Fuzzy measurable house of quality and quality function deployment for fuzzy regression estimation problem

Expert Systems with Applications 38 (2011) 14398–14406

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Fuzzy measurable house of quality and quality function deployment for fuzzy regression estimation problem Qi Wu ⇑ School of Mechanical Engineering, Southeast University, Nanjing, Jiangsu 210096, China Key Laboratory of Measurement and Control of CSE (School of Automation, Southeast University), Ministry of Education, Nanjing, Jiangsu 210096, China

a r t i c l e

i n f o

Keywords: Product design Time estimation House of quality Fuzzy measure Fuzzy support vector regression machine

a b s t r a c t In present competitive environment, it is necessary for companies to evaluate design time and effort at the early stage of product development. However, there is somewhat lacking in systemic analytical methods for product design time (PDT). For this end, this paper explores an intelligent method to evaluate the PDT. At the early development stage, designers are short of sufficient product information and have difficulty in determining PDT by subjective evaluation. Thus, a fuzzy measurable house of quality (FM-HOQ) model is proposed to provide measurable engineering information. Quality function deployment (QFD) is combined with a mapping pattern of ‘‘function ? principle ? structure’’ to extract product characteristics from customer demands. Then, a fuzzy support vector regression machine (FSVRM) model is built to fuse data and realize the estimation of PDT, which makes use of fuzzy comprehensive evaluation to simplify structure. In a word, the whole estimation method consists of four steps: time factors identification, product characteristics extraction by QFD and function mapping pattern, FSVRM learning, and PDT estimation. Finally, to illustrate the procedure of the estimation method, the case of injection mold design is studied. The results of experiments show that the fuzzy method is feasible and effective. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction As global competition increases and product life cycle shortens, companies try to employ effective management to accelerate product development. However, product development projects are often suffered with schedule overruns. In most cases, problems of overruns were due to poor estimations. That is coincident with the saying ‘‘you cannot control what you do not measure’’ (DeMarco, 1998). In the whole product development process (PDP), product design is an important phase. The control and decision of product development is based on the pre-estimation of product design time (PDT). Nevertheless, PDP always means the brand-new or modified product design. Thus the cycle time of design process cannot be measured directly. Much attention has been focused on reducing the time/cost in product design, but little systematic research has been conducted into the time estimation. Traditionally, approximate design time is determined empirically by designers in companies. With the increase of market competition and product complexity, companies require more accurate and creditable solutions. Recently, a small number of researches have dealt with the estimation of design time and effort. These existing approaches all be⇑ Address: School of Mechanical Engineering, Southeast University, Nanjing, Jiangsu 210096, China. Tel.: +86 25 51166581; fax: +86 25 511665260. E-mail address: [email protected] 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.04.095

long to the factor analytical method. Using traditional regression analysis, Bashir and Thomson (2001) propose two types of parametric models: a single-variable model based on product complexity, and a multivariable model based on product complexity and severity of requirements. As other factors have not been considered in these two models, the practicability and accuracy are suspectable. Griffin (1997a, 1997b) relates the product development cycle time to factors of project, process and team structure with a statistical method, and quantitatively analyzes the impact of the project novelty and complexity on cycle time. Nevertheless, he does not present an effective method for estimating the design time. Jacome and Lapinskii (1997) present a model for estimating effort for electronic design which takes into account three major factors: size, complexity and productivity. However, this model is applicable only for effort estimation for electronic design. Therefore, there is a demand for more systematic and general methods, which can be applied to a wide range of engineering design projects. For those nonlinear systems that have many uncertainties, there are no precise mathematic models. Fortunately, adopting intelligent technologies, such as neural network and fuzzy logic, is sometimes a good choice. Jahan-Shahi, Shayan, and Masood (2001) use multivalued fuzzy sets to model the activity time/cost estimation in flat plate processing. Based on neural networks, Seo, Park, Jang, and Wallace (2002) present an approximate method to provide the product life cycle cost in conceptual design.

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Recently, a novel machine learning technique, called support vector machine (SVM), has drawn much attention in the fields of pattern classification and regression estimation. SVM was first introduced by Vapnik (1995). It is an approximate implementation to the structure risk minimization (SRM) principle in statistical learning theory, rather than the empirical risk minimization (ERM) method. This SRM principle is based on the fact that the generalization error is bounded by the sum of the empirical error and a confidence interval term depending on the Vapnik–Chervonenkis (VC) dimension. By minimizing this bound, good generalization performance can be achieved. Compared with traditional neural networks, SVM can obtain a unique global optimal solution and avoid the curse of dimensionality. These attractive properties make SVM become a promising technique (Acır, Özdamar, & Güzelisß, 2006; Bergeron, Cheriet, Ronsky, Zernicke, & Labelle, 2005; Colliez, Dufrenois, & Hamad, 2006; Frias-Martinez, Sanchez, & Velez, 2006; Goel & Pal, 2009; Huang, Lai, Luo, & Yan, 2005; Mohammadi & Gharehpetian, in press; Osowski & Garanty, 2007; Samanta, Al-Balushi, & Al-Araimi, 2003; Übeyli, 2008; Vong, Wong, & Li, 2006; Wu, 2009; Wu, Yan, & Yang, 2008a, Wu, Yan, & Yang, 2008b). SVM was initially designed to solve pattern recognition problems (Acır et al., 2006; Frias-Martinez et al., 2006; Mohammadi & Gharehpetian, in press; Samanta et al., 2003; Übeyli, 2008). Recently, with the introduction of Vapnik’s e-insensitive loss function, SVM has been extended to function approximation and regression estimation problems (Bergeron et al., 2005; Colliez et al., 2006; Goel & Pal, 2009; Huang et al., 2005; Osowski & Garanty, 2007; Vong et al., 2006; Wu, 2009; Wu et al., 2008a, 2008b). In many real applications, the observed input data cannot be measured precisely and usually described in linguistic levels or ambiguous metrics. However, traditional support vector machine (SVM) method cannot cope with qualitative information. It is well known that fuzzy logic is a powerful tool to deal with fuzzy and uncertain data. For this end, this paper develops a time estimation method for the product remodeling design, which is based on fuzzy logic and support vector regression machine. There is a kind of nonlinear mapping relationship between engineering factors and PDT. SVRM can perform this mapping well. Fuzzy inference theory is introduced to handle the fuzzy input variables. Product characteristics are important parts of engineering factors. As the product characteristics are not available before a product design project begins, this paper attempts to extract product characteristics from customer demands using quality function deployment (QFD) and function mapping methodology. Therefore, the whole estimation method includes three steps: characteristic extraction, support vector machine learning and time estimation. The proposed FSVRM can solve the estimating problem of uncertain fuzzy system. The input and output of the proposed FSVRM are fuzzy numbers. In this paper, we put forward a new fuzzy inference theory. Based on the fuzzy inference theory and Fm-SVRM, an estimation method for product design time is proposed. The rest of this paper is organized as follows: in Section 2, PDT factors are identified firstly; Section 3 describes a new house of quality (HOQ) model and introduces a mapping methodology to extracting product characteristics; in Section 4, fuzzy support vector regression machine is presented to realize the estimation of PDT; Section 5 presents an example to illustrate the estimation method; and Section 6 is the conclusion.

2. Time factors identification The PDT estimation method requires a careful selection and identification of the design variables that are related to design time. Therefore, time factors should be confirmed before extracting product characteristics. Many research efforts have been undertaken on the factors of PDP cycle time, but few have done on the

Product Characteristics Structure Size Shape Added demands

Project Complexity Technical difficulty Parts amount Characteristic amount Uncertainty

Design Process Standardization Process control Concurrency

Design Condition Design tools Management support Available data

Design Team Collaboration Individual experience Individual skill Dedicated spirit

Original factor set

Information Process Capability Timeliness Extent

Product Design Time

Motivation Goal explicitness Goal congruence Linked rewards

Transitional factor set

Target

Fig. 1. Conceptual model of factors that influence design time.

product design time. In order to identify the PDT factors, all possible influencing elements incurred in the design process should be investigated and enumerated. Based on the models described for product development cycle time this paper proposes a conceptual model for the relationships between product design time and different factors, as shown in Fig. 1. In Fig. 1, the original factor set affects the PDT target indirectly via a transitional factor set. The transitional factor set is composed of some factors that are unobvious and difficult to be measured or evaluated. Therefore, only the original factor set is acquirable and will be taken into account in this paper. The original factor set can be sorted into four main subsets: product characteristics, design process, design condition and design team. Here the nonlinear mapping relationship between the original factor set and PDT is realized by a fuzzy support vector regression machine, which will be proposed in the latter part of the paper. In the original factor set, the factors of latter three subsets can be evaluated directly, while product characteristics must be gained by transforming customer demands before a design process begins. Different types of products have distinct product characteristics. For a specific kind of product, a list of time factors with influencing weights can be determined by analyzing pre-existing design projects. Table 1 provides a sample of time factor set chosen for the design of a gear speed reducer. 3. Product characteristics extraction QFD is a concept and mechanism for translating the voice of customers (WHATs) into quality characteristics (HOWs) through various stages of product planning, engineering and manufacturing (Prasad, 1998). In the QFD process, a matrix called the house of quality (HOQ) is used to illustrate the complex relationships between WHATs and HOWs. During the QFD transformation, the HOQ is then developed to demonstrate how the quality characteristics satisfy the customer demands (Hauser & Clausing, 1988). Initially, QFD is mainly used to improve product quality and development process. From 1990s, QFD methodology has been extended and can be applied to many specific problems (Cristiano, Liker, & White, 2001; Tsai & Chang, 2004). In this research, HOQ is employed to construct a framework, which helps us to achieve the extraction of product characteristics in the conceptual design or preliminary design stage. 3.1. Fuzzy measurable HOQ Since there are many subjective and ambiguous evaluations in QFD process, some researchers begin to integrate fuzzy logic

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Table 1 PDT factors of a gear speed reducer.

Design process

Design condition

Design team

methods with QFD (Cristiano et al., 2001). However, these existing works just partially apply fuzzy theory, such as fuzzy logic-based requirements analysis (Temponi, Yen, & Tiao, 1999), fuzzy multicriteria models for QFD (Kim, Moskowitz, Dhingra, & Evans, 2000), fuzzy ranking procedure (Zhou, 1998) and fuzzy model for deriving optimum targets (Vanegas & Labib, 2001). Using traditional HOQs, we can only obtain qualitative information of product characteristics, while measurable values will be grateful and helpful in PDT estimation. Therefore, we add fuzzy measures and fuzzy measure relationships to traditional HOQs, and propose a fuzzy measurable house of quality (FM-HOQ) model, as shown in Fig. 2.

mm Dimensionless kW N.m % ° (Degree) lm Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless

Numerical information Numerical information Numerical information Numerical information Numerical information Numerical information Numerical information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information Linguistic information

0.75 0.80 0.80 0.75 0.50 0.60 0.60 0.55 0.50 0.60 0.35 0.25 0.50 0.75 0.75 0.50 0.60 0.75 0.50 0.90 0.75 0.60

C orrelation m atrix Techn ical custom er d em an ds

Definition 1. Let Q denote the demand domain, which is a collection of all customer demands for a certain product family. Let P denote the characteristic domain, which is a collection of all technical characteristics for a certain product family. Definition 2. Let WQ denote the demand weight domain. Let WP denote the characteristic weight domain. e is a fuzzy For a given design project, demand weight set B Q subset of W , and characteristic weight set e E is a fuzzy subset of n o P e e e ~; leðq ~Þ; q ~ 2 WQ , W . B and E can be expressed as: B ¼ ½q B n o e ~Þ denotes the membership ~; leðp ~Þ; p ~ 2 W P , where leðq E ¼ ½p E

e ~ in B. value of demand weight q

B

leE ðp~Þ denotes the membership

~ in e value of characteristic weight p E. e and e The membership values of B E are represented by fuzzy e numbers defined in [0, 1]. Since domains WQ and WP are finite, B and e E can be represented respectively by such fuzzy membership e and ½ e matrices as ½ B E,

E ngineerin g characteristics

Fuzzy weights

Size (Length, Width, Height) Reduction ratio Power Transmission torque Transmission efficiency Transmission precision Manufacturing precision Airproof capability Lubrication type Reliability Modularity Disassemblability Standardization Process control Concurrency Design tools Management support Available data Collaboration Individual experience Individual skill Dedicated spirit

Weight

F uzzy relatio nship m atrix

Fuzzy measures

Factors

Product characteristics

Expression

Demand

Factor subsets

Unit

Demand

Time factor set

F uzzy transition m atrix

F uzzy w eights of characteristics F uzzy m easures of characteristics Fig. 2. The structure of FM-HOQ.

~d ; p ~i Þ; ½Rdi ¼ r di ¼ lR ðq

d ¼ 1; 2; . . . ; m;

i ¼ 1; 2; . . . ; n0 ;

ð3Þ

~d 2 W Q and p ~i 2 W P . where q e can be considered as a relationship of itself Any fuzzy set, e.g. B, on its domain. According to the synthesis theory of fuzzy relationships (Dubois & Prade, 1980), characteristic weight set e E can be e and R, i.e.: e e  R. Then, we formalized as a composition of B E¼B can have

h

i

ð1Þ

leE ðp~i Þ ¼ leB R ðp~i Þ ¼ sup min leB ðq~d Þ; lR ðq~d ; p~i Þ ; i ¼ 1; 2; . . . ; n0 :

ð2Þ

ð4Þ

where m is the number of demands and n is the number of ~d Þ and leðp ~i Þ denote the grades of membership characteristics. leðq B E of demand weights and characteristic weights. In Q  P, the relationships between fuzzy demand weights and fuzzy characteristic weights can be defined by fuzzy relationship matrix [R], as follows:

For those measurable demands and characteristics, there is a kind of mapping relationships between the demand measure sets and characteristic measure sets. Fuzzy relationship matrix [R] just reflects the degree of correlativity between demands and characteristics, but does not reveal the mapping relationships of measures.

e ¼ l ðq ~d 2 W Q ; ½ B d eB ~d Þ; d ¼ 1; 2; . . . ; m; q ~i Þ; i ¼ 1; 2; . . . ; n0 ; p ~i 2 W P ; ½e Ei ¼ leðp E

0

16d6m

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In order to deal with the different types of measures, a unified measurement scheme for demands and characteristics is established. A fuzzy measure set M consists of r subsets Mu(u = 1, 2, . . . , r), associated with r description levels. Here we consider the two domains Q and P. The fuzzy subsets M Bu of Q and fuzzy subsets M Eu of P corresponding to demand measures and characteristic measures in Mu can be represented as

i qM ; lMBu ðqM Þ ; nh i M Eu ¼ pM ; lMEu ðpM Þ ;

M Bu ¼

nh

o u ¼ 1; 2; . . . ; r; qM 2 Q ; o u ¼ 1; 2; . . . ; r; pM 2 P ;

ð5Þ ð6Þ

where lMBu ðqM Þ and lMEu ðpM Þ denote the grades of membership of demand measures and characteristic measures. The demand measure set and characteristic measure set can be represented respectively by fuzzy membership matrices [MB] and [ME], denoted as follows:

½MB ud ¼ lMBu ðqM d Þ; E

½M ui ¼ l

M MEu ðpi Þ;

u ¼ 1; 2; . . . ; r; d ¼ 1; 2; . . . ; m0 ; qM d 2 Q; 00

u ¼ 1; 2; . . . ; r; i ¼ 1; 2; . . . ; n ;

pM i

2 P;

ð7Þ ð8Þ

where m0 is the number of measurable demands, n0 is the number of measurable characteristics, m0 6 m and n00 6 n0 . The mapping relationships between measurable demands and measurable characteristics can be denoted by fuzzy transition matrix [A], expressed as M ½Adi ¼ adi ¼ lA ðqM d ; pi Þ;

d ¼ 1; 2; . . . ; m0 ; i ¼ 1; 2; . . . ; n00 :

ð9Þ

E

The characteristic measure set M can also be formalized as a composition of MB and A, i.e., ME = MBA. Thus we can have

h

i

lMEu ðpMi Þ ¼ lMBu A ðpMi Þ ¼ sup min lMBu ðqMd Þ; lA ðqMd ; pMi Þ ; 16d6m0

00

i ¼ 1; 2; . . . ; n :

ð10Þ

The fuzzy characteristic weights of e E can be used to obtain the degrees of importance by ranking. Each characteristic measure of ME obtained by computing consists of r fuzzy subsets and should be defuzzified. 3.2. FM-HOQ-based product characteristic mapping

FM-HOQ is proposed. Customer demands are composed of functional ones and technical ones. For technical demands, FM-HOQs are applied to mapping and measuring characteristics, in the decomposing idea of QFD. This is an interactive process performed by a multifunctional team. Planning FM-HOQ, design FM-HOQ and operating FM-HOQ are constructed before the design project begins. For functional demands, a mapping pattern of ‘‘function ? principle ? structure’’ is adopted (refer to the design methodology (Huang, 1992) and axiomatic design (AD) theory (Suh, 1990)). The mapping process begins from the input of function expression and forms the process of function decomposition by such three views as principle ones, functional ones and structure ones. Each function corresponding to a functional demand can be decomposed into sub-functions each of which can also be decomposed, so forth till function units. By studying the working principles of these function units, we can find the corresponding principle components. Principle components (Huang, 1992) are the carriers of relations between functions and structures. Geometrical and shape information of product parts are included in principle components. By combining these principle components, the functional tree is mapped into the structure tree. Then the product structure can be determined. For the design project of remodeling products, new functional demands are few and reference data are available. Then the principle solving process is relatively easy. The whole mapping processes of technical and functional demands form the framework of characteristic extraction, as shown in Fig. 3. There are some connections between the process of functional decomposition and that of technical decomposition. The functional decomposition process is restricted by basic engineering characteristics, while the component and process characteristics need the structure information. From the mapping framework, we can obtain the product characteristics that are defined on the list of time factors. 4. Fuzzy m-support vector regression machine (Fm-SVRM) 4.1. Preliminary definitions Suppose M 2 T(R) is triangular fuzzy number (TFN) in triangular fuzzy space, whose membership function is represented as follows:

In order to find out product characteristics in the early stage of the product development, a systematic mapping method based on

Principle view Structure info

Functional view

Structure view

Product characteristics

Customer demands

Functional demands

Engineering characteristics

Technical demands

Component characteristics Process characteristics

Planning FM-HOQ Design FM-HOQ Operating FM-HOQ Fig. 3. Characteristic extraction framework based on QFD.

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8 xaM ; aM 6 x < r M ; > < rM aM x ¼ rM ; lM ðxÞ ¼ 1; > : xbM ; r M 6 x < bM ; r M bM

ðÞ

ð11Þ

where aM 6 rM < bM, aM, rM, bM 2 R, aM 6 x < bM, x 2 R. Then we have the formulation M = (aM, rM, bM) in which rM is the center, aM is the left boundary and bM is the right boundary. The standard triangular fuzzy number is difficult to deal with input variables of SVRM, the extended version is considered and described as following: Definition 3 (Extended triangular fuzzy number (ETFN) ). M ¼ ðr M ; Dr M ; DrM Þ is extended triangular fuzzy number (ETFN) in which rM 2 R is the center, DrM = rM  aM is the left spread and DrM ¼ bM  rM is the right spread. Let A ¼ ðr A ; Dr A ; Dr A Þ and B ¼ ðrB ; DrB ; DrB Þ be two ETFNs, whose k-cuts are shown in Fig. 4. In the space T(R) of all ETFNs, we define linear operations by the extension principle: A þ B ¼ ðr A þ r B ; maxðDr A ; DrB Þ; maxðDr A ; DrB ÞÞ; kA ¼ ðkr A ; Dr A ; Dr A Þ if k P 0; kA ¼ ðkr A ; Dr A ; DrA Þ if k < 0, and A  B ¼ ðr A  r B ; maxðDr A ; Dr B Þ; maxðDr A ; Dr B ÞÞ.

where C > 0 is a penalty factor, nki ðk ¼ 1; 2; i ¼ 1; . . . ; lÞ are slack variables and v 2 (0, 1] is an adjustable regularization parameter. ðÞ Parameters C; nki ; v ; e and wj in Eq. (13) are not a fuzzy number but a crisp numbers. Problem (13) is a quadratic programming (QP) problem, whose e-insensitive tube and structure is shown in Fig. 5. By introducing Lagrangian multipliers, a Lagrangian function can be defined as follows: (see Fig. 6). " # 2 X l  1 1X 2 2 ðÞ  ðÞ ðÞ Lðw; b; e; n ; a ; b; g Þ ¼ ðn þ nki Þ kwk þ b þ C me þ 2 l k¼1 i¼1 ki 

2 X l X ðgki nki þ gki nki Þ  be k¼1 i¼1

þ

l X



a1i ðyi þ Dryi Þ  ðw  /ðrxi Þ þ b þ qðDrxi ÞÞ

i¼1

 e  n1i  þ

l X



a1i ðw  /ðrxi Þ þ b þ qðDrxi ÞÞ

i¼1

 ðyi þ Dryi Þ  e  n1i þ

l X





a2i ðyi  Dryi Þ  ðw  /ðrxi Þ þ b  qðDrxi ÞÞ

i¼1

4.2. Fv-SVRM

 e  n2i  þ

l X



a2i ðw  /ðrxi Þ þ b  qðDrxi ÞÞ

i¼1

Suppose a set of fuzzy training sample set fðxi ; yi Þgli¼1 , where xi ¼ ðr xi ; Dr xi ; Drxi Þ 2 TðRÞd and yi ¼ ðr yi ; Dr yi ; Dr yi Þ 2 TðRÞ. T(R)d is the set with d dimensional vectors in ETFN space. For computational simplicity, only symmetric triangular fuzzy numbers are taken into account, i.e. xi ¼ ðrxi ; Dr xi Þ; yi ¼ ðr yi ; Dr yi Þ where Dr xi ¼ Dr xi , r yi ¼ Dryi . We consider the approximation function f(x) = w  x + b, b 2 R, where w = (w1, w2, . . . , wd), jwj = (jw1j, jw2j, . . . , jwdj), wi 2 R and w  x denotes an inner product of w and x. In T(R), f(x) can be written as

f ðxÞ ¼ ðw  rx þ b; qðDr x ÞÞ;

w; rx ; Dr x 2 Rd ; b 2 R;

  ðyi  Dryi Þ  e  n2i ;

ð12Þ

where q(Drx) = jwjDrx. The input variables and output function f(x) are symmetric triangular fuzzy number in the Fm-SVRM, the parameters of Fm-SVRM model are crisp numbers. In the light of the idea of Fm-SVRM, the regression coefficients in T(R) can be estimated by the following constrained optimization problem:

min

w;b;e;nðÞ

s:t:

" # 2 X l  1 1X 2 2  kwk þ b þ C v e þ ðn þ nki Þ 2 l k¼1 i¼1 ki 8 ðyi þ Dr yi Þ  ðw  /ðr xi Þ þ b þ qðDr xi ÞÞ 6 e þ n1i ; > > > > > ðw  /ðrxi Þ þ b þ qðDrxi ÞÞ  ðyi þ Dryi Þ 6 e þ n1i ; > > > < ðy  Dr Þ  ðw  /ðr Þ þ b  qðDr ÞÞ 6 e þ n ; yi xi xi 2i i > ðw  /ðrxi Þ þ b  qðDrxi ÞÞ  ðyi  Dryi Þ 6 e þ n2i ; > > > > > nki ; nki P 0; k ¼ 1; 2; > > : e P 0;

Fig. 4. The k-cuts of two triangular fuzzy numbers.

Fig. 5. The e-insensitive tube of Fm-SVM.

ð13Þ

Fig. 6. The architecture of Fm-SVM.

ð14Þ

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Q. Wu / Expert Systems with Applications 38 (2011) 14398–14406 ðÞ

ðÞ

where aki ; b; gk P 0 ðk ¼ 1; 2; i ¼ 1; . . . ; lÞ are Lagrangian multipliers. Differentiating the Lagrangian function (14) with regard to ðÞ w; b; e; nki , we have 2 X l X @L ¼0)w¼ ðaki  aki Þr xi ; @w k¼1 i¼1

ð15Þ

2 X l X @L ¼0) ðaki  aki Þ ¼ b; @b k¼1 i¼1

ð16Þ

2 X l X @L ¼ 0 ) b ¼ Cm  ðaki þ aki Þ; @e k¼1 i¼1

ð17Þ

@L

ðÞ

ðÞ

¼ 0 ) gki ¼ C=l  aki :

ðÞ

@nki

Kðr xi ; r xj Þ ¼ exp 

ð18Þ

! krxi  r xj k2 ; 2r2

ð21Þ

! kDr xi  Dr xj k2 : KðDr xi ; Drxj Þ ¼ exp  2r 2

ð22Þ

Thus, the regression function (23) can be determined by

By substituting (15)–(18) into (14), we can obtain the corresponding dual form of function (13) as follows:

max   a ;a

þ

Definition 4 (Radial basis kernel function). For fuzzy training sample set fðxi ; yi Þgli¼1 ; xi 2 TðRÞd ; yi 2 TðRÞ, where T(R)d is the set of d dimensional vectors in ETFN space. In ETFN space, symmetric triangular fuzzy numbers are taken into account, i.e. xi ¼ ðr xi ; Dr xi Þ; yi ¼ ðryi ; Dr yi Þ, where Dr xi ¼ Dr xi ; r yi ¼ Dr yi . Then, Gaussian radial basis kernel function can be described as follows:

  yi  Dr yi þ qðDrxi Þ ða2i  a2i Þ

2 X l X

ðaki  aki Þ ¼ 0;

5. Case study

k¼1 i¼1

aðÞ ki 2 ½0; C=l;

2 X l X ðaki þ aki Þ 6 C m;

ð19Þ

k¼1 i¼1

P where kwk2 ¼ li;j¼1 ða1i  a1i þ a2i  a2i Þða1j  a1j þ a2j  a2j ÞKðr xi  r xj Þ ðÞ The Lagrangian multipliers aki can be determined by solving the above QP problem. Based on the Karush–Kuhn–Tucker (KKT) conditions, we have

 8  a1i yi þ Dryi  w  /ðrxi Þ  b  qðDrxi Þ  e  n1i ¼ 0; > >   > > > a1i w  /ðrxi Þ þ b þ qðDrxi Þ  yi  Dryi  e  n1i ¼ 0; > >   > > > a2i yi  Dryi  w  /ðrxi Þ  b þ qðDrxi Þ  e  n2i ¼ 0; > >  <  a2i w  /ðrxi Þ þ b  qðDrxi Þ  yi þ Dryi  e  n2i ¼ 0;   > > ðÞ ðÞ > > C=l  aki nki ¼ 0; > > > > > 2 P l > P > > : Cm  ðaki þ a Þ e ¼ 0: k¼1 i¼1

ð23Þ

where q(Drx) = jwjDrx, jwj = (jw1j, jw2j, . . . , jwdj), wi 2 R. The input variables and output function f(x) are symmetric triangular fuzzy number in the proposed Fm-SVM, the parameters of Fm-SVM are real numbers.

i¼1

s:t:

! 2 X l X ðaki  aki ÞðKðrxi ; r x Þ þ 1Þ; qðDr x Þ ; k¼1 i¼1

l  X 1 2 kwk2 þ b þ ðyi þ Dr yi  qðDrxi ÞÞða1i  a1i Þ 2 i¼1

l X

f ðxÞ ¼

ð20Þ

ki

The selection of the kernel function is important to generalization ability of support vector regression application. Many applications suggest that radial basis functions tend to perform well under general smoothness assumptions, so that they should be considered especially if no additional knowledge of the data is available.

To illustrate this time estimation method, the design of plastic injection mold is studied. Injection mold is a kind of singlepiece-designed product and the design process is usually driven by customer orders. Many product development projects involve the design process of injection mold and the pre-estimating time is meaningful for the planning, scheduling and optimization of the whole product development process. Firstly, it is necessary to obtain PDT factor values of mold characteristics by FM-HOQs and Eqs. (4) and (10) in Section 3. Take the constructing process of the planning FM-HOQ for example. Let us consider a design order for a kind of injection mold. Suppose that the customer has given us the specification of the molding product. Thus we should analyze the customer demands and extract some useful mold characteristics. The technical customer demands are taken into consideration here. Originally, some demands are expressed as quantitative information (e.g. ‘the mold life is 3000 times’), while others are described as qualitative information (e.g. ‘the molding product precision is high’). We establish a unified fuzzy measurement scheme for all these demands, and five linguistic levels are used. The importance degrees of these demands are also represented by fuzzy weight sets. Table 2 shows the processed data of customer demands.

Table 2 The technical customer demands for an injection mold. Customer demands q No.

Name

1 2 3 4 5 6 7 8 9

Molding product size Molding product precision Molding product structure Molding product shape Minimum thickness Underpropping surface dimension Molding batch Mold life Plastic type

Unit

e Weight set B

Measure set MB

mm

(0.5) (0.8) (0.3) (0.6) (0.9) (0.3) (0.5) (0.9) (0.8)

(0.25, 0.75, 0.9, 0.75, 0.25) (0, 0.25, 0.75, 0.9, 0.75) (0.75, 0.9, 0.75, 0.25, 0)

lm Dimensionless Dimensionless mm mm2 Dimensionless h Dimensionless

(0, 0.25, 0.75, 0.9, 0.75) (0, 0.1, 0.5, 0.9, 0.5) (0.1, 0.5, 0.9, 0.5, 0.1) (0, 0, 0.25, 0.75, 0.9)

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From Table 2, we have

It is remarkable that two kinds of demands (i.e. ‘molding product shape’ and ‘plastic type’) and two kinds of characteristics (i.e. ‘ejector type’ and ‘runner shape’) are immeasurable. Thus they are excluded from the fuzzy transition matrix, and the fuzzy weights and measures of the mold characteristics are then computed by Eqs. (4) and (10) as follows:

Select nine kinds of engineering characteristics of injection mold design, i.e., (p1) mold structure, (p2) cavity number, (p3) form feature number, (p4) wainscot gauge variation, (p5) injection pressure, (p6) injection capacity, (p7) ejector type, (p8) runner shape, and (p9) manufacturing precision. Then we can construct a planning FM-HOQ. According to the relative design handbooks and experience, the fuzzy relationship matrix and fuzzy transition matrix are given as follows:

The constructing process of the design FM-HOQ is similar to that of planning FM-HOQ. Then some characteristic information of the corresponding components and parts (e.g. cores, slides, mold base, cavity, gate, and so on) can be obtained. These acquired fuzzy measures of characteristics can be defuzzified into comprehensible forms (definite estimate values or linguistic descriptions). In the following text, we construct Fv-SVRM to estimate the mold design time. We place emphasis on the mold characteristics associated with design time. Some characteristics with large influencing weights are gathered to develop a time factor list, as shown in Table 3. Here, the influencing weights that indicate the influencing degree on PDT are different from the importance indexes in FM-HOQs. In our experiments, 60 sets of molds with corresponding design time are selected from past projects in a typical company. MATLAB 7.1 is used to implement the PDT estimation method. Standard PSO algorithm is used to optimize the unknown parameters of FmSVRM. The detailed characteristic data and design time of these molds compose the corresponding patterns, as shown in Table 4. We train the Fv-SVRM with 48 patterns, and the others are used for testing.

Table 3 PDT factors of mold characteristics. Mold characteristics

Unit

Expression

Weight

Structure complexity (SC)

Dimensionless

0.90

Model difficulty (MD)

Dimensionless

Wainscot gauge variation (WGV) Cavity number (CN)

Dimensionless

Mold size (height/diameter) (MS) Form feature number (FFN)

Dimensionless

Linguistic information Linguistic information Linguistic information Numerical information Numerical information Numerical information

Dimensionless

Dimensionless

0.70 0.70 0.80 0.55 0.55

Table 4 Learning and testing patterns for the Fv-SVRM. Molds

Input data

Desired PDT (h)

No.

Name

SC

MD

WGV

CN

MS

FFN

1 2 3 4 5 6 7 8 9 10 ... 59 60

Global handle Water bottle lid Medicine lid Footbath basin Litter basket Plastic silk flower Dining chair Spindling bushing Three-way pipe Hydrant shell ... Paper-lead pulley Winding tray

L H H VL L L M H H VH ... H M

L L M VL M M H VL L H ... L M

L H VL VL H M L L L M ... L VH

4 4 4 1 1 1 1 2 1 1 ... 10 12

3.10 0.56 1.50 0.50 2.10 7.10 0.50 8.07 0.45 0.30 ... 5 7.9

3 7 6 3 12 4 15 2 5 7 ... 10 2

23.0 45.5 37.0 10.0 42.5 29.5 48.0 30.0 24.5 49.0 ... 59.0 69.0

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Fig. 7. Training and testing results on the FSVM.

Table 5 Comparison of estimating results from two models. Model

1

2

3

4

5

6

7

8

9

10

11

12

Real value v-SVRM Fv-SVRM

69 70.4 70.1

59 59.2 59.2

61.5 62.4 62.2

55.5 58.1 58.4

50.5 53.8 51.9

28.5 31.7 31.1

64.5 64.8 65.2

70 71.3 71.0

39 38.9 38.9

42 43.9 43.5

67 68.5 68.2

95.5 94.3 95.3

Table 6 Error statistic of two models. Model

MAE

MAPE

MSE

v-SVRM Fv-SVRM

1.4917 1.1333

0.0307 0.0238

3.3158 2.00083

In different initial connection weights, a good many experiments are performed for each mode. The results of Fv-SVRM training and testing in a certain moment are shown in Fig. 7. In Fig. 7, we can see that there are some local minimums appearing in the training processes, and these valleys are gotten over one by one. This advantage should be owed to the added momentum factor. During the experiments, we also find that over-fitting may occur when the objective training error is excessively small. To compare the performances of the proposed model, traditional SVRM model is selected to handle the same dataset. The testing results are arranged in Table 5. It is obvious in Table 6 that the MAE, MAPE and MSE of the Fv-SVRM are somewhat smaller than those of the traditional v-SVRM. Moreover, For l training samples, the QP problem (19) consists of 4l variables, 1 linear equality constraint and 8l + 1 bound constraints. Therefore, the size of QP problem (19) is directly proportional to the number of training samples and is independent of the input dimensionality of the Fm-SVRM. The Fv-SVRM model needs fuzzification measures to obtain the training and testing set. The uncertain information is considered into modeling of FvSVRM. Therefore, the generalization of Fv-SVRM is improved. However, the duration of training on the Fm-SVRM model is longer than that of the traditional v-SVRM. This is unquestionable because the computational complexity accords with the number of constraint conditions. If the number of input variables is much more and the adjusting process of some initial parameters is considered, the computational workload of the Fv-SVRM will be acceptable. Thus it can reach the conclusion that the Fv-SVRM model is rather suitable for practical fuzzy problems. 6. Conclusions The control and decision of product development is based on the pre-estimation of PDT. However, this PDT estimation problem

is always overlooked because of the insufficiency of quantitative methods. This research attempts to develop an intelligent estimation method. In order to find out product characteristics at the early stage of product development, a fuzzy measurable house of quality (FM-HOQ) model is established. This model is applied to measure and map characteristics from customer’s technical demands, with the decomposition idea of QFD. For customer’s functional demands, a mapping pattern of ‘‘functions-principlestructure’’ is taken on. The data of product characteristics having been obtained, a new Fv-SVRM model is presented to fuse data and realize the estimation of design time. The application of the FM-HOQ model and Fv-SVRM -based intelligent estimation method to the design of injection molds indicates that the model and method are feasible. This estimation method can be used for the remodeling products at the early stages of development. The known characteristics of existing products are used to train the Fv-SVRM. Thus this trained Fv-SVRM can be approximated to the design time for a new product. For a certain kind of product, the accuracy of this method will be enhanced while more samples are added to the training set. On the other hand, there are some limitations in this estimation method. For the brand-new products, this PDT estimation method will be inapplicable. The influencing weights of linguistic variables obtained by experience or experiment are important for the FvSVRM model. If all influencing weights of linguistic and numerical variables are used as the connection weights between the input layer and fuzzifization layer, and optimized by the Fv-SVRM learning, then the Fv-SVRM model will be more reliable and these influencing weights will have further applicable value. This is a problem for future research. Anyway, the main contributions of this research can be shown as follows: (1) Develop a time estimation method that is helpful for product development. (2) Propose a kind of fuzzy measurable HOQ for mapping and analysis of product characteristics. (3) Provide a Fv-SVRM model for the MISO system with highdimensionality and mixed-type inputs.

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