CLPQR MIP (SICStus Prolog)

The predicates bb_inf/[3,5] implement a simple Branch and Bound search algorithm for Mixed Integer Linear (MIP) Optimization examples. Serious MIP is not trivial. The implementation library('clpqr/bb.pl') is to be understood as a starting point for more ambitious users who need control over branching, or who want to add cutting planes, for example.

Anyway, here is a small problem from miplib, a collection of MIP models, housed at Rice University:

NAME:         flugpl
ROWS:         18
COLUMNS:      18
INTEGER:      11
NONZERO:      46
BEST SOLN:    1201500 (opt)
LP SOLN:      1167185.73
SOURCE:       Harvey M. Wagner              
              John W. Gregory (Cray Research)
              E. Andrew Boyd (Rice University)
APPLICATION:  airline model
COMMENTS:     no integer variables are binary

% library('clpqr/examples/mip')

example(flugpl, Obj, Vs, Ints, []) :-
  Vs = [ Anm1,Anm2,Anm3,Anm4,Anm5,Anm6,
         Stm1,Stm2,Stm3,Stm4,Stm5,Stm6,
         UE1,UE2,UE3,UE4,UE5,UE6],
  Ints = [Stm6, Stm5, Stm4, Stm3, Stm2,
          Anm6, Anm5, Anm4, Anm3, Anm2, Anm1],

  Obj =    2700*Stm1 + 1500*Anm1 + 30*UE1
         + 2700*Stm2 + 1500*Anm2 + 30*UE2 
         + 2700*Stm3 + 1500*Anm3 + 30*UE3
         + 2700*Stm4 + 1500*Anm4 + 30*UE4 
         + 2700*Stm5 + 1500*Anm5 + 30*UE5
         + 2700*Stm6 + 1500*Anm6 + 30*UE6,

  allpos(Vs),
  {  Stm1 = 60, 0.9*Stm1 +1*Anm1 -1*Stm2 = 0, 
     0.9*Stm2 +1*Anm2 -1*Stm3 = 0, 0.9*Stm3 +1*Anm3 -1*Stm4 = 0, 
     0.9*Stm4 +1*Anm4 -1*Stm5 = 0, 0.9*Stm5 +1*Anm5 -1*Stm6 = 0,
     150*Stm1 -100*Anm1 +1*UE1 >= 8000,
     150*Stm2 -100*Anm2 +1*UE2 >= 9000,
     150*Stm3 -100*Anm3 +1*UE3 >= 8000,
     150*Stm4 -100*Anm4 +1*UE4 >= 10000,
     150*Stm5 -100*Anm5 +1*UE5 >= 9000,
     150*Stm6 -100*Anm6 +1*UE6 >= 12000,
     -20*Stm1 +1*UE1 =< 0, -20*Stm2 +1*UE2 =< 0, -20*Stm3 +1*UE3 =< 0, 
     -20*Stm4 +1*UE4 =< 0, -20*Stm5 +1*UE5 =< 0, -20*Stm6 +1*UE6 =< 0,
     Anm1 =< 18, 57 =< Stm2, Stm2 =< 75, Anm2 =< 18,
     57 =< Stm3, Stm3 =< 75, Anm3 =< 18, 57 =< Stm4,
     Stm4 =< 75, Anm4 =< 18, 57 =< Stm5, Stm5 =< 75,
     Anm5 =< 18, 57 =< Stm6, Stm6 =< 75, Anm6 =< 18
   }.

  allpos([]).
  allpos([X|Xs]) :- {X >= 0}, allpos(Xs).

clp(r) ?- example(flugpl, Obj, _, _, _), inf(Obj, Inf).

Inf = 1167185.7255923203

Computing the infimum under the additional constraints that Stm6, Stm5, Stm4, Stm3, Stm2, Anm6, Anm5, Anm4, Anm3, Anm2, Anm1 assume integer values at the infimum is computationally harder, but the query does not change much:

10.11.8 A Mixed Integer Linear Optimization Example