Title: Models and Optimization for A Hospital System
1Models and Optimization for A Hospital System
- LIU Liming
- IEEM/HKUST
- Presented at Ecole Nationale Superieure des Mines
de Saint-Etienne - July 2005
2IntroductionCosts, Resources, and Quality in
Healthcare
3Long 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?
41998 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
5Healthcare 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
6Complex 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
7Equity 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
8The 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
9A 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
10Proportional Allocation
- Demand rates
- Total capacity U 5
- Exact proportional capacity allocation
11Different Performances
- Average sojourn time at server 2 or 3
- Average sojourn times at server 1
- Total system average sojourn time 3
12Patients 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
13Outside 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
14Effects 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
15Research Issues
- Hospital system administrators are often
concerned with two challenging issues
- Patient flow management
- Capacity allocation
16A 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,
17Bibliography
- 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
18Part 1 Capacity/Resource Allocation
19Topics
- Problem formulation
- Extreme cases
- General allocation policy structure
- Mathematical formulation and solution
20Problem Formulation
21Patient 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
22Demand 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
23The Decision Problem
- Given the total available capacity,
- what is the optimal allocation
- so that the total average waiting time is
- minimized ?
24Constraints
- The total switching rate
-
- We must have
25Extreme Cases
26No Switching Formulation
- That is
- The optimization problem becomes
27No Switching Optimal Policy
- THEOREM Optimal allocation follows a square root
rule -
28Effective 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
29Complete 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
30A Conclusion
Simple proportional allocation is not
optimal
31General Policy Structure
32By 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
33Implications
- 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
34Mathematical Derivation
35The 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
36The First Optimization
- Switching customers optimization problem
37Solution
- This is a convex programming
- By Kuhn-Tucker condition, the optimal solution is
- where is the unique solution of
38Infeasible Solution
- Without constraint , the solution is
- What if for some j
-
39Implications
- 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
40The Capacity Allocation
- Consider allocation such that for all j
- We seek allocation to minimize waiting time
41The Second Optimization
- The optimal allocation problem becomes
Non-convex with a non-convex constraint set
42A Qualitative Property
- THEOREM For any two feasible allocations
and if i.e., a majorization
ordering, then
43Quality or Equality?
- Ignore the constraints
- The best solution is
- The worst solution is
44How 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
46Remarks
- 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)
47Conclusions
- 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
48Obligations 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)
49Questions
50Part 2 Flow Control
51Topics
- A stochastic dynamic programming model for flow
control - Switching curves, the optimal solution
- Examples
52A Stochastic Dynamic Programming Model for Flow
Control
53Two-Clinic Model
Rejection
x1
Admitting
?1
Switching
x2
?2
Figure 4
54Notations
- 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)
55The Optimal Equation
56Finite Time Cost Functions
- Define the expected discounted cost of the first
n transitions recursively as follows
and
57Convergence
- With standard dynamic programming arguments, we
have
(1)
58Theorem 1 The Properties of Vn(x1, x2)
- (i) is increasing in x1 and decreasing in x2
(2)
(3)
59Theorem 1 (continues)
(4)
60Theorem 1 (continues)
- (iii) is component-wise convex
(5)
(6)
61The 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
62The Switching Curves
63Finite 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
64Lemma 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
65Proof 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)
66Proof 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
67Discussion
- 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
68Theorem 2
- Let
- (1) is decreasing in x2
- (2) is increasing in x2
- and
(9)
(10)
69Theorem 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)
70Examples
71Example 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
72Example 1 (continues)
- The optimal control policy for type-1 patients is
described by -
Rejection thresholds
Decreasing in x2
and
Increasing in x2
Acceptance thresholds
73Example 1 Switching Curves for Type-1 Patients
Figure 5
74Example 1 (continues)
- The optimal control policy for type-2 patients is
described by
Rejection thresholds
and
Acceptance thresholds
75Example 1 Switching Curves for Type-2 Patients
Figure 6
76Example 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
77Example 2
- The second example has the following parameters
- w 20.0, U 3.4
- R1R2200.00, and C1C220.0
78Example 2 Switching Curves for Type-1 Patients
Figure 7
79Example 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
80Discussion
- 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?
81Questions