Step 1: Understand each advanced machining process listed:
We have four processes: Laser beam machining (P), Abrasive waterjet machining (Q), Ion beam machining (R), and Mechanical milling (S).
Each removes material by a fundamentally different physical mechanism, so the question is really about knowing the underlying physics of each process.
Step 2: Match Laser beam machining (P):
LBM focuses a high-intensity coherent light beam on a very small spot, raising the local temperature rapidly.
This causes the material to melt and then vaporize, so the predominant removal mechanism is Melting and vaporization, option 3. So P-3.
Step 3: Match Abrasive waterjet machining (Q):
AWJM fires a high-velocity jet of water carrying abrasive grit at the workpiece surface.
The abrasive particles strike and mechanically chip away material through repeated impact, which is Impact based erosion, option 1. So Q-1.
Step 4: Match Ion beam machining (R) and Mechanical milling (S):
IBM accelerates a stream of ions in vacuum onto the surface; individual atoms are knocked off one at a time in a process called sputtering (option 2), so R-2.
Mechanical milling uses a rotating multi-tooth cutter that physically shears layers of material off the workpiece as chips, which is Shearing, option 4, so S-4.
Final Answer:
Putting the four pairs together gives the complete match.
\[ \boxed{P\text{-}3;\ Q\text{-}1;\ R\text{-}2;\ S\text{-}4} \]