The Hybrid Fuzzy Least-Squares Regression Approach ... - IEEE Xplore

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008

The Hybrid Fuzzy Least-Squares Regression Approach to Modeling Manufacturing Processes C. K. Kwong, Y. Chen, K. Y. Chan, and H. Wong

Abstract—Uncertainty in manufacturing processes is caused both by randomness, as in material properties, and by fuzziness, as in the inexact knowledge. Previous research has seldom considered these two types of uncertainty when modeling manufacturing processes. In this paper, a hybrid fuzzy least-squares regression (HFLSR) approach to modeling manufacturing processes, which does take into consideration these two types of uncertainty, is proposed and described, and a new form of weighted fuzzy arithmetic is introduced to develop the hybrid fuzzy least-squares regression method. The proposed HFLSR approach not only features the capability of dealing with the two types of uncertainty, but also addresses the consideration of replication of responses in experiments. To investigate the effectiveness of the proposed approach to process modeling, it was applied to the modeling solder paste dispensing process. Modeling results were compared with those based on statistical regression and fuzzy linear regression. It was found that the accuracy of prediction based on the HFLSR is slightly better than that based on statistical regression and much better than that based on the Peters fuzzy regression. Index Terms—Fuzzy linear regression, hybrid fuzzy leastsquares regression (HFLSR), manufacturing process modeling, statistical regression.

I. INTRODUCTION N order to improve the quality of a process, all the key variables controlling the desired output need to be understood and optimized. This can be achieved by developing appropriate physical models or empirical models to represent the manufacturing process. Physical models are based on the physical understanding of the process and on the employment of a set of physical laws. They are attractive in modeling manufacturing processes since they provide a fundamental understanding of the relationships between the various inputs and outputs. However, a physical model may not be amenable to providing accurate analytical or numerical solutions in many cases due to the complex behavior of certain manufacturing processes. Empirical models, as opposed to physical models, are built on a basis of experimental data. Various techniques have been introduced in previous studies to develop process models based on the empirical modeling approach such as artificial neural networks, statistical regression models, fuzzy regression, and fuzzy

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Manuscript received June 5, 2006. The work described in this paper was supported by The Hong Kong Polytechnic University, Hong Kong (Project no. G-U024). C. K. Kwong, Y. Chen, and K. Y. Chan are with the Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]). H. Wong is with the Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2007.903324

logic models. Artificial neural networks have been used to develop process models for various manufacturing processes such as abrasive flow machining [15], grinding [2], and die casting [16]. They have the capability to transform a nonlinear mathematical model into a simplified black-box structure. The advantage of using neural network techniques to process modeling is that learning and generalization abilities as well as nonlinearity can be employed. Previous research has already confirmed that the artificial neural network is a powerful tool for modeling nonlinear, complex, and noisy processes. However, previous studies also found that the performance of the neural networks that have been developed is very dependent on the training algorithm that is used, and on the design of the neural network architecture, as well as on the setting of the parameters of the neural network. Statistical regression is probably the most common method used to develop process models in the manufacturing industry. It is well known that the statistical regression models are accurate only in the range in which they are developed. In conventional regression analysis, deviations between the observed values and the estimates are assumed to be due to random errors. Thus, statistical techniques are applied to perform estimation and inference in regression analysis. As a result, statistical regression models can be applied only if the given data is distributed according to a statistical model, and the relationship between dependent and independent variables is crisp. However, in some manufacturing processes, it is difficult to find probability distributions for dependent variables. The deviations of such cases are due to the indefinite structure of the system or to imprecise observations. The uncertainty in this type of regression model becomes fuzzy, not random. When the behavior of processes or systems is fuzzy, fuzzy regression seems to be more appealing for estimating the relationship between the dependent variable and independent variables [2000]. In recent years, fuzzy regression has been used in an attempt to model many manufacturing processes, especially those processes which have a high degree of fuzziness. Lai et al. [10] attempted modeling of the die-casting process using Tanaka’s fuzzy regression. Ip et al. [6] introduced the Peters fuzzy regression to develop a process model for epoxy dispensing. Modeling of transfer molding using fuzzy regression was reported by Ip et al. [7]. Kwong and Bai [9] have performed process modeling and optimization using both fuzzy regression and fuzzy linear programming approaches. Three approaches of fuzzy regression were summarized in Chang [3]. For each fuzzy regression method, numerical examples and graphical presentations were used to evaluate their characteristics and differences with statistical regression. Based on the comparative assessment, the fundamental differences between statistical regression and conventional fuzzy regression are concluded—that is, statistical regression modeling data with

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KWONG et al.: HYBRID FUZZY LEAST-SQUARES REGRESSION APPROACH

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randomness type of certainty and conventional fuzzy regression modeling data with fuzziness type of uncertainty. In order to integrate both randomness and fuzziness types of uncertainty into one regression model, a hybrid fuzzy least-squares regression (HFLSR) analysis was proposed by Chang [3], [4]. A major characteristic of the HFLSR method is to integrate the property of central tendency in the least-squares with the possibilistic property in fuzzy regression for developing regression models. A fuzzy logic modeling technique has been applied successfully to model various manufacturing processes [1], [20]. In this approach, the least mean square algorithm and the Newton gradient descent method were used in the training of fuzzy logic models. This technique is very suitable for modeling highly nonlinear manufacturing processes which involve a high degree of fuzziness. However, in most manufacturing processes, it is rare that only one type of uncertainty, the uncertainty associated with randomness or the uncertainty caused by fuzziness, exists. Therefore, modeling manufacturing processes using statistical regression or fuzzy linear regression may not yield the best modeling results as these two techniques account for randomness only or fuzziness only, respectively. To address this modeling problem, a hybrid fuzzy least-squares regression (HFLSR) approach proposed by Chang [3], [4] is introduced in this research to model manufacturing processes which can incorporate both the fuzziness and the randomness of a system in the development of a fuzzy least-squares regression model. In this research, the HFLSR was modified such that it can be used to deal with the replication of experiments that were not addressed properly in the previous studies of the fuzzy regression approach to modeling manufacturing processes. To study the effectiveness of the proposed approach to the modeling, the HFLSR was applied on modeling the process of solder paste dispensing. Results of the modeling were compared with those based on the Peters fuzzy linear regression and on statistical regression approaches, respectively. The rest of the paper is organized as follows. In the next section, the fuzzy linear regression with asymmetric triangular fuzzy numbers is described. In Section III, the principle of a hybrid fuzzy least-squares regression is described. In Section IV, comparisons of the modeling results based on HFLSR, statistical regression and fuzzy linear regression are described. Finally, conclusions are drawn in Section V. II. FUZZY LINEAR REGRESSION WITH ASYMMETRIC TRIANGULAR FUZZY COEFFICIENTS Fuzzy linear regression analysis, first introduced by Tanaka et al. [19], in which two factors, namely the degree of fit and the fuzziness of the model, are considered. The estimation problems can be transformed into linear programming (LP) based on two factors. The following shows a typical fuzzy linear regression model

such that the total systematic fuzziness is minimized while the given input-output pairs should be included in their -level set . However, it has been pointed out by described as several investigators that Tanaka’s approach is very sensitive to outliers. To tackle the deficiency mentioned above, Peters [14] proposed a new model with symmetrical triangular fuzzy number coefficients by the extension of Tanaka’s approach. In the Peters model, the dependent data y is no longer inside or outside the interval but belongs to a certain range. Outliers are compensated for by data which lies within the interval, and the estimated interval is determined by using all the data. A new variable is introduced to represent the membership degree to which the solution belongs within the set of “good solutions,” and a new fuzzy linear programming problem is formulated as follows: (2a) (2b)

(2c)

(2d) (2e) (2f) where is the width of the “tolerance” interval of datum and is the arithmetic mean of all . The parameter represents the desired value of the objective function and in most cases, will be given as zero, which makes for the desired value of the total vagueness; thus, a model as crisp as possible will be obtained. Details of the application of the Peters model of process modeling can be found in the first author’s publication [9]. Symmetric triangular fuzzy numbers are commonly used in the previous studies associated with Tanaka’s and Peters fuzzy linear regression approaches to process modeling. 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. Many residuals and the highly asymmetrical distribution of data points on both sides of the regression line would occur [19]. To make up for this deficiency, we extended the Peters approach by introducing asymmetric triare angular fuzzy coefficients. If the regression coefficients asymmetric triangular fuzzy numbers, they can be denoted as , where is the fuzzy center, is the left fuzzy spread, and is the right fuzzy spread. Thus, a Peters fuzzy linear regression model with asymmetric triangular fuzzy coefficients can be formulated as

(1) where is the estimation of using a fuzzy number after adjusting , and is the th observation of the explanatory variable, . The fuzzy coefficient is determined

(3a) (3b)

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(3c)

According to the definition of WFA, the weighted fuzzy addition of and is defined as

(3d) (3e) (3f) and are also deterwhere the regression parameters , mined in a context dependent way according to the decision maker’s experience and knowledge.

(6) When

and and

are both symmetrical fuzzy numbers, i.e., , a special case of (6) can be obtained

(7)

III. HYBRID FUZZY LEAST-SQUARES REGRESSION A major difference between fuzzy regression and statistical regression is in the way they deal with errors. Fuzzy regression deals with errors as fuzzy variables while statistical regression deals with errors as random residuals. To handle both fuzziness and randomness in developing a regression model, Chang [3], [4] proposed a hybrid fuzzy least-squares regression approach. The method uses a new form of weighted fuzzy arithmetic and the widely accepted least-squares fitting criterion.

Similarly, weighted fuzzy substation, weighted fuzzy multiplication, and weighted fuzzy division can be derived as follows:

(8)

(9)

A. Weighted Fuzzy Arithmetic In fuzzy regression analysis, particularly for engineering applications, a large number of arithmetical operations may be required for problems which involve a large number of data sets. When conventional fuzzy arithmetic is used to deal with the problems, a large number of arithmetical operations need to be involved and the fuzzy widths could add up to an unrealistically large number. Other problems of using conventional fuzzy subtraction and conventional fuzzy division in such situations have been reported by Chao and Ayyub [5]. A new definition of weighted fuzzy arithmetic is proposed in this paper in order to make up for the drawbacks of the conventional fuzzy-arithmetic. Weighted fuzzy arithmetic defines the arithmetical operations between two fuzzy numbers operating two corresponding values in each fuzzy set at the same membership level. It integrates each level operation weighted by the membership level for the entire fuzzy sets, and divides the weighted integration by the total integral of the membership function. The weighted fuzzy arithmetic uses the concept of defuzzification to convert the operation of fuzzy sets into a crisp real number. The resulting crisp number can be interpreted as a mean value of a fuzzy arithmetical operation. Conversely, the conventional definition of fuzzy arithmetic forms all possible values in a fuzzy arithmetic operation as a fuzzy set. and be triangular fuzzy numbers that can be exLet pressed as and , respectively. At membership level, the intervals of and can be expressed as

(4) and (5)

(10)

B. Analysis of Hybrid Fuzzy Least-Squares Regression In this section, an analysis of HFLSR was conducted using weighted fuzzy arithmetic. The arithmetic is used to formulate the summation of the squares of errors between the predicted and measured values

(11) , and is the sample size. Similarly, each where . observed value can be described as The objective function for the principle of the least squares is to minimize the sum of the squares of residual errors. Using the definition of weighted fuzzy arithmetic, the sum of the residual error between the predicted and the observed can be formulated as


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After rearranging the equations, the normal equations for the fuzzy centers are given as follows:

(12) 17

Therefore, the total sum of the residual errors between the can be obtained predicted and the observed , as follows:

17 .. .

.. .

.. .

.. .

.. .

17 The normal equations for the left spread can be given as

18

18 (13) In (13), the expression has unknown parameters , . In order to derive a formula for the unknown regression coefficients based on minimizing the total residual errors , the derivatives of (13) with respect unknown parameters need to be derived, set to to the unknowns. The derivatives zero, and then solved for the of (13) with respect to the fuzzy coefficients , , are given as

.. .

.. .

.. .

.. .

.. .

18 The normal equations for the right spread can be given as

19 (14) 19 .. .

.. .

.. .

.. .

.. .

(15) 19

(16)

Each of the aforementioned three sets of equations is similar to the statistical regression formulation. Therefore, they allow the use of the statistical regression method for the determination of HFLSR models. The fuzzy centers of fuzzy coefficients

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Fig. 2. Solder paste dispensing system.

TABLE I FACTORS AND THEIR LEVELS Fig. 1. Flowchart of modeling manufacturing process using hybrid fuzzy leastsquares regression.

can be obtained using the fuzzy centers of collected data. The left and the right spreads can be obtained separately, using the corresponding fuzzy spreads of the collected fuzzy data. Therefore, by solving the three sets of (17)–(19), the fuzzy centers, the left spreads, and the right spreads of triangular fuzzy coefficients respectively can be obtained. IV. MODELING OF THE SOLDER PASTE DISPENSING PROCESS USING HFLSR To investigate the effectiveness of the HFLSR approach to modeling manufacturing processes, modeling a solder paste dispenser for electronic manufacturing using HFLSR was studied. Fig. 1 shows a flowchart of modeling manufacturing processes using HFLSR. Results of the modeling are compared with those based on fuzzy linear regression and on statistical regression. A. Solder Paste Dispensing Process The highly automated electronic packaging industry uses computer-controlled solder deposition machines to place small amounts of solder paste on printed circuit boards. These solder paste spots must be positioned correctly and must contain the specified amount of solder paste such that when a surface mounted IC component is placed onto the solder paste, all of the IC’s leads align correctly with each solder spot. The dispensing machine has three axes of motion. The axes are used to place the solder paste dispenser over the desired location, and the axis is used to position the tip of the solder paste disperser at the desired height above the board. The dispensing machine works as follows. A series of coordinates describing the locations of where the solder paste is to be deposited is entered into the system’s computer program. For each of these locations, there are two programmable process parameters that are specified: 1) the amount of solder

paste, which is governed by the amount of time that the pump is engaged, and 2) the dwell time, which is the length of time the dispensing system remains over the location after the pump has been disengaged. A schematic diagram of a solder paste dispensing system is shown in Fig. 2. The continuously running motor is connected to a clutch. The output of the clutch drives a screw pump. The amount of time, that the clutch is engaged, determines the amount of solder paste deposited, and is called the shot size. The solder paste exits through the interchangeable needle, which has an inside diameter, . The different solder pastes come prepackaged in tubular containers, which are inserted in the receptacle adjacent to the motor and clutch structure. In the process of solder paste dispensing, the key quality characteristic is the diameter of the circular solder pads. In an experimental study, the target value for the diameter of the circular solder pads was set at 0.0625 inches (1.6 mm). The four significant operating parameters (factors) for the solder paste dispensing process to be studied are pressure, needle inner diameter, shot size and dwell time which are represented by , , , and , respectively. In the plan of the experiments, each factor has two levels. Table I shows the setting of each level of the factors. Table II shows the experimental results. B. Preprocessing of Experimental Data Sets In the previous studies, whatever statistic regression or the Peters fuzzy regression was applied in the modeling manufacturing processes, the mean value of the replications is commonly


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TABLE II

TABLE III PREPROCESSING RESULTS OF TABLE II

2 FACTORIAL EXPERIMENT WITH FIVE REPLICATIONS

used for the development of regression models, which can be expressed as

(20) While the HFLSR is applied to modeling the solder paste dispensing process, the values of dependent variables are fuzzy instead of crisp, which can be represented as , where can be calculated using (20), and and can be determined respectively using (21) and (22)

because the spread of the membership function also plays an important role in describing the characteristics of a fuzzy number. Defuzzification is a particular case of typicality where a fuzzy number or an equivalent possibility distribution is given a single value, which is supposed as best reflecting the amount of information conveyed by the underlying fuzzy set. There are several defuzzification methods described in literature [11], [12], [18]. In this paper, weighted fuzzy arithmetic described in Section III-A was adopted to derive the corresponding crisp values of fuzzy measured values and fuzzy predicted values. According of a triangular fuzzy number to (6), the equivalent crisp value described by can be calculated as

(21) (24)

(22) Furthermore, the values of the four operating parameters need to be normalized to [0,1] using the following equation:

Based on (24), the relative errors (Re) and variance of errors (Rv) of an empirical model can be defined as follows: (25) and

(23) By using (21)–(23), the preprocessing results of Table II are shown in Table III.

(26)

C. Model Development Although fuzzy prediction is more informative than crisp prediction, calculation of fuzzy numbers is not as straightforward as those of crisp values. In the arithmetic of triangular fuzzy numbers, only using the central values as an output is not sufficient

and are the equivalent crisp values of where fuzzy predicted values and fuzzy measured values for the th input set and of the th case, respectively. is the number of the data sets to be investigated.

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TABLE IV COMPARISONS OF MODELING RESULTS

TABLE V RELATIVE PREDICTION ERRORS AND VARIANCES

Fig. 3. Relative errors for each test using statistical regression and the HFLSR approach.

Based on the same data sets, , and , the following the Peters fuzzy regression model, using asymmetric triangular fuzzy numbers, was developed:

With the use of the 16 experimental data sets and their results, the following HFLSR model was developed using MatLab programming software package to relate solder paste diameter and the operating parameters , , , and

(27) According to (25) and (26), the relative errors and variance of fitness of HFLSR model were calculated as 4.3364 and 0.1089, respectively. Based on the same data sets, the following statistical regression can be obtained:

(28) The value of the model (28) is 0.966. The relative error and variance are 4.7067% and 0.1225, respectively, according to (25) and (26).

(29) Accordingly, the relative error and variance were calculated as 4.7234% and 0.1068, respectively. If we change the value of to 1000, to 100, the relative errors and variance of fitness become 4.7496% and 0.1722, respectively. To compare the modeling results of HFLSR with those based on statistical regression and the Peters fuzzy regression, Table IV shows their relative error and variance of fitness. From Table IV, it can be seen that the goodness of fit of HFLSR is better than those of the fuzzy regression model and of the statistical regression model. To further investigate the modeling performance of the HFLSR, four data sets were randomly selected from 16 data sets as testing data sets while the remaining 12 data sets were used to develop a fuzzy linear regression model, a statistical model and a HFLSR model. Their relative prediction errors were calculated. The above procedures were repeated ten times. Table V summarizes the relative errors and variance of errors of the three models. The relative errors for the ten test runs are depicted in Fig. 3. From Table V, it can be seen that both the prediction accuracy and variance of errors of the HFLSR model are the best among all the models. In fact, the difference of average errors between the HFLSR model and the statistical regression model is very little. This indicates that the degree of fuzziness of the solder paste dispensing process is not high.


V. CONCLUSION In this paper, a hybrid fuzzy least-squares regression approach to modeling manufacturing processes was proposed and described. It can deal with two types of uncertainties: fuzziness and randomness, which are associated with modeling manufacturing processes. A new form of weighted fuzzy arithmetic and the least-squares fitting criterion were employed to develop the HFLSR method. The method uses the definition of weighted fuzzy arithmetic and the least squares fitting criterion. To investigate the proposed approach to modeling manufacturing processes, the proposed approach was applied to modeling the solder paste dispensing process. The modeling results were compared with those based on the statistical regression model and the Peters fuzzy regression model which were developed using the same data sets. Comparisons of the results indicate that modeling performance of HFLSR in terms of prediction accuracy and variance of errors is far better than those based on fuzzy linear regression and slightly better than those based on statistical regression. Slight improvement in the latter case may be due to the fact that only little fuzziness exists in the solder paste dispensing process. However, it can be envisaged that HFLSR is more appealing in modeling manufacturing processes when uncertainties associated with substantial fuzziness and randomness coexist in a manufacturing process. Further work will involve the application of HFLSR to modeling the epoxy dispensing process for die attachment. It is believed that substantial fuzziness and randomness coexist in that process. Through the study, the effectiveness of HFLSR can be further validated.

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[14] G. Peters, “Fuzzy linear regression with fuzzy intervals,” Fuzzy Sets Syst., vol. 63, pp. 45–55, 1994. [15] K. L. Petri, R. E. Billo, and B. Bidanda, “A neural network process model for abrasive flow machining operations,” J. Manufact. Syst., vol. 17, no. 1, pp. 52–65, 1998. [16] K. D. V. Prasad and Yarlagadda, “Prediction of die casting process parameters by using an artificial neural network model for zinc alloys,” Int. J. Production Res., vol. 38, no. 1, pp. 119–139, 2000. [17] D. T. Redden and W. H. Woodall, “Further examination of fuzzy linear regression,” Fuzzy Sets Syst., vol. 79, pp. 203–211, 1996. [18] T. A. Runkler, “Selection of appropriate defuzzification methods using application specific properties,” IEEE Trans. Fuzzy Syst., vol. 5, no. 1, pp. 72–79, Jan. 1997. [19] H. Tanaka and J. Watada, “Possibilistic linear systems and their application to the linear regression model,” Fuzzy Sets Syst., vol. 272, pp. 75–289, 1988. [20] H. Xie, R. L. Mahajan, and Y. C. Lee, “Fuzzy logic models for thermally based microelectronic manufacturing processes,” IEEE Trans. Semiconduct. Manufact., vol. 8, no. 3, pp. 219–226, Mar. 1995. [21] K. K. Yen, S. Ghoshary, and G. Roig, “A linear regression model using triangular fuzzy number coefficients,” Fuzzy Sets Syst., vol. 106, pp. 167–177, 1999. C. K. Kwong received the M.Sc. degree in advanced manufacturing systems from the University of Nottingham, U.K., and the Ph.D. degree in manufacturing engineering from the University of Warwick, U.K. He is currently an Associate Professor in the Department of Industrial and Systems Engineering of the Hong Kong Polytechnic University, Hong Kong. His research interests include process modeling, new product development, and the application of computational intelligence in design and manufacturing.

Y. Chen received the B.Sc. degree in mechanical engineering from the Shenyang Institute of Aeronautical Engineering, China, the M.Sc. degree in aeronautical engineering management from Beihang University, China, and the Ph.D. degree in mechanical engineering from the Northeastern University, China. He is currently a Research Associate in the Department of Industrial and Systems Engineering of the Hong Kong Polytechnic University, Hong Kong. His research interests include the application of artificial intelligence techniques in design and manufacturing and quality function deployment.

K. Y. Chan received the M.Phil. degree in electronic engineering from the City University of Hong Kong and the Ph.D. degree in computing from the London South Bank University, U.K. He is currently a Research Associate in the Department of Applied Mathematics of the Hong Kong Polytechnic University. His research interests include computational intelligence and its applications in product design, signal processing, power systems, and operation researches.

H. Wong received the B.Sc. degree in mathematics from the Chinese University of Hong Kong, the M.Sc. degree in statistics from the University of Newcastle upon Tyne, U.K., and the Ph.D. degree in statistics from the University of Hong Kong. He is currently an Associate Professor in the Department of Applied Mathematics of the Hong Kong Polytechnic University. His research interests include statistical computations, forecasting, and conditional heteroscedastic time series modeling.