## 3 Proposed QFD Approach
### Determine the Weights of CRs Based on HFLTS-BWM  
The Best-Worst Method (BWM), proposed by [PERSON] [29], takes into account pairwise comparison matrices. It determines the best and worst attributes through expert discussions and compares them with other attributes. The entire process only requires \(2n-3\) times, simplifying the evaluation process while reducing errors and yielding more reliable results. In this paper, hesitant fuzzy linguistic term sets are introduced in combination with traditional BWM, allowing experts to evaluate customer needs using multiple semantics simultaneously. The calculation of the weights of CRs in HFLTS-BWM differs from traditional BWM in that traditional BWM constructs comparison vectors using precise values and uses mathematical programming models to obtain optimal weights, whereas HFLTSs represent expert assessment data as interval numbers, necessitating an extension of the current foundation for building a novel model to address interval weight vectors. The detailed procedure unfolds as follows:  
Step 1. In the customer demand set \(\{C_{1},C_{2},C_{3},\cdots,C_{n}\}\), determine the most critical and least essential customer demands based on the opinions of both customers and experts, denoted as \(C_{B}\) and \(C_{W}\), respectively. Step 2. Obtain expert evaluations based on the hesitant fuzzy linguistic set to determine the preference level of \(C_{B}\) compared to other demands \(C_{j}\), and construct a comparative vector \(A_{B}=(a_{B1},a_{B2},\cdots,a_{Bn})\), where \(a_{Bj}(j=1,2,\cdots,n)\) represents the degree of preference of \(C_{B}\) compared to \(C_{j}\). Similarly, the preference level of \(C_{j}\) compared to \(C_{W}\) is determined for all other demands, and a comparative vector \(A_{W}=(a_{1W},a_{2W},\cdots,a_{nW})\) is constructed, where \(a_{jW}(j=1,2,\cdots,n)\) represents the degree of preference of \(C_{j}\) compared to \(C_{W}\). Step 3. Convert the constructed comparison vectors into corresponding 2-tuple using Formula (6), and then convert the 2-tuple information into corresponding numerical values according to Formulas (9) and (10), obtaining comparison vectors \(H(A_{B})\) and \(H(A_{W})\):  
\[H(A_{B})=\left(\left[a_{B1}^{L},a_{B1}^{U}\right],\left[a_{B2}^{L},a_{B2}^{U} \right],\cdots\left[a_{Bn}^{L},a_{Bn}^{U}\right]\right), \tag{11}\]  
\[H(A_{W})=\left(\left[a_{1W}^{L},a_{1W}^{U}\right],\left[a_{2W}^{L},a_{2W}^{U} \right],\cdots\left[a_{nW}^{L},a_{nW}^{U}\right]\right), \tag{12}\]  
where \(\left[a_{Bj}^{L},a_{Bj}^{U}\right]\) signifies the level of preference of \(C_{B}\) compared to other demands \(C_{j}\), and \(\left[a_{jW}^{L},a_{jW}^{U}\right]\) signifies the level of preference of other demands \(C_{j}\) compared to \(C_{W}\). Figure 1: Flowchart of the proposed QFD approach. Step 4. Calculate the weights of each CR. Based on the following model [35], apply Lingo to obtain the interval weight vector \(\overline{w}_{j}=\big{(}\big{[}\overline{w}_{1}^{L},\overline{w}_{1}^{L}\big{]}, \big{[}\overline{w}_{2}^{L},\overline{w}_{2}^{L}\big{]},\cdots,\big{[}\overline {w}_{n}^{L},\overline{w}_{n}^{L}\big{]}\big{)}\):  
\[s.t.\left\{ \begin{array}{l}\min_{\xi}^{\xi}\\ \left|\frac{w_{j}^{L}}{w_{j}^{0}}-a_{Bj}^{L}\right|\leq\xi,\ \left|\frac{w_{j}^{ \underline{w}}}{w_{j}^{L}}-a_{Bj}^{\underline{u}}\right|\leq\xi\\ \left|\frac{w_{j}^{L}}{w_{j}^{0}}-a_{jw}^{L}\right|\leq\xi,\ \left|\frac{w_{j}^{ \underline{u}}}{w_{j}^{0}}-a_{jW}^{\underline{u}}\right|\leq\xi\\ w_{k}^{L}+\sum\limits_{j=1,j\
eq k}^{n}w_{j}^{L}\geq 1\\ w_{k}^{L}+\sum\limits_{j=1,j\
eq k}^{n}w_{j}^{L}\leq 1\\ 0\leq w_{j}^{L}\leq w_{j}^{\underline{u}},j=1,2,\cdots,n\end{array} \right. \tag{13}\]  
The proposed approach uses the idea of combined weights to construct the following optimization model to obtain the best weight \(w_{j}^{*}\) that minimizes the deviation of the interval weight:  
\[