 Hello friends, I am Mr. Sanjeev B. Naik, working as an assistant professor in mechanical engineering department, Walsh and district of technology, in this video, I am discussing about the assignment problems which occur in managerial decision making. So, at the end of the session, the learners will be able to formulate the assignment problem and also will be able to determine the optimal solution to assignment problem by over technique. Basically the assignment problem is a special type of allocation of resource problem where one-to-one allocation is required. So, it is also a special type of linear programming problem in which one-to-one allocation of resource is important and that is why the applications are peculiar. So, allocation of workers to job for example, it is a very peculiar requirement of assignment that number of workers are available which are required to assign for different jobs to be performed such seen the workers can do the jobs with different efficiency that means the time taken by each worker to perform on each job is different and that is why it is a important task for a manager to how to allocate the available workers to perform the required jobs so that the effectiveness is satisfying the objective that means if the time is considered that particular worker can perform on particular job what time he requires is an important parameter and that is called as effectiveness. So, the effectiveness is a measure of performance of particular worker when he is assigned to particular job and that measure can be a time may be a cost may be a profit may be efficiency and that is what we call as effectiveness matrix. So, while allocating one-to-one as a resource to a different task or facilities to different task then effectiveness matrix must be considered and based upon the effectiveness matrix the objective function is decided and then the best possible allocation of the workers to different jobs so that the total time is minimum or total cost is minimum if at all profit is the effectiveness then the profit must be maximum that becomes the requirement of the solution. So, similar area is workers to different machine so once again the workers may be scaled in some of the machines and naturally their efficiency will be higher. So, considering the efficiency of the worker working on different machines this will be properly allocated to particular machine so that the total efficiency which is derived should be maximum and then drivers to vehicles, salesmen to sales territories, contracts to bidders. So, these are the other major areas where one-to-one allocation is needed and that can be very well formulated as an assignment problem and can be solved by using assignment technique. Just pause the video for some time and think over all the industrial resources where one-to-one allocation is needed with specific objective to satisfy. So, just list out of such resources which are been required in the industries. So, let me consider an example where we want to formulate the assignment problem. So, it is a peculiar example where a computer center has four expert programmers which are required to develop four application programs and the head of the computer center he knows that the computer time required by each programmer to develop each program is estimated in minutes and that is been given as an effectiveness matrix over here. So, there are four programmers and all of them can do all the programs, but the efficiency being different the expertise between different they can require different times. So, programmer one can develop program A in 120 minutes whereas, if he develops program B he requires 100 minutes, see 80 minutes likewise the data about the requirement of a time to develop the program by particular programmer is given and that gives a formulation of assignment problem because this becomes an effectiveness matrix. So, naturally what the computer head is expecting that he want to allocate these programmers to develop these programs in such a way that the total time required to perform all these programs or to develop all these programs is minimum. So, that is what an objective of finding the solution. So, finding the best possible assignment of programmers to develop all these programs with minimum time and that is what an objective function to solve this assignment problem. Now the solution made available by technique is called as a Hungarian method. So, the Hungarian method plays a simple logic as I said that the expected is that the particular programmer for example, if he taken one he requires minimum time to develop program C. So, naturally if my objective function is minimizing the time I should target to allocate program or one to program C but if I give program C to one means that C cannot be given to anybody else and that is why it depends upon whether it is justifying the objective function or not and that can be calculated by particular steps by identifying the minimum element in particular row and as well as it should be minimum in column that is the best selection of minimum element. So, minimum time according to a row is not been only a helpful but it should be also minimum in its column and that is why then that becomes the best assignment of particular one to C for example and that is been obtained as a step over here. So, the first thing is that the given requirement must be n person should be allocating n jobs that means it must be square matrix. So, effectiveness matrix must be number of rows must be equal to number of columns and that is why it must be square matrix. So, here it is a square matrix as four programmers are expected to assign for four jobs or four programs. Now to identify the smallest time required by a particular programmer to particular job or particular program we subtract the smallest element from every row from the all the elements of that row. For example, in first row 80 is the minimum element. So, we are targeting this. So, targeting means we identify this 80 in terms of 0 how we subtract it from all the elements and we get here when it is subtracted from itself it automatically becomes 0 and that is why when we look to 0 in this matrix we say that this is the minimum time required for the allocation of one to C. So, this is what a step we used as subtracting the smallest element of every row we get this reduced matrix. So, for example, 80 is subtracted from 120 it is 40 then 20 0 and 10. So, this is the way we can do all these operations for four rows. Then similarly to identify it should be smallest in the column also we make the same thing that we subtract the smallest element from every column and that is why we convert this as a another reduced matrix. So, we identify there are number of zeros which are available in this matrix where we have performed row wise operation of smallest element subtraction and column wise operation of smallest element subtraction. That means all the zeros are best possible cells which are minimum in the row as well as in their column and that is why if we now assign these zeros for particular program or to develop particular program the total time required will be definitely minimum and that is what objective required. So, to make this assignment at the position of zero we follow this step that identify the reduced matrix where row and column operations are performed to convert the zeros at the smallest element and go row wise now. So, first row is identified with single zero. So, in first row there is a single zero means there is no option that we have to view program or one the job C otherwise it will be more time and that is why we make allocation means we bracket it and as the job C is already given to programmer one it is cancelled. So, nowhere else it should be given to any other programmer to assure that we draw vertical line. So, same way now we go in the second row but second row we have got two zeros any one of them can be asked allocated. So, it is option. So, we do not use it third row also there is option. So, we do not decide immediately fourth row is also two zeros. So, there is option we do not decide then we look the better allocation from column point of view. So, as far as the column considered we go one by one in the column same way and we identify single zero but here there are three zeros we do not do the assignment but in second column there is a single zero. So, this is already been assigned. So, it is been column is exhausted. So, here we do single zero is there in this column. So, we assign it and as we are considering column as a single zero we cut the row. That means fourth programmer is already given this development of program B and that is why that fourth programmer should not be used for that. So, that is why it is cut. So, in third column already assignment is done in fourth column there are two zeros. So, we do not do anything. Now, we gone row wise and column wise but the assignments done are only two but still there are certain zeros which are left out. So, now we can assign these zeros by selecting arbitrary one of the option. So, if I select this zero I discard it means I do not consider this zero and I assume that this is the only one zero available in the row and then I assign it. When I assign according to row as a single zero I cut the column. That means this program is been given to second programmer that is what decided and then I go in the third row where there is a single zero I will assign it and once again I cancel the column. So, this way now we find that all the zeros are either bracketed or cut by a line. That means we have drawn the lines over here and we have made four brackets as a four assignments that is what are the requirements. But however there is another option instead of taking this diagonal I can take this which is a cross instead of this option I can shift this bracket over here automatically this bracket will be shifted over here this called the diagonal assignment which is also the same way because it is also zero in its position that is why it will resets in the same solution and that is what I can get two solutions for this problem. One is a solution of these brackets four brackets with the four lines and that will be first solution whereas I can interchange these two brackets with this bracket because it is in the diagonal position of the zero and that is why the solution obtained by this method as the lines drawn are four means we have made the four brackets and that is why it is equal to the order of matrix the assignment is complete and we get here the optimal assignment as a first solution. Programmer one is assigned job C or developing C program programmer two is assigned this bracket that is a program A third is given D and fourth is given B and its time requirement is calculated over here which results as a 350 minutes. So this is one of the optimal solution by proper assignment of this programmers to develop program with minimum time but if I take this as another solution instead of these two brackets I shift this bracket over here and shift this bracket over here I get once again four assignments and that is another possible solution called as a suboptimal solution in which one to C two to D three to A and four to B which will also result in the same solution that means there is a flexibility provided to the manager to decide the allocation because there exists suboptimal solution. So there is two possibility solutions are there and which result in the same these are my references thank you.