A practical problem for Crew rotation set-up

Question

I work managing crew rotation for ships and we were given a project to start a better crew rotation for "core crew" (people following a fixed rotation) and "rotating personnel" (people without a steady rotation). I understand some explanation will be necessary before exposing the problem I am facing, here they are:

1. Crew rotation follows 14 days of work on a ship and 14 days of resting (days off); A person will always follow the same rotation throughout the years if he remains on the same ship;
2. They are entitled to take 30 days of vacation when they complete 12 months of work, the recess can be claimed on the 13th month and no longer than the 23rd month. The hiring date is the main driver for counting the acquiring dates of vacation;
3. If the crew is not put on programmed vacation until the 23rd day, then the company will have to make a double payment to the employee plus a hefty fine to the government and the employee must be forced to vacation immediately;
4. The 30 days of vacation will consume 2 days of rest days, so we pay that to the employee so he can remain on a 14x14 rotation without offsetting their rotation as per explained in rule 1;
5. The regular salary they receive is for 14 days onboard plus the 14 days of resting, anybody can stay no more than 7 days on board, in this case, any day counting from the 15th day through 21st will be paid in double. The reason is that we will be paying the 7 days of extra work and will be paying 7 days of the days off the person won´t use, which means a person working 21 days will only rest 7 days before returning to his regular rotation;
6. The authority overseeing the operation can issue penalties and fines if we allow the crew to stay more than 7 in a constant situation.

Given these rules, we need to have extra personnel to cover vacation and sick personnel or any absence. We are studying scenarios to have 2 types of rotators: Rotator Onshore to work exclusively on the offices in special projects and Rotator Offshore to cover the Core Crew personnel whenever necessary. The original idea we have is to have all personnel rotating between these 3 teams like this: 1 year as Core Crew, 6 months as Rotator Onshore, and 6 months as Offshore. Since they need to take a vacation between 13 and 23 months, we believe we can send them on vacation when they change from Core Crew to Rotator Offshore or between Rotators Onshore to Rotator offshore. We also received an additional challenge to allow the core crew to take vacations too, if possible.

As for the number of personnel for each position, we have teams with 2, 4, 8, and 12 crewmen. The number of extra personnel working as rotators will vary based on the quantity in each of these teams and the main driver will be the coverage of vacation, e.g. on a crew with 12 people, we will have 1 person taking a vacation every 30 days to make sure all take a vacation before the 24 months rule, in the other hand, a crew with 4 people, the vacation is more relaxed.

Here are my conundrums, how can I use mathematics to prove or test:

a. Which scenario of rotation will be ideal considering all these constraints and corporate rules to implement Core Crew > Rotation Offshore > Rotation Onshore > Core Crew?

b. How to calculate the ideal number of extra personnel to be hired?

If you need any additional information, please let me know.

I appreciate any and every help you can get me as I am clueless on how to use Math (Arithmetic, Algebra, Calculus) here. But I am positive it is possible, I just don't have the brains for it...

EDIT: This is a spreadsheet with a real situation:

Answer 1

Would organizing it in a spreadsheet that counts the total staff in each of the categories (CC, OffShore, OnShore, Rest, Vacation) during each time period (days 1-14 of the month, days 15-28 of the month, the remaining 0-3 days of the month) be a helpful first step? Maybe a good layout will enable you to see the whole picture and arrange staff without going into complicated discrete optimization? The PNG attached below shows a table violating one of the company policies (granting a vacation within the 1st year of work; i.e., before month 13), but this is to show how easily such violations can be spotted and corrected quickly by hand.