Predicting Extension of Time and Increasing Contract Price in Road Infrastructure Projects Using a Sugeno Fuzzy Logic Model

Abstract

Road infrastructure plays a crucial role in the development of countries, significantly influencing economic growth, social progress, and environmental sustainability. Major infrastructure projects are frequently challenged by substantial risks and uncertainties, leading to delays, budget overruns, and compromised quality. These issues can undermine the economic viability and efficiency of projects, making effective risk management essential for minimizing negative impacts and ensuring project success. For these reasons, a study was conducted using a Sugeno fuzzy logic system applied to completed projects. The resulting model is based on 10 project characteristics and provides highly accurate predictions for Extension of Time (EoT) and Increasing Contract Price (ICP). By utilizing this model, project management can be significantly improved through more accurate forecasting of potential delays and cost overruns. The high precision of the Sugeno fuzzy logic system enables better risk assessment and proactive decision-making, allowing project managers to implement targeted strategies to mitigate risks and optimize project outcomes.

1. Introduction

Road infrastructure is a significant factor in the development of every country, impacting economic growth, social development, and environmental sustainability [1]. Major infrastructure projects are often faced with substantial risks and uncertainties, which can lead to delays, budget overruns, and insufficient quality of the completed work. Such issues undermine the economic justification of projects and affect the overall efficiency of infrastructure development. Risk analysis is essential for preventing adverse consequences caused by Extension of Time (EoT) and Increasing Contract Price (ICP) [2]. Although risks cannot be completely eliminated, successful projects are those where risks are managed proactively, efficiently, systematically, and continuously. Risk management is a crucial area of project management [3,4,5]. The purpose of risk management is to maximize positive factors and minimize the impact of negative factors in achieving defined project goals through the application of appropriate preventive measures.

Infrastructure projects are characterized by the specificity of the route, long duration, and complexity in implementation. Due to the complexity and uniqueness of each infrastructure project, despite the existence of methodologies for project management [2,6,7], a majority of the decisions are made based on subjective assessments and personal experience. This can lead to inadequate preparation for risk mitigation or insufficient response to risk prevention. For these reasons, and due to the significant presence of risks inherent in different phases of projects, many infrastructure projects have failed to achieve the defined project goals in terms of time, cost, and quality. To successfully manage risks, a high-quality, systematic, and detailed analysis of the risks that arise in road infrastructure construction projects is necessary, as well as a thoughtful approach to the implementation of preventive measures.

Many studies have explored various methods to proactively reduce Extension of Time (EoT) and Increasing Contract Price (ICP) in project management [8,9,10,11]. These efforts range from employing different mathematical tools [2,5,9,12,13,14] to defining comprehensive risk lists [15,16,17,18]. However, many factors influence EoT and ICP [19,20,21,22], including the technical characteristics of route construction, administrative processes, logistics, and project execution [23,24,25]. Despite the advancements in risk assessment techniques and tools, the multifaceted nature of these projects often means that unexpected issues can arise, requiring a nuanced and adaptive approach to risk management. On the other hand, there are modern requirements for road infrastructure projects, which are needed for electric vehicles [26,27] and other modern technologies [28,29,30,31,32]. Effective management strategies must consider these diverse influences to mitigate delays and cost overruns effectively.

Predicting risk management outcomes has long been a major challenge in road infrastructure projects. Many studies have employed fuzzy systems to develop models aimed at preventing or mitigating project management risks [33,34,35]. These models help navigate the inherent uncertainties and complexities involved in large-scale construction projects, thereby improving decision-making processes and enhancing the likelihood of successful project completion.

Table 1. Key characteristics and research results based on a literature review with a focus on fuzzy analysis.

Authors	The Title of the Paper	Year	Methodology	Key Research Results
Alam et al. [36]	Critical Infrastructure Renewal: A Framework for Fuzzy Logic Based Risk Assessment and Microscopic Traffic Simulation Modelling	2017	Fuzzy-based delay estimation	The results indicate that the probability of bridge opening delays ranges from 18% to 30% for a delay of one hour and up to 40% for a delay of three hours, depending on the severity of the consequences causing the delay.
Topal & Atasoylu [37]	A Fuzzy Risk Assessment Model for Small Scale Construction Work	2022	Fuzzy risk assessment model	The results reveal the level of risk for each type of incident and the overall safety level of the construction site.
Rezaee Arjroody et al. [2]	Accurate estimation of cost and time utilizing risk analysis and simulation	2024	Monte Carlo simulations	This study found that an integrated modeling framework, incorporating risk analysis using methods like Delphi and Monte Carlo simulations, can improve the accuracy of time and cost estimates for road construction projects, reducing the discrepancies between estimated and actual outcomes by an average of only 4% and 6%, respectively.
Asadi et al. [34]	Project risk evaluation by using a new fuzzy model based on Elena guideline	2018	Fuzzy system	The results demonstrate that the proposed model efficiently and effectively evaluates risky projects, offering a robust tool for project risk management.
Issa, Mosaad, & Salah Hassan [38]	Evaluation and selection of construction projects based on risk analysis	2020	AHP, fuzzy risk analysis model	Five criteria and seventy factors influencing the choice of contractor were identified, and the weight and importance of each criterion was determined.
Khalilzadeh, Banihashemi, & Božanić [7]	A Step-By-Step Hybrid Approach Based on Multi-Criteria Decision-Making Methods And A Bi-Objective Optimization Model To Project Risk Management	2024	Fuzzy method	An innovative and reliable hybrid approach based on MCDM and mathematical optimization methods is proposed.
Our Study	Predicting Extension of Time and Increasing Contract Price in Road Infrastructure Projects: A Sugeno Fuzzy Logic Approach	2024	Sugeno fuzzy logic system	By utilizing the Sugeno fuzzy logic model, project management can be significantly improved through more accurate forecasting of potential delays and cost overruns.

summarizes the most important characteristics of the key papers, such as the research methodology, and the key research results on the topic of interest.

This study introduces a novel application of the Sugeno fuzzy logic approach to enhance project management in road infrastructure projects. Traditional project management often struggles with predicting delays and budget overruns, which can compromise project quality and economic viability. This research addresses these challenges by developing a Sugeno fuzzy logic model based on 10 key project characteristics from completed projects. The model offers highly accurate predictions for Extension of Time (EoT) and Increasing Contract Price (ICP), enabling project managers to forecast potential delays and cost overruns more precisely. The innovative approach allows for better risk assessment and proactive decision-making, ultimately improving project outcomes through targeted risk mitigation strategies. This study represents an advancement over similar studies that address the prediction of delays and cost overruns in infrastructure projects. While previous research often relies on traditional methods or less precise models for risk assessment, this study develops a practical model based on real project characteristics and demonstrates significantly greater accuracy in predicting EoT and ICP. Specifically, the advantage of the proposed approach lies in its ability to more accurately identify potential increases in EoT and ICP, enabling proactive decision-making and the application of targeted risk mitigation strategies. This not only improves the efficiency of project management but also minimizes the negative impacts on the economic and social aspects of these critical infrastructure investments.

In addition to the Sugeno fuzzy model, the literature also mentions a range of advanced systems, such as higher-order type-n fuzzy logic systems [39,40] and Lyapunov stability-based recurrent fuzzy systems [41,42], which offer potentially high accuracy in certain complex applications. However, their application is not always justified for all types of problems. These sophisticated systems often require a significantly complex structure, which includes multiple layers of fuzzy rules and additional feedback loops necessary for dynamic adaptation and identification in complex, nonlinear systems.

In the context of risk management in infrastructure projects, where it is essential to quickly and efficiently predict key parameters such as Extension of Time (EoT) and Increasing Contract Price (ICP), such advanced systems can be overly complex and unreliable. Instead, the Sugeno fuzzy logic system offers an optimal balance between prediction accuracy and ease of implementation. The Sugeno model is particularly suitable for situations where clear and direct outputs are needed, as it allows for the linear or nonlinear approximation of outputs based on input characteristics, making it highly effective for speed and accuracy in predictions. Its relative simplicity in terms of tuning and integration, as well as lower computational resource requirements, makes it an ideal choice for practical applications such as risk management in large infrastructure projects. This approach allows project managers to implement proactive strategies to mitigate risks and optimize project outcomes without the need for more complex and resource-intensive systems that might be unnecessary for this purpose. The application of this model is further justified by the obtained results, which, when validating the model, provide relatively accurate predictions sufficient for effective project management.

2. Methodology

This study used a Sugeno fuzzy logic system applied to 25 completed projects for model development and 3 additional completed projects for testing. The developed model incorporates 10 critical project characteristics, delivering highly precise predictions for both Extension of Time (EoT) and Increasing Contract Price (ICP). By leveraging this model, project management can achieve substantial improvements in forecasting potential delays and budget overruns.

2.1. Sugeno Fuzzy Logic System

The Sugeno fuzzy logic system, also known as a Takagi–Sugeno–Kang (TSK) model, is a type of fuzzy inference system developed to model complex, nonlinear systems. This approach is well known for its effectiveness in handling complex data. Fuzzy logic extends classical logic by allowing partial membership in sets, enabling the representation of real-world concepts that are not strictly binary (i.e., true or false). In a fuzzy logic system, each element belongs to a set to a certain degree, defined by a membership function range.

Sugeno fuzzy logic was chosen for its ability to effectively handle nonlinear and complex relationships between input and output variables, which is crucial in situations where high accuracy and model stability are required [43,44]. Unlike alternative methods, such as artificial neural networks [45,46], which can be prone to overfitting with smaller data samples, Sugeno fuzzy logic provides resistance to overfitting while maintaining a certain level of model interpretability. Additionally, for example, compared to multiple linear regression [47,48,49], which may be insufficiently accurate for complex nonlinear relationships, SFL allows for modeling such relationships in a way that is adaptable to the specificities and demands of the research problem.

2.1.1. Sugeno Fuzzy Model Structure

The Sugeno model is characterized by its rule base, where each rule is composed of a condition (antecedent) and a conclusion (consequent). The antecedents are fuzzy propositions involving linguistic variables, while the consequents are typically linear equations or constants.

A typical Sugeno fuzzy rule can be expressed as:

If x is A and y is B, then z = f(x, y)

where A and B are fuzzy sets representing the input variables x and y, and f(x, y) is a crisp function, usually a linear combination of the input variables. For instance, the consequent could be formulated as:

z = px + qy + r

where p, q, and r—constants.

2.1.2. Inference Mechanism

The inference process in a Sugeno fuzzy system involves several key steps:

Fuzzification: Crisp inputs need to be converted into degrees of membership for each fuzzy set. This process assigns a membership value between 0 and 1 to each input, reflecting its degree of belonging to a given fuzzy set.

Rule evaluation: The antecedents of the fuzzy rules are evaluated to calculate the firing strength of each rule. This is typically done using a fuzzy AND operation, such as taking the minimum or the product of the membership values of the antecedents.

Aggregation of outputs: For each rule, the output z is calculated by applying the corresponding consequent function, which is often a linear equation in the Sugeno model.

Defuzzification: The outputs from all rules are combined into a single crisp output. This is usually accomplished through a weighted average, where the weights are the normalized firing strengths of the rules.

The final output Z is determined using the formula:

$$\mathrm{Z}={\displaystyle \frac{\sum ({\mathrm{w}}_{\mathrm{i}}\times {\mathrm{Z}}_{\mathrm{i}})}{\sum {\mathrm{w}}_{\mathrm{i}}}}$$

where w_i is the firing strength of the i-th rule and z_i is the output from the i-th rule’s consequent function.

This weighted average approach ensures that the resulting output accurately reflects the contributions of all active rules [50,51,52].

It is important to emphasize the significance of hyperparameters in the Sugeno fuzzy inference system (FIS), as they play a crucial role in determining the performance, efficiency, and accuracy of the system [53,54]. The number and type of membership functions per input variable determine how many fuzzy sets (e.g., low, medium, or high) are assigned to each input variable and which type of membership functions are used (e.g., Gaussian, triangular, or trapezoidal). A larger number of membership functions can capture more nuances in the data, but this increases the complexity of the model and the risk of overfitting. The type of membership function affects the smoothness or sharpness of the transitions between fuzzy sets. Gaussian functions provide smooth transitions and are more resistant to noise, while triangular functions are simpler and less demanding in terms of computation. When it comes to the number of fuzzy rules, having more can improve the accuracy of the model by capturing finer relationships between inputs and outputs, but this increases computational complexity. As for the shape of output functions, the Sugeno FIS typically uses linear or constant output functions, where linear functions can capture more complex dependencies between inputs. Linear functions provide greater flexibility but increase the number of parameters to be adjusted. Constant functions are simpler, but they may not adequately capture complex relationships. The rule base, as a very important factor, depends on the number of rules in the system. This number depends on the combinations of membership functions of all inputs and can grow exponentially with the number of input variables and their membership functions. The method of reasoning and defuzzification has been discussed in the previous part of the chapter. Hyperparameters directly affect how the Sugeno FIS interprets input data and generates outputs. The proper tuning of these parameters can significantly improve system performance, while poor tuning can lead to inaccurate or suboptimal results. Hyperparameter tuning often requires a balance between model accuracy and computational complexity [55].

2.2. Software and Tools Used

In this study, MATLAB (v. 2024) was utilized for developing and implementing the Sugeno fuzzy logic system. MATLAB’s computational capabilities and extensive toolbox support made it an ideal choice for modeling and analyzing the data. Additionally, SPSS software (v. 28.0) was employed for statistical analysis. These tools were essential in ensuring the accuracy and reliability of the study’s findings.

3. Results

3.1. Project Dataset

The database for this research includes 28 completed projects related to the construction of European Road Corridor 10 through the Republic of Serbia. This corridor traverses Serbia from the northern border with Hungary towards the south. Near the city of Niš, Corridor 10 branches into two routes: the southern branch leading towards Greece and the capital, Athens, and the eastern branch extending towards Asia Minor via Bulgaria. The construction of the Southern branch, which spans 74.2 km, commenced in 2010 and concluded in 2024, encompassing 10 sections and resulting in 10 contracts awarded through internationally standardized tender procedures. Additionally, two contracts were executed for parallel non-commercial roads intended for the local population, whose existing roads were affected by the new highway’s construction.

The eastern branch of Corridor 10, stretching 86.5 km, also began construction in 2009 and was completed by 2024. This section involved 16 contracts, similarly finalized following international financial institutions’ rules. The projects on this route included constructing a full-profile highway, seven tunnels, and numerous bridges. The associated contracts were signed under the Conditions of Contract for Construction, Multilateral Development Bank, Harmonized Edition, June 2010. The tender procedures adhered to guidelines set by the World Bank, EBRD Procurement Rules, and the procurement guidelines effective at the time of the finance contracts. The primary procurement system employed was a two-envelope system with post-qualification, while a pre-qualification system was used for only four procurements.

Project Characteristics

In project analysis, it is crucial to thoroughly examine specific aspects that can significantly impact the execution and success of each individual project.

In order to obtain a higher degree of accuracy when determining the probability of the occurrence of certain risk events, a list containing 25 project parameters was taken into consideration.

The list of parameters in this research was created based on a review of the available literature and a specially organized expert interview called Risk Management, which was conducted during a two-day interview and discussion. A total of 14 engineers, experts from various fields with many years of experience in road infrastructure projects in Serbia and foreign countries, participated in the expert interview. All interviewed experts had also participated in some of the projects subjected to this analysis, as representatives of the employer, engineers, or technical consultants.

The following characteristics provide a comprehensive overview of the key elements essential for evaluating and managing projects. For the analyzed 28 completed projects, the following project characteristics were known:

• Accepted Contract Amount [EUR]—This represents the amount accepted in the letter of acceptance for the execution and completion of the works and the remedying of any defects. A letter of acceptance is a letter of formal acceptance, signed by the employer, of the letter of tender, including any annexed memoranda comprising agreements between and signed by both parties—employer and contractor. Values in euros were used for quantification;
• Time for Completion [days]—This represents the time for completing the works, as stated in the appendix to tender, calculated from the commencement date. The number of days was used for quantification;
• Landslides along the route [1, 2, 3, or 4]—As part of the project for the building permit, previous geological and geotechnical surveys and analysis were carried out. On the basis of these results, geological processes and phenomena were discovered, such as gully lines with erosion processes, conditionally stable slopes with occasional sliding and notch erosion, and unstable active slopes with shedding landslides and over-consolidation. Based on the types of these processes and the number of occurrences in the sections, each project was classified into the following four categories: 1—no landslides; 2—sporadic landslides (there are landslides nearby, but outside the section area); 3—moderate number of landslides (less than five landslides in the section area); 4—significant number of landslides (five or more landslides in the section area);
• Archaeological sites along the route [1, 2, 3, or 4]—For all the projects that were processed, there was an obligation to prepare an environmental and social impact assessment, as well as an environmental and social management plan, within which, among other things, immovable cultural assets were defined. Based on the preconditions obtained from Institute for Protection of Cultural Monuments and whether the statutorily protected archaeological sites will be directly affected by the construction works, each project was classified into the following four categories: 1—no recorded cultural assets; 2—existing cultural assets nearby, but outside the section area; 3—necessary preliminary archaeological research (geomagnetic methods, georadar prospecting, and conducting archaeological control probes); 4—necessary archaeological excavation work;
• Population density in the future route zone [n/km²]—For all the projects that were processed, there was an obligation to prepare an environmental and social impact assessment, as well as an environmental and social management plan, within which, among other things, demographic characteristics were defined. Population density in the future route zone was defined as the number of inhabitants divided by the zone of influence (radius of 5 km around the section);
• Difference between the highest and lowest points on the route [m]—This represents the difference between the highest and lowest points of the terrain read from the longitudinal profile in relation to the vertical alignment of the road defined in the project for the building permit. These data show the topographic characteristics of the terrain that indicate the degree of complexity of the project. Three types of terrain were defined: plain (≤50 m), hilly (50–150 m), and mountainous (≥150 m);
• Section length [km]—This represents the length of horizontal alignment from the layout plan of the road defined in the project for the building permit;
• Percent of length of embankments on route [%]—This represents the ratio of the total length of all the embankments in a section in relation to the section length, expressed as a percentage. The length of the embankments is read from the longitudinal profile in relation to the vertical alignment of the road defined in the project for the building permit;
• Percent of length of bridges on route [%]—This represents the ratio of the total length of all the bridges in a section in relation to the section length, expressed as a percentage. The length of the bridges is read from the longitudinal profile in relation to the vertical alignment of the road defined in the project for the building permit;
• Percent of length of cuts on route [%]—This represents the ratio of the total length of all the cuts in a section in relation to the section length, expressed as a percentage. The length of the cuts is read from the longitudinal profile in relation to the vertical alignment of the road defined in the project for the building permit;
• Percent of length of tunnels on route [%]—This represents the ratio of the total length of all the tunnels in a section in relation to the section length, expressed as a percentage. The length of the tunnels is read from the longitudinal profile in relation to the vertical alignment of the road defined in the project for the building permit;
• Maximum height of cuts [m]—This refers to the situation when the elevation of the ground is higher than the elevation of the road alignment, and represents the highest distance between the terrain read from the longitudinal profile perpendicular to vertical alignment, and the vertical alignment of the road read from the longitudinal profile defined in the project for the building permit. This information indicates if it is justified to build a cut instead of a tunnel or gallery;
• Maximum height of embankments [m]—This refers to the situation when the elevation of the road alignment is higher than the elevation of the ground, and represents the highest distance between the terrain read from the longitudinal profile perpendicular to vertical alignment, and the vertical alignment of the road read from the longitudinal profile defined in the project for the building permit. This information indicates if it is justified to build an embankment instead of a bridge;
• Predominant material category along the route [A-1, A-2, A-3, A-4, A-5, A-6, or A-7]—A soil classification system was developed by the American Association of State Highway and Transportation Officials (AASHTO), and is used as a guide for the classification of soils and soil aggregate mixtures for highway construction purposes. Previous geological and geotechnical surveys and analyses were carried out as part of the project for the building permit. Based on these results, a geological and geotechnical layout and longitudinal plan, as well as cross-sections, were defined. Taking these data into account, the predominant material category along the route can be determined;
• Number of collisions (box culverts, overpasses, watercourses, or utilities) [n]—Read from layout plan defined in the project for the building permit, the number of collisions with local roads (box culverts and overpasses), watercourses (bridges), railways (underpasses and overpasses), and electrical utilities (telecommunication utilities and local sewerage systems) was counted;
• Type of foundation [shallow or deep]—Shallow foundations are applied in places where the soil can adequately support the intended load and include the following types: strip foundations, footings, isolated foundations, or foundation slabs. Deep foundations are applied in places where the soil is of low resistance to considerable depth and these include the following types: pile foundations, caisson foundations, box caisson foundations, and well foundations. Based on the longitudinal profile defined in the project for the building permit, the predominant type of foundation along the route can be determined;
• Those whose contractual obligation is to prepare the project for execution [employer or contractor]—The employer is required to provide the project for the construction permit as the documentation necessary for the tender for contracts that will be signed under the Conditions of Contract for Construction, Multilateral Development Bank, Harmonized Edition, June 2010. However, the project for execution is not necessary to be provided by the employer, and if he provides it, then that saves time but increases the liability for risks related to design errors;
• Level of land expropriation completion at the time of tender announcement [%]—This represents the ratio of the number of parcels that completed land expropriation in relation to the total number of parcels for which the land expropriation needs to be completed, expressed as a percentage. The total number of parcels for which the land expropriation needs to be completed is defined in the expropriation design of the project for the building permit;
• Whether the designer is a state-owned company [yes or no]—This information indicates the expected quality of project documentation as well as the possibility of holding responsible designers accountable;
• Number of amendments and clarifications to tender documents [n]—These data indicates the number of corrections and additions to the tender documentation, which inevitably affects the duration of the tender procedure;
• Number of submitted bids [n]—In addition to indicating the interest of bidders, it also affects the time for review of all submitted bids and a higher probability of filing appeals against the decision to award the contract;
• Whether the price adjustment for changes in cost is contracted [yes or no]—According to the regulations of certain banks, the price adjustment scale is not introduced for contracts whose time for completion is less than 18 months. If the price adjustment for changes in cost applies, then the amount payable to the contractor shall be adjusted for rises or falls in the cost of labor, goods, and other inputs to the works, by the addition or deduction of the amounts determined by the formulae prescribed in the contract;
• Percent of contractual advance payment [%]—This represents the ratio of advance payment to the accepted contract amount, expressed as a percentage. This directly affects the contractor’s cash flow;
• Those whose contractual obligation is to provide borrow pits for material [employer or contractor]—These data are defined in the bill of quantities and affect who will be responsible if the risk event arises;
• Those whose contractual obligation is to provide a material disposal area [employer or contractor]—These data are defined in the bill of quantities and affect who will be responsible if the risk event arises.

Experts evaluated each project characteristic using a Likert scale ranging from 1 to 7, where the ratings reflected the degree of impact each characteristic had on EoT and ICP. Based on these evaluations, a multi-criteria analysis was conducted using the weighted linear combination (WLC) method. This approach integrates various criteria into a single score by normalizing the values of each characteristic and weighting them according to their importance. The resulting rankings of project characteristics will serve as inputs for the Sugeno fuzzy system, providing detailed insights into priority aspects for optimization and decision-making. The WLC procedure is shown below:

$${\mathrm{EoT}}_{\mathrm{norm}}={\displaystyle \frac{X-Xmin}{\mathit{Xmax}-Xmin}}$$

$${\mathrm{ICP}}_{\mathrm{norm}}={\displaystyle \frac{X-Xmin}{\mathit{Xmax}-Xmin}}$$

where EoT_norm is the normalized rating of experts for EoT and ICP_norm is the normalized rating of experts for ICP.

Based on the weighted values shown above, an overall score (E_s) can be calculated:

E_s = ω₁ × EoT_norm + ω₂ × ICP_norm

where ω₁, and ω₂ are weighting factors that reflect the relative importance of each parameter. For the given example, both factors are equally important: ω₁ = ω₂ = 1/2.

Table 2. Scores and order of project characteristics.

Project Characteristics	Score (Es)	Order
Accepted Contract Amount	11.49	1
Time for Completion	11.38	2
Landslides along the route	5.06	11
Archaeological sites along the route	4.23	13
Population density in the future route zone	4.31	12
Difference between the highest and lowest points on the route	10.88	6
Section length	10.63	8
Percent of length of embankments on route	10.55	9
Percent of length of bridges on route	10.40	10
Percent of length of cuts on route	11.04	4
Percent of length of tunnels on route	4.03	15
Maximum height of cuts	11.37	3
Maximum height of embankments	10.92	5
Predominant material category along the route	4.05	14
Number of collisions (box culvert, overpass, watercourse, utilities)	10.86	7
Type of foundation	3.89	16
Whose contractual obligation is to prepare the Project for Execution	3.88	17
Level of land expropriation completion at the time of tender announcement	3.79	18
Is the designer a state-owned company	3.74	20
Number of amendment and clarifications to tender documents	3.57	22
Number of submitted bids	3.46	24
Is the Price Adjustment for Changes in Cost contracted	3.77	19
Percent of contractual advance payment	3.60	21
Whose contractual obligation is to provide borrow pits for material	3.01	25
Whose contractual obligation to provide material disposal area	3.54	23

shows the obtained score values and the order of project characteristics.

The WLC analysis enabled the prioritization and ranking of the project’s characteristics based on their significance. This expert evaluation helps in identifying and prioritizing the critical attributes that can impact project timelines and costs, ensuring that the fuzzy logic model is informed by relevant and substantiated data. These results were then submitted to a panel of experts for validation. The experts unanimously agreed with the ranking of the project characteristics and recommended that the 10 most significant characteristics be considered in further analysis. Consequently, these top 10 characteristics will be used as inputs for the Sugeno fuzzy system, enhancing the decision-making process for future project evaluations. Concrete values from each project were utilized to develop the fuzzy model. However, the detailed project characteristics for each individual project are not disclosed. Instead, only the results of descriptive statistics are presented, providing an overview of the approximate parameters of the projects. This approach ensures that while the fuzzy model is informed by real data, the confidentiality of the source database is maintained, as it cannot be publicly published. This balance between data utilization and confidentiality is crucial for preserving the integrity and privacy of the project information. The results of the descriptive statistical analysis for these characteristics are presented in

Table 3. Descriptive statistics for selected project characteristics.

Project Characteristics	Average	Max	Min
Accepted Contract Amount [€]	27,278,833.36	74,738,676.05	3,283,504.45
Time for Completion [days]	609.82	900	120
Difference between the highest and lowest points on the route [m]	87.09	275.4	8.6
Section length [km]	7.31	21.4	0.17
Percent of length of embankments on route	50.6	98.2	0
Percent of length of bridges on route [%]	10.7	100	0
Percent of length of cuts on route [%]	20	51.1	0
Maximum height of cuts [m]	10.01	23.1	0
Maximum height of embankments [m]	6.3	12.5	0
Number of collisions (box culvert, overpass, watercourse, utilities) [n]	31.78	98	2

.

In further analysis, the values of ICP and EoT are expressed as percentages. This approach was adopted for the practical application of the model and facilitates its use in various projects, which may have variable ICP and EoT values.

Table 4. Summary of Key Data for Completed Projects.

	ICP [€]	EoT [days]
Average	41,343,706.21	~743
Min	1,537,082.86	0
Max	194,166,387.13	2075

presents the descriptive statistics for EoT and ICP, which represent the output values from the models.

3.2. Case Study of Modeling and Predicting ICP and EoT in Road Infrastructure Projects in the Republic of Serbia

In this case study, a Sugeno fuzzy logic system was used for modeling and predicting Increasing Contract Price (ICP) and Extension of Time (EoT) in road infrastructure projects, based on a set of input and output variables. This approach was chosen as suitable for this task due to its computational efficiency and its ability to provide accurate mathematical models using linear and polynomial approaches for output modeling.

The data used in this case study were collected from 28 real and highly significant road infrastructure projects in the Republic of Serbia, which has over 45,000 km of roads, of which about 1000 km are built auto roads [56]. Out of these, data from 25 completed projects were used for training the FIS, while data from 3 completed projects were used for validating the FIS.

The structure of the FIS (Figure 1), created for these purposes using Matlab 2024r, consists of 10 input variables (Table 3) and 2 outputs: Output 1: ICP; Output 2: EoT (Table 4).

This problem requires solving several different mathematical tasks. It is necessary to define input and output variables, membership functions of input variables, a set of rules, and the mathematical form of the rules, and to perform aggregation and defuzzification. Generally, for an FIS with 10 inputs and 2 outputs, the tasks can be defined as follows [57,58]:

• Defining input and output variables: ten inputs from $x_1$, $x_2, \dots , x_{10}$; two outputs: $y_1$ and $y_2$;
• Determining membership functions for each input, which involves defining membership functions that will describe the different values of the input variable in fuzzy terms (e.g., low, medium, or high). For example, if we use Gaussian membership functions, for input $x_n$ we will have:

  $${\mu }_i\left(x_n\right)=e^{-{\displaystyle \frac{{\left(x_n-Ci\right)}^2}{2\sigma i^2}}}$$

  where $c_i$ is the center and ${\sigma }_i$ is the standard deviation of the membership function. This process is repeated for all inputs.
• Defining the rule set: Rules combine input membership functions to generate outputs. Generally, a rule might look like this:

  $$y_1=a_{j1}+b_{j1}{x}_1+b_{j2}{x}_2+\dots +b_{j10}{x}_{10} (\mathrm{linear} \mathrm{function} \mathrm{for} \mathrm{the} \mathrm{first} \mathrm{output})$$

  $$y_2=a_{j2}+c_{j1}{x}_1+c_{j2}{x}_2+\dots +c_{j10}{x}_{10} (\mathrm{linear} \mathrm{function} \mathrm{for} \mathrm{the} \mathrm{second} \mathrm{output})$$

  where $a_{j1}$, $a_{j2}$, $b_{ji},$ and $c_{ji}$ are coefficients that are adjusted during training.
• Mathematical form of the rules, which can be mathematically expressed as weighted membership functions. For example:

  $$\mathrm{rule} j: {\omega }_j={\mu }_{1i}\left(x_1\right)\times {\mu }_{2i}\left(x_2\right)\dots {\mu }_{10i}\left(x_{10}\right)$$

  where ${\omega }_j$ represents the weight, or the strength of activation of rule $j$.
• Aggregation and defuzzification: Outputs are obtained by aggregating the weighted outputs of all rules:

  $$y_1={\displaystyle \frac{{\sum }_{j=1}^M{\omega }_j·(a_{j1}+b_{j1}{x}_1+\dots +b_{j10}{x}_{10})}{{\sum }_{j=1}^M{\omega }_j}}$$

  $$y_2={\displaystyle \frac{{\sum }_{j=1}^M{\omega }_j·(a_{j2}+c_{j1}{x}_1+\dots +c_{j10}{x}_{10})}{{\sum }_{j=1}^M{\omega }_j}}$$

  where M is the total number of rules.

In Figure 2, the input variables in the FIS are shown, including their membership functions and value ranges.

Gaussian membership functions were used for the inputs, which allowed for modeling uncertainty in the input data. Fuzzy numbers, particularly Gaussian ones, have been used to model input variables to effectively address the inherent uncertainty and imprecision of information available in the initial phases of project analysis and development [59,60]. For example, while the accepted contract amount and scheduled time for completion might appear as parameters that should be defined with precise values, in practice, these values are often subject to changes due to unforeseen circumstances, justifying the use of fuzzy logic. Additionally, the projects under investigation are large in every sense, making it difficult to precisely define all characteristics in advance and ensure they remain unchanged upon project completion. This complexity and scale often lead to uncertainties and variations, further complicating the task of accurate prediction and planning. Fuzzy logic proves particularly useful when the information accompanying projects is imprecise or incomplete, which is common in the initial planning phase of a project. Gaussian membership functions were selected due to their ability to model real phenomena appropriately, allowing the model to more accurately represent how uncertainty in input data affects key project output variables, contributing to more reliable predictions. Gaussian membership functions are smooth and continuous, without sharp transitions or discontinuities, which allows for more stable system behavior. Unlike, for example, triangular or trapezoidal functions, Gaussian functions do not have sharp edges, which reduces the possibility of sudden changes in the system output. This is useful in applications where gradual changes are needed, such as control systems or prediction models, as is the case in this study. These functions often better reflect real physical processes compared to other membership functions. This approach also facilitates the integration of various types of data.

The outputs (ICP and EoT) were modeled using linear functions, which facilitated precise and mathematically defined modeling of the results.

The rules were formulated as “IF-THEN” statements using mathematical functions, providing flexibility and precision in modeling. The genfis2 function was used, as detailed below:

fis = genfis2(trainingData(:, 1:10), trainingData(:, 11:12), 0.5).

Specifically, the training data are organized in a matrix where the first ten columns (A to J) represent the input data, while two columns (K and L) represent the output data. The parameter 0.5 determines the clustering parameter, which affects the number of rules generated in the FIS. This way, 25 fuzzy rules were generated, and it was clear that the system performed better with a larger amount of input data for training, meaning greater coverage of the variables by the fuzzy rules.

One example of fuzzy inference generated by the FIS for specific values is shown in Figure 3.

In Figure 4, the surface plot of the fuzzy logic system (FIS) in MATLAB is shown. The three-dimensional view allows for the analysis of dependencies between combinations of two input variables and one output variable in the Sugeno FIS. Since there are a large number of variables and their combinations, only one representative example is presented, specifically for the first input variable (Input 1: accepted contract amount) and the last input variable (Input 10: number of collisions (box culverts, overpasses, watercourses, and utility lines)) with respect to the first output variable (Output 1: ICP). The X-axis displays the first input variable with a wide range of values, the Y-axis shows the tenth input variable with a narrower range of values, and the Z-axis represents the values of the first output variable.

The colors on the surface represent different levels of the output variable, ranging from blue (low level) to yellow (high level). The surface has a complex shape with various slopes and curvatures, indicating intricate interdependencies between the input variables. Therefore, the surface is not flat, which also signifies nonlinear relationships between the input and output variables. This is expected, as nonlinearity is characteristic of many real-world systems. It can be said that the slopes and curvatures have relatively smooth transitions be-tween different parts of the surface, which means that the system is relatively stable without large, pronounced jumps or discontinuities. Monotonic trends can also be observed, where an increase in one or both input variables leads to an increase or decrease in the output variable.

The control surface below shows how the same combination of inputs affects the second output (Figure 5).

In this case as well, nonlinearity can be observed, with maximum output values occurring when both inputs are at higher levels. The surface has relatively smooth transitions between different values, indicating a relatively stable system that does not show abrupt jumps in output values. Such stability is desirable as it ensures the appropriate consistency of the system in response to changes in inputs.

Both FIS systems show smooth transitions and stability without sharp jumps. Both surfaces have clear maximum and minimum points that depend on the combinations of inputs. The second surface has a more pronounced peak and a wider range of output values, indicating stronger interaction between the inputs and greater sensitivity to changes. The first surface is more consistent with gradual changes, which may mean that it is more stable in real-world applications.

A notable feature of the Sugeno FIS is its capability to handle nonlinear systems well, thus efficiently modeling complex dependencies. The surface plot can be used for further analysis and optimization of the fuzzy logic system to achieve the desired output results.

The convergence analysis theorem for the Sugeno fuzzy logic approach involves understanding how and why this method converges towards accurate predictions for Extension of Time (EoT) and Increasing Contract Price (ICP). This analysis is crucial for verifying the reliability and stability of the model’s predictions. The optimal and careful selection of 10 project characteristics (rather than using all 25) facilitates the practical application of the model. This is particularly important as many real-world projects may not have a complete dataset for all 25 characteristics. By focusing on the most relevant 10 characteristics, the model’s prediction accuracy is enhanced. This targeted approach not only improves the precision of the model but also broadens its applicability on a global scale, regardless of the specific domain or project type. While increasing the number of real projects in the dataset could potentially refine the model’s accuracy further, the evaluation based on the three selected projects demonstrates that the model predicts ICP and EoT results with considerable accuracy. A detailed evaluation of the model is provided in the following section.

Evaluation of the FIS

The evaluation of the generated FIS was carried out using data from three real road infrastructure projects in the Republic of Serbia.

Table 5. Overview of 25 Project Characteristics Used for Evaluation, with Highlighted Fuzzy System Inputs.

Project Characteristics	The Average Value of 3 Projects
Accepted Contract Amount [€]	24,580,449.70
Time for Completion [days]	636.67
Landslides along the route [1–4]	≈1
Archaeological sites along the route [1–4]	≈3
Population density in the future route zone [number of citizens/L*5 km]	≈53
Difference between the highest and lowest points on the route [m]	51.80
Section length [km]	11.80
Percent of length of embankments on route [%]	81.83
Percent of length of bridges on route [%]	3.50
Percent of length of cuts on route [%]	14.17
Percent of length of tunnels on route [%]	0.50
Maximum height of cuts [m]	15.13
Maximum height of embankments [m]	7.53
Predominant material category along the route [1–7]	6.00
Number of collisions (box culvert, overpass, watercourse, utilities) [0–n]	≈44
Type of foundation [1-shallow foundations, 2-deep foundations]	≈2
Whose contractual obligation is to prepare the Project for Execution [1-investor, 2-contractor]	2.00
Level of land expropriation completion at the time of tender announcement [%]	55.00
Is the designer a state-owned company [0–1]	0
Number of amendments and clarifications to tender documents [0–n]	≈9
Number of submitted bids [0–n]	≈12
Is the Price Adjustment for Changes in Cost contracted [0–1]	≈1
Percent of contractual advance payment [%]	10.00
Whose contractual obligation is to provide borrow pits for material [1-investor, 2-contractor]	1.00
Whose contractual obligation to provide material disposal area [1-investor, 2-contractor]	1.00

presents the average values of 25 project characteristics for the selected three projects to provide insight into the approximate values of real-world projects. The 10 characteristics specifically chosen for use in the fuzzy system are highlighted.

In

Table 6. The evaluation results.

Projects for Evaluation	Output 1: ICP [%]		Output 2: EoT [%]
	From a Completed Project	From FIS	From a Completed Project	From FIS
Project 1	90.988	98.579	71.644	81.913
Project 2	96.829	90.996	108.219	104.387
Project 3	93.564	86.074	173.333	151.500
Average deviation	7.604%		10.206%

, the data for ICP and EoT from the real system and the outputs from the FIS are presented, along with the corresponding deviations.

Based on the obtained results, it can be concluded that the generated FIS, for the selected completed projects, makes certain deviations during prediction, which are approximately within the range of 10%. In practical terms, for project planning, this accuracy is at a fairly satisfactory level. Additional training of the FIS is a prerequisite for improving its prediction accuracy.

4. Conclusions

In conclusion, road infrastructure is a pivotal element in national development, directly impacting economic, social, and environmental outcomes. Addressing the risks and uncertainties associated with major infrastructure projects is crucial for their success. This study applied a Sugeno fuzzy logic system to predict Extension of Time (EoT) and Increasing Contract Price (ICP) for road infrastructure projects, utilizing a model derived from 10 key project characteristics. The model demonstrated a high level of accuracy, with deviations of approximately 10% for the test projects, indicating its potential for effective risk management and improved project planning. By leveraging this model, project managers can achieve more accurate forecasts of delays and cost overruns, thereby enhancing their ability to implement proactive strategies and optimize project results. Further training of the fuzzy inference system is recommended to refine its predictive accuracy and ensure even better performance in future applications.

The results of this study offer valuable insights for enhancing project management practices in road infrastructure projects. By integrating the Sugeno fuzzy logic system into project planning and risk management processes, stakeholders can benefit from more accurate predictions of potential delays and cost overruns. This approach enables project managers to proactively identify and address risks, optimize resource allocation, and refine project schedules. Implementing these predictive insights can lead to more effective mitigation strategies, reduce the likelihood of budget overruns, and improve overall project efficiency. Furthermore, the model can serve as a decision support tool for future projects, guiding strategic planning and facilitating more informed decision-making.

Despite the promising results, this study has certain limitations. The predictive accuracy of the Sugeno fuzzy logic system, while generally satisfactory with deviations of approximately 10%, may still be affected by the complexity and variability inherent in road infrastructure projects. The model’s performance is based on data from 28 significant projects, which, although substantial, may not fully capture the diversity of project conditions and challenges encountered in different regions or types of infrastructure. Additionally, the need for further training and refinement of the fuzzy inference system suggests that additional data and iterative adjustments are required to enhance its predictive capabilities. These limitations should be considered when applying the model to other projects, and ongoing validation and adaptation are necessary to ensure its reliability in diverse scenarios. Also, the geographic area and terrain topology from where the data are generated can also have a significant impact on the results.

Future research should focus on expanding the scope and dataset used to refine and validate the Sugeno fuzzy logic system further. Investigating a broader range of project types and conditions, including various geographical locations and infrastructure categories, could enhance the model’s robustness and applicability. Additionally, exploring the integration of other advanced modeling techniques, such as machine learning algorithms, artificial neural networks, or hybrid approaches like multiple linear regression, could provide deeper insights and improve predictive accuracy. Depending on the input data, the higher-order type-n fuzzy logic system and the Lyapunov stability-based recurrent fuzzy systems can be utilized in future research, and the resulting outcomes are suitable for comparison. Future research will focus on examining the risk factors that contribute to increases in ICP and EoT, as well as defining measures to mitigate these identified risks. Further studies could also examine the impact of additional variables, such as environmental factors or stakeholder influences, on project outcomes. By addressing these areas, future research can contribute to the development of more comprehensive and adaptable tools for effective project management in road infrastructure.