7.06b Slack variables: converting inequalities to equations

4. A company produces \(x _ { 1 }\) finished articles at the end of January, \(x _ { 2 }\) finished articles at the end of February, \(x _ { 3 }\) finished articles at the end of March, \(x _ { 4 }\) finished articles at the end of April. Other details for each month are as follows: The cost of storing each finished but unsold article is \(\pounds 500\) per month. Thus, for example, any article unsold at the end of January would incur a \(\pounds 500\) charge if it is stored until the end of February or a \(\pounds 1000\) charge if it is stored until the end of March. There must be no unsold stock at the end of April.
The selling price of each article is \(\pounds 4000\) and the total profit ( \(\pounds P\) ) must be maximised.

1. Rewrite \(x _ { 4 }\) in terms of the other 3 variables.
2. Show that the total cost incurred \(( \pounds C )\) is given by: $$C = 600 x _ { 1 } + 900 x _ { 2 } + 200 x _ { 3 } + 1125000$$
3. Hence, show that \(P = { } ^ { - } 600 x _ { 1 } - 900 x _ { 2 } - 200 x _ { 3 } + 2875000\).
4. Three of the constraints operating can be expressed as \(x _ { 1 } \geq 200\), \(x _ { 2 } \geq 0\) and \(x _ { 3 } \geq 0\). Write down inequalities representing two further constraints.
   (2 marks)
5. Explain why it is not appropriate to use a graphical method to solve this problem.
6. An employee of the company wishes to use the Simplex algorithm to solve the problem. He tries to generate an initial tableau with \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) as the non-basic variables. Explain why this is not appropriate and explain what he should do instead. You are not required to generate an initial tableau or to solve the problem.
   (2 marks)

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

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

5

1. Display the following linear programming problem in a Simplex tableau.
   Maximise \(\quad P = x - 2 y + 3 z\) subject to $$\begin{array} { r } x + y + z \leqslant 16 \\ x - 2 y + 2 z \leqslant 17 \\ 2 x - y + 2 z \leqslant 19 \end{array}$$ and \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\).
2. 1. The first pivot to be chosen is from the \(z\)-column. Identify the pivot and explain why this particular value is chosen.
   2. Perform one iteration of the Simplex method.
3. 1. Perform one further iteration.
   2. Interpret the tableau that you obtained in part (c)(i) and state the values of your slack variables.

3

1. Given that \(k\) is a constant, display the following linear programming problem in a Simplex tableau. $$\begin{array} { l l } \text { Maximise } & P = 6 x + 5 y + 3 z \\ \text { subject to } & x + 2 y + k z \leqslant 8 \\ & 2 x + 10 y + z \leqslant 17 \\ & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
2. 1. Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the \(x\)-column as pivot.
   2. Given that the maximum value of \(P\) has not been achieved after this first iteration, find the range of possible values of \(k\).
3. In the case where \(k = - 1\), perform one further iteration and interpret your final tableau.
   
   \includegraphics[max width=\textwidth, alt={}]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-07_2484_1707_223_155}

4 A linear programming problem involving variables \(x , y\) and \(z\) is to be solved. The objective function to be maximised is \(P = 2 x + 6 y + k z\), where \(k\) is a constant. The initial Simplex tableau is given below.

1. In addition to \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\), write down three inequalities involving \(x , y\) and \(z\) for this problem.
2. 1. By choosing the first pivot from the \(\boldsymbol { y }\)-column, perform one iteration of the Simplex method.
   2. Given that the optimal value has not been reached, find the possible values of \(k\).
3. In the case when \(k = 20\) :
   1. perform one further iteration;
   2. interpret the final tableau and state the values of the slack variables.

6

1. Display the following linear programming problem in a Simplex tableau.
   Maximise \(\quad P = 4 x + 3 y + z\) subject to $$\begin{aligned} & 2 x + y + z \leqslant 25 \\ & x + 2 y + z \leqslant 40 \\ & x + y + 2 z \leqslant 30 \end{aligned}$$ and \(x \geqslant 0 , \quad y \geqslant 0 , \quad z \geqslant 0\).
2. The first pivot to be chosen is from the \(x\)-column. Perform one iteration of the Simplex method.
3. 1. Perform one further iteration.
   2. Interpret your final tableau and state the values of your slack variables.

3. Three depots, F, G and H, supply petrol to three service stations, S, T and U. The table gives the cost, in pounds, of transporting 1000 litres of petrol from each depot to each service station. F, G and H have stocks of 540000,789000 and 673000 litres respectively.
S, T and U require 257000,348000 and 412000 litres respectively. The total cost of transporting the petrol is to be minimised. Formulate this problem as a linear programming problem. Make clear your decision variables, objective function and constraints.

9. T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne. The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit. Let \(x , y\) and \(z\) be the number of tonnes of Morning, Afternoon and Evening blend produced each week.

1. Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities.
   (4) An initial Simplex tableau for the above situation is

   Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
   \(r\)	3	2	4	1	0	0	35
   \(s\)	1	3	2	0	1	0	20
   \(t\)	2	4	3	0	0	1	24
   \(P\)	- 4	- 5	- 3	0	0	0	0

2. Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage. T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.
3. Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.

1. Four workers, Chris (C), James (J), Katie (K) and Nicky (N), are to be allocated to four tasks, 1, 2, 3 and 4. Each worker is to be allocated to one task and each task must be allocated to one worker.

The profit, in pounds, resulting from allocating each worker to each task, is shown in the table below. The profit is to be maximised.

1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total profit. You must make your method clear and show the table after each stage.
2. State which worker should be allocated to each task and the resulting total profit made.
   

3. Table 1 below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to four demand points \(1,2,3\) and 4 . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required. \begin{table}[h] \end{table} Table 2 shows an initial solution given by the north-west corner method.
Table 3 shows some of the improvement indices for this solution. \begin{table}[h] \end{table} \begin{table}[h] \end{table}

1. Explain why a zero has been placed in cell B3 in Table 2.
   (1)
2. Calculate the shadow costs and the missing improvement indices and enter them into Table 3 in your answer book.
3. Taking the most negative improvement index to indicate the entering cell, state the steppingstone route that should be used to obtain the next solution. You must state your entering cell and exiting cell.

2. The table shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to each of three demand points, 1, 2 and 3 . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required.

1. Use the north-west corner method to obtain an initial solution.
   (1)
2. Taking D1 as the entering cell, use the stepping stone method to find an improved solution. Make your route clear.
   (2)
3. Perform one further iteration of the stepping stone method to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, route, entering cell and exiting cell.
4. Determine whether your current solution is optimal, giving a reason for your answer.

1. Four bakeries, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , supply bread to four supermarkets, \(\mathrm { P } , \mathrm { Q } , \mathrm { R }\) and S . The table gives the cost, in pounds, of transporting one lorry load of bread from each bakery to each supermarket. It also shows the number of lorry loads of bread at each bakery and the number of lorry loads of bread required at each supermarket. The total cost of transportation is to be minimised.

1. Use the north-west corner method to obtain a possible solution. A partly completed table of improvement indices is given in Table 1 in the answer book.
2. Complete Table 1.
3. Taking the most negative improvement index to indicate the entering cell, use the steppingstone method once to obtain an improved solution. You must make your route clear and state your entering cell and exiting cell.
4. State the cost of your improved solution.
   

6. Four workers, A, B, C and D, are to be assigned to four tasks, 1, 2, 3 and 4. Each worker must be assigned to just one task and each task must be done by just one worker. Worker C cannot do task 4 and worker D cannot do task 1. The cost of assigning each worker to each task is shown in the table below. The total cost is to be minimised. Formulate the above situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.

6. Three warehouses, \(\mathrm { P } , \mathrm { Q }\) and R , supply washing machines to four retailers, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . The table gives the cost, in pounds, of transporting a washing machine from each warehouse to each retailer. It also shows the number of washing machines held at each warehouse and the number of washing machines required by each retailer. The total cost of transportation is to be minimised. Formulate this transportation problem as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.
You do not need to solve this problem.
(Total 7 marks)

7. Four salespersons \(A , B , C\) and \(D\) are to be sent to visit four companies 1,2,3 and 4. Each salesperson will visit exactly one company, and all companies will be visited.
Previous sales figures show that each salesperson will make sales of different values, depending on the company that they visit. These values (in \(\pounds 10000\) s) are shown in the table below.

1. Use the Hungarian algorithm to obtain an allocation that maximises the sales. You must make your method clear and show the table after each stage.
2. State the value of the maximum sales.
3. Show that there is a second allocation that maximises the sales.

4 Four workers, \(A , B , C\) and \(D\), are to be allocated, one to each of the four jobs, \(W , X , Y\) and \(Z\). The table shows how much each worker would charge for each job. \includegraphics[max width=\textwidth, alt={}, center]{9c9b1a42-8d16-446a-85a1-4c08e5e368be-3_401_846_1745_642}

1. What is the total cost of the four jobs if \(A\) does \(W , B\) does \(X , C\) does \(Y\) and \(D\) does \(Z\) ?
2. Apply the Hungarian algorithm to the table, reducing rows first. Show all your working and explain each step. Give the resulting allocation and the total cost of the four jobs with this allocation.
3. What problem does the Hungarian algorithm solve?

1 Four friends have rented a house and need to decide who will have which bedroom. The table below shows how each friend rated each room, so the higher the rating the more the room was liked. The Hungarian algorithm is to be used to find the matching with the greatest total. Before the Hungarian algorithm can be used, each rating is subtracted from 6.

1. Explain why the ratings could not be used as given in the table.
2. Apply the Hungarian algorithm, reducing rows first, to match the friends to the rooms. You must show your working and say how each matrix was formed.

6 Rowena and Colin play a game in which each chooses a letter. The table shows how many points Rowena wins for each combination of letters. In each case the number of points that Colin wins is the negative of the entry in the table. Both Rowena and Colin are trying to win as many points as possible. Colin's letter \begin{table}[h] \end{table}

1. Write down Colin's play-safe strategy, showing your working. What is the maximum number of points that Colin can win if he plays safe?
2. Explain why Rowena would never choose the letter \(W\). Rowena uses random numbers to choose between her three remaining options, so that she chooses \(X , Y\) and \(Z\) with probabilities \(x , y\) and \(z\), respectively. Rowena then models the problem of which letter she should choose as the following LP. $$\begin{array} { c l } \text { Maximise } & M = m - 1 \\ \text { subject to } & m \leqslant 2 x + 6 y + z , \\ & m \leqslant 4 x + 2 y , \\ & m \leqslant 3 y + 2 z , \\ & m \leqslant 2 x + 2 z , \\ & x + y + z \leqslant 1 \\ \text { and } & m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
3. Show how the expression \(2 x + 6 y + z\) was formed. The Simplex algorithm is used to solve the LP problem. The solution has \(x = 0.3 , y = 0.2\) and \(z = 0.5\).
4. Show that the optimal value of \(M\) is 0.6 . Colin then models the problem of which letter he should choose in a similar way. When Rowena plays using her optimal solution, from above, Colin finds that he should never choose the letter \(N\). Letting \(p , q\) and \(t\) denote the probabilities that he chooses \(P , Q\) and \(T\), respectively, Colin obtains the following equations. $$- 3 p + q - t = - 0.6 \quad - p - 2 q + t = - 0.6 \quad p - q - t = - 0.6 \quad p + q + t = 1$$
5. Explain how the equation \(- 3 p + q - t = - 0.6\) is obtained.
6. Use the third and fourth equations to find the value of \(p\). Hence find the values of \(q\) and \(t\).

4 The "Cuddly Friends Company" produces soft toys. For one day's production run it has available \(11 \mathrm {~m} ^ { 2 }\) of furry material, \(24 \mathrm {~m} ^ { 2 }\) of woolly material and 30 glass eyes. It has three soft toys which it can produce: The "Cuddly Aardvark", each of which requires \(0.5 \mathrm {~m} ^ { 2 }\) of furry material, \(2 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 3\). The "Cuddly Bear", each of which requires \(1 \mathrm {~m} ^ { 2 }\) of furry material, \(1.5 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 5\). The "Cuddly Cat", each of which requires \(1 \mathrm {~m} ^ { 2 }\) of furry material, \(1 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 2\). An analyst formulates the following LP to find the production plan which maximises profit. $$\begin{array} { l l } \text { Maximise } & 3 a + 5 b + 2 c \\ \text { subject to } & 0.5 a + b + c \leqslant 11 , \\ & 2 a + 1.5 b + c \leqslant 24 , \\ & 2 a + 2 b + 2 c \leqslant 30 . \end{array}$$

1. Explain how this formulation models the problem, and say why the analyst has not simplified the last inequality to \(a + b + c \leqslant 15\).
2. The final constraint is different from the others in that the resource is integer valued. Explain why that does not impose an additional difficulty for this problem.
3. Solve this problem using the simplex algorithm. Interpret your solution and say what resources are left over. On a particular day an order is received for two Cuddly Cats, and the extra constraint \(c \geqslant 2\) is added to the formulation.
4. Set up an initial simplex tableau to deal with the modified problem using either the big-M approach or two-phase simplex. Do not perform any iterations on your tableau.
5. Show that the solution given by \(a = 8 , b = 2\) and \(c = 5\) uses all of the resources, but that \(a = 6 , b = 6\) and \(c = 2\) gives more profit. What resources are left over from the latter solution?

3 A farmer has 40 acres of land. Four crops, A, B, C and D are available.
Crop A will return a profit of \(\pounds 50\) per acre. Crop B will return a profit of \(\pounds 40\) per acre.
Crop C will return a profit of \(\pounds 40\) per acre. Crop D will return a profit of \(\pounds 30\) per acre.
The total number of acres used for crops A and B must not be greater than the total number used for crops C and D. The farmer formulates this problem as:
Maximise \(\quad 50 a + 40 b + 40 c + 30 d\),
subject to \(\quad a + b \leqslant 20\), \(a + b + c + d \leqslant 40\).

1. Explain what the variables \(a , b , c\) and \(d\) represent. Explain how the first inequality was obtained.
   Explain why expressing the constraint on the total area of land as an inequality will lead to a solution in which all of the land is used.
2. Solve the problem using the simplex algorithm. Suppose now that the farmer had formulated the problem as:
   Maximise \(\quad 50 a + 40 b + 40 c + 30 d\),
   subject to \(\quad a + b \leqslant 20\), \(a + b + c + d = 40\).
3. Show how to adapt this problem for solution either by the two-stage simplex method or the big-M method. In either case you should show the initial tableau and describe what has to be done next. You should not attempt to solve the problem.

4 A small alpine hotel is planned. Permission has been obtained for no more than 60 beds, and these can be accommodated in rooms containing one, two or four beds. The total floor areas needed are \(15 \mathrm {~m} ^ { 2 }\) for a one-bed room, \(25 \mathrm {~m} ^ { 2 }\) for a two-bed room and \(40 \mathrm {~m} ^ { 2 }\) for a four-bed room. The total floor area of the bedrooms must not exceed \(700 \mathrm {~m} ^ { 2 }\). Marginal profit contributions per annum, in thousands of euros, are estimated to be 5 for a one-bed room, 9 for a two-bed room and 15 for a four-bed room.

1. Formulate a linear programming problem to find the mix of rooms which will maximise the profit contribution within the two constraints.
2. Use the simplex algorithm to solve the problem, and interpret your solution. It is decided that, for marketing reasons, at least 5 one-bed rooms must be provided.
3. Solve this modified problem using either the two-stage simplex method or the big-M method. You may wish to adapt your final tableau from part (ii) to produce an initial tableau, but you are not required to do so.
4. The simplex solution to the revised problem is to provide 5 one-bed rooms, 15 two-bed rooms and 6.25 four-bed rooms, giving a profit contribution of \(€ 253750\). Interpret this solution in terms of the real world problem.
5. Compare the following solution to your answer to part (iv): 8 one-bed rooms, 12 two-bed rooms and 7 four-bed rooms. Explain your findings.

4 A publisher is considering producing three books over the next week: a mathematics book, a novel and a biography. The mathematics book will sell at \(\pounds 10\) and costs \(\pounds 4\) to produce. The novel will sell at \(\pounds 5\) and costs \(\pounds 2\) to produce. The biography will sell at \(\pounds 12\) and costs \(\pounds 5\) to produce. The publisher wants to maximise profit, and is confident that all books will be sold. There are constraints on production. Each copy of the mathematics book needs 2 minutes of printing time, 1 minute of packing time, and \(300 \mathrm {~cm} ^ { 3 }\) of temporary storage space. Each copy of the novel needs 1.5 minutes of printing time, 0.5 minutes of packing time, and \(200 \mathrm {~cm} ^ { 3 }\) of temporary storage space. Each copy of the biography needs 2.5 minutes of printing time, 1.5 minutes of packing time, and \(400 \mathrm {~cm} ^ { 3 }\) of temporary storage space. There are 10000 minutes of printing time available on several printing presses, 7500 minutes of packing time, and \(2 \mathrm {~m} ^ { 3 }\) of temporary storage space.

1. Explain how the following initial feasible tableau models this problem.

   P	\(x\)	\(y\)	\(z\)	\(s 1\)	\(s 2\)	\(s 3\)	RHS
   1	- 6	- 3	- 7	0	0	0	0
   0	2	1.5	2.5	1	0	0	10000
   0	1	0.5	1.5	0	1	0	7500
   0	300	200	400	0	0	1	2000000

2. Use the simplex algorithm to solve your LP, and interpret your solution.
3. The optimal solution involves producing just one of the three books. By how much would the price of each of the other books have to be increased to make them worth producing? There is a marketing requirement to provide at least 1000 copies of the novel.
4. Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method. Briefly describe how to use the modified tableau to solve the problem. You are NOT required to perform the iterations.

4 Colin has a hobby from which he makes a small income. He makes bowls, candle holders and key fobs.
The materials he uses include wood, metal parts, polish and sandpaper. They cost, on average, \(\pounds 15\) per bowl, \(\pounds 6\) per candle holder and \(\pounds 2\) per key fob. Colin has a monthly budget of \(\pounds 100\) for materials. Colin spends no more than 30 hours per month on manufacturing these objects. Each bowl takes 4 hours, each candle holder takes 2 hours and each key fob takes half an hour.

1. Let \(b\) be the number of bowls Colin makes in a month, \(c\) the number of candle holders and \(f\) the number of key fobs. Write out, in terms of these variables, two constraints corresponding to the limit on monthly expenditure on materials, and to the limit on Colin's time. Colin sells the objects at craft fairs. He charges \(\pounds 30\) for a bowl, \(\pounds 15\) for a candle holder and \(\pounds 3\) for a key fob.
2. Set up an initial simplex tableau for the problem of maximising Colin's monthly income subject to your constraints from part (i), assuming that he sells all that he produces.
3. Use the simplex algorithm to solve your LP, and interpret the solution from the simplex algorithm. Over a spell of several months Colin finds it difficult to sell bowls so he stops making them.
4. Modify and solve your LP, using simplex, to find how many candle holders and how many key fobs he should make, and interpret your solution. At the next craft fair Colin takes an order for 4 bowls. He promises to make exactly 4 bowls in the next month.
5. Set up this modified problem either as an application of two-stage simplex, or as an application of the big-M method. You are not required to solve the problem. The solution now is for Colin to produce 4 bowls, \(6 \frac { 2 } { 3 }\) candle holders and no key fobs.
6. What is Colin's best integer solution to the problem?
7. Your answer to part (vi) is not necessarily the integer solution giving the maximum profit for Colin. Explain why.

3 Three products, A, B and C are to be made.
Three supplements are included in each product. Product A has 10 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z . Product B has 5 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 3 g per kg of supplement Z .
Product C has 12 g per kg of supplement \(\mathrm { X } , 7 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z .
There are 12 kg of supplement X available, 12 kg of supplement Y , and 9 kg of supplement Z .
Product A will sell at \(\pounds 7\) per kg and costs \(\pounds 3\) per kg to produce. Product B will sell at \(\pounds 5\) per kg and costs \(\pounds 2\) per kg to produce. Product C will sell at \(\pounds 4\) per kg and costs \(\pounds 3\) per kg to produce. The profit is to be maximised.

1. Explain how the initial feasible tableau shown in Fig. 3 models this problem. \begin{table}[h]

   P	a	b	c	s 1	s 2	s 3	RHS
   1	- 4	- 3	- 1	0	0	0	0
   0	10	5	12	1	0	0	12000
   0	5	5	7	0	1	0	12000
   0	5	3	5	0	0	1	9000

   \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{table}
2. Use the simplex algorithm to solve this problem, and interpret the solution.
3. In the solution, one of the basic variables appears at a value of 0 . Explain what this means. There is a contractual requirement to provide at least 500 kg of product A .
4. Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method. Briefly describe how the method works. You are not required to perform the iterations.

1 A furniture manufacturer is planning a production run. He will be making wardrobes, drawer units and desks. All can be manufactured from the same wood. He has available \(200 \mathrm {~m} ^ { 2 }\) of wood for the production run. Allowing for wastage, a wardrobe requires \(5 \mathrm {~m} ^ { 2 }\), a drawer unit requires \(3 \mathrm {~m} ^ { 2 }\), and a desk requires \(2 \mathrm {~m} ^ { 2 }\). He has 200 hours available for the production run. A wardrobe requires 4.5 hours, a drawer unit requires 5.2 hours, and a desk requires 3.8 hours. The completed furniture will have to be stored at the factory for a short while before being shipped. The factory has \(50 \mathrm {~m} ^ { 3 }\) of storage space available. A wardrobe needs \(1 \mathrm {~m} ^ { 3 }\), a drawer unit needs \(0.75 \mathrm {~m} ^ { 3 }\), and a desk needs \(0.5 \mathrm {~m} ^ { 3 }\). The manufacturer needs to know what he should produce to maximise his income. He sells the wardrobes at \(\pounds 80\) each, the drawer units at \(\pounds 65\) each and the desks at \(\pounds 50\) each.

1. Formulate the manufacturer's problem as an LP.
2. Use the Simplex algorithm to solve the LP problem.
3. Interpret the results.
4. An extra \(25 \mathrm {~m} ^ { 2 }\) of wood is found and is to be used. The new optimal solution is to make 44 wardrobes, no drawer units and no desks. However, this leaves some of each resource (wood, hours and space) left over. Explain how this can be possible.

1. Given that \(x\) and \(y\) are propositions, draw a 4-line truth table for \(x \Rightarrow y\), allowing \(x\) and \(y\) to take all combinations of truth values. If \(x\) is false and \(x \Rightarrow y\) is true, what can be deduced about the truth value of \(y\) ? A story has it that, in a lecture on logic, the philosopher Bertrand Russell (1872-1970) mentioned that a false proposition implies any proposition. A student challenged this, saying "In that case, given that \(1 = 0\), prove that you are the Pope."
   Russell immediately replied, "Add 1 to both sides of the equation: then we have \(2 = 1\). The set containing just me and the Pope has 2 members. But \(2 = 1\), so the set has only 1 member; therefore, I am the Pope." Russell's string of statements is an example of a deductive sequence. Let \(a\) represent " \(1 = 0\) ", \(b\) represent " \(2 = 1\) ", \(c\) represent "Russell and the Pope are 2" and \(d\) represent "Russell and the Pope are 1". Then Russell's deductive sequence can be written as \(( a \wedge ( a \Rightarrow b ) \wedge c ) \Rightarrow d\).
2. Assuming that \(a\) is false, \(b\) is false, \(a \Rightarrow b\) is true, \(c\) is true, and that \(d\) can take either truth value, draw a 2-line truth table for \(( a \wedge ( a \Rightarrow b ) \wedge c ) \Rightarrow d\).
3. What does the table tell you about \(d\) with respect to the false proposition \(a\) ?
4. Explain why Russell introduced propositions \(b\) and \(c\) into his argument.
5. Russell could correctly have started a deductive sequence: \(a \wedge [ a \Rightarrow ( ( 0.5 = - 0.5 ) \Rightarrow ( 0.25 = 0.25 ) ) ]\).
   Had he have done so could he correctly have continued it to end at \(d\) ?
   Justify your answer.
6. Draw a combinatorial circuit to represent \(( a \wedge ( a \Rightarrow b ) \wedge c ) \Rightarrow d\). 3 Floyd's algorithm is applied to the incomplete network on 4 nodes drawn below. The weights on the arcs represent journey times. \includegraphics[max width=\textwidth, alt={}, center]{4b5bc097-1052-4e44-8623-a84ceaab0289-4_400_558_347_751} The final matrices are shown below. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{final time matrix}

   \cline { 2 - 5 } \multicolumn{1}{c|}{}	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)
   \(\mathbf { 1 }\)	6	5	3	10