How do you perform row reduction on a 4x4 cost matrix in the Hungarian method?
Conditions
- The problem is a balanced assignment problem.
- The Hungarian method is being applied.
Reasoning, step by step
- Identify the minimum value in the first row.
- Subtract this minimum from all elements in the first row.
- Repeat this process for the second, third, and fourth rows.
- Record the resulting values in a new matrix.
Example
For Row A with entries [10, 12, 19, 11], the minimum is 10. Subtracting 10 from each element yields [0, 2, 9, 1].
Common misconceptions
- Subtracting the global minimum of the entire matrix from all elements.
- Forgetting to subtract the minimum from the minimum element itself, which should result in zero.
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To select independent zeros, scan the reduced matrix for rows or columns with only one available zero. Circle that zero and strike out other zeros in the same row and column.
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The presenter avoids selecting a zero when a row has two zeros because the goal is to find a set of independent zeros where each row and column has exactly one selected zero. Choosing arbitrarily from a row with multiple zeros might lead to conflicts later, so the process is postponed until the zero pattern is more constrained.
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Conditions: The cost matrix is square (same number of rows and columns).; Each job is to be assigned to exactly one person.
Column reduction is the second step of the Hungarian method. It involves taking the matrix obtained after row reduction, finding the minimum element in each column, and subtracting this minimum value from all elements in that same column.
Conditions: The matrix must have already undergone row reduction.; The problem is a balanced assignment problem.
To calculate the total cost, you must look at the original cost matrix, not the reduced matrix. Identify the positions of the circled zeros in the reduced matrix, find the corresponding entries in the original matrix, and sum those values.
Conditions: The optimal assignment has been determined from the reduced matrix.; The original cost matrix is available.
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