Journal of the Operational Research Society (2016), 1–14
© 2016 Operational Research Society Ltd. All rights reserved. 0160-5682/16 www.palgrave-journals.com/jors/
A better project performance prediction model using fuzzy time series and data envelopment analysis Mostafa Salari1* and Homayoun Khamooshi2 1
University of Calgary, Calgary, Canada; and 2George Washington University, Washington DC, USA
Earned value management (EVM) is a critical project management methodology that evaluates and predicts project performance from cost and schedule perspectives. The novel theoretical framework presented in this paper estimates future performance of a project based on the past performance data. The model benefits from a fuzzy time series forecasting model in the estimation process. Furthermore, fuzzy-based estimation is developed using linguistic terms to interpret different possible conditions of projects. Eventually, data envelopment analysis is applied to determine the superior model for forecasting of project performance. Multiple illustrative cases and simulated data have been used for comparative analysis and to illustrate the applicability of theoretical model to real situations. Contrary to EVM-based approach, which assumes the future performance is the same as the past, the proposed model can greatly assist project managers in more realistically assessing prospective performance of projects and thereby taking necessary and on-time appropriate actions. Journal of the Operational Research Society advance online publication, 6 April 2016; doi:10.1057/jors.2016.20 Keywords: earned value management; cost and schedule performance index; fuzzy time series; forecasting; performance management; data envelopment analysis
1. Introduction Earned value management (EVM) is a powerful project management technique, which grants program managers, project managers and other top-level stakeholders the ability to assess and visualize the status of project during the project life cycle. Consequently, managing projects, programs and portfolios can be achieved more effectively. In addition, EVM provides multiple project assessments if appropriately applied, and clearly identifies the opportunities to maintain control over the budget, schedule and scope of various types of projects. The project management body of knowledge (PMBOK) Guide initially defines EVM as ‘a management methodology for integrating scope, schedule, and resources for objectively measuring project performance and progress’ (Project Management Institute (PMI), 2008). To make this integration possible, EVM benefits from the measurement indices that calculate and determine schedule and cost performance all the way from initiation to completion of projects. EVM also provides a method for estimation of project total cost and duration. Owing to its simplicity in calculation and applicability in monitoring and assessment of projects, EVM has a unique position in project management arena. As such, many practitioners and researches addressed the application of EVM in different industries, disciplines and organizations (Al-Jibouri, 2003; Moselhi et al, 2004; Bagherpour et al, 2010; Baumeister *Correspondence: Mostafa Salari, Department of Civil Engineering, University of Calgary, University Drive, NW, Calgary, Alberta, T2N 1N4, Canada. E-mail:
[email protected]
and Floren, 2011). The other researchers mostly focused on the development of EVM to improve its application and efficiency. Some argued the limitation of EVM (Anbari, 2003; Jacob, 2003; Lipke, 2003). For instance, Lipke (2003) proposed the concept of earned schedule (ES) to overcome the limitation of traditional schedule performance index. Consequently, SPI(t), a new performance metric for assessment of schedule performance was developed on the basis of the ES concept. Others discussed the effectiveness of EVM implementation and improved the traditional EVM through the enhancement of this technique to monitor project progress (Cioffi, 2006; Vanhoucke and Vandevoorde, 2007; Moslemi Naeni and Salehipour, 2011; Aliverdi et al, 2013; Salari et al, 2014a,b). Project managers who are responsible for the outcome of the project need to make decisions about the future of projects. In this regard, EVM systems assist them to pursue the performance of a project effectively and provide estimate at completion for time and cost such as EAC (Estimate at completion cost) and estimate at completion for time (EAC(t)). Basically, such estimation requires the project managers to schedule and control complex activities involved in the project using a forward-looking and forward-controlling mechanism that undertakes a high level of uncertainty and vagueness affecting the project. EVM deals with the concept of estimate at completion on the basis of a deterministic approach that is suited mostly to projects with low level of uncertainty (Fleming and Koppelman, 2000). From the birth of EVM till now, many models have been presented, however, none of them has provided sufficient evidence of applicability, rigour and
2
Journal of the Operational Research Society
accuracy to be integrated with the EVM permanently. Anbari (2003) proposed six general formulas for the estimate at completion from which three were already included in the PMBOK by PMI (2000). While these models could be well used for less risky or more deterministic projects, the prediction of future project performance in terms of cost and schedule, specifically for projects with high degree of uncertainty brings about quite a new challenge, which is rarely discussed. Thus, the development of such approach can be a very beneficial tool for project managers and their decision-making processes. It will become especially useful in cases where the project managers are interested in forecasting the project performance periodically through the whole life cycle of a project or certain periods such as milestones. Such predictions help to understand, manage and monitor the future performance of a project. The main question is which period’s performance index, the current or cumulative performance measure should be used. For instance, consider a situation when the estimation of project total cost (EAC) using current period assessment cost performance indicates that the project final cost will be greater than the planned budget, but using the cumulative cost performance index (CPI) illustrates that the project has performed as planned. In such condition, it becomes significant to predict when the cost performance of project will exceed the normal condition. This problem can also be asked generally as how the future performance of a project will be through the execution stage. The same problem can occur for schedule performance. Despite extensive benefits of performance prediction addressed above, most of research and developments under the umbrella of EVM are focused on predicting the project total cost or time, and not on forecasting the project performance to predict project future cost and duration. In this regard, Anbari (2003), Lipke (2003) and Jacob and Kane (2004) developed methods for the prediction of project total time using planned value (PV), earned duration (ED) and ES metrics, respectively. Furthermore, Barraza et al (2004) applied probabilistic forecasting of project performance using stochastic S-curves. Dillibabu and Krishnaiah (2005) discussed cost estimation method in terms of effort spent on a software project. Lipke et al (2009) introduced a final time and cost forecasting method applying statistical approach. Warburton (2011) developed a model for the estimation of project final cost concerning how to exceed the convergence to the appropriate result with less variation than a typical model for the estimate at completion calculations. Feylizadeh et al (2012) discussed a fuzzy neural network to estimate at completion costs of a construction project. Azman et al (2012) studied the accuracy of preliminary cost estimates in public work departments. Recently, Elshaer (2013) studied the impact of information on forecasting project total time. He also investigated the reliability of ES method at higher levels of work breakdown structure. Caron et al (2013) proposed a Bayesian approach to improve the estimate at completion in EVM. This study focuses on periodic estimation of project performance using cost and schedule performance indexes introduced in existing practice of the EVM technique. The approach in this
1. Collecting periodic data of cost and schedule performance (i.e. CPI and SPI t)
2. Using time series models to predict the future cost and schedule performances
3. Calculating errors of prediction using three error measurement metrics including: • Mean square error (MSE) • Mean absolute error (MAE) • Mean absolute percentage error (MAPE)
4. Applying DEA to measure the efficiency of prediction of time series models based on MSE,MAE, and MAPE
5. Selection of best time series model for the prediction using the results of DEA
Figure 1 Steps of the proposed model.
paper is developed by not simply using a fuzzy time series as the prediction tool but by providing a framework that can help project managers or decision makers to assess the fuzzy time series models, and to determine the best option in each project. The predicted indices or forecasted performance measures can be applied to estimate project cost or duration at every step or phase of a project. The rest of this paper is organized as follows: the terminology and indices in EVM used in this study are defined and introduced initially. Then, the fuzzy time series models, the evaluation methods for these models, case studies, simulation model and the results obtained are discussed in the subsequent sections. Finally, discussion and conclusions are presented. We have also provided Figure 1 to introduce the major steps of the proposed model in this study.
2. EVM terminology used in the proposed model Different measures in EVM are used to assist project managers in controlling the project via accurate assessment of value earned and progress made. This function is performed by comparing the planned values against the corresponding actual. The most significant applied metrics in EVM are CPI, schedule performance metric (SPI) and estimate at completion for the duration (EACt) and estimate at completion for the cost (EAC). If SPI and CPI are less than unity they will indicate weak performance of the project from schedule and cost points of view, respectively. Moreover, CPI and SPI indicate on budget and on schedule project if they are equal to unity. Table 1 demonstrates a general scheme of metrics and other terminology used in EVM researches: The estimate at completion indices (EAC and EACt) are also the variables of interest in this paper. The EAC and EACt are
Mostafa Salari and Homayoun Khamooshi—A better project performance prediction model using fuzzy time series and data envelopment analysis
3
Table 1 The proposed metrics in EVM Abbreviations
Description
Equations
BAC SAC AP PV EV ES AC AD SPIt CPI SCI(CR) SV CV
Budget at completion Schedule at completion Actual progress (the progress based on the work performed) Planned value Earned value Earned schedule Actual cost up to the data date Actual duration up to the data date Schedule performance index Cost performance index Schedule cost index Schedule variance Cost variance
Planned progress × BAC AP × BAC - PVN *N + PVEV N + 1 - PVN ES/AD EV/AC SPI × CPI EV—PV EV—AC
*N is the longest interval in that the planned value at time N (ie PVN) is less than EV.
calculated by simply dividing BAC and SAC by CPI and SPI at each point in time. The critical question is which index value best present the future performance factor or which performance factor (PF) should be used in the denominator of both EAC and EACt. One needs to make some assumptions in order to pick the appropriate value of PF. One of these assumptions, which is also presented in a paper by Anbari (2003) assumes that performance in the future will not be the same as past performance. While we agree with Anbari’s model that future performance is going to be different from the past, in practice the most widely used PF indices are SPI and CPI or SPIt, where the latest cumulative reported performance measures are used for forecasting the project completion duration and cost. In another word, finding most reliable and practical approach for determining or predicting PF is a challenge when it is assumed that future will not necessarily be the same as the past. Hence, we argue that existing approaches provide very simplistic view of future, which could not be supported logically. We know that when the project is not performing well, the project management is expected to take action and change course to improve performance, or the performance index at different stages of the project should not be expected to be the same. So with these adjustments and corrections and varied characteristics and nature of phases it is not reasonable to assume the same performance as the past for the future periods, which is done in most of the existing approaches. Here we are going to focus on predicting the future values for SPI or CPI, which can also be used as PF in estimation of project total time and cost. In this paper, we suggest the use of fuzzy time series approach as an effective approach for estimation of project future performance, which is explained and discussed in the next section.
3. Fuzzy time series methodology Performance prediction has always played a significant role in planning and control of projects, and keeps on doing so in the future.
This prediction capability provides the project managers with a capacity to minimize the risk, improve project management performance and possibly lead to more successful delivery of a projects. There are many ways of making predictions of the future or forecasting. Fuzzy time series, as a forecasting method, provides a practical approach for predicting a variable of interest behavior in situations where neither a past trend is apparent nor a specific pattern in variations are visualized. Furthermore, it deals effectively with the data sources that are imprecise and vague. Hence it is applied in many studies for the prediction purposes in different fields (Chen and Hwang, 2000; Wang and Hsu, 2008; Wong et al, 2010). Considering the forecasting objectives, the fuzzy time series is utilized here to predict future cost and schedule performance of project. To elaborate and explain the application of fuzzy time series in EVM domain, we first introduce the basic concept of fuzzy time series based on Song and Chissom (1993a,b, 1994) primitive model. Their proposed model is comprised of seven steps: (1) determine the universe of discourse; (2) divide the universe into equal lengthy intervals (linguistic values); (3) determine fuzzy sets on the universe; (4)fuzzify the data; (5) acquire the logical relation between the fuzzified historical data; (6) calculate the predicted outputs; (7) assess and interpret the predicted outputs. Song and Chissom defined fuzzy time series as follows: Suppose U be the universe of discourse, where U = {u1,u2,u3,…,un}. A fuzzy set Ai of U can be defined by Ai ¼
fAi ðu1 Þ fAi ðu2 Þ fAi ðu3 Þ fA ðun Þ + + + + i u1 u2 u3 un
(1)
where fAi illustrates the membership function of fuzzy set Ai, fAi : U ! ½0; 1, uk is the element of fuzzy set Ai. fAi ðuk Þ is the membership degree of uk in Ai. fAi ðuk Þ 2 ½0; 1, where 1 ⩽ k ⩽ n Definition 1: (Song and Chissom (1993a,b)). Z(t) t = 0,1,2,3,…n, is a subset of R. consider Y(t) as the universe of discourse defined by fi(t), i = 0,1,2,3,…,n. if F(t) consists of fi(t), i = 0,1,2,3,…,n, F(t) is supposed to be a fuzzy time series in Z(t), t = 0,1,2,3,…n.
4
Journal of the Operational Research Society
Definition 2: (Song and Chissom (1993a,b)). If there exists a fuzzy relationship R(t − 1,t), such as: F(t) = F(t − 1) × R(t − 1,t), where × illustrates an operator. F(t) is known to be caused by F(t − 1). If F(t − 1) = Ai and F(t) = Aj. Then the relationship between F(t) and F(t − 1) is defined as fuzzy logical relationship and denoted by Ai → Aj. The × operator can be regarded as max-min or min-max. Definition 3: (Song and Chissom (1993a,b)). Suppose F(t) be a fuzzy time series. If F(t) is caused by F(t − 1), F(t − 2),…, and F(t − n). Such fuzzy logical relation is illustrated by Fðt - nÞ; ¼ ¼ :Fðt - 2Þ; Fðt - 1ÞðtÞ
(2)
A fuzzy time series with the aforementioned features is described as nth order fuzzy time series forecasting model. There are many studies in the literature that attempted to develop the Song and Chissom proposed model further (Chen, 1996; Huarng, 2001; Lee and Chou, 2004; Singh, 2007, 2009; Liu et al, 2011; Gangwar and Kumar, 2012). To assess the suitability and perfromance of the model to be used we tested three different fuzzy time series models for our forecasting puposes. It’s worth mentioning that the basic principles of all these models are based on Song and Chissom presented model. Hence, to keep the paper short, we avoid explaining each model separately. Though, the most important features of each emplyed model will be described as follows: 1. (Chen, 1996). This study benefits greatly from a simplified fuzzy-based arithmetic operations rather than the complicated min-max or max-min operations proposed in Song and Chissom (1993a). Furthermore, it is well-suited for first- and second-order fuzzy time series and is not suitable for highorder models. 2. (Singh, 2009). His proposed model further developed the approach for high-order fuzzy time series forecasting. It also proved to be well-fitted in cases of high uncertainty in time series data. 3. (Gangwar and Kumar, 2012). It is another sophisticated model focusing on high-order fuzzy logic relations in time series data. Another advantage of this model is portioning of historical data for the purpose of accuracy improvement in forecasting process. It has to be stated that because of inappropriateness of Chen (1996) model for high-order forecasting, the first order of his proposed time series model has been employed, whereas the higher order of Singh (2009) model, up to order 7, and Gangwar and Kumar (2012) model, up to order 6, have been utilized for the comparative analysis. Thus, 10 time series models have been tested for selecting the most appropriate model to be used in the estimation process of project performance.
show the accuracy of the prediction. The accuracy of forecast (outputs) is commonly evaluated in terms of mean square error (MSE), mean absolute percentage error (MAPE) and mean absolute error (MAE). Lower the MSE, MAPE or MAE indicates the better forecasting method. The MAE, MSE and MAPE are obtained as shown below: Pn ðactual valuei - forecasted valuei Þ2 Mean square error ¼ i¼1 n (3) Pn Mean absolute error ¼
i¼1
jactual valuei - forecasted valuei j n (4)
Mean absolute percentage error ¼ Pn forecasted valuei - actual valuei i¼1 actual valuei n
´ 100
ð5Þ
where, n indicates the number of errors or data points. It is worth mentioning that all of the aforementioned error estimation methods except MAPE have scaled output.
3.2. Data envelopment analysis Data envelopment analysis (DEA) is an established method used to assess performance of similar systems and to identify the relative efficiency of each. DEA is non-parametric type of programming method for evaluating a given set of decisionmaking units. There are two types of envelope models: input or output oriented. In output-oriented model, the level of all input remains constant and the model measures minimal radial increases in the outputs. Initially, Charnes et al (1978) introduced the Charnes Cooper and Roads model and then Banker et al (1984) developed the Banker Charnes Cooper model for assessment of relative efficiency in economic production systems.1 In this particular application, fuzzy time series models are compared for optimum estimation of project performance. Hence, an output-oriented DEA approach is employed to evaluate relative efficiency of time series models with respect to project performance estimation based on three error indices: MSE, MAE and MAPE. In this regard, an output-oriented model presented by Lovell and Pastor (1999) is utilized to examine the relative efficiency of time series models. The Lovell and Pastor (1999) proposed model can be shown as follows: Min eT τ s:t: Y τ ⩾ Y0
3.1. Assessment of outputs As indicated above in step 7 proposed by Song and Chissom (1993a,b), the predicted values should be evaluated in a way to
τ ⩾ 0n 1
For more detail information the reader can refer to Coelli et al (2005)
ð6Þ
Mostafa Salari and Homayoun Khamooshi—A better project performance prediction model using fuzzy time series and data envelopment analysis
In Equation (6), Y, eT and τ are the n × n output matrix, the unity n-vector and n × 1 variable matrix, respectively. This equation can measure relative efficiency of n variables using one criterion. In Equation (6), it is assumed that the variable with the highest efficiency, higher value in the criteria, gets the highest value of τ, which is equal to unity. However, most of the times, Equation (6) is used in a way to determine the relative efficiency of variables regarding more than just one criteria. In such cases, DEA model attempts to determine the order of variables in terms of efficiency ranking the best model with the highest values in all or most of the criteria. In this paper, the employed DEA model assesses the relative efficiency of time series models and provides three outputs, derived from error indices. The formulation of the model is given in Equation (7). It should be noted that to select the best model as the one with the highest values in all or most of the criteria, as in Equation (6), the inverse form of error indices (ie for instance, the reverse form of MSE is 1/MSE) are utilized in Equation (7). The inverse form of error indices are defined as IMSE, (inverse MSE), IMAE, (inverse MAE) and IMAPE (inverse IMPAE): 10 X
Min θ¼
τj
j¼1
s:t: 10 X
ðτj ´ IMSEj Þ ⩾ IMSE0
ðτj ´ IMAEj Þ ⩾ IMAE0
j¼1 10 X
ðτj ´ IMAPEj Þ ⩾ IMAPE0
j¼1
τ ⩾ 0n
Table 2 The sample projects collected data Months passed since initiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Total planned duration of project (month)
Project No. 1 Project No. 2 Project No. 3 CPI
SPIt
CPI
SPIt
CPI
SPIt
0.53 0.55 0.6 0.5 0.42 0.36 0.45 0.59 0.68 0.63 0.56 0.48 0.52 0.54 0.61 0.69 0.76 0.83
0.64 0.69 0.78 0.81 0.78 0.82 0.75 0.73 0.64 0.6 0.56 0.46 0.42 0.36 0.31 0.26 0.23 0.26
0.68 0.72 0.75 0.77 0.84 0.83 0.83 0.82 0.8 0.81 0.84 0.9 0.94 0.96 1 1.03 1.06 1.05 1.07 1.1 1.16 1.21 1.26
0.73 0.75 0.78 0.77 0.78 0.76 0.73 0.77 0.81 0.81 0.85 0.9 0.91 0.91 0.93 0.97 1.01 1.03 0.98 0.96 0.97 0.97 0.95
0.61 0.56 0.6 0.56 0.51 0.48 0.45 0.48
0.38 0.38 0.35 0.37 0.34 0.33 0.31 0.28
26
45
20
4.1. Case studies
j¼1 10 X
5
ð7Þ
In the following section, the proposed approach is applied to a number of cases to validate its application, and assess the advantages of this suggested model in contrast to existing procedures.
4. Data collection and analysis The step by step implementation procedure for cost and schedule performance estimation using fuzzy time series models is discussed thoroughly in this section. Initially, some reallife sample project data from construction industry is used for testing the model; then to reinforce the results obtained from these sample projects, the authors employed simulation to randomly generated projects data and compared the results. The following subsections provide more information on this effort.
Three projects were chosen by the authors simply because they had access to these real life projects and knew that all three of them had faced cost or schedule problems through their execution. In this paper, the historical data related to the cost and schedule performance of these construction projects is presented in Table 2. The SPIt and CPI data for each project is gathered monthly. Furthermore, the total planned duration for each project is presented in the last row of Table 2. According to formulation of fuzzy time series models, a universe of discourse and its related intervals are to be specified for each of the three employed models. These are normally determined by expert judgment or decision makers in a project and may vary from project to project. One salient point that should be considered is that the universe of discourse should be determined in a way to include all the historical data. Here, using historical data and our judgment, the required steps to formulate and establish a fuzzy time series, as Song and Chissom (1993a,b) did, are presented below: Step 1: Universe of discourse: U = [0,1.5] Step 2: Partitions of universe (linguistic values): u1 = [0,0.25] u2 = [0.25,0.5] u3 = [0.5,0.75] u4 = [0.75,1] u5 = [1,1.25] u6 = [1.25,1.5]. While for the illustrative cases in this paper, six intervals are considered, there is no strict rule for deciding the number of intervals. The number of intervals can be decided based on management recommendation bearing in mind that more
6
Journal of the Operational Research Society
Table 3 Fuzzy logical relationships of CPI in different periods (months) for each project Fuzzy relations of CPI in project No .1 1357911131517-
A3 A3 A2 A2 A3 A3 A3 A3 A4
→ → → → → → → → →
A3 A2 A2 A3 A3 A2 A3 A3 A4
246810121416-
A3 A2 A2 A3 A3 A2 A3 A3
→ → → → → → → →
A3 A2 A2 A3 A3 A3 A3 A4
Fuzzy relations of CPI in project No .2 13579111315171921-
A3 A3 A4 A4 A4 A4 A4 A4 A5 A5 A5
→ → → → → → → → → → →
A3 A4 A4 A4 A4 A4 A4 A5 A5 A5 A5
intervals yield a higher degree of sensitivity to the oscillation in historical data. In addition, it is to be noted that each interval includes the lower bound, while excludes the upper bound, which means, for instance, 0.25 belongs to u2 while 0.5 belongs to u3. Step 3: Fuzzy sets: six fuzzy sets (A1,A2,A3,A4,A5,A6) are defined as linguistic variables on U. the fuzzy variables are being introduced as: A1 A2 A3 A4 A5 A6
Very poor performance Poor performance Almost poor performance Almost good performance good performance Very good performance
Step 4: The membership grades to the fuzzy sets of linguistic values are described as follows: A1 ¼
1 0:8 0:6 0:4 0:2 0 + + + + + u1 u2 u3 u4 u5 u6
A2 ¼
0:8 1 0:8 0:6 0:4 0:2 + + + + + u1 u2 u3 u4 u5 u6
A3 ¼
0:6 0:8 1 0:8 0:6 0:4 + + + + + u1 u2 u3 u4 u5 u6
A4 ¼
0:4 0:6 0:8 1 0:8 0:6 + + + + + u1 u2 u3 u4 u5 u6
A5 ¼
0:2 0:4 0:6 0:8 1 0:8 + + + + + u1 u2 u3 u4 u5 u6
A6 ¼
0 0:2 0:4 0:6 0:8 1 + + + + + u1 u2 u3 u4 u5 u6
Step 5: The fuzzy logical relations of each project are presented in Tables 3 and 4. It should be noted that in the proposed model, steps 6 and 7, which include prediction and assessment of outputs are combined and presented in the following subsection. Here, we avoid providing details of calculation related to the prediction of
246810121416182022-
A3 A4 A4 A4 A4 A4 A4 A5 A5 A5 A5
→ → → → → → → → → → →
A3 A4 A4 A4 A4 A4 A4 A5 A5 A5 A6
Fuzzy relations of CPI in project No .3 1357-
A3 A3 A3 A2
→ → → →
A3 A3 A2 A2
246-
A3 → A3 A3 → A3 A2 → A2
future values using three models. The readers can refer to Chen (1996) Singh (2009) and Gangwar and Kumar (2012) to see the details of mathematical calculations.
4.2. Results for application of the models to three cases As explained in subsection 3.1, to assess the performance of each prediction model the MSE, MAE and MAPE are calculated by comparing the actual performance against the predicted index. The error indices (ie MSE, MAE and MAPE) for Projects No. 1, No. 2 and No. 3 are provided in Tables 5, 6 and 7. To be more specific, Table 5 presents the accuracy of forecast values of time series models for SPIt and CPI in Project No.1. Similarly, Tables 6 and 7 illustrate the results of comparison between actual and forecasted values of CPI and SPIt in case of project No. 2 and No. 3, respectively. The inverse form of three error indices presented in Tables 5, 6, and 7 are then applied in DEA model formulation shown in Equation (7) to identify the superior model among the three for estimation purposes. According to the results obtained from implementing DEA model shown in Table 8, the Chen (1996) proposed model is identified as the best for cost performance estimation of project No.1. For this project, Gangwar and Kumar (2012) model of orders 4 and 5 and Singh (2009) model of order 7 have proved to be superior to other models for schedule performance estimation. In case of projects No.2 and 3, the results of DEA model indicate that the Singh (2009) model of orders 7 and 5 have highest accuracy for schedule performance estimation respectively. Regarding the cost performance estimation in all three projects, Singh (2009) model of order 4 for project No.1, Gangwar and Kumar (2012) model of order 3 and 5 for project No.2, and Singh (2009) model of order 6 for project No. 3 are the most accurate predictors. In summary, Table 9 illustrates the best model for cost or schedule performance estimation in each project. Following the determination and recommendation of best time series models in Table 9, the future project performance can be easily be predicted and developed. Table 10 presents the next three periods and the final estimated performance according to the best time series models for all three projects.
7
Mostafa Salari and Homayoun Khamooshi—A better project performance prediction model using fuzzy time series and data envelopment analysis
Table 4 Fuzzy logical relationships of SPIt in different periods (months) for each project Fuzzy relations of SPIt in project No .1 1357911131517-
A3 A4 A4 A3 A3 A3 A2 A2 A1
→ → → → → → → → →
A3 A4 A4 A3 A3 A2 A2 A2 A2
246810121416-
A3 A4 A4 A3 A3 A2 A2 A2
→ → → → → → → →
Fuzzy relations of SPIt in project No .2
A4 A4 A3 A3 A3 A2 A2 A1
13579111315171921-
A3 A4 A4 A4 A4 A4 A4 A4 A5 A4 A4
→ → → → → → → → → → →
A3 A4 A4 A4 A4 A4 A4 A4 A5 A4 A4
246810121416182022-
A3 A4 A4 A4 A4 A4 A4 A4 A5 A4 A4
→ → → → → → → → → → →
Fuzzy relations of SPIt in project No .3
A4 A4 A4 A4 A4 A4 A4 A5 A4 A4 A4
1357-
A2 A2 A2 A2
→ → → →
A2 A2 A2 A2
A2 → A2 A2 → A2 A2 → A2
246-
Table 5 Comparison of estimation models for SPIt and CPI in project No. 1 Performance index
SPIt CPI
Metric
MSE MAE MAPE MSE MAE MAPE
Chen (1996)
Singh (2009)
Gangwar and Kumar (2012)
Orders
Orders
Orders
1
3
4
5
6
7
3
4
5
6
0.01 0.13 12.67 0.017 0.108 11.82
0.008 0.086 9.71 0.005 0.066 7.04
0.002 0.036 4.088 0.0028 0.047 4.88
0.004 0.047 5.35 0.0032 0.05 5.13
0.005 0.057 6.32 0.003 0.049 4.91
0.0017 0.032 3.55 0.0031 0.048 4.82
0.0023 0.0418 4.93 0.0029 0.047 4.94
0.007 0.07 7.608 0.0029 0.048 4.85
0.008 0.082 8.98 0.0028 0.048 4.78
0.003 0.05 5.56 0.003 0.053 5.38
Table 6 Comparison of estimation models for SPIt and CPI in project No. 2 Performance index
SPIt CPI
Metric
MSE MAE MAPE MSE MAE MAPE
Chen (1996)
Singh (2009)
Gangwar and Kumar (2012)
Orders
Orders
Orders
1
3
0.0082 0.073 15.45 0.0065 0.07 12.93
0.0097 0.085 20.56 0.0074 0.074 13.11
4
5
6
0.0022 0.0023 0.083 0.076 24.82 22.56 0.0119 0.0088 0.09 0.079 16.48 14.07
To strengthen our argument and reinforce the results obtained from three case studies, we took advantage of using simulation as well. The following subsection provides more information on this effort.
4.3. Simulated project data In this subsection, we attempt to examine the efficiency and accuracy of using fuzzy time series method, for the purpose of predicting project performance index, based on which the overall duration and cost of project will be estimated. In doing
7
0.0081 0.0029 0.082 0.052 20.12 13.36 0.0091 0.0116 0.078 0.096 13.86 16.42
3
4
5
6
0.0079 0.0074 0.00152 0.0084 0.077 0.067 0.094 0.083 16.27 11.88 18.02 20.35 0.0123 0.0127 0.01227 0.0135 0.106 0.108 0.107 0.113 19.32 19.44 18.64 17.88
so, data for 3000 projects were generated randomly taking into account the following assumptions, constraints and parameters: 1. Projects may have a scheduled duration ranging from 10 to 60 months (Approximately 1–5 years). These project durations for each project were generated randomly. 2. The data date or progress report date is a point in time within the duration of the project. These progress report dates were also determined randomly. 3. Reasonable values for cost and schedule performance matrices were assigned randomly to each progress report period. While there are many ways of coming up with a
8
Journal of the Operational Research Society
Table 7 Comparison of estimation models for SPIt and CPI in project No. 3* Performance index
Metric
(Chen, 1996)
(Singh, 2009)
Orders
Orders
MSE MAE MAPE MSE MAE MAPE
SPIt CPI
1
3
4
5
6
7
0.002453 0.039285 12.66239 0.00793 0.07642 14.16018
0.003305 0.049 16.03563 0.01198 0.095 19.3438
0.00142 0.0344 11.2833 0.00762 0.07682 17.7561
0.00064 0.02373 7.97637 0.00605 0.07293 17.4422
0.00088 0.02925 10.0403 0.00287 0.039 11.0843
0.02322 0.1524 15.3066 0.00846 0.092 17.8666
*Owing to limited data of the Project No. 3 Gangwar and Kumar (2012) cannot be applied for this project. The Chen (1996) and Singh (2009) model, however, can be utilized for this case. The reader can refer to Appendix for more information.
Table 8 DEA efficiencies for different time series models applied for three projects Time series models
Order of models Outputs of DEA model for the times series models used for cost and schedule metrics of three projects Project No.1
Chen (1996) Singh (2009)
Gangwar and Kumar (2012)
— Order 3 Order 4 Order 5 Order 6 Order 7 Order 3 Order 4 Order 5 Order 6
Project No. 2
Project No.3
SPIt
CPI
SPIt
CPI
SPIt
CPI
0.81 0.636 0.854 0.874 0.653 1 0.769 1 1 0.6458
1 0.986 0.784 0.918 0.932 0.787 0.669 0.6651 0.6937 0.7231
0.280 0.372 0.888 0.680 0.561 1 0.765 0.4666 0.3953 0.64
0.435 0.712 1 0.945 0.976 0.995 1 0.9926 1 0.9333
0.629 0.497 0.706 1 0.811 0.521 — — — —
0.782 0.573 0.624 0.635 1 0.623 — — — —
Table 9 The best time series model of DEA method and chosen based on MAPE in three projects Project Number No. 1
Performance indices CPI SPIt
No.2 CPI No.3
SPIt CPI SPIt
The best time series models in DEA
MAPE
The best chosen time series model
Chen (1996) Singh (2009), Order 7 Gangwar and Kumar (2012), Order 4 Gangwar and Kumar (2012), Order 5 Singh (2009), Order 4 Gangwar and Kumar (2012), Order 3 Gangwar and Kumar (2012), Order 5 Singh (2009), Order 7 Singh (2009), Order 6 Singh (2009), Order 5
12.93 13.36 11.88 18.02 4.88 4.94 4.78 3.55 11.084 7.976
Chen (1996) Gangwar and Kumar (2012), Order 4
reasonable algorithm to generate the set of performance indices, the authors assumed that as we move towards the completion of the project less variability one could expect for these measures. It is in harmony with real life projects in which there is less changes in values of cost and schedule metrics as the projects move forward (Christensen, 1993). To accommodate this in our simulation modelling, the
Gangwar and Kumar (2012), Order 5 Singh (2009), Order 7 Singh (2009), Order 6 Singh (2009), Order 5
project duration was randomly divided into multiple phases (two to ten) where the amount of variation is reduced as we move to later phases of the project. To implement the reduction in variation of the indices we took advantage of using mean and standard deviation (SD). After first period, the average and SD of data are calculated, then for the next (second) period the value should be ranged between average
Mostafa Salari and Homayoun Khamooshi—A better project performance prediction model using fuzzy time series and data envelopment analysis
9
Table 10 Performance estimation of three case studies Project Number
Performance indices
The best time series model for performance estimation
Forecast Periods
No. 1
CPI SPIt
Chen (1996) Gangwar and Kumar (2012), Order 4
No. 2
CPI SPIt
Gangwar and Kumar (2012), Order 5 Singh (2009), Order 7
No. 3
CPI SPIt
Singh (2009), Order 6 Singh (2009), Order 5
2 4 6 8
⩽ periods ⩽ 4 < periods ⩽ 6 < periods ⩽ 8 < periods ⩽ 10
Month 20
Month 21
Final status of project
0.87 0.21 Month 24 1.21 0.94 Month 9 0.4688 0.2873
0.87 s Month 25 1.36 0.94 Month 10 0.4615 0.2978
0.87 0.2 Month 26 1.11 0.93 Month 11 0.4506 0.3091
0.87 0.29 Final status of project 1.37 0.88 Final status of project 0.4118 0.3429
both techniques. Table 11 shows the pool of generated projects in terms of projects’ duration and number of periods:
Table 11 Details of simulated data Number of periods
Month 19
Range of projects duration 10–20
20–30
30–40
40–50
50–60
51 22 34 12
247 8 465 41
34 13 82 14
287 39 114 252
295 237 116 637
± 1/2SD. For the third period, the procedure is the same while the coefficient of SD should be changed to restrict the data more than the prior range. These coefficient can also be determined randomly or the following ordered coefficient can be used successively 1/2, 1/3, 1/4, 1/5 and so on. 4. Three randomly selected lags, number of periods away from the present and in the future, have been chosen to compare the proposed model with EVM technique. Selection of lags is of significant importance for comparing EVM and fuzzy time series, which is going to be discussed in the next section. 5. Ten fuzzy time series models introduced in Section 3 has been applied to predict each project performance and by using DEA the best model in terms of accuracy of prediction is determined. It’s worth mentioning that up to order 10 of each Gangwar and Kumar (2012) and Singh (2009) models has been utilized in the simulation process to show the fact the higher orders of time series model can also be considered for different cases. To demonstrate the processes consider the following example: let say the project duration is generated randomly and it happens to be 34 months (1) while the current date is month 19th (2). The values of cost and schedule performance indexes are provided for all 34 months (3). The months 21,22 and 29 are selected randomly to compare the prediction models. After generating the pools of projects, the procedure explained in Section 3 was applied to determine which fuzzy time series model should be selected for each specific project. Finally, the results of EVM and fuzzy time series were compared for the months which were already determined for the comparison of
5. Discussion and analysis of the findings Now, we can compare the EVM results for prediction of future project performance, duration and costs with those of fuzzy time series approach. According to one extensively used approach of EVM technique, the future performance of a project will be just like the current performance of that project (Anbari, 2003; Moslemi Naeni and Salehipour, 2011). However, as it was discussed previously in the introduction section there is no rationale behind this assumption or approach. Also it is to be noted that the further away, the period of interest that we want to predict is from present, the less reliable one expects the results to be. In this regard and to account for this, we applied different lags in both randomly generated projects and also in all of the three case studies to predict the future performance. For instance, a lag of one period means we are interested in predicting the performance for the next period. Concerning simulated projects, we randomly applied three different lags to compare EVM and fuzzy time series in terms of accuracy of prediction. Table 12 illustrates the outcomes and result of simulations: Table 12 sorts the projects according to the scheduled duration. In this table, accuracy percentages of either EVM or fuzzy time series for both SPIt and CPI are presented in the last two columns. Also to view the result from another perspective, the projects were classified according to the number of phases/ periods within the project. The idea was to evaluate the impact of reduced variability of the indexes on the accuracy of prediction models. Since the imposed restriction is a function of number of prior periods, it is expected that as the number of prior periods increases, the accuracy of time series models to improve. This is what we can see vividly in the Table 13. In addition to the simulated projects, we conducted a comparative analysis and assessed the accuracy of both EVM and fuzzy time series for the three case studies as well. Here, we extended the prediction periods (lags), to three, five or any later period in the future. Such an extension is of value to compare
10
Journal of the Operational Research Society
Table 12 Results of simulation model for generated projects with different scheduled duration Range of projects Number of generated Prediction Method Number of times each method Percentage that shows accuracy of each duration (month) projects in each Used predicted most accurate results method in terms of cost and schedule metrics specified range CPI SPIt CPI SPIt 10–20
119
20–30
761
30–40
143
40–50
692
50–60
1285
EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series
48 71 162 599 56 87 91 601 207 1078
33 86 213 548 64 79 148 544 311 974
40.33 59.66 21.28 78.71 39.16 60.83 13.15 86.84 16.10 83.89
27.73 72.27 27.98 72.02 44.76 55.24 21.38 78.62 24.21 75.79
Table 13 Results of simulation model for generated projects with different number of phases Number of phases/periods
Prediction Method Number of generated projects which are randomly Used selected to be divided to the defined periods
2 ⩽ periods ⩽ 4
914
4 < periods ⩽ 6
319
6 < periods ⩽ 8
811
8 < periods ⩽ 10
956
EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series
and evaluate the forecast error of EVM approach versus the fuzzy time series for both short and long-term horizons. In The following table, three error metrics are calculated for both EVM and fuzzy time series within the duration of project. Table 14 shows the results for all three case studies. Table 15 gives a different portray of assessment between these two approaches in which the best prediction model in each period is identified with a check mark and the alternative with an X. According to this table, EVM hardly performs better in any of these cases. One instance is the next period (1 lag) prediction of SPIt for all three projects, and that is expected as the predicted period is too close to present, making the performance less likely to change. However, in all the other predictions and more specifically longer periods, the fuzzy time series provides a more accurate prediction of both schedule and cost performance. Table 15 shows the result using checkmark presentation. To present the outcome of the analysis statistically, we simply counted the number of checks and crosses. Summary of this counting is shown in Table 16. Likewise the results obtained from simulated data projects, the results of comparing EVM and fuzzy time series in all three case projects vividly indicate that fuzzy time series performs far superior than EVM in terms of project performance prediction. Percentage-wise it
Number of times each method predicted most accurate results
Percentage that shows accuracy of each method in terms of cost and schedule metrics
CPI
SPIt
CPI
SPIt
419 495 156 163 147 664 73 883
390 524 98 221 101 710 38 918
45.84 54.16 48.91 51.09 18.12 81.88 7.63 92.37
42.66 57.33 30.72 69.28 12.46 87.54 3.97 96.03
gave better answer 86.36% of the time, whereas EVM was better only 13.66% of the cases. One salient point to note is that the determination of the best fuzzy time series model for estimation purposes is dependent on some specific factors. To start with, employing an appropriate universe of discourse is the first step of building a time series model. This can be determined by expert judgment and experience or alternatively some mathematical approach can be applied (Li et al, 2010). The process of partitioning the selected universe of discourse known as interval partitioning is another important issue the organization adopting the approach may face. The simplest approach in interval partitioning is perhaps equal-width partitioning (Sullivan and Woodall, 1994; Own and Yu, 2005). The problem of equal-width partitioning is that it does not take into consideration the distribution of real data. Hence, it may not result in best outcomes in cases where the distribution of real data is not uniform. For this issue, fuzzy clustering based partitioning has been shown to be an effective alternative solution to deal with non-uniform distributed data (Li et al, 2008). Regarding the simulation section, we could argue that it is next to impossible to consider the network characteristics such as aspect ratio, fat and short or long and thin or number of critical activities or path, and so on in generating the networks
Mostafa Salari and Homayoun Khamooshi—A better project performance prediction model using fuzzy time series and data envelopment analysis
11
Table 14 EVM and fuzzy time series prediction errors for SPIt and CPI Project Number
No. 1
Lag
Performance Index
1
CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt CPI SPIt
3 5 9 13 No. 2
1 3 5 9 13 17
No. 3
1 3 5
MSE
MAE
MAPE
EVM
Fuzzy time series
EVM
Fuzzy time series
EVM
Fuzzy time series
0.0047 0.003 0.0224 0.02005 0.0221 0.0522 0.0202 0.2142 0.0432 0.22 0.0012 0.0008 0.0080 0.0042 0.0167 0.0077 0.0465 0.0222 0.1064 0.0407 0.1426 0.0479 0.445 0.0004 0.0169 0.0025 0.0606 0.0041
0.0045 0.0073 0.0048 0.0073 0.004 0.0075 0.0067 0.0011 0.0074 0.0011 0.0027 0.0017 0.0027 0.0017 0.0027 0.0017 0.0035 0.0278 0.0041 0.0016 0.0043 0.0021 0.2875 0.0237 0.0028 0.0006 0.0028 0.0006
0.0605 0.0505 0.13 0.1306 0.1215 0.1971 0.1222 0.46 0.158 0.456 0.0309 0.0236 0.0783 0.0562 0.1166 0.0733 0.2071 0.135 0.3157 0.199 0.3766 0.215 0.065 0.02 0.13 0.0466 0.245 0.0633
0.0535 0.067 0.0546 0.067 0.0476 0.0655 0.0677 0.03 0.076 0.03 0.0484 0.0318 0.0484 0.0318 0.0484 0.0318 0.0555 0.0278 0.06 0.026 0.0616 0.0266 0.039 0.0237 0.039 0.0237 0.039 0.0237
10.8464 10.5311 23.1945 30.5971 20.3669 55.7610 19.3587 166.6379 21.1554 170.5065 3.2630 2.7193 7.7448 6.1163 11.5764 7.9503 19.7805 14.229 27.6061 20.5014 33.0154 21.9253 17.2425 6.73206 33.6877 15.6146 64.1528 17.8575
10.6585 11.8856 11.0089 11.8856 9.3912 12.066 11.1623 10.2089 11.0293 10.2089 3.4519 3.7885 5.7892 3.5517 4.7795 3.5517 5.1419 2.9734 5.2253 2.6564 5.3100 2.7216 11.084 7.9763 11.0843 7.9763 11.0843 7.9763
Table 15 Selection of best prediction model for cost and schedule performance Project Number
Error Metric
Performance Index
CPI Lag
No. 1
MSE MAE MAPE
No. 2
MSE MAE MAPE
No. 3
MSE MAE MAPE
EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series
1
3
5
9
13
17
× √ × √ × √ √ × √ × √ × × √ × √ × √
× √ × √ × √ × √ × √ × √ × √ × √ × √
× √ × √ × √ × √ × √ × √ × √ × √ × √
× √ × √ × √ × √ × √ × √ — — — — — —
× √ × √ × √ × √ × √ × √ — — — — — —
— — × √ × √ × √ × √ × √ — — — — — —
12
Journal of the Operational Research Society
Table 15: Continued
Project Number
Error Metric
Performance Index
SPIt Lag
No. 1
MSE
EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series EVM Fuzzy time series
MAE MAPE No. 2
MSE MAE MAPE
No. 3
MSE MAE MAPE
Table 16 Number of times each method predicted most accurate result Project Number Method No. 1 No. 2 No. 3
CPI SPIt Total Percentage
EVM 0 Fuzzy Time Series 17 EVM 3 Fuzzy Time Series 15 EVM 0 Fuzzy Time Series 9
3 14 3 15 3 6
3 31 6 30 3 15
8.82 91.18 16.67 83.33 16.67 83.33
randomly for the simulation. Davis (1973) in his review article long time ago argued that there is no significant relationship between network characteristics and outcome of scheduling. As such for simulating the performance of the projects we did not generate the networks for projects, but generated the performance indices (SPI and CPI) over the duration of the project for the randomly designated progress review points. While the implementation of the suggested approach may seem to be quite a challenge the improvement gained by the model is so overwhelming that we strongly recommend its use starting with a very basic model and adopting more advanced approaches as and when appropriate.
6. Conclusion Since variations and uncertainty are indispensable part of project performance, classic EVM approaches of assuming the future performance is the same as the present do not provide an accurate or reliable prediction of performance. Furthermore, the ability to anticipate the project cost and duration performance
1
3
5
9
13
17
√ × √ × √ × √ × √ × √ × √ × √ × √ ×
× √ × √ × √ × √ × √ × √ × √ × √ × √
× √ × √ × √ × √ × √ × √ × √ × √ × √
× √ × √ × √ × √ × √ × √ — — — — — —
× √ × √ × √ × √ × √ × √ — — — — — —
— — × √ × √ × √ × √ × √ — — — — — —
for future periods is a major contributor to project success and essential component of the decision-making process for project managers. This study presented an integrated model to deal with the fuzziness of project performance and provided a much needed forecasting tool to estimate the performance of project in the future periods. The historical data of cost and schedule performance of three construction project cases were employed as illustrative examples in this paper. Though all three projects are from the same industry, the total duration of these projects differs significantly illustrating the strengths of time series models in long and short term forecasting. In addition to three case studies, a simulation model has been employed to assess both EVM and fuzzy time series in terms of accuracy of prediction. The simulation model did illustrate better performance of fuzzy time series in cost and schedule performance prediction of either long-term or short-term projects. Moreover, the results of proposed approach for both three case studies and simulation model indicate that the best time series model is not necessarily the same for different projects. Finally, it is suggested to apply the model for prediction of cash flow for ongoing projects. Acknowledgements —The authors would like to thank anonymous reviewers for their constructive comments on the earlier draft of this manuscript. They also would like to sincerely thank Dr Morteza Bagherpour and Dr Mohammad Mahdi Asgari Dehabadi for their priceless support and assistance.
References Al-Jibouri SH (2003). Monitoring systems and their effectiveness for project cost control in construction. International Journal of Project Management 21(2): 145–154.
Mostafa Salari and Homayoun Khamooshi—A better project performance prediction model using fuzzy time series and data envelopment analysis
Aliverdi R, Moslemi Naeni L and Salehipour A (2013). Monitoring project duration and cost in a construction project by applying statistical quality control charts. International Journal of Project Management 31(3): 411–423. Anbari FT (2003). Earned value project management method and extensions. Project Management Journal 34(4): 12–23. Azman MA, Abdul-Samad Z and Ismail S (2012). The accuracy of preliminary cost estimates in public works department (PWD) of Peninsular Malaysia. International Journal of Project Management 31(7): 994–1005. Bagherpour M, Zareei A, Noori S and Heydari M (2010). Designing a control mechanism using earned value analysis: An application to production environment. International Journal of Advanced Manufacturing Technology 49(5–8): 419–429. Banker RD, Charnes A and Cooper WW (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30(9): 1078–1092. Barraza GA, Back WE and Mata F (2004). Probabilistic forecasting of project performance using stochastic S curves. Journal of Construction Engineering and Management 130(1): 25–32. Baumeister A and Floren A (2011). Optimizing the configuration of development teams using EVA: The case of ongoing project adjustments facing personnel restrictions. International Journal of Information Technology Project Management 2(1): 62–77. Caron F, Ruggeri F and Merli A (2013). A Bayesian approach to improve estimate at completion in earned value management. Project Management Journal 44(1): 3–16. Charnes A, Cooper WW and Rhodes E (1978). Measuring the efficiency of decision making units. European Journal of Operational Research 2(6): 429–444. Chen SM (1996). Forecasting enrollments based on fuzzy time series. Fuzzy Sets And Systems 81(3): 311–319. Chen SM and Hwang JR (2000). Temperature prediction using fuzzy time series. Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on 30(2): 263–275. Christensen DS (1993). The estimate at completion problem: A review of three studies. Project Manage Journal 24: 37–42. Cioffi DF (2006). Designing project management: A scientific notation and an improved formalism for earned value calculations. International Journal of Project Management 24(2): 136–144. Coelli TJ, Rao DSP, O’Donnell CJ and Battese GE (2005). An introduction to efficiency and productivity analysis. Springer Science & Business Media. Davis EW (1973). Project scheduling under resource constraints: Historical review and categorization of procedures. AIIE Transactions 5(4): 297–311. Dillibabu R and Krishnaiah K (2005). Cost estimation of a software product using COCOMO II.2000 model—A case study. International Journal of Project Management 23(4): 297–307. Elshaer Raafat. (2013). Impact of sensitivity information on the prediction of project’s duration using earned schedule method. International Journal of Project Management 31(4): 579–588. Feylizadeh M, Hendalianpour A and Bagherpour M (2012). A fuzzy neural network to estimate at completion costs of construction projects. International Journal of Industrial Engineering Computations 3(3): 477–484. Fleming QW and Koppelman JM (2000). Earned Value Project Management. 2nd edn. Project Management Institute: Newtown Square, PA. Gangwar SS and Kumar S (2012). Partitions based computational method for high-order fuzzy time series forecasting. Expert Systems with Applications 39(15): 12158–12164. Huarng K (2001). Heuristic models of fuzzy time series for forecasting. Fuzzy Sets and Systems 123(3): 369–386. Jacob DS (2003). Forecasting project schedule completion with earned value metrics. The Measurable News, 1: 7–9.
13
Jacob DS and Kane M (2004). Forecasting schedule completion using earned value metrics revisited. The Measurable News 1: 11–17. Lee HS and Chou MT (2004). Fuzzy forecasting based on fuzzy time series. International Journal of Computer Mathematics 81(7): 781–789. Li ST, Cheng YC and Lin SY (2008). A FCM-based deterministic forecasting model for fuzzy time series. Computers & Mathematics with Applications 56(12): 3052–3063. Li ST, Kuo SC, Cheng YC and Chen CC (2010). Deterministic vector long-term forecasting for fuzzy time series. Fuzzy Sets and Systems 161(13): 1852–1870. Lipke W (2003). Schedule is different. The Measurable News: 31–34. Lipke W, Zwikael O, Henderson K and Anbari F (2009). Prediction of project outcome: The application of statistical methods to earned value management and earned schedule performance indexes. International Journal of Project Management 27(4): 400–407. Liu TK, Chen YP and Chou JH (2011). Extracting fuzzy relations in fuzzy time series model based on approximation concepts. Expert Systems with Applications 38(9): 11624–11629. Lovell CA and Pastor JT (1999). Radial DEA models without inputs or without outputs. European Journal of operational research 118(1): 46–51. Moselhi O, Li J and Alkass S (2004). Web-based integrated project control system. Construction Management and Economics 22(1): 35–46. Moslemi Naeni L and Salehipour A (2011). Evaluating fuzzy earned value indices and estimates by applying alpha cuts. Expert Systems with Applications 38(7): 8193–8198. Own CM and Yu PT (2005). Forecasting fuzzy time series on a heuristic high-order model. Cybernetics and Systems: An International Journal 36(7): 705–717. Project Management Institute (PMI) (2000). A Guide to the Project Management Body of Knowledge: PMBOK® Guide. Project Management Institute: Newtown Square, PA. Project Management Institute (PMI) (2008). A Guide to the Project Management Body of Knowledge: PMBOK® Guide. Project Management Institute: Newtown Square, PA. Salari M, Bagherpour M and Kamyabniya A (2014a). Fuzzy extended earned value management: A novel perspective. Journal of Intelligent and Fuzzy Systems 27(3): 1393–1406. Salari M, Bagherpour M and Wang J (2014b). A novel earned value management model using Z—number. International Journal of Applied Decision Sciences 7(1): 97–119. Singh SR (2007). A simple method of forecasting based on fuzzy time series. Applied Mathematics and Computation 186(1): 330–339. Singh SR (2009). A computational method of forecasting based on high-order fuzzy time series. Expert Systems with Applications 36(7): 10551–10559. Song Q and Chissom BS (1993a). Fuzzy time series and its models. Fuzzy Sets and Systems 54(3): 269–277. Song Q and Chissom BS (1993b). Forecasting enrollments with fuzzy time series—Part I. Fuzzy Sets and Systems 54(1): 1–9. Song Q and Chissom BS (1994). Forecasting enrollments with fuzzy time series—Part II. Fuzzy Sets and Systems 62(1): 1–8. Sullivan J and Woodall WH (1994). A comparison of fuzzy forecasting and Markov modeling. Fuzzy Sets and Systems 64(3): 279–293. Vanhoucke M and Vandevoorde S (2007). A simulation and evaluation of earned value metrics to forecast the project duration. Journal of the Operational Research Society 58(10): 1361–1374. Wang CH and Hsu LC (2008). Constructing and applying an improved fuzzy time series model: Taking the tourism industry for example. Expert Systems with Applications 34(4): 2732–2738. Warburton RD (2011). A time-dependent earned value model for software projects. International Journal of Project Management 29(8): 1082–1090. Wong WK, Bai E and Chu AW (2010). Adaptive time-variant models for fuzzy-time-series forecasting. Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on 40(6): 1531–1542.
14
Journal of the Operational Research Society
Appendix In Gangwar and Kumar (2012) studies, a variable named ρ is proposed to partition the time series data using the following expression: ρ¼
ðEmax + Emin Þ 2ðEmax - Emin Þ
Where Emax and Emin are maximum and minimum of time series data, respectively. Subsequent to determination of ρ, the fuzzy relations are defined for each partition. For instance, in case of CPI and SPIt data for project No. 1, ρ is calculated as below: ρ¼ ρ¼
ð0:83 + 0:36Þ ¼ 1:26 2ð0:83 - 0:36Þ
ð0:82 + 0:23Þ ¼ 0:8898 2ð0:82 - 0:23Þ
for CPI
for SPIt
According to their model, the number of partition for CPI and SPIt will be 2 and 1, respectively. In the conclusion part of their studies, it is indicated that if ρ ⩾ N/2, (N = no. of time series data), then there will be not sufficient data in each
partitions for prediction. Here, the project No.3 has met this condition: N ¼4 N¼ 8; 2 ρ¼
ð0:61 + 0:45Þ ¼ 3:31 2ð0:61 - 0:45Þ
for CPI
ρ¼
ð0:38 + 0:28Þ ¼ 3:3 2ð0:38 - 0:28Þ
for SPIt
The model rounded ρ value to the nearest upper integer value, so the value of ρ for both SPIt and CPI will be equal to 4. This causes the fact that the aforementioned condition where ρ ⩾ (N)/(2) happens: N ρ ¼ 4⩾ ¼ 4 2 Consequently, there is no sufficient data in each partition, and the model cannot be applied in the case of project No. 3.
Received 17 February 2014; accepted 17 February 2016
