Long term policies
for operating room planning
A. Agnetis1, A. Coppi1, G. Dellino2, C. Meloni3, M. Pranzo1
1
Dept. of Information Engineering, University of Siena, Italy
2 IMT Institute for Advanced Studies, Lucca, Italy
3 Dept. of Electronics and Electrical Engineering, Polytechnic of Bari, Italy
Outline
•
•
•
•
•
•
Introduction
Problem description
Optimization models and heuristics
Case study
Preliminary results
Conclusions
2
Introduction
• Operating theatre (OT) among the most critical
resources in a hospital
– Significant impact on costs
– Affects quality of service
• Improve the efficiency of the OT management
process
• Focus on operating room (OR) planning
3
Decision problems
in OT management
Mon
Tue
Wed
Thu
Fri
OR1
MSSP
OR2
SCAP
OR3
OR4
ESSP
OR5
OR6
Gynecology
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Urology
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Day____
surgery
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General
____
surgery
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____
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____
Otolaryn____
gology
____
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Orthopedic
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surgery
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4
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Organizational complexity
vs. MSS variation
• Staffing and shift planning
– MSS fixed over time  ↑ stability, ↓ flexibility
– MSS different every week  ↓ stability, ↑ flexibility
5
Main contributions
• Alternative policies proposed to trade off efficient
management of the surgery waiting lists and
organizational complexity
• MSSP and SCAP solved through mathematical
programming models and heuristics
• Performance evaluation over one-year time horizon
• Assumptions:
– Deterministic data
– Elective patients only
6
Problem description (1)
• Input: waiting list of elective patients for each
surgical specialty
– Data for each case surgery in the list:
Waiting list – Day surgery
Surgery
code
Surgery
duration
(min)
Priority
class
Entrance time
Waiting time
(days)
Due date
6210
28
B
15/06/2010
27
15/08/2010
• Output: one-week assignment (Mon-Fri) of elective
surgeries to ORs
7
Problem description (2)
• OR sessions: morning/afternoon/full-day
• Assignment restrictions
• Objectives:
– Max ORs utilization, without overtime
– Schedule case surgeries within their due-date,
reducing patients’ waiting time
based on case surgeries duration and a score, related to:
1. case surgery waiting time and priority class
2. case surgery slack time
8
Optimization models
ILP mathematical formulations, solved by CPLEX
1. Unconstrained MSS model
Determines MSS and SCA every week, based on the
actual waiting list for each specialty
2. Constrained MSS model
Determines MSS and SCA, bounding the number of
changes in OR session assignments to different surgical
specialties w.r.t. a reference MSS
3. Fixed MSS model
Determines SCA, given an MSS
9
Unconstrained MSS model
• ‘Unconstrained’ w.r.t. long-term planning
• Constraints
– Bounds on the number of weekly OR sessions
assigned to a specialty
– Min number of ORs assigned to a specialty every day
(either half-day or full-day sessions)
– Max number of parallel OR sessions for each
specialty
– Restrictions on specialty-to-OR assignments
– Max OR session duration (no overtime)
10
Constrained MSS model
• ‘Constrained’ w.r.t. long-term planning
– Block time = # weeks during which the MSS is fixed
• Set a reference MSS
• Distance (Δ) between two MSSs: # ORs for each day
and session type assigned to different surgical
specialties in the MSSs
• One constraint added to the previous model,
bounding such a distance between the new MSS
and the reference MSS
11
Fixed MSS model
• The MSS has been already determined  OR
sessions already assigned to surgical specialties
• Assignment of case surgeries to OR sessions;
i.e., SCAP is solved
12
Heuristic methods
OR sessions = bins; Surgeries = items
MSSP
1. Candidate OR sessions (half-day/fullday) for each specialty → first-fitdecreasing rule
2. Selection of candidate sessions
assigned to OR
3. MSS is retained, discarding all surgical
cases filling it
SCAP
13
Planning policies
Unconstrained
MSS model
MSSP
SCAP
Exactly
solved
Exactly
solved
Heuristically
solved
Heuristically
solved
Constrained
MSS model
Δ = 1, block = 1
Δ = 2, block = 4
Fixed
MSS model
14
Case study: OT characteristics
Medium-size public Italian hospital (Empoli, Tuscany)
• OT = 6 operating rooms; two ORs are bigger
• 6 surgical specialties: general surgery, otolaryngology,
gynecology, orthopedic surgery, urology, day surgery
• Surgical specialty restrictions
– Gynecology must use the same OR for the whole week
– Orthopedics needs big ORs
– Two parallel OR sessions can be both assigned to general
surgery (the same for orthopedics)
• Further restrictions
– One OR quickly made available, every morning
– One OR free every afternoon
15
Case study: experimental design
• MSSP and SCAP solved every week
• Simulation over one year
• Weekly arrivals:
– nonparametric bootstrapping from the initial waiting
list
– sample size from a uniform distribution centered
around the average weekly arrival rate
• Two scenarios tested: base/stressed scenario
16
Preliminary results
Base scenario
f1
f2
17
Preliminary results
Stressed scenario
f1
f2
18
Preliminary results
• Stability of the MSS
– The unconstrained MSS model has an average
distance between two adjacent MSS of 12-13  20%
of the MSS changes from one week to the next
– Worst case: 67% changes in the MSS
– Trade-off provided by the constrained MSS model
19
Conclusions
• Long-term evaluation of alternative policies for OR
planning of elective surgeries
• Simulation on a real case study (medium-size public
hospital in Italy)
• Promising results to improve waiting lists’
management
• For future research:
– Surgeons and resources availability constraints
– Different objective functions and policies
– Uncertainty on surgery duration and surgery arrivals
20
Unconstrained MSS model
Mathematical formulation
max  Pis K is xisjwz
s
i
j
 x  1 i, s
s, j , w, z
P x  O y
s
 y  y  2 y   S
 y  y  2 y   2 w, j
w, s
 y  y   
 y  y   k w, k , s  S
  y  0 s
w
z
isjwz
j
w
z
is
isjwz
max
z
sjwz
i
sjwm
w
sjwa
sjwd
min
s
j
sjwm
sjwa
sjwd
s
sjwm
sjwd
sm
j
sjwm
sjwd
j
w
z
jNAs
(k )
p
 y  y  2 y   S
 y  1 w, j, z  d
w, s
 y  y   
 y  y   k w, k , s  S
sjwm
w
sjwa
sjwd
s
max
s
j
sjwz
s
sjwa
sjwd
sjwa
sjwd
sa
j
(k )
p
j
sjwz
xisjwz  0,1
i, s, j , w, z
y sjwz  0,1
s, j , w, z
21
Constrained MSS
• ‘Constrained’ w.r.t. long-term planning
– Block time = # weeks during which the MSS is fixed
• Set a reference MSS
• Distance (Δ) between two MSSs: # ORs for each day
and session type assigned to different surgical
specialties in the MSSs
• One constraint added to the previous model,
bounding such a distance between the new MSS
and the reference MSS
y


sjwm
 s , j , w M MSS

 y
sjwa
 s , j , w AMSS

 y
sjwm
 s , j , w DMSS
 ysjwa  2 ysjwd   N MSS  
22
Fixed MSS model
Mathematical formulation
max  Pis K is xish
i
s.t.
h
x
P x
s
ish
h
is ish
i
1
i, s
O
max
hs
xish  0,1
s, h
i, s, h
23