1000) = cR1 (2000) The second term, however, will increase the value of the objective function by d~~+ dR2(or 2dR1 since the two cost coefficients are equal) in the case where both units R1 and R2 are used, whereas in the case where only unit R1 (or R2) is used it will increase the objective function by only d ~ (or i d~2). The master problem is a MINLP problem and consists of minimizing eq 11.13 subject to constraints (1.3)-(1.6) and (11.6)-(11.12) with constraints (11.8), (II.ll), and (11.12) representing variable bounds. In addition, the master problem has the following property: Proposition. The master problem is a relaxation of the original MINLP model and provides a lower bound on the optimal solution of the original MINLP, if be 5 1 for all e. The proof appears in Appendix 11.
NLP Subproblem The NLP subproblem corresponds to the original MINLP with the values of the binary assignment variables fixed. This problem performs equipment sizing and determines the campaign lengths for a given productequipment assignment. The value of its objective function is an upper bound on the optimal solution of the original MINLP. Since the values of the binary variables are fixed, several sets of bounding constraints in the original formulation can be excluded from the constraint set of the NLP subproblem and be substituted by appropriate variable bounds. Additionally, the assignment and connectivity constraints will not be included in the NLP formulation. Consequently, the NLP subproblem will consist of minimizing objective function (1.1)subject to constraints (1.7), (1.8), (I.ll), (1.15), (1.32),and (1.17H1.28) with constraints (1.19)-(1.28) as variable bounds. The Basic Decomposition Algorithm As is evident from the description of the two subproblems, the binary assignment variables Xim&will be part of the complicating variable set used by our decomposition scheme. Figure 1shows a flow chart of the proposed de-
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2065 composition algorithm. The NLP subproblem, although nonconvex, can be efficiently solved by using MINOS 5.0 (Murtagh and Saunders, 1983). The MINLP master problem can in principle be solved via an existing MINLP solution technique. Although the size of the nonlinear constraint set and the number of the nonlinear variables in the master problem are much smaller than in the original formulation, our numerical experience shows that existing MINLP solution strategies fail to derive satisfactory results in reasonable computation time. As an alternative, the proposed decomposition scheme can be modified based on the following observation: The only nonlinearity of the master problem occurs in bilinear constraint (11.6), which also causes nonconvexity of the feasible region. As noted earlier, one of the decision levels of the general multipurpose design problem is the formation of compaigns. The campaign generation subproblem is computationally expensive, even for small problems, because it involves determination of the number of campaigns and of the campaign structure (allocation of products to campaigns and determination of campaign lengths). Since the maximum number of campaigns is equal to the number of products and each product is allowed to appear in more than one campaign, the number of alternative product assignments can grow considerably with the problem size. If the campaign lengths were fixed, the number of product-campaign assignments would be reduced, since the number of feasible product sets for each campaign would be less. In addition, the master problem would become linear. Based on this observation, the MINLP master problem can be solved through a search in the subspace of the campaign lengths. Since this subspace is continuous, a systematic search procedure must be devised. The search direction in the Tkspace can be determined as feedback from the solution of the NLP subproblem. As a result, details that are not present in the master problem (e.g., number of equipment items) will be accounted for during the solution of the master problem, and consequently, efficient campaign lengths will be computed. Therefore, the master problem will be solved iteratively as an MILP problem with the campaign lengths temporarily fixed to the values provided by the corresponding NLP subproblem. As a result of the aformentioned search strategy, the campaign lengths Tk along with the binary assignment variables Xime, will serve as the complicating variables for this problem. Furthermore, the number of campaigns K will be used as a parameter throughout the solution procedure. A flow chart of the final version of the proposed algorithm is given in Figure 2. The algorithm starts with the number of campaigns equal to some fixed number K and an initial guess for the values of the campaign lengths such that the sum of the values of Tkdoes not exceed the production horizon H. An iterative procedure is then devised that solves the MILP master problem and the NLP subproblem alternatively. For the current value of K, this procedure terminates when any one of the following criteria is met: (a) the best lower bound provided by the MILP master problem exceeds the current upper bound; (b) the current values of the campaign lengths provided by the NLP subproblem remain relatively unchanged and (c) the master problem is infeasible for a prespecified number of realizations of Tk.The complete algorithm can be summarized as follows: Step 1: Set K. Step 2: Assign initial values to Tksuch that &Tk IH.
Fix K
I
UDdate K
I : New Tk
Fix Tk
- -
Ix
Bound
Subproblem Converge ? No Terminate ? No
Yes
1
R, > R,, ( 2 ) Unchanged (3) Infeasible Master Problem
Convergence Crileria : (1)
T
Tenninution Criterion :All feasible K examined
Figure 2. Modified decomposition algorithm.
Step 3: Solve the MILP master problem. If feasible, go to step 4. Otherwise, go to step 2. If the MILP remains infeasible after a predefined number of realizations of Tk, go to step 5. Step 4: Solve the NLP subproblem. If feasible, create integer cuts to exclude previously identified assignments. The integer cuts will have the following form introduced by Crowder et al. (1983)
where SH = ((i,m,e,K)lXimek = 11,SL = ((i,m,e,k)IXimek = 0),and ISHI is the cardinality of the index set SH. Then, go to step 3. Otherwise, create integer cuts of the same form to exclude the infeasible assignment from consideration and go to step 3. If the resulting values of Tkremain approximately the same or the best lower bound provided by the MILP is greater than the current upper bound, go to step 5. Step 5: If all values of K have been examined, STOP. Otherwise, update the value of K , and go to step 2. After termination, the iterative procedure is restarted with a new value of K between its upper and lower bounds ( N and 1, respectively). The algorithm terminates when all values of K have been examined. Some preanalysis of the problem is usually required to eliminate cases in which a particular value of K will lead to infeasibility or to reduce the upper bound on the number of campaigns (which is equal to the number of products in the general case). Experience with different test problems suggests that the algorithm requires less iterations if it starts with the number of campaigns equal to its maximum value, because more degrees of freedom are given to the system. In addition, the higher the number of campaigns that are allowed, the higher the possibility that identical campaigns or campaigns with length considerably less than the length of the other campaigns in the set will be formed. In the first case, we can merge the identical campaigns into one
2066 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
equivalent campaign according to the following lemma. Lemma. A production schedule containing identical campaigns can be reduced to an equivalent schedule in which the identical campaigns have been merged into a single campaign (proof appears in the Appendix of part 1).
In the latter case, we can eliminate campaigns with insignificant length. In both cases, the value of K will be reduced and the iterative procedure can be continued with the updated value of K until one of the termination criteria is met. The algorithm can be modified to accommodate the analysis described above as follows: Step 1: Set K = K""". Step 2: Assign initial values to Tksuch that CkTkI H. Step 3a: Solve the MILP master problem. If feasible, to to step 3b; otherwise, go to step 2. If the MILP remains infeasible after a predefined number of realizations of Tk, go to step 5c. Step 3b: If the resulting production schedule contains identical campaigns, then merge these campaigns into a single campaign with length equal to the sum of the lengths of the component campaigns and go to step 5a; otherwise, go to step 4. Step 4: Solve the NLP subproblem. If feasible, create integer cuts to exclude previously identified assignment and go to step 5b. Otherwise, create integer cuts to exclude the infeasible assignment from consideration and go to step 3a. If the resulting values of Tk remain approximately the same or the best lower bound provided by the MILP is greater than the current up er bound, go to step 5c. (KP, - l),where J is the Step 5a: Set K = K number of identical campaign sets and KP, is the number of identical campaigns within set j . Go to step 4. Step 5 b If the length of some campaigns is insignificant compared to the length of other campaigns in the set, then eliminate these campaigns and set K = K - KM, where KM is the number of campaigns that have been eliminated. Update the campaign lengths Tkrand go to step 3a. Otherwise, go to step 3a. Step 5c: If K = K"", where Kminis the minimum number of campaigns for which the problem is feasible, STOP. Otherwise, set K = K - 1 and go to step 2. Since the original MINLP formulation is nonconvex, the NLP subproblem is also nonconvex, because it corresponds to the original MINLP with fixed values of the 0-1 variables. Consequently, the decomposition algorithm cannot guarantee the global optimal solution. As will be seen later, however, the proposed decomposition procedure does provide the optimal or "good" suboptimal solutions in reasonable computation time for a number of test cases.
Y
Computational Enhancements A. Campaign and Equipment Ordering Constraints. As mentioned earlier, a significant number of degenerate solutions can be created for the general multipurpose plant problem. Two sets of supplementary constraints have been introduced as part of the original MINLP formulation to eliminate part of the degeneracy. Since the campaign lengths are fixed in the MILP master problem, degeneracy can occur only when the values of the campaign lengths are equal or nearly equal. In our approach, we start the decomposition algorithm with the number of campaigns equal to the number of products and the values of the campaign lengths equal. As a result of the symmetry introduced, a large number of degenerate solutions will be generated to the first MILP. The campaign ordering constraints (1.29)-(1.31) will be used to eliminate part of the degeneracy. These constraints will not be introduced in the subsequent MILP's, since the
values of the campaign lengths will not, in general, be equal after the first iteration and the priority constraints may eliminate part of the feasible solution space. Alternatively, the size ordering constraints will be included in the constraint set of all MILP's as a means of eliminating degeneracy due to the identical nature of equipment types within the same equipment family. In the MILP master problem, the equipment priority constraints will take the following form e = 1, ..., E CAP, I CAP,,, e, e + 1 E Lf (11.14) Notice also that since the number of campaigns is reduced after each complete iteration of the decomposition algorithm, the number of binary variables is also reduced and consequently the number of degenerate solutions diminishes. B. Integer Cuts and Bounding Constraints. In addition to the ordering constraints introduced in the previous section, a bounding procedure can be devised to introduce integer cuts or tighter bounding constraints into the MILP master problem. The basic idea behind this procedure is as follows: The minimum production time TPY'" for each product in the plant can be estimated based on the derived upper and lower bounds on the batch size, the cycle time, and the number of batches of each product in each campaign. Since the campaign lengths are fixed in the master problem, the minimum production time for each product can be compared to the length of each campaign in the production schedule. If TPP, is greater than the length of a particular campaign or the sum of the lengths of a set of campaigns, then the assignment of product i to only these campaigns must be avoided, since it will result in infesible solutions. Integer cuts of a form introduced later in this section will be used to eliminate these assignments. When the campaign lengths are equal, a large number of integer cuts can be generated because of the symmetry introduced. In this case, bounding constraints involving the variable PRa can be derived to ensure that the least number of campaigns that can process product i is such that the sum of the lengths of these campaigns is greater or equal to the minimum production time for product i. The minimum production time TPP, for product i can be defined as TP,"in = npinTpin 1 (11.15) where n$" denotes the minimum number of batches of product i that can be achieved when all the production lines for product i yield the largest possible batch size; namely, nfrin
=
8, = __ 8, -
=
Q,
RmaJ
In addition, Tfindenotes the minimum cycle time of product i that can be attained when all the production lines for product i operate with the task processing times at their minimum value and the maximum allowed number of equipment groups assigned to each product task; namely,
A tighter value of the product nTinTZi"can be obtained based on the following observations:
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2067 The minimum number of batches (or the maximum batch size) is attained when each equipment group contains the largest number of maximum capacity processing units. Then, the minimum feasible cycle time corresponds to the production line in which a single maximum capacity equipment group is assigned to each product task; namely, T P = max min (time/ (11.18) mETA, eEP,,
TY
Let us denote the roduct of n? from (11.16) and from (11.18)as TPi. Alternatively, the minimum cycle time is attained when each product task is carried out in the maximum allowed number of equipment groups. This means that the largest feasible batch size is constrained by the smallest equipment group with capacity equal to V? 1 C NParNGGY eEP,,,,nL/ Sime Therefore, when the minimum cycle time is calculated by eq 11.17, the minimum number of batches must be calculated by the following formula:
P
Qi
Table I. Available Equipment Items for Example 1' upper bound on no. of units equip item capacity range, L El 500-3000 E2, E3, E4 500-3000 E5 500-3000 E6, E7, E8 500-3000 E9 500-3000 E10 500-3000 Available discrete sizes: 500-1000-2000-2500-3000 (for rounding purpose). Table 11. Size Factors ((L/kg)/Batch) and Processing Times (h/Batch) task/ size factor (processing time) for equipment equip tme E l E2,E3. E4 E5 E6, E7,E8 E9 E10 A1 1.5 (4) A2 2 (6.5) B1 1.6 (3.9) B2 2.5 (5.5) B3 1.9 (4.2) c1 1.4 (4.5) c2 2.4 (3.5) D1 2.2 (6.5) D2 1.7 (7) D3 1.8 (4.7) ~~
I
Let us denote the product of n?'" for (11.19) and TL. from (11.17) as TP:. Then, the minimum production time 'YP? can be defined as follows: T P P = min (TP!, TP?) (11.20) According to the bounding procedure described above, the minimum production time will be compared to the fixed campaign lengths and integer cuts will be introduced as a result of this comparison. In the case of equal campaign lengths, an excessive number of integer cuts may be generated because of the symmetry introduced. As an alternative, the campaign ordering constraints (1.29)-(1.31) will be introduced to the MILP master problem along with the following bounding constraint K
PRik 2
Bi
(11.21)
k=l
where the parameter Oi is defined as follows
The complete bounding procedure can be summarized as follows: Step 1: Choose initial values for the campaign lengths Tk such that CkTk IH. Step 2: Randomly select product i from the product list. If all products have been examined, STOP. Step 3: Calculate the minimum production time TPY from eq 11.20. If the campaign lengths are unequal, go to step 4; otherwise, go to step 5. Step 4: If TPPin> Tk (11.23) kE*ij
where \kii Qi = [qijlqij [1,2,...,K ) n (11.23) is true), generate integer cuts of the form (11.24) where k' E {l,2,...,KJ - qi,. Go to step 2.
Table 111. Cost Coefficients, a., for Example 1" unit tvve a. unit tvDe El 200 E6, E7, E8 E2, E3, E4 220 E9 E5 280 E10 a
a.
260 360 370
Power cost coefficient, be = 0.6.
Table IV. Production Requirements for Examule 1 yearly yearly demand, demand, product kdyear product kglyear A 300 000 C 180 000 €3 250 000 D 200 000
Step 5: If TPY'" > CkEQiTk, ai C (1,2,...,K ) , calculate parameter Bi from eq 11.22 and introduce constraint (11.21) into the MILP master problem. Go to step 2.
Examples Three test examples will be solved to illustrate the decomposition procedure. MPSX was used to solve the MILP master problems, and MINOS 5.0 was used to solve the NLP subproblems. Both codes are executed under the modeling language GAMS on an IBM 3090. Example 1. The multipurpose plant analyzed by Faqir and Karimi (1989) consists of 6 equipment families comprising 10 equipment types that are used to manufacture 4 products. The basic problem parameters (available equipment families and their capacity range, feasible equipment families for each product, unit processing times and size factors, and cost coefficients) are given in Tables I, 11, and 111, and the annual production requirements are given in Table IV. Faqir and Karimi assume that the plant consists of at most 10 equipment items. Consequently, the upper bound NFar = 1 was used for the number of units of type e. In addition, they assumed that at most one item can be used to carry out each product task. This additional restriction is incorporated in the model in two ways: (1)by setting NGET = 1 and NUimek = 1 as upper bounds on the number of equipment groups and the number of units in a group and (2) by adding the following constraint to the constraint set m E TA*j C X i m e k I1 i = 1, ..., N
2068 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 Table V. Integer Solution for Example 1 (Only Non-Zero Values)" unit type V. Ne unit type V, Ne El 1000 1 E7 500 1 E2 2000 1 E9 1000 1 E5 1000 1 E10 1000 I E6 1000 1
product-campaign
nik
Bik
TL.
A-1
502 644 504 380
599 400 357.2 526.1
6.5 5.5 4.5 7
B-1
c- 1 D-2
uCampaign length, T,: TI, 3541; T2,2659. Objective function value: $124619.
where TA*i is the set of tasks of product i that are assigned more than one feasible equipment choice (in this problem, tasks A2, B2, C1, and D1 are allowed to choose among more than one equipment type as seen in Table 111). The complete MINLP formulation involves 719 constraints and 273 (72 0-1) variables. The decomposition algorithm required two major iterations to attain the optimal (continuous with respect to the integer nonbinary variables) solution to this problem. The integer optimal solution was obtained by rounding the values of the relaxed variables to the nearest integer. The results obtained are shown in Table V. Details on the problem size and the computation time for the MILP master problem and the NLP subproblem during each of the iterations are shown in Table VI. As seen in Table V, two campaigns are formed a t the optimum, which consist of production lines used to process products A-B-C and D, respectively. Figure 3 shows the task-equipment assignment a t the optimum. The results obtained with the decomposition algorithm are the same as the ones obtained by the enumeration procedure of Faqir and Karimi. The computation times required by both methods are comparable. Notice, however, that the proposed decomposition strategy can be applied to problems with multiple units per processing stage, whereas the enumeration procedure of Faqir and Karimi is restricted to a single unit per stage. This example also shows that the proposed decomposition technique can lead to the global optimal solution even though no theoretical guarantee for attainment of the global optimum can be given. Example 2. The example problem discussed in part 1 of this paper will be examined in this section. A multipurpose plant, used to process products A,B,C and D, consists of 6 equipment families as detailed in Table VII. The feasible equipment families for each task and the
PRODUCT A
+ J + J + PRODUCI D
PRODUCT B
PRODUCI C
+ J + & CAMPAIGN
1
CAMPAIGN
2
Figure 3. Design configuration for example 1.
associated size factors and unit processing times are given in Table VIII. The unit cost coefficients and the production requirements for each product are given in Tables IX and X. Notice that there are two different equipment families that are feasible for the performance of each of the tasks B1, C2, D1, and D3. If we first assume that a single equipment unit can carry out each product task, then a feasible design cannot be found by the decomposition algorithm. In fact, a feasible design under this restrictive assumption does not exist. This can be easily seen after the solution of two problems, each corresponding to a single product plant used to process products A and B, respectively. The results show that A requires at least 4600 h to be processed, whereas B requires a t least 2000 h. This means that, in the multipurpose plant where the time horizon is 6200 h, A should be processed in the same campaign as B. But A and B are not allowed to be processed in the same campaign, because they both require equipment type P of which only one unit is available in the plant. In order to obtain a feasible design, we either have to relax selective upper bounds on the number of units and equipment sizes or increase the production horizon. Suppose that we relax the above assumption and allow at most two equipment groups to carry out product tasks that can be performed in more than one equipment unit. The complete MINLP model involves 892 constraints and 317 (92 0-1) variables. The bounding procedure succeeds in deriving tighter values of the parameter Bi for products A and B. Therefore, the bounding constraint (11.21) along with the campaign ordering constraints (1.29)-(1.31) will be introduced in the first MILP. Integer cuts of the form (11.24) will also be introduced in the subsequent MILP's
Table VI. DecomDosition Algorithm Performance I
example 1
2
3
subproblem MILPl NLPl MILP2 NLP2
no. of eqs/vars/O-1 vars 3781165172 1'781185 265j89136 1781185
MILPl NLPl MILP2 NLP2
5541221192 2021209 2631 113146 2021209
MILPl NLPl MILP2 NLP2
4051158163 2441270 289/113/49 244f 270
CPU time (s) without bounding cons 69.6 2.3 2.4 0.6 74.9 total 667 5.4 6 2 680.4 total 53.4 8.1 9 2
72.5 total
CPU time (9) with bounding cons
592 5.4 6
2 605.4 total (11% reduction) 16.8 8.1 5.4 2 32.3 total (55.4% reduction)
Ind. Eng. Chem. Res., Vol. 29,No. 10,1990 2069 Table VII. Available Equipment Items for Example 2 upper bound on no. of units equip item capacity range, L 500-3000 4 F R1, R2, R3 500-2000 1 G 500-3000 4 E l , E2, E3 500-3000 1 500-3000 1 P 500-3000 4
PRODUCT B
PRODUCl A
z
PRODUCT D
. IP
Available discrete sizes: 500-1000-2000-2500-3000.
U P
Table VIII. Size Factors ((L/kg)/Batch) and Processing Times (h/Batch) task/
WUlP tvDe
F
A1 A2
1.5 (4)
A3 B1 B2 B3
c1
C2 DI D2 D3
size factors (processing time) for equipment R1. R2. R3 C El.E2.E3 P
Z CAMPAIGN
2 (6.5)
1.6 (5)
1.5 (4)
3.6 (5.9)
2.5 (5.5) 1.4 (4.5)
1.5 (4) 1.5 i4j
1.9 (4.2) 3.7 (5.5)
2.7 (7) 2.2 (6.5)
1.6 (4.7)
1.8 (4.7)
unit type R1, R2, R3 G
a,
88 69 280
unit type E l ,. E2,. E3
P
z
a,
260 360 370
" Power cost coefficient, be = 0.6.
Table XI. Integer Solution for Example 2 (Only Non-Zero Values)" unit t w e V. N. unit tvue V, N. 2 G 2000 1 2000 F 1 El 2500 1 2000 R1 1 P 2500 1 R2 1000 nik
A-1
527 500 502 428
B-2
c-1 D-2
Bik 1326.4 1001 1002.8 1112.1
2
Table XII. Available Eauipment Items for Example 3 upper bound capacity range, L on no. of units equip item 1 F 500-5000 1 R1, R2, R3 500-2000 3 G 500-1000 1 P 500-3000 1 500-4000 a Available discrete sizes: 500-1000-2000-2500-3003500-40 00-5000.
Table XIII. Size Factors ((L/kg)/Batch) and Processing Times (h/Batch)
task/
Table X. Production Requirements for Example 2 yearly yearly demand, demand, product kg/year product kg/year A 700 000 C 380 000 B 500 000 D 400 000
product-campaign
CAMPAIGN
z
Table IX. Cost Coefficients. a., for E x a m d e 2"
F
1
Figure 4. Design configuration for example 2.
TL, 6.5 5.5 4.5 6.5
" Campaign
length, Tk: T1, 3415; T,, 2784. Objective function value: $122418.
(the calculation of Bi and the derivation of integer cuts are shown in Appendix I11 for example 3). A two-campaign configuration is then obtained with a cost of $122418 (compare to the design configuration with cost $210 310 obtained via direct solution of the problem in part 1 of this paper). The results are shown in Table XI. The decomposition algorithm required two major iterations to solve the problem with computation times reported in Table VI. As seen in Figure 4, equipment family 2 does not appear in the plant, since the tasks for which 2 is feasible (tasks C3 and D1, respectively) selected equipment type F for their processing (Fis cheaper than 2 in this problem). Notice also that products A-C and B-D share the units of equipment type F i n campaigns 1 and 2, respectively. In addition, two in-phase units are
equip type A1 A2 B1 B2 B3
c1
c2
c3 c4 c5 D1 D2 El E2
size factors (processing time) for equipment Rl,R2,R3 G 2 P 2 (3.5) 1.5 (2.5) 1.6 (3.9) 1.8 (4.1) 3.2 (3.2) 1.6 (2.9) 1.5 (4) 2.25 (3.8) 1.7 (4.5) 1.2 (3) 3 (2.9) 1.8 (5.7) 1 (3) 3 (2.5) 1 (3)
F
Table XIV. Cost Coefficients, a,, for Example 3" unit tvDe a. unit type F 200 P R1, R2, R3 220 G 280
z
a.
360 180
" Power cost coefficient, be = 0.6. Table XV. Production Requirements for Example 3 yearly yearly demand, demand, product kg/year product kgfyear A 700 000 D 548 000 B 740 000 E 200 000 C 275 OOO
used to carry out tasks A2 and D2. This is due to the fact that an increased batch size is required so that the production demands of A and D can be met, and therefore, more units are needed to carry out the corresponding product tasks. Clearly, more degrees of freedom have been introduced into the problem after the relaxation of the
2070 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 Table XVI. Integer Solution for Example 3 (Only Non-Zero Values)n unit type V. No unit type V, N, F 2000 1 G 1000 2 R1 2000 1 P 2000 1 R2 500 1 z 3500 1
product-campaign A-3 H-1
B-2 c-4
D-2 E- 1
kk
Aik
TLLk
600
1250.1 1090.3 1250.1 1111.2 1615.8
3.5 4.1 4.1 4.5 5.7 3
220 400 248 366 300
666.8
OCampaign length, Tk: TI, 901; T1, 2085; T3, 2099; T4, 1115. Objective function value: $143 254.
CIIIPAlGl I
ClMPAlGV I
(AMPAIGL 3
CAMPAIGN 4
Figure 5. Design configuration for example 3.
original assumptions, thus allowing the optimizer to form more flexible equipment configurations. Example 3. This example will be solved to illustrate the flexibility of allocating the same product to different production lines throughout the operation of a multipurpose plant. In addition, it will be used to demonstrate equipment group formation. Consider a plant that is to produce five products A, B, C, D, and E. Five possible equipment families could be used in the plant with sizes given in Table XII. The basic problem specifications are given in Tables XI11 and XIV and the annual production requirements in Table XV. The complete MINLP model involves 400 (105 binary) variables and 1147 constraints. A preanalysis of the problem data shows that product C cannot be processed in the same campaigns with the rest of the products in the plant, because it requires equipment types F, 2, and P for its processing, which are also required by products A, B, D, and E and of which a single unit is available in the plant. This observation is used to eliminate some of the binary and continuous variables and the corresponding constraints by assigning product C to a single campaign and not allowing products A, B, D, and E to be processed in that campaign. Thus, the number of constraints is reduced to 669 and the number of variables to 255 (63 binary). The decomposition algorithm required two major iterations to solve this reduced problem. Details of the sizes of the MILP and NLP subproblems and the associated computation times are given in Table VI. The values 6 = (1, 2 , 1, 2, 1) in the bounding constraint (11.21) have been used in the first MILP. Integer cuts of the form (11.24) have also been introduced in the subsequent MILP's (the calculation of 8, and the derivation of integer cuts are shown in Appendix 111). The values of the continuous variables are given in Table XVI, and the campaign structure is shown in Figure 5. The resulting design involves four campaigns. Products B and E are produced in parallel in the first campaign and
Table XVII. Results of Bounding Procedure for Example 3 DrodUCt TP! TP? TP"n 0; A 816.7 816.7 816.7 1 B 2427.2 4854.4 2427.2 2 C 928.125 1870.8 928.125 1 D 1405.6 1405.6 1405.6 2 E 600 600 600 1
B and D in the second campaign. The third and fourth campaigns involve only one product each (A and C, respectively). Restrictions on the maximum number of units for each equipment type used by A and C do not allow other products to be processed in the same campaigns as those products. Notice that product B is produced along two different production lines; in the first campaign, task B3 is carried out in unit type 2, whereas in the second campaign the same task is carried out in unit type P, both types being feasible for the performance of this task. Product B is allocated to different production lines, because products D and E , which are produced in the same campaigns as B, compete for the same equipment types with B. Furthermore, the relatively large production demand of B requires a larger fraction of the time horizon to be allocated to B, and consequently, B has to be processed in two campaigns rather than in only one. Also notice that units R1 and R2 are arranged in one equipment group for the processing of tasks Al, B2, and C2 and that one equipment group containing two identical units of type G is assigned to tasks B1 and C3. Conclusions This paper presents a solution procedure for the design of a general multipurpose batch plant. In part 1 of the series, the design problem is posed as a nonconvex mixed integer nonlinear model. The complexity of the proposed model makes the problem amenable to solution using existing MINLP solution techniques only for moderately sized problems. Consequently, an alternative solution procedure is developed that effectively decomposes the problem into two smaller subproblems: a MILP master problem that determines the best product-equipment and product-campaign assignment for fixed values of the campaign lengths and that yields a lower bound on the optimal solution of the original MINLP and a NLP subproblem that performs equipment sizing and determines the values of the campaign lengths for fixed values of the binary variables. This problem yields an upper bound on the optimal solution of the original MINLP. An iterative procedure is devised that solves an alternating sequence of MILP master problems and NLP subproblems treating the number of the campaign lengths as a parameter whose value is updated at the end of each iteration. In addition, a bounding procedure is developed tha creates integer cuts and bounding constraints to eliminate part of the problem degeneracy and, therefore, reduce the total computation time. Despite the fact that the proposed decomposition strategy does not guarantee attainment of the global optimal solution, the solutions obtained in a number of test problems can be proved to be optimal. In addition, all test problems were solved in reasonable computation time. Future work on the problem of the design of multipurpose plants should address the effects of the product changeovers on the design configuration. Although the reconfiguration of campaigns is allowed in the model developed in part l of the series, no costs have been included in the formulation to quantify the effect of campaign reconfiguration and the subsequent lost production time due to campaign setup times. This issue will be addressed in the future.
Acknowledgment This work was supported in part by the U.S.-Spain Committee for Technical Cooperation and E. I. du Pont de Nemours & Co.
Nomenclature Symbols N = number of products E = number of batch equipment types F = number of equipment families K = maximum number of campaigns H = total available production time i = index on products m = index on tasks e = index on equipment types f = index on equipment families k = index on campaigns TAi = set of tasks for product i Pi,! = set of feasible equipment types for task m of product 1
U, = set of tasks that can be executed by equipment type e Lf = set of equipment types that belong to equipment family
f Qi = yearly production requirement for product i Sime= size factor of task m of product i in equipment type e
processing time of task m of product i in equipment type e a,, be = cost coefficients for equipment type e V , = size of units of equipment type e Ne = number of units of equipment type e XImek= 0-1 assignment variable for task m of product i in equipment type e and in campaign k nik = number of batches of product i produced in campaign k Bik = batch size of product i in campaign k NUi,+ = number of units of type e that are contained in each equipment group assigned to task m of product i in campaign k NGimk= number of equipment groups assigned to task m of product i in campaign k T,, = limiting cycle time of product i in campaign k Tk = length of campaign k PRik= priority index for assignment of product i in campaign k CAP, = total capacity of equipment type e ( = V Y , ) TP?'" = minimum production time of product i ny" = minimum number of batches of product i = minimum cycle time of product i ce, de. = cost coefficients for equipment type e used in the objective function of the master problem 8, = parameter used in bounding analysis time=
NU;,,k IN~aXXi,ek i = 1, ...,N k = 1, ..., K (1.12) m E TAi e E Pi, NUimek2 Ximek i = 1, ..., N m E TAi e E Pi, k = 1, ...,K (1.13) m TLlk
time NGimb
m
k = 1, ..., K (1.15)
ETkIH
(1.17)
e
K k=l
Tk 2 nikTL,,
i = 1, ..., N
V p IV , I V r m 0 I Ne I NFm
Appendix I: Original Model Formulation
0 5 NUimekI Ntpm m E TAi
e=l
E Pi,
-TAi
0 I NGimk INGKT
E
k = 1, ..., K (1.14)
i = 1, ..., N
Ximek-
Operators [ a ] = ceiling operator, rounds a up to an integer value
min C a p e v e b e
E TAi
k = 1, ..., K e = 1, ..., E e = 1, ..., E
i = 1, ...,N m E TAi
(1.18) (1.19) (1.20)
k = 1, ..., K (1.21)
i = 1, ..., N e E Pi, k = 1, ..., K (1.22) k = 1, ..., K i = 1, ..., N n y = Tglax/T*Lik (1.23) 0 I Tk I H
k = 1, ..., K
(1.24)
2072 Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990
i = 1, ..., N
T * L , = max min (tim,/NGgy]
VCimek NUimekNGimkVe2 VfPinXimek
mETA, e E P,m
Tf;
k = 1, ..., K (1.25) k = 1, ..., K i = 1, ..., N = max (max {time}] m ETA, mEP,, (1.26)
VCimek = NUimekNGimkv, 5 NUga&NGgyVFaxXimek or
VCimek 5 N p a x v ~ X i m e k since from constraint (1.11) we get an upper bound on the product of the bounds NUKyk and NGK; xxNGimkNUimekI Ne INFax r m
k = 1, ..., K (1.27)
i = 1, ..., N
B*,k = min min { V p / S c m e } mETA, eEPIm
CPR,k 2 xpR,k+l
1
= 1, ..., N
k = 1, ..., K (1.28) k = 1, ..., K - 1
L
(1.29) PR,k 2 Xtlek PR,k I
1
Xllek
= 1, ..., N 1
e
= 1, ..., N
k = 1, ..., K (1.30) k = 1. ..., K (1.31)
Therefore, NGimkNUimekI N Fax NGgTNU&% IN F (ii) This condition states that the objective function value of problem (11) must be less than the corresponding value of problem (I). The objective function of the master problem (11) has the form
E p,1
e=l
eEP
V, 1 V,,,
e = 1, ..., E
e. e
+ 1 E L,
Appendix 11: Proof of Proposition The master problem (11) is a relaxation of the original MINLP (I) problem and consequently provides a valid lower bound on the optimal solution of the original formulation if (i) F(1) F(II), which means that every feasible solution to the original problem (I) is feasible to problem (11) and (ii) u(I1) I u(I), where F(P) and u(P) denote the set of feasible solutions and the objective value of problem P, respectively. The original formulation (I) is shown in Appendix I. (i) We must show that any feasible assignment and corresponding feasible values of the continuous variables for problem (I) are feasible in (11). First, any feasible assignment of (I) is feasible in (II), because (11) contains all the assignment and connectivity constraints of (I) that involve the binary variables. Feasibility with respect to the values of the continuous variables will be shown by aggregating constraints in (I) to yield constraints in (11). Let ntk, Blk, NUimek, NGimk, T L , ~ Tk, ) Ve, and N e denote a feasible solution to (I) with objective function value ~ ( 1 ) . Then, by aggregating constraints (1.7), (La), (1.15), and (1.18) of (I), we get constraint (11.6) in (11) Ve Qr 5 CnlkBIk 5 CnlkCNUtmek= k
h
e
aevp
(1.32)
Assuming that 0 Ibe 5 1 and be < 1 for at least one e , condition (ii) can be proved as follows
Srme
J
E
u,(VF'")~~ 5 CaeVebeN, = u(1)
where VCimek = NUimekNGimkVe(eq 11.5). Next, aggregation of constraint (1.11) in (I) with the use of definition (11.1)yields constraint (11.7) in (11)
or
CCVCimek IN,V, = CAP, i
m
Constraints (1.12) and (1.14) of (I) can be combined to yield constraint (11.9) in (11), and constraints (1.13) and (1.11) of (I) can be used to yield constraint (11.10) in (11)
e=l
In addition, if be = 1 for all e, then u(1) becomes linear (since CAP, = V Y , ) and u(1) = 411). Clearly, no linearization of the objective function u(1) is necessary in this case. In the above proof we used the following definitions and relations: (1) CAP, = VY,, (2) CAP:= = V,"NFaX, and (3) linear underestimators to approximate the term Vebe ( V y ) b e - (v?in)b, V,be I(VFin)be (V, - V p ) vyx - Vmin e
+
Appendix 111: Outline of Bounding Procedure in Example 3 First MILP.
Ind. Eng. Chem. Res., Vol. 29, No. 10, 1990 2073
Step 1: Choose Tk = 1240, k = 1, ..., 5 (H= 6200 h). Step 2: Select product B. Step 3: Calculate the minimum production time TPE'" as follows: From (11.16), njfn = Q B / ( min I N p -VE" , SBIC
R3
N y - VF" ,
e=R1
SB2e
Since the values of Tk are unequal, go to step 4 of the bounding procedure. According to the preanalysis of the problem data explained earlier, products A, B, D, and E can be assigned to campaigns 1-3, whereas product C i s assigned to campaign 4. Then, from the values of TPY in Table XVII, it is clear that product B cannot be assigned to a single campaign and product D cannot be assigned only to campaign 1. Then, the following integer cuts are introduced into the MILP master problem: 1:
2: From (11.18), TL, = max (3.9,4.1,3.2,2.9) = 4.1
3:
Therefore,
4:
3
TPB = 592 X 4.1 = 2427.2 Literature Cited
From (11.19),
740000/(
{
min 3-
3
1000 2000 3000 -3 1X 1.6' 1 X 1.18' 1 X 1.6'
=))
1 X 3.2
= 740000/625 = 1184
From (11.17), TL, = max (3.9/3,4.1/1,3.2/1,2.9/1} = 4.1 Therefore,
TPi = 1184 X 4.1 = 4854.4 Finally,
T P P = 2427.2 Since the campaign lengths T k are equal, go to step 5. Step 5: eA = [2427.2/12401 = [i.951 = 2 Similar calculations for products A, C, D, and E produce the results shown in Table XVII. Second MILP. The values of the campaign lengths provided by the NLP subproblem are Tl = 1097, T 2 = 1861, T3 = 1960, and T4 = 1282. Notice that K = 4 during the second major iteration of the algorithm.
Beale, E. M. L. Integer Programming. In The State of the Art in Numerical Analysis; Jacobs, D., Ed.; Academic Press: London, 1977. Brook, A.; Kendrick, D.; Meeraus, A. GAMS, A User's Guide; Scientific Press: Redwood City, CA, 1988. Crowder, H.; Johnson, E. L.; Padberg, M. Solving Large-Scale Zero-One Linear Programming Problems. Oper. Res. 1983,31, 803-834. Duran, M. A.; Grossmann, I. E. An Outer-Approximation Algorithm for a class of Mixed-Integer Nonlinear Programs. Math. Prog. 1986,36,307-339. Faqir, N. M.; Karimi, I. A. Design of Multipurpose Batch Planta with Multiple Production Routes. Presented at the FOCAPD Conference, Snowmass Village, CO, 1989; Paper 4. Floudas, C. A,; Aggarwal, A.; Ciric, A. R. Global Optimum search for Nonconvex NLP and MINLP problems. Comput. Chem. Eng. 1989,13 (lo), 1117-1132. Geoffrion, A. M. Generalized Benders Decomposition. J. Optim. Theory Appl. 1972,10, 237-260. Gupta, 0. K. Branch-and-Bound Experiments in Nonlinear Integer Programming. Ph.D. Dissertation, Purdue University. West Lafayette, IN, igao. Kocis, G. R.; Grossmann, I. E. Global Optimization of Nonconvex MINLP uroblems in Process Svnthesis. Presented at the AIChE Annual Meeting, New York, i987a; Paper 96b. Kocis, G. R.; Grossmann, I. E. Relaxation Strategy for the Structural Optimization of Process Flow Sheets. Znd. Eng. Chem. Res. 1987b,26, 1869-1880. Murtagh, B. A.; Saunders, M. A. MINOS 5.0 User's Guide. Technical Report SOL 83-20; Stanford University Systems Optimization Laboratory, 1983. Papageorgaki, S.; Reklaitis, G. V. Optimal Design of Multipurpose Batch Plants. 1. Problem Formulation. Znd. Eng. Chem. Res. 1990,preceding paper in this issue.
Received f o r review February 15, 1990 Revised manuscript received May 23, 1990 Accepted June 4, 1990
