A fractional LP optimum is tempting to round, but rounding is not a proof. It can leave the feasible region or land on a weaker integer point. This lesson shows both failures on the same asymmetric IP.
highlighted = computed this step
Rounded point
Rounding the LP point gives (1, 2), but x plus three y equals 7 and the limit is 6. Why: rounding can cross a constraint boundary.
x+3y=7
Nearby feasible point
The feasible point (1, 1) has z=3. Why: a nearby feasible integer point can still miss the best integer point.
z=3
True integer optimum
The true integer optimum is (0, 2) with z=4. Why: branch-and-bound proves the best point instead of guessing by rounding.
zIP=4
Diagram note
The rounded marker is intentionally outside the feasible region; the green marker is the recomputed IP optimum. Pixel positions are rounded for layout; every number shown is exact.