stochastic models for operating rooms planning mehdi lamiri, xiaolan xie, alexandre dolgui and...
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Stochastic Modelsfor Operating Rooms Planning
Mehdi LAMIRI, Xiaolan XIE, Alexandre DOLGUI and Frédéric GRIMAUD
Centre Génie Industriel et Informatique
Centre Ingénierie et Santé
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Outline
• Motivation & Problem description
• Basic model
• Monte Carlo optimization method
• Model extensions
• A column generation approach
• Conclusions and perspectives
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Problem description: Motivations
Operating rooms represent one of the most expensive sector of the hospital
Involves coordination of large number of resources
Must deal with random demand for emergent surgery and unplanned activities
Planning and scheduling operating rooms’ has become one of the major priorities of hospitals for reducing cost and improving service quality
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Problem description
How to plan elective cases when the operating rooms capacity is shared between two patients classes : elective and emergent patients
Elective patients :
Electives cases can be delayed and planned for future dates
Emergent patients :
Emergent cases arrive randomly and have to be performed in the day of arrival
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Outline
• Motivation & Problem description
• Basic model
• Monte Carlo optimization method
• Model extensions
• A column generation approach
• Conclusions and perspectives
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Basic model: operating rooms capacity
We consider the planning of a set of elective surgery cases over an horizon of H periods under uncertain demand for emergency surgery
Only aggregated capacity of the operating rooms is considered.
Tt : total operating rooms’ regular capacity
Ex: with a bloc of 5 operating rooms opened 10h each in day 1, T1 = 50 h
Exceeding the regular capacity generates overtime costs (COt)
T2
T1
1 H
TH
2
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Basic model: Emergent patients
In this work, the OR capacity needed for emergency cases for a period t is assumed to be a random variable ( wt ) based on:
- The distribution of the number of emergent patients in a given period estimated using information systems and / or by operating rooms’ manager
- The distribution of the OR time needed for emergent surgeries estimated from the historical data
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Basic model: Elective cases
At the beginning of the horizon, there are N requests for elective surgery
A plan that specifies the subset of elective cases to be performed in each period under the consideration of uncertain demand for emergency surgery
T2
T1
1 H
TH
2
Case 5
Case 12
Case 10
Case1
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Basic model: Elective cases
Each elective case i ( 1…N ) has the following characteristics :
Operating Room Time needed for performing the case i : (pi)
Estimated using information systems and/or surgeons’ expertises
A release period (Bi)
It represents hospitalisation date, date of medial test delivery
A set of costs CEit ( t = Bi …H, H+1 )
The CEit represents the cost of performing elective case i in period t
CEi,H+1 : cost of not performing case i in the current plan
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Basic model: Elective cases related cost
The cost structure is fairly general. It can represent many situations :
Hospitalization costs / Penalties for waiting time
Patient’s or surgeon’s preferences
Eventual deadlines
1 Bi H
CEit
t
1 Bi H
CEit
t
1 Bi H
CEit
t Li
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Basic model: Mathematical Model
Unplanned activities time
Planned activities time
Regular capacity
Subject to:
H
ttt
N
i
H
Btitit OCOXCEXJJ
i 11
1* Minimize
t
N
iititWt TXpWEO
t1
11
H
Btit
i
X Ni ,...,1
1 ,0itX Ni , ... ,1 1, ... ,1 Ht
(P)
Ht , ... ,1 overtime
Patient related cost Overtime cost
Decision:
-Assign case i to period t, Xit = 1
-Reject case i from plan, Xi,H+1 = 1
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Basic model: Problem complexity
The planning optimization problem is a stochastic combinatorial problem
The stochastic planning problem is strongly NP-hard The resolution time increases exponentially as the size of the problem increases
The problem is too difficult to be solved exactly within a reasonable amount of time
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Outline
• Motivation & Problem description
• Basic model
• Monte Carlo optimization method
• Model extensions
• A column generation approach
• Conclusions and perspectives
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Monte Carlo optimization or Sample Average Algorithm
Step 1. Generate randomly K different scenarios of emergency cases
Capacity needs for emergency cases
Day 1
W1 = EXP(2h)
Day 2
W2 = EXP(2h)
Day 3
W3 = EXP(2h)
Day 4
W4 = EXP(2h)
Day 5
W5 = EXP(3h)
Senario 1 2.5 1.2 0.9 3.4 4.4
Senario 2 1.4 1.8 2.3 2.1 3.5
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Monte Carlo optimization or Sample Average Algorithm
Step 2 : Estimate all performance functions according to these senarios
K
k
t
N
i
itiktt
N
i
ititWt TXpWK
TXpWEO t
1 1
,
1
1
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Monte Carlo optimization or Sample Average Algorithm
Step 3 : Solving the optimisation problems using the estimated performance functions to obtain the sample optimum solution, also called Monte Carlo optimal solution
For our problem, the Monte Carlo optimization problem can be formulated as a mixed integer program
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Monte Carlo optimization method
Subject to:
H
ttt
N
i
H
BtititKWKW OCOXCEXJJ
i 11
1
,*
, Minimize
11
H
Btit
i
X Ni ,...,1
1 ,0itX Ni , ... ,1 1, ... ,1 Ht
(1’)(P’)
(2’)
(4’)
(5’)
Ht , ... ,1 t
N
iititktk TXpWO
1Kk ,...,1
K
OT
O
K
ktk
t
1 Ht , ... ,1
0tkO(6’)Ht , ... ,1 Kk ,...,1
(3’)
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Monte Carlo optimization or Sample Average Algorithm
Step 4 : Estimate the true criterion value for the Monte Carlo optimal solution
For our problem, it is estimated with a VERY large number K’ of senarios
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Algorithm
Step 1. Generate for each time period K samples of Wt
Step 2. Formulate the Monte Carlo optimization problem (P')
Step 3. Solve the mixed integer program (P') (with CPLEX in our case)
Step 4. Evaluate the true criterion J(X*w,k ) of the resulting
Monte Carlo optimal solution X*w,k
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Convergence : Why it works?
Theorem:
(a) limK Jk(X*w,K) = J(X*), where JK(X*
w,K) is the “estimated” optimal criterion of the Monte Carlo optimization problem (P’) and J(X*) the “true” optimal criterion value
(b) N > 0, J(X*w,k) = J(X*), K > N, i.e. the Monte Carlo
optimal solution X*w,k becomes a true optimal solution.
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Convergence : why it works well?
(Exponential convergence) :
P(X*w,k is not a true optimum) exp(- ck) where X*
w,k is the Monte Carlo optimal solution.
Note : The sample criterion Jw,k(X) converges to the true criterion Jw,k(X) at rate
(Optimal convergence rate) :
The convergence rate c is maximized by the common random variable scheme used in the Monte Carlo optimization
Remark: better convergence than independent evaluation of different solutions X.
1/ k
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Computation experiments: Comments
Solutions provided by our optimization method are better than those of the deterministic method, even for small values of K (K=5)
The proposed method achieves cost reduction of 4% with K=1000, comparing to the deterministic method.
Disadvantage :
The computation time increases beyond acceptable limit as the number of elective cases N > 60.
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Outline
• Motivation & Problem description
• Basic model
• Monte Carlo optimization method
• Model extensions
• A column generation approach
• Conclusions and perspectives
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Model extensions: Multiple operating rooms
Tts : regular capacity of OR-day (s, t)
Wts : capacity needs for emergency cases in OR-day (s, t) (r.v.)
CEits : cost of assigning case i to OR-day (s, t)
COts : Overtime cost of OR-day (s, t)
Decision variables:
Xits = 1 if case i is assigned to OR-day (s, t)
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Model extensions: Overtime capacity and under utilization cost
We introduce an additional penalty cost when the overtime capacity is exceeded
Operating Room related cost
regular capacity
overtime capacity
OR workload
under use
ovetime cost overtime
capacity exceeded
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Extended model: Mathematical Model
overtime
Unplanned activities time
Planned activities time
Regular capacity
Patient related cost Overtime cost
(P) *J = Min 1 1
1 1 1i i
N M H M H
its its ts ts
i s t B s t B
J X CE X CO O
subject to :
1
ts
N
ts W ts i its ts
i
O E W p X T
1
1
1i
M H
it
s t B
X
Xits {0, 1}
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Extended model: solution methods
The Monte Carlo optimization method can be easily extended to solve the extended model
The computation time quickly goes beyond acceptable limit as the number of cases increases
Various methods such as Lagrangian relaxation and column generation have been tested.
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Outline
• Motivation & Problem description
• Basic model
• Monte Carlo optimization method
• Model extensions
• A column generation approach
• Conclusions and perspectives
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Plan for an OR-day
A “plan” is a possible assignment of patients to a particular OR-day
p : plan for a particular OR-day is defined as follows
aip = 1 if case i is in plan p
btsp = 1 if plan p is assigned to OR-day (s, t)
Cost of the plan :
, ,p ip tsp its tsp ts ts i ip ts
i t s t s i
C a b CE b CO E W p a T
Costs related to patients
assigned to the palnOvertime cost in the OR-day
related to the plan
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( ) *p p
p
J Min J Y const c Y
1
1
0 1
{ },
ip pp
tsp pp
p
a Y , i
b Y , t ,s
Y , p
Subject to:
Column formulation for the planning problem
Each OR-day receives at most one plan
Each patient is assigned at most to one selected plan
: set of all possible plans
Yp = 1, if plan p is selected and Yp = 0, otherwise
Master problem
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Master Problem
Linear master problem (LMP)
Optimal solution of the LMP
Near-optimal solution
Solution Methodology
Solve by Column Generati
on
Construct a
“good” feasible solution
Relax the integrality constraint
s
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Solving the linear master problem
simplex multipliers
i , t s
reduced cost < 0add new column
Y
N STOP
*
p pp
const c Y
1
1
0
*
*
ip pp
tsp pp
*p
a Y , i
b Y , t ,s
Y p
,
st
min
Reduced Linear Master Problem
over Ω* Ω
,p i ip ts tsp
i t s
c a b min
Pricing problem
minimizes reduced cost
0,ip tsp ia b t B
,
1tspt s
b , 0,1ip tspa b
st
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The pricing problem
The pricing problem can be decomposed into H×M sub-problems
One sub-problem for each OR-day
min ( ) its i ip ts ts i ip ts tsi i
CE a CO E W p a T
0,1 ,ipa i Subject to:
Simplex multipliers
Dynamic programming method
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Master Problem
Linear master problem (LMP)
Optimal solution of the LMP
Near-optimal solution
Solution Methodology
Solve by Column Generati
on
Construct a
“good” feasible solution
Relax the integrality constraint
s
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Constructing a near optimal solution
Step 1: Determine the corresponding patient assignment matrix (Xits) from the solution (Yp) of The Relaxed Master Problem.
Step 2: Derive a feasible solution starting from (Xits)
Step 3: Improve the solution obtained in Step 2
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Derive a feasible solution
Method I : Solving the integer master problem MP by restricting to generated columns
Method II : Complete Reassignment
Fix assignment of cases in plans with Yp = 1
Reassign myopically but optimally all other cases one by one by taking into account scheduled cases.
Method III : Progressive reassignment
Reassign each case to one OR-day by taking into account the current assignment (Xits) of all other cases, fractional or not.
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Improvement of a feasible schedule
Heuristic 1 : Local optimization of elective cases.
Reassign at each iteration the case that leads to largest improvement
Heuristic 2 : Pair-wise exchange of elective cases (EX)
Heuristic 3 : Period-based reoptimization (PB)
Re-optimize the planning of all cases assigned to a given OR-day (s, t) and all rejected cases.
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Overview of the optimization methods
Method Deriving a feasible solution Improving solutions
M1 CPLEX IP M2 Complete reassign Local opt M3 Progressive reassign Local opt M4 Progressive reassign Local opt, Period-based M5 Progressive reassign Exchange, Period-based M6 Progressive reassign Exchange, Local opt, Period-based M7
column generation
CPLEX LP
+ Dynamic
Programing Progressive reassign Local opt, Period-based, Exchange
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Computation results
The lower bound of the Column generation is very tight
Solving the integer master problem with generated columns can be very poor and it very time consuming
Progressive reassignment outperforms the complete reassignment as progressive reassignment preserves the solution structure of the column generation solution
Numerical results show that the proposed solution methods are satisfying, and the best one can successfully solve the planning problem of about 240 interventions in 12 ORs within 10 minutes while promising the gap less than 0.6%.
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Outline
• Motivation & Problem description
• Basic model
• Monte Carlo optimization method
• Model extensions
• A column generation approach
• Conclusions and perspectives
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Conclusions and perspectives
The proposed model can represent many real world constraints
Monte Carlo simulation and MIP method provide good solutions
Column generation is an efficient technique for providing provably good solutions in reasonable time for large problem.
Perspectives
Make the stochastic model realistic enough to take into account random operating times, ...
Develop exact algorithms able to solve problems with large size
Test with field data