Modeling a Mixture in Traditional Representation

When modeling mixtures, we are often faced with a large set of ingredients to choose from. A common way to formalize this type of selection problem is to assign each ingredient its own numerical parameter representing the amount of the ingredient in the mixture. A sum constraint imposed on all parameters then ensures that the total amount of ingredients in the mix is always 100%. In addition, there could be other constraints, for instance, to impose further restrictions on individual subgroups of ingredients. In BayBE’s language, we call this the traditional mixture representation.

In this example, we demonstrate how to create a search space in this representation, using a simple mixture of up to six components, which are divided into three subgroups: solvents, bases and phase agents.

Slot-based Representation

For an alternative way to describe mixtures, see our slot-based representation.

Imports

import numpy as np
import pandas as pd

from baybe.constraints import ContinuousLinearConstraint
from baybe.parameters import NumericalContinuousParameter
from baybe.recommenders import RandomRecommender
from baybe.searchspace import SearchSpace

Parameter Setup

We start by creating lists containing our substance labels according to their subgroups:

g1 = ["Solvent1", "Solvent2"]
g2 = ["Base1", "Base2"]
g3 = ["PhaseAgent1", "PhaseAgent2"]

Next, we create continuous parameters describing the substance amounts for each group. Here, the maximum amount for each substance depends on its group, i.e. we allow adding more of a solvent compared to a base or a phase agent:

p_g1_amounts = [
    NumericalContinuousParameter(name=f"{name}", bounds=(0, 80)) for name in g1
]
p_g2_amounts = [
    NumericalContinuousParameter(name=f"{name}", bounds=(0, 20)) for name in g2
]
p_g3_amounts = [
    NumericalContinuousParameter(name=f"{name}", bounds=(0, 5)) for name in g3
]

Constraints Setup

Now, we set up our constraints. We start with the overall mixture constraint, ensuring the total of all ingredients is 100%:

c_total_sum = ContinuousLinearConstraint(
    parameters=g1 + g2 + g3,
    operator="=",
    coefficients=[1] * len(g1 + g2 + g3),
    rhs=100,
)

Additionally, we require bases make up at least 10% of the mixture:

c_g2_min = ContinuousLinearConstraint(
    parameters=g2,
    operator=">=",
    coefficients=[1] * len(g2),
    rhs=10,
)

By contrast, phase agents should make up no more than 5%:

c_g3_max = ContinuousLinearConstraint(
    parameters=g3,
    operator="<=",
    coefficients=[1] * len(g3),
    rhs=5,
)

Search Space Creation

Having both parameter and constraint definitions at hand, we can create our search space:

searchspace = SearchSpace.from_product(
    parameters=[*p_g1_amounts, *p_g2_amounts, *p_g3_amounts],
    constraints=[c_total_sum, c_g2_min, c_g3_max],
)

Verification of Constraints

To verify that the constraints imposed above are fulfilled, let us draw some random points from the search space:

recommendations = RandomRecommender().recommend(batch_size=10, searchspace=searchspace)
print(recommendations)

       Base1      Base2  PhaseAgent1  PhaseAgent2   Solvent1   Solvent2
0  15.487229  19.474993     0.775300     2.654434   6.346360  55.261684
1  13.621802  12.388032     0.005418     1.615455  45.437420  26.931873
2  16.290730  19.803686     0.380844     4.403728  27.020255  32.100757
3   4.931726  10.342482     0.192480     0.242069  57.832211  26.459034
4   9.884381   5.929538     0.607034     2.978908  52.791314  27.808824
5   9.221874  13.987235     3.983641     0.608587  59.929627  12.269036
6   7.034005  19.470953     1.239950     0.394920   1.344821  70.515350
7   5.352840   9.577604     1.530765     2.543830  45.919536  35.075426
8   9.582360  12.573655     1.757240     1.338171  26.055275  48.693299
9   0.459750  15.473549     1.855240     2.571397  58.157186  21.482879

Computing the respective row sums reveals the expected result:

stats = pd.DataFrame(
    {
        "Total": recommendations.sum(axis=1),
        "Total_Bases": recommendations[g2].sum(axis=1),
        "Total_Phase_Agents": recommendations[g3].sum(axis=1),
    }
)
print(stats)

   Total  Total_Bases  Total_Phase_Agents
0  100.0    34.962223            3.429733
1  100.0    26.009834            1.620873
2  100.0    36.094416            4.784572
3  100.0    15.274207            0.434548
4  100.0    15.813920            3.585942
5  100.0    23.209109            4.592228
6  100.0    26.504958            1.634870
7  100.0    14.930444            4.074595
8  100.0    22.156015            3.095411
9  100.0    15.933299            4.426637

assert np.allclose(stats["Total"], 100)
assert (stats["Total_Bases"] >= 10).all()
assert (stats["Total_Phase_Agents"] <= 5).all()