Estimating Construction Project Duration and Costs upon Completion Using Monte Carlo Simulations and Improved Earned Value Management

Abstract

Earned value management (EVM) is widely used when monitoring and estimating operations related to construction projects. As the scope and complexity of construction projects expand, traditional EVM is sometimes ineffective and even results in conclusions that are contrary to the actual situation. Additionally, the estimate produced by EVM is a deterministic value that does not account for the uncertainty of activities involved in a construction project. This study proposes an estimation approach that combines an improved EVM, critical path method (CPM), program evaluation and review technique (PERT), and Monte Carlo simulation (MCS). The contribution is threefold. Firstly, a path-based schedule measurement approach is described using network diagrams and CPM to capture the logical relationships among activities. Secondly, the resource input categorizes activities to improve the accuracy of the duration and cost estimates. The remaining duration and cost will be estimated based on the execution performance of the activities in the same category. Thirdly, PERT and MCS approaches are used to reveal the uncertainty of estimates upon completion by replacing a deterministic value with a possible completion range. An experimental research study was used to apply the proposed approach. The result displayed that the commercial expansion project faced serious schedule delays and cost overruns. Based on the result, the project manager should focus highly on activities J, H, and G (in order of priority) and take corrective actions. In conclusion, the proposed approach demonstrates good performance when identifying deviations, estimating precise results, and determining the importance (priority) of activities that need to be controlled.

1. Introduction

Construction delays and cost overruns are significant challenges facing the construction industry. Related research shows that most construction projects are not completed within the stipulated contract period and are accompanied by significant cost overruns [1,2,3]. To avoid duration delays and cost overruns, the project’s completion duration or cost estimation results during the project implementation process comprise essential bases for revising and updating the project’s duration or cost plan [4].

A widely accepted managerial approach for project monitoring, tracking, and control is earned value management (EVM), which is typically used to comprehensively measure the progress and cost of a project [5,6]. However, the traditional EVM approach has not been adopted in all projects. A substantial number of researchers hold a negative view on applying traditional EVM and have criticized the metrics provided by it at the checkpoint [7,8,9,10,11]. Thus, identifying and improving the limitations of traditional EVM approaches would be meaningful.

Since the processes of construction projects are complex and dynamic, simply using EVM to monitor and estimate the project is insufficient. Many factors such as resource constraints, resource allocation, material procurement delay, human resource problems, and weather can cause uncertainty to the project, with the uncertainties faced by project managers known as risks [12,13]. As managing projects is not easy, a possible solution is to integrate different advanced project management approaches, which can provide more guidance to project managers and help them better manage the project.

Therefore, the objective of this study is to propose a new approach to estimate a project’s duration and cost upon completion combined with improved EVM and other project management approaches (CPM, PERT, and MCS). The goal is to provide more accurate estimates of the project’s completion duration and costs using the proposed approach, helping project managers identify deviations and make decisions based on the importance (priority) of activities. The study addresses the following research questions:

• What are the limitations of the traditional EVM approach?
• How can traditional EVM be improved to address the limitations identified in question 1?
• How can the improved EVM be combined with CPM, PERT, and MCS to propose a comprehensive monitoring and estimation approach?

Figure 1 presents a detailed flow chart of the research study. Firstly, the problems of the traditional EVM are identified. To address the problems identified and to improve the traditional EVM, activity performance, categories (resource input), and quality are considered at the activity level. A path-based schedule measurement approach is described based on CPM to capture the logical relationships among activities for improving the accuracy of the schedule variance ($SV$) and schedule performance index ($SPI$). By using improved EVM metrics at the checkpoint, three-point estimates can be obtained based on PERT. Combining the improved EVM, PERT, and CPM, the preliminary estimation approach is provided. However, the completion estimation result presented by the preliminary estimation approach is still a deterministic value without a possible completion range. Thus, integrating the preliminary estimation approach and MCS can address the issue and obtain the ultimate proposed approach. The proposed approach was put into practice via a commercial expansion project. The outcome showed that the project faced significant costs and schedule overruns. The project manager should emphasize activities J, H, and G (in order of priority) and take remedial action for those activities.

This research study has both theory and practical significance. The improved EVM model contributes to the development and extension of the EVM theory. In addition, the combination of improved EVM, PERT, and MCS contributed to the integration of approaches in the project management field. At the practical level, the proposed approach overcomes the limitations of traditional EVM, which is more in line with the actual project situation in terms of measuring schedule and cost deviation. In addition, the estimation’s result with a possible completion range provides a reference for project managers to control the project. Moreover, the importance (priority) of specific activities clearly guides project managers in making more precise decisions.

2. Literature Review

2.1. EVM

EVM originated in the U.S. Department of Defense and was first applied to managing projects in the national defense and aerospace sectors. Later, the Project Management Institute adopted it as a standard tool for project performance measurements [14]. The key metrics provided by EVM are used to evaluate schedule and cost performances throughout a project’s duration [15].

Traditional EVM operations contain three basic metrics: planned value ($PV$), earned value ($EV$), and actual cost ($AC$). Based on these metrics, four evaluation metrics, including schedule variance ($SV$), schedule performance index ($SPI$), cost variance ($CV)$, and cost performance index ($CPI$), are provided in Equations (1)–(4), respectively.

$$\mathit{SV}=\mathit{EV}-\mathit{PV}$$

(1)

$$\mathit{SPI}=\mathit{EV}/\mathit{PV}$$

(2)

$$\mathit{CV}=\mathit{EV}-\mathit{AC}$$

(3)

$$\mathit{CPI}=\mathit{EV}/\mathit{AC}$$

(4)

2.1.1. Problems of Traditional EVM

Due to the ease of implementation, EVM is becoming an increasingly popular tool for project teams to report and control project performances objectively. Traditional EVM can evaluate and monitor the overall project progress and cost, but its limitations and defects are gradually exposed. Based on the literature review, it is clear that EVM has not been universally accepted or adopted in all projects. Vargas [7] and Lukas [8] reported the negative outcome of applying traditional EVM, with the metrics provided by traditional EVM criticized by different researchers [9,10,11]. Four main limitations of traditional EVM have been identified and are listed below.

(1)

EVM does not consider engineering quality;

The three primary goals of project management are quality, cost, and schedule [16]. Since cost, duration, and quality are closely related, traditional EVM does not consider the impact of reworking operations due to the low-quality results on project duration and cost. In addition, a lack of quality monitoring operations will introduce significant risks and uncertainties to the entire project [17].

(2)

Unclear deviation sources ($SV$ and $SPI$ accuracy);

Traditional EVM is a holistic approach that ignores logical relationships and free float among activities [18]. Metrics such as $SV$ and $SPI$ are produced by a combination of critical and non-critical paths. Because there are positive and negative deviations that cancel each other out, $SV$ and $SPI$ do not reflect the actual progress of the project. Many researchers noted the struggle between $SV$ and $SPI$ [9,10], which has led to its usefulness being commonly questioned in both industry and academia.

(3)

Inaccurate completion estimates;

Traditional EVM does not distinguish the nature and category of activities. It estimates completion duration or cost based on the same performance trend at the checkpoint, which is one of the major sources of significant bias in the estimation’s results.

(4)

EVM does not consider uncertainty or give a range of possible values;

Traditional EVM assumes that the duration and cost of project activities are determined, and its metrics only reflect whether the project is delayed or overspent (ahead of schedule or saved). Nevertheless, the EVM does not specify whether deviations from planned values are (or are not) within the range of possible deviations resulting from the expected variability of the project [18]. In other words, even if the project is delayed at the checkpoint, given the inherent variability of the activity, the delay is likely to remain within the range of possible delays. In this case, the project manager may not urgently need to take corrective actions. Additionally, due to the deterministic completion estimate, traditional EVM does not provide a range of possible outcomes nor the probability of meeting the project’s objectives [19]. Without a range of probabilities, it is impossible to properly consider all possible scenarios, making it difficult for project managers or project teams to take actions.

2.1.2. Extension and Improvement of Traditional EVM

Due to the limitations of traditional EVM, a considerable amount of research has been conducted related to the possible extensions and improvements of EVM. Table 1 summarizes several studies on the extension and improvement of traditional EVM.

Dodson et al. [20] added a quality component into traditional EVM, proposing formulas for estimating the quality variance and quality performance index. However, the study did not provide any solutions for quantifying the quality. Guo et al. [21] proposed a scale for quantifying engineering quality based on the Delphi method. However, inviting experts each time was time consuming and laborious. Khesal et al. [22] represented a new EVM framework by considering a quality control index and developed easy evaluation metrics. The method was implemented in three different projects. The results indicated that adding quality metrics created more information that was more in line with the actual situation. This suggests that introducing quality for traditional EVM is helpful for performing lean monitoring and control.

According to Table 1, it is easy to find that all studies considering logical relationships or uncertainty were conducted on the activity level. Because traditional EVM is a holistic approach, it is impossible to consider logical relationships or work sequences at the project level [23]. Chang and Fei [24] indicated that a major cause of the misjudgment of EVM was the failure to consider the logical relationship and free float among activities. They suggested focusing on the performance of activities on the critical path. However, as construction is a dynamic process, the critical path may change during construction. Naeni et al. [25] presented a fuzzy-based earned value model and applied it to a real-life project. The result displayed good performances, indicating that using probability intervals instead of deterministic values is more beneficial for managers in making decisions. Acebes et al. [18] integrated EVM with activity uncertainty and implemented it into three case studies. The results showed that the proposed method could assist project managers in knowing whether the project’s deviations from planned values are within the “expected” deviations derived from the activity’s planned variability. All studies suggested that taking activity uncertainty into account in the EVM can assist project managers in making decisions with more confidence.

However, few studies considered the improvement of EVM in terms of logical relationships and resource categories among the activities. To fill this research gap, the improvement of EVM in this research study will take all factors (the last row of Table 1) into consideration based on the activity level.

Table 1. Several studies on the extension and improvement of traditional EVM.

Previous Research	Project Level	Activity Level	Cost	Schedule	Quality	Logical Relationship	Uncertainty	Resource Categories
Dodson et al. [20]	x		x	x	x
Guo et al. [21]	x		x		x
Khesal [22]	x		x	x	x
Chang and Fei [24]		x		x		x
Naeni et al. [25]		x	x	x			x
Acebes et al. [18]		x	x	x			x
This research		x	x	x	x	x	x	x

Table 1. Several studies on the extension and improvement of traditional EVM.

Previous Research	Project Level	Activity Level	Cost	Schedule	Quality	Logical Relationship	Uncertainty	Resource Categories
Dodson et al. [20]	x		x	x	x
Guo et al. [21]	x		x		x
Khesal [22]	x		x	x	x
Chang and Fei [24]		x		x		x
Naeni et al. [25]		x	x	x			x
Acebes et al. [18]		x	x	x			x
This research		x	x	x	x	x	x	x

2.2. Combine Different Project Management Approaches

A variety of approaches have been used for project monitoring, control, and estimation, such as the critical path method (CPM), critical chain project management (CCPM), program evaluation and review technique (PERT), and Monte Carlo simulation (MCS) [26,27]. An increasing number of researchers have attempted to combine different approaches in order to pursue better monitoring and estimation performances. Table 2 provides several studies on the combination of different approaches.

2.2.1. CPM and CCPM

CPM has been the most widely used approach in project planning, scheduling, and monitoring [28,29]. The main idea of this approach is to define logical relationships between activities and identify the longest path (the critical path) through the network [30]. However, CPM has been criticized because it does not clearly address interferences between activities, uncertainties, and constraints related to tasks [31,32,33]. Because one of the major factors that lead to cost and time overruns is resource constraints [34], CCPM addressed this issue based on resource dependencies, focusing on the project’s final completion by setting safety times (buffers) to eliminate potential delays in planning. CCPM can improve project performances and enables the completion of projects at one-third of the time ahead of schedule [35].

Dashcov and Komkov [28] integrated the EVM and CPM approaches, proposing a comprehensive monitoring and estimation method for large-scale projects. By using the method, it was convenient to coordinate the participants and stakeholders involved in the implementation of individual phases. However, there is still a problem that the authors only focused on the activities on the original critical path, ignoring the possible changes in the critical path during construction. Petroutsatou [36] compared CPM and CCPM, applying the two approaches to 27 infrastructure projects in Greece. The results confirmed that CCPM could perform better than CPM by achieving decreased construction times combined with accurate resource levelling in the early stages. Although CCPM is a more helpful approach, determining the critical chain is difficult in a complex project due to the close relationship with the adjustment strategy of resource conflict [37]. Furthermore, the currently popular approach in China is CPM, and CCPM still needs some time to develop and popularize in China’s construction industry. Therefore, CPM is still used in the proposed approach for the ease of implementation.

2.2.2. PERT and MCS

PERT, a technique for making and evaluating plans using network analyses, was first developed by the U.S. Navy in planning and controlling the development of the Polaris missile [38]. Unlike CPM, PERT is a probabilistic approach, assuming that the duration or cost of each activity is a random variable that follows the beta distribution [39]. The three-point estimation method of PERT is typically used in project management [40]. The three-point estimates are the optimistic point (${\mathit{T}}^{\mathit{o}}$), most likely point (${\mathit{T}}^{\mathit{m}}$), and pessimistic point (${\mathit{T}}^{\mathit{p}}$), which refer to the duration or cost when work is going well, the estimated duration or cost required under normal conditions, and the duration or cost under unfavorable conditions, respectively. Based on the definition, the mean value and standard deviation of the activity’s duration or cost can be calculated by Equations (5) and (6).

$${\mathit{T}}_{\mathit{mean}}=\frac{{\mathit{T}}^{\mathit{o}}{+4\mathit{T}}^{\mathit{m}}{+\mathit{T}}^{\mathit{p}}}{6}$$

(5)

$$\mathit{\sigma }=\frac{{\mathit{T}}^{\mathit{p}}{-\mathit{T}}^{\mathit{o}}}{6}$$

(6)

To consider uncertainty and obtain the possible value range with respect to completion duration or cost, MCS is widely used without any restrictions and enables a much broader range of uncertain eventualities to be considered in risk analyses [41,42,43]. In a study conducted by Chang and Ko [44], a new approach was developed to improve the rigor of MCS by establishing the link between parameter estimation and the assessment of individual risk sources. The developed approach was applied to eight Taiwanese sewage projects, and the estimates were more accurate. However, the financial or individual risk was evaluated by experts each time, rendering the approach hard to actualize. In Karabulut [27], both PERT and MCS were used in the construction of a villa. The estimation result by PERT and MCS showed that there was a 50% chance that the villa would be completed within 205 days, but it still faced a 50% delayed risk. However, the three-point estimates based on PERT are determined at the construction’s planning phase, ignoring the dynamic nature of construction projects, which creates a significant bias in the completion estimation’s result. To address this issue, EVM metrics can be considered as the baseline to obtain the three-point estimates at each checkpoint, enabling a more accurate completion estimation result and making the approach easy to implement without inviting experts.

In this research study, the proposed method will combine the improved EVM, CPM, PERT, and MCS approaches. The improved EVM is used to produce estimation metrics at the checkpoint, CPM is used to capture the logical relationships and free float among activities, PERT is used to present three-point estimates, and MCS is used to acquire the possible completion range of the project.

Table 2. Several studies on the extension and improvement of the traditional EVM.

Previous Research	EVM	CPM	CCPM	PERT	MCS
Dashcov and Komkov [28]	x	x
Petroutsatou [36]		x	x
Chang and Ko [44]					x
Karabulut [27]		x		x	x
This research	x	x		x	x

Table 2. Several studies on the extension and improvement of the traditional EVM.

Previous Research	EVM	CPM	CCPM	PERT	MCS
Dashcov and Komkov [28]	x	x
Petroutsatou [36]		x	x
Chang and Ko [44]					x
Karabulut [27]		x		x	x
This research	x	x		x	x

3. Proposed Estimation Approach

3.1. Estimation Principle and Checkpoint Measurement Metrics

The activities of construction projects are divided into three situations: finished activity $\mathit{i}$, in progress (ongoing) activity $\mathit{j}$, and not started activity $\mathit{x}$. The ongoing activities are further divided into finished and unfinished parts. The finished activities and the finished part of the ongoing activities will be measured at the checkpoint. Based on the performance and measurement results, the unfinished part of the ongoing and not started activities will be estimated.

At one checkpoint, metrics such as actual duration $\left(\mathit{AD}\right)$, actual cost ($\mathit{AC}$), and quality factor $\left({\mathit{Q}}^{\prime }\right)$ should be measured. ${\mathit{Q}}^{\prime }$ is defined as the degree of variance between the actual quality and the prescribed quality. The experts determine the actual quality level at one checkpoint. See [45] for the specific review process. For activity $\mathit{i}$, which is completed, the actual duration $\left({\mathit{AD}}_{\mathit{i}}\right)$ and actual cost $\left({\mathit{AC}}_{\mathit{i}}\right)$ are recorded based on facts; for activity $\mathit{j}$, which is in progress, the actual duration $\left({\mathit{AD}}_{\mathit{j}}\right)$ of activity $\mathit{j}$ can be calculated in

Table 3. Measured metrics at the checkpoint.

Measured Metrics	Finished Activities (FP = 100%)	Ongoing Activities (0% < FP < 100%)
${\mathit{AD}}_{\mathit{i}}$ and ${\mathit{AD}}_{\mathit{j}}$	Record to the facts	${\mathit{CP}-\mathrm{Max}\left[\mathit{AFT}\right(\mathit{V}}_{\mathit{aj}}\left)\right]$
${\mathit{AC}}_{\mathit{i}}$ and ${\mathit{AC}}_{\mathit{j}}$	Record to the facts	Record to the facts
${{\mathit{Q}}^{\prime }}_{\mathit{i}}$ and ${{\mathit{Q}}^{\prime }}_{\mathit{j}}$	Review confirmed	Review confirmed

, where ${\mathrm{Max}\left[\mathit{AFT}\right(\mathit{V}}_{\mathit{aj}}\left)\right]$ is the maximum value of the actual finished time for all previous activities of activity $\mathit{j}$ and ${\mathit{V}}_{\mathit{a}}$ is the set of all preceding activities of activity $\mathit{j}$. The actual cost of activity $\mathit{j}$ $\left({\mathit{AC}}_{\mathit{j}}\right)$ is recorded following the facts. Table 3 presents the measured metrics of activity $\mathit{i}$ and $\mathit{j}$ at the checkpoint, where the finished percentage $\left(\mathit{FP}\right)$ refers to the ratio of the actual progress to the total progress of the activity when the checkpoint $\left(\mathit{CP}\right)$ is reached (0% ≤ $\mathit{FP}$ ≤ 100%).

3.2. Schedule and Cost Measurement

The activity schedule variance is the difference between the time earned and the actual execution time [46]. For finished activity $\mathit{i}$, the schedule variance $\left({\mathit{SV}}_{\mathit{i}}\right)$, schedule performance index $\left({\mathit{SPI}}_{\mathit{i}}\right)$, cost variance $\left({\mathit{CV}}_{\mathit{i}}\right)$, and cost performance index $\left({\mathit{CPI}}_{\mathit{i}}\right)$ are given by Equations (7)–(10), respectively. Equations (11)–(14) also provide corresponding measurements for ongoing activity $\mathit{j}$ (0% < ${\mathit{FP}}_{\mathit{j}}$ < 100%), where ${\mathit{PD}}_{\mathit{i}}$ and ${\mathit{PD}}_{\mathit{j}}$ represent the planned duration of finished activity $\mathit{i}$ and ongoing activity $\mathit{j}$, respectively.

$${\mathit{SV}}_{\mathit{i}}{={\mathit{Q}}^{\prime }}_{\mathit{i}}{\times \mathit{PD}}_{\mathit{i}}{-\mathit{AD}}_{\mathit{i}}$$

(7)

$${\mathit{SPI}}_{\mathit{i}}{=({\mathit{Q}}^{\prime }}_{\mathit{i}}{\times \mathit{PD}}_{\mathit{i}}{)/\mathit{AD}}_{\mathit{i}}$$

(8)

$${\mathit{CV}}_{\mathit{i}}{={\mathit{Q}}^{\prime }}_{\mathit{i}}{\times \mathit{PV}}_{\mathit{i}}{-\mathit{AC}}_{\mathit{i}}$$

(9)

$${\mathit{CPI}}_{\mathit{i}}{=({\mathit{Q}}^{\prime }}_{\mathit{i}}{\times \mathit{PV}}_{\mathit{i}}{)/\mathit{AC}}_{\mathit{i}}$$

(10)

$${\mathit{SV}}_{\mathit{j}}{={\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}}{\times \mathit{PD}}_{\mathit{j}}{-(\mathit{CP}-\mathrm{Max}[\mathit{AFT}(\mathit{V}}_{\mathit{aj}}\left)\right])$$

(11)

$${\mathit{SPI}}_{\mathit{j}}=\frac{{{\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}}{\times \mathit{PD}}_{\mathit{j}}}{{\mathit{CP}-\mathrm{Max}\left[\mathit{AFT}\right(\mathit{V}}_{\mathit{aj}}\left)\right]}$$

(12)

$${\mathit{CV}}_{\mathit{j}}{={\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{PV}}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}}{-\mathit{AC}}_{\mathit{j}}$$

(13)

$${\mathit{CPI}}_{\mathit{j}}{=({\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{PV}}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}}{)/\mathit{AC}}_{\mathit{j}}$$

(14)

The path-based schedule approach is proposed to further consider the logical relationship among activities, for which its basic idea is to account for all paths in the activity on the arrow (AOA) network diagram. Suppose that a network diagram has $\mathit{n}$-many paths. Up until one checkpoint, finished, ongoing, and not started activities on path $\mathit{p}$ (1 ≤ $\mathit{p}$ ≤ $\mathit{n}$) are, respectively, recorded as ${\mathit{z}}_{\mathit{p}1}$, ${\mathit{z}}_{\mathit{p}2}$, and ${\mathit{z}}_{\mathit{p}3}$. Obviously, ${\mathit{z}}_{\mathit{p}}{=\mathit{z}}_{\mathit{p}1}{+\mathit{z}}_{\mathit{p}2}{+\mathit{z}}_{\mathit{p}3}$, where ${\mathit{z}}_{\mathit{p}}$ denotes all activities on path $\mathit{p}$. According to Equations (7) and (11), the schedule variance of path $\mathit{p}$ (${\mathit{SV}}_{\mathit{p}}$) (Equation (15)) is the sum of schedule variances for all finished (including the finished part of ongoing activities) activities on path $\mathit{p}$. Based on Equations (8) and (12), the schedule performance index of path $\mathit{p}$ (${\mathit{SPI}}_{\mathit{p}}$) (Equation (16)) is the sum of the planned duration divided by the sum of the actual duration on path $\mathit{p}$. The sub-label $\mathit{p}$ in front of activity $\mathit{i}$ and $\mathit{j}$ determines the activity’s path.

$${\mathit{SV}}_{\mathit{p}}=\underset{\mathit{i}=1}{\sum ^{{\mathit{z}}_{\mathit{p}1}}}({{\mathit{Q}}^{\prime }}_{\mathit{pi}}{\times \mathit{PD}}_{\mathit{pi}}{-\mathit{AD}}_{\mathit{pi}})+\underset{\mathit{j}=1}{\sum ^{{\mathit{z}}_{\mathit{p}2}}}{\{({\mathit{Q}}^{\prime }}_{\mathit{pj}}{\times \mathit{FP}}_{\mathit{j}}{\times \mathit{PD}}_{\mathit{pj}})-({\mathit{CP}-\mathrm{Max}[\mathit{AFT}(\mathit{V}}_{\mathit{aj}})])\}$$

(15)

$${\mathit{SPI}}_{\mathit{p}}=\frac{{\displaystyle \sum _{\mathit{i}=1}^{{\mathit{z}}_{\mathit{p}1}}}{({\mathit{Q}}^{\prime }}_{\mathit{pi}}{\times \mathit{PD}}_{\mathit{pi}})+{\displaystyle \sum _{\mathit{j}=1}^{{\mathit{z}}_{\mathit{p}2}}}{({\mathit{Q}}^{\prime }}_{\mathit{pj}}{\times \mathit{FP}}_{\mathit{j}}{\times \mathit{PD}}_{\mathit{pj}})}{{\displaystyle \sum _{\mathit{i}=1}^{{\mathit{z}}_{\mathit{p}1}}}{\mathit{AD}}_{\mathit{pi}}+{\displaystyle \sum _{\mathit{j}=1}^{{\mathit{z}}_{\mathit{p}2}}}{(\mathit{CP}-\mathrm{Max}[\mathit{AFT}(\mathit{V}}_{\mathit{aj}})])}$$

(16)

This approach addresses the issue of the failure of $\mathit{SV}$ and $\mathit{SPI}$ metrics highlighted by Vandevoorde and Vanhoucke [10]. It can also help project managers understand the actual schedule of the project and pinpoint the source of schedule deviations. Furthermore, the approach is not affected by the process of dynamic construction (even in the case of critical path changes).

Unlike the duration, the project’s cost is irrelevant to the sequence of activities and paths. The project’s total cost is the sum of the actual cost of each activity [47]. Based on this definition, the cost variance $\left({\mathit{CV}}_{\mathit{cp}}\right)$ and cost performance index $\left({\mathit{CPI}}_{\mathit{cp}}\right)$ at the checkpoint are provided by Equations (17) and (18), respectively. $\mathrm{Here}, {\mathit{z}}_1$ and ${\mathit{z}}_2$ refer to all finished and ongoing activities for the entire project.

$${\mathit{CV}}_{\mathit{cp}}=\underset{\mathit{i}=1}{\sum ^{{\mathit{z}}_1}}({\mathit{PV}}_{\mathit{i}}{\times {\mathit{Q}}^{\prime }}_{\mathit{i}}{-\mathit{AC}}_{\mathit{i}})+\underset{\mathit{j}=1}{\sum ^{{\mathit{z}}_2}}{{(\mathit{PV}}_{\mathit{j}}\times {\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}}{-\mathit{AC}}_{\mathit{j}})$$

(17)

$${\mathit{CPI}}_{\mathit{cp}}=\frac{{\displaystyle \sum _{\mathit{i}=1}^{{\mathit{z}}_1}}{({\mathit{Q}}^{\prime }}_{\mathit{i}}{\times \mathit{PV}}_{\mathit{i}})+{\displaystyle \sum _{\mathit{j}=1}^{{\mathit{z}}_2}}{({\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}}{\times \mathit{PV}}_{\mathit{j}})}{{\displaystyle \sum _{\mathit{i}=1}^{{\mathit{z}}_1}}{\mathit{AC}}_{\mathit{i}}+{\displaystyle \sum _{\mathit{j}=1}^{{\mathit{z}}_2}}{\mathit{AC}}_{\mathit{j}}}$$

(18)

3.3. Three-Point Estimation

After obtaining the schedule and cost metrics by measurement and calculation at the checkpoint, the three-point estimation for the unfinished part of ongoing and not started activities will be conducted.

Firstly, for the unfinished part of ongoing activities, if activity $\mathit{j}$ has an $\mathit{SPI}$ or $\mathit{CPI}$ that is lower than 1 (schedule delays or cost overruns), the optimistic view is that the remaining work will be performed as planned; the most likely view is that the work will follow the same trend; the pessimistic view accounts for the combined trend of $\mathit{SPI}$ and $\mathit{CPI}$. During the actual progress, both $\mathit{SPI}$ and $\mathit{CPI}$ are usually less than 1 simultaneously, resulting in a smaller denominator but more extensive duration or cost [48]. Thus, the optimistic ${(\mathit{ERD}}_{\mathit{pj}}^{\mathit{o}}, {\mathrm{ERC}}_{\mathit{j}}^{\mathit{o}})$ (Equations (19) and (24)), most likely ${(\mathit{ERD}}_{\mathit{pj}}^{\mathit{m}}, {\mathit{ERC}}_{\mathit{j}}^{\mathit{m}})$ (Equations (20) and (25)), and pessimistic ${(\mathit{ERD}}_{\mathit{pj}}^{\mathit{p}}, {\mathit{ERC}}_{\mathit{j}}^{\mathit{p}})$ (Equations (21) and (26)) estimated remaining duration or cost of ongoing activity $\mathit{j}$ can be determined. However, if ongoing activity $\mathit{j}$ has an $\mathit{SPI}$ or $\mathit{CPI}$ that is higher than 1 (schedule advances or cost savings), all three points are equal to the most likely estimation. From Equations (5) and (6), the formulas for calculating the estimated remaining mean duration or cost ${(\mathit{ERMD}}_{\mathit{pj}}, {\mathit{ERMC}}_{\mathit{j}})$ and standard deviation (${\mathit{\sigma }}_{\mathit{pj}}^{\mathit{D}}$, ${\mathit{\sigma }}_{\mathit{j}}^{\mathit{C}}$) of the ongoing activity $\mathit{j}$ are shown in Equations (22), (23), (27), and (28).

$${\mathit{ERD}}_{\mathit{pj}}^{\mathit{o}}=\frac{{\mathit{PD}}_{\mathit{pj}}{\times (1-{\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}})}{1}$$

(19)

$${\mathit{ERD}}_{\mathit{pj}}^{\mathit{m}}=\frac{{\mathit{PD}}_{\mathit{pj}}{\times (1-{\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}})}{{\mathit{SPI}}_{\mathit{j}}}$$

(20)

$${\mathit{ERD}}_{\mathit{pj}}^{\mathit{p}}=\frac{{\mathit{PD}}_{\mathit{pj}}{\times (1-{\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}})}{{\mathit{SPI}}_{\mathit{j}}{\times \mathit{CPI}}_{\mathit{j}}}$$

(21)

$${\mathit{ERMD}}_{\mathit{pj}}=\frac{{\mathit{ERD}}_{\mathit{pj}}^{\mathit{o}}{+4\times \mathit{ERD}}_{\mathit{pj}}^{\mathit{m}}{+\mathit{ERD}}_{\mathit{pj}}^{\mathit{p}}}{6}$$

(22)

$${\mathit{\sigma }}_{\mathit{pj}}^{\mathit{D}}=\frac{{\mathit{ERD}}_{\mathit{pj}}^{\mathit{p}}{-\mathit{ERD}}_{\mathit{pj}}^{\mathit{o}}}{6}$$

(23)

$${\mathit{ERC}}_{\mathit{j}}^{\mathit{o}}=\frac{{\mathit{PV}}_{\mathit{j}}{\times (1-{\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}})}{1}$$

(24)

$${\mathit{ERC}}_{\mathit{j}}^{\mathit{m}}=\frac{{\mathit{PV}}_{\mathit{j}}{\times (1-{\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}})}{{\mathit{CPI}}_{\mathit{j}}}$$

(25)

$${\mathit{ERC}}_{\mathit{j}}^{\mathit{p}}=\frac{{\mathit{PV}}_{\mathit{j}}{\times (1-{\mathit{Q}}^{\prime }}_{\mathit{j}}{\times \mathit{FP}}_{\mathit{j}})}{{\mathit{CPI}}_{\mathit{j}}{\times \mathit{SPI}}_{\mathit{j}}}$$

(26)

$${\mathit{ERMC}}_{\mathit{j}}=\frac{{\mathit{ERC}}_{\mathit{j}}^{\mathit{o}}{+4\times \mathit{ERC}}_{\mathit{j}}^{\mathit{m}}{+\mathit{ERC}}_{\mathit{j}}^{\mathit{p}}}{6}$$

(27)

$${\mathit{\sigma }}_{\mathit{j}}^{\mathit{C}}=\frac{{\mathit{ERC}}_{\mathit{j}}^{\mathit{p}}{-\mathit{ERC}}_{\mathit{j}}^{\mathit{o}}}{6}$$

(28)

For activities that have not started, the three-time estimation will be corrected based on the activity categories (more specifically, the resource input). Projects are driven by resources, especially workforce, materials, and plant input [49]. However, it is a pity that no scholars have yet incorporated activities’ categories into the improvement of EVM. To address this issue and obtain more accurate estimation results, activities are classified into different categories according to similar resource inputs; that is, activities with similar resource inputs are classified into the same categories. Activities that have not started on path $\mathit{p}$. can be divided into ${\mathit{c}}_1,{\mathit{c}}_2,\dots ,{\mathit{c}}_{\mathit{s}}$(a total number of $\mathit{s}$ categories) and ${\mathit{c}}_{\mathit{d}}$, where ${\mathit{c}}_{\mathit{d}}$ represents activities that do not belong to any previous categories. Obviously, ${\mathit{c}}_1$ + ${\mathit{c}}_2$ + … + ${\mathit{c}}_{\mathit{s}}$ + ${\mathit{c}}_{\mathit{d}}={\mathit{z}}_{\mathit{p}3}$.

If activity $\mathit{x}$ falls into category $\mathit{k}$, the three-point estimates are adjusted by ${\mathit{SPI}}_{\mathit{k}}$ or ${\mathit{CPI}}_{\mathit{k}}$. Equations (29)–(38) estimate the duration and cost of optimistic ${(\mathit{ED}}_{\mathit{pkx}}^{\mathit{o}}, {\mathit{EC}}_{\mathit{kx}}^{\mathit{o}})$, most likely ${(\mathit{ED}}_{\mathit{pkx}}^{\mathit{m}}, {\mathit{EC}}_{\mathit{kx}}^{\mathit{m}})$, pessimistic ${(\mathit{ED}}_{\mathit{pkx}}^{\mathit{p}}, {\mathit{EC}}_{\mathit{kx}}^{\mathit{p}})$, mean (${\mathit{EMD}}_{\mathit{pkx}}, {\mathit{EMC}}_{\mathit{kx}}$), and standard deviation (${\mathit{\sigma }}_{\mathit{pkx}}^{\mathit{D}}$, ${\mathit{\sigma }}_{\mathit{kx}}^{\mathit{C}}$) for not started activity $\mathit{x}$ in category $\mathit{k}$. Similarly, if the ${\mathit{SPI}}_{\mathit{k}}$ or ${\mathit{CPI}}_{\mathit{k}}$ for activities in category $\mathit{k}$ is higher than 1, all three estimates are equivalent to the most likely value.

$${\mathit{ED}}_{\mathit{pkx}}^{\mathit{o}}=\frac{{\mathit{PD}}_{\mathit{pkx}}}{1}$$

(29)

$${\mathit{ED}}_{\mathit{pkx}}^{\mathit{m}}=\frac{{\mathit{PD}}_{\mathit{pkx}}}{{\mathit{SPI}}_{\mathit{k}}}$$

(30)

$${\mathit{ED}}_{\mathit{pkx}}^{\mathit{p}}=\frac{{\mathit{PD}}_{\mathit{pkx}}}{{\mathit{SPI}}_{\mathit{k}}{\times \mathit{CPI}}_{\mathit{k}}}$$

(31)

$${\mathit{EMD}}_{\mathit{pkx}}=\frac{{\mathit{ED}}_{\mathit{pkx}}^{\mathit{o}}{+4\times \mathit{ED}}_{\mathit{pkx}}^{\mathit{m}}{+\mathit{ED}}_{\mathit{pkx}}^{\mathit{p}}}{6}$$

(32)

$${\mathit{\sigma }}_{\mathit{pkx}}^{\mathit{D}}=\frac{{\mathit{ED}}_{\mathit{pkx}}^{\mathit{p}}{-\mathit{ED}}_{\mathit{pkx}}^{\mathit{o}}}{6}$$

(33)

$${\mathit{EC}}_{\mathit{kx}}^{\mathit{o}}=\frac{{\mathit{PV}}_{\mathit{kx}}}{1}$$

(34)

$${\mathit{EC}}_{\mathit{kx}}^{\mathit{m}}=\frac{{\mathit{PV}}_{\mathit{kx}}}{{\mathit{CPI}}_{\mathit{k}}}$$

(35)

$${\mathit{EC}}_{\mathit{kx}}^{\mathit{p}}=\frac{{\mathit{PV}}_{\mathit{kx}}}{{\mathit{CPI}}_{\mathit{k}}{\times \mathit{SPI}}_{\mathit{k}}}$$

(36)

$${\mathit{EMC}}_{\mathit{kx}}=\frac{{\mathit{EC}}_{\mathit{kx}}^{\mathit{o}}{+4\times \mathit{EC}}_{\mathit{kx}}^{\mathit{m}}{+\mathit{EC}}_{\mathit{kx}}^{\mathit{p}}}{6}$$

(37)

$${\mathit{\sigma }}_{\mathit{kx}}^{\mathit{C}}=\frac{{\mathit{EC}}_{\mathit{kx}}^{\mathit{p}}{-\mathit{EC}}_{\mathit{kx}}^{\mathit{o}}}{6}$$

(38)

For activities not fitting into one of the preceding categories (${\mathit{c}}_{\mathit{d}}$), ${\mathit{SPI}}_{\mathit{k}}$ or ${\mathit{CPI}}_{\mathit{k}}$ will be replaced by the average $\mathit{SPI}$ or $\mathit{CPI}$ of all activity categories up to the checkpoint. Likewise, the formulas for calculating the relevant parameters of not started activities that do not belong to any categories are shown from Equations (39)–(48), where subscript $d$ demonstrates that the activity does not belong to any previous categories.

$${\mathit{ED}}_{\mathit{pdx}}^{\mathit{o}}=\frac{{\mathit{PD}}_{\mathit{pdx}}}{1}$$

(39)

$${\mathit{ED}}_{\mathit{pdx}}^{\mathit{m}}=\frac{{\mathit{s}\times \mathit{PD}}_{\mathit{pdx}}}{{\displaystyle \sum _{\mathit{k}=1}^{\mathit{s}}}{\mathit{SPI}}_{\mathit{k}}}$$

(40)

$${\mathit{ED}}_{\mathit{pdx}}^{\mathit{p}}=\frac{{\mathit{s}\times \mathit{PD}}_{\mathit{pdx}}}{{\displaystyle \sum _{\mathit{k}=1}^{\mathit{s}}}{\mathit{SPI}}_{\mathit{k}}{\times \mathit{CPI}}_{\mathit{k}}}$$

(41)

$${\mathit{EMD}}_{\mathit{pdx}}=\frac{{\mathit{ED}}_{\mathit{pdx}}^{\mathit{o}}{+4\times \mathit{ED}}_{\mathit{pdx}}^{\mathit{m}}{+\mathit{ED}}_{\mathit{pdx}}^{\mathit{p}}}{6}$$

(42)

$${\mathit{\sigma }}_{\mathit{pdx}}^{\mathit{D}}=\frac{{\mathit{ED}}_{\mathit{pdx}}^{\mathit{p}}{-\mathit{ED}}_{\mathit{pdx}}^{\mathit{o}}}{6}$$

(43)

$${\mathit{EC}}_{\mathit{dx}}^{\mathit{o}}=\frac{{\mathit{PV}}_{\mathit{dx}}}{1}$$

(44)

$${\mathit{EC}}_{\mathit{dx}}^{\mathit{m}}=\frac{{\mathit{s}\times \mathit{PV}}_{\mathit{dx}}}{{\displaystyle \sum _{\mathit{k}=1}^{\mathit{s}}}{\mathit{CPI}}_{\mathit{k}}}$$

(45)

$${\mathit{EC}}_{\mathit{dx}}^{\mathit{p}}=\frac{{\mathit{s}\times \mathit{PV}}_{\mathit{dx}}}{{\displaystyle \sum _{\mathit{k}=1}^{\mathit{s}}}{\mathit{CPI}}_{\mathit{k}}{\times \mathit{SPI}}_{\mathit{k}}}$$

(46)

$${\mathit{EC}}_{\mathit{dx}}=\frac{{\mathit{EC}}_{\mathit{dx}}^{\mathit{o}}{+4\times \mathit{EC}}_{\mathit{dx}}^{\mathit{m}}{+\mathit{EC}}_{\mathit{dx}}^{\mathit{p}}}{6}$$

(47)

$${\mathit{\sigma }}_{\mathit{dx}}^{\mathit{C}}=\frac{{\mathit{EC}}_{\mathit{dx}}^{\mathit{p}}{-\mathit{EC}}_{\mathit{dx}}^{\mathit{o}}}{6}$$

(48)

3.4. Preliminary Project Estimation

The preliminary project estimation refers to the estimation result of the project’s duration or cost upon completion without conducting MCS. According to Figure 1, integrating improved EVM, CPM, and PERT, the completion duration and cost of the project can be preliminarily estimated. Note that the preliminary estimation result is still a deterministic value, but it can provide project managers with an overview of the project’s completion duration and cost.

The total estimated remaining mean duration of path $\mathit{p}$ $\left({\mathit{TERMD}}_{\mathit{p}}\right)$ (Equation (49)) is the sum of the estimated mean duration of the ongoing activity (unfinished part) and not started activities. Taken absolutely, the total estimated remaining mean duration $\left(\mathit{TERMD}\right)$ (Equation (50)) of the project is the longest ${\mathit{TERMD}}_{\mathit{p}}$ with $\mathit{n}$ paths. Therefore, the total estimated duration at completion $\left(\mathit{TEMD}\right)$ is equal to the checkpoint time $\left({\mathit{CP}}_{\mathit{time}}\right)$ plus $\mathit{TREMD}$, as shown in Equation (51).

$${\mathit{TERMD}}_{\mathit{p}}{=\mathit{ERMD}}_{\mathit{pj}}+\underset{\mathit{k}=1}{\sum ^{\mathit{s}}}\underset{ \mathit{x}=1}{\sum ^{{\mathit{c}}_{\mathit{k}}}}{\mathit{EMD}}_{\mathit{pkx}}+\underset{\mathit{x}=1}{\sum ^{{\mathit{c}}_{\mathit{d}}}}{\mathit{EMD}}_{\mathit{pdx}}$$

(49)

$${\mathit{TERMD}=\mathrm{Max}\{\mathit{TERMD}}_{\mathit{p}} \mathit{p}=1, 2,\dots , \mathit{n}\}$$

(50)

$${\mathit{TEMD}=\mathit{CP}}_{\mathit{time}}+\mathit{TERMD}$$

(51)

The cost is calculated in the same way but without considering the paths. Suppose there are $\mathit{m}$ ongoing activities at the checkpoint moment. The total estimated remaining mean cost $\left(\mathit{TERMC}\right)$ (Equation (52)) is the sum of all estimated mean costs of the ongoing activities (unfinished part) and not started activities. The total estimated cost upon completion $\left(\mathit{TEMC}\right)$ is equal to the cost spent up to the checkpoint (${\mathit{CP}}_{\mathit{spent}}$) plus $\mathit{TERMC}$, which is calculated according to Equation (53).

$$\mathit{TERMC}=\underset{\mathit{j}=1}{\sum ^{\mathit{m}}}{\mathit{ERMC}}_{\mathit{j}}+\underset{\mathit{k}=1}{\sum ^{\mathit{s}}}\underset{ \mathit{x}=1}{\sum ^{{\mathit{c}}_{\mathit{k}}}}{\mathit{EMC}}_{\mathit{kx}}+\underset{\mathit{x}=1}{\sum ^{{\mathit{c}}_{\mathit{d}}}}{\mathit{EMC}}_{\mathit{dx}}$$

(52)

$${\mathit{TEMC}=\mathit{CP}}_{\mathit{spent}}+\mathit{TERMC}$$

(53)

3.5. MCS Process

The result of the preliminary estimation is still a deterministic value that fails to explain the impact of variations in each activity on the total duration or cost. A more reasonable method is to provide a probability distribution within a possible range from the perspective of statistical probability [41]. MCS will be conducted to overcome this difficulty.

Before conducting MCS, the assumption of distribution in MCS is very important. In the project management field, a previous body of the literature suggested that random variations in activity duration (cost) were generated according to triangular and beta distributions [50,51]. Historically, triangular distributions have been preferred relative to beta distributions because of their direct nature or as an approximation of beta distributions [52]. However, Kuhl et al. [53] warned against using triangular distributions without empirical data points. They argued that beta distributions were superior to triangular distributions when the distribution of the random variable was significantly skewed to the left or right. As stated by Hong et al. [54], the Beta-PERT distribution is more suitable in MCS for estimating construction activity durations. In Burciu [55], Beta-PERT is an improvement of beta distributions, which can better fit uniform, normal, and lognormal distributions. Lake et al. [56] and Jing et al. [57] clarified that Beta-PERT distributions are valuable when the data are limited because the distribution only has three inputs: most likely and minimum and maximum values. Therefore, in this research study, the duration and cost of activities are assumed to obey the Beta-PERT distribution, with the three-point estimates considered as inputs to this distribution.

After determining the distribution assumption, based on Votto et al. [58], Chen et al. [48], Bonato et al. [51], and Hendradewa [43], the specific simulation process is as follows: (1) The basic data of the project checkpoint are processed and the ongoing and not started activities that need simulation estimations are established. (2) The three-point estimates, mean value, and standard deviation of ongoing and not started activities are calculated according to Equations (19)–(48) [48,58]. (3) The random variables parameters of the activities that obey Beta-PERT distributions based on the above values are defined [48]. (4) The relationships between various variables are established and the predictor variables are defined. For the project’s duration, the total duration was simulated as a sum in the function of the distributions of the probabilities of the corresponding activities on the longest path plus checkpoint times [43,58], and these are specified according to Equations (49)–(51). For the project cost, the total cost was simulated as a sum in the function of the distributions of probabilities of all simulated activities plus the cost up to the checkpoint [48], as stated by Equations (52) and (53). (5) The simulation times are set and the simulation is run. (6) The simulation’s results are statistically processed and the probability density function (PDF) and cumulative distribution function (CDF) are produced from the processed data. (7) The sensitivity analysis and risk analysis are performed.

The MCS process is shown in Figure 2.

4. Experimental Research

4.1. Project Overview

The project came from expanding a commercial office building in Beijing, China. The project consists of 11 activities, with the AOA network diagram shown in Figure 3. The total project duration was planned to be 32 months, costing CNY 1080 thousand. The detailed project plan is shown in

Table 4. Project plan.

Activity	A	B	C	D	E	F	G	H	I	J	K
$\mathit{PD}$ (Month)	3	4	8	10	6	12	8	10	6	10	7
$\mathit{PV}$ (CNY ten thousand)	4	6	14	12	7	10	11	15	9	12	8

. Based on interviews with the project manager and production manager, it was noted that activities A and H, C and G, B and I, D and K, and E and J had similar resource inputs, but activity F was different from every previous activity due to unique materials and construction processes. The checkpoint was made once a month and determined by the project manager. In other words, at the end of each month, the inspection would be conducted, and the project manager would hold a meeting to report the actual progress and perform cost analyses.

At the end of the 10th month, the proposed approach was implemented in this project.

Table 5. Inspection results at the checkpoint.

Activity	A	B	C	D	E
$\mathit{FP}$(%)	100	100	75	65	90
${\mathit{Q}}^{\prime }$	1.1	0.9	1.0	1.1	0.9
$\mathit{AD}$	3.5	3.7	6.5	6.3	6.3
$\mathit{AC}$	5	5.8	12	8.26	6.8

provides detailed inspection results at the end of the 10th month ${(\mathit{CP}}_{\mathit{time}}=10)$. According to the table, activities A and B are completed, activities C, D, and E are in progress (the $\mathit{AC}$ of activities C, D, and E is calculated in Table 3), and activities F, G, H, I, J, and K have not started.

4.2. Checkpoint Schedule and Cost Analysis

From Equations (7)–(14),

Table 6. Metrics for finished (including the finished part) activities.

Activity	A	B	C	D	E
$\mathit{SV}$ (Month)	−0.20	−0.10	−0.50	0.85	−1.44
$\mathit{SPI}$ (%)	94.29	97.30	92.31	113.49	77.14
$\mathit{CV}$ (CNY ten thousand)	−0.60	−0.40	−1.50	0.32	−1.13
$\mathit{CPI}$ (%)	88.00	93.10	87.50	103.87	83.38

provides the schedule and cost performance’s calculation results for finished (including the finished part) activities. Table 6 shows that the duration and cost of activities A, B, C, and E are delayed, while activity D is advanced.

However, these basic metrics are insufficient for evaluating the project’s actual situation at the checkpoint. The path-based schedule approach can provide more helpful information. As shown in Figure 3, five paths are included. They are ${\mathit{P}}_{\mathit{I}}$ (A, C, F, I), ${\mathit{P}}_{\mathit{II}}$ (A, C, G, J), ${\mathit{P}}_{\mathit{III}}$ (B, D, G, J), ${\mathit{P}}_{\mathit{IV}}$ (B, E, H, J), and ${\mathit{P}}_{\mathit{V}}$ (B, E, H, K), with the activities included in these paths listed in parentheses. The planned duration of these paths is 29 months, 28 months, 32 months, 30 months, and 27 months, respectively. Obviously, ${\mathit{P}}_{\mathit{III}}$ is the original critical path.

Based on Equations (15)–(18), the $\mathit{SV}$ and $\mathit{SPI}$ for each path and $\mathit{CV}$ and $\mathit{CPI}$ at the checkpoint are calculated, as shown in

Table 7. Metrics for paths and checkpoint.

Path	${\mathit{P}}_{\mathit{I}}$	${\mathit{P}}_{\mathit{II}}$	${\mathit{P}}_{\mathit{III}}$	${\mathit{P}}_{\mathit{IV}}$	${\mathit{P}}_{\mathit{V}}$	Checkpoint
$\mathit{SV}$ (Month)	−0.7	−0.7	0.75	−1.54	−1.54	0.46
$\mathit{SPI}$ (%)	93	93	107.5	84.6	84.6	101.5
$\mathit{CV}$ (CNY ten thousand)	NA	NA	NA	NA	NA	−3.31
$\mathit{CPI}$ (%)	NA	NA	NA	NA	NA	91.26

. From the calculation’s results, the critical path (${\mathit{P}}_{\mathit{III}}$) schedule advanced. However, the significant schedule delay on ${\mathit{P}}_{\mathit{IV}}$ makes it the new critical path. If subsequent activities are executed as planned, the total duration of ${\mathit{P}}_{\mathit{IV}}$ is 31.54 months and the total duration of ${\mathit{P}}_{\mathit{III}}$ is 31.25 months.

In summary, the overall project schedule was advanced by 0.46 months and the cost was overspent by CNY 33.1 thousand up to the checkpoint.

4.3. Preliminary Estimation of Project Duration and Cost

Based on Equations (19)–(48), the three-point estimates, mean values, and standard deviations for ongoing activities (C′, D′, and E′) and not started activities (F, G, H, I, J, and K) are calculated, where C′, D′, and E′ represent the unfinished part of the related activities. For not started activities G, H, I, J, and K, the relevant parameters are calculated according to the performance of the corresponding categories; for unique activity F, it is calculated by the average performance. Details of calculation results are provided in

Table 8. Distribution parameters for ongoing activities.

Activity	C′	D′	E′
${\mathit{ERD}}^{\mathit{o}}$	2.00	2.51	1.14
${\mathit{ERD}}^{\mathit{m}}$	2.17	2.51	1.48
${\mathit{ERD}}^{\mathit{p}}$	2.48	2.51	1.77
$\mathit{ERMD}$	2.19	2.51	1.47
${\mathit{\sigma }}^{\mathit{d}}$	0.079	0.000	0.105
${\mathit{ERC}}^{\mathit{o}}$	3.50	3.29	1.33
${\mathit{ERC}}^{\mathit{m}}$	4.00	3.29	1.60
${\mathit{ERC}}^{\mathit{p}}$	4.33	3.29	2.07
$\mathit{ERMC}$	3.97	3.29	1.63
${\mathit{\sigma }}^{\mathit{c}}$	0.139	0.000	0.123

and

Table 9. Distribution parameters for not started activities.

Activity	F	G	H	I	J	K
${\mathit{ED}}^{\mathit{o}}$	12.00	8.00	10.00	6.00	10.00	6.17
${\mathit{ED}}^{\mathit{m}}$	12.64	8.67	10.61	6.17	12.96	6.17
${\mathit{ED}}^{\mathit{p}}$	13.34	9.90	12.05	6.62	15.55	6.17
$\mathit{EMD}$	12.72	8.76	10.75	6.22	12.90	6.17
${\mathit{\sigma }}^{\mathit{D}}$	0.291	0.317	0.342	0.104	0.924	0.000
${\mathit{EC}}^{\mathit{o}}$	10.00	11.00	15.00	9.00	12.00	7.70
${\mathit{EC}}^{\mathit{m}}$	10.97	12.57	17.05	9.67	14.39	7.70
${\mathit{EC}}^{\mathit{p}}$	11.45	13.62	18.08	9.94	18.66	7.70
$\mathit{EMC}$	10.89	12.48	16.88	9.60	14.70	7.70
${\mathit{\sigma }}^{\mathit{C}}$	0.242	0.437	0.513	0.156	1.109	0.000

.

After calculating the three-point estimates and mean values, the preliminary estimation of the project’s duration and cost upon completion (

Table 10. Preliminary estimation results.

Variables	${\mathit{P}}_{\mathit{I}}$	${\mathit{P}}_{\mathit{II}}$	${\mathit{P}}_{\mathit{III}}$	${\mathit{P}}_{\mathit{IV}}$	${\mathit{P}}_{\mathit{V}}$	Project
${\mathit{TEMD}}_{\mathit{p}}$ (Month)	30.98	33.80	34.14	35.05	28.26	35.05
$\mathit{TEMC}$ (CNY ten thousand)	NA	NA	NA	NA	NA	119.00

) can be obtained from Equations (49)–(53). The result demonstrates that the preliminary estimated total project duration and cost are about 35 months and CNY 1190 thousand, respectively, which are both higher than the planned values (32 months and CNY 1080 thousand). Therefore, it can be tentatively concluded that the project is more likely to be delayed or there will be an overrun in costs if no action is taken.

4.4. MCS

To further obtain the possible range of completion duration and cost, MCS will be performed on Oracle Crystal Ball, a leading spreadsheet-based application for predictive modeling, forecasting, simulation, and optimization. The MCS is executed according to the previously defined process, with 10,000 simulation times set. The statistical values and percentage points of the simulation’s results upon project completion duration and cost are presented in

Table 11. Statistical values of the simulation’s results.

Statistical Variables	${\mathit{P}}_{\mathit{I}}$	${\mathit{P}}_{\mathit{II}}$	${\mathit{P}}_{\mathit{III}}$	${\mathit{P}}_{\mathit{IV}}$	${\mathit{P}}_{\mathit{V}}$	Duration	Cost
Mean	31.06	33.84	34.16	35.11	28.39	35.11	118.99
Std	0.294	1.116	1.111	1.115	0.392	1.112	1.488
Min	30.23	30.55	30.88	31.85	27.43	31.71	114.60
25%	30.85	33.04	33.36	34.30	28.09	34.30	117.93
50%	31.06	33.84	34.16	35.12	28.36	35.12	118.95
75%	31.27	34.65	34.97	35.93	28.66	35.94	120.00
Max	32.03	37.05	37.34	38.45	29.69	38.45	123.94

and

Table 12. Percentage points of the simulation’s results.

Percentage Point	Duration	Cost
0%	31.70	114.60
10%	33.65	117.10
20%	34.11	117.72
30%	34.48	118.18
40%	34.83	118.58
50%	35.12	118.96
60%	35.44	119.36
70%	35.77	119.83
80%	36.12	120.33
90%	36.60	121.01
100%	38.47	124.00

. In addition,

Table 13. Frequency distributions of completion duration and cost.

	Duration			Cost
Number of Segments	Min	Max	Frequency	Min	Max	Frequency
1	31.70	32.38	29	114.60	115.54	61
2	32.38	33.05	269	115.54	116.48	334
3	33.05	33.73	898	116.48	117.42	1071
4	33.73	34.41	1567	117.42	118.36	2036
5	34.41	35.08	2125	118.36	119.30	2311
6	35.08	35.76	2141	119.30	120.24	1993
7	35.76	36.44	1672	120.24	121.18	1365
8	36.44	37.11	987	121.18	122.12	620
9	37.11	37.79	285	122.12	123.06	177
10	37.79	38.47	27	123.06	124.00	32

provides the frequency distributions of completion duration and cost, while Figure 4 and Figure 5 exhibit the PDF and CDF of the project’s completion duration and cost, respectively.

4.5. Analysis and Discussion

The result implies that both the total project duration and cost upon completion are subjected to the beta distribution. The estimated mean duration and cost at completion are 35.11 (months) and CNY 118.99 (units of CNY ten thousand). More importantly, there is a 90% probability that the project’s completion duration and cost fall within the range (31.70, 36.60) and (114.60, 121.01), respectively. Since the minimum boundary values are greater than or equal to the planned values, the project manager can confidently conclude that if no action is taken, there is a high probability of project delays and cost overruns. The first significant use of the proposed approach is presented here, that is, to help managers estimate the overall picture and possible ranges of the completion duration and cost of the project.

Simply determining whether a project will be delayed or overrun at completion is insufficient. The second major use of this approach is to help managers identify the sources of deviations, make decisions, and take actions on specific activities at the checkpoint based on their importance (priority).

In this case, based on Table 11, the estimated mean durations of ${\mathit{P}}_{\mathit{II}}$ (33.84 months), ${\mathit{P}}_{\mathit{III}}$ (34.16 months), and ${\mathit{P}}_{\mathit{IV}}$ (35.11 months) are all longer than the planned duration (32 months). Consequently, activities on these three paths need to be firmly focused on, especially for the activities that co-occur on multiple paths. In addition, the importance (priority) of activities is determined by the difference between the $\mathit{EMD}$ or $\mathit{EMC}$ (Table 9) and the $\mathit{PD}$ or $\mathit{PV}$ (Table 4), together with the sensitivity chart.

For the duration, the sensitivity chart (Figure 6a) shows that activities J and H significantly impact the total project’s duration. Furthermore, delayed activity J occurred on three paths with an estimated mean duration that is 2.90 months longer than planned (followed by activity G and H at 0.76 months and 0.75 months, respectively). There is no doubt that managers should focus primarily on activity J. Compared to activity H, the delayed duration of activity G is a little longer than that of activity H, but activity H is more sensitive and lies on a new critical path (P_IV). Therefore, in this case, activity H has a higher priority than activity G. For the cost, activities J, H, and G have a high sensitivity (Figure 6b). The cost overruns compared to the plan are 2.70, 1.88, and 1.48 (CNY ten thousand), respectively. Because the cost is independent of the path, activity J has the highest priority, followed by H and G.

In summary, the deviations in this case mainly originate from activities J, H, and G (in order of priority). Project managers should focus intensely on these activities, making timely decisions to ensure that the project’s schedule and cost are within controllable ranges.

5. Conclusions and Future Work

This study extends and improves the traditional EVM theory, combining CPM, PERT, and MCS techniques to propose a comprehensive approach for estimating project duration and cost upon completion. The main contribution is that it is the first study to incorporate logical relationships, work sequences, and activity categories into the improved EVM models. The path-based schedule evaluation metrics consider each path in the network diagram, addressing the problems of unclear sources of deviation, the offset of critical and non-critical paths schedules, and changes in critical paths during construction. In addition, noting the uncertainty of the activities, PERT and MCS are used to make the three-point estimates and generate fictitious project executions to obtain an empirical possible range of the project duration and cost at completion. However, Acebes et al. [18] argued that the estimated durations based on PERT are usually too optimistic. Therefore, activities are classified into different categories based on resource inputs to obtain more accurate estimations. The three-point estimates of not started activities are corrected for the performance metrics of finished activities or the finished part of ongoing activities in the same categories.

The outcome of the experimental research revealed excellent overall performances in terms of the schedule (cost) measurement, deviation identification, and completion duration (cost) estimation. At the checkpoint, the improved metrics were more in line and can help managers make accurate judgments about the schedule and cost situation of the project. The preliminary completion duration (cost) estimation based on improved EVM, CPM, and PERT helped managers understand the overall situation upon completion. In contrast, managers could make more confident judgments about project delays (cost overruns) due to the possible range based on MCS. When simulation results displayed a high probability of project delays or cost overruns, the proposed approach could identify the exact source of deviations and help managers pinpoint the specific activities that need strong attention and improvement according to their importance (priority). Suppose managers had not taken corrective actions at the checkpoint for activities J, H, and G. In that case, the project might have faced a 3-month delay and about CNY 110 thousand cost overrun, which would have been fatal for the contractor.

Applying the described approach to the experimental research validates its feasibility and usefulness, but the approach still has some limitations. For activities that have not started and do not fall into any of the categories, the average $\mathit{SPI}$ ($\mathit{CPI}$) is used to calculate the three-point estimates. However, the average performance is not a good estimator for the actual activity situation, leading to significant bias and unreliable results. Thus, the proposed approach is unsuitable for non-traditional construction projects requiring substantially more specified work. In addition, if the AOA network is very complex, it will spend more time defining the paths, making the approach more challenging in application.

In future, in order to apply this approach to various projects, a new evaluation system should be developed for not started activities that do not fall into any categories. Furthermore, the quality, environment, safety, resources, scope of work, and complex situation of interactions between different metrics should be reorganized and reconsidered to obtain more reliable and accurate estimation results.