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

Abstract

Quality function deployment (QFD) is an effective approach to satisfy the customer requirements (CRs). Furthermore, accurately prioritizing the engineering characteristics (ECs) as the core of QFD is considered as a group decision making (GDM) problem. In order to availably deal with various preferences and the vague information of different experts on a QFD team, multi-granularity 2-tuple linguistic representation is applied to elucidate the relationship and correlation between CRs and ECs without loss of information. In addition, the importance of CRs is determined using the best worst method (BWM), which is more applicable and has good consistency. Furthermore, we propose considering the relationship matrix and correlation matrix method to prioritize ECs. Finally, an example about evaluating emergency routes of metro station is proposed to illustrate the validity of the proposed methodology.

1. Introduction

In order to cope with intense global competitions, enterprises must design the highest quality products that satisfy the voice of customers (VOCs). Quality function deployment (QFD) is an effective method to map customer requirements (CRs) into engineering characteristics (ECs) in the area of product development [1] and construction industry [2]. The core of QFD is requirements conversion, moreover, the first phase in house of quality (HOQ) mapping CRs to ECs becomes an essential procedure of implementing QFD [3].

Aiming at implementing QFD successfully, plenty of CRs should be acquired, and group decision making (GDM) should be adopted [4]. QFD consists of two major steps: collecting the CRs and mapping it to ECs, both of which are performed [5,6]. This paper focuses on how the ECs in QFD can be prioritized.

There are plenty of methods to prioritize the ECs. Fuzzy set theory was widely employed to calculate the rankings of ECs under the circumstance of vagueness and impreciseness. Fuzzy multiple objective programming [7], fuzzy goal programming [8], fuzzy relationship and correlations [9], and expected value-based method [10] are proposed to prioritize ECs. In addition, Geng et al. [11] integrated the analytic network process to QFD to reflect the initial importance weights of ECs. However, the problem is that they paid little attention to the GDM method, which can aggregate different experts’ preferences. For the purpose of reaching collective decisions, we combine GDM with QFD.

Kwong et al. [6] put forward the fuzzy GDM method integrated with a fuzzy weighted average to rank ECs. Wang [12] adopted the method of aggregating technical importance rather than CRs to prioritize ECs. With respect to consensus, modified fuzzy clustering was presented so as to reach the consensus of the QFD team [13]. A two-stage GDM was proposed to simultaneously solve the two types of uncertainties (i.e., human assessment on qualitative attributes as well as input information) underlying QFD [4]. However, due to varying personal experience and knowledge, the input information of experts presented with multi-format or multi-granular linguistic preferences makes prioritizing ECs more difficult. Therefore, some scholars have focused on the GDM approach based on multi-granularity linguistic environments [14,15,16,17]. Xu [18,19] analyzed multiple formats’ preferences and provided an approach integrating information in the context of GDM. It is noteworthy that multi-granularity evaluation should be analyzed.

The correlation between CRs and ECs influencing on the relationship becomes ignored and simplified in the current study. In addition, the linguistic accuracy remains to be discussed. Considering that the 2-tuple linguistic representation can increase the information of precision [20,21]. In order to fill the gap, it is necessary that the QFD methodology is extended with a 2-tuple linguistic environment so as to lessen the loss of information and obtain accurate value of ECs. In addition, decision makers may have different knowledge and experience in the process of group decision making, and they may then adopt different linguistic labels to describe the same decision-making problems. This process is denoted as multi-granular linguistic information, which conforms to the actual decision-making process. Therefore, we allow decision makers to employ multi-granular linguistic information, i.e., the linguistic term set has different granularities.

A majority of methods deal with multi-granular linguistic information. Herrera et al. proposed the definition of a basic linguistic term set, and then different linguistic labels can be unified based on a basic linguistic term set [22]. In addition, some transformation methods based on the linguistic hierarchy and extended linguistic hierarchy were presented and applied to a plenty of decision-making problems [23,24]. Among these approaches, the method considering linguistic hierarchy is more flexible and convenient to carry out. In this paper, we adopt this method to deal with the problem of multi-granular linguistic evaluation. For determining the weight of CRs, we adopt the best–worst method (BWM) in this paper. This method has good consistency and is easier to implement [25,26]. Our contributions lie in using the BWM to determine the importance of CRs and integrate the correlations matrix with the relationship matrix based on a compromise idea, where experts can express their thoughts in different granularities.

In this paper, a GDM approach is integrated with QFD to solve different preferences and prioritize ECs. The multi-granularity 2-tuple linguistic information to reflect the attitudes of different experts is employed. This paper is organized as follows: In Section 2, a 2-tuple multi-granularity linguistic representation model, linguistic hierarchies, and a 2-tuple linguistic weighted geometric Bonferroni mean (2TLWGBM) operator are presented. In Section 3, the BWM is applied to compute the weight of CRs, and a novel GDM approach to prioritize ECs is proposed. An illustrated example about metro stations is provided in Section 4 to demonstrate the applicability of this method. Ultimately, conclusions and future research are marked in Section 5.

2. Preliminaries

In this section, we introduce some basic knowledge about QFD, 2-tuple representation and the 2TLWGBM Operator.

2.1. The Basic Knowledge on QFD

A four-phase QFD model is employed to translate the VOCs to ECs, which consists of Product Planning, Part Deployment, Process Planning, and Process and Quality control [3]. The first phase is to collect customer requirements for the product called WHATs and then to transform these needs into ECs called HOWs. This phase is so fundamental in product development that the corresponding QFD transformation matrix referred to the HOQ (Figure 1). The HOQ links customer needs to the development team’s technical responses, so we focus on this phase in order to translate different preference of customers and experts to prioritize ECs. In this paper, we first take the relationship between CRs and ECs into consideration. In order to transform the importance of CRs into ECs, the correlation of CRs and ECs is introduced to modify the initial relationship afterward.

2.2. The 2-Tuple Linguistic Representation

There are numerous formats for representing preference such as linguistic, numerical and 2-tuple linguistic representation. Compared with other forms, 2-tuple linguistic representation makes the assessment more precise and without a loss of information [20]. Next, we will introduce some basic knowledge about 2-tuple 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:

$$\Delta :[0,g]\to S\times [-0.5,0.5)$$

(1)

$$\Delta (\beta )=(s_i,\alpha ),\hspace{0.17em}\mathrm{with}\{\begin{array}{l}{s}_i,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=\mathrm{round}(\beta )\\ \alpha =\beta -i,\hspace{0.17em}\hspace{0.17em}\alpha =[-0.5,0.5)\end{array}$$

(2)

where round (∙) represents the usual round function, $s_i$ has the closest index label to $\beta$, and $\alpha$ is the value of the symbolic translation.

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:

$${\Delta }^{-1}:\to S\times [-0.5,0.5)\to [0,g]$$

(3)

$${\Delta }^{-1}(s_i,{\alpha }_i)=i+{\alpha }_i=\beta$$

(4)

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:

$$(\overline{s},\overline{\alpha })=\Delta \left(\frac{1}{n}{\displaystyle \sum _{i=1}^n{\Delta }^{-1}(r_i,{\alpha }_i)}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{s}\in S,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\alpha }\in [-0.5,0.5)$$

(5)

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:

$$d((s_i,{\alpha }_i),(s_j,{\alpha }_j))=\frac{\left|{\Delta }^{-1}(i+{\alpha }_i)-{\Delta }^{-1}(j+{\alpha }_j)\right|}{g}$$

(6)

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

$$\mathrm{sim}((s_{\pi (j)},{\alpha }_{\pi (j)}),(\overline{s},\overline{\alpha }))=1-\frac{d((s_{\pi (j)},{\alpha }_{\pi (j)}),(\overline{s},\overline{\alpha }))}{{\displaystyle \sum _{j=1}^{n}d((s_{\pi (j)},{\alpha }_{\pi (j)}),(\overline{s},\overline{\alpha }))}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\cdots ,n$$

(7)

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

$$\begin{array}{}T{F}_{t\prime }^t:l(t,n(t))\to l(t\prime ,n(t\prime ))\\ T{F}_{t\prime }^t(s_i^{n(t)},{\alpha }^{n(t)})=\Delta \left(\frac{{\Delta }^{-1}(s_i^{n(t)},{\alpha }^{n(t)})\cdot (n(t\prime )-1)}{n(t)-1}\right)\end{array}$$

(8)

where $t$ and $t\prime$ represent different levels of linguistic hierarchy.

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

There are numerous operators to aggregate information in different linguistic environments, such as hesitant fuzzy Maclaurin symmetric mean Operators [29], 2-tuple linguistic Muirhead mean operators [30], 2-tuple linguistic Neutrosophic number Bonferroni mean operators [31], and hesitant 2-tuple linguistic prioritized weighted averaging aggregation operator [32] in the context of the 2-tuple environment. In view of the Bonferroni mean (BM) operator capturing the interrelationship between input information and ranking ECs under a 2-tuple environment, so the 2TLWGBM operator [33] will be applied to prioritize the sequence of ECs. BM is defined as follows:

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

$$B{M}^{p,q}(a_1,a_2,\cdots ,a_n)={\left(\frac{1}{n(n-1)}{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n{a}_i^p{a}_j^q}\right)}^{\frac{1}{p+q}}$$

(9)

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${\displaystyle {\sum }_{i=1}^n{w}_i=1}.$ The 2TLWGBM operator is then expressed as

$$\begin{array}{l}2TLWGB{M}_w^{p,q}((r_1,a_1),(r_2,a_2),\cdots ,(r_n,a_n))\\ =\Delta \left(\frac{1}{p+q}{\left({\displaystyle \prod _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n\left(p{\left({\Delta }^{-1}(r_i,a_i)\right)}^{w_i}+q{\left({\Delta }^{-1}(r_j,a_j)\right)}^{w_j}\right)}\right)}^{\frac{1}{n(n-1)}}\right)\end{array}$$

(10)

For the sake of simplicity, it can be set$p=q=1$, the aggregation operator is indicated as

$$\begin{array}{l}2TLWGB{M}_w^{1,1}((r_1,a_1),(r_2,a_2),\cdots ,(r_n,a_n))\\ =\Delta \left(\frac{1}{2}{\left({\displaystyle \prod _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n\left({\left({\Delta }^{-1}(r_i,a_i)\right)}^{w_i}+{\left({\Delta }^{-1}(r_j,a_j)\right)}^{w_j}\right)}\right)}^{\frac{1}{n(n-1)}}\right)\hspace{0.17em}\end{array}$$

(11)

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

Best worst method (BWM) is a MCDM method possessing the advantages in aspects of reaching the consistency and simplifying the calculation with respect to AHP. The core idea of BWM is constructing comparisons relationships between the best attribute (and the worst attribute) to the other attributes. Additionally, an optimization model established ground on consistency is solved to obtain the optical weights. Owing to simple operation and calculation, the BWM is synthesized to determine the importance of CRs. The steps are listed as follows:

Step 1. CRs {CR₁, CR₂, ⋯, 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|}.

Model 1 can be transformed into a linear programming model as follows:

Model 2 is solved to obtain the optimal importance of CRs ($w_1^{*}$ , $w_2^{*}$, ⋯, $w_n^{*}$) and ξ^*. 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).

$$\text{Consistency Ratio}=\frac{{\xi }^{*}}{\text{max} \xi }=\frac{{\xi }^{*}}{\text{Consistency Index}}$$

(12)

where the max ξ is determined according to (α_BW − ξ) × (α_BW − ξ) = (α_BW + ξ) and α_BW ∈ {1, 2, ⋯, 9}. The consistency index is listed in

Table 1. Consistency index.

αBW	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

.

3.2. A Group Decision-Making Approach to Prioritize ECs

In this section, the steps of GDM for multi-granularity 2-tuple linguistic preference to prioritize ECs in QFD are given as follows:

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

Suppose the experts $E{P}_k\hspace{0.17em}(k=1,2,\cdots ,t)$ in QFD product research or design team give the relationship between $C{R}_i(i=1,2,\cdots ,n)$ and $E{C}_j(j=1,2,\cdots ,s)$ based on different multi-granularities. The kth expert’s linguistic term set and evaluation matrix respectively denoted as $S^{n{(t)}_k}=\left\{s_i^{n{(t)}_k}|i=0,1,\cdots ,n(t)-1\right\}$ and $R^k=(r_{ij}{)}_{n\times s}\hspace{0.17em}{r}_{ij}\in S^{n{(t)}_k}$, which is transformed into 2-tuple linguistic evaluation matrix ${\tilde{R}}^k=(r_{ij}^{n{(t)}_k},0{)}_{n\times s},r_{ij}^{n{(t)}_k}\in S^{n{(t)}_k}$.

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

To begin with, a basic linguistic term set $S^{n{(t)}_u}=\left\{s_i^{n{(t)}_u}|i=0,1,\cdots ,n(t)-1\right\}$ can be chosen, and the relationship matrix can then be transformed applying Equation (8) so as to make 2-tuple linguistic representation reach the same granularity. For instance, the kth expert’s judgement matrix is transformed as ${\tilde{R}}^{k\prime }=(r_{ij}^{n{(t)}_u},{\alpha }_{ij}^{n{(t)}_u}{)}_{n\times s},\hspace{0.17em}{r}_{ij}^{n{(t)}_u}\in S^{n{(t)}_u}$.

Step 5. All the evaluation matrices are aggregated.

All the evaluation matrices uniformed are aggregated with 2TLWGBM operator in virtue of Equation (10) into $R_{ij}(i=1,2,\cdots ,n\hspace{0.17em};j=1,2,\cdots ,s).$ Furthermore, the new matrix represents ultimate relationship matrix between CRs and ECs in essence.

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

After establishing the aggregation matrix, experts give the correlations among CRs and ECs and the initial HOQ can be obtained, which reflects the relationship $R_{ij}$ between $C{R}_i$ and $E{C}_j,$$i=1,2,\cdots ,n\hspace{0.17em};\hspace{0.17em}j=1,2,\cdots ,s.$It is indispensable that the QFD team estimates the correlations between CRs (i.e., $L_{i\xi }(\xi \ne i,\xi =1,2,\cdots ,n)$) and ECs (i.e., $T_{i\theta }(\theta \ne i,\theta =1,2,\cdots ,s)$) using 2-tuple based on the basic linguistic set $S_i^{n{(t)}_u}.$ Considering that the assessment result of $L_{i\xi }(\xi \ne i,\xi =1,2,\cdots ,n)$ and $T_{i\theta }(\theta \ne i,\theta =1,2,\cdots ,s)$ has an effect on the initial aggregation matrix of relationship $R_{ij}$ with respect to CRs and ECs, a higher $L_{i\xi }(\xi \ne i,\xi =1,2,\cdots ,n)$ or $T_{i\theta }(\theta \ne i,\theta =1,2,\cdots ,s)$ implies a benefit to $R_{ij}$. Consequently, the correlations are taken into account when modifying the relationships between CRs and ECs. In the process of adjustment, Equation (13) is applied to integrate $L_{i\xi }(\xi \ne i,\xi =1,2,\cdots ,n)$ and $T_{i\theta }(\theta \ne i,\theta =1,2,\cdots ,s)$ into $R_{ij}.$ Furthermore, the modified relationship is computed using the formula as follows:

$$R_{ij}^{\prime }=\Delta \left({\displaystyle \prod _{v=1}^V{\Delta }^{-1}{(s_m,{\alpha }_m)}^{{\gamma }_v}}\right),\hspace{0.17em}i=1,2,\cdots ,n\hspace{0.17em};j=1,2,\cdots ,s$$

(13)

where ${\Delta }^{-1}(r_m,a_m)$ is stemming from the set $S=\{{\Delta }^{-1}{R}_{ij}(r_{ij}^{n{(t)}_u},{\alpha }_{ij}^{n{(t)}_u}),{\Delta }^{-1}{L}_{i\xi }(r_{i\xi }^{n{(t)}_u},{\alpha }_{i\xi }^{n{(t)}_u}),$ ${\Delta }^{-1}{T}_{j\theta }(r_{j\theta }^{n{(t)}_u},{\alpha }_{j\theta }^{n{(t)}_u})\}\hspace{0.17em}\xi \ne i,\theta \ne j.$ In addition, the weight of ${\gamma }_v$ is corresponding to the proportion of ${\Delta }^{-1}(r_m,a_m)$. For the sake of reducing the impact from subjectivity, unduly high or unduly low preference values in the correlation matrices are supposed to possess a low weight under the circumstances. That means only moderated assessment giving a higher weight has a small deviation from the true value, which might be advocated in the process of evaluation. Therefore, the weight can be determined by Equation (14).

$${\gamma }_v=\frac{\mathrm{sim}((s_m,{\alpha }_m),(\overline{s},\overline{\alpha }))}{{\displaystyle \sum _{v=1}^V\mathrm{sim}((s_m,{\alpha }_m),(\overline{s},\overline{\alpha }))}}$$

(14)

Step 7. Integrated ECs priorities are determined.

On account of the inconformity of representation, the 2-tuple linguistic form of the relationship matrix, and the numerical value of CRs importance, the integrated ECs priority $S_{T{C}_j}(j=1,2,\cdots ,s)$ is calculated by Equation (11).

Step 8. Basic priority of ECs is confirmed.

The linguistic distance $d((s_{{}_{T{C}_j}}^{n(t)},{\alpha }^{n(t)})(s_{{}_{\min }}^{n(t)},{\alpha }^{n(t)}))$ can be adopted to measure the importance degree, where $(s_{{}_{\min }}^{n(t)},{\alpha }^{n(t)})$ is the minimum value of linguistic term set. Furthermore, the measurement of 2-tuple linguistic distance decides the importance of $E{C}_j(j=1,2,\cdots ,s)$. Therefore, the normative value of basic priority $bp{r}_j$ is computed as follows:

$$bp{r}_j=\frac{d((s_{{}_{E{C}_j}}^{n(t)},{\alpha }^{n(t)})(s_{{}_{\min }}^{n(t)},{\alpha }^{n(t)}))}{{\displaystyle \sum _{j=1}^{s}d((s_{{}_{E{C}_j}}^{n(t)},{\alpha }^{n(t)})(s_{{}_{\min }}^{n(t)},{\alpha }^{n(t)}))}}$$

(13)

Step 9. End.

The flow chart of the whole procedures is shown in Figure 2.

4. Case Study

4.1. Background

The Wuhan metro station is the most common two-floor island structure, which consists mainly of a platform and a station hall. The underground floor is the station hall floor. As shown in Figure 3, the metro station has four main exits and one reserved outlet for docking with the shopping mall and fire curtains are installed at each exit. Therefore, when a crowd passes through the fire curtain in the emergency evacuation process, they have reached the safe area. The station hall floor has four automatic ticket checkers and two emergency dedicated channels. In emergency situations, an automatic ticket checking machine and emergency dedicated channels are in open state. The second underground floor is the platform layer. When an emergency occurs on the platform layer, the crowd must first ascend to the station hall layer and then evacuate through the safety exit.

Taking regional $S$ as an example, we analyze the influence factors that have an effect on the evacuation route planning in this area. Five CRs and ECs are selected in order to determine the weight degree of ECs, which can be a basic of evaluating emergency routes.

$$\begin{array}{l}C{R}_1:\hspace{0.17em}\mathrm{Expected}\mathrm{evacuation}\mathrm{time}\\ C{R}_2:\hspace{0.17em}\mathrm{Crowd}\mathrm{density}\\ C{R}_3:\hspace{0.17em}\mathrm{Risk}\mathrm{level}\mathrm{in}\mathrm{the}\mathrm{region}\\ C{R}_4:\hspace{0.17em}\mathrm{Possibility}\mathrm{of}\mathrm{congestion}\\ C{R}_5:\hspace{0.17em}\mathrm{Evacuation}\mathrm{capability}\end{array}\begin{array}{l}E{C}_1:\hspace{0.17em}\mathrm{Number}\mathrm{of}\mathrm{evacuees}\mathrm{per}\mathrm{unit}\mathrm{time}\\ E{C}_2:\hspace{0.17em}\mathrm{Managerial}\mathrm{capability}\\ E{C}_3:\hspace{0.17em}\mathrm{Risk}\mathrm{level}\mathrm{of}\mathrm{disaster}\\ E{C}_4:\hspace{0.17em}\mathrm{Organizational}\mathrm{situation}\\ E{C}_5:\hspace{0.17em}\mathrm{Evacuation}\mathrm{equipment}\end{array}$$

4.2. Implementation

Step 1. The evaluation CRs relationships by passengers are shown in

Table 2. Best-to-others (BO) vector for passengers.

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

and

Table 3. Others-to-worst (OW) vector for passengers.

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

.

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:

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\min \hspace{0.17em}\xi \\ s.t.\hspace{0.17em}\left|w_2-3w_1\right|\le \xi ,\left|w_2-5w_3\right|\le \xi ,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left|w_2-9w_4\right|\le \xi ,\left|w_2-7w_5\right|\le \xi ,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left|w_1-6w_4\right|\le \xi ,\left|w_3-5w_4\right|\le \xi ,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left|w_5-4w_4\right|\le \xi ,{\displaystyle \sum _5{w}_j=1,}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{w}_j\ge 0,\hspace{0.17em}j=1,2,\cdots ,5.\end{array}$$

The parameter $\xi$ is obtained as 0.12, and the consistency ratio can be then computed using Equation (12) as 0.023, which indicates it has good consistency.

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. Evaluation matrix $R^1$ for EP₁.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
$C{R}_1$	$s_5^{7_1}$	$s_6^{7_1}$	$s_0^{7_1}$	$s_4^{7_1}$	$s_0^{7_1}$
$C{R}_2$	$s_1^{7_1}$	$s_5^{7_1}$	$s_1^{7_1}$	$s_3^{7_1}$	$s_4^{7_1}$
$C{R}_3$	$s_2^{7_1}$	$s_4^{7_1}$	$s_4^{7_1}$	$s_1^{7_1}$	$s_1^{7_1}$
$C{R}_4$	$s_3^{7_1}$	$s_5^{7_1}$	$s_1^{7_1}$	$s_6^{7_1}$	$\hspace{0.17em}{s}_5^{7_1}$
$C{R}_5$	$s_1^{7_1}$	$s_4^{7_1}$	$s_1^{7_1}$	$s_4^{7_1}$	$s_5^{7_1}$

,

Table 5. Evaluation matrix $R^2$ for EP₂.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
$C{R}_1$	$s_3^{5_2}$	$s_4^{5_2}$	$s_0^{5_2}$	$s_3^{5_2}$	$s_1^{5_2}$
$C{R}_2$	$s_1^{5_2}$	$s_3^{5_2}$	$s_2^{5_2}$	$s_2^{5_2}$	$s_3^{5_2}$
$C{R}_3$	$s_1^{5_2}$	$s_2^{5_2}$	$s_3^{5_2}$	$s_1^{5_2}$	$s_1^{5_2}$
$C{R}_4$	$s_2^{5_2}$	$s_3^{5_2}$	$s_1^{5_2}$	$s_4^{5_2}$	$s_3^{5_2}$
$C{R}_5$	$s_0^{5_2}$	$s_3^{5_2}$	$s_1^{5_2}$	$s_2^{5_2}$	$s_3^{5_2}$

and

Table 6. Evaluation matrix $R^3$ for EP₃.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
$C{R}_1$	$s_6^{9_3}$	$s_8^{9_3}$	$s_1^{9_3}$	$s_5^{9_3}$	$s_0^{9_3}$
$C{R}_2$	$s_2^{9_3}$	$s_7^{9_3}$	$s_2^{9_3}$	$s_6^{9_3}$	$s_6^{9_3}$
$C{R}_3$	$s_0^{9_3}$	$s_6^{9_3}$	$s_6^{9_3}$	$s_2^{9_3}$	$s_4^{9_3}$
$C{R}_4$	$s_3^{9_3}$	$s_7^{9_3}$	$s_2^{9_3}$	$s_8^{9_3}$	$s_6^{9_3}$
$C{R}_5$	$s_1^{9_3}$	$s_7^{9_3}$	$s_1^{9_3}$	$s_7^{9_3}$	$s_7^{9_3}$

). 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. 2-tuple linguistic evaluation matrix ${\tilde{R}}^1$ for EP₁.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
$C{R}_1$	$\left(s_5^{7_1},0\right)$	$\left(s_6^{7_1},0\right)$	$\left(s_0^{7_1},0\right)$	$\left(s_4^{7_1},0\right)$	$\left(s_0^{7_1},0\right)$
$C{R}_2$	$\left(s_1^{7_1},0\right)$	$\left(s_5^{7_1},0\right)$	$\left(s_1^{7_1},0\right)$	$\left(s_3^{7_1},0\right)$	$\left(s_4^{7_1},0\right)$
$C{R}_3$	$\left(s_2^{7_1},0\right)$	$\left(s_4^{7_1},0\right)$	$\left(s_4^{7_1},0\right)$	$\left(s_1^{7_1},0\right)$	$\left(s_1^{7_1},0\right)$
$C{R}_4$	$\left(s_3^{7_1},0\right)$	$\left(s_5^{7_1},0\right)$	$\left(s_1^{7_1},0\right)$	$\left(s_6^{7_1},0\right)$	$\left(s_5^{7_1},0\right)$
$C{R}_5$	$\left(s_1^{7_1},0\right)$	$\left(s_4^{7_1},0\right)$	$\left(s_1^{7_1},0\right)$	$\left(s_4^{7_1},0\right)$	$\left(s_5^{7_1},0\right)$

,

Table 8. 2-tuple linguistic evaluation matrix ${\tilde{R}}^2$ for EP₂.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
$C{R}_1$	$\left(s_3^{5_2},0\right)$	$\left(s_4^{5_2},0\right)$	$\left(s_0^{5_2},0\right)$	$\left(s_3^{5_2},0\right)$	$\left(s_1^{5_2},0\right)$
$C{R}_2$	$\left(s_1^{5_2},0\right)$	$\left(s_3^{5_2},0\right)$	$\left(s_2^{5_2},0\right)$	$\left(s_2^{5_2},0\right)$	$\left(s_3^{5_2},0\right)$
$C{R}_3$	$\left(s_1^{5_2},0\right)$	$\left(s_2^{5_2},0\right)$	$\left(s_3^{5_2},0\right)$	$\left(s_1^{5_2},0\right)$	$\left(s_1^{5_2},0\right)$
$C{R}_4$	$\left(s_2^{5_2},0\right)$	$\left(s_3^{5_2},0\right)$	$\left(s_1^{5_2},0\right)$	$\left(s_4^{5_2},0\right)$	$\left(s_3^{5_2},0\right)$
$C{R}_5$	$\left(s_0^{5_2},0\right)$	$\left(s_3^{5_2},0\right)$	$\left(s_1^{5_2},0\right)$	$\left(s_2^{5_2},0\right)$	$\left(s_3^{5_2},0\right)$

and

Table 9. 2-tuple linguistic evaluation matrix ${\tilde{R}}^3$ for EP₃.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
$C{R}_1$	$\left(s_6^{9_3},0\right)$	$\left(s_8^{9_3},0\right)$	$\left(s_1^{9_3},0\right)$	$\left(s_5^{9_3},0\right)$	$\left(s_0^{9_3},0\right)$
$C{R}_2$	$\left(s_2^{9_3},0\right)$	$\left(s_7^{9_3},0\right)$	$\left(s_2^{9_3},0\right)$	$\left(s_6^{9_3},0\right)$	$\left(s_6^{9_3},0\right)$
$C{R}_3$	$\left(s_0^{9_3},0\right)$	$\left(s_6^{9_3},0\right)$	$\left(s_6^{9_3},0\right)$	$\left(s_2^{9_3},0\right)$	$\left(s_4^{9_3},0\right)$
$C{R}_4$	$s_3^{9_3}$	$\left(s_7^{9_3},0\right)$	$\left(s_2^{9_3},0\right)$	$\left(s_8^{9_3},0\right)$	$\left(s_6^{9_3},0\right)$
$C{R}_5$	$\left(s_1^{9_3},0\right)$	$\left(s_7^{9_3},0\right)$	$\left(s_1^{9_3},0\right)$	$\left(s_7^{9_3},0\right)$	$\left(s_7^{9_3},0\right)$

.

Step 5. The aggregation of all the evaluation matrices in Table 9,

Table 10. The transformed 2-tuple linguistic evaluation matrix ${\tilde{R}}^{1\prime }$ for EP₁.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
$C{R}_1$	$(s_7^{9_1},-0.33)\hspace{0.17em}$	$(s_8^{9_1},0)$	$(s_0^{9_1},0)$	$(s_5^{9_1},0.33)$	$(s_0^{9_1},0)$
$C{R}_2$	$(s_1^{9_1},0.33)\hspace{0.17em}$	$(s_7^{9_1},-0.33)$	$(s_1^{9_1},0.33)$	$(s_4^{9_1},0)$	$(s_5^{9_1},0.33)$
$C{R}_3$	$(s_3^{9_1},-0.33)$	$(s_5^{9_1},0.33)$	$(s_5^{9_1},0.33)$	$(s_1^{9_1},0.33)$	$(s_1^{9_1},0.33)\hspace{0.17em}$
$C{R}_4$	$\hspace{0.17em}\hspace{0.17em}(s_4^{9_1},0)$	$(s_7^{9_1},-0.33)$	$(s_1^{9_1},0.33)$	$\left(s_8^{9_3},0\right)$	$(s_7^{9_1},-0.33)$
$C{R}_5$	$(s_1^{9_1},0.33)$	$(s_5^{9_1},0.33)\hspace{0.17em}$	$(s_1^{9_1},0.33)$	$(s_5^{9_1},0.33)$	$(s_7^{9_1},-0.33)$

and

Table 11. The transformed 2-tuple linguistic evaluation matrix ${\tilde{R}}^{2\prime }$ for EP₂.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
$C{R}_1$	$(s_6^{9_2},0)\hspace{0.17em}$	$(s_8^{9_2},0)$	$(s_0^{9_1},0)$	$(s_6^{9_2},0)\hspace{0.17em}$	$(s_2^{9_2},0)$
$C{R}_2$	$(s_2^{9_2},0)$	$(s_6^{9_2},0)\hspace{0.17em}$	$(s_4^{9_2},0)$	$(s_4^{9_1},0)$	$(s_6^{9_2},0)\hspace{0.17em}$
$C{R}_3$	$(s_2^{9_2},0)$	$(s_4^{9_2},0)$	$(s_6^{9_2},0)\hspace{0.17em}$	$(s_2^{9_2},0)$	$(s_2^{9_2},0)$
$C{R}_4$	$(s_4^{9_2},0)$	$(s_6^{9_2},0)\hspace{0.17em}$	$(s_2^{9_2},0)$	$(s_8^{9_2},0)$	$(s_6^{9_2},0)\hspace{0.17em}$
$C{R}_5$	$(s_0^{9_2},0)$	$(s_6^{9_2},0)\hspace{0.17em}$	$(s_2^{9_2},0)$	$(s_4^{9_2},0)$	$(s_6^{9_2},0)\hspace{0.17em}$

applying 2TLWGBM operator in Equation (11) into $R_{ij}$ is shown in

Table 12. The aggregation of all the evaluation matrices.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
$C{R}_1$	$(s_2^9,-0.14)$	$(s_2^9,0.05)$	$(s_0^9,0)$	$(s_2^9,-0.23)$	$(s_0^9,0)$
$C{R}_2$	$(s_1^9,0.23)$	$(s_2^9,-0.08)$	$(s_1^9,0.33)$	$(s_2^9,-0.26)$	$(s_2^9,-0.16)$
$C{R}_3$	$(s_1^9,-0.23)$	$(s_2^9,-0.23)$	$(s_2^9,-0.16)$	$(s_1^9,0.23)$	$(s_1^9,0.41)$
$C{R}_4$	$(s_2^9,-0.48)$	$(s_2^9,-0.08)$	$(s_1^9,0.23)$	$(s_2^9,0.05)$	$(s_2^9,-0.14)\hspace{0.17em}$
$C{R}_5$	$(s_1^9,-0.35)$	$(s_2^9,-0.1)$	$(s_1^9,0.1)$	$(s_2^9,-0.17)$	$(s_2^9,-0.08)\hspace{0.17em}$

.

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.

In Figure 4, the correlations between CRs and ECs are computed in the same way as the relationships between CRs and ECs are treated. Apparently, an appropriate relationship matrix should take correlations into account, so the modified relationship in virtue of Equations (5)–(7), (12), and (13) is obtained. The result is illustrated in Figure 5.

We take the relationship between $C{R}_1$ and $E{C}_1$ for an example, the process of calculation is demonstrated as follow

$$\begin{array}{l}(\overline{s},\overline{\alpha })=△(\frac{1}{3}({△}^{-1}(s_2^9,-0.14)+({△}^{-1}(s_2^9,-0.32)+({△}^{-1}(s_1^9,0.39))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=(s_2^9,-0.36)\end{array}$$

$$d((s_2^9,-0.14),(s_2^9,-0.36))=\frac{\left|{△}^{-1}(2-0.14)-{△}^{-1}(2-0.36)\right|}{9}=0.024$$

Similarly, $d((s_2^9,-0.32),(s_2^9,-0.36))=0.004,d((s_1^9,0.39),(s_2^9,-0.36))=0.028$

$$sim((s_2^9,-0.14),(s_2^9,-0.36))=1-\frac{0.024}{0.024+0.004+0.028}=0.571$$

Similarly, $sim((s_2^9,-0.32),(s_2^9,-0.36))=0.928,sim((s_1^9,0.39),(s_2^9,-0.36))=0.5$ We then compute the weight ${\gamma }_v$ by Equation (13)

$$\gamma (s_2^9,-0.14)=\frac{0.571}{0.571+0.928+0.5}=0.286$$

In the same way, $\gamma (s_2^9,-0.32)=0.464,\gamma (s_1^9,0.39)=0.25$

The modified relationship is expressed

$$R_{11}^{\prime }=△({△}^{-1}{(s_2^9,-0.14)}^{0.286}\ast {△}^{-1}{(s_2^9,-0.32)}^{0.464}\ast {△}^{-1}{(s_1^9,0.39)}^{0.25})=(s_2^9,-0.35)$$

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. The integrated ECs priority.

	$\mathit{E}{\mathit{C}}_1$	$\mathit{E}{\mathit{C}}_2\hspace{0.17em}$	$\mathit{E}{\mathit{C}}_3$	$\mathit{E}{\mathit{C}}_4$	$\mathit{E}{\mathit{C}}_5$
Priority	$(s_1^9,0.07)$	$(s_1^9,0.13)$	$(s_1^9,-0.18)$	$(s_1^9,0.11)$	$(s_1^9,-0.16)$

. 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 $(s_{{}_{\min }}^{n(t)},{\alpha }^{n(t)})=(s_{{}_0}^9,0)$. The ultimate weights of ECs are $(s_0^9,0.215)\hspace{0.17em}\hspace{0.17em}(s_0^9,0.228)\hspace{0.17em}\hspace{0.17em}(s_0^9,0.165)\hspace{0.17em}\hspace{0.17em}(s_0^9,0.223)\hspace{0.17em}\hspace{0.17em}(s_0^9,0.169)$.

Step 9. End.

4.3. Managerial Tips

The outcomes of this study are beneficial to planning and selecting the appropriate emergency routes. Moreover, the ranking result can be outlined that decision makers should be paid more attention to management ability. The result indicates that crowd density has a significant influence on emergency route evaluation. Subsequently decision makers should concentrate on these two aspects in order to design and select emergency routes.

In addition, the proposed model is sufficient robust and could be easily implemented in practices for GDM problems. DMs can choose their linguistic preference to evaluate the correlation and relationship between CRs and ECs. Furthermore, the importance of ECs can be adjusted appropriately according to the actual circumstance.

5. Conclusions and Future Research

A systematic GDM approach for prioritizing ECs in QFD under the multi-granularity 2-tuple linguistic environment is proposed in this paper. The provided method allows experts from QFD team to evaluate the relationship and correlations between CRs and ECs in accordance with their experience and preference. For the sake of guaranteeing accurate information, the 2-tuple linguistic representation addressing the vague and imprecise information is utilized. Based on the linguistic hierarchy, different granularities originating from different experts are translated into a basic linguistic term we set in advance. The BWM is applied to determine the importance of CRs, which is simple and quick to represent customers’ advice.

BM can capture inter-relationships among the aggregated information by taking the conjunction among each pairs of aggregated arguments, for instance, correlations among CRs. Therefore, the 2TLWGBM operator is applied to aggregate the evaluation matrix and the importance of CRs. In addition, correlations could have an impact on relationship between CRs and ECs. A modified matrix reflecting the influence is determined in this paper. Compared with other approaches in terms of calculating weight, a method that can lessen the subjectivity of assessment is put forward. Finally, a case study has been calculated and is presented to verify the effectiveness of the proposed method.

In this study, prioritizing ECs in QFD is extended to 2-tuple linguistic environment, in which all evaluation matrices from experts are represented by 2-tuple. For one thing, an appropriate and applicable BM operator is employed to deal with the aggregation problem, which should be suitable for accurately prioritizing ECs in QFD. Moreover, the degree of similarity is introduced to determine the weight that responds to the effect of correlations, which could obtain a more objective modified matrix.

In future research, the proposed method can be applied to supplier selection, green buildings and new product development. In addition, other GDM approaches can be integrated into QFD to rank the ECs, and consensus can be considered. A more reasonable aggregation operator should be developed and applied to QFD. In real life, plenty of problems might be complex and changeful. Establishing a dynamic HOQ is necessary.