Fuzzy Analytical Hierarchy Process Method to Determine the Project Performance in the portfolio

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 09 Issue: 10 | Oct 2022 www.irjet.net p-ISSN: 2395-0072
Fuzzy Analytical Hierarchy Process Method to Determine the Project
Performance in the portfolio.
Mr. Rohit Bhausaheb Sonkamble1, Dr. D. N. Raut2
1Student, Dept. of Production Engineering (Project Mgmt.), VJTI college of Engineering, Mumbai, India.
2Professor, Dept. of Production Engineering, VJTI college of Engineering, Mumbai, India.
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The diversity of the projects in the industry
cannot be mapped as it has a too wide scope. Also, the
human mind is vulnerable to errors while quantifying
and analyzing various factors and variables and having
sure decision-making and a trustworthy tool is essential.
studies and Scholars suggest the Analytical Hierarchy
Process (AHP) as the best method for deciding among
the complex criteria structures at different levels. The
fuzzy AHP is the nothing but an extensionoftheClassical
AHP. The study used in this paper is Fuzzy Analytical
Hierarchy process which isone of the most usedmethods
by the decision makers in various sectors. Also, Fuzzy
AHP is preferred while dealing with different decision-
making issues. This paper utilizes the F-AHP to
prioritizes the projects in the portfolio.
1.INTRODUCTION
Two researchersKer-WeiYuandChien-ChangChou
[1] had proposed ahybridfuzzyAHPmethodthathelps
to deal with the decisions based on uncertain and
multiple criteria environments, in which F-AHP
method techniques combines with the AHP with fuzzy
set theory. [2] The method proposed by Ker-Wei Yu
and Chien-Chang Chou [1] can totally concentrate on
evolution criteria along with thinking logic by the
human being. In the paper [3] results show different
categories such as safety, health, environmental,
aspects as well asphysiochemical properties showsthe
best balance of performance in every aspect.
A few assumptions we can understand made in F-
AHP are all the criteria and factors involved are
independent of each other. However, in the real world,
the relationship between criteria is complex to
understand, and there are huge chances that they can
be interdependent. To analyze the projects with no
error we need relevant elements & methods [4].
The MCDM methods have various options while
using the Fuzzy model. [5] The f-AHP model can be
helpful for most decision-makers [6]. the fuzzy AHP
method is used to control for analysis and is useful for
various multicriteria decision-making problems [7]
people think the cost of the projects does matter but
other criteria like quality of the product as a
comparison can make the quality products better than
low-cost products. [8] F-AHP method canhelp decision
makers more efficient, flexible, and realistic decisions
based on various criteria and alternatives [9].
Therefore, we can use the F-AHPmethod to analyzethe
project performance in the portfolio.
© 2022, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page 1098
Abstract –
When reaching a decision most preferable way is to go
through the analytical way instead of taking decisions
without considering all thefactors. Analyzing theproject
performance in the portfolio requires a special ability,
full knowledge and as well as years of experience
working in the industry. However, many times
measuring a verbal expressioncanbetakeninto account
for better decision making and it can offer precise
interpretation in Infront of an observer. therefore, while
making the decisions, need of a proper analyze of every
factor’s impact is essential.
numerous studies have used the F-AHP method and
proven that F-AHP can be used as a decision-making
tool to take hard decisions in real life
There are always various confusions with decision
making if there are various options available, and
studies shows that while making a decision with pre-
determined criteria is always difficult. Decision-making
issues are can be faced while analyzing the project
performance in the portfolio. The diversity of projects
and their scope with configuration, Time, Demand and
other so many factors serve as constraints when datais
incomplete and there is a lack of individual knowledge
about analyzing the performance of the project. To
have a valid performance analysis of the projects every
factor mentioned in the paper shall be considered to
perform an error-free analysis. Previous papers and
1.1 Data of the Projects
Table 1: Data here referred for the study have taken
from the portfolio of an industry which shall not be
disclosed that can cause privacy issues. Authors have
taken appropriate steps to preserve the confidentiality
of studyparticipants and the datacollected from them.

Ranking and selection of
decisions
Checking for consistency
Calculate the weight
value of the fuzzy vector
Create a
comparison matrix
Set up Triangular Fuzzy number
(TFN)
Define the Problem
Portfolio Time-line Cost Est. Revenue Resources Used
Project 1 15 Months 180 K Euros 24 K Euros 57 Persons
Project 2 12 months 150 K Euros 14 K Euros 48 Persons
Project 3 12 months 140 K Euros 21 K Euros 52 Persons
Project 4 14 months 160 K Euros 17 K Euros 40 persons
Table -1: Project Data used for the calculations (Values are Per Prototype)
Figure 1: Block diagram has six steps of F-AHP phase
Data used in the research is to be kept confidential. In
order to fully understand the implications of the study,
it is necessary to keep the data confidential. This will
help protect the participants and the research team.
This data is gathered through surveys and interviews,
and any information that could identify the
participants is carefully guarded. The sources used in
this paper were all anonymous, meaning the
researcher will not be able to identifythem. The author
feels that this keeps the data more objective and
unbiased.
Table -1: Please refer to dataset mentioned in table
format.
1.3 Materials and Methods
Analytical Hierarchy process (AHP) is a method which
breakdown the problem along with the solutions.
Mainly it was developed to support the decision
making, it does breakdowntheproblem, thesolutionof
a problem and groups them to convert it into a
hierarchical structure that gives us priority criteria.
This method uses already determined measurements
aligned with criteria to give output. The main input for
the method is given by the stakeholders or experts of
the subject therefore, there’s also a factor of
reclamation of a decision inconsistency limits along
with data validity also get considered.[10]
In this method data validity with inconsistency, and
limits are also get considered. [10] While performing
the analysis the consideration of uncertainty and the
margin of error also the doubt in giving the assessment
definitely cancause affectresults. we need to definethe
problem accordingto thecriteriaused todeterminethe
project performance. Time, Cost, features, and demand
are mainly used as determining criteria for
determining project performance.
Step 1: In order to review the project performance in
the portfolio according to the criteria used to
determine is with following factors are Timeline, Cost,
revenue & resources. Thus, due to privacy concern the
data source cannot be disclosed.
1.2 Privacy Policy
Step 2: To Create comparison matrix we shall consider
the data source available in table number1.Thematrix
referred is simple and has strong position for
consistency framework. The equation used for
determining pairwise comparison is as follows.

Equation used to determine Pairwise comparison:
…. Eq-1
calculate the eigenvector, we use the following
equation:
Where,
n = Number of criteria compared.
wi= Weight of an I criterion.
Aij =is ratio of i criterion & j criterion.
After knowing the comparison of its criteria in Table 2,
the next thing done is to normalize each column into
the Advances in Fuzzy Systems.
Matrix form by dividing each value in the column i
and row j with the largest value in column i.
…. Eq-2
…Eq-4
The largest eigenvalue is the number of times
multiplying the number of columns with the main
eigenvector (see Table 4). So, it can be obtained by the
equation.
The largest eigenvalue is the number of times
multiplying the columns with main eigen vector.
So, Equation used is as follows:
The Normalized matrix is as follows from Table 2
…. Eq-3
Step 3: The consistency index and random index
generator comparison is as follows by Saaty [10] This
value depends on the matrix order n.
Here is the equation used to calculate the value of
consistency.
First, we must recognize the value of the eigenvector
which is the weighted value of the criterion. To
Table 4: Eigenvector on criteria.
…. Eq-4
CRITERIA TIME COST REVENUE RESOURCES
TIME 1 2 4 4
COST 1/2 1 2 4
REVENUE 1/4 1/2 1 2
RESOURCES 1/4 1/4 1/2 1
Eigenvectors
TIME 0.5
COST 0.3
REVENUE 0.1
RESOURCES 0.1
Table 2: Comparison of criteria, as the weighted value of each criterion.

Table 5: Weighted sum value calculated is as follows.
The equation through which we can calculate the
consistency index also with the help of “n” -Number of
criterions.
…. Eq- 5
So, we get, λmax = 4.0 and calculating CI (Consistency
Index)
CI = (4-4) / (4-1) = 0
Hence, the matrix is consistent and the weight values
we are considering are correct.
W1 = 0.5
W2 = 0.3
W3 = 0.1
W4 = 0.1
Step 4: set up Triangular Fuzzy Number
There are three values in F-AHP scale and those are
lowest value (lower, L), middle value (median, M), and
highest value (upper,U). Please referthetableNumber
Refer ( Table Number
Step 5: In order to calculate Fuzzy synthesis value,
need to transform AHP Comparison values to F-AHP
scale value. This can be achieved by using following
formula.
�� = fuzzy synthesis value
= summing the cell value in that
column starting from column 1 in each row matrix
i = row
j = column.
Step 6: Defuzzification ordinate value.
Compare fuzzy synthesis values to get defuzzification
ordinate value.
To Calculate: V’ we can use following formula.
…. Eq – 7
To Calculate fuzzy vector weight (W’). we can use
following formula to calculate fuzzy weight value.
…. Eq – 8
Previously obtained ordinate values as below.
…. Eq – 9
Using the following formula, we can normalize the
weight vectors obtained.
…. Eq – 9
….. Eq -6

Figure 3: Graph of Fuzzy Triangle Set give above.
The calculation of each criteria weight value is given
below.
Table 9: Criteria Weight Value.
Criteria Weight
TIME 0.4826
COST 0.2867
REVENUE 0.1434
RESOURCES 0.0873
L M U
s1 0.19 0.34 7.50
s2 0.15 0.29 7.00
s3 0.11 0.20 5.17
s4 0.10 0.17 4.33
Table 6: Advances in Fuzzy Systems (Triangular Fuzzy Scale)
Table 7: TFN set of criteria; each value in the criteria comparison (see Table 6) is changed to TFN referring
to the TFN scale.
Table 8: Calculations for Synthesis values after
calculations are as follows:

Step 7: Ranking the projects in portfolio
Collecting Ordinate Values Previously obtained
…. Eq- 10
Vector Weight Values Normalization
…. Eq- 11
1.4 Discussion
While performing the calculation the help from
excel tools as well as using online AHP Portals can
be taken for ease & accuracy of calculation.
and support decision making.
Considering all the factors the results are as
follows.
1.5Result
Here with the results, we can see their project
performance in the portfolio is intheorderofProject1,
Project 4, Project 3, and Project 2 with respective
weights of 0.2899, 0.2471, 0.2374 & 0.2249.
With the data we can clearly state that in the portfolio
the project 4 which is development of a prototype has
performed well till now.
Figure 4: Ranking graph
The above figure shows that the Project number 1 has
highest weight, and it is performing well in the
portfolio.
1.6 Conclusion
Conclusion of this study article is that F-AHP system
can be implemented toanalyses and findout accurately
in the project management field.
It can also assist projectManagerto take decisions with
appropriate analysis of project performance in the
portfolio using AHP & F-AHP.
Portfolio Time-line Cost Est. revenue resources used Weights
Project 1 15 Months 180 K Euros 24 K Euros 57 Persons 0.289934663
Project 4 14 months 160 K Euros 17 K Euros 40 persons 0.247118795
Project 3 12 months 140 K Euros 21 K Euros 52 Persons 0.237492501
Project 2 12 months 150 K Euros 14 K Euros 48 Persons 0.224998953

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