$2[600]$

$[$a$][100]$ Provide the graphical solution of the following LP$.($Be sure to implement a reasonable scale $-$ that provides a READABLE GRAPH$).$ Identify the feasible region and the optimal solution.'Plot the OPTIMAL objective function as a double dashed line graph.

Maximize $Z=15x_1+15x_2$

s$.t.$

$x_1,x_{2}0$

Label the feasible extreme points starting with $[x_1,x_2]=[0,0]$ as $A,$ and continuing clockwise through $G.$

$$

$($b$)[050]$ Augment the constraint set with Slack$/$Surplus variables $---$ as required $-$ using the assigned slack$/$surplus variables. Write out the full $($augmented$)$ linear program $($LP$).$

$($c$)[050]$ What is the A matrix? What is the Right Hand Side $($RHS$)$ Vector, b$?$

$($d$)[050]$ What is the $B$ matrix when $x1,x2,x4,x6$ and $x7$ are Basic variables?

$($e$)050$ What is the maximum number of extreme points for this LP$?$

$($f$)150$ Solve the augmented linear program via the Simplex Method Using the DICTIONARY approach described in class. Clearly identify each iteration and include the SIMPLIFIED dictionary for each one. Provide the value of the objective function, and the values of ALL the variables at each iteration in the table as shown in part $[$i$].$

IMPORTANT: At each iteration, always choose the variable with the highest subscript when more than one variable could be made Basic. For example, if $x_3$ and $x_5$ can be made Basic, choose $x_5.$ Be careful to follow this requirement $-$ otherwise you will lose points since you will NOT be solving the problem with the IMPOSED requirements.

$($g$)050$ Characterize the optimal solution of $[$f$]$ as one of the following:

$($i$)$ Unique Optimal; $($ii$)$ Alternate Optima, iii$)$ Unbounded Solution.

You must state the REASON for your characterization of the solution. No reason, no credit.

$($h$)050$ On the graph of Problem $2[$a$],$ show the progression from the initial feasible solution toward the optimal solution.

$($i$)[050]$ For each FEASIBLE Extreme Point in Problem $2($a$),$ complete the following table $($add additional rows as needed$)$:

$\backslash$table$[[$Point$,x1,x2,x3,x4,x5,x6,x7,z,$Optimal? $[$Y$/$N$],$Why?$],[$A$,0\mathrm{.}0,0\mathrm{.}0,,,,,,0\mathrm{.}0,,]]$