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

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

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

"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.

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

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

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

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

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.

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

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

The Fluid Model b(t)

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

The Fluid Equations Arrival rate Departure rate

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

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.

Waiting-List Length in Hospital Model vs. Data

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 , .

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

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

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

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 .

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

Optimal Number of Beds Cu2 = 2.667 Co2 , Cu3 = 1.882 Co3, Cu4 = 4.267 Co4

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%

Waiting List Length in Hospital Optimal Solution

Comparing Optimal Solutions Fluid Model Offered-load Simulation

Optimal Number of Beds Overage

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

Optimal Periodic Reallocation The decisions: 1.) The number of periods each year (reallocation points). 2.) The length of each period. 1 2 3 4 5 6 7 8 9 10 11 12 1 3.) The number of beds in each period. RC - Reallocation cost associated with adding and removing a bed.

Optimal Periodic Reallocation Not Including Reallocation Costs

Optimal Periodic Reallocation Including Reallocation Costs

Optimal Periodic Reallocation Quarterly Reallocation

Waiting List Length in Hospital Quarterly Reallocation

Extensions Blocking Blocking ED Hospital wards Geriatric wards

K Stations in Tandem H1 H2 Hk 38

The Fluid Limit where

Numerical Experiments and Operational Insights

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

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

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.

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

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

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

THANK YOU! noazy@tx.technion.ac.il