Column Generation Approach for Operating Rooms Planning

1. Column Generation Approach for Operating Rooms Planning Mehdi LAMIRI, Xiaolan XIE and ZHANG Shuguang Industrial Engineering and Computer Sciences Division Engineering and Health Division

2. Outline • Motivation & Problem description • Problem modelling • A column generation approach • Computational results • Conclusions and perspectives

3. Problem description: Motivations • Operating rooms represent one of the most expensive sectors 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

4. 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

5. Outline • Motivation & Problem description • Problem modelling • A column generation approach • Computational results • Conclusions and perspectives

6. T2S T11 T1S T21 TH1 THS … … OR 1 OR S OR 1 OR S OR 1 OR S … … … 1 1 2 2 H H Model: Operating rooms capacities • We consider the planning of a set of elective surgery cases over an horizon of H periods (days) • In each period there are S operating rooms • For each OR-day (s, t) we have a regular capacityTts • Exceeding the regular capacity generates overtime costs (COts)

7. Model: Emergent patients • In this work, the OR capacity needed for emergency cases in OR-day (s, t) is assumed to be a random variable ( wt s ) 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 emergency surgeries estimated from the historical data

8. 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 OR-day under the consideration of uncertain demand for emergency surgery T1,2 TH,S T1,1 Case1 Case 12 Case 5 Case 10 OR-day (S, H) OR-day (1, 1) OR-day (2, 1)

9. 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 CEits ( t =Bi …H, H+1 ) • The CEitsrepresents the cost of performing elective case i in period t in OR s • CEi,H+1: cost of not performing case i in the current plan

10. Model: Elective cases related costs • 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 • ORs availabilities

11. Model : Example of planning T12 Case 2 Case 12 Case 32 T1,1 Case 16 Case 21 Case 16 Case 25 Case 10 Case 1 Case 27 Case 9 Case 1 Case 14 Case 7 Case 20 Case 5 Case 33 Case 2 OR-day (S, H) OR-day (1, 1) • Overtime costs • Cases related costs The plan must minimizes the sum of elective patient related costs and the expected overtime costs

12. Overtime cost Patient related cost Unplanned activities time Planned activities time Regular capacity overtime Mathematical Model • Decision: • Assign case i to OR-day (s, t) , Xits = 1 • Reject case i from plan, Xi,H+1,s = 1

13. Outline • Motivation & Problem description • Problem modelling • A column generation approach • Computational results • Conclusions and perspectives

14. Plan for one 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 : Costs related to patients assigned to the paln Overtime cost in the OR-day related to the plan

15. Column formulation for the planning problem •  : set of all possible plans • Yp = 1, if plan p is selected and Yp = 0, otherwise Master problem Subject to: Each patient is assigned at most to one selected plan Each OR-day receives at most one plan

16. Master problem • The master problem is an integer linear programming problem, whereas the initial formulation has a nonlinear objective: • The nonlinear quantities (expected overtime costs) are now imbedded into the columns costs • The master problem has a huge number of variables (columns) Subject to:

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

18. Linear Master problem (LMP) • The Linear Master Problem (LMP) is the same as the master problem MP except that the integrity of Yp is relaxed. • Problem LMP provides a lower bound of the master problem and hence a lower bound of the original problem. • Problem LMP can be solved by the column generation technique

19. Solving the linear master problem simplex multipliers pi , pt s Pricing problem minimizes reduced cost Reduced Linear Master Problem over Ω* Í Ω min min st st reduced cost < 0 Y add new column N STOP

20. The pricing problem • The pricing problem can be decomposed into H×S subproblems • One sub-problem for each OR-day Simplex multipliers Subject to:

21. The pricing sub-problem • The pricing sub-problem is a stochastic knapsack problem: • The capacity of the sack is a random variable • There is a penalty cost if the capacity is exceeded Subject to: Dynamic programming method

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

23. Constructing a near optimal solution • Step 1: Determine the corresponding patient assignment matrix (Xits) from the solution (Yp) of the Linear Master Problem. • Step 2: Derive a feasible solution starting from (Xits) • Step 3: Improve the solution obtained in Step 2

24. Derive a feasible solution • Method I : Solving the integer master problem MP by restricting to generated columns • Method II : Complete Reassignment Fix assignments 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.

25. Improvement of a feasible schedule • Heuristic 1 : Local optimization of elective cases. • Reassign at each iteration the case that leads to the largest improvement • Heuristic 2 : Pair-wise exchange of elective cases • Exchange the assignment of a couple of cases that leads to the largest improvement • Heuristic 3 : Period-based re-optimization • Re-optimize the planning of all cases assigned to a given OR-day (s, t) and all rejected cases.

26. Overview of the optimization methods

27. Outline • Motivation & Problem description • Problem modelling • A column generation approach • Computational results • Conclusions and perspectives

28. Computational experiments Problem instances generation • Number of periods : H = 5 • Number of operating rooms: S = 3, 6, 9, 12 • OR-day’s regular capacity : Tts= 8 hours • Capacity needed for emergency cases : Wtsis exponentially distributed with a mean of 3 hours • Overtime cost : COts = 500 € / hour • Duration of elective surgeries : pi are randomly generated from the interval [0.5, 3 hours] • Release periods : Bi are randomly generated from the set {1…H}

29. Computational experiments Problem instances generation • Patients related costs : CEits = (t- Bi) x c • c is set equal to 150 € (hospitalisation cost) • Case 1: Identical ORs • Case 2: Non-Identical ORs • ORs are equally allocated to 3 specialties, and an extra charge of 100€ is added for cases assigned to another speciality’s ORs • The number of elective cases is determined such that the workload of ORs due to elective cases is 85% of the regular capacity of the entire planning horizon.

30. Duality Gap M1 M2 M3 M4 M5 M6 M7 R=0 Nb Rooms=3 (60 cases) 9.95% 3.35% 0.86% 0.82% 0.59% 0.58% 0.50% Nb Rooms=6 (119.3 cases) 10.59% 2.22% 1.01% 0.94% 0.64% 0.62% 0.53% Nb Rooms=9 (180.9 cases) --- 1.94% 0.83% 0.74% 0.45% 0.44% 0.31% Nb Rooms=12 (239.6 cases) --- 2.30% 1.02% 0.93% 0.60% 0.59% 0.47% R=100 Nb Rooms=3 (60 cases) 0.15% 0.26% 0.14% 0.14% 0.17% 0.08% 0.05% Nb Rooms=6 (119.3 cases) 0.27% 2.39% 0.40% 0.40% 0.28% 0.24% 0.16% Nb Rooms=9 (180.9 cases) 1.54% 1.75% 0.64% 0.60% 0.31% 0.30% 0.22% Nb Rooms=12 (239.6 cases) --- 1.85% 0.64% 0.62% 0.31% 0.31% 0.19% Computational experiments: Gap Results based on 10 randomly generated instances: the average Gap Case R = 0: Identical ORs Case R = 100: extra charge of 100 € for assigning cases to another specialty's ORs

31. Computation Time M1 M2 M3 M4 M5 M6 M7 R=0 Nb Rooms=3 (60 cases) 49,3 8,7 8,2 8,6 8,2 8,8 8,7 Nb Rooms=6 (119.3 cases) >5000 53,3 51,7 52,79 53,5 53,4 54 Nb Rooms=9 (180.9 cases) --- 184 182,1 183,6 182,3 184 186,2 Nb Rooms=12 (239.6 cases) --- 436,7 438,1 433,6 435 433,8 438,8 R=100 Nb Rooms=3 (60 cases) 9,6 9,8 9,3 10,8 9,5 9,9 9,9 Nb Rooms=6 (119.3 cases) 120 71,7 70,6 77,8 72,7 72,3 72,1 Nb Rooms=9 (180.9 cases) >5000 245,7 244,9 259,4 251 246,7 247,5 Nb Rooms=12 (239.6 cases) --- 569,5 572,2 593,5 584,1 570,2 590,1 Computational experiments: computation time Results based on 10 randomly generated instances: the average Computation time (second) • Over 65% of the computation for CGP is spent on pricing problems.

32. 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 is very time consuming • Progressive reassignment outperforms the complete reassignment as progressive reassignment preserves the solution structure of the column generation solution

33. Outline • Motivation & Problem description • Problem modelling • A column generation approach • Computational results • Conclusions and perspectives

34. Model extension: Overtime capacity and under utilization cost • We introduce an additional penalty cost when the overtime capacity is exceeded Operating Room related cost overtime cost overtime capacity exceeded under use OR workload regular capacity overtime capacity

35. Model extension: Overtime capacity and under utilization cost • OR utilization cost:

36. Conclusions and perspectives • The proposed model can represent many real world constraints • Column generation is an efficient technique for providing provably good solutions in reasonable time for large problem. Future work • Make the stochastic model realistic enough to take into account random operating times, ... • Take into account other criteria such as reliability of OR plans • Develop exact algorithms able to solve problems with large size • Test with field data