LESSON LEARNED 4

European Journal of Operational Research 283 (2020) 390–403

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Innovative Applications of O.R.

Nurse scheduling with quick-response methods: Improving hospital

performance, nurse workload, and patient experience

Jan Schoenfelder a , ∗, Kurt M. Bretthauer b , P. Daniel Wright c , Edwin Coe d

a Health Care Operations/Health Information Management, Faculty of Business Administration and Economics, University of Augsburg, 86159 Augsburg,

Germany b Operations and Decision Technologies Department, Kelley School of Business, Indiana University, Bloomington, IN 47405, United States c Management and Operations Department, School of Business, Villanova University, Villanova, PA 19085, United States d CHI Franciscan Health, St. Anthony Hospital Gig Harbor, WA 98332, United States

a r t i c l e i n f o

Article history:

Received 15 June 2018

Accepted 31 October 2019

Available online 7 November 2019

Keywords:

OR in health services

Nurse scheduling

Quick-response methods

Flexibility

Mixed-integer programming

a b s t r a c t

Hospitals continue to face the challenge of providing high-quality patient care in an environment of rising

healthcare costs. In response, a great deal of attention has been given to advance planning decisions such

as nurse staffing, bed mix, scheduling, and patient flow. However, less attention has been given to incor-

porating quick-response methods in the nurse scheduling process by both anticipating and responding to

patient demand fluctuations. Therefore, in this paper, we present a model that incorporates two classes

of quick-response decisions in hospitals’ nurse scheduling: (i) adjustments to the unit assignments of

cross-trained float nurses and (ii) transfers of patients between units and off-unit admissions. Analyzing

three hospitals that are subject to different regulations with respect to patient-to-nurse ratios allows us

to draw conclusions on how these hotly debated ratios impact hospital performance, nurse workload, and

patient experience. We find that quick-response via cross-trained nurses may lead to higher total costs in

settings where an upper limit on patient-to-nurse ratios is enforced. This result has significant manage-

rial and political relevance in locations such as California. Another takeaway is that only a small number

of patient transfers or off-unit admissions provides close to the full potential benefit, thus minimizing the

negative impact on patient satisfaction and quality of care. Moreover, our proposed scheduling approach

reduces the number of undesired assigned shifts. Finally, bed and nurse capacity utilization are shown to

be important considerations when determining how and whether to use quick-response methods.

© 2019 Elsevier B.V. All rights reserved.

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1. Introduction

Public and private hospitals continue to face significant new

challenges as healthcare becomes increasingly expensive ( Keehan

et al., 2017 ) and patients become more aware and critical of the

healthcare services they receive. The combination of higher costs,

limited capacity, and changes in legislation regarding hospital

services has forced hospitals to put more emphasis on operating

efficiency ( Thompson, Nunez, Garfinkel & Dean, 2009 ) while

simultaneously maintaining or improving quality of care. One of

the often-studied, persisting problems is overcrowding. Recent

research on overcrowding discusses several negative outcomes

including ambulance diversion, patient turnaway, increased patient

∗ Corresponding author. E-mail addresses: [email protected] (J. Schoenfelder),

[email protected] (K.M. Bretthauer), [email protected] (P.D. Wright),

[email protected] (E. Coe).

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https://doi.org/10.1016/j.ejor.2019.10.047

0377-2217/© 2019 Elsevier B.V. All rights reserved.

ength-of-stay, adverse medical outcomes, patient boarding, and

ongestion in patient flow between hospital units ( Bretthauer,

eese, Pun & Coe, 2011 ; Chan, Farias, Bambos & Escobar, 2012 ;

ochran & Bharti, 2006 ; Dobson, Lee & Pinker, 2011 ; IOM, 2006 ;

hompson et al., 2009 ). In an attempt to address problems caused

y hospital overcrowding, advance-planning decisions have re-

eived a great deal of attention in the healthcare operations man-

gement and operations research literatures. However, little atten-

ion has been paid to incorporating these quick-response methods

n nurse scheduling for improving hospital performance, nurse

orking conditions, and the patient experience by anticipating and

esponding to oftentimes significant variability in patient demand.

herefore, the questions considered in this paper are: How can the

xistence of quick-response methods be considered in the nurse

cheduling process? How do differing hospital settings affect the

otential benefits provided by quick-response methods in com-

ination with nurse scheduling to reduce the negative effects of

ospital overcrowding and demand fluctuations, while providing

igh-quality patient care and consideration of nurse satisfaction?

J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403 391

Fig. 1. Hospital-wide census data and “Red Alert” days.

Fig. 2. Individual unit census data.

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We present a model that combines ideas from the nurse

cheduling and patient flow literature streams to take advantage of

flexible workforce and patient transfers between inpatient units.

hus, we fill the gap between the existing literature on quick-

esponse methods, which focuses on decisions such as optimal re-

ource sizing and allocation, cross-training levels, and patient flow

olicies, and the literature on nurse scheduling. We also provide

nsights into how an advanced scheduling process can enable ex-

sting quick-response methods to be more effective. Key findings

nd contributions to the literature include the following: (i) Quick-

esponse via cross-trained nurses will often reduce total costs, but

t can also lead to higher total costs in settings where there is

egislation enforcing limits on patient-to-nurse ratios (p-n-r), such

s in California. (ii) A very small number of patient transfers or

ff-unit admissions are necessary to gain close to the full bene-

t from this quick-response option, which is important because it

ill minimize the impact on patient experience and quality of care.

iii) Our analysis of three hospitals provides insights into the ef-

cacy of quick-response methods as a function of bed utilization

nd nurse capacity utilization. (iv) We evaluate the impact of the

mproved scheduling process on patient- and nurse-related perfor-

ance measures for different policies regarding patient-to-nurse

atios. The hospitals in our study include one medium-sized hos-

ital in California and two hospitals of similar size located in the

idwest of the United States.

To illustrate the problems associated with overcrowding and

emand fluctuations in a hospital, Fig. 1 provides sample census

ata from the aforementioned hospital in California. At this hos-

ital, when the total number of occupied inpatient beds reaches

ritical levels, for example, above 250, they may go on hospital-

ide “red alert.” In response, hospital managers divert ambulances

o other hospitals, cancel or postpone elective procedures, and ex-

edite discharges and patient transfers.

At a more detailed level, Fig. 2 provides census data for four

articular inpatient units at the same hospital: transplant, oncol-

gy/neurology, surgery, and a small medical unit. This unit level

ata illustrates four important and typical hospital characteristics:

1) the variability in census differs between units, (2) one unit

surgery) reaches maximum occupancy more frequently than the

ther units, (3) these units do not necessarily reach maximum ca-

acity at the same time, and (4) all units may not necessarily be

ull when the hospital goes on red alert. Given these observations,

uick-response methods provide a promising approach to deal with

vercrowding, red alert days, and the potentially high variability

n patient demand. Quick-response methods offer the ability to re-

ct to day-to-day demand fluctuations. We show that incorporating

uick-response decisions in nurse scheduling yields notable bene-

ts. In this paper, we present a model that combines initial nurse

chedules with two classes of quick-response decisions: (1) short-

erm adjustments to the number of nurses working a given shift in

ach unit, and (2) transfers of a limited number of qualifying pa-

ients between units and use of off-unit admissions (admissions to

n alternate non-first choice unit).

In addition to the previously mentioned challenges, the gap be-

ween unfilled nurse positions and the number of available trained

urses has led to a significant nursing shortage ( Buerhaus, Auer-

ach & Staiger, 2009 ). It has been shown that higher patient-

o-nurse ratios diminish perceived service quality and possibly

esults in harmful delays of patient treatment ( Aiken, Clarke,

loane, Sochalski & Silber, 2002 ; Driscoll et al., 2018 ; Haraden

Resar, 2004 ). Moreover, high patient-to-nurse ratios have been

inked to increasing nurse dissatisfaction and higher turnover rates

392 J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403

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( Aiken et al., 2002 ). Recently, an increasing number of states in

the United States have adopted or are considering legislation that

imposes mandatory patient-to-nurse ratios, requiring managers to

always have a sufficient number of nurses on staff. One of the

leading states in this development was California. The California

Health and Safety Code § 1276.4 set patient-to-nurse ratios for

nurses in California hospitals in 2004. Other countries have taken

action to tackle the nursing shortage as well. For example, Ger-

many is currently implementing new legislation that also puts lim-

its on patient-to-nurse ratios, which will be enforced starting in

May 2019. Consequently, hospital managers must put significant

emphasis on nurse scheduling procedures that meet staffing re-

quirements, consider nurse preferences to ensure workforce satis-

faction, and help attract new workers to the nursing profession.

Therefore, our model takes into account shift preferences, stochas-

tic patient demand, and the potential use of quick-response meth-

ods that have an immediate impact on the nurse workload and

staffing allocation in the units.

The remainder of the paper is organized as follows. We dis-

cuss the relevant literature and our contribution in Section 2 .

Section 3 presents the problem description and the developed

quick-response model. Section 4 describes the data of the hospi-

tals that participated in our study. Section 5 presents and discusses

the results, while Section 6 addresses our model validation via a

heuristic simulation. Section 7 provides concluding remarks and

opportunities for future research.

2. Healthcare literature

There are three streams of literature that are of immediate rel-

evance to this research. We build on and extend previous work

from the nurse scheduling, quick-response methods, and workforce

flexibility literature. Additionally, we apply insights from and de-

rive additional findings to the literature on patient-to-nurse ratios

and quality of care ( Aiken et al., 2002 ; Aiken, Xue, Clarke & Sloane,

2007 ).

2.1. Nurse staffing and scheduling literature

Nurse staffing and scheduling have been studied extensively

over the last couple of decades. The body of work can be

grouped into four categories of nurse planning: nurse budget-

ing, scheduling, rescheduling, and nurse-to-patient assignment

( Punnakitikashem, Rosenberger & Behan, 2008 ). Our investigated

problem falls in the categories scheduling and rescheduling of

nurses. For detailed literature surveys on all aspects of medical

staff rostering problems, please refer to Burke, De Causmaecker,

Berghe and Van Landeghem (2004) , Cheang, Li, Lim and Rodrigues

(2003) , and Ernst, Jiang, Krishnamoorthy, Owens and Sier (2004) .

Van den Bergh, Beliën, De Bruecker, Demeulemeester and De

Boeck (2013) , in their review of personnel scheduling, also dis-

cuss several nurse rostering papers. In a recent literature review,

Defraeye and Van Nieuwenhuyse (2016) offer a complete overview

of staffing and scheduling publications from 1991 to 2013 that in-

clude simulation-based performance evaluation. De Causmaecker

and Berghe (2011) provide a framework to categorize nurse ros-

tering problems according to the three categories personnel envi-

ronment, work characteristics, and optimization objective.

Recently, researchers have focused on a variety of techniques

to help managers create nurse schedules that are not necessar-

ily cost-optimal, but also take other managerial goals into con-

sideration, e.g., operating room scheduling ( Beliën & Demeule-

meester, 2008 ) and shift auctions ( De Grano, Medeiros & Eitel,

2009 ). Punnakitikashem et al. (2008) use a two-stage stochas-

tic programming model to help balance nurses’ workloads when

dealing with patient condition uncertainty. Wright, Bretthauer and

ôté (2006) study the implications of mandatory staffing ratios and

heir effect on nursing workforce management decisions and per-

ormance measures. Our model incorporates target staffing ratios

hat can be considered either mandatory or a managerial guide-

ine that can occasionally be violated, depending on legislative cir-

umstances. White, Froehle and Klassen (2011) study the effects of

ifferent patient scheduling policies in combination with capacity

nd patient flow scheduling decisions on patient waiting and treat-

ents times in outpatient clinics by means of a simulation model.

hen, Lin and Peng (2016) apply a two-stage goal programming ap-

roach to determine the smallest possible medical staff size and

ubsequently create the most desirable schedule under uncertainty.

agheri, Devin and Izanloo (2016) consider nurse assignment re-

ourse decisions – in the sense that additional nurses may be

dded to a shift on short notice – with a focus on cost mini-

ization in a single medical department. Their goal is to present

he modeling approach without deriving managerial insights. Kim

nd Mehrotra (2015) study a two-stage problem, as we do, that

nvolves staffing decisions in the first stage. In the second stage,

owever, they model the selection of weekly patterns under un-

ertain demand, whereas we extend previous research by focusing

ur attention on incorporating quick-response methods (flexible

urse assignments, patient transfers) into the scheduling decision,

s discussed below. Fügener, Pahr and Brunner (2018) consider the

cheduling of nurses that are to be cross-trained in multiple hospi-

al units over a longer time horizon to study the effects on cross-

raining intensity and continuity of care. They find that higher de-

rees of cross-training lead to improved levels of understaffing and

vertime. Recently, the results of the Second International Nurse

ostering Competition were presented in Ceschia, Dang, De Caus-

aecker, Haspeslagh and Schaerf (2019) , which asked participants

o find efficient algorithms that solve a multi-stage nurse ros-

ering problem. In contrast to our model, the multi-stage prob-

em referred to a longer planning horizon consisting of multiple

onsecutive weeks, between which some information was carried

ver. The problem neither included stochastic demand informa-

ion nor modeled multiple stages within a single scheduling time

orizon. Surprisingly, the most efficient solution algorithms were

ased on mixed integer linear programming techniques rather than

etaheuristics – the champions of the previous nurse rostering

ompetition.

.2. Quick-response literature

In service processes in general, and in healthcare services in

articular, mismatches between supply and demand prove very

ostly. Therefore, researchers and hospital managers alike have be-

un identifying ways to react to short-term demand fluctuations.

e incorporate two quick-response methods: patient transfers /

ff-unit admissions and flexible nurse assignments.

.2.1. Patient transfer and admission literature

Thompson et al. (2009) investigate the financial impact of

roactive patient transfers between hospital units as well as the

ffect it has on quality of care measures such as patient wait time

nd bed availability. They treat the number and type of staff of the

ospital as given and assume that target staffing ratios can always

e satisfied in the short-run, whereas we model the number of

vailable staff as a decision that, as we show, should be influenced

y possible patient transfers.

Dobson et al. (2011) focus on the discharge process at a single

CU unit. They develop a Markov chain model that enables them

o keep track of individual patients. Like Thompson et al. (2009) ,

hey assume that a hospital will have enough staff available at all

imes to take care of the number of beds, which represents their

nly capacity measurement. Our work considers proactive patient

J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403 393

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ransfers from a more aggregate perspective, as our focus lies on

ursing schedules for multiple units that foresee potential patient

ransferring, which might or might not occur during the scheduling

orizon depending on the demand realizations.

Batista, Vera and Pozo (2019) develop a two-stage model in one

f the most recent publications that study the multi-objective ad-

ission planning problem, which aims to make optimal decisions

ith regards to the mix of admitted patients under considera-

ion of demand and resource availability uncertainty. The proposed

odel also considers a finite set of possible scenarios but is lim-

ted to two stages. One of their key takeaways is the analysis of

he trade-off between the deviation of resource utilization from a

iven target level and the cost of service.

.2.2. Quick-Response nurse assignment literature

Nurse schedules are typically made several weeks in advance

o allow for planning reliability for the nurses. Demand variabil-

ty will often cause imbalances in supply and demand. Bard and

urnomo (2005a and 2005b ) develop a methodology for reac-

ively assigning nurses on a shift-by-shift basis. They treat the

nitial schedule as given and focus on daily short-term decisions.

ur model anticipates the possibility of quick-response decisions

hen setting up the initial schedule. Since then, nurse reschedul-

ng/rerostering has seen a spike in research interest. Bäumelt,

vo ̌rák, Šůcha and Hanzálek (2016) , for example, focus on solver

fficiency when designing a parallel algorithm solution approach

o the nurse rerostering problem.

The workplace assignment of scheduled flexible nurses is the

uick-response decisions under consideration in Campbell and Di-

by (2002) . They provide a general problem formulation and a

olution heuristic for the assignment of cross-trained workers to

ultiple departments at the beginning of a shift, which can be ap-

lied to a hospital setting.

Wright and Bretthauer (2010) extend the model from Wright et

l. (2006) to handle quick-response nurse reassignments as well as

oordination among various sources of nurse capacity. We build on

heir work and extend it in a number of ways. Most importantly,

e incorporate quick-response methods such as flexible nurses and

atient transfers in response to patient demand fluctuation when

eveloping the initial nurse schedule, while they treat the alloca-

ion and the adjustment decisions separately and do not consider

atient transfers.

.3. Workforce flexibility literature

The previous subsection already discussed a few flexible work-

orce papers where quick-response methods are an important fac-

or. Here, we discuss other studies in the workforce flexibility liter-

ture. Please refer to the paper by Easton (2011) on cross-training

erformance in different settings of scheduling flexibility for a re-

ent extensive review on workforce flexibility.

Gnanlet and Gilland (2009 and 2014 ) study sequential versus

imultaneous decision making with respect to the optimal number

f beds, nurses, and patient upgrades. They provide insights into

otential benefits that may be gained by employing cross-trained

urses and patient upgrades on a tactical level, as they determine

esource levels required to meet stochastic demand at minimum

ost. Their work does not capture the effects that come into play

hen individual nurses are scheduled and assigned to medical

nits over a given time horizon. These effects include, for example,

arget patient-to-nurse ratios in each shift, nurse availability and

hift preferences, worktime regulations, and patient length of stays

pon arrival. In our work, we incorporate these features that are

mportant on an operational level to focus on the interaction be-

ween nurse scheduling and two quick-response decisions: patient

ransfers and flexible nurse assignments.

Easton (2011) finds that “scheduling flexibility may be an im-

ortant cofactor for exploiting the benefits of cross-training in la-

or scheduling environments” in his study of the performance of

ifferent cross-training policies in service operations using an inte-

rated staffing, cross-training, scheduling, and allocation model. He

ighlights the necessity to incorporate workforce flexibility in the

cheduling process, particularly in service environments that oper-

te continuously, which is one of our contributions to the existing

iterature.

The potential benefits of using cross-trained nurses in hospitals

ave been described by several researchers (e.g., Altimier, 1995 ,

enny, Gapas & Hilton, 1995 ), and the particular value of “inte-

rated staffing-scheduling-allocation models designed for hospi-

al environments with limited nurse labor availability” for future

esearch was identified by Brusco, Futch and Showalter (1993) .

ampbell (1999) provides an analytical model to investigate the

enefit of cross-training and cross-utilization of nurses.

Hur, Mabert and Bretthauer (2004) address real-time control

ecisions in workforce scheduling that allow managers to react to

hort term mismatches between supply and demand in service op-

rations, arising mainly from demand variability and workforce ab-

ence or tardiness. While they do not focus on nursing or use the

dea of a float pool of cross-trained workers, their approach could

e modified to apply to other types of nurse schedule adjustments.

.4. Contribution to the literature

In summary, quick-response methods and workforce flexibil-

ty have been covered to some extent in the respective literature

treams. However, our work is the first to focus on the scheduling

f a mix of unit and cross-trained float nurses with the additional

ption to transfer patients between units under uncertain patient

emand in different hospital settings. Using actual data from three

ifferent hospitals in our study enables us to draw new insights

nto how much of the theoretical benefits from quick-response

ethods can be reaped when employing them in a practical set-

ing. We show that the developed model is capable of providing

ignificant benefits over the current scheduling practice, which ig-

ores the availability of quick-response, in each of the considered

ettings. These benefits include improvements especially regarding

atient experience and nurse workload performance measures. The

ollaboration with three different hospitals allows us to identify

ases where the inclusion of float nurses in the workforce actually

urts the overall hospital performance, contrary to previous find-

ngs in the literature. Finally, we provide interesting results with

egards to the relationship between resource utilization and the ef-

ectiveness of float nurses that hold true in all hospital settings.

. The nurse scheduling and quick-response model

We tackle the problem of optimally scheduling nurses in three

ifferent medical units. Such nurse schedules are typically devel-

ped some time, e.g., one month, in advance. Each unit is staffed

ith so-called unit nurses who are solely responsible for treating

atients in their respective unit. Additionally, a pool of float nurses

xists. These float nurses are cross-trained to work in either two

r three of the units.

In our interactions with the responsible schedulers, we learned

hat the schedules are currently created manually and only

hecked for violations of worktime regulations and minimum

taffing levels once entered into the administrative hospital soft-

are. The minimum staffing levels are derived by dividing the

xpected bed demand levels by the (target) patient-to-nurse ratio.

hile float nurses can be reassigned on short notice, usually

uring a meeting of the unit managers on the morning of the

orkday, they are typically pre-assigned to a specific unit in

394 J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403

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the original schedule to hit the respective target staffing level.

Once the schedule is released, unit nurses know the shifts and

units they are scheduled to work during the entire time horizon,

whereas float nurses know their assigned shifts but not necessarily

the unit they will be working in. If reassigning float nurses to units

with higher than expected occupancy is not sufficient to cover

demand in all units, managers have the options to a) transfer

patients between units, b) admit new patients to units they were

not initially intended to be placed, or c) turn away new patients.

The current scheduling practice, which we denote as the No

Foresight policy, has some obvious shortcomings. First, information

about the patient demand distribution outside of the mean is ig-

nored. Second, float nurses are initially treated like unit nurses in

the scheduling process. The possibility to assign float nurses to dif-

ferent units based on observed bed occupancy on the morning of

their workday is overlooked. Third, potential patient transfers be-

tween units, off-unit admissions, and patient turnaways are also

not taken into account.

Given the fact that demand distribution information is available

and quick-response methods have been implemented, the schedule

should be constructed in a way that anticipates and takes advan-

tage of demand distribution information and the different quick-

response options. Therefore, we formulate a multi-stage stochas-

tic programming model to improve the current scheduling process

and address each shortcoming. Information about patient demand

distribution is incorporated in the form of demand scenarios. Float

nurses are assigned to shifts, in which they may be assigned to

different units depending on the observed bed occupancy. Poten-

tial patient transfers, off-unit admissions, and turnaways are taken

into account as well, thus creating a more robust schedule. In the

following, we will call our proposed scheduling logic the Full Fore-

sight policy.

3.1. Model formulation – multi-stage stochastic program

We define the following notation that is used in our model.

Subscripts and Superscripts

i - nurse i

j - shift j

d - unit d

s - stage s

Sets

T - the set of stages when random variable realizations are

observed

N - the set of all nurses

N d - the set of all nurses capable of working in unit d

S s - the set of all shifts in stage s

S A i - the set of shifts that nurse i is available to work

D - the set of all hospital units

D i - the set of units in which nurse i can work

Stage 0 Decision Variables

x i j - 1 if nurse i works shift j , else 0

y i - number of overtime shifts assigned to nurse i

Recourse Decision Variables (Stages 1,…, ψ ) v i jd - 1 if nurse i is assigned to unit d in shift j, else 0 q j d d ′ - number of patients reallocated from unit d to unit d ’

( d ′ � = d ) in shift j n jd - number of patients turned away from unit d in shift j

when capacity does not allow taking in new patients

w jd - number of patients in unit d in shift j after patient

transfer and turning away patients

z jd - violation of constraint ( 11 ) in number of patients (vi-

olation of target patient-to-nurse ratio) in shift j in unit

d

arameters

eds d - maximum number of beds available to patients in unit

d

F j d d ′ - maximum number of patients allowed to be transferred from unit d to unit d ’ in shift j

V jd - maximum violation of patient-to-nurse ratio per nurse

in unit d in shift j

S i - maximum number of regular time shifts for nurse i

S i - minimum number of regular time shifts for nurse i

i j - 1 if nurse i prefers not to work shift j, else 0

S i - upper limit on the number of undesirable shifts as-

signed to nurse i

S i - upper limit on the number of overtime shifts assigned

to nurse i

i - regular time wage paid to nurse i per shift

ot - multiplier for each shift worked overtime (e.g., c ot = 0 . 5 for time and a half overtime)

t f

j d d ′ - cost of a patient transfer from unit d to unit d ’ in shift j

jd - cost per violation of target patient-to-nurse ratio in

shift j in unit d

jd - penalty cost for turning away a patient in shift j in unit

d

LO S d - average length of stay in unit d, rounded to the nearest

integer, in shifts

LO S d - a fraction of the average length of stay in unit d,

rounded to the nearest integer, in shifts

d - patient-to-nurse ratio in unit d

jd - the total number of patients that attempt to occupy a

bed in unit d during shift j before reallocating or turning

away patients (a random variable)

s (ω) - realization of patient demand in stage s

Also, let x and y denote appropriately dimensioned vectors of x i j nd y i . The multi-stage stochastic programming model for nurse

cheduling and quick-response decisions can be formulated as fol-

ows:

in

( ∑ i ∈ N

∑ j∈ S A i

c i x i j + ∑ i ∈ N

c ot c i y i

)

+ E [ f ( x, y, ζ1 ( ω ) ) + E [ f ( x, y, ζ2 ( ω ) ) + . . . E [ f ( x, y, ζn ( ω ) ) ] . . . ] ] (1)

ubject to

S i ≤ ∑ j∈ S A i

x i j ≤ RS i + y i i ∈ N (2)

i ≤ OS i i ∈ N (3) ∑ j∈ S A i

a i j x i j ≤ U S i i ∈ N (4)

i j + x i ( j+1 ) + x i ( j+2 ) ≤ 1 i ∈ N, j ∈ S A i (5)

i j ≥ 0 and binary i ∈ N, j ∈ S A i (6)

i ≥ 0 and integer i ∈ N (7) The function f ( x, y, ζs (ω) ) is the optimal value of the stage s

ecourse problem defined as follows:

f ( x, y, ζs ( ω ) ) = min ∑ j∈ S s

∑ d∈ D

g jd z jd + ∑ j∈ S s

∑ d∈ D

m jd n jd

+ ∑ j∈ S s

∑ d∈ D

∑ d ′ ∈ D

c t f

j d d ′ q j d d ′ (8)

J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403 395

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ubject to

jd = ω jd + ∑ d ′ ∈ D

q j d ′ d − ∑ d ′ ∈ D

q j d d ′ − n jd

− ∑ d ′ ∈ D

F LO S d −1 , f < j ∑ f =1

q ( j− f ) d d ′ j ∈ S s , d ∈ D (9)

jd ≤ Beds d j ∈ S s , d ∈ D (10)

d

∑ i ∈ N d

v i jd + z jd ≥ w jd j ∈ S s , d ∈ D (11)

V jd ∑ i ∈ N d

v i jd ≥ z jd j ∈ S s , d ∈ D (12)

∑ ∈ D i

v i jd = x i j i ∈ N, j ∈ S A i ∩ S s (13)

≤ q j d d ′ ≤ T F j d d ′ d ∈ D, d ′ ∈ D, j ∈ S s , d � = d ′ (14)

i jd ≥ 0 and binary i ∈ N, j ∈ S A i ∩ S s , d ∈ D i (15)

jd , n jd , w jd ≥ 0 and integer j ∈ S s , d ∈ D (16)

j d ′ d ≥ 0 and integer j ∈ S s , d ∈ D, d ′ ∈ D, d � = d ′ (17) The objective function ( 1 ) minimizes the combined cost of reg-

lar and overtime shifts assigned to nurses plus the expected value

f penalty costs for understaffing, turning away patients, and pa-

ient transfers. Eq. (8) defines the expected penalty costs. Con-

traints ( 2 ) and ( 3 ) restrict the number of regular and overtime

hifts that nurses are allowed to work during the scheduled time

orizon. To ensure that no nurse is assigned more undesired shifts

han he or she is willing to work, we include constraints ( 4 ). Con-

traints ( 5 ) limits the number of shifts that each nurse is allowed

o work to one out of three consecutive shifts. Eq. (8) is the ob-

ective function of the stage s recourse problem that minimizes

he expected costs of understaffing, turning away patients, and pa-

ient transfers. Eq. ( 9 ) are patient flow balance constraints. Here,

e capture the effect that patient transfers and patient turnaways

ave on the number of patients in each unit. We assume that

ransferred patients remain in their new unit for a fraction of the

verage length of stay, as patient transfers are mostly performed

n patients that are well on their way to being ready for discharge.

ence, the effect of the transfer on the number of expected pa-

ients in the involved units is captured for future shifts. If the num-

er of patients assigned to a unit exceeds the number of beds or

urse capacity in a unit, we have to turn away excessive patients.

he actual number of patients in each unit must not exceed the

umber of available beds in the unit, which is captured in con-

traints ( 10 ). Constraints ( 11 ) enforce the per-shift staffing require-

ent as determined by the patient-to-nurse ratio r d , which can be

iolated to a limited extent with an attached penalty cost. We de-

ne the allowed maximum violation of the target patient-to-nurse

atio per shift in each unit in constraints ( 12 ). Each nurse can only

e assigned to one unit per scenario, and only in shifts they are

ssigned to in the initial schedule ( x i j = 1) , as modeled in con- traints ( 13 ). We limit the number of possible patient transfers in

onstraints ( 14 ).

Patient demand parameters are modeled after historically ob-

erved demand in the respective units. Throughout the week, de-

and patterns in all three hospitals and all units show season-

lity regarding daytime and weekday/weekend. For example, bed

ccupation during weekend shifts is typically lower than during

he week, and night shifts show lower bed occupancy than day

hifts. In our model, we decide on an initial shift schedule for a

iven time horizon, in this case one week, well in advance (e.g. a

onth in advance) to stay consistent with current practice. Then,

hat one-week time horizon is split into n smaller time intervals,

ach called a stage s . Each stage spans over the shifts S s . If only

single stage is modeled, it contains every shift in the schedul-

ng horizon. The other extreme would be to assign a single shift to

ach stage. We denote the realization of patient demand in stage

as a function of the random event ω and the overall demand cenario over the entire horizon as ( ζ1 (ω) , ζ2 (ω) , . . . , ζn (ω) ) . Note hat in any given stage s ≥ 1 the information up to and includ- ng stage s is known. At the beginning of each stage, we ob-

erve the demand realization and make decisions on how to use

ur quick-response methods – assigning float nurses and trans-

erring patients. In the model, demand realizations can be higher

han, lower than, or equal to the observed mean demand of a

articular shift. As long as multiple shifts are contained in a sin-

le stage, a demand realization is assumed to impact patient de-

and in every shift within the stage. For example, in a demand

cenario “medium”/”medium”/“high” for three stages, patient de-

and would be equal to the shifts’ mean demand in all shifts up

o stage three. In all shifts within stage 3, patient demand exceeds

heir mean demand. Note that the recourse decisions in each shift

epend on the demand realization in the stage that the respec-

ive shift is part of. There is a tradeoff between information accu-

acy and computational solvability. Splitting the time horizon into

ore intervals results in more frequently updated demand obser-

ations and better decision making, but the number of possible

tates grows exponentially with each added interval, so that the

esulting optimization problem becomes harder to solve to opti-

ality. In our study presented in Section 5 , we use three stages

ith three possible demand scenarios each. The first stage contains

he shifts on Monday and Tuesday, the second stage the ones on

ednesday and Thursday, and the third stage spans from Friday to

unday.

.2. Generating No Foresight policy schedules

As one part of our study is to determine the improvements of-

ered by the proposed Full Foresight policy over the current No

oresight scheduling process, it is necessary to adapt the presented

odel in the following ways to generate No Foresight schedules.

First, we solve the model with a single scenario, in which de-

and is equal to the historical mean, to obtain the stage-0 variable

alues, the initial schedule. Then, we extract the stage-0 variable

olution values and treat them as fixed input parameters in a sec-

nd step, where we solve the model again with all the demand in-

ormation to obtain the best possible recourse decisions based on

he No Foresight schedule. It is important to note that under both

he Full Foresight policy and the No Foresight policy, we always

llow the decision maker to take quick-response actions once de-

and is realized. Hence, dynamic float nurse assignments and pa-

ient transfers are always carried out, whether they were initially

nticipated or not. This resembles the current scheduling process

hat we observed in practice.

Since the multi-stage stochastic programming model under No

oresight policies is less complex than under Full Foresight, it can

ypically be solved very close to optimality in seconds. Therefore,

ll results for No Foresight are within 0.1% of optimality. Under

ull Foresight, the results are solved to within 2% of optimality

o ensure that computation times stay within 10 min per prob-

em. Hence, the reported expected benefit from Full Foresight can

e viewed as a lower bound on the actual benefit that it pro-

ides.

396 J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403

Table 1

Summary of hospital data.

MidWest-A

(voluntary p-n-r)

MidWest-B

(voluntary p-n-r)

WestCoast

(mandatory p-n-r)

Number of Beds

(Medical/Surgical/OrthoNeuro)

39/33/ 37 33/26/29 33/44/26

Mean Census

(Medical/Surgical/OrthoNeuro)

33/28/32 28/22/25 28/35/21

Rounded Standard Deviation of Census

(Med./Surg./OrthoNeuro)

3/3/2 2/2/2 2/3/2

Percentage of Shifts at Full Bed Capacity

(Medical/Surgical/OrthoNeuro)

11%/12%/9% 12%/14%/10% 13%/8%/7%

Patient-per-Nurse Ratios ∗

(Medical/Surgical/OrthoNeuro)

6/6/6 5/5/5 5/5/5

Average Nurse Utilization at Mean

Demand and Target Patient-to-Nurse Ratio

89% 92% 82%

Mean Nurse Wage $29 $30 $42

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3.3. Deterministic equivalent formulation

We assume that there are K finitely many different possible

realizations that patient demand ω k jd

can take on in each shift j

in unit d – one for each scenario k . Weighing model accuracy and

solvability, we use three possible demand outcomes at each deci-

sion point in time. Medium demand levels in any given shift corre-

spond to the mean demand historically observed at the hospitals.

Low (and high) demand levels are assumed to be one standard de-

viation below (above) the mean demand. In our sensitivity analy-

sis, we study the implications of changes in demand with respect

to the mean demand levels and the variability in demand, as ex-

plained in Appendix B. Hence, we assume a discrete distribution

associated with the random demand, and each possible demand

realization is associated with some probability p k . The determin-

istic equivalent converts the problem into a larger linear integer

optimization problem. Please refer to the Online Appendix A for a

detailed presentation of the deterministic equivalent version of the

problem.

3.4. Minimizing the number of undesired shifts

While our model includes an upper limit on the number of as-

signed undesired shifts for each nurse in constraints ( 4 ), they are

not part of the objective function and consequently not minimized.

However, after an optimal schedule is found, we can replace the

objective function ( 1 ) with the left hand side of constraints ( 4 ) to

minimize the number of assigned undesired shifts and force the

new schedule not to exceed the original objective value by adding

a constraint that sets an upper limit on the resulting value from

( 1 ) equal to the solution value of the first run. Thus, we find the

single schedule among the ones that optimize the original cost-

oriented goal that results in the fewest assigned undesired shifts

in an attempt to improve nurse experience.

4. Hospital data and test problems

Here we describe the hospital data available to us and the ex-

perimental design we use to explore characteristics of the individ-

ual quick-response methods. Note that our goal is to understand

how different hospital settings affect the performance of the pro-

posed Full Foresight nurse scheduling process and the analyzed

quick-response methods, rather than to perform a case study and

make recommendations for a specific hospital.

4.1. Hospital data

We obtained data from three different hospitals. The data avail-

able to us for this study include nursing information and bed cen-

us information. Two of the hospitals are located in the Midwest

.S., which we refer to as “Midwest-A” and “Midwest-B”, and the

hird is on the West Coast of the U.S (referred to as “WestCoast”).

ll of them are representative of medium-sized acute care hospi-

als. Below, we discuss the characteristics of and differences be-

ween the hospitals.

One of the most important differences between the hospitals

ies in the nature of how patient-to-nurse ratios are enforced. At

estCoast, state legislation limits the patient-to-nurse ratios to a

aximum of 5 in all considered medical departments. Thus, con-

traints ( 11 ) from the model become hard constraints with z jd al-

ays equal to zero. At the MidWest hospitals A and B, patient-to-

urse ratios can be considered managerial guidelines, and exceed-

ng them is not forbidden. Interestingly, these hospitals differ with

egards to their target ratios, with MidWest-A setting the target to

, whereas management at MidWest-B aims for a ratio of 5. This

lso means that the number of nurses in the workforce pool in

oth MidWest hospitals is quite similar, even though MidWest-A

as higher mean census values across all units.

Through site visits with unit directors and nursing managers,

e obtained information regarding nurses’ availabilities, schedul-

ng preferences, and wages, as well as historical bed census,

atient-to-nurse ratios, shift length and times, and the number of

vailable beds for three acute care units: Medical, Surgical, and

rthopedics-Neurosurgery (OrthoNeuro) as described in Table 1 .

ote that detailed and extensive historical patient flow and cen-

us data was not available.

All three hospitals employ both part-time and full-time nurses.

urses are considered full-time when they work at least 40 hours

er week. Part-time nurses can be hired to work up to 40 hours

er week. Nurses are typically hired to work specific shifts (day,

vening, or night), but some are available in other shifts if needed.

owever, since scheduling them in a different shift than they were

ired for can pose an inconvenience, we consider the shifts that

urses are available but were not hired for, as “undesired” shifts.

e study the scheduling process over a one week time horizon

nd consider 8-hour shifts corresponding to the day, evening, and

ight shifts.

Nurse wages are primarily based on experience levels. For as-

igned overtime shifts, a 50% premium is added to the base rate.

loat nurses are paid a 20% premium on average. Whereas average

ages in both Midwest hospitals are comparable, we encountered

ignificantly higher average wages in the West Coast hospital.

Comparing bed census data versus the number of available

eds, we observe a noticeable difference in utilized bed capacity

etween the three hospitals at the mean demand level. Hospital

idwest-B operates at full capacity relatively often. Midwest-A, on

he other hand, faces full units less frequently. The West Coast hos-

ital has the lowest average bed capacity utilization of the three

J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403 397

Table 2

Factor variations in the full experiment.

Factor Levels considered

Patient Count Low, Medium, High, Very High

Transfer Limit 0, 1, 2, 4

Demand Variability Low, High

Nurse Pool Composition 100% Unit Nurses and 0% Float Nurses,

92%/8%, 83%/17%, 75%/25%, and 67%/33%

Foresight Level No Foresight, Full Foresight

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ospitals. Notice also that the WestCoast hospital’s surgical unit

as the largest gap between the mean census and the number of

eds. This is most likely a result of the high variability in demand

n the surgical unit (compare Fig. 2 ), which shows fairly frequent

pikes in demand even though the mean demand is relatively low

ompared to the available beds. Note that for simplicity, we have

ombined the medical and transplant units into one unit called

Medical” at the WestCoast hospital.

In the West Coast hospital, understaffing – assigning additional

atients per nurse on top of the mandated patient-to-nurse ratio

is not allowed via state legislation. Consequently, whenever pa-

ient demand reaches the scheduled nurse capacity in a unit, the

nit is considered full and has to start turning patients away. In

nticipation of this fact, staffing levels at the WestCoast hospital

re higher relative to the number of patients than in the Midwest

ospitals, where target patient-to-nurse ratios are set by hospital

anagement and may be violated temporarily without immedi-

te legal consequences. We observe lower average nurse utiliza-

ion, which we define as the nurse utilization at mean demand and

arget patient-to-nurse ratio, in the WestCoast hospital than in the

idwest hospitals. This can be at least partially attributed to the

ifferences in legislation between the states.

.2. Experimental design

We consider two different measures that govern the units’ ca-

acity levels in our model. One derives from the set of employed

urses, including their preferences and availabilities. The number

f nurses that are available for scheduling in each shift and unit

s a long-term decision. Therefore, it is a fixed input in our model.

ow much of this available capacity is actually scheduled is up to

he decision maker. The other capacity constraint is the given num-

er of beds in each unit, which in our experiments is a fixed input

s it also constitutes a strategic long-term decision. If the number

f beds cannot meet demand even after transferring patients, this

eads to turning away patients and lost revenue, which we penal-

ze in the objective function. If the number of nurses is insufficient

or a given number of patients, we can allow the patients to en-

er the hospital and incur penalties for understaffing (except in the

est Coast hospital, where this is not an option), or the hospital

an turn patients away and face lost revenue as well as associated

otential adverse medical outcomes. As long as the penalty for un-

erstaffing is lower than the patient turnaway penalty, the model

ill choose understaffing over turning away patients up to an up-

er bound on understaffing.

In our experimental design given in Table 2 , we use patient and

ed counts derived from the information available to us at each

ospital. We then consider four levels of demand (low, medium,

igh, very high) for each hospital, which determine how high the

xpected demand is in each period. On the medium demand level,

xpected patient demand is set equal to the reported mean census

t each hospital in each period, thus exhibiting realistic weekday

nd daytime seasonalities. High (7.5% increase) and very high (15%

ncrease) levels are representative of expected demand that is ob-

erved in busier times (e.g. flu season), while the low level (7.5%

ecrease) represents demand during stretches of lower patient oc-

upancy. Then, in the stochastic model, there exist three possi-

le demand realizations in each period; one equal to, one above,

nd one below the expected demand level. How far the realiza-

ions may deviate from the mean is determined by the demand

ariability. Low demand variability is set at one standard devia-

ion above and below the mean; high demand variability is two

tandard deviations above and below the mean. Demand variabil-

ty differs only slightly between the morning, evening, and night

hifts in the different units at each respective hospital.

We impose four different limits (0, 1, 2, and 4) on the allowed

umber of patients that can be reallocated each shift per unit. No

atter the limit, patient transfers are never allowed during night

hifts, as it is not common practice in hospitals to relocate patients

uring the night.

We compare two different levels of foresight: No Foresight and

ull Foresight. Under the No Foresight policy, the decision maker

gnores demand distribution information. Thus, the range of quick-

esponse float nurse assignments and patient transfers that would

e optimal in different demand outcomes is ignored when the

nitial scheduling decisions are made. This approach is compara-

le to current practice at all three hospitals in our study. Under

ull Foresight, demand distribution information and the resulting

oat nurse assignments and possible patient transfers in each sce-

ario are taken into consideration. It is important to note that float

ssignments and patient transfers are still performed under both

oresight policies. The difference is that the No Foresight policy

eads to initial scheduling decisions that only depend on expected

emand levels, where only a single choice for float nurse assign-

ents and patient transfers is initially anticipated.

Finally, we vary the composition of the overall pool of nurses.

hile some nurses are hired as unit nurses, the others work as

oat nurses whose skill set allows them to work in either two or

ll three of the units, depending on the individual nurse. In our ex-

eriments, we assume that half of the float nurses in a pool are ca-

able of working in all three units. The remaining float nurses are

ivided as evenly as possible into groups that cannot be assigned

o one of the three units. We investigate five different Nurse Pool

ompositions: 100% unit nurses / 0% float nurses, 92% / 8%, 83% /

7%, 75% / 25%, and 67% / 33%.

Overall, we generate 320 problems per hospital for a total of

60 test problems.

. Results

In this section, we discuss our findings. In particular, we inves-

igate the value of anticipating patient demand distribution infor-

ation in quick-response decisions in the nurse scheduling process

nd identify the factors that determine how beneficial this fore-

ight is in different settings. This analysis leads to insights on when

nd how quick-response should be used. As the results of numer-

cal experiments are naturally sensitive to changes in parameter

alues, we refer the reader to Appendix B for a discussion on the

hosen parameter values for turnaways, understaffing, and patient

ransfers and a sensitivity analysis.

.1. The impact of foresight on hospital performance

Averaging over all three hospitals, using Full Foresight yields a

.9% reduction of the expected overall cost (see Table 3 ) versus a

raditional scheduling approach of No Foresight. The cost reduction

s mainly due to better anticipation of patient fluctuation and po-

ential quick-response decisions which results in an initial schedule

hat is better suited to deal with demand variability. We find that

ull Foresight typically results in slightly higher overall staffing

398 J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403

Table 3

Objective value, scheduling cost, and penalty cost for different foresight policies.

Foresight level Average total objective

value (in $10 0 0)

Average scheduling

cost (in $10 0 0)

Average penalty cost

(in $10 0 0)

MidWest-A

No Foresight 128.7 89.4 39.3

Full Foresight 119.0 −7.4% ∗ 95.8 7.1% 23.3 −40.7% MidWest-B

No Foresight 137.6 96.8 40.8

Full Foresight 133.0 −3.3% 100.1 3.4% 32.8 −19.4% WestCoast

No Foresight 163.3 132.8 30.5

Full Foresight 152.0 −6.9% 142.0 6.9% 10.0 −67.0% ∗ Percentage change compared to No Foresight.

Fig. 3. Foresight benefit vs. bed capacity utilization expressed as a percentage of

shifts with at least one fully occupied unit.

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levels compared to No Foresight schedules. The corresponding in-

creased scheduling costs (see Table 3 ) are more than offset by the

reduction in expected penalty costs from patient turnaway and un-

derstaffing.

Our experiments show that the magnitude of foresight bene-

fits strongly depends on the level of bed capacity utilization in the

hospital (see Fig. 3 – note that the four data points for each hospi-

tal correspond to the low, medium, high, and very high expected

demand levels). For all three hospitals, we find the highest fore-

sight benefits in times of low demand, when units (almost) never

operate at maximum bed capacity. This is due to the fact that only

when there is free capacity there is enough room in all units to

use quick-response decisions effectively. Patient transfers are pos-

sible only when there is a free bed in the target unit, and the use

of flexible nurses in order to prepare for potential demand spikes

matters only when the hospital is not at maximum capacity, re-

gardless of whether there is a surge in demand or not. In all hos-

pitals, full units are fairly common in times of very high demand.

Naturally, nurses are then scheduled so that they can take care of

fully occupied units in a large share of the shifts - irrespective

of the applied foresight policy. Therefore, initial scheduling deci-

sions are not affected very much by the different foresight policies

in these cases, and benefits from Full Foresight are reduced when

all beds are full a large proportion of the time. This information

should be considered when managers contemplate moving to a

software-supported schedule optimization, as the implementation

of such software requires financial and time investment, training

for schedulers, and buy-in from the affected nurses.

5.2. The impact of foresight on the patient and nurse experience

In light of the ongoing debate about state-enforced patient-to-

nurse ratios, we are interested in how the different hospital set-

ings affect nurse- and patient-related outcomes, and how these

utcomes are in turn improved when our proposed Full Foresight

cheduling methodology is employed. We are particularly inter-

sted in the resulting average understaffing, the number of as-

igned undesired shifts, and the patient turnaways and transfers.

ll items except the undesired shifts are part of the penalty cost.

he undesired shifts are not penalized, but rather minimized in a

econd optimization as explained in Section 3.4 . The second opti-

ization leads to fewer assigned undesired shifts in each of the

60 test problems. Since the first optimization ignores shift desir-

bility, it would be a matter of chance if its solution already re-

ulted in the fewest number of assigned undesired shifts.

In the WestCoast hospital, patient turnaways are cut down by

ore than 68% using Full Foresight ( Table 4 ), which can be at-

ributed to the slightly increased per-shift staffing levels and the

mproved scheduling of float nurses. Here, patient-to-nurse ratios

ave been set by state legislation and are strictly enforced, so that

taffing levels translate into hard limits on how many patients can

e in a unit in any given shift. Therefore, better-informed nurse al-

ocation has an immediate impact on patient turnaways when state

egislation enforces strict patient-to-nurse ratios. Hence, nurses are

etter protected from understaffing. Due to the reduced number of

urnaways, more patients will be admitted under Full Foresight on

verage, moving the nurses’ average workload closer to the manda-

ory patient-to-nurse ratio.

In the MidWest hospitals, our proposed scheduling approach

chieves a significant reduction of the average understaffing. Here,

atient-to-nurse ratios are not mandated by law and therefore act

s target values. Note that the improvement is considerably larger

n MidWest-A, where the target level is 6, compared to MidWest-B

target level 5). There is a small trade-off between the strong im-

rovement in the nurse experience and the patient experience be-

ause the average number of turnaways marginally increases under

ull Foresight.

Across all hospitals, the Full Foresight policy achieves improve-

ents with regards to the percentage of assigned shifts that are

ndesired. As is the case with understaffing, the improvements in

his performance measure are highest where the patient-to-nurse

atio policy is the least restrictive.

With regards to the patient experience, we have to note that

higher number of average transfers results from Full Foresight

cheduling. As reported in Section 5.4 , limiting the number of al-

owed patient transfers per shift to one will only slightly dimin-

sh the overall hospital performance while ensuring a reasonable

umber of transfers. In Appendix B, we discuss that the number

f performed transfers remains relatively stable for reasonable cost

arameters. If, however, the cost of a transfer is completely ne-

lected in the model, the number of average transfers increases to

nrealistically high values as long as they are not limited on a per-

J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403 399

Table 4

Penalty cost, understaffing, undesired shifts, turnaways, and transfers under different foresight policies.

Foresight level Average penalty cost

(in $10 0 0)

Average

Understaffing ∗ Percentage of assigned

shifts that are undesired

Average number

of turnaways

Average transfers

MidWest-A

No Foresight 39.3 34.1 16.1% 11.7 8.6

Full Foresight 23.3 −40.7% 13.4 −60.7% 8.6% −46.5% 12.2 3.7% 12.6 45.9% MidWest-B

No Foresight 40.8 23.8 14.9% 21.4 11.9

Full Foresight 32.8 −19.4% 13.7 −42.6% 9.0% −39.6% 21.5 0.5% 14.0 17.1% WestCoast

No Foresight 30.5 0.0 11.5% 30.2 9.7

Full Foresight 10.0 −67.0% 0.0 — 9.2% −20.0% 9.6 −68.2% 13.2 36.6% ∗ Number of patients over patient-to-nurse-ratio summed over all shifts.

Table 5

Objective value, scheduling cost, penalty cost, turnaways, and understaffing for different nurse pool flexibilities.

Nurse pool Average objective

value (in $10 0 0)

Average scheduling

cost (in $10 0 0)

Average penalty

cost (in $10 0 0)

Average number

of turnaways

Average

understaffing

Average transfers

MidWest-A

0% FNs 130.7 92.1 38.6 14.6 28.0 15.7

8% FNs 122.8 −6.0% 95.0 3.0% 27.8 −28.0% 12.9 −11.6% 17.4 −37.9% 9.9 −36.9% 17% FNs 116.7 −10.7% 97.0 5.3% 19.7 −49.0% 11.4 −21.9% 9.6 −65.7% 6.8 −56.7% 25% FNs 115.9 −11.3% 97.5 5.9% 18.4 −52.3% 10.6 −27.4% 9.0 −67.9% 6.1 −61.1% 33% FNs 115.5 −11.6% 98.6 7.1% 16.8 −56.4% 10.3 −29.9% 7.3 −73.9% 5.8 −63.1% MidWest-B

0% FNs 140.2 95.4 44.8 23.7 24.7 13.9

8% FNs 134.7 −3.9% 100.4 5.2% 34.3 −23.4% 21.9 −7.6% 14.2 −42.5% 10.3 −25.9% 17% FNs 130.3 −7.1% 100.6 5.5% 29.7 −33.7% 21.1 −11.0% 9.6 −61.1% 8.9 −35.9% 25% FNs 131.1 −6.5% 101.5 6.4% 29.6 −33.9% 21.0 −11.4% 9.6 −61.0% 8.8 −36.7% 33% FNs 132.9 −5.2% 103.6 8.6% 29.2 −34.7% 20.8 −12.3% 9.5 −61.5% 8.7 −37.4% WestCoast

0% FNs 147.1 133.5 13.7 13.3 0.0 11.9

8% FNs 150.5 2.3% 137.7 3.1% 13.2 −3.6% 11.7 −7.1% 0.0 — 10.5 −11.8% 17% FNs 154.0 4.7% 142.2 6.5% 11.8 −13.9% 10.9 −13.5% 0.0 — 9.4 −21.0% 25% FNs 158.3 7.6% 146.5 9.7% 11.8 −13.9% 11.0 −12.7% 0.0 — 9.0 −24.4% 33% FNs 161.3 9.7% 149.7 12.1% 11.6 −15.3% 11.2 −11.1% 0.0 — 8.4 −29.4%

5

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.3. The impact of workforce flexibility on the patient and nurse

xperience

Here, we explore how different degrees of flexibility in the

ursing pool impact the results. As reported in Table 5 , the effect

f nursing flexibility on the average objective value varies by the

ospital settings, especially with respect to patient-to-nurse ratios

nd whether or not they are mandatory, as discussed below.

We find that substituting unit nurses with float nurses results

n higher scheduling costs in all three hospitals ( Table 5 ) because

oat nurses are paid higher average wages than unit nurses. How-

ver, the increased flexibility in the workforce leads to reductions

n penalty costs ( Table 5 ) that is caused by understaffing, patient

urnaway, and patient transfers between units. In two of the three

ospitals - the ones in the Midwest, where understaffing is not

rohibited by legislation - the penalty cost reduction more than

ffsets the higher scheduling costs as we move to a 17% float nurse

ool. In MidWest-A, this trend continues as we further substitute

nit nurses with float nurses. In MidWest-B, however, the addi-

ional flexibility does not warrant enough penalty cost reductions

o offset the increase in scheduling cost, so that the 33% float nurse

ool yields higher expected overall costs than the 17% float nurse

ool. At the WestCoast hospital, any float nurses in the workforce

ead to higher overall cost. The slight reduction of patient turn-

ways and patient transfers does not justify the added scheduling

xpenses. We believe that this can be attributed to the size of the

xisting nurse pool in relation to the number of beds in the West-

oast hospital. As discussed in Section 4.1 , the average nurse ca-

acity utilization in the WestCoast hospital is lower than in the

idWest hospitals, likely as a result of differences in state legisla-

ion with regards to violating patient-to-nurse ratios. Because the

estCoast hospital has a relatively higher number of unit nurses

er patient, the benefits from workforce flexibility are reduced so

uch that they do not compensate the higher wage costs of cross-

rained nurses. Another reason is the higher wages paid at the

estCoast hospital. Here, the average 20% wage premium results

n higher absolute cost increases than in the MidWest hospitals.

owever, after deflating the wages of the nurses in the WestCoast

ospital by 25%, we still found the objective value to increase by

.4% when moving from a 0% to an 8% float nurse pool.

In the two hospitals where understaffing is not prohibited by

egislation, the amount of understaffing is decreased drastically by

round 70% (MidWest-A) and 60% (MidWest-B) when float nurses

re at least 17% of the nursing pool. A higher number of float

urses can be a major contributor to maintaining the balance be-

ween supply and demand in the different units of the respective

ospitals. Moreover, the positive effects of reduced understaffing

n long-term measurements of both medical outcomes and nurse

urnover have been studied previously ( Aiken et al., 2002 ). As

hown in Appendix B, the results are sensitive with respect to the

enalty cost for understaffing. Increasing the cost parameter leads

o fewer occurrences of understaffing and lower associated penalty

osts, but the effect is more than offset by increased wage costs.

n top of reducing levels of understaffing in the two MidWest hos-

itals, we also see a decrease in the number of patients that have

o be turned away in all hospitals.

We find a strong relationship between the potential benefit

rom added flexible nurses and the nurse capacity utilization in

400 J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403

Fig. 4. Change in expected obj. value when switching from 0% float nurses to 33% float nurses.

Table 6

Objective value, scheduling cost, penalty cost, turnaways, and understaffing for different transfer limits.

Transfer limit Average objective

value (in $10 0 0)

Average scheduling

cost (in $10 0 0)

Average penalty

cost (in $10 0 0)

Average number

of turnaways

Average

understaffing

Average

transfers

MidWest-A

0 126.0 96.1 29.9 11.9 22.4 0.0

1 119.5 −5.2% 95.9 −0.2% 23.6 −21.2% 12.4 3.5% 13.6 −39.3% 10.4 2 119.0 −5.6% 95.8 −0.4% 23.2 −22.3% 12.1 1.4% 13.4 −40.4% 13.1 4 118.8 −5.8% 95.7 −0.5% 23.1 −22.8% 12.1 1.4% 13.2 −41.0% 14.2

MidWest-B

0 138.9 99.9 39.0 23.1 19.8 0.0

1 133.2 −4.1% 100.2 0.3% 33.0 −15.3% 21.8 −5.6% 13.5 −32.0% 13.3 2 132.8 −4.3% 100.1 0.2% 32.7 −16.0% 21.3 −7.8% 13.7 −30.7% 14.2 4 132.8 −4.3% 100.1 0.2% 32.7 −16.0% 21.3 −7.8% 13.8 −30.3% 14.4

WestCoast

0 161.3 142.9 18.5 18.5 0.0 0.0

1 152.4 −5.6% 142.4 −0.3% 10.1 −45.6% 9.6 −47.8% 0.0 — 11.0 2 152.0 −5.8% 142.0 −0.6% 10.0 −45.9% 9.5 −48.5% 0.0 — 13.3 4 151.7 −6.0% 141.6 −0.9% 10.1 −45.6% 9.6 −47.8% 0.0 — 15.3

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the hospital ( Fig. 4 ). Interestingly, the relationship is exactly oppo-

site to the relationship between bed utilization and Full Foresight

benefit shown in Fig. 3 . Whereas the benefit from Full Foresight

scheduling diminishes with higher utilization due to the reduction

of scheduling options (“all hands on deck”), float nurses provide

the most value when understaffing and turnaways are likely to oc-

cur.

In both Midwest hospitals, we find that switching from a nurse

pool with 0% float nurses to a pool of 33% float nurses will show

higher improvements when nurse capacity utilization is higher. We

also find this trend in the WestCoast hospital despite the fact that

it is not beneficial to switch to a more flexible nurse pool alto-

gether. Float nurses always become more useful as nurse capac-

ity utilization and consequently the likelihood of understaffing and

patient turnaways increase. In times of higher capacity utilization,

the effects of mismatches between supply and demand are more

severe in terms of incurred penalty costs, so that higher flexibility

in the workforce results in higher saved penalty costs.

It is also noteworthy that the average number of patient trans-

fers necessary to achieve optimal cost is considerably higher when

there are no float nurses. Hospital managers that want to focus on

reducing the number of patient transfers in their hospital can hire

cross-trained nurses to help them reach their goal.

5.4. The impact of patient transfers on the patient and nurse

experience

Without the ability to adjust the number of patients in each

unit (patient transfer limit set to 0), the only means of short-term

adjustment is the assignment of float nurses to units. Using this

cenario as a base case, we analyze how much benefit can be de-

ived from allowing an increasing number of patient transfers. Pa-

ient transfers are allowed only during the day or evening shifts.

ight shift transfers are disallowed in all cases. For this discussion,

e include results using the Full Foresight policy.

Allowing just one patient transfer per day and evening shift in

ach unit lowers the objective value by around 5% (see Table 6 ) on

verage. When the allowed number of transfers is increased fur-

her, the marginal benefits are small. This is in line with previous

ndings in the literature that have found sharply diminishing re-

urns of flexibility. In our experiments, we find that there is an

ncentive to conduct more than one patient transfer in only 9% of

ll shifts.

Most cost savings that result from allowing patient transfers

tem from reduced penalty costs (see Table 6 ), while scheduling

osts are only marginally affected when the number of allowed

atient transfers increases. These penalty cost reductions can be

xplained by the added flexibility provided by the option to use

atient transfers in times of high demand, which leads to drasti-

ally decreased understaffing in the two MidWest hospitals as well

s reduced patient turnaway in the MidWest-B hospital and the

estCoast hospital. Two forces act on the scheduling cost in op-

osite directions. The increased throughput increases the number

f patients that need to be taken care of, which can drive up the

otal number of shifts that are assigned over the time horizon. On

he other hand, balancing the workload of the nurses across units

y moving patients can potentially result in a lower number of

urses needed across all three units in a given shift. If, for exam-

le, a unit is occupied by 26 patients and the patient-to-nurse ra-

io is 5:1, six nurses would need to be scheduled in order to avoid

J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403 401

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nderstaffing. A patient transfer could reduce the number of pa-

ients in that unit to 25 so that one nurse less would have to be

cheduled in that particular unit. If the patient is transferred to a

nit in which a nurse can take on another patient without going

ver the patient-to-nurse ratio, that patient transfer helps to re-

uce the system-wide scheduling cost.

. Model validation via simulation and heuristic

In Section 3 , we present a multi-stage stochastic program-

ing model to determine an initial work schedule and subse-

uent quick-response recourse decisions. As explained in Section

.2 , due to problem complexity and long solution times, we are

ble to model a limited number of possible demand scenarios for

ach shift (low, medium, high demand) in our optimization model.

herefore, in this section, we present results from a patient trans-

er and nurse assignment heuristic embedded in a simulation that

andles a larger number of demand scenarios to generate more re-

listic patient demand distributions over the scheduling horizon.

e apply the heuristic in a simulation setting and present results

o compare its performance with that of the stochastic program-

ing model. It is important to note that both approaches use the

ame initial schedule obtained from the full foresight stochastic

rogramming model. In the simulation, however, quick-response

ecisions are derived based on the patient demand realizations ac-

ording to the following heuristic.

.1. Quick-response heuristic for patient transfer and float nurse

ssignment

Based on an existing initial schedule for unit and float nurses

ver a one-week horizon (as determined by the optimal x i j val-

es from the stochastic optimization model), the patient transfer

nd nurse assignment heuristic uses priority rules to determine the

est use of available flexible resources in each shift. The goal is to

inimize the costs incurred from patient turnaway, understaffing,

nd patient transfers, in hierarchical order. Following is a descrip-

ion of each step in the heuristic, carried out at the beginning of

ach shift.

Step 1.

Determine the number of patients who try to occupy a bed in

he respective units, resulting from current demand levels and re-

ent patient transfers. If in at least one unit this number is larger

han the number of available beds, identify units that have empty

eds. Depending on bed availability, perform a transfer or off-unit

ssignment from the unit with the highest number of surplus pa-

ients to the qualifying unit with the highest number of empty

eds (ignoring patient-to-nurse ratios at this point). Keep perform-

ng transfers until all surplus patients have been distributed or no

ree beds are available.

Step 2.

Calculate the patient-to-nurse ratios in each unit based on the

umber of assigned unit nurses and the number of patients cur-

ently assigned to each unit – which results in a current patient-

o-nurse ratio. Assign the first available float nurse to the unit with

he highest relative deviation (relative deviation = (current ratio – arget ratio) / (target ratio)) from the target patient-to-nurse ra-

io and recalculate the patient-to-nurse ratios. Repeat until all float

urses are assigned.

Step 3.

If any of the units are understaffed, check if units that are not

nderstaffed have empty beds available. If that is the case, perform

patient transfer to the qualifying unit with the lowest patient-to-

urse ratio. Recalculate the patient-to-nurse ratio in each unit. Re-

eat patient transfers and recalculation until either no unit is un-

erstaffed or no additional transfers are possible (because no more

mpty beds are available, the limit on the number of transfers has

een hit, or no unit with empty beds is below its target patient-

o-nurse ratio). If understaffing is not allowed, turn away the re-

aining patients until the resulting patient-to-nurse ratio does not

xceed the maximum allowed patient-to-nurse ratio. If any of the

atient turnaways performed in Step 3 was caused by a previ-

us transfer in Step 1, undo that transfer and count the patient as

urned away from the original unit to avoid double penalization. If

nderstaffing is allowed, let the remaining patients occupy beds in

he understaffed unit.

Patient demand parameters are modeled after historically ob-

erved demand in the respective units. In contrast to the modeling

pproach in Section 3 , patient demand realization updates in the

imulation take place in each individual shift. We perform repeated

ndependent runs of the simulation until the size of the confidence

ntervals of all relevant performance measures is no larger than 1%

f the mean with a likelihood of 95%.

.2. Comparison of results from the simulation heuristic and

tochastic optimization

Comparing quick-response decisions and the consequential

osts resulting from the simulation heuristic – based on initial

chedules generated by the full foresight model from Section 3 –

ith results from Section 5 , we conduct a numerical study and re-

ort results with respect to differences in performed patient trans-

ers, patient turnaways, and understaffing occurrences. The experi-

ental factors in the study are analogous to the ones presented in

ection 4 .

As discussed in Section 3 , the stochastic programming model

ssumes demand scenarios derived from actual demand for the

ake of limiting the complexity of the model. In reality, however,

emand can fluctuate more freely. More realistic demand levels

ay change from one day to another, or even from one shift to

he next. As a result, quick-response decisions such as float nurse

ssignment or patient transfers may need to deviate from the ones

rescribed by the optimal stochastic programming solution due to

he stronger refinement of the time frame. The simulation heuristic

llows us to model demand more realistically on a per-period basis

nd to track the deviation of the quick-response decisions, essen-

ially setting ψ equal to the number of periods in the scheduling orizon. On the one hand, this introduces more demand variabil-

ty into the model, which may result in higher, closer to reality

verall cost of recourse decisions and incurred penalties for patient

urnaway. On the other hand, there is (realistically) more freedom

n choosing the best recourse in each period, which may improve

verall schedule performance.

First, we are interested in how much better the overall perfor-

ance can be when quick-response decisions are made on a shift-

er-shift basis with daily updated demand information. To provide

good comparison, we first use a demand generating process that

andomly picks a low, medium, or high demand outcome for the

rst half of the week and another low, medium, or high demand

utcome for the second half of the week, replicating the scenarios

sed in the stochastic programming approach. Therefore, the un-

erlying demand is assumed to be distributed exactly the same in

his experiment as modeled in Sections 4 and 5 .

Overall, we find that the results from the simulation heuristic

atch very closely the stochastic programming approach ( Table 7 ).

he expected total cost is only 0.07% different for the simulation

euristic versus the stochastic optimization approach. The largest

ortion of this cost, the scheduling cost, however, is always the

ame in both approaches, because the simulation heuristic assumes

he same initial schedule that is generated by the full foresight

odel. Therefore, the change in the expected penalty cost is more

402 J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403

Table 7

Comparison of simulation heuristic and stochastic optimization solution.

Model Expected

total Cost

Expected

Transfers

Expected

Turnaway

Expected

penalty Cost

Simulation heuristic $151,863 14.1 9.5 $9,935

Stochastic optimization $151,974 13.2 9.6 $10,046

Table 8

Simulation heuristic results comparing alternative demand draw procedures.

Demand draw Expected

total cost

Expected

transfers

Expected

turnaway

Expected

penalty cost

(1) whole week $152,083 13.8 9.8 $10,155

(2) half week $151,863 14.1 9.5 $9,935

(3) each day $151,689 13.4 9.3 $9,761

(4) historical daily $154,910 16.0 12.5 $12,982

Table 9

Stochastic model vs. heuristic simulation using medium mean demand.

Model Expected

total cost

Expected

transfers

Expected

turnaway

Expected

penalty cost

Simulation Heuristic $147,192 12.6 4.3 $4,637

Stochastic Optimization $147,106 14.4 4.1 $4,551

Table 10

Comparison of recourse decisions from heuristic vs. model.

Model Average

obj. value

Average

transfers

Average

turnaway

Average

penalty cost

Simulation Heuristic

Recourse

$151,689 13.4 9.3 $9,761

Stochastic Optimization

Recourse

$158,459 15.2 16.1 $16,531

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relevant for this analysis. The expected penalty cost is reduced by

1.1%, averaged over all factor levels.

6.3. Comparison of alternative demand draw processes

To underline the previous finding, we present results from ad-

ditional simulation heuristic runs with alternative demand draws

( Table 8 ). In total, we use four different types of procedures for the

demand draws: (1) demand can either be low, medium, or high

for all days of the week (3 possible outcomes); (2) demand can

be either low, medium, or high for the first half of the week, then

possibly change for the remaining half of the week (9 possible out-

comes); (3) demand realizations are drawn from low, medium, or

high and can be different for each day of the week (21 possible

outcomes); (4) demand is drawn daily, but from a wider range of

historically observed values that were initially used to create the

demand scenarios (21 possible outcomes).

The results are fairly robust with respect to the different de-

mand draw procedures. In turn, this finding strengthens the results

derived from the scheduling and quick-response model of Section

3 . It shows that the use of a limited number of demand scenarios

in the scheduling and quick-response model provides a well-suited

compromise between the reduction of model complexity and the

decline in modeling accuracy.

Drawing from historical demand results in overall diminished

performance of the initial schedule. This can be attributed to the

fact that historical demand draws are different from the assumed

demand scenario properties (Patient Count and Demand Variability

in Table 2 ). When the initial schedule assumes low patient counts

and low demand variability, for example, nurse staffing levels are

naturally inadequately low and result in increased transfers and

turnaways.

If we constrain the analysis of schedule performance to sched-

ules that are generated assuming a medium mean demand count

and low or high demand variability, and we use historical daily

demand draws in the heuristic simulation, we find that the re-

sults match the ones from the stochastic programming model very

well ( Table 9 ) – which means that using a limited number of well-

chosen demand scenarios as the basis of the nurse scheduling and

quick-response model creates schedules that perform well even

when serving more realistic demand settings.

The following analyses are all performed using the demand

raw procedure ( 3 ), where demands are drawn separately for each

ay.

.4. Performance of recourse decisions

We are interested in how well the recourse decisions, which

esult from solving the stochastic programming model, perform.

ecause of the necessary reduction of the model’s complexity

hrough the aggregation of demand into scenarios, the prescribed

ecourse decisions from the stochastic optimization may not be op-

imal, but they serve the purpose of creating schedules that in-

egrate possible recourse decisions with the development of the

nitial schedule. However, taking a look at how much worse one

ould do when always following the results from the stochastic

rogramming model compared to making independent recourse

ecisions according to the heuristic in each shift after demand

as realized provides some insight into the quality of the initial

odel’s demand scenario-based recourse decisions.

Therefore, we assume daily demand realizations in the simula-

ion according to demand draw process ( 3 ) and compare the per-

ormance of our heuristic versus always taking actions that result

rom the stochastic programming solution in Table 10 . Since de-

and will fluctuate more freely in the simulation than in the un-

erlying assumed scenario setting, we use a moving average ap-

roach to determine which scenario is most closely mirrored by

he random demand stream up to each shift. That means that in

eriod p , we average demand in each unit over the last n periods

( n = | p−1 | S S | || S S | + 1 ) . If that average demand is closest to demand n scenario k , we take the recourse decisions associated with sce-

ario k in the solution of the stochastic program. In case of a tie,

he “medium” scenario value is chosen.

Indeed, following the stochastic programming recourse so-

utions results in relatively poor overall schedule performance.

hen demand streams over the individual days vary more re-

listically than in the 9-demand-scenario setting underlying the

urse scheduling and quick-response model, recourse decisions

rescribed by the model solution are based on incorrect assump-

ions of the resulting patient numbers after transfers and turn-

ways in each shift.

Based on these results, we recommend developing the initial

chedule based on the stochastic optimization model and then us-

ng the heuristic logic to find the best suitable quick-response re-

ourse decisions.

. Conclusions and future research

In this paper, we explore the use of two quick-response meth-

ds within a nurse-scheduling model to help hospitals manage pa-

ient demand fluctuations and improve performance. We present

multi-stage stochastic programming model that coordinates ini-

ial per-shift nurse scheduling decisions with quick-response de-

isions that are made after observing patient demand. The model

akes into account information about patient demand distributions,

hich allows hospitals to schedule nurses such that quick-response

ethods can be used most effectively once actual patient demand

s observed.

J. Schoenfelder, K.M. Bretthauer and P.D. Wright et al. / European Journal of Operational Research 283 (2020) 390–403 403

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Based on data from three different hospital settings, our results

how that our proposed multi-stage stochastic programming model

owers total cost. Primarily, these cost savings are a result of lower

enalty cost that hospitals incur from understaffing, patient trans-

ers, and patient turnaway. In our study, we achieve either reduced

nderstaffing or patient turnaway levels, depending on the hospi-

al setting. Average scheduling costs, on the other hand, typically

ncrease slightly under our proposed policy, as this puts hospitals

n a better position to react to potential demand surges.

Our primary contributions include an analysis of the poten-

ial benefit, or lack of, provided by costly quick-response options.

ross-trained float nurses offer management flexibility that helps

itigate the impact of demand variability on increased under-

taffing and patient turnaway, but in some instances the additional

ost caused by their wage premiums more than outweighs the

enefit. This is especially true and has significant managerial rel-

vance in settings where an upper limit on nurse workload is en-

orced. Another key takeaway is that a small number of patient

ransfers or off-unit admissions realizes almost the full potential

enefit from this quick-response option, which is important be-

ause it will minimize the associated negative impact on patient

atisfaction and quality of care. We also find bed and nurse capac-

ty utilization to be important considerations in deciding how and

hether to implement quick-response options.

This research can be extended in multiple ways. There are

dditional quick-response methods such as on-call nurses and

escheduling of surgeries, or hiring agency nurses on a short-time

asis that can be considered. Including these quick-response meth-

ds in the scheduling decision may provide further benefits and

nsights. Also, in our analysis, the number of available beds and

he total size of the nurse pool are fixed inputs with only vary-

ng degrees of the percentage of float nurses, as our focus lies on

he operational scheduling decision. Future work might focus on

he right bed mix and the number of nurses on staff that put the

ospital in the best position to take advantage of quick-response

ctions.

upplementary materials

Supplementary material associated with this article can be

ound, in the online version, at doi: 10.1016/j.ejor.2019.10.047 .

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