fuzzy logic-controlled diversity-based multi-objective memetic algorithm applied to a frequency...

Fuzzy logic-controlled diversity-based multi-objective memeticalgorithm applied to a frequency assignment problem

Eduardo Segredo n, Carlos Segura, Coromoto LeónDpto. Estadística, I. O. y Computación. Universidad de La Laguna. Avda. Astrofísico Fco. Sánchez s/n, Edif. Matemáticas, 38271, Santa Cruz de Tenerife, Spain

a r t i c l e i n f o

Article history:Received 15 August 2013Received in revised form9 November 2013Accepted 8 January 2014Available online 4 February 2014

Keywords:Parameter controlFuzzy logic controllersHyper-heuristicsDiversity preservationMemetic algorithmsFrequency assignment problem

a b s t r a c t

One of the most commonly known weaknesses of Evolutionary Algorithms (EAS) is the large dependencybetween the values selected for their parameters and the results. Parameter control approaches thatadapt the parameter values during the course of an evolutionary run are becoming more common inrecent years. The aim of these schemes is not only to improve the robustness of the controlledapproaches, but also to boost their efficiency. In this paper we investigate the application of parametercontrol schemes to address a well-known variant of the Frequency Assignment Problem (FAP). Thecontrolled EA is a highly efficient diversity-based multi-objective memetic scheme. In this work, a novelgeneral parameter control method based on Fuzzy Logic is devised. In addition, a hyper-heuristic is alsoconsidered as an established parameter control scheme. An extensive experimental evaluation of bothmethods is carried out that includes a comparison to a wide-range of fixed-parameter schemes. Theresults show that the fuzzy logic method is able to find similar or even better solutions than the hyper-heuristic and the fixed-parameter methods for several instances of the FAP. In fact, this method yieldedfrequency plans that outperform the best previously published solutions. Finally, the generality of thefuzzy logic-based scheme is demonstrated by controlling different kinds of parameters.

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1. Introduction

Many optimisation problems that arise in real world applicationsrequire the employment of approximation techniques. Among them,meta-heuristics (Glover and Kochenberger, 2003) have becomepopular in recent decades. They are high-level strategies that guidea set of heuristics in the search of an optimum. EvolutionaryAlgorithms (EAs) (Eiben and Smith, 2003) are one of the most popularstrategies belonging to this group. They are population-based algo-rithms inspired on biological evolution.

EAs have shown great promise for calculating solutions to difficultproblems. However, in some problems, EAs exhibit a tendency toconverge towards local optima, with the likelihood of this occurrencedepending on the shape of the fitness landscape (Caamaño et al.,2010). Several methods have been designed with the aim of dealingwith local optima stagnation. The reader is referred to Črepinšeket al. (2013) for an extensive survey of diversity preservationmechanisms. One of the methods that has gained some popularityin recent years is based on applying multi-objective schemes tosingle-objective optimisation problems (Segura et al., 2013a). Severalways of applying the multi-objective concepts have been devised

with diversity-based multi-objective algorithms being one of themost promising schemes (Abbass and Deb, 2003). In these schemes,a metric of the diversity introduced by each individual is used as anauxiliary objective. These schemes can better deal with strongoptima by being able to alleviate the effects of prematureconvergence.

Most popular EA variants have several components and/orparameters such as the survivor selection mechanism, or thegenetic and parent selection operators, which must be specified.In general, the performance of an EA and, consequently, the qualityof the resulting solutions, is highly dependent on these compo-nents and parameters. As a result, it is essential that the para-meters of an EA be suitably determined. However, findingappropriate parameter settings remains one of the persistentchallenges for Evolutionary Computing (Eiben and Smit, 2011).

Parameter setting strategies are commonly divided into twocategories: parameter tuning and parameter control. In parametertuning the objective is to identify the best set of values for theparameters of a given EA, which is then executed using thesevalues, which remain fixed for the duration of the run. In contrast,the aim of parameter control is to design control strategies thatselect the most suitable values for the parameters at each stage ofthe search process while the algorithm is being executed. In singleobjective optimisation, it has been empirically and theoreticallyshown that different parameter values might be optimal atdifferent stages of the search process (Srinivas and Patnaik, 1994;

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n Corresponding author. Tel.: þ34 922 319 191.E-mail addresses: [email protected] (E. Segredo), [email protected] (C. Segura),

[email protected] (C. León).

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Bäck, 1992). Therefore, it is natural to apply control strategies tomulti-objective EAs.

In this paper we devise a novel parameter control strategy basedon the use of Fuzzy Logic. Such a strategy, as well as other well-known parameter control methods, is used to control someparameters of a diversity-based multi-objective Memetic Algo-rithm (MA), which is applied to a set of real-world instances of theFrequency Assignment Problem (FAP). The MA has some compo-nents specifically tailored to deal with the FAP. It was selectedbecause it has demonstrated its efficiency against a large set ofdifferent meta-heuristics (Luna et al., 2011; Segura et al., 2013c).The contributions of this paper are as follows:

� A novel parameter control method based on fuzzy logic applicableto both continuous and discrete numeric parameters.

� First application of parameter control techniques based on fuzzylogic and hyper-heuristics in order to control the parameters of amutation operator that has been specifically designed to addressthe FAP.

� An extensive comparison of fuzzy logic-based schemes vs.hyper-heuristics as methods of parameter control applied to acomplex real-world problem.

� A broad comparison between parameter control methods andschemes with fixed parameters that highlights the benefits ofparameter control as opposed to parameter tuning.

The paper is organised as follows. In Section 2, an overview ofthe state of the art in parameter control in EAs is given. Section 3gives some background on fuzzy logic controllers, which wepropose as a parameter control method. The formal definition ofthe FAP is given in Section 4. Section 5 exposes the diversity-basedmulti-objective evolutionary engine applied herein and providessome background on related schemes. The proposed controlmethods are explained in Section 6, followed by a detailed analysisof the experimental results in Section 7. Finally, the conclusionsand future lines of work are given in Section 8.

2. State of the art of parameter control in evolutionaryalgorithms

Finding the most suitable configuration of an EA is one of themost challenging tasks in the field of Evolutionary Computation(Eiben and Smith, 2003). In order to completely define an instanceof an EA, two types of information are required (Smit and Eiben,2009):

� Symbolic—also referred to as qualitative, categoric or structureparameters—such as crossover, mutation and selection operators.

� Numeric—also referred to as quantitative or behavioural para-meters—such as the population size and the crossover andmutation rates.

For both kinds of parameters, the different elements ofthe domain are known as parameter values, and a parameter isinstantiated by assigning it a value. The main difference betweenboth types of parameters lies in their respective domains. Sym-bolic parameters, such as the crossover operator, have a finitedomain in which neither order is established nor distance metric isdefined. In contrast, numeric parameters, such as the mutationrate, have an infinite domain in which a distance metric and anorder can be defined for the values. Thus, optimisation methodscan readily be used to look for the appropriate values of thenumeric parameters of an EA. However, in the case of symbolicparameters, as noted above, distance metrics cannot be appliedbetween two values, meaning optimisation schemes are not able

to profit from the definition of these types of metrics for settingsuch parameters. In this case of this paper, we focus on controlmethods for numeric parameters.

The goal of parameter control is to design a control strategy thatselects the most suitable parameter values for every stage of thesearch process. The ideas of parameter control were first incorpo-rated in early work on EAs (Davis, 1989; Rechenberg, 1973). Never-theless, recent research has seen a marked increase in proposals formethods that achieve parameter control in EAs (Lobo et al., 2007). Infact, parameter control methods have been successfully applied to awide range of EAS and other meta-heuristics such as EvolutionStrategies (ES) (Kramer, 2010), Differential Evolution (DE) (Qin et al.,2009) and Particle Swarm Optimisation (PSO) (Zhan and Zhang,2008). Given the large number of proposals, several taxonomieshave been proposed. One of the most popular classifications (Eibenet al., 2007) considers the following types of strategies:

� Deterministic parameter control: Parameter values are altered bya deterministic rule without using any feedback from thesearch procedure.

� Adaptive parameter control: Parameter values are updated by amechanism that uses some feedback from the search process.Such a mechanism is externally supplied.

� Self-adaptive parameter control: Parameters are encoded intothe chromosome and their values are modified by the EA

variation operators.

It is worth pointing out that the majority of the work onparameter control is focused on the parameters of a ‘standard’ EA,i.e. the variation operators (mutation and crossover), the popula-tion size or combinations of all three (Eiben et al., 2007; Bäck et al.,2000). In this paper we describe the application of controltechniques to the parameters of a mutation operator specificallydesigned to address the FAP. It is the first time that theseparameters are adapted.

3. Background on fuzzy logic controllers for parameter control

Our knowledge of EAs performance has significantly increasedin recent years due to the large number of empirical analysesconducted on a wide range of applications in different areas. Itwould be desirable to profit from this human knowledge byencapsulating it within an algorithm to automate the task ofimproving the behaviour and performance of EAs. However, thissort of knowledge is usually incomplete, imprecise and/or it is notwell organised. Consequently, the application of fuzzy logic-basedmethods would seem to offer a promising approach for dealingwith this kind of knowledge.

One application of fuzzy logic is the design of Fuzzy LogicControllers (FLCs). FLCs can be used to define control approaches inwhich the incorporation of human knowledge is performedintuitively. An FLC consists of the knowledge base, the fuzzy inferenceengine and the fuzzification and defuzzification interfaces (Herreraand Lozano, 2003). The knowledge base has two different parts, adata base, which includes the definitions of the membershipfunctions of the linguistic terms for each input and outputvariable, and a rule base constituted by the collection of fuzzycontrol rules.

The main benefit of using FLCs to adapt the parameters of an EA

is that the possible values that can be assigned to certain para-meters are infinite, in contrast to other techniques that can onlyuse some values from a finite set. However, the main drawback isthat FLCs cannot be directly applied to control the symbolicparameters of an EA. Therefore, in this paper we restrict theapplication of the FLC to controlling numeric parameters. An

E. Segredo et al. / Engineering Applications of Artificial Intelligence 30 (2014) 199–212200

additional drawback is that, unlike with other systems, an errorfeedback signal does not directly exist in the fuzzy logic-basedapproach we are proposing because its aim is to optimise as muchas possible. However, our proposal has to include some way ofmeasuring performance, which makes it atypical since the desiredobjective is not known beforehand. Despite this, the term FLC iscommonly used for this kind of scheme (Xue et al., 2005), andconsequently it is the one we adopted for our fuzzy logic-basedapproach.

A considerable body of research regarding FLCs and EAs alreadyexists. For example, different EAs, as well as other meta-heuristics,have been used to optimise the design of FLCs for different applica-tions (Das Sharma et al., 2012; Hadavandi et al., 2012; Rui et al.,2010). However, in this paper, we study the reverse of this type ofapplication, and focus on the design of FLCs that adapt the para-meters of an EA, thus providing an adaptive control technique thatutilises feedback from the search process to adapt the parameters.

Several methods have been proposed for controlling the para-meters of different meta-heuristics–including EAS–through FLCs.The principle behind these schemes is to use a FLC to compute newparameter values by considering any combination of performancemeasures and current parameter values as the input to thecontrollers. This idea was first proposed in Lee and Takagi(1993). In such a scheme, the population size and mutation andcrossover rates are adapted considering the best, average andworst fitness values of the individuals in the population. Subse-quently, a large number of variants have been proposed (Gen andYun, 2006). Some of the schemes that are more closely related toour proposal are briefly summarised in this section.

Gen and Yun (2006) present a survey of several schemes thatrely on a FLC to control their internal parameters. Some of thesimplest schemes adapt the crossover and mutation rates byconsidering the average fitness of the last two generations asinput variables (Wang et al., 1997). Basically, depending on theimprovements obtained in the last generation, the crossover andmutation probabilities are modified so as to change the perturba-tion strength of the variation scheme. Using only two generationsmight not be enough, so in some schemes the average fitnessvalues of the last three generations are considered (Liu and Liu,2011). A similar idea is proposed in Liu and Lampinen (2005) toadapt the parameters of a DE approach. Specifically, two FLCs areused to adapt the mutation scale factor and the crossover rate. Inthis case, the input variables are not only made up from measuresin the objective space, but the space of the variables is alsoconsidered. Other research works reported the application of threeFLCs to adaptively adjust the parameters of a PSO algorithm (Zhangand Liu, 2005). Specifically, the inertia weight and learning factorsare adapted considering the current fitness values and the numberof generations where no improvements have been achieved. Thesystem assumes that the user can assign a quality level to thedifferent fitness values beforehand.

More advanced FLCs consider additional metrics of diversity tocarry out the decisions. For instance, in Brito et al. (2006) thefrequency of the best individual, as well as the rate of duplicateindividuals, is used to control the mutation, crossover and survivingindividual rates. In addition, an input variable that estimates thequality of the resulting fitness is used. Thus, as in some of the otherschemes described, it assumes knowledge regarding the supposedoptimal values. In Li and Maeda (2008) the diversity is calculated byconsidering the difference between the maximum and averagefitness of the population. Finally, the normalised standard deviationis considered as an input variable in Ling et al. (2012).

The feature common to most of the research described in theliterature is that the FLCs are used to adapt the parameters of themutation or crossover operators, the population size, or combina-tions of all three (Herrera and Lozano, 2003). Moreover, these FLCs

are usually tailor-made methods for a specific EA and/or para-meters, and they only make use of a single rule base. It is alsoworth noting that FLCs have been successfully used to adaptdifferent EAs applied to real-world applications, demonstratingtheir efficiency and reliability even for complex problems (Fenget al., 2006; Zhang and Liu, 2005).

From the perspective of MOEAs, it is apparent that the body ofresearch is much smaller than in the case of mono-objective EAs.However, FLCs have also been used to control the parameters ofdifferent MOEAs (Chen and Weng, 2009; Xue et al., 2005). It isimportant to note that even though in this paper a multi-objectivescheme is applied, the problem to be solved has a single objective.Thus, most of the ideas used for controlling multi-objectiveschemes with FLCs cannot be directly applied to the scheme beingproposed here.

4. The frequency assignment problem: formal definition

The FAP formulation applied herein was proposed by Luna et al.(2007). Let T ¼ ft1; t2;…; tng be a set of n transceivers, and letFi ¼ ff i1;…; f iki g �N be the set of valid frequencies that can beassigned to a transceiver tiAT , i¼1,…,n. Note that ki—the cardin-ality of Fi—is not necessarily the same for all the transceivers.Furthermore, let S¼ fs1; s2;…; smg be a set of given sectors—or cells—of cardinality m. Each transceiver tiAT is installed in exactly oneof the m sectors. From now on we denote the sector in which atransceiver ti is installed by sðtiÞAS. Finally, the matrixM¼ fðμij;sijÞgm�m

is denoted as the interference matrix. The twoelements μij and sij of a matrix entry Mði; jÞ ¼ ðμij;sijÞ are numericvalues greater than or equal to zero. The values of μij and sij

represent the mean and the standard deviation, respectively, of aGaussian probability distribution describing the Carrier-to-Interference (C/I) ratio (Walke, 2002) when sectors i and j operateon the same frequency. The higher the mean value is, the lowerthe interference is and thus the better the communication qualityis. Note that the interference matrix is defined at the sector—orcell—level because the transceivers installed in each sector servethe same area.

A solution is obtained by assigning to each transceiver tiAT oneof the frequencies from Fi. Consequently, a candidate solution—orfrequency plan—is denoted by pAF1 � F2 �⋯� Fn, wherepðtiÞAFi is the frequency assigned to the transceiver ti. Theobjective is to find a solution p that minimises the following costfunction:

CðpÞ ¼ ∑tAT

∑uAT ;ua t

Csigðp; t;uÞ ð1Þ

In order to define the function Csigðp; t;uÞ (Eq. (2)), let st and sube the sectors in which the transceivers t and u are installed, i.e.st ¼ sðtÞ and su ¼ sðuÞ, respectively. Moreover, let μst su and sst su bethe two elements of the entry Mðst ; suÞ of the interference matrixwith respect to sectors st and su.

The parameter K in Eq. (2) represents the cost associated withthe usage of the same or adjacent frequencies in the same sector.In real networks, it is unfeasible to operate with

Csigðp; t;uÞ ¼

K if st ¼ su; jpðtÞ�pðuÞjo2Ccoðμst su ;sst su Þ if stasu;μst su 40; jpðtÞ�pðuÞj ¼ 0Cadjðμst su ;sst su Þ if stasu;μst su 40; jpðtÞ�pðuÞj ¼ 10 otherwise:

8>>><>>>:

ð2Þ

more than one transceiver with the same or adjacent frequenciesserving the same sector. Thus, K is defined as a very large constant.

E. Segredo et al. / Engineering Applications of Artificial Intelligence 30 (2014) 199–212 201

Function Ccoðμ;sÞ is defined as follows:

Ccoðμ;sÞ ¼ 100 1:0�QcSH�μs

� �� ð3Þ

where

Q ðzÞ ¼Z 1

z

1ffiffiffiffiffiffi2π

p e� x2=2 dx ð4Þ

is the tail integral of a Gaussian probability distribution functionwith zero mean and unit variance, and cSH is a minimum qualitysignalling threshold. Function Q is widely used in digital commu-nication systems because it characterises the error probabilityperformance of digital signals (Simon and Alouini, 2002). Thismeans that Q ððcSH�μÞ=sÞ is the probability of the C/I ratio beinggreater than cSH, and therefore Ccoðμst su ;sst su Þ computes the prob-ability of the C/I ratio in the service area of sector st being belowthe quality threshold due to the interference caused by sector su. Ifthis probability is low, the C/I value in sector st is not likely to bedegraded by the interfering signal coming from sector su, and thusthe resulting communication quality is high. Note that this iscompliant with the definition of a minimisation problem. Incontrast, a high probability—and consequently a high cost—mostlycauses C/I to be below the minimum threshold cSH, and thus resultsin low quality communications.

Since function Q has no closed integral form, it has to be evaluatednumerically. To do so, we use the complementary error function E:

Q ðzÞ ¼ 12E

zffiffiffi2

p� �

ð5Þ

In Press et al. (1998), a numerical method is presented thatallows computing the value of E with a fractional error smallerthan 1.2�10�7. Analogously, function Cadjðμ;sÞ is defined as

Cadjðμ;sÞ ¼ 100 1:0�QcSH�cACR�μ

s

� �� ¼ 100 1:0�1

2E

cSH�cACR�μsffiffiffi2

p� ��

ð6Þ

The only difference between functions Cco and Cadj is theadditional constant cACR40 (Adjacent Channel Rejection) in thedefinition of function Cadj. This hardware specific constant mea-sures the receiver's ability to receive the desired signal in thepresence of an unwanted signal in an adjacent channel. The effectof constant cACR is that Cadjðμ;sÞoCcoðμ;sÞ. This is to be expectedsince using adjacent frequencies does not result in interference asstrong as when the same frequency is used.

5. Diversity-based multi-objective evolutionary engine

In this section we describe the meta-heuristic that is used tooptimise the aforementioned version of the FAP. This scheme wasproposed in Segredo et al. (2011) and was selected because ityielded the best frequency plans for several instances of the FAP inprevious works (Segura et al., 2013c; Segredo et al., 2011). Severalmeta-heuristics have been applied to these instances in recentyears (Luna et al., 2011) that have served to demonstrate theadequate performance of the approach selected here. However,some important parameters were hand-tuned and kept constantin previous works, so using this meta-heuristic in combinationwith parameter control methods seems very promising.

The scheme is a diversity-based multi-objective MA based onthe well-known NSGA-II (Deb et al., 2002). The only difference withrespect to the original NSGA-II is that after the variation scheme,a local search procedure is applied to each new generatedindividual. In diversity-based multi-objective schemes, a set of

objectives is calculated for each individual. The first one is theobjective associated with the problem being solved, i.e. the cost ofthe frequency plan in this case. The remaining objectives—most ofthe proposals consider only one additional objective as in our case—are measures of the diversity. Note that a measure of thepopulation diversity is not required. Instead, the additional orauxiliary objectives are measures of the diversity introduced by anindividual itself. In our proposal we tested two metrics byconsidering the promising results obtained in previous works(Segura et al., 2013c). They were defined by Toffolo and Benini(2003) and Bui et al. (2005), and are the following:

� DCN: The auxiliary objective is calculated as the distance to theclosest individual. This objective must be maximised.

� ADI: The auxiliary objective is calculated as the average distanceto all individuals. This objective must be maximised.

The genetic operators and the local search scheme are impor-tant components for the efficiency of the algorithm. The localsearch scheme is incorporated in keeping with the Lamarckianapproach (Whitley et al., 1994), i.e. the individual reflects in itsgenotype the result of the movements performed by the localsearch. The operation of the local search is detailed in Segura et al.(2013c), but basically it optimises the assignment of the frequen-cies to the transceivers located in a given sector without modifyingthe remaining network assignments.

Regarding the genetic operators, they were also specificallydesigned to address this variant of the FAP. As is normal, thescheme relies on the application of a crossover operator and amutation operator afterwards, with probabilities pc and pm,respectively. Two different crossover operators were tested, oneof them random and one that considers problem-dependentinformation. They operate as follows:

� Uniform crossover (UX): For each gene, a random variablerA ½0;1� is generated. If ro0:5, then the gene is inherited fromthe first parent; otherwise, the gene is inherited from thesecond one.

� Interference-based crossover (IX): A transceiver t is randomlyselected. Every gene associated with a transceiver that inter-feres with t or is interfered with by t, including the gene thatrepresents t, is inherited from the first parent. The remaininggenes are inherited from the second one.

After applying one of the aforementioned crossover operators, theNeighbourhood-based Mutation (NM) is applied as the mutation opera-tor. Its function is as follows. First, a transceiver t is selected at random.Then, the transceivers that interfere with t, or are interfered with by t,are included in a list called interference and are mutated with aprobability pm. The above step is repeated R times, but in thesubsequent iterations the transceiver is randomly selected fromamong those that were initially included in the interference list. Thus,this mutation operator focuses on one area of the network.

One of the main drawbacks of the application of this operator isthat two different parameters must be set. One of these parameters—pm—is continuous and the other one—R—is discrete. In addition, themost suitable values for these parameters could depend on theproblem and/or instance being solved or even on the current stageof the optimisation process, and therefore modifying them during theexecution might be beneficial. Consequently, the application of para-meter control techniques to automatically adapt these parametersought to significantly improve both the behaviour and the robustnessof the entire optimisation scheme. This idea seems to be verypromising and is addressed in detail herein.

In order to complete the definition of the diversity-based multi-objective MA, other components must be specified. The parent


selection mechanism is the Binary Tournament (Eiben and Smith,2003), whereas the individuals are encoded as arrays of n integervalues ðp1; p2;…; pnÞ, where pi is the frequency assigned to thetransceiver ti.

6. Parameter control approaches

In this section we describe in detail both of the parameter controlapproaches that are evaluated in later sections: the novel FLC proposedherein, and the hyper-heuristic used as the comparison approach. Bothprovide an external control mechanism for altering the parameters ofthe NM operator during the course of a run. Only one of theseparameters is controlled during the execution, while the other remainsconstant, so in this paper two independent studies are carried out, onefor each parameter of the NM operator.

6.1. Fuzzy logic controllers

This section describes a novel FLC introduced by the authors inthis work to control the parameters of the NM operator. Its mainnovelty lies in the incorporation of a set of different rule bases thatare enabled depending on historical information extracted fromthe optimisation process. This historical data is used to guide theadjustment of the parameter. In what follows, the parameter thatis going to be controlled is denoted by p. The pseudocode of this FLC

is shown in Algorithm 1.

Algorithm 1. FLC pseudocode.

1: Initialisation: Generate sample values for the parameter pdistributed uniformly in its corresponding rangeconsidering a certain value Δ as the difference betweentwo consecutive samples

2: for (each generated sample value of the parameter p) do3: Learning: Execute numGen generations of the

optimisation scheme with this value for the parameter p inorder to gather knowledge

4: end for5: while (NSGA-II stopping criterion is not satisfied) do6: Transformation of the parameter p. If the range of the

parameter p is different from the range [0, 1], the currentvalue of this parameter is scaled to the range [0, 1] andnamed p0

7: Calculation of input variables. Set the values for theinput variables IMP, VAR, P-IN, BEST-P-IN

8: Selection of the rule base. Select the most suitable rulebase considering the last k decisions carried out by the FLC

and the scoring function shown in Eq. (9)9: Fuzzification. Transform the crisp values of the input

variables to fuzzy sets using the fuzzification interface10: Mamdani's Fuzzy inference. Apply the fuzzy operator

AND (min), the implication method (min) and theaggregation method (max) using the selected rule base toobtain the fuzzy set of the output variable P-OUT

11: Defuzzification: Transform the fuzzy set of the outputvariable P-OUT to a crisp value Δp using the defuzzificationinterface (centroid method)

12: Parameter update: p0 ¼ p0 þΔp. The value of p0 isenclosed in the range [0, 1]

13: Transformation of the parameter p0. If the range of theparameter p is different from the range [0, 1], the currentvalue of p0 is scaled to the range of the parameter p

14: Execution: Execute numGen generations of theoptimisation scheme with the new value of p

15: end while

First, the initialisation and learning stages—lines 1–4—arecarried out. During the initialisation stage, different sample valuesare generated for the parameter p and distributed uniformly in itscorresponding range. In order to generate them, a value Δ isconsidered as the difference between two consecutive samples.Although Δ might be considered as a parameter of the FLC, it isassigned a constant value regardless of the problem instance.Then, in the learning stage, the optimisation scheme explained inSection 5 is executed for numGen generations for each of thegenerated samples in order to gather sufficient information. Oncethese two stages are complete, the FLC infers the change to beapplied over the parameter p—lines 6–13—taking into account thevalues of the input variables and the selected rule base. Then, theoptimisation scheme is executed for numGen generations—line 14—with the new value of p. This process is repeated until the globalstopping criterion of the NSGA-II is reached.

After step 13 of Algorithm 1, a continuous value for theparameter p is obtained, so in order to deal with discrete numericparameters, this value must be defined. Eq. (7) shows the functionused to transform a continuous value into a discrete one. Therandom value r is sampled from a continuous uniform distributiondefined in the range [0, 1]. Therefore, if the continuous value of pis, for instance, equal to 12.3, there is a 70% probability that thediscrete value will be 12 and a 30% probability that it will be 13:

p¼⌈p⌉ if rrp�⌊pc⌊pc if r4p�⌊pc

(ð7Þ

For the fuzzy inference process—lines 9–11—we note thatMamdani's fuzzy inference method is used. In addition, the fuzzylogic operator AND

1 uses the minimum T-norm, the implicationmethod uses the minimum T-norm, the aggregation methodapplies the maximum S-norm and the centroid algorithm isapplied as the defuzzification method. All of these componentswere selected because they are usually implemented togetherwith Mamdani FLCs. It is important to note that zero-order Takagi–Sugeno–Kang (TSK) FLCs—where the linguistic terms of the outputvariables are described by a zero order (constant) function, insteadof using membership functions—were also implemented. TheseFLCs used the weighted average as the defuzzification method. Theremaining components of the fuzzy inference process were thesame as those applied in the Mamdani FLCs exposed herein.However, the differences between the Mamdani and TSK FLCs werenot statistically significant. Consequently, only Mamdani FLCs aretaken into account in this paper.

The input variables of the FLC—line 7—are the following:

� IMP: Calculated as the improvement of the original objectivevalue of the best individual achieved by the optimisationscheme—line 14 of Algorithm 1—over the last numGen genera-tions. This input variable is normalised to delimit it to the range[0, 1].

� VAR: A measure of the diversity of the population. The higher itsvalue, the more diverse the population. The calculation of thisinput variable with no normalisation is shown in Eq. (8). Thevalues of the decision variable i of individuals j and k are givenby xj½i� and xk½i�. The total number of decision variables isrepresented by D and N is the population size. The value of varn

is normalised to enclose the variable VAR in the range [0, 1]:

varn ¼ ∑D�1

i ¼ 0∑

N�1

j ¼ 0xj½i��

1N� ∑

N�1

k ¼ 0xk½i�

!" #224

35 ð8Þ

1 Only the fuzzy logic operator AND is used in the antecedents of the fuzzy rules.


� P-IN: Defined as the current value of parameter p within therange [0, 1]:

� BEST-P-IN: Defined as that value of parameter p that has yieldedthe maximum improvement in the original objective valueconsidering the last k values of the parameter p inferred by theFLC. Its value is also in the range [0, 1].

Two different versions of the FLC are applied. The first one iscalled FUZZY-A and makes use of the input variables IMP, VAR and P-IN.The second one utilises the input variables IMP, P-IN and BEST-P-IN andis called FUZZY-B. For both FLC schemes, only one output variable isdefined, referred to as P-OUT, which represents the increment ordecrement to be applied to parameter p in order to change itsvalue. The membership functions for both the input and outputvariables are shown in Fig. 1. Due to the computational simplicityand efficiency advantage they offer, triangular-shaped member-ship functions were selected for the input and output variables.The linguistic terms represented by the membership functions—from left to right in Fig. 1—are as follows:

� Input variables IMP, VAR, and BEST-P-IN: LOW (L), MEDIUM (M) andHIGH (H).

� Input variable P-IN: LOW (L), LOW-MEDIUM-B (LMB), LOW-MEDIUM-A (LMA),MEDIUM (M), MEDIUM-HIGH-A (MHA), MEDIUM-HIGH-B (MHB) and HIGH (H).

� Output variable P-OUT: NEG-GIANT (NG), NEG-HUGE (NU), NEG-HIGH (NH),NEG-MEDIUM (NM), NEG-LOW (NL), ZERO (Z), POS-LOW (PL), POS-MEDIUM (PM),POS-HIGH (PH), POS-HUGE (PU) and POS-GIANT (PG).

For each FLC different rule bases are defined. The reason forusing different rule bases is that different fuzzy rules will beapplicable depending on the behaviour exhibited during theprevious execution. For instance, if the best results were histori-cally obtained by low values of the parameter p, the fuzzy rulesshould promote the usage of such low values. Every rule base iscomposed of different IF-THEN fuzzy rules. The left-hand side ofTable 1 shows one of the rule bases defined for the approach FUZZY-A, while the right-hand side shows another one for the schemeFUZZY-B. Only the fuzzy logic operator AND is used in the antecedentsof these fuzzy rules. In general, every fuzzy rule considers threeinput variables and one output variable. In those cases where a ‘–’

is shown, the corresponding fuzzy rule has no dependency on the

corresponding variable. The remaining rule bases are not showndue to space constraints but are similar to those shown here.2

In order to select the most suitable set of rules, in this work wepropose a novel scoring function that relies on a weighted meanthat considers historical data on both the improvement in theoriginal objective value and on the degrees of membership ofparameter p to each term defined for the input variable P-IN.

The value of k is defined as the amount of historical knowledgeconsidered by the FLC, i.e. information on the latest k decisions

0

0.2

0.4

0.6

0.8

1

0 0.5 1

Membership Functions - IMP, VAR, and BEST-P-IN

0

0.2

0.4

0.6

0.8

1

0.167 0.334 0.5 0.667 0.834 1

Membership Functions - P-IN

0

0.2

0.4

0.6

0.8

1

-0.45 -0.36 -0.27 -0.18 -0.09 0 0.09 0.18 0.27 0.36 0.45

Membership Functions - P-OUT

Fig. 1. Membership functions of the input and output variables.

Table 1Rule bases for the FUZZY-A (left-hand side) and FUZZY-B (right-hand side) schemes.

Rules Inputs Output Rules Inputs Output

ID P-IN IMP VAR P-OUT ID P-IN IMP BEST-P-IN P-OUT

1 L L – PG 1 L L L NL

2 L M – PL 2 L L M PL

3 L H – Z 3 L L H PL

4 LMB L – PG 4 L M – Z

5 LMB M – PL 5 L H – Z

6 LMB H – Z 6 LMB L – NM

7 LMA L – PG 7 LMB M – NL

8 LMA M – PL 8 LMB H – Z

9 LMA H – Z 9 LMA L – NH

10 M L – PU 10 LMA M – NL

11 M M – PL 11 LMA H – Z

12 M H – Z 12 M L – NU

13 MHA L – PH 13 M M – NL

14 MHA M – PL 14 M H – Z

15 MHA H – Z 15 MHA L – NG

16 MHB L – PM 16 MHA M – NL

17 MHB M – PL 17 MHA H – Z

18 MHB H – Z 18 MHB L – NG

19 H L L PL 19 MHB M – NL

20 H L M PL 20 MHB H – Z

21 H L H NL 21 H L – NG

22 H M – Z 22 H M – NL

23 H H – Z 23 H H – Z

2 The complete specifications for all of the rule bases designed for bothversions of the FLC are available as Online Supplementary Material.


made by the FLC is taken into account. On the other hand, d is thetotal number of decisions that the FLC has carried out, andnumTerms is the number of linguistic terms defined for the inputvariable P-IN. The score assigned to each linguistic termiA ½0;numTerms�1� is given by Eq. (9). The improvement achievedduring execution d� j of the optimisation scheme—line 14 ofAlgorithm 1—is given by γ½d� j�. In addition, the degree ofmembership of parameter p to the linguistic term i duringexecution d� j is represented by δ½i�½d� j�. Thus, the linguistic termi will be assigned a higher score if the values of parameter p havelarger degrees of membership to said linguistic term, and if, at thesame time, the values of parameter p are able to achieve higherimprovements in the original objective value. Finally, we note thatthe scoring function assigns more importance to the latestdecisions inferred by the FLC. Thus, for each linguistic term theequation represents a weighted average of its improvement,where greater importance is given to the last executions in whichvalues of the controlled parameter have a high degree of member-ship to the corresponding linguistic term.

Note that if numTerms linguistic terms are defined for thevariable P-IN, numTerms rule bases have to be implemented suchthat the FLC works with the proposed scoring function. Fig. 1 showsthat seven linguistic terms are defined for the input variable P-IN,so seven different rule bases are implemented. We tested differentnumbers of fuzzy rule bases and found that the higher the numberof rule bases, the smoother the variations of the parameter pinferred by the FLC, and thus the steadier the FLC. However, whenconsidering more than seven fuzzy rule bases, the performancestarted to degrade somewhat, as it also did with a lower number offuzzy rule bases. Thus, we opted for seven rule bases as thisyielded the best performance for the FLC. This fact also justifies theusage of seven linguistic terms for the input variable P-IN, insteadof using three linguistic terms as in the case of remaining inputvariables. For the remaining input variables, three linguistic termsare used so as to maintain the rule bases as simple as possible.

score½i� ¼∑minðk;dÞ

j ¼ 1 γ½d� j� � δ½i�½d� j� � ðminðk; dÞ� jþ1Þ∑minðk;dÞ

j ¼ 1 δ½i�½d� j� � ðminðk; dÞ� jþ1Þð9Þ

Once the scores are calculated, the fuzzy set with the maximumscore is selected. This means that those values of parameter p witha large enough degree of membership to the linguistic term shouldprovide better performance than other values. Therefore, if thelinguistic term i is selected as the most appropriate one, rule base iis enabled. This selected rule base is responsible for adapting the valueof parameter p so that it approaches the values represented by term i.For instance, assume that the current value of parameter p is 0.01 andthe most suitable rule base—considering the scoring function—is theone represented by the linguistic term HIGH of the input variable P-IN.This means that historically high values of parameter p have yieldedgood improvements in the original objective value. Thus, the rule baseto be applied in this case is precisely the one shown in the left-handside of Table 1, considering the approach FUZZY-A. If a fuzzy set for thevariable IMP, which has a large degree of membership to the term LOW,since P-IN—with value 0.01—is represented by a fuzzy set with a largedegree of membership to the term LOW, then the output fuzzy set—theone corresponding to the output variable P-OUT—will have a largedegree of membership to the linguistic term POS-GIANT (PG). Conse-quently, the value of parameter p will be considerably increased sothat it will tend towards higher values.

6.2. Hyper-heuristics

Hyper-heuristics can be defined as search methods or learningmechanisms for selecting or generating heuristics to solve compu-tational search problems (Burke et al., 2010). Hyper-heuristics

based on heuristic selection try to iteratively identify and select themost promising low-level heuristics or meta-heuristics—from a setof candidates—to solve a particular instance of a problem (Burkeet al., 2003). Hyper-heuristics can be used as parameter controlapproaches. For example, the low-level approaches could repre-sent different configurations of the same heuristic (or meta-heuristic). The hyper-heuristic then selects the configuration withthe most appropriate set of parameters at each point in the search.In fact, they can be further classified as adaptive parameter controltechniques if they receive some kind of feedback from theoptimisation process.

An extension of the hyper-heuristic approach to parameter controlfirst described by Vinkó and Izzo (2007) is implemented in order tocontrol the parameters of the NM operator. This hyper-heuristic hasbeen successfully applied in previous works (Segura et al., 2013b;Segura, 2012) and is based on using a scoring and selection strategy forchoosing the most appropriate low-level configuration of the algo-rithm to be executed. A low-level configuration in this case refers to aninstance of the optimisation scheme depicted in Section 5 with aparticular setting for one of the parameters—pm or R—of the NM

operator (all other parameters of the algorithm remaining constant).Once a strategy is selected, only that strategy is executed until a localstopping criterion is achieved. When this happens, another low-levelconfiguration is selected and executed; the final population of the lastlow-level configuration used becomes the initial population of thenew low-level configuration. This process continues until a globalstopping criterion is satisfied. The low-level configuration that must beexecuted is selected as follows.

First, the scoring strategy assigns a score to each low-level config-uration. This score estimates the improvement that each low-levelconfiguration can achieve starting from the current solution set. Thus,larger values are assigned to more promising schemes consideringtheir historical behaviour. In order to calculate this estimate, theprevious improvements to the original objective value achieved byeach configuration are used. The improvement (γ) is defined as thedifference, in terms of the original objective value, between the bestachieved individual and the best initial individual. Considering aconfiguration conf that has been executed j times, the score s(conf)proposed in Segura et al. (2010) is calculated as a weighted average ofits last k improvements:

sðconf Þ ¼∑minðk;jÞi ¼ 1 ðminðk; jÞþ1� iÞ � γ½conf �½j� i�

∑minðk;jÞi ¼ 1 i

ð10Þ

In Eq. (10), γ½conf �½j� i� represents the improvement achievedby configuration conf in execution number j� i. The adaptationlevel of the hyper-heuristic, i.e. the total amount of historicalknowledge that it considers in order to perform its decisions, canbe varied depending on the value of k. The weighted averageassigns a greater importance to the latest executions.

The score s(conf) is used to calculate a probability of selecting aparticular low-level configuration. However, the stochastic beha-viour of the low-level meta-heuristics involved may lead tovariations in the results they yield. Therefore, the probabilitycalculation also enables a fraction of selections based on a randomscheme and is implemented as follows. Specifically, the hyper-heuristic can be tuned by means of a parameter β, whichrepresents the minimum selection probability that should beassigned to a low-level configuration. If nh is the number of low-level configurations involved, a random selection based on auniform distribution is performed in β � nh percentage of the cases.Therefore, the probability of selecting each configuration conf isdefined as

probðconf Þ ¼ βþð1�β � nhÞ �sðconf Þ∑nh

i ¼ 1sðiÞ

" #ð11Þ


Two different schemes based on this hyper-heuristic areapplied in this paper.

� The first one is an elitist version—HH-ELI—which always selectsthe low-level configuration with the maximum score s(conf),besides the minimum random selections performed for eachconfiguration.

� The second one is a probabilistic version (HH-PROB). In this case,the selection probability—Eq. (11)—is proportional to the scores(conf).

7. Experimental evaluation

The experiments conducted with the diversity-based multi-objective MA described in Section 5, and the parameter controlapproaches presented in Section 6 are described at this point.The main aim of these experiments is twofold. First, to compareparameter control and parameter tuning in order to reveal thebenefits and drawbacks of adapting the values of the parametersduring the course of the optimisation process in contrast to settingthem before the run starts. Second, to demonstrate the generalityof the control techniques, and particularly of the FLC proposed inthis work, by applying them to different parameters.

Experimental method: Both the optimisation scheme and theparameter control approaches were implemented using METCO

(León et al., 2009) (Meta-heuristic-based Extensible Tool for Coop-erative Optimisation). Tests were run on a Debian GNU/Linux com-puter with four amdsopteron™ processors (model number 6164HE) at 1.7 GHz and 64 GB RAM. The compiler was the GCC 4.7.2, whilethe FLCs were implemented using the fuzzylite3.1 library (Rada-Vilela, 2013).

Since every experiment used stochastic algorithms, each execu-tion was repeated 32 times. Comparisons were performed byapplying the following statistical analysis. First, a Shapiro–Wilktest was performed in order to check whether the values ofthe results followed a normal (Gaussian) distribution or not. Ifso, the Levene test checked for the homogeneity of the variances. Ifthe samples had equal variance, an ANOVA test was done. Otherwise,the a Welch test was performed. For non-Gaussian distributions,the non-parametric Kruskal–Wallis test was used to compare themedians of the algorithms. A significance level of 5% wasconsidered.

FAP instances: The studies were conducted considering twodifferent instances representing two real cities in the USA: Seattleand Denver. The Seattle instance had n¼970 TRXs and 15 differentfrequencies to be assigned. The Denver instance was larger,consisting of n¼2612 TRXs and 18 frequencies. In both cases, theconstants used in the formal definition of the FAP exposed inSection 4 were set to K¼1 �105, cSH ¼ 6 dB and cACR ¼ 18 dB.The matrix M contains 59,169 items in the Seattle network, whileit contains 20,638 items for the Denver instance.

7.1. Analyses over the parameter pm

In this first experiment the different parameter control approacheswere applied to the parameter pm of the NM operator to solve both the

FAP instances considered. The algorithm exposed in Section 5 was alsoexecuted with different values for the parameter pm, while the value ofR was kept constant. The main aimwas to analyse the performance ofthe different parameter control approaches and to study whetherparameter control gives some benefit with regard to tuning theparameter pm.

A common parameterisation for the multi-objective MA andthe different parameter control schemes was set. Table 2 showsthe parameterisation of the diversity-based multi-objective MA

described in Section 5. Different configurations—exactly 11—weredefined by modifying the value of the parameter pm. Moreover, theauxiliary objective and the crossover operator considered for eachof the two instances were different. This is because, depending onthe instance, the most appropriate values for these componentschange.

The parameterisations of the different parameter controlapproaches are shown in Tables 3 and 4 for the hyper-heuristicsand the FLCs, respectively. Note that four different configurationsfor the HH-ELI and HH-PROB hyper-heuristics were applied by combin-ing different values for the local stopping criterion and theparameter k. In the same way, four configurations of the FUZZY-Aand FUZZY-B FLCs were defined by setting different values for thenumber of generations and the parameter k. Finally, the hyper-heuristics were applied with nh¼11 low-level configurations. Low-level configurations used the parameterisation shown in Table 2with each one using a different value for the parameter pm.

Tables 5 and 6 show the statistics for the different configura-tions of the HH-ELI and HH-PROB hyper-heuristics, the FUZZY-A andFUZZY-B FLCs and the multi-objective MA (FIXED) for the Seattle andDenver instances, respectively, including dispersion measures likethe standard deviation (SD) and the coefficient of variation (CV). Thedata in bold show, for each method, the configuration thatachieved the lowest mean of the original objective value. More-over, the remaining configurations of a given method that areshown in bold did not exhibit statistically significant differences incomparison to the method that achieved the lowest mean for theoriginal objective value, as determined by the statistical proceduredescribed earlier in this section. In contrast, the configurations of agiven method which do not appear in bold presented statisticallysignificant differences from the configuration with the lowestmean of the original objective value. In order to identify a givenapproach's particular configuration, the values of its parametersreflect the name of the approach. For example:

� HH-PROB_1500_5 is a configuration of the HH_PROB hyper-heuristicwith a local stopping criterion equal to 1500 evaluations andhistorical knowledge (k) equal to 5 decisions.

� FUZZY-A_300_2 is a configuration of the FUZZY-A FLC that uses 300generations as the local stopping criterion and historicalknowledge equal to 2 decisions.

� FIXED_0.9 is a configuration of the multi-objective MA whichapplies the NM operator with probability pm equal to 0.9.

We note the following observations. With regard to parametertuning, in the case of the Seattle instance, the configuration of theFIXED approach that obtained the lowest mean for the original

Table 2Parameterisation of the diversity-based multi-objective MA for the first experiment.

Parameter Value Parameter Value

Stopping criterion 1.5 �105 evals. Crossover rate (pc) 1Population size (N) 10 individuals Mutation rate (pm) 0, 0.1, 0.2, …, 0.9, 1Crossover operator UX (Seattle), IX (Denver) NM operator steps (R) 7Auxiliary objective DCN (Seattle), ADI (Denver)


objective value used the value 0.5 (FIXED_0.5) for the parameter pmof the NM operator, while in the case of the Denver instance thisvalue was equal to 0.8 (FIXED_0.8). This fact confirms that the mostsuitable value for a parameter changes depending on the problemand/or the instance being solved. Moreover, these configurationsexhibited statistically significant differences as compared toothers. In the case of the Seattle instance, there were statisticallysignificant differences with 3 configurations, while in the case ofthe Denver instance, there were differences with 5 configurations.We can therefore observe that the parameter pm is more sensitiveto changes in its value when the NM operator is applied to theDenver instance, so it is even more important to select theappropriate values in this particular case.

Considering the control methods applied to the Seattleinstance, their configurations did not present statistically signifi-cant differences among them. Taking into account the Denverinstance, the only statistically significant differences appearedamong the configurations of the HH-PROB hyper-heuristic and the

FUZZY-B FLC. This means that both the hyper-heuristics and the FLCsare robust enough from the point of view of their parameters.If these parameters are modified, these changes are not going togreatly determine the performance of the control strategy.

If, for each of the four control methods exposed herein, weconsider the corresponding configuration that achieved the lowestmean of the original objective value, no statistically significantdifferences appeared among them. This was the case for both theSeattle and Denver instances. As a result, we can apply hyper-heuristics or FLCs indistinctly to control the parameter pm withoutdrastically affecting the quality of either city's frequency plans.

In order to compare parameter control and parameter tuning,Tables 7 and 8 show the number of configurations for the FIXED

approach that exhibited statistically significant differences witheach of the four control schemes for Seattle and Denver, respec-tively. To carry out the statistical comparison, we used theconfiguration for each control method that obtained the lowestmean of the original objective value.

Table 3Parameterisation of the hyper-heuristics HH-ELI and HH-PROB for the first experiment.


Local stopping criterion 1.5 �103, 3 �103 evals. Minimum selection rate (β) 0.1Number of low-level configs. (nh) 11 configs. Historical knowledge (k) 2, 5

Table 4Parameterisation of the fuzzy logic controllers FUZZY-A and FUZZY-B for the first experiment.


Number of generations (numGen) 1.5 �102, 3 �102 Difference among samples (Δ) 0.1Number of fuzzy sets (numTerms) 7 Historical knowledge (k) 2, 5Range of the parameter pm [0, 1]

Table 5Control and tuning of the parameter pm – Seattle instance.

Approach Min. First Qu. Median Mean Third Qu. Max. SD CV

HH-Eli_1500_2 547.1 589.1 624.3 644.5 675.3 889.2 84.4 13.1HH-Eli_3000_2 506.0 594.9 655.1 651.7 698.8 870.8 87.2 13.4HH-Eli_1500_5 511.0 610.9 672.4 662.8 726.4 799.6 72.7 11.0HH-Eli_3000_5 525.9 595.8 637.3 646.2 686.8 896.8 79.6 12.3

HH-Prob_1500_2 530.9 629.2 668.2 669.8 708.0 855.4 74.7 11.1HH-Prob_3000_2 523.6 578.2 650.3 644.0 695.5 775.1 71.4 11.1HH-Prob_1500_5 515.6 608.2 665.4 675.2 749.5 888.4 88.8 13.1HH-Prob_3000_5 534.5 590.1 642.9 647.2 686.7 790.8 65.5 10.1

Fuzzy-A_150_2 529.7 610.2 664.1 669.5 720.8 785.2 68.8 10.3Fuzzy-A_300_2 504.1 564.5 645.2 643.1 680.0 882.8 89.4 13.9Fuzzy-A_150_5 517.4 584.6 636.7 643.0 691.0 780.0 69.0 10.7Fuzzy-A_300_5 505.3 602.2 646.3 658.7 687.8 851.9 81.3 12.3

Fuzzy-B_150_2 463.0 609.2 651.1 659.2 721.4 814.1 87.6 13.3Fuzzy-B_300_2 471.7 619.0 678.2 669.8 712.6 807.4 77.4 11.6Fuzzy-B_150_5 519.9 616.7 658.3 660.7 701.9 804.5 70.0 10.6Fuzzy-B_300_5 497.6 616.3 673.4 663.8 704.1 795.5 68.5 10.3

Fixed_0 587.9 684.6 736.3 740.4 775.3 974.4 84.4 11.4Fixed_0.1 525.5 660.7 712.7 716.2 785.0 878.6 92.2 12.9Fixed_0.2 547.9 631.1 687.4 684.8 727.3 797.8 62.4 9.1Fixed_0.3 562.7 613.0 664.9 688.7 748.5 941.1 90.4 13.1Fixed_0.4 515.2 631.8 680.9 679.3 736.2 809.2 72.1 10.6Fixed_0.5 521.3 607.7 667.4 667.3 714.3 842.6 76.7 11.5Fixed_0.6 557.9 650.9 676.6 694.1 723.6 834.7 65.4 9.4Fixed_0.7 551.8 642.1 686.6 678.0 717.4 813.2 62.2 9.2Fixed_0.8 504.1 649.2 676.9 679.7 716.1 806.0 60.8 8.9Fixed_0.9 541.5 638.0 695.4 697.0 751.5 893.5 78.1 11.2Fixed_1.0 546.6 683.5 705.9 713.6 750.8 864.6 62.9 8.8


For the Seattle instance, the HH-ELI hyper-heuristic and the FUZZY-A FLC

were able to outperform 10 configurations of the FIXED approach, whilein the case of Denver, the hyper-heuristics outperformed 10 config-urations and the FUZZY-B FLC was able to outperform 11 configurations.Moreover, we should mention that, for both instances, no configura-tion of the FIXED schemewas able to statistically outperform any controlmethod. Consequently, the advantages of using parameter controlinstead of parameter tuning are clear. With just one execution of thecontrol schemes we were able to provide similar or even bettersolutions than those obtained using the best-behaved configuration ofthe diversity-based multi-objective MA. It is important to note that inorder to find the best-behaved configuration of the multi-objective MA,we had to test 11 different parameterisations by varying the value of

the parameter pm. With this in mind, the benefits of parameter controlover parameter tuning are even higher.

Finally, we would like to mention that the best-known fre-quency plan for Seattle, which was published in Segura et al.(2013c), was improved upon by the FUZZY-B control scheme whenthe parameter pm was adapted with the original objective valuedecreasing from 486.6 to 463.0, as can be observed in Table 5.

7.2. Analyses of the parameter R

The second experiment was based on the application of thecontrol approaches to the parameter R of the NM operator in orderto solve the FAP. As in the previous section, the multi-objective MA

exposed in Section 5 was also run. Nevertheless, its configurationswere obtained by varying the parameter R while holding the valueof pm constant. The main objective was to analyse the behaviour ofthe different control schemes when adapting the parameter R.These control techniques were also compared to parameter tuning.

The same parameterisations from the previous section wereused, though in this case the value of parameter pm was heldconstant. We also tested several values for the parameter R. Table 9shows the parameterisation of the diversity-based multi-objectiveMA for this experiment. In this case, 15 different configurations ofthis approach were executed using different values for the para-meter R.

The parameterisations of the control approaches are shown inTables 10 and 11 for the hyper-heuristics and the FLCs, respectively.As in the previous experiment, four different configurations ofeach control method were applied. In the case of the hyper-heuristics, they were defined with nh¼15 low-level configurations,with each one taking on a different value for the parameter R andusing the parameterisation shown in Table 9.

Tables 12 and 13 show the statistics obtained for the Seattleand Denver instances, respectively, by the different configurationsof the hyper-heuristics, the FLCs and the multi-objective MA. Notethe following regarding the setting of parameter R: in terms of

Table 6Control and tuning of the parameter pm – Denver instance.


HH-Eli_1500_2 84,228.4 84,811.6 85,099.4 85,285.8 85,798.8 86,742.7 673.9 0.8HH-Eli_3000_2 84,019.6 84,840.6 85,113.6 85,137.3 85,373.8 86,446.6 579.7 0.7HH-Eli_1500_5 83,996.6 84,792.3 85,170.2 85,251.3 85,531.2 87,590.5 790.0 0.9HH-Eli_3000_5 83,933.5 84,840.5 85,246.2 85,229.9 85,565.8 86,405.4 625.6 0.7

HH-Prob_1500_2 83,833.4 84,918.7 85,306.4 85,397.2 85,818.7 87,083.6 805.7 0.9HH-Prob_3000_2 84,274.6 84,916.2 85,455.6 85,560.5 86,213.6 87,661.3 874.5 1.0HH-Prob_1500_5 83,789.9 84,599.5 85,013.2 85,058.1 85,380.7 87,005.8 714.7 0.8HH-Prob_3000_5 84,257.9 84,929.5 85,660.6 85,529.5 85,965.6 87,215.7 761.1 0.9

Fuzzy-A_150_2 84,442.6 84,884.3 85,218.1 85,364.2 85,783.4 86,857.3 701.7 0.8Fuzzy-A_300_2 84,277.6 84,877.2 85,139.4 85,323.3 85,812.7 88,066.0 814.3 1.0Fuzzy-A_150_5 84,194.4 84,992.8 85,356.4 85,413.8 85,690.5 86,863.9 647.3 0.8Fuzzy-A_300_5 83,594.4 84,508.5 84,979.9 85,136.9 85,445.8 87,149.7 803.8 0.9

Fuzzy-B_150_2 84,000.7 84,937.9 85,567.4 85,509.3 85,995.9 87,174.2 808.1 0.9Fuzzy-B_300_2 84,245.1 84,774.4 85,066.2 85,194.8 85,625.7 87,106.1 622.5 0.7Fuzzy-B_150_5 84,118.2 84,627.1 84,975.1 85,190.6 85,559.6 87,213.2 841.0 1.0Fuzzy-B_300_5 84,004.6 84,493.2 85,035.4 84,986.1 85,407.6 85,794.5 536.4 0.6

Fixed_0 85,243.4 86,554.4 87,230.3 87,071.1 87,572.1 88,744.8 838.7 1.0Fixed_0.1 84,765.0 85,682.4 86,444.1 86,475.8 87,050.8 88,501.1 980.1 1.1Fixed_0.2 84,552.3 85,955.6 86,276.4 86,392.5 86,867.1 89,207.3 952.2 1.1Fixed_0.3 84,432.4 85,293.9 85,796.1 85,946.2 86,550.9 88,072.8 924.0 1.1Fixed_0.4 84,447.7 85,236.0 85,824.9 85,809.0 86,155.5 87,545.7 771.5 0.9Fixed_0.5 84,055.0 84,924.4 85,552.8 85,704.0 86,210.9 87,405.7 961.4 1.1Fixed_0.6 83,969.0 84,940.1 85,429.2 85,541.8 86,073.9 88,281.3 977.2 1.1Fixed_0.7 84,075.3 84,836.5 85,478.0 85,481.3 85,950.7 87,432.7 748.9 0.9Fixed_0.8 83,400.7 84,820.1 85,295.2 85,367.0 85,750.7 87,628.2 829.2 1.0Fixed_0.9 84,441.6 85,233.9 85,569.0 85,603.4 85,816.2 87,884.8 673.6 0.8Fixed_1.0 84,292.5 85,178.5 85,409.2 85,418.6 85,683.7 87,152.4 498.5 0.6

Table 7Number of fixed configurations outperformed bythe parameter control approaches adapting theparameter pm – Seattle instance.

Approach Number of configurations

HH-Eli 10HH-Prob 9Fuzzy-A 10Fuzzy-B 3

Table 8Number of fixed configurations outperformed bythe parameter control approaches adapting theparameter pm – Denver instance.




parameter tuning, the configuration of the FIXED approach thatyielded the lowest mean for the original objective value used theNM operator with the parameter R equal to 7—FIXED_7—for theSeattle instance. In the case of Denver, FIXED_6 was the mostsuitable configuration of the multi-objective MA. Both configura-tions exposed statistically significant differences as compared toother configurations of the FIXED scheme. In the case of Seattle,there were differences with 7 configurations, while for Denver, thenumber of statistically significant differences was equal to 6.

Similar conclusions to those extracted for parameter pm can bedrawn for parameter R. The study involving parameter tuningreveals that the most appropriate value for R depends on theproblem and/or instance being solved. A statistical comparisonshows that for this particular parameter, the number of statisticaldifferences among the FIXED scheme configuration that obtainedthe lowest mean for the original objective value and other FIXED

configurations is noticeable for both instances. As a result, we canconclude that the parameter R is also sensitive to changes in itsvalue, as was the case with pm.

With regard to parameter control, and considering the Seattleinstance, the configurations did not exhibit statistically significantdifferences among them, while in the case of Denver, only onestatistically significant difference appeared between the config-urations HH-ELI_3000_2 and HH-ELI_3000_5. Once more, both hyper-heuristics and FLCs demonstrated their robustness with R beingadapted in this case, since their performance was not significantlyaffected by changes in their parameter values, as occurred whenpm was adapted.

No statistically significant differences appeared among theconfigurations that achieved the lowest mean for the originalobjective value for each of the four control schemes. This hap-pened for both instances. Consequently, not only can the para-meter pm be controlled by hyper-heuristics or FLCs indistinctly, butalso can the parameter R. Thus, we can confirm the generality ofboth control methods, which can adapt continuous and discretenumeric parameters successfully.

In order to compare parameter control and parameter tuning,Tables 14 and 15 show, for Seattle and Denver respectively, thenumber of FIXED scheme configurations that exhibited statistically

significant differences with each one the four control techniques.To perform the statistical comparison, we selected the configura-tion that obtained the lowest mean for the original objective valuefor each of the four control methods. If we consider Seattle, the HH-ELI hyper-heuristic was able to outperform 13 FIXED scheme config-urations, while remaining control approaches outperformed 10configurations. In the case of Denver, the superiority of the controltechniques is clear as they were able to outperform every FIXED

scheme configuration. As in the previous experiment, no config-uration of the FIXED scheme was able to statistically outperform anycontrol method for either instance. As was the case with theparameter pm, the benefits of adapting the parameter R instead oftuning it are also clear. A single execution of the hyper-heuristicsor the FLCs yielded frequency plans for the two cities in questionthat were similar to or even better than those provided by thebest-behaved configurations of the multi-objective MA. To find thebest-behaved configurations we had to test 15 different parame-terisations of the diversity-based multi-objective MA by modifyingthe value of R. Taking this fact into consideration, the benefits ofparameter control over parameter tuning are even higher.

Finally, we would like to note that the best-known frequencyplan for Denver, which was published in Segura et al. (2013c), wasimproved upon by the FUZZY-B control scheme when the parameterR was adapted, with the original objective value decreasing from83,340.2 to 83,280.9, as shown in Table 13.

8. Conclusions and future work

One of the most commonly known drawbacks of meta-heuristicsis that they usually have a considerable number of parameters thatmust be properly set so as to yield adequate performance. Appro-priate parameter settings are therefore a critical part of any meta-heuristic design. Parameter tuning approaches attempt to find anoptimal set of parameters that remain fixed during the course of theoptimisation procedure. In contrast, parameter control approachesattempt to adapt the values of a parameter during the course of theoptimisation based on the assumption that different values are bettersuited at different points in the search.

Table 10Parameterisation of the hyper-heuristics HH-ELI and HH-PROB for the second experiment.


Local stopping criterion 1.5 �103, 3 �103 evals. Minimum selection rate (β) 0.1Number of low-level configs. (nh) 15 configs. Historical knowledge (k) 2, 5

Table 11Parameterisation of the fuzzy logic controllers FUZZY-A and FUZZY-B for the second experiment.


Number of generations (numGen) 1.5 �102, 3 �102 Difference among samples (Δ) 1Number of fuzzy sets (numTerms) 7 Historical knowledge (k) 2, 5Range of the parameter R [1, 15]

Table 9Parameterisation of the diversity-based multi-objective MA for the second experiment.


Stopping criterion 1.5 �105 evals. Crossover rate (pc) 1Population size (N) 10 individuals Mutation rate (pm) 0.5 (Seattle), 0.8 (Denver)Crossover operator UX (Seattle), IX (Denver) NM operator steps (R) 1, 2, 3, …, 14, 15Auxiliary objective DCN (Seattle), ADI (Denver)


Table 12Control and tuning of the parameter R – Seattle instance.


HH-Eli_1500_2 496.5 591.6 636.1 634.8 675.8 738.4 61.2 9.6HH-Eli_3000_2 558.2 593.5 646.4 656.3 711.1 825.0 70.7 10.8HH-Eli_1500_5 534.3 588.7 635.6 644.2 680.7 813.5 72.1 11.2HH-Eli_3000_5 501.8 590.0 635.2 650.7 704.8 845.8 81.7 12.6

HH-Prob_1500_2 527.1 603.4 629.4 645.1 697.6 775.3 64.2 10.0HH-Prob_3000_2 540.6 611.8 649.5 674.5 750.2 819.4 80.5 11.9HH-Prob_1500_5 514.5 620.8 634.1 646.6 697.7 764.4 63.9 9.9HH-Prob_3000_5 522.6 597.9 646.1 653.9 704.6 854.6 80.4 12.3

Fuzzy-A_150_2 486.5 587.1 627.2 641.4 694.9 821.1 77.4 12.1Fuzzy-A_300_2 503.6 596.3 644.5 649.4 696.7 794.5 74.6 11.5Fuzzy-A_150_5 508.8 585.2 627.4 647.7 697.0 820.5 88.0 13.6Fuzzy-A_300_5 543.0 597.1 635.2 655.9 698.2 880.3 79.7 12.2

Fuzzy-B_150_2 473.9 563.4 617.8 638.7 701.7 832.2 95.4 14.9Fuzzy-B_300_2 516.0 582.8 654.8 650.6 706.4 834.0 82.3 12.7Fuzzy-B_150_5 496.2 607.2 642.4 639.1 683.6 809.1 65.9 10.3Fuzzy-B_300_5 520.9 590.3 623.9 642.0 701.2 794.4 73.6 11.5

Fixed_1 549.1 673.6 728.6 744.6 814.0 949.0 97.9 13.2Fixed_2 592.5 675.1 726.7 727.8 778.7 887.7 74.4 10.2Fixed_3 550.6 667.7 707.7 725.3 767.8 968.0 83.0 11.4Fixed_4 566.6 644.0 687.8 689.5 722.7 893.6 73.8 10.7Fixed_5 566.8 640.5 681.7 693.9 721.0 908.3 80.4 11.6Fixed_6 547.0 635.4 670.0 675.8 709.5 806.9 60.6 9.0Fixed_7 580.3 624.1 641.6 658.4 683.4 830.4 57.3 8.7Fixed_8 529.8 637.1 682.5 668.2 713.5 782.7 69.9 10.5Fixed_9 555.0 619.4 667.2 662.1 709.5 771.1 59.5 9.0Fixed_10 533.3 629.1 682.8 684.2 723.3 843.7 74.4 10.9Fixed_11 543.8 634.9 667.3 678.9 732.4 863.0 76.2 11.2Fixed_12 525.9 647.6 683.0 681.6 728.1 833.6 73.2 10.7Fixed_13 597.7 665.3 698.9 712.2 737.5 982.5 74.2 10.4Fixed_14 621.5 674.8 731.2 738.5 795.2 894.2 80.0 10.8Fixed_15 613.7 709.6 737.3 739.8 780.6 879.2 63.8 8.6

Table 13Control and tuning of the parameter R – Denver instance.


HH-Eli_1500_2 83,780.7 84,658.2 85,041.6 85,064.2 85,568.1 86,324.1 609.5 0.7HH-Eli_3000_2 83,876.2 84,446.8 84,856.4 84,846.1 85,196.2 86,552.9 638.7 0.8HH-Eli_1500_5 84,037.3 84,593.2 84,899.0 85,090.5 85,412.7 87,154.3 731.3 0.9HH-Eli_3000_5 83,740.6 84,784.8 84,974.7 85,184.5 85,610.9 86,783.9 654.8 0.8

HH-Prob_1500_2 83,,914.2 84,568.8 85,127.7 85,240.3 85,715.4 87,159.4 844.1 1.0HH-Prob_3000_2 84,352.3 84,667.8 84,962.4 85,177.4 85,660.6 86,766.0 691.9 0.8HH-Prob_1500_5 83,796.0 84,441.5 84,876.1 84,973.9 85,335.6 86,323.2 614.8 0.7HH-Prob_3000_5 83,548.0 84,639.6 85,026.2 84,965.3 85,269.4 86,455.1 619.5 0.7

Fuzzy-A_150_2 83,829.4 84,615.1 84,956.8 85,081.4 85,354.6 86,533.0 634.9 0.7Fuzzy-A_300_2 83,750.1 84,533.9 84,988.5 85,044.9 85,432.5 87,192.5 750.9 0.9Fuzzy-A_150_5 83,852.6 84,534.0 84,962.4 84,967.1 85,313.5 86,897.6 696.0 0.8Fuzzy-A_300_5 83,335.4 84,545.0 85,102.9 85,032.9 85,524.4 86,584.4 792.8 0.9

Fuzzy-B_150_2 83,280.9 84,446.4 85,033.5 84,881.4 85,380.0 86,181.9 694.2 0.8Fuzzy-B_300_2 83,377.4 84,515.4 84,888.5 84,915.7 85,359.1 86,106.8 592.1 0.7Fuzzy-B_150_5 83,955.2 84,549.1 85,145.9 85,228.5 85,749.9 87,100.4 831.0 1.0Fuzzy-B_300_5 83,727.2 84,481.1 84,939.1 84,942.2 85,313.6 86,433.9 660.5 0.8

Fixed_1 84,885.9 85,695.1 86,231.9 86,206.7 86,667.4 87,291.9 681.1 0.8Fixed_2 83,747.4 85,352.1 85,747.4 85,716.0 86,282.5 86,894.6 718.9 0.8Fixed_3 84,558.3 84,876.7 85,325.8 85,610.8 86,334.2 87,403.0 872.4 1.0Fixed_4 84,001.3 85,083.5 85,225.4 85,423.6 85,848.8 87,109.2 768.8 0.9Fixed_5 84,760.0 85,151.6 85,631.0 85,588.6 85,981.8 86,745.8 503.2 0.6Fixed_6 84,483.9 84,888.0 85,254.9 85,333.7 85,751.2 86,899.4 606.2 0.7Fixed_7 84,051.4 85,088.9 85,528.4 85,505.9 85,938.8 87,282.4 726.6 0.8Fixed_8 84,296.1 85,052.9 85,382.1 85,449.0 85,820.9 87,139.2 574.2 0.7Fixed_9 84,357.2 85,129.6 85,339.9 85,428.5 85,655.3 87,110.0 595.5 0.7Fixed_10 84,693.2 85,277.0 85,434.1 85,652.1 86,066.5 87,222.2 585.1 0.7Fixed_11 84,771.0 85,388.2 85,560.2 85,671.2 86,077.8 86,910.9 558.9 0.7Fixed_12 84,345.7 85,203.1 85,500.4 85,538.7 85,819.2 86,688.2 590.3 0.7Fixed_13 84,587.0 85,268.0 85,516.4 85,621.5 85,869.9 86,981.9 606.5 0.7Fixed_14 84,627.3 85,159.4 85,740.0 85,703.3 85,979.4 87,857.6 676.3 0.8Fixed_15 84,955.6 85,768.8 86,036.3 86,154.7 86,550.7 87,522.7 649.9 0.8


In this paper, a novel FLC is proposed that controls two parametersin a highly efficient meta-heuristic that is specifically designed toaddress a complex variant of the FAP. The meta-heuristic in question isa diversity-based multi-objective scheme that has reported the best-known frequency plans for several FAP instances. The FLC designedincorporates a set of different rule bases that are enabled dependingon historical information extracted from the own optimisationprocess. The scheme promotes the usage of parameter values thathave historically yielded the best performance. At the same time,other parameter values are explored so as to adapt the scheme to thevarying requirements that might arise in different optimisationstages. In addition, a well-known hyper-heuristic variant was alsoused as a parameter control scheme. One of the main differencesbetween the two methods lies in the fact that the hyper-heuristicapproach requires that a fixed set of potential values for theparameter be pre-defined by the user, whereas the fuzzy logicapproach is able to select any value within a range.

In order to show the generality of the proposals and with theaim of improving the results further, two different numeric para-meters were controlled: pm and R. These belong to the NM operator,an efficient mutation operator tailor-designed for this FAP variant. Theextensive experimental evaluation performed on two real-worldinstances of the FAP revealed that the FLCs are able to obtain similaror even better frequency plans than those obtained using hyper-heuristics or a fixed value for the parameters. The fact that betterresults are returned by some control schemes as compared to thefixed methods also highlights the advantage to be gained by adaptingthe parameter over the course of the run, i.e. in parameter controlrather than in parameter tuning. Since no statistical differences werenoted between FLCs and hyper-heuristics, they can be used indis-tinctly in order to obtain high quality frequency plans. Moreover,both approach types are quite robust from the point of view of theirparameters. Small modifications to the values of their internalparameters do not entail significant changes in their behaviour andperformance. Finally it is worth mentioning that the best-knownfrequency plans published for both of the instances considered wereimproved upon in this work by one of the control approaches basedon FLCs.

The FLCs control strategy is novel in its use of parameter control,as well as in its use of multiple rule-bases depending on feedbackfrom the optimisation. To the best of our knowledge, this is thefirst time an FLC has been used to control the parameters of the NM

operator. In addition, we have shown that the method is applicable

in general since both a continuous numeric parameter and adiscrete numeric parameter were successfully controlled.

Other numeric parameters belonging to other meta-heuristicsfrom the mono-objective and multi-objective fields might becontrolled with this approach. If a multi-objective meta-heuristicis considered, the value of the input variable IMP should becalculated by means of a multi-objective performance metric.Although two different parameters were considered, they wereadapted separately, so it would be interesting to control bothparameters simultaneously. Another promising line of researchcould be the design of a hybrid control scheme that combines FLCsand hyper-heuristics with the aim of adapting numeric andsymbolic parameters simultaneously, thus combining the advan-tages of both methods in a single control scheme.

Acknowledgments

This work was funded by the EC (FEDER) and the Spanish Ministryof Science and Innovation as part of the ’Plan Nacional de IþDþ i’,with contract number TIN2011-25448. The work of Eduardo Seg-redo was funded by grant FPU-AP2009-0457. The work was alsofunded by the HPC-EUROPA2 Project (Project number: 228398) withthe support of the European Commission—Capacities Area—Research Infrastructures.

Appendix A. Supplementary data

Supplementary data associated with this paper can be found in theonline version at http://dx.doi.org/10.1016/j.engappai.2014.01.005.

References

Abbass, H.A., Deb, K., 2003. Searching under multi-evolutionary pressures. In:Proceedings of the Fourth Conference on Evolutionary Multi-Criterion Optimi-zation, Springer-Verlag, pp. 391–404.

Bäck, T., 1992. The interaction of mutation rate, selection, and self-adaptationwithin a genetic algorithm. In: Proceedings of the 2nd Conference on ParallelProblem Solving from Nature, North-Holland, Amsterdam.

Bäck, T., Eiben, A.E., van der Vaart, N.A.L., 2000. An empirical study on gas “withoutparameters”. In: Proceedings of the 6th International Conference on ParallelProblem Solving from Nature, Springer-Verlag, London, UK, pp. 315–324.

Brito, F.H., Teixeira, A.N., Teixeira, O.N., Oliveira, R.C.L., 2006. A fuzzy intelligentcontroller for genetic algorithms' parameters. In: Jiao, L., Wang, L., Gao, X.b., Liu,J., Wu, F. (Eds.), Advances in Natural Computation, Lecture Notes in ComputerScience, vol. 4221, Springer, Berlin, Heidelberg, pp. 633–642.

Bui, L., Abbass, H., Branke, J., 2005. Multiobjective optimization for dynamicenvironments. In: The 2005 IEEE Congress on Evolutionary Computation, vol.3, pp. 2349–2356.

Burke, E.K., Hyde, M., Kendall, G., Ochoa, G., Ozcan, E., Qu, R., 2010. Hyper-Heuristics: A Survey of the State of the Art. Technical Report NOTTCS-TR-SUB-0906241418-2747. School of Computer Science and Information Technol-ogy, University of Nottingham, Computer Science.

Burke, Edmund, Kendall, Graham, Newall, Jim, Hart, Emma, Ross, Peter, Schulen-burg, Sonia, 2003. Handbook of Metaheuristics International Series in Opera-tions Research & Management Science. In: Glover, Fred, Kochenberger, Gary A.(Eds.), Hyper-Heuristics: An Emerging Direction in Modern Search Technology,vol. 57. Springer, US, pp. 457–474.

Caamaño, P., Prieto, A., Becerra, J., Bellas, F., Duro, R., 2010. Real-valued multimodalfitness landscape characterization for evolution. In: Wong, K., Mendis, B.,Bouzerdoum, A. (Eds.), Neural Information Processing. Theory and Algorithms,Lecture Notes in Computer Science, vol. 6443, Springer, Berlin, Heidelberg,pp. 567–574.

Chen, C.L., Weng, C.P., 2009. A fuzzy multi-objective genetic algorithm approach tooptimal parameter design for laser electrophotographic systems. In: Proceed-ings of the 4th IEEE Conference on Industrial Electronics and Applications,2009, ICIEA 2009, pp. 2786–2791.

Črepinšek, Matej, Liu, Shih-His, Mernik, Marjan, 2013. Exploration and Exploitationin Evolutionary Algorithms: A Survey. ACM Comput. Surv 45 (3) (35-1–35-33).

Das Sharma, K., Chatterjee, A., Rakshit, A., 2012. A Pso-Lyapunov hybrid stableadaptive fuzzy tracking control approach for vision-based robot navigation.IEEE Trans. Instrum. Meas. 61, 1908–1914.

Davis, L., 1989. Adapting operator probabilities in genetic algorithms. In: Proceed-ings of the Third International Conference on Genetic Algorithms, MorganKaufmann Publishers Inc., San Francisco, CA, USA, pp. 61–69.

Table 14Number of fixed configurations outperformed bythe parameter control approaches adapting theparameter R – Seattle instance.



Table 15Number of fixed configurations outperformed bythe parameter control approaches adapting theparameter R – Denver instance.





http://refhub.elsevier.com/S0952-1976(14)00012-8/sbref100











Deb, K., Pratap, A., Agarwal, S., Meyarivan, T., 2002. A fast and elitist multiobjectivegenetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6, 182–197.

Eiben, A.E., Smit, S.K., 2011. Parameter tuning for configuring and analyzingevolutionary algorithms. Swarm Evol. Comput. 1, 19–31.

Eiben, A.E., Smith, J.E., 2003. Introduction to Evolutionary Computing, NaturalComputing Series. Springer, Berlin, Heidelberg.

Eiben, A.E., Michalewicz, Zbigniew, Schoenauer, M., Smith, J.E., 2007. ParameterSetting in Evolutionary Algorithms, Studies in Computational Intelligence. In:Lobo, Fernando G., Lima, Cláudio F., Michalewicz, Zbigniew (Eds.), ParameterControl in Evolutionary Algorithms, vol. 54. Springer, Berlin, Heidelberg,pp. 19–46.

Feng, D., Wang, X., Fei, M., Chen, T., 2006. Tuning Parameters of PID ControllerBased on Fuzzy Logic Controlled Genetic Algorithms, pp. 635849–635849-6.

Gen, M., Yun, Y., 2006. Soft computing approach for reliability optimization: state-of-the-art survey. Reliab. Eng. Syst. Safety 91, 1008–1026.

Glover, Fred, Kochenberger, Gary A., 2003. Handbook of Metaheuristics, Interna-tional Series in Operations Research \& Management Science, vol. 57. Springer,US.

Hadavandi, E., Shavandi, H., Ghanbari, A., Abbasian-Naghneh, S., 2012. Developing ahybrid artificial intelligence model for outpatient visits forecasting in hospitals.Appl. Soft Comput. J. 12, 700–711.

Herrera, F., Lozano, M., 2003. Fuzzy adaptive genetic algorithms: design, taxonomy,and future directions. Soft Comput. 7 (8), 545–562.

Kramer, O., 2010. Evolutionary self-adaptation: a survey of operators and strategyparameters. Evol. Intell. 3, 51–65.

Lee, M.A., Takagi, H., 1993. Dynamic control of genetic algorithms using fuzzy logictechniques. In: Proceedings of the Fifth International Conference on GeneticAlgorithms, Morgan Kaufmann, pp. 76–83.

León, C., Miranda, G., Segura, C., 2009. METCO: a parallel plugin-based frameworkfor multi-objective optimization. Int. J. Artif. Intell. Tools 18, 569–588.

Li, Q., Maeda, Y., 2008. Distributed adaptive search method for genetic algorithmcontrolled by fuzzy reasoning. In: IEEE International Conference on FuzzySystems, 2008, FUZZ-IEEE 2008, pp. 2022–2027.

Ling, S.H., Nguyen, H., Leung, F.H.F., Chan, K.Y., Jiang, F., 2012. Intelligent fuzzyparticle swarm optimization with cross-mutated operation. In: 2012 IEEECongress on Evolutionary Computation (CEC'12), pp. 1–8.

Liu, J., Lampinen, J., 2005. A fuzzy adaptive differential evolution algorithm. SoftComput. 9, 448–462.

Liu, D., Liu, X., 2011. The improved genetic algorithm based on fuzzy controller withadaptive parameter adjustment. In: Zhu, M. (Ed.), Information and Manage-ment Engineering. Communications in Computer and Information Science, vol.235, Springer, Berlin, Heidelberg, pp. 491–497.

Lobo, Fernando G, Lima, Cláudio F., Michalewicz, Zbigniew, 2007. Parameter Settingin Evolutionary Algorithms, Studies in Computational Intelligence, vol. 54.Springer, Berlin, Heidelberg.

Luna, F., Blum, C., Alba, E., Nebro, A., 2007. ACO vs EAs for solving a real-worldfrequency assignment problem in GSM networks. In: Proceedings of the 2007Genetic and Evolutionary Computation Conference, pp. 94–101.

Luna, F., Estébanez, C., León, C., Chaves-González, J., Nebro, A., Aler, R., Segura, C.,Vega-Rodríguez, M., Alba, E., Valls, J., Miranda, G., Gómez-Pulido, J., 2011.Optimization algorithms for large-scale real-world instances of the frequencyassignment problem. Soft Comput. Fusion Found. Methodol. Appl. 15, 975–990.

Press, William H., Flannery, Brian P., Teukolsky, Saul A., Vetterling, William T., 1988.Numerical Recipes in C: The Art of Scientific Computing. Cambridge UniversityPress, New York, NY, USA.

Qin, A.K., Huang, V.L., Suganthan, P.N., 2009. Differential evolution algorithm withstrategy adaptation for global numerical optimization. Trans. Evol. Comput. 13,398–417.

Rada-Vilela, J., 2013. fuzzylite: a Fuzzy Logic Control Library in Cþþ .Rechenberg, I., 1973. Evolutionsstrategie: optimierung technischer systeme nach

prinzipien der biologischen evolution. Frommann-Holzboog-VerlagRui, O., Hajizadeh, A., Undeland, T.M., 2010. Parameter optimization of a fuzzy logic

controller for a power electronics boost converter using genetic algorithms. In:Proceedings of the 9th WSEAS International Conference on Artificial Intelli-gence, Knowledge Engineering, and Data Bases, World Scientific and Engineer-ing Academy and Society (WSEAS), Stevens Point, Wisconsin, USA, pp. 120–124.

Segredo, E., Segura, C., León, C., 2011. A multiobjectivised memetic algorithm for theFrequency Assignment Problem. In: 2011 IEEE Congress on EvolutionaryComputation (CEC), pp. 1132–1139.

Segura, C., 2012. Parallel Optimisation Schemes. A Hybrid Scheme based onHyperheuristics and Evolutionary Computation (Ph.D. thesis). La Laguna, Spain.

Segura, C., Miranda, G., León, C., 2010. Parallel hyperheuristics for the frequencyassignment problem. Memetic Comput. 3, 33–49.

Segura, C., Coello Coello, C.A., Miranda, G., León, C., 2013a. Using multi-objectiveevolutionary algorithms for single-objective optimization. 4OR 11, 201–228.

Segura, C., Segredo, E., León, C., 2013b. Analysing the robustness of multiobjecti-visation approaches applied to large scale optimisation problems, in: Tantar, E.,Tantar, A.A., Bouvry, P., Del Moral, P., Legrand, P., Coello Coello, C.A., Schütze, O.(Eds.), EVOLVE—A Bridge between Probability, Set Oriented Numerics andEvolutionary Computation. Studies in Computational Intelligence, vol. 447,Springer, Berlin, Heidelberg, pp. 365–391.

Segura, C., Segredo, E., León, C., 2013c. Scalability and robustness of parallelhyperheuristics applied to a multiobjectivised frequency assignment problem.Soft Comput. 17, 1077–1093.

Simon, Marvin K., Alouini, Mohamed-Slim, 2002. Digital Communication OverFading Channels: A Unified Approach to Performance Analysis. John Wiley \&Sons, Inc., New York, USA.

Smit, S.K., Eiben, A.E., 2009. Comparing parameter tuning methods for evolutionaryalgorithms, in: Proceedings of the Eleventh Congress on Evolutionary Compu-tation, IEEE Press, Piscataway, NJ, USA. pp. 399–406.

Srinivas, M., Patnaik, L., 1994. Adaptive probabilities of crossover and mutation ingenetic algorithms. IEEE Trans. Syst. Man Cybern. 24, 656–667.

Toffolo, A., Benini, E., 2003. Genetic diversity as an objective in multi-objectiveevolutionary algorithms. Evol. Comput. 11, 151–167.

Vinkó, T., Izzo, D., 2007. Learning the best combination of solvers in a distributedglobal optimization environment. In: Proceedings of Advances in GlobalOptimization: Methods and Applications (AGO), Mykonos, Greece, pp. 13–17.

Walke, B.H., 2002. Mobile Radio Networks: Networking, Protocols and TrafficPerformance. Wiley

Wang, P., Wang, G., Hu, Z., 1997. Speeding up the search process of geneticalgorithm by fuzzy logic. In: Proceedings of the 5th European Congress onIntelligent Techniques and Soft Computing, pp. 665–671.

Whitley, L.D., Gordon, V.S., Mathias, K.E., 1994. Lamarckian evolution, the baldwineffect and function optimization, in: Proceedings of the International Con-ference on Evolutionary Computation. The Third Conference on ParallelProblem Solving from Nature: Parallel Problem Solving from Nature,Springer-Verlag, London, UK, pp. 6–15.

Xue, F., Sanderson, A., Bonissone, P., Graves, R., 2005. Fuzzy logic controlled multi-objective differential evolution. In: The 14th IEEE International Conference onFuzzy Systems, 2005. FUZZ '05, pp. 720–725.

Zhan, Z.H., Zhang, J., 2008. Adaptive particle swarm optimization. In: Proceedings ofthe 6th International Conference on Ant Colony Optimization and SwarmIntelligence, Springer-Verlag, Berlin, Heidelberg. pp. 227–234.

Zhang, W., Liu, Y., 2005. Fuzzy logic controlled particle swarm for reactive poweroptimization considering voltage stability. In: The 7th International PowerEngineering Conference, 2005. IPEC 2005, pp. 1–555.





























































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