Georgia Cabinets manufactures kitchen cabinets that are sold tolocal dealers throughout the Southeast. Because of a large backlogof orders for oak and cherry cabinets, the company decided tocontract with three smaller cabinetmakers to do the final finishingoperation. For the three cabinetmakers, the number of hoursrequired to complete all the oak cabinets, the number of hoursrequired to complete all the cherry cabinets, the number of hoursavailable for the final finishing operation, and the cost per hourto 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 tocomplete all the oak cabinets and 64 hours to complete all thecherry cabinets. However, Cabinetmaker 1 only has 40 hoursavailable for the final finishing operation. Thus, Cabinetmaker 1can only complete 40/47 = 0.85,or 85%, of the oak cabinets if itworked only on oak cabinets. Similarly, Cabinetmaker 1 can onlycomplete 40/64 = 0.63, or 63%, of the cherry cabinets if it workedonly on cherry cabinets.
Formulate a linear programming model that can be used todetermine the percentage of the oak cabinets and the percentage ofthe cherry cabinets that should be given to each of the threecabinetmakers in order to minimize the total cost of completingboth projects. If the constant is "1" it must be entered in thebox.
| Let | O1 = percentage of Oak cabinets assigned tocabinetmaker 1 |
| O2 = percentage of Oak cabinets assigned tocabinetmaker 2 |
| O3 = percentage of Oak cabinets assigned tocabinetmaker 3 |
| C1 = percentage of Cherry cabinets assigned tocabinetmaker 1 |
| C2 = percentage of Cherry cabinets assigned tocabinetmaker 2 |
| C3 = percentage of Cherry cabinets assigned tocabinetmaker 3 |
| Min | O1 | + | O2 | + | O3 | + | C1 | + | C2 | + | C3 | | | |
| s.t. | | | | | | | | | | | | | | |
| O1 | | | | | + | C1 | | | | | ? | | Hours avail. 1 |
| | | O2 | | | | | + | C2 | | | ? | | Hours avail. 2 |
| | | | | O3 | | | | | + | 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 percentage of the oakcabinets and what percentage of the cherry cabinets should beassigned to each cabinetmaker? If required, round your answers tothree decimal places. If your answer is zero, enter "0".
| Cabinetmaker 1 | Cabinetmaker 2 | Cabinetmaker 3 |
|---|
| Oak | O1 = | O2 = | O3 = |
| Cherry | C1 = | C2 = | C3 = |
What is the total cost of completing both projects? If required,round your answer to the nearest dollar.
Total Cost = $
If Cabinetmaker 1 has additional hours available, would theoptimal solution change? If required, round your answers to threedecimal places. If your answer is zero, enter "0". Explain.
because Cabinetmaker 1 has of hours. Alternatively, thedual value is which means that adding one hour to this constraintwill decrease total cost by $.
If Cabinetmaker 2 has additional hours available, would theoptimal solution change? If required, round your answers to threedecimal places. If your answer is zero, enter "0". Use a minus signto indicate the negative figure. Explain.
because Cabinetmaker 2 has a of . Therefore, eachadditional hour of time for cabinetmaker 2 will reduce total costby $ per hour, up to a maximum of hours.
Suppose Cabinetmaker 2 reduced its cost to $38 per hour. Whateffect 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 = | O2 = | O3 = |
| Cherry | C1 = | C2 = | C3 = |
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 changingobjective function coefficients. This means that the linearprogram
The new optimal solution the one above but with a total cost of$ .
