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Problem 3-25 (Algorithmic) 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

January 14, 2024

Problem 3-25 (Algorithmic)

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 3 Hours required to complete all the oak cabinets474027 Hours required to complete all the cherry cabinets645136 Hours available403035 Cost per hour$34$41$52

For example, Cabinetmaker 1 estimates it will take 47 hours to complete all the oak cabinets and 64 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 40/47 = 0.85, or 85%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/64 = 0.63, or 63%, 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.Let O1 = percentage of Oak cabinets assigned to cabinetmaker 1 O2 = percentage of Oak cabinets assigned to cabinetmaker 2 O3 = percentage of Oak cabinets assigned to cabinetmaker 3 C1 = percentage of Cherry cabinets assigned to cabinetmaker 1 C2 = percentage of Cherry cabinets assigned to cabinetmaker 2 C3 = percentage of Cherry cabinets assigned to cabinetmaker 3 Min fill 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 6C3 s.t. fill in the blank 7O1 + fill in the blank 8C1 ≤ fill in the blank 9 Hours avail. 1 fill in the blank 10O2 + fill in the blank 11C2 ≤ fill in the blank 12 Hours avail. 2 fill in the blank 13O3 + fill in the blank 14C3 ≤ fill in the blank 15 Hours avail. 3 fill in the blank 16O1 + fill in the blank 17O2 + fill in the blank 18O3 = fill in the blank 19 Oak fill in the blank 20C1 + fill in the blank 21C2 + fill in the blank 22C3 = fill in the blank 23 Cherry O1, 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 1 Cabinetmaker 2 Cabinetmaker 3 Oak O1 = fill in the blank 24 O2 = fill in the blank 25 O3 = fill in the blank 26 Cherry C1 = fill in the blank 27 C2 = fill in the blank 28 C3 = 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.

because Cabinetmaker 1 has

of fill in the blank 33 hours. Alternatively, the dual value is fill in the blank 34 which means that adding one hour to this constraint will decrease total cost by $fill in the blank 35.
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.

because Cabinetmaker 2 has a

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 1 Cabinetmaker 2 Cabinetmaker 3 Oak O1 = fill in the blank 41 O2 = fill in the blank 42 O3 = fill in the blank 43 Cherry C1 = fill in the blank 44 C2 = fill in the blank 45 C3 = 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 47The change in Cabinetmaker 2’s cost per hour leads to changing

objective function coefficients. This means that the linear program

The new optimal solution

the one above but with a total cost of $ fill in the blank 51.

dealers, Georgia, kitchen, large, local, manufactures, OAK, orders, problem, sold, Southeast

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