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Simple Method of Generating Multiple Rotating Schedule Templates

Posted on 3/1/1997 by ByteBloc in category: scheduling software articles

(Reprinted with permission of the author and Emergency Medicine News, from an article in the January 1997 issue of Emergency Medicine News)

by Robert Porter, MD

Of the many difficulties facing physicians involved in shift work, one of the least written about and the most discussed is scheduling. As we all know, the greater part of this discussion revolves around whether there is an "equitable" distribution of shifts.

Some groups approach the schedule on an ad hoc basis, with shifts arbitrarily assigned one or two months in advance. This approach has the advantage of allowing staff to request specific days on or off to accommodate known meetings, CME, family events, etc. The disadvantages are that staff can never predict very far in advance when they will be working and that much time and effort are required to ensure an overall even distribution of shifts, as even subtle differences are often a source of conflict within a group.

Because of this, many groups prefer to have a "rotating" schedule that has built-in equity and that repeats itself over a period of time allowing projection of future shifts. There are an infinite number of repeating patterns possible for a given number of staff. Many of these, however, do not develop complete equity for a significant period of time. Below is described a simple method for generating multiple rotating schedule templates all of which will achieve complete equity in the theoretically shortest period of time. This method is applicable to any number of physicians working single, double or multiple-coverage shifts and can also be adapted for situations where some physicians work a different number of shifts than others.

To create a rotating template, begin with a grid of n weeks, where n is the number of physicians in the group. Within this grid assign one person one and only one of each possible shift (see figure 1), counting multiple coverage, if used, as separate shift(s). Shifts can be assigned in any pattern whatsoever as long as there is one and only one of each. It is at this point that one incorporates group preferences regarding whether there are many consecutive shifts, isolated shifts, blocks of nights, etc. It does not matter what distribution is chosen as long as one and only one of each shift is assigned. That is the key to this system.

When the pattern has been determined for one physician, it is duplicated for the next but shifted forward one week (figure 2), returning to the top of the grid when the bottom is reached. This is repeated, always shifting forwards one week, until the last physician has been assigned. This will complete filling of all available shifts (figure 3) and will always do so regardless of the initial pattern chosen. Since each physician has one and only one of each shift in the grid, it is equal, and since the grid may be repeated multiple times, it is rotating and hence predictable.

Why is this "n" week template the shortest possible one? Remember, an "equal" distribution means everyone has the same number of each of the possible day/shift combinations (counting double-coverage as a separate shift). What is the shortest time that a physician can work one of each shift? For single coverage (total of 168 hrs/week) it is simple, one physician working continually will complete one of each possible shift in 1 week. It follows that 2 physicians will require a minimum of 2 weeks and n physicians n weeks for each to complete one of each possible shift. The shortest possible rotating cycle is therefore n weeks, where n is the # of physicians.

Surprisingly, this does not change when multiple coverage is instituted. For x hours coverage/week, a single physician could not theoretically work this in less than x hours. It would take 2 physicians 2x and n physicians nx hours for each to have worked one of each shift. Dividing nx total hours by x hrs/wk still results in n weeks as the minimum time.

Of course, many groups do not utilize an exactly even distribution of clinical hours. Typically the chairman and other physicians with extensive non-clinical responsibilities will work fewer shift hours than other staff.

This can be accommodated in several ways. The simplest is when there are two or more staff needing decreased hours. In this case, those people can split a template position. For example a chairman, residency director and 8 faculty (10 people) would generate a 9 member template as above, with the chairman and residency director splitting one of the 9 slots between them. Similarly, three people could split 2 template positions.

When only one physician needs decreased hours, or several need a # of hours that cannot be achieved by an even split of one or more slots, a rotating template can still be accomplished. This is done by assigning the physicians needing decreased hours to specific shifts each week. Since for the most part, weekday shifts can be considered essentially equivalent (a Tuesday day is usually considered similar to a Thursday day) this does not create any significant inequity. The regular staff physicians then generate a rotating template as above except that their template does not rotate through those particular shifts just allocated to the administrative physicians.

To illustrated this, consider the typical case of a group with a chairman and 5 staff physicians. They utilize two 12 hour shifts daily plus an 8 hour double coverage shift weekdays. The chairman requests to work 2/3 the number of clinical hours as the staff.

First, total the # of hours/week, which is 168 + 5x8 = 208. These hours are covered by 5 2/3 staff for an average of 36.7 hrs/wk/staff. The chairman's 2/3 of this equal 24.5 hrs/wk. Next, develop a combination of shifts in one week that will give the chairman about 24 hours. Obviously two 12 hr shifts will give 24, so the chairman could be assigned to a Tues Day and Wed Night shift, which he would cover every week.

Then, start a 5 week template for the 5 remaining staff. As before, one staff is assigned to one of each of the various shifts in the 5 week block, except for Tues day and Wed N (Figure 4), and the template rolls on from there (Figure 5). No further calculations need be done, the staff will all have the proper number of hours in a completely even distribution.

If the chairman prefers to share in covering weekends, they can be added in.. In this case use a 6 week template, with the chairman assigned to two weekend days and two weekend nights in that block, just like the rest of the staff. This gives 48 hrs in 6 weeks, or 8 hrs/week. By adding two 8 hr overlap shifts/wk, say Tue and Wed, he would then achieve the desired total 24 hrs/week. The staff template would then be generated in the same manner as before.

This technique can even be used when different #s of people require different distributions although the result may then become quite complex. At our institution, we have used this method to generate a rotating schedule with 2 people doing 2/3 the weekday hrs and 1/2 the weekend hrs, one person 2/3 weekday and full weekend, one person half time but full weekends and 9 staff everything else. This template can be available on request.

Figure 1. Schedule for physician "1"
Week 1   D D       D
Week 2 D     N N N  
Week 3       D     N
Week 4 N       D D  
Week 5   N N        
days=7 1 1 1 1 1 1 1
nights=7 1 1 1 1 1 1 1

Figure 2. Schedule shifts down 1 week for physician "2"
Week 1   1-D/2-N 1-D/2-N       1-D
Week 2 1-D 2-D 2-D 1-N 1-N 1-N 2-D
Week 3 2-D     1-D/2-N 2-N 2-N 1-N
Week 4 1-N     2-D 1-D 1-D 2-N
Week 5 2-N 1-N 1-N   2-D 2-D  

Figure 3. All physicians filled in
Week 1 5-D/3-N 1-D/2-N 1-D/2-N 4-D/5-N 3-D/5-N 3-D/5-N 1-D/4-N
Week 2 1-D/4-N 2-D/3-N 2-D/3-N 5-D/1-N 4-D/1-N 4-D/1-N 2-D/5-N
Week 3 2-D/5-N 3-D/4-N 3-D/4-N 1-D/2-N 5-D/2-N 5-D/2-N 3-D/1-N
Week 4 3-D/1-N 4-D/5-N 4-D/5-N 2-D/3-N 1-D/3-N 1-D/3-N 4-D/2-N
Week 5 4-D/2-N 5-D/1-N 5-D/1-N 3-D/4-N 2-D/4-N 2-D/4-N 5-D/3-N

Figure 4. Administrative physician assigned Tue D and Wed N
Week 1     D N      
Week 2     D N      
Week 3     D N      
Week 4     D N      
Week 5     D N      
admin D=5 0 0 5 0 0 0 0
admin N=5 0 0 0 5 0 0 0

Figure 5. Physician "1" scheduled around administrative physician
Week 1   1-D A-D A-N     1-D
Week 2 1-D   A-D A-N 1-N 1-N  
Week 3     A-D 1-D A-N     1-N
Week 4 1-N   A-D A-N 1-D 1-D  
Week 5   1-N A-D 1-N A-N      
staff D=6 1 1 0 1 1 1 1
staff N=6 1 1 1 0 1 1 1

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