7.07a Simplex tableau: initial setup in standard format

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

1. Explain why this can be modelled by the inequalities $$\begin{aligned} & x + 2 y \leq 70 \\ & 3 x + 2 y \leq 90 \\ & x \geq 0 , y \geq 0 \end{aligned}$$ (2 marks)
   The tailor sells type \(A\) for \(\pounds 30\) and type \(B\) for \(\pounds 40\). All garments made are sold. The tailor wishes to maximise his total income.
2. Set up an initial Simplex tableau for this problem.
   (3 marks)
3. Solve the problem using the Simplex algorithm.
   (8 marks) Figure 4 shows a graphical representation of the feasible region for this problem. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3147dad8-2d3c-42fd-b288-7017ff1fce16-004_452_828_995_356} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
4. Obtain the coordinates of the points A, \(C\) and \(D\).
5. Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4.
   (3 marks) Answer Book (AB12)
   Graph Paper (ASG2) Items included with question papers Answer booklet

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

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

5

1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l c } \text { Maximise } & P = 3 x + 2 y + 4 z \\ \text { subject to } & x + 4 y + 2 z \leqslant 8 \\ & 2 x + 7 y + 3 z \leqslant 21 \\ & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
2. Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the \(z\)-column as pivot.
3. 1. Perform one further iteration.
   2. State whether or not this is the optimal solution, and give a reason for your answer.

3

1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l l } \text { Maximise } & P = 5 x + 8 y + 7 z \\ \text { subject to } & 3 x + 2 y + z \leqslant 12 \\ & 2 x + 4 y + 5 z \leqslant 16 \\ & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
2. The Simplex method is to be used by initially choosing a value in the \(y\)-column as a pivot.
   1. Explain why the initial pivot is 4 .
   2. Perform two iterations of your tableau from part (a) using the Simplex method.
   3. State the values of \(P , x , y\) and \(z\) after your second iteration.
   4. State, giving a reason, whether the maximum value of \(P\) has been achieved.

4 A linear programming problem involving the variables \(x , y\) and \(z\) is to be solved. The objective function to be maximised is \(P = 2 x + 3 y + 5 z\). 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 \(z\)-column, perform one iteration of the Simplex method.
   2. Explain how you know that the optimal value has not been reached.
3. 1. Perform one further iteration.
   2. Interpret the final tableau and state the values of the slack variables.

3

1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l l } \text { Maximise } & P = 4 x - 5 y + 6 z \\ \text { subject to } & 6 x + 7 y - 4 z \leqslant 30 \\ & 2 x + 4 y - 5 z \leqslant 8 \\ & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
2. The Simplex method is to be used to solve this problem.
   1. Explain why it is not possible to choose a pivot from the \(z\)-column initially.
   2. Identify the initial pivot and explain why this particular element should be chosen.
   3. Perform one iteration using your initial tableau from part (a).
   4. State the values of \(x , y\) and \(z\) after this first iteration.
   5. Without performing any further iterations, explain why \(P\) has no finite maximum value.
3. Using the same inequalities as in part (a), the problem is modified to $$\text { Maximise } \quad Q = 4 x - 5 y - 20 z$$
   1. Write down a modified initial tableau and the revised tableau after one iteration.
   2. Hence find the maximum value of \(Q\).

5 A linear programming problem involving variables \(x\) and \(y\) is to be solved. The objective function to be maximised is \(P = 4 x + 9 y\). The initial Simplex tableau is given below.

1. Write down the three inequalities in \(x\) and \(y\) represented by this tableau.
2. The Simplex method is to be used to solve this linear programming problem by initially choosing a value in the \(x\)-column as the pivot.
   1. Explain why the initial pivot has value 1.
   2. Perform two iterations using the Simplex method.
   3. Comment on how you know that the optimum solution has been achieved and state your final values of \(P , x\) and \(y\).

4 A linear programming problem involving variables \(x\) and \(y\) is to be solved. The objective function to be maximised is \(P = 3 x + 5 y\). The initial Simplex tableau is given below.

1. In addition to \(x \geqslant 0 , y \geqslant 0\), write down three inequalities involving \(x\) and \(y\) for this problem.
2. 1. By choosing the first pivot from the \(\boldsymbol { y }\)-column, perform one iteration of the Simplex method.
   2. Explain how you know that the optimal value has not been reached.
3. 1. Perform one further iteration.
   2. Interpret the final tableau and state the values of the slack variables.

4 A linear programming problem consists of maximising an objective function \(P\) involving three variables \(x , y\) and \(z\). Slack variables \(s , t , u\) and \(v\) are introduced and the Simplex method is used to solve the problem. Several iterations of the method lead to the following tableau.

1. 1. The pivot for the next iteration is chosen from the \(\boldsymbol { y }\)-column. State which value should be chosen and explain the reason for your choice.
   2. Perform the next iteration of the Simplex method.
2. Explain why your new tableau solves the original problem.
3. State the maximum value of \(P\) and the values of \(x , y\) and \(z\) that produce this maximum value.
4. State the values of the slack variables at the optimum point. Hence determine how many of the original inequalities still have some slack when the optimum is reached.

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

1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), write down two inequalities involving \(x , y\) and \(z\) for this problem.
2. 1. The first pivot is chosen from the \(\boldsymbol { x }\)-column. Identify the pivot and perform one iteration of the Simplex method.
   2. Given that the optimal value of \(P\) has not been reached after this first iteration, find the possible values of \(k\).
3. Given that \(k = 10\) :
   1. perform one further iteration of the Simplex method;
   2. interpret the final tableau.

3

1. Given that \(k\) is a constant, complete the Simplex tableau below for the following linear programming problem. Maximise $$P = k x + 6 y + 5 z$$ subject to $$\begin{gathered} 2 x + y + 4 z \leqslant 11 \\ x + 3 y + 6 z \leqslant 18 \\ x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{gathered}$$
2. Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the \(\boldsymbol { y }\)-column as pivot.
3. 1. In the case when \(k = 1\), explain why the maximum value of \(P\) has now been reached and write down this maximum value of \(P\).
   2. In the case when \(k = 3\), perform one further iteration and interpret your new tableau. \section*{Answer space for question 3}

      1. \(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { Z }\)	\(s\)	\(\boldsymbol { t }\)	value
         1	\(- k\)	-6	-5	0	0	0
         0
         0

      2. \(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { Z }\)	\(\boldsymbol { s }\)	\(\boldsymbol { t }\)	value

         \section*{Answer space for question 3}
      3. 1. \(\_\_\_\_\)

4

1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l c } \text { Maximise } & P = 3 x + 6 y + 2 z \\ \text { subject to } & x + 3 y + 2 z \leqslant 11 \\ & 3 x + 4 y + 2 z \leqslant 21 \\ \text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$
2. The first pivot to be chosen is from the \(y\)-column. Perform one iteration of the Simplex method.
3. Perform one further iteration.
4. Interpret the tableau obtained in part (c) and state the values of your slack variables.

4

1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l r } \text { Maximise } & P = 2 x + 3 y + 4 z \\ \text { subject to } & x + y + 2 z \leqslant 20 \\ & 3 x + 2 y + z \leqslant 30 \\ & 2 x + 3 y + z \leqslant 40 \\ \text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
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 your final tableau and state the values of your slack variables.

3
Maximise \(\quad P = 2 x - 3 y + 4 z\) subject to \(\quad x + 2 y + z \leqslant 20\) \(x - y + 3 z \leqslant 24\) \(3 x - 2 y + 2 z \leqslant 30\) and \(\quad x \geqslant 0 , y \geqslant 0 , z \geqslant 0\).

1. Display the linear programming problem in a Simplex tableau.
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. Perform one further iteration.
3. Interpret your final tableau and state the values of your slack variables.
   [0pt] [3 marks]

5. The tableau below is the initial tableau for a three-variable linear programming problem in \(x , y\) and \(z\). The objective is to maximise the profit, \(P\).

1. Using the information in the tableau, write down
   1. the objective function,
   2. the three constraints as inequalities.
2. Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. Make your method clear by stating the row operations you use.
3. State the final values of the objective function and each variable.

5. The initial tableau for a linear programming problem in \(x , y\) and \(z\) is shown below. The objective function to be maximised is \(P = 4 x + 2 y + k z\), where \(k\) is a positive constant.

1. Using the information in the tableau, write down the three constraints as inequalities.
2. By increasing \(x\), perform one complete iteration of the simplex algorithm to obtain tableau \(T _ { 1 }\) and state the row operations you use.
3. Given that \(T _ { 1 }\) is not optimal, find an inequality for the value of \(k\).
4. Perform a second complete iteration of the simplex algorithm to obtain tableau \(T _ { 2 }\) and state the row operations you use.
5. Given that \(T _ { 2 }\) is optimal, find a second inequality for the value of \(k\).
6. State the final value of each variable and give an expression for the final value of \(P\) in terms of \(k\).
7. Hence find the range of possible values of \(P\).

5. A linear programming problem in \(x , y\) and \(z\) is described as follows. Maximise \(P = 2 x + 3 y + z\) subject to \(\quad 2 y - 3 z \leqslant 30\) $$\begin{array} { r } - 3 x + y + z \leqslant 60 \\ x + 4 y - z \leqslant 80 \end{array}$$

1. Complete the initial tableau in the answer book for this linear programming problem.
   (3)
2. Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the simplex algorithm to obtain a new tableau, T. Make your method clear by stating the row operations you use.
   (5)
3. Write down the profit equation given by T and state the values of the slack variables given by T . The following tableau is obtained after further iterations.

   Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
   \(r\)	0	2	-3	1	0	0	30
   \(s\)	0	13	-2	0	1	3	300
   \(x\)	1	4	-1	0	0	1	80
   \(P\)	0	5	-3	0	0	2	160

4. Explain why no optimal solution can be found by applying the simplex algorithm to the above tableau.

1. A linear programming problem is defined as follows:

$$\begin{array} { l l } \text { Maximise } & P = 3 x + 3 y + 4 z \\ \text { subject to } & x + 2 y + z \leq 30 \\ & 5 x + y + 3 z \leq 60 \\ \text { and } & x \geq 0 , y \geq 0 , z \geq 0 . \end{array}$$

1. Display the problem in a Simplex Tableau.
2. Starting with a pivot chosen from the \(z\)-column, perform one iteration of your tableau.
3. Write down the resulting values of \(x , y , z\) and \(P\) and state with a reason whether or not these values give an optimal solution.
   

5. 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 the 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. Using the Simplex algorithm the following initial tableau is obtained:

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

2. Explain the purpose of the variables \(s , t\) and \(u\).
3. By increasing the value of \(y\) first, work out the next two complete tableaus.
4. Explain how you know that your final tableau gives an optimal solution and state this solution in practical terms.

5 A student is solving the following linear programming problem. $$\begin{array} { l r } \text { Minimise } & Q = - 4 x - 3 y \\ \text { subject to } & x + y \leq 520 \\ & 2 x - 3 y \leq 570 \\ \text { and } & x \geq 0 , y \geq 0 \end{array}$$ 5

1. The student wants to use the simplex algorithm to solve the linear programming problem. They modify the linear programming problem by introducing the objective function $$P = 4 x + 3 y$$ and the slack variables \(r\) and \(s\) State one further modification that must be made to the linear programming problem so that it can be solved using the simplex algorithm. 5
2. (i) Complete the initial simplex tableau for the modified linear programming problem.
   [0pt] [2 marks]

   \(P\)	\(x\)	\(y\)	\(r\)	\(S\)	value

   5 (b) (ii) Hence, perform one iteration of the simplex algorithm.

   \(P\)	\(x\)	\(y\)	\(r\)	\(s\)	value

   5
3. The student performs one further iteration of the simplex algorithm, which results in the following correct simplex tableau.

   \(P\)	\(x\)	\(y\)	\(r\)	\(s\)	value
   1	0	0	\(\frac { 18 } { 5 }\)	\(\frac { 1 } { 5 }\)	1986
   0	0	1	\(\frac { 2 } { 5 }\)	\(- \frac { 1 } { 5 }\)	94
   0	1	0	\(\frac { 3 } { 5 }\)	\(\frac { 1 } { 5 }\)	426

   5 (c) (i) Explain how the student can tell that the optimal solution to the modified linear programming problem can be determined from the above simplex tableau.
   5 (c) (ii) Find the optimal solution of the original linear programming problem.

6. A two person zero-sum game is represented by the pay-off matrix for player A, shown above.

1. Explain, with justification, why this matrix may be reduced to a \(3 \times 3\) matrix by removing option S from player A's choices.
2. Verify that there is no stable solution to the reduced game. Player A intends to make a random choice between options \(\mathrm { Q } , \mathrm { R }\) and T , choosing option Q with probability \(p _ { 1 }\), option R with probability \(p _ { 2 }\) and option T with probability \(p _ { 3 }\) Player A wants to find the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm. Player A formulates the following linear programme, writing the constraints as inequalities. Maximise \(P = V\), where \(V =\) the value of original game + 3 $$\begin{aligned} \text { subject to } & V \leqslant 4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 } \\ & V \leqslant 8 p _ { 1 } + p _ { 3 } \\ & V \leqslant 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 } \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 \\ & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , V \geqslant 0 \end{aligned}$$
3. Explain why \(V\) cannot exceed any of the following expressions $$4 p _ { 1 } + 7 p _ { 2 } + 6 p _ { 3 } \quad 8 p _ { 1 } + p _ { 3 } \quad 6 p _ { 1 } + 4 p _ { 2 } + 3 p _ { 3 }$$
4. Explain why it is necessary to use the constraint \(p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1\) The Simplex algorithm is used to solve the linear programming problem.
   Given that the optimal value of \(p _ { 1 } = \frac { 7 } { 11 }\) and the optimal value of \(p _ { 3 } = 0\)
5. calculate the value of the game to player A .
   (3) Player B intends to make a random choice between options \(\mathrm { X } , \mathrm { Y }\) and Z , choosing option X with probability \(q _ { 1 }\), option Y with probability \(q _ { 2 }\) and option Z with probability \(q _ { 3 }\)
6. Determine the optimal strategy for player B, making your working clear.

A tailor makes two types of garment, A and B. He has available 70 m² of cotton fabric and 90 m² of woollen fabric. Garment A requires 1 m² of cotton fabric and 3 m² of woollen fabric. Garment B requires 2 m² of each fabric. The tailor makes \(x\) garments of type A and \(y\) garments of type B.

1. Explain why this can be modelled by the inequalities $$x + 2y \leq 70,$$ $$3x + 2y \leq 90,$$ $$x \geq 0, y \geq 0.$$ [2 marks]

The tailor sells type A for £30 and type B for £40. All garments made are sold. The tailor wishes to maximise his total income.

1. Set up an initial Simplex tableau for this problem. [3 marks]
2. Solve the problem using the Simplex algorithm. [8 marks]

Figure 4 shows a graphical representation of the feasible region for this problem. \includegraphics{figure_4}

1. Obtain the coordinates of the points A, C and D. [4 marks]
2. Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4. [3 marks]

The tableau below is the initial tableau for a maximising linear programming problem.

1. For this problem \(x \geq 0\), \(y \geq 0\), \(z \geq 0\). Write down the other two inequalities and the objective function. [3]
2. Solve this linear programming problem. [8]
3. State the final value of \(P\), the objective function, and of each of the variables. [3]

A three-variable linear programming problem in \(x\), \(y\) and \(z\) is to be solved. The objective is to maximise the profit P. The following tableau was obtained.

1. State, giving your reason, whether this tableau represents the optimal solution. [1]
2. State the values of every variable. [3]
3. Calculate the profit made on each unit of \(y\). [2]

Flatland UK Ltd makes three types of carpet, the Lincoln, the Norfolk and the Suffolk. The carpets all require units of black, green and red wool. For each roll of carpet, the Lincoln requires 1 unit of black, 1 of green and 3 of red, the Norfolk requires 1 unit of black, 2 of green and 2 of red, and the Suffolk requires 2 units of black, 1 of green and 1 of red. There are up to 30 units of black, 40 units of green and 50 units of red available each day. Profits of £50, £80 and £60 are made on each roll of Lincoln, Norfolk and Suffolk respectively. Flatland UK Ltd wishes to maximise its profit. Let the number of rolls of the Lincoln, Norfolk and Suffolk made daily be \(x\), \(y\) and \(z\) respectively.

1. Formulate the above situation as a linear programming problem, listing clearly the constraints as inequalities in their simplest form, and stating the objective function. [4]

This problem is to be solved using the Simplex algorithm. The most negative number in the profit row is taken to indicate the pivot column at each stage.

1. Stating your row operations, show that after one complete iteration the tableau becomes

   Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
   \(r\)	\(\frac{1}{2}\)	0	\(1\frac{1}{2}\)	1	\(-\frac{1}{2}\)	0	10
   \(y\)	\(\frac{1}{2}\)	1	\(\frac{1}{2}\)	0	\(\frac{1}{2}\)	0	20
   \(t\)	2	0	0	0	\(-1\)	1	10
   \(P\)	\(-10\)	0	\(-20\)	0	40	0	1600

   [4]
2. Explain the practical meaning of the value 10 in the top row. [2]
3. 1. Perform one further complete iteration of the Simplex algorithm.
   2. State whether your current answer to part (d)(i) is optimal. Give a reason for your answer.
   3. Interpret your current tableau, giving the value of each variable. [8]