Given the following LP and its optimal final tableau:

max

$2$

$$

$1$

$+$

$$

$2$

$-$

$$

$3$

$$

$$

$.$

$$

$.$

$$

$$

$1$

$+$

$2$

$$

$2$

$+$

$$

$3$

$<=$

$8$

$-$

$$

$1$

$+$

$$

$2$

$-$

$2$

$$

$3$

$<=$

$4$

$$

$1$

$,$

$$

$2$

$,$

$$

$3$

$>=$

$0$

The optimal final tableau is shown below:

$($

a

$)$

$$Suppose that the coefficient of

$$

$1$

$$in the objective function has changed from

$2$

$$to

$2$

$+$

$$

$.$

Determine for what values of

$$

$$the current basis remains optimal.

$($

b

$)$

$$Given the option to increase the RHS of the first constraint or the second constraint by

$1$

unit, which one would you choose and why?

$($

c

$)$

$$Determine how many units

$$

$2$

$$should increase such that there would be an alternate optimal

solution with

$$

$2$

$$in the basis. How would you determine this value from the optimal tableau?

$($

d

$)$

$$Find the set of RHS vectors such that the optimal basis includes variables

$$

$1$

$$and

$$

$5$

$.$

$($

e

$)$

$$Suppose the following constraint is added to the problem:

$$

$2$

$+$

$$

$3$

$=$

$5$

$.$

$$What is the new

optimal solution? For each iteration explain which feasibility is lost.

$($

f

$)$

$$A new activity

$$

$6$

$$is proposed with a unit return of

$6$

$$and a consumption vector

$$

$6$

$=$

$($

$2$

$,$

$1$

$)$

$$

$.$

Find the new optimal solution.