Sigmoidal Mathematical Models in the Planning and Control of Rigid Pavement Works

Abstract

The objective of the research was to use sigmoidal mathematical models for the planning and control of rigid pavement works. A dataset was constructed using 140 technical files, which were then analyzed to extract the valued work schedules. These schedules contained the variables time and cost per month. Subsequently, two groups were created from the dataset: a training group comprising 80% of the data and a test group comprising the remaining 20%. Subsequently, the variables were normalized and adjusted with the proposed logistic, Von Bertalanffy, and Gompertz models using Python 3.11.13. Following the implementation of training and validation procedures, the logistic model was identified as the optimal fit, as indicated by the following metrics: R² = 0.9848, MSE = 0.0026, RMSE = 0.0506, and MAE = 0.0278. The implementation of the aforementioned model facilitates the establishment of an early warning system with a high degree of effectiveness. This system enables the evaluation of the discrepancy between the actual progress and the planned progress with an R² greater than 98%, thereby serving as a robust instrument for the adjustment and revalidation of activities before and following their execution.

1. Introduction

The construction industry exerts a significant influence on global economies, with road projects being the most in demand. These projects contribute to the enhancement of trade, the effective connection of cities, and the facilitation of the transportation of goods and people [1]. In this context, meticulous planning and effective construction control are indispensable for the success of engineering projects, which involve a multitude of resources, including labor, equipment, materials, logistics, and transportation [2]. The role of the project manager is to ensure the timely delivery of projects through cost control, the identification and measurement of deviations between planned and actual work performance, and the coordination and supervision of resources to avoid interference between the construction site configuration and work [3]. Project delays are a significant challenge, and effective planning and scheduling are critical to ensure project delivery and control costs, which vary significantly between countries due to factors such as topography, climate, wage levels, availability of materials, energy resources, and economic and institutional conditions [4]. Conventional methodologies employed in the realm of road project planning are characterized by their protracted nature and susceptibility to inaccuracies. This inherent variability gives rise to unreliable S-curves, which, under their representation, are prone to misrepresenting the project’s status at a given juncture [5]. This phenomenon often gives rise to cost overruns, which, in turn, impede the meticulous planning and modeling of construction operations. Cost estimation is imperative during the planning and scheduling stages for effective decision-making. Consequently, the development of mathematical models has been proposed as a means to enhance the precision and dependability of cost estimations in the road construction industry [6]. The utilization of planning optimization models has been put forth; nevertheless, these possess inherent limitations in their scope, as they have been developed for small-scale case studies with predefined exercises, which hinders the consideration of the dynamic nature of the construction process throughout its life cycle. Conversely, although sigmoidal models, including the logistic and Gompertz models, have been extensively employed to represent the cumulative progress of construction projects, their direct application to rigid pavement works is constrained. This phenomenon can be attributed to their inability to effectively adapt to the planning curves that are characteristic of road projects. These planning curves typically exhibit a more accelerated dynamic range compared to that of building works [2,4,7].

In response to this challenge, researchers have proposed the utilization of non-linear regression models for the analysis of multifactor data. These models facilitate the S-curve modeling of project status and subsequent prediction of project progress. S-curves are graphical representations that have gained widespread popularity in the field of project management. These representations are employed to depict the progression of costs, labor hours, percentage of work, and other metrics as a function of the elapsed time. The characteristic pattern of these curves is marked by a gradual onset, a rapid acceleration, and a subsequent decline, culminating at the point of project completion [8]. Furthermore, it encapsulates the intrinsic characteristics of projects, where the preliminary configuration necessitates resource allocation. Large-scale projects commence with a limited number of tasks but rapidly transition to concurrently managing multiple activities, which results in elevated expenses compared to the initial stages. These tools have evolved into essential instruments for project planning, control, and execution. Owners, managers, and contractors utilize these tools for various purposes, including financial forecasting, cash flow management, performance monitoring, and evaluation of project schedule compliance.

The contribution of this research is that it presents an alternative that improves the accuracy and effectiveness in the estimation of project execution time and costs, compared to traditional methods, by using models based on sigmoidal functions that allow for the efficient description and prediction of the duration of the project stages. The exhaustive review of the literature reveals that, although there are multiple models to represent the evolution of costs and progress, few address the adaptability to scenarios with high variability and dynamic conditions. In this context, the present work is part of the current trend towards digitalization and data-driven optimization in the construction industry by offering an innovative tool that improves project planning and control and can be integrated with emerging technologies such as BIM methodology and artificial intelligence. This, in turn, facilitates informed decision-making by decision-makers, optimizing the use of resources and minimizing delays, additional costs, and potential penalties.

The objective of the research was to use sigmoidal mathematical models for the planning and control of rigid pavement works, for decision-making, and projections during the execution and maintenance of works.

The sections of this paper are organized as follows: Section 1 contextualizes the situation under study. Section 2 presents a comprehensive review of the literature related to sigmoidal models applied to cost estimation and control in construction projects, highlighting previous mathematical and computational approaches. Section 3 describes in detail the materials and methods employed, including the collection and normalization of data from 140 rigid pavement projects, as well as the formulation and fitting of the proposed sigmoidal models. Section 4 presents the results obtained, showing the comparative performance of the models using statistical indicators such as R², RMSE, and AIC. Section 5 is devoted to the analysis and interpretation of the prediction results, emphasizing the robustness and versatility of the proposed model in comparison with the logistic, Von Bertalanffy, and Gompertz models. Finally, Section 6 presents the conclusions of the study, highlighting the practical implications for planning, monitoring, and cost control in rigid pavement construction projects.

2. Literature Review

In the domain of construction, methodologies have been put forth to enhance the planning of construction projects through the utilization of nonlinear models, incorporating cost and project duration as key performance indicators [9] (refer to Table 1). Enz [10] employed the logistic function to predict variations in income elasticity as economies progress, using the S-curve, in which the income elasticity of demand is equal to specific low and high-income levels.

Lu et al. [11] proposed an S-curve model to predict construction waste generation using a dataset comprising more than five million waste disposal records from 9850 projects in Hong Kong, providing contractors with a forecasting tool for the amount of waste to be generated, as well as a detailed baseline for waste management during construction.

Nashwan et al. [12] created a database of 145 road construction projects to generate atomic models for each road construction operation. These atomic models encapsulated productivity equations and influencing factors to automate job scheduling. This approach enabled him to assess various resource allocation options within a range of scenarios, facilitating the precise determination of productivity and unit cost for road activities. This, in turn, enabled the development of a road project construction schedule.

Hsieh et al. [13] proposed an S-shaped regression model for the management of working capital in construction companies in Taiwan. S-curves were utilized to regulate the progression of projects, and a fuzzy inference model was implemented to ascertain the optimal distribution of data. The findings indicated that the proposed model enabled construction companies to optimize their liquidity and profitability by strategically managing their cash and current assets during various stages of the project.

Chao and Chien [14] proposed a methodology for estimating S-curves in construction projects. This methodology is based on 101 projects and combines a polynomial function with neural networks. A comparison of the model with existing formulas was conducted, and the model was found to be accurate. The model incorporated four distinct factors as input variables: the contract amount, the duration of the contract, the nature of the work, and the geographical location. The findings indicated that the methodology reliably mitigates errors and is beneficial for owners and contractors in financial planning and schedule-based estimate verification.

Szóstak [15] proposed a polynomial model for the modeling of cumulative cost curves using third-degree polynomials, using data from 28 construction projects in various investment sectors. The costs that had been budgeted and those that were incurred were graphically represented, thus enabling the identification of significant patterns in the evolution of costs. Through meticulous analysis, it was determined that the most accurate fitting curves corresponded to higher degree polynomials (sixth degree) for the three research groups evaluated. This finding resulted in greater accuracy in the representation of the actual data.

Table 1. Models applied in the planning and control of construction projects.

Author	Model	Application
Chao et al. [16]	Artificial Neural Network	Planning and control of buildings
Hsieh et al. [13]	$y=a_k{x}^k$	Planning and control of buildings
Erzaij et al. [7]	Support Vector Machine	Planning and control of housing, schools, stadiums, and parametric port complexes
Hsieh et al. [8]	Fuzzy model	Working capital management in construction
Cristóbal et al. [5]	$f(x)={\displaystyle \frac{1}{1+e^{-\alpha (t-T_0)}}}$	Planning and monitoring of the physical progress of construction projects
Chao et al. [14]	$Pt={\displaystyle \sum _{i=1}^n{w}_{i}x{P}_{ti}}$	Planning and monitoring of the physical progress of Taiwan’s second expressway
Skitmore et al. [17]	$y=\mathrm{erf}({\displaystyle \frac{\ln x-a}{b}})$	At the beginning of a construction project, when some installments have been paid and estimates of future installments are needed
Kenley et al. [18]	$\ln \left[y/(1-y)\right]=a+b\left\{\ln \left[x/(1-x)\right]\right\}$	Net buffering for construction projects based on the logit transformation
Vahdani et al. [19]	Neuro-Fuzzy Model	Prediction model based on a new neuro-fuzzy algorithm for estimating time in construction projects
Szóstak [15]	$\begin{array}{l}y=-5.3155·x^6+2.0428·x^5+12.0040·x^4\\ -11.4940·x^3+3.7482·x^2+0.0096·x\end{array}$	Sixth-degree polynomial-based model for determining the shape and course of cost curves in construction projects

Table 1. Models applied in the planning and control of construction projects.

Author	Model	Application
Chao et al. [16]	Artificial Neural Network	Planning and control of buildings
Hsieh et al. [13]	$y=a_k{x}^k$	Planning and control of buildings
Erzaij et al. [7]	Support Vector Machine	Planning and control of housing, schools, stadiums, and parametric port complexes
Hsieh et al. [8]	Fuzzy model	Working capital management in construction
Cristóbal et al. [5]	$f(x)={\displaystyle \frac{1}{1+e^{-\alpha (t-T_0)}}}$	Planning and monitoring of the physical progress of construction projects
Chao et al. [14]	$Pt={\displaystyle \sum _{i=1}^n{w}_{i}x{P}_{ti}}$	Planning and monitoring of the physical progress of Taiwan’s second expressway
Skitmore et al. [17]	$y=\mathrm{erf}({\displaystyle \frac{\ln x-a}{b}})$	At the beginning of a construction project, when some installments have been paid and estimates of future installments are needed
Kenley et al. [18]	$\ln \left[y/(1-y)\right]=a+b\left\{\ln \left[x/(1-x)\right]\right\}$	Net buffering for construction projects based on the logit transformation
Vahdani et al. [19]	Neuro-Fuzzy Model	Prediction model based on a new neuro-fuzzy algorithm for estimating time in construction projects
Szóstak [15]	$\begin{array}{l}y=-5.3155·x^6+2.0428·x^5+12.0040·x^4\\ -11.4940·x^3+3.7482·x^2+0.0096·x\end{array}$	Sixth-degree polynomial-based model for determining the shape and course of cost curves in construction projects

3. Materials and Methods

A database was constructed by extracting the cost and time of the planning of rigid pavement execution projects, which were executed between 2020 and 2024. This database was then utilized as a source of information to analyze the relationship between cost and execution time using different sigmoidal models. Consequently, four models were proposed: The proposed model, the logistic model, the Von Bertalanffy model, and the Gompertz model were adjusted to the training database (80%) using parameters a, b, and c, which define the shape of the sigmoidal curve. These parameters were optimized to achieve the optimal alignment between the models and the real data of the projects analyzed. Finally, the sigmoidal models were evaluated with the validation data (20%) of the curves generated from the execution of the rigid pavement projects. This evaluation was conducted to ascertain their representation and adjustment capacity regarding the historical data. The objective was to select the most adequate model to describe the planning behavior of the rigid pavement projects (see Figure 1).

3.1. S-Curve

This visual representation offers a quantitative depiction of the cumulative progress of a construction project over its duration [20]. It is utilized by owners and contractors for project planning and control, providing the foundation for forecasting cash flows and establishing a benchmark for evaluating overall progress during construction. Its S-shaped distribution is well-suited for regression in construction management and social economics, representing the budget and schedule of a project [15,21]. As illustrated in Figure 2, a standard S-curve is presented, in which the X- and Y-axes denote the project duration and percentage of completion, respectively. Each data point represents the ratio of time spent and cumulative cost valued in each month of rigid pavement execution.

During the initial phase of construction execution (Phase One), the cost curves exhibit convex behavior. From a geometrical point of view, this implies that the graph of the function is completely above its tangent at any point within the interval $< 0,x_0$. Moreover, the arc of the function connecting any two points (A, B) within this interval $< 0,x_0$ lies below or, as the case may be, coincides with the chord joining these points [15].

The traditional method for estimating an S-curve is based on a schedule of the expected timing of project activities and their percentage weight in the project. For a project of $n$ activities, the percentage of progress or estimated value for each item $t$, $P_t$, is calculated by Equation (1) [14].

$$P_t={\displaystyle \sum _{i=1}^{n}vi\times p_{ti}}$$

(1)

where $vi$= percentage weight of the activity $i$ in the project, normally determined based on the contract value of the $i$; $n$ = number of project activities; $p_{ti}$ = estimated percentage of completion of the activity $i$ at $t$.

Chao and Chien [14,16] proposed the third-degree polynomial function with two parameters a, b to generalize the S-curves, as shown in Equation (2):

$$y=a{x}^3+b{x}^2+(1-a-b)x$$

(2)

where $y$ and $x$ denote standardized cumulative progress and time, respectively.

Equation (2) fulfills the boundary conditions of an S-curve with the points (x = 0, y = 0) and (x = 1, y = 1). This is adjusted using Equations (3) and (4), which are more concise compared to the logit transformation formula proposed by Kenley and Wilson [17]. This makes it a more convenient option for construction planning. It is possible to generate an S-curve with an inflection point, which connects two arcs (one convex and the other concave) within the defined limits, by properly setting the values of a and b. To obtain S-curves with varying geometrical properties, it is necessary to adjust the values of a and b at all data points by Equations (5) and (9).

$$a=(AB-DE)/(BC-E^2)$$

(3)

$$b=(CD-AE)/(BC-E^2)$$

(4)

where:

$$A={\displaystyle \sum x^{3}y-{\displaystyle \sum xy-{\displaystyle \sum x^4+{\displaystyle \sum x^2}}}}$$

(5)

$$B={\displaystyle \sum x^4-2{\displaystyle \sum x^3+{\displaystyle \sum x^2}}}$$

(6)

$$C={\displaystyle \sum x^6-2{\displaystyle \sum x^4+{\displaystyle \sum x^2}}}$$

(7)

$$D={\displaystyle \sum x^{2}y-{\displaystyle \sum xy-{\displaystyle \sum x^3+{\displaystyle \sum x^2}}}}$$

(8)

$$E={\displaystyle \sum x^5-{\displaystyle \sum x^4-{\displaystyle \sum x^3+{\displaystyle \sum x^2}}}}$$

(9)

3.2. Data Matrix

A data matrix was constructed for 140 rigid pavement projects executed in Peru during the years 2020–2024. The information was obtained by downloading the technical files available on the Electronic System of State Procurement (SEACE 3.0) platform. The evaluation of the variables was conducted following the stipulated work schedule, encompassing the execution time and the financial expenditure associated with the project. As illustrated in

Table 2. Descriptive statistics of the variables.

Variable	Count	Mean	Standard Deviation	Minimum	25%	50%	75%	Maximum
Time (Month)	718	2.87	2.13	0.00	1.00	3.00	4.00	8.00
Cost (US$)	718	2,556,345.00	3,344,270.00	0.00	446,702.70	1,669,521.00	3,311,307.00	24,470,880.00

, the descriptive statistics of the time (in months) and cost (in U.S. dollars) variables are presented, utilizing metrics such as the average, standard deviation, minimum, 25th percentile, 50th percentile, 75th percentile, and maximum.

As demonstrated in Figure 3, the database contains 24 elements derived from 140 projects, which are represented visually to illustrate the correlation between project duration and cost. The manifestation of these relationships is typically observed as an S-curve, reflecting the execution of various rigid pavement projects in Peru. In this context, each project is represented in a graph showing the relationship between standardized time (x-axis) and standardized cumulative costs (y-axis). The data points represent the actual data obtained during the execution of the projects, while the lines correspond to the patterns that vary between projects. These patterns are indicative of variations in the dynamics of cumulative costs as a function of time.

3.3. Sigmoidal Models

3.3.1. Proposed Model

The model is predicated on the logistic function, which describes the planning behavior of rigid pavement works. The implementation of this model enables managerial personnel to predict project progress, optimize resource allocation, and regulate cost over time. Furthermore, its adaptable structure enables the accommodation of diverse scenarios and conditions, thus establishing it as a pivotal instrument for the strategic planning and operational control of intricate projects. Equation (10) provides a complete description of the model, which guarantees $f(0)=0$ an upper asymptote $f(x)\to c$ according to $x\to \infty$.

$$f(x)={\displaystyle \frac{cb(e^{ax}-1)}{b(b{e}^{ax}-1)+1}}$$

(10)

It describes an asymmetric growth. The lower limit, when $\underset{x\to -\infty }{\lim }f(x)=0$ which represents the start of the project with no cumulative physical progress, and $\underset{x\to \infty }{\lim }f(x)=c/b$ which represents the maximum possible progress of the modeled process. These limits reflect a cumulative progression typical of construction projects, where one starts from an initial condition of zero and gradually advances to a final state.

Demonstrations of the limits of extremes of intervals 0 and 1.

Verification of $f(0)=0$

$$f(0)={\displaystyle \frac{cb(1-1)}{b(1-1)+1}}={\displaystyle \frac{0}{1}}=0$$

(11)

Limit when $x\to \infty$:

$$\underset{x\to \infty }{\lim }f(x)={\displaystyle \frac{cb(e^{ax}-1)}{b(b{e}^{ax}-1)+1}}=c$$

(12)

This derivative is positive for $a > 0$; therefore, the function is monotonically increasing. The maximum slope depends on $a$, while $b$ controls the curvature (asymmetry of growth).

$f(x)$ is the estimated cost of the project ($). $a$ controls the growth rate of the model, i.e., the rate at which the volume of work performed increases over time. $b$ is the form factor that adjusts the inflection point of the curve, determining the moment at which the maximum productivity of the project is reached. c represents the final value of the expected progress, corresponding to the total planned work at the end of the project, and $x$ is the project execution time (Month) (See Figure 4).

3.3.2. Logistics Model

It is a sigmoidal function used to model growth processes in resource-constrained systems, where initial growth is exponential, but it gradually decelerates as system constraints are reached [22,23], such as production capacity or material availability. This approach is particularly relevant in the field of rigid pavement construction, where factors such as availability of machinery, skilled labor, and budget constraints can influence demand responsiveness.

The logistics model was employed because it represents the growth of demand, allowing for the anticipation of not only the initial increase in demand as new projects are implemented or existing infrastructures are expanded, but also the eventual stagnation as production times become saturated. This integral perspective is imperative to circumvent issues such as overproduction, the squandering of resources, and delays in the execution of works. Equation (13) delineates this model:

$$f(x)={\displaystyle \frac{c}{1+e^{-(x-b)/a}}}$$

(13)

It describes a symmetric growth, where the inflection point x₀, occurs at 50% of c, marking the productivity peak. Its limits $\underset{x\to -\infty }{\lim }f(x)=0$ y $\underset{x\to \infty }{\lim }f(x)=c$ ideal for projects with balanced mobilization and demobilization. However, its symmetry may underestimate the initial slowness in some activities.

Demonstrations of the limits of extremes of intervals 0 and 1.

Verification of $f(0)=0$

$$f(0)=c·{\displaystyle \frac{L(0)}{1-L(0)}}=0$$

(14)

Limit when $x\to \infty$:

$$L(x)\to 1\Rightarrow f(x)\to c$$

Derived from $L(x)$:

$$L’(x)={\displaystyle \frac{e^{-(x-b)/a}}{a{(1+e^{-(x-b)/a})}^2}}\Rightarrow f\prime (x)={\displaystyle \frac{c}{1-L(0)}}·L’(x)$$

(15)

This model is symmetric concerning $x = b$, which makes it suitable for processes where the growth and deceleration phases are similar.

$f(x)$ is the valued cost of the project ($), $a$ is the parameter that reflects the rate at which costs increase over time, $b$ represents the moment in time when costs increase at the highest rate, $c$ indicates the expected total cost of the project in the long term, and $x$ is the project execution time (Month) (See Figure 5).

3.3.3. Von Bertalanffy Model

It is one of the most widely used mathematical tools for modeling the growth curve where the rate of change does not follow a constant pattern [24] and is useful in construction applications. In rigid pavement projects, project time and cost do not always increase linearly or exponentially; instead, they may experience phases of accelerated growth followed by periods of stabilization or deceleration, depending on multiple factors such as resource availability, technical complexity, and economic conditions.

In this model, time is considered the independent variable that drives project progress, while cost is the dependent variable that reflects the cumulative impact of decisions and events over time [25]. In contrast to conventional models that presuppose a direct and constant relationship between time and cost, the model captures the dynamic and nonlinear nature of this relationship [26]. It is imperative to anticipate and manage unanticipated cost escalations or schedule delays that may emerge during the various phases of a project. The model is expressed by Equation (16): The following text is intended to provide a comprehensive overview of the subject matter.

$$f(x)={\displaystyle \frac{c{e}^{a(x-b)}}{1+e^{a(x-b)}}}$$

(16)

The lower limit of this function is set at 0, while the upper limit, denoted by c, is a parameter of interest. The function’s asymmetry suggests a decelerated initiation relative to the logistic function, a phenomenon that can be attributed to the presence of initial bottlenecks. The inflection point, which is defined as the point at which the rate of change of a function is equal to zero, occurs below 50% of c. This property is particularly useful in cases where the initial stages of a project require a significant amount of time.

Demonstrations of the limits of extremes of intervals 0 and 1.

$$f(x)={\displaystyle \frac{c{e}^{a(x-b)}}{1+e^{a(x-b)}}}$$

(17)

Limit when $x\to \infty$:

$$\underset{x\to \infty }{\lim }f(x)=c$$

(18)

Derivative:

$$f’(x)={\displaystyle \frac{ac{e}^{a(x-b)}}{{(1+e^{a(x-b)})}^2}}$$

(19)

$f(x)$ is the valued cost of the project ($), $a$ is the parameter that reflects the rate at which costs increase over time, $b$ represents the moment in time when costs increase at the highest rate, $c$ indicates the expected total cost of the project in the long term, and $x$ is the project execution time (Month) (See Figure 6).

3.3.4. Gompertz Model

It is a sigmoid curve that describes asymptotic growth as the slowest at the end of a given period or the maximum of a given variable [27,28]. Concerning the construction field, it was applied to model the planning of activities for the control of rigid pavement works. It is particularly suitable for situations where the growth of a variable accelerates rapidly in the initial phases, but then gradually decelerates, approaching asymptotically an upper boundary [29]. In the context of rigid pavement construction, this behavior reflects the reality of many projects, where costs tend to increase rapidly at the beginning, due to the mobilization of resources and the execution of the first phases of the project, and then stabilize as the project progresses towards completion.

In this model, time is considered the independent variable driving project progress, while cost is the dependent variable reflecting the accumulation of expenses over time. The Gompertz model has been demonstrated to effectively capture the asymmetric nature of cost growth, thus providing a valuable instrument for the forecasting and management of cost escalations throughout the project life cycle. The Gompertz model equation, as applied to the planning and control of rigid pavements, is expressed by Equation (20).

$$f(x)=c{e}^{-e^{-(x-b)/a}}-c{e}^{-e^{(b/a)}}$$

(20)

The initial phase is characterized by a rapid acceleration, followed by a significant deceleration. The limits of this function are identical (0 and c), yet its inflection point is earlier. This adaptation is particularly relevant for projects characterized by accelerated mobilization yet also by concurrent deceleration of final activities, thereby affecting overall project progress. The lower limit (0) establishes the initial point for the delineation of milestones, while the upper limit, c, stipulates the physical objectives. The growth rate (x) is a metric of efficiency, with low values indicating delays and high values suggesting quality risks.

Demonstrations of the limits of extremes of intervals 0 and 1 are as follows.

Verification of $f(0)=0$

$$f(x)=c(e^{-e^{-(x-b)/a}}-c{e}^{-e^{(b/a)}})\Rightarrow f(0)=0$$

(21)

Limit when $x\to \infty$:

$$e^{-e^{-(x-b)/a}}\to 0\Rightarrow e^{-e^{-(x-b)/a}}\to e^{-1}\Rightarrow f(x)\to c(1-e^{-e^{b/a}})$$

(22)

Derivative:

$$f’(x)={\displaystyle \frac{c}{a}}{e}^{-(x-b)/a}-e^{{-e}^{-(x-b)/a}}$$

(23)

$f(x)$ is the valued cost of the project ($), $a$ is the parameter that reflects the rate at which costs increase over time, $b$ represents the moment in time when costs increase at the highest rate, $c$ indicates the expected total cost of the project in the long term, and $x$ is the project execution time (Month) (See Figure 7).

3.4. Evaluation of the Models

Statistical methods were used to select the optimal sigmoidal model, using 20% of the data for validation, using different performance indicators: Correlation coefficient (R) (see Equation (24)), which measures the linear relationship between the real data and the estimated data [7]. The coefficient of determination (R²) (see Equation (25)) indicates the proportion of model variability; Mean Square Error (MSE) (see Equation (26)) and Root Mean Square Error (RMSE) (see Equation (27)) allow quantifying the magnitude of the prediction errors [30]; Mean Absolute Error (MAE) (see Equation (28)), reflects the average deviation of the predictions for the actual values [19], and Akaike Information Criterion (AIC) (see Equation (29)) compares the quality of the different sigmoidal models according to their complexity and explanatory power [31].

$$R={\displaystyle \frac{{\displaystyle \sum (X_i-{\overline{X}}_i)(Y_i-{\overline{Y}}_i)}}{\sqrt{{\displaystyle \sum {(X_i-{\overline{X}}_i)}^2{\displaystyle \sum {(Y_i-{\overline{Y}}_i)}^2}}}}}$$

(24)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{(Y_i-\stackrel{\wedge }{Y_i})}^2}}{{\displaystyle \sum _{i=1}^n{(Y_i-{\overline{Y}}_i)}^2}}}$$

(25)

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(Y_i-\stackrel{\wedge }{Y_i})}^2}$$

(26)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|Y_i-\stackrel{\wedge }{Y_i}\right|}$$

(27)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(Y_i-\stackrel{\wedge }{Y_i})}^2}}$$

(28)

$$\mathrm{AIC}=2k-2\ln (L)$$

(29)

where $X_i$ is the time variable, $Y_i$ is the cost variable, $\stackrel{\wedge }{Y_i}$ is the predicted cost, and $n$ is the sample size; ${\overline{X}}_i$ and ${\overline{Y}}_i$ are the means of the variables, $k$ is the number of parameters in the model, and $L$ is the likelihood function.

3.5. Contributions and Limitations

Table 3. Contributions and limitations of sigmoidal models compared to other approaches.

Category	Description
Contributions	− Application of sigmoidal functions as robust tools for modeling the nonlinear behavior of physical progress in rigid pavement works. − Improved predictive capability concerning traditional baselines and empirical S-curves. − Allows dynamic control strategies by adjusting parameters. − High compatibility with BIM platforms and digital control systems. − Models allow professionals to make decisions in real time.
Limitations	− Requires historical data for model calibration. − Sigmoidal models are sensitive to inconsistent data. − Requires technical expertise for interpretation. − Does not consider exogenous qualitative variables directly.
Differences	− Chao et al. [16], Hsieh et al. [13]: Neural networks have high predictive ability, but are less interpretable than sigmoidal models. − Burhan et al. [7]: SVMs are effective for complex classification, but less suitable for modeling physical progress with temporal continuity. − Kenley et al. [18]: Their use of logit functions is compatible; the sigmoidal model extends possibilities with different variants. − Vahdani et al. [19]: Neuro-fuzzy models are powerful but less transparent; sigmoidal is simpler and more focused. − Szóstak [15]: Polynomial models tend to overfit; sigmoidal maintains stability and realism.

presents a structured synthesis of the contributions, limitations, and preliminary comparisons of the sigmoidal approach with leading works in the literature.

4. Results

4.1. Database

A total of 140 files were compiled according to the magnitude of the rigid pavement projects, extracting the monthly execution period and the total cost, as shown in Figure 8. A non-linear distribution was observed. The analysis revealed that each project exhibited distinct characteristics in terms of its duration and the associated costs of execution. To facilitate a meaningful comparison between projects, it was necessary to normalize the data adequately.

The set of ordered pairs (x, y), corresponding to the time and cost of the different projects, was normalized to the domain [0, 1]. Given that no amount had been executed at x = 0, the first values of each project were represented by the ordered pair (0, 0), which indicates the beginning of the project. The point (1, 1) indicates the end of the established period, and thus, the execution of the total amount allocated is complete.

4.2. Sigmoidal Models

S-curves are flatter at the beginning and at the end of the construction project, this is because they start and end rather slowly [32]. At the beginning of the construction process, human and financial resources are planned, contracts are signed for the planned works and contractual items with subcontractors, the development of the work is prepared, and simple preparatory work is carried out [33]. After the initial phase, the execution of the works begins to accelerate, which is directly reflected in the cost curve. The works are carried out on several work fronts using various specialized work brigades [34]. Contractors begin to take on an increasing number of tasks that are performed simultaneously. At the same time, the mutual execution of the works generates a much higher increase in costs compared to the initial and final phases of execution [35,36]. When the actual accrued cost falls below the reference S-curve, it is indicative of delays in project execution. Conversely, if the accumulated cost exceeds the reference S-curve, it is indicative of accelerated project progress beyond the anticipated timeline. In this context, the envelope generated by the proposed model was configured as an effective tool for monitoring and controlling temporal and economic performance.

4.2.1. Proposed Model

Figure 9a presents the behavior of the proposed model, using Equation (10), which describes the evolution of cost as a function of time. This model is characterized by its ability to accommodate a flexible structure, thereby facilitating the capture of the sigmoidal trend that was observed in the experimental data. The central curve is indicative of the optimal solution of the model, while the curves of the model reflect different simulations of the cost behavior over time for the various rigid pavement projects. The correlation between the proposed model and the experimental data, in the intermediate phase of the construction process, substantiates its aptitude to adequately represent the cost dynamics in paving projects.

The upper and lower limits, delineated by dotted lines, encompass the range of dispersion of iterations, thereby establishing an “iteration diagram” that facilitates planning. The lower curve delineates a time threshold at which initial costs remain minimal, followed by a precipitous increase in response to an acceleration in project execution. This behavior delineates a critical scheduling zone, where effective schedule management can avert sudden escalations in cost. The model enables the estimation of cost evolution and the establishment of efficiency margins within which projects should be planned. This promotes a resource allocation aligned with a technically and economically optimized execution.

4.2.2. Logistics Model

Figure 9b shows the logistic model defined by Equation (13), to model the evolution of cost as a function of time, using the modified formula of the traditional sigmoid model. This model utilizes the modified formula of the traditional sigmoid model. The model’s general behavior adequately represented the increasing trend observed in the experimental data in the intermediate sections of the construction process. The iteration curves were found to be consistently grouped around the main curve, thereby demonstrating the model’s stability and consistency under various simulation scenarios.

The upper and lower limits delineated the dispersion of the iteration curves, thereby providing a useful frame of reference for the technical planning of rigid pavement projects. The lower limit of the range indicates a scenario characterized by contained costs in the initial stages, followed by a progressive increase associated with an acceleration in the execution to meet the established deadlines.

This alternative logistic model was distinguished by its presentation of a reduced transition in the slope changes, a feature that proved advantageous in its representation of construction processes characterized by gradual variations in execution rhythms.

4.2.3. Von Bertalanffy Model

As illustrated in Figure 9c, the results obtained by applying the Von Bertalanffy model, as outlined in Equation (16), are presented. This model was employed to illustrate the progression of costs over time, incorporating a functional structure that facilitated the capture of the non-linear growth characteristic of progressive construction processes.

The model demonstrated a satisfactory representation of the observed behavior in the experimental data, particularly during the initial and intermediate phases of the construction process. The iteration curves were consistently distributed around the main curve of the model, indicating stability in the simulations under different input conditions. The upper and lower bounds delineated the dispersion of iteration trajectories. The lower limit of the range represented a scenario with low costs in the early phases of the project, followed by a sustained and gradual growth, until reaching maximum levels towards the end of the standardized period. This behavior was consistent with typical patterns of physical and financial progress in infrastructure works.

In contrast to earlier logistic models, the Von Bertalanffy model exhibited a more gradual transition between growth stages, a property that rendered it well-suited for representing constructive processes that do not necessitate abrupt accelerations. However, this same characteristic may have constrained its capacity to capture dynamics with rapid changes in slope, such as those that occur in critical phases of execution.

4.2.4. Gompertz Model

As illustrated in Figure 9d, the model’s application aimed to elucidate the progression of costs over time, as delineated by Equation (13). This endeavor incorporated an asymmetric growth behavior, a hallmark of processes characterized by fluctuating rates of progress over temporal domains. The model demonstrated a notable capacity to replicate the distribution of the experimental data, particularly during the intermediate and final phases of the construction process. The iteration curves exhibited a pronounced concentration around the model curve, indicative of a high degree of consistency in the simulations performed.

The upper and lower limits effectively delineated the range of variability observed in the iterations, thereby providing a useful frame of reference for project planning. The model’s distinctive configuration enabled the precise depiction of the initial gradual growth, subsequently followed by a steady acceleration and stabilization towards the culmination of the construction process. This progression aligns with numerous actual execution scenarios.

4.3. Validation of Sigmoidal Models

Figure 10 presents a comparison of the four models on the cost-time data corresponding to 28 rigid paving projects. These projects were used to validate the models. This evaluation enabled the analysis of the robustness, flexibility, and representability of each model under various empirical conditions. All models demonstrated a high degree of correlation between the observed data and the experimental results, following the characteristic sigmoid shape of the behavior of the accumulated cost as a function of time. However, discrepancies were identified in terms of local accuracy during the initial and accelerated transition phases of the construction process.

The proposed model, based on a modified logistic function, demonstrated remarkable performance in most cases. Its structural flexibility enabled the accurate capture of intermediate growth patterns, a notable feature that distinguishes it for its capacity to represent empirical trajectories with smooth and symmetrical transitions.

The logistic model demonstrated an adequate fit in global terms, adequately representing the displaced inflection points or curves with greater asymmetry, thereby enhancing accuracy in projects where the construction rhythm changed abruptly between phases.

The Von Bertalanffy model has been demonstrated to be a suitable representation of asymmetric behavior, characterized by gradual initiation and a swift escalation in expenses during the intermediate stages. This model demonstrated particular efficacy in projects characterized by a transition in construction dynamics to a strategy of progressively increasing yields, subsequently followed by a phase of stabilization.

The Gompertz model demonstrated particular efficacy in projects characterized by controlled construction progression, effectively representing gradual growth with an extended transition phase. However, its structural design tended to underestimate costs during intermediate phases in certain instances, particularly when there was a substantial acceleration in construction.

The four models offered acceptable results; the proposed model achieved an adequate balance between fit, simplicity, and adaptability, which positioned it as the most robust alternative to represent the temporal evolution of cost in the evaluated projects. The logistic model and Gompertz, on the other hand, were the most effective for asymmetric growth scenarios, while the proposed model and Von Bertalanffy presented a more conservative performance.

The parameters a, b, and c (see

Table 4. Parameter estimation of the proposed, logistic, Von Bertalanffy, and Gompertz sigmoidal models.

Parameter	Proposed Model	Logistics Model	Von Bertalanffy Model	Gompertz Model
a	1.5202	0.1096	9.4850	0.1682
b	−62.9902	0.4800	0.4834	0.4217
c	70.7842	1.0020	0.9991	1.0433

) were employed to obtain the models during training. These constants are utilized to create the mathematical model via an equation that can be employed in the execution of different rigid pavement projects. The factors that influence these elements include the efficiency of project execution, the availability of resources, and the complexity of the initial phases of the work. The project schedule and planning are also important factors, as they allow for efficient cash flow management and ensure that funds are available when costs are expected to be higher.

Table 5. Evaluation of sigmoidal models.

Statistician	Proposed Model	Logistics Model	Von Bertalanffy Model	Gompertz Model
R	0.9910	0.9924	0.9923	0.9917
R2	0.9821	0.9848	0.9844	0.9832
MSE	0.0026	0.0026	0.0026	0.0028
RMSE	0.0506	0.0506	0.0512	0.0531
MAE	0.0278	0.0278	0.0321	0.0311
AIC	−3386.0481	−3386.0521	−3373.5261	−3346.3187

shows the statistics of the different models, which presented high levels of fit, with R values above 0.991 and determination coefficients (R²) above 0.982. These figures indicated a strong correlation between the predicted and observed values, which reflected an adequate predictive capacity for the evolution of cost over time. The logistic model obtained the highest value of R (0.9924) and R² (0.9848), followed by the Von Bertalanffy model (R = 0.9923), the Gompertz model (R = 0.9917), and the proposed model (R = 0.9910).

The proposed model and the logistic model exhibited the lowest Root Mean Square Error (RMSE) value of 0.0506, thereby substantiating their superiority in the estimation of cost values. In turn, the proposed model also exhibited the lowest MAE (0.0278), thereby reinforcing its capacity to minimize the average absolute deviations between the simulated values and the real data. It is noteworthy that the Gompertz model, despite achieving a competitive visual fit in certain cases (see Figure 9d), reported the highest values of RMSE (0.0531) and MAE (0.0311), indicating a relatively lower statistical performance in comparison to the other models.

Finally, when considering the Akaike Information Criterion (AIC), which penalizes the complexity of the model for its goodness of fit, the logistic model presented the lowest value (AIC = −3386.0521), followed by the proposed model (AIC = −3386.0481). This finding suggests that both models are statistically equivalent and surpass the models proposed by Von Bertalanffy and Gompertz, whose AIC values were higher.

4.4. Application of Sigmoidal Models in a BIM Environment

An evaluation was conducted on the project aimed at enhancing the trafficability of the streets in the Nuevo Horizonte Sector, situated within the District of Jaén, Cajamarca, Peru (see Figure 11).

The project encompasses an area of 13,942.08 m² of rigid pavement, with a width ranging from 4.50 m to 9.50 m. This includes the pavement that will connect with the streets of the town of Jaén, thereby facilitating enhanced vehicular traffic flow. Furthermore, the minimum slope required for the lateral regions is set at 2%, ensuring the efficient drainage of surface water. This water will initially accumulate along the periphery of the sardinel, subsequently entering the triangular gutters and eventually connecting to the existing rainwater drainage network. The duration of this project is anticipated to be five months.

Subsequently, the project was modeled in the BIM Revit 2024 software (see Figure 12). The metrics were extracted, and the budget was prepared, yielding a direct cost of US$892,793.70, as shown in

Table 6. Project budget.

Description of Departure	Total (US$)
Temporary works	4345.26
Rigid floors	565,183.49
Sidewalks	140,634.43
Hammers and ramps	19,371.08
Concrete sardines	48,131.95
Works of art	56,842.21
Environmental mitigation	7251.35
Various	51,033.93
Total (Direct Cost)	892,793.70

.

Subsequently, the previously trained and validated logistic model was integrated into the Dynamo v.2.19.3 interface for Revit 2024, which allows the creation of custom nodes using Python. Two input variables were defined for this study: project cost and execution time. These variables were connected to the node of the logistic model, which facilitated the generation of the project planning S-curve (see Figure 13). This curve was subsequently employed in the execution phase.

Following the programming of the script in Dynamo, a user interface was implemented using Dynamo Player. The objective of this implementation was to facilitate the application of the script in the planning of the rigid pavement project. This interface enables users to interact intuitively with the model, entering the necessary parameters without requiring advanced programming knowledge, thereby optimizing the integration of the model in the design and execution workflow of the project (see Figure 14 and Figure 15).

5. Discussion

The research addressed a critical factor in the impediment of rigid pavement works in Peru: deficient planning. In this context, sigmoidal models were employed, which have demonstrated efficacy in the scheduling of construction projects. Kenley and Wilson [18] demonstrated that their model exhibited an adequate fit in 75% of the projects that were analyzed. In a similar vein, Castro-Lacouture et al. [37] employed fuzzy mathematical models to ascertain construction schedules and assess contingencies engendered by schedule compression and delays due to material shortages that were not foreseen. Their findings underscore the significance of prioritizing activities with minimal residual capacity for adjustment, as opposed to the mere allocation of materials to activities that are prepared to commence immediately. This finding lends credence to the hypothesis that mathematical models can be utilized to represent common patterns in construction planning.

The database used in this research consisted of 140 rigid pavement projects in Peru, associated with local regulations, construction practices, availability of materials, and particular climatic conditions with cost and time variables for training the sigmoidal models. This is similar to that presented by Chao and Chien [14], who used 101 real projects to develop a predictive model based on four factors: contract amount, duration, type of work and location, using these as input variables to estimate the S-curve parameters and thus proposed a cubic polynomial function, which showed advantages in both accuracy and simplicity over traditional formulas.

This geographical and sectoral restriction limits the extrapolation of the models to other types of infrastructure or regions with different construction dynamics. Furthermore, the data under consideration primarily concentrated on cost and duration variables, neglecting to incorporate qualitative factors such as the level of technology employed, contractor experience, or technical complexity. These qualitative factors could have a substantial impact on the temporal behavior of physical progress. To address the variability of cost data due to temporal and geographic differences, a statistical normalization process was applied. This process standardized all cost values within a range from 0 to 1. This transformation allowed for the elimination of the direct influence of monetary units and absolute disparities in the values recorded. Consequently, comparability between projects executed in different geographic and economic contexts was fostered. By focusing on the ratio of cumulative progress to total project cost, the model can capture the relative behavior of financial progress without being affected by macroeconomic factors such as inflation or regional differences in material or labor prices.

To reduce the gaps in the research literature, an alternative approach based on sigmoidal mathematical models, specifically adapted to S-curves, was proposed. This adaptation allowed the construction of a more flexible model adjusted to the real behavior observed on site, especially in the planning of road projects. The model achieved R² higher than 98%, exceeding the performance of classical models, which coincides with Cristóbal et al. [5], who pointed out that, over time, multiple mathematical formulations have been developed to estimate S-curves in construction projects, including polynomial functions, exponential and transformational approaches, which are limited and not applicable to road works projects.

In comparison with previous models, such as the support vector machine model presented by Erzaij et al. [7], our model achieved an R² of 0.94 and an MAE of 3.6000, which contrasts significantly with the proposed model, logistics model, Von Bertalanffy model, and Gompertz model in this research, which obtained a higher R² from 0.9821 to 0.9848 and a lower MAE from 0.0278 to 0.0321. Likewise, the polynomial models of Szóstak [38] and Ostojic-Skomrlj [39], with R² between 0.90 and 0.98, are lower than those obtained in the research, where an R² of 0.9821 to 0.9848 was obtained with the sigmoidal models. Vahdani et al. [19], using neuro-adaptive fuzzy logic, reported an MAE of 2.0198 and an MSE of 5.2017, higher values of error than the proposed models, where an MAE of 0.0278 to 0.0321 and an MSE of 0.0026 to 0.0028 were obtained, which are lower which indicating the reliability of the models. Models based on neural networks such as those of Chao and Chien [14,16] and Lu et al. [11] presented a low RMSE (0.0355 and 0.0552), comparable with the RMSE of sigmoidal models from 0.0506 to 0.0531, demonstrating that these have reliability levels similar to the models generated by artificial intelligence algorithms (see

Table 7. Mathematical models applied to project planning and control.

Reference	Model	R2	MAE	RMSE	MSE
Erzaij et al. [7]	Support vector machines	0.9400	3.6000	0.0700	-
Szóstak [38]	6-degree polynomial	0.9500–0.9800	-	-	-
Ostojic-Skomrlj [39]	6-degree polynomial	0.9000–0.9700	-	-	-
Vahdani et al. [19]	Neuro-adaptive fuzzy logic	-	2.0198	-	5.2017
Chao and Chien [16]	Neural Networks and Matching Progress	-	-	0.0355	-
Chao and Chien [14]	Artificial neural networks	-	-	0.0552	-
Lu et al. [11]	Artificial neural networks	-	-	-	0.0043
Szóstak [15]	Third-degree polynomials	0.9800	-	-	-
Hsieh et al. [8]	Fuzzy model	0.9400	-	-	-
Present study	Logistics Model	0.9848	0.0278	0.0506	0.0026
	Von Bertalanffy model	0.9844	0.0321	0.0512	0.0026
	Proposed model	0.9821	0.0278	0.0506	0.0026
	Gompertz model	0.9832	0.0311	0.0531	0.0028

).

Sigmoidal, polynomial, and artificial neural network models demonstrate significant discrepancies in terms of structure, requirements, and applicability in rigid pavement projects. Sigmoidal models are distinguished by their capacity to depict the temporal progression of construction activities. Their “S”-shaped structure adapts to the actual sequence of events, encompassing a gradual initiation, an accelerated intermediate phase, and a subsequent progression toward completion [11]. Their simple mathematical structure, in conjunction with moderate data requirements and high interpretability, positions them as an ideal tool for environments where decision-making must be efficient and traceable. Conversely, despite their simplicity of implementation, polynomial models are subject to limitations when extrapolating dynamic scenarios and are vulnerable to overfitting, which undermines their accuracy in real construction contexts [14]. Conversely, artificial neural networks boast high predictive capacity and adaptability; however, they require substantial data volumes and considerable computational resources. Moreover, their processes are opaque, which may impede their acceptance in engineering environments where the explanation and justification of results are paramount [16]. Consequently, sigmoidal models offer an optimal balance between adaptability, simplicity, and applicability, making them particularly well-suited for the planning and control of rigid pavement works in real construction industry contexts.

Concerning other sigmoidal models studied, such as Von Bertalanffy (R² = 0.9844) and Gompertz (R² = 0.9832), the logistic model maintains a slight superiority in all met-rics, while the model proposed in this research, with an R² of 0.9821, presents the same error values (MAE and RMSE), which confirms its robustness and power for its application in large rigid pavement projects allowing greater accuracy.

Elghaish et al. [40] presented an innovative model that integrates techniques such as Earned Value Management (EVM), Activity-Based Costing (ABC), and BIM methodology. The objective of this integration is to optimize the cost structure and establish an equitable risk and reward distribution mechanism among the parties involved in the project. This model enables automation and enhances precision in cost control and activity scheduling. This approach bears resemblance to sigmoid models, which aim to accurately represent the cumulative growth and progress of complex projects or systems over time.

Moreno et al. [41] developed a procedure and analysis using discrete event simulation to compare traditional scheduling methods with the Fixed Start Method (FSM) in repetitive projects. The findings indicate that FSM has a substantial impact on reducing variability in project completion times and enhancing schedule stability. This enhancement in predictability and time control is associated with the logic of sigmoidal models, which represent the cumulative behavior of growth processes with defined phases of acceleration, maturity, and deceleration. The FSM endeavors to standardize and optimize project execution processes, aiming to ensure consistency and predictability in outcomes. In contrast, sigmoidal models adopt a quantitative and continuous approach, elucidating the dynamics of project progress over the entire life cycle.

6. Conclusions

The findings of the research have demonstrated the applicability of sigmoidal models as effective tools for planning and progress control in rigid pavement projects. The findings indicate that the logistic and Von Bertalanffy models demonstrate optimal levels of fit, attaining coefficients of determination of 0.9848 and 0.9844, respectively. These models demonstrated a high degree of accuracy in representing the cumulative evolution of cost as a function of time, exhibiting satisfactory adaptability to the typical characteristics of S-curves.

The proposed models not only represent the dynamic behavior in construction planning but also generate a sigmoidal envelope composed of the earliest and latest time curves. This approach establishes an early warning system that identifies discrepancies between actual and planned progress, enabling the technical team to make timely corrective decisions. The presence of actual values below the curve signifies delays, while those above reflect accelerated execution.

Furthermore, the models’ capacity to delineate upper and lower limits furnishes a quantitative framework for reference, thereby enhancing the monitoring of project performance and facilitating the optimization of schedule and resource management. In construction contexts characterized by elevated uncertainty or ambiguous objectives, these models have demonstrated their efficacy as robust instruments during the planning and execution phases.

In terms of performance, the Von Bertalanffy model was effective in stable environments, while the Gompertz model and the proposed model showed adequacy in scenarios with asymmetric growth and controlled acceleration, so that the proposed model is effectively adapted to large-scale projects since its symmetry and lower error allow it to obtain greater reliability. This methodological versatility allows the analysis to be adapted to different types of projects. On the other hand, the logistic model, since it presents less dispersion, is a preferred option for general applications, opting for the different models are applied depending on the case of the project.

The sigmoidal models used in this study allow accurate modeling of the time evolution of the cost, constituting a solid methodological basis for the development of intelligent control and scheduling systems in the field of road construction engineering.

While the findings of this study demonstrate the efficacy of the proposed flexible sigmoidal model in representing the physical–financial progression in rigid pavement projects, the model necessitates precise and consistent historical data for its calibration. This may present a challenge in contexts where detailed or structured records are not readily available. Furthermore, sigmoidal models exhibit sensitivity to the presence of inconsistent or atypical data, which can adversely impact their predictive capability if rigorous debugging and validation processes are not implemented.

In light of the aforementioned limitations, subsequent research endeavors will entail the incorporation of the proposed sigmoidal models within Building Information Modeling (BIM) environments, with a particular focus on 5D platforms that consider the interplay between the dimensions of time and cost. The integration would facilitate direct linkage of the estimates of physical–financial progress with the components of the three-dimensional digital model. This linkage would, in turn, allow for the simulation of different construction scenarios, the continuous evaluation of deviations from the planned schedule and budget, and the strengthening of support for operational decision-making in real time. This methodology, which integrates mathematical modeling and digital tools, signifies a substantial advancement toward a more efficient, automated, and transparent management of road infrastructure projects, in accordance with the principles of engineering 4.0.