p1 | Amusement Park Ticket Machines

See also

This problem is sourced from 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)

See also

This formulation is sourced from 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)

See also

This formulation is sourced from 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)

See also

This formulation is sourced from 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)

See also

This formulation is sourced from 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)

See also

This formulation is sourced from 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)

See also

This formulation is sourced from 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)

See also

This formulation is sourced from 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)

See also

This formulation is sourced from 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)

See also

This formulation is sourced from 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)

See also

This formulation is sourced from 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.

\[ Min \ r + s \]