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 44 32 Hours required to complete all the cherry cabinets 61 46 34 Hours available 35 25 30 Cost per hour $36 $43 $56 For example, Cabinetmaker 1 estimates that it will take 50 hours to complete all the oak cabinets and 61 hours to complete all the cherry cabinets. However, Cabinetmaker 1 only has 35 hours available for the final finishing operation. Thus, Cabinetmaker 1 can only complete 35/50 = 0.7, or 70%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 35/61 = 0.57, or 57%, 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. 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 fill in the blank 1O1 + fill in the blank 2O2 + fill in the blank 3O3 + fill in the blank 4C1 + fill in the blank 5C2 + fill in the blank 6C3 s.t. fill in the blank 7O1 fill in the blank 8C1 ≤ fill in the blank 9 Hours avail. 1 fill in the blank 10O2 + fill in the blank 11C2 ≤ fill in the blank 12 Hours avail. 2 fill in the blank 13O3 + fill in the blank 14C3 ≤ fill in the blank 15 Hours avail. 3 fill in the blank 16O1 + fill in the blank 17O2 + fill in the blank 18O3 = fill in the blank 19 Oak fill in the blank 20C1 + fill in the blank 21C2 + fill in the blank 22C3 = fill in the blank 23 Cherry O1, O2, O3, C1, C2, C3 ≥ 0 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 = fill in the blank 24 O2 = fill in the blank 25 O3 = fill in the blank 26 Cherry C1 = fill in the blank 27 C2 = fill in the blank 28 C3 = fill in the blank 29 Total Cost = $ fill in the blank 30 If Cabinetmaker 1 has additional hours available, would the optimal solution change? If Cabinetmaker 2 has additional hours available, would the optimal solution change? 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 = fill in the blank 35 O2 = fill in the blank 36 O3 = fill in the blank 37 Cherry C1 = fill in the blank 38 C2 = fill in the blank 39 C3 = fill in the blank 40 Total Cost = $ fill in the blank 41

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 44 32 Hours required to complete all the cherry cabinets 61 46 34 Hours available 35 25 30 Cost per hour $36 $43 $56 For example, Cabinetmaker 1 estimates that it will take 50 hours to complete all the oak cabinets and 61 hours to complete all the cherry cabinets. However, Cabinetmaker 1 only has 35 hours available for the final finishing operation. Thus, Cabinetmaker 1 can only complete 35/50 = 0.7, or 70%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 35/61 = 0.57, or 57%, 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. 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 fill in the blank 1O1 + fill in the blank 2O2 + fill in the blank 3O3 + fill in the blank 4C1 + fill in the blank 5C2 + fill in the blank 6C3 s.t. fill in the blank 7O1 fill in the blank 8C1 ≤ fill in the blank 9 Hours avail. 1 fill in the blank 10O2 + fill in the blank 11C2 ≤ fill in the blank 12 Hours avail. 2 fill in the blank 13O3 + fill in the blank 14C3 ≤ fill in the blank 15 Hours avail. 3 fill in the blank 16O1 + fill in the blank 17O2 + fill in the blank 18O3 = fill in the blank 19 Oak fill in the blank 20C1 + fill in the blank 21C2 + fill in the blank 22C3 = fill in the blank 23 Cherry O1, O2, O3, C1, C2, C3 ≥ 0 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 = fill in the blank 24 O2 = fill in the blank 25 O3 = fill in the blank 26 Cherry C1 = fill in the blank 27 C2 = fill in the blank 28 C3 = fill in the blank 29 Total Cost = $ fill in the blank 30 If Cabinetmaker 1 has additional hours available, would the optimal solution change? If Cabinetmaker 2 has additional hours available, would the optimal solution change? 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 = fill in the blank 35 O2 = fill in the blank 36 O3 = fill in the blank 37 Cherry C1 = fill in the blank 38 C2 = fill in the blank 39 C3 = fill in the blank 40 Total Cost = $ fill in the blank 41

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter4: Linear Programming Models

Section: Chapter Questions

Problem 107P

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Question

	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
Hours required to complete all the oak cabinets	50	44	32
Hours required to complete all the cherry cabinets	61	46	34
Hours available	35	25	30
Cost per hour	$36	$43	$56

For example, Cabinetmaker 1 estimates that it will take 50 hours to complete all the oak cabinets and 61 hours to complete all the cherry cabinets. However, Cabinetmaker 1 only has 35 hours available for the final finishing operation. Thus, Cabinetmaker 1 can only complete 35/50 = 0.7, or 70%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 35/61 = 0.57, or 57%, 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.

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

fill in the blank 1O₁

fill in the blank 2O₂

fill in the blank 3O₃

fill in the blank 4C₁

fill in the blank 5C₂

fill in the blank 6C₃

s.t.

fill in the blank 7O₁

fill in the blank 8C₁

≤

fill in the blank 9

Hours avail. 1

fill in the blank 10O₂

fill in the blank 11C₂

≤

fill in the blank 12

Hours avail. 2

fill in the blank 13O₃

fill in the blank 14C₃

≤

fill in the blank 15

Hours avail. 3

fill in the blank 16O₁

fill in the blank 17O₂

fill in the blank 18O₃

fill in the blank 19

Oak

fill in the blank 20C₁

fill in the blank 21C₂

fill in the blank 22C₃

fill in the blank 23

Cherry

O1, O2, O3, C1, C2, C3 ≥ 0

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 = fill in the blank 24	O₂ = fill in the blank 25	O₃ = fill in the blank 26
Cherry	C₁ = fill in the blank 27	C₂ = fill in the blank 28	C₃ = fill in the blank 29

Total Cost = $ fill in the blank 30

If Cabinetmaker 1 has additional hours available, would the optimal solution change?
If Cabinetmaker 2 has additional hours available, would the optimal solution change?

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 = fill in the blank 35	O₂ = fill in the blank 36	O₃ = fill in the blank 37
Cherry	C₁ = fill in the blank 38	C₂ = fill in the blank 39	C₃ = fill in the blank 40

Total Cost = $ fill in the blank 41

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