(Solved): Q4. Sensitivity Analysis (3 marks) The simplex tableau is the final tableau obtained by the simplex ...

Q4. Sensitivity Analysis (3 marks) The simplex tableau is the final tableau obtained by the simplex method for the linear programming problem \[ \min -5 x_{1}-2 x_{2}-x_{3}-x_{4} \] subject to \[ \begin{array}{c} 2 x_{1} \quad+x_{3}+x_{4}=6 \\ 3 x_{1}+x_{2}+x_{3}+2 x_{4}=10 \\ x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0, x_{4} \geq 0 \end{array} \] (a) Find the basis \( B \) which corresponds to this tableau. (b) For what values of \( \delta \) the problem, obtained by replacing the right-hand side in (2) - (3) by \[ \left[\begin{array}{c} 6 \\ 10 \end{array}\right]+\delta\left[\begin{array}{l} 2 \\ 2 \end{array}\right], \] has the same optimal basis \( B \) ? (c) For what values of \( \delta \) the problem, obtained by replacing \( \mathbf{c}_{N} \) by \[ \left[\begin{array}{l} -1 \\ -1 \end{array}\right]+\delta\left[\begin{array}{r} -1 \\ 2 \end{array}\right], \] has the same optimal basis \( B \) ? (d) For what values of \( \delta \) the problem, obtained by replacing \( \mathbf{c}_{B} \) by \[ \left[\begin{array}{l} -5 \\ -2 \end{array}\right]+\delta\left[\begin{array}{r} 1 \\ -2 \end{array}\right] \] has the same optimal basis \( B \) ? (e) Does the optimal basis \( B \) remain optimal after the introduction of a new variable \( x_{5} \) with the objective coefficient \( -1 \) and the constraint coefficients \( \left[\begin{array}{l}3 \\ 2\end{array}\right] ? \) (f) By how much can the coefficient of \( x_{3} \) in (2) increase and decrease without changing the optimal basis \( B \) ?