Research Article Utilizing an Adaptive Grey Model for Short-Term Time Series Forecasting: A Case Study of Wafer-Level Packaging

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1 Matheatical Probles in Engineering Volue 2013, Article ID , 6 pages Research Article Utilizing an Adaptive Grey Model for Short-Ter Tie Series Forecasting: A Case Study of Wafer-Level Packaging Che-Jung Chang, 1 Der-Chiang Li, 2 Wen-Li Dai, 3 and Chien-Chih Chen 2 1 Departent of Business Adinistration, Chung Yuan Christian University, No. 200, Chung-Pei Road, Chung-Li City, Taoyuan County 32023, Taiwan 2 Departent of Industrial and Inforation Manageent, National Chen Kung University, No. 1, University Road, Tainan City 70101, Taiwan 3 Departent of Inforation Manageent, Tainan University of Technology, No. 529, Zhongzheng Road, Tainan City 71002, Taiwan Correspondence should be addressed to Der-Chiang Li; lidc@ail.ncku.edu.tw Received 7 April 2013; Revised 28 June 2013; Accepted 1 July 2013 Acadeic Editor: Dong Sun Lee Copyright 2013 Che-Jung Chang et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. The wafer-level packaging process is an iportant technology used in seiconductor anufacturing, and how to effectively control thisanufacturingsysteisthusaniportantissueforpackagingfirs.onewaytoaidinthisprocessistouseaforecastingtool. However, the nuber of observations collected in the early stages of this process is usually too few to use with traditional forecasting techniques, and thus inaccurate results are obtained. One potential solution to this proble is the use of grey syste theory, with its feature of sall dataset odeling. This study thus uses the AGM(1,1) grey odel to solve the proble of forecasting in the pilot run stage of the packaging process. The experiental results show that the grey approach is an appropriate and effective forecasting tool for use with sall datasets and that it can be applied to iprove the wafer-level packaging process. 1. Introduction Firs can gain copetitive advantages by ore effectively controlling their anufacturing systes [1]. In this context, the early stages of a anufacturing syste are especially iportant for anagers, because any slight change in the paraeters used at this tie will significantly influence final product quality and anufacturing perforance [2]. Tiely and precise inforation is thus needed for effective operations anageent [3]. However, few observations are available at the early stages of a anufacturing syste, eaning that statistical theory and data ining techniques cannot obtain accurate results [4, 5]. The introductory stage of the wafer-level packaging (WLP) process is an exaple of a sall dataset proble. The WLP process is a new, advanced technology for the packaging industry, and thus anufacturers do not have uch experience to draw its for when seeking to iprove it. However, despite the lack of data, engineers still need to understand the features of this production syste to aintain a high anufacturing yield [2]. Consequently, an appropriate forecasting tool that can deal with incoplete inforation based on sall datasets is required for effective anageent of the WLP process. Grey syste theory was proposed by Deng (1982) [6] and is used to exaine uncertain systes based on incoplete inforation [7, 8]. The ain principle of this approach is to process the data indirectly through data apping to transfor the state space, so as to extract hidden inforation in the data [9, 10]. Because of its ease of use, grey syste theory has been successfully applied in various doains [11 17]. Many studies have deonstrated that the grey odel is an effective approach for data analyses with sall saples, and it is applied in this research to the packaging process of seiconductor anufacturers, in order to solve forecasting problesinthepilotrunstage.thegreyodeldeveloped in this work is preexained by four different easureents and copared with two other popular forecasting ethods

2 2 Matheatical Probles in Engineering to exaine its flexibility and value in this context. The results showthatthegreyapproachisaneffectivetoolthatcanenable anufacturerstoakeoreaccurateforecastsbasedonliited observations collected in the early stages of the packaging process. The rest of this paper is organized as follows. Section 2 introducesthedevelopentoftheadaptivegreyodelused in this work. In Section 3, we briefly describe the proble of interest in the pilot run of the wafer-level packaging process, where the applicability of the grey approach is deonstratedinthisstudy.finally,theconclusionsarepresentedin Section Methodology While the conventional grey forecasting odel, GM(1,1), has been widely applied, it is still possible to iprove its forecasting perforance. Li et al. [18] proposed a revised grey odel, called AGM(1,1), which is one of the best and ost accurate of the existing odels. It uses the trend and potency tracking ethod [19] to for a function to deterine the background values of the grey odel. This odel can better reflect the data growth trends at different stages of the process and overcoes soe of the weaknesses of the ordinary odeling procedure in the GM(1,1), thus producing ore accurate forecasts. Therefore, this study uses this ethod to solve the forecasting proble that arises in the pilot run of the waferlevel packaging process. The AGM(1,1) odeling procedure has two ain parts, which are (1) deterining the trend and potency value and (2) building the grey forecasting odel. Both of these will be described in the following subsections Trend and Potency Tracking Method. Li and Yeh [19] proposed the trend and potency tracking ethod (TPTM), whichisananalysisethodthatusesthecharacteristicsof data to explore possible changes in data behavior in different stages of a process, and this is the key concept used in AGM(1,1) to iprove the accuracy of the conventional grey forecasting odel. The detailed procedure of the TPTM is as follows. Step 1. Given the original data series, X={x 1,x 2,...,x n },let x in be the iniu value in X and x ax the axiu one. Step 2. Figure the variations σ i of paired data (x i 1,x i ), i = 2,3,...,n, and deterine the increasing or decreasing potencies according to the data sequence. If σ i concerning the paired data (x i 1,x i ) is positive, it indicates that the trend of data at phase i is oving upward. Conversely, the oveent of the data at phase i is descending if σ i <0. Step 3. Set the biggest weight to the latest datu as the intensity of different phases, since it doinates the characteristics of the upcoing datu. The iportance w i of the datu at phase i equals i 1, i=2,3,...,n. Step 4. Let A i = σ i w i, i = 2,3,...,n,asanoperatorto strengthen the data trend and potency at different phases by ultiplying weights and variations together. Note that A i >0 TP value EDR LL p t x in q 1 CL x ax EDR UL Figure 1: An exaple of the triangular TP function. is an increasing potency (IP) and A i potency (DP). < 0 is a decreasing Step 5. Deterine the central location (CL) of existing data using the equation CL = (x in +x ax )/2. TheCListhen utilized as the ain point to conduct the asyetric doain range expansion. Step 6. Copute the average of increasing potencies (AIP) and the average of the decreasing potencies (ADP), and then use the to asyetrically extend the doain range. The upper liit of the extended doain range is EDR UL = x ax + AIP, and the lower liit is EDR LL =x in + ADP. We thus obtain an extended doain range within which to explore extrainforation about the data trend and potency. Step 7. For a triangular TP function using the CL, EDR UL, and EDR LL. Here, we set the TP value of the CL to be 1, and then we can obtain the TP values of existing data through the ratio rule of a triangle. Figure 1 illustrates an exaple and the TP value of x in is t = p/(p + q), wherep is the distance between EDR LL and x in,and q is the distance between x in and CL. The range of the TP value is between 0 and 1, and the TP value reveals that the current datu s intensity is closetothecl Adaptive Grey Forecasting Model. The background value of GM(1,1) is the ost iportant factor that affects odel construction and the final forecasting results. Li et al. [18] studied the ipact of differences in the background value on forecasting perforance and proposed an iproved grey forecasting odel, AGM(1,1), by integrating the concept of TPTM into the forula for calculating the background value. This study uses this approach to deal with forecasting probles in a sall dataset. The AGM(1,1) odel is described as follows. Step 1. Suppose the original data series is X (0) = {x (0) (1), x (0) (2),...,x (0) (n)}, n 4. Step 2. Calculate the TP values by TPTM; that is, {TP i } = {TP 1, TP 2,...,TP n },,2,...,n. Step 3. Copute the coefficient α k α k = k 2i 1 TP i k 2i 1, k 2. (1)

3 Matheatical Probles in Engineering 3 Incoing wafer Coat dielectric layer Deposit passivation layer Deposit under bup etallurgy Grind wafer back Ship to custoer Saw wafer Electrical test Reflow solder ball Drop solder ball Figure 2: Flowchart of the wafer-level packaging process. Step 4. For a new data series through the accuulating generation operator (AGO) X (1) ={x (1) (1),x (1) (2),...,x (1) (n)}, x (1) (k) = k x (0) (i), x (1) (1) =x (0) (1) ; k=2,3,...,n. Step 5. Deterine the background values z (1) (k) z (1) (k) =(1 α k )x (1) (k 1) +α k x (0) (k), α (0, 1), k=2,3,...,n. Step 6. Estiate the developing coefficient a and the grey input b fro (4) by the ordinary least-square ethod and establish the grey differential equation fro (5)to replace the source odel (2) (3) x (0) (k) +az (1) (k) =b, (4) dx (1) +ax (1) =b. (5) dt To estiate a and b, we expand (4)as Let x (0) (2) z (1) (2) 1 x (0) (3) z (1) (3) 1 [ =. ] [.. ] [ x (0) (n)] [ z (1) (n) 1] Y=[x (0) (2),x (0) (3),...,x (0) (n)] T, a =[a, b] T, z (1) (2) 1 z (1) (3) 1 B=, [.. ] [ z (1) (n) 1] a =(B T B) 1 B T Y. [ a ]. (6) b (7) Step 7. Solve (5) together with initial condition x (0) (1) = x (1) (1), and the desired forecasting output at step k+1can be obtained by (8) x (1) (k+1) =(x (0) (1) b a )e ak + b a, x (0) (k+1) = x (1) (k+1) x (1) (k). 3. Experiental Studies: Forecasting the Wafer-Level Packaging Process The applicability of the adaptive grey forecasting odel is exained here by an experient, and its details are described in the following subsection Background of Wafer-Level Packaging. Seiconductor anufacturing is an iportant link in the supply chain of consuerelectronicsandhassignificanteffectsonthequality and characteristics of the final products. The packaging ethod used in this industry is increasingly the waferlevel packaging (WLP) rather than the chip-scale packaging, as this can enable saller package sizes. Moreover, WLP can allow anufacturers to integrate the wafer fabrication, packaging, testing, and burn-in processes. In practice, WLP is a technology that packages an integrated circuit at the wafer level, instead of the traditional process of assebling individual units in packages after dicing the fro a wafer. Many different aterials are cobined on a substrate in WLP to for a chip, as shown in Figure 2, so the physical properties of this cobination depend on the rigidity of aterials, the theral expansion coefficient, and the teperature variation aong the. There are five quality inspections in the WLP process to ensure the process standards, and these exaine the passivation layer defects, diaeter size, coplanarity, etal bup defects, and the height of etal bups. Further, because the etal bup height affects the overall qualityinthepackagingprocess,thedetectionofthisisvery iportant and needs to be effectively controlled to aintain a good yield. In addition, while solder balls are generally used for surface ounting, because they have a relatively low elting point, they can also suffer fro creep and plastic behavior during ounting under coon operating conditions.thiseansthattheheightofsolderballsisunstable in the packaging process, which can adversely affect the quality of the final products. In short, if engineers can better understand the oving trend of the solder ball height and (8)

4 4 Matheatical Probles in Engineering adjust the process paraeters in a tiely fashion, then this can help iprove overall anufacturing efficiency Experiental Data and Procedure. This research eploys data collected fro a leading packaging anufacturer in Taiwan. The case copany is a wafer original equipent anufacturer, with relatively little experience of WLP production at the tie when the data was collected. The experiental tie series data were collected at the end of 2011, the period when the case copany was ipleenting a new WLP process and subjecting it to postinstallation testing. This pilot run yielded a total of thirteen observations, as shown in Table 1.Theunitofeasureentforthe height in this table is icroeters (μ). The ai of this study is to build an effective forecasting odel for the height of solder balls over tie. Four data are used to construct the forecasting odel, with the rest of the data used one at a tie fortesting,sothatatotalofninesetsofdataareobtained Coputation of the AGM(1,1) Model. To illustrate the odeling of the AGM(1,1) in ore detail, this study eploys theprecedingfourdata {192.16, , , } as an exaple to build the odel. The coputation process is shown below, and the results are listed in Table 2. (1) The original dataset is X (0) = {192.16, , , }. (2) Apply TPTM to calculate the TP values and get TP = {0, , , }. (3) Copute the coefficient α k = {0.5654, , }. (4) Use AGO to for a new series X (1) = {192.16, , , }. (5) Deterine the background values Z (1) = { , , }. (6) Calculate a =(B T B) 1 B T Y,where Y=[192.78, , ] T, B = [ ]. [ ] (9) We can get a = [ , ] T. (7) Use (8) to create the odel x (1) (k + 1) = e k and the corresponding output is predicted as x 5 = See Table Preevaluation Indexes for the Perforance of AGM(1,1) Model. To assess the reasonableness and applicability of the adaptive grey forecasting odel in the early stage of the WLP process, we first use four different easureents that are often used in pretesting odeling fitness in the grey syste theory [20] to evaluate the forecasting perforance. These are the ean relative error (MRE), absolute degree of grey incidence (ADGI), ratio of standard deviation (RSD), and Table 1: Wafer-level packaging process data (unit: icroeters). Order Height (μ) Order Height (μ) probability of sall error (PSE). We let be the su of testing saples, while x i, x i and e i are the predicted outputs, actual values and errors of the ith testing saple, respectively. The average and standard deviations of x i and e i,writtenas x, e, s x and s e, are then coputed by basic forulas in the statistics. The definitions of MRE, RSD, and PSE are presented below.sinceadgiisrelativelycoplex,itisnoteasyto define briefly here, and so please refer to an earlier study [21] for further inforation about this MRE = 1 x i x i x i, RSD = s e, s x PSE =P( e i e < s x). (10) The MRE is a easureent that is used to estiate the forecasting accuracy. The ADGI judges the closeness of the relationship between the actual and forecast values, based on the siilarity of the geoetric patterns of the sequence curves. It is often used to check whether or not a odel is able to appropriately reflect the data profile. The RSD is an indicator of the degree of discreteness of the errors, that is, the whole aplitude of the error fluctuation. The PSE shows what ratio of errors lie in the acceptable range. With these four easureent indexes, saller values of MRE and RSD represent better siulated results, while bigger ones are needed with ADGI and PSE [20, 21]. The levels used to exaine the accuracy of the odel and the critical values of the previous four indexes are shown in Table 3. These four easureents can help deterine whether a odel is accurate or not. If they fall within the range of levels 1 or 2, this suggests that the odel has high forecasting accuracy Coparison with Other Conventional Forecasting Approaches. To verify the effectiveness of the AGM(1,1) odel, we copare its results with those fro two popular forecasting techniques, the backpropagation neural network (BPNN) and support vector regression (SVR) approaches. BPNN is a neural network that has shown good forecasting ability in any contexts [22]. The learning tool used in this work is the Pythia software. We use four training saples to train a BPNN, where each saple includes one input attribute and one output attribute, such as (1, x 1 ), (2, x 2 ), (3, x 3 ),

5 Matheatical Probles in Engineering 5 Table 2: Coputation of the AGM(1,1). Order x k TP value x (1) 1 α k z (1) k x k x (1) (k + 1) = e k Table 3: Level of accuracy. Level MRE ADGI RSD PSE and (4, x 4 ). The topology of the BPNN is a structure, and after it is trained it returns the corresponding value for x 5. SVR is a nonparaetric estiation learning algorith basedonstatisticallearningtheory,whichisusedtosolve training probles with liited saples [23, 24]. Due to its good perforance, it is one of the priary ethods used in achinelearning.weuseweka3.6.9asthelearningtoolfor thesvr,andthetrainingsetisthesaeasthatusedwiththe BPNN. Since accuracy is a critical index when easuring the ability of a forecasting ethod [25], we eploy three error indexes in this work, naely, the ean square error (MSE), ean absolute error (MAE), and ean absolute percentage error (MAPE), and these are calculated as follows: MSE = 1 MAE = 1 MAPE = 1 ( x i x i ) 2, x i x i, x i x i x i 100%, (11) where is the su of testing saples and x i and x i are the predicted and actual values of the ith testing saple, respectively Experiental Results. This research uses the four pretesting easureents in Section 3.4 to assess the forecasting perforance fro various perspectives. The forecasting results of the proposed approach are shown in Table 4,andthe four easureents are as follows: MRE is , ADGI is , RSD is , and PSE is This shows that the AGM(1,1) falls within level 2 and thus has a high forecasting accuracy. Further, the MRE value, which is saller than 0.01, suggests that the risk when using this grey odel is controllable. The ADGI value is about 0.95, indicating a high siilarity Table 4: Forecasting results of AGM(1,1). Order Actual values Forecasting values between the actual and forecast results with regard to the geoetric shape, and this also explains that the proposed approach can properly reflect the data trend, with no serious inconsistencies between the forecast results and the real situation. Lastly, both the easureents of RSD and PSE are at an acceptable level, so there are neither extree values norahighleveloferrordiscreteness,whichbothindirectly confir the stability of the AGM(1,1) odel. The results shown in Table 5 indicate that the AGM(1,1) odel has better forecasting perforance and outperfors the other two ethods, with its average iproveents being over 20% for all three error indexes. In addition, the MAPE of AGM(1,1) is less than 0.2%, which falls into an acceptable range in the WLP industry. These results clearly show that the AGM(1,1) odel is a feasible tool to deal with forecasting task in the WLP process. 4. Conclusions and Discussion In recent years, the packaging technology has focused on the WLPprocess.However,duetotheliiteddataavailableat the early stages of the introduction of this technology, it is difficult for copanies to effectively control it. To iprove productionyields,itisthusnecessaryforengineerstohave an effective approach to forecast in this context, as if adverse phenoena are discovered in tie; they can be addressed in atielyanner. Thispaperpresentedanewadaptiveodelbasedongrey syste theory to solve the sall dataset forecasting proble

6 6 Matheatical Probles in Engineering Table5:Coparisonoftheforecastingethods. Methods MSE MAE MAPE (%) AGM(1,1) BPNN SVR in this context. The experiental results show that the AGM(1,1) odel can obtain satisfactory outcoes in the four pretesting easureents, outperforing the two popular forecasting techniques and thus that the grey approach has practical value for use in the packaging industries. Furtherore, one direction for future research would be to exaine how to apply this ethod to other real-world industrial cases, such as the panel and biotechnology industries. References [1] D.-C. Li, C. Wu, and F. M. Chang, Using data-fuzzification technology in sall data set learning to iprove FMS scheduling accuracy, International Advanced Manufacturing Technology,vol.27,no.3-4,pp ,2005. [2] D.-C. Li, C.-J. Chang, C.-C. Chen, and W.-C. Chen, A greybased fitting coefficient to build a hybrid forecasting odel for sall data sets, Applied Matheatical Modelling, vol.36,pp , [3] V. C. Ivǎnescu,J.W.M.Bertrand,J.C.Fransoo,andJ.P.C. Kleijnen, Bootstrapping to solve the liited data proble in production control: an application in batch process industries, the Operational Research Society,vol.57,no.1,pp.2 9, [4] Y.-S. Lin and D.-C. Li, The Generalized-Trend-Diffusion odeling algorith for sall data sets in the early stages of anufacturing systes, European Operational Research, vol. 207, no. 1, pp , [5] I. A. Gheyas and L. S. Sith, A neural network-based fraework for the reconstruction of incoplete data sets, Neurocoputing,vol.73,no.16-18,pp ,2010. [6] J. L. Deng, Control probles of grey systes, Systes and Control Letters, vol. 1, no. 5, pp , [7]N.M.Xie,S.F.Liu,Y.J.Yang,andC.Q.Yuan, Onnovel grey forecasting odel based on non-hoogeneous index sequence, Applied Matheatical Modelling, vol. 37, pp , [8] D.-C. Li, C.-J. Chang, C.-C. Chen, and W.-C. Chen, Forecasting short-ter electricity consuption using the adaptive greybased approach-an Asian case, Oega, vol. 40, no. 6, pp , [9] J.Deng, Introductiontogreysystetheory, Grey Syste,vol.1,pp.1 24,1989. [10] J. L. Deng, The Priary Methods of Grey Syste Theory, Huazhong University of Science and Technology Press, Wuhan, China, 2nd edition, [11] Y.-Y. Tai, J.-Y. Lin, M.-S. Chen, and M.-C. Lin, A grey decision and prediction odel for investent in the core copetitiveness of product developent, Technological Forecasting and Social Change,vol.78,no.7,pp ,2011. [12] L.-C. Hsu and C.-H. Wang, Forecasting integrated circuit output using ultivariate grey odel and grey relational analysis, Expert Systes with Applications, vol.36,no.2,pp , [13] R.-C. Tsaur and Y.-C. Liao, Forecasting LCD TV deand using the fuzzy grey odel GM(1,1), International Uncertainty, Fuzziness and Knowlege-Based Systes,vol.15,no.6,pp , [14] D. Yaaguchi, G.-D. Li, and M. Nagai, A grey-based rough approxiation odel for interval data processing, Inforation Sciences,vol.177,no.21,pp ,2007. [15] W. Köse, I. Teiz, and S. Erol, Grey syste approach for Econoic Order Quantity odels under uncertainty, Grey Syste,vol.23,no.1,pp.71 82,2011. [16] Z. X. Wang, An optiized Nash nonlinear grey Bernoulli odel for forecasting the ain econoic indices of high technology enterprises in China, Coputers and Industrial Engineering,vol.64,pp ,2013. [17] E. Kayacan, B. Ulutas, and O. Kaynak, Grey syste theorybased odels in tie series prediction, Expert Systes with Applications,vol.37,no.2,pp ,2010. [18] D.-C. Li, C.-W. Yeh, and C.-J. Chang, An iproved grey-based approach for early anufacturing data forecasting, Coputers and Industrial Engineering,vol.57,no.4,pp ,2009. [19] D.-C. Li and C.-W. Yeh, A non-paraetric learning algorith for sall anufacturing data sets, Expert Systes with Applications,vol.34,no.1,pp ,2008. [20] D.-C. Li, C.-J. Chang, W.-C. Chen, and C.-C. Chen, An extended grey forecasting odel for onidirectional forecasting considering data gap difference, Applied Matheatical Modelling,vol.35,no.10,pp ,2011. [21] S. F. Liu and Y. Lin, Grey Inforation: Theory and Practical Applications, Springer, London, UK, 1st edition, [22] J.W.S.Hu,Y.C.Hu,andR.R.W.Lin, Applyingneuralnetworks to prices prediction of crude oil futures, Matheatical Probles in Engineering,vol.2012,ArticleID959040,12pages, [23] J.L.WuandP.C.Chang, Atrend-basedsegentationethod and the support vector regression for financial tie series forecasting, Matheatical Probles in Engineering, vol. 2012, Article ID , 20 pages, [24] I. H. Witten and E. Frank, Data Mining: Practical Machine Learning Tools and Techniques,MorganKaufannPublishers, San Francisco, Calif, USA, 2nd edition, [25] J. T. Yokua and J. S. Arstrong, Beyond accuracy: coparison of criteria used to select forecasting ethods, International Forecasting,vol.11,no.4,pp ,1995.

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