Neutrosophic fuzzy-based graphical evaluation and review technique for construction scheduling for residential buildings

1.1 Research gap

Scheduling a construction project is a complex task that requires careful consideration of a various of challenges, including project complexity, uncertainty and variability, resource constraints, safety considerations, and coordination. ³⁸ The demand for a robust scheduling technique of managing these contingencies and developing a multi-objective schedule for complex projects has intensified GERT is an approach to project management for modelling and analysing complex projects.

Despite having proven effectiveness in various sectors, the application of the GERT to construction schedules is still limited. The construction sector heavily relies on conventional techniques like CPM, and PERT, contributing to the inadequacy of research exploring the advantages and constraints of GERT in construction scheduling.^39,40 The existing research gap offers an opportunity to further understand GERT's implementation and contribute to improvements in construction project management. This research aims to explore the application of GERT in construction scheduling and assess its effectiveness compared to traditional methods. Additionally, integrating fuzzy sets, known for handling imprecise and uncertain information, into GERT provides an extra layer of sophistication^41–44.

However, indeterminacy in construction projects arises from the inherent presence of uncertainty and variability. Uncertainty from the unpredictable nature of events, incomplete information, and unforeseen conditions impacting project outcomes. ⁴⁵ Variability, introduces fluctuations and deviations from expected patterns, such as changes in labour productivity, material quality variations, or shifts in project requirements. ⁴⁶ This combination creates a dynamic and unpredictable environment leading to indeterminacy. To comprehensively address the multifaceted spectrum of uncertainty embedded in construction projects, a more inclusive representation is needed. Neutrosophy, which embraces the coexistence of truth, falsehood, and indeterminacy, offers a framework accommodates the intricate and often ambiguous nature of uncertainties in construction projects^47–50. This exploration opens possibilities for refining construction scheduling methodologies and improving their effectiveness in handling the nuanced uncertainties. Koca et al. (2021) demonstrated the use of single-value triangular neutrosophic numbers in analysing the shortest path in real-time construction projects, suggesting its potential application to GERT as well. ⁵¹ While a symmetric representation using triangular fuzzy sets might be apt for balanced variations in project duration, the reality of construction project uncertainties often introduces asymmetry.^52,53 In situations where uncertainties are unevenly distributed, employing a trapezoidal fuzzy set offers a versatile representation to capturing intricate and potentially asymmetric variations. ⁵⁴

Expert opinions play a crucial role construction scheduling, particularly in assessing activity duration and associated probabilities. Incorporating expert opinions for neutrosophic concepts requires addressing subjectivity and minimizing biases on their responses. Achieving a balanced representation of diverse perspectives while minimizing individual biases. Establishing a consistent understanding among experts and providing clear guidelines for assessments is challenging but essential. Achieving consensus often demands iterative refinement through ongoing discussions and feedback loops to harmonize differing viewpoints. To enhance data reliability, a meticulous curation involves identifying upper and lower parameter limits. Subsequently, the data undergoes simulation through Monte Carlo analysis, an exploring uncertainties and variability, for a comprehensive perspective on potential outcomes.

This research introduces a novel integrated multimethodology termed Neutrosophic Fuzzy GERT (NF-GERT) as an innovative approach to scheduling construction projects. In this comprehensive methodology, the neutrosophic set is precisely characterized as a single-valued trapezoidal neutrosophic number (SVTNN). ⁵⁵ As construction projects become increasingly complex and subject to diverse uncertainties, this study aims to demonstrate the implementation of NF-GERT with SVTNNs to schedule construction projects of varied scales (small, medium, and large projects). To evaluate the accuracy of NF-GERT to CPM, GERT, and F-GERT, a one-way ANOVA is performed with SPSS (Statistical Package for Social Sciences). This analysis examined the disparities between estimated and actual as-built durations, giving insight into the efficiency of each scheduling approach. To precisely identify where the differences exist, a post-hoc test is conducted after ANOVA. Subsequently, a post-hoc analysis using Tukey's Honest Significant Difference (HSD) test sought out certain techniques with statistically significant variations. The findings demonstrate NF-GERT's capability to improve scheduling practices, resulting in more accurate project planning and enhanced management efficiency. Figure 1 illustrates the conceptual framework for the proposed research.

OPEN IN VIEWER

Figure 1

Conceptual framework for the proposed research.

The novel contribution of NF-GERT is its ability to further improve the effectiveness of uncertainty management in construction scheduling. While F-GERT already handles fuzzy uncertainty well, NF-GERT enhances this by incorporating neutrosophic logic, which accounts for imprecision, indeterminacy, and inconsistency in data. This added flexibility enables NF-GERT to better address complex, real-world situations where information can be incomplete or conflicting. As a result, NF-GERT offers more accurate modeling, better decision-making, and stronger risk management, making it a powerful tool for managing uncertainty in large and dynamic projects.

In recent years, the conventional GERT model has undergone modifications to model and analyse complex problems. GERT has been integrated with various theories to be adapt to diverse situations that require a stochastic technique of problem-solving. Fuzzy set theory-based GERT is one such variant of the GERT model that accounts for several types of uncertainty in project management.

Fuzzy Graphical Evaluation and Review Technique (F-GERT) is an extension of the GERT method that incorporating fuzzy set theory. F-GERT is a probabilistic network model used to represent and analyse uncertain and complex systems. It allows for more flexibility and expressiveness in modelling and analysing systems with uncertain and imprecise data. In F-GERT, replace crisp values for probabilities, durations, or other variables representing the associated uncertainty and imprecision. Fuzzy sets are mathematical tools used to represent and manipulate uncertainty or vagueness in data. F-GERT excels at representing incomplete or vague information compared to traditional GERT or other modelling approaches, offering a significant benefit of incorporating fuzzy logic in GERT. The use of F-GERT to modelling complex construction scheduling problems is relevant because it is a robust and efficient modelling tool that can handle complicated systems, providing accurate and detailed representations of ambiguous or imprecise data. Its adaptability and flexibility make it a beneficial tool for a variety of uses, facilitating decision-making and optimizing outcomes for intricate projects and systems. Fuzzy logic simplifies the use of intuitive linguistic variables, making F-GERT easier to understand and use. F-GERT more user-friendly than any other approaches to model uncertainty, especially in situations with incomplete, uncertain, or imprecise data. ¹⁷

This research aims to apply fuzzy set theory-based GERT to the scheduling of construction projects, a complicated process including many interdependent operations, limited resources, and a high degree of uncertainty.

3.1. Neutrosophic fuzzy set

In the postliminary stage, Smarandache ⁹³ conceptualized Neutrosophic Set Theory (NST), Neutrosophic Sets (NS), and Single-valued Neutrosophic Fuzzy Sets (SVNFS), fundamentally derived from Neutrosophy, an innovative discipline of Philosophy. The NS technique is a powerful tool to analyse problems that constitute incomplete information, inexactness or impressions, and inherent uncertainties requiring human creative intelligence and judgment. ⁹⁴ NS are delineated by truth membership function (T), indeterminacy membership function (I), and falsity membership function (F). ⁹⁵ Wang et al. ⁹⁶ proposed the concept of a Single-valued Neutrosophic Set (SVNS), an advancement in the conception of a NS. The SVNS is an effective tool for making quantitative judgments in uncertain and indeterminate situations. ⁹⁷ The SVNS is represented by the degree of T ( T A ~ ) , degree of I ( I A ~ ) and degree of F ( F A ~ ) to resolve problems with indeterminate ambiguous data. ⁹⁸ T A ~ , I A ~ , and F A ~ are independent of each other and are determined exceptionally. ⁹⁹

For instance, when estimating the time required for concrete curing, a project planner might initially assume a fixed duration, such as 8 days. However, this traditional approach does not account for uncertainties that may arise during the process. A more flexible approach, such as a fuzzy set, allows for a range of possible durations, like 7 to 10 days, to better reflect variations in real-world conditions. Neutrosophic sets take this a step further by considering not only the expected range but also the level of confidence and uncertainty associated with the estimate. Suppose the planner is fairly confident that the curing will be completed within 7 to 10 days under normal conditions, this confidence can be represented by a truth value (T) of 0.8. However, factors such as sudden rainfall or unexpected delays in material supply introduce uncertainty, which can be captured as an indeterminacy value (I) of 0.5. In extreme cases, if cold weather or improper curing methods significantly extend the duration beyond the expected range, the likelihood of this happening can be represented by a falsity value (F) of 0.3. This approach helps project planners incorporate both known variations and unknown uncertainties into scheduling decisions.

3.2 Neutrosophic fuzzy numbers and membership function

Neutrosophic Fuzzy Numbers (NFN) are similar to Fuzzy Numbers (FN), which are convex fuzzy subsets of the real line, R ′ .They have a peak or plateau with membership grades equal to 1, over which the members of the universe are completely in the set. The membership function of a NFS is the generalization of the indicator function in a classical set, and it is increasing towards the peak and decreasing away from it. The membership function is generally formulated from any one of the functions: piece-wise linear function, triangular (three parameters) function, trapezoidal (four parameters) function, smooth functions, such as symmetric Gaussian distribution function or generalized bell curve function or unsymmetrical sigmoidal function and quadratic or cubic polynomial curves based on the point of view of simplicity, convenience, speed, and efficiency. In this research, the durations of each activity are modelled as Single-valued Trapezoidal Neutrosophic Fuzzy Sets (SVTNFS). ¹⁰⁰ The adoption of SVTNFS to represent uncertain settings is extremely significant. SVTNFS simplifies uncertainty representation by using a single value for each linguistic term, ensuring ease of computation, and enhancing interpretability in decision-making applications. ¹⁰¹ SVTNFS is a hybrid model of the Trapezoidal Fuzzy Number (TFN) concept and the Single-valued Neutrosophic Set (SVNS) concept. ¹⁰²

Let ‘ X ’ be the universe of discourse, with a generalized element in ‘ X ’denoted by ‘ x. A SVNS ‘ A ’ in ‘X’ is characterized by a truth membership function ( T A ) , an indeterminacy membership function ( I A ) , and a falsity membership function ( F A ) . For any point x in X, T A ( x ) , I A ( x ) , F A ( x ) ϵ [ 0 , 1 ] with 0 ≤ T A ( x ) + I A ( x ) + F A ( x ) ≤ 3 for all x ∈ X . A discrete SVNS considered for this research is shown in Eq. 1.

A = ∑ i = 1 n T ( x i ) , I ( x i ) , F ( x i ) / X , ∀ x ∈ X

(1)

Let a SVTNN a ~ = ( a 1 , a 2 , a 3 , a 4 ) ; T a , I a , F a be a neutrosophic set on the real number set R , whose degree of truth membership ( T a ~ ( x ) ) , indeterminacy membership ( I a ~ ( x ) ) , and falsity membership ( I a ~ ( x ) ) , function are given in Eqs. 2–4, respectively.

T a ~ ( x ) = { ( x − a 1 ) T a ( a 2 − a 1 ) , a 1 ≤ x ≤ a 2 T a , a 2 ≤ x ≤ a 3 ( a 4 − x ) T a ( a 4 − a 3 ) , a 3 ≤ x ≤ a 4 0 , Otherwise

(2)

I a ~ ( x ) = { a 2 − x + ( I a ( x − a 1 ) ) ( a 2 − a 1 ) , a 1 ≤ x ≤ a 2 I a , a 2 ≤ x ≤ a 3 ( x − a 3 + ( I a ( a 4 − x ) ) ( a 4 − a 3 ) , a 3 ≤ x ≤ a 4 1 , Otherwise

(3)

F a ~ ( x ) = { a 2 − x + ( F a ( x − a 1 ) ) ( a 2 − a 1 ) , a 1 ≤ x ≤ a 2 F a , a 2 ≤ x ≤ a 3 x − a 3 + F a ( a 4 − x ) ) ( a 4 − a 3 ) , a 3 ≤ x ≤ a 4 1 , Otherwise

(4)

The truth membership function T a ~ ( x ) in eq. 2 expresses the confidence level that a value belongs to this range. It increases linearly from a 1 to a 2 , remains constant at T a between a 2 and a 3 , and then decreases linearly from a 3 to a 4 . Outside this range between a 1 and a 4 , the truth value is zero, indicating that values outside this range are completely excluded from being considered part of the defined uncertainty model. This function allows for the flexible representation of uncertainty in real-world scenarios, such as construction scheduling, where certain durations are more likely while others are less certain.

The indeterminacy membership function I a ~ ( x ) in eq.3 quantifies the level of uncertainty associated with a given value within the specified range. Within the interval a 2 and a 3 , the function remains constant at I a , indicating a stable degree of indeterminacy where uncertainty neither increases nor decreases. Similarly, the falsity membership function F a ~ ( x ) in eq.4 represents the degree of inconsistency of a value with the expected range. Within the interval a 2 and a 3 , the function remains constant at F a , indicating a stable level of falsity. Beyond the boundaries of a 1 and a 4 , the function reaches its maximum value of 1, representing the highest level of uncertainty, where values outside this range become entirely unpredictable.

Let a ~ = ⟨ ( a 1 , a 2 , a 3 , a 4 ) ; T a , I a , F a ⟩ and b ~ = ⟨ ( b 1 , b 2 , b 3 , b 4 ) ; T b , I b , F b ⟩ be two SVTNNs. The operations of the neutrosophic sets are exemplified in Eqs. 5–9.

3.2.1 Addition

a ~ + b ~ = ⟨ ( a 1 + b 1 , a 2 + b 2 , a 3 + b 3 , a 4 + b 4 ) , min ( T a , T b ) , max ( I a , I b ) , max ( F a , F b ) ⟩

(5)

3.2.2 Subtraction

a ~ − b ~ = ⟨ ( a 1 − b 4 , a 2 − b 3 , a 3 − b 2 , a 4 − b 1 ) , min ( T a , T b ) , max ( I a , I b ) , max ( F a , F b ) ⟩

(6)

3.2.3 Multiplication

a ~ b ~ = { ⟨ ( a 1 b 1 , a 2 b 2 , a 3 b 3 , a 4 b 4 ) ; min ( T a , T b ) , max ( I a , I b ) , max ( F a , F b ) ⟩ a 4 > 0 , b 4 > 0 ⟨ ( a 1 b 4 , a 2 b 3 , a 3 b 2 , a 4 b 1 ) ; min ( T a , T b ) , max ( I a , I b ) , max ( F a , F b ) ⟩ a 4 < 0 , b 4 > 0 ⟨ ( a 4 b 4 , a 3 b 3 , a 2 b 2 , a 1 b 1 ) ; min ( T a , T b ) , max ( I a , I b ) , max ( F a , F b ) ⟩ a 4 > 0 , b 4 < 0

(7)

3.2.4 Division

a ~ b ~ = { ⟨ ( a 1 b 4 , a 2 b 3 , a 3 b 2 , a 4 b 1 ) ; min ( T a , T b ) , max ( I a , I b ) , max ( F a , F b ) ⟩ a 4 > 0 , b 4 > 0 ⟨ ( a 4 b 4 , a 3 b 3 , a 2 b 2 , a 1 b 1 ) ; min ( T a , T b ) , max ( I a , I b ) , max ( F a , F b ) ⟩ a 4 < 0 , b 4 > 0 ⟨ ( a 4 b 1 , a 3 b 2 , a 2 b 3 , a 1 b 4 ) ; min ( T a , T b ) , max ( I a , I b ) , max ( F a , F b ) ⟩ a 4 > 0 , b 4 < 0

(8)

3.2.5 Scalar multiplication

γ a ~ = { ⟨ ( γ a 1 , γ a 2 , γ a 3 , γ a 4 ) ; T a , I a , F a ⟩ i f ( γ > 0 ) ⟨ ( γ a 4 , γ a 3 , γ a 2 , γ a 1 ) ; T a , I a , F a ⟩ i f ( γ < 0 )

(9)

where, γ is a scalar.

The score function S ( a ~ ) and the accuracy function A ( a ~ ) are key measures in neutrosophic set theory, used to evaluate the significance and reliability of a neutrosophic number. ¹⁰³ The score function, given in Eq. 10, helps convert a SVTNN into a crisp value by considering the combined effect of truth, indeterminacy, and falsity. The accuracy function, given in Eq. 11, measures the certainty or reliability of a neutrosophic number by assessing the balance between truth and falsity.

The score function S ( a ~ ) of the neutrosophic set is computed using Eq. 10.

S ( a ~ ) = 1 12 [ a 1 + a 2 + a 3 + a 4 ] × [ 2 + T a − I a − F a ]

(10)

The accuracy function A ( a ~ ) of the neutrosophic set is computed using Eq. 11.

A ( a ~ ) = 1 12 [ a 1 + a 2 + a 3 + a 4 ] × [ 2 + T a − I a + F a ]

(11)

4.1 Steps for Applying the NF-GERT Methodology in Construction Scheduling

The NF-GERT methodology for construction scheduling is implemented through the following steps:

Step 1: Developing the Work Breakdown Structure (WBS) – The first step in applying the NF-GERT methodology is to develop a comprehensive WBS for the construction project. The WBS identifies all the crucial tasks needed to complete the project and breaks it up into smaller, more manageable parts. Each task is precisely defined to ensure that no key tasks are overlooked. Furthermore, the logical sequence and interdependencies between these activities are established to create a structured workflow.

The suggested technique could be effectively presented in a multi-storey structure since high-rise building design requires numerous inspections and checkpoints for safety. It is imperative to incorporate control points in multi-storey buildings for aspects like quality and durability, standards, design, and layout. Non-compliance with these regulations always results in a plethora of issues, including financial shortfalls, quality drops, and even litigation. At each stage of implementation, the structural design and layout are inspected. Control points include checking the type and quantity of reinforcement bars for the roof slab, column, and beam, particularly at maximum load transfer points and beam-column joints; levelling the brickwork provided for walls to be the same height; inspecting shoe concreting; examining the mix ratio in the RMC; performing slump tests at the RMC and on the job site; and levelling roof concreting, which are likely to be non-conforming in certain instances. These discrepancies must be addressed to achieve the desired execution.

Conventional models do not account for the probability of occurrence associated with the activities in the building projects, making the implementation of NF-GERT appropriate. To handle issues with indeterminate ambiguous data, each activity's durations are represented as single-valued trapezoidal neutrosophic fuzzy sets denoted by truth membership degree, indeterminacy membership degree, and falsehood membership degree. This research provides a comprehensive description of the conceptual model of the suggested approach to creating a construction schedule for a multi-storey residential building. To emphasize the functionality of looping and the probability of various activities occurring, a multi-storey construction project is selected.

4.2 Case study: multi-storey (fourteen) residential building

The comprehensive contextual relevance of the proposed technique is demonstrated by employing the NF-GERT method to compute the project completion time for a fourteen-storey residential building in Chennai. Figure 4 shows a photograph of the fourteen-storey residential building in Chennai, India, considered for the study during the construction phase. The project framework is usually depicted on two levels: a WBS breaks the project into manageable tasks, and a network diagram demonstrates the technical and execution dependencies between the project's activities. ¹⁰⁶

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Figure 4.

Photograph of the construction of the fourteen-storey residential building in chennai, India.

Primarily, to construct the fourteen-storey structure with the appropriate characteristics, a Work Breakdown Structure (WBS) is established by creating an exhaustive list of the activities and interdependencies between the listed activities. The comprehensive WBS for the construction of the fourteen-storey building considered is listed in Table 1 in the electronic supplement, depicting the specifics of the input data needed to compute the expected project duration employing NF-GERT (TNFN and Crisp Value) along with the probability of occurrence (p_n) of activities.

The procedure outlining the definition of neutrosophic numbers for activities in the Work Breakdown Structure (WBS) presented in Table 1. Carefully selected experts with proficiency in construction project management are chosen, and a comprehensive background of the project's context is presented. To facilitate a collective understanding, a detailed explanation of neutrosophic fuzzy logic is provided. In this study, data were gathered from a carefully selected panel of experts within the construction sector using the Delphi technique. As described by Linstone and Turoff (1975), this method is a structured approach that collects expert opinions anonymously, helping to minimize biases such as the “bandwagon” or “halo effect”. ¹⁰⁷ A modified multi-stage Policy Delphi technique, based on the principles of non-maleficence and consensus, was utilized. Experts participated in several rounds of feedback, which allowed for the refinement of their responses and the gradual development of a consensus. This iterative process ensures that the final conclusions reflect a well-rounded expert consensus, reducing individual biases and strengthening the reliability of the findings. Experts use a numerical scale to quantify their responses to the degrees of truth ( T a ), indeterminacy ( I a ), and falsity ( F a ) in addition to the duration of each WBS activity and the probability of occurrence of the activity. The acquired data is then used to determine maximum and minimum limits, followed by the adoption of Monte Carlo simulation with 10,000 iterations to address inherent uncertainties in the expert-defined values. This simulation is employed to define neutrosophic numbers, tackling uncertainties, capture expert variability, and generate probability distributions, thereby offering a comprehensive and probabilistic understanding of complex construction project dynamics. ¹⁰⁸ This methodology improves decision-making by quantifying risks, accommodating diverse expert perspectives, and facilitating statistical analyses within the simulated dataset.^109,110 The resulting simulated data undergoes thorough analysis using basic descriptive statistics, such as mean, median, standard deviation, and interquartile range, providing a detailed overview of central tendencies and variability within the dataset. Subsequently, based on the simulation results, the membership functions of trapezoidal neutrosophic numbers are precisely defined, capturing varying levels of truth, indeterminacy, and falsity. Feedback from the initial group of experts is then sought, and their insights play a pivotal role in interpreting simulation outcomes and refining the model iteratively.

The Monte Carlo simulation is implemented in VBA (Visual Basic for Applications) to model probabilistic data assignments derived from survey responses. This approach involves generating 10,000 iterations for the variables A, B, C, D, T, I, and F, where each variable is assigned based on probability distributions obtained from survey data. The electronic supplement provides the VBA code framework that was utilized for data generation. The key aspects of the VBA implementation include:

Worksheet Initialization: Clearing existing data and setting up column headers.

For instance, consider the activity of site clearance and site levelling using mechanical means (A₀₁), which involves multiple uncertainties. This activity can be represented using a SVTNN as 〈(5, 6.5, 8.5, 10); 0.9, 0.6, 0.4〉. These values are averages obtained from the simulation, based on 10,000 iterations, ensure objectivity by utilizing data-driven probability distributions from survey responses, avoiding subjectivity in parameter selection. The trapezoidal range (5, 6.5, 8.5, 10) indicates the possible duration of the activity, where 5 days represents the minimum duration under optimal conditions, 6.5 to 8.5 days indicate the most likely duration range (lower and upper) based on normal working conditions, and 10 days is the maximum duration in case of unforeseen delays such as equipment failure or unfavourable weather. The neutrosophic components provide additional insights into the uncertainty of the estimation. The truth value (T = 0.9) suggests a high confidence level in the estimated duration based on historical data and expert judgment. The indeterminacy value (I = 0.6) accounts for uncertainties arising from unpredictable factors like soil conditions or operator efficiency. The falsity value (F = 0.4) represents the possibility that the estimated duration might be incorrect due to unexpected external disruptions. By incorporating this representation into NF-GERT, project scheduling can better accommodate uncertainties, leading to more reliable and adaptive planning. Figure 5 illustrates the geometric representation of SVTNFS for activity A₀₁ . The probability of an activity occurring in the WBS is established through the weighted average method, which addresses varying probabilities provided by different experts for a specific task. Consequently, this approach consolidates and represents the opinions of experts with higher credibility.

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Figure 5

Geometric representation of SVTNFS for A₀₁.

The WBS subsequently converted into the GERT network diagram for further analysis. Nodes are assigned numbers and outlined, with the illustration offering details about the node's inclusion in the loop and the number of activities required for the node to be actualised in the initial and subsequent occurrences. The GERT network of the fourteen-storey residential building is shown in Figure. 6. Table 1 provides the information about the various nodes depicted in the GERT network for the aforementioned building.

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Figure 6.

GERT network diagram of the multi-storey residential building.

The GERT network diagram (Figure 4) comprises loops categorized into three types: the self-loop, the recurrence loop, and the feedback loop. A self-loop occurs when one activity act as a prerequisite for the next, directed based on a certain circumstance to the same node. Activities A 04 around node 4, A 17 (node 12), A 29 (node 19), A 41 (node 26), A 53 (node 34), A 70 (node 45), A 72 (node 46) and A 137 (node 92) form self-loop.

In the case of a recurring sequence of activities, a recurrence loop depicts the set of repeated tasks. The GERT network contains eight loops representing the recurrence of tasks and nine feedback loops. The construction site is initially divided into two pours, i.e., the construction site is divided into two sections for easy handling of the construction activities based on the resources available. After the operations in pour #1 are completed, the same sequence is repeated in pour #2. Loop #1 represents this cycle for raft foundation construction, involving activities like site clearance, excavation, dewatering (if necessary), dressing of loose soil, transportation of earthwork, laying waterproofing membrane, laying PCC, steps in the construction of raft slab, main beam, and secondary beam up to the floor slab for both the pours. This cycle (loop #1) repeats for pour #2. The fourteen-storey residential building is constructed using Mivan technology (a fast-paced technique of construction that gives a construction project improved strength and durability by employing aluminium formworks). Loop #2 depicts construction activity for RCC walls and roof slabs for stilt floors and other apartment levels. On the actualization of node 52, all arcs emanating from the node (loop #2), A 83 and A 86 commences. Loop #2 occurs 14 times, initiating activities like internal plastering, building a kitchen platform, installing door and window frames, conduit laying work, installation of electrical fittings, plumbing, and firefighting interconnection work, floor, and wall tiling, and fixing of doors, windows, and ventilators, following each realization Loops #3, #4, #5, and #6 signify applying primer, wall putty, enamel paints (door, window, ventilator), and wall paints (interior) in one coat with each loop occurring twice to represent the two coats of paint applied. Loop #7 indicates the painting of the exterior wall with one coat of premium acrylic emulsion paint, applied twice. Similarly, Loop #8 denotes one coat of painting the compound wall which is painted with two coats.

The feedback loop is a controlled loop used to perform an activity iteratively, with the evaluation precondition being evaluated initially. The activities A 12 (node 9→8), A 15 (node 11→10), A 24 (node 16→15), A 27 (node 18→17), A 36 (node 23→22), A 39 (node 25→24), A 60 (node 38→37), A 64 (node 41→40) and A 67 (node 43→42) are the feedback loops in the GERT network.

The step-by-step computation process for applying the NF-GERT methodology in project duration estimation is delineated below.

Initially, each activity duration is represented as a SVTNN. The membership function for each activity is derived using Eqs. 2–4, which define the degree of truth, indeterminacy, and falsity, establishing the lower, upper, and middle bounds of the trapezoidal neutrosophic number to capture uncertainty in the time estimated by experts. The crisp value ( S i ~ ) of the time estimate for each activity is then obtained using Equation 10, which is used for defuzzification to calculate the expected duration for the activity. For instance, the crisp value for activity A₀₁ is illustrated below:

S 1 = 1 12 [ 5 + 6.5 + 8.5 + 10 ] × [ 2 + 0.9 − 0.6 − 0.4 ] = 4.75

4.5.1 Illustrating computation: section from node 1 to node 13

To illustrate the calculation of the project completion time, a part of the network (Figure 6) is examined, specifically the activity flow from node 1 to node 13 as shown in Figure 7. This section consists of nodes connected in series, parallel, and loops.

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Figure 7

Section of the GERT network from node 1 to node 13.

Node 1 → 2: Node 1 is the source node. The realization time of E ( t ) 1 → 2 is shown in Eq. 12.

E ( t ) 1 → 2 = S t ~ 01 = 4.75

(12)

Node 2 → 7: The network is further divided into small fragments since activities are connected in parallel from node 2 to node 7. This section (Node 2 → 7) also includes nodes 3, 4, and 5, which are connected in series, along with parallel connections between nodes 5 and 7. Node 4 is a probabilistic EXCLUSIVE-OR node with a self-loop, and thus it must be solved accordingly. Node 2 is a deterministic EXCLUSIVE-OR with two incident arcs, namely A₀₂ and Loop #1. The deterministic EXCLUSIVE-OR node allows the realisation of only one activity incident on the node (A₀₂ and Loop #1) while the other remains idle. Initially, only activity A₀₂ is realized, while Loop#1 is realized only after node 28 is achieved. Node 3 is a deterministic node with incident A₀₂ and emanating A₀₃. Node 4 is a probabilistic EXCLUSIVE-OR node, where A₀₄ and A₀₅ emerge forming a self-loop. If the site demands any draining procedure, then de-watering activity (A₀₄) is performed before the soil dressing activity (A₀₄) for laying the foundation. If de-watering is not performed, activity A₀₄ is omitted and A₀₅ is accomplished. The probability of transition from node 4 to 5 without transiting the loop, i.e., omitting activity A04 is p05. The probability of transition from node 4 to 5 transiting the loop is [ S t ~ 05 + ( p 04 1 − p 04 × S t ~ 04 ) ] . Activities A07 and A08 are connected in series; however, these activities occur parallel to A06, so the maximum activity duration is considered as the realization time. Additionally, node 7 has an incident activity A09 occurring parallel to all the other activities occurring between node 2 and node 7. Therefore, the realization time of section E ( t ) 2 → 7 is computed using Eq. 13.

E ( t ) 2 → 7 = ( max { ( S t ~ 02 + S t ~ 03 + [ S t ~ 05 + ( p 04 1 − p 04 × S t ~ 04 ) ] + max { S t ~ 06 S t ~ 07 + S t ~ 08 ) S t ~ 09 )

(13)

E ( t ) 2 → 7 = ( max { ( 3.39 + 4.46 + [ 3.20 + ( 0.4 1 − 0.6 × 1.56 ) ] + max { 3.25 1.19 + 1.37 ) 8.92 ) = max { 15.86 8.92 = 15.86

Node 7 → 13: Similarly, the activity flow from nodes 7 to 13 represents a fragment of the network with arcs connected in parallel. The deterministic EXCLUSIVE-OR node 8 has two incident actions, but, only one activity is accomplished at a time, holding the other activity idle. The succeeding activity has a feedback loop that ends at node 8. The workflow is as follows: after the shuttering activity (A₁₀) the placement of shutters is checked (A₁₁) the inaccuracies are adjusted (A₁₂) and the adjustments are rechecked (A₁₁). When the adjustments are perfected, the next succeeding activity, i.e., reinforcements are fixed (A₁₃). The duration of realization from node 9 to node 10 is S t ~ 11 + ( p 12 × S t ~ 12 ) + S t ~ 11 + ( p 13 × S t ~ 13 ) . Correspondingly, the expected time is computed for the other feedback loops in the network. The realization time from node 7 to 13 is determined using Eq. 14.

E ( t ) 7 → 13 = ( m a x { [ S t ~ 10 + S t ~ 11 + ( p 12 × S t ~ 12 ) + S t ~ 11 + ( p 13 × S t ~ 13 ) + S t ~ 14 + ( p 15 × S t ~ 15 ) + S t ~ 14 + ( p 16 × S t ~ 16 ) + S t ~ 18 + ( p 17 1 − p 17 × S t ~ 17 ) ] S t ~ 21 )

(14)

Similarly, the network is divided into smaller portions to solve loops and parallel arcs. Eqs. 2–4 are used to calculate the membership function of activity duration. The time estimate's crisp value is calculated using Eq. 10. The equivalent equation used to compute the project completion time E ( t ) based on NF-GERT is delineated in Eq. 15 in the electronic supplement. The estimated completion time for the multi-storey (fourteen) residential building E ( t ) is 1477.95 days ≈1478 days, however, the actual duration is 1568 days.

5.1 Sensitivity analysis

The stochastic network approach is an increasingly influential and eminent tool for analysis and elucidation of the solution to complex problems. The results are validated by comparing the accuracy of outcomes with the scheduling methods in practice, i.e., Monte Carlo simulated GERT, CPM, and PERT. The Monte Carlo simulation model is a decision support system based on the multiple-point statistical simulation technique used to analyse complex stochastic problems with uncertainties. Predominantly used in risk management projects, Monte Carlo simulation addresses uncertainties associated with project duration and cost.^60,112 The simulation technique can generate as many as 10,000 samples or more, maintaining the original complexity and uncertainty in the form of their probability distributions^113–115. Applying Monte Carlo simulations to GERT advocates for the estimation of project schedules based on a large sample rather than a single statistical calculation.

To conduct an effective simulation, this study generated 10,000 different data set. The simulated data are incorporated into the equivalent equation to determine the simulated outcomes. Descriptive statistics for the simulated outcomes of the GERT network for the fourteen-storey building are shown in Table 2.

The network is analysed using the conventional methodologies (CPM and PERT) for the fourteen-storey building. The total project completion time for the fourteen-storey building according to CPM and PERT is estimated as 1207 days and 1274 days, respectively. Additionally, the expected duration calculated using the GERT (Monte-Carlo Simulated) network technique for the fourteen-storey building is 1384.53 days (≈1385 days). Figure 8 depicts the total project completion time computed using CPM, PERT, GERT, and NF-GERT methodologies.

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Figure 8.

Total project completion time computed using CPM, PERT, GERT, and nf-GERT methodologies.

The comparison between the conventional methods (CPM and PERT), standard Monte-Carlo simulated GERT, and the proposed technique NF-GERT is depicted in a Kiviat diagram to visually demonstrate the effectiveness of the approaches as shown in Figure. 9.

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Figure 9.

Kiviat diagram for performance measure of CPM, PERT, GERT, and nf-GERT methodologies.

The Kiviat diagram (or Spider chart) shows that the proposed Neutrosophic fuzzy-based GERT (NF-GERT) technique is approximately 94% accurate compared to the real-time data. The calculated duration of 1478 days was closest to the actual project duration of 1568 days for the fourteen-storey structures, in comparison with the other existing conventional scheduling methodologies. The percentage of error for CPM is approximately 23% and for PERT is approximately 19%. The notable difference in error percentages between NF-GERT and traditional scheduling methodologies highlights the remarkable superiority of NF-GERT in delivering more reliable and precise project completion time estimates. The Kiviat diagram indicates that NF-GERT not only outperforms in precision but also offers a more reliable and effective way to handle uncertainties in practical projects.

An examination of the fluctuation in activity probabilities is conducted to understand the consequences of alterations in these probabilities. The baseline probability values for each activity are established based on initial assessments. Then, the range of values for the probabilities to be utilized in the sensitivity analysis is determined. Table 3 presents the range of values for the probability of occurrence (V₁, V₂, V₃, V₄ and V₅) of activities for the sensitivity analysis.

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Table 3

Modifications incorporated into the probability of occurrence of activities for the sensitivity analysis.

Probabilistic Node	Code (Ai)	V 1	V 2	V 3	V 4	V 5
4	A04	0.1	0.2	0.3	0.4	0.5
	A05	0.9	0.8	0.7	0.6	0.5
9	A12	0.1	0.2	0.3	0.4	0.5
	A13	0.9	0.8	0.7	0.6	0.5
11	A15	0.1	0.2	0.3	0.4	0.5
	A16	0.9	0.8	0.7	0.6	0.5
12	A17	0.1	0.2	0.3	0.4	0.5
	A18	0.9	0.8	0.7	0.6	0.5
16	A24	0.1	0.2	0.3	0.4	0.5
	A25	0.9	0.8	0.7	0.6	0.5
18	A27	0.1	0.2	0.3	0.4	0.5
	A28	0.9	0.8	0.7	0.6	0.5
19	A29	0.1	0.2	0.3	0.4	0.5
	A30	0.9	0.8	0.7	0.6	0.5
23	A36	0.1	0.2	0.3	0.4	0.5
	A37	0.9	0.8	0.7	0.6	0.5
25	A39	0.1	0.2	0.3	0.4	0.5
	A40	0.9	0.8	0.7	0.6	0.5
26	A41	0.1	0.2	0.3	0.4	0.5
	A42	0.9	0.8	0.7	0.6	0.5
34	A53	0.1	0.2	0.3	0.4	0.5
	A54	0.9	0.8	0.7	0.6	0.5
38	A60	0.1	0.2	0.3	0.4	0.5
	A61	0.9	0.8	0.7	0.6	0.5
41	A64	0.1	0.2	0.3	0.4	0.5
	A65	0.9	0.8	0.7	0.6	0.5
43	A67	0.1	0.2	0.3	0.4	0.5
	A68	0.9	0.8	0.7	0.6	0.5
45	A70	0.1	0.2	0.3	0.4	0.5
	A71	0.9	0.8	0.7	0.6	0.5
46	A72	0.1	0.2	0.3	0.4	0.5
	A73	0.9	0.8	0.7	0.6	0.5
92	A137	0.1	0.2	0.3	0.4	0.5
	A138	0.9	0.8	0.7	0.6	0.5

Subsequently, Monte Carlo simulation is employed to generate 10,000 iterations of project scenarios by sampling from the specified probability distributions. In each iteration, probabilities are adjusted within the defined range. The completion time of the construction project for each Monte Carlo simulation iteration is calculated using equation 15. This process yields a distribution of project completion times that accounts for the variability in activity probabilities.

The descriptive statistics derived from a Monte Carlo simulation, performed over 10,000 iterations, are presented in Table 4, providing an in-depth view of the distribution of completion times under various probability scenarios.

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Table 4

Descriptive statistics of completion times under various probability.

Statistical measures	Value
Mean	1502.11
Standard Deviation	75.383
Kurtosis	−1.17613
Skewness	0.05466
Minimum	1368.26
Maximum	1635.30

The histogram resulting from the Monte Carlo simulation offers a detailed perspective on the distribution of project completion times, providing key statistical measures for thorough interpretation. The mean completion time of 1502.11 days represents the statistical average, indicating a central tendency around which completion times cluster. However, the standard deviation of 75.383 days denotes a considerable degree of variability or dispersion in completion times, emphasising the inherent uncertainties associated with the project.

The negative kurtosis value of −1.1761311 indicates that the shape of the distribution of project completion times is less peaked or more spread out than a typical or “normal” distribution. Instead of having a high, narrow peak like a bell curve, the distribution is somewhat flatter. This flatter shape suggests that there is a higher chance of experiencing extreme values or outcomes that are significantly different from the average or mean completion time. The slight rightward skewness of 0.05465985 adds to this understanding. Skewness measures the asymmetry of the distribution. A slightly right-skewed distribution means that there is a longer tail on the right side of the distribution, suggesting that there is a possibility of having completion times that are later than the average or mean. So, both the negative kurtosis and rightward skewness indicate that the project completion times may vary, and there is a potential for encountering delays that exceed the anticipated symmetrical distribution.

Further analysis of the minimum and maximum completion times provides a clearer understanding of the potential project outcomes. The minimum completion time of 1368.26 days represents the shortest observed duration, while the maximum of 1635.30 days signifies the longest. This extensive range highlights the various scenarios explored in the Monte Carlo simulation, covering both optimistic and pessimistic timelines.

The 24-day difference observed between the simulated completion time (1502 days) and the calculated completion time (1478 days) suggests a slight variance in the projected project completion duration when accounting for changes in probability. This modest difference indicates a continual concordance between the outcomes derived from the Monte Carlo simulation for sensitivity analysis and the NF-GERT methodology, highlighting the adaptability and reliability of the proposed approach. However, when comparing these predictions to the actual observed as-built duration of 1568 days, it becomes apparent that the real-time scenario presented additional complexities or unforeseen factors that extended the project duration beyond the predicted average. The observed duration, falling within the range of the simulated and calculated durations, provides a degree of validation for the adaptability and robustness of the proposed novel multimethodology.

The researcher intends to demonstrate the application and reliability of NF-GERT by applying the presented technique to compute project duration for forty-five other cast-in-situ residential projects located across India. The total project time obtained using the suggested technique is compared to the total project duration obtained using the CPM methodology, considering CPM as a common approach used for scheduling construction projects. Additionally, the computed expected project completion time using NF-GERT is compared with the project duration calculated employing F-GERT to illustrate the effectiveness of the proposed method.

The construction of high-rise structures involves a significant number of repetitive activities as an essential component. The GERT network comprises loops, which depict activity recurrence. The cast-in-situ residential projects considered for validation are grouped based on the number of stories in the structure. The number of stories was employed as a criterion for grouping to demonstrate the loop feature of NF-GERT. Based on the number of stories, the projects were divided into three categories: namely, small, medium, and large. Small projects have no more than six stories, medium projects have seven to fifteen stories, and large projects have more than fifteen stories. Table 2 in the electronic supplement presents the list of projects considered for validation, along with the details of the projects such as construction area (in sq. m), number of stories, and actual project duration (construction time, in days).

The work breakdown structure (WBS) for the construction of residential buildings considered for validation is laid out, along with the relationships between the different activities. The WBS serves to construct a network diagram, which is a flowchart depicting the workflow. The project duration and the critical path for the considered projects are computed using the CPM technique. Initially, the forward pass method is used to compute early start and early finish times, while the backward pass approach is used to compute late start (LS) and late finish (LF) times. Subsequently, the activity float or slack for scheduling flexibility can then be computed using the early and late start and end dates. The critical path from the Activity-on-Arrow (A-o-A) diagram from the project manager of each construction project is determined. The WBS prepared for the CPM approach is used to construct the GERT network for all the projects being examined for validation.

To compute the project completion time ( T ~ i j ) using the F-GERT approach, the ambiguity associated connected with the activities is represented as a Trapezoidal Fuzzy Number (TFN).^116,117 For each of the forty-five projects under consideration, the realization probability ( p i j ) and activity duration as TFN are collected. The completion time ( T ~ e )is calculated as a TFN using equation Eq. 16.

T ~ e = ∑ i = 1 S ⊕ ∑ j = 1 L n T ~ i j ⊗ P i j , n = 1 , 2 , …

(16)

The trapezoidal membership function for the network's completion time is then transformed into a crisp value T e using the Center of Gravity (CoG)^118,119 technique using Eq. 17.

T e = ( t 1 + t 2 + t 3 + t 4 4 )

(17)

Then, based on the steps for applying NF-GERT explained in section 3, the project duration is computed using the proposed NF-GERT technique. When compared to the CPM, which is 80% proximate for small projects, 76% for medium projects and 72% for large projects, the proposed Neutrosophic Fuzzy based GERT (NF-GERT) technique is more accurate to the actual project duration (or completion time) with 98% significance for small projects, 95% for medium projects, and 92% for large projects, as shown in Figure. 10. To assess whether the observed improvement in accuracy using NF-GERT, compared to other methods is significant, a systematic approach through statistical hypothesis testing is essential.

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Figure 10.

Project duration (completion time) of (a) small (b) medium (c) large residential projects.

5.2.1 Statistical tests to compare accuracy

To statistically assess the accuracy of NF-GERT in comparison to other scheduling methods, a one-way ANOVA is conducted using the SPSS software package. This analysis evaluates the accuracy scores across the same set of projects by comparing the durations estimated by various scheduling methods (CPM, GERT, F-GERT, and NF-GERT) with the actual as-built durations. The ANOVA examines the mean differences between project categories (small, medium, and large), considering the variability between the methods. At a 95% confidence level, the ANOVA determines whether these mean differences are statistically significant, indicating whether NF-GERT provides a significant improvement in accuracy over the other methods.

The durations of the forty-five real-time construction projects used for to test the variance are carefully reviewed to ensure they meet the required assumptions for the analysis. ANOVA relies on three basic assumptions: independence of observations, normality of data, and homogeneity of variances. ³⁸ The durations of the 45 projects considered in the analysis are independent, ensuring that the observations within each group do not influence each other. To assess the normality of the data, the Shapiro-Wilk test which is appropriate for small sample sizes,^120,121 is conducted using the SPSS software package, providing a statistical evaluation of the normality assumption. The Shapiro-Wilk test results in Table 5 indicate that the data for all methods tested follows a normal distribution. The Shapiro-Wilk statistics (W) ranged from 0.954 to 0.966, with 45 degrees of freedom (df), indicating a close fit to normality. The corresponding p-values ranged from 0.071 to 0.204, all above the significance level of 0.05, indicating no significant deviation from normality. Therefore, it can be concluded that the data is normally distributed.

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Table 5

Normality test results for project durations using shapiro-wilk.

	Shapiro-Wilk p-value
Scheduling Technique	W	df	p-values
CPM	.962	45	.146
PERT	.966	45	.204
GERT	.957	45	.098
F-GERT	.958	45	.105
NF-GERT	.959	45	.113
As-build	.954	45	.071

Additionally, homogeneity of variances is tested through Levene's test, also performed in SPSS, to confirm that the variances across the groups are approximately equal. The Levene's test results presented in Table 6, with p-values greater than 0.05, infer that the variances across small, medium, and large project groups are homogeneous, suggesting no significant variation in variances among the groups. The Levene statistic close to 0 further indicates that the variances between the groups are very similar, meaning homogeneity of variances is satisfied. These tests ensure that the fundamental assumptions of ANOVA are met for valid statistical analysis.

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Table 6

Levene's test results for small, Medium, and large projects.

Project Size	Levene Statistic	p-value
Small	0.192	0.965
Medium	0.156	0.978
Large	0.599	0.701

An ANOVA test is conducted to identify the overall variance among the different scheduling methods, followed by a Tukey's Honest Significant Difference (HSD) post hoc test to pinpoint the specific methods contributing to the significant differences. The ANOVA results for small, medium, and large projects presented in Table 7 reveal different patterns of significance.

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Table 7

ANOVA results for small, Medium, and large projects.

Project Size	F-statistic	p-value	Result
Small	0.896	0.488	Not Significant
Medium	5.299	0.000	Significant
Large	4.037	0.002	Significant

For small projects, the p-value of 0.488 indicates no significant difference in the durations across the various scheduling methods, suggesting similar performance. However, for medium projects, the p-value of 0.000 points to a statistically significant difference between the methods, with a moderate to strong effect as indicated by the F-value of 5.299. Similarly, for large projects, the p-value of 0.002 shows a significant difference, with an F-value of 4.037, suggesting that the choice of scheduling method has a considerable impact on duration estimates for both medium and large projects. Thus, while scheduling methods may not significantly affect small projects, they do play a crucial role in determining project durations for medium and large-scale projects. Therefore, to identify which specific pairs of methods differ, a post-hoc Tukey (HSD) test is conducted. Additionally, Tukey HSD is also performed for small projects to evaluate any potential variance among the methods, despite the lack of overall significance. Table 8 shows Post-Hoc Tukey HSD test results for durations across scheduling methods in small, medium, and large projects.

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Table 8

Post-Hoc tukey HSD test results for durations.

As-build vs	Small		Medium		Large
	Mean Difference	p-value	Mean Difference	p-value	Mean Difference	p-value
CPM	148.53333	0.502	268.40000	0.001	424.93333	0.002
PERT	105.66667	0.813	195.73333	0.024	329.00000	0.035
GERT	48.40000	0.993	97.93333	0.607	177.86667	0.568
F-GERT	37.53333	0.998	76.73333	0.812	148.66667	0.739
NF-GERT	15.93333	1.000	43.13333	0.981	116.73333	0.887

From Table 8, it is observed that for small projects, GERT, F-GERT, and NF-GERT exhibit relatively smaller mean differences when compared to the As-Build schedule, in contrast to the traditional methods. Among these, NF-GERT demonstrates the smallest mean difference of 15.93 days, suggesting that it closely approximates the actual project duration, though the result is statistically insignificant (p-value = 1.000). GERT and F-GERT also show minimal deviations, with mean differences of 48.4 days and 37.53 days, respectively, and p-values nearly equal to 1. Despite the lack of statistical significance, the mean differences for the traditional methods, CPM (148.53 days) and PERT (105.67 days), are notably larger compared to GERT, F-GERT, and NF-GERT. This indicates that GERT, F-GERT, and NF-GERT may all be suitable for small projects, with NF-GERT being the most proximate to the As-Build schedule.

For medium projects, Table 8 shows that F-GERT and NF-GERT have smaller mean differences when compared to the As-Build schedule, making them more suitable for medium-scale projects. NF-GERT exhibits the smallest mean difference of 43.13 days, which suggests it closely approximates the actual project duration, although the result is statistically insignificant (p-value = 0.981). F-GERT also shows a relatively small mean difference of 76.73 days, with a p-value of 0.812, further supporting its potential as a reliable method for medium projects. On the other hand, GERT, with a mean difference of 97.93 days and a p-value of 0.607, may not adequately account for uncertainty, which limits its effectiveness for medium-scale projects. In contrast, the traditional methods, such as CPM (268.4 days) and PERT (195.73 days), have larger mean differences and demonstrate statistically significant results (p-value < 0.05). Therefore, while traditional methods show larger deviations, F-GERT and NF-GERT provide more precise approximations of the As-Build schedule, making them more appropriate for medium-scale projects.

For large projects, Table 8 indicates that NF-GERT provides the most accurate approximation to the As-Build schedule, with a mean difference of 116.73 days and a p-value of 0.887. This suggests that NF-GERT closely approximates the actual project duration and is the most robust method for large projects. In contrast, traditional methods like CPM (424.93 days) and PERT (329.00 days) show much larger mean differences, indicating that they are less effective in predicting the duration for large-scale projects. F-GERT, with a mean difference of 148.67 days and a p-value of 0.739, also provides a reasonable approximation but is still less accurate than NF-GERT. GERT, with a mean difference of 177.87 days (p-value = 0.568), is the least effective among the methods, suggesting that it does not capture the necessary uncertainty and variability in large projects as effectively as NF-GERT. Therefore, NF-GERT emerges as the most suitable and reliable method for large projects, demonstrating its robustness in approximating the As-Build schedule.

7.1 Further research

This study demonstrates the effectiveness of the NF-GERT methodology in accurately estimating project completion time by incorporating uncertainty factors such as fuzziness and randomness. However, the computational process requires additional tools due to the complexity of handling uncertain input parameters. Future research could focus on developing an integrated computational framework to streamline the application of NF-GERT in real-world construction projects.

Moreover, while this study employs single-valued neutrosophic sets, exploring alternative representations, such as interval-valued neutrosophic sets, could enhance the flexibility and accuracy of uncertainty modeling. A comparative analysis of these approaches in construction scheduling could provide valuable insights into their performance variations and decision-making implications. Investigating the applicability of interval-valued neutrosophic sets may further improve project scheduling methodologies by offering a more comprehensive representation of uncertainty.