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, t

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 1 Cabinetmaker 2 Cabinetmaker 3

Hours required to complete all the oak cabinets 47 40 27

Hours required to complete all the cherry cabinets 64 51 36

Hours available 40 30 35

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 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 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 40

total 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 47

The 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

   .