Write constraints from tableau

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 An initial simplex tableau is shown below. The variables \(s , t\) and \(u\) are slack variables.

1. For the LP problem that this tableau represents, write down the following, in terms of \(x\) and \(y\) only.
   The graph below shows the feasible region for the problem (as the unshaded region, and its boundaries), and objective lines \(P = 10\) and \(P = 20\) (shown as dotted lines). \includegraphics[max width=\textwidth, alt={}, center]{133395d2-5020-4054-a229-70168f1d0f95-7_883_1043_1272_244} The optimal solution is \(P = 23\), when \(x = 0\) and \(y = 4.6\).
2. Complete the first three rows of branch-and-bound in the Printed Answer Booklet, branching on \(y\) first, to determine an optimal solution when \(x\) and \(y\) are constrained to take integer values. In your working, you should show non-integer values to \(\mathbf { 2 }\) decimal places. The tableau entry 18.8 is reduced to 0 .
3. Describe carefully what changes, if any, this makes to the following.

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]

The tableau below is the initial tableau for a linear programming problem in \(x\), \(y\) and \(z\). The objective is to maximise the profit, \(P\). $$\begin{array}{c|c|c|c|c|c|c|c} \text{basic variable} & x & y & z & r & s & t & \text{Value} \\ \hline r & 12 & 4 & 5 & 1 & 0 & 0 & 246 \\ \hline s & 9 & 6 & 3 & 0 & 1 & 0 & 153 \\ \hline t & 5 & 2 & -2 & 0 & 0 & 1 & 171 \\ \hline P & -2 & -4 & -3 & 0 & 0 & 0 & 0 \end{array}$$ Using the information in the tableau, write down

1. the objective function, [2]
2. the three constraints as inequalities with integer coefficients. [3]

Taking the most negative number in the profit row to indicate the pivot column at each stage,

1. solve this linear programming problem. Make your method clear by stating the row operations you use. [9]
2. State the final values of the objective function and each variable. [3]

One of the constraints is not at capacity.

1. Explain how it can be identified. [1]

(Total 18 marks)

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

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
\(r\)	7	10	10	1	0	0	3600
\(s\)	6	9	12	0	1	0	3600
\(t\)	2	3	4	0	0	1	2400
\(P\)	-35	-55	-60	0	0	0	0

1. Write down the four equations represented in the initial tableau above. [4]
2. Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. State the row operations that you use. [9]
3. State the values of the objective function and each variable. [3]

(Total 16 marks)