Monitoring project duration and cost in a construction project by applying statistical quality control charts

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International Journal of Project Management 31 (2013) 411 – 423 www.elsevier.com/locate/ijproman

Monitoring project duration and cost in a construction project by applying statistical quality control charts Reza Aliverdi, Leila Moslemi Naeni, Amir Salehipour ⁎ Department of Civil Engineering, Amirkabir University of Technology, Tehran, Iran School of Industrial Engineering, Sharif University of Technology, Tehran, Iran Department of Industrial Engineering, Islamic Azad University — Garmsar Branch, Garmsar, Iran Received 25 February 2012; received in revised form 13 August 2012; accepted 14 August 2012

Abstract The earned value is a leading technique in monitoring and analyzing project performance and project progress. Although, it allows exact measurement of project progress, and can uncover any time and cost deviations from the plan, its capability in reporting accepted level of deviation is not well studied. This study presented an approach to overcome this limitation by applying statistical quality control charts to monitor earned value indices. For this purpose, project time and cost performance indices of a real construction project were monitored regularly on individual control charts. The results were quite promising, and not only competed well against traditional approaches, but also enhanced team's knowledge of project performance. At the end, it was concluded that the proposed approach improves substantially the project controlling scheme and enhances the capability of earned value technique. © 2012 Elsevier Ltd. APM and IPMA. All rights reserved. Keywords: Earned value technique; Individual control charts; Project performance indices; Schedule and cost performance indices

1. Introduction With increasing number of projects in areas of industry, construction, service, etc., and the complexity of managing and executing projects, the project management knowledge, standards and methods become more and more important. These coupled with the critically of completing projects according to the agreed scope, time and cost, have made the project management to receive special significance during recent years. Earned Value (EV) is a project management technique that measures project progress in an objective manner, and provides an early warning of performance issues, if any (PMI, 2004). EV measures project performance and progress by an integrated management of three most important elements in a project, namely cost, schedule and scope. In summary, EV ⁎ Corresponding author. E-mail addresses: [email protected] (R. Aliverdi), [email protected] (L. Moslemi Naeni), [email protected] (A. Salehipour). 0263-7863/$36.00 © 2012 Elsevier Ltd. APM and IPMA. All rights reserved. http://dx.doi.org/10.1016/j.ijproman.2012.08.005

provides indices for cost and time performances, and for project completion estimation. Although being introduced in 2000 in PMBOK® guide (PMI, 2000), the first complete guide on EV appeared in 2005 (PMI, 2005). It is widely accepted and well documented that implementing EV would bring added value to project monitoring scheme (Abba and Niel, 2010; Anbari, 2003; Blanco, 2003; Burke, 2003; Cioffi, 2006; Fleming and Koppelman, 2005; Henderson, 2003, 2004; Jacob, 2003; Jacob and Kane, 2004; Kim, et al., 2003; Lipke, 1999, 2003, 2004; McKim, et al., 2000). Several authors have improved the traditional EV by enhancing its capability in evaluating and monitoring project progress (Moslemi Naeni et al., 2011; Navon, 2005; Vandevoorde and Vanhoucke, 2005). From these, it is not surprising to see EV has been applied to many different disciplines and projects (Al-Jibouri, 2003; Antvik, 2000; Canepari and Varrone, 1986; Cass, 1998; Christensen and Ferens, 1995, 1996). In practice, project progress is evaluated by comparing EV indices and estimates against past values, against similar

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projects or according to several criteria (for instance, comparing against 1 and reporting any deviations). However, this evaluation does not usually provide additional information regarding the variation allowed for each index or estimate, provided that the status of that index remains still unchanged. Note that having such information is of interest. For instance, the Schedule Performance Index (SPI) is a measure of conformance of actual progress to schedule, and is usually evaluated by comparing the periodical SPI against 1. Unless SPI deviates from 1, everything seems under control. However, even short deviations may still contain warnings about current or future status of project which may highlight possible needs for corrective actions. Besides, in a high priority project, the small variations (still around value 1) may be more important than a project with a normal priority, however, the traditional approaches of evaluating and monitoring project progress cannot distinguish variations to this extent. In fact, the traditional approaches have not developed accordingly. Therefore, developing appropriate approaches to evaluate and monitor EV indices and estimates such that the associated variations can also be monitored would be valuable. This study targeted this goal and improved the way EV is evaluated and monitored. This includes applying statistical quality control charts to monitor statistically EV indices. Thus, performance and progress variations can also be monitored. The most important outcome of this integrated approach is that even small variations in recorded indices and estimates can be noticed and traced back. Despite benefits of this integrated approach, research targeting application of statistical methods to enhance EV is less available. Lipke discussed statistical distribution of several cost indices of EV, and discussed a method for approximating the distribution to normal distribution, in case of non-normally distributed data. The studies are important as they try to find out whether normality assumption for the indices is valid (Lipke, 2002, 2004, 2006, 2011). Christensen et al. (2003) also studied statistical distribution of cost data and reported that the data can approximately be normally distributed. Barraza et al. (2004) applied stochastic S-curves to determine forecasted project estimates. Their work included forecasting the project performance by projectizing the actual performance on plan, and by combining it with probability of completion. Later, Barraza and Bueno (2007) introduced a probabilistic project control concept by extending the performance control limit curves to derive an acceptable forecast of final project performance. Here the objective is to not exceed planned budget and schedule risk levels. Neither of these two studies considered statistical monitoring of project performance during its execution. Steyn (2008) also studied the importance of quality in projects but from another point of view, that is he studied a framework for managing quality on several specific projects. The work is more about providing a managerial framework rather than providing an analytical approach. Lipke et al. (2009) discussed forecasting final cost and duration of projects, and proposed confidence interval for this purpose. They verified the reliability of their method by applying it on 12 projects.

To the best of our knowledge, research on monitoring statistically EV indices and estimates is limited to Lipke, and Vaughn (2000), Moslemi Naeni et al. (2011), and to Leu and Lin (2008). Lipke, and Vaughn (2000) monitored several EV indices using statistical quality control charts while assuming the data are normally distributed. Moslemi Naeni et al. (2011) developed fuzzy control charts to monitor several EV indices, and provided a transformation method based on fuzzified indices. Leu and Lin (2008) improved the performance of traditional EV by implementing the statistical quality control charts. They implemented individual control charts (ImR charts) to monitor project performance data, and provided a log transformation method. However they did not perform hypotheses tests to verify auto-correlation among data. In this study, for the purpose of better reflecting the practical situations, such assumptions were replaced by statistical hypotheses tests. This innovative idea, that is combining EV and statistical quality control charts may improve the capability of EV, and can contribute to a better and more reliable project control. In fact, the study is important as it helps to find important changes in the project time and cost progress in advance. In practice, this would yield to recognizing and understanding the trend behind the project progress (those trends will help us to determine whether the project encounter delay, budget overrun, etc.), even very small trends, those that cannot be recognized if simple approaches are used. In contrast to typical evaluation methods, analyzing data statistically reveals more information, very important especially, for the case of high priority or critical projects. From these, the major contribution of this study is to develop a practical approach of combining EV and statistical quality control charts to monitor statistically EV indices, and to discover any variations. Among several advantages that the combination may bring, perhaps the most important one is understating behavior of EV indices over time, as it helps in measuring better project's performance and progress. Other contributions of the study lie in the way we applied the statistical quality control charts on project progress data. That is, we did not follow the assumptions other studies took, instead we tested statistically conditions required for developing control charts. As we will see later, the way we transform the non-normally distributed project progress data and how to understand which transformation works better are also contributions of the study. We would like to add to these contributions, application of the proposed approach on real data. From the point of applicability, it should be added that use of two statistical software will help people in charge to get more familiar on how the proposed approach can be applied in practice. We should note that the proposed approach is limited in which it cannot be applied to auto-correlated data, that is when data are statistically dependent on each other. It is worthy to mention that our approach is not limited to the underlying distributions of project progress data. Thus, as long as the data is independent, the method can be applied to projects, no matter what the underlying distribution is.

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Through the paper, our terminology is based on the PMBOK® guideline (PMI, 2004). The remaining of this study is organized as follows. Section 2 brings an introduction into EV and the associated indices and estimates. Section 3 explains briefly the statistical quality control charts. Section 4 combines EV and individual control charts and concludes with applying the developed approach on a real project experienced by authors in Iran. This case shows how the proposed approach can bring added value to the traditional project progress monitoring scheme. Besides, the superiority and effectiveness of proposed approach over traditional EV is shown by undertaking a comparative analysis. The paper ends with the conclusion. 2. Monitoring project progress by applying earned value The EV is developed to assist project teams in assessing and evaluating the project progress and performance. For this purpose, it includes several indices and estimates. By definition, EV of an activity is a measure of completed work, and represents the budgeted cost of work performed (PMI, 2005). Generally speaking, EV indicates how efficiently the project team utilizes the project resources, and is measured as what has been obtained and what was going to be obtained. Two widely used indices in evaluating project progress are Schedule Performance Index (SPI) and Cost Performance Index (CPI). Apart from these indices, several estimations are of interest, like completion cost and completion time of a project. We shall explain more SPI and CPI, as both have been studied in detail in Section 4, and refer the interested readers to “Practice standard for earned value management” (PMI, 2005) for more on EV, indices and estimates. SPI is a conformance measure of actual progress to schedule. SPI is measured as the ratio of EV to Planned Value (PV), that is . SPI ¼EV ð1Þ PV

where PV, known also as the Budgeted Cost of Work Scheduled (BCWS), is planned to be used during the project. CPI is a measure of budgetary conformance of actual cost of work performed, and is the most useful index indicating the cumulative cost efficiency of a project. CPI is the ratio of EV to Actual Cost (AC), that is . CPI ¼EV ð2Þ AC

where AC, known also as Actual Cost of Work Performed (ACWP), is an indication of the resources that have been used to achieve the actual work performed. A typical report on SPI and CPI indices has been depicted in Fig. 1. During the project, usually, EV indices are evaluated and monitored periodically on a weekly or a monthly basis. Fig. 1.a illustrated the SPI monitoring scheme, and Fig. 1.b referred to the CPI monitoring scheme. These two schemes were experienced by authors in a construction project. We shall discuss the case in more details later in Section 4.

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3. Statictical Process Control charts Statistical Process Control (SPC) is the application of statistical methods to monitor a process for the purpose of ensuring that it produces conforming product or service. Every process experiences variations. Variations can be categorized as controlled and uncontrolled. Controlled variations or common cause variations are natural to the process. In fact, they are inherent in the process, and are not considered so important in comparison with uncontrolled variations. On the other hand, uncontrolled variations or special cause variations are not typically presented in the process. It is important to find the cause of uncontrolled variations. The SPC is developed to distinguish between these two sources of process variations. For this reason, SPC includes a set of tools and techniques, among them statistical process control charts 1 are one of the most important techniques. Introduced by Shewhart in 1924, control charts have been widely applied to a variety of industries and processes (Montgomery, 2009). Although, several sophisticated control charts have been developed since 1924, Shewhart control charts are now considered as one of the primary tools for monitoring variations and shifts in processes. There are several reasons behind this, from which the most important ones are 1) simplicity of these control charts, and 2) the fact that most process measurements approximately follow statistical normal distribution. Fundamental assumptions of Shewhart control charts are: • The measurements (samples) on which control charts are applied should statistically be normally distributed, or at least can be approximated by normal distribution. • The measurements should be statistically independent of each other. That is, an observed measurement should not affect other measurements. Among Shewhart control charts, individual control charts which are known as XmR or ImR control charts can successfully be applied for the purpose of this study, that is monitoring statistically duration and cost progress of projects. The reasons are 1) the associated indices, i.e. SPI and CPI are quantifiable and can take continuous values, and 2) these EV indices and several other EV indices are observed on a weekly or monthly periods. Thus, the number of measurements (samples) is very limited. This is from the fact that in every period, a single measurement is recorded. Applying ImR control charts to SPI and CPI samples shows whether variations observed in the indices are common cause variation or special cause variation. From this and also depending on the degree of special cause variations, judgment regarding project schedule and cost performance can be made, that is whether the project performance is behind or ahead of plan (out-of-control). To further extend the idea of this study to much more practical cases, we avoid simplifying assumptions taken by previous studies like normally distributed data. Instead, routines and 1

Control charts are also known as process-behavior charts.

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a. SPI values over a 20-month period 2.5

SPI value

2

1.5

1

0.5

0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

Month

b. CPI values over a 20-month period 3 2.5

CPI value

2 1.5 1 0.5 0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

Month Fig. 1. Traditional monitoring scheme of EV indices, where project cost and schedule performances were monitored over a 20-month period. The graphs were generated by Microsoft Excel 2007. a) SPI data. b) CPI data.

approaches are considered to pre-process data in order to be practically monitored by ImR control charts. Indeed, control charts would bring added value to the traditional project progress monitoring scheme. Nevertheless, control charts have received little attention for the purpose of this study according to Section 1.

. CLIX ¼ x ¼ ∑ni¼1 xi

LCLIX ¼ x−3

 MR

ð4Þ

n

. 

ð5Þ

d2

3.1. Control limits for individual control charts The ImR control chart which monitors whether a special cause variation exists among samples of small sizes is composed of two charts. The first chart is called ‘IX’ and shows the value of each sample. The second chart or moving range, ‘MR’, monitors the variation between two consecutive samples. Eqs. (3)–(8) are those to derive control limits for IX and MR charts. UCLIX ¼ x þ 3

 MR

.  d2

ð3Þ

UCLMR ¼ D4 MR CLMR ¼ MR ¼ ∑ni¼2 MRi LCLMR ¼ D3 MR

ð6Þ . n−1

ð7Þ ð8Þ

where xi is each individual observation and MRi = |xi − xi − 1|. Note that in the case of this study, d2 = 1.128, D3 = 0, and D4 = 3.267.

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3.2. Methods to construct ImR control charts for non-normally distributed samples

Table 1 SPI and CPI measurements of the construction project observed over a 30-month period.

According to the control limits of IX and MR control charts, it is easy to follow that statistical normal distribution is not presented in the associated equations. Thus, it is not far if we say these charts are very robust for non-normal data (Wheeler, 2009, 2010). However, it is recommended against using these control charts for non-normally distributed data (Montgomery, 2009; Rashmi, 2008). Among several methods to apply Shewhart control charts for non-normally distributed data, a simple but practical method is to transform the non-normal data to normally distributed data. Another method is to derive probability control limits. Deriving probability control limits works almost perfect with any type of data as long as the probability density function can be calculated. Despite its superiority, the method is very complicated. As EV indices are to be measured and monitored by project experts who are not much familiar with advanced statistics, and with respect to the fact that ImR control charts are available for the purpose of this study, it is preferred to employ transformation. Two important transformations are Box–Cox and Johnson. In Box–Cox transformation a random variable, X, is powered to a value. This value is called λ, and is the parameter of Box–Cox transformation. Box–Cox transformation is shown in Eq. (9).   T ðX Þ ¼ X λ −1 =λ: ð9Þ

Sample no.

SPI

CPI

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.73 0.62 0.98 0.91 1.33 2.1 1.02 0.98 2.23 2.35 1.12 2.22 2.18 2.1 1.95 1.33 1.54 2.06 1.88 1.88 1.81 1.2 1.77 1.7 1.49 1.69 1.71 1.58 1.3 1.42

1 1.21 1.08 1.15 1.08 1.59 1.46 1.11 1.02 1.2 0.98 0.99 1.11 1.48 1.21 2.66 1.12 1.21 1.7 1 1.11 0.65 1.29 1.23 0.99 1.21 1.27 1 1 1.11

In Eq. (9), λ can take any value between − 5 and + 5. In case of X = 0, the transformation works according to natural logarithm. Several statistical software including Minitab can provide the best value of λ under Box–Cox transformation. One drawback of Box–Cox transformation is that it is applicable only to positive data. Despite Box–Cox, Johnson transformation can successfully be applied to even negative data. Johnson transformation is much more complicated than Box–Cox, although it can also be performed by almost all statistical software including Minitab. Similar to Box–Cox, Minitab can bring the best Johnson transformation on nonnormally distributed data. 4. Applying ImR control charts to monitor duration and cost on a construction project Following Shewhart fundamental assumptions regarding normally distributed samples, it is important to know that whether the collected samples of SPI and CPI measurements follow statistical normal distribution. Note that if other control charts rather than Shewhart control charts are implemented, understanding the statistical distribution of samples will always be a central step. The reason is that every control chart is developed according to several assumptions among which the most important one is the statistical distribution of samples. The SPI and CPI measurements (samples) have been recorded over a 30-month period from a construction project in Iran, and were reported in Table 1. The authors were responsible for monitoring project progress. For the purpose of

Table 2 Output of Easyfit's normality test over SPI samples. Kolmogorov–Smirnov Sample size 30 Statistic 0.09625 P-value 0.91887 Rank 17 α 0.2 Critical value 0.19032 Reject? No

0.1 0.21756 No

0.05 0.2417 No

0.02 0.27023 No

0.01 0.28987 No

Anderson–Darling Sample size Statistic Rank α Critical value Reject?

30 0.29289 10 0.2 1.3749 No

0.1 1.9286 No

0.05 2.5018 No

0.02 3.2892 No

0.01 3.9074 No

Chi-Square Deg. of freedom Statistic P-value Rank α Critical value Reject?

3 0.3499 0.95039 6 0.2 4.6416 No

0.1 6.2514 No

0.05 7.8147 No

0.02 9.8374 No

0.01 11.345 No

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Probability Plot of SPI Normal

99

Mean StDev N AD P-Value

Percent

95 90

1.573 0.4816 30 0.293 0.580

80 70 60 50 40 30 20 10 5 1 0.5

1.0

1.5

2.0

2.5

3.0

SPI Fig. 2. The Anderson–Darling statistical test for normality did not reject the hypothesis test of normally distributed SPI samples.

understanding the underlying statistical distribution of samples, Minitab version 15, and Easyfit version 3.4 were employed. Despite Minitab's capability for calculating and drawing control charts, its capability in determining the statistical distribution of samples is limited to only several well-known distributions. The capability even drops more when the user is to fit distributions one by one while Minitab validates the choice. On the other hand, Easyfit is developed for the purpose of determining the statistical distribution of samples. By automatically applying more than 50 different statistical distributions to fit to samples, and ranking statistically how well the applied distributions are fitted on the sample, Easyfit is one of best packages available. Ranking distribution in Easyfit is accomplished based on three statistical hypotheses tests: Kolmogorov–Smirnov, Anderson–Darling, and Chi-Square. It is worthy to mention that Easyfit performs a sensitivity analysis for each hypothesis test over different values of type I error (α). This is important as the hypothesis of, for instance, normally distributed samples may not be rejected for small α, but may be rejected for large α. In fact, when concluding the underlying distribution, the stability that is the hypothesis test is not rejected over different values of α, is of importance. According to Table 2, Easyfit reported that the hypothesis test of normally distributed SPI samples cannot be rejected according to all of three hypotheses tests. This can also be concluded by Minitab as Fig. 2 depicted this. The story was different with CPI samples (the third column of Table 1). According to the three statistical hypotheses tests performed by Easyfit over different values of type I error, that is α, Easyfit and Minitab not only rejected the hypothesis of normally distributed CPI samples (see Table 3, and Fig. 3), but also rejected the hypotheses of CPI samples being Exponentially, Weibully, and Log-Normally distributed (not shown here). The latter is important because these distributions are well studied and associated control charts are available. We

would like to emphasize that, as individual control charts (ImR) are probably the most suitable control charts for the purpose of this study, we would be interested in normally distributed CPI samples. This is mainly because despite the availability of several methods to improve robustness of ImR control charts when non-normally distributed samples exist (Wheeler, 2009, 2010), its use on non-normally distributed data is not recommended. From these, we are motivated to transform the non-normally distributed CPI measurements to normally distributed ones (Montgomery, 2009). Table 3 Output of Easyfit's normality test over CPI samples. Kolmogorov–Smirnov Sample size 30 Statistic 0.24025 P-value 0.05224 Rank 37 α 0.2 Critical value 0.19032 Reject? Yes

0.1 0.21756 Yes

0.05 0.2417 No

0.02 0.27023 No

0.01 0.28987 No

Anderson–Darling Sample size Statistic Rank α Critical value Reject?

30 2.4966 10 0.2 1.3749 Yes

0.1 1.9286 Yes

0.05 2.5018 No

0.02 3.2892 No

0.01 3.9074 No

Chi-Square Deg. of freedom Statistic P-value Rank α Critical Value Reject?

2 6.2318 0.04434 23 0.2 3.2189 Yes

0.1 4.6052 Yes

0.05 5.9915 Yes

0.02 7.824 No

0.01 9.2103 No

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Probability Plot of CPI Normal

99

Mean StDev N AD P-Value

Percent

95 90

1.207 0.3421 30 2.497 <0.005

80 70 60 50 40 30 20 10 5 1 0.5

1.0

1.5

2.0

2.5

CPI Fig. 3. The Anderson–Darling statistical test for normality rejected the hypothesis test of normally distributed CPI samples.

4.1. Transforming non-normally distributed CPI to normally distributed CPI Results of applying Box–Cox and Johnson transformations on the non-normally distributed CPI measurements were shown in Table 4. According to Minitab, the best Box–Cox transformation resulted under λ = − 0.9. Unfortunately, even this value could not result in normally distributed CPI samples according to Fig. 4. Fig. 5 illustrated the Johnson transformation on CPI samples. The figure was prepared by Minitab. According to the figure, Johnson transformation is quite capable of transforming non-normally distributed CPI samples to normally distributed samples. To conclude this section, in the construction project we implemented Johnson transformation to derive normally distributed CPI samples. 4.2. Calculating control limits Now that the SPI and CPI samples, directly or indirectly, follow normal distribution, the first condition of applying ImR control charts to monitor project progress is met. However, the second condition still has to be verified, i.e. independent samples (see second assumption of Shewhart control charts in Section 3). This condition can simply be verified in Minitab by checking autocorrelation between samples. According to this which has been depicted in Fig. 6, SPI and CPI samples are independent of each other. This means, for each index either SPI or CPI, observed measurements will not affect each other. Note that this is not a general conclusion, and the statement can be verified only for the case under study whose data is reported in Table 1. Generally speaking, the data of project performance can be either dependent or independent. If the project status is unstable, that is it experiences periods of, for instance, on schedule followed by periods of behind the schedule, etc., it is expected that the data are much more dependent on each other

compared to a stable project. This is typically because it takes a while to stick to the baseline plan after a shift happens. The opposite can be expected for a more stable project.

Table 4 The Box–Cox transformations under different λ values and the Johnson transformation on CPI measurements. Sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Box–Cox transformation

Johnson transformation

0.5

0.75

1

3

5

0 0.2 0.08 0.14 0.07 0.52 0.42 0.11 0.02 0.19 − 0.02 − 0.01 0.11 0.43 0.2 1.26 0.12 0.2 0.61 0 0.11 − 0.39 0.27 0.22 0.01 0.2 0.25 0 0 0.11

0 0.21 0.08 0.14 0.07 0.55 0.44 0.11 0.02 0.2 − 0.02 − 0.01 0.11 0.46 0.21 1.44 0.12 0.21 0.65 0 0.11 − 0.37 0.28 0.22 − 0.01 0.21 0.26 0 0 0.11

0 0.21 0.08 0.15 0.08 0.59 0.46 0.11 0.02 0.2 − 0.02 − 0.01 0.11 0.48 0.21 1.66 0.12 0.21 0.7 0 0.11 − 0.35 0.29 0.23 − 0.01 0.21 0.27 0 0 0.11

0 0.26 0.09 0.17 0.08 1.01 0.7 0.12 0.02 0.24 − 0.02 − 0.01 0.12 0.75 0.26 5.94 0.13 0.26 1.3 0 0.12 − 0.24 0.38 0.29 − 0.01 0.26 0.35 0 0 0.12

0 0.32 0.09 0.2 0.09 1.83 1.13 0.14 0.02 0.3 − 0.02 − 0.01 0.14 1.23 0.32 26.43 0.15 0.32 2.64 0 0.14 − 0.18 0.51 0.36 − 0.01 0.32 0.46 0 0 0.14

− 1.06 0.36 − 0.43 0.06 − 0.43 1.29 1.07 − 0.20 − 0.90 0.32 − 1.20 − 1.13 − 0.20 1.11 0.36 2.16 − 0.13 0.36 1.43 − 1.06 − 0.20 − 2.34 0.65 0.44 − 1.13 0.36 0.59 − 1.06 −1.06 − 0.20

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Probability Plot of CPI-Converted 0.75

Probability Plot of CPI-Converted 0.5

Normal

Normal Mean 0.1805 StDev 0.2783 N 30 AD 1.878 P-Value <0.005

95 90 80 70 60 50 40 30 20 10 5 1

99

Percent

Percent

99

Mean StDev N AD P-Value

95 90 80 70 60 50 40 30 20 10 5 1

-0.5

0.0

0.5

1.0

1.5

-0.5

0.0

0.5

Probability Plot of CPI-Converted 1

Probability Plot of CPI-Converted 3 Normal

Percent

Percent

99

Mean 0.2073 StDev 0.3422 N 30 AD 2.489 P-Value <0.005

95 90 80 70 60 50 40 30 20 10 5 0.0

0.5

1.0

1

1.5

Mean 0.4235 StDev 1.091 N 30 AD 6.058 P-Value <0.005

95 90 80 70 60 50 40 30 20 10 5 -3

-2

-1

1

2

3

4

5

6

Box-Cox Plot of CPI-Not converted

Probability Plot of CPI-Converted 5 Normal

Lower CL

99

Mean 1.236 StDev 4.797 N 30 AD 8.920 P-Value <0.005

95 90 80 70 60 50 40 30 20 10 5

Upper CL Lambda (using 95.0% confidence) Estimate -0.90

1.1 1.0

StDev

Percent

0

CPI-Converted 3

CPI-Converted 1

1

1.5

CPI-Converted 0.75

99

-0.5

1.0

CPI-Converted 0.5

Normal

1

0.1931 0.3071 30 2.160 <0.005

0.9

Lower CL Upper CL

-2.18 0.17

0.8

Rounded Value

-1.00

0.7 0.6 0.5 0.4 0.3

Limit

0.2 -10

0

10

20

30

CPI-Converted 5

-5.0

-2.5

0.0

2.5

5.0

Lambda

Fig. 4. The Box–Cox transformation on CPI samples over different values of λ.

Having in hand normally distributed and independent samples, we proceeded to calculate the control limits of ImR control charts. Here, a routine is depicting control charts, and removing out-of-control samples (usually if assignable causes are found) until no sample is required to be removed. Upon removal of each sample, control limits are calculated again. The final control limits are used to draw control charts to monitor project progress (from now on). Table 5 shows the final control limits of SPI and CPI ImR control charts. Note that as ImR control charts consists of two charts namely IX and MR, we have two sets of control limits.

Remember that SPI and CPI were sampled individually on a monthly basis. This is the main reason that probably the most suitable control charts are individual control charts. These control charts reveal whether the meaningful variation (special cause variation) in the performance of project team is detected (remember SPI and CPI show the project schedule and cost performance, which means that they reflect the performance of project team in utilizing limited project resources). Understanding this special cause variation is very important, and can alert project team before project deviates much from the plan.

R. Aliverdi et al. / International Journal of Project Management 31 (2013) 411–423

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Johnson Transformation for CPI Probability Plot for Original Data 99

P-Value for AD test

N 30 AD 2.497 P-Value < 0.005

90

Percent

Select a Transformation 0.36

50

10

0.3 0.2 0.1

Ref P

0.0 0.2

1

0.4

0.6

0.8

1.2

1.0

Z Value 0

1

2

3

(P-Value = 0.005 means < = 0.005)

Probability Plot for Transformed Data 99

N 30 AD 0.399 P-Value 0.344

Percent

90

P-Value for Best Fit: 0.344170 Z for Best Fit: 0.36 Best Transformation Type: SU Transformation function equals -0.640468 + 0.803143 * Asinh((X-1.05349) / 0.0982016)

50

10 1

-2

0

2

Fig. 5. The summary of Johnson transformation on CPI samples.

Autocorrelation Function for SPI

Autocorrelation Function for CPI-Transformed (with 5% significance limits for the autocorrelations)

1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

Autocorrelation

Autocorrelation

(with 5% significance limits for the autocorrelations)

1

2

3

4

5

6

7

8

1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 1

2

3

4

Lag

5

6

7

8

Lag

Fig. 6. Autocorrelation graphs of SPI and CPI samples show samples are statistically independent.

To show how the control limits of Table 5 can lead us to monitor statistically project progress, we have obtained 10 samples of SPI and CPI indices. Each sample referred to a monthly schedule and cost performance. The observed samples were reported in Table 6. Note that as CPI measurements for this project were previously found to be non-normally distributed, the

Table 5 Control limits of IX and MR charts.

IX chart

MR chart

UCL CL LCL UCL CL LCL

SPI

CPI-transformed

2.548 1.573 0.598 1.198 0.367 0

2.957 -0.072 -3.101 3.721 1.139 0

last column of the table, has also reported the transformed samples using Johnson transformation. Fig. 7 depicted ImR control charts to monitor project schedule and cost performance over a 10-month period. For this Table 6 SPI and CPI measurements for a 10-month period. Sample number

SPI

CPI

Transformed CPI

1 2 3 4 5 6 7 8 9 10

1.43 1.38 1.31 1.25 1.21 1.19 1.13 1.08 1.03 1

1.21 1.21 1.20 1.18 1.17 1.11 1.08 1.05 1.03 1.01

0.36 0.36 0.32 0.28 0.24 − 0.20 − 0.13 − 0.43 − 0.91 − 1.06

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I-MR Chart of SPI Individual Value

2.5

UCL=2.548

2.0 _ X=1.573

1.5 1.0

3

2

2

2

8

9

10

LCL=0.598

0.5 1

2

3

4

5

6

7

Observation

Moving Range

1.2

UCL=1.198

0.9 0.6 __ MR=0.367

0.3 0.0

2

1

2

3

4

5

6

7

8

9

2

LCL=0

10

Observation

I-MR Chart of CPI-Transformed Individual Value

UCL=2.957

2 _ X=-0.072

0 -2

LCL=-3.101

-4 1

2

3

4

5

6

7

8

9

10

Observation 4

Moving Range

UCL=3.721

3 2 __ MR=1.139

1 2

0

2

1

2

3

4

5

6

7

8

9

LCL=0

10

Observation Fig. 7. ImR control charts for monitoring project schedule and cost progress over a 10-month period.

purpose, SPI and CPI indices have been depicted separately on IX and MR control charts. Although in Fig. 7 no points are outside the control limits, but according to the Nelson's rules (Nelson, 1984) from month 7 on, SPI indices are out-of-control (this conclusion was drawn according to IX control chart and we will discuss MR control charts later in the Section). In this figure, these are highlighted in red and they mean non-random behaviors are happening in these periods. Note that points associated to previous months are also below center line; however a sequence of them in a row may imply a non-random behavior. Thus from month 7, IX control chart is alarming that something is going wrong in project execution. Here, the project team was assigned to find the assignable causes, if any. They came with “delay in delivering materials” as the cause of out-of-control indication for SPI. Remember in month 7 SPI is more than 1 and even in month 10 it is 1, which may sound quite promising. But IX control chart

warned us very early, that is, in month 7, that project execution is deviating from its typical progress (not necessarily from its schedule) more than what is accepted. Hence, even promising indices' values may proactively tell us problems in project execution. It is worthy to mention that according to Fig. 7, the project cost progress is under control. It is noticeable that cost performance measured by CPI is dropping; however, since it behaves randomly we conclude that project cost performance is under control. Let's now discuss MR charts. These charts are more complicated to interpret than IX charts; because the measurements on charts are correlated meaning they may result in pattern of runs or cycles (Montgomery, 2009). Thus, their out-of-control status as in Fig. 7 should be discussed carefully. To conclude the Section, the most important piece of information the charts provide us with is “issues might be appearing that would cause negative effects on project performance. Though they are not serious yet, but corrective

R. Aliverdi et al. / International Journal of Project Management 31 (2013) 411–423

421

Run Chart of SPI 1.4

SPI

1.3

1.2 1.1 1.0 1

2

3

4

5

6

7

8

9

10

8

9

10

8

9

10

Observation Number of runs about median: Expected number of runs: Longest run about median: Approx P-Value for Clustering: Approx P-Value for Mixtures:

2 6.0 5 0.004 0.996

Number of runs up or down: Expected number of runs: Longest run up or down: Approx P-Value for Trends: Approx P-Value for Oscillation:

1 6.3 9 0.000 1.000

Run Chart of CPI 1.20

CPI

1.15

1.10

1.05

1.00 1

2

3

4

5

6

7

Observation Number of runs about median: Expected number of runs: Longest run about median: Approx P-Value for Clustering: Approx P-Value for Mixtures:

2 6.0 5 0.004 0.996

Number of runs up or down: Expected number of runs: Longest run up or down: Approx P-Value for Trends: Approx P-Value for Oscillation:

1 6.3 9 0.000 1.000

Run Chart of CPI-Transformed 0.50

CPI-Transformed

0.25 0.00 -0.25 -0.50 -0.75 -1.00 -1.25 1

2

Number of runs about median: Expected number of runs: Longest run about median: Approx P-Value for Clustering: Approx P-Value for Mixtures:

3 2 6.0 5 0.004 0.996

4

5 6 Observation

Number of runs up or down: Expected number of runs: Longest run up or down: Approx P-Value for Trends: Approx P-Value for Oscillation:

7 3 6.3 5 0.003 0.997

Fig. 8. Run charts to monitor project duration and cost progress over a 10-month period.

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actions might be recommended”. Note that according to Table 6 and values of SPI and CPI indices, the project performance appears normal and “on the plan” at the end of this period which is considered quite promising. But how important the deviation would be is the thing that only control charts can tell us, nor SPI and CPI do, neither Fig. 1 does. Control charts provided extra information that could not be obtained if other tools such as run charts were to be applied. Fig. 8 depicted run charts for SPI and CPI measurements of Table 6. Run charts cannot judge project performances on deviations or shifts. Trends of indices over time are the only pieces of information run charts provide us with, also important, but there is no information regarding the project performance behavior. That is, how much deviation, either common cause or special cause is seen, and more importantly, how much is allowed. In fact, run charts are not developed for this purpose. For instance, from top figure in Fig. 8, it is easy to show the project was ahead of schedule in the first month. It can also be seen that the project schedule performance was good as the SPI value was about 1.43. Furthermore, project's performance gradually dropped until it reached on schedule performance in the 10th month, although, the project has been ahead of schedule in the meantime. The run charts provided no additional information regarding common cause and special cause deviations associated with project progress. 5. Conclusion Through several indices such as schedule performance index and cost performance index, earned value technique is quite capable of monitoring and analyzing project performance. Although EV has been studied well, there is a lack of study regarding an integrated approach composed of EV and statistical quality control charts. Such an approach may improve the capability of EV, and can contribute to a better and a more reliable project control. In this study individual quality control charts or ImR charts were applied to monitor SPI and CPI. This combination helps to find important changes in the project time and cost progress soon enough. Furthermore, the combination lets even very small trends, those that cannot be recognized if simple approaches are used, to be discovered. To help readers to apply the proposed approach to their own projects, ImR control charts were designed and were applied to a construction project. As results showed, the approach provided the project team with project performance behavior in the form of common cause and special cause variations together with the associated accepted levels. The latter is of importance and will bring warnings about future trends to the extent of even very small changes. It is worth mentioning that while the proposed approach can be applied successfully to other EV indices, it can be customized according to the project under execution, considering factors such as underlying distribution of EV indices' measurements (samples), and their dependency or independency. Although, the approach is not limited to the underlying distributions of indices' measurements, we should note that it is limited to only

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