Pareto Front Approximation by Ant Colony Optimization

Jaroslav Janáček and Marek Kvet

University of Žilina, Faculty of Management Science and Informatics, Univerzitná 8215/1, 010 26 Žilina, Slovak Republic

Keywords: Discrete Location Problems, Conflicting Criteria, Pareto Frontier Approximation, Ant Colony.

Abstract: The Pareto frontier of multi-objective problem solutions denotes the unique exact solution to a problem with

two or more equivalent objectives. Even when the number of problem solutions is finite, determining the

precise Pareto frontier is a difficult task. Different metaheuristics can therefore provide a user with a decent

approximation of the Pareto frontier in a reasonable amount of time, whereas the exact computational time-

intensive approaches cannot. The acceptable computational time of metaheuristics counterbalances a

solution's deviation from the Pareto frontier. This contribution describes one of a spectrum of metaheuristics

implemented with the objective of locating non-dominated solutions to the public service system design

problem involving two competing criteria. The metaheuristic minimizes the difference between the present

set of non-dominated solutions and the Pareto front by applying the ant colony optimization principle. A series

of numerical experiments with benchmarks for which the exact Pareto frontiers are known are used to evaluate

the efficacy of the proposed metaheuristic. Even though the proposed method is applicable anywhere, the

used dataset comes from an Emergency Medical Service system in Slovakia, which belongs to the generally

known wide class of public service systems.

1 INTRODUCTION

One of those crucial application spheres where many

advanced methodologies of operations research can

frequently be met is the creation of various service

systems. We are now able to address a wide range of

challenging problems that were previously

unsolvable due to huge advancements made in

numerous technical disciplines, from hardware to

cutting-edge software technology. One of such

examples is the challenging problem of establishing a

service system (Ahmadi-Javid et al, 2017, Avella,

Sassano, Vasil'ev, 2007, Brotcorne, Laporte, Semet,

2003, Current, Daskin and Schilling, 2002). Let us go

more specific about service systems.

Typically, a service system consists of a few

components that have a structure, behavior (which

may be characterized as a business process), and a

purpose (people, facilities, tools, and/or software

applications). The premise that a service system can

be regarded as a work system that generates a certain

type of specified services is an easier, but more

constrained description (Doerner et al, 2005,

Gendreau, Potvin, 2010, Gopal, 2013, Ingolfsson,

Budge, Erkut, 2008, Jánošíková, 2007, Jánošíková,

Žarnay, 2014).

In general, service systems can be split into two

sizable categories. Public service systems are based

on distinct presumptions, whereas private service

systems are typically developed to deliver the

maximum profit to their managers, founders,

shareholders, and operators regardless of the number

of clients served or equal access to given service.

Public service systems are designed to ensure that

all locals will receive services, regardless of financial

gain or loss. They are required by law to exist. These

systems comprise many different things, such as state

administration, emergency medical services (EMS),

and many more (Ahmadi-Javid et al, 2017, Brotcorne,

Laporte, Semet, 2003, Ingolfsson, Budge, Erkut,

2008, Marianov, Serra, 2002). The public service

system design problem is a member of the family of

location problems, which have been researched and

successfully resolved by numerous authors (Avella,

Sassano, Vasil'ev, 2007, Kvet, 2018).

Several factors must be taken into account while

looking for a solving method for this class of huge

location problems: The issues' combinatorial nature

suggests that mathematical programming techniques,

which have some specificities of their own, were used

to solve them. The constraint of available resources is

another significant factor. Additionally, the service

200

Janá

cek, J. and Kvet, M.

Pareto Front Approximation by Ant Colony Optimization.

DOI: 10.5220/0012262900003639

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 13th International Conference on Operations Research and Enterprise Systems (ICORES 2024), pages 200-206

ISBN: 978-989-758-681-1; ISSN: 2184-4372

must be concentrated in a number of service locations

rather than being offered everywhere. Naturally, the

customers who are served can travel to these centers

or a team can go from a service center to the

emergency location. As a result, the challenge in

designing a public service system typically lies in

determining the best network topology for service

centers to meet a particular criterion. Based on the

preliminary analyses given, the weighted p-median

problem formulation is one of the modeling

approaches that is most frequently employed.

The precise shape of the objective function and

the modeling approach itself determine whether the

problem can be solved. The so-called radial strategy

can be used in place of the commonly used location-

allocation form to tackle substantially bigger issue

instances (Avella, Sassano, Vasil'ev, 2007, Kvet,

2018). The problem is significantly simpler to solve

if the optimization criterion utilized has a min-sum

form as opposed to one where the objective function

has a min-max form with certain link-up constraints.

The second drawback is brought on by the

limitation that only one target can be optimized.

Large public service systems, like an EMS, are

known to be complex systems with a variety of

conflicting demands made by various stakeholder

groups (Arroyo et al, 2010, Grygar, Fabricius, 2019,

Janáček, Fabricius, 2021). Consequently, the primary

focus of this research study is on creating multi-

objective service systems. Only two opposing aims

will be considered for the sake of simplicity.

The primary scientific contribution of this study

is to bring a new heuristic approach to address the

issue of constructing two-objective service systems.

A so-called Pareto frontier of service system designs

must be built since a multi-criteria optimization

presumes providing a condensed set of options from

which the final system design is to be selected. It

takes a lot of time to obtain the entire Pareto frontier

(Arroyo et al, 2010, Grygar, Fabricius, 2019, Janáček,

Fabricius, 2021). Therefore, from a practical

standpoint, the creation of effective heuristics is

required. The quality of the set of solutions that is

produced is examined and empirically confirmed on

a dataset from the actual world in this study.

The structure of this article takes the following

form. Section 2 is devoted to the mathematical

formulation of the problem and the conflicting criteria

explanation. The notion of a Pareto frontier is

discussed together with the method for different

Pareto sets comparison. Section 3 provides the

readers with the principle of Pareto frontier

approximation by gradual refinement. In the fourth

section, we describe the ant colony optimization and

all the ideas behind suggested solving algorithm. The

fifth section contains the results of performed

experiments with real-world data and finally, the last

section is devoted to concluding remarks.

2 NON-DOMINATED PUBLIC

SERVICE SYSTEM DESIGNS

It is vital to clarify the Pareto frontier and give the

readers a mathematical description of the problem

before going into detail about the incremental

refinement strategy itself. Let us focus on the problem

specification, now.

Finding the stations from which ambulances are

sent to demand sites is a difficulty in EMS design. For

the mathematical formulation of the problem,

suppose that a finite set I of candidates is given. The

candidates are often selected from network

components that meet specific criteria for EMS

station location. The resulting selection of service

center locations must include exactly p entries (p is an

integer less than or equal to the cardinality of I) in

order for the given aim to take on its best value due to

several personal, technological, or other constraints.

A zero-one variable, y

∈{0, 1}, which equals one if a

center should be placed at i∈I and zero otherwise, will

be used to simulate the choice of where to place a

service center (EMS station). A vector y of location

variables y

can therefore be used to define any

solution to the corresponding p-location issue. The

following expression (1) can describe the basic

model.

() { }

min : 0, 1 , ,

yiIyp

∈





∈∈=









(1)

If one wanted to make the EMS system design

problem more general, there could be added at least

one extra objective. Under the assumption that m

denotes the cardinality of the set I and n denotes the

cardinality of the set J, the former model (1) may be

rewritten into the form of (2).

() () { }

min , : 0, 1 ,

fyp





∈=









yyy

(2)

As mentioned in the paper's introduction,

combining two criteria might provide a variety of

difficulties for the decision-making process,

especially when the aims are incompatible. We can

only concentrate on two objective functions, f

(y) and

(y), which will be referred to as so-called system and

fair criteria, respectively. To formulate them in a

Pareto Front Approximation by Ant Colony Optimization

201

mathematical way, several notations are necessary to

be introduced.

Let J represent the set of locations of system users

(service recipients). Analogically, let the symbol I

stand for the set of candidates for facility locations.

The sets I and J can be equivalent. The number of

unique users located at j from J will be represented by

the value of b

. The quantity of anticipated demands

during a specific period is one possible interpretation

of this nonnegative integer coefficient. It can be

understood as the weight of location j, though.

According to the possible center location i, the

disutility for a patient located at j will be indicated as

. Despite the benefits of integer values, the value of

need not be an integer. It is important to remember

that service requests come in at random; therefore, the

most nearby station need not be used to cover the

current demand that has emerged anywhere. From a

mathematical standpoint, it is assumed that r nearest

located centers participate in offering the service to

users, and q

signifies the possibility that the k-th

nearest center is the one that is closest and easily

accessible at the time the demand occurs. To

complete the formulations of the system and fair

objective functions, let the function min

return the k-

th smallest element from the list in the function's

parameter. The system criterion f

(y) optimizes the

average distance between system users and the closest

available center. It can be formulated by (3).

Optimization of the average distance may be achieved

by minimization of the sum of distances.

()

{}

min : {1, ..., }, 1

jkkij i

fbqdimy

=∈=



(3)

If we wanted to evaluate the average distance

AvgDist, it could be done in the following way

described by (4).

()

AvgDist



(4)

The number of users whose distance from the

closest facility exceeds the radius D is expressed by

the fair objective function value f

(

), which was

developed in accordance with formula (5). To provide

certain level of fairness (Bertsimas, Farias, Trichakis,

2011, Buzna, Koháni, Janáček, 2013).

() { }

{}

()

max 0,

min : , 1

ij i

fbsm

sm sign d i I y D

=∈=−



(5)

There is no doubt that the criteria (3) and (5) are

in direct conflict. It suggests that improving one

would inevitably make the other worse. The

aforementioned goal conflict can be resolved by

creating a full Pareto frontier of solutions, or at least,

its approximation. In other words, rather than one

final system design, a specific small subset of options

are presented. Naturally, in order to select one of the

available options and arrive at the resulting system

design, politics, negotiation, and experts must be

involved. Let us focus on Pareto frontier, now.

A Pareto frontier is often made up of a few

solutions that satisfy non-dominance for each pair of

its members. No matter what form a feasible solution

P takes, it may be evaluated using the two criteria

(P) and f

(P) in the bi-criteria optimization. The

non-dominance can be explained by the following: A

solution P is referred to as a non-dominated solution

if [f

(P), f

(P)] ≠ [f

(R), f

(R)] matches the inequality

(P) < f

(R) or f

(P) < f

(R). Then, a straightforward

explanation of the Pareto frontier is given in the

following Figure. 1.

Figure 1: Explanation of the Pareto frontier.

The red solutions do not belong to the Pareto

frontier because of being dominated by the green

ones. Since there are no solutions dominating the

green elements, both of them must be included into

the Pareto frontier (Grygar, Fabricius, 2019, Janáček,

Fabricius, 2021).

Sometimes, symbols MLM and MRM are used to

denote the most left and the most right members of

the Pareto frontier. These bordering solutions can be

computed easily making use of a simple mathematical

model solvable usually in a short time. We will

concentrate our efforts on creating a good

approximation of the Pareto frontier because it is a

challenge to obtain the entire Pareto set. The sequence

of noNDSS (non-negative integer value) solutions

…,

noNDSS

ordered according to increasing values of

will be used to represent the approximate set of non-

dominated solutions (NDSS). The bordering solutions

and

noNDSS

must be found to be very close to the

most left and the most right solutions of the Pareto

frontier in terms of the values of f

(

) and f

(

), in

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

202

order to achieve a useful approximation. Under these

presumptions, so-called area A(NDSS) computed in

accordance with expression (6), can be used to assess

the accuracy of the NDSS as the Pareto frontier

approximation. The size of the grey polygon in Figure

1 is represented by the A(NDSS) to make it easier to

comprehend.

() ( )

() ()

11 1

22 2

()

noNDSS

k noNDSS

ANDSS dif dif

dif f f

−

=−



(6)

Whenever the Pareto frontier is needed to be

approximated by a set NDSS, the quality of the

approximation must be evaluated. For this purpose,

so-called gap may be used. The gap can be defined as

follows: Under the assumption that PF denotes the

original complete Pareto frontier and NDSS denotes

its approximation, the gap can be evaluated in

percentage according to the expression (7).

()()

100*

()

ANDSS APF

gap

APF

−

(7)

3 GRADUAL REFINEMENT

SCHEME

Many alternative methods, some of which may also

be based on the decrementing neighbourhood search

algorithm, can be used to generate a Pareto frontier or

at least a good approximation of one. The schema of

its gradual refining is one of the approaches that

might be used in the development of the NDSS

(Janáček, Kvet, 2022a, Janáček, Kvet, 2022b,

Janáček, Kvet, 2022c, Kvet, Janáček, 2022).

The two-element initial NDSS of the most left and

the most right bordering solutions of the Pareto

frontier serves as the basis for the process. These

bordering solutions are simple to compute, and

getting them usually doesn't take too much effort. The

previously mentioned refining process is repeated.

This indicates that the results of processing one round

produce a set of NDSS data that may be utilized as an

input set for the subsequent inspection procedure. The

results of a process that is repeated more than once

may have different outcomes since the inbuilt

decrementing algorithm may employ random actions.

As a result, the inner cycle is nested inside a time-

controlled cycle, which keeps repeating the inner

cycle until the time limit is reached. In other words,

NDSS refining continues until a time limit prevents

algorithm performance.

The gradual refinement approach processes the

input NDSS solutions iteratively one by one. The

elements of NDSS are expected to form a sequence y

…, y

noNDSS

. If any solution y

is processed, then a

neighborhood search algorithm may be applied to

find candidates for NDSS updating. As NDSS can

change during one run of the algorithm, the solution

corresponding with the k-th position may also change.

If it happens, the algorithm is applied once again to

this new solution y

, otherwise the following solution

k+1

is processed. If k = noNDSS-1, the refinement

process terminates (Janáček, Kvet, 2022b, Kvet,

Janáček, 2022).

4 ANT COLONY OPTIMIZATION

The ant colony optimization algorithms imitate in

general cooperation of ant colony members in

searching food which is distributed in nodes of a

network of possible ants’ moves. In most of

implementations, an ant is an agent, who searches for

an improving solution in a finite set of problem

solutions equipped with a topology given by system

of neighbourhoods. The ant chooses its inspected way

in the network which vertices are represented by

individual solutions and which edges correspond to

passible moves from one solution to a neighbouring

one. As the set of feasible solution is too vast to be

able to record real path in the associated network,

only some attributes of the path inspection are taken

into account. Performance of an ant starting at a given

solution will be determined by the neighbourhood

searching strategy and buy swap operations, which

implement moves between the network vertices. The

searching strategy is determined by a combination of

two parameters thr and maxNos, where the first one

gives minimal improvement to consider a move

admissible and the second parameter gives the

number of admissible moves, from which the best one

is realized. The swap operation replaces one location

of a current solution by a location which is not

included in the solution.

Instead of recording the inspected path in detail,

we reduce the path description to the set of location

which have been subsequently included into the

starting solution. These recorded entries will be

considered in the phase of laying pheromone. A

detailed description of the ant’s search and

pheromone laying follows.

The ant starts with a given feasible solution y of

the p locations saved in the list P of the locations

Pareto Front Approximation by Ant Colony Optimization

203

selected from the set I of all possible service center

locations.

Based on thickness of the pheromone layer, the

ant randomly choices the strategy given by a pair of

the parameters thr and maxNos.

Following the chosen strategy, the ant searches

the neighbourhood of the current solution while

constantly updating the NDSS and evaluating each

found admissible swap operation applied to the

exchange of location i∈P for j∈I-P from the point of

move to a new current solution.

The admissible operation is characterized by a

decrement Idec of the A(NDSS) caused by its

performing. The inserted location j has its pheromone

layer F(j) and the resulting fitness value is Dec*F(j).

The ant’s decision on the best swap for the move to

the new solution is performed according to the result

of a sequence of comparisons. The recently appointed

swap operation with fitness value MFit is compared

to fitness CFit of a candidate and it is decided on

update according to a random trial with probability

CFit/(CFit +MFit) in favor of the candidate. The ant’s

search finishes, when either whole neighbourhood is

inspected or ImaxNos candidates are evaluated.

The ant’s search terminates with failing of finding

an admissible candidate for the move to a new current

solution. The difference between the A(NDSS) before

the ant’s search and A(NDSS) after the search denoted

by Dec is used to update the pheromone layer F(s) of

the chosen strategy s and all inserted locations j using

the following formula (8).

() ()

Fs Fs

InitArea

(8)

The final pheromone adjustment is performed

with a pheromone layer of each object according to

the expression F(s) = (1 - ρ)*F(s), where ρ is an

evaporating coefficient.

5 NUMERICAL EXPERIMENTS

Suggested ant colony optimization for Pareto fro

frontier nt approximation was explained in previous

sections of this contribution. To verify its efficiency

and accuracy, several computational analyses needed

to be performed. The content of this section is aimed

at the results of performed numerical experiments.

Let us concentrate on available software tools and

technical parameters of used machine, first.

All computational experiments reported in this

study were performed making use of Java

programming language within the NetBeans

development kit. The algorithms were run on a

common notebook equipped with the 11

Gen Intel®

Core™ i7 1165G7 2.8 GHz CPU and 40 GB RAM.

After having introduced necessary software and

hardware tools for this computational study, let us

describe solved problems and their most important

characteristics.

As far as the set of used problem instances is

concerned, we took the benchmarks from our

previous research (Grygar, Fabricius, 2019, Janáček,

Fabricius, 2021, Janáček, Kvet, 2020 , Janáček, Kvet,

2021, Janáček, Kvet, 2022a, Janáček, Kvet, 2022b,

Janáček, Kvet, 2022c, Kvet, Janáček, 2022).

Mentioned dataset represents the existing EMS

system operated by private agencies - service

providers in eight autonomous higher territorial units

in Slovakia. The list of problem instances covers the

regions of Bratislava (BA), Banská Bystrica (BB),

Košice (KE), Nitra (NR), Prešov (PO), Trenčín (TN),

Trnava (TT) and Žilina (ZA). It must be noted that all

network nodes represent both the set of candidates for

service center locating and the set of clients being

provided with urgent healthcare service as well. But

generally, the sets of candidate locations and system

users’ locations may differ. The number of users b

located in each node j from J were taken from the

official Slovak statistical analysis and the values were

rounded up to hundreds.

As the objective function f

(y) follows the concept

of so-called generalized disutility (Grygar, Fabricius,

2019, Janáček, Fabricius, 2021, Kvet, 2014), the

parameter r was set to 3. The associated probability

coefficients q

were set so that q

= 77.063, q

16.476 and q

= 100 - q

- q

. These values correspond

also to the data used in our previous research to make

the results of different methods comparable. More

details about the parameter settings suitable for the

objective function (3) can be found in (Jankovič,

2014).

Parameter D used in the fair objective (5) was set

to the value of 10.

The following summary reported in Table 1

brings the overview of used benchmarks. The

structure of the table is designed so that each row

corresponds to one solved problem. The first column

of the table is used to identify the instance by the

abbreviation of the region. The second column

denoted by |I| reports the cardinality of the set I. In

other words, there are reported the numbers of

candidates for service center locating, from which

exactly p elements are to be chosen. The values of p

are reported in the third column of the table. The right

part of Table 1 is used to summarize the most

important characteristics of the complete Pareto

fronierts. The column denoted by NoS gives the

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

204

number of solutions creating the entire Pareto set. In

the column denoted by A(PF) we provide the readers

with the size of the polygon defined by the Pareto

frontier elements as suggested by the expression (6).

Table 1: Benchmarks sizes and the exact Pareto frontiers

characteristics.

Region |I| p

)

BA 87 14 34 569039

BB 515 36 229 1002681

KE 460 32 262 1295594

NR 350 27 106 736846

PO 664 32 271 956103

TN 276 21 98 829155

TT 249 18 64 814351

ZA 315 29 97 407293

The achieved results are summarized in Table 2

and Table 3 which take the same structure. Since

many heuristic methods may perform a random trial

or they need generating random numbers, we have

performed ten runs of the algorithm for each

benchmark, and we report the average values of all

studied parameters. While Table 2 contains the results

of experiments with the first four datasets, Table 3

summarizes the second half of problem instances.

Each column of the tables corresponds to one solved

problem and each row is used to one studied

characteristic.

Table 2: Results of ant colony optimization– part 1.

Region BA BB NR KE

[s] 300.0 320.0 315.5 302.4

noNDS

33.0 206.6 250.6 102.1

Gap [%] 1.47 1.03 2.21 0.72

oTR 1227.7 1.9 1.5 9.1

oTOR 4205.1 8.9 10.8 52.2

Table 3: Results of ant colony optimization – part 2.

Region PO TN TT ZA

[s] 357.6 301.3 300.5 303.0

noNDS

263.5 94.4 62.2 92.4

[%] 2.30 0.72 0.07 0.30

oTR 1.8 32.0 56.0 15.8

oTOR 5.0 110.3 220.5 64.5

It must be noted that the computational process

was limited to five minutes of processing. Let the

symbol CT [s] denote the average computational

time. The second studied feature consists in the

average number of found solutions, which

approximate the original Pareto frontier. This result is

denoted by noNDSS. Since we do not consider it

useful to report the exact values of areas computed

according to (6), we evaluate the quality of

approximation by gap computed by (7). The column

denoted by NoTR reports the number of time runs and

NoTOR denotes the number of outer runs.

6 CONCLUSIONS

Bi-criteria optimization is crucial when the specifics

of a problem do not permit the use of a simple model

that minimizes only one objective function. A typical

example is the design of emergency medical services.

Operations researchers and other experts have

concentrated their efforts on the development of

heuristic methods that can approximate the optimal

Pareto frontier in a much shorter amount of time, as

the search for the optimal Pareto frontier has proven

to be an arduous endeavor. We provided numerical

experiment results for evaluating the quality of the

proposed algorithms. Based on the reported results, it

can be concluded that the proposed algorithm

substantially extends the state-of-the-art tools for

solving specific location problems involving the

optimization of two contradictory objectives.

Future research may concentrate on new

advanced algorithms that generate a close

approximation of the Pareto frontier or on modifying

certain existing techniques to obtain more precise

results. Application of self-learning methods to

parameter adjustment represents a fruitful research

direction.

ACKNOWLEDGEMENT

This work was financially supported by the following

research grants: VEGA 1/0216/21 “Designing of

emergency systems with conflicting criteria using

tools of artificial intelligence”, VEGA 1/0077/22

“Innovative prediction methods for optimization of

public service systems”, and VEGA 1/0654/22 “Cost-

effective design of combined charging infrastructure

and efficient operation of electric vehicles in public

transport in sustainable cities and regions”. This

paper was also supported by the Slovak Research and

Development Agency under the Contract no. APVV-

19-0441.

REFERENCES

Ahmadi-Javid, A., Seyedi, P. et al. (2017). A survey of

healthcare facility location, Computers & Operations

Research, 79, pp. 223-263.

Pareto Front Approximation by Ant Colony Optimization

205

Arroyo, J. E. C., dos Santos, P. M., Soares, M. S. and

Santos, A. G. (2010). A Multi-Objective Genetic

Algorithm with Path Relinking for the p-Median

Problem. In: Proceedings of the 12th Ibero-American

Conference on Advances in Artificial Intelligence,

2010, pp. 70–79.

Avella, P., Sassano, A., Vasil'ev, I. (2007). Computational

study of large scale p-median problems. Mathematical

Programming 109, pp. 89-114.

Bertsimas, D., Farias, V. F., Trichakis, N. (2011). The Price

of Fairness. In Operations Research, 59, 2011, pp. 17-

31.

Brotcorne, L, Laporte, G, Semet, F. (2003). Ambulance

location and relocation models. European Journal of

Operational Research, 147, pp. 451-463.

Buzna, Ľ., Koháni, M., Janáček, J. (2013). Proportionally

Fairer Public Service Systems Design. In:

Communications - Scientific Letters of the University

of Žilina 15(1), pp. 14-18.

Current, J., Daskin, M. and Schilling, D. (2002). Discrete

network location models, Drezner Z. et al. (ed) Facility

location: Applications and theory, Springer, pp. 81-118.

Doerner, K. F., Gutjahr, W. J., Hartl, R. F., Karall, M. and

Reimann, M. (2005). Heuristic Solution of an Extended

Double-Coverage Ambulance Location Problem for

Austria. Central European Journal of Operations

Research, 13(4), pp. 325-340.

Gendreau, M. and Potvin, J. (2010). Handbook of

Metaheuristics, Springer Science & Business Media.

Gopal, G. (2013). Hybridization in Genetic Algorithms.

International Journal of Advanced Research in

Computer Science and Software Engineering, vol. 3,

pp. 403–409.

Grygar, D., Fabricius, R. (2019). An efficient adjustment of

genetic algorithm for Pareto front determination. In:

TRANSCOM 2019: conference proceedings,

Amsterdam: Elsevier Science, pp. 1335-1342.

Ingolfsson, A., Budge, S., Erkut, E. (2008). Optimal

ambulance location with random delays and travel

times. Health care management science, 11(3), pp. 262-

274.

Janáček, J., Fabricius, R. (2021). Public service system

design with conflicting criteria. In: IEEE Access:

practical innovations, open solutions, ISSN 2169-3536,

Vol. 9, pp. 130665-130679.

Janáček, J. and Kvet, M. (2020). Adaptive Path-Relinking

Method for Public Service System Design. In: 38th

International Conference on Mathematical Methods in

Economics, Brno: Mendel University in Brno, 2020,

pp. 229-235.

Janáček, J., Kvet, M. (2021). Emergency Medical System

under Conflicting Criteria. In: SOR 2021 Proceedings,

pp. 629-635.

Janáček, J., Kvet, M. (2022a). Adaptive swap algorithm for

Pareto front approximation. In: ICCC 2022: 23rd

International Carpathian Control conference, Sinaia,

Romania, Danvers: IEEE, 2022, pp. 261-265.

Janáček, J., Kvet, M.. (2022b). Repeated Refinement

Approach to Bi-objective p-Location Problems. In:

INES 2022: Proccedings of the IEEE 26th International

Conference on Intelligent Engineering Systems 2022,

pp. 41-45.

Janáček, J., Kvet, M. (2022c). Pareto Front Approximation

using Restricted Neighborhood Search. In: Proceedings

of the 40th International Conference on Mathematical

Methods in Economics, 2022, Jihlava, pp. 141-147.

Jankovič, P. (2016). Calculating Reduction Coefficients for

Optimization of Emergency Service System Using

Microscopic Simulation Model. In: 17th International

Symposium on Computational Intelligence and

Informatics, pp. 163-167.

Jánošíková, Ľ. (2007). Emergency Medical Service

Planning. In: Communications - Scientific Letters of

the University of Žilina 9(2), pp. 64-68.

Jánošíková, Ľ. and Žarnay, M. (2014). Location of

emergency stations as the capacitated p-median

problem. In: Quantitative Methods in Economics

(Multiple Criteria Decision Making XVII). pp. 117-

123.

Kvet, M. (2014). Computational Study of Radial Approach

to Public Service System Design with Generalized

Utility. In: Digital Technologies 2014, Žilina, Slovakia,

pp. 198-208.

Kvet, M. (2018). Advanced radial approach to resource

location problems. In: Developments and advances in

intelligent systems and applications. Cham: Springer

International Publishing, 2018, Studies in

computational intelligence, 718, pp. 29-48.

Kvet, M., Janáček, J. (2022). Directed Search for Pareto

Front Approximation with Path-relinking Method. In:

Proceedings of the 40th International Conference on

Mathematical Methods in Economics, 2022, Jihlava,

pp. 212-217.

Marianov, V. and Serra, D. (2002). Location problems in

the public sector, Facility location - Applications and

theory (Z. Drezner ed.), Berlin, Springer, pp 119-150.

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