Allen Air Conditioning manufactures room air conditioners at plants in Houston, Phoenix, and Memphis. These are sent to regional distributors in Dallas, Atlanta, and Denver. The shipping costs vary, and the company would like to find the least-cost way to meet the demands at each of the distribution centers. Dallas needs to receive 800 air conditioners per month, Atlanta needs 600, and Denver needs 200. Houston has 850 air conditioners available each month, Phoenix has 650, and Memphis has 300. The shipping cost per unit from Houston to Dallas is $8, to Atlanta $12, and to Denver $10. The cost per unit from Phoenix to Dallas is $10, to Atlanta $14, and to Denver $9. The cost per unit from Memphis to Dallas is $11, to Atlanta $8, and to Denver $12. 14700. Required:
a. Based on the given demand and supply, the given transportation problem is
b. Before finding the initial solution, a dummy should be introduced.
c. The total cost of the optimal solution =

(A) An optimization problem

(B) Variable

(C) $14,700

Explanation:

HOUSTON, PHOENIX, MEMPHIS DALLAS, ATLANTA, DENVER

    |                     |                  |                        |               |                 |

  850               650            300                 800         600           200

Shipping Cost per Room Air Conditioner:

HOUSTON to Dallas - $8   [800 × 8 = $6400]

                   to Atlanta - $12   [50 × 12 = $600]

                   to Denver - $10  

Total shipping cost = $7,000

PHOENIX to Dallas - $10  

                 to Atlanta - $14  [250 × 14 = $3500]

                 to Denver - $9  [200 × 9 = $1800]

Total shipping cost = $5,300

MEMPHIS to Dallas - $11

                 to Atlanta - $8  [300 × 8 = $2400]

                 to Denver - $12

Total shipping cost = $2,400

The total cost of the optimal solution is:

7,000 + 5,300 + 2,400 = $14,700

How was this optimal cost gotten?

- Check the least shipping cost to Dallas, this is found in Houston and that's 8 dollars. Houston hence supplies Dallas 800 ACs and has 50 left. Remember this, in case you need more ACs.

- Check the least shipping cost to Atlanta, this is found in Memphis and that's 8 dollars. An obstruction here is that Memphis only has 300 ACs to send to Atlanta, so Atlanta needs 300 more. Keep this in mind.

- Check the least shipping cost to Denver, this is found in Phoenix and that's 9 dollars. Phoenix hence supplies Denver its needed 200 ACs and has 450 left.

- There is need to complete shipping to Atlanta and the next affordable shipping cost to Atlanta is found in Houston and that's $12. Houston should send its remaining 50 ACs to Atlanta.

- Atlanta needs 250 more ACs and Phoenix has 450 extra. Allen Air Conditioning Company can now settle for the relatively high shipping cost of Phoenix to Atlanta, to supply the remaining 250 ACs which Atlanta needs.