Minimize AbsEquality rather than enforce in OrTools

Question

I'm trying to solve the following using OR tools:

Given the following bags containing different colors of balls:

bag	red	blue	green	black
A	10	5	85	0
B	25	50	25	0
C	0	100	0	0
D	90	5	5	0
E	2	0	98	0
F	0	0	0	100

How many of each type of bag would I need to have an equal number of each color of ball?

For cases like this where there is an exact answer, the following code:

bags= [
[10,5,85,0],
[25,50,25,0],
[0,100,0,0],
[90,5,5,0],
[2,0,98,0],
[0,0,0,100]
]


bags_n = len(bags)
color_n = len(bags[0])

print(f'Bags: {bags_n}')
print(f'Colors: {color_n}')

color_count= [0] * color_n

for c in range(color_n):
  for b in bags:
    color_count[c]+= b[c]


print(color_count)
print(f'Inital total: {sum(color_count)}')
print(f'Inital equal share: {sum(color_count)//color_n}')

model = cp_model.CpModel()

weights = []

for r in range(bags_n):
  weights.append(model.NewIntVar(1,1000,f'Weight of Bag: {r}'))

total = model.NewIntVar(0, 100000, 'total')

model.Add(
    sum(flatten(
          [[bags[r][c] * weights[r] for r in range(bags_n)] for c in range(color_n)]
        )) == total
)
          
equal = model.NewIntVar(0, 10000, 'equal share')
model.AddDivisionEquality(equal, total, color_n)

for c in range(color_n):
  diff_c = model.NewIntVar(0, 1000, 'diff_'+str(c))
  model.Add(diff_c == sum([bags[r][c] * weights[r] for r in range(bags_n)]) - equal)
  model.AddAbsEquality(0, diff_c)

solver = cp_model.CpSolver()
status = solver.Solve(model)

if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE:
    print(f'Maximum of objective function: {solver.ObjectiveValue()}\n')
    for v in weights:
      print(f'{solver.Value(v)}')
    print(f'total = {solver.Value(total)}')
    print(f'equal share = {solver.Value(equal)}')
else:
    print(status)

gives back valid weights:

82 2 70 78 5 79

If I change the setup to something like

bags= [
[50,40,10],
[30,20,50],
[30,30,40],
[30,25,45],
]

The model becomes infeasible, I assume due to the fact that there are no weights that satisfy the AbsEquality for every color.

How can I change this to get me the solution closest to an even distribution even if a perfect solution is infeasable?

Do you mean an "even number" or an "equal number" of each color of balls? — Christopher Hamkins, Feb 03 '23 at 11:10
If you're aiming to get the most equal number of balls of each color, you could construct a variable that is the difference between the max number of balls of any color and the min number of balls of any color, then add as an objective to minimize this variable. — Christopher Hamkins, Feb 03 '23 at 14:11
For the abs part, you can have a look at https://github.com/google/or-tools/blob/stable/examples/python/chemical_balance_sat.py — Laurent Perron, Feb 03 '23 at 14:35
Thanks @ChristopherHamkins! I'll clean up the wording on the question too. — Chris, Feb 04 '23 at 02:14
Just this final not so serious comment: if you also allowed zero as a number of bags, you could always get an exact solution, namely 0 balls of each color ;-) — Christopher Hamkins, Feb 06 '23 at 09:43

score 1 · Accepted Answer · answered Feb 04 '23 at 02:14

Christopher Hamkins' suggestion worked great:

bags= [
[50,40,10],
[30,20,50],
[30,30,40],
[30,25,45],
]


bags_n = len(bags)
color_n = len(bags[0])

print(f'Bags: {bags_n}')
print(f'Colors: {color_n}')

color_count= [0] * color_n

for c in range(color_n):
  for b in bags:
    color_count[c]+= b[c]



    
print(color_count)
print(["{0:.0%}".format(c/sum(color_count)) for c in color_count])


model = cp_model.CpModel()

weights = []

for r in range(bags_n):
  weights.append(model.NewIntVar(1,500,f'Weight of Bag: {r}'))

max = model.NewIntVar(0,100000000,f'Max')
model.AddMaxEquality(max,
                     [sum([bags[r][c] * weights[r] for r in range(bags_n)]) for c in range(color_n)]
                     )

min = model.NewIntVar(0,100000000,f'Min')
model.AddMinEquality(min,
                     [sum([bags[r][c] * weights[r] for r in range(bags_n)]) for c in range(color_n)]
                     )
diff = model.NewIntVar(0,100000000,f'Diff')
model.Add(max - min == diff)

model.Minimize(diff)


solver = cp_model.CpSolver()
status = solver.Solve(model)

if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE:
    print(f'max = {solver.Value(max)}')
    print(f'min = {solver.Value(min)}')
    print(f'diff = {solver.Value(diff)}')

    bag_weights = [0] * bags_n

    for i,v in enumerate(weights):
      bag_weights[i] = solver.Value(v)
      print(f'{solver.Value(v)}')
    
    color_count = [0] * color_n

    for c in range(color_n):
      for i,b in enumerate(bags):
        color_count[c]+= (b[c] * bag_weights[i])


    
    print(color_count)
    print(["{0:.0%}".format(c/sum(color_count)) for c in color_count])

else:
    print(status)

Minimize AbsEquality rather than enforce in OrTools

1 Answers1