Question $3$: Mondo produces motorcycles at three plants. At each plant, the labor, raw material, and production costs $($excluding labor cost$)$ required to build a motorcycle are as shown in the table below. Each plant has sufficient machine capacity to produce up to $750$ motorcycles per week. Each of Mondo's workers can work up to $40$ hours per week and is paid $$12\mathrm{.}50$ per hour worked. Mondo has a total of $525$ workers and now owns $9,400$ units of raw material. Each week, at least $1,400$ Mondos must be produced.

Let $x_1=$ motorcycles produced at plant $1$;$x_2=$ motorcycles produced at plant $2$ ; and $x_3=$ motorcycles produced at plant $3.$

$\backslash$table$[[$Plant$,$Labor Hours,Raw Materials $($Units$),$Production Cost $($$$)],[1,20,5,50],[2,16,8,80],[3,10,7,100]]$

$($a$)$ Formulate the problem. Interpret the optimal solution from the Excel output given below. Use the Excel output to answer the following questions:

$($b$)$ What would be the new optimal solution to the problem if the production cost at plant $1$ were only $$40?$

$($c$)$ By how much would Mondo's cost change if the capacity of plant $3$ were increased by $100$ motorcycles?

$($d$)$ By how much would Mondo's cost change if it had to produce $100$ more motorcycles?

$($e$)$ By how much would Mondo's cost change if it had $9000$ units of raw materials? If I had $10500$ units of raw materials?

$\backslash$table$[[$Objective Cell $($Min$),,],[$Cell$,$Name,$\backslash$table$[[$Original$],[$Value$]],$Final Value$],[$$$K$$$1,$Total cost,$0,357750]]$

Variable Cells

$\backslash$table$[[$Cell$,$Name,$\backslash$table$[[$Original$],[$Value$]],$Final Value,Integer$],[$$$F$$$3,x1,0,350,$Contin$],[$$$G$$$3,x2,0,300,$Contin$],[$$$H$$$3,x3,0,750,$Contin$]]$

$\backslash$table$[[$Cell$,$Name,Cell Value,Formula,Status,Slack$],[$$$1$$$5,$Labor hrs$,19300,$$$1$$$5$$$K$$$5,$Not Binding,$1700],[$$$1$$$6,$Raw mat.,$9400,$$$1$$$6$$$K$$$6,$Binding,$0],[$$$1$$$7,$Demand,$1400,$$$1$$$7$$$K$$$7,$Binding,$0],[$$$1$$$8,$Plant$1$Cap,$350,$$$1$$$8$$$K$$$8,$Not Binding,$400],[,$Plant$2$Cap,$300,$$$1$$$9$$$K$$$9,$Not Binding,$450],[$$$$10,$Plant$3$Cap,$750,$$$$10$$$K$$$10,$Binding,$0]]$

$\backslash$table$[[$Variable Cells$],[$Cell$,$Name,$\backslash$table$[[$Final$],[$Value$]],\backslash$table$[[$Reduced$],[$Cost$]],\backslash$table$[[$Objective$],[$Coefficient$]],\backslash$table$[[$Allowable$],[$Increase$]],\backslash$table$[[$Allowable$],[$Decrease$]]],[$$$$$3,x1,350,0,300,1E+30,20],[$$$G$$$3,x2,300,0,280,20,92\mathrm{.}5],[$$$H$$$3,x3,750,0,225,61\mathrm{.}66666667,1E+30$