Post on 03-May-2020
Optimization of On-line Appointment Scheduling
Brian DentonEdward P. Fitts Department of Industrial and Systems Engineering
North Carolina State University
Tsinghua University, Beijing, China
May, 2012
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 1 / 40
Acknowledgements
Ayca Erdogan, School of Medicine, Stanford University
Alex Gose, NC State University
Supported by National Science Foundation: CMMI Service EnterpriseSystems Grant 0620573
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 2 / 40
Appointment Scheduling Systems
Interface between healthcareproviders and patients
Arises in many healthcarecontexts
Primary careRadiation OncologySurgeryOutpatient ProceduresChemotherapy
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 3 / 40
Scheduling Challenges
Competing criteria
Patient waiting time
Provider idle time andovertime
Complicating Factors
Uncertain service durations
Uncertain patient demandNo-showsUrgent Add-ons
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 4 / 40
Research Questions
Given a probabilistic arrival process for customer appointment requeststo a single server, in which appointments must be quoted on-line:
What is the structure of the optimal appointment schedule?How can problems be classified into easy and hard?How important is it to find optimal schedules?
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 5 / 40
Presentation Outline
Introduction
Problems
Static Appointment Scheduling
Dynamic Appointment Scheduling
Dynamic Appointment Sequencing and Scheduling
Conclusions
Other Research
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 6 / 40
Static Appointment Scheduling Problem
Problem: Schedule n customers with uncertain service times during afixed length of day, d
x1 x2 x3 x4 x5
Idling (s)
Planned Available Time (d)
Overtime (l) Waiting (w)
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 7 / 40
Common Heuristics
Mean Service Times:
a1 = 0ai = ai−1 + µi−1, ∀i
Hedging:
a1 = 0ai = ai−1 + µi−1 + κσi−1, ∀i
Ho, C., H. Lau. 1992. Minimizing Total Cost in Scheduling Outpatient Appointments,Management Science 38(12).
Cayirli, T., E. Veral. 2003. Outpatient Scheduling in Health Care: A Review ofLiterature, Production and Operations Management 12.
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 8 / 40
Literature Review
Queuing Analysis
Bailey and Welch (1952)
Jansson (1966)
Sabria and Daganzo (1989)
Heuristics
White and Pike (1964)
Soriano (1966)
Ho and Lau (1992)
Optimization
Weiss (1990)
Wang (1993)
Denton and Gupta (2003)
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 9 / 40
Two-Stage Stochastic Linear Program
First stage decisions
xi : Time allowance for customer i
Second stage decisions
wi(ω): Customer i waiting time`(ω): Server overtime w.r.t. length of session d
Random service durations:
Zi(ω): Random service time for customer i
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 10 / 40
Model Formulation
min Eω[n∑
i=2
cwi wi(ω) + c``(ω)]
s.t . w2(ω) ≥Z1(ω)− x1, ∀ω− w2(ω) + w3(ω) ≥Z2(ω)− x2,∀ω
. . . . . ....
− wn−1(ω) + wn(ω) ≥Zn−1(ω)− xn−1, ∀ω
− wn(ω) + `(ω) ≥Zn(ω) +n−1∑i=1
xi − d ,∀ω
x ≥ 0, w(ω), `(ω) ≥ 0,∀ω
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 11 / 40
Example: 6 Customers
Denton, B.T. and Gupta D., 2003, “A Sequential Bounding Approach for OptimalAppointment Scheduling,” IIE Transactions, 35, 1003-1016
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 12 / 40
Dynamic Appointment Scheduling
Problem: Up to nU customers are scheduled dynamically as theyrequest appointments. Appointment requests are probabilistic.
C1
C1 C2
C1 C2 C3
C1 C2 C3 C4
C1 C2 C3 C4 C5
C2
C5
C4
C3
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 13 / 40
Multi-stage Stochastic Program
Appointment requests are defined by a multi-stage scenario tree:
2
2
nU
nU-1
nU-1
1-q3
q3
1-qnu
qnu
nU
1
3
1-q4
3
q4
minx1{(1−q3)Q2(x1)+min
x2{q3(1−q4)Q3(x2)+· · ·+ min
xnU−1
{(nU∏i=3
)(qi )QnU (xnU−1)} · · · }}
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 14 / 40
Model Formulation: Stage j
Qj (xj , ωj ) = minw,`{
j+1∑i=2
cwi wj,i (ωj ) + c``j+1(ωj )}
s.t wj,2(ωj ) ≥ Z1(ωj )− x1
−wj,2(ωj ) + wj,3(ωj ) ≥ Z2(ωj )− x2
. . . . . ....
−wj,j (ωj ) + wj,j+1(ωj ) ≥ Zj (ωj )− xj
−wj,j+1(ωj ) + `j+1(ωj ) ≥ Zj+1(ωj ) +
j∑i=1
xi − d
wj,i (ωj ) ≥ 0 ∀i , `j (ωj ) ≥ 0.
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 15 / 40
Model Properties
Motivation for first come first serve (FCFS) appointment sequence:
Proposition
For nU = 2 with i.i.d. service durations, and identical waiting costs, theoptimal sequence is FCFS.
Counter-examples exist for non i.i.d. and nonidentical waiting costs.
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 16 / 40
Solution Methods
Variants of nested decomposition:
Fast-forward-fast-back implementation
Multi-cut method
2 variable method for master problems
Valid inequalities based on relaxations of the mean value problem
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 17 / 40
Outer Linearization
Outerlinearize the recourse function:
min{θ | θ ≥ Q(x)}
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 18 / 40
Methodology: Nested Decomposition Method
2 1- q3
1- q4
2
3 3
nU
X1
XnU
nU-1
XnU-1
nU-1
1- qnu
nU
1
X2
XnU-1
XnU
q3
q4
qnu -1
Forward
X1
X2
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 19 / 40
Methodology: Nested Decomposition Method
2 1- q3
1- q4
2
3 3
nU nU-1
nU-1
1- qnu
-1
nU
1
q3
Add optimality cut
Add Optimality cut
Add Optimality cut
q4
qnu
-1
Backward
Add Optimality cut
[ ( )]E h TxπωΘ≥ −
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 19 / 40
Multi-Cut Method
Separate cuts from master problems and subproblems (similar tomulti-cut approach proposed by Birge and Louveaux (1985)
2 1- q3 q
1- q3
2
3
3
nU
nU
1
Xq3
Cut 1
qnu -1
Cut 2
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 20 / 40
Two-variable LPs
Master problems at each stage are two-variable LPs (xj and θj )
αjxj + θj ≥ β − (α1x1 + α2x2 + . . .+ αj−1xj−1)
Solve LPs with a modified version of the algorithm proposed by Dyer(1984)
21
intersect
x
43
intersect
x
medianxintersect
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 21 / 40
Valid Inequalities
Proposition
The optimal solution to the mean value problem is x̄i = µi , ∀i .
Constraints based on mean value problem
θj ≥Qj(x , ξ̄)
Similar to valid inequalities proposed by Batun et al. (2011)
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 22 / 40
Solution Methods
Several adaptations of nested decomposition were compared:
Standard nested decomposition (ND)Multi-cut NDTwo-variable NDND with mean value valid inequalities (VI)
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 23 / 40
Comparisons of Methods
Number of Iterations CPU Time (seconds)
nU = 10 nU = 20 nU = 30 nU = 10 nU = 20 nU = 30(d=200) (d=400) (d=600) (d=200) (d=400) (d=600)
ND 244 432 438 3.42 23.26 49.68c`
cw = 10 Multi-cut ND 186 244 202 2.63 13.52 23.21Two-variable ND 254 406 362 3.56 24.06 43.59
ND with VIs 232 370 442 3.65 20.83 51.79ND 192 330 392 2.75 16.77 42.46
c`
cw = 1 Multi-cut ND 106 184 174 1.55 9.81 19.85Two-variable ND 186 290 284 2.54 16.32 31.82
ND with VIs 188 306 364 2.98 16.89 42.50ND 190 302 422 2.55 14.54 43.48
c`
cw = 0.1 Multi-cut ND 96 176 162 1.33 8.79 17.45Two-variable ND 186 290 384 2.37 15.49 42.95
ND with VIs 174 284 412 2.62 14.86 45.70
2 QuadCore Intel R© Xeon R© Processor 2.50GHz CPU, 16GB Ram, CPLEX 11.0
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 24 / 40
Value of Stochastic Solution (VSS)
Table: VSS for test instances with Zi ∼ U(20,40) and qi = 0.5 for add-onrequests.
Number of CustomersVSS (%)
(Routine, Add-on)d = 200
c`
cw = 10 c`
cw = 1 c`
cw = 0.1(0,30) 9.63 65.59 95.15
(10,30) 1.40 19.63 79.41(20,30) 0.50 23.63 80.33
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 25 / 40
Example: Scheduling an Endoscopy Suite
15
17
19
21
23
25
27
29
31
1 2 3 4 5 6 7 8 9 10 11
x i
Patients
12-0 Patients
9-3 Patients
6-6 Patients
3-9 Patients
Figure: Service times based on colonoscopy times for an outpatientendoscopy practice: Zi ∼ Lognormal(23.55,11.89), ∀i .
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 26 / 40
Example: Multi-Procedure Room Endoscopy Practice
Endoscopy Practice:2 intake rooms2 procedure rooms4 recovery roomsService timed based on empirical data
Table: Expected waiting time and overtime according to different schedules
Heuristic Stochastic Program Based Schedulec`
cw = 10 c`
cw = 1 c`
cw = 0.1 c`
cw = 10 c`
cw = 1 c`
cw = 0.1Expected total cost 975.19 111.72 253.71 878.03 104.58 162.65
Expected waiting time 15.78 16.28 10.54 5.06Expected overtime 95.94 86.17 94.05 111.97
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 27 / 40
Dynamic Appointment Sequencing and Scheduling
The appointment request sequence and the appointment arrivalsequence are not necessarily the same.
C1
C1 C2
C1 C2 C3
C1 C2 C3 C4
C1 C2 C3 C4 C5
C2
C5
C4
C3
C1
C2 C1
C2 C3 C1
C2 C4 C3 C1
C2 C4 C3 C5 C1
C2
C5
C4
C3
(A) (B)
Figure: (A) FCFS; (B) Example of the general case.
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 28 / 40
Two Stage Stochastic Integer Program
Minimize {Cost of Indirect Waiting + Eω[Direct Waiting + Overtime]}
First Stage Decisions:
Customer sequencing (binary)Service time allowances (continuous, sequence dependent)Appointment times (continuous, sequence dependent)
Second Stage Decisions:Waiting time (continuous, sequence dependent)Overtime (continuous)
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 29 / 40
Two Stage Stochastic Integer Program
First Stage Decisions:
oj,i,i ′ : binary sequencing variable where ojii ′ = 1 if customer iimmediately precedes i ′ at stage j , and oii ′j = 0 otherwise
xj,i,i ′ : time allowance for customer i given that i immediatelyprecedes i ′ at stage j
aj,i,i ′ : appointment time of customer i ′, given that i immediatelyprecedes i ′ at stage j
Second Stage Decisions:wj,i,i ′(ω) : waiting time of customer i ′ given that customer i
immediately precedes i ′ at stage j under durationscenario ω
sj,i,i ′(ω) : server idle time between customer i and i ′, given that iimmediately precedes i ′ at stage j
`j(ω) : overtime at stage j with respect to session length d
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 30 / 40
First Stage Problem
minn∑
j=1
pj [
j∑i=1
j∑i′=1
cai′aj,i,i′ ] + Q(o, x)
s.t .j+1∑i′=1
oj,i,i′ = 1,j+1∑i′=0
oj,i′,i = 1 ∀j , i = 1,2, . . . , j
j+1∑i=0
j+1∑i=0
oj,i,i′ = j + 1 ∀j
oj,i,j + oj,j,i′ − 2(oj−1,i,i′ − oj,i,i′) ≥ 0 ∀j , ∀i , i ′ < jxj,i,i′ ≤ M1oj,i,i′ , aj,i,i′ ≤ M1oj,i,i′ ∀j , i , i ′
j+1∑i′=1
xj,i,i′ =
j+1∑i′=1
aj,i,i′ −j+1∑i′=1
aj,i′,i ∀j , i
xj,i,i′ , aj,i,i′ ≥ 0, oj,i,i′ ∈ {0,1} ∀j , i , i ′,∀j
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 31 / 40
Second Stage Subproblem
Q(o, x, ω) = min Eω[
j∑i=1
j∑i′=1
(cwi′ wj,i,i′(ω) + c``j (ω)]
s.t .wj,i,i′(ω) ≤ M2(ω)oj,i,i′ ∀i , i ′, j , ωsj,i,i′(ω) ≤ M3(ω)oj,i,i′ ∀i , i ′, j , ω
−j∑
i′=1
wj,i′,i (ω) +
j∑i′=1
wj,i,i′(ω)−j∑
i′=1
sj,i,i′(ω) = Zi (ω)−j∑
i′=1
xj,i,i′ ∀i , j , ω
`j (ω) ≥j∑
i=1
j∑i′=1
sj,i,i′(ω) +
j∑i=1
Zi (ω) +
j∑i′=1
xj,0,i′ − d ∀j , ω
wj,i,i′(ω), sj,i,i′(ω), `j (ω) ≥ 0, ∀j , i , i ′, ω.
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 32 / 40
Model Properties
The addition of indirect waiting costs results in conditions under whichFCFS is not optimal:
Proposition
For nU = 2 with i.i.d. service durations if
ca2 ≥ cw
1
then the optimal sequence is LCFS.
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 33 / 40
Methodology
Compared L-shaped method and Integer L-shaped method
Fast solution to second stage subproblemsPresolveWarm startBranch-and-cut vs. dynamic searchMIP cuts (MIR, implied bound cuts, etc.)Mean value problem based valid inequalities
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 34 / 40
Computational Performance
L-Shaped MethodNo. of Class Type of CPU Time # of Iterations
Customers Customers Average Max Average Max2.1 5 Add on 449 484 192.9 202
5 Customers2.2 3 Routine + 2247.71 2546 608.7 660
2 Add on2.3 7 Add on 15000* 15000* 283 290
7 Customers2.4 4 Routine 15000* 15000* 241 247
3 Add on2.5 10 Add on 15000* 15000* 92 97
10 Customers2.6 7 Routine 15000* 15000* 93 102
3 Add on
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 35 / 40
Computational Performance
Table: Gap at the time of termination for the instances that are not solved to optimality
Problem Instance Patient Best GapSize No Type L-Shaped Method L-Shaped Method
(mean value based cuts)2.3 7 Add on 107.12% optimal
7 Patients(uniform) 2.4 4 Routine 174.62% 1.95%
3 Add on10 2.5 10Add on 240.11% 7.26%
Patients(uniform) 2.6 7 Routine 375.32% 1.99%
3 Add on3.3 7 Add on 223.32% 21.99%
7 Patients(lognormal) 3.4 4 Routine 335.02% 15.71%
3 Add on10 3.5 10Add on 338.37% 31.53%
Patients(lognormal) 3.6 7 Routine 517.307% 13.07%
3 Add on
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 36 / 40
Example 1: Structure of the Optimal Solution
Table: Examples with varying direct/indirect cost for instance 3.6 (7 routine, 3add on, lognormal service times) parameters
Instance ca cw ca cw
No Routine Routine Add-on Add-on cL Optimal Sequence CPU Time # of Iterationsave max ave max
1 0 1 0.1 0.1 10 R-R-R-R-R-R-R-A-A-A 12295.5 14980 55.2 5982 0 1 10 10 10 A-A-A-R-R-R-R-R-R-R 1174.8 1852 163.5 2093 0 1 50 50 10 A-A-A-R-R-R-R-R-R-R 418.2 613 94.9 1224 0 1 100 100 10 A-A-A-R-R-R-R-R-R-R 257.6 522 67.4 1125 0 1 250 250 10 A-A-A-R-R-R-R-R-R-R 117.2 290 36 736 0 1 500 500 10 A-A-A-R-R-R-R-R-R-R 52.5 112 18.1 367 0 1 750 750 10 A-A-A-R-R-R-R-R-R-R 28.1 48 10.3 178 0 1 1000 1000 10 A-A-A-R-R-R-R-R-R-R 19.4 30 7.1 10
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 37 / 40
Conclusions
VSS can be as high as 95% and as low as 0.5%
Large instances of dynamic scheduling problem can be solvedefficiently but sequencing and scheduling is much harder
FCFS generally optimal when probabilities of add-on customersare low and/or indirect cost of waiting is low
Placement of add on customers is frequently “all or nothing”
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 38 / 40
Other Research
Complex service systems withmultiple servers and stages ofservice
Uncertain service time,demand, and patient/providerbehavior
Applications:Hospital surgery practices
Outpatient procedure andtreatment centers
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 39 / 40
Questions?
Brian Dentonbdenton@ncsu.edu
http://www.ise.ncsu.edu/bdenton/
Brian Denton, NC State ISyE ()On-line Appointment Scheduling May, 2012 40 / 40