INTRODUCTION
The importance of designing a well organized system in order to estimate weight
values during project management comes from the fact that progress is measured
based on the distribution of weight values and is obtained through multiplying
the weight value of each task in the project as shown below:
Where:

= 
1 
W_{i} 
= 
Weight value of each task 
P_{i} 
= 
Progress amount of each task 
P_{T} 
= 
Total progress 
N 
= 
No. of tasks involved in progress measurement 
Therefore, weight values must be initialized at the start of a project and
then later the total progress must be determined periodically based on the calculated
progress of each activity. Depending on the achieved total progress, a client
can then make payments to project contractor. Thus, both the progress measurement
system and payment are highly dependent on the weight values of each activity.
In this respect, Project Weight Values (PWVs) are usually calculated as a percentage
of cost. This means that cost percentages (or manhour percentage) of each task
will be set as PWVs. However, there are many factors which may have high impact
on PWVs. These include risk, quality and other related factors whose values
hat may be assumed by top level managers. This research is looking for such
methods in order to yield appropriate weight distributions versus time and consequently
to provide more accurate progress curves. This in turn can help clients to pay
according to the actual progress made by contractors. Otherwise, due to factors
such as bad distribution of PWVs versus time, contractors may receive payments
in excess of what they deserve for their actual progress. In such cases, a small
percentage of progress may be remaining for completion at the end the project
and as a result contractors may be less interested to finalize the remaining
work. As a result, the project duration may increase more than it is necessary.
Furthermore, the contractors may also release relevant personnel to other projects
in order to decrease their overhead costs.
It is common in project control to use graphical charts, such as Scurves,
to compare planned schedule with the actual progress of a project. The Scurve
can be obtained on the basis of weight value of each task, subphases and phases
of a project. Indeed, Scurve demonstrates the cumulative progress versus time.
If the progress of each element is estimated adequately, a relatively precise
curve will be obtained. Since, payments are made with respect to the progress
of each element, this approach can be used by both the clients and contractors
to determine qualitative as well as subjective factors such as risk, quality
etc in projects. Therefore, a robust approach to set PWV is highly desirable.
By applying fuzzy logic, linguistic terms that cannot be incorporated in a mathematical
model may then be included. Another advantage of using fuzzy logic is that it
allows appropriate distribution of progress within the project life cycle. Incorporating
fuzzy logic in the decision process will allow clients to avoid unnecessary
excess payments when no associated physical work exists.
Fuzzy inference systems capable of incorporating numerous rules must be deployed
due to the fact that estimating weight value as only a percentage of cost or
manhour is not sufficient to obtain proper PWVs. Two possible applications
are presented later on in the study. On the other hand, in fuzzy rule based
systems as an effective technique in Fuzzy Logic (FL) is a convenient way for
mapping an input space to an output space. It has progressed greatly since its
introduction by Zadeh (1965) and has been established
as an important branch of human thinking and knowledge representation. FL is
based atop human knowledge on possible solutions to existing problems. On the
application level, FL can be considered as an efficient tool for transforming
and embedding linguistically expressed knowledge into useful workable and operational
mathematical algorithms (Kecman, 2001). FL is also used
whenever a mathematical model is unknown or impossible to obtain, the process
is substantially nonlinear, or there is a lack of precise sensor information.
FL is aimed at handling imprecise and approximate concepts that cannot be processed
as conveniently by other known modeling techniques. It is applied at the higher
levels of hierarchical control systems, in generic decisionmaking processes
and for enabling computers to perform precise, meaningful and reasonable operations
on vague concepts (Kecman, 2001).
The degree of progress achieved in the mathematical theory of fuzzy sets can
be clearly demonstrated by many publications in specialized journals. However,
practical engineering applications tend to be based on approaches that were
developed in the earlier stages. For example, many of the most successful commercial
engineering applications of FL, i.e., fuzzy controllers, are still based on
the study of Mamdani and Assilian (1975), where symmetric
triangular membership functions are used and logical operations for union and
intersection are performed using maximum and minimum operations, respectively.
McNeill and Freiberger (1993) found that the absence of
a clear procedure for fuzzy modeling had been always emphasized by critics of
fuzzy. FL is a precious engineering tool developed to do a good job of trading
off precision and significance (Kecman, 2001). Fuzzy models
cannot achieve the level of accuracy required by many engineering applications.
Although low accuracy can be compromised for faster computation in control applications,
this may not be acceptable in the modeling of many other engineering systems.
The system theory. must be capable of providing efficient means for model building
as well as decision making and control of the system involved. In this sense,
fuzzy modeling of engineering systems is lagging behind fuzzy control.
Many issues need to be addressed to construct a new fuzzy model for a given
system, such as definition of membership functions, obtaining fuzzy rule base,
stating the optimum expressions for performing each particular inference, deciding
on the best defuzzification technique for a given problem, reducing computational
effort in operating with fuzzy sets and improving computational accuracy of
the model (Zimmermann, 2001). In particular, the assumption
that membership functions are given and the fact that excessively voluminous
computational effort is needed for any large complex problem, have proved to
be some of the major challenges for the application of fuzzy theory. Further,
neurofuzzy models can improve existing knowledge by learning from data on available
examples and by improving the input and output membership functions of the FL
model. According to Von Altrock (1995), selection of optimum
membership functions and deduction of rules from observed data can be implicitly
carried out using artificial neural networks, which can adaptively adjust membership
functions and finetune rules to achieve better performance. A typical fuzzy
application for solving a problem is made up of three major components, namely
fuzzifier, fuzzy inference engine (fuzzy rules) and defuzzifier (McNeill
and Freiberger, 1993). Effectiveness of the fuzzy approach depends on the
successful execution of these steps.
After introducing fuzzy rule based systems it is appropriate to provide an
overview of the general methodology of AHP. Readers interested in the mathematical
foundation of this technique along with some example applications are referred
to Saaty (1980, 1990), Saaty
and Vargas (1994) and Nydick and Hill (1992). AHP
is a multicriteria decisionmaking technique suitable for problems in which
several quantitative and qualitative criteria have to be taken into account
in order to make a proper decision. The problem environment in AHP is as follows.
There are several alternative solutions for a specific problem of which a suitable
one should be selected. There are several criteria for qualifying each alternative
solution. AHP is a systematic decisionmaking tool which determines the quality
of each solution with respect to all the criteria. The decisionmaker creates
a hierarchical decomposition representation of the problem in order to employ
AHP. At the top of the hierarchy is the overall goal or prime objective that
the decisionmaker attempts to attain as closely as possible. The possible solutions
to the problem take place at the bottom of this hierarchy. The between levels
represent the progressive decomposition of the overall goal. The decisionmaker
does a pairwise comparison of all elements in each level with respect to the
elements of the immediate higher level in the hierarchy. Combination of these
comparisons will show the relative quality of solutions in the lowest level
with respect to the topmost objective.
After carefully going through the literature it was found that although there
is some research available in the literature on either FL or AHP applications
for project management, few papers related to the application of combined AHP
and FL for project management were found. Specifically, based on the extensive
review of literature, application of this approach for the estimation of PWV
in project planning and control has not been attempted. According to Dweiri
and Kablan (2006), specific applications of FL in project management are
relatively few in comparison to other application areas. They considered project
cost, time and quality as project management’s internal measures of efficiency
and presented an approach that employs Fuzzy Decision Making (FDM) to synthesize
these three measures into one measure, namely the Project Management Internal
Efficiency (PMIE). The latter represents an overall estimate of how well a project
is managed and executed. Estimation of project cost, time and quality have also
been obtained through AHP. A Multiple Attribute Decision Making (MADM) tool
in the case of fuzzy preference information has been developed by
Wang and Parkan (2005). They introduced three optimization models to obtain
relative importance weight of attributes in a MADM problem. Wang
(2005) presented a MADM model to tackle incompleteness in the preference
information.
In terms of applying fuzzy logic for project management, Noori
et al. (2008) applied a fuzzy control chart through earned value
project management. They also provided an illustrative case and presented the
related results accordingly.
After a careful examination of the literature as outlined above it appears that there is no closely related research work in this area to allow performing a comparative analysis. However, this research and the technique applied here open up a new direction for further investigation.
THE PROPOSED APPROACH
In the proposed model, manhour and work preference are considered as two major criteria for estimation of PWV. Manhour is treated as a deterministic factor through which the planner can estimate the time and cost of the project. Work preference is considered as another criterion that is captured using a fuzzy rule based mechanism. In the proposed model, the required quality and risk are used to estimate work preference value. If each activity is considered as an alternative in AHP analysis, the weight calculation process will be mathematically intractable and time consuming as any real project may contain a large number of activities, e.g., 2000. Therefore, in this study, every major phase of the project is considered as one alternative for AHP. It must be noted that the considered phases in the AHP model should be the same as in the WBS (Work Breakdown Structure) of the project. This makes it possible to calculate project progress using the weights obtained from the AHP based model.
After obtaining the weight of each major phase, the AHP process is continued
and the weights associated with subphases, activities and so on are calculated.
The summation of weights of all alternatives in a particular level in AHP will
be equal to one. Figure 1 shows the main structure of the
proposed approach.
After setting the required quality and degree of risk, work preference is determined through a fuzzy rule based system and its inferences. Therefore, the work preference is obtained as a linguistic term.

Fig. 1: 
The main structure of the proposed approach 
ILLUSTRATIVE EXAMPLE
The project that is used as an example is assumed to consist of many activities that are to be carried out within a hierarchical process. The client is seeking to confirm a set of logical weight values for payments to contractors based on both quantitative and qualitative variables. Although, the approach would be more applicable in construction type projects, the proposed approach can be conducted in engineering and consultant projects as well. In order to avoid going though many calculations and to provide a better insight the main phases of the project were also taken into account. Moreover, the items in lower levels of WBS may also be used in order to calculate their weight values.
The proposed approach for application of fuzzy rule base for calculation of
work preference is composed of three steps. In the first step, a fuzzy membership
function is chosen for the required factors: quality and degree of risk. Next,
a fuzzy inference system fuzzifies the work preference with respect to these
factors. Finally, the work preference value is defuzzified using the selected
inference system and Eq. 13 as given
by Bagherpour et al. (2007).
Figure 2 shows application of symmetric Gaussian function
selected as the membership function to fuzzify the required quality. As seen
in this diagram, the required quality can be expressed as low, medium, or high.
The horizontal and vertical axis in Fig. 2 shows degrees of required quality and membership value, respectively.
Figure 3 shows triangular membership function used to fuzzify
degree of risk. Figure 4 shows symmetric Gaussian function
chosen as a membership function for work preference. Work preference values
are obtained on the basis of quality and risk membership functions shown in
Fig. 2 and 3.
In Fig. 3, horizontal and vertical axis indicate degrees of risk and membership value, respectively.
In the Fig. 4, the horizontal and vertical axis indicate degrees of work preference and membership values.

Fig. 2: 
Membership function for the required quality 

Fig. 3: 
Membership function for risk 

Fig. 4: 
Membership function for work preference 
In order to perform fuzzy inference system, many fuzzy rules can be established
in a real project according to the circumstances within which the project is
carried out. Without loss of generality, the following fuzzy rules are considered
for the illustrative example.
• 
Rule 1: If the required quality is high and degree of risk is medium
then work preference is high 
• 
Rule 2: If the required quality is low and degree of risk is low
then work preference is low 
• 
Rule 3: If the required quality is high and degree of risk is low
then work preference is medium 
As mentioned so far, two factors, namely the required quality and degree of
risk are considered in the proposed approach to estimate work preference. In
this example, these factors have been formulated as shown in Fig.
2 and 3. Table 1 shows work preference
values obtained based on these factors, which are calculated on the basis of
Eq. 13. In Table 1,
each row and column indicate the required quality and degree of risk, respectively.
Table 1: 
Work preference obtained based on the required quality and degree of risk 

VL: Very Low, L: Low, M: Medium, H: High, VH: Very High 
Table 2: 
Specification of phases 

Table 3: 
Work preference of each phase 

Table 4: 
The relative priority of work preference 

Table 5: 
Comparison of phases with respect to work preference 

The data in Table 1 belongs to the interval (0, 10) and indicates
work preference obtained through software coded with MATLAB^{TM} Toolbox.
In order to implement AHP for the problem under consideration in this study,
we consider one level including the main phases of project and assume the hierarchy
shown in Fig. 5 to be true. Table 2 and
4 show the input data. Table 37
show the results obtained through the proposed approach. Table
7 also shows the final results of the AHP as reported by Expert Choice^{®}
software.
As shown in Table 7, phase 3 has attained the highest weight
and phases 1, 2 and 4 are in the subsequent decreasing order, respectively.
Although, according to manhour data shown in Table 2, phase
1 has the maximum weight value, phase 3 has gained the maximum weight value
in total. This is because of the fact that phase 3 has a high work preference
value due to a high degree of risk and a medium level for the required quality.
Since, the consistency index is 0.00 in this problem, validation of the results
is strongly satisfied.
Table 6: 
Comparison of phases with respect to manhour 

Table 7: 
Obtained PWV of each phase 


Fig. 5: 
Sample hierarchy structure of the project 
As indicated, although phase 1 had the maximum amount of manhour, phase 3
obtained the maximum amount of weight value amongst all the phases. That is
why phase 3 had a high degree of risk and medium degree of quality. Therefore,
the proposed model improves on its predecessors for estimating PWV in which
the weights would be estimated as a percentage. Thus the proposed approach not
only considers the manhour factor (as a commonly used approach for estimating
the PWV), but it also takes into account other important factors influencing
the PWV. Of course, its results are highly dependent on project conditions and
can vary form one project to another.
FURTHER APPLICATIONS
The approach proposed in this study has other possible practical applications as outlined below.
Project close up: Project close up is the last phase of a project during
the implementation phase of project management process. Usually due to the low
number of required manhours in comparison with other phases such as the execution
phase, the estimated manhour is usually low. However, this is the most important
phase in which contractors wish to deliver the project to their clients. Therefore,
for example during the precommissioning stage, some problems related to the
previously completed phases of the project may be encountered. The client then
tries to resolve these issues. On the other hand, contractors may unduly receive
more that 95% of the expected total payment according to the approved project
progress which is based on the weight value system. In this scenario, due to
the relatively low amount of the remaining progress and consequently inadequate
incentive associated with the remaining payment, contractors may not be not
interested enough to continue and finalize the project that has been agreed.
One logical result is the delaying of project finish date. However, by implementing
the proposed approach, a client will be able to increase the weight value of
the close up phase of the project by incorporating some linguistic and verbal
terms in the contract in addition to the usual quantitative terms. By involving
both quantitative and qualitative terms the client can ensure that the project
will be implemented safely and its progress gets distributed correctly.
Capital budgeting applications: Project cost estimation approaches may be designed from two different points of view. The first is a topdown hierarchical process in which usually mangers need to know how much money must be assigned in each phase of the project. In the second approach, usually cost estimation is done based on a bottomup hierarchical process. However, in large scale projects, due to unanticipated problems and a lack of completion of numerous items and components, costs cannot be always estimated using bottomup approaches. For instance, the material transportation costs depend on the raw material quantity required in the project which can only be calculated later on in the project when the corresponding drawings are prepared. In that case, a weighted linear programming method as outlined below must be used instead:
I 
= 
No. of phases available in the project 
W_{i} 
= 
Obtained weight value based on proposed approach 
X_{i} 
= 
Amount of budget for each phase 
L_{i} 
= 
Lower bound of assigned budget for each phase 
U_{i} 
= 
Upper bound of assigned budget for each phase 
B 
= 
Prespecified budget for the project 
The objective function ensures that the assigned budget for each phase depends
on the level of importance of the relevant phase. Thus, if one phase is more
important, higher amounts of budget will be assigned to that phase resulting
from its importance. Therefore, if in one phase of a project either too many
qualitative risks are involved or contingency activities need to be adopted,
the proposed linear programming approach can deal efficiently with such cases
and treat them mathematically. The emphasis of the first set of constraints
(bounded ones) on the budget of each phase must be estimated within an interval.
The lower bound and upper bound of such constraints are assumed to be estimated
roughly (by applying a bottomup approach) or can be determined by top level
managers (by applying a topdown approach). The last constraint ensures that
the total estimated budget for all phases of a project must be less than B,
the preassigned budget, for implementing the project. The prebudget B is obtained
from the total price of a project signed in the contract less the sum of the
marginal profits and overhead costs.
To the bests of our knowledge no related research found in which helps us to run comparative analysis accordingly. However, the study can generate open field in project management systems for the other researchers as well.
CONCLUSION
In this study, an AHP based approach for estimating the weight value of each mainphase, phase and activity in a project was presented. Two factors, i.e., manhour and work preference, were considered as the major criteria which determine the weight of each element. Without loss of generality, two factors, namely the required quality and degree of risk were considered for estimating the work preference value using a fuzzy rule based system. However, the approach proposed in this paper can be easily used to consider other factors (such as resource requirements and work importance) affecting work preference. The hierarchy of the proposed AHP is the same as the WBS of the project. The study used a numerical example for demonstrating the application of the proposed approach. If desired, the model discussed here can be further extended to capture other aspects of interest in the estimation of PWV such as: the required time, budget, etc. A reliable perspective for actual progress of a project was thus provided. It was shown that the proposed approach can be applied to other aspects of a project such as project close up and capital budgeting applications. The model can be further investigated by considering the probabilistic nature of the events that affect the actual progress of a project. The approach proposed by this study can be used an as initial step in order to implement an earned value management system.