# A Multi-Granularity 2-Tuple QFD Method and Application to Emergency Routes Evaluation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Basic Knowledge on QFD

#### 2.2. The 2-Tuple Linguistic Representation

**Definition 1 [27].**Assuming$S=\left\{{s}_{1},{s}_{2},\cdots ,{s}_{g}\right\}$is a linguistic term set and$\beta \in [0,g]$ represents the consequence of a symbolic aggregation operation. Afterwards, the 2-tuple is expressed as the equivalence to $\beta $ as follows:

**Definition 2 [27].**Let$S=\left\{{s}_{1},{s}_{2},\cdots ,{s}_{g}\right\}$be a linguistic term set and$({s}_{i},{\alpha}_{i})$be a 2-tuple. There is always a function${\Delta}^{-1}$that can be defined, such that, from a 2-tuple$({s}_{i},{\alpha}_{i}),$its equivalent numerical value$\beta \in [0,g]\subset R$ can be obtained, which is described as follows:

**Definition 3**[28]

**.**There are 2-tuples $x=\{({s}_{1},{\alpha}_{1}),({s}_{2},{\alpha}_{2}),\cdots ,({s}_{n},{\alpha}_{n})\}$. Their arithmetic mean is expressed as:

**Definition 4**[28]

**.**Let $({s}_{i},{\alpha}_{i})$ and $({s}_{j},{\alpha}_{j})$ be two 2-tuple linguistic variables. Their granularities are both $g$, and the distance between them is described as follows:

**Definition 5 [28].**Let$x=\{({s}_{1},{\alpha}_{1}),({s}_{2},{\alpha}_{2}),\cdots ,({s}_{n},{\alpha}_{n})\}$be a set of 2-tuples and$(\overline{s},\overline{\alpha})$ be the arithmetic mean of these 2-tuples. The degree of similarity is expressed as

**Definition 6 [23].**Let$LH=\underset{t}{\cup}l(t,n(t))$, which is the union of all level$t$, a linguistic hierarchy whose linguistic term set is${S}^{n(t)}=\left\{{s}_{0}^{n(t)},{s}_{1}^{n(t)},\cdots ,{s}_{n(t)-1}^{n(t)}\right\}.$ Furthermore, different granularities reflect different preferences under the circumstance of evaluating. The transformation function (TF) between level $t$ and level $t\prime $ is defined as

**Note 1.**The TF can implement the transformation between different granularities and further achieve a unified linguistic label. Without loss of generality, the transformation usually is carried out from the lower granularity to higher granularity in the process of transformation, i.e., the level $t\prime $ usually corresponds to the maximum granularity.

#### 2.3. The 2TLWGBM Operator

**Definition 7 [33].**Let$p,q\ge 0$and${a}_{i}(i=1,2,\cdots ,n)$be a series of non-negative numbers. Then the BM operator is defined as

**Definition 8 [33].**Let$x=\{({r}_{1},{a}_{1}),({r}_{2},{a}_{2}),\cdots ,({r}_{n},{a}_{n})\}$be a set of 2-tuple and$p,q\ge 0.$In addition,$w={({w}_{1},{w}_{2},\cdots ,{w}_{n})}^{T}$is the weight vector of$x,$where${w}_{i}>0\hspace{0.17em}(i=1,2,\cdots ,n)$represents the importance degree of$({r}_{i},{a}_{i})\hspace{0.17em}(i=1,2,\cdots ,n),$and${\sum}_{i=1}^{n}{w}_{i}=1}.$ The 2TLWGBM operator is then expressed as

**Note 2**. Although a majority of aggregation operators have been proposed in recent years, the 2TLWGBM operator has some merits in prioritizing ECs. On the one hand, this operator considers the relevance, which accords with the relationship and correlation between CRs and ECs. On the other hand, it is more flexible owing to the parameter $p$ and $q$, which makes it more suitable for different decision makers.

## 3. A Group Decision-Making Approach to Prioritize ECs

#### 3.1. Determine the Importance of CRs Based on BWM

**Step 1**. CRs {CR

_{1}, CR

_{2}, ⋯, CR

_{n}} are chosen, as are the best and the worst CR. The best CR is then compared with the other CRs using Number 1–9 is constructed. The best-to-others (BO) vector A

_{B}= (α

_{B1}, α

_{B2}, ⋯, α

_{Bn}) is represented where α

_{Bj}describes the preference of the best CR over CR

_{j}. Similarly, the Others-to-worst (OW) vector A

_{W}= (α

_{1W}, α

_{2W}, ⋯, α

_{nW})

^{T}is represented where α

_{jW}describes the preference of CR

_{j}over the worst CR.

**Step 2**. The optimal weights of CRs are obtained. The optimization model is established to minimize the maximum the difference {|w

_{B}− α

_{Bj}w

_{j}|} and {|w

_{j}− α

_{jW}ω

_{W}|}.

^{*}. Alternatively, the bigger ξ

^{*}demonstrates the higher consistency ratio provided by customers. The consistency ratio can be calculated by the proportion between ξ

^{*}and max ξ (Consistency Index).

_{BW}− ξ) × (α

_{BW}− ξ) = (α

_{BW}+ ξ) and α

_{BW}∈ {1, 2, ⋯, 9}. The consistency index is listed in Table 1.

#### 3.2. A Group Decision-Making Approach to Prioritize ECs

**Step 3.**Different multi-granularity linguistic preferences are obtained.

**Step 4.**Different multi-granularity linguistic preferences are unified.

**Step 5.**All the evaluation matrices are aggregated.

**Step 6.**The relationship between CRs and ECs is modified based on a compromise idea.

**Step 7.**Integrated ECs priorities are determined.

**Step 8.**Basic priority of ECs is confirmed.

## 4. Case Study

#### 4.1. Background

#### 4.2. Implementation

**Step 2.**The importance of CRs is respectively computed as 0.302, 0.359, 0.187, 0.082 and 0.070, which is determined by the average value by passengers. For example, the model by first passenger is established as follows:

**Step 3.**In order to determine the basic priority of these ECs, three experts $E{P}_{1},E{P}_{2},E{P}_{3}$ evaluate the importance of ECs according to CRs given as below (Table 4, Table 5 and Table 6). They represent preference by using the different linguistic term sets ${S}_{i}^{{7}_{1}}=\left\{{s}_{0}^{7},{s}_{1}^{7},{s}_{2}^{7},{s}_{3}^{7},{s}_{4}^{7},{s}_{5}^{7},{s}_{6}^{7}\right\}$ ${S}_{i}^{{5}_{2}}=\hspace{0.17em}\left\{{s}_{0}^{5},{s}_{1}^{5},{s}_{2}^{5},{s}_{3}^{5},{s}_{4}^{5}\right\}{S}_{i}^{{9}_{3}}=\hspace{0.17em}\left\{{s}_{0}^{9},{s}_{1}^{9},{s}_{2}^{9},{s}_{3}^{9},{s}_{4}^{9},{s}_{5}^{9},{s}_{6}^{9},{s}_{7}^{9},{s}_{8}^{9}\right\}$.

**Step 4.**Three evaluation matrices are transformed into 2-tuple representation in Table 7, Table 8 and Table 9.

**Step 5.**The aggregation of all the evaluation matrices in Table 9, Table 10 and Table 11 applying 2TLWGBM operator in Equation (11) into ${R}_{ij}$ is shown in Table 12.

**Step 6.**On the basic of different knowledge and experience, three experts adopt their own linguistic representations to evaluate correlations between CRs and ECs. These matrices are then aggregated in the same way as the fourth step. Consequently, the initial HOQ is shown in Figure 4.

**Step 7.**After obtaining the modified matrix, the importance of CRs should be integrated to reach the final relationships between CRs and ECs. The result is presented in Table 13. Therefore, the rank of integrated ECs priority is $E{C}_{2}\succ E{C}_{4}\succ E{C}_{1}\succ E{C}_{5}\succ E{C}_{3}$.

**Step 8.**The basic priority of ECs is computed according to Equations (7) and (14) and Table 13. The minimum value of linguistic term set is $({s}_{{}_{\mathrm{min}}}^{n(t)},{\alpha}^{n(t)})=({s}_{{}_{0}}^{9},0)$. The ultimate weights of ECs are $({s}_{0}^{9},0.215)\hspace{0.17em}\hspace{0.17em}({s}_{0}^{9},0.228)\hspace{0.17em}\hspace{0.17em}({s}_{0}^{9},0.165)\hspace{0.17em}\hspace{0.17em}({s}_{0}^{9},0.223)\hspace{0.17em}\hspace{0.17em}({s}_{0}^{9},0.169)$.

#### 4.3. Managerial Tips

## 5. Conclusions and Future Research

## Author Contributions

## Fundings

## Acknowledgments

## Conflicts of Interest

## References

- Yan, H.B.; Ma, T.J.; Huynh, V.N. Coping with group behaviors in uncertain quality function deployment. Decis. Sci.
**2014**, 456, 1025–1052. [Google Scholar] [CrossRef] - Ignatius, J.; Rahman, A.; Yazdani, M.; Aparauskas, J. Å.; Haron, S. H. An integrated fuzzy ANP–QFD approach for green building assessment. J. Civ. Eng. Manag.
**2016**, 224, 551–563. [Google Scholar] [CrossRef] - Franceschini, F.; Galetto, M.; Maisano, D.; Mastrogiacomo, L. Prioritisation of engineering characteristics in QFD in the case of customer requirements orderings. Int. J. Prod. Res.
**2015**, 5313, 3975–3988. [Google Scholar] [CrossRef] - Yan, H.B.; Ma, T.J. A group decision-making approach to uncertain quality function deployment based on fuzzy preference relation and fuzzy majority. Eur. J. Oper. Res.
**2015**, 2413, 815–829. [Google Scholar] [CrossRef] - Liu, C.H.; Wu, H.H. A fuzzy group decision-making approach in quality function deployment. Qual. Quant.
**2008**, 424, 527–540. [Google Scholar] [CrossRef] - Kwong, C.K.; Ye, Y.; Chen, Y.; Choy, K.L. A novel fuzzy group decision-making approach to prioritising engineering characteristics in QFD under uncertainties. Int. J. Prod. Res.
**2011**, 4919, 5801–5820. [Google Scholar] [CrossRef] - Karsak, E.E. Fuzzy multiple objective programming framework to prioritize design requirements in quality function deployment. Comput. Ind. Eng.
**2004**, 472, 149–163. [Google Scholar] [CrossRef] - Chen, L.H.; Weng, M.C. An evaluation approach to engineering design in QFD processes using fuzzy goal programming models. Eur. J. Oper. Res.
**2006**, 1721, 230–248. [Google Scholar] [CrossRef] - Kwong, C.K.; Chen, Y.; Bai, H.; Chan, D.S.K. A methodology of determining aggregated importance of engineering characteristics in QFD. Comput. Ind. Eng.
**2007**, 534, 667–679. [Google Scholar] [CrossRef] - Liu, J.; Chen, Y.Z.; Zhou, J.; Yi, X.J. An exact expected value-based method to prioritize engineering characteristics in fuzzy quality function deployment. Int. J. Fuzzy Syst.
**2016**, 184, 630–646. [Google Scholar] [CrossRef] - Geng, X.L.; Chu, X.N.; Xue, D.Y.; Zhang, Z.F. An integrated approach for rating engineering characteristics’ final importance in product-service system development. Comput. Ind. Eng.
**2010**, 594, 585–594. [Google Scholar] [CrossRef] - Wang, Y.M. A fuzzy-normalisation-based group decision-making approach for prioritising engineering design requirements in QFD under uncertainty. Int. J. Prod. Res.
**2012**, 5023, 6963–6977. [Google Scholar] [CrossRef] - Chen, L.H.; Ko, W.C.; Tseng, C.Y. Fuzzy approaches for constructing house of quality in QFD and its applications: A group decision-making method. IEEE Trans. Eng. Manag.
**2013**, 601, 77–87. [Google Scholar] [CrossRef] - Zhang, Z.F.; Chu, X.N. Fuzzy group decision-making for multi-format and multi-granularity linguistic judgments in quality function deployment. Expert Syst. Appl.
**2009**, 365, 9150–9158. [Google Scholar] [CrossRef] - Wang, X.T.; Xiong, W. An integrated linguistic-based group decision-making approach for quality function deployment. Expert Syst. Appl.
**2011**, 3812, 14428–14438. [Google Scholar] [CrossRef] - Wang, Z.Q.; Fung, R.Y.K.; Li, Y.L.; Pu, Y. A group multi-granularity linguistic-based methodology for prioritizing engineering characteristics under uncertainties. Comput. Ind. Eng.
**2016**, 91, 178–187. [Google Scholar] [CrossRef] - Bo, C.; Zhang, X.; Shao, S.; Smarandache, F. Multi-granulation neutrosophic rough sets on a single domain and dual domains with applications. Symmetry
**2018**, 10, 296. [Google Scholar] [CrossRef] - Xu, Z.S. Multiple-attribute group decision making with different formats of preference information on attributes. IEEE Trans. Syst. Man Cybern. Part B
**2007**, 376, 1500–1511. [Google Scholar] - Xu, Z.S. Group decision making based on multiple types of linguistic preference relations. Inf. Sci.
**2008**, 1782, 452–467. [Google Scholar] [CrossRef] - Martínez, L.; Herrera, F. An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges. Inf. Sci.
**2012**, 207, 1–18. [Google Scholar] [CrossRef] - Ju, Y.B.; Liu, X.Y.; Wang, A.H. Some new Shapley 2-tuple linguistic Choquet aggregation operators and their applications to multiple attribute group decision making. Soft Comput.
**2010**, 20, 4037–4053. [Google Scholar] [CrossRef] - Herrera, F.; Herrera-Viedma, E.; Martı́nez, L. A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Set. Syst.
**2000**, 114, 43–58. [Google Scholar] [CrossRef] [Green Version] - Herrera, F.; Martínez, L. A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making. IEEE Trans. Syst. Man Cybern. Part B
**2001**, 312, 227–234. [Google Scholar] [CrossRef] [PubMed] - Espinilla, M.; Liu, J. Martinez, L. An extended hierarchical linguistic model for decision-making problems. Comput. Intell.
**2011**, 27, 489–512. [Google Scholar] [CrossRef] - Rezaei, J. Best-worst multi-criteria decision-making method. Omega
**2015**, 53, 49–57. [Google Scholar] [CrossRef] - Stević, Ž.; Pamučar, D.; Kazimieras Zavadskas, E.; Ćirović, G.; Prentkovskis, O. The selection of wagons for the internal transport of a logistics company: A novel approach based on rough BWM and rough SAW methods. Symmetry
**2017**, 9, 264. [Google Scholar] [CrossRef] - Herrera, F.; Martinez, L. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst.
**2000**, 86, 746–752. [Google Scholar] - Wei, G.W.; Zhao, X.F. Some dependent aggregation operators with 2-tuple linguistic information and their application to multiple attribute group decision making. Expert Syst. Appl.
**2012**, 395, 5881–5886. [Google Scholar] [CrossRef] - Qin, J.D.; Liu, X.W.; Pedrycz, W. Hesitant fuzzy Maclaurin symmetric mean operators and its application to multiple-attribute decision making. Int. J. Fuzzy Syst.
**2015**, 174, 509–520. [Google Scholar] [CrossRef] - Qin, J.D.; Liu, X.W. 2-tuple linguistic Muirhead mean operators for multiple attribute group decision making and its application to supplier selection. Kybernetes
**2016**, 451, 2–29. [Google Scholar] [CrossRef] - Wang, J.; Wei, G.; Wei, Y. Models for green supplier selection with some 2-tuple linguistic neutrosophic number Bonferroni mean operators. Symmetry
**2018**, 10, 131. [Google Scholar] [CrossRef] - Wang, L.; Wang, Y.; Liu, X. Prioritized aggregation operators and correlated aggregation operators for hesitant 2-tuple linguistic variables. Symmetry
**2018**, 10, 39. [Google Scholar] [CrossRef] - Jiang, X.P.; Wei, G.W. Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making. J. Intell. Fuzzy Syst.
**2014**, 275, 2153–2162. [Google Scholar]

**Figure 2.**Group decision making for multi-granularity 2-tuple linguistic preference to prioritize engineering characteristics in quality function deployment.

α_{BW} | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Consistency index | 0.00 | 0.44 | 1.00 | 1.63 | 2.30 | 3.00 | 3.73 | 4.47 | 5.23 |

Passengers | Best | $\mathit{C}{\mathit{R}}_{1}$ | $\mathit{C}{\mathit{R}}_{2}$ | $\mathit{C}{\mathit{R}}_{3}$ | $\mathit{C}{\mathit{R}}_{4}$ | $\mathit{C}{\mathit{R}}_{5}$ |
---|---|---|---|---|---|---|

1 | $C{R}_{2}$ | 3 | 1 | 5 | 9 | 7 |

2 | $C{R}_{2}$ | 5 | 1 | 4 | 9 | 8 |

3 | $C{R}_{1}$ | 1 | 2 | 7 | 5 | 9 |

4 | $C{R}_{3}$ | 4 | 3 | 1 | 7 | 9 |

5 | $C{R}_{1}$ | 1 | 3 | 9 | 6 | 8 |

Passengers | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Worst | $C{R}_{4}$ | $C{R}_{4}$ | $C{R}_{5}$ | $C{R}_{5}$ | $C{R}_{3}$ |

$C{R}_{1}$ | 6 | 4 | 9 | 4 | 9 |

$C{R}_{2}$ | 9 | 9 | 8 | 7 | 7 |

$C{R}_{3}$ | 5 | 7 | 2 | 9 | 1 |

$C{R}_{4}$ | 1 | 1 | 5 | 2 | 6 |

$C{R}_{5}$ | 4 | 3 | 1 | 1 | 3 |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

$C{R}_{1}$ | ${s}_{5}^{{7}_{1}}$ | ${s}_{6}^{{7}_{1}}$ | ${s}_{0}^{{7}_{1}}$ | ${s}_{4}^{{7}_{1}}$ | ${s}_{0}^{{7}_{1}}$ |

$C{R}_{2}$ | ${s}_{1}^{{7}_{1}}$ | ${s}_{5}^{{7}_{1}}$ | ${s}_{1}^{{7}_{1}}$ | ${s}_{3}^{{7}_{1}}$ | ${s}_{4}^{{7}_{1}}$ |

$C{R}_{3}$ | ${s}_{2}^{{7}_{1}}$ | ${s}_{4}^{{7}_{1}}$ | ${s}_{4}^{{7}_{1}}$ | ${s}_{1}^{{7}_{1}}$ | ${s}_{1}^{{7}_{1}}$ |

$C{R}_{4}$ | ${s}_{3}^{{7}_{1}}$ | ${s}_{5}^{{7}_{1}}$ | ${s}_{1}^{{7}_{1}}$ | ${s}_{6}^{{7}_{1}}$ | $\hspace{0.17em}{s}_{5}^{{7}_{1}}$ |

$C{R}_{5}$ | ${s}_{1}^{{7}_{1}}$ | ${s}_{4}^{{7}_{1}}$ | ${s}_{1}^{{7}_{1}}$ | ${s}_{4}^{{7}_{1}}$ | ${s}_{5}^{{7}_{1}}$ |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

$C{R}_{1}$ | ${s}_{3}^{{5}_{2}}$ | ${s}_{4}^{{5}_{2}}$ | ${s}_{0}^{{5}_{2}}$ | ${s}_{3}^{{5}_{2}}$ | ${s}_{1}^{{5}_{2}}$ |

$C{R}_{2}$ | ${s}_{1}^{{5}_{2}}$ | ${s}_{3}^{{5}_{2}}$ | ${s}_{2}^{{5}_{2}}$ | ${s}_{2}^{{5}_{2}}$ | ${s}_{3}^{{5}_{2}}$ |

$C{R}_{3}$ | ${s}_{1}^{{5}_{2}}$ | ${s}_{2}^{{5}_{2}}$ | ${s}_{3}^{{5}_{2}}$ | ${s}_{1}^{{5}_{2}}$ | ${s}_{1}^{{5}_{2}}$ |

$C{R}_{4}$ | ${s}_{2}^{{5}_{2}}$ | ${s}_{3}^{{5}_{2}}$ | ${s}_{1}^{{5}_{2}}$ | ${s}_{4}^{{5}_{2}}$ | ${s}_{3}^{{5}_{2}}$ |

$C{R}_{5}$ | ${s}_{0}^{{5}_{2}}$ | ${s}_{3}^{{5}_{2}}$ | ${s}_{1}^{{5}_{2}}$ | ${s}_{2}^{{5}_{2}}$ | ${s}_{3}^{{5}_{2}}$ |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

$C{R}_{1}$ | ${s}_{6}^{{9}_{3}}$ | ${s}_{8}^{{9}_{3}}$ | ${s}_{1}^{{9}_{3}}$ | ${s}_{5}^{{9}_{3}}$ | ${s}_{0}^{{9}_{3}}$ |

$C{R}_{2}$ | ${s}_{2}^{{9}_{3}}$ | ${s}_{7}^{{9}_{3}}$ | ${s}_{2}^{{9}_{3}}$ | ${s}_{6}^{{9}_{3}}$ | ${s}_{6}^{{9}_{3}}$ |

$C{R}_{3}$ | ${s}_{0}^{{9}_{3}}$ | ${s}_{6}^{{9}_{3}}$ | ${s}_{6}^{{9}_{3}}$ | ${s}_{2}^{{9}_{3}}$ | ${s}_{4}^{{9}_{3}}$ |

$C{R}_{4}$ | ${s}_{3}^{{9}_{3}}$ | ${s}_{7}^{{9}_{3}}$ | ${s}_{2}^{{9}_{3}}$ | ${s}_{8}^{{9}_{3}}$ | ${s}_{6}^{{9}_{3}}$ |

$C{R}_{5}$ | ${s}_{1}^{{9}_{3}}$ | ${s}_{7}^{{9}_{3}}$ | ${s}_{1}^{{9}_{3}}$ | ${s}_{7}^{{9}_{3}}$ | ${s}_{7}^{{9}_{3}}$ |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

$C{R}_{1}$ | $\left({s}_{5}^{{7}_{1}},0\right)$ | $\left({s}_{6}^{{7}_{1}},0\right)$ | $\left({s}_{0}^{{7}_{1}},0\right)$ | $\left({s}_{4}^{{7}_{1}},0\right)$ | $\left({s}_{0}^{{7}_{1}},0\right)$ |

$C{R}_{2}$ | $\left({s}_{1}^{{7}_{1}},0\right)$ | $\left({s}_{5}^{{7}_{1}},0\right)$ | $\left({s}_{1}^{{7}_{1}},0\right)$ | $\left({s}_{3}^{{7}_{1}},0\right)$ | $\left({s}_{4}^{{7}_{1}},0\right)$ |

$C{R}_{3}$ | $\left({s}_{2}^{{7}_{1}},0\right)$ | $\left({s}_{4}^{{7}_{1}},0\right)$ | $\left({s}_{4}^{{7}_{1}},0\right)$ | $\left({s}_{1}^{{7}_{1}},0\right)$ | $\left({s}_{1}^{{7}_{1}},0\right)$ |

$C{R}_{4}$ | $\left({s}_{3}^{{7}_{1}},0\right)$ | $\left({s}_{5}^{{7}_{1}},0\right)$ | $\left({s}_{1}^{{7}_{1}},0\right)$ | $\left({s}_{6}^{{7}_{1}},0\right)$ | $\left({s}_{5}^{{7}_{1}},0\right)$ |

$C{R}_{5}$ | $\left({s}_{1}^{{7}_{1}},0\right)$ | $\left({s}_{4}^{{7}_{1}},0\right)$ | $\left({s}_{1}^{{7}_{1}},0\right)$ | $\left({s}_{4}^{{7}_{1}},0\right)$ | $\left({s}_{5}^{{7}_{1}},0\right)$ |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

$C{R}_{1}$ | $\left({s}_{3}^{{5}_{2}},0\right)$ | $\left({s}_{4}^{{5}_{2}},0\right)$ | $\left({s}_{0}^{{5}_{2}},0\right)$ | $\left({s}_{3}^{{5}_{2}},0\right)$ | $\left({s}_{1}^{{5}_{2}},0\right)$ |

$C{R}_{2}$ | $\left({s}_{1}^{{5}_{2}},0\right)$ | $\left({s}_{3}^{{5}_{2}},0\right)$ | $\left({s}_{2}^{{5}_{2}},0\right)$ | $\left({s}_{2}^{{5}_{2}},0\right)$ | $\left({s}_{3}^{{5}_{2}},0\right)$ |

$C{R}_{3}$ | $\left({s}_{1}^{{5}_{2}},0\right)$ | $\left({s}_{2}^{{5}_{2}},0\right)$ | $\left({s}_{3}^{{5}_{2}},0\right)$ | $\left({s}_{1}^{{5}_{2}},0\right)$ | $\left({s}_{1}^{{5}_{2}},0\right)$ |

$C{R}_{4}$ | $\left({s}_{2}^{{5}_{2}},0\right)$ | $\left({s}_{3}^{{5}_{2}},0\right)$ | $\left({s}_{1}^{{5}_{2}},0\right)$ | $\left({s}_{4}^{{5}_{2}},0\right)$ | $\left({s}_{3}^{{5}_{2}},0\right)$ |

$C{R}_{5}$ | $\left({s}_{0}^{{5}_{2}},0\right)$ | $\left({s}_{3}^{{5}_{2}},0\right)$ | $\left({s}_{1}^{{5}_{2}},0\right)$ | $\left({s}_{2}^{{5}_{2}},0\right)$ | $\left({s}_{3}^{{5}_{2}},0\right)$ |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

$C{R}_{1}$ | $\left({s}_{6}^{{9}_{3}},0\right)$ | $\left({s}_{8}^{{9}_{3}},0\right)$ | $\left({s}_{1}^{{9}_{3}},0\right)$ | $\left({s}_{5}^{{9}_{3}},0\right)$ | $\left({s}_{0}^{{9}_{3}},0\right)$ |

$C{R}_{2}$ | $\left({s}_{2}^{{9}_{3}},0\right)$ | $\left({s}_{7}^{{9}_{3}},0\right)$ | $\left({s}_{2}^{{9}_{3}},0\right)$ | $\left({s}_{6}^{{9}_{3}},0\right)$ | $\left({s}_{6}^{{9}_{3}},0\right)$ |

$C{R}_{3}$ | $\left({s}_{0}^{{9}_{3}},0\right)$ | $\left({s}_{6}^{{9}_{3}},0\right)$ | $\left({s}_{6}^{{9}_{3}},0\right)$ | $\left({s}_{2}^{{9}_{3}},0\right)$ | $\left({s}_{4}^{{9}_{3}},0\right)$ |

$C{R}_{4}$ | ${s}_{3}^{{9}_{3}}$ | $\left({s}_{7}^{{9}_{3}},0\right)$ | $\left({s}_{2}^{{9}_{3}},0\right)$ | $\left({s}_{8}^{{9}_{3}},0\right)$ | $\left({s}_{6}^{{9}_{3}},0\right)$ |

$C{R}_{5}$ | $\left({s}_{1}^{{9}_{3}},0\right)$ | $\left({s}_{7}^{{9}_{3}},0\right)$ | $\left({s}_{1}^{{9}_{3}},0\right)$ | $\left({s}_{7}^{{9}_{3}},0\right)$ | $\left({s}_{7}^{{9}_{3}},0\right)$ |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

$C{R}_{1}$ | $({s}_{7}^{{9}_{1}},-0.33)\hspace{0.17em}$ | $({s}_{8}^{{9}_{1}},0)$ | $({s}_{0}^{{9}_{1}},0)$ | $({s}_{5}^{{9}_{1}},0.33)$ | $({s}_{0}^{{9}_{1}},0)$ |

$C{R}_{2}$ | $({s}_{1}^{{9}_{1}},0.33)\hspace{0.17em}$ | $({s}_{7}^{{9}_{1}},-0.33)$ | $({s}_{1}^{{9}_{1}},0.33)$ | $({s}_{4}^{{9}_{1}},0)$ | $({s}_{5}^{{9}_{1}},0.33)$ |

$C{R}_{3}$ | $({s}_{3}^{{9}_{1}},-0.33)$ | $({s}_{5}^{{9}_{1}},0.33)$ | $({s}_{5}^{{9}_{1}},0.33)$ | $({s}_{1}^{{9}_{1}},0.33)$ | $({s}_{1}^{{9}_{1}},0.33)\hspace{0.17em}$ |

$C{R}_{4}$ | $\hspace{0.17em}\hspace{0.17em}({s}_{4}^{{9}_{1}},0)$ | $({s}_{7}^{{9}_{1}},-0.33)$ | $({s}_{1}^{{9}_{1}},0.33)$ | $\left({s}_{8}^{{9}_{3}},0\right)$ | $({s}_{7}^{{9}_{1}},-0.33)$ |

$C{R}_{5}$ | $({s}_{1}^{{9}_{1}},0.33)$ | $({s}_{5}^{{9}_{1}},0.33)\hspace{0.17em}$ | $({s}_{1}^{{9}_{1}},0.33)$ | $({s}_{5}^{{9}_{1}},0.33)$ | $({s}_{7}^{{9}_{1}},-0.33)$ |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

$C{R}_{1}$ | $({s}_{6}^{{9}_{2}},0)\hspace{0.17em}$ | $({s}_{8}^{{9}_{2}},0)$ | $({s}_{0}^{{9}_{1}},0)$ | $({s}_{6}^{{9}_{2}},0)\hspace{0.17em}$ | $({s}_{2}^{{9}_{2}},0)$ |

$C{R}_{2}$ | $({s}_{2}^{{9}_{2}},0)$ | $({s}_{6}^{{9}_{2}},0)\hspace{0.17em}$ | $({s}_{4}^{{9}_{2}},0)$ | $({s}_{4}^{{9}_{1}},0)$ | $({s}_{6}^{{9}_{2}},0)\hspace{0.17em}$ |

$C{R}_{3}$ | $({s}_{2}^{{9}_{2}},0)$ | $({s}_{4}^{{9}_{2}},0)$ | $({s}_{6}^{{9}_{2}},0)\hspace{0.17em}$ | $({s}_{2}^{{9}_{2}},0)$ | $({s}_{2}^{{9}_{2}},0)$ |

$C{R}_{4}$ | $({s}_{4}^{{9}_{2}},0)$ | $({s}_{6}^{{9}_{2}},0)\hspace{0.17em}$ | $({s}_{2}^{{9}_{2}},0)$ | $({s}_{8}^{{9}_{2}},0)$ | $({s}_{6}^{{9}_{2}},0)\hspace{0.17em}$ |

$C{R}_{5}$ | $({s}_{0}^{{9}_{2}},0)$ | $({s}_{6}^{{9}_{2}},0)\hspace{0.17em}$ | $({s}_{2}^{{9}_{2}},0)$ | $({s}_{4}^{{9}_{2}},0)$ | $({s}_{6}^{{9}_{2}},0)\hspace{0.17em}$ |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

$C{R}_{1}$ | $({s}_{2}^{9},-0.14)$ | $({s}_{2}^{9},0.05)$ | $({s}_{0}^{9},0)$ | $({s}_{2}^{9},-0.23)$ | $({s}_{0}^{9},0)$ |

$C{R}_{2}$ | $({s}_{1}^{9},0.23)$ | $({s}_{2}^{9},-0.08)$ | $({s}_{1}^{9},0.33)$ | $({s}_{2}^{9},-0.26)$ | $({s}_{2}^{9},-0.16)$ |

$C{R}_{3}$ | $({s}_{1}^{9},-0.23)$ | $({s}_{2}^{9},-0.23)$ | $({s}_{2}^{9},-0.16)$ | $({s}_{1}^{9},0.23)$ | $({s}_{1}^{9},0.41)$ |

$C{R}_{4}$ | $({s}_{2}^{9},-0.48)$ | $({s}_{2}^{9},-0.08)$ | $({s}_{1}^{9},0.23)$ | $({s}_{2}^{9},0.05)$ | $({s}_{2}^{9},-0.14)\hspace{0.17em}$ |

$C{R}_{5}$ | $({s}_{1}^{9},-0.35)$ | $({s}_{2}^{9},-0.1)$ | $({s}_{1}^{9},0.1)$ | $({s}_{2}^{9},-0.17)$ | $({s}_{2}^{9},-0.08)\hspace{0.17em}$ |

$\mathit{E}{\mathit{C}}_{1}$ | $\mathit{E}{\mathit{C}}_{2}\hspace{0.17em}$ | $\mathit{E}{\mathit{C}}_{3}$ | $\mathit{E}{\mathit{C}}_{4}$ | $\mathit{E}{\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

Priority | $({s}_{1}^{9},0.07)$ | $({s}_{1}^{9},0.13)$ | $({s}_{1}^{9},-0.18)$ | $({s}_{1}^{9},0.11)$ | $({s}_{1}^{9},-0.16)$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mei, Y.; Liang, Y.; Tu, Y.
A Multi-Granularity 2-Tuple QFD Method and Application to Emergency Routes Evaluation. *Symmetry* **2018**, *10*, 484.
https://doi.org/10.3390/sym10100484

**AMA Style**

Mei Y, Liang Y, Tu Y.
A Multi-Granularity 2-Tuple QFD Method and Application to Emergency Routes Evaluation. *Symmetry*. 2018; 10(10):484.
https://doi.org/10.3390/sym10100484

**Chicago/Turabian Style**

Mei, Yanlan, Yingying Liang, and Yan Tu.
2018. "A Multi-Granularity 2-Tuple QFD Method and Application to Emergency Routes Evaluation" *Symmetry* 10, no. 10: 484.
https://doi.org/10.3390/sym10100484