In general, an algorithm is expected to find the optimal solution. The performance of an algorithm can be assessed according to its global search capability: whether it finds the optimal solution and how often it finds the optimal solution in different trials. However, the optimal solutions are unknown in the majority of real problems. In practice, there is typically not enough time to find the optimal solution; thereby an approximate optimal solution is usually the target. Therefore, it is necessary to find a highly efficient algorithm.
Also in this work, the optimal solutions of the tested scenarios are unknown; thereby the global search capability of the proposed algorithms cannot be assessed according to the probability of the algorithms finding the optimum. This research aims to find the most suit-able coding strategy in terms of result quality and computational efficiency for different prob-lems. For this purpose, different coding strategies were developed. The performance of the algorithm will be assessed according to the following criteria:
• Solution quality: It refers to how good the solution of the hybrid genetic algorithm is. Since the objective is to minimize the completion timeJ, a solution with less com-pletion time is better than that with larger comcom-pletion time. To statistically evaluate the performance of the algorithm,Nrun independent runs of each algorithm are carried out.
The performance of the algorithm is assessed basing on:
– The obtained best solution ofNrun trials (minimal completion timeJmin), – The encountered worst solution ofNrun trials (maximal completion timeJmax), – The average completion timeJmean of all solutions inNrun trials,
– The standard deviationJsdof all solutions inNrun trials, – And the distribution of the completion time inNrun trials.
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Figure 5.1:Interpretation of a box plot
The distribution of the completion time obtained by the hybrid genetic algorithm is drawn as a box plot that identifies median, 25th percentile (p1), 75th percentile (p3), adjacent value, and outliers; see Figure 5.1. The value of the 75th percentile minus the 25th percentile is the inter quartile range (p3−p1), which is the height of each box.
50%of data falls inside this range. The upper adjacent value is the largest data value within the value ofp3+w(p3−p1). The lower adjacent value is the smallest data value
within the value ofp1−w(p3−p1). The default ofw= 1.5corresponds to approximately 99.3% coverage if the data are normally distributed. Outliers are extreme values that fall outside of the range of upper and lower adjacent values.
• CPU time: It refers to the average time that is needed to execute the hybrid genetic algorithm forNrun trials on a computer. In this work, the population size and the num-ber of generations are constant, i.e., the numnum-ber of the total individuals produced by the genetic algorithm is fixed. It is more efficient if an algorithm is able to provide good solutions in a shorter CPU time. Different from the time complexity of the algorithm in theory, the CPU time may vary on different computers in practice. All experimental tests are performed on a standard Windows-PC where MATLAB is a foreground appli-cation and there are not any other programs in the foreground; in addition, there is no scanning process for viruses and program update in both foreground and background processing.
• Search progress: It refers to the progress of the best solution candidates in different generations of the genetic algorithm, as local search is applied after the genetic al-gorithm terminates. The search progress of the genetic alal-gorithm is assessed in two ways:
– Evaluating the generationGFG where the final solution appears the first time. The
“final solution” refers to the best solution obtained by the genetic algorithm before performing local search. A small value ofGFG implies that fast progress can be expected and the number of generations for termination can be reduced.
– Comparison of the best solution candidates in different generations and of the shape of the distribution of the completion times inNrun trials.
To some extend, the efficiency of the genetic algorithm can be evaluated according to the search progress. The results of search progress are considered to determine the suitable number of generations for the genetic algorithm with different coding strate-gies. The convergence properties within Gmax can also be indicated by plotting the progress of the best solution candidates in different generations.
• Local search improvement: It assesses how much the solution is improved by the local search. The improvement of local search is defined as
Jimp = (JG−JHG)/JG·100 %, (5.1) where,JG is the completion time of the final solution obtained by the genetic algorithm without local search, and JHG is the completion time of the final solution obtained by the hybrid genetic algorithm with local search.
In this work, two strategies of local search (LS) will be compared: generation-elite LS where the local search is applied to the best individuals of each generation (set B);
and last-population LS where the local search is applied to all individuals in the last population. The completion time of the final solution obtained by the hybrid genetic algorithm using generation-elite LS is denoted as Jgels, and the completion time of the final solution obtained by the hybrid genetic algorithm using last-population LS is denoted as Jlpls. Compared to these two strategies, the performance gain of the generation-elite LS over the last-population LS is defined as
∆J = (Jlpls−Jgels)/Jlpls ·100 %. (5.2) If ∆J > 0, the generation-elite LS performs better than the last-population LS, other-wise the last-population LS outperforms the generation-elite LS.
Analysis of variance (ANOVA) is used to check whether the performance differences (solu-tion quality or local search improvement) of the hybrid genetic algorithm with different coding strategies are statistically significant. A one-way ANOVA test is carried out to analyze the influences of different coding strategies on the performance of the proposed hybrid genetic algorithm. The ANOVA results are considered to determine the most suitable coding strate-gies and the importance of local search for solving the investigated inspection problems.
To evaluate the difference between all developed coding strategies, the ANOVA table lists the sum of squares (SS), the degrees of freedom (DF), the mean squares (MS), the ratio of the mean squares –F statistic (F), and the significance level (Sig.Level) in each investigated scenario. If the value of Sig.Level is smaller than 0.05, the effect of the coding strategy is assessed to be statistically significant at the level of confidence of95%. Note that the value 0.05is a typical threshold to assess the significant levels, but other values are possible.
To evaluate the differences between any pair of coding strategies, the ANOVA table shows the value of Sig.Level of each pair of coding strategies. If the value of Sig.Level is smaller than0.05, the difference between these two coding strategies is statistically significant.
The effect of local search for each coding strategy is also illustrated using the ANOVA anal-ysis. For each coding strategy, the results of the genetic algorithm with and without local search are compared and analyzed by ANOVA. If the value of Sig.Level is smaller than0.05, the difference between the genetic algorithm with and without local search is statistically significant.