p3 | Summer Camp Science Experiment

See also

This problem is sourced from EquivaFormulation (instance id: 92).

Note

• The source variable type is continuous; we correct the variable type to integer.

• The cutting plane (source variation 92_e) is excluded because it is instance-specific.

• Implicit non-negativity assumptions are added for every parameter.

Description

The summer camp uses NumBeakers different types of beakers. Each beaker type i consumes FlourUsagePerBeaker[i] units of flour and SpecialLiquidUsagePerBeaker[i] units of special liquid to produce SlimeProducedPerBeaker[i] units of slime and WasteProducedPerBeaker[i] units of waste. The camp has FlourAvailable units of flour and SpecialLiquidAvailable units of special liquid available. The total waste produced must not exceed MaxWasteAllowed. The goal is to determine how many beakers of each type to use to maximize the total amount of slime produced.

Formulations

Formulation a (valid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, variation id: original).

Note

• Original formulation; no transformations.

Parameters

Name	Description	Type	Shape
NumBeakers	Number of beakers	integer	scalar
FlourAvailable	Amount of flour available	continuous	scalar
SpecialLiquidAvailable	Amount of special liquid available	continuous	scalar
MaxWasteAllowed	Maximum amount of waste allowed	continuous	scalar
FlourUsagePerBeaker	Amount of flour used by each beaker	continuous	[NumBeakers]
SpecialLiquidUsagePerBeaker	Amount of special liquid used by each beaker	continuous	[NumBeakers]
SlimeProducedPerBeaker	Amount of slime produced by each beaker	continuous	[NumBeakers]
WasteProducedPerBeaker	Amount of waste produced by each beaker	continuous	[NumBeakers]

Variables

Name	Description	Type	Shape / Indices
NumBeakersUsed	The number of beakers of type i used	integer	[NumBeakers]

Assumptions

Description	Formulation	Implicit
NumBeakers is positive (non-zero).	\(NumBeakers \geq 1\)	yes
Flour usage per beaker is non-negative for all beakers.	\(FlourUsagePerBeaker_i \geq 0 \quad \forall i\)	yes
Special liquid usage per beaker is non-negative for all beakers.	\(SpecialLiquidUsagePerBeaker_i \geq 0 \quad \forall i\)	yes
Slime produced per beaker is non-negative for all beakers.	\(SlimeProducedPerBeaker_i \geq 0 \quad \forall i\)	yes
Waste produced per beaker is non-negative for all beakers.	\(WasteProducedPerBeaker_i \geq 0 \quad \forall i\)	yes

Constraints

• The total amount of flour used by all beakers does not exceed FlourAvailable.

  \[ \sum_{i=1}^{\text{NumBeakers}} FlourUsagePerBeaker_i \cdot NumBeakersUsed_i \leq FlourAvailable \]

• The total amount of special liquid used by all beakers does not exceed SpecialLiquidAvailable.

  \[ \sum_{i=1}^{NumBeakers} SpecialLiquidUsagePerBeaker_i \cdot NumBeakersUsed_i \leq SpecialLiquidAvailable \]

• The total amount of waste produced by all beakers does not exceed MaxWasteAllowed.

  \[ \sum_{i=1}^{\text{NumBeakers}} WasteProducedPerBeaker_i \cdot NumBeakersUsed_i \leq MaxWasteAllowed \]

• The number of beakers of each type used is non-negative. (implicit)

  \[ NumBeakersUsed_i \geq 0 \quad \forall i \]

Objective

The total amount of slime produced by all beakers is maximized.

\[ Max \sum_{i=1}^{NumBeakers} SlimeProducedPerBeaker_i \cdot NumBeakersUsed_i \]

Formulation b (valid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, variation id: c).

Note

• Change the names of parameters and variables.

Parameters

Name	Description	Type	Shape
C	The quantity of waste generated by each individual beaker	continuous	[N]
E	The highest quantity of waste permitted	continuous	scalar
N	Quantity of containers	integer	scalar
X	Quantity of slime generated by every container	continuous	[N]
T	Quantity of flour utilized for each individual beaker	continuous	[N]
D	Quantity of flour that is accessible	continuous	scalar
V	Quantity of the specialized liquid utilized by individual beakers.	continuous	[N]
Z	Quantity of unique fluid that can be used	continuous	scalar

Variables

Name	Description	Type	Shape / Indices
n	The number of beakers of type i used	integer	[N]

Assumptions

Description	Formulation	Implicit
N is positive (non-zero).	\(N \geq 1\)	yes
Waste per beaker is non-negative for all beakers.	\(C_i \geq 0 \quad \forall i\)	yes
Slime per beaker is non-negative for all beakers.	\(X_i \geq 0 \quad \forall i\)	yes
Flour per beaker is non-negative for all beakers.	\(T_i \geq 0 \quad \forall i\)	yes
Liquid per beaker is non-negative for all beakers.	\(V_i \geq 0 \quad \forall i\)	yes

Constraints

• The combined quantity of unique liquid utilized by each beaker does not surpass Z.

  \[ Z \geq \sum_{i=1}^{N} V_i \cdot n_i \]

• The cumulative quantity of flour utilized by each beaker does not surpass D.

  \[ D \geq \sum_{i=1}^{N} T_i \cdot n_i \]

• The combined waste generated by every individual beaker does not surpass E.

  \[ E \geq \sum_{i=1}^{N} C_i \cdot n_i \]

• The number of beakers of each type used is non-negative. (implicit)

  \[ n_i \geq 0 \quad \forall i \]

Objective

The highest possible quantity of slime generated by each beaker is reached.

\[ Max \sum_{i=1}^{N} X_i \cdot n_i \]

Formulation c (invalid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, 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 this substitution; we applied a 2-digit base-10 substitution.

Parameters

Name	Description	Type	Shape
C	The quantity of waste generated by each individual beaker	continuous	[N]
E	The highest quantity of waste permitted	continuous	scalar
N	Quantity of containers	integer	scalar
X	Quantity of slime generated by every container	continuous	[N]
T	Quantity of flour utilized for each individual beaker	continuous	[N]
D	Quantity of flour that is accessible	continuous	scalar
V	Quantity of the specialized liquid utilized by individual beakers.	continuous	[N]
Z	Quantity of unique fluid that can be used	continuous	scalar

Variables

Name	Description	Type	Shape / Indices
n_0	Digit 0 of the number of beakers of type i used	integer	[N]
n_1	Digit 1 of the number of beakers of type i used	integer	[N]

Assumptions

Description	Formulation	Implicit
N is positive (non-zero).	\(N \geq 1\)	yes
Waste per beaker is non-negative for all beakers.	\(C_i \geq 0 \quad \forall i\)	yes
Slime per beaker is non-negative for all beakers.	\(X_i \geq 0 \quad \forall i\)	yes
Flour per beaker is non-negative for all beakers.	\(T_i \geq 0 \quad \forall i\)	yes
Liquid per beaker is non-negative for all beakers.	\(V_i \geq 0 \quad \forall i\)	yes

Constraints

• The combined quantity of unique liquid utilized by each beaker does not surpass Z.

  \[ Z \geq \sum_{i=1}^{N} V_i \cdot (n\_0_i \cdot 10^0 + n\_1_i \cdot 10^1) \]

• The cumulative quantity of flour utilized by each beaker does not surpass D.

  \[ D \geq \sum_{i=1}^{N} T_i \cdot (n\_0_i \cdot 10^0 + n\_1_i \cdot 10^1) \]

• The combined waste generated by every individual beaker does not surpass E.

  \[ E \geq \sum_{i=1}^{N} C_i \cdot (n\_0_i \cdot 10^0 + n\_1_i \cdot 10^1) \]

• Digit 0 of the beaker count is at most 9 for all beakers.

  \[ n\_0_i \leq 9 \quad \forall i \]

• Digit 1 of the beaker count is at most 9 for all beakers.

  \[ n\_1_i \leq 9 \quad \forall i \]

• Digit 0 of the beaker count is non-negative for all beakers. (implicit)

  \[ n\_0_i \geq 0 \quad \forall i \]

• Digit 1 of the beaker count is non-negative for all beakers. (implicit)

  \[ n\_1_i \geq 0 \quad \forall i \]

Objective

The highest possible quantity of slime generated by each beaker is reached.

\[ Max \sum_{i=1}^{N} X_i \cdot (n\_0_i \cdot 10^0 + n\_1_i \cdot 10^1) \]

Formulation d (valid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, variation id: f).

Note

• Substitute the objective function with a new variable and linking constraint.

Parameters

Name	Description	Type	Shape
C	The quantity of waste generated by each individual beaker	continuous	[N]
E	The highest quantity of waste permitted	continuous	scalar
N	Quantity of containers	integer	scalar
X	Quantity of slime generated by every container	continuous	[N]
T	Quantity of flour utilized for each individual beaker	continuous	[N]
D	Quantity of flour that is accessible	continuous	scalar
V	Quantity of the specialized liquid utilized by individual beakers.	continuous	[N]
Z	Quantity of unique fluid that can be used	continuous	scalar

Variables

Name	Description	Type	Shape / Indices
n	The number of beakers of type i used	integer	[N]
zed	New variable representing the objective function	continuous	scalar

Assumptions

Description	Formulation	Implicit
N is positive (non-zero).	\(N \geq 1\)	yes
Waste per beaker is non-negative for all beakers.	\(C_i \geq 0 \quad \forall i\)	yes
Slime per beaker is non-negative for all beakers.	\(X_i \geq 0 \quad \forall i\)	yes
Flour per beaker is non-negative for all beakers.	\(T_i \geq 0 \quad \forall i\)	yes
Liquid per beaker is non-negative for all beakers.	\(V_i \geq 0 \quad \forall i\)	yes

Constraints

• Constraint defining zed in terms of original variables.

  \[ zed = \sum_{i=1}^{N} X_i \cdot n_i \]

• The combined quantity of unique liquid utilized by each beaker does not surpass Z.

  \[ Z \geq \sum_{i=1}^{N} V_i \cdot n_i \]

• The cumulative quantity of flour utilized by each beaker does not surpass D.

  \[ D \geq \sum_{i=1}^{N} T_i \cdot n_i \]

• The combined waste generated by every individual beaker does not surpass E.

  \[ E \geq \sum_{i=1}^{\text{N}} C_i \cdot n_i \]

• The number of beakers of each type used is non-negative. (implicit)

  \[ n_i \geq 0 \quad \forall i \]

Objective

Maximize the new variable zed.

\[ Maximize \ zed \]

Formulation e (valid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, 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
C	The quantity of waste generated by each individual beaker	continuous	[N]
E	The highest quantity of waste permitted	continuous	scalar
N	Quantity of containers	integer	scalar
X	Quantity of slime generated by every container	continuous	[N]
T	Quantity of flour utilized for each individual beaker	continuous	[N]
D	Quantity of flour that is accessible	continuous	scalar
V	Quantity of the specialized liquid utilized by individual beakers.	continuous	[N]
Z	Quantity of unique fluid that can be used	continuous	scalar

Variables

Name	Description	Type	Shape / Indices
n	The number of beakers of type i used	integer	[N]
slack_0	Slack variable for constraint: The combined quantity of unique liquid utilized by each beaker does not surpass Z.	continuous	scalar
slack_1	Slack variable for constraint: The cumulative quantity of flour utilized by each beaker does not surpass D.	continuous	scalar
slack_2	Slack variable for constraint: The combined waste generated by every individual beaker does not surpass E.	continuous	scalar

Assumptions

Description	Formulation	Implicit
N is positive (non-zero).	\(N \geq 1\)	yes
Waste per beaker is non-negative for all beakers.	\(C_i \geq 0 \quad \forall i\)	yes
Slime per beaker is non-negative for all beakers.	\(X_i \geq 0 \quad \forall i\)	yes
Flour per beaker is non-negative for all beakers.	\(T_i \geq 0 \quad \forall i\)	yes
Liquid per beaker is non-negative for all beakers.	\(V_i \geq 0 \quad \forall i\)	yes

Constraints

• The combined quantity of unique liquid utilized by each beaker does not surpass Z. (Modified to include slack variable slack_0)

  \[ \sum_{i=1}^{N} V_i \cdot n_i + slack_0 = Z \]

• The cumulative quantity of flour utilized by each beaker does not surpass D. (Modified to include slack variable slack_1)

  \[ \sum_{i=1}^{N} T_i \cdot n_i + slack_1 = D \]

• The combined waste generated by every individual beaker does not surpass E. (Modified to include slack variable slack_2)

  \[ \sum_{i=1}^{N} C_i \cdot n_i + slack_2 = E \]

• The number of beakers of each type used is non-negative. (implicit)

  \[ n_i \geq 0 \quad \forall i \]

• The slack variable for the liquid constraint is non-negative. (implicit)

  \[ slack_0 \geq 0 \]

• The slack variable for the flour constraint is non-negative. (implicit)

  \[ slack_1 \geq 0 \]

• The slack variable for the waste constraint is non-negative. (implicit)

  \[ slack_2 \geq 0 \]

Objective

The highest possible quantity of slime generated by each beaker is reached.

\[ Max \sum_{i=1}^{N} X_i \cdot n_i \]

Formulation f (valid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, variation id: h).

Note

• Splits variables.

Parameters

Name	Description	Type	Shape
C	The quantity of waste generated by each individual beaker	continuous	[N]
E	The highest quantity of waste permitted	continuous	scalar
N	Quantity of containers	integer	scalar
X	Quantity of slime generated by every container	continuous	[N]
T	Quantity of flour utilized for each individual beaker	continuous	[N]
D	Quantity of flour that is accessible	continuous	scalar
V	Quantity of the specialized liquid utilized by individual beakers.	continuous	[N]
Z	Quantity of unique fluid that can be used	continuous	scalar

Variables

Name	Description	Type	Shape / Indices
n1	Part 1 of variable n: The number of beakers of type i used	integer	[N]
n2	Part 2 of variable n: The number of beakers of type i used	integer	[N]

Assumptions

Description	Formulation	Implicit
N is positive (non-zero).	\(N \geq 1\)	yes
Waste per beaker is non-negative for all beakers.	\(C_i \geq 0 \quad \forall i\)	yes
Slime per beaker is non-negative for all beakers.	\(X_i \geq 0 \quad \forall i\)	yes
Flour per beaker is non-negative for all beakers.	\(T_i \geq 0 \quad \forall i\)	yes
Liquid per beaker is non-negative for all beakers.	\(V_i \geq 0 \quad \forall i\)	yes

Constraints

• The combined quantity of unique liquid utilized by each beaker does not surpass Z.

  \[ Z \geq \sum_{i=1}^{N} V_i \cdot (n1_i + n2_i) \]

• The cumulative quantity of flour utilized by each beaker does not surpass D.

  \[ D \geq \sum_{i=1}^{N} T_i \cdot (n1_i + n2_i) \]

• The combined waste generated by every individual beaker does not surpass E.

  \[ E \geq \sum_{i=1}^{N} C_i \cdot (n1_i + n2_i) \]

• Part 1 of the beaker count is non-negative for all beakers. (implicit)

  \[ n1_i \geq 0 \quad \forall i \]

• Part 2 of the beaker count is non-negative for all beakers. (implicit)

  \[ n2_i \geq 0 \quad \forall i \]

Objective

The highest possible quantity of slime generated by each beaker is reached.

\[ Max \sum_{i=1}^{N} X_i \cdot (n1_i + n2_i) \]

Formulation g (valid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, 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
C	The quantity of waste generated by each individual beaker	continuous	[N]
E	The highest quantity of waste permitted	continuous	scalar
N	Quantity of containers	integer	scalar
X	Quantity of slime generated by every container	continuous	[N]
T	Quantity of flour utilized for each individual beaker	continuous	[N]
D	Quantity of flour that is accessible	continuous	scalar
V	Quantity of the specialized liquid utilized by individual beakers.	continuous	[N]
Z	Quantity of unique fluid that can be used	continuous	scalar

Variables

Name	Description	Type	Shape / Indices
n	The number of beakers of type i used	integer	[N]

Assumptions

Description	Formulation	Implicit
N is positive (non-zero).	\(N \geq 1\)	yes
Waste per beaker is non-negative for all beakers.	\(C_i \geq 0 \quad \forall i\)	yes
Slime per beaker is non-negative for all beakers.	\(X_i \geq 0 \quad \forall i\)	yes
Flour per beaker is non-negative for all beakers.	\(T_i \geq 0 \quad \forall i\)	yes
Liquid per beaker is non-negative for all beakers.	\(V_i \geq 0 \quad \forall i\)	yes

Constraints

• The combined quantity of unique liquid utilized by each beaker does not surpass Z.

  \[ Z \geq \sum_{i=1}^{N} V_i \cdot n_{i} \]

• The cumulative quantity of flour utilized by each beaker does not surpass D.

  \[ D \geq \sum_{i=1}^{N} T_i \cdot n_i \]

• The combined waste generated by every individual beaker does not surpass E.

  \[ E \geq \sum_{i=1}^{\text{N}} C_i \cdot n_{i} \]

• The scaled beaker count is non-negative for all beakers. (implicit)

  \[ n_i \geq 0 \quad \forall i \]

Objective

The highest possible quantity of slime generated by each beaker is reached.

\[ Max 2 \cdot \sum_{i=1}^{N} X_i \cdot n_{i} \]

Formulation h (invalid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, variation id: j).

Note

• Random formulation unrelated to the original problem.

Parameters

Name	Description	Type	Shape
L	The amount of time needed to cool in order to create a single tempered glass panel.	continuous	scalar
S	The amount of time needed for a regular glass pane to cool down.	continuous	scalar
P	Time needed to heat up to create a single tempered glass panel.	continuous	scalar
H	Earnings per each treated glass panel	continuous	scalar
T	Earnings from each standard-sized glass sheet	continuous	scalar
C	The duration of heating needed to make a single standard glass panel	continuous	scalar
D	The highest amount of time the heating machine can be used in a day.	continuous	scalar
V	The maximum time allotted for the cooling machine to operate each day.	continuous	scalar

Variables

Name	Description	Type	Shape / Indices
h	The quantity of tempered glass sheets needed for production	continuous	scalar
e	The quantity of standard glass sheets needed for production.	continuous	scalar

Assumptions

Description	Formulation	Implicit
The amount of time needed to cool in order to create a single tempered glass panel must be non-negative.	\(L \geq 0\)	yes
The amount of time needed for a regular glass pane to cool down must be non-negative.	\(S \geq 0\)	yes
The time needed to heat up to create a single tempered glass panel must be non-negative.	\(P \geq 0\)	yes
The earnings per each treated glass panel must be non-negative.	\(H \geq 0\)	yes
The earnings from each standard-sized glass sheet must be non-negative.	\(T \geq 0\)	yes
The duration of heating needed to make a single standard glass panel must be non-negative.	\(C \geq 0\)	yes
The highest amount of time the heating machine can be used in a day must be non-negative.	\(D \geq 0\)	yes
The maximum time allotted for the cooling machine to operate each day must be non-negative.	\(V \geq 0\)	yes

Constraints

• The combined heating time needed to create Regular and Tempered glass panes does not go beyond D.

  \[ D \geq C \cdot e + P \cdot h \]

• The combined time needed to cool both Regular and Tempered glass panels does not go beyond V.

  \[ V \geq L \times h + S \times e \]

• The number of regular panes produced is non-negative. (implicit)

  \[ e \geq 0 \]

• The number of tempered panes produced is non-negative. (implicit)

  \[ h \geq 0 \]

Objective

The aim is to increase the overall profit by maximizing the total value obtained from T multiplied by the quantity of Regular panes and H multiplied by the quantity of Tempered panes.

\[ Max \ H \times h + T \times e \]

Formulation i (invalid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, 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 amount of time needed to cool in order to create a single tempered glass panel.	continuous	scalar
S	The amount of time needed for a regular glass pane to cool down.	continuous	scalar
P	Time needed to heat up to create a single tempered glass panel.	continuous	scalar
H	Earnings per each treated glass panel	continuous	scalar
T	Earnings from each standard-sized glass sheet	continuous	scalar
C	The duration of heating needed to make a single standard glass panel	continuous	scalar
D	The highest amount of time the heating machine can be used in a day.	continuous	scalar
V	The maximum time allotted for the cooling machine to operate each day.	continuous	scalar

Variables

Name	Description	Type	Shape / Indices
h	The quantity of tempered glass sheets needed for production	continuous	scalar
e	The quantity of standard glass sheets needed for production.	continuous	scalar

Assumptions

Description	Formulation	Implicit
The amount of time needed to cool in order to create a single tempered glass panel must be non-negative.	\(L \geq 0\)	yes
The amount of time needed for a regular glass pane to cool down must be non-negative.	\(S \geq 0\)	yes
The time needed to heat up to create a single tempered glass panel must be non-negative.	\(P \geq 0\)	yes
The earnings per each treated glass panel must be non-negative.	\(H \geq 0\)	yes
The earnings from each standard-sized glass sheet must be non-negative.	\(T \geq 0\)	yes
The duration of heating needed to make a single standard glass panel must be non-negative.	\(C \geq 0\)	yes
The highest amount of time the heating machine can be used in a day must be non-negative.	\(D \geq 0\)	yes
The maximum time allotted for the cooling machine to operate each day must be non-negative.	\(V \geq 0\)	yes

Constraints

• The combined heating time needed to create Regular and Tempered glass panes does not go beyond D.

  \[ D \geq C \cdot e + P \cdot h \]

• The combined time needed to cool both Regular and Tempered glass panels does not go beyond V.

  \[ V \geq L \times h + S \times e \]

• The number of regular panes produced is non-negative. (implicit)

  \[ e \geq 0 \]

• The number of tempered panes produced is non-negative. (implicit)

  \[ h \geq 0 \]

Objective

The objective has been replaced by the solution value.

\[ Max(45.0) \]

Formulation j (invalid)

See also

This formulation is sourced from EquivaFormulation (instance id: 92, variation id: l).

Note

• Arbitrarily remove constraints from the original formulation.

Parameters

Name	Description	Type	Shape
C	The quantity of waste generated by each individual beaker	continuous	[N]
E	The highest quantity of waste permitted	continuous	scalar
N	Quantity of containers	integer	scalar
X	Quantity of slime generated by every container	continuous	[N]
T	Quantity of flour utilized for each individual beaker	continuous	[N]
D	Quantity of flour that is accessible	continuous	scalar
V	Quantity of the specialized liquid utilized by individual beakers.	continuous	[N]
Z	Quantity of unique fluid that can be used	continuous	scalar

Variables

Name	Description	Type	Shape / Indices
n	The number of beakers of type i used	integer	[N]

Assumptions

Description	Formulation	Implicit
N is positive (non-zero).	\(N \geq 1\)	yes
Waste per beaker is non-negative for all beakers.	\(C_i \geq 0 \quad \forall i\)	yes
Slime per beaker is non-negative for all beakers.	\(X_i \geq 0 \quad \forall i\)	yes
Flour per beaker is non-negative for all beakers.	\(T_i \geq 0 \quad \forall i\)	yes
Liquid per beaker is non-negative for all beakers.	\(V_i \geq 0 \quad \forall i\)	yes

Constraints

• The combined quantity of unique liquid utilized by each beaker does not surpass Z.

  \[ Z \geq \sum_{i=1}^{N} V_i \cdot n_i \]

• The number of beakers of each type used is non-negative. (implicit)

  \[ n_i \geq 0 \quad \forall i \]

Objective

The highest possible quantity of slime generated by each beaker is reached.

\[ Max \sum_{i=1}^{N} X_i \cdot n_i \]