The document discusses the least cost method (LCM) for solving unbalanced transportation problems. It outlines the process of balancing these problems by introducing dummy columns and provides step-by-step solutions for multiple transportation scenarios, along with the total transportation costs calculated for each case. The examples illustrate how LCM can efficiently determine the optimal routing and minimize transportation costs.

1. OPERATIONS RESEARCH
TRANSPORTATION PROBLEM
Least Cost Method (Unbalanced)
DR.T.SIVAKAMI
ASSISTANT PROFESSOR OF MANAGEMENT STUDIES
BON SECOURS COLLEGE FOR WOMEN,THANJAVUR
M.RASMIYA ,D.RATHIKA
BON SECOURS COLLEGE FOR WOMEN,THANJAVUR

2. LEAST COST METHOD
Least cost method is also known as matrix minimum
method in the sense we look for the row and the
column corresponding to which is minimum. This
method finds a better initial basic feasible
solution by concentrating on the cheapest routes.

3. 1. Solve the following Transportation problem by
using LCM .
M1 M2 M3 M4 M5 Supply
S1 4 2 3 2 6 8
S2 5 4 5 2 1 12
S3 6 5 4 7 7 14
Deman
d
4 4 6 8 8

4. Solution :
• The given transportation problem is unbalanced.
• To convert unbalanced into balanced.
• Introduce dummy column with the value of 4.

5. STEP 1
4 2 3 2 6 0 8
5 4 5 2 1 0 12
6 5 4 7 7 0 14
4 4 6 8 8 4

6. Step 2
4 2 3 2 6 . ..g0
5 4 5 2 1 0
6 5 4 7 7 0
4 8
12
14
4 4 6 8 8 4
4

7. 4 2 3 2 6
5 4 5 2 1
6 5 4 7 7
Step 3
8
4
12 4
14
4 4 6 8 8

8. Step 4
4 2 3 2
5 4 5 2
6 5 4 7
4
4
4
14
4 4 6 8

9. Step 5
5 5 2
6 4 7
Step 6
6 7
4 4
14
4 6 8 4
14 8
4
4 6 4
6

10. Step 7
6 7
Step 8
7
4
4
8 4
4 4
4
4

11. Total transportation cost = cell value ×Allotted value
= ( 0×4)+(1×8)+(2×4)+(2×4)+(4×6) +
(6×4)+(7×4)
= 0+8+8+8+24+24+28
= ₹ 100

12. 2. Solve the following Transportation problem by
using LCM .
R1 R2 R3 Supply
S1 4 8 8 76
S2 16 24 16 82
S3 8 16 24 77
Demand 72 102 41

13. Solution :
 The given transportation problem is unbalanced.
 To convert unbalanced into balanced.
 Introduce dummy column with the value of 20

14. Step 1
4 8 8 0 76
16 24 16 0 82
8 16 24 0 77
72 102 41 20

15. Step 2
4 8 8 0
16 24 16 0
8 16 24 0
20 76
82
77
56
72 102 41 20

16. Step 3
4 8 8
16 24 16
8 16 24
72 16 102 41
56
82
77
56

17. Step 4 Step 5
16 24 16
16 24
82
77
8
16 102 41
16
61
24
16 24
82
61
16
102 41
41
41

18. Step 6 Step 7
24
16
61
41
61
102 41
24
41
41
41

19. Total transportation cost = cell value ×Allotted value
= (0×20) + (4×56) + (8×16) + (16×41) +
(16×41) + (24×41)
= ₹ 2968

20. D1 D2 D3 D4 D5 Supply
S1 5 1 8 7 5 15
S2 3 9 6 7 8 25
S3 4 2 7 6 5 42
S4 7 11 10 4 9 35
Demand 30 20 15 10 20
3. Solve the following Transportation problem by
using LCM .

21.  The given transportation problem is unbalanced.
 To convert unbalanced into balanced.
 Introduce dummy column with the value of 22
Solution :

22. Step 1
5 1 8 7 5 0 15
3 9 6 7 8 0 25
4 2 7 6 5 0 42
7 11 10 4 9 0 35
30 20 15 10 20 22

23. Step 2
5 1 8 7 5 0. 0
3 9 6 7 8 0
4 2 7 6 5 0
7 11 10 4 9 0
15
25
42
35
30 20 15 10 20 22 7
15

24. Step 3
3 9 6 7 8 0
4 2 7 6 5 0
7 11 10 4 9 0
30 20 15 10 20 7
0
7
25 18
42
35

25. Step 4
3 9 6 7 8
4 2 7 6 5
7 11 10 4 9
2
30 20 15 10 20
20
8
42 22
35

26. Step 5
3 6 7 8
4 7 6 5
7 10 4 9
18
18
22
35
30 12 15 10 20
3

27. Step 6 Step 7
4 7 6 5
7 10 4 9
4 7 5
7 10 9
12 15 10 20
22
35 25
10
12
12 15 20
22 10
25

28. Step 8 Step 9
7 5
10 9
10
10
25
15 20 10
10 9
110
25 15
15 10

29. Step 10
10
15
15
15
Total transportation cost = cell value ×Allotted value
= (0×15) + (0×7) + (2×20) + (3×18) + (4×10) +
(4×12) + (5×10) + (9×10) + (10×15)
= ₹ 472

30. THANK U