Question

Three different aeroplane are to be assigned to carry three cargo consignments with a view to maximize profit.The profit matrix (in lakh of Rs) is as follows:

Aeroplanes	Cargo consignments	Cargo consignments	Cargo consignments
Aeroplanes	$C_1$	$C_2$	$C_3$
$A_1$	$1$	$4$	$5$
$A_2$	$2$	$3$	$3$
$A_3$	$3$	$1$	$2$

How should the cargo consignments be assigned to the aeroplane to maximise the profit?

Answer 1

$\begin{array}{cccc}& C_1& C_2& C_3\\ A_1& 1& 4& 5\\ A_2& 2& 3& 3\\ A_3& 3& 1& 2\end{array}$

First change into minimisation problem by

substracting every element from maximum value.

$\begin{array}{cccc}& C_1& C_2& C_3\\ A_1& 4& 1& 0\\ A_2& 3& 2& 2\\ A_3& 2& 4& 3\end{array}$

Applying row reduction

$\begin{array}{cccc}& C_1& C_2& C_3\\ A_1& 4& 1& 0\\ A_2& 3& 0& 0\\ A_3& 0& 2& 1\end{array}$

Applying coloumn reduction (Ref. image)

No of crossed lines $=$ rank of matrix

$=3$

optimal solution solution

$A_1$ -$C_3$

$A_2$ -$C_2$

$A_3$ -$C_1$

Maximum profit $=5+3+2$

$=10$