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	42	30
Hours required to complete all the cherry cabinets	60	48	35
Hours available	40	30	35
Cost per hour	$36	$42	$55

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 = 0.80, 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 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, and C3 = percentage of cherry cabinets assigned to cabinetmaker 3.)

Min ___ s.t.

hours available 1 ___

hours available 2 ___

hours available 3 ___

oak ___

cherry ___

O1, O2, O3, C1, C2, C3 0

(e)

Suppose Cabinetmaker 2 reduced its cost to $38 per hour. What effect would this change have on the optimal solution? Explain.

The new objective function coefficients for O2 and C2 are ___ and ___ , respectively. The optimal solution does not change and has a value of $ ___ .