Challenges in Applying System Dynamics to Address Scoping and Estimating Problems

Abstract

System dynamics (SD) is a fundamental model that facilitates research by defining and modeling a project’s scope and estimation. Scoping and estimation is a set of approaches that ensure engineering managers that a project’s accuracy and cost estimation are properly specified and mapped. Engineering management’s primary project planning responsibilities include project scoping and estimating. Insufficient project scoping and estimation criteria significantly fail the project plan. The study authors collaborated with five distinctive journal institutions to address project scoping and estimation challenges. The authors thoroughly analyzed one core research issue and five secondary research questions from each article. In terms of core issues, most project managers dislike SD modeling because of its complexity, especially if they lacked engineering, programming, mathematical analysis, or IT expertise. The first secondary question revealed that project managers’ awareness of scoping and estimating problems ranges from $78.3333\%$ to $91.6666\%$. The secondary question revealed that $83.3333\%$ to $90\%$ of managers are familiar with SD, which can help resolve scoping and estimation issues in project management. The third secondary question showed that $83.3333\%$ to $91.6666\%$ of managers are aware of SD during the project’s scoping and estimating phases. The fourth secondary question found that $83.333\%$ to $91.6666\%$ of managers can use mathematical equations and programming for SD design during the scoping and estimating phases. The authors developed a novel SD scoping and estimating model since the last question indicated that there were no downloadable models from the five journals.

1. Introduction

Scoping and estimation are methods used by managers to ensure accurate project estimation and to prevent the failure of the project’s scope [1]. Businesses must assess the feasibility of their budgets before investing in a project plan. This process of assessing the feasibility is known as scoping [2,3,4,5,6]. Estimation, on the other hand, provides a comprehensive understanding of a project’s cost, time, and effort plan, aiding both the team and project managers in making informed decisions. Scoping and estimating are interconnected aspects of engineering management that cannot be performed separately.

When a project’s scope grows beyond its initial goals or objectives, this is known as scope creep [2]. Scope creep occurs when stakeholders request changes to the project. Adapting a project’s strategy can result in uncertainty, increased resource costs, estimation errors, and difficulties in meeting deadlines. Using traditional tools such as Gantt Chart, PERT, Kanban Board, Timeline, and Critical Path methods to modify scoping and estimation might affect deadlines for completion budgets and rejection of project changes that might trigger delays or scope creep [2,3,4,5,6].

To avoid scope creep, researchers have suggested that SD can be used to solve scoping and estimating challenges [6,7,8,9,10,11]. The suggestions have shown many advantages compared to traditional tools [11,12,13,14]. One disadvantage of SD was that most project managers disliked SD modeling because of its complexity in design, especially if managers lacked engineering, programming, mathematical analysis, or IT expertise [8,10,14].

Most managers indicated that SD modeling is complicated to design. SD modeling involves mathematical programming, logical decision-making, stock understanding, flaw handling, control elements, computation models, information feedback theory, computer science, causal loop diagrams, quantitative models, feedback loops, and delays. With all these complexities, project managers avoid using SD modeling to solve scoping and estimation issues [8,9,11,15,16,17,18].

Despite the complexity, SD has other advantages. SD analyzes interconnected chains of stocks over time, revealing positive and negative feedback loops that influence system actions and stocks [17]. SD can be a useful method for assessing theories about the causes and effects of opportunities or systemic problems [8]. By reducing time and effort, SD characteristics make it simpler to handle large, complex, interconnected structures—much like project management difficulties in general [17]. SD can be used to evaluate different scenarios and approaches for system innovation or development, and it can then share the information and recommendations with decision-makers and stakeholders [8,19,20,21,22,23,24,25,26,27].

The authors were motivated to address the issue of SD being avoided by project managers as a result of its potential complexity due to its high programming and mathematical skill level. In this study, the authors proposed a brand-new SD model to solve problems with scoping and estimating. Additionally, the authors structured the research as follows. Section 1.1 is a problem statement. Section 1.2 is research questions. Section 1.3 is the importance of the study. Section 2 is the literature review. Section 3 is the research methodology applied in this study. Section 4 is the material used to conduct the study. Section 5 is the results and discussion to adapt a novel SD model and lists all the findings. Section 7 is the conclusion to summarize the findings.

1.1. Problem Statement

There is a problem with scoping and estimation during project planning. For instance, inadequate scoping and estimating during the planning phase can lead to project failures [21]. While many project managers are aware of these challenges, the majority of them lack the requisite skills and tools to solve these problems. For instance, the requirement for programming expertise hinders the use of system dynamics (SD) tools to address these difficulties [22,23,24,25,26,27]. Despite the problem of demand for advanced skills, some managers are hesitant to utilize system dynamics (SD) to tackle these issues [19,22,27]. The authors of this study realized that there is a need to solve these challenges using SD. This research introduces a new SD model proposed by the authors to address scoping and estimation challenges. Refer to Figure 1 and Figure 2 for more details.

1.2. Research Questions

Two research questions are the cruxes for this study—namely:

• Why has SD been little used to solve the scoping and estimating problem? The first question is derived based on the following statement from Section 1, paragraph 3, sentence 3: “Using traditional tools such as Gantt Chart, PERT, Kanban Board, Timeline, and Critical Path methods to modify scoping and estimation might affect deadlines for completion budgets and rejection of project changes that might trigger delays or scope creep [2,3,4,5,6]”.
• Is it possible to design a novel SD model using mathematical equations and properly define each function so that project managers would be able to understand? The second question is derived based on the following statement from Section 1, paragraph 4: “Most managers indicated that SD modeling is complicated to design. SD modeling involves mathematical programming, logical decision-making, stock understanding, flaw handling, control elements, computation models, information feedback theory, computer science, causal loop diagrams, quantitative models, feedback loops, and delays. With all these complexities, the project managers avoid using SD modeling to solve scoping and estimation issues [8,9,11,15,16,17,18]”.

The main focus of the study is on the two central questions mentioned above. The secondary questions below support the two central (core) questions listed above. The secondary questions are used for the flexibility of the research and to broaden the understanding of the scoping and estimating issues. The authors used these secondary questions to properly conduct the research and access more material, resulting in detailed answers to the two core questions as well as additional solutions to scoping and estimating issues. The secondary questions are:

• Do project managers have challenges with scoping and estimating issues during project management?
• Are managers familiar with SD, which can help solve scoping and estimating issues during project management?
• Are managers informed about SD during the scoping and estimating phases of project management?
• Can managers use mathematical equations and programming to design and program SD during the scoping and estimating phases?
• Is there any SD available (for example, from Google, journals, or compact discs) for project managers to solve scoping and estimating issues?
• Can the authors develop and implement mathematical equations for the SD model to empower project managers to overcome scoping and estimating challenges?

1.3. Importance of the Study

After the study, the manager and community will benefit from the following underpinned results of this research and work:

• Managers will be able to use the SD model instead of traditional tools such as Gantt Chart, PERT, Kanban Board, Timeline, and Critical Path methods to modify scoping and estimation, which may affect completion budget deadlines and rejection of project changes, resulting in delays or scope creep.
• Most managers will be able to create their SD model without worrying about the intricacies. It was previously stated that SD modeling is difficult to design. SD modeling entails mathematical programming, logical decision-making, stock analysis, fault management, control elements, computation models, information feedback theory, computer science, causal loop diagrams, quantitative models, feedback loops, and delays. Due to the difficulties, project managers avoid adopting SD modeling to handle scope and estimate issues. After this study, managers will acquire a good understanding of the scientific procedures needed to design an SD model.

2. Literature Review

Sadek [28] claims that project managers struggle to keep up performance in scoping and estimating. Project management, according to Tiruvengadam, Elizondo-Noriega, and Beruvides [29], offers a framework for managing organizational resources, which includes estimation and scoping. But practitioners usually start projects unaware of the cost excesses and schedule delays brought on by poor estimation and scoping [29]. According to Nakhleh [30], project failure rates remain mostly influenced by scoping and estimating, despite the growing importance of project management.

Sadek [28] researched scoping and estimating models using system dynamics. The necessity of scoping and estimating models was contextualized by the estimated information value [28]. In the United Arab Emirates, modeling was necessary to match the future vision and strategies [28]. Sadek [28] attempted to simulate the entire project lifecycle, scoping, and estimating of residential structures in the UAE during the preconstruction stage using the VENSIM system dynamics technique. The goal was to simulate dynamic scoping and estimating over time for every output [28]. A purely quantitative methodology was employed in that investigation [28]. Sadek [28] employed mathematical computation techniques and map-based applications. Sadek [28] used demo simulation and data verification modeling to ensure the accuracy of the results.

Sadek [28] used mean square error and mean absolute deviation to measure scoping and estimate accuracy. The study showed that the cost and cash flow of a residential development project in the United Arab Emirates may be precisely and dynamically projected over time using the VENSIM system dynamics technique. The work contributed significantly to theory, practice, and policy by offering the first scoping and estimating simulation for residential projects’ whole lifecycle utilizing the VENSIM system dynamics technique [28]. The results opened up opportunities in the areas of scoping and estimating models.

Sadek [28] confirmed that SD modeling needs more skill to construct. Sadek [28] intentionally avoided adding the demolition process in the SD due to mathematical complexity, many investigation processes, and various perspectives in simulating the entire project. Refer to Sadek [28], page 66. The above statement confirms the necessity for highly competent SD knowledge to manage a project utilizing SD. Hence, the authors provided this research.

System dynamics has been shown by Tiruvengadam, Elizondo-Noriega, and Beruvides [29] to be a powerful tool for modeling complex systems and the interdependencies of their components. Project managers can utilize it to identify intrinsic risks and the time-delayed implications of those risks, especially project-related expenses, so that capital can be spent more effectively. To investigate a specific facet of the extensive intersectionality between project managers and system dynamics (SD), a state-of-the-art (SAM) analysis of the extant literature was carried out, with the specific goal of evaluating only the literature with published system dynamics (SD) models. Opportunities and trends in this field of study were identified using this analysis [29].

Tiruvengadam, Elizondo-Noriega, and Beruvides [29] carefully collected and arranged the peer-reviewed papers in line with the PRISMA-P approach, which made it possible to produce descriptive statistics and a trend analysis. Subsequently, Tiruvengadam, Elizondo-Noriega, and Beruvides [29] conducted a state-of-the-art (SAM) analysis of the gathered articles. The preliminary results confirmed the growing use of system dynamics in project management, but they also highlighted the lack of realistic system dynamics (SD) models that solve project management issues, indicating the need for more study in this domain [29]. The concept of isomorphism and the similarity across systems with different origins were not employed by any of the studies that were discovered that had published system dynamics (SD) models to depict any system dynamics (SD) used for major projects [29].

Tiruvengadam, Elizondo-Noriega, and Beruvides [29] confirmed a limitation to SD simulations and expertise. Additionally, Tiruvengadam, Elizondo-Noriega, and Beruvides [29] indicated that there is a need for more research concerning SD modeling. Hence, the authors provided this study.

Odeh et al. [31] proposed a non-SD scoping and estimating model based on the diverse experiences and perspectives of practitioners. The model’s four elements (plan, review meetings, create, and rework/update) advised practitioners on scoping and estimating [31]. Odeh et al. [31] detailed specific activities that need to be finished for the project’s scoping and estimating inside each of those components. The first examination of the model showed how easy it is to use and understand. Furthermore, the model meets stakeholder expectations and effectively manages project estimating and scoping procedures [31].

Odeh et al. [31] confirmed the complexity of software (including SD). Odeh et al. [31] indicated that the company’s software maintenance and development practices have a significant impact on the quality of its software products. As a result, software development is currently confronting a considerable challenge: producing software of exceptional quality [31]. Hence, the authors provided this study.

Lane and Rouwette [32] investigated the relationship between behavioral ideas and system dynamics in detail. That was achieved by contrasting “sophisticated” and “naïve” methods of managing complex systems using a mind map and examples. Lane and Rouwette [32] presented a new and complete framework for SD-based interventions, including a full examination of behavioral consequences related to system dynamics and verification of their existence. Building on that, Lane and Rouwette [32] introduced the notion of “Behavioural system dynamics" (BehSD), which is based on the perspective on phenomena, five new constitutive axioms, and the possibility of improving practices. Lane and Rouwette [32] summarized the features and potential applications of “BehSD” in their conclusion and offered a framework for further study that adopts that perspective.

Lane and Rouwette [32] managed to construct SD modeling that worked according to expectations, but the simulation was not for scoping and estimation. As indicated in the problem statement of this study, there is a limitation to acquiring SD simulations about scoping and estimation simultaneously. Hence, the authors provided this study.

Calderon-Tellez et al. [33] looked at the contributions of system dynamics modeling to traditional project management practice. Calderon-Tellez et al. [33] provided evidence of the importance of this tactic for assisting with managerial decision-making. Key elements of that all-encompassing strategy were highlighted, and the study’s methodology was described. Calderon-Tellez et al. [33] found key explanatory structures (such as rework cycles) that were crucial to project delivery decisions after analyzing many of the published models. Furthermore, Calderon-Tellez et al. [33] investigated the extended project lifecycle and the application of the rework cycle to the process groups of the Project Management Institute. Calderon-Tellez et al. [33] acknowledged that the field was evolving. Consequently, possible future directions for system dynamics modeling in the context of project management were outlined [33]. Hence, the authors provided this study.

Calderon-Tellez et al. [33] managed to simulate SD, but the simulation was not for scoping and estimation; the SD was for the rework cycle. Additionally, Calderon-Tellez et al. [33] confirmed that SD modeling requires expertise. Refer to page 17 of Calderon-Tellez et al. [33]. The authors of this study are motivated to conduct the study due to the above information.

According to Bozzani et al. [34], given SD’s impact on the feasible impact and scale, health system constraints are becoming more widely acknowledged as a valuable complement to model-based evaluations of disease control measures. Prioritization decisions should take into account the possibility of additional scoping and estimating associated with enabling activities that are conducted in tandem with interventions to ease restrictions and achieve the desired coverage. Group model building is an interactive system dynamics simulation approach that Bozzani et al. [34] used to gather information from important stakeholders about the barriers that affect tuberculosis infection prevention and control procedures in South African primary healthcare clinics. Using that data, workable solutions were created, including the enablers required to remove current restrictions [34]. Then, at two clinics in KwaZulu–Natal, intervention and enabler scoping and estimates were developed using input prices and quantities from the published literature and local suppliers [34]. The least expensive of the recommended actions was retrofitting buildings for improved ventilation, which costs USD 1644 per year. Other lower-cost strategies included establishing appointment systems to reduce crowding (USD 9302) [34] and utilizing community sites for drug collection among stable patients undergoing antiretroviral therapy (ART) (USD 3753). Among the identified facilitators were enhanced staff training, supervision, and patient involvement programs that support behavioral change and local ownership [34].

According to Bozzani et al. [34], scoping and estimating were not feasible for several of the facilitators that the stakeholders had listed, such as obtaining building permissions or improving data, because they had shifted between health system tiers [34]. Notwithstanding this limitation, a cost-finding technique based on system dynamics modeling can be employed efficiently in economic evaluations to more accurately compute the potential cost of intervention choices in the “real world” [34]. By applying the technique to different types of interventions (for example, new diagnostics or preventive technologies), more empirical research might be able to identify interventions that do not have an impact on scoping and estimating in specific contexts regarding the amount of money needed to purchase enablers.

Nakhleh [30] states that two project components that could affect project performance are project scoping and estimation. Nakhleh conducted a correlational analysis [30]. The correlational study aimed to evaluate how scoping and estimating affect project performance [30]. The data-gathering approach involved 67 project sponsors, managers, and coordinators in Qatar [30]. The theoretical basis was the iron triangle, also referred to as the triple restriction [30]. Participants were randomly selected to answer eighteen questions using the project implementation profile instrument [30].

Using traditional multiple regression analysis, Nakhleh [30] examined the relationship between the independent and dependent variables. Nakhleh [30] reports that time and cost estimates and project performance have a significant linear correlation (F (2,63) = 24.57, p < 0.05, R = 0.66, R2 = 0.44, and adj. R2 = 0.42). The statistically verified findings of the study provided professionals and academics with a fine-grained understanding of the factors influencing a project’s performance. According to Nakhleh [30], a higher project performance rate could lead to social change by enhancing business performance, enhancing the sustainability of economies, enhancing local communities, enhancing life quality, opening up new business opportunities, and raising employment rates.

Becker and Nilsson [35] conducted a threefold study: (i) firstly, to define the characteristics of a large project; (ii) secondly, to identify factors causing inaccurate effort scoping and estimation; and (iii) finally, to understand how the identified factors impact the effort scoping and estimation process, all within the context of large-scale agile software development and from the perspective of a project team. Becker and Nilsson [35] conducted an exploratory case study to achieve the study’s aims. Data collection methods included archival research, surveys, and interviews [35]. Stata was used to do some of the data analysis [35]. According to the findings, a large project is defined by the project team as having significant complexity and a broad spectrum of needs [35]. The following issues were identified as affecting the estimation process in large projects: insufficient requirements, changes in complexity, impact in multiple areas, coordination, and necessary knowledge, with the findings indicating a negative impact on scoping and estimating accuracy. Becker and Nilsson [35] discovered that faulty effort estimates can be caused by various factors, such as requirements, complexity, coordination, input and estimation processes, management, and estimate usage.

Manenzhe, Telukdarie, and Munsamy [9] state that the complexity of employing SD in project management, scoping, and estimate challenges had led to unresolved concerns with specialist and mathematical programming for SD. As a result, many managers overlook SD [8,10,14]. Rumeser and Emsley [8] recognized three key issues regarding the complexity of SD modeling: model modification after involving stakeholders, explaining, and translating all explanations into mathematical equations for scoping and estimating models. According to Amin et al. [10], the complexity of applying SD results in a lack of quantitative models and understanding of dynamic capabilities, making it difficult to examine scoping and estimation problems. As a result, many managers overlook SD [8,10,14].

The Highway Knowledge Portal [36] suggested AASHTOWare Project Estimation: a web-based application that streamlined, simplified, and improved the organization of project-related data and project scoping and estimation management activities. According to the Highway Knowledge Portal [36], AASHTOWare Project Estimation offers various advantages over traditional scoping and estimation approaches like spreadsheets and AASHTOWare Estimator software (https://www.aashtoware.org/). According to the Highway Knowledge Portal [36], project managers rely on conventional techniques, especially when scoping and estimating are on short notice. However, automating the estimating process with AASHTOWare Project estimating does not guarantee that all errors will be reduced or consistency will be improved. To achieve success, project managers should always try to collect high-quality data for use in the scoping and estimating stages with AASHTOWare Project Estimation compared to SD simulation. Hence, the authors provided this study.

Becker and Nilsson [35] emphasized the need for effort scoping and estimating for good project planning and budget management. Despite much research on scoping and estimating, one of the most challenging aspects of project management in software development is the estimation process [35].

Out of all the statements in this literature review, little research has been conducted by a few academics and project managers to address scoping and estimation issues using SD. As a result, a new SD is necessary to solve the scope and estimation challenges. Hence, the authors provided this study.

3. Research Methodology

Three key steps were used in the research method. The steps involved in the research approach were as follows:

• Constructing a review of the literature using the following criteria:
  • Search for project scoping and estimating issues.
  • Search for why project managers mostly avoid SD modeling to solve project scoping and estimating issues.
  • Validate and confirm the problem statement in Section 1.1 using the authors’ material from items [a] and [b].
• Utilizing five project management journal organizations, compare the literature review. A total of 60 papers from each examined publication were referenced to gather information concerning estimation and scoping.
  • Gather information concerning estimation and scoping from five project management journal organizations.
  • Compare the literature review from 60 papers of project management journal organizations from Section 3[ii][a] with the materials from Section 3[i].
  • Gathering information on the main reasons why project managers fear using SD modeling to address project scoping and estimating problems was also important. A total of 60 ∗ 5 = 300 papers were gathered from the five project management journal institutes. It is important to highlight that 300 articles were excluded from the literature review section because they were utilized to confirm, support, and validate the study’s problem statement and literature review. The five journals that were utilized are:
    • Engineering Project Organization Journal: For information about the Engineering Project Organization Journal, refer to

      Table 1. The literature review collection: details of the literature search methodology.

      Institution Name for the Journal	Database or Resource	Keywords	Time Frame
      Engineering Project Organization Journal	1. Taylor Francis using the following link: https://www.tandfonline.com/ (accessed on 15 February 2024) 2. Google Scholar using the following link: https://scholar.google.com/ (accessed on 15 February 2024) 3. SciSpace using the following link: https://typeset.io/ (accessed on 15 February 2024)	1. Engineering Project Organization Journal 2. Scoping and estimation 3. System dynamics 4. Project management	From years 2019 to 2024
      Harvard Business Review	1. Harvard Business Review database using the following link: https://www.library.hbs.edu/ (accessed on 15 February 2024) 2. Harvard Business Review using the following link: https://hbr.org/ (accessed on 15 February 2024) 3. Kettering University Library using the following link: https://libguides.kettering.edu/ (accessed on 15 February 2024)	1. Harvard Business Review 2. Scoping and estimation 3. System dynamics 4. Project management	From years 2019 to 2024
      International Journal of Construction Project Management	1. Nova Science Publisher using the following link: https://novapublishers.com/ (accessed on 15 February 2024) 2. Taylor Francis using the following link: https://www.tandfonline.com/ (accessed on 15 February 2024)	1. International Journal of Construction Project Management 2. Scoping and estimation 3. System dynamics 4. Project management	From years 2019 to 2024
      International Journal of Managing Projects in Business	1. Emerald Publishing using the following link: https://www.emeraldgrouppublishing.com/ (accessed on 15 February 2024) 2. Google Scholar using the following link: https://scholar.google.com/ (accessed on 15 February 2024)	1. International Journal of Managing Projects in Business 2. Scoping and estimation 3. System dynamics 4. Project management	From years 2019 to 2024
      Journal of Engineering and Technology Management	1. Science Direct using the following link: https://www.sciencedirect.com/ (accessed on 15 February 2024) 2. Scimago Journal using the following link: https://www.scimagojr.com/ (accessed on 15 February 2024)	1. Journal of Engineering and Technology Management 2. Scoping and estimation 3. System dynamics 4. Project management	From years 2019 to 2024

      .
    • Harvard Business Review: For information about Harvard Business Review, refer to Table 1.
    • International Journal of Construction Project Management: For more information about the International Journal of Construction Project Management, refer to Table 1.
    • International Journal of Managing Projects in Business: For more information about the International Journal of Managing Projects in Business, refer to Table 1.
    • Journal of Engineering and Technology Management: For more information about the Journal of Engineering and Technology Management, refer to Table 1.

We analyzed and summarized the results of the literature review on scoping and estimating issues. Using the results of the analysis, we formulated a statement that either confirms or refutes the problem statement. We proposed a novel SD model that tackles scoping and estimation challenges.

3.1. Reasons for Choosing Five Publications

Only five journal institutions about scopes and estimations were studied, with the remaining journals left for other researchers. The rationale for choosing these five journals was that they are among those that specialize in project management and system dynamics, and the study could not include them all. Regarding scoping and estimation issues. These five journal institutions focus on project management issues, project planning, engineering management, business management difficulties, and other management-related topics like system dynamics in management.

3.2. System Dynamics Model Proposed

In this study, the authors developed an SD model to solve problems of scoping and estimation. Firstly, the authors drew a first SD model then followed with a second SD model before giving the units. Refer to Figure 1 and Figure 2 for more details. Secondly, the authors connected the two SD models as follows: (1) we made a connection from $Estimation2$ to $taproma1$, and (2) we made a connection from $Estimation1$ to $taproma2$.

NOTE: The names in Figure 1 and Figure 2 are abbreviated. For example, $cade1$ is $capacitydeterioration$, $cagro2$ means capacity growth2, and if a variable ends with 1 in the equation, then it is for the model in Figure 1.

The currency in South Africa is ZAR; hence, there are $MZAR$ used in units. $MZAR$ stands for million ZAR in Figure 1 and Figure 2. Next, the following assignments and explanations of the abbreviations used in the models and units are given:

1.

Scoping provides the most precise projection of future output amounts. Scoping is used to predict future sales volumes and calculate the expected overhead application rate [7,8,28]. In this study, scoping is defined mathematically in Equations (1) and (2) as $Scoping1$ and $Scoping2$, respectively, where $Scoping1$ and $Scoping2$ are scoping for the models in Figure 1 and Figure 2, respectively. Suffixes 1 and 2 are used to differentiate the models. In this study, the Figure 1 model and suffix 1 explain the equations, where $cagro1$ is $capacitygrowth1$, $cade1$ is $capacitydeterioration$, and $inbucap1$ is $initialbudgetcapacity$. The full explanations of these terms will be discussed in the other items of this section. For example, the explanation of $cade1$ is given in item (“2”) of this section. The same applies to the other equations.

$$\begin{array}{c}\hfill Scoping1=INTEG(cagro1-cade1,inbucap1)\end{array}$$

(1)

Units: MZAR/Year

$$\begin{array}{c}\hfill Scoping2=INTEG(cagro2-cade2,inbucap2)\end{array}$$

(2)

Units: MZAR/Year

2.

Capacity deterioration (degradation) means an immediate reduction in output, removal from service, or decrease of capacity in the system because of unanticipated failure or an issue that is beyond the project manager’s control [16,29]. In this study, capacity deterioration is defined mathematically in Equations (3) and (4) as $cade1$ and $cade2$, respectively, where $calif1$ is $capacitylifespan1$.

$$\begin{array}{c}\hfill cade1=Scoping1/calif1\end{array}$$

(3)

Units: $MZAR/Year/Year$

$$\begin{array}{c}\hfill cade2=Scoping2/calif2\end{array}$$

(4)

Units: $MZAR/Year/Year$

3.

Capacity growth determines a company’s long-term survival, performance, and profitability. Capacity growth helps to attract new employees, acquire assets, and fund investments [2,35]. In this study, capacity growth is defined mathematically in Equations (5) and (6) as $cagro1$ and $cagro2$, respectively, where $esbudsp1$ is $estimated$ $budget$ $spending1$, and $eesbudeff1$ is $estimated budget effectiveness1$.

$$\begin{array}{c}\hfill cagro1=esbudsp1\ast eesbudeff1\end{array}$$

(5)

Units: $MZAR/Year/Year$

$$\begin{array}{c}\hfill cagro2=esbudsp2\ast esbudeff2\end{array}$$

(6)

Units: $MZAR/Year/Year$

4.

Estimating budget spending is a process that requires a thorough understanding of budget preparation and planning to generate expenditure predictions and advise policymakers on the desirability and feasibility of particular budget proposals [8,36]. In this study, estimated budget spending is defined mathematically in Equations (7) and (8) as $esbudsp1$ and $esbudsp2$, respectively, where $nomaspe1$ is $nonmaterialspending1$, and $pesptoesbud1$ is $percentagespendingtoestimatedbudget1$.

$$\begin{array}{c}\hfill esbudsp1=nomaspe1\ast pesptoesbud1\end{array}$$

(7)

Units: MZAR/Year

$$\begin{array}{c}\hfill esbudsp2=nomaspe2\ast pesptoesbud2\end{array}$$

(8)

Units: MZAR/Year

5.

The term “indicated production material” refers to all lists of materials utilized in the product manufacturing process [2,31]. In this study, indicated production material is defined mathematically in Equations (9) and (10) as $inmapr1$ and $inmapr2$, respectively, where $prmadi1$ is $projectmaterialdiscontinuance1$, $taproma1$ is $targetprojectmaterials1$, $titorecout1$ is $timetorectifyoutput1$, and $Estimation1$ is an estimated budget of the indicated production material.

$$\begin{array}{c}\hfill inmapr1=MAX(0,prmadi1+(taproma1-Estimation1)/titorecout1)\end{array}$$

(9)

Units: MZAR/Year

$$\begin{array}{c}\hfill inmapr2=MAX(0,prmadi2+(taproma2-Estimation2)/titorecout2)\end{array}$$

(10)

Units: MZAR/Year

6.

Material capacity is the maximum quantity of output that a material can generate or provide at any particular time [16,30]. In this study, the indicated material capacity is defined mathematically in Equations (11) and (12) as $maca1$ and $maca2$, respectively, where $Scoping1$ is $Scoping1$, and $macapma1$ is $maximumcapacitytomaterial1$.

$$\begin{array}{c}\hfill maca1=Scoping1\ast macapma1\end{array}$$

(11)

Units: MZAR/Year

$$\begin{array}{c}\hfill maca2=Scoping2\ast macapma2\end{array}$$

(12)

Units: MZAR/Year

7.

Material spending (expenses) includes all costs related to direct materials, manufacturing overhead, and direct labor during the manufacturing process [7,8,28]. In this study, material spending is defined mathematically in Equations (13) and (14) as $maspe1$ and $maspe2$, respectively, where $maca1$ is $materialcapacity1$, and $pemacaus1$ is $percentagematerialcapacityused1$.

$$\begin{array}{c}\hfill maspe1=maca1\ast pemacaus1\end{array}$$

(13)

Units: MZAR/Year

$$\begin{array}{c}\hfill maspe2=maca2\ast pemacaus2\end{array}$$

(14)

Units:MZAR/Year

8.

Non-material spending refers to production costs that are not accounted for in regular material costs (budgeted) or manufacturing costs [7,8,28]. In this study, non-material spending is defined mathematically in Equations (15) and (16) as $nomaspe1$ and $nomaspe2$, respectively, where $Scoping1$ is $Scoping1$, and $maspe1$ is $materialspending1$.

$$\begin{array}{c}\hfill nomaspe1=Scoping1-maspe1\end{array}$$

(15)

Units: MZAR/Year

$$\begin{array}{c}\hfill nomaspe2=Scoping2-maspe2\end{array}$$

(16)

Units: MZAR/Year

9.

The percentage of material capacity used (PMCU) is the proportion of a material’s potential output that is realized [2,31]. The PMCU of a company can be measured to determine how well it is reaching its potential. In this study, PMCU is defined mathematically in Equations (17) and (18) as $pemacaus1$ and $pemacaus2$, respectively, where $inmapr1$ is $indicatedmaterialproduction1$, and $maca1$ is $materialcapacity1$.

$$\begin{array}{c}\hfill pemacaus1=WITHLOOKUP\left(ZIDZ\right(inmapr1,maca1),(Addtheinterval\left)\right)\end{array}$$

(17)

Units: $Dmnl$

$$\begin{array}{c}\hfill pemacaus2=WITHLOOKUP\left(ZIDZ\right(inmapr2,maca2),(Addtheinterval\left)\right)\end{array}$$

(18)

Units: $Dmnl$

10.

Project material discontinuance is a fault or stoppage in a material’s normal physical properties or structure caused by porosity, fractures, or inhomogeneity [8,33]. In this study, project material discontinuance is defined mathematically in Equations (19) and (20) as $prmadi1$ and $prmadi2$, respectively, where $mali1$ is $material$ $lifespan1$.

$$\begin{array}{c}\hfill prmadi1=Estimation1/mali1\end{array}$$

(19)

Units: MZAR/Year

$$\begin{array}{c}\hfill prmadi2=Estimation2/mali2\end{array}$$

(20)

Units: MZAR/Year

11.

Target project materials are materials selected by material facility operators as those that will be separated from mixed waste material to produce significant quantities of that specific material [8,36]. In this study, target project material is defined mathematically in Equations (21) and (22) as $taproma1$ and $taproma2$, respectively, where $deoura1$ is $desiredoutputratio1$.

$$\begin{array}{c}\hfill taproma1=Estimation2\ast deoura1\end{array}$$

(21)

Units: $MZAR$

$$\begin{array}{c}\hfill taproma2=Estimation1\ast deoura2\end{array}$$

(22)

Units: $MZAR$

12.

Estimation (project cost estimation) is the process of approximating expenses, indirect costs, and other types of project costs to develop a budget that meets the financial commitment necessary for a successful project [7,8,28]. In this study, estimation is defined mathematically in Equations (23) and (24) as $Estimation1$ and $Estimation2$, respectively, where $maspe1$ is $materialspending1$, $prmadi1$ is $project$ $material$ $discontinuance1$, and $inma1$ is $initialmaterial1$.

$$\begin{array}{c}\hfill Estimation1=INTEG(maspe1-prmadi1,inma1)\end{array}$$

(23)

Units: $MZAR$

$$\begin{array}{c}\hfill Estimation2=INTEG(maspe2-prmadi2,inma2)\end{array}$$

(24)

Units: $MZAR$

Testing the Functionality of the Proposed SD Model

The equations above were used to evaluate the operation of the proposed SD model and check its relationship to the SD model’s mathematical expectations. For example, Equation (1) confirmed the relation to the mathematical expectation using the results of Figure 3, Figure 4, Figure 5 and Figure 6. The mathematical expectation of Equation (1) is the proportionality of $cagro1$ and $cade1$ to $Scoping1$; if $Scoping1$ increases, then $cagro1$ and $cade1$ will also increase and vice versa. The same applies to the other equations: meaning from Equation (2) up to Equation (24). But for testing Equation (1), only Figure 3, Figure 4, Figure 5 and Figure 6 will be displayed. The explanation of the full contribution of the other equations designed for SD is discussed in Section 5.

3.3. Approach to Collecting Information

The research came to an end when a degree of data redundancy was reached, which occurred when no new or relevant material was introduced in subsequent research. All data have been captured, and the analysis has been completed. Table 1 provides details of the literature search methodology, such as databases, keywords, and time frames related to the project scoping and estimation questions.

3.4. Analysis

Based on the collection findings from Section 3.3, the authors will conduct the following analyses:

• Statistically, the authors will examine concerns with scoping and estimating issues in project management.
• The authors will conduct a statistical analysis to determine how many managers are familiar with SD, which will aid in the resolution of scoping and estimating challenges during project management.
• Statistically, the authors will examine how many managers are aware of scoping and estimation concerns throughout project management.
• Statistically, the authors will determine how many managers can design and develop SD utilizing mathematical calculations and programming.
• The authors will conduct a statistical analysis to determine how many SD models capable of handling scoping and estimating difficulties are available in the public domain (for example, from Google, media outlets, or CDs).

4. Materials Used

The study employed the following materials:

• Unlike previous applications, the authors employed the Vensim platform, which includes intuitive graphical interfaces integrated with systems for information management, legible tables, governance, and presentation via commands and animations.
• The authors employed integral equations ($INTEG$) due to their statistical advantages in problem-solving and decision-making methodologies.
• Limits equations ($ZIDZ$) help SD designers observe numbers from a distance. Subsequently, an adjustment to a variable is controlled exclusively by its most-significant values.
• The study gained access using $WITHLOOKUP$ functions, also known as lookup table functions. Lookup table functions are very useful for graphically illustrating the functional relationships between two or more variables. Lookup table functions are nonlinear and are difficult to predict.
• Both static and dynamic equations were used. The authors used static and dynamic equations to calculate the balance of the system in equilibrium.
• The study also used the $MAX$ function to select the highest (maximum) value between two numbers.
• Five journals were used in conjunction with Table 1 to get proper and new information related to studies by using the listed databases (resources), keywords, and time frames. Refer to Table 1. The authors used the listed resources, keywords, and time frames from Table 1 to answer the research questions in Section 1.2.

5. Results

The results of this study are composed of two central questions. These two central questions are supported by six secondary questions, as mentioned in Section 1.2.

5.1. Results to Answer the First Central Question from Section 1.2

The first central question from Section 1.2 was supported by first five secondary questions to get clear, detailed results.

The following results were based on the secondary research questions from Section 1.2 and the information collection procedure from Section 3.3. The first secondary question was: “Do project managers have challenges with scoping and estimating issues during project management?” A total of 60 articles per journal institution were collected and analyzed, keeping the above question in mind. Out of 60 papers from the Engineering Project Organization Journal, only 56 addressed the issue. Out of 56 articles, 48 confirmed that project managers struggle with scoping and estimating issues throughout the project lifecycle, whereas the other eight did not. Refer to

Table 2. Results: Do project managers have challenges with scoping and estimating issues?

Institution Name for the Journal	The Number of Papers Relevant to the Question	Percentage of The Number of Papers Relevant to the Question	Use of 1 Article out of 60 as an Example from the Selected Journal Institution
Engineering Project Organization Journal	48	$48/60=80\%$	The answer to the Question is Yes. Refer to [37],
Harvard Business Review	47	$47/60=78.3333\%$	The answer to the Question is Yes. Refer to [38],
International Journal of Construction Project Management	50	$50/60=83.3333\%$	The answer to the Question is Yes. Refer to [39],
International Journal of Managing Projects in Business	53	$53/60=88.3333\%$	The answer to the Question is Yes. Refer to [40]
Journal of Engineering and Technology Management	52	$52/60=86.6666\%$	The answer to the Question is Yes. Refer to [41]

and Figure 7. To address the above question, the authors reviewed 60 articles from Harvard Business Review. Only 54 of the 60 articles were related to the topic, and 47 of the 54 papers verified that project managers struggle with scoping and estimating issues throughout the project lifecycle, whereas the remaining 7 did not. Refer to Table 2 and Figure 7.

The International Journal of Construction Project Management publications were all relevant to the question. Out of 60 papers, 50 confirmed that project managers face challenges in scoping and estimating difficulties throughout project management, whereas the remaining 10 did not. Refer to Table 2 and Figure 7. From the International Journal of Project Management in Business, 53 articles confirmed that project managers have challenges in scoping and estimating difficulties throughout the project lifecycle, whereas the other three did not. Refer to Table 2 and Figure 7. From the Journal of Engineering and Technology Management, 52 of the 60 publications confirmed that project managers have difficulty with scoping and estimating concerns throughout project management, whereas the remaining 2 did not. Refer to Table 2 and Figure 7.

The second secondary question from Section 1.2 was as follows: “Are managers familiar with SD, which can help to solve scoping and estimating issues during project management?” A total 60 articles per journal institution were collected and analyzed, keeping the above question in mind. Refer to

Table 3. Results: Are managers familiar with SD, which can solve scoping and estimating issues?

Institution Name for the Journal	The Number of Papers that are Relevant to the Question	Percentage of the Number of Papers that are Relevant to the Question	Use of 1 Article out of 60 as an Example from the Selected Journal Institution
Engineering Project Organization Journal	54	$54/60=90\%$	The answer to the Question is Yes. Refer to [42]
Harvard Business Review	55	$55/60=91.6666\%$	The answer to the Question is Yes. Refer to [43]
International Journal of Construction Project Management	55	$55/60=91.6666\%$	The answer to the Question is Yes. Refer to [44]
International Journal of Managing Projects in Business	50	$50/60=83.3333\%$	The answer to the Question is Yes. Refer to [39]
Journal of Engineering and Technology Management	52	$52/60=86.6666\%$	The answer to the Question is Yes. Refer to [45]

and Figure 8. From the Engineering Project Organization Journal, all 60 articles were related to the problem. 54 confirmed that supervisors were familiar with SD, while the other six were not. Refer to Table 3 and Figure 8. To answer the question above, the authors reviewed 60 articles from Harvard Business Review. There were 60 articles about the topic. 55 of the 60 papers confirmed that managers were aware of SD, whereas the remaining five did not. Refer to Table 3 and Figure 8. The International Journal of Construction Project Management publications were pertinent to the question. Of 60 papers, 55 affirmed that managers were familiar with SD, whereas the other five did not. Refer to Table 3 and Figure 8. From the International Journal of Project Management in Business, fifty papers confirmed that supervisors were familiar with SD, whereas the remaining ten did not. Refer to Table 3 and Figure 8. From the Journal of Engineering and Technology Management, 52 of the 60 publications confirmed that managers were familiar with SD, whereas the remaining 8 did not. Refer to Table 3 and Figure 8.

The third secondary question from Section 1.2 was: “Are managers informed about scoping and estimating issues during project management?” A total of 60 papers per journal institution were collected and examined with the above question in mind. Refer to

Table 4. Results: Are managers informed about scoping and estimating issues?

Institution Name for the Journal	The Number of Papers that are Relevant to the Question	Percentage of the Number of Papers that are Relevant to the Question	Use of 1 Article out of 60 as an Example from the Selected Journal Institution
Engineering Project Organization Journal	55	$55/60=91.6666\%$	The answer to the Question is Yes. Refer to [37]
Harvard Business Review	52	$52/60=86.6666\%$	The answer to the Question is Yes. Refer to [38]
International Journal of Construction Project Management	50	$50/60=83.3333\%$	The answer to the Question is Yes. Refer to [39]
International Journal of Managing Projects in Business	55	$55/60=91.6666\%$	The answer to the Question is Yes. Refer to [40]
Journal of Engineering and Technology Management	54	$54/60=90\%$	The answer to the Question is Yes. Refer to [41]

and Figure 9. All 60 articles in the Engineering Project Organization Journal dealt with the subject. A total of 54 publications confirmed that managers were informed about scoping and estimating concerns during project management, while the other six did not. Refer to Table 4 and Figure 9. To answer the question above, the authors examined 60 articles from Harvard Business Review. There were 60 articles on the topic: 55 of the 60 articles indicated that managers were informed about scoping and estimation concerns during project management, while the other five did not. Refer to Table 4 and Figure 9. International Journal of Construction Project Management publications were pertinent to the question. Of the 60 papers examined, 50 affirmed that managers were informed about scoping and estimating issues during project management, whereas the other 10 did not. Refer to Table 4 and Figure 9. For the International Journal of Project Management in Business, 52 papers confirmed that managers were informed about scoping and estimating issues during project management, whereas the remaining 8 did not. Refer to Table 4 and Figure 9. From the Journal of Engineering and Technology Management, 55 of the 60 publications confirmed that managers were familiar with SD, whereas the remaining 5 did not. Refer to Table 4 and Figure 9.

The fourth secondary question from Section 1.2 was: “Are managers capable of designing and programming SD using mathematical equations and programming?” A total of 60 papers per journal institution were collected and examined with the above question in mind. Refer to

Table 5. Results: Are managers capable of designing and programming SD using mathematics?

Institution Name for the Journal	The Number of Papers that are Relevant to the Question	Percentage of the Number of Papers that are Relevant to the Question	Use of 1 Article out of 60 as an Example from the Selected Journal Institution
Engineering Project Organization Journal	50	$50/60=83.3333\%$	Indeed, yes. However, due to the lack of experience with design, mathematics, and programming, project managers are afraid of making mistakes when utilizing SD. Refer to [42].
Harvard Business Review	54	$54/60=90\%$	Indeed, yes. However, due to the lack of experience with design, mathematics, and programming, project managers are afraid of making mistakes when utilizing SD. Refer to [43].
International Journal of Construction Project Management	52	$52/60=86.6666\%$	Indeed, yes. However, due to the lack of experience with design, mathematics, and programming, project managers are afraid of making mistakes when utilizing SD. Refer to [44].
International Journal of Managing Projects in Business	55	$55/60=91.6666\%$	Indeed, yes. However, due to the lack of experience with design, mathematics, and programming, project managers are afraid of making mistakes when utilizing SD. Refer to [39].
Journal of Engineering and Technology Management	55	$55/60=91.6666\%$	Indeed, yes. However, due to the lack of experience with design, mathematics, and programming, project managers are afraid of making mistakes when utilizing SD. Refer to [45].

and Figure 10. All 60 articles in the Engineering Project Organization Journal dealt with the subject: 50 publications confirmed that most managers are not capable of designing and programming SD using mathematical equations and programming, while the other 10 did not. Refer to Table 5 and Figure 10. To answer the question above, the authors examined 60 articles from Harvard Business Review. There were 60 articles on the topic: 54 of the 60 articles indicated that most managers are not capable of designing and programming SD using mathematical equations and programming, while the other 6 did not. Refer to Table 5 and Figure 10.

The International Journal of Construction Project Management publications were pertinent to the question. Of 60 papers, 52 affirmed that most managers are not capable of designing and programming SD using mathematical equations and programming, whereas the other 8 did not. Refer to Table 5 and Figure 10. From the International Journal of Project Management in Business, 55 papers confirmed that most managers are not capable of designing and programming SD using mathematical equations and programming., whereas the remaining 5 did not. Refer to Table 5 and Figure 10. From the Journal of Engineering and Technology Management, 55 of the 60 publications confirmed that most managers are not capable of designing and programming SD using mathematical equations and programming., whereas the remaining 5 did not. Refer to Table 5 and Figure 10.

The fifth secondary question from Section 1.2 was: “Is there any SD model that is capable of solving scoping and estimating issues that can be freely downloaded from the public domain (for example, from Google, journals, or compact discs)?” A total of 60 papers per journal institution were collected and examined with the above question in mind. Refer to

Table 6. Results: Is there any SD model to solve scoping and estimating issues that is ready to download?

Institution Name for the Journal	The Number of Papers that are Relevant to the Question	Percentage of the Number of Papers that are Relevant to the Question	Use of 1 Article out of 60 as an Example from the Selected Journal Institution
Engineering Project Organization Journal	0	0/60 = 0%	No downloadable SD model that one can plug and play
Harvard Business Review	0	0/60 = 0%	No downloadable SD model that one can plug and play
International Journal of Construction Project Management	0	0/60 = 0%	No downloadable SD model that one can plug and play
International Journal of Managing Projects in Business	0	0/60 = 0%	No downloadable SD model that one can plug and play
Journal of Engineering and Technology Management	0	0/60 = 0%	No downloadable SD model that one can plug and play

and Figure 11. All 60 articles in the Engineering Project Organization Journal dealt with the subject. The 60 publications confirmed that no free SD model could be downloaded from the public domain. Refer to Table 6 and Figure 11. To answer the above question, the authors reviewed 60 articles from Harvard Business Review. According to the 60 articles, there were no free SD models available for download from the public domain. Refer to Table 6 and Figure 11. The International Journal of Construction Project Management publications were relevant to the question. According to 60 articles, there were not any free SD models available for download from the public domain. Refer to Table 6 and Figure 11. From the International Journal of Project Management in Business, 60 articles indicated that no free SD model could be downloaded from the public domain. Refer to Table 6 and Figure 11. Articles published articles in the Journal of Engineering and Technology Management indicated that there are no free SD models available for download from the public domain. Refer to Table 6 and Figure 11.

5.2. Results to Answer the Second Central Question from Section 1.2

The second central question from Section 1.2 was supported by the sixth secondary question to get clear, detailed results.

The sixth secondary question from Section 1.2 was: “Can the authors develop and implement mathematical equations for an SD model to empower project managers to overcome scoping and estimating challenges?” The authors designed novel SD models using materials from Section 4 and all equations from Section 3.2. Furthermore, the authors used equations from Section 3.2 to analyze the suggested SD models and their mathematical predictions. For example, Equation (1) generates the findings in Figure 3, Figure 4, Figure 5 and Figure 6 to confirm the link with the mathematical expectation. Refer to Section 3.2. After confirming the functionality of a novel SD, the authors employed the new SD to solve scoping and estimation issues.

Equation (1) was used to generate Figure 12. In Figure 12, Original Before Any Changes was treated as the current (initial) state before making any changes to a novel SD model. Figure 12 shows that the proposed model ran with no errors after connecting Figure 1 and Figure 2. Figure 12 shows the effect of Scoping1 when $initialbudgetcapacity$ ($inbucap1$) and $capacitylifespan1$ ($calif1$) are varied. Figure 12 shows that when ($inbucap1$) = 100 and ($calif1$) = 3, $cade1$ and $cagro1$ decreases exponentially from 5 $MZAR(Year\ast Year)$ to 0 and from 40 $MZAR(Year\ast Year)$ to 0 before 6 years are reached. Then, after 6 years, both remain constant at 0 $MZAR(Year\ast Year)$ until 100 years are reached. Meanwhile, $Scoping1$ decreases exponentially from 100 MZAR/Year to zero. Figure 12 shows that when ($inbucap1$) = 100 and ($calif1$) = 80, $cade1$ and $cagro1$ increase exponentially from 0 $MZAR(Year\ast Year)$ to 100 and from 0 $MZAR(Year\ast Year)$ to 30 until 100 years are reached. Meanwhile, $Scoping1$ decreases exponentially from 0 MZAR/Year to 2200 until 100 years are reached. Figure 12 shows that when ($inbucap1$) = 500 and ($calif1$) = 3, $cade1$ and $cagro1$ increase exponentially from 0 $MZAR(Year\ast Year)$ to 19 and from 0 $MZAR(Year\ast Year)$ to 2 until 100 years are reached. Meanwhile, $Scoping1$ decreases exponentially from 600 MZAR/Year to 0 in 6 years then remains constant at 0 MZAR/Year until 100 years are reached. Figure 12 shows that when ($inbucap1$) = 500 and ($calif1$) = 80, $cade1$ and $cagro1$ increase exponentially from 0 $MZAR(Year\ast Year)$ to 3 and from 0 $MZAR(Year\ast Year)$ to 5 until 100 years are reached, whereas $Scoping1$ increases exponentially from 0 to 12,500.

Equation (2) was used to generate Figure 13. Figure 13 shows that the proposed model ran with no errors after connecting Figure 1 and Figure 2. Figure 13 shows the effect of Scoping2 when $initialbudgetcapacity2$ ($inbucap2$) and $capacitylifespan2$ ($calif2$) are varied. Figure 13 shows that when ($inbucap2$) = 100 and ($calif3$) = 3, $cade2$ and $cagro2$ decrease exponentially from 5 $MZAR(Year\ast Year)$ to 0 and 30 $MZAR(Year\ast Year)$ to 0 after 3 years old, respectively; they both remain constant at 0 $MZAR(Year\ast Year)$, whereas $Scoping2$ decreases from 300 MZAR/Year to zero exponentially. Figure 13 shows that when ($inbucap2$) = 100 and ($calif2$) = 80, $cade2$ and $cagro2$ increase exponentially from 0 to 100 $MZAR(Year\ast Year)$ and 0 to 30 $MZAR(Year\ast Year)$ for 100 years old, respectively, whereas $Scoping2$ increases from 0 to 2250 MZAR/Year exponentially. Figure 13 shows that when ($inbucap2$) = 500 and ($calif2$) = 3, $cade2$ and $cagro2$ decrease exponentially from 20 $MZAR(Year\ast Year)$ to 0 and 100 to 0 for 6 years, respectively. Then, they remain constant at 0 $MZAR(Year\ast Year)$ until 100 years are reached. Meanwhile, $Scoping2$ decreases exponentially from 400 MZAR/Year to zero. Figure 13 shows that when ($inbucap2$) = 500 and ($calif2$) = 80, $cade2$ and $cagro2$ increase exponentially from 0 to 600 $MZAR(Year\ast Year)$ and 0 to 200 $MZAR(Year\ast Year)$ after 6 years old, respectively; then, both remain constant at 0 $MZAR(Year\ast Year)$. Meanwhile, $Scoping2$ increases exponentially from 0 to 20,000 MZAR/Year.

Equation (23) was used to generate Figure 14. Figure 14 shows that the proposed model ran with no errors after connecting Figure 1 and Figure 2. Figure 14 shows the effect of Estimation1 when $initialmaterial1$ ($inma1$) and $materiallifespan1$ ($mali1$) are varied. Figure 14 shows that when ($inma1$) = 0 and ($mali1$) = 0, $nonmaterial$$spending$ 1 ($nomaspe1$) and $nonmaterial$$spending$ 1 ($nomaspe1$) remain constant at 0 MZAR/Year for 100 years, whereas $project$ $material$ $discontinuance$ 1 ($prmadi1$) decreases from 2.25 MZAR/Year to zero exponentially. Figure 14 shows that when ($inma1$) = 0 and ($mali1$) = 200, ($nomaspe1$) remains constant at 0 MZAR/Year for 100 years, whereas ($prmadi1$) decreases from 2.25 MZAR/Year to zero exponentially. Figure 14 shows that when ($inma1$) = 1 and ($mali1$) = 0, ($nomaspe1$) remains constant at 0 MZAR/Year for 100 years, whereas ($prmadi1$) decreases from 2.25 to zero exponentially. Figure 14 shows that when ($inma1$) = 1 and ($mali1$) = 200, ($nomaspe1$) remains constant at 0 MZAR/Year for 100 years, whereas ($prmadi1$) decreases from 2.25 MZAR/Year to zero exponentially.

Equation (24) was used to generate Figure 15. Figure 15 shows that the proposed model ran with no errors after connecting Figure 1 and Figure 2. Figure 15 shows the effect of Estimation1 when $maximumcapacitytomaterial2$ ($macapma2$) and $initialmaterial2$ ($inma2$) are varied. Figure 15 shows that when ($macapma2$) = 0 and ($inma2$) = 0, $nonmaterialspending2$ ($nomasp2$) and $nonmaterialspending2$ ($nomaspe2$) remain constant at 0 MZAR/Year for 100 years, whereas $projectmaterialdiscontinuance2$ ($prmadi2$) remains constant at 0 MZAR/Year for 100 years. Figure 15 shows that when ($macapma2$) = 0 and ($inma2$) = 200, ($nomaspe2$) remains constant at 0 MZAR/Year for 100 years, whereas ($prmadi2$) decreases from 10 MZAR/Year to zero exponentially. Figure 15 shows that when ($macapma1$) = 2 and ($inma2$) = 0, ($nomaspe2$) decreases from 40 to 0 MZAR/Year exponentially until 6 years are reached, then remains constant until 100 years, whereas ($prmadi2$) increases from 0 MZAR/Year exponentially until 5 years are reached, then remains constant at 6 MZAR/Year until 100 years. Figure 15 shows that when ($macapma2$) = 1 and ($inma2$) = 200, ($nomaspe2$) remains constant at 10 MZAR/Year for 100 years, whereas ($prmadi2$) remains steady at 10 MZAR/Year for 100 years.

6. Discussion

The study encompassed rigorous experimentation and a thorough literature review analysis.

Table 7. Comparison of literature review with experimental results conducted in study.

Theoretical Results Found from Literature Review	Experimental Results Conducted in This Study
All five journal institutions and Section 2 included in this study indicated that no SD was applied to solve the issue of scoping and estimation [37,38,39,40,41]. Refer to Table 6 and Figure 11.	An application of an SD model was given. Refer to Section 3.2 of this study. This will help project managers understand better how to design SD models to solve scoping and estimating issues during project management and to understand the mathematical and technical equations needed in SD models.
Using traditional tools such as Gantt Chart, PERT, Kanban Board, Timeline, and Critical Path methods to modify scoping and estimation might affect deadlines for completion budgets and cause the rejection of project changes that might trigger delays or scope creep [2,3,4,5,6].	Unlike the previous application, the authors employed the Vensim platform, which includes intuitive graphical interfaces integrated with systems for information management, legible tables, governance, and presentation via commands and animations. This will help project managers better grasp how to create SD models to tackle scoping and estimating difficulties during project management as well as to create the mathematical and technical equations required for SD models. This will give project managers a better understanding of how to develop SD models to address scoping and estimating issues throughout project management as well as to develop the mathematical concepts and scientific equations required for SD models.
From Section 2, no one has ever indicated the definition and purpose of integral equations in an SD model.	The authors employed integral equations ($INTEG$) due to their offer of statistical advantages in problem-solving and decision-making methodologies. This will provide project managers with a greater understanding of how to create SD models to resolve scoping and estimating difficulties throughout the project lifecycle as well as to develop the mathematical principles and scientific equations essential for SD models.
From Section 2, there are no explanations about limits equations.	Limits equations ($ZIDZ$) help SD designers observe numbers from a distance. Subsequently, an adjustment to a variable is controlled exclusively by its most significant values. This will give administrators of projects a better understanding of how to develop SD models to handle scoping and estimating problems throughout the project’s lifespan as well as to develop the principles of mathematics and scientific formulae required for SD models.
From Section 2, static and dynamic equations were not mentioned. In this study, static and dynamic equations were used.	The authors use static and dynamic equations to calculate the balance of the SD model in equilibrium. This will provide the project team with an improved awareness of how to create SD models to address scoping and estimating issues throughout the project’s lifecycle as well as to develop the mathematical principles and scientific procedures essential for SD models.
From Section 2, no mathematical expressions were discussed.	The study also used the $MAX$ function to select the highest (maximum) value between two numbers.
Most project managers dislike SD modeling because of its complexity, especially if they lack engineering, programming, mathematical analysis, or IT expertise [8,10,14]. SD modeling necessitates mathematical programming, logical decision-making, stock understanding, flaw handling, control elements, computation models, information feedback theory, computer science, causal loop diagrams, quantitative models, feedback loops, and delays [9,11,15,18]. For a summary explanation, refer to Table 5 and Figure 10.	The authors of this paper tackle scoping and estimating problems with SD modeling. In this study, the authors developed an SD model to solve problems of scoping and estimation. Firstly, the authors drew a first SD model, followed by a second SD model, before giving the units. Refer to Section 3.2 for more details. Secondly, the authors connected the two SD models as follows: (1) making a connection from $Estimation2$ to $taproma1$ and (2) making a connection from $Estimation1$ to $taproma2$. NOTE: The names in Figure 1 and Figure 2 are abbreviated for clear visibility of the SD model drawn. For the full explanation, refer to Section 3.2. For example, $cagro2$ means capacity growth 2, and so on. The currency in South Africa is ZAR; hence, $MZAR$ is used as the units. $MZAR$ stands for million ZAR. Refer to Section 3.2.
In Section 2, from five selected journals, static and dynamic equations were not mentioned for solving scoping and estimation problems.	In this research, static and dynamic equations were used. Refer to Section 4 for static and dynamic equation functionality and advantages in SD modules.

shows the comparison between theoretical findings and experimental outcomes from five journal institutions, as stated in Section 2, Section 3.2, and Section 4. These journals specialize in civil engineering management, hospital management, business management, municipal administration, government department management, and office management. The theory findings indicated that no downloadable SD model for scoping and estimation are available in the public domain. The study experimentally created an SD model for scoping and estimation. Refer to Section 3.2. Section 3.2 explained all of the equations used in the new SD model for scoping and estimation. Then, Table 7 summarized the theoretical and experimental findings. Table 7 indicated the reasons why most managers do not like to apply SD models to run projects. Table 7 also summarized the procedure implemented to develop the new SD model. Actually, Table 7 is a summary comparison of the theoretical and experimental findings of the study from Section 1, Section 2, Section 3, Section 4, Section 5 and Section 6.

Table 7 also indicated that using traditional tools such as Gantt Chart, PERT, Kanban Board, Timeline, and Critical Path methods to modify scoping and estimation might affect deadlines for completion budgets and cause the rejection of project changes that might trigger delays or scope creep [2,3,4,5,6]. Unlike previous applications, the authors employed the Vensim platform, which includes intuitive graphical interfaces integrated with systems for information management, legible tables, governance, and presentation via commands and animations.

From Section 2, no one had ever indicated the definition and purpose of integral equations in the SD model. The authors employed integral equations ($INTEG$) due to their offer of statistical advantages in problem-solving and decision-making methodologies.

From Section 2, there were no explanations about limits equations. Limits equations ($ZIDZ$) help SD designers observe numbers from a distance. Subsequently, an adjustment to a variable is controlled exclusively by its most significant values.

From Section 2, static and dynamic equations were not mentioned. In this study, static and dynamic equations were used. The authors use static and dynamic equations to calculate the balance of the SD model in equilibrium.

From Section 2, no mathematical expressions were discussed. The study also used the $MAX$ function to select the highest (maximum) value between two numbers.

Table 7 also indicated that in Section 2, from five selected journals, static and dynamic equations were not mentioned for solving scoping and estimation problems. In this research, static and dynamic equations were used. Refer to Section 4 for static and dynamic equation functionality and advantages in SD modules.

Answers to the Research Questions

This study’s central primary research questions are:

• Do project managers have challenges with scoping and estimating issues during project management? The answer is yes.
• Are managers familiar with SD, which can help to solve scoping and estimating issues during project management? The answer is yes.
• Are managers informed about scoping and estimating issues during project management? The answer is yes.
• Are managers capable of designing and programming SD using mathematical equations and programming? The answer is yes.
• Is there any SD model that is capable of solving scoping and estimating issues that can be freely downloaded from the public domain (for example, from Google, journals, or compact discs)? The answer is no.
• Is it possible to design, program, and give mathematical equations of the SD model that will help project managers solve scoping and estimating challenges? The answer is yes. In this study, the authors provided an SD model to help managers during project management.

The primary responsibilities of project planning in engineering management encompass creating work breakdown structures, estimating costs, developing schedules, scoping, and estimation [21]. Project scoping and estimation are pivotal components of project planning. Inadequate project scope and estimation can significantly impact the project budget and schedule [46,47].

Due to the challenges of scope and estimation, the authors conducted a literature review in collaboration with five journal organizations to investigate this issue. The findings indicated that between $78.3333\%$ and $91.6666\%$ of project managers are cognizant of the challenges associated with project scoping and estimation. Refer to Table 2, Table 3, Table 4, Table 5 and Table 6.

The findings showed that there was a need to solve these challenges. Then, the authors proposed and indicated that system dynamics (SD) has the potential to tackle these issues, as $83.333\%$ to $90\%$ of project managers are aware of, but they generally shy away from using SD due to its requirement for high levels of mathematical and technical expertise. Refer to Table 2, Table 3, Table 4, Table 5 and Table 6. Furthermore, the authors have proposed a new SD model to address the challenges associated with project scoping and estimation. Refer to Figure 1 and Figure 2.

7. Conclusions

System dynamics offers fundamental ideas that enable and facilitate research in many disciplines. Furthermore, system dynamics can be extensively used to define and model project scoping and estimation. Scoping and estimating are the cornerstones of engineering management’s essential project planning responsibilities, which include schedule preparation, cost estimation, and work breakdown structure building. Inadequate project scoping and estimating criteria have a direct effect on the project cost and timeline. The authors of this study worked with five different journal institutions to answer research questions concerning project scoping and estimating. These journals were (i) the Engineering Project Organization Journal, (ii) Harvard Business Review, (iii) the International Journal of Construction Project Management, the (iv) International Journal of Managing Projects in Business, and the (v) Journal of Engineering and Technology Management. In each journal, the authors analyzed one central primary research question and five secondary research questions. The study’s primary research question is: Why is SD not used to solve the scoping and estimating problem? The following are the secondary research questions: (i) Do project managers have challenges with scoping and estimating issues during project management? (ii) Are managers familiar with SD, which can help solve scoping and estimation issues during project management? (iii) Are managers informed about SD during the scoping and estimating phases of project management? (iv) Can managers use mathematical equations and programming to design and program SD during the scoping and estimating phases? (v) Is there any SD available (for example, from Google, journals, or compact discs) for project managers to solve scoping and estimating issues?

7.1. The Findings of the Study’s Central Primary Research Question

In all five journals, the authors found that most project managers dislike SD modeling because of its complexity, especially if they lack engineering, programming, mathematical analysis, or IT expertise. SD modeling necessitates mathematical programming, logical decision-making, stock understanding, flow handling, control elements, computation models, information feedback theory, computer science, causal loop diagrams, quantitative models, feedback loops, and delays. In this study, the authors provided an SD model to help managers during project management.

7.2. The Findings of the Five Secondary Questions

According to the findings for the first secondary question, project managers’ awareness of scoping and estimate issues ranges from $78.3333\%$ to $91.6666\%$.

According to the findings for the second secondary question, $83.333\%$ to $90\%$ of managers are familiar with SD, which can help solve scoping and estimation issues during project management.

According to the findings for the third secondary question, $83.333\%$ to $91.6666\%$ of managers are informed about SD for the scoping and estimating phases of project management.

According to the findings for the fourth secondary question, $83.333\%$ to $91.6666\%$ of managers can use mathematical equations and programming to design and program SD during the scoping and estimating phases; however, most managers do not have mathematical and programming skills.

According to the findings for the fifth secondary question, the authors did not find any downloadable SD from the five journals. The authors designed and developed a novel SD model to address estimating and scoping issues. The authors tested the functionality of the new novel SD model using the proposed mathematical SD equations to confirm the relationships between the mathematical expectations of the SD outputs. This study will give administrators of projects a better understanding of how to develop SD models to handle scoping and estimating problems throughout the project’s lifespan as well as to develop the principles of mathematics and scientific formulae required for SD models.

7.3. Future Work

The authors plan to use the SD model to solve internal team challenges in an upcoming study.

7.4. Limitation of the Study

The study has the following limitations: (i) Not all journals about scoping and estimations were analyzed; only five journals were mentioned, and other journals were left for other researchers. (ii) It should be noted that while the study proposed a new SD model to address scoping and estimating issues, it does not address all issues, including planning, monitoring, and team obstacles. Every issue requires a unique SD model. (iii) The authors only used five research questions to conduct the study; other researchers could use different research questions for a specific study related to the topic. (iv) Not all aspects were included in the novel SD model; other researchers could add other aspects. (v) The findings of the study cannot answer all problems if companies have different problems with system dynamics that were not included in the study.

7.5. Recommendations from the Authors of This Study

Based on the study’s findings, practical steps for the implementation of systems dynamics for project managers or organizations can be dealt with by the consulting these recommended institutions: (i) learn-xpro.mit.edu, (ii) https://business.udemy.com (accessed on 15 February 2024), (iii) systemdynamics.org, (iv) https://www.coursera.org/ (accessed on 15 February 2024), and (v) systemdynamics.org. The authors suggest strategies to help managers overcome challenges to utilizing SD models with tools including (i) model analysis, (ii) group model building, (iii) web-based tools, (iv) documentation, and (v) core software. These institutions and technologies could mitigate many negative beliefs about the complexity of system dynamic and how particular professions require it. There are numerous organizations and tools that managers can reference for research of and attention to system dynamics.