5. Results

• Initial population: the initial population consists of individuals generated randomly in

• Crossover: new solutions are produced by matching pairs of individuals in the population and then applying a crossover operator to each chosen pair. An unmatched individual ik is matched randomly with an unmatched individual il. Thereafter, the two-point crossover operator is applied using a crossover probability to each matched pair of individuals. The two-point crossover draws two random points within a chromosome and then interchanges the two parent chromosomes between these points to produce two new offspring. The work presented in [19] shows that the results produced by the two-

• Mutation: let C ¼ c1, c2, …, cm be a chromosome represented by a binary chain where each of whose gene ci is either 0 or 1. Each gene ci is mutated through flipping this gene's allele from 0 to 1 or from 1 to 0 if the probability test is passed. The mutation probability guarantees that, theoretically, every part of the region of the search space is explored. The mutation operator adds diversity to the population while increasing the likelihood of

• Selection: based on each individual quality, the roulette method is used to determine the next population. The selection is stochastic and biased toward the best individuals. The first step is to calculate the cumulative fitness of the whole population through the sum of the fitness of all individuals. After that, the probability of selection is calculated for each

• Local search: the last part of the algorithm is the use of a local search. A fast and simple heuristic is applied for each offspring during which it seeks for the new variable-value assignment which best decreases the number of unsatisfied clauses being identified.

We evaluated the performance of the multilevel evolutionary algorithm (MLVMA) against its single variant (MA) using a set of instances taken from SATLIB. (http://www.informatik.tudarmstadt.de/AI/SATLIB). Table 1 shows the instances used in the experiment. IBM SPSS Statistics version 19 was used for statistical analysis. Due to the randomization nature of the algorithms, each problem instance was run 100 times with a cutoff parameter (max time) set to 15 min. The 100 runs were adopted because pilot runs had shown the size of the difference to be so large that 100 runs were enough for an acceptable statistical power (power > :95); this is in accordance with the suggestions given in a recent report on statistical testing of randomized algorithms [20].

The tests were carried out on a DELL machine with 800 MHz CPU and 2 GB of memory. The code was written in C and compiled with the GNU C compiler version 4.6. The list of parameters

which each gene's allele is assigned randomly the value 0 (false) or 1 (true).

point crossover are excellent especially when the problem is hard to solve.

generating individuals with better fitness values.

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= P<sup>N</sup> <sup>1</sup> f <sup>i</sup> .

individual as being PSelectioni ¼ f <sup>i</sup>

4. Experimental results

used in the experiments are as follows:

4.1. Benchmark instances

#### 5.1. Observed trends

The time development of the multilevel evolutionary algorithm against its single variant in solving the instances is shown in Figures 2–8. The plots show the 100 runs of both algorithms with a cutoff at 15 min as well as the mean of these runs. The search occurs in two phases. In the first phase, the best solution improves rapidly at first, and then flattens off as the search reaches the plateau region, marking the start of the second phase. The plateau region corresponds to a region in the search space where moves does not alter the best assignment, and

Figure 2. MLVMA versus MA: (left) 2bitadd10:cnf, (right) 2bitadd11:cnf—time development for 100 runs in 15 min.

Figure 5. MLVMA versus MA: (left) 3bitadd32:cnf, (right) 3block:cnf—time development for 100 runs in 15 min.

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Figure 6. MLVMA versus MA: (left) 4blocks:cnf, (right) 4blocksb:cnf—time development for 100 runs in 15 min.

Figure 7. MLVMA versus MA: (left) e0ddr2-10-by-5-1.cnf, (right) e0ddr2-10-by-5-4.cnf—time development for 100 runs

in 15 min.

Figure 3. MLVMA versus MA: (left) 2bitadd12:cnf, (right) 2bitcomp5:cnf—time development for 100 runs in 15 min.

Figure 4. MLVMA versus MA: (left) 2bitmax6:cnf, (right) 3bitadd31:cnf—time development for 100 runs in 15 min.

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Figure 5. MLVMA versus MA: (left) 3bitadd32:cnf, (right) 3block:cnf—time development for 100 runs in 15 min.

Figure 2. MLVMA versus MA: (left) 2bitadd10:cnf, (right) 2bitadd11:cnf—time development for 100 runs in 15 min.

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Figure 3. MLVMA versus MA: (left) 2bitadd12:cnf, (right) 2bitcomp5:cnf—time development for 100 runs in 15 min.

Figure 4. MLVMA versus MA: (left) 2bitmax6:cnf, (right) 3bitadd31:cnf—time development for 100 runs in 15 min.

Figure 6. MLVMA versus MA: (left) 4blocks:cnf, (right) 4blocksb:cnf—time development for 100 runs in 15 min.

Figure 7. MLVMA versus MA: (left) e0ddr2-10-by-5-1.cnf, (right) e0ddr2-10-by-5-4.cnf—time development for 100 runs in 15 min.

while intensifying the search by exploiting the solutions from previous levels in order to reach

Tables 2 and 3 summarize the results. M and SD represent the mean standard deviation of unsolved clauses for the MLVMA and MA algorithms. The range of solutions from each algorithm is also shown in order to analyze the overlap between solution spaces for any given instance. Statistical inferential analysis was done with an independent samples t-test which compares the difference in means between the two groups. Comparison using the non-parametric Mann-Whitney U-test gave identical results. The non-parametric effect size measure Ab<sup>12</sup> [21] was used to evaluate the relative dominance of one algorithm over the other. The Ab <sup>12</sup> effect size measure is calculated using the rank sum which is a common component in any non-parametric analysis such as the Mann-Whitney U-test [20]. Calculating Ab <sup>12</sup> is done according to the following

where R<sup>1</sup> is the rank sum of algorithm MLVMA, m is the number of observations in the first

2bitadd10:cnf 2.0 (.7) [1–3] 16.3 (2.3) [11–25] 2bitadd11:cnf 1.7 (.7) [1–4] 16.3 (3.2) [8–24] 2bitadd12:cnf 1.5 (.7) [1–3] 1.6 (.7) [1–4] 2bitcomp5:cnf 1.0 (0) [1–2] 1.0 (0.1) [1–2] 2bitmax6:cnf 1.0 (.2) [1–2] 1.0 (0.1) [1–2] 2bitadd31:cnf 132.6 (10.9) [122–216] 1106.2 (142.1) [923–2620] 3bitadd32:cnf 135.7 (11.9) [123–186] 1366.9 (179.1) [1125–1974] 3blocks 4.0 (1.8) [2–9] 7.2 (1.0) [4–9] 3blocks 8.2 (3.1) [2–14] 13.0 (1.0) [11–18] 4blocksb 5.2 (1.8) [2–8] 7.3 (0.7) [5–8]

e0ddr2-10-by-5-1 343.4 (119.0) [261–697] 10871.1 (324.5) [9895–11,527] e0ddr2-10-by-5-4 320.6 (80.8) [271–718] 10969.1 (360.1) [10,190–11,784] enddr2-10-by-5-1 371.9 (144.0) [281–1021] 12042.9 (378.1) [111,64–12,897] enddr2-10-by-5-8 358.9 (136.1) [278–967] 12241.3 (400.0) [11,169–13,446] ewddr2-10-by-5-1 399.8 (166.9) [289–1124] 12939.7 (407.9) [11,960–13,835] ewddr2-10-by-5-8 354 (107.0) [293–710] 13537.5 (423.8) [12,393–14,736]

M (SD) Range M (SD) Range

#Case MLVMA MA

Ab<sup>12</sup> ¼ ð Þ R1=m � ð Þ m þ 1 =2 =n: (1)

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better solutions.

formula:

5.2. Statistical analysis

Table 2. Statistical comparisons.

Figure 8. MLVMA versus MA: (left) enddr2-10-by-5-1.cnf, (right) enddr2-10-by-5-8.cnf—time development for 100 runs in 15 min.

occurs more specifically once the refinement reaches the finest level. The plots show that MLVMA offers a better asymptotic convergence compared to MA especially for large instances. The test cases where both algorithms reach approximately the same solution quality (with MLVMA being marginally better), the multilevel paradigm offers a cost-effective solution strategy considering the amount of time required (Figure 9).

This multilevel paradigm has two main advantages which enables the evolutionary algorithm to become much efficient. The coarsening process offers a better mechanism for performing diversification (i.e., searching different parts of the search space) and intensification (i.e., reaching better solutions within those regions). The coarsening allows the gene of each individual to represent a cluster of variables, leading the search to become guided and restricted to only those solutions in the solution space in which the variables grouped within a cluster are assigned the same value. As the size of the clusters varies from one level to another, the crossover and mutation operators are able to explore different regions in the search space

Figure 9. MLVMA versus MA: (left) ewddr2-10-by-5-1.cnf, (right) ewddr2-10-by-5-8.cnf—time development for 100 runs in 15 min.

while intensifying the search by exploiting the solutions from previous levels in order to reach better solutions.

## 5.2. Statistical analysis

occurs more specifically once the refinement reaches the finest level. The plots show that MLVMA offers a better asymptotic convergence compared to MA especially for large instances. The test cases where both algorithms reach approximately the same solution quality (with MLVMA being marginally better), the multilevel paradigm offers a cost-effective solu-

Figure 8. MLVMA versus MA: (left) enddr2-10-by-5-1.cnf, (right) enddr2-10-by-5-8.cnf—time development for 100 runs

This multilevel paradigm has two main advantages which enables the evolutionary algorithm to become much efficient. The coarsening process offers a better mechanism for performing diversification (i.e., searching different parts of the search space) and intensification (i.e., reaching better solutions within those regions). The coarsening allows the gene of each individual to represent a cluster of variables, leading the search to become guided and restricted to only those solutions in the solution space in which the variables grouped within a cluster are assigned the same value. As the size of the clusters varies from one level to another, the crossover and mutation operators are able to explore different regions in the search space

Figure 9. MLVMA versus MA: (left) ewddr2-10-by-5-1.cnf, (right) ewddr2-10-by-5-8.cnf—time development for 100 runs

tion strategy considering the amount of time required (Figure 9).

212 Machine Learning - Advanced Techniques and Emerging Applications

in 15 min.

in 15 min.

Tables 2 and 3 summarize the results. M and SD represent the mean standard deviation of unsolved clauses for the MLVMA and MA algorithms. The range of solutions from each algorithm is also shown in order to analyze the overlap between solution spaces for any given instance. Statistical inferential analysis was done with an independent samples t-test which compares the difference in means between the two groups. Comparison using the non-parametric Mann-Whitney U-test gave identical results. The non-parametric effect size measure Ab<sup>12</sup> [21] was used to evaluate the relative dominance of one algorithm over the other. The Ab <sup>12</sup> effect size measure is calculated using the rank sum which is a common component in any non-parametric analysis such as the Mann-Whitney U-test [20]. Calculating Ab <sup>12</sup> is done according to the following formula:

$$
\widehat{A}\_{12} = \left(\mathbb{R}\_1/m - (m+1)/2\right)/n. \tag{1}
$$


where R<sup>1</sup> is the rank sum of algorithm MLVMA, m is the number of observations in the first

Table 2. Statistical comparisons.


1. Random resampling with replacement from the original observations to create new data sets.

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3. Using the rank sum to calculate Ab<sup>12</sup> with Eq. (1). The three steps are then repeated 1000 times and the resulting statistic Ab<sup>12</sup> is saved to create a sampling distribution of the

The results show how MLVMA outperforms MA in 10 out of the 16 instances. MLVMA dominates MA in three instances (the 3blocks, 4blocks, and 4blocksb-instances, Ab12, is :918, :916, and :847, respectively). For the remaining three problems (2bitadd10, 2bitadd11, and 2bitadd12), there is no statistically identifiable difference between the two algorithms. However, when inspecting the time series for these instances it is clear that MLVMA reaches a solution much faster than MA. To test possible causes for the difference in solution quality, the relationship between the number of clauses and the quality of solutions provided by the two algorithms was analyzed. The relationship between the mean percentage of unsolved clauses and the number of clauses in each instance was estimated using a linear regression. The relationship between the mean percentage of unsolved clauses and the number of clauses for the MLVMA was much lower (t(15) = 3.059, = 2.041–8, 95% CI [1.163–8, 2.714–8], p = .008, r = .633) than for the MA (t(15) = 10.067, = 9.341–7, 95% CI [8.232–7, 1.04–6], p < .001, r = .937) indicating that the hierarchical paradigm is less affected by the size of the problem than the

In this chapter, a multilevel evolutionary algorithm for solving the maximum satisfiability problem is presented. During the coarsening phase, a sequence of smaller problems, each with fewer variables, is constructed. Each child level is constructed from its parent level by collapsing pairs of variables. The new formed variables are used to define a new and smaller problem and recursively iterate the coarsening process until the size of the problem reaches some desired threshold. An evolutionary algorithm is applied through several optimization levels, where the converged population at a child level will serve as the starting population for a parent level. A set of instances were used to compare the performance of the new approach. The results obtained assert the superiority of the evolutionary algorithm when combined with the multilevel paradigm and always return a better solution for the equivalent run-time compared to MA.

Noureddine Bouhmala\*, Kjell Ivar Øvergård and Karina Hjelmervik

Department of Maritime Technology and Innovation, SouthEast University, Norway

\*Address all correspondence to: noureddine.bouhmaa@usn.no

2. Calculation of the rank sum of MLVMA for each new data set.

statistic hat c 12.

6. Conclusion

Author details

standard single-level evolutionary algorithm.

Table 3. Comparing effect sizes.

data sample, and n is the number of observations in the second data sample. Calculating Ab <sup>12</sup> results in a number between 0 and 1 which represent the probability that MLVMA will yield a better solution than MA. If the two algorithms are equivalent, then Ab<sup>12</sup> ¼ :5, while a complete dominance of algorithm MLVMA over MA would entail Ab<sup>12</sup> ¼ 1.

Ab<sup>12</sup> is more easily interpreted than the more common parametric Cohen's d [22] which represents the mean difference between two groups in standard deviations for several reasons. First, Cohen's d assumes that the observed samples are normally distributed [20]. Second, when dealing with solutions to optimization problems, a researcher or a practitioner would only be interested in the single best solution given a sample of different solutions from one or more algorithms. Hence, using an effect size measure that indicates the probability that one algorithm would lead to a better solution than another (given the same amount of time) would be more informative and more easily interpretable for an optimization practitioner. The 95% confidence intervals of Ab <sup>12</sup> shown in Table 3 (where applicable) are calculated using a bootstrapping procedure [23] which is used to estimate the 95% confidence interval of Ab12. The procedure uses a computer-intensive step-by-step process that consists of the following three steps:


The results show how MLVMA outperforms MA in 10 out of the 16 instances. MLVMA dominates MA in three instances (the 3blocks, 4blocks, and 4blocksb-instances, Ab12, is :918, :916, and :847, respectively). For the remaining three problems (2bitadd10, 2bitadd11, and 2bitadd12), there is no statistically identifiable difference between the two algorithms. However, when inspecting the time series for these instances it is clear that MLVMA reaches a solution much faster than MA. To test possible causes for the difference in solution quality, the relationship between the number of clauses and the quality of solutions provided by the two algorithms was analyzed. The relationship between the mean percentage of unsolved clauses and the number of clauses in each instance was estimated using a linear regression. The relationship between the mean percentage of unsolved clauses and the number of clauses for the MLVMA was much lower (t(15) = 3.059, = 2.041–8, 95% CI [1.163–8, 2.714–8], p = .008, r = .633) than for the MA (t(15) = 10.067, = 9.341–7, 95% CI [8.232–7, 1.04–6], p < .001, r = .937) indicating that the hierarchical paradigm is less affected by the size of the problem than the standard single-level evolutionary algorithm.
