a$)$ Find the total transportation cost using Vogel$$s Approximation Method when the unit transportation costs, demands and supplies are as given below:

Destination

Origins D$1$ D$2$ D$3$ D$4$ Supply

O$1619370$

O$21152855$

O$310124770$

Demand $85355045$

b$)$ There are four jobs to be assigned to five machines. Only one job can be assigned to one machine. The amount of time in hours required for the jobs per machine are given in the following matrix:

Machines

Jobs A B C D E

$143627$

$21012111416$

$343215$

$487696$

Find an optimum assignment of jobs to the machines to minimize the total processing time and also find out for which machine no job is assigned. What is the total processing time to complete the jobs?