7.06d Graphical solution: feasible region, two variables

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.

6 Each day, a factory makes three types of widget: basic, standard and luxury. The widgets produced need three different components: type \(A\), type \(B\) and type \(C\). Basic widgets need 6 components of type \(A , 6\) components of type \(B\) and 12 components of type \(C\).
Standard widgets need 4 components of type \(A , 3\) components of type \(B\) and 18 components of type \(C\).
Luxury widgets need 2 components of type \(A , 9\) components of type \(B\) and 6 components of type \(C\).
Each day, there are 240 components of type \(A\) available, 300 of type \(B\) and 900 of type \(C\).
Each day, the factory must use at least twice as many components of type \(C\) as type \(B\).
Each day, the factory makes \(x\) basic widgets, \(y\) standard widgets and \(z\) luxury widgets.

1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), find four inequalities in \(x , y\) and \(z\) that model the above constraints, simplifying each inequality.
2. Each day, the factory makes the maximum possible number of widgets. On a particular day, the factory must make the same number of luxury widgets as basic widgets.
   1. Show that your answers in part (a) become $$2 x + y \leqslant 60 , \quad 5 x + y \leqslant 100 , \quad x + y \leqslant 50 , \quad y \geqslant x$$
   2. Using the axes opposite, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region.
   3. Find the total number of widgets made on that day.
   4. Find all possible combinations of the number of each type of widget made that correspond to this maximum number.

7 A builder needs some screws, nails and plugs. At the local store, there are packs containing a mixture of the three items. A DIY pack contains 200 nails, 200 screws and 100 plugs.
A trade pack contains 1000 nails, 1500 screws and 2500 plugs.
A DIY pack costs \(\pounds 2.50\) and a trade pack costs \(\pounds 15\).
The builder needs at least 5000 nails, 6000 screws and 4000 plugs.
The builder buys \(x\) DIY packs and \(y\) trade packs and wishes to keep his total cost to a minimum.

1. Formulate the builder's situation as a linear programming problem.
2. 1. On the grid opposite, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of an objective line.
   2. Use your diagram to find the number of each type of pack that the builder should buy in order to minimise his cost.
   3. Find the builder's minimum cost.

9 Ollyin is buying new pillows for his hotel. He buys three types of pillow: soft, medium and firm. He must buy at least 100 soft pillows and at least 200 medium pillows.
He must buy at least 400 pillows in total.
Soft pillows cost \(\pounds 4\) each. Medium pillows cost \(\pounds 3\) each. Firm pillows cost \(\pounds 4\) each.
He wishes to spend no more than \(\pounds 1800\) on new pillows.
At least \(40 \%\) of the new pillows must be medium pillows.
Ollyin buys \(x\) soft pillows, \(y\) medium pillows and \(z\) firm pillows.

1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), find five inequalities in \(x , y\) and \(z\) that model the above constraints.
2. Ollyin decides to buy twice as many soft pillows as firm pillows.
   1. Show that three of your answers in part (a) become $$\begin{aligned} 3 x + 2 y & \geqslant 800 \\ 2 x + y & \leqslant 600 \\ y & \geqslant x \end{aligned}$$
   2. On the grid opposite, draw a suitable diagram to represent Ollyin's situation, indicating the feasible region.
   3. Use your diagram to find the maximum total number of pillows that Ollyin can buy.
   4. Find the number of each type of pillow that Ollyin can buy that corresponds to your answer to part (b)(iii).
      \includegraphics[max width=\textwidth, alt={}]{1258a6d3-558a-46dc-a916-d71f71b175ff-20_2256_1707_221_153}

7 Paul is a florist. Every day, he makes three types of floral bouquet: gold, silver and bronze. Each gold bouquet has 6 roses, 6 carnations and 6 dahlias.
Each silver bouquet has 4 roses, 6 carnations and 4 dahlias.
Each bronze bouquet has 3 roses, 4 carnations and 4 dahlias.
Each day, Paul must use at least 420 roses and at least 480 carnations, but he can use at most 720 dahlias. Each day, Paul makes \(x\) gold bouquets, \(y\) silver bouquets and \(z\) bronze bouquets.

1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), find three inequalities in \(x , y\) and \(z\) that model the above constraints.
2. On a particular day, Paul makes the same number of silver bouquets as bronze bouquets.
   1. Show that \(x\) and \(y\) must satisfy the following inequalities. $$\begin{aligned} & 6 x + 7 y \geqslant 420 \\ & 3 x + 5 y \geqslant 240 \\ & 3 x + 4 y \leqslant 360 \end{aligned}$$
   2. Paul makes a profit of \(\pounds 4\) on each gold bouquet sold, a profit of \(\pounds 2.50\) on each silver bouquet sold and a profit of \(\pounds 2.50\) on each bronze bouquet sold. Each day, Paul sells all the bouquets he makes. Paul wishes to maximise his daily profit, \(\pounds P\). Draw a suitable diagram, on the grid opposite, to enable this problem to be solved graphically, indicating the feasible region and the direction of the objective line.
      (6 marks)
   3. Use your diagram to find Paul's maximum daily profit and the number of each type of bouquet he must make to achieve this maximum.
3. On another day, Paul again makes the same number of silver bouquets as bronze bouquets, but he makes a profit of \(\pounds 2\) on each gold bouquet sold, a profit of \(\pounds 6\) on each silver bouquet sold and a profit of \(\pounds 6\) on each bronze bouquet sold. Find Paul's maximum daily profit, and the number of each type of bouquet he must make to achieve this maximum.
   (3 marks) Turn over -

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?

5 Findlay is trying to get into his local swimming team. The coach will watch him swim and will then make his decision. Findlay must swim at least two lengths using each stroke and must swim at least 8 lengths in total, taking at most 10 minutes. Findlay needs to put together a routine that includes breaststroke, backstroke and butterfly. The table shows how Findlay expects to perform with each stroke. Findlay needs to work out how many lengths to swim using each stroke to maximise his expected total number of style marks.

1. Identify appropriate variables for Findlay's problem and write down the objective function, to be maximised, in terms of these variables.
2. Formulate a constraint for the total number of lengths swum, a constraint for the time spent swimming and constraints on the number of lengths swum using each stroke. Findlay decides that he will swim two lengths using butterfly. This reduces his problem to the following LP formulation: $$\begin{array} { l c } \text { maximise } & P = 2 x + y , \\ \text { subject to } & x + y \geqslant 6 , \\ & 4 x + y \leqslant 16 , \\ & x \geqslant 2 , y \geqslant 2 , \end{array}$$ with \(x\) and \(y\) both integers.
3. Use a graphical method to identify the feasible region for this problem. Write down the coordinates of the vertices of the feasible region and hence find the integer values of \(x\) and \(y\) that maximise \(P\).
4. Interpret your solution for Findlay.

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]{ccb12789-cd5f-40dc-9f10-f8bb45399580-4_919_917_322_575}

1. Obtain the four inequalities that define the feasible region.
2. Calculate the coordinates of the vertices of the feasible region, giving your values as fractions. The objective is to maximise \(P = x + 4 y\).
3. Calculate the value of \(P\) at each vertex of the feasible region. Hence write down the coordinates of the optimal point, and the corresponding value of \(P\). Suppose that the solution must have integer values for both \(x\) and \(y\).
4. Find the coordinates of the optimal point with integer-valued \(x\) and \(y\), and the corresponding value of \(P\). Explain how you know that this is the optimal solution.

6 Consider the following linear programming problem.

1. Represent the constraints graphically. Shade the regions where the inequalities are not satisfied and hence identify the feasible region.
2. Calculate the coordinates of the vertices of the feasible region, apart from the origin, and calculate the value of \(P\) at each vertex. Hence find the optimal values of \(x\) and \(y\), and state the maximum value of the objective.
3. Suppose that, additionally, \(x\) and \(y\) must both be integers. Find the maximum feasible value of \(y\) for every feasible integer value of \(x\). Calculate the value of \(P\) at each of these points and hence find the optimal values of \(x\) and \(y\) with this additional restriction. Tom wants to use the Simplex algorithm to solve the original (non-integer) problem. He reformulates it as follows.

   Maximise	\(\quad P = 5 x + 8 y\),
   subject to	\(3 x - 2 y \leqslant 12\),
   	\(3 x + 4 y \leqslant 30\),
   	\(- 3 x + 8 y \leqslant 24\),
   	\(x \geqslant 0 , y \geqslant 0\).

4. Explain why Tom needed to change the third constraint.
5. Represent the problem by an initial Simplex tableau.
6. Perform one iteration of the Simplex algorithm, choosing your pivot from the \(\boldsymbol { y }\) column. Show how the pivot row was used to calculate the other rows. Write down the values of \(x , y\) and \(P\) that result from this iteration. Perform a second iteration of the Simplex algorithm to achieve the optimum point.

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 William is making the bridesmaid dresses and pageboy outfits for his sister's wedding. He expects it to take 20 hours to make each bridesmaid dress and 15 hours to make each pageboy outfit. Each bridesmaid dress uses 8 metres of fabric. Each pageboy outfit uses 3 metres of fabric. The fabric costs \(\pounds 8\) per metre. Additional items cost \(\pounds 35\) for each bridesmaid dress and \(\pounds 80\) for each pageboy outfit. William's sister wants to have at least one bridesmaid and at least one pageboy. William has 100 hours available and must not spend more than \(\pounds 600\) in total on materials. Let \(x\) denote the number of bridesmaids and \(y\) denote the number of pageboys.

1. Show why the constraint \(4 x + 3 y \leqslant 20\) is needed and write down three more constraints on the values of \(x\) and \(y\), other than that they must be integers.
2. Plot the feasible region where all four constraints are satisfied. William's sister wants to maximise the total number of attendants (bridesmaids and pageboys) at her wedding.
3. Use your graph to find the maximum number of attendants.
4. William costs his time at \(\pounds 15\) an hour. Find, and simplify, an expression, in terms of \(x\) and \(y\), for the total cost (for all materials and William's time). Hence find, and interpret, the minimum cost solution to part (iii).

7 Consider the linear programming problem: $$\begin{array} { l l } \text { maximise } & P = 4 y - x , \\ \text { subject to } & x + 4 y \leqslant 22 , \\ & x + y \leqslant 10 , \\ & - x + 2 y \leqslant 8 , \\ \text { and } & x \geqslant 0 , y \geqslant 0 . \end{array}$$

1. Represent the constraints graphically, shading out the regions where the inequalities are not satisfied. Calculate the value of \(x\) and the value of \(y\) at each of the vertices of the feasible region. Hence find the maximum value of \(P\), clearly indicating where it occurs.
2. By introducing slack variables, represent the problem as an initial Simplex tableau and use the Simplex algorithm to solve the problem.
3. Indicate on your diagram for part (i) the points that correspond to each stage of the Simplex algorithm carried out in part (ii). \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS
   4736
   Decision Mathematics 1
   INSERT for Questions 4, 5 and 6
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   • This insert should be used to answer Questions 4, 5 and 6
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   • Write your Name, Centre Number and Candidate Number in the spaces provided at the top of this page.
   • Write your answers to Questions 4, 5 and 6
   • in the spaces provided in this insert, and attach it to your answer booklet.
   4

4. STEP	A	\(B\)	C
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5. STEP	A	\(B\)	C
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   5
6. \includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-7_661_1004_285_557}
7. Upper bound = \(\_\_\_\_\) km Lower bound = \(\_\_\_\_\) km
8. \(\_\_\_\_\) Best upper bound = \(\_\_\_\_\) km 6
9. \includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-8_668_1406_292_406} \includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-8_307_1342_1014_424}
   Least travel time = \(\_\_\_\_\) minutes Route: A- \(\_\_\_\_\) \(- D\)

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 \(\pounds 15\).

2 Consider the following linear programming problem.
Maximise $$\mathrm { P } = 6 x + 7 y$$ subject to $$\begin{aligned} & 2 x + 3 y \leqslant 9 \\ & 3 x + 2 y \leqslant 12 \\ & x \geqslant 0 \\ & y \geqslant 0 \end{aligned}$$

1. Use a graphical approach to solve the problem.
2. Give the optimal values of \(x , y\) and P when \(x\) and \(y\) are integers.

6 A company is planning its production of "MPowder" for the next three months.

• Over the next 3 months 20 tonnes must be produced.
• Production quantities must not be decreasing. The amount produced in month 2 cannot be less than the amount produced in month 1 , and the amount produced in month 3 cannot be less than the amount produced in month 2.
• No more than 12 tonnes can be produced in total in months 1 and 2.
• Production costs are \(\pounds 2000\) per tonne in month \(1 , \pounds 2200\) per tonne in month 2 and \(\pounds 2500\) per tonne in month 3.

The company planner starts to formulate an LP to find a production plan which minimises the cost of production: $$\begin{array} { l l } \text { Minimise } & 2000 x _ { 1 } + 2200 x _ { 2 } + 2500 x _ { 3 } \\ \text { subject to } & x _ { 1 } \geq 0 x _ { 2 } \geq 0 x _ { 3 } \geq 0 \\ & x _ { 1 } + x _ { 2 } + x _ { 3 } = 20 \\ & x _ { 1 } \leq x _ { 2 } \\ & \bullet \cdot \cdot \end{array}$$

1. Explain what the variables \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) represent, and write down two more constraints to complete the formulation.
2. Explain how the LP can be reformulated to: $$\begin{array} { l l } \text { Maximise } & 500 x _ { 1 } + 300 x _ { 2 } \\ \text { subject to } & x _ { 1 } \geq 0 x _ { 2 } \geq 0 \\ & x _ { 1 } \leq x _ { 2 } \\ & x _ { 1 } + 2 x _ { 2 } \leq 20 \\ & x _ { 1 } + x _ { 2 } \leq 12 \end{array}$$
3. Use a graphical approach to solve the LP in part (ii). Interpret your solution in terms of the company's production plan, and give the minimum cost.

4 An air charter company has the following rules for selling seats on a flight.

1. The total number of seats sold must not exceed 120.
2. There must be at least 100 seats sold, or the flight will be cancelled.
3. For every child seat sold there must be a seat sold for a supervising adult.
   1. Define two variables so that the three constraints can be formulated in terms of your variables. Formulate the three constraints in terms of your variables.
   2. Graph your three inequalities from part (i).
   The price for a child seat is \(\pounds 50\) and the price for an adult seat is \(\pounds 100\).
4. Find the maximum income available from the flight, and mark and label the corresponding point on your graph.
5. Find the minimum income available from a full plane, and mark and label the corresponding point on your graph.
6. Find the minimum income available from the flight, and mark and label the corresponding point on your graph.
7. At \(\pounds 100\) for an adult seat and \(\pounds 50\) for a child seat the company would prefer to sell 100 adult seats and no child seats rather than have a full plane with 60 adults and 60 children. What would be the minimum price for a child's seat for that not to be the case, given that the adult seat price remains at \(\pounds 100\) ?

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.

3 Solve the following LP problem graphically.
Maximise \(2 x + 3 y\) subject to \(\quad x + y \leqslant 11\) $$\begin{aligned} 3 x + 5 y & \leqslant 39 \\ x + 6 y & \leqslant 39 . \end{aligned}$$

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?