Periodic Vehicle Routing Problem in a Health Unit

F. Alves

1,2

, F. Alvelos

, A. M. A. C. Rocha

, Ana I. Pereira

1,2

and Paulo Leit

Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit

ecnico de Braganc¸a,

Campus de Santa Apol

onia, 5300-253 Braganc¸a, Portugal

Algoritmi R&D Centre, University of Minho, Braga, Portugal

Department of Production and Systems, Algoritmi Research Centre, University of Minho, Braga, Portugal

Keywords:

Home Health Care, Operations Research, Periodic Vehicle Routing.

Abstract:

In logistics of home health care services in the Health Units, the managers and nurses need to carry out the

schedule and the vehicles routes for the provision of care at the patients’ homes. Currently, in Portugal, these

services are increasingly used but the problem is still, usually, solved manually and without computational

resources. The increased demand for home health care due to the boost of the elderly people number entails

a high associated cost which, sometimes, does not guarantee the quality of the service. In this sense, the

periodic vehicle routing problem is a generalization of the classical vehicle routing problem in which routes

are determined for a time horizon of several days. In this work, it is provided a periodic vehicle routing

problem applied in the Health Unit in Braganc¸a. An integer linear programming formulation for the real

database, allowed to solve the problem in an efﬁcient and optimized way using the CPLEX



software.

1 INTRODUCTION

Home Health Care (HHC) is a growing medical ser-

vice in many countries (Benzarti et al., 2013). One of

the tasks of the HHC involves the management deci-

sions, that is, it is necessary to create the planning and

the routing of vehicles or nurses/doctors in the domi-

ciliary services. In Portugal, there is a large number

of elderly people who need support in their homes,

therefore the operational management problems re-

lated to home care visits are very important.

The HHC service in a health care unit has a ﬂeet

of vehicles used by nurses/doctors to travel to the pa-

tients locations. A periodic aspect is also involved,

since the jobs need to be performed, repeatedly, du-

ring a speciﬁc visits schedule. Usually, the HHC

management is solved manually by a senior nurse or

manager, who often spends a huge amount of time

to create next week’s schedule (Fikar and Hirsch,

2017). However, the scheduling and routing problems

in this area are complex procedures (Nickel et al.,

2012) due to the mathematical modeling complex-

ity and, more importantly, the absence of computa-

tional resources in the Health Unit. From a mathe-

matical point of view, this problem has special inter-

est, because it is an NP-hard problem, such as peri-

odic vehicle routing problem (PVRP). In this sense,

the PVRP is a generalization of the classical vehicle

routing problem (VRP) where routes are determined

for a planning horizon of multiple periods with some

customers demanding multiple visits (Angelelli and

Speranza, 2002; Beltrami and Bodin, 1974; Cordeau

et al., 1997). This approach is different from route

planning, in the sense that route planning performs

the scheduling for one day and this approach per-

forms for a certain period of time (Campbell and Wil-

son, 2014). Many real-world applications in trans-

portation systems require ﬁnding, for a ﬂeet of ve-

hicles, a set of minimum cost routes that satisfy or-

ders or services requested by customers/patients, over

a given planning horizon (e.g. one week). This prob-

lem is very important in real-world applications such

as the distribution for bakery companies (Pacheco

et al., 2012) and also in blood product distribution

(Hemmelmayr et al., 2009), among other services.

Some authors address and describe the HHC prob-

lem through a hybridization of constraint program-

ming and meta-heuristics including simulated anneal-

ing and tabu search (Bertels and Fahle, 2006) or ge-

netic algorithm (Alves et al., 2018). Other approaches

in the HHC problem can be found in Trautsamwieser

et al. (Trautsamwieser et al., 2011) and Nickel et

al. (Nickel et al., 2012). Thus and although there

is already an extensive operational research in HHC,

many focus only on the daily scheduling of the pro-

blem of home visits. However, in these types of health

384

Alves, F., Alvelos, F., Rocha, A., Pereira, A. and Leitão, P.

Periodic Vehicle Routing Problem in a Health Unit.

DOI: 10.5220/0007392803840389

In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 384-389

ISBN: 978-989-758-352-0

services, the scheduling and routing may be periodic

in order to group periods of home visits and to respect

regularities imposed by patients (becoming a PVRP).

Typically, the PVRP has a time horizon of T days, and

there is a demand and regularity of delivery for each

client or patient indicating how many times within the

period of T days the customer should be visited. The

solution to the PVRP is to ﬁnd the T sets of routes that

together satisfy the constraints of the requested quan-

tity and regularity and also minimize travel distances.

The applicability and versatility of the problem

has led to extensive research addressing both new

applications and solution methods. It has become a

topic with extensive studies in operational research

with extension in diverse services. However, only

limited research has been conducted into the HHC

problem.

The paper is organized as follows: the next sec-

tion, brieﬂy reviews the main deﬁnitions and the pro-

blem approach. Section 3 presents the integer linear

programming model for the PVRP. The case study,

based on real data from a Braganc¸a Health Unit, is de-

scribed in Section 4. In Section 5 the results are ana-

lyzed and discussed. Finally, some conclusions and

future work are drawn in Section 6.

2 PVRP DEFINITION

The problem addressed in this article is related to the

schedule and routes of home care visits outside the

Health Unit of Braganc¸a in a certain time horizon.

The home care service can be provided by nursing

teams of a Health Unit, which involves the scheduling

of nurses and the deﬁnition of transportation routes

to the patients’ homes. Currently, home visits are

planned manually and without computational support,

which can lead to not obtaining the optimal solution,

especially when certain constraints must be met (dis-

tances and costs involved, time windows, among o-

thers). Thus, it is necessary to develop approaches

that are able to overcome the difﬁculties that may o-

ccur in these services, such as the accumulation of

delays on routes, the deﬁnition of routes that are too

long, the planning without feasible options for repla-

cing nurses, and the difﬁculties in managing periodic

visits.

Therefore, the ﬁrst approach is to deﬁne the ge-

neral characteristics of the problem, such as the num-

ber and characterization of health professionals, the

number of available vehicles (m), the number of pa-

tients (n) and treatments they need, locations that can

be traveled and their distances, among others. These

data allow us to formulate and model the problem, in

an attempt to minimize the time spent on visits, re-

duce costs and provide support for decision making.

In this paper, an approach for the optimization of

the PVRP, which deals with home care visits in or-

der to perform treatments for patients belonging to a

Health Unit is presented. For this problem the number

of vehicles involved in the home care and the patients

requesting this type of health services is considered.

The problem in establishing periodic routes pro-

duces a schedule within a time horizon of T days,

where each patient i, in addition to the number of

visits, may have a minimum and/or maximum time

period between successive visits, e

, i.e. the num-

ber of visits combined with the time period, leads to

the set of patterns p

that corresponds to the possi-

ble patterns of visits in the time horizon T . For ex-

ample, if the planning is executed for a time horizon

of 5 days (T = 5), with a time period between visits

of at least two days (e

≥ 2), the set of patterns will

be p

{{

1, 4

}

{

1, 5

}

{

2, 5

}}

. This corresponds to

visiting the patient twice during the 5 days of the pla-

nning period and should be done on days 1 and 4, or

on days 1 and 5 or alternatively on days 2 and 5, and

no other pattern of visits is possible. The problem is to

simultaneously select the pattern of each patient and

establish the routes for each day of the time horizon.

The Figure 1 shows an example of PVRP in which

routes 1 and 2 will have to occur at different times

(because they have points in common).

Depot - Health Unit

Patients

route 1

route 2

route 3

route 4

Figure 1: Illustration of the PVRP.

Similarly, routes 3 and 4 will also have to occur in

different periods. In the PVRP, the cost of visiting

patients is independent of the vehicles that carry it,

not considering restrictions that include incompati-

bilities between vehicles and patients. The matrix

cost = (c

i j

), with rank n, reﬂects the time spent on

the route between patient i and patient j.

The PVRP consists of establishing r routes of

lower cost in order to satisfy the time horizon and cer-

tain assumptions, such that:

• Each patient belongs to exactly one route.

Periodic Vehicle Routing Problem in a Health Unit

385

• Each route starts and ends at the depot (Health

Unit of Braganc¸a)

3 FORMULATION OF THE

INTEGER LINEAR

PROGRAMMING MODEL

The PVRP can be considered as an integer linear opti-

mization problem where it is not only established the

scheduling of patients visits, but also the design of

the routes for the different days of the time horizon

in order to minimize the costs associated with all the

routes of the time period under study. Consider the

following input:

• L = {1, . . . , T } is the set of days in the planning

horizon;

• N = {1, . . . , n} is the set of patients;

• The index of a route starting or ending at the depot

has the value {0};

• M = {1, . . . , m} is the set of vehicles;

• p

is the set of patterns of patient i;

• u

is an integer corresponding to the place in the

sequence of visits of vehicle k on day l for the

patient i;

• a

is a variable to represent the s pattern that in-

cludes the patient i on day l.

The optimization problem is to ﬁnd the optimal

values for the binary decision variables such as x

i j

and y

. Thus, the formulation of PVRP is deﬁned as

follows, where the binary variables take the form:







1, if patient i is visited according to

the s pattern belonging to p

;

0, otherwise.

i j







1, if the vehicle k visits patient j

immediately after patient i on day l;

0, otherwise.

The integer programming model is given by:

minimize

∑

i=0

∑

j=0

i j

∑

k=1

∑

l=1

i j

(1)

subject to:

∑

s∈p

= 1, ∀i ∈ N (2)

∑

j=0

∑

k=1

i j

−

∑

s∈p

= 0, ∀i ∈ N; ∀l ∈ L; i 6= j (3)

∑

j=0

−

∑

j=0

i j

= 0,

∀i ∈ N;∀k ∈ M;∀l ∈ L; i 6= j

(4)

∑

j=1

0 j

≤ 1, ∀k ∈ M; ∀l ∈ L (5)

≥ u

−C(1 − x

i j

) + 1,

∀i, j ∈ N; ∀k ∈ M;∀l ∈ L; i 6= j

(6)

i j

∈

{

0, 1

}

, ∀i, j ∈ N; ∀k ∈ M; ∀l ∈ L (7)

∈

{

0, 1

}

, ∀i ∈ N;∀s ∈ p

(8)

The objective function (1) represents the minimiza-

tion of total cost, in this case the distance.

Constraints (2) ensure that each client will be

assigned to one of the admissible visit patterns,

while constraints (3) ensure that each client is visited

exactly on the days of the pattern that is assigned. In

turn, constraints (4) guarantee continuity of the route,

i.e, a vehicle entering a vertex will have to leave it.

Constraints (5) ensure that the number of available ve-

hicles is not exceeded. Finally, the sub-circuit elimi-

nation restrictions can be expressed by constraints (6).

In the presented formulation, the cost of visiting the

patients is independent of the vehicles that do it, not

considering restrictions that include incompatibilities

between vehicles and patients. Thus, in the follow-

ing section will be presented the real case study based

on the Health Unit in Braganc¸a, where the proposed

formulation will be applied.

4 CASE STUDY

In this section a real case study of a Braganc¸a Health

Unit for a time horizon of T = 5 days (one week of

home visits) is presented. In that same week, the

Braganc¸a Health Unit has ﬁve vehicles available for

home care visits (m = 5), in which they visit ﬁfteen

patients (n = 15) with the need for treatments, where

they are dispersed by their locations. Table 1 summa-

rizes the data for the problem in question.

Table 1: Problem Data Information.

Vehicles Patients Locations Time Horizon

5 15 15 5 days

According to the problem database, there are 5 ve-

hicles available for home visits at the Health Unit of

Braganc¸a, which are homogeneous in terms of capa-

cities and/or visits to patients. The number of patients

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

386

Table 2: Distances between patient locations.

HU 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

HU 0,0 15,2 18,0 22,4 25,0 20,6 11,2 21,2 26,2 32,0 25,5 33,5 15,0 11,2 32,0 30,4

1 15,2 0,0 32,6 14,6 32,2 32,2 24,8 21,0 31,6 17,8 15,6 26,4 16,6 26,4 46,9 45,4

2 18,0 32,6 0,0 34,4 20,2 23,9 16,4 36,2 36,1 47,4 43,3 50,3 23,4 9,4 21,2 13,0

3 22,4 14,6 34,4 0,0 25,0 42,7 33,5 35,4 45,0 15,0 29,2 40,3 11,2 32,0 53,2 47,2

4 25,0 32,2 20,2 25,0 0,0 41,2 31,6 46,1 50,5 40,0 47,2 57,0 15,8 25,5 41,2 29,2

5 20,6 32,2 23,9 42,7 41,2 0,0 10,0 20,6 13,9 50,0 33,5 35,4 35,4 15,8 20,0 29,2

6 11,2 24,8 16,4 33,5 31,6 10,0 0,0 20,6 19,8 42,4 30,4 35,4 25,5 7,1 22,4 25,5

7 21,2 21,0 36,2 35,4 46,1 20,6 20,6 0,0 12,2 36,4 14,1 15,0 33,5 26,9 40,3 46,1

8 26,2 31,6 36,1 45,0 50,5 13,9 19,8 12,2 0,0 48,1 26,2 24,2 40,8 26,9 33,4 42,9

9 32,0 17,8 47,4 15,0 40,0 50,0 42,4 36,4 48,1 0,0 25,0 35,4 25,5 43,0 64,0 60,4

10 25,5 15,6 43,3 29,2 47,2 33,5 30,4 14,1 26,2 25,0 0,0 11,2 32,0 35,0 52,2 55,0

11 33,5 26,4 50,3 40,3 57,0 35,4 35,4 15,0 24,2 35,4 11,2 0,0 42,4 41,2 55,2 60,8

12 15,0 16,6 23,4 11,2 15,8 35,4 25,5 33,5 40,8 25,5 32,0 42,4 0,0 22,4 43,0 36,1

13 11,2 26,4 9,4 32,0 25,5 15,8 7,1 26,9 26,9 43,0 35,0 41,2 22,4 0,0 21,2 20,0

14 32,0 46,9 21,2 53,2 41,2 20,0 22,4 40,3 33,4 64,0 52,2 55,2 43,0 21,2 0,0 15,8

15 30,4 45,4 13,0 47,2 29,2 29,2 25,5 46,1 42,9 60,4 55,0 60,8 36,1 20,0 15,8 0,0

is 15 that require and need treatments in their respec-

tive locations. Regarding the locations, it is necessary

to know the different locations/cities of each patient

belonging to the Health Unit and the respective tem-

poral distance (minutes) between each one of them.

In this way, Table 2 presents the distances (in kilo-

meters) between locations. The patients seek and

need home visits with a certain regularity in the pe-

riod of the visits (T = 5 days). Thus, Table 3 shows

the number of times each patient should be visited.

Table 3: Regularity of visits required by each patient in the

time horizon.

Period of visits they require for T = 5

Patient 1 1

Patient 2 1

Patient 3 2

Patient 4 3

Patient 5 1

Patient 6 2

Patient 7 1

Patient 8 1

Patient 9 2

Patient 10 3

Patient 11 1

Patient 12 1

Patient 13 1

Patient 14 2

Patient 15 1

According to these data, it is also possible to iden-

tify some patterns about the number of visits required

by each patient for the T period, knowing in advance

that between two or more visits a day of interval is

required. Thus, in this way it is possible to illustrate

the different patterns according to Table 4:

Based on all the data, the main objective is to ob-

tain vehicle routing/scheduling, ﬁnding the T sets of

routes that satisfy the constraints and minimizing the

Table 4: Patterns of visits according to the period T .

Possible Pattern for visits to T = 5 days

1 Visit 1, 2, 3, 4 or 5

2 Visit 1-3, 1-4, 1-5, 2-4, 2-5, 3-5

3 Visit 1-3-5

total time required to carry out the trips, treatments

and return to the starting point (Depot - Health Unit).

5 ANALYSIS AND DISCUSSION

OF RESULTS

In this section the computational results of the model

developed and proposed for the resolution of PVRP

will be presented and analyzed.

The model was coded and implemented in

the IBM



ILOG



CPLEX



Optimization Studio

that supports Optimization Programming Language

(OPL). The data of the real case under study was im-

plemented according to the periodic home care visits

approach and the results were obtained on an Intel (R)

Core i7 CPU 2.2GHz PC with 6.0 GB of RAM.

The CPLEX



took about 11 hours to reach the so-

lution. The obtained solution had the objective value

of 473 and besides the regularity and periodicity of

visits imposed by the patients in the time horizon, the

model established the route patterns according to the

objective reached. This solution indicates the mini-

mum distance to be traveled (cost) for the vehicles to

make the home visits routes, according to the deﬁned

time horizon and the regularity needed by the patients

and nurses of the Health Unit of Braganc¸a.

From Table 5, it is possible to get some statisti-

cal details, such as the number of variables used, the

average value of the target solution, among other pa-

Periodic Vehicle Routing Problem in a Health Unit

387

Depot – Health

Unit

Patients

Route 1 -

Monday

Route 2 -

Wednesday

Route 3 -

Friday

Vehicle 1

Vehicle 3

Frequency

of 3 visits

Frequency

of 2 visits

Figure 2: Representation of the PVRP solution.

rameters that allow to summarize the objective solu-

tion.

Table 5: Statistical values obtained by CPLEX



Statistic Value

Constraints 5874

Variables 6955

Objective 473

Nodes 4724907

Mean Objective 547

Figure 2, in turn, illustrates the solution obtained,

which includes the scheduling and the set of routes

for the periodicity of home visits in question.

In this sense, two of the ﬁve vehicles available

were used, where only one was used on two different

days. Another thing to mention, is that from the de-

ﬁned time horizon only 3 days were subjected to the

service of home care visits. These days allow to com-

ply with all requirements according to the periodicity

imposed. Certain patients require a higher frequency

of visits, however it is possible to check those that be-

long to more than one route and day of visits, as can

be seen in Figure 2. It is possible to conclude with the

solution provided, that all patients are visited and all

routes meet the time period between successive visits

leading to admissible patterns.

The analysis showed that the problem can be

solved by CPLEX



and provided the optimal solution

of the single problem.

6 CONCLUSIONS AND FUTURE

WORK

The PVRP are an extension of the classic vehicle

routing problems. They are more difﬁcult to solve

when compared to the problems of classic routes,

since they do not only aim to ﬁnd a set of routes,

but also the deﬁnition of a schedule of visits of the

patients that minimizes the operational costs of the

system/service within a time horizon. Since the route

problem addressed is a periodic approach that covers

a time horizon with several days, an integer linear pro-

gramming model was developed using OPL/CPLEX



implementation for its resolution and optimization.

The developed model allows to solve problems

of routes with time period between successive vi-

sits, meeting the demand and regularity of visits by

the patients. In this way it was possible to estab-

lish the home visits in a Health Unit in Braganc¸a,

optimizing the routes in the service. In this work, a

small real case of 15 patients is solved, allowing the

Health Unit to plan routes and visits using computa-

tional support. However, replicating the problem with

larger instances and/or increasing difﬁculty, may not

be feasible due to the rather long computational time

required.

The developed approach allows the planning of

a set of routes that, with the existing resources, and

guarantees the visit to all the locations with the time

period deﬁned by the managers, without incurring

additional costs or deﬁciencies in the service. In this

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

388

way, obtaining the solutions allows reducing and opti-

mizing costs and routes, improving the health service

provided and serving as a decision support system,

which does not exist today.

For future work, it is intended to use meta-

heuristics, to test this integer linear programming

model in larger instances and/or instances from the

literature, in an attempt to overcome other logistical

difﬁculties in the home health care services.

ACKNOWLEDGEMENTS

This work has been supported by COMPETE:

POCI-01-0145-FEDER-007043 and FCT - Fundac¸

para a Ci

encia e Tecnologia within the project

UID/CEC/00319/2013.

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