Engineering Characteristics Prioritization in Quality Function Deployment Using an Improved ORESTE Method with Double Hierarchy Hesitant Linguistic Information

Abstract

Quality function deployment (QFD) is a customer-driven product development technique widely utilized to translating customer requirements into engineering characteristics for maximum customer satisfaction. Nonetheless, when used in real situations, the traditional QFD method has been criticized to have many deficiencies, e.g., in expressing experts’ uncertain assessments and prioritizing engineering characteristics. In this study, we propose a new engineering characteristics prioritization approach based on double hierarchy hesitant linguistic term sets (DHHLTSs) and the ORESTE (organísation, rangement et Synthèse de données relarionnelles, in French) method to overcome the shortcomings of the traditional QFD. Specifically, the main contributions of this study to the literature are that the DHHLTSs are utilized to describe the hesitant relationship assessments between customer requirements and engineering characteristics provided by experts, and the ORESTE method is modified and used to determine the importance ranking orders of engineering characteristics. Finally, a case study and a comparison analysis are presented to illustrate the feasibility and practicability of the proposed QFD approach. The advantages of the new approach being proposed are higher flexibility in handling experts’ intricate and hesitant relationship evaluation information and effective in providing a reasonable prioritization of engineering characteristics in the practical QFD analysis.

1. Introduction

As an effective quality improvement technique, the quality function deployment (QFD) was proposed by Akao [1] in aiding the design and development of products or services for maximum customer satisfaction [2]. It can translate customer requirements (CRs), which are determined by a group of multidisciplinary experts into engineering characteristics (ECs) to facilitate cross-functional product development [3,4,5]. Through bridging the communication gap between customers and technicians, QFD can determine the most important ECs to ensure that the output meets the customer needs [6,7]. Thus, QFD plays an important role in reaching a final decision for enterprise planning new or improved products. Nowadays, QFD has been applied in many domains for the service quality improvement in radiology centers [8], the resilient strategy selection of healthcare systems, the implementation of Quality 4.0 [9], the implementation of Healthcare 4.0 [10], the formulation of sustainable supplier development programs [11], the indicator prioritization of sustainable supply chain [12,13], the mechanical structure design of subsea power devices [14], and so on.

Normally, a classical QFD model consists of four stages, i.e., scheme design, configuration of components, engineering and quality control, and manufacturing work order [15,16,17]. Especially, a relationship matrix called house of quality (HOQ) is constructed to describe the correlations between “WHATs” and “HOWs” for deriving the importance priorities of ECs [18,19]. Then, CRs are translated into ECs to narrow the differences between customers and product designers. Nevertheless, the traditional QFD method has been criticized to have many shortcomings in practical applications [20,21,22,23]. For example, crisp numbers are utilized by experts to assess the relationships between CRs and ECs. However, it is difficult for experts to express their opinions with crisp numbers because of the inherent vagueness of human cognition and limited experience in real situations. Besides, the weighted average method, a compensatory method, is applied to obtain the importance rankings of ECs in the traditional QFD, which may result in a biased ranking result of ECs.

In the QFD analysis, it is difficult for experts to assess the relationships between CRs and ECs using crisp values, because of the vagueness of human thinking and experience restriction. Instead, they are inclined to adopt linguistic terms to express their judgements [24,25,26]. Moreover, they may hesitate about their assessments because of the time pressure and the lack of data [23,27,28]. As a generalization of double hierarchy linguistic term sets, the concept of double hierarchy hesitant linguistic term sets (DHHLTSs) was introduced by Gou et al. [29] to handle the fuzzy and uncertain information in group decision-making problems. The DHHLTSs are characterized by two hierarchies of linguistic term sets. The first hierarchy expresses the basic properties of objectives or alternatives, and the second hierarchy is a detailed description or linguistic feature of the first hierarchy [30,31]. Compared with other linguistic representing methods, the benefits of DHHLTSs are that [32,33]: (1) It can provide a flexible environment for qualitative evaluation and (2) it can describe the evaluation information more intuitive and accurate. Considering these advantages, the DHHLTSs have been broadly applied to solve various decision-making problems [34,35,36]. Therefore, it is promising to adopt the DHHLTSs to handle the vague and uncertain experts’ relationship evaluation information in QFD.

On the other hand, determining the importance ranking of ECs in QFD is often considered as a multi-criteria decision making (MCDM) problem. In the literature, lots of MCDM methods have been utilized to enhance the performance of QFD [26,28,37,38]. The ORESTE (organísation, rangement et Synthèse de données relarionnelles, in French) is a general outranking MCDM method presented by Roubens [39], which can acquire high-quality results of a decision-making problem. It does not require the crisp weights of criteria in which the thresholds are calculated objectively with less subjective factors [40,41]. Meanwhile, it can show detailed distinctions between alternatives in terms of the preference relation, indifference relation, and incomparability relation [42,43]. Due to its merits, the ORESTE method and its extensions have been successfully applied to address many decision-making problems, which include rockburst risk evaluation [44], energy investment assessment [45], sustainable battery supplier selection [46], automotive development management [47], and failure mode risk analysis [43]. Hence, it is of significance to employ the ORESTE method to derive the importance priority of the identified ECs in QFD.

In this paper, we propose a new QFD approach by combining the DHHLTSs with the ORESTE method for improving the performance of traditional QFD. In this model, the DHHLTSs are adopted to assess the interrelationships between CRs and ECs for handling the ambiguous and uncertain judgments of experts. Besides, an extended ORESTE method is employed to determine the importance prioritizations of ECs. Finally, an empirical case and a comparison analysis are provided to demonstrate the feasibility and practicability of the proposed QFD model.

The rest of this paper is organized as follows: Section 2 presents a literature review of current studies related to QFD. Section 3 recalls the preliminaries of DHHLTSs and classical ORESTE method. In Section 4, a new QFD approach based on the DHHLTSs and an extended ORESTE method is proposed. Subsequently, a case study of mobile phone selection is conducted to verify the proposed QFD in Section 5. Finally, Section 6 makes a conclusion of this study and discusses potential future research directions.

2. Literature Review

In the past decades, a variety of improved QFD models have been developed to eliminate the deficiencies associated with the traditional QFD. On the one hand, many uncertainty theories, such as the fuzzy sets [4], the picture fuzzy sets [48], the interval type-2 fuzzy sets (IT2FSs) [49], the hesitant fuzzy linguistic term sets (HFLTSs) [11], the q-rung orthopair fuzzy numbers [50], the linguistic distribution assessments [51], the interval-valued intuitionistic fuzzy sets [52], and the interval-valued Pythagorean fuzzy sets [53] were employed to handle the vague relationship assessment information given by QFD team members. On the other hand, plenty of MCDM methods were adopted to facilitate obtaining the importance priorities of ECs in QFD. For example, Ping et al. [38] proposed an approach for QFD with an extended alternative queuing method (AQM) under linguistic Pythagorean fuzzy environment. Liu et al. [5] suggested a large group QFD approach based on an extended TODIM (Portuguese acronym for interactive multi-criteria decision making) method under the interval type-2 fuzzy context. Nie et al. [28] developed a QFD model using the TODIM method and linguistic distribution assessments for healthcare service quality enhancement. Wang et al. [27] put forward a method using the DHHLTSs and axiomatic design approach to enhance the performance of the classical QFD. In Mao et al. [24], a QFD approach integrating linguistic Z-numbers and the evaluation based on distance from average solution (EDAS) method was proposed to determine the prioritization of ECs. In Liu et al. [19], a QFD method integrating extended hesitant fuzzy linguistic term sets (EHFLTSs) and prospect theory was provided to overcome the limitations of the traditional QFD. In [23], a QFD approach using the proportional hesitant fuzzy linguistic term sets (PHFLTSs) and prospect theory was proposed to enhance the performance of the standard QFD.

A three-dimensional house of quality model-based interval-valued spherical fuzzy-ORESTE method was proposed by Jin et al. [54] to rank key quality characteristics. A modified multi-objective optimization by ratio analysis plus the full multiplicative form (MULTIMOORA) method based on cloud model theory was presented in [25] to improve QFD. A spherical fuzzy technique for order preference by similarity to the ideal solution (TOPSIS) method was reported in [26] for QFD. Mistarihi et al. [55] introduced fuzzy analytic network process (ANP) in QFD to determine the importance weights for engineering characteristics. Wang et al. [56] designed a hybrid methodology based on cloud model and grey relational analysis (GRA) for the technical attribute prioritization in QFD. In addition, the QFD was improved by integrating different MCDM methods in order to utilize their advantages. For instance, Chen et al. [57] proposed an integrated decision-making method for improving QFD, which integrates the HFLTSs, the decision-making trial and evaluation laboratory (DEMATEL), and the MULTIMOORA. Wu et al. [17] developed a QFD model for the product development of electric vehicles by utilizing the DEMATEL and the VIKOR (VIsekriterijumska optimizacija i KOm-promisno Resenje, in Serbian) methods. A summary of the listed QFD studies in the literature is displayed in

Table 1. Summary of QFD studies in the literature.

Reference	Methods	Advantages
Wang [4]	Fuzzy sets	Create more logical and reliable priority rankings for ECs by aggregating their technical importance ratings
Singh and Kumar [48]	Picture fuzzy sets	Measure the linguistic terms for qualitative analysis and characterize fuzziness and uncertainties more comprehensively
Efe et al. [49]	IT2FSs	Include expert opinions to describe the applications’ ambiguity more precisely
Finger and Lima-Junior [11]	HFLTSs	Support experts employ multiple linguistic terms or linguistic expressions
Efe and Efe [50]	Q-rung orthopair fuzzy numbers	Present more information about the correlations and relationships of CRs and ECs
Xiao et al. [51]	Linguistic distribution assessments	Model the vague assessments of the relationship between CRs and ECs; lessen QFD team members’ cognitive burden
Xie et al. [52]	Interval-valued intuitionistic fuzzy sets	Apply the cross-entropy to perform an objective analysis and prioritize the ECs
Haktanır and Kahraman [53]	Interval-valued Pythagorean fuzzy sets	Handle the fuzzy and vague relationship assessment information provided by QFD team members
Ping et al. [38]	Linguistic Pythagorean fuzzy; AQM	Express the assessments from experts on the relationships between CRs and ECs; determine the importance prioritization of ECs
Liu et al. [5]	Interval type-2 fuzzy sets; TODIM	Describe the imprecise and uncertain correlation assessments between CRs and ECs; derive the weights of CRs objectively
Nie et al. [28]	Linguistic distribution assessments; TODIM	Achieve a simple opinions presentation given by a large-scale of experts; reflect psychological behaviors and real desires of patients in EC prioritization
Wang et al. [27]	DHHLTSs; axiomatic design approach	Evaluate the relationships between CRs and ECs; derive the prioritization of ECs in QFD
Mao et al. [24]	Linguistic Z-numbers; EDAS	Deal with the vague evaluation information; estimate the final priority ratings of ECs
Liu et al. [19]	EHFLTSs; prospect theory	Obtain the hesitant linguistic assessment information of QFD team members; derive the ranking orders of ECs
Huang et al. [23]	PHFLTSs; prospect theory	Describe the relationships between CRs and ECs; prioritise ECs in the QFD analysis
Jin et al. [54]	Interval-valued spherical fuzzy sets; ORESTE	Improve the rationality of the ranking method; can deal with various uncertainty and hesitation
Wu et al. [25]	C-MULTIMOORA	Capture the uncertainty of experts’ assessment information and derive a credible ranking of ECs
Kutlu Gündoğdu and Kahraman [26]	TOPSIS	Allocate independently and have a greater scope for determining membership, non-membership, and hesitancy degrees
Mistarihi et al. [55]	Fuzzy ANP	Deal with vague human judgments of the intensity of preference and consider the relations among ECs
Wang et al. [56]	Cloud model; GRA	Handle uncertain information in the QFD process to produce more accurate analysis results
Chen et al. [57]	HFLTS; DEMATEL; MULTIMOORA	Analyze the causal relationships among ECs and capture their influence weights; analyze the correlations between ECs and CRs
Wu et al. [17]	DEMATEL; VIKOR	Analyze the cause-and-effect relationships among CRs and determine their influential weights; merge the dependences between CRs and ECs and derive the priorities of ECs

.

The above literature review shows that various generalized fuzzy methods have been applied in QFD to deal with imprecise and vague relationship assessments between CRs and ECs. Nevertheless, due to the increase of complexities of practical QFD problems, experts’ relationship assessment information is often qualitative description and the current fuzzy methods are not effective to describe complex linguistic expressions accurately and flexibly. Besides, many MCDM methods have been adopted to determine the importance priorities of ECs in QFD. However, few studies have been conducted to improve the QFD based on the ORESTE method. To fill these gaps, the aim of this paper is to develop a novel approach by integrating the DHHLFSs and an extended ORESTE method for deriving the importance ranking orders of ECs in QFD. The proposed QFD model is effective to describe the subjective evaluation information of experts more precisely and can offer more practical and reliable solutions to identify critical ECs for improving products or services.

3. Preliminaries

In this section, the basic concepts of the DHHLTSs and the ORESTE method that will be used in the proposed QFD approach are introduced.

3.1. The DHHLTSs

The DHHLTSs were proposed by Gou et al. [29] to handle complicated linguistic information in decision-making problems.

Definition 1

[29]. Let $\dot{S}=\left\{{\dot{s}}_t|t=-\tau ,\dots ,-1,0,1,\dots ,\tau \right\}$ and $\ddot{S}=\left\{{\ddot{s}}_a|a=-\varsigma ,\dots ,-1,0,1,\dots ,\varsigma \right\}$ be the first hierarchy and the second hierarchy linguistic term sets, respectively. Then, a DHLTS ${\dot{S}}_{\ddot{S}}$ is defined as:

$${\dot{S}}_{\ddot{S}}=\left\{{\dot{s}}_{t\langle {\ddot{s}}_a\rangle }|t=-\tau ,\dots ,-1,0,1,\dots ,\tau ;a=-\varsigma ,\dots ,-1,0,1,\dots ,\varsigma \right\}$$

(1)

where ${\dot{s}}_{t\langle {\ddot{s}}_a\rangle }$ is called the double hierarchy linguistic term (DHLT), ${\dot{s}}_t$ and ${\ddot{s}}_a$ represent the first and the second hierarchy linguistic term, respectively.

Definition 2

[29]. Let X be a fixed set, and ${\dot{S}}_{\ddot{S}}=\left\{{\dot{s}}_{t\langle {\ddot{s}}_a\rangle }|t=-\tau ,\dots ,-1,0,1,\dots ,\tau ;a=-\varsigma ,\dots ,-1,0,1,\dots ,\varsigma \right\}$ be a DHLTS. A DHHLTS on X, $H_{{\dot{S}}_{\ddot{S}}}$, is a mathematical form of

$$H_{{\dot{S}}_{\ddot{S}}}=\left\{\langle x_i,h_{{\dot{S}}_{\ddot{S}}}\left(x_i\right)\rangle |x_i\in X\right\},$$

(2)

where $h_{{\dot{S}}_{\ddot{S}}}\left(x_i\right)$ is a set of some values in ${\dot{S}}_{\ddot{S}}$ and called double hierarchy hesitant linguistic element (DHHLE).

The DHHLE can be denoted as

$$h_{{\dot{S}}_{\ddot{S}}}=\left\{{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }|{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }\in {\dot{S}}_{\ddot{S}};l=1,2,\dots ,L\right\},$$

(3)

where L is the number of linguistic terms in $h_{{\dot{S}}_{\ddot{S}}}$ and ${\dot{S}}_{{\varphi }_l}$ is the consecutive terms in ${\dot{S}}_{\ddot{S}}$.

Definition 3

[29]. Let ${\dot{S}}_{\ddot{S}}$ be a DHLTS, $h_{{\dot{S}}_{\ddot{S}}}$ be a DHHLE, and $h_{\gamma }=\left\{{\gamma }_l|{\gamma }_l\in \left[0,1\right];l=1,\dots ,L\right\}$ be a hesitant fuzzy element (HFE). The subscript ${\varphi }_l\langle {\phi }_l\rangle$ of the DHLT can express the equivalent information to the membership degree ${\gamma }_l$. They can be transformed into each other by the functions f and f⁻¹.

$$\begin{array}{l} f:\left[-\tau ,\tau \right]\times \left[-\varsigma ,\varsigma \right]\to \left[0,1\right]\\ f\left({\varphi }_l,{\phi }_l\right)=\{\begin{array}{ll}\frac{{\phi }_l+\left(\tau +{\varphi }_l\right)\varsigma }{2\varsigma \tau }={\gamma }_l,\hfill & \mathrm{if} -\tau +1\le {\varphi }_l\le \tau -1\hfill \\ \frac{{\phi }_l+\left(\tau +{\varphi }_l\right)\varsigma }{2\varsigma \tau }={\gamma }_l,\hfill & \mathrm{if} {\varphi }_l=\tau \hfill \\ \frac{{\phi }_l}{2\varsigma \tau }={\gamma }_l,\hfill & \mathrm{if} {\varphi }_l=-\tau \hfill \end{array}\end{array}$$

(4)

$$\begin{array}{l} f^{-1}:\left[0,1\right]\to \left[-\tau ,\tau \right]\times \left[-\varsigma ,\varsigma \right]\\ f^{-1}\left({\gamma }_l\right)=\{\begin{array}{ll}\tau \langle {\ddot{s}}_0\rangle ,\hfill & \mathrm{if} {\gamma }_l=1\hfill \\ \left[2\tau {\gamma }_l-\tau \right]\langle {\ddot{s}}_{\varsigma \left(2\tau {\gamma }_l-\tau -\left[2\tau {\gamma }_l-\tau \right]\right)}\rangle ,\hfill & \mathrm{if} 1<2\tau {\gamma }_l-\tau <\tau \hfill \\ 0\langle {\ddot{s}}_{\varsigma \left(2\tau {\gamma }_l-\tau \right)}\rangle ,\hfill & \mathrm{if} -1<2\tau {\gamma }_l-\tau \le 1\hfill \\ \left[2\tau {\gamma }_l-\tau \right]+1\langle {\ddot{s}}_{\varsigma \left(2\tau {\gamma }_l-\tau -\left[2\tau {\gamma }_l-\tau \right]-1\right)}\rangle ,\hfill & \mathrm{if} -\tau <2\tau {\gamma }_l-\tau \le -1\hfill \\ -\tau \langle {\ddot{s}}_0\rangle ,\hfill & \mathrm{if} {\gamma }_l=0\hfill \end{array}\end{array}$$

(5)

Furthermore, the transformation functions between the DHHLE $h_{{\dot{S}}_{\ddot{S}}}$ and the HFE $h_{\gamma }$ can be expressed as:

$$F\left(h_{{\dot{S}}_S}\right)=F\left(\left\{{\dot{s}}_{{\varphi }_l\langle \ddot{S}{\phi }_l\rangle }|{\dot{s}}_{{\varphi }_l\langle \ddot{S}{\phi }_l\rangle }\in {\dot{S}}_{\ddot{S}}\right\}\right)=\left\{{\gamma }_l|{\gamma }_l=f\left({\varphi }_l,{\phi }_l\right)\right\}=h_{\gamma }$$

(6)

$$F^{-1}\left(h_{\gamma }\right)=F^{-1}\left(\left\{{\gamma }_l|{\gamma }_l\in \left[0,1\right]\right\}\right)=\left\{{\dot{s}}_{{\varphi }_l\langle \ddot{S}{\phi }_l\rangle }|{\varphi }_l\langle \ddot{S}{\phi }_l\rangle =f^{-1}\left({\gamma }_l\right)\right\}=h_{{\dot{S}}_{\ddot{S}}}$$

(7)

Definition 4

[29]. Let ${\dot{S}}_{\ddot{S}}=\left\{{\dot{s}}_{t\langle {\ddot{s}}_a\rangle }|t=-\tau ,\dots ,-1,0,1,\dots ,\tau ;a=-\varsigma ,\dots ,-1,0,1,\dots ,\varsigma \right\}$ be a DHLTS, $h_{{\dot{S}}_{\ddot{S}}}^1$ and $h_{{\dot{S}}_{\ddot{S}}}^2$ be two DHHLEs, and λ is a real number. Then the operational laws of the DHHLEs are given as follows:

(1) 

$h_{{\dot{S}}_{\ddot{S}}}^1\oplus h_{{\dot{S}}_{\ddot{S}}}^2=F^{-1}\left(\underset{{\eta }_1\in F\left(h_{{\dot{S}}_{\ddot{S}}}^1\right),{\eta }_2\in F\left(h_{{\dot{S}}_{\ddot{S}}}^2\right)}{{\displaystyle \cup }}\left\{{\eta }_1+{\eta }_2-{\eta }_1{\eta }_2\right\}\right);$

(2) 

$h_{{\dot{S}}_{\ddot{S}}}^1\otimes h_{{\dot{S}}_{\ddot{S}}}^2=F^{-1}\left(\underset{{\eta }_1\in F\left(h_{{\dot{S}}_{\ddot{S}}}^1\right),{\eta }_2\in F\left(h_{{\dot{S}}_{\ddot{S}}}^2\right)}{{\displaystyle \cup }}\left\{{\eta }_1{\eta }_2\right\}\right);$

(3) 

$\lambda h_{{\dot{S}}_{\ddot{S}}}^1=F^{-1}\left(\underset{{\eta }_1\in F\left(h_{{\dot{S}}_{\ddot{S}}}^1\right)}{{\displaystyle \cup }}\left\{1-{\left(1-{\eta }_1\right)}^{\lambda }\right\}\right);$

(4) 

${\left(h_{{\dot{S}}_{\ddot{S}}}^1\right)}^{\lambda }=F^{-1}\left(\underset{{\eta }_1\in F\left(h_{{\dot{S}}_{\ddot{S}}}^1\right)}{{\displaystyle \cup }}\left\{{\eta }_1^{\lambda }\right\}\right).$

Definition 5

[29]. Let ${\dot{S}}_{\ddot{S}}=\left\{{\dot{s}}_{t\langle {\ddot{s}}_a\rangle }|t=-\tau ,\dots ,-1,0,1,\dots ,\tau ;a=-\varsigma ,\dots ,-1,0,1,\dots ,\varsigma \right\}$ be a DHLTS, $h_{{\dot{S}}_{\ddot{S}}}=\left\{{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }|{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }\in {\dot{S}}_{\ddot{S}};l=1,2,\dots ,L;{\varphi }_l=\left[-\tau ,\tau \right];{\phi }_l=\left[-\varsigma ,\varsigma \right]\right\}$ be a DHHLE. Then the expected value of $h_{{\dot{S}}_{\ddot{S}}}$ is defined as

$$E\left(h_{{\dot{S}}_{\ddot{S}}}\right)=\frac{1}{L}{\displaystyle \sum _{l=1}^{L}F\left({\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }\right)}$$

(8)

Besides, the variance of $h_{{\dot{S}}_{\ddot{S}}}$ is defined as:

$$v\left(h_{{\dot{S}}_{\ddot{S}}}\right)=\sqrt{\frac{1}{L}{{\displaystyle \sum _{l=1}^L\left(F\left({\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }\right)-E\left(h_{{\dot{S}}_{\ddot{S}}}\right)\right)}}^2}$$

(9)

Definition 6

[29]. Let $h_{{\dot{S}}_{\ddot{S}}}^1$ and $h_{{\dot{S}}_{\ddot{S}}}^2$ be two DHHLEs, then

(1) 

If $E\left(h_{{\dot{S}}_{\ddot{S}}}^1\right)>E\left(h_{{\dot{S}}_{\ddot{S}}}^2\right)$, then $h_{{\dot{S}}_{\ddot{S}}}^1$ is bigger than $h_{{\dot{S}}_{\ddot{S}}}^2$, denoted by $h_{{\dot{S}}_{\ddot{S}}}^1>h_{{\dot{S}}_{\ddot{S}}}^2$;

(2) 

If $E\left(h_{{\dot{S}}_{\ddot{S}}}^1\right)=E\left(h_{{\dot{S}}_{\ddot{S}}}^2\right)$, then

(a) 

if $v\left(h_{{\dot{S}}_{\ddot{S}}}^1\right)<v\left(h_{{\dot{S}}_{\ddot{S}}}^2\right)$, $h_{{\dot{S}}_{\ddot{S}}}^1$ is bigger than $h_{{\dot{S}}_{\ddot{S}}}^2$, denoted by $h_{{\dot{S}}_{\ddot{S}}}^1>h_{{\dot{S}}_{\ddot{S}}}^2$;

(b) 

if $v\left(h_{{\dot{S}}_{\ddot{S}}}^1\right)=v\left(h_{{\dot{S}}_{\ddot{S}}}^2\right)$, $h_{{\dot{S}}_{\ddot{S}}}^1$ is equivalent than $h_{{\dot{S}}_{\ddot{S}}}^2$, denoted by $h_{{\dot{S}}_{\ddot{S}}}^1=h_{{\dot{S}}_{\ddot{S}}}^2$.

Definition 7

[31]. Let ${\dot{S}}_{\ddot{S}}=\left\{{\dot{s}}_{t\langle {\ddot{s}}_a\rangle }|t=-\tau ,\dots ,-1,0,1,\dots ,\tau ;a=-\varsigma ,\dots ,-1,0,1,\dots ,\varsigma \right\}$ be a DHLTS, $H_{{\dot{S}}_{\ddot{S}}}^1=\left\{h_{{\dot{S}}_{\ddot{S}}}^{11},h_{{\dot{S}}_{\ddot{S}}}^{12},\dots ,h_{{\dot{S}}_{\ddot{S}}}^{1n}\right\}$ and $H_{{\dot{S}}_{\ddot{S}}}^2=\left\{h_{{\dot{S}}_{\ddot{S}}}^{21},h_{{\dot{S}}_{\ddot{S}}}^{22},\dots ,h_{{\dot{S}}_{\ddot{S}}}^{2n}\right\}$ be two DHHLEs. Meanwhile, $h_{{\dot{S}}_{\ddot{S}}}^{1j}=\left\{{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }^{1j}|{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }^{1j}\in {\dot{S}}_{\ddot{S}};l=1,2,\dots ,L\right\}$ and $h_{{\dot{S}}_{\ddot{S}}}^{2j}=\left\{{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }^{2j}|{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }^{2j}\in {\dot{S}}_{\ddot{S}};l=1,2,\dots ,L\right\}$. Then, the distance between $H_{{\dot{S}}_{\ddot{S}}}^1$ and $H_{{\dot{S}}_{\ddot{S}}}^2$ can be calculated by

$$d\left(H_{{\dot{S}}_{\ddot{S}}}^1,H_{{\dot{S}}_{\ddot{S}}}^2\right)={\displaystyle \sum _{j=1}^n{w}_{j}d\left(h_{{\dot{S}}_{\ddot{S}}}^{1j},h_{{\dot{S}}_{\ddot{S}}}^{2j}\right)},$$

(10)

$$d\left(h_{{\dot{S}}_{\ddot{S}}}^{1j},h_{{\dot{S}}_{\ddot{S}}}^{2j}\right)=\frac{1}{L}{\displaystyle \sum _{l=1}^L\left(\left|F\left({\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }^{1j}\right)-F\left({\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }^{2j}\right)\right|\right)},$$

(11)

where $w_i={\left(w_1,w_2,\dots ,w_n\right)}^T$ is an associated weight vector with $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^n{w}_i}=1$.

Definition 8

[29]. Let $H_{{\dot{S}}_{\ddot{S}}}=\left\{h_{{\dot{S}}_{\ddot{S}}}^1,h_{{\dot{S}}_{\ddot{S}}}^2,\dots ,h_{{\dot{S}}_{\ddot{S}}}^n\right\}$ be a DHHLE, and $w_i={\left(w_1,w_2,\dots ,w_n\right)}^T$ be their associated weights with $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _{i=1}^n{w}_i}=1$, where $h_{{\dot{S}}_{\ddot{S}}}^i=\left\{{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }^i|{\dot{s}}_{{\varphi }_l\langle {\ddot{s}}_{{\phi }_l}\rangle }^i\in {\dot{S}}_{\ddot{S}};l=1,2,\dots ,L\right\}$ and L indicates the number of DHLTS in $h_{{\dot{S}}_{\ddot{S}}}^i$. Then, the double-hierarchy hesitant linguistic weighted average (DHHLWA) operator is defined as

$$\mathrm{DHHLWA}\left(h_{{\dot{S}}_{\ddot{S}}}^1,h_{{\dot{S}}_{\ddot{S}}}^2,\dots ,h_{{\dot{S}}_{\ddot{S}}}^n\right)=\underset{i=1}{\overset{n}{\oplus }}\left(w_i{h}_{{\dot{S}}_{\ddot{S}}}^i\right)=F^{-1}\left\{\underset{{\eta }_p\in F\left(h_{{\dot{S}}_{\ddot{S}}}^i\right)}{\cup }\left[\begin{array}{l}\left(1-{\displaystyle \prod _{i=1}^n{\left(1-{\eta }_p^1\right)}^{w_i}}\right),\\ \left(1-{\displaystyle \prod _{i=1}^n{\left(1-{\eta }_p^2\right)}^{w_i}}\right),\\ \cdots ,\\ \left(1-{\displaystyle \prod _{i=1}^n{\left(1-{\eta }_p^L\right)}^{w_i}}\right)\end{array}\right]\right\}$$

(12)

3.2. The ORESTE Method

The ORESTE is an attractive method to handle the MCDM problems in which the crisp weights of criteria are difficult or unable to access. The method includes two main tasks: computing the weak ranking and building the preference/indifference/incomparability (PIR) structure. For a MCDM problem with the alternative set $A=\left\{A_1,A_2,\cdots ,A_m\right\}$ and the criteria set $C=\left\{C_1,C_2,\cdots ,C_n\right\}$, the classical ORESTE method involves the following steps:

Step 1: Calculate the global preference scores for alternatives.

The Besson’s mean rank r_j and r_j(A_i) represent a preference structure and a merit of A_i under C_j, respectively. The global preference scores D(a_ij) of the A_i regarding the C_j can be calculated by

$$D\left(a_{ij}\right)=\sqrt{\phi {\left(r_j\right)}^2+\left(1-\phi \right){\left(r_j\left(A_i\right)\right)}^2}.$$

(13)

where $\phi \left(0\le \phi \le 1\right)$ is the coefficient to obtain the relative importance between the ranking of alternatives and criteria.

Step 2: Identify the global weak rank r(a_ij).

If D(a_ij) > D(a_kv), then r(a_ij) > r(a_kv); If D(a_ij) = D(a_kv), then r(a_ij) = r(a_kv), where i, k = 1, 2, …, m and j, v = 1, 2, …, n.

Step 3: Obtain the global weak ranking.

The global weak ranking of each alternative is calculated by

$$R\left(A_i\right)={\displaystyle \sum _{j=1}^{n}r\left(a_{ij}\right)}.$$

(14)

Step 4: Compute the preference intensities.

In some cases, when two alternatives have the same weak rankings, they may have two different performances under different criteria. Therefore, their preference intensities should be computed to distinguish the incomparability relation or the indifference relation from two alternatives. The average preference intensity between A_i and A_k is defined as

$$T\left(A_i,A_k\right)=\frac{{\displaystyle \sum _{j=1}^n\max \left[r\left(a_{kj}\right)-r\left(a_{ij}\right),0\right]}}{\left(m-1\right)n^2}.$$

(15)

The net preference intensity between A_i and A_k is defined as:

$$\Delta T\left(A_i,A_k\right)=T\left(A_i,A_k\right)-T\left(A_k,A_i\right).$$

(16)

Step 5: Establish the PIR relation.

This step can subdivide the alternatives into three relationships: the preference (P), the indifference (I), and the incomparability (R) relations. The relation can be analyzed by the following rules:

(1)

If $\left|\Delta T\left(A_i,A_k\right)\right|\le \sigma$, then

(a)

if $\left|T\left(A_i,A_k\right)\right|\le \eta$ and $\left|T\left(A_k,A_i\right)\right|\le \eta$, then A_iIA_k,

(b)

if $\left|T\left(A_i,A_k\right)\right|>\eta$ or $\left|T\left(A_k,A_i\right)\right|>\eta$, then A_iRA_k.

(2)

If $\left|\Delta T\left(A_i,A_k\right)\right|>\sigma$, then

(a)

if $\frac{\min \left(T\left(A_i,A_k\right),T\left(A_k,A_i\right)\right)}{\left|\Delta T\left(A_i,A_k\right)\right|}\ge \rho$, then A_iRA_k;

(b)

if $\frac{\min \left(T\left(A_i,A_k\right),T\left(A_k,A_i\right)\right)}{\left|\Delta T\left(A_i,A_k\right)\right|}<\rho$ and $\Delta T\left(A_i,A_k\right)>0$, then A_iPA_k;

(c)

if $\frac{\min \left(T\left(A_i,A_k\right),T\left(A_k,A_i\right)\right)}{\left|\Delta T\left(A_i,A_k\right)\right|}<\rho$ and $\Delta T\left(A_k,A_i\right)>0$, then A_iPA_k.

The values of the thresholds σ, ρ, and η are determined by

$$\sigma <1/\left(m-1\right)n,\rho >\left(n-2\right)/4,\eta <\lambda /2\left(m-1\right).$$

(17)

where λ is used to show the maximal ranking difference between two indifferent alternatives.

4. The Proposed QFD Approach

In this section, we develop a new systematic QFD approach based on an extended ORESTE method with the DHHLTSs to determine the importance prioritization of ECs. Specifically, the DHHLTSs are utilized to evaluate the relationships between CRs and ECs. A modified ORESTE method is adopted to determine the importance ranking of ECs. The detailed procedure of the proposed QFD approach is depicted in Figure 1.

Given a QFD analysis problem, we suppose that there are m ECs denoted as $E{C}_i\left(i=1,2,\dots ,m\right)$ and n CRs denoted as $C{R}_j\left(j=1,2,\dots ,n\right)$. Let $w={\left(w_1,w_2,\cdots ,w_n\right)}^T$ be the weight vector of CRs, satisfying $w_j\ge 0$ and ${\displaystyle \sum _{j=1}^n{w}_j}=1$. Meanwhile, L experts $E_k\left(k=1,2,\dots ,L\right)$ are involved in evaluating the correlations between CRs and ECs. The weight of each expert is assumed to be the same. The relationship assessment matrix of the kth expert can be denoted as $A^k={\left(a_{{\dot{S}}_{{\ddot{S}}_{ij}}}^k\right)}_{m\times n}$, in which $a_{{\dot{S}}_{{\ddot{S}}_{ij}}}^k$ is a double-hierarchy hesitant linguistic (DHHL) rating of CR_j on EC_i based on the linguistic term sets $\dot{S}=\left\{{\dot{s}}_t|t=-\tau ,\dots ,-1,0,1,\dots ,\tau \right\}$ and $\ddot{S}=\left\{{\ddot{s}}_a|a=-\varsigma ,\dots ,-1,0,1,\dots ,\varsigma \right\}$.

Based on the notations and assumptions above, the proposed QFD approach for determining the prioritization of ECs is explained as follows:

Stage 1: Assess the correlations between CRs and ECs based on DHHLTSs.

Step 1: Compute the collective relationship assessment matrix.

Based on the DHHLWA operator, the collective relationship assessment matrix $A={\left(a_{ij}\right)}_{m\times n}$ can be obtained by the following equation:

$$a_{ij}=\mathrm{DHHLWA}\left(a_{{\dot{S}}_{{\ddot{S}}_{ij}}}^1,a_{{\dot{S}}_{{\ddot{S}}_{ij}}}^2,\dots ,a_{{\dot{S}}_{{\ddot{S}}_{ij}}}^L\right)=\underset{k=1}{\overset{L}{\oplus }}\left(\frac{1}{k}{a}_{{\dot{S}}_{{\ddot{S}}_{ij}}}^k\right)$$

(18)

Stage 2: Determine the importance prioritization of ECs with the ORESTE method.

The ORESTE [39] is an applicative decision-making method to select a reliable alternative. However, the traditional ORESTE may lose information because of the Besson’s mean ranking and many numerical computations. Therefore, in this study, we improve the ORESTE method and extend it to the DHHL context to determine the importance prioritization of ECs in QFD.

Step 2: Calculate the global preference scores for ECs.

The Besson’s mean rank r_j and r_j (EC_i) represent a preference structure and a merit of the EC_i under the CR_j, respectively. r_j can be determined by the weights of CRs, and r_j(EC_i) can be realized by the pairwise comparisons between DHHLEs in the matrix A. Then the global preference score D(a_ij) of the EC_i regarding the CR_j can be calculated by

$$D\left(a_{ij}\right)=\sqrt{\phi {\left(r_j\right)}^2+\left(1-\phi \right){\left(r_j\left(E{C}_i\right)\right)}^2},$$

(19)

where $\phi \left(0\le \phi \le 1\right)$ is the coefficient given by experts to measure the relative importance between the rankings of CRs and ECs. Generally, its default value is 0.5.

Step 3: Obtain the global weak ranking of ECs.

If $D\left(a_{ij}\right)$ > $D\left(a_{pq}\right)$, then $r\left(a_{ij}\right)>r\left(a_{pq}\right)$; if $D\left(a_{ij}\right)$ = $D\left(a_{pq}\right)$, then $r\left(a_{ij}\right)=r\left(a_{pq}\right)$, where i, p = 1, 2,…, m and k, q = 1, 2,…, n. Then, the global weak ranking of each EC is calculated by

$$R\left(E{C}_i\right)=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}D\left(a_{ij}\right)}.$$

(20)

Step 4: Compute the preference intensities among ECs.

If two ECs have the same global weak rankings, they may have two different performances under different CRs. Therefore, we need to compute their preference intensities, which can distinguish the incomparability relation or the indifference relation from two ECs. For this purpose, the preference intensity can be calculated by the average preference intensity and the net preference intensity. The average preference intensity of EC_i over EC_p is defined as:

$$T\left(E{C}_i,E{C}_p\right)=\frac{{\displaystyle \sum _{j=1}^n\max \left[r\left(a_{pj}\right)-r\left(a_{ij}\right),0\right]}}{\left(m-1\right)n^2}.$$

(21)

The net preference intensity of EC_i over EC_p is defined as:

$$\Delta T\left(E{C}_i,E{C}_p\right)=T\left(E{C}_i,E{C}_p\right)-T\left(E{C}_p,E{C}_i\right).$$

(22)

Step 5: Establish the PIR structure of ECs.

The PIR structure describes the priorities of ECs when they have the same global weak ranking order. This step can subdivide the ECs into three relationships: the preference (P), the indifference (I), and the incomparability (R) relations. The PIR structure of ECs can be analyzed by the following rules:

(1)

If $\left|\Delta T\left(E{C}_i,E{C}_p\right)\right|\le \sigma$, then

(a)

EC_i I EC_p, if $\left|T\left(E{C}_i,E{C}_p\right)\right|\le \eta$ and $\left|T\left(E{C}_p,E{C}_i\right)\right|\le \eta$;

(b)

EC_i R EC_p, if $\left|T\left(E{C}_i,E{C}_p\right)\right|>\eta$ or $\left|T\left(E{C}_p,E{C}_i\right)\right|>\eta$.

(2)

If $\left|\Delta T\left(E{C}_i,E{C}_k\right)\right|>\sigma$, then

(a)

EC_i R EC_p, if $\frac{\min \left(T\left(E{C}_i,E{C}_p\right),T\left(E{C}_p,E{C}_i\right)\right)}{\left|\Delta T\left(E{C}_i,E{C}_p\right)\right|}\ge \rho$;

(b)

EC_i P EC_p, if $\frac{\min \left(T\left(E{C}_i,E{C}_p\right),T\left(E{C}_p,E{C}_i\right)\right)}{\left|\Delta T\left(E{C}_i,E{C}_p\right)\right|}<\rho$ and $\Delta T\left(E{C}_i,E{C}_p\right)>0$;

(c)

EC_p P EC_i, if $\frac{\min \left(T\left(E{C}_i,E{C}_p\right),T\left(E{C}_p,E{C}_i\right)\right)}{\left|\Delta T\left(E{C}_i,E{C}_p\right)\right|}<\rho$ and $\Delta T\left(E{C}_p,E{C}_i\right)>0$.

Here, σ, ρ, and η are three different thresholds to distinguish the PIR relations. Their values are determined by the following principles [41]:

$$\sigma <1/\left(m-1\right)n,\rho >\left(n-2\right)/4,\eta <\lambda /2\left(m-1\right),$$

(23)

where λ is given by experts based on their experience to show the maximal ranking difference between two indifferent ECs.

Step 6: Determine the strong ranking of ECs.

The strong ranking of ECs is obtained according to the global weak ranking and the PIR structure. Specifically, based on the P and I relations in the PIR structure, the rank of some ECs is firstly determined, and then the full rank can be derived by combing the weak rank when the R relations exist among other ECs.

The strong ranking r(EC_i) can be determined by the sorted R(EC_i) and its PIR relation, i.e., if global weak ranking orders of two ECs R(EC_i) > R(EC_k), then r(EC_i) > r(EC_k), denoted as a natural number from 1, 2, … and the like; otherwise, R(EC_i) = R(EC_k), then r(EC_i) = r(EC_k) wherein the relationships between r(EC_i) and r(EC_k) can be determined by the PIR rules, for i, k = 1, 2, …, m.

5. Case Study

In this section, an empirical case of mobile phone selection [49] is provided to demonstrate the feasibility and effectiveness of our proposed QFD approach.

5.1. Implementation and Results

Nowadays, the mobile phone is an ordinary and indispensable wireless multipurpose tool in people’s daily life. It is not only used as a verbal communication tool but also has become multifunctional with messaging, reading books and newspapers, and sharing social media. Due to the rapid development of mobile communications technologies, mobile phone manufacturers must keep up with this development to meet customer needs. Nevertheless, compared with the requirements for actual product development, acquiring the requirements of customers for an ordinary commodity is a big challenge, because customers’ needs are changeable owing to diversity situations. They must satisfy customer needs in developing the product features. In this case, the proposed QFD approach is used to examines the CRs and DRs of the mobile phone to select the most appropriate mobile phone according to the levels of customer satisfaction.

Through the market survey and expert interview, 7 CRs $\left(C{R}_j,j=1,2,\cdots ,7\right)$ and 17 ECs $\left(E{C}_i,i=1,2,\cdots ,17\right)$ were determined for selecting a mobile phone, which are presented in

Table 2. CRs and ECs considered in the case example.

CRs	Customer Requirements	ECs	Engineering Characteristics	ECs	Engineering Characteristics
CR1	System performance	EC1	Weight	EC10	Optical zoom
CR2	Visual appeal	EC2	Thickness	EC11	Image sensor
CR3	Photograph capability	EC3	RAM capacity	EC12	Video chatting
CR4	Solidity	EC4	Internal memory	EC13	Dimension
CR5	Price	EC5	Water resistance	EC14	Screen dimension
CR6	Portability	EC6	Camera resolution	EC15	Screen type
CR7	Ease of use	EC7	Screen resolution	EC16	Battery life
		EC8	CPU’s performance score	EC17	Selfie quality
		EC9	Touch sensitivity

. Five experts from different departments were invited to assess the correlations between CRs and ECs based on the first hierarchy linguistic term set $\dot{S}$ and the second hierarchy linguistic term $\ddot{S}$. The $\dot{S}$ is denoted as:

$$\dot{S}=\left\{{\dot{s}}_{-2}=very low,{\dot{s}}_{-1}=low,{\dot{s}}_0=medium,{\dot{s}}_1=high,{\dot{s}}_2=very high\right\}$$

And the $\ddot{S}$ is determined by two situations:

(1)

if ${\dot{s}}_t\ge {\dot{s}}_0$, then

$$\begin{array}{l}{\ddot{s}}_a=\{{\ddot{s}}_{-3}=\mathrm{far} \mathrm{from},{\ddot{s}}_{-2}=\mathrm{only} \mathrm{a} \mathrm{little},{\ddot{s}}_{-1}=\mathrm{a} \mathrm{little},{\ddot{s}}_0=\mathrm{just} \mathrm{right},{\ddot{s}}_1=\mathrm{much},\\ {\ddot{s}}_2=\mathrm{very} \mathrm{much},{\ddot{s}}_3=\mathrm{entirely}\}\end{array}$$

(2)

if ${\dot{s}}_t<{\dot{s}}_0$, then

$$\begin{array}{l}{\ddot{s}}_a=\{{\ddot{s}}_{-3}=\mathrm{entirely},{\ddot{s}}_{-2}=\mathrm{very} \mathrm{much},{\ddot{s}}_{-1}=\mathrm{much},{\ddot{s}}_0=\mathrm{just} \mathrm{right},{\ddot{s}}_1=\mathrm{a} \mathrm{little},\\ {\ddot{s}}_2=\mathrm{only} \mathrm{a} \mathrm{little},{\ddot{s}}_3=\mathrm{far} \mathrm{from}\}\end{array}$$

As a consequence, the linguistic correlation assessments gathered from the five experts can be described by the double-hierarchy hesitant linguistic information. For instance, the hesitant linguistic assessments given by the first expert are presented in

Table 3. The hesitant linguistic assessments given by the first expert.

ECS	CRS
	CR1	CR2	CR3	CR4	CR5	CR6	CR7
EC1						Much medium
EC2						Much high
EC3	Much high	Just right medium			Very much high
EC4	Very much medium			Very much medium	Just right high
EC5						Much low
EC6	Only a little medium				Just right medium		A little high
EC7	Just right very high	Entirely high			Entirely medium		Much high
EC8	Much high				Very much high
EC9			A little low				Just right very high
EC10			Just right very high
EC11			A little very high
EC12		A little medium
EC13		A little medium				Just right very high
EC14		Just right very high		Entirely high		Just right very high	Just right very high
EC15		Only a little low
EC16				Just right very high	A little medium
EC17			A little medium

.

The DHHLTS is a useful tool to deal with complex qualitative information of experts since it can represent the linguistic information that is much more in line with people’s cognitions and expressions. Generally, the DHLTS consists of two hierarchy linguistic term sets. The second hierarchy linguistic term set is a linguistic feature or detailed supplementary of each linguistic term included in the first hierarchy linguistic term set. For example, the linguistic expressions “Much high”, “A little high”, and “Very much high” in Table 2 can be expressed by the DHHLEs as $\left\{{\dot{s}}_{1\langle {\ddot{s}}_1\rangle }\right\}$, $\left\{{\dot{s}}_{1\langle {\ddot{s}}_{-1}\rangle }\right\}$, and $\left\{{\dot{s}}_{1\langle {\ddot{s}}_2\rangle }\right\}$, respectively, based on the two hierarchy linguistic term sets $\dot{S}$ and $\ddot{S}$. In this way, all original linguistic correlation assessment information given by the experts can be described and saved.

In what follows, the proposed QFD method is utilized to determine the importance ranking of the given ECs.

Step 1: The original linguistic assessments for the relationships between ECs and CRs given by five experts are translated into DHHLEs to obtain the DHHL matrices $A^k={\left(a_{{\dot{S}}_{{\ddot{S}}_{ij}}}^k\right)}_{17\times 7}\left(k=1,2,\dots ,5\right)$. Taking the first expert as an example, its DHHL assessment matrix $A^1$ is presented in

Table 4. The DHHL assessment matrix $A^1$.

ECs	CRs
	CR1	CR2	CR3	CR4	CR5	CR6	CR7
EC1						${\dot{s}}_{0\langle {\ddot{s}}_1\rangle }$
EC2						${\dot{s}}_{1\langle {\ddot{s}}_1\rangle }$
EC3	${\dot{s}}_{1\langle {\ddot{s}}_1\rangle }$	${\dot{s}}_{0\langle {\ddot{s}}_0\rangle }$			${\dot{s}}_{1\langle {\ddot{s}}_2\rangle }$
EC4	${\dot{s}}_{0\langle {\ddot{s}}_2\rangle }$			${\dot{s}}_{0\langle {\ddot{s}}_{-2}\rangle }$	${\dot{s}}_{1\langle {\ddot{s}}_0\rangle }$
EC5						${\dot{s}}_{-1\langle {\ddot{s}}_{-1}\rangle }$
EC6	${\dot{s}}_{0\langle {\ddot{s}}_{-2}\rangle }$				${\dot{s}}_{0\langle {\ddot{s}}_0\rangle }$		${\dot{s}}_{1\langle {\ddot{s}}_{-1}\rangle }$
EC7	${\dot{s}}_{2\langle {\ddot{s}}_0\rangle }$	${\dot{s}}_{1\langle {\ddot{s}}_3\rangle }$			${\dot{s}}_{0\langle {\ddot{s}}_3\rangle }$		${\dot{s}}_{1\langle {\ddot{s}}_1\rangle }$
EC8	${\dot{s}}_{1\langle {\ddot{s}}_1\rangle }$				${\dot{s}}_{1\langle {\ddot{s}}_2\rangle }$
EC9			${\dot{s}}_{-1\langle {\ddot{s}}_1\rangle }$				${\dot{s}}_{2\langle {\ddot{s}}_0\rangle }$
EC10			${\dot{s}}_{2\langle {\ddot{s}}_0\rangle }$
EC11			${\dot{s}}_{2\langle {\ddot{s}}_{-1}\rangle }$
EC12		${\dot{s}}_{0\langle {\ddot{s}}_{-1}\rangle }$
EC13		${\dot{s}}_{0\langle {\ddot{s}}_{-1}\rangle }$				${\dot{s}}_{2\langle {\ddot{s}}_0\rangle }$
EC14		${\dot{s}}_{2\langle {\ddot{s}}_0\rangle }$		${\dot{s}}_{1\langle {\ddot{s}}_3\rangle }$		${\dot{s}}_{2\langle {\ddot{s}}_0\rangle }$	${\dot{s}}_{2\langle {\ddot{s}}_0\rangle }$
EC15		${\dot{s}}_{-1\langle {\ddot{s}}_2\rangle }$
EC16				${\dot{s}}_{2\langle {\ddot{s}}_0\rangle }$	${\dot{s}}_{0\langle {\ddot{s}}_{-1}\rangle }$
EC17			${\dot{s}}_{0\langle {\ddot{s}}_{-1}\rangle }$

.

Then, by using Equation (18), the collective relationship assessment matrix $A={\left(a_{ij}\right)}_{17\times 7}$ is computed as:

$$A=\left(\begin{array}{lllllllllllllllll}0\hfill & 0\hfill & 0.82\hfill & 0.67\hfill & 0\hfill & 0.33\hfill & 0.98\hfill & 0.83\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0.55\hfill & 0\hfill & 0\hfill & 0\hfill & 0.98\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.42\hfill & 0.42\hfill & 1\hfill & 0.42\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.33\hfill & 0.97\hfill & 0.92\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.45\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0.33\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0.92\hfill & 0.75\hfill & 0\hfill & 0.52\hfill & 0.75\hfill & 0.92\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.43\hfill & 0\hfill \\ 0.57\hfill & 0.83\hfill & 0\hfill & 0\hfill & 0.18\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.98\hfill & 0.98\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.68\hfill & 0.83\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.98\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right)$$

Step 2: According to the correlation-based weights of CRs given in

Table 5. Correlation-based weights of CRs.

CR1	CR2	CR3	CR4	CR5	CR6	CR7
0.196	0.094	0.091	0.111	0.239	0.094	0.175

, the Besson’s mean ranks of CRs are obtained as: r₁ = 2, r₂ = 4.5, r₃ = 6, r₄ = 7, r₅ = 1, r₆ = 4.5, and r₇ = 3.

In the same way, the Besson’s mean rank r_j (EC_i) is obtained as:

$$r_j\left(E{C}_i\right)=\left(\begin{array}{lllllllllllllllll}4.5\hfill & 4.5\hfill & 2\hfill & 2\hfill & 4.5\hfill & 3\hfill & 1.5\hfill & 2\hfill & 5\hfill & 4.5\hfill & 4.5\hfill & 4.5\hfill & 5\hfill & 6\hfill & 4.5\hfill & 5\hfill & 4.5\hfill \\ 4.5\hfill & 4.5\hfill & 3\hfill & 5.5\hfill & 4.5\hfill & 5.5\hfill & 1.5\hfill & 5\hfill & 5\hfill & 4.5\hfill & 4.5\hfill & 1\hfill & 2\hfill & 1.5\hfill & 1\hfill & 5\hfill & 4.5\hfill \\ 4.5\hfill & 4.5\hfill & 5.5\hfill & 5.5\hfill & 4.5\hfill & 5.5\hfill & 6\hfill & 5\hfill & 2\hfill & 1\hfill & 1\hfill & 4.5\hfill & 5\hfill & 6\hfill & 4.5\hfill & 5\hfill & 1\hfill \\ 4.5\hfill & 4.5\hfill & 5.5\hfill & 3\hfill & 4.5\hfill & 5.5\hfill & 6\hfill & 5\hfill & 5\hfill & 4.5\hfill & 4.5\hfill & 4.5\hfill & 5\hfill & 1.5\hfill & 4.5\hfill & 1\hfill & 4.5\hfill \\ 4.5\hfill & 4.5\hfill & 1\hfill & 1\hfill & 4.5\hfill & 2\hfill & 4\hfill & 1\hfill & 5\hfill & 4.5\hfill & 4.5\hfill & 4.5\hfill & 5\hfill & 6\hfill & 4.5\hfill & 2\hfill & 4.5\hfill \\ 1\hfill & 1\hfill & 5.5\hfill & 5.5\hfill & 1\hfill & 5.5\hfill & 6\hfill & 5\hfill & 5\hfill & 4.5\hfill & 4.5\hfill & 4.5\hfill & 1\hfill & 3.5\hfill & 4.5\hfill & 5\hfill & 4.5\hfill \\ 4.5\hfill & 4.5\hfill & 5.5\hfill & 5.5\hfill & 4.5\hfill & 1\hfill & 3\hfill & 5\hfill & 1\hfill & 4.5\hfill & 4.5\hfill & 4.5\hfill & 5\hfill & 3.5\hfill & 4.5\hfill & 5\hfill & 4.5\hfill \end{array}\right)$$

By Equation (19), the global preference score scores for ECs regarding each CR are determined as follows:

$$D\left(a_{ij}\right)=\left(\begin{array}{lllllllllllllllll}3.48\hfill & 3.48\hfill & 2\hfill & 2\hfill & 3.48\hfill & 2.55\hfill & 1.77\hfill & 2\hfill & 3.81\hfill & 3.48\hfill & 3.48\hfill & 3.48\hfill & 3.81\hfill & 4.47\hfill & 3.48\hfill & 3.81\hfill & 3.48\hfill \\ 4.5\hfill & 4.5\hfill & 3.82\hfill & 5.02\hfill & 4.5\hfill & 5.02\hfill & 3.35\hfill & 4.76\hfill & 4.76\hfill & 4.5\hfill & 4.5\hfill & 3.26\hfill & 3.48\hfill & 3.35\hfill & 3.26\hfill & 4.76\hfill & 4.5\hfill \\ 5.3\hfill & 5.3\hfill & 5.76\hfill & 5.76\hfill & 5.3\hfill & 5.76\hfill & 6\hfill & 5.52\hfill & 4.47\hfill & 4.3\hfill & 4.3\hfill & 5.3\hfill & 5.52\hfill & 6\hfill & 5.3\hfill & 5.52\hfill & 4.3\hfill \\ 5.88\hfill & 5.88\hfill & 6.29\hfill & 5.39\hfill & 5.88\hfill & 6.29\hfill & 6.52\hfill & 6.08\hfill & 6.08\hfill & 5.88\hfill & 5.88\hfill & 5.88\hfill & 6.08\hfill & 5.06\hfill & 5.88\hfill & 5\hfill & 5.88\hfill \\ 3.26\hfill & 3.26\hfill & 1\hfill & 1\hfill & 3.26\hfill & 1.58\hfill & 2.29\hfill & 1\hfill & 3.61\hfill & 3.26\hfill & 3.26\hfill & 3.26\hfill & 3.61\hfill & 4.3\hfill & 3.26\hfill & 1.58\hfill & 3.26\hfill \\ 3.26\hfill & 3.26\hfill & 5.02\hfill & 5.02\hfill & 3.26\hfill & 5.02\hfill & 5.3\hfill & 4.76\hfill & 4.76\hfill & 4.5\hfill & 4.5\hfill & 4.5\hfill & 3.26\hfill & 4.03\hfill & 4.5\hfill & 4.76\hfill & 4.5\hfill \\ 3.82\hfill & 3.82\hfill & 4.43\hfill & 4.43\hfill & 3.82\hfill & 2.24\hfill & 3\hfill & 4.12\hfill & 2.24\hfill & 3.82\hfill & 3.82\hfill & 3.82\hfill & 4.12\hfill & 3.26\hfill & 3.82\hfill & 4.12\hfill & 3.82\hfill \end{array}\right)$$

Step 3: With Equation (20), the global weak ranking of each EC is calculated as:

$$R\left(E{C}_i\right)=\left(4.22,4.22,4.05,4.09,4.22,4.07,4.12,4.03,4.25,4.25,4.25,4.22,4.27,4.35,4.22,4.22,4.25\right)$$

Step 4: According to Equations (21) and (22), the average preference intensities between the ECs which have the same global weak rankings, as well as EC₉, EC₁₀, EC₁₁, and EC₁₇ are presented in

Table 6. The average preference intensities between EC₁, EC₂, EC₅, EC₁₂, EC₁₅, and EC₁₆.

	T(EC1, ECi)	T(EC2, ECi)	T(EC5, ECi)	T(EC12, ECi)	T(EC15, ECi)	T(EC16, ECi)
EC1	/	0	0	0.0016	0.0016	0.0016
EC2	0	/	0	0.0016	0.0016	0.0016
EC5	0	0	/	0.0016	0.0016	0.0016
EC12	0.0016	0.0016	0.0016	/	0.0016	0.0033
EC15	0.0016	0.0016	0.0016	0	/	0.0033
EC16	0.0033	0.0033	0.0033	0.0033	0.0033	/

and

Table 7. The average preference intensities between EC₉, EC₁₀, EC₁₁, and EC₁₇.

	T(EC9, ECi)	T(EC10, ECi)	T(EC11, ECi)	T(EC17, ECi)
EC9	/	0.0020	0.0020	0.0020
EC10	0.0020	/	0	0
EC11	0.0020	0	/	0
EC17	0.0020	0	0	/

.

Step 5: By Equation (23), the three different thresholds are calculated as: σ = 0.0089, ρ = 1.3, η = 0.003, and the parameter λ = 0.1. Thus, the relationships between ECs are derived by their global weak rankings and PIR rules, and the PIR relations are presented in

Table 8. The PIR relations between ECs.

	EC1	EC2	EC3	EC4	EC5	EC6	EC7	EC8	EC9	EC10	EC11	EC12	EC13	EC14	EC15	EC16	EC17
EC1	/	I	>	>	I	>	>	>	<	<	<	I	<	<	I	R	<
EC2	I	/	<	<	I	<	<	<	<	<	<	I	<	<	I	R	<
EC3	<	>	/	<	<	<	<	>	<	<	<	<	<	<	<	<	<
EC4	<	<	>	/	<	<	<	>	<	<	<	<	<	<	<	<	<
EC5	I	I	>	>	/	>	>	>	<	<	<	I	<	<	I	R	<
EC6	<	<	>	<	<	/	<	>	<	<	<	<	<	<	<	<	<
EC7	<	<	>	>	<	>	/	>	<	<	<	<	<	<	<	<	<
EC8	<	<	<	<	<	<	<	/	<	<	<	<	<	<	<	<	<
EC9	>	>	>	>	>	>	>	>	/	I	I	>	<	<	>	>	I
EC10	>	>	>	>	>	>	>	>	I	/	I	>	>	>	>	>	I
EC11	>	>	>	>	>	>	>	>	I	I	/	>	>	>	>	>	I
EC12	I	I	>	>	I	>	>	>	<	<	<	/	>	<	I	R	<
EC13	>	>	>	>	>	>	>	>	>	>	>	>	/	<	>	>	>
EC14	>	>	>	>	>	>	>	>	>	>	>	>	>	/	>	>	>
EC15	I	I	>	>	I	>	>	>	<	<	<	I	<	<	/	R	<
EC16	R	R	>	>	R	>	>	>	<	<	<	R	<	<	R	/	<
EC17	>	>	>	>	>	>	>	>	I	I	I	>	<	<	>	>	/

.

Step 6: The strong ranking r(ECi) (i = 1,2,…,17) of ECs can be determined by the sorted R(EC_i) and its PIR relation, which is denoted as: r(EC₁₄) > r(EC₁₃) > r(EC₉) = r(EC₁₀) = r(EC₁₁) = r(EC₁₇) > r(EC₁) = r(EC₂) = r(EC₅) = r(EC₁₂) = r(EC₁₅) = r(EC₁₆) > r(EC₇) > r(EC₄) > r(EC₆) > r(EC₃) > r(EC₈). Therefore, EC₈ is the most important EC, which should be paid more attention for selecting the most appropriate mobile phone.

5.2. Comparative Analysis

To further verify the effectiveness of our proposed QFD approach, we conduct a comparative study with some existing methods, including the interval type-2 fuzzy (IT2F) QFD method [54], the hesitant fuzzy QFD (HF-QFD) [55], and the classical ORESTE method. The importance ranking results of ECs by using the listed methods are presented in

Table 9. ECs’ ranking under different methods.

ECs	Proposed Method		IT2F QFD Method		HF-QFD Method		ORESTE Method
	R(ECi)	Ranking	ωj	Ranking	R(ECi)	Ranking	R(ECi)	Ranking
EC1	4.22	6	0.019	16	4.22	6	4.22	6
EC2	4.22	6	0.024	15	4.22	6	4.22	6
EC3	4.05	2	0.058	6	4.05	2	4.05	3
EC4	4.09	4	0.044	8	4.11	4	4.11	5
EC5	4.22	6	0.049	7	4.22	6	4.22	6
EC6	4.07	3	0.132	3	4.07	3	4.07	4
EC7	4.12	5	0.175	1	4.11	4	4.01	1
EC8	4.03	1	0.101	4	4.04	1	4.04	2
EC9	4.25	12	0.040	9	4.25	12	4.25	12
EC10	4.25	12	0.025	14	4.25	12	4.25	12
EC11	4.25	12	0.033	10	4.25	12	4.25	12
EC12	4.22	6	0.023	14	4.22	6	4.22	6
EC13	4.27	16	0.026	11	4.58	17	4.27	16
EC14	4.35	17	0.074	5	4.30	16	4.28	17
EC15	4.22	6	0.009	17	4.22	6	4.22	6
EC16	4.22	6	0.142	2	4.22	6	4.22	6
EC17	4.25	12	0.026	11	4.25	12	4.25	12

.

As can be seen from Table 9, except for EC₇, EC₁₃, and EC₁₄, the importance prioritization of ECs yielded by the HF-QFD method is in line with the result obtained by the proposed QFD approach; except for EC₃, EC₄, EC₆, EC₇, and EC₈, the importance ranking of ECs derived by the ORESTE method is identical with the result obtained by the proposed QFD approach. As there is a good similarity between the proposed approach and the other three methods, the effectiveness and feasibility of our proposed QFD approach are validated in practical applications.

However, there are still some differences in the importance ranking orders of ECs produced by the proposed QFD approach and the other three methods. First, there is much difference between the two sets of importance priority rankings produced by the proposed approach and the IT2F QFD. The explanations of the inconsistent ranking results are summarized as follows: (1) The correlations between CRs and ECs are defined by IT2FSs in the IT2F QFD method, while they are expressed by the DHHLTSs in our proposed approach; (2) the values of the ECs are computed by relationship between CRs and ECs in the IT2F QFD method, while they are sorted by the ORESTE method in our proposed approach.

Second, the importance priorities of three ECs (EC₇, EC₁₃, and EC₁₄) obtained by the HF-QFD method are different from those determined by the proposed approach. These discrepancies may be explained by the following reasons: The experts’ relationship evaluations of the proposed approach are based on the DHHLTSs. In contrast, the hesitant fuzzy sets are used by the HF-QFD method to deal with the relationship evaluations between CRs and ECs provided by experts.

Third, the importance rankings of EC₃, EC₄, EC₆, EC₇, and EC₈ determined by the proposed QFD and the classical ORESTE method are different. This could be attributed to the fact that different methods were adopted by the compared methods to represent the relationship evaluations between CRs and ECs. In other words, the proposed QFD approach adopted the DHHLTSs, and the classical ORESTE method applied crisp numbers to depict the qualitative evaluations on “WHATs-HOWs” given by experts.

Based on the above analyses, it can be concluded that a more reasonable and credible importance ranking result of ECs can be obtained by employing the proposed QFD framework. It provides a more reasonable solution for engineers. This work has a meaningful effect on the selection process of a mobile phone, aiming at translating customer requirements into practical tactics. The results of the example illustrate the process of how to select ECs for mobile phones, which imply that to meet customer demands, the managers and designers should focus on the critical vital ECs, such as CPU performance score, RAM capacity, and camera resolution. Compared with the listed methods, the proposed QFD approach has the following advantages:

(1)

By applying the DHHLTSs, the hesitant evaluation information from each expert can be expressed more accurately and comprehensively. Thus, the propped approach can reduce information distortion and improve the accuracy of evaluation in the QFD process.

(2)

An extended ORESTE method is employed for determining the importance priority of ECs in the proposed QFD. Through calculating information contents to distinguish ECs, the ranking result given by the propped approach is more reasonable and credible.

6. Conclusions

In this paper, we developed an improved QFD methodology by combing DHHLTSs with the ORESTE method to improve the performance of traditional QFD. First, the DHHLTSs were introduced to handle the complex relationship evaluations between CRs and ECs obtained from experts. Then, the normal ORESTE method is extended with DHHLTSs to determine the importance ranking orders of ECs. Finally, we performed a case study of mobile phone selection to illustrate the feasibility and practical advantages of the proposed QFD approach. The results show that the proposed QFD can not only deal with the ambiguous and uncertain assessment information of experts, but also achieve more accurate and reliable importance ranking orders of ECs for product quality improvement.

Despite its advantages, the proposed QFD approach has several limitations that can be addressed in future researches. First, the proposed method ignores the correlations among CRs, which may exist in some situations. Hence, one orientation for future research is to take the correlations within CRs into accounts in QFD. Second, although the proposed method is designed to consider multiple experts, only five experts are involved in the case study. In the future, a large group of experts with different backgrounds is suggested to be involved to improve the proposed method. In addition, due to the complex calculation process of the proposed approach, a web-based QFD system can be established to facilitate its application in the real-world.