Fuzzy Delphi and back-propagation model for sales forecasting in PCB industry

Transcription

1 Expert Systems with Applications 30 (2006) Fuzzy Delphi and back-propagation model for sales forecasting in PCB industry Pei-Chann Chang a, *, Yen-Wen Wang b a Department of Industrial Engineering and Management, Yuan-Ze University, 135 Yuan-Dong Road, ChungLi, Taoyuan 32026, Taiwan, ROC b Department of Industrial Engineering and Management, Chin-Yun University, 229 Chien-Hsin Road, ChungLi, Taoyuan 320, Taiwan, ROC Abstract Reliable prediction of sales can improve the quality of business strategy. In this research, fuzzy logic and artificial neural network are integrated into the fuzzy back-propagation network (FBPN) for sales forecasting in Printed Circuit Board (PCB) industry. The fuzzy back propagation network is constructed to incorporate production-control expert judgments in enhancing the model s performance. Parameters chosen as inputs to the FBPN are no longer considered as of equal importance, but some sales managers and production control experts are requested to express their opinions about the importance of each input parameter in predicting the sales with linguistic terms, which can be converted into pre-specified fuzzy numbers. The proposed system is evaluated through the real world data provided by a printed circuit board company and experimental results indicate that the Fuzzy back-propagation approach outperforms other three different forecasting models in MAPE measures. q 2005 Elsevier Ltd. All rights reserved. Keywords: Sale forecasting; Fuzzy theory; Neural network; Fuzzy back-propagation network 1. Introduction The printed circuit board (PCB) industry has grown up with the rapid development of the electronic, information and communication industries recently. During the past 30 years, the PCB industry in Taiwan has been striving to improve the manufacturing techniques, increase the production equipments, and strengthen the quality control, in order to integrate the developments of up-stream, middle stream and down-stream industries. This endeavor had successfully made Taiwan be ranked top 3 in the world for the total production amount of PCB. However, the overall accomplishment of PCB industry has been decreased recently by the influence of the profit conditions of the down-stream industries such as information, communication and consuming electronic industries. To decrease a cost means to increase a profit. Hence, in order to improve the enterprise s competitiveness, the executives should be able to make correct decisions using the available information, and forecasting is viewed as an * Corresponding author. Tel.: C ; fax: C address: iepchang@saturn.yzu.edu.tw (P.-C. Chang) /$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi: /j.eswa important part of decision making. Reliable forecasting of sales can help to make an effective inventory control and a proper scheduling process to increase the usage percentage of machines, which can avoid works being held up for lack of materials. To provide appropriate decisions and help the policy maker judge correctly is the basis of the production planning, with the end of decreasing the overall costs. Thus, all enterprises are working on the exploitation of prediction methods, which decide the success and failure of an enterprise. When dealing with the problems of sales forecasting, many researchers have used hybrid artificial intelligent algorithms to forecast, and the most rewarding method is the application integrating artificial neural networks (ANNs) and fuzzy theory. This method is applied by incorporating the experience-based principal and logic-explanation capacity of fuzzy theory and the capacity of memory and error-allowance of ANNs, as well as self-learning by numerical data. This research focuses on the monthly sales forecasting of PCB and applies the fuzzy Delphi to select variables with a better and more systematic way from expert experience. These input variables will be converted into pre-specified fuzzy numbers; aggregated and then fed into the FBPN for monthly sales forecasting, with the purpose of improving

2 716 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) the forecasting accuracy and using this information to help managers in decision-making. 2. Literature review Traditional methodologies of sales forecasting can be divided as follows: 1. Qualitative method Qualitative method belongs to more subjective approaches, which transform qualitative data into quantitative estimates based on estimations and opinions. This method is best adopted when the available data are not sufficient, e.g. when a new product is introduced to the market. Generally it includes Delphi Method, Market Research, Panel Consensus, Visionary Forecast and Historical Analogy, etc. 2. Time series analysis It mainly forecasts the future demands by using the data of demands in the past. In traditional forecasting models, no matter how long the forecasting time takes, it can be done by the available information analyzed through any kinds of statistical methods. The most popular models are Moving Average, Exponential Smoothing, Box Jenkins, and Trend Projections, etc. 3. Cause and effect analysis This method adopts the technique of linear regression and attempts to investigate the cause-and-effect relationship between the predicted items and other related factors. This method can be used when the historical data can provide enough information to analyze and explain the relative factors of the forecasting items. The most commonly used models include Regression analysis and Econometric Model, etc. Although, the above three traditional methods have been proved effective, they still have certain shortcomings. Kuo and Chen (2004) believed that the traditional statistic approaches have higher performance dealing with data of seasonality and trend, but they are inappropriate for unexpected situations. Tang (2003) proposed that the Time Series and Cause and Effect analyses are obviously insufficient when dealing with more complex problems or the reciprocal function between the non-linear system and its factors. Luxh et al. (1996) claimed that many common qualitative models lack the abilities of systematic structure and judgment, which makes the final result imprecise. As discussed by Kuo and Xue (1998), the new developed Artificial Intelligent (AI) models have more flexibility and can be used to estimate the non-linear relationship, without the limits of traditional Time Series models. Therefore, more and more researchers tend to use AI forecasting models to deal with forecasting problems. Kim and Han (2000) proposed genetic algorithms (GAs) approach to feature discretization and the determination of connection weights for artificial neural networks (ANNs) to predict the stock price index. Experimental results show that GA approach to feature discretization model outperforms the other two conventional models. Chang et al. (2004) proposed a hybrid model for stock price forecasting by integrating Multiple Regression, Back Propagation neural network and an Autoregressive integrated moving average model. Versace et al. (2004) evaluate the performance of a heterogeneous mixture of neural network algorithms for predicting the exchange-traded fund DIA. A genetic algorithm is utilized to find the best mixture of neural networks, the topology of individual networks in the ensemble, and to determine the features set. Chang et al. (2005) developed an Evolving Neural Network (ENN) forecasting model by integrating Genetic Algorithms and Neural Network for sales forecasting in PCB industry. The experimental result shows that the performance of ENN is superior to traditional statistical models and Back Propagation Network. Chang and Lai (2005) proposed a hybrid system to combine the self-organizing map (SOM) of neural network with case-based reasoning (CBR) method, for sales forecasting of new released books. Delphi Method was first applied to the personnel management of pilots as early as Recently, the fuzzy concept was embedded in Delphi Method by calculating the average weights of all the factors from the worst to the best degree based on the expert s experience. Chang et al. (2000) mentioned that Fuzzy Delphi Method could be applied to deal with the fuzzy relationship of the predicted items since the fuzzy number of each factor can explain clearly how independent variables are kept in the fuzzy forecasting models. When using Fuzzy Delphi Method to select the evaluation factors, there are two points needed to be considered: (1) the correctness of the collected factors. (2) The appropriateness of selecting the expert group. Besides, Kaufmann (Kaufmann & Gupta, 1988) mentioned that this method has the following advantages: 1. To decrease the times of questionnaire survey 2. To avoid distorting the individual expert opinion 3. To clearly express the semantic structure of predicted items 4. To consider the fuzzy nature during the interview process. Fuzzy Theory was proposed in a paper written by professor L.A. Zadeh of U.C. Berkeley, which was published in the journal Information and Control in Professor Zadeh presented the concept of Fuzzy Sets, which mainly investigates the fixed-quantity method of human s subjective thinking process. After evolving for more than 30 years, fuzzy system has been widely applied to Auto-Control, Pattern Recognition, Decision Analysis, Forecasting, and Time Series Signal Process to this date. Fuzzy theory is first combined with ANNs by Lin and Lee (1991), who incorporated the traditional fuzzy

3 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) controller and ANNs to a network structure to proceed appropriate non-linear planning of unplanned control systems based on the relationship of input and output through the learning capacity of ANNs. Fuzzy theory combining with ANNs is applied in different areas and has positive performance. The following presents related literature reviews: Chen and Wang (1999) use fuzzy interpolation and fuzzy linear planning to forecast the prices of semi-conductor products. Hwang et al. (1998) applies fuzzy system to weather forecasting. Huarng (2001) did a case study to predict Taiwan s stock quotes. Xue (1994) proposed a hybrid forecasting system to solve the sales prediction problem of a current product under the complex and changeable environment. Dash et al. (1995) applied fuzzy neural network to predict the electricity quantity. The time series input parameters were hazed first by Gaussian membership function, and then proceeded the training by back propagation algorithms. Through each training process, the weights from hidden layer to output layer could be adjusted. The results showed that fuzzy neural network had less PAEs by comparing with those of the traditional ANNs. Chen (2003) applied fuzzy neural network in forecasting the wafer production time. The study first selected the parameters using backward elimination, and evaluated the importance degree of variables based on expert opinion. Then transformed the input parameters based on the subjective opinions into quantified numbers through fuzzy weights. The results showed that no matter in the aspects of Initial RMSE, Minimal RMSE or Number of epochs required, FBPN had better performance than traditional BPN. Kuo and Xue (1998) developed a forecasting model based on fuzzy algorithms to predict the unexpected promotion situations of businesses. The data was collected through expert questionnaires and using Fuzzy Delphi Method to haze the input parameters, then taking Gaussian as membership function to proceeded neural network learning. Kuo and Chen (2004) presented a decision support system for order selection problem in electronic commerce by fuzzy neural network approach supported by real-coded genetic algorithm. Both the simulation and real-life problem provided by an internationally OEM company results show that the proposed FNN can well learn the fuzzy IF-THEN rules. In conclusion, the literatures reviewed above generally had the following drawbacks: 1. The input parameters caused big changes, which made the learning process of the network ineffective. 2. Some parameters were considered vital but were not chosen as input parameters for the learning process of back propagation network. 3. When unexpected situation of data quantity occurred, such as product promotion, which could not demonstrate the trendy and seasonal features, the ANNs became inappropriate. 4. The over scale of network could prolong the executing time of training and testing and weaken the calculating ability of the network. Thus, this research will avoid the shortcomings mentioned above. 3. Problem definition 3.1. Data collection The data in this research are from an electronic company in Taiwan from 1999/2001 to 2003/2012. Monthly sales amount is considered as an objective of the forecasting model Explanations for research variables PCB industry is an industry with highly variant environment. Many factors will affect the demand of PCB products, such as the variation of macroeconomics and industry. In previous research by Chang et al. (2005), along with trend and seasonal factors considered by Winter s model, effective economical factors are chosen by the Grey Relation Analysis. However, in this research Stepwise Regression Analysis (SRA) and Fuzzy Delphi Method (FDM) will be applied to determine some main factors from many possible factors in three domains as inputs in the network. These possible factors are described in details as follows: 1. Market demand domain PCB is a main part of all electronic products; thereby PCB industry occupies a significant portion of Taiwan s manufacturing, which produces about 15% of the global production value. This research selects some computer equipments with the highest sales amount according to different product sales conditions respectively: personal computers ðf 1 1 Þ, notebooks ðf 1 2 Þ, motherboards ðf 1 3 Þ, monitors ðf 1 4 Þ and liquid crystal device (LCD) related products ðf 1 5 Þ. Where superscript represents the domain to be dealt with and subscript means the variable in this domain. 2. Macroeconomics domain The main purpose of this domain is to judge the current national economic situation by some quantified data. These indices include Gross National Product ðf 2 1 Þ, Unemployment Rate ðf 2 2 Þ, Consumer Price Index ðf 2 3 Þ, Index of Import Trade ðf 2 4 Þ and Index of Export Trade ðf 2 5 Þ, where superscript represents the domain to be dealt with and subscript means the variable in this domain. 3. Industrial production domain According to the category of Yearbook of Industrial Production Statistics, department of statistics, ministry

4 718 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) Fig. 1. Architecture of the FBPN model of economic affairs, PCB industry belongs to manufacturing. Therefore, this study lists five indexes related to manufacturing: 1. Manufacturing Product Index ðf 3 1 Þ 2. Manufacturing Sales Index ðf 3 2 Þ 3. Manufacturing Production value Index ðf 3 3 Þ 4. Semi-conduct Product Index ðf 3 4 Þ 5. PCB Production value ðf 3 5 Þ 3.3. Evaluating performance index In order to evaluate the accuracy and performance of different forecasting models, this research adopts Mean Absolute Percentage Error (MAPE) to evaluate the performance in each model. MAPE Z 1 n X n tz1 jf t KA t j A t where, F t is the expected value for period t, A t is the actual value for period t, S t the actual sales amount of PCB for period t, and n is the number of periods. The smaller the values of MAPE, the better the forecasting models will be; having smaller values means that the calculating results are closer to the historic data. 4. Methodology To provide a reliable and accurate prediction of sales, this research develops a fuzzy back-propagation network (FBPN) by integrating fuzzy logic and artificial neural network for sales forecasting in Printed Circuit Board (PCB) industry. The fuzzy back propagation network is constructed to incorporate production-control expert judgments in enhancing the model s performance. As illustrated in Fig. 1, there are three main stages in this FBPN model and each stage of the process is further described as follows: 1. Variables selection stage. Collect internal, i.e. historical data, and external data, i.e. data from three domains mentioned earlier, for all factors affecting PCB sales and then select input variables by using SRA and FDM to find the key variables for forecasting PCB industry.

5 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) Data preprocessing stage. Use Rescaled Range Analysis (R/S) to evaluate the effects of trend from serial observation according to sales appearing in the time order. The Winter s approach will be used to include the sales tendency if there exists an trend effect. 3. FBPN forecasting stage. Forecast the monthly sales of PCB industry by the FBPN model. MAPE of each model is calculated and the effectiveness of the FBPN is verified through the experimental results Variable selection stage To increase the efficiency of network learning, the following two methods were applied to select the main factors that would influence the PCB monthly sales Stepwise regression analysis (SRA) Stepwise regression analysis is applied to determine the set of independent variables that most closely affect the dependent variable. This is accomplished by the repetition of a variable selection. At each of these steps, a single variable is either entered or removed from the model. For each step, simple regression is performed using the previously included independent variables and one of the excluded variables. Each of these regressions is subjected to an F-test. If the variable small F value is greater than a user defined threshold (0.05), it is added to the model. When the variable large F value, is smaller than a user defined threshold (0.1), it is removed from the model. This general procedure is easily applied to polynomials by using powers of the independent variable as pseudo-independent variables. The statistical software SPSS for Windows 10.0 was used for stepwise regression analysis in this research Fuzzy Delphi Method (FDM) The Delphi Method was first developed by Dalkey and Helmer (1963) in corporation and has been widely applied in many management areas, e.g. forecasting, public policy analysis, and project planning. However, the conventional Delphi Method does not converge very well. Thus, the fuzzy set theory was applied in the Delphi Method to improve the effect. The FDM was used to choose the main factors, which would influence the PCB product sales quantity from all possible factors that were collected from the questionnaires in this research. The procedures of FDM are listed as follows: 1. Collect all possible factors, which may affect the PCB product monthly sales from the domain experts. This is the first questionnaire survey. 2. Conduct the second questionnaire and ask domain experts select assign a fuzzy number ranged from 1 to 10 to each factor. The fuzzy number represents the significance to the sales. 3. Fuzzify the second questionnaires that are returned by the domain experts and determine the following indices: (a) Pessimistic (Minimum) index [ A Z [ A1 C[ A2 C/C[ An n where [ Ai means the pessimistic index of the ith expert and n is the number of the experts. (b) Optimistic (Maximum) index u A Z u A1 Cu A2 C/Cu An n where u Ai means the pessimistic index of the ith expert. (c) Average (Most appropriate) index. For each interval [ Ai Cu Ai, calculate the midpoint, m Ai Zð[ Ai Cu Ai Þ=2, then find m A Zðm A1!M A2!/!m An Þ 1=n Therefore, the fuzzy number AZ(m,s R,s L ), which represents the mean, right width, and left width, respectively, for an asymmetric bell shaped function that can be determined through the above indices: s R Z [ AKm A 3 s L Z u AKm A 3 4. Finalize the significance number of each factor in the questionnaire according to the index generated in step Repeat 3 to Use the following formulas as the stopping criteria to confirm that all experts have the consentaneous significance number of each factor. dð A; BÞ Z ð 1 az0 dð A½aŠ; B½aŠÞda Z 1 2 ðb 2Kb 1 Þ K B½aŠ u jþda ð 1 az0 ðj A½aŠ L K B½aŠ L jþ Cðj A½aŠ U where A and B are the fuzzy numbers, A½ Š and B½ Š denote the membership function of fuzzy numbers. The a-cut of the fuzzy number is defined as A½aŠZfxj A½xŠR a; x2rg for 0!a%1. The distance between the two fuzzy numbers is dð A; BÞ. b 1 and b 2 are any given convenient values in order to surround both A½aŠZ 0 and B½aŠZ 0. The concept of dissemblance index of two fuzzy numbers is shown as in Fig. 2.

6 720 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) likely to be followed by decreases over all later time scales. When HZ0.5, the self-similar correlations are uncorrelated. When 0.5!H!1, the self-similar correlations at all time scales are persistent, i.e. increases at any time are more likely to be followed by increases over all later time scales. Fig. 2. The concept of dissemblance index of two fuzzy numbers Data preprocessing stage When the seasonal and trend variation is present in the time series data, the accuracy of forecasting will be influenced. This stage will use R/S analysis to detect if there is this kind of effects of serious data. If the effects are observed, Winter s exponential smoothing will be used to take the effects of seasonality and trend into consideration R/S analysis (rescaled range analysis) For eliminating possible trend influence, the rescaled range analysis, invented by Hurst (Hurst, Black, & Simaika, 1965), is used to study records in time or a series of observations in different time. Hurst spent his lifetime studying the Nile and the problems related to water storage. The problem is to determine the design of an ideal reservoir on the basis of the given record of observed discharges from the lake. The R/S analysis will be introduced as follows: Consider the XZ{x 1, x 2,., x n }, x i is the sales amount in period i, and compute M N where M N Z 1 N X t X i iz1 The standard deviation S is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P u N t ðx i KM N Þ 2 iz1 S Z N For each point i in the time series, we compute Xðt; NÞ Z Xt tz1 X i KM N R Z max Xðt; NÞK min Xðt; NÞ 1%t%N 1%t%N We computed the H coefficient as H Z Ln R =LnðaNÞ S Here az1. When 0!H!0.5, the self-similar correlations at all time scales are antipersistent, i.e. increases at any time are more Winter s exponential smoothing In order to take the effects of seasonality and trend into consideration, Winter s exponential smoothing is used to preliminarily forecast the quantity of PCB production. According to this method, three components to the model are assumed: a permanent component, a trend, and a seasonal component. Each component is continuously updated using a smoothing constant applied to the most recent observation and the last estimate. Mills (1990); Luxh et al. (1996) compared Winter s Exponential Smoothing with other forecasting methods, like ARIMA, and all showed that the Winter s method had a superior performance. In this research, we assume az0.1, bz0.1 and gz0.9. The data generating process is assumed to be of the form x t Z ða 0;t Ca 1;t ÞC t C3 t where C t seasonal factor x t a 0;t Z a Cð1KaÞða C 0;tK1 Ca 1;tK1 Þ tkn is exponentially smoothed level of the process at the end of period t x t actual monthly sales in period t N number of periods in the season (NZ12 months) a 1,tK1 trend for period tk1 a smoothing constant for a 0. The season factor, C t, is updated as follows C t Z g x t a 0;t Cð1KgÞC tkn where g is the smoothing constant for C t. For updating the trend component a 1;t Z fða 0;t Ka 0;tK1 Þ Cð1KfÞa 1;tK1 where f is the smoothing constant for a 1. Winter s forecasting model is then constructed by ^x t ða 0;t Ca 1;t ÞC t where ^x t is the estimate in time period t Fuzzy neural network forecasting stage There are many researches have been investigating the possibility of combining fuzzy theory and ANNs, among these, the concept of FBPN model is utilized to forecast the PCB amount in this research. The basic design concept of

7 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) Table 1 Notations I, h, O Present the input layer, hidden layer and output layer individually ~s ðiþ x ðiþ The fuzzy input signal in input layer ~w h ij The connection weight of input layer and hidden layer ~h _inj The input signal in the hidden layer ~h j The output signal in the hidden layer ~w h j The connection weight of hidden layer and output layer ~o _inj The input signal in the output layer ~o The output signal in the output layer O The forecasting value after defuzzification 0 d The adjusting value in the output layer D ~w o j The adjusting value of the connection weights 0 Dq d h j D ~w h ij D ~ q h j a between hidden and output layer The adjusting value of bias in the output layer The adjusting value in the hidden layer The adjusting value of the connection weights between input and hidden layer The adjusting value of bias in the hidden laye The normalized actual sales amount FBPN is very similar to that of BPN, and the system can be classified as neuron, layer, and network three parts. The difference is our FBPN takes the experts opinions into consideration and convert the qualitative data into fuzzy quantative data. These fuzzy quantative data will be input to the back-propagation network for further processing. Table 1 shows the notations that will appear in the following article. There are three main layers, input layer, hidden layer and output layer, and two training stages in our FBPN. In the feedforward stage, FBPN use the data on hand to forecast the PCB sales amount, and the forecasting error will be recalled to adjust the weights between layers in the backpropagation of error stage. The details will be described in the following: Step 0. Initial weights between layers are randomly generated. Step 1. While stopping condition is false, do step Step 2. For each training pair, do step 3 8. Feedforward stage: Step 3. Each input unit I j, which was generated by many experts, receives input signal ~s ðiþ x ðiþ and broadcasts this signal to all units in the hidden layer where ~s ðiþ is the fuzzy membership function, which is supported by the experts, and x (i) is the normalized input signal. Step 4. Sum the weighted input signals of each hidden unit h j Step 5. Apply the translation function to compute its output signals 1 ~h j Z ðh j1 ; h j2 ; h j3 Þ 1 Ce K~nh j Z ; ; ; 1 Ce K~nh j1 1 Ce K~nh j2 1 Ce K~nh j3 j Z 1;.; n; where ~n h j Z ðn h j1; n h j2; n h j3þ Z ~I h j ðkþ ~ q h j Z ði h j1kq h j3; I h j2kq h j2; I h j3kq h j1þ Step 6. Sum the weighted input signals of each output unit O i ~I o Z ði o 1; I o 2; I o 3Þ Z X ~w o j ð!þ ~h j y X w o j1h j1 ; X w o j2h j2 ; X w o j3h j3!; allj allj allj allj Step 7. Apply the translation function to compute its output signals ~o Z ðo 1 ; o 2 ; o 3 Þ Z 1 1 Ce K~no Z ; ; 1 Ce Kno 1 1 Ce Kno 2 1 Ce Kno 3 where ~n o Z ðn o 1; n o 2; n o 3Þ ZK~I o ðkþ ~ q o Z ði o 1 Kq o 3; I o 2 Kq o 2; I o 3 Kq o 1Þ Step 8.Defuzzify the output signals to the forecasting value o Z defuzzificationð ~oþ Z 1 4 ðo 1; 2o 2 ; o 3 Þ and compute its MAPE. Backpropagation of error stage: Step 9. Compare the forecasted output with the actual sales amount and compute the error term between hidden layer and output layer d o Z oð1koþðakoþ Next, calculate its weight correction term (used to update connection weights latter) D ~w o j Z ðdw o j1; Dw o j2; Dw o j3þhd o h j Z ðhd o h j1 ; hd o h j2 ; hd o h j3 Þ ~I h j Z ðij1; h Ij2; h Ij3Þ h Z X ~w h ijð!þ~xðiþ alli y X alli w h ij1xðiþ1; X alli w h ij2xðiþ2; X alli w h ij3xðiþ3! Finally, calculate its bias correction term, Dq o ZKhd o and update weights and biases. Step 10. Compute its error information term for hidden nodes. ~ d h j Zð ~ d h j1; ~ d h j2; ~ d h j3þz ~h j ð!þð1k ~h j Þð!Þ ~w o j d o yðh j1 ð1kh j1þw o j1d o ; h j2 ð1kh j2 Þw o j2d o ; h j3 ð1kh j3 Þw o j3d o Þ Then, update the information term of each hidden node.

8 722 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) Table 2 The experimental result of stepwise regression analysis Variables MAPE (%) Accuracy (%) Variables selected by stepwise regression analysis Sales in monitors ðf4 1 Þ Sales in TFT-LCD components ðf5 1 Þ Sales in PCB ðf3 1 Þ Introducing of Winter s exponential smoothing Performance Improvement (BPN), etc. In addition, various experiments will be conducted to verify the following: (1) comparisons of feature selection methods, (2) comparisons of performance measure in different forecasting methods Comparisons of variable selection methods Fig. 3. The detailed flow diagram of FBPN Step 11. Calculate its weight correction term between hidden layer and input layer D ~w h ij Z ðdw h ij1; Dw h ij2; Dw h ij3þ Z h ~ d j hð!þ~x i Z ðhd h j1x i1 ; hd h j2x i2 ; hd h j3x i3 Þ Then, calculate its bias correction term. D ~ q h j Z ðd ~ q h j1; D ~ q h j2; D ~ q h j3þ ZKh ~ d h j Z ðkhd h j3;khd h j2;khd h j1þ Finally, update weights and biases. The detailed flow diagram of FBNP is shown as in Fig. 3. The configuration of the FBPN is established as follows: Two different methods have been applied as variable selection methods in this research and they are SRA and FDM. When applied SRA in the experimental test, not considering the linear tendency, the data is input to backpropagation model for training and testing. The final error is 13.87% with an accuracy of 88.02% and the result is not very convincing. But after take the tendency into consideration, the average error is down to 7.15% with an accuracy of 93.50%. The interesting result after introducing the tendency into the data is that the error has dropped 6.72% with the accuracy improving 5.48%. Therefore, as shown in Table 2, the SRA performs much better after incorporating the tendency into the data. In Fig. 4 shows the difference before and after introducing the tendency factor into the model. (1) Number of neurons in the input layer: 5 (2) Number of neurons in the output layer: 1 (3) Single hidden layer (4) Number of neurons in the hidden layer: 5 (5) Network-learning rule: delta rule (6) Transformation function: sigmoid function (7) Learning rate: 0.1 (8) Learning times: 30, Experimental results This research develops a FBPN for sales forecasting in PCB industries and we will compare this method with other traditional methods such as Grey Forecasting (GF), Multiple Regression Analysis (MRA) and Back-propagation network Fig. 4. The performance improvement of stepwise regression analysis after considering tendency factor

9 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) Table 3 The experimental result of fuzzy Delphi Method Variables MAPE (%) Accuracy (%) Variables selected by fuzzy Delphi Method Sales in notebooks ðf2 1 Þ Sales in TFT-LCD components ðf5 1 Þ Sales in PCB ðf3 1 Þ Manufacturing product index ðf1 3 Þ Introducing of Winter s exponential smoothing Performance improvement When applied FDM, not considering the linear tendency, the data is input to back-propagation model for training and testing. The final error is 12.88% with an accuracy of 88.86% and the result is not very convincing. After take the tendency into consideration, the MAPE is dropped to 6.19% with a forecasting accuracy of 94.57%. The performance improvement after introducing tendency into the model in MAPE is 6.69% and the accuracy is also improved 5.71%. Therefore, as shown in Table 3, the FDM performs much better after incorporating the tendency into the data. In Fig. 5 shows the difference before and after introducing the tendency factor into the model. In comparisons of the effectiveness of feature selection by SRA and FDM (not considering the tendency factor). The FDM performs better and its MAPE is lower than SRA by 0.99% while the accuracy is higher by 0.84%. When take the tendency into consideration, the FDM is also superior to the SRA and its MAPE is lower by 0.96%, while the accuracy is higher by 1.07%. As shown in Fig. 6, the graph derived from FDM is much closer to the real situations, which means the error is also smaller when compared with the SRA approach. v 1t and ^b 1 is the least squares estimate, then y t ZX 0 1t ^b 1. Model 1 encompasses model 2 if a 1 Z0, a 2 Z0; in this case, models 2 is not helpful in explaining y t, conditional on model 1, and model 1 gives an unbiased forecast of y t. Four different forecasting models, i.e. Grey Forecasting (GF), Multiple Regression Analysis (MRA), BPN, and FBPN are tested by pair to pair combination using the encompassing test, the value (a 1, a 2 ) as shown in Table 4 shows which model can encompasses to the other. According to Table 4, from the point of view of encompassing test, we know in case 1 that GF and MRA is not mutually covered. This phenomenon shows that both models all include important forecasting information; and each model cannot be encompassed by one or the other. The same situation exists in the case which BPN and FBPN cannot be encompassed by one or the other. While in other cases, one can observe clearly that BPN and FBPN all can encompass the GF and MRA methods. These tests show the superiority of the neural network approaches. The conclusion that can be derived from the encompassing test: in terms of forecasting accuracy, BPN and FBPN are far superior to GF and MRA. As for the combination 1 since GF and MRA and combination 2 BPN and FBPN are not covered by each other and they will be further compared in the performance measure of the mean absolute percent of errors, i.e. MAPE Performance measure in terms of forecasting errors This research applies the MAPE as a standard performance measure for all four different models. After the intensive experimental test, the MAPEs of four different models are 15.04, 8.86, 6.19 and 3.09%. Among that, the grey forecasting has the largest errors, and then MRA, BPN, and the least is FBPN Comparisons of performance measure from different forecasting models Forecast encompassing test Forecast encompassing tests seek to evaluate whether competing forecasts may be fruitfully combined to produce a forecast superior to individual forecasts. The concept of encompassing tests is: if forecast model A cannot encompass the forecast model B, then there must has some information in the model B but not in model A, and these information will affect the forecast results. Suppose for simplicity that there are only two models, models 1 and 2. Write the encompassing regression as y t Z a 1 ~y 1t Ca 2 ~y 2t Ca 0 here, y t is a scalar variable explained by models 1 and 2, ~y it is the predicted value from model i. ~y 1t and ~y 2t are constructed from estimates of finite dimensional parameter vectors b 1 and b 2. For example, if model 1 is y t ZX 0 1t ^b 1 C Fig. 5. The performance improvement of fuzzy Delphi after considering tendency factor.

10 724 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) Fig. 6. The graph from real data, FDM and SRA Table 4 Encompassing Test for Four forecasting models Model B Model A GF MRA BPN MRA (1, 1) BPN (1, 0) (1, 0) FBPN (1, 0) (1, 0) (1, 1) As can be seen in Fig. 7, the GF has a significant up and down in the beginning and it also over estimate the data up to the end. Thus the overall MAPE is high. As for MRA, the tendency is formed and the up and down is minor compared with GF. The overall MAPE is around 0 20% and it is also a little be higher. Traditional BPN model is in a stable situation and the overall MAPE is smaller than MRA and it is around 0 10%. The same situation exist for FBPN although in the beginning it has a larger error however it converge quickly and the overall MAPE is still around 0 10%. Especially, it performs very well in the end since it is very close to the real data. Therefore, the FBPN performs the best among others (Table 5). According to the various criteria, i.e. encompassing test, MAPE, and forecasting accuracy, the best model among these four different models is FBPN with a MAPE of 3.09% and accuracy of 97.61%. Therefore, we can claim that by combining the fuzzy theory and BPN the hybrid model can be applied in forecasting the sales of PCB industry and the result is very convincing and deserve further investigation in the future for applications in other areas. Although, the GF and MRA is very powerful when the data is very scarce and they claim that with only four data points and they can be applied to forecast the future result. However, after intensive experimental test, these two methods did not perform very well especially for those non-linear and highly dynamic data. As for the fuzzy Delphi back-propagation model since it can include the opinion from various experts in PCB sales and production department. It seems the assignment of different weight to Fig. 7. The MAPE values for four different forecasting models

11 P.-C. Chang, Y.-W. Wang / Expert Systems with Applications 30 (2006) Table 5 Comparisons among four different forecasting models MAPE (%) Improvement rate (%) GF MRA BPN FBPN 3.09 the factor really improve the forecasting errors and perform much better than other models. 6. Conclusions Recently, more and more researchers and industrial practitioners are interested in applying fuzzy theory and neural network in their routine problem solving. This research combines fuzzy theory and back-propagation network into a hybrid system, which will be applied in the sales forecasting of PCB industries. The major characteristics of this FBPN include the followings: 1. Data collections. The data applied in this research are derived from the historic data from a PCB company located in Chung-Li, Taiwan, ROC. 2. The input variables in traditionally BPN network are not processed at all and they are unconditionally input into BPN for further processing during the training procedure. However, this may come out with large training errors. To correct this flaw, this research applies the SRA, step-by-step to filter out the unrelated factors and keep only those factors, which have significant effects to the sales of PCB factory. Therefore the accuracy of the forecasting results can be further improved. 3. Through the introduction of FDM, the opinion of various experts can be elucidated and incorporated into the input variables. The linguistics structure of the input variables can be designed in the questionnaires and various experts will express their personal opinion through the questionnaires. It is a very useful method for collecting data and assigning weight to various variables. The experimental results in Section 5 demonstrated the effectiveness of the FBPN that is superior to other traditional approaches. In summary, this research has the following important contribution in the sales forecasting area and these contributions might be interested to other academic researchers and industrial practitioners: 1. Feature selections. To filter out significant factors from a series of input variables, the FDM is superior to the SRA method. FDM will collect the opinion from various experts and assign different weights to these variables according to their experiences in this field. Therefore, it is very easy to extract important factors from these various variables. In contrary, gradual regression analysis may come out with a combination of various variables, which is mutually correlated. However, the effect of these selected variables may not significant enough to be included in the final inputs. The errors for input from fuzzy Delphi is 12.88% and errors from SRA is 13.87%. It is obvious to see that FDM is more effective for applications. 2. The effect of tendency. When take tendency effect into consideration, the overall errors are decreased. Tendency and seasonality are included in the time series data and these two factors will affect the accuracy of the forecasting method dramatically. This research applies the Winters trend and seasonality exponential smoothing model to forecast the sales and then convert this data as an input to the BPN model. After the training procedure, the final errors, no matter it is from FDM or SRA, are decreased significantly. Errors from gradual regression analysis decreased from to 7.15% and FDM from 12.88% down to 6.19%%. This shows the significance of including Winters trend and seasonality exponential smoothing model in the model. 3. Comparisons of different forecasting models. This research applies three different performance measures, i.e. encompassing test, forecasting errors and accuracy of forecasting to compare the FBPN with other three methods, i.e. GF, MRA and BPN. The intensive experimental results show the following: 1. In encompassing test, FBPN and BPN models are superior to GF and MRA. 2. As for MAPE, FBPN has the smallest MAPE and it is only 3.09%. Therefore, FBPN model by combining FDM and BPN model is a very powerful and effective forecasting tool that can be further applied in other field of applications since expert s opinion can be incorporated into the model. References Chang, P. C., & Lai, C. Y. (2005). A hybrid system combining selforganizing maps with case-based reasoning in wholesaler s new-release book forecasting. Expert Systems with Applications, 29(1), Chang, P. C., Wang, Y. W., & Tsai, C. Y. (2005). Evolving neural network for printed circuit board sales. Expert Systems with Applications, 29(1), Chang, P. C., Wang, Y. W., & Yang, W. N. (2004). An investigation of the hybrid forecasting models for stock price variation in Taiwan. Journal of the Chinese Institute of Industrial Engineering, 21(4), Chang, P. T., Huang, L. 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