1/15
Mathematical
modeling
Example:
Glaucoma
clinic
Scheduling and resource planning: models for a
glaucoma clinic
Future work
DAVID P HILLIPS
Department of Mathematics
United States Naval Academy
Joint work with
MAJ Marcus Colyer, M.D. (Walter Reed)
MIDN Marisa Molkenbuhr (USNA)
October 30, 2014
2/15
Mathematical
modeling
Example:
Glaucoma
clinic
So a mathematician walks into a room full of healthcare
providers...
Future work
• Mathematical modeling
• A model for the glaucoma clinic
• Future possibilities
3/15
Complexity versus simplicity
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Mathematical models provide a tool that can help guide decision makers to new solutions and analyze potential impacts
of policies.
3/15
Complexity versus simplicity
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Mathematical models provide a tool that can help guide decision makers to new solutions and analyze potential impacts
of policies.
We consider it a good principle to explain the
phenomena by the simplest hypothesis possible.
–Ptolemy
For every complex problem there is an answer that
is clear, simple, and wrong.
–H.L. Mencken
3/15
Complexity versus simplicity
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Mathematical models provide a tool that can help guide decision makers to new solutions and analyze potential impacts
of policies.
We consider it a good principle to explain the
phenomena by the simplest hypothesis possible.
–Ptolemy
For every complex problem there is an answer that
is clear, simple, and wrong.
–H.L. Mencken
4/15
Modeling cycle
Mathematical
modeling
Example:
Glaucoma
clinic
• Discuss assumptions, goals (objectives), and
limitations (constraints) with decision makers.
Future work
• Collect relevant data for the model.
• Mathematically describe the model.
• Determine algorithms that can compute solutions to the
model.
• Run computations, interpret results.
4/15
Modeling cycle
Mathematical
modeling
Example:
Glaucoma
clinic
• Discuss assumptions, goals (objectives), and
limitations (constraints) with decision makers.
Future work
• Collect relevant data for the model.
• Mathematically describe the model.
• Determine algorithms that can compute solutions to the
model.
• Run computations, interpret results.
REPEAT!!
5/15
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Problem description and
assumptions
• Patient flow through the glaucoma clinic:
1 Technicians administer Visual Fields testing (VF)
2 One of two doctors then see patients
• Two groups of patients:
• SPEC & PROC: new and repeat patients requiring VF
• EST: repeat patient not requiring VF
• Patients are already assigned doctors.
• Objectives we modeled:
• Maximize the number of patients seen
• Minimize the idle time of doctors/VF
• Minimize the wait time of patients between doctors/VF
• Minimize the average completion time
• Time required by doctors, VF is random.
6/15
Decisions
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Our model determines decision variables to achieve our
objectives
6/15
Decisions
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Our model determines decision variables to achieve our
objectives
Which doctor do patients see and when?
1 if doctor i sees patient j at time t
xijt =
0 otherwise
6/15
Decisions
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Our model determines decision variables to achieve our
objectives
Which doctor do patients see and when?
1 if doctor i sees patient j at time t
xijt =
0 otherwise
For patients requiring VF, when?
1 if patient j has a VF at time t
yjt =
0 otherwise
7/15
Optimization model in words
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
• Maximize (patients seen) - (idle time) - (wait times) -
(completion times)
7/15
Optimization model in words
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
• Maximize (patients seen) - (idle time) - (wait times) -
(completion times)
• Constraints:
• Can only schedule a patient once.
• Patients that need VF get one if seen.
• Doctors can only see one patient at a time.
• Up to three VFs at any one time.
• Patients are scheduled into time slots
8/15
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Optimization model with full
algebra (yikes!)
P P P
P
max
xpti − α p cp
p
t
i
P
s.t.
x ≤1
Pp Ppti
j
p ypjkt ≤ 1
i
P hP
P
cp ≥ t
P
x
+
(Q
+
s
)y
t
i
pti
j
pjkt
j,k
P P i
cp ≥ t i (Pi + st )xpti
cp ≤PT
P
pti ≤ 1
Pt Pi xP
pjkt ≤ 1
k
t Pj y
P
yP
≤ t i xpti
pjksP
x ≤1
Pp Ps psi
p
s ypjks ≤ 1
ypjkt , xpti ≥ 0, binary
∀i, t
∀t, k
∀p
∀p
∀p
∀p
∀p
∀p, j, k , s
∀t, ∀i
∀t, ∀j, k
∀p, j, k , t
9/15
Complications & computations
Mathematical
modeling
Example:
Glaucoma
clinic
• For a two-week schedule (4 days), ≈ 2000 − 9000
constraints, ≈ 13, 000 − 30, 000 variables
Future work
• On a 1.7 gigaherz machine with 8 GB of RAM, solves in
≈ 10 minutes
9/15
Complications & computations
Mathematical
modeling
Example:
Glaucoma
clinic
• For a two-week schedule (4 days), ≈ 2000 − 9000
constraints, ≈ 13, 000 − 30, 000 variables
Future work
• On a 1.7 gigaherz machine with 8 GB of RAM, solves in
≈ 10 minutes
• Uncertainty: We use a Monte-Carlo simulation
• Randomly generate a set of process times
• Solve the optimization model
• Analyze the average of the results
9/15
Complications & computations
Mathematical
modeling
Example:
Glaucoma
clinic
• For a two-week schedule (4 days), ≈ 2000 − 9000
constraints, ≈ 13, 000 − 30, 000 variables
Future work
• On a 1.7 gigaherz machine with 8 GB of RAM, solves in
≈ 10 minutes
• Uncertainty: We use a Monte-Carlo simulation
• Randomly generate a set of process times
• Solve the optimization model
• Analyze the average of the results
• Requires solving the model several hundred times
• Requires data on the process times!
10/15
Processing times (minutes)
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Entity
Doctor 1
n
51
Mean
17.6
Sample Error
1.2
95% Conf. Int.
[15.7,19.3]
Doctor 2
11
12.5
2.3
[8.3,14.3]
Visual Fields
60
25.9
1.0
[24.2,27.6]
11/15
Distribution fitting
Mathematical
modeling
Example:
Glaucoma
clinic
• Doctor 1: logistic distribution
Future work
• Doctor 2: exponential distribution
• VF: Inverse gaussian
Thanks to CDR David Ruth, Ph.D. for statistical help!
12/15
Mathematical
modeling
Example:
Glaucoma
clinic
So how can the model help us?
• Analyze tradeoff in appointment slot sizes
Future work
• Analyze resource allocation questions
• Analyze the impact of variance on scheduling
• Optimal “average processing time” schedules
13/15
Slot size tradeoffs
Mathematical
modeling
Example:
Glaucoma
clinic
Future work
Doctor 1:
Objective
Pts/day
10
19.2 ± 0.1
15
17.2 ± 0.2
20
16.8 ± 0.2
Wait time
11.5 ± 3.4
10.6 ± 3.5
9.8 ± 3.2
Utilization
80% ± 0.5%
68% ± 0.7%
66% ± 0.9%
Doctor 2:
Objective
Pts/day
10
20.0 ± 0.04
15
17.4 ± 0.2
20
12.9 ± 0.2
Wait time
12.1 ± 3.4
7.7 ± 5.7
6.1 ± 2.3
Utilization
65% ± 2.2%
43% ± 1.1%
24% ± 1.4%
14/15
Mathematical
modeling
Enhancing the model
• Add-on patients?
Example:
Glaucoma
clinic
Future work
• Doctors sharing patients
• Data limitations - Doctor 2 with only 11 observations
• Patient differences? SPEC versus PROC versus EST?
Even more specific?
15/15
Discussion:
Mathematical
modeling
Example:
Glaucoma
clinic
What are priorities for you all?
• Do you need to analyze the impact of
adding/subtracting resources (extra doctor, VF)?
Future work
• Do you want a dedicated tool to help with real-time
scheduling?
• Do you want strategic insights into scheduling with
uncertainty to help guide the current scheduling
process?
• Do you want to better understand the current
uncertainty in the model?
• What are your priorities with respect to idle times,
patient wait times, number of patients seen, and
overtime? Are there other objectives of interest?