## 1 Introduction

*et al.*, 2006). Nowadays, the QFD has become a powerful tool for designing and developing products or services (Huang

*et al.*, 2019). It can not only improve customer satisfaction, but also reduce cycle-time of product development, cut down production cost and enhance the performance of manufacturing process. Due to its effectiveness and benefits, the QFD method has been applied for product design and quality improvement in various areas, such as construction (Fargnoli

*et al.*, 2020; Lapinskienė and Motuzienė, 2021), manufacturing (Neramballi

*et al.*, 2020; Shi and Peng, 2020) and service (Lee

*et al.*, 2020; Park

*et al.*, 2021) industries.

*et al.*, 2019; Jia

*et al.*, 2016; Ping

*et al.*, 2020; Wu

*et al.*, 2020). On the one hand, crisp values are utilized to deal with the relationships between CRs and ECs in the traditional QFD. However, in the real world, it is often hard for experts to give accurate numerical values on the relationships between CRs and ECs due to the uncertainty and fuzziness of human perception (Aliev and Huseynov, 2014; Lorkowski

*et al.*, 2014). Instead, they prefer to use linguistic terms to express their opinions (Liu

*et al.*, 2019; Tian

*et al.*, 2019; Liu

*et al.*, 2021). Based on linguistic term sets (Zadeh, 1975) and Z-numbers (Zadeh, 2011), the concept of linguistic Z-numbers was introduced by Wang

*et al.*(2017) to express both vagueness and randomness of uncertain linguistic information. For a linguistic Z-number, the two components (i.e. restriction and reliability measure) are represented with linguistic terms. Compared to other linguistic computing methods, the linguistic Z-numbers can not only describe decision-making information more flexibly, but also avoid the distortion and loss of original information effectively (Peng and Wang, 2017; Wang

*et al.*, 2017). Hence, it is promising to employ the linguistic Z-numbers to represent experts’ uncertain and vague evaluation information on the relationships between CRs and ECs in QFD.

*et al.*, 2020). Accordingly, many MCDM methods have been adopted to improve the performance of QFD in previous researches (Mistarihi

*et al.*, 2020; Ocampo

*et al.*, 2020; Yazdani

*et al.*, 2016). As an effective MCDM method, the evaluation based on distance from average solution (EDAS) was put forward by Ghorabaee

*et al.*(2015) to address MCDM problems. The EDAS includes two measures, i.e. positive distance from average (PDA) and negative distance from average (NDA), for dealing with the desirability of alternatives (Keshavarz-Ghorabaee

*et al.*, 2018; Keshavarz Ghorabaee

*et al.*, 2017). It has simple logic and is especially useful for decision making problems with conflicting criteria (Darko and Liang, 2020). Since its introduction, the EDAS method has been broadly adopted to solve MCDM problems in many areas, which include supplier selection (Ghorabaee

*et al.*, 2016), manufacturer selection (Stević

*et al.*, 2018), health-care waste disposal technology selection (Ju

*et al.*, 2020), typhoon disaster assessment (Tan and Zhang, 2021), car selection (Yanmaz

*et al.*, 2020) and logistics centre location (Özmen and Aydoğan, 2020). Therefore, it is of vital importance to adopt the EDAS method to determine the ranking of ECs in QFD analysis.

## 2 Literature Review

*et al.*(2014) introduced analytical network process (ANP) into QFD for deciding the importance of CRs and evaluating corresponding characteristics of software quality. Jia

*et al.*(2016) proposed an integrated QFD method combining fuzzy evidence reasoning theory with fuzzy discrete Choquet integral to determine the importance value of design characteristics. Integrating QFD and decision-making trial and evaluation laboratory (DEMATEL) technique, Ramezankhani

*et al.*(2018) designed a hybrid model and applied it to supply chain performance measurement. Liu

*et al.*(2019) proposed a QFD approach by using partitioned Bonferroni mean operator and interval type-2 fuzzy sets to select a better solution from various green suppliers. Vats and Vaish (2019) employed QFD in combination with vlseKriterijumska optimisacija I kompromisno resenje (VIKOR) approach to select smart materials for thermal energy efficient architecture. Based on cloud model and grey relational analysis, Wang

*et al.*(2020) put forward an integrated QFD model to control the quality of the rotor used in air compressor. Via integrating interval type-2 fuzzy sets and DEMATEL, Yazdani

*et al.*(2020) designed a QFD approach to evaluate and rank sustainable supply chain drivers.

*et al.*(2016) developed a QFD model for the product development of electric vehicle. Combining fuzzy BWM, fuzzy maximizing deviation method and fuzzy multi-objective optimization by ratio analysis plus the full multiplicative form (MULTIMOORA), Tian

*et al.*(2018) established a hybrid QFD model to assess the performance of a smart bike-sharing program. Huang

*et al.*(2019) constructed a QFD technique to design manufacture system, in which the relationships between CRs and ECs were expressed by hesitant fuzzy linguistic term sets, the weights of CRs were derived by the best-worst method (BWM) and the ranking order of ECs were determined by the prospect theory. Lu

*et al.*(2019) presented a QFD model by combining fuzzy analytic hierarchy process (AHP) with fuzzy ANP for the design of brand revitalization. A QFD model integrating TOPSIS and EDAS method was proposed by Ping

*et al.*(2020) for a product-service system design.

## 3 Preliminaries

##### Definition 1 (Duan *et al.* 2019)*.*

*t*is a nonnegative integer. In this linguistic term set

*S*, ${s_{i}}$ and ${s_{j}}$ are required to satisfy the following characteristics:

##### Definition 2 (Peng and Wang 2017)*.*

*et al.*, 2016).

*a*is a special value obtained from experiments or subjective methods.

##### Definition 3 (Wang *et al.* 2017)*.*

*X*be a universe of discourse, ${S_{1}}=\{{s_{0}},{s_{1}},\dots ,{s_{2l}}\}$, ${S_{2}}=\{{s^{\prime }_{0}},{s^{\prime }_{1}},\dots ,{s^{\prime }_{2k}}\}$ are two finite and totally ordered uncontinuous linguistic term sets, with nonnegative integers

*l*and

*k*. Furthermore, let ${A_{\phi (x)}}\in {S_{1}}$ and ${B_{\varphi (x)}}\in {S_{2}}$. A linguistic Z-number set

*Z*in

*X*can be denoted in the following form: where ${A_{\phi (x)}}$ is a fuzzy restriction on the values that uncertain variable

*x*is allowed to take, and ${B_{\varphi (x)}}$ is a measure of reliability of ${A_{\phi (x)}}$.

*X*has only one element, the linguistic Z-number set Z is reduced to $({A_{\phi (\alpha )}},{B_{\varphi (\alpha )}})$. For convenience, ${z_{\alpha }}=({A_{\phi (\alpha )}},{B_{\varphi (\alpha )}})$ is called a linguistic Z-number.

##### Definition 4 (Wang *et al.* 2017)*.*

- (1) $\begin{array}[t]{l}{z_{i}}\oplus {z_{j}}\\ {} \hspace{1em}=\big({f^{\ast -1}}({f^{\ast }}({A_{\phi (i)}})+{f^{\ast }}({A_{\phi (j)}})),{g^{\ast -1}}\big(\frac{{f^{\ast }}({A_{\phi (i)}})\times {g^{\ast }}({B_{\varphi (i)}})+{f^{\ast }}({A_{\phi (j)}})\times {g^{\ast }}({B_{\varphi (i)}})}{{f^{\ast }}({A_{\phi (i)}})+{f^{\ast }}({A_{\phi (j)}})}\big)\big);\end{array}$
- (2) $\lambda {z_{i}}=({f^{\ast -1}}(\lambda {f^{\ast }}({A_{\phi (i)}})),{B_{\varphi (i)}})$, where $\lambda \geqslant 0$;
- (3) ${z_{i}}\otimes {z_{j}}=({f^{\ast -1}}({f^{\ast }}({A_{\phi (i)}}){f^{\ast }}({A_{\phi (j)}})),{g^{\ast -1}}({g^{\ast }}({B_{\varphi (i)}}){g^{\ast }}({B_{\varphi (j)}})))$;
- (4) ${z_{i}^{\lambda }}=({f^{\ast -1}}({f^{\ast }}({A_{\phi (i)}}){f^{\ast }}({A_{\phi (j)}})),{g^{\ast -1}}({g^{\ast }}{({B_{\varphi (i)}})^{\lambda }}))$, where $\lambda \geqslant 0$.

##### Definition 5 (Wang *et al.*, 2017)*.*

##### Definition 6 (Wang *et al.*, 2017)*.*

- (1) If ${A_{\phi (i)}}>{A_{\phi (j)}}$ and ${B_{\varphi (i)}}>{B_{\varphi (j)}}$, then ${z_{i}}$ is strictly greater than ${z_{j}}$, denoted by ${z_{i}}>{z_{j}}$;
- (2) If $S({z_{i}})\geqslant S({z_{j}})$ and $A({z_{i}})>A({z_{j}})$, then ${z_{i}}$ is greater than ${z_{j}}$, denoted by ${z_{i}}\succ {z_{j}}$;
- (3) If $S({z_{i}})=S({z_{j}})$ and $A({z_{i}})=A({z_{j}})$, then ${z_{i}}$ equals ${z_{j}}$, denoted by ${z_{i}}\sim {z_{j}}$;
- (4) If $S({z_{i}})=S({z_{j}})$ and $A({z_{i}})<A({z_{j}})$ or $S({z_{i}})<S({z_{j}})$, then ${z_{i}}$ is less than ${z_{j}}$, denoted by ${z_{i}}\prec {z_{j}}$.

##### Definition 7 (Wang *et al.*, 2017)*.*

##### (7)

\[\begin{aligned}{}d({z_{i}},{z_{j}})& =\frac{1}{2}\big(\big|{f^{\ast }}({A_{\phi (i)}})\times {g^{\ast }}({B_{\varphi (i)}})-{f^{\ast }}({A_{\phi (j)}})\times {g^{\ast }}({B_{\varphi (j)}})\big|\\ {} & \hspace{1em}+\max \big\{\big|{f^{\ast }}({A_{\phi (i)}})-{f^{\ast }}({A_{\phi (j)}})\big|,\big|{g^{\ast }}({B_{\varphi (i)}})-{g^{\ast }}({B_{\varphi (j)}})\big|\big\}\big).\end{aligned}\]##### Definition 8 (Duan *et al.*, 2019)*.*

##### (8)

\[\begin{aligned}{}& \mathrm{LZWA}({z_{i}})={\sum \limits_{i=1}^{n}}{w_{i}}{z_{i}}\\ {} & \hspace{1em}=\Bigg({f^{\ast -1}}\Bigg({\sum \limits_{i=1}^{n}}{w_{i}}{f^{\ast }}({A_{\phi (i)}})\Bigg),{g^{\ast -1}}\bigg(\frac{{\textstyle\textstyle\sum _{i=1}^{n}}{w_{i}}{f^{\ast }}({A_{\phi (i)}}){g^{\ast }}({B_{\varphi (i)}})}{{\textstyle\textstyle\sum _{i=1}^{n}}{w_{i}}{f^{\ast }}({A_{\phi (i)}})}\bigg)\Bigg),\end{aligned}\]## 4 The Proposed QFD Approach

*m*engineering characteristics ${\mathit{EC}_{i}}$ $(i=1,2,\dots ,m)$ and

*n*customer requirements ${\mathit{CR}_{j}}$ $(j=1,2,\dots ,n)$. Meanwhile,

*l*experts ${E_{k}}$ $(k=1,2,\dots ,l)$ are invited to provide their assessments for the relationships between CRs and ECs, and each expert is assigned a weight ${\lambda _{k}}$ satisfying ${\lambda _{k}}>0$ and ${\textstyle\sum _{k=1}^{l}}{\lambda _{k}}=1$ to describe his/her relative importance in the QFD analysis. Let ${Z^{k}}={[{z_{ij}^{k}}]_{m\times n}}$ be the linguistic assessment matrix of ${E_{k}}$, where ${z_{ij}^{k}}=({A_{\phi (ijk)}},{B_{\varphi (ijk)}})$ is the linguistic Z-number evaluation ${\mathit{EC}_{i}}$ with respect to ${\mathit{CR}_{j}}$ provided by ${E_{k}}$. Based on the above assumptions, the proposed QFD model is described as follows:

**Stage 1**. Evaluate the relationships between ECs and CRs using linguistic Z-numbers.

**Step 1**: Establish the collective linguistic evaluation matrix

*Z*.

##### (9)

\[\begin{aligned}{}& {z_{ij}}=\mathrm{LZWA}\big({z_{ij}^{1}},{z_{ij}^{2}},\dots ,{z_{ij}^{l}}\big)\\ {} & \hspace{1em}=\Bigg({f^{\ast -1}}\Bigg({\sum \limits_{k=1}^{l}}{\lambda _{k}}{f^{\ast }}({A_{\phi (ijk)}})\Bigg),{g^{\ast -1}}\bigg(\frac{{\textstyle\textstyle\sum _{k=1}^{l}}{\lambda _{k}}{f^{\ast }}({A_{\phi (ijk)}}){g^{\ast }}({B_{\varphi (ijk)}})}{{\textstyle\textstyle\sum _{k=1}^{l}}{\lambda _{k}}{f^{\ast }}({A_{\phi (ijk)}})}\bigg)\Bigg).\end{aligned}\]**Stage 2**. Acquire the weights of CRs by the SWARA method.

*et al.*(2010) is a powerful weighting method in solving MCDM problems. The superiority of the SWARA is that it is uncomplicated, straightforward and involves less comparisons compared with other weighing methods (Hashemkhani Zolfani

*et al.*, 2013; Karabasevic

*et al.*, 2016; Ruzgys

*et al.*, 2014; Stanujkic

*et al.*, 2017). Given its strength, it has been used to find the relative weights of evaluation criteria in many researches (Duan

*et al.*, 2019; Liu

*et al.*, 2020; Naeini

*et al.*, 2020). Therefore, the SWARA method is introduced to acquire the weights of CRs in this study. The detailed steps are listed as follows:

**Step 2**: Sort CRs in a descending order.

*n*customer requirements ${\mathit{CR}_{j}}$ $(j=1,2,\dots ,n)$ are sorted in a descending order according to their expected importance. Then, newly ranked CRs are denoted as ${\textit{CR}^{\prime }_{j}}$ $(j=1,2,\dots ,n)$.

**Step 3**: Determine the comparative importance of CRs.

**Step 4**: Calculate the CR coefficient ${k_{j}}$ by

**Step 5**: Compute the recalculated CR weights ${q_{j}}$ by

**Step 6**: Determine the final weight of each CR ${w^{\prime }_{j}}$ by

*n*customer requirement ${\mathit{CR}_{j}}$ $(j=1,2,\dots ,n)$, i.e. $w=({w_{1}},{w_{2}},\dots ,{w_{n}})$, can be derived by rearranging the weights ${w^{\prime }_{j}}$ $(j=1,2,\dots ,n)$.

**Stage 3**. Rank the ECs through EDAS method.

**Step 7**: Determine the linguistic average EC.

**Step 8**: Compute the matrices of PDA and NDA.

**Step 9**: Calculate the weighted sum of PDA and NDA for each EC.

*w*, the weighted sums of PDA and NDA for each EC are calculated by

**Step 10**: Normalize the weighted sums of PDA and NDA for each EC.

**Step 11**. Compute the important score for all ECs.

*m*ECs can be computed by

## 5 Case Study

### 5.1 Implementation and Results

##### Table 1

CRs | Customer requirements | ECs | Engineering characteristics |

${\mathit{CR}_{1}}$ | Price | ${\mathit{EC}_{1}}$ | Cost |

${\mathit{CR}_{2}}$ | Comfortability | ${\mathit{EC}_{2}}$ | Car body material |

${\mathit{CR}_{3}}$ | Safety | ${\mathit{EC}_{3}}$ | Seat material |

${\mathit{CR}_{4}}$ | Convenience | ${\mathit{EC}_{4}}$ | Car internal decoration |

${\mathit{CR}_{5}}$ | Space | ${\mathit{EC}_{5}}$ | On-board system |

${\mathit{EC}_{6}}$ | Air-conditioning system |

**Step 1:**By using Eq. (9), the individual evaluation matrices ${Z^{k}}$ $(k=1,2,\dots ,5)$ are aggregated to obtain the collective linguistic evaluation matrix $Z={({z_{ij}})_{6\times 5}}$, as shown below

##### Table 2

Experts | CRs | ${\textit{EC}_{1}}$ | ${\textit{EC}_{2}}$ | ${\textit{EC}_{3}}$ | ${\textit{EC}_{4}}$ | ${\textit{EC}_{5}}$ | ${\textit{EC}_{6}}$ |

${E_{1}}$ | ${\mathit{CR}_{1}}$ | $({s_{5}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{4}})$ |

${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{1}})$ | |

${\mathit{CR}_{3}}$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |

${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |

${\mathit{CR}_{5}}$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | |

${E_{2}}$ | ${\mathit{CR}_{1}}$ | $({s_{6}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{4}})$ |

${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{1}})$ | |

${\mathit{CR}_{3}}$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{3}})$ | |

${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |

${\mathit{CR}_{5}}$ | $({s_{0}},{s^{\prime }_{2}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{2}})$ | |

${E_{3}}$ | ${\mathit{CR}_{1}}$ | $({s_{5}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{4}})$ |

${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{1}})$ | |

${\mathit{CR}_{3}}$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{3}})$ | |

${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |

${\mathit{CR}_{5}}$ | $({s_{0}},{s^{\prime }_{0}})$ | $({s_{0}},{s^{\prime }_{0}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{1}})$ | |

${E_{4}}$ | ${\mathit{CR}_{1}}$ | $({s_{5}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{3}})$ |

${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{0}})$ | $({s_{3}},{s^{\prime }_{3}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{1}})$ | |

${\mathit{CR}_{3}}$ | $({s_{6}},{s^{\prime }_{3}})$ | $({s_{6}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{3}})$ | |

${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |

${\mathit{CR}_{5}}$ | $({s_{0}},{s^{\prime }_{2}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{2}})$ | |

${E_{5}}$ | ${\mathit{CR}_{1}}$ | $({s_{6}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{4}})$ |

${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{0}})$ | |

${\mathit{CR}_{3}}$ | $({s_{6}},{s^{\prime }_{4}})$ | $({s_{5}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{3}})$ | |

${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | |

${\mathit{CR}_{5}}$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{1}})$ | $({s_{1}},{s^{\prime }_{0}})$ | $({s_{0}},{s^{\prime }_{2}})$ |

**Step 2:**According to the opinions of experts, the five customer requirements are sorted in a descending order. As a result, we can determine the newly ranked customer requirements as: ${\mathit{CR}^{\prime }_{1}}={\mathit{CR}_{1}}$, ${\textit{CR}^{\prime }_{2}}={\mathit{CR}_{3}}$, ${\textit{CR}^{\prime }_{3}}={\mathit{CR}_{4}}$, ${\textit{CR}^{\prime }_{4}}={\mathit{CR}_{2}}$, ${\textit{CR}^{\prime }_{5}}={\mathit{CR}_{5}}$.

**Step 3:**Starting from the second customer requirement, experts are invited to assess the relative importance of the customer requirement ${\mathit{CR}^{\prime }_{j}}$ with respect to the previous customer requirement ${\mathit{CR}^{\prime }_{j-1}}$. Then the comparative importance ${\rho _{j}}$ of CRs is derived as shown in Table 3.

**Steps 4–6:**Via Eqs. (10)–(12), the CR coefficient ${k_{j}}$ $(j=1,2,\dots ,5)$, the recalculated CR weights ${q_{j}}$ $(j=1,2,\dots ,5)$ and the final weights of CRs ${w^{\prime }_{j}}$ $(j=1,2,\dots ,5)$ are calculated, respectively. The results are listed in Table 3.

##### Table 3

CRs | ${\rho _{j}}$ | ${k_{j}}$ | ${q_{j}}$ | ${w_{j}}$ |

${\mathit{CR}^{\prime }_{1}}$ | 1.00 | 1.00 | 0.47 | |

${\mathit{CR}^{\prime }_{2}}$ | 0.90 | 1.90 | 0.53 | 0.25 |

${\mathit{CR}^{\prime }_{3}}$ | 0.80 | 1.80 | 0.29 | 0.14 |

${\mathit{CR}^{\prime }_{4}}$ | 0.50 | 1.50 | 0.19 | 0.09 |

${\mathit{CR}^{\prime }_{5}}$ | 0.90 | 1.90 | 0.10 | 0.05 |

**Step 8:**Utilizing Eqs. (14) and (15) the PDA matrix ${D^{+}}={[{d_{ij}^{+}}]_{6\times 5}}$ and the NDA matrix ${D^{-}}={[{d_{ij}^{-}}]_{6\times 5}}$ are required as:

**Step 9:**Based on Eqs. (16) and (17) the weighted sum of PDA and NDA for each EC ${\textit{SP}_{i}}$ $(i=1,2,\dots ,6)$ and ${\textit{SN}_{i}}$ $(i=1,2,\dots ,6)$ are derived as presented in Table 4.

**Step 10:**The normalized values of ${\textit{SP}_{i}}$ $(i=1,2,\dots ,6)$ and ${\textit{SN}_{i}}$ $(i=1,2,\dots ,6)$ for each EC are calculated by Eqs. (18) and (19) The results are displayed in Table 4.

**Step 11:**Applying Eq. (20) the importance scores for the ECs ${\textit{IS}_{i}}$ $(i=1,2,\dots ,6)$ are obtained as shown in Table 4.

##### Table 4

ECs | ${\textit{SP}_{i}}$ | ${\textit{SN}_{i}}$ | $\overline{{\textit{SP}_{i}}}$ | $\overline{{\textit{SN}_{i}}}$ | ${\textit{IS}_{i}}$ |

${\mathit{EC}_{1}}$ | 1.45 | 0.57 | 1.00 | 0.69 | 0.85 |

${\mathit{EC}_{2}}$ | 0.40 | 1.04 | 0.28 | 0.43 | 0.36 |

${\mathit{EC}_{3}}$ | 0.08 | 0.70 | 0.06 | 0.62 | 0.34 |

${\mathit{EC}_{4}}$ | 0.26 | 1.84 | 0.18 | 0.00 | 0.09 |

${\mathit{EC}_{5}}$ | 0.18 | 1.23 | 0.12 | 0.33 | 0.23 |

${\mathit{EC}_{6}}$ | 0.08 | 0.65 | 0.06 | 0.65 | 0.35 |

### 5.2 Sensitivity Analysis

##### Table 5

ECs | Case 0 | Case 1 | Case 2 | Case 3 |

$\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}{w_{j}}& =& (0.47,0.09,\\ {} & & 0.25,0.14,0.05)\end{array}$ | $\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}{w_{j}}& =& (0.2,0.2,\\ {} & & 0.2,0.2,0.2)\end{array}$ | $\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}{w_{j}}& =& (0.1,0.35,\\ {} & & 0.1,0.1,0.35)\end{array}$ | $\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}l}{w_{j}}& =& (0.35,0.35,\\ {} & & 0.1,0.1,0.1)\end{array}$ | |

${\mathit{EC}_{1}}$ | 1 | 1 | 1 | 1 |

${\mathit{EC}_{2}}$ | 2 | 2 | 4 | 3 |

${\mathit{EC}_{3}}$ | 4 | 5 | 5 | 4 |

${\mathit{EC}_{4}}$ | 6 | 3 | 2 | 6 |

${\mathit{EC}_{5}}$ | 5 | 6 | 6 | 5 |

${\mathit{EC}_{6}}$ | 3 | 4 | 3 | 2 |

### 5.3 Comparative Analysis

*et al.*, 2016) and the classical EDAS method (Ghorabaee

*et al.*, 2015). The ranking results of the six ECs derived by the considered methods are exhibited in Fig. 2. It can be observed from Fig. 2 that the most vital EC for the considered problem remains the same (i.e. ${\textit{EC}_{1}}$) for the proposed method and the other three methods. Thus, the proposed QFD model is validated.