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Implementing Fuzzy Programming in Quality Function DeploymentMohammad KhalilzadehPhd student of Mathematical Sciences Department of Brunel University, West London,UKAmin Malayeri ^{ }Bachelor student of Industrial Engineering Department of Sharif University of Technology, Iran AbstractQuality Function Deployment (QFD) is a wellknown and customeroriented design tool that systematically translates the Voice of the Customer (VOC) into the Design and Engineering Requirements (DRs). QFD is widely used to achieve higher customer satisfaction level by integrating marketing, design engineering, manufacturing, and other related functions of an organization. Most of the existing approaches and models for QFD planning seldom consider the innate imprecision and uncertainties in the customer requirements (CRs), which are usually expressed in natural language, as well as design processes, such as illdefined or incomplete understanding of the relationships between a given set of customer requirements and design requirements, and the complexity of interdependence among DRs. For dealing with the imprecision and uncertainties in all stages of QFD planning, we apply the fuzzy environment including fuzzy AHP method for determining the relative importance weights of customer requirements, and fuzzy relationships between CRs and DRs as well as among the DRs. A fuzzy mathematical model is formulated to determine the systematic fulfilment level of each DR for increasing the customer satisfaction under the constraints such as cost and resource limitations, technical difficulty and market competition. An illustrative example is also given to illustrate how the proposed fuzzy model and optimization route can be applied to help QFD team and decision makers in a company to maximize overall customer satisfaction level. Keywords: QFD, Fuzzy AHP, Fuzzy numbers, Fuzzy linear programming, customer satisfaction IntroductionThe intense competition among organizations and the global market economics have been high impacts on demanding customer’s requirements and their needs. This accelerating competition has shortened product life cycles and has spurred the progress of technological innovations. Successful companies have made many efforts to improve their product and service quality and keep their customers satisfied. Today customer satisfaction is a crucial issue not only for success with a new product, but also critical for its very survival. New product development is a complicated process involving multiple functional teams with different perspectives. A large number of companies attempting to cope with this increasing market competition and divergence in product design have adopted Quality Function Deployment (QFD) to translate customer requirements into appropriate product and service specifications and requirements. A product is said to have quality, when it is not only defect free, but also involves more aspects such as reliability, maintainability, attractiveness, flexibility, and ease of use as perceived by the customer. A system may be well equipped to manufacture products to the required standards, but will fail to capture the market if it does not meet its customer’s requirements. The whole process of manufacturing should be customeroriented. In general, the voices of customers (VOC) are acquired through questionnaire, interviews, claims and complaint information, and sales point’s assessment. The VOCs are used to identify the customer requirements (CRs), which are qualitative representations of the end user’s requirements. QFD is a worldwide accepted tool used to match the qualitative VOCs to the quantitative design and engineering characteristics (DRs). Quality function deployment (QFD) originated in Japan in the 1970s and became increasingly popular in the western world in the 1980s. It has been successfully applied in many Japanese organizations to improve processes and to create competitive advantages (Hauser and Clausing, 1998). Today, companies are successfully using QFD as a powerful tool that addresses strategic, tactical, and operational decisions in businesses to improve product design and quality (Bossert,1991. Mizuno and Akao, 1994. Cohen, 1995). Quality function deployment is a systematic approach for ensuring that customers’ voices are deployed in the product planning, design and engineering, and manufacturing stages. The desires of customers on a product are taken into account, through conducting a survey by the marketing department and are treated as a set of customer requirements (CRs). A number of engineering design requirements (DRs) that affect CRs are also identified to maximize customer satisfaction. In general, a QFD team is organized to determine the improvement levels of DRs by analyzing the relationships between CRs and DRs as well as among the DRs, considering the cost and other organizational constraints. However, QFD still has several limits in applications and therefore many more researches should be done on this issue. QFD team members usually subjectively determine the relationships between CRs and DRs and among the DRs based on past experience, due to the lack of precise information from customer requirements which are based on linguistics and are usually nontechnical in nature (Fung et al, 1998). Moreover, information for product design is often limited and imprecise, particularly when developing an entirely new product, such that engineers usually do not have complete knowledge about the impacts of engineering characteristics on customer requirements (CRs). Park and Kim (1998) proposed a decision model for prioritizing ECs using crisp data. The procedure uses the analytic hierarchy process (AHP) to classify the relative importance of the CAs, to assign weights for the relationship between the CR and DR. Fung et al. (1998) proposed a fuzzy customer requirement inference system in which the product attributes could be mapped out. Moskowitz and Kim (1997) provided a decision support system for optimizing product designs. Temponi et al. (1999) developed a reasoning scheme for inferring requirement relationships between CRs and DRs, as well as among the DRs. Nevertheless, the development of these systems usually requires professional knowledge and experiences to establish rules and facts in ensuring that the systems can work well. Khoo and Ho (1996) used fuzzy logic theory to address the existing ambiguity of Quality Function Deployment. C.K. Kwong and H. Bai (2002) applied Analytical Hierarchy Process (AHP) to determine the importance weights of customer requirements. Yang and Fang (2003) integrated fuzzy logic into HOQ to establish a framework for prioritizing customer requirements to analyze product features and conduct product positioning. Some models are also formulated for determining the levels of engineering design requirements based on fuzzy set theory. Kim et al. (2000) proposed a fuzzy theoretical modeling approach to QFD by formulating fuzzy multiobjective models, assuming that the function relationships between CRs and DRs and that among the DRs can be identified using benchmarking data of customer competitive analysis. But it would be difficult, particularly when developing an entirely new product. Some researchers ever developed fuzzy approaches to address complex and often imprecise problems in customer requirement management by applying fuzzy sets, fuzzy arithmetic, and/or fuzzy defuzzification techniques (Zhou,1998. Wang, 1999. Shen et al, 2001. Vanegas et al, 2001). However, the interrelationships among the engineering characteristics were not properly incorporated in these models. Instead of the fuzzy approaches mentioned above, this study considers not only the inherent fuzziness in the relationships between CRs and DRs, but also those among DRs. Wasserman (1993) discussed the problem of prioritizing design requirements in QFD process and proposed an LP formulation to allocate an incremental unit cost to design requirements to maximize customer satisfaction that was represented by a technical importance weight. Two kinds of fuzzy relationships are aggregated, based on Wasserman’s study, to obtain the fuzzy normalized relationship matrix each cell of which is represented by a fuzzy number. We applied the expressions obtained by Chen and Weng (2003) for those fuzzy numbers for shortended cuts, such that fuzzy technical importance ratings for engineering design requirements can be determined in terms of cuts with less uncertainty. Under the resource limitation and the considerations of technical difficulty, and market competition, we then formulated a fuzzy LP model to determine the optimal fulfillment degrees of DRs at each cut for achieving the optimal customers’ satisfaction. An illustrative example is used to demonstrate the approach. In the following section, fuzzy concepts and fuzzy AHP method are introduced, In section 3 and 4, we extend the fuzzy AHP method and review the literature. Subsequently, fuzzy normalized relationship matrix of QFD is introduced in section 5; Fuzzy technical importance rating for each engineering design requirement at each cut is then determined. Section 6 formulates the QFD planning problem as a fuzzy model to determine the fulfillment levels of DRs to produce the maximum customer satisfaction. An illustrative example is given to demonstrate our approach in Section7. Finally, conclusions are provided in Section 8. Fuzzy sets theory and fuzzy AHPTo deal with vagueness of human thought, Zadeh (1965) first introduced the fuzzy set theory, which was oriented to the rationality of uncertainty due to imprecision or vagueness. A major contribution of fuzzy set theory is its capability of representing vague data. The theory also allows mathematical operators and programming to apply to the fuzzy domain. A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function, which assigns to each object a grade of membership ranging between zero and one. A tilde “~” will be placed above a symbol if the symbol represents a fuzzy set. Therefore, , , are all fuzzy sets.The membership function for these fuzzy sets will be denoted by µ(x/), and µ() respectively. A triangular fuzzy number, , is shown in figure (1). A triangular fuzzy number is denoted simply as (m_{1}/m_{2, }m_{2}/m_{3}) or (m_{1}, m_{2, }m_{3}). The parameters m_{1, }m_{2 }and m_{3 }respectively denote the smallest possible value, the most promising value, and the largest possible value that describe a fuzzy event. Each triangular fuzzy number has linear representations on its left and right side such that its membership function can be denoted as: A fuzzy number is always introduced by the left and right representation of each degree of membership: where l(y) and r(y) denotes the left side representation and the right side representation of a fuzzy number respectively. Many ranking methods for fuzzy numbers have been developed in the literature. These methods may give different ranking results and most methods are tedious in graphic manipulation requiring complex mathematical calculation. The algebraic operations with fuzzy numbers are given in Appendix . Many decisionmaking and problemsolving tasks are too complicated to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise. Fuzzy set theory resembles human reasoning in its use of approximate information and uncertainty to make decisions. It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems. By contrast, traditional computing demands precision down to each bit. Since knowledge can be expressed in a more natural by using fuzzy sets, many engineering and decision problems can be greatly simplified. Fuzzy set theory implements classes or groupings of data with boundaries that are not sharply defined (i.e.fuzzy). Any methodology or theory implementing “crisp” definitions such as classical set theory, arithmetic, and programming, may be “fuzzified” by generalizing the concept of a crisp set to a fuzzy set with blurred boundaries, The benefit of extending crisp theory and analysis methods to fuzzy techniques in the strength in solving realworld problems, which inevitably entail some degree of imprecision and noise in the variables and parameters measured and processed for the application. Accordingly, linguistic variables are a critical aspect of some fuzzy logic applications, where general terms such a “large”, “medium”, and “small” are each used to capture a range of numerical values. Fuzzy set theory encompasses fuzzy logic, fuzzy arithmetic, fuzzy mathematical programming, fuzzy topology, fuzzy graph theory, and fuzzy data analysis, though the term fuzzy logic is often used to describe all of these. The analytic hierarchy process (AHP) is one of the extensively used multicriteria decisionmaking methods. One of the main advantages of this method is the relative ease with which it handles multiple criteria. In addition to this, AHP is easier to understand and it can effectively handle both qualitative and quantitative data. The use of AHP does not involve cumbersome mathematics. AHP involves the principles of decomposition, pairwise comparisons, and priority vector generation and synthesis. Though the purpose of AHP is to capture the expert’s knowledge, the conventional AHP still cannot reflect the human thinking style; it is recognized that human assessment on qualitative attributes is always subjective and thus imprecise. Therefore, fuzzy AHP, a fuzzy extension of AHP, was developed to solve the hierarchical fuzzy problems. The decisionmaker can specify preferences in the form of natural language expressions about the importance of each performance attribute (hygiene, quality of meals, quality of service). The system combines these preferences using fuzzy AHP, with existing data (from industrial surveys and statistical analysis) to reemphasize attribute priorities. In the fuzzy AHP procedure, the pairwise comparisons in the judgment matrix are fuzzy numbers that are modified by the designer’s emphasis. Using fuzzy arithmetic and αcuts, the procedure calculates a sequence of weight vectors that will be used to combine the scores on each attribute. The procedure calculates a corresponding set of scores and determines one composite score that is the average of these fuzzy scores. 