Question $1.(40$ points$)$ You are provided with the following inear program:min z$=5$r $+6$y$8.$t$-+2$yS$8-$ty$>-37$z $+4$y $>28$y$<mn\; id="EditorElement\_6931675">8</mn>$I,y$>0($a$)$ On the following page, use the graphical solution method to identify the feasible region.$($b$)$ Find the feasible extreme points and caleulate their objective values.Extreme point $1$:Extreme point $2$:Extreme point $3$:Extreme point $4$:$($c$)$ Draw an isocost line that paSSes through the point $($r $=9,$y $=0)$ and find the directionof optimization.$($d$)$ Provide the optimal solution and optimal objective value.Optimal solution: r $=$Optimal objective value:Question $2.(20$ points$)$ A beverage cans manufacturer makes $3$ types of soft drink cansneeded for the beverage producers to fill soft drinks of three different volumes. The maximumavailability of the machines' time allotted per day is $90$ hours and the supply of metal islimited to $120$ kg per day. The following table provides the details of the input needed tomanufacture one batch of cans.Metal $($kg$)/$batchMachines Time $($hr$)/$batchProfit$/$batch$1$Variable Cells$$8$SansLarge Medium Small Maximum$C:$13$ Constraints$4\mathrm{.}4$$$50$The manufacturer is interested in maximizing the total profit per day. The sensitivity reportprovided below was generated using the Excel Solver to determine the optimal solution tothis problem. Using the sensitivity report, answer the following questions:$($b$)$ Optimal profit per day:$2$S$45$Cell Name Value Cost Coefficient Increase DecreaseFinal Reduced Obiective Allowable$14$$$421200$Final Shadow ConstraintCell Name Value Price RH$.$ Side$12420\mathrm{.}1052631581209$n$($a$)$ Optimal production quantity of small can per day:$540\mathrm{.}085$TI$428120$ &$571428571$LE$+30$Allowable AllowableIncrease Decrease$($c$)$ Will the current optimal solution stay optimal if the unit profit of large decreases to$40?($d$)$ Will the current optimal solution stay optimal if the unit profit of small increases to$451($e$)$ Will the current optimal solution stay optimal if we add a constraint that says theproduction quantity of medium can per day should be at least $15?($f$)$ Will the shadow price of the constraint related to the supply of metal remain valid ifwe decrease the available supply to $110$ kg per day$($g$)$ Caleulate the effect on the total profit of inereasing the supply of metal to $125$ kg perday$($h$)$ Calculate the effect on the total profit of decreasing the machine's available time to $85$hours per dayQuestion $3.(20$ points$)$ Ethan Stel, nc$.$ has thre factories that manmutacture stelcomponents for three diferent rail projects located at three diferent stes. They want todeternine bow many steel components mst be transportel from each factory to each projectsite. The demand for the stel components for the three projects, A$,$ B$,$ and C$,$ ane $250,300,$and $4500,$ espetively. The production and stipping details are as follows:Production detals:Factory Maximum capacity$2330005003001$Shipping details $($with per$-$umit shiping ost in dollars$)$:ProjectFactary A B C$178226543196$Derelop a linear pmgraming optimization nodel to determine the distrbution plan $($romfacoris to projects$)$ that minimizes the total transyportation cost. $($Do NOT solve themodel.$)$