Breakdown an integer value to an array of integer maintaining the sum

Question

I am working on a project where I need to breakdown an integer value according to an array of percentage values. My end array must contain integer value and the sum of the array must be equal to the initial integer.

Below is a fake example. We have a list of cars with some "potentials" and we need to allocate this potential to a specific postal code. the postal code allocation is dictated by some sellout information.

SELLOUTS_PER_P_CODE is dictating the weight to be put for each postal code allocation. For example, for the first car (car 1), there is a lot of weight for p_code_3 and less for p_code_2 and even less for p_code_1 so the allocation should be respectively for car 1 p_code_1=1, p_code_2=2, p_code_3=4.

Bellow is the mathematical form of the problem.

Here I am implementing this formulation using pyomo however it doesn't produce the expected result. The model doesn't take into consideration the weight factor from SELLOUTS_PER_P_CODE

from pyomo.environ import *
from pprint import pprint


def distribute(total, weights):
    scale = float(sum(weights.values())) / total
    return {k: v / scale for k, v in weights.items()}


Cars = ["car 1", "car 2", "car 3"]
Locations = ["p_code_1", "p_code_2", "p_code_3"]
POTENTIALS = {"car 1": 7, "car 2": 2, "car 3": 14}
SELLOUTS = {"p_code_1": 0.2, "p_code_2": 0.3, "p_code_3": 0.5}

SELLOUTS_PER_P_CODE = {}

for car in Cars:
    pot = POTENTIALS[car]
    scaled_sellout = distribute(pot, SELLOUTS)
    t = {(car, p_code): v for p_code, v in scaled_sellout.items()}
    SELLOUTS_PER_P_CODE.update(t)

pprint(SELLOUTS_PER_P_CODE)

model = ConcreteModel(name="Breakdown Potential to Postal Code")

model.Cars = Set(initialize=Cars)
model.Locations = Set(initialize=Locations)

model.a = Param(model.Cars, model.Locations, initialize=SELLOUTS_PER_P_CODE)
model.p = Param(model.Cars, initialize=POTENTIALS)

model.X_pos = Var(model.Cars, model.Locations, within=NonNegativeIntegers)
model.X_neg = Var(model.Cars, model.Locations, within=NonNegativeIntegers)


def objective_rule(model):
    return sum(
        (model.X_pos[i, j] - model.a[i, j] * model.p[i])
        - (model.X_neg[i, j] - model.a[i, j] * model.p[i])
        for i in model.Cars
        for j in model.Locations
    )


model.objective = Objective(rule=objective_rule, sense=minimize)


def sum_maintained_rule(model, i):
    return (
        sum(model.X_pos[i, j] for j in model.Locations)
        + sum(model.X_neg[i, j] for j in model.Locations)
        == model.p[i]
    )


model.sum_maintained = Constraint(model.Cars, rule=sum_maintained_rule)


def pyomo_postprocess(options=None, instance=None, results=None):
    model.pprint()


if __name__ == "__main__":
    opt = SolverFactory("glpk")
    results = opt.solve(model)
    results.write()
    print("\nDisplaying Solution\n" + "-" * 80)
    pyomo_postprocess(None, model, results)

And finally here is the incorrect output. Notice X_neg and X_pos for the output allocation.

Displaying Solution
--------------------------------------------------------------------------------
5 Set Declarations
    Cars : Dim=0, Dimen=1, Size=3, Domain=None, Ordered=False, Bounds=None
        ['car 1', 'car 2', 'car 3']
    Locations : Dim=0, Dimen=1, Size=3, Domain=None, Ordered=False, Bounds=None
        ['p_code_1', 'p_code_2', 'p_code_3']
    X_neg_index : Dim=0, Dimen=2, Size=9, Domain=None, Ordered=False, Bounds=None
        Virtual
    X_pos_index : Dim=0, Dimen=2, Size=9, Domain=None, Ordered=False, Bounds=None
        Virtual
    a_index : Dim=0, Dimen=2, Size=9, Domain=None, Ordered=False, Bounds=None
        Virtual

2 Param Declarations
    a : Size=9, Index=a_index, Domain=Any, Default=None, Mutable=False
        Key                   : Value
        ('car 1', 'p_code_1') : 1.4000000000000001
        ('car 1', 'p_code_2') :                2.1
        ('car 1', 'p_code_3') :                3.5
        ('car 2', 'p_code_1') :                0.4
        ('car 2', 'p_code_2') :                0.6
        ('car 2', 'p_code_3') :                1.0
        ('car 3', 'p_code_1') : 2.8000000000000003
        ('car 3', 'p_code_2') :                4.2
        ('car 3', 'p_code_3') :                7.0
    p : Size=3, Index=Cars, Domain=Any, Default=None, Mutable=False
        Key   : Value
        car 1 :     7
        car 2 :     2
        car 3 :    14

2 Var Declarations
    X_neg : Size=9, Index=X_neg_index
        Key                   : Lower : Value : Upper : Fixed : Stale : Domain
        ('car 1', 'p_code_1') :     0 :   7.0 :  None : False : False : NonNegativeIntegers
        ('car 1', 'p_code_2') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 1', 'p_code_3') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 2', 'p_code_1') :     0 :   2.0 :  None : False : False : NonNegativeIntegers
        ('car 2', 'p_code_2') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 2', 'p_code_3') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 3', 'p_code_1') :     0 :  14.0 :  None : False : False : NonNegativeIntegers
        ('car 3', 'p_code_2') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 3', 'p_code_3') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
    X_pos : Size=9, Index=X_pos_index
        Key                   : Lower : Value : Upper : Fixed : Stale : Domain
        ('car 1', 'p_code_1') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 1', 'p_code_2') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 1', 'p_code_3') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 2', 'p_code_1') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 2', 'p_code_2') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 2', 'p_code_3') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 3', 'p_code_1') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 3', 'p_code_2') :     0 :   0.0 :  None : False : False : NonNegativeIntegers
        ('car 3', 'p_code_3') :     0 :   0.0 :  None : False : False : NonNegativeIntegers

1 Objective Declarations
    objective : Size=1, Index=None, Active=True
        Key  : Active : Sense    : Expression
        None :   True : minimize : X_pos[car 1,p_code_1] - 9.8 - (X_neg[car 1,p_code_1] - 9.8) + X_pos[car 1,p_code_2] - 14.700000000000001 - (X_neg[car 1,p_code_2] - 14.700000000000001) + X_pos[car 1,p_code_3] - 24.5 - (X_neg[car 1,p_code_3] - 24.5) + X_pos[car 2,p_code_1] - 0.8 - (X_neg[car 2,p_code_1] - 0.8) + X_pos[car 2,p_code_2] - 1.2 - (X_neg[car 2,p_code_2] - 1.2) + X_pos[car 2,p_code_3] - 2.0 - (X_neg[car 2,p_code_3] - 2.0) + X_pos[car 3,p_code_1] - 39.2 - (X_neg[car 3,p_code_1] - 39.2) + X_pos[car 3,p_code_2] - 58.800000000000004 - (X_neg[car 3,p_code_2] - 58.800000000000004) + X_pos[car 3,p_code_3] - 98.0 - (X_neg[car 3,p_code_3] - 98.0)

1 Constraint Declarations
    sum_maintained : Size=3, Index=Cars, Active=True
        Key   : Lower : Body                                                                                                                                          : Upper : Active
        car 1 :   7.0 : X_pos[car 1,p_code_1] + X_pos[car 1,p_code_2] + X_pos[car 1,p_code_3] + X_neg[car 1,p_code_1] + X_neg[car 1,p_code_2] + X_neg[car 1,p_code_3] :   7.0 :   True
        car 2 :   2.0 : X_pos[car 2,p_code_1] + X_pos[car 2,p_code_2] + X_pos[car 2,p_code_3] + X_neg[car 2,p_code_1] + X_neg[car 2,p_code_2] + X_neg[car 2,p_code_3] :   2.0 :   True
        car 3 :  14.0 : X_pos[car 3,p_code_1] + X_pos[car 3,p_code_2] + X_pos[car 3,p_code_3] + X_neg[car 3,p_code_1] + X_neg[car 3,p_code_2] + X_neg[car 3,p_code_3] :  14.0 :   True

11 Declarations: Cars Locations a_index a p X_pos_index X_pos X_neg_index X_neg objective sum_maintained

Answer 1

From the problem which you posted, the parameter 'a' should be initialized with 'Locations' and not with 'Cars' and 'Locations'. Apart from that everything else looks good.