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 | 52 | 36 |
| Hours available | 40 | 30 | 35 |
| Cost per hour | $34 | $41 | $52 |
a. Formulate a linear programming model that can be usedto determine the proportion of the oak cabinets and the proportionof the 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 |
b. Solve the model formulated in part (a). Whatproportion of the oak cabinets and what proportion of the cherrycabinets should be assigned to each cabinetmaker? What is the totalcost of completing both projects? If required, round your answersfor the proportions to three decimal places, and for the total costto two decimal places.
| Cabinetmaker 1 | Cabinetmaker 2 | Cabinetmaker 3 |
|---|
| Oak | O1 = _______ | O2 = _______ | O3 = _______ |
| Cherry | C1 = _______ | C2 = _______ | C3 = _______ |
Total cost = $ __________
c. If Cabinetmaker 1 has additional hours available,would the optimal solution change? YES OR NO
Explain.
d. If Cabinetmaker 2 has additional hours available,would the optimal solution change? YES OR NO
Explain.
e. Suppose Cabinetmaker 2 reduced its cost to $38 perhour. What effect would this change have on the optimal solution?If required, round your answers for the proportions to threedecimal 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 = $ __________
Explain.
