A STOCHASTIC APPROACH TO
NURSE STAFFING AND SCHEDULING
PROBLEMS
Presented by
Sera Kahruman & Elif Ilke Gokce
Texas A&M University
INEN 689-602
Outline
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Problem definition
Nurse staffing problem
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Literature and Formulation
Computational Study
Conclusion and Future study
Nurse scheduling problem
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Formulation
Computational Study
Conclusion and Future study
Problem Definition
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Patients in a hospital today are in need of highly
skilled nursing care
In order for this care to be provided in a timely
manner, and meet rigorous quality standards, the
right number and type of nursing staff must be
available when needed
demand is unknown
health institutions have to provide service 24 hours a day over
seven days a week.
Problem Definition
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Poor staffing levels lead to:
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Burnout
Dissatisfaction
Desire to leave the job
High cost of staff replacement
More patient complications
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Injuries
Low quality of service
Nurse Skill Types
There are 3 types of nurses:
1. Registered Nurses (RNs)
2. Licensed vocational or licensed practical
nurses (LVNs,LPNs)
3. Nurse aides
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To solve this problem, we need to consider
two levels:
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Strategic level : determining the long-term regular
time nursing levels---staffing
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Tactical level : Daily work/shift assignments of the
nurses
Outline
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Problem definition
Nurse staffing problem
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Formulation
Computational Study
Conclusion and Future study
Nurse scheduling problem
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Formulation
Computational Study
Conclusion and Future study
Nurse staffing problem
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Hospital administrators determine the
regular-time nurse levels annually
So, our model should determine the levels of
RNs, LPNs(LVNs) and nurse aides for a year
Demand is unknown!!!
Use 2-stage recourse model
Literature
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Tien and Kamiyama (1982), present a list of
personnel scheduling algorithms, which are not
restricted to healthcare.
Bradley and Martin (1990), distinguish three basic
decisions in hospital personnel scheduling: staffing,
personnel scheduling and allocation
Siferd and Benton (1992), present an excellent
review of the factors influencing hospital staffing and
scheduling in the United States
Literature
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Kao, Queyranne(1985) : Budgeting Costs of
Nursing In Hospital
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Presents 8 different models
Probabilistic or deterministic, different skill types
or aggregate skill, single period or multi-period
Formulation
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Major assumption:
There are many medical units within a
hospital. We assume that each medical unit
decides its own workforce levels and we do
not allow inter-unit working schedules.
The model is same for all units, except for
different demand patterns, and parameters.
Formulation
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First stage parameters:
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ci = total expenses of the hospital for a nurse of type i for the
whole year
he = Effective working hours of a nurse for one year
ADi = minimum annual demand for each skill class nursing
hours (can also be a bound specified by hospital
regulations)
r1,i =Ratio defining the relationship of skill class 1 and i=2,3
(skill class 1 supervises the other types. So there has to be
a constraint to make sure that there are enough
supervisors)
Formulation
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First stage decision variables:
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xi : number of regular time nurses of skill type I
where i=1,2,3
txi :number of nursing hours of skill type i and
higher level skill types which can be used for the
demand of lower skill nursing hours i=1,2.
Formulation
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Second stage parameters:
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oi = cost per overtime hour of a nurse of type i
ai = cost per hour of agency nursing providing a service of
skill type i
ei,t = percentage of total working time of a nurse of skill
class i available on month t {due to vacations and etc ,
effective nursing hours can change from month to month}
Oi,t = ratio of the maximum allowed overtime to the effective
regular working hours for a nurse of skill type i, on month t
di,t = demand realization in hours for nursing skill of type i on
month t.
Formulation
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Second stage decision variables:
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yi,t= number of overtime hours that nurses of skill
type i work on month t.
zi,t= number of agency nursing hours of skill type i
used for month t.
tri :number of nursing hours of skill type i and
higher level skill types which can be used for the
demand of lower skill nursing hours i=1,2.
Formulation
min
s.t.
~ )}
c
*
x
+
E{f(x,
w
∑i=1 i i
3
he * x1 - tx1 ≥ AD1
he * x 2 + tx1 - tx 2 ≥ AD2
he * x3 + tx 2 ≥ AD3
x1 − r1, 2 * x 2 ≤ 0
x1 − r1,3 * x3 ≤ 0
x1 , x 2 , x3 ≥ 0, integer
tx1 , tx 2 ≥ 0
Formulation
where for a realization of w, f(x,ŵ) is defined as:
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i =1
t =1
~ ) = min ( (oi * y + a * z ))
f(x, w
∑∑
i
i, t
i, t
s.t. e1,t * x1 + y1,t + z1,t − tr1 ≥ d 1,t
∀t = 1,2....,12
e2,t * x 2 + y 2,t + z 2,t + tr1 − tr2 ≥ d 2,t
∀t = 1,2.....,12
e3,t * x3 + y 3,t + z 3,t + tr2 ≥ d 3,t
∀t = 1,2.....,12
y i, t − xi ,t * Oi ,t ≤ 0
all variables ≥ 0
∀i = 1,2,3 and t = 1,2,...,12
Computational Study
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Use L-shaped algorithm where master
problem is solved as a MIP.
Comparison with expected value solution,
scenario based solution and worst case
solution
Comparison with linear relaxation of master
MIP
Observe the effect of demand fluctuation
Conclusion and Future Study
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a decision making procedure which
considers all medical units at the same time
allowing staff movements from one unit to the
other one as needed.
some parameters in the model are much
likely to be probabilistic which will make the
model more realistic
Outline
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Problem definition
Nurse staffing problem
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Formulation
Computational Study
Conclusion and Future study
Nurse scheduling problem
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Formulation
Computational Study
Conclusion and Future study
INTRODUCTION
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Every hospital needs to repeatedly produce
duty rosters for its nursing staff.
The scheduling of hospital personnel is
particularly challenging because of different
staffing needs on different days and shifts.
Because of time-consuming manual
scheduling NSP has attracted much
attention.
NURSE SCHEDULING PROBLEM (NSP)
The NSP involves producing a periodic duty
roster for nursing staff, subject to a variety of
constraints.
z A key feature of real NSP is that the planned
nurse schedule usually has to be changed to
deal with unforeseen circumstances such as
staff sickness and emergencies.
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NURSE SCHEDULING PROBLEM (NSP)
Solution approaches that have been proposed to
solve NSP is classified in three main categories:
z Optimization-mathematical programming
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single-objective (Warner (1976), Rosenbloom,
E.S. andGoertzen, (1987))
multi-objective (Arthur and Ravidran,1981)
Heuristics (Cheang et al., 2003).
Artificial intelligence
NSP FORMULATION
Indices:
g = 1, 2, 3 (nurse grade index)
k = 1, 2… N1 (nurse index for grade 1 nurse)
l = 1, 2… N2 (nurse index for grade 2 nurse)
m = 1, 2… N3 (nurse index for grade 3 nurse)
j = 1, 2…21 (shift index)
j = 1, 4, 7, 10, 13, 16, 19 corresponds to 7am-3pm shifts
j = 2, 5, 8, 11, 14, 17, 20 corresponds to 3pm-11pm shifts
j = 3, 6, 9, 12, 15, 18, 21 corresponds to 11pm-7am shifts
NSP FORMULATION
First Stage Parameters:
N1: number of first grade nurses
N2: number of second grade nurses
N3: number of third grade nurses
P1kj: preference cost of first grade nurse k working at shift j
P2lj: preference cost of second grade nurse l working at shift j
P3mj: preference cost of third grade nurse m working at shift j
ADjg: average demand for g th grade nurses at shift j
NSP FORMULATION
First Stage Decision Variable:
⎧1 if the first grade nurse k is assigned to shift j as a grade g nurse
xkjg = ⎨
o/w
⎩0
yljg
⎧1 if the second grade nurse l is assigned to shift j as a grade g nurse
=⎨
o/w
⎩0
⎧1
z mj = ⎨
⎩0
if the third grade nurse i is assigned
o/w
to shift
j
~
D jg
NSP FORMULATION
Second Stage Parameters:
ocg: over time cost for nurse grade type g
ucg: unsatisfied demand cost for nurse grade type g
Djg: demand realization for nurse grade type g
NSP FORMULATION
Second Stage Decision Variable:
ox kjg
⎧1
= ⎨
⎩0
if the first grade nurse i is assigned to shift j as a grade g nurse
o/w
⎧1 if the second grade nurse i is assigned to shift j as a grade g nurse
oyljg = ⎨
o/w
⎩0
oz
mj
⎧1
= ⎨
⎩0
if the third
grade
nurse i is assigned
o/w
to shift
ujg: unsatisfied demand for nurse grade type g at shift j
j
NSP FORMULATION
N1
21
3
∑∑∑
Min
k =1 j =1 g =1
P 1 kj x kjg +
N2
21
l =1
j =1 g = 2
3
∑∑∑
P 2 lj y ljg +
N3
21
∑∑
m =1 j =1
~
P 3 mj z mj + E ( f ( x , y , z , D ) )
Subject to:
N1
∑x
k =1
kj1
≥ AD j1
N1
N2
∑x
k =1
kj 2
+ ∑ ylj 2 ≥ ADj 2
N1
∑x
k =1
21
∀ j = 1,2...21
kj 3
l =1
N2
N3
l =1
m =1
+ ∑ y lj 3 + ∑ z mj ≥ AD j 3
3
∑∑ x
j =1 g =1
∀ j = 1,2...21
kjg
=6
∀ j = 1, 2 ... 21
∀ k = 1,2...N1
NSP FORMULATION
21
3
∑∑
j =1 g = 2
21
∑
j =1
y ljg = 6
∀ l = 1 , 2 ... N
z mj = 6
∀ m = 1 , 2 ... N
xkj1 + xk( j+1)1 + xk( j+2)1 + xkj2 + xk( j+1)2 + xk( j+2)2 + xkj3 + xk( j+1)3 + xk( j+2)3 ≤1
2
3
∀ k =1,2...N1
j =1, 4,7,10,13,16,19
ylj2 + yl( j+1)2 + yl( j+2)2 + ylj3 + yl( j+1)3 + yl( j+2)3 ≤1
z mj + z m ( j +1) + z m ( j + 2) ≤ 1
x kjg ∈ {0,1} ∀ k = 1,2...N 1
∀ m = 1,2...N 2
∀ j = 1,2,...21 g = 1,2,3
z mj ∈ {0,1} ∀ m = 1,2...N 3
∀ j = 1,2,...21
∀l =1,2...N2 j =1, 4, 7,10,13,16,19
j = 1, 4, 7, 10, 13, 16, 19
y mjg ∈ {0,1} ∀ m = 1,2...N 2
∀ j = 1,2,...21 g = 2,3
NSP FORMULATION
N 3 21
N1 21 3
N 2 21 3
21 3
~
E ( f ( x, D ) = Min ∑ ∑ ∑ oc1 ox kjg + ∑ ∑ ∑ oc 2 oy ljg + ∑ ∑ oc 3 oz mj + ∑ ∑ uc g u jg
k =1 j =1 g =1
l =1 j =1 g = 2
m =1 j =1
j =1 g =1
Subject to:
N1
∑x
k =1
N1
kj1
N1
∑x
k =1
kj 2
N1
∑x
k =1
kj 3
+ ∑ oxkj1 + u j1 ≥ D j1
∀ j = 1,2...21
k =1
N2
N1
N2
l =1
k =1
l =1
+ ∑ y lj 2 + ∑ ox kj 2 + ∑ oylj 2 + u j 2 ≥ D j 2
N2
N3
N1
N2
N3
l =1
m =1
k =1
l =1
m =1
+ ∑ y lj 3 + ∑ z mj + ∑ ox kj 3 + ∑ oy lj 3 + ∑ oz mj + u j 3 ≥ D j 3
∀ j = 1,2...21
∀ j = 1,2...21
NSP FORMULATION
ox kj1 + ox kj 2 + ox kj 3 − x k ( j −1)1 − x k ( j −1) 2 − x k ( j −1) 3 ≤ 0
oy lj 2 + oy lj 3 − y l ( j −1) 2 + y l ( j −1) 3 ≤ 0
oz
mj
21
− z m ( j −1) ≤ 0
3
∑∑ ox
j =1 g =1
21
3
∑∑ oy
j =1 g = 2
21
∑ oz
j =1
mj
ljg
≤1
≤1
ox kjg ∈ {0,1} ∀ k = 1,2...N 1
oz mj ∈ {0,1} ∀ m = 1,2...N 3
∀ l = 1,2...N 2
∀ m = 1 , 2 ... N
≤1
kjg
∀ k = 1,2...N 1
3
∀ j = 2, 3...21
∀ j = 2, 3...21
∀ j = 2 , 3 ... 21
∀ k = 1,2...N1
∀ l = 1,2...N 2
∀ m = 1,2...N 3
∀ j = 1,2,...21 g = 1,2,3
∀ j = 1,2,...21
oy mjg ∈ {0,1} ∀ m = 1,2...N 2
∀ j = 1,2,...21 g = 2,3
u jg ≥ 0, int ∀ j = 1, 2...21
g = 1,2,3
COMPUTATIONAL STUDY
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L2 Algorithm
Data Set
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Number of Nurses (20-25-30)
Number of Scenarios
Number of Grade one Nurses
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Computation Time
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Solve the problem using CPLEX
CONCLUSION and FUTURE STUDY
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Improve cut in L2 Algorithm
Change demand type
Add new constraints
THANKS….