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 $3.)$