Question

I’m lost here. Here’s the problem and I think it’s NP-hard. A center is staffed with a finite number of workers with the following conditions:

1. There are 3 shifts per day with 2 people in each shift
2. Each employee works for 5 days straight and then 2 days off with only one shift per day

So the problem is: how many workers do we need if the center remains active every day and a feasible schedule?

Update:

Thanks for all the great answers. The closest I’ve come to (with a randomized brute-force algorithm) is the following:

X       3       0
1       0       3
2       3       1
2       1       3
0       1       2
0       2       1
3       0       2

I’ve simplified the problem into batches of 2 people (0-3 represent 4 batches) in the hopes of getting a feasible solution. X refers to a shift which has not been assigned (which was not the initial goal but it looks like there may not be an alternative).

Answer 1

The constraints cannot be respected exactly as expressed in the question.
That’s because the numbers don’t add up (or rather “divide up”).

Consequently, the problem should be reworded to require

• exactly 3 shifts per day
• exactly 2 workers per shift
• workers work a maximum of 5 consecutive days
• workers rest a minimum of 2 consecutive days

With the introduction of the minimum and maximum qualifiers, the minimum number of workers required is 9 (again assuming no part-time worker).
Note that although 9 appears to be a absolute minimum, given the need to cover 42 shifts per week (3 * 2 * 7) with workers who can cover a maximum of 5 shifts per week (5 work days 2 rest days = a week), there is no assurance that 9 would be sufficient given the consecutive work and/or rest day requirements.

This is how I figure…
8 workers isn’t enough, and the following 9 workers line-up, is an example of such a schedule.
To make things easy, I assigned all workers except for worker #1 and #9, to an optimal schedule of exactly a 5 days-on and 2 days-off schedule; #1 and #9 work less. Of course many other arrangements would work (maybe this is what the OP sensed when he hinted at an NP-complete problem). Also, the schedule is such that each week’s schedule is exactly the same for everyone, but that could also be changed (maybe introducing some fairness, by having all workers have a lighter week every once in a while, but this BTW can lead to some difficulties of respecting the requirement of 5 maximum work days).
The sample schedule shows two consecutive weeks to help see the consecutive work or rest days, but as said, all weeks are the same for every one.

                              Max Conseq Ws   Min Conseq Rs
Worker #1   RRWWWRW RRWWWRW         3              2
Worker #2   WWWWWRR WWWWWRR         5              2
Worker #3   WWWRRWW WWWRRWW         5              2
Worker #4   WWWRRWW WWWRRWW         5              2
Worker #5   WRRWWWW WRRWWWW         5              2
Worker #6   WRRWWWW WRRWWWW         5              2
Worker #7   RWWWWWR RWWWWWR         5              2
Worker #8   RWWWWWR RWWWWWR         5              2
Worker #9   WWRRRRW WWRRRRW         3              3

Nb of Ws    6666666 6666666

The tally at the bottom shows exactly 6 workers per day (respecting the need to cover 3 shifts with 2 workers each), the max and min columns on the right show that the maximum consecutive work and minimum consecutive rest requirements are respected.