Given the following all$-$integer linear program:

Max $3$x$1+2$x$2$

s$.$t$.3$x$1+$ x$2<=9$

x$1+3$x$2<=7$

$$x$1+$ x$2<=1$

x$1,$ x$2>=0$ and integer

$1.$Solve the problem as a linear program ignoring the integer constraints. Show that the optimal solution to the linear program gives fractional values for both x$1$ and x$2.$

$2.$What solution is obtained by rounding fractions greater than or equal to $1/2$ to the next larger number? Show that this solution is not a feasible solution.

By rounding x$1$ and x$2$ to the nearest integer, we get x$1=\_\_\_$

and x$2=\_\_.$

This point is $\_\_\_($feasible $/$ infeasible$).$

$3.$What is the solution obtained by rounding down all fractions? Is it feasible?

By rounding x$1$ and x$2$ down, we get x$1=\_\_\_$ and x$2=\_\_\_.$

This point is $\_\_\_\_($feasible $/$ infeasible$).$

If feasible, what is the objective value $=\_\_\_\_\_\_.($If it is not feasible, simply write $0$ for the obj value.$)$

Enumerate all points in the linear programming feasible region in which both x$1$ and x$2$ are integers, and show that the feasible solution obtained in the previous part is not optimal and that in fact the optimal integer is not obtained by any form of rounding.

A partially filled table is provided for your convenience. Given the possible x$1$ values, complete the missing values for x$2$ and objective values.

Alternatative x$1$ x$2$ Z

$10$

$21$

$32$

$43$

$50$

$61$

$72$

$81$

Run the model in Excel Solver, this time adding integrality condition as well. $($Make sure all decision variables can only take integer values$)$

What is the optimal solution and its objective value? $($Note: you can double$-$check your answer from the table you filled in the previous part$)$

x$1\_\_\_$

x$2\_\_\_$

Objective $\_\_\_\_$