Problem $1$: SABA toy manufacturers make four different types of stuffed animals. The following LP model has been developed for the company.

$\backslash$table$[[$Let$,x1=$ Number of stuffed bears,,,,$],[x2=$ Number of stuffed rabbits,,,,$],[x3=$ Number of stuffed dogs,,,,$],[x4=$ Number of stuffed cats,,,,$],[,5,,,,,,],[$The LP Model,PRODUCTS,,,$],[x1,x2,,x3,x4,,$Available,RESOURCES$],[$Constraint $1,3,1,,1,1,,100,$Stuffing in pounds$],[$Constraint $2,2,2,,3,2,,250,$Yards of fabric$],[$Constraint $3,4,2,,1,2,,300,$Assembly time in hours$],[$Constraint $4,1,4,,2,3.,,200,$Pieces of trim$],[$Profit$/$unit$,8\mathrm{.}00,5\mathrm{.}00,$$$,6\mathrm{.}00,$$$,4\mathrm{.}00,,,]]$

a$.$ Solve the above Linear programming problem and obtain sensitivity analysis.

DO NOT CHANGE ANY OF THE UNITS

Use your sensitivity output to answer the following questions.

b$.$ Which constraints are$($is$)$ non$-$binding and what does it mean?

Constraint $3$ is non$-$binding it means that it does not affect the optimal solution

c$.$ If the profit per unit of stuffed rabbits is increased by $$2,$ will the optimal solution change?

What would be the new value of the objective function? Give the new value here

No$,$ the optimal solution will not change because it can be increased by up to $5$ without changing the solution.

The new objective function would be $560\mathrm{.}526316$

d$.$ As you note, at optimum no stuffed cats $($X$4)$ are being produces.

What should the profit per unit of stuffed cats $($X$4),$ so it becomes profitable to produce it$?$ Explain how you arrived at the value.

The profit per unit for stuffed cats should be $4\mathrm{.}842105263$

The allowable increase for stuffed cats is $0\mathrm{.}842105263$ therefore increasing it by that much should make it profitable to produce.

e$.$ Fifty yards of fabric can be obtained at $1\mathrm{.}20$ above the current cost.

Should the additional yards of fabric be obtained at the quoted cost? Why or why not? Fully explain.