Operations research : Assignment problem (One's method) presentation

Assign the five jobs to five machines so as
to minimize the total cost

Find the minimum element of each row in
the assignment matrix (say ai)
and write it on the right hand side of the
matrix, as follows:

Then divide each element of ith row of the
matrix by ai. These operations
create ones to each rows, and the matrix
reduces to following matrix.

Now find the minimum element of each column
in assignment matrix (say bj), and write it below
that column. Then divide each element of jth
column of the matrix by bj

The minimum number of lines
required to pass through all the ones
of thematrix is 5

So, the complete assignment is
possible

• we can assign the ones and the solution is
(1,5)
(2,3)
(3,4)
(4,1)
(5,2)
and minimum cost is 24

A company has 5 jobs to be done. The following
matrix shows the return of assigning ith
machine to the jth job. Assign the five jobs to
the five machines so as to maximize the total
return

step 1. Find the maximum element of each
row in the assignment matrix(say ai) and
write it on the right hand side of the
matrix, as follows:

Then divide each element of ith row of the
matrix by ai. These operations
create ones to each rows, and the matrix
reduces to following matrix

• step 2.
• Since we can not make assignment in terms of
the ones, now find the maximum element of
each column in assignment matrix ( say bj), and
write it below that column.
• Then divide each element of jth column of the
matrix By bj
• These operations create ones to each columns,
and matrix reduces to following matrix , feasible
assignment can not be secured from step (1) and
(2) so, go to step 3

step 3. Draw the minimum number of lines
equired to pass through all
the ones of the matrix. Clearly the four lines are
passing through all the ones. Since the number
of lines is not equal to the order of the matrix,
optimal assignment has not been reached

step 4.
• Select the largest element (say
• dij) out of those which do not lie on any of the
lines in the above matrix. It is in the third
column dij=0.83. Then divide each element of
the third column, which dij is on it. This
operation creates a new one to this column

Now, we can assign the ones, it is
based on priority rule. Priority rule is
assigning one on the rows which have
greatest element on the right hand
side, respectively.

The details of this program are as follows:
• Machine 3 assigns to job 4 profit 14
• so the optimal assignment has been reached, and
total profit according to this plan is
10+5+14+14+7=50.

• References
http://www.m-hikari.com/ams/ams-2012/ams-
45-48-2012/basirzadehAMS45-48-2012.pdf
Thanks to
• Hadi Basirzadeh
• Department of Mathematics
• Shahid Chamran University, Ahvaz, Iran
• basirzad@scu.ac.ir

Operations research : Assignment problem (One's method) presentation

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