 Hello friends, I am Mr. Sanjeev B. Naik working as an assistant professor in mechanical engineering department, Walsh and history of technology, Sholapur. In this video, I am discussing about the solution to unbalanced assignment problems. At the end of the session, learners will be able to determine the optimal solution to assignment unbalanced problem. So, understanding of unbalanced problem is that, whenever the number of facilities to be allocated to number of tasks are not equal. For example, if the number of workers are not equal to number of jobs to be performed, then it is called as unbalanced assignment problem. That means, when you construct the effectiveness matrix of performing all the jobs by all the workers and if the workers are not equal to the jobs, then number of rows of such matrix will not be equal to number of columns and that is why it is not a square matrix. So, whenever the effectiveness matrix is not a square such that the facilities are not equal to tasks, then it is called as unbalanced assignment problem. So, pause the video for sometime and just recall the Hungarian method that I have explained in my earlier video which can be used to solve the assignment problem. Let me consider an example of unbalanced assignment problem. Here a manager has got four jobs to be performed, but staff available is only three critical staff. That means jobs are four whereas the availability of facility like clerical staff are only three and all the staff differ in the efficiency and that efficiency is measured as a time taken by particular staff to perform the particular job or to complete the particular job and that has been shown over here. So, this is an assignment effectiveness matrix in which a clerical staff is said to be men x, y, z. So, three clerical staff is there and there are four jobs. So, this is a time taken by a particular man to perform particular job as a ten hours. So, if x clerk is assigned a job e, then he requires ten hours to perform it. Similarly, if y is given the same job, he may take twenty cents. So, that is what the effectiveness matrix is given in the time taken by particular staff to complete particular job. However, if you find here the number of staff available is a three whereas the jobs are four and that is why it is an unbalanced assignment problem. So, the first requirement to solve it by Hungarian method is that we have to see that the number of men as they are less than number of jobs, it is unbalanced assignment problem and we have to convert into balanced problem. So, first step of Hungarian method is that it must be square matrix. That means number of rows must be equal to number of column and then we have to add a shortfall as the column is less. That is why we have to add one column and column refers for staff and that is why it is a dummy staff and that is a dummy person. We introduce in this matrix to balance it and all the elements of that particular column must be zero. So, this is the way first step is performed to balance the given problem as it was unbalanced. One column was less means one person is less, we add that person as a dummy as a false and all these elements are zero. So, all these four elements to perform the jobs is zero that we assign it. So, this is the first step to make it balanced. Once it is balanced then we apply Hungarian method to identify the smallest element in the row because our target is to assign that smallest time element. So, here we find that every row has got zero in it and that is what a smallest element. And that is why if you subtract this smallest element from all these elements it remains same. So, all these elements will remain same in this step also. So, whatever unbalanced problem we have converted is a balanced and we have got effectiveness matrix that itself gives the same matrix. When we do this operation that subtract the smallest element of every row from all the elements of respective row. That means I am subtracting zero from all these it remains same. And that is why the first step of subtracting smallest element row wise is here. Now, we know that we have to also identify the smallest element in the column. And that is why we have to subtract the smallest element of every column from all the elements of that corresponding column. So, here if you see first column the smallest element is 10. So, I want to subtract this from all so that I identify this is the smallest element in the particular column. Where I should allocate the assignment that will result in minimum time. So, 10 being smaller I will subtract it and I will get the next matrix which is called the reduced matrix. So, look here 10 minus 10 is zero, 14 minus 10 is 4, 36 minus 10 is 26 and 19 minus 39. That means smallest element of every column is subtracted from all the elements of corresponding column. So, this way I perform the operation of step 3 and I get this reduced matrix. So, now in this matrix we have seen that the minimum element which is identified as a zero by considering row as well as column. And that is why these are the best possible positions of the minimum element cells where I should allocate particular man to particular job. So, that the time required is minimum and that is been performed by step number 4. So, what is the assignment we want to do? Go row wise and find out single zero. So, here there is no single zero there are two zeroes means either I can assign a job to x or demi person. So, as option is there I do not utilize it. Second row also two zeroes I do not do it. Third row two zeroes I do not do it, but fourth row here is a zero which I can assign. Here is a zero which I can assign. As it is considered row wise I cancel the column means this demi person is already given the job D. So, he should not be given any other job to make that I draw this line vertical line. So, this is the first assignment I have done after going through all the rows. Now, I repeat the procedure through column. Now, I go in the first column in the first column there is a single zero means there is no option. Any other position is not there which is minimum only this is the position. So, I can grab this opportunity of allocating the person. So, this is I am allocating with column wise that means I am taking man x to be allocating job A. So, A job is given to x means it cannot be given now to anybody else and that is why I can draw the horizontal line. So, what is understanding the person x is given job A and when it is given job A, job A should not be given to any other person to make that sure I draw this horizontal line. Then I go in second column, second column I have got single zero once again. So, I allocate it allocate means I make a bracket and I am fixing this person y for job C. And as a job C is given to y it should not be given to anybody else. So, I draw horizontal line then similarly going the third column once again a single zero we can allocate that zero. As it is single there is no option when there is no option we can directly bracket it and we decide it. What we have decided the Z man should be given job B and as job B is given to Z. So, it is been drawn with horizontal line means B should not be given to anybody else and that is what we draw horizontal line. Then we go to the last column. So, last column is already allocated. So, no zero is there to make the allocation. So, that way we have made this allocation when we have went row wise we have done this one and these three allocations we have made column wise because there was no option for that. So, now you just look that any other zero is left out whether it is bracketed or not cut by a line we identified but there is no zero. So, you just look that this is the only row which does not have any line but there is no zero left out and that is why the assignment is complete. Now to see the assignment is complete we count on the lines 1, 2, 3 and 4 or we just see that the brackets made are 1, 2, 3 and 4. That means if the brackets made are 4 or number of lines drawn equal to 4 which is a size of matrix or order of matrix. We require allocation 1 to 1 as a for 4 jobs and 4 workers. That means we require 4 assignments of particular worker to particular job which is being completed over here. That is why we conclude that the solution to this problem is as the lines drawn are 4 equal to order of matrix that is 4. The optimal assignment is obtained by this way that we have assigned a to x, b to z, c to y and d to demi person. So, this is what we have drawn the brackets. So, this is what assignment and that is why the optimal assignment is obtained over here as a to x, b to z, c to y and d to demi person. When it is demi person which is not there. So, this demi person is not at all there. That is why the job allocated to this person will not be completed. That is why it will be idle. Only 3 men are there so 3 jobs will be done but the way they have been done is with minimum time. So, if you just add this 10 plus 7 plus 21 is the total time required to complete 3 jobs by 3 workers and that is 38 man hours. This is what a minimum solution by which out of 4 3 jobs can be completed because it is unbalanced assignment problem. The fourth job will be idle and that is what we get the answer as a optimal solution. These are my references.