1. Noa Zychlinski* Avishai Mandelbaum*, Petar Momcilovic**, Izack Cohen*
Analysis of Hospital Networks
via Time-varying Fluid Models
with Blocking
Noa Zychlinski*
Avishai Mandelbaum*, Petar Momcilovic**, Izack Cohen*
* The Faculty of Industrial Engineering and Management
Technion - Israel Institution of Technology
** Department of Industrial and Systems Engineering
University of Florida
January 2018

2. Motivation Medical concern Financial concern ‘Bed-Blocking’ Problem
Hospital
wards
Geriatric wards
Geriatric wards
Blocking
Blocking
Medical concern
Financial concern

5. Life expectancy at birth
Introduction
Life expectancy at birth
From: Israel Central Bureau of Statistics

6. "בכל יום תוחלת החיים שלנו עולה ב- 5 שעות"
(דו"ח משרד הבריאות, ינואר 2017)
Each day life expectancy increases in five hours.
Even if you don’t enjoy this talk, don’t feel that you wasted your time, because in the one hour of my presentation, your life expectancy will have increased by 15 minutes.

7. Introduction Elderly percent from the total population 1960-2030
Twice the increase in total population
Total
From: Israel Ministry of health, Information & Computing Department
Processed by: CBS

8. Life expectancy in nursing:
Introduction
1/5 of admissions to EDs are made by elderly patients (65+).
1/4 of elderly admissions to ED are re-admissions.
1/3 of the hospitalizations are elderly patients.
Life expectancy in nursing:
18 months
36 months
In 1995
in 2015

9. Geriatric Institution
Patient Flow Chart
Hospital
Geriatric Institution
1.5 months
5.5 months
1 months

10. Introduction Average waiting time for a vacant bed
Source: our Partner hospital

11. Research Objectives
Collecting and analyzing patient flow data between hospitals and geriatric institutions.
Developing a fluid (mathematical) model to describe the patient flow in the system.
Offering ways to improve the performance of the system.

12. The Fluid Model Arrival rate to Station 1 at time
Inpatient ward
Geriatric ward
Arrival rate to Station 1 at time
Routing probability from Station 1 to 2
Treatment rate at Station
Number of beds at Station

13. The Fluid Model Number of patients at Station
Number of arrivals to Station 1 that have not completed their treatment at Station 1 at time
Number of patients that have completed treatment at Station 1, but not at Station 2 at time

14. The Fluid Model
b(t)

15. The Fluid Equations Arrival rate – Departure rate Blocked beds
Unblocked beds in Station 1
Occupied unblocked beds in Station 1

16. The Fluid Equations
Arrival rate
Departure rate

17. Network Modeling Routing probability from Station 1 to Station
Readmission rate from Station
Mortality rate at Station

18. Data
Patient flow data from a district comprising four general hospitals and three geriatric hospitals .
Waiting lists for geriatric wards, including individual waiting times from our Partner Hospital.

19. Waiting-List Length in Hospital Model vs. Data

20. Estimating the Optimal Number of Geriatric Beds
Objective:
To minimize the total cost of operating the system
– Planning horizon (usually 3-5 years)
Decision variables:
– Number of beds at Ward ,

21. Cost construction Estimating the Optimal Number of Geriatric Beds
per bed per day
overage cost
per bed per day
underage cost

22. Estimating the Optimal Number of Geriatric Beds
The objective function:
Blocked patients
Empty beds
Analytically intractable

23. Estimating the Optimal Number of Beds
Underage cost
Overage cost
t [days]

24. Estimating the Optimal Number of Geriatric Beds
We use the approach of Jennings et al. (1997) and treat as a continuous variable. We let
denote the decreasing rearrangement of on the interval

25. Estimating the Optimal Number of Beds
Theorem:
The optimal number of beds, which minimizes is:
When =0
When =0

26. Optimal Number of Beds
Cu2 = Co2 , Cu3 = Co3, Cu4 = Co4

27. Optimal Number of Beds 51%, 53% and 69% 67%, 74% and 88%
Increasing the current number of beds by
25%, 35% and 33% in
Rehabilitation, Mechanical Ventilation and Skilled Nursing Care
Overage and underage cost reduction of
51%, 53% and 69%
Waiting list length reduction of
67%, 74% and 88%

28. Waiting List Length in Hospital
Optimal Solution

29. Comparing Optimal Solutions
Fluid Model
Offered-load
Simulation

30. Optimal Number of Beds
Overage

31. Periodic Reallocation of Beds
t [days]
Underage cost
Overage cost

32. Optimal Periodic Reallocation
The decisions:
1.) The number of periods each year (reallocation points).
2.) The length of each period.
3.) The number of beds in each period.
RC - Reallocation cost associated with adding and removing a bed.

33. Optimal Periodic Reallocation Not Including Reallocation Costs

34. Optimal Periodic Reallocation Including Reallocation Costs

35. Optimal Periodic Reallocation Quarterly Reallocation

36. Waiting List Length in Hospital Quarterly Reallocation

37. Extensions Blocking Blocking ED Hospital wards Geriatric wards

38. K Stations in Tandem H1 H2 Hk 38

39. The Fluid Limit
where

40. Numerical Experiments and Operational Insights

41. Line-Length Effect – Infinite Capacities
Eick, S., Massey, W., Whitt, W. (1993)

42. Line-Length Effect – Finite Capacities
Identical stations
H = ∞

43. Numerical Experiments
Case 1: No waiting room (H = 0)
Case 2: An infinite sized waiting room (H = ∞)
In the following slides we compare two networks:
The first with no waiting room before station 1 and the second with unlimited waiting room.
This waiting room plays an important role in the performance of the entire network.

44. Total Number of Customers in Each Station
k=8 identical stations
Case 1
No waiting room
H = 0
Case 2
An infinite sized waiting room H = ∞
Average sojourn time is 20% shorter t t

45. The Bottleneck Effect on Performance
Bottleneck = Station 8
Case 1
No waiting room
H = 0
Case 2
An infinite sized waiting room H = ∞

46. The Bottleneck Effect on Performance
Bottleneck = Station 8
Case 1
No waiting room
H = 0
Case 2
An infinite sized waiting room H = ∞

47. THANK YOU!