Solve a linear programming problem

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Bill Foster

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Has some problems

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Bill Foster 6 years, 6 months ago

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Name	Status	Author	Last Modified
Solve a linear programming problem	Has some problems	Newcastle University Mathematics and Statistics	16/11/2020 10:03
Adrian's copy of Solve a linear programming problem	draft	Adrian Shepherd	16/11/2020 10:02
Denis's copy of Adrian's copy of Solve a linear programming problem	draft	Denis Flynn	16/11/2020 10:02
Luis's copy of Solve a linear programming problem	draft	Luis Hernandez	16/11/2020 10:03

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Question statement

{business_long} ({business_short}) makes two products: {product_1s} and {product_2s}.

Existing orders require {business} to make at least {xconstraint} {product_1s} and {yconstraint} {product_2s} each day.

Each unit of {product_1s} produced uses {resource1x} {resource_1_units} {resource_1} and {resource2x} {resource_2_units} {resource_2}.

Each unit of {product_2s} produced uses {resource1y} {resource_1_units} {resource_1} and {resource2y} {resource_2_units} {resource_2}.

However, the resources are limited: {business} can only obtain {totalresource1} {resource_1_units} {resource_1} and {totalresource2} {resource_2_units} {resource_2} each day.

Each sale of a unit of {product_1s} gives £{profitx} profit. Each sale of a unit of {product_2s} gives £{profity} profit.

Give any introductory information the student needs.

			Name	Type	Generated Value

xconstraint	integer	4
yconstraint	integer	10
resource1x	integer	26
resource1y	integer	62
resource2x	integer	350
resource2y	integer	2800
r1	integer	20
r2	integer	9
scalingfactor1	decimal	dec("1.1")
scalingfactor2	decimal	dec("0.06")
totalresource1	decimal	dec("2.50e+3")
totalresource2	decimal	dec("8.82e+4")

			Name	Type	Generated Value

boundx	decimal	dec("172")
boundy	integer	39
constraint1xvalue	decimal	dec("72")
constraint1yvalue	integer	39
constraint2xvalue	decimal	dec("172")
constraint2yvalue	integer	30

			Name	Type	Generated Value

			profitx	integer	4
			profity	integer	15

			Name	Type	Generated Value

intersection_points	list	Nested 5×2 list
possible_points	list	[ 1, 2, 4 ]
profits	list	List of 5 items
best_point	number	4
best_x_float	decimal	dec("2.997260273972602380769230769230769230769e+1")
best_y_float	decimal	dec("2.775342465753424657534246575342465753425e+1")
best_profit_float	number	536.1917808219
best_x	decimal	dec("29")
best_y	decimal	dec("27")
best_profit	number	521
best_lines	list	[ 0, 1 ]
line_names	list	List of 4 items

			Name	Type	Generated Value

business_long	string	Amalgamated Mines and Sheep, I
business_short	string	AMS
business	string	AMS
resource_1	string	wool
resource_1_units	string	kg
resource_2	string	iron
resource_2_units	string	tonnes
product_1	string	bulletproof tea cosy
product_1s	string	bulletproof tea cosies
product_2	string	double-edged cardigan
product_2s	string	double-edged cardigans

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
​
random(4..10)

JME function reference

Description

Minimum production of the first product

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

constraint1xvalue
constraint2xvalue
intersection_points
possible_points
totalresource1
totalresource2

Statement
Advice
"Part a)" → "Gap 0." - Minimum accepted value
"Part a)" → "Gap 0." - Maximum accepted value

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
Part b) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
  Add
4. Gap Gap 3. Number entry
  Add
5. Gap Gap 4. Number entry
  Add
6. Gap Gap 5. Number entry
  Add
Part c) Gap-fill
1. Gap Mathematical expression
  Add
Part d) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Let $x$ , $y$ be the number of units of {product_1s} and {product_2s} produced, respectively.

What are the constraints on production resulting from the existing orders?

$x \geq$ Gap 0.

$y \geq$ Gap 1.

Advice

a)

The existing orders give two constraints:

\begin{align}
x &\geq \var{xconstraint} \\
y &\geq \var{yconstraint}
\end{align}

b)

The limits on the two resources, and the amounts of each resource used to produce a unit of each product, give two further constraints:

\begin{align}
\simplify{{resource1x}x + {resource1y}y} &\leq \var{totalresource1} \\
\simplify{{resource2x}x + {resource2y}y} &\leq \var{totalresource2}
\end{align}

c)

The objective function to be maximised is the total profit earned, that is, the sum of the profits earned from each unit made:

\[ \simplify{{profitx}x + {profity}y} \]

d)

The solution graph goes here

It is clear that the minimum profit is given by producing enough {product_1s} and {product_2s} so that the existing demand is met and no more. You can see this by moving the objective function line down to the intersection of the two minimum constraint lines - there are no other points inside the feasible region under the objective line.

Next, drag the objective function line upwards until there are no points above the objective line in the feasible region. At this point, the objective line touches the intersection of the lines representing the constraints for {line_names[best_lines[0]]} and {line_names[best_lines[1]]}.

You can find the coordinates of this point by solving the equations for these two lines:

\[ \simplify{{resource1x}x + {resource1y}y = {totalresource1}} \]\[ \simplify{{resource2x}x + {resource2y}y = {totalresource2}} \]

\[ \simplify{{resource2x}x + {resource2y}y = {totalresource2}} \]\[ x = \var{xconstraint} \]\[ y = \var{yconstraint} \]

The solution to this system of equations is $x = \var{best_x}$, $y = \var{best_y}$, giving a profit of

\[ \text{Profit} = \simplify[]{{profitx}*{best_x} + {profity}*{best_y}} = \text{£}\var{best_profit} \]

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
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Gap-fill
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Gap-fill
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This question is used in the following exams:

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Taxonomy: mathcentre

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Taxonomy: Context

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Checkpoint description

Bill Foster 6 years, 6 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

d)

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Solve a linear programming problem

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 6 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

d)

Generated value: `integer`