A novel earned value management model using Z

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Abstract: The earned value management (EVM) model is an essential ... earned schedule (ES) in order to overcome the restrictions of the standard schedule.
Int. J. Applied Decision Sciences, Vol. 7, No. 1, 2014

A novel earned value management model using Z-number Mostafa Salari* Department of Industrial Engineering, Sharif University of Technology, Azadi Street, Tehran, Iran E-mail: [email protected] *Corresponding author

Morteza Bagherpour Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran E-mail: [email protected]

John Wang Department of Information and Operation Management, Montclair State University, Montclair, NJ 07043, USA Fax: (973) 655-7678 E-mail: [email protected] Abstract: The earned value management (EVM) model is an essential technique for managing and forecasting project features such as scheduling and cost performances indexes. This paper presents a novel fuzzy earned-value model based on Z-number theory incorporating both the impreciseness of real life conditions and a degree of reliability through considering an expert judgment process. The latter factor has not been used by other researchers in the field. The proposed model provides a reliable assessment for the progress performance of a project and its ‘at completion’ cost in an uncertain environment. Finally, an illustrative case demonstrates the applicability of the proposed model in real life projects. Keywords: earned value; cost control; estimation process; Z-number; applied decision. Reference to this paper should be made as follows: Salari, M., Bagherpour, M. and Wang, J. (2014) ‘A novel earned value management model using Z-number’, Int. J. Applied Decision Sciences, Vol. 7, No. 1, pp.97–119. Biographical notes: Mostafa Salari is currently a Master student of Industrial Engineering in Sharif University of Technology. His research interests include project scheduling, earned value management and risk management.

Copyright © 2014 Inderscience Enterprises Ltd.

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M. Salari et al. Morteza Bagherpour is an Assistant Professor in Iran University of Science and Technology. He has many publications in project management area of research. Furthermore, he has experiences in teaching project planning and control, project management, simulation study, engineering economics, computer and its applications for BSc, MSc, and PhD students in industrial engineering, Faculty of Industrial Engineering, Iran University of Science and Technology. John Wang is a Professor in the Department of Information and Operations Management at Montclair State University, USA. Having received a scholarship award, he came to the USA and completed his PhD in Operation Research from Temple University. He has published over 100 refereed papers and seven books. He has also developed several computer-software programs based on his research findings. He is a member of the Institute for Operation Research and the Management Science (INFORMS), Information Resources Management Association (IRMA), The Decision Science Institute (DSI), and The Production and Operation Management Society.

1

Introduction

The earned value management (EVM) is a powerful technique that allows program managers, project managers and other top-level stakeholders to visualise the status of project during the project life cycle. Consequently, the management of projects, programs, and portfolios can be achieved more efficiently. Furthermore, EVM provides project assessments, if appropriately applied, and clearly quantifies 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” (PMI, 2008). However, in spite of the proven applicability of implementing EVM in real life projects, limited research has been carried out into the practical use of EVM so far. Lipke (1999) provided a novel ratio for appropriately managing cost and scheduling in projects. His study further introduced the earned schedule (ES) in order to overcome the restrictions of the standard schedule performance index (SPI) [previously addressed by Lipke (2003)]. Other studies, then, attempted to develop the reliabilities behind ES (Henderson, 2003, 2004). In addition to what Lipke performed, other researchers have proposed different metrics to address the limitations of SPI (Anbari, 2003; Jacob, 2003; Jacob and Kane, 2004). Vandevoorde and Vanhoucke (2006) carried out a study to evaluate different proposed metrics in EVM and proved that ES can be regarded as the most reliable and applicable method, not only in the assessment of project schedule performance, but also in the estimation of project completion time. In other study, Lipke et al. (2009) proposed a cost and duration estimation model using statistical approaches. However, the aforementioned research mostly concentrated on the development of the proposed indices in standard EVM, whilst the other type of studies in the literature attempted to implement the EVM methodology in different systems as well as organisations. In this category, Kim et al. (2003) provided a model for the effective implementation of the EVM methodology in different types of projects. Moselhi et al. (2004) developed a web-based model aimed at time and cost management in construction projects. Furthermore, Owen (2007) studied the application of EVM in research and development projects. In other study, Bagherpour et al. (2010)

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designed a control mechanism to discuss the application of EVM in production environments under fuzzy conditions. Recently, there is an increasing trend of the incorporation of the uncertainty in decision making practices (Farzipoor, 2011; Houska et al., 2012; Khalili-Damghani et al., 2012) such as credit evaluation (Aouam et al., 2009), supplier assessment in a supply chain system (Amid et al., 2006; Jajimoggala et al., 2011) and sourcing selection (Parthiban et al., 2009). However, there is only a limited number of studies focusing on the uncertainty of real-life situations in the EVM area of research (Ponz-Tienda et al., 2012; Naeni et al., 2011; Moslemi Naeni and Salehipour, 2011). Naeni et al. (2011) and Moslemi Neini and Salehipour (2011) provided a well-organised model incorporating the uncertain nature of performance assessment against the backdrop of the progress of a project. The basic concept of their model relied on the utilisation of human decisions determined by analysing linguistic terms in order to determine how the project is actually progressing. Despite all these prior efforts, the existing models in the literature still suffer from lacking the reliability of expert input for evaluating the progress of a project. The core purpose of this paper is to address the aforementioned lacunae in the previous models. The proposed model in this paper considers both the uncertainties of the linguistic terms utilised by experts (to evaluate the progress of a project) and the degree of reliability on the employed terms. This paper is organised as follows. First, the statement of the problem is presented. Then, the EVM principles are explained followed by the introduction of fuzzy theory, the novel notion of a Z-number and its application into the EVM. Section 6 presents a simple illustrative case to clarify the applicability of the proposed model. The next section provides an analytical comparison of the previous approaches and the proposed model. Finally, in Section 8, concluding remarks are provided.

2

Problem statement

There are many situations in real life projects where the amount of work (or the quantity of work) for an activity is unknown or imprecise. For example, in oil and gas projects, the amount of drilling process needed to be carried out per day is unknown. An oil well may not be correctly located on the initial drilling plan. Other examples come from the field of construction projects. In this kind of project, there are various types of task modules performed in order to achieve satisfactory conclusions, such as ‘pour foundation’. These task modules are normally dependent on different conditions. Consequently, the exact amount of work to derive an acceptable result from a structural engineering point of view is unknown. On the other words, the process is highly dependent on labour, material, machinery, and equipment. In such cases, it would be better to assess the percentage of work performed using linguistic terms. However, there is another challenge related to linguistic terms employed in the demonstration of progress. This challenge can be clearly indicated in the following question: “What is the degree of the reliability of an assigned linguistic term”? In spite of copious research previously conducted in the fuzzy modelling of EVM, the answer to this question is never discussed. The degree of reliability is an important issue in this context in order to deal with uncertainty and vagueness in real life projects. For example in the aforementioned oil and gas projects, in addition to the uncertain amount of work required for the drilling operation, the degree of the reliability of human judgment about the percentage of work

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performed also has to be incorporated into uncertainty modelling. The basic and fundamental idea of this proposed model consists of two modelling parts. They are the modelling of the impreciseness available in the assessment of how well the activities are progressing and the reliability of the judgment employed in evaluating progression forms. This idea to the knowledge of authors has never been considered before in the literature.

3

The EVM technique

The EVM assists project managers to carry out more appropriate measurement and evaluation of the progress of a project and the accompanying project performance indexes. The core of this concept is an evaluation based on the metrics demonstrating project performance from the cost and scheduling perspectives. The EVM mainly presents the efficiency of a project in terms of resource utilisation and it is defined as the amount of budget allocated to the work performed (the value of work done) against the actual costs. Table 1 provides a list of existing methods for the calculation of EVM1. Among the methods introduced in Table 1, the percent complete method is also included in this paper due to its simplicity and applicability in different types of projects. In the percent complete method, a person who is in charge of measurement in each measurement period estimates the percentage of the progress for activities undertaken. These judgments about the percentage of the progress of activities may be made subjectively, if there were no objective indicators to guide the estimation. Naeni et al. (2011) addressed this uncertainty in their research and provided fuzzy-based indicators using an analysis of linguistic terms to make the measurement of the progress of activities more accurate and reliable. However, as mentioned in the problem statement, the reliability of the utilised linguistic terms is still under investigation. Table 1

Different techniques for measuring progress

Product of activity

Number of measurement periods throughout activity duration

Tangible

1 or 2

More than 2

Fixed formula

Weighted milestone Percent complete

Intangible

4

Apportioned effort level of effort

Utilisation of Z-number in the measurement of earned value

Fuzzy sets were initially introduced by Zadeh (1965) to deal with the vagueness that is a pervasive phenomenon of real life cases. However, his novel notion of the Z-number has a greater capability to express the uncertain conditions of real life situations. Generally, the Z-number is an evolution of typical fuzzy sets. Prior to introducing the Z-number, the basic concept and definition of fuzzy sets are introduced below. Definition 1: A is a fuzzy set on a universe X and is illustrated as below (Zadeh, 1965): A = { x, μ A ( x ) | x ∈ X }

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where μA: X → [0, 1] is the membership function of A. The membership value, μA(x), describes the belongingness of x ∈ X in A. A fuzzy number can be described using different shapes. Among of the various shapes, triangular and trapezoidal shapes are the most common employed ones used due to the ease of representation and calculation. Definition 2: B = (b1 , b2 , b3 ) demonstrates a triangular fuzzy number where the membership function of this number is determined as below: ⎧0 ⎪ x −b 1 ⎪ ⎪ b2 − b1 μB ( x) = ⎨ ⎪ b3 − x ⎪ b3 − b2 ⎪ ⎩0

x < b1 b1 < x < b2

(1) b2 < x < b3 b3 < x

Definition 3: A = (a1 , a2 , a3 , a4 ) defines a trapezoidal fuzzy number where the membership function of this number is determined as follows: ⎧0 ⎪ x−a 1 ⎪ ⎪ a2 − a1 ⎪ μ A ( x) = ⎨1 ⎪ a −x ⎪ 3 ⎪ a4 − a3 ⎪0 ⎩

x < a1 a1 < x < a2 a2 < x < a3

(2)

a3 < x < a4 x > a4

Definition 4: A Z-number is an ordered pair of fuzzy numbers ( A , B ). The first component A has the same role as the fuzzy restriction plays, and the second component B illustrates the reliability of the first component (Zadeh, 2011). Figure 1 shows a possible illustration of a Z-number. Due to the impreciseness that is evident in the utilisation of linguistic terms, it is reasonable to employ fuzzy sets to provide objective indicators for the appropriate classification of these terms (Naeni et al., 2011). However, fuzzy sets face the restriction of not taking into account the degree of reliability of the linguistic terms employed. Therefore, the new notion of the Z-number will be applied to effectively overcome this limitation. In the proposed calculation of EVM, the first component of the Z-number relates to the fuzzy illustration of linguistic terms, and the second component is associated with the degree of reliability of the first component. The next example makes the idea of using a Z-number more comprehensible. Let us consider the actual progress (AP) of an activity is determined according to the following parts.

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Figure 1

A simple Z-number

Z = ( A , B )

μ B ( x )

μA ( x)

A

a1

B

a2

a3

a4

b1

b2

b3

The first part: the AP of activity is ‘very high’. The second part: the degree of the reliability of the above assigned term for the progress of the activity is ‘approximately high’. Figure 2 µ

Membership function of fuzzy numbers related to linguistic terms indicating actual progress

Very Low

Low

Less than Half

Half

M ore than Half

High

Very High

1

0.1

Table 2

0.2

Very low Low Less than half Half More than half Very high

0.4

0.5

0.6

0.7

0.8

The fuzzy numbers assigned to each linguistic term of Figure 2

Linguistic terms

High

0.3

Allocated fuzzy numbers (0, 0, 0.1, 0.2) (0.1, 0.15, 0.25, 0.3) (0.2, 0.3, 0.4, 0.5) (0.4, 0.45, 0.55, 0.6) (0.5, 0.6, 0.7, 0.8) (0.7, 0.75, 0.85, 0.9) (0.8, 0.9, 1, 1)

0.9

1

Progress

A novel earned value management model using Z-number Figure 3 µ

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Membership function of fuzzy numbers related to linguistic terms indicating degree of reliability

Very Low

Approximately Approximately Medium High Low Low High

Very High

1

0.1

Table 3

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Degree of reliabilty

The fuzzy numbers assigned to each linguistic term of Figure 3

Linguistic terms

Assigned fuzzy numbers

Very low

(0, 0, 0.1, 0.24)

Low

(0.1, 0.22, 0.35)

Approximately low

(0.21, 0.33, 0.48)

Medium

(0.35, 0.5, 0.62)

Approximately high

(0.52, 0.61, 0.8)

High

(0.64, 0.79, 0.9)

Very high

(0.78, 0.89, 1, 1)

Obviously, these parts cannot contribute toward the calculation of the earned value (EV) without converting them into number. The first part indicates the AP of activity based on linguistic terms and shall refer to the first component of a Z-number to determine the fuzzy illustration of linguistic terms. The second part presents the degree of the reliability of the first part and relates the first term to the second component of Z-number. Consequently, the aforementioned parts that relate to the AP of an activity shall transform into a Z-number and make the computation of EVM possible. However, the terms ‘very high’ and ‘approximately high’ utilised in the first and second components are subjective terms and cannot be considered directly as Z-number components. Therefore, the application of the objective indicators is required in advance to transform these subjective terms into objective indices. Figure 2 and Figure 3 show these fuzzy-based indicators with their corresponding linguistic terms. Figure 2 depicts the linguistic terms associated with the evaluation of the progress of a project. The horizontal axis of Figure 2 has a scale of [0-1] and refers to the AP. Figure 3 shows the evaluation of linguistic terms as applied to determining the degree of the reliability of expert judgment. Similar to the AP

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scale, the degree of the reliability is expressed as a scale of [0-1] in Figure 3. Table 2 and Table 3 show the details of the transformation associated with Figure 2 and Figure 3, respectively. For instance, according to Figure 2 and Table 2, the linguistic terms ‘very high’ utilised in the first part of the previous example are equivalent to [0.8, 0.9, 1, 1]. In addition to the first part, the linguistic term ‘approximately high’ employed in the second part, equals [0.2, 0.4, 0.5, 0.7]. Note that Figure 2 and Figure 3, provided as an example here, can be completely changed in different projects and situations.

4.1 Transformation of a Z-number into a normal fuzzy number From a mathematical standpoint, it is important to convert a Z-number into a standard computable form. Therefore, a method of transforming a Z-number into a classical fuzzy number is presented in this section. However, since the concept of Z-numbers is fairly novel, research addressing Z-number is rare. Among them, Kang et al. (2012) presented a new method of transforming a Z-number into a fuzzy number based on the fuzzy expectation of the fuzzy sets. Their model is utilised in this paper due to its simplicity and being straightforward to employ. Let us assume a Z-number as Z = ( A , B ) where the left and the right parts describe the restriction and reliability, respectively. Let A = {〈 x, μ A ( x)〉 | x ∈ [0, 1]} and B = {〈 x, μB ( x)〉 | x ∈ [0, 1]} where μ A and μB are trapezoid membership functions. The procedure to apply method is given as follows: 1

Transform the second part (reliability) into a crisp value (Kang et al., 2012):

α=

∫ xμ ( x)dx ∫ μ ( x)dx B

(3)

B

where



indicates an algebraic integration.

2

Add the weight of the second part (α) to the first part. The weighted Z-number is illustrated as Z α = {〈 x, μ Aα ( x)〉 | μ Aα ( x) = α μ A ( x)} Figure 4 demonstrates the novel Z α .

3

Convert the weighted Z-number into a normal fuzzy number by multiplying A α . Z ′ = α × A α = ( α × a1 , α × a2 , α × a3 , α × a4 )

α by (4)

Eventually, the initial Z-number is transformed to a normal fuzzy number (the reader can refer to Kang et al. (2012) for more details and proof of the above theorem). The aforementioned procedure can then be applied to convert a Z-number-based AP into a normal fuzzy number in order to make the utilisation of AP possible in the relevant calculations.

A novel earned value management model using Z-number Figure 4

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The weighted Z-number

µ (x )

A

1

A α

a

a1

Figure 5

a3

a2

a4

x

The normal fuzzy number transformed from the Z-number

µ

Z ′

1

α × a1

α × a2

α × a3

α × a4

x

4.2 New calculations of EVM Generally, the EVM shows the amount of the value earned in comparison with the money paid for each individual activity. The following equation calculates EV for an activity: ki = j EV AP i × BACi = ( EV1i , EV2i , EV3i , EV4i )

(5)

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where the AP stands for the AP of each individual activity. The AP is initially provided in the form of a Z-number as mentioned earlier or referred in the next section. Then, the obtained result is transformed into a typical trapezoidal fuzzy number. The BACi is the budget at completion of activity i. In addition to the calculation of the EV for each activity, the equation (6) calculates the total EV at each measurement period as follows k= EV

n

∑ i =1

k i = ⎛⎜ EV ⎜ ⎝

n



n

EV1i ,

i =1



n

EV2i ,

i =1

∑ i =1

n

EV3i ,

∑ EV

4i

i =1

⎞ ⎟⎟ ⎠

(6)

where n denotes the total number of the project activities.

5

Z-number-based EV indices and estimation approach

The indexes in the EVM evaluate the project performance from different point of views. They also estimate the completion cost and duration of the project. This section presents the development of such indices.

5.1 Cost performance index One of the most employed indexes in the EVM is the cost performance index (CPI). CPI mainly assesses the project performance from the aspect of cost by comparing the actual value earned and the actual amount spent. The CPI can be calculated as follows.

CPI =

EV AC

(7)

where, the AC stands for the actual cost of the performed works. Equation (8) presents the novel calculation of the CPI using the new method for the EV. k k = EV = ⎛⎜ EV1 , EV2 , EV3 , EV4 ⎞⎟ = ( CPI1 , CPI 2 , CPI 3 , CPI 4 ) CPI AC ⎝ AC AC AC AC ⎠

(8)

5.2 Schedule performance index The SPI evaluates the behaviour of the project by comparing the AP against the planned one. In other words, the SPI is calculated as a proportion of the EV to the planned value (PV) as shown below: SPI =

EV PV

(9)

There are inefficiencies in using the SPI for measuring project schedule performance. Normally, the SPI does calculations based on cost units. Therefore, it is not meaningful enough to employ it individually for the evaluation of the schedule performance. Moreover, the SPI leads to the value 1 at the end of project in deterministic calculation. Lipke (2003) discussed the ineffectiveness of a typical SPI in his study and proposed the ES for the assessment of project performance from the schedule aspect. Generally, the ES can be considered as a time equivalent of the EV. Figure 6 and equation (10) demonstrate the basic concept and calculation of the ES, respectively (Lipke, 2003).

A novel earned value management model using Z-number Figure 6

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The ES position in comparison with the EV

C ost PV

EV

N

N+1

T im e

ES

⎛ ( EV − PVN ) ⎞ ES = N + ⎜ ⎟ ⎝ ( PVN +1 − PVN ) ⎠

(10)

where the longest time interval that the PV is less than the EV has been determined as N. PVN and PVN+1 are PV at time N and N + 1, respectively (Vandevoorde and Vanhoucke, 2005). Equation (11) illustrates the new formulation for the ES using the novel calculation for the EV presented in prior sections. ⎛ k ⎞ j = N + ⎜ ( EV − PVN ) ⎟ = ( ES1 , ES2 , ES3 , ES4 ) ES ⎜ ( PV − PV ) ⎟ N +1 N ⎠ ⎝

(11)

The concept of the ES also incorporates the calculation of the SPI and provides a novel SPI on the basis of time units (i.e, SPIt). The SPIt addresses the available restrictions of the SPI. It is calculated as the proportion of the ES to the actual duration (AD), i.e., ES SPI t = . Equation (12) demonstrates the other formulation of the SPIt using the AD fuzzy-based ES.

k t = ⎛⎜ ES1 , ES2 , ES3 , ES4 ⎞⎟ = ( SPI t1 , SPI t 2 , SPI t 3 , SPI t 4 ) SPI ⎝ AD AD AD AD ⎠

(12)

5.3 Estimating costs at completion using the Z-number There are different methods available for estimating the cost of projects (EAC). Among them, a general model considers the project performance and the EV simultaneously (Christensen, 1996; Zwikael et al., 2000).

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M. Salari et al. EAC = AC +

BAC − EV PI

(13)

where PI stands for project performance index, AC illustrates the AC of the project up to the data date, and the BAC indicates the total amount of activities cost (i.e., n

BAC =

∑ BAC ). i

Different metrics and methods are employed to determine the best

i =1

value of the PI (Christensen, 1993; Zorriassatine and Bagherpour, 2009). Conventionally, there are four common kinds of the PI (Christensen, 1996): •

cost performance index (CPI)



schedule performance index (SPIt)2



composite index: W1 × SPI + W2 × CPI3



schedule cost index: SCI = SPI × CPI.

However, the determination of the best value for the PI leads to uncertainty because it cannot be established which of the above suggested values is the most appropriate one for a specific project. In this case, it is suggested the person who is in charge of measurement utilises one of the offered metrics and uses the degree of reliability of the selected metric. For instance, the schedule cost index (SCI) can be employed as the PI for the cost estimation of a project according to the following parts. •

first part: The SCI is selected as the performance index.



second part: The reliability of this selection is approximately high.

The first part determines the value of the PI and it can be considered as the fuzzy restriction component of a Z-number. The second part may be interpreted as reliability component of a Z-number. Figure 3 can easily show objective indicators for the evaluation of the reliability (of the second part). For instance, according to Figure 3, the term ‘approximately high’ related to the second part has been assigned to the following fuzzy number: [0.2, 0.4, 0.5, 0.7]. Consequently, the aforementioned parts related to the value of the PI can be transformed into a Z-number (i.e., the first and the second parts form two components of a Z-number). Note that the four suggested kinds of the PI should be calculated as fuzzy numbers in order to make it possible to consider them as the first component of a Z-number. Subsequently, to determine the PI in the form of a Z-number, it should be transformed into a fuzzy-based number to facilitate the calculation of EAC (refer to Section 5). Hence, the EV and PI can be obtained as fuzzy numbers. The EAC will provide a fuzzy number as below: k k = AC + BAC − EV = ( EAC1 , EAC2 , EAC3 , EAC4 ) EAC j PI

(14)

5.4 Time forecasting using the Z-number One important part of the EVM purpose is associated with the estimation of the total project efforts to manage and assess the temporal aspects. The three most important available methods in the estimation of the project duration are:

A novel earned value management model using Z-number •

the PV method (Anbari, 2003)



the earned duration (ED) method (Jacob, 2003; Jacob and Kane, 2004)



the ES method (Lipke, 2003; Henderson, 2004)

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Out of the three introduced methods, the ES is the most wide-used one for the estimation of project duration (Vandevoorde and Vanhoucke, 2005). Thus, the concept of the ES in terms of time estimation has been included in the proposed model. The following equation demonstrates fuzzy-based EACt. j k t = AD + PD − ES = ( EACt1 , EACt 2 , EACt 3 , EACt 4 ) EAC j PI

(15)

where the index t in EACt notation indicates the proposed estimation associated with the duration of the project. AD stands for actual duration up to the data date, and PD demonstrates the total planned duration of the project in terms of scheduling process. j has the similar role in both EACt and EAC. Thus, the introduced method to PI j in the EAC can be employed for the calculation of the determine the value of the PI j is selected from potential EACt similarly. This means the value of fuzzy-based PI alternatives that will be introduced in Section 8.

5.5 Interpretation of fuzzy cost and time forecasting k and EAC k t is the next necessary step of the proposed model. The interpretation of EAC As fuzzy illustration provides a range of crisp numbers instead of presenting a certain k and EAC k t numbers can be value, it is important to determine how the fuzzy-based EAC evaluated and described. The following figures and tables make this assessment possible. According to Figure 7 and Table 4, five scenarios are available for determining the state k in comparison with the BAC. Furthermore, Figure 8 and Table 5 explain of the EAC k t against the planned duration of the project. For five potential scenarios of EAC k t and BAC is in accordance with scenario 4 of instance, if the position of the EAC Figure 2 then the cost estimation indicates that the “final cost of the project is almost over the planned budget”. Table 4 Scenario

k Interpretation of scenarios related to the EAC k state in EAC comparison with BAC

Explanation of performed estimation

1

BAC > EAC4

Final cost of project is under the planned budget

2

EAC3 < BAC < EAC4

Final cost of project is almost under the planned budget

3

EAC2 < BAC < EAC3

Final cost of project is equal to the planned budget

4

EAC1 < BAC < EAC2

Final cost of project is almost over the planned budget

5

BAC > EAC1

Final cost of project is over the planned budget

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Figure 7

Table 5

k (see online version for colours) Different scenarios of EAC

kt Interpretation of scenarios related to the EAC k t state in EAC comparison with BAC

Explanation of performed estimation

1

PD > EACt4

Final duration of project is under the planned duration

2

EACt3 < PD < EACt4

Final duration of project is almost under the planned duration

Scenario

3

EACt2 < PD < EACt3

Final duration of project is equal to the planned duration

4

EACt1 < PD < EACt2

Final duration of project is almost over the planned duration

5

PD > EACt1

Final duration of project is over the planned duration

A novel earned value management model using Z-number Figure 8

6

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k t (see online version for colours) Different scenarios of EAC

Case study

In this section, an illustrative case study is presented which demonstrates the applicability of the proposed model. This case has been extracted from a construction-based study including a house-building project. According to Figure 9, the elements in the first level of the work breakdown structure (WBS) are ‘concrete’, ‘framing’, ‘plumbing’, ‘electrical’, ‘interior’ and ‘roofing’. Each of the elements consists of three work packages. Similarly, the work packages can be divided into separate activities. However, to keep the case study simple, the WBS is limited to the work packages level (i.e., second level of WBS). Consequently, the progress, the degree of the reliability on measured

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progress and the BAC shown in Table 6 are related to the work packages of the project. It is scheduled that the project is to be concluded at 14 month. Figure 9

WBS of case study House Building Project

concrete

Framing

Plumbing

Electrical

Interior

Roofing

Pour Foundation

Frame Exterior Walls

Install Water Lines

Install Wiring

Install Drywall

Install Felts

Install Patio

Frame Interior Walls

Install Gas Lines

Install Outlets/ Switches

Install Carpet

Install Shingles

Stairway

Install Roofing Trusses

Install Bath and Kitchen Fixture

Install Fixture

Painting

Install Vents

Table 6

The information of work packages

Work package

Progress

Degree of reliability on assigned progress

BAC($)

Pour foundation

Completed

---------

8,000

Install patio

Very high

Approximately high

3,400

Stairway

Half

Low

2,500

Not started

---------

2,500

Frame interior walls

Low

Very high

3,000

Install roofing trusses

Not started

---------

1,250

Frame exterior walls

Install water lines

Low

High

1,900

Install gas lines

Low

Approximately high

2,300

Not started

---------

850

Very low

Very high

950

Not started

---------

1,350

Very low

Approximately low

1,200

Install bath and kitchen fixture Install wiring Install outlets/switches Install fixtures Install drywalls

Half

Medium

2,400

Not started

---------

3,200

Paintings

Less than half

Low

5,600

Install felt

More than half

Low

3,600

Very low

Approximately low

2,600

Not started

---------

2,900

Install carpets

Install shingles Install vents

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A novel earned value management model using Z-number

Table 7 demonstrates the PV and the AC of the project at each month up to the data date (i.e., month 8). Initially, the information provided in Table 6 should be transformed into a Z-number in order to make further calculation possible. Section 4 completely described the method of converting linguistic terms into a Z-number and transforming the Z-number into a typical fuzzy number, respectively. Table 8 shows the results of such a transformation. Table 7

The AC and planned value division of the budget throughout the project

Month

1

2

3

4

5

6

7

AC

3,500

4,000

2,600

4,360

2,300

2,600

3,100

PV

4,000

4,000

3,500

2,500

2,500

3,100

3,600

8

9

10

11

12

13

14

2,300

3,500

5,500

5,500

5,000

2,000

AC

1,950

PV

2,500

Table 8

Work packages transformation from Z-numbers into fuzzy numbers

Work package Pour foundation Install patio Stairway

Assigned Z-number to the AP of Work package

Transformation of Z-number into a fuzzy number

Completed

---------

((0.8, 0.9, 1, 1), (0.52, 0.61, 0.8))

(0.64, 0.72, 0.8, 0.8)

((0.4, 0.45, 0.55, 0.6), (0.1, 0.22, 0.35)) (0.19, 0.21, 0.25, 0.28)

Frame exterior walls

Not started

Frame interior walls

---------

((0.1, 0.15, 0.25, 0.3), (0.78, 0.89, 1, 1)) (0.09, 0.14, 0.24, 0.28)

Install roofing trusses

Not started

---------

Install water lines

((0.1, 0.15, 0.25, 0.3), (0.64, 0.79, 0.9)) (0.09, 0.13, 0.22, 0.26)

Install gas lines

((0.1, 0.15, 0.25, 0.3), (0.52, 0.61, 0.8))

(0.08, 0.12, 0.2, 0.24)

Not started

---------

((0, 0, 0.1, 0.2), (0.78, 0.89, 1, 1))

(0, 0, 0.95, 0.19)

Not started

---------

((0, 0, 0.1, 0.2), (0.21, 0.33, 0.48))

(0, 0, 0.06, 0.11)

Install bath and kitchen fixture Install wiring Install outlests/switches Install fixtures Install drywalls

((0.4, 0.45, 0.55, 0.6), (0.35, 0.5, 0.62)) (0.28, 0.32, 0.38, 0.42)

Install carpets

Not started

---------

Paintings

((0.2, 0.3, 0.4, 0.5), (0.1, 0.22, 0.35))

(0.09, 0.14, 0.19, 0.24)

Install felt

((0.5, 0.6, 0.7, 0.8), (0.1, 0.22, 0.35))

(0.23, 0.28, 0.33, 0.38)

((0, 0, 0.1, 0.2), (0.21, 0.33, 0.48))

(0, 0, 0.06, 0.11)

Not started

---------

Install shingles Install vents

The fuzzy-based actual progresses resulting from Z-numbers incorporated in the calculation of the project EV [see equation (6)] can be presented as: k= EV

n

k ∑ EV i =1

i

= (1, 2789.33, 1,3817.31, 1,5584.78, 1, 6760.06)

(0.37, 0.44, 0.46, 0.49)

Approximately high Medium

kt SPI

j k t = AD + PD − ES EAC j PI

j Fuzzy value of PI

(0.25, 0.32, 0.38, 0.44)

j Degree of reliability on selected PI

j PI

k t × CPI k SPI

k k = AC + BAC − EV EAC j PI

Result of formula

k t = (13.8, 14.1, 15.6, 16.3) EAC

k = (107,843, 118,311, 130,395, 155,369) EAC

Table 9

Formula

114 M. Salari et al.

Estimation of project cost and duration

A novel earned value management model using Z-number

115

It is now possible to determine the CPI based on fuzzy calculation using equation (8): k k = EV = ⎛⎜ EV1 , EV2 , EV3 , EV4 ⎞⎟ = (0.52, 0.56, 0.63, 0.68) CPI AC ⎝ AC AC AC AC ⎠ k t ) is obtained as below: The ES and the SPI based on time calculation ( SPI

⎛ k ⎞ j = N + ⎜ ( EV − PVN ) ⎟ = ( ES1 , ES2 , ES3 , ES4 ) = (6.14, 7.26, 7.56, 8.15) ES ⎜ ( PV − PV ) ⎟ N +1 N ⎠ ⎝ k t = ⎛⎜ ES1 , ES2 , ES3 , ES4 ⎞⎟ = (0.76, 0.9, 0.94, 1.01) SPI ⎝ AD AD AD AD ⎠ k t and CPI k in their Naeni et al. (2011) obtained a fuzzy-based evaluation for SPI research. Their proposed method was partly applied in this paper to assess the state of k cost and schedule performance. According to their presented approach, the CPI k indicates that “the project is behind budget” and SPI t demonstrates that “the project is approximately behind schedule”. As previously mentioned in Section 5.3, four potential k and EAC kt . alternatives can be used as performance index for the calculation of EAC The degree of reliability on each selected alternative can also be employed to deal with the vagueness of choosing the best alternatives. Table 9 illustrates these values using the proposed method. Following computations are given as an example to illustrate the detail of calculations k: behind EAC k = ⎛⎜ 24, 410 + 49,500 − 12, 789.33 , 24, 410 + 49,500 − 13,817.3 , EAC 0.44 0.38 ⎝ 49,500 − 15,584.78 49,500 − 16, 760.06 ⎞ 24, 410 + , 24, 410 + ⎟ 0.32 0.25 ⎠ = (107,843, 118,311, 130,395, 155,369)

Scenario 5 introduced in Table 4 implies “the final cost of the project is over the planned budget”. Similarly, the different scenarios obtained in Table 5 can be used to interpret the k t According to scenario 4 in Table 5, “the final duration of the project is value of EAC almost over the planned duration”.

7

Discussion

In this section of the paper, the implications of the proposed approach from both managerial and researcher viewpoints have been discussed. Furthermore, a comparative analysis between the available method and the proposed approach is presented.

7.1 Managerial implication The evaluation of the different aspects of projects based on group judgments have been employed in different project management practices. For instance, Zeng et al. (2007)

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utilised experts’ judgment for the risk assessment in a construction project. They suggested that a risk assessment group should be formed for the identification of potential risks and the determination of risks magnitude. Such teamwork can greatly assist project managers or decision makers to benefit from the incorporation of experts distinctive viewpoints. The core idea of Z-number is based on the evaluation of the assigned number to a specific phenomenon. A team of experts can perform the evaluation process. For instance, in the case of risk identification and the determination of risk magnitude, the judgment of experts may be considered as the second component of a Z-number and further calculation can be easily performed. Hence, the utilisation of the concept of Z-number facilitates the idea of employing the team attitude in decision-making process.

7.2 Researcher implication Fuzzy reasoning techniques have demonstrated its practical functionality in handling the ill-defined and complicated problems arising in different projects (Zeng et al., 2004, 2005, 2007). However, such techniques can be developed using the notion of Z-number in order to insert the reliability factor in their related calculations. The incorporation of the reliability aspect in the fuzzy reasoning techniques provides a novel and promising prospect of their application. This new perspective can be viewed in the context of performing a comprehensive analysis for detecting the influence of the degree of the reliability in associated computations. Particularly, the assessment of the influence of different expert judgments in the progress measurement of a projects can be considered as a new field of interest for practitioners and researchers.

7.3 Comparative analysis The proposed model in this paper is compared with the existing works in the literature which are taken from different aspects in order to demonstrate validity and superiority over existing ones (see Table 10). Table 10

Comparison of the proposed model with different models in the literature Features

EVM-related works

Detecting degree of Considering Fuzzy Fuzzy reliability on PI for degree of Fuzzy calculation assessment computation of measurement reliability on of EVM of EVM k kt measured of progress EAC and EAC metrics metrics progress (forecasting features)

Noori et al. (2008)

×

×





×

Naeni et al. (2011)



×





×

Moslemi Naeni and Salehipour (2011)



×



×

×

Ponz-Tienda et al. (2012)

×

×





×

The proposed model











A novel earned value management model using Z-number

8

117

Conclusion remarks and further recommendation

The novel notion of a Z-number has been efficiently applied in this paper to provide an appropriate measurement of the progress of a project under uncertain conditions. The SPIt and CPI are also obtained based on a new progress evaluation system. In addition to the above advantages, the forecasting of the project completion time and cost has also been presented. The proposed model can assist project managers to assess the progress of a project effectively since it incorporates the bias of expert judgment in a progress calculation and presents the fuzzy-based assessment of EVM indices much more realistically. Further recommendation may focus on applying the Z-number to financial performance indexes and invoice control systems.

Acknowledgements The authors would like to thank Dr. Hashem Salari and Mr. M.M. Asgary for their valuable and helpful comments and providing project data.

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Notes 1 2 3

The reader can refer to ‘practice standard for earned value management’ (PMI, 2008) to achieve more details. Note that Christensen (1996) introduced the SPI as one of the potential alternative for the PI. In this paper, we use SPIt due to its mentioned advantages over the SPI. W1 and W2 are modified weights and their values indicate the importance of each index in comparison with the other one.