INFORMATICAInformatica1822-88440868-49520868-4952Vilnius UniversityINFOR47610.15388/22-INFOR476Research ArticleNew Product Design Using Chebyshev’s Inequality Based Interval-Valued Intuitionistic Z-Fuzzy QFD Methodhttps://orcid.org/HaktanirElifelif.haktanir@altinbas.edu.tr12∗
E. Haktanır is currently a lecturer at Altinbas University. She received her MSc and PhD degrees in industrial engineering from Istanbul Technical University. Her research interests are fuzzy decision making, multi-criteria decision making, statistical decision making, quality control and management, and new product development. She is an organization committee member of International Conference on Intelligent and Fuzzy Systems, INFUS. Her refereed articles have appeared in a variety of journals including Computers & Industrial Engineering, Journal of Intelligent & Fuzzy Systems, Journal of Multiple-Valued Logic & Soft Computing.
KahramanCengizkahramanc@itu.edu.tr1
C. Kahraman received his MSc and Phd degrees in industrial engineering from Istanbul Technical University. He is on the editorial board of some journals such as International Journal of Computational Intelligence Systems (Atlantis Press), Journal of Enterprise Information Management (Emerald), New Mathematics and Natural Computation (World Scientific), and Human and Ecological Risk Assessment (Taylor And Francis). He has also been the guest editor of special issues of some international journals such as Information Sciences (Elsevier), Journal of Enterprise Information Management (Emerald), International Journal of Approximate Reasoning (Elsevier), Human and Ecological Risk Assessment, and Stochastic Environmental Research and Risk Assessment. He is the editor of the Springer books Fuzzy Applications in Industrial Engineering, Fuzzy Multi-Criteria Decision Making: Theory and Applications with Recent Developments, Fuzzy Engineering Economics with Applications, Intelligence Systems in Environmental Management: Theory and Applications, Computational Intelligence Systems in Industrial Engineering, Fuzzy Statistical Decision-Making – Theory and Applications, Production Engineering and Management under Fuzziness, Fuzzy Logic in Its 50th Year: New Developments, Directions and Challenges, Supply Chain Management Under Fuzziness Recent Developments and Techniques, Intelligent Techniques in Engineering Management Theory and Applications, and Intelligent Decision Making in Quality Management Theory and Applications.
In Quality function deployment (QFD) approach, customers tend to express their needs in linguistic terms rather than exact numerical values and these needs generally contain vague and imprecise information. To overcome this challenge and to use the method more effectively for complex customer-oriented design problems, this paper introduces a novel intuitionistic Z-fuzzy QFD method based on Chebyshev’s inequality (CI) and applies it for a new product design. CI provides the assignment of a more objective reliability function. The reliability value is based on the maximum probability obtained from CI. Then, the expected values of lower and upper bounds of interval-valued intuitionistic fuzzy (IVIF) numbers are determined. A competitive analysis among our firm and competitor firms and an integrative analysis for the different functions of QFD is presented. The proposed Z-fuzzy QFD method is applied to the design and development of a hand sanitizer for struggling with COVID-19.
quality function deploymentinterval-valued intuitionistic fuzzy setsZ-fuzzy numbersChebyshev’s inequalitynew product designIntroduction
With each passing day, customers’ expectations of the product that they are planning to purchase are increasing. Today, manufacturers and service providers must meet customer demands at the maximum level in order to be successful and maintain their continuity. Their competitive advantage depends on the aesthetic success of the product they offer for sale as well as the technical features. Customers generally expect the product to be affordable, durable, easy to use and appealing to the eye. However, it is difficult, even impossible sometimes, for the producers to meet all these demands at the same time due to economical and timewise limitations. Companies must first prioritize customer needs in order to determine the best product they can produce using their competencies and the maximum customer demands they can respond to. One of the most used methods for this purpose is Quality Function Deployment (QFD).
House of Quality (HOQ) is a special and mostly used part of QFD which is named for its shape that reminds of a house with a roof on top. A classical HOQ consists of some parts in matrix form such as customer demands (CDs), customer evaluations (CEs) of those demands, technical descriptors (TDs), relationship matrix between CDs and TDs, and correlation matrix among TDs. In some recent studies, new matrices are added eligibly to the common parts such as technical difficulty and direction of improvement of TDs, and competitive analysis for both CDs and TDs. The HOQ matrices are generally constructed by an effort of a team of experts and multiple customers. Since humans tend to express their thoughts and ideas linguistically rather than exact and precise numbers, this brings vagueness and impreciseness to the design and development process. To overcome this obstacle and deal with complex problems more realistically, the fuzzy set theory has been applied successfully for decades.
The fuzzy set theory was introduced in the literature by Zadeh (1965) as ordinary fuzzy sets which are represented by an x value and its membership degree. Later, in 1986, intuitionistic fuzzy sets (IFSs) have been developed as a generalization of Zadeh’s ordinary fuzzy sets by Atanassov (1986) which involve the degrees of membership and non-membership together with experts’ hesitancies for an x value. Later, neutrosophic sets are introduced in the literature by Smarandache (1998) which consist of three components truthiness, indeterminacy, and falsity where these components can be assigned independently. Pythagorean fuzzy sets are developed by Yager (2013) and allowed the squared sum of the membership and non-membership degrees to be at most one. Picture fuzzy sets (PiFS) have been developed by Cuong (2015) in order to define a fuzzy set by membership, non-membership, and hesitancy degrees so that their squared sum is at most equal to one. As an extension of PiFs, Kutlu Gündoğdu and Kahraman (2019) developed the spherical fuzzy sets that the squared sum of three components (membership, non-membership, and hesitancy degrees) to be between zero and one. One of the latest extensions of intuitionistic fuzzy sets is circular intuitionistic fuzzy sets developed by Atanassov (2020). They add the uncertainty of the membership and non-membership degrees by defining a circle with radius “r” for these values.
In this paper IVIFSs are employed in the proposed QFD method taking into consideration the reliability of the assigned IVIF numbers. The reliability in this method is handled by Z-fuzzy numbers developed by Zadeh (2011). Z-fuzzy number is an ordered pair of fuzzy numbers where the first component is a real-valued uncertain variable as a restriction on the values. The second component is a measure of reliability for the first component. Z- fuzzy numbers are used to make computations with fuzzy numbers which are not totally reliable. A Z-fuzzy number can represent the information about an uncertain variable, whose first component represents a value of the variable, and the second component represents an idea of uncertainty or probability. In other words, the second component shows how sure the decision maker is with the first component (Yaakob and Gegov, 2015). Chebyshev’s inequality is employed to calculate the maximum probability to determine the expected values of lower and upper bounds of the IVIF number in the first component. Thus, we obtain more realistic and objective results compared to classical Z-fuzzy approaches.
The advantage of our study and its contribution to the literature can be explained as follows. In most of the Z-fuzzy number studies, sufficient details on how to construct the reliability function are not presented. This study scientifically explains how to create the reliability function and integrate it into the restriction function with the help of Chebyshev’s theory. Obtaining the extreme values in IVIF numbers through the integration of reliability factor is realized by using probability theory. Therefore, this paper offers a very different Z-fuzzy number idea from Zadeh’s classical Z-fuzzy proposal. The advantage of our method is that it presents the QFD approach under intuitionistic fuzziness with all its aspects such as technical difficulty, competitive analysis through CDs and TDs.
The rest of this study is organized as follows. Section 2 presents a literature review on fuzzy QFD (F-QFD). Section 3 gives the preliminaries for intuitionistic Z-fuzzy numbers based on Chebyshev’s inequality. Section 4 develops the intuitionistic Z-fuzzy QFD method based on Chebyshev’s inequality. Section 5 illustrates the application of the proposed model on a new hand sanitizer design and development. Section 6 concludes the paper with discussions and future directions.
Literature Review
A literature review on F-QFD based on Scopus database gives a list of 185 publications. Figure 1 shows the distribution of the F-QFD publications with respect to years.
Distribution of the F-QFD publications with respect to years.
Document type distributions of F-QFD publications.
Document type distributions of F-QFD publications.
After the first study on F-QFD was published in 1998, the highest publication rate was attained in 2019 with 25 studies.
As given in Fig. 2, most of the F-QFD studies are in article form which is followed by conference papers and book chapters.
F-QFD has been applied to many subject areas. Figure 3 shows the frequencies of these publications. Engineering, computer science, and business, management and accounting are the most frequently applied subjects, respectively.
Some representative F-QFD studies are presented in Table 1 together with the type of fuzzy sets used, integrated methods, and application areas.
We can conclude at the end of the literature review that TFNs are used more than other types of fuzzy numbers. The most integrated methods with F-QFD are AHP, ANP, TOPSIS, FMEA, and DM, respectively. The most used extensions of ordinary fuzzy sets with F-QFD are IFNs, HFNs, T2FNs and SFNs, respectively. The application areas of F-QFD are quite different from delivery drone design to choosing the ideal gas fuel at wastewater treatment plants. A focused application area of F-QFD is not observed in this comprehensive literature review.
Chebyshev’s Inequality Based IV-Intuitionistic Z-Fuzzy Numbers
Some representative F-QFD studies.
Authors (year)
Type of fuzzy sets
Integrated methods
Application area
1
Haktanır et al. (2021)
SFNs
–
Delivery drone design
2
Lee and Park (2021)
TFNs
–
Prioritization of work activities of construction for safety
3
Efe et al. (2020)
IT2FNs
TOPSIS
Mobile phone selection
4
Baskar et al. (2020)
TFNs
DM, ISM, ANP, VIKOR, FMEA
Sesame seed separator development
5
Kang (2020)
TFNs
RST
Aesthetic product design
6
Bhuvanesh Kumar and Parameshwaran (2020)
TFNs
FMEA, AHP
Prioritizing lean tools for manufacturing industries
7
Ocampo et al. (2020)
TFNs
AHP, DEMATEL, ANP
Sustainable product design
8
Wang et al. (2020)
TFNs
GDM
Supply chain collaborative quality design of large complex products
9
Aouag et al. (2020)
TFNs
DEMATEL
Enhancement of value stream mapping application process
10
Büyüközkan et al. (2020)
TFNs
AHP
Customer oriented multifunctional power bank design
11
Kutlu Gündoğdu and Kahraman (2020)
SFNs
–
Linear delta robot technology development
12
Seker (2020a)
TFNs
AHP
Retail chain
13
Li et al. (2020)
TFNs
GOA, DM, ML
Analysis and extraction of consumer information for the evaluation of design requirement
14
Büyüközkan and Uztürk (2020)
IVIFNs
MCDM
Smart fridge design
15
Seker (2020b)
TFNs
–
Smart phone product design
16
Fan et al. (2020)
IFNs
ANP
Optimal selection of design scheme in cloud environment
17
Haktanır (2020)
IVPFSs
COPRAS
Prioritization of competitive suppliers
18
Deveci et al. (2019)
IVIFNs
PCA
Evaluation of service quality in public bus transportation
19
Kayapınar and Erginel (2019)
TFNs
SERVQUAL, MODM
Designing the airport service
20
Haktanır and Kahraman (2019)
IVPFSs
–
Solar photovoltaic technology development
21
Beheshtinia and Farzaneh Azad (2019)
TFNs
SERVQUAL, KANO
Budget constraint for hotel services
22
Lu et al. (2019)
TFNs
AHP, ANP
Design of brand revitalisation
23
Bilişik et al. (2019)
TFNs
–
Passenger satisfaction evaluation of public transportation in Istanbul
24
Ma et al. (2019a)
TFNs
FMEA
Identification of to-be-improved components for redesign of complex products and systems
25
Wang et al. (2019)
TFNs
AHP, MAM
Design and implementation of a hand training device
26
Wang (2019)
IFNs
AHP
Product design: case study on touch panels
27
Senthilkannan and Parameshwaran (2019)
TFNs
DM, AHP, FMEA, TOPSIS
Performance analysis and quality improvement in paper industry
28
Piengang et al. (2019)
TFNs
AHP, VIKOR
An APS software selection methodology
29
Ma et al. (2019b)
TFNs
FMEA
Identifying function components for product redesign
30
Fitriana et al. (2019)
TpFNs
DMM
Measurement and proposal of improving marketing process to improve the quality of aftersales in OV agency
31
Yazdani et al. (2019)
IVTFNs
GRA
Multi attribute decision support model in a supply chain
32
Jafarzadeh et al. (2018)
TFNs
DEA
Project portfolio selection
33
Shuofang et al. (2018)
TFNs
EGM
Study methods of design elements
34
Osorio-Gómez and Manotas-Duque (2018)
TFNs
TOPSIS
Dispatching prioritization in maritime transportation considering operational risk
35
Osiro et al. (2018)
HFNs
–
Selecting supply chain sustainability metrics
36
De Almeida et al. (2018)
TFNs
ANP
New defense product development
37
Bhuvanesh Kumar and Parameshwaran (2018)
TFNs
FMEA
Selection of lean tools in a manufacturing organization
38
Milunovic Koprivica and Filipovic (2018)
TFNs
–
Improvement of boiler (house electric water heater)
39
Yu et al. (2018)
IVIFNs
CIM
Process of designing steering wheel for electric vehicles
40
Babbar and Amin (2018)
TpFNs
–
Supplier selection and order allocation in beverages industry
41
Liu et al. (2018)
TFNs
EGM, AHP
The importance of customer requirements and design elements and the correlation among various design elements
42
Amaladhasan et al. (2018)
TFNs
TOPSIS
Analysis and prioritisation of eco drivers in supply chain
43
Kang et al. (2018)
TFNs
EGM, KANO, AHP
New product development
44
Vongvit et al. (2017)
TFNs
TRIZ
Methodology for product development involving design of a 5-axis CNC machine from a 3-axis CNC machine
45
Liu et al. (2017)
TFNs
DSM
Process optimization of customer collaborative design
46
Chiadamrong and Tham (2017)
TFNs
SEM, MOLPM
Supply chain management strategy development
47
Akbaş and Bilgen (2017)
TFNs
TOPSIS, ANP, AHP
Choosing the ideal gas fuel at wastewater treatment plants
48
Keshteli and Davoodvandi (2017)
TFNs
AHP, TOPSIS
Ceramic and tile industry of Iran
49
Haq and Boddu (2017)
TFNs
AHP, TOPSIS
Analysis of enablers for the implementation of leagile supply chain management
50
Vinodh et al. (2017)
TFNs
–
Sustainable design of consumer electronics products
51
Çevik Onar et al. (2016)
HFNs
AHP, TOPSIS
Computer workstation selection
52
Rattawut (2016)
TFNs
AHP
Mini-CNC milling machine retrofit
53
Hakim et al. (2016)
TFNs
MOGP
Selecting processes in business process reengineering
54
Chowdhury and Quaddus (2016)
TFNs
MPOM
Sustainable service design
55
Chen (2016)
TFNs
DT
Green design quality management in industrial chain
56
Büyüközkan and Güleryüz (2015)
TFNs
GDM
IT planning in collaborative product development
57
Dat et al. (2015)
TFNs
TOPSIS
Market segment evaluation and selection
58
Xiao et al. (2015)
TpFNs
–
Identification of software non-functional requirement
59
Mohanraj et al. (2015)
TFNs
VSM
Framework for value stream mapping in an Indian camshaft manufacturing organization
60
Raut and Mahajan (2015)
TFNs
AHP
Construction industry
61
Noorul Haq and Boddu (2015)
TFNs
TOPSIS
Leanness in supply chain
62
Roghanian and Alipour (2014)
TFNs
AHP, PROMETHEE
Achieving lean attributes for competitive advantages development
63
Zaim et al. (2014)
TFNs
ANP
Product development
64
Jamalnia et al. (2014)
TpFNs
MOGP
Global facility location-allocation problem
65
Palanisamy and Zubar (2013)
TFNs
MM, ANP
Vendor ranking
66
Taylan (2013)
TFNs
GRA, FIS
Determining multi attribute customer preferences of edible oil
67
Yang et al. (2013)
TFNs
–
Design for remanufacturing
68
Tavana et al. (2013)
TFNs
ANP
Balanced scorecard
69
Nejatian and Zarei (2013)
TFNs
TOPSIS
Improving organizational agility
70
Bevilacqua et al. (2012)
TFNs
–
Characterizing customers rating of extra virgin olive oil
71
Chang (2012)
TFNs
TRIZ
Teaching quality improvement
72
Lee et al. (2012)
TFNs
FDM
Customer needs and technology analysis in new product development
73
Vinodh and Chintha (2011)
TFNs
–
Enabling sustainability
74
Chen and Huang (2011)
TFNs
–
Knowledge management
75
Kavosi and Mavi (2011)
TFNs
TOPSIS, AHP
Product design and development (pen company in Iran)
76
Khademi-Zare et al. (2010)
TFNs
TOPSIS, AHP
Ranking the strategic actions of Iran mobile cellular telecommunication
77
Yang et al. (2010)
TFNs
DMAIC, FMEA
Problem selection in the 6σ definition stage
78
Liu (2009)
TFNs
FMEA
Extension fuzzy QFD from product planning to part deployment
79
Juan et al. (2009)
TFNs
PROMETHEE
Housing refurbishment contractor selection
80
Celik et al. (2009)
TFNs
AHP, FAD
Routing of shipping investment decisions in crude oil tanker market
81
Mousavi et al. (2008)
TFNs
TOPSIS
Bridge scheme selection
82
Su and Lin (2008)
TFNs
TRIZ
Service quality improvement
83
Wang et al. (2007)
TFNs
–
Customizing positioning of logistics service products of 3PLS
84
Kahraman et al. (2006)
TFNs
ANP, AHP
Improving product design and quality in a Turkish company producing PVC window and door systems
85
Hong and Wang (2005)
TFNs
–
Developing an integrated service strategy
86
Tsai et al. (2003)
TFNs
–
Enhancing manufacturing strategic planning
87
Sohn and Choi (2001)
TFNs
–
Supply chain management with reliability consideration
88
Verma et al. (1998)
TFNs
–
Facilitating strategic product planning, early design decision-making and parameter target setting
Integrated methods abbreviations: Analytic Hierarchy Process (AHP), Analytic Network Process (ANP), Choquet Integral Method (CIM), COmplex PRoportional ASsessment (COPRAS), Data Envelopment Analysis (DEA), Data Mining Methods (DMM), Decision Making Trial and Evaluation Laboratory (DEMATEL), Decision Tree (DT), Define-Measure-Analyze-Improve-Control (DMAIC), Delphi Method (DM), Design Structure Matrix (DSM), Evaluation Grid Method (EGM), Failure Mode and Effects Analysis (FMEA), Fuzzy Axiomatic Design (FAD), Fuzzy Delphi Method (FDM), Fuzzy Inference System (FIS), Grey Decision-Making Approach (GDM), Grey Relational Analysis (GRA), Group Decision Making Approach (GDM), Group-Organization Approach (GOA), Interpretive Structural Modelling (ISM), KANO, Machine Learning (ML), Mathematical Modelling (MM), Morphological Analysis Method (MAM), Multi-Objective Decision Model (MODM), Multi-Objective Goal Programming (MOGP), Multi-Objective Linear Programming Model (MOLPM), Multi-Phased 0-1 Optimization Model (MPOM), Multiple-Criteria Decision-Making (MCDM), Preference Ranking Organization METHod for Enrichment Evaluation (PROMETHEE), Principal Component Analysis (PCA), Rough Set Theory (RST), Service Quality (SERVQUAL), Structural Equation Modelling (SEM), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Theory of Inventive Problem Solving (TRIZ), Value Stream Mapping (VSM), VIekriterijumsko KOmpromisno Rangiranje (VIKOR).
In this section, we first present the preliminaries of single-valued intuitionistic fuzzy (SVIF) and IVIF sets with some of their arithmetic operations. Then, ordinary Z-fuzzy numbers are introduced. And finally, Chebyshev’s inequality-based interval-valued intuitionistic Z-fuzzy numbers are developed.
Preliminaries
Ordinary fuzzy sets are defined as in Eq. (1) (Zadeh, 1965):
A˜={(x,μ(x))|x∈X},
where the universe is X, and 0⩽μ(x)⩽1.
Intuitionistic fuzzy sets (IFSs) are defined as in Eq. (2) (Atanassov, 1986):
A˜={⟨u,(μA(u),vA(u)⟩|u∈U},
where μA:U→[0,1], vp:U→[0,1] and 0⩽μA(u)+vA(u)⩽1. For any IFS A˜ and u∈U, πA=1−μA(u)−vA(u) gives the hesitancy degree.
The addition, multiplication of two SVIF numbers, multiplication by a scalar, and power operations on SVIF numbers are presented as in Eqs. (3)–(6), respectively (Atanassov, 1994): A˜⊕B˜=(μA+μB−μAμB,vAvB),A˜⊗B˜=(μAμB,vA+vB−vAvB),αA˜=(1−(1−μA)α,vAα),A˜α=(μAα,1−(1−vA)α), where α is a real value and α>00$]]>.
The score function of SVIF numbers is presented in Eq. (7) (Zhang et al., 2012):
SA(x)=1−vA(x)2−μA(x)−vA(x).
Let closed subintervals be represented by D⊆[0,1]. An IVIFS A˜ over X is defined as in Eq. (8) (Büyüközkan and Uztürk, 2020):
A˜={⟨x,μA(x),vA(x)⟩|x∈X},
where
μA˜→D⊆[0,1],vA˜(x)→D⊆[0,1]
with the condition 0⩽supμA˜(x)+supvA˜(x)⩽1, ∀x∈X.
The lower and upper end points are represented by the symbols μA˜L(x), μA˜U(x), vA˜L(x), and vA˜U(x), respectively. Then, an IVIFS A˜ is given by Eq. (9) (Büyüközkan and Uztürk, 2020):
A˜={⟨x,[μA˜L(x),μA˜U(x)],[vA˜L(x),vA˜U(x)]⟩|x∈X},
where 0⩽μA˜U(x)+vA˜U(x)⩽1, μA˜L(x)⩾0, vA˜L(x)⩾0.
For any x, the hesitancy degree can be computed by Eq. (10):
πA˜(x)=1−μA˜(x)−vA˜(x)=([1−μA˜U(x)−vA˜U(x)],[1−μA˜L(x)−vA˜L(x)]).
For convenience, let μA˜(x)=[μL,μU], vA˜(x)=[vL,vU], so A˜=([μL,μU],[vL,vU]).
Let A˜=([μL,μU],[vL,vU]) be an IVIF number. The following score function is proposed for defuzzifying A˜ (Karasan and Kahraman, 2019):
I(A˜)=μL+μU+(1−vL)+(1−vU)+μL×μU−(1−vL)×(1−vU)4.
Classical Z-Fuzzy Numbers
A Z-fuzzy number is defined by Zadeh (2011) as an ordered pair of fuzzy numbers, (A˜,R˜) which includes a restriction function A˜ and a reliability function R˜ representing the reliability level of the restriction function. If a fuzzy number is not totally reliable, Z-fuzzy numbers can provide a systematic approach to increase the reliability of that fuzzy number.
A Z-fuzzy number can be defined as in Fig. 4.
A Z-fuzzy number.
The expected value of a fuzzy set is calculated as in Eq. (12) (Zadeh, 2011):
EA˜(x)=∫xxμA˜(x)dx,
where A˜ is defined as A˜={⟨x,μA˜(x)⟩|x∈X}, and μA˜:X→[0,1].
Consider a Z-fuzzy number Z=(A˜,R˜), which is described as in Fig. 4. Let A˜={⟨x,μA˜(x)⟩|μ(x)∈[0,1]} and R˜={⟨x,μR˜(x)⟩|μ(x)∈[0,1]} (Zadeh, 2011).
The triangular fuzzy reliability function can be converted into a classical number by Eq. (13):
α=∫xμR˜(x)dx∫μR˜(x)dx.
Then, the result of Eq. (13) is integrated with the trapezoidal fuzzy restriction function as in Eq. (14):
Z˜α={⟨x,μA˜α(x)⟩|μA˜α(x)=αμA˜(x),μ(x)∈[0,1]}.
After applying Eq. (14), the Z-fuzzy number becomes a single ordinary fuzzy number as in Fig. 5.
Z-fuzzy number converted into a single ordinary fuzzy number.
In the next section, ordinary Z-fuzzy numbers will be extended by a new approach using Chebyshev’s inequality. In this approach, reliability component of the Z-fuzzy number is calculated more objectively based on Chebyshev’s probability terms.
Chebyshev’s Inequality Based IV-Intuitionistic Z-Fuzzy Numbers
Chebyshev’s inequality provides the maximum probability between two points with a given mean and variance as illustrated in Fig. 6 when the distribution of the considered data is not known. Let’s assume that μ=E(X)∈R and σ=sd(X)∈(0,∞), where X is a random variable.
Chebyshev’s inequality is given in Eq. (15):
P(|X−μ|⩾kσ)⩽1k2,k>0,0,\]]]>
where k determines the distance from the population mean as in Fig. 6.
Chebyshev’s inequality.
Assume that n number of linguistic evaluations is given as A˜={E1,E2,…,En}, each is represented by an interval-valued intuitionistic fuzzy number. Let the arithmetic mean of the lower and upper values of the membership degrees be μx‾L and μx‾U, respectively. Similarly, let the lower and upper values of non-membership degrees be vx‾L, and vx‾U, respectively. Then let the standard deviation of the lower and upper values of the membership degrees be μσL and μσU, respectively, whereas let the lower and upper values of non-membership degrees be vσL, and vσU, respectively.
Next operation is to find k value in Eq. (15) in a way that the maximum reliability Rmax of the lower and upper values of membership and non-membership degrees is obtained. In this operation the k value must satisfy that x‾−kS=0 and/or x‾+kS=1. Then maximum reliability is calculated by Rmax=1−1/k2 for each lower and upper values of membership and non-membership degrees to be RmaxμL, RmaxμU, RmaxvL, and RmaxvU, respectively. Thus, the maximum reliability level becomes maximum between RmaxμL and RmaxμU and between RmaxvL and RmaxvU. Then the expected value of the IVIF number is obtained by Eqs. (16)–(19): E[μL]=[(x‾μL−kμLSμL)×RmaxμL,(x‾μL+kμLSμL)×RmaxμL]=[μLL,μLU],E[μU]=[(x‾μU−kμUSμU)×RmaxμL,(x‾μU+kμUSμU)×RmaxμL]=[μUL,μUU],E[vL]=[(x‾vL−kvLSvL)×RmaxvL,(x‾vL+kvLSμL)×RmaxvL]=[vLL,vLU],E[vU]=[(x‾vU−kvUSvU)×RmaxvL,(x‾vU+kvUSvU)×RmaxvL]=[vUL,vUU]. The IVIF number ([E[μL],E[μU]],[E[vL],E[vU]]) is converted to a SVIF number by Eq. (20) for membership interval and Eq. (21) for non-membership interval, respectively. D([E[μL],E[μU]])=E[μLL]+E[μLU]+(1−E[vLL])+(1−E[vLU])+E[μLL]×E[μLU]−(1−E[vLL])×(1−E[vLU])4,D([E[vL],E[vU]])=E[μUL]+E[μUU]+(1−E[vUL])+(1−E[vUU])+E[μUL]×E[μUU]−(1−E[vUL])×(1−E[vUU])4. Thus, SVIF number (D(E[μ]),D(E[v])) is obtained.
Intuitionistic Z-Fuzzy QFD Based on Chebyshev’s Inequality
In this section, we present our novel Chebyshev’s inequality based intuitionistic Z-fuzzy QFD approach. The proposed approach requires the number of experts to be ne and the number of customers to be nc that we interviewed. The steps of the proposed approach are composed of two phases and 10 steps in total, each is presented in detail below. The phase of customer demands (CDs) and technical descriptors (TDs) relation analysis and the phase of competitive analysis are the two main phases of the approach.
Phase 1 – CD&TD Relation Analysis
Step 1: Let nc number of customers define the linguistic CDs and assign the linguistic customer evaluations using the scale in Table 2. The total number of CDs is T. Then, translate the linguistic customer evaluations into IVIF values by using Table 2 and aggregate by using Eqs. (20)–(21). Here, customers’ weights (wc) can be assigned differently. This is realized by Eqs. (22)–(25) which require the weighted mean and the weighted standard deviation of the assigned customer evaluations, respectively. This is applied for each element of T number of CDs. Please note that after the aggregation operations, the IVIF values are turned into SVIF values which is to decrease the vagueness. x‾t=∑i=1ncwcixiμLnc,St=∑i=1ncwci(xiμL−x‾)2(M−1)M∑i=1ncwci,t=1,2,…,T,x‾t=∑i=1ncwcixiμUnc,St=∑i=1ncwci(xiμU−x‾)2(M−1)M∑i=1ncwci,t=1,2,…,T,x‾t=∑i=1ncwcixivLnc,St=∑i=1ncwci(xivL−x‾)2(M−1)M∑i=1ncwci,t=1,2,…,T,x‾t=∑i=1ncwcixivUnc,St=∑i=1ncwci(xivU−x‾)2(M−1)M∑i=1ncwci,t=1,2,…,T, where nc is the number of customers; M is the number of non-zero weights; wci is the weight of customer i; xiμL, xiμU, xivL, xivU are the corresponding lower and upper membership and non-membership degrees of customer evaluations, respectively.
Step 2: Let the ne number of experts define the TDs. The total number of TDs is S. Then translate their linguistic assessments for the CD-TD relationship matrix into IVIF numbers by using Table 2. Experts’ weights (we) can be assigned differently depending on our trust in their experiences. Next, aggregate each IVIF relation to a SVIF number by using Eqs. (20)–(21). Eqs. (26)–(29) are used to calculate the weighted mean and the weighted standard deviation of the assigned relations, respectively. This is applied for each element of S number of TDs. Please note that after the aggregation operations, the IVIF values are turned into SVIF values which is to decrease the vagueness. x‾s=∑i=1neweixiμLne,St=∑i=1newei(xiμL−x‾)2(M−1)M∑i=1newei,s=1,2,…,S,x‾s=∑i=1newexiμUne,St=∑i=1newei(xiμU−x‾)2(M−1)M∑i=1newei,s=1,2,…,S,x‾s=∑i=1newexivLne,St=∑i=1newei(xivL−x‾)2(M−1)M∑i=1newei,s=1,2,…,S,x‾s=∑i=1neweixivUne,St=∑i=1newei(xivU−x‾)2(M−1)M∑i=1newei,s=1,2,…,S.
Linguistic and corresponding numerical scale for the weights of criteria.
Medium Low Importance (MLI) / Medium Low Satisfactory (MLS) / Medium Low Relation (MLR) / Medium Low Difficulty (MLD)
([0.3,0.4],[0.5,0.6])
Approximately Equal Importance (AEI) / Approximately Equal Satisfactory (AES) / Approximately Equal Relation (AER) / Approximately Equal Difficulty (AED)
([0.4,0.5],[0.4,0.5])
Medium High Importance (MHI) / Medium High Satisfactory (MHS) / Medium High Relation (MHR) / Medium High Difficulty (MHD)
([0.5,0.6],[0.3,0.4])
High Importance (HI) / High Satisfactory (HS) / High Relation (HR) / High Difficulty (HD)
([0.6,0.7],[0.2,0.3])
Very High Importance (VHI) / Very High Satisfactory (VHS) / Very High Relation (VHR) / Very High Difficulty (VHD)
([0.7,0.8],[0.1,0.2])
Absolutely High Importance (AHI) / Absolutely High Satisfactory (AHS) / Absolutely High Relation (CHR) / Absolutely High Difficulty (AHD)
([0.8,0.9],[0.0,0.1])
Step 3: Let the experts determine the level of technical difficulty of the TDs by using the scale given in Table 2. The weights of the experts are accepted to be the same as Step 2 and similar calculations are applied to find the aggregated SVIF values for each TDs’ technical difficulty as in Step 2.
Step 4: Construct the correlation matrix among TDs based on the IVIF scale presented in Table 3. In this matrix two types of correlations are considered: positive and negative. Positive correlations and negative correlations are indicated by PC and NC, respectively. PC means that two TDs move to the same direction whereas NC means that two TDs move to the opposite directions whenever the value of one of these two TDs is changed. When there exists no correlation, the cell includes no linguistic value in the correlation matrix. The differences between PCs and NCs are obtained by Eq. (31).
Step 5: Obtain the Chebyshev’s inequality-based absolute priority degree (AP˜C) for each TD as in Eq. (30):
AP˜ijC={(⨁i=1TCE˜iC⊗RM˜jC)⊗(1+CC˜jC)}⊘(1+RTDF˜jC),(j=1,2,…,S),
where CE˜C: aggregated linguistic customer evaluations of CDs; RM˜C: aggregated linguistic terms in the relationship matrix; and CC˜C: the aggregated correlation correction factor. CC˜jC in Eq. (30) is calculated by Eq. (31).
CC˜jC=(nccj/(S−1))×(PC‾˜j⊖NC‾˜j),
where −1˜⩽CC˜jC⩽+1˜; nccj: correlation number of TDj with the other TDs; PC‾˜j: average value of the PCs for the considered TDj; and NC‾˜j: average value of the NCs for the considered TDj.
IVIF correlation scale.
Linguistic term for positive or negative correlations
Very Low Positive Correlation (VLPC) or Very Low Negative Correlation (VLNC)
([0.1,0.2],[0.7,0.8])
Low Positive Correlation (LPC) or Low Negative Correlation (LNC)
([0.2,0.3],[0.6,0.7])
Medium Low Positive Correlation (MLPC) or Medium Low Negative Correlation (MLNC)
([0.3,0.4],[0.5,0.6])
Approximately Equal Positive Correlation (AEPC) or Approximately Equal Negative Correlation (AENC)
([0.4,0.5],[0.4,0.5])
Medium High Positive Correlation (MHPC) or Medium High Negative Correlation (MHNC)
([0.5,0.6],[0.3,0.4])
High Positive Correlation (HPC) or High Negative Correlation (HNC)
([0.6,0.7],[0.2,0.3])
Very High Positive Correlation (VHPC) or Very High Negative Correlation (VHNC)
([0.7,0.8],[0.1,0.2])
Absolutely High Positive Correlation (AHPC) or Absolutely High Negative Correlation (AHNC)
([0.8,0.9],[0.0,0.1])
Relative technical difficulty (RTDF˜C) in Eq. (30) is calculated as in Eq. (32):
RTDF˜jC=TDF˜jC⊘(⨁j=1STD˜FjC),
where technical difficulty (TDF˜C) indicates the difficulty of an organization to reach the planned level of TD. Our objective is to decrease the impact of TDs whose technical difficulties are bigger. Smaller AP˜j are caused by bigger TDF˜jC values.
Fuzzy relative absolute priority (RAP˜ijC) values are found by Eq. (33):
RAP˜ijC=AP˜ij⊘(⨁j=1SAP˜ij),i=1,2,…,T.
Since division and subtraction operations for SVIF numbers are not clearly defined in the literature, defuzzification is employed for these arithmetic operations in our calculations.
Step 6: Rank the TDs regarding their RAP˜ijC values. The highest RAP˜ijC shows the TD with the highest priority for the product developers to consider in the new product design and development phase.
Phase 2 – Competitive Analysis
Step 7: Determine the customers’ linguistic assessments for the competitive analysis through CDs assigned by nc number of customers using the IVIF scale given in Table 2. To locate the position of our company among the competitors whose number is y, the customer assessments should be first aggregated with regarding the corresponding CDs. Next, the distances between our company and other companies (D˜O−CℓCD) are calculated by using Eq. (34):
D˜O−CℓCD=⨁i=1T(κO−CℓCD×diCD(O,Cℓ)×CE˜iC),ℓ=1,…,y;i=1,…,T,
where O and Cℓ represent our company and competitor ℓ, respectively. CE˜i is the aggregated customer evaluations with respect to the corresponding CDi.
κO−CellCD in Eq. (32) is defined as in Eq. (35):
κO−CℓCD=+1,ifOis better thanCℓ,−1,ifCℓis better thanO,0,ifOis equal toCℓ,ℓ=1,…,ydiCD(O,Cℓ) in Eq. (34) is calculated by Eq. (36):
diCD(O,Cℓ)=12(μO−μCℓ)2+(vO−vCℓ)2+((1−μO−vO)−(1−μCℓ−vCℓ))2,ℓ=1,…,y;i=1,…,T.
Step 8: Find the linguistic customer assessments of the competitive analysis through TDs assigned by ne number of experts using the IVIF scale given in Table 2. To locate the position of our company among the competitors, the expert assessments should be first aggregated with regarding the corresponding TDj. Next, the distances between our company and other companies (D˜O−CℓTD) are calculated by using Eq. (37):
D˜O−CℓTD=⨂j=1S(κO−CℓTD×djTD(O,Cℓ)×AP˜ijC),ℓ=1,…,y;i=1,…,T;j=1,…,S,
where O and Cℓ represent our company and competitor ℓ, respectively.
κO−CℓTD in Eq. (37) is defined as in Eq. (38):
κO−CℓTD=+1,ifOis better thanCℓ,−1,ifCℓis better thanO,0,ifOis equal toCℓ,ℓ=1,…,ydjTD(O,Cℓ) in Eq. (37) is calculated by Eq. (39):
djTD(O,Cℓ)=12(μO−μCℓ)2+(vO−vCℓ)2+((1−μO−vO)−(1−μCℓ−vCℓ))2,ℓ=1,…,y;j=1,…,S.
Step 9: Calculate our company’s combined performance rating score (CPR˜) to locate the position of our firm among the competitors regarding engineering assessments and customer ratings together as in Eq. (40):
CPR˜=χD˜O−CℓCD⊕(1−χ)D˜O−CℓTD,ℓ=1,…,y,
where χ and (1−χ) are the coefficients of importance of CDs and TDs, respectively.
Step 10: Find the location of our company relative to the other competitive firms as in Fig. 7. Larger positive distance between our company and Cℓ indicates that our company is in a more advantageous position than Cℓ. At the other negative side, bigger distance between our company and Cℓ indicates that our company is in a more disadvantageous position than Cℓ. The relative location of our company is determined by the indicators in Table 4.
Scale to indicate the position of our company.
Indicators.
Our company
Distance between O−Cℓ
Better than Cℓ
Positive
Worse than Cℓ
Negative
Equal to Cℓ
Zero
Application: Hand Sanitizer Design and Development
COVID-19 is a contagious disease, first identified in China, in December 2019 and has since spread worldwide, leading to an ongoing pandemic. Centres for Disease Control and Prevention recommend washing the hands with soap and water for at least 20 seconds to prevent the spread of the virus and minimize the risk of getting infected. However, in many cases especially at public places, they are mostly not available. In such situations, hand sanitizers with at least 60% of alcohol are the most suggested solutions. Hand sanitizers (Fig. 8) are generally liquid, gel or foam form of agents applied on the hands to remove viruses/bacteria/microorganisms.
Hand sanitizer representation.
In this section an application on hand sanitizer design and development will be presented in steps to illustrate the proposed novel intuitionistic Z-fuzzy QFD approach based on Chebyshev’s inequality.
To determine the CDs for hand sanitizer, a questionnaire was designed to ask their expectations from this product. This questionnaire was distributed to the e-mail addresses of the customers of one of the largest markets in İstanbul. The total number of the customers was 2078 and 219 of them replied. Based on these responses, the following CDs from a hand sanitizer product were determined: Easy storage, compact package, nice smell, fast absorption and/or drying, moisturizing formula, aesthetic design, powerful formula, environmentally friendly and cruelty free, easy and convenient use, and no hard chemicals. After determining these CDs from the customers, we gathered a small focus group to interview and discuss with them the importance degrees of these CDs. Then we asked a chemical cleaning supplies producer in İstanbul how these CDs can be met by which TDs. The producer firm determined the following TDs: Active ingredients, hazardous ingredients, colour, fragrance, package design, and compliance with laws. The relations between these CDs and TDs can be seen in Table 8.
Now the steps of the proposed intuitionistic Z-fuzzy QFD approach based on Chebyshev’s inequality will be given in details in the following.
Phase 1 – CD&TD Relation Analysis
Step 1: Linguistic CDs are defined, and linguistic customer evaluations are assigned by three customers using the scale in Table 2. Customers’ weights are assigned to be wc1=3, wc2=2, and wc3=1, based on the scale in Table 5. Then, the linguistic customer evaluations are translated into IVIF numbers by using Table 2 and aggregated by using Eqs. (20)–(21). The linguistic CDs and corresponding evaluations are given in Table 6 with their aggregated SVIF representations. These are calculated based on the weighted mean and the weighted standard deviation of the assigned customer evaluations by using Eqs. (22)–(25). Please note that after the aggregation operations, the IVIF numbers are turned into SVIF numbers which is to decrease the vagueness.
Scale for experience level of customers and experts.
Degree of experience
Corresponding numerical score
Very experienced
3
Quite experienced
2
Slightly experienced
1
CDs, linguistic customer evaluations, and aggregated SVIF values.
Customer demands
Linguistic customers evaluations
Aggregated SVIF customer evaluations
Easy storage, compact package
HI, AEI, LI
(0.37, 0.31)
Nice smell
MLI, VHI, AEI
(0.36, 0.32)
Fast absorption and/or drying
AHI, HI, MHI
(0.47, 0.19)
Moisturizing formula
AHI, MHI, HI
(0.46, 0.20)
Aesthetic design
VLI, AEI, VHI
(0.23, 0.35)
Powerful formula
VHI, VHI, AHI
(0.53, 0.24)
Environmentally friendly and cruelty free
VLI, MHI, HI
(0.25, 0.26)
Easy and convenient use
LI, AEI, HI
(0.31, 0.27)
No hard chemicals
MHI, AHI, HI
(0.44, 0.22)
To have a better understanding with the calculations, a sample calculation is given in Table 7 showing the aggregation operation for the customer demand “Easy Storage, Compact Package” evaluated by three customers.
Step 2: TDs are defined by three experts where their weights are we1=1, we2=2, and we3=1 depending on the scale given in Table 5. Then their linguistic assessments for the CD-TD relationship matrix are translated into IVIF numbers by using Table 2. Later, each IVIF relation is aggregated to a SVIF number by using Eqs. (20)–(21). These are calculated based on the weighted mean and the weighted standard deviation of the values in the relationship matrix by using Eqs. (26)–(29). Table 8 presents this linguistic relationship matrix between CDs and TDs, and their aggregated SVIF correspondences.
Sample calculations of linguistic CD translation into SVIF value.
k values are found by trial-and-error and interpolation methods.
Linguistic relationship matrix between CDs and TDs, and their aggregated SVIF correspondences.
Technical descriptors ∖Customerdemands
Active ingredients
Hazardous ingredients
Colour
Fragrance
Package design
Compliance with laws
Easy storage, compact package
AHR, AHR, VHR
(0.57, 0.16)
Nice smell
LR, VLR, VLR
ALR, VLR, ALR
AHR, AHR, VHR
(0.26, 0.51)
(0.21, 0.54)
(0.57, 0.16)
Fast absorption and/or drying
AHR, VHR, HR
ALR, LR, AER
(0.51, 0.24)
(0.23, 0.42)
Moisturizing formula
HR, MHR, VHR
ALR, LR, VLR
(0.44, 0.28)
(0.24, 0.48)
Aesthetic design
HR, MHR, MLR
VHR, AHR, AHR
(0.38, 0.30)
(0.56, 0.19)
Powerful formula
AHR, HR, VHR
AER, MLR, MHR
HR, MHR, HR
(0.48, 0.23)
(0.37, 0.40)
(0.44, 0.31)
Environmentally friendly and cruelty free
AER, HR, VHR
AER, VHR, AHR
VHR, AHR, HR
(0.39, 0.28)
(0.40, 0.22)
(0.50, 0.22)
Easy and convenient use
AHR, VHR, AHR
(0.54, 0.20)
No hard chemicals
LR, AER, MLR
VHR, AHR, VHR
LR, MLR, VLR
HR, AER, VHR
(0.31, 0.41)
(0.53, 0.21)
(0.28, 0.45)
(0.42, 0.27)
To have a better understanding with the calculations, a sample calculation is given in Table 9 showing the aggregation operation for the relation between the CD “Nice Smell” and the TD “Active Ingredients” evaluated by three experts.
Step 3: The level of technical difficulty of the TDs are determined by using the scale given in Table 2 by the three experts. The weights are accepted to be the same as in Step 2 and similar calculations are applied to find the aggregated SVIF numbers for each TDs’ technical difficulty. Table 10 shows the linguistic technical difficulty of each TD and their corresponding aggregated SVIF value.
Sample calculation of linguistic TD’s translation into SVIF value.
k values are found by trial-and-error and interpolation methods.
Step 4: The linguistic correlation matrix among TDs is constructed by the experts as given in Fig. 9 by using the scale given in Table 2. In this way the directions of the correlations which can be positive or negative have been determined. These directions of improvements are represented with “+” and “−” signs to show whether the TD is needed to be increased or decreased, respectively. In Fig. 9, each cell shows three assessments from three experts. The blank cells in Fig. 9 indicate no correlation between the considered two TDs.
Linguistic technical difficulties of TDs and their aggregated SVIF correspondences.
Technical descriptors
Active ingredients
Hazardous ingredients
Colour
Fragrance
Package design
Compliance with laws
Linguistic technical difficulty
AHD, VHD, AHD
VHD, AHD, AHD
ALD, VLD, ALD
AED, MLD, MHD
AED, MLD, MHD
HD, MHD, VHD
Aggregated SVIF technical difficulty
(0.54, 0.20)
(0.56, 0.19)
(0.21, 0.54)
(0.37, 0.40)
(0.23, 0.44)
(0.44, 0.28)
Step 5: We obtained the Chebyshev’s inequality based absolute priority degrees for each TD by using Eq. (30) as given in Table 11.
Linguistic and SVIF correlation matrices.
Absolute priorities of TDs.
Active ingredients
Hazardous ingredients
Colour
Fragrance
Package design
Compliance with laws
Absolute priority
0.50
0.30
0.34
0.44
0.37
0.43
To better explain this step, a sample calculation is given below for TD “active ingredients”.
First, we multiplied each SVIF customer evaluation value with the corresponding cell in the relation matrix for TD “active ingredients” by using Eq. (4) and then summed these values up by using Eq. (3). Results are shown in Table 12. We added up each SVIF value separately to the summation of the previous ones by applying Eq. (3) successively. The summation result is found to be (0.68, 0.01). Next, we defuzzified this value with Eq. (7) and the result is found as 0.76, where 0.76=1−0.012−0.68−0.01.
Results of SVIF multiplication of customer evaluations by relation matrix of Active Ingredients.
Customer demands
SVIF customer evaluations
SVIF relation matrix of active ingredients
Multiplied SVIF values
Easy storage, compact package
(0.37, 0.31)
Nice smell
(0.36, 0.32)
(0.26, 0.51)
(0.09, 0.67)
Fast absorption and/or drying
(0.47, 0.19)
(0.51, 0.24)
(0.24, 0.38)
Moisturizing formula
(0.46, 0.20)
(0.44, 0.28)
(0.20, 0.42)
Aesthetic design
(0.23, 0.35)
Powerful formula
(0.53, 0.24)
(0.48, 0.23)
(0.25, 0.41)
Environmentally friendly and cruelty free
(0.25, 0.26)
(0.39, 0.28)
(0.10, 0.47)
Easy and convenient use
(0.31, 0.27)
No hard chemicals
(0.44, 0.22)
(0.31, 0.41)
(0.14, 0.63)
Total
(0.68, 0.01)
Next, to find the correlation correction factor for TD “active ingredients”, first we defuzzified the SVIF correlation values. Then applied Eq. (31) as (4/5)×(0.53+0.53+0.563−0.61)=−0.06, where ncc1=4, S=6. Then, we defuzzified all the SVIF technical difficulty values of TDs and divided the technical difficulty of TD “active ingredients” to all technical difficulty’s summation as 0.63/(0.63+0.65+0.37+0.49+0.42+0.56)=0.20. This gives us the relative technical difficulty of “active ingredients”, given in Eq. (32).
Finally, we applied Eq. (30) as follows:
AP1=0.76+(1+(−0.06))(1+0.20)=0.50.
Step 6: We calculated the relative absolute priorities by using Eq. (33) as shown in Table 13. The TD with the highest relative absolute priority is found as TD “Active Ingredients” with RAP= 0.21 which means that it needs to be taken into consideration promptly by the product developers.
Phase 2- Competitive Analysis
Relative absolute priorities of TDs.
Active Ingredients
Hazardous Ingredients
Colour
Fragrance
Package Design
Compliance with laws
Relative absolute priority
0.21
0.13
0.14
0.18
0.16
0.18
Step 7: First, we collected the linguistic customer assessments for the competitive analysis through CDs assigned by three customers using the IVIF scale given in Table 2. Their linguistic assessments are shown in Fig. 11 and their corresponding aggregated SVIF values are given in Fig. 12. Next, to determine our company’s position among the competitors, we applied Eq. (34) and the results of the computations are given in Table 14. The scores of SVIF customers’ assessments are found by Eq. (7). κO−C1CD and κO−C1CD are calculated by Eq. (35). diCD(O,C1) and diCD(O,C2) are found by Eq. (36). Here, O represents Our Company, C1 represents Company 1 and C2 represents Company 2.
Results of competitive analysis through CDs.
CDs
Score of O
Score of C1
Score of C2
κO−C1CD
κO−C2CD
diCD(O,C1)
diCD(O,C2)
CEiC
DO−C1CD
DO−C2CD
Easy storage, compact package
0.62
0.45
0.60
1
1
0.44
0.41
0.52
0.23
0.22
Nice smell
0.49
0.40
0.55
1
−1
0.41
0.39
0.52
0.21
−0.20
Fast absorption and/or drying
0.45
0.43
0.62
1
−1
0.41
0.44
0.60
0.25
−0.26
Moisturizing formula
0.60
0.60
0.50
0
1
0.43
0.45
0.60
0.00
0.27
Aesthetic design
0.45
0.62
0.49
−1
−1
0.45
0.39
0.46
−0.21
−0.18
Powerful formula
0.60
0.50
0.60
1
0
0.48
0.43
0.62
0.30
0.00
Environmentally friendly and cruelty free
0.62
0.45
0.43
1
1
0.44
0.45
0.50
0.22
0.22
Easy and convenient use
0.40
0.45
0.62
−1
−1
0.47
0.50
0.51
−0.24
−0.26
No hard chemicals
0.60
0.62
0.49
−1
1
0.41
0.42
0.58
−0.24
0.24
Total
0.52
0.05
In order to better explain the operations used in this table, a sample calculation is presented below for CD “Easy Storage, Compact Package”.
Score ofO=1−0.232−0.52−0.23=0.62,Score ofC1=1−0.422−0.29−0.42=0.45,Score ofC2=1−0.202−0.47−0.20=0.60,κ1O−C1CD=1,(0.62>0.45),κ1O−C2CD=1,(0.62>0.60),d1CD(O,C1)=12((0.52−0.29)2+(0.23−0.42)2+((1−0.52−0.23)−(1−0.29−0.42))2)=0.44,d1CD(O,C2)=12((0.52−0.47)2+(0.23−0.20)2+((1−0.52−0.23)−(1−0.47−0.20))2)=0.41,CE1=1−0.312−0.37−0.31=0.52,D1O−C1CD=1×0.44×0.52=0.23,D1O−C2CD=1×0.41×0.52=0.22.0.45),\\ {} & {\kappa _{1O-{C_{2}}}^{\textit{CD}}}=1,\hspace{1em}(0.62>0.60),\\ {} & {d_{1}^{\textit{CD}}}(O,{C_{1}})\\ {} & \hspace{1em}=\displaystyle \sqrt{\frac{1}{2}\big({(0.52-0.29)^{2}}+{(0.23-0.42)^{2}}+{\big((1-0.52-0.23)-(1-0.29-0.42)\big)^{2}}\big)}=0.44,\\ {} & {d_{1}^{\textit{CD}}}(O,{C_{2}})\\ {} & \hspace{1em}=\displaystyle \sqrt{\frac{1}{2}\big({(0.52-0.47)^{2}}+{(0.23-0.20)^{2}}+{\big((1-0.52-0.23)-(1-0.47-0.20)\big)^{2}}\big)}=0.41,\\ {} & {\textit{CE}_{1}}=\frac{1-0.31}{2-0.37-0.31}=0.52,\\ {} & {D_{1O-{C_{1}}}^{\textit{CD}}}=1\times 0.44\times 0.52=0.23,\\ {} & {D_{1O-{C_{2}}}^{\textit{CD}}}=1\times 0.41\times 0.52=0.22.\end{aligned}\]]]>
Step 8: First, we collected the experts’ linguistic assessments for the competitive analysis through TDs assigned by three experts using the IVIF scale given in Table 2. Their linguistic assessments are shown in Fig. 11 and their corresponding aggregated SVIF values are given in Fig. 12. Next, to determine our company’s position among the competitors, we applied Eq. (37) and the results of the computations are given in Table 15. The scores of SVIF experts’ assessments are found by Eq. (7). κO−C1TD and κO−C2TD are calculated by Eq. (38). diTD(O,C1) and diTD(O,C2) are found by Eq. (39).
Results of competitive analysis through TDs.
TDs
Score of O
Score of C1
Score of C2
κO−C1TD
κO−C2TD
djTD(O,C1)
djTD(O,C2)
APijC
DO−C1TD
DO−C2TD
Active ingredients
0.56
0.50
0.57
1
−1
0.41
0.45
0.50
0.21
−0.23
Hazardous ingredients
0.42
0.52
0.41
−1
1
0.48
0.50
0.30
−0.15
0.15
Colour
0.56
0.53
0.48
1
1
0.48
0.44
0.34
0.16
0.15
Fragrance
0.57
0.47
0.52
1
1
0.46
0.49
0.44
0.20
0.21
Package design
0.41
0.42
0.50
−1
−1
0.50
0.49
0.37
−0.19
−0.18
Compliance with laws
0.40
0.48
0.56
−1
−1
0.35
0.46
0.43
−0.15
−0.20
Total
0.08
−0.09
In order to better understand the operations used in this table, a sample calculation is presented below for TD “Active Ingredients”.
Score ofO=1−0.272−0.42−0.277=0.56,Score ofC1=1−0.372−0.37−0.37=0.50,Score ofC2=1−0.242−0.43−0.24=0.57,κ1O−C1TD=1,(0.56>0.50),0.50),\end{aligned}\]]]>κ1O−C2TD=−1,(0.56<0.57),d1TD(O,C1)=12((0.42−0.37)2+(0.27−0.37)2+((1−0.42−0.27)−(1−0.37−0.37))2)=0.41,d1TD(O,C2)=12((0.42−0.43)2+(0.27−0.24)2+((1−0.42−0.27)−(1−0.43−0.24))2)=0.45,AP1=0.50,D1O−C1TD=1×0.41×0.50=0.21,D1O−C2TD=−1×0.45×0.50=−0.23.
Step 9: We obtained the combined performance rating score (CPR˜) of our company to determine our position among the competitors by using Eq. (40). Here, we accepted the importance coefficient of CD as χ=0.40 and importance coefficient of TD as (1−χ)=0.60 which means we assigned more weight to the experts’ views compared to the customers. CPRs among O−C1 and O−C2 are found as follows:
CPRO−C1=(0.40×0.52)+(0.60×0.05)=0.24,CPRO−C2=(0.40×0.08)+(0.60×−0.09)=−0.02.
Step 10: We determined the relative position of our company on a scale as in Fig. 10. Since CPRO−C1 found to be a positive number 0.24, it means O is better than C1 on the scale and the negative value −0.02 for CPRO−C2 shows that C2 is better than O considering the competitive advantage. But since it is a very small number, we can accept our company equals to C2.
Scale indicating the location of our company.
As mentioned above, the whole linguistic HOQ matrix and the whole aggregated SVIF HOQ matrix are given in Figs. 11 and 12, respectively.
Linguistic HOQ.
Aggregated SVIF HOQ.
Conclusion
In the literature, the QFD approach has been an effective tool to incorporate customer voice into product design and development. The voice of customer is often included in the QFD approach in linguistic expressions that contain a certain degree of ambiguity. It has been seen that this uncertainty has been modelled mostly with the help of fuzzy sets in the literature. More than ten extensions of ordinary fuzzy sets have been proposed to the literature, each aiming to model human thoughts in a more detailed and accurate way through membership functions. Our review revealed that the most used extension in QFD approach is intuitionistic fuzzy sets and the most often integrated decision-making tool is AHP method. In most of the QFD studies the reliability to the assigned fuzzy values of QFD parameters are not considered. The purpose of this study was to develop a novel approach integrating the reliability with the assigned fuzzy values of QFD method based on the principles of the probability theory. The contribution of our method to the literature is the presentation of a new reliability integrated QFD approach under intuitionistic fuzziness with all its aspects such as technical difficulty, competitive analysis through CDs and TDs. Intuitionistic Z-fuzzy numbers have been developed and successfully applied to represent the uncertainty in linguistic terms of CDs and TDs. Chebyshev’s inequality allowed us to objectively obtain the degree of reliability of the restriction function, which is subjectively determined in the previous studies. This study also proposed a model that successfully integrates parts of the QFD approach that are often considered separately in the literature. This model comprehensively integrated customer evaluations, relationship matrix, correlation matrix, and technical difficulties of TDs, to calculate the absolute priority degrees of TDs. One limitation of our study is that IVIF division and subtraction operations are not precisely defined in the literature which forces us to use defuzzification when these operations are needed.
For further research we suggest IVPF, IVSF or IVPiF sets to be used in our model instead of IVIF sets. Besides, aggregation operators can be differentiated by using intuitionistic fuzzy Einstein aggregation operators such as the intuitionistic fuzzy Einstein weighted geometric (IFEWG) operator, or the intuitionistic fuzzy Einstein ordered weighted geometric (IFEOWG) operator. Alternatively, the linguistic intuitionistic fuzzy weighted partitioned Heronian mean (LIFWPHM) operator or the linguistic intuitionistic fuzzy partitioned geometric Heronian mean (LIFPGHM) operator can be used.
ReferencesAkbaş, H., Bilgen, B. (2017). An integrated fuzzy QFD and TOPSIS methodology for choosing the ideal gas fuel at WWTPs. , 125, 484–497.Amaladhasan, S., Parthiban, P., Dhanalakshmi, R. (2018). Analysis and prioritisation of eco drivers in supply Chain. , 31(3), 336–362.Aouag, H., Soltani, M., Mouss, M.D. (2020). Enhancement of value stream mapping application process through using fuzzy DEMATEL and fuzzy QFD approaches: a case study considering economic and environmental perspectives. . 16(3), 1002–1023.Atanassov, K.T. (1986). Intuitionistic fuzzy sets. , 20(1), 87–96.Atanassov, K.T. (1994). New operations defined over the intuitionistic fuzzy sets. , 61, 137–142.Atanassov, K.T. (2020). Circular intuitionistic fuzzy sets. , 39(5), 5981–5986.Babbar, C., Amin, S.H. (2018). A multi-objective mathematical model integrating environmental concerns for supplier selection and order allocation based on fuzzy QFD in beverages industry. , 92, 27–38.Baskar, C., Parameshwaran, R., Nithyavathy, N. (2020). Implementation of fuzzy-based integrated framework for sesame seed separator development. , 24(10), 7715–7734.Beheshtinia, M.A., Farzaneh Azad, M. (2019). A fuzzy QFD approach using SERVQUAL and kano models under budget constraint for hotel services. , 30(7–8), 808–830.Bevilacqua, M., Ciarapica, F.E., Marchetti, B. (2012). Development and test of a new fuzzy-QFD approach for characterizing customers rating of extra virgin olive oil. , 24(1), 75–84.Bhuvanesh Kumar, M., Parameshwaran, R. (2018). Fuzzy integrated QFD, FMEA framework for the selection of lean tools in a manufacturing organisation. , 29(5), 403–417.Bhuvanesh Kumar, M., Parameshwaran, R. (2020). A comprehensive model to prioritize lean tools for manufacturing industries: a fuzzy FMEA, AHP and QFD-based approach. , 37(2), 170–196.Bilişik, Ö.N., Şeker, Ş., Aydın, N., Güngör, N., Baraçlı, H. (2019). Passenger satisfaction evaluation of public transportation in İstanbul by using fuzzy quality function deployment methodology. , 44(3), 2811–2824.Büyüközkan, G., Güleryüz, S. (2015). Extending fuzzy QFD methodology with GDM approaches: An application for IT planning in collaborative product development. , 17(4), 544–558.Büyüközkan, G., Uztürk, D. (2020). Smart fridge design with interval-valued intuitionistic fuzzy QFD. , 1029, 1170–1179.Büyüközkan, G., Güler, M., Mukul, E. (2020). An integrated fuzzy QFD methodology for customer oriented multifunctional power bank design. , 279, 73–91.Celik, M., Cebi, S., Kahraman, C., Er, I.D. (2009). An integrated fuzzy QFD model proposal on routing of shipping investment decisions in crude oil tanker market. , 36(3 PART 2), 6227–6235.Chang, W. (2012). A new perspective on EFL teaching: applying fuzzy QFD in TRIZ for teaching quality improvement. , 2(2), 43–53.Chen, R. (2016). Green design quality management in industrial Chain using fuzzy decision tree and QFD. , 19(3), 345–365.Chen, C., Huang, S. (2011). Implementing KM programmes using fuzzy QFD. , 22(4), 387–406.Chiadamrong, N., Tham, T.T. (2017). An integrated approach with SEM, fuzzy-QFD and MLP for supply chain management strategy development. , 28(1), 84–125.Chowdhury, M.M.H., Quaddus, M.A. (2016). A multi-phased QFD based optimization approach to sustainable service design. , 171, 165–178.Cuong, B. (2015). Picture fuzzy sets. , 30(4), 409.Çevik Onar, S., Büyüközkan, G., Öztayşi, B., Kahraman, C. (2016). A new hesitant fuzzy QFD approach: an application to computer workstation selection. , 46, 1–16.Dat, L.Q., Phuong, T.T., Kao, H., Chou, S., Nghia, P.V. (2015). A new integrated fuzzy QFD approach for market segments evaluation and selection. , 39(13), 3653–3665.De Almeida, M.F.L., Silva Da Luz, C.E., De Andrade Martins, G. (2018). Fuzzy quality function deployment (fuzzy-QFD) applied to new defense product development. Paper presented at the Towards Sustainable Technologies and Innovation. In: .Deveci, M., Öner, S.C., Canıtez, F., Öner, M. (2019). Evaluation of service quality in public bus transportation using interval-valued intuitionistic fuzzy QFD methodology. , 33, 100387.Efe, B., Yerlikaya, M.A., Efe, Ö.F. (2020). Mobile phone selection based on a novel quality function deployment approach. , 24(20), 15447–15461.Fan, J., Yu, S., Yu, M., Chu, J., Tian, B., Li, W., Chen, C. (2020). Optimal selection of design scheme in cloud environment: a novel hybrid approach of multi-criteria decision-making based on F-ANP and F-QFD. , 38(3), 3371–3388.Fitriana, R., Kurniawan, W., Anwar, M.R. (2019). Measurement and proposal of improving marketing process to improve the quality of aftersales services with fuzzy quality function deployment and data mining methods in OV agency. , 528(1), 012072.Hakim, A., Gheitasi, M., Soltani, F. (2016). Fuzzy model on selecting processes in business process reengineering. , 22(6), 1118–1138.Haktanır, E. (2020). Prioritization of competitive suppliers using an interval-valued Pythagorean fuzzy QFD & COPRAS methodology. , 34(1/2), 177–199.Haktanır, E., Kahraman, C. (2019). A novel interval-valued Pythagorean fuzzy QFD method and its application to solar photovoltaic technology development. , 132, 361–372.Haktanır, E., Kahraman, C., Kutlu Gündoğdu, F. (2021). Delivery drone design using spherical fuzzy quality function deployment. , 392, 399–430.Haq, A.N., Boddu, V. (2017). Analysis of enablers for the implementation of leagile supply chain management using an integrated fuzzy QFD approach. , 28(1), 1–12.Hong, W., Wang, H. (2005). Fuzzy QFD approach to developing an integrated service strategy. , 7(3), 120–132.Jafarzadeh, H., Akbari, P., Abedin, B. (2018). A methodology for project portfolio selection under criteria prioritisation, uncertainty and projects interdependency – combination of fuzzy QFD and DEA. , 110, 237–249.Jamalnia, A., Mahdiraji, H.A., Sadeghi, M.R., Hajiagha, S.H.R., Feili, A. (2014). An integrated fuzzy QFD and fuzzy goal programming approach for global facility location-allocation problem. , 13(2), 263–290.Juan, Y., Perng, Y., Castro-Lacouture, D., Lu, K. (2009). Housing refurbishment contractors selection based on a hybrid fuzzy-QFD approach. , 18(2), 139–144.Kahraman, C., Ertay, T., Büyüközkan, G. (2006). A fuzzy optimization model for QFD planning process using analytic network approach. , 171(2), 390–411.Kang, X. (2020). Aesthetic product design combining with rough set theory and fuzzy quality function deployment. , 39(1), 1131–1146.Kang, X., Yang, M., Wu, Y., Ni, B. (2018). Integrating evaluation grid method and fuzzy quality function deployment to new product development. , 2018, 2451470.Karasan, A., Kahraman, C. (2019). A novel intuitionistic fuzzy DEMATEL – ANP – TOPSIS integrated methodology for freight village location selection. , 36, 1335–1352.Kavosi, M., Mavi, R.K. (2011). Fuzzy quality function deployment approach using TOPSIS and analytic hierarchy process methods. , 7(3), 304–324.Kayapınar, S., Erginel, N. (2019). Designing the airport service with fuzzy QFD based on SERVQUAL integrated with a fuzzy multi-objective decision model. , 30(13–14), 1429–1448.Keshteli, R.N., Davoodvandi, E. (2017). Using fuzzy AHP and fuzzy TOPSIS in fuzzy QFD: a case study in ceramic and tile industry of Iran. , 20(2), 197–216.Khademi-Zare, H., Zarei, M., Sadeghieh, A., Saleh Owlia, M. (2010). Ranking the strategic actions of Iran mobile cellular telecommunication using two models of fuzzy QFD. , 34(11), 747–759.Kutlu Gündoğdu, F., Kahraman, C. (2019). Spherical fuzzy sets and spherical fuzzy TOPSIS method. , 36(1), 337–352.Kutlu Gündoğdu, F., Kahraman, C. (2020). A novel spherical fuzzy QFD method and its application to the linear delta robot technology development. , 87, 103348.Lee, G.H., Park, S.H. (2021). Fuzzy QFD-based prioritization of work activities of construction for safety. , 12(1), 1–8.Lee, Z., Pai, C., Yang, C. (2012). Customer needs and technology analysis in new product development via fuzzy QFD and Delphi. , 9(1), 1–15.Li, S., Tang, D., Wang, Q., Zhu, H. (2020). Analysis and extraction of consumer information for the evaluation of design requirement depending on consumer involvement. , 77, 342–353.Liu, A., Hu, H., Zhang, X., Lei, D. (2017). Novel two-phase approach for process optimization of customer collaborative design based on fuzzy-QFD and DSM. , 64(2), 193–207.Liu, H. (2009). The extension of fuzzy QFD: From product planning to part deployment. , 36(8), 11131–11144.Liu, S., Zhang, Y., Lai, Y., Wang, M. (2018). A novel method of design elements based on EGM and fuzzy QFD. , 22(5), 408–420.Lu, C., Lin, L., Yeh, H. (2019). A multi-phased FQFD for the design of brand revitalisation. , 30(7–8), 848–871.Ma, H., Chu, X., Li, Y. (2019a). An integrated approach to identify function components for product redesign based on analysis of customer requirements and failure risk. , 36(2), 1743–1757.Ma, H., Chu, X., Xue, D., Chen, D. (2019b). Identification of to-be-improved components for redesign of complex products and systems based on fuzzy QFD and FMEA. , 30(2), 623–639.Milunovic Koprivica, S., Filipovic, J. (2018). Application of traditional and fuzzy quality function deployment in the product development process. , 30(2), 98–107.Mohanraj, R., Sakthivel, M., Vinodh, S., Vimal, K.E.K. (2015). A framework for VSM integrated with fuzzy QFD. , 27(5), 616–632.Mousavi, S.M., Malekly, H., Hashemi, H., Mojtahedi, S.M.H. (2008). A two-phase fuzzy decision making methodology for bridge scheme selection. In: , pp. 415–419.Nejatian, M., Zarei, M.H. (2013). Moving towards organizational agility: are we improving in the right direction? , 14(4), 241–253.Noorul Haq, A., Boddu, V. (2015). An integrated fuzzy QFD and TOPSIS approach to enhance leanness in supply Chain. , 7(2), 171–188.Ocampo, L.A., Labrador, J.J.T., Jumao-as, A.M.B., Rama, A.M.O. (2020). Integrated multiphase sustainable product design with a hybrid quality function deployment – multi-attribute decision-making (QFD-MADM) framework. , 24, 62–78.Osiro, L., Lima-Junior, F.R., Carpinetti, L.C.R. (2018). A group decision model based on quality function deployment and hesitant fuzzy for selecting supply chain sustainability metrics. , 183, 964–978.Osorio-Gómez, J.C., Manotas-Duque, D.F. (2018). Fuzzy QFD and TOPSIS for dispatching prioritization in maritime transportation considering operational risk. Best Practices in Manufacturing Processes: Experiences from Latin America, 97–116.Palanisamy, P., Zubar, H.A. (2013). Hybrid MCDM approach for vendor ranking. , 24(6), 905–928.Piengang, F.C.N., Beauregard, Y., Kenné, J. (2019). An APS software selection methodology integrating experts and decisions-maker’s opinions on selection criteria: a case study. , 6(1), 1594509.Rattawut, V. (2016). Integration of Fuzzy-QFD and AHP base on a fuzzy scale for mini-CNC milling machine retrofit. In: , pp. 89–94.Raut, R.D., Mahajan, V.C. (2015). A new strategic approach of fuzzy-quality function deployment and analytical hierarchy process in construction industry. , 20(2), 260–290.Roghanian, E., Alipour, M. (2014). A fuzzy model for achieving lean attributes for competitive advantages development using AHP-QFD-PROMETHEE. , 10(3), 68.Seker, S. (2020a). Fuzzy AHP-QFD methodology and its application to retail chain. , 1029, 1189–1197.Seker, S. (2020b). Fuzzy quality function Deployment Method for smart phone product design. , 279, 57–71.Senthilkannan, N., Parameshwaran, R. (2019). Performance analysis and quality improvement using fuzzy MCDM and lean tools in a paper industry. , 12(3), 205–229.Shuofang, L., Yang, Z., Yuchung, L., Minghong, W. (2018). Study methods of design elements based on EGM and fuzzy QFD. In: , pp. 83–86.Smarandache, F. (1998). . American Research Press, pp. 105.Sohn, S.Y., Choi, I.S. (2001). Fuzzy QFD for supply chain management with reliability consideration. , 72(3), 327–334.Su, C., Lin, C. (2008). A case study on the application of fuzzy QFD in TRIZ for service quality improvement. , 42(5), 563–578.Tavana, M., Mousavi, N., Golara, S. (2013). A fuzzy-QFD approach to balanced scorecard using an analytic network process. , 5(4), 331–363.Taylan, O. (2013). A hybrid methodology of fuzzy grey relation for determining multi attribute customer preferences of edible oil. , 13(5), 2981–2989.Tsai, C., Lo, C., Chang, A.C. (2003). Using fuzzy qfd to enhance manufacturing strategic planning. , 20(1), 33–41.Verma, D., Chilakapati, R., Fabrycky, W.J. (1998). Analyzing a quality function deployment matrix: an expert system-based approach to identify inconsistencies and opportunities. , 9(3), 252–262.Vinodh, S., Chintha, S.K. (2011). Application of fuzzy QFD for enabling sustainability. , 4(4), 313–322.Vinodh, S., Manjunatheshwara, K.J., Karthik Sundaram, S., Kirthivasan, V. (2017). Application of fuzzy quality function deployment for sustainable design of consumer electronics products: a case study. , 19(4), 1021–1030.Vongvit, R., Kongprasert, N., Fournaise, T., Collange, T. (2017). Integration of fuzzy-QFD and TRIZ methodology for product development. In: , pp. 326–329.Wang, C. (2019). Integrating a novel intuitive fuzzy method with quality function deployment for product design: case study on touch panels. , 37(2), 2819–2833.Wang, D., Yu, H., Wu, J., Meng, Q., Lin, Q. (2019). Integrating fuzzy based QFD and AHP for the design and implementation of a hand training device. , 36(4), 3317–3331.Wang, F., Li, X., Rui, W., Zhang, Y. (2007). A fuzzy QFD-based method for customizing positioning of logistics service products of 3PLS. In: , pp. 3326–3329.Wang, H., Fang, Z., Wang, D., Liu, S. (2020). An integrated fuzzy QFD and grey decision-making approach for supply chain collaborative quality design of large complex products. , 140, 106212.Xiao, S., Wu, J., He, E., Yang, Z. (2015). Identification of software NFR based on the fuzzy-QFD model. , 9(11), 145–154.Yaakob, A.M., Gegov, A. (2015). Fuzzy rule based approach with z-numbers for selection of alternatives using TOPSIS. In: 2015 IEEE International Conference on Fuzzy Systems.Yager, R.R. (2013). Pythagorean fuzzy subsets. In: , Edmonton, Canada, pp. 57–61.Yang, M., Li, Y., Liu, Y., Gao, X. (2010). A method for problem selection in the 6σ definition stage. , 139–141, 1485–1489.Yang, S., Ong S, K., c. Nee A, Y. (2013). Design for remanufacturing – a fuzzy-QFD approach. In: , pp. 655–661.Yazdani, M., Kahraman, C., Zarate, P., Onar, S.C. (2019). A fuzzy multi attribute decision framework with integration of QFD and grey relational analysis. , 115, 474–485.Yu, L., Wang, L., Bao, Y. (2018). Technical attributes ratings in fuzzy QFD by integrating interval-valued intuitionistic fuzzy sets and Choquet integral. , 22(6), 2015–2024.Zadeh, L.A. (1965). Fuzzy Sets. , 8(3), 338–353.Zadeh, L.A. (2011). A note on Z-numbers. , 181(14), 2923–2932.Zaim, S., Sevkli, M., Camgöz-Akdağ, H., Demirel, O.F., Yesim Yayla, A., Delen, D. (2014). Use of ANP weighted crisp and fuzzy QFD for product development. , 41(9), 4464–4474.Zhang, Z., Yang, J., Ye, Y., Hu, Y., Zhang, Q. (2012). A type of score function on intuitionistic fuzzy sets with double parameters and its application to pattern recognition and medical diagnosis. , 29, 4336–4342.