برآورد روابط کاربردی برای گسترش کارکرد کیفیت تحت عدم قطعیت

Product planning is one of four important processes in new product development (NPD) using quality function deployment (QFD), which is a widely used customer-driven approach. In our opinion, the first problem to be solved is how to incorporate both qualitative and quantitative information regarding relationships between customer requirements (CRs) and engineering characteristics (ECs) as well as those among ECs into the problem formulation. Owing to the typical vagueness or imprecision of functional relationships in a product, product planning is becoming more difficult, particularly in a fuzzy environment. In this paper, an asymmetric fuzzy linear regression approach is proposed to estimate the functional relationships for product planning based on QFD. Firstly, by integrating the least-squares regression into fuzzy linear regression, a pair of hybrid linear programming models with asymmetric triangular fuzzy coefficients are developed to estimate the functional relationships for product planning under uncertainties. Secondly, using the basic concept of fuzzy regression, asymmetric triangular fuzzy coefficients are extended to asymmetric trapezoidal fuzzy coefficients, and another pair of hybrid linear programming models with asymmetric trapezoidal fuzzy coefficients is proposed. The main advantage of these hybrid-programming models is to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression. Next, the illustrated example shows that trapezoidal fuzzy number coefficients have more flexibility to handle a wider variety of systematic uncertainties and ambiguities that cannot be modeled efficiently using triangular number fuzzy coefficients. Both asymmetric triangular and trapezoidal fuzzy number coefficients can be applicable to a much wider variety of design problems where uncertain, qualitative, and fuzzy relationships are involved than when symmetric triangular fuzzy numbers are used. Finally, future research direction is also discussed.

Being able to perform new product development (NPD) in a short lead time and at a minimum cost is one of core factors for improving competitiveness in the global market. As far as product planning and development decisions are concerned, the use of quality function deployment (QFD) has gained extensive international support. QFD is a widely used customer-driven design and manufacturing tool originated in Japan in the late 1960s [1]. Generally QFD utilizes four sets of matrices called houses of quality (HOQ) to relate the customer requirements (CRs) to product planning, parts deployment, process planning and manufacturing operations [11]. When organizations direct their efforts towards meeting the customer requirements (CRs), internal conflict minimizes, development cycle time shortens, market penetration increases, product quality improves, and customer satisfaction increases, resulting in higher revenues. HOQ matrices have been frequently used in the industry to help design team undergo product planning [10], i.e., capture the CRs by assessing customer preferences, convert those attributes into engineering characteristics (ECs) and then determine the target levels for ECs of new/improved products to match or exceed performance of all competitors in the target market with limited organizational resources. It is a complex decision process with multiple variables to determine the target levels. In practice, it is normally accomplished in a subjective, ad hoc manner, or a heuristicway, such as using prioritization-based methods to yield feasible design, rather than an optimal one. In order to enhance the QFD methodology, developing more reasonable and effective modeling approach for product planning to determine the target values for ECs of a product, towards the maximum degree of customer satisfaction within limited recourses is usually the focus in the HOQ. And some progress has been made along this line. For product planning modeling approach, see [4,6–9,15,16,19,23–25,28]. In our opinion, the first problem to be solved in the product planning modeling based on HOQ is to incorporate both qualitative and quantitative information of relationships between CRs and ECs as well as those among ECs into the problem formulation. However, the literature has not paid enough attention to this aspect. In most of product planning models and methods mentioned above, the HOQ was usually analyzed in a fairly simplistic manner, namely functional relationships in product planning were determined based on organizational judgment using engineering knowledge. Unfortunately, owing to the typical vagueness or imprecision of these functional relationships, it is difficult to identify them using engineering knowledge. Especially when a given HOQ contains large number of CRs and ECs, many trade-offs have to be made among the degrees of customer satisfaction as well as among the implicit or explicit relationships, and it will become more difficult to determine them using engineering knowledge. The inherent vagueness or impreciseness of functional relationships in product planning arises mainly from these aspects: (a) The QFD process involves various inputs often in the form of linguistic data, e.g., human perception, judgment on market benchmarking, or evaluation on importance of CRs, which are highly subjective andvague [3,21,22]. Thus it seems more appropriate to treat these inputs as linguistic variables expressed as fuzzy numbers. (b) Formal mechanisms for translating CRs (which are generally qualitative) into ECs (which are generally quantitative) are lacking. In real-life design, there are many CRs for a product, each CR can be translated into multiple ECs, and conversely a certain EC may affect multiple CRs. In general, these CRs tend to be translated into ECs in a subjective, qualitative and non-technical way, which should be expressed in more quantitative and technical terms. So the functional relationships between CRs and ECs are often vague or imprecise. (c) Besides, the ECs have correlations between them, for instance the EC “door seal resistance” has positive impacts on the EC “noise resistance”. However, owing to the uncertainties in the design process, data available for product design are often limited and inaccurate. So the relationships of a given EC with other ECs are often not fully comprehended, and it is difficult or unnecessary to identify the complete relationships during product planning, especially when developing an entirely new product, certain vagueness is often inevitable. The inherent fuzziness of functional relationships in product planning makes the use of the fuzzy regression justifiable. Possibilistic fuzzy regression analysis firstly proposed by Tanaka et al. [17,18], in which two factors, namely the degree of fit and the fuzziness of the model, are considered. Estimation problems can be transformed into linear programming models based on the two factors [26]. In conventional regression analysis, deviations between the observed values and the estimates are assumed to be random errors. Thus, statistical techniques are applied to perform estimation and inference in regression analysis. However, the deviations are sometimes due to the indefinite structure of the system or imprecise observations. The uncertainty in this type of regression model becomes fuzzy, not random. Hence, fuzzy regression will be more appealing than conventional regression tools in estimating functional relationships in product planning [13]. Kim et al. [13] first suggested using fuzzy regression to estimate functional relationships in the field of QFD, and they proposed a fuzzy multi-criteria modeling approach for product planning, in which fuzzy linear regression with symmetric triangular fuzzy number was used to investigate the functional relationships. However, symmetric triangular fuzzy coefficients are not flexible and efficient enough for estimating these complicated functional relationships.When fuzzy regression with symmetric coefficients is applied, the regression line obtained may not be the best fit because of the existence of a large number of outliers and high residuals. There are data sets that generate scatter plots in which the data do not fall symmetrically on both sides of the regression line [27]. Therefore, by extending symmetric triangular fuzzy coefficients to asymmetric triangular ones and integrating the least-squares regression into fuzzy linear regression, a pair of hybrid asymmetric linear programming models are proposed for estimating the functional relationships between CRs and ECs as well as those among ECs for product planning. The limitations of the symmetric triangular fuzzy coefficients are remedied by such an extension. Triangular fuzzy numbers are most widely used in fuzzy regression because they are easy to handle arithmetically and interpretate intuitively for practical purposes. However, when a larger value is given to the degree of fit, the fuzzy linear regression using triangular fuzzy coefficients tends to yield large unnecessary fuzziness and estimated parameters with too large aspiration, which leads to the fuzzy predictive interval too fuzzier and has no operational definition or interpretation. This fact will be shown in Section 5. In our opinion, the properties of trapezoidal fuzzy numbers may be more powerful and practical than triangular fuzzy numbers as coefficients in fuzzy regression. Unfortunately, both research and applications using trapezoidal fuzzy numbers in fuzzy regression have been rare. Therefore, theidea of using asymmetric triangular fuzzy number coefficients is further extended to trapezoidal fuzzy number coefficients in fuzzy regression to estimate functional relationship for product planning under uncertainties. The rest of the paper is organized as follows. In the next section, the general model of product planning is formulated and illustrated. In Section 3, the estimating of functional relationships in product planning under uncertainties using fuzzy linear regression theory with asymmetric triangular fuzzy coefficients is discussed. In Section 4, triangular fuzzy number coefficients are extended to trapezoidal fuzzy number coefficients. Further, in Section 5, the case study illustrates how the functional relationships to the quality improvement problem of emulsification dynamite packing-machine can be obtained using asymmetric triangular and trapezoidal fuzzy number coefficients, respectively. Finally, conclusions are presented in Section 6.

In this paper, in order to overcome limitations of symmetric fuzzy linear regression, an asymmetric fuzzy linear regression approach is proposed to estimate the functional relationships for product planning based on QFD. Firstly, by integrating the least-squares regression into fuzzy linear regression, a pair of hybrid linear programming models with asymmetric triangular fuzzy coefficients are developed to estimate the functional relationships for product planning under uncertainties. Based on the basic idea of fuzzy regression, asymmetric triangular fuzzy coefficients can be extended to the case of asymmetric trapezoidal fuzzy coefficients, and another pair of hybrid linear programming models with asymmetric trapezoidal fuzzy coefficients is proposed to estimate the functional relationships for product planning. The proposed approach integrates both the properties of central tendency in least-squares regression and the possibilistic properties in fuzzy regression, to give a more central tendency. Asymmetric triangular or trapezoidal fuzzy coefficients have more flexibility and can handle a wider variety of systematic uncertainties and ambiguities that cannot be modeled efficiently using symmetric triangular fuzzy coefficients. The extension of symmetric triangular fuzzy coefficients to asymmetric triangular and trapezoidal fuzzy coefficients increases the flexibility of the linear fuzzy regression, and would be applicable to a much wider variety of design problems where functional relationships are involved in an uncertain, qualitative and fuzzy way than using symmetric triangular fuzzy coefficients.The illustrated example shows that when a larger h value is selected, the linear fuzzy regression with the triangular fuzzy coefficients tends to have larger fuzziness and some coefficients with larger aspirations accordingly. Such limitations can be avoided efficiently by using trapezoidal fuzzy coefficients in linear fuzzy regression. Therefore, trapezoidal fuzzy coefficients perform better than triangular fuzzy coefficients in estimating the functional relationships in product planning when the h value is large. Furthermore, the illustrated example also suggests the direction for future research. Since fuzzy linear regression approach can be reduced to a linear programming problem, when it is applied to estimate the functional relationships in product planning, some coefficients tend to become non-crisp generally. Therefore in order to overcome this characteristic of linear programming and obtain more diverse spread coefficients, a non-linear fuzzy regression approach, such as quadratic programming, should be further considered.