7.06b Slack variables: converting inequalities to equations

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.

6. A linear programming problem in \(x\) and \(y\) is described as follows. Minimise \(C = 2 x + 3 y\) subject to $$\begin{aligned} x + y & \geqslant 8 \\ x & < 8 \\ 4 y & \geqslant x \\ 3 y & \leqslant 9 + 2 x \end{aligned}$$

1. Add lines and shading to Diagram 1 in the answer book to represent these constraints.
2. Hence determine the feasible region and label it R .
3. Use the objective line (ruler) method to find the exact coordinates of the optimal vertex, V, of the feasible region. You must draw and label your objective line clearly.
4. Calculate the corresponding value of \(C\) at V . The objective is now to maximise \(2 x + 3 y\), where \(x\) and \(y\) are integers.
5. Write down the optimal values of \(x\) and \(y\) and the corresponding maximum value of \(2 x + 3 y\). A further constraint, \(y \leqslant k x\), where \(k\) is a positive constant, is added to the linear programming problem.
6. Determine the least value of \(k\) for which this additional constraint does not affect the feasible region.

8. Charlie needs to buy storage containers. There are two different types of storage container available, standard and deluxe. Standard containers cost \(\pounds 20\) and deluxe containers cost \(\pounds 65\). Let \(x\) be the number of standard containers and \(y\) be the number of deluxe containers. The maximum budget available is \(\pounds 520\)

1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint. Three further constraints are: $$\begin{aligned} x & \geqslant 2 \\ - x + 24 y & \geqslant 24 \\ 7 x + 8 y & \leqslant 112 \end{aligned}$$
2. Add lines and shading to Diagram 1 in the answer book to represent all four constraints. Hence determine the feasible region and label it R . The capacity of a deluxe container is \(50 \%\) greater than the capacity of a standard container. Charlie wishes to maximise the total capacity.
3. State an objective function, in terms of \(x\) and \(y\).
4. Use the objective line method to find the optimal vertex, V, of the feasible region. You must make your objective line clear and label the optimal vertex V.
5. Calculate the exact coordinates of vertex V.
6. Determine the number of each type of container that Charlie should buy. You must make your method clear and calculate the cost of purchasing the storage containers. Write your name here
   Final output \(\_\_\_\_\) (b)
   6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-22_807_1426_121_267} \captionsetup{labelformat=empty} \caption{Figure 5
   [0pt] [The total weight of the network is 384]}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-24_2651_1940_118_121} \includegraphics[max width=\textwidth, alt={}, center]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-25_2261_50_312_36} \section*{Q uestion 7 continued} (c) \(\_\_\_\_\) (d) \section*{Diagram 2} (Total 12 marks)
   □ 8.
   \includegraphics[max width=\textwidth, alt={}]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-26_1570_1591_260_189}
   Diagram 1 \section*{Q uestion 8 continued}
   \includegraphics[max width=\textwidth, alt={}]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-28_2646_1833_116_118}

5. \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. Write down the inequalities that form region \(R\).
2. Find the exact coordinates of the vertices of the feasible region. The objective is to maximise \(P\), where \(P = 2 x + 3 y\)
3. Use point testing to find the optimal vertex, V, of the feasible region. The objective is changed to maximise \(Q\), where \(Q = 2 x + \lambda y\) Given that \(\lambda\) is a constant and V is still the only optimal vertex of the feasible region,
4. find the range of possible values of \(\lambda\).

6. \begin{figure}[h] \end{figure} Figure 4 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The vertices of the feasible region are \(A ( 4,7 ) , B ( 5,3 ) , C ( - 1,5 )\) and \(D ( - 2,1 )\).

1. Determine the inequality that defines the boundary of \(R\) that passes through vertices \(A\) and \(C\), leaving your answer with integer coefficients only. The objective is to maximise \(P = 5 x + y\)
2. Find the coordinates of the optimal vertex and the corresponding value of \(P\). The objective is changed to maximise \(Q = k x + y\)
3. If \(k\) can take any value, find the range of values of \(k\) for which \(A\) is the only optimal vertex.

7 A company repairs and sells computer hardware, including monitors, hard drives and keyboards. Each monitor takes 3 hours to repair and the cost of components is \(\pounds 40\). Each hard drive takes 2 hours to repair and the cost of components is \(\pounds 20\). Each keyboard takes 1 hour to repair and the cost of components is \(\pounds 5\). Each month, the business has 360 hours available for repairs and \(\pounds 2500\) available to buy components. Each month, the company sells all of its repaired hardware to a local computer shop. Each monitor, hard drive and keyboard sold gives the company a profit of \(\pounds 80 , \pounds 35\) and \(\pounds 15\) respectively. The company repairs and sells \(x\) monitors, \(y\) hard drives and \(z\) keyboards each month. The company wishes to maximise its total profit. 7

1. Find five inequalities involving \(x , y\) and \(z\) for the company's problem.
   [0pt] [3 marks]
   7
2. (i) Find how many of each type of computer hardware the company should repair and sell each month.
   7 (b) (ii) Explain how you know that you had reached the optimal solution in part (b) (i).
   7 (b) (iii) The local computer shop complains that they are not receiving one of the types of computer hardware that the company repairs and sells. Using your answer to part (b) (i), suggest a way in which the company's problem can be modified to address the complaint.
   [0pt] [1 mark]

4. The manager of a factory is planning the production schedule for the next three weeks for a range of cabinets. The following constraints apply to the production schedule.

• The total number of cabinets produced in week 3 cannot be fewer than the total number produced in weeks 1 and 2
• At most twice as many cabinets must be produced in week 3 as in week 2
• The number of cabinets produced in weeks 2 and 3 must, in total, be at most 125

The production cost for each cabinet produced in weeks 1,2 and 3 is \(\pounds 250 , \pounds 275\) and \(\pounds 200\) respectively.
The factory manager decides to formulate a linear programming problem to find a production schedule that minimises the total cost of production. The objective is to minimise \(250 x + 275 y + 200 z\)

1. Explain what the variables \(x , y\) and \(z\) represent.
2. Write down the constraints of the linear programming problem in terms of \(x , y\) and \(z\). Due to demand, exactly 150 cabinets must be produced during these three weeks. This reduces the constraints to $$\begin{gathered} x + y \leqslant 75 \\ x + 3 y \geqslant 150 \\ x \geqslant 25 \\ y \geqslant 0 \end{gathered}$$ which are shown in Diagram 1 in the answer book.
   Given that the manager does not want any cabinets left unfinished at the end of a week,
3. 1. use a graphical approach to solve the linear programming problem and hence determine the production schedule which minimises the cost of production. You should make your method and working clear.
   2. Find the minimum total cost of the production schedule.

3. Donald plans to bake and sell cakes. The three types of cake that he can bake are brownies, flapjacks and muffins. Donald decides to bake 48 brownies and muffins in total.
Donald decides to bake at least 5 brownies for every 3 flapjacks.
At most \(40 \%\) of the cakes will be muffins.
Donald has enough ingredients to bake 60 brownies or 45 flapjacks or 35 muffins.
Donald plans to sell each brownie for \(\pounds 1.50\), each flapjack for \(\pounds 1\) and each muffin for \(\pounds 1.25\) He wants to maximise the total income from selling the cakes. Let \(x\) represent the number of brownies, let \(y\) represent the number of flapjacks and let \(z\) represent the number of muffins that Donald will bake. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.

4. \begin{figure}[h] \end{figure} Figure 4 shows three of the six constraints for a linear programming problem in \(x\) and \(y\) The unshaded region and its boundaries satisfy these three constraints.

1. State these three constraints as simplified inequalities with integer coefficients. The variables \(x\) and \(y\) represent the number of orange fish and the number of blue fish, respectively, that are to be kept in an aquarium. The number of fish in the aquarium is subject to these three further constraints
   • there must be at least one blue fish
   • the orange fish must not outnumber the blue fish by more than ten
   • there must be no more than five blue fish for every orange fish
   • Write each of these three constraints as a simplified inequality with integer coefficients.
   • Represent these three constraints by adding lines and shading to Diagram 1 in the answer book, labelling the feasible region, \(R\)
   The total value (in pounds) of the fish in the aquarium is given by the objective function $$\text { Maximise } P = 3 x + 5 y$$
2. 1. Use the objective line method to determine the optimal point of the feasible region, giving its coordinates as exact fractions.
   2. Hence find the maximum total value of the fish in the aquarium, stating the optimal number of orange fish and the optimal number of blue fish. \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Please check the examination details below before entering your candidate information}

      Candidate surname								Other names
      Centre Number				Candidate Number
      □			□

      \end{table} \section*{Pearson Edexcel Level 3 GCE} \section*{Friday 17 May 2024} Afternoon \section*{Further Mathematics} Advanced Subsidiary
      Further Mathematics options
      27: Decision Mathematics 1
      (Part of options D, F, H and K) \section*{D1 Answer Book} Do not return the question paper with the answer book.
      1. \(\begin{array} { l l l l l l l l l l l } 4 & 6.5 & 7 & 1.3 & 2 & 5 & 1.5 & 6 & 4.5 & 6 & 1 \end{array}\) 2.
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_435_815_392_463}
      \section*{Diagram 1} Use this diagram only if you need to redraw your activity network. \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_442_820_2043_458} Copy of Diagram 1

      VJYV SIHI NI JIIYM ION OC

      V346 SIHI NI JLIYM ION OC

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      Key: \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-13_1217_1783_451_236} \captionsetup{labelformat=empty} \caption{Diagram 2}
      \end{figure} 3. \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-14_2463_1240_339_465}
      Shortest route from A to M:
      Length of shortest route from A to M:
      
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-16_3038_2264_0_0}
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-17_1103_1247_397_512}
      \section*{Diagram 1} \section*{There is a copy of Diagram 1 on page 11 if you need to redraw your graph.}

      VJYV SIHI NI JIIIM ION OC

      V341 S1H1 NI JLIYM ION OA

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      Use this diagram only if you need to redraw your graph. \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-19_1108_1252_1606_509} Copy of Diagram 1

1. Jonathan makes two types of information pack for an event, Standard and Value.

Each Standard pack contains 25 posters and 500 flyers.
Each Value pack contains 15 posters and 800 flyers.
He must use at least 150000 flyers.
Between \(35 \%\) and \(65 \%\) of the packs must be Standard packs.
Posters cost 20p each and flyers cost 4p each.
Jonathan wishes to minimise his costs.
Let x and y represent the number of Standard packs and Value packs produced respectively.
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. \section*{(Total for Question 5 is 5 marks)} TOTAL IS 40 MARKS

8. Susie is preparing for a triathlon event that is taking place next month. A triathlon involves three activities: swimming, cycling and running. Susie decides that in her training next week she should

• maximise the total time spent cycling and running
• train for at most 39 hours
• spend at least \(40 \%\) of her time swimming
• spend a total of at least 28 hours of her time swimming and running

Susie needs to determine how long she should spend next week training for each activity. Let

• \(x\) represent the number of hours swimming
• \(y\) represent the number of hours cycling
• \(z\) represent the number of hours running
  1. Formulate the information above as a linear programming problem. State the objective and list the constraints as simplified inequalities with integer coefficients.
     (5)

Susie decides to solve this linear programming problem by using the two-stage Simplex method.

Set up an initial tableau for solving this problem using the two-stage Simplex method. As part of your solution you must show how

The following tableau \(T\) is obtained after one iteration of the second stage of the two-stage Simplex method.

b.v.	\(x\)	\(y\)	\(z\)	\(s _ { 1 }\)	\(\mathrm { S } _ { 2 }\)	\(S _ { 3 }\)	Value
\(y\)	0	1	0	1	0	1	11
\(s _ { 2 }\)	0	0	5	-2	1	-5	62
\(x\)	1	0	1	0	0	-1	28
\(P\)	0	0	-1	1	0	1	11

Obtain a suitable pivot for a second iteration. You must give reasons for your answer.

Starting from tableau \(T\), solve the linear programming problem by performing one further iteration of the second stage of the two-stage Simplex method. You should make your method clear by stating the row operations you use.

3. The table below shows the stock held at each supply point and the stock required at each demand point in a standard transportation problem. The table also shows the cost, in pounds, of transporting the stock from each supply point to each demand point. The problem is partially described by the linear programming formulation below.
Let \(x _ { i j }\) be the number of units transported from i to j $$\begin{aligned} & \text { where } \quad i \in \{ A , B , C , D \} \\ & \quad j \in \{ Q , R , S \} \text { and } x _ { i j } \geqslant 0 \\ & \text { Minimise } P = 23 x _ { A Q } + 18 x _ { A R } + 12 x _ { A S } + 8 x _ { B Q } + 10 x _ { B R } + 14 x _ { B S } \\ & \quad + 11 x _ { C Q } + 14 x _ { C R } + 21 x _ { C S } + 19 x _ { D Q } + 15 x _ { D R } + 11 x _ { D S } \end{aligned}$$

1. Write down, as inequalities, the constraints of the linear program.
2. Use the north-west corner method to obtain an initial solution to this transportation problem.
3. Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear.
4. Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and the

3. The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H , to three sales points, \(\mathrm { A } , \mathrm { B }\) and C . It also shows the stock held at each supply point and the amount required at each sales point.
A minimum cost solution is required.

1. Explain why it is necessary to add a dummy demand point.
2. On Table 1 in the answer book, insert appropriate values in the dummy demand column, D. After finding an initial feasible solution and applying one iteration of the stepping-stone method, the table becomes

   	\(A\)	\(B\)	\(C\)	\(D\)
   \(E\)	21
   \(F\)	19	13
   \(G\)		6	23
   \(H\)	5			18

3. Starting with GD as the next entering cell, perform two further iterations of the stepping-stone method to obtain an improved solution. You must make your method clear by showing your routes and stating the

1. Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S.

Each task must be assigned to just one worker and each worker can do only one task.
Worker B cannot be assigned to task Q and worker D cannot be assigned to task R.
The amount, in pounds, that each worker would earn when assigned to each task is shown in the table below. The Hungarian algorithm can be used to find the maximum total amount that would be earned by the four workers.

1. 1. Explain how to modify the table so that the Hungarian algorithm could be applied.
   2. Modify the table as described in (a)(i).
2. Formulate the above situation as a linear programming problem. You must define the decision variables and make the objective function and constraints clear.
   

2. A company has three factories, \(\mathrm { A } , \mathrm { B }\) and C . It supplies mattresses to three shops, \(\mathrm { D } , \mathrm { E }\) and F . The table shows the transportation cost, in pounds, of moving one mattress from each factory to each shop. It also shows the number of mattresses available at each factory and the number of mattresses required at each shop. A minimum cost solution is required.

1. Use the north-west corner method to obtain an initial solution.
2. Show how the transportation algorithm is used to solve this problem. You must state, at each appropriate step, the

1. Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S.

Each worker must be assigned to at most one task and each task must be done by just one worker. The amount, in pounds, that each worker would earn while assigned to each task is shown in the table below. The Hungarian algorithm is to be used to find the maximum total amount which may be earned by the four workers.

1. Explain how the table should be modified.
2. Reducing rows first, use the Hungarian algorithm to obtain an allocation which maximises the total earnings, stating how each table was formed.
3. Formulate the problem as a linear programming problem. You must define your decision variables and make your objective function and constraints clear.
   

6 An online magazine consists of an editorial, articles, reviews and advertisements.
The magazine must have a total of at least 12 pages. The editorial always takes up exactly half a page. There must be at least 3 pages of articles and at most 1.5 pages of reviews. At least a quarter but fewer than half of the pages in the magazine must be used for advertisements. Let \(x\) be the number of pages used for articles, \(y\) be the number of pages used for reviews and \(z\) be the number of pages used for advertisements. The constraints on the values of \(x , y\) and \(z\) are: $$\begin{aligned} & x + y + z \geqslant 11.5 \\ & x \geqslant 3 \\ & y \leqslant 1.5 \\ & 2 x + 2 y - 2 z + 1 \geqslant 0 \\ & 2 x + 2 y - 6 z + 1 \leqslant 0 \\ & y \geqslant 0 \end{aligned}$$

1. (a) Explain why \(x + y + z \geqslant 11.5\).
   (b) Explain why only one non-negativity constraint is needed.
   (c) Show that the requirement that at least one quarter of the pages in the magazine must be used for advertisements leads to the constraint \(2 x + 2 y - 6 z + 1 \leqslant 0\). Advertisements bring in money but are not popular with the subscribers. The editor decides to limit the number of pages of advertisements to at most four.
2. Graph the feasible region in the case when \(z = 4\) using the axes in the Printed Answer Booklet. To be successful the magazine needs to maximise the number of subscribers.
   The editor has found that when \(z \leqslant 4\) the expected number of subscribers is given by \(P = 300 x + 400 y\).
3. (a) What is the maximum expected number of subscribers when \(z = 4\) ?
   (b) By first considering the feasible region for \(z = k\), where \(k \leqslant 4\), find an expression for the maximum number of subscribers in terms of \(k\). \section*{END OF QUESTION PAPER} \section*{OCR} \section*{Oxford Cambridge and RSA}

7. A tailor makes two types of garment, \(A\) and \(B\). He has available \(70 \mathrm {~m} ^ { 2 }\) of cotton fabric and \(90 \mathrm {~m} ^ { 2 }\) of woollen fabric. Garment \(A\) requires \(1 \mathrm {~m} ^ { 2 }\) of cotton fabric and \(3 \mathrm {~m} ^ { 2 }\) of woollen fabric. Garment \(B\) requires \(2 \mathrm {~m} ^ { 2 }\) of each fabric. The tailor makes \(x\) garments of type \(A\) and \(y\) garments of type \(B\).

1. Explain why this can be modelled by the inequalities $$\begin{aligned} & x + 2 y \leq 70 \\ & 3 x + 2 y \leq 90 \\ & x \geq 0 , y \geq 0 \end{aligned}$$ The tailor sells type \(A\) for \(\pounds 30\) and type \(B\) for \(\pounds 40\). All garments made are sold. The tailor wishes to maximise his total income.
2. Set up an initial Simplex tableau for this problem.
3. Solve the problem using the Simplex algorithm. Figure 4 shows a graphical representation of the feasible region for this problem. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-008_686_1277_1319_453} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
4. Obtain the coordinates of the points A, \(C\) and \(D\).
5. Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4. 6689 Decision Mathematics 1 (New Syllabus) Order of selecting edges
   Final tree
   (b) Minimum total length of cable
   Question 4 to be answered on this page
   (a) \(A\)
   • Monday (M) \(B\) ◯
   • Tuesday (Tu) \(C \odot\)
   • Wednesday (W) \(D\) ◯
   • Thursday (Th) \(E\) -
   • Friday (F)
     (b)
     (c)
   Question 5 to be answered on this page
   Key
   (a) Early
   Time
   Late
   Time \includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_433_227_534_201} \(F ( 3 )\) \includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_117_222_1016_992}
   H(4) K(6)
   (b) Critical activities
   Length of critical path \(\_\_\_\_\) (c) \includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_492_1604_1925_266} Question 6 to be answered on pages 4 and 5
   (a)
   1. SAET \(\_\_\_\_\)
   2. SBDT \(\_\_\_\_\)
   3. SCFT \(\_\_\_\_\)
   
   (b) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-012_691_1307_893_384} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure} (c) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-013_699_1314_167_382} \captionsetup{labelformat=empty} \caption{Diagram 2}
   \end{figure} Flow augmenting routes
   (d) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-013_693_1314_1368_382} \captionsetup{labelformat=empty} \caption{Diagram 3}
   \end{figure} (e) \(\_\_\_\_\)

8 Each day, a factory makes three types of hinge: basic, standard and luxury. The hinges produced need three different components: type \(A\), type \(B\) and type \(C\). Basic hinges need 2 components of type \(A , 3\) components of type \(B\) and 1 component of type \(C\). Standard hinges need 4 components of type \(A , 2\) components of type \(B\) and 3 components of type \(C\). Luxury hinges need 3 components of type \(A\), 4 components of type \(B\) and 5 components of type \(C\). Each day, there are 360 components of type \(A\) available, 270 of type \(B\) and 450 of type \(C\). Each day, the factory must use at least 720 components in total.
Each day, the factory must use at least \(40 \%\) of the total components as type \(A\).
Each day, the factory makes \(x\) basic hinges, \(y\) standard hinges and \(z\) luxury hinges.
In addition to \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\), find five inequalities, each involving \(x , y\) and \(z\), which must be satisfied. Simplify each inequality where possible.

8 A factory packs three different kinds of novelty box: red, blue and green. Each box contains three different types of toy: \(\mathrm { A } , \mathrm { B }\) and C . Each red box has 2 type A toys, 3 type B toys and 4 type C toys.
Each blue box has 3 type A toys, 1 type B toy and 3 type C toys.
Each green box has 4 type A toys, 5 type B toys and 2 type C toys.
Each day, the maximum number of each type of toy available to be packed is 360 type A, 300 type B and 400 type C. Each day, the factory must pack more type A toys than type B toys.
Each day, the total number of type A and type B toys that are packed must together be at least as many as the number of type C toys that are packed. Each day, at least \(40 \%\) of the total toys that are packed must be type C toys.
Each day, the factory packs \(x\) red boxes, \(y\) blue boxes and \(z\) green boxes.
Formulate the above situation as 6 inequalities, in addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), simplifying your answers.

8 [Figure 2, printed on the insert, is provided for use in this question.]
A company makes two types of boxes of chocolates, executive and luxury.
Every hour the company must make at least 15 of each type and at least 35 in total.
Each executive box contains 20 dark chocolates and 12 milky chocolates.
Each luxury box contains 10 dark chocolates and 18 milky chocolates.
Every hour the company has 600 dark chocolates and 600 milky chocolates available.
The company makes a profit of \(\pounds 1.50\) on each executive box and \(\pounds 1\) on each luxury box.
The company makes and sells \(x\) executive boxes and \(y\) luxury boxes every hour.
The company wishes to maximise its hourly profit, \(\pounds P\).

1. Show that one of the constraints leads to the inequality \(2 x + 3 y \leqslant 100\).
2. Formulate the company's 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 hourly profit.