Post on 18-Oct-2020
Optimization Online
Chemotherapy appointment scheduling under uncertaintyusing mean-risk stochastic integer programming
Michelle M Alvarado · Lewis Ntaimo
Submitted: August 24, 2016
Abstract Oncology clinics are often burdened with
scheduling large volumes of cancer patients for chemother-
apy treatments under limited resources such as the num-
ber of nurses and chairs. These cancer patients require a
series of appointments over several weeks or months and
the timing of these appointments is critical to the treat-
ment’s effectiveness. Additionally, the appointment du-
ration, the acuity levels of each appointment, and the
availability of clinic nurses are uncertain. The timing
constraints, stochastic parameters, rising treatment costs,
and increased demand of outpatient oncology clinic ser-
vices motivate the need for efficient appointment sched-
ules and clinic operations. In this paper, we develop
three mean-risk stochastic integer programming (SIP)
models, referred to as SIP-CHEMO, for the problem
of scheduling individual chemotherapy patient appoint-
ments and resources. These mean-risk models are pre-
sented and an algorithm is devised to improve compu-
tational speed. Computational results were conducted
using a simulation model and results indicate that the
risk-averse SIP-CHEMO model with the expected ex-
cess mean-risk measure can decrease patient waiting
times and nurse overtime when compared to determinis-
tic scheduling algorithms by 42% and 27% respectively.
Keywords Health care · oncology clinics · patient
service · chemotherapy scheduling · mean-risk
stochastic programming
michelle.alvarado@tamu.edu · ntaimo@tamu.edu
Industrial and Systems EngineeringTexas A&M University3131 TAMUCollege Station, TX USA 77840-3131
1 Introduction
Reports have shown that cancer costs in the U.S. ex-
ceeded $124 billion in 2010 and are expected to increase
27% by 2020 [13]. The demand for oncology services is
projected to increase by 48% between 2005 and 2020
[25]. Chemotherapy is a common treatment method
for cancer patients. Chemotherapy treatments are often
administered orally or intravenously at outpatient on-
cology clinics. Cancer patients receiving chemotherapy
treatment require a series of appointments over several
weeks or months and the timing of these appointments
is critical to the treatment’s effectiveness. When the ap-
pointment is not scheduled on the dates recommended
by the physician, the treatment’s dose intensity can
be diminished [12] and the cancer patient’s mortality
risk increases [3]. The timing constraints along with
rising costs and demand motivate the need for efficient
chemotherapy appointment schedules so that patients
may receive treatment when needed at a fair price.
Oncology clinics have the challenging problem of
scheduling large volumes of patient appointments us-
ing limited clinic resources. Resources such as chairs
and nurses are needed to effectively manage patients.
Chemotherapy nurses have the flexibility of managing
multiple patients, but these assignments are restricted
by acuity levels and new patient starts. The appoint-
ment scheduling problem for oncology clinics is stochas-
tic in nature and deterministic models do not suffi-
ciently capture the scheduling process. Patient requests
for appointments, treatment duration, and resource avail-
ability are all examples of stochastic parameters in on-
cology clinic scheduling.
In the last decade research has developed in the area
of scheduling chemotherapy appointments. Initial ap-
proaches implemented various classification approaches,
2 Michelle M Alvarado, Lewis Ntaimo
and more recently, researchers have started using opti-
mization approaches. However, none of the approaches
have considered uncertainty in the problem’s param-
eters such as appointment duration. SIP is a proven
optimization method for modeling decision problems
involving uncertainty. In this paper, three SIP mod-
els, termed SIP-CHEMO, are developed to address the
complexities of the chemotherapy scheduling problem.
Some of the SIP-CHEMO models also include mean-
risk measures in order to better reflect the inherent
“risk” in the decision problem. This research develops
the first optimization model for scheduling chemother-
apy appointments that incorporates uncertainty in prob-
lem parameters and considers risk.
The decision model and solution approach for on-
cology clinic scheduling presented in this paper makes
several contributions for management science. Specifi-
cally, the model: 1) specifies patient appointment sched-
ules; 2) specifies clinic resource schedules; 3) consid-
ers the inherit uncertainty in appointment duration,
acuity levels, and nurse availability; 4) models risk to
the patient’s health status due to deviations from the
physicians recommended start dates; 5) models risk of
having a scheduling conflict (e.g. overtime or overlap-
ping appointments) due to uncertainty; 6) is adapt-
able to the management’s level of risk for each patient.
Numerical results based on data from a real oncology
clinic show that using mean-risk SIP models to schedule
chemotherapy appointments can generate more efficient
schedules that benefit patients by reducing waiting time
and nurse overtime by while increasing patient through-
put.
The rest of this paper is organized as follows: Section
2 provides a review of recent literature on chemother-
apy scheduling. Section 3 describes the chemotherapy
scheduling problem and Section 4 gives an overview of
mean-risk SIP problems. The mean-risk SIP model for-
mulations are presented in Section 5 along with solution
approaches. A real application setting is described in
Section 6 along with computational experiments. Sec-
tion 7 contains a summary and concluding remarks.
2 Literature review
In the last decade research has developed in the area
of scheduling of chemotherapy appointments, some of
which were classification approaches and others were
optimization approaches. Classification approaches clas-
sify the patients, medications, or resources to develop
scheduling rules, templates, or algorithms. Optimiza-
tion approaches model the problem with an objective
function and constraints. A review of the literature for
both of these approaches is presented in this section,
along with a review of mean-risk SIP.
2.1 Classification chemotherapy scheduling methods
A number of oncology clinics have worked to improve
the scheduling of chemotherapy appointments using var-
ious classification approaches. Some clinics created sched-
ules by classifying nurse tasks [11] or acuity levels [9]
while others have used drug [6] or patient types [4].
Although none of these classification techniques used
optimization or uncertainty, they were simple methods
that were successfully implemented in practice. These
works provide guidance on the key aspects of the deci-
sion problem (acuity levels, resource availability, treat-
ment duration, etc.). In addition, many clinics have
noted considerable success using next-day scheduling
[6,11,17]. Next-day, or split-scheduling, method implies
that patient arrives one day for blood work and returns
the next day to receive their chemotherapy treatment.
Repeated success of next-day scheduling systems [6]
motivated the decision to limit the scope of our problem
to only the drug infusion appointment.
One study found success through the implemen-
tation of a five-level acuity rating system to address
scheduling problems [9]. After treatment lengths were
also incorporated in the scheduling template, the new
system resulted in improved patient satisfaction scores.
Another study by Ahmed, Elmekkawy, and Bates [2]
developed several scenarios to match resource schedules
with the clinics arrival pattern of patients. The scenario
with the best simulation performance was used to create
a scheduling template that increased throughput and
increased resource utilization without requiring more
resources.
2.2 Optimization chemotherapy scheduling methods
In the past few years, researchers started using opti-
mization models to address the chemotherapy schedul-
ing problem. A Chemo Smartbook scheduling system
was developed as an innovative software approach that
offered customized, flexible scheduling and considered
patient time preferences, appointments from different
departments, system capacity, nurse workload, and staff
schedules [20]. Later, an inverse optimization model was
developed to determine nurses’ preferences in order to
create better schedules in the Chemo Smartbook [5].
A multi-period time horizon approach to address
the problem of scheduling patients and resources for an
oncology outpatient clinic was developed by Turkcan,
Zeng, and Lawley [25]. The objectives were to minimize
Chemotherapy appointment scheduling under uncertainty 3
the treatment delay, patient waiting times, and staff
overtime while simultaneously maximizing the staff uti-
lization. The model by Turkcan, Zeng, and Lawley [25]
is most closely related to the one presented in this
paper, but it did not include mean-risk measures or
uncertain problem parameters.
Algorithms for scheduling chemotherapy regimens
were developed by Sevinc, Sanli, and Goker [23] with
the goal of maintaining the treatment regimen spec-
ifications, minimizing patient waiting time, and opti-
mizing chair utilization. This was one of the few pa-
pers to consider lab appointments along with infusion
appointments. The main contribution of Sevinc, Sanli,
and Goker [23] was that this work addressed infusion
appointment cancellations and delays due to poor lab-
oratory test results. Another study also addressed the
appointment scheduling problem in an outpatient on-
cology clinic [19]. Their work is one of the few that con-
sider the oncologist consultation in the problem setting.
Recently, a dynamic optimization model was devel-
oped by Hahn-Goldberg et al. [8] to schedule chemother-
apy appointments. Their work considered uncertainty
through real-time requests for appointments and uncer-
tainty due to last-minute scheduling changes. This work
used a scheduling template and an online optimization
in a novel technique they refer to as dynamic template
scheduling. Gocgun and Puterman [7] used simulation
and Markov decision processes (MDP) to dynamically
schedule chemotherapy patient appointments. To the
best of our knowledge, this paper is the first to use
mean-risk SIP for chemotherapy appointment schedul-
ing.
2.3 Mean-risk SIP
For optimal decision-making under uncertainty, this pa-
per uses mean-risk SIP. Mean-risk stochastic program-
ming was first developed for financial risk analysis and
began with the axiomatic principles of stochastic domi-
nance, a form of stochastic ordering [18]. In a two-stage
mean-risk SIP, the first-stage decision variables repre-
sent the “here and now” decisions while the second-
stage decisions represent the “recourse” decisions made
after uncertainty is realized. Historically, SIP used the
expected value of the first-stage objective function, which
is appropriate for the risk-neutral case or when the
law of large number can be applied. But in certain
applications it may be more appropriate to explicitly
model risk within its objective. Mean-risk SIP mod-
els represent risk using both the expected value and
a mean-risk measure in the objective function to more
accurately reflect the inherent uncertainty in a problem.
This paper models the risk associated with the patient’s
health status due to delayed treatment and the risk
of having scheduling conflicts (e.g. overtime) due to
uncertainty.
Deviation measures such as expected excess (EE)
and absolute semideviation (ASD) measure deviation
from a target. For ASD, the target is the expected value.
Structural and algorithmic properties of two-stage stochas-
tic linear programs (SLP) with deviation measures are
derived in [10]. Similar results for excess probability
are obtained in [16]. Risk aversion for SLP is addressed
in [1] with a focus on convexity properties and sub-
gradient decomposition. Stochastic mixed-integer pro-
grams with risk functionals based on the semideviation
and value-at-risk (VaR) were studied by Markert and
Schultz [14] and in a thesis by Tiedemann [24]. Schultz
and Tiedemann studied SIP based on excess proba-
bilities [21] and conditional value-at-risk (CVaR) [22].
CVaR is computed as the conditional expectation of
losses that exceed the value-at-risk.
SIP has never been applied to decision-making in
oncology clinics. Perez et al. [15] is an example of SIP
applied to nuclear medicine department, but the com-
plexities and constraints for that problem setting are
quite different than those seen in oncology clinics. This
paper further extends upon this idea and is the first
to apply mean-risk SIP to chemotherapy appointment
scheduling.
3 Chemotherapy scheduling problem
description
This section describes the chemotherapy scheduling pro-
cess, uncertainty in the problem parameters, and rele-vant performance measures.
3.1 Scheduling Process
Information acquired through visits and communica-
tion with an outpatient oncology clinic provided valu-
able insight into the constraints and objectives of the
chemotherapy problem. Once a patient is diagnosed
with cancer, an oncologist prescribes a unique treatment
regimen, or series of chemotherapy appointments, to
each cancer patient based on the patient’s current state
of health. A treatment regimen describes the frequency
of appointments (days), the prescribed chemotherapy
drugs, the expected appointment duration, and the acu-
ity level for each appointment. An example treatment
regimen in Table 1 shows a patient with five appoint-
ments in the first week, then follow-up treatments on
days 8 and 15 during in a three week cycle. The drugs
the patient receives in each treatment may vary from
4 Michelle M Alvarado, Lewis Ntaimo
one appointment to another. The appointment duration
is the total time that the appointment is expected to
take from the time the patient arrives to the clinic until
the patient is discharged from the clinic. The acuity
level is a relative measure of the nurse’s attention re-
quired by a patient during an appointment. In addition,
the physician will also recommend a start date for the
treatment regimen, which specifies the day in which
the first appointment should begin. Treatment regimens
depend on the patient’s type of cancer, stage of the
cancer growth, and current health. Therefore treatment
regimens are unique to each individual patient.
Days Drugs Appt. AcuityDuration Levels
1 CISplatin, Etoposide, 8 hours 1Bleomycin
2-5 CISplatin, Etoposide 7 hours 26-78 Bleomycin 1 hour 3
9-1415 Bleomycin 1 hour 3
16-21
Table 1: Example chemotherapy treatment regimen.
The treatment regimen prescribed by the oncolo-
gist is sent to a scheduler to determine the appoint-
ment schedule and to allocate clinic resources for each
appointment in the treatment regimen. The scheduler
must immediately schedule all appointments in the treat-
ment regimen to guarantee the availability of the later
appointments. To maximize treatment effectiveness, theseappointments should be scheduled as close to the state
date recommended by the oncologist as possible. De-
lay from the recommended start date is referred to as
type I delay. The scheduler must make a chemother-
apy scheduling decision, which allocates a specific date,
time, and set of clinic resources (e.g., chair and nurse)
to each appointment in the patient’s treatment regi-
men. The chemotherapy scheduling decision problem
determines when to schedule all of the appointments in
the chemotherapy patient’s treatment regimen and to
determine which resources to allocate to the patient at
each appointment.
Chemotherapy chairs and nurses are both assigned
to a patient for the entire duration of their chemother-
apy treatment. It generally takes around 15 to 30 min-
utes to start the chemotherapy drug infusion for each
patient. This process is called a patient start. During a
patient start, the nurse is primarily dedicated to start-
ing the drug infusion of that patient. As a result, each
nurse is limited to one new start during each time slot.
Chemotherapy treatments are well-known for caus-
ing nausea and the cancer weakens the immune system,
both of which can severely deteriorate a patient’s state
of health. The side-effects can occur suddenly during
chemotherapy administration. Depending on the type
and intensity of the treatment, the assigned nurse must
pay close attention to the patient in order to moni-
tor the patient’s condition and reactions to these side-
effects. However, it is possible for each nurse to si-
multaneously monitor the chemotherapy treatments of
several patients at the same time. Yet, we still assume
only one of those patients can be in the patient start
process.
It is crucial that the nurses are not over-utilized
since they must be available to assist patients expe-
riencing adverse reactions to the chemotherapy drugs.
To account for this, the concept of acuity levels is used.
Acuity levels are assigned a value of say 1, 2, or 3, where
an acuity level of 3 (or the largest number used) rep-
resents the maximum attention required by the patient
from the nurse. Each nurse can monitor several patients
at once provided that the sum of the acuity levels for all
patients is less than or equal to a pre-determined max-
imum acuity level for that nurse. The pre-determined
maximum acuity level can be determined by the opinion
of management or the charge nurse.
Figure 1 provides an example of limitations associ-
ated with scheduling a patient appointment using acu-
ity levels and patient starts when scheduling a nurse.
This example assumes one nurse and 15 minute time
slots. Patient A begins treatment during time slot 1
and continues for 60 minutes (four time slots) with an
increased acuity level in the final two time slots. Patient
B begins treatment during time slot 2 and continues
for 45 minutes (three time slots) with a constant acuity
level. Therefore, the nurse has a patient start during
time slots 1 and 2. This single nurse could not have
started both patients in the same time slot. The acuity
levels of each patient are summed to compute the total
acuity. The nurse can handle multiple patients as long
as the total acuity does not exceed a pre-determined
maximum acuity level (e.g. 5).
Fig. 1 Acuity Level and Patient Start Example
Chemotherapy appointment scheduling under uncertainty 5
3.2 Problem Uncertainty
There are several stochastic parameters associated with
the chemotherapy scheduling problem. The side-effects
of chemotherapy drugs can influence both the treat-
ment duration and acuity level during an appointment.
If a patient is very sick, the patient may require more
attention from the nurse and in some cases, treatment
may be paused to allow the patient time to recover.
This translates to a higher acuity level and a longer
appointment duration. Additionally, some patients
take longer to begin treatment because of small veins for
the infusion needle or a clogged port-a-catheter, among
other things. Due to these variations, the acuity level
and appointment duration of an appointment are two
stochastic parameters.
The number of nurses on duty in a given day
is the third stochastic parameter. We consider nurse
availability stochastic because they are often the lim-
iting resource, as was the case in the oncology clinic
collaborating on this research. When a nurse is unex-
pectedly unavailable on a particular day (e.g., when
a nurse calls in sick to work), then an understaffed
clinic will have difficulty adjusting to the workload for
the day. Therefore, nurse availability is assumed to be
stochastic in the decision problem to account for the
possibility that a nurse may not be able to complete
their assigned responsibilities.
3.3 Performance Measures
The scheduling decision models were evaluated via a
simulation model from both the management’s perspec-
tive and the patient’s perspective. From the patient’s
perspective, the type I delay, type II delay, and system
time are measured. From the management’s perspec-
tive, the throughput, nurse overtime, nurse overtime+
were measured. See Table 2 for definitions.
4 Mean-risk SIP Notation
SIP was selected for the chemotherapy scheduling prob-
lem because of the uncertainty in the appointment du-
ration, acuity levels, and nurse availability. The mean-
risk SIP approach was chosen for this problem for two
reasons. First, risk in this problem is the probability
of a diminished health outcome due to treatment de-
lays. When appointments do not begin on the recom-
mended start date, then the treatments become less
effective and delays pose risk to the patient’s health
status. The models developed in this research are risk-
averse because scheduling decisions will be made on
Patient PerspectiveType I Delay Time (days) between the first
scheduled appointment start dateand the state date recommendedby the oncologist
Type II Delay Time (minutes) between the pa-tient arriving to waiting room andthe patient being called by thenurse to start the appointment
System Time Time (minutes) the patient is atthe oncology clinic from arrival todeparture
Management PerspectiveThroughput Number of appointments in the
oncology clinic each dayNurse Overtime Time (minutes) that the nurse
must work beyond normal clinicoperating hours
Nurse Overtime+ Nurse overtime (minutes) withzero (0) entries excluded
Table 2: Performance Measures
the assumption that there is a reluctance to take risks
with the patient’s health status. Second, mean-risk SIP
allows for the management to consider different mean-
risk measures (e.g. EE or ASD) and define different risk
levels for each patient. The risk levels for a patient is
best captured through the inclusion of a suitable weight
factor, λ, in the mean-risk SIP model.
In the proposed two-stage SIP formulation, the first-
stage scheduling decisions are made ‘here-and-now’ for
each patient before observing future uncertainty. The
second-stage decisions represent the “recourse” deci-
sions made after uncertainty is realized. Each stage has
its own objective function, which lends itself easily to
modeling multi-objective problems.
A mean-risk two-stage SIP [18] can be stated as
follows:
SIP: Min E[f(x, ω)] + λD[f(x, ω)],
s.t. Ax ≥ b
x ∈ Rn1 × Zn′1 ,
(1)
where x is the first-stage decision vector and f(x, ω)) =
c>x+Q(x, ω). The vector c ∈ Rn1 (where n1 = n1 +n′1)
is the first-stage cost vector, b ∈ Rm1 is the right-hand
side, A ∈ Rm1×n1 is the first-stage constraint matrix,
and ω is a multi-variate discrete random variable with
an outcome (scenario) ω ∈ Ω with probability of oc-
currence pω. The random variable ω is defined on the
probability space, (Ω,A,P) where Ω is the set of all
possible outcomes, A is the set of events, and P is the
probability measure. E : F → R denotes the expected
value, where F is the space of all real random cost
variables f : Ω → R satisfying E[|f(ω)|] <∞. Modeling
problems using only the expectation in the objective
6 Michelle M Alvarado, Lewis Ntaimo
makes the formulation risk-neutral. To introduce risk,
a risk measure D : F 7→ R is used resulting in the
so-called mean-risk stochastic program, where λ > 0
is a suitable weight factor that quantifies the trade
off between expected cost and risk. D measures the
dispersion (variability) of the random variable f(x, ω).
Common risk measures in the literature include CVaR,
EE, and ASD.
For any outcome (scenario) ω, the recourse function
Q(x, ω) is given by the following standard second-stage
subproblem:
Q(x, ω) =Min q(ω)>y
s.t. W (ω)y ≥ r(ω)− T (ω)x
y ∈ Rn2 × Zn′2 .
(2)
The vector q(ω) ∈ Rn2 (where n2 = n2 + n′2) is the
second-stage cost vector, W (ω) ∈ Rm2×n2 is the re-
course matrix, r(ω) ∈ Rm2 is the right-hand side, and
T (ω) ∈ Rm2×n1 is the technology matrix. A scenario
defines the realization of the stochastic problem data
q(ω), r(ω),W (ω), T (ω).CVaR is the most commonly used mean-risk mea-
sure and minimizes the expectation of the worst out-
comes; in this case, the scenarios with the largest devi-
ations from the recommended start date. Using CVaR
does not allow the oncology management to set a target
value for the scheduling decisions. Instead, this paper
develops mean-risk SIP models for EE and ASD where
the scheduling decisions minimize the expected value
of the excess above a target value, which can be in-
terpreted as the number of days the patient’s scheduledeviates from the recommended start date. Next, the
extension to the EE and ASD mean-risk SIP models
are defined.
Given a target η ∈ R and λ > 0, EE [14] is defined
as
φEEη (x) = E[maxf(x, ω)− η, 0].
EE is the expected value of the excess over a target
η ∈ R. Substituting D := φEEη in (1) results in SIP
with EE as follows:
Min x∈X E[f(x, ω)] + λφEEη (x). (3)
Using EE, the management can select a target for the
objective function. For example, in the formulation to
follow a target of 2 days implies that the first appoint-
ment should deviate no more than 2 days from the
recommended start date. Assuming a finite number of
scenarios ω ∈ Ω, each with probability of occurrence
p(ω), λ ≥ 0, and a target level η ∈ R, problem (3) is
equivalent to the following formulation [14]:
SIP-EE:
Min c>x+∑ω∈Ω
p(ω)q(ω)>y(ω) + λ∑ω∈Ω
p(ω)ν(ω)
(4)
s.t. T (ω)x+W (ω)y(ω) ≥ r(ω), ∀ω ∈ Ω− c>x− q(ω)>y(ω) + ν(ω) ≥ −η, ∀ω ∈ Ω
x ∈ X, y(ω) ∈ Zn2+ × Rn
′2
+ , ν(ω) ∈ R+,∀ω ∈ Ω.
The ASD model is obtained by replacing the target
value in EE with the expected (mean) value E[f(x, ω)]
and is given as φASD(x) = E[maxf(x, ω)−E[f(x, ω)], 0].ASD reflects the expected value of the excess over the
mean value. Setting D := φASD in (1), results in the
following SIP with semideviation:
Min x∈X E[f(x, ω)] + λφASD(x). (5)
Similarly to the EE problem, note that
φASD(x) ≡ E[maxf(x, ω),E[f(x, ω)]] − E[f(x, ω)],
give the deterministic equivalent program (DEP) for-
mulation for ASD. Using ASD, the management does
not need to select a target for the objective function.
Instead, the formulation to follow will minimize the
expected value of the excess above the mean deviation
from the recommended start date. Given λ ∈ [0, 1],
problem (5) is equivalent to the following formulation
[14] :
SIP-ASD:
Min (1− λ)c>x+ (1− λ)∑ω∈Ω
p(ω)q(ω)>y(ω) +
λ∑ω∈Ω
p(ω)ν(ω) (6)
s.t. T (ω)x+W (ω)y(ω) ≥ r(ω), ∀ω ∈ Ω− c>x− q(ω)>y(ω) + ν(ω) ≥ 0, ∀ω ∈ Ω
− c>x−∑ω∈Ω
p(ω)q(ω)>y(ω) + ν(ω) ≥ 0, ∀ω ∈ Ω
x ∈ X, y(ω) ∈ Zn2+ × Rn
′2
+ , ν(ω) ∈ R,∀ω ∈ Ω.
5 SIP-CHEMO Models
The notation for the chemotherapy scheduling problem
is defined in this section. The risk-neutral (RN) for-
mulation is modeled first because it is the simplest of
the SIP-CHEMO models. Then extensions to both the
EE and ASD formulations are made. Collectively, these
models are referred to as the SIP-CHEMO models. Fi-
nally, solution approaches and algorithms for solving
the SIP-CHEMO models are presented.
Chemotherapy appointment scheduling under uncertainty 7
5.1 Problem definition and notation
This section defines the notation for the SIP-CHEMO
models. Consider a new patient whose oncologist has
recommended a unique treatment regimen and start
date. The availability of chemotherapy chair and nurse
resources as well as the current schedule of appoint-
ments are known. An appointment schedule for this new
patient is needed. The schedule should specify the start
date, time slot, chair assignment, and nurse assignment
for each appointment in the treatment regimen.
The chemotherapy scheduling problem assumes a
finite planning horizon. Let set D be the days in the
planning horizon where D is the last day of the planning
horizon. The nurses expected to be on duty on day d
are given by the set Jd and the chemotherapy chairs
available on day d are given by the set Kd. The number
of nurses working on day d is Jd. All chair and nurse
resources are assumed to have the same properties and
are therefore interchangeable.
Each day in the planning horizon is divided into
time slots of equal length and the same number of time
slots exist each day, which are specified by the set S.
The size of the set S is S. The set Sd is the set of
time slots available on day d while Sdk is the set of
time slots available on day d for chair k. Note that⋃k∈Kd Sdk = Sd, Sdk ⊆ Sd, Sdk ⊆ S, and Sd ⊆ S.
First-StageD: Set of days in the planning horizon, indexed by
dJd: Set of nurses expected to work on day d,
indexed by jKd: Set of available chairs on day d, indexed by kS: Set of time slots for the clinic’s operating
hours, indexed by sSd: Set of available time slots on day d, indexed by
sSdk: Set of available time slots on day d for chair k,
indexed by sT : Set of days in the treatment regimen, indexed
by tUd1 : u|u = max(1, s− rt + 1)...max(1, s), u ∈ Sd
Second-StageΩ : Set of scenarios, indexed by ω
Jd(ω): Set of nurses working on day d for scenario ω,indexed by j
Udk1 (ω): u|u = max(1, s − rt(ω) + 1)...max(1, s), u ∈Sdk
Udk2 (ω): u|u = max(1, S − rt(ω) + 2)...S, u ∈ Sdk
Table 3: Sets for the SIP-CHEMO Models
Each patient has a unique treatment regimen and
the set T specifies which days the patient has an ap-
pointment. The size of set T , |T | = n, specifies the
number of appointments in the patient’s treatment reg-
imen. Consider T = t1, t2, t3 = 1, 8, 15 where the
patient has three treatments specified by t1, t2, and t3respectively and n = 3. Note that t2 − t1 = 8 − 1
= 7 indicates that the second appointment should be
seven days after the first appointment. Set T should be
defined such that t1 = 1 and tn is the length of the
treatment regimen. All sets used in the SIP-CHEMO
models are defined in Table 3.
The expected acuity level for appointment t of the
treatment regimen is given by at. Let amax be the pre-
determined maximum acuity level for each nurse. The
number of time slots expected to be needed for appoint-
ment t of the treatment regimen is rt. The treatment
start date recommended by the oncologist is specified
by dstart. This date must be part of the planning hori-
zon such that dstart ∈ D. The penalty for each day
(either early or late) is δdelay.
First-StageD : = maxd|d ∈ D last day of the planning
horizonS: = maxs|s ∈ S last time slot of the clinic’s
operating hoursJd: = |Jd| the number of nurses working on day dT : = maxt|t ∈ T the last day, or length, of
treatment regimen cycleat: Acuity level on day t of the treatment regimen,
at ∈ 1, 2, 3amax: Maximum acuity level per nurse in one time
slotbjds: Acuity on day d of existing patients for nurse
j in slot sdstart: Treatment start day recommended by the on-
cologistrt: Number of time slots needed for appointment
t of the treatment regimenδdelay: Penalty for each day of treatment delayδslots : Penalty for time slot sδslot: Penalty for each additional time slotδα: Penalty for α overtime variableδβ : Penalty for β excess acuity variableδγ : Penalty for γ new start variableδδ: Penalty for δ overlap variablenjds: = 1 if nurse j is starting an existing patient on
day d during time slot s, 0 otherwiseqds: Sum of the acuity levels of existing patients on
day d in slot sSecond-StageJd(ω): =|Jd(ω)| number of nurses working on day d
in scenario ωat(ω): Acuity level on day t of their treatment regi-
men in scenario ωrt(ω): Number of time slots needed for appointment
t in scenario ωods(ω): = Jd(ω)∗amax, the maximum acuity level load
that the nurses can handle on day d in slot sin scenario ω
Table 4: Parameters for the SIP-CHEMO Models
8 Michelle M Alvarado, Lewis Ntaimo
The current schedule of appointments is known, so
let bjds be the sum of the acuity levels of the patient(s)
assigned to nurse j during slot s on day d. The acuity
across all nurses on day d in slot s is qds. If a nurse
j is assigned to start a patient on day d in slot s, let
njds = 1, otherwise njds = 0.
There are three types of uncertainty considered in
this problem formulation. A scenario is the realization
of an outcome for acuity level, appointment duration,
and number of nurses on duty for each appointment in
the patient’s treatment regimen. The set Ω represents a
finite set of scenarios indexed by ω. First, recall that the
expected acuity level for appointment t of the treatment
regimen is at, but the actual acuity level given the
realization of scenario ω is at(ω). Note that 1 ≤ at(ω) ≤amax. Second, recall that the number of time slots ex-
pected to be needed for appointment t of the treatment
regimen is rt, but the actual number of time slots used
for the realization of scenario ω is rt(ω). Third, the
number of nurses on duty may decrease because a nurse
may be unable to work that day. Jd(ω) is the set of
nurses working on day d in scenario ω. The number of
nurses available during the realization of any scenario
ω is assumed to be less than or equal to the number of
nurses originally scheduled to work, therefore Jd(ω) ⊆Jd,∀ω ∈ Ω. All of the parameters used in the SIP-
CHEMO models are in Table 4.
5.2 Risk-neutral formulation
There are three first-stage decisions that need to be
made here-and-now (see Table 5). Let xd be a binarydecision variable that indicates if the first appointment
in the treatment regimen begins on day d, also known as
the start date. Let ydt
ks be a binary decision variable that
indicates if appointment t of the patient’s treatment
regimen is scheduled for day d in chair k during time
slot s. Finally, let vdjs be a binary decision variable
that indicates if nurse j is assigned to start the patient
during slot s on day d. The decision variable for the
chair assignment is separated from the decision variable
of the nurse assignment because they have different
constraints for subsequent time slots.
xd: xd = 1 if the first treatment is on day d, xd = 0otherwise.
ydt
ks: ydt
ks = 1 if the tth treatment starts in chair k
during slot s on day d, ydt
ks = 0 otherwise.vdjs: vdjs = 1 if nurse j is scheduled to start the
patient during slot s on day d, vdjs = 0 otherwise.
Table 5: Risk-Neutral First-Stage Decision Variables
The first-stage formulation is stated in (7). The first-
stage objective (7a) is to 1) minimize the deviation from
the recommended start date of the first appointment,
2) fill the time slots with the highest priority, 3) min-
imize the expected value of the second stage objective
function. It is expected that δdelay ≥ δslots ∀s because
moving the appointment backwards or forwards one day
has larger consequences than moving the appointment
backward or forward one time slot. By defining δslots
appropriately, one can encourage appointments to be
scheduled early in the day, late in the day, or even
consider patient preferences for certain times of the day.
Constraint (7b) links the xd decision variable and
the ydt
ks decision variable by forcing agreement on the
start date of the first treatment in the treatment regi-
men. Constraint (7c) is necessary to ensure that the rest
periods between appointments is consistent with the
recommendation the oncologist made for the patient’s
treatment regimen. Constraint (7d) forces the require-
ment that each appointment in the patient’s treatment
regimen is scheduled. If this constraint is not satisfied,
then the problem is infeasible and the planning horizon
should be extended. Constraint (7e) requires that the
sum of the acuity levels of all nurses assigned to a nurse
during any given time slot is less than or equal to amax.
Constraint (7f) links the ydt
ks decision variable to the
vdjs decision variable such that all patients scheduled to
start must have a nurse assigned. Constraint (7g) limits
the number of new patient starts for a nurse during
a time slot to one or fewer. Constraints (7h)-(7j) are
binary constraints.
5.3 Second-Stage
αd(ω): (overtime variable) number of overtime slots forthe clinic on day d in scenario ω
βds (ω): (excess acuity variable) excess acuity above themaximum for all nurses during slot s on day din scenario ω
γdjs(ω): (new start variable) indicates if nurse j is unableto start an assigned patient on day d in slot s inscenario ω
δdks(ω): (overlap variable) indicates if an appointmentoverlaps an existing appointment in chair k onday d in slot s in scenario ω
Table 6: Risk-Neutral Second-Stage Decision Variables
The second-stage decision variables are the recourse
decision variables. There are four types of scheduling
conflicts that can occur from the realization of uncer-
tainty: overtime, excess acuity, new starts, and appoint-
ment overlaps. Each scheduling conflict is modeled as
Chemotherapy appointment scheduling under uncertainty 9
a second-stage decision variable given in Table 6. First,
an increase in the appointment duration can cause over-
time for the clinic. Let αd(ω) be a continuous decision
variable that indicates the number of overtime slots
caused by the realization of scenario ω on day d. Second,
an increase in acuity level or a decrease in the number
of nurses can cause the maximum acuity level for a time
slot to be exceeded. Let βds (ω) be a continuous decision
variable that indicates the amount of excess acuity in
time slot s on day d in scenario ω.
Third, a decrease in the number of nurses on duty
can cause scheduling problems with the schedule for
starting patient appointments if the nurse who does not
come in to work was assigned to start a patient’s ap-
pointment that day. Let γdjs(ω) be a continuous decision
variable that indicates if a nurse j is not able to start the
assigned patient during day d in slot s under scenario
ω. Fourth, an increase in appointment duration can
cause the appointment to overlap another appointment
already scheduled. Let δdks(ω) be a continuous decision
variable that indicates if an appointment overlaps an
existing appointment in chair k on day d in time slot
s for scenario ω. Each of these four continuous deci-
sion variables has an associated penalty of δα, δβ , δγ , δδ
respectively.
The second-stage formulation is stated in (8). The
second-stage objective (8a) minimizes scheduling con-
flicts for overtime, excess acuity, new starts, and ap-
pointment overlaps by minimizing the sum of all second-
stage decision variables with their respective penalties
δα, δβ , δγ , and δδ. Constraint (8b) determines the num-
ber of overtime slots for the clinic in scenario ω, which
may occur if rt(ω) > rt. In the second-stage, patientsassigned to nurses that are unable to come to work need
to be re-allocated to other nurses on duty. Because some
nurses may be unavailable for some scenarios, the indi-
vidual nurse acuity is no longer limited to amax. Instead,
the sum of acuity levels of all patients scheduled for each
time slot is less than or equal to the collective maximum
acuity ods(ω) = Jd(ω) ∗ amax of all nurses on duty. The
collective acuity requirement is used because the nurses
that are available must work together to handle the
patients who had been assigned to the absent nurse.
Constraint (8c) determines if any time slots have
excess acuity for scenarios in which at(ω) > at and\or
Jd(ω) < Jd. Constraint (8d) determines if any nurses
that are unable to work (e.g., scenarios in which Jd(ω) <
Jd) have been assigned to start a patient’s appoint-
ment. Constraint (8e) determines if the new appoint-
ment overlaps any existing appointments, which can
occur in scenarios where rt(ω) > rt. Finally, constraints
(8f) - (8i) define all second-stage variables to be non-
negative. In summary, the RN SIP-CHEMO model de-
fined in (7) and (8) is a two-stage model with binary
first-stage decision variables and continuous second-stage
decision variables.
5.4 Expected excess formulation
The RN SIP-CHEMO problem ((7) and (8)) is refor-
mulated as a deterministic equivalent formulation for
EE in (4). The adapted model, EE, is given as problem
(9) where a new decision variable ν(ω) is introduced.
The objective is stated in (9a) which now has one ad-
ditional summation for the expected value of the new
decision variable multiplied by λ. Several constraints
are unmodified as indicated by (9b) and (9c). However,
two additional constraints are needed: the complicating
constraint (9d) and the non-negative constraint (9e) for
the new decision variable.
5.5 Absolute semideviation formulation
The RN SIP-CHEMO problem ((7) and (8)) was refor-
mulated as the deterministic equivalent formulation for
ASD (6). The adapted model, ASD, is given as problem
(10) where a new decision variable ν(ω) is also intro-
duced. The objective (10a) for ASD now has the original
objective (7a) multiplied by 1 − λ and one additional
summation for the expected value of the new decision
variable multiplied by λ. Several constraints are unmod-
ified as indicated by (10b) and (10c). Three additional
constraints are needed: two complicating constraints
(10d) and (10e) and one constraint for the unbounded,
continuous decision variable ν(ω) (10f).
5.6 Solution approaches
When solving SIP-CHEMO, there are several ways to
keep the problem tractable. Three of these approaches
are: generating only necessary constraints, using a small
number of scenarios, and branching using the xd deci-
sion variable. The first approach only generates the nec-
essary constraints in the second-stage formulation. Note
that overtime and overlapping appointments can only
be caused when rt(ω) > rt. Therefore, only generate
constraints (8b) and (8e) for such scenarios. Similarly,
excess acuity can only exist when at(ω) > at, therefore,
only generate constraints (8c) for such scenarios.
The second approach to simplifying SIP-CHEMO
involves using only a limited number of scenarios so that
the set Ω is relatively small. SIP-CHEMO is suitable for
this approach because all three stochastic parameters
can reasonably be limited to two or three scenarios.
10 Michelle M Alvarado, Lewis Ntaimo
Min∑d∈D
[δdelay ∗ |d− dstart|xd +∑
t∈t|t∈T,t≤d
∑k∈Kd
∑s∈Sdk
δslots ∗ ydt
ks] + E[f(x, y, v, ω)] (7a)
s.t. xd −∑k∈Kd
∑s∈Sdk
yd1
ks = 0, ∀d ∈ D (7b)
− xd +∑k∈Kd
∑s∈S(d+t−1),k
y(d+t−1)t
ks ≥ 0, ∀d ∈ 1...(D − T + 1),∀t ∈ T, d ≥ t (7c)
∑d∈d|d∈D,d≥t
∑k∈Kd
∑s∈Sdk
ydt
ks = 1,∀t ∈ T (7d)
−∑u∈Ud
1
at ∗ vdju ≥ bjds − amax, ∀d ∈ D, ∀j ∈ Jd, ∀s ∈ S (7e)
∑j∈Jd
vdjs −∑
t∈t|t∈T,t≤d
∑k∈k|k∈Kd,s∈Sdk
ydt
ks = 0, ∀d ∈ D,∀s ∈ Sd (7f)
− vdjs ≥ njds − 1, ∀d ∈ D,∀j ∈ Jd, ∀s ∈ Sd (7g)
xd ∈ 0, 1, ∀d ∈ D (7h)
ydt
ks ∈ 0, 1, ∀d ∈ D,∀k ∈ Kd, ∀s ∈ Sdk, ∀t ∈ T, d ≥ t (7i)
vdjs ∈ 0, 1, ∀d ∈ D, ∀s ∈ Sd, ∀j ∈ Jd, (7j)
where for each outcome (scenario)ω ∈ Ω of ω
Min f(x, y, v, ω) =∑d∈D
[δα ∗ αd(ω) + δβ∑s∈S
βds (ω) + δγ∑s∈Sd
∑j∈Jd\Jd(ω)
γdjs(ω)
+ δδ∑k∈Kd
∑s∈S\Sdk
δdks(ω)] (8a)
s.t. αd(ω) ≥∑
t∈t|t∈T,t≤d
∑k∈K
∑s∈Udk
2(ω)
(S − rt(ω)− s+ 3) ∗ ydt
ks, ∀d ∈ D (8b)
βds (ω) ≥ qds − ods(ω) +∑
t∈t|t∈T,t≤d
∑k∈K
∑u∈Udk
1(ω)
at(ω) ∗ ydt
ku, ∀d ∈ D, ∀s ∈ S (8c)
γdjs(ω) ≥ vdjs + njds, ∀d ∈ D, ∀s ∈ Sd, ∀j ∈ Jd\Jd(ω) (8d)
δdks(ω) ≥∑
t∈t|t∈T,t≤d
∑u∈Udk
1(ω)
ydt
ku, ∀d ∈ D,∀k ∈ Kd, ∀s ∈ S\Sdk (8e)
αd(ω) ≥ 0, ∀d ∈ D (8f)
βds (ω) ≥ 0, ∀d ∈ D, ∀s ∈ S (8g)
γdjs(ω) ≥ 0, ∀d ∈ D,∀s ∈ Sd, ∀j ∈ Jd\Jd(ω) (8h)
δdks(ω) ≥ 0, ∀d ∈ D, ∀k ∈ Kd, ∀s ∈ S\Sdk. (8i)
The acuity levels can only take three values and it is
assumed that only one nurse will call in sick on any
given day and thus Jd(ω)| = Jd(ω), Jd(ω) − 1 and
|Jd(ω)| = 2. The third stochastic parameter, treatment
duration, is discrete and bounded between zero and S.
In the realistic setting, if the size of each time slot s is
reasonably large (e.g., 15 or 30 minutes), then the treat-
ment regimen may only change by a few time slots and
thus be limited to a few (e.g., three to five) scenarios
as well.
The third and final approach is to separate the de-
cision problem using the treatment regimen and set D.
When a potential start date is selected from set D,
then the spacing between appointments, as determined
by the treatment regimen in set T (constraints (7c)),
reduces the scope of days in set D to size |T |. This
approach is similar to a branch-and-cut approach in
which one chooses a start date d by setting xd = 1. One
can then determine the following appointment dates
using set T and thereby reduce |D| = |T |. Observe
then that there is a need to only create variables ydt
ks
for d = di and t = ti when di and ti correspond to
element i in sets D and T respectively.
The objective increases as one selects d farther from
dstart. We have developed an algorithm,MinAlg(), that
first checks xd = dstart, then searches values in the
Chemotherapy appointment scheduling under uncertainty 11
EE: Min∑d∈D
[δdelay ∗ |d− dstart|xd +∑
t∈t|t∈T,t≤d
∑k∈Kd
∑s∈Sdk
δslots ∗ ydt
ks]
+∑ω∈Ω
p(ω) ∗∑d∈D
[δα ∗ αd(ω) +∑s∈S
δβ ∗ βds (ω) +∑s∈Sd
∑j∈Jd\Jd(ω)
δγ ∗ γdjs(ω)
+∑k∈Kd
∑s∈S\Sdk
δδ ∗ δdks(ω)] + λ ∗∑ω∈Ω
p(ω)ν(ω) (9a)
s.t. Constraints (7b), (7d)− (7j) (9b)
Constraints (8b)− (8i) (9c)
−∑d∈D
[δdelay ∗ |d− dstart|xd +∑
t∈t|t∈T,t≤d
∑k∈Kd
∑s∈Sdk
δslots ∗ ydt
ks]
−∑d∈D
[δα ∗ αd(ω) + δβ∑s∈S
βds (ω) + δγ∑s∈Sd
∑j∈Jd\Jd(ω)
γdjs(ω)
+ δδ∑k∈Kd
∑s∈S\Sdk
δdks(ω)] + ν(ω) ≥ −η,∀ω ∈ Ω (9d)
ν(ω) ≥ 0, ∀ω ∈ Ω (9e)
ASD: Min (1− λ)∑d∈D
[δdelay ∗ |d− dstart|xd +∑
t∈t|t∈T,t≤d
∑k∈Kd
∑s∈Sdk
δslots ∗ ydt
ks]
+ (1− λ) ∗∑ω∈Ω
p(ω)∑d∈D
[δα ∗ αd(ω) + δβ∑s∈S
βds (ω) (10a)
+ δγ∑s∈Sd
∑j∈Jd\Jd(ω)
γdjs(ω) + δδ∑k∈Kd
∑s∈S\Sdk
δdks(ω)] + λ ∗∑ω∈Ω
p(ω)ν(ω)
s.t. Constraints (7b), (7d)− (7j) (10b)
Constraints (8b)− (8i) (10c)
−∑d∈D
[δdelay ∗ |d− dstart|xd +∑
t∈t|t∈T,t≤d
∑k∈Kd
∑s∈Sdk
δslots ∗ ydt
ks]
−∑d∈D
[δα ∗ αd(ω) +∑s∈S
δβ ∗ βds (ω) +∑s∈Sd
∑j∈Jd\Jd(ω)
δγ ∗ γdjs(ω)
+∑k∈Kd
∑s∈S\Sdk
δδ ∗ δdks(ω)] + ν(ω) ≥ 0, ∀ω ∈ Ω (10d)
−∑d∈D
[δdelay ∗ |d− dstart|xd +∑
t∈t|t∈T,t≤d
∑k∈Kd
∑s∈Sdk
δslots ∗ ydt
ks]
−∑ω∈Ω
p(ω)∑d∈D
[δα ∗ αd(ω) +∑s∈S
δβ ∗ βds (ω) +∑s∈Sd
∑j∈Jd\Jd(ω)
δγ ∗ γdjs(ω)
+∑k∈Kd
∑s∈S\Sdk
δδ ∗ δdks(ω)] + ν(ω) ≥ 0, ∀ω ∈ Ω (10e)
ν(ω) free, ∀ω ∈ Ω (10f)
neighborhood (e.g., dstart + 1, dstart − 1, etc.) to find
the start date d that results in the minimum objective
value. Furthermore, this approach eliminates the need
for constraint (7c) because rest days have already been
excluded.
Next, pseudocode is used to describe the algorithm
MinAlg() that identifies the best solution, referred to
as x∗ for simplicity. The algorithm assumes there is a
global minimum value for the SIP-CHEMO problem
instance near dstart. It first finds the solution using
xdstart = 1. Afterwards, the algorithm searches a few
days before and after the initial dstart value until find-
ing maxFail worse objective values in each direction.
When the maxFail worse objective values are found
after (before) the dstart value, then posStop (negStop)
becomes true. The algorithm is driven by searching for a
sets of days D ∈ D that can provide a feasible solution.
The following three methods are used in the algorithm
are used in the MinAlg() algorithm:
12 Michelle M Alvarado, Lewis Ntaimo
1. inSetD(D): returns true if for each d ∈ D, then
d ∈ D; returns false otherwise.
2. getDHat(d): returns set D of size |T | where d1 = d.
3. solve(D): returns (x∗, obj∗) after solving an SIP-
CHEMO problem instance (e.g., problem RN in (7))
using D = D where x∗ is the solution and obj∗ is
the objective value.
Next, the algorithm is stated using pseudocode. The
left arrow ← is used to denote assignment, & is the
“and” operator, ! is the “not” operator, and == is
the “equal to” operator. The steps of the MinAlg()
algorithm are stated in Algorithm 1. The MingAlg()
algorithm identifies the optimal solution x∗ and optimal
objective value obj∗ to problem (7).
1 obj∗ ←∞, negStop← false, posStop←false, done← false, fail← 0, d← dstart, D ←getDHat(d);
2 while !done do
3 if inSetD(D) then
4 (x, obj) ← solve(D);5 if obj < obj∗ then6 obj∗ ← obj, x∗ ← x;7 else8 fail← fail + 1;9 if fail ≥ maxFail & posStop == false
then10 posStop← true;11 fail← 0;12 d← dstart;
13 else if fail ≥ maxFail &negStop == false;
14 then15 negStop← true;
16 end
17 else18 if !posStop then19 d← d+ 1;20 else if !negStop then21 d← d− 1;
22 end23 if posStop & negStop then24 done← true;25 else
26 D ← getDHat(d);27 end
28 end29 return x∗, obj∗;
Algorithm 1: MinAlg()
6 Application
The SIP-CHEMO models were analyzed based on data
from a real outpatient oncology clinic. This section first
describes the real oncology clinic setting at Baylor Scott
& White Hospital and then provides details on the
design of experiments, presents computational results,
and discusses the implications in management science.
The outpatient oncology clinic at Baylor Scott &
White Hospital in Temple, Texas, USA operates five
days a week for nine hours each day. The clinic typically
has one charge nurse and four to eight registered nurses
on duty at any given time. There are 17 chemotherapy
chairs that are regularly used in the oncology clinic for
scheduling purposes. The clinic treats an average of 23.5
patients each day.
Baylor Scott & White’s oncology clinic provided
historical data from a five-month period. The database
contained 505 sample patients. On average there were
around four appointments in each patient’s treatment
regimen, but actual values ranged from 1 to 21 appoint-
ments . The maximum acuity a single registered nurse
could have was assumed to be five (amax = 5).
To evaluate the SIP-CHEMO models, the authors
developed a simulation model of the oncology clinic
called DEVS-CHEMO. DEVS-CHEMO gives system
performance results on the type I delay, type II de-
lay, system time, throughput, and nurse overtime. All
experiments were conducted using a four-month plan-
ning horizon and simulated the oncology clinic opera-
tions for one month. A warm-up period scheduled 170
patients. During the simulation, five to six additional
appointment requests occurred each day which resulted
in around 276 patients each month. For scheduling pur-
poses, time slots were assumed to be 30 minutes each
because the clinic currently uses time slots of this length.
With nine operating hours, there were 18 time slots in
each day.
Creating scenarios is an important part of the exper-
imental design for the SIP-CHEMO models. For each
scheduling problem solved, there were 12 scenarios. The
12 scenarios were generated from combining three out-
comes of appointment duration, two outcomes of acuity
levels, and two outcomes of number of nurses. An ex-
ample of these outcomes is shown in Table 7. The three
outcomes of stochastic appointment duration were equally
weighted and were dependent on the type of drug(s)
used in the treatment regimen. If historical data on a
specific drug had at least one hundred data points in the
historical database, then the appointment duration was
generated using a distribution. Otherwise, the appoint-
ment duration was sampled from the existing pool of
data values. The number of time slots was then found
by dividing the appointment duration by 30 minutes
and rounding to the nearest integer value.
The two acuity level outcomes were sampled from a
distribution where an acuity level of 1 occurred 70% of
the time, a value of 2 occurred 20% of the time, and a
Chemotherapy appointment scheduling under uncertainty 13
Treatment No. Outcome Appointment Duration (slots) Probability
1;8;101 4;3;4 0.332 5;3;3 0.333 3;4;5 0.33
Treatment No. Outcome Acuity Probability
1;8;101 1;1;2 0.502 1;3;1 0.50
Days Outcome No. of Nurses Probability1 5;7;6 0.902 4;6;5 0.10
Table 7: Example SIP-CHEMO Outcomes
value of 3 occurred 10% of the time. Registered nurses
were assumed to have a 10% probability of taking a
vacation or sick day. This assumption came from the
Bureau of Labor statistics by citing the average sick
and vacation time for a ten-year employee. Thus, the
original number of nurses was assumed to be available
90% of the time and there was a 10% probability of
having one less nurse. There were 12 outcomes because
3× 2× 2 = 12 and combining the outcomes from Table
7 results in the 12 scenarios in Table 8 for a start date
on day eight (x8 = 1).
The penalties in the SIP-CHEMO objective func-
tions (7a), (9a), and (10a) were determined by con-
verting the units of each variable into acuity. Recall
that one time slot has a maximum acuity amax and one
day has S time slots. Therefore, 1 slot = amax and 1
day = S ∗ amax. Then the excess acuity penalty δβ =
1 because the β decision variable already represents
acuity. Next the new start penalty δγ = amax and
overlapping time slot penalty δδ = amax because the γ
and δ decision variables are both indicators for a time
slot. The overtime decision variable also represents time
slots and thus one could use δα = amax. However, early
experiments revealed that this penalty did not signifi-
cantly impact overtime. Therefore, the penalty measure
for α was set to a half-day with δα = 0.5∗amax∗S. Since
the x decision variable represents one day, then δdelay =
S ∗ amax. Note that the y decision variable represents
a time slot. In the SIP-CHEMO models, later time
slots were penalized more heavily and thus the model
rewarded appointments that started early in the day.
Since this is a penalty term used to avoid unnecessary
gaps between appointments, one-tenth of the value was
used and thus δslots = 0.1 ∗ s ∗ amax.
6.1 Design of experiments
The first set of experiments implemented five schedul-
ing models and compared their performance. The sec-
ond set of experiments examined the impact that the
risk factor λ had on scheduling performance for the
EE and ASD SIP-CHEMO models. Finally, the third
set of experiments analyzed how the target value of η
impacted the EE SIP-CHEMO model.
The first research question for SIP-CHEMO requires
a comparison of five scheduling methods. When patients
call the scheduler to get an appointment schedule, pa-
tients provide their recommended treatment regimen
and start date. In the real oncology clinic, the sched-
uler uses a scheduling algorithm called the as-soon-as-
possible (ASAP) algorithm that selects the first avail-
able appointment slots. The ASAP algorithm uses only
the availability of the chemotherapy chairs to schedule
the patient’s appointments and ignores the availability
of the registered nurses. The Individual algorithm de-
veloped in the DEVS-CHEMO simulation model con-
siders both the availability of both the registered nurse
and chairs. The first experiment compared the ASAP
and Individual algorithms to three SIP-CHEMO mod-
els: RN, EE, and ASD. These models used λ = 0.5 in
the first experiment and the EE model used a target
value of two days with η = 2 ∗ S ∗ amax. This value
was chosen to indicate that moving more than two days
from the recommended start date can cause risk to the
patient’s health status.
The second research question examined how the value
of λ in the EE and ASD scheduling models impacts the
system performance. When λ = 0, the decision-maker
is risk-neutral and when λ = 1, then decision-maker
is risk-averse. The labels of ASD 05 and ASD 10 were
used to label the ASD mean-risk SIP-CHEMO model
simulation runs with λ = 0.5 and λ = 1.0 respectively.
Similarly, the names of EE 05 and EE 10 were used to
label the EE mean-risk SIP-CHEMO model simulation
runs. The EE model used η = 1 ∗ S ∗ amax.
Finally, the third research question focused on how
the value of η impacts the system performance. Recall
that η is the target value for the EE mean-risk SIP-
CHEMO model. In all simulation runs, λ = 0.5 and the
value of η changes with η = 0.0, η = 1 ∗ S ∗ amax, and
η = 2∗ S ∗amax in the simulation runs labeled EE-Eta0,
EE-Eta1, and EE-Eta2 respectively.
14 Michelle M Alvarado, Lewis Ntaimo
ω Prob. Days Treatment No. Appt. Dur. Acuity No. of Nurses1 0.15 8;15;17 1;8;10 4;3;4 1;1;2 5;7;62 0.15 8;15;17 1;8;10 5;3;3 1;3;1 5;7;63 0.15 8;15;17 1;8;10 3;4;5 1;1;2 5;7;64 0.15 8;15;17 1;8;10 4;3;4 1;3;1 5;7;65 0.15 8;15;17 1;8;10 5;3;3 1;1;2 5;7;66 0.15 8;15;17 1;8;10 3;4;5 1;3;1 5;7;67 0.02 8;15;17 1;8;10 4;3;4 1;1;2 4;6;58 0.02 8;15;17 1;8;10 5;3;3 1;3;1 4;6;59 0.02 8;15;17 1;8;10 3;4;5 1;1;2 4;6;510 0.02 8;15;17 1;8;10 4;3;4 1;3;1 4;6;511 0.02 8;15;17 1;8;10 5;3;3 1;1;2 4;6;512 0.02 8;15;17 1;8;10 3;4;5 1;3;1 4;6;5
Table 8: Example SIP-CHEMO Scenarios with x8 = 1
There were 10 replications of each DEVS-CHEMO
simulation. During each simulation run, one of the re-
spective scheduling models was used to schedule each
new patient. The DEP for each SIP-CHEMO model
was directly solved using CPLEX 12. The experiments
were all conducted on a Dell Precision T7500 with an
Intel(R) Xeon(R) processor running at 2.4 GHz with
12.0 GB RAM.
6.2 Computational results
The first set of experiments compared five scheduling
models (Table 9). All SIP-CHEMO models (RN, EE,
and ASD) outperformed the ASAP and Individual al-
gorithms for several performance measures captured
in the DEVS-CHEMO simulation model such as total
throughput (Figure 2), system time (Figure 3), type II
delay, nurse overtime+, and nurse overtime (Figure 4).
The EE model had the highest total throughput (473
appointments) and the lowest type II delay (16 min-
utes), and system time (208 minutes). The RN model
had the lowest nurse overtime+ (95 minutes) and nurse
overtime (31 minutes). The EE model minimized the
risk of a delay in the start date and the risk of schedul-
ing conflicts, especially in the more extreme cases be-
cause the target value was equivalent to two days. As
a result, EE made more efficient daily scheduling deci-
sions that achieved lower system performance measures
for system time and type II delay. The RN model does
not consider risk to the patient’s health status, but
does minimize the expected value of the incidence of
scheduling conflicts occurring in the second stage. As a
result, the RN model achieved low system performance
measures in several categories (though not as low as
EE), but most significantly in nurse overtime and nurse
overtime+. This is because nurse overtime scheduling
conflicts occurs less frequently than the others (e.g.
excess acuity, overlap, and new patient starts) and thus
are considered more in the RN model than the EE or
450
455
460
465
470
475
ASAP Individual RN EE ASD
Model
Num
ber
of A
ppoi
ntm
ents
Fig. 2 Average Throughput Versus Scheduling Model
ASD models which only consider risk above a target or
mean value.
200
205
210
215
220
225
ASAP Individual RN EE ASD
Model
Min
utes
Fig. 3 Average System Time Versus Scheduling Model
Chemotherapy appointment scheduling under uncertainty 15
ExperimentsPerformance Measures ASAP Individual RN EE ASDTotal Throughput (appts.) 467.5 458.4 471.8 473.4 472.1Nurse Overtime+ (min.) 116.96 108.71 94.96 98.33 99.64Nurse Overtime (min.) 46.91 45.19 30.64 34.17 34.97Type I Delay (days) 1.36 1.63 1.55 1.54 1.54Type II Delay (min.) 28.46 18.69 16.57 16.44 16.89System Time (min.) 221.39 211.39 208.02 207.97 209.00Sim. Run Time (sec.) 1.46 1.32 67.78 86.00 102.84
Table 9: Performance Results for Scheduling Models
0
30
60
90
120
ASAP ASD EE Individual RN
Model
Min
utes Nurse Overtime+
Nurse OvertimeType II Delay
Fig. 4 Average Time-Based Performance Measures VersusScheduling Model
ASAP has the lowest type I delay (1.36 days) be-
cause the algorithm schedules patients as quickly as
possible and ignores the nurse resource. Aside from the
ASAP algorithm, the SIP-CHEMO models have the
lowest type I delay (Figure 5) with 1.5 days.
1.2
1.3
1.4
1.5
1.6
1.7
ASAP Individual RN EE ASD
Model
Day
s
Fig. 5 Average Type I Delay Versus Scheduling Model
The SIP-CHEMO models have simulation run times
of one minute compared to the two algorithms which
took less than 1.5 seconds each. The run times for the
SIP-CHEMO models increase as the number of con-
straints increases in the RN, EE, and ASD models which
took an average of 68, 86, and 103 seconds to run,
respectively. EE had the best performance measures for
the most categories, specifically the EE model held type
I delay within 0.2 days of the ASAP algorithm while
making improvements in other performance measures
such as throughput (increased 1%), nurse overtime+
(reduced 16%), nurse overtime (reduced 27%), system
time (reduced 6%), and type II delay (reduced 42%)
when compared to the ASAP algorithm. RN had the
lowest system performance measures for nurse overtime
and nurse overtime+ and were statistically more sig-
nificant than the values for all other models, including
ASAP and EE. Thus, it can be concluded that the SIP-
CHEMO models supersede the decisions made using
just the DEVS-CHEMO simulation model scheduling
algorithms. The standard deviation and 90% confidence
intervals for each performance measure for the five mod-
els is in Table 12 of the Appendix.
The second set of experiments investigated the im-
pact of the risk factor λ on the system performance.
Table 10 contains the averages for each performance
measure. For the EE models, the results of λ = 0.5 or
λ = 1.0 were not statistically significant. The through-
put only differed by 2 appointments (<1%) and type
I delay by 0.01 days (1%). For the ASD models, the
most risk-averse case ASD 10 was the most conserva-
tive in scheduling decisions, especially for minimizing
scheduling conflicts (second-stage objective). This re-
sulted in lower nurse overtime+ (by 20 minutes), nurse
overtime (by 12 minutes), type II delay (by 5 minutes),
and system time (by 5 minutes) than ASD 05. This
was possible by moving appointments further from the
recommended start date, as evidenced by lower total
throughput (by 37 appointments) and higher type I
delay (by 1.0 day).
The third set of experiments investigated the impact
of the EE target value, η, on system performance for
the EE SIP-CHEMO model (Table 11). The results
indicate that there is very little difference in the system
performance results based on η. The type I delay was
exactly 1.54 days and total throughput was around 473
16 Michelle M Alvarado, Lewis Ntaimo
Performance Measures EE 05 EE 10 ASD 05 ASD 10Total Throughput (appts.) 474.8 476.7 472.1 434.8Nurse Overtime+ (min.) 99.85 99.78 99.64 79.73Nurse Overtime (min.) 35.61 34.31 34.97 22.74Type I Delay (days) 1.54 1.55 1.54 2.57Type II Delay (min.) 17.25 16.80 16.89 11.97System Time (min.) 209.77 207.77 209.00 204.85
Table 10: Performance Results for the λ Experiments
appointments for all three experiments. These results
indicate that regardless of whether the target was rep-
resented as zero, one, or two days from the physician’s
recommended start date, all three EE models were able
to schedule the patients close to the recommended date
unless necessary to move the first appointment to an-
other date, such as when there was an infeasible start
date due to the cyclic nature of all appointments in the
treatment regimen. The system time was between 208
and 210 minutes while the nurse overtime was 34 to
35 minutes. Results indicate that η did not substan-
tially influence the average system performance mea-
sures across the three models. However, it was noted
that EE Eta2 was able to achieve slightly lower nurse
overtime, type II delay, and system time performance
measures. This is likely because EE Eta2’s higher tar-
get value emphasized eliminating the risk associated
with more extreme scenarios of scheduling conflicts and
thus made scheduling decisions that generated more
balanced nurse work schedules on a daily basis.
6.3 Discussion
The SIP-CHEMO models presented in this paper make
several contributions to management science. Specifi-
cally, the SIP-CHEMO models determine patient ap-
pointment schedules, clinic resource schedules, and con-
sider uncertainty in appointment duration, acuity lev-
els, and nurse availability. The models also adapt to the
management’s level of risk for each patient. The RN
model can be used for the risk-neutral case or the ASD
and EE models can be used for risk-averse preferences
with λ = 1 indicating the most risk-averse manager.
Finally, the SIP-CHEMO models also account for risk
to the patient’s health status due to deviations from
the physicians recommended start dates as well as the
risk of having scheduling conflicts (e.g. overtime and
overlapping appointments) due to uncertainty in the
appointment duration, acuity levels, and nurse avail-
ability.
Computational results showed that the three SIP-
CHEMO models, RN, EE, and ASD, were found to out-
perform the current scheduling algorithms for several
performance measures. The SIP-CHEMO models gen-
erate efficient schedules, increase throughput, reduce
patient waiting times, and reduce nurse overtime. When
comparing the EE and ASD models, the most risk-
averse EE model had the highest total throughput and
low type I delay. However, the most risk-averse ASD
model had the lowest nurse overtime, type II delay,
and system time. The results of the analysis indicate
that healthcare managers can use SIP-CHEMO models
to optimize the scheduling of clinic resources and pa-
tient appointments to improve clinic performance by in-
creasing total throughput and decreasing patient wait-
ing time (type II delay), appointment duration (system
time), and nurse overtime with minimal impact on the
deviation from the physician recommended start date
(type I delay). Since patients arrive one-at-a-time, the
SIP-CHEMO models can still generate solutions in just
a few seconds per patient using the deterministic equiv-
alent formulation, so they are practical to use in a real
clinic setting. However, limitations of the SIP-CHEMO
models include the requirement of subjective input from
healthcare managers (e.g. setting the risk factor and
penalty values) who may not have expertise in opti-
mization modeling or using an optimization solver such
as CPLEX.
7 Summary and areas of future work
The main contribution of this research is the devel-
opment of three SIP-CHEMO models for scheduling
chemotherapy patients, chairs, and nurses under un-
certainty. SIP-CHEMO determines an optimal appoint-
ment schedule for a new chemotherapy patient who
has been prescribed a unique treatment regimen and
recommended start date. The appointment duration,
acuity levels, and nurse resource availability are as-
sumed to be stochastic. A risk-neutral formulation for
the chemotherapy decision problem was first developed.
The first-stage decisions determined an appointment
time and resource assignment while minimizing the type
I delay and appointment start times. The second-stage
objective minimized clinic overtime, excess acuity as-
signments, conflicts with new patient starts, and con-
flicts with overlapping appointment times.
Chemotherapy appointment scheduling under uncertainty 17
Performance Measures EE-Eta0 EE-Eta1 EE-Eta2Total Throughput (appts.) 472.6 474.8 473.4Nurse Overtime+ (min.) 98.13 99.85 98.33Nurse Overtime (min.) 34.82 35.61 34.17Type I Delay (days) 1.54 1.54 1.54Type II Delay (min.) 16.79 17.25 16.44System Time (min.) 209.23 209.77 207.97
Table 11: Performance Results for the η Experiments
The risk-neutral formulation was extended to in-
clude two mean-risk measures. EE aims to minimize
expected value of the excess over a target value while
ASD minimizes the expected value of the excess over
the mean value. The MinAlg() algorithm was devel-
oped to improve the solution speed of the SIP-CHEMO
models by branching on feasible start dates. The SIP-
CHEMO optimization models are the first optimiza-
tion models for the chemotherapy decision problem of
scheduling chemotherapy patients, chairs, and nurses
that consider uncertain problem parameters and risk.
SIP-CHEMO is important because a risk-averse opti-
mization model was developed that outperforms the
original scheduling algorithms to achieve better overall
system performance. Using the SIP-CHEMO models,
the throughput increased by 1%, waiting time (type II
delay) decreased by 41-42%, system time decreased by
6%, nurse overtime+ decreased by 15-19%, and nurse
overtime decreased by 25-35% when compared to the
current ASAP scheduling algorithm.
There are three avenues of future research for SIP-
CHEMO. First, the SIP-CHEMO models could be re-
formulated to make the amount of time allocated to
each appointment part of the decision problem. Second,
one could develop additional SIP-CHEMO models us-
ing mean-risk measures from the class of quantile mea-
sures such as quantile deviation (QDEV). Also, the SIP-
CHEMO models take longer to solve (using CPLEX)
than the algorithms. Although steps have been taken
to simplify these models, current implementations still
solve the deterministic equivalent formulation. One fu-
ture direction would be to implement a decomposition
method such as the Fenchel Disjunctive Decomposi-
tion to further improve the solution speed for the SIP-
CHEMO models.
Acknowledgements The authors wish to thank WilliamCarpentier, MD for providing access to the Baylor Scott &White oncology clinic and historical patient data. The au-thors are also grateful to Theresa Kelley, RN, and ValerieOxley, RN, for sharing expert knowledge in outpatient on-cology clinic operations at Baylor Scott & White Hospital inTemple, TX.
8 Appendix
This Appendix contains Table 12 which gives a sum-
mary of performance results for the scheduling algo-
rithms.
18 Michelle M Alvarado, Lewis Ntaimo
Algorithm Performance Measure (units) AVG STDEV 90% CIASAP Total Throughput (appts.) 467.5 16.4 (464.8,470.2)
Nurse Overtime+ (min.) 116.96 16.36 (114.27,119.65)Nurse Overtime (min.) 46.91 8.45 (45.52,48.30)Type I Delay (days) 1.36 0.08 (1.34,1.37)Type II Delay (min.) 28.46 7.81 (27.17,29.74)System Time (min.) 221.39 10.16 (219.71,223.06)
Individual Total Throughput (appts.) 458.4 14.5 (456.1,460.8)Nurse Overtime+ (min.) 108.71 11.88 (106.75,110.66)Nurse Overtime (min.) 45.19 7.42 (43.97,46.41)Type I Delay (days) 1.63 0.16 (1.61,1.66)Type II Delay (min.) 18.69 4.73 (17.91,19.47)System Time (min.) 211.39 8.21 (210.04,212.74)
RN Total Throughput (appts.) 471.8 20.5 (468.4,475.1)Nurse Overtime+ (min.) 94.96 12.21 (92.95,96.96)Nurse Overtime (min.) 30.64 5.83 (29.68,31.60)Type I Delay (days) 1.55 0.10 (1.53,1.56)Type II Delay (min.) 16.57 3.87 (15.93,17.21)System Time (min.) 208.02 7.09 (206.85,209.19)
EE Total Throughput (appts.) 473.4 18.8 (470.3,476.4)Nurse Overtime+ (min.) 98.33 13.55 (96.10,100.56)Nurse Overtime (min.) 34.17 7.47 (32.94,35.40)Type I Delay (days) 1.54 0.11 (1.52,1.55)Type II Delay (min.) 16.44 4.28 (15.73,17.14)System Time (min.) 207.97 7.70 (206.71,209.24)
ASD Total Throughput (appts.) 472.1 18.1 (469.2,475.1)Nurse Overtime+ (min.) 99.64 14.69 (97.22,102.05)Nurse Overtime (min.) 34.97 7.04 (33.81,36.12)Type I Delay (days) 1.54 0.12 (1.52,1.56)Type II Delay (min.) 16.89 5.31 (16.02,17.77)System Time (min.) 209.00 8.05 (207.68,210.32)
Nurse Overtime+ excludes zero entries
Table 12: Performance Results for Scheduling Algorithms
Chemotherapy appointment scheduling under uncertainty 19
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