1. Lorenzo Ros Mª Victoria de la Fuente Norina Szander
19th International working seminar on Production Economics. Innsbruck.
An approximate algorithm for optimal home care schedule – The case of Zalaegerszeg.
Lorenzo Ros
Mª Victoria de la Fuente
Norina Szander

2. The aim of this paper
To present an algorithm to be used for scheduling home-care services address the increase of efficiency in the use of resources.
The development of exact mathematical model has been the traditional method used, it has proven very difficult to find optimal solutions, due to complexity.
Other mathematical models as B&B and heuristics facilitate the search of solutions simplifying the resolving of real problems.
Many of the contributions have focused on finding the solution using algorithms and calculations carried out in stages (usually 2).

3. A description of the algorithm
The algorithm is based on B&B applied together with decision rules to a combined problem of transport, capacity and assignment, facilitating the service home-care in patients in the shortest possible time and with the lowest possible cost. ( ) i T
Max ( ) o N
Max
Variables definition:
Ti = Type of transport, according to their costs and availability,
No = Nurses available.
Rk = Existing routes.
Pjk = Patient assigned to route k.
HClj = Home-care issued to each patient.
Ci = Cost assigned to each transport route (Rk)
QSn = Special home-care service that requires less than 10 minutes.
Algorithm constraints:
Quantity of transport
Quantity of nurses available:
Maximum patients service:
Maximum time available for each nurse:
Maximum number of routes combination:
Maximum number of patients to visit ( ) i T
Qty. = [ ]

4. The stages of the algorithm 1
Stage1: The grouping of patients by route
Different transport routes are defined by the Central Health Care, such routes are decided in relation to the location of the patients.
Stage 2: Calculation load of daily home-care services
Step 1. Calculation of load by each patient
Step 2. Calculation by each route
Stage 3: Assignment nurses to routes
Step1. Selection of the route with the greatest load and assignment of vehicle No ; No is assigned if:
Step 2. If …………………..…… the division of the route´s load ( ) jk
Load P = å d
m=1 mj HC ( ) k R
Load = å b j jk 1 P mj d HC
m=1 ( ) k R
Load
< o N
Capacity ( ) k R
Load
> o N
Capacity ( ) k R
Load 1 = o N
Capacity ( ) k R
Load 2 = - 1

5. The stages of algorithm 2
Step 3. If the home-care required is greater than the capacity of the nurse:
The division of the orders are carried out in the following way:
Step 4. In the event of home-care load assigned to a route being less than the capacity of the allocated nurse, but failing to meet C-6:
Step 5. Complete nurses allocation to routes with the greatest load. Allocation continues with the next largest load, steps 1, 2, 3 and 4 are carried out. Procedure until home-care loads greater than 10 minutes have been processed. Allocation of non assigned orders. ( ) jk
Load P
> o N
Capacity ( ) jk
Load P = 1 + 2 ( ) jk
Load P 1 = o N
Capacity ( ) jk
Load P 2 = å
non assigned mj HC ( ) k R
Load = 1 + 2 ( ) k R
Load 1 = jk P
max
qty. ( ) k R
Load 2 = å
non assigned jk P

6. The stages of algorithm 3
Stage 4: Calculation of cost
Stage 5: Calculation of the efficiencies in the assignment of the means of road transport.
Step 1. Calculation of the efficiency for each selected Ti.
Step 2. If the efficiencies calculated by each Ti are equal or greater than 0.6 (consider as enough by the CHC) the assignment is definitive.
Step 3. If any efficiency calculation produces a result less than 0.6, the assignment of transport Ti, should be redone (returning to stage 3). i d =
Capacity.nurses.to.visit.in Rk (wik)
Load.route.assigned.to Ti in Rk

7. Approaches to health care at Zalaegerszeg area.
When a patient, requires a home care service, the nurse always go from an Origin to a Destination. Usually, there will be several possible transport system. In Majority of the cases these transport system will differ in type of allowable speed, road requirements, etc. Therefore, travelling time will vary depending on the chosen transport.
Citizens
Age 65+
Age 0-14
Aging index
2001
8.061
9.092
88,70%
2008
9.271
7.515
123,40%
2009
9.545
7.460
127,90%
2010
9.718
7.404
131,30%
2011
9.888
7.312
135,20%
Residential
98 (32,56%)
Day-care
130 (43,2%)
Home-care
73 (24,25%)

8. Frequency of currents tasks in the Urban Functional area of the study.
Monday to Friday
Monday to Sunday
Tuesday to Friday
1 x per day 34 7 1
1 per week 4
2 per day 5 3
2 per week 13
3 per day
3 per week
Total 73

9. Example of a daily nurse timetable

10. Example of a daily nurse timetable with idle time

11. Example of assignment of patients to nurses.
Monday
Number of patients
Time care
Travel time / inactive.
Nurse 7 2
2,0
6,0
Nurse 3 5
3,9
4,1
Nurse 4
5,0
3,0
Nurse 6
2,8
5,3
Nurse 8
3,5
4,6
Nurse 1 6
5,1
2,9
Nurse 9
5,2
Nurse 2 7
3,4
Nurse 5
Nurse 10
3,3
8,7
5,6
3,7

12. Aggregate data as result of efficiency in the assignment of the routes to nurses
Monday
Number of patients
Time care
Travel time/inactive.
Nurse 1
7,2
5,8
2,2
Nurse 2
6,0
3,9
4,1
Nurse 3
4,8
3,8
4,2
Nurse 4
5,2
2,8
Nurse 5
6,4
3,4
4,6
Nurse 6
5,0
5,3
Nurse 7
2,6
2,5
5,5
Nurse 8
4,7
Nurse 9
Nurse 10
7,4
3,6
8,5

13. Conclusions 1
The development of the exact mathematical model has been the traditional way to study these VRP problems, however it is prove very difficult to reach optimal solutions.
due to the complexity of both the real problem as well as the mathematical model.
The use of alternatives such as B&B and heuristics facilitate the search for solutions reducing the time taken in calculations, and simplifying real problems models.
The algorithm develops a solution for assignment of home-care services, assignment made related to the capacity, categorising each possible transport in order to minimize the inefficiencies of transport
Reducing nurses transport time.

14. Conclusions 2
The use of algorithm in a real environment did have as consequences
Service to the patients was improved by reducing lead times.
Minimising the underutilisation of home-care resources, reducing the associated costs (time).
Algorithm deals with type of patient as well as home-care services and the requirements of the service to be provided.
The application of the algorithm in a real situation has show that at first the algorithm does not always reach to an optimal result. However after a second or third iteration,
Maximise the activities assigned to each nurse.
Increase the efficiencies of nurses for the Central Health Care office.
Minimize the time derived from the home-care service.

15. Thanks for your attention