Ind. Eng. Chem. Res. 1996, 35, 2065-2070
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RESEARCH NOTES A Computational Approach for the Modeling/Decomposition Strategy in the MINLP Optimization of Process Flowsheets with Implicit Models Zdravko Kravanja† and Ignacio E. Grossmann*,‡ Faculty of Chemistry and Chemical Engineering, University of Maribor, Maribor SLO-2000, Slovenia, and Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
This paper describes a computational approach for performing topology optimization in process flowsheets that are described by implicit models using the modeling/decomposition strategy. The key point in the proposed approach is a procedure for the generation of the MILP master problem that would correspond to the approximation one would have obtained with an equationoriented system (e.g., PROSYN-MINLP). Two numerical examples are presented. Introduction The objective of this paper is to address the application of MINLP for the topology optimization of process flowsheets when they are described by implicit models, as is the case in process simulators. Most of the previous work has assumed that the MINLP optimization model is given in equation form (e.g., see Grossmann, 1990; Grossmann and Kravanja, 1995). A good example of a computer code that assumes the problem to be given in that form is DICOPT++ (Viswanathan and Grossmann, 1990). However, when applied to process flowsheets, MINLP models tend to experience two major problems: (a) singularities in the equations of process units with zero flows that are not selected; (b) large effect of nonconvexities in the master problem caused by the linearization of units at zero flows. To circumvent these problems, Kocis and Grossmann (1989) proposed a modeling/decomposition (M/D) strategy. This strategy is based on the outer-approximation/ equality relaxation algorithm for MINLP but with the key difference that the NLP subproblems only involve equations for the existing part of the superstructure. The motivation is to reduce the dimensionality of the NLP and to minimize the problem with singularities and nonconvexities that arise with zero flows. The NLP subproblems are solved alternatively with the MILP master problems in which special care is taken in the modeling of interconnection nodes. The M/D strategy has been successfully implemented in PROSYN-MINLP (Kravanja and Grossmann, 1990, 1994) which currently has the capability of synthesizing process flowsheets that are described by relatively simple process models. In order to consider complex process models, it is clearly desirable to be able to perform the MINLP optimization in sequential process simulators (Diaz and Bandoni, 1996). Three major attempts have been made at developing implementations of MINLP for topology optimization in process simulators. First, Diwekar et al. (1992a) developed an implementation of the M/D strategy in the public version of Aspen. This was a useful study in terms of establishing whether, in * Author to whom correspondence should be addressed. † University of Maribor. ‡ Carnegie-Mellon University.
S0888-5885(95)00424-6 CCC: $12.00
principle, such a technique can be applied to process simulators. However, while a relatively complex example was solved, that study relied on the restrictive assumption that the user would specify as an input the mathematical structure of the MILP master problem. Also, the NLP optimization step was performed in the full space of the variables, which is not only computationally expensive but defeats the main purpose of the M/D strategy which is to reduce the burden of the NLP optimization step. One of the assumptions in the M/D strategy is that an explicit representation of equations of interconnection nodes (mixers and splitters) and process units is given. This allows one to construct an MILP master problem using linearizations and relaxations based on Lagrange multipliers even if the corresponding NLP subproblems only include the equations of the specific flowsheet at hand. The difficulty in a sequential process simulator is that the equations are given in implicit form and therefore the optimizer only “sees” the problem in a reduced space. Therefore, it is generally unclear on how to recover or construct an MILP master problem using the information from the reduced NLP subproblems which are given implicitly by a “black box” computation. To circumvent this problem, in a subsequent development Diwekar et al. (1992b) proposed the use of auxiliary variables and equations for input/output equations in order to develop linear approximations for the MILP master problem. Last, in a recent paper Renaume et al. (1995) adopted a strategy similar to that of Diwekar et al. (1992b), but with the advantage of making use of the analytical derivatives that are available in their process simulator. One drawback with the approach proposed by Diwekar et al. (1992b) is that the auxiliary variables and equations for the input/output equations have to be postulated. One reason this may not be a trivial task is that the proposed simplified models may not bear a close relationship with the actual models of the simulator. In this paper we show that an alternative computational approach can be developed that relies on the idea of generating derivatives for the input/output relations and applying a linear programming problem to determine the appropriate relaxation of the equations. Provided the derivative information in the full space is exact, the proposed approach produces a © 1996 American Chemical Society
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performance identical to that of the M/D strategy as if the problem had been solved in equation form. Two examples will be presented to illustrate the proposed approach. Proposed Approach The basic idea in the proposed approach is to use the M/D strategy but separating as much as possible the close relationship between the NLP subproblem and the MILP master problem. On the one hand, the objective is to make a simulator look more like an equation-based system for the sake of the MILP master problem, while, on the other hand, the objective is to perform the NLP optimization as is currently done in process simulators, allowing for the possibility of using gradient-based methods or even direct-search methods if needed. To simplify the presentation of the equations, we use the generalized disjunctive programming representation by Turkay and Grossmann (1996) for describing the M/D strategy. Let us assume first that the MINLP was to be given in the equation form:
min Z )
∑i ci + f(x)
s.t.
[ ][
g(x) e 0
Yi ¬Yi hi(x) ) 0 ∨ Bix ) 0 ri(x) e 0 ci ) 0 ci ) γi
]
i∈D
(P1)
Ω(Y) ) True x ∈ Rn, c g 0, Y ∈ {True, False}m The above model involves the following three types of variables: x and ci are the continuous variables (the former correspond to flows, pressures, and temperatures, while the latter are used to exclusively represent fixed charges); the Boolean variables, Yi, that are associated with the existence of units and are used to determine whether a unit is present (Yi ) True) or absent (Yi ) False) in the superstructure. The first set of inequalities represent global inequalities that hold irrespective of the discrete choices. The set of disjunctions, D, apply for the processing units. If a process unit exists (Yi ) True), then the equations and constraints describing that unit are enforced and a fixed charge is applied; otherwise, (¬Yi ) False) a subset of continuous variables and the fixed charge are set to zero. We define Bi ) [bjT], such that bjT ) ejT if xj ) 0, and bjT ) 0T if xj * 0. In this way only a subset of the variables x is forced to zero (typically flows). The NLP subproblems arise from a fixed choice of the Boolean variables Yil. However, due to the structure of the implicit models in a process simulator, the equations hi(x) ) hi(u,v) ) 0 for Yil ) True are eliminated by expressing dependent variables v in terms of decision variables u of dimension q , n; that is,
hi(u,v) ) 0 w v ) φi(u)
(1)
Therefore, the NLP subproblem as it arises in a process simulator for fixed Boolean variables Yil is given as follows:
min ZU ) s.t.
∑i ci + f(u,φ(u))
g(u,φ(u)) e 0 ri(u,φi(u)) e 0 ci ) γi Bix ) 0 ci ) 0
}
}
for Yil ) True
(P2)
for Yil ) False
u ∈ Rq, c g 0 The difficulty with problem (P2) is that, since it is given in the reduced space u, it cannot be directly used to construct an MILP master problem corresponding to (P1) in the full space of x. To circumvent this problem, the requirement in the proposed approach is to obtain derivatives for linearizing each module of the process units and interconnection nodes in terms of input and output variables and design variables (we assume these correspond to all original variables x). The input/output variables will typically include individual component flows, pressures, and temperatures, while the design variables will typically include equipment parameters such as sizes or related specifications. In general, the derivatives could be obtained by perturbations of each module at the solution of the NLP, or ideally they might be available in analytical form from the simulator (e.g., see Renaume et al., 1995). Since these linearizations will be used as a basis to construct the MILP master problem, an important additional question is how to decide on the relaxation of the equations into inequalities as is needed in the MILP master for the outerapproximation algorithm (Kocis and Grossmann, 1987). The reason this is a nontrivial question is due to the fact that no multipliers are available from the NLP for the relevant equations that enter the master problem. The key idea in this work is to take advantage of the linear representation of the MILP master and solve the following LP for the equations of a flowsheet in order to obtain dual information that can be used to decide on how to relax the equations:
min ZLP ) Roa +
∑i ci
s.t. Roa g f(xl) + ∇f(xl)T(x - xl) g(xl) + ∇g(xl)T(x - xl) e 0 ∇hi(xl)T(x - xl) ) 0 ri(xl) + ∇ri(xl)T(x - xl) e 0 ci ) γi Bix ) 0 ci ) 0
}
}
for Yil ) True
(P3)
for Yil ) False
x ∈ R n,
cg0
The solution of the above problem will then yield multipliers λil of each of the equations hi(x) ) 0 for Yil ) True. Also, both (P2) and (P3) will have the same optimal objective function value. In this way, given the solution of an initial flowsheet and appropriate subproblems, the MINLP in equation form (P1) can be
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996 2067
replaced by
min ZL ) Roa +
3. Iterate between NLP and MILP steps until convergence is achieved. There are of course a number of other detailed items involved in the above strategy such as the modeling of interconnection nodes. These are discussed in Kravanja and Grossmann (1990, 1994). The important point, however, of the above solution approach is that, if the derivatives are exact, the optimization problem with an implicit model becomes equivalent as if it had been given in equation form as in problem (P1).
∑i ci
s.t. Roa g f(xl) + ∇f(xl)T(x - xl)
[
g(xl) + ∇g(xl)T(x - xl) e 0
}
l ) 1, ..., L
]
Yi Tik[∇hi(xl)T(x - xl)] e 0 l ∈ KLi ∨ ri(xl) + ∇ri(xl)T(x - xl) e 0 l ∈ KLi ci ) γi ¬Yi
[ ] Bix ) 0 ci ) 0
Small Example
i ∈ D (P4)
The proposed approach will be illustrated with the following small example which mimics the situation of a sequential process simulator. The superstructure is shown in Figure 1. The following MINLP model (2) is considered for the superstructure in Figure 1.
min Z ) c1 + c2 + c3 + c4 + 6u1 + 7u2 + 8u3 + 9u4 + 2x1
Ω(Y) ) True x ∈ Rn, c g 0, Y ∈ {True, False}m
s.t.
where given L major iterations, the linearization set KLi ) {l|Yil ) True, l ) 1, ..., L} can be defined for linearizations generated for disjunctions i in which their corresponding Boolean variable is True. Also, Tik is a diagonal matrix with elements tiik ) {sign(λil)}. Problem (P4) can, in turn, also be solved as an MILP problem as shown by Turkay and Grossmann (1996).
min ZL ) Roa + s.t. Roa g f(xl) + ∇f(xl)T(x - xl) g(xl) + ∇g(xl)T(x - xl) e 0
∑i γiyi
}
l ) 1, ..., L (P5)
Tik∇hi(xl)Tx e Tik[-hi(xl) + l T l
i
∇hi(x ) x ]yi l ∈ KL , i ∈ D Ay e a Roa ∈ R1, x ∈ Rn, y ∈ {0, 1}m In summary, the proposed steps of the M/D strategy for the MINLP optimization of process flowsheets with implicit models would be as follows: 1. Initialization and decomposition step. (a) The superstructure is partitioned into an initial flowsheet and subsystems that are to be optimized (see Kocis and Grossmann, 1989; Kravanja and Grossmann, 1990). (b) The NLP (P2) for the initial flowsheet is solved. The linearizations corresponding to input/output relation in each unit are derived at the NLP solution point to construct problem (P3). The LP in (P3) is solved to determine the corresponding Lagrange multipliers λi. (c) The subsystems are suboptimized with an NLP subproblem similar to (P2) using information of multipliers and flows of the initial flowsheet (Kocis and Grossmann, 1989). Linearizations are derived and the corresponding LP as in (P3) is solved to obtain multipliers λi. 2. Construct MILP master problem (P5), and solve to predict a new flowsheet.
x2 + x4 ) x1 x6 ) x3 + x5 x7 + x9 ) x6
[ [ [ [
][ ][ ][ ][
x11 ) x8 + x10
Y1 ¬Y1 x3 ) 0 x3 ) 0.6 log(x2 + u1) x2 e 25 ∨ x2 ) 0 u1 ) 0 1 e u1 e 15 c1 ) 0 c1 ) 6 Y2 ¬Y2 x5 ) 0 x5 ) 0.7 log(x4 + u2) x4 e 25 ∨ x4 ) 0 u2 ) 0 1 e u2 e 15 c2 ) 0 c2 ) 7 Y3 ¬Y3 x8 ) 0 x8 ) 0.8 log(x7 + u3) x7 e 25 ∨ x7 ) 0 u3 ) 0 1 e u3 e 1.5 c3 ) 0 c3 ) 8
] ] ] ]
(2)
Y4 ¬Y4 x10 ) 0 x10 ) 0.9 log(x9 + u4) x9 e 25 ∨ x9 ) 0 u4 ) 0 1 e u4 e 1.5 c4 ) 0 c4 ) 9 Y1 ∨ Y2 Y3 ∨ Y4 x g 0, c g 0, x1 e 25, x11 g 1
This is a simple problem involving 11 flow variables x, 4 design variables u, and 4 Boolean variables Y. All
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Figure 1. Superstructure for the small example. Table 1. Results: Progress of Iterations in the Small Example iteration 1
cost/lower bound
NLP flowsheet (units 1, 3) LP, multipliers S1 ) -6, M1 ) -275.84, S2 ) -275.84, M2 ) -1203.46, unit 1 ) -275.84, unit 2 ) -1203.46 NLP unit 2 LP, multiplier unit 1 ) -275.84 NLP unit 4 LP, multiplier unit 4 ) -1203.46 MILP iteration 2 NLP flowsheet (units 2, 4) LP, multipliers unit 2 ) -25.702, unit 4 ) -86.75 MILP Optimum: x1 ) x4 ) 7.996, x5 ) x6 ) x9 ) 1.538, x10 ) x11 ) 1.0, u2 ) 1.0, u4 ) 1.50
91.504
-465.42 -782.29 49.223
cost/ lower bound 52.492
Figure 2. Superstructure for example 2. Notation: MS, singlechoice mixer; SS, single-choice splitter; M, multiple-choice mixer; COMP, compressor; HEAT, heater; COOL, cooler; RCT, reactor; VALV, valve; PROD, main product; S, multiple-choice splitter; ByPROD, byproduct.
the corresponding Lagrange multipliers. The search is stopped since the lower bound of the MILP master (including the integer cut) is 61.05 which exceeds the current upper bound, 52.492. It is important to note that the results shown in Table 1 are identical to the ones when the MINLP model (P1) is solved in equation form with the M/D strategy. Flowsheet Example
61.05
f STOP cost ) 52.492
the units involve nonlinear input/output relationships. Although the model is given in equation form, we will solve the NLP subproblems as a black box or implicit model in the space of decision variables as would be done in a process simulator. Below we highlight the major steps in the procedure (see Table 1): 1. (a) The superstructure is partitioned into the initial flowsheet consisting of units 1 and 3 (Y1 ) True, Y2 ) False, Y3 ) True, Y4 ) False) and the subsystems consisting of units 2 and 4. (b) The NLP is solved for the initial flowsheet having as decision variables u1, u3, and x1. The cost is Z ) 91.504. Linearizations are derived for the input/output of units 1 and 3. The Lagrange multipliers are determined by solving the LP based on the above linearizations. (c) The units 2 and 4 are suboptimized fixing the flows x4 ) x1 and x9 ) x6, respectively, and assigning the prices to streams 4 and 5 and 9 and 10 with the multipliers found in the previous LP. Linearizations are derived for each unit and an LP is solved for each of them to obtain the Lagrange multipliers of the equations. 2. The MILP master problem is derived by including the linearizations obtained in step 1 and by converting the equations into inequalities (e) based on the sign of the Lagrange multipliers. 3. The solution of the MILP predicts as the new structure the one involving units 2 and 4. The lower bound for the cost predicted by this master problem is Z ) 49.23. 4. The NLP for the flowsheet with units 2 and 4 is solved. Note that this flowsheet has x1 and u2 and u4 as decision variables. The optimum cost is Z ) 52.492 and represents the new upper bound. Linearizations are derived for the input/output of units 2 and 4, and an LP is solved to determine the Lagrange multipliers of these equations. 5. The MILP master problem is augmented with the linearizations of step 4 which are relaxed according to
A larger example problem that is representative of a flowsheet (Kocis and Grossmann 1987; Kravanja and Grossmann, 1990) will be considered in this section. The basic steps of the M/D strategy and the OA algorithm are the same as the ones described above. Note that after each NLP step an LP has been solved for the corresponding linear approximation of the NLP subproblem to get the multipliers which were used for equality relaxation and the evaluation of weights for the penalty function (in the suboptimization also for inlet/ outlet costs). PROSYN-MINLP was used to “simulate” an actual process simulator as well as steps of the MINLP in the sequential process simulator. Following are the basic features of this example problem: (A) It illustrates the M/D strategy and the MINLP basic steps. (B) It has modular network structure. According to the M/D strategy, all nodes are divided into a group of process unit nodes and a group of interconnection nodes. (C) All modules for process units and multiple-choice nodes in PROSYN-MINLP were rewritten so that their linearizations would exactly represent linearizations generated for the MILP phase in sequential simulators. Only single-choice node models remain unchanged since they can be used in the MILP as they are. (D) Each unit is described by input/output equations. The vector of continuous variables typically consists of input and output stream variables (total flowrates, component flowrates, temperatures, and pressures), design variables (volumes, areas, etc.), performance variables (split fractions, conversions, consumption of utilities, etc.), costs, and unit parameters. Note that a unit parameter can be any of these variables. (E) The same modifications to handle unstructured nonconvexities have been used as described in a recent paper about the modified OA/ER algorithm by Kravanja and Grossmann (1994): (a) deactivation of linearizations, (b) use of the penalty function, (c) global convexity test and validation of the outer approximation. The superstructure is shown in Figure 2, and the problem data are shown in Table 2. It involves 16 alternative flowsheets. The alternatives for producing
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996 2069 Table 2. Flowsheet Synthesis Problem Data feedstock or (by)product FEED1 FEED2 PROD e 1 kmol/s ByPROD operating time
composition
cost ($/kmol)
60%A 25%B 15%D 65%A 30%B 5%D >90%C
0.0245 0.0294 0.2614 0.0163 Figure 3. Initial flowsheet for example 2.
8500 h/yr
utilities
costs
electricity steam cooling water
$0.03 kW/h $8.0/1 × 106 kJ $0.7/1 × 106 kJ
chemical product C from A and B (D is inert) are either of two feedstocks, either single-stage or double-stage compression of the reactor feed, either of two adiabatic reactors, and either of single-stage or double-stage compression of recycling stream. The objective for this synthesis problem is the maximization of the annual profit. The size of the NLP problem is about 160 model equations and 190 module input/output relation equations with about 380 variables. The superstructure contains 8 binary variables and the following 15 decision variables:
FEED1 total flowrate FEED2 total flowrate COMP1 outlet pressure COMP2 outlet pressure COMP3 outlet pressure COMP5 outlet pressure COOL1 cooling duty COOL2 cooling duty COOL3 cooling duty COOL4 cooling duty HEAT1 heating duty RCT1 volume RCT2 volume VALV1 pressure drop S1 purge fraction, in which the outlet pressures of COMP3 and COMP1 must equal the one of COMP4 or COMP6 and the total flowrate of PROD is set to 1 kmol/s. The steps of the proposed approach are as follows: (1) (a) The superstructure has been partitioned into the initial flowsheet (Figure 3) y1 ) {1, 0, 0, 1, 1, 0, 0, 1} to be optimized and into the remaining units (Figure 4) y ) {0, 1, 1, 0, 0, 1, 1, 0} to be suboptimized using a Lagrangian scheme. (b) I. NLP for the initial flowsheet (Figure 3) at y1 ) {1, 0, 0, 1, 1, 0, 0, 1}:
Figure 4. Substructures.
svector of the continuous variables x1, optimal solution: -558 000$/yr Linearization for the first NLP; Solve the LP on the above linearizations: sObtain the Lagrange multipliers. (c) The suboptimization NLP for the remaining units (Figure 4) at y ) {0, 1, 1, 0, 0, 1, 1, 0} with fixed outlet variables for FEED2, inlet variables for RCT2, inlet variables and outlet pressure for COMP1 and COMP4, and the multipliers found at previous LP: svector of the continuous variables x1, suboptimal solution: 2 929 000$/yr Linearization for the suboptimization NLP; Solve the LP on the above linearizations: sObtain the Lagrange multipliers. (2) Generation of the I. MILP master problem which now consists of: slinearizations of units in/out equations, slinear equations for single choice mixers and splitters, soriginal linear logical constraints (they are actually presented in single choice nodes) soriginal linear inequality constraints (production, specifications), sand linear in/output relation between the units and interconnection nodes. Note that based on the Lagrangian multipliers from the previous LPs the linearization is properly relaxed and weights for the penalty function are also now evaluated. (3) I. MILP phase: spredicts new topology at y2 ) {0, 1, 0, 1, 1, 0, 1, 0} sinitial point for the next NLP at the upper bound +2 985 000 $/yr. (4) II. NLP for topology y2 ) {0, 1, 0, 1, 1, 0, 1, 0}: svector of the continuous variables x2 and optimal solution: +2 139 000 $/yr Linearization for the second NLP; LP on the above linearizations: sThe Lagrange multipliers. (5) II. MILP phase: Linearizations of II. NLP unit in/out equations are added to the MILP master problem. Again, the linearizations are first relaxed based on the sign of the
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that would be obtained had the problem been solved in equation form. Therefore, provided exact derivatives are supplied for the input/output equations for each unit, the proposed method will be identical to the equation-based representation of the M/D strategy. It is clear that the availability of derivatives will increase the feasibility of implementing MINLP optimization in process simulators with the proposed approach. It is clear, however, that the ideas presented in this paper must be fully tested with an actual implementation in a process simulator. Figure 5. Final optimal flowsheet for exampe 2.
Lagrangian multipliers from the previous LP and new weights for penalty function are evaluated. spredicts new topology at y3 ) {0, 1, 0, 1, 1, 0, 0, 1} sinitial point for the next NLP at the solution of +2 334 000 $/yr. (6) III. NLP for topology y3 ) {0, 1, 0, 1, 1, 0, 0, 1}: svector of the continuous variables x3 and optimal solution: +2 098 000 $/yr The III. NLP solution is not improved; therefore, the procedure is stopped. The optimal solution is the one that is found in II. NLP (Figure 5) with the following optimal decision variables: FEED2 total flowrate COMP2 outlet pressure COOL1 cooling duty COMP3 (COMP4) outlet pressure COOL2 cooling duty HEAT1 heating duty RCT1 volume VALV1 pressure drop COOL3 cooling duty S1 purge fraction
3.81 kmol/s 0.775 MPa 6.67 × 108 MJ/yr 8.12 MPa 0 MJ/yr 0 MJ/yr 27.23 m3 0 MPa 9.84 × 108 MJ/yr 0.037
Various convexity tests have been performed in this example (see Kravanja and Grossmann, 1994). Since linearizations are based on input/output model equations which are now more dense in terms of the nonlinearities and nonconvexities that are involved, the convexity test has appeared to be very important to reduce the undesirable effects of the nonconvexities. The convexity test by which all linearizations infeasible to the last Karush-Kuhn-Tucker point are temporarily dropped from the MILP master problem, seem to be by far the best choice. Conclusions This paper has presented a computational strategy that allows the application of the M/D strategy in process flowsheets that are described by implicit models. As was shown, the key idea relies on producing an MILP master problem that is, in principle, identical to the one
Acknowledgment The authors are grateful to Aspen-Technology and to the U.S.-Slovenian Scientific Exchange Program for the financial support for this work. Literature Cited Diaz, M. S.; Bandoni, J. A. A Mixed-Integer Optimization Strategy for a Large Scale Chemical Plant in Operation. Comput. Chem. Eng. 1996, 20, 531-545. Diwekar, U. M.; Grossmann, I. E.; Rubin, E. S. An MINLP Process Synthesizer for a Sequential Modular Simulator. Ind. Eng. Chem. Res. 1992a, 31, 313-322. Diwekar, U. M.; Frey, C. M.; Rubin, E. S. Synthesizing Optimal Flowsheets. Application to IGCC System Environmental Control. Ind. Eng. Chem. Res. 1992b, 31, 1927-1936. Grossmann, I. E. MINLP Optimization Algorithms and Strategies for Process Synthesis. In Proceedings of FOCAPD ‘89; Sirrola et al., Eds.; Elsevier: New York, 1990; p 105. Grossmann, I. E.; Kravanja, Z. Mixed-Integer Nonlinear Programming Techniques for Process Systems Engineering. Comput. Chem. Eng. Suppl. 1995, S189-S204. Kocis, G. R.; Grossmann, I. E. Relaxation Strategy for the Structural Optimization of Process Flowsheets. Ind. Eng. Chem. Res. 1987, 26, 1869-1880. Kocis, G. R.; Grossmann, I. E. Modelling and Decomposition Strategy for the MINLP Optimization of Process Flowsheets. Comput. Chem. Eng. 1989, 13, 797-819. Kravanja, Z.; Grossmann, I. E. PROSYNsan MINLP Process Synthesizer. Comput. Chem. Eng. 1990, 14, 1363-1378. Kravanja, Z.; Grossmann, I. E. New Developments and Capabilities in PROSYNsan Automated Topology and Parameter Process Synthesizer. Comput. Chem. Eng. 1994, 18, 1097-1114. Renaume, J. M.; Joulia, X.; Koehret, B. Development of a Process Sysnthesizer in a Modular Environment. Comput. Chem. Eng. Suppl. 1995, 19, S33-S38. Turkay, M.; Grossmann, I. E. Logic-Based MINLP Algorithms for the Optimal Synthesis of Process Networks. Comput. Chem. Eng. 1996, 20, 959-978. Viswanathan, J.; Grossmann, I. E. A Combined Penalty Function and Outer Approximation Method for MINLP Optimization. Comput. Chem. Eng. 1990, 14, 769-782.
Received for review July 11, 1995 Revised manuscript received March 22, 1996 Accepted March 23, 1996X IE950424F X Abstract published in Advance ACS Abstracts, May 1, 1996.
