Q.Explain why an assignment problem is regarded as a special case of the transportation problem. Using the cost matrix of Worked Example 4 (three workers, three jobs), state the implied supply at each worker-row and the implied demand at each job-column.

An assignment problem is a transportation problem whose supplies and demands are all forced to equal exactly 1, on a square matrix, rather than being general positive quantities.

A general transportation problem allows any positive supply $a_i$ at each of $m$ sources and any positive demand $b_j$ at each of $n$ destinations, with $m$ and $n$ not necessarily equal. An assignment problem is the extreme special case of this structure where:

• the number of sources equals the number of destinations, $m=n$ (a square cost matrix), and
• every supply and every demand equals exactly 1, since each worker can do only one job and each job needs only one worker.

Applied to Worked Example 4's cost matrix, each worker row ($W_1,W_2,W_3$) is treated as though it has a supply of exactly 1 unit of 'capacity', and each job column ($J_1,J_2,J_3$) is treated as though it has a demand of exactly 1 unit. This means every $x_{ij}$ can only be 0 (not assigned) or 1 (assigned) — never a fractional or larger quantity, unlike a general transportation allocation. Because every supply and demand is the same trivial value (1), a specialised algorithm (the Hungarian Method) that works directly with 0/1 assignments is far more efficient than running a general transportation method (NWC/LCM/VAM followed by MODI) on a table where every row and column total is 1.

Thinking an assignment problem needs entirely separate logic unrelated to the transportation problem — in fact it is the extreme case where every supply and demand equals exactly 1, which is precisely why a specialised one-to-one algorithm is both possible and faster than a general transportation method here.