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A mathematical programming approach for scheduling physicians in the emergency room

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Health Care Management Science 3 (2000) 193–200 193
A mathematical programming approach for scheduling physiciansin the emergency room
Huguette Beaulieu
a
, Jacques A. Ferland
b
, Bernard Gendron
b,
∗
and Philippe Michelon
c
a
Imperial Oil Limited, 111 St. Clair Ave. W., Toronto, Ontario, Canada M5W 1K3
b
D´ epartement d’informatique et de recherche op´ erationnelle, Universit ´ e de Montr ´ eal, P.O. Box 6128, Succursale Centre-Ville, Montr ´ eal, Qu´ ebec, Canada H3C 3J7
c
Laboratoire d’informatique d’Avignon, Universit ´ e d’Avignon et des Pays de Vaucluse, B.P. 1228, F-84911 Avignon, France
Received 27 July 1999; accepted 4 February 2000
Preparing a schedule for physicians in the emergency room is a complex task, which requires taking into account a large number of (often conﬂicting) rules, related to various aspects: limits on the number of consecutive shifts or weekly hours, special rules for nightshifts and weekends, seniority rules, vacation periods, individual preferences,
. . .
In this paper, we present a mathematical programmingapproach to facilitate this task. The approach models the situation in a major hospital of the Montr´eal region (approximately 20 physiciansare members of the working staff). We show that the approach can signiﬁcantly reduce the time and the effort required to construct asix-month schedule. A human expert, member of the working staff, typically requires a whole dedicated week to perform this task, withthe help of a spreadsheet. With our approach, a schedule can be completed in less than one day. Our approach also generates betterschedules than those produced by the expert, because it can take into account simultaneously more rules than any human expert can do.
Keywords:
health administration, emergency physician scheduling, mathematical programming
1. Introduction
Preparing a schedule for physicians in the emergencyroom is a complex task, which requires taking into accounta large number of (often conﬂicting) rules, related to var-ious aspects: limits on the number of consecutive shiftsor weekly hours, special rules for night shifts and week-ends, seniority rules, vacation periods, individual prefer-ences,
...
In this paper, we present a mathematical pro-gramming approach to facilitate this task. The approachmodels the situation in a major hospital of the Montr´eal re-gion (Sacr´e-Coeur Hospital, where approximately 20 physi-cians are members of the working staff at the emergencyroom).The paper is organized as follows. First, our contribu-tions are highlighted in section 2, which provides a reviewof the literature on methods for scheduling health care per-sonnel. Then, section 3 gives an overview of the problemfaced by the planner in our case study. The mathematicalprogramming model is presented in section 4. In section 5,we describe the solution method based on the model, wegive a brief outline of its implementation (the interestedreader is referred to [2] for further details), and we com-pare the schedules it produces with those proposed by ahuman expert (a member of the working staff who has beenin charge of the scheduling task for many years). In theconclusion, we summarize our work and propose exten-sions.
∗
Corresponding author. E-mail: gendron@iro.umontreal.ca.
2. Literature review
Methods for generating workforce schedules are typi-cally divided into cyclic and non-cyclic approaches. Cyclictechniques proceed by deﬁning ﬁxed sequences of shifts,which are then assigned equally (or almost) among work-ers. Such techniques are well-adapted to situations wherethe same schedules can accommodate all workers. Eventhen, they must account for vacations and days-off, and inthe presence of many seniority rules and individual prefer-ences, as in our case (see the next section), they are usuallyof no help. Non-cyclic methods must then be considered,which generally fall into two categories: (1) those requiringhuman expertise and the use of a spreadsheet (which helpsthe planner to balance the schedules among categories of workers); (2) optimization approaches. These latter meth-ods have two main advantages over the former ones: ﬁrst,they require very little human intervention and can there-fore be (almost) fully computerized; second, when properlydesigned, they can handle many more rules simultaneouslythan any human expert can do, even with the help of aspreadsheet. However, mastering them is by no means atrivial task, and their application to a speciﬁc case mightrequire several years of development.Much of the research on scheduling health care person-nel has been devoted to the case of hospital nurses [18].A recent account of the nurse scheduling literature is givenin [9]. An example of a non-cyclic method based on humanexpertise and the use of a spreadsheet is given in [20], whileclassical optimization approaches are described in [14,23].Progress in computer technology and software tools has
©
Baltzer Science Publishers BV
194
H. Beaulieu et al. / A mathematical programming approach for scheduling physicians
recently provided a number of successful applications of optimization methods to nurse scheduling problems (see inparticular [4,5,8,10,16,22,24] and the references therein).Surprisingly, the problem of scheduling physicians in theemergency room has not attracted much attention. In a re-cent survey on the topic, which covers the situation in sixhospitals located in the Montr´eal area, Lapierre and Carter[13] highlight the basic difference between physicians andnurses: the former are not employed by the hospitals, con-trary to the latter. Therefore, in the case of nurse schedul-ing, both maximizing personnel satisfaction and minimiz-ing salary cost are often considered as two objectives toachieve simultaneously. In the case of emergency physi-cian scheduling, maximizing satisfaction only matters, asphysician retention is the most critical issue faced by hos-pital administrations (according to Lapierre and Carter). Inaddition, nurse schedules must adhere to collective unionagreements, while emergency physician schedules are moredriven by personal preferences. In general, planning theschedules for emergency physicians requires satisfying avery large number of (often conﬂicting) rules. Examplesof such rules are given in the next section, which describesour case study (in light of Lapierre and Carter’s survey, itappears very representative of the situation in other majorhospitals).We are aware of only two applications of operationsresearch methods to emergency physician scheduling prob-lems: [21], which presents a methodology based on sto-chastic models, and [11,12], where a cyclic method is im-proved by the use of modern heuristic techniques. To thebest of our knowledge, our work is the ﬁrst to present amathematical programming approach for scheduling physi-cians in the emergency room. Note that some commercialsoftware packages for emergency physician scheduling areavailable [1,6,15]. After a careful examination of the doc-umentation available from their Web sites, it seemed to usthat these software packages cannot handle all the rules of our problem. We should point out, however, that a fairevaluation of the capabilities of these software packages todeal with our problem would require extensive experiments.
3. Problem overview
At the Sacr´e-Coeur Hospital of Montr´eal, schedulesfor emergency physicians are established once every sixmonths, with a special schedule being also planned for thetwo-week Christmas period. A human expert, member of the working staff, is in charge of the task (for many years,the same person has been responsible for generating theschedules). The staff is composed of 20 physicians, in-cluding 15 working full-time. Among them, there can beup to ﬁve “young” physicians (with less than three yearsof experience). These ﬁgures are only indicative, as thesituation is changing every year.Approximately two months before the beginning of thenext planning horizon, each physician submits a list of indi-vidual preferences for the next period; these include vaca-tions and days-off, number of weekly hours, desired shiftsor sequences of shifts (for example, some physicians pre-fer to work three consecutive nights, whenever they haveto work at least one, while others prefer to work only onenight at a time),
...
These personal preferences are insertedinto a spreadsheet ﬁle representing a typical schedule: thedays are along the horizontal axis and the physicians alongthe vertical one. The planner works directly on this sheet,starting by manually ﬁxing the weekends, then the nights,and ﬁnally the other shifts. When assigning shifts to physi-cians, the planner attempts to respect a large set of rules,including individual preferences, but also ergonomic rules(such as after 2 or 3 nights worked, a physician shouldhave the beneﬁt of 2 or 3 days of rest), seniority rules(most senior physicians work fewer weekends and nightsthan others),
...
The use of the spreadsheet is essential inthis context: it facilitates the task of balancing the numberof shifts of each type (for example, day, evening or night)among physicians (according to their seniority). This isdone by compiling statistics for each type of shift, statisticswhich include not only the days of the current planninghorizon, but also those of the previous one (to account forpossible unbalances in the previous schedule).The process of generating the schedule by successivemanual assignments and corrections guided by the statisticsrequires a whole dedicated week. This time is consideredexcessive by the planner. Moreover, because of the highrisk of errors inherent to this type of manual approach, theplanning time might be further increased (for example, mis-takes can be discovered after the distribution of the “ﬁnal”schedule, which can force the planner to go back to hisworking table).However, the most annoying problem faced by the plan-ner is the following: when attempting to assign a particularshift, it frequently happens that none of the available physi-cians can be assigned without violating important rules.The planner usually reacts by “backtracking” over his pre-vious assignments. This process can be very long, andafter a few trials, the planner will often be satisﬁed with anassignment that still violates some rules.The challenge that we were facing when we started thisproject was therefore to design a solution approach thatrespects more rules than the human expert usually doesand that requires signiﬁcantly less time. Our approach isbased on a mathematical programming formulation of theproblem, which we describe in the next section.
4. The model
The model is an abstract representation of the rules of the problem, written in mathematical language. Rules aretranslated into constraints, which are linear inequalities builtaround variables. Before providing a detailed description of the model, we give an overview of the basic rules appliedat the Sacr´e-Coeur Hospital of Montr´eal.
H. Beaulieu et al. / A mathematical programming approach for scheduling physicians
195
We ﬁrst distinguish whether the rules are
compulsory
(e.g., rules that must absolutely be enforced) or
ﬂexible
(e.g., rules that can occasionnaly be violated, at the cost of losing some “quality”). Demand rules are the most basic inthe ﬁrst category. They deﬁne how many physicians shouldwork at different periods of a day and which responsibil-ities are attached to particular shifts. Each day is dividedinto three periods of eight hours: day, evening and night.Three physicians (two on weekends or holidays) work dur-ing day and evening shifts, including one exclusively incharge of traumas (“heavy” emergencies). “Trauma” shiftsare considered heavier than “regular” shifts (which mostlyinvolve the treatment of “light” cases and patients in sta-bilized condition). At night, there is only one night shift,the physician assuming the responsibilities of trauma andregular shifts. Three days per week, one physician works afour-hour shift, the “follow-up” shift, when he receives byappointment patients that have recently been treated at theemergency room. Other compulsory rules include: vaca-tions, days-off, or particular shifts requested by the physi-cians, and the basic ergonomic rule: “there must be at least16 hours between the end of one shift and the beginning of another one”.Flexible rules can be divided into two categories:
er-gonomic rules
which aim at improving the “quality” of theschedule of each physician, and
distribution rules
whichaim at distributing the assignment of particular types of shifts among physicians, sometimes according to their se-niority. Flexible rules are frequently conﬂicting with eachother, so they cannot always be satisﬁed simultaneously.Hence, some of these are regarded as “goals” to be reached,allowing for small deviations. Our approach exploits thisfeature. Indeed, the objective of the model is to minimizeall deviations, which amounts to ﬁnding an
efﬁcient solu-tion
(e.g., such that we cannot ﬁnd another solution withsmaller or equal deviations and with at least one strictlysmaller deviation). Hence, in the operations research jar-gon, the model is a special case of
multi-objective integer programming
(e.g., it includes several objectives, speciﬁedby the deviations, and all variables must take on integervalues) [17,19].
4.1. Notation and variables
The following notation is required to formulate themodel:1. Set of
physicians
I
.2. Set of
days of the planning period
J
=
{1,2,
...
,
n
}.We assume day 1 is a Monday, day
n
is a Sundayand
n
28 (otherwise, some constraints related toweekend shifts cannot be modeled). In addition, wealso consider the set
J
P
=
{
−
m
+
1,
...
,
−
1,0}, whichincludes
m
days of the previous planning period (in ourapplication, we used
m
=
5). The assignments duringthese days are required to guarantee the continuity of the planning.3. Set of
shifts
K
. In the model description, we considerseveral subsets of this set, namely:
K
D
, the day shifts,
K
E
, the evening shifts,
K
N
, the night shifts,
K
R
, theregular shifts, and
K
T
, the trauma shifts.Three different types of variables are used to formulatethe model. The
assignment variables
are decision variablesto indicate whether or not physician
i
is assigned to shift
k
on day
j
:
x
ijk
=
1, physician
i
is assigned to shift
k
on day
j
,0, otherwise.A second type of variables is required to formulate con-straints associated with rules involving sequences of con-secutive shifts to be followed by days-off. These variablesare called
succession variables
. They are speciﬁed shortlywhen we introduce this type of constraints.Finally,
deviation variables
are used to capture positiveand negative deviations from the targets in the constraintsassociated with the so-called “goal” rules. As mentionedabove, the objective of the model is speciﬁed in terms of these deviation variables.
4.2. Constraints
The constraints of the model are partitioned into four cat-egories according to the types of rules to which they cor-respond:
compulsory constraints
,
ergonomic constraints
,
distribution constraints
, and
goal constraints
. It is worthnoting that most of the constraints we introduce can be eas-ily modiﬁed and that additional constraints can be includedwithin the same framework to account for any speciﬁc ap-plication.
4.2.1. Compulsory constraints
1. One physician must be assigned to each shift of theperiod:
i
∈
I
x
ijk
=
1,
j
∈
J
,
k
∈
K
(
j
),where
K
(
j
) is the set of shifts to be completed duringday
j
. In order to simplify the notation, in the remain-der we assume that
K
(
j
)
=
K
,
j
∈
J
. Note that, inorder to satisfy these constraints, we might add to set
I
a “dummy” physician, who completes shifts that can-not be assigned to the regular members of the workingstaff.2. A physician cannot be assigned to more than one shiftper day:
k
∈
K
x
ijk
1,
i
∈
I
,
j
∈
J.
3. A physician assigned to an evening shift cannot beassigned to a day shift of the day after:
k
∈
K
E
x
i
(
j
−
1)
k
+
k
∈
K
D
x
ijk
1,
i
∈
I
,
j
∈
J.
196
H. Beaulieu et al. / A mathematical programming approach for scheduling physicians
4. A physician assigned to a night shift must not be as-signed to a shift of another type on the next day:
k
∈
K
N
x
i
(
j
−
1)
k
+
k/
∈
K
N
x
ijk
1,
i
∈
I
,
j
∈
J.
Note that a physician can be assigned to the nightshift of two consecutive days. Note also that the lastthree constraints imply the satisfaction of the basic er-gonomic rule mentioned above: “there must be at least16 hours between the end of one shift and the begin-ning of another one”.5. Vacations, days-off, particular shifts requested byphysicians:
x
ijk
=
1 (0),
i
∈
I
,
j
∈
J
,
k
∈
K
,where
J
⊆
J
and
K
⊆
K
are used to represent eitherworking days (or vacations and days-off) requested bya physician, or shifts a physician requested (not) beingassigned to.
4.2.2. Ergonomic constraints
1. Upper limits on the number of weekly (or monthly)hours of certain types of shifts:
j
∈
J
k
∈
K
h
k
x
ijk
U
(
J
,
K
),
i
∈
I
,where
h
k
representsthe numberof hours correspondingto shift
k
;
J
⊆
J
and
K
⊆
K
capture weeks or months,and the types of shifts (for example, night or follow-upshifts, or even all shifts), respectively; and
U
(
J
,
K
) isthe upper limit on the number of hours correspondingto days
J
and shifts
K
.2. Limited number of successive working days:
j
l
=
j
−
d
k
∈
K
x
ilk
d
,
i
∈
I
,
j
∈
J
,where
d
m
denotes the admissible maximum numberof successive working days (typically,
d
=
4 is used,but a limit of ﬁve days can be used during vacationtimes, when four or more physicians are away at thesame time).3. Consecutive weekend periods (evening, day or night)are assigned together to the same physician, trauma andregular shifts alternating between physicians (note thatweekends include Friday evenings and Friday nights):
k
∈
K
N
x
ijk
=
k
∈
K
N
x
i
(
j
+
1)
k
=
k
∈
K
N
x
i
(
j
+
2)
k
,
i
∈
I
,
j
∈
{5,12,
...
,
n
−
2},
k
∈
K
E
∩
K
T
x
ijk
=
k
∈
K
E
∩
K
R
x
i
(
j
+
1)
k
=
k
∈
K
E
∩
K
T
x
i
(
j
+
2)
k
,
i
∈
I
,
j
∈
{5,12,
...
,
n
−
2},
k
∈
K
E
∩
K
R
x
ijk
=
k
∈
K
E
∩
K
T
x
i
(
j
+
1)
k
=
k
∈
K
E
∩
K
R
x
i
(
j
+
2)
k
,
i
∈
I
,
j
∈
{5,12,
...
,
n
−
2},
k
∈
K
D
∩
K
T
x
ijk
=
k
∈
K
D
∩
K
R
x
i
(
j
+
1)
k
,
i
∈
I
,
j
∈
{6,13,
...
,
n
−
1},
k
∈
K
D
∩
K
R
x
ijk
=
k
∈
K
D
∩
K
T
x
i
(
j
+
1)
k
,
i
∈
I
,
j
∈
{6,13,
...
,
n
−
1}
.
4. After working a weekend, a physician should not work on the next Monday:
k
∈
K
{
x
i
(
j
−
1)
k
+
x
ijk
}
1,
i
∈
I
,
j
∈
{1,8,
...
,
n
−
6}
.
5. Whenever he works a night shift, a physician requeststo work three consecutive night shifts:
k
∈
K
N
{
x
ijk
−
x
i
(
j
+
1)
k
}
=
0,
i
∈
˜
I
,
j
∈
{2,9,
...
,
n
−
5},
k
∈
K
N
{
x
ijk
−
x
i
(
j
−
1)
k
−
x
i
(
j
+
2)
k
}
=
0,
i
∈
˜
I
,
j
∈
{2,9,
...
,
n
−
5},where
˜
I
⊆
I
is the set of physicians who require towork three consecutive nights whenever they work one(some physicians prefer to work one or two nights suc-cessively, while others do not express any preferencein this respect). Note that the constraints need only bedeﬁned for Tuesdays, as corresponding constraints forweekend days are already taken into account by theconstraints on weekends, deﬁned above.6. If a physician is assigned to a night shift after complet-ing any shift type on the day before, then he must haveat least two days off before being assigned to a shiftof any other type than a night shift. We have to intro-duce succession variables
s
1
ij
to formulate constraintsassociated with this rule. These variables are speciﬁedthrough the following constraints:
s
1
ij
−
k
∈
K
x
i
(
j
−
2)
k
0,
i
∈
I
,
j
∈
J
,
s
1
ij
−
k
∈
K
N
x
i
(
j
−
1)
k
0,
i
∈
I
,
j
∈
J
,
s
1
ij
−
k
∈
K
x
i
(
j
−
2)
k
−
k
∈
K
N
x
i
(
j
−
1)
k
+
k
∈
K
N
x
ijk
−
1,
i
∈
I
,
j
∈
J
,
s
1
ij
+
k
∈
K
N
x
ijk
1,
i
∈
I
,
j
∈
J.
H. Beaulieu et al. / A mathematical programming approach for scheduling physicians
197
Now, it is easy to verify that
s
1
ij
=
1, if physician
i
is working on day
j
−
2,if he is completing a night shift onday
j
−
1, and if he is not assignedto a night shift on day
j
,0, otherwise.Then, the rule is veriﬁed by adding the constraints:
s
1
i
(
j
−
1)
+
k
∈
K
x
i
(
j
−
3)
k
+
k
∈
K
N
x
i
(
j
−
2)
k
+
k/
∈
K
N
x
i
(
j
−
1)
k
+
k
∈
K
x
ijk
3,
i
∈
I
,
j
∈
J.
7. If a physician completes a night shift on three con-secutive days, then he must be off for the next threeconsecutive days. We must introduce a second type of succession variables
s
2
ij
speciﬁed through the follow-ing constraints:
s
2
ij
−
k
∈
K
N
x
i
(
j
−
3)
k
0,
i
∈
I
,
j
∈
J
,
s
2
ij
−
k
∈
K
N
x
i
(
j
−
2)
k
0,
i
∈
I
,
j
∈
J
,
s
2
ij
−
k
∈
K
N
x
i
(
j
−
1)
k
0,
i
∈
I
,
j
∈
J
,
s
2
ij
−
k
∈
K
N
{
x
i
(
j
−
3)
k
+
x
i
(
j
−
2)
k
+
x
i
(
j
−
1)
k
}
+
k
∈
K
N
x
ijk
−
2,
i
∈
I
,
j
∈
J
,
s
2
ij
+
k
∈
K
N
x
ijk
1,
i
∈
I
,
j
∈
J.
It is easy to verify that
s
2
ij
=
1, if physician
i
is assigned to a night shifton days
j
−
3,
j
−
2,
j
−
1, but not onday
j
,0, otherwise.Then, the rule is veriﬁed by adding the constraints:
s
1
i
(
j
−
2)
+
s
2
i
(
j
−
2)
+
k
∈
K
N
{
x
i
(
j
−
5)
k
+
x
i
(
j
−
4)
k
+
x
i
(
j
−
3)
k
}
+
k
∈
K
{
x
i
(
j
−
2)
k
+
x
i
(
j
−
1)
k
+
x
ijk
}
5,
i
∈
I
,
j
∈
J.
Note that the last two sets of constraints, in conjunctionwith compulsory constraints 4, imply the rule: “after
n
3 consecutive night shifts, any physician shouldhave at least
n
days off”.We have modeled a number of other ergonomic con-straints, using similar arguments (see [2] for further de-tails). These constraints include: “after
n
3 consecutivenight shifts, any physician should have at least 14 dayswithout night shifts”; “after coming back from vacation,any physician should have at least two days without nightor trauma shifts”; “in a sequence of four consecutive days,there should be no more than three consecutive evenings”.
4.2.3. Distribution constraints
All distribution constraints take the form:
j
∈
J
k
∈
K
x
ijk
(
)
F
i
(
J
,
K
),
i
∈
˜
I
,where
F
i
(
J
,
K
) is a minimal (maximal) frequency for anysubset of shift types
K
⊆
K
and for any subset of days
J
⊆
J
, that can be speciﬁed by any physician
i
∈
˜
I
⊆
I
.Some of these distribution constraints take seniority intoaccount, as is the case for weekend shifts: “young” physi-cians (with less than three years of experience) should work during two weekends every month (if possible, these week-ends should not be consecutive), while other physiciansshould work during at least one weekend per month (theremaining weekend shifts are assigned evenly among “old”physicians). Night shifts also obey seniority rules: physi-cians with more than four years of experience are assignedapproximately 10 night shifts for the next six months, thosewith less than four but more than three years of experienceare assigned 12 night shifts, etc.
...
, up to a maximum of 18 night shifts assigned to physicians with less than oneyear of experience (with the exception of newcomers, whoare not assigned any night shift).
4.2.4. Goal constraints
1. A physician should work a speciﬁed number of hoursper week:
j
∈
J
(
l
)
k
∈
K
h
k
x
ijk
+
uw
il
−
vw
il
=
TW
i
,
i
∈
I
,
l
∈
{1,2,
...
,
n/
7},where
TW
i
is the target number of weekly hours spec-iﬁed for physician
i
,
J
(
l
) is the subset of days in week
l
,
uw
il
and
vw
il
are the deviation variables.2. Certain types of shifts (night, evening and follow-upshifts) must be fairly distributed among physicians:
j
∈
J
x
ijk
+
uk
ik
−
vk
ik
=
TK
ik
,
i
∈
I
,
k
∈
K
,where
TK
ik
is the target number of shifts of type
k
∈
K
⊆
K
required by physician
i
,
uk
ik
and
vk
ik
are thedeviation variables. Note that
TK
ik
can be establishedby taking into account the assignments of type
k
shiftsduring the previous planning periods (for example, inour application, we use a planning period of 28 days,but take into account the last ﬁve periods). In thisway, we can allow for some degree of unfairness fora speciﬁc physician during a speciﬁc planning period,but still distribute the shifts fairly among physiciansover an horizon including several planning periods.

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