Open Access Published by De Gruyter August 23, 2023

Application of adaptive improved DE algorithm based on multi-angle search rotation crossover strategy in multi-circuit testing optimization

Wenchang Wu

From the journal Journal of Intelligent Systems

https://doi.org/10.1515/jisys-2022-0269

Abstract

This study based on the standard differential evolution (DE) algorithm was carried out to address the issues of control parameter imprinting, mutation process, and crossover process in the standard DE algorithm as well as the issue of multidimensional circuit testing optimization. A rotation control vector was introduced to expand the search range in the poor strategy to the circumference range of the individual and the parent target individual, and a rotation crossover operator and a binomial poor operator were combined. Finally, an improved adaptive DE algorithm based on a multi-angle search rotation crossover strategy was obtained. The research will improve the DE algorithm to optimize the testing of multidimensional circuits. It can be noted that the improved average precision value is 0.9919 when comparing the precision recall curves of the DE algorithm before and after the change, demonstrating a significant improvement in accuracy and stability. The fitness difference of the 30-dimensional problem is discovered to be between 0.25 × 10³ and 0.5 × 10³ by comparing the box graphs of the 30-dimensional problem with that of the 50-dimensional problem. On the 50-dimensional problem, when calculating the F4–F10 function, the fitness difference of the improved DE algorithm is 0.2 × 10⁴–0.4 × 10⁴. In summary, the improved DE algorithm proposed in this study compensates for the shortcomings of traditional algorithms in complex problem calculations and has also achieved significant optimization results in multidimensional circuit testing.

Keywords: differential evolution algorithm; circuit test; global optimization; parameter adaptation

1 Introduction

The importance of using optimization techniques in numerous industries has grown, and multidimensional circuit research is currently humming along. This technology is extremely important in many different fields. The complexity of integrated circuits has constantly expanded as a result of technological advancements, and numerous issues with the manufacturing and operating processes have also come to light. For example, the time for algorithm-solving problems, whether it is easy to implement, and convergence speed are all criteria for measuring the advantages and disadvantages of algorithms in solving multidimensional circuit problems [1]. Therefore, in the testing of multidimensional integrated circuits, it is crucial to ensure the safe and reliable operation of the circuit, which results in stricter requirements for efficient and stable detection methods in relevant fields [2]. The structural performance of integrated circuits is currently the subject of increased research, whereas multidimensional circuit testing is the subject of very less research due to technological constraints. In terms of current multidimensional circuit testing, traditional testing methods have exposed many shortcomings, such as being unable to meet the difficulty of testing and difficulty in implementing a scheduling model for testing [3]. It is required to provide more potent algorithms to control the corresponding circuit testing process in order to address the testing issues associated with multidimensional integrated circuits. Since its debut at the beginning of the twenty-first century, the differential evolution (DE) algorithm has been extensively used in the field of intelligent optimization [4]. This algorithm was born on the basis of Darwin’s theory of “survival of the fittest, survival of the fittest.” It is a robust global optimization algorithm, and it has the advantages of simplicity, efficiency, stability, etc. [5]. Due to its benefits, the DE algorithm is frequently used in multidimensional circuit testing, but there are some drawbacks to the standard DE algorithm, including a lack of local search capability, sensitivity to the values of mutation control parameters and crossover probability factors, a lack of fixed rules for parameter selection, and the existence of multiple compilation crossover strategies, which makes it extremely challenging to select the right mutation strategies. In response to the aforementioned issues, research is conducted to improve the mutation and crossover operators of the algorithm, and an adaptive adjustment strategy is introduced to solve the problem of difficult control parameter selection. Finally, an adaptive improved DE algorithm based on multi-angle search rotation crossover strategy is designed to better optimize multidimensional circuit testing methods. The innovation points of the research are as follows: on the one hand, an adaptive improved DE algorithm based on multi-angle search rotation crossover strategy was designed to solve various problems existing in the standard DE algorithm, and on the other hand, the improved DE algorithm was applied to the optimization of multidimensional circuit testing, providing reference for future testing techniques. The rest structure of this study is mainly divided into four parts. The analysis of domestic and foreign research on DE algorithms and their optimization comes first. The second step is to create a multi-angle search rotation crossover strategy-based adaptive enhanced DE algorithm. The third step entails using the enhanced algorithm to enhance multidimensional circuit testing. The final step is to reach pertinent conclusions.

2 Related works

Through domestic and foreign research, it is found that the DE algorithm has been widely used in all walks of life. Mahua Rakshit introduced an optimized DE algorithm to reduce the computational complexity of the carrier. The experimental results showed that the introduction of this method significantly improved the computational efficiency, and compared with other algorithms, the method had better peak to average power ratio suppression and bit error rate performance, while also greatly reducing the complexity [6]. Swetha et al. proposed a metaheuristic natural heuristic hybrid algorithm of the opposing ocean predator algorithm (OMPA) and Harris Hawkes optimization algorithm (HHO). This algorithm is jointly implemented as OMPA–HHO on the standard IEEE 57 bus system. Simulation results verify the scalability, repeatability, and robustness of the proposed algorithm. This algorithm can be considered as a superior algorithm in complex optimization problems [7]. When the load frequency of a wind power-integrated power system is easily perturbed, Pan and Lu proposed to develop a load frequency control model with time delay and uncertainty by applying DE algorithm. The simulation trials demonstrate the system’s stability, and the modified genetic algorithm also demonstrates the method’s effectiveness and logic [8]. Shaikh et al. applied an improved moth flame optimization technology to different test case of transmission lines in order to correctly manage the transmission network. The comparative experimental results show that the proposed method can converge to the optimal value quickly and smoothly, and its solution quality and feasibility are better than those of other algorithms, proving its effectiveness and ability [9]. The workshop scheduling problem (job scheduling on the plant) with fuzzy execution time and fuzzy completion time was studied by Wang et al. They also suggested a new selection method to improve the conventional DE approach. Examples were used to demonstrate the improved DE algorithm’s effectiveness, and a comparison of the proposed algorithm to other algorithms revealed that it can produce more better workable solutions [10].

When investigating the watermarking issue, Ladan Salimi’s experimental group suggested a new watermarking technique based on the DE algorithm and wavelet transform approach. This method achieves greater peak signal to noise ratio in the reconstruction stage when compared to the watermarking technique before improvement, demonstrating its superiority in the majority of situations [11]. When Pan et al. studied the DE algorithm, they found that the algorithm is sensitive to the control parameters associated with it, so they proposed an adaptive update mechanism to control this possible problem. The optimized DE algorithm also has better performance than algorithms such as JDE/IWPSO (a multi-target tracking algorithm and a Particle swarm optimization algorithm with inertia factor) [12]. Wang et al. proposed a fuzzy relative entropy method to achieve a more comprehensive modeling of job shop scheduling problems with fuzzy processing time and completion time. This method can evaluate the quality of feasible solutions after comparing actual and ideal values. The experimental results show that the performance of this method is superior to that of some other latest algorithms [13]. When evaluating multidimensional circuits, Nagamani et al. proposed using genetic algorithms to model and assess it. The simulation results demonstrate that the algorithm may be successfully applied to multidimensional circuit testing and that circuit testing efficiency can be somewhat increased [14]. Jameil et al. proposed a new Design Under Test algorithm to improve and optimize the low-power, high-speed circuit test architecture. The results showed that the addition of this algorithm reduced latency by about 60% and power consumption by nearly 76%, truly achieving high speed and low power consumption [15].

Through previous studies, it has been found that DE algorithms have been used in various fields, and they have shown extremely strong performance. And they are also widely used in circuit testing, which greatly improves the speed of testing and reduces the power consumption of the circuit. Among them, references [12,14] and [15] all have the same problem, that is, the setting of several parameters in the proposed DE algorithm is very sensitive, and the values lack rigorous prior theory to refer to. Although significant progress has been made in fields related to optimization, significant advancement in the field of artificial intelligence requires not only the imitation of physical changes and biological functions and laws but also the investigation of novel forms of intelligence through the integration of artificial brain functions. The mutation operator and crossover operator were specifically improved, and adaptive strategies to self-adjust based on historical data were used to address the issues of insufficient local search ability, sensitive parameter settings, and challenging operator selection in the standard DE algorithm. Finally, an adaptive improved DE algorithm based on multi-angle search rotation crossover strategy was constructed.

3 Construction of an adaptive improved DE algorithm based on multi-angle search rotation crossover strategy

3.1 Establishment of standard difference evolution algorithm

DE is a population-based evolutionary algorithm. At the same time, as a global optimization algorithm, this algorithm has the advantages of simplicity, high efficiency, and strong stability [16]. When searching for information, this algorithm mainly relies on the heuristic search of the group, and each individual in the group can be mapped to a set of solution vectors. Moreover, the DE algorithm has a lot in common with the genetic algorithm when it is running. Both algorithms involve operations such as mutation, crossover, and selection, although the precise actions will vary owing to various settings and are less complex than genetic algorithms.

The DE algorithm mainly includes four steps: initialization, differential mutation, crossover, and selection. Each individual in the population is initialized, and individuals in the population were then randomly selected, differencing and scaling any two of them. The individual used in the experiment is obtained by cross-operating the parent individual and the individual produced by the mutation operation after it has been randomly joined with another member of the population [17]. The individuals are then arranged from small to large according to their fitness values, and the individual with the highest fitness value is chosen to enter the next iteration until the termination condition is satisfied, at which point the iteration loop is jumped out of. This process is performed by comparing the fitness values of the parent individuals and the experimental individuals.

In the process of individual initialization, the encoding method selects the real number encoding form. First generate a random population, and the population size is ( NP , D ) , where NP represents the size of the population and the D number of variables involved in decision-making. The first individual in the generated population is shown in the following formula:

(1) x i = { x i 1 , x i 2 … x i j … x i D } , i ∈ ( 1 , NP ) , j ∈ ( 1 , D ) ,

where x i j represents i the value of the first dimension of the x i j ∈ ( L j , U j ) first individual, j . L j and U j , respectively, represent the upper and lower bounds of the variables involved in the decision and are uniformly initialized in them.

(2) x i j = rand j ( 0 , 1 ) ⋅ ( U j − L j ) + L j , i ∈ ( 1 , NP ) , j ∈ ( 1 , D ) ,

where rand j ( 0 , 1 ) is a function that can randomly generate a number in the range of (0,1). After the initialization is completed, the mutation operation is performed on the individual, and usually, the mutation operation DE / a / b in the expression of the DE algorithm is adopted. Among them, DE represents the DE algorithm, in which a represents a way of selecting mutant individuals. There are two commonly used methods rand [18]. b represents the number of individuals performing mutation operations in the algorithm, as shown in the following formulas:

(3) V → i , G = X → r 1 , G + F × ( X → r 2 , G − X → r 3 , G ) ,

(4) V → i , G = X → best , G + F × ( X → r 2 , G − X → r 3 , G ) ,

where the individual V i G → representing the mutation operation F is the mutation factor. r 1 , r 2 , r 3 is any integer between (1, NP) and is not equal to each other. X → best G represents the optimal individual in the mutation operation, and the schematic diagram of the mutation operation is shown in Figure 1.

Figure 1

Schematic diagram of mutation operation.

In Figure 1, when performing the mutation operation, it is necessary to judge the new individuals generated by the mutation operation to determine whether the individual is in the search space. In the space, the next step can be performed. If it is not in the search space, the generated mutants need to be further processed with boundary conditions [19].

When performing crossover operations on mutant individuals, the commonly used crossover operations in DE algorithm are binomial crossover and exponential crossover. The binomial crossover is shown in the following formula:

(5) u i G j = v i G j , rand ( 0 , 1 ) ≤ C R or j = j rand x i G j , otherwise ,

where the value u i G j represents the first dimension of the CR , and j represents the crossover probability.

When performing the crossover operation on the experimental individuals, first randomly select an integer in the range of (1, D) n as the initial position of the crossover operation. Then, Q is taken L ∈ ( 1 , D ) , L ≠ n as the number of mutant individuals occupying the position of the target individual, and the experimental individuals generated by the crossover operation are shown in the following formula:

(6) u i G j = v i G j , j ≤ n , n > D , < n + 1 > D , … , < n + L − 1 > D x i G j , otherwise ,

where the pair D modulo is expressed.

Finally, when the individual is selected, the experimental individual and the target individual are compared first, and whether to enter the next generation is judged by comparing the fitness value between the two individuals. The experimental individual will replace the corresponding target individual and enter the next generation if the fitness value of the target individual is greater than or equal to the fitness value of the experimental individual. The target individual enters the next generation if the experimental person’s fitness value is higher. The selection operation of the minimization problem is shown in the following formula:

(7) x i G j = u i G j , f u i G ≤ f x i G x i G j , otherwise ,

where f x i G represents the fitness value of the target individual, and the f u i G represents the fitness value of the experimental individual. The operating flow chart of the DE algorithm is shown in Figure 2.

Figure 2

Flow chart of the DE algorithm.

As shown in Figure 2, when the DE algorithm runs, the data is initialized first. The result is judged first, and if the termination condition is satisfied, the optimal individual termination program is output. If it is not satisfied, jump to mutation, crossover, and operation, and then perform boundary processing on it. The data processed by the boundary conditions are then used to generate the objective function, and the selection procedure is then carried out. If the conditions are satisfied, determine whether they are, if not, skip to the mutation crossover operation and enter another loop, if they are, output the ideal individual, and the program is finished.

Although the DE algorithm is very mature, it still has several major flaws. One is that the algorithm is easy to fall into the local optimal solution because the function to be solved is complex during operation. The second is that when solving high-dimensional functions, its convergence speed is slow and it is easy to be premature [20]. Due to the existence of these defects, optimization and improvement are still required.

3.2 Design of an adaptive improved DE algorithm based on multi-angle search rotation crossover strategy

This research will improve the control parameters, operator, population structure, and hybrid algorithm. The control parameters in the DE algorithm mainly include crossover probability, scaling factor, and population size. The changes of these parameters will have a great impact on the operation of the algorithm. Therefore, in order to improve the algorithm, it is necessary to start with these control parameters.

Due to the nature of population size, relatively few improvements have been made to it. Most of the improvements are based on the adaptive degree of scale, and the calculation formula of population size reduction is shown in the following formulas:

(8) x i = x NP 2 + i , f x NP 2 + i < f ( x i ) ∩ G = G R x i , otherwise ,

(9) NP ( G + 1 ) = NP ( G ) 2 , G = G R NP ( G ) , otherwise .

Formulas (8) and (9) show that the majority of the population’s individuals remain stable, with only a small number of persons declining. In order to prevent future population evolution and increase population variety, the population must be kept in a stable state. In addition, it was found that the search status and a distribution of the population distribution also have a large impact on the size of the population, so the algorithm can also be improved in this regard. The study found that the algorithm has a dynamic-state monitoring system that can be used to track evolving individuals in real time.

In previous studies, most of the improved parameter adaptation improvements of the algorithms are carried out for the two parameters of scaling factor F and crossover probability CR .

As shown in Figure 3, μ , δ is the Cauchy distribution curves when they take different values. First, the new scaling factor F i and the new crossover probability are obtained by means of the Cauchy distribution CR i , as shown in the following formula:

(10) F i = rand c i ( μ F , 0.1 ) CR i = rand n i ( μ CR , 0.1 ) ,

where rand c i represents the random number in the Cauchy distribution, and r and n i represents the random number in the normal distribution. Among them, μ F and μ CR update method is shown in the following formulas:

(11) μ F G + 1 = ( 1 − c ) × μ F G + c × mean L ( S F ) μ CR G + 1 = ( 1 − c ) × μ CR G + c × mean A ( S CR ) ,

(12) mean L ( S F ) = ∑ F ∈ S F F 2 ∑ F ∈ S F F ,

where c is a real number randomly generated between (0, 1) and S CR , S F is the high-quality solution generated by the previous generation of individuals after processing, and mean A is a function used to calculate the arithmetic mean. A weighted analysis is necessary for each individual in order to guarantee that the control parameters of each variable stay independent throughout evolution [21]. The weight calculation is shown in the following formula:

(13) w i G = ∣ f ( u i G + 1 ) − f max ∣ ∑ i = 1 NP ∣ f ( u i G + 1 ) − f max ∣ .

Figure 3

Schematic diagram of Cauchy distribution.

As the weighting formula in formula (13), the improved formula of each individual’s control parameters is shown in the following formula:

(14) F w j G + 1 = ∑ i = 1 NP w i G × F i j G F i j G + 1 = N ( F w j G + 1 , δ 1 ) F w j G + 1 = ∑ i = 1 NP w i G × F i j G F i j G + 1 = N ( F w j G + 1 , δ 1 ) ,

where N means that it obeys the normal distribution: δ 1 = δ 2 = 0.8 − ( 1 − ( G / G max ) 2 ) 2 . In the DE algorithm, there are mainly three operators: mutation, crossover, and selection. This research will improve the mutation operator. In order to avoid the phenomenon that the difference vector is prone to premature maturity, a new mutation operator will be proposed, as shown in the following formula:

(15) V i G = X i G + F i ⋅ ( X p best , G − X i G ) + F i ⋅ ( X r 1 i G − X r 2 i G ) ,

where X p best , G means that the individuals in the population are randomly selected, and p% outstanding individuals are selected, where p ∈ ( 0 , 1 ) . For the adaptive schemes in these mutation operations, many schemes have strong detection ability. And it is beneficial to the development of the algorithm. To enhance the DE algorithm’s functionality and adapt to various environmental factors and evolutionary processes, various mutation schemes will be used. The speed at which the algorithm converges and how well it handles complex function issues can both be greatly improved by the enhancement of these approaches.

Some scholars have proposed that the two crossover operators, binomial crossover and exponential crossover, can only generate a super-rectangular fixed point, which is also the problem that leads to the limitation of DE algorithm search [22,23]. In the process of improving the crossover operator, orthogonal design can be added to the crossover operator. Orthogonal design can greatly increase a capacity inside the algorithm, making it better able to handle different factors. The crossover operator can also be improved and optimized by the eigenvector method, the principle of which is to project each donor vector onto the eigenvector. Relying on the eigenvectors of the covariance matrix of a single solution enables the principle of cross-rotation invariance. The crossover behavior in this improved method is rotation invariant and does not vary as the crossover operator in the eigenvector changes. At the same time, every little modification in operation can be successfully implemented using the improved operator of this method.

Crossover procedures frequently fail to consider the interaction between variables, according to prior research. Additionally, there will be a lot of search blind spots because of the algorithm’s restrictions. In response to this problem, this study proposes a hybrid linkage scheme to replace the traditional scheme. When extracting information, the hybrid linkage method uses the perturbation method to automatically extract specific problems. The proposal of this scheme greatly improves the performance of the algorithm [24]. The flow chart of the improved DE algorithm is shown in Figure 4.

Figure 4

Flow chart of the improved DE algorithm.

As shown in Figure 4, in this study, the population size and operation operators in the DE algorithm were optimized and improved. Moreover, in previous studies, a single DE algorithm always has more or less problems. Therefore, hybrid algorithms have appeared in a large number of studies to improve the performance of the entire algorithm. The specific steps of the improved DE algorithm shown in Figure 4 are as follows: first, initialization processing is carried out, and then N − 1 floating-point coordinate information is set according to the threshold. After individual evaluation, the interaction coefficient of each floater is calculated, and compilation, mutation, and crossover operations are carried out. Second, the experimental carrier is determined using random traces and mutation probability. Subsequently, a selection operation is performed. It is decided whether the difference in the interaction is bigger than the threshold after computing the cross-coefficient of the experimental vector and choosing the best value. If so, it is possible to figure out the float’s coordinates. Otherwise, the individual evaluation step is returned and the cycle is repeated. Based on the aforementioned methods, the following assumptions can be proposed: the improved DE algorithm proposed in the study has higher accuracy than the current mainstream algorithms.

4 Result analysis of the improved DE algorithm in multi-dimensional circuit test optimization

The threshold in the experiment must be predetermined, and it is typically set at 0.5. This threshold separates the positive and negative situations of the projected value. The relevant sample is now categorized as a positive example when the expected value is greater than or equal to 0.5. In a similar manner, the comparable sample is labeled as a negative example when the projected value is less than 0.5. Through the constant change of the threshold value, the PRecision and Recall values also change continuously, and the precision recall (PR) curve is formed according to different test results. The PR curves of the test results of the improved DE algorithm and the traditional DE algorithm are shown in Figure 5.

Figure 5

PR curves of the two algorithms: (a) improved DE algorithm PR curve and (b) traditional DE algorithm PR curve.

As shown in Figure 5, Figure 5(a) is the PR curve of the improved DE algorithm, and Figure 5(b) is the PR curve of the traditional DE algorithm. It can be seen from Figure 5 that the improved recommendation algorithm is superior to the traditional recommendation algorithm in terms of accuracy and detection speed. The upgraded DE algorithm’s average precision (AP) value is 0.9919, which is much higher than the recommended algorithm’s (0.9237) AP value before the change. Comparatively, there is a better balance between recall rates and superior performance. In this experiment, the CEC2017 standard test set is used to verify the performance of the improved algorithm. The functions used for verification can be divided into four functions: unimodal function, multimodal function, hybridization function, and hybrid function. To more scientifically verify the advantages of the improved DE algorithm proposed in the study in terms of convergence speed and optimization performance, the study uses the variants of five existing DE algorithms to compare with them. The six algorithms are as follows: Java Agent Development Framework (JADE), Based on Adaptive Sine Differential Evolution Algorithm (SinDE), Differential Evolution Algorithm Based on Two-Stage Strategy (TSDE), Adaptive Guided Differential Evolution (AGDE), and Adaptive Explicit Time Delay Estimation (EFADE) [25,26]. Compared with DE algorithm, the improvement of JADE algorithm is to use the mutation strategy of current-to-lead, archive population and non-archive, and adaptive parameter adjustment of scale factors and crossover probabilities, which can directly call the mixing of the original signal and is fast [27]. The AGDE algorithm has great exploration and balancing search capabilities, and its outcomes in locating the ideal solution are also successful [28]. The benefits of the EFADE algorithm are strong universality, excellent reliability, and quick calculation times. For a comparative examination of boxplots, the modified DE method and the aforementioned five techniques are used [29]. Figure 6 is a boxplot of the improved DE algorithm when processing a 30-dimensional function.

Figure 6

Performance analysis of 30-dimensional function boxplot: (a) boxplots of 30-dimensional functions F12–F15 and (b) boxplots of 30-dimensional functions F21–F25.

As shown in Figure 6, Figure 6(a) shows the boxplots of the six algorithms when calculating the F12–F15 function in a 30-dimensional problem; Figure 6(b) is a 30-dimensional problem, and the six algorithms are calculating the F21-boxplots for the F25 function. In the 30-dimensional problem, the fitness difference of the improved DE algorithm is significantly lower than that of the other five algorithms. When calculating the F12–F15 function, the fitness difference is in the region of 0.25 × 10³ to 0.5 × 10³. When calculating the F21–F25 function, the fitness difference is in the region of 0.7 × 10³ to 1 × 10³. This demonstrates that when calculating multimodal functions, the modified DE method outperforms the other five algorithms in terms of accuracy and anti-interference capability. And when calculating the hybrid function, the other five algorithms’ calculation accuracy is still much inferior to that of the enhanced DE algorithm. In comparison, the computing performance of the AGDE and SinDE algorithms is second. At the same time, the improved DE algorithm also has a strong effect when calculating the mixing function. Among the ten mixed functions calculated, the improved DE algorithm is only weaker than the AGDE algorithm in the calculation of two functions. And especially in the F30 function calculation, the calculation effect has an order of magnitude improvement compared with other algorithms. The analysis shows the powerful performance of the improved DE algorithm in solving complex problems. Figure 7 shows the comparative analysis of the boxplots of the six algorithms in the 50-dimensional problem.

Figure 7

Performance analysis of the 50-dimensional function boxplot: (a) boxplots of 50-dimensional functions F4–F10 and (b) boxplots of 50-dimensional functions F23–F26.

As shown in Figure 7, Figure 7(a) shows the boxplots of the six algorithms when calculating the F4–F10 function in the 50-dimensional problem. Figure 7(b) is the 50-dimensional problem, and the six algorithms are calculating the F23-boxplots for the F26 function. The multimodal function, the mixing function, and the hybridization function are all calculated using the improved DE algorithm, according to a recent research of the 30-dimensional problem. It has been demonstrated that this algorithm is better at optimizing difficult low-dimensional function problems. As can be seen from Figure 7, in the calculation of the F4–F10 function, the fitness difference of the improved DE algorithm is between 0.2 × 10⁴ and 0.4 × 10⁴; in the calculation of the F23–F26 function, the adaptation of the improved DE algorithm. The degree difference is between 2.0 × 10² and 3.9 × 10², which is the lowest fitness difference compared to other algorithms. It demonstrates that the modified DE method is still the best among the six algorithms when dealing with larger-dimensional situations. The complexity of the problem and the difficulty of the calculation both rise as the problem’s dimension increases. However, while the dimension increases, the computing power of the improved DE algorithm has not been weakened, and it still shows a strong optimization ability. It also demonstrates the remarkable durability of this approach and the effective and high-dimensional optimization effect because it climbs rather than falls when compared to other algorithms.

By using the improved DE algorithm together with JADE, SinDE, TSDE, AGDE, and EFADE in the analysis of ED problems, the effect of new DE variants on ED problems is analyzed. In this experiment, 100 populations were set, and the evolutionary generations were set to 500 and 1,000 generations, respectively. Twenty simulation experiments are carried out independently for each algorithm, and the simulation results are shown in Table 1.

Table 1

Simulation results of variants

Algorithm	Standard deviation	Median number	Maximum value	Minimum
Improved DE	137.11	120706.73	120977.34	120554.48
EFADE	747.36	126413.54	126589.17	125413.69
TSDE	180.12	120577.46	121478.36	120164.96
AGDE	349.19	122169.44	122574.38	121579.91
SinDE	901.33	122739.62	124351.22	122280.16
JADE	637.88	125344.65	125587.12	124465.36

As shown in Table 1, from the variant simulation results, the standard deviation of the improved DE algorithm is 137.11. Except for the standard deviation of the TSDE algorithm, which is 180.12, the standard deviations of the remaining four algorithms are all above 300. This also reflects that the improved DE algorithm has better performance and stronger stability. In order to further analyze the overall performance of the six algorithms, the study used F1 values for testing, and the results are shown in Figure 8.

Figure 8

F1 value results of six algorithms.

As illustrated in Figure 8, the study’s enhanced DE algorithm consistently achieves F1 values above 0.945 over the course of 300 iterations, with an average F1 value of 0.979. This shows that the enhanced DE algorithm has the best overall performance throughout the entire procedure. The average F1 values of other algorithms are SinDE, TSDE, AGDE, JADE, and EFADE, respectively, from high to low. Continue to analyze these six algorithms, and set the population size to 100 and the population evolution generation to 1,000. The convergence result of its consumption function is shown in Figure 9.

Figure 9

Consumption function convergence results.

As shown in Figure 9, the graph of the total consumption convergence function of the six algorithms when the population evolves for 1,000 generations is obtained. It can be seen from the figure that the total consumption of the improved DE algorithm reaches the lowest value of 1.21 × 10⁵ when it evolves to 400 generations, which is the lowest among the six algorithms. Among them, the total consumption of the EFADE algorithm reaches the lowest value of 1.275 × 10⁵ when it evolves to 300 generations, which is the highest among the six algorithms. The modified DE method has the highest convergence accuracy and the fastest convergence speed when the convergence functions of the six algorithms are compared. As a result, the analysis is completed, and Figure 10 shows the boxplot of the system running for 1,000 generations and the time complexity histograms of the six algorithms.

Figure 10

Time complexity histogram and system boxplot: (a) time complexity histogram and (b) boxplots of 40 sets of systems from 30 independent runs under 1,000 generations.

As shown in Figure 10, Figure 10(a) is the time complexity histogram of the six algorithms, and Figure 10(b) is the test boxplot of the six algorithms independently solving the ED problem. The time complexity diagram in Figure 10(a) shows that the enhanced DE method has a time complexity of 39 s when it runs for 1,000 generations and a time complexity of 28 s when it runs for 500 generations. Among them, the complexity of the TSDE algorithm is the highest when running for 1,000 generations. Comparing the six algorithms, it is found that the improved DE algorithm has the lowest time complexity. The boxplot of Figure 10(b) shows that the improved DE algorithm’s total consumption is approximately 1.208 × 10⁵, while the budget method’s total consumption is higher than 1.21 × 10⁵. And the EFADE algorithm has the largest overall consumption of the six methods, at roughly 1.259 × 10⁵. This demonstrates that the enhanced DE method has the best processing efficiency for resolving the ED problem.

5 Conclusion

To solve the optimization problem of standard DE algorithm and circuit testing in integrated circuits, an adaptive improved DE algorithm based on multi-angle search rotation crossover strategy was constructed in this study. The research results showed that compared with the PR curves before and after improvement, the AP value of the improved DE algorithm was 0.9919, which was significantly higher than the 0.9237 of the preimprovement DE algorithm. When calculating 30-dimensional problems and F12–F15 functions, the fitness difference is between 0.25 × 10³ and 0.5 × 10³. When calculating the F21–F25 function, the difference of fitness is between 0.7 × 10³ and 1 × 10³. The enhanced DE algorithm’s fitness difference in the computation of the 50-dimensional issue and the F4–F10 function is between 0.2 × 10⁴ and 0.4 × 10⁴. The enhanced DE method is still the best of the six when dealing with higher-dimensional issues, as shown by the fitness difference of the algorithm while calculating the F23–F26 function, which ranges between 2.0 × 10² and 3.9 × 10². Through the time complexity histogram, it was found that the time complexity of the improved DE algorithm was 39 s when running 1,000 generations, and 28 s when running 500 generations, which is significantly lower than the other five algorithms. In conclusion, the study’s revised DE algorithm improves circuit testing accuracy significantly while using less energy by addressing the computational weaknesses of conventional methods in complicated issues. The research does still have some flaws, though. The multi-angle search rotation crossover strategy-based adaptive enhanced DE algorithm that has been suggested should not be restricted to multidimensional line testing optimization. This approach can be used to solve a variety of optimization issues in both continuous and discrete domains in future study.

Author contributions: Wenchang Wu: writing—original draft, writing—review and editing, methodology, formal analysis.
Conflict of interest: No conflict of interest exits in this manuscript.
Data availability statement: The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

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Received: 2022-11-21

Revised: 2023-05-06

Accepted: 2023-06-12

Published Online: 2023-08-23

This work is licensed under the Creative Commons Attribution 4.0 International License.