---
tocdepth: 3
---

# p1 | Amusement Park Ticket Machines

```{seealso}
This problem is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47).
```

```{note}
- The source variable type is continuous; we correct the variable type to integer.
- The cutting plane (source variation `47_e`) is excluded because it is instance-specific.
- Implicit non-negativity assumptions are added for every parameter.
```


## Description

An amusement park is installing cash-based machines and card-only machines. A cash-based machine can process CashMachineProcessingRate people per hour, while a card-only machine can process CardMachineProcessingRate people per hour. The cash-based machine needs CashMachinePaperRolls rolls of paper per hour, while the card-only machine requires CardMachinePaperRolls rolls of paper per hour. The amusement park needs to be able to process at least MinPeopleProcessed people per hour but can use at most MaxPaperRolls paper rolls per hour. Additionally, the number of card-only machines must not exceed the number of cash-based machines. The objective is to minimize the total number of machines in the park.

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

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47, variation id: original).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `CashMachineProcessingRate` | Processing rate of a cash-based machine in people per hour | continuous | *scalar* |
| `CardMachineProcessingRate` | Processing rate of a card-only machine in people per hour | continuous | *scalar* |
| `CashMachinePaperRolls` | Number of paper rolls used per hour by a cash-based machine | continuous | *scalar* |
| `CardMachinePaperRolls` | Number of paper rolls used per hour by a card-only machine | continuous | *scalar* |
| `MinPeopleProcessed` | Minimum number of people that must be processed per hour | continuous | *scalar* |
| `MaxPaperRolls` | Maximum number of paper rolls that can be used per hour | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `NumCashMachines` | The number of cash-based machines | integer | *scalar* |
| `NumCardMachines` | The number of card-only machines | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The processing rate of a cash-based machine must be non-negative. | $CashMachineProcessingRate \geq 0$ | yes |
| The processing rate of a card-only machine must be non-negative. | $CardMachineProcessingRate \geq 0$ | yes |
| The number of paper rolls used per hour by a cash-based machine must be non-negative. | $CashMachinePaperRolls \geq 0$ | yes |
| The number of paper rolls used per hour by a card-only machine must be non-negative. | $CardMachinePaperRolls \geq 0$ | yes |
| The minimum number of people that must be processed per hour must be non-negative. | $MinPeopleProcessed \geq 0$ | yes |
| The maximum number of paper rolls that can be used per hour must be non-negative. | $MaxPaperRolls \geq 0$ | yes |


#### Constraints

- The total number of people processed per hour by cash-based and card-only machines must be at least MinPeopleProcessed.

  $$ CashMachineProcessingRate \cdot NumCashMachines + CardMachineProcessingRate \cdot NumCardMachines \geq MinPeopleProcessed $$

- The total number of paper rolls used per hour by cash-based and card-only machines must not exceed MaxPaperRolls.

  $$ NumCashMachines \times CashMachinePaperRolls + NumCardMachines \times CardMachinePaperRolls \leq MaxPaperRolls $$

- The number of card-only machines must not exceed the number of cash-based machines.

  $$ NumCardMachines \leq NumCashMachines $$

- The number of cash-based machines must be non-negative. _(implicit)_

  $$ NumCashMachines \geq 0 $$

- The number of card-only machines must be non-negative. _(implicit)_

  $$ NumCardMachines \geq 0 $$

#### Objective

Minimize the total number of machines in the park.

$$ Min \ NumCashMachines + NumCardMachines $$

### Formulation `b` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47, variation id: c).
```

```{note}
- Change the names of parameters and variables.
- Source typo $U \leq A \cdot K + s \cdot r$ corrected to $U \leq A \cdot s + K \cdot r$ to match Gurobi code and problem description.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `Y` | The quantity of paper rolls consumed every hour by a machine that operates on cash. | continuous | *scalar* |
| `U` | The lowest amount of individuals that need to be handled per hour. | continuous | *scalar* |
| `A` | The number of individuals processed by a cash-operated machine within an hour. | continuous | *scalar* |
| `V` | The highest quantity of paper rolls that can be utilized within a single hour | continuous | *scalar* |
| `K` | The number of individuals a card-only machine can handle in one hour. | continuous | *scalar* |
| `W` | Quantity of paper rolls utilized hourly by a machine that only accepts cards | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `r` | The quantity of machines that exclusively accept cards | integer | *scalar* |
| `s` | The quantity of machines that operate using cash. | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The quantity of paper rolls consumed every hour by a machine that operates on cash must be non-negative. | $Y \geq 0$ | yes |
| The lowest amount of individuals that need to be handled per hour must be non-negative. | $U \geq 0$ | yes |
| The number of individuals processed by a cash-operated machine within an hour must be non-negative. | $A \geq 0$ | yes |
| The highest quantity of paper rolls that can be utilized within a single hour must be non-negative. | $V \geq 0$ | yes |
| The number of individuals a card-only machine can handle in one hour must be non-negative. | $K \geq 0$ | yes |
| The quantity of paper rolls utilized hourly by a machine that only accepts cards must be non-negative. | $W \geq 0$ | yes |


#### Constraints

- The minimum number of individuals handled within an hour by machines accepting both cash and cards must be U.

  $$ U \leq A \cdot s + K \cdot r $$

- The quantity of machines that only accept cards should not surpass the quantity of machines that accept cash.

  $$ s \geq r $$

- The combined amount of paper rolls used per hour by machines that accept cash and those that only accept cards should not surpass V.

  $$ V \geq r \times W + s \times Y $$

- The number of cash machines must be non-negative. _(implicit)_

  $$ s \geq 0 $$

- The number of card machines must be non-negative. _(implicit)_

  $$ r \geq 0 $$

#### Objective

Reduce the overall quantity of machines at the park.

$$ Min \ r + s $$

### Formulation `c` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47, 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 typo $U \leq A \cdot K + (s_0*10^0 + s_1*10^1) \cdot (r_0*10^0 + r_1*10^1)$ corrected to $U \leq A \cdot (s_0*10^0 + s_1*10^1) + K \cdot (r_0*10^0 + r_1*10^1)$ to match Gurobi code and problem description.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `Y` | The quantity of paper rolls consumed every hour by a machine that operates on cash. | continuous | *scalar* |
| `U` | The lowest amount of individuals that need to be handled per hour. | continuous | *scalar* |
| `A` | The number of individuals processed by a cash-operated machine within an hour. | continuous | *scalar* |
| `V` | The highest quantity of paper rolls that can be utilized within a single hour | continuous | *scalar* |
| `K` | The number of individuals a card-only machine can handle in one hour. | continuous | *scalar* |
| `W` | Quantity of paper rolls utilized hourly by a machine that only accepts cards | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `s_0` | Digit 0 of the The quantity of machines that operate using cash. | integer | *scalar* |
| `s_1` | Digit 1 of the The quantity of machines that operate using cash. | integer | *scalar* |
| `r_0` | Digit 0 of the The quantity of machines that exclusively accept cards | integer | *scalar* |
| `r_1` | Digit 1 of the The quantity of machines that exclusively accept cards | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The quantity of paper rolls consumed every hour by a machine that operates on cash must be non-negative. | $Y \geq 0$ | yes |
| The lowest amount of individuals that need to be handled per hour must be non-negative. | $U \geq 0$ | yes |
| The number of individuals processed by a cash-operated machine within an hour must be non-negative. | $A \geq 0$ | yes |
| The highest quantity of paper rolls that can be utilized within a single hour must be non-negative. | $V \geq 0$ | yes |
| The number of individuals a card-only machine can handle in one hour must be non-negative. | $K \geq 0$ | yes |
| The quantity of paper rolls utilized hourly by a machine that only accepts cards must be non-negative. | $W \geq 0$ | yes |


#### Constraints

- The minimum number of individuals handled within an hour by machines accepting both cash and cards must be U.

  $$ U \leq A \cdot (s_0*10^0 + s_1*10^1) + K \cdot (r_0*10^0 + r_1*10^1) $$

- The quantity of machines that only accept cards should not surpass the quantity of machines that accept cash.

  $$ (s_0*10^0 + s_1*10^1) \geq (r_0*10^0 + r_1*10^1) $$

- The combined amount of paper rolls used per hour by machines that accept cash and those that only accept cards should not surpass V.

  $$ V \geq (r_0*10^0 + r_1*10^1) \times W + (s_0*10^0 + s_1*10^1) \times Y $$

- Digit s_0 must be non-negative. _(implicit)_

  $$ s_0 \geq 0 $$

- Digit s_1 must be non-negative. _(implicit)_

  $$ s_1 \geq 0 $$

- Digit r_0 must be non-negative. _(implicit)_

  $$ r_0 \geq 0 $$

- Digit r_1 must be non-negative. _(implicit)_

  $$ r_1 \geq 0 $$

- Digit s_0 must be at most 9. _(implicit)_

  $$ s_0 \leq 9 $$

- Digit s_1 must be at most 9. _(implicit)_

  $$ s_1 \leq 9 $$

- Digit r_0 must be at most 9. _(implicit)_

  $$ r_0 \leq 9 $$

- Digit r_1 must be at most 9. _(implicit)_

  $$ r_1 \leq 9 $$

#### Objective

Reduce the overall quantity of machines at the park.

$$ Min \ (r_0*10^0 + r_1*10^1) + (s_0*10^0 + s_1*10^1) $$

### Formulation `d` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47, variation id: f).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `Y` | The quantity of paper rolls consumed every hour by a machine that operates on cash. | continuous | *scalar* |
| `U` | The lowest amount of individuals that need to be handled per hour. | continuous | *scalar* |
| `A` | The number of individuals processed by a cash-operated machine within an hour. | continuous | *scalar* |
| `V` | The highest quantity of paper rolls that can be utilized within a single hour | continuous | *scalar* |
| `K` | The number of individuals a card-only machine can handle in one hour. | continuous | *scalar* |
| `W` | Quantity of paper rolls utilized hourly by a machine that only accepts cards | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `r` | The quantity of machines that exclusively accept cards | integer | *scalar* |
| `s` | The quantity of machines that operate using cash. | integer | *scalar* |
| `zed` | New variable representing the objective function | continuous | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The quantity of paper rolls consumed every hour by a machine that operates on cash must be non-negative. | $Y \geq 0$ | yes |
| The lowest amount of individuals that need to be handled per hour must be non-negative. | $U \geq 0$ | yes |
| The number of individuals processed by a cash-operated machine within an hour must be non-negative. | $A \geq 0$ | yes |
| The highest quantity of paper rolls that can be utilized within a single hour must be non-negative. | $V \geq 0$ | yes |
| The number of individuals a card-only machine can handle in one hour must be non-negative. | $K \geq 0$ | yes |
| The quantity of paper rolls utilized hourly by a machine that only accepts cards must be non-negative. | $W \geq 0$ | yes |


#### Constraints

- Constraint defining zed in terms of original variables.

  $$ zed = s + r $$

- The minimum number of individuals handled within an hour by machines accepting both cash and cards must be U.

  $$ U \leq A \cdot s + K \cdot r $$

- The quantity of machines that only accept cards should not surpass the quantity of machines that accept cash.

  $$ s \geq r $$

- The combined amount of paper rolls used per hour by machines that accept cash and those that only accept cards should not surpass V.

  $$ V \geq r \times W + s \times Y $$

- The number of cash machines must be non-negative. _(implicit)_

  $$ s \geq 0 $$

- The number of card machines must be non-negative. _(implicit)_

  $$ r \geq 0 $$

#### 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: 47, variation id: g).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `Y` | The quantity of paper rolls consumed every hour by a machine that operates on cash. | continuous | *scalar* |
| `U` | The lowest amount of individuals that need to be handled per hour. | continuous | *scalar* |
| `A` | The number of individuals processed by a cash-operated machine within an hour. | continuous | *scalar* |
| `V` | The highest quantity of paper rolls that can be utilized within a single hour | continuous | *scalar* |
| `K` | The number of individuals a card-only machine can handle in one hour. | continuous | *scalar* |
| `W` | Quantity of paper rolls utilized hourly by a machine that only accepts cards | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `r` | The quantity of machines that exclusively accept cards | integer | *scalar* |
| `s` | The quantity of machines that operate using cash. | integer | *scalar* |
| `slack_0` | Slack variable for constraint: The minimum number of individuals handled within an hour by machines accepting both cash and cards must be U. | continuous | *scalar* |
| `slack_1` | Slack variable for constraint: The quantity of machines that only accept cards should not surpass the quantity of machines that accept cash. | continuous | *scalar* |
| `slack_2` | Slack variable for constraint: The combined amount of paper rolls used per hour by machines that accept cash and those that only accept cards should not surpass V. | continuous | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The quantity of paper rolls consumed every hour by a machine that operates on cash must be non-negative. | $Y \geq 0$ | yes |
| The lowest amount of individuals that need to be handled per hour must be non-negative. | $U \geq 0$ | yes |
| The number of individuals processed by a cash-operated machine within an hour must be non-negative. | $A \geq 0$ | yes |
| The highest quantity of paper rolls that can be utilized within a single hour must be non-negative. | $V \geq 0$ | yes |
| The number of individuals a card-only machine can handle in one hour must be non-negative. | $K \geq 0$ | yes |
| The quantity of paper rolls utilized hourly by a machine that only accepts cards must be non-negative. | $W \geq 0$ | yes |


#### Constraints

- The minimum number of individuals handled within an hour by machines accepting both cash and cards must be U. (Modified to include slack variable slack_0)

  $$ A * s + K * r - slack_0 = U $$

- The quantity of machines that only accept cards should not surpass the quantity of machines that accept cash. (Modified to include slack variable slack_1)

  $$ r + slack_1 = s $$

- The combined amount of paper rolls used per hour by machines that accept cash and those that only accept cards should not surpass V. (Modified to include slack variable slack_2)

  $$ s * Y + r * W + slack_2 = V $$

- The number of cash machines must be non-negative. _(implicit)_

  $$ s \geq 0 $$

- The number of card machines must be non-negative. _(implicit)_

  $$ r \geq 0 $$

- Slack variable slack_0 must be non-negative. _(implicit)_

  $$ slack_0 \geq 0 $$

- Slack variable slack_1 must be non-negative. _(implicit)_

  $$ slack_1 \geq 0 $$

- Slack variable slack_2 must be non-negative. _(implicit)_

  $$ slack_2 \geq 0 $$

#### Objective

Reduce the overall quantity of machines at the park.

$$ Min \ r + s $$

### Formulation `f` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47, variation id: h).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `Y` | The quantity of paper rolls consumed every hour by a machine that operates on cash. | continuous | *scalar* |
| `U` | The lowest amount of individuals that need to be handled per hour. | continuous | *scalar* |
| `A` | The number of individuals processed by a cash-operated machine within an hour. | continuous | *scalar* |
| `V` | The highest quantity of paper rolls that can be utilized within a single hour | continuous | *scalar* |
| `K` | The number of individuals a card-only machine can handle in one hour. | continuous | *scalar* |
| `W` | Quantity of paper rolls utilized hourly by a machine that only accepts cards | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `r1` | Part 1 of variable r: The quantity of machines that exclusively accept cards | integer | *scalar* |
| `r2` | Part 2 of variable r: The quantity of machines that exclusively accept cards | integer | *scalar* |
| `s1` | Part 1 of variable s: The quantity of machines that operate using cash. | integer | *scalar* |
| `s2` | Part 2 of variable s: The quantity of machines that operate using cash. | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The quantity of paper rolls consumed every hour by a machine that operates on cash must be non-negative. | $Y \geq 0$ | yes |
| The lowest amount of individuals that need to be handled per hour must be non-negative. | $U \geq 0$ | yes |
| The number of individuals processed by a cash-operated machine within an hour must be non-negative. | $A \geq 0$ | yes |
| The highest quantity of paper rolls that can be utilized within a single hour must be non-negative. | $V \geq 0$ | yes |
| The number of individuals a card-only machine can handle in one hour must be non-negative. | $K \geq 0$ | yes |
| The quantity of paper rolls utilized hourly by a machine that only accepts cards must be non-negative. | $W \geq 0$ | yes |


#### Constraints

- The minimum number of individuals handled within an hour by machines accepting both cash and cards must be U.

  $$ U \leq A (s1 + s2) + K (r1 + r2) $$

- The quantity of machines that only accept cards should not surpass the quantity of machines that accept cash.

  $$ s1 + s2\geq r1 + r2 $$

- The combined amount of paper rolls used per hour by machines that accept cash and those that only accept cards should not surpass V.

  $$ V \geq (r1 + r2)\times W + (s1 + s2)\times Y $$

- Part 1 of cash machine count must be non-negative. _(implicit)_

  $$ s1 \geq 0 $$

- Part 2 of cash machine count must be non-negative. _(implicit)_

  $$ s2 \geq 0 $$

- Part 1 of card machine count must be non-negative. _(implicit)_

  $$ r1 \geq 0 $$

- Part 2 of card machine count must be non-negative. _(implicit)_

  $$ r2 \geq 0 $$

#### Objective

Reduce the overall quantity of machines at the park.

$$ Min \ r1 + r2+ s1 + s2 $$

### Formulation `g` (valid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47, variation id: i).
```

```{note}
- Re-scale the objective function.
- Source was missing objective re-scaling; applied arbitrary scaling factor of 2.
```


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `Y` | The quantity of paper rolls consumed every hour by a machine that operates on cash. | continuous | *scalar* |
| `U` | The lowest amount of individuals that need to be handled per hour. | continuous | *scalar* |
| `A` | The number of individuals processed by a cash-operated machine within an hour. | continuous | *scalar* |
| `V` | The highest quantity of paper rolls that can be utilized within a single hour | continuous | *scalar* |
| `K` | The number of individuals a card-only machine can handle in one hour. | continuous | *scalar* |
| `W` | Quantity of paper rolls utilized hourly by a machine that only accepts cards | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `r` | The quantity of machines that exclusively accept cards | integer | *scalar* |
| `s` | The quantity of machines that operate using cash. | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The quantity of paper rolls consumed every hour by a machine that operates on cash must be non-negative. | $Y \geq 0$ | yes |
| The lowest amount of individuals that need to be handled per hour must be non-negative. | $U \geq 0$ | yes |
| The number of individuals processed by a cash-operated machine within an hour must be non-negative. | $A \geq 0$ | yes |
| The highest quantity of paper rolls that can be utilized within a single hour must be non-negative. | $V \geq 0$ | yes |
| The number of individuals a card-only machine can handle in one hour must be non-negative. | $K \geq 0$ | yes |
| The quantity of paper rolls utilized hourly by a machine that only accepts cards must be non-negative. | $W \geq 0$ | yes |


#### Constraints

- The minimum number of individuals handled within an hour by machines accepting both cash and cards must be U.

  $$ U \leq A \cdot s + K \cdot r $$

- The quantity of machines that only accept cards should not surpass the quantity of machines that accept cash.

  $$ s \geq r $$

- The combined amount of paper rolls used per hour by machines that accept cash and those that only accept cards should not surpass V.

  $$ V \geq r \times W + s \times Y $$

- The number of cash machines must be non-negative. _(implicit)_

  $$ s \geq 0 $$

- The number of card machines must be non-negative. _(implicit)_

  $$ r \geq 0 $$

#### Objective

Reduce the overall quantity of machines at the park.

$$ Min \ 2 \cdot (r + s) $$

### Formulation `h` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47, variation id: j).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `L` | The fewest amount of containers required | continuous | *scalar* |
| `V` | The smallest quantity of oil units required to be transported to the port. | continuous | *scalar* |
| `G` | The quantity of oil that each container is capable of holding. | continuous | *scalar* |
| `Y` | Quantity of oil that each truck is capable of carrying | continuous | *scalar* |
| `K` | The highest permitted proportion of trucks to containers | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `c` | The quantity of containers utilized | integer | *scalar* |
| `p` | The quantity of trucks utilized | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The fewest amount of containers required must be non-negative. | $L \geq 0$ | yes |
| The smallest quantity of oil units required to be transported to the port must be non-negative. | $V \geq 0$ | yes |
| The quantity of oil that each container is capable of holding must be non-negative. | $G \geq 0$ | yes |
| The quantity of oil that each truck is capable of carrying must be non-negative. | $Y \geq 0$ | yes |
| The highest permitted proportion of trucks to containers must be non-negative. | $K \geq 0$ | yes |


#### Constraints

- The maximum amount of trucks allowed is half the number of containers used.

  $$ K \times c \geq p $$

- A minimum of 15 containers must be utilized.

  $$ L \leq c $$

- A minimum of 2000 units of oil must be delivered to the port, with the calculation being 30 units per container and 40 units per truck.

  $$ V \leq Y \times p + G \times c $$

- The number of containers must be non-negative. _(implicit)_

  $$ c \geq 0 $$

- The number of trucks must be non-negative. _(implicit)_

  $$ p \geq 0 $$

#### Objective

The goal is to reduce the overall requirement for containers and trucks to a minimum.

$$ Min c + p $$

### Formulation `i` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47, 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 |
|---|---|---|---|
| `L` | The fewest amount of containers required | continuous | *scalar* |
| `V` | The smallest quantity of oil units required to be transported to the port. | continuous | *scalar* |
| `G` | The quantity of oil that each container is capable of holding. | continuous | *scalar* |
| `Y` | Quantity of oil that each truck is capable of carrying | continuous | *scalar* |
| `K` | The highest permitted proportion of trucks to containers | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `c` | The quantity of containers utilized | integer | *scalar* |
| `p` | The quantity of trucks utilized | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The fewest amount of containers required must be non-negative. | $L \geq 0$ | yes |
| The smallest quantity of oil units required to be transported to the port must be non-negative. | $V \geq 0$ | yes |
| The quantity of oil that each container is capable of holding must be non-negative. | $G \geq 0$ | yes |
| The quantity of oil that each truck is capable of carrying must be non-negative. | $Y \geq 0$ | yes |
| The highest permitted proportion of trucks to containers must be non-negative. | $K \geq 0$ | yes |


#### Constraints

- The maximum amount of trucks allowed is half the number of containers used.

  $$ K \times c \geq p $$

- A minimum of 15 containers must be utilized.

  $$ L \leq c $$

- A minimum of 2000 units of oil must be delivered to the port, with the calculation being 30 units per container and 40 units per truck.

  $$ V \leq Y \times p + G \times c $$

- The number of containers must be non-negative. _(implicit)_

  $$ c \geq 0 $$

- The number of trucks must be non-negative. _(implicit)_

  $$ p \geq 0 $$

#### Objective

The objective has been replaced by the solution value.

$$ Min(20.0) $$

### Formulation `j` (invalid)

```{seealso}
This formulation is sourced from [EquivaFormulation](https://huggingface.co/datasets/humainlab/EquivaFormulation) (instance id: 47, variation id: l).
```

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


#### Parameters

| Name | Description | Type | Shape |
|---|---|---|---|
| `Y` | The quantity of paper rolls consumed every hour by a machine that operates on cash. | continuous | *scalar* |
| `U` | The lowest amount of individuals that need to be handled per hour. | continuous | *scalar* |
| `A` | The number of individuals processed by a cash-operated machine within an hour. | continuous | *scalar* |
| `V` | The highest quantity of paper rolls that can be utilized within a single hour | continuous | *scalar* |
| `K` | The number of individuals a card-only machine can handle in one hour. | continuous | *scalar* |
| `W` | Quantity of paper rolls utilized hourly by a machine that only accepts cards | continuous | *scalar* |


#### Variables

| Name | Description | Type | Shape / Indices |
|---|---|---|---|
| `r` | The quantity of machines that exclusively accept cards | integer | *scalar* |
| `s` | The quantity of machines that operate using cash. | integer | *scalar* |


#### Assumptions

| Description | Formulation | Implicit |
|---|---|---|
| The quantity of paper rolls consumed every hour by a machine that operates on cash must be non-negative. | $Y \geq 0$ | yes |
| The lowest amount of individuals that need to be handled per hour must be non-negative. | $U \geq 0$ | yes |
| The number of individuals processed by a cash-operated machine within an hour must be non-negative. | $A \geq 0$ | yes |
| The highest quantity of paper rolls that can be utilized within a single hour must be non-negative. | $V \geq 0$ | yes |
| The number of individuals a card-only machine can handle in one hour must be non-negative. | $K \geq 0$ | yes |
| The quantity of paper rolls utilized hourly by a machine that only accepts cards must be non-negative. | $W \geq 0$ | yes |


#### Constraints

- The combined amount of paper rolls used per hour by machines that accept cash and those that only accept cards should not surpass V.

  $$ V \geq r \times W + s \times Y $$

- The number of cash machines must be non-negative. _(implicit)_

  $$ s \geq 0 $$

- The number of card machines must be non-negative. _(implicit)_

  $$ r \geq 0 $$

#### Objective

Reduce the overall quantity of machines at the park.

$$ Min \ r + s $$