operational research in topic assignment problem

1. ASSIGNMENT PROBLEM-
MAXIMUM
 There are problems where certain facilities
have to be assigned to a number of jobs so as
to maximize the overall performance of the
assignment.
 The problem can be converted into a
minimization problem in the following ways
and then Hungarian method can be used for
its solution.
I. Change the signs of all values given in the table.
II. Select the highest element in the entire assignment table and
subtract all the elements of the table from the highest element.

2. Example : A marketing manager has five salesmen and
sales districts. Considering the capabilities of the
salesmen and the nature of districts, the marketing
manager estimates that sales per month (in hundred
rupees) for each salesman in each district would be as
follows. Find the assignment of salesmen to districts
that will result in maximum sales.

3. Step 1: The given maximization problem is converted into minimization problem by subtracting from the
highest sales value (i.e., 41) with all elements of the given table.
Max value

4. HUNGARIAN
METHODStep 2: Reduce the matrix by selecting the smallest value in each row and
subtracting from
other values in that corresponding row.
the smallest value:
row 1, is 1,row 2 is 1,row 3 is 0, row 4 is 0 and row 5 is 1.

5. HUNGARIAN
METHODStep 3: Reduce the new matrix given in the following table by selecting the smallest value in
each column and subtract from other values in that corresponding column.
In column A, the smallest value is 0, column B is 2,
column C is 0, column D is 5 and column E is 0.

6. HUNGARIAN
METHODStep 4: Draw minimum number of lines possible to cover all the zeros in the matrix given in Table
Check whether number of lines drawn is equal to the order of the matrix, i.e., 3 ≠ 5. Therefore
optimally is not reached.

7. HUNGARIAN
METHODStep 4: Take the smallest element of the matrix that is not covered by single line, which is 3. Subtract 3 from
all other values that are not covered and add 3 at the intersection of lines. Leave the values which are
covered by single line.
4
Select the least uncovered element, i.e., 4 and subtract it from other uncovered elements, add it to the elements at
intersection of line and leave the elements that are covered with single line unchanged,

8. HUNGARIAN
METHOD Now, number of lines drawn = Order of matrix, hence optimality is reached.

9. HUNGARIAN
METHOD There are two alternative assignments due to presence of zero elements in cells (4, C), (4, D), (5,
C) and (5, D).