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in Operations Research by (48.0k points)
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Determine an initial basic feasible solution to the following transportation problem by using (i) NorthWest Corner rule (ii) least-cost method.

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Total supply = 25 + 35 + 40 = 100 = Σai

Total requirement = 30 + 25 + 45 = 100 = Σbj 

Since Σai = Σbj the given transportation problem is balanced and we can find an initial basic feasible solution.

(i) North West Corner Rule 

First allocation:

Second allocation:

Third allocation:

Fourth allocation:

We first allot 5 units to cell (S2 , D3 ). Then balance 40 units we allot to cell (S3 , D3 ). 

The final allotment is given as follows

Transportation schedule: 

S1 → D1 , S2 → D1 , S2 → D2 , S2 → D3 , S3 → D3 

(i.e) x11 = 25, x 21= 5, x22 = 25, x23 = 5, x33 = 40 

Total cost is = (25 × 9) + (5 × 6) + (25 × 8) + (5 × 4) + (40 × 9) 

= 225 + 30 + 200 + 20 + 360 

= 835 

Hence the minimum cost is Rs. 835 by NWC method. 

(ii) Least cost method (LCM) 

First allocation:

Second allocation:​​​​​​​

Third allocation:

Fourth allocation:​​​​​​​

We first allot 15 units to cell (S3 , D1) since it has the least cost. Then we allot the balance 15 units to cell (S1 , D1 ). 

The final allotment is given as follows

Transportation schedule: 

S1 → D1 , S1 → D3 , S2 → D3 , S3 → D1 , S3 → D2 

(i.e) x11 = 15, x13 = 10, x23 = 35, x31 = 15, x32 = 25 

Total cost is = (15 × 9) + (10 × 5) + (35 × 4) + (15 × 7) + (25 × 6) 

= 136 + 50 + 140 + 105 + 150 

= 580 

The optimal cost by LCM is Rs. 580.

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