Stochastic Models for Operating Rooms Planning Mehdi LAMIRI, Xiaolan XIE, Alexandre DOLGUI and Frédéric GRIMAUD Centre Génie Industriel et Informatique Centre Ingénierie et Santé

- 2 - Outline Motivation & Problem description Basic model Monte Carlo optimization method Model extensions A column generation approach Conclusions and perspectives

- 3 - Problem description: Motivations n Operating rooms represent one of the most expensive sector of the hospital n Involves coordination of large number of resources n Must deal with random demand for emergent surgery and unplanned activities n Planning and scheduling operating rooms’ has become one of the major priorities of hospitals for reducing cost and improving service quality

- 4 - Problem description n How to plan elective cases when the operating rooms capacity is shared between two patients classes : elective and emergent patients n Elective patients : Electives cases can be delayed and planned for future dates n Emergent patients : Emergent cases arrive randomly and have to be performed in the day of arrival

- 5 - Outline Motivation & Problem description Basic model Monte Carlo optimization method Model extensions A column generation approach Conclusions and perspectives

- 6 - Basic model: operating rooms capacity n We consider the planning of a set of elective surgery cases over an horizon of H periods under uncertain demand for emergency surgery n Only aggregated capacity of the operating rooms is considered. T t : total operating rooms’ regular capacity Ex: with a bloc of 5 operating rooms opened 10h each in day 1, T 1 = 50 h n Exceeding the regular capacity generates overtime costs (CO t ) T2T2 T1T1 1 H THTH 2

- 7 - Basic model: Emergent patients n In this work, the OR capacity needed for emergency cases for a period t is assumed to be a random variable ( w t ) 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

- 8 - Basic model: Elective cases n 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 T2T2 T1T1 1 H THTH 2 Case 5 Case 12 Case 10 Case1

- 9 - Basic model: Elective cases n Each elective case i ( 1…N ) has the following characteristics : Operating Room Time needed for performing the case i : (p i )  Estimated using information systems and/or surgeons’ expertises A release period (B i )  It represents hospitalisation date, date of medial test delivery A set of costs CE it ( t = B i …H, H+1 )  The CE it represents the cost of performing elective case i in period t  CE i,H+1 : cost of not performing case i in the current plan

Basic model: Elective cases related cost n 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 Bi H CE it t 1 Bi Bi H t 1 Bi Bi H t Li Li

Basic model: Mathematical Model Unplanned activities time Planned activities time Regular capacity Subject to: (P) overtime Patient related costOvertime cost Decision: -Assign case i to period t, Xit = 1 -Reject case i from plan, X i,H+1 = 1

Basic model: Problem complexity n The planning optimization problem is a stochastic combinatorial problem n 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

Outline Motivation & Problem description Basic model Monte Carlo optimization method Model extensions A column generation approach Conclusions and perspectives

Monte Carlo optimization or Sample Average Algorithm n Step 1. Generate randomly K different scenarios of emergency cases Capacity needs for emergency cases Day 1 W 1 = EXP(2h) Day 2 W 2 = EXP(2h) Day 3 W 3 = EXP(2h) Day 4 W 4 = EXP(2h) Day 5 W 5 = EXP(3h) Senario Senario

Monte Carlo optimization or Sample Average Algorithm n Step 2 : Estimate all performance functions according to these senarios

Monte Carlo optimization or Sample Average Algorithm n 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

Monte Carlo optimization method Subject to: (1’)(P’) (2’) (4’) (5’) (6’) (3’)

Monte Carlo optimization or Sample Average Algorithm n 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

Algorithm Step 1. Generate for each time period K samples of W t 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

Convergence : Why it works? Theorem: (a) lim K  J k (X * w,K ) = J(X * ), where J K (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.

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 J w,k (X) converges to the true criterion J w,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.

Computation experiments: Comments n Solutions provided by our optimization method are better than those of the deterministic method, even for small values of K (K=5) n The proposed method achieves cost reduction of 4% with K=1000, comparing to the deterministic method. n Disadvantage : The computation time increases beyond acceptable limit as the number of elective cases N > 60.

Outline Motivation & Problem description Basic model Monte Carlo optimization method Model extensions A column generation approach Conclusions and perspectives

Model extensions: Multiple operating rooms n T ts : regular capacity of OR-day (s, t) n W ts : capacity needs for emergency cases in OR-day (s, t) (r.v.) n CE its : cost of assigning case i to OR-day (s, t) n CO ts : Overtime cost of OR-day (s, t) n Decision variables: X its = 1 if case i is assigned to OR-day (s, t)

Model extensions: Overtime capacity and under utilization cost n 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

Extended model: Mathematical Model overtime Unplanned activities time Planned activities time Regular capacity Patient related costOvertime cost

Extended model: solution methods n The Monte Carlo optimization method can be easily extended to solve the extended model n The computation time quickly goes beyond acceptable limit as the number of cases increases n Various methods such as Lagrangian relaxation and column generation have been tested.

Outline Motivation & Problem description Basic model Monte Carlo optimization method Model extensions A column generation approach Conclusions and perspectives

Plan for an OR-day n A “plan” is a possible assignment of patients to a particular OR-day n p : plan for a particular OR-day is defined as follows n a ip = 1 if case i is in plan p n b tsp = 1 if plan p is assigned to OR-day (s, t) n Cost of the plan : Costs related to patients assigned to the paln Overtime cost in the OR-day related to the plan

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 n  : set of all possible plans n Y p = 1, if plan p is selected and Y p = 0, otherwise Master problem

Master Problem Linear master problem (LMP) Optimal solution of the LMP Near-optimal solution Solution Methodology Solve by Column Generation Construct a “good” feasible solution Relax the integrality constraints

Solving the linear master problem simplex multipliers  i,  t s reduced cost < 0 add new column Y N STOP st min Reduced Linear Master Problem over Ω*  Ω min Pricing problem minimizes reduced cost st

The pricing problem n The pricing problem can be decomposed into H×M sub- problems One sub-problem for each OR-day Subject to: Simplex multipliers Dynamic programming method

Master Problem Linear master problem (LMP) Optimal solution of the LMP Near-optimal solution Solution Methodology Solve by Column Generation Construct a “good” feasible solution Relax the integrality constraints

Constructing a near optimal solution n Step 1: Determine the corresponding patient assignment matrix (X its ) from the solution (Y p ) of The Relaxed Master Problem. n Step 2: Derive a feasible solution starting from (X its ) n Step 3: Improve the solution obtained in Step 2

Derive a feasible solution n Method I : Solving the integer master problem MP by restricting to generated columns n 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. n Method III : Progressive reassignment Reassign each case to one OR-day by taking into account the current assignment (X its ) of all other cases, fractional or not.

Improvement of a feasible schedule n Heuristic 1 : Local optimization of elective cases. Reassign at each iteration the case that leads to largest improvement n Heuristic 2 : Pair-wise exchange of elective cases (EX) n 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.

Overview of the optimization methods

Computation results n The lower bound of the Column generation is very tight n Solving the integer master problem with generated columns can be very poor and it very time consuming n Progressive reassignment outperforms the complete reassignment as progressive reassignment preserves the solution structure of the column generation solution n 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%.

Outline Motivation & Problem description Basic model Monte Carlo optimization method Model extensions A column generation approach Conclusions and perspectives

Conclusions and perspectives n The proposed model can represent many real world constraints n Monte Carlo simulation and MIP method provide good solutions n Column generation is an efficient technique for providing provably good solutions in reasonable time for large problem. Perspectives n Make the stochastic model realistic enough to take into account random operating times,... n Develop exact algorithms able to solve problems with large size n Test with field data