An integrated approach to identify function components for product redesign based on analysis of customer requirements and failure risk

Abstract

Product redesign strategy can effectively shorten design lead time and reduce production cost of new variants development. Identification of function components is the basis of product redesign. In the existing methods to identify the function components, customer requirements are primarily considered while the failure knowledge, a critical information to improve product reliability, is often ignored. The objective of this research is to identify the to-be-improved components considering both customer requirements and product reliability. First, a two-stage fuzzy quality function deployment (QFD) is used to calculate the importance weight of each component considering customer requirements. Second, the fuzzy failure mode effects and analysis (FMEA) is adopted to measure the failure risk of each component. Different from traditional FMEA, the failure causality relationships are analyzed in this work to provide a means of making use of failure information more effectively for constructing a directed failure causality relationship diagram. Fuzzy permanent function is developed to quantify the failure risk of each component. Then, a modification necessity index is introduced to model the degree of modification necessity for each component considering customer requirements and failure risk. Finally, the optimal set of function components that need to be modified is identified by 0-1 objective programming and constrained optimization. A case study for identification of the function components for the operation device of a crawler crane is implemented to demonstrate the effectiveness of the developed approach.

Keywords

Identification of function component fuzzy QFD FMEA fuzzy permanent function 0-1 objective programming

1 Introduction

With the advances in design and manufacturing technologies, functions and structures of mechanical products are becoming more complex. In addition, customers also expect these products to maintain high reliability and long life-span under working environments [12, 21]. These bring great challenges to product design with new variants. New variants are usually developed by adjusting or modifying the defective function components (FCs) of the existing designs [27]. The ultimate goal of product redesign is to create new products with high quality by improving the selected target FCs.

Many researches on product redesign mainly focus on how to optimize the FCs of the existing products for resolving the conflicts between the original design and customer requirements (CRs) [5, 13]. As the basis of product redesign, identification of FCs has been rarely researched. Recently, the mostly used method for identifying the FCs is QFD technology, which usually applies the house of quality (HoQ) for rating the existing FCs considering customer requirements [4, 23]. For example, QFD and similar techniques have been employed to estimate the importance weights of components [12], subsystems [14] and component parameters [29]. As the main input of QFD, the CRs are generally extracted by user surveys or online-interviews to focus groups and identify key FCs [20, 29]. Smith et al. built a component factor table based on QFD to analyze the impact of FC on CRs and to identify the important component for a given redesign task [21]. Liu presented a nonlinear programming model to calculate the ratings of the design requirements (DRs) without knowing their exact membership functions [22]. Chen and Weng proposed a fuzzy goal programming model to determine the fulfillment levels of the DRs considering business and the preemptive priorities between goals [13]. Delice and Güngör proposed a fuzzy mixed-integer goal programming model to calculate a combination of optimal DR values. Different from the existing fuzzy goal programming models, the values of the DRs in the proposed model are taken as discrete [7]. Yang and Chen developed a fuzzy linear programming model to determine the optimal level of product engineering characteristics, where the objective function is the overall customer satisfaction and the cost constraint is fuzzified [30]. From these researches, CRs are primarily considered to increase customer satisfaction degree of new variants. However, the CRs are usually obtained by marketing surveys without considering the product reliability. Although QFD is a popular tool to achieve CRs and shorten development lead-time, it is not effective to improve product reliability and to identify hidden quality problems of subsequent processes in the early product development stage, thus leading to many product failure problems after products are released to market. Furthermore, because the traditional QFD is used to obtain the FCs only from CRs, only customer satisfaction can be improved by modifying these components. But these methods based on the traditional QFD cannot be used to identify and modify the components with high failure risk. To improve the reliability of new product, both the components with low customer satisfaction and the components with high failure risk need to be identified and modified.

Various failure data collected during the long operation periods of products provides reference for many engineering decision-making problems [11 , 27]. To eliminate or reduce the chance of failure, Stone et al. developed a function failure design method (FFDM) that performs failure analysis and offers substantial reliability improvement [15]. Liu et al. identified the key factors of product design through mutual assessments and investigations by QFD and design FMEA for the package design of semi-finished products [18]. Tan developed a customer-focused methodology for the built-in reliability by combining the FMEA and QFD [6]. To prevent product failures in the early design stage, Chen and Ko developed a fuzzy linear programming model by combing QFD and FMEA to determine the fulfillment levels of part characteristics [12]. Design risk, which was treated as the constraint in the model, was incorporated into QFD process by FMEA. Ko adopted a 2-tuple linguistic computational approach to treat the assessments of the three risk indices of FMEA and developed a new assessment model to overcome the drawbacks of conventional RPN calculation [24]. Efe et al. provided an integrated approach to overcome the drawbacks of traditional FMEA using an intuitionistic fuzzy multi-criteria decision making method and a linear mathematical programming model [2]. The researches discussed above demonstrate that the product failure data is accessible and achievable for the product improvement. These researches also provide valuable references for design improvement using failure data. In these methods, however, the causality relationships among failure modes of FCs were not considered. During the long operation periods, one failure mode may result in other failure modes. For example, the leakage of the hydraulic system of a crane can cause the blockage of the main boom tip. The analysis on design risk of failure modes without considering their failure causality relationships (FCRs) often leads to inaccurate analysis results. To address this problem, Rao and Gandhi [16], and Jangra et al. [10] both analyzed the FCRs using crisp number such as 1-3-9 to quantify the relationship strength. However, fuzzy decision-making information was not considered in the two researches for measure the FCRs and risk of failure modes. Actually, the subjective and imprecise linguistic terms are usually used by decision makers to describe their judgements on the strength of FCRs. Furthermore, the above researches merely focused on the severity of each failure mode, without considering the detectability and occurrence possibility of each failure mode. To solve the above problems, an integrated approach is developed in this research to identify the FCs by combining fuzzy QFD and FMEA considering the FCRs among failure modes.

In addition, linguistic terms are usually used by decision makers to describe their judgements in the identification process. Due to the various experience and subjective preference of these experts, evaluation results are not precise. Therefore, it is important to incorporate the uncertainty and fuzziness of information into decision-making. To deal with the fuzzy uncertain information, fuzzy set theory and related methods, are employed in this research to deal with fuzzy judgements of decision makers.

The rest of this article is organized as follows. In Section 2, the new method to identify FCs for product redesign is explained. In Section 3, a case study for identification of the FCs for a crawler crane is implemented to demonstrate the effectiveness of the developed approach. In Section 4, discussions and conclusions are presented.

2 Methodology

The identification process is mainly conducted in 3 steps: (1) Calculation of the importance weight of each component using fuzzy QFD, (2) Quantification of the failure risk of each component using fuzzy FMEA and a directed failure causality relationship diagram (DFCRD) model, (3) Identification of the FCs using 0-1 objective programming model. In the proposed approach, a modification necessity index (MNI) is introduced to model the degree of modification necessity for each FC of the existing product considering CRs and failure risk. The MNI is defined as follows: $MNI = W_{C} * CRI + W_{F} * FRI$ (1) where, CRI represents the importance index of each component calculated by QFD based on CRs, and the FRI represents the failure risk of each component quantified by FMEA based on failure data. The procedure to calculate CRI is described in Section 2.1, and the procedure to calculate FRI is described in Section 2.2. The W_C and W_F in Equation (1) are the weighting factors of CRI and FRI respectively (W_C + W_F = 1). Finally, a 0-1 objective programming model is built to obtain the optimal set of FCs for product redesign after the MNI is determined. In the model, the redesign cost, time and other engineering constraints are integrated to make the identification results more accordant to engineering practice.

Fig. 1.

The proposed approach.

2.1 Calculation of the importance weight of each component

The house of quality (HoQ) is used to calculate the CRI through a two-stage QFD. Before the deployment process of HoQ, the importance weights of CRs are calculated using fuzzy pairwise comparison (FPC) and fuzzy logarithmic least-square method (FLLSM).

2.1.1 Calculation of the importance weight of each customer requirement (CR)

The FPC method is employed to obtain the importance weights of CRs in this research. In this method, the relative importance ${\tilde{r}}_{i ξ}$ between two CRs CR_i and CR_ξ (i = 1, 2, …, I; ξ = 1, 2, …, I) is described by one of the following linguistic terms: “SL (significantly less important)”, “L (less important)”, “E (equally important)”, “M (more important)”, and “SM (significantly more important)”. These linguistic terms are then transformed into triangular fuzzy numbers (TFNs) in the form of ${\tilde{r}}_{i ξ} = (r_{i ξ l}, r_{i ξ m}, r_{i ξ u})$ , where r_iξl, r_iξm and R_gju represent the lower, medium, upper values of a TFN, respectively. The five linguistic terms are defined as (1, 1, 3), (1, 3, 5), (3, 5, 7), (5, 7, 9), and (7, 9, 9), respectively. The calculation process of the importance weight of each CR is given as follows.

Step 1. A I × I fuzzy relative importance matrix $\tilde{r}$ is constructed by each design engineer through comparing the I CRs as shown in Equation (2).

$[\begin{matrix} 1 & (r_{12 l}, r_{12 m}, r_{12 u}) & \dots & (r_{1 Il}, r_{1 Im}, r_{1 Iu}) \\ (r_{21 l}, r_{21 m}, r_{21 u}) & 1 & \dots & (r_{2 Il}, r_{2 Im}, r_{2 Iu}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (r_{I 1 l}, r_{I 1 m}, r_{I 1 u}) & (r_{I 2 l}, r_{I 2 m}, r_{I 2 u}) & \dots & 1 \end{matrix}]$ (2)

where (r_iξl, r_itξm, r_iξu) = (1/r_ξiu, 1/r_ξim, 1/r_ξiul) when i ≠ ξ, and r_iξl, r_iξm, r_iξu = (1, 1, 1) when i = ξ. Suppose each designer gives a fuzzy comparison matrix ${\tilde{A}}^{k}$ (k = 1, 2, …, K, K is the total number of designers).

Step 2. The final fuzzy comparison matrix $\tilde{A}$ of the expert group can be obtained using Equation (3) [3, 9] ${\tilde{r}}_{ij} = [{(\prod_{k = 1}^{K} r_{ijl}^{k})}^{\frac{1}{K}}, {(\prod_{k = 1}^{K} r_{ijm}^{k})}^{\frac{1}{K}}, {(\prod_{k = 1}^{K} r_{iju}^{k})}^{\frac{1}{K}}]$ (3)

Step 3. Execute the consistency check for fuzzy comparison matrix $\tilde{A}$ consisting of TFNs. The consistency index (CI) is calculated using Equation (4) [3] $CI = \frac{λ_{\max} - I}{I - 1}$ (4) where, λ_max is the largest eigenvector of $\tilde{A}$ , which can be calculated in Step 4; I is the dimension of matrix $\tilde{A}$ . Then, consistency ratio (CR) of $\tilde{A}$ is obtained using Equation (5) [17, 19] $CR = \frac{CI}{RI (I)}$ (5) where RI (I) is a random index for CI, which can be determined using Table 1 [5]. When CR < 0.1, the consistency of comparison matrix $\tilde{A}$ can be accepted.

Step 4. Calculation of λ_max for fuzzy comparison matrix $\tilde{A}$ . The TFN ${\tilde{r}}_{ij}$ in $\tilde{A}$ can be represented using a confidence factor α as ${\tilde{r}}_{ij} = [r_{ij l} + α (r_{ijm} - r_{ij l}), r_{iju} - α (r_{iju} - r_{ijm})]$ , α ∈ [0, 1] [8]. Then, the crisp ${\hat{r}}_{ij}^{α}$ can be further obtained using Equation (6) according to a expert satisfaction factor μ, μɛ [0, 1] ${\hat{r}}_{ij}^{α} = μ r_{iju}^{α} + (1 - μ) r_{ijl}^{α}$ (6) When α and μ are fixed, fuzzy comparison matrix $\tilde{A}$ can be transformed into crisp matrix $\hat{A}$ . Then $λ_{max}^{α}$ of $\hat{A}$ is obtained. By adjusting α and μ, $λ_{max}^{α}$ and consistency ratio CR of ${\hat{A}}^{α}$ can be calculated under different decision-making environments [8]. Therefore, the consistency of fuzzy comparison matrix $\tilde{A}$ can be checked.

Step 5. Calculation of the quantitative importance weights of CRs. In this step, the FLLSM is employed in this work to calculate the triangular fuzzy importance degree of CR_k, i.e., ${\tilde{w}}_{i} = (w_{il}, w_{im}, w_{iu})$ . The optimization model of the FLLSM is defined in Equation (7) [25]. The w_i = (w_il, w_im, w_iu) is obtained by optimization using Equation (7) $\begin{matrix} Minf & = & \sum_{i = 1}^{I} \sum_{ξ = 1, ξ \neq i}^{I} ({({lnw}_{il} - {lnw}_{ξ u} - {lnr}_{i ξ l})}^{2} \\ + {({lnw}_{im} - {lnw}_{ξ m} - {lnr}_{i ξ m})}^{2} \\ + {({lnw}_{iu} - {lnw}_{ξ l} - {lnr}_{i ξ u})}^{2}), \end{matrix}$ $s . t . {\begin{matrix} w_{il} + \sum_{ξ = 1, ξ \neq i}^{I} w_{ξ u} \geq 1, \\ w_{iu} + \sum_{ξ = 1, ξ \neq i}^{I} w_{ξ l} \leq 1, \\ \sum_{i = 1}^{I} w_{im} = 1, \\ \sum_{j = 1}^{J} w_{ξ l} + w_{iu} = 2, \\ 0 < w_{ξ l} \leq w_{ξ m} \leq w_{iu} < 1 . \end{matrix}$ (7)

The TFN ${\tilde{w}}_{i} = (w_{il}, w_{im}, w_{iu})$ can be defuzzified by Equation (8) to obtain the crisp function weight $w_{i} = \frac{w_{il} + 2 w_{im} + w_{iu}}{4}, i = 1, 2, \dots, I .$ (8)

2.1.2 Calculation of the importance weight of each component using fuzzy QFD

In the QFD process, the CRs are first used to determine the function requirements (FRs), and then the FRs are used to determine the CRI measures of FCs, as shown in Fig. 2.

Fig. 2.

The two stages of QFD processes.

The interrelationship value ${\tilde{R}}_{ih}$ between the ith CR and the hth FR is quantified by linguistic terms of “SW (significantly weak)”, W (weak)”, “M (medium)”, “S (strong)” and “SS (significantly strong)”. The interrelations ${\tilde{r}}_{hH}$ between FRs, and the relations ${\tilde{R}}_{hj}$ between the hth FR and the jth FC are also quantified using linguistic terms. What’s more, the interrelations ${(\tilde{r})}_{j J}^{'}$ between FCs are quantified according to their functional or physical relationships, respectively as shown in Tables 1 and 2. The 5 linguistic terms are defined by (1,1,3), (1,3,5), (3,5,7), (5,7,9), and (7,9,9).

Table 1

Random index RI (I)

I	3	4	5	6	7	8	9
RI (I)	0.58	0.90	1.12	1.24	1.32	1.41	1.4

Table 2

Functional interrelation strength of any two FCs

Linguistic terms	TFN	Description
SS	(7,9,9)	FC A and FC B realize together the same sub-function
S	(5,7,9)	FC A realize one main function, while FC B realize the sub-function of the main function
M	(3,5,7)	FC A and FC B realize together the same main function
W	(1,3,5)	FC A and FC B realize different sub-functions of the same main function
SW	(1,1,3)	FC A and FC B realize together different main functions

In Fig. 2 (a), w_i represents the importance score of CR_i where ∑w_i = 1. g_h is the fuzzy importance weight of the hth FR, which is calculated by Equation (9) ${\tilde{g}}_{h} = \sum_{i = 1}^{I} w_{i} . {\tilde{R}}_{i h}^{'}$ (9) where ${\tilde{R}}_{i h}^{'}$ is the normalized relationship value between the ith CR and the hth FR quantified by linguistic terms considering the correlations among FRs. ${\tilde{R}}_{i h}^{'}$ can be determined by Equation (10) ${\tilde{R}}_{i h}^{'} \frac{\sum_{s = 1}^{H} {\tilde{R}}_{is} \cdot {\tilde{r}}_{sh}}{\sum_{h = 1}^{H} \sum_{s = 1}^{H} {\tilde{R}}_{is} \cdot {\tilde{r}}_{sh}}$ (10)

In Fig. 2(b), the

{\tilde{CRI}}_{j}

is determined by translating the FRs to FCs considering the interrelations between FCs in Tables 2 and 3, as expressed by Equation (11)

Table 3

Physical interrelation strength of any two FCs

	No.	Connection type	Connection parameters	TFN
Static connection	(1)	Single-fastener connection	Type of pin/bolt/rivet, bolt hole depth	(3,5,7)
	(2)	Multiple-fastener connection	Type of pin/bolt/rivet, bolt hole depth	(5,7,9)
	(3)	Sliding connection	Size and location of connection slot	(3,5,7)
	(4)	Welding connection	Welding surface dimensions, welding process parameters	(5,7,9)
Dynamic connection	(5)	Bearing connection	Bearing type, shaft diameter, etc.	(7,9,9)
	(6)	Gearing connection	Gearing type, gear ratio, center distance, etc.	(7,9,9)
	(7)	Belting connection	Belting type, center distance, transmission ratio, belt length, etc.	(5,7,9)
	(8)	Sprocket connection	Chain type, gear teeth number, chain length, transmission ratio, etc.	(5,7,9)

{\tilde{CRI}}_{j} = \sum_{h = 1}^{H} {\tilde{g}}_{h} \cdot {\tilde{R}}_{i h}^{'}

(11) where

{\tilde{R}}_{i h}^{'}

is the normalized relationship value between the hth FR and the jth FC quantified by linguistic terms considering the interrelations among FCs.

{\tilde{R}}_{i h}^{'}

can be determined by Equation (12)

{\tilde{R}}_{i h}^{'} = \frac{\sum_{t = 1}^{J} {\tilde{R}}_{h t} . {(\tilde{r})}_{t h}^{,}}{\sum_{h = 1}^{J} \sum_{t = 1}^{J} {\tilde{R}}_{h t} . {(\tilde{r})}_{t h}^{,}}

(12)

In Equation (11), different interval values $[{({\tilde{CRI}}_{j})}_{L}^{α}, {({\tilde{CRI}}_{j})}_{U}^{α}]$ can be obtained for fuzzy ${\tilde{QII}}_{j}$ with different α levels, α ∈ (0, 1) . For each α level, a $[{({\tilde{CRI}}_{j})}_{L}^{α}, {({\tilde{CRI}}_{j})}_{U}^{α}]$ is obtained. Finally, the crisp CRI_j can be calculated by aggregating the $({\tilde{CRI}}_{j})^{α}$ at each α level by Equation (13) [1] $X = \frac{\sum_{i} [1 / 2 (X_{L}^{α i} + X_{U}^{α i})] . {\bar{μ}}_{i}^{'}}{\sum_{i} {\bar{μ}}_{i}^{'}}$ (13) where the fuzzy number $\tilde{X}$ can be defuzzied by Equation (13) to obtain a crisp number X, α_i is the ith level of α, and ${\bar{μ}}_{i}^{'}$ is the membership degree of $1 / 2 (X_{L}^{α_{i}} + X_{U}^{α_{i}})$ belonging to the fuzzy number $\tilde{X}$ determined by α cut and extension principles [1, 13].

2.2 Quantification of the failure risk of each component

In the product usage period, the FCs in products may fail in various modes. For one FC, FCRs among failure modes need to be considered. In this research, the failure risk of each FC is analyzed using the DFCRD model. The main potential failure modes of one FC are modeled as vertices in the graph, while their FCRs are modeled as edges. The fuzzy risk priority numbers (RPNs) of failure modes are calculated as the weights of vertices, while the fuzzy quantification values of FCRs are determined as the weights of edges in DFCRD. The failure risk, i.e., FRI_j, of FC_j is obtained based on the fuzzy permanent function of the failure causality matrix (FCM) of DFCRD. Suppose FC_j has V_j failure modes denoted as ${FM}_{j}^{1}, \dots, {FM}_{j}^{v}, \dots, {FM}_{j}^{V_{j}}$ as shown in Fig. 3, the process to calculate FRI is explained in the following steps.

Step 1. Create the DFCRD topology

Fig. 3.

The DFCRD topology of FC_j.

The DFCRD model is described as a graph G = (V, E), which is composed of two sets, V and E. The elements of V are the vertices of the graph G, and the elements of E are the edges. Here, the V_j vertices represent ${FM}_{j}^{1}, \dots, {FM}_{j}^{v}, \dots, {FM}_{j}^{V_{j}}$ and the directed edges represent the FCRs among these failure modes.

Step 2. Quantify the FCR to determine the weight of edge

The directed edge of DFCRD represents the causality relationship between failure modes, and the weight of the edge denotes the strength of this causality relationship. Usually, the assessment of the weight of the causality relationship is subjective and qualitatively described in natural language. Therefore, the linguistic terms and TFNs are used to describe the FCRs for reflecting this imprecise nature.

Step 3. Calculate the fuzzy RPN as the weight of vertex

The fuzzy RPN of each failure mode can be assigned as the weight of the vertex. The RPN in this work is calculated using Equation (14) [28, 31] $\tilde{RPN} = {\tilde{O}}^{{\tilde{w}}_{1}} * {\tilde{S}}^{{\tilde{w}}_{2}} * {\tilde{D}}^{{\tilde{w}}_{3}}$ (14) where, the occurrence index (O), the severity index (S) and the detectability index (D) can be obtained by TFNs. $\tilde{W} = ({\tilde{w}}_{1}, {\tilde{w}}_{2}, {\tilde{w}}_{3})$ is the fuzzy weight of $\tilde{O}$ , $\tilde{S}$ and $\tilde{D}$ . For calculating the fuzzy α levels of cuts $[R_{L}^{α}, R_{U}^{α}]$ of $\tilde{RPN}$ , the model defined in Equation (14) is transformed into the model defined in Equation (15) ${\begin{matrix} R_{L}^{α} = Min exp (\frac{w_{1} \ln (O)_{L}^{α} + w_{2} \ln (S)_{L}^{α} + w_{3} \ln (D)_{L}^{α}}{\sum_{i = 1}^{3} w_{i}}), \\ s . t . (w_{i})_{L}^{α} \leq w_{i} \leq (w_{i})_{U}^{α}, \\ R_{U}^{α} = Max exp (\frac{w_{1} \ln (O)_{U}^{α} + w_{2} \ln (S)_{U}^{α} + w_{3} \ln (D)_{U}^{α}}{\sum_{i = 1}^{3} w_{i}}), \\ s . t . (w_{i})_{L}^{α} \leq w_{i} \leq (w_{i})_{U}^{α} . \end{matrix}$ (15)

By calculating the $R_{L}^{α}$ and $R_{U}^{α}$ in Equation (15), the fuzzy RPN of each failure mode can be obtained [31].

Step 4. Build the failure causality matrix (FCM)

According to the DFCRD, the FCM of FC_j is built as shown in Equation (16) $FCM ({FC}_{j}) = [\begin{matrix} {\tilde{R}}_{j}^{1} & {\tilde{η}}_{12} & \begin{matrix} \dots & {\tilde{η}}_{1 V_{j}} \end{matrix} \\ {\tilde{η}}_{21} & {\tilde{R}}_{j}^{2} & \begin{matrix} \dots & {\tilde{η}}_{2 V_{j}} \end{matrix} \\ \begin{matrix} ⋮ \\ {\tilde{η}}_{V_{j} 1} \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{η}}_{V_{j} 2} \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ {\tilde{R}}_{j}^{V_{j}} \end{matrix} \end{matrix} \end{matrix}]$ (16) where ${\tilde{R}}_{j}^{v}$ represents the vertex weight of the vth failure mode in DFCRD model for FC_j, and ${\tilde{η}}_{v θ}$ represents the edge weight of causality relationship between ${FM}_{j}^{v}$ and ${FM}_{j}^{θ}$ , quantified by the linguistic terms and TFNs.

Step 5. Define and calculate the FRI

Since the permanent function of matrix does not contain any negative sign and thus no information will be lost, in this research, the FRI_j is defined as the fuzzy permanent function of the FCM of FC_j as shown in Equation (17). The permanent function can characterize a FC based on the subgroups of Equation (17) and the failure risks of these subgroups.

$\begin{matrix} {\tilde{FRI}}_{j} & = & \prod_{v = 1}^{V_{j}} {\tilde{R}}_{j}^{v} + \sum_{σ} \sum_{ς} \sum_{τ} \dots \sum_{V_{j}} ({\tilde{η}}_{σ ς} {\tilde{η}}_{ς σ}) \\ \cdot {\tilde{R}}_{j}^{τ} \cdot {\tilde{R}}_{j}^{φ} \dots {\tilde{R}}_{j}^{V_{j}} \\ + \sum_{σ} \sum_{ς} \sum_{τ} \dots \sum_{V_{j}} ({\tilde{η}}_{σ ς} {\tilde{η}}_{ς τ} {\tilde{η}}_{τ σ} \\ + {\tilde{η}}_{σ τ} {\tilde{η}}_{τ ς} {\tilde{η}}_{ς σ}) \cdot {\tilde{R}}_{j}^{φ} \cdot {\tilde{R}}_{j}^{χ} \dots {\tilde{R}}_{j}^{V_{j}} \\ + (\sum_{σ} \sum_{ς} \sum_{τ} \dots \sum_{V_{j}} ({\tilde{η}}_{σ ς} {\tilde{η}}_{ς σ}) ({\tilde{η}}_{τ φ} {\tilde{η}}_{φ τ}) \\ \cdot {\tilde{R}}_{j}^{χ} \cdot {\tilde{R}}_{j}^{δ} \dots {\tilde{R}}_{j}^{V_{j}} \\ + \sum_{σ} \sum_{ς} \sum_{τ} \dots \sum_{V_{j}} ({\tilde{η}}_{σ ς} {\tilde{η}}_{ς τ} {\tilde{η}}_{τ φ} {\tilde{η}}_{φ σ} \\ + {\tilde{η}}_{σ φ} {\tilde{η}}_{φ τ} {\tilde{η}}_{τ ς} {\tilde{η}}_{ς σ}) \cdot {\tilde{R}}_{j}^{χ} {\tilde{R}}_{j}^{δ} \dots {\tilde{R}}_{j}^{V_{j}}) + \dots \end{matrix}$ (17)

All the failure risk aspects (i.e., the subgroups in Equation (17) are analyzed and measured in a comprehensive manner to achieve the failure risk considering all these failure subgroups [10, 16]. Furthermore, the failure causality relations between failure modes are integrated into the subgroups in Equation (17). Therefore, no failure information is lost for analyzing the failure risk of each FC.

The terms in Equation (17) are arranged in V_j + 1 subgroups. Equation (17) is explained as follows:

The first subgroup is a series of RPN values of failure modes that FC_j may occur (i.e. $\prod_{v = 1}^{V_{j}} {\tilde{R}}_{j}^{v}$ ).

The second subgroup is not required, because no failure is the cause and effect for itself.

The third subgroup contains a set of two-failure modes causality loops (i.e., ${\tilde{η}}_{σ ς} {\tilde{η}}_{ς σ}$ ), and the RPN of the remaining V_j - 2 failure modes.

The fourth subgroup is a set of three-failure modes causality loops (i.e. ${\tilde{η}}_{σ ς} {\tilde{η}}_{ς τ} {\tilde{η}}_{τ σ}$ or ${\tilde{η}}_{σ τ} {\tilde{η}}_{τ ς} {\tilde{η}}_{ς σ}$ ) and the RPN of the remaining V_j - 3 failure modes.

Similarly, other terms in Equation (17) can be defined.

By calculating the ${({\tilde{FRI}}_{j})}_{L}^{α}$ and ${({\tilde{FRI}}_{j})}_{U}^{α}$ using Equation (17), the fuzzy ${\tilde{FRI}}_{j}$ of FC_j can be obtained. Finally, the crisp FRI_j can be calculated by aggregating the $({\tilde{FRI}}_{j})^{α}$ at each α level using Equation (13).

2.3 Identification of the optimal set of FCs

After the CRI_j and FRI_j are respectively calculated in Sections 2.1 and 2.2, the MNI_j is consequently obtained using Equation (1) for FC_j. However, the redesign cost, time and other engineering factors is also need to be considered since they have significant influence on design decision-making. In this section, a 0-1 objective programming model is built to obtain the optimal set of FCs for redesign. Firstly, the parameters and variables of the programming model are given as follows.

P_i: the priority weight of the ith objective, which can be determined using FPC method in Section 2.1.1;

$d_{i}^{+}$ : the positive deviation of the ith objective;

$d_{i}^{-}$ : the negative deviation of the ith objective;

x_j: a 0-1 variable, which means whether the jth FC is selected as the FC or not;

r_ij: the amount of the ith quantitative engineering factors of the jth components, these quantitative engineering factors refer to redesign cost, time and so on;

R_i: the limits of the ith quantitative constraints, i.e. the maximum design cost and time, (i = 2, …, t);

w_ij: the weight of the jth component in terms of the ith qualitative engineering factor (i = t + 1, …, T), the qualitative engineering factors refer to convenience of maintenance services, resource utilization rate, etc.

The 0-1 objective programming model is constructed as Equation (18).

$\begin{matrix} \min P_{1} d_{1}^{-} + \sum_{i = 2}^{t} P_{i} (\frac{d_{i}^{-}}{R_{i}} + \frac{d_{i}^{+}}{R_{i}}) + \sum_{i = t + 1}^{T} P_{i} (d_{i}^{-}) \\ s . t . {\begin{matrix} \sum_{j = 1}^{J} {MNI}_{j} x_{j} + d_{1}^{-} - d_{1}^{+} = 1, \\ \sum_{j = 1}^{J} r_{ij} x_{j} + d_{i}^{-} - d_{i}^{+} = R_{i}, i = 2, 3, \dots, t, \\ \sum_{j = 1}^{J} w_{ij} x_{j} + d_{i}^{-} - d_{i}^{+} = 1, i = t + 1, \dots, T \\ x_{j} \in {0, 1}, j = 1, 2, \dots, J, d_{i}^{-}, d_{i}^{+} \geq 0, i = 1, 2, \dots, T \end{matrix} \end{matrix}$ (18)

Here, W_ij is also determined using FPC method in Section 2.1.1, which will be further explained in our case study. Besides, since the optimization model is defined by an objective programing model with only 0-1 decision variables, the Lingo software system is used to solve the constrained optimization problem.

3 Case study

A case study to identify the FCs in design modification of a crawler crane is presented to demonstrate the effectiveness of the proposed approach. The data for this case study were provided Sany Heavy Industry (Shanghai Branch), a large hoisting products manufacture in Shanghai. The company was planning to launch a new type of cranes aiming at improving reliability and customer satisfaction through product design modifications. At the earlier stages, the FCs of the main body of the crane (called “operation device”) were required to be identified since the given design tasks do not require changing all the components. The operation device consists of 10 components including 01, rotational gear; 02, main boom base; 03, load limit device; 04, hydraulic systems of upper-lift; 05, upper lift mast; 06, jib lubbing device; 07, luffing jib boom; 08, luffing jib tip; 09, fixed jib; 10, cantilever mast. The failure modes of components, summarized in Table 4, were obtained from the failure knowledge repository.

Table 4
Failure modes of components for the operation device

Components Failure modes

01 ${FM}_{1}^{1}$ , Rotational bearing vibration; ${FM}_{1}^{2}$ , RG slow motion; ${FM}_{1}^{3}$ , RG abnormal pressure; ${FM}_{1}^{4}$ , Swing brake valve unable to lock

02 ${FM}_{2}^{1}$ , MBB fracture; ${FM}_{2}^{2}$ , MBB vibration; ${FM}_{2}^{3}$ , Anchor bolt looseness; ${FM}_{2}^{4}$ , Anchor bolt breakage

03 ${FM}_{3}^{1}$ , Weight sensor failure; ${FM}_{3}^{2}$ , No display of amplifier circuits; ${FM}_{3}^{3}$ , Actuator damage

04 ${FM}_{4}^{1}$ , Hydraulic shock; ${FM}_{4}^{2}$ , HS leakage; ${FM}_{4}^{3}$ , High pressure ball-valve spun off; ${FM}_{4}^{4}$ , Pressure reducing valve stuck; ${FM}_{4}^{5}$ , Pressure overload of hydraulic system

05 ${FM}_{5}^{1}$ , Mast broken; ${FM}_{5}^{2}$ , Mast deformation; ${FM}_{5}^{3}$ , Failure of performing; ${FM}_{5}^{4}$ , Mast vibration; ${FM}_{5}^{5}$ , Inaccurate placement

06 ${FM}_{6}^{1}$ , Luffing cylinder sinking; ${FM}_{6}^{2}$ , Rotation locating error; ${FM}_{6}^{3}$ , JLD gnawing rail; ${FM}_{6}^{4}$ , Structure fracture; ${FM}_{6}^{5}$ , Luffing turntable vibration; ${FM}_{6}^{6}$ , JLD of UL vibration

…… ……

Components	Failure modes
01	${FM}_{1}^{1}$ , Rotational bearing vibration; ${FM}_{1}^{2}$ , RG slow motion; ${FM}_{1}^{3}$ , RG abnormal pressure; ${FM}_{1}^{4}$ , Swing brake valve unable to lock
02	${FM}_{2}^{1}$ , MBB fracture; ${FM}_{2}^{2}$ , MBB vibration; ${FM}_{2}^{3}$ , Anchor bolt looseness; ${FM}_{2}^{4}$ , Anchor bolt breakage
03	${FM}_{3}^{1}$ , Weight sensor failure; ${FM}_{3}^{2}$ , No display of amplifier circuits; ${FM}_{3}^{3}$ , Actuator damage
04	${FM}_{4}^{1}$ , Hydraulic shock; ${FM}_{4}^{2}$ , HS leakage; ${FM}_{4}^{3}$ , High pressure ball-valve spun off; ${FM}_{4}^{4}$ , Pressure reducing valve stuck; ${FM}_{4}^{5}$ , Pressure overload of hydraulic system
05	${FM}_{5}^{1}$ , Mast broken; ${FM}_{5}^{2}$ , Mast deformation; ${FM}_{5}^{3}$ , Failure of performing; ${FM}_{5}^{4}$ , Mast vibration; ${FM}_{5}^{5}$ , Inaccurate placement
06	${FM}_{6}^{1}$ , Luffing cylinder sinking; ${FM}_{6}^{2}$ , Rotation locating error; ${FM}_{6}^{3}$ , JLD gnawing rail; ${FM}_{6}^{4}$ , Structure fracture; ${FM}_{6}^{5}$ , Luffing turntable vibration; ${FM}_{6}^{6}$ , JLD of UL vibration
……	……

Note: Only partial failure data are provided.

3.1 Calculate the importance weight of each component

The CRs of the operation device include CR₁, lifting weight; CR₂, lifting speed; CR₃, boom luffing; CR₄, lifting height; CR₅, purchase cost; CR₆, transportation convenience; CR₇, disassembly/assembly convenience; CR₈, working stability. A total of 6 customers and 4 experts from areas of crawler crane development (2), technology support (1) and management (1) involved in the design process and related evaluations activities.

Before implementing the QFD process, the FPC method in Section 2.1.1 is employed to obtain the importance weights. The fuzzy comparison matrix $\tilde{A}$ of the expert group is obtained by Step 1 and Step 2 in Section 2.1.1 as Table 5. To save space, only partial data are provided in Table 5.

Table 5
The fuzzy comparison matrix $\tilde{A}$ of the expert group

CR CR₁ CR₂ CR₃ … CR₈

CR₁ (1,1,1) (7/2,5,17/3) (3,5,13/2) … (2/3,3/2,3)

CR₂ (3/17,1/5,2/7) (1,1,1) (5/2,5,20/3) … (1,3/2,2)

CR₃ (2/13,1/5,1/3) (1,1,1) … (2/3,1,2)

… … … … … …

CR₈ (1/3,2/3,3/2) (1/2,2/3,1) (1/2,1,3/2) … (1,1,1)

CR	CR₁	CR₂	CR₃	…	CR₈
CR₁	(1,1,1)	(7/2,5,17/3)	(3,5,13/2)	…	(2/3,3/2,3)
CR₂	(3/17,1/5,2/7)	(1,1,1)	(5/2,5,20/3)	…	(1,3/2,2)
CR₃	(2/13,1/5,1/3)		(1,1,1)	…	(2/3,1,2)
…	…	…	…	…	…
CR₈	(1/3,2/3,3/2)	(1/2,2/3,1)	(1/2,1,3/2)	…	(1,1,1)

Then, the consistency check was executed according to Steps 3 and 4. Without loss of generality, the confidence factor α and satisfaction factor μ were both set as 0.5 by designers, then consistency ratio CR was obtained as 0.086 < 0.1, which means the consistency of fuzzy comparison matrix can be accepted. Finally, according to Step 5 in Section 2.1.1, the weights of the eight CRs were calculated as 0.2, 0.15, 0.15, 0.16, 0.11, 0.08, 0.06 and 0.09, respectively.

To implement the QFD process, CRs were first used to determine the FRs based on the HoQ matrix given in Fig. 4(a). The defuzzied weights of FRs were calculated using Equations (9, 13). Then the eight FRs were used to calculate the CRI of each FC realizing the 8 FRs in Fig. 4(b). The evaluations on the interrelations among FRs and FCs in Fig. 4(a) and (b) were obtained by decision makers. The CRI of each FC was calculated using Equations (9–13), and the normalized CRI_ms were listed in Fig. 4(b).

Fig. 4.

The two stages of QFD processes for the operation device.

3.2 Quantify the failure risk of each component

Step 1. Create the DFCRD for each FC

The DFCRD model for each FC was created considering the causality relationships among failure modes. To save space, only the DFCRDs of FC₁∼FC₆ are shown in Fig. 5. For the rest of this case study, only the parts related with FC₆ are shown in the calculation processes, tables and figures.

Fig. 5.

The DFCRD model for each FC of the operation device.

Step 2. Determine the weights of edges

The FCRs of FC₆ shown in Fig. 5 were quantified by experts, and the edge weights are given in Table 6.

Table 6

The weights of edges in the DFCRD of FC₆

FM	${FM}_{6}^{1}$	${FM}_{6}^{2}$	${FM}_{6}^{3}$	${FM}_{6}^{4}$	${FM}_{6}^{5}$	${FM}_{6}^{6}$
${FM}_{6}^{1}$	—	H	0	0	VH	0
${FM}_{6}^{2}$	M	—	0	M	0	0
${FM}_{6}^{3}$	H	M	—	H	0	0
${FM}_{6}^{4}$	0	0	H	—	0	0
${FM}_{6}^{5}$	0	0	0	M	—	M
${FM}_{6}^{6}$	H	M	0	0	0	—

Step 3. Calculate the weights of vertices

The judgments in TFN form on S, O, D (α = 0) are shown in Table 7, and their weights are evaluated as (0.3, 0.5, 07), (0.1, 0.3, 0.5) and (0.1, 0.1, 0.3). The fuzzy RPNs (α = 0) were then calculated using Equation (15), as shown in Table 7.

Table 7

The judgments on S, O, D and their weights (α = 0)

FM	$\tilde{s}$	$\tilde{o}$	$\tilde{D}$	RPN
${FM}_{6}^{1}$	(3,5,7)	(1,3,5)	(3,5,7)	(3.7,8.3)
${FM}_{6}^{2}$	(5,7,9)	(3,5,7)	(1,3,5)	(2.9,6.2)
${FM}_{6}^{3}$	(1,3,5)	(3,5,7)	(3,5,7)	(2.6,6.6)
${FM}_{6}^{4}$	(1,3,5)	(5,7,9)	(3,5,7)	(3.0,7.0)
${FM}_{6}^{5}$	(3,5,7)	(1,3,5)	(3,5,7)	(3.7,7.6)
${FM}_{6}^{6}$	(3,5,7)	(3,5,7)	(1,3,5)	(5.1,8.4)

Step 4. Calculate the FRI of a component

FC₆ is taken as an example to calculate the FRI₆ based on Equation (17) as follows $\begin{matrix} \tilde{{FII}_{6}} = \prod_{v = 1}^{6} {\tilde{R}}_{6}^{v} \\ + ({\tilde{e}}_{6}^{12} {\tilde{e}}_{6}^{21} \cdot \prod_{v = 3}^{6} {\tilde{R}}_{6}^{v} + {\tilde{e}}_{6}^{34} {\tilde{e}}_{6}^{43} {\tilde{R}}_{6}^{1} {\tilde{R}}_{6}^{2} {\tilde{R}}_{6}^{5} {\tilde{R}}_{6}^{6}) \\ + ({\tilde{e}}_{6}^{15} {\tilde{e}}_{6}^{56} {\tilde{e}}_{6}^{61} \cdot \prod_{v = 2}^{4} {\tilde{R}}_{6}^{v} + {\tilde{e}}_{6}^{24} {\tilde{e}}_{6}^{43} {\tilde{e}}_{6}^{32} {\tilde{R}}_{6}^{1} {\tilde{R}}_{6}^{5} {\tilde{R}}_{6}^{6}) \\ + ({\tilde{e}}_{6}^{15} {\tilde{e}}_{6}^{54} {\tilde{e}}_{6}^{43} {\tilde{e}}_{6}^{31} \\ \cdot {\tilde{R}}_{6}^{2} {\tilde{R}}_{6}^{6} + {\tilde{e}}_{6}^{15} {\tilde{e}}_{6}^{56} {\tilde{e}}_{6}^{62} {\tilde{e}}_{6}^{21} {\tilde{R}}_{6}^{3} {\tilde{R}}_{6}^{4}) \\ + ({\tilde{e}}_{6}^{15} {\tilde{e}}_{6}^{54} {\tilde{e}}_{6}^{43} {\tilde{e}}_{6}^{32} {\tilde{e}}_{6}^{21} \cdot {\tilde{R}}_{6}^{6}) \end{matrix}$

The membership function of ${\tilde{FII}}_{6}$ was obtained using fuzzy arithmetic operations and the results are shown in Table 8.

Table 8

Membership functions of ${\tilde{FII}}_{6}$ with different α levels

α	${\tilde{FRI}}_{6}$	α	${\tilde{FRI}}_{6}$
0	(7765.4, 36212.3)	0.6	(14670.9, 25112.7)
0.2	(9986.6, 32431.6)	0.8	(16793.2, 22134.3)
0.4	(12255.7, 28462.4)	1	(19337.9, 19337.9)

Step 5. Calculate the FRIs of all components

The defuzzied was obtained based on Equation (13). Similarly, the FRIs and s of other components were calculated. The normalized FRI_m based on of each FC is shown in Table 9.

Table 9

The FRI_ms and MNIs of all components

	01	02	03	04	05	06	07	08	09	10
FRI _m	0.12	0.09	0.06	0.10	0.11	0.17	0.09	0.10	0.13	0.05
Order of FRI_m	3	6	7	5	4	1	6	5	2	8
MNI	0.105	0.075	0.065	0.11	0.13	0.15	0.08	0.11	0.105	0.08
Order of MNI	4	6	7	3	2	1	5	3	4	5

3.3 Identify the optimal set of FCs for redesign

After the normalized CRI_m and FRI_m of each FC are obtained through Sections 3.1 and 3.2, the MNI of each component can be calculated using Equation (1), as shown in Table 9 when W_C = W_F = 0.5. To construct the 0-1 objective programming model for identifying the optimal set of FCs, the redesign cost and cycle time of each FC were investigated from the manufacturer of crawler crane as listed in Table 10. In terms of the convenience of maintenance services of components, the weight of each FC in Table 10 was determined by comparing the degrees of maintenance convenience of any two components using FPC method, the weight for maintenance convenience of each FC was determined using Equations (7, 8). Besides, the maximum redesign cost and cycle time were selected as 150,000 Yuan and 110 Days, respectively.

Table 10
The redesign cost, time and weight for maintenance convenience of each FC

01 02 03 04 05 06 07 08 09 10

Redesign cost (1000 Yuan) 32 27 24 57 55 26 42 39 16 46

Cycle time (Days) 25 18 10 23 40 31 35 27 11 38

Weight for maintenance convenience 0.09 0.05 0.13 0.08 0.07 0.17 0.12 0.07 0.15 0.07

	01	02	03	04	05	06	07	08	09	10
Redesign cost (1000 Yuan)	32	27	24	57	55	26	42	39	16	46
Cycle time (Days)	25	18	10	23	40	31	35	27	11	38
Weight for maintenance convenience	0.09	0.05	0.13	0.08	0.07	0.17	0.12	0.07	0.15	0.07

Considering the above three engineering constraints in Table 10 and the MNI of each FC in Table 9, a 0-1 objective programming model was constructed according to Equation (18) as follows $\begin{matrix} \min 0 . 42 d_{1}^{-} + \frac{0 . 27 d_{2}^{+}}{150} + \frac{0 . 19 d_{3}^{+}}{90} + 0 . 12 d_{4}^{-}, \\ s . t . {\begin{matrix} \sum_{j = 1}^{10} {MNI}_{j} x_{j} + d_{1}^{-} - d_{1}^{+} = 1, \\ \sum_{j = 1}^{10} c_{j} x_{j} + d_{2}^{-} - d_{2}^{+} = 150, \\ \sum_{j = 1}^{10} t_{j} x_{j} + d_{3}^{-} - d_{3}^{+} = 110, \\ \sum_{j = 1}^{10} w_{j} x_{j} + d_{4}^{-} - d_{4}^{+} = 1, \\ x_{j} \in {0, 1}, j = 1, 2, \dots, 10, d_{i}^{-}, d_{i}^{+} \geq 0, i = 1, 2, 3, 4 . \end{matrix} \end{matrix}$ where, c_j is the redesign cost of FC_j, t_j is the redesign cycle time of FC_j, and w_j is the weight for maintenance convenience of FC_j. 0.42, 0.27, 0.19 and 0.12 are the weights of the four constraint objectives, determined using FPC method.

In the model, the redesign cost and cycle time are the quantitative engineering constraints, and thus their positive deviation were selected into the objective function. The convenience of maintenance services is a qualitative constraint, and thus the negative deviation was selected into the objective function. Then Lingo software system was used to solve the 0-1 objective programming model and obtain the optimal solution X^* = [1 0 0 0 1 0 1 0 0 0]. Thus, the optimal set of FCs was consisting of FC₁, FC₅ and FC₇.

4 Discussions and conclusions

4.1 Discussions

4.1.1 Comparison with traditional failure risk indices

To compare FRI based on fuzzy permanent function with traditional failure risk index (FRI) of a component, the FRI₁ developed by Rao and Gandhi [16] based on crisp permanent function and the FRI₂ developed by Chen and Ko [12] based on fuzzy ordered weighted geometric averaging (FOWGA) operator were calculated as shown in Table 11, respectively. The FRI₁ was developed using crisp number such as 1-3-9 to quantify the FCR between failure modes and the risk level of each failure mode, while the FRI₂ was calculated using FOWGA operator didn’t consider the FCRs. To compare the three indices,

Table 11
Calculation results of different risk indices

	FRI _m	FRI ₁ m	FRI ₂ m	$\tilde{FRI} (α = 0)$
01	0.12	0.11	0.08	(6192.6, 23834.2)
02	0.09	0.10	0.14	(4373.3, 18571.5)
03	0.06	0.08	0.13	(2772.2, 13263.5)
04	0.1	0.08	0.16	(4834.3, 21022.2)
05	0.11	0.13	0.14	(5485.3, 22269.4)
06	0.17	0.15	0.11	(7965.4, 36212.3)
07	0.09	0.06	0.08	(4162.7, 17696.6)
08	0.1	0.12	0.05	(4724.2, 19526.8)
09	0.13	0.14	0.07	(6826.4, 25115.2)
10	0.05	0.07	0.04	(2533.3, 11961.2)

the defuzzied and normalized FRI_m, FRI_1m and FRI_2m, and the fuzzy FRI at α = 0 are shown in Table 11. The visual comparison among the results of the three indices is illustrated in Fig. 6.

Fig. 6.

Comparison of the three risk indices.

From Table 11 and Fig. 6, the following conclusions are obtained by comparing these indices.

By comparing FRI_m, FRI_1m and $\tilde{FRI}$ , we can find $\tilde{FRI}$ can reflect the fuzzy and vague evaluation information of designers. The defuzzied and normalized FRI_m and FRI_1m of components are roughly the same, as shown in Fig. 6. Thus, we concluded that the results of the proposed approach are acceptable. On the other hand, although FRI_1m reflects the failure effect of failure modes of each component, the fuzzy evaluation results of $\tilde{FRI}$ are closer to engineering practice since designers cannot always provide crisp assessment on the strength of FCR and the severity index, occurrence index and detectability index of each failure mode.

By comparing FRI_m and FRI_2m, we can find obvious difference for the failure priority of each component. For example, FC₂ is higher than FC₁ and FC₆ using the FRI_2m, but FC₁ and FC₆ are obvious higher than FC₂ because FC₁ and FC₆ have more FCRs among failure modes than FC₂. This difference comes from that the FRI_2m only considers the failure mode of one specific component, while FRI_m considers failure risk of each failure mode and FCRs among these failure modes, thus reflecting the failure priority of each component more comprehensively.

4.1.2 Comparison with traditional identification methods

The traditional approach to identify the FCs based on QFD developed by Smith et al. [21] was employed for the comparative study. In the traditional QFD, the ranking order of CRIs of components was used to identify the FCs, which means the components with higher CRIs were identified as the FCs. The values of CRI_m in Fig. 4(b), FRI_m and MNI in Table 9 are visually shown in Fig. 7.

Fig. 7.

Comparison of the three ranking indices.

By comparison, there are many obvious differences between the ranking orders achieved based on MNI and CRI_m, such as the ranking order of FC₁ jumped from the fifth of CRI_m to the fourth of MNI, the ranking of FC₂ jumped from the eighth of CRI_m to the sixth of MNI and other ranking order changes.

The above differences are caused by two reasons. The first reason is the FRIs of components considering design risk, and the second reason is the weights W_C and W_F in calculating the MNI_s. FRIs reveal the failure interactions among failure modes of components. Taken FC₆ as an example, because it its failure modes have more influences on each other than FC₅, the failure risk level of FC₆ is larger than FC₅ by FRI_m ranking as shown in Fig. 7. Obviously, the ranking positions of other components change for the same reason. Besides, the weights W_C and W_F in calculating the MNIs can also result in the changes of ranking positions, which will be further discussed in Section 4.1.3.

4.1.3 Sensitive analysis for W_C and W_F

In the MNIs of components, the W_C and W_F are predetermined by decision makers based on history data or design experience. Different decision makers may give different assessments on the two weights, which may affect the MNIs. Furthermore, the identification results of function components using the 0-1 objective programming model changes with the change of MNIs of components. In order to verify the robustness of the proposed approach, sensitivity analysis was carried out through changing the W_C and W_F as shown in Table 12. Because WC + WF = 1, when WF = 0, the MNIs are the same as the CRIm in Fig. 7. When WF = 1, the MNIs are the same as the FIIM in Fig. 7. Besides, the larger WF also means that the design engineers take design risk into account more seriously when redesigning the crane crawler to improve product reliability. The influence of the MNIs with different W_C and W_F in identifying the function components is shown in Table 12. When 0, 0.2 and 0.4 were selected as the value of WF, all the identification results of function components were FC₁, FC₅ and FC₆. When 0.6, 0.8 and 1.0 were selected as the value of WF, however, the optimal set of function components was consisting of FC₁, FC₅ and FC₇. On the whole, the influence of the two weights to the identification results is insignificant.

Table 12
Deviations of the identification results when W_C and W_F are changed

01 02 03 04 05 06 07 08 09 10

WF = 0.0, WC = 1.0 MNI 0.09 0.06 0.07 0.12 0.15 0.13 0.07 0.12 0.08 0.11

X ^* 1 0 0 0 1 1 0 0 0 0

WF = 0.2, WC = 0.8 MNI 0.096 0.066 0.068 0.116 0.142 0.138 0.074 0.116 0.09 0.098

X ^* 1 0 0 0 1 1 0 0 0 0

WF = 0.4, WC = 0.6 MNI 0.102 0.072 0.066 0.112 0.134 0.146 0.078 0.112 0.1 0.086

X ^* 1 0 0 0 1 1 0 0 0 0

WF = 0.6, WC = 0.4 MNI 0.108 0.078 0.064 0.108 0.126 0.154 0.082 0.108 0.11 0.074

X ^* 1 0 0 0 1 0 1 0 0 0

WF = 0.8, WC = 0.2 MNI 0.114 0.084 0.062 0.104 0.118 0.162 0.086 0.104 0.12 0.062

X ^* 1 0 0 0 1 0 1

WF = 1.0, WC = 0.0 MNI 0.12 0.09 0.06 0.1 0.11 0.17 0.09 0.1 0.13 0.05

X ^* 1 0 0 0 1 0 1 0 0 0

		01	02	03	04	05	06	07	08	09	10
WF = 0.0, WC = 1.0	MNI	0.09	0.06	0.07	0.12	0.15	0.13	0.07	0.12	0.08	0.11
	X ^*	1	0	0	0	1	1	0	0	0	0
WF = 0.2, WC = 0.8	MNI	0.096	0.066	0.068	0.116	0.142	0.138	0.074	0.116	0.09	0.098
	X ^*	1	0	0	0	1	1	0	0	0	0
WF = 0.4, WC = 0.6	MNI	0.102	0.072	0.066	0.112	0.134	0.146	0.078	0.112	0.1	0.086
	X ^*	1	0	0	0	1	1	0	0	0	0
WF = 0.6, WC = 0.4	MNI	0.108	0.078	0.064	0.108	0.126	0.154	0.082	0.108	0.11	0.074
	X ^*	1	0	0	0	1	0	1	0	0	0
WF = 0.8, WC = 0.2	MNI	0.114	0.084	0.062	0.104	0.118	0.162	0.086	0.104	0.12	0.062
	X ^*	1	0	0	0	1	0	1
WF = 1.0, WC = 0.0	MNI	0.12	0.09	0.06	0.1	0.11	0.17	0.09	0.1	0.13	0.05
	X ^*	1	0	0	0	1	0	1	0	0	0

4.2 Conclusions

In the traditional QFD methods for identifying the FCs for redesign, since only customer requirements are considered, these methods have difficulty to ensure the reliability of products. As an important part of products operation data, the failure data collected during the operation stage can be used to guide products’ redesign. Therefore, the CRs and failure data can be integrated to determine the FCs’ importance using fuzzy QFD and FMEA. The main contributions of this research are summarized as follows:

An integrated approach to identify FCs for product resign is proposed. A new index, i.e. MNI, for measuring the degree of modification necessity of each FC is proposed based on their effects on CRs and failure risk degree. Then, the optimal set of FCs for redesign is identified using a 0-1 objective programming model.

By considering the failure causality relations among failure modes of one specific component, a DFCRD model can be constructed. In this model, the failure modes are modeled as vertices, and importance factors of these vertices are calculated by fuzzy RPN. The failure causality relationships are modeled as the directed edges, and the edge weight is quantified using fuzzy set theory. Moreover, fuzzy permanent function is developed to comprehensively measure the failure effects on product reliability considering the failure risk of each failure mode and FCRs among failure modes.

To cope with the subjective and imprecise linguistic terms used by decision makers to describe their judgements, integrated fuzzy approaches, such as TFNs, FPC and FLLSM and so on, are employed in this work to quantify the functional/physical interrelations among FCs, to determine the weight of each component in each qualitative engineering factor and to calculate the importance weight of each objective in the 0-1 objective programming model.

Through the engineering case study, the feasibility and effectiveness of the proposed method for solving product redesign problems are demonstrated. The critical success factors and benefits include three aspects as comparisons and discussions in Section 4.1: (1) Both the customer satisfaction and design risk are considered during the identification process; (2) Failure causality relations are analyzed to provide a means of making use of failure information more effectively for constructing a DFCRD model; (3) Fuzzy set theory and related techniques are explored to deal with imprecise evaluation linguistic terms used in calculation process.

Future work will extend the developed approach for the redesign of the function components. The developed methods can be further improved by considering various data collected in product operation process, such as product performance degradation data and testing data under different testing environments for identification of the FCs for product redesign.

Footnotes

Acknowledgments

This project is supported by National Natural Science Foundation of China (No. 51875345, No. 51475290, No. 51505480), Natural Science Foundation of Jiangsu Province (No. BK20150197).

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An integrated approach to identify function components for product redesign based on analysis of customer requirements and failure risk

Abstract

Keywords

1 Introduction

2 Methodology

2.1.1 Calculation of the importance weight of each customer requirement (CR)

4.1 Discussions

4.1.1 Comparison with traditional failure risk indices

Table 11 Calculation results of different risk indices

Footnotes

Acknowledgments

References

Table 11
Calculation results of different risk indices