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Labor Scheduling with Employee Turnover and Absenteeism
Fred F. Easton
Robert H. Brethen Operations Management Institute, School of Management
Syracuse University
Syracuse, NY 13244-2130
(315) 443-3463; ffeaston@syr.edu
John C. Goodale
Department of Management, College of Business
Ball State University
Muncie, IN 47306
(765) 285-5323; jgoodale@bsu.edu
Working Paper SLSDL_MS_031502.doc
March 15, 2002
This project was partially supported by a research grant from the Center for Creation and
Management of Digital Ventures at Syracuse University
Labor Scheduling with Employee Turnover and Absenteeism
Abstract
Most labor staffing and scheduling models presume that all employees scheduled for duty
reliably report for work at the beginning of their shift. For industries with even moderate
turnover or absenteeism, this assumption may be quite costly. We present a profit-oriented labor
scheduling model that accounts for the day-to-day flux of employees and capacity induced by
voluntary resignations, new hires, experience curves, and absenteeism. The proposed model also
anticipates revenue losses due to reneging by customers whose patience decays exponentially
with queue time. Our computational studies suggest that firms with comparatively high
transaction volumes, long transaction times, and/or relatively tight profit margins may
experience significant benefit from this approach. Compared with conventional labor scheduling
models, the proposed method boosts average expected profits by more than 10 percent in certain
operating environments.
2
Labor Scheduling with Employee Turnover and Absenteeism
1. Introduction
A host of direct and indirect costs arise from the wake of each employee who voluntarily
leaves an organization. Obvious expenses include the employer's recruiting, hiring, and training
costs for a replacement employee. Until the vacancy is filled, employers may also face
additional overtime costs, reduced productivity, increased customer queue times, lost sales and
business opportunities, and the likelihood of additional turnover due to the extra work shouldered
by coworkers of the departing employees (Herman, 1997; McConnel, 1999; Richardson, 1999).
Turnover is not just expensive; it is pervasive, arising in virtually all professions. Twenty
percent of all new school-teachers and forty-four percent of all new lawyers quit within three
years (Cooper, 2000; Flaherty, 1999). The average annual turnover rate (the number of annual
resignations divided by the average workforce size) among call center employees is 31 percent
(Karr, 1999), although larger operations (500 or more agents) average 61 percent/year. Zuber
(2001), citing declining turnover rates for limited-service restaurant workers, reported that 1999
turnover averaged 123 percent in this industry.
Voluntary turnover problems also tend to be persistent and difficult to eliminate. Leonard's
(1998) survey of human resource professionals found that 55 percent took measures to improve
turnover, but only 10 percent reported noticeable improvements. Furthermore, many firms report
that it takes 75 - 90 days to fill a vacant position (Fitz-enz, 1997; Matson, 1999), followed by
weeks or months of training before a newly-hired employee becomes proficient.
While turnover results in permanent losses of human capital, unexpected absences due to
illness or personal matters consume 2.5 percent of all scheduled work hours in the U.S. service
3
sector (Bureau of Labor Statistics, 2001). Some industries, however, suffer much higher
absenteeism rates. Call center managers, for example, report losing 12 - 16 percent of all
scheduled work hours to absenteeism (Call Center Ops, 2001).
The need to protect service delivery systems from turnover and absenteeism is well
established. Turnover planning models have been developed for many industries and
professions, including banking (Jones et al, 1973), engineering (Lapp & Thompson, 1974), law
enforcement (Leeson, 1981), and the armed services (Charnes et al., 1972; Collins, et al, 1983;
Eiger et al., 1988). Typically, these models anticipate the effects of turnover on existing staff and
estimate the number of new employees that should be recruited into the organization each year at
each grade to satisfy projected future staffing needs (Bartholomew, Forbes & McClean, 1991).
In this paper, however, we focus on the day-to-day operational impact of turnover and
absenteeism, and developing techniques that mitigate their impact through short-term staffing
and scheduling decisions. Our premise is that both service demand and employee availability are
random variables. We model the day-to-day flux of employee resignations and new hires as a
Markov process, then derive estimates for the probabilities of realizing different incumbency
levels. From the underlying employee survivor function, we predict workforce experience levels
and proficiencies. Modeling the service delivery system as a multi-server queue, we estimate
revenue losses due to reneging when customer patience decays exponentially with waiting time.
Finally, we devise a labor staffing and scheduling model that integrates workforce incumbency
probabilities, experience levels, and random employee absenteeism with the object of
maximizing expected profit under stochastic demand and impatient customers. The model
determines the nominal workforce size, and how those employees should be deployed over time,
to compensate for anticipated turnover and absenteeism.
4
To identify operating environments likely to benefit from this model, we compare its
solutions with those from a conventional profit-oriented labor staffing and scheduling model that
assumes stochastic service demand and deterministic labor supply. Our experiments reveal that
even with modest turnover rates and brief job vacancies, conventional labor scheduling models
consistently misstate ideal staffing levels and overstate expected profits and service levels. This
limitation is more pronounced as a firm's average service rate or profit margin decreases, and as
either its average turnover rate, mean job vacancy duration, absenteeism, or training time
increases. In several operating environments, the proposed model improved expected profit by
more than 10 percent over that earned with a conventional labor staffing and scheduling model.
The rest of the paper is organized in the following manner. In Section 2, we develop a
framework for estimating workforce incumbency levels and experience levels. In Section 3, we
characterize a multi-server queuing system with customers whose patience decays exponentially
with queue time. In Section 4, we describe our labor staffing and scheduling model for
stochastic service demand, impatient customers, and stochastic labor availability. In section 5,
we describe our experiments to compare the proposed model with conventional techniques and
discuss the results. Our summary of the project appears in Section 6.
2. Modeling turnover, absenteeism, and experience levels
Our goal in this research is to account for the stochastic nature of the labor supply in staffing
and scheduling decisions for systems with random arrivals and impatient customers. We adopt
the premise that employee resignations are in general random, independent events and that on
average, fraction Tof the total workforce will voluntarily leave the organization each year. We
assume that once a vacancy occurs, the employer initiates efforts to recruit a replacement. After
a random interval, a replacement worker arrives to staff the vacant position. We assume the new
5
worker's proficiency improves gradually with training and experience, approaching that of
veteran staffers after a certain period of time. Finally, we assume that a random fraction of the
employees scheduled for duty will fail to report due to unexpected illness. In the following
subsections, we develop models to estimate: (1) the short-term impact of turnover on the firm's
labor supply, (2) the effects of turnover on aggregate experience levels and productive capacity,
and (3) the effects of absenteeism.
2.1 Turnover, recruitment, and workforce dynamics.
Let W be the nominal workforce size during the period of interest (W is an output of our
proposed labor scheduling model, described in Section 4). In this section we estimate the
probabilities that w = 0, …, W of those positions are occupied. If employees independently and
randomly decide when to resign, the number of resignations/day should resemble a Poisson
process. Let γ = T /365 denote the expected fraction of the available workforce that resigns on a
given day. For γ to remain stationary1, the number of voluntary departures will depend on the
number of occupied positions. The Poisson probability that exactly k employees will resign
during interval t, given w positions are occupied, is:
(
)
!)|( kewtwkP wt
k
γ
γ
−
=. (1)
When an employee resigns, a search for a qualified replacement begins. Let v be the number
of days until the position is filled, and assume that v is exponentially distributed with mean v.
At any instant 0 < w < W of the authorized positions might be occupied, depending on the flux
of resignations and new recruits, with W-w active searches underway. On average vwW /)( −
vacant positions will be filled each day, so the Poisson probability that exactly k new recruits
1 Due to the increased workload and job stress for surviving workers, Tsui, Pearce, Porter, & Tripoli (1997) posit
that the ratio of resignations to occupied positions tends to increase with the number of job vacancies.
6
report for duty during some interval t is given by equation (1), substituting vwW /)( − for
parameters γw. The probability of exactly one voluntary resignation during interval h is:
0.h as h!1)()|1( 1→≈= −wewhwP wh
γγ
γ
(2)
The probability that exactly one new recruit arrives during interval h also depends on the state
variable w and the number of authorized positions W. For state w, the probability that exactly
one new recruit arrives during interval h is h(W-w)/ v as h→0.
The birth-death diagram in Figure 1 illustrates the flux of existing workers resigning and new
workers recruited into the system. The nodes represent the number of occupied positions,
ranging from 0 to W. The values within each node denote the probability of occupying that
state. The directed arcs pointing to the right are the state-dependent probabilities of exactly one
new recruit reporting for duty during interval h. The directed arcs pointing left are the state-
dependent probabilities of exactly one employee resignation occurring during the same interval.
(please insert Figure 1 about here)
To maintain a stable workforce size, the flow leaving each state in Figure 1 must equal the
flow into that state. The graph of this system is a tree, so it is reversible (Kelly, 1979) and its
state probabilities can be estimated from the probability flux between any pair of adjacent states
(Nelson, 1993). For each state, the following relationships must hold:
State Flow out = Flow in (3)
1 vWPP 01 =
γ
(3.1)
2
(
)
vWPP 12 12 −=
γ
(3.2)
3
(
)
vWPP 23 23 −=
γ
(3.3)
…
W vPWP WW 1
1−
=
γ
(3.W)
Rearranging and substituting P0 we have for state:
7
1 vWPP
γ
01 =, (4.1)
2
()
(
)
()
2
012 12
1
)2(1
v
WW
PvWPP
γ
γ
∗
−
=−= , (4.2)
3
(
)
(
)
()
3
023 123
21
3
2
v
WWW
P
v
W
PP
γ
γ
∗∗
−
−
=
−
=, (4.3)
and in general,
()
() ()
=
−
=kk
kv
k
W
P
vkkW
W
PP
γγ
1
!!
!
00 . (4)
Since the state probabilities summed from 0 to W must equal 1, we require:
()
1
1
0
0∑==
W
kk
v
k
W
P
γ
. (5.1)
Exploiting the binomial expansion of (1+x)W (Beyer, 1981, p. 64), P0 simplifies to:
()
1
1
0
0
1
1
1
−
−
=
+=
=∑
W
W
kkv
v
k
W
P
γ
γ
(5.2)
For a nominal workforce size W, the average number of employees available for duty is:
∑
=
=W
k
k
kPW
0
. (6)
In Table 1, we illustrate the effects of turnover at different average turnover rates T and average
vacancy durations v. For example, with a nominal workforce W = 10, the chance of being fully
staffed (i.e., P(k=10)) is 85 percent when annual turnover is 20% and positions remain vacant for
an average of 30 days. The likelihood of at least one vacant position increases with turnover and
the average duration of a vacancy. For T = 60% and v = 60 days, for example, the model
predicts the organization will operate short-handed more than 60 percent of the time.
(please insert Table 1 about here)
8
2.2 Turnover and Aggregate Workforce Experience Levels
To maintain a stable workforce, new employees must be recruited to replace those who
voluntarily leave the organization. Until they gain experience, those recruits may not be as
productive as veteran workers. To illustrate, suppose newly-hired employees achieve full
productivity after τ days of experience, and that during training, they process transactions at
average rate µ1. After τ periods of experience, their average service rate increases to µ2. Since
average employee tenure is 1/ T , and employees randomly and independently decide when they
will resign, the probability that an employee will remain with the organization for at least τ days
may follow the exponential distribution. Under this distribution, the likelihood of remaining
with the firm for at least τ varies inversely with bothT and τ.
While Bartholomew et al (1991) suggest that each employee cohort may have its own
characteristic turnover rate, we shall assume for now this rate is identical for all employees. Let
(1-φ) be the fraction of employees who resign during the interval [0,τ]. Expressed as a survivor
function, the expected fraction of new hires who reach full productivity (or the fraction of
veteran employees likely to still be working τ periods from now), is:
τ
τφ
T
exp −
=>= )( (7)
For example, with T = 0.6/yr and τ = 2 months, φ is about 90 percent. During any one training
cycle, we should expect (1-φ)W resignations. If the organization plans to maintain a workforce
of size W, on average (1-φ)W new employees will always be in training. The average processing
time per transaction µ-1 depends on the expected mix of experienced and inexperienced
employees in the workforce, so:
21
)1(
1
µφµφ
µ
+−= . (8)
9
2.3 Absenteeism
Even those employees who fully intend to continue their relationship with the organization
occasionally get sick or must attend to personal business during the times they are scheduled to
work. In 2000, U.S. industry lost 2.0% of all scheduled work hours due to unscheduled absences
(Bureau of Labor Statistics, 2001), and 2.5% in the service sector. For industries like call
centers, however, anecdotal reports of 12 - 16 percent absenteeism are not unusual (Call Center
Ops, 2001). CCH (2000) reports that 41 percent of the HR professionals responding to their
annual survey now view unscheduled absenteeism as a serious problem in their organizations.
Furthermore, few of the managers surveyed expect absenteeism to improve in the near future.
Mitchell (2001) observed that the direct costs of absenteeism are a significant component of
total payroll expense. For example, employee benefit plans often indemnify workers from
incidental absence and short-term illness. However, unscheduled absenteeism also adversely
affects productivity, especially when compounded by employee turnover. Let A be the ratio of
scheduled labor hours lost to unplanned absenteeism divided by total scheduled hours. Although
H workers may have been originally scheduled for duty at a particular time, turnover and
absenteeism will reduce the expected number of workers who actually report for duty to
WWAH )1( −.
3. Expected loss due to reneging in M/M/S
Most workforce management systems use M/M/S queueing models (Erlang C) to estimate
customer delays for different arrival rate/staffing level combinations (Durr, 1994; Cleveland,
1999). M/M/S assumes queued customers patiently advance in line once every 1/sµ time units,
on average. In practice, their patience often decays exponentially with average waiting time
10
(Sasser et al, 1991; Whitt, 1999b) and some customers may renege (abandon the queue) before
service begins.
While reneging customers speed up the line for others still in the queue, their behavior
reduces the effective arrival rate (λeff )and sales/period. We can estimate λeff and lost revenue
from the definition for server utilization. Let Fk = probability that the M/M/S: reneging system is
occupied by k customers. With s servers each operating at mean service rate µ, server utilization
Ueff is:
−+
=+
== ∑∑∑∑ ==
∞
+==
s
j
j
s
i
i
sj
j
s
i
i
eff FF
s
i
F
s
s
F
s
i
s0010
eff 1U
µ
λ
, (9)
so the effective arrival rate λeff must equal Ueffsµ.
Reneging behavior impacts total revenue, and is therefore an important consideration for
staffing and scheduling decisions. Unfortunately, the recent literature contains surprisingly few
references on this topic. Boots and Tijms (1999) estimate loss probabilities for systems where
customers renege after waiting a fixed time period. For systems that serve a population of
humans, however, it is more likely that patience varies from customer to customer. Gross and
Harris (1998, p. 94) outlined a single-server model where the propensity to renege increases
exponentially with queue time. Below, we extend that notion to the multi-server case.
If customers are willing to wait an average of α-1 before reneging (exponentially distributed),
the probability that queue time exceeds T is P(q>T) ≈ e(-αT). From the relationship between the
Poisson and the exponential, αTe-(αT) is the probability that exactly one customer reneges during
interval T. As T approaches zero, this probability approaches αh. The birth-death diagram in
Figure 2 illustrates the dynamics of an M/M/S queue with reneging. Each circle represents a
possible system state, corresponding to the number of customers in the system. The arcs
11
pointing right indicate the arrival of a new customer, with the probability of exactly one Poisson
arrival during a brief instant of time shown above the arc. The arcs pointing left represent the
flow of customers departing a given state, which could be due to a Poisson service completion
(at rates up to sµ) or a reneging customer. Departures from state k>s occur if any of the
customers at queue positions 1 to k-s renege during interval h, so the departure probabilities for
state k must include the chance of a single service completion plus the summed reneging
probabilities from any queue position up to and including position k-s.
(please insert Figure 2 about here)
The birth-death diagram forms a tree, so the system is reversible. If the system is stationary,
conservation of flow requires:
State Flow out = Flow in (10)
1 F1µ = F0λ
2 F22µ = F1λ
3 F33µ = F2λ
… …
s Fssµ = Fs-1λ
s+1 Fs+1(sµ+α) = Fsλ
s+2 Fs+2(sµ+2α) = Fs+1λ
... ...
The birth-death diagram for the system has two distinct sections: the portion from state 0 to state
s, where the probabilities of a single service completion steadily increase with occupancy; and
the portion from state s to ∞, where all servers are busy and departures may be due to either a
service completion or a reneging customer. For a stationary system, the probability of occupying
state k in an M/M/s system with reneging is F(k), where:
12
∞≤<
+
=
−+
≤<
=
=
∏
−
=
−
−
ksfor
!)(
sk0for ,
!
1
01
01
sk
i
s
k
k
k
iss
F
sks
F
k
k
F
k
F
F
αµ
λ
µ
λ
αµ
λ
µ
λ
µ
λ
(11)
The state probabilities must sum to 1.0. Thus:
0.1
!!
011
0
00 1
=
+
+
=+= ∑∑∏∑∑∑ =
∞
+=
−
=
∞
==
∞
+=
s
isj
sj
i
si
k
s
isj
jik issi
FFFF
αµ
λ
µ
λ
µ
λ
(12)
and therefore:
1
011
0!!
−
=
∞
+=
−
=
+
+
=∑∑∏
s
isj
sj
k
si
kssi
F
αµ
λ
µ
λ
µ
λ
(13)
The infinite sum in (13) converges to zero rapidly. While a closed form solution for (13)
may exist, we find that than an efficient approximation technique such as Newton's method and
the relationship in (12) can be used to estimate F0 with adequate precision.
4. Employee Staffing and Scheduling Decisions
Many service organizations with turnover and absenteeism problems face consumer demands
that vary from hour-to-hour and day-to-day. Their production capacity is difficult to store, and
they risk lost sales when their services cannot be produced upon demand. Furthermore, their
service capacity decisions, in the form of employee work schedules, are usually made long
before that demand is realized.
Employee scheduling problems have been modeled as deterministic generalized set covering
problems (Bailey, 1985; Dantzig, 1954) and deterministic goal programs (Andrews & Parsons,
13
1989; Baker, 1976; Keith, 1979; Mabert & Watts, 1982). In both models, target staffing
requirements for each period are exogenous parameters. In their labor scheduling models for
stochastic demand (SLS/D), Thompson (1995b) and Easton & Rossin (1996) integrated labor
requirements planning with staffing and scheduling decisions, explicitly accounting for the
stochastic nature of service demand.
All of these models assume deterministic employee availability. Since turnover and
absenteeism reduce the likelihood that the system will be fully staffed, their solutions may
overstate labor expenses. Because some employees scheduled for duty at a particular time may
have resigned or be absent, customer queues at those times may be longer, and more customers
may renege, than these models anticipate. Thus, existing labor staffing and scheduling models
may also overestimate revenues.
These limitations motivate our proposed model for labor staffing and scheduling decisions
under stochastic demand and labor supply (SLS/DL). The model assumes that each completed
transaction earns an incremental contribution (in dollars) before labor expense, and that customer
arrivals/hour and hourly service rates per employee are random variables. In addition, the model
assumes customers have limited patience and will abandon the queue after waiting an
exponentially distributed amount of time. What makes this model distinctive, however, is that it
adjusts the workforce size and the number of workers scheduled for duty at a particular time to
account for turnover, the mix of trainees and veteran employees, and absenteeism. To fully
describe our labor staffing and scheduling model for variable demand and turnover, we rely on
the following notation:
14
___________________________________________________________________________
Model Parameters
Workforce Characteristics
T = average annual turnover rate (number of voluntary resignations/yr
÷average workforce size), Poisson-distributed
v = average vacancy duration (days), distributed exponential;
A = average absenteeism rate; total hours lost due to unexpected absences
÷total scheduled and paid hours;
τ = length of the training period for new employees;
µ1 = mean service rate for employees in training (Poisson-distributed);
µ2 = mean service rate for experienced employees (Poisson-distributed);
µ = aggregate service rate (see equations (7) and (8)).
Customer and Market Characteristics
λi = average customer arrival rate forecasted for period i (Poisson-distributed);
α-1 = average customer patience, exponentially distributed.
Ri = average contribution per completed customer transaction, before labor
costs, for period i.
Schedule Characteristics
I = set of time periods in planning horizon, indexed i=1,…,I;
K = set of allowable work schedules for employees, indexed k=1,…,K;
aik = 1 if i is a working period in schedule k, 0 otherwise, for i=1,…,I;
k=1,…,K;
Ck = Wage cost to assign one employee to schedule k.
Decision Variables and Consequence Variables
Xk = number of employees assigned to schedule k, ∀ k ∈ K;
Pk = probability of exactly k incumbent employees (equations (4.W) and (5.6));
W = nominal workforce size
i
w = expected number of employees who actually report for duty in period i;
Fk = probability of exactly k customers in M/M/S: reneging system (equations
(11) and (13))
λeffi = effective customer arrival rate for period i, adjusted for impatient
customers who abandon the queue before initiating service (equation (9) ).
_____________________________________________________________________
15
The object of our stochastic labor scheduling model with labor turnover and absenteeism
(SLS/DL) is to maximize expected profits. Using the above notation, we determine the number
of employees who should be assigned to each feasible work schedule that will:
SLS/DL: Maximize ∑∑ ∈∈
−=
Kk
kk
Ii
ii XC
W
W
ReffZ
λ
(13)
subject to:
∑
∈
=
Kk
k
XW (14)
∑
=
=W
j
j
jPW
1
(15)
∑
∈
=
Kk
kiki xaw , ∀ i ∈ I ; (16)
()
−= W
W
Aww i
i1, ∀ i ∈ I; (17)
;,1eff 0
0
iIiFwFk i
iw
kk
i
k
w
k
∈∀
−+= ∑∑ =
=
µλ
(18)
0xk≥, and integer, k∈K. (19)
The objective function (13) computes the expected contribution for the solution by
determining expected revenues and expenses. The first term calculates the product of the
effective number of customer arrivals and the marginal contribution per arrival. The second term
computes the expected labor expenses for the solution, after adjusting for turnover (the
ratio /WW is the fraction of occupied positions). Note that labor expense includes the expected
wages for employees with unplanned absences.
Equation (14) is a consequence variable that determines the nominal workforce size.
Equation (15) adjusts this total for expected turnover. Equation (16) calculates the nominal
16
number of employees scheduled for duty during period i. Lacking more complete information
about individual workers, we assume that employee resignations and absences are dispersed
uniformly throughout the workforce. Equation (17) adjusts the number of employees scheduled
for duty during each period to reflect uniform turnover and absenteeism.
Equation (18) determines the effective arrival rate, adjusting the nominal arrival rate for
customers who abandon the queue. This relationship is adapted from equation (9). The state
probabilities Fk used in equation (18) are based on those described in equations (10) and (13).
SLS/DL is similar to SLS/D when p(W) = 1.0 and A = 0.0 (i.e., when turnover and
absenteeism are zero). Both models include non-linear and discontinuous terms, and it is often
difficult to obtain provably optimal solutions. Fortunately, meta-heuristics like simulated
annealing (Brusco & Jacobs, 1998) and hybrid genetic algorithms (Easton & Mansour, 1999)
work well with this class of problems. However, unless labor scheduling decisions that account
for employee turnover and absenteeism provide significant economic benefits for decision
makers, there is little reason to deploy such models. In the next section, we hypothesize the
types of operating environments where the advantages of SLS/DL are likely to be noteworthy.
We then describe our experiments to ascertain the validity of those hypotheses.
5. SLS/DL Model Assessment
Turnover and absenteeism reduce effective service capacity. As the intensities of these
factors increase, labor staffing and scheduling models that anticipate their impact should
outperform approaches that ignore them. However, other operating factors may either heighten
or lessen their economic impact. For example, if scheduling policies and consumer demand
patterns interact to provide significant surplus labor, employee turnover or absenteeism may
actually reduce operating expenses without adversely affecting revenues.
17
To help identify the market and service delivery system characteristics where SLS/DL is
likely to provide a significant performance advantage over SLS/D, we simulated 1024 different
operating environments. The characteristics of these labor staffing and scheduling environments
are summarized in Table 2. For each environment, we modeled the labor scheduling problem as
both an SLS/D problem and as an SLS/DL problem, then obtained near-optimal solutions for
each. Finally, we computed the performance improvement attributable to the SLS/DL model.
(please insert Table 2 about here)
SLS/DL addresses four aspects of labor supply uncertainty that have been ignored in earlier
labor scheduling models: turnover rate, average vacancy time, the time for new employees to
achieve proficiency, and absenteeism. To assess the effectiveness of SLS/DL, we simulated
labor environments with turnover rates T = 0.0, 0.2, and 0.6; job vacancy times of v = 30 and v
= 60 days; and absenteeism rates of A = 0.00 and 0.05/scheduled hour. The turnover rates are
similar to call center averages reported by Karr (1999). The average job vacancy times are
similar to those reported by Fitz-enz (1997). The simulated absenteeism rates straddle the US
service sector average (BLS 2001), but are probably much lower that those in some industries.
Task complexity affects the level of training required for new hires, and in high-turnover
labor environments may have a significant impact on productivity. Therefore, our simulations
included two levels for task complexity and new-hire training: 0 days with initial productivity of
100% of veteran employees, and 20 days, with initial productivity averaging 60%.
Systems with a small number of high-speed servers tend to have shorter mean queue times
than systems with identical capacity that use more, but slower, servers (Hillier & Lieberman,
1995). With more workers subject to resignation and absenteeism, systems with slower servers
may lose more revenue than fast-server systems due to increases in average customer queue
18
times and reneging rates. Thus these slower-server systems are more likely to benefit from the
SLS/DL model than systems with high service rates. Guided by earlier studies (Andrews &
Parsons, (1993) and Thompson (1997)), we simulated systems with average service rates of µ =
4/hr and 8/hr per employee.
Potential gains attainable with SLS/DL may be influenced by customer arrival
characteristics. Like Thompson (1995a, 1997), we simulated daily demand patterns with
different numbers of demand peaks per day (unimodal and trimodal), different amplitudes (COV
= 0.25 and 0.50), and arrival rates (400/hr and 800/hr). Customer patience may also influence
the improvements attainable with the SLS/DL model. Less patient customers are more apt to
renege when queue times increase due to turnover and absenteeism. Therefore, we simulated
systems with average customer patience of 120 seconds and 240 seconds. Finally, as profit
margins narrow, workforce utilization levels are often driven higher. Systems with high worker
utilization may be more vulnerable to loss from absenteeism and turnover, because they have
smaller capacity buffers. In our study, we simulated average revenue/transaction values of $7.00
and $14.15 ( adapted from Andrew and Parsons, 1993).
Other parameters:
Full- and part-time employees. We allowed solutions with a mix of full- and part-time
employees, provided at least 50% of all employees were full-timers (Easton & Rossin,
1991)
Hourly wage rate. We assumed hourly wage rates for full- and part-time employees were
$15.00 and $12.00, respectively.
Expected orders/arrival: We assumed 80% of all transactions produced income. The
balance were transactions such as status inquiries or information requests.
19
We conducted a full-factorial experiment by first solving SLS/D problems for the 128
different combinations of service delivery features and market characteristics described in Table
2. These solutions, obtained with Easton & Mansour's (1999) distributed genetic algorithm,
provided benchmarks for assessing the improvements attainable with SLS/DL. The solution
procedure converged to answers that were on average within 0.08 percent of a computed bound.
SLS/D ignores turnover and absenteeism, so a solution for a problem with 60 percent annual
turnover is identical to one with zero turnover. To complete our benchmarks, we "priced" each
SLS/D solution with the SLS/DL objective function assuming the different turnover rates, job
vacancy times, absenteeism levels, and training times described in Table 2. Finally, we solved
each problem as an SLS/DL problem and compared its expected profit with the profit for the
corresponding benchmark SLS/D solution.
We then regressed the experimental factors against SLS/DL performance to identify the
environmental characteristics where SLS/DL could be expected to provide significant benefit
over existing labor scheduling methods. The results, reported in Table 3, indicate that all 10
attributes play a significant role in explaining the improvements obtained with SLS/DL. As
expected, the regression coefficients confirm that increasing absenteeism and employee turnover
seriously degrade SLS/D schedule performance, creating opportunities for SLS/DL to find
alternative solutions with significantly greater profits.
(Please insert Table 3 about here)
Under high turnover and absenteeism, SLS/DL returned an average of 4.9% more profit than
the SLS/D solutions (see Table 4). However, SLS/DL achieved better than average results in
operating environments with comparatively high customer arrival rates, low profit margins, and
more patient customers. We also found larger than average profit increases when new workers
20
require more initial training and are less productive that veteran employees. In Table 4, we "drill
down" through the various market attributes, demand attributes, and service delivery attributes to
zero in on the operating characteristics where SLS/DL provided the greatest benefit.
(please insert Table 4 about here)
In Table 4.A, we isolate the market attributes where SLS/DL produces above-average
improvements. As the regression coefficients in Table 3 indicate, the SLS/DL's performance
improves with customer arrival rates, decreases with revenue per order, and improves with
customer patience. With standard labor scheduling models, worker utilization tends to increase
with transaction volume. The loss of a busy worker to absenteeism or turnover has a greater
impact on revenue than the loss of a worker with lower utilization. A similar effect occurs with
tighter profit margins (or less revenue/order). To maximize profitability, SLS/D tries to squeeze
surplus capacity from the system. It is thus more vulnerable to revenue losses when absenteeism
and turnover increase queue times. With greater average customer patience, SLS/D allows
longer queues. When absenteeism and turnover deplete capacity, queued customers tend to wait
longer than expected and are more likely to renege, thereby reducing revenue and profits.
Table 4.B focuses on the pattern of service demand, decomposing the column of Table 4.A
with the greatest average improvement. With multiple daily demand peaks, work rules about
minimum shift length make it difficult to avoid significant overstaffing. These systems thus
tend to be less vulnerable to absenteeism and turnover, leaving few opportunities for SLS/DL to
improve on the SLS/D solutions. As the cell averages in Table 4.B and the regression
coefficients of Table 3 show, SLS/DL's advantages decrease with the number of daily demand
peaks. On the other hand, SLS/DL apparent performance advantages increase slightly with
21
demand "peakiness." Doubling the amplitude of demand leads to small, but significant
performance improvements.
In Table 4.C we decompose the column from Table 4.B with the greatest gains to examine
how service delivery system characteristics influence SLS/DL's performance. Generally, we find
SLS/DL's advantages are more pronounced in slower systems, which tend to have more
employees subject to turnover and absenteeism. Thus SLS/D solutions for slower systems tend
to lose a larger number of servers than fast systems with comparable capacity, leading to a
greater percentage increase in average queue times (Hillier & Lieberman, 1995). Increased task
complexity (training time), coupled with higher absenteeism and turnover, reduces the effective
service rate. The SLS/D solutions tend to have longer than expected queues when absenteeism
and turnover strike, reducing revenues as customers become impatient and renege.
6. Discussion
This paper characterized the impact of employee turnover and absenteeism on workforce
availability and integrated those results into a stochastic labor staffing and scheduling model we
labeled SLS/DL. We completed a series of experiments to help isolate the operating
environments most likely to benefit from this approach. Our preliminary tests indicate that firms
with low turnover, low absenteeism, short average vacancy times, comparatively high unit
revenues, and multiple daily demand peaks benefit least from the SLS/DL model. In contrast,
firms with comparatively long transaction times, high customer arrival rates, relatively tight
profit margins, more patient customers, and greater task complexity (training time) should
experience significant benefits from this approach.
These results suggest a number of directions for future research. For example, we assumed
interchangeable employees with common skill sets, scheduling preferences, availability and
22
reliability. In future research, we hope to apply this methodology to systems with extant,
heterogeneous employees. We also assumed single stage processes in fairly large-scale systems.
For smaller, lower volume operations or systems with a sequence of operations, it may be more
difficult to realize performance advantages of the magnitude reported here. Finally, SLS/DL
considers only one strategy for coping with turnover -- overstaffing. Using other coping
strategies, such as temporary workers, overtime, reassigning cross-trained employees, etc., may
provide additional benefits.
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Figure 1: Birth-death diagram for worker flux under state-dependent resignations and
recruitment
P0 P1 P2 PW-1 PW
. . .
hW/ vh
(W
-1
)/
v h
(W
-2
)/
vh2
/
vh1
/
v
hγ h2γ h3γ h (W-1)
γhWγ
______________________________________________________________________________
Figure 2.
Birth-death diagram for M/M/s with reneging
S S+1 S+2S-1 …
0 …
state
λh λh
λ
h
λ
h λh
µh sµh s
µ
h+
α
h s
µ
h+2
α
h sµh+3αh
Table 1: State probabilities for number of employees available for duty when nominal workforce size W = 10
Average vacancy v= 30 days Average vacancy v = 60 days
Number of
Available
Employees T = 0.2/yr T = 0.4/yr T=0.6/yr T = 0.2/yr T = 0.4/yr T=0.6/yr
k Pk Pk P
k P
k P
k P
k
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000001
2 0.000000 0.000000 0.000000 0.000000 0.000000 0.000008
3 0.000000 0.000000 0.000001 0.000000 0.000005 0.000067
4 0.000000 0.000001 0.000012 0.000001 0.000058 0.000477
5 0.000001 0.000023 0.000148 0.000023 0.000531 0.002901
6 0.000026 0.000350 0.001505 0.000350 0.004036 0.014706
7 0.000628 0.004263 0.012207 0.004263 0.024549 0.059639
8 0.011463 0.038901 0.074260 0.038901 0.112007 0.181403
9 0.139466 0.236650 0.301167 0.236650 0.340686 0.367845
10 0.848417 0.719811 0.610699 0.719811 0.518127 0.372954
w= Avg.
Workforce
Size =
9.8356 9.6712 9.50685 9.6712 9.3425 9.0137
Table 2. Experimental factors and factor levels
Experimental Factor
Number
of Levels
Factor Levels
Employee Characteristics
Turnover rate
3 0, 20%, and 60% per year
Average job vacancy time
2 30 and 60 days
Absenteeism rate
2 0 and 5.0%
Service Delivery Features
Average service rate for
fully-trained employees
2 µ2=8 and 4 transactions per hour
Initial training
period/trainee productivity
2 0 days @ 100% µ2 and 20 days @ 60% µ2
Market Characteristics
Average arrival rate
2 λ=400 and 800 arrivals/hour,
Daily demand pattern
2 Unimodal and trimodal sinusoidal patterns
Amplitude of demand
2 Coefficient of variation (COV) of 0.25 and 0.50; sine
function amplitudes of 0.353 and 0.706
Unit contribution before
labor
2 $7.00 and $14.15 per order (80% of all arrivals are
orders, 20% are non-revenue producing)
Average customer
patience
2 120 seconds and 240 seconds, exponentially
distributed
Table 3: Regression Coefficients for Environmental Factors Influencing SLS/DL Performance
Attribute Levels Coefficients t Statistic P-value
Average arrival
rate/hr
400/hr 800/hr 1.215E-05 9.0082 1.02E-18
Demand Amplitude
0.353 0.706 -0.0070 -4.6058 4.63E-06
Number of Demand
Peaks/day
1 3 -0.0019 -15.2140 3.12E-47
Average Revenue
$7.00/order $14.15/order -0.0006 -8.5501 4.48E-17
Average Patience
120 sec 240 sec 1.192E-05 2.6524 0.008117
Average Service Rate
4/hr per
employee
8/hr per
employee
-0.0021 -15.6720 9.95E-50
Mean Training
Time/Proficiency
0 days @
100%
20 days @
60%
-0.0121 -8.9388 1.83E-18
Average Turnover
Rate
0.2/year 0.6/year 0.0482 35.7120 1.9E-181
Average Vacancy
Time
30 days 60 days 0.0004 22.4406 7.42E-91
Mean Absenteeism
0%
scheduled
hrs/day
5%
scheduled
hrs/day
0.2987 27.6843 4.5E-126
Table 4: Drill-down Relationships: Environmental Factors and SLS/DL Performance
A. Market Attributes
Arrival rate 400/hr (80% orders) 800/hr (80% orders)
Revenue/Order $7 $14.15 $7 $14.15
Avg. Patience 120s 240s 120s 240s 120s 240s 120s 240s
Tbar Vbar Absent SLS/DL Profit Increase (%)
0% 0.19 0.17 0.08 0.09 0.19 0.28 0.12 0.12
30
days 5% 0.97 0.98 0.78 0.82 1.27 1.57 1.06 1.09
0% 0.43 0.41 0.25 0.28 0.51 0.62 0.33 0.34
0.2/yr
60
days 5% 1.47 1.53 1.19 1.23 1.92 2.28 1.58 1.64
0% 0.95 0.96 0.63 0.67 1.14 1.39 0.83 0.84 30
days 5% 2.29 2.42 1.89 1.96 2.94 3.47 2.49 2.59
0% 2.59 2.72 1.88 1.96 3.14 3.65 2.42 2.51
0.6/yr
60
days 5% 4.53 4.82 3.78 3.91 5.68 6.52 4.81 5.05
Arrivals=800/hr, Revenue=$7/order, Patience=240s.
Demand Peaks/day 1 3
Demand Amplitude low high low high
Tbar Vbar Absent SLS/DL Profit Increase (%)
0.0% 0.41 0.31 0.24 0.17 30
days 5.0% 1.88
2.10 1.39 0.91
0.0% 0.71 0.77 0.57 0.43
0.2/yr
60
days 5.0% 2.67
3.03 2.05 1.36
0.0% 1.65 1.73 1.26 0.94 30
days 5.0% 4.04
4.51 3.18 2.14
0.0% 4.20 4.53 3.38 2.51
0.6/yr
60
da
y
s
5.0%
7.48
8.21
6.09
4.28
Arrivals=800/hr, Revenue=$7/order, Patience=240s
Peaks/day =1, Amplitude = high
Service Rate 4/hr 8/hr
Training time/Initial
Proficiency
0 day/
100%
20 day/
60%
0 day/
100%
20 day
/60%
Tbar Vbar Absent SLS/DL Profit Increase (%)
0.0% 0.32 0.56 0.11 0.24 30 days
5.0% 2.42 2.76 1.45 1.78
0.0% 0.84 1.19 0.44 0.62
0.2/yr
60 days
5.0% 3.46 3.88 2.20 2.58
0.0% 1.68 2.75 0.86 1.62 30 days
5.0% 4.64 6.09 3.04 4.28
0.0% 4.72 6.26 2.95 4.17
0.6/yr
60 days
5.0% 8.51 10.40 6.15 7.79
B.
Demand
Attributes
C. Service
Delivery
Attributes
