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The assignment problem is a problem of assigning a number of jobs/tasks to a number of persons/machines (assignees) so that the total cost or time is minimum. So main theme of the assignment problem is to assign โbest person for the jobโ from a set of workers with varying degrees of skills.
The Assignment Problem
๏ถ Basic assumptions:
- Number of jobs/tasks and the number of persons are the same.
- Each person is assigned with exactly one job.
- Each job/task is to be performed by exactly one person.
๏ถ General assignment model:
Let there are ๐๐ workers and ๐๐ jobs. Let ๐๐๐๐๐๐ be the cost of assigning worker ๐๐ (=1,2,โฆ,๐๐) to job ๐๐
(=1,2,โฆ,๐๐) (๐๐๐๐๐๐ are the pay off elements of the assignment problem, which may also represent
time, distance instead of cost depending upon the problem). This assignment problem can be represented by the following table (matrix) with ๐๐ ร ๐๐ entries (cells):
Jobs
Assignment Model
1 2 โฆ n
Workers
We generally call the above matrix as cost matrix (the entries of the matrix may represent cost, time, distance, etc.)
For an assignment problem with ๐๐ workers and ๐๐ jobs, there are ๐๐! possible ways of making assignments. For example, for an assignment problem with 5 workers and 5 jobs, there are 5! = 120 possible ways of making assignment of jobs to the workers. The objective of the assignment problem is to determine the best assignment out of those large numbers of possibilities so that the total cost/time is minimum.
๐๐๐๐ 1 ๐๐๐๐ 2 โฆ^ ๐๐๐๐๐๐
The assignment problem actually is a special class of transportation problem in which workers represent sources, and jobs represent the destinations. The supply amount of each source as well as the demand amount of each destination exactly equals to 1.
๏ถ Solution procedure:
Note: If a number is added to or subtracted from all the entries of any row or any column
of a cost matrix, then the optimal solution (i.e. assignment) wonโt change.
The Hungarian Method:
To obtain an optimum assignment the following steps are to be performed to the given ๐๐ ร ๐๐ cost matrix.
Step 1. Subtract the smallest entry of each row from all the entries of the row. Similarly subtract each columnโs minimum from all the entries of the corresponding column.
Step 2: Optimal assignment can be obtained by making assignment to the obtained zero element position (as our problem is a minimization problem).
However, the zeros created in step 1 may not yield a feasible assignment directly, in such case we have to proceed as follows to create more zero entries.
Step 2(a). (i) Using minimum number of lines through appropriate rows and columns, cover all the zero entries of the last reduced matrix. If the minimum number of covering lines is ๐๐, then optimal assignment is possible, so make the assignment. If not, then go the next step.
(ii) Choose the smallest entry not covered by any line. Subtract this smallest value from each uncovered entry (i.e. from the element without a line through them), and add it to each entry that lies in the intersection of two lines. Return to step 2(a).
- Minimum cost (or time, distance, etc.) is then obtained by considering the original values of the cells in which the assignments have been made.
Example 1. A construction company has four large bulldozers located at four different garages. The bulldozers are to be moved to four different construction sites. The distances in miles between the bulldozers and the construction sites are given below
revenue or profit is then obtained by considering the original values of the cells in which the assignments have been made.
Ex. 4. A manufacturing company has four zones A, B, C, D. Since the zones are not equally rich in sales potential, it is estimated that a particular sales engineer operating in a particular zone will bring the following sales (in Rs.):
Zone A: 4,20,000, Zone B: 3,36,000, Zone C: 2,94,000, Zone D: 4,62,
There are four sales engineers P, Q, R, S having different sales ability. Their yearly sales are proportional to 14, 9, 11 and 8 respectively. Find the optimum assignment to maximize the sales, and also find the maximum sales.
