1. Describe the transportation problem with its general mathematical formulation.
2. Explain the following:
3. Explain the following with the help of an example:
4. Discuss the similarities and differences between the stepping stone method and the MODI method in solving a transportation problem.
1. A factory has three warehouses W1, W2 and W3 which supply
to four stores S1, S2, S3 and S4. Monthly capacities of the warehouses
are W1 = 100 units, W2 = 40 units and W3 = 60 units. Monthly demands
at the stores are S1 = 30 units, S2 = 50 units, S3 = 65 units and S4=
55 units.
The shipping cost in terms of rupees from warehouses to stores is as
given below:
Stores | ||||
---|---|---|---|---|
Warehouses | S1 | S2 | S3 | S4 |
W1 | 14 | 16 | 12 | 20 |
W2 | 12 | 14 | 10 | 8 |
W3 | 10 | 16 | 8 | 15 |
The problem here is to determine the optimum distribution for the factory to minimize shipping costs.
(a)
Stores | ||||
---|---|---|---|---|
Factories | 1 | 2 | 3 | Supply |
1 | 2 | 7 | 4 | 5 |
2 | 3 | 3 | 1 | 8 |
3 | 5 | 4 | 7 | 7 |
4 | 1 | 6 | 2 | 14 |
Demand | 7 | 9 | 18 | 34 |
(b)
Stores | |||||||
---|---|---|---|---|---|---|---|
Factories | A | B | C | D | E | F | Supply |
1 | 1 | 2 | 1 | 4 | 5 | 1 | 30 |
2 | 3 | 3 | 2 | 1 | 4 | 3 | 50 |
3 | 4 | 2 | 5 | 9 | 6 | 2 | 75 |
4 | 3 | 1 | 7 | 3 | 4 | 6 | 20 |
Demand | 20 | 40 | 30 | 10 | 50 | 25 |
(c)
X | Y | Z | a_{i} | |
---|---|---|---|---|
A | 8 | 7 | 3 | 60 |
B | 3 | 8 | 9 | 70 |
C | 11 | 3 | 5 | 80 |
b_{j} | 50 | 80 | 80 |
(d)
D1 | D2 | D3 | D4 | a_{i} | |
---|---|---|---|---|---|
O1 | 10 | 7 | 3 | 6 | 3 |
O2 | 1 | 6 | 8 | 3 | 5 |
O3 | 7 | 4 | 5 | 3 | 7 |
b_{j} | 3 | 2 | 6 | 4 | 15 |
(e)
D1 | D2 | D3 | a_{i} | |
---|---|---|---|---|
O1 | 7 | 3 | 3 | 4 |
O2 | 3 | 1 | 4 | 1 |
O3 | 4 | 3 | 6 | 5 |
b_{j} | 2 | 3 | 5 | 10 |
(f)
D1 | D2 | D3 | D4 | D5 | D6 | a_{i} | |
---|---|---|---|---|---|---|---|
O1 | 9 | 12 | 9 | 6 | 9 | 10 | 5 |
O2 | 7 | 3 | 7 | 7 | 5 | 5 | 6 |
O3 | 6 | 5 | 9 | 11 | 3 | 11 | 2 |
O4 | 6 | 8 | 11 | 2 | 2 | 10 | 2 |
b_{j} | 4 | 4 | 6 | 2 | 4 | 2 |
(g)
D1 | D2 | D3 | D4 | a_{i} | |
---|---|---|---|---|---|
O1 | 4 | 3 | 0 | 5 | 24 |
O2 | 1 | 2 | 6 | 1 | 17 |
O3 | 3 | 6 | 2 | 3 | 19 |
b_{j} | 15 | 19 | 18 | 8 | 60 |
3. Solve the following transportation problems by Stepping Stone Method and MODI method:
(a)
Distributor | ||||
---|---|---|---|---|
Factory | 1 | 2 | 3 | Inventory |
1 | 2 | 1 | 5 | 10 |
2 | 7 | 3 | 4 | 25 |
3 | 6 | 5 | 3 | 20 |
Order | 15 | 22 | 18 | 55 |
(b)
Plant | Market | Available | |||
---|---|---|---|---|---|
A | B | C | D | ||
X | 19 | 30 | 50 | 10 | 7 |
Y | 70 | 30 | 40 | 60 | 9 |
Z | 40 | 8 | 70 | 20 | 18 |
Required | 5 | 8 | 7 | 14 |
(c)
Plant | Market | Available | |||
---|---|---|---|---|---|
A | B | C | D | ||
X | 14 | 9 | 18 | 6 | 11 |
Y | 10 | 11 | 7 | 16 | 13 |
Z | 25 | 20 | 11 | 34 | 19 |
Required | 6 | 10 | 12 | 15 |
(d)
Plant | Market | Available | |||
---|---|---|---|---|---|
A | B | C | D | ||
X | 10 | 22 | 0 | 20 | 8 |
Y | 15 | 20 | 12 | 8 | 13 |
Z | 20 | 12 | 10 | 15 | 11 |
Required | 5 | 11 | 8 | 8 |
(e)
D1 | D2 | D3 | D4 | D5 | Total | |
---|---|---|---|---|---|---|
O1 | 12 | 4 | 9 | 5 | 9 | 55 |
O2 | 8 | 1 | 6 | 6 | 7 | 45 |
O3 | 1 | 12 | 4 | 7 | 7 | 30 |
O4 | 10 | 15 | 6 | 9 | 1 | 50 |
Total | 40 | 20 | 50 | 30 | 40 |
(f)
D1 | D2 | D3 | Total | |
---|---|---|---|---|
O1 | 2 | 4 | 1 | 40 |
O2 | 6 | 3 | 2 | 50 |
O3 | 4 | 5 | 6 | 20 |
O4 | 3 | 2 | 1 | 30 |
O5 | 5 | 2 | 5 | 10 |
Total | 50 | 60 | 40 | 150 |
4. Solve the following transportation problem.
Furnaces | Mills | |||||
---|---|---|---|---|---|---|
M1 | M2 | M3 | M4 | M5 | Total | |
F1 | 4 | 2 | 3 | 2 | 6 | 8 |
F2 | 5 | 4 | 5 | 2 | 1 | 12 |
F3 | 6 | 5 | 4 | 7 | 3 | 14 |
Requirement | 4 | 4 | 6 | 8 | 8 |