Q$5.$ Consider the following LP problem, in which $x$ and $Y$ denote the number of units of products $x$ and $Y$ to produce, respectively.

Maximize profit $=$$$4x+$$$5Y$

Subject to the constraints

$x+2Y10($labor available, in hours$)$

$6x+6Y36($material available, in pounds$)$

$8x+4Y40($storage available, in square feet$)$

$x,Y0$

The Excel sensitivity report for the problem is shown below. Calculate and explain what will be the effect of each of the given changes. Each question is independent of the other questions.

Varable Cells

$\backslash$table$[[$Cell$,$Name,Final,Reduced,Objective,Allowable,Allowrable$],[$Salue$,$Cost,Coeflicient,Increase,Decrease,,$],[$SBS$4$ Solution value X$,200,0\mathrm{.}00,400,100,150,],[$SCS$4$ Solution value Y$,400,0\mathrm{.}00,5\mathrm{.}00,300,100,]]$

Constraints

$\backslash$table$[[$Cell$,$Name,$\backslash$table$[[$Final$],[$Value$]],\backslash$table$[[$Shadow$],[$Price$]],\backslash$table$[[$Constraint$],[$R$.$H$.$ Side$]],\backslash$table$[[$Allowable$],[$Increase$]],\backslash$table$[[$Allowable$],[$Decrease$]]],[$SDS$7,$Labor,$10\mathrm{.}00,1\mathrm{.}00,10\mathrm{.}00,2\mathrm{.}00,2\mathrm{.}00],[$SDS$8,$Material,$36\mathrm{.}00,0\mathrm{.}50,36\mathrm{.}00,4\mathrm{.}00,6\mathrm{.}00],[$SDS$9,$Storage,$32\mathrm{.}00,0\mathrm{.}00,40\mathrm{.}00,1E+30,8\mathrm{.}00]]$

$($a$).$ Write the optimal solution and objective function value of the problem.

$($b$).$ You acquire $2$ additional pounds of material.

$($c$).$ You acquire $1\mathrm{.}5$ additional hours of labor.

$[2]$

$($d$).$ You give up $1$ hour of labor and get $1\mathrm{.}5$ pounds of material. What will be the effect of this combined change?

$[2]$