7.06a LP formulation: variables, constraints, objective function

8. \begin{figure}[h] \end{figure} 4. \begin{figure}[h] \end{figure} 5. \begin{figure}[h] \end{figure} 6. \(\begin{array} { l l l l l l l l l l } 30 & 11 & 21 & 53 & 50 & 39 & 16 & 4 & 60 & 43 \end{array}\) 7.

7. A company makes two types of wooden bookcase, the Manhattan and the Brooklyn. The pieces of wood used for each bookcase go through three stages. They must be cut, assembled and packaged. The table below shows the time, in hours, needed to complete each of the three stages for a single bookcase, and the profit made, in pounds, when each type of bookcase is sold. The table also shows the amount of time, in hours, that is available each week for each of the three stages. Shortest route: \(\_\_\_\_\) Length of shortest route: \(\_\_\_\_\) 3. \section*{Diagram 1} \section*{Grid 1} 4. \(\begin{array} { l l l l l l l l l l l } 180 & 80 & 250 & 115 & 100 & 230 & 150 & 95 & 105 & 90 & 390 \end{array}\) \(\begin{array} { l l l l l l l l l l l } 180 & 80 & 250 & 115 & 100 & 230 & 150 & 95 & 105 & 90 & 390 \end{array}\) 5.

1. (a) \includegraphics[max width=\textwidth, alt={}, center]{e7f89fa1-0afa-4aec-a430-14ec98f487c8-22_616_1477_735_230}
   

\begin{figure}[h] \end{figure} 7. \begin{figure}[h] \end{figure}

2. A restaurant sells two sizes of pizza, small and large. The restaurant owner knows that, each evening, she needs to make

• at least 85 pizzas in total
• at least twice as many large pizzas as small pizzas

In addition, at most \(80 \%\) of the pizzas must be large.
Each small pizza costs \(\pounds 2\) to make and each large pizza costs \(\pounds 3\) to make.
The restaurant owner wants to minimise her costs. Let \(x\) represent the number of small pizzas made each evening and let \(y\) represent the number of large pizzas made each evening. Formulate the information above as a linear programming problem. State the objective and list the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.

7. A farmer has 100 acres of land available that can be used for planting three crops: A, B and C . It takes 2 hours to plant each acre of crop A, 1.5 hours to plant each acre of crop B and 45 minutes to plant each acre of crop C . The farmer has 138 hours available for planting. At least one quarter of the total crops planted must be crop A.
For every three acres of crop B planted, at most five acres of crop C will be planted.
The farmer expects a profit of \(\pounds 160\) for each acre of crop A planted, \(\pounds 75\) for each acre of crop B planted and \(\pounds 125\) for each acre of crop C planted. The farmer wishes to maximise the profit from planting these three crops.
Let \(x , y\) and \(z\) represent the number of acres of land used for planting crop A, crop B, and crop C respectively.

1. Formulate this information as a linear programming problem. State the objective, and list the constraints as simplified inequalities with integer coefficients. The farmer decides that all 100 acres of available land will be used for planting the three crops.
2. Explain why the maximum total profit is achieved when \(- 7 x + 10 y\) is minimised. The farmer's decision to use all 100 acres reduces the constraints of the problem to the following: $$\begin{aligned} x & \geqslant 25 \\ 3 x + 8 y & \geqslant 300 \\ x + y & \leqslant 100 \\ 5 x + 3 y & \leqslant 252 \\ y & \geqslant 0 \end{aligned}$$
3. Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region, \(R\).
4. 1. Determine the exact coordinates of each of the vertices of \(R\).
   2. Apply the vertex method to determine how the 100 acres should be used for planting the three crops.
   3. Hence find the corresponding maximum expected profit.

5. Michael and his team are making toys to give to children at a summer fair. They make two types of toy, a soft toy and a craft set. Let \(x\) be the number of soft toys they make and \(y\) be the number of craft sets they make.
Each soft toy costs \(\pounds 3\) to make and each craft set costs \(\pounds 5\) to make. Michael and his team have a budget of \(\pounds 1000\) to spend on making the toys for the summer fair.

1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint. Two further constraints are: $$\begin{gathered} y \leqslant 2 x \\ 4 y - x \geqslant 210 \end{gathered}$$
2. Add lines and shading to Diagram 1 in the answer book to represent all of these constraints. Hence determine the feasible region and label it R . Michael's objective is to make as many toys as possible.
3. State the objective function.
4. Determine the exact coordinates of each of the vertices of the feasible region, and hence use the vertex method to find the optimal number of soft toys and craft sets Michael and his team should make. You should make your method clear.

7. Ian plans to produce two types of book, hardbacks and paperbacks. He will use linear programming to determine the number of each type of book he should produce. Let \(x\) represent the number of hardbacks Ian will produce. Let \(y\) represent the number of paperbacks Ian will produce. Each hardback takes 1 hour to print and 15 minutes to bind.
Each paperback takes 35 minutes to print and 24 minutes to bind.
The printing machine must be used for at least 14 hours. The binding machine must be used for at most 8 hours.

1. 1. Show that the printing time restriction leads to the constraint \(12 x + 7 y \geqslant k\), where \(k\) is a constant to be determined.
   2. Write the binding time restriction in a similar simplified form. Ian decides to produce at most twice as many hardbacks as paperbacks.
2. Write down an inequality to model this constraint in terms of \(x\) and \(y\).
3. Add lines and shading to Diagram 1 in the answer book to represent the constraints found in (a) and (b). Hence determine, and label, the feasible region R. Ian wishes to maximise \(\mathrm { P } = 60 x + 36 y\), where P is the total profit in pounds.
4. 1. Use the objective line (ruler) method to find the optimal vertex, V, of the feasible region. You must draw and clearly label your objective line and the vertex V .
   2. Determine the exact coordinates of V. You must show your working.
5. Given that P is Ian's expected total profit, in pounds, find the number of each type of book that he should produce and his maximum expected profit.

3. \begin{figure}[h] \end{figure} Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The equations of two of the lines have been given.

1. Determine the inequalities that define the feasible region.
2. Find the exact coordinates of the vertices of the feasible region. The objective is to maximise \(P\), where \(P = k x + y\).
3. For the case \(k = 2\), use point testing to find the optimal vertex of the feasible region.
4. For the case \(k = 2.5\), find the set of points for which \(P\) takes its maximum value.

7. A theatre company is planning to sell two types of ticket, standard and premier. The theatre company has completed some market research and has used this to form the following constraints.

• They will sell at most 450 tickets.
• They will sell at least three times as many standard tickets as premier tickets.
• At most \(85 \%\) of all the tickets sold will be standard.

The theatre wants to maximise its profit. The profit on each standard ticket sold is \(\pounds 5\) and the profit on each premier ticket sold is \(\pounds 8\) Let \(x\) represent the number of standard tickets sold and \(y\) represent the number of premier tickets sold. Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.
(Total 6 marks)

5. A school awards two types of prize, junior and senior. The school decides that it will award at least 25 junior prizes and at most 60 senior prizes.
Let \(x\) be the number of junior prizes that the school awards and let \(y\) be the number of senior prizes that the school awards.

1. Write down two inequalities to model these constraints.
   (2) Two further constraints are $$\begin{aligned} & 2 x + 5 y \geqslant 250 \\ & 5 x - 3 y \leqslant 150 \end{aligned}$$
2. Add lines and shading to Diagram 1 in the answer book to represent all four of these constraints. Hence determine the feasible region and label it \(R\). The cost of a senior prize is three times the cost of a junior prize. The school wishes to minimise the cost of the prizes.
3. State the objective function, giving your answer in terms of \(x\) and \(y\).
4. Determine the exact coordinates of the vertices of the feasible region. Hence use the vertex method to find the number of junior prizes and the number of senior prizes that the school should award. You should make your working clear.

5. A clothing shop sells a particular brand of shirt, which comes in three different sizes, small, medium and large. Each month the manager of the shop orders \(x\) small shirts, \(y\) medium shirts and \(z\) large shirts.
The manager forms constraints on the number of each size of shirts he will have to order.
One constraint is that for every 3 medium shirts he will order at least 5 large shirts.

1. Write down an inequality, with integer coefficients, to model this constraint. Two further constraints are $$x + y + z \geqslant 250 \text { and } x \leqslant 0.2 ( x + y + z )$$
2. Use these two constraints to write down statements, in context, that describe the number of different sizes of shirt the manager will order. The cost of each small shirt is \(\pounds 6\), the cost of each medium shirt is \(\pounds 10\) and the cost of each large shirt is \(\pounds 15\) The manager must minimise the total cost of all the shirts he will order.
3. Write down the objective function. Initially, the manager decides to order exactly 150 large shirts.
4. 1. Rewrite the constraints, as simplified inequalities with integer coefficients, in terms of \(x\) and \(y\) only.
   2. Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region \(R\).
5. Use the objective line method to find the optimal vertex, \(V\), of the feasible region. You must make your objective line clear and label \(V\).
6. Write down the number of each size of shirt the manager should order. Calculate the total cost of this order. Later, the manager decides to order exactly 50 small shirts and exactly 75 medium shirts instead of 150 large shirts.
7. Find the minimum number of large shirts the manager should order and show that this leads to a lower cost than the cost found in (f).

6. \begin{figure}[h] \end{figure} The graph in Figure 2 is being used to solve a linear programming problem in \(x\) and \(y\). The three constraints have been drawn on the graph and the rejected regions have been shaded out. The three vertices of the feasible region \(R\) are labelled \(\mathrm { A } , \mathrm { B }\) and C .

1. Determine the inequalities that define \(R\).
   (2) The objective function, \(P\), is given by $$P = a x + b y$$ where \(a\) and \(b\) are positive constants.
   The minimum value of \(P\) is 8 and the maximum value of \(P\) occurs at C .
2. Find the range of possible values of \(a\). You must make your method clear.
   (5)

8. A bakery makes three types of doughnut. These are ring, jam and custard. The bakery has the following constraints on the number of doughnuts it must make each day.

• The total number of doughnuts made must be at least 200
• They must make at least three times as many ring doughnuts as jam doughnuts
• At most \(70 \%\) of the doughnuts the bakery makes must be ring doughnuts
• At least a fifth of the doughnuts the bakery makes must be jam doughnuts

It costs 8 pence to make each ring doughnut, 10 pence to make each jam doughnut and 14 pence to make each custard doughnut. The bakery wants to minimise the total daily costs of making the required doughnuts. Let \(x\) represent the number of ring doughnuts, let \(y\) represent the number of jam doughnuts and let z represent the number of custard doughnuts the bakery makes each day.

1. Formulate this as a linear programming problem stating the objective and listing the constraints as simplified inequalities with integer coefficients. On a given day, instead of making at least 200 doughnuts, the bakery requires that exactly 200 doughnuts are made. Furthermore, the bakery decides to make the minimum number of jam doughnuts which satisfy all the remaining constraints. Given that the bakery still wants to minimise the total cost of making the required doughnuts, use algebra to
2. 1. calculate the number of each type of doughnut the bakery will make on that day,
   2. calculate the corresponding total cost of making all the doughnuts. \section*{END}

8. A headteacher is deciding how to allocate prizes to the students who are leaving at the end of the school year. There are three categories of prize: academic, sport, and leadership.

• Each academic prize costs \(\pounds 14\), each sport prize costs \(\pounds 8\), and each leadership prize costs \(\pounds 12\). The total amount available to spend on all prizes is \(\pounds 976\)
• For every 5 academic prizes there must be at least 2 leadership prizes
• At least half the prizes must be academic
• \(20 \%\) of the prizes must be for sport

The headteacher wishes to maximise the total number of prizes.
Let \(x , y\) and \(z\) represent the number of academic, sport and leadership prizes respectively.

1. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective and listing the constraints as simplified inequalities with integer coefficients. Given that the headteacher awards 16 sport prizes,
2. calculate the corresponding number of leadership prizes that the headteacher awards. You must show your working.

5. The head of a Mathematics department needs to order three types of paper. The three types of paper are plain, lined and graph. All three types of paper are sold in reams. (A ream is 500 sheets of paper.)
Based on the last academic year the head of department formed the following constraints.

• At least half the paper must be lined
• No more than \(15 \%\) of the paper must be graph paper
• The ratio of plain paper to graph paper must be \(5 : 2\)

The cost of each ream of plain, lined and graph paper is \(\pounds 5 , \pounds 12\) and \(\pounds 15\) respectively. The head of department has at most \(\pounds 834\) to spend on paper. The head of department wants to maximise the total number of reams of paper ordered.
Let \(x , y\) and \(z\) represent the number of reams of plain paper, lined paper and graph paper ordered respectively.

1. Formulate this information as a linear programming problem in \(x\) and \(y\) only, stating the objective and listing the constraints as simplified inequalities with integer coefficients. The head of department decides to order exactly 42 reams of lined paper and still wishes to maximise the total number of reams of paper ordered.
2. Determine
   1. the total number of reams of paper to be ordered,
   2. the number of reams of graph paper to be ordered.

2. Chris has been asked to design a badge in the shape of a triangle XYZ subject to the following constraints.

• Angle \(Y\) should be at least three times the size of angle \(X\)
• Angle \(Z\) should be at least \(50 ^ { \circ }\) larger than angle \(X\)
• Angle \(Y\) must be at most \(120 ^ { \circ }\)

Chris has been asked to maximise the sum of the angles \(X\) and \(Y\).
Let \(x\) be the size of angle \(X\) in degrees.
Let \(y\) be the size of angle \(Y\) in degrees.
Let z be the size of angle \(Z\) in degrees.
Formulate this information as a linear programming problem in \(x\) and \(y\) only. State the objective and list the constraints as simplified inequalities with integer coefficients. You are not required to solve this problem.

7. You are in charge of buying new cupboards for a school laboratory. The cupboards are available in two different sizes, standard and large.
The maximum budget available is \(\pounds 1800\). Standard cupboards cost \(\pounds 150\) and large cupboards cost \(\pounds 300\).
Let \(x\) be the number of standard cupboards and \(y\) be the number of large cupboards.

1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint.
   (2) The cupboards will be fitted along a wall 9 m long. Standard cupboards are 90 cm long and large cupboards are 120 cm long.
2. Show that this constraint can be modelled by $$3 x + 4 y \leqslant 30$$ You must make your reasoning clear. Given also that \(y \geqslant 2\),
3. explain what this constraint means in the context of the question. The capacity of a large cupboard is \(40 \%\) greater than the capacity of a standard cupboard. You wish to maximise the total capacity.
4. Show that your objective can be expressed as $$\text { maximise } 5 x + 7 y$$
5. Represent your inequalities graphically, on the axes in your answer booklet, indicating clearly the feasible region, R.
6. Find the number of standard cupboards and large cupboards that need to be purchased. Make your method clear.

6. \begin{figure}[h] \end{figure} Edgar has recently bought a field in which he intends to plant apple trees and plum trees. He can use linear programming to determine the number of each type of tree he should plant. Let \(x\) be the number of apple trees he plants and \(y\) be the number of plum trees he plants. Two of the constraints are $$\begin{aligned} & x \geqslant 40 \\ & y \leqslant 50 \end{aligned}$$ These are shown on the graph in Figure 6, where the rejected region is shaded out.

1. Use these two constraints to write down two statements that describe the number of apple trees and plum trees Edgar can plant. Two further constraints are $$\begin{aligned} 3 x + 4 y & \leqslant 360 \\ x & \leqslant 2 y \end{aligned}$$
2. Add two lines and shading to Diagram 1 in your answer book to represent these inequalities. Hence determine the feasible region and label it R . Edgar will make a profit of \(\pounds 60\) from each apple tree and \(\pounds 20\) from each plum tree. He wishes to maximise his profit, P.
3. Write down the objective function.
4. Use an objective line to determine the optimal point of the feasible region, R . You must make your method clear.
5. Find Edgar's maximum profit.

6. \begin{figure}[h] \end{figure} Lethna is producing floral arrangements for an awards ceremony.
She will produce two types of arrangement, Celebration and Party.
Let \(x\) be the number of Celebration arrangements made.
Let \(y\) be the number of Party arrangements made.
Figure 6 shows three constraints, other than \(x , y \geqslant 0\) The rejected region has been shaded.
Given that two of the three constraints are \(y \leqslant 30\) and \(x \leqslant 60\),

1. write down, as an inequality, the third constraint shown in Figure 6. Each Celebration arrangement includes 2 white roses and 4 red roses.
   Each Party arrangement includes 1 white rose and 5 red roses.
   Lethna wishes to use at least 70 white roses and at least 200 red roses.
2. Write down two further inequalities to represent this information.
   (3)
3. Add two lines and shading to Diagram 1 in the answer book to represent these two inequalities.
4. Hence determine the feasible region and label it R . The times taken to produce each Celebration arrangement and each Party arrangement are 10 minutes and 4 minutes respectively. Lethna wishes to minimise the total time taken to produce the arrangements.
5. Write down the objective function, T , in terms of \(x\) and \(y\).
6. Use point testing to find the optimal number of each type of arrangement Lethna should produce, and find the total time she will take.

8. A chemical company produces two products \(X\) and \(Y\). Based on potential demand, the total production each week must be at least 380 gallons. A major customer's weekly order for 125 gallons of \(Y\) must be satisfied. Product \(X\) requires 2 hours of processing time for each gallon and product \(Y\) requires 4 hours of processing time for each gallon. There are 1200 hours of processing time available each week. Let \(x\) be the number of gallons of \(X\) produced and \(y\) be the number of gallons of \(Y\) produced each week.

1. Write down the inequalities that \(x\) and \(y\) must satisfy.
   (3) It costs \(\pounds 3\) to produce 1 gallon of \(X\) and \(\pounds 2\) to produce 1 gallon of \(Y\). Given that the total cost of production is \(\pounds C\),
2. express \(C\) in terms of \(x\) and \(y\).
   (1) The company wishes to minimise the total cost.
3. Using the graphical method, solve the resulting Linear Programming problem. Find the optimal values of \(x\) and \(y\) and the resulting total cost.
4. Find the maximum cost of production for all possible choices of \(x\) and \(y\) which satisfy the inequalities you wrote down in part (a).

8. Class 8 B has decided to sell apples and bananas at morning break this week to raise money for charity. The profit on each apple is 20 p , the profit on each banana is 15 p . They have done some market research and formed the following constraints.

• They will sell at most 800 items of fruit during the week.
• They will sell at least twice as many apples as bananas.
• They will sell between 50 and 100 bananas.

Assuming they will sell all their fruit, formulate the above information as a linear programming problem, letting \(a\) represent the number of apples they sell and \(b\) represent the number of bananas they sell. Write your constraints as inequalities.
(Total 7 marks)

7. \begin{figure}[h] \end{figure} A company is going to hire out two types of car, standard and luxury. Let \(x\) be the number of standard cars it should buy.
Let \(y\) be the number of luxury cars it should buy. Figure 6 shows three constraints, other than \(x , y \geqslant 0\) Two of these are \(x \geqslant 20\) and \(y \geqslant 8\)

1. Write, as an inequality, the third constraint shown in Figure 6. The company decides that at least \(\frac { 1 } { 6 }\) of the cars must be luxury cars.
2. Express this information as an inequality and show that it simplifies to $$5 y \geqslant x$$ You must make the steps in your working clear. Each time the cars are hired they need to be prepared. It takes 5 hours to prepare a standard car and it takes 6 hours to prepare a luxury car. There are 300 hours available each week to prepare the cars.
3. Express this information as an inequality.
4. Add two lines and shading to Diagram 1 in the answer book to illustrate the constraints found in parts (b) and (c).
5. Hence determine the feasible region and label it R . The company expects to make \(\pounds 80\) profit per week on each car.
   It therefore wishes to maximise \(\mathrm { P } = 80 x + 80 y\), where P is the profit per week.
6. Use the objective line (ruler) method to find the optimal vertex, V, of the feasible region. You must clearly draw and label your objective line and the vertex V.
7. Given that P is the expected profit, in pounds, per week, find the number of each type of car that the company should buy and the maximum expected profit.

8. \begin{figure}[h] \end{figure} A company makes two types of garden bench, the 'Rustic' and the 'Contemporary'. The company wishes to maximise its profit and decides to use linear programming. Let \(x\) be the number of 'Rustic' benches made each week and \(y\) be the number of 'Contemporary' benches made each week. The graph in Figure 6 is being used to solve this linear programming problem.
Two of the constraints have been drawn on the graph and the rejected region shaded out.

1. Write down the constraints shown on the graph giving your answers as inequalities in terms of \(x\) and \(y\). It takes 4 working hours to make one 'Rustic' bench and 3 working hours to make one 'Contemporary' bench. There are 120 working hours available in each week.
2. Write down an inequality to represent this information. Market research shows that 'Rustic' benches should be at most \(\frac { 3 } { 4 }\) of the total benches made each week.
3. Write down, and simplify, an inequality to represent this information. Your inequality must have integer coefficients.
4. Add two lines and shading to Diagram 1 in your answer book to represent the inequalities of (b) and (c). Hence determine and label the feasible region, R. The profit on each 'Rustic' bench and each 'Contemporary' bench is \(\pounds 45\) and \(\pounds 30\) respectively.
5. Write down the objective function, P , in terms of \(x\) and \(y\).
6. Determine the coordinates of each of the vertices of the feasible region and hence use the vertex method to determine the optimal point.
7. State the maximum weekly profit the company could make.
   (Total 16 marks)

6. Harry wants to rent out boats at his local park. He can use linear programming to determine the number of each type of boat he should buy. Let \(x\) be the number of 2 -seater boats and \(y\) be the number of 4 -seater boats.
One of the constraints is $$x + y \geqslant 90$$

1. Explain what this constraint means in the context of the question. Another constraint is $$2 x \leqslant 3 y$$
2. Explain what this constraint means in the context of the question. A third constraint is $$y \leqslant x + 30$$
3. Represent these three constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R . Each 2 -seater boat costs \(\pounds 100\) and each 4 -seater boat costs \(\pounds 300\) to buy. Harry wishes to minimise the total cost of buying the boats.
4. Write down the objective function, C , in terms of \(x\) and \(y\).
5. Determine the number of each type of boat that Harry should buy. You must make your method clear and state the minimum cost.

5. A linear programming problem in \(x\) and \(y\) is described as follows. Maximise \(\quad P = 2 x + 3 y\) subject to $$\begin{aligned} x & \geqslant 25 \\ y & \geqslant 25 \\ 7 x + 8 y & \leqslant 840 \\ 4 y & \leqslant 5 x \\ 5 y & \geqslant 3 x \\ x , y & \geqslant 0 \end{aligned}$$

1. Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it R .
2. Use the objective line method to find the optimal vertex, V, of the feasible region. You must clearly draw and label your objective line and the vertex V.
3. Calculate the exact coordinates of vertex V. Given that an integer solution is required,
4. determine the optimal solution with integer coordinates. You must make your method clear.

8. A manufacturer of frozen yoghurt is going to exhibit at a trade fair. He will take two types of frozen yoghurt, Banana Blast and Strawberry Scream. He will take a total of at least 1000 litres of yoghurt.
He wants at least \(25 \%\) of the yoghurt to be Banana Blast. He also wants there to be at most half as much Banana Blast as Strawberry Scream. Each litre of Banana Blast costs \(\pounds 3\) to produce and each litre of Strawberry Scream costs \(\pounds 2\) to produce. The manufacturer wants to minimise his costs. Let \(x\) represent the number of litres of Banana Blast and \(y\) represent the number of litres of Strawberry Scream. Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.
(Total 6 marks)