**"I was also hoping to not rely on "helper" formulas"**
An advantage of the array formulas is that I could use a defined name to refer to each formula and it would then only be evaluated when the result is needed for a further formula. I didn't do this because my part solution is probably sufficiently obscure without hiding the calculation!

All the matrices do is define a set of equations that state that the overall manning level (known) is the sum of the individual crew levels. I solve the equations (using matrix inversion) to determine the crew levels. The mathematics is more advanced but it allows the solution to be expressed more simply - which doesn't help a bit if you are not familiar with the mathematics

**"I … note that it has negative values (S24, T24 etc) which is not correct"**
The calculation does not take into consideration the possibility of demobilising part of roster crew from site early so the manning levels calculated do not provide a valid solution. I do not know whether further manual input is needed to set the level and timing of such demobilisation or whether some rules could be introduced (I had the former in mind).

**Note**
Although I chose a top-down approach in which the equations determining the required manning levels per roster crew were formulated and solved, this problem, by its nature can be solved by accumulating the levels week by week,

- A newly rostered crew will be manned at a level to meet the overall needs.
- The second or third week will generally be the same as the previous week.
- The fourth week will be zero, with the newly-rostered team picking up the slack.

This would require some arithmetic modulo 4 to control the increment to the next week's figures.

Life 'down amongst the weeds' can also get somewhat hairy. I don't know which approach will turn out to be the 'simpler' overall.

*p.s. I have attached a file that allows the user to determine the level of demobilisation required for the crew in their final week.*