Discussion
Consider the transportation problem
Min. \(z = \sum_{i=1}^{m}\sum_{j=1}^{n} c_{ij}x_{ij}\)
Subject to :
\(\sum_{j=1}^{n} x_{ij} = a_{i}, i = 1, 2, ....., m,\)
\(\sum_{i=1}^{m} x_{ij} = b_{j}, j = 1, 2, ....., n\) and \(x_{ij} \ge 0\)
The existence of a feasible solution of the transportation problem is possible :
(I) if \(\sum_{i=1}^{m} a_{i} = \sum_{j=1}^{n} b_{j}\)
(II) if and only if \(\sum_{i=1}^{m} a_{i} = \sum_{j=1}^{n} b_{j}\)
Then which of the following is true ?
Answer: C
Both statements are true.
If total supply equals total demand, a feasible solution always exists: for example \(x_{ij} = a_{i}b_{j}/T\) with \(T = \sum a_{i} = \sum b_{j}\) satisfies every row and column constraint and is non-negative. So (I) holds.
Conversely, with equality constraints, summing all row constraints gives \(\sum\sum x_{ij} = \sum a_{i}\) and summing all column constraints gives \(\sum\sum x_{ij} = \sum b_{j}\), so a feasible solution forces \(\sum a_{i} = \sum b_{j}\). Hence the condition is necessary as well as sufficient, and (II) holds too.
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