7.06e Sensitivity analysis: effect of changing coefficients

8 [Figure 2, printed on a separate sheet, is provided for use in this question.]
A bakery makes two types of pizza, large and medium.
Every day the bakery must make at least 40 of each type.
Every day the bakery must make at least 120 in total but not more than 400 pizzas in total.
Each large pizza takes 4 minutes to make, and each medium pizza takes 2 minutes to make. There are four workers available, each for five hours a day, to make the pizzas. The bakery makes a profit of \(\pounds 3\) on each large pizza sold and \(\pounds 1\) on each medium pizza sold.
Each day, the bakery makes and sells \(x\) large pizzas and \(y\) medium pizzas.
The bakery wishes to maximise its profit, \(\pounds P\).

1. Show that one of the constraints leads to the inequality $$2 x + y \leqslant 600$$
2. Formulate this situation as a linear programming problem.
3. On Figure 2, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and an objective line.
4. Use your diagram to find the maximum daily profit.
5. The bakery introduces a new pricing structure in which the profit is \(\pounds 2\) on each large pizza sold and \(\pounds 2\) on each medium pizza sold.
   1. Find the new maximum daily profit for the bakery.
   2. Write down the number of different combinations that would give the new maximum daily profit.

5 The feasible region of a linear programming problem is determined by the following: $$\begin{aligned} y & \geqslant 20 \\ x + y & \geqslant 25 \\ 5 x + 2 y & \leqslant 100 \\ y & \leqslant 4 x \\ y & \geqslant 2 x \end{aligned}$$

1. On Figure 1 opposite, draw a suitable diagram to represent the inequalities and indicate the feasible region.
2. Use your diagram to find the minimum value of \(P\), on the feasible region, in the case where:
   1. \(P = x + 2 y\);
   2. \(P = - x + y\). In each case, state the corresponding values of \(x\) and \(y\).

5 The feasible region of a linear programming problem is defined by $$\begin{aligned} x + y & \leqslant 60 \\ 2 x + y & \leqslant 80 \\ y & \geqslant 20 \\ x & \geqslant 15 \\ y & \geqslant x \end{aligned}$$

1. On the grid opposite, draw a suitable diagram to represent these inequalities and indicate the feasible region.
2. In each of the following cases, use your diagram to find the maximum value of \(P\) on the feasible region. In each case, state the corresponding values of \(x\) and \(y\).
   1. \(P = x + 4 y\)
   2. \(P = 4 x + y\)

6 [Figure 1, printed on the insert, is provided for use in this question.]
A factory makes two types of lock, standard and large, on a particular day.
On that day:
the maximum number of standard locks that the factory can make is 100 ;
the maximum number of large locks that the factory can make is 80 ;
the factory must make at least 60 locks in total;
the factory must make more large locks than standard locks.
Each standard lock requires 2 screws and each large lock requires 8 screws, and on that day the factory must use at least 320 screws. On that day, the factory makes \(x\) standard locks and \(y\) large locks.
Each standard lock costs \(\pounds 1.50\) to make and each large lock costs \(\pounds 3\) to make.
The manager of the factory wishes to minimise the cost of making the locks.

1. Formulate the manager's situation as a linear programming problem.
2. On Figure 1, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of the objective line.
3. Find the values of \(x\) and \(y\) that correspond to the minimum cost. Hence find this minimum cost.

5 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{197624b2-ca67-4bad-9c2c-dc68c10be0fd-04_1118_816_404_662}

1. Write down four inequalities that define the feasible region. The objective is to maximise \(P = 5 x + 3 y\).
2. Using the graph or otherwise, obtain the coordinates of the vertices of the feasible region and hence find the values of \(x\) and \(y\) that maximise \(P\), and the corresponding maximum value of \(P\). The objective is changed to maximise \(Q = a x + 3 y\).
3. For what set of values of \(a\) is the maximum value of \(Q\) equal to 3?

4 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{f2b85dfb-49df-4ea5-b118-9b95f0b27bad-03_1025_826_374_657}

1. Write down inequalities that define the feasible region.
2. Find the coordinates of the four vertices of the feasible region. The objective is to maximise \(P\), where \(P = x + 2 y\).
3. Find the values of \(x\) and \(y\) that maximise \(P\), and the corresponding maximum value of \(P\). The objective is changed to minimise \(Q\), where \(Q = 2 x - y\).
4. Find the minimum value of \(Q\) and describe the set of feasible points for which \(Q\) takes this value.
5. Show that there are no points in the feasible region for which the value of \(P\) is the same as the value of \(Q\).

3 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{7ca6d572-d776-4ad7-a0ed-9ec43c975585-03_908_915_392_614}

1. Write down the inequalities that define the feasible region. The objective is to maximise \(P _ { 1 } = x + 6 y\).
2. Find the values of \(x\) and \(y\) at the optimal point, and the corresponding value of \(P _ { 1 }\). The objective is changed to maximise \(P _ { k } = k x + 6 y\), where \(k\) is positive.
3. Calculate the coordinates of the optimal point, and the corresponding value of \(P _ { k }\) when the optimal point is not the same as in part (ii).
4. Find the range of values of \(k\) for which the point identified in part (ii) is still optimal.

1 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{cec8d4db-4a72-43a3-88f3-ff9df2a11d2c-2_885_873_388_635}

1. Write down the inequalities that define the feasible region. The objective is to maximise \(P _ { m } = x + m y\), where \(m\) is a positive, real-valued constant.
2. In the case when \(m = 2\), calculate the values of \(x\) and \(y\) at the optimal point, and the corresponding value of \(P _ { 2 }\).
3. (a) Write down the values of \(m\) for which point \(A\) is optimal.
   (b) Write down the values of \(m\) for which point \(B\) is optimal.

3 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{372c062a-793f-4fb8-a769-957479f5fce7-05_846_833_365_614} The vertices of the feasible region are \(A ( 3.5,2 ) , B ( 1.5,3 ) , C ( 0.5,1.5 ) , D ( 1,0.5 )\).
The objective is to maximise \(P = x + 3 y\).

1. Find the coordinates of the optimum vertex and the corresponding value of \(P\).
2. Find the optimum point if \(x\) and \(y\) must both have integer values. The objective is changed to maximise \(P = x + k y\).
3. If \(k\) is positive, explain why the optimum point cannot be at \(C\) or \(D\).
4. If \(k\) can take any value, find the range of values of \(k\) for which \(A\) is the optimum point.

6 A recipe for jam states that the weight of sugar used must be between the weight of fruit used and four thirds of the weight of fruit used. Georgia has 10 kg of fruit available and 11 kg of sugar.

1. Define two variables and formulate inequalities in those variables to model this information.
2. Draw a graph to represent your inequalities.
3. Find the vertices of your feasible region and identify the points which would represent the best mix of ingredients under each of the following circumstances.
   (A) There is to be as much jam as possible, given that the weight of jam produced is the sum of the weights of the fruit and the sugar.
   (B) There is to be as much jam as possible, given that it is to have the lowest possible proportion of sugar.
   (C) There is to be as much jam as possible, given that it is to have the highest possible proportion of sugar.
   (D) Fruit costs \(\pounds 1\) per kg, sugar costs 50 p per kg and the objective is to produce as much jam as possible within a budget of \(\pounds 15\).

6 A manufacturing company holds stocks of two liquid chemicals. The company needs to update its stock levels. The company has 2000 litres of chemical A and 4000 litres of chemical B currently in stock. Its storage facility allows for no more than a combined total of 12000 litres of the two chemicals. Chemical A is valued at \(\pounds 5\) per litre and chemical B is valued at \(\pounds 6\) per litre. The company intends to hold stocks of these two chemicals with a total value of at least \(\pounds 61000\). Let \(a\) be the increase in the stock level of A, in thousands of litres ( \(a\) can be negative).
Let \(b\) be the increase in the stock level of B , in thousands of litres ( \(b\) can be negative).

1. Explain why \(a \geqslant - 2\), and produce a similar inequality for \(b\).
2. Explain why the value constraint can be written as \(5 a + 6 b \geqslant 27\), and produce, in similar form, the storage constraint.
3. Illustrate all four inequalities graphically.
4. Find the policy which will give a stock value of exactly \(\pounds 61000\), and will use all 12000 litres of available storage space.
5. Interpret your solution in terms of stock levels, and verify that the new stock levels do satisfy both the value constraint and the storage constraint.

6 Jean knits items for charity. Each month the charity provides her with 75 balls of wool.
She knits hats and scarves. Hats require 1.5 balls of wool each and scarves require 3 balls each. Jean has 100 hours available each month for knitting. Hats require 4 hours each to make, and scarves require 2.5 hours each. The charity sells the hats for \(\pounds 7\) each and the scarves for \(\pounds 10\) each, and wants to gain as much income as possible. Jean prefers to knit hats but the charity wants no more than 20 per month. She refuses to knit more than 20 scarves each month.

1. Define appropriate variables, construct inequality constraints, and draw a graph representing the feasible region for this decision problem.
2. Give the objective function and find the integer solution which will give Jean's maximum monthly income.
3. If the charity drops the price of hats in a sale to \(\pounds 4\) each, what would be an optimal number of hats and scarves for Jean to knit? Assuming that all hats and scarves are sold, by how much would the monthly income drop?

6 A company manufactures two types of potting compost, Flowerbase and Growmuch. The weekly amounts produced of each are constrained by the supplies of fibre and of nutrient mix. Each litre of Flowerbase requires 0.75 litres of fibre and 1 kg of nutrient mix. Each litre of Growmuch requires 0.5 litres of fibre and 2 kg of nutrient mix. There are 12000 litres of fibre supplied each week, and 25000 kg of nutrient mix. The profit on Flowerbase is 9 p per litre. The profit on Growmuch is 20 p per litre.

1. Formulate an LP to maximise the weekly profit subject to the constraints on fibre and nutrient mix.
2. Solve your LP using a graphical approach.
3. Consider each of the following separate circumstances.
   (A) There is a reduction in the weekly supply of fibre from 12000 litres to 10000 litres. What effect does this have on profit?
   (B) The price of fibre is increased. Will this affect the optimal production plan? Justify your answer.
   [0pt] (C) The supply of nutrient mix is increased to 30000 kg per week. What is the new profit? [1]

6 Ian the chef is to make vegetable stew and vegetable soup for distribution to a small chain of vegetarian restaurants. The recipes for both of these require carrots, beans and tomatoes. 10 litres of stew requires 1.5 kg of carrots, 1 kg of beans and 1.5 kg of tomatoes.
10 litres of soup requires 1 kg of carrots, 0.75 kg of beans and 1.5 kg of tomatoes. Ian has available 100 kg of carrots, 70 kg of beans and 110 kg of tomatoes.

1. Identify appropriate variables and write down three inequalities corresponding to the availabilities of carrots, beans and tomatoes.
2. Graph your inequalities and identify the region corresponding to feasible production plans. The profit on a litre of stew is \(\pounds 5\), and the profit on a litre of soup is \(\pounds 4\).
3. Find the most profitable production plan, showing your working. Give the maximum profit. Ian can buy in extra tomatoes at \(\pounds 2.50\) per kg .
4. What extra quantity of tomatoes should Ian buy? How much extra profit would be generated by the extra expenditure? \section*{END OF QUESTION PAPER} \section*{OCR}

6. The manager of a new leisure complex needs to maximise the Revenue \(( \pounds R )\) from providing the following two weekend programmes. The following restrictions apply to each weekend.
No more than 90 participants can be accommodated.
There must be at most 40 adults.
A maximum of 600 person-hours of windsurfing can be offered.
A maximum of 300 person-hours of sailing can be offered.

1. Formulate the above information as a linear programming problem, listing the constraints as inequalities and stating the objective function \(R\).
2. On graph paper, illustrate the constraints, indicating clearly the feasible region.
3. Solve the problem graphically, stating how many adults and how many children should be accepted each weekend and what the revenue will be. The manager is considering buying more windsurfing equipment at a cost of \(\pounds 2000\). This would increase windsurfing provision by \(10 \%\).
4. State, with a reason, whether such a purchase would be cost effective.

5 Greetings cards are sold in luxury, standard and economy packs.
The table shows the cost of each pack and number of cards of each kind in the pack. Alice needs 25 cards, of which at least 8 must be handmade cards, at least 8 must be cards with flowers and at least 4 must be cards with animals.

1. Explain why Alice will need to buy at least two packs of cards. Alice does not want to spend more than \(\pounds 12\) on the cards.
2. (a) List the combinations of packs that satisfy all Alice's requirements.
   (b) Which of these is the cheapest? Ben offers to buy any cards that Alice buys but does not need. He will pay 12 pence for each handmade card and 5 pence for any other card. Alice does not want her net expenditure (the amount she spends minus the amount that Ben pays her) on the cards to be more than \(\pounds 12\).
3. Show that Alice could now buy two luxury packs. Alice decides to buy exactly 2 packs, of which \(x\) are luxury packs, \(y\) are standard packs and the rest are economy packs.
4. Give an expression, in terms of \(x\) and \(y\) only, for the number of cards of each type that Alice buys. Alice wants to minimise her net expenditure.
5. Find, and simplify, an expression for Alice's minimum net expenditure in pence, in terms of \(x\) and \(y\). You may assume that Alice buys enough cards to satisfy her own requirements.
6. Find Alice's minimum net expenditure.

5 Corey is training for a race that starts in 18 hours time. He splits his training between gym work, running and swimming.

• At most 8 hours can be spent on gym work.
• At least 4 hours must be spent running.
• The total time spent on gym work and swimming must not exceed the time spent running.

Corey thinks that time spent on gym work is worth 3 times the same time spent running or 2 times the same time spent swimming. Corey wants to maximise the worth of the training using this model.

1. Formulate a linear programming problem to represent Corey's problem. Your formulation must include defining the variables that you are using. Suppose that Corey spends the maximum of 8 hours on gym work.
2. 1. Use a graphical method to determine how long Corey should spend running and how long he should spend swimming.
   2. Describe why this solution is not practical.
   3. Describe how Corey could refine the LP model to make the solution more realistic.

5 A baker makes three types of jam-and-custard doughnuts.

• Each batch of type X uses 6 units of jam and 4 units of custard.
• Each batch of type Y uses 7 units of jam and 3 units of custard.
• Each batch of type Z uses 8 units of jam and 2 units of custard.

The baker has 360 units of jam and 180 units of custard available. The baker has plenty of doughnut batter, so this does not restrict the number of batches made. From past experience the baker knows that they must make at most 30 batches of type X and at least twice as many batches of type Y as batches of type Z . Let \(x =\) number of batches of type X made \(y =\) number of batches of type Y made \(z =\) number of batches of type Z made.

1. Set up an LP formulation for the problem of maximising the total number of batches of doughnuts made. The baker finds that type Z doughnuts are not popular and decides to make zero batches of type Z .
2. Use a graphical method to find how many batches of each type the baker should make to maximise the total number of batches of doughnuts made.
3. Give a reason why this solution may not be practical. The baker finds that some of the jam has been used so there are only \(k\) units of jam (where \(k < 360\) ).
   There are still 180 units of custard available and the baker still makes zero batches of type Z .
4. Find the values of \(k\) if exactly one of the other (non-trivial) constraints is redundant. Express your answer using inequalities.

7 A linear programming problem is
Maximise \(P = 4 x + y\) subject to $$\begin{aligned} 3 x - y & \leqslant 30 \\ x + y & \leqslant 15 \\ x - 3 y & \leqslant 6 \end{aligned}$$ and \(x \geqslant 0 , y \geqslant 0\)

1. Use a graphical method to find the optimal value of \(P\), and the corresponding values of \(x\) and \(y\). An additional constraint is introduced.
   This constraint means that the value of \(y\) must be at least \(k\) times the value of \(x\), where \(k\) is a positive constant.
2. 1. Determine the set of values of \(k\) for which the optimal value of \(P\) found in part (a) is unchanged.
   2. Determine, in terms of \(k\), the values of \(x , y\) and \(P\) in the cases when the optimal solution is different from that found in part (a).

4. \begin{figure}[h] \end{figure} Figure 5 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region.

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 \(P = 2 x + 3 y\).
3. Use point testing at each vertex to find the optimal vertex, \(V\), of the feasible region and state the corresponding value of \(P\) at \(V\).
   (3) The objective is changed to maximise \(Q = 2 x + k y\), where \(k\) is a constant.
4. Find the range of values of \(k\) for which the vertex identified in (c) is still optimal.
   (2)

7. \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 shown in Figure 3. Given that \(k\) is a positive constant,

1. determine, in terms of \(k\) where necessary, the inequalities that define \(R\). The objective is to maximise \(P = 5 x + k y\) Given that the value of \(P\) is 38 at the optimal vertex of \(R\),
2. determine the possible value(s) of \(k\). You must show algebraic working and make your method clear.
   (Total 11 marks)

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.

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)

7. \begin{figure}[h] \end{figure} Figure 5 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 and the three intersection points, \(A\), \(B\) and \(C\), are shown. The coordinates of \(C\) are \(\left( \frac { 35 } { 4 } , \frac { 15 } { 4 } \right)\) The objective function is \(P = x + 3 y\) When the objective is to maximise \(x + 3 y\), the value of \(P\) is 24
When the objective is to minimise \(x + 3 y\), the value of \(P\) is 10

1. 1. Find the coordinates of \(A\) and \(B\).
   2. Determine the inequalities that define \(R\). An additional constraint, \(y \geqslant k x\), where \(k\) is a positive constant, is added to the linear programming problem.
2. Determine the greatest value of \(k\) for which this additional constraint does not affect the feasible region.