7.06b Slack variables: converting inequalities to equations

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.

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 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.

2 A baker can make apple cakes, banana cakes and cherry cakes.
The baker has exactly enough flour to make either 30 apple cakes or 20 banana cakes or 40 cherry cakes. The baker has exactly enough sugar to make either 30 apple cakes or 40 banana cakes or 30 cherry cakes. The baker has enough apples for 20 apple cakes, enough bananas for 25 banana cakes and enough cherries for 10 cherry cakes. The baker has an order for 30 cakes. The profit on each apple cake is 4 p , on each banana cake is 3 p and on each cherry cake is 2 p . The baker wants to maximise the profit on the order.

1. The availability of flour leads to the constraint \(4 a + 6 b + 3 c \leqslant 120\). Give the meaning of each of the variables \(a , b\) and \(c\) in this constraint.
2. Use the availability of sugar to give a second constraint of the form \(X a + Y b + Z c \leqslant 120\), where \(X , Y\) and \(Z\) are numbers to be found.
3. Write down a constraint from the total order size. Write down constraints from the availability of apples, bananas and cherries.
4. Write down the objective function to be maximised.
   [0pt] [You are not required to solve the resulting LP problem.]

3 Maggie is a personal trainer. She has twelve clients who want to lose weight. She decides to put some of her clients on weight loss programme \(X\), some on programme \(Y\) and the rest on programme \(Z\). Each programme involves a strict diet; in addition programmes \(X\) and \(Y\) involve regular exercise at Maggie's home gym. The programmes each last for one month. In addition to the diet, clients on programme \(X\) spend 30 minutes each day on the spin cycle, 10 minutes each day on the rower and 20 minutes each day on free weights. At the end of one month they can each expect to have lost 9 kg more than a client on just the diet. In addition to the diet, clients on programme \(Y\) spend 10 minutes each day on the spin cycle and 30 minutes each day on free weights; they do not use the rower. At the end of one month they can each expect to have lost 6 kg more than a client on just the diet. Because of other clients who use Maggie's home gym, the spin cycle is available for the weight loss clients for 180 minutes each day, the rower for 40 minutes each day and the free weights for 300 minutes each day. Only one client can use each piece of apparatus at any one time. Maggie wants to decide how many clients to put on each programme to maximise the total expected weight loss at the end of the month. She models the objective as follows. $$\text { Maximise } P = 9 x + 6 y$$

1. What do the variables \(x\) and \(y\) represent?
2. Write down and simplify the constraints on the values of \(x\) and \(y\) from the availability of each of the pieces of apparatus.
3. What other constraints and restrictions apply to the values of \(x\) and \(y\) ?
4. Use a graphical method to represent the feasible region for Maggie's problem. You should use graph paper and choose scales so that the feasible region can be clearly seen. Hence determine how many clients should be put on each programme.

5 An online shopping company selects some of its parcels to be checked before posting them. Each selected parcel must pass through three checks, which may be carried out in any order. One person must check the contents, another must check the postage and a third person must check the address. The parcels are classified according to the type of customer as 'new', 'occasional' or 'regular'. The table shows the time taken, in minutes, for each check on each type of parcel. The manager in charge of checking at the company has allocated each type of parcel a 'value' to represent how useful it is for generating additional income. In suitable units, these values are as follows. $$\text { new } = 8 \text { points } \quad \text { occasional } = 7 \text { points } \quad \text { regular } = 4 \text { points }$$ The manager wants to find out how many parcels of each type her department should check each hour, on average, to maximise the total value. She models this objective as $$\text { Maximise } P = 8 x + 7 y + 4 z .$$

1. What do the variables \(x , y\) and \(z\) represent?
2. Write down the constraints on the values of \(x , y\) and \(z\). The manager changes the value of parcels for regular customers to 0 points.
3. Explain what effect this has on the objective and simplify the constraints.
4. Use a graphical method to represent the feasible region for the manager's new problem. You should choose scales so that the feasible region can be clearly seen. Hence determine the optimal strategy. Now suppose that there is exactly one hour available for checking and the manager wants to find out how many parcels of each type her department should check in that hour to maximise the total value. The value of parcels for regular customers is still 0 points.
5. Find the optimal strategy in this situation.
6. Give a reason why, even if all the timings and values are correct, the total value may be less than this maximum. \section*{Question 6 is printed overleaf.}

6 Consider the following LP problem.

1. Since \(a \leqslant 6\) it follows that \(6 - a \geqslant 0\), and similarly for \(b\) and \(c\). Let \(6 - a = x\) (so that \(a\) is replaced by \(6 - x ) , 8 - b = y\) and \(10 - c = z\) to show that the problem can be expressed as $$\begin{array} { l l } \text { Maximise } & 2 x - 4 y + 5 z , \\ \text { subject to } & 3 x + 2 y - z \leqslant 14 , \\ & 2 x - 4 z \leqslant 7 , \\ & - 4 x + y \leqslant 4 , \\ \text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$
2. Represent the problem as an initial Simplex tableau. Perform two iterations of the Simplex algorithm, showing how each row was obtained. Hence write down the values of \(a , b\) and \(c\) after two iterations. Find the value of the objective for the original problem at this stage.
   [0pt] [10]

5 Roland Neede, the baker, is making cupcakes. He makes three sizes of cupcake: miniature, small and standard. Miniature cupcakes are sold in boxes of 24 and each cupcake uses 3 units of topping and 2 decorations. Small cupcakes are sold in boxes of 20 and each cupcake uses 5 units of topping and 3 decorations. Standard cupcakes are sold in boxes of 12 and each cupcake uses 7 units of topping and 4 decorations. Roland has no restriction on the amount of cake mix that he uses but he only has 5000 units of topping and 3000 decorations available. Cupcakes are only sold in complete boxes, and Roland assumes that he can sell all the boxes of cupcakes that he makes. Irrespective of size, each box of cupcakes sold will give Roland a profit of \(\pounds 1\). Roland wants to maximise his total profit. Let \(x\) denote the number of boxes of miniature cupcakes, \(y\) denote the number of boxes of small cupcakes and \(z\) denote the number of boxes of standard cupcakes that Roland makes.

1. Construct an objective function, \(P\), to be maximised.
2. By considering the number of units of topping used, show that \(18 x + 25 y + 21 z \leqslant 1250\).
3. Construct a similar constraint by considering the number of decorations used, simplifying the coefficients so that they are integers with no common factor.
4. Set up an initial Simplex tableau to represent Roland's problem.
5. Perform one iteration of the Simplex algorithm, choosing a pivot from the \(x\) column. Explain how the choice of pivot row was made and show how each row was calculated.
6. Write down the values of \(x , y\) and \(z\) from the first iteration of the Simplex algorithm. Hence find the maximum profit that Roland can make, remembering that cupcakes can only be sold in complete boxes. Calculate the number of units of topping and the number of decorations that are left over with this solution.
7. The constraint from the number of units of topping can be rewritten as \(18 P + 7 y + 3 z \leqslant 1250\). Form a similar expression for the constraint from the number of decorations. Use this to find the number of boxes of small cupcakes which maximises the profit when there are no decorations left over. Find the solution which gives the maximum profit using all the topping and all the decorations, and find the values of \(x , y\) and \(z\) for this solution. {}

5 Consider the linear programming problem:

1. Using slack variables, \(s \geqslant 0\) and \(t \geqslant 0\), express the two non-trivial constraints as equations.
2. Represent the problem as an initial Simplex tableau.
3. Explain why the pivot element must be chosen from the \(x\)-column and show the calculations that are used to choose the pivot.
4. Perform one iteration of the Simplex algorithm. Show how you obtained each row of your tableau and write down the values of \(x , y , z\) and \(P\) that result from this iteration. State whether or not this is the maximum feasible value of \(P\) and describe how you know this from the values in the tableau.

2 A landscape gardener is designing a garden. Part of the garden will be decking, part will be flowers and the rest will be grass. Let d be the area of decking, f be the area of flowers and g be the area of grass, all measured in \(\mathrm { m } ^ { 2 }\). The total area of the garden is \(120 \mathrm {~m} ^ { 2 }\) of which at least \(40 \mathrm {~m} ^ { 2 }\) must be grass. The area of decking must not be greater than the area of flowers. Also, the area of grass must not be more than four times the area of decking. Each square metre of grass will cost \(\pounds 5\), each square metre of decking will cost \(\pounds 10\) and each square metre of flowers will cost \(\pounds 20\). These costs include labour. The landscape gardener has been instructed to come up with the design that will cost the least.

1. Write down a constraint in d , f and g from the total area of the garden.
2. Explain why the constraint \(\mathrm { g } \leqslant 4 \mathrm {~d}\) is required.
3. Write down a constraint from the requirement that the area of decking must not be greater than the area of flowers.
4. Write down a constraint from the requirement that at least \(40 \mathrm {~m} ^ { 2 }\) of the garden must be grass and write down the minimum feasible values for each of \(d\) and \(f\).
5. Write down the objective function to be minimised.
6. Write down the resulting LP problem, using slack variables to express the constraints from parts (ii) and (iii) as equations.
   (You are not required to solve the resulting LP problem.)

6 Sandie makes tanning lotions which she sells to beauty salons. She makes three different lotions using the same basic ingredients but in different proportions. These lotions are called amber, bronze and copper. To make one litre of tanning lotion she needs one litre of fluid. This can either be water or water mixed with hempseed oil. One litre of amber lotion uses one litre of water, one litre of bronze lotion uses 0.8 litres of water and one litre of copper lotion uses 0.5 litres of water. Any remainder is made up of hempseed oil. Sandie has 40 litres of water and 7 litres of hempseed oil available.

1. By defining appropriate variables \(a , b\) and \(c\), show that the constraint on the amount of water available can be written as \(10 a + 8 b + 5 c \leqslant 400\).
2. Find a similar constraint on the amount of hempseed oil available. The tanning lotions also use two colourants which give two further availability constraints. Sandie wants to maximise her profit, \(\pounds P\). The problem can be represented as a linear programming problem with the initial Simplex tableau below. In this tableau \(s , t , u\) and \(v\) are slack variables.

   \(P\)	\(a\)	\(b\)	\(c\)	\(s\)	\(t\)	\(u\)	\(v\)	RHS
   1	-8	-7	-4	0	0	0	0	0
   0	10	8	5	1	0	0	0	400
   0	0	2	5	0	1	0	0	70
   0	2	4	1	0	0	1	0	176
   0	5	1	3	0	0	0	1	80

3. Use the initial Simplex tableau to write down two inequalities to represent the availability constraints for the colourants.
4. Write down the profit that Sandie makes on each litre of amber lotion that she sells.
5. Carry out one iteration of the Simplex algorithm, choosing a pivot from the \(a\) column. Show the operations used to calculate each row. After a second iteration of the Simplex algorithm the tableau is as given below.

   \(P\)	\(a\)	\(b\)	\(c\)	\(s\)	\(t\)	\(u\)	\(v\)	RHS
   1	0	0	14.3	0	2.7	0	1.6	317
   0	0	0	-16	1	-3	0	-2	30
   0	0	1	2.5	0	0.5	0	0	35
   0	0	0	-9.2	0	-1.8	1	-0.4	18
   0	1	0	0.1	0	-0.1	0	0.2	9

6. Explain how you know that the optimal solution has been achieved.
7. How much of each lotion should Sandie make and what is her maximum profit? Why might the profit be less than this?
8. If none of the other availabilities change, what is the least amount of water that Sandie needs to make the amounts of lotion found in part (vii)?

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]

5 John is reviewing his lifestyle, and in particular his leisure commitments. He enjoys badminton and squash, but is not sure whether he should persist with one or both. Both cost money and both take time. Playing badminton costs \(\pounds 3\) per hour and playing squash costs \(\pounds 4\) per hour. John has \(\pounds 11\) per week to spend on these activities. John takes 0.5 hours to recover from every hour of badminton and 0.75 hours to recover from every hour of squash. He has 5 hours in total available per week to play and recover.

1. Define appropriate variables and formulate two inequalities to model John's constraints.
2. Draw a graph to represent your inequalities. Give the coordinates of the vertices of your feasible region.
3. John is not sure how to define an objective function for his problem, but he says that he likes squash "twice as much" as badminton. Letting every hour of badminton be worth one "satisfaction point" define an objective function for John's problem, taking into account his "twice as much" statement.
4. Solve the resulting LP problem.
5. Given that badminton and squash courts are charged by the hour, explain why the solution to the LP is not a feasible solution to John's practical problem. Give the best feasible solution.
6. If instead John had said that he liked badminton more than squash, what would have been his best feasible solution?

1 Consider the following LP.
Maximise \(x + y\) subject to \(2 x + y < 44\) \(2 x + 3 y < 60\) \(10 x + 11 y < 244\) Part of a graphical solution is produced below and in your answer book.
Complete this graphical solution in your answer book. \includegraphics[max width=\textwidth, alt={}, center]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-2_1316_1346_916_356}

4 In a factory, two types of motor are made. Each motor of type X takes 10 man hours to make and each motor of type Y takes 12 man hours to make. In each week there are 200 man hours available. To satisfy customer demand, at least 5 of each type of motor must be made each week.
Once a motor has been started it must be completed; no unfinished motors may be left in the factory at the end of each week. When completed, the motors are put into a container for shipping. The volume of the container is \(7 \mathrm {~m} ^ { 3 }\). A type X motor occupies a volume of \(0.5 \mathrm {~m} ^ { 3 }\) and a type Y motor occupies a volume of \(0.3 \mathrm {~m} ^ { 3 }\).

1. Define appropriate variables and from the above information derive four inequalities which must be satisfied by those variables.
2. Represent your inequalities on a graph and shade the infeasible region. The profit on each type X is \(\pounds 100\) and on each type Y is \(\pounds 70\).
3. The weekly profit is to be maximised. Write down the objective function and find the maximum profit.
4. Because of absenteeism, the manager decides to organise the work in the factory on the assumption that there will be only 180 man hours available each week. Find the number of motors of each type that should now be made in order to maximise the profit.

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}

4 Two products are to be made from material that is supplied in a single roll, 20 m long and 1 m wide. The two products require widths of 47 cm and 32 cm respectively. Two ways of cutting lengths of material are shown in the plans below. \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-5_408_1538_520_269} \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-5_403_1533_952_274}

1. Given that there should be no unnecessary waste, draw one other cutting plan that might be used for a cut of length \(z\) metres.
2. Write down an expression for the total area that is wasted in terms of \(x , y\) and \(z\). All of the roll is to be cut, so \(x + y + z = 20\).
   There needs to be a total length of at least 20 metres of the material for the first product, the one requiring width 47 cm .
3. Write this as a linear constraint on the variables. There needs to be a total length of at least 24 metres of the material for the second product, the one requiring width 32 cm .
4. Write this as a linear constraint on the variables.
5. Formulate an LP in terms of \(x\) and \(y\) to minimise the area that is wasted. You will need to use the relationship \(x + y + z = 20\), together with your answers to parts (ii), (iii) and (iv).
6. Solve your LP graphically, and interpret the solution.

7. A leisure company owns boats of each of the following types: 2-person boats which are 4 metres long and weigh 50 kg .
4-person boats which are 3 metres long and weigh 20 kg .
8-person boats which are 14 metres long and weigh 100 kg .
The leisure company is willing to donate boats to a local sports club to accommodate up to 40 people at any one time. However, storage facilities mean that a maximum combined length of the boats must not be more than 75 metres. Also, it must be possible to transport all the boats on a single trailer which has a maximum load capacity of 600 kg . The club intends to hire the boats out to help with the cost of maintaining them. It plans to charge \(\pounds 10 , \pounds 12\) and \(\pounds 8\) per day, for the 2 -, 4 - and 8 -person boats respectively and wishes to maximise its daily revenue ( \(\pounds R\) ). Let \(x , y\) and \(z\) represent the number of 2-, 4- and 8-person boats respectively given to the club.

1. Model this as a linear programming problem simplifying your expressions so that they have integer coefficients.
   (4 marks)
2. Show that the initial tableau, when using the simplex algorithm, can be written as:

   Basic Variable	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	Value
   \(s\)	1	2	4	1	0	0	20
   \(t\)	4	3	14	0	1	0	75
   \(u\)	5	2	10	0	0	1	60
   \(R\)	\({ } ^ { - } 10\)	\({ } ^ { - } 12\)	\({ } ^ { - } 8\)	0	0	0	0

3. Explain the purpose of the variables \(s\), \(t\) and \(u\).
4. By increasing the value of \(y\) first, work out the next two complete tableaus.
5. Explain how you know that your final tableau gives an optimal solution and state this solution in practical terms. Sheet for answering question 3
   NAME \section*{Please hand this sheet in for marking}
   3. \includegraphics[max width=\textwidth, alt={}, center]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-08_2017_1051_462_244}
      \section*{Please hand this sheet in for marking}
   4. \(F \quad \bullet\)
      H •
      I •
      J •
      Complete matching:
      F •
      \section*{Sheet for answering question 5} NAME \section*{Please hand this sheet in for marking}
      \includegraphics[max width=\textwidth, alt={}]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-10_2398_643_248_1224}
      Sheet for answering question 6
      NAME \section*{Please hand this sheet in for marking}
   5. \(\_\_\_\_\)
   6. \(\_\_\_\_\)
   7. \(\_\_\_\_\)
   8. \includegraphics[max width=\textwidth, alt={}, center]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-11_592_1292_1078_312}
      Sheet for answering question 6 (cont.) \includegraphics[max width=\textwidth, alt={}, center]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-12_595_1299_351_312} \includegraphics[max width=\textwidth, alt={}, center]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-12_597_1298_1409_308}

6. A company makes lighting sets to be sold to stores for use during the Christmas period. As the product is only required at this time of year, all manufacturing takes place during September, October and November. The sets are delivered to stores at the end of each of these months. Any sets that have been made but do not need to be delivered at the end of each of September and October are put into storage which the company must pay for. Let \(x , y\) and \(z\) be the number of sets manufactured in September, October and November respectively. The demand for lighting sets and the relevant costs are shown in the table below. The company must be able to meet the demand at the end of each month and there must be no unsold articles at the end of November.

1. 1. Express \(z\) in terms of \(x\) and \(y\).
   2. Hence, find an expression for the total costs incurred in terms of \(x\) and \(y\). The company wishes to minimise its total costs by modelling this situation as a linear programming problem.
2. Find as inequalities the constraints that apply in addition to \(x \geq 800\) and \(y \geq 0\).
   (2 marks)
3. On graph paper, illustrate these inequalities and label clearly the feasible region.
   (4 marks)
4. Use your graph to solve the problem. You must state how many sets should be produced in each month and the total costs incurred by the company.
   (3 marks)

7. A fitness centre runs introductory courses aimed at the following groups of customers: Pensioners, who will be charged \(\pounds 4\) for a 2 -hour session.
Other adults, who will be charged \(\pounds 10\) for a 4 -hour session.
Children, who will be charged \(\pounds 2\) for a 1 -hour session.
Let the number of pensioners, other adults, and children be \(x , y\) and \(z\) respectively.
Regulations state that the number of pensioners, \(x\), must be at most 5 more than the number of adults, \(y\). There must also be at least twice as many adults, \(y\), as there are children, \(z\). The centre is able to supervise up to 40 person-hours each day at the centre and wishes to maximise the revenue \(( \pounds R )\) that can be earned each day from these sessions. You may assume that the places on any courses that the centre runs will be filled.

1. Modelling this situation as a linear programming problem, write down the constraints and objective function in terms of \(x , y\) and \(z\). Using the Simplex algorithm, the following initial tableau is obtained.
2. \(\_\_\_\_\)