A linear programming computer package is needed. 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 Min Hours required to complete all the oak cabinets Hours required to complete all the cherry cabinets Hours available oak Cost per hour s.t. hours available 1 cherry hours available 2 hours available 3 50 01, 02, 03, C1, C2, C3 ≥ 0 60 01 02 03 C1 C2 C3 total cost $ 40 $36 42 For example, Cabinetmaker 1 estimates 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. 48 (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 01 = percentage of oak cabinets assigned to cabinetmaker 1, 02= percentage of oak cabinets assigned to cabinetmaker 2, 03= 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, and C3= percentage of cherry cabinets assigned to cabinetmaker 3.) 30 $42 30 35 35 $55 (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? What is the total cost (in $) of completing both projects? (Round your percentage values to one decimal place.) % % % % % %
A linear programming computer package is needed. 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 Min Hours required to complete all the oak cabinets Hours required to complete all the cherry cabinets Hours available oak Cost per hour s.t. hours available 1 cherry hours available 2 hours available 3 50 01, 02, 03, C1, C2, C3 ≥ 0 60 01 02 03 C1 C2 C3 total cost $ 40 $36 42 For example, Cabinetmaker 1 estimates 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. 48 (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 01 = percentage of oak cabinets assigned to cabinetmaker 1, 02= percentage of oak cabinets assigned to cabinetmaker 2, 03= 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, and C3= percentage of cherry cabinets assigned to cabinetmaker 3.) 30 $42 30 35 35 $55 (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? What is the total cost (in $) of completing both projects? (Round your percentage values to one decimal place.) % % % % % %
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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Transcribed Image Text:A linear programming computer package is needed.
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.
Min
s.t.
Hours required to complete
all the oak cabinets
Hours required to complete
all the cherry cabinets
oak
Hours available
hours available 1
cherry
hours available 2
Cost per hour
hours available 3
8 8 8 8 8 8
01, 02, 03, C1, C2, C3 ≥ 0
total cost
Cabinetmaker 1
$
50
60
LUJLL
40
For example, Cabinetmaker 1 estimates 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.
$36
(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 01 = percentage of oak cabinets assigned to cabinetmaker 1, 02 = percentage of oak cabinets assigned to cabinetmaker 2, 03 = 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, and C3= percentage of cherry cabinets assigned to cabinetmaker 3.)
Cabinetmaker 2
42
48
30
$42
Cabinetmaker 3
30
35
35
$55
(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? What is the total cost (in $) of completing both projects? (Round your
percentage values to one decimal place.)
%
%
%
%
%
%
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Transcribed Image Text:(c) If Cabinetmaker 1 has additional hours available, would the optimal solution change? Explain.
O Yes, each additional hour of time for cabinetmaker 1 will reduce the total cost by $5 per hour.
O Yes, each additional hour of time for cabinetmaker 1 will reduce the total cost by $-1.75 per hour.
O No, cabinetmaker 1 has a slack of 26.458 hours, so increasing cabinetmaker 1's time will not reduce costs.
O No, cabinetmaker 1 has a slack of 37.5 hours, so increasing cabinetmaker 1's time will not reduce costs.
(d) If Cabinetmaker 2 has additional hours available, would the optimal solution change? Explain.
O Yes, each additional hour of time for cabinetmaker 2 will reduce the total cost by $5 per hour.
O Yes, each additional hour of time for cabinetmaker 2 will reduce the total cost by $-1.75 per hour.
No, cabinetmaker 2 has a slack of 26.458 hours, so increasing cabinetmaker 2's time will not reduce costs.
O No, cabinetmaker 2 has a slack of 37.5 hours, so increasing cabinetmaker 2's time will not reduce costs.
(e) Suppose Cabinetmaker 2 reduced its cost to $38 per hour. What effect would this change have on the optimal solution? Explain.
The new objective function coefficients for O2 and C2 are
and
respectively. The optimal solution ---Select---
and has a value of $
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