1 Introduction

Construction scheduling is a critical component in project management, where optimized schedules significantly influence project success. Researchers have explored various optimization techniques, such as linear programming historically [1, 2]. Multi-objective linear programming has been particularly effective for handling resource-constrained problems like construction costs, project duration, resource idle time, and delivery schedules [3, 4]. Particle swarm optimization has also been proposed to address resource constraints in construction scheduling [5] while metaheuristic algorithms have been widely applied across diverse scheduling challenges in construction, including job-shop scheduling, concrete precast production optimization, resource leveling, and building performance assessment [6-9]. Recent studies integrate Building Information Modeling (BIM) with Multi-Objective Optimization (MOO) algorithms to enhance decision-making in construction management [10, 11]. This approach aims to synergize information systems with optimization methods to improve building design and management processes. Baghalzadeh Shishehgarkhaneh et al. [12] propose a framework combining BIM and the Fire Hawk Optimizer (FHO) to optimize project scheduling by maximizing quality while minimizing time, cost, risk, and environmental impact. Also, different analytical and numerical methods have been used by researchers to model and solve various engineering problems [13-23].

Mehmet Yılmaz and Tayfun Dede [24] investigate the time–cost trade-off dilemma in construction projects, employing non-dominant sorting (NDS) within optimization algorithms such as Rao-1 and Rao-2. They assess these methods against PSO, ACO, GA, and TLBO for small to medium-scale problems to gauge their effectiveness. Wa et al. [25] introduce an enhanced non-dominated sorting genetic algorithm (INSGA-II) tailored for time and cost optimization in construction, demonstrating its effectiveness in managing complex scheduling challenges under diverse resource constraints. Abhilasha Panwar and Kumar Neeraj Jha [26] propose a multi-objective scheduling framework (MOSM) utilizing a non-dominated sorting genetic algorithm to optimize time, cost, resources, and environmental impact concurrently, adapting solutions based on project-specific priorities. These approaches underscore the evolving use of advanced optimization techniques in improving construction scheduling efficiency and effectiveness. So far, there has been little to no research on the Whale Optimisation Algorithm (WOA) for optimizing construction schedules. Therefore, in this study, we aim to focus on WOA. Prior to that, we will explore the application of WOA in scheduling in other domains.

Understanding the stability principles of the structure is essential for developing a cohesive construction schedule. These rules are typically stored in a well-defined matrix format known as Directed DSM (Design Structure Matrix). By utilizing the rules encoded in this matrix, the WOA can effectively generate desired construction schedules that adhere to structural stability requirements. The algorithm introduced in this research endeavors to create construction schedules from the ground up, employing a fitness function. This necessitates utilizing the stability matrix derived from the Building Information Modeling (BIM) of the project. The main goal of this Whale Optimization Algorithm (WOA) is to enhance the constructability of the project to its fullest extent. This entails finding project schedules that ensure all elements feasible and secure only when its two end structural supports (such as columns or beams) were installed previously. The core operations of the WOA utilized in this research include encircling, shrinking, and hunting and evaluation of fitness based on the defined constructability criteria. These operations are pivotal in iteratively refining and optimizing the construction schedule to align with the project's structural stability requirements.

2 Model and Geometry

The currently developing algorithm is designed to support only specific structural elements from the IFC file format in BIM projects, specifically columns (IfcColumn) and beams (IfcBeam). To streamline calculations, these beams and columns are represented as lines with bounding boxes around them, as illustrated in Figure 3. According to the IFC standard [62, 63], dimensions can be simplified to their start and end points, eliminating the necessity to account for all intricate details. In this study, the assumption is made that if two elements intersect within their respective bounding box regions, they are considered to be physically linked. Using data extracted from the IFC and stability rules are presented in Table 1.

3 Optimization Algorithm

Optimization and finding optimal states are of great importance, especially in engineering design and modeling, and there are various algorithms for optimization [64-66]. The Whale Optimization Algorithm (WOA), inspired by the hunting behavior and feeding patterns of whales, is a powerful method for solving optimization problems. Utilizing concepts such as random search and evolutionary operations, this algorithm optimizes a wide range of fields including engineering, computer science, and civil engineering, parameter optimization of machine learning algorithms, industry, medicine, and energy system optimization. Known for its high efficiency, execution speed, and adaptable implementation, WOA is recognized as an effective and well-established tool for optimizing complex problems in various practical applications and research endeavors.

3.1 WOA Main Operations

The Whale Optimization Algorithm (WOA) mimics the behavior of a whale swarm navigating through the search space to identify the best global solution. WOA includes three key operations: encircling, shrinking, and hunting. During the exploitation phase, activities like encircling and shrinking are employed, while hunting is characteristic of the exploration phase. For a multi-dimensional optimization problem, the update process for the 𝑖th whale in the 𝑡th generation with WOA is as follows [67]:

$$$$ \mathrm{Encircling}\ \mathrm{operation}:{x}_{i,j}^{t+1}={x}_{best,j}^t-A.{d}_{i,j}^t. $$$$ (2)

$$$$ \mathrm{Shrinking}\ \mathrm{operation}:{x}_{i,j}^{t+1}={x}_{best,j}^t-{e}^{bl}.\cos \left(2\pi l\right).{d^{\prime}}_{i,j}^t. $$$$ (3)

$$$$ \mathrm{Hunting}\ \mathrm{operation}:{x}_{i,j}^{t+1}={x}_{k,j}^t-A.{d^{best}}_{i,j}^t. $$$$ (4)

$$$$ A=2\left(1-\frac{t}{t_{max}}\right).\left(2r-1\right), $$$$ (5)

where t represents the current iteration number, $$$ {t}_{max} $$$ is the maximize number of iterations. $$$ {x}_{best,j}^t $$$ is the position vector of the best solution obtained so far and $$$ x $$$ is the position vector. $$$ b $$$ is a constant for defining the shape of the logarithmic spiral, $$$ l $$$ is a random number in [−1,1]. $$$ {x}_k^t $$$ represents a random position vector, selected as a random whale from the current population. Three distances are defined as follows:

$$$$ {d}_{i,j}^t=\left|2r.{x}_{best,j}^t-{x}_{i,j}^t\right|. $$$$ (6)

$$$$ {d^{\prime}}_{i,j}^t=\left|{x}_{best,j}^t-{x}_{i,j}^t\right|. $$$$ (7)

$$$$ {d^{best}}_{i,j}^t=\left|2r.{x}_{k,j}^t-{x}_{i,j}^t\right|. $$$$ (8)

The WOA updates individuals using different equations depending on the probability 𝑝. If 𝑝 is less than 0.5 and the absolute value of 𝐴 is less than 1, Equation (2) is used for updating. When 𝑝 is less than 0.5 but $$$ \mid A\mid $$$ is greater than or equal to 1, Equation (4) is applied. For probabilities 𝑝 of 0.5 or higher, Equation (3) is utilized to update the individuals. The Pseudo code of WOA algorithm is shown in Figure 4.

3.3 Fitness Function

In this research, the focus is on optimizing the constructability of project sequences using WOA. Although multi-objective WOAs can measure multiple variables, this study prioritizes a single objective: the constructability of the project schedule. Ensuring that the project schedule is fully constructible is paramount, meaning that all scheduled components must be feasible for installation. The constructability score, expressed as a percentage, is calculated by dividing the number of elements that comply with the constraints in the Matrix of Constructability Constraints (MoCC) by the total number of elements.

To compute this score, the MoCC, which details the rules and constraints for each 3D BIM element based on their geometry and interdependencies, is utilized. Using the Matrix of Whales (MoW), a function reads the whale position sequence to identify the elements scheduled for installation in each time unit. For each scheduled element, the prerequisites listed in the MoCC are checked against the MoW to ensure they have been installed earlier. Once all conditions are fulfilled, the element is considered feasible for construction. The constructability percentage is determined by dividing the count of feasible elements by the total number of elements and multiplying the result by 100. This score forms the basis for selecting genomes (schedules) in the subsequent steps. All the details regarding the fitness function described in this paper are summarized in Figure 5.

3.4 Whale Position Validation

It must be mentioned that it is highly probable that the newly position of a whale after the encircling, shrinking, and hunting operations may result in invalid project schedules. To tackle this problem, the methodology includes a “Validation Function,” which is formally defined in Equation (12).

$$$$ {\displaystyle \begin{array}{ll}& \forall i\in \left\{1,2,\dots, n\right\}\kern0.36em \exists j\left\{1,2,\dots, k\right\}:{w}_{i,j}=1\\ {}& \mathrm{and}\kern0.48em \forall i\in \left\{1,2,\dots, n\right\}\&j\left\{1,2,\dots, k\right\}\&0<c<k:{w}_{i,j}\\ {}& \kern1em =1\&{w}_{i,j+c}={w}_{i,j-c}=0\\ {}& \mathrm{or}\kern0.84em \forall i\in \left\{1,2,\dots, n\right\}:\sum \limits_{j=1}^k{w}_{i,j}=1,\end{array}} $$$$ (12)

where w_i,j is the value of element i in time-unit j in the MoW, n is total number of elements, k is total number of time-unit steps and c is any random number between [0, k]. The Validation Function examines MoW to identify elements that are either not scheduled for installation or have multiple installation times. In the former case, this function randomly assigns installation times within the project duration. In the latter case, it preserves the first installation and removes the duplicates. For example, the new MoW and its validated version are illustrated in Figure 6. In this instance, the second installation attempt of the first element is canceled, and the second element is rescheduled randomly for installation during the third time period.

Further details and explanations about this algorithm are available in Refs [7, 68-73].

4 Research Validation

The study's algorithm incorporates numerous factors and parameters. To ensure its efficacy across different scenarios and validate its practicality, it is essential to test various combinations of these factors rigorously. The chosen validation approach, “Experimental Validation and Design,” aligns with discussed methodologies [74]. Parameters and their ranges for validation are sourced from relevant literature, detailed further in the following section.

4.1 Experimental Design

During an experiment, intentional adjustments to one or more process variables are made to observe their impact on corresponding response variables. Design of experiments (DOE) serves as a powerful method for analyzing collected data and deriving reliable, unbiased conclusions while minimizing the number of required experiments. The general structure of the laboratory design is explained and available in Refs [75-79]. In this research, several variables have been identified for intentional modification. The complexity of the input 3D BIM is adjusted by varying the number of elements and connection types. Additionally, parameters related to the Whale Optimization Algorithm (WOA) are manipulated, including the population size of whales and the range of construction durations. These adjustments are crucial for examining their effects on the outcomes and validating their impact on the proposed methodologies.

Three distinct 3D models have been developed in this study to depict varying levels of complexity: simple, moderate, and complex BIMs. The focus of this research is exclusively on structural models and architectural elements; components such as piping, equipment, and HVAC systems are omitted from these models. Complexity levels of the models are determined by evaluating factors such as the quantity of structural elements, overall model size, and the types of connections between these structural elements. Screenshots illustrating the three different BIM inputs used in the methodology are displayed in Figure 7.

Figure 7 depicts three distinct models: (a) a straightforward structural model featuring 42 elements comprising 18 columns and 24 beams. Model (b) represents a more intricate structure with 42 columns and 58 beams, totaling 100 elements. The final model, (c), represents a typical turbine building structure with 274 elements, including 102 columns, 146 girders, and 26 joists this makes it among the intricate steel structures encountered in the construction industry. Derived from standard turbine building models used in power plants, this model spans the spectrum of complexity for 3D steel structures, ranging from the simplest Figure 7a to the most complex Figure 7c, as outlined in the detailed descriptions of each 3D model input.

Various researchers have suggested different optimal sizes for population parameter. Recommended population sizes range from as low as 16 [80] to approximately 20–30 [67, 81] and occasionally extend to larger ranges like 50–100 population size. Taking into account the findings of other researchers, we have chosen a population size range of 20–100. The adjustments made to the WOA parameters are detailed in Table 2.

TABLE 2. Whale optimization algorithm parameter change set.

WOA parameters	1st	2nd	3rd	4th	5th	6th	7th
Population size	20	30	50	20	30	50	100
Duration range	10% ± 20%	15% ± 20%	20% ± 20%	20% ± 10%	10% ± 10%	15% ± 10%	25% ± 10%

In order to showcase the benefits of the proposed methodology for creating project schedules, we conducted experiments using 21 distinct sets of parameters. These 21 scenarios included three variations in model complexity, as specified in Figure 7, for each of the seven WOA parameter sets detailed in Table 2. The adjustments in both WOA parameters and model complexities resulted in 21 distinct executions of the algorithm, aimed at validating its efficacy across various situations.

The goal of these experiments is to produce several full construction sequences, matching the size of the population, that satisfy the constructability and stability criteria of the model, as calculated in MoCC. Achieving this across various experimental configurations would validate the algorithm's capability to automatically generate reliable construction project schedules.

5 Results and Discussion

As described earlier, different sets of inputs were generated and fed into the proposed algorithm to assess their effectiveness in generating complete and feasible project schedules. The constructability objective, as previously explained, involves arranging project elements in a way that ensures the stability of both individual components and the overall project model. This stability is governed by the MoCC, which was calculated beforehand. Table 3 provides further details on how these different runs achieve the research objectives. The methodology was implemented on a laptop featuring the following specifications: CPU—Intel(R) Core(TM) i7-4720HQ @ 2.60GHz, RAM—16 GB, OS—Windows 11 Pro.

In Table 3, the “Run Name” field begins with labels such as Simple, Moderate, or Complex, corresponding to the input model complexities defined earlier and depicted in Figure 7 as models (a–c), respectively. Moreover, in this field, the label W denotes the number of whales in each population for the specific run, while the label T signifies the mean of the initial duration range as outlined in Table 2. The study aims to generate stable and feasible project schedules for any 3D model. Through maximizing the constructability score, the WOA algorithm targets reaching 100% in each iteration. Some experiments achieve the maximum score more quickly than others. This variation in calculation speed is influenced by several parameters, with the most significant being the input 3D model complexity and the population size.

The correlation coefficient between DEP and the calculation duration per iteration, detailed in Table 3, is +0.9, indicating a strong positive relationship. This highlights that augmenting DEP—whether through increasing 3D model elements, expanding initial duration ranges, or enlarging population sizes—leads to extended calculation times per population. Table 4 further illustrates these dynamics: DEP significantly impacts iteration calculation time (> 90%) but has a minor effect on total calculation duration (< 40%). Conversely, the number of 3D model elements moderately influences both calculation rounds and durations (> 40% and < 90%), while population size and average initial duration marginally decrease calculation rounds and total time but slightly increase each iteration's calculation duration.

Moreover, each iteration's calculation duration increases nearly in proportion to DEP increments. It also scales about half as much with changes in 3D model elements, population size, or average initial duration. Interestingly, the total number of iteration rounds correlates to half the increase in 3D model elements but decreases in relation to a third of the increase in population size or initial duration. Changes in DEP have a moderate impact on the total number of calculation rounds but can extend the overall computation time by around 30%. According to findings in Table 4, project duration scales by 76% with adjustments in 3D model elements and decreases by 10% in relation to increases in population size or initial duration.

Table 4 compiles the computed correlations, highlighting how WOA parameters significantly influence the computational demands to achieve desired outcomes. These correlation coefficients play a crucial role in refining WOA parameters for future research. Table 4 further details the varying speeds at which experiments reach their final scores, revealing a consistent trend of rapid initial score increases followed by gradual improvements as scheduling proceeds.

The methodology's developed tool not only generates dependable construction schedules but also offers a 4-dimensional depiction of the construction sequence, showing the evolution of the 3D model over time. This visual representation not only clarifies the construction process but also assesses schedule reliability and constructability.

6 Conclusion

The project schedule is essential for managing time, cost, and quality in Architecture, Engineering, and Construction (AEC) projects. Creating these schedules is challenging, relying on planners' expertise and understanding of project geometries and stability. Recently, integrating project information into a 3D model (Building Information Modeling or BIM) has gained importance. This paper introduces a method to extract information from BIM to develop construction sequences using a computer application and the Whale Optimization Algorithm (WOA). This approach ensures the structure remains stable during installation. The authors validated this method through 21 experiments, successfully generating stable construction schedules, demonstrating a new application of WOA in construction projects. The primary aim was to derive construction schedules from the geometric data within the project's BIM, and this objective was successfully demonstrated. The key contribution of this research lies in defining essential functions for the WOA to generate construction schedules based on a project's BIM. This study validates the use of geometric data embedded in structural BIM to sequence element installations feasibly, suggesting future potential for improving various construction processes. Future studies could also involve conducting detailed analyses to understand how changes in parameters affect schedule configurations, such as the influence of 3D model complexity on schedules and determining optimal gene numbers, population size, and initial durations.

Author Contributions

S. M. Golmaei: conceptualization, software, formal analysis, writing – original draft, investigation. J. Vahidi: supervision, methodology, writing – review and editing, investigation, validation. Morteza Jamshidi: supervision, writing – review and editing, formal analysis, data curation.

Conflicts of Interest

The authors declare no conflicts of interest.