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

Engineering Applications of Artificial Intelligence 30 (2014) 199–212

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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Fuzzy logic-controlled diversity-based multi-objective memetic algorithm applied to a frequency assignment problem Eduardo Segredo n, Carlos Segura, Coromoto León Dpto. Estadística, I. O. y Computación. Universidad de La Laguna. Avda. Astrofísico Fco. Sánchez s/n, Edif. Matemáticas, 38271, Santa Cruz de Tenerife, Spain

art ic l e i nf o

a b s t r a c t

Article history: Received 15 August 2013 Received in revised form 9 November 2013 Accepted 8 January 2014 Available online 4 February 2014

One of the most commonly known weaknesses of Evolutionary Algorithms (EAS) is the large dependency between the values selected for their parameters and the results. Parameter control approaches that adapt the parameter values during the course of an evolutionary run are becoming more common in recent years. The aim of these schemes is not only to improve the robustness of the controlled approaches, but also to boost their efficiency. In this paper we investigate the application of parameter control schemes to address a well-known variant of the Frequency Assignment Problem (FAP). The controlled EA is a highly efficient diversity-based multi-objective memetic scheme. In this work, a novel general parameter control method based on Fuzzy Logic is devised. In addition, a hyper-heuristic is also considered as an established parameter control scheme. An extensive experimental evaluation of both methods is carried out that includes a comparison to a wide-range of fixed-parameter schemes. The results show that the fuzzy logic method is able to find similar or even better solutions than the hyperheuristic and the fixed-parameter methods for several instances of the FAP. In fact, this method yielded frequency plans that outperform the best previously published solutions. Finally, the generality of the fuzzy logic-based scheme is demonstrated by controlling different kinds of parameters. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Parameter control Fuzzy logic controllers Hyper-heuristics Diversity preservation Memetic algorithms Frequency assignment problem

1. Introduction Many optimisation problems that arise in real world applications require the employment of approximation techniques. Among them, meta-heuristics (Glover and Kochenberger, 2003) have become popular in recent decades. They are high-level strategies that guide a set of heuristics in the search of an optimum. Evolutionary Algorithms (EAs) (Eiben and Smith, 2003) are one of the most popular strategies belonging to this group. They are population-based algorithms inspired on biological evolution. EAs have shown great promise for calculating solutions to difficult problems. However, in some problems, EAs exhibit a tendency to converge towards local optima, with the likelihood of this occurrence depending on the shape of the fitness landscape (Caamaño et al., 2010). Several methods have been designed with the aim of dealing with local optima stagnation. The reader is referred to Črepinšek et al. (2013) for an extensive survey of diversity preservation mechanisms. One of the methods that has gained some popularity in recent years is based on applying multi-objective schemes to single-objective optimisation problems (Segura et al., 2013a). Several ways of applying the multi-objective concepts have been devised

n

Corresponding author. Tel.: þ 34 922 319 191. E-mail addresses: [email protected] (E. Segredo), [email protected] (C. Segura), [email protected] (C. León). 0952-1976/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engappai.2014.01.005

with diversity-based multi-objective algorithms being one of the most promising schemes (Abbass and Deb, 2003). In these schemes, a metric of the diversity introduced by each individual is used as an auxiliary objective. These schemes can better deal with strong optima by being able to alleviate the effects of premature convergence. Most popular EA variants have several components and/or parameters such as the survivor selection mechanism, or the genetic and parent selection operators, which must be specified. In general, the performance of an EA and, consequently, the quality of the resulting solutions, is highly dependent on these components and parameters. As a result, it is essential that the parameters of an EA be suitably determined. However, finding appropriate parameter settings remains one of the persistent challenges for Evolutionary Computing (Eiben and Smit, 2011). Parameter setting strategies are commonly divided into two categories: parameter tuning and parameter control. In parameter tuning the objective is to identify the best set of values for the parameters of a given EA, which is then executed using these values, which remain fixed for the duration of the run. In contrast, the aim of parameter control is to design control strategies that select the most suitable values for the parameters at each stage of the search process while the algorithm is being executed. In single objective optimisation, it has been empirically and theoretically shown that different parameter values might be optimal at different stages of the search process (Srinivas and Patnaik, 1994;

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Bäck, 1992). Therefore, it is natural to apply control strategies to multi-objective EAs. In this paper we devise a novel parameter control strategy based on the use of Fuzzy Logic. Such a strategy, as well as other wellknown parameter control methods, is used to control some parameters of a diversity-based multi-objective Memetic Algorithm (MA), which is applied to a set of real-world instances of the Frequency Assignment Problem (FAP). The MA has some components specifically tailored to deal with the FAP. It was selected because it has demonstrated its efficiency against a large set of different 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 applicable to both continuous and discrete numeric parameters.

 First application of parameter control techniques based on fuzzy

 

logic and hyper-heuristics in order to control the parameters of a mutation operator that has been specifically designed to address the FAP. An extensive comparison of fuzzy logic-based schemes vs. hyper-heuristics as methods of parameter control applied to a complex real-world problem. A broad comparison between parameter control methods and schemes with fixed parameters that highlights the benefits of parameter control as opposed to parameter tuning.

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

to profit from the definition of these types of metrics for setting such parameters. In this case of this paper, we focus on control methods for numeric parameters. The goal of parameter control is to design a control strategy that selects the most suitable parameter values for every stage of the search process. The ideas of parameter control were first incorporated in early work on EAs (Davis, 1989; Rechenberg, 1973). Nevertheless, recent research has seen a marked increase in proposals for methods that achieve parameter control in EAs (Lobo et al., 2007). In fact, parameter control methods have been successfully applied to a wide range of EAS and other meta-heuristics such as Evolution Strategies (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 taxonomies have been proposed. One of the most popular classifications (Eiben et al., 2007) considers the following types of strategies:

 Deterministic parameter control: Parameter values are altered by  

a deterministic rule without using any feedback from the search procedure. Adaptive parameter control: Parameter values are updated by a mechanism that uses some feedback from the search process. Such a mechanism is externally supplied. Self-adaptive parameter control: Parameters are encoded into the chromosome and their values are modified by the EA variation operators.

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

3. Background on fuzzy logic controllers for parameter control 2. State of the art of parameter control in evolutionary algorithms Finding the most suitable configuration of an EA is one of the most challenging tasks in the field of Evolutionary Computation (Eiben and Smith, 2003). In order to completely define an instance of an EA, two types of information are required (Smit and Eiben, 2009):

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

 Numeric—also referred to as quantitative or behavioural parameters—such as the population size and the crossover and mutation rates. For both kinds of parameters, the different elements of the domain are known as parameter values, and a parameter is instantiated by assigning it a value. The main difference between both types of parameters lies in their respective domains. Symbolic parameters, such as the crossover operator, have a finite domain in which neither order is established nor distance metric is defined. In contrast, numeric parameters, such as the mutation rate, have an infinite domain in which a distance metric and an order can be defined for the values. Thus, optimisation methods can readily be used to look for the appropriate values of the numeric parameters of an EA. However, in the case of symbolic parameters, as noted above, distance metrics cannot be applied between two values, meaning optimisation schemes are not able

Our knowledge of EAs performance has significantly increased in recent years due to the large number of empirical analyses conducted on a wide range of applications in different areas. It would be desirable to profit from this human knowledge by encapsulating it within an algorithm to automate the task of improving the behaviour and performance of EAs. However, this sort of knowledge is usually incomplete, imprecise and/or it is not well organised. Consequently, the application of fuzzy logic-based methods would seem to offer a promising approach for dealing with this kind of knowledge. One application of fuzzy logic is the design of Fuzzy Logic Controllers (FLCs). FLCs can be used to define control approaches in which the incorporation of human knowledge is performed intuitively. An FLC consists of the knowledge base, the fuzzy inference engine and the fuzzification and defuzzification interfaces (Herrera and Lozano, 2003). The knowledge base has two different parts, a data base, which includes the definitions of the membership functions of the linguistic terms for each input and output variable, and a rule base constituted by the collection of fuzzy control 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 parameters are infinite, in contrast to other techniques that can only use some values from a finite set. However, the main drawback is that FLCs cannot be directly applied to control the symbolic parameters of an EA. Therefore, in this paper we restrict the application of the FLC to controlling numeric parameters. An

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additional drawback is that, unlike with other systems, an error feedback signal does not directly exist in the fuzzy logic-based approach we are proposing because its aim is to optimise as much as possible. However, our proposal has to include some way of measuring performance, which makes it atypical since the desired objective is not known beforehand. Despite this, the term FLC is commonly used for this kind of scheme (Xue et al., 2005), and consequently it is the one we adopted for our fuzzy logic-based approach. A considerable body of research regarding FLCs and EAs already exists. For example, different EAs, as well as other meta-heuristics, have been used to optimise the design of FLCs for different applications (Das Sharma et al., 2012; Hadavandi et al., 2012; Rui et al., 2010). However, in this paper, we study the reverse of this type of application, and focus on the design of FLCs that adapt the parameters of an EA, thus providing an adaptive control technique that utilises feedback from the search process to adapt the parameters. Several methods have been proposed for controlling the parameters of different meta-heuristics–including EAS–through FLCs. The principle behind these schemes is to use a FLC to compute new parameter values by considering any combination of performance measures and current parameter values as the input to the controllers. This idea was first proposed in Lee and Takagi (1993). In such a scheme, the population size and mutation and crossover rates are adapted considering the best, average and worst fitness values of the individuals in the population. Subsequently, a large number of variants have been proposed (Gen and Yun, 2006). Some of the schemes that are more closely related to our proposal are briefly summarised in this section. Gen and Yun (2006) present a survey of several schemes that rely on a FLC to control their internal parameters. Some of the simplest schemes adapt the crossover and mutation rates by considering the average fitness of the last two generations as input variables (Wang et al., 1997). Basically, depending on the improvements obtained in the last generation, the crossover and mutation probabilities are modified so as to change the perturbation strength of the variation scheme. Using only two generations might not be enough, so in some schemes the average fitness values of the last three generations are considered (Liu and Liu, 2011). A similar idea is proposed in Liu and Lampinen (2005) to adapt the parameters of a DE approach. Specifically, two FLCs are used to adapt the mutation scale factor and the crossover rate. In this case, the input variables are not only made up from measures in the objective space, but the space of the variables is also considered. Other research works reported the application of three FLCs to adaptively adjust the parameters of a PSO algorithm (Zhang and Liu, 2005). Specifically, the inertia weight and learning factors are adapted considering the current fitness values and the number of generations where no improvements have been achieved. The system assumes that the user can assign a quality level to the different fitness values beforehand. More advanced FLCs consider additional metrics of diversity to carry out the decisions. For instance, in Brito et al. (2006) the frequency of the best individual, as well as the rate of duplicate individuals, is used to control the mutation, crossover and surviving individual rates. In addition, an input variable that estimates the quality of the resulting fitness is used. Thus, as in some of the other schemes described, it assumes knowledge regarding the supposed optimal values. In Li and Maeda (2008) the diversity is calculated by considering the difference between the maximum and average fitness of the population. Finally, the normalised standard deviation is considered as an input variable in Ling et al. (2012). The feature common to most of the research described in the literature is that the FLCs are used to adapt the parameters of the mutation or crossover operators, the population size, or combinations of all three (Herrera and Lozano, 2003). Moreover, these FLCs

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are usually tailor-made methods for a specific EA and/or parameters, and they only make use of a single rule base. It is also worth noting that FLCs have been successfully used to adapt different EAs applied to real-world applications, demonstrating their efficiency and reliability even for complex problems (Feng et al., 2006; Zhang and Liu, 2005). From the perspective of MOEAs, it is apparent that the body of research is much smaller than in the case of mono-objective EAs. However, FLCs have also been used to control the parameters of different MOEAs (Chen and Weng, 2009; Xue et al., 2005). It is important to note that even though in this paper a multi-objective scheme is applied, the problem to be solved has a single objective. Thus, most of the ideas used for controlling multi-objective schemes with FLCs cannot be directly applied to the scheme being proposed here.

4. The frequency assignment problem: formal definition The FAP formulation applied herein was proposed by Luna et al. (2007). Let T ¼ ft 1 ; t 2 ; …; t n g be a set of n transceivers, and let F i ¼ ff i1 ; …; f iki g  N be the set of valid frequencies that can be assigned to a transceiver t i A T, i¼ 1,…,n. Note that ki—the cardinality of Fi—is not necessarily the same for all the transceivers. Furthermore, let S ¼ fs1 ; s2 ; …; sm g be a set of given sectors—or cells —of cardinality m. Each transceiver t i A T is installed in exactly one of the m sectors. From now on we denote the sector in which a transceiver ti is installed by sðt i Þ A S. Finally, the matrix M ¼ fðμij ; sij Þgmm is denoted as the interference matrix. The two elements μij and sij of a matrix entry Mði; jÞ ¼ ðμij ; sij Þ are numeric values greater than or equal to zero. The values of μij and sij represent the mean and the standard deviation, respectively, of a Gaussian probability distribution describing the Carrier-toInterference (C/I) ratio (Walke, 2002) when sectors i and j operate on the same frequency. The higher the mean value is, the lower the interference is and thus the better the communication quality is. Note that the interference matrix is defined at the sector—or cell—level because the transceivers installed in each sector serve the same area. A solution is obtained by assigning to each transceiver t i A T one of the frequencies from Fi. Consequently, a candidate solution—or frequency plan—is denoted by p A F 1  F 2  ⋯  F n , where pðt i Þ A F i is the frequency assigned to the transceiver ti. The objective is to find a solution p that minimises the following cost function: CðpÞ ¼ ∑

∑

t A T u A T;u a t

C sig ðp; t; uÞ

ð1Þ

In order to define the function C sig ðp; t; uÞ (Eq. (2)), let st and su be 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 be the two elements of the entry Mðst ; su Þ of the interference matrix with respect to sectors st and su. The parameter K in Eq. (2) represents the cost associated with the usage of the same or adjacent frequencies in the same sector. In real networks, it is unfeasible to operate with 8 K > > > < C ðμ ; s Þ co st su st su C sig ðp; t; uÞ ¼ > ð μ ; s C st su Þ > adj s s t u > : 0

if st ¼ su ; jpðtÞ  pðuÞj o 2 if st a su ; μst su 4 0; jpðtÞ  pðuÞj ¼ 0 if st a su ; μst su 4 0; jpðtÞ  pðuÞj ¼ 1

otherwise: ð2Þ

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

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Function C co ðμ; sÞ is defined as follows:  c  μ SH C co ðμ; sÞ ¼ 100 1:0  Q

s

ð3Þ

where Z

1

Q ðzÞ ¼ z

1 2 pffiffiffiffiffiffie  x =2 dx 2π

ð4Þ

is the tail integral of a Gaussian probability distribution function with zero mean and unit variance, and cSH is a minimum quality signalling threshold. Function Q is widely used in digital communication systems because it characterises the error probability performance of digital signals (Simon and Alouini, 2002). This means that Q ððcSH  μÞ=sÞ is the probability of the C/I ratio being greater than cSH , and therefore C co ðμst su ; sst su Þ computes the probability of the C/I ratio in the service area of sector st being below the quality threshold due to the interference caused by sector su. If this probability is low, the C/I value in sector st is not likely to be degraded by the interfering signal coming from sector su, and thus the resulting communication quality is high. Note that this is compliant with the definition of a minimisation problem. In contrast, a high probability—and consequently a high cost—mostly causes C/I to be below the minimum threshold cSH , and thus results in low quality communications. Since function Q has no closed integral form, it has to be evaluated numerically. To do so, we use the complementary error function E:   1 z Q ðzÞ ¼ E pffiffiffi ð5Þ 2 2 In Press et al. (1998), a numerical method is presented that allows computing the value of E with a fractional error smaller than 1.2  10  7. Analogously, function C adj ðμ; sÞ is defined as  c  c  SH ACR  μ C adj ðμ; sÞ ¼ 100 1:0  Q s    1 cSH  cACR  μ pffiffiffi ¼ 100 1:0  E ð6Þ 2 s 2 The only difference between functions C co and C adj is the additional constant cACR 4 0 (Adjacent Channel Rejection) in the definition of function C adj . This hardware specific constant measures the receiver's ability to receive the desired signal in the presence of an unwanted signal in an adjacent channel. The effect of constant cACR is that C adj ðμ; sÞ o C co ðμ; sÞ. This is to be expected since using adjacent frequencies does not result in interference as strong 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 to optimise the aforementioned version of the FAP. This scheme was proposed in Segredo et al. (2011) and was selected because it yielded the best frequency plans for several instances of the FAP in previous works (Segura et al., 2013c; Segredo et al., 2011). Several meta-heuristics have been applied to these instances in recent years (Luna et al., 2011) that have served to demonstrate the adequate performance of the approach selected here. However, some important parameters were hand-tuned and kept constant in previous works, so using this meta-heuristic in combination with parameter control methods seems very promising. The scheme is a diversity-based multi-objective MA based on the well-known NSGA-II (Deb et al., 2002). The only difference with respect to the original NSGA-II is that after the variation scheme, a local search procedure is applied to each new generated individual. In diversity-based multi-objective schemes, a set of

objectives is calculated for each individual. The first one is the objective associated with the problem being solved, i.e. the cost of the frequency plan in this case. The remaining objectives—most of the proposals consider only one additional objective as in our case —are measures of the diversity. Note that a measure of the population diversity is not required. Instead, the additional or auxiliary objectives are measures of the diversity introduced by an individual itself. In our proposal we tested two metrics by considering 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 the closest individual. This objective must be maximised. ADI: The auxiliary objective is calculated as the average distance to all individuals. This objective must be maximised.

The genetic operators and the local search scheme are important components for the efficiency of the algorithm. The local search scheme is incorporated in keeping with the Lamarckian approach (Whitley et al., 1994), i.e. the individual reflects in its genotype the result of the movements performed by the local search. The operation of the local search is detailed in Segura et al. (2013c), but basically it optimises the assignment of the frequencies to the transceivers located in a given sector without modifying the remaining network assignments. Regarding the genetic operators, they were also specifically designed to address this variant of the FAP. As is normal, the scheme relies on the application of a crossover operator and a mutation operator afterwards, with probabilities pc and pm, respectively. Two different crossover operators were tested, one of them random and one that considers problem-dependent information. They operate as follows:

 Uniform crossover (UX): For each gene, a random variable



r A ½0; 1 is generated. If r o 0:5, then the gene is inherited from the first parent; otherwise, the gene is inherited from the second one. Interference-based crossover (IX): A transceiver t is randomly selected. Every gene associated with a transceiver that interferes with t or is interfered with by t, including the gene that represents t, is inherited from the first parent. The remaining genes are inherited from the second one.

After applying one of the aforementioned crossover operators, the Neighbourhood-based Mutation (NM) is applied as the mutation operator. 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 a probability pm. The above step is repeated R times, but in the subsequent iterations the transceiver is randomly selected from among 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 is that two different parameters must be set. One of these parameters— pm—is continuous and the other one—R—is discrete. In addition, the most suitable values for these parameters could depend on the problem and/or instance being solved or even on the current stage of the optimisation process, and therefore modifying them during the execution might be beneficial. Consequently, the application of parameter control techniques to automatically adapt these parameters ought to significantly improve both the behaviour and the robustness of the entire optimisation scheme. This idea seems to be very promising and is addressed in detail herein. In order to complete the definition of the diversity-based multiobjective MA, other components must be specified. The parent

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

selection mechanism is the Binary Tournament (Eiben and Smith, 2003), whereas the individuals are encoded as arrays of n integer values ðp1 ; p2 ; …; pn Þ, where pi is the frequency assigned to the transceiver ti. 6. Parameter control approaches In this section we describe in detail both of the parameter control approaches that are evaluated in later sections: the novel FLC proposed herein, and the hyper-heuristic used as the comparison approach. Both provide an external control mechanism for altering the parameters of the NM operator during the course of a run. Only one of these parameters is controlled during the execution, while the other remains constant, so in this paper two independent studies are carried out, one for each parameter of the NM operator. 6.1. Fuzzy logic controllers This section describes a novel FLC introduced by the authors in this work to control the parameters of the NM operator. Its main novelty lies in the incorporation of a set of different rule bases that are enabled depending on historical information extracted from the optimisation process. This historical data is used to guide the adjustment of the parameter. In what follows, the parameter that is going to be controlled is denoted by p. The pseudocode of this FLC is shown in Algorithm 1. Algorithm 1. 1:

FLC

pseudocode.

Initialisation: Generate sample values for the parameter p distributed uniformly in its corresponding range

considering a certain value Δ as the difference between two consecutive samples 2: for (each generated sample value of the parameter p) do 3: Learning: Execute numGen generations of the optimisation scheme with this value for the parameter p in order to gather knowledge 4: end for 5: while (NSGA-II stopping criterion is not satisfied) do 6: Transformation of the parameter p. If the range of the parameter p is different from the range [0, 1], the current value of this parameter is scaled to the range [0, 1] and named p0 7: Calculation of input variables. Set the values for the input variables IMP, VAR, P-IN, BEST-P-IN 8: Selection of the rule base. Select the most suitable rule base 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 interface 10: Mamdani's Fuzzy inference. Apply the fuzzy operator AND (min), the implication method (min) and the aggregation method (max) using the selected rule base to obtain the fuzzy set of the output variable P-OUT 11: Defuzzification: Transform the fuzzy set of the output

12: 13:

14: 15:

variable P-OUT to a crisp value Δp using the defuzzification interface (centroid method) Parameter update: p0 ¼ p0 þ Δp . The value of p0 is enclosed in the range [0, 1] Transformation of the parameter p0 . If the range of the parameter p is different from the range [0, 1], the current value of p0 is scaled to the range of the parameter p Execution: Execute numGen generations of the optimisation scheme with the new value of p end while

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First, the initialisation and learning stages—lines 1–4—are carried out. During the initialisation stage, different sample values are generated for the parameter p and distributed uniformly in its corresponding range. In order to generate them, a value Δ is considered as the difference between two consecutive samples. Although Δ might be considered as a parameter of the FLC, it is assigned a constant value regardless of the problem instance. Then, in the learning stage, the optimisation scheme explained in Section 5 is executed for numGen generations for each of the generated samples in order to gather sufficient information. Once these two stages are complete, the FLC infers the change to be applied over the parameter p—lines 6–13—taking into account the values of the input variables and the selected rule base. Then, the optimisation scheme is executed for numGen generations—line 14 —with the new value of p. This process is repeated until the global stopping criterion of the NSGA-II is reached. After step 13 of Algorithm 1, a continuous value for the parameter p is obtained, so in order to deal with discrete numeric parameters, this value must be defined. Eq. (7) shows the function used to transform a continuous value into a discrete one. The random value r is sampled from a continuous uniform distribution defined in the range [0, 1]. Therefore, if the continuous value of p is, for instance, equal to 12.3, there is a 70% probability that the discrete value will be 12 and a 30% probability that it will be 13: ( ⌈p⌉ if r r p  ⌊pc p¼ ð7Þ ⌊pc if r 4 p  ⌊pc For the fuzzy inference process—lines 9–11—we note that Mamdani's fuzzy inference method is used. In addition, the fuzzy logic operator AND1 uses the minimum T-norm, the implication method uses the minimum T-norm, the aggregation method applies the maximum S-norm and the centroid algorithm is applied as the defuzzification method. All of these components were selected because they are usually implemented together with Mamdani FLCs. It is important to note that zero-order Takagi– Sugeno–Kang (TSK) FLCs—where the linguistic terms of the output variables are described by a zero order (constant) function, instead of using membership functions—were also implemented. These FLCs used the weighted average as the defuzzification method. The remaining components of the fuzzy inference process were the same as those applied in the Mamdani FLCs exposed herein. However, the differences between the Mamdani and TSK FLCs were not statistically significant. Consequently, only Mamdani FLCs are taken into account in this paper. The input variables of the FLC—line 7—are the following:





IMP: Calculated as the improvement of the original objective value of the best individual achieved by the optimisation scheme—line 14 of Algorithm 1—over the last numGen generations. This input variable is normalised to delimit it to the range [0, 1]. VAR: A measure of the diversity of the population. The higher its value, the more diverse the population. The calculation of this input variable with no normalisation is shown in Eq. (8). The values of the decision variable i of individuals j and k are given by xj ½i and xk ½i. The total number of decision variables is represented by D and N is the population size. The value of varn is normalised to enclose the variable VAR in the range [0, 1]: 2 " !#2 3 D1 N 1 N 1 1 n 4 5 ð8Þ var ¼ ∑ ∑ xj ½i   ∑ xk ½i N i¼0 j¼0 k¼0

1

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

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

1

Membership Functions - P-IN 1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.5

1

0

0.167

0.334

0.5

0.667

0.834

1

Membership Functions - P-OUT

1 0.8 0.6 0.4 0.2

0 -0.45 -0.36 -0.27 -0.18 -0.09

0

0.09 0.18 0.27 0.36 0.45

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

 P-IN: Defined as the current value of parameter p within the 

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

Two different versions of the FLC are applied. The first one is called 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 and is called FUZZY-B. For both FLC schemes, only one output variable is defined, referred to as P-OUT, which represents the increment or decrement to be applied to parameter p in order to change its value. The membership functions for both the input and output variables are shown in Fig. 1. Due to the computational simplicity and efficiency advantage they offer, triangular-shaped membership 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) and

(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). HIGH

For each FLC different rule bases are defined. The reason for using different rule bases is that different fuzzy rules will be applicable depending on the behaviour exhibited during the previous execution. For instance, if the best results were historically obtained by low values of the parameter p, the fuzzy rules should promote the usage of such low values. Every rule base is composed of different IF-THEN fuzzy rules. The left-hand side of Table 1 shows one of the rule bases defined for the approach FUZZYA, while the right-hand side shows another one for the scheme FUZZY-B. Only the fuzzy logic operator AND is used in the antecedents of these fuzzy rules. In general, every fuzzy rule considers three input variables and one output variable. In those cases where a ‘–’ is shown, the corresponding fuzzy rule has no dependency on the

Table 1 Rule bases for the

FUZZY-A

(left-hand side) and

Rules

Inputs

ID

P-IN

IMP

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

L

L

L

M

L

H

LMB

L

LMB

M

LMB

H

LMA

L

LMA

M

LMA

H

M

L

M

M

M

H

MHA

L

MHA

M

MHA

H

MHB

L

MHB

M

MHB

H

– – – – – – – – – – – – – – – – – –

H

L

L

PL

H

L

M

PL

H

L

H

NL

H

M

H

H

VAR

– –

FUZZY-B

(right-hand side) schemes.

Output

Rules

Inputs

P-OUT

ID

P-IN

IMP

BEST-P-IN

P-OUT

PG

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

L

L

L

NL

L

L

M

PL

L

L

H

PL

L

M

L

H

LMB

L

LMB

M

LMB

H

LMA

L

LMA

M

LMA

H

M

L

M

M

M

H

MHA

L

MHA

M

MHA

H

MHB

L

MHB

M

MHB

H

H

L

H

M

H

H

PL Z PG PL Z PG PL Z PU PL Z PH PL Z PM PL Z

Z Z

Output

– – – – – – – – – – – – – – – – – – – –

Z Z NM NL Z NH NL Z NU NL Z NG NL Z NG NL Z NG NL Z

corresponding variable. The remaining rule bases are not shown due to space constraints but are similar to those shown here.2 In order to select the most suitable set of rules, in this work we propose a novel scoring function that relies on a weighted mean that considers historical data on both the improvement in the original objective value and on the degrees of membership of parameter p to each term defined for the input variable P-IN. The value of k is defined as the amount of historical knowledge considered by the FLC, i.e. information on the latest k decisions

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

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

made by the FLC is taken into account. On the other hand, d is the total number of decisions that the FLC has carried out, and numTerms is the number of linguistic terms defined for the input variable P-IN. The score assigned to each linguistic term i A ½0; numTerms  1 is given by Eq. (9). The improvement achieved during execution d j of the optimisation scheme—line 14 of Algorithm 1—is given by γ ½d  j. In addition, the degree of membership of parameter p to the linguistic term i during execution d j is represented by δ½i½d  j. Thus, the linguistic term i will be assigned a higher score if the values of parameter p have larger degrees of membership to said linguistic term, and if, at the same time, the values of parameter p are able to achieve higher improvements in the original objective value. Finally, we note that the scoring function assigns more importance to the latest decisions inferred by the FLC. Thus, for each linguistic term the equation represents a weighted average of its improvement, where greater importance is given to the last executions in which values of the controlled parameter have a high degree of membership to the corresponding linguistic term. Note that if numTerms linguistic terms are defined for the variable P-IN, numTerms rule bases have to be implemented such that the FLC works with the proposed scoring function. Fig. 1 shows that seven linguistic terms are defined for the input variable P-IN, so seven different rule bases are implemented. We tested different numbers of fuzzy rule bases and found that the higher the number of rule bases, the smoother the variations of the parameter p inferred by the FLC, and thus the steadier the FLC. However, when considering more than seven fuzzy rule bases, the performance started to degrade somewhat, as it also did with a lower number of fuzzy rule bases. Thus, we opted for seven rule bases as this yielded the best performance for the FLC. This fact also justifies the usage of seven linguistic terms for the input variable P-IN, instead of using three linguistic terms as in the case of remaining input variables. For the remaining input variables, three linguistic terms are used so as to maintain the rule bases as simple as possible. score½i ¼

∑minðk;dÞ γ ½d  j  δ½i½d  j  ðminðk; dÞ  jþ 1Þ j¼1

δ½i½d  j  ðminðk; dÞ  j þ 1Þ ∑minðk;dÞ j¼1

ð9Þ

Once the scores are calculated, the fuzzy set with the maximum score is selected. This means that those values of parameter p with a large enough degree of membership to the linguistic term should provide better performance than other values. Therefore, if the linguistic term i is selected as the most appropriate one, rule base i is enabled. This selected rule base is responsible for adapting the value of 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 and the most suitable rule base—considering the scoring function—is the one represented by the linguistic term HIGH of the input variable P-IN. This means that historically high values of parameter p have yielded good improvements in the original objective value. Thus, the rule base to be applied in this case is precisely the one shown in the left-hand side of Table 1, considering the approach FUZZY-A. If a fuzzy set for the variable 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 large degree of membership to the term LOW, then the output fuzzy set—the one corresponding to the output variable P-OUT—will have a large degree of membership to the linguistic term POS-GIANT (PG). Consequently, the value of parameter p will be considerably increased so that it will tend towards higher values. 6.2. Hyper-heuristics Hyper-heuristics can be defined as search methods or learning mechanisms for selecting or generating heuristics to solve computational search problems (Burke et al., 2010). Hyper-heuristics

205

based on heuristic selection try to iteratively identify and select the most promising low-level heuristics or meta-heuristics—from a set of candidates—to solve a particular instance of a problem (Burke et al., 2003). Hyper-heuristics can be used as parameter control approaches. For example, the low-level approaches could represent different configurations of the same heuristic (or metaheuristic). The hyper-heuristic then selects the configuration with the most appropriate set of parameters at each point in the search. In fact, they can be further classified as adaptive parameter control techniques if they receive some kind of feedback from the optimisation process. An extension of the hyper-heuristic approach to parameter control first described by Vinkó and Izzo (2007) is implemented in order to control the parameters of the NM operator. This hyper-heuristic has been successfully applied in previous works (Segura et al., 2013b; Segura, 2012) and is based on using a scoring and selection strategy for choosing the most appropriate low-level configuration of the algorithm to be executed. A low-level configuration in this case refers to an instance of the optimisation scheme depicted in Section 5 with a particular 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 local stopping criterion is achieved. When this happens, another low-level configuration is selected and executed; the final population of the last low-level configuration used becomes the initial population of the new low-level configuration. This process continues until a global stopping criterion is satisfied. The low-level configuration that must be executed is selected as follows. First, the scoring strategy assigns a score to each low-level configuration. This score estimates the improvement that each low-level configuration can achieve starting from the current solution set. Thus, larger values are assigned to more promising schemes considering their historical behaviour. In order to calculate this estimate, the previous improvements to the original objective value achieved by each configuration are used. The improvement (γ) is defined as the difference, in terms of the original objective value, between the best achieved individual and the best initial individual. Considering a configuration conf that has been executed j times, the score s(conf) proposed in Segura et al. (2010) is calculated as a weighted average of its last k improvements: sðconf Þ ¼

∑minðk;jÞ i ¼ 1 ðminðk; jÞ þ 1  iÞ  γ ½conf ½j  i ∑minðk;jÞ i¼1 i

ð10Þ

In Eq. (10), γ ½conf ½j i represents the improvement achieved by configuration conf in execution number j  i. The adaptation level of the hyper-heuristic, i.e. the total amount of historical knowledge that it considers in order to perform its decisions, can be varied depending on the value of k. The weighted average assigns a greater importance to the latest executions. The score s(conf) is used to calculate a probability of selecting a particular low-level configuration. However, the stochastic behaviour of the low-level meta-heuristics involved may lead to variations in the results they yield. Therefore, the probability calculation also enables a fraction of selections based on a random scheme and is implemented as follows. Specifically, the hyperheuristic can be tuned by means of a parameter β, which represents the minimum selection probability that should be assigned to a low-level configuration. If nh is the number of lowlevel configurations involved, a random selection based on a uniform distribution is performed in β  nh percentage of the cases. Therefore, the probability of selecting each configuration conf is defined as " # sðconf Þ probðconf Þ ¼ β þ ð1  β  nh Þ  ð11Þ h ∑ni ¼ 1 sðiÞ

206

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

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

 The first one is an elitist version—HH-ELI—which always selects



the low-level configuration with the maximum score s(conf), besides the minimum random selections performed for each configuration. The second one is a probabilistic version (HH-PROB). In this case, the selection probability—Eq. (11)—is proportional to the score s(conf).

7. Experimental evaluation The experiments conducted with the diversity-based multiobjective MA described in Section 5, and the parameter control approaches presented in Section 6 are described at this point. The main aim of these experiments is twofold. First, to compare parameter control and parameter tuning in order to reveal the benefits and drawbacks of adapting the values of the parameters during the course of the optimisation process in contrast to setting them before the run starts. Second, to demonstrate the generality of the control techniques, and particularly of the FLC proposed in this work, by applying them to different parameters. Experimental method: Both the optimisation scheme and the parameter control approaches were implemented using METCO (León et al., 2009) (Meta-heuristic-based Extensible Tool for Cooperative Optimisation). Tests were run on a Debian GNU/Linux computer with four amds opteron™ processors (model number 6164 HE) at 1.7 GHz and 64 GB RAM. The compiler was the GCC 4.7.2, while the FLCs were implemented using the fuzzylite3.1 library (RadaVilela, 2013). Since every experiment used stochastic algorithms, each execution was repeated 32 times. Comparisons were performed by applying the following statistical analysis. First, a Shapiro–Wilk test was performed in order to check whether the values of the results followed a normal (Gaussian) distribution or not. If so, the Levene test checked for the homogeneity of the variances. If the 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 the medians of the algorithms. A significance level of 5% was considered. FAP instances: The studies were conducted considering two different instances representing two real cities in the USA: Seattle and Denver. The Seattle instance had n ¼ 970 TRXs and 15 different frequencies to be assigned. The Denver instance was larger, consisting of n ¼2612 TRXs and 18 frequencies. In both cases, the constants used in the formal definition of the FAP exposed in Section 4 were set to K¼ 1  105, cSH ¼ 6 dB and cACR ¼ 18 dB. The matrix M contains 59,169 items in the Seattle network, while it contains 20,638 items for the Denver instance.

instances considered. The algorithm exposed in Section 5 was also executed with different values for the parameter pm, while the value of R was kept constant. The main aim was to analyse the performance of the different parameter control approaches and to study whether parameter control gives some benefit with regard to tuning the parameter pm. A common parameterisation for the multi-objective MA and the different parameter control schemes was set. Table 2 shows the parameterisation of the diversity-based multi-objective MA described in Section 5. Different configurations—exactly 11—were defined by modifying the value of the parameter pm. Moreover, the auxiliary objective and the crossover operator considered for each of the two instances were different. This is because, depending on the instance, the most appropriate values for these components change. The parameterisations of the different parameter control approaches are shown in Tables 3 and 4 for the hyper-heuristics and the FLCs, respectively. Note that four different configurations for the HH-ELI and HH-PROB hyper-heuristics were applied by combining different values for the local stopping criterion and the parameter k. In the same way, four configurations of the FUZZY-A and FUZZY-B FLCs were defined by setting different values for the number of generations and the parameter k. Finally, the hyperheuristics were applied with nh ¼11 low-level configurations. Lowlevel configurations used the parameterisation shown in Table 2 with each one using a different value for the parameter pm. Tables 5 and 6 show the statistics for the different configurations of the HH-ELI and HH-PROB hyper-heuristics, the FUZZY-A and FUZZY-B FLCs and the multi-objective MA (FIXED) for the Seattle and Denver instances, respectively, including dispersion measures like the standard deviation (SD) and the coefficient of variation (CV). The data in bold show, for each method, the configuration that achieved the lowest mean of the original objective value. Moreover, the remaining configurations of a given method that are shown in bold did not exhibit statistically significant differences in comparison to the method that achieved the lowest mean for the original objective value, as determined by the statistical procedure described earlier in this section. In contrast, the configurations of a given method which do not appear in bold presented statistically significant differences from the configuration with the lowest mean of the original objective value. In order to identify a given approach's particular configuration, the values of its parameters reflect the name of the approach. For example: FAP

  

HH-PROB_1500_5 is a configuration of the HH_PROB hyper-heuristic with a local stopping criterion equal to 1500 evaluations and historical knowledge (k) equal to 5 decisions. FUZZY-A_300_2 is a configuration of the FUZZY-A FLC that uses 300 generations as the local stopping criterion and historical knowledge equal to 2 decisions. FIXED_0.9 is a configuration of the multi-objective MA which applies the NM operator with probability pm equal to 0.9.

7.1. Analyses over the parameter pm In this first experiment the different parameter control approaches were applied to the parameter pm of the NM operator to solve both the Table 2 Parameterisation of the diversity-based multi-objective

MA

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

for the first experiment.

Parameter

Value

Parameter

Value

Stopping criterion Population size (N) Crossover operator Auxiliary objective

1.5  105 evals. 10 individuals UX (Seattle), IX (Denver) DCN (Seattle), ADI (Denver)

Crossover rate (pc) Mutation rate (pm) NM operator steps (R)

1 0, 0.1, 0.2, …, 0.9, 1 7

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

Table 3 Parameterisation of the hyper-heuristics

HH-ELI

and

207

for the first experiment.

HH-PROB

Parameter

Value

Parameter

Value

Local stopping criterion Number of low-level configs. (nh)

1.5  103, 3  103 evals. 11 configs.

Minimum selection rate (β) Historical knowledge (k)

0.1 2, 5

Table 4 Parameterisation of the fuzzy logic controllers

FUZZY-A

and

FUZZY-B

Parameter

for the first experiment.

Value 2

1.5  10 , 3  10 7 [0, 1]

Number of generations (numGen) Number of fuzzy sets (numTerms) Range of the parameter pm

2

Parameter

Value

Difference among samples (Δ) Historical knowledge (k)

0.1 2, 5

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

Min.

First Qu.

Median

Mean

Third Qu.

Max.

SD

CV

HH-Eli_1500_2 HH-Eli_3000_2 HH-Eli_1500_5 HH-Eli_3000_5

547.1 506.0 511.0 525.9

589.1 594.9 610.9 595.8

624.3 655.1 672.4 637.3

644.5 651.7 662.8 646.2

675.3 698.8 726.4 686.8

889.2 870.8 799.6 896.8

84.4 87.2 72.7 79.6

13.1 13.4 11.0 12.3

HH-Prob_1500_2 HH-Prob_3000_2 HH-Prob_1500_5 HH-Prob_3000_5

530.9 523.6 515.6 534.5

629.2 578.2 608.2 590.1

668.2 650.3 665.4 642.9

669.8 644.0 675.2 647.2

708.0 695.5 749.5 686.7

855.4 775.1 888.4 790.8

74.7 71.4 88.8 65.5

11.1 11.1 13.1 10.1

Fuzzy-A_150_2 Fuzzy-A_300_2 Fuzzy-A_150_5 Fuzzy-A_300_5

529.7 504.1 517.4 505.3

610.2 564.5 584.6 602.2

664.1 645.2 636.7 646.3

669.5 643.1 643.0 658.7

720.8 680.0 691.0 687.8

785.2 882.8 780.0 851.9

68.8 89.4 69.0 81.3

10.3 13.9 10.7 12.3

Fuzzy-B_150_2 Fuzzy-B_300_2 Fuzzy-B_150_5 Fuzzy-B_300_5

463.0 471.7 519.9 497.6

609.2 619.0 616.7 616.3

651.1 678.2 658.3 673.4

659.2 669.8 660.7 663.8

721.4 712.6 701.9 704.1

814.1 807.4 804.5 795.5

87.6 77.4 70.0 68.5

13.3 11.6 10.6 10.3

Fixed_0 Fixed_0.1 Fixed_0.2 Fixed_0.3 Fixed_0.4 Fixed_0.5 Fixed_0.6 Fixed_0.7 Fixed_0.8 Fixed_0.9 Fixed_1.0

587.9 525.5 547.9 562.7 515.2 521.3 557.9 551.8 504.1 541.5 546.6

684.6 660.7 631.1 613.0 631.8 607.7 650.9 642.1 649.2 638.0 683.5

736.3 712.7 687.4 664.9 680.9 667.4 676.6 686.6 676.9 695.4 705.9

740.4 716.2 684.8 688.7 679.3 667.3 694.1 678.0 679.7 697.0 713.6

775.3 785.0 727.3 748.5 736.2 714.3 723.6 717.4 716.1 751.5 750.8

974.4 878.6 797.8 941.1 809.2 842.6 834.7 813.2 806.0 893.5 864.6

84.4 92.2 62.4 90.4 72.1 76.7 65.4 62.2 60.8 78.1 62.9

11.4 12.9 9.1 13.1 10.6 11.5 9.4 9.2 8.9 11.2 8.8

objective value used the value 0.5 (FIXED_0.5) for the parameter pm of the NM operator, while in the case of the Denver instance this value was equal to 0.8 (FIXED_0.8). This fact confirms that the most suitable value for a parameter changes depending on the problem and/or the instance being solved. Moreover, these configurations exhibited statistically significant differences as compared to others. In the case of the Seattle instance, there were statistically significant differences with 3 configurations, while in the case of the Denver instance, there were differences with 5 configurations. We can therefore observe that the parameter pm is more sensitive to changes in its value when the NM operator is applied to the Denver instance, so it is even more important to select the appropriate values in this particular case. Considering the control methods applied to the Seattle instance, their configurations did not present statistically significant differences among them. Taking into account the Denver instance, the only statistically significant differences appeared among the configurations of the HH-PROB hyper-heuristic and the

FUZZY-B FLC. This means that both the hyper-heuristics and the FLCs are robust enough from the point of view of their parameters. If these parameters are modified, these changes are not going to greatly determine the performance of the control strategy. If, for each of the four control methods exposed herein, we consider the corresponding configuration that achieved the lowest mean of the original objective value, no statistically significant differences appeared among them. This was the case for both the Seattle and Denver instances. As a result, we can apply hyperheuristics or FLCs indistinctly to control the parameter pm without drastically 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 with each of the four control schemes for Seattle and Denver, respectively. To carry out the statistical comparison, we used the configuration for each control method that obtained the lowest mean of the original objective value.

208

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

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

Min.

First Qu.

Median

Mean

Third Qu.

Max.

SD

CV

HH-Eli_1500_2 HH-Eli_3000_2 HH-Eli_1500_5 HH-Eli_3000_5

84,228.4 84,019.6 83,996.6 83,933.5

84,811.6 84,840.6 84,792.3 84,840.5

85,099.4 85,113.6 85,170.2 85,246.2

85,285.8 85,137.3 85,251.3 85,229.9

85,798.8 85,373.8 85,531.2 85,565.8

86,742.7 86,446.6 87,590.5 86,405.4

673.9 579.7 790.0 625.6

0.8 0.7 0.9 0.7

HH-Prob_1500_2 HH-Prob_3000_2 HH-Prob_1500_5 HH-Prob_3000_5

83,833.4 84,274.6 83,789.9 84,257.9

84,918.7 84,916.2 84,599.5 84,929.5

85,306.4 85,455.6 85,013.2 85,660.6

85,397.2 85,560.5 85,058.1 85,529.5

85,818.7 86,213.6 85,380.7 85,965.6

87,083.6 87,661.3 87,005.8 87,215.7

805.7 874.5 714.7 761.1

0.9 1.0 0.8 0.9

Fuzzy-A_150_2 Fuzzy-A_300_2 Fuzzy-A_150_5 Fuzzy-A_300_5

84,442.6 84,277.6 84,194.4 83,594.4

84,884.3 84,877.2 84,992.8 84,508.5

85,218.1 85,139.4 85,356.4 84,979.9

85,364.2 85,323.3 85,413.8 85,136.9

85,783.4 85,812.7 85,690.5 85,445.8

86,857.3 88,066.0 86,863.9 87,149.7

701.7 814.3 647.3 803.8

0.8 1.0 0.8 0.9

Fuzzy-B_150_2 Fuzzy-B_300_2 Fuzzy-B_150_5 Fuzzy-B_300_5

84,000.7 84,245.1 84,118.2 84,004.6

84,937.9 84,774.4 84,627.1 84,493.2

85,567.4 85,066.2 84,975.1 85,035.4

85,509.3 85,194.8 85,190.6 84,986.1

85,995.9 85,625.7 85,559.6 85,407.6

87,174.2 87,106.1 87,213.2 85,794.5

808.1 622.5 841.0 536.4

0.9 0.7 1.0 0.6

Fixed_0 Fixed_0.1 Fixed_0.2 Fixed_0.3 Fixed_0.4 Fixed_0.5 Fixed_0.6 Fixed_0.7 Fixed_0.8 Fixed_0.9 Fixed_1.0

85,243.4 84,765.0 84,552.3 84,432.4 84,447.7 84,055.0 83,969.0 84,075.3 83,400.7 84,441.6 84,292.5

86,554.4 85,682.4 85,955.6 85,293.9 85,236.0 84,924.4 84,940.1 84,836.5 84,820.1 85,233.9 85,178.5

87,230.3 86,444.1 86,276.4 85,796.1 85,824.9 85,552.8 85,429.2 85,478.0 85,295.2 85,569.0 85,409.2

87,071.1 86,475.8 86,392.5 85,946.2 85,809.0 85,704.0 85,541.8 85,481.3 85,367.0 85,603.4 85,418.6

87,572.1 87,050.8 86,867.1 86,550.9 86,155.5 86,210.9 86,073.9 85,950.7 85,750.7 85,816.2 85,683.7

88,744.8 88,501.1 89,207.3 88,072.8 87,545.7 87,405.7 88,281.3 87,432.7 87,628.2 87,884.8 87,152.4

838.7 980.1 952.2 924.0 771.5 961.4 977.2 748.9 829.2 673.6 498.5

1.0 1.1 1.1 1.1 0.9 1.1 1.1 0.9 1.0 0.8 0.6

Table 7 Number of fixed configurations outperformed by the parameter control approaches adapting the parameter pm – Seattle instance. Approach

Number of configurations

HH-Eli HH-Prob Fuzzy-A Fuzzy-B

10 9 10 3

Table 8 Number of fixed configurations outperformed by the parameter control approaches adapting the parameter pm – Denver instance. Approach

Number of configurations

HH-Eli HH-Prob Fuzzy-A Fuzzy-B

10 10 8 11

For the Seattle instance, the HH-ELI hyper-heuristic and the FUZZY-A FLC were able to outperform 10 configurations of the FIXED approach, while in the case of Denver, the hyper-heuristics outperformed 10 configurations and the FUZZY-B FLC was able to outperform 11 configurations. Moreover, we should mention that, for both instances, no configuration of the FIXED scheme was able to statistically outperform any control method. Consequently, the advantages of using parameter control instead of parameter tuning are clear. With just one execution of the control schemes we were able to provide similar or even better solutions than those obtained using the best-behaved configuration of the diversity-based multi-objective MA. It is important to note that in order 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 control over parameter tuning are even higher. Finally, we would like to mention that the best-known frequency plan for Seattle, which was published in Segura et al. (2013c), was improved upon by the FUZZY-B control scheme when the parameter pm was adapted with the original objective value decreasing 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 the control approaches to the parameter R of the NM operator in order to solve the FAP. As in the previous section, the multi-objective MA exposed in Section 5 was also run. Nevertheless, its configurations were obtained by varying the parameter R while holding the value of pm constant. The main objective was to analyse the behaviour of the different control schemes when adapting the parameter R. These control techniques were also compared to parameter tuning. The same parameterisations from the previous section were used, though in this case the value of parameter pm was held constant. We also tested several values for the parameter R. Table 9 shows the parameterisation of the diversity-based multi-objective MA for this experiment. In this case, 15 different configurations of this approach were executed using different values for the parameter R. The parameterisations of the control approaches are shown in Tables 10 and 11 for the hyper-heuristics and the FLCs, respectively. As in the previous experiment, four different configurations of each control method were applied. In the case of the hyperheuristics, they were defined with nh ¼15 low-level configurations, with each one taking on a different value for the parameter R and using the parameterisation shown in Table 9. Tables 12 and 13 show the statistics obtained for the Seattle and Denver instances, respectively, by the different configurations of the hyper-heuristics, the FLCs and the multi-objective MA. Note the following regarding the setting of parameter R: in terms of

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

Table 9 Parameterisation of the diversity-based multi-objective

MA

209

for the second experiment.

Parameter

Value

Parameter

Value

Stopping criterion Population size (N) Crossover operator Auxiliary objective

1.5  105 evals. 10 individuals UX (Seattle), IX (Denver) DCN (Seattle), ADI (Denver)

Crossover rate (pc) Mutation rate (pm) NM operator steps (R)

1 0.5 (Seattle), 0.8 (Denver) 1, 2, 3, …, 14, 15

Table 10 Parameterisation of the hyper-heuristics

HH-ELI

and

HH-PROB

for the second experiment.

Parameter

Value

Parameter

Value

Local stopping criterion Number of low-level configs. (nh)

1.5  103, 3  103 evals. 15 configs.

Minimum selection rate (β) Historical knowledge (k)

0.1 2, 5

Table 11 Parameterisation of the fuzzy logic controllers

FUZZY-A

and

FUZZY-B

for the second experiment.

Parameter

Value

Parameter

Value

Number of generations (numGen) Number of fuzzy sets (numTerms) Range of the parameter R

1.5  102, 3  102 7 [1, 15]

Difference among samples (Δ) Historical knowledge (k)

1 2, 5

parameter tuning, the configuration of the FIXED approach that yielded the lowest mean for the original objective value used the NM operator with the parameter R equal to 7—FIXED_7—for the Seattle instance. In the case of Denver, FIXED_6 was the most suitable configuration of the multi-objective MA. Both configurations exposed statistically significant differences as compared to other configurations of the FIXED scheme. In the case of Seattle, there were differences with 7 configurations, while for Denver, the number of statistically significant differences was equal to 6. Similar conclusions to those extracted for parameter pm can be drawn for parameter R. The study involving parameter tuning reveals that the most appropriate value for R depends on the problem and/or instance being solved. A statistical comparison shows that for this particular parameter, the number of statistical differences among the FIXED scheme configuration that obtained the lowest mean for the original objective value and other FIXED configurations is noticeable for both instances. As a result, we can conclude that the parameter R is also sensitive to changes in its value, as was the case with pm. With regard to parameter control, and considering the Seattle instance, the configurations did not exhibit statistically significant differences among them, while in the case of Denver, only one statistically significant difference appeared between the configurations HH-ELI_3000_2 and HH-ELI_3000_5. Once more, both hyperheuristics and FLCs demonstrated their robustness with R being adapted in this case, since their performance was not significantly affected by changes in their parameter values, as occurred when pm was adapted. No statistically significant differences appeared among the configurations that achieved the lowest mean for the original objective value for each of the four control schemes. This happened for both instances. Consequently, not only can the parameter pm be controlled by hyper-heuristics or FLCs indistinctly, but also can the parameter R. Thus, we can confirm the generality of both control methods, which can adapt continuous and discrete numeric parameters successfully. In order to compare parameter control and parameter tuning, Tables 14 and 15 show, for Seattle and Denver respectively, the number of FIXED scheme configurations that exhibited statistically

significant differences with each one the four control techniques. To perform the statistical comparison, we selected the configuration that obtained the lowest mean for the original objective value for each of the four control methods. If we consider Seattle, the HHELI hyper-heuristic was able to outperform 13 FIXED scheme configurations, while remaining control approaches outperformed 10 configurations. In the case of Denver, the superiority of the control techniques is clear as they were able to outperform every FIXED scheme configuration. As in the previous experiment, no configuration of the FIXED scheme was able to statistically outperform any control method for either instance. As was the case with the parameter pm, the benefits of adapting the parameter R instead of tuning it are also clear. A single execution of the hyper-heuristics or the FLCs yielded frequency plans for the two cities in question that were similar to or even better than those provided by the best-behaved configurations of the multi-objective MA. To find the best-behaved configurations we had to test 15 different parameterisations of the diversity-based multi-objective MA by modifying the value of R. Taking this fact into consideration, the benefits of parameter control over parameter tuning are even higher. Finally, we would like to note that the best-known frequency plan for Denver, which was published in Segura et al. (2013c), was improved upon by the FUZZY-B control scheme when the parameter R was adapted, with the original objective value decreasing from 83,340.2 to 83,280.9, as shown in Table 13.

8. Conclusions and future work One of the most commonly known drawbacks of meta-heuristics is that they usually have a considerable number of parameters that must be properly set so as to yield adequate performance. Appropriate parameter settings are therefore a critical part of any metaheuristic design. Parameter tuning approaches attempt to find an optimal set of parameters that remain fixed during the course of the optimisation procedure. In contrast, parameter control approaches attempt to adapt the values of a parameter during the course of the optimisation based on the assumption that different values are better suited at different points in the search.

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

Min.

First Qu.

Median

Mean

Third Qu.

Max.

SD

CV

HH-Eli_1500_2 HH-Eli_3000_2 HH-Eli_1500_5 HH-Eli_3000_5

496.5 558.2 534.3 501.8

591.6 593.5 588.7 590.0

636.1 646.4 635.6 635.2

634.8 656.3 644.2 650.7

675.8 711.1 680.7 704.8

738.4 825.0 813.5 845.8

61.2 70.7 72.1 81.7

9.6 10.8 11.2 12.6

HH-Prob_1500_2 HH-Prob_3000_2 HH-Prob_1500_5 HH-Prob_3000_5

527.1 540.6 514.5 522.6

603.4 611.8 620.8 597.9

629.4 649.5 634.1 646.1

645.1 674.5 646.6 653.9

697.6 750.2 697.7 704.6

775.3 819.4 764.4 854.6

64.2 80.5 63.9 80.4

10.0 11.9 9.9 12.3

Fuzzy-A_150_2 Fuzzy-A_300_2 Fuzzy-A_150_5 Fuzzy-A_300_5

486.5 503.6 508.8 543.0

587.1 596.3 585.2 597.1

627.2 644.5 627.4 635.2

641.4 649.4 647.7 655.9

694.9 696.7 697.0 698.2

821.1 794.5 820.5 880.3

77.4 74.6 88.0 79.7

12.1 11.5 13.6 12.2

Fuzzy-B_150_2 Fuzzy-B_300_2 Fuzzy-B_150_5 Fuzzy-B_300_5

473.9 516.0 496.2 520.9

563.4 582.8 607.2 590.3

617.8 654.8 642.4 623.9

638.7 650.6 639.1 642.0

701.7 706.4 683.6 701.2

832.2 834.0 809.1 794.4

95.4 82.3 65.9 73.6

14.9 12.7 10.3 11.5

Fixed_1 Fixed_2 Fixed_3 Fixed_4 Fixed_5 Fixed_6 Fixed_7 Fixed_8 Fixed_9 Fixed_10 Fixed_11 Fixed_12 Fixed_13 Fixed_14 Fixed_15

549.1 592.5 550.6 566.6 566.8 547.0 580.3 529.8 555.0 533.3 543.8 525.9 597.7 621.5 613.7

673.6 675.1 667.7 644.0 640.5 635.4 624.1 637.1 619.4 629.1 634.9 647.6 665.3 674.8 709.6

728.6 726.7 707.7 687.8 681.7 670.0 641.6 682.5 667.2 682.8 667.3 683.0 698.9 731.2 737.3

744.6 727.8 725.3 689.5 693.9 675.8 658.4 668.2 662.1 684.2 678.9 681.6 712.2 738.5 739.8

814.0 778.7 767.8 722.7 721.0 709.5 683.4 713.5 709.5 723.3 732.4 728.1 737.5 795.2 780.6

949.0 887.7 968.0 893.6 908.3 806.9 830.4 782.7 771.1 843.7 863.0 833.6 982.5 894.2 879.2

97.9 74.4 83.0 73.8 80.4 60.6 57.3 69.9 59.5 74.4 76.2 73.2 74.2 80.0 63.8

13.2 10.2 11.4 10.7 11.6 9.0 8.7 10.5 9.0 10.9 11.2 10.7 10.4 10.8 8.6

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

Min.

First Qu.

Median

Mean

Third Qu.

Max.

SD

CV

HH-Eli_1500_2 HH-Eli_3000_2 HH-Eli_1500_5 HH-Eli_3000_5

83,780.7 83,876.2 84,037.3 83,740.6

84,658.2 84,446.8 84,593.2 84,784.8

85,041.6 84,856.4 84,899.0 84,974.7

85,064.2 84,846.1 85,090.5 85,184.5

85,568.1 85,196.2 85,412.7 85,610.9

86,324.1 86,552.9 87,154.3 86,783.9

609.5 638.7 731.3 654.8

0.7 0.8 0.9 0.8

HH-Prob_1500_2 HH-Prob_3000_2 HH-Prob_1500_5 HH-Prob_3000_5

83,,914.2 84,352.3 83,796.0 83,548.0

84,568.8 84,667.8 84,441.5 84,639.6

85,127.7 84,962.4 84,876.1 85,026.2

85,240.3 85,177.4 84,973.9 84,965.3

85,715.4 85,660.6 85,335.6 85,269.4

87,159.4 86,766.0 86,323.2 86,455.1

844.1 691.9 614.8 619.5

1.0 0.8 0.7 0.7

Fuzzy-A_150_2 Fuzzy-A_300_2 Fuzzy-A_150_5 Fuzzy-A_300_5

83,829.4 83,750.1 83,852.6 83,335.4

84,615.1 84,533.9 84,534.0 84,545.0

84,956.8 84,988.5 84,962.4 85,102.9

85,081.4 85,044.9 84,967.1 85,032.9

85,354.6 85,432.5 85,313.5 85,524.4

86,533.0 87,192.5 86,897.6 86,584.4

634.9 750.9 696.0 792.8

0.7 0.9 0.8 0.9

Fuzzy-B_150_2 Fuzzy-B_300_2 Fuzzy-B_150_5 Fuzzy-B_300_5

83,280.9 83,377.4 83,955.2 83,727.2

84,446.4 84,515.4 84,549.1 84,481.1

85,033.5 84,888.5 85,145.9 84,939.1

84,881.4 84,915.7 85,228.5 84,942.2

85,380.0 85,359.1 85,749.9 85,313.6

86,181.9 86,106.8 87,100.4 86,433.9

694.2 592.1 831.0 660.5

0.8 0.7 1.0 0.8

Fixed_1 Fixed_2 Fixed_3 Fixed_4 Fixed_5 Fixed_6 Fixed_7 Fixed_8 Fixed_9 Fixed_10 Fixed_11 Fixed_12 Fixed_13 Fixed_14 Fixed_15

84,885.9 83,747.4 84,558.3 84,001.3 84,760.0 84,483.9 84,051.4 84,296.1 84,357.2 84,693.2 84,771.0 84,345.7 84,587.0 84,627.3 84,955.6

85,695.1 85,352.1 84,876.7 85,083.5 85,151.6 84,888.0 85,088.9 85,052.9 85,129.6 85,277.0 85,388.2 85,203.1 85,268.0 85,159.4 85,768.8

86,231.9 85,747.4 85,325.8 85,225.4 85,631.0 85,254.9 85,528.4 85,382.1 85,339.9 85,434.1 85,560.2 85,500.4 85,516.4 85,740.0 86,036.3

86,206.7 85,716.0 85,610.8 85,423.6 85,588.6 85,333.7 85,505.9 85,449.0 85,428.5 85,652.1 85,671.2 85,538.7 85,621.5 85,703.3 86,154.7

86,667.4 86,282.5 86,334.2 85,848.8 85,981.8 85,751.2 85,938.8 85,820.9 85,655.3 86,066.5 86,077.8 85,819.2 85,869.9 85,979.4 86,550.7

87,291.9 86,894.6 87,403.0 87,109.2 86,745.8 86,899.4 87,282.4 87,139.2 87,110.0 87,222.2 86,910.9 86,688.2 86,981.9 87,857.6 87,522.7

681.1 718.9 872.4 768.8 503.2 606.2 726.6 574.2 595.5 585.1 558.9 590.3 606.5 676.3 649.9

0.8 0.8 1.0 0.9 0.6 0.7 0.8 0.7 0.7 0.7 0.7 0.7 0.7 0.8 0.8

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

Table 14 Number of fixed configurations outperformed by the parameter control approaches adapting the parameter R – Seattle instance. Approach

Number of configurations

HH-Eli HH-Prob Fuzzy-A Fuzzy-B

13 10 10 10

Table 15 Number of fixed configurations outperformed by the parameter control approaches adapting the parameter R – Denver instance. Approach

Number of configurations

HH-Eli HH-Prob Fuzzy-A Fuzzy-B

15 15 15 15

In this paper, a novel FLC is proposed that controls two parameters in a highly efficient meta-heuristic that is specifically designed to address a complex variant of the FAP. The meta-heuristic in question is a diversity-based multi-objective scheme that has reported the bestknown frequency plans for several FAP instances. The FLC designed incorporates a set of different rule bases that are enabled depending on historical information extracted from the own optimisation process. The scheme promotes the usage of parameter values that have historically yielded the best performance. At the same time, other parameter values are explored so as to adapt the scheme to the varying requirements that might arise in different optimisation stages. In addition, a well-known hyper-heuristic variant was also used as a parameter control scheme. One of the main differences between the two methods lies in the fact that the hyper-heuristic approach requires that a fixed set of potential values for the parameter be pre-defined by the user, whereas the fuzzy logic approach is able to select any value within a range. In order to show the generality of the proposals and with the aim of improving the results further, two different numeric parameters were controlled: pm and R. These belong to the NM operator, an efficient mutation operator tailor-designed for this FAP variant. The extensive experimental evaluation performed on two real-world instances of the FAP revealed that the FLCs are able to obtain similar or even better frequency plans than those obtained using hyperheuristics or a fixed value for the parameters. The fact that better results are returned by some control schemes as compared to the fixed methods also highlights the advantage to be gained by adapting the parameter over the course of the run, i.e. in parameter control rather than in parameter tuning. Since no statistical differences were noted between FLCs and hyper-heuristics, they can be used indistinctly in order to obtain high quality frequency plans. Moreover, both approach types are quite robust from the point of view of their parameters. Small modifications to the values of their internal parameters do not entail significant changes in their behaviour and performance. Finally it is worth mentioning that the best-known frequency plans published for both of the instances considered were improved upon in this work by one of the control approaches based on 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 feedback from the optimisation. To the best of our knowledge, this is the first time an FLC has been used to control the parameters of the NM operator. In addition, we have shown that the method is applicable

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in general since both a continuous numeric parameter and a discrete numeric parameter were successfully controlled. Other numeric parameters belonging to other meta-heuristics from the mono-objective and multi-objective fields might be controlled with this approach. If a multi-objective meta-heuristic is considered, the value of the input variable IMP should be calculated by means of a multi-objective performance metric. Although two different parameters were considered, they were adapted separately, so it would be interesting to control both parameters simultaneously. Another promising line of research could be the design of a hybrid control scheme that combines FLCs and hyper-heuristics with the aim of adapting numeric and symbolic parameters simultaneously, thus combining the advantages of both methods in a single control scheme.

Acknowledgments This work was funded by the EC (FEDER) and the Spanish Ministry of Science and Innovation as part of the ’Plan Nacional de I þ D þi’, with contract number TIN2011-25448. The work of Eduardo Segredo was funded by grant FPU-AP2009-0457. The work was also funded by the HPC-EUROPA2 Project (Project number: 228398) with the support of the European Commission—Capacities Area— Research Infrastructures.

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