The assignment problem is a special case of transportation problem where the number of sources and destinations are equal. Here, jobs represent sources and machines represent destinations.

The mathematical form of assignment problem is minimize Z = \(\overset { n }{ \underset { i=1 }{ \Sigma } } \overset { n }{ \underset { j=1 }{ \Sigma } } { { C }_{ ij } }{ x }_{ ij }\) subject to the
Constraints \(\overset { n }{ \underset { i=1 }{ \Sigma } }{ x }_{ ij } = 1, j=1,2,.....n\)
\(\overset { n }{ \underset { j=1 }{ \Sigma } }{ x }_{ ij } = 1, i=1,2,.....n\) and x_ij = 0 or 1 for all i, j

To assign the different jobs to the different machines (one job per machine) to minimize the overall cost is known as assignment problem.

If the total supply = total demand, then the given problem is a balanced transportation problem.

A feasible solution to a transportation problem is a set of non negative values x_ij (i = 1, 2, m, j = 1, 2, ....... n) that satisfies the constraints.
If a basic feasible solution to a transportation problem contains exactly m + n - l allocations. in independent positions, it is called a non degenerate basic feasible solution. Here m is the number of rows and n is the number of columns in a transportation problem.

The objective function is minimize Z = \(\overset { m }{ \underset { i=1 }{ \Sigma } } \overset { n }{ \underset { j=1 }{ \Sigma } } { { C }_{ ij } }{ x }_{ ij }\) subject to the constraints
\(\overset { n }{ \underset { j=1 }{ \Sigma } }{ x }_{ ij } = a_i, i=1,2,.....m\) (Supply constraints)
\(\overset { m }{ \underset { i=1 }{ \Sigma } }{ x }_{ ij } = b_j, j=1,2,.....n\) (demand constraints)
x_ij ≥, 0 for all i, j (non-negative restrictions)

A transportation problem is to determine the amount to be transported from each origin to each destinations such that the total transportation cost is minimized.