Overview
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 1Cabinetmaker 2Cabinetmaker 3Hours required to complete all the oak cabinets504430Hours required to complete all the cherry cabinets604333Hours available402530Cost per hour$32$43$59
For example, Cabinetmaker 1 estimates it will take 50 hours to complete all the oak cabinets and 60 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 40/50 = 0.8, or 80%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/60 = 0.67, or 67%, 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. LetO1 = percentage of Oak cabinets assigned to cabinetmaker 1O2 = percentage of Oak cabinets assigned to cabinetmaker 2O3 = percentage of Oak cabinets assigned to cabinetmaker 3C1 = percentage of Cherry cabinets assigned to cabinetmaker 1C2 = percentage of Cherry cabinets assigned to cabinetmaker 2C3 = percentage of Cherry cabinets assigned to cabinetmaker 3 Minfill in the blank 1O1+fill in the blank 2O2+fill in the blank 3O3+fill in the blank 4C1+fill in the blank 5C2+fill in the blank 6C3s.t.fill in the blank 7O1+fill in the blank 8C1?fill in the blank 9Hours avail. 1fill in the blank 10O2+fill in the blank 11C2?fill in the blank 12Hours avail. 2fill in the blank 13O3+fill in the blank 14C3?fill in the blank 15Hours avail. 3fill in the blank 16O1+fill in the blank 17O2+fill in the blank 18O3=fill in the blank 19Oakfill in the blank 20C1+fill in the blank 21C2+fill in the blank 22C3=fill in the blank 23CherryO1, 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 1Cabinetmaker 2Cabinetmaker 3OakO1 = fill in the blank 24O2 = fill in the blank 25O3 = fill in the blank 26CherryC1 = fill in the blank 27C2 = fill in the blank 28C3 = 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. YesNoYes because Cabinetmaker 1 has slacksurplusdual valuedual value of fill in the blank 33 . Therefore, each additional hour of time for cabinetmaker 1 will reduce total cost by $ fill in the blank 34 per hour, up to a maximum of fill in the blank 35hours.
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. YesNoYes because Cabinetmaker 2 has a slacksurplusdual valuedual value 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 40total 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 1Cabinetmaker 2Cabinetmaker 3OakO1 = fill in the blank 41O2 = fill in the blank 42O3 = fill in the blank 43CherryC1 = fill in the blank 44C2 = fill in the blank 45C3 = 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 onetwothreetwo objective function coefficients. This means that the linear program solution given can be used again.needs to be rerun.needs to be rerun. The new optimal solution differs fromis the same asis the same as the one above but with a total cost of $ fill in the blank 51.