Problem 3-25 (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 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
fill in the blank 1
O1 +
fill in the blank 2
O2 +
fill in the blank 3
O3 +
fill in the blank 4
C1 +
fill in the blank 5
C2 +
fill in the blank 6
C3
s.t.
fill in the blank 7
O1 +
fill in the blank 8
C1 ≤
fill in the blank 9
Hours avail. 1
fill in the blank 10
O2 +
fill in the blank 11
C2 ≤
fill in the blank 12
Hours avail. 2
fill in the blank 13
O3 +
fill in the blank 14
C3 ≤
fill in the blank 15
Hours avail. 3
fill in the blank 16
O1 +
fill in the blank 17
O2 +
fill in the blank 18
O3 =
fill in the blank 19
Oak
fill in the blank 20
C1 +
fill in the blank 21
C2 +
fill in the blank 22
C3 =
fill in the blank 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 =
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
What is the total cost of completing both projects? If required, round your answer to the nearest dollar.
Total Cost = $
fill in the blank 30
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
.
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
fill in the blank 38
. Therefore, each additional hour of time for cabinetmaker 2 will reduce cost by a total of $
fill in the blank 39
per hour, up to an overall maximum of
fill in the blank 40
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 =
fill in the blank 41
O2 =
fill in the blank 42
O3 =
fill in the blank 43
Cherry C1 =
fill in the blank 44
C2 =
fill in the blank 45
C3 =
fill in the blank 46
What is the total cost of completing both projects? If required, round your answer to the nearest dollar.
Total Cost = $
fill in the blank 47
The change in Cabinetmaker 2’s cost per hour leads to changing
objective function coefficients. This means that the linear program
The new optimal solution
the one above but with a total cost of $
fill in the blank 51
.
