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“Nursevendor Problem”: Personnel Staffing in thePresence of Endogenous AbsenteeismLinda V. Green, Sergei Savin, Nicos Savva

To cite this article:Linda V. Green, Sergei Savin, Nicos Savva (2013) “Nursevendor Problem”: Personnel Staffing in the Presence of EndogenousAbsenteeism. Management Science 59(10):2237-2256. http://dx.doi.org/10.1287/mnsc.2013.1713

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MANAGEMENT SCIENCEVol. 59, No. 10, October 2013, pp. 2237–2256ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2013.1713

© 2013 INFORMS

“Nursevendor Problem”: Personnel Staffing in thePresence of Endogenous Absenteeism

Linda V. GreenGraduate School of Business, Columbia University, New York, New York 10027, lvg1@columbia.edu

Sergei SavinThe Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104, savin@wharton.upenn.edu

Nicos SavvaLondon Business School, London NW1 4SA, United Kingdom, nsavva@london.edu

The problem of determining nurse staffing levels in a hospital environment is a complex task because ofvariable patient census levels and uncertain service capacity caused by nurse absenteeism. In this paper, we

combine an empirical investigation of the factors affecting nurse absenteeism rates with an analytical treatmentof nurse staffing decisions using a novel variant of the newsvendor model. Using data from the emergencydepartment of a large urban hospital, we find that absenteeism rates are consistent with nurses exhibiting anaversion to higher levels of anticipated workload. Using our empirical findings, we analyze a single-periodnurse staffing problem considering both the case of constant absenteeism rate (exogenous absenteeism) as wellas an absenteeism rate that is a function of the number of nurses scheduled (endogenous absenteeism). We pro-vide characterizations of the optimal staffing levels in both situations and show that the failure to incorporateabsenteeism as an endogenous effect results in understaffing.

Key words : healthcare; hospitals; nursing; newsvendor; econometrics; organizational studies; manpowerplanning

History : Received October 24, 2011; accepted December 12, 2012, by Yossi Aviv, operations management.Published online in Articles in Advance June 14, 2013.

1. Introduction and Literature ReviewIn recent years hospitals have been faced with increas-ing pressure from their major payers—federal andstate governments, insurers, and large employers—to cut costs. Because nursing personnel account fora very large portion of expenses, the response inmany instances has been reductions of the nursingstaff. Nurse workloads have been further increasedby shorter hospital lengths of stay and increasinguse of outpatient procedures, resulting in sicker hos-pitalized patients who require more nursing care.The adverse impact of these changes has been docu-mented by a number of studies and include increasesin medical errors, delays for patients waiting forbeds in emergency rooms, and ambulance diversions(Needleman et al. 2002, Aiken et al. 2002, Cho et al.2003). In response, a number of state legislatures,for example, Victoria in Australia and California inthe United States, have mandated minimum nursestaffing levels (Gordon et al. 2008).

Establishing the appropriate nursing level for a par-ticular hospital unit during a specific shift is com-plicated by the need to make staffing decisions wellin advance (e.g., six to eight weeks) of that shift, as

well as labor constraints dealing with the number ofconsecutive and weekend shifts worked per nurse,vacation schedules, personal days, and preferences(see, e.g., Miller et al. 1976, Wright et al. 2006). Addingto these complexities is the prevalence of nurse absen-teeism. According to the U.S. Bureau of Labor Statis-tics (2008), in 2008 U.S. nurses exhibited the highestnumber of incidents of illness or injury involvingdays away from work, 7.8 per 100 FTEs, substantiallyhigher than the national average of 2.1 incidents per100 FTEs. The goal of this paper is to construct amodel for nurse staffing that includes the impact ofabsenteeism and investigate whether and how clini-cal units should take it into consideration when theymake staffing level decisions. To do this, we must firstunderstand whether and how absenteeism is affectedby staffing levels themselves. Because, to the best ofour knowledge, there is no literature that adequatelyaddresses this linkage, we first conduct an empiricalhospital-based study to understand how to incorpo-rate absenteeism into an analytical model.

Most studies on nurse absenteeism find that it ispositively related to levels of work-related stress(Shamian et al. 2003) and nurse workload (Bryant

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et al. 2000, Tummers et al. 2001, McVicar 2003, Unruhet al. 2007). However, at least one study examiningabsenteeism among trainee nurses finds a negativecorrelation between nurse absenteeism and work-load (Parkes 1982). Nearly all of the existing studiesof nurse absenteeism use qualitative or self-reportedmeasures of workload and cross-sectional analysesof this phenomenon (see Rauhala et al. 2007 foran exception). In particular, they compare the long-run absenteeism behavior of nurses across differentclinical units, rather than tracking the same set ofnurses over time. As such, these studies are of limitedvalue to managers responsible for day-to-day staffingdecisions.

The first part of our paper presents a longitudi-nal investigation of the link between nurse absen-teeism and workload using data from the emergencydepartment (ED) of a large New York City hospi-tal. Rather than relying on subjective self-reportedworkload measures, we use patient census values tocalculate nurse-to-patient ratios that are treated asproxies for the workload experienced by nurses work-ing a particular shift. Nurse shortages and other orga-nizational limitations (such as union rules) providethe necessary exogenous variation in staffing deci-sions that we exploit to identify the impact of work-load on absenteeism. We hypothesize that fluctuationsin nurse workload due to irregularities in schedulingand/or unpredictable demand have a direct impacton absenteeism. An anticipated increase in work-load could result in an increase in nurses’ motiva-tion to help their fellow nurses and, therefore, indecreased absenteeism, or, given generally high work-loads and adverse working conditions, in increasedabsenteeism. In either case, the resulting behaviormay result in optimal nurse staffing levels that aredifferent from those predicated by models that donot consider this workload-induced behavior. There-fore, our empirical investigation aims to provide sup-port for one of these specific hypotheses so that wecan incorporate the appropriate assumption into ouranalytical modeling. Our main finding is that absen-teeism increases when there is a higher anticipatedworkload. Specifically, we find that for our data setwith an average absenteeism rate of 703%, an extrascheduled nurse is associated with an average reduc-tion in the absenteeism rate of 006%. As such, ourpaper is related to the growing body of literatureon the effects of workload on system productivity(KC and Terwiesch 2009, Powell and Schultz 2004,Powell et al. 2012, Schultz et al. 1999).

The second goal of our paper is to develop a modelof optimal staffing in service environments withworkload-dependent absenteeism. Extant literatureeither ignores absenteeism or treats it as an exoge-nous phenomenon (Bassamboo et al. 2010, Easton and

Goodale 2005, Harrison and Zeevi 2005, and Whitt2006 provide examples of call-center staffing, whereasFry et al. 2006 and He et al. 2012 provide examplesof firefighter and hospital operating room staffing,respectively). The uncertain supply of service capac-ity created by nurse absenteeism connects our workwith a stream of literature focused on inventory plan-ning in the presence of unreliable supply/stochasticproduction yield (Yano and Lee 1995) with two impor-tant distinctions. First, the overwhelming majority ofpapers on stochastic supply yields model them asbeing either additive or multiplicative (Henig andGerchak 1990, Ciarallo et al. 1994, Bollapragada andMorton 1999, Gupta and Cooper 2005, Yang et al.2007), a justifiable approach in manufacturing set-tings. The supply uncertainty in our model has abinomial structure, a more appropriate choice in per-sonnel staffing settings. Binomial yield models area relative rarity in the stochastic yield literature,perhaps due to their limited analytical tractability(Grosfeld-Nir and Gerchak 2004, Fadiloglu et al.2008). Most importantly, to the best of our knowl-edge, our analysis is the first to introduce and analyzeendogenous stochastic yields.

Using our model we characterize the optimalstaffing levels under exogenous and endogenousabsenteeism. We show that the failure to incorpo-rate absenteeism as an endogenous effect results inunderstaffing, which leads to a higher-than-optimalabsenteeism rate and staffing costs. Specifically, formodel parameters that closely match the hospitalwe study, we find that ignoring the endogenousnature of absenteeism can lead to a staffing costincrease of 2% to 3%. In addition to the cost impact,understaffing associated with ignoring absenteeismmay result in an increase in medical errors, partic-ularly in the pressured and sensitive environmentof an ED. Considering that nursing costs is one ofthe biggest components of overall hospital operatingcosts, more accurate nurse staffing based on endoge-nous absenteeism constitutes a substantial opportu-nity for hospitals to simultaneously reduce costs andimprove quality of care.

Finally, we show that despite understaffing, theexogenous-absenteeism model will appear to be self-consistent in the sense that the assumed exoge-nous absenteeism rate will be equal to the observed(endogenous) absenteeism rate. This is particularlyworrisome for staffing managers because it impliesthat it is impossible to tell whether the model is wellspecified just by examining the observed absenteeismrate. In this regard, our paper contributes to the lit-erature on model specification errors in operations(e.g., Cachon and Kök 2007, Cooper et al. 2006, Leeet al. 2012, Mersereau 2012). In this literature, as inour paper, the model specification error cannot be

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detected by studying data as the misspecified modelwill produce consistent outcomes. To the best of ourknowledge, our paper is the first to study model spec-ification error in the context of staffing, and the firstin the model specification literature to start from anempirical observation.

2. Endogeneity in Nurse AbsenteeismRates: An Empirical Study

Our study is based on nurse absenteeism and patientcensus data from the ED of a large New YorkCity hospital. Nurses employed in this unit arefull-time employees, each working on average 3.25shifts per week. The unit uses two primary nursingshifts: the day shift starts at 8:00 a.m. and ends at8:00 p.m.; the night shift starts at 8:00 p.m. and ends at8:00 a.m. Another (evening) shift is also operated from12:00 p.m. to 12:00 a.m. For each shift, for a periodof 10 months starting on July 1, 2008 (304 day shifts,304 evening shifts, and 303 night shifts), we collectedthe following data: the number of nurses scheduled,the number of nurses absent, the number of patientvisits, and the patient census data recorded every twohours. The resulting descriptive statistics for threeshifts are presented in Table 1.

In our analysis of absenteeism, we limit our atten-tion to the nurses on the day and night shifts becausethe evening shift is fundamentally different from theother two. First, the nurses working on this shift donot work on the other two shifts. Thus, it is less likelythat they are informed about the nurse staffing sched-ules for the day and night shifts. Second, the eveningshift consists of fewer nurses who are more experi-enced and exhibit less absenteeism than the other twoshifts, as shown in Table 1. However, we do take into

Table 1 Descriptive Statistics for Nurse and Patient Data

Measure Mean Std. dev. Minimum Maximum

Day shiftNurses scheduled 11.4 1.07 8 16Absenteeism rate 0.0762 0.0799 0 0.4Patient visits 141 20.1 77 188Average census 116 17.1 56.5 158Maximum census 136 20.8 64 182

Night shiftNurses scheduled 10.5 0.849 9 14Absenteeism rate 0.0707 0.0829 0 0.4Patient visits 66.0 9.45 40 95Average census 102 14.2 54.3 142Maximum census 127 20.4 57 174

Evening shiftNurses scheduled 3.63 0.756 2 5Absenteeism rate 0.0589 0.119 0 0.5Patient visits 137 16.2 75 196Average census 125 18.6 58.2 164Maximum census 137 20.7 64 182

account the evening shift when measuring workloadbecause the evening shift overlaps with both the dayshift and the night shift.

The nurse scheduling process starts several weeksbefore the actual work shift when the initial scheduleis established. This initial schedule often undergoes anumber of changes and corrections due to, for exam-ple, family illnesses, medical appointments, and juryduty obligations, which may continue until the daybefore the actual shift. In our study, we have usedthe final schedules, that is, the last schedules in effectbefore any “last minute” absenteeism is reported forthe shift. We record as absenteeism any event where anurse does not show up for work without giving suf-ficiently advance notice for the schedule to be revised.In the clinical unit we study, nurses are allowed touse up to 10 personal days per year, which do notrequire any significant advance notice. To compen-sate for absenteeism, the management of the ED useseither agency nurses or nurses from the previous shiftto work overtime. Therefore, the number of nursespresent to work during any given shift is generallyequal to the number of nurses in the final schedule.1

The average patient census during a shift variessubstantially from day to day. Some of this varia-tion (5202% for day shifts and 3206% for night shifts)can be explained by day-of-week and weekly fixedeffects. Furthermore, the patient census exhibits sig-nificant serial autocorrelation (� = 3006%) with thevalues recorded during the previous shift. The num-ber of nurses scheduled for a particular type of shift,for example, day shift on a Wednesday, is highlyvariable. Approximately 25% of the variation in thenumber of nurses scheduled can be explained by day-of-week and weekly fixed effects (adjusted R2 = 2705%for the day shift and adjusted R2 = 2401% for the nightshift). Also, after controlling for fixed effects, the num-ber of nurses scheduled for a shift shows little depen-dence on either the average patient census during thatshift or on the average census values for the 14 pre-vious shifts, which correspond to one calendar week.This indicates that the unit’s nurse staffing policy oneach shift does not seem to be affected in any sig-nificant way by the realized patient census, over andabove the day-of-week fixed effects.

Our discussions with the ED nurse manager indi-cated that there are two main factors driving the sig-nificant variations in the number of scheduled nurses.First, personnel scheduling is subject to numerousconstraints (e.g., union rules) that often prevent the

1 We note that because of budget constraints, the ED is only allowedto use agency/overtime nurses to cover for absenteeism and isnot allowed to make use of such expensive resources for routinestaffing. Inquiries with staffing managers in other hospital settingshave suggested that this practice is by no means unique to thisspecific ED.

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manager from assigning the number of nurses desiredfor a particular shift. Second, as mentioned earlier,initial schedules often undergo a series of changesbefore they are finalized. Such scheduling variationsare not desirable from the point of view of managingthe match between the demand for nursing servicesand the supply of nursing capacity, but they providean opportunity to examine how absenteeism rates arerelated to the numbers of scheduled nurses.

2.1. Nurse Workload and Absenteeism:Empirical Results

We model the phenomenon of nurse absenteeism asfollows. We treat all nurses as being identical andindependent decision makers and focus on a groupof yt nurses scheduled to work during a particu-lar shift t (t = 1 for the first shift in the data set,t = 2 for the second shift, etc., up to t = 607). Fornurse n, n= 11 0 0 0 1 yt , the binary variable Yn1 t denotesher decision to be absent from work (Yn1 t = 1), orto be present (Yn1 t = 0). We assume that this absen-teeism decision is influenced by a number of factorsexpressed by the vector xt which include parametersrelated to workload as well as fixed effects such asthe day of the week or the shift. Each nurse comparesthe utility she receives from being absent from workto the utility she receives from going to work. Thedifference in these utility values is given by U ∗

n1 t =

x′t� + �n1 t , where �n1 t are, for each n and t, inde-

pendent and identically distributed random variableswith mean zero. Although the utility difference U ∗

n1 t isan unobservable quantity, we can potentially observeeach nurse’s decision to show up for work. The deci-sion is such that Yn1 t = 1 if U ∗

n1 t > 0, and Yn1 t = 0otherwise. Assuming that �n1 t follow the standard-ized logistic distribution (the standard normal distri-bution), we obtain the logit (probit) model (Greene2005). Note that our empirical data do not record theattendance decisions of individual nurses. Rather, wemeasured the aggregate absenteeism behavior of agroup of nurses scheduled for a particular shift. Con-sequently, we treat all nurses scheduled for a givenshift as a homogeneous group and build the model forthe corresponding group behavior. We examine theimpact of relaxing this assumption in §2.2. We focuson the maximum-likelihood-based logit estimation ofthe probability of absenteeism �t during shift t.

Because our goal is to study how the nurse absen-teeism rate is affected by workload, we need to mea-sure and quantify nurse workload for each shift.We use the nurse-to-patient ratio as a proxy for theworkload nurses experience during a particular shift.For shift t, we define the nurse-to-patient ratio vari-able, denoted as NPRt , as the ratio of the numberof nurses working during a particular shift and thepatient census averaged over the duration of that

shift. Therefore, the number of nurses present dur-ing each 24-hour period varies as follows: Between8:00 a.m. and 12:00 p.m., it is equal to the numberof nurses scheduled for the day shift (yt); between12:00 p.m. and 8:00 p.m., it is equal to the numberof nurses scheduled for the day shift (yt) plus thenumber of nurses scheduled for the evening shift (et);between 8:00 p.m. and 12:00 a.m., it is equal to thenumber of nurses scheduled for the evening shift (et)plus the number of nurses scheduled for the nightshift; and between 12:00 a.m. and 8:00 a.m., it isequal to the number of nurses scheduled for the nightshift (yt). Thus, we estimate NPRt as follows:

NPRt =yt +

23et

Ct

for the day shift1

NPRt =y+

13et

Ct

for the night shift1

(1)

where Ct is the patient census averaged over the dura-tion of shift t.

In making their attendance decisions for shift t,nurses may be influenced by the anticipated work-load for shift t. The impact of anticipated workloadarises because nurses are informed in advance of theirschedule and are aware of how many (and which)other nurses are scheduled to work on the same shift.Because nurses anticipate a certain patient censusE6Ct7, consistent with their past experience of work-ing in the ED, nurses form an expectation about theanticipated workload for that shift. Naturally, if fewer(more) nurses are scheduled on that particular shiftthan the nurses deem appropriate, they will anticipatea higher (lower) workload than normal. The groupattendance data do not present a measurement chal-lenge because nurses scheduled for the same shift aresubjected to the same anticipated workload value.

The anticipated workload can have a dampen-ing effect on absenteeism through the “pressure-to-attend” mechanism (Steers and Rhodes 1978) or canenhance absenteeism by encouraging “withdrawalbehavior” (Hill and Trist 1955, Hobfoll 1989). To thebest of our knowledge, the impact of anticipatedworkload on absenteeism has not been previouslystudied. We test this potential impact in our setting byincluding in the vector of covariates xt the anticipatedvalue of nurse-to-patient ratio,

ENPR1t =

yt +23et

E6Ct7for the day shift1

ENPR1t =

yt +13et

E6Ct7for the night shift1

(2)

where et is the number of nurses scheduled in theevening shift which overlaps with 2

3 ( 13 ) of the dura-

tion of the day (night) shift in question. Day- and

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night-shift nurses are fully informed about the sched-ule for their shifts, but it is not clear that they wouldbe as familiar with the schedule of the evening shiftstaffed by a different pool of nurses. Motivated by thisobservation, we estimate two models based on alter-native definitions of the expected nurse-to-patientratio. In the first definition (ENPR1

t of Equation (2)),we use the exact number of evening nurses sched-uled (et), whereas in the second definition (ENPR2

t ),we use the average value of et (averaged over allevening shifts in our sample). The latter formula-tion reflects the situation where day- and night-shiftnurses do not know precisely how many eveningnurses will be present but form a rational expectationabout this value. In other words, in the second model,day- and night-shift nurses behave as if they ignoreany variation in the number of nurses scheduled forthe evening shift that overlaps with their own shift.The expected patient census values E6Ct7 are com-puted by averaging the patient census over all shiftsin our sample. This formulation reflects an assump-tion that nurses, when making their attendance deci-sions, use a mental model that captures any potentialdifference occurring on different days/shifts with afixed effect and, therefore, focus on expected patientcensus. We also assume that the nurses form rationalexpectations about the patient census consistent withempirically observed patient census data. In additionto the models based on (2), we have also estimatedseveral alternative variants, which we discuss in 2.2.

In addition to the anticipated nurse-to-patient ratio(ENPRi

t1 i = 112), the vector of covariates xt includes anumber of controls. In particular, we include a day-of-the-week dummy variable to capture any systematicvariation in absenteeism across days, a day/night-shift fixed effect to capture variations between dayand night shifts, and a weekly fixed effect to cap-ture any systematic variations that remain constantover one week and affect absenteeism, but are other-wise unobservable. Also, we include a holiday fixedeffect that takes the value of 1 on national public hol-idays and 0 on any other day. This last variable isdesigned to deal with a potential endogeneity prob-lem because nurses may be inherently reluctant towork on some select days, such as public holidays.These days are known to the management of theclinical unit which tries to accommodate the nurses’aversion by staffing fewer nurses on such days. Nev-ertheless, the nurses that are scheduled to work onthese “undesirable” days are still more likely to beabsent, irrespective of the chosen staffing levels. Byincluding the holiday variable we are trying to explic-itly account for this effect. There may exist othercorrelated variables that we omit, but to the extentthat they do not vary drastically over a period ofone week, the weekly fixed effect should capture the

influence of those variables. Finally, to account forthe possibility that absenteeism might be a delayedresponse to past workloads, we include the values of14 lagged nurse-to-patient ratios NPRt−j , j = 11 0 0 0 114,which correspond to one calendar week. Because thenumber of past shifts we use is rather arbitrary, weconducted our statistical analysis for several differentvalues to make sure the results are not sensitive to thenumber we choose.

Specifically, the models we estimate are

logit4�t5 = �i0 +�i

ENPR × ENPRit +

14∑

j=1

�iNPR1 j × NPRt−j

+

7∑

d=2

�iDAY1d × DAYd1 t +

44∑

f=2

�iW1f ×Wf 1 t

+�iDAYSHIFT × DAYSHIFTt

+�iHOLIDAY × HOLIDAYt1 (3)

where �t is the probability that a nurse is absentin shift t; i = 112 refers to the definition of ENPRi

used, DAYd1 t and Wf 1 t are the day and week fixedeffects, and DAYSHIFTt and HOLIDAYt are the shiftand holiday fixed effects. The estimation results forEquation (3) are presented in Table 2. Model I usesthe first definition of the anticipated nurse-to-patientratio (ENPR1

t ), whereas Model III uses the second def-inition (ENPR2

t ). To test whether the observed effectof anticipated nurse-to-patient ratio on absenteeismis robust, we also estimated the restricted versionsof Models I and III (which we denote as Models IIand IV), where we omit the 14 lagged nurse-to-patientratio variables. If the lagged nurse-to-patient ratiosare not related to absenteeism (i.e., if �i

NPR1 j = 0 for allj = 11 0 0 0 114), omitting these variables will not intro-duce any bias even if the lagged nurse-to-patient vari-ables are correlated with the variables included inthe model. Model II uses the first definition of theanticipated nurse-to-patient ratio (ENPR1

t ), whereasModel IV uses the second definition (ENPR2

t ).As can be seen from Table 2, the anticipated nurse-

to-patient ratio has a significant effect (at the 5%or 10% level) on absenteeism rates in Models I, III,and IV. In Model II the p-value of the anticipatednurse-to-patient ratio is 1007%. The more nursesscheduled for a particular shift, the less likely eachnurse is to be absent. In particular, according to thefirst model, we estimate that the marginal effect ofstaffing an extra nurse (calculated at the mean val-ues of all remaining independent variables and usingthe expected patient census value of 109) on the indi-vidual absenteeism rate is around 0.572% = 00623/109.In other words, the absenteeism rate would decreasefrom its average value of 7034% to about 6078% when

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Table 2 Estimation Results for Logit Models

Model I Model II Model III Model IV

Coefficient Marginal effect Coefficient Marginal effect Coefficient Marginal effect Coefficient Marginal effectVariable (robust error) (robust error) (robust error) (robust error) (robust error) (robust error) (robust error) (robust error)

ENPR1 −10001∗ −00623∗ −80534 −005334506695 4003525 4502915 4003305

ENPR2 −15024∗∗ −00946∗∗ −13025∗∗ −00827∗∗

4603555 4003935 4508925 4003675NPR1 −00248 −000154 −00474 −000294

4300615 4001905 4300715 4001915NPR2 20463 00153 30024 00188

4304385 4002145 4304595 4002155NPR3 −00492 −000306 −00712 −000442

4209895 4001865 4209825 4001855NPR4 30314 00206 30673 00228

4302655 4002035 4302755 4002035NPR5 −20620 −00163 −20917 −00181

4300205 4001885 4300235 4001885NPR6 50912∗ 00368∗ 50837∗ 00362∗

4302495 4002015 4302515 4002015NPR7 00518 000322 00615 000382

4301555 4001965 4301575 4001965NPR8 10556 000968 10570 000974

4302455 4002025 4302545 4002025NPR9 −40112 −00256 −40252 −00264

4209985 4001865 4300025 4001865NPR10 50656∗ 00352∗ 50746∗ 00357∗

4302055 4001995 4302075 4001985NPR11 −30320 −00207 −30263 −00203

4209805 4001865 4209915 4001865NPR12 20462 00153 20707 00168

4302715 4002045 4302765 4002035NPR13 −20934 −00183 −20724 −00169

4300165 4001885 4300185 4001875NPR14 40934∗ 00307∗ 40823 00299

4209875 4001855 4209785 4001845DAYSHIFT 00112 0000694 00233 000145 00211 000130 00327∗∗ 000203∗∗

4001515 400009375 4001435 400008935 4001635 40001015 4001555 400009595HOLIDAY −00853∗ −000382∗∗∗ −00787∗∗ −000364∗∗∗ −00848∗ −000380∗∗∗ −00783∗ −000362∗∗∗

4004405 40001345 4004015 40001315 4004405 40001355 4004015 40001315Week FE Yes Yes Yes Yes Yes Yes Yes YesDay-of-Week FE Yes Yes Yes Yes Yes Yes Yes YesConstant −20381∗ −20171∗∗∗ −10834 −10726∗∗

4104265 4007215 4104495 4007525Observations 6,481 6,631 6,481 6,631Log-likelihood −1,654 −1,690 −1,653 −1,688LR-test 102.7 92.35 105.3 95.13

Notes. Standard errors are shown in parentheses. Marginal effects are calculated at the mean values of the independent variables. For the dummy variables,the marginal effect shows the difference caused by changing the variable from 0 to 1.

∗, ∗∗, and ∗∗∗ denote statistical significance at the 10%, 5%, and 1% confidence levels, respectively.

an extra nurse is added to the schedule. The log-likelihood is (marginally) higher and the coefficientof the anticipated nurse-to-patient ratio is larger inModels III and IV, where the variation in the numberof scheduled evening nurses is ignored. This mightsuggest that when nurses decide whether to show upfor work, they place greater emphasis on the numberof nurses working in their shift (i.e., ENPR2

t ) ratherthan both the number of nurses working in their shift

and the evening shift that overlaps with their own(i.e., ENPR1

t ).As with any empirical finding, one might argue that

the relationship we find between absenteeism andanticipated workload is a result of reverse causality(i.e., it is not absenteeism that reacts to anticipatedworkload but instead staffing, and thus workload,that reacts to absenteeism). However, in the ED studysite, we know that absenteeism was not considered

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by the nurse manager making staffing decisions.More generally, in discussions with managers respon-sible for nurse staffing in other hospitals, absen-teeism patterns were not tracked or used in staffingdecisions.

Note that the lag 6, lag 10, and lag 14 (lag 6 andlag 10) of the nurse-to-patient ratio variables havepositive coefficients in Model I (Model III) which areindividually significant at the 10% level. This seems toimply that the probability of a nurse being absent onany shift would increase if the shift occurring three,five, or seven days ago had a higher nurse-to-patientratio. Although these lags are individually significant,the Wald test statistic and the likelihood ratio teststatistic for joint significance of all 14 lagged workloadvariables in Model I as well as Model III reject thehypothesis (even at the 10% level) that lagged nurse-to-patient ratios have any joint explanatory power.In the absence of any plausible explanation as to whya lighter workload on a similar shift occuring three,five, or seven days ago might increase absenteeism,and in light of the weakness of this statistical relation-ship, we are inclined to treat this result as spurious.

Interestingly, the holiday variable’s effect is signif-icant (at 10% confidence level) and negative, thussuggesting that nurses are about 308% less likely tobe absent on public holidays. Week fixed effects arejointly significant (at the 1% level) in all four mod-els. One of the effects that week dummies seem todetect reasonably well is the impact of weather (inparticular, heavy snow conditions) on absenteeism.For example, week 35 of our data set includes March2–4, 2009. During six day and night shifts correspond-ing to these dates, snow on the ground in New YorkCity was recorded to be more than five inches,2 thelevel identified by the New York Metropolitan Trans-portation Authority as the one at which the publictransportation disruptions are likely to set in.3 Theweek-35 fixed effect is positive and significant (at the5% level) in Models II and IV with a marginal effectequal to 8075% in Model I and 9080% in Model IV.Only three other days in our data set had as muchsnow on the ground (December 3, December 21, andJanuary 20). Turning to the impact of day-of-weekfixed effect, there is some (weak) evidence that nursesare more likely to be absent on weekends. They arealso more likely to be absent during a day shift,when conflicting family obligations is often cited asan important reason behind nurse absenteeism (Erick-son et al. 2000, Nevidjon and Erickson 2001) are likelyto be more prevalent.

2 http://www.accuweather.com/.3 http://www.mta.info/news/stories/?story=173 (last accessed Jan-uary 23, 2011).

2.2. Verification TestsTo check the validity of our model estimation pro-cedure, we have also conducted several verificationtests as described below. (Estimation details are avail-able from the authors.)

2.2.1. Alternative Specifications. To ensure the ro-bustness of our results, we estimated a number ofalternative modeling specifications. Namely, we esti-mated the models of Equation (3) under the probitspecification. We also estimated variants of our mod-els that use month fixed effect variables instead ofweek fixed effect variables. The results were almostidentical in terms of variable significance, model sig-nificance, and magnitude of marginal effects. Themodel of Equation (3) assumes that any differencebetween the night shift and the day shift is com-pletely captured by the dummy variable DAYSHIFT.However, it is possible that the two shifts are inher-ently different, and so are their best-fit coefficients.To test the hypothesis that the coefficients for twoshifts differ, we fitted an unrestricted binary choicemodel to the data for each of the two shifts sepa-rately and compared the fit with the restricted modelby constructing a likelihood ratio test. The test doesnot reject the hypothesis that the coefficients are thesame at the 5% confidence level in all models. Finally,we estimated models with alternative definitions ofthe expected nurse-to-patient ratio. More specifically,when estimating the expected patient census duringthe upcoming shift in Equation (2), nurses (a) distin-guish between day and night shifts, but not betweendays of the week; (b) distinguish between days of theweek, but not between day shifts or night shifts; and(c) distinguish between both the shift type and theday of the week. Our main finding that higher (lower)anticipated nurse-to-patient ratios decrease (increase)nurse absenteeism is robust to these alternative mod-eling specifications.

2.2.2. Nurse Heterogeneity and Aggregation Bias.The aggregate nature of our data does not permitan exact characterization of how workload impactsthe absenteeism behavior of each individual nurse.Indeed, it would be interesting to measure whetheraversion to anticipated increased workloads is a com-monly occurring nurse characteristic or limited to arelatively small subset of nurses. However, our aggre-gate results are not invalidated by the lack of sucha characterization. Through a simulation study, wefind that estimating a homogeneous logit model whennurses are in fact heterogeneous does produce biasedestimation coefficients. However, we find that thebias on the ENPR coefficient is positive, that is, itwould bias our (negative) coefficient toward zero andnot away from it. Therefore, our estimate should betreated as a lower bound on how workload affects

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nurse absenteeism. Furthermore, we find that formost reasonable assumptions about nurse heterogene-ity, the magnitude of the bias is small compared to theestimated value of the coefficient. This result is simi-lar to that of Allenby and Rossi (1991), whose analysisof marketing data lead the authors to conclude that inmost realistic settings there exists no significant aggre-gation bias in logit models.

2.2.3. Within-Shift Correlation. One of the as-sumptions we have made when estimating our mod-els is that, conditional on the vector of covariates xt ,the nurses are independent decision makers. In thissection, we relax this assumption and assume thatnurses can exhibit within-shift correlation. We esti-mate a model within the generalized estimating equa-tions framework, which allows the individual nursedecisions Yn1 t to exhibit within shift correlation. Thecorrelation structure estimated is of the “exchange-able form,” that is, Corr4Yn1 t1Ym1 l5 = � for t = l, andis 0 otherwise. The least-square estimation of thesemodels yields very similar results to those presentedin Table 2, and the estimated correlation � is aroundnegative 1%.

2.3. Discussion and Implicationsof the Empirical Study

Our empirical investigation has uncovered a linkbetween absenteeism and workload at the shift level.We find that nurses are rational and forward lookingin their decision whether to attend or to be absentfrom work. When an extra nurse is added to theschedule, the absenteeism rate decreases on averagefrom 7.34% to 6.78%, a relative decrease of 7.8%. Thisparticular behavior, where nurses are more likely notto show up when the workload per nurse is alreadyhigh, if not appropriately countered, will exacerbatean already difficult situation for the hospital by cre-ating more workload for the nurses that do show up.Such a positive feedback in workload (i.e., high work-load generates even higher workload) can have direconsequences for patient safety (see Needleman et al.2002) unless mitigating actions take place. However,there is a silver lining to our finding in the sense thatthe endogeneity of absenteeism might represent anopportunity for hospitals. In the presence of endoge-nous absenteeism, scheduling an extra nurse might bemore beneficial and even more cost effective than hos-pital managers might realize because this extra nurseincreases the probability that all of her colleagues willshow up for work, therefore reducing the costs associ-ated with absenteeism. Failing to account for this fun-damental additional marginal benefit, which is purelydue to the endogeneity of absenteeism, might lead tochronic understaffing.

To further investigate the implications of absen-teeism and, in particular, its endogenous nature with

regard to staffing level decisions, we construct a styl-ized model of nurse staffing. The aim of our modelis to generate managerial insights on the impact ofnurse absenteeism, in general, and the endogenousnature of absenteeism, in particular, as it affects thedecision of how many nurses to staff. Although ourmodel is parsimonious, we believe that an appropri-ately calibrated version of it can be used by nurs-ing management in making tactical staffing decisions.In particular, by periodically running the model forall types of shifts (which differ in the distribution ofpatient census and in nurse absenteeism propensity),the nurse manager can decide how many nurses theunit will need for each shift. Thus, coupled with ros-tering considerations, the model can help the managerdecide the appropriate aggregate staffing level for theunit. In addition to providing insight on the aggre-gate level of permanent nurses needed by observinghow many agency or overtime nurses are required onaverage to cover for last minute mismatches betweendemand and supply, our model can help to deter-mine how many such “flexible” nurses the hospitalwill need to maintain adequate nurse staffing levels.

3. Endogenous Nurse Absenteeism:Implications for Nurse Staffing

We begin our nurse staffing model by outlining thekey assumptions. We assume that a clinical unit usesthe primary nursing care (PNC) mode of nursingcare delivery (Seago 2001), which was used in theED we studied. Under the PNC mode, the nursingstaff includes only registered nurses (as opposed tolicensed practical nurses or unlicensed nursing per-sonnel) who provide all direct patient care through-out the patients’ stay in the clinical unit. The nursestaffing process starts several weeks in advance of theactual shift for which planning is performed. It is thenthat a hospital staff planner needs to decide how manynurses (y) to schedule for that particular shift. Becauseof the phenomenon of absenteeism, the actual numberof nurses who show up for work on that shift, N , isuncertain. We model N as a binomial random variableB4y11 − �4y55, where �4y5 is the probability that anyscheduled nurse will be absent from work:

Prob4N =k �y1�4y55

=p4k3y1�4y55=

y!

k!4y−k5!4�4y55y−k41−�4y55k

for 0≤k≤y,

0 otherwise.

(4)

We assume that the clinical unit follows a policy ofspecifying, for each value of the average patient cen-sus during a shift, C, a target integer number ofnurses T =R4C5 required to provide adequate patient

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care during a particular shift. We assume that Ctakes on discrete values and that R4C5 is a monotoneincreasing function with R405 = 0. A simple exam-ple of R4C5 is provided by a “ratio” approach underwhich R4C5 = ��C�, with � ∈ 60117 representing amandated nurse-to-patient ratio. Alternately, if a clin-ical unit is modeled as a queueing system in whichpatients generate service requests and nurses playthe role of servers, as was done in Yankovic andGreen (2011), R4C5 can take a more complex form toensure that certain patient service performance mea-sures, such as the expected time patients wait to beserved, conform to prespecified constraints.

At the time of the nurse staffing decision, we assumethat the decision maker uses a known probability den-sity function of the average patient census C duringthe shift for which personnel planning is conducted,Prob4C = n5 = pC4n5, n ∈ N+,

∑�

n=0 pC4n5= 1. We treatthe demand uncertainty expressed by the patient cen-sus C and the supply uncertainty expressed by N asbeing independent and assume that the realized val-ues of C and N become known shortly before thebeginning of the shift. Any nursing shortage 4R4C5−

N5+ is covered by either hiring agency nurses or ask-ing nurses who have just completed their shift to stayovertime. We further assume that nurses that do showup are paid $wr per shift, whereas nurses that do notshow up are paid $wn, where wr ≥wn. Setting wr =wn

represents a clinical unit where nurses are paid the fullwage whether or not they actually show up for work.This setting is consistent with the PNC mode of nurs-ing care delivery where nurses are salaried employeeswho can only be scheduled to work on a fixed numberof shifts per week. When a scheduled nurse does notshow up for work, she cannot be rescheduled in lieuof the shift she missed. Thus, in effect, nurses are paidfor each shift for which they are scheduled and arenot penalized for being absent, as long as their absen-teeism does not exceed the annual limit of 10 personaldays. In contrast, setting wn = 0 represents a settingwhere nurses receive no pay when absent. In addi-tion, we assume that if more nurses show up for workthan required given the number of patients present(N >R4C5) they all have to be paid and cannot be senthome. The per-shift cost of extra/overtime nurses is$we, where we ≥wr .

The goal of the decision maker is to choose a nursestaffing level y that minimizes the expected cost W4y5of meeting the target R4C5:

W4y5 = wny+ 4wr −wn5EN 6N � y7

+weEC1N 64R4C5−N5+ � y71 (5)

where EN denotes expectation taken with respect tothe number of nurses who show up for work andEC1N denotes expectation taken with respect to both

the number of patients and the number of nurses whoshow up for work. Note that because there is a one-to-one correspondence in our model between the patientdemand C and the number of required nurses T ,we can recast the calculation of the expectation withrespect to the demand value in terms of an equivalentcalculation over the distribution of T using the corre-sponding probability distribution function. In partic-ular, let Sn be the set of average patient census valuesall corresponding to the same number of requirednurses n: Sn = 4C ∈N+ �R4C5= n5. Then the probabil-ity distribution for T is given by Prob4T = n5= pT 4n5,n ∈ N+, pT 4n5 =

∑

l∈SnpC4l5,

∑�

n=0 pT 4n5 = 1. In turn,the cost minimization based on (5) becomes

miny∈N+

(

wny+ 4wr −wn541 −�4y55y

+weET 1N 64T −N5+ � y7)

0 (6)

Note that with no absenteeism (�4y5 = 0), the num-ber of nurses showing up for work N is equal to thenumber of scheduled nurses y, and the nurse staffingproblem reduces to a standard newsvendor modelwith the optimal staffing level given by

y∗

0 = min(

y ∈N+

∣

∣

∣

∣

FT 4y5≥ 1 −wr

we

)

1 (7)

with FT 4y5 =∑y

n=0 pT 4n5 being the cumulative den-sity function of the demand function evaluated at y,and the value 1 − wr/we playing the role of the crit-ical newsvendor fractile. Below we present an anal-ysis of the staffing decision (6) starting with thecase of exogenous absenteeism, which we use as abenchmark.

3.1. Optimal Nurse Staffing UnderExogenous Absenteeism Rate

Consider a clinical unit that experiences an endoge-nous nurses’ absenteeism rate �4y5, but treats it asexogenous. For example, the schedule planner usesthe average value of all previously observed dailyabsenteeism rates, �ave. The cost function to be mini-mized under this approach is given by

Wave4y5 = y4wr 41 −�ave5+wn�ave5

+we

y∑

k=0

�∑

n=0

4n− k5+pT 4n5p4k3y1�ave5

= wy+we

y∑

k=0

q4k5p4k3y1�ave51 (8)

where w = wr 41 − �ave5 + wn�ave is the effective costper scheduled nurse, and

q4k5=

�∑

n=0

4n− k5+pT 4n5 (9)

represents an expected nursing shortage given that kregular nurses show up for work. The optimal staffinglevel in this case is expressed by the following result.

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Proposition 1. (a) The minimizer of (8) is given by

y∗

ave = min(

y ∈N+

∣

∣

∣

∣

y∑

k=0

FT 4k5p4k3y1�ave5

≥ 1 −wr 41 −�ave5+wn�ave

we41 −�ave5

)

1 (10)

and is a nonincreasing function of wr/we and wn/we.(b) Consider two cumulative distribution functions for

the required number of nurses T , F 1T 4k5 and F 2

T 4k5 suchthat F 1

T 4k5≥ F 2T 4k5 for all k ∈N+, and let y∗1 i

ave be the opti-mal staffing levels corresponding to F i

T 4k5, i = 112. Then,y∗11

ave ≤ y∗12ave .

We relegate all proofs to the appendix. Notethat (10) represents a generalization of the expressionfor the optimal staffing levels without absenteeism (7).As in the no-absenteeism setting, it is never optimal todecrease staffing levels when the target nursing levelincreases or when the cost advantage associated withearlier staffing becomes more pronounced. While thisbehavior of the optimal policy is intuitive, the depen-dence of the optimal staffing levels on the value ofthe absenteeism rate is not as straightforward. In par-ticular, depending on the interplay between the ratiosof the cost parameters wr/we and wn/we, the charac-teristics of the target nursing level distribution, andthe absenteeism rate, the increase in the absenteeismrate can increase or decrease the optimal staffing level.The following result describes the properties of theoptimal staffing levels in general settings.

Proposition 2. (a) There exists �uave such that the opti-

mal staffing level y∗ave is a nonincreasing function of the

absenteeism rate �ave for all �ave ∈ 6�uave1 17.

(b) For

wn≤we

(⌈

F −1T

(

1−wr

we

)⌉)

pT

(⌈

F −1T

(

1−wr

we

)⌉)

1 (11)

there exists � lave such that the optimal staffing level y∗

ave isa nondecreasing function of the absenteeism rate �ave forall �ave ∈ 601� l

ave7.

Figure 1 Optimal Staffing Level as a Function of the Absenteeism Rate for Different Values of the Cost Ratio w/we for w = wr = wn and theEmpirical Targeted Nursing Level Distribution

Absenteeism rate, �ave Absenteeism rate, �ave

0.1 0.2 0.3 0.4 0.5

10

11

12

13

14

15= 0.5we

wr = 0.7we

wr

0.1 0.2 0.3 0.4 0.5

2

4

6

8

10

Opt

imal

sta

ffin

g le

vel,

y ave*

Opt

imal

sta

ffin

g le

vel,

y ave*

A more detailed characterization of the optimalstaffing levels can be obtained for some target nursinglevel distributions, for example, for a discrete uniformdistribution.

Corollary 1. Let

FT 4k5=

k+ 1Tmax + 1

for 0 ≤ k ≤ Tmax1

1 k ≥ Tmax0

(12)

Then, for w/we ≥ 14 , the optimal nurse staffing level is

given by

y∗

ave =

⌈(

Tmax

1 −�ave−

Tmax + 141 −�ave5

2

w

we

)⌉

1 (13)

and is a nondecreasing (nonincreasing) function of �ave for�ave ≤ �u

ave (�ave >�uave), where

�uave

=max(

011−max(

0124Tmax +15wr

Tmaxwe−4Tmax +154wr −wn5

))

0

(14)

To illustrate the monotonicity properties of the opti-mal staffing levels formalized in Proposition 2, weuse the distribution for the number of required nursesobtained from our empirical data for the averagepatient census using 1-to-10 nurse-to-patient ratio. Forthis distribution, Figure 1 shows the dependence ofthe optimal staffing level on the absenteeism rate forw =wr =wn. For a given value of the cost ratio w/we,there exists a critical value of the absenteeism rate�u

ave for which the optimal response to an increasein absenteeism switches from staffing more nurses tostaffing fewer. Note that irrespective of the distribu-tion for targeted nursing level, for high values of theabsenteeism rate or high values of the cost ratio w/we

(to be precise, for �ave ≥ 1 − w/we), it is more cost-effective not to staff any nurses in advance and to

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rely exclusively on the extra/overtime mechanismsof supplying the nursing capacity. For low values ofthe absenteeism rate and low values of the cost ratiow/we, higher absenteeism can induce an increase instaffing levels, as it is cheaper to counter the increasedabsenteeism by staffing more nurses. However, as thecost ratio w/we increases, it becomes more cost effec-tive to staff fewer nurses, relying increasingly on theextra/overtime supply mechanism.

3.2. Endogenous Absenteeism: Optimal StaffingIn the endogenous absenteeism setting, the expectedstaffing cost (6) becomes

W4y5= ywr − 4wr −wn5a4y5+weL4y1�4y551 (15)

wherea4y5= y�4y5 (16)

is the expected number of absent nurses, and

L4y1�4y55=

y∑

k=0

q4k5p4k3y1�4y551 (17)

where q4k5 is defined by (9), p4k3y1�4y55 is the prob-ability mass function of the binomial distribution,where k nurses show up for work when y are sched-uled and �4y5 is the (endogenous) probability of anurse being absent. Note that for general absenteeismrate function �4y5, the increasing marginal property ofthe “exogenous” staffing cost function (8) with respectto the number of scheduled nurses may not hold.Below we formulate a sufficient condition for thisproperty to be preserved under endogenous absen-teeism. First, for a given distribution of the targetednursing level pT 4k51 k ≥ 0, we introduce the followingquantity:

�T 4y5 = 1 − min(

11(

∑�

k=y−2 pT 4k5

ypT 4y− 15+ pT 4y− 25

)1/4y−15)

1

y ∈N+1 y ≥ 20 (18)

As shown below, (18) represents one of the bounds onthe absenteeism rate function that ensures the opti-mality of the greedy-search approach to finding theoptimal nurse staffing level.

Proposition 3. Let �4x5 ∈ C2401�5, 0 ≤ �4x5 ≤ 1 bea nonincreasing, convex function defined on x ≥ 0. Con-sider an endogenous absenteeism setting characterized bythe absenteeism rate �4y5 for y ∈ N+ scheduled nurses.Then, the optimal staffing level is given by

y∗= min

(

y ∈N+

∣

∣

∣

∣

L4y+ 11�4y+ 155−L4y1�4y55

≥ −wr

we

41 −�4y55−wn

we

�4y5

+wr −wn

we

y4�4y+ 15−�4y55

)

(19)

and is a nonincreasing function of wr/we, provided that

�4y5≤ min(

2y1�T 4y5

)

(20)

andd2a4y5

dy2≤ 0 (21)

for any y ≥ y∗. In addition, consider two cumulative dis-tribution functions for the required number of nurses T ,F 1T 4k5 and F 2

T 4k5 such that F 1T 4k5 ≥ F 2

T 4k5 for all k ∈ N+,and let y∗1 i be the optimal staffing levels corresponding toF iT 4k5, i = 112. Then, y∗11 ≤ y∗12, provided that (20) holds

for any y ≥ y∗12.

Proposition 3 essentially describes a greedy-searchalgorithm for finding the optimal number of nurses toschedule. Starting from a chosen number of nurses y,the nursing manager can start adding (or subtracting)nurses as long as the total costs continue decreasing.At y∗, where the costs stop decreasing, the nursingmanager can stop the search. While this is a one-dimensional optimization problem, we would arguethat the greedy-search approach is useful because thesums over the probability mass of the patient census(see (9)) and nurses present (see (17)) required to cal-culate the costs of this problem can be time consum-ing to estimate.

The fact that such a greedy-search approach is opti-mal requires the sufficient condition (20) that states,intuitively, that the increasing marginal shape of thestaffing cost function with respect to the numberof scheduled nurses is preserved under endogenousabsenteeism if the absenteeism rate is not too high.Thus, (15) is not too different from the cost func-tion in (6). More specifically, this sufficient conditionrequires that the absenteeism rate function is lim-ited from above by two separate bounds. The firstbound implies that the expected number of absentnurses does not exceed two irrespective of the numberof nurses actually scheduled for work. The sufficientcondition (21) requires that the expected number ofabsent nurses exhibits nonincreasing returns to scale.

To study the endogenous absenteeism case further,we use a parametric specification consistent with ourempirical findings, in particular with the logit modelspecification. We specify that

�4y5=1

1 + e�+�y1 (22)

where both � and � are positive constants. Theassumption about positive values for these absen-teeism rate parameters is plausible in a wide rangeof settings; � > 0 implies that the absenteeismrate declines with the number of scheduled nurses,whereas � > 0 ensures that the absenteeism rate is

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not too high even when the number of schedulednurses is low and the anticipated workload is high.In particular, evaluating the best-fit logit model in (3),using the estimates reported in Table 2, we obtain �=

−�1ENPR/E6Ct7 = 00092, with �1

ENPR = −10001, E6Ct7 =

10900. For the average absenteeism rate to match oursample average of 7034%, we set �= 10533. Note thatthe endogenous absenteeism rate �4y5 characterizedby the logistic function given by (22) with � ≥ 0 and�≥ 0 is a monotone decreasing convex function. Thus,the result of Proposition 3 is ensured by the followingrestrictions on the values of � and �:

Lemma 1. For �4y5 = 1/41 + e�+�y5, with �, � ≥ 0,�e1+�+2� ≥

12 implies (20), and �≤ 2 implies (21).

In the ED we studied, the estimated values of � =

10533 and �= 00092 satisfy Lemma 1. In particular, themaximum value of the product of number of sched-uled nurses y and the estimated absenteeism ratecalculated using these values is equal to 00804, wellbelow 2. The second bound on the right-hand sideof (20) takes the form of an effective absenteeism ratefunction that depends exclusively on the distributionof the targeted nursing level. Note that �T 4y5 ≥ 0 ifand only if

pT 4y− 15∑�

k=y−1 pT 4k5≥

1y0 (23)

The expression on the left-hand side of (23) is thehazard rate function for the distribution of the tar-geted nursing level. Thus, (23) stipulates that thebound described by (20) is meaningful only in set-tings where such a hazard rate evaluated at y exceeds1/4y+ 15. For the absenteeism rate function givenby (22), the constraint �4y5 ≤ �T 4y5 implies, in thesame spirit as Lemma 1, the lower-bound restrictionon the values of � and �: �4y5 ≤ �T 4y5 ⇔ � + �y ≥

log441 −�T 4y55/�T 4y55. Figure 2 compares the absen-teeism rate (22) computed for �= 10533 and �= 00092,with the effective absenteeism rate �T 4y5 from (18)

Figure 2 Absenteeism Rate �4y 5 Computed for �= 10533 and�= 00092 and the Effective Absenteeism Rate �T 4y 5

Computed Using the Empirical Targeted NursingLevel Distribution

Staffing level y

Abs

ente

eism

rat

e

3025201510

0.2

0.4

0.6

0.8

1.0

5

� (y)

�T (y)

computed using the empirical targeted nursing leveldistribution. Note that the sufficient condition ofProposition 3 (�4y5 ≤ �T 4y5) is satisfied for virtuallyany staffing level above 10 nurses, which is approx-imately equal to the expected number of requirednurses in our setting.

3.3. Endogenous Absenteeism: Implications ofModel Misspecification on Staffing Decisions

In this section, we compare the optimal nurse staffinglevels with those made by a clinical unit that incor-rectly treats the absenteeism rate as exogenous anduses a trial-and-error procedure under which theassumed exogenous absenteeism rate is updated everytime a new staffing decision in made. In this lattercase, which we label “misspecified-with-learning,” theclinical unit selects staffing level yML such that

yML= min

(

y ∈N+

∣

∣

∣

∣

y∑

k=0

FT 4k5p4k3y1�4yML55

≥ 1 −wr 41 −�4yML55+wn�4y

ML5

we41 −�4yML55

)

1 (24)

with �4y5 denoting the endogenous absenteeism rate.Note that (24) reflects a self-consistent way of select-ing the staffing level; yML is the best staffing decisionin the setting where the absenteeism rate is exogenousand determined by �4yML5. In other words, a clinicalunit assuming that the absenteeism rate is given byconstant value �4yML5 will respond by scheduling yML

nurses and, as a result, will observe exactly the samevalue of the absenteeism rate, even if the true absen-teeism process is endogenous and described by �4y5.An intuitive way of rationalizing the choice of yML isto consider a sequence of exogenous staffing levels yn,n ∈N+, such that

yn+1 =min(

y∈N+

∣

∣

∣

∣

y∑

k=0

FT 4k5p4k3y1�4yn55

≥1−wr 41−�4yn55+wn�4yn5

we41−�4yn55

)

1 n∈N+0 (25)

Equation (25) reflects a sequence of repeated adjust-ments of staffing levels, starting with some y0, eachbased on the value of the absenteeism rate observedafter the previously chosen staffing level is imple-mented. In this updating scheme, yML can be thoughtof as the limit, limn→� yn, if such a limit exists. Notethat for a general demand distribution FT 4k5 and ageneral absenteeism rate function �4y5, the set ofstaffing levels E satisfying (24) may be empty or maycontain multiple elements. The analysis of existenceand uniqueness of yML is further complicated by thediscrete nature of staffing levels. In the following dis-cussion, we bypass this analysis and assume thatthere exists at least one staffing level satisfying (24).

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Figure 3 Worst-Case Performance Gaps Between the Optimal Staffing Policy and the Misspecified-with-Learning (ML) and Exogenous Policies asFunctions of the Average Absenteeism Rate �ave Under the Empirical Targeted Nursing Level Distribution for wn/wr = 1100510

0.02 0.04 0.06 0.08 0.10 0.12 0.14

0.01

0.02

0.03

0.04

0.05

0.06

�ave

Max performance gap,W(y) – (W(y*)ˆW(y)ˆ

= 1wr

wn

y = yave

y = yML

0.02 0.04 0.06 0.08 0.10 0.12 0.14�ave

Max performance gap,W(y) – (W(y*)ˆW(y)ˆ

= 0.5wr

wn

y = yave

y = yML

0.004

0.006

0.008

0.010

0.012

0.02 0.04 0.06 0.08 0.10 0.12 0.14�ave

Max performance gap,W(y) – (W(y*)ˆW(y)ˆ

= 0wr

wn

y = yave

y = yML

0.001

0.002

0.003

0.004

0.005

As the following result shows, even if E contains mul-tiple elements, each of them is bounded from aboveby the optimal endogenous staffing level in settingswhere the expected number of absentees decreaseswith the number of scheduled nurses.

Proposition 4. Suppose that the conditions of Proposi-tion 3 hold and that the set of staffing levels satisfying (24),E, is nonempty. Then, yML ≤ y∗, for any yML ∈ E, providedthat, at the optimal staffing level y∗, the expected num-ber of absent nurses decreases with the number of nursesscheduled, a4y∗ + 15 < a4y∗5.

Proposition 4 implies that ignoring the endogenousnature of absenteeism can lead to understaffing in set-tings where both the endogenous absenteeism rate andthe expected number of absent nurses decline with thenumber of scheduled nurses. Figure 3 illustrates theresults of the numerical experiment designed to quan-tify a potential cost impact of using heuristic staffingpolicies for realistic values of problem parameters. Inour study, we have varied the cost ratio wr/we fromwr/we = 005 to wr/we = 009. The lower limit of thisinterval, wr/we = 005, corresponds to the setting inwhich use of agency nurses carries a 100% cost pre-mium, a realistic upper bound on the values encoun-tered in practice. The upper limit, wr/we = 009, reflectsthe use of overtime to compensate for absenteeism,with we at about 10% premium with respect to wr .In the ED we studied, absent nurses were paid at thesame rate as the nurses who showed up for work (sothat wn = wr ). To investigate the effect of lower com-pensation levels for absent nurses, we have includedthe cost ratios wn = 005wr and wn = 0. As the absen-teeism rate function, we have used �4y5 = 1/41 + e�y5with the value of � varied to explore different averageabsenteeism rates calculated as

�ave =

ymax∑

y=ymin

pT 4y5

1 + e�y1 (26)

where ymin = 6 and ymax = 16 reflect the smallest andlargest possible targeted nursing level realizations,and pT 4y5 reflects the empirical distribution for the

required number of nurses. In our study, we havecompared the performance of three staffing policies:the optimal policy described in Proposition 3, theML policy described by (24), and the exogenous pol-icy that assumes that the absenteeism rate does notdepend on the staffing level and is given by (26). Fig-ure 3 shows the worst-case performance gaps (calcu-lated over the range of cost ratios wr/we ∈ 600510097)of the ML and the exogenous policies as functions of�ave for three ratio levels wn/wr = 11005, and 0. Ournumerical results indicate that in the settings wherethe average absenteeism rate �ave is small, ML andexogenous policies represent good approximations forthe optimal staffing policies. For example, in the EDwe studied, the average absenteeism rate was 7034%,and the corresponding worst-case performances forthese two policies were between 2% and 3% forwn =wr . However, as �ave increases, so does theworst-case performance gap for both policies: In par-ticular, in the same setting, the worst-case perfor-mance gaps approximately double to 4% (6%) for theexogenous (ML) policy when the average absenteeismrate reaches 15%. As it turns out, a reduction in theamount of hourly compensation paid to absent nursessignificantly closes these performance gaps for bothpolicies: For wn = 0, the maximum performance gapsdrop below 1%. Figure 4 displays similarly definedperformance gaps obtained by focusing on the aver-age absenteeism rate of 7.34% we have observed, andby varying the value of � within the 95% confidenceinterval around the estimate �= 001398 obtained inModel III, namely, � ∈ 6001398 − 1096 × 005831001398 +

1096 × 005837= 600025510025417. The values of the per-formance gaps in Figure 4 are similar to those inFigure 3, displaying, as expected, a substantial dropas the value of � shifts toward the lower limit of theconfidence interval.

4. DiscussionNurses constitute one of the most important resourcesin hospitals, both in terms of cost and clinical out-comes. Therefore, any insights into how to deploy

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Figure 4 Worst-Case Performance Gaps Between the Optimal Staffing Policy and the Misspecified-with-Learning (ML) and Exogenous Policies asFunctions of � for �ave = 7034% Under the Empirical Targeted Nursing Level Distribution for wn/wr = 1100510

Max performance gap,W(y) – (W(y*)ˆW(y)ˆ

= 1wr

wn

y = yave

y = yML

0.05 0.250.200.150.10

0.005

0.010

0.015

0.020

0.025

0.030

�

Max performance gap,W(y) – (W(y*)ˆW(y)ˆ

= 0.5wr

wny = yave

y = yML

0.002

0.004

0.006

0.008

0.05 0.250.200.150.10�

Max performance gap,W(y) – (W(y*)ˆW(y)ˆ

= 0wr

wn

y = yave

y = yML

0.0005

0.0010

0.0015

0.0020

0.05 0.250.200.150.10�

nurses more effectively are of great interest to hospi-tal managers. This paper focuses on the issue of nurseabsenteeism, a problem that has vexed hospital man-agers for a long time. Using data from a large urbanhospital ED, we find that nurse absenteeism is exac-erbated when fewer nurses are scheduled for a partic-ular shift. This finding highlights the need for hospi-tal managers to use better methods to identify nursestaffing levels that are adequate to handle the antic-ipated workload. Our study relies on aggregate datafrom a single department and thus does not permita detailed investigation of the impact of workload atthe individual nurse level. We leave such an extensionto future research. It is, nevertheless, the first study todemonstrate that staffing decisions have an impact onshift-level worker absenteeism—a fact that seems notto have been examined in prior staffing literature.

We analytically examine the implications of absen-teeism, both exogenous and endogenous, on opti-mal staffing policies. To do this, we develop anextension to the single-period newsvendor model,which explicitly accounts for uncertainty in patientcensus and in the number of nurses that showup for work. Our work suggests that the presenceof endogenous absenteeism gives rise to systematicunderstaffing, which, in turn, has important practi-cal consequences for hospitals. First, as our modeldemonstrates, endogenous absenteeism gives rise tonoticeable cost increases even in settings with lowabsenteeism rates, as long as absenteeism exhibits asubstantial degree of endogeneity and as long as mon-etary compensation for absent nurses is comparableto that of nurses who show up for work. For modelparameters that represent the hospital we study, wefind that the cost of ignoring the endogenous natureof absenteeism can be about 2% to 3%. Second, suchchronic understaffing harms patients, especially in thelife-and-death setting of an ED. Third, it is likely to bea contributing factor to widely reported nurse job dis-satisfaction (Aiken et al. 2002). Our research points toan important opportunity for cash-constrained hospi-tals to improve quality of patient care as well as nurseworking conditions, while reducing operating costs.

Turning to the specific context of our analysis, notethat our assumption about the unlimited availabilityof extra/overtime nursing capacity may not be validin some clinical environments. In such environmentsit may be impossible to replace absent nurses at areasonable cost or in reasonable time, and the endo-geneity of absenteeism can lead to significant under-staffing with the possibility of serious deterioration ofservice quality and longer ED delays. In other clini-cal units the use of agency nurses who may be lessfamiliar with the unit can lead to similar declines inquality of patient care and an increase in the rateof medical errors. Thus, an accurate understandingof the nature of nurse absenteeism and the use of amodel that accurately incorporates this phenomenonin determining appropriate staffing levels is impera-tive to ensuring high-quality patient care.

Other than adjusting staffing levels, endogenousabsenteeism may be addressed with a number of com-plementary initiatives. Of particular value is the useof methodologies that lead to better matching of sup-ply and demand through more effective allocationof nurses across units/shifts (Wang and Gupta 2012)and more accurate forecasting of patient census. Alsoof value are interventions that target the organiza-tional culture of burnout (Maslach et al. 2012). Futureresearch should investigate and compare the efficacyand cost effectiveness of such interventions.

AcknowledgmentsThe authors thank Dr. Robert Green, associate director ofEmergency Medicine, New York–Presbyterian Hospital, andGermaine Nelson, director of nursing, emergency roomadministration, New York–Presbyterian Hospital, for theircooperation in obtaining the required data and informationto conduct this research.

AppendixBelow we present outlines of the proofs of the analyticalresults.

Proof of Proposition 1. Note that

Wave4y+ 15−Wave4y5 = w+we

(

L4y+ 11�ave5−L4y1�ave5)

= w+weãL4y1�ave50

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We establish the result of the proposition by showingthat ãL4y1�ave5 ≤ 0 and ã2L4y1�ave5 = L4y + 21�ave5 −

2L4y + 11�ave5 + L4y1�ave5 ≥ 0. First, note that q4k + 15 −

q4k5 = −∑�

n=k+1 pT 4n5 ≤ 0 and q4k + 25 − 2q4k + 15 + q4k5 =

pT 4k+15≥ 0. Then, ãL4y1�ave5=∑y+1

k=0 q4k54p4k3y+11�ave5−p4k3y1�ave55, where we have used p4y + 13y1�ave55 ≡ 0.Next, using

p4k3y+ 11�ave5 = 41 −�ave5p4k− 13y1�ave5

+�avep4k3y1�ave51 k = 01 0 0 0 1 y1 (27)

we get

y+1∑

k=0

q4k5(

p4k3y+11�ave5−p4k3y1�ave5)

= 41−�ave5y∑

k=0

4q4k+15−q4k55p4k3y1�ave5≤01 (28)

where we have usedy+1∑

k=0

q4k5p4k− 13y1�ave5=

y+1∑

k=0

q4k+ 15p4k3y1�ave50

Note that p4k − 13y1�ave5 ≡ 00 From p4y + 23y1�ave5 ≡ 0and (27), we get

ã2L4y1�ave5 = 41−�ave52(y+2∑

k=0

4q4k+15−q4k55

·(

p4k−13y1�ave5−p4k3y1�ave5)

)

0 (29)

Note thaty+2∑

k=0

(

q4k+ 15− q4k5)

p4k− 13y1�ave5

=

y+2∑

k=0

4q4k+ 25− q4k+ 155p(

k3y1�ave

)

1

where we have used p4−13y1�ave5 ≡ p4y + 13y1�ave5 ≡

p4y+ 23y1�ave5≡ 0. Thus, we obtain for

ã2L4y1�ave5= 41 −�ave52( y∑

k=0

(

q4k+ 25− 2q4k+ 15+ q4k5)

· p4k3y1�ave5

)

≥ 00

Furthermore, note that ãL4y1�ave5≥ −w/we is equivalent toy∑

k=0

FT 4k5p4k3y1�ave5≥ 1 −w/4we41 −�ave550

Now, consider F 1T 4k5 and F 2

T 4k5 such that F 1T 4k5 ≥ F 2

T 4k5for any k ∈ N+. Then, ãL14y1�ave5 ≥ ãL24y1�ave5 for anyy ∈N+ and, respectively, y∗11

ave = min4y ∈ N+ � ãL14y1�ave5 ≥

−w/we5≤ min4y ∈N+ �ãL24y1�ave5≥ −w/we5= y∗12ave . �

Proof of Proposition 2. First, we introduce G4y1�5 =∑y

k=041 − FT 4k55p4k3y1�5 and note that

G4y+ 11�5 = 41 −�5y+1∑

k=1

41 − FT 4k55p4k− 13y1�5

+�y∑

k=0

41 − FT 4k55p4k3y1�51 (30)

where we have used (27). Then,

G4y+11�5 ≤ 41−�5y∑

k=0

41−FT 4k55p4k3y1�5

+�y∑

k=0

41−FT 4k55p4k3y1�5=G4y1�50 (31)

Furthermore,

¡G

¡�= y

(y−1∑

k=0

41 − FT 4k554y− 15!

k!4y− 1 − k5!4�y−1−k41 −�5k5

)

− y

( y−1∑

k−1=0

41 − FT 4k55y!

4k− 15!4y− 1 − 4k− 155!

· 4�y−1−4k−1541 −�5k−15

)

= y

(y−1∑

k=0

4FT 4k+ 15− FT 4k55p4k3y− 11�5)

≥ 00 (32)

The optimality condition (10) can be rewritten as

y∗

ave =min(

y∈N+

∣

∣

∣

∣

G4y1�ave5≤wr −wn

we

+wn

we41−�ave5

)

0

Since y is a discrete variable and �ave is a continu-ous parameter, there exist a set of values � i

ave ∈ 60117,i = 11 0 0 0 1 Imax, with �1

ave = 0 and �Imaxave = 1, with the optimal

y∗ave4�ave5 remaining constant in each interval 4�i

ave1�i+1ave 5,

i = 11 0 0 0 1 Imax − 1, and exhibiting finite jumps at � iave, i.e.,

y∗ave4�

iave −05 6= y∗

ave4�iave +05, i = 21 0 0 0 1 Imax −1. Note that the

sign of the difference y∗ave4�

iave + 05 − y∗

ave4�iave − 05 is com-

pletely determined by the sign of

H4� iave5 =

¡G4min4y∗ave4�

iave − 051 y∗

ave4�iave + 0551�i

ave5

¡�iave

−wn

we41 −� iave5

20 (33)

Indeed, suppose that y∗ave4�

iave + 05 > y∗

ave4�iave − 05. Then,

we should have G4y∗ave4�

iave − 051�i

ave − 05 ≤ 4wr −wn5/we +

wn/4we41 − 4�iave − 0555 and G4y∗

ave4�iave − 051�i

ave + 05 >4wr −wn5/we +wn/4we41 − 4�i

ave + 0555. Combining these twoexpressions, we get

G4y∗

ave4�iave − 051�i

ave + 05−G4y∗

ave4�iave − 051�i

ave − 05

>wn

we41 − 4�iave + 055

−wn

we41 − 4� iave − 055

(34)

or

¡G4y∗ave4�

iave − 051�i

ave − 05¡�ave

>wn

we41 − 4� iave − 0552

0

Similarly, y∗ave4�

iave + 05 < y∗

ave4�iave − 05 implies that

¡G4y∗ave4�

iave + 051�i

ave − 05¡�ave

<wn

we41 − 4� iave − 0552

0

Note that according to (32),

¡G4y∗1�iave5

¡�iave

= y∗

(y∗−1∑

k=0

pT 4k+ 15p4k3y∗− 11�i

ave5

)

1

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so that (33) can be expressed as

H4�iave5= y

(y−1∑

k=0

pT 4k+15p4k3y−11�iave5

)

−wn

we41−� iave5

21

where y = min4y∗ave4�

iave − 051 y∗

ave4�iave + 055. Note that for

� iave → 0, y∗

ave4�iave5 → y∗

0 , as expressed in (7), so that y∗0 =

�F −1T 41 −wr/we5�. Then,

H4�iave → 05=

(⌈

F −1T

(

1 −wr

we

)⌉)

pT

(⌈

F −1T 41 −

wr

we

5

⌉)

−wn

we

0

Thus,(

�F −1T 41 −

wr

we

5

⌉)

pT

(⌈

F −1T 41 −

wr

we

5

⌉)

≥wn

we

implies that there exist � lave such that y∗

ave is a nondecreas-ing function of �ave for �ave ≤ � l

ave. On the other hand,� i

ave ≥ 4we −wr 5/4wn +we −wr 5 implies that y∗ave4�

iave5 = 0,

and for �iave → 4we −wr 5/4wn +we −wr 5, y∗

ave4�iave5 → 0, and

H4� iave → 4we −wr 5/4wn +we −wr 55 = −4wn +we −wr 5

2/4wewn5 < 0. Thus, there exist �u

ave such that y∗ave is a nonin-

creasing function of �ave for �ave ≥ �uave. �

Proof of Corollary 1. Under the discrete uniformdemand distribution specified by (12), the sum in theexpression for the optimal staffing level (10) becomes

y∑

k=0

FT 4k5p4k3y1�ave5

=

y41−�ave5+1Tmax +1

for y≤Tmax1

1Tmax +1

Tmax∑

k=0

4k+15p4k3y1�ave5

+

y∑

k=Tmax+1

p4k3y1�ave5 for y≥Tmax +10

(35)

Note that for y = Tmax, (35) becomes

Tmax41 −�ave5+ 1Tmax + 1

= 1 −�aveTmax

Tmax + 1≥ 1 −

w

we41 −�ave51

as long as w/we ≥ �ave41 − �ave5Tmax/4Tmax + 15. The supre-mum of the right-hand side of this expression is 1

4 (for�ave = 005 and Tmax → �), so that this expression is impliedby w/we ≥

14 . Thus, under this condition, the optimal

staffing level does not exceed Tmax and, consequently,

y∗

ave = min(

y ∈N+

∣

∣

∣

∣

y41 −�ave5+ 1Tmax + 1

≥ 1 −w

we41 −�ave5

)

=

⌈(

Tmax

1 −�ave−

Tmax + 141 −�ave5

2

w

we

)⌉

0

Furthermore, differentiating the expression under the “ceil-ing” function on the right-hand side with respect to �ave,we get

141 −�ave5

3

((

Tmax − 4Tmax + 15wr −wn

we

)

41 −�ave5

−24Tmax + 15wr

we

)

1

which is nonnegative (nonpositive) if and only if �ave ≤ �uave

(�ave ≥ �uave). �

Proof of Proposition 3. Using L4y1�4y55 =∑y

k=0 q4k5 ·

p4k3y1�4y55, we have W4y + 15 − W4y5 = wr − 4wr − wn5 ·

ãa4y5 + weãL4y1�4y55, where ãL4y1�4y55 = L4y + 11�4y + 155− L4y1�4y55, and ãa4y5 = a4y + 15− a4y5. We pro-ceed by identifying sufficient conditions for ãL4y1�4y55≤0,ã2L4y1�4y55 = ãL4y + 11�4y + 155 − ãL4y1�4y55 ≥ 0, and4wr − wn5ã

2a4y5 = 4wr − wn54ãa4y + 15 − ãa4y55 ≤ 0. Since�4y5 is continuous and twice differentiable, it follows imme-diately that a4y5 is also continuous and twice differentiable,and therefore a sufficient condition for 4wr −wn54ãa4y+15−ãa4y55 ≤ 0 is given by 4wr − wn54d

2/dy25a4y5 ≤ 00 Now,ãL4y1�4y55 can be written as

y+1∑

k=0

q4k5p4k3y+11�4y+155−y∑

k=0

q4k5p4k3y1�4y+155

+

y∑

k=0

q4k5p4k3y1�4y+155−y∑

k=0

q4k5p4k3y1�4y550 (36)

We are now going to consider separately the first two andthe last two terms in Equation (36) and show that both arenonpositive. The first two terms can be expressed as −41 −

�4y + 155∑y

k=041 − FT 4k55p4k3y1�4y + 155, which is nonposi-tive. Next, we examine the second two terms, which can beexpressed as

y∑

k=0

q4k5∫ y+1

y

¡p4k3y1�4s55

¡sds

=

∫ y+1

yds

d�4s5

ds

y∑

k=0

q4k5¡p4k3y1�4s55

¡�4s50 (37)

Note that

y∑

k=0

q4k5¡p4k3y1�4s55

¡�4s5

=

y∑

k=0

q4k5

(

y!

k!4y− k5!4y− k5�4s5y−k−141 −�4s55k

− k�4s5y−k41 −�4s55k−1)

= yy−1∑

k=0

q4k5p4k3y− 11�4s55

− yy∑

k=1

q4k5p4k− 13y− 11�4s55

= yy−1∑

k=0

41 − FT 4k55p4k3y− 11�4s551 (38)

so that (37) becomes

y∑

k=0

q4k54p4k3y1�4y+ 155− p4k3y1�4y555

= y∫ y+1

yds

d�4s5

ds

y−1∑

k=0

41 − FT 4k55p4k3y− 11�4s550

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A sufficient condition for this last expression to be negativeis d�4s5/ds ≤ 0. Thus, ãL4y1�4y55≤ 0. Furthermore, considerã2L4y1�4y55, expressed as

−41 −�4y+ 255y+1∑

k=0

41 − FT 4k55p4k3y+ 11�4y+ 255

+ 41 −�4y+ 155y∑

k=0

41 − FT 4k55p4k3y1�4y+ 155

+ 4y+ 15∫ y+2

y+1ds

d�4s5

ds

y∑

k=0

41 − FT 4k55p4k3y1�4s55

− y∫ y+1

yds

d�4s5

ds

y−1∑

k=0

41 − FT 4k55p4k3y− 11�4s550 (39)

Focusing on the first two terms in (39), we get

−41 −�4y+ 255y+1∑

k=0

41 − FT 4k55p4k3y+ 11�4y+ 255

+ 41 −�4y+ 155y∑

k=0

41 − FT 4k55p4k3y1�4y+ 155

= 41 −�4y+ 255( y∑

k=0

41 − FT 4k55p4k3y1�4y+ 255

−

y+1∑

k=0

41 − FT 4k55p4k3y+ 11�4y+ 255)

+

y∑

k=0

41 − FT 4k55441 −�4y+ 155p4k3y1�4y+ 155

− 41 −�4y+ 255p4k3y1�4y+ 25550 (40)

The first term in (40) is equivalent to

41 −�4y+ 2552( y∑

k=0

pT 4k+ 15p4k3y1�4y+ 255)

and is nonnegative. The second term in (40) is equal to

−

y∑

k=0

41 − FT 4k55∫ y+2

y+1

¡441 −�4s55p4k3y1�4s555

¡sds

or

−

y∑

k=0

41 − FT 4k55∫ y+2

y+1

¡441 −�4s55p4k3y1�4s555

¡�4s5

d�4s5

dsds0

Focusing on the last two terms in (39), we obtain

4y+15∫ y+2

y+1ds

d�4s5

ds

y∑

k=0

41−FT 4k55p4k3y1�4s55

−y∫ y+2

y+1ds

d�4s5

ds

y−1∑

k=0

41−FT 4k55p4k3y−11�4s55

+y∫ y+2

y+1ds

d�4s5

ds

y−1∑

k=0

41−FT 4k55p4k3y−11�4s55

−y∫ y+1

yds

d�4s5

ds

y−1∑

k=0

41−FT 4k55p4k3y−11�4s55

=

∫ y+2

y+1ds

d�4s5

ds

y∑

k=0

41−FT 4k55

·44y+15p4k3y1�4s55−yp4k3y−11�4s555

+y∫ y+2

y+1ds∫ s

s−1d�

[y−1∑

k=0

41−FT 4k55¡

¡�

·

[

d�4�5

d�p4k3y−11�4�55

]]

0 (41)

Thus,∫ y+2

y+1ds

d�4s5

ds

y∑

k=0

41−FT 4k55

(

−¡41−�4s55p4k3y1�4s55

¡�4s5

+4y+15p4k3y1�4s55−yp4k3y−11�4s55)

+y∫ y+2

y+1ds∫ s

s−1d�

[y−1∑

k=0

41−FT 4k55

·¡

¡�

[

d�4�5

d�p4k3y−11�4�55

]]

=−2∫ y+2

y+1

ds

�4s5

d�4s5

ds

y∑

k=0

41−FT 4k55p4k3y1�4s55

·4y41−�4s55−k−�4s55

+y∫ y+2

y+1ds∫ s

s−1d�

y−1∑

k=0

41−FT 4k55d2�4�5

d�2

+y∫ y+2

y+1ds∫ s

s−1d�×

(

d�

d�

)2 1�4�541−�4�55

·

[y−1∑

k=0

41−FT 4k55p4k3y−11�4�55

·44y−1541−�4�55−k5

]

0 (42)

A sufficient condition for the second line of (42) to be non-negative is d2�4�5/d�2 ≥ 0. Since d�4s5/ds ≤ 0, a sufficientcondition for the first line of (42) to be nonnegative is that,for given y,

y∑

k=0

41 − FT 4k55p4k3y1�54y41 −�5− k−�5≥ 01 (43)

for any � ∈ 6�4y + 251�4y + 157, and for the third lineof (42) is that for given y,

∑y−1k=041 − FT 4k55p4k3y − 11�5 ·

44y − 1541 − �5 − k5 ≥ 0, for any � ∈ 6�4y51�4y + 157. Thislast condition is equivalent to the condition, for given y,∑y

k=041 − FT 4k55p4k3y1�54y41 − �5 − k5 ≥ 0, for any� ∈ 6�4y+ 151�4y+ 257. We next derive sufficient a conditionfor (43). Note that

y∑

k=0

41 − FT 4k55p4k3y1�54y41 −�5− k−�5

=

y−2∑

k=0

41 − FT 4k55p4k3y1�54y41 −�5− k−�5

+ 41 − FT 4y− 155�y41 −�5y−141 − 4y+ 15�5

+ 41 − FT 4y5541 −�5y4−4y+ 15�51 (44)

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Green, Savin, and Savva: “Nursevendor” Problem2254 Management Science 59(10), pp. 2237–2256, © 2013 INFORMS

so that, for 4y + 15� ≤ 2, y41 − �5 − k − � ≥ 0 for all k =

01 0001 y− 2, and

y−2∑

k=0

41 − FT 4k55p4k3y1�54y41 −�5− k−�5

≥ 41 − FT 4y− 255y−2∑

k=0

p4k3y1�54y41 −�5− k−�5

= 41 − FT 4y− 255( y∑

k=0

p4k3y1�54y41 −�5− k−�5

−�y41 −�5y−141 − 4y+ 15�5

− 41 −�5y4−4y+ 15�5)

= 41 − FT 4y− 255(

−� −�y41 −�5y−141 − 4y+ 15�5

−41 −�5y4−4y+ 15�5)

0 (45)

Thus, the expression in (44) is nonnegative if 4FT 4y5 −

FT 4y − 25541 − �5y4y + 15+ 4FT 4y − 15− FT 4y − 255y41 − �5y−1 ·

44y + 15� − 15 ≥ 1 − FT 4y − 25. The left-hand side of thisexpression can be rearranged as

4FT 4y5− FT 4y− 25541 −�5y4y+ 15

+ 4FT 4y− 15− FT 4y− 255y41 −�5y−144y+ 15� − 15

= 4pT 4y5+ pT 4y− 15541 −�5y4y+ 15

+ pT 4y− 15y41 −�5y−144y+ 15� − 15

= pT 4y541−�5y4y+15+pT 4y−1541−�5y(

1+y2�

1−�

)

≥ 41 −�5y4pT 4y54y+ 15+ pT 4y− 1550 (46)

Thus, 4FT 4y5 − FT 4y − 25541 − �5y4y + 15 + 4FT 4y − 15 −

FT 4y − 255y41 − �5y−144y + 15� − 15 ≥ 1 − FT 4y − 25 as long as41 −�5y4pT 4y54y+ 15+ pT 4y− 155≥ 1 − FT 4y− 25 or

� ≤ 1 −

(

1 − FT 4y− 254y+ 15pT 4y5+ pT 4y− 15

)1/y

0

Combining this expression with 4y + 15� ≤ 2 and � ∈

6�4y+251�4y+157, and noting that for � = 0 the monotonic-ity of ãL4y1�4y55 is assured, we obtain the final sufficientcondition

�4y+ 15 ≤ min(

2y+ 1

1

1 − min(

11(

1−FT 4y−254y+15pT 4y5+pT 4y−15

)1/y))

= min(

2y+ 1

1�T 4y+ 15)

0 (47)

Given that ãL4y1�4y55 is a monotone function of y if�4y5 is a nonincreasing convex function of y and if (47)is satisfied, we establish the monotonicity of the optimalstaffing level y∗ with respect to changes in wr/we andFT 4k5, following the same arguments used in the proof ofProposition 1. �

Proof of Lemma 1. Note that for �1� ≥ 0, the func-tion a4x5 = x�4x5 = x/41 + e�+�x5 defined on continuous setx ≥ 0 has a unique global maximum x∗, which satisfies thefirst-order optimality condition e−� = e�x

∗

4�x∗ − 15. Thus,the maximum value for this function can be expressedas x∗�4x∗5 = 41/�54�x∗/41 + e�+�x∗

55 = 4�x∗ − 15/�. This lastexpression does not exceed 2 if and only if �x∗ ≤ 2�+ 1,which is equivalent to e−� ≤ 2�e2�+1 ⇔ 2�e�+2�+1 ≥ 1.This condition ensures that the maximum of y�4y5 cannotexceed 2 for all integer values of y as well. Furthermore,d2a/dy2 = �2�4y541−�4y554�41−2�4y55−25, which is alwaysnonpositive for �≤ 2. �

Proof of Proposition 4. Under (20), the optimal staffinglevel y∗ satisfies (19), which can be re-expressed as y∗ =

min4y ∈ N+ � wr 41 − �4y55+wn�4y5− 4wr −wn5y4�4y + 15−

�4y55+weãL4y1�4y55 ≥ 05, where the last two terms insidethe bracket are

−4wr −wn5y4�4y+ 15−�4y55−we41 −�4y+ 155

·

y∑

k=0

41 − FT 4k55p4k3y1�4y+ 155

+wey∫ y+1

yds

d�4s5

ds

y−1∑

k=0

41 − FT 4k55p4k3y− 11�4s55

= −we41 −�4y+ 155y∑

k=0

41 − FT 4k55p4k3y1�4y+ 155

+ y∫ y+1

yds

d�4s5

ds

[

we

(y−1∑

k=0

41−FT 4k55p4k3y−11�4s55)

− 4wr −wn5

]

0 (48)

Thus,

wr 41−�4y∗55+wn�4y∗5−we41−�4y∗

+155

·

y∗

∑

k=0

41−FT 4k55p4k3y∗1�4y∗

+155

+y∗

∫ y∗+1

y∗

dsd�4s5

ds

[

we

(y∗−1∑

k=0

41−FT 4k55p4k3y∗−11�4s55

)

−4wr −wn5

]

≥0

or

y∗

∑

k=0

41 − FT 4k55p4k3y∗1�4y∗

+ 155

≤

(

wr − 4wr −wn5�4y∗5

+ y∗

∫ y∗+1

y∗

dsd�4s5

ds

·

[

we

(y∗−1∑

k=0

41 − FT 4k5

)

p4k3y∗− 11�4s555

−4wr −wn5

])

4we41 −�4y∗+ 1555−10 (49)

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Now, consider an element of E, yML, which satisfies

yML= min

(

y ∈N+

∣

∣

∣

∣

y∑

k=0

41 − FT 4k55p4k3y1�4yML55

≤wr − 4wr −wn5�4y

ML5

we41 −�4yML55

)

1

so that

yML∑

k=0

41 − FT 4k55p4k3yML1�4yML55≤

wr − 4wr −wn5�4yML5

we41 −�4yML550

Below we show by contradiction that, if a4y∗ + 15 < a4y∗5,then yML ≤ y∗. Suppose that yML > y∗ ⇔ yML ≥ y∗ + 1. This,in turn, implies that �4yML5≤ �4y∗ + 15 and

y∗

∑

k=0

41 − FT 4k55p4k3y∗1�4yML55

>wr − 4wr −wn5�4y

ML5

we41 −�4yML550 (50)

Now, since for ∀ s ∈ 6y∗1y∗ + 17, it follows that s ≤ yML and�4yML5≤ �4s5, we can use (31), (32), and (50) to get

y∗−1∑

k=0

41 − FT 4k55p4k3y∗− 11�4s55

≥

y∗

∑

k=0

41 − FT 4k55p4k3y∗1�4s55

≥

y∗

∑

k=0

41 − FT 4k55p4k3y∗1�4yML55

>wr − 4wr −wn5�4y

ML5

we41 −�4yML550 (51)

Furthermore, since d�4y5/dy ≤ 0, (49) implies that

wr − 4wr −wn5�4yML5

we41 −�4yML55

<y∗

∑

k=0

41 − FT 4k55p4k3y∗1�4y∗

+ 155

≤

(

wr − 4wr −wn5�4y∗5

+ y∗

∫ y∗+1

y∗

dsd�4s5

ds

·

[

we

(y∗−1∑

k=0

41 − FT 4k55p4k3y∗− 11�4s55

)

− 4wr −wn5

])

4we41 −�4y∗+ 1555−1

≤wr −4wr −wn5�4y

∗5+y∗4�4y∗ +15−�4y∗554wn/41−�4yML555

we41−�4y∗ +155

≤ 4wr − 4wr −wn5�4y∗+ 15+ y∗4�4y∗

+ 15−�4y∗55

· 4wn/41 −�4yML5555 · 4we41 −�4y∗+ 1555−10 (52)

Equation (52) is equivalent to

�4yML5

1 −�4yML5≤

�4y∗ + 151 −�4y∗ + 15

+y∗4�4y∗ + 15−�4y∗5541/41 −�4yML555

41 −�4y∗ + 1551

and

�4yML541 −�4y∗+ 155 ≤ �4y∗

+ 1541 −�4yML55

+ y∗4�4y∗+ 15−�4y∗551

so that

�4yML5≤ �4y∗+ 15+ y∗4�4y∗

+ 15−�4y∗550

Note that this contradicts �4yML5≥ 0. �

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