Annals of Operations Research

, Volume 42, Issue 1, pp 169–191 | Cite as

Symbolic integration of logic in MILP branch and bound methods for the synthesis of process networks

  • Ramesh Raman
  • Ignacio E. Grossmann
Article

Abstract

This paper deals with the branch and bound solution of process synthesis problems that are modelled as mixed-integer linear programming (MILP) problems. The symbolic integration of logic relations between potential units in a process network is proposed in the LP based branch and bound method to expedite the search for the optimal solution. 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. The important feature of this approach is that it does not require additional constraints in the MILP and the logic can be systematically generated for process networks. Strategies for performing the integration are proposed that use the disjunctive and conjunctive normal form representations of the logic, respectively. Computational results will be presented to illustrate that substantial savings can be achieved.

Keywords

Normal Form Computational Result Additional Variable Additional Constraint Form Representation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M.J. Andrecovich and A.W. Westerberg, An MILP formulation for heat-integrated distillation sequences synthesis, AIChE J. 31(1985)1461–1474.CrossRefGoogle Scholar
  2. [2]
    E. Balas and C.H. Martin, Pivot and complement — A heuristic for 0–1 programming, Manag. Sci. 26(1980)86–96.CrossRefGoogle Scholar
  3. [3]
    A. Barr and E.A. Feigenbaum (eds.),Handbook of Artificial Intelligence, 3 vols. (William Kaufmann, Los Altos, CA, 1981).Google Scholar
  4. [4]
    E.M.L. Beale,Integer Programming. The State of the Art in Numerical Analysis, ed. D. Jacobs (Academic Press, London, 1977) pp. 408–448.Google Scholar
  5. [5]
    J.F. Benders, Partitioning procedures for solving mixed-variables programming problems, Numer. Math. 4(1962)238–252.CrossRefGoogle Scholar
  6. [6]
    T.M. Cavalier, P.M. Pardalos and A.L. Soyster, Modelling and integer programming techniques applied to propositional calculus, Comp. Oper. Res. 17(1990)561–570.CrossRefGoogle Scholar
  7. [7]
    V. Chandru and J.N. Hooker, Extended Horn sets in propositional logic, Working Paper 88-89-39, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh (1989).Google Scholar
  8. [8]
    M. Davis and H. Putnam, A computing procedure for quantification theory, ACM 17(1960)201–215.CrossRefGoogle Scholar
  9. [9]
    M.A. Duran and I.E. Grossmann, An Outer-Approximation algorithm for a class of mixed-integer nonlinear programs, Math. Progr. 36(1986)307–339.Google Scholar
  10. [10]
    R.S. Garfinkel and G.L. Nemhauser,Integer Programming (Wiley, New York, 1972).Google Scholar
  11. [11]
    A.M. Geoffrion, Generalized Benders decomposition, J. Optim. Theory 10(1972)237–260.CrossRefGoogle Scholar
  12. [12]
    I.E. Grossmann, MINLP optimization strategies and algorithms for process synthesis,Proc. FOCAPD Meeting, eds. Siirola et al. (Elsevier, New York, 1990) pp. 105–132.Google Scholar
  13. [13]
    O.K. Gupta, Branch and bound experiments in nonlinear integer programming, Ph.D. Thesis, Purdue University (1980).Google Scholar
  14. [14]
    J.N. Hooker, H. Yan, I.E. Grossmann and R. Raman, Logic cuts for processing networks with fixed charges (1993), Comp. Oper. Res., to appear.Google Scholar
  15. [15]
    R.E. Jeroslow and J. Wang, Solving propositional satisfiability problems, Ann. Math. AI 1(1990)167–187.Google Scholar
  16. [16]
    G.R. Kocis and I.E. Grossmann, A modelling decomposition strategy for MINLP optimization of process flowsheets, Comp. Chem. Eng. 13(1989)797–819.CrossRefGoogle Scholar
  17. [17]
    S. Nabar and L. Schrage, Modelling and solving nonlinear integer programming problems,Annual AIChE Meeting, Chicago (1991).Google Scholar
  18. [18]
    G.L. Nemhauser and L.A. Wolsey,Integer and Combinatorial Optimization (Wiley-Interscience, New York, 1988).Google Scholar
  19. [19]
    OSL,OSL User's Manual (IBM Corp., Kingston, New York, 1991).Google Scholar
  20. [20]
    R. Raman and I.E. Grossmann, Relation between MILP modelling and logical inference for chemical process synthesis. Comp. Chem. Eng. 15(1991)73–84.CrossRefGoogle Scholar
  21. [21]
    R. Raman and I.E. Grossmann, Integration of qualitative knowledge in MINLP optimization for process synthesis, Comp. Chem. Eng. 16(1992)155–171.CrossRefGoogle Scholar
  22. [22]
    J. Singhal, R.E. Marsten and T. Morin, Fixed order branch and bound methods for mixed-integer programming: the ZOOM system, Working Paper, Management Information Science Department, The University of Arizona, Tucson, AZ (1987).Google Scholar
  23. [23]
    SCICON,SCICONIC/VM User Guide (SCICON Ltd., London, 1986).Google Scholar
  24. [24]
    T.J. Van Roy and L.A. Wolsey, Solving mixed integer programs by automatic reformulation, Oper. Res. 35(1988)45–57.CrossRefGoogle Scholar
  25. [25]
    H.P. Williams,Model Building in Mathematical Programming (Wiley, Chichester, 1985).Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • Ramesh Raman
    • 1
  • Ignacio E. Grossmann
    • 1
  1. 1.Department of Chemical EngineeringCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations