Refer to the MOLP problem presented below.

Lee is the owner of a Mining Company, which operates two different coal mines in Wythe and Giles counties in Southwest Virginia. Due to increased commercial and residential development in the primary areas served by these mines, Lee is anticipating an increase in demand for coal in the coming year. Specifically, her projections indicate a $24-$ton increase in the demand for high$-$grade coal, a $20-$ton increase in the demand for medium$-$grade coal, and a $80-$ton increase in the demand for low$-$grade coal. To handle this increase in demand, Lee must schedule extra shifts of workers at the mines. It costs $$42,000$ per month to run an extra shift of workers at the Wythe county mine and $$34,000$ per month at the Giles mine. Only one additional shift can be scheduled each month at each mine. The amount of coal that can be produced in a month's time at each mine by a shift of workers is summarized in the following table.

Type of Coal Wythe Mine Giles Mine

High grade $12$ tons $4$ tons

Medium grade $4$ tons $4$ tons

Low grade $10$ tons $20$ tons

Unfortunately, the methods used to extract coal from these mines produce toxic water that enters the local groundwater aquifers. At the Wythe mine, running an extra shift will generate approximately $750$ gallons of toxic water per month, whereas the mine in Giles county will generate about $1,200$ gallons of toxic water. Although these amounts are within EPA $($Environmental Protection Agency$)$ guidelines, Lee is concerned about the environment and doesn't want to create any more pollution than is absolutely necessary. Additionally, although the company follows all OSHA safety guidelines, company records indicate that approximately $0\mathrm{.}10$ life$-$threatening accidents occur per shift each month at the Wythe mine whereas $0\mathrm{.}50$ accidents occur per shift each month at the Giles mine. Lee knows that mining is a hazardous occupation, but she cares about the health and welfare of her workers and wants to keep the number of life$-$threatening accidents to a minimum.

Define the variables of interest as follows.

X$1=$ number of months to schedule an extra shift at the Wythe county mine

X$2=$ number of months to schedule an extra shift at the Giles county mine

Define the three objective functions of interest as follows.

Minimize: $42$X$1+34$X$2($production costs $($in $$1,000))$

Minimize: $750$X$1+1,200$X$2($toxic water produced $($in gallons$))$

Minimize: $0\mathrm{.}1$X$1+0\mathrm{.}5$X$2($life$-$threatening accidents$)$

Define the constraints of interest as follows.

$12$X$1+4$X$2>=24($high$-$grade coal required$)$

$4$X$1+4$X$2>=20($medium$-$grade coal required$)$

$10$X$1+20$X$2>=80($low$-$grade coal required$)$

X$1,$ X$2>=0($nonnegativity conditions$)$

Considering each of the objective functions as it's own minimization problem subject to the constraints described above we arrive at the following solutions.

The total production cost is minimized at

X$1=0\mathrm{.}5$

and

X$2=4\mathrm{.}5$

resulting in a cost of $$174,000\mathrm{.}00.$

The volume of toxic water produced is minimized at

X$1=2\mathrm{.}0$

and

X$2=3\mathrm{.}0$

resulting in $5,100$ gallons of toxic water.

The number of life threatening accidents is minimized at

X$1=8\mathrm{.}0$

and

X$2=0\mathrm{.}0$

resulting in $0\mathrm{.}8$ life threatening accidents.

These results suggest the following target values and goals for total production, gallons of toxic water produced, and number of life threatening accidents.

Goal $1$: The total production cost should be approximately $$174,000\mathrm{.}00.$

Goal $2$: The gallons of toxic water produced should be approximately $5,100.$

Goal $3$: The number of life$-$threatening accidents should be approximately $0\mathrm{.}8.$

In the solutions shown above, each result in more than the required amount of one or more types of coal being produced, as summarized in the following table.

Excess Production

Solution Shown in: High$-$Grade Coal Medium$-$Grade Coal Low$-$Grade Coal

high$-$grade coal $0$ tons $0$ tons $15$ tons

medium$-$grade coal $12$ tons $0$ tons $0$ tons

low$-$grade coal $72$ tons $12$ tons $0$ tons

$($a$)$

Formulate an LP model that could be solved to find the solution that minimizes the maximum amount of excess coal produced. $($Hint: Use a MINIMAX objective rather than a MAX$()$ function.$)$

MIN: Q

Subject to:

high$-$grade coal $0<=$ 



Correct: Your answer is correct.

$<=$ Q

medium$-$grade coal $0<=$ 



Correct: Your answer is correct.

$<=$ Q

low$-$grade coal $0<=$ 



Correct: Your answer is correct.

$<=$ Q

X$1,$ X$2>=0$

$($b$)$

Implement your model in a spreadsheet and solve it$.$ What is the optimal solution? $($Round your answers to two decimal places.$)$

$($X$1,$ X$2)=$ 





Correct: Your answer is correct.





$($c$)$

Revise your model to find the solution that minimizes the maximum percentage of excess coal produced. What is the optimal solution? $($Round your answers to two decimal places.$)$

$($X$1,$ X$2)=$