Estimating the functional relationships for quality function deployment under uncertainties

Fuzzy Sets and Systems 157 (2006) 98 – 120 www.elsevier.com/locate/fss

Estimating the functional relationships for quality function deployment under uncertainties Richard Y.K. Funga , Yizeng Chena,∗ , Jiafu Tangb a Department of Manufacturing Engineering & Engineering Management, City University of Hong Kong, Tat Chee Avenue,

Kong Loon, Hong Kong b Key Laboratory of Process Industrial Automation of MOE, School of Information Science & Engineering,

Northeastern University (NEU), Shenyang, Liaoning, PR China Received 27 November 2003; received in revised form 25 May 2005; accepted 30 May 2005 Available online 21 June 2005

Abstract 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

∗ Corresponding author.

E-mail address: [email protected] (Y. Chen). 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.05.032

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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. © 2005 Elsevier B.V. All rights reserved. Keywords: Product planning; Quality function deployment; House of quality; Fuzzy linear regression; Least-squares regression; Asymmetric triangular fuzzy number; Trapezoidal fuzzy number

1. Introduction 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 heuristic way, 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 and

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vague [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, the

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idea 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.

2. Product planning model The basic concept of the first HOQ in the product design is to translate CRs into ECs. Assume that in a product design, m CRs denoted by CRi (i = 1, 2, . . . , m), n ECs denoted by ECj (j = 1, 2, . . . , n) and l competitors denoted by Compr (r = 1, 2, . . . , l) are considered. Let yi (i = 1, 2, . . . , m) be the customer perception of the degree of satisfaction of CRi , and xj (j = 1, 2, . . . , n) be the level of attainment of ECj with 0
xj
1(j = 1, 2, . . . , n). The product planning process based on the first HOQ is to determine a set of x1 , x2 , . . . , xn for ECs of the new/improved product to match or exceed the degree of overall customer satisfaction of all competitors in the target market with limited organizational resources. Let S present the degree of overall customer satisfaction for (y1 , y2 , . . . , ym ), the process of determining attainment level of target values of ECs for a new/improved product can be formulated as an optimization problem as follows: 
 max S = S(y1 , y2 , . . . , ym ) − max Sr (1a) r=1,...,l

s.t. yi = fi (X) = fi (x1 , x2 , . . . xn ),

i = 1, . . . , m,

xj = gj (Xj ) = gj (x1 , . . . , xj −1 , xj +1 , . . . , xn ), qe (X)
0,

e = 1, . . . , p,

yimin
yi
yimax , 0
xj
1,

i = 1, . . . , m,

j = 1, . . . , n,

(1b) j = 1, . . . , n,

(1c) (1d) (1e) (1f)

where fi (i = 1, 2, . . . , m) expresses the functional relationship between yi and xj , j = 1, 2, . . . , n, i.e. yi = fi (X) = fi (x1 , x2 , . . . xn ), and gj (j = 1, 2, . . . , n) represents functional relationship between xj and other levels of attainment of ECs, i.e. xj = gj (Xj ) = gj (x1 , . . . , xj −1 , xj +1 , . . . , xn ). Sr is the degree of overall customer satisfaction of the rth competitor, qe (X) is the eth organizational resource constraint and p is number of organizational resource constraints.

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The degree of overall customer satisfaction S(y1 , y2 , . . . , ym ) can be obtained by aggregating the degrees of customer satisfaction with individual CRs [20], i.e. S(y1 , y2 , . . . , ym ) =

m 

wi si (yi ),

(2)

i=1

where si (yi ) is an individual value function on CRi , and wi is the relative weight of CRi and can be determined using existing method such as simple rating methods or more elaborate methods based on the pairwise comparisons of attributes [2]. The value of yi (i = 1, 2, . . . , m) indicates the degree of satisfaction of CRi in comparison with the competitors and can be scaled in such a way that si (yimin ) = 0 and si (yimax ) = 1 and determined by si (yi ) = (yi − yimin )/(yimax − yimin ).

(3)

If the value of objective function is positive, i.e. S  > 0, then the target determination problem determines a set of x1 , x2 , . . . , xn with maximum degree of overall customer satisfaction. Else S  < 0, it determines a set of x1 , x2 , . . . , xn to minimize the difference of the customer satisfaction from any of the competitors. There are multiple resources required for supporting the design of a product, including technical engineers, advanced equipment, tools and other facilities. At the level of strategic planning, owing to the uncertainties in the design process, such as ill-defined or incomplete understanding of the relationship between the CRs and the ECs, as well as those among the ECs, these types of resources can be expressed involving the concept of fuzziness generally. And these resources constraints can be taken into account according to Tang et al. [20] and Fung et al. [8]. Now the problem at hand is how to determine the coefficients in fi (i = 1, 2, . . . , m) and gj (j = 1, 2, . . . , n) in the product planning model given in (1). In this paper, those coefficients are identified specifically using fuzzy linear regression with asymmetric triangular fuzzy coefficients (see Section 3) and trapezoidal fuzzy coefficients (see Section 4), respectively. 3. Estimating functional relationships using asymmetric triangular fuzzy coefficients 3.1. Model formulation First of all, the degree of customer satisfaction of CRi is assumed to be a linear combination of the level of attainment of individual ECs, and then consider a fuzzy linear function Y˜ i = fi (X, A˜ i ) = A˜ i0 + A˜ i1 x1 + · · · + A˜ in xn ,

(4)

where Y˜ i is the fuzzy output of the degree of customer satisfaction of CRi , X = (x0 , x1 , . . . , xn )T is the real-valued input vector of the level of attainment of ECs with x0 = 1, and A˜ i = (A˜ i0 , A˜ i1 , . . . , A˜ in ) is a vector of fuzzy numbers. The regression analysis problem in HOQ can be defined as: given a number of l non-fuzzy input–output pairs (xr , yir ), r = 1, . . . , l, a set of fuzzy parameters A˜ i0 , A˜ i1 , . . . , A˜ in will be determined such that (4) best fits the given data points in a sense of goodness of fit under some criteria whereby xr is the set of level of attainment of ECs of Compr , xr = (x0r , x1r , . . . , xj r , . . . , xnr ). Moreover, x0r = 1∀r, and xj r is the level of attainment of the ECj of Compr and yir is the degree of customer satisfaction of the CRi of Compr . If A˜ ij has triangular membership functions, it can be uniquely

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

103


A~ij(aij)

1

0 a ij(1)

aijL

aij(2)

aijR

aij(3)

Fig. 1. Asymmetric triangular fuzzy coefficient A˜ ij .

(1)

(2)

(3)

(1)

(2)

(3)

(2)

defined by three parameters in terms of (aij , aij , aij ) with aij
aij
aij . Here aij is the peak point (2)

(1)

(3)

value that satisfies
A˜ ij (aij ) = 1. aij and aij represent the lower limit and the upper limit, respectively. (2) (1) (3) The peak point value a describes the most possible value of A˜ ij , while a and a represent the ij

ij

ij

precision of A˜ ij . The left-side spread aijL and the right-side aijR of A˜ ij (See Fig. 1), respectively, can be calculated as (2)

(1)

(3)

(2)

aijL = aij − aij

(5)

and aijR = aij − aij .

(6)

The basic idea of fuzzy regression is to minimize the fuzziness of the linear fuzzy model that includes all the given input–output pairs in its h-level set as follows [19]: yir ∈ [fi (xr )]h ,

r = 1, 2, . . . , l,

(7)

where [fi (xr )]h is the h-level set of the fuzzy output fi (xr ) from the linear fuzzy model fi (X, A˜ i ) corresponding to the input vector xr . Since h-level set of fuzzy numbers are intervals, the determination of the linear fuzzy model fi (X, A˜ i ) can be viewed as determining a linear interval model with interval coefficients that includes all the given input–output pairs. Interval arithmetic is used for the calculation of the linear interval model (and for the calculation of the h-level set of linear model). When all the fuzzy coefficients A˜ ij (j = 1, 2, . . . , n) of the linear fuzzy model fi (X, A˜ i ) given in (4) are asymmetric triangular, according to fuzzy arithmetic on triangular fuzzy numbers, the fuzzy output from the linear model fi (X, A˜ i ) is also calculated as an asymmetric triangular fuzzy number. If we define  xj r if xj r  0, + (8) xj r = 0 otherwise

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and xj−r

 =

if xj r  0, otherwise

0 −xj r

(9)

for j = 1, 2, . . . , n, r = 1, 2, . . . , l, then xj+r and xj−r are all non-negative and satisfy that xj r = xj+r − xj−r .

(10) (1)

(2)

Thus if we denote the fuzzy output fi (xr ) by its lower limit fi (xr ), peak point fi (xr ) and upper limit (3) (1) (2) (3) fi (xr ) as fi (xr ) = (fi (xr ), fi (xr ), fi (xr )), then by the sum and product operation of triangular (1) (2) (3) fuzzy numbers, we can obtain fi (xr ), fi (xr ) and fi (xr ) as follows, respectively: (1) fi (xr )

=

n  j =0

(2) fi (xr )

=

n  j =0

(3)

fi (xr ) =

n  j =0

(1) aij xj+r

−

n  j =0

(3)

aij xj−r ,

(11)

(2)

aij xj r , (3)

aij xj+r −

(12) n  j =0

(1)

aij xj−r .

(13) (1)

(2)

(3)

From (11)–(13), the 0
h < 1 set of fi (xr ) = (fi (xr ), fi (xr ), fi (xr )) is calculated as the following interval: (2)

(1)

(2)

(3)

[fi (xr )]h = [hfi (xr ) + (1 − h)fi (xr ), hfi (xr ) + (1 − h)fi (xr )].

(14)

Therefore the inclusion relation yir ∈ [fi (xr )]h , r = 1, 2, . . . , l, can be rewritten as (2)

(1)

(2)

(3)

hfi (xr ) + (1 − h)fi (xr )
yir
hfi (xr ) + (1 − h)fi (xr ),

r = 1, 2, . . . , l.

(15)

The total spreads in a multi-input fuzzy output function is given by Z=

l  r=1

(3) (1) (fi (xr ) − fi (xr ))

n l  

=

r=1 j =0

(3)

(1)

(aij − aij )(xj+r + xj−r ).

(16)

Henceforth, with asymmetric triangular fuzzy number, the functional relationships fi (i = 1, 2, . . . , m) can be obtained by solving the linear programming model LPfi − 1(i = 1, 2, . . . , m): min

Z=

n  j =0

(3) (aij

s.t. h

n  j =0

(1) − aij )

l 

|xj r |

 (2)

(17a)

r=1

aij xj r + (1 − h) 

n  j =0

(1)

aij xj+r −

n  j =0

 (3)

aij xj−r 
yir ,

r = 1, 2, . . . , l,

(17b)

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

h

n  j =0

(1)

 (2)

aij xj r + (1 − h)  (2)

(3)

aij
aij
aij ,

n  j =0

(3)

aij xj+r −

n  j =0

105

 (1)

aij xj−r   yir ,

r = 1, 2, . . . , l,

j = 0, 1, . . . , n.

(17c)

(17d)

Similarly, one can determine functional relationships among the degree of attainment of target level of ECs using fuzzy linear regression. Assume that ˜ j = gj (Xj , A˜ j ) = A˜ j 0 + A˜ j 1 x1 + · · · + A˜ j,j −1 xj −1 + A˜ j,j +1 xj +1 + · · · + A˜ j n xn , X

(18)

where X˜ j is the fuzzy output of the degree of attainment of target level of the ECj , Xj = (x0 , x1 , . . . , xj −1 , xj +1 , . . . , xn )T is the real-valued input vector of the level of attainment of ECs with x0 = 1, and A˜ j = (A˜ j 0 , A˜ j 1 , . . . , A˜ j,j −1 , A˜ j,j +1 , . . . , A˜ j n ) is a set of asymmetric fuzzy triangular number to be determined by solving the following linear programming model LPgj − 1(j = 1, 2, . . . , n): min

Z=

n l   (3) (1) (aj u − aj u ) |xur |

(19a)

r=1

u=0 u =j

s.t.

h

n  u=0 u =j

h

n  u=0 u =j

(1)

  (2) aj u xur + (1 − h)    (2) aj u xur + (1 − h)  (2)

(3)

aj u
aj u
aj u ,

n  u=0 u =j

n  u=0 u =j

(1)

+ aj u xur −

(3)

+ aj u xur −

n  u=0 u =j

n  u=0 u =j

 (3) −  aj u xur 
xj r ,

r = 1, 2, . . . , l,

(19b)

r = 1, 2, . . . , l,

(19c)

 (1) −  aj u xur   xj r ,

u = 0, 1, . . . , j − 1, j + 1, . . . , n.

(19d)

(2) (1) (3) Obviously, if we select aij = (aij + aij )/2, i.e. all the fuzzy coefficients A˜ ij (j = 1, 2, . . . , n) are symmetric triangular fuzzy coefficients, LPfi − 1 will be reduced to the symmetric triangular case. In order to determine the asymmetric triangular fuzzy coefficients of LPfi − 1 given in (17), according to Ishibuchi and Nii [12], a hybrid method using fuzzy regression combined with least-squares regression (2) is introduced. First, we have to determine the peak point set aˆ i by using least-squares regression as follows: (2)

aˆ i

= (XT X)−1 XTYi ,

(20)

where X and Yi contain the explanatory and dependent variables, respectively. Subsequently, we uti(1) (3) lize fuzzy regression to determine the parameter sets ai and ai . Henceforth, the above programming model LPfi − 1 given in (17) is transformed into the following hybrid linear programming model

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Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

LPfi − 2(i = 1, 2, . . . , m): min

Z=

n  j =0

(3)

(1)

(aij − aij )

s.t. h

n  j =0

h

n  j =0

l 

|xj r |

r=1

 (2)

aˆ ij xj r + (1 − h)   (2)

aˆ ij xj r + (1 − h) 

(1)

(2)

(3)

aij
aˆ ij
aij ,

(21a)

n  j =0 n  j =0

(1)

aij xj+r −

(3)

aij xj+r −

n  j =0 n  j =0

 (3)

aij xj−r 
yir ,

r = 1, 2, . . . , l,

(21b)

r = 1, 2, . . . , l,

(21c)

 (1)

aij xj−r   yir ,

j = 0, 1, . . . , n.

(21d)

Correspondingly, the linear programming model LPgj −1 given in (19) is transformed into the following hybrid linear programming model LPgj − 2(j = 1, 2, . . . , n): min

Z=

n  u=0 u =j

(3)

(1)

(aj u − aj u )

s.t. h

n  u=0 u =j

h

n  u=0 u =j

l 

|xur |

(22a)

r=1

  (2) aˆ j u xur + (1 − h) 

n 

n 

 (3) −  aj u xur 
xj r ,

r = 1, 2, . . . , l,

(22b)

 n n    (3) + (1) −  + (1 − h)  aj u xur − aj u xur   xj r ,

r = 1, 2, . . . , l,

(22c)

u=0 u =j

(1)

+ aj u xur −

u=0 u =j



(2)

aˆ j u xur

(1)

(2)

u=0 u =j

(3)

aj u
aˆ j u
aj u ,

u=0 u =j

u = 0, 1, . . . , j − 1, j + 1, . . . , n.

(22d)

Therefore the functional relationships fi (i = 1, 2, . . . , m) and gj (j = 1, 2, . . . , n) with asymmetric triangular fuzzy numbers resulted from the pair of hybrid linear programming models LPfi − 2 given in (21) and LPgj − 2 given in (22) are obtained as y˜i = f˜i (X) =

n  (1) (2) (3) (aij , aˆ ij , aij )xj ,

i = 1, 2, . . . , m,

(23)

j =0

n  (1) (2) (3) x˜j = g˜ j (X ) = (aj u , aˆ j u , aj u )xu , j

u=0 u =j

j = 1, 2, . . . , n.

(24)

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

107


A~i(ai)

1 h2 h1 0

* * (ai(1))h2 (ai(1))h1

aˆi(2)

(3)

* (ai )h1

* (ai(3))h2

Fig. 2. The case when h2  h1 .

3.2. The effects of h on the solution As we can see in Section 3.1, the value of h determines the range of the possibility distribution of the fuzzy parameters, so it is important to select a suitable value for h in fuzzy regression. Moskowitz and Kim [14] studied the relationship between the h value, membership function shape, and the spreads of fuzzy parameters in fuzzy linear regression with symmetric fuzzy numbers, and developed a systematic approach to assess the proper h parameter values. In the same manner, now let us discuss the effect of h on the optimal solution to LPfi − 2 given in (21). The effect of h on the optimal solution to LPfi − 2 can be assessed from the following theorem. Theorem. If we denote the optimal solution of the linear programming model LPfi − 2 with regard to (1) (2) (3) h1 as (A˜ i )∗h1 = ((ai )∗h1 , aˆ i , (ai )∗h1 ) and Zh∗1 , and then the optimal solution of the linear programming model LPfi − 2 with regard to h2 can be obtained as 
h1 − h2 (2) 1 − h1 (1) ∗ (2) h1 − h2 (2) 1 − h1 (3) ∗ ∗ (A˜ i )h2 = , aˆ + (a ) , aˆ , aˆ + (a ) 1 − h2 i 1 − h2 i h 1 i 1 − h2 i 1 − h2 i h1 1 − h1 ∗ Zh∗2 = Z . (25) 1 − h 2 h1 Proof. First, let us consider the case of h2  h1 (See Fig. 2), then 1 − h2
1 − h1 . Let
h1 and
h2 denote the feasible region of the linear programming model LPfi − 2 given in (21) with h1 and h2 , respectively. (1) (2) (3) It follows from ((ai )∗h1 , aˆ i , (ai )∗h1 ) ∈
h1 that


h1 − h2 (2) 1 − h1 (1) ∗ (2) h1 − h2 (2) 1 − h1 (3) ∗ aˆ + (a ) , aˆ , aˆ + (a ) 1 − h2 i 1 − h2 i h 1 i 1 − h2 i 1 − h2 i h 1

 ∈
h 2 .

(26)

It can be concluded that
h1 ⊇
h2 , since 1 − h2
1 − h1 . The objective function value associated with


h1 − h2 (2) 1 − h1 (1) ∗ (2) h1 − h2 (2) 1 − h1 (3) ∗ aˆ + (a ) , aˆ , aˆ + (a ) 1 − h2 i 1 − h2 i h1 i 1 − h2 i 1 − h2 i h1



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Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

is given as

 n l  1 − h1  (3) (1) |xj r | ((ai )∗h1 − (aij )∗h1 ) 1 − h2 r=1 j =0  n l   1 − h1 (3) (1) = ((aij )h1 − (aij )h1 ) min |xj r | 1 − h2 (a(1) ,ˆa(2) ,a(3) )∈
h r=1 i i i 1 j =0  n l   1 − h1 (3) (1) ((aij )h1 − (aij )h1 )
min |xj r | . 1 − h2 (a(1) ,ˆa(2) ,a(3) )∈
h i

i

i

2

j =0

(27)

r=1

Therefore
 h1 − h2 (2) 1 − h1 (1) ∗ (2) h1 − h2 (2) 1 − h1 (3) ∗ aˆ + (a ) , aˆ , aˆ + (a ) 1 − h2 i 1 − h2 i h1 i 1 − h2 i 1 − h2 i h1 is an optimal solution of the linear programming model LPfi − 2 with regard to h2 . (1) (2) (3) Next, let us consider the case of h2 < h1 . Given the optimal solution (A˜ i )∗h2 = ((ai )∗h2 , aˆ i , (ai )∗h2 ), the optimal solution with regard to h1 can be obtained from the above proof as
h2 − h1 (2) 1 − h2 (1) ∗ (2) (1) ∗ (A˜ i )h1 = (ai )∗h1 = aˆ + (a ) , aˆ , 1 − h1 i 1 − h1 i h2 i  h2 − h1 (2) 1 − h2 (3) ∗ (3) ∗ (ai )h1 = . (28) aˆ + (a ) 1 − h1 i 1 − h1 i h2 Thus, the optimal solution with regard to h2 can be obtained as follows: 
h1 − h2 (2) 1 − h1 (1) ∗ (2) h1 − h2 (2) 1 − h1 (3) ∗ ∗ ˜ . aˆ + (a ) , aˆ , aˆ + (a ) (Ai )h2 = 1 − h2 i 1 − h2 i h 1 i 1 − h2 i 1 − h2 i h 1

(29)

The theorem is proved.  According to (5) and (6), the theorem given in (25) can also be described as 
1 − h1 L ∗ (2) 1 − h1 R ∗ ∗ (A˜ i )h2 = , (a ) , aˆ , (a ) 1 − h 2 i h 1 i 1 − h2 i h 1 1 − h1 ∗ Zh∗2 = Z . 1 − h 2 h1

(30)

The theorem above means that the values of left spread aijL and right spread aijR of each A˜ ij and objective function Z becomes (1 − h1 )/(1 − h2 ) times simultaneous when the h value changes from h1 to h2 . It is obvious from the theorem that the optimal solution for h can be easily obtained from the optimal solution for h = 0, i.e. 
−h (2) 1 1 (1) ∗ (2) −h (2) (3) ∗ ∗ ˜ (Ai )h2 = (a ) , aˆ , (a ) , aˆ + aˆ + 1−h i 1 − h i h1 i 1 − h i 1 − h i h1 1 Zh∗ = Z∗. (31) 1−h 0

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

109

4. Extension to trapezoidal fuzzy number coefficients Triangular fuzzy numbers are easy to handle arithmetically and they have an intuitive interpretation. However, according to (31), when a larger value is given to h, the fuzzy linear regression with triangular fuzzy number coefficients tend to yield large unnecessary fuzziness and some estimated parameters with large aspirations, which are difficult to interpret at times. Furthermore, from Section 3, the inclusion relation yir ∈ [fi (xr )]h is used as a constraint condition and h must satisfy 0
h < 1 in fuzzy regression with triangular (both symmetric and asymmetric) fuzzy coefficients. Since h-level set for h = 1, the (2) linear fuzzy model fi (X, A˜ i ) with triangular fuzzy coefficients is the same as its center Yi , the inclusion (2) relation yir ∈ [fi (xr )]h reduced to the equality relation yir = fi (xr ). This means that all the input– (2) output pairs should be on the same hyper-plane yir = fi (xr ). If the given input–output pairs do not satisfy this requirement, the linear programming models LPfi − 2 given in (21) and LPgj − 2 given in (22) have no solution. In general, this requirement is not satisfied [5,14,19]. In order to deal with above deficiencies, we further extend asymmetric triangular fuzzy number coefficients to the case of trapezoidal fuzzy number coefficients because trapezoidal fuzzy numbers have higher capability to model more varieties of fuzziness than triangular fuzzy numbers. In this section, we still follow the basic idea of fuzzy regression: to minimize the fuzziness of the linear fuzzy model that includes all the given data. If each fuzzy coefficient A˜ ij in (4) has trapezoidal membership functions, it can be (1) (2) (2) (3) (1) (2) (2) (3) uniquely determined by quadruples (aij , aij , a¯ ij , aij ) of crisp numbers with aij
aij
a¯ ij
aij (see Fig. 3). When all the fuzzy coefficients A˜ ij are trapezoidal, the linear fuzzy model fi (X, A˜ i ) is also trapezoidal. (1) (2) (2) (3) If we denote fi (xr ) as (fi (xr ), fi (xr ), f¯i (xr )fi (xr )), then by the sum and product operations on (1) (2) (2) (3) trapezoidal fuzzy numbers, fi (xr ), fi (xr ), f¯i (xr ) and fi (xr ) are given as follows, respectively: =

n  j =0

(1)

(3)

(aij xj+r − aij xj−r ),

(32)

1
A~ij (aij)

(1) fi (xr )

0 aij(1)

aij(2)

aij(2)

(3)

aij

Fig. 3. Asymmetric trapezoidal fuzzy coefficient A˜ i .

110

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120 (2) fi (xr )

n  (2) (2) = (aij xj+r − a¯ ij xj−r ),

(33)

n  (2) (2) (a¯ ij xj+r − aij xj−r ),

(34)

j =0

(2) f¯i (xr ) =

(3)

fi (xr ) =

j =0

n  j =0

(3)

(1)

(aij xj+r − aij xj−r ).

(35)

The h-level set of the linear fuzzy model fi (xr , A˜ i ) can be expressed as (2) (1) (2) (3) [fi (xr )]h = [hfi (xr ) + (1 − h)fi (xr ), hf¯i (xr ) + (1 − h)fi (xr )].

(36)

On the other hand, according to Ishibuchi and Nii [12], the total fuzziness of the linear model fi (xr , A˜ i ) with trapezoidal fuzzy numbers can be defined as l  (3) (1) (2) (2) Z= [fi (xr ) − fi (xr ) + f¯i (xr ) − fi (xr )]

=

r=1 n l   r=1 j =0

(3)

(1)

(2)

(2)

(aij − aij + a¯ ij − aij )(xj+r + xj−r ).

(37)

Henceforth, the fuzzy regression problem aiming to determine the functional relationships fi (i = 1, 2, . . . , m) with trapezoidal fuzzy numbers is transformed into a linear program LPfi − 3(i = 1, 2, . . . , m): min

Z=

n  j =0

(3)

(1)

(2)

(2)

(aij − aij + a¯ ij − aij )

l 

|xj r |

(38a)

r=1

s.t. h

n  j =0

h

n  j =0

(1)

(2)

(2)

(aij xj+r − a¯ ij xj−r ) + (1 − h) (2)

(2)

(a¯ ij xj+r − aij xj−r ) + (1 − h) (2)

(2)

(3)

aij
aij
a¯ ij
aij , (1)

(2)

n  (1) (3) (aij xj+r − aij xj−r )
yir ,

r = 1, 2, . . . , l, (38b)

n  (3) (1) (aij xj+r − aij xj−r )  yir ,

r = 1, 2, . . . , l, (38c)

j =0

j =0

j = 0, 1, . . . , n. (2)

(38d)

(3)

The constraint condition aij
aij
a¯ ij
aij , j = 0, 1, . . . , n is to keep the trapezoidal shape of each (1)

(2)

fuzzy coefficient (See Fig. 3). Since the h-level set of the linear fuzzy model fi (xr ) = (fi (xr ), fi (xr ), (3) (2) f¯i (xr ), fi (xr )) is an interval even if h = 1, the inclusion relation yir ∈ [fi (xr )]h is always feasible for any data set, namely the above linear programming problem always has a feasible solution. More

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

111

important, while a larger value of h is selected, unnecessary large fuzziness can be avoided efficiently due to this characteristic of trapezoidal fuzzy numbers. Similarly, one can determine function relationships gj (j = 1, 2, . . . , n) with trapezoidal fuzzy coefficients by solving the following linear programming model LPgj − 3(j = 1, 2, . . . , n): min

Z=

n l   (3) (1) (2) (2) (aj u − aj u + a¯ j u − aj u ) |xur |

(39a)

r=1

u=0 u =j

s.t. h

n  u=0 u =j

h

n  u=0 u =j

(1)

(2)

(2)

+ − (aj u xur − a¯ ij xur ) + (1 − h)

(2)

(2)

+ − (a¯ j u xur − aij xur ) + (1 − h)

(2)

(2)

(3)

aj u
aj u
a¯ j u
aj u ,

n  (1) + (3) − (aj u xur − aij xur )
xj r ,

r = 1, 2, . . . , l, (39b)

n  (3) + (1) − (aj u xur − aij xur )  xj r ,

r = 1, 2, . . . , l, (39c)

u=0 u =j

u=0 u =j

u = 0, 1, . . . , j − 1, j + 1, . . . , n.

(39d)

In order to guarantee that LPfi − 3 given in (38) is equivalent to LPfi − 2 given in (21) when we select (2) (2) aij = a¯ ij , j = 0, 1, 2, . . . , n, a hybrid algorithm using fuzzy regression combined with least-squares regression is applied here in the same manner as in the case with asymmetric triangular fuzzy numbers. (2) Let the estimates obtained using least-squares regression be aˆ ij (j = 0, 1, 2, . . . , n) for LPfi − 3, then (2)

(2)

we let the mean value of two peak points of each trapezoidal fuzzy coefficients, i.e. aij and a¯ ij is equal (2)

to aˆ ij , which can be expressed as (2)

(2)

(2)

aij + a¯ ij = 2aˆ ij ,

j = 0, 1, 2, . . . , n.

(40)

The linear programming models LPfi − 3 given in (38) is then transformed into the following hybrid linear programming model LPfi − 4(i = 1, 2, . . . , m): min

Z=

n l   (3) (1) (2) (2) (aij − aij + a¯ ij − aij ) |xj r |

(41a)

r=1

j =0

s.t. h

n  j =0

h

n  j =0

(2)

(2) (aij xj+r

(2) − a¯ ij xj−r ) + (1 − h)

n  (1) (3) (aij xj+r − aij xj−r )
yir ,

r = 1, 2, . . . , l, (41b)

(2) (a¯ ij xj+r

(2) − aij xj−r ) + (1 − h)

n  (3) (1) (aij xj+r − aij xj−r )  yir ,

r = 1, 2, . . . , l, (41c)

(2)

(2)

aij + a¯ ij = 2aˆ ij , (1)

(2)

(2)

(3)

aij
aij
a¯ ij
aij ,

j =0

j =0

j = 0, 1, . . . , n, j = 0, 1, . . . , n.

(41d) (41e)

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Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

Similarly, the linear programming model LPgj − 3 given in (39) is transformed into the following hybrid linear programming model LPgj − 4(j = 1, 2, . . . , n): min

Z=

n  u=0 u =j

(3)

(1)

(2)

(2)

(aj u − aj u + a¯ j u − aj u )

l 

|xur |

(42a)

r=1

s.t h

n  u=0 u =j

h

n  u=0 u =j

(2) + (aj u xur

(2) − − a¯ ij xur ) + (1 − h)

(2)

(2)

+ − (a¯ j u xur − aij xur ) + (1 − h)

(2)

(2)

(2)

aj u + a¯ j u = 2aˆ j u , (1)

(2)

(2)

(3)

aj u
aj u
a¯ j u
aj u ,

n 

(1)

(3)

+ − (aj u xur − aij xur )
xj r ,

r = 1, 2, . . . , l, (42b)

n  (3) + (1) − (aj u xur − aij xur )  xj r ,

r = 1, 2, . . . , l, (42c)

u=0 u =j

u=0 u =j

u = 0, 1, . . . , j − 1, j + 1, . . . , n, u = 0, 1, . . . , j − 1, j + 1, . . . , n. (2)

(42d) (42e)

(2)

Obviously, in LPfi −4, if we select aij = a¯ ij , j = 0, 1, . . . , n, then the constraints and the expression of Z become equivalent to those in LPfi − 2 given in (21), so the trapezoidal fuzzy number is reduced to the asymmetric triangular case. This means that the pair of linear programming models LPfi − 4 and LPgj − 4 can be viewed as an extension of LPfi − 2 and LPgj − 2. Hence, the functional relationships fi (i = 1, 2, . . . , m) and gj (j = 1, 2, . . . , n) with trapezoidal fuzzy numbers resulted from the pair of hybrid linear programming models LPfi − 4 given in (41) and LPgj − 5 given in (42) can be expressed as y˜i = f˜i (X) =

n  (1) (2) (2) (3) (ai0 , ai0 , a¯ i0 , ai0 )xj ,

i = 1, 2, . . . , m,

(43)

j =0

x˜j = g˜ j (Xj ) =

n  (1) (2) (2) (3) (aj u , aj u , a¯ j u , aj u )xu ,

j = 1, 2, . . . , n.

(44)

u=0 u =j

Because the peak points of trapezoidal fuzzy number coefficients in LPfi − 4and LPgj − 4 are not fixed any more, so the effect of the h value on the solution for LPfi − 4 and LPgj − 4 become more complicated than that in LPfi − 2 and LPgj − 2. And the theorem in Section 3 does not hold in such cases. 5. Case study—empirical application To demonstrate the performance of fuzzy linear regression techniques with asymmetric triangular and trapezoidal fuzzy coefficients in product planning, respectively, a case study of product development of emulsification dynamite packing-machine is cited.

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

CRs W CR1 y CR2 y2 CR3 y3 CR4 y4

Unites Comp1 1 Comp2 Comp3 Comp4 Comp5 min Max

EC2 x2

+ EC3 x3

+ EC4 x4

+ EC5 x5

EC6 x6

EC7 x7

• •

Correlation

EC1 EC2 EC3 EC4 EC5 EC6 EC7 eighs 10.46 0.28 0.16 0.10

EC1 x1

• • Relation

•

•

• •

•

• •

mm-2 1 8 12 9 10 6 15

mm-2 N 7 6 9 8 10 4 12

58 65 60 62 65 55 70

ns-1 90 85 70 85 75 60 100

HRC 55 50 50 45 55 40 60

dB 75 68 55 80 70 50 90

2. • m 1.9 1.7 1.8 1.7 1.6 1.5 2

Comp1 3.4 3.1 2 1.6 48.60

Comp2 4 3 3.7 3.7 66.05

Benchmarking information Comp3 Comp4 Comp5 1.9 3.7 3.6 1.8 2.9 3.9 4.3 1.8 3.5 3.3 3.7 4 34.90

54.30

67.70

min 1 1 1 1

max 5 5 5 5 S (%)

Engineering Measures

ECs

113

Fig. 4. House of Quality of emulsification dynamite packing-machine.

5.1. Building a HOQ for the emulsification dynamite packing-machine A corporation is developing a new type of emulsification dynamite packing-machine. According to the survey in the marketplace and feedback from users, four major CRs are identified to represent the biggest concerns of the customers. They are “improve the quality of packing dynamite’’ (CR1 ), “increase efficiency of packing dynamite’’ (CR2 ), “reduce the packing noise’’ (CR3 ) and “increase the rigidity of the machine’’(CR4 ). Their relative weights are determined by analytic hierarchy process (AHP) [2]. The customer perception of the degree of satisfaction of each CR has been scaled from 1(worst) to 5(best) by benchmarking technique. Based on the design team’s experience and expert knowledge on this product, seven ECs are determined, i.e., “improving the precision of the molding of the clip’’ (EC1 ), “improving the precision of the dynamite packing’’ (EC2 ), “increasing the control force of the dynamite packing’’ (EC3 ), “improving the efficiency of the dynamite packing’’ (EC4 ), “increasing the hardness of the pressing hammer’’ (EC5 ), “reducing the noise of the cam power transmission’’ (EC6 ), and “reducing the height of the machine bed’’ (EC7 ). These ECs are measured in units of mm−2 , mm−2 , kgf,
m−1 , HRC, dB and m, respectively. The negative and positive sign on ECs means the design team hope to reduce and increase the target values of ECs. In the mean time, five main competitors, i.e. Comp1 (our corporation), Comp2 , Comp3 , Comp4 , Comp5 are selected. Engineering data have been collected from the company and its main competitors. According to design experience and engineering knowledge, the design team identified the relationship between the CRs and the ECs as well as those among the ECs, which are marked by the symbol “•’’. The HOQ of emulsification dynamite packing-machine is shown in Fig. 4. 5.2. Normalizing the target values of ECs The level of attainment xj (j = 1, 2, . . . , n) used in fuzzy regression can be obtained by normalizing the target values of ECs, i.e. changing lj (j = 1, . . . , n), the current target value of the ECj , into the level

114

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

of attainment xj (j = 1, . . . , n), such that 0
xj
1(j = 1, . . . , n). Target values of ECs for a product can be positive or negative [11]. With the positive ones, the performance of EC is positively proportional to the target value of EC, and vice versa with the negative ones. For example, the design team hopes to reduce the energy required to close the door of a car, and to increase the water resistance of the door. The two categories of target values of ECs can be normalized according to Eqs. (45) and (46), respectively: xj = xj =

ljmax − lj ljmax − ljmin lj − ljmin ljmax − ljmin

,

(45)

,

(46)

where ljmax and ljmin can be determined by the consideration of competition requirement and technology feasibility [28]. For the first category of ECs, ljmax is maximum target value of the ECj to match competitors’ performance, and ljmin is the minimum obtainable. Whereas, for the second category of ECs, ljmin is minimum target value of the ECj to match competitors’ performance, and ljmax is the maximum obtainable. As indicated in Fig. 4, EC3 , EC4 and EC5 are positive, and EC1 , EC2 , EC6 and EC7 are negative. Thus, the target values of EC3 , EC4 and EC5 of five competitors are normalized by using (45), and the target values of EC1 , EC2 , EC6 and EC7 are normalized by using (46), which are given as follows:   0.44 0.63 0.20 0.75 0.75 0.38 0.20  0.78 0.75 0.67 0.63 0.50 0.55 0.60     X=  0.33 0.38 0.33 0.25 0.50 0.88 0.40  .  0.67 0.50 0.47 0.62 0.25 0.25 0.60  0.56 0.25 0.67 0.38 0.75 0.50 0.80 5.3. Comparative study Once the HOQ of emulsification dynamite packing-machine is established, the problem at hand is to obtain the coefficients in the linear relationships between the four CRs and the seven ECs as well as those among the ECs by solving fuzzy linear regression models with asymmetric triangular and trapezoidal fuzzy coefficients. In this subsection, a comparative study is made from an illustrative point of view. The identification of the linear relationship between y1 and x1 , x2 and x3 (see Fig. 4) is used to perform the comparison of the fuzzy linear regression with asymmetric triangular and trapezoidal fuzzy coefficients. In order to examine how h influence the solution, several values for h are selected and the corresponding results for LPfi − 2 in (21) and LPfi − 4 in (41) are summarized in Tables 1 and 2, respectively. From Table 1, it is easy to verify that the theorem proposed in Section 3 is correct, i.e. the values of left spread aijL and right spread aijR of each A˜ ij and objective function Z becomes (1 − h1 )/(1 − h2 ) times simultaneously when the h value changes from h1 to h2 . It is also clearly shown in Table 1 that when a larger value of h is selected, some estimated parameters with very large aspiration and larger unnecessary systematic fuzziness are obtained. For example, if h is quality to 0.99, A˜ 10 described by (1) (2) (3) (1) (2) (3) (a10 , a10 , a10 ) and A˜ 12 described by (a12 , a12 , a12 ) are given as (−42.2692, 1.2872, 32.2128) and

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

115

Table 1 The influence of the h on solutions with asymmetric triangular fuzzy numbers h=0 (1) a10 (2) a10 (3) a10 (1) a11 (2) a11 (3) a11 (1) a12 (2) a12 (3) a12 (1) a13 (2) a13 (3) a13

Z

h = 0.3

h = 0.5

h = 0.8

h = 0.99

0.8516

0.6650

0.4161

−0.8906

−42.2692

1.2872

1.2872

1.2872

1.2872

1.2872

1.5965

1.7290

1.9057

2.8335

32.2128

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

−0.8939

−0.8939

−0.8939

−0.8939

−0.8939

−0.8939

−0.8939

−0.8939

−0.8939

−0.8939

−0.6934

−0.6074

−0.4928

0.1087

19.1590

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

4.2273

6.0392

8.4549

21.1371

422.7426

(−0.8939, −0.8939, 19.1590), respectively. The total systematic fuzziness, i.e., Z value, is 422.7426, which is 100 times that of h = 0, i.e., 4.2273. Correspondingly, this will result in the predictive fuzzy interval being too wide to provide any useful information for decision-making. For example, if x1 , x2 (1) (2) (3) and x3 are set as 0.78, 0.75 and 0.67, respectively, the estimation of y˜1 described by (y1 , y1 , y1 ) is given as (−39.4802, 4.0762, 50.0415). Obviously, it is a bad prediction that cannot be interpreted appropriately and make no sense for practical purpose. However, as shown in Table 2, while the linear fuzzy regression with trapezoidal fuzzy number coefficients is applied, the estimation of y˜1 described by (1) (2) (2) (3) (y1 , y1 , y¯1 , y1 ) is given as (3.4158,3.8655,4.2869,4.7848) and Z value is 8.7115 when the same value of h is selected, which are more reasonable and appreciable. As we can see, when a larger value of h is selected, the linear fuzzy regression with triangular fuzzy number coefficients tends to produce the estimated parameters with very large aspiration and large unnecessary systematic fuzziness, which leads to the predictive fuzzy interval too wide to provide any useful information. Such problem can be avoided effectively when trapezoidal fuzzy numbers as fuzzy coefficients are used. This is because the trapezoidal fuzzy numbers have higher capability to model more varieties of fuzziness than triangular fuzzy numbers, and h-level set of the linear fuzzy regression with trapezoidal is an interval even if h = 1. The extension of asymmetric triangular fuzzy coefficients to trapezoidal fuzzy coefficients further increases the flexibility of the linear fuzzy regression. Generally, when fuzzy linear regression is used to estimate functional relationships between CRs and ECs as well as those among ECs, the value of h is subjectively pre-selected by a design team according to their available engineering knowledge. Tanaka and Watada [19] suggested that the selection of the h value should be based on the sufficiency of the collected data set. When the data set is sufficiently large, h = 0 should be used and is increased along with the decreasing volume of the collected data. On the

116

Richard Y.K. Fung et al. / Fuzzy Sets and Systems 157 (2006) 98 – 120

Table 2 The influence of the h on solutions with trapezoidal fuzzy numbers h=0 (1) a10 (2) a10 (2) a¯ 10 (3) a10 (1) a11 (2) a11 (2) a¯ 11 (3) a11 (1) a12 (2) a12 (2) a¯ 12 (3) a12 (1) a13 (2) a13 (2) a¯ 13 (3) a13

Z

h = 0.3

h = 0.5

h = 0.8

h = 0.99

1.2872

1.2872

0.6268

0.7531

0.8517

1.2872

1.2872

1.0765

0.9635

0.8517

1.2872

1.2872

1.4979

1.6109

1.7227

1.5965

1.7290

1.6950

1.6823

1.7227

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

5.8297

−0.8939

−0.8939

−0.8939

−0.8940

−0.8940

−0.8939

−0.8939

−0.8939

−0.8940

−0.8940

−0.8939

−0.8939

−0.8939

−0.8938

−0.8938

−0.6934

−0.6074

−0.4928

−0.8938

−0.8938

−5.9238

−3.6211

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

−1.6235

12.1124

7.6023

8.4549

7.7845

8.7115

other hand, Moskowitz and Kim [14] also suggested that when we are pessimistic about the data that we collect, a large value of h is needed. As we know, it would be very difficult to obtain sufficient data set for the implementation of QFD in most cases because of commercial secrets of competitors or high cost of data collection, such as the expensive conduction of engineering experiment to determine the target values of ECs of main competitors. Therefore, a larger value of h should be selected for the estimation of functional relationships in QFD. In such cases, the performance of trapezoidal fuzzy coefficients would be better than triangular fuzzy coefficients. As shown in Fig. 4, because the volume of obtained data set for product planning of emulsification dynamite packing-machine is small, a larger value of h should be selected to estimate the relationships between the four CRs and the seven ECs as well as those among the seven ECs. If the value of h is specified as 0.9, by solving LPfi − 2 and LPgj − 2, or LPfi − 4 and LPgj − 4, i = 1, 2, . . . , m, j = 1, . . . , n, the fuzzy coefficients of f1 , f2 , f3 , f4 and g2 , g4 , g6 can be determined in the form of asymmetric triangular or trapezoidal fuzzy coefficients as shown in Tables 3 and 4, respectively. Because x1 , x3 , x5 and x7 are not correlated with other ECs (see Fig. 4), therefore g1 , g3 , g5 and g7 are zero. From Table 3 one can see that the aspirations of some triangular fuzzy coefficients are extremely wide, such as (−7.42, 1.29, 7.49), (−12.58, 2.35, 9.05), (− 7.21, 1.25, 11.33) and (0.75, 0.75, 5.31) and so on, which are not useful, obviously. However, the aspirations of trapezoidal fuzzy coefficients in Table 4 are relatively small and more reasonable for practical purpose. Therefore, by incorporating the functional

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117

Table 3 The assessed fi and gj with asymmetric triangular fuzzy numbers (h = 0.9)

Intercept x1 x2 x3 x4 x5 x6 x7

Intercept x1 x2 x3 x4 x5 x6 x7

y1

y2

y3

y4

(−7.42, 1.29, 7.49) (5.83, 5.83, 5.83) (−0.89, −0.89, 3.12) (−1.62, −1.62, −1.62)

(−0.14, −0.14, −0.14)

(−6.13, 0.99, 9.98)

(−7.21, 1.25, 11.33)

(−12.58, 2.35, 9.05) (1.74, 1.74, 8.58) (1.93, 1.93, 1.93)

(4.12, 4.12, 4.12)

x2

x4

x6

(−2.75, 0.11, 0.64)

(−0.41, 0.48, 0.48)

(−0.77, 0.98, 3.25)

(3.87, 3.87, 3.87)

(0.63, 0.63, 2.94) (0.75, 0.75, 5.31)

(−3.79, −0.90, −0.33) (−0.53, −0.53, −0.53)

Table 4 The assessed fi and gj with trapezoidal fuzzy numbers (h = 0.9) y1 Intercept (0.85, 0.85, 1.72, 1.72) x1 (5.83, 5.83, 5.83, 5.83) x2 (−0.89, −0.89, −0.89, −0.49) x3 (−1.62, −1.62, −1.62, −1.62) x4 x5 x6 x7 x2

y2

y3

(−0.14, −0.14, −0.14, −0.14) (0.54, 0.54, 1.44, 1.44)

y4 (0.75, 0.75, 1.75, 1.75)

(1.70, 1.70, 3.00, 3.00) (1.64, 1.64, 1.84, 1.84) (1.93, 1.93, 1.93, 1.93)

x4

(4.12, 4.12, 4.12, 4.12)

(3.87, 3.87, 3.87, 3.87)

x6

Intercept (0.11, 0.11, 0.11, 0.14) (0.39, 0.48, 0.48, 0.48) (0.70, 0.70, 1.27, 1.27) x1 x2 (0.52, 0.52, 0.75, 0.75) x3 x4 (−0.15, −0.15, 1.64, 1.64) (−0.90, −0.90, −0.90, −0.84) x5 x6 (−0.53, −0.53, −0.53, −0.53) x7

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relationships described in Table 4 into the product-planning model given in (1), the product planning model for emulsification dynamite packing-machine can be obtained. The same phenomenon aforementioned is also observed when we apply fuzzy linear regression to model a spin flash drying process. The goal of the project is to establish an empirical model between the key quality characteristic of the process, i.e., granularity of drying material (
m), and the three significant process parameters, namely the radius of classifier (mm), the amount of blast air (m3 /s) and the amount of pumping air (m3 /s). Since only 10 modeling data were acquired based on engineering experiment, so a higher h value is selected. When triangular fuzzy number coefficients are used, the obtained process model cannot be used for predictive and auto-controlling purposes when h  0.7 because some fuzzy coefficients have larger aspirations, vice versa, the results are more appreciable when trapezoidal fuzzy number coefficients are used. Therefore, whether fuzzy linear regression is applied in QFD or other engineering problem, when the volume of sample data set is small, in order to establish a well-estimated fuzzy linear regression model to support a better prediction and provide more useful information for decision makers, we recommend that a larger value of h should be selected. Accordingly, trapezoidal fuzzy coefficients seem more appealing than triangular fuzzy coefficients in such cases. It should be pointed out that in Table 3 some coefficients are crisp, and in Table 4 the coefficients tend to be triangular and some times become crisp because of the characteristic of linear programming. In fact, when fuzzy linear regression analysis is applied to the estimated functional relationships in product planning, the more non-crisp coefficients we have, the better result we have since the inherent fuzziness of functional relationships in product planning can be interpreted by many coefficients. Therefore, in order to overcome this crisp characteristic of linear programming, non-linear fuzzy regression approaches, such as quadratic programming, should be further considered to estimate functional relationships in product planning.

6. Conclusions 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.

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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.

Acknowledgements The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU1028/99E), the National Nature Science Foundation of Chinese (Project No. NSFC70471028), and the funds of ‘Program for New Century Excellent Talents in University (NCET) of MOE’. References [1] Y. Akao, Quality Function Deployment: Integrating Customer Requirements into Product Design (translated by Glenn Mazur), Productivity Press, Cambridge, MA, 1990. [2] R.L. Armacost, P. Componation, M. Mullens, W. Swart, An AHP framework for prioritizing customer requirements in QFD: An industrialized housing application, IIE Trans. 26 (4) (1994) 72–79. [3] L.K. Chan, M.L. Wu, Quality function deployment: A literature review, European J. Oper. Res. 143 (2002) 463–497. [4] Y. Chen, J. Tang, R.Y.K. Fung, Z. Ren, Fuzzy regression-based mathematical programming model for quality function deployment, Internat. J. Prod. Res. 42 (5) (2004) 1009–1027. [5] J.P. Dunyaka, D. Wunsch, Fuzzy regression by fuzzy number neural networks, Fuzzy Sets and Systems 112 (2000) 371–380. [6] R.Y.K. Fung, D.S.T. Law, W.H. Ip, Design targets determination for inter-dependent product attributes in QFD using fuzzy inference, Internat. J. Integr. Manuf. Systems 10 (6) (1999) 376–383. [7] R.Y.K. Fung, K. Popplewell, J. Xie, An intelligent hybrid system for customer requirements analysis and product attribute targets determination, Internat. J. Prod. Res. 36 (1) (1998) 13–34. [8] R.Y.K. Fung, J. Tang, Y. Tu, D. Wang, Product design resource optimization using a non-linear fuzzy quality function deployment model, Internat. J. Prod. Res. 40 (3) (2002) 585–599. [9] R.Y.K. Fung, J. Tang, P.Y. Tu, Y. Chen, Modeling of quality function deployment planning with resource allocation, Res. Engrg. Design 14 (2003) 247–255. [10] J.A. Harding, K. Popplewell, R.Y.K. Fung, A.R. Omar, An intelligent information framework for market driven product design, Comput. Ind. 44 (1) (2001) 51–65. [11] J.R. Hauser, D. Clausing, The house of quality, Harvard Business Rev. May–June (1988) 63–73. [12] H. Ishibuchi, M. Nii, Fuzzy regression using asymmetric fuzzy coefficients and fuzzified neural networks, Fuzzy Sets and Systems 119 (2001) 273–290. [13] K.J. Kim, H. Moskowitz, A. Dhingra, G. Evans, Fuzzy multicriteria models for quality function deployment, European J. Oper. Res. 121 (2000) 504–518. [14] H. Moskowitz, K.J. Kim, On assessing the H value in fuzzy linear, Fuzzy Sets and Systems 58 (1993) 303–327.

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