Question

Steinwelt Piano manufactures uprights and consoles in two plants, Plant I and Plant II. The output of Plant I is at most 300/month, whereas the output of Plant II is at most 250/month. These pianos are shipped to three warehouses that serve as distribution centers for the company. To fill current and projected orders, Warehouse A requires a minimum of 200 pianos/month, Warehouse B requires at least 150 pianos/month, and Warehouse C requires at least 200 pianos/month. The shipping cost of each piano from Plant I to Warehouse A, Warehouse B, and Warehouse C is $25, $25, and $45, respectively, and the shipping cost of each piano from Plant II to Warehouse A, Warehouse B, and Warehouse C is $45, $35, and $15, respectively. Use the method of this section to determine the shipping schedule that will enable Steinwelt to meet the warehouses' requirements while keeping the shipping costs to a minimum.

Plant I to Warehouse A		pianos
Plant I to Warehouse B		pianos
Plant I to Warehouse C		pianos
Plant II to Warehouse A		pianos
Plant II to Warehouse B		pianos
Plant II to Warehouse C		pianos

What is the minimum cost?
$

Answer 1

Facts of the Scenario:

1. It is balanced transportation problem as because demand of warehouse is equal to supply of both plant I and Plant II

Demand of warehouse A+ Demand of warehouse B+ Demand of warehouse C = Supply of Plant I + Supply of Plant II

= 200+ 150 +200 = 300+250 equals to 550

	Supply	A	B	C
Plant I	300	25	25	45
Plant II	250	45	35	15
	550	200	150	200	Demand

Step One: Choose Least tranportation cost in all Boxes , here it is in column C and PLant II, and check the demand is lower than the supply, if yes then complete all the supply for that warehouse. So, here Demand of C is 200 and supply by plant II is 250 so we can fill demand of Warehouse C from Plant II

		Supply	A	B	C
Plant I	300	300	25	25	45
Plant II	50	250	45	35	15
	350	550	200	150	200	Demand
	Balance of Demand and Supply is 350

Step Two: Now Apply the first rule of least cost in remaining matrix , here we come across of two least transportation cost with two deifferent warehouse, if this ties happens then apply rule where the demand is high and fulfill all its demand, here in the scenario both warehouse A and warehouse B having least cost 25, but demand of warehouse is more as compared to warehouse B i.e A>B. So complete all demend of Warehouse A from Plant I because as per rule already stated if supply is more than demand complete all its demand.

		Supply	A	B	C
Plant I	100	300	25	25	45
Plant II	50	250	45	35	15
	150	550	200	150	200	Demand
	Balance of Demand and Supply is 150

Step Three: Now Again Check the least cost and balance demand to be completed:

so as per figure : Complete demand of warehouse B from Plant I (100 pianos) and from Plant II( 50 Pianos)

		Supply	A	B	C
Plant I	0	300	25	25	45
Plant II	0	250	45	35	15
	0	550	200	150	200	Demand
	Balance of Demand and Supply is 0

Now Least Transporatation Cost is:

Supply by Plant I = 200 for warehouse A + 100 for Warehouse B

Supply by Plant II = 50 for warehouse B + 200 for Warehouse C

Total Cost = 200*25+ 100*25+ 50*35+200*15 = 12250

Total Minimum cost is $12250