Content uploaded by Mostafa Salari
Author content
All content in this area was uploaded by Mostafa Salari
Content may be subject to copyright.
I
nt. J.
A
pplied Decision Sciences, Vol.
7
, No. 1, 2014 97
Copyright © 2014 Inderscience Enterprises Ltd.
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: mostafa.salari.ind@gmail.com
*Corresponding author
Morteza Bagherpour
Department of Industrial Engineering,
Iran University of Science and Technology,
Narmak, Tehran, Iran
E-mail: Bagherpour@iust.ac.ir
John Wang
Department of Information and Operation Management,
Montclair State University,
Montclair, NJ 07043, USA
Fax: (973) 655-7678
E-mail: j.john.wang@gmail.com
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.
98
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)
A
novel earned value management model using Z-numbe
r
99
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
100
M
. Salari et al.
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
1 or 2 More than 2
Tangible Fixed formula Weighted milestone
Percent complete
Intangible Apportioned effort level of effort
4 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
Ax
μ
xxX=∈
A
novel earned value management model using Z-numbe
r
101
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: 123
(, , )Bbbb=
demonstrates a triangular fuzzy number where the
membership function of this number is determined as below:
1
1
12
21
3
23
32
3
0
()
0
B
xb
xb bxb
bb
μxbxbxb
bb
bx
<
⎧
⎪−
⎪<<
−
⎪
=⎨−
⎪<<
⎪−
⎪<
⎩
(1)
Definition 3: 1234
(, , , )
A
aa a a=
defines a trapezoidal fuzzy number where the
membership function of this number is determined as follows:
1
1
12
21
23
3
34
43
4
0
() 1
0
A
xa
xa axa
aa
μxaxa
axaxa
aa
xa
<
⎧
⎪−
⎪<<
−
⎪
⎪
=<<
⎨
⎪−
⎪<<
−
⎪
⎪>
⎩
(2)
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.
102
M
. Salari et al.
Figure 1 A simple Z-number
()
Ax
μ
()
Bx
μ
a1 a2 a3 a4b1 b2 b3
A
B
(,)ZAB=
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 Membership function of fuzzy numbers related to linguistic terms indicating actual
progress
1
Very
Low Low Less than Half More than Half High
Very
High
Progress
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
µ
Half
Table 2 The fuzzy numbers assigned to each linguistic term of Figure 2
Linguistic terms Allocated fuzzy numbers
Very low (0, 0, 0.1, 0.2)
Low (0.1, 0.15, 0.25, 0.3)
Less than half (0.2, 0.3, 0.4, 0.5)
Half (0.4, 0.45, 0.55, 0.6)
More than half (0.5, 0.6, 0.7, 0.8)
High (0.7, 0.75, 0.85, 0.9)
Very high (0.8, 0.9, 1, 1)
A
novel earned value management model using Z-numbe
r
103
Figure 3 Membership function of fuzzy numbers related to linguistic terms indicating degree of
reliability
Approximately
High
Medium
Approximately
Low
Very
Low
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Degree of reliabilty
µ
1
Low High
Very
High
Table 3 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
104
M
. Salari et al.
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
AB= where
the left and the right parts describe the restriction and reliability, respectively. Let
{ , ()| [0,1]}
A
Axμxx=〈 〉 ∈
and { , ()| [0,1]}
B
Bxμxx=〈 〉 ∈
where
A
μ
and
B
μ
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):
()
()
B
B
x
μ
xdx
μ
xdx
=∫
∫
α
(3)
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 { , ()| () ()}
AA A
Zx
μ
xμxμx=〈 〉 =
αα
α
α
Figure 4 demonstrates the novel
.
Z
α
3 Convert the weighted Z-number into a normal fuzzy number by multiplying
α
by
.
A
α
()
1234
,,,
Z
Aaaaa
′=×= × × × ×
α
ααααα
(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-numbe
r
105
Figure 4 The weighted Z-number
1
a
x
µ(x)
a1a2
a3a4
A
A
α
Figure 5 The normal fuzzy number transformed from the Z-number
x
µ
Z
′
1
1
a
α
×
2
a
α
×3
a
α
×
4
a
α
×
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:
k
j
()
1234
,,,
i
iiiiii
EV AP BAC EV EV EV EV=× = (5)
106
M
. Salari et al.
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
kk
1234
11111
,,,
nnnnn
iiiii
iiiii
EV EV EV EV EV EV
=====
⎛⎞
==
⎜⎟
⎜⎟
⎝⎠
∑∑∑∑∑
(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.
EV
CPI
A
C
= (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
()
1234
1234
,,, , , ,
EV EV EV EV EV
CPI CPI CPI CPI 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:
EV
SPI 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-numbe
r
107
Figure 6 The ES position in comparison with the EV
Time
EV
NN+1
ES
C
os
t
PV
()
()
1
N
NN
EV PV
ES N PV PV
+
⎛⎞−
=+
⎜⎟
−
⎝⎠
(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.
j
k
()
()
()
1234
1
,,,
N
NN
EV PV
ES N ES ES ES ES
PV PV
+
⎛⎞
−
=+ =
⎜⎟
⎜⎟
−
⎝⎠ (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.,
.
t
ES
SPI
A
D
= Equation (12) demonstrates the other formulation of the SPIt using the
fuzzy-based ES.
k
()
1234
1234
,,, , , ,
ttt t t
ES ES ES ES
SPI SPI SPI SPI 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).
108
M
. Salari et al.
BAC EV
EAC AC 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.,
1
).
n
i
i
BAC BAC
=
=∑ Different metrics and methods are employed to determine the best
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
j
()
1234
,,,
BAC EV
EAC AC EAC EAC EAC EAC
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-numbe
r
109
• the PV method (Anbari, 2003)
• the earned duration (ED) method (Jacob, 2003; Jacob and Kane, 2004)
• the ES method (Lipke, 2003; Henderson, 2004)
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.
k
j
j
()
1234
,,,
ttt t t
PD ES
EAC AD EAC EAC EAC EAC
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
PI has the similar role in both EACt and EAC. Thus, the introduced method to
determine the value of the
j
PI in the EAC can be employed for the calculation of the
EACt similarly. This means the value of fuzzy-based
j
PI is selected from potential
alternatives that will be introduced in Section 8.
5.5 Interpretation of fuzzy cost and time forecasting
The interpretation of
k
EAC and
k
t
EAC is the next necessary step of the proposed model.
As fuzzy illustration provides a range of crisp numbers instead of presenting a certain
value, it is important to determine how the fuzzy-based
k
EAC and
k
t
EAC numbers can be
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
of the
k
EAC in comparison with the BAC. Furthermore, Figure 8 and Table 5 explain
five potential scenarios of
k
t
EAC against the planned duration of the project. For
instance, if the position of the
k
t
EAC and BAC is in accordance with scenario 4 of
Figure 2 then the cost estimation indicates that the “final cost of the project is almost over
the planned budget”.
Table 4 Interpretation of scenarios related to the
k
E
AC
Scenario
k
E
AC state in
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
110
M
. Salari et al.
Figure 7 Different scenarios of
k
E
AC (see online version for colours)
Table 5 Interpretation of scenarios related to the
k
t
EAC
Scenario
k
t
EAC state in
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
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-numbe
r
111
Figure 8 Different scenarios of
k
t
E
AC (see online version for colours)
6 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
112
M
. Salari et al.
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 Build ing
Project
concrete ElectricalPlumbingFraming RoofingInterior
Pour
Foundation
Install
Patio
Install Bath
and Kitchen
Fixture
Install Gas
Lines
Install Water
Lines
Install
Roofing
Trusses
Frame
Interior
Walls
Frame
Exterior
Walls
Stairway Painting
Install
Carpet
Install
Drywall
Install
Fixture
Install
Outlets/
Switches
Install
Wiring
Install Vents
Install
Shingles
Install Felts
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
Frame exterior walls Not started --------- 2,500
Frame interior walls Low Very high 3,000
Install roofing trusses Not started --------- 1,250
Install water lines Low High 1,900
Install gas lines Low Approximately high 2,300
Install bath and kitchen fixture Not started --------- 850
Install wiring Very low Very high 950
Install outlets/switches Not started --------- 1,350
Install fixtures Very low Approximately low 1,200
Install drywalls Half Medium 2,400
Install carpets Not started --------- 3,200
Paintings Less than half Low 5,600
Install felt More than half Low 3,600
Install shingles Very low Approximately low 2,600
Install vents Not started --------- 2,900
A
novel earned value management model using Z-numbe
r
113
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
AC 1,950
PV 2,500 2,300 3,500 5,500 5,500 5,000 2,000
Table 8 Work packages transformation from Z-numbers into fuzzy numbers
Work package Assigned Z-number to the
AP of Work package
Transformation of
Z-number into a
fuzzy number
Pour foundation Completed ---------
Install patio ((0.8, 0.9, 1, 1), (0.52, 0.61, 0.8)) (0.64, 0.72, 0.8, 0.8)
Stairway ((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)
Install bath and kitchen fixture Not started ---------
Install wiring ((0, 0, 0.1, 0.2), (0.78, 0.89, 1, 1)) (0, 0, 0.95, 0.19)
Install outlests/switches Not started ---------
Install fixtures ((0, 0, 0.1, 0.2), (0.21, 0.33, 0.48)) (0, 0, 0.06, 0.11)
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)
Install shingles ((0, 0, 0.1, 0.2), (0.21, 0.33, 0.48)) (0, 0, 0.06, 0.11)
Install vents Not started ---------
The fuzzy-based actual progresses resulting from Z-numbers incorporated in the
calculation of the project EV [see equation (6)] can be presented as:
kk
1
(1, 2789.33, 1,3817.31, 1,5584.78, 1, 6760.06)
n
i
i
EV EV
=
==
∑
114
M
. Salari et al.
Table 9 Estimation of project cost and duration
Formula
j
PI
Degree of reliability on selected
j
PI
Fuzzy value of
j
PI
Result of formula
k
k
j
B
AC EV
EAC AC PI
−
=+
k
k
t
SPI CPI×
Approximately high (0.25, 0.32, 0.38, 0.44)
k
(107,843,118, 311,130, 395, 155,369)EAC =
k
j
j
t
PD ES
EAC AD PI
−
=+
k
t
SPI
Medium (0.37, 0.44, 0.46, 0.49)
k
(13.8, 14.1,15.6, 16.3)
t
EAC =
A
novel earned value management model using Z-numbe
r
115
It is now possible to determine the CPI based on fuzzy calculation using equation (8):
k
k
1234
, , , (0.52, 0.56, 0.63, 0.68)
EV EV EV EV EV
CPI AC AC AC AC AC
⎛⎞
== =
⎜⎟
⎝⎠
The ES and the SPI based on time calculation
k
()
t
SPI is obtained as below:
j
k
()
()
()
k
1234
1
1234
, , , (6.14, 7.26, 7.56, 8.15)
, , , (0.76, 0.9, 0.94, 1.01)
N
NN
t
EV PV
ES N ES ES ES ES
PV PV
ES ES ES ES
SPI AD AD AD AD
+
⎛⎞
−
=+ = =
⎜⎟
⎜⎟
−
⎝⎠
⎛⎞
==
⎜⎟
⎝⎠
Naeni et al. (2011) obtained a fuzzy-based evaluation for
k
t
SPI and
k
CPI in their
research. Their proposed method was partly applied in this paper to assess the state of
cost and schedule performance. According to their presented approach, the
k
CPI
indicates that “the project is behind budget” and
k
t
SPI demonstrates that “the project is
approximately behind schedule”. As previously mentioned in Section 5.3, four potential
alternatives can be used as performance index for the calculation of
k
EAC and
k
.
t
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
behind
k
:EAC
k
49,500 12,789.33 49,500 13,817.3
24, 410 , 24, 410 ,
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)
EAC −−
⎛
=+ +
⎜
⎝
−−
⎞
++
⎟
⎠
=
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
value of
k
t
EAC According to scenario 4 in Table 5, “the final duration of the project is
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)
116
M
. Salari et al.
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
EVM-related
works
Features
Fuzzy
measurement
of progress
Considering
degree of
reliability on
measured
progress
Fuzzy
calculation
of EVM
metrics
Fuzzy
assessment
of EVM
metrics
Detecting degree of
reliability on PI for
computation of
k
EAC and
k
t
EAC
(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-numbe
r
117
8 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.
References
Amid, A., Ghodsypour, S.H. and O’Brien, C. (2006) ‘Fuzzy multi objective linear model for
supplier selection in a supply chain’, International Journal of Production Economics,
Vol. 104, No. 2, pp.394–407.
Anbari, F.T. (2003) ‘Earned value project management method and extensions’, Project
Management Journal, Vol. 34, No. 4, pp.12–23.
Aouam, T., Lamrani, H., Aguenaou, S. and Diabat, A. (2009) ‘A benchmark based AHP model
for credit evaluation’, International Journal of Applied Decision Sciences, Vol. 2, No. 2,
pp.151–166.
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, Vol. 49, Nos. 5–8, pp.419–429.
Christensen, D.S. (1993) ‘The estimate at completion problem – a review of three studies’, Project
Management Journal, Vol. 24, No. 1, pp.37–42.
Christensen, D.S. (1996) ‘Project advocacy and the estimate at completion problem’, The Journal
of Cost Analysis, Vol. 13, No. 1, pp.35–60.
Farzipoor, S.R. (2011) ‘A decision model for selecting the best entry modes via data envelopment
analysis’, International Journal of Applied Decision Sciences, Vol. 4, No. 3, pp.213–229.
Henderson, K. (2003) ‘Earned schedule: a breakthrough extensions to earned value theory? A
retrospective analysis of real project data’, The Measurable News, Vol. 1, No. 2, pp.13–23.
Henderson, K. (2004) ‘Further developments in earned schedule’, The Measurable News,
Vol. 1, No. 1, pp.15–22.
Houska, M., Domeova, L. and Berankova, M. (2012) ‘The conversion of cost and benefit criteria in
the TOPSIS method’, International Journal of Applied Decision Sciences, Vol. 5, No. 4,
pp.329–341.
Jacob, D.S. (2003) ‘Forecasting project schedule completion with earned value metrics’, The
Measurable News, Vol. 1, No. 1, pp.7–9.
Jacob, D.S. and Kane, M. (2004) ‘Forecasting schedule completion using earned value metrics
revisited’, The Measurable News, Vol. 1, No. 2, pp.11–17.
118
M
. Salari et al.
Jajimoggala, S., Rao, V.V.S.K. and Beela, S. (2011) ‘Supplier evaluation using hybrid multiple
criteria decision making approach’, International Journal of Applied Decision Sciences,
Vol. 4, No. 3, pp.260–279.
Kang, B., Wei, D., Li, Y. and Deng, Y. (2012) ‘A method of converting Z-number to
classical fuzzy number’, Journal of Information and Computational Science, Vol. 9, No. 3,
pp.703–709.
Khalili-Damghani, K., Taghavi-Fard, M. and Abtahi, A.R. (2012) ‘A fuzzy two-stage DEA
approach for performance measurement: real case of agility performance in dairy supply
chains’, International Journal of Applied Decision Sciences, Vol. 5, No. 4, pp.293–317.
Kim, E.H., Wells Jr., W.G. and Duffey, M.R. (2003) ‘A model for effective implementation of
earned value management methodology’, International Journal of Project Management,
Vol. 21, No. 5, pp.375–382.
Lipke, W. (1999) ‘Applying management reserve to software project management’, Journal of
Defense Software Engineering, pp.17–21.
Lipke, W. (2003) ‘Schedule is different’, The Measurable News, Vol. 31, No. 4, pp.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, Vol. 27, No. 4,
pp.400–407.
Moselhi, O., Li, J. and Alkass, S. (2004) ‘Web-based integrated project control system’,
Construction Management and Economics, Vol. 22, No. 1, pp.35–46.
Moslemi Naeni, L. and Salehipour, A. (2011) ‘Evaluating fuzzy earned value indices and estimates
by applying alpha cuts’, Expert Systems with Applications, Vol. 38, No. 7, pp.8193–8198.
Naeni, L.M., Shadrokh, S. and Salehipour, A. (2011) ‘A fuzzy approach for the earned value
management’, International Journal of Project Management, Vol. 29, No. 6, pp.764–772.
Noori, S., Bagherpour, M. and Zareei, A. (2008) ‘Applying fuzzy control chart in earned value
analysis : a new application’, World Applied Sciences Journal, Vol. 3, No. 4, pp.684–690.
Owen, J.K. (2007) ‘Implementing earned value management in a R&D environment’, 51st AACE
International Annual Meeting, Nashville, TN, USA, 15–18 July.
Parthiban, P., Punniyamoorthy, M., Ganesh, K. and Janardhana, G.R. (2009) ‘A hybrid model for
sourcing selection with order quantity allocation with multiple objectives under fuzzy
environment’, International Journal of Applied Decision Sciences, Vol. 2, No. 3, pp.275–298.
PMI (2008) A Guide to the Project Management Body of Knowledge, Project Management
Institute, Newtown Square.
Ponz-tienda, J.L., Pellicer, E. and Yepes, V. (2012) ‘Complete fuzzy scheduling and fuzzy earned
value management in construction projects’, Journal of Zhejiang University SCIENCE A,
Vol. 13, No. 1, pp.56–68.
Vandevoorde, S. and Vanhoucke, M. (2006) ‘A comparison of different project duration
forecasting methods using earned value metrics’, International Journal of Project
Management, Vol. 24, No. 4, pp.289–302.
Zadeh, L.A. (1965) ‘Fuzzy sets’, Information and Control, Vol. 8, No. 3, pp.338–353.
Zadeh, L.A. (2011) ‘A note on Z-numbers’, Information Sciences, Vol. 181, No. 14, pp.2923–2932.
Zeng, J., An, M. and Chan, A.H.C. (2005) ‘A fuzzy reasoning decision making approach based
multi-expert judgement for construction project risk analysis’, in Khosrowshahi, F. (Ed.): 21st
Annual ARCOM Conference, 7–9 September, SOAS, University of London. Association of
Researchers in Construction Management, pp.841–852.
Zeng, J., An, M. and Smith, N.J. (2007) ‘Application of a fuzzy based decision making
methodology to construction project risk assessment’, International Journal of Project
Management, Vol. 25, No. 6, pp.589–600.
A
novel earned value management model using Z-numbe
r
119
Zeng, J., An, M., Hin, A., Chan, C. and Lin, Y. (2004) ‘A methodology for assessing risks in the
construction process’, in Proceedings of the 20th International Conference on Construction
Management-ARCOM‘2004. UNSPECIFIED, pp.1165–1174.
Zorriassatine, F. and Bagherpour, M. (2009) ‘A new method for estimating project weight values’,
Journal of Applied Sciences, Vol. 9, No. 5, pp.917–923.
Zwikael, O., Globerson, S. and Raz, T. (2000) ‘Evaluation of models for forecasting the final cost
of a project’, Project Management Journal, Vol. 31, No. 1, pp.53–57.
Notes
1 The reader can refer to ‘practice standard for earned value management’ (PMI, 2008) to
achieve more details.
2 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.
3 W1 and W2 are modified weights and their values indicate the importance of each index in
comparison with the other one.
