This document describes a transshipment problem involving supplying raw materials from multiple suppliers to multiple plants. It provides the supply amounts from each supplier, demand amounts for each plant, and transportation costs between suppliers and warehouses/plants. An initial solution is generated using the Vogel approximation method. This solution is then checked for optimality using the MODI method by calculating costs of occupied and unoccupied cells in the transportation table. The MODI method confirms the solution is optimal, with a total cost of 63,100 rupees.

1. Transshipment
Problem

2. Question
• A multi-plant organization sources its raw materials from two
suppliers S1 and S2 located in different plants P1,P2 and P3
located in different cities . The organization maintains two
different raw material warehouses W1 and W2 to receive and
stock raw materials from the two suppliers and then ship
them to the three plants.
• The supply values of S1 and S2 are 1000 units and 1500 units
respectively
• The demands of the plants P1 , P2 and P3 are 800,1200 and
1000 units respectively
• The cost of transportation (in rupees) per unit b/w different
sources and destination combinations are shown in table

3. Table given:
P1 P2 P3 W1 W2 SUPPLY
S1 ∞ ∞ ∞ 17 16 1000
S2 ∞ ∞ ∞ 11 13 1500
W1 15 8 20 0 13
W2 18 10 9 14 0
DEMAND 800 1200 1000

4. • This is an unbalanced problem because demand and supply
are not equal
• Total demand = 3000
• Total supply = 2500
• So we introduce one supplier S3 (as a dummy row) with value
500 units.
• Value of S3 corresponding to warehouses – infinity

5. P1 P2 P3 W1 W2 SUPPLY
S1 ∞ ∞ ∞ 17 16 1000
S2 ∞ ∞ ∞ 11 13 1500
S3 ∞ ∞ ∞
0 0
500
W1 15 8 20 0 13 3000
W2 18 10 9 14 0 3000
DEMAND 800 1200 1000 3000 3000
After introducing the new row S3

6. InitialSolution:byvogalapproximationmethod
P1 P2 P3 W1 W2
S1 - - - 500 500
S2 - - - 1500 -
S3 - - - - 500
W1 800 1200 - 1000 -
W2 - - 1000 - 2000
Check
M+N-1=5+5-1=9 =9

7. NowweneedtochecktheoptimalityofthesolutionbyModi
Method
Costofoccupiedcellsandunoccupiedcells
P1 P2 P3 W1 W2 Ui
S1 - - - 17 16 16
S2 - - - 11 - 10
S3 - - - - 0 0
W1 15 8 - 0 - -1
W2 - - 9 - 0 0
Vi 16 9 9 1 0
P1 P2 P3 W1 W2 Ui
S1 ∞ ∞ ∞ - - 16
S2 ∞ ∞ ∞ - 13 10
S3 ∞ ∞ ∞ 0 - 0
W1 - - 20 - 13 -1
W2 18 10 - 14 - 0
Vi 16 10 9 1 0

8. P1 P2 P3 W1 W2
S1 ∞ ∞ ∞ - -
S2 ∞ ∞ ∞ - 3
S3 ∞ ∞ ∞ -1 -
W1 - - 29 - 14
W2 2 1 - 13 -
Net evaluation table for initial feasible
solution:

9. P1 P2 P3 W1 W2
S1 - - - 500 500
S2 - - - 1500 -
S3 - - - ø 500
W1 800 1200 - 1000 -
W2 - - 1000 - 2000
Introducing ø

10. Cost
P1 P2 P3 W1 W2
S1 E 1000
S2 1500
S3 500
W1 800 1200 1000
W2 1000 2000

11. P1 P2 P3 W1 W2 Ui
S1 17 16 16
S2 11 10
S3 0 -1
W1 15 8 0 -1
W2 9 0 0
vi 16 9 9 1 0
Check again with MODI method to
find if the solution is optimal
Occupied cells:

12. Unoccupied cells:
P1 P2 P3 W1 W2 Ui
S1 ∞ ∞ ∞ 16
S2 ∞ ∞ ∞ 13 10
S3 ∞ ∞ ∞ 0 -1
W1 20 13 -1
W2 18 10 14 0
16 9 9 1 0

13. Net Evaluation table:
all Cij’ > 0 hence this time the solution is
optimal
P1 P2 P3 W1 W2
S1 ∞ ∞ ∞
S2 ∞ ∞ ∞ 3
S3 ∞ ∞ ∞ 1
W1 12 14
W2 2 1 13

14. S1
W2
S2
S3
W1
P1
P2
P31000
1500
500 800
1200
1000
1000
2000
SHIPPING PLAN
Total cost – 63100

15. THANKYOU