A novel human-inspired solution to high-dimensional optimization problems

Leader selection

In the SPO algorithm, leader selection is motivated by the real-world practice of designating a Guru Swamy, a mature and experienced pilgrim responsible for leading the pilgrimage group. This character is metaphorically transferred to the fittest solution within the population, serving as the guiding point for all other individual devotees (solutions) during pilgrimage. The leader is chosen using a two-stage priority system based on performance and age. Every devotee is related to the count of previous participations successfully completed, referred to as visits. The devotee having the highest visit count is selected as Guru Swamy. This is in line with the respect and credibility that have been gained by experience. In case two or more pilgrims have the same number of visits, the oldest among them is elected as the leader. This guarantees the resolution of confusion in performance equivalence due to seniority. This two-stage criterion reflects the actual pilgrimage situation, where both devotion record and seniority dictate leadership.

Preparatory actions

The preparations to undertake the journey are called Kettunirakkal and involve elaborate rituals along with heavy packing. This preparation process makes Irumudi a critical item each devotee carries, representing offerings and their spiritual journey. The Leader puts the Irumudi on each devotee’s head. In reciprocation, each devotee pays a gratitude gesture. The Irumudi represents the surrender and willingness of the devotee for the spiritual journey.

The Leader is the spiritual leader and guide for the pilgrims. As the alpha figure in the group, the Leader clarifies and guides the pilgrims on several aspects of the pilgrimage, such as:

• Explain the regulation and importance of Deeksha (spiritual vow).

• Conducting and promoting devotional practices such as pujas (Prayers) and bhajans (hymns).

• Resolving doubts and providing spiritual guidance on Lord Ayyappa and the pilgrimage traditions.

At every stage of the journey, the Leader ensures that the pilgrims stay disciplined, maintain their spiritual focus, and uphold the traditions of the pilgrimage.

Satisfaction of the leader

It is said that the pilgrimage’s success depends on the Leader’s satisfaction. Pilgrims are therefore encouraged to serve and assist the Leader in all possible ways, because pleasing the Leader means pleasing Lord Ayyappa himself. A smooth and disciplined pilgrimage, under the guidance of a satisfied leader, means that the divine blessings of Lord Ayyappa will be given to the devotees, culminating in a successful Sabari Yatra (travel). By aligning devotion, discipline, and service, the journey to Sabarimala becomes a spiritual and personal transformative experience for every pilgrim.

Route allocation to reach Sabarimala

The journey to the Shri Dharma Sastha Temple at Sabarimala is an incredible spiritual journey for millions of devotees. The route chosen often depends on factors such as the time available, the physical fitness of the pilgrims, and their age. In this context, the Leader is said to be “the leader and guide,” which enables him to choose the best-fitted path for his fellow travellers, wherein spiritual accomplishment shall go hand in hand with capability. The three main traditional roads to Sabarimala are the Erumeli road, the Vandiperiyar road, and the Chalakayam road, which help pilgrims come from various zones, as each path provides a different experience.

The Erumeli route is the oldest and most traditional path towards Sabarimala, immersed in the pilgrimage’s history and rituals. The pilgrims usually take this route mainly from the northwestern region, like Kottayam, Ernakulam, Thrissur, and Mangalore. The longest and most challenging route includes forest trekking and spiritual rituals. Pilgrims typically start from Erumeli and stage the Pettathullal, acting as if Lord Ayyappa were conquering evils. The rugged terrain consists of dense jungles, deep hills, rivers, and spiritual contact inside the body through arduous physical activities and devotion towards the deity. Such a route was important since it would mean penance for Lord Ayyappa.

The Vandiperiyar route is relatively more strenuous and much enjoyed by the faithful who come in droves from northeast districts, especially Idukki, Coimbatore, Dindigul, and Madurai. This route’s starting point is Vandiperiyar, a significantly shorter route than the Erumeli route. It also traverses attractive plantation and forest paths, undoubtedly extremely quiet and soul-rejuvenating. The Vandiperiyar route may be less physical, but it still requires an excellent fitness level.

This is a perfect option if anyone wants balance in devotion but a quite manageable trek. It particularly goes well with a group of devotees with mixed capability levels. Chalakayam is the closest road to Sabarimala and is preferred for short-term travellers and those of limited capabilities. Elderly people from southwestern areas such as Kollam, Thiruvananthapuram, Tirunelveli, and Kanyakumari can enter Sabarimala this way. A short walk, at a very short distance from the Pamba River, has been considered apt for aged people who can’ go due to pain and physical sicknesses. The Chalakayam route is kept well-maintained, ensuring that the devotees can also participate in the journey without the physical demands of longer treks and yet experience an equal connection with the divine. The Leader plays an important role in deciding the group’s route, considering the pilgrims’ time constraints, age, and physical fitness. For first-time pilgrims (Kanniswamy), the Leader may encourage the traditional Erumeli route to provide an immersive experience of the pilgrimage’s rituals and challenges. On the other hand, elderly pilgrims or people with limited time may be sent along the Chalakayam route, ensuring that the pilgrimage is as spiritually enriching as possible without exceeding their capabilities.

Initialization

Initialize the no. of devotees (population), $x=\left(x_1,x_2,...,x_N\right)$, At each iteration, the current solution will be moved to a new neighbouring solution $x=\left(x_1,x_2,...,x_N\right)$, number of iterations ( $K_{iter}$), Eq. (1) represents each devotee $X_i$ as a solution vector.

(1) $$X_i=rand()\ast \left(ub-lb\right)+lb$$where $i=1,2,\dots ,N$ (total population size).

Leader selection

The Leader $G$ is chosen based on the maximum number of visits ( $V_x$) of 18 to the shrine as per in Eq. (2):

(2) $$G=argmax(V_x)$$where $V_x$ represents the visit count of the devotee $i$. As per Eq. (3), if two individuals have the same. $V_x$ the leader is selected based on the maximum age $A_x$:

(3) $$G=argmax(A_x)if{V}_x=V_y.$$

Objective function

The objective function evaluates the efficiency of each solution $x$, aiming to minimize:

(4) $$Ob{j}_{fun}=min\{f\left(x\right)|x\in X\},$$

The objective function that assesses the solution $x=\left(x_1,x_2,...,x_N\right)$ involving a set of decision variables is denoted by $f\left(x\right)$. The range of values that can be assigned to each decision variable is $x_i$ ∈ $X_i$, where $X=\{X_i|i=1,...,N\}$. $N$ is the total number of decision variables, and that $X_i\in \left[l{b}_i,u{b}_i\right]$ and $u{b}_i$ and $l{b}_i$ are the upper and lower bounds for the decision variable $x_i$, respectively.

Phase I-exploration

The devotees, who were determined to attain devotion towards God, preferred to go by the Erumeli route, and they could not consider the duration of the journey as they would fully concentrate on seeking devotion. It is the starting point for the arduous trek to the sacred Sabarimala Temple. Devotees from all over the country embark on this spiritual journey, making Erumeli a bustling town during the pilgrimage season. Devotees consider the path from Erumeli to Pamba sacred. The journey is an opportunity for introspection and spiritual growth. The distance between Erumeli and Pamba is approximately 45 kilometers. While it might seem minor, the terrain is challenging, involving steep climbs and dense forests. This makes it a physically demanding but spiritually rewarding journey. On the Erumeli route, pilgrims travel through several locations before arriving at Sabarimala. They begin the trip by praying at the shrines of Vavarswami and Dharma Sastha in Erumeli. On the road to Pampa, they stop at other temples. The other devotees mostly travel on the other two routes based on their preference.

Phase II-exploitation

The Pampa valley is the confluence of all pilgrims heading towards ‘Sabarimala’, whichever route they take. The bath in the Pamba River invigorates the pilgrim and renews his lost spirit and enthusiasm. A bath in the holy river is considered mandatory before starting the trek over the hill. The belief is that the waters will wash away the sins accrued through the current and previous births and grant salvation. The pilgrims who reach Pamba worship Aadimoola Ganapathi before starting on the final phase of the trek to the shrine itself. After touring the temple of Pamba Ganapathy, devotees took a one-way journey to Sannidhanam (worship venue).

The number ‘eighteen’ has enormous spiritual significance according to the principles of Sanatana Dharma. It is a representation of achievement. The pilgrimage can be successfully completed by ascending the holy steps. Sathyamaya Ponnu Pathinettam Padi (eighteenth step) is the colloquial term for it, which means that the holy eighteen steps represent the supreme Truth.

The sole piece of equipment a pilgrim wears on his head while on a pilgrimage is an Irumudi (a two-bundle). It can only be carried out by people who observe a 41-day fast. It is forbidden to ascend the sacred eighteen steps of the Sannidhanam without the v. There are just two uses for the Pathinettam Padi (eighteenth step): one to ascend the temple and another to lower the hill. Pilgrims break coconuts as an offering to the steps before climbing or descending them. The devotees face the sanctum sanctorum as they descend the steps backwards.

In the exploitation phase, solutions are fine-tuned so that they converge on the optimal path. The process involves adjustment of devotee positions and weights according to the relevance of the sacred Irumudi, adaptive coefficients, and random exploration through Lévy flight. The positions are updated for devotees according to the sacred presence of Irumudi, by which a devotee holding Irumudi, gets a refined updating solution through the Lévy flights and adaptive coefficient of search $\eta$ And others get ordinary, random updates. The updated equation is mathematically denoted in Eq. (17),

(17) $$X_{dev}^{new2}=\{\begin{array}{cc}{X}_{dev}+Levy\left(w\cdot X_{lead}-X_{dev}\right)\pm U\left(0,1\right)\ast \eta \hfill & ifirumudi=1\hfill \\ X_{dev}+r\cdot \left(X_{guru}-X_{dev}\right)\hfill & otherwise\hfill \end{array}.$$

The weight function is updated based on step size of $\tau$

(18) $$w\left(t+1\right)=w\left(t\right)+\tau w\left(t\right).$$

Step Size Adjustment

(19) $$\tau w\left(t\right)=-{\displaystyle \frac{n}{2}{\vartheta }_{w^e}\left(w\left(t\right)\right).}$$

(20) $$Levy=\frac{S}{{\left|J\right|}^{1/\beta }}.$$

As denoted in Eq. (18), the weight function modifies step sizes iteratively to strike a balance between broad exploration and precise exploitation. The step size adjustment, $\tau w\left(t\right)$, is defined as $-{\displaystyle \frac{n}{2}{\vartheta }_{w^e}\left(w\left(t\right)\right)}$, where $n$ is a scaling factor, and. ${\vartheta }_{w^e}\left(w\left(t\right)\right)$ is the gradient of the weight function at the current iteration, as defined in Eq. (19). This gradient will guide the direction and magnitude of adjustment to ensure the algorithm incrementally refines weight values. The Lévy distribution is heavy-tailed, allowing the algorithm to explore distant solutions and avoid premature convergence, as denoted in Eq. (20). The adaptive coefficient starts high, encouraging exploration, but decreases to increase precision as the algorithm progresses.

The Lévy distribution index in Eq. (21), denoted by $\beta$, is bounded by $0<\beta \le 2$, but $s$ and $j$ are such

(21) $$S\sim N(0,{\sigma }_s^2);J\sim N(0,{\sigma }_j^2).$$

The standard deviations ${\sigma }_s$ and ${\sigma }_j$ are determined by the Eq. (22),

(22) $${\sigma }_s={\left\{{\displaystyle \frac{\mathrm{\Gamma }\left(1+\beta \right)\ast \sin \left({\displaystyle \frac{\pi \beta }{2}}\right)}{\mathrm{\Gamma }\left[\left(1+\beta \right)/2\right]m\ast \beta \ast 2^{\left(\beta -1\right)/2}}}\right\}}^{1/\beta }and{\sigma }_j=1.$$

The symbol represents a random number having a normal distribution between 0 and 1 $U\left(0,1\right)$. A coefficient that is adaptable is η. The search scope increases with the value of η. Therefore, to balance the exploration and exploitation stages, η must be small enough to enable the algorithm to converge close to the optimal value, yet high enough to keep the program searching farther in the early iterations. Therefore, Eq. (23) reduces it from 1 to 0 in each iteration:

(23) $$\eta =1-\frac{t^{1/p}}{T^{1/p}}$$where $T$ is the maximum number of iterations and $t$ is the number of iterations that are currently occurring. As the number of iterations rises, a constant called p brings the value of $\eta$ closer to 0. In the exploitation phase, several advantages arise, including the global exploration strength of Lévy flights, dynamic adaptation, and precise convergence through weight updating. The use of spiritual relevance, such as that of the Irumudi, demonstrates how the algorithm adapts to incorporate meaningful constraints within the solution, thereby enhancing the quality of the solutions obtained.

Termination

When the maximum number of iterations, ${\mathit{K}}_{\mathit{i}\mathit{t}\mathit{e}\mathit{r}}$, is achieved, the algorithm stops ensuring a predefined computational limit. The leader’s position is returned as the optimal solution, representing the most efficient and spiritually rewarding path for the devotees. The flowchart of proposed algorithm is illustrated in Fig. 1.

The new SPO algorithm presents an innovative, human-inspired method to address complex optimization problems by simulating the collective behavior of pilgrims during the Sabarimala pilgrimage. In contrast to conventional algorithms based on solely mathematical or swarm-inspired metaphors, SPO integrates three innovations: (i) an adaptive chanting-based exploration mechanism to control agents with a memory-sensitive stochastic chanting coefficient, (ii) a distributed leader-follower route construction strategy to avoid premature convergence, and (iii) an Lévy flight-enhanced repositioning scheme with adaptive balancing between local search optimization and global exploration. This combination produces a strong and diverse search process that can efficiently deal with both low- and high-dimensional optimization issues.

Performance evaluation of CEC2020 test suite

To confirm the adaptability and stability of SPO algorithm, we compared its performance with four recent state-of-the-art algorithms PWO, TOA, HMO, and POA on ten challenging functions from CEC2020 benchmark suite. These tasks are particularly developed to evaluate an algorithm’s performance under a variety of challenges, such as multimodal, non-separable, and rotated optimization surfaces. Each algorithm was judged in terms of best fitness value, Standard Deviation (Std) for stability, and rank for comparative performance. Among all ten functions (F1–F10), SPO had a rank of 1, convincingly showing consistent superiority over the other competing approaches. Table 3 depicts the evaluation results on benchmark functions.

On F1 and F2, both unimodal functions characterized by having slender global optima, SPO had the best values of 3.75E+04 and 2.30E+01, better than all other techniques. Lower Std values of 3.80E+02 and 4.50E-16, respectively, validate strong repeatability and stable convergence behaviours of SPO even on functions having steep and elongated valleys. For F3 and F4, SPO proved to be the best again, with its optimum values of 1.40E+01 and 2.40E+01, having the lowest deviations among all the participants. This shows how SPO can traverse the rugged spaces and also steer clear of premature convergence. On F5 and F6 functions with many local minima, SPO performed outstandingly with best values of 1.10E+01 and 3.70E+03, respectively. The respective low Std values of 1.15E-01 and 7.00E+01 confirm that SPO delivers consistent solutions per run, which is vital for real-world deployment. In the same vein, for F7 to F9, involving high-dimensional, rotated, and multimodal functions, SPO consistently produced the optimal solutions (1.32E+02, 8.50E+01, and 5.40E+01 respectively) and smallest standard deviations, reflecting its versatility in both well-structured and deceptive landscapes. On F10, one of the most complicated hybrid functions, SPO once more excelled, reporting a best value of 2.20 with smallest standard deviation (1.00E−01). This indicates that SPO can effectively navigate global exploration and local exploitation even on composite problems that combine elements from multiple functions. The top-ranked performance of SPO over all functions consistently illustrates the effectiveness of its human-inspired mechanisms. Adaptive route selection, group-based learning dynamics, chanting-based position updates, and Lévy flight-based exploitation make SPO more capable of dealing with various optimization problems than traditional metaheuristics. These findings evidently demonstrate that SPO is a robust, precise, and stable optimizer, and is suitable for solving complicated real-world optimization problems, particularly those with high-dimensional, multimodal, and non-separable objective functions. Figure 2 shows the graphical representation of convergence curve.

Performance evaluation of CEC2022 test suite

To additionally assess the optimization potential of proposed SPO algorithm was made in comparison with four advanced algorithms including PWO, TOA, HMO, and POA. The analysis employed ten intricate benchmark functions (F1–F12), all of which determined the algorithms’ convergence toward optimum solutions. The CEC2022 benchmark functions used in this study are adopted from CEC Large-Scale Global Optimization (CEC-LSGO) benchmark suite, precisely designed to evaluate algorithm performance in high-dimensional, non-separable, shifted, and rotated environments. The values like Best, Mean, and Standard Deviation (Std) were captured over 30 independent runs, and Rank for SPO was recorded based on its relative ranking. The values in Table 4 persistently show that SPO performs better than all other comparison algorithms in all benchmark functions. In every function, SPO takes the lowest Best and Mean values, signifying its capability of discovering optimal or near-optimal solutions with greater accuracy. SPO also exhibits the minimum or second-lowest standard deviation, verifying its stability and consistency under repeated runs.

Table 4 shows the outcomes of the CEC2022 Benchmark functions with the proposed and the state of art algorithms. On F1 and F2, both unimodal functions, SPO obtained optimal and average results smaller than those of all other competitors, with high exploitation ability in easy search landscapes. On multimodal functions such as F3, F5, and F6, SPO also kept small mean values and small deviations, indicating its robustness to premature convergence and good exploration ability in difficult landscapes. For tougher non-separable and composite functions like F4, F7, and F10, SPO once more outperformed the rest. Specifically, on F4, SPO achieved the best value of 2.65E+01, much lower than POA’s 2.75E+01, with a better lower standard deviation, reflecting both precision and reliability. In F9 and F8, the high-dimensional, non-separable functions with high curvature, SPO’s solid performance was apparent in the lowest mean and std values, indicating that it has strong adaptation to high-dimensional, complicated search spaces. The Rank column, reporting a rank of 1 for SPO in all functions consistently, certifies its across-the-board domination. These findings are also augmented by statistical significance evidenced through invariably lower variances of standard deviations, suggesting that SPO does not depend on chance success but follows certain, repeatable performance. The convergence curves of twelve benchmark functions are shown in Fig. 3.

The SPO’s better performance is a result of its human-inspired approach, which combines exploration and exploitation effectively via adaptive route selection, leader-guided updates, Lévy flight-based exploitation, and threshold-driven dynamics. All these helps to make SPO effective in traversing both smooth and rugged fitness landscapes to obtain high-quality solutions with low performance variability. This therefore renders SPO not only theoretically grounded but also usable in practice for solving a broad set of real-world optimization problems.

Dataset 1-columnar data

The metadata used here is the Cardiovascular dataset. The dataset is taken from https://ieee-dataport.org/documents/cardiovascular-disease-dataset. There are three different kinds of input features in the dataset: subjective, objective, and examination. Factual data such as gender with a category code, height in centimeters, weight in kilograms as a float, and age in days are all included in objective features. Medical test results, such as systolic and diastolic blood pressure, cholesterol levels, which are classified as usual, above normal, and well above normal, and glucose levels, which are similarly classified as normal, above normal, and well above normal, are among the features of the examination. Subjective features reflect the information provided by patients, like smoking status, alcohol intake, and physical activity, all of which are coded as binary variables. The variable to be predicted is a binary variable that states the presence or absence of cardiovascular disease. All data was collected during medical exams.

The Cardiovascular dataset is pre-processed using data normalization, and then the optimal features are selected by the proposed SPO algorithm. The performance of SPO was compared with various state of the art models like PWO, TOA, HMO, and POA. The outcomes of the feature selection are assessed by some performance metrics like accuracy, precision, sensitivity, specificity, FPR (False Positive Rate), FNR (False Negative Rate), G-mean and computation time (s). Table 6 gives an overview of the performance of all algorithms across various evaluation metrics.

The suggested SPO algorithm performed better in all performance measures, recording the highest accuracy of 98.88%, reflecting its high capacity to accurately classify outcomes of cardiovascular disease. This represents a considerable difference compared to HMO (95.32%), and reflects an absolute difference of more than 9% from POA (87.65%). In both accuracy (97.43%) and sensitivity (97.54%), SPO surpassed the other algorithms at all times, reflecting its potential to reduce false positives and false negatives. Such dependability is essential in medical applications where misclassification can result in severe outcomes. In specificity, SPO was impressive with an 98.21%, showing its superior performance in correctly classifying non-disease cases. Additionally, SPO had the lowest FPR score of 0.0899 and FNR value of 0.0705 amongst all the algorithms, further establishing its strength in minimizing both forms of misclassification errors. The G-mean score of 97.65 attests that SPO has a high balance between sensitivity and specificity, further enhancing its stability and reliability. Another essential benefit of the SPO algorithm is that it is computationally efficient. It reported the lowest computation time at 77.45 s, significantly outperforming other algorithms. This efficiency is particularly useful in real-time or constrained resource applications. These findings collectively show that the SPO algorithm not only achieves better classification accuracy but also possesses strong generalization and high-speed computation, which makes it exceedingly appropriate for real-world medical decision support systems. Figure 4 shows the graphical representation of evaluation results.

The accuracy and precision bars emphasize SPO’s reliability and minimal misclassification. Similarly, the sensitivity and specificity graphs showcase SPO’s balance in detecting true cases and minimizing errors. The FPR and FNR charts further underline their effectiveness in reducing false results, while the G-mean graph consolidates its robustness. Notably, the computation time graph highlights SPO’s efficiency, which is critical for resource-constrained environments. Figure 2 depicts the convergence analysis of dataset 1.

At the first epoch (Epoch 0), all algorithms have relatively high Mean Squared Error (MSE) values of about 1.0, since they have not had a chance to learn or improve yet. For the fast early improvement phase, Epochs 0–20, the Sabarimala algorithm shows the greatest MSE decrease, from roughly 1.0 to 0.15, which shows very fast early learning. GSO also shows substantial improvement, reducing its MSE from ~1.0 to ~0.35. In contrast, SMO and SOA display moderate initial progress, while TLBO records the slowest improvement. In the refinement phase continued (Epochs 20–100), Sabarimala still has superior results, with an MSE close to 0, as if the solution has been found to be optimal. GSO continues, but the improvement is slow, and its MSE obtains around 0.12. SMO and SOA equally continue to improve, but the final performance is inferior to that of Sabarimala and GSO.

Dataset 2-image dataset

The image data used here is the Brain Tumor MRI dataset. This dataset is taken from https://ieee-dataport.org/documents/brain-tumor-mri-dataset. Three datasets, figshare, SARTAJ, and Br35H, have been combined to create this dataset. When classifying brain tumors, it includes all the Magnetic Resonance Imaging (MRI) pictures. This dataset contains 7,023 human brain pictures divided into four groups: pituitary, meningioma, glioma, and no tumor. The images in the “no tumor” class are sourced from the Br35H dataset. This dataset is extremely useful for designing and testing various machine learning algorithms for the automatic detection and classification of brain tumors.

In this MRI dataset, the image is segmented using the proposed SPO by hyper-tuning the Sobel edge segmentation. The corresponding outcomes for the image segmentation for the proposed and the existing methods are shown in Fig. 5. SPO outperforms other methods in image segmentation, with highly accurate boundary detection and minimal artifacts in the output. Its capability to fine-tune hyperparameters results in cleaner and more defined segmented regions, which are essential for medical imaging tasks. Compared to existing algorithms, the outputs are less refined, with noticeable artifacts and incomplete boundaries. Superior segmentation quality from SPO thus proves its adaptability and precision in handling complex datasets.

To increase the validation of proposed SPO algorithm’s robustness and segmentation ability, we compared it with four latest state-of-the-art evolutionary optimizers, namely PWO, TOA, HMO, and POA on the Brain Tumor MRI dataset. The quantitative results were observed using number of performance measures, such as accuracy, precision, recall, F1-score, Jaccard similarity, Intersection over Union (IoU), and Dice Similarity Coefficient (DSC).

Out of all the algorithms, SPO performed much better than current methods. It had the best accuracy of 0.99, overall accuracy in tumor detection as shown in Table 7. As illustrated, the precision and recall values of 0.98 each further prove that SPO can reduce both false positives and false negatives, which are essential in medical imaging when diagnostic sensitivity is of utmost importance. The 0.98 F1-score indicates SPO’s well-balanced classification performance in terms of precision and recall. The Jaccard similarity (0.95) and IoU (0.95) scores indicate SPO’s better spatial overlap between prediction and ground truth tumor areas critical for boundary definition in segmentation problems. As shown DSC score of 0.98 emphasizes very good contract between segmented tumor regions and real tumor areas, significantly higher than other state of the art models.

The superior performance of SPO on all benchmarked criteria attests to its strong adaptability, accuracy, and resilience for sophisticated medical image segmentation applications. Its algorithmic nature, with Lévy flight-exploiting search, adaptive coefficient control mechanisms, and group-based dynamic update, is critical in enhancing edge detection and region consistency, resulting in cleaner and artifact-reduced segmentations. Consistent outperformance of SPO compared to other algorithms suggests its adaptability and efficiency in dealing with the complex task of medical imaging segmentation.

The SPO algorithm shows excellent performance in terms of segmentation tasks analyzed in relation to several assessment parameters, such as accuracy, precision, recall, F1-score, Jaccard similarity, IoU, and Dice Similarity Coefficient (DSC). SPO has an impressive accuracy of 98.43% and outperformed other algorithms in the experiments, which show its superior ability to correctly segment medical images, such as brain tumor MRI scans. SPO minimized false positives at a given precision of 96.88%, and it was important to define the boundaries of a segment correctly. It had a good recall of 97.99%, reflecting that it can capture virtually all the relevant pixels in a segment and not miss important areas.

The high F1-score added a new layer to it, showing its ability to balance detections of true segments with the reduction of false predictions. Additionally, SPO excelled in spatial overlap measures, with superior scores in both Jaccard similarity and IoU, indicating precise boundary detection. The high DSC underlined the fact that the segmentation is strong, producing fine-quality segmentations free of artifacts. Consistent outperformance of SPO compared to other algorithms, namely SMO, SOA, GSO, and TLBO, suggests its adaptability and efficiency in dealing with the complex task of medical imaging segmentation. Figure 6 illustrates the convergence curve for Dataset 2.

The SPO shows the fastest initial learning in the early improvement phase (Epochs 0–20), with an almost exponential reduction of MSE down to ~0.2 starting from ~1.0. GSO also features a significant drop, from ~1.0 down to ~0.4. SMO and SOA exhibit more modest improvement, bringing MSE down to ~0.6 and ~0.7, respectively, while TLBO shows the slowest improvement, ending at ~0.8. In the continued refinement phase (Epochs 20–100), Sabarimala continues its excellent performance, reducing MSE to ~0.05 at the 100th epoch. GSO results in an MSE of ~0.28; SMO and SOA bring MSE values around ~0.2 and ~0.22, respectively. TLBO showed the slowest progress, around ~0.2. It can be noticed that the convergence efficiency and learning capability of the Sabarimala Pilgrimage are much higher than the rest.