Ilp - count number of times variables received a value

Question

In an ILP, is it possible to have a variable whose value will be the number of variables with value N?

N is a bounded integer, with lower bound 1.

Thank you

Answer 1

This achieves the goal. It is written in pyomo but should be fairly easy to translate to other frameworks.

Code:

# magic number counter

import pyomo.environ as pyo

M = 100
magic_number=7

m = pyo.ConcreteModel()

m.I = pyo.Set(initialize=[1,2,3,4])

m.x    = pyo.Var(m.I, domain=pyo.NonNegativeIntegers, bounds=(1, M))
m.magic = pyo.Var(m.I, domain=pyo.Binary)

# obj:  max sum of x, plus some sugar for magic numbers's

m.obj = pyo.Objective(expr=sum(m.x[i] + 0.1*m.magic[i] for i in m.I), sense=pyo.maximize)

# constraints

m.sum_limit = pyo.Constraint(expr=pyo.sum_product(m.x) <= 19)

@m.Constraint(m.I)
def linking_1(m, i):
    return m.x[i] <= magic_number + (1 - m.magic[i]) * M

@m.Constraint(m.I)
def linking_2(m, i):
    return m.x[i] >= magic_number * m.magic[i]

solver = pyo.SolverFactory('glpk')
soln = solver.solve(m)
print(soln)
m.display()

print(f"\nmagic numbers {magic_number}'s produced: {pyo.value(pyo.sum_product(m.magic))}")

Output:

Problem: 
- Name: unknown
  Lower bound: 19.2
  Upper bound: 19.2
  Number of objectives: 1
  Number of constraints: 10
  Number of variables: 9
  Number of nonzeros: 21
  Sense: maximize
Solver: 
- Status: ok
  Termination condition: optimal
  Statistics: 
    Branch and bound: 
      Number of bounded subproblems: 5
      Number of created subproblems: 5
  Error rc: 0
  Time: 0.005478858947753906
Solution: 
- number of solutions: 0
  number of solutions displayed: 0

Model unknown

  Variables:
    x : Size=4, Index=I
        Key : Lower : Value : Upper : Fixed : Stale : Domain
          1 :     1 :   7.0 :   100 : False : False : NonNegativeIntegers
          2 :     1 :   4.0 :   100 : False : False : NonNegativeIntegers
          3 :     1 :   7.0 :   100 : False : False : NonNegativeIntegers
          4 :     1 :   1.0 :   100 : False : False : NonNegativeIntegers
    magic : Size=4, Index=I
        Key : Lower : Value : Upper : Fixed : Stale : Domain
          1 :     0 :   1.0 :     1 : False : False : Binary
          2 :     0 :   0.0 :     1 : False : False : Binary
          3 :     0 :   1.0 :     1 : False : False : Binary
          4 :     0 :   0.0 :     1 : False : False : Binary

  Objectives:
    obj : Size=1, Index=None, Active=True
        Key  : Active : Value
        None :   True : 19.200000000000003

  Constraints:
    sum_limit : Size=1
        Key  : Lower : Body : Upper
        None :  None : 19.0 :  19.0
    linking_1 : Size=4
        Key : Lower : Body   : Upper
          1 :  None :    0.0 :   0.0
          2 :  None : -103.0 :   0.0
          3 :  None :    0.0 :   0.0
          4 :  None : -106.0 :   0.0
    linking_2 : Size=4
        Key : Lower : Body : Upper
          1 :  None :  0.0 :   0.0
          2 :  None : -4.0 :   0.0
          3 :  None :  0.0 :   0.0
          4 :  None : -1.0 :   0.0

magic numbers 7's produced: 2.0