1 Introduction
2 Literature Review
3 Preliminaries
Definition 1 (Duan et al. 2019).
Definition 2 (Peng and Wang 2017).
Definition 3 (Wang et al. 2017).
Definition 4 (Wang et al. 2017).
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(1) $\begin{array}[t]{l}{z_{i}}\oplus {z_{j}}\\ {} \hspace{1em}=\big({f^{\ast -1}}({f^{\ast }}({A_{\phi (i)}})+{f^{\ast }}({A_{\phi (j)}})),{g^{\ast -1}}\big(\frac{{f^{\ast }}({A_{\phi (i)}})\times {g^{\ast }}({B_{\varphi (i)}})+{f^{\ast }}({A_{\phi (j)}})\times {g^{\ast }}({B_{\varphi (i)}})}{{f^{\ast }}({A_{\phi (i)}})+{f^{\ast }}({A_{\phi (j)}})}\big)\big);\end{array}$
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(2) $\lambda {z_{i}}=({f^{\ast -1}}(\lambda {f^{\ast }}({A_{\phi (i)}})),{B_{\varphi (i)}})$, where $\lambda \geqslant 0$;
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(3) ${z_{i}}\otimes {z_{j}}=({f^{\ast -1}}({f^{\ast }}({A_{\phi (i)}}){f^{\ast }}({A_{\phi (j)}})),{g^{\ast -1}}({g^{\ast }}({B_{\varphi (i)}}){g^{\ast }}({B_{\varphi (j)}})))$;
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(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).
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(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 ${z_{i}}>{z_{j}}$;
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(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 ${z_{i}}\succ {z_{j}}$;
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(3) If $S({z_{i}})=S({z_{j}})$ and $A({z_{i}})=A({z_{j}})$, then ${z_{i}}$ equals ${z_{j}}$, denoted by ${z_{i}}\sim {z_{j}}$;
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(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
(9)
\[\begin{aligned}{}& {z_{ij}}=\mathrm{LZWA}\big({z_{ij}^{1}},{z_{ij}^{2}},\dots ,{z_{ij}^{l}}\big)\\ {} & \hspace{1em}=\Bigg({f^{\ast -1}}\Bigg({\sum \limits_{k=1}^{l}}{\lambda _{k}}{f^{\ast }}({A_{\phi (ijk)}})\Bigg),{g^{\ast -1}}\bigg(\frac{{\textstyle\textstyle\sum _{k=1}^{l}}{\lambda _{k}}{f^{\ast }}({A_{\phi (ijk)}}){g^{\ast }}({B_{\varphi (ijk)}})}{{\textstyle\textstyle\sum _{k=1}^{l}}{\lambda _{k}}{f^{\ast }}({A_{\phi (ijk)}})}\bigg)\Bigg).\end{aligned}\]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 |
Table 2
Experts | CRs | ${\textit{EC}_{1}}$ | ${\textit{EC}_{2}}$ | ${\textit{EC}_{3}}$ | ${\textit{EC}_{4}}$ | ${\textit{EC}_{5}}$ | ${\textit{EC}_{6}}$ |
${E_{1}}$ | ${\mathit{CR}_{1}}$ | $({s_{5}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{4}})$ |
${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{1}})$ | |
${\mathit{CR}_{3}}$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |
${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |
${\mathit{CR}_{5}}$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | |
${E_{2}}$ | ${\mathit{CR}_{1}}$ | $({s_{6}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{4}})$ |
${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{1}})$ | |
${\mathit{CR}_{3}}$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{3}})$ | |
${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |
${\mathit{CR}_{5}}$ | $({s_{0}},{s^{\prime }_{2}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{2}})$ | |
${E_{3}}$ | ${\mathit{CR}_{1}}$ | $({s_{5}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{4}})$ |
${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{1}})$ | |
${\mathit{CR}_{3}}$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{5}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{3}})$ | |
${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |
${\mathit{CR}_{5}}$ | $({s_{0}},{s^{\prime }_{0}})$ | $({s_{0}},{s^{\prime }_{0}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{1}})$ | |
${E_{4}}$ | ${\mathit{CR}_{1}}$ | $({s_{5}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{3}})$ |
${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{0}})$ | $({s_{3}},{s^{\prime }_{3}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{1}})$ | |
${\mathit{CR}_{3}}$ | $({s_{6}},{s^{\prime }_{3}})$ | $({s_{6}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{3}})$ | |
${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{1}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{3}})$ | |
${\mathit{CR}_{5}}$ | $({s_{0}},{s^{\prime }_{2}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{2}})$ | |
${E_{5}}$ | ${\mathit{CR}_{1}}$ | $({s_{6}},{s^{\prime }_{4}})$ | $({s_{2}},{s^{\prime }_{4}})$ | $({s_{3}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{3}})$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{3}},{s^{\prime }_{4}})$ |
${\mathit{CR}_{2}}$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{0}},{s^{\prime }_{1}})$ | $({s_{4}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{2}})$ | $({s_{4}},{s^{\prime }_{0}})$ | |
${\mathit{CR}_{3}}$ | $({s_{6}},{s^{\prime }_{4}})$ | $({s_{5}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{3}})$ | |
${\mathit{CR}_{4}}$ | $({s_{2}},{s^{\prime }_{3}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | $({s_{1}},{s^{\prime }_{4}})$ | $({s_{0}},{s^{\prime }_{4}})$ | |
${\mathit{CR}_{5}}$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{1}},{s^{\prime }_{2}})$ | $({s_{1}},{s^{\prime }_{1}})$ | $({s_{3}},{s^{\prime }_{1}})$ | $({s_{1}},{s^{\prime }_{0}})$ | $({s_{0}},{s^{\prime }_{2}})$ |
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 |
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 |