Assignment Model in Operation Research – Examples and types

If you had multiple jobs to be performed on different machines, how would you do it? How would you assign a particular job to a machine? Similarly for 10 different sales jobs, how would you assign the most apt role to every person according to his capabilities?

Let’s say there are 2 jobs (J1 and J2) to be performed on 2 machines (M1 and M2). J1 on M1, J1 on M2, J2 on M1 and J2 on M2 will be the combinations. Now, which of the combinations will you choose to perform the jobs? While dealing with such problems, less cost, more efficiency and many other factors come into play.

But what if there are multiple jobs to be performed on multiple machines? For this, a Hungarian mathematician developed a method called the assignment model, which can be used to solve such problems. (Note that this method is useful when the number of sources equals the number of destinations and the capacity & demand value is 1 unit).

Example 1: A departmental head has 4 subordinates and 4 tasks for completion. The subordinates differ in their work capabilities and the tasks differ in their work contents and intrinsic difficulties. His estimate of time for each subordinate and each task is given in the matrix below:

Solution:

The task is to assign 1 job to 1 person so that the total number of hours are minimized. So, the first step in the assignment model would be to deduce all the numbers by the smallest number in the row. Hence, the smallest number becomes 0 and then we can target the zeroes to arrive at a conclusion.

The following numbers in bold, will be reduced to 0

As we completed this operation row wise, now we will identify the smallest number in the column and deduce it to 0. While doing the operation column wise, make sure you identify the columns which don’t have a zero. In the above table, the column of subordinate 2 does not have any 0.   

Hence, we have reached the optimal solution. Now we have to assign the tasks to respective subordinates.

From the above table we can conclude that task A will be performed by subordinate I and task D will be performed by subordinate IV. Since, IV has been assigned D, it can’t be assigned C. Therefore, task C will be assigned to II. And similarly, task C has been assigned to II, hence task B will be assigned to III.

Therefore, the total number of hours to perform all the tasks by all subordinates will be 77 hours.