Similar to most metaheuristic methods, EO utilizes initialized candidate solutions to commence the optimization process. The initial concentration is described below.

(1)Xi=Xmin+randi(Xmax−Xmin),i=1,2,⋯,Nwhere X_i is the concentration of the i-th initial particle; X_max and X_min denote the upper and lower boundaries of the exploration space, respectively; N signifies the scale of the population under consideration.

The four particles with the optimal fitness and the mean of these four particles are selected to create the equilibrium pool. X_{eq_pool} and X_{eq_ave} are expressed below, respectively.

(2)Xeq_pool={Xeq1,Xeq2,Xeq3,Xeq4,Xeq_ave}

(3)Xeq_ave=Xeq1+Xeq2+Xeq3+Xeq44

The exponential factor F plays a crucial role in maintaining the delicate exploration-exploitation balance. This factor is determined by Eq. (4).

(4)F=e−λ(t−t0)where λ represents a randomly generated variable within the range of [0,1]; t is a non-linear computation, formulated as follows.

(5)t=(1−ItT)(a2ItT)where It corresponds to the current iteration count and T corresponds to the predefined maximum iteration count; a₂ stands as a constant parameter. t₀ is expressed below.

(6)t0=1λln⁡(−a1sign(r−0.5)[1−e−λt])+twhere a₁ takes on a value of 2 and a₂ takes on a value of 1; sign(r − 0.5) effectively determines the direction of development; r symbolizes a random variable confined within the range of [0,1]. Substituting Eq. (6) into Eq. (4), we get:

(7)F=a1sign(r−0.5)[1−e−λt]

The parameter responsible for regulating the extent of exploitation is known as the generation rate G, and its definition is provided below.

(8)G=G0e−λ(t−t0)=G0Fwhere

(9)G0=GCP(Xeq−λX)

(10)GCP={0.5r1,r2≥GP0,r2<GPwhere r₁ and r₂ represent random values of [0,1], GCP signifies the generation rate control parameter, calculated through the generation probability GP.

Thus, the update method of EO is below.

(11)X=Xeq+(X−Xeq).F+GλV(1−F)

Regarding metaheuristic techniques, the exploration-exploitation balance is essential. As can be observed from Eq. (11), which F is employed to influence concentration changes to reach equilibrium, while the equilibrium candidate is a randomly chosen candidate from the equalization pool. Once it gets stuck in a local optimum, the best solution will prematurely stagnate during the search. Thus, to avoid this problem, the adaptive search for the random candidate is carried out, and we design an adaptive update formula to instead of Eq. (11). The novel updating equation is below.

(12)X=ω×Xeq+(X−Xeq).F+GλV(1−F)where ω is an inertia weight.

Inspired by the PSO, the inertia weight mechanism is broadly employed in metaheuristic approaches. For instance, Ding et al. [18] revise the update equation of the salp swarm algorithm (SSA) by introducing an adaptive inertia weight factor. Wang et al. [19] use an inertia weight to modify the leader position of the SSA.

In this study, a novel adaptive inertial weight coefficient is designed and mathematically expressed as follows.

(13)ω=ωmin+(ωmax−ωmin)×11+ab(It−T2)where ω_max represents the upper limit of the weight; ω_min signifies the lower limit of the weight; a and b are constants.

To deal with the problem that EO tends to become stuck in local optimum and thus loses population diversity, this work employ the Cauchy mutation to improve the population diversity. The Cauchy distribution exhibits a narrower peak at its origin and a wider spread towards both extremes. By using the Cauchy mutation, a larger perturbation can be generated near the particle of the current variation, thereby allowing the Cauchy distribution function to search all areas of the solution space. This approach effectively leverages the mutation effects at the distribution’s edges to optimize individuals, thereby achieving the global optimum while simultaneously preserving population diversity. The Cauchy distribution function is described below.

(14)Ft(x)=12+1πarctan⁡(xt)where t > 0.

Based on the Cauchy distribution function, we can further update the position of the particles. The update equation is below.

(15)X′=X+X×Cauchy(0,1)where Cauchy (0,1) represents a randomly generated value drawn from the Cauchy distribution, X denotes the position of the current iteration, X′ signifies the position of the mutated particle.

In the standard EO, the update mechanism can readily become ensnared in local optima, leading to a decline in its capacity for global exploration. To address the limitations, this work introduces the position update equation of the sine cosine algorithm (SCA) [20] with inertia weight factor to perform adaptive search of the particles. The novel position update equation is below.

(16)X″={X+r3×sin⁡(r4)×|ω×r5×gbest−X|,r6<0.5X+r3×cos⁡(r4)×|ω×r5×gbest−X|,r6≥0.5

(17)r3=a(1−ItT)where a is set to 2; r₃ linearly decreases from 2 to 0. r₄ is derived from a uniform random distribution that encompasses the interval [0,2π]. r₅ and r₆ signfy random numbers in [0,1]. gbest is the global optimum solution. X signifies the position of the current iteration. X″ represents the updated position.

The developed SCEO algorithm updates the particle positions and does not affect the framework of the basic EO. Firstly, it injects an adaptive inertial weight to enhance the navigation of a stochastic equilibrium candidate. During iterations, the inertia weight continually adjusts the position of the equilibrium candidate and hopes to move it to a more prospective region. Next, the entire particle is updated again using the adaptive SCA to promote the search capability. The sine-cosine function decides the exploration horizon of the search individual’s next position by controlling the direction of the particle’s search. Once stuck in a local optimum, the search direction is changed, effectively avoiding search stagnation. Finally, population diversity is ensured by the addition of the Cauchy mutation mechanism in case of increased ability. By harnessing the distinctive characteristics of the Cauchy distribution, which exhibits a significant shift from a peak at the origin to an elongated distribution at its extremities, substantial perturbations are strategically induced near particles throughout the iterative process. This enables the algorithm to explore all areas and increase the particles diversity. Based on the above analysis, the flowchart of SCEO is given in Fig. 1.

The computational complexity of an optimization approach is an essential criterion for analyzing performance. In this paper, the Big-O notation is adopted to denote complexity. This complexity is contingent upon key factors such as the particle numbers (N), the dimensions (d) and the iterations (t), and (f) is the fitness evaluation.

O(SCEO) = O(initialization) + O(fitness evaluation) + O(memory saving) + O(concentration update) + O(SCA updating strategy) + O(Cauchy mutation).

Consequently, the computational complexity of SCEO can be elucidated below.

O(SCEO)=O(N×d+t×f×d+t×N+t×N×d+N(1+t+t×d)+t×N×d)≅O(t×f×d+t×N×d).

The computational complexity of EO is expressed as O(t × f × d + t × N × d). In essence, it can be summarized that the SCEO maintains the same computational complexity as the basic EO algorithm.

In this subsection, we delve into a theoretical exploration of SCEO to substantiate its efficiency. The methodology employed for this proof is outlined in the subsequent steps.

We classify SCEO into three stages: adaptive inertial weight position update, adaptive sine cosine position update, and Cauchy mutation update.

Phase 1: From Eq. (12), during the iteration, ωX_eq, F, λ, V, GCP are constants held constant, then 1−FλV is also a constant, let GCP=a1, 1−FλV=a2, (t + 1)th: X_eq = n₁, (t + 2)th: X_eq = n₂.

Thus, the problem is elaborated below:

(18)X(t+2)−(F−λa1a2)2X(t)=n1(F−λa1a2)(1−F+a1a2)+n2(1−F+a1a2)

Let F−λa1a2=A, we get

(19)X(t+2)−A2X(t)=n1A(1−F+a1a2)+n2(1−F+a1a2)

The characteristic equation of Eq. (19) is

(20)λ2−A2=0

Clearly, Δ=4A2≥0, the following two cases need to be considered.

(1) Δ=0. Only when A=0, Δ=0. At this point, the characteristic roots are λ1=λ2=0. The general solution is

(21)X(t)=p0+p1λ1t+p2tλ2t=p0where p0 is the constant coefficient to be determined.

(2) Δ>0, where A≠0. The characteristic roots are λ1,2=±A2. The general solution is

(22)X(t)=p0+p1λ1t+p2λ2twhere p0, p1, p2 are constant coefficients to be determined.

When t→∞, X(t) has a limit and converges to a finite value, it implies that the iteration converges. If X(t) of the above two cases is made to converge, the conditions need to be satisfied ∥λ1∥<1, ∥λ2∥<1.

When Δ=0, λ1=λ2=0, the convergence condition is satisfied. In this case, X(t) converges.

When Δ>0, the convergence region should satisfy:

(23){4A2>0|±A2|<1

From Eq. (23), the convergence region is −1<A<1. According to Eqs. (7) and (10), F−λa1a2∈(−1,1), so A∈(−1,1) in the convergence range. In this case, X(t) converges.

In summary, Phase 1 converges.

From Eq. (16), it can be seen that Phase 2 has the same search pattern as Phase 1, and it can be proved in the same way that Phase 2 also converges.

Phase 3: From Eq. (15), Cauchy(0,1) is constants held constant, let (t + 1)th: Cauchy(0,1)=V1, (t + 2)th: Cauchy(0,1)=V2.

A second-order linear differential equation is obtained, and the problem be expressed in the following form:

(24)X(t+2)−(1+V1)(1+V2)X(t)=0

The characteristic equation of Eq. (24) is

(25)λ2−(1+V1)(1+V2)=0

Clearly, Δ=4(1+V1)(1+V2). Because V1>0, V2>0, Δ>0.

When Δ>0, the characteristic roots are λ1,2=±(1+V1)(1+V2). The general solution is

(26)X(t)=s1λ1t+s2λ2twhere s1, s2 are constant coefficients.

When t→∞, X(t) has a limit and converges to a limited value, it implies that the iteration converges. If X(t) is made to converge, the conditions need to be satisfied ∥λ1∥<1, ∥λ2∥<1.

Because limt→∞V1=0, limt→∞V2=0, limt→∞(λ1,2)=limt→∞(±(1+V1)(1+V2))=±1.

It satisfies the convergence condition. In this case, X(t) converges. In summary, Phase 3 converges.

Overall, the SCEO algorithm converges.

To ascertain the efficacy of the SCEO, 15 typical benchmark functions are applied. These functions are meticulously detailed in Table 1. According to Table 1, they have various features, which contain 7 unimodal functions and 8 multi-modal functions.

To thoroughly assess the performance of SCEO, this paper compares the SCEO algorithm with five novel EO variants, including the basic EO [9], LWMEO [15], m-EO [16], IS-EO [21], and IEO [17]. To ensure equitable evaluation, a consistent population scale of 30 and a uniform maximum iteration count of 500 are maintained across all tests. Each optimizer is subjected to 30 independent runs, all conducted under identical conditions with a fixed function dimension of 100. Mean minimum (Mean) values and their corresponding standard deviations (Std) are employed as key indicators to gauge the performance of all methods. Furthermore, the Wilcoxon signed rank test is adopted to provide a comprehensive assessment. The findings, including Friedman’s rank and the Wilcoxon rank-sum test, are succinctly presented in Table 2. The symbols “+/−/=” in Table 2 signify that the SCEO algorithm demonstrates superior, inferior, or similar performance compared to other algorithms, respectively. The “N/A” notation indicates instances where no statistical significance is observed.

Based on Table 2, it reveals that the SCEO finds the theoretical best solution on all cases except f₄. For this function, although the theoretical optimal values are not achieved, the SCEO is still better than the other competitors. Compared with the basic EO and IEO, SCEO performs better across 12 test functions. With respect to LWMEO and IS-EO, SCEO gets better results on all of the benchmark problems. For f₂, SCEO and m-EO provide similar results. SCEO is superior to m-EO on 12 functions, and on the rest 3 functions the two algorithms get similar results. For f₈, f₁₀, and f ₁₄, SCEO, m-EO and IEO obtains similar results. SCEO resembles basic EO on 2 functions and is inferior to basic EO on f₈. Based on the Friedman’s rank test, it can be observed that SCEO ranks first followed by m-EO, EO, IEO, LWMEO, and IS-EO, which reveals that SCEO provides the best performance. The p-values consistently remain below the threshold of 0.05 within the 100-dimensional environments, which reveals that the distinctions between the SCEO and the other methods are statistically significant. In summary, SCEO presents more competitive performance than other comparative algorithms for 15 traditional benchmark functions.

To prove the validity of the SCEO, it is compared with some advanced metaheuristics, including the enhanced SSA (RDSSA) [22], the improved grey wolf optimizer (OGWO) [23], the modified moth-flame optimization algorithm (DMMFO) [24], adaptive salp swarm optimization (ASSO) [25], and improved particle swarm optimization (BPSO) [26]. 15 test functions in Table 1 are adopted and the function’s dimension is configured to 100. The Mean and Std values for the six metaheuristics are meticulously documented in Table 3. Additionally, the outcomes of Friedman’s rank test and the Wilcoxon rank-sum test are presented at the base of the table. In Table 3, the symbols “+/−/=” are employed to signify instances where the SCEO algorithm demonstrates superior, inferior, or comparable performance to other algorithms, respectively. The notation “N/A” is employed when no statistically significant difference is observed.

From the data in Table 3, for f₈, f₁₀, f₁₄, RDSSA, ASSO, and SCEO obtain the theoretical optimal solution. Compared with DMMFO, OGWO, and BPSO SCEO reaches better performance on all of its cases. With respect to RDSSA, and ASSO, SCEO obtains better on 12 cases and has similar results on 3 cases. According to the Friedman mean ranks acquired by eight approaches on 15 test functions, it can be shown that SCEO is first, followed by RDSSA, ASSO, OGWO, BPSO, and DMMFO. The p-values consistently remain below the threshold of 0.05 within the 100-dimensional environments, which reveals that the differences observed between SCEO and the alternative methods hold substantial statistical significance.

Table 4 presents the average runtime duration (measured in seconds) of seven different techniques across the set of 15 benchmark functions. Notably, Table 4 highlights that EO boasts the shortest average runtime compared to the other techniques. Importantly, the average runtime of SCEO closely aligns with that of EO across all scenarios. This observation underscores the fact that the devised SCEO achieves performance enhancements without compromising runtime efficiency.

In this subsection, we investigate the convergence behavior of each method by visually showing the convergence process of these methods. The 15 benchmark test problems are employed from Table 1, and the dimension is set to 100. Fig. 2 displays the convergence curves of eleven methods across selected cases.

Observing Fig. 2, it becomes evident that SCEO exhibits a gradual convergence rate in the initial phases of the search process. At the later stage, the convergence speed is accelerated and the convergence curve is smoother. This behavior is attributed to a strategic trade-off: during the early stages, the algorithm intentionally prioritizes preserving population diversity, thus modestly compromising early convergence. Subsequently, the algorithm transitions from broad global exploration to focused local search, boosting solution accuracy by exploiting previously explored areas. According to the experimental results, SCEO finds better solutions quicker than other all compared algorithms across more than half of the cases tested. Notably, SCEO not only accelerates convergence but also demonstrates heightened accuracy in solution outcomes when contrasted with alternative algorithms. Overall, these results underscore the pronounced convergence advantages of SCEO over other EO variants and the broader spectrum of compared algorithms.

This subsection comprehensively evaluates the performance of SCEO employing the CEC2017 test suite. The outcomes of SCEO are meticulously compared against the basic EO, various EO variants, and a spectrum of advanced metaheuristics including the EO, m-EO, LWMEO, IS-EO, IEO, RDSSA, OGWO, DMMFO, ASSO, BPSO, and VC-SSA [18]. For this experiment, the upper limit of evaluations is prudently established at D × 10⁴. Consistency is maintained in terms of parameter settings for both the compared methods and SCEO, mirroring those employed in Section 4.2. All methods are subjected to 30 independent runs, and the functional dimension remains consistent at 100. The performance of the 13 methods on the CEC2017 benchmark function with 100 dimensions is summarized in Table 5.

According to Table 5, SCEO performs better to other comparative algorithms across 15 functions and is inferior to the other algorithms on 14 functions. With respect to m-EO, IS-EO, RDSSA, OGWO, DMMFO, and ASSO, SCEO is superior on all functions. Compared with BPSO, SCEO gets better and worse results on 24 and 5 functions. For F15 and F19, SCEO is worse compared with the EO. SCEO beats IEO in 18 cases and is inferior in 11 cases. SCEO is better than LWMEO on 25 functions, but SCEO is inferior on F11, F13, F15, and F18. For F15, F17, and F19, IEO and BPSO have better results than SCEO. For F11, LWMEO and BPSO are superior to SCEO. According to the average ranking, SCEO obtains the first rank, followed by IEO, EO, LWMEO, BPSO, VC-SSA, m-EO, OGWO, DMMFO, RDSSA, IS-EO, ASSO. SCEO outperforms EO on 27 functions and underperforms on 2 functions. SCEO outperforms VC-SSA on all functions. Overall, these data reveal that SCEO is significantly superior to its competitors on the CEC2017 test suite.

Recently, many metaheuristics-based MRPP techniques have been developed. Wang et al. [27] use a modified SSA variant to tackle the MRPP problem. Wu et al. [28] harness an modified ant colony optimization (ACO) for the MRPP task. Wang et al. [29] introduce an improved SSA and apply the novel SSA variant to the MRPP problem. Luan et al. [30] propose an MRPP method based on a hybrid genetic algorithm (HGA).

In this subsection, we adopt the SCEO to plan the best collision-avoidance path. Acknowledging the intricacies of the MRPP problem, an adaptation function with the route length and obstacle avoidance is established to assess the mass of the produced paths based on the SCEO algorithm in the MRPP problem. The fitness function is described below.

(27)fit=kL+(1−k)Ewhere k is the weight, L is the route length, E is the path collision penalty function.

In this subsection, we utilize the developed SCEO for the MRPP task. The SCEO is compared with some classical optimization techniques such as GWO [31], ABC [32], PSO [10], FA [11], and VC-SSA [18]. It is examined in the three maps provided in [33]. The general parameter settings are the same as those in the original work. The shortest path lengths generated by the six algorithms on the three maps are shown in Table 6. The trajectory routes for six algorithms are shown in Figs. 3–5.

From the data in Table 6, it can be viewed that the path planning task SCEO-based plans the shortest path for collision avoidance on all three terrains compared with the other approaches, which illustrates that the SCEO can get satisfying paths for simple and complex environmental settings in the MRPP task. The trajectory routes produced by all algorithms are described in detail below.

Figs. 3a–3f displays the comparison of the trajectory routes of the six algorithms in the first map. As can be noticed from the figure, all methods can plan a safe route because this environment is a simple terrain. Compared with other algorithms, SCEO has the shortest path. PSO, ABC, and GWO map an identical path, which is smooth but not optimal. FA, VC-SSA, and SCEO design an alternative trajectory route. The route of VC-SSA has more inflection points. FA has a smooth path but not its optimal path. Figs. 4a–4f depict the trajectory routes of the second environment map. Upon examining the figures, it becomes evident that SCEO yields the shortest path among the showcased options. SCEO, VC-SSA, ABC, and FA produce similar trajectory routes, while GWO and PSO provide another trajectory routes. The route of ABC gets stuck in local optimum, which results in redundancy of the produced trajectories, and thus the path is the longest. The paths of VC-SSA, GWO, PSO, FA are smooth but not optimal. The best trajectories of environment map are shown in Figs. 5a–5f. GWO, ABC, and PSO provide a trajectory route through the outside of the danger area, which avoids collision with the obstacle area but increases the fuel burn of the robot. The route of FA has more inflection points and increases the path length. The route of VC-SSA is smoother but not optimal. In conclusion, SCEO is the most favorable algorithmic tool, and is the superior algorithm among all the techniques.