Step 1: Determine Bond Orders
Using Molecular Orbital Theory, the bond orders are: $O_{2}^{2-} = 1$, $O_{2}^{-} = 1.5$, $O_{2} = 2$, and $O_{2}^{+} = 2.5$.
Step 2: Comparison
The values show a clear increasing sequence from 1 to 2.5.
Step 3: Arrangement
The correct increasing order is $O_{2}^{2-}<O_{2}^{-}<O_{2}<O_{2}^{+}$.
Final Answer: (a)