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
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 cabinetmaker1 | |||||||||||||||
| C2 = percentage of Cherry cabinets assigned to cabinetmaker2 | |||||||||||||||
| C3 = percentage of Cherry cabinets assigned to cabinetmaker3 | |||||||||||||||
| Min | fill in the blank O1 | fill in the blank O2 | fill in the blank O3 | fill in the blank C1 | fill in the blank C2 | fill in the blank C3 | |||||||||
| s.t. | |||||||||||||||
| fill in the blank O1 | fill in the blank C1 | ≤ | fill in the blank | Hours avail. 1 | |||||||||||
| fill in the blank O2 | fill in the blank C2 | ≤ | fill in the blank | Hours avail. 2 | |||||||||||
| fill in the blank 13O3 | fill in the blank C3 | ≤ | fill in the blank | Hours avail. 3 | |||||||||||
| fill in the blank O1 | fill in the blank O2 | fill in the blank 18O3 | fill in the blank | Oak | |||||||||||
| fill in the blank C1 | fill in the blank C2 | fill in the blank C3 | fill in the blank | 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 | O2 = fill in the blank | O3 = fill in the blank |
| Cherry | C1 = fill in the blank | C2 = fill in the blank | C3 = fill in the blank |
What is the total cost of completing both projects? If required, round your answer to the nearest dollar.
Total Cost = $ fill in the blank
C. 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 ______hours. Alternatively, the dual value is____ which means that adding one hour to this constraint will decrease total cost by $________.
D .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.
e. 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 | O2 = fill in the blank | O3 = fill in the blank |
| Cherry | C1 = fill in the blank | C2 = fill in the blank | C3 = fill in the blank |
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 ________
objective function coefficients. This means that the linear program ________
The new optimal solution ______________
the one above but with a total cost of $_________
