A Government Organization Consists of N Employees.

A government organization consists of n employees.  


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1. A government organization consists of n employees. Requests from petitioners are submitted
to a designated official, and then are forwarded from one bureaucrat to another until they
reach the head of the organization who either approves or declines. When a bureaucrat
receives a petition, they forward it to one of their colleagues, and for each pair of bureaucrats it
is known whether one can send documents to the other and vice versa (nobody besides the
head makes any decisions other than forwarding the petitions). Forwarding between the
officials is done with a courier, so there is a limit uij – the maximum number of petitions in an
hour that bureaucrat i can forward to bureaucrat j. You are performing an audit of this
organization. While trying to eliminate the useless paperwork forwarding all together is not
realistic, you at least want to know how many petitions in an hour can be processed by the
current system.
(a) Formulate the problem of determining the maximum rate at which the petitions can be
processed as a standard maximum flow problem.
(b) Now suppose that since each employer is very busy with other equally important duties,
there is also a limit on the number of petitions per hour that they can process: ai . Formulate
the problem of determining the maximum rate at which the petitions can be processed as a
standard maximum flow problem.
(c) Suppose that n = 5 and the relevant data is given below. Draw the graphs that correspond to
your models in 2a and 2b