Classical metaheuristic algorithms such as the GA, PSO, and ACO have laid the foundation for TCP studies. A genetic algorithm-based method for time-constrained TCP in which APFD is used as the evaluation metric was proposed [8]. A search-based TCP method that uses the GA, a greedy algorithm, and the hill-climbing algorithm was introduced. The experimental results demonstrated that the GA performed optimally [17]. Two value-based TCP techniques were proposed; both techniques involve the use of the GA. The results showed that the GA outperformed existing state-of-the-art TCP technologies [18]. PSO was used to automatically prioritize test cases on the basis of modified software units. Empirical results demonstrated that the PSO effectively and efficiently ranked the test cases within test suites and placed them in new optimal positions [10]. In [19], a three-stage method was proposed to address TCP by using a multiobjective PSO (MOPSO) algorithm to optimize both fault coverage and execution time. In [20], an improved TCP method based on quantum PSO was introduced, which outperformed the bat algorithm, grey wolf optimizer, and PSO in experimental evaluations. By reordering test suites under time constraints, ACO was applied to TCP as a method for solving time-constrained prioritization issues [9]. In [21], a hybrid of ACO and GA was proposed, which generates test cases on the basis of priority assigned by test factors and then uses ACO to compute the optimal sequence with the shortest execution time and highest fault rate. For solving requirement-based TCP, an ACO-based solution was introduced in [22], and two implementation methods were presented. The experimental results show that this solution has a strong global optimization ability.

The aforementioned methods typically involve the use of a single algorithmic framework for TCP. As the non-free-lunch theorem [23] demonstrates, standalone heuristic algorithms have inherent limitations when optimization challenges are addressed.

In recent years, a series of novel metaheuristic algorithms have been introduced into TCP and have demonstrated significant potential to address increasingly complex testing environments. In [24], an approach for optimal test case prioritization based on the firefly algorithm was proposed. The overall APFD results indicate that the firefly algorithm is a promising competitor in TCP applications. In [25], an asexual reproduction repair mechanism for discretizing the continuous cuckoo search algorithm was introduced. In [26], a TCP method that is based on an improved Harris hawks optimization algorithm with two optimization targets, namely, the average defect detection rate and the overall execution time, was proposed. In [27], the gray wolf optimization algorithm (GWO) was improved by using cyclic chaotic functions for TCP. In [28], a TCP method based on an improved whale optimization algorithm (WOA) was developed, and a multidimensional directed search space was established to better apply the WOA to TCP. Single heuristic algorithms have inherent limitations in solving optimization problems. In [29], a novel hybrid metaheuristic method for prioritizing and optimizing test cases was introduced. The proposed hybrid algorithm comprehensively considers factors such as the code coverage, fault discovery rate, and execution time, thereby effectively addressing the challenges posed by large-scale test cases and dynamically evolving systems.

Fusion algorithms demonstrate advantages in increasing TCP efficiency. However, their reliance on disparate single-algorithm principles significantly increases method complexity after integration, which consequently increases the computational overhead.

In addition to classical and novel metaheuristic algorithms, numerous studies have applied machine learning, reinforcement learning, and deep learning techniques to TCP in recent years. In [30], a systematic review of machine learning applications in test case selection and prioritization was conducted. In [31], the scalability and accuracy of TCP technology in continuous integration environments were highlighted, and the use of machine learning algorithms to adapt to evolving test data and execution patterns by many contemporary approaches was reported. In [32], a semantic-aware two-stage TCP framework that uses information retrieval techniques for initial sorting and filtering of test cases, followed by fine-grained sorting on the basis of semantic similarity using pretrained Siamese language models, was proposed. Most of the aforementioned approaches use machine learning to mitigate issues in various scenarios.

However, these approaches often require sustained feedback mechanisms and involve computationally intensive operations such as model training and parameter tuning, thereby leading to high training costs and complex implementation.

During this stage, the aquila hovers at high altitude and uses its acute vision to observe and identify potential prey zones. It then dives vertically to select the optimal hunting zone. The mathematical model is expressed as follows:(4)Xi(t+1)=Xbest(t)×(1−tT)+(XM(t)−Xbest(t)×rand1)where X_i(t + 1) represents the position of the i-th individual at iteration t + 1; X_best(t) represents the best solution found by iteration t (the elite individual), which indicates the approximate prey location; and X_M(t) indicates the mean position of all the individuals at the t-th iteration, which can be calculated by Eq. (5). (5)XM(t)=1n∑i=1NXi(t)

In Eq. (5), X_i(t) denotes the position of the i-th individual at the t-th iteration.

During this stage, the aquila hovers above the target prey and meticulously searches the designated hunting area while preparing to strike. The mathematical model that characterizes this behavior is formulated as follows:(6)Xi(t+1)=Xbest(t)×Levy(Dim)+XR(t)+(y−x)×rand2where X_i(t + 1) and X_best(t) are defined as in Eq. (4); Levy(Dim) denotes the Lévy flight distribution function, which can be calculated using Eq. (7). (7)Levy(Dim)=(s×u×σ)/|v|1/β

In Eq. (7), s and β are constants with values of 0.01 and 1.5, respectively; u and v denote normally distributed random numbers within the interval [0, 1]; and σ can be determined via Eq. (8). (8)σ=τ(1+β)×sin⁡(πβ/2)τ(1+β)×β×2(β−1)/2

In Eq. (8), τ represents the gamma function.

X_R(t) represents a randomly selected individual from the current iteration, and y and x describe the spiral flight trajectory of the aquila during exploration, which can be computed using Eqs. (9) and (10), respectively. (9)y=(r1+U×D1)×cos⁡(θ) (10)x=(r1+U×D1)×sin⁡(θ) (11)θ=3π/2−ω×D1where r₁ has a value range of [1, 20], which indicates the predetermined number of search cycles; U and ω are constant parameters set to 0.0056 and 0.005, respectively; and D₁ denotes an integer value within the range [1, Dim].

During this stage, upon precise prey localization, the aquila executes a rapid vertical descent to conduct a preliminary attack while monitoring the response of the target. The mathematical model for this behavior is formulated as follows:(12)Xi(t+1)=(Xbest(t)−XM(t))×α−rand3+((UB−LB)×rand4+LB)×δwhere X_i(t + 1), X_best(t), and X_M(t) are defined as in Eq. (4); both α and δ in the AO are assigned values of 0.1 and function as exploitation tuning parameters; and LB and UB denote the lower and upper bounds, respectively, of the defined search space of the given problem.

During this stage, the aquila accurately hunts the prey according to its stochastic ground movements, which is modeled mathematically as follows:(13)Xi(t+1)=QF(t)×Xbest(t)−(G1×Xi(t)×rand5)−G2×Levy(Dim)+G1×rand6where X_i(t + 1) and X_best(t) are defined as in Eq. (4), X_i(t) and Levy(Dim) are defined as in Eq. (5), and QF(t) represents the search-strategy-balancing quality function defined in Eq. (14). (14)QF(t)=t2×rand7−1(1−T)2

The parameter G₁ = 2 × rand 8 − 1, characterizes the tracking trajectory of the aquila during the aerial pursuit of escaping prey and decreases linearly from 1 to −1 during iterative optimization. The parameter G₂ = 2 × (1 − t/T), represents the flight slope of the aquila during the aerial pursuit of escaping prey and decreases linearly from 2 to 0 during iterative optimization.

Parameters rand1 through rand8 in the aforementioned equations denote uniformly distributed random variables sampled from the closed interval [0, 1].

Since the original AO operates in a continuous space while TCP is a combinatorial permutation problem, we adapt MCIAO using the SPV rule to bridge this gap: (1) Solution Representation: Each individual is maintained as a D-dimensional real-valued vector X→ = [x₁, x₂,..., x_D], where D is the number of test cases. (2) Decoding Mechanism: A valid execution sequence is generated by sorting the components of X→ in an ascending order, the indices of the sorted values determine the test case order. For example, given X→ = [1.5, 0.8, 2.3, 0.4] corresponding to {TC₁, TC₂, TC₃, TC₄}, sorting yields the index sequence [4, 2, 1, 3] (since 0.4 < 0.8 < 1.5 < 2.3), producing the test order [TC₄, TC₂, TC₁, TC₃]. (3) Fitness Evaluation: The decoded permutation is evaluated using APFD, APBC, or APDC metrics, and the resulting fitness value guides the continuous position updates in the subsequent AO iterations. (4) Constraint Handling: The SPV rule inherently guarantees solution legality—ensuring each test case appears exactly once without duplicates or omissions—thus eliminating the need for explicit constraint handling.

These adaptations ensure that MCIAO effectively operates in the discrete combinatorial search space of TCP while maintaining the optimization capabilities of the original metaheuristic framework.

Like most SI algorithms, the original AO initializes the population using randomly generated data, which leads to an uneven distribution of initial individuals and compromised population diversity. This makes the algorithm prone to premature convergence in later iterations, thus ultimately degrading its search efficiency and optimization capabilities. In light of this, we build upon the cosine transform method by integrating logistic mapping and sine mapping to generate a novel logistic–sine–cosine [34] composite chaotic mapping as an improvement strategy. The mathematical formulation is expressed as follows:(15)xi+1=cos⁡(π(4rxi(1−xi)+(1−r)sin⁡(πxi)−0.5)),r∈[0,1]

To further demonstrate the advantages of this composite chaotic mapping, Fig. 1 shows a comparison of the distributions of the three individual mappings and random initialization in the [0, 1] interval. The results conclusively show that the logistic–sine–cosine mapping has a more uniform distribution than logistic mapping, sine mapping, and random initialization do and can better cover the search space to obtain a well-distributed initial solution position. The flat frequency distribution observed in Fig. 1 provides quantitative evidence that the proposed strategy prevents initial clustering, thereby ensuring a diverse population from the outset.

In both the narrowed exploration and narrowed exploitation phases of the AO, the Lévy flight strategy was introduced for random searching because of its fixed step size. However, this approach fails to consider the dynamic environmental adaptation of aquila during predation, which makes balancing large-scale coarse-grained exploration with small-scale fine-grained exploitation challenging. Consequently, the search process lacks directional guidance and hierarchical coordination, which potentially causes the algorithm to overlook regions that contain optimal solutions.

The Cauchy distribution is a continuous probability distribution. Under the condition that its cumulative distribution function can be calculated as an inverse function, the inverse transform method can be used to generate random numbers that follow a uniform distribution [35]. The inverse function is defined as follows:(16)step=a+b∗tan⁡(pi∗(p−1/2) (17)p=Randn(1,Dim)

During the narrowed exploration and narrowed exploitation phases, the random walk behavior is regulated by the location parameter a and the scale parameter b. The location parameter is set to a = 0 to center the perturbation around the current solution. The scale parameter b = 0.01 is empirically chosen to balance the heavy-tailed nature of the Cauchy distribution—which enables occasional large jumps for global exploration—with sufficiently small step sizes to support local refinement. This configuration leverages the exploratory strength of Cauchy-based perturbations while maintaining effective local search capability, thereby facilitating stable and effective convergence toward high-quality solutions. Improved equations, namely, Eqs. (18) and (19), are obtained by substituting these values into Eqs. (6) and (13). (18)Xi(t+1)=Xbest(t)×(0.01×tan⁡((pi×(p−1/2))))+XR(t)+(y−x)×rand2 (19)Xi(t+1)=QF(t)×Xbest(t)−(G1×Xi(t)×rand5)−G2×(0.01×tan⁡((pi×(p−1/2))))+G1×rand6

In the exploration phase of AO, a fixed-step uniform random search is conducted, which demonstrates inadequate responsiveness to the dynamic characteristics of the solution space. As a result, the search range cannot be dynamically adjusted according to the complexity of the solution space, which ultimately causes either extensive regional omissions or redundant searches.

The random walk strategy is a commonly used approach in search and optimization algorithms. During its execution, the system randomly selects a candidate solution from the search space according to predefined rules to participate in individual updates, thereby improving the comprehensive exploration of the solution space while preventing the population from becoming stuck in local optima [36]. Drawing inspiration from mutation mechanisms, in this study, a random walk approach, namely, the mutated random walk approach, which extends conventional random walk strategies, is introduced. This approach dynamically modulates individual movement characteristics by adjusting the step size, direction parameters, and path constraints while incorporating a mutation mechanism to generate new solutions, thereby increasing the adaptability and diversity of the population and significantly improving the global exploration performance of the algorithm.

The new individuals generated after mutation are calculated using Eq. (20):(20)Xi(t+1)=Xi(t)+rand∗(X(n(1),:)−X(n(2)))where: n = rand i([1 Dim], 2). Introducing the mutated random walk into Eq. (4) yields:(21)Xi(t+1)=Mutation(Xbest(t)×(1−tT)+(XM(t)−Xbest(t)×rand1))

The modified random walk strategy is introduced into Eq. (18):(22)Xi(t+1)=Mutation(Xbest(t)×Distribution(Dim,Index_fly)+XR(t)+(y−x)×rand2)

The mutated random walk operates entirely in the continuous domain. After updating the continuous position vector using Eqs. (21) or (22), the SPV rule described in Section 3.2.1 is applied to convert the continuous vector into a valid permutation of test cases. This two-step mechanism guarantees that any movement in the continuous space—regardless of its magnitude or direction—always yields a legitimate test case sequence without requiring additional repair operators or constraint handling.

The inspiration for specular reflection learning stems from the reflection of light, and its model is illustrated in Fig. 2 [37]. This method involves leveraging the principle of symmetry to generate boundary-constrained opposite solutions along multiple mirror directions. By replacing original individual positions with these opposite solutions, it effectively guides individuals to migrate toward high-resource regions.

The AO predominantly uses the current optimal individual to guide the search process during late iterations, which tends to decrease the population diversity and may lead to premature convergence to a local optimum. To overcome this limitation, the specular reflection learning strategy can be used to perturb the position of the best individual in the later stages. This mechanism not only enriches the diversity of the population but also better emulates the natural habitat selection behavior of aquila birds, thereby enabling broader solution space exploration while robustly preventing entrapment in local optima and increasing convergence accuracy. According to the principle of symmetry in specular reflection learning, the opposite point x⌣ of x can be calculated using Eq. (23). (23)x⌣=(0.5λ+0.5)×(ZL+ZU)−λx

By integrating specular reflection learning with AO, Eq. (23) can be transformed into Eq. (24). (24)Xi(t+1)=(0.5×rand+0.5)×(UB+LB)−rand×Xi(t)

In the context of TCP, specular reflection learning operates in the continuous domain. The opposite solution generated by Eq. (24), which is computed as a symmetric point with respect to the current best solution, is mapped to a test case permutation via the SPV rule (Section 3.2.1), yielding a sequence that is structurally distinct from the existing candidates. This mechanism introduces targeted diversity during later iterations when the population tends to concentrate around local optima.

A schematic diagram of the MCIAO_TCP is shown in Fig. 3. The process begins by extracting relevant software programs from the datasets for testing. A test suite is then developed to generate test cases. TCP sequencing is performed using a search-based optimization approach, where the proposed improved algorithm MCIAO is used to formulate strategies while incorporating evaluation metrics such as APFD, APBC, and APDC. The final step yields prioritized test cases.

The MCIAO algorithm is developed by applying the improvement strategies described in Sections 3.2.2–3.2.5 to the AO; its workflow is illustrated in detail in Algorithm 1.

The performance of the MCIAO is evaluated on the CEC2017 and CEC2020 benchmark functions. Four metrics are defined as follows: Mean (average fitness) reflects convergence accuracy and optimization capability; Std (standard deviation) indicates robustness and stability; Min (best achieved fitness) represents the potential to reach near-global optima; and Epoch (number of iterations to convergence) measures convergence speed and computational efficiency. Table 3 reports the complete numerical results. To improve readability, only the summarized winning counts are discussed here, while the full results are provided in Appendix A.1, Table A1, and Appendix A.2, Table A2.

The results from Table 3 show that on CEC2017, MCIAO achieves the best (lowest) Mean on 22 out of 29 functions (75.9%), the best Min on 20 out of 29 functions (69.0%), and the lowest Std on 16 out of 29 functions (55.2%), indicating consistently high solution quality and robustness across independent runs. In terms of convergence speed, MCIAO attains the fastest Epoch on 7 out of 29 functions (24.1%); on the remaining functions, its Epoch values remain competitive (mostly well below the maximum iteration limit of 500). On CEC2020, MCIAO achieves the best (lowest) Mean on 7 out of 10 functions (70%), including F₁, F₃, F₅, F₆, F₇, F₉, and F₁₀, and the best Min on 5 functions (50%), notably on F₁, F₃, F₆, F₇, and F₉, indicating its strong capability to locate high-quality solutions. In terms of convergence speed, MCIAO attains the fastest Epoch on 9 out of 10 functions (90%), with only F₆ where GWO converges slightly faster. Regarding stability, MCIAO achieves the lowest Std on F₁ and F₉; on the remaining functions, its Std values are generally comparable to the best competitors. Overall, MCIAO exhibits a strong balance between solution accuracy, stability, and convergence efficiency on both test suites.

To visually compare the MCIAO with the reference algorithms, we plot partial convergence curves, as shown in Figs. 4 and 5, using simulation data from Appendix A.1, Table A1, and Appendix A.2, Table A2.

As shown in Fig. 4, MCIAO achieves the fastest convergence speed and highest solution accuracy among all the tested functions F₄–F₇, F₁₆, and F₂₀. PSO exhibits rapid initial convergence but premature stagnation, often trapped in local optima. BKA and SSA show moderate performance with somewhat lower final precision. WOA and GWO demonstrate average convergence behavior, with visible degradation on complex landscapes like F₁₆ and F₂₀. LSHADE stagnates prematurely on F₇ and F₂₀, while iCSPM converges slowly across all test cases. GOOSE performs poorly, failing completely on F₇. Overall, MCIAO demonstrates superior exploration–exploitation balance, leading to robust optimization performance.

As shown in Fig. 5, which presents results on selected functions including multimodal functions F₂ and F₃, hybrid function F₆, and composition function F₉, MCIAO consistently achieves the fastest convergence and lowest final fitness among all algorithms. While PSO exhibits rapid initial convergence on F₂ and F₆, it suffers from premature stagnation on F₃ and F₉, demonstrating inconsistent performance. WOA, GWO, and SSA show moderate convergence but significantly lower final accuracy. LSHADE and iCSPM, despite occasional competitiveness, generally converge slower or stagnate earlier than MCIAO. GOOSE performs poorest, failing to converge on F₉. MCIAO’s robust exploration–exploitation balance ensures its adaptability across diverse landscapes.

The Wilcoxon rank-sum test was employed to investigate whether the observed differences between MCIAO and the competing algorithms are statistically reliable. In this study, the significance level was fixed at 0.05. When the calculated p-value is smaller than 0.05, the corresponding difference is regarded as statistically significant; otherwise, the two methods are considered to exhibit comparable performance from a statistical perspective. Tables 4 and 5 summarize the statistical comparison results obtained from the Wilcoxon rank-sum test on CEC2017 and CEC2020 benchmark suites. For clarity, only the numbers of statistically significant and non-significant cases are reported in the main manuscript, while the detailed p-values are provided in Appendix B.1 Table A3, and Appendix B.2 Table A4.

The results confirm that MCIAO exhibits statistically significant superiority over the majority of competitors. Specifically, MCIAO significantly outperforms PSO and LSHADE on all 29 functions (100%), GWO on 28 functions (96.6%), BKA on 27 functions (93.1%), AO and SSA on 26 functions (89.7%), iCSPM on 25 functions (86.2%), GOOSE on 23 functions (79.3%), HHO on 21 functions (72.4%), and WOA on 19 functions (65.5%). These results demonstrate that MCIAO achieves consistently better solution quality with high statistical confidence across the majority of test functions.

The results demonstrate that MCIAO exhibits statistically significant superiority over the majority of competing algorithms. Specifically, MCIAO achieved significantly better results (p < 0.05) in 10 out of 10 functions against LSHADE and iCSPM, indicating complete dominance over these advanced continuous optimizers. It also outperformed PSO, GWO, AO, and GOOSE on 9 out of 10 functions (90%), and BKA on 8 functions (80%). Strong statistical differences were also observed against SSA and HHO (7 out of 10 functions each). In comparison with WOA, which emerged as the most competitive counterpart, MCIAO showed significant superiority in 5 out of 10 functions, while the remaining cases (p ≥ 0.05) suggest comparable performance in specific complex landscapes. Notably, there were no instances where any competitor significantly outperformed MCIAO. These results confirm the consistent statistical advantage and robustness of MCIAO on the CEC2020 benchmark suite.

To evaluate the actual computational efficiency, we measured the average wall-clock execution time (in seconds) over 30 independent runs on four representative CEC2020 functions: F₂ (multimodal), F₃ (multimodal), F₆ (hybrid), and F₉ (composition). All experiments were conducted under identical hardware and software conditions. Table 6 reports the results.

As shown, MCIAO achieves the fastest execution time on the hybrid function F₆ (0.035 s) and maintains competitive overall efficiency. Although it is slightly slower than PSO and GOOSE on some functions, this trade-off is compensated by its better solution quality and faster convergence (Sections 5.1.1–5.1.3). Moreover, MCIAO outperforms BKA, SSA, HHO, and iCSPM in execution time by 9%–72%. These results confirm that MCIAO offers superior optimization performance without compromising practical efficiency, making it suitable for real-world applications requiring both accuracy and speed.

To evaluate the effects of different improvement strategies, in this study, comparative experiments involving the MCIAO, the AO, and the following four variants are conducted: AO-1 (improved solely with the logistic–sine–cosine chaotic mapping strategy), AO-2 (improved only with the Cauchy inverse cumulative distribution flight strategy), AO-3 (modified exclusively using the mutated random walk strategy), and AO-4 (optimized only with the specular reflection learning strategy). Table 7 summarizes the number of benchmark functions on which each variant achieves the best Mean and Std values. The detailed numerical results of the ablation experiments are provided in Appendix C.1, Table A5.

As shown in Table 7, the MCIAO achieves the highest convergence accuracy, with the best average performance on 9 of the 10 functions. In contrast, AO-3 performs the worst on multiple functions (e.g., F₁, F₅, and F₇), which indicates its tendency to fall into local optima; AO-1, AO-2, AO-4, and AO perform similarly on some functions but are outperformed overall by the MCIAO. In terms of stability, the MCIAO demonstrates the smallest Std for most functions, which remains within a reasonable range without abnormal fluctuations, thus indicating its superior stability and reduced likelihood of extreme deviations. The Stds of AO-3 on F₁, F₅, and F₇ are extremely large, thus indicating highly unstable performance. The stability of AO-1, AO-2, AO-4, and the AO varies with the function, but overall, it is inferior to that of the MCIAO.

In summary, the ablation results in Table 7 show that each component of MCIAO addresses a specific aspect of the blind-search limitation: AO-1 mitigates initial blindness by ensuring diverse starting points; AO-2 and AO-3 jointly mitigate exploratory blindness through adaptive, long-range jumps that replace undirected movements; and AO-4 prevents convergence blindness in later stages by perturbing solutions near local optima. The full MCIAO algorithm, which integrates all four strategies, achieves the best performance, confirming the synergistic effect of these components.

To more intuitively compare the performance of the hybrid strategy–improved algorithm and single strategy–improved algorithms, we plot partial convergence curves using simulation data from Appendix C.1, Table A5, as shown in Fig. 6.

As shown in Fig. 6, for the multimodal function and hybrid functions tested (F₂, F₃, F₆, F₉,), the MCIAO achieves the fastest convergence and the lowest final fitness value, significantly outperforming all others. The AO-3 and AO-4 are the closest competitors but still fall short, especially in later convergence stages. The AO and AO-1 algorithms consistently show the slowest convergence and poorest accuracy. Furthermore, the AO generally has slower convergence and lower accuracy and often becomes trapped in local optima or has suboptimal search efficiency. Both the individual and hybrid improvement approaches demonstrate measurable performance improvements over the original algorithm.

To provide deeper insights into the search behavior of MCIAO, we adopted the Convergence Driver Score (CDS) from the EvoMapX framework [47]. CDS quantifies the contribution of each search operator to the overall convergence by tracking fitness improvements during optimization. Four representative operators were evaluated on two CEC2020 functions (F₃ and F₉). Table 8 reports the average CDS and normalized contributions.

As shown in Table 8, all four operators contribute almost equally (about 24%–26%), indicating a well-balanced exploration-exploitation strategy. The Cauchy-based operators together account for approximately 50% of the convergence events, confirming their central role in search guidance. Mutated random walk ensures global exploration, while specular reflection learning provides non-negligible local refinement in later iterations. This balanced operator attribution explains the robust optimization performance of MCIAO across diverse function landscapes.

In the boxplots in Fig. 7, the performance of the five open-source datasets selected in Section 4.2 are compared across several evaluation metrics, with APFD as the key metric.

As shown in Fig. 7, the MCIAO consistently demonstrates superior performance across all five projects. In each case, it achieves the highest median APFD values–reaching up to 98.5% in Chart and 99% in Time–accompanied by compact data distributions, minimal outliers, and exceptional stability. Specifically, the median APFD of MCIAO substantially exceeds the overall average of all algorithms in every project: by approximately 2.5% in Chart (96% average), 7% in Closure (83.5% average), 5% in Lang (90% average), 8% in Math (83% average), and 2.3% in Time (96.7% average). These results underscore the robustness and consistent leading performance of MCIAO in comparison to the other methods evaluated.

In the boxplots in Fig. 8, the performance of all the algorithms on the five open-source datasets selected in Section 4.2 is compared in terms of the APBC evaluation metric.

As shown in Fig. 8, the MCIAO consistently achieves the best overall performance across all projects in terms of APBC metrics, leading in both median value and stability. It is followed closely by GWO and WOA in most projects, which also exhibit stable and high performance. Notably, in the Closure project, MCIAO performs comparably to BKA and GWO, though BKA shows greater fluctuation and lower stability. In all cases, HHO consistently yields the poorest results. Furthermore, the APBC value of MCIAO exceeds the overall average of all algorithms in each project: by approximately 3% in Chart (95% average), 2.4% in Closure (87.6% average), 5.2% in Lang (89.8% average), 6.3% in Math (83.7% average), and 2.2% in Time (95.8% average), underscoring its robust and superior effectiveness compared to the other evaluated algorithms.

In the boxplots in Fig. 9, the performance on the five open-source datasets selected in Section 4.2 is compared across several evaluation metrics, with the APDC score as the key performance indicator.

As shown in Fig. 9, the MCIAO demonstrates the best overall APDC performance across all projects, consistently achieving the highest median values and superior stability. It is followed by GWO and WOA in the Chart project, while HHO consistently yields the lowest results. The APDC value of MCIAO exceeds the overall algorithm average in each project: by approximately 2.9% in Chart (94.6% average), 7.8% in Closure (82.2% average), 5.9% in Lang (88.1% average), 6.5% in Math (83.9% average), and 5.1% in Time (92.9% average), further confirming its robust and leading effectiveness.

Although code structure and complexity may affect algorithm performance across projects, the MCIAO consistently outperforms the other algorithms, which demonstrates its robustness and adaptability to software projects with diverse characteristics.

To further evaluate MCIAO on discrete optimization, we conducted TCP experiments using APFD, APBC, and APDC. Fig. 10 presents the convergence curves of all compared algorithms on the Chart and Time instances for each metric.

As shown in Fig. 10, MCIAO consistently achieves the fastest convergence and the highest final values across all three metrics and both projects. On Chart, MCIAO reaches APFD above 0.95 within 200 iterations, significantly outperforming PSO, GWO, and others. On Time, MCIAO converges to APDC ≈ 0.98 after only 150 iterations. PSO shows rapid early progress but premature stagnation; BKA and SSA exhibit moderate convergence but lower final accuracy; GOOSE and HHO perform poorly, often staying below 0.85.

The convergence curves in Fig. 10 reveal a strong practical correlation among APFD, APBC, and APDC across all studied TCP instances. The relative rankings of all algorithms remain highly consistent across the three metrics, and test sequences achieving higher structural coverage (APBC and APDC) consistently yield superior fault detection (APFD), demonstrating that coverage improvements serve as a reliable proxy for fault detection effectiveness. This strong empirical correlation validates the practical efficacy of MCIAO and confirms that optimizing for coverage metrics directly translates to improved fault detection performance.

Overall, the TCP experiments confirm that MCIAO maintains its convergence speed and solution quality on discrete optimization tasks.