Fuzzy linear programming models for NPD using a four-phase QFD activity process based on the means-end chain concept

European Journal of Operational Research 201 (2010) 619–632

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Fuzzy linear programming models for NPD using a four-phase QFD activity process based on the means-end chain concept Liang-Hsuan Chen a,*, Wen-Chang Ko a,b a b

Department of Industrial and Information Management, National Cheng Kung University, Tainan 701, Taiwan, ROC Department of Information Management, Kun Shan University, Tainan County, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 10 July 2007 Accepted 11 March 2009 Available online 21 March 2009 Keywords: Fuzzy sets Fuzzy linear programming Quality function deployment (QFD) Means-end Chain (MEC) Risk analysis

a b s t r a c t Quality function deployment (QFD) is a customer-driven approach in processing new product development (NPD) to maximize customer satisfaction. Determining the fulfillment levels of the ‘‘hows”, including design requirements (DRs), part characteristics (PCs), process parameters (PPs) and production requirements (PRs), is an important decision problem during the four-phase QFD activity process for new product development. Unlike previous studies, which have only focused on determining DRs, this paper considers the close link between the four phases using the means-end chain (MEC) concept to build up a set of fuzzy linear programming models to determine the contribution levels of each ‘‘how” for customer satisfaction. In addition, to tackle the risk problem in NPD processes, this paper incorporates risk analysis, which is treated as the constraint in the models, into the QFD process. To deal with the vague nature of product development processes, fuzzy approaches are used for both QFD and risk analysis. A numerical example is used to demonstrate the applicability of the proposed model. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Quality function deployment (QFD) is a useful customer-driven product development tool that uses a series of structured management processes to translate the customers’ needs into efficient communication through the various stages of product planning, design, engineering, and manufacturing. With short life-cycles and dynamic competition in global markets, the major challenge of any product-oriented firm is how to efficiently design, develop, and manufacture new products that will be preferred more by customers than those offered by competitors. QFD provides a comprehensive and systematic approach to new product development (NPD), ensuring that new products meet customers’ expectations. Since its introduction in the late 1960s, QFD has been successfully applied in many industries to improve product design, decision making processes, and customer satisfaction (Cristiano et al., 2001a; Lager, 2005). A typical QFD process consists of four phases that relate the customer requirements (CRs) (or voice of customer) to product design requirements (DRs) (phase 1), translate the settings of DRs into critical parts characteristics (PCs) (phase 2), determine critical process parameters (PPs) (phase 3), and finally establish production requirements (PRs) (phase 4). For an NPD project, a QFD team is organized to implement all four phases of the QFD process to improve customer satisfaction. Cristiano et al. (2001b) argued that QFD has been widely accepted largely due to the logical ordering of the relationships and the fact that the four phases reflect the product development processes. However, most of the existing literature has only focused on the first phase of QFD (Chen and Weng, 2003, 2006; Kwong et al., 2007; Chen and Ko, 2008), and so the result has very limited application in a practical NPD project. This situation motivated this study to consider all four phases of the QFD activity process in an NPD. A typical QFD diagram contains information on ‘‘whats”, ‘‘hows”, the relationship between the ‘‘what” and the ‘‘how”, and the relationship between the ‘‘hows” themselves. For example, in phase 1, a QFD team has to collect and treat a set of ‘‘whats”, i.e., CRs, create a number of appropriate ‘‘hows”, i.e., DRs that affect the CRs, and then determine the relationship strength between the CRs and DRs and the relationships between the DRs themselves. Based on these inputs, the importance ratings of the ‘‘hows” will be calculated. This result is the first-stage output since it leads towards the relevant decision making later, such as the fulfillment level of DRs, resource allocation, and the following QFD analyses. In traditional QFD, the relationship strength is expressed using a point system such as 1-3-9 or 1-5-9, indicating linguistic expressions such as ‘‘weak”, ‘‘moderate”, and ‘‘strong”, respectively. Nevertheless, QFD members usually do not have * Corresponding author. E-mail address: [email protected] (L.-H. Chen). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.03.010

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Fig. 1. A means-end chain model for the four phases of the QFD process.

sufficient knowledge and information about the influence of engineering responses on the ‘‘hows”, due to the lack of information or language hedge from the ‘‘whats” (Chen and Weng, 2003). These considerations have made the application of fuzzy approaches significant in addressing diversified and imprecise problems in QFD (Chan and Wu, 2002, 2005; Chen and Weng, 2003, 2006; Kwong and Bai, 2003; Chen et al., 2006; Kahraman et al., 2006; Chen and Ko, 2008). In addition, in the QFD process, each phase is closely related since the outcome from one phase applies the decision of the previous phase. However, how the outcomes from the latter phase make the decision of the former phase applicable is not mentioned in the existing research (Chan and Wu, 2002; Myint, 2003). In this study, considering the consistency of the QFD decision making, the concept of meansend chain (MEC) is applied to develop the QFD four-phase process to deal with the issue mentioned above. A means-end chain model (Gutman, 1982) is concerned with the relationships between attributes of a product, consequences (or benefits) accrued from the product, and customer values (satisfaction level). The attributes are created to provide the customer with certain benefits which reinforce the value to the customer. These concepts are applied in the four phases of the QFD process, and a conceptual model is shown in Fig. 1. From Fig. 1, the chain starts with the first phase of the QFD and the customer satisfaction (value) is achieved by the DRs’ characteristics (i.e. design planning) which fit the CRs expectations. However, the determination of the DRs’ characteristics should be realized in the next QFD process. Therefore, the role of the DRs is translated into phase 2 as the ‘‘consequences”. Then, the PCs should be created to make the DRs satisfactory in phase 2. In addition, the outcomes of PCs should make the DRs applicable in phase 1 for finally realizing the customer satisfaction. This concept is also applied to the other phases to build up the decision models in the QFD process. To decrease the risk of new product development, the risk analysis of the ‘‘hows” is usually necessary during QFD activities, and we adopted failure mode and effect analysis (FMEA) for this. FMEA is a systematic technique for identifying, prioritizing, and acting on potential failure modes before failures occur (Stamatis, 1995). Several studies have described the application of FMEA in the QFD process (Tan, 2003; Al-Mashari et al., 2005). However, the studies above were limited to only descriptive analyses. The methods of carrying out the aggregation of the QFD and FMEA are not mentioned, and the uncertainty is not considered at the product development stage, although Almannai et al. (2008) incorporated QFD and FMEA to propose a decision tool in the manufacturing system design and execution phases. In this study, we extend Chen and Weng’s fuzzy model(2003) by introducing risk analysis and FMEA into the existing fuzzy QFD approach. However, Chen and Weng’s fuzzy model (2003) only focused on the QFD phase 1, so that its application to a practical NPD project is limited. This paper proposes four-phase linked fuzzy QFD linear programming models for NPD projects based on the MEC concept. The proposed methodology adopted FMEA to deal with the potential risk of the ‘‘hows” in QFD phase 1, the outcomes of risk evaluation are applied to the next phase as the constraint factors of ends (or the ‘‘whats”, e.g., DRs in phase 2) in determining the achievement levels of means (or the ‘‘hows”, e.g., PCs in phase 2). Unlike the existing literature (Myint, 2003; Almannai et al., 2008), the risk evaluation results of the product planning stage (phase 1) are joined with the design stage (phase 2) to make the decision outcomes more applicable. This potential risk evaluation is also applied to the other phases for the engineering and manufacturing activities of an NPD project. The objective of this paper is to propose the linked fuzzy linear programming models to determine the fulfillment levels of ‘‘hows” with the aim of achieving the determined contribution levels of ‘‘whats” in QFD processes for customer satisfaction. In addition, to deal with the potential risk of NPD, this paper incorporates FMEA into QFD processes, and treats it as the constraint factor in the model. In the following two sections, the approaches of fuzzy QFD and risk analysis are introduced. In Section 4, a set of fuzzy linear programming models are developed to determine the achievement of the ‘‘hows” in each phase of the QFD process, considering the constraints of MEC, risk, and budget. An example of a semiconductor packing case is presented to demonstrate our approach in Section 5. Finally, the concluding remarks are provided in Section 6, as well as the limitations of the study and suggestions for future research. 2. Fuzzy QFD To implement the QFD process, a relationship matrix, also called an HOQ, is usually used for each phase to construct the relationships between the ‘‘what” and the ‘‘how” to determine the achievement priority or level of output variables. In practice, the major goal in phase 1 of the QFD process is to determine the achievement priority or level of DRs based on the importance of each CR, the relationships between CRs and DRs, and the relationships between the DRs themselves. Referring to the results from the first stage of the QFD process, similar work is performed with DRs–PCs in phase 2, PCs–PPs in phase 3, and PPs–PRs in the final phase. Since the four phases are linked, in phase 1 we denote R1,ij as the relationship level in terms of the score between CRi and DRj, and r1,jJ is the correlation score between DRj and DRJ. Similar notations are also used in the other phases. Considering the relationships between the DRs themselves in phase 1 of QFD, and considering the imprecise nature of the relationships, Chen and Weng (2003) proposed a fuzzified formulation based on Wasserman (1993) to calculate the normalized fuzzy relationship value between CRs and DRs as

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PJ e R 1;if  ~r 1;fj e 0 ¼ P f¼1 R ; 1;ij J PJ e ~ f¼1 R 1;if  r 1;fj j¼1

ð1Þ

e 1;if and ~r 1;fj are described by linguistic terms and defined as the fuzzy subsets of [0, 1], i = 1, . . . , I; j = 1, . . . , J. A fuzzy set can be fully and where R e 1;if at the a level, a 2 [0, 1], can be denoted uniquely represented by its a-cuts (Klir and Yuan, 2003). For example, the a-cut of the fuzzy set R by its lower and upper bounds as ½ðR1;if ÞLa ; ðR1;if ÞUa , which is defined by ðR1;if ÞLa ¼ inf x fxjle ðxÞ P ag, and ðR1;if ÞUa ¼ supx fxjle ðxÞ P ag, where R 1;if R 1;if le ðxÞ is the membership degree of x belonging to Re 1;if . R 1;if

Based on a-cuts and the extension principle (Zadeh, 1978; Zimmermann, 1991), the membership function of the fuzzy normalized relae u;if and ~r u;fj ; u ¼ 1; . . . ; 4. Chen and Weng (2003) further proposed tionship can be defined by the lower and upper bounds of each a-cut of R a modified formulation to obtain a more precise representation of the fuzzy normalized relationship, in which the lower and upper bounds of the membership function at each a-cut are formulated as

mðR0u;ij ÞLa

PJ

¼ PJ

m¼1 m–j

mðR0u;ij ÞUa ¼ PJ

L L f¼1 ðRu;if Þa ðr u;fj Þa PJ U U L L f¼1 ðRu;if Þa ðr u;fm Þa þ f¼1 ðRu;if Þa ðr u;fj Þa

PJ

m¼1 m–j

;

and

ð2aÞ

PJ

U U f¼1 ðRu;if Þa ðr u;fj Þa PJ L L U U f¼1 ðRu;if Þa ðr u;fm Þa þ f¼1 ðRu;if Þa ðr u;fj Þa

PJ

ð2bÞ

:

Once the modified lower and upper bounds of a-cuts of the fuzzy normalized relationship are obtained, the fuzzy technical importance rating f u;j ; u ¼ 1; . . . ; 4, for the four phases can be determined in the form of a-cuts, expressed as W

" # I I h i X X L U L U 0 0 ðW u;j Þa ¼ ðW u;j Þa ; ðW u;j Þa ¼ ku;i  mðRu;ij Þa ; ku;i  mðRu;ij Þa ; i¼1

ð3Þ

i¼1

P where ku,i represents the importance score of the ith ‘‘what” in the uth phase, and i ku;i ¼ 1; u ¼ 1; . . . ; 4. The fuzzy technical importance f u;j is then employed to find the optimal fulfillment level of each ‘‘how” to maximally fulfill the ‘‘whats” in the relevant phase of rating W the QFD process.

3. Fuzzy FMEA and risk analysis To reduce risk, designers usually perform risk analysis of the new product development process. This study adopts failure mode and effect analysis (FMEA) to perform risk analysis of the ‘‘hows” in the relevant QFD process. FMEA is a forward-looking approach using the potential failure modes to assess the potential risks and verify the likely cause in order to prevent them. 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 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 (F), and the detectability index (D), as

RPN ¼ S  F  D:

ð4Þ

The three indexes in (4) are defined on the same scale, such as the 10-point system, to identify the various levels of risk. In general, the design functions (or the ‘‘hows”) of a product’s component 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, F, and D, formulated as (4). Usually, an assessment of those indices is subjective and qualitatively described, so sometimes it is difficult to apply conventional FMEA to determine the rating of failure indices. Some researchers have applied fuzzy set theory to deal with the FMEA problems in a subjective and qualitative way, as mentioned above (Pillay and Wang, 2003; Guimarães and Lapa, 2004; Sharma et al., 2005). In this study, we perform risk analysis of DRs and PCs by using FMEA in the first two stages of QFD, corresponding to design FMEA and process FMEA, respectively. To describe 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 (or PC) can be expressed as follows:

g u;j ¼ maxðe e u;/ Þ ; RPN S u;n  e F u;u  D j n;u;/

j ¼ 1; 2; . . . ; J u ; u ¼ 1; 2;

ð5Þ

e u;/ are defined as the fuzzy subsets of [0, 1]. The RPN g u;j ; u ¼ 1; 2 is the fuzzy risk of the DRj and PCj in phases 1 and 2, where e S u;n ; e F u;u ; and D respectively, and is determined as the maximum of the three indices’ product from all sets of S, F, and D, if there is more than one set, to represent the fuzzy risk priority level of the DRj or the PCj. Furthermore, considering the different importance of each failure index, various g u;j . This study uses a fuzzy ordered weighted geometric averaging (FOWGA) operator (Xu indices can have different weights to determine RPN and Da, 2003) to modify (5). A FOWGA operator can be used to aggregate m (>1) fuzzy sets as (Xu and Da, 2003)

~1 ; a ~2 ; . . . ; a ~v ; . . . ; a ~V Þ ¼ f ða

V Y

~v Þwv ; ðb

v ¼1

~v is the vth largest element among V fuzzy numbers a ~v ; and ~v , v = 1, . . . , V; wv is the weight of the b where b FOWGA, (6) can be reformulated as

ð6Þ PV

v ¼1 wv ¼ 1; wv 2 ½0; 1. Following

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g u;j ¼ f ðe e RPN S; e F ; DÞ u;j ¼ max n;u;/

3 Y

~v Þwv ; ðb u;j

ð7Þ

v ¼1

~v is the vth largest set of the ðe e / Þ ; u ¼ 1; 2. In (7), the weighting vector of w = (w1, w2, w3)T can be determined according to the where b Sn ; e F u; D u g u;j , its membership function can be constructed by deriving designers’ or QFD team members’ experiences or knowledge. To determine RPN g u;j as: the lower and upper bounds of the a-cuts of RPN

ðRPN u;j ÞLa ¼ max n;u;/

3 h Y

v ðbv Þw u;j

v ¼1

iL a

and ðRPN u;j ÞUa ¼ max n;u;/

3 h Y

v ðbv Þw u;j

v ¼1

iU a

;

u ¼ 1; 2:

ð8Þ

g u;j is built-up, the defuzzified value can be obtained by the following formula: Once the membership function of RPN

Pn

RPN0u;j

¼

k

h io  0u;ðaÞj 1=2 ðRPN u;j ÞLak þ ðRPNu;j ÞUak  l P 0 ; l u;ðaÞj

u ¼ 1; 2;

ð9Þ

L g u;j , i.e., l  0u;ðaÞj is the membership degree of 1=2½ðRPNu;j ÞLa þ ðRPN u;j ÞUa  belonging to the fuzzy number RPN  0ðaÞj ¼ l where l f u;j ð1=2½ðRPNu;j Þak þ RPN U g ðRPNu;j Þ Þ, and k is the number of a-cuts. The defuzzified value of RPN u;j represents the fuzzy risk score of the means (i.e. DRj in phase 1

ak

or PCj in phase 2), and will be applied to the next phase to obtain the fuzzy risk ratings due to the consideration of design FMEA and process FMEA. The fuzzy risk rating of each PC and PP in phases 2 and 3 can be determined based on the fuzzy normalized relationship by

" # J J h i X X L U L U 0 0 0 0 e ð Ri uþ1;g Þa ¼ ðRiuþ1;g Þa ; ðRiuþ1;g Þa ¼ RPNu;j  mðRuþ1;jg Þa ; RPNu;j  mðRuþ1;jg Þa ; j¼1

ð10Þ

j¼1

e uþ1;g is the fuzzy risk rating of the gth ‘‘how” in the (u + 1)th phase, u = 1, 2. The fuzzy risk rating of each PC is used as a constraint where Ri factor in decision model of phase 2 to determine the fulfillment level of each PC in the design stage to satisfy the design requirements and finally maximize customers’ satisfaction. The same concept is also applied to phases 3 and 4. Unlike phases 2 and 3, the fuzzy risk rating of PRs in phase 4 of the QFD is obtained based on the fulfillment degrees of PPs. This is because the higher fulfillment degrees of PPs that should be achieved in phase 3 indicate the lower process capability at the current status, and therefore imply a high potential process risk of the associated PPs. The resulting fulfillment level of PPh in phase 3 is a fuzzy number ~ x3;h , which is obtained by the proposed models in the following section, and can also be denoted by its lower and upper bounds as x3;h is obtained, the defuzzified ½ðx3;h ÞLa ; ðx3;h ÞUa  for use in risk evaluation. Similar to (9) in the FMEA model, once the membership function of ~ value can be determined by

h io P n  03;ðaÞ k 1=2 ðx3;h ÞLak þ ðx3;h ÞUak  l h P 0 Lh ¼ ; l 3;ðaÞ

ð11Þ

h

 03;ðaÞ is the membership degree of 1=2½ðx3;h ÞLa þ ðx3;h ÞUa  belonging to the fuzzy number x~3;h , and ak denotes the kth a-cut. The defuzzwhere l k k h ified value of ~ x3;h represents the risk level of PPs in phase 3, and is applied to the final phase to determine the fuzzy risk rating of PRs with the consideration of the fuzzy normalized relationship. Similar to (10), the formula is expressed as

" # H H h i X X L U L U 0 0 e ð Ri 4;n Þa ¼ ðRi4;n Þa ; ðRi4;n Þa ¼ ðLh Þ  mðR4;hn Þa ; ðLh Þ  mðR4;hn Þa : h¼1

ð12Þ

h¼1

4. Fuzzy linear models of the QFD four-phase process In this section, a series of fuzzy linear models are proposed to formulate the decision models in the four-phase QFD application of the NPD. Each phase of QFD processes can be formulated with a fuzzy linear programming model to determine the contribution level of each ‘‘how” under the relevant constraint conditions in order to achieve the maximum total satisfaction of ‘‘whats”. Based on the MEC concept, each phase is closely linked with the former one to ensure that the customer requirements can be realized all the QFD activities for the NPD project. In addition, as described previously, to deal with the risk in NPD, this study incorporates FMEA into QFD processes, and treats it as the constraint factor in the model. A general model for phases 2–4 is also presented in this section, since they have a similar nature in the model formulation. In the following, the pairs in parentheses in Sections 4.1 and 4.2 represent the ‘‘whats” and the ‘‘hows”. 4.1. Phase 1: (CRs, DRs) Using the fuzzy technical importance ratings of DRs, Chen and Weng (2003) proposed a fuzzy linear programming model in which the decision variable, x1,j, is defined as a percentage to denote the fulfillment level percentage of the DR1,j, j = 1, . . . , J, i.e., x1,j 2 [0, 1]. 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 increased unit cost to e 1;j , to reflect its fuzzy nature in the design planning stage. With the fulfillment percentage achieve the fulfillment level is a fuzzy number, C of DR1,j, a corresponding percentage of increased unit cost is required to enhance the characteristics of the product or service. The total increased unit cost cannot exceed a budget constraint at this stage. In addition, various DRs are prioritized based on their impacts on customer satisfaction; business competition and technological difficulties are also considered in the model. If the pth DR is preferred to the qth DR in terms of customer satisfaction, a constraint is added to the model (see (13a.2) and (13b.2)). e 1;j are placed in the model to find the lower f 1;j and C To solve the fuzzy linear programming model, the lower and upper bounds of a-cuts of W and upper bounds of x1,j and the customer satisfaction at each a-cut, respectively. Thus, a fuzzy linear programming model is formulated as

L.-H. Chen, W.-C. Ko / European Journal of Operational Research 201 (2010) 619–632

ðZ 1 ÞLa ¼ max

J X ðW 1;j ÞLa  x1;j

623

ð13aÞ

j¼1

s:t:

J X ðC 1;j ÞUa  x1;j 6 B1 ;

ð13a:1Þ

j¼1

ðZ 1 ÞUa ¼ max

ðW 1;p ÞLa  x1;p  ðW 1;q ÞUa  x1;q P 0;

ð13a:2Þ

0 6 ej 6 x1;j 6 g1;j 6 1;

ð13a:3Þ

8j;

J X ðW 1;j ÞUa  x1;j

ð13bÞ

j¼1

s:t:

J X ðC 1;j ÞLa  x1;j 6 B1 ;

ð13b:1Þ

j¼1

ðW 1;p ÞUa  x1;p  ðW 1;q ÞLa  x1;q P 0; 0 6 ej 6 x1;j 6 g1;j 6 1; 8j;

ð13b:2Þ ð13b:3Þ

p; q 2 f1; 2; . . . ; Jg; where B1 is the budget limitation, and ðZ 1 ÞLa and ðZ 1 ÞUa represent the lower and upper bounds, respectively, of objective values, i.e., customer satisfaction in phase 1 of the QFD process, at each a-cut. Eqs. (13a.1) and (13b.1) delimit the budget for the lower and upper costs at each a level. As described above, the preemptive priority among DRs is presented in Eqs. (13a.2) and (13b.2). As to Eqs. (13a.3) and (13b.3), 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 total satisfaction of CRs and the fuzzy achievement levels of DRs can be obtained. 4.2. Phase 2: (DRs, PCs), phase 3: (PCs, PPs) and phase 4: (PPs, PRs) 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 first. In the 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 the QFD process (Myint, 2003). However, the achievement levels of DRs in phase 1 will affect the PCs’ deployment since DRs are treated as the ‘‘whats” in phase 2. Therefore, we propose a weighted-average equation to determine the importance score by using both the importance ratings and the achievement levels of DRs in phase 1, as follows: 0 k2;j

¼

1 2

Ph k

i  1;ðaÞj ðW 1;j ÞLak ðx1;j ÞLak þ ðW 1;j ÞUak ðx1;j ÞUak  l P ; l 1;ðaÞj

j ¼ 1; . . . ; J;

ð14Þ

f 1;j ~x1;j determined by the  1;ðaÞj denotes the membership degree of 1=2½ðW 1;j ÞLa ðx1;j ÞLa þ ðW 1;j ÞUa ðx1;j ÞUa  belonging to the fuzzy number W where l k k k k a-cut and extension principle. The normalized importance score of each DR is then expressed as 0

k2;j k2;j ¼ P 0 ; j k2;j

j ¼ 1; 2; . . . ; J:

ð15Þ

The general form of Eqs. (14) and (15) can be expressed as the following equations for the importance scores in each phase of the QFD process: 1 0

ku;m ¼ 2

P

L L k ½ðW u1;m Þak ðxu1;m Þak

 u1;ðaÞv þ ðW u1;m ÞUak ðxu1;m ÞUak   l P ; l u1;ðaÞv

and

ð16Þ

0

ku;m ku;m ¼ P 0 ; m ku;m

u ¼ 2; 3; 4;

ð17Þ

where m depends on the number of ‘‘how” factors in each phase, and ak denotes the kth a-cut. Considering the concept of MEC, the role of the ‘‘attributes”, DRs, is translated into ‘‘consequences” from the former phase to phase 2, as described in Fig. 1 for the MEC model. Meanwhile, the new ‘‘attributes”, PCs, should be created to ensure that the desired ends, DRs, can meet the internal customer’s expectations (satisfaction of design planning). The contribution to the customers’ satisfaction by DRj in phase hP i G e 0 Þ~ mð R x2;g , which should be greater than or equal to the 1 due to the design satisfaction of PCs in phase 2 can be expressed as k2;j g¼1

2;jg

f 1;j ~ x1;j , in phase 1 to realize customer satisfaction (see (18a.1) and (18b.1)). Using FMEA for the risk analysis of DRs, contribution of DRj, W the fuzzy risk ratings of PCs are obtained by (10) and used as the constraint in the proposed model (see (18a.2) and (18b.2)). Moreover, the budget limitation, preemptive priority, and technological ability for PCs are also taken into account (see (18a.3)–(18a.5) and (18b.3) and (18b.5)). Taking the above considerations as the constraints and using the PCs’ fuzzy importance rating of (3) as well as the decision variables, x2,g, as the objective function, a fuzzy linear programming model is formulated in the a-cut form as

ðZ 2 ÞLa ¼ max

G X ðW 2;g ÞLa  x2;g ; g¼1

s:t:

k2;j

" G X g¼1

ð18aÞ #

mðR02;jg ÞLa  x2;g P W 1;j x1;j ;

j ¼ 1; . . . ; J;

ð18a:1Þ

624

L.-H. Chen, W.-C. Ko / European Journal of Operational Research 201 (2010) 619–632 G X ðRi2;g ÞUa  x2;g 6 Ha2 ;

ð18a:2Þ

g¼1 G X ðC 2;g ÞUa  x2;g 6 B2 ;

ð18a:3Þ

g¼1

ðZ 2 ÞUa ¼ max

ðW 2;p ÞLa  x2;p  ðW 2;q ÞUa  x2;q P 0;

ð18a:4Þ

0 6 x2;g 6 g2;g 6 1;

ð18a:5Þ

8g;

G X ðW 2;g ÞUa  x2;g ;

ð18bÞ

g¼1

"

s:t:

k2;j

G X

# mðR02;jg ÞUa

 x2;g P W 1;j x1;j ;

j ¼ 1; . . . ; J;

ð18b:1Þ

g¼1 G X ðRi2;g ÞLa  x2;g 6 Ha2 ;

ð18b:2Þ

g¼1 G X ðC 2;g ÞLa  x2;g 6 B2 ;

ð18b:3Þ

g¼1

ðW 2;p ÞUa  xu;p  ðW 2;q ÞLa  x2;q P 0;

ð18b:4Þ

0 6 x2;g 6 g2;g 6 1;

ð18b:5Þ

8g;

p; q 2 f1; 2; . . . ; Gg: In Model (18), the objective function is to find the maximum satisfaction of design requirements in phase 2. The values of ðZ 2 ÞLa and ðZ 2 ÞUa represent the lower and upper bounds of objective values, respectively, at each a-cut. In the constraints, Ha2 is a threshold of risk; B2 is a e 2;g and g2,g are the increased unit cost to achieve the fulfillment level and technological difficulty of the PCs, budgetary limitation; and C respectively. These parameters are determined by the QFD team. For simplification and application purposes, the crisp contribution level f 1;j ~x1;j calcuof each DR is taken as the action by the QFD team and is adopted in the model. In addition, W1,jx1,j is the defuzzified value of W 0 lated by (14), so that W1,jx1,j is equal to k2;j . Since phases 2–4 have the similar model structure based on the MEC concept and considering the risk analysis, the general model formulating phases 2–4 can be expressed as

ðZ u ÞLa ¼ max

V X

ðW u;v ÞLa  xu;v ;

v ¼1

s:t:

ku;w

" V X v ¼1

ð19aÞ #

mðR0u;wv ÞLa  xu;v P W u1;w xu1;w ;

w ¼ 1; 2; . . . ; W;

V X ðRiu;v ÞUa  xu;v 6 Hau ;

ð19a:1Þ ð19a:2Þ

v ¼1 V X ðC u;v ÞUa  xu;v 6 Bu ;

ð19a:3Þ

v ¼1

ðW u;p ÞLa  xu;p  ðW u;q ÞUa  xu;q P 0;

ðZ u ÞUa ¼ max

0 6 xu;v 6 gu;v 6 1;

8v ;

p; q 2 f1; 2; . . . ; Vg;

u ¼ 2; 3; 4;

V X ðW u;v ÞUa  xu;v ;

v ¼1

s:t:

ð19a:4Þ ð19a:5Þ

ku;w

" V X v ¼1

ð19bÞ #

mðR0u;wv ÞUa  xu;v P W u1;w xu1;w ;

V X ðRiu;v ÞLa  xu;v 6 Hau ;

w ¼ 1; 2; . . . ; W;

ð19b:1Þ ð19b:2Þ

v ¼1 V X ðC u;v ÞLa  xu;v 6 Bu ;

ð19b:3Þ

v ¼1

ðW u;p ÞUa  xu;p  ðW u;q ÞLa  xu;q P 0; 0 6 xu;v 6 gu;v 6 1; 8v ; p; q 2 f1; 2; . . . ; Vg; L

ð19b:4Þ ð19b:5Þ

u ¼ 2; 3; 4: U

In Model (19), the values of ðZ u Þa and ðZ u Þa represent the lower and upper bounds of objective values in phase u, u = 2, 3, 4, at each a-cut, in which xu,v is the decision variable. The MEC constraint is represented by (19a.1) and (19b.1) for the lower and upper bounds at the a level,

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625

where w is the number of ‘‘whats” in phase u (or ‘‘hows” in phase u  1); V is the number of ‘‘hows” in phase u; the importance score ku,w can e u;v and f u;v ~xu;v calculated by (16), so that Wu1,vxu1,v is equal to k0 ; C be obtained by (16) and (17); Wu1,vxu1,v is the defuzzified value of W u;v gu,v are the increased unit cost to achieve the fulfillment level and technological difficulty of the ‘‘hows” in uth phase, respectively. Using the fuzzy risk rating of (10) as u = 2 or 3 (or (12) as u = 4), (19a.2) and (19b.2) show the risk evaluation constraint at the a level in uth phase. Budget limitation is considered by (19a.3) and (19b.3); the preemptive priority constraint is shown in (19a.4) and (19b.4) for the lower and upper bounds. 4.3. Solution procedure The major concept of the proposed four-phase models is to translate customers’ requirements into design planning, parts design, engineering and manufacturing stages, considering risk analysis, MEC concept, and the relevant constraints for NPD. Specifically, those models are closely linked to finally achieve the maximum customer satisfaction. In summary, the solution procedure is described as follows: Step 0: Build up an HOQ for each phase, and determine the associated constraint parameters. Step 1: Find the lower and upper bounds of x1,j and obtain the maximum customer satisfaction in phase 1 at each a-cut by using the ae 1;j ; ej and g1,j in Model (13). f 1;j and relevant constraints, such as C cuts of W e uþ1;g , as u = 1. g u;j and Ri Step 2: Evaluate the potential risk of DRj and determine RPN f 2;g . f 1;j to k0 and find the a-cuts of W Step 3: Transfer W 2;j

e 2;g and relevant constraints, such as C e 2;g ; g in Model (19) as u = 2 for finding the lower and upper f 2;g ; Ri Step 4: Use the a-cuts of W 2;g bounds of x2,g and the DRs’ satisfaction at each a-cut. e uþ1;g as u = 2. g u;g and Ri Step 5: Evaluate the potential risk of PCg and determine the a-cuts of RPN f 3;h . f 2;g to k0 and find the a-cuts of W Step 6: Transfer W 3;g e 3;h and relevant constraints, such as C e 3;h ; g in Model (19) as u = 3 for finding the lower and upper f 3;h ; Ri Step 7: Use the a-cuts of W 3;h bounds of x3,h and the PCs’ satisfaction at each a-cut. e 3;h . x 3;h and determine the a-cuts of Ri Step 8: Evaluate the potential risk of PPh by the a-cuts of e f 4;n . f 3;h to k0 and find the a-cuts of W Step 9: Transfer W 3;h e 4;n and relevant constraints, such as C e 4;n ; g in Model (19) as u = 4 for finding the lower and upper f 4;n ; Ri Step 10: Use the a-cuts of W 4;n bounds of x4,n and the PPs’ satisfaction at each a-cut.

5. An illustrative example To show the applicability of the proposed models, a semiconductor packing case of the turbo thermal ball grid array (T2-BGA) package is used. A heat slug is inserted into the molding compound of a plastic BGA (PBGA), since T2-BGA has the advantages of high heat dissipation and better electrical characteristics, although there are some problems because of the complex package structure (Chen et al., 2002). 5.1. Phase 1 Based on the characteristics of T2-BGA, the 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. 2. 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 ‘‘molding flow” (DR5). First, e 1;if between CRs and DRs as well as the relationships ~r 1;fj between DRs themselves are determined. These relationthe fuzzy relationships R ships are denoted as ‘‘weak”, ‘‘moderate”, or ‘‘strong”, which are translated into the triangular fuzzy numbers as a three-element set, (0, .2, .4), (.3, .5, .7), (.6, .8, 1), respectively, with the membership functions shown in Fig. 3. The middle value in the three-element set repree 1;if ¼ ð:6; :8; 1Þ is formulated as sents the most likely one with the membership degree equivalent to 1, such as lstrong(.8) = 1. For example, R

Fig. 2. Phase 1 for a T2-BGA package.

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Fig. 3. The membership functions of linguistic relationships in the HOQs.

leR 1;if ðR1;if Þ ¼ fðR1;if  :6Þ=ð:8  :6Þ; :6 6 R1;if 6 :8; and ð1  R1;if Þ=ð1  :8Þ; :8 6 R1;if 6 1:g: Considering the linguistic relationships in Figs. 4 and 5, the fuzzy normalized relationships at each a-cut can be calculated using (2). To e 1;if and ~r 1;fj should be determined beforehand based on their membership functions. do this, the lower and upper bounds of each a-cut of R The lower and upper bounds of each a-cut can be represented as functions of a (2[0, 1]). For example, the a-cut of the membership function (.6, .8, 1) of ‘‘strong” is expressed as ½ðR1;if ÞLa ; ðR1;if ÞUa  ¼ ½:6 þ :2a; 1  :2a. Without any biases, the a levels in this paper are evenly distributed in [0, 1]. Let af denote the fth a level, and af = f/k, f 2 {1, 2, . . . , k}, such that the distance between two adjacent a levels is equal, i.e. f 1;j Þ ; j ¼ 1; 2; . . . ; J, at each a level are obaf  af1 = 1/k, f P 1. The lower and upper bounds of the fuzzy technical importance rating ð W a tained by (3) to determine the importance priority of each DR, as listed in Table 1. They are applied to Model (13) to solve the fuzzy linear model of phase 1. We suppose that the budget limit (B1) is 1 unit and the QFD team has prioritized DRs as DR4  DR1  DR2 and DR3  DR5 for the design preference, where ‘‘” means ‘‘is more preferred than”. Table 2 lists the associated data for Model (13). Applying fuzzy technical importance ratings, increased cost, and the associated constraints to Model (13) at several a levels, the fulfillment level of each DR and the total customer satisfaction degree (the objective function value) can be obtained. In the fuzzy sense, the a level can be interpreted as the confidence degree (Bondia and Picó, 2003; Wu, 2003). In this example, the settings of a levels less than 0.8

Fig. 4. Phase 2 for a T2-BGA package.

Fig. 5. Phase 3 for a T2-BGA package.

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L.-H. Chen, W.-C. Ko / European Journal of Operational Research 201 (2010) 619–632 Table 1 The fuzzy technical importance ratings (%) at different a levels.

a

W L1;1

W U1;1

W L1;2

W U1;2

W L1;3

W U1;3

W L1;4

W U1;4

W L1;5

W U1;5

1 .5 0

16.9 8.4 0

16.9 25.3 33.7

20.8 15.1 10.6

20.8 27.6 35.1

5.9 4.0 2.5

5.9 8.1 10.0

37.0 28.7 20.0

37.1 45.6 54.0

8.2 3.5 0

8.2 13.9 20.4

Table 2 The cost, business competition, and technical difficulty of each DR. Fulfillment level

e 1;j C

a-Cut

ej

g1,j

x1,1 x1,2 x1,3 x1,4 x1,5

(.2, .3, .4) (.8, .9, 1) (.4, .5, .6) (.3, .4, .5) (.1, .2, .3)

[.2 + .1a, .4  .1a] [.8 + .1a, 1  .1a] [.4 + .1a, .6  .1a] [.3 + .1a, .5  .1a] [.1 + .1a, .3  .1a]

.1 .1 .1 .1 .5

1 1 1 1 1

will produce infeasible regions in solving the fuzzy linear model. From the properties of a fuzzy number, this means the confidence level of fuzzy inputs should be higher to avoid producing wider ranges of a-cuts. Table 3 shows the outcomes of the total customer satisfaction degree and fulfillment levels of each DR for Model (13) based on three a levels (.8, .9, 1.). The decision variable x1,j = 100% denotes a complete fulfillment of DRj. In phase 1, DR4 should have the better quality level in order to achieve the CRs’ satisfaction degree. On the other hand, DR2 is the smallest, due to the high incremental cost, as shown in Table 2. 5.2. Phase 2 The outcomes from phase 1 of the QFD, including the fuzzy technical importance rating and fulfillment level of each DR, are applied to (14) and normalized by (15) to determine the importance score in phase 2 of the QFD. They are calculated as .13, .042, .082, .663 and .083 for the five DRs, respectively. In addition, the DRs are used to design FMEA to evaluate the potential failure modes and effects for design risk analysis. A risk analysis of the T2-BGA package is listed in Table 4, in which the three failure indices are described in linguistic terms as ‘‘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, 1, 2), (1, 2.5, 2.5, 4), (3, 5, 5, 7), (6, 7.5, 7.5, 9), and (8, 8, 10, 10), respectively. e / , and Sn; e F u; D To obtain the fuzzy risk priority number (RPN), the upper and lower bounds of a-cuts of the three fuzzy failure indices e their weights should be determined beforehand. Suppose that the set of weights is w = {.7, .21, .09}, determined by the QFD team. The fuzzy RPN of each DR is constructed by (8) and then defuzzified by (9) for risk ratings in phase 2 of the QFD process. These 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), namely ‘‘profile of heat slug” (PC1), ‘‘leg shape of heat slug” (PC2), ‘‘plating quality of heat slug” (PC3), ‘‘jointibility between attached material and molding compound” (PC4), and ‘‘jointibility among attached material, heat slug, and substrate” (PC5). Similar to phase 1, considering the relationships between DRs and PCs and the relationships between the PCs themselves in Fig. 4, as well as the membership functions of linguistic relationships in Fig. 3, the fuzzy normalized relationship at each a-cut can be obtained using (2). Then, f 2;g Þ ; g ¼ 1; 2; . . . ; G, at each a level, are obtained by (3) to determine the lower and upper bounds of the fuzzy importance rating of PCs, ð W a f 2;g Þ at different a levels are listed in Table 5. The fuzzy risk ratings of the importance priority of each PC. The lower and upper bounds of ð W a each PC at each a-cut can be calculated by (10) and are listed in Table 6. The resulting fuzzy importance and risk level ratings of PCs with the associated constraints listed in Table 7 are applied to Model (18) at several a levels to solve the total DR satisfaction degree (the objective function value) and the fulfillment level of each PC. For phase 2, the set of DRs’ importance scores produced by (14) is {.064, .021, .041, .330, .041}. The budget limit (B2) and the threshold of risk (Ha2) are 1.5 and 25 units, respectively. In addition, suppose that the QFD team considers the preemptive priority of PCs as PC5  PC2 for the design process. In solving Model (18), a levels less than 0.8 will produce infeasible regions. Applying three a levels, i.e., .8, .9, and 1., to Model (18), the total DR satisfaction degree and the fulfillment level of each PC can be obtained, as shown in Table 8. The fulfillment level of decision variable x2,g = 100% denotes the complete fulfillment of PCg. In Table 8, the fulfillment levels of PC1 and PC5 represent the best performance levels in order to achieve the DR’s satisfaction degree in this case. Regarding the aggregation of the concept of MEC’s into the QFD process, the design planning satisfaction degree (achieved level) of DRj in phase 2 should be greater than or equal to the contribution of DRj (effort of the attributes or means) in phase 1 to realize customer satisfaction. As a simple demonstration, considering DR5 at a = 1.0. The satisfaction degree is equal to .062 by calculating either hP i hP i L U L U 0 0 0 0 or k2;5 k2;5 g mðR2;5g Þa  x2;g g mðR2;5g Þa  x2;g , since mðR2;5g Þa ¼ mðR2;5g Þa . Then, the fulfillment level ascribed to the PCs’ contribution

Table 3 The outcomes of CRs satisfaction degree and fulfillment level of each DR in phase 1.

a

Z L1

Z U1

xL1;1

xU1;1

xL1;2

xU1;2

xL1;3

xU1;3

xL1;4

xU1;4

xL1;5

xU1;5

1 .9 .8

.514 .378 .3

.514 .584 .682

.256 .147 .177

.256 .483 .844

.1 .1 .1

.1 .1 .1

.667 .9 1

.667 .583 .429

1 .731 .527

1 1 1

.5 .5 .5

.5 .5 .5

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Table 4 Design FMEA of the T2-BGA package. 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

M

H

Package looks defect

H

Plating solution contaminated

M

Profile inspection Profile inspection Surface inspection

L

EMC overflow

Packing/ transportation Heat slug forming

M

M L

Heat slug attached material

Molding failure

Package profile defect

VH

Turbulence

M

Simulation

H

Height of heat slug

Plating layer peeling off Plating thickness inconsistent

Package looks defect Substrate damage

M

Plating solution contaminated Stress of the H.S.

L

R

EMC overflow

H

L

Package looks defect Short circuit

M M

Heat slug height out of spec. Unsuitable packaging Burr

Surface inspection Stress measurement H.S.height inspection Eye inspection

L

Surface inspection

R

Unusual height

H

M

L

H M 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

Table 5 The fuzzy importance ratings of PCs (%) for various values of a.

a

W L2;1

W U2;1

W L2;2

W U2;2

W L2;3

W U2;3

W L2;4

W U2;4

W L2;5

W U2;5

1 .5 0

39.7 25.6 15.7

39.7 58.4 79.4

12.3 8.0 4.6

12.3 17.3 22.4

3.0 1.9 1.1

3.0 4.2 5.3

3.6 1.8 .6

3.6 6.5 10.8

14.7 6.4 1.1

14.7 26.5 41.0

Table 6 The fuzzy risk ratings of PCs for various values of a.

a

RiL2;1

RiU2;1

RiL2;2

RiU2;2

RiL2;3

RiU2;3

RiL2;4

RiU2;4

RiL2;5

RiU2;5

1 .5 0

7.17 4.87 3.20

7.17 10.09 13.26

8.94 6.01 3.73

8.94 12.35 15.86

2.62 1.70 .96

2.62 3.66 4.68

4.45 2.49 1.10

4.45 7.24 11.11

5.12 3.38 2.09

5.12 7.33 9.87

Table 7 The cost and technical ability for each PC. Fulfillment level

e 2;g C

a-cut

g2,g

x2,1 x2,2 x2,3 x2,4 x2,5

(.2, .3, .4) (.2, .3, .4) (.4, .5, .6) (.3, .4, .5) (.3, .4, .5)

[.2 + .1a, .4  .1a] [.2 + .1a, .4  .1a] [.4 + .1a, .6  .1a] [.3 + .1a, .5  .1a] [.3 + .1a, .5  .1a]

1 .85 .8 1 1

Table 8 The outcomes of DRs satisfaction degree and fulfillment level of each PC in phase 2.

a

Z L2

Z U2

xL2;1

xU2;1

xL2;2

xU2;2

xL2;3

xU2;3

xL2;4

1 .9 .8

.693 .627 .555

.693 .748 .801

1 1 1

1 1 1

.85 .85 .774

.85 .85 .85

.321 .311 .524

.321 .8 .68

.961 .873 .57

is ^ x1;5 ¼ k2;5

hP

L 0 g mðR2;5g Þa

xU2;4 .961 .276 0

xL2;5

xU2;5

1 1 1

1 1 1

i  x2;g =W 1;5 ¼ :756 with W1,5 = .082, which is greater than x1,5 (=.5) determined in phase 1, as shown in Table 3. The

resulting fulfillment level is acceptable because the possible range of x1,5 is in [.5, 1].

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5.3. Phases 3 and 4 Similar to phase 2, the use of the fuzzy importance rating and fulfillment level of each PC in phase 2 determines the importance score of each PC in phase 3 by (16) and (17). They are obtained as .581, .15, .022, .031 and .216 for the five PCs, respectively. FMEA is then carried out for PCs to evaluate the potential failure modes and effects for process risk. An analysis output of the T2-BGA package is listed in Table 9. Based on the table, the fuzzy RPN of each PC is constructed by (8) and then defuzzified by (9) for risk ratings in phase 3. They are calculated as 7.227, 6.776, 6.631, 6.631, and 6.219 for the five PCs, respectively. In addition, five key process parameters (PPs), namely ‘‘heat slug picked-up mode” (PP1), ‘‘glue pattern” (PP2), ‘‘density control of plating solution” (PP3), ‘‘heat slug cleaning” (PP4), and ‘‘heat slug feeding positioning” (PP5), are created in order to deploy QFD phase 3, which is shown in Fig. 5. Referring to the relationships between PCs and PPs and those between the PPs themselves in Fig. 5 and the membership functions of linguistic relationships in Fig. 3, each a-cut of the fuzzy normalized relationship can be obtained using (2). Similar to phase 2, applying (3) and (10), the a-cuts of the fuzzy importance rating and the fuzzy risk rating of each PP can be determined. Both results and the associated constraint conditions (see Table 10), are applied to Model (19) as u = 3 at several a levels to solve the fuzzy linear model to determine the importance priority of each PP. The fulfillment level of each PP and the total PC satisfaction degree (the objective function value) can then be obtained. The set of PCs’ importance scores produced by (16), as u = 3, is {.399, .103, .015, .021, .148}. In this phase, the budget limit (B3) and the threshold of risk (Ha3) are 2 and 35 units, respectively. In addition, suppose that the QFD team does not consider the preemptive priority of PPs in this phase. In solving Model (19), a levels less than 0.8 will produce infeasible regions in phase 3. The total PC satisfaction degree and the fulfillment level of each PP are shown in Table 11, in which the fulfillment levels of PP2 and PP5 represent the best performance level that should be attained to achieve the PC’s satisfaction degree in this case. Applying the fuzzy importance rating and fulfillment level of each PP in phase 3 to (16) and (17) can obtain the importance score of each PP in phase 4. They are calculated as .161, .151, .069, .079 and .54 for the five PPs, respectively. In addition, a higher fulfillment degree of a PP indicates a lower process capability and implies a higher process risk for this PP at present. The risk level is determined from the fulfillment degree of PPs by (11) for risk ratings in phase 4 of the QFD process. They are calculated as .159, .228, .182, .203, and .228 for the five Table 9 Process FMEA of the T2-BGA package. PCk Process function

Potential failure mode

Potential effects of failure

S

Potential causes of failure

O

Detecting method

D

Profile of heat slug

Distortion of the heat slug Warpage of the heat slug

EMC overflows

H

Transportation

M

L

EMC overflows

H

Inappropriately picked-up location Inappropriately picked-up stress

M

Profile inspection Profile inspection

M

Profile inspection

M

Burrs

M

L

Containment

H

Ab-alignment

M

Turbulence

L

Ultrasonicwaves scan Ultrasonicwaves scan Ultrasonicwaves scan Simulation

Plating solution contaminated Stress concentration

L

Eye inspection

R

M

Stress measurement H. S. height inspection Liquid inspection

M

Leg shape of heat slug

Plating quality of heat slug

Popcorn

Reliability test failed

Plating layer peeling off Plating thickness inconsistent

Plating color inconsistent

H

Heat slug surface looks defective Substrate damage

VH

EMC overflow

H

Heat slug surface looks defective

H

H

Jointibility between attached material and molding compound

Popcorn

Reliability test failed

H

Jointibility among attached material, heat slug and substrate

Popcorn

Reliability test failed

H

H

L L L

Heat slug height out of spec. Plating solution concentration out of spec. Recipe error of plating solution

L

L

Liquid inspection

M

Contaminant

M

M

Turbulence

L

Ultrasonicwave scan Simulation

Temperature profile

M

Temperature adjustment

L

M

M M

L

Table 10 The cost, technical ability, budget limitation, and risk threshold of each PP. Fulfillment level

e C 3;h

a-cut

g3,h

x3,1 x3,2 x3,3 x3,4 x3,5

(.2, .3, .4) (.2, .3, .4) (.4, .5, .6) (.3, .4, .5) (.3, .4, .5)

[.2 + .1a, .4  .1a] [.2 + .1a, .4  .1a] [.4 + .1a, .6  .1a] [.3 + .1a, .5  .1a] [.3 + .1a, .5  .1a]

1 .85 .8 1 1

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Table 11 The outcomes of the PCs satisfaction degree and fulfillment level of each PP in phase 3.

a

Z L3

Z U3

xL3;1

xU3;1

xL3;2

xU3;2

xL3;3

xU3;3

xL3;4

xU3;4

xL3;5

xU3;5

1 .9 .8

.848 .776 .725

.848 .921 1.046

.7 .7 .7

.7 .7 .7

1 1 1

1 1 1

.8 .8 .8

.8 .8 .8

.9 .9 .854

.9 .9 .9

1 1 1

1 1 1

PPs, respectively. Furthermore, six key production requirements (PRs), namely ‘‘heat slug picked-up jig detection” (PR1), ‘‘clean inspection of heat slug” (PR2), ‘‘glue pattern and volume inspection” (PR3), ‘‘quality specification of plating solution” (PR4), ‘‘fool-proof device of heat slug feeding positioning” (PR5), and ‘‘heat slug profile inspection” (PR6), are created to deploy the QFD final phase, which is shown in Fig. 6. In the figure, Lh denotes the risk level of PP4,h. Similarly, various a-cuts of the fuzzy normalized relationship can be obtained using (2). In Fig. 6, PPs’ importance scores are calculated by (16) and (17), as u = 4, and then those of the fuzzy importance rating and the fuzzy risk ratings of each PR can be calculated by (2) and (12), respectively. In phase 4, the budget limit (B4) and the threshold of risk (Ha4) are 2 and 1 units, respectively. It is noted that Ha4 is much less than Ha2 or Ha3, due to the different scale system. The associated constraints for the decision model of this phase are listed in Table 12. In addition, suppose that the QFD team considers the preemptive priority of PRs as PR6  PR2 for the production process. Model (19), as u = 4, is employed at several a levels to determine the total satisfaction of PPs and the fulfillment level of each PR. In this phase, a levels less than 0.9 will produce infeasible regions. This means that the confidence level (P0.9) should be adopted for phase 4 in this QFD process. The outcomes of the final phase in QFD for the NPD are shown in Table 13, in which the fulfillment levels of PR4 and PR5 represent the best performance level needed to achieve the PC’s satisfaction degree in this case. The verification of the MEC concept at phases 3 and 4 of the QFD application can be done. For example, the process parameters satisfaction degree (achieved level) of PPh in phase 4 should be greater than or equal to the contribution of PPh (effort of the attributes or means) in phase 3 to realize parts design satisfaction. Similarly, considering the first PP in the example, when a = 1.0, the satisfaction degree is hP i hP i L U 0 0 equal to .1478 by calculating either k4;1 n mðR4;1n Þa  x4;n or k4;1 n mðR4;1n Þa  x4;n . The fulfillment level ascribed to the PPs’ contribution hP i L 0 is then ^ x3;1 ¼ k4;1 n mðR4;1n Þa  x4;n =W 3;1 ¼ :758 with W3,1 = .195, which is greater than x3,1 (=.7) determined in phase 3, if referring to Table 11. The resulting fulfillment level is acceptable because the possible range of x3,1 is in [0, 1] (see Table 10).

Fig. 6. The phase 4 for a T2-BGA package.

Table 12 The cost and technical ability of each PR. Fulfillment level

e 4;n C

a-cut

g4,n

x4,1 x4,2 x4,3 x4,4 x4,5 x4,6

(.2, .3, .4) (.4, .5, .6) (.2, .3, .4) (.3, .4, .5) (.1, .2, .3) (.2, .3, .4)

[.2 + .1a, .4  .1a] [.4 + .1a, .6  .1a] [.2 + .1a, .4  .1a] [.3 + .1a, .5  .1a] [.1 + .1a, .3  .1a] [.2 + .1a, .4  .1a]

.95 1 1 1 1 .95

Table 13 The outcomes of the PPs satisfaction degree and fulfillment level of each PR in phase 4.

a

Z L4

Z U4

xL4;1

xU4;1

xL4;2

xU4;2

xL4;3

xU4;3

xL4;4

xU4;4

xL4;5

xU4;5

xL4;6

xU4;6

1 .9

.924 .872

.924 1.004

.95 .95

.95 .95

.9

.9 1

1 .97

1 1

1 1

1 1

1 1

1 1

.867 .95

.867 .95

1

L.-H. Chen, W.-C. Ko / European Journal of Operational Research 201 (2010) 619–632

Z1

Z2

1

Z3

Membership degree

Membership degree

1 0.95 0.9 0.85 0.8 0.2

0.4

0.6

0.8

% ; Zi

(a) α ≥ 0.8

1

1.2

Z1

Z2 Z3

631

Z4

0.975 0.95 0.925 0.9 0.2

0.4

0.6

0.8

1

1.2

% ; Zi

(b) α ≥ 0.9

Fig. 7. The membership functions of the satisfaction degree at each QFD processes.

5.4. Discussion The proposed models are illustrated through the example of the semiconductor packaging product development problem. Model (13) (as u = 1) and Model (19) (as u = 2, 3, and 4) are solved using Lingo 9.0 software in a Microsoft Windows XP environment running on a laptop with a 1.5 GHz processor and 512 MB of RAM (random access memory). The efficiency is satisfactory since the software runtime is negligible for each a-cut. Particularly, unlike the existing literature, this paper adds fuzzy FMEA in phases 2 and 3 as well as the risk evaluation in phase 4 to the proposed fuzzy linear models to deal with the potential risk problem in an NPD project. From the applications, a levels less than 0.8 and 0.9 will produce infeasible regions in solving fuzzy linear models for the first three and final phases, respectively. Furthermore, according to the MEC concept in Fig. 1, the performance level of the ‘‘value” of each QFD activity is based on the fulfillment degree of the ‘‘hows” to satisfy the requirement of the ‘‘whats”. The satisfaction of the ‘‘whats” in a phase should be greater than or equal to that of the corresponding ‘‘hows” in the previous phase to finally realize that the most important value (customers’ satisfaction) can be achieved in the four-phase QFD process. The relevant verifications have been discussed and exemplified in the above sections. By setting a levels to produce feasible regions, Fig. 7 demonstrates the membership function of the objective function value in each phase representing the satisfaction degree of the ‘‘whats” achieved by the ‘‘hows” in the QFD process. In the figure, the trend of the satisfaction degree of each phase in the QFD process shows that the illustrated NPD case conforms to the MEC concept in the proposed models. 6. Concluding remarks Determining the fulfillment levels of the ‘‘hows”, including DRs, PCs, PPs, and PRs, is an important task in the four phases of the QFD process when it is applied to new product development. However, most related studies only focus on the priority or achievement levels of DRs in phase 1. Unlike previous research, the four-phase decision process is considered in this paper. Based on the previous mathematical programming model for phase 1 under a fuzzy environment, this paper considers the close link between the four phases using the MEC concept to build up a set of fuzzy linear programming models for QFD applications to determine the fulfillment levels of the ‘‘hows” in the four-phase process. To deal with the potential risk problems during NPD, risk analysis of DRs and PCs, namely fuzzy FMEA, is taken into account as the constraint factor in the phase 2 and 3 models. As for the risk analysis of PPs, the risk characteristics from PPs’ fulfillment levels are applied in the phase 4 model. In addition, considering MEC, the satisfaction of the ‘‘whats” determined in one phase should be greater than or equal to the contribution of the corresponding the ‘‘hows” in the previous phase to realize the satisfaction of the former phase. Furthermore, the satisfaction degree of each phase of the QFD process should conform to the MEC concept to ensure that the most important value, customers’ satisfaction in phase 1, is satisfied. The applicability of the proposed models in practice was demonstrated with a numerical example. The resulting ranges of satisfaction degree of each phase as well as those of fulfillment levels of the ‘‘hows” of each phase can provide the QFD team with useful information for new product development. However, the usefulness of the outcomes from the proposed models should depend on the validities of the relationships in HOQ, parameters, and variables (‘‘hows”) in each phase. To this end, the QFD team members should make more efforts in the NPD problem formulation. In addition, besides the MEC concept, other considerations, such as Kano’s concept, could be modeled in QFD problems in future research.

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