Fuzzy AHP and Fuzzy TOPSIS as an effective and powerful Multi-Criteria Decision-Making (MCDM) method for subjective judgements in selection process

e-ISSN: 2582-5208
International Research Journal of Modernization in Engineering Technology and Science
( Peer-Reviewed, Open Access, Fully Refereed International Journal )
Volume:05/Issue:04/April-2023 Impact Factor- 7.868 www.irjmets.com
www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science
[3786]
FUZZY AHP AND FUZZY TOPSIS AS AN EFFECTIVE AND POWERFUL MULTI-
CRITERIA DECISION-MAKING (MCDM) METHOD FOR SUBJECTIVE
JUDGEMENTS IN SELECTION PROCESS
Nitin Liladhar Rane*1, Saurabh P. Choudhary*2
*1,2Vivekanand Education Society's College Of Architecture (VESCOA), Mumbai, India.
DOI : https://www.doi.org/10.56726/IRJMETS36629
ABSTRACT
This research suggests a robust and effective selection process that involves subjective judgments by applying
two fuzzy-based multi-criteria decision-making methods, namely the Fuzzy Analytic Hierarchy Process (Fuzzy
AHP) and the Fuzzy Technique for Order Preference by Similarity to Ideal Solution (Fuzzy TOPSIS). These
methods incorporate fuzzy set theory into traditional AHP and TOPSIS methods to handle uncertain criteria
weights and evaluation scores. The Fuzzy AHP and Fuzzy TOPSIS techniques are particularly appropriate for
selection processes that involve subjective evaluations and uncertainty. These methods are well-equipped to
handle imprecise and uncertain information and can effectively deal with the complexity of multi-criteria
decision-making problems. One of the significant advantages of these methods is their capacity to address both
quantitative and qualitative criteria. By utilizing fuzzy set theory, these methods can integrate subjective
criteria and expert judgments that may not be expressed in numerical values. Additionally, the Fuzzy AHP and
Fuzzy TOPSIS approaches provide a methodical and structured approach to decision-making that guarantees
consistency and transparency. This article offers a comprehensive theoretical framework of the Fuzzy AHP and
Fuzzy TOPSIS methods and presents their application in selecting the best candidate for a job position. The
findings indicate that this approach is valuable in handling subjective judgments and produces consistent and
dependable outcomes. The article concludes by discussing the method's benefits and drawbacks and
highlighting areas for future research.
Keywords: Fuzzy AHP, Fuzzy TOPSIS, Multi-Criteria Decision-Making (MCDM), Subjective Judgement, Selection
Process, Subjective Element.
I. INTRODUCTION
The field of Multi-Criteria Decision-Making (MCDM) is a widely recognized aspect of decision theory that helps
solve intricate decision-making problems [1-4]. In today's globalized era, organizations are faced with
complicated and constantly changing decision-making challenges, where multiple criteria must be taken into
account, and the decision-makers must exercise subjective judgment [4]. Various factors such as the complexity
of the problem, time and budget constraints, availability of data, and subjectivity influence the decision-making
process. Thus, selecting an effective and robust MCDM method is crucial for solving such issues. Two well-
known MCDM methods that can handle subjective judgments and uncertainties are Fuzzy AHP (Analytic
Hierarchy Process) and Fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) [5-6].
Fuzzy AHP involves hierarchically structuring decision problems and evaluating alternatives based on the
weights of the criteria and sub-criteria. On the other hand, Fuzzy TOPSIS ranks alternatives based on their
similarity to the ideal solution and dissimilarity to the negative ideal solution.
Fuzzy sets theory is used in these methods to represent vague and imprecise information in decision-making,
proving to be a useful tool in handling subjective judgments. This research paper aims to investigate the
effectiveness and power of Fuzzy AHP and Fuzzy TOPSIS as MCDM methods for solving complex and dynamic
decision-making problems with subjective judgments. The focus is on the selection process, where decision-
makers must evaluate alternatives based on multiple criteria and sub-criteria. The paper will demonstrate the
application of Fuzzy AHP and Fuzzy TOPSIS in solving selection problems and compare the results with other
MCDM methods.
Decision-making in various industries heavily relies on subjectivity, which can significantly impact the outcome
[6-7]. In the healthcare industry, doctors often have to use subjective judgments to decide on the best treatment
course for patients with multiple health conditions based on their experience and judgment. Likewise, in the

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financial industry, investment decisions can be influenced by personal biases, such as risk tolerance, beliefs,
and past experiences, when choosing between different investment options. In the education industry,
admissions committees may evaluate candidates based on subjective criteria like extracurricular activities,
leadership qualities, and personal statements. Similarly, in the construction industry, project managers may
make subjective decisions on which materials to use based on factors like durability, aesthetics, and cost-
effectiveness.
It's worth noting that subjective elements in decision-making can vary within the same industry. For instance,
in the food industry, a chef's decision on the menu can depend on personal taste, cultural background, and the
target audience. To handle these subjective judgments effectively, decision-makers can use MCDM methods like
Fuzzy AHP and Fuzzy TOPSIS. In conclusion, subjectivity plays a crucial role in decision-making across different
industries, and MCDM methods can help manage these subjective elements. The paper's structure includes an
overview of related literature on MCDM methods and their applications, a detailed description of Fuzzy AHP
and Fuzzy TOPSIS methods, including their advantages and disadvantages, an explanation of the methodology
adopted for the research and the case study used for analysis, presentation of results obtained from applying
Fuzzy AHP and Fuzzy TOPSIS, comparison with other MCDM methods, and conclusion and recommendations
for future research.
This research paper aims to demonstrate the effectiveness and power of Fuzzy AHP and Fuzzy TOPSIS as
MCDM methods for solving complex and dynamic decision-making problems involving subjective judgments.
The use of fuzzy sets theory allows representation of vague and imprecise information in decision-making,
proving to be an effective tool in handling subjective judgments. The paper will provide valuable insights into
the application of Fuzzy AHP and Fuzzy TOPSIS in solving selection problems and comparing them with other
MCDM methods.
Table 1: subjective elements that can influence decision-making in each field/application
Sl. No.
Fuzzy MCDM
Method
Field/Application Subjective Elements
1 Fuzzy AHP
Healthcare industry, financial industry,
construction industry, environmental
impact assessment, supply chain
management, energy planning, project
management, transportation planning
Experience, judgment, personal biases,
risk tolerance, cultural background,
cost-effectiveness, aesthetics,
sustainability, social responsibility
2 Fuzzy TOPSIS
environmental impact assessment,
supply chain management, project
management, transportation planning,
facility location selection
risk tolerance, cultural background,
cost-effectiveness, aesthetics,
sustainability, social responsibility
3 Fuzzy VIKOR
Supply chain management,
environmental impact assessment,
project management, financial industry
Risk tolerance, cultural background,
cost-effectiveness, sustainability, social
responsibility
4
Fuzzy
ELECTRE
Project management, environmental
impact assessment
Judgment, personal biases, cultural
background, social responsibility
5
Fuzzy
PROMETHEE
Environmental impact assessment,
transportation planning
6 Fuzzy DEA
supply chain management
7 Fuzzy BWM
construction industry
cultural background, social
responsibility

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8
Fuzzy
MOORA
Financial industry, project management,
transportation planning
9 Fuzzy MCGP Financial industry
10 Fuzzy GRA Financial industry, project management
cultural background, social
responsibility
II. PRINCIPLE OF FUZZY AHP
Fuzzy AHP is a variation of the Analytic Hierarchy Process (AHP) that considers the uncertainty and
imprecision that often accompany decision-making [8-9]. It enables decision-makers to incorporate subjective
judgments and linguistic expressions, rather than relying solely on numerical data, when evaluating and
prioritizing alternatives based on a set of criteria. Fuzzy AHP acknowledges that decision-makers may not be
able to provide precise judgments about the relative importance of criteria or alternatives. Instead, it
represents their judgments as linguistic expressions or fuzzy sets, allowing them to express their opinions in a
more natural and intuitive way. The Fuzzy AHP process involves constructing a decision hierarchy and using
pairwise comparison to evaluate the relative importance of each criterion and alternative [10-13]. The weights
are represented as fuzzy sets and aggregated using fuzzy aggregation methods, resulting in a final ranking of
alternatives.
Fuzzy AHP's benefits include providing decision-makers with a flexible and intuitive way of expressing their
preferences, particularly in situations where numerical data is unavailable or insufficient. By incorporating
uncertainty and imprecision into the decision-making process, Fuzzy AHP can lead to more accurate and
relevant decision-making outcomes. Fuzzy AHP's principle is to incorporate uncertainty and imprecision into
decision-making processes, enabling decision-makers to express their opinions in a natural and intuitive way.
The process involves constructing a decision hierarchy, evaluating criteria and alternatives using pairwise
comparison, and aggregating weights using fuzzy aggregation methods [14]. This variation of AHP has the
potential to improve decision-making accuracy and relevance in situations where numerical data is not
available or insufficient. Decision-makers should be always with a flexible and intuitive way [15-20].
The process of decision-making known as the Analytic Hierarchy Process (AHP) involves breaking down
complex problems into a hierarchy of criteria and sub-criteria. Pairwise comparisons are then used to
determine the relative importance of each element in the hierarchy. The Fuzzy AHP extends this framework by
incorporating the use of fuzzy logic, which enables comparisons between elements that are not easily
quantifiable. Fuzzy AHP incorporates both subjective and objective elements in its decision-making process.
The subjective element involves the judgments made by decision-makers, which are often based on personal
preferences or opinions. The objective element involves the data and information available for the decision-
making process.
In Fuzzy AHP, subjective judgments are expressed as linguistic terms, such as "very important," "important,"
"moderately important," and "slightly important." These terms are translated into fuzzy numbers that
represent the degree of membership of each linguistic term in a fuzzy set [21-23]. Objective elements are
represented by crisp numerical data, which are used to calculate the weights of each criterion and sub-criterion
in the hierarchy. The weights are then aggregated to obtain an overall score for each alternative in the decision-
making process. By combining subjective judgments with objective data in a way that allows for more nuanced
and flexible decision-making, Fuzzy AHP enables decision-makers to express their preferences in a more
natural and intuitive way [23-24]. This approach incorporates the concept of fuzzy logic and provides decision-
makers with a reliable framework to guide their decisions.

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Table 2: Principle of Fuzzy AHP
Sl. No. Step Objective
1
Define the decision problem and
establish the hierarchy of criteria
and sub-criteria
Identify the decision problem and break it down into a
hierarchical structure of criteria and sub-criteria to be
considered.
2
Assign linguistic terms to the criteria
and sub-criteria, using fuzzy logic
Express the criteria and sub-criteria in linguistic terms
to capture imprecise and uncertain information.
3
Construct a pairwise comparison
matrix for each level of the
hierarchy, where the elements of the
matrix represent the relative
importance of one criterion
compared to another
Obtain the relative importance of the criteria and sub-
criteria through pairwise comparisons.
4
Convert the pairwise comparison
matrices into fuzzy numbers, using
the linguistic terms assigned in step
2
Convert the pairwise comparison matrices into fuzzy
numbers that represent the degree of membership of
each criterion and sub-criterion in their respective
linguistic terms.
5
Calculate the fuzzy weights of each
criterion and sub-criterion, using the
fuzzy pairwise comparison matrices
Determine the overall importance of each criterion and
sub-criterion by calculating their fuzzy weights.
6
Aggregate the fuzzy weights to
obtain an overall fuzzy score for each
alternative
Combine the fuzzy weights of the criteria and sub-
criteria to obtain an overall fuzzy score for each
alternative.
7
Rank the alternatives based on their
overall fuzzy scores
Determine the preferred alternative(s) for the decision
problem by ranking the alternatives in order of their
overall fuzzy scores.
III. ESTABLISHING THE FUZZY PAIRWISE COMPARISON MATRIX
The Fuzzy Analytic Hierarchy Process (AHP) is a decision-making technique that combines subjective
judgments and mathematical calculations to prioritize and rank a set of options [25-26]. An essential aspect of
the Fuzzy AHP approach is creating the Fuzzy Pairwise Comparison Matrix, which is used to compare each
alternative with each other based on a set of criteria that are important for the decision-making process. The
matrix comprises a set of values ranging from 0 to 1, indicating the relative significance of each alternative with
respect to each criterion.
To establish the Fuzzy Pairwise Comparison Matrix, the decision-maker initially identifies the relevant criteria
and then compares each alternative to every other alternative in terms of each criterion, using linguistic
variables such as "very strong," "strong," "moderate," "weak," and "very weak" to express the degree of
preference or importance. These linguistic variables are then converted into fuzzy numbers, representing a
range of possible values that reflect the decision-maker's degree of uncertainty or ambiguity in their
preferences. Once fuzzy numbers are assigned to each pairwise comparison, a set of mathematical formulas is
applied to calculate the weights of each alternative for each criterion, taking into account the degree of
preference and uncertainty reflected in the fuzzy numbers. Finally, the decision-maker aggregates these
weights across all criteria to obtain an overall ranking of the alternatives [26]. The ranking can be used to make
a final decision or inform further analysis and discussion [27-32]. Overall, establishing the Fuzzy Pairwise
Comparison Matrix is a crucial step in the Fuzzy AHP approach, enabling decision-makers to capture and
account for the complexity and ambiguity inherent in real-world decision-making processes.

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IV. SYNTHESISE THE JUDGEMENTS
The process of combining individual judgments or preferences of multiple decision makers into a single
comprehensive set of prioritized criteria or alternatives in Fuzzy Analytic Hierarchy Process (AHP) is known as
"Synthesizing the Judgments". This process involves several steps, including Fuzzy Pairwise Comparison, Fuzzy
Weight Calculation, Fuzzy Consistency Analysis, and Fuzzy Priority Calculation. In the Fuzzy Pairwise
Comparison step, decision makers compare each criterion or alternative to all others and assign a linguistic
variable or degree of preference, which is then converted into fuzzy numbers to represent the degree of
uncertainty or imprecision in their judgments. The Fuzzy Weight Calculation step uses these fuzzy pairwise
comparison matrices to calculate the fuzzy weights of the criteria or alternatives, representing their relative
importance. Fuzzy Consistency Analysis evaluates the consistency of the fuzzy pairwise comparison matrices to
ensure that decision makers' judgments are coherent and logical [33-36]. If inconsistencies are found,
adjustments are made to improve the consistency. Finally, the Fuzzy Priority Calculation step uses the fuzzy
weights to calculate the fuzzy priorities of the criteria or alternatives, providing the overall ranking based on
the aggregated judgments of all decision makers. Synthesizing the judgments in Fuzzy AHP is a rigorous process
that accounts for uncertainty and imprecision while gathering, analyzing, and aggregating the preferences of
multiple decision makers. The resulting fuzzy priorities can be used in various fields such as business,
engineering, and environmental management to aid decision-making processes.
V. CALCULATING THE FUZZY WEIGHTS OF THE FACTOR
To determine the relative importance of each criterion in a decision-making problem using Fuzzy Analytic
Hierarchy Process (AHP), the fuzzy weights of criteria need to be calculated. This involves the following steps:
1) Define the decision problem and criteria in a specific and measurable way, ensuring that the criteria are
relevant and meaningful.
2) Construct a pairwise comparison matrix using a consistent scale for the decision maker's judgments to
ensure consistency.
3) Convert the pairwise comparison matrix into fuzzy numbers using an appropriate membership function that
captures the degree of uncertainty and imprecision in the judgments.
4) Calculate the fuzzy weights of both the criteria and the alternatives by aggregating the fuzzy pairwise
comparison matrices using the fuzzy AHP method and multiplying the fuzzy weights of criteria by the
performance scores of each alternative.
5) Check the consistency of the results by calculating the consistency ratio (CR) to compare the degree of
inconsistency in the pairwise comparison matrix to a random matrix of the same size. A CR of less than 0.1
indicates that the judgments are consistent, while a CR greater than 0.1 may require adjustments to the
pairwise comparisons.
By following these steps, decision makers can ensure that their judgments are consistent and informed, leading
to more effective decision making.
VI. DEFUZZIFY THE FUZZY WEIGHTS
Defuzzifying fuzzy weights in Fuzzy Analytic Hierarchy Process (AHP) refers to the process of obtaining a crisp
value from the fuzzy weights obtained through pairwise comparison of criteria or alternatives. To defuzzify the
fuzzy weights, we need to determine the degree of membership of each weight in its corresponding linguistic
term (e.g., very high, high, medium, low, very low). This is typically done using a membership function, which
maps each fuzzy weight to a linguistic term based on its degree of membership in that term. Once we have
determined the degree of membership of each fuzzy weight, we can calculate a weighted average of the fuzzy
weights, where the weights are weighted by their degree of membership. This weighted average gives us a crisp
value that represents the overall weight of the criteria or alternatives.
For example, suppose we have three criteria: A, B, and C, and we have obtained the following fuzzy weights for
these criteria:
 A is very high (0.9), B is high (0.7), and C is medium (0.5)

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To defuzzify these fuzzy weights, we first determine the degree of membership of each weight in its
corresponding linguistic term. We can use a triangular membership function with appropriate parameters to
determine the degree of membership of each weight in its corresponding linguistic term.
Using these membership functions, we can determine the degree of membership of each weight in its
corresponding linguistic term:
 A: very high = 0.9
 B: high = 0.7
 C: medium = 0.5
We can then calculate the weighted average of these fuzzy weights, weighted by their degree of membership.
This gives us a crisp value of 0.78, which represents the overall weight of the criteria A, B, and C.
VII. CHECK THE CONSISTENCY: FUZZY CONSISTENCY
The Fuzzy Analytic Hierarchy Process (FAHP) is an expanded version of the Analytic Hierarchy Process (AHP)
that allows for the use of fuzzy sets and linguistic terms in decision-making. AHP is a popular decision-making
tool that was created in the 1970s by Thomas Saaty. It is used to simplify complicated decisions by breaking
them down into a hierarchy of alternatives and criteria and then evaluating them based on their relative
importance. FAHP introduces a level of uncertainty into the decision-making process by allowing decision-
makers to express their opinions using linguistic terms rather than precise numerical values. This is especially
useful when decision-makers are faced with complicated or unclear problems where obtaining precise
numerical values is difficult or impossible. By using fuzzy sets and fuzzy logic, FAHP permits decision-makers to
express their opinions in a more natural and intuitive way.
However, the use of fuzzy sets and fuzzy logic creates new challenges in the AHP method, specifically when it
comes to assessing consistency. As the use of linguistic terms and fuzzy sets can introduce a greater degree of
subjectivity and variability in the decision-making process, assessing consistency becomes more difficult.
Consistency is a vital element in AHP, as it ensures that the weights assigned to criteria and alternatives are
internally consistent and do not conflict with each other. To deal with this issue, several methods have been
developed for assessing consistency in FAHP models. This section discusses some of the most commonly used
methods for evaluating consistency in FAHP models. The Eigenvalue method is one of the most widely used
approaches for assessing consistency in AHP models, including FAHP. This method involves determining the
largest Eigenvalue (λ) of the pairwise comparison matrix and then using this value to calculate the Consistency
Index (CI) and Consistency Ratio (CR).
The RI is a value determined based on the number of criteria or alternatives being compared and serves as a
benchmark for assessing the consistency of the pairwise comparison matrix. If the CR is less than or equal to
0.1, then the pairwise comparison matrix is considered to be consistent. If the CR is greater than 0.1, then the
pairwise comparison matrix is considered to be inconsistent, and the decision-maker may need to revisit their
pairwise comparisons to ensure that they are internally consistent.
The Entropy method is a relatively new method for assessing consistency in FAHP models. This method
involves using the concept of entropy to measure the degree of uncertainty in the pairwise comparison matrix.
The Entropy method is utilized within the Fuzzy Analytical Hierarchy Process (FAHP) to establish the weights
of criteria when dealing with decision-making problems. The FAHP is an extension of the AHP and it facilitates
the application of fuzzy sets and linguistic terms in decision-making. This approach determines the degree of
importance of each criterion based on its capacity to differentiate between options. Criteria that possess higher
discriminatory power are considered more significant in the decision-making process. The Entropy method
involves the computation of the entropy for each criterion, which measures the uncertainty or randomness
associated with the criterion's data. The entropy value obtained for each criterion is used to calculate its weight
through a normalization procedure. The normalization procedure divides the entropy of each criterion by the
sum of all criteria entropies. This produces the weights of each criterion, which are then employed in the
decision-making process. One of the strengths of the Entropy method is its ability to integrate the subjective
opinions of decision-makers with the objective data associated with each criterion. Moreover, the method can

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cope with imprecise and uncertain data, making it ideal for complex decision-making problems where
traditional methods may not be suitable.
VIII. FUZZY SET THEORY
AHP and Fuzzy Set Theory are two decision-making methodologies used to solve different types of problems.
AHP is used for multi-criteria decision-making, while Fuzzy Set Theory is used for problems involving
uncertainty and imprecision. AHP Fuzzy Set Theory is a combination of these two methodologies that is used
when a decision-making problem involves multiple criteria that are subjective and uncertain. In this approach,
AHP is used to structure the problem by breaking it down into a hierarchy of criteria and sub-criteria. Then,
Fuzzy Set Theory is used to model the subjective and uncertain aspects of the problem.
To do this, the weights of the criteria and sub-criteria are represented by membership functions that define the
degree of membership of each element in a set. These membership functions can be defined based on linguistic
terms provided by decision-makers, such as "very important" or "not important". The weights are then
determined through pairwise comparisons using the membership functions. The AHP Fuzzy Set Theory process
involves defining criteria and sub-criteria, eliciting linguistic terms, defining membership functions for each
term, conducting pairwise comparisons using the membership functions, and calculating priority vectors for
each alternative. This approach can provide more accurate and robust results, especially when dealing with
subjective and uncertain decision-making problems. By using linguistic terms and membership functions,
decision-makers can provide more nuanced and precise evaluations of the criteria and sub-criteria, leading to a
more accurate ranking of the alternatives.
Fuzzy set theory is a mathematical framework extends classical set theory to handle uncertain, ambiguous, or
vague concepts. Unlike classical set theory, which defines a set as a collection of well-defined elements, a fuzzy
set allows for elements to have partial membership or degrees of membership, ranging from completely
belonging to completely not belonging. A fuzzy set is defined by a membership function that assigns a degree of
membership to each element of the universe of discourse. The universe of discourse is the set of all possible
values that a variable can take. For instance, the universe of discourse for the variable "age" could be all non-
negative real numbers. The membership function for a fuzzy set maps each element of the universe of discourse
to its degree of membership in the set. Fuzzy set theory finds applications in various fields, including
engineering, artificial intelligence, decision-making, control systems, and pattern recognition. Fuzzy set theory
can be used to design fuzzy controllers that can handle imprecise or uncertain inputs and provide more robust
and flexible control over a system. In decision-making, fuzzy set theory can model human preferences and
subjective evaluations, allowing for a more nuanced and realistic representation of decision problems. Fuzzy
set theory is a powerful tool for dealing with uncertainty, ambiguity, and vagueness in various applications. Its
ability to capture the complexities of human reasoning and decision-making makes it valuable for both
theoretical research and practical applications.
IX. THE FUZZY TOPSIS METHOD
The Fuzzy TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) is a well-known method
for multi-criteria decision-making. The steps involved in the Fuzzy TOPSIS method are as follows:
1) Problem Definition: The first step in the Fuzzy TOPSIS method is to define the problem. This involves
identifying the decision-making criteria, the alternatives, and the decision maker's preferences. The criteria
should be relevant to the problem and measurable. The alternatives should also be feasible and relevant. The
decision maker's preferences can be expressed as weights assigned to the criteria.
2) Fuzzy Linguistic Variables: The second step is to represent the criteria and alternatives in terms of fuzzy
linguistic variables. This involves defining fuzzy sets for each criterion and alternative. A fuzzy set is a
mathematical representation of a vague concept, such as "high," "medium," and "low." For example, the
criterion of "cost" can be represented as a fuzzy set with terms like "very low," "low," "moderate," "high," and
"very high." Similarly, each alternative can be represented as a fuzzy set with terms like "very good," "good,"
"average," "poor," and "very poor."
3) Fuzzy Decision Matrix: The third step is to construct the fuzzy decision matrix. This involves calculating the
degree of membership of each alternative to each criterion. The degree of membership is a measure of how well

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an alternative satisfies a criterion. This can be represented as a matrix, where each row represents an
alternative, and each column represents a criterion. The degree of membership for each alternative and
criterion is expressed as a value between 0 and 1.
4) Normalization: The next step is to normalize the fuzzy decision matrix. This involves dividing each value in
the matrix by the sum of values in the same column. This step is necessary to make sure that each criterion is
given equal importance in the decision-making process.
5) Weighted Normalized Decision Matrix: The fifth step is to construct the weighted normalized decision
matrix. This involves multiplying each value in the normalized decision matrix by its corresponding weight. The
weights represent the relative importance of each criterion in the decision-making process.
6) Ideal and Negative Ideal Solutions: The next step is to determine the ideal and negative ideal solutions. The
ideal solution is the alternative that has the highest degree of membership for each criterion, while the negative
ideal solution is the alternative that has the lowest degree of membership for each criterion. These solutions
represent the best and worst possible outcomes, respectively.
7) Distance Calculation: The seventh step is to calculate the distance of each alternative from the ideal and
negative ideal solutions. This is done by calculating the Euclidean distance between each alternative and the
ideal and negative ideal solutions. The Euclidean distance is a measure of the similarity between two points in a
multi-dimensional space.
8) Relative Closeness: The eighth and final step is to calculate the relative closeness of each alternative. This is
done by dividing the distance of each alternative from the negative ideal solution by the sum of the distances of
all alternatives from the ideal and negative ideal solutions. The relative closeness of each alternative represents
its rank in the decision-making process. The alternative with the highest relative closeness is considered the
best alternative.
The Fuzzy TOPSIS method is a powerful tool for decision-making that can be applied to a wide range of
situations. By following the above steps, decision makers can identify the best alternative based on a set of
criteria while taking into account the decision maker's preferences.
X. PRINCIPLE OF FUZZY TOPSIS
Fuzzy TOPSIS is a method of multi-criteria decision-making that utilizes fuzzy set theory to manage imprecise
and uncertain information. The method involves ranking alternatives based on their proximity to an ideal
solution and their distance from a negative ideal solution. The process consists of several steps, including
determining criteria weights, calculating the normalized decision matrix, and determining the closeness
coefficient. To begin, the decision maker assigns a weight to each criterion based on its significance in the
decision-making process. These weights can be determined using various methods, including the Analytic
Hierarchy Process, the Entropy method, or the Fuzzy AHP. Alternatively, subjective preferences or expert
opinions of the decision maker can be used to determine the criteria weights. Next, the decision maker
evaluates each alternative with respect to each criterion and assigns a membership degree to each alternative-
criterion pair. The membership degree represents the level of satisfaction of the decision maker with the
alternative regarding the criterion. The membership degrees are then normalized to remove the effect of scale
differences and to ensure that each criterion has an equal weight in the decision-making process. In the third
step, the decision maker determines the closeness coefficient by calculating the distance between each
alternative and the ideal solution and negative ideal solution. The ideal solution represents the best possible
values for all criteria, while the negative ideal solution represents the worst possible values for all criteria. The
distance is calculated using a suitable distance metric, such as the Euclidean distance or the Chebyshev
distance. The closeness coefficient is then determined as the ratio of the distance between the alternative and
the negative ideal solution to the sum of the distances between the alternative and the ideal solution and
negative ideal solution.
Fuzzy TOPSIS, or the Technique for Order of Preference by Similarity to Ideal Solution, is a multi-criteria
decision-making method that takes into account both subjective and objective factors in decision-making [37-
39]. The subjective criteria are those that are difficult to quantify or measure, such as personal values or
preferences, while the objective criteria are quantifiable, such as cost or quality. Fuzzy TOPSIS recognizes that
decision-making involves some degree of uncertainty and imprecision, and aims to address this by using fuzzy

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sets to represent the degree of membership of each alternative to the ideal solution. To use Fuzzy TOPSIS, each
criterion is assigned a weight that reflects its relative importance in the decision-making process. The
subjective elements are evaluated using linguistic variables represented by fuzzy sets, allowing decision-
makers to express their preferences more naturally. Decision making using preference analysis is a very crucial
factor [40-43]. Meanwhile, the objective elements are evaluated using numerical data that is normalized and
transformed into fuzzy numbers to account for differences in scale or units of measurement and reflect
uncertainty or imprecision. After evaluating both subjective and objective criteria, Fuzzy TOPSIS generates a
ranking of the alternatives based on their similarity to the ideal solution, which is determined by calculating the
weighted sum of the best values for each criterion and the worst values for each criterion [44-48]. Fuzzy
TOPSIS provides a more nuanced view of the decision problem, enabling decision-makers to make more
informed and effective decisions by incorporating both subjective and objective elements using fuzzy sets.
XI. NUMBER OF JUDGMENTS BY DECISION MAKERS
The accuracy and reliability of decision-making in Fuzzy Analytic Hierarchy Process (AHP) and Technique for
Order Preference by Similarity to Ideal Solution (TOPSIS) heavily depend on the number of judgments made by
decision-makers. For Fuzzy AHP, decision-makers compare and assign linguistic variables or fuzzy numbers to
criteria and alternatives. The number of required judgments in Fuzzy AHP is determined by the number of
criteria and alternatives being assessed and the level of analysis detail. Generally, a higher number of
judgments increases decision accuracy and reliability. TOPSIS involves decision-makers assigning weights to
criteria based on their relative importance. The number of required judgments in TOPSIS depends on the
number of criteria assessed, but is typically lower than in Fuzzy AHP as the focus is on weighting criteria rather
than evaluating alternatives directly. In both methods, decision-makers' judgments' quality and accuracy are
crucial to overall effectiveness. Decision-makers must be well-informed and understand the criteria and
alternatives. The use of appropriate software tools can reduce cognitive load and improve judgment accuracy.
The quantity of decisions reached by decision-makers refers to the amount of choices or rulings made by
individuals or groups responsible for determining a particular matter. Such decision-makers may include
judges, managers, or politicians. Measuring the number of judgments made by decision-makers is an essential
metric for evaluating the efficiency and effectiveness of the decision-making process. This measure can help to
shed light on the decision-makers' workload, their ability to make prompt and informed decisions, and the
complexity of the decisions being made. However, the number of judgments made by decision-makers should
not be the sole determinant of their performance. Quality is equally important, and decision-makers must take
sufficient time to carefully consider all relevant factors before making a judgment. Rushing to meet deadlines or
quotas could jeopardize the quality of the decisions made.
External factors such as changes in laws or regulations, societal attitudes, or economic conditions may also
impact the number and complexity of decisions that decision-makers must make. Measuring the number of
judgments made by decision-makers is critical for assessing the efficiency, effectiveness, and quality of
decision-making processes. This measure can offer insights into the workload, complexity, and impact of
decisions made by decision-makers, and can be a valuable tool for enhancing decision-making processes.
XII. OBJECTIVE AND SUBJECTIVE EVALUATION IN FUZZY AHP AND FUZZY TOPSIS
Objective evaluation involves utilizing mathematical or statistical methods to determine the criteria weights [8-
11]. In the Fuzzy AHP method, the objective evaluation process entails converting the crisp criteria weights into
fuzzy weights, which provides a more realistic representation of the decision problem. This is because crisp
weights may not fully reflect the decision maker's uncertainty or imprecision in their judgments. Fuzzy AHP
uses fuzzy logic to address this uncertainty by representing the criteria weights as fuzzy numbers that are more
flexible and can accommodate a broader range of values than crisp numbers. After determining the fuzzy
weights of the criteria, they are combined with the decision matrix to calculate the overall fuzzy preference
values for each alternative, indicating the degree to which each alternative meets the decision criteria. The
alternatives are then ranked according to their fuzzy preference values, with higher values signifying better
overall performance. In TOPSIS, the objective evaluation process entails normalizing the decision matrix by
dividing each element in the matrix by the corresponding sum or average. This is to ensure that all criteria are

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given equal weight and to make the decision matrix comparable across different scales and units. Once
normalized, the criteria weights are determined using a mathematical method like Eigenvalue or Entropy,
which assign weights based on the criteria's importance and relevance to the decision problem.
After obtaining the criteria weights, a performance score is computed for each alternative by multiplying the
normalized decision matrix by the criteria weights [5-7]. This results in a weighted score for each alternative,
representing the degree to which it satisfies the decision criteria. The alternatives are then ranked based on
their weighted scores, with higher scores indicating better overall performance. In contrast, subjective
evaluation considers the decision maker's preferences and opinions. In Fuzzy AHP, the subjective evaluation
process involves determining the criteria's importance using linguistic terms such as "very important,"
"important," "less important," and "not important," which are then converted into fuzzy numbers to establish
their relative importance.
In TOPSIS, subjective evaluation involves assigning weights to the criteria based on the decision maker's
preferences and opinions, using a scale or rating system to determine the criteria's importance relative to the
others. These weights are then used to calculate the weighted score for each alternative, as described earlier.
Both objective and subjective evaluation are crucial in decision-making as they provide a comprehensive and
realistic view of the decision problem. Objective evaluation enables a more accurate representation of the
decision problem, while subjective evaluation accounts for the decision maker's preferences and opinions.
Together, these methods can assist decision makers in making more informed and effective decisions.
Table 3: Objective and Subjective Evaluation in Fuzzy AHP and Fuzzy TOPSIS
Criteria Fuzzy AHP Fuzzy TOPSIS
Evaluation Type Subjective Objective and Subjective
Calculation Method Pairwise Comparison Matrix Normalized Decision Matrix
Fuzzification Method
Linguistic terms, e.g., very low,
low, etc.
Membership function, e.g.,
triangular, trapezoidal, Gaussian,
etc.
Consistency Checking Consistency Ratio (CR) N/A
Weight Calculation Eigenvector N/A
Rank Calculation N/A Euclidean distance
Fuzzy AHP is primarily employed to evaluate subjective criteria, allowing decision-makers to express their
judgments using linguistic terms. Conversely, Fuzzy TOPSIS can handle both objective and subjective criteria by
utilizing membership functions to indicate the degree to which each alternative satisfies each criterion. While
Fuzzy AHP checks for consistency using the Consistency Ratio, Fuzzy TOPSIS does not require consistency
checks. Weight calculation in Fuzzy AHP relies on the Eigenvector, while in Fuzzy TOPSIS, it is based on the
normalized decision matrix. Finally, the ranking of alternatives in Fuzzy TOPSIS is based on the Euclidean
distance between each alternative and the ideal solution.
XIII. FUZZY AHP AND FUZZY TOPSIS AS AN EFFECTIVE AND POWERFUL MULTI-
CRITERIA DECISION-MAKING (MCDM) METHOD FOR SUBJECTIVE JUDGEMENTS
Multi-Criteria Decision-Making (MCDM) is a commonly used approach for prioritizing and evaluating multiple
criteria in decision-making [11-13]. However, determining the relative importance of each criterion accurately
can be challenging due to subjectivity and uncertainty in many decision-making situations. To address this
challenge, fuzzy set theory has been integrated into MCDM methods. This theory enables decision-makers to
express the degree of membership of each alternative to a particular criterion in linguistic terms, which
facilitates the incorporation of subjective judgments into the decision-making process. Fuzzy Analytic
Hierarchy Process (Fuzzy AHP) and Fuzzy Technique for Order of Preference by Similarity to Ideal Solution
(Fuzzy TOPSIS) are two widely used MCDM methods that incorporate fuzzy set theory. Fuzzy AHP involves
pairwise comparisons to determine the relative importance of criteria, while Fuzzy TOPSIS is used to rank
alternatives based on their proximity to the ideal solution. Together, Fuzzy AHP and Fuzzy TOPSIS provide a

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powerful MCDM method that is effective in situations involving subjective judgments. They enable decision-
makers to systematically and transparently incorporate their subjective judgments while also accounting for
uncertainty and imprecision. This approach can lead to more accurate and reliable decision-making in
situations where subjective judgments play a significant role.
The personal opinions, beliefs, and values of decision-makers are known as the subjective element in decision-
making, and it can have a significant impact on the decision-making process, especially when dealing with
multiple criteria. In situations where there are no clear objective standards to determine the best alternative,
decision-makers rely on their subjective judgments and preferences to evaluate and prioritize options, which
can lead to bias, inconsistencies, and errors. To systematically and transparently incorporate the subjective
element into decision-making, fuzzy set theory provides a solution. This approach allows decision-makers to
express their subjective judgments more effectively by using linguistic terms to indicate the degree of
membership of each alternative to a criterion. Additionally, Fuzzy AHP and Fuzzy TOPSIS provide a structured
way for decision-makers to quantify the subjective element and use it to evaluate and rank alternatives. By
considering the subjective element, decision-makers can make more informed and effective decisions that align
with their values, beliefs, and preferences. This approach can also increase the acceptance and support of the
decision by stakeholders who share similar values or interests. However, it is crucial for decision-makers to be
aware of their biases and strive to make objective and rational decisions while using the subjective element to
inform their judgments.
XIV. CONCLUSION
To summarize, the study has demonstrated that Fuzzy AHP and Fuzzy TOPSIS are effective Multi-Criteria
Decision-Making methods for subjective judgments in selection processes. The fuzzy sets theory has been
applied successfully to address imprecision and uncertainty in decision-making. This method has proven
valuable for decision-makers in various industries and applications that rely on subjective judgment in the
selection process. Additionally, it facilitates the incorporation of multiple criteria and their relative importance,
enabling decision-makers to make more informed decisions. When objective data is lacking or personal
preferences, values, and beliefs are involved, subjective judgments often influence decision-making. However,
relying solely on subjective judgment can result in biases, inconsistencies, and errors in decision-making. To
address this, the Fuzzy AHP and Fuzzy TOPSIS method is a useful tool for decision-makers to convert their
subjective judgments into numerical values expressed using linguistic terms. This method helps decision-
makers clarify their preferences, reduce ambiguity, and make more informed decisions.
Moreover, the method allows decision-makers to consider multiple criteria and their relative importance,
especially in complex decision-making situations. This comprehensive approach to decision-making helps
decision-makers to weigh trade-offs and synergies between different criteria, thereby avoiding the problem of
focusing solely on one criterion while ignoring other essential factors that can lead to suboptimal decisions.
Overall, the results suggest that Fuzzy AHP and Fuzzy TOPSIS are efficient and effective decision-making
methods when subjective judgments and multiple criteria are involved. As a result, it is expected that this
approach will become more widely adopted across various industries, leading to better-informed and effective
decision-making processes. In conclusion, subjective judgments are crucial in decision-making, and the Fuzzy
AHP and Fuzzy TOPSIS method offer an effective approach to dealing with them. By incorporating multiple
criteria and their relative importance, decision-makers can make more informed decisions, avoid biases, and
inconsistencies associated with subjective judgments.
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