Configuration optimization of product-service system design requirements based on hesitant information axiom

Abstract

Product-service system (PSS) has attracted attention of manufacturers to shift from product-providing to solution-providing, which is a marketable set of products and services. The existing researches emphasize the fulfillment of individualized customer requirements through different PSS configurations. The PSS planning phase is of high importance in generating conceptual schemes, which translates customer requirements (CRs) to design requirements (DRs). In this paper, a systematic decision-making approach based on QFD is put forward aiming to configure the PSS design requirements (DRs). To address the uncertainty and hesitancy in QFD modeling, a hesitant fuzzy linguistic term sets (HFLTSs) is applied to elicit the experts’ linguistic preferences in evaluating the importance of CRs and the relationships between CRs and DRs. To dealing with the group decision-making problems concerning the HFLTSs, the min-upper operator and the max-lower operator assemble the experts’ evaluation results into a linguistic interval, and then the numerical results can be obtained by using the 2-tuple linguistic representation model and the interval preference degree computation. A non-linear 0-1 programming model is proposed to select the target DRs’ specifications for maximizing customer satisfaction under cost constraint. In order to objectively determine the satisfaction degree of each optional specification of DR, the information axiom is introduced to construct the objective function via information content computation. To deal with the qualitative DRs, HFLTSs and information axiom are combined and hesitant information axiom (HIA) is proposed. Finally, a DRs optimization model is established using HIA and the imprecision method. A case study is carried out to demonstrate the effectiveness of the optimal PSS planning approach developed.

Keywords

Product-service system (PSS)design requirement information axiom hesitant fuzzy linguistic term set non-linear programming

1 Introduction

Traditional design thinking frequently focuses on providing products that meet customer needs, but lacks systematic consideration of products and services. Product Service System (PSS) combines tangible products and intangible services to meet customer needs, while considering the full supply-side life cycle stage and increasing productivity, reducing resource consumption and adverse environmental impacts [1].The concept of PSS is also referred to as Functional (total care) Products [2], Technical Product Service System [3], Industrial Product-Service Systems (IPS2) [4], Service Engineering [5] or Through-life Engineering Services [6].

The PSS concept reflects the willingness of manufacturing companies to provide a systemic solution, including products and services [7]. The early stage of PSS development, conceptual design, plays a critical role in delivering value to customers and faces the challenge of stakeholders’ requirements. Conceptual design is a design stage in which the requirements of design problem are transformed into schematic descriptions of design solution concepts [8]. The commonly used methods for PSS conceptual design are Service/product engineering (SPE) [5], TRIZ theory [9], quality function deployment (QFD) [10], and various configuration approaches have been developed. Wang et al. [11] introduced a modular development framework of PSS and built a PSS configuration method based on the product–service ontology. Cui et al. [12] considered PSS configuration as a multi-classification problem and provided personalized optimal central air conditioning PSS configuration by optimizing support vector machines to reduce the dimensionality of customer requirements. All of these studies pay attention to the PSS conceptual design and emphasize the realization of individualized customer needs through various configurations. Among them, QFD is a cross-functional planning tool used to assist the development team in understanding the voice of customer and reducing design conflict.

Once the customer requirements have been defined, it is also critical to effectively transform them into design requirements [13]. Design requirements (DRs), also known as engineering characteristics, are the voice of design engineers. They can be further transformed into parts characteristics, process operation, and production requirements. QFD for PSS is the use of QFD methods to assist engineers in properly identifying, evaluating and describing customer requirements in the context of PSS development [14 –18].

The challenges of QFD for PSS are both product DRs and service DRs exist in house of quality, and they have different characteristics. QFD evaluation information in PSS is more ambiguous and complex. Besides quantification of DRs satisfaction when constructing the optimal model is also a problem to be solved. Moreover, QFD brings together cross-functional teams and fosters communication and cooperation among multidisciplinary development teams.

For ambiguity and uncertainty information in QFD of PSS, fuzzy sets frequently utilized to solve uncertain information existing in the linguistic evaluation, such as fuzzy set [21], intuitionistic fuzzy set [22], rough set [23] and 2-tuple linguistic representation model [24]. The traditional fuzzy theory utilizes a single membership, which can constrain the ability to capture a wide range of expert preferences feely in QFD process. Intuitionistic fuzzy set and rough set consider the non-membership degree, however they ignore the hesitancy parameter and fail to capture experts’ hesitation between different linguistic values effectively [25]. Rodriguez et al. [26] proposed the hesitant fuzzy linguistic term set (HFLTSs) based on hesitant fuzzy sets to improve the richness of linguistic elicitation. HFLTSs is a new topic and is used as a modeling tool in QFD less frequently. This paper introduces the HFLTSs into the QFD for PSS planning in order to express the linguistic evaluation hesitancy as clearly as possible.

DRs optimization plays a vital role in a PSS development due to its critical impact on successive design activities as well as the novelty and competence of the final product. Although there have been many studies on DRs optimization in the product planning field, there are few studies on configuration optimization for DRs of PSS. The key to configuring an optimization model for DRs specifications is how to quantify the satisfaction of the different options. The divergences between product DRs and service DRs exacerbate the challenge. It is imprecise to assess customer satisfaction because the satisfaction level is always a range especially for service attributes. Information axiom (IA), which is one of the two axioms in Axiomatic Design (AD), provides illumination for estimating customer satisfaction via information content computation. IA considers that the scheme with the minimum information content is optimal, and the information content is measured by the degree of satisfaction between design and the scope of the system [27]. It is frequently used to select the best design solution that satisfies the independence axiom [28, 29].

Most product service system design requirements are qualitative. For quantitative DRs, traditional information axiom can be used. For qualitative DRs, the optional specifications are tough to be estimated due to the fuzziness and hesitation in the process of evaluation. The method of hesitation information axiom (HIA) is proposed to calculate the information content. A nonlinear 0-1 programming model based on HIA is constructed to configure DRs values to maximize customer satisfaction under cost constraints by using the compensation relationship between DRs.

In addition, QFD is a group decision function complete by an expert group. It is necessary to concentrate on the integration of linguistic information. Considering that HFLTSs have different lengths and hard to be operated, a new approach based on the min-upper operator and max-lower operator is proposed in this paper to aggregate multiple HFLTSs directly to simplify the operation.

This paper aims to develop an improved QFD approach by integrating the HFLTSs and the nonlinear 0-1 programming model, in order to optimize configuration of product-service system design requirements. The contributions of this paper are as follows. First, HFLTSs is employed to manage the ambiguity and hesitancy of QFD information for PSS. The 2-tuple linguistic representation model is utilized to compute with HFLTSs. Second, the concept of information content is employed to describe the DRs’ design objectives. A nonlinear 0-1programming model is constructed to configure the optimal PSS design requirements with the minimum information content. Third, the information axiom is extended to HFLTSs and the method of hesitation information axiom (HIA) is proposed to calculate the information content for qualitive design requirements.

The remaining part of this paper is organized as follows. Section 2 presents a literature review about QFD and HFLTSs, optimization of design requirements, and information axiom. Section 3 gives the framework of PSS configuration and the basic definitions of HFLTSs and 2-tuple linguistic representation. The process for determining the importance of DRs is detailed in Section 4. Section 5 proposes DRs optimization model based on HIA and non-linear programming. Section 6 demonstrates an illustrative example of the proposed approach. Finally, the concluding remarks are given in Section 7.

2 Literature review

2.1 QFD and HFLTSs

QFD as an effective customer-driven design tool has been used successfully to translate CRs into DRs in product or service design. Determining CRs’ importance ratings and translating into the importance of DRs are crucial tasks, but they may be influenced by the uncertainty of the evaluation information [30 –33]. Considering the ambiguity and fuzziness that exist in QFD, the combination of fuzzy set theory [34 –37], rough set [38, 39] and 2-tuple fuzzy linguistic representation model [24 , 40–42] with QFD are effective countermeasures in response to this issue.

Determining the importance weights for the CRs and DRs are essential because they significantly affect the target values set for the DRs [43]. The studies about the fuzzy theory used in these fields can be listed as follows. Wang et al. [44] applied fuzzy arithmetic to figure out the importance of DRs in QFD. Liu et al. [45] computed the importance of DRs without knowing their membership functions using fuzzy weighted aggregation. Kwong et al. [46] presented an approach to derive the importance of DRs, and they considered the fuzzy relations between CRs and DRs as well as fuzzy correlations among DRs based on the fuzzy expert systems approach. Besides, rough QFD is also a common solution for dealing with uncertain information. For example, li et al. [47] presented a novel approach to acquire the DRs in product planning based on rough set theory. Li et al. [38] applied the rough set to identify the correlations among DRs. Grey relational analysis (GRA) is another method proposed for such problems with fewer data in an uncertain environment, and several studies focus on the grey QFD. Wu et al. [48] combined GRA with QFD as a practical method to analyze CRs. Wu et al. [49] formulated QFD by GM (1, N) and GM (0, N) models to improve product quality. Song et al. [50] identified the importance of DRs by combining rough set and GRA to build a rough-grey relational analysis approach.

In the assessing process of QFD, the experts usually hesitate among several possible linguistic values and require richer expressions to give out their opinions. HFLTSs are more suitable to be used in this condition due to their distinguished efficiency and flexibility in modeling uncertainty and vagueness in the decision-making process. Several decision-making approaches using HFLTSs have been studied in the literature. Wei et al. [51] introduced several aggregation operators of HFLTSs and gave a comparison about them. Yu et al. [52] proposed some formulates of unbalanced HFLTSs for dealing with multi-criteria group decision-making problems in an unbalanced HFLTSs situation. Hesamian et al. [53] proposed the similarity measures between HFLTSs and a method to rank HFLTSs. Wu et al. [54] dealt with the consistency and consensus of hesitant fuzzy linguistic preference relations in group decision-making problems.

HFLTSs begins to be applied in the QFD approach. Cevik et al. [31] considered the relations between CRs & DRs and correlations among DRs via HFLTSs in fuzzy QFD, and HFLTSs were aggregated by using the fuzzy envelope and the OWA operator. Nevertheless, determining the parameters of the fuzzy membership function in the fuzzy envelope is fairly complicated. Osiro et al. [25] combined HFLTSs and QFD to select supply chain sustainability metrics. The method of distance measures between HFLTSs was used to prioritize the requirements. The distance was measured by adding extend linguistic terms to the HFLTSs having fewer numbers of linguistic terms, and it may introduce new uncertainty in the process of extension. Huang et al. [55] proposed a novel QFD approach using proportional hesitant fuzzy linguistic term sets (PHFLTSs) and prospect theory. HFLTSs has become an effective method to deal with the uncertainty existing in the QFD.

Several related studies have attempted to apply QFD methodology to PSS planning. Song et al. [56] proposed a systematic evaluation approach for eliciting and assessing IPS2 requirements in QFD. Geng et al. [57] proposed a systematic decision-making approach for the optimal PSS planning based on QFD, and the optimal fulfillment levels of engineering characteristics were determined by a non-linear programming model. Song et al. [58] optimized the configuration of product-extension service based on QFD and established a multi-objective configuration model. From the literature review, we can conclude that HFLTSs has not been applied in the PSS planning field and selecting the DR’s optimal specification for PSS is rarely studied.

2.2 Optimization of design requirements

Through constructing the optimization model based on QFD, we can identify the fulfillment levels or the befitting selection of DRs among the optional values to achieve specific goals. The common objective function of the programming model is to maximize customer satisfaction. Customer satisfaction is often determined by the DRs’ contribution via the importance and fulfillment levels of DRs [59 –61]. Table 1 shows the QFD related researches using optimization models.

Table 1
QFD related researches using optimization models

Type Authors Optimisation model Objective

Determine the fulfillment level or the target level of attributes Chaudhuri and Bhattacharyya (2009) Integer programming Determine the improvement levels of the attributes in technical characteristics

Alptekin and Karsak (2011) Linear programming Determine the target values for e-learning product characteristics

Geng, Chu, Xue, and Zhang (2011) Non-linear programming Maximize the fulfillment levels of PSS engineering characteristics

Kahraman, Ertay, and Büyüközkan (2006) Mixed integer linear programming Optimize the improvement levels of product technical requirements

Delice and Güngör (2009) Mixed integer linear programming Determine the fulfillment levels of the continuous design requirements

Chen, Su, Tu, Lin, and Chang (2016), Chen and Weng (2006) Fuzzy goal programing model Determine the fulfillment levels of design requirements

Chen, Ko, and Yeh (2017) Additive fuzzy goal programing model Determine and modify the fulfillment levels of design requirements

Miao, Liu, Chen, Zhou, and Ji (2017) Two uncertain chance-constrained programming (CCP) models Determine the target levels of design attributes

He, Song, Wu, Xu, Zheng, and Ming (2017) Nonlinear programming model considering KANO Determine the target levels of engineering characteristics

Select the optimal attributes or optimize attributes from a set of feasible solutions Luo, Kwong, and Tang (2010) An integrated optimization model Achieve the optimal target settings of engineering characteristics

Wang and Chen (2012) Linear integer programming Select optimal module mix

Karsak, Sozer, and Alptekin (2003) 0–1 goal programming Determine the product technical requirements which will be taken into account in the design phase

Chowdhury and Quaddus (2016) 0–1 nonlinear optimization model Find the optimal portfolio of service strategies

Lee, Kang, Yang, and Lin (2010) Multi-choice 0-1 goal programming model Select the most suitable engineering characteristics to be considered in the product development

Raharjo, Xie, and Brombacher (2006) 0–1 goal programming Select quality characteristics

Song and Chan (2015) Multi-objective optimization model Configure the product-extension service and find the optimized service characteristics from a large number of feasible solutions

Delice and Güngör (2011), Delice and Güngör (2013) Mixed integer goal programming Optimize alternative values for the design requirements of a product

Type	Authors	Optimisation model	Objective
Determine the fulfillment level or the target level of attributes	Chaudhuri and Bhattacharyya (2009)	Integer programming	Determine the improvement levels of the attributes in technical characteristics
	Alptekin and Karsak (2011)	Linear programming	Determine the target values for e-learning product characteristics
	Geng, Chu, Xue, and Zhang (2011)	Non-linear programming	Maximize the fulfillment levels of PSS engineering characteristics
	Kahraman, Ertay, and Büyüközkan (2006)	Mixed integer linear programming	Optimize the improvement levels of product technical requirements
	Delice and Güngör (2009)	Mixed integer linear programming	Determine the fulfillment levels of the continuous design requirements
	Chen, Su, Tu, Lin, and Chang (2016), Chen and Weng (2006)	Fuzzy goal programing model	Determine the fulfillment levels of design requirements
	Chen, Ko, and Yeh (2017)	Additive fuzzy goal programing model	Determine and modify the fulfillment levels of design requirements
	Miao, Liu, Chen, Zhou, and Ji (2017)	Two uncertain chance-constrained programming (CCP) models	Determine the target levels of design attributes
	He, Song, Wu, Xu, Zheng, and Ming (2017)	Nonlinear programming model considering KANO	Determine the target levels of engineering characteristics
Select the optimal attributes or optimize attributes from a set of feasible solutions	Luo, Kwong, and Tang (2010)	An integrated optimization model	Achieve the optimal target settings of engineering characteristics
	Wang and Chen (2012)	Linear integer programming	Select optimal module mix
	Karsak, Sozer, and Alptekin (2003)	0–1 goal programming	Determine the product technical requirements which will be taken into account in the design phase
	Chowdhury and Quaddus (2016)	0–1 nonlinear optimization model	Find the optimal portfolio of service strategies
	Lee, Kang, Yang, and Lin (2010)	Multi-choice 0-1 goal programming model	Select the most suitable engineering characteristics to be considered in the product development
	Raharjo, Xie, and Brombacher (2006)	0–1 goal programming	Select quality characteristics
	Song and Chan (2015)	Multi-objective optimization model	Configure the product-extension service and find the optimized service characteristics from a large number of feasible solutions
	Delice and Güngör (2011), Delice and Güngör (2013)	Mixed integer goal programming	Optimize alternative values for the design requirements of a product

Table 1 shows the comprehensive review of the related literature, and the existing optimization models for DRs usually have the goal of maximizing customer satisfaction or minimizing deviations from customer satisfaction. For the type of determining the fulfillment levels of DRs, customer satisfaction is always represented by the utility function contributed from the fulfillment levels of DRs and the importance of DRs. For the type of selecting the optimal specifications, the deviations from customer satisfaction are usually defined by customer satisfaction or dissatisfaction coefficients, which are determined using subjective approaches, e.g. KANO model. The purpose of this paper is to find the optimal specifications of PSS DRs to satisfy the personalized customer requirements. How to measure the satisfaction degree of the optional DR specification is a difficult task in the PSS planning process. The application of IA provides us with a new idea to construct the objective function for minimizing the information content.

2.3 Information Axiom

AD is a methodology about how to describe design objects and use fundamental principles to evaluate relations between expected functions and means. In the literature review, there are various studies related to the usage of IA to determine the most appropriate alternative. Suh et al. [62] presented IA to solve the criteria with crisp design ranges or random system range. The traditional IA can help to evaluate decision-making problems with quantitative criteria [63]. In order to capture the impreciseness and vagueness in decision-making situations, fuzzy information axiom has been paid much attention due to its extensive application. Kulak et al. [64] proposed FIA for selecting transportation companies, in which both the system range and design range were considered as fuzzy variables. Kulak et al. [65] adopted a multi-attribute IA approach including both objective and fuzzy criteria and applied it to the equipment selection problem. Celik et al. [66] used FIA to investigate a systematic evaluation model on docking facilities in the shipbuilding industry. Akay et al. [67] presented interval-type-2 FIA, a new methodology for solving conceptual selection problems, which extended the FIA to incorporate interval type-2 fuzzy sets. Chen et al. [68] developed a decision-making approach based on IA under hybrid uncertain environments and the design range and system range were expressed as the fuzzy variable and the random variable respectively to model the hybrid uncertainties. Celik et al. [69] utilized FIA and fuzzy techniques to confront the decision-making problems with incomplete information by using the similarity between order performance and ideal solution (TOPSIS). Goren et al. [70] proposed a fuzzy axiomatic design approach with risk factors and provided a new multi-attribute decision-making tool. Kahraman et al. [71] used intuitive fuzzy sets to extend IA to fuzzy environments in an attempt to select search algorithms. Few studies consider the hesitation in determining the design range and common range for IA, and investigates the combination of HFLTSs and IA. In this paper, IA is extended to the hesitant fuzzy environment to deal with the decision-makers hesitation in evaluating DRs’ specifications.

3 The framework of PSS design requirements configuration and HFLTSs related theory

In this paper, PSS design requirements configuration consists of the process of translating CRs into DRs and the process of selecting appropriate DRs’ specifications. The objective of this research is to configure PSS design requirements to maximize customer satisfaction within the cost constraint. Figure 1 shows the framework of DRs configuration.

Fig. 1

The framework of PSS design requirements configuration.

First, acquisition of customer requirements is initial and critical step when obtaining the fundamental information on PSS planning in QFD. There are several methods to identify requirements, expectations and complaints of customers, for example customer panels, focused group discussions, in-depth customer observation and complaint and compliment database. The list of customer demand is obtained through a survey questionnaire and then focused on group discussions and brainstorming organized in the client company. In addition, selecting a small customer group implement in-depth customer observation, which is combined with the complaint and compliment in history. After that, the initial CRs list is obtained. And the importance of them is obtained by HFLTSs related methods. The expert group will make decisions on DRs that are more strongly associated with CRs based on their historical experience and product design database.

Second, hesitant QFD which introduces HFLTSs in the QFD approach is proposed and used to determine the importance of DRs. The importance degrees of DRs are important inputs of the configuration model.

Third, a non-linear programming model is established to select the optimal DRs’ configurations. The total information content of all design requirements is calculated as the objective function. HIA is put forward to compute the information contents of the qualitative DRs.

3.1 Concept and basic operations of HFLTSs

In many real situations of decision-making, the use of linguistic information is more reasonable and appropriate to reflect the uncertainty and ambiguity due to the nature of human habits. One common approach to deal with such problems is the fuzzy linguistic approach. However, the fuzzy linguistic approach cannot cope with the situation when decision-makers hesitant about several linguistic values. HFLTSs allow decision-makers to use different expressions freely. To solve decision-makers’ hesitation in obtaining the CRs importance and the relationships between CRs and DRs in PSS planning, HFLTSs are used in QFD. The related concepts of HFLTSs are introduced in this section.

Definition 1 [26]. Let S = {s₀, s₁ … , s_g} be a linguistic term set, an HFLTS, H_S, is an ordered finite subset of the consecutive linguistic terms of S.

The empty HFLTS and the full HFLTS for a linguistic variable are defined as follows:

The empty HFLTS: H_S = {},

The full HFLTS: H_S = S.

Example 1. Let S be a linguistic term set, S = {s₀ : nothing, s₁: very low, s₂: low, s₃: medium, s₄: high, s₅: very high, s₆: perfect}. Two experts give different HFLTSs aiming at the same evaluation object: $H_{S}^{1} = {s_{2}, s_{3}, s_{4}}, H_{S}^{2} = {s_{2}, s_{3}} .$

Definition 2 [26]. The upper bound, $H_{S^{+}}$ , and lower bound, $H_{S^{-}}$ , of the HFLTSs, HS, are defined as: $H_{S^{+}} = max (s_{i}) = s_{j}, s_{i} \in H_{S} and \forall i, s_{i} ⩽ s_{j}$ $H_{S^{-}} = min (s_{i}) = s_{j}, s_{i} \in H_{S} and \forall i, s_{i} ⩾ s_{j}$

Definition 3 [26]. The envelope of the HFLTSs, env(H_S), is a linguistic interval whose limits are obtained using upper bound (max) and lower bound (min), hence:

$env (H_{S}) = [H_{S^{-}}, H_{S^{+}}], H_{S^{-}} ⩽ H_{S^{+}}$ (1)

Example 2. Let S be a linguistic term set, S = {s₀: nothing, s₁: very low, s₂: low, s₃: medium, s₄: high, s₅: very high, s₆: perfect}, and H_S = {s₂, s₃, s₄} be an HFLTSs of S, its envelope is: $H_{S^{-}} (s_{2} : low, s_{3} : medium, s_{4} : high) = low$ $H_{S^{+}} (s_{2} : low, s_{3} : medium, s_{4} : high) = high$ $env = (H_{S}) = [low, high] = [s_{2}, s_{4}]$

HFLTSs allow experts to use different expressions freely based on the fuzzy linguistic approach and context-free grammar. Context-free grammar can help experts to generate comparative linguistic expressions. It contains unary relation (lower than, greater than, at least, at most), binary relation (between, or), and conjunction relation (and). Let E_GH be a function that transforms linguistic expressions obtained by the context-free grammar. The comparative linguistic expressions can be converted into HFLTSs using the following:

E_GH (s_i) = {s_i|s_i ∈ S};

E_GH (at most s_i) = {s_j|s_j ∈ S and s_j ≤ s_i};

E_GH (lower than s_i) = {s_j|s_j ∈ S and s_j < s_i};

E_GH (at least s_i) = {s_j|s_j ∈ S and s_j ≥ s_i};

E_GH (greater than s_i) = {s_j|s_j ∈ S and s_j > s_i};

E_GH (between s_i and s_j) = {s_k|s_k ∈ Sand s_i ≤ s_j ≤ s_j}.

E_GH (s_iors_j … s_k) = {s_i, s_j, …, s_k}.

Considering that HFLTSs have different lengths and hard to be operated, a new approach based on the min-upper operator and max-lower operator is proposed in this paper to aggregate multiple HFLTSs directly to simplify the operation. The min-upper operator and the max-lower operator [26] are used to find a balance between both pessimistic approximation and the optimistic approximation. The min-upper operator obtains the worst of the maximum linguistic terms and the max-lower operator obtains the best of the minimum linguistic terms. The two operators assemble the experts’ evaluation results into a linguistic interval which can represent the core information.

Let V = {ϑ_i|1 ⩽ i ⩽ v} be a set of objects, E = {E_k|1 ⩽ k ⩽ l} be a group of experts, S = {s₀, s₁ … , s_g} be a linguistic term set and ${H_{S}^{k} (ϑ_{i}) i = 1, 2, \dots v, k = 1, 2, \dots l}$ be a set of HFLTSs given by E_k on ϑ_i. $H_{S^{+}} (ϑ_{i}) = {H_{S^{+}}^{k} (ϑ_{i}) k = 1, 2, \dots, l, i = 1, 2, \dots, v}$ and $H_{S^{-}} (ϑ_{i}) = {H_{S^{-}}^{k} (ϑ_{i}) k = 1, 2, \dots, l, i = 1, 2, \dots, v}$ is the upper bound and lower bound respectively for each HFLTSs. The aggregation result of evaluations on ϑ_i, i.e., $H_{S}^{k} (ϑ_{i}) (k = 1, 2, \dots l)$ can be obtained by the min-upper and the max-lower operator, as follows:

$\begin{matrix} H (J_{i}) = [min {min {H_{S^{+}} (J_{i})}, max {H_{S^{-}} (J_{i})}}, \\ max {min {H_{S^{+}} (J_{i})}, max {H_{S^{-}} (J_{i})}}] \\ = [s_{i}, s_{j}], i ⩽ j \end{matrix}$ (2)

3.2 Computing with HFLTSs using the 2-tuple fuzzy linguistic representation model

The 2-tuple linguistic representation model proposed by Herrera et al. [72] is a technique to compute with words, which deals with linguistic information by introducing a new parameter called symbolic translation. This computational technique can compute with words without loss of information. The concept and basic operations of the 2-tuple fuzzy linguistic representation model are as follows.

Definition 4 [72]. Let S = {s₀, s₁ … , s_g} be a linguistic term set, and β ∈ [0, g] be a value representing the result of a symbolic aggregation operation, then the 2-tuple that expresses the equivalent information to β is obtained with the following function: $Δ [0, g] \to S \times [- 0.5, 0.5)$ $Δ (β) = (s_{i}, α), with {\begin{matrix} s_{i}, i = round (β) \\ α = β - i, α \in [- 0.5, 0.5) \end{matrix}$

where round(·) is the usual round operation, s_i has the closest index label to “β”, and “α” is the value of the symbolic translation.

It is noteworthy that Δ has an inverse function with Δ^-1, Δ^-1 : S × [0.5, - 0.5) → [0, g] and Δ^-1 (s_i, α) = i + α = β. In this way, the 2-tuple returns its equivalent numerical value β ∈ [0, g] and the calculation step is carried out accurately. To accomplish the process of computing with words, the 2-tuple linguistic representation model is used to transfer linguistic intervals into interval-valued 2-tuples as follows: [s_i, s_j] ⇒ [(s_i, 0) , (s_j, 0)] , s_i, s_j ∈ S, i ⩽ j.

An interval-valued 2-tuple is composed of two linguistic terms and two crisp numbers, denoted by [(s_i, α₁) , (s_j, α₂)], where i ⩽ j and α₁ ⩽ α₂. The interval-valued 2-tuple linguistic variable can be converted into an interval value by the following function [73].

$\begin{matrix} Δ^{- 1} ([(s_{i}, α_{1}), (s_{j}, α_{2})]) \\ = [i / g + α_{1}, j / g + α_{2}] \\ = [β_{1}, β_{2}], β_{1}, β_{2} \in [0, 1], β_{1} ⩽ β_{2} \end{matrix}$ (3)

Once we obtain the linguistic intervals by the min-upper operator and the max-lower operator, the conversion between linguistic intervals and numeric intervals can be realized by Eq. (3).

4 Determining the importance of PSS design requirements

PSS planning is a process translating CRs into PSS DRs which contain product DRs and service DRs. The first step is obtaining CRs and evaluating CRs’ importance. The second step is identifying DRs and computing DRs’ importance considering the relationships between CRs and DRs. HFLTSs and its related aggregation approaches are used to cope with the uncertainty and hesitation existed in the above process.

4.1 Compute the importance of CRs

Through the process of customer requirement acquisition, a total number of m CRs is obtained by customer investigations and C_i (i = 1, 2, …, m) is the i-th customer requirement. The importance degrees of CRs are evaluated by each expert E_k (k = 1, 2, …, t) who has rich experience in the QFD team. The semantic evaluation information is modeled by HFLTSs. The steps of determining CRs’ importance are as follows.

Step 1: Experts estimate CRs’ importance according to a given linguistics term set S, and the evaluation information is expressed by HFLTSs. Aggregate all the experts’ evaluation information into linguistic intervals using the min-upper operator and max-lower operator by Eq. (2). The linguistic interval $[s_{p}^{i}, s_{q}^{i}] (p ⩽ q)$ indicates the aggregated result for the importance degree of C_i.

Step 2: Transfer the linguistic interval $[s_{p}^{i}, s_{q}^{i}]$ into the interval-valued 2-tuple $[(s_{p}^{i}, α_{p}^{i}), (s_{q}^{i}, α_{q}^{i})]$ .

The interval-valued 2-tuples can be converted into a set of numeric intervals $β_{i} = [β_{1}^{i}, β_{2}^{i}]$ by the function Δ^-1 using Eq. (3).

Step 3: β_i is still an interval number which is hard to be compared. In response to this problem, the preference degree between interval numbers is introduced to transform the fuzzy importance degree into a crisp importance degree which indicates the relative importance degree of CR.

Definition 5 [74]. Let A = [a₁, a₂] and B = [b₁, b₂] be two interval values, the preference degree of A over B (or A > B) is defined as:

$P (A > B) = \frac{max (0, a_{2} - b_{1}) - max (0, a_{1} - b_{2})}{(a_{2} - a_{1}) + (b_{2} - b_{1})}$ (4)

And the preference degree of B over A (or B > A) is defined as:

$P (B > A) = \frac{max (0, b_{2} - a_{1}) - max (0, b_{1} - a_{2})}{(a_{2} - a_{1}) + (b_{2} - b_{1})}$ (5)

Apparently, P(A > B)+P(B > A)=1 and P(A > B)=P(B > A)=0.5 when A = B, i.e., a₁ = b₁ and a₂ = b₂.

For a vector of interval values $a_{i} = [a_{i}^{-}, a_{i}^{+}]$ , i = 1, 2, …, e. The relative importance u_i of each alternative can be computed as follows [75].

$u_{i} = \frac{\sum_{i = 1}^{e} P (a_{i} > a_{j}) + \frac{e}{2} - 1}{e (e - 1)}, i = 1, 2, \dots, e$ (6)

Once the fuzzy importance degrees of CRs are obtained from Step 1-3, we can calculate the relative importance cw_i (i = 1, 2, …, m) of C_i.

${cw}_{i} = \frac{\sum_{j = 1}^{m} P (β_{i} > β_{j}) + \frac{m}{2} - 1}{m (m - 1)}, i, j = 1, 2, \dots, m$ (7)

4.2 Compute the importance of DRs

The experts identify n DRs, which have a strong associated with CRs, according to their professional experience and product design database. And F_j (j = 1, 2, …, n) is the j-th DR. The importance degrees of DRs can be computed depending on the importance degrees of CRs and the relationships between CRs and DRs. The relationships between C_i and F_j evaluated by E_k (k = 1, 2, …, t) is denoted as ${\tilde{r}}_{ij}^{k}$ , which is also expressed by HFLTSs. The quantization process of ${\tilde{r}}_{ij}^{k}$ is the same as that of CRs’ importance. We can transform the relationships between CRs and DRs to interval values. The interval-valued assessment results establish the fuzzy relationship matrix $\tilde{R}$ , which is given as $\tilde{R} = [{\tilde{r}}_{ij}] = (\begin{matrix} [r_{11}, t_{11}] & [r_{12}, t_{12}] & \dots & [r_{1 n}, t_{1 n}] \\ [r_{21}, t_{21}] & [r_{22}, t_{22}] & \dots & [r_{2 n}, t_{2 n}] \\ ⋮ & ⋮ & ⋮ & ⋮ \\ [r_{m 1}, t_{m 1}] & [r_{m 2}, t_{m 2}] & \dots & [r_{mn}, t_{mn}] \end{matrix})$

According to the importance cw_i (i = 1, 2, …, m) and relationship matrix $\tilde{R}$ , the DR’s importance degree ${\tilde{w}}_{j}$ can be computed by the following equation.

${\tilde{w}}_{j} = \sum_{i = 1}^{m} {cw}_{i} \cdot {\tilde{r}}_{ij}, j = 1, 2, \dots, n$ (8)

${\tilde{w}}_{j}$ is an interval value and can be transformed into crisp relative importance by Eq. (7). The importance degrees of DRs are the important input of the configuration model for PSS design requirements.

5 Optimizing the PSS design requirements based on hesitant information axiom and non-linear programming

5.1 Determine the information content of DR using hesitant information axiom

5.1.1 The concept of fuzzy information axiom

Information axiom indicates that the best design scheme has the smallest information content. Information content is determined by the probability of satisfying the design goals. Assume the probability of success for a given DR is p, then the information content I is - log ₂p, where p is determined by the design range and the system range. Design range is the expected design goal, which usually has an upper value and a lower value. The system range indicates the distribution of the real system. The intersection of the design range and the system range is called the common range. The information axiom also can be expressed as

$I = {log}_{2} {\frac{system range}{common range}}$ (9)

For qualitative DRs in PSS planning, evaluating the design range and system range is a difficult problem. Fuzzy information axiom has been proposed to cope with this problem. The area enclosed by the membership function curve of the design range is the fuzzy design range, and the area enclosed by the membership function curve of the system range is the fuzzy system range. The area enclosed by their intersection is the fuzzy common range. For example, if the membership function belongs to a triangular distribution, Fig. 2 shows the corresponding fuzzy design range, fuzzy system range, and fuzzy common range. Accordingly, the fuzzy information content can be calculated as follows.

$I = {log}_{2} {\frac{fuzzy system range}{fuzzy common range}}$ (10)

Fig. 2

Fuzzy design range, fuzzy common range and fuzzy system range of triangular membership function.

For a certain scheme, it is ideal for a DR when its evaluation value coincides with the target value. In this case, the fuzzy system range coincides with its fuzzy design range, and the information content is zero. When the evaluation value and the target value do not coincide, the information content is infinite, and the scheme is not ideal for the DR.

The equation of computing information content showed in Eq. (10) has limitations in certain conditions. An example is taken to illustrate this. For a benefit-type DR, suppose its design range is 50–80. One of its optional specifications is 60, then the system range is 50–60 and the public range is 50–60. In this case, the public range is equal to the system range, and information content is zero. Suppose another optional specification is 70, then its system range is 50 70 and the common range is also 50 70. The information content in this case also equals zero. However, the latter optional specification (70) is better than the former (60). Therefore, we need to make some modifications to Eq. (10), and the revised fuzzy information axiom can be expressed as follows.

$I = {log}_{2} {\frac{fuzzy design range}{fuzzy common range}}$ (11)

5.1.2 Hesitant information axiom for calculating the information contents of qualitative DRs

In the traditional fuzzy information axiom, experts always use one linguistic value to evaluate the design goal (design range) and actual level (system range) concerning a certain DR. However, experts may hesitant from selecting appropriate values in a given linguistic set. In this paper, HFLTSs is combined with information axiom for the first time, which has the advantage to cope with the above situation. The design range of DRs and the system range of each optional specification are all obtained and expressed using HFLTSs. According to Section 3, the hesitant fuzzy information can be translated into interval numbers finally by means of the min-upper and max-lower operators and 2-tuple linguistic representation model related operators. That is to say, the calculation of HIA is an operation with numeric interval values.

DRs can be divided into the benefit-type (the bigger, the better), the cost-type (the smaller, the better), the interval-type (the optimal value is an interval number), and the target-type (the optimal value is a determined number). Different DR types should be calculated using different calculation methods. Assume that the evaluation value of a certain qualitative DR’s optional specification expressed in the numeric interval be a = [a^l, a^u], the different methods for computing the interval information content are given as follows.

(1) For the benefit-type DRs

Let the maximum value among all optional specifications be X^max and the minimum value be X^min, the acceptable range, i.e., the design range of the DR be L = X^max- X^min. The common range equals to (a^l + a^u - 2X^min)/2, and the information content is calculated as follows.

$I = {log}_{2} (\frac{2 L}{a^{l} + a^{u} - 2 X^{min}})$ (12)

(2) For the cost-type DRs

Let the maximum value among all optional specifications be X^max and the minimum value be X^min, the acceptable range, i.e., the design range of the DR be L = X^max- X^min. The common range equals to (2X^max–a^l–a^u)/2, and the information content is calculated as follows.

$I = {log}_{2} (\frac{2 L}{2 X^{max} - (a^{l} + a^{u})})$ (13)

(3) For the interval-type DRs

Let the expected value of an interval-type DR be [X^max, X^min]. The common range is the intersection of the optimal interval and the evaluation value and can be expressed as [X^min, X^max] ∩ [a^l, a^u]. The information content is calculated as follows.

$I = {log}_{2} (\frac{L}{[X^{min}, X^{max}] \cap [a^{l}, a^{u}]})$ (14)

(4) For the target-type DRs

The expected value of a target-type DR is a certain numerical value. Assume the optimal expected value be X^∘pt, the acceptable range, i.e., the design range of the DR be L = X^max- X^min and the common range be L - (|X^opt - X^l| + |X^u - X^opt|). The information content is calculated as follows.

$I = {log}_{2} (\frac{L}{L - (| X^{opt} - a^{l} | + | a^{u} - X^{opt} |)})$ (15)

5.2 Minimizing the total information content of DRs using the non-linear programming

In the process of PSS design requirements optimization, the total information content of all DRs is taken into account as the objective to measure customer satisfaction. Besides, the cost constraint is considered because of the limited company resources. The design ranges of DRs and DRs’ optional specifications are given by the experts with rich experience and knowledge. A 0-1 non-linear programming model is established to minimize the total information content.

Assume DR F_j (1 ⩽ j ⩽ n) has t optional specifications, where n indicates there are n different DRs. $I_{j}^{p}$ (1 ⩽ p ⩽ t) indicates the information content of p-th specification of F_j. w_j (1 ⩽ j ⩽ n) indicates the crisp relative importance degree of F_j calculated in Section 4. $D_{j}^{p}$ indicates the cost of the p-th specification of F_j. B is the cost constraint. $X_{j}^{p}$ is a binary decision variable. When $X_{j}^{p} = 1$ , the p-th specification of F_j is selected. Conversely, when $X_{j}^{p} = 0$ , the p-th specification of F_j is not selected. Then the DRs configuration model by minimizing the information content is established as follows.

$\begin{matrix} min y_{s} = \sum_{j} \sum_{p} wj (I_{j}^{p} X_{j}^{p}) \\ s . t . \sum_{p} X_{j}^{p} = 1, for \forall j \\ \sum_{j} \sum_{p} D_{j}^{p} X_{j}^{p} ⩽ B \\ X_{j}^{p} \in {0, 1} \end{matrix}$ (16)

There exist compensation relationships among DRs. The total information content for different compensation levels can be calculated using the method of imprecision, which considers the relationships among DRs to improve the reliability and accuracy of the results. The 0-1 non-linear programming model of DRs configuration can be expressed as follows. $\begin{matrix} \begin{matrix} min y_{s} = {[\sum_{j} \sum_{p} wj {(I_{j}^{p} X_{j}^{p})}^{s} / \sum_{j} w_{j}]}^{1 / s}, \\ s \in (- \infty, + \infty) \end{matrix} \end{matrix}$

$\begin{matrix} s . t . \sum_{p} X_{j}^{p} = 1, for \forall j \\ \sum_{j} \sum_{p} D_{j}^{p} X_{j}^{p} ⩽ B \\ X_{j}^{p} \in {0, 1} \end{matrix}$ (17)

Where s indicates the compensation factor among DRs. When s = -∞, the level of compensation is the lowest, i.e., there is no compensation relationship among DRs. In this case, the total information content is limited by the DR having the lowest performance. When s = 0, there exist complete compensation relationships among DRs. The DR with higher performance has a compensation relationship with the lower DR. In this case, the total information content is the average of all DR’s information contents. When s = 1, the total content is the weighted average of all DRs’ information contents. When s =+ ∞, the total information is determined by the DR with the highest performance. According to the method of imprecision, the optimization objectives can be expressed as follows.

$y_{s} = max_{j} (I_{j}^{p} X_{j}^{p}), if s = - \infty$ (18)

$y_{s} = (\prod_{j} I_{j}^{p} X_{j}^{p})^{1 / \sum_{j} w_{j}}, if s = 0$ (19)

$y_{s} = \sum_{j} \sum_{p} w_{j} (I_{j}^{p} X_{j}^{p}) / \sum_{j} w_{j}, if s = 1$ (20)

$y_{s} = min_{j} (I_{j}^{p} X_{j}^{p}), if s = + \infty$ (21)

6 A case study

Taking loader-PSS configuration optimization as an example, the feasibility and applicability of the proposed approach are illustrated. The loader is a kind of construction machinery widely used in highway, railway, and other construction projects, as shown in Fig. 3. As one of the world’s top 500 companies, company W manufactures world-class loaders and provides services to its products in the Asia Pacific region. With the increase in customer requirements, company W pays attention not only to the products, but also the lifecycle services. Product and services should be planned in an integrated way in the conceptual design phase. In this section, the proposed approach is applied to the loader PSS configuration optimization.

Fig. 3

A picture of the loader.

W company’s product user groups include loader drivers, special equipment maintenance personnel, on-site safety officers and engineering project leaders. Through research with such user groups, the initial customer requirements can be collected from multiple perspectives, and then the major CRs can be obtained by cluster analysis. The initial identified CRs are quick to start, easy to start, fast acceleration, stabile acceleration, flexible gear shift, high climbing ability, flexible turn, low failure rate, high mechanical properties stability, high driving stability, long life of components, easy updating of components, easy repairing of component, fewer exhaust emissions, comfortable usability, high-qualified after-sale service and so on. The major CRs obtained by the K-J method and cluster analysis include Strong motivation (C₁), High flexibility and efficiency (C₂), Easy to control (C₃), High stability and reliability (C₄), Timely service response (C₅), and Low energy consumption (C₆).

This company uses the modular design method to develop solutions, and product and service modules are predefined. Through analyzing the customer requirements, DRs of the predefined modules are decided by experts. The relevant DRs obtained in the QFD include Engine’s rated power (F₁), Maximum breakout force (F₂), Buckets capacity (F₃), Maximum unloading height (F₄), Total cycling time (F₅), Minimum turning radius (F₆), Variable speed controlling modes (F₇), Preventive maintenance strategies (F₈), Service response model (F₉), Service efficiency (F₁₀), Energy-saving programs (F₁₁) and Exhaust emission standards (F₁₂).

6.1 Calculating the importance of CRs

In order to evaluate the importance of CR, a five-member expert group was established, consisting of a senior manager, a project controller, a product manager, a mechanical design engineer and a user experience designer. The education and professional experience of the experts are shown in Table 2.

Table 2
The education and professional experience of the experts

Experts Expert 1 Expert 2 Expert 3 Expert 4 Expert 5

Gender Male Female Female Male Male

Education Graduate Postgraduate Graduate Graduate Postgraduate

Work experience 11 years 9 years 12 years 15 years 17 years

Position Senior manager Project controller Product manager Mechanical design engineer Operations director

Experts	Expert 1	Expert 2	Expert 3	Expert 4	Expert 5
Gender	Male	Female	Female	Male	Male
Education	Graduate	Postgraduate	Graduate	Graduate	Postgraduate
Work experience	11 years	9 years	12 years	15 years	17 years
Position	Senior manager	Project controller	Product manager	Mechanical design engineer	Operations director

And the linguistic term set S is {s₀: nothing, s₁: very low, s₂: low, s₃: medium, s₄: high, s₅: very high, s₆: perfect}. The evaluation information given by the five experts is listed in Table 3. The aggregated importance of the six CRs is obtained by using Eq. (2). They are {[s₂, s₃], [s₄, s₅], [s₁, s₃], [s₂, s₆], [s₂, s₄], [s₂, s₅]}. Then the 2-tuple linguistic representation model related approaches are used to transfer the linguistic intervals into the interval-valued 2-tuple. Finally, the numeric interval values are derived by the function Δ^-1 with Eq. (3). The final result of the CRs’ importance is shown in Table 4.

Table 3

Evaluation values of CRs given by E_k (k = 1, 2, 3, 4, 5)

	C₁	C₂	C₃	C₄	C₅	C₆
E₁	{s₁,s₂}	{s₅}	{s₃,s₄}	{s₄}	{s₂}	{s₄,s₅}
E₂	{s₃}	{s₄,s₅}	{s₁,s₂}	{s₃}	{s₂}	{s₃}
E₃	{s₂}	{s₃,s₄,s₅}	{s₁}	{s₄}	{s₃}	{s₂}
E₄	{s₂}	{s₄}	{s₃}	{s₁,s₂}	{s₄}	{s₃}
E₅	{s₃}	{s₄}	{s₃}	{s₆}	{s₃,s₄}	{s₅}

Table 4

Aggregation results of evaluation values and relative importance of CRs

	C₁	C₂	C₃	C₄	C₅	C₆
Linguistic intervals	[s₂,s₃]	[s₄,s₅]	[s₁,s₃]	[s₂,s₆]	[s₂,s₄]	[s₂,s₅]
Interval-valued 2-tuple	[(s₂,0),(s₃,0)]	[(s₄,0),(s₅,0)]	[(s₁,0),(s₃,0)]	[(s₂,0),(s₆,0)]	[(s₂,0),(s₄,0)]	[(s₂,0),(s₅,0)]
Numeric intervals	[0.33,0.50]	[0.67,0.83]	[0.17,0.50]	[0.33,1.00]	[0.33,0.67]	[0.33,0.83]
Importance of CRs	0.13	0.23	0.11	0.2	0.14	0.18

According to the aggregation results in Table 4, applying the formula of interval preference degree in Eqs. (4,7) to compute the relative importance of CRs. For instance, the importance of C₁ (cw₁) can be acquired as follows. $\begin{matrix} P (C_{1} > C_{1}) \\ = \frac{max (0, 0.5 - 0.33) - max (0, 0.33 - 0.5)}{(0.5 - 0.33) + (0.5 - 0.33)} = 0 . 5 \\ P (C_{1} > C_{2}) \\ = \frac{max (0, 0.5 - 067) - max (0, 0.33 - 0.83)}{(0.5 - 0.33) + (0.83 - 0.67)} = 0 \\ P (C_{1} > C_{3}) \\ = \frac{max (0, 0.5 - 0.17) - max (0, 0.33 - 0.5)}{(0.5 - 0.33) + (0.5 - 0.17)} = 0.66 \\ P (C_{1} > C_{4}) \\ = \frac{max (0, 0.5 - 0.33) - max (0, 0.33 - 1)}{(0.5 - 0.33) + (1 - 0.33)} = 0.21 \\ P (C_{1} > C_{5}) \\ = \frac{max (0, 0.5 - 0.33) - max (0, 0.33 - 0.67)}{(0.5 - 0.33) + (0.67 - 0.33)} \\ = 0.33 \\ P (C_{1} > C_{6}) \\ = \frac{max (0, 0.5 - 0.33) - max (0, 0.33 - 0.83)}{(0.5 - 0.33) + (0.83 - 0.33)} \\ = 0.25 \\ {cw}_{1} \\ = \frac{(0.5 + 0 + 0.66 + 0.21 + 0.33 + 0.25) + \frac{6}{2} - 1}{6 \times (6 - 1)} \\ = 0.13 \end{matrix}$

Similarly, the other relative importance of CRs can be derived. cw_i =(0.13, 0.23, 0.11, 0.20, 0.14, 0.18). The results are shown in the last column in Table 4.

6.2 Calculating the importance of DRs

In order to maximize customer satisfaction, the above-mentioned group of experts decided on 12 design requirements based on professional experience, which are more relevant to CRs. These DRs constructed by Engine rated power (F₁), Maximum breakout force (F₂), Bucket capacity (F₃), Maximum unloading height (F₄), Total cycling time (F₅), Minimum turning radius (F₆), Variable speed controlling modes (F₇), Preventive maintenance strategies (F₈), Service response time (F₉), Service efficiency (F₁₀), Energy-saving programs (F₁₁), and Exhaust emission standards (F₁₂).

The evaluation of relationships between CRs and DRs is a crucial process in QFD. Five experts’ evaluation information expressed by HFLTSs is listed in Tables 5 –9. HFLTSs information is handled in the same way as in Section 6.1. The group decision-making information is first aggregated with Eq.(2). Then, the linguistic intervals are transformed into the numeric interval values with Eq.(3) and they are shown in Table 10.

Table 5
Evaluation of relationships between CRs and DRs given by E₁

C₁ C₂ C₃ C₄ C₅ C₆

F₁ {s₁,s₂ } {s₃} {s₄} {s₄,s₅} {s₀} {s₆}

F₂ {s₃} {s₃} {s₃,s₄} {s₄} {s₂} {s₄}

F₃ {s₂} {s₄} {s₂} {s₃} {s₃} {s₅}

F₄ {s₄} {s₄,s₅,s₆} {s₂} {s₀} {s₆} {s₁}

F₅ {s₄} {s₂} {s₀} {s₃} {s₃} {s₆}

F₆ {s₁} {s₂} {s₃} {s₄} {s₃,s₄} {s₃}

F₇ {s₅} {s₃} {s₄} {s₃} {s₂} {s₃,s₄}

F₈ {s₃} {s₀} {s₄} {s₃,s₄,s₅} {s₃} {s₄}

F₉ {s₁} {s₃} {s₂} {s₁,s₂,s₃} {s₆} {s₃}

F₁₀ {s₅} {s₃} {s₅} {s₁} {s₃} {s₀}

F₁₁ {s₃} {s₄} {s₂} {s₁} {s₅} {s₃,s₄}

F₁₂ {s₄} {s₂} {s₀} {s₅} {s₂} {s₃}

	C₁	C₂	C₃	C₄	C₅	C₆
F₁	{s₁,s₂ }	{s₃}	{s₄}	{s₄,s₅}	{s₀}	{s₆}
F₂	{s₃}	{s₃}	{s₃,s₄}	{s₄}	{s₂}	{s₄}
F₃	{s₂}	{s₄}	{s₂}	{s₃}	{s₃}	{s₅}
F₄	{s₄}	{s₄,s₅,s₆}	{s₂}	{s₀}	{s₆}	{s₁}
F₅	{s₄}	{s₂}	{s₀}	{s₃}	{s₃}	{s₆}
F₆	{s₁}	{s₂}	{s₃}	{s₄}	{s₃,s₄}	{s₃}
F₇	{s₅}	{s₃}	{s₄}	{s₃}	{s₂}	{s₃,s₄}
F₈	{s₃}	{s₀}	{s₄}	{s₃,s₄,s₅}	{s₃}	{s₄}
F₉	{s₁}	{s₃}	{s₂}	{s₁,s₂,s₃}	{s₆}	{s₃}
F₁₀	{s₅}	{s₃}	{s₅}	{s₁}	{s₃}	{s₀}
F₁₁	{s₃}	{s₄}	{s₂}	{s₁}	{s₅}	{s₃,s₄}
F₁₂	{s₄}	{s₂}	{s₀}	{s₅}	{s₂}	{s₃}

Table 6

Evaluation of relationships between CRs and DRs given by E₂

	C₁	C₂	C₃	C₄	C₅	C₆
F₁	{s₂}	{s₂,s₃}	{s₄}	{s₅}	{s₃}	{s₄}
F₂	{s₃}	{s₄}	{s₄}	{s₆}	{s₄}	{s₂}
F₃	{s₀}	{s₂}	{s₂}	{s₃}	{s₅}	{s₄}
F₄	{s₄}	{s₃,s₄}	{s₄}	{s₀}	{s₄}	{s₂}
F₅	{s₄}	{s₂}	{s₂}	{s₃,s₄}	{s₃}	{s₅}
F₆	{s₁}	{s₃}	{s₃}	{s₃}	{s₂}	{s₂}
F₇	{s₄}	{s₃}	{s₂}	{s₃}	{s₃}	{s₅}
F₈	{s₃}	{s₄}	{s₂}	{s₂,s₃}	{s₁}	{s₄}
F₉	{s₂}	{s₃}	{s₂}	{s₂,s₃}	{s₄}	{s₃}
F₁₀	{s₄}	{s₃}	{s₅}	{s₃}	{s₄}	{s₂}
F₁₁	{s₀,s₁}	{s₂}	{s₃}	{s₅}	{s₅}	{s₂}
F₁₂	{s₄}	{s₅}	{s₃}	{s₂}	{s₀,s₁}	{s₄}

Table 7

Evaluation of relationships between CRs and DRs given by E₃

	C₁	C₂	C₃	C₄	C₅	C₆
F₁	{s₂,s₃}	{s₂,s₃}	{s₄}	{s₄}	{s₂}	{s₅}
F₂	{s₃,s₄}	{s₃}	{s₃,s₄}	{s₄}	{s₂}	{s₄}
F₃	{s₂}	{s₄}	{s₂}	{s₃}	{s₃}	{s₄,s₅}
F₄	{s₄,s₅}	{s₅,s₆}	{s₃}	{s₀,s₁}	{s₅}	{s₁}
F₅	{s₃}	{s₃}	{s₀}	{s₃}	{s₃}	{s₆}
F₆	{s₁,s₂}	{s₂}	{s₃}	{s₃}	{s₃,s₄}	{s₄}
F₇	{s₅}	{s₃,s₄}	{s₄}	{s₄}	{s₂}	{s₃,s₄}
F₈	{s₃}	{s₁}	{s₃}	{ s₄,s₅}	{s₂}	{s₄}
F₉	{s₂}	{s₃}	{s₂,s₃}	{ s₂,s₃}	{s₅}	{s₃}
F₁₀	{s₅}	{s₂,s₃}	{s₄}	{s₁}	{s₄}	{s₁}
F₁₁	{s₃}	{s₄}	{s₂}	{s₁}	{s₅}	{s₄}
F₁₂	{s₃}	{s₂,s₃}	{s₀}	{s₅}	{s₂}	{s₃}

Table 8

Evaluation of relationships between CRs and DRs given by E₄

	C₁	C₂	C₃	C₄	C₅	C₆
F₁	{s₃}	{s₃}	{s₂}	{s₅}	{s₂}	{s₄}
F₂	{s₃,s₄}	{s₂}	{s₂}	{s₃}	{s₂}	{s₂,s₃}
F₃	{s₁}	{s₂}	{s₂}	{s₄}	{s₃}	{s₅}
F₄	{s₄}	{s₄,s₅}	{s₃}	{s₃}	{s₅}	{s₂}
F₅	{s₅}	{s₃}	{s₂}	{s₃}	{s₃}	{s₅}
F₆	{s₁,s₂}	{s₃}	{s₄}	{s₃}	{s₂}	{s₃}
F₇	{s₆}	{s₅}	{s₁}	{s₅}	{s₃,s₄}	{s₂}
F₈	{s₁}	{s₃}	{s₄}	{s₁,s₂}	{s₂}	{s₄}
F₉	{s₃}	{s₃}	{s₁}	{s₂,s₃}	{s₅}	{s₄}
F₁₀	{s₄}	{s₂}	{s₄}	{s₃}	{s₄}	{s₃}
F₁₁	{s₂,s₃}	{s₄}	{s₃}	{s₃}	{s₅}	{s₅}
F₁₂	{s₅}	{s₀}	{s₃}	{s₅}	{s₂}	{s₅,s₆}

Table 9

Evaluation of relationships between CRs and DRs given by E₅

	C₁	C₂	C₃	C₄	C₅	C₆
F₁	{s₂}	{s₃, s₄}	{s₃, s₄}	{s₅}	{s₀}	{s₅}
F₂	{s₃}	{s₄}	{s₃}	{s₄}	{s₂, s₃}	{s₄}
F₃	{s₂}	{s₃, s₄}	{s₃}	{s₄}	{s₃}	{s₅}
F₄	{s₄}	{s₄, s₅}	{s₂}	{s₀}	{s₆}	{s₁, s₂}
F₅	{s₄}	{s₃}	{s₀}	{s₄}	{s₃}	{s₆}
F₆	{s₂}	{s₂}	{s₃}	{s₄}	{s₃}	{s₃}
F₇	{s₄, s₅}	{s₃}	{s₄}	{s₃}	{s₂, s₃}	{s₃, s₄}
F₈	{s₃}	{s₀}	{s₃, s₄}	{s₃, s₄}	{s₃}	{s₄}
F₉	{s₁}	{s₃}	{s₂}	{s₁, s₂}	{s₆}	{s₃}
F₁₀	{s₄, s₅}	{s₃}	{s₅}	{s₁}	{s₃}	{s₁}
F₁₁	{s₄}	{s₃}	{s₂}	{s₁}	{s₅}	{s₃, s₄}
F₁₂	{s₃, s₄}	{s₂}	{s₀, s₁}	{s₅}	{s₃}	{s₂}

Table 10

Fuzzy relationship matrix between CRs and DRs

	C₁	C₂	C₃	C₄	C₅	C₆
F₁	[0.33,0.50]	[0.50,0.50]	[0.33,0.33]	[0.67,0.83]	[0.00,0.50]	[0.67,1.00]
F₂	[0.50,0.50]	[0.33,0.67]	[0.33,0.67]	[0.50,1.00]	[0.33,0.67]	[0.33,0.67]
F₃	[0.00,0.33]	[0.33,0.67]	[0.33,0.50]	[0.50,0.67]	[0.50,0.83]	[0.67,0.83]
F₄	[0.67,0.67]	[0.67,0.83]	[0.33,0.67]	[0.00,0.50]	[0.67,1.00]	[0.17,0.33]
F₅	[0.50,0.83]	[0.33,0.50]	[0.00,0.33]	[0.50,0.67]	[0.50,0.50]	[0.83,1.00]
F₆	[0.17,0.33]	[0.33,0.50]	[0.50,0.67]	[0.50,0.67]	[0.33,0.50]	[0.33,0.67]
F₇	[0.67,1.00]	[0.50,0.83]	[0.17,0.67]	[0.33,0.50]	[0.33,0.50]	[0.33,0.87]
F₈	[0.17,0.50]	[0.00,0.67]	[0.33,0.67]	[0.67,1.00]	[0.17,0.50]	[0.67,0.67]
F₉	[0.17,0.50]	[0.50,0.50]	[0.17,0.33]	[0.33,0.33]	[0.67,1.00]	[0.50,0.67]
F₁₀	[0.67,0.83]	[0.33,0.50]	[0.67,0.83]	[0.17,0.50]	[0.50,0.67]	[0.00,0.50]
F₁₁	[0.17,0.33]	[0.33,0.67]	[0.33,0.50]	[0.17,0.83]	[0.83,0.83]	[0.33,0.83]
F₁₂	[0.50,0.83]	[0.00,0.83]	[0.00,0.50]	[0.33,0.83]	[0.33,0.50]	[0.33,0.83]

According to the importance of CRs and the relationship matrix between CRs and DRs, the fuzzy importance vector of F_j (j = 1,2,…,12) are computed in the QFD as ([0.45,0.63], [0.38,0.71], [0.40,0.65], [0.42,0.66], [0.46,0.64], [0.38,0.56], [0.39,0.72], [0.34,0.68], [0.41,0.54], [0.34,0.60], [0.34,0.68], [0.24,0.74]). By using the formula of interval preference degree in Eq. (4,7), the relative importance of DRs are obtained as w_j=(0.089,0.088,0.085,0.087,0.091,0.074,0.090,0.083, 0.075,0.076,0.083,0.080)^T, j = 1,2, …,12

6.3 Determining the information content with hesitant information axiom

Table 11 shows the optional specifications and costs of all DRs for the loader, the mark (+) and (–) indicates the benefit-type and cost-type DR respectively.

Table 11
Optional specifications and costs of DRs

DRs Specification/cost(RMB)

F₁(+) 92kW/30000 118kW/40000 130kW/45000

F₂(+) 70KN/21730 96KN/25520 125KN/28200 70KN/21730

F₃(+) 2.2 m³/6200 2.5 m³/8130 3.0 m³/10600

F₄(+) 2454mm/590 2800mm/7450 2950mm/9210 2454mm/590 2800mm/7450

F₅(–) 9s/9030 10s/8550 12s/7120

F₆(–) 5670mm/8250 6250mm/7600 6450mm/6820 6580mm/6540 6950mm/6100

F₇(+) mechanical control /8150 hydraulic control/10860 electro-hydraulic control /12200 automatic transmission/13590

F₈(+) strategy1/7600 strategy 2/9000 strategy 3/10720

F₉(–) Service response time 12h/12920 Service response time 24h/10730 Service response time 48h/9500

F₁₀(–) service completion time 1h/10400 service completion time 3h/8980 service completion time 6h/7310

F₁₁(+) project1/12240 project 2/13050 project 3/15300

F₁₂(+) GB 1/7820 GB 2/8350 GB 3/9100 GB 1/7820

DRs	Specification/cost(RMB)
F₁(+)	92kW/30000	118kW/40000	130kW/45000
F₂(+)	70KN/21730	96KN/25520	125KN/28200	70KN/21730
F₃(+)	2.2 m³/6200	2.5 m³/8130	3.0 m³/10600
F₄(+)	2454mm/590	2800mm/7450	2950mm/9210	2454mm/590	2800mm/7450
F₅(–)	9s/9030	10s/8550	12s/7120
F₆(–)	5670mm/8250	6250mm/7600	6450mm/6820	6580mm/6540	6950mm/6100
F₇(+)	mechanical control /8150	hydraulic control/10860	electro-hydraulic control /12200	automatic transmission/13590
F₈(+)	strategy1/7600	strategy 2/9000	strategy 3/10720
F₉(–)	Service response time 12h/12920	Service response time 24h/10730	Service response time 48h/9500
F₁₀(–)	service completion time 1h/10400	service completion time 3h/8980	service completion time 6h/7310
F₁₁(+)	project1/12240	project 2/13050	project 3/15300
F₁₂(+)	GB 1/7820	GB 2/8350	GB 3/9100	GB 1/7820

For quantitative DRs, the traditional IA is employed to compute the information content. The design ranges of quantitative DRs are given out by designers: Engine rated power (F₁) is 72kW∼130kW, Maximum breakout force (F₂) is 65kN∼150kN, Bucket capacity (F₃) is 2m³∼3m³, Maximum unloading height (F₄) is 2400mm∼3500mm, Total cycling time (F₅) is 7s∼13s, Minimum turning radius (F₆) is 5500mm∼7000mm, Service response time (F₉) is 9h∼48h, Service efficiency (F₁₀) is 0.5h∼6.5h. According to the above data and the information in Table 11, the computed results of the information content of the quantitative DRs are listed in Table 14.

Table 14

Information contents of DRs’ optional specifications

DRs	Specification/the information content
F₁(+)	$A_{1}^{1}$ /1.536	$A_{1}^{2}$ /0.334	$A_{1}^{3}$ /0
F₂(+)	$A_{2}^{1}$ /4.088	$A_{2}^{2}$ /1.456	$A_{2}^{3}$ /0.503	$A_{2}^{4}$ /0.000
F₃(+)	$A_{3}^{1}$ /2.322	$A_{3}^{2}$ /1.000	$A_{3}^{3}$ /0.000
F₄(+)	$A_{4}^{1}$ /4.348	$A_{4}^{2}$ /1.459	$A_{4}^{3}$ /1.000	$A_{4}^{4}$ /0.875	$A_{4}^{5}$ /0.325
F₅(–)	$A_{5}^{1}$ /0.585	$A_{5}^{2}$ /1.000	$A_{5}^{3}$ /2.585
F₆(–)	$A_{6}^{1}$ /0.174	$A_{6}^{2}$ /1.000	$A_{6}^{3}$ /1.448	$A_{6}^{4}$ /1.837	$A_{6}^{5}$ /4.907
F₇(+)	$A_{7}^{1}$ /1.414	$A_{7}^{2}$ /0.679	$A_{7}^{3}$ /2	$A_{7}^{4}$ /1.417
F₈(+)	$A_{8}^{1}$ /1.582	$A_{8}^{2}$ /2.582	$A_{8}^{3}$ /0.584
F₉(–)	$A_{9}^{1}$ /0.000	$A_{9}^{2}$ /0.553	$A_{9}^{3}$ /2.667
F₁₀(–)	$A_{10}^{1}$ /0.126	$A_{10}^{2}$ /0.778	$A_{10}^{3}$ /3.585
F₁₁(+)	$A_{11}^{1}$ /2.323	$A_{11}^{2}$ /1.323	$A_{11}^{3}$ /0.515
F₁₂(+)	$A_{12}^{1}$ /0.264	$A_{12}^{2}$ /1.587	$A_{12}^{3}$ /0.587	$A_{12}^{4}$ /0.586

For qualitative DRs, the information contents depend on the experts’ evaluation information. The linguistic term set S={s₀: nothing, s₁: very low, s₂: low, s₃: medium, s₄: high, s₅: very high, s₆: perfect}. Five experts’ evaluation information about Variable speed controlling modes (F₇), Preventive maintenance strategies (F₈), Energy-saving programs (F₁₁), and Exhaust emission standards (F₁₂) are given in Table 11 ( $A_{j}^{p}$ indicates the p-th specification of F_j).

We use the 2-tuple linguistic representation model related approaches to transfer the evaluation information in Table 12 into numeric interval values as shown in Table 13.

Table 12

Five experts’ HFLTSs evaluation values about optional specifications of qualitative DRs

Specifications		E₁	E₂	E₃	E₄	E₅
F₇	$A_{7}^{1}$	{s₂, s₃}	{s₁,s₂}	{s₃}	{s₄}	{s₂}
	$A_{7}^{2}$	{s₄}	{s₃}	{s₅}	{s₃}	{s₂}
	$A_{7}^{3}$	{s₁}	{s₂}	{s₁}	{s₁,s₂}	{s₃}
	$A_{7}^{4}$	{s₃}	{s₂}	{s₂}	{s₃}	{s₃}
F₈	$A_{8}^{1}$	{s₂,s₃}	{s₃}	{s₄}	{s₂}	{s₂}
	$A_{8}^{2}$	{s₂}	{s₃}	{s₃}	{s₃}	{s₂,s₃}
	$A_{8}^{3}$	{s₅}	{s₃}	{s₄}	{s₅}	{s₃,s₄}
F₁₁	$A_{11}^{1}$	{s₁,s₂}	{s₂}	{s₃}	{s₂,s₃}	{s₁}
	$A_{11}^{2}$	{s₂,s₃}	{s₄}	{s₃}	{s₂}	{s₃}
	$A_{11}^{3}$	{s₅}	{s₃}	{s₆}	{s₄}	{s₅}
F₁₂	$A_{12}^{1}$	{s₃}	{s₃}	{s₂}	{s₃}	{s₄}
	$A_{12}^{2}$	{s₁}	{s₂}	{s₃}	{s₂}	{s₁}
	$A_{12}^{3}$	{s₂,s₃}	{s₃}	{s₄}	{s₂}	{s₂}
	$A_{12}^{4}$	{s₃}	{s₄}	{s₂}	{s₄}	{s₃}

Table 13

Evaluation intervals about optional specifications of qualitative DRs

Specifications of DRs	F ₇				F ₈
	$A_{7}^{1}$	$A_{7}^{2}$	$A_{7}^{3}$	$A_{7}^{4}$	$A_{8}^{1}$	$A_{8}^{2}$	$A_{8}^{3}$
intervals	[0.167,0.667]	[0.333,0.833]	[0.167,0.5]	[0.333,0.5]	[0.333,0.667]	[0.333,0.5]	[0.5,0.833]
Specifications of DRs	F ₁₂	F ₁₁
	$A_{12}^{1}$	$A_{12}^{2}$	$A_{12}^{3}$	$A_{12}^{4}$	$A_{11}^{1}$	$A_{11}^{2}$	$A_{12}^{4}$
intervals	[0.5,0.667]	[0.167,0.5]	[0.333,0.667]	[0.333,0.667]	[0.167,0.5]	[0.333,0.667]	[0.333,0.667]

According to the evaluation results in Table 13, the information contents of the optional DR specifications can be calculated using HIA with Eqs. (12–15). The information contents are listed in Table 14. Take $A_{7}^{1}$ as an example, the system range of $A_{7}^{1}$ is (0.833-0.167)=0.666, the common range is (0.167 + 0.667-2×0.167)/2 = 0.25, and information content is log₂(0.666/0.25)=1.414.

6.4 DR configuration based on the non-linear programming model

According to three different customer clusters, the corresponding cost constraints are B₁ = 140,000 RMB, B₂ = 160,000 RMB, B₃ = 180,000 RMB. Assume the compensation factor s = 1, we can establish the non-linear programming model using Eq.(17). Lingo 12.0 is used to solve the model. The configuration results under different cost constraints indicate that DR tends to be configured with the better specifications with the increase of cost, especially those with lower information contents. Engine rated power (F₁), Maximum breakout force (F₂), Buckets capacity (F₃), and Preventive maintenance strategies (F₈) belong to this category. Their configured specifications become better as the cost increases. This result is consistent with the actual situation. Figure 4 shows the comparison of configuration results under different cost constraints.

Fig. 4

Comparison of configuration results under different cost constraints.

Assume the cost is 180,000 RMB, we can obtain the configuration results under different compensation factors, i.e., s=–2, s=–1, s = 1, s = 2. The results are listed in Table 15 and shown in Fig. 5.

Table 15

Configuration results under different compensation factors

s	F₁	F₂	F₃	F₄	F₅	F₆	F₇	F₈	F₉	F₁₀	F₁₁	F₁₂
s=–2	$A_{1}^{1}$	$A_{2}^{1}$	$A_{3}^{1}$	$A_{4}^{1}$	$A_{5}^{3}$	$A_{6}^{5}$	$A_{7}^{2}$	$A_{8}^{2}$	$A_{9}^{3}$	$A_{10}^{3}$	$A_{11}^{1}$	$A_{12}^{2}$
s=–1	$A_{1}^{1}$	$A_{2}^{1}$	$A_{3}^{1}$	$A_{4}^{1}$	$A_{5}^{3}$	$A_{6}^{5}$	$A_{7}^{3}$	$A_{8}^{2}$	$A_{9}^{3}$	$A_{10}^{3}$	$A_{11}^{1}$	$A_{12}^{2}$
s=1	$A_{1}^{3}$	$A_{2}^{4}$	$A_{3}^{3}$	$A_{4}^{5}$	$A_{5}^{1}$	$A_{6}^{1}$	$A_{7}^{2}$	$A_{8}^{3}$	$A_{9}^{1}$	$A_{10}^{1}$	$A_{11}^{3}$	$A_{12}^{1}$
s=2	$A_{1}^{1}$	$A_{2}^{2}$	$A_{3}^{2}$	$A_{4}^{5}$	$A_{5}^{1}$	$A_{6}^{1}$	$A_{7}^{1}$	$A_{8}^{1}$	$A_{9}^{2}$	$A_{10}^{1}$	$A_{11}^{2}$	$A_{12}^{1}$

They show that DRs with higher importance tend to be configured with the specifications having lower information content with the compensation factor increasing. DRs with lower importance tend to be configured with the specifications having higher information content. For example, as the compensation factor changes from –1 to 1, the configured specification of Total cycling time (F₅) changes from 12s with information content 2.585 to 9s with information content 0.585. It is consistent with the qualitative analysis results.

6.5 Comparison analysis

In this section, two kinds of comparison analysis will be performed and discussed, to validate the proposed QFD approach. First, the proposed programming model is tested by comparing with traditional programming model based on utility function, and the results are analyzed. Second, the proposed QFD method is compared with other existing QFD methods, and the ranking performances are discussed.

6.5.1 Comparison with traditional programing model based on the utility function

To further highlight the advantages of the proposed method, we will compare the proposed programming model based on HIA with the traditional programming model based on the utility function. Both of the objective functions are customer satisfaction. In the proposed approach, customer satisfaction is defined by information content. In the comparative approach, customer satisfaction is defined by the commonly used utility function, which is contributed by the importance of DRs via the DRs’ realization degrees [24 , 60]. Assume $Z_{j}^{p}$ indicates the value of p-th specification of DR_j, then the realization degree of benefit-type DR can be formulated as

$z_{j}^{p} = \frac{Z_{j}^{p} - min_{t} Z_{j}^{p}}{max_{t} Z_{j}^{p} - min_{t} Z_{j}^{p}}, j = 1, 2, \dots, n$ (22)

Fig. 5

Comparison of configuration results under different compensation levels.

The realization degree of cost-type DR can be formulated as

$z_{j}^{p} = \frac{max_{t} Z_{j}^{p} - Z_{j}^{p}}{max_{t} Z_{j}^{p} - min_{t} Z_{j}^{p}}, j = 1, 2, \dots, n$ (23)

The comparative objective function can be expressed as

$\begin{matrix} max y_{s} = (\sum_{j} \sum_{p} w_{j} (z_{j}^{p} X_{j}^{p})^{s} / \sum_{j} w_{j})^{1 / s}, \\ s \in (- \infty, + \infty) \end{matrix}$ (24)

For qualitative DRs, we use the proposed decision-making approach based on HFLTSs to evaluate the specifications’ performance. Take the specifications of F₁ for example, Table 16 shows the calculating process of estimating the performance using HFLTSs by ten experts. For the compensation factor s = 1 and the cost constraint is 160,000 RMB, the configuration results of the proposed method and the comparison method are listed in Table 17. According to the results of the comparative analysis in Table 17, Maximum breakout force (F₂) is configured with 150KN, Maximum unloading height (F₄) is configured with 3278 mm, Variable speed controlling modes (F₇) is configured with hydraulic control for the proposed approach. The configuration results are better than those obtained by the comparative approach. Comparative analysis shows that the proposed approach can configure DRs with higher performance under the same cost constraint. In other words, the proposed approach can configure PSS solutions with higher customer satisfaction. The result verifies the advantage and applicability of the proposed approach.

Table 16

The calculation process of estimating the performance of F₁

	E₁	E₂	…	E₉	E₁₀	The aggregation result	The interval number	The performance value
130kW	{s₀,s₁,}	{s₂, s₃}	…	{s₁,s₂,s₃}	{s₄, s₅}	[s₄, s₅]	[0.67,0.83]	0.833
118kW	{s₁}	{s₂, s₃}	…	{s₃,s₄,s₅}	{s₂, s₃}	[s₂, s₃]	[0.33,0.67]	0.389
92kW	{s₁, s₂}	{s₃, s₄}	…	{s₄, s₅}	{s₃}	[s₂, s₄]	[0.33,0.50]	0.278

Table 17

Configuration results of the proposed method and the comparison method

DRs	The proposed method	The comparison method	DRs	The proposed method	The comparison method
F₁	$A_{1}^{3}$	$A_{1}^{3}$	F₇	$A_{7}^{2}$	$A_{7}^{1}$
F₂	$A_{2}^{4}$	$A_{2}^{2}$	F₈	$A_{8}^{3}$	$A_{8}^{3}$
F₃	$A_{3}^{3}$	$A_{3}^{3}$	F₉	$A_{9}^{1}$	$A_{9}^{1}$
F₄	$A_{4}^{5}$	$A_{4}^{3}$	F₁₀	$A_{10}^{1}$	$A_{10}^{1}$
F₅	$A_{5}^{1}$	$A_{5}^{1}$	F₁₁	$A_{11}^{3}$	$A_{11}^{3}$
F₆	$A_{6}^{1}$	$A_{6}^{1}$	F₁₂	$A_{12}^{1}$	$A_{12}^{1}$

6.5.2 Comparison with other existing QFD method

Based on the case of loader-PSS configuration optimization, this section compares the proposed HFLTSs-QFD method with triangular fuzzy QFD (TF-QFD), rough QFD and intuitionistic fuzzy QFD (IF-QFD). The effectiveness and superiority of the proposed method are demonstrated in customer requirement acquisition and CRs-DRs correlation mapping by analyzing three aspects of result similarity, difference and possible reasons for the difference. The linguistic terms inputs of the four methods are shown in Table 18. The importance and ranking results of design requirements for HFLTSs-QFD, TF-QFD, and IF-QFD were obtained by the evaluation and method calculation of the expert group as shown in Table 19, and the results of Rough QFD calculation as shown in Table 20.

Table 18
Four method linguistic terms

Linguistic terms HFLTSs-QFD TF-QFD Rough QFD IF-QFD

Nothing s₀ (0,0,0.25) s₀ (0.1,0.3)

Very Low s₁ (0,0.25,0.5) s₁ (0.2,0.5)

Low s₂ (0.25,0.5,0.75) s₂ (0.4,0.6)

Medium s₃ (0.5,0.5,0.5) s₃ (0.5,0.4)

High s₄ (0.55,0.6,0.75) s₄ (0.6,0.2)

Very High s₅ (0.7,0.75,1) s₅ (0.7,0.2.5)

Perfect s₆ (0.75,1,1) s₆ (0.8,0.1)

Linguistic terms	HFLTSs-QFD	TF-QFD	Rough QFD	IF-QFD
Nothing	s₀	(0,0,0.25)	s₀	(0.1,0.3)
Very Low	s₁	(0,0.25,0.5)	s₁	(0.2,0.5)
Low	s₂	(0.25,0.5,0.75)	s₂	(0.4,0.6)
Medium	s₃	(0.5,0.5,0.5)	s₃	(0.5,0.4)
High	s₄	(0.55,0.6,0.75)	s₄	(0.6,0.2)
Very High	s₅	(0.7,0.75,1)	s₅	(0.7,0.2.5)
Perfect	s₆	(0.75,1,1)	s₆	(0.8,0.1)

Table 19

Three methods DRs’ importance and ranking

DRs	HFLTSs-QFD		TF-QFD		IF-QFD
	Importance	Ranking	Importance	Ranking	Importance	Ranking
F₁	0.089	3	0.092	3	0.196	2
F₂	0.088	4	0.087	4	0.073	4
F₃	0.085	6	0.083	6	0.066	5
F₄	0.087	5	0.086	5	0.063	6
F₅	0.09	1	0.131	1	0.203	1
F₆	0.074	12	0.041	12	0.008	12
F₇	0.09	2	0.112	2	0.185	3
F₈	0.083	7	0.082	7	0.051	8
F₉	0.075	11	0.063	11	0.017	11
F₁₀	0.076	10	0.071	10	0.047	9
F₁₁	0.083	8	0.078	8	0.05	7
F₁₂	0.08	9	0.074	9	0.041	10

Table 20

Rough QFD samples’ importance

	DRs	Importance
Category 1	F₅, F₇, F₁, F₂, F₆	0.4
Category 2	F₈, F₉, F₁₀	0.33
Category 3	F₁₁, F₁₂	0.15
Category 4	F₃, F₄	0.12

In terms of the input linguistic terms, five experts were invited to evaluate the importance of CRs according to Table 18. The inputs of HFLTSs consist of interval values, while the inputs of TF and IF are fuzzy numbers. The rough set input is either a scale or a score.

According to Table 19, the ranking result of the proposed method is consistent with the TF-QFD method, and different from the IF-QFD. Rough sets classify DRs according to the input linguistic terms and group similar indicators into one category. And the result of the rough set is the importance of each category.

According to Fig. 6, DRs with the same ranking result are obtained with significant different importance. In particular, F₅, F₁, F₇ ranked in the top three, respectively, and the importance obtained by IF-QFD calculation was higher than the other methods. Relatively, the 11^th and 12^th ranked F₆ and F₉ obtained lower importance, which is computed by IF-QFD. Intuitionistic fuzzy and triangular fuzzy can deal with the ambiguity and uncertainty in the decision-making

Fig. 6

Three methods DRs’ importance.

process of decision makers, but ignore parameters that can characterize the hesitancy of decision makers.

Through the comparison experiments of the four methods, the differences and advantages of the four QFD methods can be found. It also confirms that the HFLTSs-QFD method proposed in this paper is feasible and effective, and more in line with the actual decision-making scenarios.

7 Conclusions

Manufacturing companies are striving to increase customer satisfaction and market position, and begin to turn to PSS providing. This paper concentrates on the PSS conceptual design phase. The main contribution of this paper is carrying on research on the optimization of design requirements based on QFD in the PSS environment.

Although fuzzy QFD has been widely applied in literature, fuzzy QFD considering hesitance has not been concentrated in the PSS field. Given the advantage of HFLTSs in providing experts greater flexibility in eliciting linguistic preferences, hesitant QFD is used to determine the input information in PSS planning. The proposed group decision-making approach based on HFLTSs requires no external parameters and don’t need to compute the fuzzy envelope or add terms. Another contribution of this paper is combining HFLTSs with IA for the first time and constructing a non-linear 0-1 programming model whose objective function is described based on HIA. For qualitative DRs, the approaches for computing the interval information content are proposed aiming at the benefit-type DRs, the cost-type DRs, the interval-type DRs, and the target-type DRs respectively. By constructing the 0-1 nonlinear programming model, the total information content of DRs configuration is minimized, which realizes the optimized configuration of PSS. In this model, the compensation relationships among DRs are considered by using the method of imprecision.

While this research makes some contributions, it indicates that interdependencies among product DRs and service DRs should be taken into consideration. Furthermore, it is necessary to develop a much more aggregation operators to reflect the real situation of cross-functional group decision team. The aggregation operators’ influence on the proposed QFD approach will be further explored in future researches. In the future, the influence of aggregation operators will be further explored.

Footnotes

Acknowledgments

The authors would like to acknowledge funding support from the National Natural Science Foundation Committee (NSFC) of China (No. 72271164) and The Humanities and Social Sciences Research Planning Fund of the Ministry of Education (No. 19YJA630021), as well as the contributions from all collaborators with in the projects mentioned.

References

Mitake

, Noto

, Kimita

and Shimomura

, A Context Extracting Method for Value-in-Use Enhancement, Procedia CIRP 64 (2017), 312–317.

Teresa

A.-R.

, Graham

and Bengt-Olof

, The design of functional (total care) products, Journal of Engineering Design 15 (2004), 515–540.

Aurich

J.C.

, Fuchs

and DeVries

M.F.

, An Approach to Life Cycle Oriented Technical Service Design, CIRP Annals 53 (2004), 151–154.

Roy

and Cheruvu

, A competitive framework for industrial product-service systems, International Journal of Internet Manufacturing and Services 2 (2009), 4–29.

Sakao

, Shimomura

, Sundin

and Comstock

, Modeling design objects in CAD system for Service/Product Engineering, Computer-Aided Design 41 (2009), 197–213.

Roy

, Shaw

, Erkoyuncu

J.A.

and Redding

, Through-life engineering services, Measurement and Control 46 (2013), 172–175.

Morelli

, Product-service systems, a perspective shift for designers: A case study: the design of a telecentre, Design Studies 24 (2003), 73–99.

Chakrabarti

and Bligh

T.P.

, A scheme for functional reasoning in conceptual design, Design Studies 22 (2001), 493–517.

Kim

and Yoon

, Developing a process of concept generation for new product-service systems: a QFD and TRIZ-based approach, Service Business 6 (2012), 323–348.

10.

, Lee

and Park

, Development of an integrated product - service roadmap with QFD, International Journal of Service Industry Management 19 (2008), 621–638.

11.

Wang

P.P.

, Ming

X.G.

, Li

, Kong

F.B.

, Wang

and Wu

Z.Y.

, Modular Development of Product Service Systems, Concurrent Engineering 19 (2011), 85–96.

12.

Cui

and Geng

, Product Service System Configuration Based on a PCA-QPSO-SVM Model, Sustainability 13 (2021), 9450–9472.

13.

Hakanen

, Helander

and Valkokari

, Servitization in global business-to-business distribution: The central activities of manufacturers, Industrial Marketing Management 63 (2017), 167–178.

14.

Arai

and Shimomura

, Service CAD System –Evaluation and Quantification, CIRP Annals 54 (2005), 463–466.

15.

Fargnoli

and Sakao

, Uncovering differences and similarities among quality function deployment-based methods in Design for X: Benchmarking in different domains, Quality Engineering 29 (2017), 690–712.

16.

Hara

, Arai

and Shimomura

, A CAD system for service innovation: integrated representation of function, service activity, and product behaviour, Journal of Engineering Design 20 (2009), 367–388.

17.

Fargnoli

and Haber

, A practical ANP-QFD methodology for dealing with requirements’ inner dependency in PSS development, Computers & Industrial Engineering 127 (2019), 536–548.

18.

Haber

, Fargnoli

and Sakao

, Integrating QFD for product-service systems with the Kano model and fuzzy AHP, Total Quality Management & Business Excellence 31 (2018), 929–954.

19.

Aurich

J.C.

, Mannweiler

and Schweitzer

, How to design and offer services successfully, CIRP Journal of Manufacturing Science and Technology 2 (2010), 136–143.

20.

Regan

W.J.

, The service revolution, Journal of Marketing 27 (1963), 57–62.

21.

Wang

Y.-M.

and Chin

K.-S.

, Technical importance ratings in fuzzy QFD by integrating fuzzy normalization and fuzzy weighted average, Computers & Mathematics with Applications 62 (2011), 4207–4221.

22.

, Jin

and Wang

, A new MCDM method combining QFD with TOPSIS for knowledge management system selection from the user’s perspective in intuitionistic fuzzy environment, Applied Soft Computing 21 (2014), 28–37.

23.

Zhai

L.Y.

, Khoo

L.P.

and Zhong

Z.W.

, Towards a QFD-based expert system: A novel extension to fuzzy QFD methodology using rough set theory, Expert Systems with Applications 37 (2010), 8888–8896.

24.

W.-C.

, Construction of house of quality for new product planning: A 2-tuple fuzzy linguistic approach, Computers in Industry 73 (2015), 117–127.

25.

Osiro

, Lima-Junior

F.R.

and Carpinetti

L.C.R.

, A group decision model based on quality function deployment and hesitant fuzzy for selecting supply chain sustainability metrics, Journal of Cleaner Production 183 (2018), 964–978.

26.

Rodriguez

R.M.

, Martinez

and Herrera

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

27.

Suh

N.P.

, Axiomatic design theory for system, Research in Engineering Design 10 (1998), 189–209.

28.

Kahraman

, Cebi

, Onar

S.C.

and Oztaysi

, A novel trapezoidal intuitionistic fuzzy information axiom approach: An application to multicriteria landfill site selection, Engineering Applications of Artificial Intelligence 67 (2018), 157–172.

29.

Ijadi Maghsoodi

, Mosavat

, Hafezalkotob

and Hafezalkotob

, Hybrid hierarchical fuzzy group decision-making based on information axioms and BWM: Prototype design selection, Computers & Industrial Engineering 127 (2019), 788–804.

30.

Yan

H.-B.

and Ma

, A group decision-making approach to uncertain quality function deployment based on fuzzy preference relation and fuzzy majority, European Journal of Operational Research 241 (2015), 815–829.

31.

Cevik Onar

, Büyüközkan

, Öztayşi

and Kahraman

, A new hesitant fuzzy QFD approach: An application to computer workstation selection, Applied Soft Computing 46 (2016), 1–16.

32.

Liu

H.-T.

and Cheng

H.-S.

, An improved grey quality function deployment approach using the grey TRIZ technique, Computers & Industrial Engineering 92 (2016), 57–71.

33.

Aydin

, Seker

, Deveci

, Ding

and Delen

, A linear programming-based QFD methodology under fuzzy environment to develop sustainable policies in apparel retailing industry, Journal of Cleaner Production 387 (2023), 135887–135902.

34.

Allehashemi

, Amin

S.H.

and Zolfaghari

, A proposed multi-objective model for cellphone closed-loop supply chain optimization based on fuzzy QFD, Expert Systems with Applications 210 (2022), 118577–118592.

35.

Lee

A.H.I.

, Kang

H.-Y.

, Yang

C.-Y.

and Lin

C.-Y.

, An evaluation framework for product planning using FANP, QFD and multi-choice goal programming, International Journal of Production Research 48 (2009), 3977–3997.

36.

Kwong

C.K.

and Bai

, A fuzzy AHP approach to the determination of importance weights of customer requirements in quality function deployment, Journal of Intelligent Manufacturing 13 (2002), 367–377.

37.

Torkayesh

A.E.

, Yazdani

and Ribeiro-Soriano

, Analysis of industry 4.0 implementation in mobility sector: An integrated approach based on QFD, BWM, and stratified combined compromise solution under fuzzy environment, Journal of Industrial Information Integration 30 (2022), 100406–100420.

38.

Y.-L.

, Tang

J.-F.

, Chin

K.-S.

, Han

and Luo

X.-G.

, A rough set approach for estimating correlation measures in quality function deployment, Information Sciences 189 (2012), 126–142.

39.

Zhu

G.-N.

, Design concept evaluation considering information reliability, uncertainty, and subjectivity: An integrated rough-Z-number-enhanced MCGDM methodology, Advanced Engineering Informatics 54 (2022), 101796–101808.

40.

Lizarelli

F.L.

, Osiro

, Ganga

G.M.D.

, Mendes

G.H.S.

and Paz

G.R.

, Integration of SERVQUAL, Analytical Kano, and QFD using fuzzy approaches to support improvement decisions in an entrepreneurial education service, Applied Soft Computing 112 (2021), 107786–107801.

41.

, Liu

, Qin

and Herrera

, An interval type-2 fuzzy Kano-prospect-TOPSIS based QFD model: Application to Chinese e-commerce service design, Applied Soft Computing 111 (2021), 107665–107681.

42.

Zhang

, Zhang

, Gong

, Dincer

and Yüksel

, An integrated hesitant 2-tuple Pythagorean fuzzy analysis of QFD-based innovation cost and duration for renewable energy projects, Energy 248 (2022), 123561–123574.

43.

Büyüközkan

, Ertay

, Kahraman

and Ruan

, Determining the importance weights for the design requirements in the house of quality using the fuzzy analytic network approach, International Journal of Intelligent Systems 19 (2004), 443–461.

44.

Wang

, Fuzzy outranking approach to prioritize design requirements in quality function deployment, International Journal of Production Research 37 (1999), 899–916.

45.

Liu

S.-T.

, Rating design requirements in fuzzy quality function deployment via a mathematical programming approach, International Journal of Production Research 43 (2005), 497–513.

46.

Kwong

C.K.

, Chen

, Bai

and Chan

D.S.K.

, A methodology of determining aggregated importance of engineering characteristics in QFD, Computers & Industrial Engineering 53 (2007), 667–679.

47.

, Tang

, Luo

, Yao

and Xu

, A quantitative methodology for acquiring engineering characteristics in PPHOQ, Expert Systems with Applications 37 (2010), 187–193.

48.

H.H.

, Liao

A.Y.H.

and Wang

P.C.

, Using grey theory in quality function deployment to analyse dynamic customer requirements, The International Journal of Advanced Manufacturing Technology 25 (2004), 1241–1247.

49.

H.-H.

, Applying grey model to prioritise technical measures in quality function deployment, The International Journal of Advanced Manufacturing Technology 29 (2005), 1278–1283.

50.

Song

, Ming

and Han

, Prioritising technical attributes in QFD under vague environment: a rough-grey relational analysis approach, International Journal of Production Research 52 (2014), 5528–5545.

51.

Wei

, Zhao

and Tang

, Operators and Comparisons of Hesitant Fuzzy Linguistic Term Sets, IEEE Transactions on Fuzzy Systems 22 (2014), 575–585.

52.

, Zhang

, Zhong

and Sun

, Extended TODIM for multi-criteria group decision making based on unbalanced hesitant fuzzy linguistic term sets, Computers & Industrial Engineering 114 (2017), 316–328.

53.

Hesamian

and Shams

, Measuring Similarity and Ordering based on Hesitant Fuzzy Linguistic Term Sets, Journal of Intelligent & Fuzzy Systems 28 (2015), 983–990.

54.

and Xu

, Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations, Omega 65 (2016), 28–40.

55.

Huang

, You

X.-Y.

, Liu

H.-C.

and Si

S.-L.

, New approach for quality function deployment based on proportional hesitant fuzzy linguistic term sets and prospect theory, International Journal of Production Research 57 (2018), 1283–1299.

56.

Song

, Ming

, Han

and Wu

, A rough set approach for evaluating vague customer requirement of industrial product-service system, International Journal of Production Research 51 (2013), 6681–6701.

57.

Geng

, Chu

, Xue

and Zhang

, A systematic decision-making approach for the optimal product–service system planning, Expert Systems with Applications 38 (2011), 11849–11858.

58.

Song

and Chan

F.T.S.

, Multi-objective configuration optimization for product-extension service, Journal of Manufacturing Systems 37 (2015), 113–125.

59.

Chen

L.-H.

and Weng

M.-C.

, A fuzzy model for exploiting quality function deployment, Mathematical and Computer Modelling 38 (2003), 559–570.

60.

Wasserman

G.S.

, On How to Prioritize Design Requirements during the Qfd Planning Process, IIE Transactions 25 (1993), 59–65.

61.

W.-C.

and Chen

L.-H.

, An approach of new product planning using quality function deployment and fuzzy linear programming model, International Journal of Production Research 52 (2013), 1728–1743.

62.

Suh

N.P.

, The Principles of Design, New York, 1990.

63.

Gunasekera

J.S.

and Ali

A.F.

, A three-step approach to designing a metal-forming process, The Journal of The Minerals, Metals & Materials Society 47 (1995), 22–25.

64.

Kulak

and Kahraman

, Fuzzy multi-attribute selection among transportation companies using axiomatic design and analytic hierarchy process, Information Sciences 170 (2005), 191–210.

65.

Kulak

, Durmusoğlu

M.B.

and Kahraman

, Fuzzy multi-attribute equipment selection based on information axiom, Journal of Materials Processing Technology 169 (2005), 337–345.

66.

Celik

, Kahraman

, Cebi

and Er

I.D.

, Fuzzy axiomatic design-based performance evaluation model for docking facilities in shipbuilding industry: The case of Turkish shipyards, Expert Systems with Applications 36 (2009), 599–615.

67.

Akay

, Kulak

and Henson

, Conceptual design evaluation using interval type-2 fuzzy information axiom, Computers in Industry 62 (2011), 138–146.

68.

Chen

, Chu

, Sun

, Li

and Su

, An Information Axiom based decision making approach under hybrid uncertain environments, Information Sciences 312 (2015), 25–39.

69.

Celik

, Cebi

, Kahraman

and Er

I.D.

, Application of axiomatic design and TOPSIS methodologies under fuzzy environment for proposing competitive strategies on Turkish container ports in maritime transportation network, Expert Systems with Applications 36 (2009), 4541–4557.

70.

Gören

H.G.

and Kulak

, A New Fuzzy Multi-criteria Decision Making Approach: Extended Hierarchical Fuzzy Axiomatic Design Approach with Risk Factors, in: Decision Support Systems III – Impact of Decision Support Systems for Global Environments, 2014, pp. 141–156.

71.

Kahraman

, Cevik Onar

, Cebi

and Oztaysi

, Extension of information axiom from ordinary to intuitionistic fuzzy sets: An application to search algorithm selection, Computers & Industrial Engineering 105 (2017), 348–361.

72.

Herrera

and Martínez-López

, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Trans Fuzzy Syst 8 (2000), 746–752.

73.

Zhang

, The multiattribute group decision making method based on aggregation operators with interval-valued 2-tuple linguistic information, Mathematical and Computer Modelling 56 (2012), 27–35.

74.

Wang

Y.-M.

, Yang

J.-B.

and Xu

D.-L.

, A preference aggregation method through the estimation of utility intervals, Computers & Operations Research 32 (2005), 2027–2049.

75.

, Priority Method Based on Possibility and Error Analysis for Interva lNumber Complementary Judgement Matrix, Journal of PLA University of Science and Technology 04 (2003), 96–98.