Step 1: Group the mechanisms by category first:
Instead of going process by process, it helps to first sort the four removal mechanisms into two families: thermal (melting/vaporization) and mechanical (impact, sputtering, shearing).
This narrows down the matching before we even look at the specific process names.
Step 2: Identify the one purely thermal process:
Among P, Q, R, S only Laser beam machining (P) works by concentrated heat energy, so it must pair with the thermal mechanism Melting and vaporization (3).
That immediately fixes P-3 and rules out P being paired with 1, 2, or 4.
Step 3: Separate the two mechanical-erosive processes:
Abrasive waterjet machining (Q) and Ion beam machining (R) both remove material mechanically but at very different scales.
AWJM works at the macro/particle scale, so its abrasive grains chip material away by direct Impact based erosion (1): Q-1.
IBM works at the atomic scale in vacuum, knocking off individual atoms through Sputtering (2): R-2.
Step 4: Assign the leftover pair by elimination:
Mechanical milling (S) is the only conventional process in the list, and its cutter removes material as continuous chips through plastic deformation, which is Shearing (4).
With 3, 1 and 2 already used, 4 is the only mechanism left for S, confirming S-4 both by reasoning and by elimination.
Final Answer:
Grouping by thermal versus mechanical mechanism confirms the same pairing found directly.
\[ \boxed{P\text{-}3;\ Q\text{-}1;\ R\text{-}2;\ S\text{-}4} \]