Unrelated Maschines

Proof

model as Minimum-cost bipartite matching

1. Add dummy jobs with cost 0 to have $n\cdot m$ jobs. ($n\cdot m-n$ jobs)

2. Every machine has $n$ jobs.

3. Connect every Job $j$ to slot $k$ of machine $i$ with cost $k\cdot p_{ij}$

4. Run the jobs in the slots per machine in reveres order.