Problem 8-25 (Algorithmic) Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the

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Problem 8-25 (Algorithmic) Georgia Cabinets manufactures kitchencabinets 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 todo the final finishing operation. For the three cabinetmakers, thenumber of hours required to complete all the oak cabinets, thenumber of hours required to complete all the cherry cabinets, thenumber of hours available for the final finishing operation, andthe cost per hour to perform the work are shown here:
Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3 Hours required tocomplete all the oak cabinets 50 44 30 Hours required to completeall the cherry cabinets 60 43 33 Hours available 40 25 30 Cost perhour $32 $43 $59
For example, Cabinetmaker 1 estimates that it will take 50 hoursto complete all the oak cabinets and 60 hours to complete all thecherry cabinets. However, Cabinetmaker 1 only has 40 hoursavailable for the final finishing operation. Thus, Cabinetmaker 1can only complete 40/50 = 0.8, or 80%, of the oak cabinets if itworked only on oak cabinets. Similarly, Cabinetmaker 1 can onlycomplete 40/60 = 0.67, or 67%, of the cherry cabinets if it workedonly on cherry cabinets.
Formulate a linear programming model that can be used todetermine the proportion of the oak cabinets and the proportion ofthe 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 to cabinetmaker 1O2 = 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 inthe 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 theblank 8 C1 ≤ fill in the blank 9 Hours avail. 1 fill in the blank10 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 theblank 15 Hours avail. 3 fill in the blank 16 O1 + fill in the blank17 O2 + fill in the blank 18 O3 = fill in the blank 19 Oak fill inthe 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 ofthe cherry cabinets should be assigned to each cabinetmaker? Whatis the total cost of completing both projects? If required, roundyour answers for the proportions to three decimal places, and forthe 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 theoptimal solution change?
If Cabinetmaker 2 has additional hours available, would theoptimal solution change? ___________
Suppose Cabinetmaker 2 reduced its cost to $38 per hour. Whateffect would this change have on the optimal solution? If required,round your answers for the proportions to three decimal places, andfor the total cost to two decimal places.
Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3
Oak O1 = ____ O2 = _______ O3 = ________
Cherry C1 = ___________ C2 = __________ C3 = ________
Total Cost = $ ___________