Quality function deployment (QFD) is a customer-driven product development technique that converts customer requirements (CRs) into design attributes (DAs) of a product and service. Nevertheless, in real situations, the traditional QFD method has been found that possesses some deficiencies, such as the accuracy assessment of relationships between CRs and DAs, and the inter-relationships among DAs. To fill in the above gaps, this study develops a new QFD approach by a CoCoSo-based ranking method under Pythagorean fuzzy environment. To begin with, an extended Pythagorean fuzzy decision-making trial and evaluation laboratory (DEMATEL) method is proposed to identify the relationships within DAs. Second, the aggregation method of the weighted average method and objective penalty function are propounded to construct the programming models for calculating the importance of DAs under Pythagorean fuzzy setting. Third, a new CoCoSo-based ranking method for Pythagorean triangular fuzzy numbers (PTrFNs) is proposed to obtain the ranking of DAs. Lastly, a case regarding “Ping An Health” mobile medical App is carried out to verify the effectiveness and superiority of the proposed QFD approach. The results show that the top DA is perceptibility. Therefore, perceptibility should be focus on firstly in the “Ping An Health” App design, such as system fluency, interface comfort and network stability. Additionally, the results show that the new QFD can express experts’ hesitant assessment information, deal with the interrelations among DAs, and yield more precise rankings of DAs in QFD.
Quality function deployment (QFD)([1]) proposed based on reasoning and deduction is an effective tool to couple customer requirements (CRs) with technical attributes (TAs)/design attributes (DAs) of a product ([2]). Nowadays, the applications and fields of QFD have been expanded in light of it distinct advantageous in the aspect of quality management of industry engineering, such as manufacturing ([3, 4]), supply chain management ([5, 6]), product design ([7]), and transportation ([8]).
A traditional QFD model is based on a tool called “house of quality (HoQ)”, which mainly including the following blocks: CRs, importance of CRs, DAs, correlations between CRs and DAs, interrelationships among DAs and importance and priorities of DAs (see in Fig. 1)([9]). In particular, obtaining and prioritizing the importance of DAs play a key role in QFD model. It guides the production or design team in resource allocation and the subsequent QFD analyses and makes a company achieve higher customer satisfaction. However, there are many limitations of the traditional QFD in existing researches, which are summarized as below: (1) The traditional QFD method processes the input information, such as the importance of CRs and the correlation matrix among CRs and DAs, by a definite number, which cannot express the uncertainty and ambiguity of QFD experts’ assessments ([10]). (2) The traditional QFD determines the ranking of DAs without considering their interrelationships ([10, 11]). (3) In the traditional QFD methods, there is a large amount of calculation in obtaining the ranking results ([12]). The aforementioned defects bring various of challenges for managers to solve practical QFD problems.
The house of quality (HoQ).
Most scholars utilized the concepts of fuzzy sets ([13]) and intuitionistic fuzzy sets (IFS) ([14]) to deal with the uncertainty and fuzziness in the evaluation process. For intuitionistic fuzzy sets, the membership degree and non-membership degree are crisp values, and the sum of membership degree and non-membership degree is less than or equal to 1. However, in the evaluation process, the language given by users and experts is much more fuzzy, so Yager proposed Pythagorean fuzzy set (PFS) ([15]) to deal with more complex and uncertain problems through enlarging the range of information representation. The Pythagorean fuzzy set composed of Pythagorean membership degree satisfies that the square sum of membership degree and non-membership degree is less than or equal to 1. The main advantages of the PFS are as follows: (1) The Pythagorean Fuzzy Set (PFS) is a developed IFS to increase the freedom of professionals to express their opinions and the reliability of decision making; (2) The PFS enables a larger domain to express the uncertainty and ambiguity, and is more powerful on the issues of uncertainty. Since its introduction, scholars investigated it in the aspects of theories and applications, and received numerous importance achievements. In terms of theoretical research, scholars have discussed the Pythagorean fuzzy operational rules ([16]), the distance measures ([17]), and the properties of the Pythagorean operator ([18]). Afterwards, interval Pythagorean fuzzy set has been developed and received a multitude of attentions in the theoretical developments and methodology constructions in different perspectives ([19, 20]). In the present, some researchers have applied the Pythagorean fuzzy set in QFD method ([21, 22]). Haktanir and Kahraman ([21]) used Pythagorean fuzzy set in QFD to select suppliers, in which the weighted fuzzy operator and defuzzification are used to rate technical attributes of QFD. Liao et al. ([22]) developed a score function to rate technical attributes in evaluating of robot design. Therefore, it is meaningful to utilize the PFS to describe the uncertainty and ambiguity of expects’ assessments in QFD.
DEMATEL method ([23]) is proposed to solve the complex relationships among attributes in multi-criteria decision making (MCDM). At present, scholars have applied this method in the field of fuzzy McDM, from fuzzy DEMATEL ([24]) to intuitionistic fuzzy DEMATEL ([25]). And it has been applied in various realistic fields, such as green supply chain ([25]), risk assessment ([26])and so on. Furthermore, the DEMATEL method can intuitively reflect the strength of the influence relationship among TAs in QFD model. On the other hand, the combined compromise solution (CoCoSo) method was developed by Yazdani et al. ([27]) by integrating an additive weighted model and an exponentially weighted model. Compared with other MCDM methods, the CoCoSo method has the following merits ([28]): First, it can obtain the suitable alternative without counterintuitive phenomena and division by zero problems; Second, the deletion or addition of alternatives has a minimum impact on final results. The CoCoSo method encourages the accuracy of a decision-making system and has a better resolution in distinguishing the considered alternatives. Due to these features, the CoCoSo becomes an effective tool to solve MCDM problem and has been widely used in various fields ([29, 30]). Therefore, it is expected to utilize the DEMATEL and the CoCoSo methods to derive a more precise importance ranking of ECs in QFD.
More recently, various fuzzy methods are utilized to build up different QFD approach for solving actual problems, which are confirmed that the fuzzy theory-based approaches possess stronger applicability in the analysis of QFD model within uncertainty. For instance, Vangeas and Labib ([32]) first used fuzzy weighted average in QFD, and they developed a model to obtain the optimal solution of TAs by the fuzzy weighted average. Afterwards, the fuzzy weighted average is widely combined with many different methods, such as Chen et al. ([12]) integrated fuzzy weighted average method and fuzzy expected value operator in QFD; Wang and Chin ([33]) integrated fuzzy normalization and fuzzy weighted average in QFD. Since it is hard to get the membership function of TAs accurately, the defuzzification method of fuzzy expected value operator is developed and is used widely. However, the complexity of solving practical problems using the traditional QFD method described above is very high, which increases the error of the final result. ([34]) is to convert a constraint optimization problem into an unconstrained optimization problem. In these traditional exact penalty function methods, the constraint penalty parameter need to be continuously increased, which makes it not efficient in practical computing of Matlab. In order to deal with this situation and simplify the calculation process, the penalty function with objective penalty parameter called objective penalty function ([35]) is proposed. Therefore, this paper combines the weighted average and objective penalty function to simplify the calculation in traditional QFD technique.
By means of the mentioned analysis and discussions, a new CoCoSo ranking-based QFD approach in Pythagorean fuzzy environment is proposed in this paper. The highlights and contributions of this study can be outlined as follows: (1) The DEMATEL method is extended under the Pythagorean fuzzy environment, which is applied to describe the inter-relationships of DAs in QFD.
(2) An integrated method by merging the weighted averaging and objective penalty function is proposed to simply the original model in the traditional QFD for improving the performance of QFD method.
(3) The CoCoSo method is extended under the Pythagorean fuzzy environment, which is developed to developed to obtain the prioritization of DAs in QFD.
In Section 2, the definitions of Pythagorean fuzzy set are sorted out and the definitions of left set, right set, cut set and weighted Hamming distance of Pythagorean fuzzy number are proposed. In Section 3, a new CoCoSo-based ranking method for PTrFNs is proposed. In Section 4, a new QFD approach by a CoCoSo-based ranking method in Pythagorean fuzzy environment is proposed. In Section 5, this paper takes “Ping An Health Cloud” Mobile App as an example to show the advantages and practicabilities of the proposed QFD approach. Finally, conclusions are made in Section 5.
Preliminaries
In this section, we briefly show basic concepts of the Pythagorean fuzzy set ([36]) and the objective penalty function.
Pythagorean fuzzy set
The Pythagorean fuzzy set is proposed by Dick et al.[36] to deal with vague and fuzzy information provided by decision makers in decision problems.
Definition 1. ([36]) Let U be a universe of discourse. A Pythagorean fuzzy set in U is given by
where, μA : A → [0, 1] denotes the degree of membership and νA : A → [0, 1] denotes the degree of non-membership of x ∈ U to the set A, which satisfied . What’s more, denotes the degree of indeterminacy.
Let is a Pythagorean triangular fuzzy number (PTrFN), the membership function and non-membership function are defined as follows:
Furthermore, referring to interval Pythagorean fuzzy set ([21]) and intuitionistic fuzzy set ([37, 38]), we sorts out the following concepts of the left set, the right set, the cut set and the weighting Hamming distance regarding Pythagorean fuzzy set.
Definition 2. Let A = {< x, μA (x) , νA (x) > |x ∈ U} is a Pythagorean fuzzy set. Then, the left and right set of membership degree are denoted by and , respectively. And their membership function are follows:
The left and right set of membership degree are denoted by and , respectively. And their non-membership function are follows:
Definition 3. Let A = {< x, μA (x) , νA (x) > |x ∈ U} is a Pythagorean fuzzy set in finite universe U, for any α ∈ [0, 1] and β ∈ [0, 1], and that satifies 0 ≤ α2 + β2 ≤ 1, then
is called the <α, β >- cut set, and <α, β> is called the confidence degree. Similarly, the α- cut set and the β- cut set are defined as follows:
Let is a PTrFN, the α- cut set and the β- cut set are closed, which are indicated as follows respectively:
Definition 4. Let X is the direct product space of {X1, X2, . . . , Xn}, and {A1, A2, . . . , An} are Pythagorean fuzzy sets in {X1, X2, . . . , Xn}, respectively. F is the n from X to Y, y = F (x1, x2, . . . , xn). And then F (A1, A2, . . . , An) is the Pythagorean fuzzy sets in Y, which are defined as follows:
where,
and
Definition 5. Let , be two Pythagorean triangular fuzzy numbers, δ is a real number, then the operational rules of Pythagorean triangular fuzzy numbers are defined as follows:
Definition 6. Let and be two Pythagorean triangular fuzzy numbers, μA and μB are the membership functions of A and B, uA and uB are the non-membership functions of A and B. Then, the Hamming distance between A and B is defined as follows:
where, β ∈ [0, 1] is a risk coefficient and denotes experts’ uncertain preference information. β ∈ (0.5, 1] shows that experts prefer positive information and β ∈ [0, 0.5) shows that experts prefer negative information.
Inspired by Li [39], the definitions of value and ambiguity regarding PTrFNs are generalized as follows.
Definition 7. Suppose that is a PTrFN, is the α-cut set of and is the β-cut set of . Then, the values of the membership function and the non-membership function for are defined as:
and the ambiguities of the membership function and the non-membership function for are defined as:
where, is a non-decreasing function on the interval , which represents different weight given to different α- cut set. In addition, . Similarly, is a non-increasing function on the interval and represents different weight given to different β- cut set. Additionally, .
Theorem 1.Let and be two PTrFNs, and , , then
Proof. Based on Eq. (15-18) and , , then
Definition 8. Suppose that is a PTrFN, is the α-cut set of and is the β-cut set of . Then, the value index and the ambiguity index for are respectively defined as:
where λ ∈ [0, 1] is a weight coefficient. The closer λ is to 1, the more positive information decision makers prefer, that is, decision makers pay attention to the satisfaction of uncertainty. On the contrary, the closer λ to 0 indicates that decision makers prefer negative information, that is, decision makers pay more attention to the degree of dissatisfaction with uncertainty.
Theorem 2. Let and be two PTrFNs, and , , then
Proof. By Eqs. (19-20), we can get
Then, based on Theorem 1, we can obtain
Theorem 3.Let be a PTrFN, is the value index of and is the ambiguity index of . Suppose that is an non-negative PFrTN,
(1) If , then and are continuous non-decreasing and non-increasing functions of the parameter λ ∈ [0, 1], respectively;
(2) If and , then and are continuous non-increasing and non-decreasing functions of the parameter λ ∈ [0, 1], respectively.
Li′s ranking rule ([41]) Let and be two PTrFNs, and are the membership functions of and , and are the non-membership functions of and . Let
and then,
(a) If , then ;
(b) If , then .
Where, λ ∈ [0, 1] denotes experts’ uncertain preference information. λ ∈ (0.5, 1] shows that experts prefer positive information and λ ∈ [0, 0.5) shows that experts prefer negative information.
Example 1. Given three PTrFNs A1 = < (3, 4, 5) ;0.8, 0.4 > , A2 = < (3, 4, 8) ;0.8, 0.4 > , and A3 =< (3, 5, 6) ; 0.75, 0.3>, the ranking results obtained by Li’s method are given as follows:
therefore A1 = A2 < A3 (∀ λ ∈ [0, 1]). This is not reasonable.
Objective penalty function
Objective penalty function ([40]) is a useful tool to solve the constraint programming models. It can convert a constraint programming problem into an unconstrained programming problem in order to easily solve. Considering the following nonlinear constrained optimization problem:
where gi : Rn → R, i = 1, 2, . . . , n. The feasible set of (P′) is denoted by {x ∈ Rn|gi (x) ≤0, i = 1, 2, . . . , n}. To solve the above constrained optimization problem (P′), Meng et al. ([35]) studied the following objective penalty function:
where M ∈ R is an objective penalty parameter. Therefore, the constrained optimization problem (P′) can be converted into the following unconstrained optimization problem:
A new CoCoSo-based ranking method for PTrFNs
The combined compromise solution (CoCoSo) method is a novel method proposed by Yazdani et al.[27] to solve multi-criteria decision-making (MCDM) problem. It applies to three different aggregation strategies to form a complete measure and has the advantage of high flexible. Due to its advantages, the new ranking method for PTrFNs based on CoCoSo method is proposed, which will combine their value index, ambiguity index and the Hamming distance.
Definition 10. Let be a PTrFN, the ratio of value index and ambiguity index is defined as
Theorem 4.Let be a PTrFN, and is an non-negative PTrFN.
(1) If , then is a continuous non-decreasing function of the parameter λ ∈ [0, 1].
(2) If and , then is a continuous non-increasing function of the parameter λ ∈ [0, 1].
Proof. (1) If , then . By Eqs. (15-18), it has that , . Then,
(2) If , then . By Eqs. (15-18), it has that , .
Ranking rule 1 Let and be two PTrFNs, and are the membership functions of and , uA and are the non-membership functions of and . Then,
(a) If , then ,
(b) If , then ,
where, λ ∈ [0, 1] denotes experts’ uncertain preference information. λ ∈ (0.5, 1] shows that experts prefer positive information and λ ∈ [0, 0.5) shows that experts prefer negative information.
Example 2. Given three PTrFNs A1 = < (3, 4, 5) ;0.8, 0.4 > , A2 = < (3, 4, 8) ;0.8, 0.4 > , and A3 =< (3, 5, 6) ; 0.75, 0.3>. By the Eq.(24),
(1) For A1 and A2, if λ ∈ [0, 0.1429), then A2 > A1; if λ ∈ [0.1429, 1], then A1 > A2;
(2) For A1 and A3, if λ ∈ [0, 1], then A3 > A1;
(3) For A2 and A3, if λ ∈ [0, 1], then A3 > A2,
Definition 11. Let be a PTrFN, the Hamming distance-based dominance index is defined as:
where denotes the Hamming distance between and the negative solution , and denotes the Hamming distance between and the ideal solution .
Theorem 5.The Hamming distance-based dominance index fulfills the following properties:
(1) ;
(2) if and only if ;
(3) if and only if .
In addition, the ideal and negative ideal ratings are seen as the points of reference. In general, the ratings that have closer Hamming distances from are more favorable and advantageous. In contrast, the ratings that have farther Hamming distances from are more recommendable. Therefore, the Hamming distance-based dominance index is proposed.
Ranking rule 2 Let and be two PTrFNs, and are the membership functions of and , and are the non-membership functions of and . Then,
(a) If , then ;
(b) If , then ,
where λ ∈ [0, 1] denotes experts’ uncertain preference information. λ ∈ (0.5, 1] shows that experts prefer positive information and λ ∈ [0, 0.5) shows that experts prefer negative information.
Example 3. Given three PTrFNs A1 = < (3, 4, 5) ;0.8, 0.4 > , A2 = < (3, 4, 8) ;0.8, 0.4 > , and A3 =< (3, 5, 6) ; 0.75, 0.3>. Therefore, the ideal solution and the negative solution are given as follows:
Then, by Eq.(14), it has
and by Eq.(25),
(1) For C (A1) and C (A2), if λ ∈ [0, 1], then A2 > A1;
(2) For C (A1) and C (A3), if λ ∈ [0, 1], then A3 > A1;
(3) For C (A2) and C (A3), if λ ∈ [0, 0.6259), then A3 > A2, if λ ∈ [0.6259, 1], then A2 > A3
therefore, the ranking results obtained by Ranking rule 2 are given as follows: A3 > A2 < A1 (∀ λ ∈ [0, 0.6259)) or A3 > A2 < A1 (∀ λ ∈ [0, 0.6259)).
Based on the above definitions, let be m PTrFNs. The steps of the CoCoSo-based ranking method are as follows:
Step 1. Using Eqs.(15)-(18) and (19)-(20), the value index and the ambiguity index are obtained, respectively.
Step 2. For the given weight coefficient λ ∈ [0, 1], using Eq.(24) to obtain the ratio of the value index and the ambiguity index .
Step 3. Obtaining the ideal solution and the negative solution among , then utilizing Eq.(14) to calculate and
Step 4. For the given weight coefficient λ ∈ [0, 1], using Eq.(25) to obtain the Hamming distance-based the dominance index .
Step 5. Then, three relative ranking indexes of can be calculated by three aggregation strategies, which are shown as follows respectively:
where ɛ ∈ [0, 1] is a compromise coefficient. The closer ɛ is to 1, decision makers pay more attention to the external characteristics of a PTFN in the collective. The closer ɛ is to 0, decision makers pay more attention to the intrinsic properties of a PTFN (its value index and the ambiguity index).
Step 6. The final ranking index can be calculated by the following Eq.(29):
According to non-increasing order of , PFTNs can be ranked.
Example 4. Given three PTrFNs A1 = < (3, 4, 5) ;0.8, 0.4 > , A2 = < (3, 4, 8) ;0.8, 0.4 > , and A3 =< (3, 5, 6) ; 0.75, 0.3>. Then, the ranking result by the new CoCoSo-based ranking method and other different ranking rules can be obtained (see in Table 1).
The ranking results of A1, A2 and A3 in different ranking rules
ɛ
λ
Li’s ranking rule
Ranking rule 1
Ranking rule 2
The new CoCoSo-based
ranking method
0.5
0.1
A3 > A2 = A1
A3 > A2 > A1
A3 > A2 > A1
A3 > A2 > A1
0.5
0.2
A3 > A2 = A1
A3 > A1 > A2
A3 > A2 > A1
A3 > A2 > A1
0.5
0.3
A3 > A2 = A1
A3 > A1 > A2
A3 > A2 > A1
A3 > A2 > A1
0.5
0.4
A3 > A2 = A1
A3 > A1 > A2
A3 > A2 > A1
A3 > A2 > A1
0.5
0.5
A3 > A2 = A1
A3 > A1 > A2
A3 > A2 > A1
A3 > A2 > A1
0.5
0.6
A3 > A2 = A1
A3 > A1 > A2
A3 > A2 > A1
A3 > A2 > A1
0.5
0.7
A3 > A2 = A1
A3 > A1 > A2
A2 > A3 > A1
A2 > A3 > A1
0.5
0.8
A3 > A2 = A1
A3 > A1 > A2
A2 > A3 > A1
A2 > A3 > A1
0.5
0.9
A3 > A2 = A1
A3 > A1 > A2
A2 > A3 > A1
A2 > A3 > A1
0.5
1
A3 > A2 = A1
A3 > A1 > A2
A2 > A3 > A1
A2 > A3 > A1
An improved QFD approach
An improved QFD approach is proposed in this section, which integrates PTrFNs, DEMATEL method and the proposed CoCoSo-based ranking method for PTrFNs. The specific steps are as follows in Fig. 3.
The framework of the proposed QFD approach.
The rankings of DAs by taking different β.
For a QFD problem, suppose that there are m customer requirements CRi (i = 1, 2, . . . , m) and n design attributes DAj (j = 1, 2, . . . , n), respectively. Let wi (i = 1, 2, . . . , m) be the weights of CRs. In addition, p experts Ek (k = 1, 2, . . . , p) are invited to assess the relationships between CRs and DAs. The evaluation linguistic and according Pythagorean triangular fuzzy numbers are shown in Table 2.
Linguistic and according Pythagorean triangular fuzzy numbers
Linguistic variables
Pythagorean triangular fuzzy numbers
None (O)
<(0, 0, 0) ;1, 0>
Very Weak (VW)
<(0, 0, 0.3) ; ωVW, uVW>
Weak (W)
<(0, 0.25, 0.5) ; ωW, uW>
Moderate (M)
<(0.25, 0.5, 0.75) ; ωM, uM>
Strong (S)
<(0.5, 0.75, 1) ; ωS, uS>
Very Strong (VS)
<(0.7, 1, 1) ; ωVS, uVS>
1 *ω represents the maximum satisfaction.2 *u represents the minimum dissatisfaction.
Let be the correlation matrix of the expert Ek, where is the correlation degree between CRi and DAj provided by the expert Ek. Based on these assumptions, the main steps of the proposed approach are presented below:
Phase 1. Determining the interrelationships among DAs by an extended DEMATEL method
Step 1 . Determining the initial individual influence matrices between DAs.
The initial individual influence matrices of experts regarding the interrelationships among DAs can be defined by
where, denotes the direct influence degree between CRi and CRj by the expert E2s, which is a Pythagorean triangular fuzzy number and denoted by . In addition, , which means there is no influence between DAi and DAi. Furthermore, plus-minus sign indicates an increase or decrease in the influence relationship.
Step 2 . Determining the total influence matrix between DAs. First, based on the initial individual matrices , the direct influence matrix regarding the interrelationships between DAs can be calculated by
where, In addition, can be divided into 3 n × n matrices, namely and , which are shown as follows:
Then, the total influence matrix can be determined by the following 3 n × n matrices, namely RA, RB and RC, which are calculated by
where and .
Step 3 . Determining the final interrelationship matrix among DAs.
The interrelationship matrix between DAs also can be determined by
where and Similarly, can be divided into 3 matrices, namely , and .
Then, the final interrelationship matrix between DAs can be divided into 3 matrices, namely MA, MB, MC, which are obtained by the following equations:
where and .
Phase 2. Calculating the Pythagorean fuzzy importance of DAs
Step 4 . Conducting two pairs of the unconstraint programming models
First, based on the fuzzy weighted average method and the Pythagorean fuzzy extension principle, the Pythagorean fuzzy importance of DAs can be obtained by the following nonlinear programming models:
where and . . . , n), and are the crisp universal sets of the weights of CRs and the relationships between CRs and DAs, respectively.
Then, based on the theory of cut sets, the α- cut sets and the β- cut sets of Wi and Uij are expressed as follows:
v
Therefore, the programming model (P1) can be converted a pair of the following models (P1a) and (P1b), which are built as follows:
where, α′ ∈ [0, 1] . In addition, , that is 0 ≤ α ≤ ωYj.
Similarly, the programming models (P2) can be converted a pair of constraint programming models (P2a) and (P2b), which are built as follows:
where, β′ ∈ [0, 1] . In addition, , that is uYj ≤ β ≤ 1.
In order to simplify the calculation process, by using the objective penalty function method, the constrained programming models (P1a) , (P1b) , (P2a) and (P2b) can be transformed into the unconstrained programming models (P1a (φ)) , (P1b (φ)) , (P2a (φ)) and (P2b (φ)) respectively, which are shown as follows:
where, φ is a penalty parameter, and
Step 5 . Determining the Pythagorean fuzzy importance of DAs
According to the unconstrained optimization problems with penalty parameters (P1a) , (P1b) , (P2a) , (P2b) and the definition of Pythagorean fuzzy cut set, when α′ = β′, it is obvious that the optimal solution of (P1a) is equal to the optimal solution of (P2a) and the optimal solution of (P1b) is equal to the optimal solution of (P2b), that is
and
Therefore, α′ = β′ = 0 and α′ = β′ = 1 are taken to solve only the models of (P1a (M)) and (P1b (M)), respectively. And then, L0 (Yj) = L1 (Yj) can be obtained by solving the unconstrained optimization problems (P1a (M)) (or (P2a (M))), R0 (Yj) = R1 (Yj) can be obtained by solving the unconstrained optimization problems (P1b (M)) (or (P2b (M))). LωYj (Yj) = RωYj (Yj) = LuYj (Yj) = RuYj (Yj) = a (Yj) can be obtained by solving the unconstrained optimization problems P1a (M) (or P1b (M), or P2a (M), or P2b (M)). Where, and . In addition, let , , LωYj (Yj) = RωYj (Yj) = LuYj (Yj) = RuYj (Yj) = a (Yj), and the Pythagorean fuzzy importance of Yj is denoted as , therefore, its membership function and the non-membership function are shown as follows:
Phase 3. Obtaining the rankings of DAs by an new CoCoSo-based ranking method
Step 6 . Calculating the value and the ambiguity indexes of DAs.
The value indexes V (Yj) (j = 1, 2, . . . , n) and the ambiguity indexes A (Yj) (j = 1, 2, . . . , n) of DAs can be calculated by
where λ ∈ [0, 1] is a weight coefficient.
Step 7 . Computing the ratios of the value index and the ambiguity index regarding DAs.
The ratios of the value index and the ambiguity index R (Yj, λ) (j = 1, 2, . . . , n) regarding DAs can be calculated by
Step 8 . Determining the Hamming distances between the ideal solutions and DAs.
First, the positive ideal solution Y+ and the negative solution Y- regarding DAs are determined as follows, respectively:
Then, the Hamming distances dH (Yj, Y+) (j = 1, 2, . . . , n) between Y+ and DAs are computed by
and the Hamming distances dH (Yj, Y-) (j = 1, 2, . . . , n) between Y- and DAs are computed by
Step 9 . Obtaining the dominance indexes of DAs.
The dominance indexes C (Yj, λ) (j = 1, 2, . . . , n) of DAs are computed by
Step 10 . Determining three relative ranking indexes of DAs. C (Yj, λ) (j = 1, 2, . . . , n) Three relative ranking indexes K1 (Yj), K2 (Yj) and K3 (Yj) (j = 1, 2, . . . , n) of DAs can be calculated by the following expressions, respectively:
where ɛ ∈ [0, 1] is a compromise coefficient.
Step 11 . Determining the final ranking indexes of DAs.
Finally, the final ranking indexes K (Yj, λ, ɛ) (j = 1, 2, . . . , n) of DAs can be obtained by
The larger the value of K (Yj, λ, ɛ) and the higher the ranking order of DAj.
Illustrative example
In this section, a case study of mobile medical App design is performed to proof the efficacy and accuracy of the proposed QFD approach.
Problem description
With the increasing improvement of living standard, people’s demand for health information is increasing. The development of the Internet has changed the way people seek health and medical information, and the medical App has become an important channel for the public to understand medical and health knowledge. In this section, a mobile medical App- “Ping An Health”, which is a wholly-owned subsidiary of Ping An Group of China, is adopted to verify the feasibility and effectiveness of the proposed model. With the increasing demand for health care, for the developers of “Ping An Health” App, its performance and quality need to continually improved for satisfy customer requirements. However, due to limited resources, it is difficult to optimize all aspects. Therefore, the goal of this study is to prioritize App’s DAs through CRs.
Implementation
To design and optimize “Ping An Health” App, it is necessary to obtain CRs. The steps of that are as follows: First, crawling online reviews. We use Python software to crawl online reviews regarding “Ping An Health” App from the app store. Second, pre-processing the crawled online comments. It mainly includes deleting repeated and invalid comments, modifying wrong sentences, and deleting special symbols. Third, extracting high-frequency words and classifying them. Then, ten customers requirements CRm (m = 1, 2, . . . , 10) are obtained as shown in Table 3. Based on the CRs, refering to the SERVQUAL model, seven design attributes DAn (n = 1, 2, . . . , 7) are obtained and their symbolic representation and definitions are shown in Table 4. In the evaluation process, the relationships among CRs and DAs is conducted by five experts Ek (k = 1, 2, . . . , 5). These experts possess relevant professional knowledge and have worked in medical field for more than three years. Here, the pre-defined Pythagorean fuzzy set is used to assess the weights of CRs, which are shown as follows: . Furthermore, the pre-defined Pythagorean fuzzy set U* is adopted to express the relationships between CRs and DAs, which are shown as follows: . As a result of questionnaire survey and expert interview, the weights of CRs and the relationships between CRs and DAs are determined. Next, the detailed steps of the proposed improved QFD approach are shown as follows.
Comprehensive, Weight loss, Departments, Complete, Health care, Beauty, Health...
Timely Feedback
CR7
In time, Timely, Speed...
Authenticity of information
CR8
Conscience, Trust, Accuracy...
Articles are easy to understand
CR9
Knowledge, Popular science, Articles, Practical...
Network stabilization
CR10
Internet, Traffic, Online...
Design attributes (DAs) and their definitions
Design attributes (DAs)
Symbol
Definitions
Perceptibility
DA1
Perceptibility refers to the user’s perception of the quality of the App’s hardware and software, as well as the external perception. Including system fluency, interface comfort, network stability and so on.
Reliability
DA2
Reliability refers to the ability of an App to fulfill its promise to users. It includes curative effect, extensive medical coverage, information authenticity, advertising rationality and so on.
Economy
DA3
Economy refers to the fact that users spend the least cost or gain the most benefit for the services and products they enjoy on the App platform. Including pricing rationality, user profitability and so on.
Assurance
DA4
Assurance refers to the knowledge, reputation, and ability to express trustworthiness of the App platform. Including physician authority, platform confidentiality and so on.
Informedness
DA5
Informaedness refers to the fact that the information provided by the App platform meets users’ needs. It includes content diversity, intelligibility of tweets, content clarity, information attribution, etc.
Empathy
DA6
Empathy refers to the ability of App platform providers to care about users and provide personalized services for users. Including service friendliness, feedback timeliness, product service satisfaction, service personalization, etc.
Usablity
DA7
Usability refers to the degree to which users can achieve their goals efficiently, effectively, and satisfactorily. Including platform convenience, content practicability, etc.
Phase 1. Determining the interrelationships among DAs by an extended DEMATEL method
Step 1 . Utilizing Eq. (30) to obtain the initial individual influence matrices of experts regarding the interrelationships among DAs. As an example, the initial individual influence matrix given by expert E1 is exhibited in Table 5.
Step 2 . Eqs. (31)-(32) is employed to determine the total influence matrix R = (rjk) n×n between DAs, as shown in Table 6.
Step 3 . Using Eqs. (33)-(34) to determine the final interrelationship matrix M = (mjk) n×n among DAs, as shown in Table 7.
The initial individual influence matrix among DAs given by E1
DA1
DA2
DA3
DA4
DA5
DA6
DA7
DA1
<N;1,0> (+)
<W;0.8,0.3> (+)
<M;0.8,0.3> (-)
<M;0.7,0.4> (+)
<M;0.85,0.2> (+)
<M;0.9,0.15> (+)
<VS;0.85,0.25> (+)
DA2
<VW;0.9,0.2> (+)
<N;1,0> (+)
<W;0.65,0.5> (+)
<VS;0.9,0.2> (+)
<VW;0.8,0.3> (+)
<W;1,0> (+)
<S;0.9,0.1> (+)
DA3
<VS;0.8,0.3> (-)
<M;0.95,0.2> (-)
<N;1,0> (+)
<VW;0.8,0.15> (+)
<W;0.75,0.4> (-)
<W;0.8,0.3> (-)
<S;0.8,0.3> (+)
DA4
<S;0.75,0.5> (+)
<S;0.9,0.15> (+)
<W;1,0> (-)
<N;1,0> (+)
<VW;0.9,0.15> (+)
<VW;0.85,0.2> (+)
<VS;1,0> (+)
DA5
<VW;0.85,0.35> (+)
<VW;1,0> (+)
<VW;0.85,0.3> (-)
<N;0.7,0.35> (+)
<N;1,0> (+)
<VW;0.7,0.4> (+)
<S;0.7,0.45> (+)
DA6
<N;0.9,0.25> (+)
<VW;0.8,0.3> (+)
<M;0.7,0.45> (-)
<VW;0.8,0.25> (+)
<N;0.9,0.2> (+)
<N;1,0> (+)
<S;0.9,0.15> (+)
DA7
<S;0.95,0.1> (+)
<S;0.85,0.2> (+)
<M;1,0> (-)
<M;0.95,0.15> (+)
<VW;1,0> (+)
<S;0.8,0.25> (+)
<N;1,0> (+)
The total influence matrix among DAs
DA1
DA2
DA3
DA4
DA5
DA6
DA7
DA1
< (0,0,0);1,0> (+)
< (0.1,0.35,0.6); 0.8,0.3> (+)
< (0.15,0.35,0.61); 0.75,0.35> (-)
< (0.25,0.5,0.75); 0.7,0.4> (+)
< (0.15,0.4,0.65); 0.85,0.2> (+)
< (0.05,0.3,0.55); 0.9,0.15> (+)
< (0.62,0.9,1); 0.7,0.45> (+)
DA2
< (0.1,0.2,0.42); 0.8,0.3> (+)
< (0,0,0);1,0> (+)
< (0,0.25,0.5); 0.65,0.5> (-)
< (0.66,0.95,1.0); 0.85,0.25> (+)
< (0,0,0.24); 0.75,0.35> (+)
< (0.15,0.4,0.65); 0.7,0.5> (+)
< (0.62,0.9,1); 0.9,0.15> (+)
DA3
< (0.49,0.75,0.95); 0.8,0.3> (-)
< (0.1,0.2,0.42); 0.85,0.25> (-)
< (0,0,0);1,0> (+)
< (0,0,0.3); 0.7,0.45> (+)
< (0,0.25,0.5); 0.75,0.4> (-)
< (0.1,0.35,0.6); 0.8,0.3> (-)
< (0.4,0.65,0.9); 0.65,0.4> (+)
DA4
< (0.58,0.85,1); 0.75,0.5> (+)
< (0.66,0.95,1); 0.8,0.35> (+)
< (0,0.25,0.5); 0.7,0.45> (-)
< (0,0,0);1,0> (+)
< (0.05,0.1,0.39); 0.7,0.35> (+)
< (0,0,0.24); 0.85,0.2> (+)
< (0.62,0.9,1); 0.85,0.25> (+)
DA5
< (0.15,0.3,0.57); 0.75,0.35> (+)
< (0.2,0.4,0.66); 0.85,0.3> (+)
< (0,0,0.3); 0.85,0.3> (-)
< (0.05,0.1,0.21); 0.7,0.35> (+)
< (0,0,0);1,0> (+)
< (0.05,0.1,0.39); 0.7,0.4> (+)
< (0.44,0.7,0.9); 0.7,0.45> (+)
DA6
< (0.05,0.1,0.21); 0.9,0.25> (+)
< (0.05,0.1,0.39); 0.75,0.4> (+)
< (0.15,0.4,0.65); 0.7,0.45> (-)
< (0,0,0.24); 0.8,0.25> (+)
< (0.05,0.1,0.33); 0.9,0.2> (+)
< (0,0,0);1,0> (+)
< (0.49,0.75,0.95); 0.9,0.15> (+)
DA7
< (0.15,0.35,0.61); 0.8,0.3> (+)
< (0.54,0.8,1); 0.75,0.3> (+)
< (0.15,0.3,0.51); 0.85,0.25> (+)
< (0.25,0.5,0.75); 0.85,0.2> (+)
< (0,0,0.3); 0.85,0.3> (+)
< (0.54,0.8,1); 0.8,0.25> (+)
< (0,0,0);1,0> (+)
The final interrelationship matrix M among DAs
DA1
DA2
DA3
DA4
DA1
< (0,0,0);1,0>
< (0.0909,0.2881,0.7278);0.80,0.30>
< (0.1515,0.3230,0.7370);0.75,0.35>
< (0.2075,0.4215,0.8308);0.70,0.50>
DA2
< (0.0909,0.2881,0.7278);0.80,0.30>
< (0,0,0);1,0>
< (0.0431,0.2110,0.6370);0.65,0.50>
< (0.3264,0.5571,0.8973);0.80,0.35>
DA3
< (0.1515,0.3230,0.7370);0.75,0.35>
< (0.0431,0.2110,0.6370);0.65,0.50>
< (0,0,0);1,0>
< (0.0227,0.1667,0.5943);0.70,0.45>
DA4
< (0.2075,0.4215,0.8308);0.70,0.50>
< (0.3264,0.5571,0.8973);0.80,0.35>
< (0.0227,0.1667,0.5943);0.70,0.45>
< (0,0,0);1,0>
DA5
< (0.0741,0.2038,0.5892);0.75,0.35>
< (0.0612,0.1592,0.5457);0.75,0.35>
< (0.0080,0.1053,0.4741);0.75,0.40>
< (0.0394,0.1200,0.4785);0.70,0.35>
DA6
< (0.0470,0.1920,0.5769);0.90,0.25>
< (0.0745,0.2184,0.6326);0.70,0.50>
< (0.0710,0.2405,0.6190);0.70,0.45>
< (0.0313,0.1220,0.5221);0.80,0.25>
DA7
< (0.2060,0.4440,0.9220);0.70,0.45>
< (0.3001,0.5477,1.0000);0.75,0.30>
< (0.1475,0.3450,0.8176);0.65,0.40>
< (0.2505,0.5009,0.9491);0.85,0.25>
DA5
DA6
DA7
DA1
< (0.0741,0.2038,0.5892);0.75,0.35>
< (0.0470,0.1920,0.5769);0.90,0.25>
< (0.2060,0.4440,0.9220);0.70,0.45>
DA2
< (0.0612,0.1592,0.5457);0.75,0.35>
< (0.0745,0.2184,0.6326);0.70,0.50>
< (0.3001,0.5477,1.0000);0.75,0.30>
DA3
< (0.0080,0.1053,0.4741);0.75,0.40>
< (0.0710,0.2405,0.6190);0.70,0.45>
< (0.1475,0.3450,0.8176);0.65,0.40>
DA4
< (0.0394,0.1200,0.4785);0.70,0.35>
< (0.0313,0.1220,0.5221);0.80,0.25>
< (0.2505,0.5009,0.9491);0.85,0.25>
DA5
< (0,0,0);1,0>
< (0.0351,0.1012,0.4512);0.70,0.40>
< (0.1151,0.2378,0.6719);0.70,0.45>
DA6
< (0.0351,0.1012,0.4512);0.70,0.40>
< (0,0,0);1,0>
< (0.2446,0.4371,0.8669);0.80,0.25>
DA7
< (0.1151,0.2378,0.6719);0.70,0.45>
< (0.2446,0.4371,0.8669);0.80,0.25>
< (0,0,0);1,0>
Phase 2. Calculating the Pythagorean fuzzy importance of DAs
Step 4 . According to pre-defined Pythagorean fuzzy sets W* and U*, the weights of CRs and the relationships between CRs and DAs are determined by experts, which are shown in Tables 8 and 9, respectively. Then, the unconstrained programming models can be constructed by using (P1a (φ)) , (P1b (φ)) , (P2a (φ)) and (P2b (φ)).
The weights of CRs
CRs
The weights(Wi)
CRs
The weights(Wi)
CR1
< (0.675,0.875,1);0.8,0.2>
CR6
< (0.25,0.425,0.6);0.8,0.2>
CR2
< (0.6375,0.8313,1);0.8,0.25>
CR7
< (0.15,0.3062,0.4875);0.75,0.3>
CR3
< (0.4125,0.5938,0.775);0.9,0.2>
CR8
< (0.475,0.65,0.825);0.8,0.25>
CR4
< (0.3625,0.5563,0.75);0.75,0.3>
CR9
< (0.75,0.95,1);0.8,0.25>
CR5
< (0.275,0.4437,0.6125);0.8,0.25>
CR10
< (0.2625,0.4437,0.625);0.8,0.2>
The relationships between CRs and DAs
DA1
DA2
DA3
DA4
CR1
< (0.0996, 0.3529, 0.9132); 0.7, 0.5>
< (0.1117, 0.3348, 0.8327); 0.65, 0.5>
< (0.0409, 0.2224, 0.7428); 0.65, 0.5>
< (0.0913, 0.2897, 0.7845); 0.7, 0.5>
CR2
< (0.1004, 0.3558, 0.9420); 0.7, 0.5>
< (0.1083, 0.3321, 0.8576); 0.65, 0.5>
< (0.0389, 0.2176, 0.7428); 0.65, 0.5>
< (0.0879, 0.2856, 0.8078); 0.7, 0.5>
CR3
< (0, 0.0320, 0.3507); 0.7, 0.5>
< (0.0130, 0.0872, 0.4548); 0.65, 0.5>
< (0.0217, 0.0929, 0.4209); 0.65, 0.5>
< (0.0297, 0.1196, 0.4685); 0.7, 0.5>
CR4
< (0.0151, 0.0982, 0.4699); 0.7, 0.5>
< (0.0279, 0.1377, 0.5312); 0.65, 0.5>
< (0.0301, 0.1326, 0.5170); 0.65, 0.5>
< (0.0566, 0.1959, 0.5852); 0.7, 0.5>
CR5
< (0.0027, 0.0473, 0.3720); 0.7, 0.5>
< (0.0169, 0.1086, 0.4832); 0.65, 0.5>
< (0.0236, 0.0984, 0.4310); 0.65, 0.5>
< (0.0330, 0.1290, 0.4815); 0.7, 0.5>
CR6
< (0.0127, 0.1218, 0.5229); 0.7, 0.5>
< (0.0135, 0.1248, 0.5254); 0.65, 0.5>
< (0.0126, 0.1004, 0.4912); 0.65, 0.5>
< (0.0348, 0.1527, 0.5472); 0.7, 0.5>
CR7
< (0.0348, 0.1547, 0.5878); 0.7, 0.5>
< (0.0355, 0.1430, 0.5396); 0.65, 0.5>
< (0.0195, 0.1235, 0.5467); 0.65, 0.5>
< (0.0604, 0.1958, 0.5994); 0.7, 0.5>
CR8
< (0.0127, 0.0880, 0.4449); 0.7, 0.5>
< (0.0255, 0.1375, 0.5398); 0.65, 0.5>
< (0.0217, 0.1020, 0.4750); 0.65, 0.5>
< (0.0330, 0.1280, 0.5080); 0.7, 0.5>
CR9
< (0.0063, 0.0929, 0.4700); 0.7, 0.5>
< (0.0126, 0.1206, 0.5254); 0.65, 0.5>
< (0.0133, 0.0999, 0.4728); 0.65, 0.5>
< (0.0163, 0.1183, 0.5070); 0.7, 0.5>
CR10
< (0.0191, 0.1086, 0.5040); 0.7, 0.5>
< (0.0362, 0.1491, 0.5254); 0.65, 0.5>
< (0.0277, 0.1327, 0.5466); 0.65, 0.5>
< (0.0350, 0.1392, 0.5451); 0.7, 0.5>
DA5
DA6
DA7
CR1
< (0.0357, 0.1417, 0.5929); 0.7, 0.45>
< (0.0598, 0.2117, 0.6865); 0.7, 0.5>
< (0.1366, 0.3955, 0.9681); 0.65, 0.45>
CR2
< (0.0342, 0.1410, 0.6114); 0.7, 0.45>
< (0.0574, 0.2101, 0.7107); 0.7, 0.5>
< (0.1425, 0.4115, 1); 0.65, 0.45>
CR3
< (0.0106, 0.0573, 0.3434); 0.7, 0.45>
< (0.0067, 0.0575, 0.3619); 0.7, 0.5>
< (0.0295, 0.1335, 0.5485); 0.65, 0.45>
CR4
< (0.0189, 0.0879, 0.4283); 0.7, 0.45>
< (0.0226, 0.1105, 0.4702); 0.7, 0.5>
< (0.0421, 0.1802, 0.6296); 0.65, 0.45>
CR5
< (0.0121, 0.0635, 0.3566); 0.7, 0.45>
< (0.0099, 0.0733, 0.3889); 0.7, 0.5>
< (0.0295, 0.1272, 0.5401); 0.65, 0.45>
CR6
< (0.0087, 0.0654, 0.3892); 0.7, 0.45>
< (0.0099, 0.0955, 0.4745); 0.7, 0.5>
< (0.0410, 0.1943, 0.6769); 0.65, 0.45>
CR7
< (0.0171, 0.0773, 0.4120); 0.7, 0.45>
< (0.0289, 0.1213, 0.5206); 0.7, 0.5>
< (0.0540, 0.1950, 0.6753); 0.65, 0.45>
CR8
< (0.0128, 0.0623, 0.3691); 0.7, 0.45>
< (0.0179, 0.0878, 0.4357); 0.7, 0.5>
< (0.0218, 0.1216, 0.5611); 0.65, 0.45>
CR9
< (0.0063, 0.0563, 0.3621); 0.7, 0.45>
< (0.0047, 0.0708, 0.4052); 0.7, 0.5>
< (0.0313, 0.1557, 0.6097); 0.65, 0.45>
CR10
< (0.0161, 0.0768, 0.4342); 0.7, 0.45>
< (0.0087, 0.0657, 0.4154); 0.7, 0.5>
< (0.0740, 0.2550, 0.7598); 0.65, 0.45>
Step 5 . Let α′ = β′ = 0 and α′ = β′ = 1, the Pythagorean fuzzy importance of DAs can be obtained as showns:
Y1 = < (0.0269, 0.1609, 0.6358) ;0.7, 0.5 > ,
Y2 = < (0.0361, 0.1781, 0.639) ;0.7, 0.5 > ,
Y3 = < (0.0221, 0.1384, 0.5813) ;0.7, 0.5 > ,
Y4 = < (0.0435, 0.1812, 0.6252) ;0.7, 0.5 > ,
Y5 = < (0.0159, 0.0866, 0.4657) ;0.65, 0.45 > ,
Y6 = < (0.0193, 0.115, 0.5259) ;0.7, 0.5 > ,
Y7 = < (0.0574, 0.2324, 0.7578) ;0.65, 0.45 > ,
Phase 3. Obtaining the rankings of DAs by an new CoCoSo-based ranking method
Step 6 . Let λ = 0.5, by using Eqs. (37)-(38), the value indexes V (Yj) (j = 1, 2, . . . , n) and the the ambiguity indexes A (Yj) (j = 1, 2, . . . , n) of DAs are obtained as shown in Table 10.
Step 7 . Utilizing Eq. (39), the ratios of the value index and the ambiguity index R (Yj) (j = 1, 2, . . . , n) of DAs can be calculated as shown in Table 10.
Step 8 . Based on Eqs. (42)-(43), the Hamming distances between the positive ideal solution and DAs dH (Yj, Y+) (j = 1, 2, . . . , n), and the Hamming distances between the negative ideal solution and DAs dH (Yj, Y-) (j = 1, 2, . . . , n) are obtained as shown in Table 10.
Step 9 . According to Eq. (44), the dominance indexes of DAs C (Yj, λ) (j = 1, 2, . . . , n) are shown in Table 10.
Step 10 . Using Eqs. (45)-(47), three relative ranking indexes K1 (Yj), K2 (Yj) and K3 (Yj) (j = 1, 2, . . . , n) of DAs are shown in Table 10.
The index results of DAs in Step 6-10
DA1
DA2
DA3
DA4
DA5
DA6
DA7
V (Yj)
0.0502
0.0538
0.0442
0.0541
0.0302
0.0381
0.0667
A (Yj)
0.0188
0.0186
0.0172
0.0180
0.0136
0.0156
0.0212
R (Yj)
0.0493
0.0528
0.0435
0.0532
0.0298
0.0375
0.0654
dH (Yj, Y+)
0.0523
0.0453
0.068
0.0453
0.1018
0.0837
0.008
dH (Yj, Y-)
0.0535
0.0606
0.0379
0.0605
0.0041
0.0222
0.0979
C (Yj, λ)
0.5057
0.5722
0.3579
0.5718
0.0387
0.2096
0.9245
K1 (Y1)
0.1580
0.1780
0.1143
0.1780
0.0195
0.0704
0.2819
K2 (Y2)
14.7200
16.5578
10.7060
16.5608
2.0017
6.6752
26.0813
K3 (Y3)
0.5606
0.6314
0.4054
0.6314
0.0693
0.2497
0.9999
Step 11 . Let ɛ = 0.5, according to Eq. (48), the final ranking indexes K (Yj, λ, ɛ) (j = 1, 2, . . . , n) are computed as: K (Y1, λ, ɛ) =6.2387, K (Y2, λ, ɛ) =7.0190, K (Y3, λ, ɛ) =4.5335, K (Y4, λ, ɛ) =7.0201, K (Y5, λ, ɛ) =0.8362, K (Y6, λ, ɛ) =2.8212, K (Y7, λ, ɛ) =11.0654. The ranking orders of DAs are: DA7 ≻ DA4 ≻ DA2 ≻ DA1 ≻ DA3 ≻ DA6 ≻ DA5. Thus, the top DA is perceptibility. Therefore, perceptibility should be focus on firstly in the “Ping An Health” App design, such as system fluency, interface comfort and network stability.
Sensitivity analysis
Sensitive analysis is carried out for observing the effects of parameters β, γ, ɛ on the final ranking results in this subsection. In the process of each sensitive analysis, we maintain the target parameter varied while other parameters unchanged. Let the initial values of parameters be β = 0.5, γ = 0.5, ɛ = 0.5, respectively.
(1) Effects of risk coefficient β on the ranking results.
When γ = 0.5 and ɛ = 0.5, the compromise ratio of DAs by different β are shown in Fig. 3 (x-axis denoted the weight coefficient β). And there are two permutations of ranking orders for the seven DAs, that is,
(2) Effects of weight coefficient λ on the ranking results.
When β = 0.5 and ɛ = 0.5, the compromise ratio of DAs by different γ are shown in Fig. 4 (x-axis denoted the weight coefficient γ). And there are three permutations of ranking orders for the seven DAs, that is,
(3) Effects of compromise coefficient ɛ on the ranking results.
When γ = 0.5 and β = 0.5, the compromise ratio of DAs by different ɛ are shown in Fig. 5 (x-axis denoted the weight coefficient ɛ). And there are three permutations of ranking orders for the seven DAs, that is,
From Figs. 4–5, the following results can be obtained: (1) If the values of γ and ɛ are fixed, we can find that the ranking orders of DAs are different with the different values of β. This implies that experts’ uncertain preference affects the ranking orders of DA2 and DA4; (2) If the values of β and ɛ are fixed, we can find that the ranking orders of DAs are different with the different values of γ. This implies that the more positive the expert is, the higher the DA2 ranking; (3) If the values of β and γ are fixed, we can find that ɛ in the new CoCoSo-based ranking method has little influence on the ranking orders of DAs. This implies that the new CoCoSo-based ranking method has great robustness and stability.
The rankings of DAs by taking different γ.
The rankings of DAs by different ɛ.
Comparison analysis
To verify the effectiveness and benefits of the proposed QFD approach, we conduct a comparison analysis with some existing methods, which includes the traditional method ([10]), the fuzzy expected operator ([12]) and the fuzzy TOPSIS method ([42]). By these methods in same case, the rankings of the seven DAs in the four considered methods are obtained as shown in Table 11.
Comparative rankings of ECs for the considered case study
From Table 11, it is obvious that DA7 is the most worthy of attention and DA5 are the least worthy of attention based on the three comparative methods and the proposed method. What’s more, the ranking orders of DA1, DA3, DA5, DA6 and DA7 are same with the three comparative methods and the proposed novel method. In addition, the ranking acquired by the proposed novel method is consistent with the ranking by the fuzzy TOPSIS method. Therefore, it proves that the validity and effectiveness of the proposed novel method to priority DAs in QFD. But, there are differences between the ranking obtained by the traditional QFD method, the fuzzy expected operator and the proposed novel method. In the traditional QFD method and the fuzzy expected operator method, DA4 ranks before DA2. On the contrary, DA2 ranks before DA4 in the proposed novel method. The main reason for the discrepancy is the distortion and loss information in the domain experts’ assessment. In the traditional QFD, the fuzziness of information is not taken into account. And in the fuzzy expected value operator, the nonmembership degree are not taken into account.
Compared with these methods, the proposed QFD method has the following highlighted features: (1) First, using Pythagorean triangular fuzzy numbers, experts’ hesitant assessment information can be expressed more accurately to priority DAs in QFD. This is something that the traditional QFD method cannot handle. (2) Second, the extended DEMATEL method is used to consider the interrelationships among DAs, which improves the accuracy of the results with respect to the ranking of DAs in QFD. (3) Third, utilizing the combined method of the weighted average and objective penalty function to solve the constraint programming models, the calculation process is simplified. (4) Forth, by the extended compromise ratio method with weighted Hamming distance, the problem that the TOPSIS method can not approach the positive ideal solution while staying away from the negative ideal solution, can be solved. (5) Fifth, the proposed ranking method can better reflect the preference information of domain experts, the different of the preference coefficient can lead to the different ranking of DAs. The detailed comparisons among the proposed QFD approach and other considered QFD methods are described in Table 12.
Comparision with other considered QFD methods
Methods
Expressing experts’ hesitance information
Handling the interrelations among DAs
Using different aggregation strategies in prioritizing DAs
Considering experts’ preference information in prioritizing DAs
In this paper, a new QFD approach by a CoCoSo-based ranking method in Pythagorean fuzzy environment is proposed. In the first phase, an extended DEMATEL method, called PTrF-DEMATEL method, is proposed to determine the interrelationships among DAs. In the second phase, the aggregation method of the weighted average method and objective penalty function is proposed to construct programming models to obtain the Pythagorean fuzzy importance of DAs. Then, a new CoCoSo-based ranking method for PTrFNs is utilized to obtain the rankings of DAs by considering different preferences of QFD experts. Finally, a case study of “Ping An Health” App design is introduced to verify the effectiveness and benifits of the proposed QFD method. The results shown that the proposed method can better describe the uncertainty and vagueness of domain experts’ evaluations and obtain more accurate rankings by considering different preference information of experts. In this case, the top DA is perceptibility. Therefore, perceptibility should be focus on firstly in the “Ping An Health” App design, such as system fluency, interface comfort and network stability.
The new QFD approach may provide more efficient and accurate solutions that can identify and solve real-world QFD problems more quickly. Additionally, it may lead to new research directions, i.e. the applying in QFD by newly MCDM methods or hybrid MCDM methods. However, there are still some limitations of the proposed QFD approach. First, the interrelations among CRs are not taken into account in the proposed QFD approach; Second, the experts from various departments usually have different backgrounds, knowledge structures and interests. As a result, their assessments in correlations between CRs and ECs are often inconsistent. But the proposed QFD method cannot consider this case. Third, the proposed QFD approach is complex which may induce inconvenience while executing it in real applications. Therefore, in the future, further exploration can be made in the following aspects: First, the interrelations among CRs will be considered by the cooperative game theory; Second. the consensus model can be introduced to reach consensus in correlation assessments between CRs and ECs for improving the performance of QFD method; Third, in the future, it is important to develop a software tool which enables practitioners to implement the proposed QFD algorithm easily.
Acknowledgments
The authors are grateful to the associate editor and the anonymous referees for their valuable comments and suggestions that significantly improved the paper.
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