An evaluation approach to engineering design in QFD processes using fuzzy goal programming models

Transcription

1 European Journal of Operational Research 7 (006) 0 48 Production Manufacturing and Logistics An evaluation approach to engineering design in QFD processes using fuzzy goal programming models Liang-Hsuan Chen a * Ming-Chu Weng b a Department of Industrial and Information Management National Cheng Kung University Tainan Taiwan 70 ROC b Department of Industrial Management Kun Shan University of Technology Tainan County 700 Taiwan ROC Received 9 August 00; accepted 5 October 004 Available online 8 December 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 004 Elsevier B.V. All rights reserved. Keywords: Quality function deployment (QFD); Fuzzy numbers; Fuzzy goal programming. 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.: /540; fax: address: lhchen@mail.ncku.edu.tw (L.-H. Chen) /$ - see front matter Ó 004 Elsevier B.V. All rights reserved. doi:0.06/j.ejor

2 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 0 48 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 (998) and Trappey et al. (996) 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 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. (998) proposed a fuzzy inference system of customer requirements which allowed the product attributes to be mapped out. Moskowitz and Kim (997) 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. (000) 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. (00) Vanegas and Labib (00) Wang (999) and Zhou (998) 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 (99) 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. 00; King 987; Park and Kim 998; Trappey et al. 996; Wasserman 99; Wang 999; Zhou 998). 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 (00). 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 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

3 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 0 48 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.. 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.. 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 (99) 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 R 0 ij ¼ k¼ P R ikr kj n n j¼p k¼ R ðþ ikr kj where R 0 ij normalized relationship between customer requirement i and engineering design requirement j i =...m j =...n R ik quantified relationship between customer requirement i and engineering design requirement k i =...m k =...n quantified relationship between design requirements kj =...n. r kj The conventional method to quantify the relationships is accomplished using a --9 or -5-9 scale to denote weak medium and strong relationships between CRs and DRs (Fung et al. 00). 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 represent the vagueness of the relationship. Three fuzzy numbers are denoted as R e 0 ij R e ik and ~c kj which correspond to R 0 ij R ik and r kj respectively and can be defined as follows: er 0 e ij ¼ k¼r ik ~c kj n j¼p e ðþ k¼r ik ~c kj r jn Degree of Importance Engineering Design Requirements DR DRj DRn CR k Customer Wants CRi k i R ij CRm k m Fig.. QFD relationship matrix.

4 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 0 48 er ik ¼fðR ik l erik ðr ik ÞÞ j R ik R ik g 8i k ~c kj ¼fðr kj l ~ckj ðr kj ÞÞ j r kj c kj g 8k j ðþ where l ðr ik Þ and l ~ckj ðr kj Þ denote the associated membership functions. The above formulation is not easy erik to solve since it contains the multiplication and addition of two fuzzy numbers in the numerator and denominator. For dealing with this we first use the a-cut approach to represent R e ik and~c kj as the several crisp interval values under different a levels which can be expressed in the following forms: ðr ik Þ a ¼ minfr ik R ik j l ðr ik Þ P ag maxfr ik R ik j l ðr ik Þ P ag ¼½ðR ik Þ L R ik erik erik a ðr ikþ U a Š ðc kj Þ a ¼ minfr kj c kj j l ~ckj ðr kj Þ P ag maxfr kj k kj j l ~ckj ðr kj Þ P ag ¼½ðc kj Þ L r a ðc kj kjþ U a Š: r kj R ik The above crisp interval values ½ðR ik Þ L a ðr ikþ U a Š and ½ðc kjþ L a ðc kjþ U a Š can be considered as the corresponding ranges of R e ik and ~c kj respectively under a confidence degree. Based on ZadehÕs extension principle (Zadeh 978) the membership function of fuzzy normalized relationship R e 0 ij can be defined as ( P ) n l 0 ðr 0 er ij Þ¼sup min l ðr ik Þl ~ckj ðr kj Þ 8kj R 0 ij R;r erik ij ¼ k¼ P R ikr kj n k¼ R : ð4þ ikr kj To find the membership function of l 0 it suffices to find the lower and upper bonds of the a-cuts of R e 0 er ij ij which can be solved as (Kao and Liu 000) ðr 0 ij ÞL a ¼ min R0 ij ¼ k¼ R ikr kj j¼ k¼ R ikr kj j¼ ðr ik Þ L a 6 R ik 6 ðr ik Þ U a 8k ðc kj Þ L a 6 r kj 6 ðc kj Þ U a 8k j ð5aþ ðr 0 ij ÞU a ¼ max R0 ij ¼ k¼ P R ikr kj n j¼ k¼ R ikr kj ðr ik Þ L a 6 R ik 6 ðr ik Þ U a 8k ðc kj Þ L a 6 r kj 6 ðc kj Þ U a 8k j: Mathematically the lower and upper bounds of a-cuts of R e 0 ij ð R e 0 ij ÞL a ðr 0 ij ÞL a ¼ k¼ ðr ikþ L a ðc kjþ L a j¼ k¼ ðr ikþ U a ðc kjþ U a ðr 0 ij ÞU a ¼ k¼ ðr ikþ U a ðc kjþ U a j¼ k¼ ðr ikþ L a ðc : kjþ L a ð5bþ and ð R e 0 ij ÞU a can be reformulated as ð6þ Solving Eq. (6) gives us a set of solutions with the possible extreme ranges at each a-cut. For improving the outcomes Chen and Weng (00) have proposed new formulations to find more accurate ranges mðr 0 ij ÞL a and mðr 0 ij ÞU a which are formulated as follows:

5 4 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 0 48 mðr 0 ij ÞL a ¼ l¼ l6¼j k¼ ðr ikþ L a ðc kjþ L a k¼ ðr ikþ U a ðc klþ U a þ k¼ ðr ikþ L a ðc kjþ L a mðr 0 ij ÞU a ¼ k¼ ðr ikþ U a ðc kjþ U a l¼ k¼ ðr ikþ L a ðc klþ L a þ k¼ ðr ikþ U a ðc : kjþ U a 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 ]. The new a-cuts of fuzzy normalized relationship are applied to formulate the fuzzy technical importance ratings of DRs. The fuzzy technical importance rating of design requirement j ew j is determined by the fuzzy weighted average of each fuzzy weight of customer requirement and the jth fuzzy normalized relationship shown as Eq. (8). The rating of ew j is used to measure the overall impact of the jth design requirement on customer satisfaction. In other words the fuzzy set of ew j represents the overall customer satisfaction that can be achieved by the jth DR. ð7þ ew j ¼ P m i¼ mð e R 0 ij Þ ek i P m i¼ e K i ð8þ where ek i ew j fuzzy weight of customer requirement i i =...m fuzzy technical importance rating for engineering design requirement j j =...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 ewj can be obtained using Eq. (9). Vanegas and Labib (00) have also proposed a similar formulation. ðw j Þ L a ¼ min P m i¼ mðr0 ij ÞL a k i P m i¼ k i ðk i Þ L a 6 k i 6 ðk i Þ U a i ¼...m ð9aþ ðw j Þ U a ¼ max P m i¼ mðr0 ij ÞU a k i P m i¼ k i ðk i Þ L a 6 k i 6 ðk i Þ U a i ¼...m: ð9bþ. 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

6 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) (Fung et al. 00; King 987; Park and Kim 998; Trappey et al. 996; Wasserman 99; Wang 999; Zhou 998). 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 x j be the fulfillment level of engineering design requirement j j =...n.ifx j = 00% it denotes complete fulfillment of the objective targets for the jth DR; ew j is the descriptive of the overall impact of the fulfillment of the jth DR on customer satisfaction; ec j represents the fuzzy cost required to the jth DR; and et j denotes the fuzzy technical difficulty to the jth DR. Furthermore considering business competition a company usually desires some fulfillment levels (x j ) of engineering design requirement better than its competitors (l j ) i.e. x j P l j. The model is then formulated as follows: max Xn ew j x j min Xn ec j x j j¼ j¼ x j P l j j ¼...n 0 6 x j l j 6 : min Xn j¼ et j x j ð0þ.. 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 G min s and s represent the lower and upper bounds of the aspiration of G s as the goal level of customer satisfaction. The design team would be completely dissatisfied with a design (x) at which G s ðxþ 6 G min s while the design would be completely satisfied if G s ðxþ P s where x denotes the variable vector. While if G p represents the cost or technical difficulty with the smaller-thebetter characteristic the design team would be completely satisfied with a design (x) at which G p ðxþ 6 G min p but it would be completely dissatisfied if G p ðxþ P p. Here G s (x)/g p (x) is the achievement degree of the sth/pth goal at x. The degree of satisfaction can be formulated linearly as (Zimmermann ) 8 0 if G s ðxþ 6 G min s >< G l s ðxþ ¼ s ðxþ G min s if G min s G min s 6 G s ðxþ 6 s s >: if G s ðxþ P s or 8 >< l p ðxþ ¼ >: if G p ðxþ 6 G min p p G pðxþ p G min p if G min p 6 G p ðxþ 6 p 0 if G p ðxþ P p : Based on the above formulations the lower and upper bounds of the aspiration level of each goal i.e. G min and should be predetermined. However determining the two bounds is not easy because Eq. (0)

7 6 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 0 48 contains more than one goal and each goal has fuzzy coefficients. For doing this a three-step solution procedure is developed as follows: Step : 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 : 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 : 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. 987)... 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 G G and G be the goals of customer satisfaction cost expenditure and technical difficulty respectively. Suppose that G and G are considered more important than G 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 00). To illustrate the three fuzzy goals are ranked as Priority level : G and G. Priority level : G. Denoted as membership functions the preemptive priority structure is represented as l ðxþ P l ðxþ l ðxþ P l ðxþ:.. Fuzzy coefficients in FGP Based on the three fuzzy goals and their preemptive priority structure the overall model can be formulated as follows: ez ¼ max X ~l ðxþ ¼ h¼ ~l h ðxþ e j¼w j x j

8 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) ~l ðxþ ¼ Gmax e j¼c j x j ~l ðxþ ¼ Gmax e j¼t j x j ~l ðxþ ~l ðxþ ~l ðxþ ~l ðxþ ~l i ðxþ 6 ~l i ðxþ P 0 i ¼ x j P l j j ¼...n x j l j 6 x j l j P 0 ðþ 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 978). The membership function of the objective value can be defined as l ez ðzþ ¼sup w;c;t ( ) min l ewj ðw j Þl ec j ðc j Þl etj ðt j Þ 8j j z ¼ X l h ðxþ ðþ where w candt are the element values of fuzzy coefficients and z is the objective value. Applying Eq. () the membership function of l ez can be determined based on membership degrees of all fuzzy coefficients. Similar to Eq. (4) we separate Model () into two crisp sub-problems to find the lower and upper bounds of l ez by specifying the a-cuts of all fuzzy coefficients as follows: h¼ ðzþ L a ¼ min Z ðw j Þ L a 6 w j 6 ðw j Þ U a 8j ðc j Þ L a 6 c j 6 ðc j Þ U a 8j ðt j Þ L a 6 t j 6 ðt j Þ U a 8j; ðzþ U a ¼ max Z ðw j Þ L a 6 w j 6 ðw j Þ U a 8j ðc j Þ L a 6 c j 6 ðc j Þ U a 8j ðt j Þ L a 6 t j 6 ðt j Þ U a 8j: ðaþ ðbþ

9 8 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 0 48 And therefore the full form is formulated as follows: ðzþ L a ¼ min ðw jþ L a 6wj6ðW jþu a ;8j ðc jþ L a 6cj6ðCjÞU a ;8j ðt jþ L a 6tj6ðT jþu a ;8j max X j¼ l ðxþ ¼ w jx j l ðxþ ¼ Gmax j¼ c jx j l ðxþ ¼ Gmax j¼ t jx j l ðxþ P l ðxþ l ðxþ P l ðxþ l i ðxþ 6 l i ðxþ P 0 i ¼ x j P l j j ¼...n x j l j 6 x j l j P 0; h¼ l h ðxþ ð4aþ ðzþ U a ¼ max ðw jþ L a 6wj6ðW jþu a ;8j ðc jþ L a 6cj6ðCjÞU a ;8j l ðxþ ¼ ðt jþ L a 6tj6ðT jþu a ;8j max X j¼ w jx j l ðxþ ¼ Gmax h¼ j¼ c jx j l ðxþ ¼ Gmax j¼ t jx j l ðxþ P l ðxþ l ðxþ P l ðxþ l i ðxþ 6 l i ðxþ P 0 i ¼ x j P l j j ¼...n x j l j 6 x j l j P 0: l h ðxþ ð4bþ

10 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 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. (4) can be simplified to conventional linear programming models shown as Eq. (5). The membership function l ez can then be constructed from ½ðZÞ L a ðzþu a Š at different a levels. ðzþ L X a ¼ max h¼ l h ðxþ j¼ l ðxþ ¼ ðw jþ L a x j l ðxþ ¼ Gmax j¼ ðc jþ U a x j l ðxþ ¼ Gmax j¼ ðt jþ U a x j l ðxþ P l ðxþ l ðxþ P l ðxþ l i ðxþ 6 l i ðxþ P 0 i ¼ x j P l j j ¼...n x j l j 6 x j l j P 0; ðzþ U a X ¼ max h¼ l h ðxþ j¼ l ðxþ ¼ ðw jþ U a x j l ðxþ ¼ Gmax j¼ ðc jþ L a x j l ðxþ ¼ Gmax j¼ ðt jþ L a x j l ðxþ P l ðxþ l ðxþ P l ðxþ l i ðxþ 6 l i ðxþ P 0 i ¼ x j P l j j ¼...n x j l j 6 x j l j P 0: ð5aþ ð5bþ

11 40 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) Illustrations 4.. 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 99). The design includes four customer requirements (CRs) and five design requirements (DRs). Fig. illustrates the HOQ. In the figure four CRs are easy to hold (CR) does not smear (CR) point lasts (CR) and does not roll (CR4) while the important engineering design requirements contain length of pencil (DR) time between sharpening (DR) least dust generated (DR) hexagonal (DR4) and minimal erasure residue (DR5). In addition cost and technical difficulty of DRs are also incorporated in the planning processes. Firstly the fuzzy relationships ( e R ik ) between CRs and DRs those (~c kj ) among DRs and the relative importance ( ek i ) of the four CRs must be determined to derive the fuzzy importance ratings of the five 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 for different descriptions. Each group contains seven linguistic terms. For example the descriptions of relationship strength are weakest weak fairly weak Strongest Strong Customer Wants CR CR CR Relative Importance CR4 Estimated Cost Technical Difficulty Engineering Design Requirements DR DR DR DR4 DR5 Fairly strong Medium Fairly weak Weak Weakest Fig.. QFD matrix for a writing instrument. Table 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 Very unimportant Very high Very difficult Weak Unimportant High Difficult Fairly weak Fairly unimportant Fairly high Fairly difficult Medium Medium Medium Medium Fairly strong Fairly important Fairly low Fairly easy Strong Important Low Easy Strongest Very important Very low Very easy

12 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 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. 99). Their definitions are ( ) ( ) ( ) ( ) ( ) ( ) and ( ) respectively as shown in Fig.. Using the linguistic terms in Table to represent the various relationships in the QFD matrix as shown in Fig. the fuzzy normalized relationship mðr e 0 ij Þ can be calculated using Eq. (7). For obtaining mð R e 0 ij Þ the upper and lower bounds of the a-cuts of R e ik and those of ~c kj should be determined beforehand based on their membership functions. The membership function of a trapezoidal fuzzy number is defined by linear functions. As an illustration suppose that R e ik is assessed as fairly strong (es) and the membership function of the fuzzy number es ¼ð0:50:60:70:8Þ can be expressed as 8 >< ðr ik 0:5Þ=ð0:6 0:5Þ 0:5 6 R ik 6 0:6 l es ðr ik Þ¼ 0:6 6 R ik 6 0:7 >: ð0:8 R ik Þ=ð0:8 0:7Þ 0:7 6 R ik 6 0:8: Then the a-cut of the above membership function is h i ðr ik Þ L a ðr ikþ U a ¼½0:5þ0:a0:8 0:aŠ: 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) are smaller than those when Eq. (6) is used. For comparison purposes four membership functions mðr e 0 4 Þ mð R e 0 44 Þ R e 0 4 and R e 0 44 are shown in Fig. 4. This justifies the derived formulations in Appendix A. The fuzzy technical importance rating of the jth engineering design requirement ew j can be obtained using Eq. (9) to determine the priority of each design requirement. The membership functions ew to ew 5 are shown in Fig. 5. Both ew and ew 5 rank the highest with a range of % while ew ranks the lowest with a range of. 6.%. These ratings are then used to formulate the customer satisfaction function. The objective of a QFD planning is not only to maximize customer satisfaction (G ) but also to minimize cost (G ) and technical difficulty (G ) 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.. Cost and technical difficulty are evaluated and illustrated in Fig. membership degree Weakest Fairly Fairly weak strong Weak Medium Strong Strongest strength scale Fig.. The membership functions of linguistic terms for relationships.

13 4 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 0 48 Fig. 4. The membership functions of normalized relationship degrees. α ~ W W ~ W ~ = W~ technical importance degree Fig. 5. The membership functions of technical importance ratings. W ~ 4 using the seven linguistic scales of Table. For formulating fuzzy goal programming models the range of aspiration levels of G h i.e. ½G min h h Š should be specified beforehand. Suppose that the minimum fulfillment levels of the five engineering design requirements x...x 5 are required as and 0.5 respectively. Following the solution procedure in Section. the model can be formulated as follows: max X ðw i Þ U a¼0 x i ¼ 0:6x þ 0:x þ 0:408x þ 0:5x 4 þ 0:408x 5 min X ðc i Þ L a¼0 x i ¼ 0:7x þ 0:5x þ 0:4x þ 0:x 4 þ 0:x 5 min X ðt i Þ L a¼0 x i ¼ 0:x þ 0:4x þ 0:7x þ 0:x 4 þ 0:4x 5 x P 0: x P 0:5 x P 0: x 4 P 0:7 x 5 P 0:5 x i 6 i ¼...5: The ranges of aspiration levels of the three goals are determined as [ ] [0.68.] and [0.7.7] respectively. And the full model Eq. (5) can be constructed subject to the preemptive priority structure and the required fulfillment levels of DRs. ð6þ

14 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) Table The ranges for the three fuzzy goals and the fulfillment levels of ex and ex at different possibility a values a ½ðl Þ L a ; ðl Þ U a Š ½ðl Þ L a ; ðl Þ U a Š ½ðl Þ L a ; ðl Þ U a Š ½ðx Þ L a ; ðx Þ U a Š ½ðx Þ L a ; ðx Þ U a Š α ~ µ = ~ µ ~ µ satisfaction degree Fig. 6. The membership functions of goal values. Solving the model (5) of the example using different a-cuts i.e. a = the satisfaction degree for each goal and their sum as well as the fulfillment levels of all DRs can be acquired at each a level. Table lists the ranges of three fuzzy goals and those of the fulfillment levels of ex and ex at different possibility levels. Based on the ranges Fig. 6 depicts the membership functions of satisfaction degree of the three goals. The satisfaction degree of G is greater than those of G and G for which G locates in the interval [ ] and G as well as G in [ ] although G has the same priority as G. Obviously achieving the cost objective (G ) is easier than that of customer satisfaction (G ) in this example. In Fig. 7 x and x are determined as being fuzzy ex and ex respectively while x x 4 and x 5 are crisp with the fulfillment level of 0% 00% 00% respectively. As described before the decision variable x j = 00% denotes complete fulfillment of the jth DR. This means that the DRs x 4 and x 5 should have the best quality level in order to achieve the total satisfaction degree. Particularly x and x are smaller than the others in the example due to the low technical importance ratings and high estimated costs if referring to Figs. and 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

15 44 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 0 48 X X fulfillment level Fig. 7. The membership functions of decision variables x and x while x = 0. and x 4 = x 5 =. proposed approaches. Consider for instance the second DR in the example i.e. ex in Fig. 7 with the ranges at the possibility levels a =.0 and a = 0 being [ ] and [ ] respectively. In the fuzzy sense it is definitely possible that the fulfillment level of x is in [ ]; 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 Picó 00; Chang and Lee 996; Mon et al. 995; Wu 00). 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 ] 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 995). Among them the centroid method is the commonly used and hence has been adopted in this paper. The centroid method for defuzzifying a fuzzy number ex i is formulated as (Klir and Yuan 995). P m k¼ ^x i ¼ l ex i ðx ðiþ k ÞxðiÞ k P m k¼ l ex i ðx ðiþ k Þ ð7þ where ex i is defined on a finite universal set and l ex i ðx ðiþ k Þ is the membership degree (possibility level) of the kth element (fulfillment level in this paper) x ðiþ k in ex i. Consider again ex in Fig. 7. Applying the interpolation method based on the a-cuts (possibility levels) renders 5 fulfillment levels x ðþ k k =...5 for ex ; these 5 fulfillment levels and the corresponding possibility levels give the defuzzified value ^x ¼ 0:6 in Eq. (7). 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 (5). The a value and the resulting range of each fuzzy technical importance rating in Model (9) are taken as the input of Model (5). Obviously if a designer requires more information to decide the fulfillment level of DRs more a-cuts are needed. As listed in Table 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 as shown in Figs. 6 and 7. Therefore the number of a-cuts is critical to the proposed approach.

16 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 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 membership function of ex in Fig. 7 will somewhat change if the number of a values is small. For illustrations we can construct the membership functions of ex by using three different numbers of a-cuts i.e. (a = 0) (a = 00.5) and 6 (a = ) in Fig. 8. Although these membership functions differ from the membership function of ex in Fig. 7 the actual differences are not significant especially those under and 6 a-cuts. Similarly substituting 5 elements from the membership functions in Fig. 8(a) (c) into Eq. (7) gives respectively the defuzzified values and all of which are close to the defuzzified value 0.6 in Fig. 7. α X ~ 0 (a) fulfillment level α X ~ (b) fulfillment level α X ~ (c) fulfillment level Fig. 8. The membership functions of ex based on three different numbers of a-cuts. (a) The membership function based on two a-cuts (a = 0 ). (b) The membership function based on three a-cuts (a = ). (c) The membership function based on six a-cuts (a = ).

17 46 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) 0 48 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 = ) or six (a = ) 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: R 0 ij ¼ k¼ R ikr kj j¼ k¼ R ikr kj ¼ l¼ l6¼j k¼ P R ikr kj n k¼ R ikr kl þ k¼ R ikr kj ða:þ where 0 6 ðr ik Þ L a 6 R ik 6 ðr ik Þ U a 6 8k i ¼...m 0 6 ðc kj Þ L a 6 r kj 6 ðc kj Þ U a 6 8kj: Let / ¼ k¼ R ikr kj and u ¼ l¼ k¼ R ikr kl. l6¼j Then Eq. (A.) is expressed as f ð/þ ¼ / u þ / :

18 L.-H. Chen M.-C. Weng / European Journal of Operational Research 7 (006) Since f 0 u ð/þ ¼ ðu þ /Þ P 0 f(/) is an increasing function and X ðr ik Þ L a ðc kjþ L a 6 / 6 X ðr ik Þ U a ðc kjþ U a : k k Therefore and min f ð/þ ¼ k¼ ðr ikþ L a ðc kjþ L a u þ k¼ ðr ikþ L a ðc kjþ L a k¼ max f ð/þ ¼ ðr ikþ U a ðc kjþ U a u þ k¼ ðr ikþ U a ðc : kjþ U a Furthermore X n l¼ l6¼j X n k¼ ðr ik Þ L a ðc klþ L a 6 u 6 Xn l¼ l6¼j X n k¼ ðr ik Þ U a ðc klþ U a such that the new lower and upper bounds of a-cuts of e R 0 ij mðr0 ij ÞL a and mðr0 ij ÞU a can be formulated as mðr 0 ij ÞL a ¼ min f ð/þ ¼ P l¼ l6¼j n k¼ ðr ikþ L a ðc kjþ L a k¼ ðr ikþ U a ðc klþ U a þ k¼ ðr ikþ L a ðc kjþ L a mðr 0 ij ÞU a ¼ max f ð/þ ¼ k¼ ðr ikþ U a ðc kjþ U a l¼ k¼ ðr ikþ L a ðc klþ L a þ k¼ ðr ikþ U a ðc : kjþ U a l6¼j References Bondia J. Picó J. 00. Analysis of linear systems with fuzzy parametric uncertainty. Fuzzy Sets and Systems 5 8. Chang P.-T. Lee E.S A generalized fuzzy weighted least-squares regression. Fuzzy Sets and Systems Chen L.-H. Tsai F.C. 00. Fuzzy goal programming with different importance and priorities. European Journal of Operational Research Chen L.-H. Weng M.C. 00. A fuzzy model for exploiting quality function deployment. Mathematical and Computer Modelling Chen S.-J. Hwang C.-L. Hwang F.P. 99. Fuzzy Multiple Attribute Decision Making Methods and Applications. Springer- Verlag. Fung R.Y.K. Popplewell K. Xie J An intelligent hybrid system for customer requirements analysis and product attribute targets determination. International Journal of Production Research 6 () 4. Fung R.Y.K. Tang J. Tu Y. Wang D. 00. Product design resources optimization using a non-linear fuzzy quality function deployment model. International Journal of Production Research 40 () Kao C. Liu S.T Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets and Systems Kim K.-J. Moskowitz H. Dhingra A. Evans G Fuzzy multicriteria models for quality function deployment. European Journal of Operational Research King B Better Designs in Half the Time: Implementing QFD in America Goal/QPC. Methuen MA. Klir G.J. Yuan B Fuzzy Sets and Fuzzy Logic Theory and Applications. Prentice Hall Englewood Cliffs NJ.

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