Question

I’m looking at the standard definition of the assignment problem as defined here

My question is to do with the two constraints (latex notation follows):

\sum_{j=1}^n(x_{ij}) = 1 for all i = 1, ... , n
\sum_{i=1}^n(x_{ij}) = 1 for all j = 1, ... , n

Specifically, why the second constraint required? Doesn’t the first already cover all pairs of x_{ij}?

Answer 1

Consider the matrix x_ij with the i ranging over the rows, and j ranging over the columns.

The first equation says that for each i (that is, for each row!) the sum of values in that row equals 1.

The second equations says thta for each j (that is, for each column!) the sum of values in that column equals 1.