Consistent Staffing for Long-Term CareThrough On-Call Pools

(Authors’ names blinded for peer review)

Nursing homes managers are increasingly striving to ensure consistency of care, defined as minimizing the

number of unique nurse aides who care for a resident during at least one shift over the course of one month.

Unfortunately, managers often struggle to provide consistent care, primarily due to last-minute nurse aide

absences and choosing to staff these absences with aides from an external rental agency. We are the first to

study the use of an “on-call” pool — aides on staff who may be called in to work in the event of absences —

as an operational strategy to improve consistency. We provide structural results for the relationship between

the number of aides in the on-call pool, staffing cost, and inconsistency level. We also show that a “restricted”

on-call pool — in which each slot in the pool is restricted to nurse aides from different units — outperforms an

“open,” or unrestricted, on-call pool. We further demonstrate that converting full-time positions to part-time

positions can improve consistency of care if the part-time aides’ on-call pool participation rate is sufficiently

high. Numerical results for a typical facility demonstrate that an on-call pool can (a) reduce the staffing

costs due to absences by 24% while also (slightly) reducing the inconsistency level, or (b) significantly reduce

the inconsistency level without increasing costs.

Key words : health care; absenteeism; nursing homes; stochastic models; Schur-convexity

1

1. Introduction

Recent studies aimed at improving care in nursing homes have identified “consistent as-

signment” — having the same nurse aides care for the same residents over time — as an

important pillar of effective person-centered care paradigms. National organizations and

individual thought leaders have advocated consistent assignment as a means to elevate

both the care that residents receive and the job satisfaction of nurse aides. Nurse aides,

who often have the title of Certified Nursing Assistant (CNA), provide the majority of

care hours in nursing homes through tasks such as feeding, bathing, and dressing. The pur-

ported benefits of consistent assignment are many. For example, residents do not have to

re-explain preferences to new caregivers and also become more comfortable when receiving

care. Also, nurse aides tend to develop more meaningful relationships with residents and

can more quickly detect and respond to resident health issues. Some states have even used

consistent assignment goals as part of pay-for-performance programs (Roberts et al. 2015).

The Advancing Excellence in America’s Nursing Homes Campaign (AE, www.

nhqualitycampaign.org), a coalition that includes 62% of nursing homes in the United

States, promotes consistent assignment as one of four “organizational goals.” The AE Cam-

paign defines consistency as the number of unique caregivers assigned to each resident in

one month, and encourages a goal of 12 or fewer. It even provides a free spreadsheet tool

for nursing homes to calculate and track consistency. The Massachusetts Office of Medi-

caid provided a $2.8 million prize pool for nursing homes that used the AE Campaign’s

consistency tracking tool and achieved the AE Campaign’s stated consistency goal (Harris

2012).

Yet, despite the prominence of consistent assignment as a quality goal for nursing homes,

little attention has been given to the underlying operational challenges that pursuit of this

goal entails, specifically how staffing policies may have to be dramatically altered to reach

the patient-centered goal of consistent assignment. Even the AE Campaign’s own “Create

Improvement” guidelines primarily focus on improving staff communication, and do not

address staffing policies such as on-call pools. In fact, based on a survey by Roberts et al.

(2015), we are the first to connect patient-centered consistency metrics such as that of the

AE Campaign to tactical managerial staffing decisions.

Practitioners’ literature does address consistent assignment of course, but the most

common view, for example in Castle (2013), tends to focus on the staff-centered metric

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articulated by Quality Partners of Rhode Island (2007): They measure consistency under

the assumption that each full-time aide has an assigned unit of residents. The consistency

score is then calculated as the percentage of shifts scheduled to be worked by full-time

aides in their assigned unit, and the target score is usually set at 80% or 85%.

Unfortunately, in our conversations with managers of nursing homes that had a nearly

100% consistency score using the staff-centered metric, we still found significant concern

and frustration about consistency of care. By comparing planned schedules to realized

schedules for each shift over several months, we tied this discrepancy to absenteeism — i.e.,

aides “calling off” a short time before the start of a shift, which occurs with approximately

a 5% probability for each aide for each shift (Castle 2013). Nurse aide call-offs led to

reduced patient-centered consistency, that was reflected in a steady stream of complaints

from residents’ families: Up to forty different aides could care for a resident over all shifts in

a month, even though the planned schedule could have a perfect staff-centered consistency

measure.

To try to reconcile this conflict we study “on-call” pools — in which full-time and part-

time nurse aides who are on a nursing home’s staff volunteer to be available to work if

a scheduled nurse aide is absent — as an operational strategy. On-call pools may also

be referred to as “float” pools, but we use the terminology “on-call pool” as “float pool”

sometimes refers to nurses or nurse aides who are scheduled to work a shift in a unit which

is only assigned immediately before the start of a shift. We provide structural results for

the relationship between the on-call pool size, cost, and consistency metrics in the presence

of absenteeism. We also illustrate how the use of part-time aides can lower inconsistency

in a system with random participation by aides in on-call pools. Compared to a staffing

policy common in practice, we show that utilization of an on-call pool can simultaneously

increase consistency and reduce staffing costs. Our findings complement Lu and Lu (2016)

who empirically find that laws limiting overtime by staff in nursing homes harm quality

partially due to an increased reliance on rental agency staff.

Specifically, we make the following contributions:

1. Model and Framework: To the best of our knowledge, we are the first to consider a

metric based on the unique number of servers who provide service over some time horizon,

an important metric for long-term care facilities. To do so, our model includes a structure

with nursing home residents organized into units with assigned staff, random absences

3

among staff, a random staffing requirement, and rules for assigning on-call pool nurse aides

to units when absences occur.

2. Structural Results:

(a) Accounting for overtime wages, bonuses paid for enlisting in the on-call pool, and

the cost of hiring nurse aides from a rental agency, we establish the convexity of the staffing

cost in the number of aides in the on-call pool. We also prove that the optimal pool size

is of critical fractile type relating the cumulative distribution function of the number of

absences to a ratio incorporating the cost of overtime aides, the on-call pool bonus amount,

and the cost of rental agency aides.

(b) We show that the expected level of inconsistency due to absences is convex and

non-increasing in the on-call pool size. In other words, there are decreasing returns of

on-call pool size on the consistency level.

(c) We analyze two types of on-call pools: open, in which any aide may enlist in any

on-call pool slot, and restricted, in which the home units of the nurse aides in the on-call

pool must be represented as evenly as possible. The expected inconsistency level due to

absences improves as the on-call pool slots are more evenly distributed over the different

units — a property that we prove using the concept of Schur-convexity. Consequently, the

restricted on-call pool outperforms the open pool in terms of consistency.

(d) We show that it may be possible for a nursing home to replace one full-time aide

with two part-time aides and improve the overall consistency of care, in particular if the

on-call pool participation rate of the part-time aides is sufficiently high.

3. Managerial Insights from Numerical Study:

(a) For a facility with the national median size of 100 beds, we show that it is possible

for the implementation of an on-call pool to reduce staffing costs due to absences by

24% while simultaneously reducing the inconsistency due to absences (i.e., the reduction

in consistency due to absences) by 23%. Alternately, a nursing home may reduce the

inconsistency level due to absences by 70% without increasing staffing costs.

(b) Enacting an on-call pool may allow a nursing home to transfer approximately

$60,000 per year in wages “in house” — moving them from rental staffing agencies to

overtime wages and on-call pool bonuses for staff at the nursing home.

(c) For the same on-call pool size, inconsistency due to absences is initially lower for

smaller units than for larger units (i.e., 2 nurse aides per shift compared to 4) but becomes

4

lower for larger units as the on-call pool size increases. Thus the use of an on-call pool,

and its size, can influence whether larger or small units are preferred, from a consistency

of care perspective.

The remainder of the paper is organized as follows. Section 2 reviews relevant literature.

Section 3 introduces the nursing home industry motivating our model. We define our model

for the operation of the on-call pool and establish its structural properties in Section 4.

Numerical analysis showing the benefits of an on-call pool for a typical nursing home follows

in Section 5. We conclude with a summary of findings in Section 6. All proofs appear in

the appendix.

2. Related Literature

Respondents to a survey conducted by Miller et al. (2010) named workforce issues as the

foremost challenge facing long-term care providers; not surprisingly, nurse aide staffing

policies have received attention in practitioners’ literature. Castle and Ferguson (2010)

provide an overview of quality measures for nursing homes, including the important five-

star rating system of the Centers for Medicare and Medicaid Services. Castle and Engberg

(2008) show that higher staffing levels are weakly associated with better quality of care, but

they also noted the importance of the use of rented staff from staffing agencies, the ratio of

registered nurses to other caregivers, and turnover rates. Castle (2009) also demonstrates

that the use of agency staff has a significant negative association with quality of care.

Weech-Maldonado et al. (2004) propound the importance of using full-time registered

nurses instead of part-time registered nurses for quality outcomes. And Lu and Lu (2016)

find that laws prohibiting mandatory overtime result in the increased use of contract nurses

and nurse trainees, harming quality of care.

Burgio et al. (2004) compare nursing homes practicing “permanent assignment” — an

alternate term for consistent assignment — to those practicing “rotating assignment.”

Measuring the “percentage of time all residents in the facility were cared for by their

most frequently assigned” nurse aide as the “permanency rate,” the mean permanency

rate was found to be only only 50% for the facilities with permanent assignment and 26%

for those with rotating assignment. They found significantly higher ratings of residents’

personal appearance and hygiene and higher reported job satisfaction among nurse aides for

permanent assignment schemes, but otherwise found few statistically significant differences

5

in quality of care outcomes. Castle (2013) shows that nursing homes that practice consistent

assignment experience lower turnover and absence rates, and Castle and Ferguson-Rome

(2014) link absenteeism to decreases in various measures of care quality. Defining consistent

assignment as in the widely promoted metric by Quality Partners of Rhode Island (2007),

they find that 68% of nursing homes attempt to use consistent assignment while only 28%

of nursing homes achieve the recommended 85% consistency rate.

To the best of our knowledge, we are the first to study consistent assignment of health

care servers over time in the operations management literature; we are also the first to study

the relationship between absenteeism and consistent assignment. Researchers have studied

staffing decisions in hospitals and other service settings. Green et al. (2013) empirically

demonstrate that nurse absences at an urban hospital increase in anticipation of higher

workloads; they solve a single shift newsvendor problem to avoid understaffing.

Wang and Gupta (2014) use Schur-convexity to show that nursing costs are minimized

by assigning nurses to home units in hospitals in a way that maximizes heterogeneity of

absentee rates among nurses within each unit. Using a closed queueing model, de Vericourt

and Jennings (2011) demonstrate that staffing levels based on simple nurse-to-patient ratios

are insufficient for achieving a desired probability of excessive delay; it is necessary to also

account for the total number of patients in the system. Outside of healthcare staffing,

Fry et al. (2006) study the staffing decision for firefighters incorporating unplanned and

extended absences, as well as calendar constraints related to work rules.

Researchers have also studied the value of workforce flexibility in healthcare and other

settings. Wright and Bretthauer (2010) show the value of workforce flexibility (from the

use of rental staffing agencies) in a hospital. In the context of call centers, Bhandari et al.

(2008) propose an algorithm to compute the number of permanent operators, the number

of temporary operators, and the threshold number of customers in the system for when

to use the temporary servers. Kesavan et al. (2014) show that using part-time and tempo-

rary workers in retail stores initially improves financial performance but eventually harms

performance if they comprise too much of the workforce.

As a stochastic model, worker absences bear similarities to random yields in production

planning. Yano and Lee (1995) present the binomial distribution as the simplest model of

random yields and review related literature.

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3. Nursing Home Industry Overview

The Centers for Medicare & Medicaid Services project nursing care facilities and continuing

care retirement communities to be a $176.1 billion industry in 2016, with $90.8 billion of

this amount by Medicare and Medicaid (Keehan et al. 2015). Out-of-pocket payments are

projected to constitute $52.6 billion of this spending, while the remaining $32.7 billion is

divided among private insurance, other government insurance programs, and other third-

party payers. In its National Study of Long-Term Care Providers, the Centers for Disease

Control and Prevention’s 2013 report lists 15,700 nursing homes in the United States serv-

ing 1,383,700 residents, while 22,200 residential care communities serve 713,300 residents

(Harris-Kojetin et al. 2013). This study reports that nursing homes employ 952,100 full-

time equivalent nursing workers, of whom 65% are nurse aides. Aides provide the majority

of care in nursing homes; nurse aides perform an average of 2.46 hours of care per resident

per day, while registered nurses and licensed practical nurses provide 0.52 and 0.85 hours,

respectively. According to the Nursing Home Compare dataset (Centers for Medicare and

Medicaid Services 2015), the median facility size is exactly 100 beds, and the national bed

occupancy rate is 82%. Of all nursing homes, 69% have for-profit ownership, 24% are

not-for-profit, and 7% are owned by various government agencies. Approximately 70% of

nursing home residents live in facility with at least 100 beds, and 15% live in a facility

with at least 200 beds.

Long Term Care Facilities surveys of the Pennsylvania Department of Health (2015)

report a total of 701 nursing homes with 88,063 licensed beds that serve 79,297 residents

in Pennsylvania. The median price per day for a private room — which includes nursing

care, meals, and utilities — is $301. The median daily Medicare reimbursement rate across

facilities is $446, which may reflect both additional services received by residents and a

willingness by Medicare to be a higher payor. The numbers of full-time and part-time nurse

aides employed by the facilities are 29,029 and 11,342, respectively. Herzenberg (2015)

reports the median hourly wage of a nurse aide in Pennsylvania as $13.01.

We interviewed managers and analyzed nurse aide schedules of two non-profit nursing

homes in southwestern Pennsylvania, each of which has between 100 and 200 licensed beds

and between 50 and 100 nurse aides on staff. Both had previously adopted a consistent

assignment regime in which each nurse aide almost exclusively served one unit of either

approximately 15 or 30 residents. Both facilities reported between 2.7 and 3.0 nurse aide

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Table 1 Notation

Symbol Description

y Staffing level; i.e., the number of nurse aides scheduled or on-call for one shiftu Number of units (i.e., groups of residents) in the nursing homeq Number of nurse aides required to be scheduled per unit per shift (exogenously determined)Rk Random number of nurse aides required for unit kgU (l) Probability mass function for Rk, l ∈ {0,1,2, . . .}gT (m) Probability mass function for the total requirement over all units, m∈ {0,1,2, . . .}y0 Minimum number of aides to schedule for one shift (y0 = q ∗u)γ Probability that any one scheduled nurse aide is absent on any one shiftNS Random number of scheduled nurse aides (out of y0) who show up to work a shiftwc Marginal cost of an on-call aide working one shift (compared to a scheduled aide)wa Marginal cost of a rental agency aide working one shift (compared to a scheduled aide)b Bonus paid to on-call aide who is not called ind Cost of cancelling a scheduled shift for a nurse aide

C(y) Expected cost of staffing due to absences for one shift with staffing level yLO(y) Expected inconsistency level for one unit for one shift with an open on-call pool and staffing level yLR(y) Expected inconsistency level for one unit for one shift with a restricted on-call pool and staffing level yκ Number of shifts in span of time over which inconsistency level is measured (e.g., one month)Sk Random variable representing net staffing shortage in unit k on one shift

Hρk (y) Random variable for aides in on-call pool (having y aides and sign-up rule ρ) with unit k as home unitσk Net staffing shortage for unit kηj The home unit of on-call pool aide jxk The number of on-call pool aides for some arbitrary unit k on a sample pathχk The home unit corresponding to xk

B(k;n,p) Binomial probability mass functionH(k;N,K,n) Hypergeometric probability mass function

∆f(y) First forward difference operator; i.e., ∆f(y) = f(y+ 1)− f(y)

staffing hours per patient per day. But the two nursing homes adopted quite different

approaches towards using part-time aides: part-time aides comprised approximately 15%

of aides on staff at one nursing home and 40% at the other.

4. The Operation of On-Call Pools

Residents of the nursing home are grouped into u symmetric “units,” which are also some-

times called “communities,” “neighborhoods,” or “households.” For each shift the number

of aides required to be scheduled for each unit is q; q is exogenously determined and in-

sensitive to small changes in the resident census. Choosing q is typically a longer-term

strategic decision related to facility layout; we consider that decision to be outside the

scope of the paper.

The total number of nurse aides required to be scheduled per shift is then y0 = q ∗ u.

However, we allow the actual number of nurses required in each unit for a shift to vary

randomly immediately before the start of a shift: Extra aides might be needed for various

reasons, including a large number of admissions from hospitals, an outbreak of an illness

among residents, or an unannounced inspection. In contrast, nursing home managers and

8

consultants have explained to us that the staffing requirement on a unit would almost

never decrease at the last minute. Nevertheless, to generalize our model and further its

applicability to other settings, we allow the staffing requirement of each unit to increase

or decrease randomly at the start of each shift.

For convenience, we denote the probability mass function of the binomial distribution

as

B(k;n,γ) =

n!

k!(n−k)!γn−k(1− γ)k for 0≤ k≤ n,

0 otherwise.

and the probability mass function of the hypergeometric distribution as

H(k;N,K,n) =

(Kk)(

N−Kn−k )

(Nn)for max{0, n+K −N} ≤ k≤min{n,K},

0 otherwise.

We also denote max{x,0} by x+ and max{−x,0} by x−.

The sequence of events for any shift is as follows:

1. The nursing home enlists a total of y ≥ y0 nurse aides who volunteer to work during

the shift under consideration; y0 of the aides are “scheduled” and the remaining y − y0aides are “on-call.”

2. Of the y0 scheduled nurse aides, a total of NS aides show up to work. Each aide is

absent with probability γ; these absences are independent and identically distributed.

3. A random number Rk of nurse aides are needed as the actual staffing requirement

in each unit k for a shift. Rk follows an independent probability mass function gU(l), l ∈{0,1,2, . . .}, and reflects any last-minute changes to the patient census or other institutional

tasks during the shift that would increase or decrease the number of staff needed in the

unit. The random staffing requirement over all units for the shift,∑u

k=1Rk, follows the

probability mass function gT (m), m ∈ {0,1,2, . . .}, which is the convolution of the unit

independent probability mass functions identical to gU(l).

4. We define Sk as the random variable representing the net staffing shortage in unit k

considering the random staffing requirement and absences by scheduled aides in that unit.

More precisely,

Pr {Sk = i}=

q∑j=0

gU(q+ i− j)B(j; q, γ) (1)

9

and

Pr

{u∑k=1

Sk = i

}=

y0∑j=0

gT (y0 + i− j)B(j;y0, γ). (2)

A value of Sk < 0 represents a staffing surplus in unit k. When the staffing requirement is

deterministically q aides, Sdetk ∼Binomial(q, γ). The total number of positions that need

to be filled across all units is then∑u

k=1 (Sk)+; these are replaced as follows (from highest

to lowest priority):

Reallocated scheduled aides: min{∑u

k=1 (Sk)+ ,∑u

k=1 (Sk)−} scheduled aides are not

needed in the unit for which they were scheduled and are reallocated to other units.

If∑u

k=1Rk <NS, then the shifts for NS −∑u

k=1Rk nurse aides are cancelled at a cost

of d per shift.

On-call aides: min{y− y0, (

∑uk=1Sk)

+}on-call aides are called in to work at an ad-

ditional cost of wc per aide per shift beyond what a scheduled aide would have cost.

The remaining [y−∑u

k=1Rk− y0 +NS]+

aides who are not called in receive a bonus

b, b≥ 0, per aide per shift for being on call.

Rental agency aides: (∑u

k=1Sk− (y− y0))+ rental agency aides are employed for the

shift at an additional cost of wa per aide per shift beyond what a scheduled aide would

have cost.

All notation is presented in Table 1. Our analysis separates the consistency of care metric

for the day, evening, and night shifts, and our model can be applied to any one of these

shifts. We assume that wc ≤ wa; i.e., it is less expensive to use an on-call aide than an

agency aide. Otherwise, the optimal solution would be to have no on-call pool. We also

assume that the number of aides willing to participate in on-call pools is relatively large

— a realistic assumption if aides are allowed to sign up for on-call pool slots of other shifts

— and any aide participating in the pool is equally likely to have any unit as his or her

“home” unit. Furthermore, our modeling emphasis on marginal cost allows us to ignore

paid sick leave for a call off as a constant value by which the costs of every strategy shifts.

We now must consider the system dynamics within each of the u symmetric units to

express the consistency of care metric. Each unit has a set of nurse aides — who may be full-

time or part-time employees — assigned to it; this set of nurse aides has sufficient capacity

to staff the unit. Reflecting what we have observed in practice, each aide is assigned to

only one unit. These aides provide a baseline level of inconsistency; this is the minimum

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inconsistency level for the unit and best consistency score that residents can experience for

the specified shift or shifts. For example, a unit which requires q = 4 nurse aides for each

day shift might have seven full-time nurse aides to cover all day shifts in a month, which

corresponds to a baseline inconsistency level of seven. Treating the baseline inconsistency

level as fixed, we focus on the expected number of additional aides that work in the unit

given a total staffing level y, defining this as the inconsistency due to absenteeism, or

inconsistency level. If the unit experiences six absences in a month and relies on rental

agency aides as substitutes (i.e., y = y0), the inconsistency level due to absenteeism is

six. (From our observations of nursing home schedules it is unlikely that the same rental

agency aide will substitute for the same unit on multiple occasions in one month.) The

total inconsistency level, which corresponds to the AE Campaign’s metric for consistency

of care, is then 13 aides.

We let Hρk (y) ∈ {0,1, . . . , y − y0} represent the random number of aides in the on-call

pool (out of the y − y0 aides in the on-call pool) that have unit k as their home unit

given an on-call pool with y aides and some enlistment rule ρ. Aides from the on-call pool

that serve in their native unit do not increase the inconsistency level. But each reallocated

scheduled aide and each rental agency aide increases the inconsistency level by one; we

assume that the probability that these aides would serve in the same unit outside their

home unit multiple times (within the interval over which consistency of care is measured)

is sufficiently low. Likewise, all aides from the on-call pool that serve in units different

from their home unit each increase the inconsistency level by one.

Because a nursing home prefers to have an aide work a scheduled shift rather than

dismiss the scheduled aide and bring in an on-call aide to work at an overtime rate,

the reallocated scheduled aides have the highest priority for filling shortages. Contingent

upon this, on-call aides with a shortage in their native units are assigned to their native

units as long as all surplus scheduled aides can be reallocated to satisfy other shortages.

Thus, min{∑u

k=1 min{Hρk (y), S+

k

}, (∑u

k=1Sk)+}

on-call aides serve in their native unit.

The expected inconsistency level Lρ(y) experienced by each of the u symmetric units is

the expected difference between the total number of shortages and the number of on-call

aides serving in their native unit; i.e.,

Lρ(y) = E

[u∑k=1

(Sk)+−min

{u∑k=1

min{Hρk (y), S+

k

},

(u∑k=1

Sk

)+}]/u. (3)

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The expected inconsistency due to absences over κ shifts can then be computed as κLρ(y);

e.g., κ= 30 for the day, evening, or night shift over a 30-day month.

We characterize two types of on-call pools based on the rules governing which aides may

join on-call pools: open (ρ=O) and restricted (ρ=R). In the open on-call pool, any aide

may fill any of the y− y0 positions, which means that each unit is equally likely to be the

home unit of the aide filling any position and

Pr{HOk (y) = i

}=B(i;y− y0,1/u)

for any k ∈ {1,2, . . . , u}. When the staffing requirement is deterministic, we can write the

inconsistency level from the perspective of any one unit accounting for the possible values

of Sk and HOk (y) as

LdetO (y) =

y−y0∑i=0

B(i;y− y0,1/u)

q∑j=i

B(j; q, γ)(j− i), (4)

where the index i refers to the number of on-call aides in the pool from the unit and the

index j refers to the number of absences within the unit.

With a restricted on-call pool, aides are equally likely to come from any unit subject to

the restriction that no unit may have more than one more aide in the on-call pool than

any other unit; each unit has either b(y − y0)/uc or b(y − y0)/uc+ 1 aides in the on-call

pool. For the restricted on-call pool,

Pr{HRk (y) = i

}=

(y− y0)/u−b(y− y0)/uc if i= b(y− y0)/uc+ 1,

1− (y− y0)/u+ b(y− y0)/uc if i= b(y− y0)/uc,

0 otherwise.

The expected inconsistency level with a deterministic staffing requirement is

LdetR (y) =

q∑i=b(y−y0)/uc+1

B(i; q, γ)

(i−⌊y− y0u

⌋−(y− y0u−⌊y− y0u

⌋))

=

q∑i=b(y−y0)/uc+1

B(i; q, γ)

(i− y− y0

u

), (5)

which we can alternatively write as

LdetR (y) =

q∑i=0

B(i; q, γ)

(i− y− y0

u

)+

. (6)

12

One detail from practice different from our model assumptions can occur if the same

rental agency aide covers for an aide who calls off for a multiple-day absence. Excluding

the absences following the initial call-off when estimating γ can help ensure that LO(y) and

LR(y) accurately represent the consistency metric. Another potentially relevant practical

detail that we ignore is that any bonus pay (b) would be considered nondiscretionary, and

thus may affect each employee’s “regular rate” of pay, which is used to determine the

overtime pay rate. For more details, see Rotman (2012).

4.1. Cost of Absenteeism

We define the cost of absenteeism as the difference between the staffing cost with absen-

teeism and the cost without absenteeism. Thus, minimizing the cost of absenteeism also

minimizes the total staffing cost. The expected facility-wide cost of absenteeism C(y) for

a single shift is

C(y) =dE

[(u∑k=1

Sk

)−]+ bE

(y− y0−( u∑k=1

Sk

)+)+

+wcE

[min

{y− y0,

(u∑k=1

Sk

)+}]+waE

[(u∑k=1

Sk− (y− y0)

)+],

which using (2) we can alternatively express as

C(y) =−1∑

i=−y0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)(b(y− y0)− di)

+

y−y0∑i=0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)(wci+ b(y− y0− i))

+

∞∑i=y−y0+1

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)(wa(i− y+ y0) +wc(y− y0)) . (7)

The following result demonstrates the optimal staffing policy for our model, which has

the structure of a newsvendor model with a piecewise linear cost function. For analytical

ease, we define the first forward difference function for any function f(·) as

∆f(y) = f(y+ 1)− f(y).

Proposition 1. The minimizer of (7) is given by

y∗ = min

(y ∈N+|y≥ y0,

y−y0∑i=0

y0∑j=0

gT (y0 + i− j)B(j;y0, γ)≥ wa−wcb+wa−wc

). (8)

13

When the staffing requirement is not subject to random variation, we can state the

cost-minimizing staffing level y∗det more succinctly using only the cumulative distribution

function for a binomial random variable.

Corollary 1. When Rk = q for every unit k, k= 1, . . . , u, the minimizer of (7) is given

by

y∗det = min

(y ∈N+|y≥ y0,

y−y0∑j=0

B(j;y0, γ)≥ wa−wcb+wa−wc

). (9)

4.2. Inconsistency Due to Absenteeism

The complexity of the system’s operational dynamics — particularly the reallocation of

scheduled aides — leads us to use a sample path approach as described by Lindvall (2002)

to demonstrate properties of the inconsistency function. We define a sample path using

two pieces of information:

1. The net staffing shortages σ = {σ(1), . . . , σ(u)} with σ(k) being the shortage in unit

(k) (or surplus of(σ(k))−

aides for σ(k) < 0) which is mapped to one of the u units upon

realization of the sample path. This allows us to take the expectation — which we denote

with the subscript σ — of the u! combinations of assignments of net shortage quantities

{σ(k), . . . , σ(k)} to the u units.

2. The home units of on-call pool aides η= {η1, . . . , ηy−y0} with ηi representing the home

unit — using the same frame of reference (i.e., indexing) for units as in σ — of each aide

i= 1, . . . , y− y0 in the on-call pool.

We can rewrite (3) on a sample path as

Lρ(y;η,σ) =

∑uk=1

(σ(k))+−min

{∑uk=1 min

{∑y−y0i=1 1{ηi = k}, σ+

(k)

},(∑u

k=1 σ(k))+}

u,

(10)

which can be alternatively written as

Lρ(y;η,σ) =

∑uk=1

(σ(k))+

u−∑y−y0

i=1

∑uk=1 1{ηi = k}1

{σ(k) >

∑i−1j=1 1{ηj = k}

}1{∑u

k=1 σ(k) >∑i−1

j=1

∑ul=1 1{ηj = l}

}u

(11)

14

Thus, the value of a marginal aide in the on-call pool is

∆Lρ(y;η,σ) =Lρ(y+ 1;η, ηy−y0+1,σ)−Lρ(y;η,σ)

=−

∑uk=1 1{ηy−y0+1 = k}1

{σ(k) >

∑y−y0j=1 1{ηj = k}

}1{∑u

k=1 σ(k) >∑y−y0

j=1

∑ul=1 1{ηj = l}

}u

(12)

We can then characterize the expected change in the inconsistency level due to one addi-

tional aide in the on-call pool — with expectation taken over the assignment of units from

η and the home unit ηy−y0+1 of the marginal aide in the on-call pool — as

∆Eη,ηy−y0+1 [Lρ(y;η,σ)] =

−u∑k=1

E

[1

{ηy−y0+1 = k,σ(k) >

y−y0∑j=1

1{ηj = k} ,u∑k=1

σ(k) >

y−y0∑j=1

u∑l=1

1{ηj = l}

}]/u.

(13)

Because the home unit of any on-call aide (i.e., ηi) depends on the on-call pool enlist-

ment rule, we present separate results for the open and restricted on-call pools. We then

show that the restricted on-call pool outperforms the open on-call pool in terms of the

inconsistency.

Proposition 2. For a system with an open on-call pool, the inconsistency level LO(y)

is convex and non-increasing in the staffing level y.

Corollary 2. When the staffing requirement is deterministic, enlisting one additional

aide in an open on-call pool reduces the inconsistency level by

−∆LdetO (y) =

∑qj=1B(j; q, γ)

∑j−1i=0 B(i;y− y0,1/u)

u. (14)

We analyze the restricted on-call pool in a similar fashion, and take advantage of the

properties of the restricted on-call pool that allow us to write the value of a marginal aide

in the on-call pool more explicitly in our proofs.

Proposition 3. For a system with a restricted on-call pool, the inconsistency level

LR(y) is convex and non-increasing in the staffing level y.

If the staffing requirement is not subject to randomness, we can more precisely charac-

terize the relationship between the inconsistency and the staffing level.

15

Corollary 3. When the staffing requirement is deterministic for a system with a re-

stricted on-call pool and a staffing level y, enlisting one additional on-call aide reduces the

inconsistency level by

−∆LdetR (y) =

q∑i=b y−y0u c+1

B(i; q, γ)/u. (15)

Taken together, Propositions 1, 2, and 3 could inform a manager’s choice of the on-call

pool size considering both cost and consistency. Because inconsistency is non-increasing in

y, a manager seeking to minimize costs subject to a constraint on the inconsistency level

could find the value of y that minimizes costs and compare this with the lowest value of

y for which the inconsistency is below the maximum allowable level. (This latter value

could be found by bisection search over y using (3) for inconsistency.) Alternatively, if a

manager could assign a monetary (non-negative) penalty value to the inconsistency level,

the resulting cost function — i.e., the sum of the staffing cost and the inconsistency penalty

— would be convex; the sum of a convex function and a non-negative value multiplied by

a convex function. This could then be optimized using standard techniques.

We next slightly modify (3) to study how the allocation of on-call nurse aides across the

various units affects the expected inconsistency due to absences. Instead of a sample path

defined by the home unit ηj of each aide in the on-call pool, we use x = {x(1), . . . , x(u)}

where x(k) is the number of on-call pool aides for some arbitrary unit k that is mapped

to a specific unit on a sample path with the variable χk ∈ {1, . . . , u}. Specifically, we now

have

Lρ(x;σ) = Eχ

[u∑k=1

(σk)+−min

{u∑k=1

u∑l=1

min{σ+k , x(l)

}1{χl = k} ,

(u∑k=1

σk

)+}]/u, (16)

with

Lρ(x) = Eσ [Lρ(x;σ)] . (17)

We use the concepts of majorization and Schur-convexity to assess the performance in

terms of the expected inconsistency level due to absences of different on-call pool enlistment

rules. We use the notation x[1] to represent the largest element in x, x[2] to represent the

second-largest element, and so on, and use the notation x�x′ to say that x majorizes x′.

16

Table 2 Additional Notation for Section 4.3

Symbol Description

µ Number of full-time aides required for one unit across κ shifts if only full-time aides may be usedδ Number of part-time aides employed in the unitτ Ratio of scheduled hours per week for a part-time aide to full-time aideα Probability that any one unscheduled full-time aide enlists in on-call pool for one shiftβ Probability that any one unscheduled part-time aide enlists in on-call pool for one shift

C(δ) Expected value of consistency metric with δ part-time aidesφ Probability that native aide would get on-call slot if native aide is available

Proposition 4. The expected inconsistency level due to absences Lρ(x) is Schur-convex

in x; i.e., Lρ(x)≥Lρ(x′) if x�x′.

Because the restricted on-call pool produces an allocation that cannot be majorized (i.e.,

x[1] ≤ x[u] + 1 for the restricted on-call pool, we have the following result:

Corollary 4. The restricted on-call pool minimizes the expected inconsistency due to

absences Lρ(x).

4.3. Workforce Mix and Consistency of Care

We now consider a variation on our model that accounts for random participation in the

on-call pool by nurse aides and a decision to choose the mix of part-time and full-time

workers. Employing a higher percentage of full-time aides means that fewer unique aides

are needed to cover all shifts in one month. However, part-time aides typically are eager to

participate in an on-call pool, thus having additional part-time aides in a unit may increase

the probability that at least one native aide enlists in the on-call pool for a shift. We focus

our analysis on the restricted on-call pool model due to the improved consistency it offers

compared to the open model. New notation introduced in this section is listed in Table 2.

We assume that all shifts over the course of κ scheduling periods can be served by µ

full-time nurse aides. In other words, without absenteeism the patients would experience

µ unique full-time aides over the course of κ shifts. Each full-time aide may be replaced,

however, by part-time aides who work a fraction τ , 0< τ < 1, of the hours of a full-time

aide. The decision variable in this model is then the number of part-time aides to employ

in one unit, which we denote by δ, 0≤ δ≤ µ/τ .

For analytical tractability, we assume that the fraction of the q scheduled aides on each

shift who are part-time aides is constant and equivalent to the proportion of total scheduled

shifts assigned to part-time aides. That is, (µ− τδ)/µ of all shifts are worked by full-time

aides, and the remaining τδ/µ shifts are worked by part-time aides. Thus, the number of

17

full-time aides not working on a given shift (and thus eligible to volunteer for the on-call

pool) is µ− τδ− (µ− τδ)q/µ, and the number of part-time aides eligible is δ− τδq/µ. For

example, if µ= 6 full-time aides are required to cover all shifts and if part-time aides are

scheduled for τ = 0.5 as many hours as full-time aides, choosing δ = 2 corresponds to a

system with two part-time aides and µ− τδ = 5 full-time aides. To simplify our analysis,

we only consider a system with y− y0 ≤ u; there may only be up to one aide on-call per

unit. For the cases that we consider, y− y0 ≤ u includes all values for the on-call pool size

that reduce inconsistency while not increasing staffing costs compared to a system without

an on-call pool.

Given this constraint, it becomes necessary to define rules for choosing among nurse

aides in the event that multiple aides from different units wish to enlist in the on-call pool

on a shift. In such a case, specifically when y − y0 < u, we assume that the unit under

consideration has the opportunity to provide a nurse aide to the on-call pool on a fraction

φ of all shifts. On the remaining fraction (1− φ) of shifts, enough aides from outside the

unit have volunteered for the on-call pool slot and have priority over any native aide. In

practice, a system in which on-call pool enlistment priority rotates among different units

corresponds to this model. For example, suppose one on-call pool position is open to aides

in two different units, Unit A and Unit B. A nurse aide from Unit A may occupy the on-call

pool slot if Unit A has priority over Unit B on that shift or if Unit B has priority but no

aide from Unit B enlists. If the probability that no aide from a specific unit enlists in the

on-call pool is 50%, then φ= 75%.

We evaluate the expected number of unique nurse aides who provide care in a unit over

κ scheduled shifts when the unit has δ part-time aides on staff:

C(δ) = µ− τδ+ δ+κE[(S−H)+

],

which accounts for the µ− τδ full-time aides, δ part-time aides, and effect of absences over

κ shifts. As before, we assume that non-native aides who substitute for absent aides are

different each time that a substitution is required.

With (S−H)+, we only need to consider cases when S ≥H (recall H ≤ 1):

C(δ) = µ− τδ+ δ+κ

q∑i=1

(iPr (S = i)−Pr (H = 1)) . (18)

18

Each full-time aide who is not scheduled volunteers for the on-call pool with independent

and identical probability α, and each part-time aide who is not scheduled volunteers with

independent and identical probability β. Thus, the probability that the on-call pool includes

a native aide is

Pr (H = 1) = φ(

1− (1−β)δ−τδqµ (1−α)µ−τδ−

(µ−τδ)qµ

),

which we substitute into (18) to get the overall inconsistency level:

C(δ) = µ− τδ+ δ+κ

q∑i=1

B(i; q, γ)(i−φ

(1− (1−β)δ−

τδqµ (1−α)µ−τδ−

(µ−τδ)qµ

)).

Because∑q

i=1 iB(i; q, γ) =∑q

i=0 iB(i; q, γ) = qγ is the expected value of a binomial random

variable and∑q

i=1B(i; q, γ) = 1−B(0; q, γ), we rewrite this expression as

C(δ) =µ− τδ+ δ+κqγ−κ (1−B(0; q, γ))φ(

1− (1−β)δ−τδqµ (1−α)µ−τδ−

(µ−τδ)qµ

)=µ− τδ+ δ+κqγ−κφ (1−B(0; q, γ))

+κφ (1−B(0; q, γ)) (1−β)δ−τδqµ (1−α)µ−τδ−

(µ−τδ)qµ . (19)

Using (19) we can evaluate the effect of the number of part-time workers on consistency.

Proposition 5. The overall inconsistency level may decrease in the number of part-

time workers δ; specifically, δ close to zero and a sufficiently high part-time worker on-call

pool participation rate β ensure C(δ) decreases in δ.

Proposition 5 provides the managerial insight that converting a full-time position into

multiple part-time positions can actually decrease the expected number of unique nurse

aides caring for residents. Thus, subject to constraints on the number of hours worked that

correspond to part-time status, managers should hire part-time aides who are willing to

participate in the on-call pool at a relatively high rate. The increasing use of part-time

aides in on-call pools has another significant benefit; part-time aides in the on-call pool

typically would not be working enough hours to earn overtime pay.

5. Numerical Results

Motivated by our collaboration with a nursing home to implement an on-call pool, we in-

troduce model parameters representing typical nursing homes in Section 5.1. In Section 5.2

we study the impact of the size of the on-call pool on both the cost of absenteeism and

inconsistency level for facilities with different numbers of beds and with different numbers

of residents per unit. We analyze the workforce mix decision in Section 5.3.

20

$2,496 less paid to workers who are absent. As shown in Figure 1, an on-call pool of one

nurse aide minimizes the expected monthly cost, and produces a savings of $609, or 24% of

costs due to absences. Using an on-call pool of three aides for the staff of 16 aides provides

a solution that has approximately the same cost as the system without an on-call pool, but

keeps most of the payments “in-house” — to aides on staff in the forms of on-call wages

and bonus pay.

Figure 1 also shows that the monthly inconsistency level due to absences is convex and

non-increasing in the on-call pool size. Each of the first u aides in the on-call pool reduces

the inconsistency level for the day or evening shift by a constant amount for restricted

pools: 1.39 fewer unique aides each month per on-call aide for a large-unit facility and

0.37 per aide for a small-unit facility. As indicated by Proposition 4, the restricted on-call

pool outperforms the open on-call pool. The maximum difference between restricted and

open systems occurs when y = y0 + u. This difference in performance shows that nursing

homes should manage on-call pool enlistment so that participation on any shift is spread

as evenly as possible across the units.

The on-call pool sizes in Figure 1 that are identified as cost-neutral (i.e., approximately

the same as when y= y0) can significantly reduce the expected monthly inconsistency level.

A restricted on-call pool with three aides reduces the inconsistency due to absences by 70%

for a facility with large units and 37% for a facility with small units. The inconsistency level

due to absences is initially higher for large unit facilities compared to small unit facilities,

but eventually becomes lower for larger pool sizes. For example, the inconsistency level is

lower for large units than small units when a restricted on-call pool is operated with at

least three aides. The additional staff needed for a large unit compared to the small unit

means that the overall consistency score — i.e., the number of unique aides per resident

per month — will initially be lower for the small unit. However, the value of having small

units diminishes when absences are considered, because it is easier to replace an absent

aide with a “native” aide from the on-call pool when the unit size is larger (and thus there

are fewer units).

For a system without an on-call pool, it is possible to decrease both the cost and in-

consistency by adding one. That the dominated strategy of relying solely on rental agency

aides is commonly used in practice reflects the challenge that many managers have making

decisions given the stochastic nature of absences and overtime wages combined with the

22

per month that is lower than if there were δ = 0 part-time aides on staff. In other words,

replacing one full-time aide with two part-time aide both increases the number of unique

caregivers on staff by 1.0 and lowers the inconsistency level due to absences by at least 1.0.

As the full-time aide participation rate increases, the minimum part-time aide participation

rate also must increase because part-time aides must be available for more shifts to get

the same reduction in the inconsistency level. The required value of β is 50-65% higher for

the small unit facility than the large unit facility, which reflects the lower probability of

an on-call nurse to be native with the same on-call pool size.

6. Concluding Remarks

Nurse aides play a critical role in the delivery of care to nursing home residents. Nursing

home managers make concerted efforts to avoid having a high number of unique aides

caring for each resident; i.e., to attain high consistency of care. This paper analyzes both

staffing cost and care consistency when nurse aides are subject to random absences on

each shift. One key feature of our model is the explicit accounting for the expected number

of unique nurse aides who care for a resident in a month — a metric that is increasingly

promoted by advocates of patient-centered care in nursing homes.

We provide structural results for decisions related to the on-call pool’s size and organi-

zation, and show numerically that an on-call pool can reduce both the staffing cost due to

absences and improve consistency of care. We also show that using part-time aides in con-

junction with on-call pools can further improve consistency of care if the part-time aides’

on-call pool participation rate is sufficiently high. We recognize that our assumption of a

constant fraction of part-time and full-time aides working on each shift and eligible for the

on-call pool may not be strictly implementable. However, we believe that this model still

provides important insights on how adding part-time nurse aides to a unit can actually

decrease the total number of unique nurse aides with whom residents in that unit interact

in a month. Our work points to an opportunity for resource-constrained nursing homes to

improve nurse aide job satisfaction, nurse aide pay, the personal comfort of nursing home

residents, the medical care that they receive, and nurse aide staffing costs.

Future work can examine nurse aide scheduling policies in more detail to guide decisions

about pay differentials for weekend and night shifts and promises made to employees to

satisfy labor rules or provide schedule predictability. For example, some nursing homes

23

require nurse aides to work every other weekend. Other nursing homes have nurse aides

who have full-time status but only work extended shifts on weekends. These questions for

future research are especially important when connected to the overall staffing decision.

Appendix

Proof of Proposition 1. The first forward difference of the cost of absences ∆C(y) =C(y+ 1)−C(y) can

be written using (7) and cancelling identical terms:

∆C(y) =

y−y0+1∑i=0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)(wci+ b(y− y0 + 1− i))

+∞∑

i=y−y0+2

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)(wa(i− y− 1 + y0) +wc(y− y0 + 1))

−y−y0∑i=0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)(wci+ b(y− y0− i))

−∞∑

i=y−y0+1

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)(wa(i− y+ y0) +wc(y− y0)) .

Collecting terms,

∆C(y) =b

y−y0∑i=0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)+

(y0∑j=0

gT (y+ 1− j)B(j;y0, γ)

)wc(y− y0 + 1)

−

(y0∑j=0

gT (y+ 1− j)B(j;y0, γ)

)(wa +wc(y− y0))

+∞∑

i=y−y0+2

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)(−wa +wc)

=b

y−y0∑i=0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)− (wa−wc)

y0∑i=y−y0+1

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

).

Next, we demonstrate that C(y) is convex in y by showing that ∆C(y)≤∆C(y+ 1) for any y ≥ y0, which

is equivalent to

b

y−y0∑i=0

(∞∑j=0

gT (y0 + i− j)B(j;y0, γ)

)− (wa−wc)

∞∑i=y−y0+1

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)

≤by−y0+1∑i=0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)− (wa−wc)

∞∑i=y−y0+2

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

).

Rearranging terms, we have

−b

(y0∑j=0

gT (y+ 1− j)B(j;y0, γ)

)≤ (wa−wc)

(y0∑j=0

gT (y+ 1− j)B(j;y0, γ)

),

which reduces to the relationship, wa ≥wc− b, that we have assumed regarding the costs.

Due to the convexity of C(y) in y, we know that

y∗ = min(y ∈N+|y≥ y0,∆C(y)≥ 0

)

24

minimizes (7). We rewrite ∆C(y)≥ 0 as

b

y−y0∑i=0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)≥ (wa−wc)

∞∑i=y−y0+1

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)

b

y−y0∑i=0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)≥ (wa−wc)

(1−

y−y0∑i=0

(y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

))

(y−y0∑i=0

y0∑j=0

gT (y0 + i− j)B(j;y0, γ)

)(b+wa−wc)≥wa−wc

y−y0∑i=0

y0∑j=0

gT (y0 + i− j)B(j;y0, γ)≥ wa−wcb+wa−wc

.

Thus the minimizer of (7) is given by

y∗ = min

(y ∈N+|y≥ y0,

y−y0∑i=0

y0∑j=0

gT (y0 + i− j)B(j;y0, γ)≥ wa−wcb+wa−wc

). �

Proof of Proposition 2. We compare sample paths to show that

∆Eη,ω [LO(y;η,σ)]≤∆Eη,ψ,ω [LO(y;η ∪ψ,σ)] , (20)

where ω is the value being coupled representing the home unit for the marginal aide in the on-call pool

(i.e., ηy−y0+1 = ω on the left-hand side of (20) and ηy−y0+2 = ω on the right-hand side). The two sample

paths being compared would also share η, and ηy−y0+1 on the right-hand side of (20) could take any value

ψ ∈ {1, . . . , u}. To illustrate with an example for y − y0 = 3, the sample path could have η = {6,2,1} and

both ηy−y0+1 on the left-hand side and ηy−y0+2 on the right-hand side equal to 4. We show that (20) holds

because it holds for the sample paths being paired, which — by substitution using (13) and because ηy−y0+1

on the left-hand side of (20) is equivalent to ηy−y0+2 on the right-hand side — is equivalent to

1

{σ(k) >

y−y0∑j=1

1{ηj = k} ,u∑k=1

σ(k) >

y−y0∑j=1

u∑l=1

1{ηj = l}

}

≥1

{σ(k) >

y−y0+1∑j=1

1{ηj = k} ,u∑k=1

σ(k) >

y−y0+1∑j=1

u∑l=1

1{ηj = l}

},

which holds because∑y−y0+1

j=1 1{ηj = k} ≥∑y−y0

j=1 1{ηj = k} and∑y−y0+1

j=1

∑u

l=1 1{ηj = l} ≥∑y−y0j=1

∑u

l=1 1{ηj = l} on any sample path. Because this comparison holds on all pairings for the home units

ηy−y0+1 and ηy−y0+2 of the marginal aides, it holds in expectation. �

Proof of Corollary 2. Taking the first forward difference of LdetO (y) using (4), we have

∆LdetO (y) =LdetO (y+ 1)−LO(y)

=

y−y0+1∑i=0

B(i;y− y0 + 1,1/u)

q∑j=i

B(j; q, γ)(j− i)

−y−y0∑i=0

B(i;y− y0,1/u)

q∑j=i

B(j; q, γ)(j− i). (21)

25

By the definition of the binomial distribution, we note that

B(i;y− y0 + 1,1/u) =u− 1

uB(i;y− y0,1/u) +

1

uB(i− 1;y− y0,1/u),

which we substitute into (21) to get

∆LdetO (y) =

y−y0+1∑i=0

(u− 1

uB(i;y− y0,1/u) +

1

uB(i− 1;y− y0,1/u)

) q∑j=i

B(j; q, γ)(j− i)

−y−y0∑i=0

B(i;y− y0,1/u)

q∑j=i

B(j; q, γ)(j− i).

Noting that B(y− y0 + 1;y− y0,1/u) = 0 and subtracting terms,

∆LdetO (y) =

y−y0∑i=0

−1

uB(i;y− y0,1/u)

q∑j=i

B(j; q, γ)(j− i)

+

y−y0+1∑i=0

1

uB(i− 1;y− y0,1/u)

q∑j=i

B(j; q, γ)(j− i).

Because B(i;y− y0,1/u) = 0 for i < 0, we can change the summation index to get

∆LdetO (y) =

y−y0∑i=0

−1

uB(i;y− y0,1/u)

q∑j=i

B(j; q, γ)(j− i)

+

y−y0∑i=0

1

uB(i;y− y0,1/u)

q∑j=i+1

B(j; q, γ)(j− i− 1).

We combine terms, noting that B(j; q, γ)(j− i) = 0 for j = i, to get

∆LdetO (y) =

y−y0∑i=0

−1

uB(i;y− y0,1/u)

q∑j=i+1

B(j; q, γ)(j− i− j+ i+ 1)

=−∑y−y0

i=0 B(i;y− y0,1/u)∑q

j=i+1B(j; q, γ)

u.

Changing the order of summation, we have

∆LdetO (y) =−∑q

j=1B(j; q, γ)∑j−1

i=0 B(i;y− y0,1/u)

u. �

Proof of Proposition 3. Due to the operational characteristics of the restricted on-call pool, we can write

(13) as

∆Eη,ηy−y0+1[LR(y;η,σ)] =−

∑u

k=1 1{σ(k) >

⌊y−y0u

⌋}u

∑uk=1 σ(k)−1−

∑uk=1 min

{σ+(k),b y−y0u c

}∑i=0

H

(i;u− 1,

u∑k=1

1

{σ(k) >

⌊y− y0u

⌋}− 1, y− y0−u

⌊y− y0u

⌋)/u, (22)

where∑uk=1 1{σ(k)>b y−y0u c}

urepresents the probability that the home unit of the marginal aide in the on-call

pool has a shortage when there are y − y0 + 1 aides in the on-call pool. The remainder of the expression

represents the probability that the marginal aide is not blocked by reallocated scheduled aides (for the case

that the marginal aide’s home unit does have a shortage). The marginal aide is blocked if the number of

26

on-call aides who could potentially fill shortages in their home units is greater than or equal to the number

of scheduled aides who need to be reassigned due to low demand. The number of on-call aides serving in

their home units can be calculated for the restricted on-call pool using the knowledge that at least⌊y−y0u

⌋aides in the on-call pool belong to each unit. Specifically, y − y0 − u

⌊y−y0u

⌋units have

⌊y−y0u

⌋+ 1 aides in

the on-call pool and the remaining units have⌊y−y0u

⌋aides in the on-call pool. Thus, the number of on-call

aides that could serve in their home units among those indexed u⌊y−y0u

⌋+ 1, . . . , y− y0 is a hypergeometric

random variable.

To illustrate, if 0 < y − y0 < u the number of on-call aides with a shortage in their home unit is a hy-

pergeometric random variable with parameters defined as the number of units u− 1 (not counting the unit

with the marginal on-call aide), the number of units with a shortage (again, for the case that the marginal

aide’s home unit has a shortage), and the number of aides in the on-call pool y− y0. For the marginal aide

to serve in his or her home unit, the value of this random variable must be less than or equal to the net

staffing shortage (∑u

k=1 σ(k)) minus one (to allow space for the marginal aide). If y− y0 ≥ u, then we must

also account for the min{σ+(k),⌊y−y0u

⌋}on-call aides in each unit k that are already assigned to their home

unit when considering if it is still possible for the marginal aide to be assigned to the aide’s home unit.

We use a sample path coupling approach as described in the proof of Proposition 2 to show that

∆Eη,ω [LR(y;η,σ)]≤∆Eη,ψ,ω [LR(y+ 1;η ∪ψ,σ)] , (23)

where the coupling notation is the same as in (20).

If⌊y−y0u

⌋=⌊y−y0+1

u

⌋, then substituting (22) into (23) and replacing

⌊y−y0+1

u

⌋with

⌊y−y0u

⌋gives the

condition∑uk=1 σ(k)−1−

∑uk=1 min

{σ+(k),b y−y0u c

}∑i=0

H

(i;u− 1,

u∑k=1

1

{σ(k) >

⌊y− y0u

⌋}− 1, y− y0−u

⌊y− y0u

⌋)

≥

∑uk=1 σ(k)−1−

∑uk=1 min

{σ+(k),b y−y0u c

}∑i=0

H

(i;u− 1,

u∑k=1

1

{σ(k) >

⌊y− y0u

⌋}− 1, y− y0 + 1−u

⌊y− y0u

⌋),

which holds by the properties of the hypergeometric distribution; i.e., the probability that the value of a

hypergeometric random variable is less than or equal to some value is non-increasing in the number of draws

when all other parameters remain the same.

If⌊y−y0u

⌋=⌊y−y0+1

u

⌋− 1, then substituting (22) into (23) and replacing

⌊y−y0+1

u

⌋with

⌊y−y0u

⌋+ 1 gives

the condition ∑u

k=1 1{σ(k) >

⌊y−y0u

⌋}u

∑uk=1 σ(k)−1−

∑uk=1 min

{σ+(k),b y−y0u c

}∑i=0

H

(i;u− 1,

u∑k=1

1

{σ(k) >

⌊y− y0u

⌋}− 1, y− y0−u

⌊y− y0u

⌋)

≥∑u

k=1 1{σ(k) >

⌊y−y0u

⌋+ 1}

u

∑uk=1 σ(k)−1−

∑uk=1 min

{σ+(k),b y−y0u c+1

}∑i=0

H

(i;u− 1,

u∑k=1

1

{σ(k) >

⌊y− y0u

⌋+ 1

}− 1, y− y0 + 1−u

⌊y− y0 + 1

u

⌋).

27

The relationship⌊y−y0u

⌋=⌊y−y0+1

u

⌋−1 implies that y−y0−u

⌊y−y0u

⌋= u−1 and y−y0 +1−u

⌊y−y0+1

u

⌋= 0;

i.e., the hypergeometric random variable on the left-hand side of the inequality has u− 1 “draws,” and the

hypergeometric random variable on the right-hand side has zero “draws.” The case with u−1 draws implies

that∑u

k=1 min{σ+(k),⌊y−y0u

⌋+ 1}− 1 of the on-call aides indexed 1, . . . , y− y0 could serve in the home unit

(with one subtracted to account for the marginal on-call aide, which is assumed to be able to serve in his or

her home unit due to the conditional probability used to formulate (22)). Thus, returning to the left hand

side, we have

∑uk=1 σ(k)−1−

∑uk=1 min

{σ+(k),b y−y0u c

}∑i=0

H

(i;u− 1,

u∑k=1

1

{σ(k) >

⌊y− y0u

⌋}− 1, u− 1

)

=1

{u∑k=1

σ(k)− 1−u∑k=1

min

{σ+(k),

⌊y− y0u

⌋}>

u∑k=1

1

{σ(k) >

⌊y− y0u

⌋}− 1

}

=1

{u∑k=1

σ(k) >u∑k=1

min

{σ+(k),

⌊y− y0u

⌋+ 1

}},

because 1{σ(k) >

⌊y−y0u

⌋}implies that min

{σ+(k),⌊y−y0u

⌋}=⌊y−y0u

⌋, and thus

u∑k=1

(min

{σ+(k),

⌊y− y0u

⌋}+1

{σ(k) >

⌊y− y0u

⌋})=

u∑k=1

min

{σ+(k),

⌊y− y0u

⌋+ 1

}.

The case with none drawn can be analyzed as

∑uk=1 σ(k)−1−

∑uk=1 min

{σ+(k),b y−y0u c+1

}∑i=0

H

(i;u− 1,

u∑k=1

1

{σ(k) >

⌊y− y0u

⌋+ 1

}− 1,0

),

which is equal to one if the summation includes zero; i.e.,

1

{u∑k=1

σ(k)− 1−u∑k=1

min

{σ+(k),

⌊y− y0u

⌋+ 1

}≥ 0

}

=1

{u∑k=1

σ(k) >u∑k=1

min

{σ+(k),

⌊y− y0u

⌋+ 1

}}.

We then have the condition∑u

k=1 1{σ(k) >

⌊y−y0u

⌋}u

1

{u∑k=1

σ(k) >u∑k=1

min

{σ+(k),

⌊y− y0u

⌋+ 1

}}

≥∑u

k=1 1{σ(k) >

⌊y−y0u

⌋+ 1}

u1

{u∑k=1

σ(k) >u∑k=1

min

{σ+(k),

⌊y− y0u

⌋+ 1

}},

which holds because 1{σ(k) >

⌊y−y0u

⌋}≥ 1

{σ(k) >

⌊y−y0u

⌋+ 1}

for k= 1, . . . , u. �

Proof of Corollary 3. We again analyze the first forward difference of the inconsistency level using (5):

∆LdetR (y) =LdetR (y+ 1)−LdetR (y)

=

q∑i=b(y−y0+1)/uc+1

B(i; q, γ)

(i− y− y0 + 1

u

)−

q∑i=b(y−y0)/uc+1

B(i; q, γ)

(i− y− y0

u

).

28

We must consider the two cases of b(y− y0 + 1)/uc= b(y− y0)/uc and b(y− y0 + 1)/uc= 1 + b(y− y0)/uc.When b(y− y0 + 1)/uc= b(y− y0)/uc, we observe that

∆LdetR (y) =

q∑i=b(y−y0)/uc+1

B(i; q, γ)

((i− y− y0 + 1

u

)−(i− y− y0

u

))

=

q∑i=b(y−y0)/uc+1

B(i; q, γ)

(−(y− y0 + 1) + (y− y0)

u

)

=

q∑i=b(y−y0)/uc+1

B(i; q, γ)

(−1

u

).

Now we turn to the other case, when b(y − y0 + 1)/uc = 1 + b(y − y0)/uc. Using the relationship that

(y− y0)/u−b(y− y0)/uc= (u− 1)/u when b(y− y0 + 1)/uc= 1 + b(y− y0)/uc, we have

∆LdetR (y) =−q∑

i=⌊

(y−y0+1)u

⌋+1

B(i; q, γ)/u−B(⌊

(y− y0)

u

⌋+ 1; q, γ

)(⌊(y− y0)

u

⌋+ 1− y− y0

u

),

and note that ⌊(y− y0)

u

⌋+ 1− y− y0

u= 1−

(y− y0u−⌊

(y− y0)

u

⌋)=

1

u.

Thus, if b(y− y0 + 1)/uc= 1 + b(y− y0)/uc,

∆LdetR (y) =−q∑

i=⌊

(y−y0+1)u

⌋+1

B(i; q, γ)/u−B(⌊

(y− y0)

u

⌋+ 1; q, γ

)(1

u

)

=−q∑

i=⌊

(y−y0)u

⌋+2

B(i; q, γ)/u−B(⌊

(y− y0)

u

⌋+ 1; q, γ

)/u.

Therefore, for either case,

∆LdetR (y) =−q∑

i=⌊

(y−y0)u

⌋+1

B(i; q, γ)/u. �

Proof of Proposition 4. Our proof relies on a switching argument on a sample path based on inconsistency

defined as in (16). Specifically, we take an allocation vector x with x[1] ≥ x[u] + 2 and show that substituting

x[i]−1 for x[i] and x[j] +1 for x[j] with x[i] ≥ x[j] +2 is a non-increasing transformation of Lρ(x). This process

can be repeated until the vector x becomes x′, which can be any vector that is majorized by x.

Without loss of generality we let i= 1 and j = u for notational convenience. Assignments {χ[2], . . . , χ[u−1]}and staffing shortage values σ are known for the sample path. Without loss of generality σ1 and σu are the

possible shortage values for the units corresponding to x[1] and x[u] (i.e., {χ[1], χ[u]} ∈ {{1, u},{u,1}}) and

χ[k] = k for k = 2, . . . , u. Calculating the expected inconsistency level involves taking the expectation over

χ[1] and χ[u]; i.e., which number of on-call pool aides is matched to which unit staffing shortage value.

With this characterization of a sample path, (16) becomes

Lρ(x[1], x[u];σ,x,{χ[2], . . . , χ[u−1]}) =

Eχ[1],χ[u]

u∑k=1

(σk)+−min

u∑k=1

u∑l=1

min{σ+k , x(l)

}1{χl = k} ,

(u∑k=1

σk

)+/u, (24)

29

which can be rewritten to account for the two equally likely scenarios:

Lρ(x[1], x[u];σ,x,{χ[2], . . . , χ[u−1]}) =u∑k=1

(σk)+−

min{∑u

k=1 min{σ+k , x[k]

}, (∑u

k=1 σk)+}

2u

−min

{min

{σ+1 , x[u]

}+ min

{σ+u , x[1]

}+∑u−1

k=2 min{σ+k , x[k]

}, (∑u

k=1 σk)+}

2u. (25)

We now show that

Lρ(x[1], x[u];σ,x,{χ[2], . . . , χ[u−1]})≥Lρ(x[1]− 1, x[u] + 1;σ,x,{χ[2], . . . , χ[u−1]}), (26)

which by substituting (25) and reducing the inequality is equivalent to

min

min{σ+1 , x[1]

}+ min

{σ+u , x[u]

}+u−1∑k=2

min{σ+k , x[k]

},

(u∑k=1

σk

)+

+ min

min{σ+1 , x[u]

}+ min

{σ+u , x[1]

}+u−1∑k=2

min{σ+k , x[k]

},

(u∑k=1

σk

)+

≤min

min{σ+1 , x[1]− 1

}+ min

{σ+u , x[u] + 1

}+u−1∑k=2

min{σ+k , x[k]

},

(u∑k=1

σk

)+

+ min

min{σ+1 , x[u] + 1

}+ min

{σ+u , x[1]− 1

}+u−1∑k=2

min{σ+k , x[k]

},

(u∑k=1

σk

)+ (27)

For notational simplicity we define G := (∑u

k=1 σk)+−

∑u−1k=2 min

{σ+k , x[k]

}. We then recognize that

min{

min{σ+1 , x[1]− 1

}+ min

{σ+u , x[u] + 1

},G}

= min{

min{σ+1 , x[1]

}+ min

{σ+u , x[u]

},G}

−1{x[1] ≤ σ+

1 , x[u] ≥ σ+u ,G≥min

{σ+1 , x[1]

}+ min

{σ+u , x[u]

}}+1

{x[u] <σ

+u , x[1] >σ

+1 ,G>min

{σ+1 , x[1]

}+ min

{σ+u , x[u]

}}and

min{

min{σ+1 , x[u] + 1

}+ min

{σ+1 , x[1]− 1

},G}

= min{

min{σ+1 , x[u]

}+ min

{σ+u , x[1]

},G}

+1{x[u] <σ

+1 , x[1] >σ

+u ,G>min

{σ+1 , x[u]

}+ min

{σ+u , x[1]

}}−1

{x[1] ≤ σ+

u , x[u] ≥ σ+1 ,G≥min

{σ+1 , x[u]

}+ min

{σ+u , x[1]

}},

which by substitution into (27) and subtraction of identical terms provides the inequality

0≤−1{x[1] ≤ σ+

1 , x[u] ≥ σ+u ,G≥min

{σ+1 , x[1]

}+ min

{σ+u , x[u]

}}+1

{x[u] <σ

+u , x[1] ≤ σ+

1 ,G>min{σ+1 , x[1]

}+ min

{σ+u , x[u]

}}+1

{x[u] <σ

+1 , x[1] >σ

+u ,G>min

{σ+1 , x[u]

}+ min

{σ+u , x[1]

}}−1

{x[1] ≤ σ+

u , x[u] ≥ σ+1 ,G≥min

{σ+1 , x[u]

}+ min

{σ+u , x[1]

}}, (28)

which is equivalent to

0≤−1{x[1] ≤ σ+

1 , x[u] ≥ σ+u ,G≥ x[1] +σ+

u

}+1

{x[u] <σ

+u , x[1] >σ

+1 ,G> x[u] +σ+

1

}+1

{x[u] <σ

+1 , x[1] >σ

+u ,G> x[u] +σ+

u

}−1

{x[1] ≤ σ+

u , x[u] ≥ σ+1 ,G≥ x[1] +σ+

1

}. (29)

30

Because x[1] ≥ x[u] + 2,

1{x[u] <σ

+1 , x[1] >σ

+u ,G> x[u] +σ+

u

}≥ 1

{x[1] ≤ σ+

1 , x[u] ≥ σ+u ,G≥ x[1] +σ+

u

}and

1{x[u] <σ

+u , x[1] >σ

+1 ,G> x[u] +σ+

1

}≥ 1

{x[1] ≤ σ+

u , x[u] ≥ σ+1 ,G≥ x[1] +σ+

1

},

which implies that the inequality defined in (26) holds. Thus, any allocation x with the property that

x[1] ≤ x[u] + 1 minimizes Lρ(x) as the allocation minimizes Lρ(x;σ) on any sample path σ. �

Proof of Proposition 5. Taking the derivative of (19) with respect to the number of part-time workers:

dC(δ)

dδ=(1− τ) +κφ (1−B(0; q, γ))

[(1−β)(1−

τqµ )δ ln(1−β)

(1− τq

µ

)(1−α)(µ−τδ)(1−

τqµ )

+ (1−β)(1−τqµ )δ(1−α)(µ−τδ)(1−

qµ ) ln(1−α)(−τ)

(1− q

µ

)].

Combining terms,

dC(δ)

dδ=1− τ +κφ (1−B(0; q, γ)) (1−β)(1−

τqµ )δ(1−α)(µ−τδ)(1−

τqµ )(

ln(1−β)

(1− τq

µ

)− ln(1−α)τ

(1− q

µ

)). (30)

It is sufficient to show that C ′(0)≤ 0, which is expressed using (30) as

C ′(0) =1− τ +κφ (1−B(0; q, γ)) (1−α)µ−τq(

ln(1−β)

(1− τq

µ

)− ln(1−α)τ

(1− q

µ

)).

Thus, C ′(0)≤ 0 corresponds to

1− τ ≤−κφ (1−B(0; q, γ)) (1−α)µ−τq(

ln(1−β)

(1− τq

µ

)− ln(1−α)τ

(1− q

µ

)).

Rearranging terms and noting that κ≥ 0, 1−B(0; q, γ)≥ 0, and 1−α≥ 0,

1− τκφ (1−B(0; q, γ)) (1−α)µ−τq

≤− ln(1−β)

(1− τq

µ

)+ ln(1−α)τ

(1− q

µ

),

which is equivalent to

1− τκφ (1−B(0; q, γ)) (1−α)µ−τq

− ln(1−α)τ

(1− q

µ

)≤− ln(1−β)

(1− τq

µ

),

and thus

− 1− τφκ (1−B(0; q, γ)) (1−α)µ−τq

+ ln(1−α)τ

(1− q

µ

)≥ ln(1−β)

(1− τq

µ

).

Since 0≤ τ ≤ 1 and q≤ µ, 0≤ τq/µ≤ 1. Thus, 0≤ 1− τq/µ≤ 1 and

− 1− τ

κφ (1−B(0; q, γ)) (1−α)µ−τq(

1− τq

µ

) +ln(1−α)τ

(1− q

µ

)1− τq

µ

≥ ln(1−β).

Because the exponential function is increasing,

e− 1−τκφ(1−B(0;q,γ))(1−α)µ−τq(1− τq

µ )+

ln(1−α)τ(1− qµ )

1− τqµ ≥ 1−β,

31

which is equivalent to

β ≥ 1− e− 1−τκφ(1−B(0;q,γ))(1−α)µ−τq(1− τq

µ )+

ln(1−α)τ(1− qµ )

1− τqµ . (31)

Because ln(1−α)≤ 0, κφ≥ 0, 0≤ q/µ≤ 1, and all other terms are between 0 and 1,

0≤ 1− e− 1−τκφ(1−B(0;q,γ))(1−α)µ−τq(1− τq

µ )+

ln(1−α)τ(1− qµ )

1− τqµ ≤ 1.

From (31), we see that the value of β necessary for C ′(0)≤ 0 is decreasing in κ and increasing in α. Naturally,

the part-time aides have more chances to participate in the pool if the inconsistency level is measured over

more shifts. Also, it is easier for part-time nurse aides to affect the inconsistency level when the full-time

aides have a lower participation level, and a thus a lower part-time participation rate is required. �

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