Effect of parameter changes

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 \(P = 4 x + y + k z\), 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. \(\_\_\_\_\)

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 \(P\).