Tools for reformulating logical forms into zero-one mixed integer programs

Abstract A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Programming (ILP) formulation or a Mixed Integer Programming (MIP) formulation is presented. A reformulation procedure which uses the extended reverse Polish notation to represent a compound logical form is then described. The syntax of an LP modelling language is extended to incorporate statements in propositional logic forms with linear algebraic forms whereby 0–1 MIP models can be automatically formulated. A prototype user interface by which logical forms can be reformulated and the corresponding MIP constructed and analysed within an existing Mathematical Programming modelling system is illustrated. Finally, the steps to formulate a discrete optimisation model in this way are illustrated by means of an example.

[1]  Joxan Jaffar,et al.  Constraint logic programming , 1987, POPL '87.

[2]  H. P. Williams.  Linear and integer programming applied to the propositional calculus , 1987 .

[3]  James F. Allen.  Maintaining knowledge about temporal intervals , 1983, CACM.

[4]  Gautam Mitra,et al.  UIMP: User Interface for Mathematical Programming , 1982, TOMS.

[5]  Bruce G. Buchanan,et al.  The MYCIN Experiments of the Stanford Heuristic Programming Project , 1985 .

[6]  Brian W. Kernighan,et al.  AMPL: a mathematical programming language , 1989 .

[7]  Pascal Van Hentenryck.  Constraint satisfaction in logic programming , 1989, Logic programming.

[8]  H. P. Williams.  Experiments in the formulation of integer programming problems , 1974 .

[9]  Gautam Mitra,et al.  Analysis of mathematical programming problems prior to applying the simplex algorithm , 1975, Math. Program..

[10]  H. P. Williams.  Logic applied to integer programming and integer programming applied to logic , 1995 .

[11]  John N. Hooker,et al.  A quantitative approach to logical inference , 1988, Decis. Support Syst..

[12]  Alonzo Church,et al.  Introduction to Mathematical Logic , 1991 .

[13]  M GeoffrionArthur.  The SML Language for Structured Modeling , 1992 .

[14]  Jean-Louis Laurière,et al.  A Language and a Program for Stating and Solving Combinatorial Problems , 1978, Artif. Intell..

[15]  Gautam Mitra,et al.  Computer-Assisted Mathematical Programming (Modelling) System: CAMPS , 1988, Comput. J..

[16]  Alain Colmerauer,et al.  Opening the Prolog III universe , 1987 .

[17]  J. Meeraus A. Bisschop,et al.  ON THE DEVELOPMENT OF A GENERAL ALGEBRAIC MODELING SYSTEM IN A STRATEGIC PLANNING ENVIRONMENT , 1982 .

[18]  K. I. M. McKinnon,et al.  Constructing integer programming models by the predicate calculus , 1989 .