Question

The following table summarizes the supply, demand and cost information for four factors S₁, S₂, S₃, S4 Shipping goods to three warehouses D₁, D₂, D₃.

	D1	D2	D3	Supply
S1	2	7	14	5
S2	3	3	1	8
S3	5	4	7	7
S4	1	6	2	14
Demand	7	9	18

Find an initial solution by using north west corner rule. What is the total cost for this solution?

Answer 1

Let ‘a_i‘ denote the supply and ‘b_j‘ denote the demand.

Then total supply = 5 + 8 + 7 + 14 = 34 and Total demand = 7 + 9 + 18 = 34

`sum"a"_"j" = sum"b"_"j"`.

So the problem is a balanced transportation problem and we can find a basic feasible solution, by North-west comer rule.

First allocation:

	D1	D2	D3	Supply (aj)
S1	(5)2	7	14	5/0
S2	3	3	1	8
S3	5	4	7	7
S4	1	6	2	14
Demand (bj)	7/2	9	18

Second allocation:

	D1	D2	D3	(aj)
S2	(2)3	3	1	8/6
S3	5	4	7	7
S4	1	6	2	14
(bj)	2/0	9	18

Third allocation:

	D2	D3	(aj)
S2	(6)3	1	6/0
S3	4	7	7
S4	6	2	14
(bj)	9/3	18

Fourth allocation:

	D2	D3	(aj)
S2	(3)4	7	7/4
S4	6	2	14
(bj)	3/0	18

Fifth allocation:

	D3	(aj)
S2	(4)7	4/0
S4	(14)2	14/0
(bj)	18/14/0

We first allow 4 units to cell (S₃, D₃) and then the balance 14 units to cell (S₄, D₃).

Thus we get the following allocations:

	D1	D2	D3	Supply
S1	(5)2	7	14	5
S2	(2)3	(6)3	1	8
S3	5	(3)4	(4)7	7
S4	1	6	(14)2	14
Demand	7	9	18

The transportation schedule:

S₁ → D₁

S₂ → D₁

S₂ → D₂

S₃ → D₂

S₃ → D₃

S₄ → D₃

i.e x₁₁ = 5

x₂₁ = 2

x₂₂ = 6

x₃₂ = 3

x₃₃ = 4

x₄₃ = 14

Total cost = (5 × 2) + (2 × 3) + (6 × 3) + (3 × 4) + (4 × 7) + (14 × 2)

= 10 + 6+ 18 + 12 + 28 + 28

= 102

Thus the initial basic solution is got by NWC method and minimum cost is ₹ 102.