Engineering characteristics prioritization for quality function deployment by using CRITIC and cumulative prospect theory with extended hesitant fuzzy linguistic term sets

Abstract

Quality Function Deployment (QFD) is a powerful approach for improving product quality that can transform customer requirements (CRs) into engineering characteristics (ECs) during product manufacturing. The limitations of traditional QFD methods lead to imprecise quantification of CRs and difficulty in accurately mapping customer needs. To address these issues, this paper introduces an innovative QFD approach that integrates extended hesitant fuzzy linguistic term sets (EHFLTSs), CRITIC, and cumulative prospect theory. The method expresses the subjectivity and hesitancy of decision makers when evaluating the relationship between ECs and CRs using EHFLTSs, considering the conflicts among CRs. The CRITIC is used to comprehensively evaluate the comparison strength and conflict between indicators, and the cumulative prospect theory is utilized to derive the prioritization of ECs. A case study is presented to demonstrate the effectiveness of the proposed approach.

Keywords

Extended hesitant fuzzy linguistic term set cumulative prospect theory quality function deployment CRITIC

1 Introduction

Quality function development (QFD) is a methodology that transforms customer requirements (CRs) into quantifiable engineering characteristics (ECs) for product manufacturing by constructing a linguistic data system with the quality house, and analyzing the degree of correlation between the desired quality from CRs and ECs of the product [1]. Therefore, QFD has been adopted in various industries, including machinery [2], automobile [3], service industry [4], electronics [5] and other fields, with significant achievements made in many fields [6].

The quality house, a component of quality function development, facilitates the transformation of customer needs into ECs by employing a matrix framework to express their relative importance [7]. In practice, enterprises typically prioritize quality control efforts on characteristics with higher degrees of importance using the principle of critical minority and secondary majority. This approach ensures cost control while maximizing economic benefits and returns for the enterprise [8]. Exploring the construction of an appropriate quality house is an imperative inquiry. The quality house is built on customer evaluations and perspectives, which are transformed into key CRs. Consequently, constructing a quality house necessitates handling linguistic information, while acknowledging the inherent ambiguity and hesitancy in human language that inevitably leads to incomplete information acquisition [9]. Therefore, in order to maximize the objectivity of the evaluations obtained, multiple experts are usually chosen to conduct evaluations in groups.

However, in the evaluation of the relationship between CRs and product ECs, experts often employ linguistic terms to describe positive, negative, and strong relationships. As a result, product development has evolved into a group linguistic decision-making process. Given the diverse perspectives of experts, there arises an imperative need to consolidate evaluation information from multiple sources for comprehensive analysis. Currently, approaches have been proposed to address the multiple attribute group language decision-making problem in QFD. Akbas et al. [10] introduced a comprehensive framework for fuzzy quality function unfolding along with a similarity-based ideal solution selection method that enables obtaining an all-encompassing ranking of ECs. Yang et al. [11] developed a novel group decision-making method, combining proportional hesitant fuzzy linguistic term sets (HFLTS) to obtain the priority sort of ECs in QFD. Fang et al. [12] developed an integrated QFD theory utilizing rough set and cloud models in uncertain information. Wang et al. [13] proposed a novel approach that integrates cloud models with grey correlation analysis to address the multi-criteria decision-making problem in QFD. Yang et al. [14] introduced an evaluation model for probabilistic linguistic term sets using grey correlation analysis. Li et al. [15] developed an extended probabilistic linguistic approach relying on fuzzy QFD to sort the importance of ECs according to the CRs in open design. Li et al. [16] suggested employing a binary linguistic QFD model to manage inaccurate and fuzzy evaluation information. Wang et al. [17] utilized a binary transformation function approach to deal with ambiguities and multi-granularity in quality planning simultaneously. Rodrı'guez et al. [18] proposed a hesitant fuzzy linguistic operator, which deals with the situation of expert hesitation. Building upon the work of Rodrı'guez, Wei [19] further proposed an extended hesitation fuzzy linguistic operator that can construct the concatenation set of expert hesitation fuzzy evaluations, addressing evaluative information loss when handling the subjectivity and hesitation of experts in group linguistic decision-making. Chen [20] proposed a integrated multiple criteria decision-making QFD approach, integrating the hesitant fuzzy linguistic term set. He et al. [21] developed a comprehensive method to achieve effective design resilient solutions for SSCs while considering both CRs and risk factors. In addition, methods such as image fuzzy linguistic set and enriched evaluation preference ranking method have also contributed to the multi-attribute decision-making problem in QFD [22 –25]. When evaluating the relationship between ECs and CRs, decision makers may use complex language to express their opinions, and these expressions may be vague and uncertain. In order to clearly quantify this complex language expression, Wang [26] proposed the conception of extended hesitant fuzzy linguistic term sets (EHFLTSs), and EHFLTSs can be formed by the union of HFLTSs, which allows it to flexibly adapt to the group language decision model.

The group linguistic decision-making process can be regarded as a multi-attribute decision-making problem (MADM). In the MADM problem, it is very important to determine the attribute weights. When experts evaluate the relationship between CRs and ECs, it is challenging to accurately calculate the weights of CRs due to the different preferences of DMs. The CRITIC can determine the objective weight of the MADM problem. CRITIC compare the strength of the assessment indicators and the contradictions between the indicators, which can assess the objective weights of the indicators [27, 28]. In addition to the preferences of experts, the psychological factors of expert decision-making will also affect the final decision. Tversky [29] extended the prospect theory to the cumulative prospect theory, which can well capture the psychological behavior of decision makers in the assessment procedures. The combination of the cumulative prospect theory and MADM methods to solve uncertain problems has emerged as a prominent research focus within the decision-making domain.

In the current QFD research, two significant challenges are being faced. Firstly, accurately assessing the relationship between CRs and ECs is challenging [30, 31]. Secondly, QFD-based product development is a process that involves group members with varying levels of heterogeneity. To address mentioned issues, this paper proposes a novel QFD method that leverages the advantages of EHFLTSs to express DM’s subjectivity and hesitancy, CRITIC can comprehensively measure the contrast intensity between indicators and express the conflict between indicators. Additionally, it incorporates cumulative prospect theory to consider psychological factors in decision making. Table 1 presents a list of studies on the QFD integrated with different decision-making methods.

Table 1
Studies on the QFD integrated with different decision-making methods

Reference Information representation Weighted CRs Fuzziness of indicators Hesitancy of decision-makers

ours EHFLTS √ √ √

Hong, Z. X., et al. [2] IRN √ √ ×

Yang, Y. X., et al. [14] PLTSs √ × ×

de Lima, B. P., et al. [25] HFLTS × √ √

Liu, H. C., et al. [29] IT2FSs √ √ ×

Shi, H., et al. [30] DHHLTSs × × √

Wu, T., et al. [4] IT2FL √ × √

Jin, C. X., et al.[8] IVSF-ORESTE √ √ ×

He, L. N., et al. [21] Kano model √ × ×

Avikal, S., et al. [3] Fuzzy Kano model × √ ×

Wang, X., et al. [13] GRA-cloud model √ √ ×

Ping, Y. J., et al. [24] PFLS × √ ×

Li, S. P., et al. [15] Probabilistic language modeling √ × ×

Akbas, H. and B. Bilgen [10] FANP × √ ×

Li, X. B. and Z. He [16] B2LTS √ × ×

Wang, Z. Q., et al. [17] multi-granularity linguistic terms √ √ ×

Reference	Information representation	Weighted CRs	Fuzziness of indicators	Hesitancy of decision-makers
ours	EHFLTS	√	√	√
Hong, Z. X., et al. [2]	IRN	√	√	×
Yang, Y. X., et al. [14]	PLTSs	√	×	×
de Lima, B. P., et al. [25]	HFLTS	×	√	√
Liu, H. C., et al. [29]	IT2FSs	√	√	×
Shi, H., et al. [30]	DHHLTSs	×	×	√
Wu, T., et al. [4]	IT2FL	√	×	√
Jin, C. X., et al.[8]	IVSF-ORESTE	√	√	×
He, L. N., et al. [21]	Kano model	√	×	×
Avikal, S., et al. [3]	Fuzzy Kano model	×	√	×
Wang, X., et al. [13]	GRA-cloud model	√	√	×
Ping, Y. J., et al. [24]	PFLS	×	√	×
Li, S. P., et al. [15]	Probabilistic language modeling	√	×	×
Akbas, H. and B. Bilgen [10]	FANP	×	√	×
Li, X. B. and Z. He [16]	B2LTS	√	×	×
Wang, Z. Q., et al. [17]	multi-granularity linguistic terms	√	√	×

This paper is organized as follows: Section II provides an overview of EHFLTSs and cumulative prospect theory. Section III describes the proposed QFD approach. A case study is presented in Section IV to demonstrate the validity of the proposed approach and conduct a comparative analysis. Section V concludes with some remarks.

2 The proposed approach

This section proposes an innovative QFD method that combines EHFLTS and cumulative prospect theory to derive the weights of CRs and evaluate the significance of ECs. The proposed framework is illustrated in Fig. 1.

Fig. 1

Framework of the proposed QFD method.

Step 1: Aggregate opinions of experts. The opinions are grouped and summarized, and the evaluation information EC_i provided by expert G_r for customer requirement CR_j,denoted as $h_{s}^{r} ({EC}_{i}, {CR}_{j})$ , can be expressed as:

$h_{s}^{r} ({EC}_{i}, {CR}_{j}) = ⋃_{k = 1}^{l_{r}} h_{s}^{rl} ({EC}_{i}, {CR}_{j})$ (1)

Step 2: Summarize the different groups and construct the matrix. Every element of entire EHFL matrix can be determined by the following equation: $\begin{matrix} h_{sij} = EHFLHA \\ (h_{sij}^{1} ({EC}_{i}, {CR}_{j}), \dots, h_{sij}^{R} ({EC}_{i}, {CR}_{j})) \\ = \oplus_{r = 1}^{R} (ω_{r} h_{s}^{σ (r)}) = \oplus_{r = 1}^{R} (\cup_{s_{β_{r}} \in h_{s}^{σ (r)}} {ω_{r} s_{β r}}) \\ = \cup_{s_{β_{q}} \in h_{s}^{σ (1)}, \dots, s_{β_{q}} \in h_{s}^{σ (R)}} {ω_{1} s_{β_{1}} \oplus \dots \oplus ω_{R} s_{β_{R}}} \\ = \cup_{s_{β r} \in h_{s}^{σ (1)}, \dots, s_{β r} \in h_{s}^{σ (R)}} {s_{\bar{β}}} \end{matrix}$ (2)

Where the linguistic weighting vectors ω = (ω₁, …, ω_R), $\sum_{r = 1}^{R} ω_{r} = 1$ ,ω_r ∈ [0, 1], $\bar{β} = \sum_{r = 1}^{R} ω_{r} β_{r}$ ,is the rth largest linguistic weighting independent variable, ${\bar{h}}_{s}^{r} = {Rw}_{r} h_{s}^{r}$ ,w = (w₁, …, w_R) is ${h_{s}^{r}}$ the weighting vector of $\sum_{r = 1}^{R} w_{r} = 1$ ,w_r ∈ [0, 1].

The entire EHFL matrix Hs is denoted as: $H_{s} = [\begin{matrix} h_{s 11} & h_{s 12} & \dots & h_{s 1 n} \\ h_{s 21} & h_{s 22} & \dots & h_{s 2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ h_{sm 1} & h_{sm 2} & \dots & h_{smn} \end{matrix}]$ (3)

Where h_sij denotes the evaluation of correlation between EC_i and CR_j.

Step 3: Obtain CRs weights through CRITIC method. In this section, we will provide an explanation computing the weights of CRs using the CRITIC method within EHFLTS. The steps are as follows:

(1) Convert h_sij to a numerical value φ_ij by binary semantic model, and apply the Equation (4) to obtain the score matrix to expand the hesitant fuzzy group decision matrix φ_s = (φ_sij) _m×n, $k_{ij} = \frac{\hat{S} (φ_{ij}) - {min}_{i} \hat{S} (φ_{ij})}{{max}_{i} \hat{S} (φ_{ij}) - {min}_{i} \hat{S} (φ_{ij})}$ (4)

Where i = 1, …, n,j = 1, …, m, $\hat{S} (φ_{ij})$ represent the numerical value of φ_ij,k_ijrepresents the standardized score value of φ_ij.

(2) Use the Equation (5) to determine the correlation coefficient between CRs. $γ_{je} = \frac{\sum_{i = 1}^{m} (k_{ij} - {\bar{k}}_{j}) \cdot (k_{ie} - {\bar{k}}_{e})}{\sqrt{\sum_{i = 1}^{m} {(k_{ij} - {\bar{k}}_{j})}^{2} \cdot \sum_{i = 1}^{m} {(k_{ie} - {\bar{k}}_{e})}^{2}}}$ (5)

Where ${\bar{k}}_{j} = \frac{1}{m} \sum_{i = 1}^{m} k_{ij}$ and ${\bar{k}}_{e} = \frac{1}{m} \sum_{i = 1}^{m} k_{ie}$ , γ_je denotes the correlation coefficient between the j-th and eth CRs.

(3) Employ Equation (6) to measure the standard deviation of each CR. $σ_{a} = \sqrt{\frac{1}{m - 1} \sum_{i = 1}^{m} {(k_{ij} - {\bar{k}}_{j})}^{2}}$ (6)

Where σ_a indicates the standard deviation of the j-th CR, j = 1, …, n.

(4) Calculate the deviation degree of each CR with Equation (7). $τ_{j} = σ_{j} \cdot \sum_{e = 1}^{n} (1 - γ_{je})$ (7)

Where τ_jdenotes the standard deviation degree of the j-th CR,j = 1, …, n.

(5) The CR weights are calculated based on Equation (8). $ψ_{j} = \frac{τ_{j}}{\sum_{j = 1}^{n} τ_{j}}$ (8)

Where ψ_j indicates the weights of the j-th attributes.

Step 4: Choose the positive ideal solution and the negative ideal solution as the reference points. The positive ideal solution is obtained from the following equation: $\begin{matrix} P = {P_{1}, P_{2}, \dots, P_{n}} \\ = {max_{1 ⩽ i ⩽ m} (h_{si 1}), \dots, max_{1 ⩽ i ⩽ m} (h_{\sin})} \end{matrix}$ (9)

The negative ideal solution is obtained from the following equation: $\begin{matrix} N = {N_{1}, N_{2}, \dots, N_{n}} \\ = {min_{1 ⩽ i ⩽ m} (h_{si 1}), \dots, min_{1 ⩽ i ⩽ m} (h_{\sin})} \end{matrix}$ (10)

Step 5: Calculate the distance between h_sij and the reference points. Use d (h_sij, P_j) to denote the distance between h_sij and the positive reference point P_j, and d (h_sij, P_j) is defined as: $d (h_{sij}, P_{j}) = {(\frac{1}{l} \sum_{q = 1}^{l} {(\frac{I (s_{h_{sij}}^{q}) - I (s_{P_{j}}^{q})}{g})}^{2})}^{\frac{1}{2}}$ (11)

Use d (h_sij, N_j) to denote the distance between h_sij and the negative reference point N_j, and d (h_sij, N_j) is defined as: $d (h_{sij}, N_{j}) = {(\frac{1}{l} \sum_{q = 1}^{l} {(\frac{I (s_{h_{sij}}^{q}) - I (s_{N_{j}}^{q})}{g})}^{2})}^{\frac{1}{2}}$ (12)

Where I (s_i) = s_i,g = 2t. t is determined by the linguistic term set S ={ s_-t, …, s_t }. l is the number of elements in h_s, if l (h_sij) ≠ l (N_j),l = max{ l (h_sij) , l (N_j) }.

Then, collecting the above results, the distance sets of EC_i to the reference points are obtained. $d ({EC}_{i}, P) = {d (h_{si 1}, P_{1}), \dots, d (h_{sin}, P_{n})}$ (13) $d ({EC}_{i}, N) = {d (h_{si 1}, N_{1}), \dots, d (h_{sin}, N_{n})}$ (14)

Step 6: Calculate the prospect function values.

The value of the prospect function from the following equations: $v^{+} (d (h_{sij}, N_{j})) = {(d (h_{sij}, N_{j}))}^{α}, h_{sij} ⩾ N_{j}$ (15) $v^{-} (d (h_{sij}, P_{j})) = - θ {(d (h_{sij}, P_{j}))}^{β}, h_{sij} ⩽ P_{j}$ (16)

Where α (0 < α < 1)and β (0 < β < 1) are concave degree of the functions, θ (θ > 1)shows the sensitivity parameter for decision makers to avoid losses, α, β and θare the parameters.

Step 7: Calculate the weight function of prospect function values. The weight function of prospect function values from the following equations: $π_{ij}^{+} = ψ_{j}^{η} / {(ψ_{j}^{η} + {(1 - ψ_{j})}^{η})}^{\frac{1}{η}}$ (17) $π_{ij}^{-} = ψ_{j}^{ς} / {(ψ_{j}^{ς} + {(1 - ψ_{j})}^{ς})}^{\frac{1}{ς}}$ (18)

Where η (0 < η < 1)and ς (0 < ς < 1) are the weighting function curvature function, generally, η = 0.61, 1pt1pt1ptς = 0.69.

Step 8: Calculate the values of entire prospect function. The values of entire prospect function from the following equation: $V_{i} = \sum_{j = 1}^{n} v_{ij}^{+} π_{ij}^{+} + \sum_{j = 1}^{n} v_{ij}^{-} π_{ij}^{-}$ (19)

Step 9: Sort ECs in descending order by the values of V_i.

According to the cumulative prospect function, the value of V_i and the priority of EC are positively correlated. Therefore, the most important ECs can be obtained by sorting the values of V_i in descending order.

3 Case study and comparative analysis

3.1 Case study

In this section, the proposed method is used to a building material equipment enterprise to prove the effectiveness of the developed QFD model. QFD has important status in the mainframe manufacturing production of building material equipment enterprises, addressing current challenges such as controlling production conditions and balancing economic cost with quality level. This case study provides a detailed computational process of the proposed QFD methodology. Five decision makers (DM1, DM2,..., DM5) were invited to participate in the interview and choose the significant ECs based on their experience and knowledge. Through the investigation of expert interviews, six CRs are selected as shown in Table 2, and five ECs are selected as shown in Table 3. Divide the five decision makers into two groups: group 1 consists of decision makers 1, 2, and 3, while group 2 consists of decision makers 4 and 5. Each expert has the same weight. Depending on the number of experts in each group, the weight vectors for both groups are determined to be w = (0.6, 0.4).

Table 2
Customer Requirements

CRs Description

Easy maintenance (CR₁) Simple maintenance operation of the equipment

Stability (CR₂) Equipment can operate stably most of the time

Productivity (CR₃) The highest possible capacity when the machine is running

Difficulty in use (CR₄) Operation is as simple as possible

Energy consumption (CR₅) Lowest possible energy consumption when the equipment is in operation

Heat resistance (CR₆) The heat is as unobtrusive as possible to the operation of the equipment

CRs	Description
Easy maintenance (CR₁)	Simple maintenance operation of the equipment
Stability (CR₂)	Equipment can operate stably most of the time
Productivity (CR₃)	The highest possible capacity when the machine is running
Difficulty in use (CR₄)	Operation is as simple as possible
Energy consumption (CR₅)	Lowest possible energy consumption when the equipment is in operation
Heat resistance (CR₆)	The heat is as unobtrusive as possible to the operation of the equipment

Table 3

Engineering Characteristics

ECs	Description
Reliability (EC₁)	Completion of specified functions in a specified period of time
Durability (EC₂)	Capable of running for long periods of time
Accuracy (EC₃)	Random error size
Performance stability (EC₄)	System Stability Indicators
Availability (EC₅)	System uptime

The proposed QFD method involves the following steps to prioritize the ECs:

Step 1: Combine the linguistic assessments of EC_i relative to CR_j provided by each group of experts. The collective EHFL correlation matrices H₁ _s and H₂ _s were established by means of Eqs (1-2), as indicated in Table 4. The assessment of the correlation between ECs and CRs can be regarded as EHFLTS.

Table 4

Collective EHFL correlation matrix for each group

groups	Ecs	CRs
		CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
G₁	EC₁	s₀, s_0.6	s_0.6, s_1.2	s_–0.6, s₀	s_0.6, s_1.2	s_–0.6, s₀	s_1.2, s_1.8
	EC₂	s₀	s_1.2, s_1.8	s_–0.6, s₀	s_1.2, s_1.8	s_–1.2, s_–0.6	s_1.2, s_1.8
	EC₃	s_1.2, s_1.8	s_1.2, s_1.8	s₀, s_0.6	s_–1.2, s_–0.6	s₀	s_1.8
	EC₄	s_1.2, s_1.8	s_–1.8, s_–1.2	s_1.2, s_1.8	s_–0.6, s₀	s_–1.2, s_–0.6	s_0.6, s_1.2
	EC₅	s_–1.2	s₀, s_0.6	s_–1.8, s_–1.2	s₀, s_0.6	s_0.6, s_1.2	s_–1.2, s_–0.6
G₂	EC₁	s_0.4	s_0.4, s_0.8	s_–0.4, s₀	s_1.2	s_–0.4, s₀	s_0.8, s_1.2
	EC₂	s₀, s_–0.4	s_1.2	s_–0.4	s_1.2	s_–0.8, s_–0.4	s_0.4, s_0.8
	EC₃	s_0.4, s_0.8	s_1.2	s_0.4	s_–0.4, s₀	s₀	s_1.2
	EC₄	s_1.2	s_–0.8, s_–0.4	s_0.8	s_–0.4, s₀	s_–0.4, s₀	s_0.8
	EC₅	s_–0.4, s₀	s_0.4	s_–0.8	s₀, s_0.4	s_0.8	s_–0.8, s_–0.4

Step 2: Using Equation (3), the two sets of relationship matrices are aggregated in order to establish the entire EHFL correlation matrix Hs, as presented in Table 5. The OWA weighting vector is derived by utilizing the normal distribution approach [33] as ω = (0.5, 0.5).

Table 5

Overall EHFL correlation matrix Hs

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
EC₁	s_0.4, s₁	s₁, s_1.4, s_1.6, s₂	s_–1, s_–0.6, s_–0.4, s₀	s_1.8, s_2.4	s_–1, s_–0.6, s_–0.4, s₀	s₂, s_2.4, s_2.6, s₃
EC₂	s_–0.4, s₀	s_2.4, s₃	s_–1, s_–0.4	s_2.4, s₃	s_–2, s_–1.6, s_–1.4, s_–1	s_1.6, s₂, s₂.₂, s_2.6
EC₃	s_1.6, s₂, s_2.2, s_2.6	s_2.4, s₃	s_0.4, s₁	s_–1.6, s_–1.2, s_–1, s_–0.6	s₀	s₃
EC₄	s_2.4, s₃	s_–2.6, s_–2.2, s_–2, s_–1.6	s₂, s_2.6	s_–1, s_–0.6, s_–0.4, s₀	s_–1.6, s_–1.2, s_–1, s_–0.6	s_1.4, s₂
EC₅	s_–1.6, s_–1.2	s_0.4, s₁	s_–2.6, s_–2	s₀, s_0.4, s_0.6, s₁	s_1.4, s₂	s_–2, s_–1.6, s_–1.4, s_–1

Step 3: The CRITIC method is used to obtain the CR weights through Equations (4–8). The calculation results are shown in Table 6 to Table 11.

Table 6

Converting assessment information into a score matrix

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
EC₁	0.700	1.500	0.667	2.100	–0.500	2.500
EC₂	–0.200	2.700	–0.700	2.700	–1.500	2.100
EC₃	2.100	2.700	0.700	–1.100	0.000	3.000
EC₄	2.700	1.933	2.300	–0.500	–1.100	1.700
EC₅	–2.400	0.700	–2.300	0.500	1.700	–1.500

Table 7

Standardized score matrix

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
EC₁	0.608	0.400	0.645	0.842	0.313	0.889
EC₂	0.431	1.000	0.348	1.000	0.000	0.800
EC₃	0.882	1.000	0.652	0.000	0.469	1.000
EC₄	1.000	0.617	1.000	0.158	0.125	0.711
EC₅	0.000	0.000	0.000	0.421	1.000	0.000

Table 8

Correlation coefficient between CRs

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
CR₁	1.000	0.611	0.963	–0.461	–0.625	0.787
CR₂	0.611	1.000	0.438	–0.036	–0.725	0.810
CR₃	0.963	0.438	1.000	–0.359	–0.644	0.702
CR₄	–0.461	–0.036	–0.359	1.000	–0.327	0.021
CR₅	–0.625	–0.725	–0.644	–0.327	1.000	–0.749
CR₆	0.787	0.810	0.702	0.021	–0.749	1.000

Table 9

Standard deviation for each CR

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
σ _j	0.396	0.424	0.375	0.430	0.389	0.395

Table 10

Deviation for each CR

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
τ _j	1.475	1.656	1.463	2.649	3.142	1.354

Table 11

Weights of CRs

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
ψ _j	0.126	0.141	0.125	0.226	0.268	0.115

Step 4: Positive reference point P_j and negative reference point N_j are determined by Equations (9-10) as respectively: $\begin{matrix} P = {(s_{1 . 6} {, s}_{2} {, s}_{2 . 2} {, s}_{2 . 6}), (s_{2 . 4} {, s}_{3}) \\ (s_{2}, s_{2.6}), (s_{2.4}, s_{3}), \\ (s_{1.4}, s_{2}), (s_{3})} \end{matrix}$ $\begin{matrix} N = {(s_{- 1.6}, s_{- 1.2}), (s_{- 2.6}, s_{- 2.2}, s_{- 2}, s_{- 1.6}), \\ (s_{- 2.6}, s_{- 2}), (s_{- 1.6}, s_{- 1.2}, s_{- 1}, s_{- 0.6}) \\ (s_{- 2}, s_{- 1.6}, s_{- 1.4}, s_{- 1}), (s_{- 2}, s_{- 1.6}, s_{- 1.4}, s_{- 1})} \end{matrix}$

Step 5: Determine the distance between h_sij and P_j by Equation (11) and (13), and the distance between h_sij and N_j by Equation (12) and (14). The outcomes are presented in Table 12.

Table 12

Distance between h_sijand the reference point

distances	ECs	CRs
		CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
d (h_sij, P_j)	EC₁	0.211	0.228	0.493	0.100	0.393	0.103
	EC₂	0.369	0.000	0.500	0.000	0.560	0.162
	EC₃	0.000	0.000	0.267	0.659	0.288	0.000
	EC₄	0.130	0.826	0.000	0.560	0.493	0.222
	EC₅	0.568	0.333	0.767	0.393	0.000	0.752
d (h_sij, N_j)	EC₁	0.350	0.600	0.277	0.560	0.167	0.667
	EC₂	0.200	0.826	0.267	0.659	0.000	0.600
	EC₃	0.568	0.826	0.500	0.000	0.257	0.752
	EC₄	0.684	0.000	0.767	0.100	0.067	0.560
	EC₅	0.000	0.493	0.000	0.267	0.560	0.000

Step 6: We can derive the prospect function of gain from Equation (15), and the loss by Equation (16). The results are shown in Tables 13-14.

Table 13

Value function of the prospect functionv⁺

distances	Ecs	CRs
		CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
v⁺	EC₁	0.397	0.638	0.324	0.600	0.207	0.700
	EC₂	0.243	0.845	0.313	0.693	0.000	0.638
	EC₃	0.608	0.845	0.543	0.000	0.303	0.779
	EC₄	0.715	0.000	0.792	0.132	0.092	0.600
	EC₅	0.000	0.537	0.000	0.313	0.600	0.000

Table 14

Value function of the prospect functionv^-

distances	ECs	CRs
		CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
v^-	EC₁	–0.573	–0.612	–1.208	–0.297	–0.990	–0.304
	EC₂	–0.936	0.000	–1.223	0.000	–1.350	–0.452
	EC₃	0.000	0.000	–0.703	–1.560	–0.752	0.000
	EC₄	–0.374	–1.901	0.000	–1.350	–1.208	–0.599
	EC₅	–1.368	–0.856	–1.781	–0.990	0.000	–1.752

Step 7: The gain and loss weights of CRs are calculated according to the Equations (17-18), as shown in Tables 15-16.

Table 15

Probability weights as CRs gains

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
EC₁	0.208	0.196	0.187	0.231	0.244	0.181
EC₂	0.208	0.196	0.187	0.231	0.244	0.181
EC₃	0.208	0.196	0.187	0.231	0.244	0.181
EC₄	0.208	0.196	0.187	0.231	0.244	0.181
EC₅	0.208	0.196	0.187	0.231	0.244	0.181

Table 16

Probability weights as CRs losses

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
EC₁	0.195	0.209	0.194	0.276	0.306	0.185
EC₂	0.195	0.209	0.194	0.276	0.306	0.185
EC₃	0.195	0.209	0.194	0.276	0.306	0.185
EC₄	0.195	0.209	0.194	0.276	0.306	0.185
EC₅	0.195	0.209	0.194	0.276	0.306	0.185

Step 8: The overall foreground function value of ECs is determined by the Equation (19) as follows: $V_{1} = - 0.330$ $V_{2} = - 0.365$ $V_{3} = - 0.188$ $V_{4} = - 0.864$ $V_{5} = - 1.065$

Step 9: By ranking the values in descending order, the five ECs are ranked: ${EC}_{3} > {EC}_{1} > {EC}_{2} > {EC}_{4} > {EC}_{5}$

Hence, EC₃ emerges as the foremost critical EC. In the manufacturing of products, manufacturers should focus on EC₃.

3.2 Comparative analysis

In this Section, relevant QFD methods are used to analyze the case of the previous section, including conventional QFD, rough cloud QFD [12], hesitant fuzzy QFD [22], and AHP-QFD [25], and the outcomes are presented in Table 17.

Table 17
Ranking of different methods

Conventional-QFD Rough cloud QFD Hesitant fuzzy QFD AHP-QFD Proposed method

EC₁ 1 2 2 3 2

EC₂ 3 4 3 2 3

EC₃ 1 1 1 1 1

EC₄ 5 3 4 4 4

EC₅ 4 5 5 5 5

	Conventional-QFD	Rough cloud QFD	Hesitant fuzzy QFD	AHP-QFD	Proposed method
EC₁	1	2	2	3	2
EC₂	3	4	3	2	3
EC₃	1	1	1	1	1
EC₄	5	3	4	4	4
EC₅	4	5	5	5	5

Firstly, in contrast to conventional QFD method, significant differences are observed between their respective outcomes. For instance, the conventional QFD method ranks EC₁ and EC₃ together as top priorities, while proposed method places EC₃ at the forefront, which demonstrates the effectiveness of proposed method. Moreover, unlike conventional approaches that cannot distinguish the priority between EC₁ and EC₃ or provide consistent rankings for EC₄ and EC₅ across different methods, solely relying on numerical values for evaluation purposes, our approach takes into account expert hesitation levels.

Second, it can be observed that the first two EC priority rankings of the two methods are consistent, with a slight difference in the priority ranking of EC₂ and EC₄. Additionally, the conventional QFD method, fuzzy QFD, and the proposed method consider EC₂ to have a higher priority than EC₄, highlighting the impact generated by EC₂ over EC₄ in terms of satisfying customer satisfaction. This variation may be attributed to differences in expert experience and preference, as well as the fact that the rough cloud QFD method does not account for the interaction between CR and the psychological behavior of experts.

Besides, a comparative study between hesitant QFD and AHP-QFD reveals that while both methods utilize hesitant fuzzy term sets, AHP-QFD differs slightly from all other methods concerning the priority ranking of EC₁ and EC₂. This difference stems from AHP-QFD that each expert is rational. Further analysis shows that although hesitant QFD yields similar results to proposed method, it calculates CR weights differently, using fuzzy measures without considering data conflict. In contrast, CRITIC accounts for these factors when calculating weights.

Finally, to corroborate the efficacy of the purposed approach, the ECs composite distances of different methods under CRs are calculated with Equation (20). $C_{i} = \frac{d (h_{sij}, N_{j})}{d (h_{sij}, P_{j}) + d (h_{sij}, N_{j})}$ (20)

To exemplify the feasibility and effectiveness of the proposed method, EC₃ is considered to serve as an illustration. The composite distances of EC₃ is calculated as shown in the Table 18. The CR₁, CR₂, CR₃ and CR₆ represent revenue-oriented customer needs, and CR₄ and CR₅ correspond to loss-oriented customer needs. The different ways of calculating C₁ are 1, 0.8195, 0.8817, 0.8477, 0.9024. The proposed method exhibits the highest composite distances among the compared methods, which prove that the description of the relationship between CRs and EC₃ is closest to the ideal solution when CR is revenue-oriented customer need. Based on the comparison of composite distances, it is evident that the proposed method yields the largest distances for CR₁, CR₂, CR₆, while the smallest distance is observed for CR₄. Therefore, these results indicate that proposed method surpasses other methods by satisfying the CRs.

Table 18

Composite distances of EC₃

	CR₁	CR₂	CR₃	CR₄	CR₅	CR₆
PM	1	1	0.6519	0	0.4716	1
Conventional QFD	0.8195	0.7333	0.3919	0.2871	0.3876	0.6866
Rough cloud QFD	0.8817	0.8755	0.7383	0.1875	0.5086	0.8545
Hesitant fuzzy QFD	0.8477	0.9274	0.5629	0.4237	0.5771	0.9014
AHP QFD	0.9024	0.8923	0.5280	0.1543	0.7238	0.8829

Calculate the composite distances of different methods for all ECs, the results are shown in the Fig. 2. Figure 2(a) shows the composite distances of EC₁, CR₁, CR₂, CR₃ and CR₅ represent revenue-oriented customer needs, while CR₄ and CR₆ represent loss-oriented customer needs. As can be seen from Fig. 2(a), compared with other methods, the composite distances of CR₁, CR₂, CR₃ and CR₅ are most close to 1, which can more truly reflect the relationship between EC₁ and CRs. Figure 2(b) displays the composite distances of EC₂, the PM is close to revenue-oriented CRs of the most closed to 1, which is significantly higher than composite distances obtained by other methods. It can be seen from the Fig. 2(d) and(e), the PM calculates the closest composite distances to 1 for the revenue-oriented customer needs of EC₄ and EC₅. Therefore, the proposed method can describe the relationship between the ECs and CRs accurately.

Fig. 2

Composite distances for EC_S calculated under different methods.

3.3 Sensitivity analysis

Given the diverse risk preferences of decision-makers, this subsection conducts numerical experiments on employing the curvature parameters of value function (η) and weight function (ζ) as variables.

The ranking of ECs alternatives obtained from sensitivity analysis with respect to the two scenarios are presented in Figs. 3-4. The conclusion can be drawn from the graph that despite the continuous fluctuation of curvature parameters, the ranking of ECs has remained constant,EC₃ > EC₁ > EC₂ > EC₄ > EC₅. In summary, the sensitivity analysis indicates that the parameters of the cumulative prospect theory have virtually no impact on the final ranking of ECs. Therefore, the deduction results of the PM are reliable.

Fig. 3

Sensitivity analysis in the case of the parameter η changes.

Fig. 4

Sensitivity analysis in the case of the parameter ζ changes.

4 Conclusion

In this paper, a new decision-making method is proposed for EC priority rankings in QFD, which captures the subjectivity and hesitancy regarding the relationship between ECs and CRs while preserving the evaluations content of DM. This allows us to obtain the evaluation aggregation information of EC and CR. Subsequently, the CR weights are obtained by considering the conflicting nature between the CRITIC evaluation indicators. Finally, we incorporate the psychological factors of DMs, such as cumulative prospect theory, to determine the ranking of ECs.

Future research can explore two aspects: First, increasing the number of experts and considering their heterogeneity by assigning different weights; Secondly, in many cases involving QFD problems, the CR weight information may be entirely unknown, rendering the PM insufficient. Therefore, it is recommended to expand the approach in future research to address this limitation.

Abbreviations

B2LTS

basic 2-tuple linguistic term set

customer requirement

DHHLTSs

double hierarchy hesitant linguistic term sets

decision maker

engineering characteristics

EHFL

extended hesitant fuzzy linguistic

EHFLTS

extended hesitant fuzzy linguistic terms sets

FANP

fuzzy analytic hierarchy process

IVSF

interval-valued spherical fuzzy

IRN

interval rough number

IT2FSs

interval type-2 fuzzy sets

MADM

multi-attribute decision-making

PFLS

probabilistic fuzzy language term sets

PHFLTSs

probabilistic hesitant fuzzy language term sets

PLTSs

probabilistic language term sets

proposed method

QFD

quality function deployment

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 51705386); China Scholarship Council (No.201606955091).

References

Cristiano

J.J.

, et al., Key factors in the successful application of quality function deployment (QFD), IEEE Transactions on Engineering Management 48(1) (2001), 81–95.

Hong

Z.X.

, et al., Quality characteristic extraction for complex products with multi-granular fuzzy language based on the triple bottom lines of sustainability, Computers & Industrial Engineering 167 (2022).

Avikal

, et al., QFD and Fuzzy Kano model based approach for classification of aesthetic attributes of SUV car profile, Journal of Intelligent Manufacturing 31(2) (2020), 271–284.

, et al., An interval type-2 fuzzy Kano-prospect-TOPSIS based QFD model: Application to Chinese e-commerce service design, Applied Soft Computing 111 (2021).

Hsu

C.H.

, et al., Deploying Industry 4.0 Enablers to Strengthen Supply Chain Resilience to Mitigate Ripple Effects: An Empirical Study of Top Relay Manufacturer in China, IEEE Access 10 (2022), 114829–114855.

Yazdani

, et al., Integrated QFD-MCDM framework for green supplier selection, Journal of Cleaner Production 142 (2017), 3728–3740.

Chan

L.K.

and Wu

M.L.

, A systematic approach to quality function deployment with a full illustrative example, Omega-International Journal of Management Science 33(2) (2005), 119–139.

Jin

C.X.

, et al., Prioritization of key quality characteristics with the three-dimensional HoQ model-based interval-valued spherical fuzzy-ORESTE method, Engineering Applications of Artificial Intelligence 104 (2021).

Kong

M.M.

, et al., New Operations on Generalized Hesitant Fuzzy Linguistic Term Sets for Linguistic Decision Making, International Journal of Fuzzy Systems 21(1) (2019), 243–262.

10.

Akbas

and Bilgen

, An integrated fuzzy QFD and TOPSIS methodology for choosing the ideal gas fuel at WWTPs, Energy 125 (2017), 484–497.

11.

Yang

, et al., Large-scale group decision-making for prioritizing engineering characteristics in quality function deployment under comparative linguistic environment, Applied Soft Computing 127 (2022).

12.

Fang

, et al., A New Method for Quality Function Deployment Based on Rough Cloud Model Theory, IEEE Transactions on Engineering Management 69(6) (2022), 2842–2856.

13.

Wang

, et al., Technical attribute prioritization in QFD based on cloud model and grey relational analysis, International Journal of Production Research 58(19) (2020), 5751–5768.

14.

Yang

Y.X.

, et al., Multi-attribute decision making method based on probabilistic linguistic term sets and its application in the evaluation of emergency logistics capacity, Journal of Intelligent & Fuzzy Systems 42(3) (2022), 2157–2168.

15.

S.P.

, et al., Rating engineering characteristics in open design using a probabilistic language method based on fuzzy QFD, Computers & Industrial Engineering 135 (2019), 348–358.

16.

X.B.

and He

, Determining importance ratings of patients’ requirements with multi-granular linguistic evaluation information, International Journal of Production Research 55(14) (2017), 4110–4122.

17.

Wang

Z.Q.

, et al., A group multi-granularity linguistic-based methodology for prioritizing engineering characteristics under uncertainties, Computers & Industrial Engineering 91 (2016), 178–187.

18.

Rodriguez

R.M.

, Luis

and Herrera

, Hesitant Fuzzy Linguistic Term Sets for Decision Making, IEEE Transactions on Fuzzy Systems 20(1) (2012), 109–119.

19.

Wei

C.P.

, et al., A Novel Linguistic Group Decision-Making Model Based on Extended Hesitant Fuzzy Linguistic Term Sets, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 23(3) (2015), 379–98.

20.

Chen

Y.F.

, et al., A new integrated MCDM approach for improving QFD based on DEMATEL and extended MULTIMOORA under uncertainty environment, Applied Soft Computing 105 (2021).

21.

L.N.

, et al., A novel Kano-QFD-DEMATEL approach to optimise the risk resilience solution for sustainable supply chain, International Journal of Production Research 59(6) (2021), 1714–1735.

22.

Zeng

W.Y.

, et al., Weighted hesitant fuzzy linguistic term sets and its application in group decision making, Journal of Intelligent & Fuzzy Systems 37(1) (2019), 1099–1112.

23.

Zhang

Z.M.

and Wu

, Hesitant fuzzy linguistic aggregation operators and their applications to multiple attribute group decision making, Ibid 26(5) (2014), 2185–2202.

24.

Ping

Y.J.

, et al., A new integrated approach for engineering characteristic prioritization in quality function deployment, Advanced Engineering Informatics 45 (2020).

25.

B.P.

, Lima, et al., New hybrid AHP-QFD-PROMETHEE decision-making support method in the hesitant fuzzy environment: An application in packaging design selection, Journal of Intelligent & Fuzzy Systems 42(4) (2022), 2881–2897.

26.

Wang

, Extended hesitant fuzzy linguistic term sets and their aggregation in group decision making, International Journal of Computational Intelligence Systems 8(1) (2015), 14–33.

27.

Abdel-Basset

and Mohamed

, A novel lithogenic TOPSIS- CRITIC model for sustainable supply chain risk management, Journal of Cleaner Production 247 (2020).

28.

Rostamzadeh

, et al., Evaluation of sustainable supply chain risk management using an integrated fuzzy TOPSIS- CRITIC approach, Journal of Cleaner Production 175 (2018), 651–669.

29.

Liu

H.C.

, et al., An integrated behavior decision-making approach for large group quality function deployment, Information Sciences 582 (2022), 334–348.

30.

Shi

, et al., Engineering Characteristics Prioritization in Quality Function Deployment Using an Improved ORESTE Method with Double Hierarchy Hesitant Linguistic Information, Sustainability 14(15) (2022).

31.

Song

C.Y.

, et al., Environmental quality evaluation based on the TODIM method with normal wiggly hesitant fuzzy set, Soft Computing 27(12) (2023), 8161–8173.

32.

Tversky

, Kahneman

, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty (1992), 297–323.

33.

Z.S.

, An overview of methods for determining OWA weights, International Journal of Intelligent Systems 20(8) (2005), 843–865.

Engineering characteristics prioritization for quality function deployment by using CRITIC and cumulative prospect theory with extended hesitant fuzzy linguistic term sets

Abstract

Keywords

1 Introduction

3.1 Case study

Table 17 Ranking of different methods Conventional-QFD Rough cloud QFD Hesitant fuzzy QFD AHP-QFD Proposed method EC1 1 2 2 3 2 EC2 3 4 3 2 3 EC3 1 1 1 1 1 EC4 5 3 4 4 4 EC5 4 5 5 5 5

Abbreviations

Footnotes

Acknowledgments

References

Table 17
Ranking of different methods

Conventional-QFD Rough cloud QFD Hesitant fuzzy QFD AHP-QFD Proposed method

EC₁ 1 2 2 3 2

EC₂ 3 4 3 2 3

EC₃ 1 1 1 1 1

EC₄ 5 3 4 4 4

EC₅ 4 5 5 5 5