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Generating Different Shift Patterns
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March 2011
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Articles on Creating Shift Patterns
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Introduction
All of us would like to think that there is a repeatable pattern of shifts that would only have to be done once. Yet that pattern seems to elude everyone. We never seem to be able to take the same schedule, say from this time last year and repeat it again. Later, when you see how many ways you could schedule staff, then it isn't such a surprise that the schedule is never the same, ever.
Seven day working. If a department is operating every day, a very common requirement is to have your staff work 5 days per week and have 2 days off per week. Sometimes the rule in many companies is to have staff work 10 shifts in 2 weeks, and others use an average such as 'an average of 5 shifts/week' or 'an average of 40 hours/week. The maths of 5 shifts/week is fairly straight forward. There are 21 different ways to allocate 5 shifts/week. The table below shows each of the possible ways to do it.

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Mathematics of Shift Patterns
If you wanted 15 staff at work each day, you could use a table such as the one above to schedule 21 people to work each week and have an end result of 15 staff each day. The staff would usually rotate the shifts to equalise days off, and hence you have a rolling shift roster repeating every 21 weeks.

You can select various combinations to give differing numbers of staff on duty, but there are only 21 options to choose from. If you want a varying number of staff through the week, you can select those options that give you the correct result.

If you want 5 staff on duty, then below is an example using 7 staff and selecting 7 weeks from the above 21 weeks of shifts.
This is one possible selection, however, the total number of possible selections is literally without limit. There will be Millions of permutations and there are numerous ways of combining the weeks of shift patterns. Below is an example of using week 13 twice to give 5 staff on duty.
With this plethora of possible shift patterns that match a requirement of staffing in multiples of 5, i.e. 5, 10, 15, 20, etc. the main difficulty is deciding which selection to use. A second problem concerns intermediate staffing levels, i.e. staffing to have 3 on duty or 7 on duty. This requires a separate method of generating shift patterns. Another problem area concerns holidays and how they are accommodated in a shift pattern.
There are many more problem areas to be covered in generating the most appropriate shift pattern, problems such as; absence, training, back up, variable staffing levels, skills mix, etc. These all require special techniques and usually some training in these techniques. Once you know the mathematics behind Staff Rostering and how to use these techniques, then, generating Shift Patterns becomes extremely quick and relatively easy.
Shift Pattern where staff work a two shift system
A similar table to the first one can be made for staff working a 2 shift system of Earlies and Lates, which is normal in many industries. Now, instead of the pattern giving 21 different options/weeks, there are 672 different options/weeks of shifts. You would not need to repeat any of the different weeks for 13 years. No wonder it's difficult to simply use last years schedule.
The first 32 weeks are shown below, this is the number of different shift patterns based on the first line of the above diagram. Each line from the above diagram can be similarly represented..
For those of you that regularly use a shift pattern that goes something like Early-Late-Early-Late, you will be pleased to see so many different ways to do it. There are millions of ways to combine shifts, the difficult part is not finding an appropriate shift pattern, but deciding which is the best one for you, and this requires a totally different way of generating shift patterns.

Shift Patterns for longer periods
Often staff rosters can be created for longer periods such as 2 weeks (10 shifts) or 4 weeks (20 shifts). Whenever I create a new staff roster it is for a year at a time and this requires a totally different mathematical approach to solve this problem. If you did a similar table for staff working 20 shifts in 28 days, the table would be 3,000,000,000,000 lines long, which represents a period of time, 10 times longer than the Universe has been in existence. The paper used for the table would circle the Earth a thousand times. You would only have to get to line 650 before your member of staff retires after 50 years continuous service.

One more mathematical fact. If you tried to schedule 2 staff for one week, then the total number of ways you could schedule 2 staff to work 5 days per week, 2 days off, all earlies, isn't doubled, it's squared. There are 441 different ways to do it, and it would be 9 years before you had to repeat it.
Computer Models & Shift Patterns
When operating 365 days/ year our computer models generate shift patterns with incredible speed to match any staffing requirement. Usually several shift patterns are generated for different times of the year. For instance there is usually a Summer roster and a Xmas-New Year roster. Once a 'fair' copy is available, where everyone is working 5 shifts a week, then it is time to let the staff have a look at it. What the staff always do is to re-arrange the shifts between themselves, so the roster becomes almost self-rostering, but there are rules to obey. Because the roster is prepared for a year at a time, then the staff have all their shifts allocated to them and they can't just take a shift or day off. If they did that, they would be working one shift too few over the year. Instead they have to follow 2 basic rules when swapping shifts or taking a day off.

Rule 1. Swapping Shifts. They can swap shifts on any day, with someone else, as long as that person is working. Then they are still working the same number of shifts per year.

Rule 2. Swapping a shift for a day off. They can swap a shift for a day off with someone, as long as they swap another day off elsewhere for a shift with the same person.

Hence, if a person is working a shift on a Monday and they want a day off. They need to find someone with a day off on Monday, who will swap for it for a shift. Then they can swap the shift for the day off. However, this means that the first person is working one shift less and the second person is working one shift more. So, they need to find another day, where the second person has a shift and the first person has a day off, and swap. Then the system is balanced and everyone is working the right number of shifts. With the computer models covering 365 days, this swap can be at any time, it doesn't have to be in the same week or even month, it can be in 3 months time. Of course, sometimes staff will request their day off as part on their Annual Holiday Entitlement, but swapping shifts gives them far more flexibility without comproming the productivity of the company. Also, being able to swap shifts frequently and with minimal effort often reduces absence on those days when the staff cannot come to work for whatever reason.
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