A declarative interface to LP/MIP packages

B-Prolog provides a declarative and easy-to-use interface to LP/MIP packages through which LP/MIP problems can be described declaratively. The following gives an example program that uses the interface.

       go:-
           Vars=[X1,X2,X3],
           lp_domain(X1,0,40),         % 0 =< X1 =< 40
           -X1+X2+X3 $=< 20,           % constraints
           X1-3*X2+X3 $=< 30,
           Profit=X1+2*X2+3*X3,        % objective function
           lp_solve(Vars,max(Profit)), % call the LP solver
           format("sol(~w,~f)~n",[Vars,Profit]).

Three new operators are introduced: $=, $>=, and $=< for expressing equality and disequality constraints.^13.4 The call lp_solve(Vars,max(Profit)) calls the LP solver to find a valuation for Vars that maximizes Profit and satisfies all the constraints on the variables. A MIP problem is similar to a LP problem except that some variables are required to be integers. The built-in lp_integers(Vars) is provided to declare integer variables. For MIP problems, the same built-in lp_solve/2 is used to call the MIP solver. The interface decides which solver to call based on whether or not there are integer variables. More examples can be found in the directory examples/lp.

The following built-ins are provided to communicate with LP/MIP packages:

• lp_domain(Xs,L,U): The variables in the list Xs are in the range of L..U, where L and U are either integers or floats. If the domain of a variable is not declared, it is assumed to be in the range of $0..\infty$.

• Exp1 R Exp2 where R is $=, $>=, or $=<: An equality or disequality constraint, where Exp1 and Exp2 are two arithmetic expressions made up of integers, floats, variables, and the following arithmetic functions: + (addition), - (subtraction), * (multiplication), / (division), and sum. The expression sum(List) denotes the sum of the elements in List.

• lp_integers(Xs): The variables in the list Xs are required to have integer values. A problem becomes a MIP one if at least one of the variables is an integer variable.

• lp_solve(Xs,Obj) where Obj is either min(Exp) or max(Exp): Call the LP/MIP solver to find a valuation for the variables Xs such that the objective function is optimal and the accumulated constraints are all satisfied. This call fails or succeeds with some error message if the solver fails to solve the problem.

• lp_solve(Xs): Solve the problem as constraint satisfaction problem with no objective function. It is equivalent to lp_solve(Xs,min(0)).