How does the Hungarian Method find an initial feasible solution?
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The Hungarian Method finds an initial feasible solution by first performing row reduction, where the smallest value in each row is subtracted from every element in that row, ensuring each row contains at least one zero. Next, it performs column reduction, where the smallest value in each column is subtracted from every element in that column, ensuring each column also has at least one zero. After this, the method involves covering all zeros in the matrix with the minimum number of horizontal and vertical lines. If the number of lines equals the number of rows or columns, an optimal solution is possible. If not, the method adjusts the matrix by subtracting the smallest uncovered element from all uncovered elements and adding it to the elements at the intersections of the lines. This process continues until the number of lines equals the number of rows or columns, at which point an optimal assignment can be derived from the zeros in the matrix.
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The Hungarian Method finds an initial feasible solution by reducing the cost matrix. It first subtracts the smallest value in each row from all the elements in that row, followed by subtracting the smallest value in each column from all the elements in that column. This process creates zeros in the matrix, setting up the conditions to identify optimal assignments.