Discussion
Consider the following transportation problem :
| Destinations | |||||
|---|---|---|---|---|---|
| Sources | D1 | D2 | D3 | D4 | |
| S1 | 6 | 4 | 1 | 5 | 14 |
| S2 | 8 | 9 | 2 | 7 | 16 |
| S3 | 4 | 3 | 6 | 2 | 5 |
| 6 | 10 | 15 | 4 | ||
The initial transportation cost by Vogel's method is :
Answer: D
Vogel's method gives an initial cost of 114.
Supply 14 + 16 + 5 = 35 equals demand 6 + 10 + 15 + 4 = 35, so the problem is balanced. At each step, penalty = difference of the two lowest costs in a row or column; allocate to the cheapest cell of the line with the largest penalty.
- Largest penalty 5 (row S2): cell S2-D3, cost 2, allocate 15; D3 done, S2 has 1 left.
- Largest penalty 3 (column D4): cell S3-D4, cost 2, allocate 4; D4 done, S3 has 1 left.
- Largest penalty 2 (row S1, tie with column D1): cell S1-D2, cost 4, allocate 10; D2 done, S1 has 4 left.
- Only D1 remains (demand 6): S1 gives 4 at cost 6, S2 gives 1 at cost 8, S3 gives 1 at cost 4.
\[ \begin{aligned} \text{Cost} &= 15(2) + 4(2) + 10(4) \\ &\quad + 4(6) + 1(8) + 1(4) \\ &= 30 + 8 + 40 + 24 + 8 + 4 = 114 \end{aligned} \]
Breaking the tie in step 3 by column D1 instead leads to the same total. 124, 128 and 152 are what cruder starting rules such as the north-west corner method give.
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