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Minimise $Z=\sum_{j=1}^{n} \sum_{i=1}^{m} c_{i j} x_{i j}$
Subject to $\sum_{i=1}^{m} x_{i j}=b_{j}, j=1,2, \ldots, n$
$\sum_{j=1}^{n} x_{i j}=b_{j}, i=1,2, \ldots, m$ is a $L P P$ with number of constr
Options:
Subject to $\sum_{i=1}^{m} x_{i j}=b_{j}, j=1,2, \ldots, n$
$\sum_{j=1}^{n} x_{i j}=b_{j}, i=1,2, \ldots, m$ is a $L P P$ with number of constr
Solution:
1405 Upvotes
Verified Answer
The correct answer is:
$\mathrm{m}+\mathrm{n}$
Constraints will be
$$
\mathrm{x}_{11}+\mathrm{x}_{21}+\ldots .+\mathrm{x}_{\mathrm{ml}}=\mathrm{b}_{1}
$$
$\mathrm{x}_{12}+\mathrm{x}_{22}+\ldots \ldots \mathrm{x}_{\mathrm{m} 2}=\mathrm{b}_{2}$
$\mathrm{x}_{\ln }+\mathrm{x}_{2 \mathrm{n}}+\ldots .+\mathrm{x}_{\mathrm{mn}}=\mathrm{b}_{\mathrm{n}}$
$\mathrm{x}_{11}+\mathrm{x}_{12}+\ldots+\mathrm{x}_{1 \mathrm{n}}=\mathrm{b}_{1}$
$\mathrm{x}_{21}+\mathrm{x}_{22}+\ldots+\mathrm{x}_{2 \mathrm{n}}=\mathrm{b}_{2}$
$\mathrm{x}_{\mathrm{ml}}+\mathrm{x}_{\mathrm{m} 2}+\ldots+\mathrm{x}_{\mathrm{mn}}=\mathrm{b}_{\mathrm{n}}$
So, total number of constraints $=\mathrm{m}+\mathrm{n}$
$$
\mathrm{x}_{11}+\mathrm{x}_{21}+\ldots .+\mathrm{x}_{\mathrm{ml}}=\mathrm{b}_{1}
$$
$\mathrm{x}_{12}+\mathrm{x}_{22}+\ldots \ldots \mathrm{x}_{\mathrm{m} 2}=\mathrm{b}_{2}$
$\mathrm{x}_{\ln }+\mathrm{x}_{2 \mathrm{n}}+\ldots .+\mathrm{x}_{\mathrm{mn}}=\mathrm{b}_{\mathrm{n}}$
$\mathrm{x}_{11}+\mathrm{x}_{12}+\ldots+\mathrm{x}_{1 \mathrm{n}}=\mathrm{b}_{1}$
$\mathrm{x}_{21}+\mathrm{x}_{22}+\ldots+\mathrm{x}_{2 \mathrm{n}}=\mathrm{b}_{2}$
$\mathrm{x}_{\mathrm{ml}}+\mathrm{x}_{\mathrm{m} 2}+\ldots+\mathrm{x}_{\mathrm{mn}}=\mathrm{b}_{\mathrm{n}}$
So, total number of constraints $=\mathrm{m}+\mathrm{n}$
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