Largest and smallest area triangles on imprecise points

Assume we are given a set of parallel line segments in the plane, and wish to place point on each segment such that resulting maximizes or minimizes area largest smallest triangle set. We analyze complexity four computational problems, show three them admit polynomial-time algorithms, while fourth is NP-hard. Specifically, maximizing can be done $O(n^2)$ time (or $O(n \log n)$ for unit segments); minimizing $O(n^2 time; NP-hard; but time. also discuss what extent our results generalized polygons with $k>3$ sides.