I am trying to derive an upper bound on the sum of the size of the common prefixes of an (arbitrary) element with all incomparable elements with lower rank from an (arbitrary) antichain. This means that for choosing the greatest element and the maximal antichain this sum is actually 0, as the greatest element is comparable with all elements from the poset. I have updated the problem definition accordingly and I would be very grateful for any further feedback.

Thank you very much for your comments @GerhardPaseman and @HughThomas! Of course, your are both correct and choosing the greatest element and the largest antichain will not only suggest and upper bound, but be the exact solution to the problem as I originally stated it. In fact, that is exactly what I did to derive my current upper bound. Your comments made me realize that I forgot to add an important additional restriction on the elements from the antichain that I need to consider: