All$-$Integer Graphical Solution. Consider the following all$-$integer linear program.

LO $2,3$

Max $5$x$\_(1)+8$x$\_(2)$

s$.$t$.$

$6$x$\_(1)+5$x

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$1$x$\_(1)+2$x

x$\_(1),$x$\_(2)>=0$ and integer

a$.$ Graph the constraints for this problem. Use dots to indicate all feasible

integer solutions.

b$.$ Find the optimal solution to the LP Relaxation. Round down to find a

feasible integer solution.

c$.$ Find the optimal integer solution. Is it the same as the solution obtained in

part $($b$)$ by rounding down?

Do this for part b$.$ Run Excel with the Solver tool without declaring the variables as interger. Get the solution for x$1$ and x$2.$ On the answer report find the values for x$1$ and x$2$ and in a nearby cell type, "Wow, these values are not integers." Then type in another nearby cell on the answer report, $"$x$1$ and and x$2$ rounded down to the nearest integers would be $....$ and $...."($You fiill in those values.$)$

Before you do part c$,$ take the rounded values for x$1$ and x$2$ and go back to the sheet with the problem set up and plug in these rounded values as the initial values for x$1$ and x$2.$ Then a few rows below this type in a cell, "Hey, when I put in thsse rounded down values for the variables I see the profit is $...."($Fill in the value for profit.$)$

Follow part c$.$ Get the new Answer Report. Compare the integer solution with what you rounded the values to and that max value.

Remember, sometimes we want only integers for a solution. Rounding may not be the best thing to do$.$ So$,$ I am asking if the rounding in part b is better than the integer solution in part c$.$ Say if it is better to round or not in this case.