MIP Objective with xsum()

Question

With mip I want to involve cardinality in the objective function. I don't understand why the following doesn't give a solution where all lists in x have precisely four binary variables set.

from mip import Model, xsum, maximize, BINARY

model = Model()

x = [[model.add_var(var_type=BINARY) for _ in range(6)] for _ in range(5)]

def f(x):
  return xsum([4 <= xsum(v) for v in x]) - xsum([4 < xsum(v) for v in x])                                                                    #2*x[0] + 3*x[1] - 4*x[2]

model.objective = maximize(f(x))

model.optimize()

for v in x:
  print([a.x for a in v])

Thanks for any hint!

Answer 1

I have expressed the problem in google ortool's linear solver.

from ortools.linear_solver import pywraplp

solver = pywraplp.Solver("penalty_obj",pywraplp.Solver.SCIP_MIXED_INTEGER_PROGRAMMING)

x = [[solver.BoolVar("") for _ in range(6)] for _ in range(5)]

xsum = [solver.IntVar(lb = 0, ub = 6, name = "") for _ in range(len(x))]

for i in range(len(xsum)):
    solver.Add(xsum[i] == sum(x[i]))
    
grt_eq_4 = [solver.BoolVar("") for _ in range(len(xsum))]
grt_4 = [solver.BoolVar("") for _ in range(len(xsum))]

for i in range(len(grt_eq_4)):
    # if sum is >= 4 => grt_eq_4 == 1
    solver.Add(xsum[i] - (100 * grt_eq_4[i]) <= 4 - 1)
    
for i in range(len(grt_4)):
    # if sum is > 4 (or sum >= 5) => grt_4 == 1
    solver.Add(xsum[i] - (100 * grt_4[i]) <= 5 - 1)
    
solver.Maximize(sum(grt_eq_4) - sum(grt_4))
solver.Solve()

solver.Objective().Value()

Objective function return 5 which is length of x