Question

DryIce, Inc., is a manufacturer of air conditioners. The company anticipates nationwide demand for the next year to be 180,000 units in the South, 120,000 units in the Midwest, 110,000 units in the East, and 100,000 units in the West. Managers at DryIce are designing the manufacturing network and have selected four potential sites --- New York, Atlanta, Chicago and San Diego. Plants could have a capacity of 200,000 units. The annual fixed costs at the four locations are shown in the following table, along with the cost of producing and shipping an air conditioner to each of the four markets. Where should Dryice build its factories and what will be the quantities to be produced at each location and shipped to each market? How does the solution change if you are required to locate a plant in each location? Comment on the solution

	east	south	midwest	west	fixed costs(200k)
new york	211	232	240	300	6,000,000
atlanta	232	212	230	280	5,500,000
chicago	238	230	215	270	5,600,000
san diego	299	280	270	225	6,100,000
total demand	110,000	180,000	120,000	100,000

Answer 1

The Vogel's Approximation Method or VAM is an iterative procedure calculated to find out the initial feasible solution of the transportation problem.

Our initial matrix we have

	East	South	Mid West	West	Plant max capacity	Fixed Cost
New York	211	232	240	300	2,00,000	6000000
Atlanta	232	212	230	280	2,00,000	5500000
Chigago	238	230	215	270	2,00,000	5600000
San Diego	299	280	270	225	2,00,000	6100000
Total Demand	110000	180000	120000	100000

We follow the follwoing steps to find our first solution

	East	South	Mid West	West	Plant max capacity	Iteration 1	Fixed Cost
New York	211	232	240	300	2,00,000	21	6000000
Atlanta	232	212	230	280	2,00,000	20	5500000
Chigago	238	230	215	270	2,00,000	15	5600000
San Diego	299	280	270	225	2,00,000	45	6100000
Total Demand	110000	180000	120000	100000
Iteration 1	21	20	15	45

As complete demand of colmun "West " is fulfilled so we remove it

	East	South	Mid West	Plant max capacity	Iteration 2	Fixed Cost
New York	211	232	240	2,00,000	21	6000000
Atlanta	232	212	230	2,00,000	20	5500000
Chigago	238	230	215	2,00,000	15	5600000
San Diego	299	280	270	1,00,000	10	6100000
Total Demand	110000	180000	120000
Iteration 2	21	20	15

We make allocation to new york at EAST

	South	Mid West	Plant max capacity	Iteration 3	Fixed Cost
New York	232	240	90,000	8	6000000
Atlanta	212	230	2,00,000	18	5500000
Chigago	230	215	2,00,000	15	5600000
San Diego	280	270	1,00,000	10	6100000
Total Demand	180000	120000
Iteration 3	20	15

Making allocation at Atlanta Chicago

In last allocation we allocate to Chicago Middle west

We can summarise our allocation as

The above table shows most optimised way in which we can do allocations.

Where should Dryice build its factories and what will be the quantities to be produced at each location and shipped to each market?

We can evaluate the cost beared at each location to answer question

Place	Fixed cost	Variable cost	Total cost (10^4)
New York	6000000	211*110000+232*90000	5009
Atlanta	5500000	212*180000+230*20000	4776
Chigago	5600000	215*120000+230*80000	4980
San Diego	6100000	225*100000+270*100000	5560

So we can conclude he should make his plant at ATLANTA. He will be shipping to south and middle west if he is working his plant with maximum capacity. In case he want to keep his cost to minimum he should only ship to south.

In case he is interested to make plant in every location, the respective plant capacity is given in table below