A linear programming computer package is needed.

Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the Southeast. Because of a large backlog of orders for oak and cherry cabinets, the company decided to contract with three smaller cabinetmakers to do the final finishing operation. For the three cabinetmakers, the number of hours required to complete all the oak cabinets, the number of hours required to complete all the cherry cabinets, the number of hours available for the final finishing operation, and the cost per hour to perform the work are shown here.

Cabinetmaker $1$ Cabinetmaker $2$ Cabinetmaker $3$

Hours required to complete

all the oak cabinets $504230$

Hours required to complete

all the cherry cabinets $604835$

Hours available $403035$

Cost per hour $$36$ $$42$ $$55$

For example, Cabinetmaker $1$ estimates it will take $50$ hours to complete all the oak cabinets and $60$ hours to complete all the cherry cabinets. However, Cabinetmaker $1$ only has $40$ hours available for the final finishing operation. Thus, Cabinetmaker $1$ can only complete $40/50=0\mathrm{.}80,$ or $80\%,$ of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker $1$ can only complete $40/60=0\mathrm{.}67,$ or $67\%,$ of the cherry cabinets if it worked only on cherry cabinets.

$($a$)$

Formulate a linear programming model that can be used to determine the percentage of the oak cabinets and the percentage of the cherry cabinets that should be given to each of the three cabinetmakers in order to minimize the total cost of completing both projects. $($Let O$1=$ percentage of oak cabinets assigned to cabinetmaker $1,$ O$2=$ percentage of oak cabinets assigned to cabinetmaker $2,$ O$3=$ percentage of oak cabinets assigned to cabinetmaker $3,$ C$1=$ percentage of cherry cabinets assigned to cabinetmaker $1,$ C$2=$ percentage of cherry cabinets assigned to cabinetmaker $2,$ and C$3=$ percentage of cherry cabinets assigned to cabinetmaker $3.)$

Min

s$.$t$.$ hours available $1$

hours available $2$

hours available $3$

oak

cherry

O$1,$ O$2,$ O$3,$ C$1,$ C$2,$ C$3>=0$

$($b$)$

Solve the model formulated in part $($a$).$ What percentage of the oak cabinets and what percentage of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost $($in $$)$ of completing both projects? $($Round your percentage values to one decimal place.$)$

O$1\%$ O$2\%$ O$3\%$ C$1\%$ C$2\%$ C$3\%$ total cost $

$($c$)$

If Cabinetmaker $1$ has additional hours available, would the optimal solution change? Explain.

Yes, each additional hour of time for cabinetmaker $1$ will reduce the total cost by $$5$ per hour. Yes, each additional hour of time for cabinetmaker $1$ will reduce the total cost by $$-1\mathrm{.}75$ per hour. No$,$ cabinetmaker $1$ has a slack of $26\mathrm{.}458$ hours, so increasing cabinetmaker $1\text{'}$s time will not reduce costs. No$,$ cabinetmaker $1$ has a slack of $37\mathrm{.}5$ hours, so increasing cabinetmaker $1\text{'}$s time will not reduce costs.

$($d$)$

If Cabinetmaker $2$ has additional hours available, would the optimal solution change? Explain.

Yes, each additional hour of time for cabinetmaker $2$ will reduce the total cost by $$5$ per hour. Yes, each additional hour of time for cabinetmaker $2$ will reduce the total cost by $$-1\mathrm{.}75$ per hour. No$,$ cabinetmaker $2$ has a slack of $26\mathrm{.}458$ hours, so increasing cabinetmaker $2\text{'}$s time will not reduce costs. No$,$ cabinetmaker $2$ has a slack of $37\mathrm{.}5$ hours, so increasing cabinetmaker $2\text{'}$s time will not reduce costs.

$($e$)$

Suppose Cabinetmaker $2$ reduced its cost to $$38$ per hour. What effect would this change have on the optimal solution? Explain.

The new objective function coefficients for O$2$ and C$2$ are and $,$ respectively. The optimal solution

and has a value of $ $.$