In case, if the given problem is unbalanced one, you have to add appropriate number of rows or columns with zero as assignment cost [dummy row or dummy column] and make the problem as balanced assignment model.
Variations in Assignment
Problem- Case 2: Unbalanced Models
1. An Assignment problem is known as balanced one, if the number of rows equals to the number of columns.
2. When the number of rows is not equal to number of
columns, the assignment models are called as unbalanced assignment problems. 3. In case,
if the given problem is unbalanced one, you have to add appropriate number of
rows or columns with zero as assignment cost [dummy row or dummy column] and
make the problem as balanced assignment model. Exercise-1 A university wants to allocate the four subjects
and six teachers claim that they have the required competencies!! /knowledge!!
to teach all the subjects. The dean believes that the failure in the course is
reflection of faculty member’s performance! Allocate the subject to appropriate faculty
members.
Solution You may notice that the number of rows (6) not
equal to the number of columns (4); thus, the given problem is an example of
unbalanced assignment models. We add two dummy columns (2 dummy subjects) with
zero as number failures. The modified problem becomes a balanced one and the
initial table is given below;
Now we can use Hungarian algorithm, which requires
a balanced one to apply. The following table is obtained after column
reduction, since row reduction will give the same table above.
1. The first
straight line is drawn at Column-5 and second is at Column-6, which are having
six zeros each.
2. The next
line is drawn at Row-3 3. Fourth
line is drawn at Row-1; fifth at Row-4; the last line at Row-5 4. Since the
number of straight lines equal to number of rows/columns, optimum number of zeros is
available to make the final allocation.
The next
step is to consider the zeros and making the allocations. Since, every row is
having zeros; we do the column wise search;
1. Column-1,
Column-3 & Column-4 are having unique zero, we mark them for allocation and
cross out the respective rows in which the zero appear.
2. In the
process, we get unique zero in Column-2 and strike out the Row-5
3. Now we
have tie at Row-2 and Row-6, which can be broken arbitrarily.
4. We
allocate Row-2 to Column 5 (E) and Row-6 to Column-6 (F). Tags : Operations Management - Transportation / Assignment & Inventory Management
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