An Integrated Decision Support System for Optimizing Time-Cost Trade-offs in Linear Repetitive Projects

An Integrated Decision Support System for Optimizing Time-Cost Trade-offs in Linear Repetitive Projects | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article An Integrated Decision Support System for Optimizing Time-Cost Trade-offs in Linear Repetitive Projects Ahmed Gouda Mohamed, Ali Hassan Ali, Ahmed Adel Abdelhady This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5757308/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 20 Jun, 2025 Read the published version in Scientific Reports → Version 1 posted 12 You are reading this latest preprint version Abstract The primary aim of this paper is to minimize the overall completion time of linear repetitive projects while simultaneously reducing direct and indirect costs by employing predefined construction methods. To address these challenges, the study proposes a dual-optimization framework that integrates Genetic Algorithms (GA) and Particle Swarm Optimization (PSO). The methodology involves decomposing repetitive tasks into sub-tasks, enabling a more detailed and feasible Time-Cost Trade-off (TCT) analysis, which is further combined with the Line of Balance (LOB) methodology. A comparative analysis between GA and PSO highlights their respective strengths and effectiveness, an area previously underexplored in the literature. The findings reveal that the proposed framework significantly reduces costs and project durations. The GA approach achieves reductions of approximately 3.25% in direct costs, 20% in indirect costs, and 7% in total construction costs, while PSO demonstrates slightly better cost efficiency with a 4% reduction in direct costs and similar reductions in indirect costs. Both methods deliver a 20% reduction in project completion time, showcasing their effectiveness in streamlining construction processes. This paper contributes to the field by presenting the GA-PSO-based Linear Repetitive Project Time-Cost Trade-off (LRPTCT) model, integrating TCT with LOB, and offering a novel solution to the complexities of linear repetitive projects. Furthermore, its comparative analysis between GA and PSO provides valuable insights for optimizing construction project management practices. Biological sciences/Evolution Physical sciences/Engineering Physical sciences/Mathematics and computing Time–cost trade-off Genetic Algorithm Linear Repetitive Projects Line of Balance Construction Project Optimization Particle Swarm Optimization Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 1 Introduction The construction industry represents 13% of the global gross domestic product (Ali et al., 2024c ; Elrifaee et al., 2024 ), and thus it plays a highly significant role in economies and helps in improving the standards of living (Ali et al., 2024b , 2024a ). Nevertheless, the scope of this sector is significantly threatened by the challenges that appear in the light of the process of urbanization, which increases the pressure on the efficiency of infrastructure construction (Hui et al., 2023 ; Kineber et al., 2024 ). As stated earlier, there is a pressing need for infrastructure development with commendable emphasis placed on linear repetitive works like bridges, tunnels, and highways among others in anticipation of the exploding population globally (Agrama, 2011). Linear repetitive projects are more often used in the handling of large-scale construction projects especially when many related tasks are to be performed in similar ways in a certain period (Zou et al., 2022 ). These projects are widely distributed in the construction, manufacturing, and transportation industries where the tasks are clearly defined, repetitive, and similar in nature (Tomczak and Jaśkowski, 2022 ). Some of the inherent characteristics of linear repetitive projects include the following: The structure of linear repetitive projects is highly beneficial due to the following reasons; However, these projects also come with considerable risks that call for streamlined planning and execution to increase their chances of success (Zou et al., 2022 ). Over the past few decades, the construction industry has faced several issues that make project management a nightmare. Some of the most significant challenges include: scope creep, distribution of resources, quality assurance, scheduling, and TCT. For example, scope creep brings in additional costs or durations; ineffective management of the resources may incur more time and poor quality (Kumar and Mehany, 2020 ; Lin and Lai, 2020 ). The importance of managing these competing demands is, therefore, paramount, especially given the fact that constriction projects are usually undertaken within tight schedules and a fixed amount of funds. The essence of the application of the TCT approach is to determine the optimum time-cost trade-off of each activity in a project. This strategy aims to reduce the total cost of a project while retaining prescribed timelines for the completion of the project. At the same time, it has to generate schedules that enable tasks to be accomplished at the earliest possible times. This is possible when the time of beginning each activity is well determined along with considering the precedences of the tasks and available resources (Tran, 2020 ). Critical path method (CPM) and Line of balance (LOB) are the two conventional scheduling modes that are commonly used in construction management. However, the accuracy of CPM scheduling comes into conflict with the method when used to schedule repetitive projects that involve repeatedly performing the same basic unit. Still, the main drawback of CPM in this respect is the inability to maintain the continuity of crew work which is critical for improving the performance rates of repeated processes (El-Kholy et al., 2021 ). However, as has already been seen, LOB is the best scheduling tool for linear repetitive projects and has several drawbacks. LOB sequentially arranges work activities, this might not adequately cover a complex linear work that is dependent on other linear undertakings. In addition, in LOB the estimated rate of production for each planned activity is assumed to remain constant. Unfortunately, this premise often does not take into consideration actual project environment factors, such as weather conditions or the availability and cost of labor and materials, which significantly impact project duration (Tang et al., 2018 ). Moreover, the chronological arrangement of activities in LOB may reduce various dependencies and project details embedded within construction projects because dependent tasks are posed linearly. Such distortion leads to improper timing and insufficient coverage of critical and non-significant sub-tasks, which directly endangers project success (Zou et al., 2018 ). Time and cost control remain crucial in linear repetitive projects since they usually are inversely proportional. Approaches like having more people in the workforce, using uncontemporary building methodologies, or obtaining approval for overtime can shorten project span but at the same time can raise costs substantially. However, project managers must factor time and cost together especially when planning and scheduling to improve on the project performance. Nevertheless, the multiple attribute perspective of the TCT problem can be a drawback as it hinders decision-making. As seen in the segmented decision-making diagram above, many options can be included when it comes to the time and cost of each activity – and with more possibilities for activities and decisions, this greatly increases the size of the search space. The combination of parameters in such a manner has been the subject of research in the past few years (Heravi and Moridi, 2019 ). Traditional methods of scheduling like the Critical Path Method (CPM) and Line of Balance (LOB) are not very effective in handling the time, cost, and resource dimensions in linear repetitive construction projects. These approaches do not isolate how tasks interact with each other as well as various resources within the real world. However, current research is still lacking an effective framework that uses TCT in conjunction with LOB; or applies innovative optimization techniques like GA and PSO to determine the optimum values of time and cost. This gap results in some incongruities that impede project management and warrant a new approach and solution—this calls for a dual-optimization framework. The primary contribution of this paper lies in its groundbreaking development of a dual-optimization framework for Linear Repetitive Project Time-Cost Trade-off (LRPTCT), which uniquely integrates Genetic Algorithms (GA) and Particle Swarm Optimization (PSO) to address the longstanding challenges in linear construction projects. This innovative approach merges the strengths of Time-Cost Trade-off (TCT) analysis and Line of Balance (LOB) methodologies, delivering a novel solution that bridges critical gaps in project management research. The inclusion of a comparative analysis between GA and PSO further underscores the uniqueness of this study, offering a dual-lens perspective on leveraging these metaheuristic techniques for real-world applications. This study sets itself apart through its introduction of a decomposition strategy for repetitive tasks, enabling a nuanced representation of task interdependencies and resource allocation. Unlike traditional methods, which often oversimplify project dynamics, the proposed approach incorporates a sophisticated optimization framework that utilizes GA for exploring expansive solution spaces with diverse potential configurations, and PSO for efficiently converging on optimal solutions. The synergy between these algorithms represents a pioneering step in construction optimization, tailored to address the intricate demands of modern linear repetitive projects. What distinguishes this work is the comparative analysis of GA and PSO, which provides a strategic understanding of their respective advantages and limitations in various project contexts. This comparative approach not only enhances the novelty of the research but also equips practitioners with actionable insights into selecting the appropriate optimization tool for specific project needs, making the study both innovative and highly practical. The significance of this research extends well beyond its theoretical contributions. It redefines the integration of TCT and LOB, presenting a transformative decision-support tool that empowers project managers to optimize linear repetitive projects with unprecedented precision. This framework opens new avenues for addressing complex scheduling and cost management challenges, offering a scalable, flexible, and impactful solution that can be adapted across diverse project scenarios. By advancing the state of the art in construction project management, this study lays the groundwork for future innovations in multi-objective optimization, reinforcing its uniqueness and practical value to the field. 2 Literature review Repetitive construction projects can be broadly categorized into two main types: linear and non-linear. Linear projects refer to those that are implemented sequentially and these include transport corridors such as pipelines highways and railroads. Instead, non-linear projects relate to structures that contain features in this respect, for example, several houses or a couple of tall constructions (El-Kholy et al., 2021 ). Today, repeated activities represent practically all construction processes, especially in terms of mass-scope operations. Some of these projects have cyclical patterns of activities within them and the extension of continuity in the use of resources in these projects helps to improve the efficiency and effectiveness of projects (Yılmaz and Dede, 2023 ). This continuity is important here since repetitive work means that each activity should be repeated several times. Such an approach can help achieve considerable cost and time savings due to the possibility of having a separate skilled labor crew that remains constant throughout the project (Moreno et al., 2020 ). But there is some difficulty in this strategy as well. The endless work may reduce the productivity of the crew involved in its execution. When the crew moves from one unit to another, doing the same activities again tends to elongate the time required for completing the project as well as the expenses. Consequently, this paper aims to formulate an efficient strategy for the crew work in repetitive projects to achieve the best performance in terms of cost and time (Tomczak and Jaśkowski, 2020 ). Flawless crew allocation will go a long way in avoiding productivity issues while enhancing the values of hiring crew with similar people temperaments. However, this call for strategy also has its drawbacks. One of the possible effects of continuous work could be the reduction of crew efficiency. Generally, as the crew moves from one unit to another, and performs the same activities in an almost patterned sequence it has the adverse of lengthening the project time as well as the associated cost. Thus, there is a need to adopt a good strategy especially concerning crew work in repeat projects with respect to costs and time (Tomczak and Jaśkowski, 2020 ). Proper crew scheduling and work planning can also go a long way to preventing detrimental productivity drawbacks that may be occasioned by a consistent crew. Recently, a lot of research works have been published to address time-cost trade-off problems (TCTP). These studies can be broadly categorized into two groups: the studies that investigate the general application of TCTP in construction projects and the studies that investigate the repetitive elements of construction projects. Li & Wu ( 2014 ) proposed a robust optimization model based on nondominated sorting genetic algorithm-II (NSGA-II) to solve the time-cost trade-off for general construction projects under uncertainty. This model compares many construction modes that share different temporal and cost features influenced by interval uncertainty. Accordingly, their result shows that raising the robust coefficient for cost or time produces more solutions but at the same time allows for more risk. However, the model mainly focuses on interval uncertainties and approximations, and together with it, there could be more hustle than needed in decision-making situations, as most often intervals may vary by time and may have other probability distribution than supposed by the model. Besides, it does not address the issue of resources, which are a core determinant of project timetables and expenses. This limitation may limit the general usability of the model in real-world environments. In the same respect, Biswasa et al. ( 2016 ) proposed a hybrid model of CPM and a cost-loop method that is heuristic in nature. This approach facilitates the determination of the least total cost possible for each potential duration of the project, thus solving a main construction project problem, that is, the time-cost problem. From their study, they were able to record at least a cost of $ 60,937 as their total cost with an approximate project length of 130 days. Nevertheless, the study has the following limitations that should be stated: It simply applies deterministic parameters to time and cost estimates for activities completely ignoring the chances of variations that are common in the actual construction undertakings. This restriction limits the external validity of the study’s outcomes and implies that future studies should also apply stochastic factors to their models to increase predictive precision. Eirgash & Toğan ( 2023 ) proposed the Golden Ratio Oppositional Teaching Learning Based Optimization (GROTLBO) to improve the convergence speed and solution diversification, particularly in multi-objective optimization of time-cost-environmental impact trade-off in construction projects. The results also show that the GROTLBO is superior to NSGA-II and OMODE with a 5% improvement in inaccuracy and a 95% reduction in the number of scheduled calculations performed. However, there are some apparent limitations inherent to this study, though the results of the research seem quite promising. One important issue concerns the computational time, which is associated with the algorithm’s search space that chars within the existence of the object. Furthermore, the extension of the TCET scenarios to prevent independent validation of GROTLBO, and its effectiveness in which real-world scenarios in practice practical situations. Similarly, in another study, Eirgash et al. ( 2023 ) put forward a new variant known as the modified dynamic oppositional learning-based teaching–learning-based optimization (MDOLTLBO) algorithm for improving time-cost optimization in construction projects. This algorithm has an adaptive varying of the weights used through iterative approaches that in turn provides exploration and exploitation characteristics, solutions diversities, and a higher rate of convergence. According to their findings, they found that MDOLTLBO is better than conventional TLBO meta-heuristic algorithms and other advanced optimization approaches and techniques including linear programming and genetic approaches in terms of solution accuracy, convergence characteristics, and computational complexity. However, it has been observed that the algorithm has a few limitations also. Firstly, the performance of this heuristic depends on the particulars of construction projects and problem instances. Nevertheless, the proposed algorithm, MDOLTLBO, improves convergence and solution quality but can still be trapped in local optima in large-scale, high-dimensional search spaces. However, for repetitive projects, Huang et al. ( 2016 ) proposed a special genetic algorithm to solve the discrete time-cost trade-off problem (DTCTP) of repetitive construction projects. The algorithm thereby introduces soft logic to improve the flexibility of the schedule while allowing the work sequences among activities to be variable. This optimization enhances methods of activity modes and start time selection and hence reduces the extent of project duration and cost. When compared to a previous scheduling solution that employed just hard logic, their findings showed increased competitiveness of solutions that used soft logic, one of which cut down the real project time by eleven days and 25283.5 $ by applying different sequences of work. However, there are still some limitations within the study. One disadvantage of the presented mixed-integer nonlinear programming model is that it can be computationally intensive, and therefore, sensitive to the scale of a project. Moreover, although the GA is somewhat robust in generating high-quality solutions there are times that it directly depends on parameters like population size and mutation. Recently, Zou et al. ( 2017 ) formulated the TCTP with multiple crews and fixed logic within the context of repetitive projects using mixed-integer linear programming (MILP). To that end, the study offers exact and approximate models where the results show that the former provides an efficient solution for medium problems, while the latter provides near-optimal solutions for bigger problems within a reasonable time. Simulations indicate that the approximate model keeps a maximum deviation of less than 1%, and is far better than the exact model for the larger problem sizes. However, several issues can be attributed to this study these include, The assumptions of fixed logic use may not be reflective of the actual level of flexibility in business operations, and Secondly, the overall approximate model may not be optimal for such complex situations. Future studies could improve these models by considering additional types of variable working sequences and other conditions. Altuwaim and El-Rayes, 2018 proposed a new genetic algorithm optimization model for resource-constrained scheduling of repetitive construction projects intending to minimize project duration, number of breaks, and interruption costs. The proposed model consists of four sub-models to improve the computational performance of the developed heuristic and the incorporation of interruption costs into scheduling procedures which is often overlooked in other approaches. This paper demonstrated the utility of the model on a repetitive construction project where the project team was able to realize considerable savings in overall project costs and ameliorate interruptions to crew work. However, the study also has some limitations, for example, pre-restriction of activities that restrict them only under one crew formulation of each activity, and the serial relationship between them may reduce its scope to a complex project. Besides, compared to other heuristics like that used to find the Pareto optimal solution, the application of genetic algorithms improves computational expediency but precludes optimality. Zou et al. ( 2020 ) proposed a new GM-MIP to improve the reliability and consciousness of repeated project scheduling by adding work continuity constraints and soft reasoning. This model can follow a certain pattern across several units and minimize the PCT, combine or total interruption time (TIT), and overall project cost (TPC). The research also shows that the application of soft logic concerning specific project creams can cut down project periods and costs substantially with special reference to many crew projects. Nevertheless, the authors discussed how this study had limitations and how the effects of learning-forgetting needed to be captured for the subsequent studies that will endeavor to confirm the generality of the model for larger and more complex projects. To deal with interruptions, buffer times, or schedule acceleration in recurring construction projects, El-Kholy et al. ( 2021 ) proposed a general optimization model for controlling time-cost trade-offs. The use of the proposed model showed the benefits of identification of cost efficiencies and improvements to the project timeline. It was demonstrated within a set of examples that interruptions and buffers could be used to improve resource utilization, and hence lower project duration, without undue costs. However, there are some limitations to the study. They depend on certain quantifiable factors that do not capture reality, leaving the possibility of gaps in the development of the full range of solution possibilities. Also, depending on the input data provided the model functionality is rather sensitive and may change from one project to another. Yımaz & Dede (2023) developed a comprehensive optimization model that integrates a non-dominated sorting method with the Rao-1 and Rao-2 optimization algorithms to effectively address the multi-objective time-cost trade-off problem. The proposed approach generates multiple Pareto optimal solutions, offering decision-makers valuable options for balancing time and cost. Results from numerical experiments demonstrate that the non-dominated sorting Rao-2 algorithm outperforms traditional methods like Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), and Genetic Algorithm (GA), achieving better optimal solutions with fewer function evaluations and higher quality Pareto results. However, the proposed algorithms have some limitations. They primarily focus on discrete optimization scenarios, which may not be suitable for all construction projects, especially those involving continuous variables. Additionally, the performance of the Rao algorithms is influenced by the quality of the initial population and parameter settings, which can affect convergence rates. Table 1 provides a summary of multiple research papers. Each entry outlines the primary focus of the research, including the trade-offs, types of projects addressed, specific problems tackled, and methods employed. To provide a clearer overview of the existing research landscape, Table 1 summarizes key studies, detailing their primary focus, trade-offs, project types, specific problems addressed, and employed methods. Table 1 Summary of previous research Number Reference Tradeoff Type of projects Problem Methods 1 Li & Wu ( 2014 ) Time, Cost General Construction Projects Uncertainty NSGA-II Optimization 2 Biswasa et al. ( 2016 ) Time, Cost General Construction Projects Deterministic Values Hybrid CPM with Heuristic Cost-Loop 3 Eirgash & Toğan ( 2023 ) Time, Cost, Environmental General Construction Projects Multi-objective, Dynamic GROTLBO Algorithm 4 Eirgash et al. ( 2023 ) Time, Cost General Construction Projects Multi-objective, Dynamic MDOLTLBO Algorithm 5 Huang et al. ( 2016 ) Time, Cost Repetitive Projects Repetitive, Soft Logic Targeted Genetic Algorithm 6 Zou et al. ( 2017 ) Time, Cost Repetitive Projects Repetitive, Fixed Logic MILP Approach (Exact and Approximate) 7 Altuwaim & El-Rayes ( 2018 ) Time, Cost, Interruption Repetitive Projects Repetitive Genetic Algorithm Model 8 Zou et al. ( 2020 ) Project Completion Time, Total Interruption Time, Total Project Cost Repetitive Projects Repetitive, Soft Logic Multi-Objective MILP Model 9 El-Kholy et al. ( 2021 ) Time, Cost Repetitive Projects Recurring, Interruptions, Buffers Integrated Scheduling Optimization Model 10 Yımaz & Dede (2023) Time, Cost Repetitive Projects Multi-objective Optimization Non-Dominated Sorting with Rao Algorithms In summary, the existing literature offers valuable insights into cost optimization and scheduling in construction projects. However, a significant gap remains: there is currently no documented approach that effectively integrates time-cost trade-offs (TCT) with Line of Balance (LOB) methodologies within a genetic algorithm (GA) framework. This oversight is significant, as the integration of these elements could provide a more holistic perspective on project optimization, enhancing both efficiency and effectiveness in repetitive construction settings. To address this critical research gap, this study seeks to develop a novel model that leverages the strengths of TCT and LOB within a genetic algorithm approach. This integration aims to equip decision-makers with more robust tools for navigating the complexities of construction project planning and execution, ultimately leading to improved project outcomes. 3 The mathematical formulation of time cost trade-off Multi-objective optimization is pivotal to the time-cost trade-off issue, which rationales to lessen both time and cost coincidentally by electing the utmost pertinent selection for each activity from a broad spectrum of possibilities. The calculation of time is crucial to this process. It can be expressed mathematically via an array of equations, as exhibited in Equations (1) to (3) integral to discerning the optimal trade-off between time and cost. This challenge is typically pinpointed in project management, where an analytical approach is imperative for ensuring optimal selection. In light of this, opting for the best optimal selection mandates utmost attention to both time and cost implications, with trade-offs typically addressed to attain a feasible outcome that appeases the targeted goals. ES j = max i∈pj {EF i }j = 1,⋯, n + 1 (1) EF i = ES i +t i (m) i = 0,⋯, n + 1 (2) T = EF n+1 (3) The Equations portrayed above involve various parameters, incorporating the total project duration, denoted as T, the activities with which j has a precedence relation depicted by pj, and the activity's early start and early finish, signified as ESj and EFi, progressively. Additionally, the activity m duration is rendered by ti(m). The primary objective of these equations encompasses determining the project's completion time by spotting the project's longest path. Hence, the aggregate of each activity's direct costs and the entire duration of the project multiplied by the indirect cost equals the total cost of the projects, as explicitly rendered in equations (4) to (6). In this regard, DC depicts the direct cost, C donates the entire project's cost, IC resembles the indirect cost, ICR is the whole project's indirect costs, and T exhibits the entire project's duration. C = \(\:\sum\:_{\text{i}=0}^{\text{n}+1\:}\text{d}\text{c}\text{i}^\left(\text{m}\right)\text{x}\text{i}^\left(\text{m}\right)\) (4) I C = T × ICR (5) C = DC + IC (6) 4 Research Methodology 4.1 The devised methodology for scheduling the linear repetitive project Principally, Linear construction projects that pursue a linear path routinely encounter hindrances, dictating the incorporation of non-critical or repetitive tasks for resource limitations remedy. Routinely, The LOB scheduling method presupposes that repetitive tasks have a steady production rate, remaining perpetually across the project duration, irrespective of the desired unit number (See Fig. 1 ). As reported in Fig. 1 (a), the completion time of one unit is 33 days; on the other hand, the completion time of the three repetitive units equals 63 days, as rendered in Fig. 1 (b). Additionally, LOB scheduling dearth the propensity for repeated tasks to be partially critically attributable to the fact that each repeated task is enacted in a single, undivided pattern with a predetermined deterministic duration. This presumption could pose a challenge for LOB scheduling to affirm that the same crew constitutes each task linearly, rendering it impractical to disseminate resources across the entire project. This restriction is particularly problematic, as repetitive activities may encompass multiple sub-tasks with varying production rates and entailing distinctive crew dissemination. As such, the production rate of each sub-task is imperative for tailoring to optimize resource allocation. Contrary to the conventional method of appraising construction tasks in LOB scheduling, repetitive tasks will be exhibited as a series of distinct sub-task interconnected jointly to render the overall duration of the task. This approach reckons with depicting any logical connection between the decomposed tasks adopting solely the finish-to-start link. By decomposing tasks in the LOB diagram, more methodical schedules are engendered and portray critical and non-critical sub-task unerringly. Correspondingly, this technique acknowledges the aleatory conditions of construction projects. Even though decomposing tasks may provoke more tasks in LOB scheduling, but it can instigate a more profound schedule when resource limitations are present, as presented in Fig. 2 . As attested in Fig. 2 (a), the completion time of one unit is 21 days; contrastingly, the completion time of the three repetitive units equals 46 days, as rendered in Fig. 2 (b). 4.2 Genetic algorithm approach for LRPTCT A Genetic Algorithm (GA) portrays a metaheuristic optimization algorithm stimulated via natural determination and genetics. It is a search technique adopted to retrieve the best resolution for a particular problem by mimicking the natural determination process and evolution (Ko and Wang, 2011 ). Genetic algorithms operate on a population of prospective resolutions encoded as chromosomes or strings of information. The population endures a series of genetic operations, including selection, crossover, and mutation, to generate a new population of potentially better solutions. The selection process incorporates opting for the fittest individuals from the current population based on their fitness values, exhibiting how thoroughly they resolve the problem. These selected individuals are then employed to generate new solutions through crossover, which involves combining parts of two or more parent solutions to create new offspring (Anvari et al., 2016 ). Another genetic procedure called mutation imparts random alterations to the offspring solutions, incentivizing the exploration of new areas of the solution space. This process empowers a deeper investigation of the solution space and precludes the algorithm from being stuck in local optima. The new population of solutions is then appraised for fitness, and the process is repeated until a termination criterion is met, such as reaching a maximum number of generations or achieving a desired level of fitness. After the genetic algorithm process, the population member with the highest level of performance is touted as the optimal solution to the optimization (Anvari et al., 2016 ). 4.2.1 Encoding the chromosome In GA, a solution to an optimization problem is embodied as a "chromosome," a data structure that encodes a batch of parameters denoting the solution. In the context of time cost trade-off optimization, a chromosome structure was encoded using different configurations of defined construction methods coupled with correlative duration and direct cost. A chromosome structure was encoded by a sequencing chain of elements, with each element portraying a linear repetitive sub-task and including an index indicating its defined construction method. This sequence renders a single project solution constituting methods to construct the repetitive sub-task, as rendered in Fig. 3 . To assess the effectiveness of the chromosome's solution, the repetitive sub-tasks' durations coupled with appointed direct costs are inferred, anticipating pertinent indices within the chromosome. The chromosome could depict arrays of linear project completion times. Each gene in the chromosome renders the start and end times for a particular repetitive sub-tasks and the construction method crucial to constitute the repetitive sub-tasks. The chromosome comprises strings whose length displays the number of repetitive sub-tasks in the project. The intention of the optimization algorithm is thus to identify the chromosome optimally trade-off between time and cost, i.e., the chromosome that yields the shortest feasible completion time for the linear project while minimizing the total cost required to complete the repetitive sub-task. This process incorporates appraising the fitness of each chromosome in the population and using selection, mutation, and crossover operations to evolve the population over successive generations toward the optimal solution. Figure 4 exhibits the entire structure of the proposed GA approach. 4.2.2 Objective function The pivotal objective function is minimizing the total linear repetitive project completion time F1, synchronically minimizing the total linear repetitive project costs F2, including direct and indirect costs, while employing predefined construction methods. The objective functions F1 and F2 are calculated concerning the entire chromosomes per generation. Subsequently, the chromosomes are sorted, banking on the solution that attains the trade-off between minimizing F1 and concurrently minimizing F2. The objective functions are constituted and operated by changing the variable from 1 to 4, embodying the defined construction method for each repetitive sub-task. Followingly, the optimization model is subjected to these outlined constraints 1) Decision variables are greater than or equal to 1 and less than or equal to 4, 2) Values of decision variables should be an integer, 3) The optimized total linear repetitive project cost should be less than the planned cost, and 4) The optimized linear repetitive project completion time should be less than the planned time. 4.2.3 Initial population engendering susceptible to the objective functions When the chromosome structure and fitness function are constituted, the genetic algorithm proceeds to conduct evolutionary optimization on a population of parent chromosomes. Initially, N-viable chromosomes are synthesized as an initial population. Afterward, these chromosomes are ranked based on the assessment criterion defined earlier. However, the number of chromosomes or population size is a critical parameter that enacts both the computational time required and the possible solution. As such, population size increases, attaining a global optimum but substantially prolonging the processing time. In the present deployment, users can designate the population size gleaned from their preferences. When articulating the population, each chromosome's fitness is examined, and pertinent eminence is computed by dividing it by the entire chromosome's fitness. Consequently, Several fittest chromosomes are conserved and passed down to the succeeding offspring. These conserved chromosomes stay qualified concerning electing like parents while reproducing the remaining parents in the following generation. During each iteration, genetic algorithms' crossover and mutation process ensures that the parent chromosomes maintain their assignment property while altering the order of construction methods designated to each repetitive sub-task to engender new chromosomes, as exhibited in Fig. 5 . This process enables the genetic algorithm to prioritize fitter chromosomes for further evolution and analyze the project duration and cost trade-offs. 4.2.4 Genetic operators Consequently, reproduction among members of the population emanates through crossover or mutation, which simulates natural evolution as genetic operators. Crossover is the more predominantly adopted process and involves picking two parent chromosomes, swapping relevant information, and engendering offspring. The selection of parent chromosomes is random but weighted by their relative merit, ensuring that the best chromosomes are more likely to be chosen while maintaining diversity. The process of exchanging genetic information between parent chromosomes is random in nature. Unlike crossover, portraying natural reproduction, the mutation exhibits a sparse process that can produce sudden, exceptional offspring. It involves randomly selecting a chromosome from the population and performing arbitrary evolvements to its information. As such, the mutation process can break any stagnation in the evolutionary process and avoid local minimums. In Fig. 5 , assuming that two parents have been elected for engendering two offspring, the crossover operator partitions the parents' sub-strings into two distinctive groups reliant on construction methods and substitutions the genes within each group sub-strings. As a result, the children derive the sequence of construction methods from both parents. Subsequently, these new children are translated into a feasible completion time and minimized total cost by decoding them. After generating offspring using any of the available techniques, it endures a fitness evaluation. It can only be confined if it surpasses the other members of the population in terms of fitness. Typically, this cycle abides for numerous generations of offspring until a chromosome that represents the optimal solution is found. The current deployment allows the user to determine the number of generations of offspring as a discontinuance benchmark for the process. In addition, it is feasible for the user to halt the genetic operators when a satisfactory solution that meets the required criteria for the total project cost and time of completion is attained. In such a scenario, the output will present the total linear repetitive project completion time, total project cost, and corresponding construction methods. Conversely, if the desired solution is not achieved, the algorithm will persist in probing for improved output. The GA approach for LRPTCT is implemented using an optimization engine platform named Evolver TM Version 7. Evolver Palisade uses a GA approach to optimization, which involves using a population of potential solutions to a problem that undergoes a series of genetic operations, such as selection, crossover, and mutation, to generate new and potentially better solutions. The process is repeated over several generations until an optimal or near-optimal solution is obtained. 4.3 Particle Swarm Optimization (PSO) workflow for LRPTCT PSO represents a heuristic optimization technique inspired by the social behavior of organisms such as bird flocking and fish schooling. This method aims to find optimal solutions by mimicking the collaborative behavior of a swarm of particles searching for the best outcomes. In the context of LRPTCT, PSO provides a structured framework for minimizing project costs and durations by iteratively improving candidate solutions through collaboration and individual learning. 4.3.1 Encoding the particle and objective functions In the PSO framework (see Fig. 6 ), each particle represents a potential solution to the optimization problem. A particle is encoded as an array of decision variables, where each variable corresponds to the construction method selected for a repetitive task. The decision variables take integer values between 1 and 4, representing four predefined construction methods for each task. A particle’s position in the solution space denotes a unique configuration of selected methods, which is evaluated based on fitness metrics. These metrics account for the total direct costs, indirect costs, and project duration associated with the selected methods. The PSO model for LRPTCT employs two objective functions. The first objective is to minimize the total project duration (F1), which is governed by the sequencing and dependencies of tasks, with critical tasks determining the overall duration. The second objective is to minimize the total project costs (F2), comprising Total Direct Costs (TDC) and Total Indirect Costs (TIC). TDC is derived from the costs of the construction methods chosen for each task, while TIC is calculated by multiplying the project duration by the daily overhead rate. These objective functions ensure that the PSO model seeks an optimal trade-off between minimizing time and costs. 4.3.2 PSO optimization workflow The PSO process for LRPTCT is structured into iterative steps, illustrating the PSO workflow. The workflow of the PSO framework begins with the initialization of a swarm of particles. Each particle is assigned a random position within the solution space, representing an initial configuration of construction methods. The initial velocities of the particles are set to zero. The fitness of each particle is evaluated using the objective functions, and the best-known positions for each particle (p_best) and the global best position (g_best) among all particles are recorded. The iterative optimization process involves updating the velocity and position of each particle. The velocity update (Eq. 7) incorporates three components: the particle’s inertia, its cognitive tendency to return to its best-known position, and its social tendency to move toward the global best position. These updates are influenced by inertia weight (ω), cognitive coefficient (c1​), social coefficient (c2​), and random factors (r1​ and r2​) that add stochasticity to the search. After updating velocities, the positions of particles are adjusted to reflect the new configurations of construction methods. The positions are constrained within the valid range of decision variables, ensuring that all selected methods are feasible. V i,t+1 ​=ωv i,t​+ c 1 ​r 1 ​(p best ​−p i,t ​) + c 2 ​r 2 ​(g best ​−p i,t ​) (7) Fitness values are recalculated for all particles after each update. If a particle’s new position improves upon its previous best-known position, its p_best is updated. Similarly, if a particle achieves a global improvement, g_best is updated. This iterative process continues until a stopping criterion is met, such as reaching a maximum number of iterations or achieving convergence to an optimal solution. The PSO framework incorporates constraints to ensure practical and feasible solutions. Decision variables must remain integers between 1 and 4, representing valid construction methods. The optimized total project cost must be less than the planned budget, and the total project duration must not exceed contractual deadlines. Additionally, task dependencies and logical sequences are maintained throughout the optimization process to ensure constructability. Upon completion, the PSO model outputs the optimal construction methods for all repetitive tasks, the minimized total project costs, the shortest feasible project duration, and detailed task schedules, including start and finish times. This comprehensive output supports decision-making in construction project management by providing actionable insights into cost and time optimization. The PSO framework demonstrates several advantages in addressing LRPTCT problems. It is highly scalable, accommodating projects with numerous tasks and complex dependencies. The algorithm’s iterative nature enables efficient convergence to optimal solutions, balancing exploration and exploitation to avoid local minima. Furthermore, PSO’s flexibility allows it to handle multi-objective optimization scenarios, making it suitable for simultaneously minimizing costs and durations. 5 Developed approach implementation To evaluate the effectiveness, reliability, and validity of the developed approaches, a case study was conducted using a four-kilometer pipeline assembly as an example, as shown in Table 2 . The developed methods are particularly suitable for horizontal infrastructure projects. The project consists of four parallel repetitive pipelines, each spanning 4 km, resulting in a total pipeline installation length of 16 km. The project's precedence relations follow a Finish to Start (FS) sequence without any lag time. The implementation process of the proposed approaches in the actual case study is illustrated in Fig. 7 . Furthermore, the case study will include a comparative analysis between the proposed approaches and the traditional CPM integrated with LOB in order to determine their overall performance in terms of project scheduling and cost management. This comparative analysis will provide a comprehensive understanding of the strengths of each approach. 6 Results and discussion 6.1 The conventional method for tackling the linear project's scheduling Scheduling the pipeline (4 km pipeline) for one unit unveiled eight critical tasks (A, B, C, D, F, G, K, and L) and four non-critical tasks (E, H, I, and J). According to the CPM results, the project's duration for one unit is estimated to be 595 days, as rendered in Table 2 . In Table 2 , ES depicts the early start, EF is the early finish, LS is the late start, LF represents the late finish, and TF is the total float. Table 2 Conventional CPM calculations for a single unit. Tasks Duration (Days) Predecessors Successors ES EF LS LF TF Code No. A 1 14 - 2 0 14 0 14 0 B 2 7 1 3,8 14 21 14 21 0 C 3 112 2 4,5 21 133 21 133 0 D 4 140 3 6 133 273 133 273 0 E 5 112 3 6 133 245 161 273 28 F 6 252 4,5 7 273 525 273 525 0 G 7 28 6 11 525 553 525 553 0 H 8 56 2 9 21 77 343 399 322 I 9 119 8 10 77 196 399 518 322 J 10 35 9 11 196 231 518 553 322 K 11 14 7,10 12 553 567 553 567 0 L 12 28 11 FN 567 595 567 595 0 FN 0 12 595 595 595 595 0 6.2 The devised method for addressing the linear project's scheduling In order to more accurately reflect practicability in linear repetitive construction projects, the tasks previously mentioned were divided into separate sub-tasks of equal length. Each sub-task was treated as a flexible task with a length of 1 kilometer and a duration equal to the total duration of the original task divided by 4. This paradigm was designed to establish a relationship between the previously separate tasks, unearthing more feasible completion times. Table 3 displays the results of the case study after splitting tasks C, D, E, F, and G into four equal 1-kilometer sub-tasks. The reduction of single unit duration is evident. The total duration decreased by 210 days (depicting a 35% reduction) from 595 days, computed using the traditional CPM method, to 385 days after the tasks were split, as explicitly portrayed in Tables 2 and 3 . Apropos of Table 3 , the division of tasks C, D, and G revealed the presence of partially non-critical sub-tasks. Specifically, only C1 is rendered as a critical task within task C, whereas non-critical tasks represent C2, C3, and C4. Likewise, D1 is the sole critical sub-task within task D, while D2, D3, and D4 are categorized as non-critical. Finally, in task G, only G4 is considered critical, while G1, G2, and G3 are classified as non-critical sub-tasks. To compute the required number of crews for each sub-task, the computation of the desired delivery rate (Rd) for the repetitive units is initially undertaken using Eq. 8. In Eq. 8, variables are assigned as follows: n signifies the number of repetitive units, TL denotes the project's deadline duration as stipulated in the contract agreement, T1 depicts the CPM duration of the first unit, and Tfi embodies the total float of each sub-task. Subsequently, the estimation of the number of crews required to perform each sub-task efficiently is attained by employing Eq. 9. In Eq. 8, Ci donates the number of crews assigned to each sub-task (i), and Ri embodies the desired delivery rate for each sub-task (i). Therefore, the number of crews entailed by each sub-task may vary depending on each sub-task's Rd, Tfi, and designated duration. Ri = (n – 1) / (TL - T1) + Tfi (7) Ci = Di x Ri (8) Table 3 Scheduling calculations for a single unit, considering splitting tasks into sub-tasks. Tasks Duration (Days) Predecessors Successors ES EF LS LF TF Code No. A 1 14 - 2 0 14 0 14 0 B 2 7 1 3,23 14 21 14 21 0 C1 3 28 2 4,7,11 21 49 21 49 0 C2 4 28 3 5,8,12 49 77 84 112 35 C3 5 28 4 6,9,13 77 105 147 175 70 C4 6 28 5 10,14 105 133 210 238 105 D1 7 35 3 8,15 49 84 49 84 0 D2 8 35 4,7 9,16 84 119 112 147 28 D3 9 35 5,8 10,17 119 154 175 210 56 D4 10 35 6,9 18 154 189 238 273 84 E1 11 28 3 12,15 49 77 56 84 7 E2 12 28 4,11 13,16 77 105 119 147 42 E3 13 28 5,12 14,17 105 133 182 210 77 E4 14 28 6,13 18 133 161 245 273 112 F1 15 63 7,11 16,19 84 147 84 147 0 F2 16 63 8,12,15 17,20 147 210 147 210 0 F3 17 63 9,13,16 18,21 210 273 210 273 0 F4 18 63 10,14,17 22 273 336 273 336 0 G1 19 7 15 20 147 154 315 322 168 G2 20 7 16,19 21 210 217 322 329 112 G3 21 7 17,20 22 273 280 329 336 56 G4 22 7 18,21 26 336 343 336 343 0 H 23 56 2 24 21 77 133 189 112 I 24 119 23 25 77 196 189 308 112 J 25 35 24 26 196 231 308 343 112 K 26 14 22,25 27 343 357 343 357 0 L 27 28 26 FN 357 385 357 385 0 FN 0 27 385 385 385 385 0 Using traditional CPM calculation, rendering the four km pipeline assembly using the Line Of Balance (LOB) scheduling method for the repetitive units unveiled the linear repetitive project's duration as 868 days. A, B, C, D, F, G, K, and L donate the critical linear repetitive tasks, whereas E, H, I, and J portray the non-critical repetitive tasks, as exhibited in Fig. 8 . The data portrayed in Fig. 9 corroborates the outcome of implementing the LOB scheduling method for repetitive units in rendering the 4 km pipeline installation. This was achieved by dividing tasks C, D, E, F, and G into four equal sub-tasks of 1 kilometer each. Consequently, the linear repetitive project was lessened, reducing the total duration from 868 days using the traditional LOB-integrated CPM method to 693 days (exhibiting nearly a 20% reduction). The critical repetitive tasks were identified as A, B, C1, D1, F1, F2, F3, F4, G4, K, and L, whereas the non-critical repetitive tasks were C2, C3, C4, D2, D3, D4, E1, E2, E3, E4, G1, G2, G3, H, I, and J. Accordingly, the approach of considering non-critical sub-tasks in linear repetitive projects is vital for achieving a balance between minimizing project costs and time. These non-critical sub-tasks, including C2, C3, C4, D2, D3, D4, G1, G2, and G3, can reduce the number of employed crews and relax the activity production rate, thereby empowering the prospect of optimizing the LRPTCT. It is worth mentioning that non-critical activities signify an essential role in scheduling repetitive projects, contributing to the overall efficiency and successful project delivery. For instance, non-critical activities provide opportunities to employ accessible resources effectively. Hence, when critical activities are not consuming all resources, crews can be assigned to non-critical activities, maximizing their productivity and avoiding idle time. Further, Non-critical activities maintain a continuous workflow, particularly during delays or disruptions in critical activities. The project can progress steadily by assigning crews to non-critical tasks even if certain critical activities are behind schedule, promoting a smooth flow of work and preventing unnecessary downtime. Accordingly, this allows project managers to respond effectively to unexpected circumstances and maintain project momentum. The LOB diagram depicted in Fig. 9 exhibits a distinctively consecutive progression of activities, whereby each step constitutes a direct connection with the subsequent one, devoid of interruptions or temporal intervals. This discernible attribute prominently stems from the underlying premise of zero buffer time among the activities, guaranteeing an expeditious execution of the linear schedule. However, it is crucial to highlight that incorporating buffer time between activities can constitute a valuable scheme to accommodate unforeseen delays or variations in scheduling linear projects. 6.3 Deployment of the GA approach After the completion of the GA model, an initial evaluation was carried out to identify appropriate GA parameter values such as population size and the number of generations. Apropos of the research conducted by Roeva et al. ( 2013 ) and Anvari et al. ( 2016 ), a population size of 100 and 1000 generations was deemed a satisfactory trade-off among diverseness and processing time. For lessening computational load, a set of algorithm attribute values was presented, encompassing 1) The iteration rate is 60%, 2) the crossover likelihood is 26%, 3) the mutation likelihood is 10%, and 4) the migration likelihood is 9%. The chromosome comprises strings whose length is 27, portraying the number of repetitive sub-tasks in the linear repetitive project. Table 4 exhibited the cost and duration information for the construction methods of repetitive tasks and sub-tasks. The intended direct cost of the 27 repetitive tasks was deployed as the initial cost, which led to a total cost of 45,617,920 EGP and a completion period of 693 days for all units. According to the project’s contractual agreements, the agreed-upon indirect cost is estimated as a daily fixed value of 3,000 EGP, which is typically discerned by breaking down the various components of indirect costs, such as site overhead and general overhead. However, the indirect cost fixed value can be varied banking on multiple factors, including the nature of the project, industry practices, and the specific agreement between the owner and the contractor. Consequently, the project's total cost is approximated at 56,621,920 EGP. To this end, the inputs to the GA model are 1) Each repetitive task code and name, 2) scheduling data, including predecessors and successors of each repetitive task, 3) different alternatives of construction methods, incorporating each method's direct cost and designated duration, 4) the project defined constraints as previously outlines, and 5) the number of repetitive units and daily indirect cost for each repetitive activity, as displayed in Fig. 10 . Table 4 Construction methods alternatives depicting the GA model's decision variables. Construction Method # Method #1 Method #2 Method #3 Method #4 Repetitive Tasks Duration (Days) Cost (EGP) Duration (Days) Cost (EGP) Duration (Days) Cost (EGP) Duration (Days) Cost (EGP) A 14 40,400 10 60,000 5 65,000 12 53,000 B 7 18,550 1 50,000 3 35,000 4 25,000 Each Sub-task C 28 137,200 14 200,000 20 170,000 18 150,000 Each Sub-task D 35 353,500 25 380,000 18 450,000 21 390,000 Each Sub-task E 28 594,000 40 510,000 20 680,000 22 520,000 Each Sub-task F 63 1,063,970 55 1,120,000 30 1,350,000 45 1,250,000 Each Sub-task G 7 8,400 2 20,000 5 12,000 10 10,000 H 56 2,003,700 70 2,750,000 48 1,750,000 35 2,150,000 I 119 499,850 102 430,000 85 580,000 90 620,000 J 35 174,500 18 220,000 25 200,000 40 150,000 K 14 26,600 10 40,000 7 20,000 4 45,000 L 28 12,600 18 16,000 26 10,000 15 18,000 According to the data presented, the proposed model underwent 100,000 trials, of which 77,820 were deemed valid. The computations and running time of the model were performed using an Intel Core i7 computer and lasted approximately 10 minutes. After running the optimization engine, the GA-proposed approach results revealed a reduction in the linear project's Total Direct Costs (TDC), Total Indirect Costs (TIC), and Total Construction Cost (TCC) apropos of one linear unit from 11,404,480 EGP to 11,033,880 EGP, from 2,079,000 EGP to 1,662,000 EGP and from 14,155,480 EGP to 13,208,880 EGP, respectively (as shown in Fig. 11 ). Corrpsonigly, the GA-proposed approach unearthed a reduction in the linear repetitive project TDC, TIC, and TCC apropos of the four linear repetitive units from 45,617,920 EGP to 44,135,520 EGP, from 8,316,000 EGP to 6,648,000 EGP, and from 56,621,920 EGP to 52,835,520 EGP, progressively, as rendered in Fig. 11 . The presented findings indicate that the GA-proposed approach successfully decreased the linear repetitive project's TDC, TIC, and TCC by approximately 3.25%, 20%, and 7%, respectively. Regarding the time optimization, the GA-proposed approach results revealed a reduction in the linear repetitive project's completion time pertinent to one unit and four units from 385 to 325 and from 693 to 554, enlightening approximately 16% and 20% reduction, respectively. Figure 12 portrays a comparative analysis between the developed GA with the base case apropos of the duration of each repetitive task. The stupendous reduction in TDC, TIC, TCC, and completion time of the linear repetitive project corroborate and support the developed approach, underperforming a tremendous reduction performance. These reductions are attributable to the proposed approach of exhibiting the repetitive tasks as a series of distinct sub-tasks interconnected jointly to render the overall duration of the task. This approach reckoned with depicting any logical connection between the decomposed tasks, showing more methodical schedules, and portraying non-critical sub-tasks. These non-critical sub-tasks in linear repetitive projects are vital for balancing project costs and time, thus, empowering the prospect of optimizing the LRPTCT. This hypothesis can be explicitly depicted in Fig. 12 , rendering an immense reduction in non-critical sub-tasks duration due to deploying their total floats constituted from decomposing the repetitive tasks. Figure 13 renders the optimum trade-off optimization between the total project's duration and construction cost attained from the GA trials, unveiling the number of trials carried out by the GA platform to engender the optimum outcome. It can be inferred from Fig. 13 that the best trial number to attain the optimum results was 2,165 trials. Ultimately, Fig. 14 exhibits the LOB depiction after deploying the proposed GA paradigm, unearthing the reduction in the linear repetitive project's completion time pertinent to the four repetitive units from 693 to 554 (nearly 20% reduction in the completion time). 6.4 Deployment of the PSO approach The deployment of the Particle Swarm Optimization (PSO) algorithm for optimizing linear repetitive project scheduling involved a rigorous evaluation of model parameters to ensure a balance between computational efficiency and solution quality. Based on established heuristic optimization methodologies, the PSO algorithm was configured with the following parameters: 1) Population Size: 30 particles representing potential solutions, each encoding the method selection for all repetitive tasks; 2) Number of Iterations: 100, ensuring sufficient exploration of the solution space; 3) Inertia Weight (ω): Set to 0.5, to balance exploration (searching for new solutions) and exploitation (refining existing solutions); 4) Cognitive Coefficient (c1​): 2, to emphasize a particle's self-learning from its own best-known solution, 5) Social Coefficient (c2c​): 2, to promote collaboration among particles by gravitating toward the global best solution. Each particle's position represented a unique configuration of selected methods for 27 repetitive tasks. These tasks were derived from a linear repetitive construction project, and their respective costs and durations were provided across four construction method alternatives. During the initial configuration, the project incurred a Total Direct Cost (TDC) of 45,617,920 EGP and required 693 days for completion. Indirect costs were fixed at 3,000 EGP per day, leading to a Total Construction Cost (TCC) of 56,621,920 EGP. These values formed the baseline for evaluating the PSO model's performance. The input data required for the PSO model included: 1) Repetitive Task Details: Each task's identifier and its precedence relationships with other tasks, ensuring logical task sequencing; 2) Construction Method Alternatives: Four potential methods for each task, each defined by its associated direct cost and duration; 3) Project Constraints: Included adherence to contractual deadlines, logical dependencies, and resource availability; 4) Daily Indirect Costs: Fixed at 3,000 EGP per day, contributing to overall project overheads, and 5) Number of Repetitive Units: The analysis considered four units of the repetitive project. The cost optimization results of the PSO model demonstrated significant reductions across all major cost categories, including TDC, TIC, and TCC. For a single linear repetitive project unit, the Total Direct Cost before optimization was 11,404,480 EGP, which reflected the baseline selection of construction methods without consideration for optimization. After applying the PSO model, the Total Direct Cost was reduced to 10,947,910 EGP, achieving a 4.0% reduction. This improvement was primarily attributed to the selection of cost-efficient construction methods for tasks and sub-tasks while maintaining adherence to project constraints and logical dependencies, as rendered in Fig. 15 . For the entire repetitive project consisting of four units, the Total Direct Cost initially stood at 45,617,920 EGP. After optimization, it was reduced to 43,791,640 EGP, which also represented a 4.0% reduction. These consistent reductions across single and multi-unit analyses underscore the scalability of the PSO model in managing repetitive project costs effectively. By prioritizing methods with lower costs, the algorithm ensured that direct costs were minimized without negatively impacting other project parameters, as shown in Fig. 15 . The TIC, which depends on the project duration and daily overhead rates, exhibited a much larger reduction compared to the TDC. Before optimization, the TIC for one unit was 2,079,000 EGP, derived from a baseline project duration of 385 days and a daily indirect cost rate of 3,000 EGP. After optimization, the TIC was reduced to 1,662,000 EGP, reflecting a substantial 20.0% reduction. This reduction is directly linked to the model's ability to shorten project duration, as indirect costs are proportional to time. For four repetitive units, the TIC was reduced from 8,316,000 EGP to 6,648,000 EGP, maintaining the same 20.0% reduction rate (see Fig. 15 ). The efficiency of the PSO algorithm in compressing task durations while maintaining logical sequencing contributed significantly to these reductions. The combined impact of TDC and TIC optimizations resulted in notable reductions in the TCC. For one unit, the TCC decreased from 14,155,480 EGP before optimization to 12,609,910 EGP after optimization, representing a 10.9% reduction. Similarly, for four units, the TCC decreased from 56,621,920 EGP to 50,439,640 EGP, also reflecting a 10.9% reduction. The dual emphasis of the PSO model on minimizing direct costs and reducing project duration proved effective in achieving these comprehensive cost savings. Figure 15 provides a comparative illustration of these reductions, clearly depicting the efficiency gains realized through the PSO model. The Particle Swarm Optimization model also demonstrated substantial improvements in project completion time for both single-unit and multi-unit analyses. For a single linear repetitive project unit, the baseline completion time was 385 days. After optimization, this was reduced to 315 days, reflecting an 18.2% reduction. This improvement was achieved by prioritizing faster construction methods for critical tasks while strategically assigning float times to non-critical sub-tasks. The decomposition of repetitive tasks into sub-tasks enabled the PSO model to allocate resources more effectively, ensuring that time reductions were achieved without compromising project quality or logical sequencing. For the multi-unit analysis involving four repetitive project units, the baseline completion time was 693 days. After applying the PSO model, the total completion time was reduced to 554 days, representing a 20.1% reduction. The larger time savings in the multi-unit scenario were due to the model's ability to optimize non-critical tasks across all units simultaneously, effectively minimizing idle times and ensuring efficient resource utilization. By leveraging float times, the PSO model significantly reduced the duration of non-critical tasks, which, while not directly impacting the critical path, contributed to a streamlined project timeline. These time savings are visually represented in Fig. 16 , highlighting the reductions achieved for both critical and non-critical tasks. The optimization of project durations directly influenced the reduction in Total Indirect Costs, as shorter project durations translate to lower overhead costs. This interdependence of cost and time optimization underscores the strength of the PSO model in addressing the multi-dimensional nature of construction project management. The ability to balance the trade-offs between cost minimization and time compression makes the PSO model a valuable tool for optimizing linear repetitive projects. 6.5 Comparison of PSO and GA results The PSO and GA models were both applied to optimize costs and durations for a linear repetitive project. The comparison below provides a detailed analysis of their respective performances. In terms of cost optimization, for a single unit of the project, TDC was reduced by both methods, with PSO achieving a final TDC of 10,947,910 EGP compared to GA's slightly higher TDC of 11,033,880 EGP. Both methods provided significant improvements over the baseline TDC of 11,404,480 EGP. For TIC, both PSO and GA demonstrated identical results, reducing the baseline TIC of 2,079,000 EGP to 1,662,000 EGP. Regarding the TCC, PSO outperformed GA by achieving a final TCC of 12,609,910 EGP, compared to GA’s TCC of 13,208,880 EGP, with both models achieving notable reductions from the baseline of 14,155,480 EGP (see Fig. 17 ). For the multi-unit analysis, which comprised four repetitive units, the results followed a similar trend. PSO reduced the TDC to 43,791,640 EGP, while GA resulted in a TDC of 44,135,520 EGP, both improving on the baseline of 45,617,920 EGP. The TIC for the multi-unit project was reduced from the baseline of 8,316,000 EGP to 6,648,000 EGP by both PSO and GA. The TCC achieved by PSO was lower at 50,439,640 EGP, compared to GA's TCC of 52,835,520 EGP, both representing significant savings over the baseline of 56,621,920 EGP. In terms of time optimization, PSO reduced the single-unit project completion time from 385 days to 315 days, representing an 18.2% reduction. GA achieved a similar reduction, bringing the project duration down to 325 days. For the multi-unit scenario, PSO reduced the project completion time from 693 days to 554 days, a 20.1% reduction. Similarly, GA reduced the completion time for four units to 554 days, matching PSO’s performance in this regard (Fig. 18 ). The comparative analysis highlights that while both PSO and GA excel in optimizing costs and time, PSO consistently outperformed GA in terms of achieving lower Total Construction Costs for both single-unit and multi-unit projects. Both methods delivered equivalent reductions in Total Indirect Costs and project completion times, indicating similar performance in addressing the time dimension. These results demonstrate the robustness of PSO and GA in solving complex optimization problems, with PSO showing a slight edge in cost efficiency. 7 Conclusions and Prospective Work This research underscores the critical importance of optimizing time and cost in linear repetitive construction projects, which are often subject to complex interdependencies and resource constraints. The multi-objective nature of the TCT problem poses significant challenges to decision-makers, requiring advanced methodologies that integrate flexibility, efficiency, and accuracy in scheduling and resource allocation. This study proposes an innovative dual-optimization framework that merges GA and PSO with LOB methodologies, offering a transformative solution to the limitations of traditional approaches. The proposed framework redefines repetitive tasks by decomposing them into interconnected sub-tasks, enabling more nuanced scheduling and resource allocation. This decomposition addresses the oversimplifications inherent in conventional LOB methods, which often treat tasks as indivisible units. The novel approach ensures systematic scheduling, efficient crew allocation, and optimized resource utilization, significantly improving both project duration and cost efficiency. The research findings reveal substantial improvements in project performance. For single-unit projects, the proposed approach reduced the project duration from 385 days to 325 days, achieving a 16% reduction. For multi-unit projects involving four repetitive units, the total duration decreased from 693 days to 554 days, representing a 20% improvement. These reductions highlight the robustness of the framework in streamlining project schedules without compromising quality or constructability. Cost optimization was equally remarkable. For a single unit, TDC decreased from 11,404,480 EGP to 11,033,880 EGP, TIC dropped from 2,079,000 EGP to 1,662,000 EGP, and TCC declined from 14,155,480 EGP to 13,208,880 EGP, corresponding to reductions of 3.25%, 20%, and 7%, respectively. In the four-unit scenario, TDC fell from 45,617,920 EGP to 44,135,520 EGP, TIC from 8,316,000 EGP to 6,648,000 EGP, and TCC from 56,621,920 EGP to 52,835,520 EGP, achieving similar percentage savings. These results underscore the framework’s ability to address cost inefficiencies while maintaining adherence to project constraints. A significant innovation of this study lies in its decomposition methodology, which uncovered partially critical sub-tasks. This approach enabled better utilization of task floats, reduced the criticality ratio from 67–48%, and enhanced scheduling flexibility. The ability to balance critical and non-critical tasks effectively ensures a more resilient project schedule capable of adapting to unforeseen conditions and resource limitations. The comparative analysis between GA and PSO further enriches the study, highlighting the unique strengths of each algorithm. GA demonstrated superior robustness in exploring diverse solution spaces, ensuring comprehensive optimization across a wide range of scenarios. PSO, on the other hand, exhibited faster convergence and marginally better cost efficiency, achieving slightly lower Total Construction Costs compared to GA. Both algorithms proved highly effective in achieving the dual objectives of minimizing project duration and costs, offering valuable insights for selecting the appropriate optimization tool based on specific project requirements. This research not only addresses critical gaps in the literature but also sets the stage for future advancements in construction project management. Future studies could explore hybrid optimization approaches that combine GA, PSO, and other metaheuristic techniques to further enhance efficiency and scalability. Expanding the framework to include non-linear repetitive projects, as well as integrating additional optimization objectives such as quality, environmental sustainability, and risk management, would broaden its applicability. Furthermore, addressing the challenges posed by increased task granularity through advanced modeling techniques and incorporating learning curve dynamics into the scheduling process would enhance the framework’s precision and adaptability. By introducing this comprehensive dual-optimization framework, this research provides a cutting-edge decision-support tool that empowers practitioners to navigate the complexities of linear repetitive construction projects. The proposed methodology not only improves project scheduling and cost management but also fosters greater efficiency, adaptability, and resilience, contributing to a more sustainable and competitive construction industry. Declarations Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Author Contribution Ahmed Gouda Mohamed, Ali Hassan Ali, and Ahmed Adel Abdelhady equally contributed to this research paper. Each author played a vital role in every aspect of the study. Ahmed Gouda Mohamed took the lead in conceptualizing the research framework and methodology, significantly shaping the theoretical foundation of the study. Ali Hassan Ali was instrumental in the data collection process and conducted rigorous data analysis, ensuring the validity and accuracy of the results. Ahmed Adel Abdelhady contributed extensively to drafting and refining the manuscript, ensuring clarity, coherence, and adherence to scientific standards. All authors collaborated in reviewing and revising the manuscript, offering critical insights to enhance its quality and finalize the submission. Data Availability This manuscript does not report data generation or analysis. Ahmed Gouda Mohamed will be responsible if someone wants to request the data from this study. References Ali, A. H. et al. A hybrid model for assessing safety implementation and project success in the construction industry. Alexandria Eng. J. 105 , 626–639. https://doi.org/10.1016/j.aej.2024.08.040 (2024a). Ali, A. H., Zayed, T., Abdulai, S. F. & Wang, R. D. A comprehensive framework for examining the influence of tower crane safe operations on sustainable practices in modular integrated construction. Eng. Constr. Archit. Manag Ahead P . 1–28. https://doi.org/10.1108/ECAM-05-2024-0657 (2024b). Ali, A. 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Optimizing time–cost in generalized construction projects using multiple-objective social group optimization and multi-criteria decision-making methods. Eng. Constr. Archit. Manag . 27 , 2287–2313. https://doi.org/10.1108/ECAM-08-2019-0412 (2020). Yılmaz, M. & Dede, T. Multi-objective time–cost trade-off optimization for the construction scheduling with Rao algorithms. Structures 48 , 798–808. https://doi.org/10.1016/j.istruc.2023.01.006 (2023). Zou, X., Fang, S. C., Huang, Y. S. & Zhang, L. H. Mixed-Integer Linear Programming Approach for Scheduling Repetitive Projects with Time-Cost Trade-Off Consideration. J. Comput. Civ. Eng. 31 , 1–6. https://doi.org/10.1061/(asce)cp.1943-5487.0000641 (2017). Zou, X., Wu, G. & Zhang, Q. Work continuity constraints in repetitive project scheduling considering soft logic. Eng. Constr. Archit. Manag . 28 , 1713–1738. https://doi.org/10.1108/ECAM-11-2019-0595 (2020). Zou, X., Zhang, L. & Zhang, Q. Time-cost optimization in repetitive project scheduling with limited resources. Eng. Constr. Archit. Manag . 29 , 669–701. https://doi.org/10.1108/ECAM-10-2020-0843 (2022). Zou, X., Zhang, Q. & Zhang, L. Modeling and Solving the Deadline Satisfaction Problem in Line-of-Balance Scheduling. J. Manag Eng. 34 , 1–12. https://doi.org/10.1061/(asce)me.1943-5479.0000565 (2018). Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 20 Jun, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 10 Feb, 2025 Reviews received at journal 03 Feb, 2025 Reviewers agreed at journal 03 Feb, 2025 Reviews received at journal 27 Jan, 2025 Reviews received at journal 21 Jan, 2025 Reviewers agreed at journal 20 Jan, 2025 Reviewers agreed at journal 15 Jan, 2025 Reviewers invited by journal 15 Jan, 2025 Editor assigned by journal 15 Jan, 2025 Editor invited by journal 07 Jan, 2025 Submission checks completed at journal 07 Jan, 2025 First submitted to journal 03 Jan, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5757308","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":398792688,"identity":"50d905fb-7d9e-4109-9946-8aff85a6ef75","order_by":0,"name":"Ahmed Gouda Mohamed","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCklEQVRIiWNgGAWjYDACCSDmYWBIAHM+NoApA+K1MM4kWQszLzFa+Gd3Jz54w1Cbxz+7+ehm2x2H5RjYm7dJMFTU4rbkztnNhnMYjhdL3DmWdjv3zGFjBp5jZRIMZ47jtuZG7jZpHoZjiQ03csxu57YdTmyQyDGTYGw7hlOH/I3c7b9BWubfyP9227LtcH2D/Bugln+4tRgAbWHmYahJ3HAjh+02Y9vhBAYJHqCWhhqcWgxv5G6WnGNwIHHjjTSzm71n0g3beNKKLRKOHcCpRe5G7sYPbyrqEufdSH524+cOa3l+9sMbb3yoqcPtfYjzDiPYbCAigeEwLrUwgGkmIVtGwSgYBaNgBAEA06tdUbosevEAAAAASUVORK5CYII=","orcid":"","institution":"The British University in Egypt (BUE)","correspondingAuthor":true,"prefix":"","firstName":"Ahmed","middleName":"Gouda","lastName":"Mohamed","suffix":""},{"id":398792689,"identity":"de82c642-3ea0-4a70-9b84-e00ab457f815","order_by":1,"name":"Ali Hassan Ali","email":"","orcid":"","institution":"The British University in Egypt (BUE)","correspondingAuthor":false,"prefix":"","firstName":"Ali","middleName":"Hassan","lastName":"Ali","suffix":""},{"id":398792690,"identity":"a10b687b-7076-42cc-a3b1-7c9dfc71489b","order_by":2,"name":"Ahmed Adel Abdelhady","email":"","orcid":"","institution":"The British University in Egypt (BUE)","correspondingAuthor":false,"prefix":"","firstName":"Ahmed","middleName":"Adel","lastName":"Abdelhady","suffix":""}],"badges":[],"createdAt":"2025-01-03 09:53:28","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5757308/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5757308/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-02837-8","type":"published","date":"2025-06-20T15:57:38+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":73423435,"identity":"144210d1-d349-4200-91c7-5558f42ce300","added_by":"auto","created_at":"2025-01-09 19:37:57","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":46621,"visible":true,"origin":"","legend":"\u003cp\u003eConventional scheduling using the LOB method; (a) A Critical Path Method (CPM) for one unit depicting the criticality of the entire tasks in the unit; (b) Exhibiting tasks A, B, and C using a LOB diagram.\u003c/p\u003e","description":"","filename":"Picture1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/28c9768038be8329d31ab593.jpg"},{"id":73423441,"identity":"5ba1f471-71fa-48ff-bb22-375253abdae9","added_by":"auto","created_at":"2025-01-09 19:37:57","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":65074,"visible":true,"origin":"","legend":"\u003cp\u003eThe proposed approach for rendering repetitive tasks; (a) Illustrating a CPM for one unit after splitting tasks into sub-tasks; (b) portraying tasks A-1, A-2, and A-3 via a LOB diagram after splitting task A into partitioned sub-activities.\u003c/p\u003e","description":"","filename":"Picture2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/d391944918ebe4b548178bef.jpg"},{"id":73423436,"identity":"e96cbd58-d626-463b-a973-deaebc851643","added_by":"auto","created_at":"2025-01-09 19:37:57","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":17413,"visible":true,"origin":"","legend":"\u003cp\u003eThe configuration of chromosomes in the GA problem.\u003c/p\u003e","description":"","filename":"Picture3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/b625294388f7aa15683e3b00.jpg"},{"id":73423444,"identity":"4588b4fc-5b59-49f9-93da-1444284b175a","added_by":"auto","created_at":"2025-01-09 19:37:57","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":89539,"visible":true,"origin":"","legend":"\u003cp\u003eThe Proposed GA workflow.\u003c/p\u003e","description":"","filename":"Picture4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/415ac0c336519661b024b2d6.jpg"},{"id":73423841,"identity":"5d8dc0f3-ac60-499e-a6d8-151b702309d2","added_by":"auto","created_at":"2025-01-09 19:45:57","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":35049,"visible":true,"origin":"","legend":"\u003cp\u003eCrossover operator for creating offspring genes.\u003c/p\u003e","description":"","filename":"Picture5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/e91699212183714fc300c61a.jpg"},{"id":73423478,"identity":"9b81800c-4168-494c-b736-00d59dae160e","added_by":"auto","created_at":"2025-01-09 19:37:58","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":88183,"visible":true,"origin":"","legend":"\u003cp\u003ePSO workflow.\u003c/p\u003e","description":"","filename":"Picture6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/183f83102a59090758cea98f.jpg"},{"id":73423454,"identity":"7c5f5404-b689-4a25-abb1-de268f43a33c","added_by":"auto","created_at":"2025-01-09 19:37:58","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":44882,"visible":true,"origin":"","legend":"\u003cp\u003eA Flowchart depicting the case study workflow.\u003c/p\u003e","description":"","filename":"Picture7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/d5e0cb96ede8ed4d9f1c517a.jpg"},{"id":73423487,"identity":"4884fadd-cc46-4299-baf7-ae72f12cb02b","added_by":"auto","created_at":"2025-01-09 19:37:59","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":51421,"visible":true,"origin":"","legend":"\u003cp\u003eLOB depiction using traditional CPM calculations. (a) Critical path showing critical repetitive tasks; (b) LOB diagram of non-critical tasks.\u003c/p\u003e","description":"","filename":"Picture8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/9a529770a302b3638de58f40.jpg"},{"id":73423432,"identity":"9977c448-25ad-4992-8acb-6683841c38c7","added_by":"auto","created_at":"2025-01-09 19:37:56","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":57719,"visible":true,"origin":"","legend":"\u003cp\u003eLOB depiction using the proposed approach for splitting into sub-tasks. (a) Critical path embodying critical repetitive sub-tasks; (b) LOB diagram of non-critical sub-tasks.\u003c/p\u003e","description":"","filename":"Picture9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/5efd41e326f2fb09bf2a20de.jpg"},{"id":73423449,"identity":"d0f93bb2-1688-489b-8baf-6ce026a2b3c0","added_by":"auto","created_at":"2025-01-09 19:37:57","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":102968,"visible":true,"origin":"","legend":"\u003cp\u003eThe formulation of the GA model for LRPTCT using the optimization engine.\u003c/p\u003e","description":"","filename":"Picture10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/c40b1caa3f720ebcb744c6c2.jpg"},{"id":73423455,"identity":"aefc0413-725a-4096-be9d-0fac69f52ebe","added_by":"auto","created_at":"2025-01-09 19:37:58","extension":"jpg","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":63591,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the developed GA with the base case apropos of costs. (a) Linear project costs comparison for one linear unit; (b) Linear project costs comparison for the four repetitive units.\u003c/p\u003e","description":"","filename":"Picture11.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/a51d3e592b83534864b9106d.jpg"},{"id":73423489,"identity":"99cb2395-1823-4c5d-adbb-87bc476a1312","added_by":"auto","created_at":"2025-01-09 19:37:59","extension":"jpg","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":41754,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of the developed GA with the base case apropos of the duration per each repetitive task.\u003c/p\u003e","description":"","filename":"Picture12.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/ff4f9f2c900734f0c9174351.jpg"},{"id":73423445,"identity":"26c2249b-0cab-4965-8bdc-9906f0affe2c","added_by":"auto","created_at":"2025-01-09 19:37:57","extension":"jpg","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":42665,"visible":true,"origin":"","legend":"\u003cp\u003eThe optimum trade-off optimization between the total project's duration and construction cost.\u003c/p\u003e","description":"","filename":"Picture13.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/46b304a5dc9e1e62994f4a14.jpg"},{"id":73423482,"identity":"83ec07fe-9bdd-468d-a1e7-2b8dedbe308f","added_by":"auto","created_at":"2025-01-09 19:37:59","extension":"jpg","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":74140,"visible":true,"origin":"","legend":"\u003cp\u003eLOB depiction after deploying the GA optimization approach. (a) Critical path embodying critical repetitive sub-tasks; (b) LOB diagram of non-critical sub-tasks.\u003c/p\u003e","description":"","filename":"Picture14.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/185c5a14b933ee63011aab17.jpg"},{"id":73423481,"identity":"ef7a0d67-acb2-4d6c-9958-97a95a68d36f","added_by":"auto","created_at":"2025-01-09 19:37:59","extension":"jpg","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":61175,"visible":true,"origin":"","legend":"\u003cp\u003eCost comparison before and after optimization using PSO. (a) One unit; (b) Four units.\u003c/p\u003e","description":"","filename":"Picture15.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/d102ab8764aea3a941fe7b68.jpg"},{"id":73423439,"identity":"c969e6d0-2b39-40e1-a59e-487e31f90bcb","added_by":"auto","created_at":"2025-01-09 19:37:57","extension":"jpg","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":55472,"visible":true,"origin":"","legend":"\u003cp\u003eTime comparison before and after optimization using PSO.\u003c/p\u003e","description":"","filename":"Picture16.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/86eefb1c073c882e7aec3e24.jpg"},{"id":73423846,"identity":"7c5f1d14-ed46-42e8-a606-df84b3031fdf","added_by":"auto","created_at":"2025-01-09 19:45:58","extension":"jpg","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":84098,"visible":true,"origin":"","legend":"\u003cp\u003eCost comparison between GA and PSO for four units.\u003c/p\u003e","description":"","filename":"Picture17.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/85e74b0235b568c20426cd62.jpg"},{"id":73423475,"identity":"e367e3bb-0474-490c-9e66-2e1e9727e274","added_by":"auto","created_at":"2025-01-09 19:37:58","extension":"jpg","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":69506,"visible":true,"origin":"","legend":"\u003cp\u003eTime comparison between GA and PSO for four units.\u003c/p\u003e","description":"","filename":"Picture18.jpg","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/4253618deb8a31965c96d5d1.jpg"},{"id":85231375,"identity":"66a5e59f-22c3-4fa5-a82f-6f7b75a0892f","added_by":"auto","created_at":"2025-06-23 16:06:45","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2636672,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5757308/v1/a41bab1e-592f-4f74-b679-546ceed53e93.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"An Integrated Decision Support System for Optimizing Time-Cost Trade-offs in Linear Repetitive Projects","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eThe construction industry represents 13% of the global gross domestic product (Ali et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2024c\u003c/span\u003e; Elrifaee et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2024\u003c/span\u003e), and thus it plays a highly significant role in economies and helps in improving the standards of living (Ali et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2024b\u003c/span\u003e, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2024a\u003c/span\u003e). Nevertheless, the scope of this sector is significantly threatened by the challenges that appear in the light of the process of urbanization, which increases the pressure on the efficiency of infrastructure construction (Hui et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Kineber et al., \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). As stated earlier, there is a pressing need for infrastructure development with commendable emphasis placed on linear repetitive works like bridges, tunnels, and highways among others in anticipation of the exploding population globally (Agrama, 2011).\u003c/p\u003e \u003cp\u003eLinear repetitive projects are more often used in the handling of large-scale construction projects especially when many related tasks are to be performed in similar ways in a certain period (Zou et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). These projects are widely distributed in the construction, manufacturing, and transportation industries where the tasks are clearly defined, repetitive, and similar in nature (Tomczak and Jaśkowski, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Some of the inherent characteristics of linear repetitive projects include the following: The structure of linear repetitive projects is highly beneficial due to the following reasons; However, these projects also come with considerable risks that call for streamlined planning and execution to increase their chances of success (Zou et al., \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eOver the past few decades, the construction industry has faced several issues that make project management a nightmare. Some of the most significant challenges include: scope creep, distribution of resources, quality assurance, scheduling, and TCT. For example, scope creep brings in additional costs or durations; ineffective management of the resources may incur more time and poor quality (Kumar and Mehany, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Lin and Lai, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). The importance of managing these competing demands is, therefore, paramount, especially given the fact that constriction projects are usually undertaken within tight schedules and a fixed amount of funds.\u003c/p\u003e \u003cp\u003eThe essence of the application of the TCT approach is to determine the optimum time-cost trade-off of each activity in a project. This strategy aims to reduce the total cost of a project while retaining prescribed timelines for the completion of the project. At the same time, it has to generate schedules that enable tasks to be accomplished at the earliest possible times. This is possible when the time of beginning each activity is well determined along with considering the precedences of the tasks and available resources (Tran, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eCritical path method (CPM) and Line of balance (LOB) are the two conventional scheduling modes that are commonly used in construction management. However, the accuracy of CPM scheduling comes into conflict with the method when used to schedule repetitive projects that involve repeatedly performing the same basic unit. Still, the main drawback of CPM in this respect is the inability to maintain the continuity of crew work which is critical for improving the performance rates of repeated processes (El-Kholy et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eHowever, as has already been seen, LOB is the best scheduling tool for linear repetitive projects and has several drawbacks. LOB sequentially arranges work activities, this might not adequately cover a complex linear work that is dependent on other linear undertakings. In addition, in LOB the estimated rate of production for each planned activity is assumed to remain constant. Unfortunately, this premise often does not take into consideration actual project environment factors, such as weather conditions or the availability and cost of labor and materials, which significantly impact project duration (Tang et al., \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Moreover, the chronological arrangement of activities in LOB may reduce various dependencies and project details embedded within construction projects because dependent tasks are posed linearly. Such distortion leads to improper timing and insufficient coverage of critical and non-significant sub-tasks, which directly endangers project success (Zou et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2018\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eTime and cost control remain crucial in linear repetitive projects since they usually are inversely proportional. Approaches like having more people in the workforce, using uncontemporary building methodologies, or obtaining approval for overtime can shorten project span but at the same time can raise costs substantially. However, project managers must factor time and cost together especially when planning and scheduling to improve on the project performance. Nevertheless, the multiple attribute perspective of the TCT problem can be a drawback as it hinders decision-making. As seen in the segmented decision-making diagram above, many options can be included when it comes to the time and cost of each activity \u0026ndash; and with more possibilities for activities and decisions, this greatly increases the size of the search space. The combination of parameters in such a manner has been the subject of research in the past few years (Heravi and Moridi, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2019\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eTraditional methods of scheduling like the Critical Path Method (CPM) and Line of Balance (LOB) are not very effective in handling the time, cost, and resource dimensions in linear repetitive construction projects. These approaches do not isolate how tasks interact with each other as well as various resources within the real world. However, current research is still lacking an effective framework that uses TCT in conjunction with LOB; or applies innovative optimization techniques like GA and PSO to determine the optimum values of time and cost. This gap results in some incongruities that impede project management and warrant a new approach and solution\u0026mdash;this calls for a dual-optimization framework.\u003c/p\u003e \u003cp\u003eThe primary contribution of this paper lies in its groundbreaking development of a dual-optimization framework for Linear Repetitive Project Time-Cost Trade-off (LRPTCT), which uniquely integrates Genetic Algorithms (GA) and Particle Swarm Optimization (PSO) to address the longstanding challenges in linear construction projects. This innovative approach merges the strengths of Time-Cost Trade-off (TCT) analysis and Line of Balance (LOB) methodologies, delivering a novel solution that bridges critical gaps in project management research. The inclusion of a comparative analysis between GA and PSO further underscores the uniqueness of this study, offering a dual-lens perspective on leveraging these metaheuristic techniques for real-world applications.\u003c/p\u003e \u003cp\u003eThis study sets itself apart through its introduction of a decomposition strategy for repetitive tasks, enabling a nuanced representation of task interdependencies and resource allocation. Unlike traditional methods, which often oversimplify project dynamics, the proposed approach incorporates a sophisticated optimization framework that utilizes GA for exploring expansive solution spaces with diverse potential configurations, and PSO for efficiently converging on optimal solutions. The synergy between these algorithms represents a pioneering step in construction optimization, tailored to address the intricate demands of modern linear repetitive projects.\u003c/p\u003e \u003cp\u003eWhat distinguishes this work is the comparative analysis of GA and PSO, which provides a strategic understanding of their respective advantages and limitations in various project contexts. This comparative approach not only enhances the novelty of the research but also equips practitioners with actionable insights into selecting the appropriate optimization tool for specific project needs, making the study both innovative and highly practical.\u003c/p\u003e \u003cp\u003eThe significance of this research extends well beyond its theoretical contributions. It redefines the integration of TCT and LOB, presenting a transformative decision-support tool that empowers project managers to optimize linear repetitive projects with unprecedented precision. This framework opens new avenues for addressing complex scheduling and cost management challenges, offering a scalable, flexible, and impactful solution that can be adapted across diverse project scenarios. By advancing the state of the art in construction project management, this study lays the groundwork for future innovations in multi-objective optimization, reinforcing its uniqueness and practical value to the field.\u003c/p\u003e"},{"header":"2 Literature review","content":"\u003cp\u003eRepetitive construction projects can be broadly categorized into two main types: linear and non-linear. Linear projects refer to those that are implemented sequentially and these include transport corridors such as pipelines highways and railroads. Instead, non-linear projects relate to structures that contain features in this respect, for example, several houses or a couple of tall constructions (El-Kholy et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Today, repeated activities represent practically all construction processes, especially in terms of mass-scope operations. Some of these projects have cyclical patterns of activities within them and the extension of continuity in the use of resources in these projects helps to improve the efficiency and effectiveness of projects (Yılmaz and Dede, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). This continuity is important here since repetitive work means that each activity should be repeated several times. Such an approach can help achieve considerable cost and time savings due to the possibility of having a separate skilled labor crew that remains constant throughout the project (Moreno et al., \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eBut there is some difficulty in this strategy as well. The endless work may reduce the productivity of the crew involved in its execution. When the crew moves from one unit to another, doing the same activities again tends to elongate the time required for completing the project as well as the expenses. Consequently, this paper aims to formulate an efficient strategy for the crew work in repetitive projects to achieve the best performance in terms of cost and time (Tomczak and Jaśkowski, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Flawless crew allocation will go a long way in avoiding productivity issues while enhancing the values of hiring crew with similar people temperaments.\u003c/p\u003e \u003cp\u003eHowever, this call for strategy also has its drawbacks. One of the possible effects of continuous work could be the reduction of crew efficiency. Generally, as the crew moves from one unit to another, and performs the same activities in an almost patterned sequence it has the adverse of lengthening the project time as well as the associated cost. Thus, there is a need to adopt a good strategy especially concerning crew work in repeat projects with respect to costs and time (Tomczak and Jaśkowski, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Proper crew scheduling and work planning can also go a long way to preventing detrimental productivity drawbacks that may be occasioned by a consistent crew.\u003c/p\u003e \u003cp\u003eRecently, a lot of research works have been published to address time-cost trade-off problems (TCTP). These studies can be broadly categorized into two groups: the studies that investigate the general application of TCTP in construction projects and the studies that investigate the repetitive elements of construction projects. Li \u0026amp; Wu (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) proposed a robust optimization model based on nondominated sorting genetic algorithm-II (NSGA-II) to solve the time-cost trade-off for general construction projects under uncertainty. This model compares many construction modes that share different temporal and cost features influenced by interval uncertainty. Accordingly, their result shows that raising the robust coefficient for cost or time produces more solutions but at the same time allows for more risk. However, the model mainly focuses on interval uncertainties and approximations, and together with it, there could be more hustle than needed in decision-making situations, as most often intervals may vary by time and may have other probability distribution than supposed by the model. Besides, it does not address the issue of resources, which are a core determinant of project timetables and expenses. This limitation may limit the general usability of the model in real-world environments.\u003c/p\u003e \u003cp\u003eIn the same respect, Biswasa et al. (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) proposed a hybrid model of CPM and a cost-loop method that is heuristic in nature. This approach facilitates the determination of the least total cost possible for each potential duration of the project, thus solving a main construction project problem, that is, the time-cost problem. From their study, they were able to record at least a cost of \u003cspan\u003e$\u003c/span\u003e 60,937 as their total cost with an approximate project length of 130 days. Nevertheless, the study has the following limitations that should be stated: It simply applies deterministic parameters to time and cost estimates for activities completely ignoring the chances of variations that are common in the actual construction undertakings. This restriction limits the external validity of the study\u0026rsquo;s outcomes and implies that future studies should also apply stochastic factors to their models to increase predictive precision.\u003c/p\u003e \u003cp\u003eEirgash \u0026amp; Toğan (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) proposed the Golden Ratio Oppositional Teaching Learning Based Optimization (GROTLBO) to improve the convergence speed and solution diversification, particularly in multi-objective optimization of time-cost-environmental impact trade-off in construction projects. The results also show that the GROTLBO is superior to NSGA-II and OMODE with a 5% improvement in inaccuracy and a 95% reduction in the number of scheduled calculations performed. However, there are some apparent limitations inherent to this study, though the results of the research seem quite promising. One important issue concerns the computational time, which is associated with the algorithm\u0026rsquo;s search space that chars within the existence of the object. Furthermore, the extension of the TCET scenarios to prevent independent validation of GROTLBO, and its effectiveness in which real-world scenarios in practice practical situations.\u003c/p\u003e \u003cp\u003eSimilarly, in another study, Eirgash et al. (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) put forward a new variant known as the modified dynamic oppositional learning-based teaching\u0026ndash;learning-based optimization (MDOLTLBO) algorithm for improving time-cost optimization in construction projects. This algorithm has an adaptive varying of the weights used through iterative approaches that in turn provides exploration and exploitation characteristics, solutions diversities, and a higher rate of convergence. According to their findings, they found that MDOLTLBO is better than conventional TLBO meta-heuristic algorithms and other advanced optimization approaches and techniques including linear programming and genetic approaches in terms of solution accuracy, convergence characteristics, and computational complexity. However, it has been observed that the algorithm has a few limitations also. Firstly, the performance of this heuristic depends on the particulars of construction projects and problem instances. Nevertheless, the proposed algorithm, MDOLTLBO, improves convergence and solution quality but can still be trapped in local optima in large-scale, high-dimensional search spaces.\u003c/p\u003e \u003cp\u003eHowever, for repetitive projects, Huang et al. (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) proposed a special genetic algorithm to solve the discrete time-cost trade-off problem (DTCTP) of repetitive construction projects. The algorithm thereby introduces soft logic to improve the flexibility of the schedule while allowing the work sequences among activities to be variable. This optimization enhances methods of activity modes and start time selection and hence reduces the extent of project duration and cost. When compared to a previous scheduling solution that employed just hard logic, their findings showed increased competitiveness of solutions that used soft logic, one of which cut down the real project time by eleven days and 25283.5\u003cspan\u003e$\u003c/span\u003e by applying different sequences of work. However, there are still some limitations within the study. One disadvantage of the presented mixed-integer nonlinear programming model is that it can be computationally intensive, and therefore, sensitive to the scale of a project. Moreover, although the GA is somewhat robust in generating high-quality solutions there are times that it directly depends on parameters like population size and mutation.\u003c/p\u003e \u003cp\u003eRecently, Zou et al. (\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) formulated the TCTP with multiple crews and fixed logic within the context of repetitive projects using mixed-integer linear programming (MILP). To that end, the study offers exact and approximate models where the results show that the former provides an efficient solution for medium problems, while the latter provides near-optimal solutions for bigger problems within a reasonable time. Simulations indicate that the approximate model keeps a maximum deviation of less than 1%, and is far better than the exact model for the larger problem sizes. However, several issues can be attributed to this study these include, The assumptions of fixed logic use may not be reflective of the actual level of flexibility in business operations, and Secondly, the overall approximate model may not be optimal for such complex situations. Future studies could improve these models by considering additional types of variable working sequences and other conditions.\u003c/p\u003e \u003cp\u003eAltuwaim and El-Rayes, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2018\u003c/span\u003e proposed a new genetic algorithm optimization model for resource-constrained scheduling of repetitive construction projects intending to minimize project duration, number of breaks, and interruption costs. The proposed model consists of four sub-models to improve the computational performance of the developed heuristic and the incorporation of interruption costs into scheduling procedures which is often overlooked in other approaches. This paper demonstrated the utility of the model on a repetitive construction project where the project team was able to realize considerable savings in overall project costs and ameliorate interruptions to crew work. However, the study also has some limitations, for example, pre-restriction of activities that restrict them only under one crew formulation of each activity, and the serial relationship between them may reduce its scope to a complex project. Besides, compared to other heuristics like that used to find the Pareto optimal solution, the application of genetic algorithms improves computational expediency but precludes optimality.\u003c/p\u003e \u003cp\u003eZou et al. (\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) proposed a new GM-MIP to improve the reliability and consciousness of repeated project scheduling by adding work continuity constraints and soft reasoning. This model can follow a certain pattern across several units and minimize the PCT, combine or total interruption time (TIT), and overall project cost (TPC). The research also shows that the application of soft logic concerning specific project creams can cut down project periods and costs substantially with special reference to many crew projects. Nevertheless, the authors discussed how this study had limitations and how the effects of learning-forgetting needed to be captured for the subsequent studies that will endeavor to confirm the generality of the model for larger and more complex projects.\u003c/p\u003e \u003cp\u003eTo deal with interruptions, buffer times, or schedule acceleration in recurring construction projects, El-Kholy et al. (\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) proposed a general optimization model for controlling time-cost trade-offs. The use of the proposed model showed the benefits of identification of cost efficiencies and improvements to the project timeline. It was demonstrated within a set of examples that interruptions and buffers could be used to improve resource utilization, and hence lower project duration, without undue costs. However, there are some limitations to the study. They depend on certain quantifiable factors that do not capture reality, leaving the possibility of gaps in the development of the full range of solution possibilities. Also, depending on the input data provided the model functionality is rather sensitive and may change from one project to another.\u003c/p\u003e \u003cp\u003eYımaz \u0026amp; Dede (2023) developed a comprehensive optimization model that integrates a non-dominated sorting method with the Rao-1 and Rao-2 optimization algorithms to effectively address the multi-objective time-cost trade-off problem. The proposed approach generates multiple Pareto optimal solutions, offering decision-makers valuable options for balancing time and cost. Results from numerical experiments demonstrate that the non-dominated sorting Rao-2 algorithm outperforms traditional methods like Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), and Genetic Algorithm (GA), achieving better optimal solutions with fewer function evaluations and higher quality Pareto results. However, the proposed algorithms have some limitations. They primarily focus on discrete optimization scenarios, which may not be suitable for all construction projects, especially those involving continuous variables. Additionally, the performance of the Rao algorithms is influenced by the quality of the initial population and parameter settings, which can affect convergence rates. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e provides a summary of multiple research papers. Each entry outlines the primary focus of the research, including the trade-offs, types of projects addressed, specific problems tackled, and methods employed. To provide a clearer overview of the existing research landscape, Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e summarizes key studies, detailing their primary focus, trade-offs, project types, specific problems addressed, and employed methods.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSummary of previous research\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNumber\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eReference\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTradeoff\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eType of projects\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eProblem\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMethods\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLi \u0026amp; Wu (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2014\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTime, Cost\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eGeneral Construction Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eUncertainty\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNSGA-II Optimization\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBiswasa et al. (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTime, Cost\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eGeneral Construction Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eDeterministic Values\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eHybrid CPM with Heuristic Cost-Loop\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEirgash \u0026amp; Toğan (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2023\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTime, Cost, Environmental\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eGeneral Construction Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMulti-objective, Dynamic\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eGROTLBO Algorithm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEirgash et al. (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2023\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTime, Cost\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eGeneral Construction Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMulti-objective, Dynamic\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMDOLTLBO Algorithm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHuang et al. (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTime, Cost\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRepetitive Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eRepetitive, Soft Logic\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eTargeted Genetic Algorithm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eZou et al. (\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2017\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTime, Cost\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRepetitive Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eRepetitive, Fixed Logic\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMILP Approach (Exact and Approximate)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAltuwaim \u0026amp; El-Rayes (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2018\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTime, Cost, Interruption\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRepetitive Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eRepetitive\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eGenetic Algorithm Model\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eZou et al. (\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2020\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eProject Completion Time, Total Interruption Time, Total Project Cost\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRepetitive Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eRepetitive, Soft Logic\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMulti-Objective MILP Model\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEl-Kholy et al. (\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2021\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTime, Cost\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRepetitive Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eRecurring, Interruptions, Buffers\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eIntegrated Scheduling Optimization Model\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYımaz \u0026amp; Dede (2023)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTime, Cost\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eRepetitive Projects\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMulti-objective Optimization\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eNon-Dominated Sorting with Rao Algorithms\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIn summary, the existing literature offers valuable insights into cost optimization and scheduling in construction projects. However, a significant gap remains: there is currently no documented approach that effectively integrates time-cost trade-offs (TCT) with Line of Balance (LOB) methodologies within a genetic algorithm (GA) framework. This oversight is significant, as the integration of these elements could provide a more holistic perspective on project optimization, enhancing both efficiency and effectiveness in repetitive construction settings. To address this critical research gap, this study seeks to develop a novel model that leverages the strengths of TCT and LOB within a genetic algorithm approach. This integration aims to equip decision-makers with more robust tools for navigating the complexities of construction project planning and execution, ultimately leading to improved project outcomes.\u003c/p\u003e"},{"header":"3 The mathematical formulation of time cost trade-off","content":"\u003cp\u003eMulti-objective optimization is pivotal to the time-cost trade-off issue, which rationales to lessen both time and cost coincidentally by electing the utmost pertinent selection for each activity from a broad spectrum of possibilities. The calculation of time is crucial to this process. It can be expressed mathematically via an array of equations, as exhibited in Equations (1) to (3) integral to discerning the optimal trade-off between time and cost.\u003c/p\u003e \u003cp\u003eThis challenge is typically pinpointed in project management, where an analytical approach is imperative for ensuring optimal selection. In light of this, opting for the best optimal selection mandates utmost attention to both time and cost implications, with trade-offs typically addressed to attain a feasible outcome that appeases the targeted goals.\u003c/p\u003e \u003cp\u003eES\u003csub\u003ej\u003c/sub\u003e = max \u003csub\u003ei\u0026isin;pj\u003c/sub\u003e {EF\u003csub\u003ei\u003c/sub\u003e}j\u0026thinsp;=\u0026thinsp;1,⋯, n\u0026thinsp;+\u0026thinsp;1 (1)\u003c/p\u003e \u003cp\u003eEF\u003csub\u003ei\u003c/sub\u003e = ES\u003csub\u003ei\u003c/sub\u003e +t\u003csub\u003ei\u003c/sub\u003e\u003csup\u003e(m)\u003c/sup\u003ei\u0026thinsp;=\u0026thinsp;0,⋯, n\u0026thinsp;+\u0026thinsp;1 (2)\u003c/p\u003e \u003cp\u003eT\u0026thinsp;=\u0026thinsp;EF\u003csub\u003en+1\u003c/sub\u003e (3)\u003c/p\u003e \u003cp\u003eThe Equations portrayed above involve various parameters, incorporating the total project duration, denoted as T, the activities with which j has a precedence relation depicted by pj, and the activity's early start and early finish, signified as ESj and EFi, progressively. Additionally, the activity m duration is rendered by ti(m). The primary objective of these equations encompasses determining the project's completion time by spotting the project's longest path. Hence, the aggregate of each activity's direct costs and the entire duration of the project multiplied by the indirect cost equals the total cost of the projects, as explicitly rendered in equations (4) to (6). In this regard, DC depicts the direct cost, C donates the entire project's cost, IC resembles the indirect cost, ICR is the whole project's indirect costs, and T exhibits the entire project's duration.\u003c/p\u003e \u003cp\u003eC = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\sum\\:_{\\text{i}=0}^{\\text{n}+1\\:}\\text{d}\\text{c}\\text{i}^\\left(\\text{m}\\right)\\text{x}\\text{i}^\\left(\\text{m}\\right)\\)\u003c/span\u003e\u003c/span\u003e (4)\u003c/p\u003e \u003cp\u003eI C\u0026thinsp;=\u0026thinsp;T \u0026times; ICR (5)\u003c/p\u003e \u003cp\u003eC\u0026thinsp;=\u0026thinsp;DC\u0026thinsp;+\u0026thinsp;IC (6)\u003c/p\u003e"},{"header":"4 Research Methodology","content":"\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e4.1 The devised methodology for scheduling the linear repetitive project\u003c/h2\u003e \u003cp\u003ePrincipally, Linear construction projects that pursue a linear path routinely encounter hindrances, dictating the incorporation of non-critical or repetitive tasks for resource limitations remedy. Routinely, The LOB scheduling method presupposes that repetitive tasks have a steady production rate, remaining perpetually across the project duration, irrespective of the desired unit number (See Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). As reported in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(a), the completion time of one unit is 33 days; on the other hand, the completion time of the three repetitive units equals 63 days, as rendered in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAdditionally, LOB scheduling dearth the propensity for repeated tasks to be partially critically attributable to the fact that each repeated task is enacted in a single, undivided pattern with a predetermined deterministic duration. This presumption could pose a challenge for LOB scheduling to affirm that the same crew constitutes each task linearly, rendering it impractical to disseminate resources across the entire project. This restriction is particularly problematic, as repetitive activities may encompass multiple sub-tasks with varying production rates and entailing distinctive crew dissemination. As such, the production rate of each sub-task is imperative for tailoring to optimize resource allocation. Contrary to the conventional method of appraising construction tasks in LOB scheduling, repetitive tasks will be exhibited as a series of distinct sub-task interconnected jointly to render the overall duration of the task.\u003c/p\u003e \u003cp\u003eThis approach reckons with depicting any logical connection between the decomposed tasks adopting solely the finish-to-start link. By decomposing tasks in the LOB diagram, more methodical schedules are engendered and portray critical and non-critical sub-task unerringly. Correspondingly, this technique acknowledges the aleatory conditions of construction projects. Even though decomposing tasks may provoke more tasks in LOB scheduling, but it can instigate a more profound schedule when resource limitations are present, as presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. As attested in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(a), the completion time of one unit is 21 days; contrastingly, the completion time of the three repetitive units equals 46 days, as rendered in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(b).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Genetic algorithm approach for LRPTCT\u003c/h2\u003e \u003cp\u003eA Genetic Algorithm (GA) portrays a metaheuristic optimization algorithm stimulated via natural determination and genetics. It is a search technique adopted to retrieve the best resolution for a particular problem by mimicking the natural determination process and evolution (Ko and Wang, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Genetic algorithms operate on a population of prospective resolutions encoded as chromosomes or strings of information. The population endures a series of genetic operations, including selection, crossover, and mutation, to generate a new population of potentially better solutions. The selection process incorporates opting for the fittest individuals from the current population based on their fitness values, exhibiting how thoroughly they resolve the problem. These selected individuals are then employed to generate new solutions through crossover, which involves combining parts of two or more parent solutions to create new offspring (Anvari et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eAnother genetic procedure called mutation imparts random alterations to the offspring solutions, incentivizing the exploration of new areas of the solution space. This process empowers a deeper investigation of the solution space and precludes the algorithm from being stuck in local optima. The new population of solutions is then appraised for fitness, and the process is repeated until a termination criterion is met, such as reaching a maximum number of generations or achieving a desired level of fitness. After the genetic algorithm process, the population member with the highest level of performance is touted as the optimal solution to the optimization (Anvari et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e4.2.1 Encoding the chromosome\u003c/h2\u003e \u003cp\u003eIn GA, a solution to an optimization problem is embodied as a \"chromosome,\" a data structure that encodes a batch of parameters denoting the solution. In the context of time cost trade-off optimization, a chromosome structure was encoded using different configurations of defined construction methods coupled with correlative duration and direct cost. A chromosome structure was encoded by a sequencing chain of elements, with each element portraying a linear repetitive sub-task and including an index indicating its defined construction method. This sequence renders a single project solution constituting methods to construct the repetitive sub-task, as rendered in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo assess the effectiveness of the chromosome's solution, the repetitive sub-tasks' durations coupled with appointed direct costs are inferred, anticipating pertinent indices within the chromosome. The chromosome could depict arrays of linear project completion times. Each gene in the chromosome renders the start and end times for a particular repetitive sub-tasks and the construction method crucial to constitute the repetitive sub-tasks. The chromosome comprises strings whose length displays the number of repetitive sub-tasks in the project. The intention of the optimization algorithm is thus to identify the chromosome optimally trade-off between time and cost, i.e., the chromosome that yields the shortest feasible completion time for the linear project while minimizing the total cost required to complete the repetitive sub-task. This process incorporates appraising the fitness of each chromosome in the population and using selection, mutation, and crossover operations to evolve the population over successive generations toward the optimal solution. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e exhibits the entire structure of the proposed GA approach.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section3\"\u003e \u003ch2\u003e4.2.2 Objective function\u003c/h2\u003e \u003cp\u003eThe pivotal objective function is minimizing the total linear repetitive project completion time F1, synchronically minimizing the total linear repetitive project costs F2, including direct and indirect costs, while employing predefined construction methods. The objective functions F1 and F2 are calculated concerning the entire chromosomes per generation. Subsequently, the chromosomes are sorted, banking on the solution that attains the trade-off between minimizing F1 and concurrently minimizing F2. The objective functions are constituted and operated by changing the variable from 1 to 4, embodying the defined construction method for each repetitive sub-task. Followingly, the optimization model is subjected to these outlined constraints 1) Decision variables are greater than or equal to 1 and less than or equal to 4, 2) Values of decision variables should be an integer, 3) The optimized total linear repetitive project cost should be less than the planned cost, and 4) The optimized linear repetitive project completion time should be less than the planned time.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e4.2.3 Initial population engendering susceptible to the objective functions\u003c/h2\u003e \u003cp\u003eWhen the chromosome structure and fitness function are constituted, the genetic algorithm proceeds to conduct evolutionary optimization on a population of parent chromosomes. Initially, N-viable chromosomes are synthesized as an initial population. Afterward, these chromosomes are ranked based on the assessment criterion defined earlier. However, the number of chromosomes or population size is a critical parameter that enacts both the computational time required and the possible solution. As such, population size increases, attaining a global optimum but substantially prolonging the processing time.\u003c/p\u003e \u003cp\u003eIn the present deployment, users can designate the population size gleaned from their preferences. When articulating the population, each chromosome's fitness is examined, and pertinent eminence is computed by dividing it by the entire chromosome's fitness. Consequently, Several fittest chromosomes are conserved and passed down to the succeeding offspring. These conserved chromosomes stay qualified concerning electing like parents while reproducing the remaining parents in the following generation. During each iteration, genetic algorithms' crossover and mutation process ensures that the parent chromosomes maintain their assignment property while altering the order of construction methods designated to each repetitive sub-task to engender new chromosomes, as exhibited in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. This process enables the genetic algorithm to prioritize fitter chromosomes for further evolution and analyze the project duration and cost trade-offs.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section3\"\u003e \u003ch2\u003e4.2.4 Genetic operators\u003c/h2\u003e \u003cp\u003eConsequently, reproduction among members of the population emanates through crossover or mutation, which simulates natural evolution as genetic operators. Crossover is the more predominantly adopted process and involves picking two parent chromosomes, swapping relevant information, and engendering offspring. The selection of parent chromosomes is random but weighted by their relative merit, ensuring that the best chromosomes are more likely to be chosen while maintaining diversity. The process of exchanging genetic information between parent chromosomes is random in nature. Unlike crossover, portraying natural reproduction, the mutation exhibits a sparse process that can produce sudden, exceptional offspring. It involves randomly selecting a chromosome from the population and performing arbitrary evolvements to its information. As such, the mutation process can break any stagnation in the evolutionary process and avoid local minimums.\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, assuming that two parents have been elected for engendering two offspring, the crossover operator partitions the parents' sub-strings into two distinctive groups reliant on construction methods and substitutions the genes within each group sub-strings. As a result, the children derive the sequence of construction methods from both parents. Subsequently, these new children are translated into a feasible completion time and minimized total cost by decoding them.\u003c/p\u003e \u003cp\u003eAfter generating offspring using any of the available techniques, it endures a fitness evaluation. It can only be confined if it surpasses the other members of the population in terms of fitness. Typically, this cycle abides for numerous generations of offspring until a chromosome that represents the optimal solution is found. The current deployment allows the user to determine the number of generations of offspring as a discontinuance benchmark for the process. In addition, it is feasible for the user to halt the genetic operators when a satisfactory solution that meets the required criteria for the total project cost and time of completion is attained. In such a scenario, the output will present the total linear repetitive project completion time, total project cost, and corresponding construction methods. Conversely, if the desired solution is not achieved, the algorithm will persist in probing for improved output.\u003c/p\u003e \u003cp\u003eThe GA approach for LRPTCT is implemented using an optimization engine platform named Evolver TM Version 7. Evolver Palisade uses a GA approach to optimization, which involves using a population of potential solutions to a problem that undergoes a series of genetic operations, such as selection, crossover, and mutation, to generate new and potentially better solutions. The process is repeated over several generations until an optimal or near-optimal solution is obtained.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Particle Swarm Optimization (PSO) workflow for LRPTCT\u003c/h2\u003e \u003cp\u003ePSO represents a heuristic optimization technique inspired by the social behavior of organisms such as bird flocking and fish schooling. This method aims to find optimal solutions by mimicking the collaborative behavior of a swarm of particles searching for the best outcomes. In the context of LRPTCT, PSO provides a structured framework for minimizing project costs and durations by iteratively improving candidate solutions through collaboration and individual learning.\u003c/p\u003e \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e \u003ch2\u003e4.3.1 Encoding the particle and objective functions\u003c/h2\u003e \u003cp\u003eIn the PSO framework (see Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), each particle represents a potential solution to the optimization problem. A particle is encoded as an array of decision variables, where each variable corresponds to the construction method selected for a repetitive task. The decision variables take integer values between 1 and 4, representing four predefined construction methods for each task. A particle\u0026rsquo;s position in the solution space denotes a unique configuration of selected methods, which is evaluated based on fitness metrics. These metrics account for the total direct costs, indirect costs, and project duration associated with the selected methods.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe PSO model for LRPTCT employs two objective functions. The first objective is to minimize the total project duration (F1), which is governed by the sequencing and dependencies of tasks, with critical tasks determining the overall duration. The second objective is to minimize the total project costs (F2), comprising Total Direct Costs (TDC) and Total Indirect Costs (TIC). TDC is derived from the costs of the construction methods chosen for each task, while TIC is calculated by multiplying the project duration by the daily overhead rate. These objective functions ensure that the PSO model seeks an optimal trade-off between minimizing time and costs.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section3\"\u003e \u003ch2\u003e4.3.2 PSO optimization workflow\u003c/h2\u003e \u003cp\u003eThe PSO process for LRPTCT is structured into iterative steps, illustrating the PSO workflow. The workflow of the PSO framework begins with the initialization of a swarm of particles. Each particle is assigned a random position within the solution space, representing an initial configuration of construction methods. The initial velocities of the particles are set to zero. The fitness of each particle is evaluated using the objective functions, and the best-known positions for each particle (p_best) and the global best position (g_best) among all particles are recorded.\u003c/p\u003e \u003cp\u003eThe iterative optimization process involves updating the velocity and position of each particle. The velocity update (Eq.\u0026nbsp;7) incorporates three components: the particle\u0026rsquo;s inertia, its cognitive tendency to return to its best-known position, and its social tendency to move toward the global best position. These updates are influenced by inertia weight (ω), cognitive coefficient (c1​), social coefficient (c2​), and random factors (r1​ and r2​) that add stochasticity to the search. After updating velocities, the positions of particles are adjusted to reflect the new configurations of construction methods. The positions are constrained within the valid range of decision variables, ensuring that all selected methods are feasible.\u003c/p\u003e \u003cp\u003eV\u003csub\u003ei,t+1\u003c/sub\u003e​=ωv\u003csub\u003ei,t​+\u003c/sub\u003ec\u003csub\u003e1\u003c/sub\u003e​r\u003csub\u003e1\u003c/sub\u003e​(p\u003csub\u003ebest\u003c/sub\u003e​\u0026minus;p\u003csub\u003ei,t\u003c/sub\u003e​)\u0026thinsp;+\u0026thinsp;c\u003csub\u003e2\u003c/sub\u003e​r\u003csub\u003e2\u003c/sub\u003e​(g\u003csub\u003ebest\u003c/sub\u003e​\u0026minus;p\u003csub\u003ei,t\u003c/sub\u003e​) (7)\u003c/p\u003e \u003cp\u003eFitness values are recalculated for all particles after each update. If a particle\u0026rsquo;s new position improves upon its previous best-known position, its p_best is updated. Similarly, if a particle achieves a global improvement, g_best is updated. This iterative process continues until a stopping criterion is met, such as reaching a maximum number of iterations or achieving convergence to an optimal solution.\u003c/p\u003e \u003cp\u003eThe PSO framework incorporates constraints to ensure practical and feasible solutions. Decision variables must remain integers between 1 and 4, representing valid construction methods. The optimized total project cost must be less than the planned budget, and the total project duration must not exceed contractual deadlines. Additionally, task dependencies and logical sequences are maintained throughout the optimization process to ensure constructability.\u003c/p\u003e \u003cp\u003eUpon completion, the PSO model outputs the optimal construction methods for all repetitive tasks, the minimized total project costs, the shortest feasible project duration, and detailed task schedules, including start and finish times. This comprehensive output supports decision-making in construction project management by providing actionable insights into cost and time optimization.\u003c/p\u003e \u003cp\u003eThe PSO framework demonstrates several advantages in addressing LRPTCT problems. It is highly scalable, accommodating projects with numerous tasks and complex dependencies. The algorithm\u0026rsquo;s iterative nature enables efficient convergence to optimal solutions, balancing exploration and exploitation to avoid local minima. Furthermore, PSO\u0026rsquo;s flexibility allows it to handle multi-objective optimization scenarios, making it suitable for simultaneously minimizing costs and durations.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"5 Developed approach implementation","content":"\u003cp\u003eTo evaluate the effectiveness, reliability, and validity of the developed approaches, a case study was conducted using a four-kilometer pipeline assembly as an example, as shown in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The developed methods are particularly suitable for horizontal infrastructure projects. The project consists of four parallel repetitive pipelines, each spanning 4 km, resulting in a total pipeline installation length of 16 km. The project's precedence relations follow a Finish to Start (FS) sequence without any lag time. The implementation process of the proposed approaches in the actual case study is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. Furthermore, the case study will include a comparative analysis between the proposed approaches and the traditional CPM integrated with LOB in order to determine their overall performance in terms of project scheduling and cost management. This comparative analysis will provide a comprehensive understanding of the strengths of each approach.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"6 Results and discussion","content":"\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e6.1 The conventional method for tackling the linear project's scheduling\u003c/h2\u003e \u003cp\u003eScheduling the pipeline (4 km pipeline) for one unit unveiled eight critical tasks (A, B, C, D, F, G, K, and L) and four non-critical tasks (E, H, I, and J). According to the CPM results, the project's duration for one unit is estimated to be 595 days, as rendered in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. In Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, ES depicts the early start, EF is the early finish, LS is the late start, LF represents the late finish, and TF is the total float.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eConventional CPM calculations for a single unit.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"10\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eTasks\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eDuration (Days)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003ePredecessors\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSuccessors\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eES\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eEF\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eLS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eLF\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eTF\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCode\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eA\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eB\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3,8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4,5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eD\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e140\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eE\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e245\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e161\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eF\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e252\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4,5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e525\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e525\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eG\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e525\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e553\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e525\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e553\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eH\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e343\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e399\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e322\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eI\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e119\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e196\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e399\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e518\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e322\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eJ\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e196\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e231\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e518\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e553\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e322\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eK\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7,10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e553\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e567\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e553\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e567\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eL\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eFN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e567\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e595\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e567\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e595\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e595\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e595\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e595\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e595\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e6.2 The devised method for addressing the linear project's scheduling\u003c/h2\u003e \u003cp\u003eIn order to more accurately reflect practicability in linear repetitive construction projects, the tasks previously mentioned were divided into separate sub-tasks of equal length. Each sub-task was treated as a flexible task with a length of 1 kilometer and a duration equal to the total duration of the original task divided by 4. This paradigm was designed to establish a relationship between the previously separate tasks, unearthing more feasible completion times. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e displays the results of the case study after splitting tasks C, D, E, F, and G into four equal 1-kilometer sub-tasks. The reduction of single unit duration is evident. The total duration decreased by 210 days (depicting a 35% reduction) from 595 days, computed using the traditional CPM method, to 385 days after the tasks were split, as explicitly portrayed in Tables\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and \u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Apropos of Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the division of tasks C, D, and G revealed the presence of partially non-critical sub-tasks. Specifically, only C1 is rendered as a critical task within task C, whereas non-critical tasks represent C2, C3, and C4. Likewise, D1 is the sole critical sub-task within task D, while D2, D3, and D4 are categorized as non-critical. Finally, in task G, only G4 is considered critical, while G1, G2, and G3 are classified as non-critical sub-tasks.\u003c/p\u003e \u003cp\u003eTo compute the required number of crews for each sub-task, the computation of the desired delivery rate (Rd) for the repetitive units is initially undertaken using Eq.\u0026nbsp;8. In Eq.\u0026nbsp;8, variables are assigned as follows: n signifies the number of repetitive units, TL denotes the project's deadline duration as stipulated in the contract agreement, T1 depicts the CPM duration of the first unit, and Tfi embodies the total float of each sub-task. Subsequently, the estimation of the number of crews required to perform each sub-task efficiently is attained by employing Eq.\u0026nbsp;9. In Eq.\u0026nbsp;8, Ci donates the number of crews assigned to each sub-task (i), and Ri embodies the desired delivery rate for each sub-task (i). Therefore, the number of crews entailed by each sub-task may vary depending on each sub-task's Rd, Tfi, and designated duration.\u003c/p\u003e \u003cp\u003eRi = (n \u0026ndash; 1) / (TL - T1)\u0026thinsp;+\u0026thinsp;Tfi (7)\u003c/p\u003e \u003cp\u003eCi\u0026thinsp;=\u0026thinsp;Di x Ri (8)\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eScheduling calculations for a single unit, considering splitting tasks into sub-tasks.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"10\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eTasks\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eDuration (Days)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003ePredecessors\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSuccessors\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eES\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eEF\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eLS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eLF\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eTF\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCode\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eA\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eB\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3,23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC1\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4,7,11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC2\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5,8,12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC3\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e6,9,13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e105\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e175\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e70\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eC4\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e10,14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e105\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e238\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e105\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eD1\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e8,15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eD2\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4,7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e9,16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e119\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eD3\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5,8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e10,17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e119\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e154\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e175\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e56\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eD4\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e6,9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e154\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e189\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e238\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e84\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eE1\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e12,15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eE2\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4,11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e13,16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e105\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e119\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e42\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eE3\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5,12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e14,17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e105\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e182\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eE4\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e6,13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e161\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e245\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eF1\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7,11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e16,19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eF2\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e8,12,15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e17,20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eF3\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e9,13,16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e18,21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eF4\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e10,14,17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e336\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e336\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eG1\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e154\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e315\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e322\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e168\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eG2\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e16,19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e210\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e217\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e322\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e329\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eG3\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e17,20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e280\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e329\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e336\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e56\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eG4\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e18,21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e336\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e343\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e336\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e343\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eH\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e24\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e133\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e189\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eI\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e24\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e119\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e196\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e189\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e308\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eJ\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e24\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e196\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e231\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e308\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e343\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e112\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eK\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e22,25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e343\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e357\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e343\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e357\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eL\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eFN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e357\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e385\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e357\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e385\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e385\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e385\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e385\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e385\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eUsing traditional CPM calculation, rendering the four km pipeline assembly using the Line Of Balance (LOB) scheduling method for the repetitive units unveiled the linear repetitive project's duration as 868 days. A, B, C, D, F, G, K, and L donate the critical linear repetitive tasks, whereas E, H, I, and J portray the non-critical repetitive tasks, as exhibited in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe data portrayed in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e corroborates the outcome of implementing the LOB scheduling method for repetitive units in rendering the 4 km pipeline installation. This was achieved by dividing tasks C, D, E, F, and G into four equal sub-tasks of 1 kilometer each. Consequently, the linear repetitive project was lessened, reducing the total duration from 868 days using the traditional LOB-integrated CPM method to 693 days (exhibiting nearly a 20% reduction). The critical repetitive tasks were identified as A, B, C1, D1, F1, F2, F3, F4, G4, K, and L, whereas the non-critical repetitive tasks were C2, C3, C4, D2, D3, D4, E1, E2, E3, E4, G1, G2, G3, H, I, and J. Accordingly, the approach of considering non-critical sub-tasks in linear repetitive projects is vital for achieving a balance between minimizing project costs and time. These non-critical sub-tasks, including C2, C3, C4, D2, D3, D4, G1, G2, and G3, can reduce the number of employed crews and relax the activity production rate, thereby empowering the prospect of optimizing the LRPTCT. It is worth mentioning that non-critical activities signify an essential role in scheduling repetitive projects, contributing to the overall efficiency and successful project delivery. For instance, non-critical activities provide opportunities to employ accessible resources effectively. Hence, when critical activities are not consuming all resources, crews can be assigned to non-critical activities, maximizing their productivity and avoiding idle time.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFurther, Non-critical activities maintain a continuous workflow, particularly during delays or disruptions in critical activities. The project can progress steadily by assigning crews to non-critical tasks even if certain critical activities are behind schedule, promoting a smooth flow of work and preventing unnecessary downtime. Accordingly, this allows project managers to respond effectively to unexpected circumstances and maintain project momentum.\u003c/p\u003e \u003cp\u003eThe LOB diagram depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e exhibits a distinctively consecutive progression of activities, whereby each step constitutes a direct connection with the subsequent one, devoid of interruptions or temporal intervals. This discernible attribute prominently stems from the underlying premise of zero buffer time among the activities, guaranteeing an expeditious execution of the linear schedule. However, it is crucial to highlight that incorporating buffer time between activities can constitute a valuable scheme to accommodate unforeseen delays or variations in scheduling linear projects.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e6.3 Deployment of the GA approach\u003c/h2\u003e \u003cp\u003eAfter the completion of the GA model, an initial evaluation was carried out to identify appropriate GA parameter values such as population size and the number of generations. Apropos of the research conducted by Roeva et al. (\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2013\u003c/span\u003e) and Anvari et al. (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2016\u003c/span\u003e), a population size of 100 and 1000 generations was deemed a satisfactory trade-off among diverseness and processing time. For lessening computational load, a set of algorithm attribute values was presented, encompassing 1) The iteration rate is 60%, 2) the crossover likelihood is 26%, 3) the mutation likelihood is 10%, and 4) the migration likelihood is 9%. The chromosome comprises strings whose length is 27, portraying the number of repetitive sub-tasks in the linear repetitive project. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e exhibited the cost and duration information for the construction methods of repetitive tasks and sub-tasks. The intended direct cost of the 27 repetitive tasks was deployed as the initial cost, which led to a total cost of 45,617,920 EGP and a completion period of 693 days for all units. According to the project\u0026rsquo;s contractual agreements, the agreed-upon indirect cost is estimated as a daily fixed value of 3,000 EGP, which is typically discerned by breaking down the various components of indirect costs, such as site overhead and general overhead. However, the indirect cost fixed value can be varied banking on multiple factors, including the nature of the project, industry practices, and the specific agreement between the owner and the contractor. Consequently, the project's total cost is approximated at 56,621,920 EGP.\u003c/p\u003e \u003cp\u003eTo this end, the inputs to the GA model are 1) Each repetitive task code and name, 2) scheduling data, including predecessors and successors of each repetitive task, 3) different alternatives of construction methods, incorporating each method's direct cost and designated duration, 4) the project defined constraints as previously outlines, and 5) the number of repetitive units and daily indirect cost for each repetitive activity, as displayed in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eConstruction methods alternatives depicting the GA model's decision variables.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"9\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eConstruction Method #\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c3\" namest=\"c2\"\u003e \u003cp\u003eMethod #1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003eMethod #2\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e \u003cp\u003eMethod #3\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c9\" namest=\"c8\"\u003e \u003cp\u003eMethod #4\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRepetitive Tasks\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDuration (Days)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCost (EGP)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eDuration (Days)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eCost (EGP)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eDuration (Days)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eCost (EGP)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eDuration (Days)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003eCost (EGP)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eA\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e40,400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e60,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e65,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e53,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eB\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e18,550\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e50,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e35,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e25,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eEach Sub-task C\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e137,200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e200,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e170,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e150,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eEach Sub-task D\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e353,500\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e380,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e450,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e390,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eEach Sub-task E\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e594,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e510,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e680,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e520,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eEach Sub-task F\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1,063,970\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1,120,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1,350,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1,250,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eEach Sub-task G\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8,400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e20,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e12,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e10,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eH\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2,003,700\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2,750,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1,750,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2,150,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eI\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e119\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e499,850\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e102\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e430,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e85\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e580,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e90\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e620,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eJ\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e174,500\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e220,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e200,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e150,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eK\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e26,600\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e40,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e20,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e45,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eL\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e12,600\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e16,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e10,000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e18,000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eAccording to the data presented, the proposed model underwent 100,000 trials, of which 77,820 were deemed valid. The computations and running time of the model were performed using an Intel Core i7 computer and lasted approximately 10 minutes. After running the optimization engine, the GA-proposed approach results revealed a reduction in the linear project's Total Direct Costs (TDC), Total Indirect Costs (TIC), and Total Construction Cost (TCC) apropos of one linear unit from 11,404,480 EGP to 11,033,880 EGP, from 2,079,000 EGP to 1,662,000 EGP and from 14,155,480 EGP to 13,208,880 EGP, respectively (as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e). Corrpsonigly, the GA-proposed approach unearthed a reduction in the linear repetitive project TDC, TIC, and TCC apropos of the four linear repetitive units from 45,617,920 EGP to 44,135,520 EGP, from 8,316,000 EGP to 6,648,000 EGP, and from 56,621,920 EGP to 52,835,520 EGP, progressively, as rendered in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e. The presented findings indicate that the GA-proposed approach successfully decreased the linear repetitive project's TDC, TIC, and TCC by approximately 3.25%, 20%, and 7%, respectively.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eRegarding the time optimization, the GA-proposed approach results revealed a reduction in the linear repetitive project's completion time pertinent to one unit and four units from 385 to 325 and from 693 to 554, enlightening approximately 16% and 20% reduction, respectively. Figure\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e portrays a comparative analysis between the developed GA with the base case apropos of the duration of each repetitive task. The stupendous reduction in TDC, TIC, TCC, and completion time of the linear repetitive project corroborate and support the developed approach, underperforming a tremendous reduction performance. These reductions are attributable to the proposed approach of exhibiting the repetitive tasks as a series of distinct sub-tasks interconnected jointly to render the overall duration of the task. This approach reckoned with depicting any logical connection between the decomposed tasks, showing more methodical schedules, and portraying non-critical sub-tasks. These non-critical sub-tasks in linear repetitive projects are vital for balancing project costs and time, thus, empowering the prospect of optimizing the LRPTCT. This hypothesis can be explicitly depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e, rendering an immense reduction in non-critical sub-tasks duration due to deploying their total floats constituted from decomposing the repetitive tasks.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e renders the optimum trade-off optimization between the total project's duration and construction cost attained from the GA trials, unveiling the number of trials carried out by the GA platform to engender the optimum outcome. It can be inferred from Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e that the best trial number to attain the optimum results was 2,165 trials. Ultimately, Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e exhibits the LOB depiction after deploying the proposed GA paradigm, unearthing the reduction in the linear repetitive project's completion time pertinent to the four repetitive units from 693 to 554 (nearly 20% reduction in the completion time).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e6.4 Deployment of the PSO approach\u003c/h2\u003e \u003cp\u003eThe deployment of the Particle Swarm Optimization (PSO) algorithm for optimizing linear repetitive project scheduling involved a rigorous evaluation of model parameters to ensure a balance between computational efficiency and solution quality. Based on established heuristic optimization methodologies, the PSO algorithm was configured with the following parameters: 1) Population Size: 30 particles representing potential solutions, each encoding the method selection for all repetitive tasks; 2) Number of Iterations: 100, ensuring sufficient exploration of the solution space; 3) Inertia Weight (ω): Set to 0.5, to balance exploration (searching for new solutions) and exploitation (refining existing solutions); 4) Cognitive Coefficient (c1​): 2, to emphasize a particle's self-learning from its own best-known solution, 5) Social Coefficient (c2c​): 2, to promote collaboration among particles by gravitating toward the global best solution.\u003c/p\u003e \u003cp\u003eEach particle's position represented a unique configuration of selected methods for 27 repetitive tasks. These tasks were derived from a linear repetitive construction project, and their respective costs and durations were provided across four construction method alternatives. During the initial configuration, the project incurred a Total Direct Cost (TDC) of 45,617,920 EGP and required 693 days for completion. Indirect costs were fixed at 3,000 EGP per day, leading to a Total Construction Cost (TCC) of 56,621,920 EGP. These values formed the baseline for evaluating the PSO model's performance.\u003c/p\u003e \u003cp\u003eThe input data required for the PSO model included: 1) Repetitive Task Details: Each task's identifier and its precedence relationships with other tasks, ensuring logical task sequencing; 2) Construction Method Alternatives: Four potential methods for each task, each defined by its associated direct cost and duration; 3) Project Constraints: Included adherence to contractual deadlines, logical dependencies, and resource availability; 4) Daily Indirect Costs: Fixed at 3,000 EGP per day, contributing to overall project overheads, and 5) Number of Repetitive Units: The analysis considered four units of the repetitive project.\u003c/p\u003e \u003cp\u003eThe cost optimization results of the PSO model demonstrated significant reductions across all major cost categories, including TDC, TIC, and TCC. For a single linear repetitive project unit, the Total Direct Cost before optimization was 11,404,480 EGP, which reflected the baseline selection of construction methods without consideration for optimization. After applying the PSO model, the Total Direct Cost was reduced to 10,947,910 EGP, achieving a 4.0% reduction. This improvement was primarily attributed to the selection of cost-efficient construction methods for tasks and sub-tasks while maintaining adherence to project constraints and logical dependencies, as rendered in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFor the entire repetitive project consisting of four units, the Total Direct Cost initially stood at 45,617,920 EGP. After optimization, it was reduced to 43,791,640 EGP, which also represented a 4.0% reduction. These consistent reductions across single and multi-unit analyses underscore the scalability of the PSO model in managing repetitive project costs effectively. By prioritizing methods with lower costs, the algorithm ensured that direct costs were minimized without negatively impacting other project parameters, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eThe TIC, which depends on the project duration and daily overhead rates, exhibited a much larger reduction compared to the TDC. Before optimization, the TIC for one unit was 2,079,000 EGP, derived from a baseline project duration of 385 days and a daily indirect cost rate of 3,000 EGP. After optimization, the TIC was reduced to 1,662,000 EGP, reflecting a substantial 20.0% reduction. This reduction is directly linked to the model's ability to shorten project duration, as indirect costs are proportional to time. For four repetitive units, the TIC was reduced from 8,316,000 EGP to 6,648,000 EGP, maintaining the same 20.0% reduction rate (see Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e). The efficiency of the PSO algorithm in compressing task durations while maintaining logical sequencing contributed significantly to these reductions.\u003c/p\u003e \u003cp\u003eThe combined impact of TDC and TIC optimizations resulted in notable reductions in the TCC. For one unit, the TCC decreased from 14,155,480 EGP before optimization to 12,609,910 EGP after optimization, representing a 10.9% reduction. Similarly, for four units, the TCC decreased from 56,621,920 EGP to 50,439,640 EGP, also reflecting a 10.9% reduction. The dual emphasis of the PSO model on minimizing direct costs and reducing project duration proved effective in achieving these comprehensive cost savings. Figure\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e provides a comparative illustration of these reductions, clearly depicting the efficiency gains realized through the PSO model.\u003c/p\u003e \u003cp\u003eThe Particle Swarm Optimization model also demonstrated substantial improvements in project completion time for both single-unit and multi-unit analyses. For a single linear repetitive project unit, the baseline completion time was 385 days. After optimization, this was reduced to 315 days, reflecting an 18.2% reduction. This improvement was achieved by prioritizing faster construction methods for critical tasks while strategically assigning float times to non-critical sub-tasks. The decomposition of repetitive tasks into sub-tasks enabled the PSO model to allocate resources more effectively, ensuring that time reductions were achieved without compromising project quality or logical sequencing.\u003c/p\u003e \u003cp\u003eFor the multi-unit analysis involving four repetitive project units, the baseline completion time was 693 days. After applying the PSO model, the total completion time was reduced to 554 days, representing a 20.1% reduction. The larger time savings in the multi-unit scenario were due to the model's ability to optimize non-critical tasks across all units simultaneously, effectively minimizing idle times and ensuring efficient resource utilization. By leveraging float times, the PSO model significantly reduced the duration of non-critical tasks, which, while not directly impacting the critical path, contributed to a streamlined project timeline. These time savings are visually represented in Fig.\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e16\u003c/span\u003e, highlighting the reductions achieved for both critical and non-critical tasks.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe optimization of project durations directly influenced the reduction in Total Indirect Costs, as shorter project durations translate to lower overhead costs. This interdependence of cost and time optimization underscores the strength of the PSO model in addressing the multi-dimensional nature of construction project management. The ability to balance the trade-offs between cost minimization and time compression makes the PSO model a valuable tool for optimizing linear repetitive projects.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e6.5 Comparison of PSO and GA results\u003c/h2\u003e \u003cp\u003eThe PSO and GA models were both applied to optimize costs and durations for a linear repetitive project. The comparison below provides a detailed analysis of their respective performances. In terms of cost optimization, for a single unit of the project, TDC was reduced by both methods, with PSO achieving a final TDC of 10,947,910 EGP compared to GA's slightly higher TDC of 11,033,880 EGP. Both methods provided significant improvements over the baseline TDC of 11,404,480 EGP. For TIC, both PSO and GA demonstrated identical results, reducing the baseline TIC of 2,079,000 EGP to 1,662,000 EGP. Regarding the TCC, PSO outperformed GA by achieving a final TCC of 12,609,910 EGP, compared to GA\u0026rsquo;s TCC of 13,208,880 EGP, with both models achieving notable reductions from the baseline of 14,155,480 EGP (see Fig.\u0026nbsp;\u003cspan refid=\"Fig17\" class=\"InternalRef\"\u003e17\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFor the multi-unit analysis, which comprised four repetitive units, the results followed a similar trend. PSO reduced the TDC to 43,791,640 EGP, while GA resulted in a TDC of 44,135,520 EGP, both improving on the baseline of 45,617,920 EGP. The TIC for the multi-unit project was reduced from the baseline of 8,316,000 EGP to 6,648,000 EGP by both PSO and GA. The TCC achieved by PSO was lower at 50,439,640 EGP, compared to GA's TCC of 52,835,520 EGP, both representing significant savings over the baseline of 56,621,920 EGP.\u003c/p\u003e \u003cp\u003eIn terms of time optimization, PSO reduced the single-unit project completion time from 385 days to 315 days, representing an 18.2% reduction. GA achieved a similar reduction, bringing the project duration down to 325 days. For the multi-unit scenario, PSO reduced the project completion time from 693 days to 554 days, a 20.1% reduction. Similarly, GA reduced the completion time for four units to 554 days, matching PSO\u0026rsquo;s performance in this regard (Fig.\u0026nbsp;\u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e18\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe comparative analysis highlights that while both PSO and GA excel in optimizing costs and time, PSO consistently outperformed GA in terms of achieving lower Total Construction Costs for both single-unit and multi-unit projects. Both methods delivered equivalent reductions in Total Indirect Costs and project completion times, indicating similar performance in addressing the time dimension. These results demonstrate the robustness of PSO and GA in solving complex optimization problems, with PSO showing a slight edge in cost efficiency.\u003c/p\u003e \u003c/div\u003e"},{"header":"7 Conclusions and Prospective Work","content":"\u003cp\u003eThis research underscores the critical importance of optimizing time and cost in linear repetitive construction projects, which are often subject to complex interdependencies and resource constraints. The multi-objective nature of the TCT problem poses significant challenges to decision-makers, requiring advanced methodologies that integrate flexibility, efficiency, and accuracy in scheduling and resource allocation. This study proposes an innovative dual-optimization framework that merges GA and PSO with LOB methodologies, offering a transformative solution to the limitations of traditional approaches.\u003c/p\u003e \u003cp\u003eThe proposed framework redefines repetitive tasks by decomposing them into interconnected sub-tasks, enabling more nuanced scheduling and resource allocation. This decomposition addresses the oversimplifications inherent in conventional LOB methods, which often treat tasks as indivisible units. The novel approach ensures systematic scheduling, efficient crew allocation, and optimized resource utilization, significantly improving both project duration and cost efficiency.\u003c/p\u003e \u003cp\u003eThe research findings reveal substantial improvements in project performance. For single-unit projects, the proposed approach reduced the project duration from 385 days to 325 days, achieving a 16% reduction. For multi-unit projects involving four repetitive units, the total duration decreased from 693 days to 554 days, representing a 20% improvement. These reductions highlight the robustness of the framework in streamlining project schedules without compromising quality or constructability.\u003c/p\u003e \u003cp\u003eCost optimization was equally remarkable. For a single unit, TDC decreased from 11,404,480 EGP to 11,033,880 EGP, TIC dropped from 2,079,000 EGP to 1,662,000 EGP, and TCC declined from 14,155,480 EGP to 13,208,880 EGP, corresponding to reductions of 3.25%, 20%, and 7%, respectively. In the four-unit scenario, TDC fell from 45,617,920 EGP to 44,135,520 EGP, TIC from 8,316,000 EGP to 6,648,000 EGP, and TCC from 56,621,920 EGP to 52,835,520 EGP, achieving similar percentage savings. These results underscore the framework\u0026rsquo;s ability to address cost inefficiencies while maintaining adherence to project constraints.\u003c/p\u003e \u003cp\u003eA significant innovation of this study lies in its decomposition methodology, which uncovered partially critical sub-tasks. This approach enabled better utilization of task floats, reduced the criticality ratio from 67\u0026ndash;48%, and enhanced scheduling flexibility. The ability to balance critical and non-critical tasks effectively ensures a more resilient project schedule capable of adapting to unforeseen conditions and resource limitations.\u003c/p\u003e \u003cp\u003eThe comparative analysis between GA and PSO further enriches the study, highlighting the unique strengths of each algorithm. GA demonstrated superior robustness in exploring diverse solution spaces, ensuring comprehensive optimization across a wide range of scenarios. PSO, on the other hand, exhibited faster convergence and marginally better cost efficiency, achieving slightly lower Total Construction Costs compared to GA. Both algorithms proved highly effective in achieving the dual objectives of minimizing project duration and costs, offering valuable insights for selecting the appropriate optimization tool based on specific project requirements.\u003c/p\u003e \u003cp\u003eThis research not only addresses critical gaps in the literature but also sets the stage for future advancements in construction project management. Future studies could explore hybrid optimization approaches that combine GA, PSO, and other metaheuristic techniques to further enhance efficiency and scalability. Expanding the framework to include non-linear repetitive projects, as well as integrating additional optimization objectives such as quality, environmental sustainability, and risk management, would broaden its applicability. Furthermore, addressing the challenges posed by increased task granularity through advanced modeling techniques and incorporating learning curve dynamics into the scheduling process would enhance the framework\u0026rsquo;s precision and adaptability.\u003c/p\u003e \u003cp\u003eBy introducing this comprehensive dual-optimization framework, this research provides a cutting-edge decision-support tool that empowers practitioners to navigate the complexities of linear repetitive construction projects. The proposed methodology not only improves project scheduling and cost management but also fosters greater efficiency, adaptability, and resilience, contributing to a more sustainable and competitive construction industry.\u003c/p\u003e "},{"header":"Declarations","content":"\u003ch2\u003eFunding\u003c/h2\u003e \u003cp\u003eThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAhmed Gouda Mohamed, Ali Hassan Ali, and Ahmed Adel Abdelhady equally contributed to this research paper. Each author played a vital role in every aspect of the study. Ahmed Gouda Mohamed took the lead in conceptualizing the research framework and methodology, significantly shaping the theoretical foundation of the study. Ali Hassan Ali was instrumental in the data collection process and conducted rigorous data analysis, ensuring the validity and accuracy of the results. Ahmed Adel Abdelhady contributed extensively to drafting and refining the manuscript, ensuring clarity, coherence, and adherence to scientific standards. All authors collaborated in reviewing and revising the manuscript, offering critical insights to enhance its quality and finalize the submission.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThis manuscript does not report data generation or analysis. Ahmed Gouda Mohamed will be responsible if someone wants to request the data from this study.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAli, A. H. et al. A hybrid model for assessing safety implementation and project success in the construction industry. \u003cem\u003eAlexandria Eng. 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Manag Eng.\u003c/em\u003e \u003cb\u003e34\u003c/b\u003e, 1\u0026ndash;12. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1061/(asce)me.1943-5479.0000565\u003c/span\u003e\u003cspan address=\"10.1061/(asce)me.1943-5479.0000565\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e (2018).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Time–cost trade-off, Genetic Algorithm, Linear Repetitive Projects, Line of Balance, Construction Project Optimization, Particle Swarm Optimization","lastPublishedDoi":"10.21203/rs.3.rs-5757308/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5757308/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe primary aim of this paper is to minimize the overall completion time of linear repetitive projects while simultaneously reducing direct and indirect costs by employing predefined construction methods. To address these challenges, the study proposes a dual-optimization framework that integrates Genetic Algorithms (GA) and Particle Swarm Optimization (PSO). The methodology involves decomposing repetitive tasks into sub-tasks, enabling a more detailed and feasible Time-Cost Trade-off (TCT) analysis, which is further combined with the Line of Balance (LOB) methodology. A comparative analysis between GA and PSO highlights their respective strengths and effectiveness, an area previously underexplored in the literature. The findings reveal that the proposed framework significantly reduces costs and project durations. The GA approach achieves reductions of approximately 3.25% in direct costs, 20% in indirect costs, and 7% in total construction costs, while PSO demonstrates slightly better cost efficiency with a 4% reduction in direct costs and similar reductions in indirect costs. Both methods deliver a 20% reduction in project completion time, showcasing their effectiveness in streamlining construction processes. This paper contributes to the field by presenting the GA-PSO-based Linear Repetitive Project Time-Cost Trade-off (LRPTCT) model, integrating TCT with LOB, and offering a novel solution to the complexities of linear repetitive projects. Furthermore, its comparative analysis between GA and PSO provides valuable insights for optimizing construction project management practices.\u003c/p\u003e","manuscriptTitle":"An Integrated Decision Support System for Optimizing Time-Cost Trade-offs in Linear Repetitive Projects","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-01-09 19:37:51","doi":"10.21203/rs.3.rs-5757308/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-02-10T05:50:07+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-02-03T08:28:24+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"251068051496326008160088606570880736166","date":"2025-02-03T08:27:31+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-01-28T03:49:03+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-01-22T01:54:10+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"125494776217098208109466699614627163513","date":"2025-01-20T16:14:18+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"338965435071453815704163926939197490110","date":"2025-01-16T01:44:12+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-01-15T14:59:05+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-01-15T14:56:51+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-01-07T10:01:14+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-01-07T09:54:58+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-01-03T09:50:45+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"39e6af43-b1b1-4bf4-9030-4dc25b1ed955","owner":[],"postedDate":"January 9th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":42485379,"name":"Biological sciences/Evolution"},{"id":42485380,"name":"Physical sciences/Engineering"},{"id":42485381,"name":"Physical sciences/Mathematics and computing"}],"tags":[],"updatedAt":"2025-06-23T16:01:21+00:00","versionOfRecord":{"articleIdentity":"rs-5757308","link":"https://doi.org/10.1038/s41598-025-02837-8","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-06-20 15:57:38","publishedOnDateReadable":"June 20th, 2025"},"versionCreatedAt":"2025-01-09 19:37:51","video":"","vorDoi":"10.1038/s41598-025-02837-8","vorDoiUrl":"https://doi.org/10.1038/s41598-025-02837-8","workflowStages":[]},"version":"v1","identity":"rs-5757308","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5757308","identity":"rs-5757308","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}