Models and Optimization for A Hospital System

1. Models and Optimization for A Hospital System LIU Liming IEEM/HKUST Presented at Ecole Nationale Superieure des Mines de Saint-Etienne July 2005

2. Introduction:Costs, Resources, and Quality in Healthcare

3. Long Waiting Times • Two types of waiting • Wait at the clinics for treatments • Wait for appointments/scheduled operations • Long type 1 waiting time, a sign of poor service quality • Long type 2 waiting time, a sign of crises in public healthcare services • Under funding?

4. 1998 National Healthcare Expenditures/GDP Source: OECD and HKHA • 14.2% in USA • 10.5% Germany • 9.2% in Canada • 7.2% in Japan • 6.9% in UK • More than 5% in Hong Kong The challenge: growth outpaces GDP growth

5. Healthcare Characteristics • Technology intensive: drugs, equipments • Highly trained personnel • 8 to 12 years of post secondary education & training • Demand intensive • Aids epidemic, about 36.1 million people living with aids or HIV infection worldwide at the end of 2000. • Hepatitis virus carriers in Hong Kong account for more than 1/3 of the total population

6. Complex Operations • The operations of a hospital system are very complicated involving: • multiple hospitals and clinics • specialists, nurses, and technicians • treatments equipments and testing facilities • operation rooms, recovery rooms, and inpatient wards for overnight pre and post treatment stays

7. Equity in Resource Allocation • Equity is a fundamental principle for healthcare policy makers • In US, CDC’s HIV prevention budget was allocated to states according to number of aids cases reported • Hong Kong Hospital Authority uses a population-based allocation scheme

8. The Waiting Times at HKHA Specialist Clinics • 16 specialty clinics throughout Hong Kong • Social and economic development changes the distribution of population • Maximum clinic-wide average waiting time was 9.5 times of the overall system-wide average waiting time and 40+ times of the best average waiting time (in the mid 90’s) • This motivated this research

9. A 3-Clinic System A system with 3 M/M/1 servers Region 2 Completion Region 1 Clinic 2 Clinic 1 Region 3 Completion Clinic 3 Completion Figure 1

10. Proportional Allocation • Demand rates • Total capacity U = 5 • Exact proportional capacity allocation

11. Different Performances • Average sojourn time at server 2 or 3 • Average sojourn times at server 1 • Total system average sojourn time = 3

12. Patients Can Switch to the Clinic with a Shorter Waiting Time Region 2 Completion Region 1 Clinic 2 Clinic 1 Patient flows Region 3 Completion Clinic 3 Completion Figure 2

13. Outside of the System Rejection Region 2 Region 1 Completion Clinic 2 Clinic 1 Rejection Patient flows Region 3 Completion Clinic 3 Completion Rejection Figure 3

14. Effects and Implications • Shorter waiting time for patients who switch • Shorter average waiting time • Persistent inter-regional patient flow makes it difficult to assess accurately the performance and capacity requirements of the regional service center • Overall system performance may deteriorate • Resources can be saved to serve the rest of the population better • Need to study the trade-off

15. Research Issues • Hospital system administrators are often concerned with two challenging issues - Patient flow management - Capacity allocation

16. A Multi-Site Service Network Model • N clinics and N regions • Poisson demand from region j, • Exponential service time at clinic j, • Identical services in all clinics and switching is allowed/feasible • A patient is serviced at only one clinic • Available capacity U,

17. Bibliography • Healthcare is a major area for OR: Anh and Hornberger, MS 1996 (organ allocation); So and Tang, MS 2000 (reimbursement); Liu and Liu, IIE Trans. 1998 (appointment); Zon and Kommer, Healthcare MS 1999 & McKillop, U of Waterloo 1997 (patient flows); • Flow control: Hajek, IEEE Trans. Control 1984; Xu & Shanthikumar, OR 1993; Naor, Econometrica 1969 • General: Ross (Stochastic Processes 1983, Dynamic Programming 1983); Topkis, OR 1978; Glasserman and Yao (Monotone Structure in Discrete Event Systems) 1994 • Mingpao Daily Feb. 25, 2001 • Joint works with Chao, Shang, and Zheng

18. Part 1: Capacity/Resource Allocation

19. Topics • Problem formulation • Extreme cases • General allocation policy structure • Mathematical formulation and solution

20. Problem Formulation

21. Patient Behavior • Patients prefer shorter waiting time • It is more convenient to receive service from the “home” service center • Patients are not homogenous • Some will switch server for shorter waiting time • Some will choose to stay regardless

22. Demand Model • Poisson demand stream from a region • total rate at region • stay • shopping for shortest queue • Once joined a service center, a customer stays until served

23. The Decision Problem Given the total available capacity, what is the optimal allocation so that the total average waiting time is minimized ?

24. Constraints • The total “switching rate” • We must have

25. Extreme Cases

26. No Switching: Formulation • That is • The optimization problem becomes

27. No Switching: Optimal Policy • THEOREM: Optimal allocation follows a square root rule

28. Effective Use of Resources • Example: N =3, U =5, • Optimal allocation • Average waiting time at station 1 = 2.4049 • Average waiting times at 2 and 3 = 3.2 (+33%) • Total average waiting time = 3.2/2+2.4/2=2.8 comparing to 3 for proportional allocation

29. Complete Switching • That is and no constraints • THEOREM: The optimal solution is to build one facility and allocation all budget there, that is Region 1 has the largest original demand rate

30. A Conclusion Simple proportional allocation is not optimal

31. General Policy Structure

32. By Intuition --- • What is the structure of the optimal allocation policy ? • ANSWER: “ONE BIG, many small” • For hospital management • One large system-wide hospital located at the region with largest population • Many satellite hospitals to meet the minimum needs of individual regions

33. Implications • Resource allocation according to proportional rule may be politically easier to implement, but is not cost-effective • It is more cost-effective to have more imbalanced allocation • Cost-effective allocation may serve equity better

34. Mathematical Derivation Non-convex NP

35. The Demand Allocation • Question: For fixed , how do “switching customers” choose station for service ? • Let be the “switching rate” to station j • The total average waiting time

36. The First Optimization • Switching customers’ optimization problem

37. Solution • This is a convex programming • By Kuhn-Tucker condition, the optimal solution is where is the unique solution of

38. Infeasible Solution • Without constraint , the solution is • What if for some j

39. Implications • Such infeasible solution implies that for a given is already too high • In other words, if an allocation leads to an infeasible switching solution, it is not a good design • The optimal allocation is in the set for which all corresponding switching rates are non-negative

40. The Capacity Allocation • Consider allocation such that for all j • We seek allocation to minimize waiting time

41. The Second Optimization • The optimal allocation problem becomes Non-convex with a non-convex constraint set

42. A Qualitative Property • THEOREM: For any two feasible allocations and if i.e., a majorization ordering, then

43. Quality or Equality? • Ignore the constraints • The best solution is • The worst solution is

44. How to Find the Optimal Solution? • Kuhn-Tucker condition does not seem to help because: • Solution obtained is local optimum • Can be either local minimum or maximum • It is more likely a local maximum -- bad solution • We have a difficult problem by the standard method

45. The Optimal Allocation Rule THEOREM: Let . We have where

46. Remarks • The first facility with the largest initial traffic has the largest allocation . • All the switch traffic goes to the first facility • Each facility is first allocated a capacity equals its total demand rate • The remaining capacity is then allocated according to a square-root rule (ratio)

47. Conclusions • Resources allocation based on proportionality is not optimal, even though it is often politically easier to implement • The optimal allocation policy for minimizing waiting time has the structure of one big and many small • This structure can be observed in practice in healthcare industry, such as some HMO’s

48. Obligations and Efficiency • The small units are used to fulfill the basic obligations of the service provider • While the large center represents the effort to run the system efficiently for the benefit of the customers as well as the provider • The optimal allocation is also better for equity (in terms of waiting time)

49. Questions

50. Part 2: Flow Control