Use the graphical solution and$/$or Algebra $($NOT EXCEL$)$ to answer questions $8-12$:

Question $8$: Let $c$ be the profit per unit of Product $1.$ Currently, $c=30.$ Assume that the profit from one

unit of Product $2$ is fixed at its current value of $$20.$ Determine the lower and upper limits for $c$ so that

the current optimal solution remains optimal.

Question $9$: Assume that it is possible to increase the processing time for machine $1$ from the

current level of $80$ hours a week to $100$ hours a week.

Use the graph in Figure $2($below$)$ to determine the optimal solution:

which of the points A$-$G represents the optimal mix?

What is the optimal value of $x$ and $y?$

What is the optimal objective value?

Figure $2$: Graphical Solution: Capacity Expansion

Page $2$

OIM $310$

$H{W}_3$

Question $10$: How much will you be willing to pay to increase the capacity of machine $1$ from $80$

hours a week to $100$ hours a week? Explain.

Question $11$: Consider the original formulation of the problem as you determined in question #$1$:

One of the managers examined your LP model and noted that you did not take into account the

requirement to run a quality assurance process for product $1$ and $2.$ She told you that it takes $10$

minutes to complete the quality assurance of a product $($regardless if is product $1$ or $2)$ and that

there are $10$ hours a week available for performing the quality assurance. Would this new constraint

change the optimal product mix that you determined in question $5($and which you verified in question

$?$ If yes, how would the optimal solution change? If not, explain why the new constraint does not

change the optimal product mix.

Question $12$: Consider the original formulation of the problem as you determined in question #$1$:

One of the managers examined your LP model and noted that you did not take into account the

requirement that a total of at least $60$ products will be produced each week. Would this new

constraint change the optimal product mix that you determined in question $5($and which you verified in

question $6)?$ If yes, how would the optimal solution change? If not, explain why the new constraint

does not change the optimal product mix.