# 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.

```{admonition} Slot-based Representation
:class: seealso
For an alternative way to describe mixtures, see our
[slot-based representation](/examples/Mixtures/slot_based.md).
```

## Imports


```python
import numpy as np
import pandas as pd
```


```python
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:


```python
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:


```python
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%:


```python
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:


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

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


```python
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:


```python
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:


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

           Base1      Base2  PhaseAgent1  PhaseAgent2   Solvent1   Solvent2
    0  11.738578   6.886905     1.189723     3.573333  34.245040  42.366420
    1   2.385397  10.616448     1.702108     2.095700  66.955990  16.244358
    2  12.250538  13.503466     1.198818     3.123361   2.280450  67.643368
    3   6.288515   7.824844     3.083884     1.135495  77.788632   3.878630
    4  10.723116   8.361606     2.134417     1.057947  21.423914  56.299000
    5  16.638572  17.438264     3.349855     1.595435  24.873821  36.104053
    6  15.599627  14.297017     3.617621     0.631540  55.127226  10.726968
    7  10.090574  18.668147     2.104922     1.374698  23.457960  44.303698
    8  18.518312   8.195068     2.580252     1.911285  54.672477  14.122606
    9   1.651443  15.022637     0.031753     4.320184  29.640800  49.333183


Computing the respective row sums reveals the expected result:


```python
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    18.625483            4.763057
    1  100.0    13.001844            3.797808
    2  100.0    25.754004            4.322179
    3  100.0    14.113359            4.219379
    4  100.0    19.084722            3.192364
    5  100.0    34.076836            4.945290
    6  100.0    29.896644            4.249162
    7  100.0    28.758721            3.479620
    8  100.0    26.713380            4.491537
    9  100.0    16.674080            4.351938



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