7.07c Interpret simplex: values of variables, slack, and objective

8. A company makes three sizes of lamps, small, medium and large. The company is trying to determine how many of each size to make in a day, in order to maximise its profit. As part of the process the lamps need to be sanded, painted, dried and polished. A single machine carries out these tasks and is available 24 hours per day. A small lamp requires one hour on this machine, a medium lamp 2 hours and a large lamp 4 hours. Let \(x =\) number of small lamps made per day, $$\begin{aligned} & y = \text { number of medium lamps made per day, } \\ & z = \text { number of large lamps made per day, } \end{aligned}$$ where \(x \geq 0 , y \geq 0\) and \(z \geq 0\).

1. Write the information about this machine as a constraint.
2. 1. Re-write your constraint from part (a) using a slack variable \(s\).
   2. Explain what \(s\) means in practical terms. Another constraint and the objective function give the following Simplex tableau. The profit \(P\) is stated in euros.

      Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	Value
      \(r\)	3	5	6	1	0	50
      \(s\)	1	2	4	0	1	24
      \(P\)	- 1	- 3	- 4	0	0	0

3. Write down the profit on each small lamp.
4. Use the Simplex algorithm to solve this linear programming problem.
5. Explain why the solution to part (d) is not practical.
6. Find a practical solution which gives a profit of 30 euros. Verify that it is feasible.

7. A maximisation linear programming problem in \(x , y\) and \(z\) is to be solved using the Simplex method. The tableau after the 1st iteration is shown below.

1. State the column that contains the pivot value for the 1st iteration. You must give a reason for your answer.
2. By considering the equations represented in the above tableau, formulate the linear programming problem in \(x , y\) and \(z\) only. State the objective and list the constraints as inequalities with integer coefficients.
3. Taking the most negative number in the profit row to indicate the pivot column, perform the 2nd iteration of the Simplex algorithm, to obtain a new tableau, T . Make your method clear by stating the row operations you use.
4. 1. Explain, using T, how you know that an optimal solution to the original linear programming problem has not been found after the 2nd iteration.
   2. State the values of the basic variables after the 2nd iteration. A student attempts the 3rd iteration of the Simplex algorithm and obtains the tableau below.

      b.v.	\(x\)	\(y\)	\(z\)	\(s _ { 1 }\)	\(S _ { 2 }\)	\(\mathrm { S } _ { 3 }\)	Value
      z	0	0	1	\(\frac { 1 } { 2 }\)	\(- \frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 43 } { 2 }\)
      \(x\)	1	0	0	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 8 }\)	\(- \frac { 1 } { 8 }\)	\(\frac { 57 } { 4 }\)
      \(y\)	0	1	0	\(- \frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 3 } { 4 }\)	\(\frac { 9 } { 2 }\)
      \(P\)	0	1	0	\(\frac { 5 } { 4 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 7 } { 8 }\)	\(\frac { 361 } { 4 }\)

5. Explain how you know that the student's attempt at the 3rd iteration is not correct.

6 Kai mixes hot drinks using coffee and steamed milk.
The amounts ( ml ) needed and profit ( \(\pounds\) ) for a standard sized cup of four different drinks are given in the table. The table also shows the amount of the ingredients available. Kai makes the equivalent of \(w\) standard sized americanos, \(x\) standard sized cappuccinos, \(y\) standard sized flat whites and \(z\) standard sized lattes. He can make different sized drinks so \(w , x , y , z\) need not be integers. Kai wants to find the maximum profit that he can make, assuming that the customers want to buy the drinks he has made.

1. What is the minimum value of X for it to be worthwhile for Kai to make cappuccinos? Kai makes no cappuccinos.
2. Use the simplex algorithm to solve Kai's problem. The grids in the Printed Answer Booklet should have at least enough rows and columns and there should be at least enough grids to show all the iterations needed. Only record the output from each iteration, not any intermediate stages.
   Interpret the solution and state the maximum profit that Kai can make.

6 Jack is making pizzas for a party. He can make three types of pizza:

• There is enough dough to make 30 pizzas.
• Type Z pizzas use vegan cheese. Jack only has enough vegan cheese to make 2 type Z pizzas.
• At least half the pizzas made must be suitable for vegetarians.
• Jack has enough onions to make 50 type X pizzas or 20 type Y pizzas or 20 type Z pizzas or some mixture of the three types.

Suppose that Jack makes \(x\) type X pizzas, \(y\) type Y pizzas and \(z\) type Z pizzas.

1. Formulate the constraints above in terms of the non-negative, integer valued variables \(x , y\) and \(z\), together with non-negative slack variables \(s , t , u\) and \(v\). Jack wants to find out the maximum total number of pizzas that he can make.
2. 1. Set up an initial simplex tableau for Jack's problem.
   2. Carry out one iteration of the simplex algorithm, choosing your pivot so that \(x\) becomes a basic variable. When Jack carries out the simplex algorithm his final tableau is:

      \(P\)	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	\(v\)	RHS
      1	0	0	0	0	0	\(\frac { 3 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(28 \frac { 4 } { 7 }\)
      0	0	0	0	1	0	\(- \frac { 3 } { 7 }\)	\(- \frac { 2 } { 7 }\)	\(1 \frac { 3 } { 7 }\)
      0	0	0	1	0	1	0	0	2
      0	1	0	0	0	0	\(\frac { 5 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(14 \frac { 2 } { 7 }\)
      0	0	1	1	0	0	\(- \frac { 2 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(14 \frac { 2 } { 7 }\)

3. Use this final tableau to deduce how many pizzas of each type Jack should make. Jack knows that some of the guests are vegans. He decides to make 2 pizzas of type \(Z\).
4. 1. Plot the feasible region for \(x\) and \(y\).
   2. Complete the branch-and-bound formulation in the Printed Answer Booklet to find the number of pizzas of each type that Jack should make.
      You should branch on \(x\) first. \section*{END OF QUESTION PAPER}

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.

3 The Simplex tableau below is optimal. 3

1. Deduce the range of values that \(k\) must satisfy.
   3
2. Write down the value of the variable \(s\) which corresponds to the optimal value of \(P\).

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.