In: Advanced Math
Problem 8-25 (Algorithmic)
Georgia Cabinets manufactures kitchen cabinets that are sold to
local dealers throughout the...
Problem 8-25 (Algorithmic)
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 |
45 |
41 |
34 |
Hours required to complete all the cherry cabinets |
63 |
44 |
31 |
Hours available |
35 |
25 |
30 |
Cost per hour |
$33 |
$41 |
$60 |
For example, Cabinetmaker 1 estimates that it will take 45 hours
to complete all the oak cabinets and 63 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 35/45 = 0.78, or 78%, of the oak cabinets if it
worked only on oak cabinets. Similarly, Cabinetmaker 1 can only
complete 35/63 = 0.56, or 56%, 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 |
O1 |
+ |
O2 |
+ |
O3 |
+ |
C1 |
+ |
C2 |
+ |
C3 |
|
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s.t. |
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O1 |
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C1 |
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|
≤ |
|
Hours avail. 1 |
|
|
|
O2 |
|
|
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|
+ |
C2 |
|
|
≤ |
|
Hours avail. 2 |
|
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O3 |
|
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|
+ |
C3 |
≤ |
|
Hours avail. 3 |
|
O1 |
+ |
O2 |
+ |
O3 |
|
|
|
|
|
|
= |
|
Oak |
|
|
|
|
|
|
|
C1 |
+ |
C2 |
+ |
C3 |
= |
|
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 = |
O2 = |
O3 = |
Cherry |
C1 = |
C2 = |
C3 = |
Total Cost = $
- If Cabinetmaker 1 has additional hours available, would the
optimal solution change?
Yes
Explain.
The input in the box below will not be graded, but may be reviewed
and considered by your instructor.
- If Cabinetmaker 2 has additional hours available, would the
optimal solution change?
Yes
Explain.
The input in the box below will not be graded, but may be reviewed
and considered by your instructor.
- 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.
|
Cabinetmaker 1 |
Cabinetmaker 2 |
Cabinetmaker 3 |
Oak |
O1 = |
O2 = |
O3 = |
Cherry |
C1 = |
C2 = |
C3 = |
Total Cost = $
Explain.
The input in the box below will not be graded, but may be reviewed
and considered by your instructor.