Scheduling Emergency Room Physicians

Health Care Management Science 4, 347-360, 2001
© 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.
Scheduling Emergency Room Physicians
MICHAEL W. CARTER
Mechanical and Industrial Engineering, University of Toronto, 5 King's Coltege Road, Toronto, ON, Canada M5S 3G8
E-mail: [email protected]
SOPHIE D.LAPIERRE*
CRT and Department of Math, and Industrial Engineering, Ecole Polytechnique, CP 6079 Succ. Centre-ville, Montreal, QC, Canada H3C 3A7
E-mail: [email protected]
Received 27 July 1999; Revised 18 April 2001
Abstract. This paper introduces the problem of scheduling emergency room physicians. We interviewed physicians from six hospitals
in the greater Montreal, Canada area, in order to understand the emergency room scheduling problem. Extracting the real scheduling
problem is difficult because physician working conditions are based on informal mutual cooperation which is usually not documented.
We present the characteristics of the scheduling problem and the scheduling techniques currently used in the six emergency rooms
we analyzed. Using the scheduling problems of Charles-Lemoyne Hospital and the Jewish General Hospital, we show how to modify a hospital's existing scheduling rules to develop techniques which produce better schedules and reduce the time needed to build
them.
Keywords: health care, staff scheduling, cyclic scheduling, scheduling physicians, emergency rooms, tabu search
1. Introduction
Emergency rooms (ERs) are exciting and challenging places
for physicians to work. However, they are also stressful
workplaces. A recent survey of Canadian ER physicians
indicated that 24.5% are not satisfied with their jobs [26].
ER work is inherently stressful; but, in addition, shift work
is well known to be more demanding than regular daytime
work [12]. Recent studies have shown that the scheduling
of shifts has a "significant impact on shift workers physiologically, psychologically and socially" [18,19]. Consequently, scheduling is very important for ER physicians and
good schedules can both reduce physician attrition rates and
improve recruiting.
Among the various groups of physicians, scheduling ER
physicians is one of the most challenging scheduling problems. Physicians usually work full-time in the ER, whereas
most other specialties require physicians to work in a hospital only a few times per month and spend the rest of their
time taking care of their own patients. The ER is open 24 h
per day, 7 days a week, which requires physicians to work
different shifts and to work as much on nights and on weekends as during the day. Many other specialties also require
night and weekend shift availability, but physicians are often
simply "on call" and may only spend a small portion of this
time at the hospital.
In this paper, we attempt to extract the characteristics of
a generic ER physician scheduling problem from six hospitals located in greater Montreal, Canada. Although the situations encountered in these hospitals might not be exactly
* Corresponding author.
the same in other Canadian provinces and countries, the diversity of ER practices found in this study makes it a good
start for providing a more general understanding of the ER
scheduling problem. The purpose of this paper is to investigate some of the issues involved in modeling the ER scheduling problem, and some of the trade-offs that we discovered.
We also discuss some automated solution techniques (based
on a method described in another paper) to illustrate some of
the practical issues.
In section 2, we provide a literature review of physician and other staff scheduling problems. In section 3, we
present the principal schedule requirements and constraints
we encountered from our discussions with several physicians working at the ERs we visited. In section 4, we present
the methods used by the physicians to establish schedules
and compare them with other methods found in the research
literature. In section 5, we analyze schedules prepared for
Charles-Lemoyne Hospital and the Jewish General Hospital, and show how to develop techniques which produce
better schedules and reduce the time needed to build them.
We will see that a large number of scheduling rules do not
necessarily result in a high satisfaction rate among the physicians. A wise selection of scheduling rules and an appropriate scheduling technique that adequately reflects the physicians' organizational culture will facilitate physician recruitment and lead to high physician retention rate. Physicians
should be aware that automating the staff scheduling process
is non-trivial. Firsdy, there are very few commercially available packages, and secondly, every hospital has very different rules.
W.M. CARTER, S.D. LAPIERRE
348
2. Literature review
Little work has been done on the problem of scheduling
physicians in a hospital. Previous research on scheduling
problems in health care has focused instead on nursing staff
[30]. The physician scheduling problem is different from
nurse scheduling in several ways. Nurses usually work under a union agreement or written rules; while physicians typically have no formal scheduling rules. For example, when a
full-time nurse takes a vacation, the hospital must either find
a replacement (from part-time, casual, overtime or agency
staff) or work short staffed. In the ER, when physicians take
a vacation, there may be part-time people who can pick up
extra shifts. Often, however, the doctors themselves trade
off extra shifts with their colleagues. When one is away, the
other full-timers pick up the slack. The ER cannot normally
run short staffed. Therefore, nurse scheduling is often performed in two steps. First, a schedule is generated to satisfy
the union agreement for the regular staff, minimizing staff
shortages and surplus. Then, shortages are filled using float
nurses, agencies, and overtime. The physician schedule is
a one step process, where physician demand must be met
minimizing violations to the scheduling rules.
In the recent years, a few commercial physician scheduling software packages have been developed and successfully used by physicians. "Tangier Emergency Physician
Scheduling Software," by Peake Software Laboratories, and
"Epsked," by ByteBloc Medical Software, are the only two
packages capable of generating full schedules for emergency
physicians [1]. "Docs for Windows," by Acme Express [33],
and "Physician Scheduler 4.0," by Sana-Med, are also commercial packages for scheduling physicians and are used
by emergency physicians. More than a thousand copies of
these products have been sold to emergency departments.
But, judging from their published features and from software demo versions, there is room for improvement. Unfortunately, the proprietary nature of these packages prevents
the research community from benefiting from their experience.
Much work has been done on staff scheduling since the
first paper on the topic was written by Dantzig [9]. The dominant technique for generating acyclic health care schedules
is mathematical programming. Early work on nurse scheduling using mathematical programming was led by Miller et
al. [28] and Warner [39]. Since there are often conflicting
scheduling objectives, Arthur and Ravindran [2] proposed a
multiple objective model which they solved using a heuristic. Berrada et al. [4] improved the heuristic to solve the
nurse scheduling problem. Finally, Jaumard et al. [16,17]
presented exact methods to solve the multi-objective nurse
scheduling problem. These multi-objective models can be
modified to solve the ER scheduling problem. Beaulieu et
al. [3] showed that the model and algorithm of Berrada et
al. [4] works very well on an ER physician scheduling problem if constraints on shift rotations are not included in the
model. However, more research needs to be done to improve
the quality of scheduling with these methods when shift rotations are complex.
The first work on cyclic scheduling in health care was
performed by Maier-Rothe and Wolfe [27]. Early work was
developed around the single shift schedule with the principal objective being to minimize the number of employees
(see, for example. Bums and Carter [7]). The first papers
on multiple shift scheduling only accounted for a limited
number of rules (see, for example, Morris and Showalter
[29]) and such basic cyclic schedules can be solved easily
without a computer [22]. But Siferd and Benton [34] note
that new social concems call for other scheduling strategies and techniques. Shiftwork can also have a negative
impact on our sleep pattems (circadian rhythms) and there
has been considerable research on designing "ergonomic
rules" to minimize the adverse effects. Ergonomic issues
are discussed in more detail in section 3.1. When these rules
are taken into account, constructing a cyclic schedule becomes a more difficult task [25]. Hung [15], Schwarzenau
et al. [32], Lapierre et al. [21] and Lau [23,24] have developed algorithms for generating cyclic schedules which meet
ergonomic criteria, but more research is needed in that direction.
Smith [35], and Smith et al. [36] suggest interactive software to help nurses construct their own schedules, enabling
the scheduler to analyze tradeoffs in scheduling conflicting
requests. Taylor and Huxley [37] and Ecker et al. [10] also
suggest this approach in other contexts. However, some suggest a different approach to improve the quality of schedules: the "self-scheduling" approach [14,40], which refers
to letting the workers make their own schedule instead of the
manager. The basic assumption is that centralized scheduling, whether computer-assisted or not, does not work. If the
scheduling task is the responsibility of everyone instead of
one person, then the schedule will be of better quality. For
more information on physician staffing decisions at an ER,
see Blake et al. [5] and Vassilacopoulos [38].
3. Problem description
In this section, we first describe the scheduling problems we
encountered in six hospitals located in the greater Montreal
area, Canada. We then present a generic scheduling model
that we constructed based on these instances. We discuss
some methodologies used to solve these scheduling problems in section 4.
3.1. General context
We first describe the characteristics of ERs in Canadian hospitals. We then give details of how many physicians work
at an ER, how many shifts each of them works per week
(on average), how shifts are assigned to each physician and,
finally, how vacation time and days off are managed.
The six hospitals visited are all funded by the province
of Quebec's health insurance. Four of them are "teaching"
SCHEDULING EMERGENCY ROOM PHYSICIANS
hospitals with a university affiliation, and two of them are
designated trauma centers (out of four trauma centers in the
province). The physicians working at the ERs are not paid
by the hospital but are reimbursed directly by the govemment health insurance program on a fee-for-service basis (for
more information on the Canadian health system, see Evans
[11]). When a patient arrives at an emergency department,
they will be seen immediately by a triage nurse who will assign a priority level. Patients with the most serious problems
will be seen immediately and the least serious cases will wait
until the higher priority cases have been treated.
Each of the study ERs employs from 3 to 16 full-time
physicians and up to 17 part-time physicians, for a total
full-time equivalent of between 10 and 16 physicians. The
physicians are either emergency specialists or family physicians. A full-time physician normally works 3.5 eight-hour
shifts per week on average (28 h per week) and each parttime physician works between 1 and 2.25 eight-hour shifts
per week. A typical pattern might be three days on and
three days off. When more than one physician is working in the ER, tasks can be divided into ambulance cases,
walk-in cases, and cases in the follow-up clinic. The ambulance cases represent a heavier workload than walk-in cases
or ones in the follow-up clinic, since patients arriving by ambulance tend to have more serious problems.
The assignment of shifts to physicians involves a number
of criteria. Most doctors prefer working weekdays rather
than weekends, day shifts rather than evening shifts, and
evenings rather than nights. In certain hospitals, compensation for years of service in the ER is given by scheduling
fewer night and weekend shifts. Shifts are assigned equitably among physicians of a given seniority group.
Several rules generally govern the design of a physician's
schedule:
(1) There must be at least 16 h between the end of one shift
and the beginning of the next shift. A violation of this
rule is called a backward shift rotation.
(2) A work stretch is normally two to four consecutive days.
Some hospitals will assign blocks of five consecutive
days if there is a "break" of at least 24 but less than 36 h
between the end of one shift and the beginning of the
next shift within that block (often referred to as a forward shift rotation).
(3) Days off are normally grouped in blocks of at least two
consecutive days.
(4) Two days off are usually sufficient after working day or
evening shifts, but at least three days off should be assigned after night shifts.
(5) People prefer to have Saturday and Sunday shifts assigned together.
(6) Different shift types should be assigned equitably.
Blocks of night shifts should be spread as evenly as possible in the schedule to help the physicians avoid fatigue.
349
(7) Finally, weekends off should be evenly distributed over
the schedule.
The scheduling rules may be hard - i.e., absolutely required - or soft - i.e., desirable characteristics - depending on
the preferences of the hospital and the amount of fiexibility
the physicians have. A hard rule at one hospital might be a
soft rule at another.
In addition to the hospital rules, certain schedulers allow individual physicians to express preferences concerning
their schedules. For example, special requests for days off
may be considered for physicians working at other clinics
or for physicians whose religion precludes them from working on specific days. Hospitals that do not accept requests
for days off typically produce their schedule several weeks
or even months in advance so that the physicians can plan
other activities around their ER schedule or exchange shifts
with other physicians. The rationale for ERs that do not consider requests for days off is to minimize the work required
for the scheduler and to let the physicians cooperate informally to accommodate personal issues. When the schedule
includes a long stretch of days off, such as for a vacation,
physicians prefer day shifts and lighter shifts when they retum to the ER. Sometimes, requests by individuals actually
make it easier for the scheduler. For example, at one hospital, a physician only wanted to work night shifts. However, this special request created unexpected problems when
the physician left the group. The scheduler had to ask all
of the remaining physicians to work extra night shifts and
several of them decided to quit. This hospital has since
changed its policy and no longer accepts special shift requests, even if they seem favorable. Any doctor who wants
special shifts must do so by exchanging shifts with colleagues. Finally, scheduling rules during popular holiday
seasons, such as Christmas, New Year's Eve, and summer
are different than for the rest of the year. Days off are usually distributed among physicians such that each physician
can have either Christmas or New Year's off. During the
summertime, requests for days off are coordinated based on
seniority or priority (at one hospital, physicians are given
priority for summer vacation on a rotating basis). Some hospitals schedule these directly, while others leave it up to the
individuals.
Another important consideration for shift scheduling is
the disruption in circadian rhythms associated with phaseshifting in sleep/wakefulness cycles, and interference with
social and family life [31]. The effects of shiftwork can vary
widely among shift workers depending on several variables
including individual factors (e.g., age, personality traits,
physiological characteristics), as well as working situations
(e.g., work loads, shift schedules) and social conditions (e.g.,
number and age of children, housing, commuting) [8]. Peter Knauth presents an extensive survey of the literature on
designing shift systems [18]. He acknowledges that there is
"no single optimum shift system" and that "all systems have
advantages and drawbacks". However, based on a number
of intervention studies on shift workers in Germany, Knauth
350
has developed a set of "ergonomic guidelines" on the creation of shift schedules [18,19].
(1) minimizing permanent night shifts;
(2) reducing the number of successive night shifts to a
maximum of two or three;
(3) reducing the number of successive evening shifts to a
maximum of four;
(4) avoiding short intervals of time off (less than 11 h) between two shifts;
(5) shift systems including work on weekends should provide some free weekends with at least two consecutive
days off;
(6) long work sequences followed by four- to seven-day
mini-vacations should be avoided;
(7) forward rotations are preferred;
(8) individual schedules with few changes over time are
preferred;
(9) shift lengths are adjusted according to task intensity;
(10) shorter night shifts should be considered;
(11) a very early start time for the morning shift should be
avoided; and
(12) preference should be given to flexible working time
arrangements among workers.
According to Knauth, these rules are based on what seems to
be tbe current consensus among shift work researchers, although there are other opinions. For example, some people
argue for relatively long stretches of night shifts to allow the
body to adjust to the new circadian rhythm. Most researchers
in that field would agree, with the proviso that the workers
maintain the habit of sleeping during the day on their days
off. Unfortunately, in surveys, they discovered most people do not do this. On their off-days, they want to socialize
with friends who probably do not work at night. A recent
assessment of shift systems in Germany showed that many
of them do not satisfy these ergonomic guidelines [13]. For
a discussion of the pros and cons of these guidelines, refer
to Knauth [18].
3.2. Problem definition
The problems encountered at each of the six hospitals studied are different but nevertheless share a number of similarities. In this section, we present a problem framework which
includes the common aspects of the scheduling problems,
as well as options that may be selected to meet the specific
needs of the hospital and/or its physicians. We present the
degree of difficulty associated with each option, based on our
knowledge of scheduling techniques. We have included a
mathematical representation of some of the constraints when
it improves clarity.
W.M. CARTER, S.D. LAPIERRE
5.2.7. Decision variables
There are two ways to define the decision variables:
• A spreadsheet approach
I
i
0
if shift type i is assigned
to physician k on day j ,
otherwise.
The spreadsheet approach refers to the tool that most
physicians use to make the ER schedule: they list the K
physicians' names in rows, list the J dates to be staffed
in columns, and then assign a shift of type i to each column. These decision variables are most appropriate when
the schedules are prepared manually on paper or within a
spreadsheet. It is easy to visualize the solution. The figures
presented in sections 4 and 5 use a spreadsheet format to
display sample schedules.
• A mathematical programming formulation
I
I
0
if shift type i is assigned
to physician k on day j ,
otherwise.
This notation is obviously equivalent to the spreadsheet
approach but is a little easier to describe constraints mathematically.
3.2.2. Constraints satisfying the physicians' requirements
The most important constraint is to staff the exact number of
physicians per day and shift type. This is an equality constraint since there cannot be a surplus or a shortage. The
constraint can be written as:
v/, j ,
k=\
where c,y is the number of physicians required on shift type i
on day j . For the ER scheduling problem, we encountered
three cases characterizing c,^ 's. For a fixed value of i:
• Simple case: Cjj 's have the same pattern for every day j .
• Extension: c,y 's have two patterns: one for week days
and one for weekends.
• Second extension: c,y 's have more than two different patterns.
The variation of values for cij 's can lead to more complex
solution techniques.
3.2.3. Constraints on individual schedules
It is relatively easy to make a schedule but it is very difficult
to make a good schedule for every physician. The next set of
scheduling constraints concems the physicians' individual
schedules. These constraints are among the major reasons
why scheduling ER physicians is difficult. The fundamental
constraints are the following:
• There must be a minimum of 16 h between the end of one
shift and the beginning of the next shift, i.e., Xi^j^_k +
SCHEDULING EMERGENCY ROOM PHYSICIANS
351
Xi-,.p.k ^ 1 for all shifts /] on day j \ and /2 on day 72
with less than 16 h between them.
• There must be no more than IV"^" consecutive working
days (maximum work stretch) before a day off, i.e..
j=n
* Simple case: each physician must work precisely
shifts (only full-time physicians), i.e..
* Extension: each physician works A' shifts, but two or
more part-time physicians can split the schedule of one
full-time physician.
* Extension: each physician, k, works N^ shifts (people may be different due to administrative duties, seniority, covering for people on vacation, etc.), i.e..
1= 1
forn = 1 , . . . , 7 = 4 for most ERs.
where
* Simple case: consecutive working days must have the
same type of shift.
• Equitable distribution of shift types among physicians,
A',-, or for each physician.
* Extension: a change of shift type is allowed within a
block of consecutive working days.
* Extension: if there is a change of shift type, there can
be up to W'"^^ consecutive working days.
* Simple case: each physician works precisely A^,- shifts
of type I, i.e., ^ ^ ^ , xijk = Ni Vi, k.
* Nights, extension: there must be no more than
consecutive nights, W^^^^^ < W""^".
* Nights, extension: no change of shift type is allowed
within a block of consecutive working days that contains night shifts.
• There must be at least Wmin consecutive working days
before a day off, Wmin = 2 for most ERs.
* Nights, extension: There must be at least ^
secutive working nights before a day off.
con-
There must be at least Omin consecutive days off after a
working period, Omin = 2 for most ERs.
* Nights, extension: There must be at least O^f^ ' consecutive days off after a working period on night shift,
Omin
>
* Extension: Each physician k works precisely A^,*
shifts of type ;. For example, senior physicians may
have a reduced number of night shifts, weekend shifts
and/or evening shifts i.e., E J = I ^ijk = M* V(, k.
* Extension: This rule can be made more flexible by
putting upper and lower limits on the number of shifts
of type i that physician k must work.
• Availability of physicians.
* Simple case: schedules are constructed as if every
physician were always available.
* Extension: physicians can request days off on a small
number of days per scheduling period, i.e., Xjjk — 0
for some i, i,k.
* Extension: physicians can request days off on a large
number of days per scheduling period.
3.2.5. Schedule equilibrium
Define:
1 if physician k works
on the weekend in week /,
otherwise,
^min.
• Weekend shifts are treated as either:
* Simple case: treated the same as weekdays (no distinction).
* Extension: If a physician works shift type / on either
day on the weekend, then s/he automatically works
shift type ( on the other weekend day, i.e.,
Xijk + Xi.j+\.k = 2v where y € {0,1}.
* Extension, extended weekend: If a physician works an
evening or night shift on Friday, then s/he automatically works the same shift type on Saturday and Sunday. The reasoning behind this variation is that people
also get three day weekends off. Many physicians feel
that working Friday evening with Saturday and Sunday off does not really constitute a weekend off.
3.2.4. Shift assignments among physicians
• Constraints on the total number of shifts worked by each
physician during the scheduling period J.
given y = 1 is Monday, j = 11 is Sunday V/ and L = {J/I)
is the number of weeks.
• Distribution of weekend shifts.
* Simple case: Ignore the distribution.
* Extension: No one works more than C weekends in a
row i.e.,
"+C+1
E
y^ik^C
/=«
Vn = 1 , . . . , L - C - 1 .
* Extension: No one works more than A out of any consecutive B weekends, i.e.,
n+B
l=n
W.M, CARTER. S.D. LAPIERRE
352
• Distribution of night shifts.
* Simple case: Ignore the distribution.
MDl
* Extension: Night shifts should be spread uniformly
across the schedule i.e., for some period of time M
(e.g., any 3 week period) each physician should have a
maximum of Z^^l^ night shifts:
MD2
n+M
E
n = 1 , . . . , 7 — M; i = "nights".
MD3
M
Tu
8
8
16
16
W
8
8
16
16
0
MD4
MD5
MD6
Th
0
F
Su
M
16
16
16
0
8
0
0
0
0
8
0
MD7
MD8
Sa
8
8
16
• Overall distribution of shifts
* Simple case: Ignore the distribution.
* Extension: All shift types should be equally distributed
over the schedule (similar to "nights").
3.2.6. Objective
• If possible, find a feasible schedule.
• Otherwise, respect the hard constraints and minimize the
"cost" of violating soft constraints according to a set of
weights reflecting the physicians' schedule preferences.
* Simple case: All physicians' preferences are given the
same weight.
* Extension: Physicians' preferences can be weighted
differently.
4. Scheduling methods
4.1. Existing scheduling methods
The existing scheduling tools are quite simple. Four of the
six hospitals use an electronic spreadsheet to create and edit
their schedules; none of these sites use scheduling software
to generate the schedule. We can classify the existing schedule types into three categories:
(1) the acyclic schedule;
(2) the cyclic schedule without rotation;
(3) the cychc schedule with rotation.
For each category, we present the technique used to prepare the schedule and the issues typically considered by the
hospitals using that type of schedule.
4.1.1. The acyclic schedule
The acyclic schedule is created from scratch each period and,
in general, no two physicians have exactly the same schedule. This was the most common schedule we encountered at
the ERs we visited. An example is shown in figure 1. The
three hospitals having this type of schedule create it using a
spreadsheet without any specialized scheduling software to
generate the initial solution. This type of schedule allows the
scheduler to make allowances for vacations, a few personal
preferences, requests for days off and full weekend working
Figure 1. The acyclic schedule. The numbers indicate the starting time of
the shift, based on a 24-h clock (where 0 denotes midnight).
assignments. The quality of the schedule depends on the person creating it. Two of the hospitals start by assigning weekends, then night shifts, and finally, the other shifts. The third
hospital asks the physicians to produce their own schedules,
beginning with the most senior staff member. The scheduler
then improves the schedule by making shift exchanges. It
is usually not possible to respect every scheduling requirement, so the scheduler tries to produce a fair schedule while
satisfying the most important constraints for the physicians
(hard constraints) and minimizing the soft constraint violations. This type of schedule takes a lot of time to develop.
Based on the experience of the schedulers at the study hospitals, we estimate that it takes about 40 h for an experienced
scheduler building a three month schedule with 22 physicians, or a six month schedule with 14 physicians.
4.1.2. The cyclic schedule without rotation
A cyclic schedule is constructed such that each physician
constantly repeats the same shift pattern. Two of the hospitals used this type of schedule. In the simplest case, the
period lasts 7 or 14 days, in which case the physician always
works at the same time, on the same days of the week. When
the period is not a multiple of 7, then every physician has
to work on less popular days and the schedule is equitable
assuming that each physician's sequence includes an equal
number of each shift type, as shown in figure 2. Notice that
this scheduling technique can have physicians working a different total number of shifts, to account for part-time or fulltime status. The drawback of using a cycle length which is
not a multiple of 7 is that (1) weekends are often broken, i.e.,
only one day out of two is a working day and (2) an equal
number of physicians has to be assigned every day, i.e., the
number of physicians is not reduced on less busy days.
One hospital uses a period of three weeks to assign
weekday shifts, but weekends are assigned separately, in a
fair manner without seniority rules. Another hospital uses
a 37 day cycle, with the advantage that the physicians work
equally on every day of the week. This type of schedule
does not take much time to prepare, usually about two hours.
However, it does not account for requests for days off or any
personal preferences. In order to accommodate special re-
SCHEDULING EMERGENCY ROOM PHYSICIANS
.MDl
DI
D2
8
8
16
D4
D6
16
16
8
D8
D9
DIO
0
0
0
D12
0
0
8
0
8
Dll
16
0
16
0
MD6
D7
16
0
16
0
D5
16
MD4
MD5
D3
8
iMD2
MD3
353
8
16
8
MD7
0
MD8
8
8
0
8
D13
8
16
16
8
16
16
0
Figure 2. The cyclic schedule without rotation.
M
Tu
W
Schedule A
8
8
8
Schedule B
16
Schedule C
0
16
0
Schedule F
0
F
Sa Su
16
16
8
8
0
0
0
16
Schedule D
Schedule E
Th
8
8
16
16
0
Schedule G
Figure 3. The cyclic schedule with rotation.
quests for days off and vacations, both hospitals ask that the
physicians exchange shifts among themselves.
4.1.3. The cyclic schedule with rotation
A cyclic schedule with rotation is constructed by building a
single, long pattern for one physician. Every doctor follows
the same pattern, but with different, equally-spaced starting
points, and when they get to the end, they go back to the beginning. For example, in figure 3, the first physician will
work schedule A in week 1, then schedule B in week 2,
and so on. The second physician will work schedule B in
week 1, schedule C in week 2, and so on. This schedule cannot accommodate personal preferences and special requests
for days off, but it has the advantage of being equitable. One
hospital uses this method and the scheduler makes changes
to the original schedule to respect requests for days off and
outside work arrangements as much as possible. A few fulltime schedules are split between two physicians to accommodate the part-timers. Once the basic cycle has been established, it takes about three to 10 h to make a three month
schedule for 16 physicians with most of the time being taken
up by changing the basic schedule to accommodate days off.
4.2. Schedule assessment
As we review the hospitals' rules and methods for making
schedules, certain patterns emerge. Cyclic schedules, with
or without rotation, take less time to prepare than the acyclic
ones; but the acyclic schedules tend to be of better quality
in terms of meeting special requests. An interesting observation was that a less appealing schedule characteristic,
for example a broken assignment on weekends, was acceptable as long as physicians thought it simplified the scheduler's job or that it reduced the time it took to develop the
schedule. As long as the scheduler appeared to be fair, the
physicians were happy with their schedules. The important
factors when considering schedule type and schedule preparation technique are the time needed to produce the schedule, fairness, and quality in terms of how well the schedule
adheres to the hospital's scheduling rules. The schedulers
working with acyclic schedules are interested in software or
some technique that maintains the same quality but reduces
the time it takes to build the schedule. The groups using
cyclic schedules, with or without rotation, seem to be interested in a technique that would improve the quality of the
schedule while not increasing the time spent on making it.
There was considerable variation in the number of rules
employed by hospital schedulers. We were a little surprised
to discover that a few hospitals seem to be quite content
with a schedule which meets only a few scheduling rules
and has no seniority rules. Other hospitals with a large number of rules and preferences have difficulty producing satisfactory schedules. It appears that the rules which are taken
into account are part of an evolutionary process directed by
the physicians themselves. A hospital develops rules over
time based on the (not always compatible) goals of retaining
physicians and recruiting new ones.
5. Two applications
Developing ER scheduling tools is not an easy task as one
has to consider group preferences and how a hospital's
scheduhng rules originated in order to decide which rules
should be integrated into the software. We present two cases
illustrating how one can approach schedule development in
a logical manner.
5.7. Charles-Lemoyne Hospital
Charles-Lemoyne Hospital (CLH) is the major trauma center
of Montreal's southshore suburbs, the second busiest ER in
354
W.M. CARTER, S.D. LAPIERRE
the province of Quebec. We describe the ER schedule as
it was when we first started working with the group, and
then present our analysis of their revised schedule which was
later solved using a computer approach called Tabu Search
[20,21].
5.7.7. Existing schedule
CLH is the hospital with the simplest scheduling rules. This
group used to have a larger set of scheduling rules seven
years ago, including individual preferences and seniority
rules. The departure of several physicians in a short period
of time forced the group to change their approach and find
ways to develop appealing schedules in order to facilitate the
recruiting of new physicians. Physicians at CLH has appear
to be very satisfied with their current scheduling approach.
This appreciation can be measured by a very low turnover
rate, making CLH one of the few hospitals that does not have
ER recruiting problems.
CLH uses a cyclic schedule without rotation, with a cycle
of 37 days. Each full-time physician works the following
rotation:
spaced in the schedule. The fact that the schedule does not
account for days off and vacation requests is not much of a
problem since the part-time physicians are flexible and make
exchanges readily.
Physicians have only one real complaint with the existing
schedule: they want more regular full weekends off. Since
they work one day out of two, physicians should have, on
average, one weekend off every second weekend. Because
the cycle length of the schedule is not a multiple of 7, and
since the underlying pattern is a six day cycle (3 on and
3 off), there are broken weekends and several consecutive
working weekends. On average, physicians have only one
full free weekend every three weeks and they can work up
to five consecutive weekends. Physicians would therefore
like to have a schedule which retains the positive aspects of
their existing schedule, yet allows for full weekends as well.
However, the physicians are interested in such an improved
schedule only if it does not require more time on their part
to prepare.
5.7.2. Schedule analysis
When developing a scheduling tool for CLH, there are two
main features that we can try to improve: weekends and vacations. The current cycle length of 37 days makes it imwhere the X's represent days off and the numbers represent possible to improve weekend schedules. For example, in a
the shift start time. Part-time schedules follow the pattern of fixed cyclic rotation, a two day off pattern will have seven
the full-time schedule. Each part-time physician works one different start days each time it rotates through the schedof the following rotations:
ule every 37 days. Two of the seven start days [FridaySaturday] and [Sunday-Monday] contain only one weekend
day off. Only one of the patterns has [Saturday-Sunday]
XXXXnnXXXXOOXXXXX
(2/3 time),
off. When the cycle is divisible by seven, it is at least possible to ensure that all weekends off include Saturday and
or
Sunday. We explained the mechanisms of cyclic schedul7XXXXXSXXXXX\2XXXXX\6
ing with rotation to the physician who produces the schedXXXXXnXXXXXOXXXXXX
(l/3time).
ule. He was pleased to discover that this technique could
Currently, five physicians work full-time, six work 2/3 of the not only take care of assigning full weekends at a time, but
time, and nine work 1/3 of the time, for an equivalent num- would also allow for different physician requirements on difber of 12 full-time physicians. Most full-time physicians and ferent days of the week, a feature not possible with the old
those who work 2/3 of the time do not work outside the ER, technique. This made us realize that the fact that the same
but all 1 /3 physicians work as general practitioners at private number of physicians were assigned on weekends as during
the week was a result of the current scheduling technique;
clinics.
The schedule makes no allowance for vacations or days it was not necessarily based on visit frequencies since there
off. Three-month schedules are made up and given out at are fewer patients visiting the ER on weekends. However,
least a month in advance. In order to get time off, the physi- CLH was not willing to change the staffing requirements
cians must find another physician with whom they can ex- right away.
change shifts or who is willing to take over a shift entirely.
To generate a new schedule, we looked at different shift
Clearly, this is a sub-optimal strategy as substitution is al- rotations to see if it was possible to improve the schedule
most guaranteed to create ergonomically poor schedules. We with respect to the circadian rhythm and vacation. Accordwould, in most cases, prefer to have vacations and days off ing to the recommendations of Knauth [18], CLH has a good
incorporated directly in the automated process.
basic work cycle:
The physicians like their schedules for several reasons.
The cyclic schedule is fair to all the physicians, making unity
(1) short work stretches;
in the group quite strong. Second, the schedule is easy to re(2) rapid shift rotation;
member because it is repetitive. Third, the schedule respects
(3)
forward shift rotation;
the circadian rhythm since the shift changes are forward rotations ones and are fairly rapid. Night shifts are evenly
(4) enough rest days after night shifts.
777XXX888XXX121212XXX161616
xxxnnnxxxoooxxxx,
355
SCHEDULING EMERGENCY ROOM PHYSICIANS
Could we improve this? A short rotation, such as the following, provides better spread for the evening shifts:
vacation time and making vacation time exchanges before
giving the schedules to physicians.
We now consider the issue of how to incorporate full
weekends off while keeping a good work stretch. There are
two possible approaches to this problem: (1) start with a
schedule which has good work cycles but does not include
full weekends off, and then improve it to account for weekends; (2) build a schedule which accounts for weekends and
then improve it to produce decent work cycles. Both approaches have advantages; but when making schedules manually, (1) seems easier. Again, just by hand, we improved the
schedule in figure 4 by exchanging shifts to obtain the schedule shown in figure 5. More precisely, we start by moving
one shift from a broken weekend to the other broken weekend (whichever is better) since if there is a weekend where
a physician only works Saturday, there is someone else who
only works on Sunday. This may create a single day off between two work stretches. So, we remove the last shift of the
work stretch before the day off or the first working day after
the day off and move it up or down the same column in the
matrix to attach it to another work stretch until we obtain at
least two days off between work stretches. We did not move
the night shifts since four working nights in a row were not
acceptable. This schedule has almost no broken weekends
but it has several working weekends in a row. It also has a
few unacceptable work stretches. If we want a good quality
schedule, better exchange techniques have to be developed
to build the schedule.
777XXX121212XXX1717I7XXX
The physicians agreed that such a rotation would be better, acknowledging the fact that the current stretch of
161616XXX171717XXX000 is quite demanding.
The current cyclic method does not integrate vacations
into the schedule. A simple way to facilitate vacation time
would be to intensify the working cycle in order to set up a
bank of vacation days. The key is to compress the time between day and evening shifts. The vacation time after night
shifts, or before a backward rotation, needs to be kept as is
or even increased. An intensified working period could be:
777XX121212XX171717XXXX888XX
161616XX000XXXXXX.
However, this schedule done by hand does not have the same
nice properties - three work days followed by three off-days
- and a full cyclic schedule needs to be built in order to allow work intensification. An example of a schedule for 12
physicians working on an intense rotadon is shown in figure 4. Note that intensifying the workload over a longer period of time instead of one rotation could extend the six-day
vacation to a two week or more vacation period. One advantage of planning vacations this way is that it provides a
constant working rhythm for physicians, which is difficult
to obtain by exchanging shifts with other physicians. However, if the vacation period does not occur at a convenient
time, the physician having the better vacation period might
not be willing to trade. Also, since every physician has a
tight schedule, it is more difficult to exchange shifts, making it more difficult to obtain vacation at the right time than
it would be in a schedule without vacation stretches. This
difficulty can be probably overcome by centrally managing
M Tu W
Schl
7
Sch2
7
12 12
Sch4
0
Scho
17
0
SchlO
16
8
0
8
7
8
12
12 12
0
16
0
8
8
8
8
16
16
16 16 16
0
12
0
0
8
7
12 12 12
0
8
8
12
16
17 17
0
17 17 17
8
0
7
8
7
7
7
7
12 12
8
8
7
0
7
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8
17 17 17
17 17 17
M Tu W Th F Sa Su
7
8
7
Sa Su
0
16 16 16
8
F
17 17 17
7
Schll
Schl2
0
Th
17 17 17
7
Sch8
Sch9
12 12 12
0
Sch6
Sch7
F Sa Su M Tu W
7
16 16
Sch3
Th
Based on our analysis, Lapierre et al. [21 ] developed a set
of Tabu Search metaheuristics to find a good cyclic schedule for CLH. Tabu Search works by assigning a score to the
schedule - which represents constraint violations or distance
from the ideal schedule - and then iteratively modifying the
schedule to improve the score. For example, if a score of
one is assigned to a broken weekend, a work stretch of more
than four days, and a single day off, the schedules in figures 4
and 5 would be given a total score of 12 and 3, respectively,
reflecting the fact that lower scores are better. The modifi-
8
7
8
8
7
12
16 16 16
7
12 12 12
16 16 16
0
0
0
Figure 4. A basic cyclic schedule for Charles-Lemoyne Hospital.
7
12 12
0
17
7
16 16
17 17
0
0
W.M. CARTER, S.D. LAPIERRE
356
M
Schl
7
Sch2
7
W Th F Sa Su M Tu W
12
Sch4
0
0
0
8
12 12
0
16
0
12 17
8
7
12 12
12
16
0
0
16
12
0
0
0
8
8
12 12
16
16
17 17
0
7
8
7
7
12
12
7
7
12 12 12
16 16 16
0
0
0
17
8
8
7
7
16 16
8
16 16 16 16
8
8
8
17 17
7
7
7
8
7
0
Schll
Schl2
8
17 17 17
7
16
8
F Sa Su
7
8
12 12
8
Th
8
7
8
16 16 16
8
12
SchlO
8
7
F Sa Su M Tu W
0
7
7
8
0
0
17 17
Sch8
Sch9
16
Th
17 17 17 17
17 17 17 17
Sch6
Sch7
12 12 12
7
16 16
Sch3
Sch5
Tu
17
0
0
17
0
Figure 5.An improved cyclic schedule for Charles-Lemoyne Hospital.
M Tu W Th F Sa Su M Tu W
Schl
12 12 12
Sch2
16
Sch3
7
7
Sch4
Sch5
8
Sch8
8
Sch9
0
7
12 12
17 17
0
0
8
8
7
7
7
12 12 12
12
0
7
12
8
17
7
17 17
17
17
0
0
12
8
16
8
16
8
16
16 16
8
8
16 16
16 16
7
12
17 17
17 17
0
12 12
7
0
12 12
17 17
7
8
F Sa Su
12 12 12
7
17
7
0
0
8
8
7
8
7
17 17
16 16
8
8
Tu \\' Th
17 17
16 16
8
7
7
8
16 16
12 12
0
0
0
8
8
F Sa Su U
7
16
0
0
0
0
SchlO
0
16
7
17
Sch7
Schl 2
16
16 16
Sch6
Schll
17 17
Th
8
0
7
0
0
7
Figure 6. The new cyclic schedule used at CLH.
cations to the schedule are similar to those done manually to
move from the schedule in figure 4 to the one in in figure 5.
In Tabu Search, we look for the best possible exchange, even
if it makes the current solution worse! We then make backward moves "tabu", in order to prevent the heuristic from
cycling (i.e., looking at the same schedules over and over
again). This enables the heuristic to move away from a local optimal solution, and provides a framework for looking
for new, better answers. With the development of such Tabu
Search heuristics, we were able to obtain the schedule shown
in figure 6, which is much better than the best schedule we
could find by hand (see figure 5).
One of the most valuable benefits of having a computerized solution is that the user can perform analyses experimenting with a variety of different rules and constraints.
They can develop an understanding of some of the tradeoffs involved. Generally, whenever we improve one criteria,
something else will suffer. At CLH, we produced schedules
with individual vacation times built into the rotation. How-
ever, the physicians decided that they preferred to have a
schedule witbout vacation time, since the resulting schedules were much more compact, which made it difficult to
exchange shifts among physicians.
The physicians were also interested in extending the
length of the weekend so that it started on Friday at 16:00
instead of Friday midnight. Most people do not consider that
they had the weekend off if they worked the Friday evening
shift. Unfortunately, the resulting schedules were judged to
be of very poor quality.
In the old manual system, the planning period was three
months long. Moreover, the shifts were always assigned
"3 shifts on and 3 shifts off". Several physicians work part
time, and three part time physicians were combined to cover
the load of one full time person. The workload was distributed equitably by giving each of them one of the three shifts
in a row. Under the new system, weekend assignments are
much better, but the "3 on; 3 off" pattern is gone. In order to
enable them to continue to provide equitable schedules to the
SCHEDULING EMERGENCY ROOM PHYSICIANS
part timers, we instituted a 36 week planning period, three
weeks by 12 physicians. For example, consider the schedule
shown in figure 6. Over a 36 week period, every physician
will work every type of shift exactly three times (e.g., 8AM
Monday). When we split the shifts among three part timers,
we can ensure that each person gets exactly one of each shift
type on each day. The assignments were also adjusted so
that, when a part time physician works on the weekend, they
work both days together, which provides more full weekends off. Labb6 [20] also developed a Tabu Search heuristic
to split a full-time schedule into part-time schedules. Physicians still exchange shifts to organize their vacation period.
The best schedule found by the heuristic was implemented in June 1998. This schedule is better than the former
one based on the ergonomic criteria presented in section 2.
As predicted by the theory, the majority of physicians are
pleased with the new schedule. It is popular since it assigns
one free weekend every two weeks, although the tradeoff is
a less regular schedule. Consequendy, it has become more
important to print individual schedules, to ensure that physicians do not miss a shift assignment (i.e., under the old system, the rotation was consistent, and easy to remember).
5.2. Jewish General Hospital
In this section, we present the case of the Jewish General
Hospital (JGH), a university affiliated hospital. This hospital has one of the busiest ER's in the province of Quebec
in terms of cases requiring hospitalization. We present the
ER schedule, the most complex of the schedules we encountered, and then present an approach for developing scheduling software for that group.
5.2.7. Existing schedule
JGH has had a large increase in the number of ER visits
in recent years, due in large part to a strong proactive "no
wait" policy. To sustain this growth, the hospital hired several physicians. Most of them are full-time ER physicians
at JGH, although some also work in the ERs of nearby hospitals. The physician developing the schedules began doing
this task ten years ago, when the ER was relatively small,
after physicians complained that the schedules made by clerical staff were "unfair". Like CLH, physicians appeared to
be very satisfied with the schedules as indicated by their low
attrition rate and the ease of recruiting physicians.
The JGH schedules are acyclic. Each physician works a
different number and type of shifts based on his/her seniority and the amount of administrative and teaching duty s/he
has. On average, a full time physician works three shifts per
week. The current scheduling technique is primarily selfscheduling:
• The scheduler looks at each three month period, notes
statutory holidays, and determines the number of physicians required each day. This number does not change
frequendy (perhaps one to three times a year).
357
• The physicians give their days off and vacation requirements to the scheduler - physicians are allowed up to 7
weeks of vacation per year.
• The scheduler runs a small program that takes the vacation requirements, the physicians' requests for days off,
and their status in order to compute the number of shifts
and shift types for each physician for the three month period.
• The scheduler then builds a matrix of weekly shift assignments for each physician. Those weeks requested for
vacation are blocked out and shifts are assigned in order
to balance weekly workload as much as possible (i.e., try
to give each physician 3 shifts per week) unless an unbalanced workload has been specifically requested. Because
of vacation requirements, a perfectly balanced workload
is not always possible.
• Based on the balancing matrix and the shift type requirements, the scheduler first prepares the schedule for the
physicians working at other hospitals.
• Most of the regular JGH physicians make their own
schedule, starting with the most senior physician. Each
physician has to respect the balancing matrix and shift
type requirements, as well as previously entered schedules. A program written in Microsoft Excel facilitates
this task.
• Four remaining physicians (the scheduler plus three volunteers) then make their schedules together. If there are
not enough remaining physicians on a given day, the
scheduler has the right to modify the self-made schedule as long as the requested days off and vacation periods
are respected. The scheduler has written sophisticated
macros in Excel that help to respect the matrix requirements and days off requested by each physician. Over
the years, the scheduler has come to know the physicians
and has learned which of their preferences and which
scheduling rules can be violated when fine-tuning the final schedule.
The resulting schedules are generally satisfactory. An example is shown in figure 7 where "OC" means the physician is "On Call" for a 24.h period; Not every physician
has their desired schedule, particularly the less senior physicians, but the system does give the impression of fairness.
However, the scheduling task takes time; even with the programs and interfaces written in Microsoft Excel, around 40 h
are needed to prepare a three-month schedule and it takes
about six weeks to complete since physicians do not work
every day and the input of all of them is necessary for the
self-scheduling. The scheduler would really love to have an
automated system to relieve him of the workload. Moreover,
JGH would be in serious trouble if the scheduler decided to
leave since the task has become too complex for a newcomer
to easily take over.
W.M. CARTER, S.D. LAPIERRE
358
M Tu
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MD3
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MDll
MD12
Tu
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M
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0
OC
MD22
F
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MD9
MD19
Th
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MDIO
MD16
w
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MD13
M Tu
8
MD6
MD7
Su
8
MD4
MD5
Sa
16
0
16
16
8
8
0
OC
16
Figure 7. An example of a schedule at the Jewish General Hospital.
5.2.2. Schedule analysis
The schedule preparation process at JGH is complex but the
physicians really like the results. Developing a model and
solution approach that will incorporate all of their scheduling rules is challenging. Seniority rules and religious constraints make it impossible to use a cyclic schedule. Moreover, the sequential method of having physicians self schedule in seniority order is relatively easy to do manually; but
we did not feel that it was amenable or appropriate for a
computer solution. In particular, since people pick their own
schedule, the solution is highly constrained. The approach
tries to find a feasible answer without moving many assignments. In a computerized approach, it would be preferable
to try to describe physician preferences in more general,
flexible terms to give the algorithm some room to search
for good solutions. Therefore, we decided to develop a
set of rules that we could use as the basis for an algorithm.
As a first approximation, we generated a cyclic schedule to get some appreciation for what the physicians believe
would be a good general solution, and still meet the JGH requirements. With the metaheuristic algorithm developed for
CLH, we generated a cyclic schedule for 15 full-time equivalent physicians (without the "on-calls"). We presented the
schedule shown in figure 8 in order to validate the general
scheduling rules for JGH. Of course this schedule is unacceptable for JGH, but it was easier to discuss one model
schedule with the physicians rather than 22 individual schedules. The physicians were pleased to learn we could generate
schedules that takes account of circadian rhythm, a feature
that is not considered in the current system. They gave us
their feedback on our scheduling rules in order to adjust the
algorithm's weighting systems. We refer to these rules as the
average preferences for JGH physicians.
We then used the algorithm developed by Buzon [6] to
generate individual schedules for three months while respecting the physicians' requests for days off and vacation requirements but using the average preferences for each
physician. This allowed us to adjust the algorithm in order to
generate balanced schedules for both the full-time and parttime physicians.
Finjtlly, we adjusted the algorithm in order to generate schedules taking each physician's individual preferences
into consideration. We received very positive comments on
the resulting schedules by the scheduler at JGH. He then
gave us additional information on individual preferences for
each physician. We are currently working on "fine tuning"
the algorithm to generate the desired schedules but we can
already expect to generate better schedules than the actual
ones.
SCHEDULING EMERGENCY ROOM PHYSICIANS
359
M Tu W Th F Sa Su M Tu
Schl
8
Sch2
8
16 16
OC
12 12 12
Sch4
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Sch5
16 16 16
Sch6
0
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w
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OC
Tu
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OC
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8
Sch9
Th
16 16
OC
Sch3
Sch7
w
16 16 16
8
8
OC
0
0
0
Figure 8. A cyclic schedule for the Jewish General Hospital.
We believe that the first step of developing a cyclic schedule was critical to the development of a new solution at JGH.
The quality of a schedule is very personal and subjective.
The involvement of the physicians to achieve some form
of group consensus is necessary before trying to design an
acyclic pattern.
6. Conclusion
We presented the problem of scheduling emergency room
physicians and examined the characteristics of the schedules encountered in six hospitals located in greater Montreal,
Canada. Extracting the real scheduling problem is difficult
because physicians' working conditions are based on informal mutual cooperation which is usually not documented.
The scheduling differences at the various hospitals require
different approaches when developing scheduling software.
We showed how to computerize the scheduling problem of
Charles-Lemoyne Hospital, an ER with simple scheduling
rules, and the Jewish General Hospital, an ER with complex
scheduling rules.
Acknowledgements
Physicians We wish to thank Renee Amilcar, Pratt & Whitney Canada, for her help in starting this project. For her
undergrad senior thesis, she studied the scheduling problem
of four emergency services: Hopital Le Gardeur, Hopital
Maisonneuve-Rosemont, Hopital Sacre-Coeur, and former
Hopital St-Michel. We wish to thank Dr. Alphonse Montminy, Charles-Lemoyne Hospital, and Dr. Bernard Unger,
Jewish General Hospital, for sharing their scheduling expertise with us. We also thank Michel Gendreau, Patrick Soriano, and two anonymous referees for their helpful comments
on this paper. The authors were supported by the Natural
Sciences and Engineering Research Council of Canada under grant numbers OGP0001359 and OGP0184219.
References
[1] American College of Emergency Physicians. Directory of Software
in Emergency Medicine, Dallas, TX ( h t t p : / / w w w . a c e p . o r g )
(1998).
[2] J.L. Arthur and A. Ravindran. A multiple objective nurse scheduling
model. IIE Transactions 13 (1981) 55-60.
[3] H. Beaulieu, J.A. Ferland, B. Gendron and P. Michelon, A mathematical programming approach for scheduling physicians in the emergency room, Health Care Management Science 3 (2000) 193-200.
[4] I. Berrada, J.A. Ferland and P. Michelon, A multi-objective approach to nurse scheduling with both hard and soft constraints, SocioEconomic Planning Sciences 30 (1996) 183-193.
[5] J. Blake, M.W. Carter and S. Richardson, An analysis of emergency
room wait time issues via computer simulation, INFOR 34(4) (1996)
263-273.
[6] I. Buzon, La confection des horaires de travail des medecins
d'urgence resolue a l'aide de la recherche avec tabou. Master's thesis, Ecole Polytechnique, Montreal, Canada (2000).
[7] R.N. Bums and M.W. Carter, Work force size and single shift schedules with variable demands. Management Science 31 (1985) 599-607.
[8] G. Costa, The impact of shift and night work on health. Applied Ergonomics 27 (1996) 9-16.
[9] G. Dantzig, A comment on Edie's U-affic delays at toll booths. Operations Research 2 (1954) 339-341.
[10] K. Ecker, J.N.D. Gupta and G. Schmidt, A framework for decision
support systems for scheduling problems, European Journal of Operational Research 101 (1997) 452-462.
[11] R. Evans, We'll take care of it for you, Daedalus 117(4) (1988) 155189.
[12] S. Folkard, Shiftwork and performance, in: Biological Rhythms, Sleep
and Shiftwork, eds. L.C. Johnson, D.I. Tepas, W.P. Colquhoun and
M.J. CoUigan (MTP Press, 1981) pp. 283-306.
[13] A. Gissel and P. Knauth, Assessment of shift systems in the German
industry and sector; a computer application of the Besiak procedure,
Intemational Journal of Industrial Ergonomics 21 (1998) 233-242.
360
[14] N, Hoffart and S. Willdermood, Self-scheduling in five med/surg
units: a comparison, Nursing Management 28(4) (1997) 42-45.
[15] R. Hung, Shiftwork scheduling with phase-delay feature. International Joumal of Production Research 35 (1997) 1961-1968.
[16] B, Jaumard, G, Caron and P, Hansen, The assignment problem with
seniority and job priority constraints. Operations Research 47(3)
(1999)449-453,
[17] B. Jaumard, F, Semet and T Vovor, A generalized linear programming model for nurse scheduling, European Joumal of Operational
Research 107 (1998) 1-18,
[18] P, Knauth, The design of shift systems. Ergonomics 36 (1993) 15-28,
[19] P, Knauth, Design better shift systems. Applied Ergonomics 27(1)
(1996) 39-44,
[20] S, Labbe, La confection automatise d'horaire pour les medecins en
salle d'urgence. Master's thesis, Ecole des HEC, Montreal, Canada
(1998),
[21] S,D, Lapierre, P, Soriano. 1, Buzon, S, Labbe and M, Gendreau, Cyclic
schedules for emergency room physicians. Working paper for the
Centre for Research on Transportation, Montreal, Canada (2001),
[22] G, Laporte, The art and science of designing rotating schedules. Journal of Operational Research Society 50 (1999) 1011-1017,
[23] H,C, Lau, Preference-based scheduling via constraint satisfaction,
in: Optimization Techniques and Applications, eds, K,L, Phua, CM,
Wang, W,Y, Yeong, T,Y, Leong, H,T, Loh, K,C, Tan and RS, Chou
(World Scientific, 1992) pp, 546-554,
[24] H,C, Lau, Manpower scheduling with shift change constraints,
Proc. 5th Int. Symp. Algorithms and Computation (ISAAC), Lecture Notes in Computer Science, Vol, 834 (Springer, 1995) pp, 616624,
[25] H,C, Lau, On the complexity of manpower shift scheduling. Computers and Operations Research 23 (1996) 93-102,
[26] S, Lloyd, D, Streiner and S, Shannon, Bumout, depression, life and
job satisfaction among Canadian emergency physicians. The Joumal
of Emergency Medicine 12 (1994) 559-565,
[27] C, Maier-Rothe and H. Wolfe, Cyclical scheduling and allocation of
nursing staff, Socio-Economic Planning Sciences 7 (1973) 471-487,
[28] M,E, Miller, W.P. Pierskalla and G,J, Rath, Nurse scheduling using
mathematical programming. Operations Research 24 (1976) 857—870,
W,M, CARTER, S,D. LAPIERRE
[29] J,G, Morris and M,J, Showalter, Simple approaches to shift, days off
and tour scheduling problems. Management Science 29 (1983) 942950.
[30] W,P, Pierskalla and D,J, Brailer, Applications of operations research
in health care delivery, in: Handbooks in OR and MS, Vol, 6, eds, S,M,
Pollock, M,H, Rothkopf and A, Bamett (Elsevier, North-Holland,
1994) pp, 469-505.
[31] R,R, Rosa and M,J, Colligan, Plain language about shiftwork. Department of Health and Social Services, National Institute for Occupational Safety and Health, Cincinnati, OH (1997),
[32] P, Schwarzenau, P, Knauth, E, Kiesswelter, W, Brockmann and J,
Rutenfranz, Algorithms for the computerized construction of shift
systems which meet ergonomic criteria. Applied Ergonomics 17
(1986) 169-176,
[33] D. Scipione, V, Zavadsky, X, Dong, L, Scipione, F, Sullivan, T, Bctlach, C, Coker, J, Scipione, J, Cook and P, Scipione, Optimizing
staff scheduling by Monte-Carlo simulation, ProcAnnu Symp Comptit
ApplMed Care (1992) pp, 678-681,
[34] S,P, Siferd and W,C, Benton, Workforce staffing and scheduling: hospital nursing specific models, European Joumal of Operations Research 60 (1992) 233-246,
[35] L,D, Smith, The application of an interactive algorithm to develop
cyclical rotational schedules for nursing personnel, INFOR 14 (1976)
53-70,
[36] L,D, Smith, A, Wiggins and D, Bird, Post-implementation experience
with computer assisted nurse scheduling in a large hospital, INFOR
17(1979)309-321,
[37] P,E, Taylor and S,J, Huxley, A break from tradition for the San Francisco Police: Patrol officer scheduling using an optimization-based
decision support system. Interfaces 19 (1989) 4—24,
[38] G, Vassilacopoulos, Allocating doctors to shifts in an accident and
emergency department, Joumal of the Operational Research Society
36(1985)517-523,
[39] D,M, Wamer, Scheduling nursing personnel according to nurses preferences: a mathematical approach. Operations Research 5 (1976)
842-856,
[40] P,G. Zimmermann, Self-scheduling in the emergency department,
Joumal of Emergency Nursing 21(1) (1995) 58-61,