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

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:

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.

1. 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.

1. 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

3. 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 = 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

4. 
   Total Cost = $  fill in the blank 30

5. If Cabinetmaker 1 has additional hours available, would the optimal solution change?
    
   Explain.
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   

6. If Cabinetmaker 2 has additional hours available, would the optimal solution change?
    
   Explain.
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   

7. 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 = fill in the blank 35	O2 = fill in the blank 36	O3 = fill in the blank 37
   Cherry	C1 = fill in the blank 38	C2 = fill in the blank 39	C3 = fill in the blank 40

8. 
   Total Cost = $  fill in the blank 41
   Explain.
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.