Non Emergency Patients Transport

A Mixed Integer Linear Programming

José A. Oliveira, João Ferreira, Luis Dias, Manuel Figueiredo and Guilherme Pereira

Algoritmi Research Centre, University of Minho, Guimarães, Portugal

Keywords: Non Emergency Patients Transport, Team Orienteering Problem, Mixed Integer Linear Programming,

AMPL, NEOS Server.

Abstract: This work presents a model and a heuristic to solve the non-emergency patients transport (NEPT) service

issues given the new rules recently established in Portugal. The model follows the same principle of the

Team Orienteering Problem by selecting the patients to be included in the routes attending the maximum

reduction in costs when compared with individual transportation. This model establishes the best sets of

patients to be transported together. The model was implemented in AMPL and a compact formulation was

solved using NEOS Server. A heuristic procedure based on iteratively solving problems with one vehicle

was presented, and this heuristic provides good results in terms of accuracy and computation time.

1 INTRODUCTION

In 2012, Portugal published several official

documents about the non-emergency patient

transport (NEPT) service in “Diário da República”

(DRE). The legislative motivation was based on a

requirement laid down in the Memorandum of

Understanding signed between the Portuguese

Government and the International Monetary Fund,

the European Central Bank and the European Union

to reduce the cost of NEPT services (DRE, 2011).

This legislation mandates the minimum

requirements for the quality and safety of care

delivered to patients by NEPT services, (DRE,

2012a).

In 2011, the Northern Department of Health

(ARSN, 2011) reported the implementation of a

computer system to optimize the management of

NEPT services. The published documents state that

the computer system "will allow greater accuracy in

terms of prescription and simultaneously ensures

your organization rationally, promoting the transport

of multiple users whenever appropriate and

possible." The Northern Department of Health

expected to achieve a reduction of transport costs in

the region of €3 million related to a reduction of

20% of costs.

The current paper is composed of five sections.

After the introduction, Section 2 presents a

description of the problem, the main topics referred

by the law, and a brief literature review. Section 3

describes the model and presents the mathematical

formulation. Section 4 includes a discussion of the

heuristic and its results. Section 5 summarizes the

main conclusions of this work.

2 TRANSPORT PROBLEM

2.1 NEPT Definition

In light of the legislation in 2014 in Portugal (DRE,

2012b-h), and with regard to access by the users to

the services of the National Health Service (NHS),

NEPT is considered the associated transport system

for the health care system, where the origin or

destination are the medical centres and services

within the NHS, private entities or social entities

with a contract or agreement for the provision of

health care under the following conditions:

consultations for inpatient or outpatient surgery,

diagnostic procedures and therapeutic treatments,

transporting the patient after discharge from hospital

(with prior prescription) and transporting the patient

after discharge from the emergency room (with prior

prescription).

2.2 Transport Prescription

Prescription of transportation is solely the

262

A. Oliveira J., Ferreira J., Dias L., Figueiredo M. and Pereira G..

Non Emergency Patients Transport - A Mixed Integer Linear Programming.

DOI: 10.5220/0005214902620269

In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 262-269

ISBN: 978-989-758-075-8

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

responsibility of the attending physician, who shall

record the following information in the support

doctor system or equivalent system: the clinical

justification, or reasons for needing transport, and

verification of the economic condition. Where there

is a need to make the ambulance transport, the

following is required: the justification of the mode

of transport; the conditions under which the

transport should occur, particularly if the patient

requires ventilation, oxygen, wheelchair or is sick in

bed or isolated, the justification of the need for a

companion, and the justification of the need for

monitoring a health professional. After the shipping

prescription by the physician is completed, a

member of the administrative services staff validates

the economic condition of the patient and proceeds

to request transportation.

The requisition of transport satisfies the criteria

of minimizing the distance between the place of

origin, which must match the address from which

the transportation is effected, and the place of

destination, which must take into account the

location of the nearest place of origin. The NEPT is

performed by ambulance or by ambulette (simple

vehicle for patient transport - SVPT), which is a

passenger car, with a maximum capacity of five or

nine people, for the NEPT service whose clinical

condition does not impose the need for health care

during the transport.

The non-emergency patient transport is carried

out, where possible, in SVPT, taking into account

the need to optimize the capacity of the vehicle

against the following criteria: a) Grouping of clients,

regardless of origin, within the same route; b)

intended for health facility and preferably in the

same county or geographical area; c) users for the

same time period for consultation or treatment.

The attending physician shall justify the use of

individual transport ambulances, in a reasoned

manner. To further the grouping of users, the NEPT

may allow deviations of less than 10 km or 30

minutes journey, considering the travelling of the

first patient. The first patient assumes a critical

importance to define the route and to define the cost

of the transportation service. The first patient should

be the most distant patient to destination. A transport

on SVPT may carry a single patient in cases where

there are no other patients in the same time period or

along the same route, but this is an exception, and an

effort must be made to carry additional patients

where possible.

2.3 Transport Organization

Transportation must be ordered and scheduled at

least 48 hours before the journey. In individual

situations of an exceptional nature where this time

limit cannot be observed since no agreement has

been authorized between the requesting entity and

the carrier, the time limit of tacit acceptance and

approval of daily transportation requests is

determined by the computer system, at 15 hours and

30 minutes.

Requests for transportation are grouped

according to the schedules of supply of care to

which patients are intended, according to the

following guidelines:

• If the destination is within the geographic

boundaries of the patient’s county of origin or within

a radius of 10 km, these requests are grouped at one

hour intervals between delivery of the first and the

last patient;

• If the destination is outside the patient’s county of

origin, the interval between the delivery of the first

patient and the last can be two or four hours,

depending on the distance which, according to

geographical features, can be a range of 100 km to

130 km. This time is to be determined by the entity

responsible for organizing the transport.

The law states that the rule of the maximum

deviation should be applied to the "Optimization of

routes". Patients can be grouped for a journey,

regardless of parish or municipality, provided that

there is no deviation more than 10 km or 30 minutes,

provided the previous guidelines are observed.

The costs resulting from NEPT are the

responsibility of the requesting entity. Thus it

becomes important for the requesting entity to

optimize and streamline the process of transporting

non-emergency patients. It is not known how the

requesting entity must make the optimization of

NEPT, so this study hopes to be an important and

timely contribution.

2.4 Literature Review

The health care industry is rife with problems of

management and organization which have been

studied over the past several decades (Stiver et al.,

1982, Begur, Miller and Weaver, 1997). The

research attention to such problems is increasing

and, in the Western world, results from increased

demand for health care and the need to keep the

social cost of health care as low as possible. The

increased demand for health care has two vectors

justifying its growth: the democratization of access

to healthcare in developed societies and an aging

population. The increased demand for health care

causes transporting patients an important problem in

NonEmergencyPatientsTransport-AMixedIntegerLinearProgramming

263

the logistics of the health systems, since it is a

significant portion of operating costs. As far as

health logistic problems are concerned, an effective

transport service is now becoming an extremely

complex problem that has to be solved efficiently, so

it requires the best solution methodologies. Bellamy

et al. (2003) defines non-emergency transport needs

including ordinary situations in which a patient

simply cannot get to and from a healthcare facility

under their own power.

According to Wilken et al. (2005), patient

transportation is a critical part in providing

healthcare services. The authors discuss non-

emergency transportation in rural southern Illinois,

and they point out the importance of this issue.

Many individuals do not have the funds to pay for

public transportation and often public transportation

is not available or easily accessible to them so they

may miss regular doctor’s appointments, treatments,

and so on. As a consequence, the patient may

become more ill and then must be transported to a

medical facility by emergency transportation. Safei

(2011) studied the non-emergency medical

transportation services available in rural British

Columbia, and he reported the success of the

"Connections service" in enhancing rural and

northern communities’ access to healthcare services,

in particular among those with limited means and

resources.

Health authorities and health managers rely on

knowledge and state of the art technology to address

the logistic of health systems. Today, information

systems provide support for making the decision,

allowing gains in effectiveness and efficiency.

Transport in health care is a subject which has been

studied in the literature of optimization of logistic

processes for many years. Vaisblat and Albert

(2013) carried out one study which focused on the

scheduling of a special fleet serving the needs of

patients. Hains et al. (2011) addressed the issue of

safety and quality in NEPT. Recently Díaz-Parra et

al. (2014) published a state of the art review on the

problem of transport which included several variants

of problems, mathematical formulas, and techniques

used in problem solving.

One of the most studied routing problems is the

vehicle routing problem (VRP), which basically

aims to define a set of vehicle routes which are

optimized to visit a series of well-defined locations.

This problem presents a large number of variants

which address more specific situations. Berbeglia et

al. (2010) refer to the static or dynamic nature of

routing problems. In the static case, all the

information is known in advance and does not

change during the construction and implementation

phases. In the dynamic case, the available

information is updated (changed) during both the

construction phase and the implementation phase, by

virtue of new user requests. In these problems, the

proposed solution is a solution strategy that can be

changed with the passage of time. Typically,

dynamic situations of this kind occur in transport at

the request of users with special needs, which need

to be sent to the car which will transport them to

their destination. The dynamic aspect of this

problem stems from the fact that transportation

requests sometimes arise on the same day they need

to be met: this type of problem is called a DARP

(Dial-A-Ride Problem).

Both the static and dynamic versions of DARP

have received significant contributions, such as a

review of Cordeau and Laporte (2007) about models

and algorithms. Psaraftis (1988) examined a single

route of this problem in which clients request a

service to be made available as soon as possible.

Whenever a new order is entered, the system updates

the proceedings and attempts to accommodate the

new request on an existing, but only partially

complete, route. Meanwhile, Madsen et al. (1995)

presented an algorithm to a real case of the dynamic

DARP with multiple vehicles that met up to 300

requests daily to transport people with special needs.

One VRP variant is the “Vehicle routing

problems with profits” (Archetti et al., 2013). In this

type of problem, two different decisions have to be

made simultaneously—which which customers to

serve and how to sequence them in one or several

routes. In general, a profit is associated with each

customer that makes the customer more or less

attractive. The majority of real-world applications

require systems that are more flexible in order to

overcome some imposed constraints that may lead to

the selection of customers. To deal with the selection

of customers, the Team Orienteering Problem (TOP)

models can be used. The main difference between

the TOP and the VRP is related to the fact that not

all the TOP vertices of the graph (clients) must be

visited, as in the VRP. In the TOP, each customer

has an associated profit, and the routes have

maximum durations or distances. The choice of

customers is made by balancing their profits and

their contributions to the route duration or distance.

The objective is to maximize the total reward

collected by all routes while satisfying the time

limit.

The TOP is a fairly recent concept, first

suggested by Butt and Cavalier (1994) under the

name Multiple Tour Maximum Collection Problem.

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Later, Chao et al. (1996) formally introduced the

problem and designed one of the most frequently

used sets of benchmark instances. TOP has recieved

significant attention from the scientific community

(Vansteenwegen, Souffriau, Oudheusden, 2011;

Archetti, Speranza and Vigo, 2013) either in

presenting exact solution methodology (Archetti,

Bianchessi and Speranza, 2013) or in approximate

solution methodology (Hu and Lim, 2014).

Vansteenwegen and his team maintain a repository

of public instances (The Orienteering Problem: Test

Instances, 2014).

Gutiérrez-Jarpa et al. (2009) studied the problem

with fixed delivery and optional collections, utilizing

a mixed solution which uses VRP for delivery and

TOP for collection. The authors studied the

particular case of a single vehicle and presented a

new branch-and-cut method that allows the system

to solve larger instances. The method can solve

instances which include up to 90 vertices. The

authors refer to the need to extend the investigation

to cases with multiple vehicles and the development

of heuristics to solve large scale instances. Despite

the great practical interest that this modelling system

has for reverse logistics, the authors report they

found only study, Gribkovskaia et al. (2008) that

have applied tabu search to the single vehicle pickup

and delivery problem with selective pickups.

3 THE MODEL

The real problem studied in this article concerns the

non-emergency transport of patients from their

homes to the hospital and from the hospital back to

their homes. Currently, in Portugal the shuttle

typically collects patients from their homes to the

hospital for treatment and back again. This system

creates the situation – “many (origins)-to-one

(target)-to-many (destinations).” The way to

organize this transport is not clearly established in

law, which means that money is being wasted on the

waiting time of the vehicle, and patients often waste

time waiting for their transportation.

We decided by modelling the NEPT problem

with the TOP point of view. What will happen is that

given a list of non-urgent patients for whom

transportation was requested and given a fleet of

vehicles available with a capacity of eight seats, it is

the allocation of transport services to maximize the

occupancy of the vehicle and minimizing the

distance travelled. Patients who can not be included

in the routes of these vehicles will be transported in

specially requested for this ambulances service.

3.1 Mathematical Model

Since the group of patients is a severely constrained

situation, it is our choice to model the real problem

presented by the NHS as a Team Orienteering

Problem, solving for the set of available vehicles.

Vertices not included in the routes of the problem

are the users who will make the path by individual

transport.

Our model is based on the Team Orienteering

Problem and we follow the mathematical model

presented by Labadie et al (2012). We define the

search of different paths from a common start point

(i=1) to a common ending point (i=n). If we want

the start point to be the same ending point, we use

the same coordinates for both points.

We have established the following variables:

- set of binary variables that is equal to 1 if arc

(i, j) is selected in the path k and 0 otherwise.

- set of binary variables that is equal to 1 if

vertex j is in path k and 0 otherwise.

w - set of binary variables that is equal to 1 if

vertex j is the first vertex in the path after the start

point.

W - is the maximum value for the length of the

path that is a function of the distance of the first

vertex to terminal vertex and an allowed increase.

- a variable the controls a sequential number for

the vertices in the path.

TPS - is an auxiliary variable to linearize the

product of



To define the objective function we define three

parcels.

TIC is the Total Individual Cost, equivalent

to transporting each patient individually.

CIP is the

Cost of Individual Transport for patients who are

transported together with some other patient. This

parcel is the main savings when a patient is not

transported individually.

PC is Additional Patient

Cost (incremental cost) related to patients who are

transported together in same vehicle with the first

patient. This parcel is to pay the additional

deviations to collect patients in the route of the first

patient. By law, this cost is nowadays 20% of the

cost of the first patient in the route. In a solution

with an individual transport for all patients, CIP and

APC are equal to zero.

iin

TIC c c









()

mn n

jn ij

kj i

CIP c c x









NonEmergencyPatientsTransport-AMixedIntegerLinearProgramming

265



121

mnn

iin ij

kji

PC s c c TPS









Figure 1 explains these calculations considering

the transportation of two patients: A and B. COST1

is the solution cost using two vehicles/routes, both

starting at S and ending at E; COST2 is the solution

cost using only one vehicle, where A is first patient

in the route; while COST3 is the solution cost using

one vehicle, where B is the first patient in the route.

It is possible to establish the following relations:

COST1=TIC;

CIP(A)=a1+a2; CIP(B)=b1+b2;

COST2=TIC-CIP(B)+APC(B);

COST2=TIC-CIP(B)+20%CIP(A);

COST3=TIC-CIP(A)+APC(A);

COST3=TIC-CIP(B)+20%CIP(B).

Obviously, APC(A) and APC(B) depends on

which is the first patient in the route where they are

included.

Figure 1: Cost parcelles.

The mathematical formulation of the Mixed

Integer Linear Programming (MILP) is presented



min TIC CIP APC

(1)

subject to:

1, , 1; 1, ,

ij j

yj nk m











(2)

2, , ; 1, ,

ij ji

xj nk m











(3)

11,,

jin

xk m











(4)

12,,1













(5)

1, ,

wxk m

(6)

1, ,

jn j

WcxdMaxk m





 





(7)

1, ,

ij ij

cx W k m











(8)



(9)



1, , 1;

12,,;

1, ,

j i ij ij

XxM x j n



  





(10)

max

1, ,

wyLk m





 





(11)

2, , 1

1, ,

ij i

ij j

kkk

ij i j

TPS w

TPS y j n

TPS w y

























(12)

Expression (1) represents the objective function

to be minimized. It is intended to diminish the total

transport cost removing individual transportation as

much as possible, paying the necessary deviations to

collect patients in shared routes.

In terms of constraints, expression (2) assign

visited patients to only one route, and in expression

(5) patients could be visited by only one route.

Expression (3) ensures the flow conservation in each

node. Expression (4) ensures that a vehicle starts the

route from node 1. Expressions (6) (7) and (8)

establish the first patient in the route and calculate

the maximum length for the route according to the

distance from the first patient to the destination. The

law allows an increase in the length of the route to

collect additional patients, but this is currently

limited (

dMax ) to 10 km or 30 minutes.

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Expressions (9) and (10) eliminate sub tours, and the

capacity of vehicle is verified in expression (11).

Expression (12) linearizes the objective function.

3.2 NEOS Server Experiments

The model was implemented in AMPL language and

submitted to the NEOS Server to evaluate the

quality of solutions provided by this compact

formulation. Sixty-four Euclidean instances were

randomly created to perform the computational

experiments. The instances are divided into two sets,

based on the capacity of the vehicle: four places and

eight places available to transport the patients. The

size of the instances varies from twenty to one

hundred patients, and from three to ten vehicles.

Table 1: Experimental results (capacity = 4).

vehicles x capacity

nodes 3x4 4x4 7x4 10x4

1272.65 1184.0 1055.93 1055.93

1840.37 1693.42 1383.03

gap 5.43%

1299.10

gap 34.16%

2671.87 2498.09 2048.70

gap 20.73%

1776.02

gap 70.69%

3478.46 3248.65 2709.00

gap 17.63%

2362.49

gap 47.45%

4311.22 4074.27

gap 3.42%

3455.71

gap 17.67%

2987.28

gap 36.01%

5043.11 4802.52

gap 3.29%

4131.28

gap 12.73%

3718.51

gap 31.35%

5818.70 ----

mem_error

4931.43

gap 12.31%

4507.76

gap 30.52%

100

----

mem_error

7155.63

gap 4.55%

----

mem_error

8198.99

gap 77.61%

Table 2: Experimental results (capacity = 8).

vehicles x capacity

nodes 3x8 4x8 7x8 10x8

1245.22 1156.65 1048.91 1048.91

1812.94 1665.99 1376.02 1299.10

gap 30.13%

2561.95 2362.82 1915.61

gap 23. 26%

1706.82

gap 66.38%

3238.03 2974.04 2362.52

gap 31. 87%

2075.69

gap 68.38%

4003.97 3702.97 2932.47

gap 34. 77%

2523.03

gap 83.67%

4693.05 4389.54

gap 6.88%

3682.41

gap 36. 55%

3135.39

gap 81.95%

5450.52 ----

mem_error

4332.84

gap 32. 28%

3686.95

gap 70.91%

100

6785.59 6505.46

gap 13.98%

----

mem_error

----

mem_error

Using the NEOS Server with AMPL/

Gurobi/MINTO/scip/XpressMP, the MILP could not

find a solution for all instances. Experiments with

instances of different sizes were performed to find

the maximum number of vertices that it is possible

to solve optimally. Memory errors (“mem_error”)

were reported when the solution exceed 3GB of

memory limit. Also, it is only possible to dispose a

maximum of eight hours of computation with the

NEOS Server. When the maximum time was

achieved, it reported the best solution founded and

the correspondent gap. Tables 1-2 present these

results.

Apparently solving instances with vehicles with

larger capacity becomes easier and it was possible to

solve an instance with one hundred patients and

three vehicles.

4 HEURISTIC PROCEDURE

As expected, the compact model could not be used

to solve for large instances using the NEOS Server.

However, the NEOS Server can solve the large

instance’s terms of vertices using only one vehicle

(Orienteering Problem - OP). Considering this

situation, we developed a heuristic procedure to

solve the TOP that is based in successive OP

solutions. Iteratively, to the remaining unvisited

vertices, we solve the problem using the compact

formulation with one vehicle.

Figure 2: Heuristic Solution (100 nodes_10x8).

Figure 2 presents the solution obtained with

heuristic procedure to solve the large instance (one

hundred patients with ten vehicles each with

capacity of eight patients). The patients not included

in these ten routes must be transported individually.

In terms of computational time, 198 seconds was

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267

required to produce the 10 routes.

To confirm the results obtained with this

heuristic, we solved the 32 instances with capacity

equal 8. Table 3 compares the results obtained with

NEOS Server (“NeosS” line) and heuristic (“heur”

line) in this set of instances.

Table 3: Experimental results (capacity = 8).

vehicles x capacity

nodes 3x8 4x8 7x8 10x8

NeosS

heur

1245.22

1156.65

1048.91

NeosS

heur

1812.94

1665.99

1376.02

1299.10*

1311,43

NeosS

heur

2561.95

2362.82

1915.61*

1930,27

1706.82*

1748,65

NeosS

heur

3238.03

3240.36

2974.04

2994.45

2362.52*

2382.93

2075.69*

2118,02

NeosS

heur

4003.97

3702.97

3707.29

2932.47*

2917.65

2523.03*

2478.83

NeosS

heur

4693.05

4389.54*

4389.54

3682.41*

3624.17

3135.39*

3163.34

NeosS

heur

5450.52

----

5141.44

4332.84*

4240,82

3686.95*

3627,75

100

NeosS

heur

6785.59

6505.46*

6403.38

----

5398.40

----

4801.58

The heuristic obtained the optimal solution in 13

instances, and obtained better or equal result than

NEOS Server in 23 of 32 instances, representing

around 70%. We recall that for some instances we

present the values obtained by the NEOS Server at

the end of available computation time. These

instances are signalized with an asterisk “*” .

5 CONCLUSIONS

This work presents a model and a heuristic to solve

the problems posed by the non-emergency patient

transport in Portugal, given the new rules recently

established. The model follows the same principle of

the Team Orienteering Problem to select the patients

to be included and the routes providing the

maximum reduction in the costs. This approach is

different from VRP strategies because some vertices

are not visited. Particularly in this problem a patient

that it is not visited by the routes means that the

patient must be transported individually. Indeed, this

model establishes the best sets of patients that

should be transported jointly.

In this study, several Euclidean instances were

generated to test our approach. The model was

implemented in AMPL and our compact formulation

was used to solve the instances using the NEOS

Server. Instances with one hundred patients and ten

vehicles with a capacity for eight patients each could

not be solved within available computation time

provided by the NEOS Server.

A heuristic procedure based on iteratively

solving problems with one vehicle was presented,

and this heuristic provides good results in terms of

accuracy and computation time. Taking into account

the knowledge provided by this study, a greedy

heuristic and a genetic algorithm will be developed

to solve this problem.

In this work, it is assumed the transport is the

type “1 to many to 1”, meaning the all patients have

the same destination. For further work, we will study

the situation of several destinations. Also, this study

assumes an equal due date for all patients, but future

work will consider different due dates for patients.

Finally, our model was tested with real instances

with distances provide by Google Maps and

generates promising preliminary results.

ACKNOWLEDGEMENTS

This work has been partially supported by the

project GATOP - Genetic Algorithms for Team

Orienteering Problem (Ref PTDC/EME-GIN/

120761/2010), financed by national funds by FCT /

MCTES, and co-funded by the European Social

Development Fund (FEDER) through the

COMPETE - Programa Operacional Fatores de

Competitividade (POFC) Ref FCOMP-01-0124-

FEDER-020609.

This work has been partially supported by FCT –

Fundação para a Ciência e Tecnologia within the

Project Scope: PEst-OE/EEI/UI0319/2014

The authors would like to thank the NEOS

Server at the University of Wisconsin in Madison

for providing support to this study.

The authors would like to thank the anonymous

reviewers for their valuable comments and

suggestions to improve the paper.

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