Goods manufactured at $3$ plants, A$,$ B and C are required to be transported to sales outlets X$,$ Y and Z$.$ the Unit costs of transporting the goods from the plants to the outlets are given in the talble below

You are required to:

$($i$)$ Compute the initial allocation by north$-$west corner rule

$($ii$)$ Compute the initial allocation by vogel$$s approximation method and check whether it is optional

$($iii$)$ State your analysis on the optionality of allocation under North$-$West corner rule and vogel$$s approximation method