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. - 2 - ORAHS 06, Wroclaw Outline Motivation & Problem description Problem modelling A column generation approach Computational results Conclusions and perspectives

3. - 3 - ORAHS 06, Wroclaw Problem description: Motivations n Operating rooms represent one of the most expensive sectors 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. - 4 - ORAHS 06, Wroclaw 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. - 5 - ORAHS 06, Wroclaw Outline Motivation & Problem description Problem modelling A column generation approach Computational results Conclusions and perspectives

6. - 6 - ORAHS 06, Wroclaw Model: Operating rooms capacities n We consider the planning of a set of elective surgery cases over an horizon of H periods (days) n In each period there are S operating rooms n For each OR-day (s, t) we have a regular capacity T ts n Exceeding the regular capacity generates overtime costs (CO ts ) 12H12H OR 1 OR S … OR 1 OR S … OR 1 OR S … … T 11 T 1S T 21 T 2S T HS T H1

7. - 7 - ORAHS 06, Wroclaw Model: Emergent patients n In this work, the OR capacity needed for emergency cases in OR-day (s, t) is assumed to be a random variable ( w t 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. - 8 - ORAHS 06, Wroclaw 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 OR-day under the consideration of uncertain demand for emergency surgery T 1,2 T 1,1 OR-day (1, 1) OR-day (S, H) T H,S OR-day (2, 1) Case 5 Case 12 Case 10 Case1

9. - 9 - ORAHS 06, Wroclaw 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 its ( t = B i …H, H+1 )  The CE its represents the cost of performing elective case i in period t in OR s  CE i,H+1 : cost of not performing case i in the current plan

10. - 10 - ORAHS 06, Wroclaw Model: Elective cases related costs 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 ORs availabilities

11. - 11 - ORAHS 06, Wroclaw Model : Example of planning T 12 T 1,1 Case 5 Case 1 Case 10 Case 16 Case 7 Case 9 Case 12 Case 21 OR-day (1, 1) OR-day (S, H) Case 2 Case 16 Case 14 Case 20 Case 1 Case 25 Case 33 Case 2 Case 32 Case 27 The plan must minimizes the sum of elective patient related costs and the expected overtime costs n Overtime costs n Cases related costs

12. - 12 - ORAHS 06, Wroclaw Mathematical Model Unplanned activities time Planned activities time Regular capacity overtime Patient related cost Overtime cost Decision: - Assign case i to OR-day (s, t), X its = 1 - Reject case i from plan, X i,H+1,s = 1

13. - 13 - ORAHS 06, Wroclaw Outline Motivation & Problem description Problem modelling A column generation approach Computational results Conclusions and perspectives

14. - 14 - ORAHS 06, Wroclaw Plan for one 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

15. - 15 - ORAHS 06, Wroclaw 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

16. - 16 - ORAHS 06, Wroclaw 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 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 n The master problem has a huge number of variables (columns) Master problem

17. - 17 - ORAHS 06, Wroclaw 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

18. - 18 - ORAHS 06, Wroclaw Linear Master problem (LMP) n The Linear Master Problem (LMP) is the same as the master problem MP except that the integrity of Y p is relaxed. n Problem LMP provides a lower bound of the master problem and hence a lower bound of the original problem. n Problem LMP can be solved by the column generation technique

19. - 19 - ORAHS 06, Wroclaw 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

20. - 20 - ORAHS 06, Wroclaw The pricing problem n The pricing problem can be decomposed into H×S subproblems One sub-problem for each OR-day Subject to: Simplex multipliers

21. - 21 - ORAHS 06, Wroclaw The pricing problem Subject to:

22. - 22 - ORAHS 06, Wroclaw The pricing problem n 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

23. - 23 - ORAHS 06, Wroclaw 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

24. - 24 - ORAHS 06, Wroclaw Constructing a near optimal solution n Step 1: Determine the corresponding patient assignment matrix (X its ) from the solution (Y p ) of the Linear Master Problem. n Step 2: Derive a feasible solution starting from (X its ) n Step 3: Improve the solution obtained in Step 2

25. - 25 - ORAHS 06, Wroclaw Derive a feasible solution n Method I : Solving the integer master problem MP by restricting to generated columns n 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. 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.

26. - 26 - ORAHS 06, Wroclaw Improvement of a feasible schedule n Heuristic 1 : Local optimization of elective cases. Reassign at each iteration the case that leads to the largest improvement n Heuristic 2 : Pair-wise exchange of elective cases Exchange the assignment of a couple of cases that leads to the largest improvement n 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.

27. - 27 - ORAHS 06, Wroclaw Overview of the optimization methods

28. - 28 - ORAHS 06, Wroclaw Outline Motivation & Problem description Problem modelling A column generation approach Computational results Conclusions and perspectives

29. - 29 - ORAHS 06, Wroclaw Computational experiments Problem instances generation Number of periods : H = 5 Number of operating rooms: S = 3, 6, 9, 12 OR-day’s regular capacity : T ts = 8 hours Capacity needed for emergency cases : W ts is exponentially distributed with a mean of 3 hours Overtime cost : CO ts = 500 € / hour Duration of elective surgeries : p i are randomly generated from the interval [0.5, 3 hours] Release periods : B i are randomly generated from the set {1…H}

30. - 30 - ORAHS 06, Wroclaw Computational experiments Problem instances generation n Patients related costs : CE its = (t- B i ) x c  c is set equal to 150 € (hospitalisation cost) n Case 1: Identical ORs n 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 n 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.

31. - 31 - ORAHS 06, Wroclaw Computational experiments: Gap Duality Gap M1M2M3M4M5M6M7 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% 0.17%0.08%0.05% Nb Rooms=6 (119.3 cases) 0.27%2.39%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% 0.19% Case R = 0: Identical ORs Case R = 100: extra charge of 100 € for assigning cases to another specialty's ORs Results based on 10 randomly generated instances: the average Gap

32. - 32 - ORAHS 06, Wroclaw Computation Time M1M2M3M4M5M6M7 R=0 Nb Rooms=3 (60 cases) 49,38,78,28,68,28,88,7 Nb Rooms=6 (119.3 cases) >500053,351,752,7953,553,454 Nb Rooms=9 (180.9 cases) ---184182,1183,6182,3184186,2 Nb Rooms=12 (239.6 cases) ---436,7438,1433,6435433,8438,8 R=100 Nb Rooms=3 (60 cases) 9,69,89,310,89,59,9 Nb Rooms=6 (119.3 cases) 12071,770,677,872,772,372,1 Nb Rooms=9 (180.9 cases) >5000245,7244,9259,4251246,7247,5 Nb Rooms=12 (239.6 cases) ---569,5572,2593,5584,1570,2590,1 Computational experiments: computation time Over 65% of the computation for CGP is spent on pricing problems. Results based on 10 randomly generated instances: the average Computation time (second)

33. - 33 - ORAHS 06, Wroclaw 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 is very time consuming n Progressive reassignment outperforms the complete reassignment as progressive reassignment preserves the solution structure of the column generation solution

34. - 34 - ORAHS 06, Wroclaw Outline Motivation & Problem description Problem modelling A column generation approach Computational results Conclusions and perspectives

35. - 35 - ORAHS 06, Wroclaw Model extension: 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 overtime cost overtime capacity exceeded

36. - 36 - ORAHS 06, Wroclaw Conclusions and perspectives n The proposed model can represent many real world constraints n Column generation is an efficient technique for providing provably good solutions in reasonable time for large problem. Future work n Make the stochastic model realistic enough to take into account random operating times,... n Take into account other criteria such as reliability of OR plans n Develop exact algorithms able to solve problems with large size n Test with field data