Men over-estimate their value. How many male sci-fi writers imagine a post-apocalypse scenario where intelligent men and “young breeding females” are locked in an ark for generations, destined to revive the species…? Hold up there, Son. Your penis will not save the human race. You cannot fuck your way to a better tomorrow.
Only select women… and sperm.
There will be no men on the Ark. Women work co-operatively and are far less violent. Sperm can be frozen and transported easily. Men would consume resources without providing a useful womb (assuming there are no artificial wombs). The women can easily be trained to fill all science and technical positions. You can’t afford to have a man-rage incident that would jeopardize lives or the longevity of the mission.
With frozen sperm you can also bypass all inbreeding issues for generations to come, maybe indefinitely. They can easily select for sex while they screen for genetic problems. All pregnancies will be artificially inseminated anyway. The oldest verified mother was nearly 67, so there is no reason to worry about age. Exercise, science, and hormone treatments may be able to extend that age even more.
You certainly wouldn’t allow a couple to have TWO children with the same zygote donors, and you wouldn’t leave such decisions up to random Friday night dates and mood lighting. If your entire population is women, they could pair off into couples (or triples) to co-parent. Some women might actually enjoy pregnancy while others would delay or abstain, but with an all female crew you’ve doubled your capability for pregnancies.
If you also bring frozen eggs, the scientists and engineers are not necessarily reusing their own DNA. The 1st generation could have maximum diversity with every individual being no genetic relation to any other – although you would still screen for genetic disorders so I’m not sure how useful this would be (maybe very useful if you suspect future generations will not have the same invitro or screening technology).
They should not create any male births until the crisis has safely passed (when they reach a new planet, or however your ark works). Men carry less genetic information – most people are surprised by how much less – and the whole point of a breeding program is to be able to safely select for recessive genes. You need two complete pairs. Y chromosomes would compromise that.
People actually might not feel as bad about being left behind if they believed they would live on in some way, by contributing to a future human colony. I would hard sell the sperm/egg bank idea upfront to the general population, and tell everyone they are accepted. It would be up to future generations to decide if and when to use the gene bank anyway (and they might actually use it as spare parts for gene-splicing).