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 51 36 Hours available 40 30 35 Cost per hour $34 $41 $52 For example, Cabinetmaker 1 estimates 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 = 0.85, or 85%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/64 = 0.63, 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 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. If the constant is "1" it must be entered in the box. Let O1 = percentage of Oak cabinets assigned to cabinetmaker 1 O2 = percentage of Oak cabinets assigned to cabinetmaker 2 O3 = percentage of Oak cabinets assigned to cabinetmaker 3 C1 = percentage of Cherry cabinets assigned to cabinetmaker 1 C2 = percentage of Cherry cabinets assigned to cabinetmaker 2 C3 = percentage 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 ____20C1 + _____21C2 + ____22C3 = ____ 23 Cherry O1, O2, O3, C1, C2, C3 ≥ 0 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. If your answer is zero, enter "0". Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3 Oak O1 = O2 = O3 = Cherry C1 = C2 = C3 = What is the total cost of completing both projects? If required, round your answer to the nearest dollar. Total Cost = $ ______ 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.

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 51 36 Hours available 40 30 35 Cost per hour $34 $41 $52 For example, Cabinetmaker 1 estimates 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 = 0.85, or 85%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/64 = 0.63, 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 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. If the constant is "1" it must be entered in the box. Let O1 = percentage of Oak cabinets assigned to cabinetmaker 1 O2 = percentage of Oak cabinets assigned to cabinetmaker 2 O3 = percentage of Oak cabinets assigned to cabinetmaker 3 C1 = percentage of Cherry cabinets assigned to cabinetmaker 1 C2 = percentage of Cherry cabinets assigned to cabinetmaker 2 C3 = percentage 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 20C1 + _21C2 + 22C3 = 23 Cherry O1, O2, O3, C1, C2, C3 ≥ 0 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. If your answer is zero, enter "0". Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3 Oak O1 = O2 = O3 = Cherry C1 = C2 = C3 = What is the total cost of completing both projects? If required, round your answer to the nearest dollar. Total Cost = $ ____ 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.

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter2: Introduction To Spreadsheet Modeling

Section: Chapter Questions

Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...

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Related questions

Question

	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	51	36
Hours available	40	30	35
Cost per hour	$34	$41	$52

For example, Cabinetmaker 1 estimates 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 = 0.85, or 85%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/64 = 0.63, 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 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. If the constant is "1" it must be entered in the box.

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

____20C₁

_____21C₂

____22C₃

____ 23

Cherry

O₁, O₂, O₃, C₁, C₂, C₃ ≥ 0

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. If your answer is zero, enter "0".

Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3

Oak O₁ = O₂ = O₃ =

Cherry C₁ = C₂ = C₃ =

What is the total cost of completing both projects? If required, round your answer to the nearest dollar.

Total Cost = $ ______
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.

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

___because Cabinetmaker 1 has

of ____ hours. Alternatively, the dual value is ____ which means that adding one hour to this constraint will decrease total cost by $____.
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

of _____ . Therefore, each additional hour of time for cabinetmaker 2 will reduce cost by a total of $ per hour, up to an overall maximum of ------total hours.
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. If your answer is zero, enter "0".

Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3

Oak O₁ = O₂ = O₃ =

Cherry C₁ = C₂ = C₃ =

What is the total cost of completing both projects? If required, round your answer to the nearest dollar.

Total Cost = $

The change in Cabinetmaker 2’s cost per hour leads to changing