7.06b Slack variables: converting inequalities to equations

3 Neil is refurbishing a listed building. There are two types of paint that he can use for the inside walls. One costs \(\pounds 1.45\) per \(\mathrm { m } ^ { 2 }\) and the other costs \(\pounds 0.95\) per \(\mathrm { m } ^ { 2 }\). He must paint the lower half of each wall in the more expensive paint. He has \(350 \mathrm {~m} ^ { 2 }\) of wall to paint. He has a budget of \(\pounds 400\) for wall paint. The more expensive paint is easier to use, and so Neil wants to use as much of it as possible. Initially, the following LP is constructed to help Neil with his purchasing of paint.
Let \(x\) be the number of \(\mathrm { m } ^ { 2 }\) of wall painted with the expensive paint.
Let \(y\) be the number of \(\mathrm { m } ^ { 2 }\) of wall painted with the less expensive paint. $$\begin{array} { l l } \text { Maximise } & P = x + y \\ \text { subject to } & 1.45 x + 0.95 y \leqslant 400 \\ & y - x \leqslant 0 \\ & x \geqslant 0 \\ & y \geqslant 0 \end{array}$$

1. Explain the purpose of the inequality \(y - x \leqslant 0\).
2. The formulation does not include the inequality \(x + y \geqslant 350\). State what this constraint models and why it has been omitted from the formulation.
3. Use the simplex algorithm to solve the LP. Pivot first on the "1" in the \(y\) column. Interpret your solution. The solution shows that Neil needs to buy more paint. He negotiates an increase in his budget to \(\pounds 450\).
4. Find the solution to the LP given by changing \(1.45 x + 0.95 y \leqslant 400\) to \(1.45 x + 0.95 y \leqslant 450\), and interpret your solution. Neil realises that although he now has a solution, that solution is not the best for his requirements.
5. Explain why the revised solution is not optimal for Neil. In order to move to an optimal solution Neil needs to change the objective of the LP and add another constraint to it.
6. Write down the new LP and the initial tableau for using two-stage simplex to solve it. Give a brief description of how to use two-stage simplex to solve it. \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-5_497_558_269_751}
   1. Solve the route inspection problem in the network above, showing the methodology you used to ensure that your solution is optimal. Show your route.
   2. Floyd's algorithm is applied to the same network to find the complete network of shortest distances. After three iterations the distance and route matrices are as follows.

      \cline { 2 - 6 } \multicolumn{1}{c|}{}	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)
      \(\mathbf { 1 }\)	48	24	28	11	15
      \(\mathbf { 2 }\)	24	8	4	11	16
      \(\mathbf { 3 }\)	28	4	8	7	12

2. A supplier has three warehouses, \(A , B\) and \(C\), at which there are 42,26 and 32 crates of a particular cereal respectively. Three supermarkets, \(D , E\) and \(F\), require 29, 47 and 24 crates of the cereal respectively. The supplier wishes to minimise the cost in meeting the requirements of the supermarkets. The cost, in pounds, of supplying one crate of the cereal from each warehouse to each supermarket is given in the table below. Formulate this information as a linear programming problem.

1. State your decision variables.
2. Write down the objective function in terms of your decision variables.
3. Write down the constraints, explaining what each one represents.

7. A distributor has six warehouses. At one point the distributor needs to move 25 lorries from warehouses \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\) to warehouses \(W _ { \mathrm { A } } , W _ { \mathrm { B } }\) and \(W _ { \mathrm { C } }\) for the minimum possible cost. The transportation tableau below shows the unit cost, in tens of pounds, of moving a lorry between two warehouses, and the relevant figures regarding the number of lorries available or required at each warehouse.

1. Write down the initial solution given by the north-west corner rule.
2. Obtain improvement indices for the unused routes.
3. Use the stepping-stone method to find an improved solution and state why it is degenerate.
4. Placing a zero in cell \(( 2,2 )\), show that the improved solution is optimal and state the transportation pattern.
5. Find the total cost of the optimal solution. \section*{Please hand this sheet in for marking}

   Stage	State	Destination	Cost	Total cost
   \multirow[t]{3}{*}{1}	Marquee	Deluxe Cuisine
   	Castle	Deluxe Castle Cuisine
   	Hotel	Deluxe Cuisine Hotel
   \multirow[t]{3}{*}{2}	Church	Marquee Castle Hotel
   	Castle	Marquee Castle
   	Registry Office	Marquee Castle Hotel
   3	Home	Castle Church Registry

   \section*{Please hand this sheet in for marking}

   1. 	A	B	\(C\)	D	\(E\)	\(F\)	\(G\)	\(H\)
      A	-	85	59	31	47	52	74	41
      B	85	-	104	73	51	68	43	55
      C	59	104	-	54	62	88	61	45
      D	31	73	54	-	40	59	65	78
      E	47	51	62	40	-	56	71	68
      \(F\)	52	68	88	59	56	-	53	49
      \(G\)	74	43	61	65	71	53	-	63
      \(H\)	41	55	45	78	68	49	63	-

   2. 	A	\(B\)	\(C\)	D	\(E\)	\(F\)	\(G\)	\(H\)
      A	-	85	59	31	47	52	74	41
      B	85	-	104	73	51	68	43	55
      C	59	104	-	54	62	88	61	45
      D	31	73	54	-	40	59	65	78
      E	47	51	62	40	-	56	71	68
      \(F\)	52	68	88	59	56	-	53	49
      G	74	43	61	65	71	53	-	63
      \(H\)	41	55	45	78	68	49	63	-

2. A school entrance examination consists of three papers - Mathematics, English and Verbal Reasoning. Three teams of markers are to mark one style of paper each. The table below shows the average time, in minutes, taken by each team to mark one script for each style of paper. It is desired that the scripts are marked as quickly as possible.
Formulate this information as a linear programming problem.

1. State your decision variables.
2. Write down the objective function in terms of your decision variables.
3. Write down the constraints, explaining what each one represents.

7. Mrs. Hartley organises the tennis fixtures for her school. On one day she has to send a team of 10 players to a match against school \(A\) and a team of 6 players to a match against school \(B\). She has to select the two teams from a squad that includes 7 players who live in village \(C\), 5 players who live in village \(D\) and 8 players who live in village \(E\). Having a small budget, Mrs. Hartley wishes to minimise the total amount spent on travel. The table below shows the cost, in pounds, for one player to travel from each village to each of the schools they are competing against.

1. Use the north-west corner rule to find an initial solution to this problem.
2. Obtain improvement indices for this initial solution.
3. Use the stepping-stone method to obtain an optimal solution and state the pattern of transportation that this represents. \section*{Please hand this sheet in for marking}

   Stage	State	Action
   \multirow[t]{2}{*}{1}	G	GI
   	H	HI
   \multirow[t]{3}{*}{2}	D	DG DH
   	E	EG \(E H\)
   	F	FG FH
   \multirow[t]{3}{*}{3}	A	AD \(A E\) \(A F\)
   	B	BD BE \(B F\)
   	C	CD CE CF
   4	Home	Home-A Home-B Home-C

   \section*{Please hand this sheet in for marking}
   1. \includegraphics[max width=\textwidth, alt={}, center]{4e50371b-0c1c-4b4e-b21d-60858ae160df-8_662_1025_529_440}
   2. Sheet for answering question 6 (cont.)

4. A furniture manufacturer has three workshops, \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\). Orders for rolls of fabric are to be placed with three suppliers, \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\). The supply, demand and cost per roll in pounds, according to which supplier each workshop uses, are given in the table below. Starting with the north-west corner method of finding an initial solution, find an optimal transportation pattern which minimises the total cost. State the final solution and its total cost.
(11 marks)

5. A car-hire firm has six branches in a region. Three of the branches, \(A , B\) and \(C\), have spare cars, whereas the other three, \(D , E\) and \(F\), require cars. The total number of cars required is equal to the number of cars available. The table below shows the cost in pounds of sending one car from each branch with spares to each branch needing more cars and the number of cars available or required by each branch.

1. Use the north-west corner method to obtain a possible pattern of moving cars and find its cost. The firm wishes to minimise the cost of redistributing the cars.
2. Calculate shadow costs for the pattern found in part (a) and improvement indices for each unoccupied cell.
3. State, with a reason, whether or not the pattern found in part (a) is optimal.

5. A carpet manufacturer has two warehouses, \(W _ { 1 }\) and \(W _ { 2 }\), which supply carpets for three sales outlets, \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\). At one point \(S _ { 1 }\) requires 40 rolls of carpet, \(S _ { 2 }\) requires 23 rolls of carpet and \(S _ { 3 }\) requires 37 rolls of carpet. At this point \(W _ { 1 }\) has 45 rolls in stock and \(W _ { 2 }\) has 40 rolls in stock. The following table shows the cost, in pounds, of transporting one roll from each warehouse to each sales outlet: The company's manager wishes to supply the 85 rolls that are in stock such that transportation costs are kept to a minimum.

1. Use the north-west corner rule to obtain an initial solution to the problem.
2. Calculate improvement indices for the unused routes.
3. Use the stepping-stone method to obtain an optimal solution.

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.

5 A garden centre sells mixed packs of flower bulbs.
Each pack contains bulbs which produce flowers of two different colours. The cost, in \(\pounds\), of a pack of each colour combination is shown in the table below.

1. Represent the information in the table as a network in which the vertices are the colours and the arcs are the packs available, weighted using the costs.
2. Construct a minimum spanning tree for the network. Taylor wants to buy at most four different packs of bulbs and to ensure that the packs include bulbs capable of producing all four flower colours. Taylor wants to minimise the total cost of these packs.
3. Determine whether or not buying the packs represented by the solution to part (b) solves Taylor's problem.
4. Represent Taylor's problem as an LP formulation, in which the variables are the number of packs of each type.

6 Beth wants to buy some tokens for use in a game.
Each token is either a silver token or a gold token.
Silver tokens and gold tokens have different points values in the game.
Silver tokens have a value of 1.5 points each.
Gold tokens have a value of 4 points each.
Beth already has 2 silver tokens and 1 gold token.
She also has \(\pounds 10\) that can be spent on buying more tokens.
Silver tokens can be bought for \(\pounds 2\) each.
Gold tokens can be bought for \(\pounds 6\) each.
After buying some tokens, Beth has \(x\) silver tokens and \(y\) gold tokens.
She now has a total of at least 5 tokens and no more than 8 tokens.

1. Set up an LP formulation in \(x\) and \(y\) for the problem of maximising the points value of tokens that she finishes with.
2. Use a graphical method to determine how many tokens of each type Beth should buy to maximise the points value of her tokens.

6 Tamsin is planning how to spend a day off. She will divide her time between walking the coast path, visiting a bird sanctuary and visiting the garden centre. Tamsin has given a value to each hour spent doing each activity. She wants to decide how much time to spend on each activity to maximise the total value of the activities. Tamsin's requirements are that she will spend:

• a total of exactly 6 hours on the three activities
• at most 3.5 hours walking the coast path
• at least as long at the bird sanctuary as at the garden centre
• at least 1 hour at the garden centre.
  1. 1. Explain why the maximum total value of the activities done is achieved when \(3 x + y\) is maximised.
     2. Show how the requirement that she spends at least as long at the bird sanctuary as at the garden centre leads to the constraint \(x + 2 y \geqslant 6\).
     3. Explain why there is no need to require that \(y \geqslant 0\).
  2. Represent the constraints graphically and hence find a solution to Tamsin's problem.

8 A sweet shop sells three different types of boxes of chocolate truffles. The cost of each type of box and the number of truffles of each variety in each type of box are given in the table below. Narendra wants to buy some boxes of truffles so that in total he has at least 20 milk chocolate, 10 plain chocolate, 16 white chocolate and 12 nutty chocolate truffles.

1. Explain why Narendra needs to buy at least four boxes of truffles.
2. Narendra decides that he will buy exactly four boxes. Determine the minimum number of Assorted boxes that Narendra must buy.
3. For your answer in part (ii),
   Narendra finds that the sweet shop has sold out of Assorted boxes, but he then spots that it also sells small boxes of milk chocolate truffles and small boxes of nutty chocolate truffles. Each small box contains 4 truffles (all of one variety) and costs \(\pounds 0.50\). He decides to buy \(x\) boxes of No Nuts and \(y\) boxes of Speciality, where \(x + y < 4\), so that he has at least 10 plain chocolate and 16 white chocolate truffles. He will then buy as many small boxes as he needs to give a total of at least 20 milk chocolate and 12 nutty chocolate truffles.
4. (a) Set up constraints on the values of \(x\) and \(y\).
   (b) Represent the feasible region graphically.
   (c) Hence determine the cheapest cost for Narendra. www.ocr.org.uk after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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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. 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.

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).

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.

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).