Question

the steps in solving transportation problem using Least Cost Method.

Answer 1

Least Cost Method for Solving Transportation problem.

In Least cost method,the allocation starts from the cell which is having lower cost compared to others.

Steps are illustrated with an example.

Question.Unit cost of transportation from station to Plant is given in each cell. Calculate the total transportation cost using least cost method.

	Plant 1	Plant 2	Plant 3	Plant 4	Supply
Station A	10	22	10	20	8
Station B	15	20	12	8	13
Station C	20	12	10	15	11
Demand	7	10	6	9

	Plant 1	Plant 2	Plant 3	Plant 4	Supply
Station A	10	22	10	20	8
Station B	15	20	12	8	13
Station C	20	12	10	15	11
Demand	7	10	6	9

Steps

1. First check whether the problem given is balanced transportation problem. Balanced transportation problem means Supply =Demand

Here Supply=8+13+11=32

Demand =7+10+6+9=32

Supply=Demand=32,

So Balanced Transportation Problem

2. Check whether the values given in each cell is cost, If so the problem is minimization of cost.

3. Next select the cell where we can start allocating. Selection is based on the cell with lesser cost      Here it is Cell Station B –Plant 4 with cost Rs 8

4. Once the cell is selected for allocation, look for how many units can be allocated.

In this example Station B can supply of 13 but Plant 4 requires only 9, so the minimum value is allocated which is 9 here

Rule is minimum of Supply /Demand unit is allotted

	Plant 1	Plant 2	Plant 3	Plant 4	Supply
Station A	10	22	10	20	8
Station B	15	20	12	8 (9)	13 4
Station C	20	12	10	15	11
Demand	7	10	6	9 0

So the demand of Plant 4 is met and we can avoid plant 4 for further allocation.

Station B can further supply 4(13-9) units

	Plant 1	Plant 2	Plant 3	Supply
Station A	10	22	10	8
Station B	15	20	12	4
Station C	20	12	10	11
Demand	7	10	6

5. Now again look for the cell with lower cost, here it is 10.Since there are many cells with cost 10.We can select the cell where we can allocate maximum units.

If we take

Station A Plant 1, 7 can be allocated

Station A Plant 3, 6 can be allocated

Station C Plant 3, 6 can be allocated

Note: Whenever there is a tie between two cells for a same cost, cells which can be allocated maximum unit is taken. If that value is also tied, arbitrary selection will be made

So, we will select the Station A Plant 1 and allocated 7 units. So Demand of Plant 1 is met. So no need to take Plant 1 for further steps

	Plant 1	Plant 2	Plant 3	Supply
Station A	10 (7)	22	10	8 1
Station B	15	20	12	4
Station C	20	12	10	11
Demand	7 0	10	6

6. Repeat the steps

Again repeat the steps for identifying least cost cell and here it is Station C Plant 3 as cost is less and more number can be allocated

	Plant 2	Plant 3	Supply
Station A	22	10	1
Station B	20	12	4
Station C	12	10(6)	11 5
Demand	10	6 0

A plant 3 demand is met completely, so avoided further.

7. Repeat the steps

	Plant 2	Supply
Station A	22 (1)	1
Station B	20 (4)	4
Station C	12 (5)	5
Demand	10

Since no more plants available Allocate as per the supply requirement

So allocations are

Sl No:	Cell	No: of Units allotted	Cost of Transportation	Total Cost
1	Station B Plant 4	9	8	9*8=72
2	Station A Plant 1	7	10	70
3	Station C Plant 3	6	10	60
4	Station A Plant 2	1	22	22
5	Station B Plant 2	4	20	80
6	Station C Plant 2	5	12	60

8. Calculate the total cost

So total transportation cost is 72+70+60+22+80+60=364

Note:

To check the feasibility, check whether the allocations are made to n+m-1 cells

Where, n is number of rows, m is number of columns.

Here it is 3+4-1=6

Note: In case profit matrix is given in each cells .Convert the matrix into cost matrix by subtracting all elements by the largest value in profit matrix to obtain cost matrix and do the above steps.