Problem 3-25

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	50	42	30
Hours required to complete all the cherry cabinets	60	48	35
Hours available	40	30	35
Cost per hour	$36	$42	$55

For example, Cabinetmaker 1 estimates that 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 = 0.80, or 80%, 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 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
	C 3 = percentage of Cherry cabinets assigned to cabinetmaker 3

1. Min

   fill 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

   Oak

   fill 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

   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? If required, round your answers to three decimal places.

	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
Oak	O 1 = fill in the blank 24	O 2 = fill in the blank 25	O 3 = fill in the blank 26
Cherry	C 1 = fill in the blank 27	C 2 = fill in the blank 28	C 3 = fill in the blank 29

b. What is the total cost of completing both projects? If required, round your answer to the nearest dollar. Total Cost = $ _____

c. If Cabinetmaker 1 has additional hours available, would the optimal solution change? If required, round your answers to three decimal places. If your answer is zero, enter "0". Explain. because Cabinetmaker 1 has of fill in the blank 33 hours. Alternatively, the dual value is fill in the blank 34 which means that adding one hour to this constraint will decrease total cost by $ fill in the blank 35 .

d. If Cabinetmaker 2 has additional hours available, would the optimal solution change? If required, round your answers to three decimal places. If your answer is zero, enter "0". Use a minus sign to indicate the negative figure. Explain. because Cabinetmaker 2 has a dual value of fill in the blank 37. Therefore, each additional hour of time for cabinetmaker 2 will reduce cost by a total of $fill in the blank 38 per hour, up to an overall maximum of _____ total hours.

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 to three decimal places.

	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
Oak	O 1 = fill in the blank	O 2 = fill in the blank	O 3 = fill in the blank
Cherry	C 1 = fill in the blank	C 2 = fill in the blank	C 3 = fill in the blank

What is the total cost of completing both projects? If required, round your answer to the nearest dollar. Total Cost = $ Explain. The change in Cabinetmaker 2s cost per hour leads to changing _______ objective function coefficients. This means that the linear program _________ The new optimal solution is ________ before but with a total cost of $ __________.