Recipe

Check whether one design dominates another

Use when: you have two feasible designs' objective values and need a yes/no verdict on whether one beats the other outright

Goal: a confirmed dominates/does-not-dominate answer, plus (checking every pair) the full Pareto set when the design set is small

Quick recipe

1
Check $f_i(x)\le f_i(y)$ for every objective $i$ .

Example

$X=(3,6)$ , $Y=(3,4)$ : $f_1(X)=3\le3=f_1(Y)$ ✓ and $f_2(X)=6>4=f_2(Y)$ ✗.
- any coordinate has $f_i(x)>f_i(y)$ — $x$ does not dominate $y$ ; stop here.
- every coordinate satisfies $f_i(x)\le f_i(y)$ — continue to the next step.
Why: domination requires $x$ to never be worse than $y$ on any single objective. One failure ends the check.
2
Among the coordinates that tied or passed, verify $f_i(x)<f_i(y)$ for at least one $i$ .
- at least one strict inequality holds — $x$ dominates $y$ .
- every coordinate ties exactly — $x$ does not dominate $y$ (they are simply incomparable, not equal winners).
Why: a tie on every objective means the two points are equally good, not that one has beaten the other.
3
A point is Pareto-optimal exactly when no other feasible point dominates it. Run the first two steps for every OTHER candidate against it, not just one.

Why: checking against a single rival only tells you about that one comparison. Pareto optimality is a claim about every other feasible point at once.

Common mistakes

Checking only one objective and concluding domination from it alone.
Treating a tie on every objective as domination, rather than as "incomparable."
Stopping after one candidate fails to dominate a point, and declaring it Pareto-optimal without checking the rest.