Abstract

Setting the control parameters of a genetic algorithm to obtain good results is a long-standing problem. We define an experiment design and analysis method to determine relative importance and effective settings for control parameters of any evolutionary algorithm, and we apply this method to a classic binary-encoded genetic algorithm (GA). Subsequently, as reported elsewhere, we applied the GA, with the control parameter settings determined here, to steer a population of cloud-computing simulators toward behaviors that reveal degraded performance and system collapse. GA-steered simulators could serve as a design tool, empowering system engineers to identify and mitigate low-probability, costly failure scenarios. In the existing GA literature, we uncovered conflicting opinions and evidence regarding key GA control parameters and effective settings to adopt. Consequently, we designed and executed an experiment to determine relative importance and effective settings for seven GA control parameters, when applied across a set of numerical optimization problems drawn from the literature. This paper describes our experiment design, analysis, and results. We found that crossover most significantly influenced GA success, followed by mutation rate and population size and then by rerandomization point and elite selection. Selection method and the precision used within the chromosome to represent numerical values had least influence. Our findings are robust over 60 numerical optimization problems.

1  Introduction

We aim to devise tools that system engineers can use to explore model-based designs for low-probability, costly failure scenarios (Taleb, 2010). As a first approach, we are investigating guided random search techniques that can steer a population of model simulators toward parameter combinations that yield degraded performance or system collapse. We identified genetic algorithms (GAs) as a search technique that might be well suited for our problem. GAs can find good solutions within a large, ill-defined search space and can be readily adapted for a wide variety of search problems (Mitchell, 1998).

The classic binary-encoded GA we adopted exhibits a number of control parameters (see Section 2), such as population size, selection method, number of elite individuals, generations at which to rerandomize the population, number of crossover points when swapping chromosomes between pairs of individuals, and mutation rate. We consulted the literature and found conflicting advice about relative importance and effective settings for these GA control parameters (see Section 3). Further, we found only a small collection of studies (none definitive) attempting to guide selection of GA control settings. Some authors (DeJong, 2007) advise that one should experiment with a GA in the intended application and use those experiments to determine the most effective control settings. Other researchers (Bartz-Beielstein et al., 2005) provide techniques that can be used to sequentially search the parameter space of evolutionary algorithms to find optimal control settings for individual problems. While sequential parameter optimization (SPO) might prove effective for a wide range of numerical optimization problems, sequentially searching the GA in our intended application is not feasible because the cloud-computing simulators we are exploring require significant computation time. Instead, we designed and conducted an experiment and analysis to search, across a set of 60 numerical optimization problems, for the most important GA control parameters and effective settings to use. While a classic binary-encoded GA may not exhibit the best performance on numerical optimization problems, we show that a sufficient set of numerical optimization test problems can give significant insight into the effect of various GA control parameters.

This paper makes two main contributions: (1) We define an experiment design and analysis method that can be adapted to determine relative importance and effective settings for control parameters in any evolutionary algorithm (see Section 4), and (2) for a classic binary-encoded GA we determine the relative importance and effective settings for seven control parameters (see Section 5). Our findings are robust over 60 numerical optimization problems. The GA control settings discovered using our method proved effective when applied to steer a population of cloud-computing simulators in search of potential failure scenarios (Mills et al., 2013). Our method can also be used to discover effective initial parameter settings for SPO when applied to individual evolutionary algorithms used to solve specific problems. Finding effective starting parameter settings could shorten search time when using SPO.

2  Genetic Algorithm under Test

GAs are a subclass of evolutionary algorithms (EAs). EAs comprise a collection of heuristic methods that use techniques such as mutation, recombination, and selection (inspired by genetics and biological evolution) to search for optimal solutions to difficult problems. For our application, we adopted a classic GA (Mitchell, 1998) that encodes problem variables as fixed-length bit strings, which enable simple crossover and mutation operations. Some EAs represent problem variables as vectors of floating-point numbers. To achieve good performance, floating-point encoding requires altering the typical crossover and mutation operators used by classic GAs (Janikow and Michalewicz, 1991). Ongoing research on EAs (Ali et al., 2005; Rahnamayan et al., 2008; Sahin, 2011) continues to investigate the best formulation for crossover and mutation operators when using floating-point representation.

2.1  Fixed-Length Binary Encoding

The classic GA in our experiments uses simple binary encoding, which can represent the full range of variable types (Boolean, integer, and floating-point) required for our cloud-computing simulator. A user first identifies the variables for a problem and specifies the minimum and maximum values and precision (or quantization) for each variable and then the GA computes the number of bits per variable and thus the chromosome (i.e., bit string) length required to encode all problem variables. The GA includes a control parameter, precision scaling, that can increase or decrease the quantization originally specified by a user.

In most of the numerical optimization problems explored in this paper, the GA generates binary encoding of quantized floating-point variables (some variables are integers). The number of bits required to encode quantized variables can allow more values than necessary. In such cases variable values are derived by linear scaling. For example, 10 values require at least 4 bits (3 bits are too few), which can encode as many as 16 values. In this example, each binary value encodes 0.625 (10/16) of a real value.

While simple binary encoding can lead to Hamming walls, where single-bit mutations cannot transform a bit pattern into a neighboring bit pattern and thus can contribute to premature convergence in numerical optimization problems, our application (searching for failure scenarios in cloud-computing simulations) was not hindered by such issues because we sought general failure outcomes rather than specific optimal numerical values. For that reason, we were not deterred by the fact that our adopted GA uses simple binary encoding instead of Gray encoding, which would eliminate Hamming walls. We discuss this issue further in Section 4, where we identify additional GA control parameters that could be studied, using the same method we describe.

2.2  Control Parameters

The GA begins by generating a random population of individuals, where each individual consists of an appropriate-length bit string representing values for every variable of a problem to be solved. The population size is a control parameter of the GA. The GA evaluates the fitness of many populations of individuals over time, where each population is called a generation. The population of individuals for generation is created through some transformation of individuals composing generation n.

After completing each generation, the GA considers whether or not the population should be rerandomized, which involves randomly regenerating all or part of the next generation. The GA includes a control parameter, reboot proportion, which determines how many generations must be completed before a population is rerandomized.

Whenever the population of individuals is created for an upcoming generation, some number of the most fit individuals (i.e., the elite) from the previous generation can be included unchanged. The GA has a control parameter, elite selection percentage, which defines how many individuals from generation n will be placed unaltered into generation . Such elite individuals can be placed into a population whether or not the remaining individuals are generated randomly or by transforming individuals from the previous generation.

The GA also includes a control parameter, selection method, that defines the algorithm used to select individuals from generation n for inclusion into a candidate pool, where some individuals from generation n may be included multiple times, whereas others may not be included at all. Given a pair of individuals chosen randomly from the candidate pool, a GA control parameter, number of crossover points, determines how bits will be swapped (or recombined) among the pair. Subsequently, the GA iterates over each bit representing each individual in the population of recombined individuals while deciding whether the bit should be inverted. A GA control parameter, mutation rate, specifies the probability that any given bit will be inverted.

3  Related Work

We first consulted the seminal investigation of behavior in classic GAs (DeJong, 1975), where DeJong studied a small set of numerical optimization problems. In his study DeJong reported that the best solutions are obtained when GA control parameters are set as follows: a population size of 50; a .60 probability that pairs of individuals crossover at a single point; a mutation probability of .001, with the most elite individual surviving to the next generation while the remainder of the population is replaced with transformed individuals from the previous generation. Other researchers (Grefenstette, 1986; Goldberg, 1989; Schaffer et al., 1989; Spears and DeJong, 1992) soon followed with additional studies of the best settings to adopt. Unfortunately, the findings of these studies are not always in agreement. For example, Grefenstette (1986) found that mutation rates above .05 were generally harmful but that failing to use mutation leads to poor performance, while using elitism is helpful and population size should be kept in the range of 30 to 100. Goldberg (1989) found that mutation can replace key genetic material that might be lost through crossover, but Tate and Smith (1993) disputed this view, pointing out that high mutation rates can be disruptive when most of the population is replaced with each generation. Schaffer et al. (1989) found an interaction between crossover, mutation rate, and population size, specifically suggesting that small populations are quite sensitive to mutation rate and less sensitive to crossover rate. Further, they found that with larger populations, low rates of mutation and crossover prove most effective.

Studies continue (Baeck, 1996; Odetayo, 1997; Digalakis and Margaritis, 2001; Charbonneau, 2002; Rojas et al., 2002; Boyabatli and Sabuncuoglu, 2004; Núñez-Letamendia, 2007; Diaz-Gomez and Hougen, 2009; Arenas et al., 2010; Kapoor et al., 2011) up to the present, and with continued inconsistencies among findings. For example, Rojas et al. (2002) found that the most important GA control parameters are selection method, mutation rate, and population size, but the number of crossover points (one or two) does not have much influence, while the probability of crossover does. Arenas et al. (2010) found that crossover and mutation must be combined to have the best outcomes, but mutation rate must remain within a narrow range because too much mutation prevents convergence to a good solution and too little mutation leads to premature convergence. Charbonneau (2002) argued that mutation rate should adapt, increasing with decreasing population diversity and decreasing with increasing diversity. Digalakis and Margaritis (2001) found that a population size between 200 and 250 is optimal combined with a crossover rate of 70%–75%, and absence of mutation leads to poor outcomes, whereas a small mutation rate improves results. Boyabatli and Sabuncuoglu (2004) reported that crossover does not have significant influence on outcomes, whereas high mutation rate provides better results. Baeck (1996) agreed with Charbonneau that mutation rate should be variable but argued that crossover must be used in order to find a global optimum. Kapoor et al. (2011) suggested that mutation plays a critical role for simple problems with few parameters, but crossover is important for complex problems. In general, they concluded that one should combine high crossover rate with low mutation rate and a correctly sized population. Núñez-Letamendia (2007) found that while various combinations of crossover and mutation probabilities lead to optimum solutions for different problems, the best outcomes usually (but not always) occur when combing high crossover probability with low mutation rate. All in all, the results reported in the literature provide a rather confusing picture.

We found two retrospective articles (DeJong, 1999, 2007) from the researcher who initiated investigation of control parameters in GAs. DeJong (1999) reported a general lack of theory to guide selection of population size, use of selection strategies, and choice of representation. He observed that a few theoretical models exist in the evolutionary strategies research community for finding optimal mutation rates, but those theories have been unable to explain anomalous results from experiments, some of which find benefits from adaptation in both mutation rate and recombination mechanism. In 2007, DeJong took a broader view, considering what was known with respect to the wider community of EAs. He observed that while it appears that adapting mutation rate online during execution provides advantages, most EAs are deployed with a default set of static parameter values that have been found quite robust in practice. (The method we describe in Section 4 provides a rigorous means to determine effective values to use as default settings for control parameters.) DeJong also observed that larger population size increases parallelism, which aids in solving complex problems, but there is diminishing return to increasing population size. He reported that choosing a selection method is difficult owing to interactions with population size.

DeJong (2007) also noted a promising approach involving periodic restarts of an EA search, using historical information to adapt control parameters in a type of metasearch to find the best control parameters that yield optimal outcomes for individual problems. Bartz-Beielstein et al. (2005) defined a specific method, SPO, for conducting such a metasearch. Despite the existence of methods such as SPO, DeJong believes that EAs pretuned with default parameter values for particular problem classes will continue providing better performance than EAs that attempt to dynamically adapt too many control parameters for specific problems. So, even after more than 30 years of research, the question of best settings for EA and GA control parameters has no widely agreed-upon answer.

A critical review of the previous research on GA control parameters suggests some reasons that might underlie the current state of uncertainty. First, published studies examine performance of GAs only under a small number of numerical optimization problems, ranging from one (Boyabatli and Sabuncuoglu, 2004) to 14 (Digalakis and Margaritis, 2001). These represent an unacceptably small sample, providing little robustness in results. This shortcoming does not appear in studies (Ali et al., 2005; Rahnamayan et al., 2008; Sahin, 2011) that compare the effectiveness and efficiency of different proposed EAs. EA comparison studies typically use more than 50 problems, and those problems are drawn from the same set (Adorio and Diliman, 2005) of difficult numerical optimization problems. Second, published GA studies examine different sets of control parameters over different ranges of values. Most studies also examine more values for some control parameters than for others, which can lead to unbalanced results. Third, since 1990 very few GA studies attempt to replicate the results of prior studies. Fourth, only one study (Arenas et al., 2010) used analysis of variance (ANOVA) to determine the statistical significance of each control parameter studied.

Kleijnen (2010) described the state of the art in design and analysis of computer experiments, where the main concepts are derived from the statistical design of experiments (Box et al., 2005). Bartz-Beielstein (2006) embodied those concepts in a three-phase approach to find optimal parameter settings for EAs when applied to solve specific optimization problems. Phase one used highly fractionated (resolution III) two-level experiment designs to identify the most important set of control parameters from among a potentially large set. Phase two used less fractionated (resolution IV and V) two-level experiment designs to assess interactions among important control parameters and to identify regions of the parameter space that might be fertile areas over which to seek optimal settings. Phase three used SPO (Bartz-Beielstein et al., 2005) to search for optimal settings for control parameters when applied to solve some specific optimization problem. SPO amounts to a metasearch for optimal control parameter settings that yield the best performance on some specific optimization problem. SPO metasearches must be undertaken for each new optimization problem of interest.

We also adopt fundamental concepts from statistical design of experiments, but we use only the equivalent of the second phase of the Bartz-Beielstein approach. In particular, we use high-resolution designs to simultaneously identify relative importance of control parameters, to assess interactions among control parameters, and to find the most effective control parameter settings (from among those considered). We adopt this one-phase approach because (1) we are performing screening/sensitivity analysis instead of optimization; (2) sequential parameter optimization is not feasible in our intended application because the cloud-computing simulators we plan to investigate require significant computation time; and (3) the GA we investigate has only a handful (k = 7) of control parameters.

We use four levels per control parameter instead of two levels, as is typical in statistical experiment designs. For the GA we investigate, which has only seven control parameters, using four levels enables us to search across 16,384 parameter combinations rather than the 128 combinations possible with a two-level design. Further, while we cannot directly investigate the GA in the context of our intended cloud-computing application, we do investigate the performance of the GA across a large variety and number of test problems. Exploring substantially more parameter combinations across a wide variety and large number of test problems somewhat offsets our inability to use SPO. On the other hand, the specialized analysis methods typically used with two-level experiment designs cannot always be applied directly to four-level designs. Thus, as described in Sections 4 and 5, we introduce several analysis innovations that enable us to extract essential information from our experiment results.

In Section 4, we outline an experiment design and analysis method that can be used to obtain statistically significant results robust over many numerical optimization problems. Our method can be used to identify effective default parameter settings for EAs, enabling good performance on many search problems. Our proposed method is intended to complement rather than replace SPO. Specifically, our method can be used for applications, such as ours, where SPO metasearch is infeasible. When compared with the two (pre-SPO) phases of the Bartz-Beielstein approach, our one-phase method considers substantially more parameter combinations across a large variety and number of test problems. For this reason, where SPO metasearch is feasible, our method could be used to identify effective starting values for EA control parameters. Section 4 discusses our proposed experiment design and analysis method in the context of a classic GA, but the method appears general enough to be applied to study and tune control parameters of other search algorithms.

4  Experiment Design and Analysis Method

Robust and rigorous statistical experiments naively require full factorial (FF) designs, which examine each parameter under study at every possible value and then analyze the influence of the parameters on outcomes. For most experiments, an FF design proves impractical because even a handful of parameters can encompass an infeasible space of possibilities. For example, suppose (as in our case) a system under study is controlled by seven parameters. Assuming each parameter can be represented within a 32-bit integer, the space of possibilities is of O().

A natural first step to reduce the search space is to limit the number of values at which to examine each parameter. For example, in our case, we examine each parameter at only four values (i.e., levels), which reduces the search space to . Restricting parameters to take on a reduced set of values has obvious limitations: only a small number of parameter values are explored, and extrapolating from the results assumes that a model behaves monotonically in the range between chosen values. On the other hand, adopting a reduced-level design provides some advantages (Box et al., 2005): (1) It requires relatively few runs per parameter, (2) identifies promising directions for future experiments (and may be augmented with thorough local explorations), (3) fits naturally into a sequential strategy (such as SPO), and (4) forms the basis for further reduction in parameter combinations through use of fractional factorial designs.

Even with only a few levels, a full factorial design can still be prohibitively expensive. For example, in our case, an FF exploration of parameter combinations for 60 numerical optimization problems would require about 1 million executions of the GA, and running each GA execution through 500 generations would require executing over 60 billion function evaluations (assuming an average population size). To further limit the number of experiment executions, one can adopt orthogonal fractional factorial (OFF) experiment designs or the geometrically equivalent orthogonal Latin hypercube (OLH) designs (Box et al., 2005).

OFF and OLH experiment designs sample a full factorial design space in a balanced and orthogonal fashion. Balance ensures good effect estimates by reducing an estimator’s bias and variability. Orthogonality ensures good estimates of two-term interactions. Balance is achieved by ensuring that each level of every factor occurs an equal number of times in the selected sample. Orthogonality is achieved by ensuring that each pair of levels occurs an equal number of times across all experiment parameters in the selected sample. OFF and OLH designs exhibit good space-spanning properties, which aid screening, sensitivity, and comparative analyses. On the other hand, highly fractionated OFF or OLH designs can have poor space-filling properties, which are necessary for optimization analyses. As explained later, we adopt a OFF design, which achieves a reasonably space-filling (1/16th) sample of the full experiment space. If the number of factors and levels were to increase significantly, we would need to increase the number of experiment runs or else adopt some alternative experiment design method. For example, Cioppa and Lucas (2007) describe an algorithm for generating “nearly” orthogonal Latin hypercube designs that exhibit improved space-filling properties but at the cost of inducing small correlations, which can complicate the analysis.

By ensuring balance and orthogonality, OFF designs achieve two desirable properties, given the limits of the selected sample size. First, main effects estimates are representative of main effects that would be found in a full factorial experiment. This means that a list of experiment parameters, ranked by main effects, tends to be close to the true ordering that would result from an FF experiment. Second, main effects estimates exhibit an uncertainty as small as possible, given the sample size. More specifically, running an OFF experiment design provides an estimate of main effects with uncertainty on the order of , where n is the number of experiment runs, v is the number of values used per parameter, and is the standard deviation among the underlying observations. This formula simply adapts the standard uncertainty estimate for main effects in two-level OFF experiment designs (Box et al., 2005) to account for the fact that our design uses v levels per factor, which means there are observations at each level rather than .

Using OFF principles as a basis, we defined an experiment design and analysis method encompassing seven steps: (1) factor identification and level selection, (2) problem set selection, (3) OFF experiment design, (4) experiment execution and data collection, (5) per problem factor analysis, (6) per problem factor interaction analysis, and (7) results summarization. We explain each step in turn, illustrating with details from our study of the GA that was described in Section 2.

The first step entails identification of the experiment factors (i.e., control parameters) along with selection of levels (i.e., values) at which each factor will be examined. As experiment factors, we used the seven control parameters from a classic GA (see Section 2) and we selected four levels for each factor (see Table 1). We selected population sizes (factor x1) between 50 and 200 because those values encompassed the range found to provide good performance in previous studies (see Section 3).

Table 1:
Experiment factors (control parameters) and levels (parameter values).
FactorLevel 1Level 2Level 3Level 4
Population size (50 100 150 200 
Selection method (SUS T (T (T (
Elite selection % (
Reboot proportion (0.1 0.2 0.4 
No. of crossover points (
Mutation rate (Adaptive 0.001 0.0055 0.01 
Precision scaling (1/2 
FactorLevel 1Level 2Level 3Level 4
Population size (50 100 150 200 
Selection method (SUS T (T (T (
Elite selection % (
Reboot proportion (0.1 0.2 0.4 
No. of crossover points (
Mutation rate (Adaptive 0.001 0.0055 0.01 
Precision scaling (1/2 

The GA included only two selection methods (factor x2), stochastic universal sampling (SUS) (Baker, 1987) and paired tournament (also known as 2-tournament) with replacement (T) (Mitchell, 1998); we investigated both. The SUS algorithm contained no tunable parameters. In 2-tournament a pair of evaluated individuals is drawn randomly (uniform distribution) from the previous generation, and then with some probability r the more fit of those individuals is selected for inclusion in the recombination and mutation processes that will form the next generation. The lesser fit individual of the pair is selected otherwise. As shown in Table 1, we used three values for r. Had the GA included other available selection methods (e.g., q-tournament selection, roulette wheel selection, or rank selection), we could have included two of them (in place of T with multiple values for r) as additional parameter values in our experiment. Had we been able to explore such a larger set of selection methods, a subsequent experiment iteration could have been designed to explore any tunable parameter values associated with the best-performing selection method. (Experiment iteration leverages the ability of OFF designs to support thorough local explorations.) Since specific GA selection methods are likely to have different tunable parameters, exploring them when exploring heterogeneous selection methods would be impractical.

To probe the utility of elitism, we varied elite selection percentage (factor x3) from 0% to 8% of the population size. Including a value of 0 allows elite selection to be disabled, which permits us to investigate findings (see Section 3) that elitism is one key to GA success. By exploring a range of other elite selection percentages, one could establish whether too much elite selection impairs the ability of a GA to converge to good solutions. Subsequent experiment iterations could be conducted to search more precisely for effective elite selection percentages, starting from the best value found in our experiment.

To investigate the influence of population rerandomization, we ranged reboot proportion (factor x4) from 0 to 0.4 of the generations executed. When reboot proportion is set to 0, the population chromosomes are randomized only once, at initiation of the GA. When reboot proportion is set to 0.1, chromosomes for nonelite members of the population will be randomized nine more times after the initial randomization, once after incremental completion of each 10% of the total number of generations. Similarly, when reboot proportion is set to 0.2 (or 0.4), nonelite members of the population will be randomized 4 (or 2) times subsequent to the initial randomization, that is, after incremental completion of each 20% (or 40%) of the total number of generations.

For crossover (factor x5) we specified different numbers of crossover points, ranging from 0 to 3. When the number of crossover points was 0, no crossover occurred; otherwise, for each pair of recombining individuals, the locations of the specified number (1, 2, or 3) of crossover points were selected randomly and then the indicated bits were swapped. While the GA did allow a probabilistic crossover threshold, to avoid overweighting investigation of crossover techniques compared with other factors, we did not complicate crossover by adding a probabilistic threshold or by considering uniform crossover, where each bit has some probability of being swapped. If desired, these more complicated schemes could be investigated in a subsequent OFF experiment, where other factors are fixed to the best levels discovered here.

For mutation rate (factor x6) we specified three levels with rates ranging from small (0.001) to high (0.01). We reserved the remaining level for an adaptive algorithm (Charbonneau, 2002) that adjusts the mutation rate between 0.001 and 0.1, lowering the rate when a population exhibits a wide spread in fitness values and raising the rate when the spread is narrow. Adopting these settings enabled us to investigate the various conflicting findings about mutation (see Section 3).

We also investigated the influence of problem-variable discretization on GA performance by varying precision scaling (factor x7) from 1/2 to 4, including the value 1. A precision scaling value of 1 means the GA performs a search with variable granularity as determined by the user when specifying the encoding for a problem. A precision scaling value of 1/2 means variable granularity is set to be twice as coarse (requiring fewer chromosome bits) as specified by the user, while values of 2 and 4 mean that variable granularity is set two or four times, respectively, as fine (requiring more chromosome bits) as the user specified. Investigating this factor allowed us to evaluate whether increasing or decreasing the size of a search space would improve the performance of the GA.

Readers familiar with GAs will note that while we explored a wide range of GA control parameters, there are other such parameters that could have been investigated. In some cases, our investigation was restricted by the range of available parameters in the GA. For example, the GA offered only two of the possible selection methods sometimes available in GAs. Similarly, GAs often offer Gray encoding to forestall Hamming cliffs (see Section 2.1), or other encoding methods like permutation encoding or real value encoding. The GA used only simple binary encoding. In other cases, we chose to limit the specific control parameters investigated in order to maintain a balanced experiment that did not place too much emphasis on any particular control parameter. For example, the GA offered the possibility of setting a threshold for comparing against a uniformly sampled random variate to determine whether crossover would take place for specific pairs of individuals. We elected to design our experiments so that crossover always took place unless it was disabled. Had additional experiment factors (control parameters) been of interest, OFF experiment design techniques provide suitable means to include them. One means to expand the number of control parameters in an experiment would simply be to include them within the experiment factors. For example, had the GA offered them, additional selection methods could have been added as parameter values (levels), replacing the multiple r values we used for 2-tournament selection.

In cases where particular control parameters identify algorithms with tunable variables, OFF experiment design can be applied sequentially. For example, various selection methods have unique tunable variables. To establish appropriate settings for those variables, an experimenter could first run an OFF experiment design, fixing selection method, to determine which other control parameter values yield best performance on a set of test problems. Subsequently, the experimenter could design OFF experiments to explore a range of tunable values for each selection method while fixing other control parameters to values determined in a preceding experiment. A similar approach could be used to investigate potential unbalanced control parameters. For example, an experimenter could first set the crossover threshold so that crossover (if enabled) would always occur and then run a subsequent experiment that varies the crossover threshold while fixing other parameters to the values that performed best in a preceding experiment. Similarly, an experimenter could follow up any OFF experiment with a related experiment with a wider range of parameter settings for each control parameter. The second experiment would establish whether the results from the first experiment were robust (the same) over the increased range of value settings.

Step two in our method is to select a set of numerical optimization problems to evaluate when the GA is configured with various combinations of level settings for the seven factors identified in Table 1. A full factorial () exploration of all combinations would require executing the GA 16,348 times for each selected problem. (Perhaps this explains why previous studies were limited to 14 or fewer problems?) To increase the robustness of our results, we aimed for about the same number (55–58) of problems used in recent studies comparing various EAs (see Section 3). We selected 53 test problems from that set, where problems ranged from relatively simple (two parameters) to complex (100 parameters). We chose seven additional problems, two from the search literature (Ingber, 1993; Brent, 2002) and five from the statistics literature (Box et al., 1990, 2005; Saltelli et al., 2004), bringing the total number of test problems to 60 (see Appendix  A for a specific list with sources). Using 60 test problems extended the scope of our results substantially compared to previous studies of GA control parameters found in the literature.

Step three applies an OFF design to significantly reduce the number of required experiment runs. We chose a OFF design, reducing the number of parameter combinations per GA run to 1,024, which, assuming an average population size, requires about 64 million function evaluations for each test problem and just over 3.8 billion for all 60 problems. Using a OFF design yielded good estimates of main effects with an uncertainty of 0.088 while expending only 6.25% of the computational resources required for an FF experiment. (See Appendix  B for a comparison of results from a OFF design versus results from an FF design for three of our test problems.)

Step four executes the OFF design against each test problem and collects the resulting data. For each problem the GA is run once under each of the 1,024 parameter combinations identified by the OFF design. For a given parameter combination the GA is executed for 500 generations and the output (y) is recorded as the maximum value discovered by the GA. Let n (= 1,024) be the total number of parameter combinations, k (= 7) the number of factors, and (= 4) the number of levels within each factor. The output for each problem is a table containing n rows, where each row includes columns: a level setting for each of the factors and the resulting y value.

Step five analyzes the results for each test problem. This requires computing for each factor the absolute (Ei) and relative (REi) effects and the statistical significance (. A multidimensional factor analysis plot (e.g., Figure 1) reveals the relative factor importance and the most effective setting for each factor. Computing (Ei) requires determining , the average y values for factor xi at level setting . Let S be the set of all OFF parameter combinations and ,
formula
1
computes the average value of y when xi = , and then . , where
formula
2
To measure statistical significance we used analysis of variance (ANOVA) by computing , where is the reference F distribution with and degrees of freedom and
formula
3
Figure 1:

Factor analysis plot for numerical optimization problem 2: maximizing the percentage of nondefective springs.

Figure 1:

Factor analysis plot for numerical optimization problem 2: maximizing the percentage of nondefective springs.

After computing Ei, REi, and for each factor, we plot the results, as shown, for example, in Figure 1 for numerical optimization problem 2. The figure plots 28 points: mean for each level () of each factor (xi). The dashed horizontal line reports . The bottom of the plot (just above the x-axis) reports the Ei, REi, and for each factor.

The factor exhibiting largest Ei has most influence on GA success for the problem. We highlight this factor, mutation rate (x6), with a dashed rectangle in Figure 1. Ordering Ei values from high to low reveals the relative importance of each factor, as we indicate by labeling the rank-ordered factors from 1 (most important) to 7 (least important) in Figure 1. Where Ei values are tied, we order the ranking based on visual inspection of the plot.

Large effects are not necessarily statistically significant, but the values adjudicate that question. We scaled to a percentage, which means that the probability of type I error is when and when . On our plots, we mark with a * symbol and with a ** symbol beneath values associated with statistically significant factors. In Figure 1 statistically significant factors include population size (x1), reboot proportion (x4), number of crossover points (x5), mutation rate (x6), and precision scaling (x7).

The plot also identifies the best setting for each factor, as represented by the maximum . In Figure 1 the best for each factor is circled: population size (200), selection method (2-tournament, ), elite selection percent (0), reboot proportion (0), number of crossover points (3), mutation rate (adaptive), and precision scaling (1/2 as fine).

Step six of our method computes two-term interactions for all pairs of experiment factors for each test problem. We computed (6 × 7 =) 42 two-term interactions for each of the 60 test problems. Figure 2 shows four cells extracted from the upper left corner of a 7 × 7 matrix containing a per factor interaction analysis for numerical optimization problem 1, maximizing chemical yield. Each row of the matrix corresponds to one factor and contains seven columns. One column in each row (the cell appearing on the matrix diagonal) identifies the factor for which two-term interactions are analyzed, and the six remaining columns show the interaction of that factor with each of the other six factors. The extracted submatrix shows interaction analyses for population size (x1) and selection method (x2).

Figure 2:

Four cells extracted from the upper left corner of a matrix containing a per-factor interaction analysis for numerical optimization problem 1, maximizing chemical yield. Two of the cells contain two-term interaction analysis plots for population size (x1) and selection method (x2). The upper left cell establishes the y-axis scale for the interaction analysis plots. The upper right cell plots five curves. The first curve is the main effects plot for x1. Each additional curve reports the effect of varying x1 while holding x2 constant, first at level 1, then at level 2, and so on, to level 4. The number under each curve reports the related effect. The number in the upper right corner of the cell is the overall interaction effect, computed by subtracting the smallest of the four interactions from the largest. The lower left cell depicts the interaction analysis plots when varying x2 while holding x1 constant at each of its four levels.

Figure 2:

Four cells extracted from the upper left corner of a matrix containing a per-factor interaction analysis for numerical optimization problem 1, maximizing chemical yield. Two of the cells contain two-term interaction analysis plots for population size (x1) and selection method (x2). The upper left cell establishes the y-axis scale for the interaction analysis plots. The upper right cell plots five curves. The first curve is the main effects plot for x1. Each additional curve reports the effect of varying x1 while holding x2 constant, first at level 1, then at level 2, and so on, to level 4. The number under each curve reports the related effect. The number in the upper right corner of the cell is the overall interaction effect, computed by subtracting the smallest of the four interactions from the largest. The lower left cell depicts the interaction analysis plots when varying x2 while holding x1 constant at each of its four levels.

The cell at the upper right of Figure 2 shows five curves. The first curve is the main effects plot for population size. Each data point is the average of 256 experiment data points, where population size is set to each of its four levels. Each data point on the second curve gives the average of only 64 data points, where population size varies across its four levels while holding selection method to its first level. The third through fifth curves vary population size while holding selection method to its second, then third, and finally fourth level. The numbers below each curve report the size of each interaction effect. The number at the upper right corner of a cell represents the overall interaction effect (IEij), which is computed by subtracting the smallest of the four interactions from the largest. Note that interaction effects are not necessarily symmetric for pairs of factors. For example, Figure 2 reports an interaction effect of 0.044 for and 0.027 for .

The seventh, and final, step in our method summarizes and analyzes results across all test problems. This step is described in the next section.

5  Results and Discussion

Given Ei, REi, , IEij, and related factor analysis and factor interaction plots representing GA performance on each of the 60 test problems, we summarized the analysis results using five main techniques: (1) factor significance matrix, (2) main effects rank histograms, (3) results summary table, (4) factor rank/most effective level table, and (5) factor interaction summary. We describe each of these in turn, illustrating them with data collected from our experiment, and we discuss the findings revealed by these techniques.

Figure 3 shows a factor significance matrix for the 60 test problems solved by the GA. A visual scan of the matrix reveals that factors x1, x4, x5, and x6 had most statistically significant influence on GA success. Factors x2, x3, and x7 had less influence. The last column in Figure 3 provides a quantitative verification of these visual impressions by reporting the percentage of test problems on which each factor was statistically significant.

Figure 3:

Factor significance matrix with 420 cells, one row for each of seven factors, and one column for each of 60 test problems (see Appendix  A for descriptions). Black cells indicate the factor had significant effect () for the identified problem. A sixty-first column gives the percentage of test problems on which each factor was statistically significant ().

Figure 3:

Factor significance matrix with 420 cells, one row for each of seven factors, and one column for each of 60 test problems (see Appendix  A for descriptions). Black cells indicate the factor had significant effect () for the identified problem. A sixty-first column gives the percentage of test problems on which each factor was statistically significant ().

We note that GAs attempt to balance two competing processes: exploitation and exploration. Exploitation leverages good problem solutions in a given generation by selecting them as the basis for solutions forming the next generation and by adding (optionally) the very best (elite) solutions unchanged to the next generation. Exploration expands the scope of a search by transforming the best solutions in a given generation into recombined and mutated variants and by occasionally rerandomizing an entire population except for any elite solutions from the previous generation. Each generation can explore a population of solutions in parallel; thus larger population sizes appear likely to increase the ability of a GA to exploit good solutions as well as to explore mutated and recombined variants. The factor significance matrix shown in Figure 3 suggests that exploration (i.e., crossover, mutation, and population rerandomization) has more influence on GA success than does exploitation (i.e., selection and elitism). Further, the matrix indicates that population size somewhat influences GA success.

The 60 test problems can be categorized using the scheme from Jamil and Yang (2013), which classifies optimization functions based on continuity, differentiability, scalability, separability, and modality. Over 93% (56) of the test functions are continuous and differentiable; only the three Step functions and the Corana function are discontinuous and nondifferential. Just over half (35) of the test functions are scalable, though we did not make use of this property in our experiments. Sixty percent (36) of the problems are nonseparable, and 65% (39) are multimodal. For each problem the type field in Figure 3 reports separability and modality, using two bit positions. The first bit position denotes separability (separable = 1) and the second bit position denotes modality (unimodal = 1). Examining Figure 3 closely reveals no specific pattern with respect to problem type or number of dimensions. This means that the significance of factor influence applies across the 60 test problems regardless of function type or dimension.

The factor significance matrix in Figure 3 also shows that no factors (or only one) significantly influenced GA performance for three problems: problem 36, Multimod (Mm) with 30 variables, problem 47, Goldstein-Price (Gp) with two variables, and problem 54, Dekkers and Aarts (Dk) with two variables. To gain more insight, we consulted the factor analysis plots for each of these problems.

For the Multimod problem, three levels of each factor successfully found the theoretical best solutions, but one level for each factor did not. Because three levels in each factor led to successful solutions, the factors could not be distinguished as statistically significant; however, no crossover performed worse than crossover (at any level), and the lowest level of fixed mutation rate performed worse than higher mutation rates (including adaptive mutation). For the Goldstein-Price problem, only population size (x1) exceeded the higher significance threshold (), and precision scaling (x7) exceeded the lower threshold (). The smallest population size (50) led to poor solutions, as did the finest (four times) precision scaling. The exploration factors (rerandomization point, crossover, and mutation rate) each exhibited , suggesting influence but not establishing statistical significance. For the Dekkers and Aarts problem, elite selection percentage (x3) exceeded , with no elite section performing worst. Population size, selection method, and crossover exhibited influence but not to the level of statistical significance. This analysis reveals that altering factor levels did influence GA success on these problems but not sufficiently to pass our chosen level () for statistical significance. We concluded that there was nothing particular about these three problems that prevented the GA control parameters from influencing the success of the GA. Examining these problems in detail reinforced our decision to establish as the threshold for statistical significance.

Figure 4 shows main effects rank histograms for the seven control parameters across the 60 test problems. The histograms reveal that crossover (x5) had low relative influence on main effects for very few problems; in contrast, precision scaling (x7) had high relative influence for very few problems. The relative influence of mutation rate (x6) and population size (x1) varied more evenly across the problems, with mutation rate exhibiting moderate relative influence (rank of 3) among a cluster of problems. Similarly, the relative influence of selection method (x2) varied fairly evenly across the problems while showing increased influence (rank of 2) among a cluster of problems. The relative influence for reboot proportion tended to clump in the middle (ranks 3 and 4) for most problems, exhibiting high relative influence on very few problems. For one problem (Multimod), all factors exhibited the same ranking. The reason for this was discussed earlier.

Figure 4:

Main effects rank histograms for seven factors. The seven cells in each histogram represent the frequency with which the factor exhibited most (1) to least (7) influence on main effects across the 60 test problems. The factor histograms are ordered, top to bottom, by average rank from highest (3 for factor x5, crossover) to lowest (5.3 for factor x7, precision scaling).

Figure 4:

Main effects rank histograms for seven factors. The seven cells in each histogram represent the frequency with which the factor exhibited most (1) to least (7) influence on main effects across the 60 test problems. The factor histograms are ordered, top to bottom, by average rank from highest (3 for factor x5, crossover) to lowest (5.3 for factor x7, precision scaling).

The test problems can be divided roughly into three categories, based on problem dimensionality: (1) 31 low-dimensional () problems, (2) 22 moderate-dimensional () problems, and (3) 7 high-dimensional () problems. The relative influence of crossover, mutation rate, and population size tended to moderate on the highest-dimensional (100) problems, while the influence of elite selection percentage (and, to some degree, selection method) increased for those problems. Analysis of the main effects rank histograms, while providing only relative information, suggests that (if possible) a future iteration of a set of test problems would benefit from extension to include about ten additional problems with high dimensionality. Readers should bear this information in mind when considering our findings. In general, analysis of the main effects rank histograms supports our argument in Section 3 that conflicting conclusions from previous studies of GA control parameters likely arose in part because those studies considered an insufficient number of problems (at most 14 and usually fewer). Our analysis also suggests the need to increase the number of high-dimensional problems included in the set of numerical optimization problems often used to evaluate evolutionary search algorithms.

The average ranks of the seven factors, shown by vertical lines in Figure 4, indicate their influence on main effects, that is, the degree to which the factor influences the ability of the GA to find good numerical solutions. First, factors x5 (crossover) and x6 (mutation rate) most influenced the main effects produced across the test problems. These two factors also proved among the most statistically significant factors (see Figure 3). Second, factor x3 (elite selection percentage) showed next most influence on main effects. Examining x3 on the factor analysis plots for each problem revealed that the main effects of this factor arose when comparing a positive elite selection percentage against no elite selection. From this, we infer that including some amount of elite selection has a large influence on the ability of the GA to find good solutions to numerical optimization problems. On the other hand, as discussed earlier when considering Figure 3, the influence of elite selection percentage is statistically significant in just over half of the problems. Third, factors x1, x2, and x4 showed comparable (i.e., average rank of about 4) levels of influence on main effects. Whereas x1 (population size) and x4 (reboot proportion) were also among the most statistically significant influences on GA success (see Figure 3), x2 (selection method) was not. Examination of x2 on the factor analysis plots for each problem showed that the main effects arose when comparing 2-tournament selection with and against SUS and 2-tournament selection with . From this, we infer that 2-tournament selection with insufficient selection pressure (i.e., r probabilities that are too low) hampered the ability of the GA to find good solutions to numerical optimization problems. Fourth, factor x7 (precision scaling) appeared to have least influence on the ability of the GA to find good solutions.

Comparing Figure 3 (statistical significance) to Figure 4 (size of effect) gives a different ordering of factors influencing GA success. For this reason, we decided to rank factors based on a combination of statistical significance and relative effect. Next, we present and discuss quantitative summaries encompassing both these characteristics, leading to a relative ranking of factor importance.

Table 2 provides a quantitative summary of relative effect, statistical significance, and most effective setting, across all 60 test problems, for each factor. The quantitative measures of significance () provide a precise ranking of factors from most to least influential: number of crossover points (x5), mutation rate (x6), reboot proportion (x4), population size (x1), elite selection percentage (x3), selection method (x2), and then, least influential, precision scaling (x7).

Table 2:
Average standardized relative effect, , and the numbers and percentages of 60 test problems on which each factor significantly influenced GA success. Also, the numbers and percentages of test problems on which each level led the GA to the best answer. In some cases, more than one level led to the best answer (), so the related percentages could sum to more than 100.
SignificantMost Effective Level on Individual Functions
FactorLevel 1Level 2Level 3Level 4
Population size (0.71 39 (65%) (0%) (3%) 12 (20%) 52 (87%) 
Selection method (0.22 31 (52%) 35 (58%) (13%) 11 (18%) 14 (23%) 
Elite selection % (0.43 33 (55%) 14 (23%) 12 (20%) 15 (25%) 26 (43%) 
Reboot proportion (0.35 48 (80%) 24 (40%) (7%) (3%) 46 (77%) 
No. of crossover points (0.87 54 (90%) (0%) 14 (23%) (13%) 46 (77%) 
Mutation rate (0.65 49 (82%) 37 (62%) (7%) 10 (20%) 12 (20%) 
Precision scaling (0.24 25 (43%) 25 (42%) 16 (27%) (8%) 18 (30%) 
SignificantMost Effective Level on Individual Functions
FactorLevel 1Level 2Level 3Level 4
Population size (0.71 39 (65%) (0%) (3%) 12 (20%) 52 (87%) 
Selection method (0.22 31 (52%) 35 (58%) (13%) 11 (18%) 14 (23%) 
Elite selection % (0.43 33 (55%) 14 (23%) 12 (20%) 15 (25%) 26 (43%) 
Reboot proportion (0.35 48 (80%) 24 (40%) (7%) (3%) 46 (77%) 
No. of crossover points (0.87 54 (90%) (0%) 14 (23%) (13%) 46 (77%) 
Mutation rate (0.65 49 (82%) 37 (62%) (7%) 10 (20%) 12 (20%) 
Precision scaling (0.24 25 (43%) 25 (42%) 16 (27%) (8%) 18 (30%) 

Table 2 also reports the average standardized relative effect, , for each factor across all test problems. Recall (Section 4) that relative effect measures how much difference a factor makes in the best outcome found for individual numerical optimization problems. This measure, which is independent of statistical significance, gauges the influence of a factor on effective outcomes.

To compute for a given factor i, we first standardized REi. Let N (= 60) be the number of test problems and K (= 7) be the number of GA control parameters. For each numerical optimization problem ) we can define the set, Ef, of relative effects for each GA control parameter k, where . We then standardized REi
formula
4
Subsequently, we averaged over all N test problems
formula
5
When computed for each factor, this yields the values shown in the column of Table 2.

The values in Table 2 quantify the influence of each factor on the ability of the GA to achieve the best outcomes over all the test problems. As the table shows, number of crossover points (x5), population size (x1), and mutation rate (x6) have most influence on relative effect, followed by elite selection percentage (x3) and reboot proportion (x4). Precision scaling (x7) and selection method (x2) have relatively little influence. Comparing the statistically significant influence of factors () versus their influence on relative effect () finds general agreement, though population size ranks second on relative effect but only fourth on statistical significance, and reboot proportion ranks third on statistical significance but only fifth on relative effect.

Finally, Table 2 identifies the most effective level settings found for each factor: population size = 200, selection method = SUS, elite selection percentage = 8%, reboot proportion = 0.4, number of crossover points = 3, mutation rate = adaptive, and precision scaling = 1/2 as fine as specified by the user. Differences in the percentages for each level setting for a given factor provide a measure of how much changing the level influenced GA performance.

Summary data from Table 2 can be extracted to produce a factor rank/most effective level table (Table 3), which answers the two main questions addressed by our experiment design and analysis: What is the relative importance of the GA control parameters evaluated? and What is the most effective level setting (among those examined) to use for each control parameter? Table 3 reports factor rank and most effective level setting for each GA control parameter. Because rankings differed somewhat when based solely on either statistical significance or relative effect, we chose to rank the factors on D, the Euclidean distance of each factor from an ideal outcome, which is the point (1, 1) on a Cartesian plot of X against Y . Using the data from Table 3, we show such a plot as Figure 5.

Figure 5:

Cartesian plot of (x-axis) versus (y-axis) for seven control parameters and for the ideal point (1, 1).

Figure 5:

Cartesian plot of (x-axis) versus (y-axis) for seven control parameters and for the ideal point (1, 1).

Table 3:
GA control parameters ordered by increasing distance () from ideal (see Figure 5) and partitioned into four groups based on relative differences in . The table also reports: the fraction of 60 test problems on which each factor had statistically significant influence on GA success, the average relative effect, , and the most effective level (setting).
Most Effective
RankFactorsDLevel (Setting)
No. of crossover points (0.90 0.87 0.16 4 (3 points) 
Mutation rate (0.82 0.65 0.39 1 (adaptive) 
Population size (0.65 0.71 0.45 4 (0.4) 
Reboot proportion (0.80 0.35 0.68 4 (200) 
Elite selection % (0.55 0.43 0.73 4 (8%) 
Selection method (0.52 0.22 0.92 1 (SUS) 
 Precision scaling (0.43 0.24 0.95 1 (1/2 as fine) 
Most Effective
RankFactorsDLevel (Setting)
No. of crossover points (0.90 0.87 0.16 4 (3 points) 
Mutation rate (0.82 0.65 0.39 1 (adaptive) 
Population size (0.65 0.71 0.45 4 (0.4) 
Reboot proportion (0.80 0.35 0.68 4 (200) 
Elite selection % (0.55 0.43 0.73 4 (8%) 
Selection method (0.52 0.22 0.92 1 (SUS) 
 Precision scaling (0.43 0.24 0.95 1 (1/2 as fine) 

Figure 5 shows a grouping of three factors—crossover (x5), mutation (x6), and population size (x1)—closest to the ideal point (1, 1). The next closest factor appears to be reboot proportion (x4), which is followed by another grouping of three factors—elite selection percentage (x3), precision scaling (x7), and selection method (x2)—that appear farthest from the ideal point. Computing the Euclidean distance between each factor and the ideal point yields the values reported in the column labeled D in Table 3, which we used to rank the influence of each factor across all 60 test problems in order of increasing distance from (1, 1).

The D values in Table 3 identify a clear ranking: (1) number of crossover points showed most influence, followed by (2) mutation rate and (3) population size, and then by (4) reboot proportion and (5) elite selection percentage. Selection method and precision scaling proved least influential. Evidently, two of the exploration control parameters (crossover and mutation rate) had large statistical significance and also substantial influence on relative effect. The other exploration parameter (reboot proportion) had a statistically significant influence on 80% of the test problems, but the relative effect of the parameter was not large. Population size, which affects both exploration and exploitation, influenced GA outcomes substantially and was statistically significant on about two-thirds of the problems. The exploitation control parameters (elite selection percentage and selection method) exhibited only modest influence on relative effect and were statistically significant on only about one-half of the problems.

Figure 6 shows a factor interaction summary, which plots interaction effects, standardized and averaged across all test problems, for each of the 42 two-term interactions. For each two-term interaction (), we standardized the interaction effect (using the same method as shown in Equation (4) but substituting IEij for REi) to compute for each of the test problems. We then averaged those 60 values using the method shown in Equation (5) to obtain , representing the average interaction effect for () across the entire set of test problems.

Figure 6:

Standardized relative interaction effect, , for each of 42 two-term pairs of factors (y-axis) versus pair identifiers () (x-axis). The pair identifiers are sorted from largest to smallest . Two rectangles highlight the smallest and largest interaction effects. Ovals identify three subgroups among the largest interactions. A sparse matrix reports numerical values for the 14 largest . Pairs of the largest 14 two-term interactions are averaged, , and reported in a table of seven values, sorted from largest to smallest.

Figure 6:

Standardized relative interaction effect, , for each of 42 two-term pairs of factors (y-axis) versus pair identifiers () (x-axis). The pair identifiers are sorted from largest to smallest . Two rectangles highlight the smallest and largest interaction effects. Ovals identify three subgroups among the largest interactions. A sparse matrix reports numerical values for the 14 largest . Pairs of the largest 14 two-term interactions are averaged, , and reported in a table of seven values, sorted from largest to smallest.

These values are plotted on the y-axis of Figure 6 versus the pair identifiers () on the x-axis, which is sorted from largest to smallest . The plot identifies two groups, largest interactions and smallest interactions, as well as largest interactions divided into three subgroups. The numerical values for are extracted from the group of largest interactions and reported in a sparse matrix. Since those 14 values correspond to seven pairs of pairs – {(), ()}, we average the and for each of these pairs of pairs and report those seven averages as a sorted table.

In comparing the seven values representing in Figure 6 to the values in Table 3, we see that the values for the three largest main effects (x5, x1, and x6) exceed the values for the largest standardized interaction effects. This implies that those main effects are more influential on GA success than two-term interactions. On the other hand, the seven two-term interactions exceed the main effects values for the other factors. For that reason, we decided to examine those seven two-term interactions in detail by reviewing the per factor interaction analyses for those interactions across the test problems.

The largest two-term interaction occurs between selection method (x2) and elite selection percentage (x3). In about two-thirds of the test problems, combining no elite selection with 2-tournament (T) selection when leads to poor outcomes. This implies that elite selection can compensate somewhat for the low selection pressure of T (), while lack of elite selection allows the low selection pressure to cause the GA to perform poorly. The importance of this interaction is low because SUS performed better than 2-tournament selection with any value for r.

The next three largest two-term interactions involve precision scaling (x7) combined with reboot proportion (x4), mutation rate (x6), and number of crossover points (x5). In general, each of these interactions arises on about half the test problems. The precise nature of these interactions is somewhat muddled, since they vary with problem (but not dimension) and also among specific combinations of levels. For example, rerandomizing frequently (every 10% of generations) combined with precision scaling of 1, and rerandomizing somewhat frequently (every 20% of generations) combined with precision scaling of 2 (twice as fine), seem to be combinations that can yield bad GA outcomes, depending on problem type. On the other hand, rerandomizing never or infrequently (every 40% of generations) combined with coarsening (1/2 as fine) or highly increasing (four times as fine) precision scaling seems to mute this interaction. Similarly, low or moderate mutation rates combined with increases in precision scaling yielded poor results on specific problems. On the other hand, coarsening the precision scaling combined with having less than three crossover points led to bad outcomes on specific problems. Using three crossover points appeared to compensate somewhat for coarsening precision scaling. We concluded that these interactions are not particularly important because (1) the nature of the interactions varied, (2) the interactions appeared on only about half the test problems, and (3) the interactions varied on specific problems rather than by problem dimension.

On about half the problems, two other two-term interactions arose for some combinations of reboot proportion (x4) with mutation rate (x6) or number of crossover points (x5). A combination of frequent rerandomization (after every 50 generations) and moderate mutation rate (0.0055) yielded bad outcomes on specific problems. Similarly, on selected problems, a combination of no rerandomization and one or two crossover points led to poor GA outcomes. For some problems, three crossover points compensated for lack of rerandomization. We concluded that these interactions are not particularly important, since the specific interactions varied, appeared on only about half the test problems, and varied on specific problems rather than by problem dimension.

The final large two-term interaction concerned number of crossover points (x5) and mutation rate (x6). We found that the combination of one crossover point with adaptive mutation performed poorly on 21 of the test problems. The importance of this interaction is low because three crossover points performed best on 77% of the test problems.

In summary, only seven parameter interactions showed somewhat large effects when averaged across the test problems. For the preceding reasons, we concluded that none of the interactions is particularly important and that the success of the GA was driven primarily by number of crossover points, mutation rate, and population size. Earlier work by Schaffer et al. (1989) found an interaction among crossover, mutation rate, and population size. While our experiments found an interaction between crossover and mutation rate, we found no significant interactions between population size and any other control parameter for the 60 problems we examined. Interactions between population size and crossover, or between population size and mutation rate, did appear in our experiments, but the size of those interactions was small.

Next, we consider the four rightmost columns (Level 1 to Level 4) in Table 2, which provide additional insights about specific GA control parameters. First, note that the GA performed much better when the population rerandomized after 200 generations (best level on 77% of the problems) or when there was no population rerandomization (best level on 40% of the problems) than when population rerandomization occurred every 50 or 100 generations. While showing that population rerandomization can improve GA performance, this result also indicates that rerandomizing a population too frequently destroys the ability of the GA to converge to good problem solutions. This raises the question of determining the best rerandomization frequency, which remains for investigation in a future expansion of this experiment, where reboot proportion might be set to levels 0, 0.4, 0.5, and 0.6. If such experiments reveal no additional insights, then perhaps the GA had already converged to good solutions somewhere between 100 and 200 generations, so that rerandomization points occurring at or beyond 200 generations would have no effect on GA performance. Investigating this question would require future experiments where reboot proportion would be set to levels 0, 0.25, 0.3, and 0.35.

Second, the highest percentage value (8%) we used for elite selection yielded substantially better GA performance than lower percentages (2% and 4%), which performed indistinguishably from no elite selection. While Grefenstette (1986) also found elitism to be useful, his study moved only a single elite individual to the next generation. Our results suggest few benefits accrue from moving only a single elite individual from one generation to the next. In fact, our results suggest that a substantially larger proportion of elite individuals is required for a GA to reap significant benefits. On the other hand, the degree of exploration may decrease as elite selection percentage increases, leading to diminishing returns and to less effective outcomes. This raises the question of determining the best elite selection percentage, which remains for future investigation, where elite selection percentage might be set to levels 0, 8%, 10%, and 12%.

Third, the data provide some evidence that coarsening parameter discretization can improve GA success rate. Specifically, a precision scaling of 1/2 (as fine as specified by the user) was the best level setting on 42% of the test problems. Apparently, for these problems, coarsening parameter discretization reduces the search space, which can aid the ability of a GA to converge to a higher fitness value. This result might, however, be due to the fact that the maximum or minimum values were integers for 36 of the test problems. Also, this finding might not hold for GA applications outside the domain of numerical optimization. Other coding questions could also be investigated. For example, additional experiments could be conducted to compare the effectiveness of simple binary encoding against other encoding methods, e.g., Gray encoding, permutation encoding, and real-value encoding; however, including additional encoding methods would require extending the GA, which currently supports only simply binary encoding.

Finally, while the study reported here encompasses 60 test problems, we caution readers not to overinterpret our results. Our findings hold only for the GA algorithm described in Section 2, and only for the range of levels investigated in our experiments, and only for the 60 numerical optimization problems on which we evaluated the GA. Within these bounds, we observe that our results suggest some general findings with respect to control parameters for a classic binary-encoded GA applied to numerical optimization.

We found crossover and mutation most influential in GA success. This suggests that GA control parameters associated with exploration have more influence on GA success than control parameters associated with exploitation. Further, we found that crossover is an essential element for successful GA performance, since disabling crossover exhibited poor GA success in our experiments. Three crossover points yielded much better GA success than fewer crossover points. We suspect that increasing the number of crossover points will yield diminishing returns at some point, because collections of bits representing key genetic material would be sliced up and lost. Determining the threshold for diminishing returns requires additional experimentation. More experiments are also needed to determine the influence of crossover probability on GA success, because our experiments used probabilities of only 0 and 1. In addition, further experiments could be conducted to compare the effectiveness of uniform crossover against slicing crossover; however, including uniform crossover would entail extending the GA, which currently supports only slicing crossover. Finally, as argued by Baeck (1996), Charbonneau (2002), and DeJong (2007), we found adaptive mutation, when compared to fixed mutation rates, led to substantially better GA performance.

Regarding population size, we observed that 200 individuals led to much better GA success than lower population sizes. This confirms the findings of Digalakis and Margaritis (2001), the researchers who investigated GA control settings over 14 different numerical optimization problems. As suggested by DeJong (2007), we suspect that increasing population size further will lead to diminishing returns, but confirming this (and establishing a threshold) requires additional experimentation.

We also observed that rerandomizing a population too frequently can destroy the ability of a GA to converge to good solutions. Our experiments found that rerandomizing after 40% of the intended number of generations led to better GA success than more frequent rerandomization, but more experiments are needed to determine the effects of higher and intermediate rerandomization thresholds.

Considering selection method, we observed that SUS led to much better GA success than a 2-tournament selection algorithm, regardless of the r value. Overall, we found selection method to be less important to GA success than number of crossover points, mutation rate, rerandomization points, and population size. These findings conflict somewhat with those of Rojas et al. (2002), who studied six test problems and three selection methods (roulette wheel, elitist roulette wheel, and deterministic) and reported selection method among the most important GA control parameters. We suspect that SUS is a robust selection method that provides reasonable utility in classic GAs, though further experiments comparing SUS to q-tournament selection () and other proposed selection methods might prove informative. Conducting such comparisons would require extending the GA, which currently provides only two selection methods: SUS and 2-tournament.

6  Conclusions and Future Work

We defined an experiment design and analysis method to determine the relative importance and most effective setting (among those studied) for each control parameter in a GA. The method is general enough to adapt and apply to determine the same information for a wide variety of search techniques. The method can also be used to determine effective starting parameter values for use in metasearch techniques, such as sequential parameter optimization. We demonstrated the method applied to a classic binary-encoded GA, evaluated against 60 numerical optimization problems, providing findings spanning four times as many problems as any previously reported study of GA control parameters. We confirmed some findings from previous studies in the literature and raised questions about others. We found that two exploration control parameters (i.e., crossover and mutation) most significantly influence GA success, followed by population size and population rerandomization points, while exploitation control parameters (i.e., elitism and selection method) showed less influence. We determined that adapting mutation rate based on population diversity provided better GA success than selecting a fixed mutation rate.

Based on our experiment, we identified the need for further study of various forms of crossover, including crossover probability and uniform crossover. We also outlined future experiments to investigate whether GA success would be improved by rerandomizing a population more frequently. We noted that future experiments might focus on determining the population size at which improvement in GA success begins to level off as well as on comparing additional selection and encoding methods. The experiment design and analysis method we defined can be used to conduct such follow-on studies.

Mills et al. (2013) applied the GA described in Section 2 to search for failure and performance degradation scenarios in a parallel population of cloud-computing simulators. That application used the findings reported in this paper to select settings for six GA control parameters: population size = 200, selection method = stochastic uniform sampling, elite selection percentage = 8%, reboot proportion = 0.4, number of crossover points = 3, and mutation rate = adaptive. (Precision scaling was set at 1, since that parameter had least influence on GA success.) While the cloud-computing search problem differed substantially from the numerical optimization problems used to calibrate the GA control parameters, the parameterized GA proved quite effective at finding failure and performance degradation scenarios in a cloud-computing simulator.

Acknowledgments

The authors thank the anonymous reviewers for many constructive comments and suggestions, which improved the substance and readability of the manuscript. Any residual errors remain the responsibility of the authors.

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Appendix A:  List of Numerical Optimization Problems

Tables 4–7 each list 15 of the 60 numerical optimization problems used in this experiment. For each problem, the tables show the numerical identifier (ID) we assigned, the function name and an abbreviation, the number of parameters (Dim) in the problem, the problem type, the theoretical maximum (or minimum) value where known, the best value found by the GA, and the source for the problem. The overwhelming majority (56) of the test set problems are continuous and differentiable; only the three Step functions and the Corana function are discontinuous and nondifferentiable. Using the scheme from Jamil and Yang (2013), the test problems are classified in the Type column using two bit positions. The first bit position denotes separability (separable = 1 and nonseparable = 0), and the second bit position denotes modality (unimodal = 1 and multimodal = 0). A mathematical description for many of these problems can be found in Adorio and Diliman (2005).

Table 4:
Test problems 1–15.
No. ofMax (Min)GA Best
IDFunctionDimensionsTypeValueValueSource
Chemical Yield (Cy) 11 Unknown 89.7505 Box et al. 2005  
Defective Springs (Ds) 11 100 99.9982 Box and Bisgard 1990  
Chemical Reactor (Cr) 11 100 100 Box et al. 2005  
Morris10 (Mo) 10 11 Unknown 112.9689 Saltelli et al%. 2004  
Morris20 (Mo) 20 11 Unknown 129.894 Saltelli et al%. 2004  
SchafferF6 (Sf) 01 (0) Adorio and Diliman 2005 
ShekelM10 (Sh) 00 (−10.536284) −10.53628 Adorio and Diliman 2005 
ShekelM5 (Sh) 00 (−10.1532) −10.1532 Adorio and Diliman 2005 
ShekelM7 (Sh) 00 (−10.403) −10.40282 Adorio and Diliman 2005 
10 Axis Parallel Hyper Ellipsoid (Ax) 15 11 (0) 9.4e-5 Rahnamayan et al. 2008  
11 Axis Parallel Hyper Ellipsoid (Ax) 30 11 (0) 1.21e-3 Rahnamayan et al. 2008  
12 Quartic without Noise (Qn) 100 11 (0) 1.841e-3 Rahnamayan et al. 2008  
13 Quartic with Normal Noise (Qn) 100 10 Unknown −3.842 Rahnamayan et al. 2008  
14 Quartic with Uniform Noise (Qn) 100 11 (0) 1.67e-1 Rahnamayan et al. 2008  
15 Sphere Model (Sp) 15 11 (0) 1.3e-5 Rahnamayan et al. 2008  
No. ofMax (Min)GA Best
IDFunctionDimensionsTypeValueValueSource
Chemical Yield (Cy) 11 Unknown 89.7505 Box et al. 2005  
Defective Springs (Ds) 11 100 99.9982 Box and Bisgard 1990  
Chemical Reactor (Cr) 11 100 100 Box et al. 2005  
Morris10 (Mo) 10 11 Unknown 112.9689 Saltelli et al%. 2004  
Morris20 (Mo) 20 11 Unknown 129.894 Saltelli et al%. 2004  
SchafferF6 (Sf) 01 (0) Adorio and Diliman 2005 
ShekelM10 (Sh) 00 (−10.536284) −10.53628 Adorio and Diliman 2005 
ShekelM5 (Sh) 00 (−10.1532) −10.1532 Adorio and Diliman 2005 
ShekelM7 (Sh) 00 (−10.403) −10.40282 Adorio and Diliman 2005 
10 Axis Parallel Hyper Ellipsoid (Ax) 15 11 (0) 9.4e-5 Rahnamayan et al. 2008  
11 Axis Parallel Hyper Ellipsoid (Ax) 30 11 (0) 1.21e-3 Rahnamayan et al. 2008  
12 Quartic without Noise (Qn) 100 11 (0) 1.841e-3 Rahnamayan et al. 2008  
13 Quartic with Normal Noise (Qn) 100 10 Unknown −3.842 Rahnamayan et al. 2008  
14 Quartic with Uniform Noise (Qn) 100 11 (0) 1.67e-1 Rahnamayan et al. 2008  
15 Sphere Model (Sp) 15 11 (0) 1.3e-5 Rahnamayan et al. 2008  
Table 5:
Test problems 16–30.
No. ofMax (Min)GA Best
IDFunctionDimensionsTypeValueValueSource
16 Sphere Model (Sp) 30 11 (0) 6.9e-5 Rahnamayan et al. 2008  
17 Sphere Model (Sp) 60 11 (0) 3.15e-3 Rahnamayan et al. 2008  
18 Beale (Be) 01 (0) Adorio and Dilliman 2005 
19 Bohachevsky (Bo) 00 (0) Adorio and Dilliman 2005 
20 Griewank (Gk) 00 (0) Adorio and Dilliman 2005 
21 Griewank (Gk) 10 00 (0) 1.769e-2 Adorio and Dilliman 2005 
22 Michalewitz (Mz) 15 00 (−14.49) −13.5745 Adorio and Dilliman 2005 
23 Michalewitz (Mz) 30 00 (−28.98) −27.428176 Adorio and Dilliman 2005 
24 Michalewitz (Mz) 60 00 (−57.96) −53.464618 Adorio and Dilliman 2005 
25 Plateau (Pl) 00 (5) Ingber 1993  
26 Step (St) 15 11 (0) Rahnamayan et al. 2008  
27 Step (St) 30 11 (0) Rahnamayan et al. 2008  
28 Step (St) 60 11 (0) Rahnamayan et al. 2008  
29 Paviani (Pa) 10 00 (−45.7784) −45.7784 Adorio and Dilliman 2005 
30 Levy (Le) 15 00 (0) −1.92e-1 Rahnamayan et al. 2008  
No. ofMax (Min)GA Best
IDFunctionDimensionsTypeValueValueSource
16 Sphere Model (Sp) 30 11 (0) 6.9e-5 Rahnamayan et al. 2008  
17 Sphere Model (Sp) 60 11 (0) 3.15e-3 Rahnamayan et al. 2008  
18 Beale (Be) 01 (0) Adorio and Dilliman 2005 
19 Bohachevsky (Bo) 00 (0) Adorio and Dilliman 2005 
20 Griewank (Gk) 00 (0) Adorio and Dilliman 2005 
21 Griewank (Gk) 10 00 (0) 1.769e-2 Adorio and Dilliman 2005 
22 Michalewitz (Mz) 15 00 (−14.49) −13.5745 Adorio and Dilliman 2005 
23 Michalewitz (Mz) 30 00 (−28.98) −27.428176 Adorio and Dilliman 2005 
24 Michalewitz (Mz) 60 00 (−57.96) −53.464618 Adorio and Dilliman 2005 
25 Plateau (Pl) 00 (5) Ingber 1993  
26 Step (St) 15 11 (0) Rahnamayan et al. 2008  
27 Step (St) 30 11 (0) Rahnamayan et al. 2008  
28 Step (St) 60 11 (0) Rahnamayan et al. 2008  
29 Paviani (Pa) 10 00 (−45.7784) −45.7784 Adorio and Dilliman 2005 
30 Levy (Le) 15 00 (0) −1.92e-1 Rahnamayan et al. 2008  
Table 6:
Test problems 31–45.
No. ofMax (Min)GA Best
IDFunctionDimensionsTypeValueValueSource
31 Levy (Le) 30 00 (0) 6.1e-1 Rahnamayan et al. 2008  
32 Levy (Le) 60 00 (0) 8.473 Rahnamayan et al. 2008  
33 Corana (Co) 10 (0) Adorio and Dilliman 2005 
34 Hosaki (Hk) 00 (−2.3458) −2.3458 Ali et al. 2005 
35 Gear (Gr) 00 (2.7e-12) 2.7e-12 Adorio and Dilliman 2005 
36 Multimod (Mm) 30 01 (0) Adorio and Dilliman 2005 
37 Easom (Ea) 10 (−1) −1 Adorio and Dilliman 2005 
38 Camel 3-hump (Cm) 00 (0) Ali et al. 2005 
39 Camel 6-hump (Cm) 00 (−1.0316285) −1.031628 Ali et al. 2005 
40 Hartman (Ha) 00 (−3.86) −3.86 Adorio and Dilliman 2005 
41 Hartman (Ha) 00 (−3.32) −3.32 Adorio and Dilliman 2005 
42 Schwefel (Sw) 10 (0) Adorio and Dilliman 2005 
43 Kowalik (Kw) 00 (0.00030748610) 5.58e-4 Adorio and Dilliman 2005 
44 Watson (Wa) 01 (≈2.28767e-3) 2.413e-3 Brent 2002  
45 Zettl (Zt) 00 (−0.003791) −0.03791 Adorio and Dilliman 2005 
No. ofMax (Min)GA Best
IDFunctionDimensionsTypeValueValueSource
31 Levy (Le) 30 00 (0) 6.1e-1 Rahnamayan et al. 2008  
32 Levy (Le) 60 00 (0) 8.473 Rahnamayan et al. 2008  
33 Corana (Co) 10 (0) Adorio and Dilliman 2005 
34 Hosaki (Hk) 00 (−2.3458) −2.3458 Ali et al. 2005 
35 Gear (Gr) 00 (2.7e-12) 2.7e-12 Adorio and Dilliman 2005 
36 Multimod (Mm) 30 01 (0) Adorio and Dilliman 2005 
37 Easom (Ea) 10 (−1) −1 Adorio and Dilliman 2005 
38 Camel 3-hump (Cm) 00 (0) Ali et al. 2005 
39 Camel 6-hump (Cm) 00 (−1.0316285) −1.031628 Ali et al. 2005 
40 Hartman (Ha) 00 (−3.86) −3.86 Adorio and Dilliman 2005 
41 Hartman (Ha) 00 (−3.32) −3.32 Adorio and Dilliman 2005 
42 Schwefel (Sw) 10 (0) Adorio and Dilliman 2005 
43 Kowalik (Kw) 00 (0.00030748610) 5.58e-4 Adorio and Dilliman 2005 
44 Watson (Wa) 01 (≈2.28767e-3) 2.413e-3 Brent 2002  
45 Zettl (Zt) 00 (−0.003791) −0.03791 Adorio and Dilliman 2005 
Table 7:
Test problems 46–60.
No. ofMax (Min)GA Best
IDFunctionDimensionsTypeValueValueSource
46 Ackley (Ak) 10 00 (0) 3.196e-3 Adorio and Diliman 2005 
47 Goldstein-Price (Gp) 00 (3) Adorio and Diliman 2005 
48 Rosenbrock (Rk) 15 01 (0) 9.568 Ali et al. 2005  
49 Powell (Pw) 01 (0) 3.82e-28 Ali et al. 2005  
50 Shubert Problem (Sb) 10 (−186.7309) −186.7309 Ali et al. 2005  
51 Periodic (Pe) 10 (0.9) 0.9 Ali et al. 2005  
52 Salomon Problem (Sm) 10 00 (0) 0.099873 Ali et al. 2005  
53 Sinusoidal Problem (Sn) 10 10 (−3.5) −3.499856 Ali et al. 2005  
54 Dekkers and Aarts (Dk) 10 (−24777) −24,776.518 Ali et al. 2005  
55 McCormick (Mk) 00 (−1.9133) −1.913223 Ali et al. 2005  
56 Colville (Cv) 00 (0) Rahnamayan et al. 2008  
57 Zakharov (Zh) 15 00 (0) 0.0748 Rahnamayan et al. 2008  
58 Branin's (Br) 00 (0.3979) 0.397888 Rahnamayan et al. 2008  
59 Perm (Pm) 10 10 (0) 1.2e-5 Rahnamayan et al. 2008  
60 Neumaier3 (Nu) 10 00 (−210) −209.71224 Ali et al. 2005  
No. ofMax (Min)GA Best
IDFunctionDimensionsTypeValueValueSource
46 Ackley (Ak) 10 00 (0) 3.196e-3 Adorio and Diliman 2005 
47 Goldstein-Price (Gp) 00 (3) Adorio and Diliman 2005 
48 Rosenbrock (Rk) 15 01 (0) 9.568 Ali et al. 2005  
49 Powell (Pw) 01 (0) 3.82e-28 Ali et al. 2005  
50 Shubert Problem (Sb) 10 (−186.7309) −186.7309 Ali et al. 2005  
51 Periodic (Pe) 10 (0.9) 0.9 Ali et al. 2005  
52 Salomon Problem (Sm) 10 00 (0) 0.099873 Ali et al. 2005  
53 Sinusoidal Problem (Sn) 10 10 (−3.5) −3.499856 Ali et al. 2005  
54 Dekkers and Aarts (Dk) 10 (−24777) −24,776.518 Ali et al. 2005  
55 McCormick (Mk) 00 (−1.9133) −1.913223 Ali et al. 2005  
56 Colville (Cv) 00 (0) Rahnamayan et al. 2008  
57 Zakharov (Zh) 15 00 (0) 0.0748 Rahnamayan et al. 2008  
58 Branin's (Br) 00 (0.3979) 0.397888 Rahnamayan et al. 2008  
59 Perm (Pm) 10 10 (0) 1.2e-5 Rahnamayan et al. 2008  
60 Neumaier3 (Nu) 10 00 (−210) −209.71224 Ali et al. 2005  

Appendix B:  Results from a OFF Experiment versus Results from an FF Experiment for Three Test Problems

Earlier in this paper, we stated that an orthogonal fractional factorial experiment design samples a subset of the parameter combinations from a full factorial experiment. We also stated that OFF designs give estimates for main effects, where estimate accuracy is influenced by the number of samples (n), the number of levels (), and the variance () in the underlying observations. Main effects estimates directly determine the relative rank among factors as well as the best level for each factor; thus the accuracy of such estimates influences findings and conclusions from OFF experiments. Further, uncertainty in main effects estimates generated by OFF experiments also influences the computation of statistical significance using ANOVA. This latter point stands to reason because ANOVA compares variation within groups of data points versus variation among groups. Sampling methods typically exhibit increased variance over measurements taken over an entire population. This variance increase directly influences the computation of ANOVA statistics. This appendix provides some data from our experiments regarding the effects of uncertainty arising when applying a OFF experiment to a sample from a FF experiment.

We selected three of the 60 test problems on which to compare results from the OFF design versus results from an FF experiment. We chose a problem from each of the complexity categories among our numerical optimization problems: (1) problem 3, Chemical Reactor (Cr) with five dimensions; (2) problem 11, Axis Parallel Hyper Ellipsoid (Ax) with 30 dimensions; and (3) problem 24, Michalewitz (Mz) with 60 dimensions. For each factor of each problem, we compared the estimated effect (Ei) including the derived factor ranking and most effective level and the ANOVA statistic (). Our comparison demonstrates that (1) OFF experiment designs do indeed produce estimates; (2) the main effects estimates are fairly accurate, as are the derived factor rankings and best levels; and (3) the estimates are somewhat less accurate, because for statistically significant factors in an FF experiment the measures converge toward 100 owing to decreased variance among the data points used to compute main effects. This decrease in variance leads to increased statistical significance, even for small differences in main effects.

Table 8 compares results for main effects estimates (Ei), , factor rank, and best level for the Chemical Reactor problem when using a OFF design and an FF experiment. Tables 9 and 10 provide similar results for two other problems: Axis Parallel Hyper Ellipsoid and Michalewitz.

Table 8:
Problem 3: Chemical Reactor (Cr) with five dimensions. Comparison of main effects (), statistical significance (), rank, and best level from a OFF experiment versus a FF experiment.
   Rank Best Level 
Factor OFF FF OFF FF OFF FF OFF FF 
 0.04 0.03 99.99 100 
 0.01 0.01 56.29 100 
 0.01 0.02 53.68 100 
 0.06 0.01 100 99.86 
 0.07 0.05 100 100 
 0.07 0.06 100 100 
 0.05 0.03 100 100 
   Rank Best Level 
Factor OFF FF OFF FF OFF FF OFF FF 
 0.04 0.03 99.99 100 
 0.01 0.01 56.29 100 
 0.01 0.02 53.68 100 
 0.06 0.01 100 99.86 
 0.07 0.05 100 100 
 0.07 0.06 100 100 
 0.05 0.03 100 100 
Table 9:
Problem 11: Axis Parallel Hyper Ellipsoid (Ax) with 30 dimensions. Comparison of main effects (), statistical significance (), rank, and best level from a OFF experiment versus a FF experiment.
   Rank Best Level 
Factor OFF FF OFF FF OFF FF OFF FF 
 34.30 37.11 77.91 100 
 152.26 147.25 100 100 
 315.59 319.37 100 100 
 82.49 90.05 100 100 
 62.12 65.17 99.78 100 
 90.77 69.50 100 100 
 20.48 2.59 26.56 5.11 
   Rank Best Level 
Factor OFF FF OFF FF OFF FF OFF FF 
 34.30 37.11 77.91 100 
 152.26 147.25 100 100 
 315.59 319.37 100 100 
 82.49 90.05 100 100 
 62.12 65.17 99.78 100 
 90.77 69.50 100 100 
 20.48 2.59 26.56 5.11 
Table 10:
Problem 24: Michalewitz (Mz) with 60 dimensions. Comparison of main effects (), statistical significance (), rank, and best level from a OFF experiment versus a FF experiment.
   Rank Best Level 
Factor OFF FF OFF FF OFF FF OFF FF 
 5.19 5.18 100 100 
 7.41 7.23 100 100 
 14.39 14.46 100 100 
 1.54 2.70 79.97 100 
 5.24 5.35 100 100 
 9.98 9.83 100 100 
 0.93 1.10 39.58 100 
   Rank Best Level 
Factor OFF FF OFF FF OFF FF OFF FF 
 5.19 5.18 100 100 
 7.41 7.23 100 100 
 14.39 14.46 100 100 
 1.54 2.70 79.97 100 
 5.24 5.35 100 100 
 9.98 9.83 100 100 
 0.93 1.10 39.58 100 

Table 8 shows that most main effects estimates from the OFF experiment are fairly close to the FF results for the Chemical Reactor problem. An exception is the estimate for factor x4 (reboot proportion), where the OFF design indicates a substantially larger effect than the FF experiment. This difference also leads to some differences in factor rankings, which are derived from relative differences in main effects. The top two factors (x5 and x6) remain the same, but the difference in main effects estimate for x4 leads to reordering in the ranking of three factors. The OFF and FF experiments produced the same best level setting for all but one factor, x2 (selection method). Finally, in the FF experiment, the values are at or near 100% for all factors, while factors x2 and x3 are nearer 50% in the OFF experiment. We explore these differences more thoroughly later in this appendix.

Table 9 shows that most main effects estimates from the OFF experiment are close to the FF results for the Axis Parallel Hyper Ellipsoid problem. Results from the OFF experiment report somewhat higher estimated effects for two factors: mutation rate (x6) and precision scaling (x7). Small differences in main effects estimates lead to factors x4 and x6 swapping ranks 3 and 4. The OFF and FF experiments produced the same best level setting for all but one factor, population size (x1), where the OFF experiment reported the size of 150 to be best, while the FF experiment found 200 to be best. Most of the values were near 100%, except for x1 and x7. In the case of x1, the variance in the data from the OFF design was sufficient to lower . In the case of x7, the estimated effect for the FF experiment was sufficiently low that the rightly reflects a lack of statistical significance.

Table 10 shows that most main effects estimates from the OFF experiment are quite close to the FF results for the Michalewitz problem. In fact, the OFF and FF experiments produced identical rankings and best levels for each of the seven factors. As is typically the case, the FF experiment reported values at 100%, while the variance in the data acquired from the OFF experiment design led to lower values for two factors, x4 and x7.

Readers should bear in mind that the preceding comparisons consider only three of the 60 test problems. The overall analysis averages data from all 60 test problems. Such averaging tends to allow minor estimate errors in individual problems to offset each other (unless there is some hidden bias in the procedure). This is another justification for ensuring that EAs are evaluated on a sufficiently large sample of search problems.

To better understand the reasons underlying differences in main effects estimates in OFF and FF experiments, we delve more deeply into the Chemical Reactor problem. Recall that uncertainty in main effects estimates is a function of the number of samples (n), the number of levels (), and the variance () in the underlying observations. For our experiments, n and are fixed, which yields an uncertainty estimate of 0.088 . This implies that uncertainty in main effects estimates is driven by , which is unknown for our experiment. Though is unknown, the confidence intervals around the main effects estimates for our experiments may give some indication of the relative nature of .

Figure 7 shows a factor analysis plot from the OFF experiment for the Chemical Reactor problem. The plot contains the same type of information we described when discussing Figure 1, but here we added vertical bars through the estimated mean for each level () of each factor (xi). The vertical bars indicate the 95% confidence interval, which implies that there is a small chance that the true mean falls outside the interval. The larger the interval, the greater the uncertainty in the data used to make the estimate. Figure 8 shows a factor analysis plot (including 95% confidence intervals) from the FF experiment for the Chemical Reactor problem.

Figure 7:

Factor analysis plot from OFF experiment for numerical optimization problem 3: maximizing the output of a chemical reactor process. The 95% confidence intervals are given by vertical bars through each estimated mean for each level () of each factor (xi).

Figure 7:

Factor analysis plot from OFF experiment for numerical optimization problem 3: maximizing the output of a chemical reactor process. The 95% confidence intervals are given by vertical bars through each estimated mean for each level () of each factor (xi).

Figure 8:

Factor analysis plot from FF experiment for numerical optimization problem 3: maximizing the output of a chemical reactor process. The 95% confidence intervals are given by vertical bars through each estimated mean for each level () of each factor (xi).

Figure 8:

Factor analysis plot from FF experiment for numerical optimization problem 3: maximizing the output of a chemical reactor process. The 95% confidence intervals are given by vertical bars through each estimated mean for each level () of each factor (xi).

Recall Table 8, which shows that the main effects estimates produced by the OFF experiment varied somewhat from the results of the FF experiment. For example, the estimated effect found by the OFF experiment for factor x4 was substantially larger than the effect shown by the FF results. The large variance associated with the OFF data for that factor indicates that the main effects estimate would tend to be more uncertain. Comparing Figures 7 and 8, and their associated confidence intervals, reveals why the main effects estimates from the OFF experiment differed from the FF results. For example, the mean value for x4 level 3 was much lower when estimated from the data sampled in the OFF experiment than it was when computed from the FF data. Further, mean values from the FF results exhibited much less uncertainty.

The uncertainty in main effects estimates produced by an OFF design cannot be eliminated simply by iterating the OFF design while varying the random number seed so as to generate independent samples that can be averaged. To demonstrate this, we iterated the OFF experiment 16 times for the Chemical Reactor problem, producing 16,384 datasets, which is equivalent in number to the 16,384 datasets produced by the FF experiment. Subsequently, for each factor, we averaged the main effects estimates, values, and ranks across the 16 OFF experiment iterations and took the mode of the best levels from the 16 iterations. We then compared these results against the FF results (see Table 11).

Table 11:
Problem 3: Chemical Reactor (Cr) with five dimensions. Comparison of main effects (), statistical significance (), and rank averaged across 16 iterations of a OFF experiment versus a FF experiment. The best level for each factor of the OFF experiment was determined by taking the mode across 16 iterations.
   Rank Best Level 
 Average  Average  Average  Average  
Factor 16 OFFs FF 16 OFFs FF 16 OFFs FF 16 OFFs FF 
 0.05 0.03 100 100 
 0.02 0.01 80.92 100 
 0.02 0.02 88.83 100 
 0.04 0.01 99.95 99.86 
 0.04 0.05 99.31 100 
 0.07 0.06 100 100 
 0.05 0.03 99.98 100 
   Rank Best Level 
 Average  Average  Average  Average  
Factor 16 OFFs FF 16 OFFs FF 16 OFFs FF 16 OFFs FF 
 0.05 0.03 100 100 
 0.02 0.01 80.92 100 
 0.02 0.02 88.83 100 
 0.04 0.01 99.95 99.86 
 0.04 0.05 99.31 100 
 0.07 0.06 100 100 
 0.05 0.03 99.98 100 

As Table 11 shows, the main effects estimates and ranks for each factor taken from the averaged results of the OFF experiment differ somewhat from the FF results. Similarly, though there is largely agreement in the best levels for each factor, the best level for factor x4 differs. This difference is of no consequence because, as discussed earlier, levels 1 and 4 for factor x4 yield comparable effectiveness across the 60 test problems.

Figure 9 provides an overview of changes in factor rankings across the 16 iterations of the OFF design for the Chemical Reactor problem. Mutation rate (x6) is consistently top ranked on all iterations, while selection method (x2) and elite selection percentage (x3) are consistently among the lowest ranked. The ranks of other factors fluctuate across the iterations, as dictated by relative changes in main effects estimates. This example provides some insight into how results from OFF experiments can lead to differences in factor ranking when compared with FF results. Further, the example suggests that OFF designs provide confident estimates of the most and least influential factors for any specific problem while introducing uncertainty about the true ordering of factors exhibiting intermediate influence. Analyzing factor rankings across a large set of problems allows uncertainty on individual problems to be averaged, leading to reasonably confident factor rankings.

Figure 9:

Main effects estimates (y-axis) for seven control parameters on 16 iterations (x-axis) of problem 3, Chemical Reactor (Cr) with five dimensions. For each iteration, the parameter identifiers (1 for x1 to 7 for x7) are sorted from largest (top) to smallest (bottom) main effect (Ei).

Figure 9:

Main effects estimates (y-axis) for seven control parameters on 16 iterations (x-axis) of problem 3, Chemical Reactor (Cr) with five dimensions. For each iteration, the parameter identifiers (1 for x1 to 7 for x7) are sorted from largest (top) to smallest (bottom) main effect (Ei).

The main conclusions from this limited comparison of results from a OFF design against FF results are (1) OFF experiment designs do indeed produce estimates, (2) the main effects estimates are fairly accurate, as are the derived factor rankings and best levels, and (3) the estimates are somewhat less accurate, because in an FF experiment measures converge toward 100 owing to decreased variance among the data points used to compute main effects. We also demonstrated that OFF designs produce confident rankings of the most and least influential factors for individual test problems. Confidence in the rankings of factors with intermediate influence relies on averaging results across a large collection of varied test problems.