---
tocdepth: 3
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# p5 | Garden Bed Soil Hydration

```{seealso}
This problem is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217).
```

```{note}
- The source variable type is continuous; we correct the variable type to integer.
- The cutting plane (source variation `217_e`) is excluded because it is instance-specific.
- Implicit non-negativity assumptions are added for every parameter.
```


## Description

Both subsoil and topsoil need to be added to a garden bed. The objective is to minimize the total amount of water required to hydrate the garden bed, where each bag of subsoil requires WaterSubsoil units of water per day and each bag of topsoil requires WaterTopsoil units of water per day. The total number of bags of subsoil and topsoil combined must not exceed MaxTotalBags. Additionally, at least MinTopsoilBags bags of topsoil must be used, and the proportion of topsoil bags must not exceed MaxTopsoilProportion of all bags.

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

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, variation id: original).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `WaterSubsoil` | Amount of water required to hydrate one bag of subsoil per day | continuous | *scalar* |
| `WaterTopsoil` | Amount of water required to hydrate one bag of topsoil per day | continuous | *scalar* |
| `MaxTotalBags` | Maximum number of bags of topsoil and subsoil combined | integer | *scalar* |
| `MinTopsoilBags` | Minimum number of topsoil bags to be used | integer | *scalar* |
| `MaxTopsoilProportion` | Maximum proportion of bags that can be topsoil | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `SubsoilBags` | The number of subsoil bags | integer | *scalar* |
| `TopsoilBags` | The number of topsoil bags | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The amount of water required to hydrate one bag of subsoil per day must be non-negative. | $WaterSubsoil \geq 0$ | yes |
| The amount of water required to hydrate one bag of topsoil per day must be non-negative. | $WaterTopsoil \geq 0$ | yes |
| The maximum number of bags of topsoil and subsoil combined must be non-negative. | $MaxTotalBags \geq 0$ | yes |
| The minimum number of topsoil bags to be used must be non-negative. | $MinTopsoilBags \geq 0$ | yes |
| The maximum proportion of bags that can be topsoil must be non-negative. | $MaxTopsoilProportion \geq 0$ | yes |


#### Constraints

- The total number of subsoil and topsoil bags combined must not exceed MaxTotalBags.

  $$ SubsoilBags + TopsoilBags \leq MaxTotalBags $$

- At least MinTopsoilBags bags of topsoil must be used.

  $$ TopsoilBags \geq MinTopsoilBags $$

- The proportion of topsoil bags must not exceed MaxTopsoilProportion of all bags.

  $$ TopsoilBags \leq MaxTopsoilProportion \times ( TopsoilBags + SubsoilBags ) $$

- The number of subsoil bags must be non-negative.

  $$ SubsoilBags \geq 0 $$

- The number of topsoil bags must be non-negative.

  $$ TopsoilBags \geq 0 $$

#### Objective

Total water required is the sum of (WaterSubsoil * number of subsoil bags) and (WaterTopsoil * number of topsoil bags). The objective is to minimize the total water required.

$$ Min \left( WaterSubsoil \times SubsoilBags + WaterTopsoil \times TopsoilBags \right ) $$

### Formulation `b` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, variation id: c).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `P` | The smallest quantity of topsoil bags that should be employed | integer | *scalar* |
| `B` | The quantity of water needed to moisturize one bag of topsoil daily. | continuous | *scalar* |
| `D` | The highest quantity of bags containing a mixture of topsoil and subsoil | integer | *scalar* |
| `Z` | Quantity of water needed to keep one bag of subsoil hydrated on a daily basis | continuous | *scalar* |
| `K` | The highest percentage of bags that can contain topsoil. | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `d` | The quantity of bags containing topsoil | integer | *scalar* |
| `h` | The quantity of underground bags | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The smallest quantity of topsoil bags that should be employed must be non-negative. | $P \geq 0$ | yes |
| The quantity of water needed to moisturize one bag of topsoil daily must be non-negative. | $B \geq 0$ | yes |
| The highest quantity of bags containing a mixture of topsoil and subsoil must be non-negative. | $D \geq 0$ | yes |
| The quantity of water needed to keep one bag of subsoil hydrated on a daily basis must be non-negative. | $Z \geq 0$ | yes |
| The highest percentage of bags that can contain topsoil must be non-negative. | $K \geq 0$ | yes |


#### Constraints

- The percentage of topsoil bags should not go beyond K out of all bags.

  $$ K \times ( d + h ) \geq d $$

- The combined amount of subsoil bags and topsoil bags must not surpass D.

  $$ D \geq d + h $$

- A minimum of P bags of topsoil is required to be utilized.

  $$ P \leq d $$

- The number of subsoil bags must be non-negative.

  $$ h \geq 0 $$

- The number of topsoil bags must be non-negative.

  $$ d \geq 0 $$

#### Objective

The total amount of water needed consists of the combined value of (Z multiplied by the quantity of subsoil bags) and (B multiplied by the quantity of topsoil bags). The aim is to reduce the overall water consumption as much as possible.

$$ Min \left( B \times d + Z \times h \right ) $$

### Formulation `c` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, 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.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `P` | The smallest quantity of topsoil bags that should be employed | integer | *scalar* |
| `B` | The quantity of water needed to moisturize one bag of topsoil daily. | continuous | *scalar* |
| `D` | The highest quantity of bags containing a mixture of topsoil and subsoil | integer | *scalar* |
| `Z` | Quantity of water needed to keep one bag of subsoil hydrated on a daily basis | continuous | *scalar* |
| `K` | The highest percentage of bags that can contain topsoil. | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `h_0` | Digit 0 of the The quantity of underground bags | integer | *scalar* |
| `h_1` | Digit 1 of the The quantity of underground bags | integer | *scalar* |
| `d_0` | Digit 0 of the The quantity of bags containing topsoil | integer | *scalar* |
| `d_1` | Digit 1 of the The quantity of bags containing topsoil | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The smallest quantity of topsoil bags that should be employed must be non-negative. | $P \geq 0$ | yes |
| The quantity of water needed to moisturize one bag of topsoil daily must be non-negative. | $B \geq 0$ | yes |
| The highest quantity of bags containing a mixture of topsoil and subsoil must be non-negative. | $D \geq 0$ | yes |
| The quantity of water needed to keep one bag of subsoil hydrated on a daily basis must be non-negative. | $Z \geq 0$ | yes |
| The highest percentage of bags that can contain topsoil must be non-negative. | $K \geq 0$ | yes |


#### Constraints

- The percentage of topsoil bags should not go beyond K out of all bags.

  $$ K \times ( (d_0*10^0 + d_1*10^1) + (h_0*10^0 + h_1*10^1) ) \geq (d_0*10^0 + d_1*10^1) $$

- The combined amount of subsoil bags and topsoil bags must not surpass D.

  $$ D \geq (d_0*10^0 + d_1*10^1) + (h_0*10^0 + h_1*10^1) $$

- A minimum of P bags of topsoil is required to be utilized.

  $$ P \leq (d_0*10^0 + d_1*10^1) $$

- Digit 0 of subsoil bag count is non-negative. _(implicit)_

  $$ h\_0 \geq 0 $$

- Digit 1 of subsoil bag count is non-negative. _(implicit)_

  $$ h\_1 \geq 0 $$

- Digit 0 of topsoil bag count is non-negative. _(implicit)_

  $$ d\_0 \geq 0 $$

- Digit 1 of topsoil bag count is non-negative. _(implicit)_

  $$ d\_1 \geq 0 $$

- Digit 0 of subsoil bag count is at most 9. _(implicit)_

  $$ h\_0 \leq 9 $$

- Digit 1 of subsoil bag count is at most 9. _(implicit)_

  $$ h\_1 \leq 9 $$

- Digit 0 of topsoil bag count is at most 9. _(implicit)_

  $$ d\_0 \leq 9 $$

- Digit 1 of topsoil bag count is at most 9. _(implicit)_

  $$ d\_1 \leq 9 $$

#### Objective

The total amount of water needed consists of the combined value of (Z multiplied by the quantity of subsoil bags) and (B multiplied by the quantity of topsoil bags). The aim is to reduce the overall water consumption as much as possible.

$$ Min \left( B \times (d_0*10^0 + d_1*10^1) + Z \times (h_0*10^0 + h_1*10^1) \right ) $$

### Formulation `d` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, variation id: f).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `P` | The smallest quantity of topsoil bags that should be employed | integer | *scalar* |
| `B` | The quantity of water needed to moisturize one bag of topsoil daily. | continuous | *scalar* |
| `D` | The highest quantity of bags containing a mixture of topsoil and subsoil | integer | *scalar* |
| `Z` | Quantity of water needed to keep one bag of subsoil hydrated on a daily basis | continuous | *scalar* |
| `K` | The highest percentage of bags that can contain topsoil. | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `d` | The quantity of bags containing topsoil | integer | *scalar* |
| `h` | The quantity of underground bags | integer | *scalar* |
| `zed` | New variable representing the objective function | continuous | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The smallest quantity of topsoil bags that should be employed must be non-negative. | $P \geq 0$ | yes |
| The quantity of water needed to moisturize one bag of topsoil daily must be non-negative. | $B \geq 0$ | yes |
| The highest quantity of bags containing a mixture of topsoil and subsoil must be non-negative. | $D \geq 0$ | yes |
| The quantity of water needed to keep one bag of subsoil hydrated on a daily basis must be non-negative. | $Z \geq 0$ | yes |
| The highest percentage of bags that can contain topsoil must be non-negative. | $K \geq 0$ | yes |


#### Constraints

- Constraint defining zed in terms of original variables.

  $$ zed = Z * h + B * d $$

- The percentage of topsoil bags should not go beyond K out of all bags.

  $$ K \times ( d + h ) \geq d $$

- The combined amount of subsoil bags and topsoil bags must not surpass D.

  $$ D \geq d + h $$

- A minimum of P bags of topsoil is required to be utilized.

  $$ P \leq d $$

- The number of subsoil bags must be non-negative.

  $$ h \geq 0 $$

- The number of topsoil bags must be non-negative.

  $$ d \geq 0 $$

#### Objective

Minimize the new variable zed.

$$ Minimize \ zed $$

### Formulation `e` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, variation id: g).
```

```{note}
- Introduce slack variables to convert inequality constraints into equality constraints.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `P` | The smallest quantity of topsoil bags that should be employed | integer | *scalar* |
| `B` | The quantity of water needed to moisturize one bag of topsoil daily. | continuous | *scalar* |
| `D` | The highest quantity of bags containing a mixture of topsoil and subsoil | integer | *scalar* |
| `Z` | Quantity of water needed to keep one bag of subsoil hydrated on a daily basis | continuous | *scalar* |
| `K` | The highest percentage of bags that can contain topsoil. | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `d` | The quantity of bags containing topsoil | integer | *scalar* |
| `h` | The quantity of underground bags | integer | *scalar* |
| `slack_0` | Slack variable for constraint: The percentage of topsoil bags should not go beyond K out of all bags. | continuous | *scalar* |
| `slack_1` | Slack variable for constraint: The combined amount of subsoil bags and topsoil bags must not surpass D. | continuous | *scalar* |
| `slack_2` | Slack variable for constraint: A minimum of P bags of topsoil is required to be utilized. | continuous | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The smallest quantity of topsoil bags that should be employed must be non-negative. | $P \geq 0$ | yes |
| The quantity of water needed to moisturize one bag of topsoil daily must be non-negative. | $B \geq 0$ | yes |
| The highest quantity of bags containing a mixture of topsoil and subsoil must be non-negative. | $D \geq 0$ | yes |
| The quantity of water needed to keep one bag of subsoil hydrated on a daily basis must be non-negative. | $Z \geq 0$ | yes |
| The highest percentage of bags that can contain topsoil must be non-negative. | $K \geq 0$ | yes |


#### Constraints

- The percentage of topsoil bags should not go beyond K out of all bags. (Modified to include slack variable slack_0)

  $$ d + slack_0 = K * (d + h) $$

- The combined amount of subsoil bags and topsoil bags must not surpass D. (Modified to include slack variable slack_1)

  $$ h + d + slack_1 = D $$

- A minimum of P bags of topsoil is required to be utilized. (Modified to include slack variable slack_2)

  $$ d - slack_2 = P $$

- The number of subsoil bags must be non-negative.

  $$ h \geq 0 $$

- The number of topsoil bags must be non-negative.

  $$ d \geq 0 $$

- Slack variable for topsoil proportion constraint must be non-negative. _(implicit)_

  $$ slack\_0 \geq 0 $$

- Slack variable for total bags constraint must be non-negative. _(implicit)_

  $$ slack\_1 \geq 0 $$

- Slack variable for min topsoil constraint must be non-negative. _(implicit)_

  $$ slack\_2 \geq 0 $$

#### Objective

The total amount of water needed consists of the combined value of (Z multiplied by the quantity of subsoil bags) and (B multiplied by the quantity of topsoil bags). The aim is to reduce the overall water consumption as much as possible.

$$ Min \left( B \times d + Z \times h \right ) $$

### Formulation `f` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, variation id: h).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `P` | The smallest quantity of topsoil bags that should be employed | integer | *scalar* |
| `B` | The quantity of water needed to moisturize one bag of topsoil daily. | continuous | *scalar* |
| `D` | The highest quantity of bags containing a mixture of topsoil and subsoil | integer | *scalar* |
| `Z` | Quantity of water needed to keep one bag of subsoil hydrated on a daily basis | continuous | *scalar* |
| `K` | The highest percentage of bags that can contain topsoil. | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `d1` | Part 1 of variable d: The quantity of bags containing topsoil | integer | *scalar* |
| `d2` | Part 2 of variable d: The quantity of bags containing topsoil | integer | *scalar* |
| `h1` | Part 1 of variable h: The quantity of underground bags | integer | *scalar* |
| `h2` | Part 2 of variable h: The quantity of underground bags | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The smallest quantity of topsoil bags that should be employed must be non-negative. | $P \geq 0$ | yes |
| The quantity of water needed to moisturize one bag of topsoil daily must be non-negative. | $B \geq 0$ | yes |
| The highest quantity of bags containing a mixture of topsoil and subsoil must be non-negative. | $D \geq 0$ | yes |
| The quantity of water needed to keep one bag of subsoil hydrated on a daily basis must be non-negative. | $Z \geq 0$ | yes |
| The highest percentage of bags that can contain topsoil must be non-negative. | $K \geq 0$ | yes |


#### Constraints

- The percentage of topsoil bags should not go beyond K out of all bags.

  $$ K \times ( d1 + d2+ h1 + h2) \geq d1 + d2 $$

- The combined amount of subsoil bags and topsoil bags must not surpass D.

  $$ D \geq d1 + d2+ h1 + h2 $$

- A minimum of P bags of topsoil is required to be utilized.

  $$ P \leq d1 + d2 $$

- Part 1 of topsoil bags must be non-negative.

  $$ d1 \geq 0 $$

- Part 2 of topsoil bags must be non-negative.

  $$ d2 \geq 0 $$

- Part 1 of subsoil bags must be non-negative.

  $$ h1 \geq 0 $$

- Part 2 of subsoil bags must be non-negative.

  $$ h2 \geq 0 $$

#### Objective

The total amount of water needed consists of the combined value of (Z multiplied by the quantity of subsoil bags) and (B multiplied by the quantity of topsoil bags). The aim is to reduce the overall water consumption as much as possible.

$$ Min \left( B \times (d1 + d2)+ Z \times (h1 + h2)\right ) $$

### Formulation `g` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, variation id: i).
```

```{note}
- Re-scale the objective function.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `P` | The smallest quantity of topsoil bags that should be employed | integer | *scalar* |
| `B` | The quantity of water needed to moisturize one bag of topsoil daily. | continuous | *scalar* |
| `D` | The highest quantity of bags containing a mixture of topsoil and subsoil | integer | *scalar* |
| `Z` | Quantity of water needed to keep one bag of subsoil hydrated on a daily basis | continuous | *scalar* |
| `K` | The highest percentage of bags that can contain topsoil. | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `d` | The quantity of bags containing topsoil | integer | *scalar* |
| `h` | The quantity of underground bags | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The smallest quantity of topsoil bags that should be employed must be non-negative. | $P \geq 0$ | yes |
| The quantity of water needed to moisturize one bag of topsoil daily must be non-negative. | $B \geq 0$ | yes |
| The highest quantity of bags containing a mixture of topsoil and subsoil must be non-negative. | $D \geq 0$ | yes |
| The quantity of water needed to keep one bag of subsoil hydrated on a daily basis must be non-negative. | $Z \geq 0$ | yes |
| The highest percentage of bags that can contain topsoil must be non-negative. | $K \geq 0$ | yes |


#### Constraints

- The percentage of topsoil bags should not go beyond K out of all bags.

  $$ K \times ( d + h ) \geq d $$

- The combined amount of subsoil bags and topsoil bags must not surpass D.

  $$ D \geq d + h $$

- A minimum of P bags of topsoil is required to be utilized.

  $$ P \leq d $$

- The number of subsoil bags must be non-negative.

  $$ h \geq 0 $$

- The number of topsoil bags must be non-negative.

  $$ d \geq 0 $$

#### Objective

The total amount of water needed, scaled by a factor of 2. The aim is to reduce the overall water consumption as much as possible.

$$ Min \ 2 \times \left( B \times d + Z \times h \right ) $$

### Formulation `h` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, variation id: j).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `Q` | The smallest quantity of antique bottles that must be manufactured | continuous | *scalar* |
| `D` | Volume of an antique bottle measured in milliliters | continuous | *scalar* |
| `O` | The smallest proportion of standard bottles compared to antique bottles. | continuous | *scalar* |
| `J` | Volume of a typical bottle measured in milliliters | continuous | *scalar* |
| `A` | The overall quantity of wine that is in milliliters and ready for use. | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `z` | The quantity of standard bottles needed for production | continuous | *scalar* |
| `g` | The quantity of old bottles to manufacture | continuous | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The smallest quantity of antique bottles that must be manufactured must be non-negative. | $Q \geq 0$ | yes |
| The volume of an antique bottle measured in milliliters must be non-negative. | $D \geq 0$ | yes |
| The smallest proportion of standard bottles compared to antique bottles must be non-negative. | $O \geq 0$ | yes |
| The volume of a typical bottle measured in milliliters must be non-negative. | $J \geq 0$ | yes |
| The overall quantity of wine that is in milliliters and ready for use must be non-negative. | $A \geq 0$ | yes |


#### Constraints

- The quantity of standard bottles must be a minimum of O times the quantity of vintage bottles.

  $$ O \times g \leq z $$

- A minimum of Q bottles with a vintage must be manufactured.

  $$ Q \leq g $$

- The combined quantity of vine used in vintage and regular bottles should not surpass A milliliters.

  $$ A \geq z \times J + g \times D $$

- The number of regular bottles produced must be non-negative.

  $$ z \geq 0 $$

- The number of vintage bottles produced must be non-negative.

  $$ g \geq 0 $$

#### Objective

Optimize the overall quantity of bottles manufactured.

$$ Max \left( z + g \right) $$

### Formulation `i` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, 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 |
|---|---|---|---|
| `Q` | The smallest quantity of antique bottles that must be manufactured | continuous | *scalar* |
| `D` | Volume of an antique bottle measured in milliliters | continuous | *scalar* |
| `O` | The smallest proportion of standard bottles compared to antique bottles. | continuous | *scalar* |
| `J` | Volume of a typical bottle measured in milliliters | continuous | *scalar* |
| `A` | The overall quantity of wine that is in milliliters and ready for use. | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `z` | The quantity of standard bottles needed for production | continuous | *scalar* |
| `g` | The quantity of old bottles to manufacture | continuous | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The smallest quantity of antique bottles that must be manufactured must be non-negative. | $Q \geq 0$ | yes |
| The volume of an antique bottle measured in milliliters must be non-negative. | $D \geq 0$ | yes |
| The smallest proportion of standard bottles compared to antique bottles must be non-negative. | $O \geq 0$ | yes |
| The volume of a typical bottle measured in milliliters must be non-negative. | $J \geq 0$ | yes |
| The overall quantity of wine that is in milliliters and ready for use must be non-negative. | $A \geq 0$ | yes |


#### Constraints

- The quantity of standard bottles must be a minimum of O times the quantity of vintage bottles.

  $$ O \times g \leq z $$

- A minimum of Q bottles with a vintage must be manufactured.

  $$ Q \leq g $$

- The combined quantity of vine used in vintage and regular bottles should not surpass A milliliters.

  $$ A \geq z \times J + g \times D $$

- The number of regular bottles produced must be non-negative.

  $$ z \geq 0 $$

- The number of vintage bottles produced must be non-negative.

  $$ g \geq 0 $$

#### Objective

The objective has been replaced by the solution value.

$$ Max(300.0) $$

### Formulation `j` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 217, variation id: l).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `P` | The smallest quantity of topsoil bags that should be employed | integer | *scalar* |
| `B` | The quantity of water needed to moisturize one bag of topsoil daily. | continuous | *scalar* |
| `D` | The highest quantity of bags containing a mixture of topsoil and subsoil | integer | *scalar* |
| `Z` | Quantity of water needed to keep one bag of subsoil hydrated on a daily basis | continuous | *scalar* |
| `K` | The highest percentage of bags that can contain topsoil. | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `d` | The quantity of bags containing topsoil | integer | *scalar* |
| `h` | The quantity of underground bags | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The smallest quantity of topsoil bags that should be employed must be non-negative. | $P \geq 0$ | yes |
| The quantity of water needed to moisturize one bag of topsoil daily must be non-negative. | $B \geq 0$ | yes |
| The highest quantity of bags containing a mixture of topsoil and subsoil must be non-negative. | $D \geq 0$ | yes |
| The quantity of water needed to keep one bag of subsoil hydrated on a daily basis must be non-negative. | $Z \geq 0$ | yes |
| The highest percentage of bags that can contain topsoil must be non-negative. | $K \geq 0$ | yes |


#### Constraints

- The percentage of topsoil bags should not go beyond K out of all bags.

  $$ K \times ( d + h ) \geq d $$

- The number of subsoil bags must be non-negative.

  $$ h \geq 0 $$

- The number of topsoil bags must be non-negative.

  $$ d \geq 0 $$

#### Objective

The total amount of water needed consists of the combined value of (Z multiplied by the quantity of subsoil bags) and (B multiplied by the quantity of topsoil bags). The aim is to reduce the overall water consumption as much as possible.

$$ Min \left( B \times d + Z \times h \right ) $$