Ilp - count number of times variables received a value - algorithm

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