Models and Optimization for A Hospital System

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Models and Optimization for A Hospital System

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Title: 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
IntroductionCosts, 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, CDCs 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 90s)
  • 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 3.2/22.4/22.8
    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 HMOs

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
51
Topics
  • A stochastic dynamic programming model for flow
    control
  • Switching curves, the optimal solution
  • Examples

52
A Stochastic Dynamic Programming Model for Flow
Control
53
Two-Clinic Model
Rejection
x1
Admitting
?1
Switching
x2
?2
Figure 4
54
Notations
  • A type-j patient is one from region j
  • w unit waiting cost
  • Cj switching cost for a type-j patient
  • Rj rejection cost for type-j patient
  • ? discount rate
  • (x1, x2) state of the system
  • V(x1, x2) the total expected discounted cost
    starting right after a transition with the state
    (x1, x2)

55
The Optimal Equation
  • Setting we have

56
Finite Time Cost Functions
  • Define the expected discounted cost of the first
    n transitions recursively as follows

and
57
Convergence
  • With standard dynamic programming arguments, we
    have

(1)
58
Theorem 1 The Properties of Vn(x1, x2)
  • (i) is increasing in x1 and decreasing in x2

(2)
(3)
59
Theorem 1 (continues)
  • is supermodular

(4)
60
Theorem 1 (continues)
  • (iii) is component-wise convex

(5)
(6)
61
The Proof of Theorem 1
  • Theorem 1 is true for
  • Using induction, we only need to prove Theorem 1
    is true for assuming it is true for

62
The Switching Curves
63
Finite Time Switching Curves
  • Lemma 1 Suppose Theorem 1 holds for Vn(x1,x2).
    There exist two functions
  • such that
  • is decreasing in x2
  • (2) is increasing in x2 and

64
Lemma 1 (continues)
  • (3) To minimize Vn1 (x1, x2),
  • it is optimal to
  • (i) reject a type-1 patient
  • if
  • (ii) refer a type-1 patient to clinic 2
  • if
  • (iii) accept a type-1 patient to clinic 1
  • if

65
Proof of Lemma 1
  • Let
  • It is optimal to reject a type-1 patient at state
    (x1,x2) if and only if
  • Let
  • or
  • if the set on the RHD of (8) is empty.

(7)
(8)
66
Proof of Lemma 1 (continues)
  • Since is component-wise convex, fn is
    increasing in both x1 and x2.
  • as given by (8) is thus well defined
    and is decreasing in x2 due to the
    supermodularity of
  • Since x1 gt is equivalent to fn gt 0,
  • is indeed the threshold for rejecting the
    type-1 patient.
  • (ii) and (iii) can be proven similarly

67
Discussion
  • Lemma 1 provides the switching curves for type-1
    patient in finite time.
  • Similar switching curves can be constructed for
    type-2 patients
  • Properties similar to those in Theorem 1 exist
    for V(x1,x2), so we have
  • Theorem 2 for type-1 patients and similar results
    for type-2 patients

68
Theorem 2
  • Let
  • (1) is decreasing in x2
  • (2) is increasing in x2
  • and

(9)
(10)
69
Theorem 2 (continues)
  • (3) It is optimal at (x1,x2) to
  • (i) reject a type-1 patient
  • if
  • (ii) refer a type-1 patient to clinic 2
  • if
  • (iii) accept a type-1 patient to clinic 1
  • if

(11)
(12)
(13)
70
Examples
71
Example 1
  • Let
  • the arrival rates from regions 1 and 2 be
  • the service rates at clinics 1 and 2 be
  • The cost parameters are
  • w 10.0 C1 C2 20.0 R1 R2 200.0
  • Set the discount factor to 0.01

72
Example 1 (continues)
  • The optimal control policy for type-1 patients is
    described by

Rejection thresholds
Decreasing in x2
and
Increasing in x2
Acceptance thresholds
73
Example 1 Switching Curves for Type-1 Patients
Figure 5
74
Example 1 (continues)
  • The optimal control policy for type-2 patients is
    described by

Rejection thresholds
and
Acceptance thresholds
75
Example 1 Switching Curves for Type-2 Patients
Figure 6
76
Example 1 Performance
  • 4.06 of type-1 and 10.85 of type-2 patients are
    rejected
  • 8.6 of type-1 patients are referred to clinic 2,
    and 6.66 of type-2 patients are referred to
    clinic 1.
  • The effective traffic intensities are
  • For clinic 1 and clinic 2, respectively
  • The average sojourn times are
  • 40 and 25.64
  • The average queue lengths are
  • 39 and 75.92

77
Example 2
  • The second example has the following parameters
  • w 20.0, U 3.4
  • R1R2200.00, and C1C220.0

78
Example 2 Switching Curves for Type-1 Patients
Figure 7
79
Example 2 Performance
  • 4.6 of type-1 and 4.5 of type-2 patients are
    rejected.
  • 18.6 of type-1 patients are referred to clinic
    2, and 12.2 of type-2 patients are referred to
    clinic 1.
  • The effective traffic intensities are
  • For clinic 1 and clinic 2, respectively
  • The average sojourn times are
  • 4.567 and 3.1423
  • The average queue lengths are
  • 4.5981 and 4.7738

80
Discussion
  • Without switching sojourn times in example 2
    become 6.25 and 4.1667
  • Different allocation
  • in example 2, then sojourn times becomes
    (without switching) 5.5617 and 4.5413
  • How do we find an optimal solution consider both
    flow control and capacity allocation?
  • Can we extend the flow control to multi-node and
    how?

81
Questions
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