Fog computing plays an important role in industrial and information process. The programs in fog computing such as iFogSim toolkit usually contain some infeasible paths (paths cannot be covered), which makes it impossible to compare algorithm in models that require covering all paths. In this paper, we proposed a mathematical model to build automated test case generation based on path coverage (ATCG-PC) in fog computing programs as a single-objective problem. Single-objective helps to reduce the cost of evaluation functions which is proportional to the number of test cases. When infeasible paths are contained in tested programs, algorithms can also be compared in this model. In this paper, classical differential evolution (DE) is used to solve the ATCG-PC. However, it is difficult for DE to use generated test cases covering remaining paths in the ATCG-PC of fog computing. Therefore, we proposed a test-case-path relationship matrix to empower differential evolution (RP-DE). Experiment results show that RP-DE uses significantly less test cases and achieves higher path coverage rate than compared state-of-the-art algorithms.

Multimodal multiobjective optimization problems, characterized by multiple solutions mapping to identical objective vectors, are ubiquitous in real-world applications. Despite their prevalence, most existing multimodal multiobjective evolutionary algorithms (MMOEAs) predominantly focus on identifying global Pareto sets, often overlooking the equally significant local Pareto sets. While some algorithms attempt to address local Pareto sets, their performance in the objective space remains suboptimal. The inherent challenge lies in the fact that a single strategy cannot effectively tackle problems with and without local Pareto fronts. This study proposes a novel approach that first detects the presence of local Pareto fronts using a neural network, thereby enabling adaptive adjustments to the algorithm’s selection strategy and search scope. Based on this detection mechanism, we design a microscale searching multimodal multiobjective evolutionary algorithm (MMOEA_MS). Through extensive experiments on twenty-two benchmark problems, MMOEA_MS demonstrates superior performance in identifying local Pareto fronts and outperforms existing algorithms in the objective space. This study highlights the effectiveness of MMOEA_MS in solving multimodal multiobjective optimization problems with diverse Pareto front characteristics, thereby addressing key limitations of current methodologies.

The brain storm optimization (BSO) algorithm has received increased attention in the field of evolutionary computation. While BSO has been applied in numerous industrial scenarios due to its effectiveness and accessibility, there are few theoretical analysis results about its running time. Running-time analysis can be conducted through the estimation of the upper bounds of the expected first hitting time to evaluate the efficiency of BSO. This study estimates the upper bounds of the expected first hitting time on six single individual BSO variants (BSOs with one individual) based on the average gain model. The theoretical analysis indicates the following results. (1) The time complexity of the six BSO variants is $\mathrm{O}\left(\sqrt{n}\right)$ in equal coefficient linear functions regardless of the presence or absence of the disrupting operator, where n is the number of the dimensions. Moreover, the coefficient of the upper bounds on the expected first hitting time shows that the single individual BSOs with the disrupting operator require fewer iterations to obtain the target solution than the single individual BSOs without the disrupting operator. (2) The upper bounds on the expected first hitting time of single individual BSOs with the standard normally distributed mutation operator are lower than those of BSOs with the uniformly distributed mutation operator. (3) The upper bounds on the expected first hitting time of single individual BSOs with the $U\left(-\frac{1}{2},\frac{1}{2}\right)$ mutation operator are approximately twice those of BSOs with the $U(-1,1)$ mutation operator. The corresponding numerical results are also consistent with the theoretical analysis results.