# Uncertain Quality Function Deployment Using a Hybrid Group Decision Making Model

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Preliminaries

#### 3.1. Hesitant Fuzzy Linguistic Term Sets

**Definition**

**1.**

_{S}is an ordered finite subset of the consecutive linguistic terms of S. The empty and full HFLTSs for a linguistic variable $\vartheta $ are defined as ${H}_{S}\left(\vartheta \right)=\varnothing $ and ${H}_{S}\left(\vartheta \right)=S$, respectively.

**Definition**

**2.**

_{H}= (V

_{N}, V

_{T}, I, P), where V

_{N}indicates a set of nonterminal symbols, V

_{T}is a set of terminal symbols, I is the starting symbol, and P denotes the production rules. The elements of G

_{H}are defined as follows:

**Definition**

**3.**

_{GH}be a function that transforms the comparative linguistic expressions obtained by the context-free grammar G

_{H}into an HFLTS H

_{S}of the linguistic term set S. The linguistic expressions generated by G

_{H}using the production rules can be converted into HFLTSs according to the following ways:

- ${E}_{GH}\left(\mathrm{lower}\text{}\mathrm{than}\text{\hspace{0.17em}}{s}_{i}\right)=\left\{{s}_{k}|{s}_{k}\in S\text{}\mathrm{and}\text{}{s}_{k}{s}_{i}\right\};$
- ${E}_{GH}\left(\mathrm{greater}\text{}\mathrm{than}\text{\hspace{0.17em}}{s}_{i}\right)=\left\{{s}_{k}|{s}_{k}\in S\text{}\mathrm{and}\text{}{s}_{k}{s}_{i}\right\};$
- ${E}_{GH}\left(\mathrm{at}\text{}\mathrm{least}\text{\hspace{0.17em}}{s}_{i}\right)=\left\{{s}_{k}|{s}_{k}\in S\text{}\mathrm{and}\text{}{s}_{k}\ge {s}_{i}\right\};$
- ${E}_{GH}\left(\mathrm{at}\text{}\mathrm{most}\text{\hspace{0.17em}}{s}_{i}\right)=\left\{{s}_{k}|{s}_{k}\in S\text{}\mathrm{and}\text{}{s}_{k}\le {s}_{i}\right\};$
- ${E}_{GH}\left(\mathrm{between}\text{\hspace{0.17em}}{s}_{i}\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}}{s}_{j}\right)=\left\{{s}_{k}|{s}_{k}\in S\text{}\mathrm{and}\text{}{s}_{i}\le {s}_{k}\le {s}_{j}\right\}.$

**Definition**

**4.**

_{S}), is a linguistic interval whose limits are determined by its upper bound ${H}_{S}^{+}$ and lower bound ${H}_{S}^{-}$, shown as follows:

#### 3.2. Interval 2-Tuple Linguistic Model

**Definition**

**5.**

**Definition**

**6.**

- ${\tilde{a}}_{1}\otimes {\tilde{a}}_{2}=\left[\left({r}_{1},{\alpha}_{1}\right),\left({t}_{1},{\epsilon}_{1}\right)\right]\otimes \left[\left({r}_{2},{\alpha}_{2}\right),\left({t}_{2},{\epsilon}_{2}\right)\right]=\Delta \left[{\Delta}^{-1}\left({r}_{1},{\alpha}_{1}\right)\cdot {\Delta}^{-1}\left({r}_{2},{\alpha}_{2}\right),{\Delta}^{-1}\left({t}_{1},{\epsilon}_{1}\right)\cdot {\Delta}^{-1}\left({t}_{2},{\epsilon}_{2}\right)\right];$
- ${\tilde{a}}_{1}\oplus {\tilde{a}}_{2}=\left[\left({r}_{1},{\alpha}_{1}\right),\left({t}_{1},{\epsilon}_{1}\right)\right]\oplus \left[\left({r}_{2},{\alpha}_{2}\right),\left({t}_{2},{\epsilon}_{2}\right)\right]=\Delta \left[{\Delta}^{-1}\left({r}_{1},{\alpha}_{1}\right)+{\Delta}^{-1}\left({r}_{2},{\alpha}_{2}\right),{\Delta}^{-1}\left({t}_{1},{\epsilon}_{1}\right)+{\Delta}^{-1}\left({t}_{2},{\epsilon}_{2}\right)\right];$
- ${\tilde{a}}^{\lambda}={\left(\left[\left(r,\alpha \right),\left(t,\epsilon \right)\right]\right)}^{\lambda}=\Delta \left[{\left({\Delta}^{-1}\left(r,\alpha \right)\right)}^{\lambda},{\left({\Delta}^{-1}\left(t,\epsilon \right)\right)}^{\lambda}\right];$
- $\lambda \tilde{a}=\lambda \left[\left(r,\alpha \right),\left(t,\epsilon \right)\right]=\Delta \left[\lambda {\Delta}^{-1}\left(r,\alpha \right),\lambda {\Delta}^{-1}\left(t,\epsilon \right)\right].$

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

## 4. QFD Using Hesitant 2-Tuples and QUALIFLEX Method

#### 4.1. Assess the Relationships between WHATs and HOWs

_{k}over the relationship between CR

_{i}and DR

_{j}. Based on these assumptions and notations, the steps of dealing with the uncertain CR-DR relationship assessments are presented as follows:

**Step**

**1:**

_{H}. After converting into corresponding HFLTSs according to the transformation function E

_{GH}, every QFD team member’s hesitant linguistic expressions ${h}_{ij}^{k}$ can be transformed into linguistic intervals $env\left({h}_{ij}^{k}\right)=\left[{\underset{\_}{h}}_{ij}^{k},{\overline{h}}_{ij}^{k}\right]$ by calculating the envelope of each HFLTS (as in Definition 4). Then, the linguistic intervals are represented using the interval 2-tuple linguistic approach and translated into $\left[\left({r}_{ij}^{k},0\right),\left({t}_{ij}^{k},0\right)\right]$. As a result, the hesitant linguistic assessment information of QFD team members can be expressed by interval 2-tuple assessments as follows:

**Step**

**2:**

#### 4.2. Determine the Importance Weights of CRs

- A weak ranking: ${H}_{1}=\left\{{w}_{i}\ge {w}_{l}\right\}$;
- A strict ranking: ${H}_{2}=\left\{{w}_{i}-{w}_{l}\ge {\gamma}_{l}\right\}\left({\gamma}_{l}>0\right)$;
- A ranking of differences: ${H}_{3}=\left\{{w}_{i}-{w}_{l}\ge {w}_{p}-{w}_{q}\right\}\left(l\ne p\ne q\right)$;
- A ranking with multiples: ${H}_{4}=\left\{{w}_{i}\ge {\gamma}_{l}{w}_{l}\right\}\left(0\le {\gamma}_{l}\le 1\right)$;
- An interval form: ${H}_{5}=\left\{{\gamma}_{i}\le {w}_{i}\le {\gamma}_{i}+{\epsilon}_{i}\right\}\left(0\le {\gamma}_{i}\le {\gamma}_{i}+{\epsilon}_{i}\le 1\right)$.

**Step**

**3:**

_{j}will be. Thus, a reasonable weight vector of CRs should be determined so as to make all the grey relational grades $\xi \left({\tilde{r}}_{j},{\tilde{r}}^{*}\right)\left(j=1,2,...,n\right)$ as larger as possible, which means to maximize the grey relational grade vector $\mathrm{\Gamma}\left(w\right)=\left(\xi \left({\tilde{r}}_{1},{\tilde{r}}^{*}\right),\xi \left({\tilde{r}}_{2},{\tilde{r}}^{*}\right),...,\xi \left({\tilde{r}}_{n},{\tilde{r}}^{*}\right)\right)$ under the condition $w\in H$. As a result, we can reasonably form the following multiple objective optimization model:

#### 4.3. Determine the Ranking Order of DRs

**Step**

**4:**

**Step**

**5:**

**Step**

**6:**

**Step**

**7:**

## 5. Illustrative Example

#### 5.1. Implementation

_{1}), expected profit (CR

_{2}), competitive intensity (CR

_{3}), capital required (CR

_{4}), and level of technology utilization (CR

_{5}), and the company business strengths (DRs) are selected as relative cost position (DR

_{1}), delivery reliability (DR

_{2}), technological position (DR

_{3}), and management strength and depth (DR

_{4}). Each member of the QFD team analyzes the match between the market segment features and the company’s business strengths (WHATs–HOWs), and judges the relationships between them by means of grammar-free expressions over a seven-point linguistic term set S:

_{GH}. Then, the linguistic intervals are yielded by calculating the envelope of each obtained HFLTS and the interval 2-tuple relationship matrix ${\tilde{R}}_{k}\left(k=1,2,...,5\right)$ of every QFD team member is subsequently constructed. For instance, the interval 2-tuple relationship matrix of TM

_{1}${\tilde{R}}_{1}$ is presented in Table 2. By implementing the ITOWA operator, the collective assessments regarding the relationship judgements between CRs and DRs are taken as the collective interval 2-tuple relationship matrix $\tilde{R}={\left[{\tilde{r}}_{ij}\right]}_{5\times 4}$, as shown in Table 3. Note that the ITOWA operator weights are derived using the argument-dependent approach [44].

- P
_{1}= (DR_{1}, DR_{2}, DR_{3}, DR_{4}), P_{2}= (DR_{1}, DR_{2}, DR_{4}, DR_{3}), P_{3}= (DR_{1}, DR_{3}, DR_{2}, DR_{4}), - P
_{4}= (DR_{1}, DR_{3}, DR_{4}, DR_{2}), P_{5}= (DR_{1}, DR_{4}, DR_{2}, DR_{3}), P_{6}= (DR_{1}, DR_{4}, DR_{3}, DR_{2}), - P
_{7}= (DR_{2}, DR_{1}, DR_{3}, DR_{4}), P_{8}= (DR_{2}, DR_{1}, DR_{4}, DR_{3}), P_{9}= (DR_{2}, DR_{3}, DR_{1}, DR_{4}), - P
_{10}= (DR_{2}, DR_{3}, DR_{4}, DR_{1}), P_{11}= (DR_{2}, DR_{4}, DR_{1}, DR_{3}), P_{12}= (DR_{2}, DR_{4}, DR_{3}, DR_{1}), - P
_{13}= (DR_{3}, DR_{1}, DR_{2}, DR_{4}), P_{14}= (DR_{3}, DR_{1}, DR_{4}, DR_{2}), P_{15}= (DR_{3}, DR_{2}, DR_{1}, DR_{4}), - P
_{16}= (DR_{3}, DR_{2}, DR_{4}, DR_{1}), P_{17}= (DR_{3}, DR_{4}, DR_{1}, DR_{2}), P_{18}= (DR_{3}, DR_{4}, DR_{2}, DR_{1}), - P
_{19}= (DR_{4}, DR_{1}, DR_{2}, DR_{3}), P_{20}= (DR_{4}, DR_{1}, DR_{3}, DR_{2}), P_{21}= (DR_{4}, DR_{2}, DR_{1}, DR_{3}), - P
_{22}= (DR_{4}, DR_{2}, DR_{3}, DR_{1}), P_{23}= (DR_{4}, DR_{3}, DR_{1}, DR_{2}), P_{24}= (DR_{4}, DR_{3}, DR_{2}, DR_{1}).

_{i}$\left(i=1,2,...,5\right)$. Considering the first permutation P

_{1}for example, the results of the concordance/discordance index are shown in Table 4. In Step 6, we utilize Equation (20) to compute the weighted concordance/discordance index ${\varphi}^{\rho}\left({\mathrm{DR}}_{\xi},{\mathrm{DR}}_{\zeta}\right)$ for each pair of $\left({\mathrm{DR}}_{\varphi},{\mathrm{DR}}_{\eta}\right)$ in the permutation ${P}_{\rho}$, and the results are indicated in Table 5. In Step 7, the comprehensive concordance/discordance index ${\varphi}^{\rho}\left(\rho =1,2,...,24\right)$ is calculated by applying Equation (21) for each permutation ${P}_{\rho}$. The computation results are given as follows:

_{3}because ${\varphi}^{3}=0.3183$ gives the maximum value, and the final priory order of the four DRs is ${\mathrm{DR}}_{1}\succ {\mathrm{DR}}_{3}\succ {\mathrm{DR}}_{2}\succ {\mathrm{DR}}_{4}$. Therefore, the most important company business strength for the considered case study is “relative cost position (DR1)”, which should be given the highest priority for selecting the optimal market segment, followed by DR3, DR2, and DR4.

#### 5.2. Comparisons and Discussions

- Different types of uncertainties in the implementation of QFD, such as imprecision, uncertainty and hesitation, can be well modeled via the hesitant 2-tuple linguistic term sets. The QFD team members can use more flexible and richer expressions to express their subjective judgments.
- By using the ITOWA operator, the proposed method can relieve the influence of unfair judgments concerning the relationships between CRs and DRs on the QFD analysis results, through assigning very low weights to those “false” or “biased” opinions.
- The proposed approach is able to deal with QFD problems in which the information about CR weights is incompletely known. Under the condition of incomplete weight information, a multiple objective programming model can be established to solve the optimal weights of CRs.
- The proposed methodology can get a more reasonable and credible ranking of DRs by using the modified QUALIFLEX approach, which makes the QFD analysis results certain and facilitates product planning decision-making.
- The proposed model is suitable to solve complicated QFD problems with comprehensive CRs and limited DRs, since the number of CRs has little effect upon the implementation efficiency of the proposed method.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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WHATs (CRs) | Team Members | HOWs (DRs) | |||
---|---|---|---|---|---|

DR_{1} | DR_{2} | DR_{3} | DR_{4} | ||

CR_{1} | TM_{1} | Greater than MH | ML | Between L and M | M |

TM_{2} | H | M | M | At least ML | |

TM_{3} | Between H and VH | M | ML | M | |

TM_{4} | H | MH | Less than M | M | |

TM_{5} | H | At most MH | M | Between ML and MH | |

CR_{2} | TM_{1} | At least H | Greater than MH | Between MH and VH | H |

TM_{2} | VH | H | H | Greater than H | |

TM_{3} | Greater than H | VH | VH | Between MH and VH | |

TM_{4} | H | At least H | Greater than MH | H | |

TM_{5} | Between MH and VH | H | H | At least H | |

CR_{3} | TM_{1} | Greater than M | H | At least H | Between MH and VH |

TM_{2} | MH | Between MH and VH | VH | H | |

TM_{3} | H | H | Greater than H | H | |

TM_{4} | At least H | VH | VH | At most H | |

TM_{5} | VH | Greater than MH | VH | VH | |

CR_{4} | TM_{1} | Less than H | M | Greater than MH | MH |

TM_{2} | H | Between ML and MH | H | H | |

TM_{3} | MH | M | H | Less than H | |

TM_{4} | M | At most MH | MH | At most MH | |

TM_{5} | At most H | Less than MH | Between MH and VH | M | |

CR_{5} | TM_{1} | Between MH and VH | MH | At most H | Between L and ML |

TM_{2} | H | H | MH | ML | |

TM_{3} | H | Less than H | H | L | |

TM_{4} | At least MH | M | M | At most M | |

TM_{5} | H | MH | Between MH and VH | ML |

WHATs | HOWs | |||
---|---|---|---|---|

DR_{1} | DR_{2} | DR_{3} | DR_{4} | |

CR_{1} | [(s_{5}, 0), (s_{6}, 0)] | [(s_{2}, 0), (s_{2}, 0)] | [(s_{1}, 0), (s_{3}, 0)] | [(s_{3}, 0), (s_{3}, 0)] |

CR_{2} | [(s_{5}, 0), (s_{6}, 0)] | [(s_{5}, 0), (s_{6}, 0)] | [(s_{4}, 0), (s_{6}, 0)] | [(s_{5}, 0), (s_{5}, 0)] |

CR_{3} | [(s_{4}, 0), (s_{6}, 0)] | [(s_{5}, 0), (s_{5}, 0)] | [(s_{5}, 0), (s_{6}, 0)] | [(s_{4}, 0), (s_{6}, 0)] |

CR_{4} | [(s_{0}, 0), (s_{4}, 0)] | [(s_{3}, 0), (s_{3}, 0)] | [(s_{5}, 0), (s_{6}, 0)] | [(s_{4}, 0), (s_{4}, 0)] |

CR_{5} | [(s_{4}, 0), (s_{6}, 0)] | [(s_{4}, 0), (s_{4}, 0)] | [(s_{0}, 0), (s_{5}, 0)] | [(s_{1}, 0), (s_{2}, 0)] |

WHATs | HOWs | |||
---|---|---|---|---|

DR_{1} | DR_{2} | DR_{3} | DR_{4} | |

CR_{1} | ∆[0.833, 0.884] | ∆[0.884, 0.448] | ∆[0.448, 0.543] | ∆[0.543, 0.322] |

CR_{2} | ∆[0.876, 0.994] | ∆[0.994, 0.839] | ∆[0.839, 0.949] | ∆[0.949, 0.833] |

CR_{3} | ∆[0.791, 0.935] | ∆[0.935, 0.833] | ∆[0.833, 0.949] | ∆[0.949, 0.994] |

CR_{4} | ∆[0.426, 0.709] | ∆[0.709, 0.29] | ∆[0.290, 0.551] | ∆[0.551, 0.782] |

CR_{5} | ∆[0.782, 0.884] | ∆[0.884, 0.614] | ∆[0.614, 0.667] | ∆[0.667, 0.614] |

P_{1} | CR_{1} | CR_{2} | CR_{3} | CR_{4} | CR_{5} |
---|---|---|---|---|---|

${\varphi}_{i}^{1}\left({\mathrm{FM}}_{1},{\mathrm{FM}}_{2}\right)$ | 0.131 | 0.059 | −0.030 | 0.069 | 0.147 |

${\varphi}_{i}^{1}\left({\mathrm{FM}}_{1},{\mathrm{FM}}_{3}\right)$ | 0.250 | 0.063 | −0.165 | −0.141 | 0.103 |

${\varphi}_{i}^{1}\left({\mathrm{FM}}_{1},{\mathrm{FM}}_{4}\right)$ | 0.119 | 0.063 | 0.034 | 0.008 | 0.250 |

${\varphi}_{i}^{1}\left({\mathrm{FM}}_{2},{\mathrm{FM}}_{3}\right)$ | 0.119 | 0.004 | −0.135 | −0.210 | −0.044 |

${\varphi}_{i}^{1}\left({\mathrm{FM}}_{2},{\mathrm{FM}}_{4}\right)$ | −0.012 | 0.004 | 0.065 | −0.061 | 0.103 |

${\varphi}_{i}^{1}\left({\mathrm{FM}}_{3},{\mathrm{FM}}_{4}\right)$ | −0.130 | 0.000 | 0.200 | 0.149 | 0.147 |

P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | P_{6} | P_{7} | P_{8} | P_{9} | P_{10} | P_{11} | P_{12} |

0.0708 | 0.0708 | 0.0250 | 0.0250 | 0.0908 | 0.0908 | −0.0708 | −0.0708 | −0.0458 | −0.0458 | 0.0200 | 0.0200 |

0.0250 | 0.0908 | 0.0708 | 0.0908 | 0.0708 | 0.0250 | −0.0458 | 0.0200 | −0.0708 | 0.0200 | −0.0708 | −0.0458 |

0.0908 | 0.0250 | 0.0908 | 0.0708 | 0.0250 | 0.0708 | 0.0200 | −0.0458 | 0.0200 | −0.0708 | −0.0458 | −0.0708 |

−0.0458 | 0.0200 | 0.0458 | 0.0659 | −0.0200 | −0.0659 | 0.0250 | 0.0908 | −0.0250 | 0.0659 | −0.0908 | −0.0659 |

0.0200 | −0.0458 | 0.0659 | 0.0458 | −0.0659 | −0.0200 | 0.0908 | 0.0250 | 0.0659 | −0.0250 | −0.0659 | −0.0908 |

0.0659 | −0.0659 | 0.0200 | −0.0200 | −0.0458 | 0.0458 | 0.0659 | −0.0659 | 0.0908 | −0.0908 | 0.0250 | −0.0250 |

0.0708 | 0.0708 | 0.0250 | 0.0250 | 0.0908 | 0.0908 | −0.0708 | −0.0708 | −0.0458 | −0.0458 | 0.0200 | 0.0200 |

0.0250 | 0.0908 | 0.0708 | 0.0908 | 0.0708 | 0.0250 | −0.0458 | 0.0200 | −0.0708 | 0.0200 | −0.0708 | −0.0458 |

P_{13} | P_{14} | P_{15} | P_{16} | P_{17} | P_{18} | P_{19} | P_{20} | P_{21} | P_{22} | P_{23} | P_{24} |

−0.0250 | −0.0250 | 0.0458 | 0.0458 | 0.0659 | 0.0659 | −0.0908 | −0.0908 | −0.0200 | −0.0200 | −0.0659 | −0.0659 |

0.0458 | 0.0659 | −0.0250 | 0.0659 | −0.0250 | 0.0458 | −0.0200 | −0.0659 | −0.0908 | −0.0659 | −0.0908 | −0.0200 |

0.0659 | 0.0458 | 0.0659 | −0.0250 | 0.0458 | −0.0250 | −0.0659 | −0.0200 | −0.0659 | −0.0908 | −0.0200 | −0.0908 |

0.0708 | 0.0908 | −0.0708 | 0.0200 | −0.0908 | −0.0200 | 0.0708 | 0.0250 | −0.0708 | −0.0458 | −0.0250 | 0.0458 |

0.0908 | 0.0708 | 0.0200 | −0.0708 | −0.0200 | −0.0908 | 0.0250 | 0.0708 | −0.0458 | −0.0708 | 0.0458 | −0.0250 |

0.0200 | −0.0200 | 0.0908 | −0.0908 | 0.0708 | −0.0708 | −0.0458 | 0.0458 | 0.0250 | −0.0250 | 0.0708 | −0.0708 |

−0.0250 | −0.0250 | 0.0458 | 0.0458 | 0.0659 | 0.0659 | −0.0908 | −0.0908 | −0.0200 | −0.0200 | −0.0659 | −0.0659 |

0.0458 | 0.0659 | −0.0250 | 0.0659 | −0.0250 | 0.0458 | −0.0200 | −0.0659 | −0.0908 | −0.0659 | −0.0908 | −0.0200 |

HOWs | QFD | Fuzzy QFD | Linguistic QFD | Proposed Approach | |||
---|---|---|---|---|---|---|---|

W_{j} | Ranking | ${\tilde{\mathit{W}}}_{\mathit{j}}$ | Ranking | ${\widehat{\mathit{W}}}_{\mathit{j}}$ | Ranking | ||

DR_{1} | 7.221 | 1 | (0.267, 0.475, 0.724) | 1 | s_{5.06} | 1 | 1 |

DR_{2} | 6.303 | 3 | (0.231, 0.415, 0.659) | 3 | s_{4.32} | 3 | 3 |

DR_{3} | 7.035 | 2 | (0.253, 0.448, 0.689) | 2 | s_{4.60} | 2 | 2 |

DR_{4} | 5.919 | 4 | (0.217, 0.400, 0.641) | 4 | s_{4.05} | 4 | 4 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Z.-L.; You, J.-X.; Liu, H.-C.
Uncertain Quality Function Deployment Using a Hybrid Group Decision Making Model. *Symmetry* **2016**, *8*, 119.
https://doi.org/10.3390/sym8110119

**AMA Style**

Wang Z-L, You J-X, Liu H-C.
Uncertain Quality Function Deployment Using a Hybrid Group Decision Making Model. *Symmetry*. 2016; 8(11):119.
https://doi.org/10.3390/sym8110119

**Chicago/Turabian Style**

Wang, Ze-Ling, Jian-Xin You, and Hu-Chen Liu.
2016. "Uncertain Quality Function Deployment Using a Hybrid Group Decision Making Model" *Symmetry* 8, no. 11: 119.
https://doi.org/10.3390/sym8110119