We find the basic feasibility solution for a transportation problem using Vogel's Approximation Method.

1. EXPLAIN HOW DO YOU FIND THE BASIC
FEASIBLE SOLUTION FOR A
TRANSPORTATION PROBLEM USING VAM
WITH A NUMERICAL EXAMPLE
Quantitative methods, Jeyasree SB, Aug - 2024

2. CONSIDER THE TRANSPORTATION PROBLEM GIVEN
BELOW. SOLVE THIS PROBLEM BY VOGEL’S
APPROXIMATION METHOD.
Origin Destination Supply
D1 D2 D3 D4
O1 3 1 7 4 300
O2 2 6 5 9 400
O3 8 3 3 2 500
Demand 250 350 400 200

3. BEFORE STARTING THE SOLUTION
Check whether TP is balanced.
Σ Demand = 250 + 350 + 400 + 200 = 1200
Σ Supply = 300 + 400 + 500 = 1200
Σ Demand = Σ Supply
Hence Balanced.
Cost Matrix is given and it is a Minimization problem.

4. STEP 1: Identifying the least and the second least
values to find the penalties in each row and column.
Origin Destination Supply
D1 D2 D3 D4
O1 3 1 7 4 300 2
O2 2 6 5 9 400 3
O3 8 3 3 2 500 1
Demand 250 350 400 200
1 2 2 2

5. STEP 2: Identifying the highest penalty and choosing
the least cost cell of the row or column of the penalty.
Origin Destination Supply
D1 D2 D3 D4
O1 3 1 7 4 300 2
O2 2 6 5 9 400 3
O3 8 3 3 2 500 1
Demand 250 350 400 200
1 2 2 2

6. STEP 3: Assign the minimum value (supply or demand)
to the least value cell.
Origin Destination Supply
D1 D2 D3 D4
O1 3 1 7 4 300 2
O2 2 6 5 9 400 3
O3 8 3 3 2 500 1
Demand 250 350 400 200
1 2 2 2

7. STEP 3: Assign the minimum value (supply or
demand) to the least value cell.
Origin Destination Supply
D1 D2 D3 D4
O1 300 2
O2 400 3
O3 500 1
Demand 250 350 400 200
1 2 2 2
3
2
8
1
6
3
7
5
3
4
9
2

8. STEP 3: Assign the minimum value (supply or
demand) to the least value cell.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 2
O2 250 3
O3 - 500 1
Demand 250 350 400 200
1 2 2 2
3
2
8
1
6
3
7
5
3
4
9
2
400

9. STEP 3: Assign the minimum value (supply or
demand) to the least value cell.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 2
O2 250 150 3
O3 - 500 1
Demand 250 350 400 200
1 2 2 2
3
2
8
1
6
3
7
5
3
4
9
2
400

10. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300
O2 250 150
O3 - 500
Demand 250 350 400 200
-
3
2
8
1
6
3
7
5
3
4
9
2
400

11. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 3
O2 250 150 1
O3 - 500 1
Demand 250 350 400 200
- 2 2 2
3
2
8
1
6
3
7
5
3
4
9
2
400

12. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 3
O2 250 150 1
O3 - 500 1
Demand 250 350 400 200
- 2 2 2
3
2
8
1
6
3
7
5
3
4
9
2
400

13. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 3
O2 250 150 1
O3 - 500 1
Demand 250 350 400 200
- 2 2 2
3
2
8
1
6
3
7
5
3
4
9
2
400

14. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 300 3
O2 250 150 1
O3 - 500 1
Demand 250 350 400 200
- 2 2 2
3
2
8
1
6
3
7
5
3
4
9
2

15. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 3
O2 250 150 1
O3 - 500 1
Demand 250 50 400 200
- 2 2 2
3
2
8
1
6
3
7
5
3
4
9
2
400
350

16. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 150
O3 - 500
Demand 250 50 400 200
-
3
2
8
1
6
3
7
5
3
4
9
2
400
350

17. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 150 1
O3 - 500 1
Demand 250 50 400 200
- 3 2 7
3
2
8
1
6
3
7
5
3
4
9
2
400
350

18. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 150 1
O3 - 500 1
Demand 250 50 400 200
- 3 2 7
3
2
8
1
6
3
7
5
3
4
9
2
400
350

19. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 150 1
O3 - 500 1
Demand 250 50 400 200
- 3 2 7
3
2
8
1
6
3
7
5
3
4
9
2
400
350

20. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 150 1
O3 - 200 500 1
Demand 250 50 400 200
- 3 2 7
3
2
8
1
6
3
7
5
3
4
9
2
400
350

21. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150
O3 - 200 300
Demand 250 50 400 200
- -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
500

22. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 1
O3 - 200 300 0
Demand 250 50 400 200
- 3 2 -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
500

23. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 1
O3 - 200 300 0
Demand 250 50 400 200
- 3 2 -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
500

24. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 1
O3 - 200 300 0
Demand 250 50 400 200
- 3 2 -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
500

25. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 1
O3 - 50 200 250 0
Demand 250 50 400 200
- 3 2 -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300

26. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - - 150
O3 - 50 200 250
Demand 250 50 400 200
- - -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300

27. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - - 150 5
O3 - 50 200 250
Demand 250 50 400 200
- - 3 -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300

28. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - - 150 5
O3 - 50 200 250
Demand 250 50 400 200
- - 3 -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300

29. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - - 150 5
O3 - 50 200 250
Demand 250 50 400 200
- - 3 -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300

30. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 - 150 5
O3 - 50 200 250
Demand 250 50 400 200
- - 3 -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300

31. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 - 150 -
O3 - 50 200 250
Demand 250 50 250 200
- - -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300
400

32. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 - 150 -
O3 - 50 200 250
Demand 250 50 250 200
- - -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300
400

33. STEP 4: Repeat the process until all the values are
assigned.
Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 - 150 -
O3 - 50 250 200 250 -
Demand 250 50 250 200
- - - -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300
400

34. Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 - 150 -
O3 - 50 250 200 250 -
Demand 250 50 250 200
- - - -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300
400

35. Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 - 150 -
O3 - 50 250 200 250 -
Demand 250 50 250 200
- - - -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300
400
Step 5 : Checking if all the conditions are satisfied
Total no. of allotments = 6.
M + n – 1 = 4 + 3 – 1 = 6.
The above solution satisfies all the conditions and is a Basic Feasibility solution.

36. Origin Destination Supply
D1 D2 D3 D4
O1 - 300 - - 300 -
O2 250 - 150 - 150 -
O3 - 50 250 200 250 -
Demand 250 50 250 200
- - - -
3
2
8
1
6
3
7
5
3
4
9
2
400
350
300
400
Total cost of BFS = (250 × 2)+(300 × 1)+(50 × 3)+(150 × 5)+(250 × 3)+(200 × 2)
= 500 + 300 +150 + 750 + 750 + 400
= 2850 currency units (c.u.)

37. Thank You
https://byjus.com/maths/vogels-approximation-method/#:~:text=Step
%201%3A%20Identify%20the%20two,min(supply%2C%20demand).