1. A TWO-LEVEL RESOLUTION APPROACH FOR
THE STOCHASTIC OR PLANNING PROBLEM
Elena Tànfani, Angela Testi
Department of Economics and Quantitative Methods (DIEM)
University of Genova (Italy)
Rene Alvarez
Centre for Research in Healthcare Engineering
Department of Mechanical and Industrial Engineering
University of Toronto (Canada)
ORAHS 2010 – Genova, Italy
2. Outline
• The problem addressed
• Modelling approach:
• First level deterministic model
• Second level stochastic problem and individual chance constraints
• Robust solutions & safety slacks
• log normal case
• Montecarlo simulation
• Application to a case study: preliminary results
• Conclusions and further work
ORAHS 2010 – Genova, Italy
3. Outline
• The problem addressed
• Modelling approach:
• First level deterministic model
• Second level stochastic problem
• Robust solutions & safety slacks
• log normal case
• Montecarlo simulation
• Application to a case study: preliminary results
• Conclusions and further work
ORAHS 2010 – Genova, Italy
4. Problem addressed
• We deal with the Operating Rooms (ORs) planning
problem
• We focus our attention on hospital surgery
departments made up of:
• n surgical specialties
• m ORs
• a given planning horizon (usually a week)
ORAHS 2010 – Genova, Italy
5. Assumptions
1. Demand greater than capacity in the planning
horizon
2. Block scheduling system
3. Block times could not be split among specialties
4. Emergency patients use dedicated urgent surgery
rooms
ORAHS 2010 – Genova, Italy
6. Operating Rooms Planning & Scheduling
CMPP (Case Mix Planning Problem)
Available OR capacity (block times) should be split among different surgical specialties
MSSP (Master Surgical Schedule Problem)
Each specialty is assigned to a particular block time during the planning horizon
SCAP (Surgical Case Assignment Problem)
Sub-sets of patients are assigned to each block time and sequenced
ORAHS 2010 – Genova, Italy
7. Outline
• The problem addressed
• Modelling approach:
• First level deterministic model
• Second level stochastic problem
• Robust solutions & safety slacks
• log normal case
• Montecarlo simulation
• Application to a case study: preliminary results
• Conclusions and further work
ORAHS 2010 – Genova, Italy
8. CMMP model
• Q: How many block times to each specialty?
• The solution to the CMPP is determined by means of a
Mini-Max programming model
• Objective: leveling the resulting weighted waiting list of the
specialties belonging to the department
• Point of view: hospital
• The solution to the CMPP, is used as input of the MSSP
and SCAP model (demand constraints)
ORAHS 2010 – Genova, Italy
9. A Mini-Max Model for the CMPP (1)
Minimize max{( hw − sw y w ) β w }
w∈W
∑y
w
w =Q
yw ≥ l w ∀w
yw ≤ u w ∀w
yw ≥ yw+1 ∀w
y w ≥ 0, int
Where
yw are the integer variables (#of block times assigned to specialty w )
hw is the waiting list length of specialty w
sw is the average service rate of specialty w
βw is the average urgency coefficient of patients belonging to specialty w
Q total number of block times available in the planning horizon
lw, uw lower and upper bound on the number of block times to specialty w
ORAHS 2010 – Genova, Italy
10. A Mini-Max MIP Model for the CMPP (2)
Minimize σ
∑y
w
w =Q
yw ≥ l w ∀w
yw ≤ u w ∀w
yw ≥ yw+1 ∀w
y w ≥ 0, int
σ w = ( hw − sw yw ) βw ∀w
σ w ≤ σ ∀w
σ w ,σ ≥ 0
ORAHS 2010 – Genova, Italy
11. MMSP & SCAP model
• Q: Which day of the week are block times assigned
to each sub-specialty?
• Q: Which patients are assigned to each block time?
• The MSSP & SCAP model is formulated as a Chance
constrained stochastic model
• Objective: minimizing the weighted waiting time of admitted and
still waiting patients
• Point of view: patients
ORAHS 2010 – Genova, Italy
12. A 0-1 programming model for MSSP&SCAP
• Variables:
⎧1 if patient i is assigned to OR k on day t
xikt = ⎨
⎩0 otherwise
⎧1 if specialty w is assigned to OR k on day t
y wkt =⎨
⎩0 otherwise
ORAHS 2010 – Genova, Italy
13. MSSP&SCAP: Deterministic version
n c b n
Min ∑∑∑ x ikt (t + d i ) β i + ∑ [(1 − xikt )( b + 1 + d i ) β i ]
i =1 k =1 t =1 i =1
c b
∑∑ x
k =1 t =1
ikt ≤ 1 ∀i = 1, 2,...,n
c
∑∑ ∑ xikt = 0 ∀h = 1, 2,...,5
i∈I h k =1 t∈Th
∑x
i∈I w
ikt − Pywkt ≤ 0 ∀k = 1, 2,...,c;∀t = 1, 2,...,b;∀w = 1, 2,...,m
m
∑y
w=1
wkt =1 ∀k = 1, 2,...,c; ∀t = 1, 2,...,b
c
∑y
k =1
wkt ≤ ewt ∀t = 1, 2,...,b; ∀w = 1, 2,...,m
c b
∑∑ ywkt ≤ yw ∀w = 1, 2,...,m; (Yw= solution of CMPP)
k =1 t =1
n
∑xikt pi ≤ qkt ∀k = 1, 2,...,c; ∀t = 1, 2,...,b
i=1
x ikt ∈ {0,1} y wkt ∈ {0,1}
ORAHS 2010 – Genova, Italy
14. Outline
• The problem addressed
• Modelling approach:
• First level deterministic model
• Second level stochastic problem
• Robust solutions & safety slacks
• log normal case
• Montecarlo simulation
• Application to a case study: preliminary results
• Conclusions and further work
ORAHS 2010 – Genova, Italy
15. Deterministic versus stochastic model
• The solutions of the deterministic model are feasible with
respect to each OR block length constraints, i.e. no
overtime will occur
• What will happen if random durations are introduced?
ORAHS 2010 – Genova, Italy
16. Deterministic versus stochastic model
n
∑x
i =1
p ≤ qkt
ikt i ∀k = 1, 2,...,c;∀t = 1, 2,...,b
n
∑x
i =1
ξ ≤ qkt
ikt i ∀k = 1, 2,...,c; ∀t = 1, 2,...,b
where
qkt Is the length of OR block k in day t
pi Is the Expected Operating Time (EOT) of patient i
ξi Is the stochastic operating time of each patient i with a
mean expected duration μi and standard deviation σi
ORAHS 2010 – Genova, Italy
17. The stochastic problem
• The aim is to make decisions feasible with an high
probability level
⎛n ⎞
Ρ⎜ ∑xiktξi ≤ qkt ⎟ ≥ (1 − p*)
⎝ i=1 ⎠
p* = allowable overtime probability for each block
ORAHS 2010 – Genova, Italy
18. Outline
• The problem addressed
• Modelling approach:
• First level deterministic model
• Second level stochastic problem
• Robust solutions & safety slacks
• log normal case
• Montecarlo simulation
• Application to a case study: preliminary results
• Conclusions and further work
ORAHS 2010 – Genova, Italy
19. For those bloks where the probability of overtime is
greater than (1‐p*), we calculate safety slacks to be
used in a new run of the DETERMINISTIC MODEL
Deterministic model Slack time
(BASE SOLUTION) ITERATION (STOCHASTIC SOLUTION)
STOP CRITERIUM – ROBUST SOLUTION
ORAHS 2010 – Genova, Italy
20. Outline
• The problem addressed
• Modelling approach:
• First level deterministic model
• Second level stochastic problem
• Robust solutions & safety slacks
• log normal case
• Montecarlo simulation
• Application to a case study: preliminary results
• Conclusions and further work
ORAHS 2010 – Genova, Italy
22. Outline
• The problem addressed
• Modelling approach:
• First level deterministic model
• Second level stochastic problem
• Robust solutions & safety slacks
• log normal case
• Montecarlo simulation
• Application to a case study: preliminary results
• Conclusions and further work
ORAHS 2010 – Genova, Italy
23. Montecarlo simulation (other distributions)
1. Randomly generate N samples of operating time for each patient i :
εi1, εi2, … εin,…, εiN
2. Using the xikt values of the first level deterministic problem, compute
the duration of the schedule for each block time kt and simulation
run r
n
Lkt = ∑ ε ir xikt
r
i =1
3. Calculate the (1-α) percentile point of the series for each block time
kt
4. If this (1-α) percentile point is greater than qkt, a safety slack time
δkt is calculated
μψ kt + z1−α σψ kt
e − qkt = δ kt
ORAHS 2010 – Genova, Italy
24. Outline
• The problem addressed
• Modelling approach:
• First level deterministic model
• Second level stochastic problem
• Robust solutions & safety slacks
• (log normal case)
• Montecarlo simulation (others)
• Application to a case study: preliminary results
• Conclusions and further work
ORAHS 2010 – Genova, Italy
25. Case study
• Data from Surgery Department, San Martino Public
Hospital, Genova
• 6 Surgical subspecialties (SS)
• # teams available varying between 0 and 3
• Lower and upper bound varying between 2 and 12
• 400 patients on the waiting lists
• 6 ORs
• 6 Hours - block length
• 5 Days planning horizon
• Q=30 block times
ORAHS 2010 – Genova, Italy
26. Case study
Surgery Department, San Martino Public Hospital, Genova
ORAHS 2010 – Genova, Italy
27. Operating times distributions
• Operating times are EOT Mean Standard
assumed to follow Group Deviation
LOGNORMAL distributions
1 1.5 0.3
2 2.0 0.5
3 2.5 0.7
4 3.0 0.9
5 3.5 1.0
6 4.0 1.1
ORAHS 2010 – Genova, Italy
30. Robust final solution
Monday Thursday Wednesday Thursday Friday 75 patients
SS 3
1. xxx [A1-2-3]
SS 5
1. xxx [A2-1-2]
SS 1
1. xxx [A1-2-3]
SS 1
1. xxx [A2-2-3]
SS 5
1. xxx [A1-6-3]
have been
OR1 2. xxx [A1-6-2] 2. xxx [A2-6-2]
3. xxx [A2-2-1.5]
2. xxx [A2-2-2] 2. xxx [A2-2-2.5] 2. xxx [B-6-2.5] scheduled.
SS 2 SS 3 SS 2 SS 1 SS 1
OR2
1. xxx [A1-1-3]
2. xxx [A1-6-1.5]
1. xxx [A2-6-2]
2. xxx [A2-3-1.5]
1. xxx [A2-2-2]
2. xxx [A2-6-2]
1. xxx [A2-6-2]
2. xxx [B-1-1.5]
1. xxx [A2-6-3]
2. xxx [B-6-1.5]
11 patients
3. xxx [A2-6-1.5] 3. xxx [B-2-1.5] 3. xxx [B-2-1.5] less than the
SS 2
1. xxx [A1-6-4]
SS 1
1. xxx [A1-2-3]
SS 1
1. xxx [A2-1-1.5]
SS 2
1. xxx [A2-6-2.5]
SS 2
1. xxx [A2-1-3.5]
starting base
OR3 2. xxx [A2-5-1.5] 2. xxx [A2-4-1.5] 2. xxx [A2-2-1.5]
3. xxx [A2-6-1.5]
2. xxx [B-2-1.5]
3. xxx [B-2-1.5]
2. xxx [B-1-1.5] solution.
SS 6 SS 2 SS 3 SS 3 SS 1
1. xxx [A1-6-3] 1. xxx [A2-2-2] 1. xxx [A2-2-2] 1. xxx [A2-2-2.5] 1. xxx [A2-6-2.5]
OR4 2. xxx [A2-6-2] 2. xxx [A2-2-1.5] 2. xxx [B-6-1.5] 2. xxx [A2-1-1.5] 2. xxx [B-6-1.5]
3. xxx [A2-2-1.5] 3. xxx [B-6-1.5] 3. xxx [A2-2-1.5] 3. xxx [B-6-1.5]
SS 4 SS 1 SS 4 SS 2 SS 3
1. xxx [A1-6-3] 1. xxx [A1-6-2] 1. xxx [A2-6-2] 1. xxx [A2-2-2.5] 1. xxx [A2-6-3]
OR5 2. xxx [A2-1-2] 2. xxx [A2-6-2] 2. xxx [A2-6-1.5] 2. xxx [A2-6-2.5] 2. xxx [A2-6-2.5]
3. xxx [A2-2-1.5] 3. xxx [B-2-1.5]
SS 1 SS 2 SS 6 SS 1 SS 1
1. xxx [A1-6-3] 1. xxx [A1-2-2] 1. xxx [A2-3-2.5] 1. xxx [A2-1-2] 1. xxx [A2-1-3]
OR6 2. xxx [A1-1-1.5] 2. xxx [A2-3-1.5] 2. xxx [A2-2-1.5] 2. xxx [A2-6-1.5] 2. xxx [B-1-2]
3. xxx [A2-6-1.5] 3. xxx [B-2-1.5] 3. xxx [B-2-1.5]
ORAHS 2010 – Genova, Italy
31. Outline
• The problem addressed
• Modelling approach:
• First level deterministic model
• Second level stochastic problem
• Robust solutions & safety slacks
• (log normal case)
• Montecarlo simulation (others)
• Application to a case study: preliminary results
• Conclusions and further work
ORAHS 2010 – Genova, Italy
32. Conclusions
• When assuming random surgery times, an optimal
deterministic solution could potentially violate the capacity
constraints of the problem
• Our proposal is to improve the deterministic optimal
solution in a way that the probability of having overtime is
minimized by including safety slack times for each OR block
combination
• The proposed framework can, therefore, satisfy the main
concern of healthcare managers and be more easily
accepted with respect to a deterministic solution that
doesn’t verify the effect of realization of random variables
pertaining surgery duration
ORAHS 2010 – Genova, Italy
33. Future work
• It is necessary to perform an extensive
computational experimentation aimed at showing
the model convergence under different real life
instances
• Some future work could test local search
metaheuristic approaches to find deterministic
solutions to be used in the second level stochastic
model
ORAHS 2010 – Genova, Italy
34. A TWO-LEVEL RESOLUTION APPROACH FOR
THE STOCHASTIC OR PLANNING PROBLEM
Elena Tànfani, Angela Testi
Department of Economics and Quantitative Methods (DIEM)
University of Genova (Italy)
Rene Alvarez
Centre for Research in Healthcare Engineering
Department of Mechanical and Industrial Engineering
University of Toronto (Canada)
ORAHS 2010 – Genova, Italy
