*The smallest element in the first row is, for example, 69.*Therefore, we substract 69 from each element in the first row.

*The smallest element in the first row is, for example, 69.*Therefore, we substract 69 from each element in the first row.

In the matrix formulation, we are given a nonnegative n×n matrix, where the element in the i-th row and j-th column represents the cost of assigning the j-th job to the i-th worker.

We have to find an assignment of the jobs to the workers, such that each job is assigned to one worker and each worker is assigned one job, such that the total cost of assignment is minimum.

The Hungarian method finds a perfect matching and a potential such that the matching cost equals the potential value. In fact, the Hungarian method finds a perfect matching of tight edges: an edge ) which has the property that the edges oriented from T to S form a matching M.

Initially, y is 0 everywhere, and all edges are oriented from S to T (so M is empty).

Step 4: Create additional zeros First, we find that the smallest uncovered number is 6.

We subtract this number from all uncovered elements and add it to all elements that are covered twice.

We consider an example where four jobs (J1, J2, J3, and J4) need to be executed by four workers (W1, W2, W3, and W4), one job per worker.

The matrix below shows the cost of assigning a certain worker to a certain job.

In this simple example there are three workers: Paul, Dave, and Chris.

One of them has to clean the bathroom, another sweep the floors and the third washes the windows, but they each demand different pay for the various tasks.

## Comments Hungarian Method Solving Assignment Problem

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