MODELING AND OPTIMIZATION OF CLOSED-LOOP HEALTHCARE SYSTEMS WITH GENERAL DISTRIBUTION SERVICE TIMES Closed queuing system of a healthcare department
** TransSolutions LLC., FortWorth TX, 76155, USA * Western New England University, Department of IE & EM, Springfield MA, 01119, USA
What-If scenarios
The flow chart of the healthcare optimization algorithm
PH1
Assumptions:
Morteza Assadi* (
[email protected]) Mohammadsadegh Mobin**, (
[email protected]) S. Hossein Cheraghi**, (
[email protected]) Zhaojun Li**, (
[email protected])
Motivations: ●In the context of healthcare systems studies, the assumption of generally distributed service times has been usually avoided or those systems have been analyzed by means of simulation experiments. ●The fact that the capacity (number of beds) in a healthcare system is limited and sometimes they are working at the full capacity, made them good candidates to be modeled as closed queuing systems. ●These assumptions make the model realistic and complex all the more. ●The limited capacity assumption along with the non-exponential distributions for service times increases the complexity of the problem to the extent that finding a closed form solution becomes very difficult if not impossible.
PH2
Study the behavior of the optimal configuration of the health care system with respect to the changes in:
● Health care process can be divided into various phases. ● In each phase, the length of stay (LOS) of patients is generally distributed. ● Total number of beds in the system (K) is constant. ●After receiving the required service in a phase, the patients may leave the system or proceed into another phase for further treatment. ●The system operates at full capacity which means that upon any departure of patients from the system, another patient is admitted into the first phase (PH1). This assumption entails the application of closed queuing system.
●The mean LOSs ●Transfer probabilities.
These changes in real world health care departments may include: ● New technologies which process the patients faster, ● Reduce the need for further treatments in other phases, ● New diseases resistant to the current medicine which increases the treatment duration.
PH3
PH1
Example: Three-phase closed-loop healthcare system LOS
Abstract
Solid surface: Optimal number of beds Transparent surface: The mean number of patients
Optimization Model: ● Mean Number of Patients (MNP) in each phase (node): 𝑀𝑁𝑃𝑖 = 𝑓𝑖 𝑔1 𝑡 , … , 𝑔𝐿 𝑡 ; 𝑚1 , … , 𝑚𝐿 𝑔𝑖 𝑡 : the distribution of length of stay of patients at the 𝑖 𝑡ℎ node. 𝑚𝑖 : the number of servers (beds) in the 𝑖 𝑡ℎ node (a decision variable).
Current Configuration
𝝁
𝑉𝑎𝑟
# of Beds
Utilization
Optimal Configuration
MNP
# of Beds
Utilization
MNP
PH1
Diagnosis & Acute Care
5
2
90
63.98%
57.58
75
95.125
76.10
PH2
Rehabilitation
41
11
55
68.68%
37.77
50
93.66
50.29
PH3
Longer Stay
540
45
65
99.50%
115.08
85
94.33
83.89
PH2
PH3
Note: In a closed-loop health care system, minimizing the number of patients waiting for service results in the optimization of total idle capacity assuming that resources (beds) can be moved between nodes. Objective: 𝑴𝒊𝒏 𝐌𝐚𝐱 { 𝑴𝑵𝑷𝒊 − 𝒎𝒊 } = 𝑴𝒂𝒙{ 𝒇𝒊 𝒈𝟏 𝒕 , … , 𝒈𝑳 𝒕 ; 𝒎𝟏 , … , 𝒎𝑳 − 𝒎𝒊 } 𝟏≤𝒊≤𝑳
𝟏≤𝒊≤𝑳
s.t.
𝑳 𝒊=𝟏 𝒎𝒊
= 𝑲 , 𝟏 ≤ 𝒎𝒊 ≤ 𝑲
PH1 PH2
Objectives: 1-Analyzing a health care system in which the service times are generally distributed. Because of the complexities which arise when the health care systems possess non-exponentially distributed service times, in the literature these type of systems are approached using simulation. 2-Developing an optimization algorithm by which the optimal number of beds in each phase of the health care process can be determined when the service times are generally distributed, 3-Gaining insight into the influence of the changes in the length of stay of patients as well as the transfer probabilities in between the phases on the optimal configuration of the beds in the health care system.
Results: ●An algorithm is developed by which the optimal configuration (number of beds in each phase) of a healthcare system with generally distributed service time and limited capacity could be identified. ●This algorithm is verified by comparing its performance to the analytical exact solutions of two healthcare models given in the literature. ●The comparison of the results given by our algorithm and the theoretical results given in the literature depicted the good performance of our algorithm. ●The behavior of optimal configuration of the health care system was analyzed with respect to the changes in transfer probabilities or mean LOS of patients in different phases. ●The new approach could also help the health care managers in order to get insight into the different technological improvements or introduction of new diseases which can affect length of stay (LOS) of patients or the need for further treatments thereby affecting the optimal configuration of the beds throughout the healthcare system.
●Objective function can be replace by: 𝑀𝑖𝑛 Max 1 − 𝑈𝑖 = Max {1 − ℎ𝑖 (𝑔1 𝑡 , … , 𝑔𝐿 𝑡 ; 𝑚1 , … , 𝑚𝐿 } 1≤𝑖≤𝐿
1≤𝑖≤𝐿
● ℎ𝑖 : The function which represents the relation between the utilization of the phases based on the service time distributions and the number of servers throughout the system. ●There is no closed form solution given for the functions 𝑓𝑖 . and ℎ𝑖 . for closed queuing systems with generally distributed service times.
Optimization algorithm verification
Solution algorithm
Performance of developed optimization algorithm is compared with the analytical solutions of two models given in the literature: ● Analyzing a closed queuing geriatric healthcare system.
●The rationale of the developed optimization algorithm: to change the number of beds (servers) in each phase step by step to ensure that resource utilization maximized resulting in minimum wait time for patients. ●Initialize the system: All the K beds (servers) are almost equally distributed among the L phases (nodes). ●Performance Evaluation: Using MEBOTT (Modified Extended Bottleneck Analysis) to find the 𝑀𝑁𝑃𝒊 and utilization of each phase. ●The disparity between the utilization of the phases is evaluated by computing the sum of squared errors from the mean of utilizations. ●Identifying the phases with the lowest and highest utilizations. ●One of the beds is removed from the phase with the lowest utilization (reduce the idle capacity) and is added to the phase with the highest utilization (reduce workload). ●Performance evaluation of this new configuration (using MEBOTT) ●The disparity between the utilization of the phases is measured. ●If the disparity between the utilization of the phases is reduced, the original configuration is replaced by this new configuration and the process is repeated. ●The iterations continue until either the utilization of the nodes get close enough to each other or further adjustments will no longer reduce the utilizations disparity.
LOS
Chaussalet et al (2006) result
Proposed optimization algorithm result
𝝁
Expected number of patient
Expected number of bed
# of Beds
Utilization
PH1
Acute Care
9
177.67
178
177
95.2 %
PH2
Rehabilitation
67
92.58
92
93
94.49 %
PH3
Longer Stay
863
202.74
203
203
94.79 %
● Analyzing the LOS of stroke-related patients aged over 65 years. LOS
Vasilakis & Marshall (2005) result
𝝁
Expected number of bed
PH3
Proposed optimization algorithm result # of Beds
Utilization
PH1
Short stay
10.8
3083
3088
98.28 %
PH2
Medium stay
36.7
973
938
98 %
PH3
Long Stay
653.1
156
150
98.14 %
In both cases, the optimal configuration (number o beds) and the utilization of phases are obtained by the developed algorithm. The optimal configurations show small idle capacity ratio in both case studies.
● The increments in the mean LOS in a phase results in growth of the optimal number of beds in that phase compensated for by decreasing the number of beds in other phases. ● As the probability of transfer from one phase to another phase increases, it results in the increments in the optimal number of beds in all the phases in front.
Main References ● Assadi, Morteza. "Modeling and optimization of closed-loop systems with generally distributed failure/service times." 2008, PhD dissertation, Wichita State University. ● Chaussalet T.J., Xie H., Millard P.H., 2006, A closed queuing network approach to the analysis of patient flow in health care systems, Methods of Information in Medicine, Vol. 45, pp. 492-497. ● Vasilakis C., Marshall A.H., 2005, Modeling nationwide hospital length of stay: opening the black box, Journal of the Operational Research Society, Vol. 56, pp. 862-869.
