Question

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	47	40	27
Hours required to complete all the cherry cabinets	64	52	36
Hours available	40	30	35
Cost per hour	$34	$41	$52

a. 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.

Let	O1 = proportion of Oak cabinets assigned to cabinetmaker 1
	O2 = proportion of Oak cabinets assigned to cabinetmaker 2
	O3 = proportion of Oak cabinets assigned to cabinetmaker 3
	C1 = proportion of Cherry cabinets assigned to cabinetmaker 1
	C2 = proportion of Cherry cabinets assigned to cabinetmaker 2
	C3 = proportion of Cherry cabinets assigned to cabinetmaker 3

Min

__________O1

+

__________O2

+

__________O3

+

__________C1

+

__________C2

+

__________C3

s.t.

__________O1

__________C1

≤

__________

Hours avail. 1

__________O2

+

__________C2

≤

__________

Hours avail. 2

__________O3

+

__________C3

≤

__________

Hours avail. 3

__________O1

+

__________O2

+

__________O3

=

__________

Oak

__________C1

+

__________C2

+

__________C3

=

__________

Cherry

O1, O2, O3, C1, C2, C3 ≥ 0

b. 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
Oak	O1 = _______	O2 = _______	O3 = _______
Cherry	C1 = _______	C2 = _______	C3 = _______

Total cost = $ __________

c. If Cabinetmaker 1 has additional hours available, would the optimal solution change? YES OR NO
Explain.

d. If Cabinetmaker 2 has additional hours available, would the optimal solution change? YES OR NO
Explain.

e. 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
Oak	O1 = _______	O2 = _______	O3 = _______
Cherry	C1 = _______	C2 = _______	C3 = _______

Total cost = $ __________

Explain.

Answer 1

Let O₁ = proportion of Oak cabinets assigned to cabinetmaker 1; O₂ = proportion of Oak cabinets assigned to cabinetmaker 2; O₃ = proportion of Oak cabinets assigned to cabinetmaker 3; C₁ = proportion of Cherry cabinets assigned to cabinetmaker 1; C₂ = proportion of Cherry cabinets assigned to cabinetmaker 2; C₃ = proportion of Cherry cabinets assigned to cabinetmaker 3

b.)

c.) Yes, as all hours of Cabinetmaker 1 has been used up and his cost is minimum.

d.) No, as the hours already available are left

e.)