Given is the LP problem

$mamize,z=3x_1-2x_2$

subject $to3x_1+x_{2}10$

$,x_1+2x_{2}12$

$,x_1,x_{2}0$

$($a$)$ Solve this problem with the Management Scientist software. Print the solution,

compare it with the solution in the lecture notes and state whether both are

equal.

$($b$)$ Test the correctness of the non$-$zero 'Reduced Cost' value, i$.$e$.$

Inform yourself what the meaning of the Reduced Cost is using the lecture

notes.

Which variable had a non$-$zero Reduced Cost in the solution?

Increase the coefficient of this variable in your function $z$ by slightly less

than the Reduced Cost and run the problem and note the optimal solution

$($i$.$e$.x_1,x_2).$

Increase in your function $z$ the coefficient of this variable by exactly the

Reduced Cost and run the problem and note the optimal solution $($i$.$e$.x_1,x_2).$

Increase in your function $z$ the coefficient of this variable by slightly more

than the Reduced Cost and run the problem and note the optimal solution

$($i$.$e$.x_1,x_2).$

Give the basic variables and non$-$basic variables for the optimal solutions in

each of these $3$ tests.

$($c$)$ Test the correctness of the non$-$zero slack value, i$.$e$.$

Inform yourself what the meaning of a non$-$zero slack value is using the

lecture notes.

Which slack variable had a non$-$zero slack value in the solution?

Lower the right hand side of the corresponding constraint by slightly less

than the slack value and run the problem and note the optimal solution $($i$.$e$.$

$x_1,x_2).$

Lower in the corresponding constraint the right hand side by exactly the

slack value and run the problem and note the optimal solution $($i$.$e$.x_1,x_2).$

Lower in the corresponding constraint the right hand side by slightly more

than the slack value and run the problem and note the optimal solution $($i$.$e$.$

$x_1,x_2).$

Which of the constraints were binding in each of these $3$ tests?

$($d$)$ Test the correctness of the non$-$zero 'Dual Price' values, i$.$e$.$

Using the lecture notes inform yourself what the meaning of the Dual Price

related to a constraint is$.$

Which constraint had a non$-$zero Dual Price?

Increase the $b_i$ of this constraint by $b_i=0\mathrm{.}1,$ solve the optimization problem

again, compare $\frac{z_{\text{n}\text{e}\text{w}}^{*}-z_{\text{o}\text{l}\text{d}}^{*}}{}{b}_i$ with the dual price.

Do the same experiment with $b_i=-0\mathrm{.}1$ and report your findings.

Do more such experiments to find the maximum $b_i$ and the minimum

$($negative$)b_i$ giving the same ratio $\frac{z_{\text{n}\text{e}\text{w}}^{*}-z_{\text{o}\text{l}\text{d}}^{*}}{}{b}_i$ as for small $b_i.$