modelling activities at a neurological rehabilitation unit richard wood jeff griffiths janet...
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Modelling Activities at a Neurological Rehabilitation Unit
Richard WoodJeff Griffiths
Janet Williams
Neurological Injury
• An injury to the brain that has occurred since birth• ABI = TBI + Non-TBI
• Typical patient pathway:
A + E
ICU
Neuro Rehab
Home
LT care
District Hospital Specialist Unit
Rookwood Hospital
• Cardiff• Treatment provided by a multidisciplinary team
• Annual demand: 375• 21 beds• Average LOS: 5 months• Annual throughput: 50• Average bed cost per day: £480• Average cost of patient episode: £72,000
Highly sought after
Expensive
150 days
Time between arrivals
Service time
Average LOS = 5 months
Queuing System
Q
Bed 1
Bed 2
Bed 21
EXIT
Average time = 2.75 days
Prob
abili
ty d
ensi
ty
Prob
abili
ty d
ensi
ty
Home
Nursing home
Other hospital
Demand / referrals
Queuing System (simple)
Q
µ
µ
µ
EXITDemand / referrals
λ
Length of stay
M | M | 21
Queuing System (bed-blocking)
Q
Coxk
EXITDemand / referrals
λ
Active
M | Ck + M | 21Exp
Coxk Exp
Coxk Exp
Blocked
Queuing System (balking and reneging)
Q
Coxk
EXITDemand
Active
MB+R | Ck + M | 21Exp
Coxk Exp
Coxk Exp
Blocked
Referrals
Balking Reneging
Queuing System (patient groups)
Q
Coxk1
EXIT
Demand
Active MB+R | Ck1 + M | r1
Exp
Coxk1 Exp
Blocked
Referrals
Balking
Reneging
Q
Coxkp
EXITExp
Coxkp ExpReneging
Referrals
MB+R | Ckp + M | rp
EXIT
∑ 𝑟 𝑖=21
Required Output
1. Steady-state results2. Performance measures3. What if? analysis
Activ
e LO
SBl
ocke
d LO
SCART Analysis Queuing
System
1 2 43
1
2
3
4
1 2 3 4
1 2 3 4
149 days
100 days 231 days
86 days 162 days 175 days 255 days
72 days 136 days 122 days 178 days
14 days 26 days 53 days 77 days
6 beds
3 beds
3 beds
9 beds
Solution
Balking Reneging
Results
Probability of reneging 0.62
Mean bed occupancy 20.8 patients
Annual throughput 51 patients/year
Mean queue length 10 referrals
Mean waiting time 29 days
Annual cost £3.64m
Validated against data
What-if AnalysisMeasure Original
modelReduce delays to discharge (50% / 100%)
One-third increase in older patients
Increase/decrease number of beds
Reneging probability0.62 0.58 / 0.45 0.65
Annual throughput51 57 / 60 51
Annual cost£3.64m £3.62m / £3.57m £3.68m
.... can we use the model to assess other meaningful what-if scenarios?
Better for patients
More costly
Effect of Treatment Intensity on LOS
• Length of stay is dependent on the number of hours of therapy each week• More therapy = quicker recovery
• To incorporate this concept within the model:• Service rates in queuing system must be dependent on treatment
intensity
Queuing System
Q
Coxk1
EXIT
Demand
Active MB+R | Ck1 + M | r1
Exp
Coxk1 Exp
Blocked
Referrals
Balking
Reneging
Q
Coxkp
EXITExp
Coxkp ExpReneging
Referrals
MB+R | Ckp + M | rp
EXIT
∑ 𝑟 𝑖=21
Active Length of Stay
Active LOS Treatment Intensity
Aver
age
Activ
e LO
S
Scaled Active LOS
Mean + Variance Mean Variance
For a particular patient group:
• Treatment intensity cannot be directly controlled
• Dependent on treatment timetables
Prob
abili
ty d
ensi
ty
Prob
abili
ty d
ensi
ty
Scheduling Treatment
Each week:• Demand set for each patient• Supply determined by availability of staff• Demand fitted to supply (excess demand)
Aim: automate scheduling process to rapidly evaluate the effects of changes to….• Staff skill-mix and availability• Patient demand and availabilityon average treatment intensity....• For each patient group
Automated scheduling program
Excel/VBA Multi-objective hierarchical
combinatorial optimisation problem
Heuristics Purpose-built alogrithms
to target constraint violations
Meta-heuristics Simulated annealing Tabu search
Intensity vs LOS
Fit 1-over-x relations to data• Constrain to known LOS for
typical treatment intensity
What-if scenario• Change to timetable variables• E.g. Increases intensity for PG4• Reduces active LOS• Scale service rates in Coxian
distribution to reflect this• Solve system
PG 1
PG 3
PG 2
PG 4
What-if Analysis
1. More group sessions• Amend schedule, run program, find avg LOSs from intensities, solve system• 3 extra patients per year, 2 days fewer waiting time, reduced reneging
2. Composition of workforce (budget cuts)• Retain number of FTEs, but skew towards lower bands• Leads to lower treatment intensity (since staff cannot lead sessions)• Thus: wasted resources, longer LOS, fewer patients per year
Automated Scheduling Program
• Used since January 2011
• Before• 8 hours each week
• After• More time for clinical work• Better solution• Performance measures• Audit data
Results of dry-run (3 trial average)
By-hand Program
Objective function value
(normalised)1 0.54
Demanded sessions scheduled
(avg per patient) 85% 86%
Sessions with neither
primary/secondary therapist
(avg per patient)41% 21%
SchedulingThe scheduling work has released at least 4 hours a week of qualified
physiotherapists who would otherwise be involved in scheduling the patient treatments for the following week
The automated computer scheduling creates a fairer system for patients as it takes into account what treatment the patient received the previous week
ModellingThe service modelling work has been a real asset in that it has opened the eyes
of the operational service managers to the issues regarding patient flowThese insights are now used on a regular basis in waiting list management and
admissions meetingsThe research work has had a huge impact in how we utilise our resourcesThe investment from the department in support of the research has been
well worth it
[email protected]@hotmail.co.uk
Scheduling physiotherapy treatment in an inpatient setting
Operations Research for Health Care (2012)
Modelling activities at a neurological rehabilitation unit
European Journal of Operational Research (2013)
Optimising resource management in neurorehabilitation
NeuroRehabilitation (In press)