Problem 8-25 (Algorithmic)
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 | 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 hoursto complete all the oak cabinets and 63 hours to complete all thecherry cabinets. However, Cabinetmaker 1 only has 35 hoursavailable for the final finishing operation. Thus, Cabinetmaker 1can only complete 35/45 = 0.78, or 78%, of the oak cabinets if itworked only on oak cabinets. Similarly, Cabinetmaker 1 can onlycomplete 35/63 = 0.56, or 56%, of the cherry cabinets if it workedonly on cherry cabinets.
- Formulate a linear programming model that can be used todetermine the proportion of the oak cabinets and the proportion ofthe cherry cabinets that should be given to each of the threecabinetmakers in order to minimize the total cost of completingboth projects.
| Let | O1 = proportion of Oak cabinets assigned tocabinetmaker 1 |
| O2 = proportion of Oak cabinets assigned tocabinetmaker 2 |
| O3 = proportion of Oak cabinets assigned tocabinetmaker 3 |
| C1 = proportion of Cherry cabinets assigned tocabinetmaker 1 |
| C2 = proportion of Cherry cabinets assigned tocabinetmaker 2 |
| C3 = proportion 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 proportion of theoak cabinets and what proportion of the cherry cabinets should beassigned to each cabinetmaker? What is the total cost of completingboth projects? If required, round your answers for the proportionsto three decimal places, and for the total cost to two decimalplaces.
| Cabinetmaker 1 | Cabinetmaker 2 | Cabinetmaker 3 |
|---|
| Oak | O1 = | O2 = | O3 = |
| Cherry | C1 = | C2 = | C3 = |
Total Cost = $
- If Cabinetmaker 1 has additional hours available, would theoptimal solution change?
Yes
Explain.
The input in the box below will not be graded, but may be reviewedand considered by your instructor.
- If Cabinetmaker 2 has additional hours available, would theoptimal solution change?
Yes
Explain.
The input in the box below will not be graded, but may be reviewedand considered by your instructor.
- Suppose Cabinetmaker 2 reduced its cost to $39 per hour. Whateffect would this change have on the optimal solution? If required,round your answers for the proportions to three decimal places, andfor 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 reviewedand considered by your instructor.
