Enhancing analogy-based software cost estimation using Grey Wolf Optimization algorithm

Similarity function

The similarity function measures the similarity degree between two different projects and is a critical component in the ABE method. Since the similarity functions use different structures to measure the distance between projects, selecting various types may affect the projects selected as the nearest neighbors. Therefore, selecting the similarity function is of great importance. Among many different similarity functions, the Euclidean distance-based similarity and the Manhattan distance-based similarity are the most popular methods (AlMutlaq, Jawawi & Arbain, 2023; Bardsiri et al., 2013; Benala & Mall, 2018; Dashti et al., 2022; Huang & Chiu, 2006; Khatibi Bardsiri & Hashemi, 2016; Nasr & Mohebbi, 2023; Shahpar, Bardsiri & Bardsiri, 2021).

The similarity function is a fundamental component of the ABE method, as it determines the degree of resemblance between a new project and historical projects. This function quantifies how closely the features of two projects align, thereby forming the basis for selecting the most similar historical projects, known as the nearest neighbors.

In its standard form, the similarity function can be expressed as follows (Eq. (1)): (1) $$sim\left(p.p^{\prime }\right)=f\left(Lsim\left(f_1.f_1^{\prime }\right).Lsim\left(f_2.f_2^{\prime }\right).\dots .Lsim\left(f_n.f_n^{\prime }\right)\right).$$

In this equation, the terms p and p′ represent the prospective project and an existing project from the dataset, respectively (Amazal, Idri & Abran, 2019). The features of these projects are denoted by $f_i$ ,and $f_i^{\prime }$, where n is the total number of features. The function $Lsim()$ computes the similarity for each pair of features, and the results are aggregated using $f()$, which synthesizes these individual similarity scores into a single comprehensive similarity value for the project pair.

The function operates in two main stages. First, for every feature of the projects, the similarity between the feature values of the current project and prior ones is computed using $Lsim\left(f_{\mathrm{i}},f_{\mathrm{i}}^{\prime }\right)$. This process evaluates how closely each feature of the new project aligns with its counterpart in the historical dataset. The resulting similarity scores for all features are then combined using the $f()$, function, which generates a unified similarity score reflecting the overall resemblance between the projects (Gautam & Singh, 2017).

In our study, we extended this standard formulation by introducing two additional parameters, w and δ, alongside a feature weighting mechanism. The parameter w represents the weight assigned to each feature, taking values between 0 and 1 to account for the varying importance of features in the similarity calculation. Meanwhile, δ is a small constant added to the denominator to avoid division by zero and ensure numerical stability. These enhancements allow the similarity function to dynamically adjust to the characteristics of different datasets, making it more robust and effective in practice.

By incorporating these refinements, the similarity function not only ensures a more precise comparison of projects but also enhances the reliability of the ABE process, contributing to improved software effort estimation accuracy.

Euclidean distance: The Euclidean distance is a widely used similarity measure that calculates the straight-line distance between two projects in a multi-dimensional space. Mathematically, it is defined as Eq. (2):

(2) $$\mathrm{s}\mathrm{i}\mathrm{m}\left(\mathrm{p}.{\mathrm{p}}^{\prime }\right)={\displaystyle \frac{1}{\left[\sqrt{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{i}}\mathrm{D}\mathrm{i}\mathrm{s}\left({\mathrm{f}}_{\mathrm{i}}.{\mathrm{f}}_{\mathrm{i}}^{\prime }\right)+\mathrm{\delta }}\right]}\delta =0.0001}$$

$$\{\begin{array}{c}{\left({\mathrm{f}}_{\mathrm{i}}-{\mathrm{f}}_{\mathrm{i}}^{\prime }\right)}^{2}iffeaturesarenumeric\\ 1iffeaturesarenumericand{\mathrm{f}}_{\mathrm{i}}={\mathrm{f}}_{\mathrm{i}}^{\prime }\\ 0iffeaturesarenumericand{\mathrm{f}}_{\mathrm{i}}\ne {\mathrm{f}}_{\mathrm{i}}^{\prime }\end{array}$$where $\mathrm{p}$ and ${\mathrm{p}}^{\prime }$ are the two projects, ${\mathrm{f}}_{\mathrm{I}}$ and ${\mathrm{f}}_{\mathrm{i}}^{\prime }$ represent the ith feature of projects $\mathrm{p}$ and ${\mathrm{p}}^{\prime }$, respectively. ${\mathrm{w}}_{\mathrm{i}}\in \left[0.1\right]$ is the weight of that feature. Furthermore, $\mathrm{\delta }$ is a small constant to prevent the denominator from being zero, and n represents the number of total features. The $Dis()$ function represents the mathematical operation performed to calculate the distance or similarity. $Dis()$ computes the squared differences between corresponding features of the projects, followed by summation and normalization.

Manhattan distance: This similarity is defined based on the Manhattan distance, which is the sum of total absolute value distances for each feature pair and is represented in Eq. (3).

(3) $$\mathrm{s}\mathrm{i}\mathrm{m}\left(\mathrm{p}.{\mathrm{p}}^{\prime }\right)={\displaystyle \frac{1}{\left[{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{i}}\mathrm{D}\mathrm{i}\mathrm{s}\left({\mathrm{f}}_{\mathrm{i}}.{\mathrm{f}}_{\mathrm{i}}^{\prime }\right)+\mathrm{\delta }\right]}\delta =0.0001}$$

$$\{\begin{array}{c}\left|{\mathrm{f}}_{\mathrm{i}}-{\mathrm{f}}_{\mathrm{i}}^{\prime }\right|\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{c}\\ 1\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{f}}_{\mathrm{i}}={\mathrm{f}}_{\mathrm{i}}^{\prime }\\ 0\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{f}}_{\mathrm{i}}\ne {\mathrm{f}}_{\mathrm{i}}^{\prime }.\end{array}$$

As with the Euclidean distance, p, p′, f_i, f_i′, w_i, and δ retain the same definitions.

There are other types of similarity criteria in the background, such as maximum distance-based similarity, Minkowski distance-based similarity (Angelis & Stamelos, 2000), and rank mean similarity, which is the mean of ranking value of each feature of the project (Walkerden & Jeffery, 1997). Table 1 summarizes some similarity functions employed in previous studies.

Some background studies have compared the performance of different similarity functions; however, as Table 1 illustrates, the Euclidean and Manhattan distance-based similarity are the most popular ones due to the simple geometrical distance definition they provide for the distance between two points in a k-dimension Euclidean space.

Moreover, studies such as those (Angelis & Stamelos, 2000) concluded that the Euclidean, Manhattan, and maximum distance-based similarities provide almost the same results, probably affected by the dataset selection. This claim is also confirmed in a study performed by Huang & Chiu (2006). In total, no solution for this problem determines when to prefer what similarity function so far. The present article employs the Euclidean and Manhattan similarity functions based on the results of Table 1.

According to the similarity function in relations Eqs. (1) and (2), it seems different features may have different significances for the similarity functions. For example, “function points (FPs)” are more important than the “programming language” in many cost models. Furthermore, numerous researchers state that there is a high potential for improving the precision of ABE by assigning proper weights to proper features. In this regard, multiple studies have determined the optimal weight for each feature (feature weighting). Some of the most important ones are introduced in the related studies section.

K-nearest neighbors

The KNN algorithm is pivotal in analogy-based SCE. After calculating distances, KNN identifies the ‘k’ nearest projects (analogies) that are most similar to the unseen project. Choosing the appropriate “k” is vital for accurate results. The parameter ‘k’ denotes the count of closest neighbors considered in the estimation (Ali & Gravino, 2021). A small ‘k’ can lead to classifications that are highly sensitive to noise or outliers. In contrast, a large ‘k’ may result in overly generalized decisions, missing important local data details. Therefore, it is crucial to assess the data’s features and the specific project requirements to determine the optimal “k” value. This optimal value balances capturing significant patterns and minimizing the impact of noise.

In this study, we selected k = 3 after conducting preliminary experiments and reviewing similar studies in the field. Several studies have demonstrated that choosing k = 3 provides a balance between minimizing noise sensitivity and preserving the local structure of the data , leading to improved estimation accuracy (Bardsiri et al., 2013; Benala & Mall, 2018; Dashti et al., 2022). This value was chosen as it ensures that the algorithm captures meaningful patterns without overfitting or underfitting the model.

Solution function

Solution function is a critical component in ABE methods, responsible for determining the predicted value (e.g., effort) based on the characteristics of similar projects. It aggregates the outcomes of selected analogies to produce a single estimation value. Commonly used solution functions in ABE include closest analogy, mean, and median, as they balance the influence of outliers and ensure a robust estimate of software development effort (Jorgensen & Shepperd, 2006). Furthermore, addressing noise in historical datasets and assigning appropriate weights to project features are essential for enhancing solution function reliability (Madari & Niazi, 2019). Research shows that choosing a suitable solution adaptation technique in ABE significantly impacts accuracy (Phannachitta et al., 2017). However, selecting an appropriate solution function relies on the data distribution and the particular demands of the estimation task (Rashid et al., 2025). In the following, different types of solution functions employed in software cost estimation are presented.

• Closest analogy method estimates cost by identifying the single most similar historical project and using its cost value as the estimate. This approach assumes that the most similar project provides the best reference for the new project.

• Mean method involves averaging the cost values of several similar projects to obtain an estimate. This technique helps to smooth out anomalies or variations in individual projects, providing a more balanced estimate.

• Median method calculates the cost by finding the middle value of the cost values from the selected similar projects. This approach reduces the impact of outliers and extreme values, offering a more robust estimate in the presence of skewed data.

Model training

In this model, the effort (cost) feature in the datasets is considered the target feature or dependent feature, and the rest of the features are classified as independent features for software development cost estimation. In the training section, whole projects are initially divided into three different sets, i.e., training (60%), validation (20%), and testing (20%). This split is a widely adopted practice in machine learning and SCE studies as it ensures a balanced distribution of data for model training. The chosen 60/20/20 split aligns with recommendations from previous research, which suggest that this proportion offers a good balance between training and evaluation while maintaining the reliability of the results (Bardsiri et al., 2013; Karimi & Gandomani, 2021). In this setup, the training set is used to train the model, while the validation set plays a crucial role in hyperparameter tuning and model selection, ensuring that the trained model generalizes well before final evaluation. The test set remains completely unseen during training and validation and is solely used for final performance assessment. By maintaining this clear separation between training, validation, and testing, we ensure that the model’s predictive performance is evaluated on entirely new data, reducing the risk of overfitting. The first two sets are used to train and optimize the model, and the third set is used to test the model. The training and testing projects are compared to the base projects for the appropriate weights to be found from the training projects and to evaluate the precision of the estimation model through the testing projects.

Initially, when the model is being trained, the weight of features could be set either equally (all zeros or ones, for example) or randomly. It should be noted that the random value must be chosen from interval [0, 1].

In this part, one project is selected as the target project in each iteration and undergoes Euclidean or Manhattan similarity functions. Every time the project goes through these functions, a series of optimized weights are generated in the interval [0, 1] and are assigned to the independent features. The optimized weights are generated using the grey wolf algorithm and are injected into the model. Then, the comparison operation is performed between the selected project and the training datasets for the KNN to be found.

When the most similar projects are found using the similarity measures defined in Eqs. (2) and (3), the amount of cost is calculated through the solution function—which can be mean or median—for the selected project. In fact, the most similar projects refer to the projects in the historical dataset that exhibit the highest similarity to the new project based on their features. The captured cost is evaluated using our intended evaluation criterion, i.e., MRE. This trend is repeated until there is no project to be selected in the training datasets. The architecture of the training part is shown in Fig. 2.

Model testing

In this part, test datasets are used to determine the precision performance of the model trained in the previous part. This part is mainly similar to the previous part. The only difference is that this part uses the optimized weights generated in the model training part, and the grey wolf algorithm is not executed again.

In this part, a project is selected from test datasets and sent to the similarity functions; then, the generated optimal weights in the model testing part are put alongside each feature as coefficients and, like the previous method, the number of KNN and the solution function is executed as well. Figure 3 shows the architecture of the testing part.

Evaluation criteria

Evaluation criteria are essential for the experimental validation of cost estimation methods. In this study, we evaluated the performance of the proposed model using MMRE and PRED(q), which are widely adopted in ABE research. These metrics were selected because they specifically measure the relative estimation error, making them more suitable for assessing the accuracy of effort estimation models (Idri, Azzahra Amazal & Abran, 2015; Jorgensen & Shepperd, 2006). While absolute error-based metrics such as mean absolute error (MAE) and mean squared error (MSE) are commonly used in regression-based models, they do not account for the proportionality of the error in relation to the actual effort values, which is crucial in ABE studies. Additionally, R² is more applicable in models where the goal is to measure variance explained by independent variables, whereas in ABE, the focus is on minimizing relative estimation errors to improve prediction accuracy. In the following, each evaluation criterion is introduced separately.

The magnitude of relative error (MREi): Error criterion in the class of relative error-based criteria, known as MREi, is defined as Eq. (11):

(11) $$\mathrm{M}\mathrm{R}{\mathrm{E}}_{\mathrm{i}}=\left|{\displaystyle \frac{{\mathrm{C}}_{\mathrm{i}}-{\hat{\mathrm{C}}}_{\mathrm{i}}}{{\mathrm{C}}_{\mathrm{i}}}}\right|$$where ${\mathrm{C}}_{\mathrm{I}}$ indicates the actual cost of the ith project, and ${\hat{\mathrm{C}}}_{\mathrm{I}}$ represents the estimated cost for the ith project.

The mean magnitude of relative error (MMRE): MMRE is defined as Eq. (12):

(12) $$\mathrm{M}\mathrm{M}\mathrm{R}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}\times \sum _{\mathrm{i}=1}^{\mathrm{n}}\left|{\displaystyle \frac{{\mathrm{C}}_{\mathrm{i}}-{\hat{\mathrm{C}}}_{\mathrm{i}}}{{\mathrm{C}}_{\mathrm{i}}}}\right|=\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n}\left(\mathrm{M}\mathrm{R}\mathrm{E}\right)}$$where $\mathrm{n}$ indicates the number of projects to be estimated, ${\mathrm{C}}_{\mathrm{I}}$ indicates the actual cost of the ith project, and ${\hat{\mathrm{C}}}_{\mathrm{I}}$ represents the estimated cost for the ith project. Low amounts of MMRE indicate a low level of estimation error.

Prediction at level q (PRED(q)): PRED is the percentage of the predictions lying in a specific percentage of the actual cost (Eq. (13)):

(13) $$\mathrm{P}\mathrm{R}\mathrm{E}\mathrm{D}\left(\mathrm{q}\right)={\displaystyle \frac{1}{\mathrm{n}}\times \sum _{\mathrm{i}=1}^{\mathrm{n}}\mathrm{\delta }\left(\mathrm{M}\mathrm{R}{\mathrm{E}}_{\mathrm{i}}-\mathrm{q}\right)}$$

$$\mathrm{\delta }\left(\mathrm{x}\right)=\{\begin{array}{c}1x\le 0\\ 0x>0\end{array}$$where q is a predefined threshold. PRED(q) calculates the percentage of predictions whose MRE values are equal to or less than q. Most of the articles set q to 0.25 or 0.30. We set q to 0.25 because the comparison articles set the same value for q (Ahadi & Jafarian, 2016; Beiranvand & Zare Chahooki, 2023; ul Hassan et al., 2022).

The median magnitude of relative error (MdMRE): The MdMRE is the median of the MREs, which is shown in Eq. (14) and is calculated using the median function.

(14) $$\mathrm{M}\mathrm{d}\mathrm{M}\mathrm{R}\mathrm{E}=\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left(\mathrm{M}\mathrm{R}\mathrm{E}\right).$$

Datasets

We used four different, freely available datasets in this study, which are vastly utilized in the research community of software effort estimation. We believe utilizing different datasets may help provide a more transparent evaluation. Research has shown that leveraging multiple datasets in software effort estimation can provide more robust and generalizable outcomes (Rahman, Goncalves & Sarwar, 2023). The utilized datasets in this study include Desharnais, Albrecht, China, and Maxwell. Table 2 provides the details of these datasets, including the number of features and their related information.

The Desharnais dataset contains data from 81 real-world software projects, making it one of the most commonly used datasets in this domain. It provides essential attributes such as the actual effort required for project completion (measured in person-months), the size of the development team, the function point’s number (FP) as a measure of software functionality, and the total development time in months (Karimi, Gandomani & Mosleh, 2023). These attributes are particularly relevant for estimating software development efforts. The Albrecht dataset, introduced by Albrecht and Gaffney in 1983, originates from IBM software projects. It has been pivotal for FPA and software project estimation, exploring the relationship between software size and the effort required for development. The dataset includes information about the number of function points, source lines of code (SLOC), and actual effort in person-hours, which are key metrics for assessing software project performance. The China dataset includes information on 499 software projects executed by Chinese companies (Corazza et al., 2013). It includes 19 distinct attributes that focus primarily on functional components such as input, output, query, file, and interface. These components are integral to calculating function points, a critical metric for estimating both software size and development effort. The substantial size and extensive features of this dataset render it highly valuable for assessing the generalizability of SCE models (Wen et al., 2012). The Maxwell dataset comprises 62 industrial software projects, predominantly developed in Finland by major commercial banks (Ali & Gravino, 2021). This dataset provides insights into real-world software development processes and includes attributes such as project size, measured in terms of lines of code (LoC), qualitative metrics for project complexity, and the total effort required for each project (measured in person-hours). Its focus on industrial applications enhances its relevance to real-world scenarios. All datasets and developed codes used in this study are available in the Supplemental Files to ensure the reproducibility of our work.

These datasets indicate an exciting set of software projects, including a unique software company or different companies’ data, illustrating various application domains and projects’ features.

Experiment design

In SCE problems, data preprocessing is necessary before injecting it into the model. Otherwise, the training quality will decrease dramatically. For this, we use the min-max equation for the values of the independent features to lie in the interval [0, 1], so-called normalization (Eq. (15)). Normalization makes all features have a similar impact on the dependent feature.

(15) $$X^{\prime }={\displaystyle \frac{X-X_{min}}{X_{max}-X_{min}}}$$where $X$ represents the original value, $X_{min}$ and $X_{max}$ denote the minimum and maximum values of the feature, respectively, and $X^{\prime }$ and is the normalized value (Krishnan et al., 2021). This approach has been extensively employed in research on SCE (Wani, Giri & Bashir, 2019).

In the next step of the model training, we should divide the datasets into training and testing datasets. For this, we use the cross-validation method. Here, we employ a three-fold cross-validation technique, which is widely used in SCE studies (Amazal, Idri & Abran, 2014; Huang & Chiu, 2006; Li, Xie & Goh, 2009a; Rahman et al., 2023). Datasets in this method are threefold base, training, and testing datasets, which are selected randomly from the intended dataset. In this process, the data is partitioned into three groups: two groups are utilized as training datasets, while the remaining group is designated as the testing dataset. This division is repeated three times to ensure that each group serves as the testing dataset once (Guo et al., 2023). Table 3 shows the three-fold cross-validation.

About the cross-validation, it should be noted that the number of projects in each set is the same. The performance is measured for two separate sequences in each step, whose mean is considered as the resulting step. The final results are determined by averaging the mean values across the three steps, ensuring consistency and reliability in the performance evaluation.

We use the Manhattan and Euclidean similarity functions in this study. We set k = 3 for the number of nearest neighbors because most articles have set this value for k (Alsaadi & Saeedi, 2022; Benala & Mall, 2018; Dashti et al., 2022). We use the mean and median for the solution function. To design the experiment, we systematically evaluated different combinations of the control parameters—similarity function, k value, and solution function—to identify the optimal configuration. Finally, we select the fittest resulting model to be compared to other algorithms.

In the GWO algorithm, the encoding scheme represents the feature weights in ABE as a vector comprising continuous values bounded within the range [0, 1] (Shahpar, Bardsiri & Bardsiri, 2021). Each element of the vector corresponds to the weight assigned to a particular feature in the similarity function, ensuring consistency with the normalized data. The fitness function evaluates the quality of each candidate solution based on its ability to minimize the mean magnitude of relative error (MMRE). The fitness value for each wolf is calculated as Eq. (16):

(16) $$Fitness={\displaystyle \frac{1}{n}\sum _{i=1}^n\left|{\displaystyle \frac{ActualEffor{t}_i-EstimatedEffor{t}_i}{ActualEffor{t}_i}}\right|}$$where $ActualEffort$ is the observed effort for a project, $EstimatedEffort$ is the predicted effort, and n is the total number of projects in the training set.

For GWO, the population size was set to 8 as it produced the best results in our preliminary experiments. This value was chosen based on empirical testing to balance exploration and exploitation while ensuring computational efficiency. The number of iterations was set to 200, based on our preliminary experiments, which indicated that this value provides a balance between convergence speed and computational cost. A higher number of iterations showed diminishing improvements in accuracy, while lower iterations led to insufficient convergence. This approach ensures that the optimization process runs efficiently without unnecessary computational overhead. The convergence parameters (a, A, and C) were determined following the approach outlined in Mirjalili, Mirjalili & Lewis (2014). The parameter a linearly decreases from 2 to 0 over iterations, allowing the algorithm to transition from exploration to exploitation. The coefficient A and C is dynamically adjusted using the Eqs. (6) and (7). These adjustments help the algorithm maintain an effective balance between global and local search, preventing premature convergence to local optima.

A standardized experimental setup was implemented to ensure the results’ reliability. All algorithms were tested on the same datasets, under identical preprocessing and evaluation conditions. Each algorithm was executed 30 times to account for the stochastic nature of metaheuristic optimization, and the reported results represent the mean performance over these runs to ensure statistical reliability. For a fair comparison, we adopted the default parameter settings for baseline methods from their original research articles, as these configurations have been optimized and widely validated in prior studies. This ensures that each algorithm operates under its best-recommended settings without arbitrary modifications that could introduce bias. Furthermore, all experiments were conducted on the same hardware and software environment to eliminate any discrepancies due to computational resources. The datasets used in this study remained unchanged across all experiments, ensuring that each algorithm was evaluated on identical data partitions. Table 4 summarizes the key parameters and their values used in this study to provide a clear overview of the experimental settings.

Execution time analysis

To provide a fair comparison of computational efficiency, we measured the execution times of all AI algorithms under identical hardware and software conditions. Table 5 presents these times, demonstrating that GWO-ABE maintains a competitive efficiency performance.

The results indicate that while traditional ABE is the fastest due to its simplicity, evolutionary algorithms such as GA, PSO, and ABC require longer execution times due to their iterative optimization processes. GWO effectively balances accuracy and computational efficiency, outperforming several metaheuristic algorithms while maintaining a competitive execution speed.

This observation is consistent with previous research on GWO, which demonstrated superior exploration and exploitation balance, better convergence speed, and robustness in solving optimization problems compared to PSO and DE (Mirjalili, Mirjalili & Lewis, 2014; Shial, Sahoo & Panigrahi, 2023). GWO’s ability to dynamically balance exploration and exploitation through adaptive parameter tuning allows it to avoid local optima more effectively, leading to higher accuracy with relatively low computational cost compared to other evolutionary techniques.

Efficiency comparison

For comparing different utilized algorithms in this field, it should be considered that datasets, evaluation criteria, and sampling distances are selected the same, so the results are reliable. This research attempts to compare the performance of the proposed model with other methods. One consideration is that the evaluation criteria of the estimator functions are the same to perform a sound comparison. As mentioned before, the value of k is set to three. The performance of the model is initially tested with the base model ABE and then tested with evolutionary algorithms used in this field, including PSO-ABE, GE-ABE, DABE, ABC-ABE, and LEMABE. These methods were selected as they are frequently cited in the literature and have been extensively analyzed in the related work section. All mentioned estimation models are trained using historical datasets, ensuring consistency in evaluation, while the algorithm parameters are set automatically based on their recommended configurations from previous studies.

The required experiments were conducted on a PC equipped with a Ryzen 7900 CPU, an Nvidia RX 580 8GB GPU, and 16GB of RAM. Additionally, the model was implemented using MATLAB 2022. Tables 6–9 display the comparative results across various datasets.

As can be seen from the above tables, the amount of MMRE and MdMRE is lower in the GWO-ABE algorithm compared to the other algorithms. For a more transparent indication, in Figs. 4 to 7, we show two methods according to the Euclidean similarity of different sets. Next, we evaluate these two with the Manhattan similarity, whose results are shown in Figs. 8 to 11.

In Figs. 8 to 11, it can be seen that the proposed algorithm has managed to yield better results compared to the other algorithms. It should be noted in the charts that the proposed algorithm achieves better results in the Euclidean distance compared to the Manhattan distance. As mentioned before, the PRED (0.25) is another evaluation criterion, which is investigated. For the resulting information from this criterion to be represented with more clarity, we express it using the Euclidean and Manhattan distances as well as mean and mean solution functions separately for different datasets in Figs. 12 to 15.

As can be seen from Figs. 12 to 15, the PRED (0.25) criterion in the proposed algorithm has managed to yield more desirable results.

Desharnais dataset

According to the main results (Tables 6–9), when the Euclidean distance measure was applied, DABE exhibited the lowest MMRE (0.31061), closely followed by GWO (0.31265), suggesting superior prediction accuracy compared to other methods. For MdMRE, PSO-ABE demonstrated the lowest value (0.16667), indicating its proficiency in generating accurate predictions as well. Conversely, a higher PRED value signifies better prediction performance. ABC-ABE and GWO yielded the highest PRED values (0.5671 and 0.57319, respectively), showcasing their effectiveness in prediction. These observations imply that DABE excels at reducing mean errors, PSO-ABE handles median errors proficiently, and ABC-ABE and GWO identify the largest share of acceptable predictions.

Under the Manhattan distance measure, DABE again showcased the lowest MMRE (0.39899), closely trailed by LEMABE (0.4326), suggesting their effectiveness in generating precise predictions based on the Manhattan distance measure. For MdMRE, GA-ABE and ABC-ABE demonstrated the lowest value (0.3142), followed by GWO (0.3634), indicating their proficiency in accurate predictions. ABC-ABE exhibited the highest PRED value (0.61111), closely followed by GWO (0.59524), showcasing their effectiveness in prediction according to this metric.

Overall, DABE and LEMABE excelled in MMRE, while GA-ABE, ABC-ABE, and GWO performed well in MdMRE and PRED.

Maxwell dataset

In the context of the Euclidean distance measure, ABC-ABE stood out with the lowest MMRE value of 0.29997. Following closely is LEMABE with an MMRE of 0.45036, indicating its effectiveness in generating accurate predictions using this measure. About MdMRE ABC-ABE showcased the lowest MdMRE at 0.2254, closely trailed by LEMABE at 0.25833. These results underscore the proficiency of both algorithms in generating precise predictions based on the median. For PRED, GWO demonstrated the highest PRED value at 0.71569, closely followed by ABC-ABE at 0.53186. These results highlight the strength of GWO and ABC-ABE in terms of prediction according to this metric. A higher PRED value signifies better prediction performance, making GWO and ABC-ABE stand out in this regard under the Euclidean distance measure.

In the realm of the Manhattan distance measure, ABC-ABE exhibited remarkable performance, displaying the lowest MMRE at 0.28048, closely followed by LEMABE at 0.38598, showcasing their effectiveness in accurate predictions using this distance measure. ABC-ABE demonstrated exceptional proficiency, securing the lowest MdMRE of 0.19511, closely trailed by LEMABE at 0.21429, indicating their aptitude in generating precise predictions based on the median. ABC-ABE also emerged as a frontrunner, displaying the highest PRED value at 0.6973, highlighting its strength in prediction according to this measure under the Manhattan distance measure. While GWO still performed well in certain criteria, Maxwell highlights scenarios where ABC-ABE may edge out other approaches.

China dataset

In the China dataset, when analyzing the results using the Euclidean distance measure, In the context of MMRE, GWO exhibited the lowest MMRE value of 0.6083, showcasing its remarkable predictive performance. Conversely, PSO-ABE displayed the poorest performance in MMRE with the highest value of 0.56695 among all methods. Shifting the focus to MdMRE, GWO again showcased the most impressive performance with the lowest MdMRE value of 0.3182. On the contrary, PSO-ABE displayed the least effective performance in MdMRE with the highest value of 0.33986, emphasizing its comparatively poorer predictive capability. Considering PRED, GWO outperformed all other methods with the highest PRED value of 0.5288, suggesting its strong predictive abilities. Conversely, GA-ABE exhibited the weakest performance in PRED with the lowest value of 0.1328, indicating its limitations in prediction according to this metric.

In the examination of results using the Manhattan distance measure, Within the domain of MMRE, GWO emerged as the frontrunner with the lowest MMRE value of 0.6862, denoting its exceptional predictive accuracy. Conversely, ABE demonstrated the least effective performance in MMRE, displaying the highest value of 1.8934, marking it as the weakest performer in this regard among all methods considered. Shifting the focus to MdMRE, GWO once again exhibited the most outstanding performance, boasting the lowest MdMRE value of 0.5556. Conversely, ABE showed the poorest performance in MdMRE, having the highest value of 1.2063, underlining its limited predictive capability based on the median. Considering PRED, GWO stood out with the highest PRED value of 0.2130, affirming its robust predictive abilities. Conversely, PSO-ABE demonstrated the weakest performance in PRED, showcasing the lowest value of 0.085213, suggesting its limitations in prediction according to this metric.

Albrecht dataset

In the evaluation of project cost estimation using the Euclidean distance measure for the Albrecht datasets, GWO exhibited superior predictive accuracy, achieving the lowest MMRE (0.36446) and MdMRE (0.3795) values, showcasing its proficiency in generating precise predictions, especially based on the median. Conversely, ABE displayed the highest MMRE (0.73303), indicating the least accurate predictions. Moreover, GWO demonstrated the highest PRED value (0.55556), implying robust prediction capabilities, while PSO-ABE exhibited the lowest PRED value (0.38889), suggesting relatively weaker prediction performance.

In the assessment of project cost estimation employing the Manhattan distance measure for the Albrecht datasets, GWO demonstrated superior predictive accuracy, displaying the lowest MMRE (0.38648) and MdMRE (0.44444) values, indicating precise median-based predictions. Conversely, PSO-ABE exhibited the highest MMRE (0.80701), signifying less accurate predictions. GWO also stood out with the highest PRED value (0.55195), denoting strong prediction capabilities, while ABE showcased the weakest predictive performance with the lowest PRED value (0.0125).

In light of these observations, Table 10 provides a concise overview of which method ranked best for each metric (MMRE, MdMRE, PRED) under Euclidean and Manhattan distance measures. While some algorithms such as ABC-ABE and DABE dominate specific metrics in particular datasets (e.g., Desharnais or Maxwell), GWO stands out in the China and Albrecht datasets for most or all metrics.

As part of a deeper comparison, we computed how GWO-ABE’s performance improves (or deteriorates) against each competing method (ABE, GA-ABE, PSO-ABE, DABE, LEMABE, ABC-ABE) under the Euclidean+Mean setting. We then averaged these percentage improvements across all four datasets. Tables 11 through 14 depict the results for each dataset in metrics MMRE, MdMRE, and PRED; Table 15 compiles the final overall improvement when summing up all comparisons.

As shown in Tables 11–14, each dataset reveals unique comparative insights when evaluating GWO-ABE against other ABE-based methods under the same configuration. In most cases particularly for Desharnais, China, and Albrecht GWO-ABE demonstrates significant reductions in MMRE and MdMRE, as well as increases in PRED(0.25). Nonetheless, there are instances (e.g., with LEMABE in the Maxwell dataset or ABC-ABE in certain metrics) where the competing methods achieve lower errors or higher PRED values. These variations confirm that dataset characteristics, algorithmic design, and optimization strategies all contribute to the final performance.

To provide a consolidated view of how GWO-ABE compares with other ABE-based methods under identical conditions, we aggregated the percentage improvements in MMRE, MdMRE, and PRED(0.25) across all datasets and competing methods. These improvements were first computed for each dataset (Desharnais, Maxwell, China, Albrecht) relative to each algorithm (ABE, GA-ABE, PSO-ABE, DABE, LEMABE, and ABC-ABE), then averaged to yield a single value per metric. Table 15 compiles the final overall improvement when summing up all comparisons.

As shown in Table 15, GWO-ABE yields a moderate but consistent reduction in error rates, indicated by its positive average improvements of 15.63% for MMRE and 3.64% for MdMRE. More strikingly, its improvement in PRED(0.25) exceeds 100%, suggesting that, on average, GWO-ABE more than doubles the percentage of acceptable predictions compared to the other methods. This marked performance in PRED(0.25) can be attributed to the algorithm’s capacity for assigning more suitable weights to key project features, thus improving the accuracy of identifying projects with similar attributes.

While certain methods occasionally surpass GWO in specific metrics or datasets, the summarized findings suggest that GWO-ABE remains highly competitive overall. It achieves moderate reductions in average error (MMRE and MdMRE) and exhibits a substantial increase in PRED. Consequently, GWO-ABE appears to be a particularly promising enhancement to analogy-based cost estimation, combining robust error minimization with a high likelihood of generating practically usable predictions.