Problem $8-25($Algorithmic$)$

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$474027$Hours required to complete all the cherry cabinets$645236$Hours available$403035$Cost per hour$$34$$$41$$$52$

For example, Cabinetmaker $1$ estimates that it will take $47$ hours to complete all the oak cabinets and $64$ 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 $40/47=0\mathrm{.}85,$ or $85\%,$ of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker $1$ can only complete $40/64=0\mathrm{.}63,$ or $63\%,$ of the cherry cabinets if it worked only on cherry cabinets.

Formulate a linear programming model that can be used to determine the proportion of the oak cabinets and the proportion 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.

LetO$1=$ proportion of Oak cabinets assigned to cabinetmaker $1$O$2=$ proportion of Oak cabinets assigned to cabinetmaker $2$O$3=$ proportion of Oak cabinets assigned to cabinetmaker $3$C$1=$ proportion of Cherry cabinets assigned to cabinetmaker $1$C$2=$ proportion of Cherry cabinets assigned to cabinetmaker $2$C$3=$ proportion of Cherry cabinets assigned to cabinetmaker $3$

Minfill in the blank $1$O$1+$fill in the blank $2$O$2+$fill in the blank $3$O$3+$fill in the blank $4$C$1+$fill in the blank $5$C$2+$fill in the blank $6$C$3$s$.$t$.$fill in the blank $7$O$1$fill in the blank $8$C$1$fill in the blank $9$Hours avail. $1$fill in the blank $10$O$2+$fill in the blank $11$C$2$fill in the blank $12$Hours avail. $2$fill in the blank $13$O$3+$fill in the blank $14$C$3$fill in the blank $15$Hours avail. $3$fill in the blank $16$O$1+$fill in the blank $17$O$2+$fill in the blank $18$O$3=$fill in the blank $19$Oakfill in the blank $20$C$1+$fill in the blank $21$C$2+$fill in the blank $22$C$3=$fill in the blank $23$CherryO$1,$ O$2,$ O$3,$ C$1,$ C$2,$ C$30$

Solve the model formulated in part $($a$).$ What proportion of the oak cabinets and what proportion of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects? If required, round your answers for the proportions to three decimal places, and for the total cost to two decimal places.

Cabinetmaker $1$Cabinetmaker $2$Cabinetmaker $3$OakO$1=$ fill in the blank $24$O$2=$ fill in the blank $25$O$3=$ fill in the blank $26$CherryC$1=$ fill in the blank $27$C$2=$ fill in the blank $28$C$3=$ fill in the blank $29$

Total Cost $=$ $$$fill in the blank $30$

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

YesNo

Explain.

The input in the box below will not be graded, but may be reviewed and considered by your instructor.

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

YesNo

Explain.

The input in the box below will not be graded, but may be reviewed and considered by your instructor.

Suppose Cabinetmaker $2$ reduced its cost to $$38$ per hour. What effect would this change have on the optimal solution? If required, round your answers for the proportions to three decimal places, and for the total cost to two decimal places.

Cabinetmaker $1$Cabinetmaker $2$Cabinetmaker $3$OakO$1=$ fill in the blank $35$O$2=$ fill in the blank $36$O$3=$ fill in the blank $37$CherryC$1=$ fill in the blank $38$C$2=$ fill in the blank $39$C$3=$ fill in the blank $40$

Total Cost $=$ $$$fill in the blank $41$

Explain.

The input in the box below will not be graded, but may be reviewed and considered by your instructor.