---
tocdepth: 3
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# p2 | Electricity Generation Experiment

```{seealso}
This problem is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74).
```

```{note}
- The source variable type is continuous; we correct the variable type to integer.
- The cutting plane (source variation `74_e`) is excluded because it is instance-specific.
- Implicit non-negativity assumptions are added for every parameter.
```


## Description

A scientist is conducting NumExperiments different experiments to produce electricity. Each experiment i produces ElectricityProduced[i] units of electricity and requires specific amounts of NumResources types of resources as defined by ResourceRequired[j][i]. The laboratory has ResourceAvailable[j] units of each resource available. The scientist aims to determine the number of each experiment to conduct in order to maximize the total electricity produced.

## Formulations
### Formulation `a` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: original).
```

```{note}
- Original formulation; no transformations.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `NumExperiments` | Number of experiment types | integer | *scalar* |
| `NumResources` | Number of resource types | integer | *scalar* |
| `ResourceAvailable` | Amount of resource j available | continuous | `[NumResources]` |
| `ResourceRequired` | Amount of resource j required for experiment i | continuous | `[NumResources, NumExperiments]` |
| `ElectricityProduced` | Amount of electricity produced by experiment i | continuous | `[NumExperiments]` |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `ConductExperiment` | The number of times each experiment is conducted | integer | `[NumExperiments]` |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The electricity produced by each experiment must be non-negative. | $ElectricityProduced_i \geq 0 \quad \forall i$ | yes |
| The resource required by each experiment must be non-negative. | $ResourceRequired_{j,i} \geq 0 \quad \forall j, i$ | yes |
| The amount of each resource available must be non-negative. | $ResourceAvailable_j \geq 0 \quad \forall j$ | yes |
| The number of resource types must be positive. | $NumResources \geq 1$ | yes |
| The number of experiment types must be positive. | $NumExperiments \geq 1$ | yes |


#### Constraints

- For each resource, the total amount required across all experiments does not exceed the available amount.

  $$ \sum_{i=1}^{NumExperiments} ResourceRequired_{j,i} \cdot ConductExperiment_i \leq ResourceAvailable_j \quad \forall j \in \{1, \ldots, NumResources\} $$

- The number of times each experiment is conducted must be non-negative. _(implicit)_

  $$ ConductExperiment_i \geq 0 \quad \forall i $$

#### Objective

Maximize the total electricity produced by conducting the experiments.

$$ Max \sum_{i=1}^{NumExperiments} ConductExperiment_{i} \times ElectricityProduced_{i} $$

### Formulation `b` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: c).
```

```{note}
- Change the names of parameters and variables.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `N` | Number of resource types | integer | *scalar* |
| `Y` | The quantity of resource j that is accessible. | continuous | `[N]` |
| `A` | The quantity of electrical energy generated from trial i | continuous | `[M]` |
| `I` | Resource j quantity needed for experiment i. | continuous | `[N, M]` |
| `M` | Number of experiment types | integer | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `j` | The frequency at which each experiment is performed. | integer | `[M]` |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The electrical energy generated from each trial must be non-negative. | $A_i \geq 0 \quad \forall i$ | yes |
| The resource quantity needed for each experiment must be non-negative. | $I_{j,i} \geq 0 \quad \forall j, i$ | yes |
| The quantity of each accessible resource must be non-negative. | $Y_j \geq 0 \quad \forall j$ | yes |
| The number of resource types must be positive. | $N \geq 1$ | yes |
| The number of experiment types must be positive. | $M \geq 1$ | yes |


#### Constraints

- For each resource, the total amount required across all experiments does not exceed the available amount.

  $$ \sum_{i=1}^{M} I_{k,i} \cdot j_i \leq Y_k \quad \forall k \in \{1, \ldots, N\} $$

- The frequency at which each experiment is performed must be non-negative. _(implicit)_

  $$ j_i \geq 0 \quad \forall i $$

#### Objective

Increase the overall electrical output by conducting the experiments to their full potential.

$$ Max \sum_{i=1}^{M} j_{i} \times A_{i} $$

### Formulation `c` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: d).
```

```{note}
- Substitute integer variables with base-10 representations.
- This formulation is marked invalid because it is only a reformulation at the instance-level, not the formulation-level.
- Source variation was missing the substitution; we applied a 2-digit base-10 substitution.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `N` | Number of resource types | integer | *scalar* |
| `Y` | The quantity of resource j that is accessible. | continuous | `[N]` |
| `A` | The quantity of electrical energy generated from trial i | continuous | `[M]` |
| `I` | Resource j quantity needed for experiment i. | continuous | `[N, M]` |
| `M` | Number of experiment types | integer | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `j_0` | Digit 0 of the frequency at which each experiment is performed. | integer | `[M]` |
| `j_1` | Digit 1 of the frequency at which each experiment is performed. | integer | `[M]` |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The electrical energy generated from each trial must be non-negative. | $A_i \geq 0 \quad \forall i$ | yes |
| The quantity of each accessible resource must be non-negative. | $Y_j \geq 0 \quad \forall j$ | yes |
| The resource quantity needed for each experiment must be non-negative. | $I_{j,i} \geq 0 \quad \forall j, i$ | yes |
| The number of resource types must be positive. | $N \geq 1$ | yes |
| The number of experiment types must be positive. | $M \geq 1$ | yes |


#### Constraints

- For each resource, the total amount required across all experiments does not exceed the available amount.

  $$ \sum_{i=1}^{M} I_{k,i} \cdot (j\_0_i \cdot 10^0 + j\_1_i \cdot 10^1) \leq Y_k \quad \forall k \in \{1, \ldots, N\} $$

- The upper bound on digit 0 of each experiment frequency (must be at most 9).

  $$ j\_0_i \leq 9 \quad \forall i $$

- The upper bound on digit 1 of each experiment frequency (must be at most 9).

  $$ j\_1_i \leq 9 \quad \forall i $$

- Digit 0 of each experiment frequency must be non-negative. _(implicit)_

  $$ j\_0_i \geq 0 \quad \forall i $$

- Digit 1 of each experiment frequency must be non-negative. _(implicit)_

  $$ j\_1_i \geq 0 \quad \forall i $$

#### Objective

Increase the overall electrical output by conducting the experiments to their full potential.

$$ Max \sum_{i=1}^{M} (j\_0_i \cdot 10^0 + j\_1_i \cdot 10^1) \times A_{i} $$

### Formulation `d` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: f).
```

```{note}
- Substitute the objective function with a new variable and linking constraint.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `N` | Number of resource types | integer | *scalar* |
| `Y` | The quantity of resource j that is accessible. | continuous | `[N]` |
| `A` | The quantity of electrical energy generated from trial i | continuous | `[M]` |
| `I` | Resource j quantity needed for experiment i. | continuous | `[N, M]` |
| `M` | Number of experiment types | integer | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `j` | The frequency at which each experiment is performed. | integer | `[M]` |
| `zed` | New variable representing the objective function | continuous | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The electrical energy generated from each trial must be non-negative. | $A_i \geq 0 \quad \forall i$ | yes |
| The quantity of each accessible resource must be non-negative. | $Y_j \geq 0 \quad \forall j$ | yes |
| The resource quantity needed for each experiment must be non-negative. | $I_{j,i} \geq 0 \quad \forall j, i$ | yes |
| The number of resource types must be positive. | $N \geq 1$ | yes |
| The number of experiment types must be positive. | $M \geq 1$ | yes |


#### Constraints

- Constraint defining zed in terms of original variables.

  $$ zed = \sum_{i=1}^{M} A_i \cdot j_i $$

- For each resource, the total amount required across all experiments does not exceed the available amount.

  $$ \sum_{i=1}^{M} I_{k,i} \cdot j_i \leq Y_k \quad \forall k \in \{1, \ldots, N\} $$

- The frequency at which each experiment is performed must be non-negative. _(implicit)_

  $$ j_i \geq 0 \quad \forall i $$

#### Objective

Maximize the new variable zed.

$$ Maximize \ zed $$

### Formulation `e` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: g).
```

```{note}
- Introduce slack variables to convert inequality constraints into equality constraints.
- Source variation was missing the slack variables; we introduced them.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `N` | Number of resource types | integer | *scalar* |
| `Y` | The quantity of resource j that is accessible. | continuous | `[N]` |
| `A` | The quantity of electrical energy generated from trial i | continuous | `[M]` |
| `I` | Resource j quantity needed for experiment i. | continuous | `[N, M]` |
| `M` | Number of experiment types | integer | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `j` | The frequency at which each experiment is performed. | integer | `[M]` |
| `s` | Slack variable for constraint: For each resource, the total amount required across all experiments does not exceed the available amount. | continuous | `[N]` |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The electrical energy generated from each trial must be non-negative. | $A_i \geq 0 \quad \forall i$ | yes |
| The quantity of each accessible resource must be non-negative. | $Y_j \geq 0 \quad \forall j$ | yes |
| The resource quantity needed for each experiment must be non-negative. | $I_{j,i} \geq 0 \quad \forall j, i$ | yes |
| The number of resource types must be positive. | $N \geq 1$ | yes |
| The number of experiment types must be positive. | $M \geq 1$ | yes |


#### Constraints

- For each resource, the total amount required across all experiments plus slack equals the available amount.

  $$ \sum_{i=1}^{M} I_{k,i} \cdot j_i + s_k = Y_k \quad \forall k \in \{1, \ldots, N\} $$

- The frequency at which each experiment is performed must be non-negative. _(implicit)_

  $$ j_i \geq 0 \quad \forall i $$

- The slack variable for each resource constraint must be non-negative. _(implicit)_

  $$ s_k \geq 0 \quad \forall k $$

#### Objective

Increase the overall electrical output by conducting the experiments to their full potential.

$$ Max \sum_{i=1}^{M} j_{i} \times A_{i} $$

### Formulation `f` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: h).
```

```{note}
- Splits variables.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `N` | Number of resource types | integer | *scalar* |
| `Y` | The quantity of resource j that is accessible. | continuous | `[N]` |
| `A` | The quantity of electrical energy generated from trial i | continuous | `[M]` |
| `I` | Resource j quantity needed for experiment i. | continuous | `[N, M]` |
| `M` | Number of experiment types | integer | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `j1` | Part 1 of variable j: The frequency at which each experiment is performed. | integer | `[M]` |
| `j2` | Part 2 of variable j: The frequency at which each experiment is performed. | integer | `[M]` |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The electrical energy generated from each trial must be non-negative. | $A_i \geq 0 \quad \forall i$ | yes |
| The quantity of each accessible resource must be non-negative. | $Y_j \geq 0 \quad \forall j$ | yes |
| The resource quantity needed for each experiment must be non-negative. | $I_{j,i} \geq 0 \quad \forall j, i$ | yes |
| The number of resource types must be positive. | $N \geq 1$ | yes |
| The number of experiment types must be positive. | $M \geq 1$ | yes |


#### Constraints

- For each resource, the total amount required across all experiments does not exceed the available amount.

  $$ \sum_{i=1}^{M} I_{k,i} \cdot (j1_i + j2_i) \leq Y_k \quad \forall k \in \{1, \ldots, N\} $$

- Part 1 of each experiment frequency must be non-negative. _(implicit)_

  $$ j1_i \geq 0 \quad \forall i $$

- Part 2 of each experiment frequency must be non-negative. _(implicit)_

  $$ j2_i \geq 0 \quad \forall i $$

#### Objective

Increase the overall electrical output by conducting the experiments to their full potential.

$$ Max \sum_{i=1}^{M} (j1_{i} + j2_{i}) \times A_{i} $$

### Formulation `g` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: i).
```

```{note}
- Re-scale the objective function.
- Source variation scaled the variables and objective by a factor of 1/10. This makes the reformulation invalid when the decision variables are integer. We replace the source transformation by simply scaling the objective by a factor of 2.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `N` | Number of resource types | integer | *scalar* |
| `Y` | The quantity of resource j that is accessible. | continuous | `[N]` |
| `A` | The quantity of electrical energy generated from trial i | continuous | `[M]` |
| `I` | Resource j quantity needed for experiment i. | continuous | `[N, M]` |
| `M` | Number of experiment types | integer | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `j` | The frequency at which each experiment is performed. | integer | `[M]` |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The electrical energy generated from each trial must be non-negative. | $A_i \geq 0 \quad \forall i$ | yes |
| The quantity of each accessible resource must be non-negative. | $Y_j \geq 0 \quad \forall j$ | yes |
| The resource quantity needed for each experiment must be non-negative. | $I_{j,i} \geq 0 \quad \forall j, i$ | yes |
| The number of resource types must be positive. | $N \geq 1$ | yes |
| The number of experiment types must be positive. | $M \geq 1$ | yes |


#### Constraints

- For each resource, the total amount required across all experiments does not exceed the available amount.

  $$ \sum_{i=1}^{M} I_{k,i} \cdot  j_i \leq Y_k \quad \forall k \in \{1, \ldots, N\} $$

- The scaled experiment count for each experiment must be non-negative. _(implicit)_

  $$ j_i \geq 0 \quad \forall i $$

#### Objective

Increase the overall electrical output by conducting the experiments to their full potential.

$$ Max 2 \cdot \sum_{i=1}^{M} j_{i} \times A_{i} $$

### Formulation `h` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: j).
```

```{note}
- Random formulation unrelated to the original problem.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `F` | Proportion of cat paw treats in the initial blend | continuous | *scalar* |
| `S` | Profit earned per each kilogram of the initial blend | continuous | *scalar* |
| `R` | Quantity of cat paw treats in stock | continuous | *scalar* |
| `Z` | Revenue generated for each kilogram of the alternate blend | continuous | *scalar* |
| `W` | Gold shark treats in stock by weight. | continuous | *scalar* |
| `M` | The proportion of feline paw treats in the second blend | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `v` | The amount of the second blend in kilograms | continuous | *scalar* |
| `n` | The amount of the initial blend in kilograms | continuous | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The proportion of cat paw treats in the initial blend must be non-negative. | $F \geq 0$ | yes |
| The profit earned per each kilogram of the initial blend must be non-negative. | $S \geq 0$ | yes |
| The quantity of cat paw treats in stock must be non-negative. | $R \geq 0$ | yes |
| The revenue generated for each kilogram of the alternate blend must be non-negative. | $Z \geq 0$ | yes |
| The gold shark treats in stock by weight must be non-negative. | $W \geq 0$ | yes |
| The proportion of feline paw treats in the second blend must be non-negative. | $M \geq 0$ | yes |


#### Constraints

- The combined weight of cat paw treats in both blends should not surpass R kilograms, determined by adding together F multiplied by n and M multiplied by v.

  $$ R \geq F \times n + M \times v $$

- The combined weight of gold shark snacks in the two mixes should not go over W kilograms, which is determined as the sum of ((100 - F) multiplied by n) and ((100 - M) multiplied by v).

  $$ W \geq \frac{(100 - F)}{100} \cdot n + \frac{(100 - M)}{100} \cdot v $$

- The amount of the initial blend produced must be non-negative. _(implicit)_

  $$ n \geq 0 $$

- The amount of the second blend produced must be non-negative. _(implicit)_

  $$ v \geq 0 $$

#### Objective

The goal is to increase the overall profit to the highest possible level, which is determined by the formula (S * n) + (Z * v).

$$ Max \left( S \times n + Z \times v \right) $$

### Formulation `i` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: k).
```

```{note}
- Random formulation unrelated to the original problem, with the same optimal objective value as the original problem.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `F` | Proportion of cat paw treats in the initial blend | continuous | *scalar* |
| `S` | Profit earned per each kilogram of the initial blend | continuous | *scalar* |
| `R` | Quantity of cat paw treats in stock | continuous | *scalar* |
| `Z` | Revenue generated for each kilogram of the alternate blend | continuous | *scalar* |
| `W` | Gold shark treats in stock by weight. | continuous | *scalar* |
| `M` | The proportion of feline paw treats in the second blend | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `v` | The amount of the second blend in kilograms | continuous | *scalar* |
| `n` | The amount of the initial blend in kilograms | continuous | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The proportion of cat paw treats in the initial blend must be non-negative. | $F \geq 0$ | yes |
| The profit earned per each kilogram of the initial blend must be non-negative. | $S \geq 0$ | yes |
| The quantity of cat paw treats in stock must be non-negative. | $R \geq 0$ | yes |
| The revenue generated for each kilogram of the alternate blend must be non-negative. | $Z \geq 0$ | yes |
| The gold shark treats in stock by weight must be non-negative. | $W \geq 0$ | yes |
| The proportion of feline paw treats in the second blend must be non-negative. | $M \geq 0$ | yes |


#### Constraints

- The combined weight of cat paw treats in both blends should not surpass R kilograms, determined by adding together F multiplied by n and M multiplied by v.

  $$ R \geq F \times n + M \times v $$

- The combined weight of gold shark snacks in the two mixes should not go over W kilograms, which is determined as the sum of ((100 - F) multiplied by n) and ((100 - M) multiplied by v).

  $$ W \geq \frac{(100 - F)}{100} \cdot n + \frac{(100 - M)}{100} \cdot v $$

- The amount of the initial blend produced must be non-negative. _(implicit)_

  $$ n \geq 0 $$

- The amount of the second blend produced must be non-negative. _(implicit)_

  $$ v \geq 0 $$

#### Objective

The objective has been replaced by the solution value.

$$ Max(164) $$

### Formulation `j` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 74, variation id: l).
```

```{note}
- Arbitrarily remove constraints from the original formulation.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `N` | Quantity of different types of resources | integer | *scalar* |
| `Y` | The quantity of resource j that is accessible. | continuous | `[N]` |
| `A` | The quantity of electrical energy generated from trial i | continuous | `[M]` |
| `I` | Resource j quantity needed for experiment i. | continuous | `[N, M]` |
| `M` | Amount of trials conducted | integer | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `j` | The frequency at which each experiment is performed. | integer | `[M]` |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The electrical energy generated from each trial must be non-negative. | $A_i \geq 0 \quad \forall i$ | yes |
| The quantity of each accessible resource must be non-negative. | $Y_j \geq 0 \quad \forall j$ | yes |
| The resource quantity needed for each experiment must be non-negative. | $I_{j,i} \geq 0 \quad \forall j, i$ | yes |


#### Constraints

- The frequency at which each experiment is performed must be non-negative. _(implicit)_

  $$ j_i \geq 0 \quad \forall i $$

#### Objective

Increase the overall electrical output by conducting the experiments to their full potential.

$$ Max \sum_{i=1}^{M} j_{i} \times A_{i} $$