Q$7)$ Sensitivity Analysis Interpretation $(20$ pts$)$

A company has solved a linear programming model to maximize profit. The following sensitivity report is obtained from the optimal solution:

$-$ Constraint $1($Labor availability$)$:

Shadow price $\backslash (=\backslash$$ $8/\backslash )$ unit

Allowable increase $\backslash (=25\backslash )$ units

Allowable decrease $\backslash (=0\backslash )$ units

$-$ Decision variable $\backslash (\backslash$mathbf$\{$x$\}\_\{\backslash$mathbf$\{2\}\}\backslash )$ :

Objective coefficient $\backslash (=\backslash$$ $45\backslash )$

Allowable range $\backslash (=[\backslash$$ $40,\backslash$$ $55]\backslash )$

$-$ Decision variable x$4$:

Reduced cost $\backslash (=\backslash$$ $10\backslash )$

$-$ Current optimal profit: $\backslash (\backslash$$ $12,500\backslash )$

$-$ Decision variable $\backslash (\backslash$mathbf$\{$x$\}\_\{3\}\backslash )$ is the only variable with a value of zero in the current optimal solution.

$1.$ How would you interpret the shadow price of Constraint $1?$ What is the importance of the allowable increase range? How can the company use this information in decision$-$making?

$2.$ If the objective coefficient of $\backslash (\backslash$mathrm$\{$x$\}\_\{2\}\backslash )$ increases, under what circumstances would the optimal solution remain unchanged? When might it change?

$3.$ Given that the reduced cost of x $4$ is $\backslash (\backslash$$ $10\backslash ),$ what does this indicate about its value in the optimal solution? How might this insight influence the company's operational strategy?

$4.$ Since $\backslash ($ x$\_\{3\}\backslash )$ has a value of zero in the current solution, what would need to happen for this variable to become active in the optimal solution?

$5.$ If the right$-$hand side of Constraint $1$ is reduced by $10$ units, what can be said about the validity of the current solution? Can we use the current shadow price to estimate the impact on the objective value?