This construct the integer program for a tiling problem; it can be easily adapted to exact covers, and allowing reuse of tiles

get_milp.sage

def milp(self, solver=None):
    tiles = {}
    for P in self.pieces():
        tiles[P.frozenset()] = [Q.frozenset() for Q in P.isometric_copies(self._box, orientation_preserving=not(self._reflection))]
    
    p = MixedIntegerLinearProgram(solver=solver)
    x = p.new_variable(binary=True)
    for i in self.space():
        p.add_constraint(p.sum(x[T, P]  for T in tiles for P in tiles[T] if i in P) <= 1) 
    for T in tiles:
        p.add_constraint(p.sum(x[T, P] for P in tiles[T]) <= 1)
    p.set_objective(p.sum(x.values()))
    
    return p, x