3-25 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 50 4. 42 30 all the oak cabinets Hours required to complete 60 481 35 all the cherry cabinets Hours available Cost per hour 40 30 35 $36 $42 $55 For example, Cabinetmaker 1 estimates that 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 -0.80, 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. a. 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. Let 01 - percentage of Oak cabinets assigned to cabinetmaker 1 a. 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. Let O1-percentage of Oak cabinets assigned to cabinetmaker 1 02-percentage of Oak cabinets assigned to cabinetmaker 2 03-percentage of Oak cabinets assigned to cabinetmaker 3 C-percentage of Cherry cabinets assigned to cabinetmaker 1 C2 percentage of Cherry cabinets assigned to cabinetmaker 2 Ca percentage of Cherry cabinets assigned to cabinetmaker 3 Min ) s.t. 50 X 42 X 30 X 60 X 48 X 01 02 03 C 1 Cz 01 50 02 42 03 30 C: 60 Ca 48 C C 35 X Hours 40 avail. 1 Hours 30 avail. 2 Hours 35 35 avail. 3 Prednu 1 look 01 02 03 1 Cz C 1 Oak 1 Cherry 000CC, C3 20 b. 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. Cabinetmaker 1 Cabinetmaker 21 Cabinetmaker 3 Oak 0- 27.083 X 02- 00-72.917 x Cherry C- 0C2- 62.5 X C3- 37.5 X What is the total cost of completing both projects? If required, round your answer to the nearest dollar. Total Cost - $ 3,673 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. No because Cabinetmaker 1 has surplus X of adding one hour to this constraint will decrease total cost by $ 40 x hours. Alternatively, the dual value is 0 which means that 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. answer is zero, enter "0". Use a minus sign to indicate the negative figure. Explain. Yes because Cabinetmaker 2 has a dual value of -1.75 cost by a total of $ -1.75 X per hour, up to an overall maximum of Therefore, each additional hour of time for cabinetmaker 2 will reduce 11.143 X 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. Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3 Oak 01- 0.778 x 02- 0.222 X 03- 0X 0C2- 0.361 X C- 0.639 X Cherry C:1 What is the total cost of completing both projects? If required, round your answer to the nearest dollar. Total Cost $ 3,368 X Explain. D The change in Cabinetmaker 2's cost per hour leads to changing two solution given can be used again. - X objective function coefficients. This means that the linear program The new optimal solution is not the same as X before but with a total cost of $ Feedback