Compulers them. Engng, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Vol. I?. No. 9, pp. 909-927, 1993 Printed in Great Britain. All rights reserved 0098-1354/93 $6.00 + 0.00 zyxwvut Copyright 0 1993 Pcrgamon Press Ltd SYMBOLIC INTEGRATION OF LOGIC IN MIXED-INTEGER LINEAR PROGRAMMING TECHNIQUES FOR PROCESS SYNTHESIS R. RAMAN and I. E. GROSSMANN? Department of ChemicalEngineering, Carnegie Mellon University,Pittsburgh, PA 15213, U.S.A. zyxwvutsrqponmlk (Received 10 February 1992; final revision received 18 December 1992; received for publicarion 7 January 1993) Abstract-This paper deals with the branch and bound solution of synthesis problems that are modeled as mixed-integer linear programming (MILP) problems. Logic relations between potential units in a superstructure are considered through symbolic integration within the numerical basedbranch and bound scheme. The objective of this integration is to reduce the number of nodes that must be enumerated by using the logic to decide on the branching of variables, and to determine by symbolic inference whether additional variables can be fixed at each node. Two different strategies for performing the integration are proposed that use the disjunctive and conjuctive normal form representations of the logic, respectively. The paper also addresses the question of how to systematically generate the logic for process flowsheet superstructures. Computational results are presented to compare the performance of the proposed methods and a variant that includes violated logic inequalities in the model with the cases when all logic inequalities are included in or excluded from the model. Discrete decisions constitute an important element in process design and synthesis problems (e.g. deciding what units to integrate in a flowsheet). Mathematical programming approaches make use of 0-l binary variables to model these decisions. Together with continuous variables, one can model design and synthesis problems through an objective function and a set of constraints representing material, heat bal- ances and specifications. When these involve nonlin- earities, a mixed-integer nonlinear program (MINLP) results. Depending on the assumptions and the super- structure representation, the synthesis problem can often be posed as a mixed-integer linear program- ming (MILP) problem. The modeling step is crucial to the success of the synthesis method and a discus- sion on this topic can be found in Kocis and Gross- mann (1989) and Grossmann (1990). Another major step involves the solution of the MI(N)LP optimization model. The most common solution technique for the MILP problem is the well known LP-based branch and bound algorithm (see Beale, 1977; Nemhauser and Wolsey, 1988) which is implemented in most computer codes (e.g. LINDO, ZOOM, MPSX, SCICONIC). Recently these methods have been improved by the use of prepro- cessing techniques that can reduce dimensionality, and cutting planes that can reduce the integrality gap tTo whom all correspondence should be addressed. of the MILP (e.g. see Van Roy and Wolsey, 1987). An alternative method for MILP is the Benders de- composition method (Benders, 1962) which also re- quires the branch and bound method for solving the master problem. Similarly, the Generalized Benders Decomposition (GBD) (Geoffrion, 1972) and the Outer Approximation (OA) (Duran and Grossmann, 1986) algorithms for MINLP problems require that the corresponding MILP master problem be solved by branch and bound. Finally, the solution of the MINLP can also be performed directly with a branch and bound method where NLP subproblems are solved at each node of the tree (see Beale, 1977; Gupta, 1980; Nabar and Schrage, 1990). From the above, it is clear that the branch and bound method plays a central role in the solution of the mixed- integer optimization problems. Since in many synthesis problems the number of &-I variables that is required can be rather large, the potential size of the tree that must be examined by a branch and bound method can become a major bottleneck in the computations. On the other hand, knowledge about the synthesis problem in the form of heuristics and logic of relations among units can potentially provide information about the design space and make the problem easier to solve. Raman and Grossmann (1991a) showed how both logic relations and heuristics expressed in the form of propositional logic can be represented in terms of linear inequalities involving O-1 variables. Based on this representation, Raman and Grossmann (1992) 909