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European Journal of Operational Research 172 (2006) 230–248 www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

An evaluation approach to engineering design in QFD processes using fuzzy goal programming models Liang-Hsuan Chen a

a,*

, Ming-Chu Weng

b

Department of Industrial and Information Management, National Cheng Kung University, Tainan, Taiwan 701, ROC b Department of Industrial Management, Kun Shan University of Technology, Tainan County 71003, Taiwan, ROC Received 9 August 2002; accepted 5 October 2004 Available online 8 December 2004

Abstract Quality function deployment (QFD) is a product development process used to achieve higher customer satisfaction: the engineering characteristics affecting the product performance are designed to match the customer requirements. From the viewpoint of QFDs designers, product design processes are performed in uncertain environments, and usually more than one goal must be taken into account. Therefore, when dealing with the fuzzy nature in QFD processes, fuzzy approaches are applied to formulate the relationships between customer requirements (CRs) and engineering design requirements (DRs), and among DRs. In addition to customer satisfaction, the cost and technical difficulty of DRs are also considered as the other two goals, and are evaluated in linguistic terms. Fuzzy goal programming models are proposed to determine the fulfillment levels of the DRs. Differing from existing fuzzy goal programming models, the coefficients in the proposed model are also fuzzy in order to expose the fuzziness of the linguistic information. Our model also considers business competition by specifying the minimum fulfillment levels of DRs and the preemptive priorities between goals. The proposed approach can attain the maximal sum of satisfaction degrees of all goals under each confidence degree. A numerical example is used to illustrate the applicability of the approach. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Quality function deployment (QFD); Fuzzy numbers; Fuzzy goal programming

1. Introduction Quality function deployment (QFD) is a systematic method for translating the voice of customers into a final product through various product planning, engineering and manufacturing stages in order to achieve *

Corresponding author. Tel.: +886 6 2757575/53140; fax: +886 6 2362162. E-mail address: [email protected] (L.-H. Chen).

0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.10.004

L.-H. Chen, M.-C. Weng / European Journal of Operational Research 172 (2006) 230–248

231

higher customer satisfaction. The method includes both ‘‘customer requirement management’’ and ‘‘product development’’ systems, which begin by sampling the desires and preferences of consumers of a product through marketing surveys or interviews, and organizes them as a set of customer requirements (CRs). A group of engineering design requirements (DRs) affecting CRs are then identified, analyzed and improved in order to maximize customer satisfaction. By analyzing the relationships among DRs and between CRs and DRs, while considering cost and technical constraints as well as organizational strategies, QFD team members are responsible for determining the fulfillment levels of DRs. In the conventional QFD approach, decisions are achieved imprecisely in an uncertain environment because customer requirements tend to be subjective and qualitative. In addition, data availability for product design is often limited, inaccurate, or vague—particularly when developing an entirely new product. Therefore, engineers usually do not have the full knowledge necessary to map CRs onto the relevant DRs. Some authors, such as Park and Kim (1998) and Trappey et al. (1996), have presented some modified methods for assigning the relationship ratings between CRs and DRs, instead of a conventional relationship rating scale, such as the three point levels of 1, 3, and 9. However, these methods still use crisp measurement data with the result that ambiguous relationships cannot be captured. Some researchers have applied fuzzy theory in order to quantitatively formulate the problem for optimizing the improvements of DRs. Fung et al. (1998) proposed a fuzzy inference system of customer requirements which allowed the product attributes to be mapped out. Moskowitz and Kim (1997) presented a decision support system for optimizing product designs. The development of these systems usually requires professional knowledge and experience to establish rules and facts in ensuring that the system works well. Kim et al. (2000) used a fuzzy theoretical modeling approach to QFD by developing fuzzy multi-objective models, under the assumption that the function relationships among DRs and between CRs and DRs could be recognized based on the benchmarking data set of customer competitive analysis. Justifying this assumption in a general situation is difficult, particularly when developing an entirely new product. Some researchers, such as Shen et al. (2001), Vanegas and Labib (2001), Wang (1999), and Zhou (1998), developed some fuzzy approaches, for example, fuzzy sets, fuzzy arithmetic, and/or defuzzification techniques, to address complex and often imprecise problems in customer requirement management. However, in these models the interrelationships among the engineering design requirements (DRs) were not properly considered. In this study, we consider both the inherent fuzziness in the relationships among DRs and those between CRs and DRs. An aggregation of the two kinds of fuzzy relationships based on WassermanÕs (1993) study is performed to obtain the fuzzy normalized relationship matrix, containing a fuzzy number in each cell. Using the matrix and fuzzy weights of CRs, the fuzzy importance ratings of DRs are determined, after which the customer satisfaction function is formulated. In addition, some authors emphasized the need of conducting cost consideration and/or technical difficulty in the models in accordance with the QFD planning effort (Fung et al., 2002; King, 1987; Park and Kim, 1998; Trappey et al., 1996; Wasserman, 1993; Wang, 1999; Zhou, 1998). Therefore, this paper incorporates the costs and technical difficulties of DRs into the models so as to formulate three objectives for maximizing customer satisfaction, minimizing cost, and minimizing technical difficulties. Fuzzy goal programming models are formulated to achieve the objectives in terms of the fulfillment levels of DRs. Moreover, due to some organizational strategies and constraints in QFD processes, the design team may have a preference order, i.e., a preemptive priority structure, to achieve the goals. For this reason, we adopt a preemptive priority structure into the formulations based on the study of Chen and Tsai (2001). Different from existing models, the goals and coefficients in the proposed models are fuzzy, with the objective of achieving maximal total satisfaction of all goals. In the following section, a fuzzy approach is introduced to determine the fuzzy normalized relationship matrix of QFD and fuzzy technical importance ratings for DRs. Section 3 formulates the QFD planning problem as fuzzy goal programming problems with conflicting objectives and a preemptive priority. This paper applies the concept of a-cut and the extension principle to transform the fuzzy model into a series of conventional crisp linear programming models to find the fulfillment levels of DRs so as to produce

232


the maximal total satisfaction degree of all goals. Section 4 demonstrates our approach using an example, and discusses the findings. Finally, we give our conclusion in Section 5. 2. A fuzzy QFD approach The QFD employs the matrix, called House Of Quality (HOQ), to establish the relationships between CRs and DRs, as shown in Fig. 1. Two dimensions, customer wants and engineering design requirements, are included in the matrix. We place a triangular-shaped matrix over the engineering design requirements to indicate the correlation between engineering design requirements. Wasserman (1993) proposed a normalized transformation on the relationship values contained in the relationship matrix to account for the dependency effects among DRs, as described by the following equation, Pn R r Pn ik kj , R0ij ¼ Pn k¼1 ð1Þ j¼1 k¼1 Rik rkj where R0ij Rik rkj

normalized relationship between customer requirement i and engineering design requirement j, i = 1, 2, . . . , m, j = 1, 2, . . . , n, quantified relationship between customer requirement i and engineering design requirement k, i = 1, 2, . . . , m, k = 1, 2, . . . , n, quantified relationship between design requirements, k,j = 1, 2, . . . , n.

The conventional method to quantify the relationships is accomplished using a 1-3-9 or 1-5-9 scale to denote weak, medium, and strong relationships between CRs and DRs (Fung et al., 2002). However, in practice the relationships are usually vague and imprecise, and can be described in linguistic terms. In this study, the relationships are represented as linguistic terms, and fuzzy set theorems are employed to repree0 , R e ik , and ~c , which corresent the vagueness of the relationship. Three fuzzy numbers are denoted as R kj ij spond to R0ij , Rik and rkj, respectively, and can be defined as follows: Pn e R ik ~ckj 0 e , ð2Þ R ij ¼ Pn k¼1 Pn e ckj j¼1 k¼1 R ik ~

r jn

Degree of Importance

Customer Wants

CR1

k1

CRi

ki

Engineering Design Requirements DR1 DR j

R ij

CRm k m Fig. 1. QFD relationship matrix.

DR n


e ik ¼ fðRik ,l R

ðRik ÞÞ j Rik 2 Rik g 8i, k, e R ik ~ckj ¼ fðrkj ,l~ckj ðrkj ÞÞ j rkj 2 ckj g 8k, j,

233

ð3Þ

where l ðRik Þ and l~ckj ðrkj Þ denote the associated membership functions. The above formulation is not easy e R ik to solve, since it contains the multiplication and addition of two fuzzy numbers in the numerator and e ik , and ~c as the several denominator. For dealing with this, we first use the a-cut approach to represent R kj crisp interval values under different a levels, which can be expressed in the following forms: L U ðRik Þa ¼ minfRik 2 Rik j l ðRik Þ P ag, maxfRik 2 Rik j l ðRik Þ P ag ¼ ½ðRik Þa ,ðRik Þa , e e R ik R ik Rik Rik L U ðckj Þa ¼ minfrkj 2 ckj j l~ckj ðrkj Þ P ag, maxfrkj 2 kkj j l~ckj ðrkj Þ P ag ¼ ½ðckj Þa ,ðckj Þa : rkj

rkj

L

U

L

U

The above crisp interval values, ½ðRik Þa ,ðRik Þa and ½ðckj Þa ,ðckj Þa , can be considered as the corresponding e ik and ~c , respectively, under a confidence degree. Based on ZadehÕs extension principle (Zadeh, ranges of R kj e 0 , can be defined as 1978), the membership function of fuzzy normalized relationship, R ij ( ) P n 0 0 k¼1 Rik rkj l 0 ðRij Þ ¼ sup min l ðRik Þ,l~ckj ðrkj Þ, 8k,jRij ¼ Pn Pn : ð4Þ e e R ij R ik R;r j¼1 k¼1 Rik rkj e0 0 , it suffices to find the lower and upper bonds of the a-cuts of R , ij e R ij which can be solved as (Kao and Liu, 2000) Pn R r L Pn ik kj ðR0ij Þa ¼ min R0ij ¼ Pn k¼1 j¼1 k¼1 Rik rkj

To find the membership function of l

s:t:

ð5aÞ L

U

ðRik Þa 6 Rik 6 ðRik Þa L

U

ðckj Þa 6 rkj 6 ðckj Þa

8k, 8k, j,

Pn 0 k¼1 Rik rkj P ¼ max R ¼ ðR0ij ÞU n Pn ij a j¼1 k¼1 Rik rkj s:t:

ð5bÞ L

U

ðRik Þa 6 Rik 6 ðRik Þa L

U

ðckj Þa 6 rkj 6 ðckj Þa

8k, 8k, j:

e 0 , ðR e 0 ÞL and ð R e 0 ÞU , can be reformulated as Mathematically, the lower and upper bounds of a-cuts of R ij ij a ij a Pn L L L k¼1 ðRik Þa ðckj Þa , ðR0ij Þa ¼ Pn P n U U j¼1 k¼1 ðRik Þa ðckj Þa ð6Þ Pn U U k¼1 ðRik Þa ðckj Þa 0 U ðRij Þa ¼ Pn Pn : L L j¼1 k¼1 ðRik Þa ðckj Þa Solving Eq. (6) gives us a set of solutions with the possible extreme ranges at each a-cut. For improving the L outcomes, Chen and Weng (2003) have proposed new formulations to find more accurate ranges, mðR0ij Þa 0 U and mðRij Þa , which are formulated as follows:

234


mðR0ij ÞLa

Pn

L L k¼1 ðRik Þa ðckj Þa Pn U U L L k¼1 ðRik Þa ðckl Þa þ k¼1 ðRik Þa ðckj Þa

¼ Pn Pn l¼1 l6¼j

mðR0ij ÞU a

¼

, ð7Þ

Pn

U U k¼1 ðRik Þa ðckj Þa Pn Pn Pn L L U U l¼1 k¼1 ðRik Þa ðckl Þa þ k¼1 ðRik Þa ðckj Þa l6¼j

:

Appendix A lists the derivative processes of the new formulations. The ranges produced by Eq. (7) are obviously smaller than those by Eq. (6), such that more accurate representations can be obtained. Comparisons between the above two equations are made in the illustrated example in Section 4. The support of the fuzzy number is the subset of the universe of discourse [0, 1]. The new a-cuts of fuzzy normalized relationship are applied to formulate the fuzzy technical importance ratings of DRs. ej , is determined by the fuzzy weighted The fuzzy technical importance rating of design requirement j, W average of each fuzzy weight of customer requirement and the jth fuzzy normalized relationship, shown as e j is used to measure the overall impact of the jth design requirement on customer Eq. (8). The rating of W e j represents the overall customer satisfaction that can be satisfaction. In other words, the fuzzy set of W achieved by the jth DR. ej¼ W

Pm

e0

i¼1 mð R ij Þ Pm e i¼1 K i

ei K

ð8Þ

,

where ei K ej W

fuzzy weight of customer requirement, i, i = 1, 2, . . . , m, fuzzy technical importance rating for engineering design requirement, j, j = 1, 2, . . . , n.

The above formulation is also difficult to solve, since several fuzzy numbers are included. Similarly, the calculations can be performed via a-cuts of fuzzy numbers. At a specific possibility level a, the lower and upper bounds of the a-cuts of l Wej can be obtained using Eq. (9). Vanegas and Labib (2001) have also proposed a similar formulation. Pm 0 L i¼1 mðRij Þa k i L Pm ðW j Þa ¼ min i¼1 k i ð9aÞ s:t: L

U

ðK i Þa 6 k i 6 ðK i Þa , ðW

U j Þa

Pm ¼ max

i ¼ 1,2, . . . ,m,

0 U i¼1 mðRij Þa Pm i¼1 k i

ki

s:t: L

U

ðK i Þa 6 k i 6 ðK i Þa ,

ð9bÞ

i ¼ 1,2, . . . ,m:

3. Formulations In addition to customer satisfaction emphasized by the conventional QFD, some authors also highlighted the need to conduct the cost and/or technical difficulty considerations in the QFD planning effort


235

(Fung et al., 2002; King, 1987; Park and Kim, 1998; Trappey et al., 1996; Wasserman, 1993; Wang, 1999; Zhou, 1998). Similarly, the cost and technical difficulty are also represented in fuzzy terms in order to coincide with the fuzzy nature in the design stage. In this paper, a fuzzy goal programming model is formulated to assist the design team in selecting a mix of DRs to produce the maximal sum of satisfaction degrees of all goals. Three goals are considered for maximizing customer satisfaction, minimizing cost, and minimizing technical difficulty. First, some notations are specified as follows. Let xj be the fulfillment level of engineering design requirement, j, j = 1, 2, . . . , n. If xj = 100%, it denotes complete fulfillment of the objective targets e j is the descriptive of the overall impact of the fulfillment of the jth DR on customer satfor the jth DR; W e j represents the fuzzy cost required to the jth DR; and Te j denotes the fuzzy technical difficulty to isfaction; C the jth DR. Furthermore, considering business competition, a company usually desires some fulfillment levels (xj) of engineering design requirement better than its competitors (lj), i.e., xj P lj. The model is then formulated as follows: n n n X X X e j xj , min e j xj , min W Te j xj C max j¼1

j¼1

j¼1

s:t: xj P lj , j ¼ 1,2, . . . ,n,

ð10Þ

0 6 xj , lj 6 1: 3.1. Aspiration levels of goal According to the above formulation, determining the goal values precisely is difficult for the design team, since the customersÕ satisfaction, cost, and technical difficulty are not easy to measure exactly. These goals usually conflict with each other. For dealing with this, the design team first determines the aspiration levels for each goal, and then finds a set of solutions to achieve the maximum satisfaction degree of all goals in total. Let Gmin and Gmax represent the lower and upper bounds of the aspiration of s s Gs as the goal level of customer satisfaction. The design team would be completely dissatisfied with a max design (x) at which Gs ðxÞ 6 Gmin , where s , while the design would be completely satisfied if Gs ðxÞ P Gs x denotes the variable vector. While, if Gp represents the cost or technical difficulty with the smaller-thebetter characteristic, the design team would be completely satisfied with a design (x) at which max Gp ðxÞ 6 Gmin p , but it would be completely dissatisfied if Gp ðxÞ P Gp . Here Gs(x)/Gp(x) is the achievement degree of the sth/pth goal at x. The degree of satisfaction can be formulated linearly as (Zimmermann, 1978, 1983) 8 0 if Gs ðxÞ 6 Gmin > s , > < min ðxÞGs ls ðxÞ ¼ GGsmax if Gmin 6 Gs ðxÞ 6 Gmax , s s Gmin > s s > : 1 if Gs ðxÞ P Gmax s or

lp ðxÞ ¼

8 1 > > >
> > :

min Gmax p Gp

0

if Gp ðxÞ 6 Gmin p , if Gmin 6 Gp ðxÞ 6 Gmax p p , if Gp ðxÞ P Gmax p :

Based on the above formulations, the lower and upper bounds of the aspiration level of each goal, i.e., Gmin and Gmax, should be predetermined. However, determining the two bounds is not easy, because Eq. (10)

236


contains more than one goal and each goal has fuzzy coefficients. For doing this, a three-step solution procedure is developed as follows: Step 1: Set each fuzzy coefficient as the upper (lower) bound at the a-level = 0 for the goal having the maximum (minimum) target, such as customerÕs satisfaction (cost). This obtains the largest (smallest) crisp value of each coefficient. Step 2: Solve the problem for each single goal, i.e., remove the other goals, under the system constraints. The optimal solution set and the goal value can be obtained in this step, which are supported by all the resources. The determined goal values are considered as the upper (lower) bound of the aspiration level. Step 3: Place the solution set of one goal into other goals to determine their goal values. Find the lower (upper) bounds of each goal with the maximum (minimum) target using the smallest (largest) goal values determined in this step. Once the two bounds of aspiration level are obtained, this study uses an additive model to sum up the goals for finding the maximal overall satisfaction degree (Tiwari et al., 1987). 3.2. Preemptive priority structure for goals A design team usually has a preemptive priority in achieving goals. For example, increasing customer satisfaction may be the main purpose in the QFD process. However, cost expenditure and technical difficulty are also taken into account in the design stage. Let G1, G2 and G3 be the goals of customer satisfaction, cost expenditure, and technical difficulty, respectively. Suppose that G1 and G2 are considered more important than G3 such that two priority levels are recommended in the QFD process. For simplifying the computational efforts, a recently proposed model has been adopted in this study (Chen and Tsai, 2001). To illustrate, the three fuzzy goals are ranked as Priority level 1: G1 and G2. Priority level 2: G3. Denoted as membership functions, the preemptive priority structure is represented as l1 ðxÞ P l3 ðxÞ, l2 ðxÞ P l3 ðxÞ: 3.3. Fuzzy coefficients in FGP Based on the three fuzzy goals and their preemptive priority structure, the overall model can be formulated as follows: e ¼ max Z

3 X

~h ðxÞ l

h¼1

s:t: ~1 ðxÞ ¼ l

Pn

min e j¼1 W j xj G1 Gmax Gmin 1 1

,


~2 ðxÞ ¼ l ~3 ðxÞ ¼ l

Gmax 2

Pn

e

j¼1 C j xj max G2 Gmin 2

Gmax 3 Gmax 3

Pn

e

j¼1 T j xj Gmin 3

237

,

,

~3 ðxÞ, ~1 ðxÞ l l ~2 ðxÞ l ~3 ðxÞ, l

ð11Þ

~i ðxÞ 6 1, l ~i ðxÞ P 0, i ¼ 1,2,3, l xj P lj , j ¼ 1, . . . ,n, xj ,

lj 6 1,

xj ,

lj P 0,

where ‘‘’’ means that a fuzzy number ‘‘dominates’’ the other fuzzy number, since partial ordering usually exists between fuzzy numbers. Note that the coefficients of the above formulation are fuzzy, such that the solutions are difficult to obtain. For solving this problem, we transform the model with fuzzy coefficients to a family of conventional crisp mathematical programming models by applying the a-cut approach and ZadehÕs extension principle (Zadeh, 1978). The membership function of the objective value can be defined as ( leZ ðzÞ ¼ sup min l Wej ðwj Þ,le ðc Þ,leT j ðtj Þ, 8j j z ¼ Cj j w;c;t

3 X

) lh ðxÞ ,

ð12Þ

h¼1

where w, c, and t are the element values of fuzzy coefficients, and z is the objective value. Applying Eq. (12), the membership function of leZ can be determined based on membership degrees of all fuzzy coefficients. Similar to Eq. (4), we separate Model (11) into two crisp sub-problems to find the lower and upper bounds of leZ by specifying the a-cuts of all fuzzy coefficients as follows: ðZÞLa ¼ min Z s:t: L

U

ðW j Þa 6 wj 6 ðW j Þa ,

8j,

ðC j ÞLa 6 cj 6 ðC j ÞU a ,

8j,

ðT j ÞLa

8j;

6 tj 6

ðT j ÞU a ,

ð13aÞ

U

ðZÞa ¼ max Z s:t: L

U

ðW j Þa 6 wj 6 ðW j Þa , L ðC j Þa L

6 cj 6

U ðC j Þa , U

ðT j Þa 6 tj 6 ðT j Þa ,

8j, 8j, 8j:

ð13bÞ

238


And therefore, the full form is formulated as follows: ðZÞLa ¼

min

U ðW j ÞL a 6wj 6ðW j Þa ;8j U ðC j ÞL 6c 6ðC Þ j j a ;8j a U ðT j ÞL a 6tj 6ðT j Þa ;8j

3 X

max

lh ðxÞ

h¼1

s:t: Pn l1 ðxÞ ¼ l2 ðxÞ ¼ l3 ðxÞ ¼

min j¼1 wj xj G1 , Gmax Gmin 1 1 P Gmax nj¼1 cj xj 2 , Gmax Gmin 2 2 P Gmax nj¼1 tj xj 3 , Gmax Gmin 3 3

ð14aÞ

l1 ðxÞ P l3 ðxÞ, l2 ðxÞ P l3 ðxÞ, li ðxÞ 6 1, li ðxÞ P 0,

i ¼ 1,2,3,

j ¼ 1, . . . ,n,

x j P lj ,

xj , lj 6 1, xj , lj P 0; ðZÞU a ¼

max

U ðW j ÞL a 6wj 6ðW j Þa ;8j U ðC j ÞL 6c 6ðC Þ j j a ;8j a U ðT j ÞL a 6tj 6ðT j Þa ;8j

s:t: l1 ðxÞ ¼ l2 ðxÞ ¼ l3 ðxÞ ¼

max

lh ðxÞ

h¼1

Pn

min j¼1 wj xj G1 , Gmax Gmin 1 1 Pn Gmax j¼1 cj xj 2 , Gmax Gmin 2 2 Pn Gmax j¼1 tj xj 3

Gmax Gmin 3 3

l1 ðxÞ P l3 ðxÞ, l2 ðxÞ P l3 ðxÞ, li ðxÞ 6 1, li ðxÞ P 0, x j P lj ,

3 X

i ¼ 1,2,3,

j ¼ 1, . . . ,n,

xj , lj 6 1, xj , lj P 0:

,

ð14bÞ


239

When the importance ratings, costs, and technical difficulties vary, the minimum of Z occurs when the coefficients of importance ratings are set to their lower bounds and the coefficients of costs and technical difficulties are set to their upper bounds; otherwise, the maximum of Z occurs. Therefore, the mathematical formulations in Eq. (14) can be simplified to conventional linear programming models, shown as Eq. (15). L U The membership function leZ can then be constructed from ½ðZÞa ,ðZÞa at different a levels. L

ðZÞa ¼ max

3 X

lh ðxÞ

h¼1

s:t:

Pn

l1 ðxÞ ¼ l2 ðxÞ ¼ l3 ðxÞ ¼

L min j¼1 ðW j Þa xj G1 Gmax Gmin 1 1 Pn U Gmax ðC j Þa x j j¼1 2

,

, Gmax Gmin 2 2 Pn U Gmax j¼1 ðT j Þa xj 3 Gmax Gmin 3 3

,

ð15aÞ

l1 ðxÞ P l3 ðxÞ, l2 ðxÞ P l3 ðxÞ, li ðxÞ 6 1, li ðxÞ P 0, i ¼ 1,2,3, xj P lj , j ¼ 1, . . . ,n, xj , xj ,

lj 6 1, lj P 0;

U

ðZÞa ¼ max

3 X

lh ðxÞ

h¼1

s:t:

Pn

l1 ðxÞ ¼ l2 ðxÞ ¼ l3 ðxÞ ¼

U min j¼1 ðW j Þa xj G1 , Gmax Gmin 1 1 Pn L Gmax j¼1 ðC j Þa xj 2 , Gmax Gmin 2 2 P Gmax nj¼1 ðT j ÞLa xj 3 , Gmax Gmin 3 3

l1 ðxÞ P l3 ðxÞ, l2 ðxÞ P l3 ðxÞ, li ðxÞ 6 1, li ðxÞ P 0, i ¼ 1,2,3, xj P lj , j ¼ 1, . . . ,n, xj , xj ,

lj 6 1, lj P 0:

ð15bÞ

240


4. Illustrations 4.1. A writing instrument example In order to demonstrate the feasibility of the proposed models, a simple example of a writing instrument from a related work is adopted in this section (Wasserman, 1993). The design includes four customer requirements (CRs) and five design requirements (DRs). Fig. 2 illustrates the HOQ. In the figure, four CRs are ‘‘easy to hold’’ (CR1), ‘‘does not smear’’ (CR2), ‘‘point lasts’’ (CR3), and ‘‘does not roll’’ (CR4), while the important engineering design requirements contain ‘‘length of pencil’’ (DR1), ‘‘time between sharpening’’ (DR2), ‘‘least dust generated’’ (DR3), ‘‘hexagonal’’ (DR4), and ‘‘minimal erasure residue’’ (DR5). In addition, cost and technical difficulty of DRs are also incorporated in the planning processes. e ik ) between CRs and DRs, those (~c ) among DRs, and the relative Firstly, the fuzzy relationships ( R kj e i ) of the four CRs must be determined to derive the fuzzy importance ratings of the five importance ( K DRs. Owing to the imprecise design information available in the early design stage, it is difficult to assess the relationship of the specified design variables in design planning accurately. Therefore, linguistic terms are used to describe the strengths of relationship among DRs and between CRs and DRs, the relative importance of the four CRs, and the estimated cost and technical difficulty of each DR. In this paper, four groups of linguistic terms are defined in Table 1 for different descriptions. Each group contains seven linguistic terms. For example, the descriptions of relationship strength are ‘‘weakest’’, ‘‘weak’’, ‘‘fairly weak’’,

Strongest Strong Fairly strong

Engineering Design Relative Requirements ImportDR1 DR2 DR3 DR4 DR5 ance CR1 Customer CR2 Wants CR3 CR4 Estimated Cost Technical Difficulty

Medium Fairly weak Weak Weakest

Fig. 2. QFD matrix for a writing instrument.

Table 1 The linguistic scales used by design team Linguistic scale for relationship strengths

Linguistic scale for relative importance

Linguistic scale for estimated cost

Linguistic scale for technical difficulty

Weakest Weak Fairly weak Medium Fairly strong Strong Strongest

Very unimportant Unimportant Fairly unimportant Medium Fairly important Important Very important

Very high High Fairly high Medium Fairly low Low Very low

Very difficult Difficult Fairly difficult Medium Fairly easy Easy Very easy


241

‘‘medium’’, ‘‘fairly strong’’, ‘‘strong’’, and ‘‘strongest’’. For the subsequent fuzzy operations, these linguistic terms should be translated to fuzzy numbers. Seven trapezoidal fuzzy numbers are used to represent each group of linguistic terms according to the conversion scale (Chen et al., 1992). Their definitions are (0, 0, 0.1, 0.2), (0.1, 0.2, 0.2, 0.3), (0.2, 0.3, 0.4, 0.5), (0.4, 0.5, 0.5, 0.6), (0.5, 0.6, 0.7, 0.8), (0.7, 0.8, 0.8, 0.9), and (0.8, 0.9, 1.0, 1.0), respectively, as shown in Fig. 3. Using the linguistic terms in Table 1 to represent the various relationships in the QFD matrix, as shown e 0 Þ, can be calculated using Eq. (7). For obtaining mð R e 0 Þ, in Fig. 2, the fuzzy normalized relationship, mð R ij ij e ik and those of ~c should be determined beforehand, based the upper and lower bounds of the a-cuts of R kj on their membership functions. The membership function of a trapezoidal fuzzy number is defined by linear e ik is assessed as ‘‘fairly strong’’ ( e functions. As an illustration, suppose that R S ), and the membership function of the fuzzy number e S ¼ ð0:5,0:6,0:7,0:8Þ can be expressed as 8 > < ðRik 0:5Þ=ð0:6 0:5Þ, 0:5 6 Rik 6 0:6, leS ðRik Þ ¼ 1, 0:6 6 Rik 6 0:7, > : ð0:8 Rik Þ=ð0:8 0:7Þ, 0:7 6 Rik 6 0:8: Then, the a-cut of the above membership function is h i L U ðRik Þa ,ðRik Þa ¼ ½0:5 þ 0:1a,0:8 0:1a: Once the a-cuts of all relationships are determined, they are placed into the equations for obtaining the upper and lower bounds of a-cuts of fuzzy normalization relationships. As mentioned before, the ranges produced by applying Eq. (7)0 are smaller than those 0when Eq. (6) is used. For comparison purposes, four e Þ, mð R e 0 Þ, R e 0 , and R e , are shown in Fig. 4. This justifies the derived formembership functions, mð R 14 44 14 44 mulations in Appendix A. e j , can be obtained The fuzzy technical importance rating of the jth engineering design requirement, W e 1 to using Eq. (9) to determine the priority of each design requirement. The membership functions, W e e e e W 5 , are shown in Fig. 5. Both W 3 and W 5 rank the highest, with a range of 16.2–40.8%, while W 1 ranks the lowest, with a range of 3.1–16.2%. These ratings are then used to formulate the customer satisfaction function. The objective of a QFD planning is not only to maximize customer satisfaction (G1), but also to minimize cost (G2) and technical difficulty (G3), subject to other organizational constraints, such as the fulfillment level of engineering design requirements. These goals have the preemptive priority structure that is the same as that described in Section 3.2. Cost and technical difficulty are evaluated and illustrated in Fig. 2,

Fairly weak

Weakest

membership degree

Weak

1

Fairly strong Strongest Strong Medium

0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

strength scale Fig. 3. The membership functions of linguistic terms for relationships.

242


Fig. 4. The membership functions of normalized relationship degrees.

α

~ W2

~ W1

1

~ ~ W3= W5 ~ W4

0.8 0.6 0.4 0.2 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

technical importance degree Fig. 5. The membership functions of technical importance ratings.

using the seven linguistic scales of Table 1. For formulating fuzzy goal programming models, the range of max aspiration levels of Gh, i.e., ½Gmin h ,Gh , should be specified beforehand. Suppose that the minimum fulfillment levels of the five engineering design requirements, x1, . . . , x5, are required as 0.2, 0.5, 0.3, 0.7, and 0.5, respectively. Following the solution procedure in Section 3.1, the model can be formulated as follows: X U max ðW i Þa¼0 xi ¼ 0:62x1 þ 0:311x2 þ 0:408x3 þ 0:325x4 þ 0:408x5 X min ðC i ÞLa¼0 xi ¼ 0:7x1 þ 0:5x2 þ 0:4x3 þ 0:1x4 þ 0:2x5 X L min ðT i Þa¼0 xi ¼ 0:1x1 þ 0:4x2 þ 0:7x3 þ 0:1x4 þ 0:4x5 s:t: x1 P 0:2, x2 P 0:5,

ð16Þ

x3 P 0:3, x4 P 0:7, x5 P 0:5, xi 6 1, i ¼ 1, . . . ,5: The ranges of aspiration levels of the three goals are determined as [0.245, 1.615], [0.68, 3.1], and [0.7, 2.17], respectively. And the full model, Eq. (15), can be constructed, subject to the preemptive priority structure and the required fulfillment levels of DRs.


243

Table 2 e 2 and X e 3 at 11 different possibility a values The ranges for the three fuzzy goals and the fulfillment levels of X a

½ðl1 ÞLa ; ðl1 ÞU a



½ðx2 ÞLa ; ðx2 ÞU a

½ðx3 ÞLa ; ðx3 ÞU a

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.18 0.20 0.23 0.25 0.27 0.29 0.31 0.34 0.36 0.39 0.41

0.37 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56

0.18 0.20 0.23 0.25 0.27 0.29 0.31 0.34 0.36 0.39 0.41

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.76 0.78 0.81 0.83 0.85 0.87 0.89 0.90 0.92 0.94 0.95

0.72 0.70 0.67 0.65 0.62 0.60 0.57 0.55 0.52 0.50 0.47

0.87 0.85 0.83 0.81 0.80 0.78 0.76 0.74 0.72 0.70 0.69

α

µ~1 = µ~3

0.72 0.70 0.67 0.65 0.62 0.60 0.57 0.55 0.52 0.50 0.47

0.79 0.79 0.78 0.78 0.77 0.75 0.74 0.73 0.71 0.67 0.67

1 1 1 1 1 1 1 1 1 1 1

µ~2

1 0.8 0.6 0.4 0.2 0 0.1

0.2

0.3

0.4 0.5 0.6 satisfaction degree

0.7

0.8

0.9

Fig. 6. The membership functions of goal values.

Solving the model (15) of the example using 11 different a-cuts, i.e., a = 0, 0.1, 0.2, . . . , 1, the satisfaction degree for each goal and their sum, as well as the fulfillment levels of all DRs, can be acquired at each a e 2 and X e 3 at 11 diflevel. Table 2 lists the ranges of three fuzzy goals and those of the fulfillment levels of X ferent possibility levels. Based on the ranges, Fig. 6 depicts the membership functions of satisfaction degree of the three goals. The satisfaction degree of G2 is greater than those of G1 and G3 for which G2 locates in the interval [0.37, 0.87] and G1 as well as G3 in [0.18, 0.72], although G1 has the same priority as G2. Obviously, achieving the cost objective (G2) is easier than that of customer satisfaction (G1) in this example. e 2 and X e 3 , respectively, while x1, x4, and x5 are crisp In Fig. 7, x2 and x3 are determined as being fuzzy, X with the fulfillment level of 20%, 100%, 100%, respectively. As described before, the decision variable xj = 100% denotes complete fulfillment of the jth DR. This means that the DRs, x4 and x5, should have the best quality level in order to achieve the total satisfaction degree. Particularly, x1 and x2 are smaller than the others in the example, due to the low technical importance ratings and high estimated costs, if referring to Figs. 2 and 5. 4.2. Discussion Echoing a common belief that imprecise input data generally produce imprecise output in a decisionmaking problem, our example illustrates that the fulfillment levels of some DRs are fuzzy due to the use of imprecise information, and their ranges at different possibility levels can be obtained by applying the

244


X2

X3

1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

fulfillment level Fig. 7. The membership functions of decision variables x2 and x3, while x1 = 0.2 and x4 = x5 = 1.

e 2 in Fig. 7, with the proposed approaches. Consider, for instance, the second DR in the example, i.e., X ranges at the possibility levels a = 1.0 and a = 0 being [0.5, 0.67] and [0.5, 0.79], respectively. In the fuzzy sense, it is definitely possible that the fulfillment level of x2 is in [0.5, 0.67]; this fulfillment level will never exceed 0.79 or fall below 0.5. With the fuzzy sense, the possibility level can be interpreted as the confidence degree (Bondia and Pico´, 2003; Chang and Lee, 1996; Mon et al., 1995; Wu, 2003). A designer can adopt a fulfillment level of one DR from the range produced under an acceptable confidence degree (say 0.7), such that only one a-cut is needed for the proposed approach. Alternatively, if a designer desires to utilize more information from QFD processes, the fuzzy fulfillment level can be defuzzified into a real number in [0, 1], which can then be considered to be the action (fulfillment level) to be taken by the designer. Several defuzzification methods have been developed in the fuzzy control area, such as the centroid method, the center of maxima method, and the mean of maxima method (Klir and Yuan, 1995). Among them, the centroid method is the commonly used, and hence has been e i is formulated as (Klir adopted in this paper. The centroid method for defuzzifying a fuzzy number X and Yuan, 1995). Pm ^xi ¼

ðiÞ ðiÞ ðx Þxk k¼1 le Xi k Pm ðiÞ ðx Þ k¼1 le Xi k

ð17Þ

,

e i is defined on a finite universal set, and l ðxk Þ is the membership degree (possibility level) of the where X Xi e ðiÞ e i . Consider again X e 2 in Fig. 7. Applying the interpolakth element (fulfillment level in this paper) xk in X ð2Þ tion method, based on the 11 a-cuts (possibility levels), renders 15 fulfillment levels xk , k = 1, . . . , 15, for e 2 ; these 15 fulfillment levels and the corresponding possibility levels give the defuzzified value ^x2 ¼ 0:62 in X Eq. (17). In this paper, the a-cut approach is employed in order to determine the fulfillment level of each DR in an imprecise environment. The proposed approach mainly consists of two sequent models, i.e., Models (9) and (15). The a value and the resulting range of each fuzzy technical importance rating in Model (9) are taken as the input of Model (15). Obviously, if a designer requires more information to decide the fulfillment level of DRs, more a-cuts are needed. As listed in Table 2, different a values lead to different ranges of satisfaction degrees of the goals and also those of the fulfillment levels of the DRs. The membership functions for fuzzy goals and fuzzy fulfillment levels are constructed by piecewise linear segments based on different a values and the resulting ranges in Table 2, as shown in Figs. 6 and 7. Therefore, the number of a-cuts is critical to the proposed approach. ðiÞ


245

From Fig. 6, the variability of membership function of each goal is not significant if different numbers of a values are adopted, since their piecewise linear segments connect smoothly. However, as an example, the e 2 in Fig. 7 will somewhat change if the number of a values is small. For illustramembership function of X e 2 by using three different numbers of a-cuts, i.e., 2 tions, we can construct the membership functions of X (a = 0, 1), 3 (a = 0, 0.5, 1), and 6 (a = 0, 0.2, 0.4, 0.6, 0.8, 1), in Fig. 8. Although these membership functions e 2 in Fig. 7, the actual differences are not significant, especially differ from the membership function of X those under 3 and 6 a-cuts. Similarly, substituting 15 elements from the membership functions in Fig. 8(a)–(c) into Eq. (17) gives, respectively, the defuzzified values 0.6126, 0.6176, and 0.6158, all of which are close to the defuzzified value 0.62 in Fig. 7.

α

~ X2

1

0 0

0.2

(a)

0.4

0 .6

0.8

1

fulfillment level

α

~ X2

1

0.5

0 0

0.2

(b)

0.4

0.6

0.8

1

fulfillment level

α

~ X2

1 0.8 0.6 0.4 0.2 0

(c)

0

0.2

0.4

0.6

0.8

1

fulfillment level

e 2 based on three different numbers of a-cuts. (a) The membership function based on two a-cuts Fig. 8. The membership functions of X (a = 0, 1). (b) The membership function based on three a-cuts (a = 0, 0.5, 1). (c) The membership function based on six a-cuts (a = 0, 0.2, 0.4, 0.6, 0.8, 1).

246


In summary, for application purposes, a designer can perform one a-cut model-solving process with an acceptable degree of confidence. If more information is required, more a-cuts are usually needed, and the interpolation method as well as the defuzzification method is used. With the fuzzy nature of design in the early stage of product planning, a small number of a-cuts, say four (a = 0, 0.33, 0.67, 1) or six (a = 0, 0.2, 0.4, 0.6, 0.8, 1), usually can suffice for the designersÕ needs, such as the illustrated example.

5. Conclusions Ambiguity exists in the QFD planning, since the assessments are imprecise in the relationships between CRs and DRs as well as among DRs, the relative importance between CRs, cost, and technical difficulty. Due to the impreciseness in a QFD process, fuzzy approaches are applied in this paper to determine the required fulfillment levels of DRs for achieving the maximum satisfaction degree of several goals in total in the product design stage. Three goals are considered: maximizing customer satisfaction, minimizing cost, and minimizing technical difficulty with respect to each DR. The coefficients in the three goal formulations are allowed to be fuzzy, and the satisfaction of each goal is also fuzzy. In addition, the minimum fulfillment degree of each DR can be delimited, and the preemptive priority structure for the goals can be required. In general, crisp values can be considered as special conditions of fuzzy numbers. Therefore, through the applications of fuzzy goal programming models on the QFD processes, the formulations in this study can allow a QFD planning team to make various kinds of assessments under an uncertain environment. The applicability of our formulations is demonstrated by a simple example from the existing study. Only a few a-cuts are required to construct the membership functions of fuzzy goals and those of fuzzy fulfillment levels of DRs in the example. The resulting ranges of satisfaction degree of each goal and the possible ranges of the fulfillment levels of DRs can provide a QFD team with more useful information. For applications, a designer can perform one a-cut model-solving process with an acceptable degree of confidence. Alternatively, performing more a-cuts, the fuzzy fulfillment levels of DRs can be defuzzified into real numbers, which can be done by designers.

Appendix A The new formulations for the upper and lower bounds of fuzzy normalized relationship can be derived as follows: Pn Pn 0 k¼1 Rik rkj k¼1 Rik r kj Pn Rij ¼ Pn Pn ¼ Pn Pn , ðA:1Þ R r R l¼1 ik rkl þ j¼1 k¼1 ik kj k¼1 k¼1 Rik r kj l6¼j

where L

U

0 6 ðRik Þa 6 Rik 6 ðRik Þa 6 1,

8k, i ¼ 1, . . . ,m,

8k,j: 0 6 ðckj ÞLa 6 rkj 6 ðckj ÞU a 6 1, Pn Pn Pn Let / ¼ k¼1 Rik rkj and u ¼ l¼1 k¼1 Rik rkl . l6¼j

Then Eq. (A.1) is expressed as f ð/Þ ¼

/ : uþ/


247

Since f 0 ð/Þ ¼

u ðu þ /Þ2

P 0,

f(/) is an increasing function and X X L L U U ðRik Þa ðckj Þa 6 / 6 ðRik Þa ðckj Þa : k

k

Therefore, min f ð/Þ ¼

Pn

L

Pn

U

L

k¼1 ðRik Þa ðckj Þa Pn L L u þ k¼1 ðRik Þa ðckj Þa

and max f ð/Þ ¼

U

k¼1 ðRik Þa ðckj Þa : Pn U u þ k¼1 ðRik ÞU a ðckj Þa

Furthermore, n X n n X n X X L L U U ðRik Þa ðckl Þa 6 u 6 ðRik Þa ðckl Þa , l¼1 l6¼j

k¼1

l¼1 l6¼j

k¼1

e 0 , mðR0 ÞL and mðR0 ÞU , can be formulated as such that the new lower and upper bounds of a-cuts of R ij a ij a ij Pn L L k¼1 ðRik Þa ðckj Þa , mðR0ij ÞLa ¼ min f ð/Þ ¼ Pn Pn Pn U U L L l¼1 k¼1 ðRik Þa ðckl Þa þ k¼1 ðRik Þa ðckj Þa l6¼j

U mðR0ij Þa

Pn

U U k¼1 ðRik Þa ðckj Þa P L L n U U k¼1 ðRik Þa ðckl Þa þ k¼1 ðRik Þa ðckj Þa

¼ max f ð/Þ ¼ Pn Pn l¼1 l6¼j

:

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