chemotherapy appointment scheduling under uncertainty ... · scheduling large volumes of cancer...

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

[email protected] · [email protected]

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


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


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


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.


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


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


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.


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


EE: Min∑d∈D


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


t∈t|t∈T,t≤d

∑k∈Kd

∑s∈Sdk

δslots ∗ ydt

ks]

−∑d∈D


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


t∈t|t∈T,t≤d

∑k∈Kd

∑s∈Sdk

δslots ∗ ydt

ks]

+ (1− λ) ∗∑ω∈Ω

p(ω)∑d∈D


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


t∈t|t∈T,t≤d

∑k∈Kd

∑s∈Sdk

δslots ∗ ydt

ks]

−∑d∈D



∑j∈Jd\Jd(ω)

δγ ∗ γdjs(ω)

+∑k∈Kd

∑s∈S\Sdk

δδ ∗ δdks(ω)] + ν(ω) ≥ 0, ∀ω ∈ Ω (10d)

−∑d∈D


t∈t|t∈T,t≤d

∑k∈Kd

∑s∈Sdk

δslots ∗ ydt

ks]

−∑ω∈Ω

p(ω)∑d∈D



∑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:


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


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.


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


Model

Min

utes

Fig. 3 Average System Time Versus Scheduling Model


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


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


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.


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.


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


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