Causal Reasoning in Construction Process Scheduling

Abstract

This paper introduces an advanced framework for modeling and scheduling construction processes using causal inference techniques, with particular emphasis on capturing complex technological and organizational interdependencies. By integrating causal calculus and counterfactual reasoning, the study demonstrates how construction schedules can be analyzed and optimized not only through temporal relationships but also through explicit cause–effect structures. A matrix-based scheduling methodology is presented, incorporating diagonal and reverse-diagonal time couplings consistent with the Time Coupling Method (TCM). The computational procedure is detailed, including the determination of earliest and latest event times, identification of the critical path, and computation of activity floats. Based on an in-depth examination of technological and organizational constraints, eight theorems are formulated and proven, establishing the fundamental properties of a scheduling approach that embeds causal mechanisms. The findings indicate that the integration of causal inference into construction planning enables more accurate identification of factors influencing project duration, enhances synchronization of dependent activities, and minimizes conflicts and idle times. This causally informed framework strengthens decision-making by allowing practitioners to predict the consequences of modifications in project execution strategies. The developed models constitute a robust foundation for future research on leveraging causal inference algorithms and artificial intelligence to advance construction process management.

1. Introduction

Causal inference represents a rapidly developing and interdisciplinary field of research, with applications spanning both the natural and engineering sciences. The present work is situated within this methodological paradigm, whose theoretical foundations were established by J. Pearl and D. Mackenzie [1]. These authors introduced a formal calculus of causality based on two complementary languages: a graphical language, enabling the modeling of causal relations through directed acyclic graphs (DAGs), and a symbolic language, analogous to algebraic notation, which allows for the precise formulation of causal queries. A central element of this framework is the intervention operator do(), used to articulate counterfactual research questions aimed at determining the consequences of hypothetical actions.

A key component of causal inference is the use of counterfactual expressions, which enable questions of the form “what would happen if the conditions of the process were altered?” to be rigorously addressed. Owing to its predictive and decision-oriented nature, causal inference constitutes a valuable analytical tool in engineering contexts, where understanding the influence of individual variables on final outcomes is of fundamental importance.

Construction scheduling is one of the fundamental processes of construction project management, determining not only the efficiency of project implementation, but also its ultimate economic and technical success [2,3,4]. In view of the growing complexity of modern construction projects and increasing requirements for planning precision, traditional scheduling methods, based mainly on deterministic time models, prove insufficient to cope with the uncertainty and variability characteristic of the construction environment [2,5,6]. Causal inference, as a relatively new field in the context of construction project management, offers promising prospects for improving the accuracy and reliability of planning processes [7,8]. Unlike conventional methods based on statistical correlations, causal inference allows the identification of actual cause-and-effect relationships between variables affecting schedule performance. This approach allows for a deeper understanding of the mechanisms behind delays, prediction of the effects of potential disruptions, and optimization of resource allocation based on proven causal relationships [9].

Despite significant advances in project management methodologies, the construction industry continues to struggle with delays and budget overruns, which, according to industry reports, affect up to 70% of projects [10,11]. Existing scheduling tools, such as the critical path method (CPM) and the program evaluation and review technique (PERT), while effective in relatively stable environments, have limited adaptability to dynamically changing project circumstances [12,13]. Previous research in scheduling has focused mainly on algorithmic optimization and incorporating uncertainty through probabilistic methods [14]. However, there is a lack of comprehensive research using modern causal inference techniques to identify and model the actual mechanisms that influence the implementation of construction schedules. This research gap is a significant limitation in the development of more effective and resilient project planning systems.

2. Previous Research

The origins and development of the concept of causality have been extensively discussed in the literature, with individual publications contributing significantly to both the theoretical and methodological foundations of causal analysis. A historical overview of causal thought was presented, among others, by Hoover [15], Peters et al. [16], Losee [17], and Mumford and Anjum [18], whereas contemporary approaches to causal identification based on observational data and the problem of transporting causal conclusions across populations were outlined by Pearl and Bareinboim [19].

Hoover [15] examines causality in the context of economics—particularly macroeconomics—demonstrating how causal relations can be reconstructed through economic modeling. The author highlights the limitations of traditional statistical and econometric tools, which often capture correlations rather than genuine causal mechanisms. Hoover argues that understanding economic processes requires formulating causal models grounded in economic theory and in an analysis of the data-generating mechanisms.

Peters, Janzing, and Schölkopf [16] present the formal foundations of modern causal inference, integrating them with machine-learning methodologies. Their work discusses issues such as identifiability, structural causal models (SCMs), algorithms for causal structure discovery, and methods for assessing the quality of causal models. The publication makes a significant contribution to bridging causal theory with artificial intelligence, including both supervised and unsupervised learning approaches.

Losee [17] offers a broad historical study of theories of causality—from metaphysical and philosophical perspectives (Aristotle, Hume), through the development of logic and the natural sciences, to contemporary scientific frameworks. The author traces how the concept of cause evolved across historical periods and the role it played in the development of scientific reasoning. The work situates modern causal models within a wider philosophical and methodological context.

Mumford and Anjum [18] present an interpretation of causality grounded in dispositionalism, emphasizing the notion of causal powers. The authors criticize the reduction of causality to purely statistical dependencies and argue that causation should instead be understood as a property of entities and processes that possess the capacity to generate effects under appropriate conditions. Their publication offers an alternative philosophical perspective, highlighting the dynamic and context-dependent nature of causal relations.

A contemporary methodological outlook—including causal calculus and the identification of intervention effects from observational data—is provided by Pearl and Bareinboim [19]. The authors introduce the concept of transportability, referring to the conditions under which causal conclusions can be transferred across different populations or research environments. Their work expands Pearl’s do-calculus by characterizing the circumstances in which findings from experiments or observational studies can be generalized to other groups or systems. This constitutes an essential component of modern causal methodology, particularly in analyses based on large-scale data.

In recent years, a growing interest in the application of causal inference methods in civil engineering and construction has been observed. The rapid development of data acquisition techniques, the increasing availability of large engineering datasets, and the need for interpretable and robust decision-support models have significantly contributed to the adoption of causal approaches—including causal inference, causal discovery, and Bayesian networks—not only in theoretical studies but also in practical engineering applications [20]. These methods have found particular relevance in several domains:

• Building energy performance modelling and evaluation—Causal methods allow for distinguishing between mere correlations and true causal relationships between design parameters and energy use. This enables more reliable prediction of the effects of design interventions and supports the optimisation of energy-efficient solutions. Recent work proposes a four-step causal inference framework tailored to building design and performance analysis [21].
• Risk analysis, structural reliability, and infrastructure asset management—Causal–probabilistic models, such as Bayesian networks, are increasingly used to represent the propagation of failures, update knowledge about structural condition, and support maintenance and rehabilitation decisions. Studies published in the past few years demonstrate the value of causal modelling for predicting degradation and structural reliability [22].
• Schedule risk management and delay analysis in construction projects—Bayesian networks and related causal modelling techniques have been applied to identify critical factors contributing to project delays and to simulate potential mitigation strategies (e.g., alternative resource allocation). This allows project managers to better prioritise preventive measures and assess the impacts of interventions [22].
• Safety management and accident prevention on construction sites—Causal inference methods, including the integration of causal models with information extracted from unstructured accident data, are increasingly employed to analyse root causes of “struck-by” incidents, estimate risk levels, and design more effective preventive actions [23,24].
• Decision-support tools integrating BIM and digital twins—The integration of causal models with Building Information Modelling (BIM) and digital twin technologies enables the development of more interpretable monitoring and prediction tools for the behaviour and performance of building and infrastructure assets. Recent literature highlights the importance of combining expert knowledge (priors) with observational data within causal frameworks [20].
• Causal inference is increasingly used in the context of designing energy-efficient buildings and monitoring systems [21]. Studies using a Bayesian approach to estimate the life cycle costs of bridges show how probabilistic models (which can be easily extended to Bayesian networks) can be used for LCCA and uncertainty modeling [25]. In addition, ref. [26] describes system dynamics and causal loop diagrams, which, although not formal “causal inference” in the statistical sense, create a cause-and-effect framework useful for modeling mechanisms affecting LCC (e.g., cost-maintenance feedback loops).

Over the past two years, causal inference has been used in research related to construction project management, particularly in areas such as the analysis of the impact of management actions, modeling the relationships between risk factors, and building causal models to predict the behavior of construction systems. One of the most direct examples of this trend is the work of Zhitian Zhang and colleagues (2025), who presented quantitative causal inference in construction safety management [27]. This study applied a Structural Causal Model to quantitatively analyze the impact of management measures on employee safety behavior, revealing the mechanisms of influence of individual managerial actions on behavior, along with their stability and significance. The study [27] was based on data collected in a seven-week survey and analysis of changes and causal diagrams, which made it possible to identify the actual effects of managerial interventions in the construction environment.

In addition to analyzing the impact of health and safety practices, there are also studies in the literature that attempt to discover general causal models in structural and civil engineering. An example is the 2024 study [28], published in Defence Technology, which is one of the first studies to construct complete causal models for structural phenomena and construction engineering, using classes of causal discovery algorithms (such as PC, FCI, and GES) to identify causal relationships in technical data on the strength of structural elements and the behavior of materials under load. This study represents an important step towards the formal modeling of cause-and-effect relationships in engineering processes, with potential usefulness in the forecasting and quality control of construction projects [28].

One approach that has emerged in technical literature is the concept of uncertainty handling in schedule planning, where probabilistic methods and simulation-based models are combined with formal causal structures to select optimal schedules that take risk into account [29,30]. Acebes and co-authors proposed a method that allows for the integration of random uncertainty in the analysis of alternative schedules, which makes it possible to identify the schedules with the highest probability of meeting deadlines under conditions of risk (aleatory uncertainty) [30]. Although this approach does not use a formal causal framework, it is an example of research aimed at understanding which schedule is actually most resistant to risk factors, which is closer to cause-and-effect analysis than to classical deterministic models.

The applications of causal inference methods also extend beyond scheduling and safety—although not always directly in construction, there are examples in related engineering fields that illustrate practical approaches to identifying causal structures. An example is the 2024 article by Lingyun You and co-authors [31], which proposed a framework combining the classical structural model with a potential outcomes approach to assess the long-term effects of asphalt pavement maintenance policies, linking causal methodology with practical infrastructure engineering problems. This approach can serve as inspiration for the adoption of causal techniques in the analysis of decisions concerning maintenance and construction and infrastructure optimization.

An important element in the development of causal inference methods is also their integration with machine learning techniques and automatic data analysis. An example of this is the proposal to use large language models (LLMs) and natural language processing techniques to automatically infer the causes of construction delays by analyzing progress reports [32]. Such approaches, although still at the preprint stage, indicate the broad potential for integrating AI techniques with causal analysis in the practice of construction project management.

Overall, contemporary studies in civil engineering and construction demonstrate not only the growing adoption of causal inference methods but also their increasing relevance in contexts where interpretability, robustness, and reliable prediction of intervention effects are essential. These characteristics position causal inference as a promising and valuable approach for advancing engineering decision-making. Although the literature is still limited in terms of strictly harmonogram and planning applications, existing works from 2024 and 2025 point to the growing popularity of causal approaches and their potential benefits for advanced construction project management.

3. Problem Statement

Despite growing interest in the application of causal inference in various fields of science, its use in construction project management accounts for only a small percentage. Existing research in the construction sector is characterized by significant methodological limitations that prevent the full potential of causal inference from being exploited, e.g., in scheduling practice. Previous work has focused mainly on identifying static causal relationships, ignoring the dynamic and temporal nature of construction processes, where causal effects manifest themselves through complex temporal couplings between different phases of project implementation. One of the gaps in the current state of knowledge is the lack of an integrated approach combining causal inference with temporal coupling methods (TCM), which are necessary to understand the propagation of disturbances in construction schedules. None of the existing studies address the fundamental problem of modeling causal temporal relationships between interdependent construction activities.

The presented issue concerns the modeling of the process of performing a complex of construction works, taking into account specific technological dependencies. It may concern the condition of simultaneous commencement or completion of works by work teams on successive work fronts in a complex of facilities. In essence, it concerns the parallel execution of technologically successive works on work sites, taking into account additional restrictions, e.g., the simultaneous commencement of works on fronts as a result of the division of the contractor’s work group into specialized teams. This complicates the work plan and detailed schedule. The analysis of dependencies and the elimination of conflicts between works can be carried out using causal inference.

The technique of causal inference essentially involves manipulating the effects of events based on their causes using mathematical tools. Currently, the approach to finding answers to many questions, using artificial intelligence and data analysis, paves the way for causal analysis. A new approach has been proposed, referred to as “the science of causal inference” [1].

Research on modeling construction processes and scheduling using the LSM (Linear Scheduling Method), VSM (Velocity Scheduling Method), LOB (Line of Balance), and approaches based on optimization algorithms has been extensively developed in the literature. Key contributions in this domain were made by, among others, Lucko [33], Fewings [34], O’Brien [35], Russell [36], and Senouci [37].

Lucko and Peña Orozco [33] conducted analyses related to the modeling of time buffers in linear scheduling by employing singularity functions. Their work enabled a more precise formulation of methods for calculating different types of float in linear projects, which in turn increased the accuracy of detecting activity conflicts, analyzing critical paths, and optimizing crew flow in linear projects such as roadworks or utility networks.

Fewings and Henjewele [34] presented an integrated approach to construction project management encompassing process structures, work organization models, tools for visualizing workflow, and mechanisms for process improvement. Their work also includes practical applications of value stream mapping (VSM) and sequential scheduling tools in the context of enhancing resource efficiency and reducing execution cycles.

The classical monograph by O’Brien [35] represents one of the earliest comprehensive treatments of scheduling techniques in construction. The author compiled and systematized methods such as CPM, PERT, network scheduling, resource leveling techniques, and cyclic methods, including concepts related to LOB. This work became a foundation for the subsequent development of deterministic and probabilistic scheduling and remains a key reference point for researchers focusing on the structural aspects of planning.

Russell and Wong [36] proposed a new generation of planning structures by defining more flexible and functional models for scheduling repetitive processes. They focused on the analysis of repetitive tasks in space and time, introducing enhanced graphical representations and mathematical models that enable the identification of potential resource conflicts and crew-to-crew interferences. Their work formed the basis for the development of modern LSM tools and velocity-based planning.

Senouci and Eldin [37] developed one of the first applications of dynamic programming to linear project scheduling, particularly for nonserial structures. The proposed methodology enables optimizing the sequence of activities and determining work trajectories that minimize project duration or cost. By applying dynamic techniques, the authors demonstrated the possibility of reducing the number of conflicts, allocating resources more effectively, and performing computations for complex dependency structures—constituting a breakthrough in the modeling of linear projects.

In the present study, a specific case of scheduling construction works is considered. The analysis focuses on TCMs (Time Coupling Methods), in which time couplings generate temporal dependencies (i.e., start and finish times of construction activities). TCMs constitute a group of analytical tools used in planning, modeling, and controlling the execution of construction works. Their essence lies in representing causal–temporal relationships between tasks by defining couplings that determine permissible start and finish times. These couplings make it possible not only to define the sequence of activities but also to model technological, organizational, and resource constraints that influence the dynamics of the schedule.

In TCMs, each task is linked to other tasks through specific types of temporal relations that generate downstream consequences within the schedule. These couplings represent:

• Technological dependencies (e.g., concrete can be placed only after reinforcement has been installed),
• Organizational constraints (e.g., a crew may begin work only after the work front becomes available),
• Resource limitations (e.g., equipment availability),
• Environmental and formal/administrative factors.

As a result, TCMs generate causal temporal consequences that define the maximally admissible structure of the execution sequence.

The application of time-coupling methods makes it possible to:

• Determine the most realistic project execution trajectory arising from actual technological interdependencies,
• Analyze the impact of changes in individual tasks on the entire schedule—crucial for plan revisions and what-if analyses,
• Model chains of delays and assess their propagation to subsequent activities,
• identify the critical path and potential time reserves,
• Optimize the sequence of work with respect to resource and execution constraints,
• Obtain a basis for process-level causal analysis, in which schedule changes are attributed to dependencies resulting from temporal couplings.

From the perspective of contemporary causal methodology, TCMs can be interpreted as a formal mechanism for modeling time-dependent causal relations, for example:

• Through couplings enabling statements such as: “The completion of A is a causal precondition for the start of B.”
• Through lag parameters that model technological mechanisms, e.g., “After concrete placement, at least 48 h must elapse before formwork for the next storey can begin.”
• Through the critical path, which reflects a chain of necessary causes whose violation affects the entire project.

In this way, TCM can be incorporated into causal-effect analyses within the framework of structural causal models, particularly in the context of delay propagation assessment, counterfactual what-if scenario analysis, and validation of technological and organizational assumptions.

A typical practical example of applying TCM is the analysis of the impact of material delivery delays on the construction schedule. In cases of disruptions in the supply of essential materials—such as reinforcing steel—Finish-to-Start (FS) and Start-to-Start (SS) dependencies allow for automatic recalculation of the start and finish times of subsequent tasks. A delay in steel delivery shifts reinforced-concrete works and, through SS relations, also affects the start of installation works that require prior execution of structural elements.

Another area where TCM improves planning accuracy is the scheduling of precast assembly. In such projects, assembly tasks are tightly linked to the delivery status of prefabricated components through FS relations, as well as to the site’s logistical capabilities—particularly vertical transportation—through SS relations that capture parallel operational dependencies. This enables realistic modeling of assembly sequences and early identification of potential temporal clashes.

TCMs are also used in variant analyses related to assessing the efficiency of construction technologies. The introduction of new execution methods—such as faster formwork systems—allows for the reduction in technological lags between successive work cycles, which directly shortens the overall project duration. By modifying time-coupling parameters, it becomes possible to quickly estimate the effects of technological changes and select the most advantageous scheduling variant.

All these examples confirm that TCM constitutes an effective tool for analyzing dynamic temporal relations and optimizing construction processes under material, operational, and organizational uncertainty.

4. Basic Assumptions

The causal inference technique was applied to a selected example from the TCM group of methods, using diagonal and inverse couplings [38,39,40]. A temporal coupling is the time interval between individual robots. The basic types of temporal couplings are listed below:

• Coupling between means of implementation (S_S)—the difference between the earliest (latest) start date of a given task and the earliest (latest) completion date of the preceding task of the same type;
• Coupling between objects (S_F)—the difference between the earliest (latest) start date of a given task and the earliest (latest) completion date of the preceding task performed in a given object;
• Diagonal and inverse couplings (applies to works with the same sum of indices in the matrix notation, i.e., consecutive works on consecutive objects).

In the specific case of scheduling a complex of works linked by technological dependencies, we can synchronize them, ensuring their parallel execution, taking into account the impact of diagonal and reverse couplings. This applies to the dependencies between the earliest and latest start and end dates of the works.

In the assumptions of the method of organizing works, taking into account the impact of diagonal and reverse couplings [29,38,39,40,41,42,43,44], single-stage works form so-called diagonals, the duration of which is equal to the duration of critical works. Although there may be several critical works (with the same duration) in a diagonal, only one critical work is taken into account when determining the duration of the complex of works.

The time required to complete the complex of works can be presented in the form of:

$$T={\displaystyle \sum _{q=1}^{n+m-1}{t}_{ijq}}$$

(1)

where:

• T—duration of complex implementation with critical works, taking into account diagonal and reverse diagonal connections,
• t_ijq—duration of critical work of the j-th type on the i-th work front, determining the duration of the q-th diagonal,
• n + m − 1—number of diagonals.

5. General Characteristics of the Scheduling Method

The condition of parallel execution of successive technological works on construction sites can be ensured by taking into account diagonal and reverse couplings in the adopted matrix model. It allows for the organization of numerical data and the performance of calculations. Numerical data for scheduling, taking into account the adopted technological conditions, i.e., for example, the sequence of works on construction sites, can be functions or numbers. In the calculation process for determining the start and end dates of works, three main stages can be distinguished. In the first stage (

Table 1. Stage I—matrix with fixed rows in place of the first row.

		Types of Work
		1	2	…	j	…	…	m − 1	m
Work fronts	U	tk1	tk2	Stage I	line	fixed	(U)	tk,m−1	tkm
	2	t21	t22					t2,m−1	t2,m
	j
	…
	n	tn,1	tn,2					tn,m−1	tn,m

), the first series of intermediate matrices is constructed, in which all rows of the input matrix are determined in sequence. Next, prospective matrices are determined by indicating the branches that should be developed in the simultaneously constructed graph tree. These are matrices in which the Marginal Minimum Possible (MMP) indicator of the duration of the complex of works has reached its lowest value. A full description of the method with the calculation algorithm can be found in [29,38,39,40,41,42].

In the matrix model, robots with the same sum of element indices form diagonals linked by technological and organizational dependencies. This applies to successive works in the technological case on successive work fronts. The diagonals determine the time interval in which they should be performed. Robots with different execution times may appear in the diagonal, but the robot with the longest duration is the critical robot that determines the dimension of the diagonal, i.e.,

$${\left\{t_{1,n}, t_{2, n-1},\dots t_{m,1}\right\}}^{\underset{\max }{\to }}{t}_{i,j,q}$$

(2)

where

• n—number of rows in the matrix (work fronts, objects),
• m—number of columns in the matrix (types of work),
• q—critical work q in the diagonal of the matrix.

The remaining work may be performed at any time between $t_{i,j,g}^{wr}$ and $t_{i,j,q}^{pz}$ so they will have time reserves.

In the second stage of the solution (

Table 2. Stage II—matrix built in intermediate phases.

		Types of Work
		1	2	…	j	…	…	m − 1	m
Work fronts	${\mathrm{i}}_{\mathrm{F}1}^{\mathrm{p}}$
	${\mathrm{k}}_{\mathrm{F}2}^{\mathrm{p}}$
	U			STAGE II line determined in phase k
	…
	n − 1
	n	tn,1	tn,2					tn,m−1	tn,m

where ${\mathrm{i}}_{\mathrm{F}\mathrm{k}}^{\mathrm{p}}$—perspective lines determined in intermediate phases.

), a series of matrices is constructed, establishing two, three, or n − 2 rows (working fronts) depending on the number of rows in the initial matrix. The second stage should therefore be divided into phases in which the first rows are obtained from the matrices identified as promising in the previous stages and phases. The next row is obtained from the subsequent rows of the output matrix. The analysis of the sequence of steps is analogous to that in the first stage. It should be remembered that if the Marginal Possible Minima (MPM) determined in the second stage are greater than or equal to the MPM obtained in the previous stage, then from that moment on, the branches from the previous stage must also be considered promising. Only when we have considered all the competing matrices, indicating which branches of the variant tree should be developed, can we move on to the final stage of the solution.

In the third stage of the solution, no intermediate matrices are introduced, and the formed matrices are final variants that unambiguously determine the position of all rows, i.e., they determine all partial complexes (partial fronts). At the same time, the conditional MPM indicator transitions to the real time of implementation of the complex of works T.

The final matrices have n − 2 rows determined in the previous stages of the solution as prospective. In one branch of the graph developed to the end, we have only two possible combinations. The number of final matrices to be solved is therefore a double multiple (shown in the previous stages) of the prospective branches.

6. Calculation of Time Characteristics

The solution consists in determining the earliest and latest dates for the commencement and completion of works, the critical path, numerical values of dependencies in the earliest and latest dates, time reserves, and the minimum time required for the completion of the entire complex. Using the given symbols, the earliest dates for the commencement and completion of works are determined using the following formulas:

$${\mathrm{t}}_{\mathrm{ij}}^{\mathrm{wr}} = \max 〈{\mathrm{t}}_{\mathrm{i}+1,\mathrm{j}-1}^{\mathrm{wr}}; {\mathrm{t}}_{\mathrm{i}-1,\mathrm{j}+1}^{\mathrm{wr}}〉$$

(3)

$$t_{ij}^{wz} = t_{ij}^{wr} + t_{ij}$$

(4)

This means that when determining the earliest start dates for work j on front i, diagonal coupling is applied from work in the immediately preceding partial pipeline and, at the same time, in the immediately following partial complex of front i + 1, works j − 1 and the reverse diagonal coupling is applied from works in the next partial pipeline and simultaneously in the immediately preceding partial front i − 1 of works j + 1. The earliest completion dates for works j on front i are determined by adding up the earliest start dates for these works and the duration of their implementation.

The latest completion dates for works j on front i are determined taking into account the diagonal coupling from works in the immediately preceding partial front and simultaneously in the immediately following partial stream j + 1, as well as the reverse diagonal coupling from works in the immediately following partial front i + 1 and simultaneously in the immediately preceding partial stream j − 1.

$$t_{ij}^{pz} = \min 〈t_{i-1,j+1}^{pz}; t_{i+1,j-1)}^{pz}〉$$

(5)

$$t_{ij}^{pr} = t_{ij}^{pz} - t_{ij}$$

(6)

The dependencies between the means of implementation for the earliest deadlines are determined as the difference between the earliest completion dates for a given task j on front i − 1 and the earliest start dates for that task on front i.

$${\mathrm{S}}_{\mathrm{ij}}^{\mathrm{sw}} = {\mathrm{t}}_{\mathrm{i}-1,\mathrm{j}}^{\mathrm{wz}} - {\mathrm{t}}_{\mathrm{i},\mathrm{j}}^{\mathrm{wr}}$$

(7)

Similarly, the links between fronts for the earliest dates are determined as the difference between the earliest start dates for the given works j on front i and the earliest completion dates for the preceding works j − 1 on the partial front i.

$${\mathrm{S}}_{\mathrm{ij}}^{\mathrm{fw}} = {\mathrm{t}}_{\mathrm{ij}}^{\mathrm{wr}} - {\mathrm{t}}_{\mathrm{i},\mathrm{j}-1}^{\mathrm{wz}}$$

(8)

For the latest dates, the coupling between implementation measures is determined from the relationship:

$${\mathrm{S}}_{\mathrm{ij}}^{\mathrm{sp}} = {\mathrm{t}}_{\mathrm{i}+1,\mathrm{j}}^{\mathrm{pr}} - {\mathrm{t}}_{\mathrm{i},\mathrm{j}}^{\mathrm{pz}}$$

(9)

Similarly, the dependencies between work fronts for the latest completion dates are determined using the formula:

$${\mathrm{S}}_{\mathrm{ij}}^{\mathrm{fp}} = {\mathrm{t}}_{\mathrm{i},\mathrm{j}+1}^{\mathrm{pr}} - {\mathrm{t}}_{\mathrm{i},\mathrm{j}}^{\mathrm{pz}}$$

(10)

Time reserves are the difference between the latest and earliest start dates for the works, as well as between the latest and earliest completion dates for the works:

$${\mathrm{R}}_{\mathrm{ij}} = {\mathrm{t}}_{\mathrm{ij}}^{\mathrm{pz}} - {\mathrm{t}}_{\mathrm{ij}}^{\mathrm{wr}} - {\mathrm{t}}_{\mathrm{ij}} = {\mathrm{t}}_{\mathrm{ij}}^{\mathrm{pr}} - {\mathrm{t}}_{\mathrm{ij}}^{\mathrm{wr}} = {\mathrm{t}}_{\mathrm{ij}}^{\mathrm{pz}} - {\mathrm{t}}_{\mathrm{ij}}^{\mathrm{wz}}$$

(11)

where

• R—time reserve.

A full description of the calculation method and numerical examples can be found in [40,41].

7. Causal Reasoning

It should be noted that when using the scheduling method described above, the tasks will be characterized by parallelism in their execution within set time intervals. They will constitute a group of tasks with a parallel-stream structure, commonly found in the implementation process. The effects of intuitive construction process management can cause disruptions, collisions, unexpected downtime, and other phenomena.

It is necessary to define the basic parameters of the works, i.e., the start and end dates and the values of time couplings, as well as to determine the sequence of works in order to synchronize them rationally, taking into account the optimization criterion in the form of a time function.

Due to the number of practical advantages of the method of organizing work, a set of properties can be formulated.

7.1. Properties of the Method

In the case of zero diagonal and reverse diagonal couplings for late and early terms:

$$t_{ij}^{wr} = t_{i-1, j+1}^{wr} = t_{i+1, j-1}^{wr}$$

(12)

and

$$t_{ij}^{pr} = t_{i-1, j+1}^{pr} = t_{i+1, j-1}^{pr}$$

(13)

Definition:

$$\underset{x\in W}{\forall } \underset{N\subset W}{\exists } f(x_1) = a \wedge f(x_2) = b \wedge f(x_3)=c\Rightarrow a = b = c$$

(14)

and

$$\underset{y\in P}{\forall } \underset{W\subset P}{\exists } f(y_1) = d \wedge f(y_2) = e \wedge f(y_3) = f \Rightarrow d = e = f$$

(15)

where:

• $N=\left\{x_1, x_2,x_3\right\}$.
• $W=\left\{y_1,y_2,y_3\right\}$.

This dependency imposes a practical condition on the course of construction works, consisting in setting time limits, i.e., start and end dates, for works related to neighboring work fronts and for subsequent technological construction processes. Due to the varying duration of the works (as a result of the different sizes of the work fronts and labor intensity), some of them will be critical, while others will provide time reserves. Knowledge of these values will allow the process manager to make the right decisions.

An analysis of the relationships between robots, given the accepted technological limitations (e.g., the commencement of various types of work on multiple work fronts limited by material delivery dates or vertical transport) or organizational limitations (e.g., the availability of work groups, etc.), allows for the formulation of a number of properties.

7.2. Claim 1

A matrix of task durations T with zero diagonal and inverse diagonal couplings is given. If the minimum number of critical tasks has been determined, it is equal to the sum of the types of tasks (columns in the matrix, m) and partial fronts n (minus one), and the maximum number is equal to their product.

Proof. 

• Let us mark:
• q—number of critical tasks,
• m—number of types of work,
• n—number of work fronts,
• T—work duration matrix.

In the matrix, m + n − 1 diagonals q can be designated (i.e., works with the same sum of indices in the matrix).

For the earliest and latest dates:

$$\underset{x\in W}{\forall } \underset{N\subset W}{\exists } f(x_1) = a \wedge f(x_2) = b \wedge f(x_3) = c \Rightarrow a = b = c$$

(16)

where:

• $N=\left\{x_1,x_2,x_3\right\}$

Therefore, in each diagonal q, no less than one critical operation can be determined for T in diagonal q:

$$\left\{t_{ij},t_{i-1,j+1},t_{i+1,j-1}\right\} = \max t_{ijq}$$

(17)

so

$$\min {\displaystyle \sum _{q=1}^{n=m-1}{t}_{ij1} = T}$$

(18)

Number of elements in the matrix: m × n = L

$$\underset{y\in P}{\forall } \underset{Y\subset P}{\exists } f(y_1) = a \wedge f(y_2) = b\dots f(y_m) = c \wedge f(y_n) = d \Rightarrow a=b=c=d$$

(19)

i.e., the values of the matrix elements are equal, so all elements are critical:

$$m\times n=L$$

(20)

□

Conclusion: In the process of construction works, the security of timely contract performance depends on the number of critical works in the complex. The minimum number of critical works reduces the risk of failure to meet the deadline specified in the contract. Disruptions occurring in the implementation process affect the time of performance of works, in particular critical works, which have a direct impact on the deadline for completion of construction works.

7.3. Claim 2

A matrix of work durations with designated diagonals is given. If the minimum number of critical tasks has been determined, the duration of the complex of tasks is equal to the sum of the durations of critical tasks in the diagonals of the matrix.

Proof. 

Assuming a fixed number of diagonals q in matrix T, equal to m + n − 1, the largest (i.e., critical) value a can be determined in each diagonal q:

$$\underset{y\in N}{\forall } \underset{B\subset N}{\exists }f(y_1) = a \Rightarrow \max \left\{t_{i-1,j+1},t_{ij,j+1},t_{i+1,j-1}\right\}$$

(21)

Duration of the complex of works:

$$T = {\displaystyle \sum _{q=1}^{n+m-1}{a}_q,}$$

(22)

where:

• a—the value of the critical work in the diagonal q is equal to the sum of the largest values in the diagonals of the matrix (critical works). □

Conclusion: When planning construction works, the leading process is usually determined, i.e., the sequence of works with the longest duration. These are usually critical works that determine the success of the project—the completion of the task within the contractual deadline. By summing up the duration of critical works and monitoring their progress, we ensure that the construction complex is completed in accordance with the agreed work schedule.

7.4. Claim 3

If a matrix of work durations with designated diagonals is given, then non-critical tasks have time reserves that can be treated as planned downtime. These reserves are the same for early and late deadlines.

Proof. 

In an m × n matrix, the number of diagonals q is set to m + n − 1. In each diagonal, the largest value corresponding to the critical work a can be determined.

For this work S^sw and S^sp and S^tw and S^fp are equal to zero.

$$\underset{z\in M}{\forall } \underset{Y\subset M}{\exists } f(z_1) = a \wedge f(z_2) = a\dots \wedge f(z_n) = a$$

(23)

where:

• $Y=\left\{z_1, z_2,z_3\right\}$

The set y contains critical task a and non-critical tasks with values less than a. Possibilities:

$$z_1 = a \Rightarrow S^{sw}=S^{sp}=S^{fw}=S^{fp}=0$$

(24)

$$z_1 <a \Rightarrow S^{sw}=S^{fp}>0$$

(25)

and

$$S^{sp}=S^{fw}>0$$

(26)

For non-critical tasks, the values of time couplings are greater than zero. In extreme situations, when non-critical tasks begin at the moment $t_{ij}^{wr}$, or they are terminated $t_{ij}^{pz}$, the time reserve values are the same, because:

$$t_{ij}^{wr} + t_{ij} + \Delta = t_{ij}^{pz}$$

(27)

and

$$t_{ij}^{pz} - t_{ij}^{wr} - \Delta = t_{ij}$$

(28)

where:

• $∆$—time reserve value for the robot t_ij. □

Conclusion: For non-critical works, unavoidable downtime is a loss in the event of an unexpected phenomenon. It causes collisions with other works and freezes funds. Therefore, awareness of the time reserves that arise and the ability to determine their value allows decisions to be made during the implementation process and technical solutions to be prepared to minimize losses, i.e., planning auxiliary work, organizing additional tasks, preparing additional materials and structures on alternative work fronts.

7.5. Claim 4

If diagonals in the RF system (ordinates—working fronts) have been designated in the work duration matrix, then after transposition into the RP system (ordinates—types of work), the time characteristics of the work remain unchanged.

Proof. 

In the matrix for determining time parameters, the diagonals are influenced by diagonal and inverse diagonal couplings.

For diagonal couplings:

$$\underset{z\in M}{\forall } \underset{X\subset M}{\exists } f(z_1) = c \wedge f(z_2) = c\dots \wedge f(z_n) = c$$

(29)

where:

• X = {z₁, z₂,…, z_n}

For diagonal feedback loops:

$$\underset{y\in M}{\forall } \underset{Z\subset M}{\exists }f(y_1) = c \wedge f(y_2) = c\dots \wedge f(y_n) = c$$

(30)

where:

• Z = {y₁, y₂,…, y_n}. □

Conclusion: After transposing the matrix, e.g., from the RF system (working fronts) to the RP system (types of work) or vice versa, the values of the function f(z) and $f_{wr}^{(y)}$ remain unchanged—time characteristics of works $t_{ij}^{wr}$ i $t_{ij}^{pz}$, will have the same values in the diagonals q of the T matrix. In practice, this does not affect the readability of the calculation results, which will form the basis for the graphical representation of the work plan.

7.6. Claim 5

If diagonals with designated critical tasks have been established in the work duration matrix, performing the preceding tasks at later dates and the following tasks at earlier dates ensures that the links between the means of implementation are minimized.

Evidence: In the construction work complex, critical works q are determined in each of the diagonals, which have no time reserves. The remaining non-critical works have time reserves and their implementation may take place in the time interval between the earliest start date $t_{ij}^{wr}$ and the latest date for its completion $t_{ij}^{pz}$.

$$\underset{n\in Z}{\forall } \underset{Y\subset W}{\exists }f(n_1) = a \wedge f(n_2) = b\dots \wedge f(n_n) = g$$

(31)

where:

• Y = {a, b,…, g}

In the set y, you can find values for which:

$$n_1=a⟹{\mathrm{S}}^{\mathrm{sw}}={\mathrm{S}}^{\mathrm{sp}}=0$$

(32)

Conclusion: The coupling between the S^sw implementation measures for early deadlines and S^sw for late deadlines will be minimal, equal to zero in extreme cases. This ensures that the practical requirement for continuity of work of a given type (e.g., earthworks, foundation works, assembly works, and others) is met.

7.7. Claim 6

If diagonals with designated critical tasks have been established in the work duration matrix, performing tasks that precede early deadlines and follow late deadlines ensures that the links between work fronts are minimized.

Proof. 

In a construction project, in accordance with the accepted assumptions, critical tasks can be identified in each of the diagonals. The remaining non-critical tasks have time reserves and can be carried out at any time between $t_{ij}^{wr}$ and $t_{ij}^{pz}$, i.e., the earliest start date and the latest completion date.

$$\underset{w\in Y}{\forall } \underset{X\subset Y}{\exists }f(w_1) = x$$

(33)

and

$$\underset{x\in W}{\forall } \underset{X\subset W}{\exists } f(x_1) = y$$

(34)

i.e., that you can find values of x and y for which:

$${\mathrm{S}}^{\mathrm{fw}}={\mathrm{S}}^{\mathrm{fp}}=0$$

(35)

□

Conclusion: Interactions between work fronts, in the case of preliminary works for early deadlines and subsequent works for late deadlines, are minimal. In construction practice, meeting the above conditions allows for the minimization of downtime on work fronts during the performance of various types of works (e.g., earthworks, foundation works, construction works, and others).

7.8. Claim 7

If diagonals with critical tasks have been established in the task duration matrix, increasing the intensity of a given type of task (in a partial pipeline) will shorten the completion time of the task complex only if the partial pipeline includes a critical task.

Proof. 

In matrix notation, it is possible to divide into rows, columns, and diagonals. The RP layout—working front ordinates—means that the columns of the matrix correspond to types of work—partial pipelines. A change in the intensity of work based on a decision by the implementation process manager may affect the completion time of the complex of works. The completion time will be shortened if the critical work is part of a partial pipeline.

$$\underset{z\in W}{\forall } \underset{X\subset W}{\exists }f(z_1) = x \wedge f(z_2) = q \Rightarrow q \in Z \vee q \notin Z$$

(36)

If

$q\in Z$—this critical work belongs to the diagonal q and to the set of elements of the matrix column z;

$q\in Z$—this critical job belongs to the diagonal q, but is not an element of the set of elements of the column of the matrix z. □

Conclusion: The practice of constructing buildings shows that increasing the intensity of certain types of work, e.g., some finishing works, does not shorten the construction time. This applies to works in a complex that do not include critical work. This is a common organizational mistake. During the construction process, only those types of work that include critical work should be intensified, as only critical work determines the time of completion of the work. Shortening the duration of other work may cause downtime on the work fronts and interruptions in the work of work groups and equipment, as well as an increase in material stocks.

7.9. Claim 8

If diagonals q have been determined in the work duration matrix, with critical tasks designated within them, then changing the order of tasks on the work fronts causes a change in the duration of the complex of tasks.

Evidence: As a result of changing the order of work on the work fronts, the layout of diagonals in the work duration matrix and the set of critical tasks changes.

$$\underset{x\in T}{\forall } \underset{X\subset W}{\exists }f(x_1) = T_1 \wedge f(x_2) = T_2 \vee ,\dots , \vee f(x_n) = T_n$$

(37)

where:

• X = {x₁, x₂,…, x_n}

Using the function f(x_n), it is possible to determine the total Tn corresponding to the completion date of the complex of works. For each value of the argument from the set, it is possible to obtain different durations of works T, in the case of different sequences of works on the fronts.

Conclusion: By applying a procedure to find a rational sequence for carrying out work on different fronts, it is possible to find a sequence that will ensure the shortest completion time for a complex of construction works, e.g., by using task sequencing algorithms.

8. Discussion and Conclusions

The article presents an example of the application of causal inference techniques.

An analysis was conducted of the technological and organizational dependencies affecting the collision-free nature of construction processes. Eight statements were formulated and proven concerning the properties of organizational methods ensuring collision-free coordination of construction works, taking into account technological and organizational consequences. A methodology for determining the time parameters of works, i.e., the earliest and latest dates for the commencement and completion of works, was presented. The developed algorithms allow for the determination of work deadlines. It should be noted that implementation plans are subject to constant modification in practice due to changing conditions in the environment in which the implementation takes place. Most often, they are a dynamic instrument that allows changes to be managed in a conscious and orderly manner. This can be achieved through time couplings that actively ensure the determination of time parameters for construction works. Causal inference makes it possible to answer the question about the properties of construction work schedule variants.

The authors’ future research will focus on integrating causal inference methods and Time Coupling Methods with modern computational technologies such as artificial intelligence, causal structure discovery algorithms, Big Data, and real-time monitoring systems (BIM 4D/5D, IoT, 3D scanning). The combination of causal models with machine learning can enable automatic identification of causal relationships and dynamic updating of schedules. In turn, integration with big data analysis and agent-based simulations will allow for more accurate prediction of delays, analysis of counterfactual scenarios, and assessment of the impact of technological and organizational changes. The inclusion of multi-criteria optimization methods and Digital Twin-type digital platforms may ultimately lead to the development of intelligent, adaptive scheduling systems capable of predicting and minimizing risks in complex construction projects.

Recent advances in large language models (LLMs) have opened new perspectives for supporting causal inference in construction project analysis, particularly in the context of schedule planning and control. Although LLMs are fundamentally correlation-based models and do not perform causal inference in a formal sense, they have demonstrated significant potential as auxiliary tools for extracting domain knowledge from unstructured textual data, such as progress reports, site diaries, and delay claims. By identifying recurring patterns, potential causal factors, and implicit dependencies, LLMs can facilitate the construction of preliminary causal structures that support subsequent formal analysis using established causal frameworks, including Bayesian networks or structural causal models.

From the perspective of construction scheduling, the integration of LLMs with formal causal methods represents a promising hybrid approach. In such settings, LLMs serve as an intelligent interface for knowledge elicitation and scenario exploration, while causal reasoning and validation remain grounded in rigorous statistical or graph-based models. This combination may enhance the interpretability and scalability of delay analysis and risk assessment in complex construction projects. However, further research is required to ensure methodological rigor, particularly with respect to causal validation, robustness against bias in textual data, and the reliable integration of LLM-generated insights into algorithmic scheduling and decision-support systems.