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Fuzzy linear programming models for new product design using QFD with FMEA
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Applied Mathematical Modelling 33 (2009) 633–647 www.elsevier.com/locate/apm
Fuzzy linear programming models for new product design using QFD with FMEA Liang-Hsuan Chen *, Wen-Chang Ko Department of Industrial and Information Management, National Cheng Kung University, Tainan, Taiwan, ROC Received 30 November 2006; received in revised form 21 November 2007; accepted 26 November 2007 Available online 4 December 2007
Abstract Quality function deployment (QFD) is a customer-driven approach in processing new product developments in order to maximize customer satisfaction. Determining the fulfillment levels of design requirements (DRs) and parts characteristics (PCs) is an important decision problem during QFD activity processes for new product development. Unlike the existing literature, which mainly focuses on the determination of DRs, this paper proposes fuzzy linear programming models to determine the fulfillment levels of PCs under the requirement to achieve the determined contribution levels of DRs for customer satisfaction. In addition, considering the design risk, this paper incorporates failure modes and effect analysis (FMEA) into QFD processes, which is treated as the constraint in the models. To cope with the vague nature of product development processes, fuzzy approaches are used for both FMEA and QFD. The illustration of the proposed models is performed with a numerical example to demonstrate the applicability in practice. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Fuzzy models; Operations management; Planning and control; Quality function deployment (QFD)
1. Introduction Quality function deployment (QFD) is a customer-driven product development tool, considered as a structured management approach for efficiently translating customer needs into design requirements and parts deployment, as well as manufacturing plans and controls in order to achieve higher customer satisfaction. QFD has been successfully introduced in many industries to improve processes, customer satisfaction, and competitive advantages [1,2]. A general QFD process consists of four phases in order to relate the voice of customer to product design requirements (phase 1), and then translate this into parts characteristics (phase 2), manufacturing operations (phase 3), and production requirements (phase 4). In practice, at the design stage of new product development, a QFD team is organized to implement the first and second phases of QFD processes. The two phases are closely related at the design stage, since the outcomes from the latter phase should make the decisions from the former phase applicable. However, most of the existing research only focuses on *
Corresponding author. Tel.: +886 6 2757575x53140; fax: +886 6 2362162. E-mail address: [email protected] (L.-H. Chen).
0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.11.029
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the first phase of QFD in order to maximize customer satisfaction. Thus, the study that considers the two phases in QFD becomes necessary. For the development at the first phase, the relation matrices between customer requirements (CRs) and design requirements (DRs) and among the DRs are constructed, and the relationships are usually evaluated subjectively by ambiguous or vague judgments, due to the lack of information about the influence of engineering responses on CRs [3,4]. However, those relationships are often treated as crisp variables or linguistic scales [1,2]. For example, the relation strength by which a DR affects a CR is expressed by a scale system, i.e., 1–3–9, or 1–5–9, indicating linguistic expressions such as ‘‘weak”, ‘‘moderate”, and ‘‘strong”, respectively. In order to make the improvements, fuzzy approaches are adopted by several researchers to address diversified and imprecise problems between CRs and DRs and among the DRs [5–10]. However, these existing studies still suffer some drawbacks in approaches, as mentioned by Chen and Weng [3,4] in their fuzzy mathematical programming model for QFD. Furthermore, in order to decrease the risk of new product design, the risk analysis of DRs is necessary during the design level of new product development, and the outcomes are applied to phase 2 as the constraints in determining the achievement levels of parts characteristics (PCs). In this respect, failure mode and effect analysis (FMEA) is a systematic technique for identifying, prioritizing and acting on potential failure modes before the failures occur, so this study also adopts FMEA for the risk analysis in the early stages of new product development. Stamatis [11] described the methods of FMEA and its applications. Ginn et al. [12] proposed a methodology for investigating the interaction between QFD and FMEA by cross-functional and multi-disciplined teamwork. Tan [13] developed a customer-focused methodology for the built-in reliability to maximize customers’ satisfaction based on the constrained resources by combining the FMEA and QFD. However, their studies are only limited to descriptive analyses for obtaining the quality and resource benefits. The methods to carry out the aggregation of QFD and FMEA are not mentioned, and the uncertainty at the product design stage is not considered, either. In this study, we extend Chen and Weng’s fuzzy model [4] by introducing FMEA into the existing fuzzy QFD approach and linking phase 1 to phase 2 of QFD in determining the achievement levels of DRs and part characteristics (PCs) to maximize the customers’ satisfaction. In the following two sections, the approaches of fuzzy QFD and FMEA are introduced. In Section 4, a fuzzy linear programming model is developed to determine the achievement degrees of PCs, constrained by the need of DRs in phase 1 and the risk ratings of PCs according to the design of FMEA on DRs. A semiconductor packing example is presented to demonstrate our approach in Section 5. Finally, the concluding remarks are provided. 2. The approach of fuzzy QFD For implementing QFD processes, a relation matrix, also called a House of Quality (HOQ), is usually used for each phase to construct the input–output relationships in determining the achievement priority or level of output variables. In practice, the major work in phase 1 of the QFD processes is to determine the achievement priority or level of DRs based on the importance of each CR, the relationships between CRs and DRs as well as among the DRs. Based on the results from the first QFD process, the similar work is performed with DRs and PCs in phase 2. Fig. 1 demonstrates the relation matrices of the two phases. Obviously, from the figure the fulfillment levels of PCs should make the DRs applicable in meeting the customers’ satisfaction. In Fig. 1, R1,ij denotes the relation level in terms of score between the ith CR and jth DR, and r1,jn is the correlation score between the jth and nth DRs in the firstPphase of QFD. The notations k and W represent the importance score and rating in the two matrices, where k ¼ 1. For symbolizations, the subscripts, 1 and 2, are added to the associated variables, such as R, r, k, and W, to denote phase 1 and phase 2 of QFD, respectively, in Fig. 1 and the equations hereafter. Considering the correlations among DRs at the first stage of QFD, Wasserman [14] proposed a formulation to calculate the normalized relationship value between CRs and DRs as PJ R1;if r1;fj ; ð1Þ R01;ij ¼ PJ f¼1 PJ j¼1 f¼1 R1;if r 1;fj
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Fig. 1. The relationship matrices for phases 1 and 2 of the QFD processes.
P where R01;ij is the normalized relationship value between CRi and DRj, i = 1, . . . , I; j = 1, . . . , J; j R01;ij ¼ 1 for each i. As mentioned before, the relationships are quantified as crisp values to denote the relationship degrees between CRs and DRs, as well as the correlation among DRs. For reflecting the imprecise nature of the relationships, Chen and Weng [4] proposed a fuzzified quantitative formulation as follows: PJ e 1;if ~r1;fj R 0 e ; ð2Þ R 1;ij ¼ PJ f¼1 PJ e r1;fj j¼1 f¼1 R 1;if ~ e 1;if and ~r1;fj are described by linguistic terms and defined as the fuzzy subsets of [0, 1]. Actually, a fuzzy where R e 1;if at the a set can be fully and uniquely represented by its a-cuts [15]. For example, the a-cut of the fuzzy set R L U level, a 2 [0, 1], can be denoted by its lower and upper bounds as ½ðR1;if Þa ; ðR1;if Þa , which is defined by ðR1;if ÞLa ¼ inf fxjle ðxÞ P ag; R 1;if x2½0;1
ð3aÞ
and
U
ðR1;if Þa ¼ sup fxjle ðxÞ P ag; x2½0;1
ð3bÞ
R 1;if
e 1;if . where le ðxÞ is the membership degree of x belonging to R R 1;if Based on a-cuts and the extension principle [15,16], the membership functions of the fuzzy normalized relae u;if and ~ru;fj , tionship in phases 1 and 2 can be defined by the lower and upper bounds of each a-cut of R u = 1, 2. In addition, Chen and Weng [4] further proposed a modified formulation to obtain a more precise representation of the fuzzy normalized relationship, of which the lower and upper bounds of the membership function at each a-cut are formulated as PJ L L L f¼1 ðR1;if Þa ðr1;fj Þa 0 ; ð4aÞ mðR1;ij Þa ¼ PJ PJ PJ U U L L m¼1 f¼1 ðR1;if Þa ðr 1;fm Þa þ f¼1 ðR1;if Þa ðr1;fj Þa m–j U mðR01;ij Þa
¼
PJ
U U f¼1 ðR1;if Þa ðr1;fj Þa PJ PJ PJ L L U U m¼1 f¼1 ðR1;if Þa ðr 1;fm Þa þ f¼1 ðR1;if Þa ðr1;fj Þa m–j
:
ð4bÞ
Once the modified lower and upper bounds of a-cuts of the fuzzy normalized relationship are obtained, the e 1;j for the jth DR can be determined in the form of a-cuts, expressed as fuzzy technical importance rating W
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" e 1;j Þ ¼ ½ðW ðW a
L U 1;j Þa ; ðW 1;j Þa
¼
I X
k 1;i
L mðR01;ij Þa ;
i¼1
I X
# k 1;i
U mðR01;ij Þa
;
ð5Þ
i¼1
which will be employed to find out the optimal fulfillment level of each DR in phase 1 of the QFD process. Using the fuzzy technical importance ratings of DRs, Chen and Weng [4] proposed a fuzzy linear programming model in which the decision variable, x1,j, is defined in percentage to denote the level of fulfillment percentage of the DRj, j = 1, . . . , J, i.e., x1,j 2 [0, 1]. The x1,j = 0 implies that the DR has a basic design requirement so that no more efforts and resources are needed. In their model, the increment unit cost to e 1;j , to reflect its fuzzy nature at the design plan stage. achieve the fulfillment level is used as a fuzzy number, C With the fulfillment percentage of DRj, a corresponding percentage of increment unit cost is required to enhance the characteristic of product or service. The total incremental unit cost cannot exceed a budget constraint in this stage. Besides, the impacts on customer satisfaction of various DRs are prioritized, and business competition and technological difficulties are also considered in the model. If the sth DR is preferred to the pth e 1;p x1;p P 0 is needed. In order to solve the e 1;s x1;s W DR in terms of customer satisfaction, a constraint of W e 1;j are placed in the e 1;j and C fuzzy linear programming model, the lower and upper bounds of a-cuts of W model to find the corresponding values of x1,j and determine the lower and upper bounds of the customer satisfaction at each a-cut, respectively. Thus, a fuzzy linear programming model is formulated as L
ðZ 1 Þa ¼ max
J X
L
ðW 1;j Þa x1;j ;
j¼1
s:t:
J X
U
ðC 1;j Þa x1;j 6 B1 ;
j¼1 L
U
ðW 1;s Þa x1;s ðW 1;p Þa x1;p P 0; 0 6 ej 6 x1;j 6 g1;j 6 1
8j;
s; p 2 f1; 2; . . . ; J g; J X U U ðZ 1 Þa ¼ max ðW 1;j Þa x1;j ;
ð6aÞ
j¼1
s:t:
J X
ðC 1;j ÞLa x1;j 6 B1 ;
j¼1 U
L
ðW 1;s Þa x1;s ðW 1;p Þa x1;p P 0; 0 6 ej 6 x1;j 6 g1;j 6 1 8j; s; p 2 f1; 2; . . . ; J g; ðZ 1 ÞLa
ð6bÞ ðZ 1 ÞU a
where B1 is the budget limitation, and as well as represent the lower and upper bounds of objective values, i.e., customer satisfaction in phase 1 of the QFD process, at each a-cut, respectively. In addition, ej and g1,j denote the possible range of the fulfillment level of one DR, indicating the minimum required level due to the business competition and the maximum level due to technical difficulty, respectively. From the above model, the achievement levels of DRs at different a levels can be obtained. It is worth noting that, for maximizing the Model (6), the fulfillment levels of some DRs determined in Model (6a) for the lower bound of the customer satisfaction may be greater than those in Model (6b) for the upper bound of the customer satisfaction. However, these results of the fulfillment degree still provide an interval for building up the membership functions of fulfillment degree of DRs in the fuzzy sense. To avoid the confusions, the optimal achievement levels ðLÞ ðUÞ of DRs determined for the lower and upper bounds of the customer satisfaction are denoted as x1;j and x1;j , respectively. Similar cases are applied to PCs in phase 2. To develop the fuzzy mathematical model for phase 2 of the QFD process, the importance scores of the DRs in the second phase should be determined. In the existing literature, the importance ratings W1,j of the DRs in phase 1 are adopted as the importance scores k2,j for the deployment of phase 2 in QFD processes [17]. However, the achievement levels of DRs will affect the PCs’ deployment. Therefore, we proposed a
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weighted-average equation to determine the importance score by using both the importance ratings and the achievement levels of DRs in phase 1, which is formulated as follows: P L ðLÞ U ðUÞ 1 l l ½ðW 1;j Þal ðx1;j Þal þ ðW 1;j Þal ðx1;j Þal l 2 0 P ; j ¼ 1; . . . ; J : ð7Þ k 2;j ¼ l ll L
ðLÞ
U
ðUÞ
l denotes the membership degree of 1=2½ðW 1;j Þal ðx1;j Þal þ ðW 1;j Þal ðx1;j Þal belonging to In the above equation, l e 1;j~x1;j determined by the a-cut and extension principle. Furthermore, the normalized the fuzzy number W importance score of each DR is expressed as k 02;j k 2;j ¼ PJ 0 : j¼1 k 2;j
ð8Þ
Similar with (4) in phase 1, the fuzzy normalized relationship between DRs and PCs in phase 2 is formulated in terms of the lower and upper bounds as Pk L L L l¼1 ðR2;jl Þa ðr2;lk Þa ; ð9aÞ mðR02;jk Þa ¼ Pk Pk P U U k L L m¼1 l¼1 ðR2;jl Þa ðr 2;lm Þa þ l¼1 ðR2;jl Þa ðr 2;lk Þa m–j U mðR02;jk Þa
¼ Pk
m¼1 m–j
Pk
U U l¼1 ðR2;jl Þa ðr2;lk Þa P L L k U U l¼1 ðR2;jl Þa ðr 2;lm Þa þ l¼1 ðR2;jl Þa ðr 2;lk Þa
Pk
:
Then, the PCs’ fuzzy importance ratings are calculated below: " # J J X X L U L U 0 0 e 2;k Þ ¼ ½ðW 2;k Þ ; ðW 2;k Þ ¼ ðW k 2;j mðR2;jk Þa ; k 2;j mðR2;jk Þa : a a a j¼1
ð9bÞ
ð10Þ
j¼1
3. The approach of fuzzy FMEA For the reduction of design risk, designers usually take risk analysis of DRs into account so as to build up the requirements of parts characteristics (PCs) in phase 2 of the QFD process during the design stage of new product development. Failure mode and effect analysis (FMEA) is an effective approach to provide information for making risk management decisions. This study also adopts this approach to make risk analysis of each DR under the fuzzy environment. A conventional form of FMEA includes (i) the design function of parts, (ii) the potential failure mode (categories of failure), (iii) the potential effects of failure (measured by the severity index), (iv) the potential causes of failure (measured by the occurrence (frequency) index), (v) the detection method (measured by the detectability index), and (vi) the risk priority number (RPN). The RPN is used to evaluate the risk level of a part’s failure mode in design stage, and is determined by the multiplication of three characteristic failure mode indexes, i.e., the severity of the potential failure (S), the frequency of potential failure (O), and the detectability index (D), respectively, as RPN ¼ S O D:
ð11Þ
The three indices in (11) are defined on the same scale level, such as the 10-point system, to identify the various levels of risk situation. Tables 1–3 present the scales for the S, O and D indices [18]. In general, the design function (or DR in QFD) of a part may have more than one failure mode in FMEA, and each failure mode may be described by more than one effect of failure. In addition, one effect of failure may result from more than one cause of failure, which could be detected by several detection methods. The detectable degree of each cause of failure is measured by the detectability index (D). Each failure mode is evaluated by RPN in terms of the three indices, i.e., S, O, and D. Usually, an assessment of those indices is subjective and qualitatively described in natural language. With regard to this, sometimes it is difficult to apply conventional FMEA to determine the rating of failure indices. For example, the frequency of potential failure
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Table 1 Scales for severity Severity
Rating
Remote Low
1 2 3
Moderate
4 5 6
High
7 8
Very high
9 10
Table 2 Scales for occurrence Frequency of occurrence
Rating
Possible failure rate
Remote Low
1 2 3
<1:20,000 1:20,000 1:10,000
Moderate
4 5 6
1:2000 1:1000 1:200
High
7 8
1:100 1:20
9 10
1:10 1:2
Very high
Table 3 Scales for detectability Detectability
Rating
Probability of detectability (%)
Remote Low
1 2 3
86–100 76–85 66–75
Moderate
4 5 6
56–65 46–55 36–45
High
7 8
26–35 16–25
9 10
6–15 0–5
Very high
could be evaluated to be close to 1:50. The existing measurement system cannot give a suitable rating for this situation. Appropriately, such a measurement can be described by a linguistic term, such as ‘‘medium high”. Some researchers have proposed fuzzy set theory to deal with the above problems [18–21]. Xu et al. [19] presented a fuzzy-logic-based method and an assessment expert system for FMEA of a diesel engine’s turbocharger system. Pillay and Wang [18] used a fuzzy rules base and grey relation theory to determine the RPN by multiplying the factor scores of the probability of failure, severity and detectability. It is necessary to have adequate experts’ knowledge to build up the fuzzy rule base for reasoning to obtain an acceptable RPN. However, it could be difficult to figure out the suitable inference system, because integrating the related experiences is not easy during the design stage.
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In this study, we make the risk analysis of DRs by performing FMEA at the design stage. For describing the assessments meaningfully, the three failure indices are evaluated by linguistic terms and fuzzy set theorems are employed to demonstrate the fuzziness of the risk analysis. The fuzzified RPN of each DR can be expressed as follows:
es D e tÞ ; Sr O ðRPNÞj ¼ maxð e j
j ¼ 1; 2; . . . ; J ;
r;s;t
ð12Þ
e s , and D e t are defined as the fuzzy subsets of [0, 1]. The ðRPNÞj is determined as the maximum of where e S r, O the three indices’ product from all sets of S, O, and D, if there is more than one set, to represent the fuzzy risk priority level of the jth DR. Furthermore, considering the different importance of each failure index, various indices can have a different weights to determine ðRPNÞj . According to the characteristic of the multiplication of the three indices in (12), this study uses a fuzzy ordered weighted geometric averaging (FOWGA) operator [22] to modify Eq. (12). A FOWGA operator can be used to aggregate m (>1) fuzzy sets, and the formulation is expressed as [22] a2 ; . . . ; ~ am Þ ¼ f ð~ a1 ; ~
m Y
ð~ bi Þ w i ;
ð13Þ
i¼1
P where ~ bi is the ith largest one in m fuzzy numbers ~ak , k = 1, . . . , m; wi is the weight of the ~bi and mi¼1 wi ¼ 1, wi 2 [0, 1]. Following FOWGA, (12) can be reformulated as
e DÞ e ¼ max ðRPNÞj ¼ f ð e S; O; j r;s;t
3 Y ð~ bi Þwj i ;
ð14Þ
i¼1
e s; D e t Þ. In Eq. (14), the weighting vector of w = (w1, w2, w3)T can be where ~ bi is the ith largest set of the ð e S r; O determined according to designers’ or QFD team members’ experiences. For determining ðRPNÞj , its membership function can be constructed by deriving the lower and upper bounds of the a-cuts of ðRPNÞj as ðRPNj ÞLa ¼ max r;s;t
U
ðRPNj Þa ¼ max r;s;t
3 Y ½ðbi Þwj i La
ð15aÞ
and
i¼1 3 Y w U ½ðbi Þj i a :
ð15bÞ
i¼1
Once the membership function of ðRPNÞj is built-up, its defuzzified value can be obtained by the following formulation: P L U 0i f1=2½ðRPNj Þai þ ðRPNj Þai g l 0 P 0 ; ð16Þ ðRPNj Þ ¼ i i il
0i is the membership where l degree of 1=2½ðRPNj ÞLai þ ðRPNj ÞU ai belonging to the fuzzy number ðRPNÞj . The defuzzified value of ðRPNÞj represents the risk level of the jth DR, and is applied to phase 2 for parts design deployment in determining the fuzzy risk rating of PCs with the consideration of the fuzzy normalized relationship. The formulation is expressed as " # J J X X L U 0 L 0 U 0 0 e k Þ ¼ ½ðRik Þ ; ðRik Þ ¼ ðRPNj Þ mðR Þ ; ðRPNj Þ mðR Þ : ð17Þ ð Ri a
a
2;jk a
a
j¼1
2;jk a
j¼1
The fuzzy risk rating of each PC is used in phase 2 to determine the fulfillment level of each PC in the design level to satisfy the design requirements and finally maximize customers’ satisfaction. 4. Fuzzy linear model with FMEA From Model (6), the fulfillment levels of DRs are obtained in phase 1 of the QFD processes in order to maximize customer satisfaction. Extending Model (6) to phase 2 by including the risk analysis of DRs via
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FMEA, this study proposed a fuzzy linear programming model to determine the fulfillment level of the PCs in phase 2. Similar with Model (6), the notation of x2,k is defined as the fulfillment level of the kth PC in the proposed model. By performing FMEA on DRs, the fuzzy risk ratings of PCs are obtained by Eq. (17) and used as the constraint in the proposed model. In addition, the determination of the fulfillment levels of the PCs has to be done that the DRs can achieve customer satisfaction. The contribution to the customers’ satisfaction by the jth DR due to the design satishP i K e 0 Þ~x2;k , which should be greater than or equal to mð R faction of PCs in phase 2 can be expressed as k 2;j 2;jk k¼1 e 1;j~x1;j , in phase 1 to realize the customer satisfaction. Moreover, the budget the contribution of the jth DR, W limitation, preemptive priority, and technological ability for PCs are also taken into account in the proposed model. Taking the above considerations as the constraints and using the PCs’ fuzzy importance rating of Eq. (10) as well as the decision variables, x2,k, as the objective function, a fuzzy linear programming model is formulated in the a-cut form as ðZ 2 ÞLa ¼ max
K X k¼1
s:t:
ðW 2;k ÞLa x2;k ; "
k 2;j
K X
# L mðR02;jk Þa
x2;k P W 1;j x1;j ; j ¼ 1; . . . ; J ;
k¼1 K X
U
ðRik Þa x2;k 6 H ;
k¼1 K X
ðC 2;k ÞU a x2;k 6 B2 ;
k¼1
ðW 2;s ÞLa x2;s ðW 2;p ÞU a x2;p P 0; 8k;
0 6 x2;k 6 g2;k 6 1 s; p 2 f1; 2; . . . ; Kg; U
ðZ 2 Þa ¼ max
K X
U
ðW 2;k Þa x2;k ;
k¼1
"
s:t: k 2;j
ð18aÞ
K X
# mðR02;jk ÞU a
x2;k P W 1;j x1;j ; j ¼ 1; . . . ; J ;
k¼1 K X
L
ðRik Þa x2;k 6 H ;
k¼1 K X
L
ðC 2;k Þa x2;k 6 B2 ;
k¼1 U
L
ðW 2;s Þa x2;s ðW 2;p Þa x2;p P 0; 0 6 x2;k 6 g2;k 6 1 s; p 2 f1; 2; . . . ; Kg:
8k; ð18bÞ
In the above model, the objective function is to find the maximum satisfaction of design requirement in phase L U 2. The values of ðZ 2 Þa and ðZ 2 Þa represent the lower and upper bounds of objective values, respectively, at each a-cut. Similar with the cases in Model (6), the optimal fulfillment levels of PCs determined in Model ðLÞ ðUÞ L U (18) for ðZ 2 Þa and ðZ 2 Þa are represented as x2;k and x2;k , respectively. In the constraints, H is a threshold e 2;k and g2,k are the increment unit cost to achieve the fulfillment level of risk; B2 is a budgetary limitation; C and technological difficulty of the PCs, respectively. These parameters are determined by the QFD team. In
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addition, for simplification and application purposes, the crisp contribution level of each DR is taken as the e 1;j~x1;j action by the QFD team and is adopted in the model. In Model (18), W1,jx1,j is a defuzzified value of W 0 and calculated by Eq. (7), so that W1,jx1,j is equal to k 2;j . 5. An illustrative example In this section, a semiconductor packing case of the turbo thermal ball grid array (T2-BGA) package is used to exemplify the applicability of the proposed models. Since a heat slug is inserted into the molding compound of a plastic BGA (PBGA), T2-BGA has the advantages of high heat dissipation and good ground shielding to reduce the electromagnetic interference (EMI) and electromagnetic compatibility (EMC) effects between adjacent traces; however, it also produces some problems due to the more complex structure of the package when compared to PBGA [23]. Fig. 2 displays the cross-sections of PBGA and T2-BGA. Based on the above considerations of T2-BGA, a QFD team collects five customer requirements (CRs) and proposes five design requirements (DRs) in the earlier design process. The associated house of quality (HOQ) is shown in Fig. 3. In phase 1, the CRs are ‘‘package profile” (CR1), ‘‘thermal performance” (CR2), ‘‘electrical performance” (CR3), ‘‘reliability” (CR4), and ‘‘co-planarity” (CR5), and the five DRs are ‘‘heat slug exposed area” (DR1), ‘‘heat slug attached material” (DR2), ‘‘height of heat slug” (DR3), ‘‘copper pattern” (DR4), and e 1;if between CRs and DRs as well as the relevant relations ‘‘molding flow” (DR5). Firstly, the fuzzy relations R ~r1;fj among DRs are determined. These relationships are denoted as ‘‘weak”, ‘‘moderate”, or ‘‘strong”, which
Chip
Gold wire
Solder ball PBGA
Heat slug
Molding compound
T2-BGA
Fig. 2. The cross-sections of PBGA and T2-BGA.
Fig. 3. The phase 1 for a T2-BGA package.
Substrate
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are translated into the triangular fuzzy numbers as a 3-element set, (0, 0.2, 0.4), (0.3, 0.5, 0.7), (0.6, 0.8, 1), e 1;if ¼ ð0:6; 0:8; 1Þ is formulated as respectively, with membership functions in Fig. 4. As an example, R 8 < ðR1;if 0:6Þ ; 0:6 6 R1;if 6 0:8; ð0:80:6Þ le ðR1;if Þ ¼ ð1R Þ R 1;if 1;if : ; 0:8 6 R1;if 6 1: ð10:8Þ The middle value in the 3-element set represents the most likely one with the membership degree equivalent to 1, such as lstrong (0.8) = 1. In this study, the system of 0.2–0.5–0.8 is used to express the most likely value of the linguistic terms on weak–moderate–strong. Considering the linguistic relationships in Figs. 3 and 4, the fuzzy normalized relationship at each a-cut can be calculated using Eq. (4). For doing this, the lower and upper bounds of each e 1;if and ~r1;fj should be determined beforehand based on their membership functions. Actually, the a-cut of R lower and upper bounds of each a-cut can be represented as the functions of a (2[0, 1]). For example, the L U a-cut of the membership function (0.6, 0.8, 1) of ‘‘strong” is expressed as ½ðR1;if Þa ; ðR1;if Þa ¼ ½0:6 þ 0:2a; 1 0:2a. Without any biases, the a levels in this paper are evenly distributed in [0, 1]. Let am denote the mth a level, and am = m/n, m 2 {0, 1, . . . , n}, such that the distance between each two adjacent a levels is equal, i.e. am am1 = 1/n, m P 1. e 1;j Þ ; j ¼ 1; 2; . . . ; J, at each a The lower and upper bounds of the fuzzy technical importance rating ð W a level, is obtained by Eq. (5) to determine the importance priority of each DR, as listed in Table 4. They are applied to Model (6) to solve the fuzzy linear model. The budgetary cost is limited to 1 unit, i.e., B1 = 1.0, in Model (6); meanwhile, we suppose that the QFD team has prioritized DRs as DR4 DR1 DR2 and DR3 DR5 for the design preference, where ‘‘” means ‘‘is more preferred than”. Table 5 lists the associated data for the fuzzy linear Model (6). Applying fuzzy technical importance ratings, fuzzy increment cost, and the associated constraints to Model (6) at several a levels, the fulfillment level of each DR and the total customer satisfaction degree (the objective function value) can be obtained. In fact, with the fuzzy sense, the a level can be interpreted as the confidence
Fig. 4. The membership functions of linguistic relationship in the HOQs.
Table 4 The lower and upper bounds (%) of fuzzy technical importance ratings at different a levels a
W L1;1
WU 1;1
W L1;2
WU 1;2
W L1;3
WU 1;3
W L1;4
WU 1;4
W L1;5
WU 1;5
1 0.8 0.6 0.4 0.2 0
16.9 13.5 10.1 6.7 3.4 0
16.9 20.2 23.6 27.0 30.3 33.7
20.8 18.3 16.1 14.1 12.2 10.6
20.8 23.4 26.2 29.1 32.1 35.1
5.9 5.1 4.3 3.6 3.0 2.5
5.9 6.7 7.6 8.6 9.5 10.0
37.0 34.0 30.0 27.0 24.0 20.0
37.1 40.5 43.9 47.3 50.6 54.0
8.2 6.2 4.4 2.7 1.3 0
8.2 10.4 12.7 15.2 17.7 20.4
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Table 5 The cost, business competition, and technical difficulty for each DR Fulfillment level
e 1;j C
a-cut
ej
g1,j
x1,1 x1,2 x1,3 x1,4 x1,5
(0.2, 0.3, 0.4) (0.8, 0.9, 1) (0.4, 0.5, 0.6) (0.3, 0.4, 0.5) (0.1, 0.2, 0.3)
[0.2 + 0.1a, 0.4–0.1a] [0.8 + 0.1a, 1–0.1a] [0.4 + 0.1a, 0.6–0.1a] [0.3 + 0.1a, 0.5–0.1a] [0.1 + 0.1a, 0.3–0.1a]
0.1 0.1 0.1 0.1 0.5
1 1 1 1 1
degree [24,25]. In this example, the settings of a levels less than 0.8 will produce infeasible regions in solving the fuzzy linear model. From the property of a fuzzy number, this means the confidence level of fuzzy inputs should be higher to avoid producing the wider ranges of a-cuts. Table 6 shows the outcomes of the total customer satisfaction degree and fulfillment levels of each DR for the Model (6) based on three a levels (0.8, 0.9, 1). The decision variable x1,j = 100% denotes complete fulfillment of the jth DR. In this example, DR4 should have the better quality level in order to achieve the CRs’ satisfaction degree. On the other hand, DR2 is smaller than the others, due to the high incremental cost, as shown in Table 5. In Table 6, it is also ðLÞ ðUÞ noted that x1;3 is greater than x1;3 at a level of.9. As described before, for maximizing the model, the optimal values of some decision variables for Z L1 may be greater than those for Z U 1 . However, the maximum customer L is greater than Z at the level, as desired for the models. satisfaction Z U 1 1 From the outcomes from phase 1 of QFD, the fuzzy technical importance rating and fulfillment level of each DR are applied to (7) and normalized by (8) to determine the importance score in phase 2 of the QFD. They are calculated as 0.13, 0.042, 0.082, 0.663 and 0.083 for the five DRs, respectively. Furthermore, the DRs from phase 1 of the QFD processes are used to design FMEA for evaluating the potential failure modes and effects for risk analysis. An analysis of the T2-BGA package is listed in Table 7, in which the three failure indices are described as linguistic terms by ‘‘R (remote)”, ‘‘L (low)”, ‘‘M (moderate)”, ‘‘H (high)”, and ‘‘VH (very high)”. For the following resolution procedure, these linguistic terms should be translated to fuzzy numbers. Their definitions are (0, 0, 0.1, 0.2), (0.1, 0.25, 0.25, 0.4), (0.3, 0.5, 0.5, 0.7), (0.6, 0.75, 0.75, 0.9), and (0.8, 0.8, 1, 1), respectively, as shown in Fig. 5. For obtaining fuzzy risk priority number (RPN), the upper and lower bounds of a-cuts of the three fuzzy e s; D e t , and their weights should be determined beforehand. Suppose that the set of weights failure indices e S r; O is w = {0.7, 0.21, 0.09}, determined by the QFD team. The fuzzy RPN of each DR is constructed by (15) and then defuzzified by (16) for risk ratings in phase 2 of the QFD processes. They are calculated as 6.631, 8.365, 7.227, 6.631, and 6.219 for the five DRs, respectively. The QFD team investigates the part characteristics of the principal parts or subsystem to deploy in phase 2 of the QFD. There are five part characteristics (PCs), including ‘‘profile of heat slug” (PC1), ‘‘leg shape of heat slug” (PC2), ‘‘plating quality of heat slug” (PC3), ‘‘jointability between attached material and molding compound” (PC4), and ‘‘jointability among attached material, heat slug and substrate” (PC5), as shown in Fig. 6. Similar with phase 1 of the QFD, considering the relationships between DRs and PCs as well as among the PCs in Fig. 6 and the membership functions of linguistic relationships in Fig. 4, the fuzzy normalized relationship at each a-cut can be obtained using (9). The lower and upper bounds of the fuzzy importance rating e 2;k Þ , k = 1, 2, . . . , K, at each a level, are obtained by (10) to determine the importance priority of of PCs, ð W a e 2;k Þ at different a levels are listed in Table 8. The fuzzy risk rateach PC. The lower and upper bounds of ð W a ings of each PC at each a-cut can be calculated by (17) and are presented in Table 9. The resulting fuzzy importance and risk level ratings of PCs with the associated constraints listed in Table 10 are applied to Model (18) at several a levels to solve the fuzzy linear model. The fulfillment level of each PC Table 6 Outcomes of the total customer satisfaction degree and fulfillment level of each DR a
Z L1
ZU 1
x1;1
ðLÞ
x1;1
ðUÞ
x1;2
ðLÞ
x1;2
ðUÞ
x1;3
ðLÞ
x1;3
ðUÞ
x1;4
ðLÞ
x1;4
ðUÞ
x1;5
ðLÞ
x1;5
ðUÞ
1 0.9 0.8
0.514 0.378 0.300
0.514 0.584 0.682
.256 0.147 0.177
0.256 0.483 0.844
0.1 0.1 0.1
0.1 0.1 0.1
0.667 0.900 1
0.667 0.583 0.429
1 0.731 0.527
1 1 1
0.500 0.500 0.500
0.500 0.500 0.500
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Table 7 T2-BGA package FMEA DRj Design function of part
Potential failure mode
Potential effects of failure
S
Potential causes of failure
O
Detection method
D
Heat slug exposed area
Distortion of the heat slug Warpage of the heat slug Plating layer peeling off
EMC overflow
H
Packing/Transportation
M
Profile inspection
L
EMC overflow
H
Heat slug forming
M
Profile inspection
M
Package looks defective
H
plating solution contaminated
M
Surface inspection
L
Heat slug attached material
Molding failure
Package profile defective
VH
Turbulence
M
Simulation
H
Height of heat slug
Plating layer peeling off Plating thickness inconsistent
Package looks defective Substrate damage
M
Plating solution contaminated
L
Surface inspection
R
H
Stress of the HS
M
Stress measurement
H
EMC overflow
H
L
Package looks defective Short circuit
M M
L L
HS Height inspection Eyes inspection Surface inspection
M
Unusual height
Heat slug height out of spec. Unsuitable packaging Burr
R R
Copper pattern
Popcorn
Reliability test failed
H
Pattern distribution
M
Reliability test
M
Molding flow
Popcorn
Reliability test failed
H
Turbulence
M
Simulation
L
Fig. 5. The membership functions of linguistic terms in the FMEA.
and the total DR satisfaction degree (the objective function value) can then be obtained. In phase 2 of the QFD processes, the budgetary cost is limited to 1.5 unit, i.e., B2 = 1.5 in (18); the risk threshold H is bounded to 25 in this case. For phase 2, the set of DRs’ importance scores by (7) is {0.064, 0.021, 0.041, 0.330, 0.041}. In addition, suppose that the QFD team concerns the preemptive priority of PCs as PC5 PC2 for the design process. In solving Model (18), the settings of a levels less than 0.8 will produce infeasible regions, as happened in phase 1. This means that the same confidence level (P0.8) should be adopted for both phases in this QFD design. Appling three a levels (0.8, 0.9, 01) to Model (18), the total DR satisfaction degree and the fulfillment level of each PC are shown in Fig. 7 and Table 11, respectively. In Fig. 7, the most possible satisfaction degree of DR is around 69%. The fulfillment level of decision variable x2,k = 100%, which denotes complete fulfillment of the kth PC. In Table 11, the fulfillment levels of PC1 and PC5 represent the best performance level in order to achieve the DR’s satisfaction degree in this case. In addition, similar with the outcomes by Model ðLÞ ðUÞ (6), x2;4 is greater than x2;4 at a levels of 0.8 and 0.9, as shown in Table 11. As described previously, the satisfaction of the jth DR in phase 2 should be greater than or equal to the contribution of the jth DR in phase 1 in order to realize customer satisfaction. For simplification, considering
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Fig. 6. The Phase 2 for a T2-BGA package. Table 8 The lower and upper bounds (%) of fuzzy importance ratings of PCs in various a a
W L2;1
WU 2;1
W L2;2
WU 2;2
W L2;3
WU 2;3
W L2;4
WU 2;4
W L2;5
WU 2;5
1 0.8 0.6 0.4 0.2 0
39.7 33.5 28.1 23.3 19.2 15.7
39.7 46.7 54.3 62.5 71.0 79.4
12.3 10.4 8.8 7.2 5.8 4.6
12.3 14.2 16.2 18.3 20.4 22.4
3.0 2.5 2.1 1.7 1.4 1.1
3.0 3.4 3.9 4.4 4.9 5.3
3.6 2.8 2.1 1.5 1.0 .6
3.6 4.6 5.8 7.2 8.8 10.8
14.7 11.0 7.8 5.2 2.9 1.1
14.7 19.0 23.8 29.2 35.0 41.0
Table 9 The fuzzy risk ratings of PCs in various a a
RiL1
RiU 1
RiL2
RiU 2
RiL3
RiU 3
RiL4
RiU 4
RiL5
RiU 5
1 0.8 0.6 0.4 0.2 0
7.17 6.17 5.27 4.48 3.80 3.20
7.17 8.27 9.46 10.72 12.01 13.26
8.94 7.69 6.54 5.50 4.56 3.73
8.94 10.26 11.64 13.06 14.48 15.86
2.62 2.23 1.87 1.53 1.23 .96
2.62 3.03 3.45 3.87 4.28 4.68
4.45 3.58 2.83 2.17 1.60 1.10
4.45 5.45 6.61 7.92 9.43 11.11
5.12 4.36 3.69 3.09 2.56 2.09
5.12 5.95 6.86 7.83 8.84 9.87
Table 10 The cost and technical ability for each PC Fulfillment level
e 2;k C
a-cut
g2,k
x2,1 x2,2 x2,3 x2,4 x2,5
(0.2, 0.3, 0.4) (0.2, 0.3, 0.4) (0.4, 0.5, 0.6) (0.3, 0.4, 0.5) (0.3, 0.4, 0.5)
[0.2 + 0.1a, 0.4–0.1a] [0.2 + 0.1a, 0.4–0.1a] [0.4 + 0.1a, 0.6–0.1a] [0.3 + 0.1a, 0.5–0.1a] [0.3 + 0.1a, 0.5–0.1a]
1 0.85 0.8 1 1
the hfifth DR in the iexample, hPas a = 1.0, the isatisfaction degree is equal to 0.062 by calculating either P L U L U 0 0 mðR Þ x or k mðR0 Þ ¼ mðR02;5k Þa at a = 1.0. The fulfillment level 2;5 2;5 k k mðR2;5k Þa x2;5 , since 2;5k a hP 2;5k a i L 0 ascribed to the PCs’ contribution is then ^x1;5 ¼ k 2;5 k mðR2;5k Þa x2;5 =W 1;5 ¼ 0:756, where W1,5 = 0.082,
k 2;5
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Fig. 7. The membership function of DR satisfaction.
Table 11 The ranges for the fulfillment levels of x2,k in the second stage QFD process ðLÞ
ðUÞ
ðLÞ
ðUÞ
ðLÞ
ðUÞ
ðLÞ
ðUÞ
ðLÞ
ðUÞ
a
x2;1
x2;1
x2;2
x2;2
x2;3
x2;3
x2;4
x2;4
x2;5
x2;5
1 0.9 0.8
1 1 1
1 1 1
0.850 0.850 0.774
0.850 0.850 0.850
0.321 0.311 0.524
0.321 0.800 0.68
0.961 0.873 0.570
0.961 0.276 0
1 1 1
1 1 1
which is greater than x1,5 (=0.5) determined in phase 1, if referring to Table 6. The resulting fulfillment level is acceptable because the possible range of x1,5 is in [0.5, 1] (see Table 5). 6. Concluding remarks Determining the fulfillment levels of DRs and PCs is an important task in the first two phases of the QFD processes, when QFD is applied to the new product design and development. However, most related studies only focus on the priority or achievement levels of DRs in phase 1. Unlike the existing research works, both phases 1 and 2 are considered in this paper. Based on the previous mathematical programming model for phase 1 under the fuzzy environment, this paper considers the close link between the two phases to build up a fuzzy linear programming model for phase 2 in determining the fulfillment levels of PCs. For reducing the design risk, risk analysis of DRs, namely a fuzzy FMEA, is taken into account in the phase 2 model. In addition, the satisfaction of the jth DR determined in phase 2 is greater than or equal to the contribution of the jth DR in phase 1 in order to realize customer satisfaction. The proposed model is illustrated with a numerical example to demonstrate the applicability in practice. The resulting ranges of satisfaction degree of phases 1 and 2 as well as the possible ranges of fulfillment levels of DRs and PCs can provide the QFD team with useful information for new product design and development. Acknowledgements The authors appreciate the anonymous reviewer’s useful comments to make this paper more comprehensive. This research was financially supported by the National Science Council, Republic of China, under Contract NSC95-2416-H-006-028-MY3.
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Applied Mathematical Modelling 33 (2009) 633–647 www.elsevier.com/locate/apm
Fuzzy linear programming models for new product design using QFD with FMEA Liang-Hsuan Chen *, Wen-Chang Ko Department of Industrial and Information Management, National Cheng Kung University, Tainan, Taiwan, ROC Received 30 November 2006; received in revised form 21 November 2007; accepted 26 November 2007 Available online 4 December 2007
Abstract Quality function deployment (QFD) is a customer-driven approach in processing new product developments in order to maximize customer satisfaction. Determining the fulfillment levels of design requirements (DRs) and parts characteristics (PCs) is an important decision problem during QFD activity processes for new product development. Unlike the existing literature, which mainly focuses on the determination of DRs, this paper proposes fuzzy linear programming models to determine the fulfillment levels of PCs under the requirement to achieve the determined contribution levels of DRs for customer satisfaction. In addition, considering the design risk, this paper incorporates failure modes and effect analysis (FMEA) into QFD processes, which is treated as the constraint in the models. To cope with the vague nature of product development processes, fuzzy approaches are used for both FMEA and QFD. The illustration of the proposed models is performed with a numerical example to demonstrate the applicability in practice. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Fuzzy models; Operations management; Planning and control; Quality function deployment (QFD)
1. Introduction Quality function deployment (QFD) is a customer-driven product development tool, considered as a structured management approach for efficiently translating customer needs into design requirements and parts deployment, as well as manufacturing plans and controls in order to achieve higher customer satisfaction. QFD has been successfully introduced in many industries to improve processes, customer satisfaction, and competitive advantages [1,2]. A general QFD process consists of four phases in order to relate the voice of customer to product design requirements (phase 1), and then translate this into parts characteristics (phase 2), manufacturing operations (phase 3), and production requirements (phase 4). In practice, at the design stage of new product development, a QFD team is organized to implement the first and second phases of QFD processes. The two phases are closely related at the design stage, since the outcomes from the latter phase should make the decisions from the former phase applicable. However, most of the existing research only focuses on *
Corresponding author. Tel.: +886 6 2757575x53140; fax: +886 6 2362162. E-mail address: [email protected] (L.-H. Chen).
0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.11.029
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the first phase of QFD in order to maximize customer satisfaction. Thus, the study that considers the two phases in QFD becomes necessary. For the development at the first phase, the relation matrices between customer requirements (CRs) and design requirements (DRs) and among the DRs are constructed, and the relationships are usually evaluated subjectively by ambiguous or vague judgments, due to the lack of information about the influence of engineering responses on CRs [3,4]. However, those relationships are often treated as crisp variables or linguistic scales [1,2]. For example, the relation strength by which a DR affects a CR is expressed by a scale system, i.e., 1–3–9, or 1–5–9, indicating linguistic expressions such as ‘‘weak”, ‘‘moderate”, and ‘‘strong”, respectively. In order to make the improvements, fuzzy approaches are adopted by several researchers to address diversified and imprecise problems between CRs and DRs and among the DRs [5–10]. However, these existing studies still suffer some drawbacks in approaches, as mentioned by Chen and Weng [3,4] in their fuzzy mathematical programming model for QFD. Furthermore, in order to decrease the risk of new product design, the risk analysis of DRs is necessary during the design level of new product development, and the outcomes are applied to phase 2 as the constraints in determining the achievement levels of parts characteristics (PCs). In this respect, failure mode and effect analysis (FMEA) is a systematic technique for identifying, prioritizing and acting on potential failure modes before the failures occur, so this study also adopts FMEA for the risk analysis in the early stages of new product development. Stamatis [11] described the methods of FMEA and its applications. Ginn et al. [12] proposed a methodology for investigating the interaction between QFD and FMEA by cross-functional and multi-disciplined teamwork. Tan [13] developed a customer-focused methodology for the built-in reliability to maximize customers’ satisfaction based on the constrained resources by combining the FMEA and QFD. However, their studies are only limited to descriptive analyses for obtaining the quality and resource benefits. The methods to carry out the aggregation of QFD and FMEA are not mentioned, and the uncertainty at the product design stage is not considered, either. In this study, we extend Chen and Weng’s fuzzy model [4] by introducing FMEA into the existing fuzzy QFD approach and linking phase 1 to phase 2 of QFD in determining the achievement levels of DRs and part characteristics (PCs) to maximize the customers’ satisfaction. In the following two sections, the approaches of fuzzy QFD and FMEA are introduced. In Section 4, a fuzzy linear programming model is developed to determine the achievement degrees of PCs, constrained by the need of DRs in phase 1 and the risk ratings of PCs according to the design of FMEA on DRs. A semiconductor packing example is presented to demonstrate our approach in Section 5. Finally, the concluding remarks are provided. 2. The approach of fuzzy QFD For implementing QFD processes, a relation matrix, also called a House of Quality (HOQ), is usually used for each phase to construct the input–output relationships in determining the achievement priority or level of output variables. In practice, the major work in phase 1 of the QFD processes is to determine the achievement priority or level of DRs based on the importance of each CR, the relationships between CRs and DRs as well as among the DRs. Based on the results from the first QFD process, the similar work is performed with DRs and PCs in phase 2. Fig. 1 demonstrates the relation matrices of the two phases. Obviously, from the figure the fulfillment levels of PCs should make the DRs applicable in meeting the customers’ satisfaction. In Fig. 1, R1,ij denotes the relation level in terms of score between the ith CR and jth DR, and r1,jn is the correlation score between the jth and nth DRs in the firstPphase of QFD. The notations k and W represent the importance score and rating in the two matrices, where k ¼ 1. For symbolizations, the subscripts, 1 and 2, are added to the associated variables, such as R, r, k, and W, to denote phase 1 and phase 2 of QFD, respectively, in Fig. 1 and the equations hereafter. Considering the correlations among DRs at the first stage of QFD, Wasserman [14] proposed a formulation to calculate the normalized relationship value between CRs and DRs as PJ R1;if r1;fj ; ð1Þ R01;ij ¼ PJ f¼1 PJ j¼1 f¼1 R1;if r 1;fj
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635
Fig. 1. The relationship matrices for phases 1 and 2 of the QFD processes.
P where R01;ij is the normalized relationship value between CRi and DRj, i = 1, . . . , I; j = 1, . . . , J; j R01;ij ¼ 1 for each i. As mentioned before, the relationships are quantified as crisp values to denote the relationship degrees between CRs and DRs, as well as the correlation among DRs. For reflecting the imprecise nature of the relationships, Chen and Weng [4] proposed a fuzzified quantitative formulation as follows: PJ e 1;if ~r1;fj R 0 e ; ð2Þ R 1;ij ¼ PJ f¼1 PJ e r1;fj j¼1 f¼1 R 1;if ~ e 1;if and ~r1;fj are described by linguistic terms and defined as the fuzzy subsets of [0, 1]. Actually, a fuzzy where R e 1;if at the a set can be fully and uniquely represented by its a-cuts [15]. For example, the a-cut of the fuzzy set R L U level, a 2 [0, 1], can be denoted by its lower and upper bounds as ½ðR1;if Þa ; ðR1;if Þa , which is defined by ðR1;if ÞLa ¼ inf fxjle ðxÞ P ag; R 1;if x2½0;1
ð3aÞ
and
U
ðR1;if Þa ¼ sup fxjle ðxÞ P ag; x2½0;1
ð3bÞ
R 1;if
e 1;if . where le ðxÞ is the membership degree of x belonging to R R 1;if Based on a-cuts and the extension principle [15,16], the membership functions of the fuzzy normalized relae u;if and ~ru;fj , tionship in phases 1 and 2 can be defined by the lower and upper bounds of each a-cut of R u = 1, 2. In addition, Chen and Weng [4] further proposed a modified formulation to obtain a more precise representation of the fuzzy normalized relationship, of which the lower and upper bounds of the membership function at each a-cut are formulated as PJ L L L f¼1 ðR1;if Þa ðr1;fj Þa 0 ; ð4aÞ mðR1;ij Þa ¼ PJ PJ PJ U U L L m¼1 f¼1 ðR1;if Þa ðr 1;fm Þa þ f¼1 ðR1;if Þa ðr1;fj Þa m–j U mðR01;ij Þa
¼
PJ
U U f¼1 ðR1;if Þa ðr1;fj Þa PJ PJ PJ L L U U m¼1 f¼1 ðR1;if Þa ðr 1;fm Þa þ f¼1 ðR1;if Þa ðr1;fj Þa m–j
:
ð4bÞ
Once the modified lower and upper bounds of a-cuts of the fuzzy normalized relationship are obtained, the e 1;j for the jth DR can be determined in the form of a-cuts, expressed as fuzzy technical importance rating W
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" e 1;j Þ ¼ ½ðW ðW a
L U 1;j Þa ; ðW 1;j Þa
¼
I X
k 1;i
L mðR01;ij Þa ;
i¼1
I X
# k 1;i
U mðR01;ij Þa
;
ð5Þ
i¼1
which will be employed to find out the optimal fulfillment level of each DR in phase 1 of the QFD process. Using the fuzzy technical importance ratings of DRs, Chen and Weng [4] proposed a fuzzy linear programming model in which the decision variable, x1,j, is defined in percentage to denote the level of fulfillment percentage of the DRj, j = 1, . . . , J, i.e., x1,j 2 [0, 1]. The x1,j = 0 implies that the DR has a basic design requirement so that no more efforts and resources are needed. In their model, the increment unit cost to e 1;j , to reflect its fuzzy nature at the design plan stage. achieve the fulfillment level is used as a fuzzy number, C With the fulfillment percentage of DRj, a corresponding percentage of increment unit cost is required to enhance the characteristic of product or service. The total incremental unit cost cannot exceed a budget constraint in this stage. Besides, the impacts on customer satisfaction of various DRs are prioritized, and business competition and technological difficulties are also considered in the model. If the sth DR is preferred to the pth e 1;p x1;p P 0 is needed. In order to solve the e 1;s x1;s W DR in terms of customer satisfaction, a constraint of W e 1;j are placed in the e 1;j and C fuzzy linear programming model, the lower and upper bounds of a-cuts of W model to find the corresponding values of x1,j and determine the lower and upper bounds of the customer satisfaction at each a-cut, respectively. Thus, a fuzzy linear programming model is formulated as L
ðZ 1 Þa ¼ max
J X
L
ðW 1;j Þa x1;j ;
j¼1
s:t:
J X
U
ðC 1;j Þa x1;j 6 B1 ;
j¼1 L
U
ðW 1;s Þa x1;s ðW 1;p Þa x1;p P 0; 0 6 ej 6 x1;j 6 g1;j 6 1
8j;
s; p 2 f1; 2; . . . ; J g; J X U U ðZ 1 Þa ¼ max ðW 1;j Þa x1;j ;
ð6aÞ
j¼1
s:t:
J X
ðC 1;j ÞLa x1;j 6 B1 ;
j¼1 U
L
ðW 1;s Þa x1;s ðW 1;p Þa x1;p P 0; 0 6 ej 6 x1;j 6 g1;j 6 1 8j; s; p 2 f1; 2; . . . ; J g; ðZ 1 ÞLa
ð6bÞ ðZ 1 ÞU a
where B1 is the budget limitation, and as well as represent the lower and upper bounds of objective values, i.e., customer satisfaction in phase 1 of the QFD process, at each a-cut, respectively. In addition, ej and g1,j denote the possible range of the fulfillment level of one DR, indicating the minimum required level due to the business competition and the maximum level due to technical difficulty, respectively. From the above model, the achievement levels of DRs at different a levels can be obtained. It is worth noting that, for maximizing the Model (6), the fulfillment levels of some DRs determined in Model (6a) for the lower bound of the customer satisfaction may be greater than those in Model (6b) for the upper bound of the customer satisfaction. However, these results of the fulfillment degree still provide an interval for building up the membership functions of fulfillment degree of DRs in the fuzzy sense. To avoid the confusions, the optimal achievement levels ðLÞ ðUÞ of DRs determined for the lower and upper bounds of the customer satisfaction are denoted as x1;j and x1;j , respectively. Similar cases are applied to PCs in phase 2. To develop the fuzzy mathematical model for phase 2 of the QFD process, the importance scores of the DRs in the second phase should be determined. In the existing literature, the importance ratings W1,j of the DRs in phase 1 are adopted as the importance scores k2,j for the deployment of phase 2 in QFD processes [17]. However, the achievement levels of DRs will affect the PCs’ deployment. Therefore, we proposed a
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637
weighted-average equation to determine the importance score by using both the importance ratings and the achievement levels of DRs in phase 1, which is formulated as follows: P L ðLÞ U ðUÞ 1 l l ½ðW 1;j Þal ðx1;j Þal þ ðW 1;j Þal ðx1;j Þal l 2 0 P ; j ¼ 1; . . . ; J : ð7Þ k 2;j ¼ l ll L
ðLÞ
U
ðUÞ
l denotes the membership degree of 1=2½ðW 1;j Þal ðx1;j Þal þ ðW 1;j Þal ðx1;j Þal belonging to In the above equation, l e 1;j~x1;j determined by the a-cut and extension principle. Furthermore, the normalized the fuzzy number W importance score of each DR is expressed as k 02;j k 2;j ¼ PJ 0 : j¼1 k 2;j
ð8Þ
Similar with (4) in phase 1, the fuzzy normalized relationship between DRs and PCs in phase 2 is formulated in terms of the lower and upper bounds as Pk L L L l¼1 ðR2;jl Þa ðr2;lk Þa ; ð9aÞ mðR02;jk Þa ¼ Pk Pk P U U k L L m¼1 l¼1 ðR2;jl Þa ðr 2;lm Þa þ l¼1 ðR2;jl Þa ðr 2;lk Þa m–j U mðR02;jk Þa
¼ Pk
m¼1 m–j
Pk
U U l¼1 ðR2;jl Þa ðr2;lk Þa P L L k U U l¼1 ðR2;jl Þa ðr 2;lm Þa þ l¼1 ðR2;jl Þa ðr 2;lk Þa
Pk
:
Then, the PCs’ fuzzy importance ratings are calculated below: " # J J X X L U L U 0 0 e 2;k Þ ¼ ½ðW 2;k Þ ; ðW 2;k Þ ¼ ðW k 2;j mðR2;jk Þa ; k 2;j mðR2;jk Þa : a a a j¼1
ð9bÞ
ð10Þ
j¼1
3. The approach of fuzzy FMEA For the reduction of design risk, designers usually take risk analysis of DRs into account so as to build up the requirements of parts characteristics (PCs) in phase 2 of the QFD process during the design stage of new product development. Failure mode and effect analysis (FMEA) is an effective approach to provide information for making risk management decisions. This study also adopts this approach to make risk analysis of each DR under the fuzzy environment. A conventional form of FMEA includes (i) the design function of parts, (ii) the potential failure mode (categories of failure), (iii) the potential effects of failure (measured by the severity index), (iv) the potential causes of failure (measured by the occurrence (frequency) index), (v) the detection method (measured by the detectability index), and (vi) the risk priority number (RPN). The RPN is used to evaluate the risk level of a part’s failure mode in design stage, and is determined by the multiplication of three characteristic failure mode indexes, i.e., the severity of the potential failure (S), the frequency of potential failure (O), and the detectability index (D), respectively, as RPN ¼ S O D:
ð11Þ
The three indices in (11) are defined on the same scale level, such as the 10-point system, to identify the various levels of risk situation. Tables 1–3 present the scales for the S, O and D indices [18]. In general, the design function (or DR in QFD) of a part may have more than one failure mode in FMEA, and each failure mode may be described by more than one effect of failure. In addition, one effect of failure may result from more than one cause of failure, which could be detected by several detection methods. The detectable degree of each cause of failure is measured by the detectability index (D). Each failure mode is evaluated by RPN in terms of the three indices, i.e., S, O, and D. Usually, an assessment of those indices is subjective and qualitatively described in natural language. With regard to this, sometimes it is difficult to apply conventional FMEA to determine the rating of failure indices. For example, the frequency of potential failure
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Table 1 Scales for severity Severity
Rating
Remote Low
1 2 3
Moderate
4 5 6
High
7 8
Very high
9 10
Table 2 Scales for occurrence Frequency of occurrence
Rating
Possible failure rate
Remote Low
1 2 3
<1:20,000 1:20,000 1:10,000
Moderate
4 5 6
1:2000 1:1000 1:200
High
7 8
1:100 1:20
9 10
1:10 1:2
Very high
Table 3 Scales for detectability Detectability
Rating
Probability of detectability (%)
Remote Low
1 2 3
86–100 76–85 66–75
Moderate
4 5 6
56–65 46–55 36–45
High
7 8
26–35 16–25
9 10
6–15 0–5
Very high
could be evaluated to be close to 1:50. The existing measurement system cannot give a suitable rating for this situation. Appropriately, such a measurement can be described by a linguistic term, such as ‘‘medium high”. Some researchers have proposed fuzzy set theory to deal with the above problems [18–21]. Xu et al. [19] presented a fuzzy-logic-based method and an assessment expert system for FMEA of a diesel engine’s turbocharger system. Pillay and Wang [18] used a fuzzy rules base and grey relation theory to determine the RPN by multiplying the factor scores of the probability of failure, severity and detectability. It is necessary to have adequate experts’ knowledge to build up the fuzzy rule base for reasoning to obtain an acceptable RPN. However, it could be difficult to figure out the suitable inference system, because integrating the related experiences is not easy during the design stage.
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In this study, we make the risk analysis of DRs by performing FMEA at the design stage. For describing the assessments meaningfully, the three failure indices are evaluated by linguistic terms and fuzzy set theorems are employed to demonstrate the fuzziness of the risk analysis. The fuzzified RPN of each DR can be expressed as follows:
es D e tÞ ; Sr O ðRPNÞj ¼ maxð e j
j ¼ 1; 2; . . . ; J ;
r;s;t
ð12Þ
e s , and D e t are defined as the fuzzy subsets of [0, 1]. The ðRPNÞj is determined as the maximum of where e S r, O the three indices’ product from all sets of S, O, and D, if there is more than one set, to represent the fuzzy risk priority level of the jth DR. Furthermore, considering the different importance of each failure index, various indices can have a different weights to determine ðRPNÞj . According to the characteristic of the multiplication of the three indices in (12), this study uses a fuzzy ordered weighted geometric averaging (FOWGA) operator [22] to modify Eq. (12). A FOWGA operator can be used to aggregate m (>1) fuzzy sets, and the formulation is expressed as [22] a2 ; . . . ; ~ am Þ ¼ f ð~ a1 ; ~
m Y
ð~ bi Þ w i ;
ð13Þ
i¼1
P where ~ bi is the ith largest one in m fuzzy numbers ~ak , k = 1, . . . , m; wi is the weight of the ~bi and mi¼1 wi ¼ 1, wi 2 [0, 1]. Following FOWGA, (12) can be reformulated as
e DÞ e ¼ max ðRPNÞj ¼ f ð e S; O; j r;s;t
3 Y ð~ bi Þwj i ;
ð14Þ
i¼1
e s; D e t Þ. In Eq. (14), the weighting vector of w = (w1, w2, w3)T can be where ~ bi is the ith largest set of the ð e S r; O determined according to designers’ or QFD team members’ experiences. For determining ðRPNÞj , its membership function can be constructed by deriving the lower and upper bounds of the a-cuts of ðRPNÞj as ðRPNj ÞLa ¼ max r;s;t
U
ðRPNj Þa ¼ max r;s;t
3 Y ½ðbi Þwj i La
ð15aÞ
and
i¼1 3 Y w U ½ðbi Þj i a :
ð15bÞ
i¼1
Once the membership function of ðRPNÞj is built-up, its defuzzified value can be obtained by the following formulation: P L U 0i f1=2½ðRPNj Þai þ ðRPNj Þai g l 0 P 0 ; ð16Þ ðRPNj Þ ¼ i i il
0i is the membership where l degree of 1=2½ðRPNj ÞLai þ ðRPNj ÞU ai belonging to the fuzzy number ðRPNÞj . The defuzzified value of ðRPNÞj represents the risk level of the jth DR, and is applied to phase 2 for parts design deployment in determining the fuzzy risk rating of PCs with the consideration of the fuzzy normalized relationship. The formulation is expressed as " # J J X X L U 0 L 0 U 0 0 e k Þ ¼ ½ðRik Þ ; ðRik Þ ¼ ðRPNj Þ mðR Þ ; ðRPNj Þ mðR Þ : ð17Þ ð Ri a
a
2;jk a
a
j¼1
2;jk a
j¼1
The fuzzy risk rating of each PC is used in phase 2 to determine the fulfillment level of each PC in the design level to satisfy the design requirements and finally maximize customers’ satisfaction. 4. Fuzzy linear model with FMEA From Model (6), the fulfillment levels of DRs are obtained in phase 1 of the QFD processes in order to maximize customer satisfaction. Extending Model (6) to phase 2 by including the risk analysis of DRs via
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FMEA, this study proposed a fuzzy linear programming model to determine the fulfillment level of the PCs in phase 2. Similar with Model (6), the notation of x2,k is defined as the fulfillment level of the kth PC in the proposed model. By performing FMEA on DRs, the fuzzy risk ratings of PCs are obtained by Eq. (17) and used as the constraint in the proposed model. In addition, the determination of the fulfillment levels of the PCs has to be done that the DRs can achieve customer satisfaction. The contribution to the customers’ satisfaction by the jth DR due to the design satishP i K e 0 Þ~x2;k , which should be greater than or equal to mð R faction of PCs in phase 2 can be expressed as k 2;j 2;jk k¼1 e 1;j~x1;j , in phase 1 to realize the customer satisfaction. Moreover, the budget the contribution of the jth DR, W limitation, preemptive priority, and technological ability for PCs are also taken into account in the proposed model. Taking the above considerations as the constraints and using the PCs’ fuzzy importance rating of Eq. (10) as well as the decision variables, x2,k, as the objective function, a fuzzy linear programming model is formulated in the a-cut form as ðZ 2 ÞLa ¼ max
K X k¼1
s:t:
ðW 2;k ÞLa x2;k ; "
k 2;j
K X
# L mðR02;jk Þa
x2;k P W 1;j x1;j ; j ¼ 1; . . . ; J ;
k¼1 K X
U
ðRik Þa x2;k 6 H ;
k¼1 K X
ðC 2;k ÞU a x2;k 6 B2 ;
k¼1
ðW 2;s ÞLa x2;s ðW 2;p ÞU a x2;p P 0; 8k;
0 6 x2;k 6 g2;k 6 1 s; p 2 f1; 2; . . . ; Kg; U
ðZ 2 Þa ¼ max
K X
U
ðW 2;k Þa x2;k ;
k¼1
"
s:t: k 2;j
ð18aÞ
K X
# mðR02;jk ÞU a
x2;k P W 1;j x1;j ; j ¼ 1; . . . ; J ;
k¼1 K X
L
ðRik Þa x2;k 6 H ;
k¼1 K X
L
ðC 2;k Þa x2;k 6 B2 ;
k¼1 U
L
ðW 2;s Þa x2;s ðW 2;p Þa x2;p P 0; 0 6 x2;k 6 g2;k 6 1 s; p 2 f1; 2; . . . ; Kg:
8k; ð18bÞ
In the above model, the objective function is to find the maximum satisfaction of design requirement in phase L U 2. The values of ðZ 2 Þa and ðZ 2 Þa represent the lower and upper bounds of objective values, respectively, at each a-cut. Similar with the cases in Model (6), the optimal fulfillment levels of PCs determined in Model ðLÞ ðUÞ L U (18) for ðZ 2 Þa and ðZ 2 Þa are represented as x2;k and x2;k , respectively. In the constraints, H is a threshold e 2;k and g2,k are the increment unit cost to achieve the fulfillment level of risk; B2 is a budgetary limitation; C and technological difficulty of the PCs, respectively. These parameters are determined by the QFD team. In
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addition, for simplification and application purposes, the crisp contribution level of each DR is taken as the e 1;j~x1;j action by the QFD team and is adopted in the model. In Model (18), W1,jx1,j is a defuzzified value of W 0 and calculated by Eq. (7), so that W1,jx1,j is equal to k 2;j . 5. An illustrative example In this section, a semiconductor packing case of the turbo thermal ball grid array (T2-BGA) package is used to exemplify the applicability of the proposed models. Since a heat slug is inserted into the molding compound of a plastic BGA (PBGA), T2-BGA has the advantages of high heat dissipation and good ground shielding to reduce the electromagnetic interference (EMI) and electromagnetic compatibility (EMC) effects between adjacent traces; however, it also produces some problems due to the more complex structure of the package when compared to PBGA [23]. Fig. 2 displays the cross-sections of PBGA and T2-BGA. Based on the above considerations of T2-BGA, a QFD team collects five customer requirements (CRs) and proposes five design requirements (DRs) in the earlier design process. The associated house of quality (HOQ) is shown in Fig. 3. In phase 1, the CRs are ‘‘package profile” (CR1), ‘‘thermal performance” (CR2), ‘‘electrical performance” (CR3), ‘‘reliability” (CR4), and ‘‘co-planarity” (CR5), and the five DRs are ‘‘heat slug exposed area” (DR1), ‘‘heat slug attached material” (DR2), ‘‘height of heat slug” (DR3), ‘‘copper pattern” (DR4), and e 1;if between CRs and DRs as well as the relevant relations ‘‘molding flow” (DR5). Firstly, the fuzzy relations R ~r1;fj among DRs are determined. These relationships are denoted as ‘‘weak”, ‘‘moderate”, or ‘‘strong”, which
Chip
Gold wire
Solder ball PBGA
Heat slug
Molding compound
T2-BGA
Fig. 2. The cross-sections of PBGA and T2-BGA.
Fig. 3. The phase 1 for a T2-BGA package.
Substrate
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are translated into the triangular fuzzy numbers as a 3-element set, (0, 0.2, 0.4), (0.3, 0.5, 0.7), (0.6, 0.8, 1), e 1;if ¼ ð0:6; 0:8; 1Þ is formulated as respectively, with membership functions in Fig. 4. As an example, R 8 < ðR1;if 0:6Þ ; 0:6 6 R1;if 6 0:8; ð0:80:6Þ le ðR1;if Þ ¼ ð1R Þ R 1;if 1;if : ; 0:8 6 R1;if 6 1: ð10:8Þ The middle value in the 3-element set represents the most likely one with the membership degree equivalent to 1, such as lstrong (0.8) = 1. In this study, the system of 0.2–0.5–0.8 is used to express the most likely value of the linguistic terms on weak–moderate–strong. Considering the linguistic relationships in Figs. 3 and 4, the fuzzy normalized relationship at each a-cut can be calculated using Eq. (4). For doing this, the lower and upper bounds of each e 1;if and ~r1;fj should be determined beforehand based on their membership functions. Actually, the a-cut of R lower and upper bounds of each a-cut can be represented as the functions of a (2[0, 1]). For example, the L U a-cut of the membership function (0.6, 0.8, 1) of ‘‘strong” is expressed as ½ðR1;if Þa ; ðR1;if Þa ¼ ½0:6 þ 0:2a; 1 0:2a. Without any biases, the a levels in this paper are evenly distributed in [0, 1]. Let am denote the mth a level, and am = m/n, m 2 {0, 1, . . . , n}, such that the distance between each two adjacent a levels is equal, i.e. am am1 = 1/n, m P 1. e 1;j Þ ; j ¼ 1; 2; . . . ; J, at each a The lower and upper bounds of the fuzzy technical importance rating ð W a level, is obtained by Eq. (5) to determine the importance priority of each DR, as listed in Table 4. They are applied to Model (6) to solve the fuzzy linear model. The budgetary cost is limited to 1 unit, i.e., B1 = 1.0, in Model (6); meanwhile, we suppose that the QFD team has prioritized DRs as DR4 DR1 DR2 and DR3 DR5 for the design preference, where ‘‘” means ‘‘is more preferred than”. Table 5 lists the associated data for the fuzzy linear Model (6). Applying fuzzy technical importance ratings, fuzzy increment cost, and the associated constraints to Model (6) at several a levels, the fulfillment level of each DR and the total customer satisfaction degree (the objective function value) can be obtained. In fact, with the fuzzy sense, the a level can be interpreted as the confidence
Fig. 4. The membership functions of linguistic relationship in the HOQs.
Table 4 The lower and upper bounds (%) of fuzzy technical importance ratings at different a levels a
W L1;1
WU 1;1
W L1;2
WU 1;2
W L1;3
WU 1;3
W L1;4
WU 1;4
W L1;5
WU 1;5
1 0.8 0.6 0.4 0.2 0
16.9 13.5 10.1 6.7 3.4 0
16.9 20.2 23.6 27.0 30.3 33.7
20.8 18.3 16.1 14.1 12.2 10.6
20.8 23.4 26.2 29.1 32.1 35.1
5.9 5.1 4.3 3.6 3.0 2.5
5.9 6.7 7.6 8.6 9.5 10.0
37.0 34.0 30.0 27.0 24.0 20.0
37.1 40.5 43.9 47.3 50.6 54.0
8.2 6.2 4.4 2.7 1.3 0
8.2 10.4 12.7 15.2 17.7 20.4
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Table 5 The cost, business competition, and technical difficulty for each DR Fulfillment level
e 1;j C
a-cut
ej
g1,j
x1,1 x1,2 x1,3 x1,4 x1,5
(0.2, 0.3, 0.4) (0.8, 0.9, 1) (0.4, 0.5, 0.6) (0.3, 0.4, 0.5) (0.1, 0.2, 0.3)
[0.2 + 0.1a, 0.4–0.1a] [0.8 + 0.1a, 1–0.1a] [0.4 + 0.1a, 0.6–0.1a] [0.3 + 0.1a, 0.5–0.1a] [0.1 + 0.1a, 0.3–0.1a]
0.1 0.1 0.1 0.1 0.5
1 1 1 1 1
degree [24,25]. In this example, the settings of a levels less than 0.8 will produce infeasible regions in solving the fuzzy linear model. From the property of a fuzzy number, this means the confidence level of fuzzy inputs should be higher to avoid producing the wider ranges of a-cuts. Table 6 shows the outcomes of the total customer satisfaction degree and fulfillment levels of each DR for the Model (6) based on three a levels (0.8, 0.9, 1). The decision variable x1,j = 100% denotes complete fulfillment of the jth DR. In this example, DR4 should have the better quality level in order to achieve the CRs’ satisfaction degree. On the other hand, DR2 is smaller than the others, due to the high incremental cost, as shown in Table 5. In Table 6, it is also ðLÞ ðUÞ noted that x1;3 is greater than x1;3 at a level of.9. As described before, for maximizing the model, the optimal values of some decision variables for Z L1 may be greater than those for Z U 1 . However, the maximum customer L is greater than Z at the level, as desired for the models. satisfaction Z U 1 1 From the outcomes from phase 1 of QFD, the fuzzy technical importance rating and fulfillment level of each DR are applied to (7) and normalized by (8) to determine the importance score in phase 2 of the QFD. They are calculated as 0.13, 0.042, 0.082, 0.663 and 0.083 for the five DRs, respectively. Furthermore, the DRs from phase 1 of the QFD processes are used to design FMEA for evaluating the potential failure modes and effects for risk analysis. An analysis of the T2-BGA package is listed in Table 7, in which the three failure indices are described as linguistic terms by ‘‘R (remote)”, ‘‘L (low)”, ‘‘M (moderate)”, ‘‘H (high)”, and ‘‘VH (very high)”. For the following resolution procedure, these linguistic terms should be translated to fuzzy numbers. Their definitions are (0, 0, 0.1, 0.2), (0.1, 0.25, 0.25, 0.4), (0.3, 0.5, 0.5, 0.7), (0.6, 0.75, 0.75, 0.9), and (0.8, 0.8, 1, 1), respectively, as shown in Fig. 5. For obtaining fuzzy risk priority number (RPN), the upper and lower bounds of a-cuts of the three fuzzy e s; D e t , and their weights should be determined beforehand. Suppose that the set of weights failure indices e S r; O is w = {0.7, 0.21, 0.09}, determined by the QFD team. The fuzzy RPN of each DR is constructed by (15) and then defuzzified by (16) for risk ratings in phase 2 of the QFD processes. They are calculated as 6.631, 8.365, 7.227, 6.631, and 6.219 for the five DRs, respectively. The QFD team investigates the part characteristics of the principal parts or subsystem to deploy in phase 2 of the QFD. There are five part characteristics (PCs), including ‘‘profile of heat slug” (PC1), ‘‘leg shape of heat slug” (PC2), ‘‘plating quality of heat slug” (PC3), ‘‘jointability between attached material and molding compound” (PC4), and ‘‘jointability among attached material, heat slug and substrate” (PC5), as shown in Fig. 6. Similar with phase 1 of the QFD, considering the relationships between DRs and PCs as well as among the PCs in Fig. 6 and the membership functions of linguistic relationships in Fig. 4, the fuzzy normalized relationship at each a-cut can be obtained using (9). The lower and upper bounds of the fuzzy importance rating e 2;k Þ , k = 1, 2, . . . , K, at each a level, are obtained by (10) to determine the importance priority of of PCs, ð W a e 2;k Þ at different a levels are listed in Table 8. The fuzzy risk rateach PC. The lower and upper bounds of ð W a ings of each PC at each a-cut can be calculated by (17) and are presented in Table 9. The resulting fuzzy importance and risk level ratings of PCs with the associated constraints listed in Table 10 are applied to Model (18) at several a levels to solve the fuzzy linear model. The fulfillment level of each PC Table 6 Outcomes of the total customer satisfaction degree and fulfillment level of each DR a
Z L1
ZU 1
x1;1
ðLÞ
x1;1
ðUÞ
x1;2
ðLÞ
x1;2
ðUÞ
x1;3
ðLÞ
x1;3
ðUÞ
x1;4
ðLÞ
x1;4
ðUÞ
x1;5
ðLÞ
x1;5
ðUÞ
1 0.9 0.8
0.514 0.378 0.300
0.514 0.584 0.682
.256 0.147 0.177
0.256 0.483 0.844
0.1 0.1 0.1
0.1 0.1 0.1
0.667 0.900 1
0.667 0.583 0.429
1 0.731 0.527
1 1 1
0.500 0.500 0.500
0.500 0.500 0.500
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Table 7 T2-BGA package FMEA DRj Design function of part
Potential failure mode
Potential effects of failure
S
Potential causes of failure
O
Detection method
D
Heat slug exposed area
Distortion of the heat slug Warpage of the heat slug Plating layer peeling off
EMC overflow
H
Packing/Transportation
M
Profile inspection
L
EMC overflow
H
Heat slug forming
M
Profile inspection
M
Package looks defective
H
plating solution contaminated
M
Surface inspection
L
Heat slug attached material
Molding failure
Package profile defective
VH
Turbulence
M
Simulation
H
Height of heat slug
Plating layer peeling off Plating thickness inconsistent
Package looks defective Substrate damage
M
Plating solution contaminated
L
Surface inspection
R
H
Stress of the HS
M
Stress measurement
H
EMC overflow
H
L
Package looks defective Short circuit
M M
L L
HS Height inspection Eyes inspection Surface inspection
M
Unusual height
Heat slug height out of spec. Unsuitable packaging Burr
R R
Copper pattern
Popcorn
Reliability test failed
H
Pattern distribution
M
Reliability test
M
Molding flow
Popcorn
Reliability test failed
H
Turbulence
M
Simulation
L
Fig. 5. The membership functions of linguistic terms in the FMEA.
and the total DR satisfaction degree (the objective function value) can then be obtained. In phase 2 of the QFD processes, the budgetary cost is limited to 1.5 unit, i.e., B2 = 1.5 in (18); the risk threshold H is bounded to 25 in this case. For phase 2, the set of DRs’ importance scores by (7) is {0.064, 0.021, 0.041, 0.330, 0.041}. In addition, suppose that the QFD team concerns the preemptive priority of PCs as PC5 PC2 for the design process. In solving Model (18), the settings of a levels less than 0.8 will produce infeasible regions, as happened in phase 1. This means that the same confidence level (P0.8) should be adopted for both phases in this QFD design. Appling three a levels (0.8, 0.9, 01) to Model (18), the total DR satisfaction degree and the fulfillment level of each PC are shown in Fig. 7 and Table 11, respectively. In Fig. 7, the most possible satisfaction degree of DR is around 69%. The fulfillment level of decision variable x2,k = 100%, which denotes complete fulfillment of the kth PC. In Table 11, the fulfillment levels of PC1 and PC5 represent the best performance level in order to achieve the DR’s satisfaction degree in this case. In addition, similar with the outcomes by Model ðLÞ ðUÞ (6), x2;4 is greater than x2;4 at a levels of 0.8 and 0.9, as shown in Table 11. As described previously, the satisfaction of the jth DR in phase 2 should be greater than or equal to the contribution of the jth DR in phase 1 in order to realize customer satisfaction. For simplification, considering
L.-H. Chen, W.-C. Ko / Applied Mathematical Modelling 33 (2009) 633–647
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Fig. 6. The Phase 2 for a T2-BGA package. Table 8 The lower and upper bounds (%) of fuzzy importance ratings of PCs in various a a
W L2;1
WU 2;1
W L2;2
WU 2;2
W L2;3
WU 2;3
W L2;4
WU 2;4
W L2;5
WU 2;5
1 0.8 0.6 0.4 0.2 0
39.7 33.5 28.1 23.3 19.2 15.7
39.7 46.7 54.3 62.5 71.0 79.4
12.3 10.4 8.8 7.2 5.8 4.6
12.3 14.2 16.2 18.3 20.4 22.4
3.0 2.5 2.1 1.7 1.4 1.1
3.0 3.4 3.9 4.4 4.9 5.3
3.6 2.8 2.1 1.5 1.0 .6
3.6 4.6 5.8 7.2 8.8 10.8
14.7 11.0 7.8 5.2 2.9 1.1
14.7 19.0 23.8 29.2 35.0 41.0
Table 9 The fuzzy risk ratings of PCs in various a a
RiL1
RiU 1
RiL2
RiU 2
RiL3
RiU 3
RiL4
RiU 4
RiL5
RiU 5
1 0.8 0.6 0.4 0.2 0
7.17 6.17 5.27 4.48 3.80 3.20
7.17 8.27 9.46 10.72 12.01 13.26
8.94 7.69 6.54 5.50 4.56 3.73
8.94 10.26 11.64 13.06 14.48 15.86
2.62 2.23 1.87 1.53 1.23 .96
2.62 3.03 3.45 3.87 4.28 4.68
4.45 3.58 2.83 2.17 1.60 1.10
4.45 5.45 6.61 7.92 9.43 11.11
5.12 4.36 3.69 3.09 2.56 2.09
5.12 5.95 6.86 7.83 8.84 9.87
Table 10 The cost and technical ability for each PC Fulfillment level
e 2;k C
a-cut
g2,k
x2,1 x2,2 x2,3 x2,4 x2,5
(0.2, 0.3, 0.4) (0.2, 0.3, 0.4) (0.4, 0.5, 0.6) (0.3, 0.4, 0.5) (0.3, 0.4, 0.5)
[0.2 + 0.1a, 0.4–0.1a] [0.2 + 0.1a, 0.4–0.1a] [0.4 + 0.1a, 0.6–0.1a] [0.3 + 0.1a, 0.5–0.1a] [0.3 + 0.1a, 0.5–0.1a]
1 0.85 0.8 1 1
the hfifth DR in the iexample, hPas a = 1.0, the isatisfaction degree is equal to 0.062 by calculating either P L U L U 0 0 mðR Þ x or k mðR0 Þ ¼ mðR02;5k Þa at a = 1.0. The fulfillment level 2;5 2;5 k k mðR2;5k Þa x2;5 , since 2;5k a hP 2;5k a i L 0 ascribed to the PCs’ contribution is then ^x1;5 ¼ k 2;5 k mðR2;5k Þa x2;5 =W 1;5 ¼ 0:756, where W1,5 = 0.082,
k 2;5
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Fig. 7. The membership function of DR satisfaction.
Table 11 The ranges for the fulfillment levels of x2,k in the second stage QFD process ðLÞ
ðUÞ
ðLÞ
ðUÞ
ðLÞ
ðUÞ
ðLÞ
ðUÞ
ðLÞ
ðUÞ
a
x2;1
x2;1
x2;2
x2;2
x2;3
x2;3
x2;4
x2;4
x2;5
x2;5
1 0.9 0.8
1 1 1
1 1 1
0.850 0.850 0.774
0.850 0.850 0.850
0.321 0.311 0.524
0.321 0.800 0.68
0.961 0.873 0.570
0.961 0.276 0
1 1 1
1 1 1
which is greater than x1,5 (=0.5) determined in phase 1, if referring to Table 6. The resulting fulfillment level is acceptable because the possible range of x1,5 is in [0.5, 1] (see Table 5). 6. Concluding remarks Determining the fulfillment levels of DRs and PCs is an important task in the first two phases of the QFD processes, when QFD is applied to the new product design and development. However, most related studies only focus on the priority or achievement levels of DRs in phase 1. Unlike the existing research works, both phases 1 and 2 are considered in this paper. Based on the previous mathematical programming model for phase 1 under the fuzzy environment, this paper considers the close link between the two phases to build up a fuzzy linear programming model for phase 2 in determining the fulfillment levels of PCs. For reducing the design risk, risk analysis of DRs, namely a fuzzy FMEA, is taken into account in the phase 2 model. In addition, the satisfaction of the jth DR determined in phase 2 is greater than or equal to the contribution of the jth DR in phase 1 in order to realize customer satisfaction. The proposed model is illustrated with a numerical example to demonstrate the applicability in practice. The resulting ranges of satisfaction degree of phases 1 and 2 as well as the possible ranges of fulfillment levels of DRs and PCs can provide the QFD team with useful information for new product design and development. Acknowledgements The authors appreciate the anonymous reviewer’s useful comments to make this paper more comprehensive. This research was financially supported by the National Science Council, Republic of China, under Contract NSC95-2416-H-006-028-MY3.
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