The document discusses the assignment problem, which involves assigning people, jobs, machines, etc. to minimize costs or maximize profits. It provides an example of assigning 4 men to 4 jobs to minimize total cost, walking through the Hungarian method steps. It also discusses how to handle imbalance by adding dummy rows or columns, and how to convert a maximization problem to minimization.

1. Abu Bashar

2.  It involves assignment of people to projects,
jobs to machines, workers to jobs and teachers
to classes etc., while minimizing the total
assignment costs.
 One of the important characteristics of
assignment problem is that only one job (or
worker) is assigned to one machine (or project).
 An assignment problem is a special type of linear
programming problem where the objective is to
minimize the cost or time of completing a
number of jobs by a number of persons.

3.  This method was developed by D. Konig, a
Hungarian mathematician and is therefore
known as the Hungarian method of
assignment problem.
 In order to use this method, one needs to
know only the cost of making all the possible
assignments.
 Each assignment problem has a matrix (table)
associated with it. Normally, the objects (or
people) one wishes to assign are expressed in
rows, whereas the columns represent the
tasks (or things) assigned to them.
 The number in the table would then be the
costs associated with each particular
assignment.

4.  Though assignment problem finds
applicability in various diverse business
situations, we discuss some of its main
application areas:
 In assigning machines to factory orders.
 In assigning sales/marketing people to sales
territories.
 In assigning contracts to bidders by
systematic bid-evaluation.
 In assigning teachers to classes.
 In assigning accountants to accounts of the
clients.

6.  Mathematically the assignment problem can be
expressed as
 The objective function is
 Minimize C11X11 + C12X12 + ------- +
CnnXnn.

7.  Step 1. Determine the cost table from the given
problem.
 (i) If the no. of sources is equal to no. of
destinations, go to step 3.
 (ii) If the no. of sources is not equal to the no.
of destination, go to step2.
 Step 2. Add a dummy source or dummy
destination, so that the cost table becomes a
square matrix. The cost entries of the dummy
source/destinations are always zero.
 Step 3. Locate the smallest element in each row
of the given cost matrix and then subtract the
same from each element of the row.

8.  Step 4. In the reduced matrix obtained in the step 3,
locate the smallest element of each column and then
subtract the same from each element of that column.
Each column and row now have at least one zero.
 Step 5. In the modified matrix obtained in the step 4,
search for the optimal assignment as follows:
 (a) Examine the rows successively until a row with a
single zero is found. Enrectangle this row ( )and cross
off (X) all other zeros in its column. Continue in this
manner until all the rows have been taken care of.
 (b) Repeat the procedure for each column of the
reduced matrix.
 (c) If a row and/or column has two or more zeros and
one cannot be chosen by inspection then assign
arbitrary any one of these zeros and cross off all other
zeros of that row / column.
 (d) Repeat (a) through (c) above successively until the
chain of assigning ( ) or cross (X) ends.

9.  Step 6. If the number of assignment (􀀀 ) is equal
to n (the order of the cost matrix), an optimum
solution is reached.
 If the number of assignment is less than n(the
order of the matrix), go to the next step.
 Step7. Draw the minimum number of horizontal
and/or vertical lines to cover all the zeros of the
reduced matrix.
 Step 8. Develop the new revised cost matrix as
follows:
 (a)Find the smallest element of the reduced matrix
not covered by any of the lines.
 (b)Subtract this element from all uncovered
elements and add the same to all the elements
laying at the intersection of any two lines.
 Step 9. Go to step 6 and repeat the procedure until
an optimum solution is attained.

10.  A job has four men available for work on four
separate jobs. Only one man can work on any
one job. The cost of assigning each man to each
job is given in the following table. The objective
is to assign men to jobs such that the total cost
of assignment is minimum.

11.  Step 1
 Identify the minimum element in each row
and subtract it from every element of that
row.

12.  Step 2
 Identify the minimum element in each column
and subtract it from every element of that
column.

13.  Make the assignment for the reduced matrix
obtain from steps 1 and 2 in the following
way:

14.  Draw the minimum number of vertical and
horizontal lines necessary to cover all the
zeros in the reduced matrix obtained from
last step

15.  Select the smallest element from all the
uncovered elements. Subtract this smallest
element from all the uncovered elements and
add it to the elements, which lie at the
intersection of two lines. Thus, we obtain
another reduced matrix for fresh assignment.

16. Since the number of assignments is equal to the number
of rows (& columns), this is the optimal solution.
The total cost of assignment = A1 + B4 + C2 + D3
Substitute the values from original table: 20 + 17 + 24
+ 17 = 78.

19.  Some assignment problems entail
maximizing the profit, effectiveness, or layoff
of an assignment of persons to tasks or of
jobs to machines.
 The conversion is accomplished by
subtracting all the elements of the given
effectiveness matrix from the highest
element.
 It turns out that minimizing opportunity loss
produces the same assignment solution as
the original maximization problem.

20.  Five different machines can do any of the five
required jobs, with different profits resulting
from each assignment as given below:

21.  Here, the highest element is 62. So we
subtract each value from 62.
The maximum profit through this
assignment is 214.

23.  It is an assignment problem where the
number of persons is not equal to the
number of jobs.
 If the number of persons is less than the
number of jobs then we introduce one or
more dummy persons (rows) with zero values
to make the assignment problem balanced.
 Likewise, if the number of jobs is less than
the number of persons then we introduce one
or more dummy jobs (columns) with zero
values to make the assignment problem
balanced