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:

1. 	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
   Hours required to complete all the oak cabinets	47	40	32
   Hours required to complete all the cherry cabinets	60	43	36
   Hours available	40	25	30
   Cost per hour	$38	$42	$54

For example, Cabinetmaker 1 estimates that it will take 47 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 40/47 = 0.85, or 85%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/60 = 0.67, 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 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

1. 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.

1. 	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
   Oak	O1 =	O2 =	O3 =
   Cherry	C1 =	C2 =	C3 =

   
   Total Cost = $
2. If Cabinetmaker 1 has additional hours available, would the optimal solution change? Yes or no

d.If Cabinetmaker 2 has additional hours available, would the optimal solution change? Yes or No

e.Suppose Cabinetmaker 2 reduced its cost to $39 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.

1. 	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
   Oak	O1 =	O2 =	O3 =
   Cherry	C1 =	C2 =	C3 =

   
   Total Cost = $  

Answer 1

Final answers are highlighted in yellow

If the cabinetmaker 1 has additional hours then the optimal solution will not change as already in the optimal solution cabinetmaker 1's full hours are not used.

If the cabinetmaker 2 has additional hours then the optimal solution will change as already in the optimal solution cabinetmaker 1's full hours are used.

The optimal solution remains same but the total cost has reduced as rate for cabinetmaker 2 has reduced. The solution did not change as hours for both maker 2 and maker 3 has been fully used up.