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

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.

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.

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.

OPEN IN VIEWER

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: h sij = EHFLHA ( h sij 1 ( EC i , CR j ) , … , h sij R ( EC i , CR j ) ) = ⊕ r = 1 R ( ω r h s σ ( r ) ) = ⊕ r = 1 R ( ∪ s β r ∈ h s σ ( r ) { ω r s β r } ) = ∪ s β q ∈ h s σ ( 1 ) , … , s β q ∈ h s σ ( R ) { ω 1 s β 1 ⊕ … ⊕ ω R s β R } = ∪ s β r ∈ h s σ ( 1 ) , … , s β r ∈ h s σ ( R ) { s β ¯ }(2)

Where the linguistic weighting vectors ω = (ω₁, …, ω _R ), ∑ r = 1 R ω r = 1 ,ω _r  ∈ [0, 1], β ¯ = ∑ r = 1 R ω r β r ,is the rth largest linguistic weighting independent variable, h ¯ s r = Rw r h s r ,w = (w₁, …, w _R ) is { h s r } the weighting vector of ∑ r = 1 R w r = 1 ,w _r  ∈ [0, 1].

The entire EHFL matrix Hs is denoted as: H s = [ h s 11 h s 12 … h s 1 n h s 21 h s 22 … h s 2 n ⋮ ⋮ ⋮ ⋮ h sm 1 h sm 2 … h smn ](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 = S ˆ ( φ ij ) - min i S ˆ ( φ ij ) max i S ˆ ( φ ij ) - min i S ˆ ( φ ij )(4)

Where i = 1, …, n,j = 1, …, m, S ˆ ( φ ij ) represent the numerical value of φ _ij ,k _ij represents the standardized score value of φ _ij .

(2) Use the Equation (5) to determine the correlation coefficient between CRs. γ je = ∑ i = 1 m ( k ij - k ¯ j ) · ( k ie - k ¯ e ) ∑ i = 1 m ( k ij - k ¯ j ) 2 · ∑ i = 1 m ( k ie - k ¯ e ) 2(5)

Where k ¯ j = 1 m ∑ i = 1 m k ij and k ¯ e = 1 m ∑ 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 = 1 m - 1 ∑ i = 1 m ( k ij - 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 · ∑ e = 1 n ( 1 - γ je )(7)

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

(5) The CR weights are calculated based on Equation (8). ψ j = τ j ∑ 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: P = { P 1 , P 2 , … , P n } = { max 1 ⩽ i ⩽ m ( h si 1 ) , … , max 1 ⩽ i ⩽ m ( h sin ) }(9)

The negative ideal solution is obtained from the following equation: N = { N 1 , N 2 , … , N n } = { min 1 ⩽ i ⩽ m ( h si 1 ) , … , min 1 ⩽ i ⩽ m ( h sin ) }(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 ) = ( 1 l ∑ q = 1 l ( I ( s h sij q ) - I ( s P j q ) g ) 2 ) 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 ) = ( 1 l ∑ q = 1 l ( I ( s h sij q ) - I ( s N j q ) g ) 2 ) 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 ) , … , d ( h sin , P n ) }(13) d ( EC i , N ) = { d ( h si 1 , N 1 ) , … , 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 ) 1 η(17) π ij - = ψ j ς / ( ψ j ς + ( 1 - ψ j ) ς ) 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 = ∑ j = 1 n v ij + π ij + + ∑ 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.

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