# assignment problem

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```					       UNIT II
TOPIC
ASSIGNMENT PROBLEM

gaurav sonkar
Gaurav Sonkar

ASSIGNMENT PROBLEM
The assignment problem is a particular case of
transportation problem in which the objective is to
assign a number of resources to an equal number of
activities so as to minimize total cost or maximize
total profit.
Typical examples of decision making situations where
assignment model can be used are:
workers to machines
sales personnel to different sales areas
products to factories
jobs to machines
classes to rooms                                  2
A GENERAL ASSIGNMENT PROBLEM LOOKS
Gaurav Sonkar

LIKE:

1      2        ……….            n
1    C11   C12      …………..         C1n   1
2    C21   C22      ……………          C2n   1
:     .     .         .              .
.     .         .              .
……………
Cn1   Cn2                     Cnn
n                     .                  1
1           1   ………..            1

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Gaurav Sonkar

HOW TO SOLVE AN A.P.

Step I:
Row Operation:
Subtracting smallest element in each row from all other elements of
that row.
Column Operation:
Subtracting smallest element in each column from all other elements
of that column.

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Step II: MAKING ASSIGNMENTS:
Gaurav Sonkar

Search for an optimal assignment in the finally
modified cost matrix as follows:
Examine the first row. If there is only one zero in it,
then enclose this zero in a box ( ) cross (X) all the
zero of that column passing through the enclosed
zero.
Similarly done with other rows.
If any row has more than one zero, then do not
touch that row & passes on the next row.
Repeat the same procedure with columns.
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Step III:                              Gaurav Sonkar

 If each row and each column of the reduced matrix has
exactly one enclosed zero, then the enclosed zero yield an
optimal assignment.
 If not, then go to next step.
Step IV:
 Draw minimum number of horizontal and/or vertical lines
to cover all the zeros of the reduced matrix.
 Since the assignment is not optimal, the number of lines
will be less than n (i.e. order of matrix).
 In order to move towards optimality, generates more zeros
as follows:
 Find the smallest of the elements of the reduced matrix not
covered by any of the lines. Let this element be α.
 Subtract α from each of the element not covered by the
lines and add α to the element at the intersection of these
lines.
 Do not change the remaining elements                       6
Gaurav Sonkar

Step V:
Go to step II and repeat the procedure till an
optimal assignment is achieved.

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ILLUSTRATION 1       Gaurav Sonkar

performed. The subordinates differ in efficiency and the tasks
differ in their intrinsic difficulty. His estimate of the times each
man would take to perform each task is given in the effectiveness
matrix below. How should the task be allocated, one to a man, so
as to minimize the total man hours?

I        II        III          IV
A        8        26        17            11
B        13       28         4            26
C        38       19        18            15
D        19       26        24            10

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SOLUTION       Gaurav Sonkar

Step I:
Row Operation:
Subtracting smallest element in each row from all other elements of
that row.
I      II     III     IV              I            II   III   IV

A                                   A
B                                   B
C                                   C
D                                   D
Column Operation:
Subtracting smallest element in each column from all other
elements of that column.                                                9
Step II: MAKING ASSIGNMENTS:         Gaurav Sonkar

I    II    III           IV

A    0    14    9            3
B    9    20    0           22
C   23     0    3            0
D    9    12    14           0
Since every row and every column have one assignment,
hence the current solution is optimal.

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, THE OPTIMAL SOLUTION OBTAINED IS
Gaurav Sonkar

AS UNDER

TOTAL MAN HOURS
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ILLUSTRATION 2
Gaurav Sonkar

A department has four subordinates and four tasks to be
performed. The subordinates differ in efficiency and the
tasks differ in their intrinsic difficulties. The estimates of
time (in man-hours) each man would take to perform
SUBORDINATES
JOBS
I        II     III              IV
A       18       26      17               11
B       13       28      14               26
C       38       19      18               15
D       29       26      24               10

How should the tasks to be allocated to men so as to
optimize the total man-hours?                   12
SOLUTION       Gaurav Sonkar

STEP I:
Subtracting the smallest element of each row from all the
elements of that row, we obtain the reduced matrix (a) given below:
Further subtracting the smallest element of each column from all
the elements of that column, we obtain the reduced matrix (b)
given below:

I     II    III    IV                    I       II   III   IV
A     7     15     6      10        A           7       11   5     0
B     0     15     1      13        B           0       11   0     13
C     23     4     3      0         C         23        0    2     0
D     19    16     14     0         D         19        12   13    0

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STEP II:                                      Gaurav Sonkar

Examine the rows of matrix (b) successively until a row with exactly
one zero is found, enclose that zero in a box and cross all the other
zero in its column. Thus we obtain matrix (c)

I     II     III          IV
A     7      11     5             0
B     0      11     0            13
C     23     0      2             0
D     19    12     13             0

Further examine the columns of matrix (c) successively until a column
with exactly one unmarked zero is found, enclose that zero in a box
and cross all the other zero in its row
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STEP III:                              Gaurav Sonkar

Matrix (c) does not provide an optimal solution, since fourth row as
well as third column does not have an enclosed zero.
STEP IV:
Now we draw minimum number of horizontal and / or vertical lines to
cover all the zeros of the reduced matrix (c)
I     II    III    IV
A      7    11      5     0
B      0    11      0     13
C     23     0      2     0
D     19    12     13     0

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The smallest element in matrix (d) not covered by any of the lines above is 5.
Gaurav Sonkar

Subtracting this element from all uncovered elements and adding the same to the
elements lying at the intersection of these lines, we obtain matrix (e) given
below:

I       II     III     IV
A       2       6       0      0
B       0      11       0      18
C      23       0       2       5
D      14       7       8      0

STEP V:
Following the procedure of enclosing and crossing the zeros as in
(step II) in matrix (e).
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STEP VI:                      Gaurav Sonkar

Since each row and each column in matrix (g) has
exactly one enclosed zero, we have attained the
following optimal assignment schedule:
A     III, B    I, C  II, D   IV
The minimum total time for this assignment
schedule is:
Z = 17+13+19+10= 59 man-hours.

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