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Radial Basis Function Neural Network Method of Determining Functional Relationships for Quality Function Deployment

LI Xin，HUANG Lu-cheng

School of Economics and Management, Beijing University of Technology, P. R. China, 100124

Abstract: Quality Function Deployment (QFD) is a systematic approach that captures customer requirements and translates them, through house of quality (HOQ), into engineering characteristics of product. As the functional relationships between customer requirements and engineering characteristics in QFD are uncertain, unclear and fuzzy, Radial Basis Function (RBF) to determine the functional relationships for QFD is presented, and a QFD functional relationships model based on RBF is proposed. According to RBF neural network can realize the nonlinear mapping space from the input space to the output, and can obtain the optimal relationships pattern of the input and output, the customer requirements and engineering characteristics in QFD constituted the input and output of the RBF Neural Network respectively, the optimal relationships are constructed through the neural network training. A case study of natural lighting products development is provided to illustrate the application of the presented method.

Keywords: functional relationships, house of quality, quality function deployment, radial basis function 1 Introduction

Manufacturing companies are facing intense competition in the global markets. They realize that efficient design and manufacture of products preferred by customers at competitive costs within shorter timeframe over those offered by competitors is crucial to their survival. A widespread practice in industry to cope with global competition is the adoption of Quality Function Deployment (QFD), which is well known as a customer-driven product development method originated in Japan in the late 1960s[1]-[2]. QFD is a planning and problem solving tool that is gaining growing acceptance for translating customer requirements into engineering characteristics of a product. The method can help engineers to have better understanding of how engineering characteristics were generated and how their target values setting were determined. Typically, a QFD system can be broken down into four inter-linked phases to fully deploy the customer requirements phase by

Supported by the National Natural Science Foundation of China (70639002)

phase [3]-[5]. The first phase of QFD, usually called house of quality, is of fundamental and strategic importance in the QFD system, since it links the “voice of the customer” to the “voice of the technician” through which process and production plans can be developed in the other phases of the QFD system.

The difficulty of building the house of quality (HOQ) is how to determine the functional relationships between customer requirements and engineering characteristics. Many researchers have contributed to propose effective models of HOQ since 1960s, but most models assume these relationships are simply linear. The reference[6] put forward a mathematical programming model to obtain the relationships between customer requirements and engineering characteristics. The reference[7] proposed a multi-objective fuzzy model to ascertain the above relationships. However, the above methods used less data variables, and overlooked a number of influencing factors in the process of mathematical modeling, so the research methods are not feasible in certain circumstances.

The reference[8] used symmetric triangle fuzzy parameter to ascertain the relationships between customer requirements and engineering characteristics. The reference[9] combined the fuzzy regression theory with symmetric triangular fuzzy coefficients into QFD system to determine the relationships between customer requirements and engineering characteristics. Subsequently reference[10] extended the symmetric triangular fuzzy number into asymmetric triangular fuzzy number and trapezoidal fuzzy number, then integrated least square regression into fuzzy linear regression model, and put forward the two mixed asymmetric fuzzy linear programming model to determine the relationships. The reference[11] presented fuzzy expert systems approach to determine the relationships. Those previous studies have used fuzzy numbers, but determining the membership degree of fuzzy numbers is difficult and not precise, so the results will be unreliable in some conditions. The reference[12] combined the rough set method into QFD, and used some related methods of relative reduction and relative core in rough set theory to determine the relationships, as well as fully exploited the experience and knowledge of QFD team experts. However, the calculation of this method is complex and relatively difficult to implement.

2009 International Conference on Management Science & Engineering (16th) September 14-16, 2009 Moscow, Russia

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Through the analysis mentioned above we can see that the functional relationships between customer requirements and engineering characteristics are uncertain, nonlinear and fuzzy. And to determine the relationships should fully make use of the experience and knowledge of QFD team experts. Artificial neural networks are the technology to simulate characteristics of mankind’s neural network system, and exhibit many advantages such as the strong ability of nonlinear approaching, no need of mechanism model, self-learning and so on. Radial basis function (RBF) neural network is a kind of good feed-forward neural networks, and has best approximation and the global optimum characteristics. RBF neural network can realize the nonlinear mapping from the input Rn space to the output Rm space, and can obtain the optimal relationships pattern between the input and the output. And to determine the functional relationships between customer requirements and engineering characteristics, in essence, is to find out the non-linear mapping relation from Rn space to Rm space. Therefore, this paper uses RBF neural network to determine the optimal relationships between customer requirements and engineering characteristics through the neural network training, and establishes a QFD functional relationships model based on RBF neural network.

This paper is organized as follows. Section 2 describes the structure and principle of RBF neural network. Section 3 presents RBF method of determining functional relationships for QFD. Section 4, a case study of natural lighting products development is provided to illustrate the application of the presented method. Finally, Section 5 concludes the paper.

2 Structure and principle of RBF neural network

RBF neural networks have strong biology background. In the cortex region of human brain, overlapping jurisdictions and the feeling of local regulation are its characteristics. Based on that, a RBF neural network structure was firstly proposed by J. Moody C. Darken in the 1980s [13]. RBF networks have some good qualities, which are based on the theory of function approximation [14]-[15]. Chen and other researchers [16] proved that Radial Basis Function neural networks can be simplified up to three layers to approximate any kinds of process. Fig. 1 and Fig. 2 illustrate the structure of this thinking, RBF neural network being made up of three layers: the input signal is transferred to hidden layer through input layer; the hidden layer is made of radiate function such as Gaussian function and the output layer is usually a simple linear function [17].

The input and output of ith hidden unit is represented by the following model respectively [18]:

qik = i

j

qjji bxw 1)1( 2 ×∑ − (1)

Fig. 1 RBF neural network structure

Fig. 2 The input and output of hidden layer

( )( ) ( ) ⎟⎠⎞⎜

⎝⎛ ×−−=−=

22 11expexp iq

iqi

qi bXwkr (2)

Here jiw1 is the weight between the ith hidden unit

and the jth input, qjx is the jth input vector. The vector

ib1 of the RBF can adjust the sensitivity of the function,

but people usually use the constant C in actual work. The most relationship between the C and ib1 is

ii C

b 8326.01 = . Therefore, in this case, the output of ith

hidden is defined as:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −×−=

2

21

8326.0expi

qiq

i C

Xwr (3)

The output of output layer is defined as:

∑ ×==

n

iii

q wry1

2 (4)

Here iw2 is the weight between the ith hidden unit and the output. 3 RBF method of determining functional relationships for QFD

Through the structure and principle of RBF neural network, we can see that the RBF neural network is known to learn data by measuring the Euclidean distance to realize the non-linear mapping, and then obtains the optimal relation model between input and output. Therefore, this paper uses it to obtain the functional relationships between customer requirements and engineering characteristics.

3.1 The Construction of functional relationships model based on RBF 3.1.1 Obtain the functional relationships data

Assume that a product contains n items QFD

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customer requirements ni CRCRCRCR ,,,,, 21 , and then determine m items engineering characteristics

mj ECECECEC ,,,,, 21 according to the customer requirements. In general, the engineering characteristics in QFD are the most important characteristics of product to meet all customer requirements.

After determining customer requirements and engineering characteristics, in accordance with the construction process of HOQ, we need to identify the functional relationships between customer requirements and engineering characteristics. In essence, the correlation coefficient between any of the customer requirements and each engineering characteristic in the HOQ are the contribution measure of each engineering characteristic to meet the customer requirements [19]. So in order to obtain the correlation coefficient between customer requirements and engineering characteristics, QFD team need evaluate the relationship between each customer requirement and each engineering characteristic according to the actual situation, and collect the evaluation data. The functional relationships sample set composed by evaluation data is defined as { }qklP EEEECCCCU ,,,,,,,,, 2121= ,

where { }pnpipPp CCCCC ,,,, 21= ,

{ }kmkjkkk EEEEE ,,,,, 21= , piC denotes the measurement importance of the customer p to the requirement iCR , kjE denotes the strength ratings of the expert k scoring the relationship between engineering characteristic jEC and each customer requirement, l is customers number, q is experts number. 3.1.2 Data preprocessing

When QFD team conducting customer requirements market research, there are differences that customers know the product characteristics; At the same time, QFD team organizing the engineering technical experts to evaluate functional relationships between customer requirements and engineering characteristics, the extent of each engineering technical experts familiar with the product also has some differences; Therefore, in order to increase the effectiveness of the access data, we need give a certain value to the customers and experts in accordance with the extent of each customer and engineering experts being familiar with product respectively, then we preprocess the above evaluation data sample set using the following formula:

∑ ω

∑ ω=

=

=l

pp

l

ppip

i

CC

1

1' ( ni ,,2,1 L= ) (5)

Where pω is the weight of customer p ;

∑ ω

∑ ω=

=

=q

kk

q

kkjk

j

EE

1

1' ( mj ,,2,1= ) (6)

Where kω is the weight of engineering technical expert k . After that, we obtain the treatment sample set { }'''

2'1

'''2

'1

' ,,,,,,,,, mjni EEEECCCCU = . The training sample set of the functional relationships model is defined as

{ }mjni TTTTRRRRX ,,,,,,,,,,, 2121= , where

∑=

=

n

ii

ii

C

CR

1

'

',

∑=

=

m

jj

jj

E

ET

1

'

'

(7)

3.1.3 The construction of QFD functional relationships model based on RBF

According to the idea of production rule, QFD is defined as: if we can satisfy the customer requirement iCR , then take the engineering characteristic jEC . The degree of the functional

relationship is ),( ECCRR . This paper adopts the RBF neural network with three layers, which includes input layer, hidden layer and output layer. The unit numbers of input layer and output layer are n and m respectively, where n is the number of customers, m is the number of engineering characteristics. The QFD functional relationships model based on RBF is shown in Fig. 3.

Fig.3 QFD functional relationships model based on RBF

In the Fig. 3, X is the input vector, which denotes customer requirements CR ; Y is the output of neuron EC , its value is jT in sample set X , the

connected weights between X and CR neurons are iR in sample set X . The nonlinear function of hidden layer is given as the Gaussian function.

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

δ

μ−−=φ

22exp)(

j

jj

xx lj ,,3,2,1= (8)

Where x is the input vector, jμ is the mean value of

the j th hidden unit, and jδ is the covariance of the

j th hidden unit. The k th output can be calculated as

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∑ φ==

m

jjkjk xwxy

1)()( mk ,,3,2,1= (9)

Where kjw is the weight between the k th hidden unit and the j th output unit.

After fully utilizing the knowledge and experience of QFD team experts to obtain the sample set X of functional relationships between customer requirements and engineering characteristics, we use QFD functional relationship model based on RBF to get the relationships pattern among data in sample set X through the network training, and then to obtain the optimal relationships between customer requirements and engineering characteristics.

3.2 RBF method of determining functional relationships

The main process for structuring RBF method to determining functional relationships is shown as follows: 3.2.1 Determining the customer requirements

QFD theory emphasizes the guiding role of customer requirements for product development, and considers customer requirements throughout the process of product development and design. As customer-driven product development method, to obtain the customer requirements is the most critical and difficult step in QFD process [20].

QFD team determining customer requirements usually in accordance with the right steps and scientific methods, the main steps are shown as follows: ① Determining the reasonable survey object; ② Choosing the appropriate methods of investigation; ③ Implementing market research; ④ Organizing and analyzing the customer requirements. Generally speaking, customer to the ideal product demand is the lack of systematic, vague and not clearly defined, and has many redundancies. Therefore, after QFD team in accordance with the above steps to obtain the customer requirements, QFD team should deal with the redundancy of customer requirements in order to obtain the customer requirements of product planning HOQ, and then calculates the importance ratings of customer requirements. 3.2.2 Determining the engineering characteristics

According to the above-mentioned customer requirements have been identified, QFD team firstly uses the “brainstorming” method to determine the four existence relationships of mutually exclusive, non-related, positive correlation and negative correlation among the engineering characteristics, which compose the selected set of engineering characteristics; Secondly, filters the mutually exclusive relations of engineering characteristics to obtain only non-related, positive correlation and negative correlation among the engineering characteristics, which compose the filtered set of engineering characteristics; Finally, when analyzes the importance rating of engineering characteristics, it is also necessary to consider the cost, time, resource

utilization and feasibility requirements in order to finally determine the engineering characteristics of product planning HOQ [21]. 3.2.3 The design and training of RBF neural network model

(1) After determining the customer requirements and engineering characteristics, QFD team needs evaluate the relationship between each customer requirement and each engineering characteristic according to the actual situation, and collects the evaluation data which compose the functional relationships sample set, and then utilizes the formula (5), (6) and (7) to preprocess evaluation data sample set in order to obtain the training sample set X of functional relationship model.

(2) According to the numbers of customer requirements and engineering characteristics, the numbers of input layer neurons and output layer neurons in the QFD functional relationship model based on RBF are determined. The training of functional relationship network model is shown as follows: step 1: Non-teachers learning to determine the weight 1W between input layer and hidden layer; step 2: Teachers learning to determine the weight 2W between hidden layer and output layer. The threshold 1b is determined by constantC , which needs trial and error to find the best C value.

To determine the numbers of hidden layer neurons, the method used in this paper is from 0 neurons start training, by examining the output error to make the network automatically increase the number of neurons in each cycle, making the maximum error of the network generate the corresponding input vector as a weight

vector iW1 , creating a new hidden layer neurons, then checking for the new network error, repeating the process until it reaches the error requirement. When the error is reduced to the requirements, the system stops studying. At this point the network weights and threshold vector are fixed, which compose the internal knowledge of system.

(3) Using test samples to test the best trained network. The purpose of network training is that the network weights of all neurons get to the optimal value in the process of training, i.e. obtain the optimal relation model between customer requirements and engineering characteristics.

According to the above idea, we can draw a picture to visualize the process for structuring RBF method to determine functional relationships. It is shown in Fig. 4. 4 Case study

In order to show the idea of this paper directly, and verify the feasibility of the QFD functional relationships model based on RBF, this part we will take the research and development of natural lighting products to illustrate the application of the former content.

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Fig.4 The process for structuring RBF method to determining functional relationships

Tab.1 The network training sample set

1CR 2CR 3CR 4CR 5CR 6CR 1EC 2EC 3EC 4EC 5EC

1 0.2000 0.1000 0.1000 0.0000 0.1000 0.0000 0.1538 0.2308 0.3077 0.2308 0.0769 2 0.0000 0.2000 0.1000 0.0000 0.1000 0.0000 0.0667 0.1333 0.2667 0.2667 0.2667 3 0.1000 0.0000 0.2000 0.1000 0.0000 0.1000 0.0909 0.1818 0.2727 0.3636 0.0909 4 0.0000 0.0000 0.1000 0.2000 0.1000 0.1000 0.5000 0.1250 0.1250 0.1250 0.1250 5 0.1000 0.1000 0.0000 0.1000 0.2000 0.0000 0.1250 0.1250 0.1250 0.1250 0.5000 6 0.0000 0.0000 0.1000 0.0000 0.0000 0.2000 0.0833 0.1667 0.2500 0.1667 0.3333 7 0.2250 0.2075 0.1750 0.1500 0.1925 0.1575 0.0300 0.1150 0.2900 0.3100 0.2500

4.1 Determining the customer requirements and engineering characteristics

Natural lighting products have been successfully applied to the Olympic venues and other public places during the Beijing Olympic Games, and achieve a very good lighting effects. An enterprise company adopts QFD strategy to improve the quality of natural lighting products in order to further promote the implementation of the products. By the market research and customers’ feedback to determine customer requirements for products mainly include: high brightness lighting ( 1CR ), uniform illumination ( 2CR ), good lighting effects ( 3CR ), can be isolated from the ultraviolet part of sunlight ( 4CR ), easy to operate ( 5CR ), can adjust the brightness of light ( 6CR ).

Then QFD team identifies the corresponding engineering characteristics in accordance with the above-mentioned customer requirements as follows: the coating of mining mask ( 1EC ), the performance of mining mask ( 2EC ), lighting catheter-ray materials ( 3EC ), diffusers texture ( 4EC ), dimming control device ( 5EC ).

4.2 The construction and training of RBF neural network model

QFD team organizes technical experts to evaluate the relationships between customer requirements and engineering characteristics, and in accordance with market research and customers’ feedback. The functional relationships sample set composed by evaluation data is defined as { }892110721 ,,,,,,,,, EEEECCCCU kP= , and then according to the formula (5), (6) and (7) to

preprocess the data of functional relationships sample set, and finally the network training sample set is shown in Tab. 1. 4.2.1 The construction and training process of the model

The construction and training process of the model is shown as follows:

(1) Network initialization. In this paper, the RBF neural network being made up of three layers has six input nodes, the number of hidden layer neurons determined by examining the network goal errors to dynamically increase the network nodes, and five output nodes.

(2) Network training. The 1-6 group data of sample set are training data, where iCR (i = 1, 2, 3, 4, 5, 6) is the input of network, the corresponding jEC (j = 1, 2, 3, 4, 5) is the output of network, the mean square error is 10-5, the spread value is 0.15. The error curve of constructing RBF network is shown in Fig. 5.

(3) After network training, we use the remaining data to test the generalization of the network. The result is shown in Fig. 6. Accordingly to the result, we can conclude that the RBF neural network model has a high credibility.

(4) To further verify the accuracy of the above method, we compare RBF neural network model and BP neural network model. BP neural network is the most popular methods in the field of simulation. In this paper, a single hidden layer BP neural network model is constructed for determining the relationships between customer requirements and engineering characteristics, in which the input layer neurons are 6 nodes, and the output layer neurons are 5 nodes, the number of hidden layer neurons is adjusted between 3 and 8 at random, and the right number is decided by the performance of BP

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neural network. When the number of hidden layer is 5, the average relative error is the smallest. So we choose 6-5-5 topology for training. Functions of Matlab BP toolbox are used to train the network, among of them the active function of hidden layer is tansig function, and the active function of output layer is purelin function respectively. The smallest error is 10-6. The results are shown in Fig. 7 and Fig. 8.

Fig.5 The error curve of constructing RBF network

Fig.6 The simulation curve of RBF network

4.2.2 The comparison results Accordingly to the above figures, we can see that

BP network need 7 steps to achieve the error requirement 10-6, while the RBF network need 5 steps to achieve the error requirement 10-6. In other words, the training speed of RBF network is faster than BP network. In addition, in the process of network training, BP network is not stable enough, and sometimes needs many steps to achieve the error requirements, while RBF network is stable, and its numerous training simulation curve is similar to actual results.

As can be seen from the case, the use of RBF network to obtain the functional relationships between

customer requirements and engineering characteristics is feasible and effective.

Fig.7 The error performance curve of BP network training

Fig.8 The simulation curve of BP network

5 Conclusion

In this paper, RBF network is used to obtain the functional relationships between customer requirements and engineering characteristics, at the same time the BP network is used to obtain the relationships. Through the case, we can see that BP network training time is longer than RBF network, while the RBF network has better stability than BP network; but how to reduce the noise of evaluation sample set, and better improve the generalization performance of RBF network, it need further study.

QFD is an effective way which develops and designs new product through the HOQ. There are, however, difficulties to determine the functional relationships between customer requirements and engineering characteristics through establishing the HOQ. As the functional relationships between customer requirements and engineering characteristics are uncertain, unclear and fuzzy, RBF neural network to

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determining the relationships is presented, and establishes a functional relationship for QFD model based on RBF. An application case shows that the model is feasible and effective with faster calculate speed, high simulation precision and a certain degree of practicality, and also can fully mine the knowledge and experience of the QFD team experts.

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