Multi-Criteria Service System Designing Using Tabu Search Method

Marek Kvet and Jaroslav Janáček

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

Keywords: Multi-Criteria Location Problems, Pareto Front, Tabu Search Heuristics, Algorithm Efficiency Improvement.

Abstract: Designing a good public service system that provides a geographical region with service through a specified

number of service centers is a very difficult task, particularly when multiple quality evaluation criteria are

applied. A Pareto front of public service system designs is a very useful instrument for any designer who must

consider multiple requests from public representatives. Due to the computational difficulty of determining the

Pareto front, a number of heuristic approaches have been developed. One of these techniques, gradual

refinement, proved to be quite effective, but its performance could be enhanced by eliminating the repetition

of some rudimentary swap routines. This contribution focuses on the application of tabu search features to

enhance and increase the efficacy of the gradual refinement process by suspending the routines' few useful

applications. The resulting metaheuristic is validated through numerical experimentation using benchmarks,

and the approximations of the Pareto front are compared to the exact Pareto fronts.

1 INTRODUCTION

Establishing a new service system, improving an

existing one, or solving other similar issue involving a

public service system involves figuring out the best

locations for service centers, stations, or facilities that

are stocked with the tools, personnel, or other

resources required to meet customers’ demands. It

goes without saying that the combinatorial nature of

the aforementioned challenges necessitates the use of

various mathematical modeling techniques, software

development expertise, or other advanced abilities. As

a result, while making strategic decisions,

professionals in operations research cannot be

disregarded. Because of enormous and quick

advancements being made in practically all relevant

domains, we are able to quickly and effectively

produce good results for significant problem instances

(Ahmadi-Javid et al, 2017, Current, Daskin and

Schilling, 2002, Ingolfsson, Budge, Erkut, 2008).

Speaking of the designing of service systems, it

must be recognized that we do not study private

service systems in this study because they are

primarily focused on maximizing profit regardless of

the number of users covered by the system or the

degree of equity in service accessibility. As a result,

we exclusively focus on public service systems, the

existence of which is typically guaranteed by

legislation. Public service systems are designed to

ensure that all local citizens will receive services,

regardless of financial gain or loss (Jánošíková, 2007,

Jánošíková, Žarnay, 2014). According to science, the

discrete network location problem family includes the

public service system design problem, which has been

researched and successfully resolved by numerous

authors (Brotcorne, Laporte, Semet, 2003, Doerner et

al, 2005, Marianov, Serra, 2002). The weighted p-

median problem is the most concrete form of the

problem, and one of the most popular modeling

notions is what comes next.

Common mathematical models may have a

number of serious drawbacks, one of which is the

limitation that only one objective function can be

maximized/minimized. Large service systems are

complicated systems with a variety of competing

demands made by various stakeholder groups

involved in the decision-making process, and not all

of them lend themselves to abstraction. Therefore,

multi-objective service system optimization is the

focus of attention. Only two opposing aims will be

taken into account for the sake of simplicity in this

paper. Another obstacle connected with large

mathematical models consists in the complexity of

most exact methods, which usually disables their

application for practice. On the other hand, experts

have found a solution also for such a situation.

As well as existing conventional exact methods

based usually on the branch and bounds principle,

Kvet, M. and Janá

cek, J.

Multi-Criteria Service System Designing Using Tabu Search Method.

DOI: 10.5220/0012262800003639

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 193-199

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

193

newer heuristic algorithms, metaheuristics, and more

advanced evolutionary approaches to the optimization

issues have been created. Verifying that the best

answer identified is the optimal solution is a particular

weakness of practically all exact approaches. Some

modeling strategies, such as the radial approach

(Avella, Sassano, Vasil'ev, 2007, Kvet, 2014, Kvet,

2018), can greatly speed up the associated solving

process. Nonetheless, the time it takes to verify these

accurate methods is sometimes prohibitive. However,

heuristic approaches allow us to get a decent answer

in much less time. Furthermore, public service system

design can address bi-criteria location challenges,

which present a similar challenge with the same

precise techniques of slow performance. The Pareto

front is a unique collection of solutions that must be

looked for when there are two or more objectives to be

optimized. Since completing the complete

inextensible Pareto front requires a lot of effort

(Grygar, Fabricius, 2019, Janáček, Fabricius, 2021),

academics have focused on developing approximate

methods and efficient heuristics (Arroyo et al, 2010,

Gendreau, Potvin, 2010, Gopal, 2013).

This contribution is focused on application of tabu

search features to enhance the gradual refinement

process developed especially to approximate the

original Pareto front. The main goal is to increase its

efficiency by suspending the little useful applications

of associated routines. Obviously, suggested

algorithm has been experimentally verified and the

obtained results are reported here.

The structure of this paper is organized according

to the following scheme: The main goal of the first

section was to introduce the problem and to place it

into a wider scientific context. The second section

discusses the Pareto front of bi-criteria location

problem solutions. In section 3, we introduce the

neighborhood search with tabu moves. The fourth

section summarizes the numerical experiments. Here,

we provide the readers with several comments on the

computational study. The last section is dedicated to

the conclusions and future research directions.

2 PARETO FRONT OF

BI-CRITERIA LOCATION

PROBLEM SOLUTIONS

A discrete location problem can be concisely

described as the task to select p locations from the set

of m candidate locations so that a given criterion

value is minimal. Thus the set of all feasible problem

solutions Y can be defined as Y={y: y⊂{1, …, m},

y=p}.

As concerns quantified criteria of the individual

elements of Y, they depend on the sort of the real

problem formulated as the location problem. In the

case of private service system design, the objective is

often minimal total cost of service distribution from

service centers to the customers. The total cost is

usually proportional to the sum of weighted distances

from customers to the closest service center.

Considering a public service system design, the

situation is more complex due to more points of view

at the system utility. In principle, the applied criteria

can be divided into two classes called system criteria

and fairness criteria. The system criterion minimizes

disutility perceived by an average system user and the

fairness criterion minimizes disutility perceived by

the worst situated minority of the system users. The

system criterion can be represented by an average

response time of the system subject to the assumption

that a user’s demand is satisfied from the nearest

available service center. The fairness criterion can be

represented by the number of users’ demands, which

are situated outside a radius R from the nearest

located service center.

Taking into account random occurrence of the

users’ demands and limited capacity of the service

centers, the nearest available center need not mean the

nearest center due to possible occupancy of the

nearest center. This situation can be modelled by

series q

, q

, …, q

of probability values, where q

expresses the probability that the k-th nearest service

center is the nearest available one. If t

denotes time

necessary for transport of service from a possible

service center location i to a user located at location

j∈{1, …, n} and if b

denotes frequency of the

demand occurrence at a user’s location j, then the

system objective function f

(y) can be defined by (1).

()

{}

min :

kj k ij

fqbti

=∈



(1)

In formula (1), the min

operation performed on a

set of values returns the k-minimum value from the

set.

The fairness criterion can be expressed by (2), see

(Bertsimas, Farias, Trichakis, 2011, Buzna, Koháni,

Janáček, 2013).

()

{}

(

)

max 0, min :

jij

fbsign tRi

=−∈



(2)

The criteria f

and f

are in conflict, which means

that a decrease in one of them is paid for by an

increase in the other. It follows that there is no

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

194

optimal solution, but a usable result of the two-

criterion problem can be seen in determining such a

set PF of solutions that satisfy clauses (3) and (4).

() () () ()

11 2 2

For each , there exists :

and

ff ff≤≤

∈∈xY y

yx yx

(3)

() () () ()

11 2 2

For each pair , , it holds that

either and

or and

ff ff

∈

<≥

≥<

yz yz

(4)

Such a set PF is called a Pareto front. If two

solutions x and y satisfy (3), it is said that solution y

dominates solution x.

As determination of exact Pareto front demands

extremely big portion of computational time, our

attention is focused on approximation of the Pareto

front by a set of noNDSS non-dominated solutions,

which will be denoted by symbol NDSS. The used

implementation of NDSS will be kept in the form of

ordered sequence of feasible solutions y

, …, y

noNDSS

for which the following inequalities hold f

) <

) < ... < f

noNDSS

) and f

) > f

) > … >

noNDSS

). Furthermore, the first and last members of

NDSS must correspond to the first and last bordering

members of the Pareto front, i.e. the solutions which

have the minimal and maximal value of f

respectively. These properties of NDSS enable fast

decision on arbitrary element y of Y concerning its

suitability for improving the current approximation.

That can be used for construction of procedure

Update(NDSS, y), which returns the value of “true” if

y improves the current NDSS and it returns the value

of “false” otherwise. At the same time, the procedure

updates NDSS inserting the admissible y.

The procedure starts with determination of such

k∈{1, …, noNDSS-1} that f

) ≤ f

(y) and f

(y) <

k+1

). If such k does not exist, y is dominated by

noNDSS

and the result of the procedure is “false”. If k

is found, then either f

(y) ≥ f

) or f

(y) < f

). In

the former case, y is dominated by y

and the

procedure returns “false”. In the latter case, y is

included into NDSS, which can be accompanied by

exclusion of some original members, when f

(y) <

k+1

) holds. In this case, the procedure returns the

value of “true” and updated NDSS.

Proximity of NDSS to PF can be measured by so-

called NDSS-Area, which is computed according to

(5). The complementary constants f

dif

and f

dif

can

be computed by (6) and (7) respectively.

()( )

noNDSS

NDSS Area f dif f dif

−



(5)

11 1

()-( )

k noNDSS

fdif f f= yy

(6)

22 2

()-()

fdif f f

= yy

(7)

Each update of the NDSS by a new solution y is

followed by a reduction of the NDSS-Area and the

associated value is bounded from below the PF-Area.

3 NEIGHBORHOOD SEARCH

WITH TABU MOVES

The neighbourhood search algorithm has proved to be

a massive source of feasible solutions, which

represent candidates for NDSS improving. In general,

the neighbourhood of a given current solution is

defined by a set of permitted operations, which can be

used to modify the current solution keeping feasibility

of the operation result. Each feasible result of a

permitted operation is considered to be an element of

the neighbourhood.

The neighbourhood search algorithm comes from

an initial solution declared as the starting current

solution and searches element-by-element through

the neighbourhood of the current solution. If the used

searching strategy yields n admissible solution, then

this solution is declared to be the new current solution

and the neighbourhood search is continued with the

new neighbourhood. If the opposite case occurs, the

simple neighbourhood search algorithm terminates

and returns the last current solution as the result.

Various strategies can be applied to the

neighbourhood search. The two most known ones are

the first or best admissible strategies. The first

admissible strategy provides the first solution found

that is better than the current one and the best

admissible strategy provides the best admissible

solution of the current neighbourhood. These

strategies can be generalized using parameters called

MaxNos and Threshold. The parameter Threshold

gives minimal difference between objective function

values of the inspected and current solutions for the

inspected solution to be considered admissible. The

parameter MaxNos gives the number of admissible

solutions, which must be met during the

neighbourhood inspection to be allowed to stop the

inspection prematurely. The best of the found

admissible solutions is used as the new current

solution. If the parameter Threshold equals to zero

and the parameter MaxNos takes the value of one,

then the associated strategy reduces to the first

admissible strategy. If the parameter MaxNos is set to

a bigger value than the number of the neighbourhood

Multi-Criteria Service System Designing Using Tabu Search Method

195

elements, then the strategy behaves as the best

admissible strategy.

In this paper, we focus on the neighbourhood

search algorithm with the generalized strategy and

with the only one permitted operation represented by

so called swap operation. The swap operation

replaces one service center location i of the current

solution y

curr

by a possible service center location j,

which is not included in the current solution. The

resulting solution is denoted as swap(y

curr

, i, j). The

inspected solution admissibility is evaluated by the

NDSS-Area decrease caused by insertion of the

solution into NDSS.

The neighbourhood search algorithm was

embedded into the process of NDSS improvement in

the following way.

The process starts with NDSS consisting of

exactly two solutions representing the border

solutions of the exact Pareto front, i.e. the solutions

with the minimal and maximal values of the function

. Then, the process continues with selecting an

element of the current NDSS in some order and

applying the neighbourhood search algorithm to the

selected solution. During the run of the algorithm, the

NDSS is updated whenever such solution is inspected,

which is not dominated by any solution of the current

NDSS. This process continues until the given

computational time limit is exceeded.

The process in the above described form cannot

avoid repeating the neighbourhood search algorithm

with the same starting solution. Repeating the

algorithm reduces the efficiency of the process

because it only produces candidates that have already

been inspected once. This drawback evoked an idea

of prevent the algorithm from inspecting the series of

current solutions, which was already inspected. For

the purpose, tabu approach taken from the tabu search

approach was implemented here. Time limited

prohibition (tabu) is imposed on both locations of the

performed swap operation so that each possible center

location i is connected with two time instants In(i) and

Out(i) initialized by the value of –Exp, where Exp is

the time of prohibition expiration.

When swap operation i for j should be performed

at current time t, then the clauses t – Out(i) ≥ Exp and

t – In(j) ≥ Exp are verified. If the clauses are satisfied,

the swap operation is performed and the attributes

Out(i) and In(j) are updated by t. The whole process

of NDSS improvement can be described by following

sequence of steps.

0. Initialize NDSS, set up the parameters

MaxNos, Threshold, Exp and time limit T. Set

In(i) and Out(i) at the value if –

Exp for all

possible locations and set t=0.

1. If CPU < T then continue with the step 2,

otherwise terminate and return the current

NDSS.

2. Set k=1 and continue with the step 3.

3. If k < noNDSS, then select y

from the current

NDSS and go to the step 4, otherwise go to

the step 1.

4. {Application of the neighborhood search

algorithm to y

} Substitute y

for the current

solution y

curr

and continue with the step 5.

5. Define set C of location not contained in y

curr

by C = {1, …, m} – y

curr

, Area0 =

NDSS_Area, Nos = 0, BestDecrement = 0 and

continue with the step 6.

6. While Nos < MaxNos choose step-by-step a

pair (i, j), where i∈y

curr

and j ∈ C and

define y = swap(y

curr

, i, j).

If Updated(NDSS, y), then compute Area1 =

NDSS_Area, Decrement = Area0 – Area1.

If Decrement > Threshold then perform Nos

= Nos+1, Area0 = Area1 and if Decrement >

BestDecrement, then set BestDecrement =

Decrement, i

best

= i and j

best

= j.

After processing of the step 6 has finished,

continue with the step 7.

7. If BestDecrement > 0, then redefine y

curr

swap(y

curr

, i

best

, j

best

), In(j

best

) = t, Out(i

best

) = t

, t =t+1 and continue with step 5.

Otherwise, check whether the solution at the

k-th position of the current NDSS has

changed. If it stays the same, set k=k+1.

Continue with the step 3.

4 NUMERICAL EXPERIMENTS

This section is used to report the performed numerical

experiments aimed at verifying the efficiency of

suggested approach.

4.1 Benchmarks and Solving Tools

As far as the technical support like hardware and

software tools is concerned, we used the

programming language Java within the NetBeans

IDE 8.2 environment. The experiments were run on a

common PC equipped with the 11

Gen Intel®

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

As the input dataset for the reported

computational study, we made use of commonly used

benchmarks described in (Grygar, Fabricius, 2019),

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

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

196

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

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

2022c, Kvet, Janáček, 2022), the origin of which

comes from the road network of Slovakia, through

which the urgent medical care is provided by the

emergency agencies. The list of higher territorial

units, frequently referred to as self-governing regions,

contains 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 inhabitants being

provided with service.

As the objective function f

expressed by (1)

follows from the concept of so-called generalized

disutility, the parameter r was set to 3. The

coefficients q

were set so that q

= 77.063, q

16.476 and q

= 100 - q

- q

. These values were

obtained from a simulation model the details of which

are discussed in (Jankovič, 2016). Parameter R in the

fair objective function described by the formula (2)

was set to the value of 10 in accordance with previous

experiments.

The basic characteristics of benchmarks are

summarized in Table 1. The column denoted by m

reports the cardinality of the set of candidates I, from

which exactly p center locations are to be chosen. The

complete exact inextensible Pareto front is reported

by two values. While the number of its elements is

referred to by NoS, the last column of the table

denoted by PF-Area contains the area of the complete

Pareto front PF computed according to (5).

Table 1: Benchmarks sizes and the exact Pareto fronts

characteristics.

ion m

rea

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

4.2 Results of Experiments

This subsection is devoted to the results of numerical

experiments. The experiments should reveal a

dependence of proximity of NDSS and PF on

expiration “time” Exp. If the value of Exp = 0, no tabu

is imposed on the swap operations. If Exp reaches the

value of p almost each exchange operation is

prohibited. Therefore, we have performed the

experiments in such a way that the parameter Exp was

set according to the expression Exp = coeff*p. The

coeff could vary from 0 (no tabu) to 0.8.

Each run of the algorithm was restricted to five

minutes of computation. This time threshold of five

minutes was chosen on purpose to keep the

comparability of the newly obtained results with the

results of previously developed heuristic approaches.

This is also the reason, why the required computation

time is not reported in the following table.

As far as the quality of the Pareto front

approximation is concerned, it was necessary to find

a suitable metric to compare two sets possibly with

different cardinality. As mentioned in previous

sections, a good metric is the area formed by the

members of PF and NDSS respectively. The area can

be computed easily by the expression (5). To avoid

reporting and comparing high values of areas, a

simpler coefficient called gap can be used. Generally,

gap can be understood as a relative difference

between two values. In our case, it can be expressed

by (8).

100*

NDSS Area PF Area

gap

PF Area

−

(8)

The following Table 2 and Table 3 summarize the

obtained results. Both tables keep the same structure.

Each row corresponds to one setting of coeff, which

is used to determine the value of Exp. Each table

contains the results of experiments performed for half

of benchmarks. In the tables, the values of gap are

reported.

Table 2: Results of numerical experiments – part 1.

coeff

her territorial unit

BA BB KE NR

0 2.075 1.373 3.031 7.485

0.1 2.005 0.535 3.490 7.483

0.2 2.203 2.483 4.622 6.249

0.3 2.257 3.168 4.607 7.543

0.4 1.485 1.794 5.481 6.218

0.5 1.468 2.906 3.535 0.649

0.6 0.334 4.535 4.004 0.749

0.7 1.699 5.807 2.972 0.764

0.8 0.416 11.449 8.090 2.695

Based on the reported results we can see that the

suggested heuristic approach is sensitive to the

parameters settings like many other approximate

approaches (Janáček, Kvet, 2021, Janáček, Kvet,

2022a, Janáček, Kvet, 2022b). On the other hand, the

achieved values of gaps are very promising and they

show, that the tabu search-based method is able to

produce such a Pareto front approximation that shows

a satisfactory level of accuracy.

Multi-Criteria Service System Designing Using Tabu Search Method

197

Table 3: Results of numerical experiments – part 2.

coeff

Higher territorial unit

PO TN TT ZA

0 3.203 4.376 0.652 2.068

0.1 3.205 4.376 0.652 2.068

0.2 3.232 4.996 0.652 0.573

0.3 3.373 3.294 0.704 1.488

0.4 3.610 4.795 0.859 0.361

0.5 0.745 3.851 1.158 0.033

0.6 4.793 5.552 0.091 0.045

0.7 14.459 0.724 0.091 0.116

0.8 19.177 1.115 0.091 0.302

5 CONCLUSIONS

This research paper was intended to develop such

heuristic approach to Pareto front approximation that

incorporates the basics of tabu search principle.

Methods for approximating the Pareto front are

required whenever there are multiple contradictory

objectives to be optimized simultaneously. In this

manner, we have attempted to extend the state-of-the-

art approaches for solving bi-criteria location

problems.

The achieved results show that the suggested tabu

search can produce a very precise approximation of

the original Pareto front of service system designs in

acceptably short computational time. Such a great

accuracy makes it suitable for practical applications.

Obviously, we cannot omit the sensitivity of the

method to the parameter value. Therefore, future

research could be aimed at finding possible ways of

finding proper value, for which the best possible

results could be achieved.

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.

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