Problem 8-25 (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 1 Cabinetmaker 2 Cabinetmaker 3 Hours required to complete all the oak cabinets 50 44 30 Hours required to complete all the cherry cabinets 60 43 33 Hours available 40 25 30 Cost per hour $32 $43 $59

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 = 0.8, 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 proportion of the oak cabinets and the proportion 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 = proportion of Oak cabinets assigned to cabinetmaker 1 O2 = proportion of Oak cabinets assigned to cabinetmaker 2 O3 = proportion of Oak cabinets assigned to cabinetmaker 3 C1 = proportion of Cherry cabinets assigned to cabinetmaker 1 C2 = proportion of Cherry cabinets assigned to cabinetmaker 2 C3 = proportion of Cherry cabinets assigned to cabinetmaker 3

Min fill in the blank 1 O1 + fill in the blank 2 O2 + fill in the blank 3 O3 + fill in the blank 4 C1 + fill in the blank 5 C2 + fill in the blank 6 C3 s.t. fill in the blank 7 O1 fill in the blank 8 C1 fill in the blank 9 Hours avail. 1 fill in the blank 10 O2 + fill in the blank 11 C2 fill in the blank 12 Hours avail. 2 fill in the blank 13 O3 + fill in the blank 14 C3 fill in the blank 15 Hours avail. 3 fill in the blank 16 O1 + fill in the blank 17 O2 + fill in the blank 18 O3 = fill in the blank 19 Oak fill in the blank 20 C1 + fill in the blank 21 C2 + fill in the blank 22 C3 = fill in the blank 23 Cherry O1, O2, O3, C1, C2, C3 0

Solve the model formulated in part

(a). What proportion of the oak cabinets and what proportion of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects? If required, round your answers for the proportions to three decimal places, and for the total cost to two decimal places.

Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3

Oak O1 = ____________ O2 = ___________O3 = __________

Cherry C1 = ________ C2 = __________ C3 =_________

Total Cost = $ __________

If Cabinetmaker 1 has additional hours available, would the optimal solution change?

If Cabinetmaker 2 has additional hours available, would the optimal solution change? ___________

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 for the proportions to three decimal places, and for the total cost to two decimal places.

Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3

Oak O1 = ____ O2 = _______ O3 = ________

Cherry C1 = ___________ C2 = __________ C3 = ________

Total Cost = $ ___________