INFORMATICAInformatica1822-88440868-49520868-4952Vilnius UniversityINFOR51510.15388/23-INFOR515Research ArticleA Novel Z-Fuzzy AHP&EDAS Methodology and Its Application to Wind Turbine Selectionhttps://orcid.org/0000-0002-0913-8351TüysüzNurdanyildizn18@itu.edu.tr12∗
N. Tüysüz received her BSc in industrial engineering in 2013 from Konya Selçuk University and her MSc in industrial engineering from Istanbul University, in 2017. She has been a PhD candidate in Department of Industrial Engineering at Istanbul Technical University since 2018. She is currently working as a research assistant at Istanbul Gelisim University. Her research interests include fuzzy sets and their extensions, decision theory and system simulation.
C. Kahraman is a full professor at Istanbul Technical University. His research areas are engineering economics, quality management, statistical decision making, multicriteria decision making, and fuzzy decision making. He published about 300 international journal papers and about 200 conference papers. He became the guest editor of many international journals and the editor of many international books from Springer. He is a member of editorial boards of 20 international journals. He is the chair of INFUS International Conferences on fuzzy and intelligent systems ZS, Yager RR, some geometric aggregation operators based on intuitionistic fuzzy sets.
Modelling the reliability information in decision making process is an important issue to inclusively reflect the thoughts of decision makers. The Evaluation Based on Distance from Average Solution (EDAS) and Analytic Hierarchy Process (AHP) are frequently used MCDM methods, yet their fuzzy extensions in the literature are incapable of representing the reliability of experts’ fuzzy preferences, which may have important effects on the results. The first goal of this study is to extend the EDAS method by using Z-fuzzy numbers to reinforce its representation ability of fuzzy linguistic expressions. The second goal is to propose a decision making methodology for the solution of fuzzy MCDM problems by using Z-fuzzy AHP method for determining the criteria weights and Z-fuzzy EDAS method for the selection of the best alternative. The contribution of the study is to present an MCDM based decision support tool for the managers under vague and imprecise data, which also considers the reliability of these data. The applicability of the proposed model is presented with an application to wind energy investment problem aiming at the selection of the best wind turbine. Finally, the effectiveness and competitiveness of the proposed methodology is demonstrated by making a comparative analysis with the Z-fuzzy TOPSIS method. The results show that the proposed methodology can not only represent experts’ evaluation information extensively, but also reveal a logical and consistent sequence related to wind turbine alternatives using reliability information.
We face decision-making processes at every moment of our lives. In the decision-making process, people express their knowledge and thoughts via their personal opinions and comments. Decision makers (DMs) often use expressions containing doubt and uncertainty in their judgments. Expressions such as “not very clear”, “likely”, etc., show the uncertainty of human thought and are frequently used in daily or business life. Zadeh (1965) introduced fuzzy set theory in order to model this ambiguity and subjectivity of human judgments and to use linguistic terms in the decision-making process. Thus, fuzzy set theory enables DMs to incorporate their uncertain information in the decision model.
DMs who have knowledge and experience are often not exactly sure of their assessments when they are making a decision. The probability of correct diagnosis of even a doctor is not one hundred percent (Xian et al., 2019). For example, one doctor can say “you likely have anemia”. In the medical world, tests and investigations can be performed to confirm this diagnosis. However, in many fields that need decision-making, subjective judgments cannot be confirmed in that way. Moreover, when quantitative data are used in decision making, they are treated to be exactly accurate since the sources’ reliability level is not questioned. However, it would not be correct to assume the numerical data with 100% certainty due to factors such as the concept of time and measurement accuracy. The possible variations that may occur in numerical data can be modelled with different extensions of fuzzy set theory. However, when qualitative data consisting of uncertain judgments is used in decision making, it would be most logical to explicitly ask people about their confidence level in their judgments. In these cases, the reliability of the experts’ fuzzy judgments must be considered and incorporated to the decision model. As a result, it is clear that restrictive information must be integrated with reliability information especially when linguistic expressions, which represent subjective judgments, are employed in the decision model.
After the introduction of fuzzy set theory, fuzzy versions of classical multi criteria decision making (MCDM) methods have emerged to capture the DMs’ uncertain expressions (Chatterjee et al., 2018a). These methods have been expanded by ordinary fuzzy sets and their several extensions, such as type-2 fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, Pythagorean fuzzy sets, and neutrosophic sets, to find the best representation of human thinking structure. Although the extensions of fuzzy sets are highly beneficial and suited to deal with vague information, their capabilities are limited to represent the reliability of the assigned fuzzy data. In order to overcome this limitation and to reach more accurate and effective results, reliability information must be incorporated into the decision processes.
Z-fuzzy numbers have been proposed by Zadeh (2011) in order to deal with the vagueness and impreciseness of membership functions by incorporating a reliability function to the evaluation system as a complementary element. This can be commented as a similar effort by Zadeh to his type-2 fuzzy sets for preventing the criticisms that membership functions themselves are not fuzzy. Thus, the requirement of reliability information in the decision-making can be satisfied by the use of Z-fuzzy numbers. Z-fuzzy numbers reflect the uncertainty in DMs’ mind through a reliability function, which express how confident they are about their evaluations. In the doctor example, whereas the word “anemia” represents restrictive information, the word “likely” represents reliability information.
Evaluation Based on Distance from Average Solution (EDAS) is one of the recently developed MCDM methods. The EDAS method has been integrated with various fuzzy set extensions to better define the DMs’ uncertain judgments. However, these versions of the EDAS method such as intuitionistic fuzzy EDAS or picture fuzzy EDAS do not fully include the reliability information. To the best knowledge of the authors, the EDAS method has not been extended with Z-fuzzy numbers by any researcher. In the literature, there is only one paper trying to use linguistic Z-numbers in EDAS method, different from our study, for quality function deployment (Mao et al., 2021). In this study, EDAS method is extended to Z-fuzzy EDAS method using ordinary Z-fuzzy numbers to strengthen the reliability degree of the given decisions.
Main objectives of the study are as follows:
The first aim of the study is to extend the traditional EDAS method to Z-fuzzy EDAS for the solution of MCDM problems under vagueness and impreciseness, which takes the reliability of the experts’ data into account.
The second aim of this study is to integrate Z-fuzzy AHP method with Z-fuzzy EDAS method in order to use the criteria weights obtained from AHP in the Z-fuzzy EDAS method for ranking the alternatives.
The proposed methodology is applied to a wind turbine technology selection problem to present its practicality and efficiency. A comparative analysis is performed by using the same data with the Z-fuzzy TOPSIS method.
This study contributes to the literature in four aspects:
First, a novel Z-fuzzy EDAS has been developed for the first time by formulating it step by step using Z-fuzzy numbers. Thus, the literature gap on Z-fuzzy MCDM methods will be filled.
Second, to the best of our knowledge, a methodology integrating Z-fuzzy numbers and AHP & EDAS methods has not been developed.
Third, all steps of the Z-fuzzy EDAS method have been performed by Z-fuzzy numbers which prevents the loss of information existing in the fuzzy data.
Finally, the proposed approach has been applied to a renewable energy problem in the literature illustrating how to use the proposed methodology step by step.
The rest of the paper is organized as follows. Section 2 presents a literature review on EDAS and Z-fuzzy MCDM. Section 3 includes the preliminaries of Z-fuzzy numbers. Section 4 presents the proposed Z-fuzzy AHP method and Section 5 gives the steps of the proposed Z-fuzzy EDAS method. Section 6 presents the application on wind turbine technology selection. Section 7 gives a comparative analysis using Z-fuzzy AHP&TOPSIS methodology. The last section presents the conclusions and future research directions.
Literature Review on EDAS and Z-Fuzzy MCDM
Decision making problems arise when there is a need for comparison or selection from a set of alternatives, taking into account the impact of multiple conflicting criteria. For this purpose, various multiple criteria decision making (MCDM) methods are constructed to determine the best alternative with respect to all relevant criteria (Chatterjee et al., 2018b). Decisions taken in daily life or business life may have different degrees of difficulty due to the factors such as the considered criteria, the relationship between them and the number of alternatives. However, when DMs need to evaluate the alternatives by considering many criteria; many factors such as the number of criteria and alternatives, criteria weights and conflicts between criteria further complicate the problem and need to be evaluated with more comprehensive methods. Therefore, multi-criteria decision making (MCDM) methods are used in order to get more accurate decisions in solving more complex decision problems.
EDAS method has been introduced to the literature by Keshavarz Ghorabaee et al. (2015) as a MCDM method. It is based on the measurement of the positive and negative distances from the average solution rather than calculating the negative ideal solution (NIS) and positive ideal solution (PIS) as in TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) (Chatterjee and Kar, 2016) and VIKOR (Vise Kriterijumska Optimizacija I Kompromisno Resenje) methods. Thus, unlike the TOPSIS and VIKOR methods, EDAS offers a solution based on how far the alternatives are from the average solution instead of PIS and NIS.
After the introduction of EDAS method to the literature, it has been used in many application areas such as supplier selection, project selection, personnel selection, material selection and drug selection. Due to the fact that fuzzy set theory in decision making better defines human thoughts, various fuzzy extensions of EDAS method have been used more frequently than classical EDAS method in the literature. Table 1 presents the classical, stochastic, neutrosophic, and fuzzy EDAS papers published in the literature and their application areas in historical order.
Papers in the literature on EDAS method.
Year
Authors
Extension of EDAS
Application area
2015
Keshavarz Ghorabaee et al.
Crisp EDAS
Inventory classification
2016
Keshavarz Ghorabaee et al.
Fuzzy EDAS
Supplier selection
2017
Kahraman et al.
Intuitionistic EDAS
Solid waste disposal site selection
2017a
Keshavarz Ghorabaee et al.
Stochastic EDAS
Performance evaluation of bank branches
2017
Stanujkic et al.
Interval grey valued EDAS
Contractor selection
2017b
Keshavarz Ghorabaee et al.
Interval type-2 fuzzy EDAS
Supplier selection with respect to environmental criteria
2017c
Keshavarz Ghorabaee et al.
Interval type-2 fuzzy EDAS
Evaluation of subcontractors
2017
Peng and Liu
Single valued neutrosophic EDAS
Evaluation of software development project
2018
Stević et al.
Fuzzy EDAS
Carpenter manufacturer selection
2018
Feng et al.
Hesitant fuzzy EDAS
Project selection
2018c
Chatterjee et al.
Crisp EDAS
Material selection
2018
Keshavarz Ghorabaee et al.
Dynamic fuzzy EDAS
Evaluation of subcontractors
2018
Karabasevic et al.
Crisp EDAS
Personnel Selection
2018
Liang et al.
Integrated EDAS-ELECTRE method
Cleaner Production Evaluation
2018
Ilieva
Interval type-2 fuzzy EDAS
An illustrative example
2018
Karaşan and Kahraman
Interval-valued neutrosophic EDAS
Prioritization of the united nations national sustainable development goals
2018
Kutlu Gündoğdu et al.
Hesitant fuzzy EDAS
Hospital selection
2019
Karaşan et al.
Interval-valued neutrosophic EDAS
Ranking of social responsibility projects
2019
Zhang et al.
Picture 2-tuple linguistic EDAS
Green supplier selection
2019
Schitea et al.
Intuitionistic EDAS
Selection of hydrogen collection site
2019
Kundakcı
Crisp EDAS
Steam boiler selection
2019
Wang et al.
2-tuple linguistic neutrosophic EDAS
Safety assessment of construction project
2019
Stević et al.
Fuzzy EDAS
Supplier selection
2020
Yanmaz et al.
Interval-valued Pythagorean Fuzzy EDAS
Car selection
2020
Han and Wei
Neutrosophic EDAS
Investment evaluation
2020
Liang
Intuitionistic Fuzzy EDAS
Selection of energy-saving design projects
2020
He et al.
Pythagorean 2-tuple linguistic sets based EDAS
Construction project selection
2020
Darko and Liang
q-rang orthopair fuzzy EDAS
Mobile payment platform selection
2020
Li et al.
q-rung orthopair fuzzy EDAS
Refrigerator selection
2020
Mishra et al.
Intuitionistic fuzzy EDAS
Disposal method selection
2020
Tolga and Basar
Fuzzy EDAS
Hydroponic system evaluation
2021
Wei et al.
Probabilistic EDAS
Supplier selection
2021
Chinram et al.
Intuitionistic fuzzy EDAS
Geographical site selection for construction
2021
Özçelik and Nalkıran
Trapezoidal bipolar Fuzzy numbers based EDAS
Medical device selection
2021
Jana and Pal
Bipolar fuzzy EDAS
Construction company selection
2021
Mao et al.
Z-fuzzy EDAS
Ranking of engineering characteristics in quality function deployment
2022
Mitra
Crisp EDAS
Selection of cotton fabric
2022
Batool et al.
EDAS method under Pythagorean probabilistic hesitant fuzzy information
Drug selection for coronavirus disease
2022
Garg and Sharaf
Spherical fuzzy EDAS
Supplier selection and industrial robot selection
2022
Mishra et al.
Fermatean fuzzy EDAS
Evaluation of sustainable third-party reverse logistics providers
2022
Naz et al.
2-tuple linguistic T-spherical fuzzy EDAS
Selecting of the best COVID-19 vaccine
2022
Liao et al.
Probabilistic hesitant fuzzy EDAS
Evaluation of the commercial vehicles and green suppliers
2022
Demircan and Acarbay
Neutrosophic fuzzy EDAS
Vendor selection
2022
Rogulj et al.
Intuitionistic fuzzy EDAS
Prioritization of historic bridges
2022
Huang et al.
2-tuple spherical linguistic EDAS
Selection of the optimal emergency response solution
2022
Polat and Bayhan
Fuzzy EDAS
Supplier selection
2022
Su et al.
Probabilistic uncertain linguistic EDAS
Green finance evaluation of enterprises
2023
Akram et al.
Linguistic Pythagorean fuzzy EDAS
Selection of waste management technique
A literature review on MCDM studies using Z-fuzzy numbers.
Year
Authors
MCDM method’s used Z-fuzzy number
Application areas
2012a
Kang et al.
A proposed approach
Vehicle selection
2013
Azadeh et al.
AHP
Weighing the performance evaluation factors of universities
2014
Xiao
A proposed approach
Evaluation of cloths
2015
Sahrom and Dom
AHP and DEA
Risk assessment
2015
Yaakob and Gegov
TOPSIS
Stock selection
2016
Azadeh and Kokabi
DEA
Portfolio selection
2016
Sadi-Nezhad and Sotoudeh-Anvari
DEA
Efficiency assessment
2016
Yaakob and Gegov
TOPSIS
Stock selection
2017
Peng and Wang
A proposed approach
ERP selection
2017a
Khalif et al.
TOPSIS
Performance assessment
2017b
Khalif et al.
TOPSIS
Staff selection
2017
Wang et al.
TODIM
Evaluation of medical inquiry applications
2018
Karthika and Sudha
AHP
Risk assessment
2018
Forghani et al.
TOPSIS
Supplier selection
2018
Chatterjee and Kar
COPRAS
Renewable energy selection
2018
Aboutorab et al.
Best-worst method
Supplier development problem
2018
Peng and Wang
MULTIMOORA
Evaluation of potential areas of air pollution
2018
Shen and Wang
VIKOR
Selection of economic development plan
2018
Akbarian Saravi et al.
DEA
Evaluation of biomass power plants location
2018
Kahraman and Otay
AHP
Power plant location selection
2019
Gardashova
TOPSIS
Vehicle selection
2019
Wang and Mao
TOPSIS
Supplier selection
2019
Xian et al.
TOPSIS
Numerical examples on investment and medical diagnosis
2019
Kahraman et al.
AHP
Evaluation of law offices
2019
Krohling et al.
TODIM and TOPSIS
Case studies from literature
2019
Shen et al.
MABAC
Selection of economy development program
2020
Yildiz and Kahraman
AHP
Prioritization of social sustainable development factors
2020
Qiao et al.
PROMETHEE
Travel plan selection
2020
Das et al.
VIKOR
Prioritizing risk of hazards for crane operations.
2020
Jiang et al.
DEMATEL
Hospital performance measurement
2020
Mohtashami and Ghiasvand
DEA
Evaluation of banks and financial institutes
2020
Liu et al.
ANP and TODIM
Evaluation of suppliers for the nuclear power industry
2020a
Tüysüz and Kahraman
AHP
Evaluation of social sustainable development factors
2020b
Tüysüz and Kahraman
CODAS
Supplier selection
2021
Akhavein et al.
DEMATEL and VIKOR
Evaluation of projects
2021
Zhu and Hu
DEMATEL
Evaluation of sustainable value propositions for smart product-service systems
2021
Wang et al.
DEMATEL
Evaluation of human error probability for cargo loading operations.
2021
Mao et al.
EDAS
Ranking of engineering characteristics in quality function deployment
2021
Sergi and Ucal Sari
AHP and WASPAS
Evaluation of public services
2021
Karaşan et al.
DEMATEL
Blockchain risk assessment
2022
Peng et al.
MULTIMOORA
Hotel selection
2022
İlbahar et al.
DEMATEL and VIKOR
Evaluation of hydrogen energy storage systems
2022
Sari and Tüysüz
AHP and TOPSIS
Covid-19 risk assessment of occupations
2022
Liu et al.
ELECTRE II
Selection of logistics provider
2022
Rahmati et al.
SWARA and WASPAS
Prioritization of financial risk factors
2022
Gai et al.
MULTIMOORA
Green supplier selection
2022
RezaHoseini et al.
AHP and DEA
Performance evaluation of sustainable projects
2022
Božanić et al.
MABAC
Selection of the best contingency strategy
Table 1 shows that the classical EDAS method has been developed by many extensions of ordinary fuzzy sets such as type-2 fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets. However, since it was only put forward in 2015, there is still a gap in the literature about the method and its usage areas.
Since the fuzzy versions of the EDAS method proposed so far do not fully reflect the reliability information, another possible extension of the classical EDAS method is realized in this study through Z-fuzzy numbers, which represent the natural language with better descriptive ability. Thus, apart from the fuzzy extensions in Table 1, the EDAS method has been extended with Z-fuzzy numbers, which are composed of trapezoidal restriction function and triangular fuzzy reliability function.
After Z-fuzzy numbers were introduced to the literature, they have been integrated with several MCDM methods such as AHP (Azadeh et al., 2013; Sergi and Sari, 2021; Tüysüz and Kahraman, 2020a; Kahraman and Otay, 2018), TOPSIS (Krohling et al., 2019), VIKOR (Shen and Wang, 2018), and WASPAS (Sergi and Sari, 2021). Table 2 presents the Z-fuzzy number integrated MCDM methods based on their publication years.
As can be seen in Table 2, Z-fuzzy numbers are integrated with different MCDM methods, and they are used in different application areas. However, there is still a significant literature gap regarding the combined use of Z-fuzzy numbers and MCDM methods. This study contributes to fill this literature gap by integrating the EDAS method with Z-fuzzy numbers.
Z-Fuzzy Numbers: Preliminaries
DMs are often not 100% confident in their assignments for membership degrees. Hence, in addition to assigning a membership degree/function μA˜(x), it makes sense to also assign a reliability degree μB˜(x) so that DMs can reflect their confidence to the membership. The corresponding pairs (μA˜(x), μB˜(x)) are known as a Z-fuzzy number which was introduced by Zadeh (2011).
A Z-fuzzy number is an ordered pair of fuzzy numbers Z(A˜,B˜), as given in Fig. 1. The first component A˜ is a restriction function whereas the second component B˜ is a measure of reliability for the first component.
A simple Z-fuzzy number, Z(A˜,B˜).
The concept of a Z-fuzzy number is intended to provide a basis for computation with ordinary fuzzy numbers which are not reliable.
Let a fuzzy set A˜ be defined on a universe X, which may be given as: A˜={⟨x,μA˜(x)⟩|xϵX} where μA˜:X→[0,1] is the membership function A˜. The membership value μA˜(x) describes the degree of belongingness of x∈X in A˜. The Fuzzy Expectation of a fuzzy set is given in Eq. (1):
EA(x)=∫xxμA(x)dx,
which is not the Expectation of Probability Space.
(Converting Z-fuzzy number to Regular Fuzzy Number, Kang et al., 2012b).
Consider a Z-fuzzy number Z=(A,˜B˜), which is described by Fig. 1. The figure on the left is the part of restriction, and the figure on the right is the part of reliability. Let A˜={⟨x,μA˜(x)⟩|μ(x)∈[0,1]} and B˜={⟨x,μB˜(x)⟩|μ(x)∈[0,1]}, μA˜(x) is a trapezoidal membership function, μB˜(x) is a triangular membership function.
(1) Convert the reliability function into a crisp number using Eq. (2):
α=∫xμB˜(x)dx∫μB˜(x)dx,
where ∫ denotes an algebraic integration.
Alternatively, the defuzzification equation (a1+2∗a2+2∗a3+a4)/6 for symmetrical trapezoidal fuzzy numbers and (a1+2∗a2+a3)/4 for symmetrical triangular fuzzy numbers can be used.
(2) Weigh the restriction function with the crisp value of the reliability function (α). The weighted restriction number is denoted in Eq. (3).
Z˜α={⟨x,μA˜α(x)⟩|μA˜α(x)=αμA˜(x),μ(x)∈[0,1]}.
(3) Convert the weighted restriction number to ordinary fuzzy number using Eq. (4):
Z˜′={⟨x,μZ˜′(x)⟩|μZ˜′(x)=μA˜(xα),μ(x)∈[0,1]},Z˜′ has the same Fuzzy Expectation with Z˜α, and they are equal with respect to Fuzzy Expectation, which can be denoted by Fig. 2.
Ordinary fuzzy number converted from Z-fuzzy number.
(4) If the restriction function and reliability function are defined as in Fig. 3, the calculations are modified as follows:
Let A˜δ={⟨x,(μA˜(x);δ)⟩|μ(x)∈[0,1]} and B˜β={⟨x,(μB˜(x);β)⟩|μ(x)∈[0,1]}, μA˜δ(x) is a trapezoidal membership function, μB˜β(x) is a triangular membership function.
A simple Z˜δ,β number, Z˜δ,β=(A˜δ,B˜β).
In this case, restriction and reliability functions are given in Eqs. (5)–(6), respectively. The reliability membership function in Eq. (6) is substituted into the defuzzification formula Eq. (2); so that, Eq. (7) is obtained. μA˜δ(x)=x−a1a2−a1δ,ifa1⩽x⩽a2,δ,ifa2⩽x⩽a3,a4−xa4−a3δ,ifa3⩽x⩽a4,0,otherwise,μB˜β(x)=x−b1b2−b1β,ifb1⩽x⩽b2,b3−xb3−b2β,ifb2⩽x⩽b3,0,otherwise. Thus, we have
α=∫xμB˜β(x)dx∫μB˜β(x)dx.
Then, the weighted Z˜δ,β number can be denoted as in Eq. (8):
Z˜δ,βα={⟨x,μA˜αδ(x)⟩|μA˜αδ(x)=∫xμB˜β(x)dx∫μB˜β(x)dxμA˜δ(x),μ(x)∈[0,1]}.
The ordinary fuzzy number converted from Z-fuzzy number can be given as in Eq. (9):
Z˜δ,β′={⟨x,μz˜′δ(x)⟩|μz˜′δ(x)=μA˜δ(x∫μB˜β(x)dx∫xμB˜β(x)dx),μ(x)∈[0,1]}.
Z-Fuzzy AHP
The AHP method is one of the most widely used MCDM methods to calculate the criteria weights and there are several versions of it (Chatterjee and Kar, 2017). Due to the nature, it is usual for DMs to have hesitation while making pairwise comparisons, and in these situations, it is expected that they will not be absolutely sure about their evaluations. These preferences can be included in the decision methods by modelling the DMs’ thinking structure under the concept of Z-fuzzy numbers. Therefore, in this study, to obtain criteria weights, it is suggested to collect DMs’ judgments using Z-fuzzy numbers integrated AHP method rather than commonly used fuzzy versions of AHP method.
To calculate criteria weights, the steps of the Z-fuzzy AHP method are presented in the following:
Step 1. Determine the criteria set of the decision problem. Fig. 4 can be used to establish the hierarchical structure of goal, main criteria and sub-criteria. Level 1 of the hierarchy represents a goal whereas Level 2 and Level 3 are composed of main-criteria and sub-criteria, respectively.
Step 2. Determine the linguistic terms and their corresponding Z-fuzzy restriction and reliability numbers. Collect the linguistic pairwise comparison evaluations from each DM for the main criteria and sub-criteria by using questionnaires. Then, Z-fuzzy pairwise comparison matrices are constructed based on these evaluations. Each DM can use Z-fuzzy linguistic scales given in Tables 3–4 for his/her assessments, respectively.
Hierarchical structure for criteria.
Let each decision maker (DMk) assign an independent assessment for any pairwise comparison as shown in Eq. (10):
ZDMk=(A˜,B˜)=((a1DMk,a2DMk,a3DMk),(b1DMk,b2DMk,b3DMk)).
Triangular restriction scale for pairwise comparisons of criteria.
Linguistic terms
Abbreviation
Restriction function
Equally Important
EI
(1,1,1;1)
Slightly Important
SLI
(1,1,3;1)
Moderately Important
MI
(1,3,5;1)
Strongly Important
STI
(3,5,7;1)
Very Strongly Important
VSTI
(5,7,9;1)
Certainly Important
CI
(7,9,10;1)
Absolutely Important
AI
(9,10,10;1)
Step 3. Calculate the consistency ratio (CR) of each Z-fuzzy pairwise comparison matrix obtained by the DMs’ assessments. Defuzzify the restriction functions of Z-fuzzy numbers in the pairwise comparison matrix using Eq. (2) and obtain the crisp pairwise comparison matrix. Apply Saaty’s classical consistency procedure and check if CR is less than 0.1, which is accepted as the consistency limit in the literature (Saaty, 1980).
Triangular reliability scale.
Linguistic terms
Abbreviation
Reliability function
Certainly Reliable
CR
(1,1,1;1)
Very Strongly Reliable
VSR
(0.8,0.9,1;1)
Strongly Reliable
SR
(0.7,0.8,0.9;1)
Very Highly Reliable
VHR
(0.6,0.7,0.8;1)
Highly Reliable
HR
(0.5,0.6,0.7;1)
Fairly Reliable
FR
(0.4,0.5,0.6;1)
Weakly Reliable
WR
(0.3,0.4,0.5;1)
Very Weakly Reliable
VWR
(0.2,0.3,0.4;1)
Strongly Unreliable
SU
(0.1,0.2,0.3;1),
Absolutely Unreliable
AU
(0,0.1,0.2;1)
Step 4. Apply the aggregation procedure for DMs’ Z-fuzzy assessments. Each element of restriction and reliability functions of Z-fuzzy assessments is aggregated by using geometric mean and one Z-fuzzy decision matrix is obtained.
Assume three DMs assign the following terms:
Z˜DM1=(A˜,B˜)=((a1DM1,a2DM1,a3DM1),(b1DM1,b2DM1,b3DM1)),Z˜DM2=(A˜,B˜)=((a1DM2,a2DM2,a3DM2),(b1DM2,b2DM2,b3DM2)),Z˜DM3=(A˜,B˜)=((a1DM3,a2DM3,a3DM3),(b1DM3,b2DM3,b3DM3)).
Aggregation of these three DMs’ assessments is made by using the geometric mean operator given in Eqs. (11)–(12):
Z˜Agg=(A˜Agg,B˜Agg)=c˜11c˜12…c˜1mc˜21c˜22…c˜2m⋮⋮⋱⋮c˜m1c˜m2…c˜mm,
where
c˜ij=(a1,ijDM1∗a1,ijDM2∗a1,ijDM33,a2,ijDM1∗a2,ijDM2∗a2,ijDM33,a3,ijDM1∗a3,ijDM2∗a3,ijDM33),(b1,ijDM1∗b1,ijDM2∗b1,ijDM33,b2,ijDM1∗b2,ijDM2∗b2,ijDM33,b3,ijDM1∗b3,ijDM2∗b3,ijDM33),i=1,2,…,m;j=1,2,…,m.
Step 5. Calculate the alpha (α) from the reliability components of the aggregated pairwise comparison matrix by using Eq. (13). The reciprocal reliability values are the multiplicative inverse of the calculated α values.
αij=(b1,ijDM1∗b1,ijDM2∗b1,ijDM33+2∗b2,ijDM1∗b2,ijDM2∗b2,ijDM33+b3,ijDM1∗b3,ijDM2∗b3,ijDM33)4,i=1,2,…,m;j=1,2,…,m.
Step 6. Convert the Z-fuzzy numbers (Z˜Agg) to ordinary fuzzy numbers (O˜) using the matrix obtained in Step 5 by using Eqs. (14) and (15):
O˜=o˜11o˜12…o˜1mo˜21o˜22…o˜2m⋮⋮⋱⋮o˜m1o˜m2…o˜mm,
where
o˜ij=a1,ijDM1∗a1,ijDM2∗a1,ijDM33αij,a2,ijDM1∗a2,ijDM2∗a2,ijDM33αij,a3,ijDM1∗a3,ijDM2∗a3,ijDM33αij.
Step 7. Apply the ordinary fuzzy AHP method using Buckley’s method (Buckley, 1985).
Step 7.1. Calculate the geometric mean vector (GM˜) whose elements are given in Eqs. (16)–(17). Thus, m×1 matrix is obtained from m×m matrix.
GM˜=g˜11g˜21⋮g˜m1,
where
g˜i1=∏j=1m(a1,ijDM1∗a1,ijDM2∗a1,ijDM33αij)m,∏j=1m(a2,ijDM1∗a2,ijDM2∗a2,ijDM33αij)m,∏j=1m(a3,ijDM1∗a3,ijDM2∗a3,ijDM33αij)m,i=1,2,…,m.
Step 7.2. Sum the values in GM˜ vector using Eq. (18):
S˜=∑i=1m(∏j=1m(a1,ijDM1∗a1,ijDM2∗a1,ijDM33αij)m),∑i=1m(∏j=1m(a2,ijDM1∗a2,ijDM2∗a2,ijDM33αij)m),∑i=1m(∏j=1m(a3,ijDM1∗a3,ijDM2∗a3,ijDM33αij)m).
Step 7.3. Apply fuzzy division operation to obtain relative fuzzy weights vector (R˜) of criteria as given in Eqs. (19)–(20):
R˜=r˜11r˜21⋮r˜m1=g˜11/S˜g˜21/S˜⋮g˜m1/S˜,
where
r˜i1=∏j=1m(a1,ijDM1∗a1,ijDM2∗a1,ijDM33αij)m∑i=1m∏j=1m(a3,ijDM1∗a3,ijDM2∗a3,ijDM33αij)m,∏j=1m(a2,ijDM1∗a2,ijDM2∗a2,ijDM33αij)m∑i=1m∏j=1m(a2,ijDM1∗a2,ijDM2∗a2,ijDM33αij)m,∏j=1m(a3,ijDM1∗a3,ijDM2∗a3,ijDM33αij)m∑i=1m∏j=1m(a1,ijDM1∗a1,ijDM2∗a1,ijDM33αij)m,i=1,2,…,m.
Step 7.4. Defuzzify the relative fuzzy weights vector (R˜) using Eq. (21):
dj=∏j=1m(a1,ijDM1∗a1,ijDM2∗a1,ijDM33αij)m∑i=1m∏j=1m(a3,ijDM1∗a3,ijDM2∗a3,ijDM33αij)m+2∗∏j=1m(a2,ijDM1∗a2,ijDM2∗a2,ijDM33αij)m∑i=1m∏j=1m(a2,ijDM1∗a2,ijDM2∗a2,ijDM33αij)m+∏j=1m(a3,ijDM1∗a3,ijDM2∗a3,ijDM33αij)m∑i=1m∏j=1m(a1,ijDM1∗a1,ijDM2∗a1,ijDM33αij)m∗4−1,j=1,2,…,m.
Step 7.5. Normalize the defuzzified weights to satisfy ∑wj=1 using Eq. (22). Thus, the weights of the criteria are obtained as crisp values.
wj=dj∑j=1mdj,j=1,2,…,m.
Step 8. Apply Steps 3–6 for the other Z-fuzzy pairwise comparison matrices of DMs for the sub-criteria under each main criterion and obtain the weight of each sub-criterion j´, j´=1,2,…,p.
wjj´˙wherej=1,2,…,mandj´=1,2,…,pfor eachj.Step 9. Combine the local sub-criteria weights (wjj´˙) and main criteria weights (wj) in order to obtain global criteria weights (wjj´G) as in Eq. (23).
wjj´G=wj∗wjj´˙,j=1,2,…,mandj´=1,2,…,pfor eachj.
Z-Fuzzy EDAS
The first fuzzy EDAS method is introduced by Keshavarz Ghorabaee et al. (2016) for the solution of MCDM problems under uncertainty. It is integrated with various fuzzy set extensions to model the vagueness and impreciseness. In this study, due to the fact that these extensions cannot completely combine the reliability information with the EDAS method, it is extended to Z-fuzzy EDAS method by using ordinary Z-fuzzy numbers. This method allows to define the DMs’ preferences over the alternatives with their degree of confidence, which creates a more comprehensive and flexible decision-making environment. Z-Fuzzy EDAS method is presented as follows:
Step 1. Determine the evaluation criteria C=(C1,C2,…,Cm) and alternatives A=(A1,A2,…,An) for the decision problem.
Step 2. Construct the fuzzy decision matrix (D˜) using Z-fuzzy numbers, shown as in Eq. (24):
D˜=[x˜ij]n×m=A1A2⋮Anx˜11x˜12…x˜1mx˜21x˜22…x˜2m⋮⋮⋱⋮x˜n1x˜n2…x˜nm,
where x˜ij⩾0 and it denotes the Z-fuzzy performance value of ith alternative on jth criterion
(i∈{1,2,…,n}andj∈{1,2,…,m}).
Z-fuzzy linguistic restriction scale presented in Table 5 and the reliability scale in Table 4 are used for DMs’ assessments in the decision matrix.
Step 3. Aggregate the Z-fuzzy evaluation matrices of all DMs. Aggregation of three DMs’ assessments is made by using the geometric mean given in Eqs. (25)–(26):
Z˜D˜Agg=x˜11x˜12…x˜1mx˜21x˜22…x˜2m⋮⋮⋱⋮x˜n1x˜n2…x˜nm,
where
x˜ij=(a1,ijDM1∗a1,ijDM2∗a1,ijDM33,a2,ijDM1∗a2,ijDM2∗a2,ijDM33,a3,ijDM1∗a3,ijDM2∗a3,ijDM33),(b1,ijDM1∗b1,ijDM2∗b1,ijDM33,b2,ijDM1∗b2,ijDM2∗b2,ijDM33,b3,ijDM1∗b3,ijDM2∗b3,ijDM33),i=1,2,…,n;j=1,2,…,m.
Z-fuzzy restriction scale for evaluation of alternatives.
Linguistic terms
Abbreviation
Restriction function
Very Poor
VP
(1/4,1/2,1/2,1;1)
Poor
P
(1/2,1,1,3;1)
Medium Poor
MP
(1,3,3,5;1)
Fair
F
(3,5,5,7;1)
Medium Good
MG
(5,7,7,9;1)
Good
G
(7,9,9,10;1)
Very Good
VG
(9,10,10,10;1)
Step 4. Calculate the Z-fuzzy average values (AV˜) by using Eqs. (27)–(28): AV˜=[AV˜j]1×m=AV˜1AV˜2…AV˜j,AV˜j=∑i=1nX˜ijn,∀j,j=1,2,…,m.
Step 5. Calculate the Z-fuzzy positive distance from average (PDA˜) and Z-fuzzy negative distance from average (NDA˜) for each alternative by employing Eqs. (29)–(32): PDA˜=[PDA˜ij]n×m,NDA˜=[NDA˜ij]n×m,PDA˜ij=max(0,(x˜ij−AV˜j))AV˜jM,NDA˜ij=max(0,(AV˜j−x˜ij))AV˜jM,for benefit criteria,PDA˜ij=max(0,(AV˜j−x˜ij))AV˜jM,NDA˜ij=max(0,(x˜ij−AV˜j))AV˜jM,for cost criteria, where PDA˜ij and NDA˜ij represent the Z-fuzzy positive and negative distances from average value of ith alternative according to jth criterion, respectively.
To determine max(0,(x˜ij−AV˜j)), Z-fuzzy numbers are defuzzified as in Eqs. (33)–(34) and compared with each other. aj=(a1,ij+2∗a2,ij+2∗a3,ij+a4,ij)6,∀j,for restriction function,bj=(b1,ij+2∗b2,ij+b3,ij)4,∀j,for reliability function. After determining the max(0,(x˜ij−AV˜j)), we still continue with Z-fuzzy numbers. Then, max(0,(x˜ij−AV˜j)) is divided by AV˜j using Z-fuzzy numbers.
Step 6. Use the criteria weights obtained by Z-fuzzy AHP method in Section 4 and calculate the weighted summation of PDA˜ and NDA˜ shown as in Eqs. (35)–(36): SP˜i=∑j=1mwj∗PDA˜ij,SN˜i=∑j=1mwj∗NDA˜ij, where wj=(w1,w2,…,wm) and it is the weight of jth criterion.
wj(0<wj<1) denotes the weight of jth criterion and ∑j=1mwj=1.
Step 7. Transform the obtained Z-fuzzy SP˜i and SN˜i values to positive values if there is any negative value among them for all alternatives shown as in Eqs. (37)–(40). Thus, we obtain the shifted SP˜i and SN˜i values, SSP˜i and SSN˜i, respectively.
For restriction function: SSP˜iRes=SP˜iRes+maxi|(SP˜ia1Res)|,if anya1<0,SSN˜iRes=SN˜iRes+maxi|(SN˜ia1Res)|,if anya1<0. For reliability function: SSP˜iRel=SP˜iRel+maxi|(SP˜ib1Rel)|,if anyb1<0,SSN˜iRel=SN˜iRel+maxi|(SN˜ib1Rel)|,if anyb1<0.
Step 8. Normalize the Z-fuzzy SSP˜i and SSN˜i values by using Eqs. (41)–(44).
For restriction function
NSP˜iaRes=(SSP˜ia1maxi(SP˜iRes),SSP˜ia2maxi(SP˜iRes),SSP˜ia3maxi(SP˜iRes),SSP˜ia4maxi(SP˜iRes))
and
NSN˜iaRes=(1,1,1,1)−(SSN˜ia4maxi(SN˜iRes),SSN˜ia3maxi(SN˜iRes),SSN˜ia2maxi(SN˜iRes),SSN˜ia1maxi(SN˜iRes))
for reliability function
NSP˜ibRel=(SSP˜ib1maxi(SP˜iRel),SSP˜ib2maxi(SP˜iRel),SSP˜ib3maxi(SP˜iRel))
and
NSN˜ibRel=(1,1,1)−(SSN˜ib3maxi(SN˜iRel),SSN˜ib2maxi(SN˜iRel),SSN˜ib1maxi(SN˜iRel)).
Step 9. Calculate the Z-fuzzy appraisal score (AS˜i=(ASiaRes,ASibRel)) of alternatives, as shown in Eqs. (45)–(46): ASiaRes=12(NSP˜iaRes+NSN˜iaRes),ASibRel=12(NSP˜ibRel+NSN˜ibRel).
Step 10. Convert the Z-fuzzy AS˜i to ordinary fuzzy number using Definition 2.
Step 11. Transform the ordinary fuzzy AS˜i to a crisp number using Eq. (2).
Step 12. Rank the alternatives according to the decreasing values of crisp ASi. The alternative which has the highest ASi is the best choice among the alternatives.
Fig. 5 shows the flowchart of the methodology which integrates Z-fuzzy AHP and Z-fuzzy EDAS methods. The proposed methodology aims at finding the weights of the criteria to be used in wind turbine selection (Z-fuzzy AHP) and also ranking the alternatives (Z-fuzzy EDAS) according to these criteria.
Proposed Z-fuzzy AHP&EDAS methodology.
Application: Wind Turbine Selection
Wind power is one of the fastest growing renewable energy alternatives. Due to the increasing energy demand, investments toward renewable energy sources are getting more importance day by day. Wind energy is the most widely used renewable energy source in Turkey (Kahraman and Kaya, 2010). According to the March 2022 TEİAŞ (Turkish Electricity Transmission Corporation) report, there are 355 wind power plants, and approximately 10861 megawatts of energy are produced from the wind in Turkey (TEİAŞ, 2022). In order to produce energy efficiently from the wind, the turbine characteristics of the power plant to be established have great importance. Therefore, the selection of wind turbines in a wind energy investment is extremely important for investors. There are many types of wind turbines according to their characteristics. In order to produce energy efficiently from the wind, the right wind turbine should be selected by the DMs according to the wind characteristics of the region to be established. In addition, the problem should be considered as a MCDM problem since many factors should be evaluated together in wind turbine selection. The MCDM studies of wind turbine selection in the literature are quite limited (Supciller and Toprak, 2020). Studies related to wind turbine selection can be found in Supciller and Toprak (2020) and Pang et al. (2021).
The proposed Z-fuzzy AHP&EDAS methodology is applied for the selection of the best alternative among wind turbines in the Aegean region of Turkey. For this purpose, in Step 1, the alternatives and criteria have been determined. There are five wind turbine alternatives represented by A1, A2, A3, A4 and A5 and six criteria which are reliability (C1), technical characteristics (C2), performance (C3), cost factors (C4), availability (C5) and maintenance (C6) (Cevik Onar et al., 2015). In Step 2, decision matrices have been constructed by three DMs using the linguistic terms given in Tables 4 and 5. Three DMs’ pairwise comparison matrices for the criteria are presented in Tables 6–8.
Pairwise comparisons of the criteria by DM1.
DM1
C1
C2
C3
C4
C5
C6
C1
(EI, CR)
(CI, VSR)
(STI, HR)
(SLI, VSR)
(VSTI, VHR)
(CI, VSR)
C2
(1/CI, VSR)
(EI, CR)
(1/MI, SR)
(1/VSTI, FR)
(1/MI, SR)
(MI, FR)
C3
(1/STI, HR)
(MI, SR)
(EI, CR)
(1/MI, VHR)
(SLI, VSR)
(STI, VHR)
C4
(1/SLI, VSR)
(VSTI, FR)
(MI, VHR)
(EI, CR)
(STI, FR)
(CI, VSR)
C5
(1/VSTI, VHR)
(MI, SR)
(1/SLI, VSR)
(1/STI, FR)
(EI, CR)
(STI, WR)
C6
(1/CI, VSR)
(1/MI, FR)
(1/STI, VHR)
(1/CI, VSR)
(1/STI, WR)
(EI, CR)
λmax=6.6085, Consistency index (CI) = 0.1216, Consistency ratio (CR) = 0.097.
Pairwise comparisons of the criteria by DM2.
DM2
C1
C2
C3
C4
C5
C6
C1
(EI, CR)
(VSTI, VHR)
(MI, FR)
(EI, SR)
(STI, SR)
(VSTI, HR)
C2
(1/VSTI, VHR)
(EI, CR)
(1/STI, VHR)
(1/CI, HR)
(1/SLI, VSR)
(SLI, VSR)
C3
(1/MI, FR)
(STI, VHR)
(EI, CR)
(1/STI, FR)
(MI, FR)
(MI, HR)
C4
(EI, SR)
(CI, HR)
(STI, FR)
(EI, CR)
(VSTI, VHR)
(CI, VHR)
C5
(1/STI, SR)
(SLI, VSR)
(1/MI, FR)
(1/VSTI, VHR)
(EI, CR)
(MI, FR)
C6
(1/VSTI, HR)
(1/SLI, VSR)
(1/MI, HR)
(1/CI, VHR)
(1/MI, FR)
(EI, CR)
λmax=6.5761, Consistency index (CI) = 0.1152, Consistency ratio (CR) = 0.092.
Pairwise comparisons of the criteria by DM3.
DM3
C1
C2
C3
C4
C5
C6
C1
(EI, CR)
(AI, SR)
(VSTI, VHR)
(MI, WR)
(VSTI, VHR)
(STI, HR)
C2
(1/AI, SR)
(EI, CR)
(1/SLI, SR)
(1/STI, VHR)
(EI, VSR)
(1/SLI, SR)
C3
(1/VSTI, VHR)
(SLI, SR)
(EI, CR)
(1/MI, VSR)
(MI, FR)
(SLI, FR)
C4
(1/MI, WR)
(STI, VHR)
(MI, VSR)
(EI, CR)
(CI, HR)
(VSTI, HR)
C5
(1/VSTI, VHR)
(EI, VSR)
(1/MI, FR)
(1/CI, HR)
(EI, CR)
(1/SLI, VSR)
C6
(1/STI, HR)
(SLI, SR)
(1/SLI, FR)
(1/VSTI, HR)
(SLI, VSR)
(EI, CR)
λmax=6.5962, Consistency index (CI) = 0.1192, Consistency ratio (CR) = 0.095.
Applying the Z-fuzzy AHP method in Section 4 the criteria weights have been obtained as in Table 9.
Criteria weights obtained by Z-fuzzy AHP method.
Reliability
Technical char.
Performance
Cost factors
Availability
Maintenance
0.353
0.046
0.118
0.355
0.074
0.053
After the DMs have compared the criteria, the evaluations of the alternatives according to the criteria have been collected. Tables 10–12 show the Z-fuzzy decision matrices including the linguistic evaluations of three DMs.
Z-fuzzy decision matrix of DM1.
C1
C2
C3
C4
C5
C6
A1
(MG, SR)
(VP, HR)
(VG, SR)
(F, HR)
(MG, FR)
(P, SR)
A2
(VG, FR)
(F, VHR)
(P, SU)
(G, VHR)
(P, WR)
(VG, SR)
A3
(MG, HR)
(MG, HR)
(G, HR)
(VG, FR)
(MP, SU)
(G, HR)
A4
(G, HR)
(G, SR)
(F, WR)
(P, SR)
(VG, HR)
(F, SU)
A5
(P, SR)
(VG, HR)
(VP, FR)
(G, HR)
(MG, HR)
(VG, HR)
Z-fuzzy decision matrix of DM2.
C1
C2
C3
C4
C5
C6
A1
(F, VHR)
(MP, VHR)
(MG, HR)
(G, SR)
(MG, SR)
(F, FR)
A2
(G, SR)
(G, WR)
(F, VWR)
(G, WR)
(P, HR)
(G, FR)
A3
(MP, SU)
(G, VSR)
(VG, FR)
(G, HR)
(G, HR)
(MG, SR)
A4
(VG, FR)
(VG, HR)
(G, HR)
(VP, HR)
(VG, SU)
(G, HR)
A5
(F, HR)
(G, SR)
(P, HR)
(MG, FR)
(MG, VHR)
(G, VSR)
Z-fuzzy decision matrix of DM3.
C1
C2
C3
C4
C5
C6
A1
(MP, HR)
(F, SR)
(G, FR)
(MG, SU)
(G,SU)
(MP, HR)
A2
(MG, WR)
(MG, FR)
(MP, HR)
(VG, FR)
(VP, VHR)
(VG, VHR)
A3
(G, FR)
(MG, SR)
(G, SR)
(G, SR)
(F, SU)
(F, WR)
A4
(F, HR)
(VG, FR)
(MG, FR)
(P, WR)
(G, SR)
(MG, SR)
A5
(MP, VHR)
(G, FR)
(F, CR)
(MG, SU)
(F, HR)
(G, WR)
In Step 3, the individual evaluations of DMs are aggregated by using geometric mean method given by Eqs. (25)–(26). The obtained aggregated matrix is presented in Table 13.
In Step 4, using the aggregated evaluations and Eqs. (27)–(28), the Z-fuzzy average values are calculated for both the restriction and reliability functions separately, and the resulting values are shown in Table 14.
In Step 5, Z-fuzzy PDA˜ and NDA˜ values are obtained for each alternative using Eqs. (29)–(34) and they are shown in Tables 15–16, respectively.
In Step 6, the criteria weights obtained in Section 4 by using Z-fuzzy AHP method are employed to find SP˜i and SN˜i values. They are given in Tables 17–18, respectively.
In Step 8, Z-fuzzy SSP˜i and SSN˜i values are normalized for both restriction and reliability functions separately by using Eqs. (41)–(44). The obtained NSP˜i and NSN˜i values are given in Tables 21–22, respectively.
In Step 10, Z-fuzzy AS˜i values are converted to ordinary fuzzy numbers using Definition 2. The obtained trapezoidal fuzzy numbers are shown in Table 24.
Trapezoidal fuzzy AS˜i values converted from Z-fuzzy AS˜i.
Trapezoidal fuzzyAS˜i values of alternatives
A1
(0.094, 0.307, 0.307, 0.462)
A2
(0.089, 0.227, 0.227, 0.380)
A3
(0.061, 0.214, 0.214, 0.321)
A4
(0.265, 0.389, 0.389, 0.574)
A5
(0, 0.226, 0.226, 0.39)
In Step 11, trapezoidal fuzzy AS˜i values are transformed to crisp numbers using Eq. (2). In Step 12, alternatives are ranked according to the decreasing values of crisp ASi. Crisp ASi values and ranking of the alternatives are presented in Table 25. A4 which has the highest ASi is the best choice among five alternatives. Based on the computed ASi values, the ranking of the alternatives is A4 > A1 > A2 > A5 >A3. These results show that alternative A4 is the best choice among the wind turbine alternatives according to the determined criteria.
Crisp ASi values.
Alternative
Crisp ASi
A1
0.2926
A2
0.2306
A3
0.2024
A4
0.4044
A5
0.2106
In order to investigate the importance of reliability information, the reliability judgments regarding all DMs’ evaluations have been accepted as “certainly reliable” when applying the Z-fuzzy EDAS method without changing the criteria weights. Then, Z-fuzzy EDAS method has been re-applied. The obtained ASi values are presented in the Table 26.
Crisp ASi values (DMs’ reliability judgments accepted as (1,1,1)).
Alternative
Crisp ASi
A1
0.2431
A2
0.2751
A3
0.3071
A4
0.5332
A5
0.2220
According to these results, when the reliability information is neglected (accepted as (1,1,1) for all evaluations), the ranking of all alternatives except for the alternatives A4 and A2 has changed. A4 alternative has been found as the best alternative again. Although the best alternative does not change, this difference shows that the reliability information should not be neglected. The fact that the ranking of the best alternative (A4) remains the same can be interpreted as the DMs stated their restriction judgments quite dominantly when comparing the alternative A4 with the other alternatives.
Similarly, while the Z-fuzzy AHP method has been applied to find the criteria weights, the reliability information has been accepted as “certainly reliable”, and the criteria weights have been recalculated. The obtained criteria weights are presented in Table 27.
Criteria weights obtained by Z-fuzzy AHP method (DMs’ reliability judgements accepted as (1,1,1)).
Reliability
Technical char.
Performance
Cost factors
Availability
Maintenance
0.396
0.049
0.119
0.328
0.066
0.042
Table 27 shows that the ranking of cost factor and reliability factor, which are in the first two rankings, have changed when compared to previous results (Table 9). Among the six criteria, only the rankings of the performance and availability factors have not changed. These results support the obtained result regarding the importance of reliability information as in the EDAS method.
Comparative Analysis Using Z-Fuzzy AHP&TOPSIS Methodology
To compare the results, the Z-fuzzy TOPSIS methodology proposed by Yaakob and Gegov (2016) is used. Z-fuzzy TOPSIS is one of the first fuzzy extensions which is performed by Z-fuzzy numbers in MCDM methodology. TOPSIS method was developed by Yoon and Hwang (1981). It is one of the most commonly used MCDM methodology by researchers in the literature. TOPSIS method allows to reach the solution by using the distances of the alternatives from the positive and negative ideal solutions.
Z-fuzzy TOPSIS methodology consists of the following steps; (i) construction of Z-fuzzy decision matrix, (ii) conversion of Z-fuzzy numbers to ordinary fuzzy numbers, (iii) normalization procedure, (iv) weighing the normalized decision matrix, (v) calculation of distances from positive and negative ideal solutions, and (vi) calculation of closeness coefficients (Yaakob and Gegov, 2016).
Table 28 presents the results of Z-fuzzy AHP&TOPSIS methodology and it shows the distances from positive and negative ideal solutions (d∗ and d−), and closeness coefficients (CC*), respectively. Based on the computed CC* values, the ranking of the alternatives is obtained as A4 > A1 > A2 > A3 > A5.
Results of Z-fuzzy TOPSIS methodology.
d∗
d−
CC*
A1
5.5530
1.4140
0.2030
A2
5.5551
1.3976
0.2010
A3
5.5572
1.3945
0.2006
A4
5.4034
1.5714
0.2253
A5
5.6267
1.3523
0.1938
According to the results obtained by the Z-fuzzy TOPSIS method, the ranking of the alternatives, except alternatives 3 and 5, is the same as the methodology proposed in this study. The comparison of the rankings can be seen in Table 29.
Comparison of Z-fuzzy EDAS and Z-fuzzy TOPSIS.
Alternatives
Ranking of Z-fuzzy EDAS
Ranking of Z-fuzzy TOPSIS
A1
2
2
A2
3
3
A3
5
4
A4
1
1
A5
4
5
EDAS method considers the positive and negative distances from the average solution rather than calculating the negative and positive ideal solutions as in TOPSIS method. According to the results of both methods, the closeness coefficients in Z-fuzzy TOPSIS are composed of quite closer values whereas appraisal scores in Z-fuzzy EDAS indicate larger differences between alternatives. In general, it can be concluded that the proposed method is consistent since the rankings of two methods are quite similar. The only difference is between alternatives A3 and A5. The first three best alternatives are the same in both methods.
As a result of the comparative analysis, obtaining similar results with the Z-fuzzy TOPSIS method shows the consistency and competitiveness of the proposed method.
Conclusion
Extensions of ordinary fuzzy sets are quite successful in modelling the uncertainty in the decision-making process. However, they do not exactly represent the reliability information inherent in the solutions. The reliability information of the evaluations is very important as it can have significant impacts on the obtained results. The Z-fuzzy numbers introduced by Zadeh (2011) allow the reliability of the DMs’ judgments to be included in the decision models. In this study, a novel Z-fuzzy EDAS method is introduced to the literature. Then, an integrated usage of Z-fuzzy AHP and Z-fuzzy EDAS method is proposed to the field for the first time to deal with uncertain expressions of DMs in real life decision making problems. The inclusion of the reliability information of the DMs in the decision model makes the decision making process more realistic in both daily and business decisions as in the case of renewable energy investment decisions.
The importance of renewable energy sources has increased considerably with the concern of leaving a sustainable world to future generations in recent years. In this study, the selection of a suitable wind turbine problem has been handled by considering the multiple factors affecting the decision. Criteria weights to be used in alternative selection have been calculated by using Z-fuzzy AHP method which has also been integrated to Z-fuzzy EDAS method. Z-fuzzy numbers integrated AHP method offers a more realistic solution by reflecting the DMs’ hesitancy in pairwise comparisons to the proposed Z-fuzzy AHP&EDAS methodology. After defining the criteria weights, three DMs have evaluated the five alternatives using Z-fuzzy EDAS method. All the DMs’ evaluations have been expressed by Z-fuzzy numbers in both methods, and all steps of the Z-fuzzy EDAS method have been performed by Z-fuzzy numbers. The proposed methodology allows DMs to express both restriction and reliability information about criteria and alternatives. In order to show the effects of reliability component on the decision system, the reliability information of all evaluations have been made “certainly reliable” and the calculations have been re-performed, then the results were compared with the proposed method. It is concluded from this analysis that the difference in the ranking results displays the importance of consideration of the reliability information. Therefore, the proposed methodology offers a more reliable evaluation system to DMs, including their degree of confidence to their assessments.
In order to show the robustness and stability of the proposed method, the obtained results have been compared with the results of the Z-Fuzzy AHP&TOPSIS methodology. It can be stated that the suggested methodology is an effective and useful method for researchers who want to make decisions based on distances from average solution rather than the distance from positive and negative ideal solutions. For further research, other MCDM approaches integrated with Z-fuzzy numbers can be used and compared with the results of this paper.
Although there are many fuzzy versions of the AHP method in the literature, its integration with Z-fuzzy numbers is limited. This research gap in the literature can be filled with increased application of Z-fuzzy AHP method, then importance and advantages of Z-fuzzy numbers can be further analysed. In addition, other fuzzy set extensions such as fermatean fuzzy sets or picture fuzzy sets can be used in the improvement of Z-fuzzy numbers. Then, in future research, it can be suggested to combine these extensions of Z-fuzzy numbers with different MCDM methods to expand the related literature.
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