Research Article Precise and Accurate Job Cycle Time...
15
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 496826, 14 pages http://dx.doi.org/10.1155/2013/496826 Research Article Precise and Accurate Job Cycle Time Forecasting in a Wafer Fabrication Factory with a Fuzzy Data Mining Approach Toly Chen and Richard Romanowski Department of Industrial Engineering and Systems Management, Feng Chia University, 100 Wenhwa Road, Seatwen, Taichung 407, Taiwan Correspondence should be addressed to Toly Chen; [email protected] Received 5 November 2012; Revised 18 February 2013; Accepted 12 March 2013 Academic Editor: Sabrina Senatore Copyright © 2013 T. Chen and R. Romanowski. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Many data mining methods have been proposed to improve the precision and accuracy of job cycle time forecasts for wafer fabrica- tion factories. is study presents a fuzzy data mining approach based on an innovative fuzzy backpropagation network (FBPN) that determines the lower and upper bounds of the job cycle time. Forecasting accuracy is also significantly improved by a combination of principal component analysis (PCA), fuzzy c-means (FCM), and FBPN. An applied case that uses data collected from a wafer fab- rication factory illustrates this fuzzy data mining approach. For this applied case, the proposed methodology performs better than six existing data mining approaches. 1. Introduction To forecast the job cycle time in a wafer fabrication factory is difficult because each wafer fabrication factory is a compli- cated production system with idiosyncratic features such as changing demand, a variety of product types and priorities, equipment unreliability, unbalanced capacity, job reentry into machines, alternative machines, sequence-dependent setup times, and shiſting bottlenecks. An average job cycle time is several months with hundreds of hours of standard deviation. Many studies have shown that accurately predicting the cycle/ completion times for such large systems is very difficult [2–4]. ere are two approaches to forecasting the cycle time of a job. e first approach, the input-output relationship approach, is to determine certain factors (e.g., average waiting time, queuing length, utilization, future release plans, bottle- necks, etc.) that influence the job cycle time, and then to apply different approaches (e.g., multiple linear regression (MLR), artificial neural networks (ANN), etc.) to model the relation- ship between the job cycle time and these factors, in order to forecast the cycle time of a new job. e second approach, the time-series approach, is to treat fluctuations in the job cycle time as a type of time series. Many approaches, for example, moving average (MA), weighted moving average (WMA), exponential smoothing (ES), MLR, ANN, autoregressive inte- grated moving average (ARIMA), and others, can be applied to forecast the cycle time of a new job. e cycle time of a job is usually predicted when the job is released into the factory, but several months are required for the job to complete all oper- ations. In such a case, the production controller may need to forecast the cycle time of job number +10000 according to the historical data of jobs 1∼. is is a difficult problem for time-series methods; the present study contributes a useful approach to this problem. Existing approaches to job cycle time estimation for a wafer fabrication factory can be classified into six categories: statistical analysis, production simulation (PS), ANN, case- based reasoning (CBR), fuzzy modeling methods, and hybrid approaches [5]. Many approaches are in fact special types of data mining methods. Table 1 summarizes references related to these categories. Data mining methods that have been extensively applied to job cycle time forecasting include clustering, genetic algorithms (GA), ANNs, decision trees, and hybrid approaches. e relevant references are reviewed below. Backus et al. [1] compared the performances of clustering, K-nearest neighbors (i.e., CBR), classification and regression tree (CART), and ANN; CART gave the best performance
Transcript of Research Article Precise and Accurate Job Cycle Time...
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 496826 14 pageshttpdxdoiorg1011552013496826
Research ArticlePrecise and Accurate Job Cycle Time Forecasting in a WaferFabrication Factory with a Fuzzy Data Mining Approach
Toly Chen and Richard Romanowski
Department of Industrial Engineering and Systems Management Feng Chia University 100 Wenhwa Road SeatwenTaichung 407 Taiwan
Correspondence should be addressed to Toly Chen tcchenfcuedutw
Received 5 November 2012 Revised 18 February 2013 Accepted 12 March 2013
Academic Editor Sabrina Senatore
Copyright copy 2013 T Chen and R RomanowskiThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Many data miningmethods have been proposed to improve the precision and accuracy of job cycle time forecasts for wafer fabrica-tion factoriesThis study presents a fuzzy datamining approach based on an innovative fuzzy backpropagation network (FBPN) thatdetermines the lower and upper bounds of the job cycle time Forecasting accuracy is also significantly improved by a combinationof principal component analysis (PCA) fuzzy c-means (FCM) and FBPN An applied case that uses data collected from awafer fab-rication factory illustrates this fuzzy data mining approach For this applied case the proposed methodology performs better thansix existing data mining approaches
1 Introduction
To forecast the job cycle time in a wafer fabrication factoryis difficult because each wafer fabrication factory is a compli-cated production system with idiosyncratic features such aschanging demand a variety of product types and prioritiesequipment unreliability unbalanced capacity job reentry intomachines alternative machines sequence-dependent setuptimes and shifting bottlenecks An average job cycle time isseveralmonths with hundreds of hours of standard deviationMany studies have shown that accurately predicting the cyclecompletion times for such large systems is very difficult [2ndash4]
There are two approaches to forecasting the cycle timeof a job The first approach the input-output relationshipapproach is to determine certain factors (eg averagewaitingtime queuing length utilization future release plans bottle-necks etc) that influence the job cycle time and then to applydifferent approaches (eg multiple linear regression (MLR)artificial neural networks (ANN) etc) to model the relation-ship between the job cycle time and these factors in order toforecast the cycle time of a new jobThe second approach thetime-series approach is to treat fluctuations in the job cycletime as a type of time series Many approaches for examplemoving average (MA) weighted moving average (WMA)
exponential smoothing (ES)MLRANN autoregressive inte-grated moving average (ARIMA) and others can be appliedto forecast the cycle time of a new jobThe cycle time of a job isusually predicted when the job is released into the factory butseveral months are required for the job to complete all oper-ations In such a case the production controller may needto forecast the cycle time of job number 119896+10000 according tothe historical data of jobs 1 sim 119896 This is a difficult problem fortime-series methods the present study contributes a usefulapproach to this problem
Existing approaches to job cycle time estimation for awafer fabrication factory can be classified into six categoriesstatistical analysis production simulation (PS) ANN case-based reasoning (CBR) fuzzymodelingmethods and hybridapproaches [5] Many approaches are in fact special types ofdata mining methods Table 1 summarizes references relatedto these categories Data mining methods that have beenextensively applied to job cycle time forecasting includeclustering genetic algorithms (GA) ANNs decision treesand hybrid approaches The relevant references are reviewedbelow
Backus et al [1] compared the performances of clusteringK-nearest neighbors (ie CBR) classification and regressiontree (CART) and ANN CART gave the best performance
2 Mathematical Problems in Engineering
Table 1 References relevant to the six categories
Category Related referencesStatisticalanalysis
Pearn et al [2] Chien et al [4] and Tai et al [16]
PSSivakumar and Chong [17] Okumura [18] and Veegeret al [19]
ANNChen [5] Chang and Hsieh [9] Sha and Hsu [10]Chang et al [8] Backus et al [1] Chien et al [4]Meidan et al [6] and Tirkel [15]
CBR Chang et al [20] Chiu et al [21] and Chang et al [22]
Fuzzymodeling
FBPN [5] Chang et al [8] kM-FBPN [11] look-aheadSOM-FBPN [3 23 24] post-classifying FBPN-BPN[13 14] and bi-directional classifying FCM-FBPN-RBF[25]
Hybridmethods
SOM-WM [26] SOM-BPN [27] kM-FBPN [11]FCM-FBPN [28 29] look-ahead SOM-FBPN[3 23 24] post-classifying FBPN-BPN [13 14] andbi-directional classifying FCM-FBPN-RBF [25]
of that experiment Meidan et al [6] focused on the choiceof key factors in cycle time forecasting They comparedmultiple linear regression (ie statistical analysis) ANN anda selective naive Bayesian classifier (SNBC) they found thattheir ANN had the best performance
Wang andMendel [7] proposed a fuzzymodelingmethodcalled the WM method the first step of that method waspartitioning the range of each input variable into several fuzzyintervals Chang et al [8] modified the first step of the WMmethod with a simple GA and proposed an evolving fuzzyrule (EFR) approach to predict the cycle time of a job in awafer fabrication factory Their EFR approach outperformedCBR and ANN in terms of forecasting accuracy
A BPN is an effective tool for modeling complex physicalsystems described by sets of different equations BPNs areuseful for prediction control and design purposes they canoffer both effectiveness (high forecasting accuracy) and effi-ciency (short execution time) Chang and Hsieh [9] Changet al [8] and Sha andHsu [10] all predicted the cycle times ofjobs in wafer fabrication factories with backpropagation net-works (BPNs) that each had a single hidden layerThese BPNsdelivered better average forecasting accuracy than statisticalanalysis approaches as measured by root mean squared error(RMSE) For example an improvement of about 40 inRMSE was achieved in Chang et al [8] Chen [5] constructeda fuzzy backpropagation network (FBPN) that incorporatedexpert opinions to modify the inputs of the FBPN ChenrsquosFBPN surpassed a crisp BPN especially with respect to effi-ciency Chen [11] incorporated the job release plan of a waferfabrication factory into a BPN and constructed a ldquolook-aheadrdquo BPN for the same purpose which led to an averagereduction of 12 in RMSE Beeg [12] and Chen et al [13 14]predicted the cycle time of a job in a ramping-up wafer fab-rication factory Chen et al used a BPN-based method Beegtried to figure out the impact of utilization for the cycle timeTirkel [15] applied two data mining approachesmdashdecisiontrees and ANNmdashand concluded that ANN was more accu-rate
Chang and Liao [26] combined a self-organizingmap andthe Wang and Mendel method (SOM-WM) a job was classi-fied using a SOM before the job cycle time was predictedby the WM method Chen [11] constructed a look-ahead k-means-fuzzy backpropagation network (kM-FBPN) for thesame purpose and discussed the effects of using differentlook-ahead functions More recently Chen et al [23] pro-posed a look-ahead SOM-FBPN approach for job cycle timeforecasting in a semiconductor factory a set of fuzzy infer-ence rules was developed to evaluate the achievability ofa cycle time forecast Subsequently Chen et al [3] addeda selective time allowance to the look-ahead SOM-FBPNapproach to determine internal due dates Further Chen [24]showed that a combined SOM and FBPN could be improvedby feeding back the forecasting error by the FBPN to adjustthe classification results of the SOM Chen et al [13 14] pro-posed a postclassification FBPN-BPN approach in which ajob was not preclassified the job was classified after the cycletime had been predicted Experimental results showed thatthe post-classification approachwas better than somepreclas-sification approaches in some cases In order to combine theadvantages of preclassifying and post-classifying approachesChen [25] proposed a bidirectional classifying approach thatcombined FCM FBPN and a radial basis function network(RBF) in which jobs are not only preclassified but also post-classified Instead of bi-directional classification Chien et al[4] used nonlinear regression equations and then related theforecasting error to some factory conditions and job attri-butes with a backpropagation network (BPN) to improve theforecasting accuracy
Table 2 summarizes the dataminingmethods in this fieldThe advantages and disadvantages of these methods are com-pared in Table 3
To improve the precision and accuracy of job cycle timeforecasting in a wafer fabrication factory this study presentsa fuzzy data mining approach that emphasizes both accuracyand precision
(1) Accuracy the forecast value should be as close aspossible to the actual value
(2) Precision a narrow interval containing the actualvalue should be established
Chen [30] defined several methods to measure the precisionand accuracy of a fuzzy forecasting method and noted thatprecision has seldom been considered However past studieshave confirmed that precision and accuracy are closelyrelated Song and Chissom [31] proposed a first-order time-variant model to forecast enrollment and considered theeffect of the model basis on the forecasting precision Specifi-cally if the space of feasible solutions is narrowed it becomesmore likely that the actual value can be found [30 32]
The fuzzy data mining approach has the following inno-vative characteristics
(1) Some factors used to estimate the cycle time aredependent on each other which may cause problemsin classifying jobs and in fitting the relationshipbetween the job cycle time and these factors To solvethis problem PCA FCM and FBPN are combined to
Mathematical Problems in Engineering 3
Table 2 Data mining methods for job cycle time forecasting
Category ExamplesClustering Backus et al [1] Meidan et al [6]GA Chang et al [8]Decisiontrees CART [1] SNBC [6] and Tirkel [15]
ANNChen [5] Chang and Hsieh [9] Sha and Hsu [10]Chang et al [8] Backus et al [1] Chien et al [4]Meidan et al [6] and Tirkel [15]
Hybridapproaches
SOM-WM [26] kM-FBPN [11 29] look-aheadSOM-FBPN [3 23 24] post-classifying FBPN-BPN[13 14] and bi-directional classifyingFCM-FBPN-RBF [25]
predict the job cycle time which is an innovation inthis field
(2) The job cycle time is predicted by an effective FBPNapproach For the job cycle time there are upper andlower bounds that constrain the forecast to be withina possible range The FBPN approach from Chen andWangrsquos FBPN research [33] has beenmodified for thispurpose In Chen and Wangrsquos FBPN approach twononlinear programming (NP) models are solved todetermine the upper and lower bounds of the jobcycle time However the NP models involve compli-cated constraints and therefore are difficult to solveIn addition the NP models will become too huge ifmany jobs are to be considered To solve these pro-blems in this study the upper and lower bounds aredetermined in a simpler way
The differences between the proposedmethodology and vari-ous previous methods are summarized in Table 4
The remainder of this paper is organized as followsSection 2 introduces the proposed methodology which iscomposed of six steps Section 3 demonstrates the proposedmethod with an illustrative example then presents anothercase with data collected from a real wafer fabrication factoryand makes comparisons with some existing approachesFinally concluding remarks anddirections for future researchare given in Section 4
2 Methodology
The fuzzy data mining procedure consists of several steps
Step 1 Replacing parameters using PCA
Step 2 Classifying jobs using fuzzy c-means (FCM)
Step 3 Forecasting the cycle times of jobs in each categoryusing a FBPN
Step 4 Updating the upper and lower bounds
Step 5 Evaluating and comparing the forecasting precisionand accuracy of the proposed methodology
Variable replacement using PCA
Classifying jobs using FCM
Forecasting the job cycle times using FBPN
Is the improvement
negligible
Stop
Yes
No
Updating the upper and lower bounds
Evaluating the forecasting precision and accuracy
Figure 1 The flowchart of the proposed methodology
Step 6 If the improvement in the forecasting precisionor accuracy becomes negligible stop otherwise return toStep 3
A flow chart of the proposed methodology is shown inFigure 1 These steps can be compared to the basic steps ofdata mining as shown in Table 5
21 Step 1 Variable Replacement Using PCA First PCA isused to replace the inputs to the FBPN A series of linearcombinations of the original variables forms a new variableso that these new variables are as unrelated to each other aspossible Although there are more advanced applications ofPCA in this study PCA is used to enhance the efficiency ofFBPN training PCA consists of the four following steps
(1) Raw data standardization to eliminate dimensionaldifferences and large numerical differences the orig-inal variables are standardized [13 14]
119909lowast
119895119894=
119909119895119894minus 119909119894
120590119894
119909119894=
sum119899
119895=1119909119895119894
119899
120590119894=
radicsum119899
119895=1(119909119895119894minus 119909119894)2
119899 minus 1
(1)
4 Mathematical Problems in Engineering
Table 3 Advantages and disadvantages of various data mining approaches for job cycle time forecasting
Approach Advantages Disadvantages
Clustering (i) Easiest to use(ii) Can be used with other methods
(i) Not accurate(ii) No method to determine the optimal number of clusters
GA (i) Accurate(ii) Can be used with other methods Time-consuming iterations
Decision trees (i) Handles missing values well(ii) Robust to outliers High variance of tree structure
ANN (i) Accurate(ii) Robust to outliers
(i) Difficult to use(ii) Time-consuming training(iii) Possible overfitting
Hybrid approaches Very accurate (i) Very difficult to use(ii) Very time consuming
Table 4 The differences between the proposed methodology and the previous methods
Method Variable replacement Job pre-classification Job post-classification Determining the boundsSOM-WM kM-FBPNLook-ahead SOM-FBPN No Yes No Not discussed
Post-classifying FBPN-BPN No No Yes Not discussedBi-directional classifyingFCM-FBPN-RBF No Yes Yes Not discussed
BPN-NP No No No DifficultThe proposed methodology Yes Yes Yes Easy
Table 5 The proposed methodology compared to data mining
Data mining steps The proposed methodologyApplication domainidentification
Job cycle time forecasting for a waferfabrication factory
Target dataset selection The most recent PROMIS data of a majorproduct type with normal priority
Data preprocessing PCA partial normalization
Data mining
(i) FCM for job classification(ii) FBPN to model the relationshipbetween the job cycle time and six jobparameters
Knowledge extraction Training data (34 of the collected data)Knowledge application Testing data (the remaining data)
Knowledge evaluation Forecasting precision (the average range)and accuracy (MAE MAPE RMSE)
lowastPROMIS production management information system MAE mean abso-lute error MAPE mean absolute percentage error
where 119909119895119894is the 119894th attribute of job 119895 119895 = 1 sim 119899 119909
119894
and 120590119894indicate the mean and standard deviation of
variable i respectively(2) Establishment of the correlation matrix 119877 is as fol-
lows
119877 =1
119899 minus 1119883lowast119879
119883lowast
(2)
where119883lowast is the standardized data matrix The eigen-
values and eigenvectors of 119877 are calculated and rep-resented as 120582
1sim 120582119898and 119906
1sim 119906119898 respectively 120582
1ge
1205822ge sdot sdot sdot ge 120582
119898
(3) Determination of the number of principal compo-nents the variance contribution rate is calculated as
120578119902=
120582119902
sum119898
119903=1120582119903
sdot 100 (3)
and the accumulated variance contribution rate is
120578Σ(119901) =
119901
sum
119902=1
120578119902 (4)
Choose the smallest 119875 value such that 120578Σ(119901) ranges-
from 85 to 90 A Pareto analysis chart can beused to show the percentage of variability that can beattributed to each principal component
(4) Formation of the following matrices
119880119898times119901
= [1199061 1199062 119906
119901]
119885119899times119901
= 119883lowast
119899times119898119880119898times119901
(5)
119885119899times119901
= [119911119895119902] (119895 = 1 sim 119899 119902 = 1 sim 119901) are the com-
ponent scoresThese scores contain the coordinates ofthe original data in the new coordinate systemdefinedby the principal componentsThey will be used as thenew inputs to the FBPN
22 Step 2 Classifying Jobs Using FCM After PCA examplesare classified by FCM If a crisp clustering method is appliedinstead then it is possible that some clusters will have veryfew examples FCM avoids the problems of crisp clusteringbecause in FCM an example belongs to multiple clusters to
Mathematical Problems in Engineering 5
different degrees Similarly in probability theory the naıveBayesmethod provides a probability that each itembelongs toa class However the classification of jobs in FCM can includesubjective judgments
FCM classifies jobs byminimizing the following objectivefunction
Min119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896) (6)
where119870 is the required number of categories n is the numberof jobs120583
119895(119896)indicates the degree ofmembership that job 119895 has
in category k 119890119895(119896)
measures the distance from job 119895 to thecentroid of category 119896 119898 isin [1infin) is a parameter to adjustthe fuzziness and is usually set to 2 The procedure of FCM isas follows
(1) Normalize the input data
(2) Produce a preliminary clustering result
(3) Iterations calculate the centroid of each category as
119911(119896)
= 119911(119896)119902
119896 = 1 sim 119870
119911(119896)119902
=sum119899
119895=1120583119898
119895(119896)119911119895119902
sum119899
119895=1120583119898119895(119896)
119896 = 1 sim 119870 119902 = 1 sim 119901
120583119895(119896)
=1
sum119870
119892=1(119890119895(119896)
119890119895(119892)
)2(119898minus1)
119895 = 1 sim 119899 119896 = 1 sim 119870
119890119895(119896)
= radic
119901
sum
119902=1
(119911119895119901
minus 119911(119896)119901
)2
119895 = 1 sim 119899 119896 = 1 sim 119870
(7)
where 119911(119896)
is the centroid of category 119896 120583(119905)119895(119896)
is thedegree ofmembership that job 119895has in category 119896 afterthe 119905th iteration
(4) Remeasure the distance from each job to the centroidof each category and then recalculate the correspond-ing membership
(5) Stop if the following condition is met Otherwisereturn to Step (3)
max119896
max119895
10038161003816100381610038161003816120583(119905)
119895(119896)minus 120583(119905minus1)
119895(119896)
10038161003816100381610038161003816lt 119889 (8)
where 119889 is a real number representing the member-ship convergence threshold
The performance of FCM is affected by the settings of theinitial values initialization can be repeated multiple timesin order to find the optimal solution Finally the separate
distance test (119878 test) proposed by Xie and Beni [34] can beapplied to determine the optimal number of categories119870
Min 119878
subject to 119869119898
=
119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896)
1198902
min = min1198961 = 1198962
(
119901
sum
119902=1
(119911(1198961)119902
minus 119911(1198962)119902
)2
)
119878 =119869119898
119899 times 1198902min 119870 isin 119885
+
(9)
The119870 value thatminimizes 119878 determines the optimal numberof categories
23 Step 3 Forecasting the Cycle Times of Jobs in EachCategory Using a FBPN All categories in the present studyare considered by the FBPN There are different types ofFBPNs for example Nomura et al [35] Chen [5] Lin [36]and Chen and Wang [33] Among them Chen and WangrsquosFBPN is unique since it is aimed at optimizing precision Forthis reason it is chosen for this study This FBPN is a proventechnology that has incorporated many features of priorartificial neural networks which have solved a wide variety ofproblems characterized by sets of different equations Somespecial problems can benefit from specialized tools such ascompositional pattern-producing networks cascading neu-ral networks and dynamic neural networks but for thepresent job cycle time problem a well-trained FBPN withan optimized structure can still produce very good resultsAll parameters in the FBPN are expressed in triangularfuzzy numbers (TFNs) There are various types of fuzzynumbers with different shapes among them the triangulartype is easily implemented andhas been universally applied tonumerous applications (eg [37 38]) Although the proposedmethodology uses TFNs it can be easilymodified to use othertypes of fuzzy numbers such as trapezoidal fuzzy numbers orbounded LR-type fuzzy numbers
The FBPN is configured as follows
(1) Inputs there are 119895 sets of inputs each set consists ofthe new factors determined by PCA from the 119895th jobThese factors have to be partially normalized so thattheir values fall within [01 09] [13 14]
(2) Single hidden layer generally one hidden layer givesthe best convergence results for the FBPN
(3) For simplicity the number of neurons in the hiddenlayer is twice the number in the input layer Anincrease in the number of hidden-layer nodes lessensthe output errors for the training examples butincreases the errors for novel examples This phe-nomenon is often called ldquooverfittingrdquo Some researchhas considered the connections between the complex-ity of a FBPN the performance for the training dataand the number of examples noteworthy researchincludes Akaikersquos information criterion (AIC) [39]
6 Mathematical Problems in Engineering
and Rissanenrsquos minimum description length (MDL)[40]
(4) Output the (normalized) cycle time forecast of theexample
The procedure for determining the parameter valuesis now described After pre-classification a portion of theadopted examples in each category is fed as ldquotraining exam-plesrdquo into the FBPN to determine the parameter valuesTwo phases are involved at the training stage First in theforward phase inputs are multiplied with weights summedand transferred to the hidden layer Then activated signalsare outputted from the hidden layer as
ℎ119897= (ℎ1198971 ℎ1198972 ℎ1198973)
=1
1 + 119890minus119899ℎ
119897
= (1
1 + 119890minus119899ℎ
1198971
1
1 + 119890minus119899ℎ
1198972
1
1 + 119890minus119899ℎ
1198973
)
(10)
where
119899ℎ
119897= (119899ℎ
1198971 119899ℎ
1198972 119899ℎ
1198973)
= 119868ℎ
119897(minus) 120579ℎ
119897
= (119868ℎ
1198971minus 120579ℎ
1198973 119868ℎ
1198972minus 120579ℎ
1198972 119868ℎ
1198973minus 120579ℎ
1198971)
119868ℎ
119897= (119868ℎ
1198971 119868ℎ
1198972 119868ℎ
1198973)
= sum
all 119896119908ℎ
119896119897sdot 119909119896
= (sum
all 119896min (119908
ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896)
sum
all 119896119908ℎ
1199091198962119909119896 sum
all 119896max (119908ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896))
(11)
ℎ119897is the output from hidden-layer node 119897 119897 = 1 sim 119871 120579ℎ
119897is
the threshold for screening out weak signals by hidden-layernode 119897119908ℎ
119896119897is theweight of the connection between input node
119896 and hidden-layer node 119897 119896 = 1 sim 119901 119897 = 1 sim 119871 119909119894119895is
the 119896th input 119896 = 1 sim 119901 The remaining parameters aretransition variables (minus) and (times) denote fuzzy subtraction andmultiplication respectively
ℎ119897s are also transferred to the output layer with the same
procedure Finally the output of the FBPN is generated asfollows
119900119895= (1199001198951 1199001198952 1199001198953)
=1
1 + 119890minus119899119900
= (1
1 + 119890minus119899119900
1
1
1 + 119890minus119899119900
2
1
1 + 119890minus119899119900
3
)
(12)
where
119899119900
= (119899119900
1 119899119900
2 119899119900
3) = 119868119900
(minus) 120579119900
= (119868119900
1minus 120579119900
3 119868119900
2minus 120579119900
2 119868119900
3minus 120579119900
1)
(13)
119868119900
= (119868119900
1 119868119900
2 119868119900
3)
=sum
all 119897119908119900
119897(times) ℎ119897
cong (sum
all 119897min (119908
119900
1198971ℎ1198971 119908119900
1198973ℎ1198973)
sum
all 119897119908119900
1198972ℎ1198972 sum
all 119897max (119908119900
1198971ℎ1198971 119908119900
1198973ℎ1198973))
(14)
119900119895is the network output which is the normalized value of the
cycle time forecast of job 119895 120579119900 is the threshold for screeningout weak signals by the output node 119908119900
119897is the weight of
the connection between hidden-layer node 119897 and the outputnode 119897 = 1 sim 119871
Subsequently in the backward phase the training of theFBPN is decomposed into three subtasks determining thecenter value and upper and lower bounds of the parameters
First to determine the center of each parameter (such as119908ℎ
1198961198972 120579ℎ1198972 1199081199001198972 and 120579
119900
2) the FBPN is treated as a crisp network
Algorithms applicable for this purpose include the gradi-ent descent algorithms the conjugate gradient algorithmsthe Levenberg-Marquardt algorithm and others In thisstudy the Levenberg-Marquardt algorithm is applied TheLevenberg-Marquardt algorithm was designed for trainingwith second-order speed without having to compute theHes-sian matrix It uses approximation and updates the networkparameters in a Newton-like way as described below
The network parameters are placed in vector 120573 = [119908ℎ
11
119908ℎ
119901119871 120579ℎ
1 120579
ℎ
119871 119908119900
1 119908
119900
119871 120579119900
] The network output 119900119895
can be represented with 119891(x119895120573) The objective function of
the FBPN is to minimize RMSE or equivalently the sum ofsquared error (SSE)
SSE (120573) =
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573))2
(15)
The Levenberg-Marquardt algorithm is an iterative pro-cedure In the beginning the user should specify the initialvalues of the network parameter vector 120573 It is a commonpractice to set 120573T = (1 1 1) In each step the parametervector 120573 is replaced by a new estimate 120573 + 120575 where120575 = [Δ119908
ℎ
11 Δ119908
ℎ
119901119871 Δ120579ℎ
1 Δ120579
ℎ
119871 Δ119908119900
1 Δ119908
119900
119871 Δ120579119900
] Thenetwork output becomes 119891(x
119895120573 + 120575) it is approximated by
its linearization as
119891 (x119895120573 + 120575) asymp 119891 (x
119895120573) + J
119895120575 (16)
where
J119895=
120597119891 (x119895120573)
120597120573(17)
Mathematical Problems in Engineering 7
is the gradient vector of 119891 with respect to 120573 Substituting (16)into (15) gives
SSE (120573 + 120575) asymp
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573) minus J
119895120575)2
(18)
When the network reaches the optimal solution the gradientof SSE with respect to 120575 will be zero Taking the derivative ofSSE(120573+120575)with respect to 120575 and setting the result to zero gives
(JTJ) 120575 = JT (119873 (CT119895) minus 119891 (x
119895120573)) (19)
where J is the Jacobian matrix containing the first derivativeof network error with respect to the weights and biasesEquation (19) includes a set of linear equations that can besolved for 120575
24 Step 4 Determining the Upper Bounds of Parameters inthe FBPN In Chen andWang [33] and Chen and Lin [41] anNPmodel is constructed to adjust the connectionweights andthresholds in the FBPN such anNPmodel is not easy to solveThe proposedmethodology only makes iterative adjustmentsto the threshold on the output nodeThisway ismuch simplerand can also achieve good results
Substituting (13) into (12) gives
1199001198952
=1
1 + 119890minus119899119900
1198952
=1
1 + 119890minus(119868119900
1198952minus120579119900
2)
=1
1 + 119890120579119900
2minus119868119900
1198952
(20)
Therefore
ln(1
1199001198952
minus 1) = 120579119900
2minus 119868119900
1198952 (21)
So
119868119900
1198952= 120579119900
2minus ln(
1
1199001198952
minus 1) (22)
Assume that the adjustment made to the output nodethreshold is denoted as Δ120579
119900
= 120579119900
3minus 120579119900
2 After adjustment the
output from the new FBPN 1199001198953 determines the upper bound
of the cycle time
1199001198953
=1
1 + 119890minus119899119900
1198953
(23)
where119899119900
1198953= 119868119900
1198953minus 120579119900
3= 119868119900
1198953minus (120579119900
2+ Δ120579119900
) (24)
Substituting (24) into (23) gives
1199001198953
=1
1 + 119890minus(119868119900
1198953minus120579119900
2minusΔ120579119900)
(25)
and substituting (22) into (25) gives
1199001198953
=1
1 + 119890minus(120579119900
2minusln(11199001198952minus1)minus1205791199002minusΔ120579119900)
=1
1 + 119890ln(11199001198952minus1)+Δ120579119900
=1
1 + 119890Δ120579119900
(11199001198952
minus 1)
(26)
Actual cycle time
Upper bound 1Upper bound 2
Upper bound 3
Upper bound 5Upper bound 4
Upper bound 6
Final result
Figure 2 Iterative reduction of the upper bound
Obviously the maximum ofΔ120579119900 determines the lowest upper
bound since 1199001198953is the upper bound of the cycle time 119900
1198953ge
119873(CT119895)
1
1 + 119890ln(11199001198952minus1)+Δ120579119900
ge 119873(CT119895) (27)
Δ120579119900
le ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) (28)
Equation (28) holds for all jobs so
Δ120579119900
le min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(29)
According to (26) the optimal value of Δ120579119900 should be set to
the maximum possible value
Δ120579119900lowast
= min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(30)
Then the FBPNrsquos optimization results depend on theinitial conditions and therefore are different at every iterationAssume that the optimal value of 119900
1198953in the 119905th iteration is
denoted by 1199001198953(119905) then after some iterations
1199001198953
(all iterations) = min119905
1199001198953
(119905) (31)
In this way the upper bound of the cycle time is decreasedgradually (see Figure 2) Another merit of this approach isthat it does not rely on the parameters of the FBPN
Theorem 1 Theupper bound can be found by considering onlyjobs with119873(CT
119895) ge 1199001198952
8 Mathematical Problems in Engineering
Proof Divide jobs into two categories 1198621
(jobs with119873(CT
119895) lt 1199001198952) and 119862
2(jobs with 119873(CT
119895) ge 1199001198952) Equation
(30) becomes
Δ120579119900lowast
= min(min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)))
(32)
If119873(CT119895) lt 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) gt 0 (33)
Therefore
min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) gt 0 (34)
In contrast if119873(CT119895) ge 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) le 0
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) le 0
(35)
Finally
Δ120579119900lowast
= min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(36)
Theorem 1 is proved
25 Step 4 Determining the Lower Bounds of Parameters in theFBPN In a similar way the threshold on the output node canbe modified to determine the lower bound of the cycle timeforecast so that the actual value will be greater than the lowerbound of the network output The optimal value of Δ120579
119900 canbe obtained as
Δ120579119900lowast
= max119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(37)
Theorem2 The lower bound can be found by considering onlyjobs with 119873(CT
119895) lt 1199001198952
The proof of Theorem 2 resembles that of Theorem 1
Proof Assume that the optimal value of 1199001198951
in the 119905threplication is indicated with 119900
1198951(119905) then after some iterations
1199001198951
(all iterations) = max119905
1199001198951
(119905) (38)
In this way the upper bound of the cycle time is increasedgradually Δ120579
119900lowastdoes not rely on the parameters of the FBPN
3 Application and Analyses
Todemonstrate the application of the proposedmethodologyan illustrative example containing the data of 40 jobs (seeTable 6) was used
We standardize the data and obtain the correlationmatrix119877 The eigenvalues and eigenvectors of 119877 are calculated as
1205821= 266 120582
2= 115 120582
3= 094
1205824= 083 120582
5= 025 120582
6= 002
(39)
1199061=
[[[[[[[
[
020
052
016
056
007
059
]]]]]]]
]
1199062=
[[[[[[[
[
047
minus019
068
minus019
minus049
005
]]]]]]]
]
1199063=
[[[[[[[
[
minus027
031
minus032
003
minus085
minus002
]]]]]]]
]
1199064=
[[[[[[[
[
081
minus002
minus057
006
minus004
minus015
]]]]]]]
]
1199065=
[[[[[[[
[
minus012
minus072
minus007
065
minus018
009
]]]]]]]
]
1199066=
[[[[[[[
[
002
029
028
047
003
minus078
]]]]]]]
]
(40)
respectively The variance contribution rates are
1205781= 46 120578
2= 20 120578
3= 16
1205784= 14 120578
5= 4 120578
6= 0
(41)
Summing up 120578119902rsquos we obtain
120578Σ(1) = 46 120578
Σ(2) = 65 120578
Σ(3) = 81
120578Σ(4) = 95 120578
Σ(5) = 100 120578
Σ(6) = 100
(42)
A Pareto analysis chart is used to compare the percentageof variability explained by each principal component (seeFigure 3) There is a clear break in the amount of varianceaccounted for by each component between the first andsecond components However that component by itselfexplains less than 50 of the variance so more components
Mathematical Problems in Engineering 9
Table 6 The illustrative example
119895 1199091198951
1199091198952
1199091198953
1199091198954
1199091198955
1199091198956
CT119895
1 24 1261 181 781 112 092 9352 24 1263 181 762 127 090 9583 24 1220 176 761 127 089 10474 23 1282 178 802 127 094 10115 23 1303 180 780 175 093 10686 23 1281 183 782 175 093 11437 23 1242 184 741 163 089 11038 24 1262 182 681 139 086 12509 22 1260 182 701 98 086 118110 22 1260 179 700 257 087 119411 24 1301 163 722 99 084 126012 22 1221 184 641 131 082 124013 23 1323 159 740 247 087 118014 24 1362 181 782 191 095 122715 24 1261 181 762 219 091 123616 23 1321 177 801 219 096 121517 22 1343 180 822 219 097 122818 24 1321 177 762 54 093 126619 25 1343 179 781 54 096 128520 25 1300 180 740 54 092 127221 22 1320 181 721 54 091 131022 24 1321 182 742 49 092 126523 23 1262 165 680 201 080 130824 22 1240 161 722 103 082 133125 23 1183 183 661 53 082 129426 23 1282 184 701 53 088 131427 22 1202 177 680 248 084 132128 23 1202 178 681 248 085 135329 24 1202 185 701 82 086 122630 23 1202 158 721 98 081 130131 24 1343 181 760 67 094 128032 24 1381 185 801 67 097 128633 22 1362 156 780 67 091 125234 23 1282 179 782 223 092 121435 23 1320 180 782 176 093 125136 25 1340 176 801 462 097 122237 23 1320 182 781 168 095 118738 22 1361 181 781 141 094 120539 22 1381 179 781 95 097 112040 23 1363 178 802 179 097 1133
are considered To meet the requirement 120578Σ(119901) ge 85sim90
p is chosen as 3 We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so we reduce the issue to these threecomponents in order to visualize the data
The component scores are recalculated as the new inputsto the FCM-FBPN (see Table 7) These scores contain thecoordinates of the original data in the new coordinate systemdefined by the principal components
Subsequently jobs are classified using FCM based on thenew variables The results of the 119878 test are summarized inTable 8 In this case the optimal number of job categories is 5
Table 7 New inputs to the FCM-FBPN
1199111198951
1199111198952
1199111198953
minus056 091 minus019
minus013 087 minus034
051 057 minus037
minus097 minus010 020
minus087 minus020 minus026
minus075 014 minus051
057 056 minus066
130 118 minus055
155 031 047
137 minus087 minus104
111 minus059 091
304 063 minus020
051 minus244 minus002
minus194 012 minus043
minus030 035 minus129
minus162 minus084 minus048
minus204 minus124 minus017
minus087 077 089
minus192 134 064
minus058 170 034
022 023 129
minus062 131 073
254 minus126 minus016
239 minus164 120
302 157 014
089 121 066
256 minus074 minus119
219 minus013 minus154
161 190 minus042
272 minus123 087
minus127 099 071
minus256 107 078
minus037 minus244 247
minus060 minus051 minus082
minus106 minus027 minus017
minus254 minus136 minus341
minus131 minus002 minus018
minus132 minus063 067
minus177 minus058 132
minus213 minus066 012
Table 8 The results of the S test
Number of categories (119870) 119869119898
1198902
min 119878
2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043
However therewill be some categories with very few jobs Forthis reason the second best solution is used that is 4 cate-gories A common practice is to set a threshold of member-
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
Table 1 References relevant to the six categories
Category Related referencesStatisticalanalysis
Pearn et al [2] Chien et al [4] and Tai et al [16]
PSSivakumar and Chong [17] Okumura [18] and Veegeret al [19]
ANNChen [5] Chang and Hsieh [9] Sha and Hsu [10]Chang et al [8] Backus et al [1] Chien et al [4]Meidan et al [6] and Tirkel [15]
CBR Chang et al [20] Chiu et al [21] and Chang et al [22]
Fuzzymodeling
FBPN [5] Chang et al [8] kM-FBPN [11] look-aheadSOM-FBPN [3 23 24] post-classifying FBPN-BPN[13 14] and bi-directional classifying FCM-FBPN-RBF[25]
Hybridmethods
SOM-WM [26] SOM-BPN [27] kM-FBPN [11]FCM-FBPN [28 29] look-ahead SOM-FBPN[3 23 24] post-classifying FBPN-BPN [13 14] andbi-directional classifying FCM-FBPN-RBF [25]
of that experiment Meidan et al [6] focused on the choiceof key factors in cycle time forecasting They comparedmultiple linear regression (ie statistical analysis) ANN anda selective naive Bayesian classifier (SNBC) they found thattheir ANN had the best performance
Wang andMendel [7] proposed a fuzzymodelingmethodcalled the WM method the first step of that method waspartitioning the range of each input variable into several fuzzyintervals Chang et al [8] modified the first step of the WMmethod with a simple GA and proposed an evolving fuzzyrule (EFR) approach to predict the cycle time of a job in awafer fabrication factory Their EFR approach outperformedCBR and ANN in terms of forecasting accuracy
A BPN is an effective tool for modeling complex physicalsystems described by sets of different equations BPNs areuseful for prediction control and design purposes they canoffer both effectiveness (high forecasting accuracy) and effi-ciency (short execution time) Chang and Hsieh [9] Changet al [8] and Sha andHsu [10] all predicted the cycle times ofjobs in wafer fabrication factories with backpropagation net-works (BPNs) that each had a single hidden layerThese BPNsdelivered better average forecasting accuracy than statisticalanalysis approaches as measured by root mean squared error(RMSE) For example an improvement of about 40 inRMSE was achieved in Chang et al [8] Chen [5] constructeda fuzzy backpropagation network (FBPN) that incorporatedexpert opinions to modify the inputs of the FBPN ChenrsquosFBPN surpassed a crisp BPN especially with respect to effi-ciency Chen [11] incorporated the job release plan of a waferfabrication factory into a BPN and constructed a ldquolook-aheadrdquo BPN for the same purpose which led to an averagereduction of 12 in RMSE Beeg [12] and Chen et al [13 14]predicted the cycle time of a job in a ramping-up wafer fab-rication factory Chen et al used a BPN-based method Beegtried to figure out the impact of utilization for the cycle timeTirkel [15] applied two data mining approachesmdashdecisiontrees and ANNmdashand concluded that ANN was more accu-rate
Chang and Liao [26] combined a self-organizingmap andthe Wang and Mendel method (SOM-WM) a job was classi-fied using a SOM before the job cycle time was predictedby the WM method Chen [11] constructed a look-ahead k-means-fuzzy backpropagation network (kM-FBPN) for thesame purpose and discussed the effects of using differentlook-ahead functions More recently Chen et al [23] pro-posed a look-ahead SOM-FBPN approach for job cycle timeforecasting in a semiconductor factory a set of fuzzy infer-ence rules was developed to evaluate the achievability ofa cycle time forecast Subsequently Chen et al [3] addeda selective time allowance to the look-ahead SOM-FBPNapproach to determine internal due dates Further Chen [24]showed that a combined SOM and FBPN could be improvedby feeding back the forecasting error by the FBPN to adjustthe classification results of the SOM Chen et al [13 14] pro-posed a postclassification FBPN-BPN approach in which ajob was not preclassified the job was classified after the cycletime had been predicted Experimental results showed thatthe post-classification approachwas better than somepreclas-sification approaches in some cases In order to combine theadvantages of preclassifying and post-classifying approachesChen [25] proposed a bidirectional classifying approach thatcombined FCM FBPN and a radial basis function network(RBF) in which jobs are not only preclassified but also post-classified Instead of bi-directional classification Chien et al[4] used nonlinear regression equations and then related theforecasting error to some factory conditions and job attri-butes with a backpropagation network (BPN) to improve theforecasting accuracy
Table 2 summarizes the dataminingmethods in this fieldThe advantages and disadvantages of these methods are com-pared in Table 3
To improve the precision and accuracy of job cycle timeforecasting in a wafer fabrication factory this study presentsa fuzzy data mining approach that emphasizes both accuracyand precision
(1) Accuracy the forecast value should be as close aspossible to the actual value
(2) Precision a narrow interval containing the actualvalue should be established
Chen [30] defined several methods to measure the precisionand accuracy of a fuzzy forecasting method and noted thatprecision has seldom been considered However past studieshave confirmed that precision and accuracy are closelyrelated Song and Chissom [31] proposed a first-order time-variant model to forecast enrollment and considered theeffect of the model basis on the forecasting precision Specifi-cally if the space of feasible solutions is narrowed it becomesmore likely that the actual value can be found [30 32]
The fuzzy data mining approach has the following inno-vative characteristics
(1) Some factors used to estimate the cycle time aredependent on each other which may cause problemsin classifying jobs and in fitting the relationshipbetween the job cycle time and these factors To solvethis problem PCA FCM and FBPN are combined to
Mathematical Problems in Engineering 3
Table 2 Data mining methods for job cycle time forecasting
Category ExamplesClustering Backus et al [1] Meidan et al [6]GA Chang et al [8]Decisiontrees CART [1] SNBC [6] and Tirkel [15]
ANNChen [5] Chang and Hsieh [9] Sha and Hsu [10]Chang et al [8] Backus et al [1] Chien et al [4]Meidan et al [6] and Tirkel [15]
Hybridapproaches
SOM-WM [26] kM-FBPN [11 29] look-aheadSOM-FBPN [3 23 24] post-classifying FBPN-BPN[13 14] and bi-directional classifyingFCM-FBPN-RBF [25]
predict the job cycle time which is an innovation inthis field
(2) The job cycle time is predicted by an effective FBPNapproach For the job cycle time there are upper andlower bounds that constrain the forecast to be withina possible range The FBPN approach from Chen andWangrsquos FBPN research [33] has beenmodified for thispurpose In Chen and Wangrsquos FBPN approach twononlinear programming (NP) models are solved todetermine the upper and lower bounds of the jobcycle time However the NP models involve compli-cated constraints and therefore are difficult to solveIn addition the NP models will become too huge ifmany jobs are to be considered To solve these pro-blems in this study the upper and lower bounds aredetermined in a simpler way
The differences between the proposedmethodology and vari-ous previous methods are summarized in Table 4
The remainder of this paper is organized as followsSection 2 introduces the proposed methodology which iscomposed of six steps Section 3 demonstrates the proposedmethod with an illustrative example then presents anothercase with data collected from a real wafer fabrication factoryand makes comparisons with some existing approachesFinally concluding remarks anddirections for future researchare given in Section 4
2 Methodology
The fuzzy data mining procedure consists of several steps
Step 1 Replacing parameters using PCA
Step 2 Classifying jobs using fuzzy c-means (FCM)
Step 3 Forecasting the cycle times of jobs in each categoryusing a FBPN
Step 4 Updating the upper and lower bounds
Step 5 Evaluating and comparing the forecasting precisionand accuracy of the proposed methodology
Variable replacement using PCA
Classifying jobs using FCM
Forecasting the job cycle times using FBPN
Is the improvement
negligible
Stop
Yes
No
Updating the upper and lower bounds
Evaluating the forecasting precision and accuracy
Figure 1 The flowchart of the proposed methodology
Step 6 If the improvement in the forecasting precisionor accuracy becomes negligible stop otherwise return toStep 3
A flow chart of the proposed methodology is shown inFigure 1 These steps can be compared to the basic steps ofdata mining as shown in Table 5
21 Step 1 Variable Replacement Using PCA First PCA isused to replace the inputs to the FBPN A series of linearcombinations of the original variables forms a new variableso that these new variables are as unrelated to each other aspossible Although there are more advanced applications ofPCA in this study PCA is used to enhance the efficiency ofFBPN training PCA consists of the four following steps
(1) Raw data standardization to eliminate dimensionaldifferences and large numerical differences the orig-inal variables are standardized [13 14]
119909lowast
119895119894=
119909119895119894minus 119909119894
120590119894
119909119894=
sum119899
119895=1119909119895119894
119899
120590119894=
radicsum119899
119895=1(119909119895119894minus 119909119894)2
119899 minus 1
(1)
4 Mathematical Problems in Engineering
Table 3 Advantages and disadvantages of various data mining approaches for job cycle time forecasting
Approach Advantages Disadvantages
Clustering (i) Easiest to use(ii) Can be used with other methods
(i) Not accurate(ii) No method to determine the optimal number of clusters
GA (i) Accurate(ii) Can be used with other methods Time-consuming iterations
Decision trees (i) Handles missing values well(ii) Robust to outliers High variance of tree structure
ANN (i) Accurate(ii) Robust to outliers
(i) Difficult to use(ii) Time-consuming training(iii) Possible overfitting
Hybrid approaches Very accurate (i) Very difficult to use(ii) Very time consuming
Table 4 The differences between the proposed methodology and the previous methods
Method Variable replacement Job pre-classification Job post-classification Determining the boundsSOM-WM kM-FBPNLook-ahead SOM-FBPN No Yes No Not discussed
Post-classifying FBPN-BPN No No Yes Not discussedBi-directional classifyingFCM-FBPN-RBF No Yes Yes Not discussed
BPN-NP No No No DifficultThe proposed methodology Yes Yes Yes Easy
Table 5 The proposed methodology compared to data mining
Data mining steps The proposed methodologyApplication domainidentification
Job cycle time forecasting for a waferfabrication factory
Target dataset selection The most recent PROMIS data of a majorproduct type with normal priority
Data preprocessing PCA partial normalization
Data mining
(i) FCM for job classification(ii) FBPN to model the relationshipbetween the job cycle time and six jobparameters
Knowledge extraction Training data (34 of the collected data)Knowledge application Testing data (the remaining data)
Knowledge evaluation Forecasting precision (the average range)and accuracy (MAE MAPE RMSE)
lowastPROMIS production management information system MAE mean abso-lute error MAPE mean absolute percentage error
where 119909119895119894is the 119894th attribute of job 119895 119895 = 1 sim 119899 119909
119894
and 120590119894indicate the mean and standard deviation of
variable i respectively(2) Establishment of the correlation matrix 119877 is as fol-
lows
119877 =1
119899 minus 1119883lowast119879
119883lowast
(2)
where119883lowast is the standardized data matrix The eigen-
values and eigenvectors of 119877 are calculated and rep-resented as 120582
1sim 120582119898and 119906
1sim 119906119898 respectively 120582
1ge
1205822ge sdot sdot sdot ge 120582
119898
(3) Determination of the number of principal compo-nents the variance contribution rate is calculated as
120578119902=
120582119902
sum119898
119903=1120582119903
sdot 100 (3)
and the accumulated variance contribution rate is
120578Σ(119901) =
119901
sum
119902=1
120578119902 (4)
Choose the smallest 119875 value such that 120578Σ(119901) ranges-
from 85 to 90 A Pareto analysis chart can beused to show the percentage of variability that can beattributed to each principal component
(4) Formation of the following matrices
119880119898times119901
= [1199061 1199062 119906
119901]
119885119899times119901
= 119883lowast
119899times119898119880119898times119901
(5)
119885119899times119901
= [119911119895119902] (119895 = 1 sim 119899 119902 = 1 sim 119901) are the com-
ponent scoresThese scores contain the coordinates ofthe original data in the new coordinate systemdefinedby the principal componentsThey will be used as thenew inputs to the FBPN
22 Step 2 Classifying Jobs Using FCM After PCA examplesare classified by FCM If a crisp clustering method is appliedinstead then it is possible that some clusters will have veryfew examples FCM avoids the problems of crisp clusteringbecause in FCM an example belongs to multiple clusters to
Mathematical Problems in Engineering 5
different degrees Similarly in probability theory the naıveBayesmethod provides a probability that each itembelongs toa class However the classification of jobs in FCM can includesubjective judgments
FCM classifies jobs byminimizing the following objectivefunction
Min119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896) (6)
where119870 is the required number of categories n is the numberof jobs120583
119895(119896)indicates the degree ofmembership that job 119895 has
in category k 119890119895(119896)
measures the distance from job 119895 to thecentroid of category 119896 119898 isin [1infin) is a parameter to adjustthe fuzziness and is usually set to 2 The procedure of FCM isas follows
(1) Normalize the input data
(2) Produce a preliminary clustering result
(3) Iterations calculate the centroid of each category as
119911(119896)
= 119911(119896)119902
119896 = 1 sim 119870
119911(119896)119902
=sum119899
119895=1120583119898
119895(119896)119911119895119902
sum119899
119895=1120583119898119895(119896)
119896 = 1 sim 119870 119902 = 1 sim 119901
120583119895(119896)
=1
sum119870
119892=1(119890119895(119896)
119890119895(119892)
)2(119898minus1)
119895 = 1 sim 119899 119896 = 1 sim 119870
119890119895(119896)
= radic
119901
sum
119902=1
(119911119895119901
minus 119911(119896)119901
)2
119895 = 1 sim 119899 119896 = 1 sim 119870
(7)
where 119911(119896)
is the centroid of category 119896 120583(119905)119895(119896)
is thedegree ofmembership that job 119895has in category 119896 afterthe 119905th iteration
(4) Remeasure the distance from each job to the centroidof each category and then recalculate the correspond-ing membership
(5) Stop if the following condition is met Otherwisereturn to Step (3)
max119896
max119895
10038161003816100381610038161003816120583(119905)
119895(119896)minus 120583(119905minus1)
119895(119896)
10038161003816100381610038161003816lt 119889 (8)
where 119889 is a real number representing the member-ship convergence threshold
The performance of FCM is affected by the settings of theinitial values initialization can be repeated multiple timesin order to find the optimal solution Finally the separate
distance test (119878 test) proposed by Xie and Beni [34] can beapplied to determine the optimal number of categories119870
Min 119878
subject to 119869119898
=
119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896)
1198902
min = min1198961 = 1198962
(
119901
sum
119902=1
(119911(1198961)119902
minus 119911(1198962)119902
)2
)
119878 =119869119898
119899 times 1198902min 119870 isin 119885
+
(9)
The119870 value thatminimizes 119878 determines the optimal numberof categories
23 Step 3 Forecasting the Cycle Times of Jobs in EachCategory Using a FBPN All categories in the present studyare considered by the FBPN There are different types ofFBPNs for example Nomura et al [35] Chen [5] Lin [36]and Chen and Wang [33] Among them Chen and WangrsquosFBPN is unique since it is aimed at optimizing precision Forthis reason it is chosen for this study This FBPN is a proventechnology that has incorporated many features of priorartificial neural networks which have solved a wide variety ofproblems characterized by sets of different equations Somespecial problems can benefit from specialized tools such ascompositional pattern-producing networks cascading neu-ral networks and dynamic neural networks but for thepresent job cycle time problem a well-trained FBPN withan optimized structure can still produce very good resultsAll parameters in the FBPN are expressed in triangularfuzzy numbers (TFNs) There are various types of fuzzynumbers with different shapes among them the triangulartype is easily implemented andhas been universally applied tonumerous applications (eg [37 38]) Although the proposedmethodology uses TFNs it can be easilymodified to use othertypes of fuzzy numbers such as trapezoidal fuzzy numbers orbounded LR-type fuzzy numbers
The FBPN is configured as follows
(1) Inputs there are 119895 sets of inputs each set consists ofthe new factors determined by PCA from the 119895th jobThese factors have to be partially normalized so thattheir values fall within [01 09] [13 14]
(2) Single hidden layer generally one hidden layer givesthe best convergence results for the FBPN
(3) For simplicity the number of neurons in the hiddenlayer is twice the number in the input layer Anincrease in the number of hidden-layer nodes lessensthe output errors for the training examples butincreases the errors for novel examples This phe-nomenon is often called ldquooverfittingrdquo Some researchhas considered the connections between the complex-ity of a FBPN the performance for the training dataand the number of examples noteworthy researchincludes Akaikersquos information criterion (AIC) [39]
6 Mathematical Problems in Engineering
and Rissanenrsquos minimum description length (MDL)[40]
(4) Output the (normalized) cycle time forecast of theexample
The procedure for determining the parameter valuesis now described After pre-classification a portion of theadopted examples in each category is fed as ldquotraining exam-plesrdquo into the FBPN to determine the parameter valuesTwo phases are involved at the training stage First in theforward phase inputs are multiplied with weights summedand transferred to the hidden layer Then activated signalsare outputted from the hidden layer as
ℎ119897= (ℎ1198971 ℎ1198972 ℎ1198973)
=1
1 + 119890minus119899ℎ
119897
= (1
1 + 119890minus119899ℎ
1198971
1
1 + 119890minus119899ℎ
1198972
1
1 + 119890minus119899ℎ
1198973
)
(10)
where
119899ℎ
119897= (119899ℎ
1198971 119899ℎ
1198972 119899ℎ
1198973)
= 119868ℎ
119897(minus) 120579ℎ
119897
= (119868ℎ
1198971minus 120579ℎ
1198973 119868ℎ
1198972minus 120579ℎ
1198972 119868ℎ
1198973minus 120579ℎ
1198971)
119868ℎ
119897= (119868ℎ
1198971 119868ℎ
1198972 119868ℎ
1198973)
= sum
all 119896119908ℎ
119896119897sdot 119909119896
= (sum
all 119896min (119908
ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896)
sum
all 119896119908ℎ
1199091198962119909119896 sum
all 119896max (119908ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896))
(11)
ℎ119897is the output from hidden-layer node 119897 119897 = 1 sim 119871 120579ℎ
119897is
the threshold for screening out weak signals by hidden-layernode 119897119908ℎ
119896119897is theweight of the connection between input node
119896 and hidden-layer node 119897 119896 = 1 sim 119901 119897 = 1 sim 119871 119909119894119895is
the 119896th input 119896 = 1 sim 119901 The remaining parameters aretransition variables (minus) and (times) denote fuzzy subtraction andmultiplication respectively
ℎ119897s are also transferred to the output layer with the same
procedure Finally the output of the FBPN is generated asfollows
119900119895= (1199001198951 1199001198952 1199001198953)
=1
1 + 119890minus119899119900
= (1
1 + 119890minus119899119900
1
1
1 + 119890minus119899119900
2
1
1 + 119890minus119899119900
3
)
(12)
where
119899119900
= (119899119900
1 119899119900
2 119899119900
3) = 119868119900
(minus) 120579119900
= (119868119900
1minus 120579119900
3 119868119900
2minus 120579119900
2 119868119900
3minus 120579119900
1)
(13)
119868119900
= (119868119900
1 119868119900
2 119868119900
3)
=sum
all 119897119908119900
119897(times) ℎ119897
cong (sum
all 119897min (119908
119900
1198971ℎ1198971 119908119900
1198973ℎ1198973)
sum
all 119897119908119900
1198972ℎ1198972 sum
all 119897max (119908119900
1198971ℎ1198971 119908119900
1198973ℎ1198973))
(14)
119900119895is the network output which is the normalized value of the
cycle time forecast of job 119895 120579119900 is the threshold for screeningout weak signals by the output node 119908119900
119897is the weight of
the connection between hidden-layer node 119897 and the outputnode 119897 = 1 sim 119871
Subsequently in the backward phase the training of theFBPN is decomposed into three subtasks determining thecenter value and upper and lower bounds of the parameters
First to determine the center of each parameter (such as119908ℎ
1198961198972 120579ℎ1198972 1199081199001198972 and 120579
119900
2) the FBPN is treated as a crisp network
Algorithms applicable for this purpose include the gradi-ent descent algorithms the conjugate gradient algorithmsthe Levenberg-Marquardt algorithm and others In thisstudy the Levenberg-Marquardt algorithm is applied TheLevenberg-Marquardt algorithm was designed for trainingwith second-order speed without having to compute theHes-sian matrix It uses approximation and updates the networkparameters in a Newton-like way as described below
The network parameters are placed in vector 120573 = [119908ℎ
11
119908ℎ
119901119871 120579ℎ
1 120579
ℎ
119871 119908119900
1 119908
119900
119871 120579119900
] The network output 119900119895
can be represented with 119891(x119895120573) The objective function of
the FBPN is to minimize RMSE or equivalently the sum ofsquared error (SSE)
SSE (120573) =
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573))2
(15)
The Levenberg-Marquardt algorithm is an iterative pro-cedure In the beginning the user should specify the initialvalues of the network parameter vector 120573 It is a commonpractice to set 120573T = (1 1 1) In each step the parametervector 120573 is replaced by a new estimate 120573 + 120575 where120575 = [Δ119908
ℎ
11 Δ119908
ℎ
119901119871 Δ120579ℎ
1 Δ120579
ℎ
119871 Δ119908119900
1 Δ119908
119900
119871 Δ120579119900
] Thenetwork output becomes 119891(x
119895120573 + 120575) it is approximated by
its linearization as
119891 (x119895120573 + 120575) asymp 119891 (x
119895120573) + J
119895120575 (16)
where
J119895=
120597119891 (x119895120573)
120597120573(17)
Mathematical Problems in Engineering 7
is the gradient vector of 119891 with respect to 120573 Substituting (16)into (15) gives
SSE (120573 + 120575) asymp
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573) minus J
119895120575)2
(18)
When the network reaches the optimal solution the gradientof SSE with respect to 120575 will be zero Taking the derivative ofSSE(120573+120575)with respect to 120575 and setting the result to zero gives
(JTJ) 120575 = JT (119873 (CT119895) minus 119891 (x
119895120573)) (19)
where J is the Jacobian matrix containing the first derivativeof network error with respect to the weights and biasesEquation (19) includes a set of linear equations that can besolved for 120575
24 Step 4 Determining the Upper Bounds of Parameters inthe FBPN In Chen andWang [33] and Chen and Lin [41] anNPmodel is constructed to adjust the connectionweights andthresholds in the FBPN such anNPmodel is not easy to solveThe proposedmethodology only makes iterative adjustmentsto the threshold on the output nodeThisway ismuch simplerand can also achieve good results
Substituting (13) into (12) gives
1199001198952
=1
1 + 119890minus119899119900
1198952
=1
1 + 119890minus(119868119900
1198952minus120579119900
2)
=1
1 + 119890120579119900
2minus119868119900
1198952
(20)
Therefore
ln(1
1199001198952
minus 1) = 120579119900
2minus 119868119900
1198952 (21)
So
119868119900
1198952= 120579119900
2minus ln(
1
1199001198952
minus 1) (22)
Assume that the adjustment made to the output nodethreshold is denoted as Δ120579
119900
= 120579119900
3minus 120579119900
2 After adjustment the
output from the new FBPN 1199001198953 determines the upper bound
of the cycle time
1199001198953
=1
1 + 119890minus119899119900
1198953
(23)
where119899119900
1198953= 119868119900
1198953minus 120579119900
3= 119868119900
1198953minus (120579119900
2+ Δ120579119900
) (24)
Substituting (24) into (23) gives
1199001198953
=1
1 + 119890minus(119868119900
1198953minus120579119900
2minusΔ120579119900)
(25)
and substituting (22) into (25) gives
1199001198953
=1
1 + 119890minus(120579119900
2minusln(11199001198952minus1)minus1205791199002minusΔ120579119900)
=1
1 + 119890ln(11199001198952minus1)+Δ120579119900
=1
1 + 119890Δ120579119900
(11199001198952
minus 1)
(26)
Actual cycle time
Upper bound 1Upper bound 2
Upper bound 3
Upper bound 5Upper bound 4
Upper bound 6
Final result
Figure 2 Iterative reduction of the upper bound
Obviously the maximum ofΔ120579119900 determines the lowest upper
bound since 1199001198953is the upper bound of the cycle time 119900
1198953ge
119873(CT119895)
1
1 + 119890ln(11199001198952minus1)+Δ120579119900
ge 119873(CT119895) (27)
Δ120579119900
le ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) (28)
Equation (28) holds for all jobs so
Δ120579119900
le min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(29)
According to (26) the optimal value of Δ120579119900 should be set to
the maximum possible value
Δ120579119900lowast
= min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(30)
Then the FBPNrsquos optimization results depend on theinitial conditions and therefore are different at every iterationAssume that the optimal value of 119900
1198953in the 119905th iteration is
denoted by 1199001198953(119905) then after some iterations
1199001198953
(all iterations) = min119905
1199001198953
(119905) (31)
In this way the upper bound of the cycle time is decreasedgradually (see Figure 2) Another merit of this approach isthat it does not rely on the parameters of the FBPN
Theorem 1 Theupper bound can be found by considering onlyjobs with119873(CT
119895) ge 1199001198952
8 Mathematical Problems in Engineering
Proof Divide jobs into two categories 1198621
(jobs with119873(CT
119895) lt 1199001198952) and 119862
2(jobs with 119873(CT
119895) ge 1199001198952) Equation
(30) becomes
Δ120579119900lowast
= min(min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)))
(32)
If119873(CT119895) lt 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) gt 0 (33)
Therefore
min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) gt 0 (34)
In contrast if119873(CT119895) ge 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) le 0
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) le 0
(35)
Finally
Δ120579119900lowast
= min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(36)
Theorem 1 is proved
25 Step 4 Determining the Lower Bounds of Parameters in theFBPN In a similar way the threshold on the output node canbe modified to determine the lower bound of the cycle timeforecast so that the actual value will be greater than the lowerbound of the network output The optimal value of Δ120579
119900 canbe obtained as
Δ120579119900lowast
= max119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(37)
Theorem2 The lower bound can be found by considering onlyjobs with 119873(CT
119895) lt 1199001198952
The proof of Theorem 2 resembles that of Theorem 1
Proof Assume that the optimal value of 1199001198951
in the 119905threplication is indicated with 119900
1198951(119905) then after some iterations
1199001198951
(all iterations) = max119905
1199001198951
(119905) (38)
In this way the upper bound of the cycle time is increasedgradually Δ120579
119900lowastdoes not rely on the parameters of the FBPN
3 Application and Analyses
Todemonstrate the application of the proposedmethodologyan illustrative example containing the data of 40 jobs (seeTable 6) was used
We standardize the data and obtain the correlationmatrix119877 The eigenvalues and eigenvectors of 119877 are calculated as
1205821= 266 120582
2= 115 120582
3= 094
1205824= 083 120582
5= 025 120582
6= 002
(39)
1199061=
[[[[[[[
[
020
052
016
056
007
059
]]]]]]]
]
1199062=
[[[[[[[
[
047
minus019
068
minus019
minus049
005
]]]]]]]
]
1199063=
[[[[[[[
[
minus027
031
minus032
003
minus085
minus002
]]]]]]]
]
1199064=
[[[[[[[
[
081
minus002
minus057
006
minus004
minus015
]]]]]]]
]
1199065=
[[[[[[[
[
minus012
minus072
minus007
065
minus018
009
]]]]]]]
]
1199066=
[[[[[[[
[
002
029
028
047
003
minus078
]]]]]]]
]
(40)
respectively The variance contribution rates are
1205781= 46 120578
2= 20 120578
3= 16
1205784= 14 120578
5= 4 120578
6= 0
(41)
Summing up 120578119902rsquos we obtain
120578Σ(1) = 46 120578
Σ(2) = 65 120578
Σ(3) = 81
120578Σ(4) = 95 120578
Σ(5) = 100 120578
Σ(6) = 100
(42)
A Pareto analysis chart is used to compare the percentageof variability explained by each principal component (seeFigure 3) There is a clear break in the amount of varianceaccounted for by each component between the first andsecond components However that component by itselfexplains less than 50 of the variance so more components
Mathematical Problems in Engineering 9
Table 6 The illustrative example
119895 1199091198951
1199091198952
1199091198953
1199091198954
1199091198955
1199091198956
CT119895
1 24 1261 181 781 112 092 9352 24 1263 181 762 127 090 9583 24 1220 176 761 127 089 10474 23 1282 178 802 127 094 10115 23 1303 180 780 175 093 10686 23 1281 183 782 175 093 11437 23 1242 184 741 163 089 11038 24 1262 182 681 139 086 12509 22 1260 182 701 98 086 118110 22 1260 179 700 257 087 119411 24 1301 163 722 99 084 126012 22 1221 184 641 131 082 124013 23 1323 159 740 247 087 118014 24 1362 181 782 191 095 122715 24 1261 181 762 219 091 123616 23 1321 177 801 219 096 121517 22 1343 180 822 219 097 122818 24 1321 177 762 54 093 126619 25 1343 179 781 54 096 128520 25 1300 180 740 54 092 127221 22 1320 181 721 54 091 131022 24 1321 182 742 49 092 126523 23 1262 165 680 201 080 130824 22 1240 161 722 103 082 133125 23 1183 183 661 53 082 129426 23 1282 184 701 53 088 131427 22 1202 177 680 248 084 132128 23 1202 178 681 248 085 135329 24 1202 185 701 82 086 122630 23 1202 158 721 98 081 130131 24 1343 181 760 67 094 128032 24 1381 185 801 67 097 128633 22 1362 156 780 67 091 125234 23 1282 179 782 223 092 121435 23 1320 180 782 176 093 125136 25 1340 176 801 462 097 122237 23 1320 182 781 168 095 118738 22 1361 181 781 141 094 120539 22 1381 179 781 95 097 112040 23 1363 178 802 179 097 1133
are considered To meet the requirement 120578Σ(119901) ge 85sim90
p is chosen as 3 We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so we reduce the issue to these threecomponents in order to visualize the data
The component scores are recalculated as the new inputsto the FCM-FBPN (see Table 7) These scores contain thecoordinates of the original data in the new coordinate systemdefined by the principal components
Subsequently jobs are classified using FCM based on thenew variables The results of the 119878 test are summarized inTable 8 In this case the optimal number of job categories is 5
Table 7 New inputs to the FCM-FBPN
1199111198951
1199111198952
1199111198953
minus056 091 minus019
minus013 087 minus034
051 057 minus037
minus097 minus010 020
minus087 minus020 minus026
minus075 014 minus051
057 056 minus066
130 118 minus055
155 031 047
137 minus087 minus104
111 minus059 091
304 063 minus020
051 minus244 minus002
minus194 012 minus043
minus030 035 minus129
minus162 minus084 minus048
minus204 minus124 minus017
minus087 077 089
minus192 134 064
minus058 170 034
022 023 129
minus062 131 073
254 minus126 minus016
239 minus164 120
302 157 014
089 121 066
256 minus074 minus119
219 minus013 minus154
161 190 minus042
272 minus123 087
minus127 099 071
minus256 107 078
minus037 minus244 247
minus060 minus051 minus082
minus106 minus027 minus017
minus254 minus136 minus341
minus131 minus002 minus018
minus132 minus063 067
minus177 minus058 132
minus213 minus066 012
Table 8 The results of the S test
Number of categories (119870) 119869119898
1198902
min 119878
2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043
However therewill be some categories with very few jobs Forthis reason the second best solution is used that is 4 cate-gories A common practice is to set a threshold of member-
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Table 2 Data mining methods for job cycle time forecasting
Category ExamplesClustering Backus et al [1] Meidan et al [6]GA Chang et al [8]Decisiontrees CART [1] SNBC [6] and Tirkel [15]
ANNChen [5] Chang and Hsieh [9] Sha and Hsu [10]Chang et al [8] Backus et al [1] Chien et al [4]Meidan et al [6] and Tirkel [15]
Hybridapproaches
SOM-WM [26] kM-FBPN [11 29] look-aheadSOM-FBPN [3 23 24] post-classifying FBPN-BPN[13 14] and bi-directional classifyingFCM-FBPN-RBF [25]
predict the job cycle time which is an innovation inthis field
(2) The job cycle time is predicted by an effective FBPNapproach For the job cycle time there are upper andlower bounds that constrain the forecast to be withina possible range The FBPN approach from Chen andWangrsquos FBPN research [33] has beenmodified for thispurpose In Chen and Wangrsquos FBPN approach twononlinear programming (NP) models are solved todetermine the upper and lower bounds of the jobcycle time However the NP models involve compli-cated constraints and therefore are difficult to solveIn addition the NP models will become too huge ifmany jobs are to be considered To solve these pro-blems in this study the upper and lower bounds aredetermined in a simpler way
The differences between the proposedmethodology and vari-ous previous methods are summarized in Table 4
The remainder of this paper is organized as followsSection 2 introduces the proposed methodology which iscomposed of six steps Section 3 demonstrates the proposedmethod with an illustrative example then presents anothercase with data collected from a real wafer fabrication factoryand makes comparisons with some existing approachesFinally concluding remarks anddirections for future researchare given in Section 4
2 Methodology
The fuzzy data mining procedure consists of several steps
Step 1 Replacing parameters using PCA
Step 2 Classifying jobs using fuzzy c-means (FCM)
Step 3 Forecasting the cycle times of jobs in each categoryusing a FBPN
Step 4 Updating the upper and lower bounds
Step 5 Evaluating and comparing the forecasting precisionand accuracy of the proposed methodology
Variable replacement using PCA
Classifying jobs using FCM
Forecasting the job cycle times using FBPN
Is the improvement
negligible
Stop
Yes
No
Updating the upper and lower bounds
Evaluating the forecasting precision and accuracy
Figure 1 The flowchart of the proposed methodology
Step 6 If the improvement in the forecasting precisionor accuracy becomes negligible stop otherwise return toStep 3
A flow chart of the proposed methodology is shown inFigure 1 These steps can be compared to the basic steps ofdata mining as shown in Table 5
21 Step 1 Variable Replacement Using PCA First PCA isused to replace the inputs to the FBPN A series of linearcombinations of the original variables forms a new variableso that these new variables are as unrelated to each other aspossible Although there are more advanced applications ofPCA in this study PCA is used to enhance the efficiency ofFBPN training PCA consists of the four following steps
(1) Raw data standardization to eliminate dimensionaldifferences and large numerical differences the orig-inal variables are standardized [13 14]
119909lowast
119895119894=
119909119895119894minus 119909119894
120590119894
119909119894=
sum119899
119895=1119909119895119894
119899
120590119894=
radicsum119899
119895=1(119909119895119894minus 119909119894)2
119899 minus 1
(1)
4 Mathematical Problems in Engineering
Table 3 Advantages and disadvantages of various data mining approaches for job cycle time forecasting
Approach Advantages Disadvantages
Clustering (i) Easiest to use(ii) Can be used with other methods
(i) Not accurate(ii) No method to determine the optimal number of clusters
GA (i) Accurate(ii) Can be used with other methods Time-consuming iterations
Decision trees (i) Handles missing values well(ii) Robust to outliers High variance of tree structure
ANN (i) Accurate(ii) Robust to outliers
(i) Difficult to use(ii) Time-consuming training(iii) Possible overfitting
Hybrid approaches Very accurate (i) Very difficult to use(ii) Very time consuming
Table 4 The differences between the proposed methodology and the previous methods
Method Variable replacement Job pre-classification Job post-classification Determining the boundsSOM-WM kM-FBPNLook-ahead SOM-FBPN No Yes No Not discussed
Post-classifying FBPN-BPN No No Yes Not discussedBi-directional classifyingFCM-FBPN-RBF No Yes Yes Not discussed
BPN-NP No No No DifficultThe proposed methodology Yes Yes Yes Easy
Table 5 The proposed methodology compared to data mining
Data mining steps The proposed methodologyApplication domainidentification
Job cycle time forecasting for a waferfabrication factory
Target dataset selection The most recent PROMIS data of a majorproduct type with normal priority
Data preprocessing PCA partial normalization
Data mining
(i) FCM for job classification(ii) FBPN to model the relationshipbetween the job cycle time and six jobparameters
Knowledge extraction Training data (34 of the collected data)Knowledge application Testing data (the remaining data)
Knowledge evaluation Forecasting precision (the average range)and accuracy (MAE MAPE RMSE)
lowastPROMIS production management information system MAE mean abso-lute error MAPE mean absolute percentage error
where 119909119895119894is the 119894th attribute of job 119895 119895 = 1 sim 119899 119909
119894
and 120590119894indicate the mean and standard deviation of
variable i respectively(2) Establishment of the correlation matrix 119877 is as fol-
lows
119877 =1
119899 minus 1119883lowast119879
119883lowast
(2)
where119883lowast is the standardized data matrix The eigen-
values and eigenvectors of 119877 are calculated and rep-resented as 120582
1sim 120582119898and 119906
1sim 119906119898 respectively 120582
1ge
1205822ge sdot sdot sdot ge 120582
119898
(3) Determination of the number of principal compo-nents the variance contribution rate is calculated as
120578119902=
120582119902
sum119898
119903=1120582119903
sdot 100 (3)
and the accumulated variance contribution rate is
120578Σ(119901) =
119901
sum
119902=1
120578119902 (4)
Choose the smallest 119875 value such that 120578Σ(119901) ranges-
from 85 to 90 A Pareto analysis chart can beused to show the percentage of variability that can beattributed to each principal component
(4) Formation of the following matrices
119880119898times119901
= [1199061 1199062 119906
119901]
119885119899times119901
= 119883lowast
119899times119898119880119898times119901
(5)
119885119899times119901
= [119911119895119902] (119895 = 1 sim 119899 119902 = 1 sim 119901) are the com-
ponent scoresThese scores contain the coordinates ofthe original data in the new coordinate systemdefinedby the principal componentsThey will be used as thenew inputs to the FBPN
22 Step 2 Classifying Jobs Using FCM After PCA examplesare classified by FCM If a crisp clustering method is appliedinstead then it is possible that some clusters will have veryfew examples FCM avoids the problems of crisp clusteringbecause in FCM an example belongs to multiple clusters to
Mathematical Problems in Engineering 5
different degrees Similarly in probability theory the naıveBayesmethod provides a probability that each itembelongs toa class However the classification of jobs in FCM can includesubjective judgments
FCM classifies jobs byminimizing the following objectivefunction
Min119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896) (6)
where119870 is the required number of categories n is the numberof jobs120583
119895(119896)indicates the degree ofmembership that job 119895 has
in category k 119890119895(119896)
measures the distance from job 119895 to thecentroid of category 119896 119898 isin [1infin) is a parameter to adjustthe fuzziness and is usually set to 2 The procedure of FCM isas follows
(1) Normalize the input data
(2) Produce a preliminary clustering result
(3) Iterations calculate the centroid of each category as
119911(119896)
= 119911(119896)119902
119896 = 1 sim 119870
119911(119896)119902
=sum119899
119895=1120583119898
119895(119896)119911119895119902
sum119899
119895=1120583119898119895(119896)
119896 = 1 sim 119870 119902 = 1 sim 119901
120583119895(119896)
=1
sum119870
119892=1(119890119895(119896)
119890119895(119892)
)2(119898minus1)
119895 = 1 sim 119899 119896 = 1 sim 119870
119890119895(119896)
= radic
119901
sum
119902=1
(119911119895119901
minus 119911(119896)119901
)2
119895 = 1 sim 119899 119896 = 1 sim 119870
(7)
where 119911(119896)
is the centroid of category 119896 120583(119905)119895(119896)
is thedegree ofmembership that job 119895has in category 119896 afterthe 119905th iteration
(4) Remeasure the distance from each job to the centroidof each category and then recalculate the correspond-ing membership
(5) Stop if the following condition is met Otherwisereturn to Step (3)
max119896
max119895
10038161003816100381610038161003816120583(119905)
119895(119896)minus 120583(119905minus1)
119895(119896)
10038161003816100381610038161003816lt 119889 (8)
where 119889 is a real number representing the member-ship convergence threshold
The performance of FCM is affected by the settings of theinitial values initialization can be repeated multiple timesin order to find the optimal solution Finally the separate
distance test (119878 test) proposed by Xie and Beni [34] can beapplied to determine the optimal number of categories119870
Min 119878
subject to 119869119898
=
119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896)
1198902
min = min1198961 = 1198962
(
119901
sum
119902=1
(119911(1198961)119902
minus 119911(1198962)119902
)2
)
119878 =119869119898
119899 times 1198902min 119870 isin 119885
+
(9)
The119870 value thatminimizes 119878 determines the optimal numberof categories
23 Step 3 Forecasting the Cycle Times of Jobs in EachCategory Using a FBPN All categories in the present studyare considered by the FBPN There are different types ofFBPNs for example Nomura et al [35] Chen [5] Lin [36]and Chen and Wang [33] Among them Chen and WangrsquosFBPN is unique since it is aimed at optimizing precision Forthis reason it is chosen for this study This FBPN is a proventechnology that has incorporated many features of priorartificial neural networks which have solved a wide variety ofproblems characterized by sets of different equations Somespecial problems can benefit from specialized tools such ascompositional pattern-producing networks cascading neu-ral networks and dynamic neural networks but for thepresent job cycle time problem a well-trained FBPN withan optimized structure can still produce very good resultsAll parameters in the FBPN are expressed in triangularfuzzy numbers (TFNs) There are various types of fuzzynumbers with different shapes among them the triangulartype is easily implemented andhas been universally applied tonumerous applications (eg [37 38]) Although the proposedmethodology uses TFNs it can be easilymodified to use othertypes of fuzzy numbers such as trapezoidal fuzzy numbers orbounded LR-type fuzzy numbers
The FBPN is configured as follows
(1) Inputs there are 119895 sets of inputs each set consists ofthe new factors determined by PCA from the 119895th jobThese factors have to be partially normalized so thattheir values fall within [01 09] [13 14]
(2) Single hidden layer generally one hidden layer givesthe best convergence results for the FBPN
(3) For simplicity the number of neurons in the hiddenlayer is twice the number in the input layer Anincrease in the number of hidden-layer nodes lessensthe output errors for the training examples butincreases the errors for novel examples This phe-nomenon is often called ldquooverfittingrdquo Some researchhas considered the connections between the complex-ity of a FBPN the performance for the training dataand the number of examples noteworthy researchincludes Akaikersquos information criterion (AIC) [39]
6 Mathematical Problems in Engineering
and Rissanenrsquos minimum description length (MDL)[40]
(4) Output the (normalized) cycle time forecast of theexample
The procedure for determining the parameter valuesis now described After pre-classification a portion of theadopted examples in each category is fed as ldquotraining exam-plesrdquo into the FBPN to determine the parameter valuesTwo phases are involved at the training stage First in theforward phase inputs are multiplied with weights summedand transferred to the hidden layer Then activated signalsare outputted from the hidden layer as
ℎ119897= (ℎ1198971 ℎ1198972 ℎ1198973)
=1
1 + 119890minus119899ℎ
119897
= (1
1 + 119890minus119899ℎ
1198971
1
1 + 119890minus119899ℎ
1198972
1
1 + 119890minus119899ℎ
1198973
)
(10)
where
119899ℎ
119897= (119899ℎ
1198971 119899ℎ
1198972 119899ℎ
1198973)
= 119868ℎ
119897(minus) 120579ℎ
119897
= (119868ℎ
1198971minus 120579ℎ
1198973 119868ℎ
1198972minus 120579ℎ
1198972 119868ℎ
1198973minus 120579ℎ
1198971)
119868ℎ
119897= (119868ℎ
1198971 119868ℎ
1198972 119868ℎ
1198973)
= sum
all 119896119908ℎ
119896119897sdot 119909119896
= (sum
all 119896min (119908
ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896)
sum
all 119896119908ℎ
1199091198962119909119896 sum
all 119896max (119908ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896))
(11)
ℎ119897is the output from hidden-layer node 119897 119897 = 1 sim 119871 120579ℎ
119897is
the threshold for screening out weak signals by hidden-layernode 119897119908ℎ
119896119897is theweight of the connection between input node
119896 and hidden-layer node 119897 119896 = 1 sim 119901 119897 = 1 sim 119871 119909119894119895is
the 119896th input 119896 = 1 sim 119901 The remaining parameters aretransition variables (minus) and (times) denote fuzzy subtraction andmultiplication respectively
ℎ119897s are also transferred to the output layer with the same
procedure Finally the output of the FBPN is generated asfollows
119900119895= (1199001198951 1199001198952 1199001198953)
=1
1 + 119890minus119899119900
= (1
1 + 119890minus119899119900
1
1
1 + 119890minus119899119900
2
1
1 + 119890minus119899119900
3
)
(12)
where
119899119900
= (119899119900
1 119899119900
2 119899119900
3) = 119868119900
(minus) 120579119900
= (119868119900
1minus 120579119900
3 119868119900
2minus 120579119900
2 119868119900
3minus 120579119900
1)
(13)
119868119900
= (119868119900
1 119868119900
2 119868119900
3)
=sum
all 119897119908119900
119897(times) ℎ119897
cong (sum
all 119897min (119908
119900
1198971ℎ1198971 119908119900
1198973ℎ1198973)
sum
all 119897119908119900
1198972ℎ1198972 sum
all 119897max (119908119900
1198971ℎ1198971 119908119900
1198973ℎ1198973))
(14)
119900119895is the network output which is the normalized value of the
cycle time forecast of job 119895 120579119900 is the threshold for screeningout weak signals by the output node 119908119900
119897is the weight of
the connection between hidden-layer node 119897 and the outputnode 119897 = 1 sim 119871
Subsequently in the backward phase the training of theFBPN is decomposed into three subtasks determining thecenter value and upper and lower bounds of the parameters
First to determine the center of each parameter (such as119908ℎ
1198961198972 120579ℎ1198972 1199081199001198972 and 120579
119900
2) the FBPN is treated as a crisp network
Algorithms applicable for this purpose include the gradi-ent descent algorithms the conjugate gradient algorithmsthe Levenberg-Marquardt algorithm and others In thisstudy the Levenberg-Marquardt algorithm is applied TheLevenberg-Marquardt algorithm was designed for trainingwith second-order speed without having to compute theHes-sian matrix It uses approximation and updates the networkparameters in a Newton-like way as described below
The network parameters are placed in vector 120573 = [119908ℎ
11
119908ℎ
119901119871 120579ℎ
1 120579
ℎ
119871 119908119900
1 119908
119900
119871 120579119900
] The network output 119900119895
can be represented with 119891(x119895120573) The objective function of
the FBPN is to minimize RMSE or equivalently the sum ofsquared error (SSE)
SSE (120573) =
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573))2
(15)
The Levenberg-Marquardt algorithm is an iterative pro-cedure In the beginning the user should specify the initialvalues of the network parameter vector 120573 It is a commonpractice to set 120573T = (1 1 1) In each step the parametervector 120573 is replaced by a new estimate 120573 + 120575 where120575 = [Δ119908
ℎ
11 Δ119908
ℎ
119901119871 Δ120579ℎ
1 Δ120579
ℎ
119871 Δ119908119900
1 Δ119908
119900
119871 Δ120579119900
] Thenetwork output becomes 119891(x
119895120573 + 120575) it is approximated by
its linearization as
119891 (x119895120573 + 120575) asymp 119891 (x
119895120573) + J
119895120575 (16)
where
J119895=
120597119891 (x119895120573)
120597120573(17)
Mathematical Problems in Engineering 7
is the gradient vector of 119891 with respect to 120573 Substituting (16)into (15) gives
SSE (120573 + 120575) asymp
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573) minus J
119895120575)2
(18)
When the network reaches the optimal solution the gradientof SSE with respect to 120575 will be zero Taking the derivative ofSSE(120573+120575)with respect to 120575 and setting the result to zero gives
(JTJ) 120575 = JT (119873 (CT119895) minus 119891 (x
119895120573)) (19)
where J is the Jacobian matrix containing the first derivativeof network error with respect to the weights and biasesEquation (19) includes a set of linear equations that can besolved for 120575
24 Step 4 Determining the Upper Bounds of Parameters inthe FBPN In Chen andWang [33] and Chen and Lin [41] anNPmodel is constructed to adjust the connectionweights andthresholds in the FBPN such anNPmodel is not easy to solveThe proposedmethodology only makes iterative adjustmentsto the threshold on the output nodeThisway ismuch simplerand can also achieve good results
Substituting (13) into (12) gives
1199001198952
=1
1 + 119890minus119899119900
1198952
=1
1 + 119890minus(119868119900
1198952minus120579119900
2)
=1
1 + 119890120579119900
2minus119868119900
1198952
(20)
Therefore
ln(1
1199001198952
minus 1) = 120579119900
2minus 119868119900
1198952 (21)
So
119868119900
1198952= 120579119900
2minus ln(
1
1199001198952
minus 1) (22)
Assume that the adjustment made to the output nodethreshold is denoted as Δ120579
119900
= 120579119900
3minus 120579119900
2 After adjustment the
output from the new FBPN 1199001198953 determines the upper bound
of the cycle time
1199001198953
=1
1 + 119890minus119899119900
1198953
(23)
where119899119900
1198953= 119868119900
1198953minus 120579119900
3= 119868119900
1198953minus (120579119900
2+ Δ120579119900
) (24)
Substituting (24) into (23) gives
1199001198953
=1
1 + 119890minus(119868119900
1198953minus120579119900
2minusΔ120579119900)
(25)
and substituting (22) into (25) gives
1199001198953
=1
1 + 119890minus(120579119900
2minusln(11199001198952minus1)minus1205791199002minusΔ120579119900)
=1
1 + 119890ln(11199001198952minus1)+Δ120579119900
=1
1 + 119890Δ120579119900
(11199001198952
minus 1)
(26)
Actual cycle time
Upper bound 1Upper bound 2
Upper bound 3
Upper bound 5Upper bound 4
Upper bound 6
Final result
Figure 2 Iterative reduction of the upper bound
Obviously the maximum ofΔ120579119900 determines the lowest upper
bound since 1199001198953is the upper bound of the cycle time 119900
1198953ge
119873(CT119895)
1
1 + 119890ln(11199001198952minus1)+Δ120579119900
ge 119873(CT119895) (27)
Δ120579119900
le ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) (28)
Equation (28) holds for all jobs so
Δ120579119900
le min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(29)
According to (26) the optimal value of Δ120579119900 should be set to
the maximum possible value
Δ120579119900lowast
= min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(30)
Then the FBPNrsquos optimization results depend on theinitial conditions and therefore are different at every iterationAssume that the optimal value of 119900
1198953in the 119905th iteration is
denoted by 1199001198953(119905) then after some iterations
1199001198953
(all iterations) = min119905
1199001198953
(119905) (31)
In this way the upper bound of the cycle time is decreasedgradually (see Figure 2) Another merit of this approach isthat it does not rely on the parameters of the FBPN
Theorem 1 Theupper bound can be found by considering onlyjobs with119873(CT
119895) ge 1199001198952
8 Mathematical Problems in Engineering
Proof Divide jobs into two categories 1198621
(jobs with119873(CT
119895) lt 1199001198952) and 119862
2(jobs with 119873(CT
119895) ge 1199001198952) Equation
(30) becomes
Δ120579119900lowast
= min(min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)))
(32)
If119873(CT119895) lt 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) gt 0 (33)
Therefore
min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) gt 0 (34)
In contrast if119873(CT119895) ge 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) le 0
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) le 0
(35)
Finally
Δ120579119900lowast
= min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(36)
Theorem 1 is proved
25 Step 4 Determining the Lower Bounds of Parameters in theFBPN In a similar way the threshold on the output node canbe modified to determine the lower bound of the cycle timeforecast so that the actual value will be greater than the lowerbound of the network output The optimal value of Δ120579
119900 canbe obtained as
Δ120579119900lowast
= max119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(37)
Theorem2 The lower bound can be found by considering onlyjobs with 119873(CT
119895) lt 1199001198952
The proof of Theorem 2 resembles that of Theorem 1
Proof Assume that the optimal value of 1199001198951
in the 119905threplication is indicated with 119900
1198951(119905) then after some iterations
1199001198951
(all iterations) = max119905
1199001198951
(119905) (38)
In this way the upper bound of the cycle time is increasedgradually Δ120579
119900lowastdoes not rely on the parameters of the FBPN
3 Application and Analyses
Todemonstrate the application of the proposedmethodologyan illustrative example containing the data of 40 jobs (seeTable 6) was used
We standardize the data and obtain the correlationmatrix119877 The eigenvalues and eigenvectors of 119877 are calculated as
1205821= 266 120582
2= 115 120582
3= 094
1205824= 083 120582
5= 025 120582
6= 002
(39)
1199061=
[[[[[[[
[
020
052
016
056
007
059
]]]]]]]
]
1199062=
[[[[[[[
[
047
minus019
068
minus019
minus049
005
]]]]]]]
]
1199063=
[[[[[[[
[
minus027
031
minus032
003
minus085
minus002
]]]]]]]
]
1199064=
[[[[[[[
[
081
minus002
minus057
006
minus004
minus015
]]]]]]]
]
1199065=
[[[[[[[
[
minus012
minus072
minus007
065
minus018
009
]]]]]]]
]
1199066=
[[[[[[[
[
002
029
028
047
003
minus078
]]]]]]]
]
(40)
respectively The variance contribution rates are
1205781= 46 120578
2= 20 120578
3= 16
1205784= 14 120578
5= 4 120578
6= 0
(41)
Summing up 120578119902rsquos we obtain
120578Σ(1) = 46 120578
Σ(2) = 65 120578
Σ(3) = 81
120578Σ(4) = 95 120578
Σ(5) = 100 120578
Σ(6) = 100
(42)
A Pareto analysis chart is used to compare the percentageof variability explained by each principal component (seeFigure 3) There is a clear break in the amount of varianceaccounted for by each component between the first andsecond components However that component by itselfexplains less than 50 of the variance so more components
Mathematical Problems in Engineering 9
Table 6 The illustrative example
119895 1199091198951
1199091198952
1199091198953
1199091198954
1199091198955
1199091198956
CT119895
1 24 1261 181 781 112 092 9352 24 1263 181 762 127 090 9583 24 1220 176 761 127 089 10474 23 1282 178 802 127 094 10115 23 1303 180 780 175 093 10686 23 1281 183 782 175 093 11437 23 1242 184 741 163 089 11038 24 1262 182 681 139 086 12509 22 1260 182 701 98 086 118110 22 1260 179 700 257 087 119411 24 1301 163 722 99 084 126012 22 1221 184 641 131 082 124013 23 1323 159 740 247 087 118014 24 1362 181 782 191 095 122715 24 1261 181 762 219 091 123616 23 1321 177 801 219 096 121517 22 1343 180 822 219 097 122818 24 1321 177 762 54 093 126619 25 1343 179 781 54 096 128520 25 1300 180 740 54 092 127221 22 1320 181 721 54 091 131022 24 1321 182 742 49 092 126523 23 1262 165 680 201 080 130824 22 1240 161 722 103 082 133125 23 1183 183 661 53 082 129426 23 1282 184 701 53 088 131427 22 1202 177 680 248 084 132128 23 1202 178 681 248 085 135329 24 1202 185 701 82 086 122630 23 1202 158 721 98 081 130131 24 1343 181 760 67 094 128032 24 1381 185 801 67 097 128633 22 1362 156 780 67 091 125234 23 1282 179 782 223 092 121435 23 1320 180 782 176 093 125136 25 1340 176 801 462 097 122237 23 1320 182 781 168 095 118738 22 1361 181 781 141 094 120539 22 1381 179 781 95 097 112040 23 1363 178 802 179 097 1133
are considered To meet the requirement 120578Σ(119901) ge 85sim90
p is chosen as 3 We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so we reduce the issue to these threecomponents in order to visualize the data
The component scores are recalculated as the new inputsto the FCM-FBPN (see Table 7) These scores contain thecoordinates of the original data in the new coordinate systemdefined by the principal components
Subsequently jobs are classified using FCM based on thenew variables The results of the 119878 test are summarized inTable 8 In this case the optimal number of job categories is 5
Table 7 New inputs to the FCM-FBPN
1199111198951
1199111198952
1199111198953
minus056 091 minus019
minus013 087 minus034
051 057 minus037
minus097 minus010 020
minus087 minus020 minus026
minus075 014 minus051
057 056 minus066
130 118 minus055
155 031 047
137 minus087 minus104
111 minus059 091
304 063 minus020
051 minus244 minus002
minus194 012 minus043
minus030 035 minus129
minus162 minus084 minus048
minus204 minus124 minus017
minus087 077 089
minus192 134 064
minus058 170 034
022 023 129
minus062 131 073
254 minus126 minus016
239 minus164 120
302 157 014
089 121 066
256 minus074 minus119
219 minus013 minus154
161 190 minus042
272 minus123 087
minus127 099 071
minus256 107 078
minus037 minus244 247
minus060 minus051 minus082
minus106 minus027 minus017
minus254 minus136 minus341
minus131 minus002 minus018
minus132 minus063 067
minus177 minus058 132
minus213 minus066 012
Table 8 The results of the S test
Number of categories (119870) 119869119898
1198902
min 119878
2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043
However therewill be some categories with very few jobs Forthis reason the second best solution is used that is 4 cate-gories A common practice is to set a threshold of member-
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Table 3 Advantages and disadvantages of various data mining approaches for job cycle time forecasting
Approach Advantages Disadvantages
Clustering (i) Easiest to use(ii) Can be used with other methods
(i) Not accurate(ii) No method to determine the optimal number of clusters
GA (i) Accurate(ii) Can be used with other methods Time-consuming iterations
Decision trees (i) Handles missing values well(ii) Robust to outliers High variance of tree structure
ANN (i) Accurate(ii) Robust to outliers
(i) Difficult to use(ii) Time-consuming training(iii) Possible overfitting
Hybrid approaches Very accurate (i) Very difficult to use(ii) Very time consuming
Table 4 The differences between the proposed methodology and the previous methods
Method Variable replacement Job pre-classification Job post-classification Determining the boundsSOM-WM kM-FBPNLook-ahead SOM-FBPN No Yes No Not discussed
Post-classifying FBPN-BPN No No Yes Not discussedBi-directional classifyingFCM-FBPN-RBF No Yes Yes Not discussed
BPN-NP No No No DifficultThe proposed methodology Yes Yes Yes Easy
Table 5 The proposed methodology compared to data mining
Data mining steps The proposed methodologyApplication domainidentification
Job cycle time forecasting for a waferfabrication factory
Target dataset selection The most recent PROMIS data of a majorproduct type with normal priority
Data preprocessing PCA partial normalization
Data mining
(i) FCM for job classification(ii) FBPN to model the relationshipbetween the job cycle time and six jobparameters
Knowledge extraction Training data (34 of the collected data)Knowledge application Testing data (the remaining data)
Knowledge evaluation Forecasting precision (the average range)and accuracy (MAE MAPE RMSE)
lowastPROMIS production management information system MAE mean abso-lute error MAPE mean absolute percentage error
where 119909119895119894is the 119894th attribute of job 119895 119895 = 1 sim 119899 119909
119894
and 120590119894indicate the mean and standard deviation of
variable i respectively(2) Establishment of the correlation matrix 119877 is as fol-
lows
119877 =1
119899 minus 1119883lowast119879
119883lowast
(2)
where119883lowast is the standardized data matrix The eigen-
values and eigenvectors of 119877 are calculated and rep-resented as 120582
1sim 120582119898and 119906
1sim 119906119898 respectively 120582
1ge
1205822ge sdot sdot sdot ge 120582
119898
(3) Determination of the number of principal compo-nents the variance contribution rate is calculated as
120578119902=
120582119902
sum119898
119903=1120582119903
sdot 100 (3)
and the accumulated variance contribution rate is
120578Σ(119901) =
119901
sum
119902=1
120578119902 (4)
Choose the smallest 119875 value such that 120578Σ(119901) ranges-
from 85 to 90 A Pareto analysis chart can beused to show the percentage of variability that can beattributed to each principal component
(4) Formation of the following matrices
119880119898times119901
= [1199061 1199062 119906
119901]
119885119899times119901
= 119883lowast
119899times119898119880119898times119901
(5)
119885119899times119901
= [119911119895119902] (119895 = 1 sim 119899 119902 = 1 sim 119901) are the com-
ponent scoresThese scores contain the coordinates ofthe original data in the new coordinate systemdefinedby the principal componentsThey will be used as thenew inputs to the FBPN
22 Step 2 Classifying Jobs Using FCM After PCA examplesare classified by FCM If a crisp clustering method is appliedinstead then it is possible that some clusters will have veryfew examples FCM avoids the problems of crisp clusteringbecause in FCM an example belongs to multiple clusters to
Mathematical Problems in Engineering 5
different degrees Similarly in probability theory the naıveBayesmethod provides a probability that each itembelongs toa class However the classification of jobs in FCM can includesubjective judgments
FCM classifies jobs byminimizing the following objectivefunction
Min119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896) (6)
where119870 is the required number of categories n is the numberof jobs120583
119895(119896)indicates the degree ofmembership that job 119895 has
in category k 119890119895(119896)
measures the distance from job 119895 to thecentroid of category 119896 119898 isin [1infin) is a parameter to adjustthe fuzziness and is usually set to 2 The procedure of FCM isas follows
(1) Normalize the input data
(2) Produce a preliminary clustering result
(3) Iterations calculate the centroid of each category as
119911(119896)
= 119911(119896)119902
119896 = 1 sim 119870
119911(119896)119902
=sum119899
119895=1120583119898
119895(119896)119911119895119902
sum119899
119895=1120583119898119895(119896)
119896 = 1 sim 119870 119902 = 1 sim 119901
120583119895(119896)
=1
sum119870
119892=1(119890119895(119896)
119890119895(119892)
)2(119898minus1)
119895 = 1 sim 119899 119896 = 1 sim 119870
119890119895(119896)
= radic
119901
sum
119902=1
(119911119895119901
minus 119911(119896)119901
)2
119895 = 1 sim 119899 119896 = 1 sim 119870
(7)
where 119911(119896)
is the centroid of category 119896 120583(119905)119895(119896)
is thedegree ofmembership that job 119895has in category 119896 afterthe 119905th iteration
(4) Remeasure the distance from each job to the centroidof each category and then recalculate the correspond-ing membership
(5) Stop if the following condition is met Otherwisereturn to Step (3)
max119896
max119895
10038161003816100381610038161003816120583(119905)
119895(119896)minus 120583(119905minus1)
119895(119896)
10038161003816100381610038161003816lt 119889 (8)
where 119889 is a real number representing the member-ship convergence threshold
The performance of FCM is affected by the settings of theinitial values initialization can be repeated multiple timesin order to find the optimal solution Finally the separate
distance test (119878 test) proposed by Xie and Beni [34] can beapplied to determine the optimal number of categories119870
Min 119878
subject to 119869119898
=
119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896)
1198902
min = min1198961 = 1198962
(
119901
sum
119902=1
(119911(1198961)119902
minus 119911(1198962)119902
)2
)
119878 =119869119898
119899 times 1198902min 119870 isin 119885
+
(9)
The119870 value thatminimizes 119878 determines the optimal numberof categories
23 Step 3 Forecasting the Cycle Times of Jobs in EachCategory Using a FBPN All categories in the present studyare considered by the FBPN There are different types ofFBPNs for example Nomura et al [35] Chen [5] Lin [36]and Chen and Wang [33] Among them Chen and WangrsquosFBPN is unique since it is aimed at optimizing precision Forthis reason it is chosen for this study This FBPN is a proventechnology that has incorporated many features of priorartificial neural networks which have solved a wide variety ofproblems characterized by sets of different equations Somespecial problems can benefit from specialized tools such ascompositional pattern-producing networks cascading neu-ral networks and dynamic neural networks but for thepresent job cycle time problem a well-trained FBPN withan optimized structure can still produce very good resultsAll parameters in the FBPN are expressed in triangularfuzzy numbers (TFNs) There are various types of fuzzynumbers with different shapes among them the triangulartype is easily implemented andhas been universally applied tonumerous applications (eg [37 38]) Although the proposedmethodology uses TFNs it can be easilymodified to use othertypes of fuzzy numbers such as trapezoidal fuzzy numbers orbounded LR-type fuzzy numbers
The FBPN is configured as follows
(1) Inputs there are 119895 sets of inputs each set consists ofthe new factors determined by PCA from the 119895th jobThese factors have to be partially normalized so thattheir values fall within [01 09] [13 14]
(2) Single hidden layer generally one hidden layer givesthe best convergence results for the FBPN
(3) For simplicity the number of neurons in the hiddenlayer is twice the number in the input layer Anincrease in the number of hidden-layer nodes lessensthe output errors for the training examples butincreases the errors for novel examples This phe-nomenon is often called ldquooverfittingrdquo Some researchhas considered the connections between the complex-ity of a FBPN the performance for the training dataand the number of examples noteworthy researchincludes Akaikersquos information criterion (AIC) [39]
6 Mathematical Problems in Engineering
and Rissanenrsquos minimum description length (MDL)[40]
(4) Output the (normalized) cycle time forecast of theexample
The procedure for determining the parameter valuesis now described After pre-classification a portion of theadopted examples in each category is fed as ldquotraining exam-plesrdquo into the FBPN to determine the parameter valuesTwo phases are involved at the training stage First in theforward phase inputs are multiplied with weights summedand transferred to the hidden layer Then activated signalsare outputted from the hidden layer as
ℎ119897= (ℎ1198971 ℎ1198972 ℎ1198973)
=1
1 + 119890minus119899ℎ
119897
= (1
1 + 119890minus119899ℎ
1198971
1
1 + 119890minus119899ℎ
1198972
1
1 + 119890minus119899ℎ
1198973
)
(10)
where
119899ℎ
119897= (119899ℎ
1198971 119899ℎ
1198972 119899ℎ
1198973)
= 119868ℎ
119897(minus) 120579ℎ
119897
= (119868ℎ
1198971minus 120579ℎ
1198973 119868ℎ
1198972minus 120579ℎ
1198972 119868ℎ
1198973minus 120579ℎ
1198971)
119868ℎ
119897= (119868ℎ
1198971 119868ℎ
1198972 119868ℎ
1198973)
= sum
all 119896119908ℎ
119896119897sdot 119909119896
= (sum
all 119896min (119908
ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896)
sum
all 119896119908ℎ
1199091198962119909119896 sum
all 119896max (119908ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896))
(11)
ℎ119897is the output from hidden-layer node 119897 119897 = 1 sim 119871 120579ℎ
119897is
the threshold for screening out weak signals by hidden-layernode 119897119908ℎ
119896119897is theweight of the connection between input node
119896 and hidden-layer node 119897 119896 = 1 sim 119901 119897 = 1 sim 119871 119909119894119895is
the 119896th input 119896 = 1 sim 119901 The remaining parameters aretransition variables (minus) and (times) denote fuzzy subtraction andmultiplication respectively
ℎ119897s are also transferred to the output layer with the same
procedure Finally the output of the FBPN is generated asfollows
119900119895= (1199001198951 1199001198952 1199001198953)
=1
1 + 119890minus119899119900
= (1
1 + 119890minus119899119900
1
1
1 + 119890minus119899119900
2
1
1 + 119890minus119899119900
3
)
(12)
where
119899119900
= (119899119900
1 119899119900
2 119899119900
3) = 119868119900
(minus) 120579119900
= (119868119900
1minus 120579119900
3 119868119900
2minus 120579119900
2 119868119900
3minus 120579119900
1)
(13)
119868119900
= (119868119900
1 119868119900
2 119868119900
3)
=sum
all 119897119908119900
119897(times) ℎ119897
cong (sum
all 119897min (119908
119900
1198971ℎ1198971 119908119900
1198973ℎ1198973)
sum
all 119897119908119900
1198972ℎ1198972 sum
all 119897max (119908119900
1198971ℎ1198971 119908119900
1198973ℎ1198973))
(14)
119900119895is the network output which is the normalized value of the
cycle time forecast of job 119895 120579119900 is the threshold for screeningout weak signals by the output node 119908119900
119897is the weight of
the connection between hidden-layer node 119897 and the outputnode 119897 = 1 sim 119871
Subsequently in the backward phase the training of theFBPN is decomposed into three subtasks determining thecenter value and upper and lower bounds of the parameters
First to determine the center of each parameter (such as119908ℎ
1198961198972 120579ℎ1198972 1199081199001198972 and 120579
119900
2) the FBPN is treated as a crisp network
Algorithms applicable for this purpose include the gradi-ent descent algorithms the conjugate gradient algorithmsthe Levenberg-Marquardt algorithm and others In thisstudy the Levenberg-Marquardt algorithm is applied TheLevenberg-Marquardt algorithm was designed for trainingwith second-order speed without having to compute theHes-sian matrix It uses approximation and updates the networkparameters in a Newton-like way as described below
The network parameters are placed in vector 120573 = [119908ℎ
11
119908ℎ
119901119871 120579ℎ
1 120579
ℎ
119871 119908119900
1 119908
119900
119871 120579119900
] The network output 119900119895
can be represented with 119891(x119895120573) The objective function of
the FBPN is to minimize RMSE or equivalently the sum ofsquared error (SSE)
SSE (120573) =
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573))2
(15)
The Levenberg-Marquardt algorithm is an iterative pro-cedure In the beginning the user should specify the initialvalues of the network parameter vector 120573 It is a commonpractice to set 120573T = (1 1 1) In each step the parametervector 120573 is replaced by a new estimate 120573 + 120575 where120575 = [Δ119908
ℎ
11 Δ119908
ℎ
119901119871 Δ120579ℎ
1 Δ120579
ℎ
119871 Δ119908119900
1 Δ119908
119900
119871 Δ120579119900
] Thenetwork output becomes 119891(x
119895120573 + 120575) it is approximated by
its linearization as
119891 (x119895120573 + 120575) asymp 119891 (x
119895120573) + J
119895120575 (16)
where
J119895=
120597119891 (x119895120573)
120597120573(17)
Mathematical Problems in Engineering 7
is the gradient vector of 119891 with respect to 120573 Substituting (16)into (15) gives
SSE (120573 + 120575) asymp
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573) minus J
119895120575)2
(18)
When the network reaches the optimal solution the gradientof SSE with respect to 120575 will be zero Taking the derivative ofSSE(120573+120575)with respect to 120575 and setting the result to zero gives
(JTJ) 120575 = JT (119873 (CT119895) minus 119891 (x
119895120573)) (19)
where J is the Jacobian matrix containing the first derivativeof network error with respect to the weights and biasesEquation (19) includes a set of linear equations that can besolved for 120575
24 Step 4 Determining the Upper Bounds of Parameters inthe FBPN In Chen andWang [33] and Chen and Lin [41] anNPmodel is constructed to adjust the connectionweights andthresholds in the FBPN such anNPmodel is not easy to solveThe proposedmethodology only makes iterative adjustmentsto the threshold on the output nodeThisway ismuch simplerand can also achieve good results
Substituting (13) into (12) gives
1199001198952
=1
1 + 119890minus119899119900
1198952
=1
1 + 119890minus(119868119900
1198952minus120579119900
2)
=1
1 + 119890120579119900
2minus119868119900
1198952
(20)
Therefore
ln(1
1199001198952
minus 1) = 120579119900
2minus 119868119900
1198952 (21)
So
119868119900
1198952= 120579119900
2minus ln(
1
1199001198952
minus 1) (22)
Assume that the adjustment made to the output nodethreshold is denoted as Δ120579
119900
= 120579119900
3minus 120579119900
2 After adjustment the
output from the new FBPN 1199001198953 determines the upper bound
of the cycle time
1199001198953
=1
1 + 119890minus119899119900
1198953
(23)
where119899119900
1198953= 119868119900
1198953minus 120579119900
3= 119868119900
1198953minus (120579119900
2+ Δ120579119900
) (24)
Substituting (24) into (23) gives
1199001198953
=1
1 + 119890minus(119868119900
1198953minus120579119900
2minusΔ120579119900)
(25)
and substituting (22) into (25) gives
1199001198953
=1
1 + 119890minus(120579119900
2minusln(11199001198952minus1)minus1205791199002minusΔ120579119900)
=1
1 + 119890ln(11199001198952minus1)+Δ120579119900
=1
1 + 119890Δ120579119900
(11199001198952
minus 1)
(26)
Actual cycle time
Upper bound 1Upper bound 2
Upper bound 3
Upper bound 5Upper bound 4
Upper bound 6
Final result
Figure 2 Iterative reduction of the upper bound
Obviously the maximum ofΔ120579119900 determines the lowest upper
bound since 1199001198953is the upper bound of the cycle time 119900
1198953ge
119873(CT119895)
1
1 + 119890ln(11199001198952minus1)+Δ120579119900
ge 119873(CT119895) (27)
Δ120579119900
le ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) (28)
Equation (28) holds for all jobs so
Δ120579119900
le min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(29)
According to (26) the optimal value of Δ120579119900 should be set to
the maximum possible value
Δ120579119900lowast
= min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(30)
Then the FBPNrsquos optimization results depend on theinitial conditions and therefore are different at every iterationAssume that the optimal value of 119900
1198953in the 119905th iteration is
denoted by 1199001198953(119905) then after some iterations
1199001198953
(all iterations) = min119905
1199001198953
(119905) (31)
In this way the upper bound of the cycle time is decreasedgradually (see Figure 2) Another merit of this approach isthat it does not rely on the parameters of the FBPN
Theorem 1 Theupper bound can be found by considering onlyjobs with119873(CT
119895) ge 1199001198952
8 Mathematical Problems in Engineering
Proof Divide jobs into two categories 1198621
(jobs with119873(CT
119895) lt 1199001198952) and 119862
2(jobs with 119873(CT
119895) ge 1199001198952) Equation
(30) becomes
Δ120579119900lowast
= min(min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)))
(32)
If119873(CT119895) lt 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) gt 0 (33)
Therefore
min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) gt 0 (34)
In contrast if119873(CT119895) ge 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) le 0
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) le 0
(35)
Finally
Δ120579119900lowast
= min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(36)
Theorem 1 is proved
25 Step 4 Determining the Lower Bounds of Parameters in theFBPN In a similar way the threshold on the output node canbe modified to determine the lower bound of the cycle timeforecast so that the actual value will be greater than the lowerbound of the network output The optimal value of Δ120579
119900 canbe obtained as
Δ120579119900lowast
= max119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(37)
Theorem2 The lower bound can be found by considering onlyjobs with 119873(CT
119895) lt 1199001198952
The proof of Theorem 2 resembles that of Theorem 1
Proof Assume that the optimal value of 1199001198951
in the 119905threplication is indicated with 119900
1198951(119905) then after some iterations
1199001198951
(all iterations) = max119905
1199001198951
(119905) (38)
In this way the upper bound of the cycle time is increasedgradually Δ120579
119900lowastdoes not rely on the parameters of the FBPN
3 Application and Analyses
Todemonstrate the application of the proposedmethodologyan illustrative example containing the data of 40 jobs (seeTable 6) was used
We standardize the data and obtain the correlationmatrix119877 The eigenvalues and eigenvectors of 119877 are calculated as
1205821= 266 120582
2= 115 120582
3= 094
1205824= 083 120582
5= 025 120582
6= 002
(39)
1199061=
[[[[[[[
[
020
052
016
056
007
059
]]]]]]]
]
1199062=
[[[[[[[
[
047
minus019
068
minus019
minus049
005
]]]]]]]
]
1199063=
[[[[[[[
[
minus027
031
minus032
003
minus085
minus002
]]]]]]]
]
1199064=
[[[[[[[
[
081
minus002
minus057
006
minus004
minus015
]]]]]]]
]
1199065=
[[[[[[[
[
minus012
minus072
minus007
065
minus018
009
]]]]]]]
]
1199066=
[[[[[[[
[
002
029
028
047
003
minus078
]]]]]]]
]
(40)
respectively The variance contribution rates are
1205781= 46 120578
2= 20 120578
3= 16
1205784= 14 120578
5= 4 120578
6= 0
(41)
Summing up 120578119902rsquos we obtain
120578Σ(1) = 46 120578
Σ(2) = 65 120578
Σ(3) = 81
120578Σ(4) = 95 120578
Σ(5) = 100 120578
Σ(6) = 100
(42)
A Pareto analysis chart is used to compare the percentageof variability explained by each principal component (seeFigure 3) There is a clear break in the amount of varianceaccounted for by each component between the first andsecond components However that component by itselfexplains less than 50 of the variance so more components
Mathematical Problems in Engineering 9
Table 6 The illustrative example
119895 1199091198951
1199091198952
1199091198953
1199091198954
1199091198955
1199091198956
CT119895
1 24 1261 181 781 112 092 9352 24 1263 181 762 127 090 9583 24 1220 176 761 127 089 10474 23 1282 178 802 127 094 10115 23 1303 180 780 175 093 10686 23 1281 183 782 175 093 11437 23 1242 184 741 163 089 11038 24 1262 182 681 139 086 12509 22 1260 182 701 98 086 118110 22 1260 179 700 257 087 119411 24 1301 163 722 99 084 126012 22 1221 184 641 131 082 124013 23 1323 159 740 247 087 118014 24 1362 181 782 191 095 122715 24 1261 181 762 219 091 123616 23 1321 177 801 219 096 121517 22 1343 180 822 219 097 122818 24 1321 177 762 54 093 126619 25 1343 179 781 54 096 128520 25 1300 180 740 54 092 127221 22 1320 181 721 54 091 131022 24 1321 182 742 49 092 126523 23 1262 165 680 201 080 130824 22 1240 161 722 103 082 133125 23 1183 183 661 53 082 129426 23 1282 184 701 53 088 131427 22 1202 177 680 248 084 132128 23 1202 178 681 248 085 135329 24 1202 185 701 82 086 122630 23 1202 158 721 98 081 130131 24 1343 181 760 67 094 128032 24 1381 185 801 67 097 128633 22 1362 156 780 67 091 125234 23 1282 179 782 223 092 121435 23 1320 180 782 176 093 125136 25 1340 176 801 462 097 122237 23 1320 182 781 168 095 118738 22 1361 181 781 141 094 120539 22 1381 179 781 95 097 112040 23 1363 178 802 179 097 1133
are considered To meet the requirement 120578Σ(119901) ge 85sim90
p is chosen as 3 We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so we reduce the issue to these threecomponents in order to visualize the data
The component scores are recalculated as the new inputsto the FCM-FBPN (see Table 7) These scores contain thecoordinates of the original data in the new coordinate systemdefined by the principal components
Subsequently jobs are classified using FCM based on thenew variables The results of the 119878 test are summarized inTable 8 In this case the optimal number of job categories is 5
Table 7 New inputs to the FCM-FBPN
1199111198951
1199111198952
1199111198953
minus056 091 minus019
minus013 087 minus034
051 057 minus037
minus097 minus010 020
minus087 minus020 minus026
minus075 014 minus051
057 056 minus066
130 118 minus055
155 031 047
137 minus087 minus104
111 minus059 091
304 063 minus020
051 minus244 minus002
minus194 012 minus043
minus030 035 minus129
minus162 minus084 minus048
minus204 minus124 minus017
minus087 077 089
minus192 134 064
minus058 170 034
022 023 129
minus062 131 073
254 minus126 minus016
239 minus164 120
302 157 014
089 121 066
256 minus074 minus119
219 minus013 minus154
161 190 minus042
272 minus123 087
minus127 099 071
minus256 107 078
minus037 minus244 247
minus060 minus051 minus082
minus106 minus027 minus017
minus254 minus136 minus341
minus131 minus002 minus018
minus132 minus063 067
minus177 minus058 132
minus213 minus066 012
Table 8 The results of the S test
Number of categories (119870) 119869119898
1198902
min 119878
2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043
However therewill be some categories with very few jobs Forthis reason the second best solution is used that is 4 cate-gories A common practice is to set a threshold of member-
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
different degrees Similarly in probability theory the naıveBayesmethod provides a probability that each itembelongs toa class However the classification of jobs in FCM can includesubjective judgments
FCM classifies jobs byminimizing the following objectivefunction
Min119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896) (6)
where119870 is the required number of categories n is the numberof jobs120583
119895(119896)indicates the degree ofmembership that job 119895 has
in category k 119890119895(119896)
measures the distance from job 119895 to thecentroid of category 119896 119898 isin [1infin) is a parameter to adjustthe fuzziness and is usually set to 2 The procedure of FCM isas follows
(1) Normalize the input data
(2) Produce a preliminary clustering result
(3) Iterations calculate the centroid of each category as
119911(119896)
= 119911(119896)119902
119896 = 1 sim 119870
119911(119896)119902
=sum119899
119895=1120583119898
119895(119896)119911119895119902
sum119899
119895=1120583119898119895(119896)
119896 = 1 sim 119870 119902 = 1 sim 119901
120583119895(119896)
=1
sum119870
119892=1(119890119895(119896)
119890119895(119892)
)2(119898minus1)
119895 = 1 sim 119899 119896 = 1 sim 119870
119890119895(119896)
= radic
119901
sum
119902=1
(119911119895119901
minus 119911(119896)119901
)2
119895 = 1 sim 119899 119896 = 1 sim 119870
(7)
where 119911(119896)
is the centroid of category 119896 120583(119905)119895(119896)
is thedegree ofmembership that job 119895has in category 119896 afterthe 119905th iteration
(4) Remeasure the distance from each job to the centroidof each category and then recalculate the correspond-ing membership
(5) Stop if the following condition is met Otherwisereturn to Step (3)
max119896
max119895
10038161003816100381610038161003816120583(119905)
119895(119896)minus 120583(119905minus1)
119895(119896)
10038161003816100381610038161003816lt 119889 (8)
where 119889 is a real number representing the member-ship convergence threshold
The performance of FCM is affected by the settings of theinitial values initialization can be repeated multiple timesin order to find the optimal solution Finally the separate
distance test (119878 test) proposed by Xie and Beni [34] can beapplied to determine the optimal number of categories119870
Min 119878
subject to 119869119898
=
119870
sum
119896=1
119899
sum
119895=1
120583119898
119895(119896)1198902
119895(119896)
1198902
min = min1198961 = 1198962
(
119901
sum
119902=1
(119911(1198961)119902
minus 119911(1198962)119902
)2
)
119878 =119869119898
119899 times 1198902min 119870 isin 119885
+
(9)
The119870 value thatminimizes 119878 determines the optimal numberof categories
23 Step 3 Forecasting the Cycle Times of Jobs in EachCategory Using a FBPN All categories in the present studyare considered by the FBPN There are different types ofFBPNs for example Nomura et al [35] Chen [5] Lin [36]and Chen and Wang [33] Among them Chen and WangrsquosFBPN is unique since it is aimed at optimizing precision Forthis reason it is chosen for this study This FBPN is a proventechnology that has incorporated many features of priorartificial neural networks which have solved a wide variety ofproblems characterized by sets of different equations Somespecial problems can benefit from specialized tools such ascompositional pattern-producing networks cascading neu-ral networks and dynamic neural networks but for thepresent job cycle time problem a well-trained FBPN withan optimized structure can still produce very good resultsAll parameters in the FBPN are expressed in triangularfuzzy numbers (TFNs) There are various types of fuzzynumbers with different shapes among them the triangulartype is easily implemented andhas been universally applied tonumerous applications (eg [37 38]) Although the proposedmethodology uses TFNs it can be easilymodified to use othertypes of fuzzy numbers such as trapezoidal fuzzy numbers orbounded LR-type fuzzy numbers
The FBPN is configured as follows
(1) Inputs there are 119895 sets of inputs each set consists ofthe new factors determined by PCA from the 119895th jobThese factors have to be partially normalized so thattheir values fall within [01 09] [13 14]
(2) Single hidden layer generally one hidden layer givesthe best convergence results for the FBPN
(3) For simplicity the number of neurons in the hiddenlayer is twice the number in the input layer Anincrease in the number of hidden-layer nodes lessensthe output errors for the training examples butincreases the errors for novel examples This phe-nomenon is often called ldquooverfittingrdquo Some researchhas considered the connections between the complex-ity of a FBPN the performance for the training dataand the number of examples noteworthy researchincludes Akaikersquos information criterion (AIC) [39]
6 Mathematical Problems in Engineering
and Rissanenrsquos minimum description length (MDL)[40]
(4) Output the (normalized) cycle time forecast of theexample
The procedure for determining the parameter valuesis now described After pre-classification a portion of theadopted examples in each category is fed as ldquotraining exam-plesrdquo into the FBPN to determine the parameter valuesTwo phases are involved at the training stage First in theforward phase inputs are multiplied with weights summedand transferred to the hidden layer Then activated signalsare outputted from the hidden layer as
ℎ119897= (ℎ1198971 ℎ1198972 ℎ1198973)
=1
1 + 119890minus119899ℎ
119897
= (1
1 + 119890minus119899ℎ
1198971
1
1 + 119890minus119899ℎ
1198972
1
1 + 119890minus119899ℎ
1198973
)
(10)
where
119899ℎ
119897= (119899ℎ
1198971 119899ℎ
1198972 119899ℎ
1198973)
= 119868ℎ
119897(minus) 120579ℎ
119897
= (119868ℎ
1198971minus 120579ℎ
1198973 119868ℎ
1198972minus 120579ℎ
1198972 119868ℎ
1198973minus 120579ℎ
1198971)
119868ℎ
119897= (119868ℎ
1198971 119868ℎ
1198972 119868ℎ
1198973)
= sum
all 119896119908ℎ
119896119897sdot 119909119896
= (sum
all 119896min (119908
ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896)
sum
all 119896119908ℎ
1199091198962119909119896 sum
all 119896max (119908ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896))
(11)
ℎ119897is the output from hidden-layer node 119897 119897 = 1 sim 119871 120579ℎ
119897is
the threshold for screening out weak signals by hidden-layernode 119897119908ℎ
119896119897is theweight of the connection between input node
119896 and hidden-layer node 119897 119896 = 1 sim 119901 119897 = 1 sim 119871 119909119894119895is
the 119896th input 119896 = 1 sim 119901 The remaining parameters aretransition variables (minus) and (times) denote fuzzy subtraction andmultiplication respectively
ℎ119897s are also transferred to the output layer with the same
procedure Finally the output of the FBPN is generated asfollows
119900119895= (1199001198951 1199001198952 1199001198953)
=1
1 + 119890minus119899119900
= (1
1 + 119890minus119899119900
1
1
1 + 119890minus119899119900
2
1
1 + 119890minus119899119900
3
)
(12)
where
119899119900
= (119899119900
1 119899119900
2 119899119900
3) = 119868119900
(minus) 120579119900
= (119868119900
1minus 120579119900
3 119868119900
2minus 120579119900
2 119868119900
3minus 120579119900
1)
(13)
119868119900
= (119868119900
1 119868119900
2 119868119900
3)
=sum
all 119897119908119900
119897(times) ℎ119897
cong (sum
all 119897min (119908
119900
1198971ℎ1198971 119908119900
1198973ℎ1198973)
sum
all 119897119908119900
1198972ℎ1198972 sum
all 119897max (119908119900
1198971ℎ1198971 119908119900
1198973ℎ1198973))
(14)
119900119895is the network output which is the normalized value of the
cycle time forecast of job 119895 120579119900 is the threshold for screeningout weak signals by the output node 119908119900
119897is the weight of
the connection between hidden-layer node 119897 and the outputnode 119897 = 1 sim 119871
Subsequently in the backward phase the training of theFBPN is decomposed into three subtasks determining thecenter value and upper and lower bounds of the parameters
First to determine the center of each parameter (such as119908ℎ
1198961198972 120579ℎ1198972 1199081199001198972 and 120579
119900
2) the FBPN is treated as a crisp network
Algorithms applicable for this purpose include the gradi-ent descent algorithms the conjugate gradient algorithmsthe Levenberg-Marquardt algorithm and others In thisstudy the Levenberg-Marquardt algorithm is applied TheLevenberg-Marquardt algorithm was designed for trainingwith second-order speed without having to compute theHes-sian matrix It uses approximation and updates the networkparameters in a Newton-like way as described below
The network parameters are placed in vector 120573 = [119908ℎ
11
119908ℎ
119901119871 120579ℎ
1 120579
ℎ
119871 119908119900
1 119908
119900
119871 120579119900
] The network output 119900119895
can be represented with 119891(x119895120573) The objective function of
the FBPN is to minimize RMSE or equivalently the sum ofsquared error (SSE)
SSE (120573) =
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573))2
(15)
The Levenberg-Marquardt algorithm is an iterative pro-cedure In the beginning the user should specify the initialvalues of the network parameter vector 120573 It is a commonpractice to set 120573T = (1 1 1) In each step the parametervector 120573 is replaced by a new estimate 120573 + 120575 where120575 = [Δ119908
ℎ
11 Δ119908
ℎ
119901119871 Δ120579ℎ
1 Δ120579
ℎ
119871 Δ119908119900
1 Δ119908
119900
119871 Δ120579119900
] Thenetwork output becomes 119891(x
119895120573 + 120575) it is approximated by
its linearization as
119891 (x119895120573 + 120575) asymp 119891 (x
119895120573) + J
119895120575 (16)
where
J119895=
120597119891 (x119895120573)
120597120573(17)
Mathematical Problems in Engineering 7
is the gradient vector of 119891 with respect to 120573 Substituting (16)into (15) gives
SSE (120573 + 120575) asymp
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573) minus J
119895120575)2
(18)
When the network reaches the optimal solution the gradientof SSE with respect to 120575 will be zero Taking the derivative ofSSE(120573+120575)with respect to 120575 and setting the result to zero gives
(JTJ) 120575 = JT (119873 (CT119895) minus 119891 (x
119895120573)) (19)
where J is the Jacobian matrix containing the first derivativeof network error with respect to the weights and biasesEquation (19) includes a set of linear equations that can besolved for 120575
24 Step 4 Determining the Upper Bounds of Parameters inthe FBPN In Chen andWang [33] and Chen and Lin [41] anNPmodel is constructed to adjust the connectionweights andthresholds in the FBPN such anNPmodel is not easy to solveThe proposedmethodology only makes iterative adjustmentsto the threshold on the output nodeThisway ismuch simplerand can also achieve good results
Substituting (13) into (12) gives
1199001198952
=1
1 + 119890minus119899119900
1198952
=1
1 + 119890minus(119868119900
1198952minus120579119900
2)
=1
1 + 119890120579119900
2minus119868119900
1198952
(20)
Therefore
ln(1
1199001198952
minus 1) = 120579119900
2minus 119868119900
1198952 (21)
So
119868119900
1198952= 120579119900
2minus ln(
1
1199001198952
minus 1) (22)
Assume that the adjustment made to the output nodethreshold is denoted as Δ120579
119900
= 120579119900
3minus 120579119900
2 After adjustment the
output from the new FBPN 1199001198953 determines the upper bound
of the cycle time
1199001198953
=1
1 + 119890minus119899119900
1198953
(23)
where119899119900
1198953= 119868119900
1198953minus 120579119900
3= 119868119900
1198953minus (120579119900
2+ Δ120579119900
) (24)
Substituting (24) into (23) gives
1199001198953
=1
1 + 119890minus(119868119900
1198953minus120579119900
2minusΔ120579119900)
(25)
and substituting (22) into (25) gives
1199001198953
=1
1 + 119890minus(120579119900
2minusln(11199001198952minus1)minus1205791199002minusΔ120579119900)
=1
1 + 119890ln(11199001198952minus1)+Δ120579119900
=1
1 + 119890Δ120579119900
(11199001198952
minus 1)
(26)
Actual cycle time
Upper bound 1Upper bound 2
Upper bound 3
Upper bound 5Upper bound 4
Upper bound 6
Final result
Figure 2 Iterative reduction of the upper bound
Obviously the maximum ofΔ120579119900 determines the lowest upper
bound since 1199001198953is the upper bound of the cycle time 119900
1198953ge
119873(CT119895)
1
1 + 119890ln(11199001198952minus1)+Δ120579119900
ge 119873(CT119895) (27)
Δ120579119900
le ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) (28)
Equation (28) holds for all jobs so
Δ120579119900
le min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(29)
According to (26) the optimal value of Δ120579119900 should be set to
the maximum possible value
Δ120579119900lowast
= min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(30)
Then the FBPNrsquos optimization results depend on theinitial conditions and therefore are different at every iterationAssume that the optimal value of 119900
1198953in the 119905th iteration is
denoted by 1199001198953(119905) then after some iterations
1199001198953
(all iterations) = min119905
1199001198953
(119905) (31)
In this way the upper bound of the cycle time is decreasedgradually (see Figure 2) Another merit of this approach isthat it does not rely on the parameters of the FBPN
Theorem 1 Theupper bound can be found by considering onlyjobs with119873(CT
119895) ge 1199001198952
8 Mathematical Problems in Engineering
Proof Divide jobs into two categories 1198621
(jobs with119873(CT
119895) lt 1199001198952) and 119862
2(jobs with 119873(CT
119895) ge 1199001198952) Equation
(30) becomes
Δ120579119900lowast
= min(min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)))
(32)
If119873(CT119895) lt 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) gt 0 (33)
Therefore
min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) gt 0 (34)
In contrast if119873(CT119895) ge 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) le 0
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) le 0
(35)
Finally
Δ120579119900lowast
= min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(36)
Theorem 1 is proved
25 Step 4 Determining the Lower Bounds of Parameters in theFBPN In a similar way the threshold on the output node canbe modified to determine the lower bound of the cycle timeforecast so that the actual value will be greater than the lowerbound of the network output The optimal value of Δ120579
119900 canbe obtained as
Δ120579119900lowast
= max119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(37)
Theorem2 The lower bound can be found by considering onlyjobs with 119873(CT
119895) lt 1199001198952
The proof of Theorem 2 resembles that of Theorem 1
Proof Assume that the optimal value of 1199001198951
in the 119905threplication is indicated with 119900
1198951(119905) then after some iterations
1199001198951
(all iterations) = max119905
1199001198951
(119905) (38)
In this way the upper bound of the cycle time is increasedgradually Δ120579
119900lowastdoes not rely on the parameters of the FBPN
3 Application and Analyses
Todemonstrate the application of the proposedmethodologyan illustrative example containing the data of 40 jobs (seeTable 6) was used
We standardize the data and obtain the correlationmatrix119877 The eigenvalues and eigenvectors of 119877 are calculated as
1205821= 266 120582
2= 115 120582
3= 094
1205824= 083 120582
5= 025 120582
6= 002
(39)
1199061=
[[[[[[[
[
020
052
016
056
007
059
]]]]]]]
]
1199062=
[[[[[[[
[
047
minus019
068
minus019
minus049
005
]]]]]]]
]
1199063=
[[[[[[[
[
minus027
031
minus032
003
minus085
minus002
]]]]]]]
]
1199064=
[[[[[[[
[
081
minus002
minus057
006
minus004
minus015
]]]]]]]
]
1199065=
[[[[[[[
[
minus012
minus072
minus007
065
minus018
009
]]]]]]]
]
1199066=
[[[[[[[
[
002
029
028
047
003
minus078
]]]]]]]
]
(40)
respectively The variance contribution rates are
1205781= 46 120578
2= 20 120578
3= 16
1205784= 14 120578
5= 4 120578
6= 0
(41)
Summing up 120578119902rsquos we obtain
120578Σ(1) = 46 120578
Σ(2) = 65 120578
Σ(3) = 81
120578Σ(4) = 95 120578
Σ(5) = 100 120578
Σ(6) = 100
(42)
A Pareto analysis chart is used to compare the percentageof variability explained by each principal component (seeFigure 3) There is a clear break in the amount of varianceaccounted for by each component between the first andsecond components However that component by itselfexplains less than 50 of the variance so more components
Mathematical Problems in Engineering 9
Table 6 The illustrative example
119895 1199091198951
1199091198952
1199091198953
1199091198954
1199091198955
1199091198956
CT119895
1 24 1261 181 781 112 092 9352 24 1263 181 762 127 090 9583 24 1220 176 761 127 089 10474 23 1282 178 802 127 094 10115 23 1303 180 780 175 093 10686 23 1281 183 782 175 093 11437 23 1242 184 741 163 089 11038 24 1262 182 681 139 086 12509 22 1260 182 701 98 086 118110 22 1260 179 700 257 087 119411 24 1301 163 722 99 084 126012 22 1221 184 641 131 082 124013 23 1323 159 740 247 087 118014 24 1362 181 782 191 095 122715 24 1261 181 762 219 091 123616 23 1321 177 801 219 096 121517 22 1343 180 822 219 097 122818 24 1321 177 762 54 093 126619 25 1343 179 781 54 096 128520 25 1300 180 740 54 092 127221 22 1320 181 721 54 091 131022 24 1321 182 742 49 092 126523 23 1262 165 680 201 080 130824 22 1240 161 722 103 082 133125 23 1183 183 661 53 082 129426 23 1282 184 701 53 088 131427 22 1202 177 680 248 084 132128 23 1202 178 681 248 085 135329 24 1202 185 701 82 086 122630 23 1202 158 721 98 081 130131 24 1343 181 760 67 094 128032 24 1381 185 801 67 097 128633 22 1362 156 780 67 091 125234 23 1282 179 782 223 092 121435 23 1320 180 782 176 093 125136 25 1340 176 801 462 097 122237 23 1320 182 781 168 095 118738 22 1361 181 781 141 094 120539 22 1381 179 781 95 097 112040 23 1363 178 802 179 097 1133
are considered To meet the requirement 120578Σ(119901) ge 85sim90
p is chosen as 3 We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so we reduce the issue to these threecomponents in order to visualize the data
The component scores are recalculated as the new inputsto the FCM-FBPN (see Table 7) These scores contain thecoordinates of the original data in the new coordinate systemdefined by the principal components
Subsequently jobs are classified using FCM based on thenew variables The results of the 119878 test are summarized inTable 8 In this case the optimal number of job categories is 5
Table 7 New inputs to the FCM-FBPN
1199111198951
1199111198952
1199111198953
minus056 091 minus019
minus013 087 minus034
051 057 minus037
minus097 minus010 020
minus087 minus020 minus026
minus075 014 minus051
057 056 minus066
130 118 minus055
155 031 047
137 minus087 minus104
111 minus059 091
304 063 minus020
051 minus244 minus002
minus194 012 minus043
minus030 035 minus129
minus162 minus084 minus048
minus204 minus124 minus017
minus087 077 089
minus192 134 064
minus058 170 034
022 023 129
minus062 131 073
254 minus126 minus016
239 minus164 120
302 157 014
089 121 066
256 minus074 minus119
219 minus013 minus154
161 190 minus042
272 minus123 087
minus127 099 071
minus256 107 078
minus037 minus244 247
minus060 minus051 minus082
minus106 minus027 minus017
minus254 minus136 minus341
minus131 minus002 minus018
minus132 minus063 067
minus177 minus058 132
minus213 minus066 012
Table 8 The results of the S test
Number of categories (119870) 119869119898
1198902
min 119878
2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043
However therewill be some categories with very few jobs Forthis reason the second best solution is used that is 4 cate-gories A common practice is to set a threshold of member-
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
and Rissanenrsquos minimum description length (MDL)[40]
(4) Output the (normalized) cycle time forecast of theexample
The procedure for determining the parameter valuesis now described After pre-classification a portion of theadopted examples in each category is fed as ldquotraining exam-plesrdquo into the FBPN to determine the parameter valuesTwo phases are involved at the training stage First in theforward phase inputs are multiplied with weights summedand transferred to the hidden layer Then activated signalsare outputted from the hidden layer as
ℎ119897= (ℎ1198971 ℎ1198972 ℎ1198973)
=1
1 + 119890minus119899ℎ
119897
= (1
1 + 119890minus119899ℎ
1198971
1
1 + 119890minus119899ℎ
1198972
1
1 + 119890minus119899ℎ
1198973
)
(10)
where
119899ℎ
119897= (119899ℎ
1198971 119899ℎ
1198972 119899ℎ
1198973)
= 119868ℎ
119897(minus) 120579ℎ
119897
= (119868ℎ
1198971minus 120579ℎ
1198973 119868ℎ
1198972minus 120579ℎ
1198972 119868ℎ
1198973minus 120579ℎ
1198971)
119868ℎ
119897= (119868ℎ
1198971 119868ℎ
1198972 119868ℎ
1198973)
= sum
all 119896119908ℎ
119896119897sdot 119909119896
= (sum
all 119896min (119908
ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896)
sum
all 119896119908ℎ
1199091198962119909119896 sum
all 119896max (119908ℎ
1198961198971119909119896 119908ℎ
1198961198973119909119896))
(11)
ℎ119897is the output from hidden-layer node 119897 119897 = 1 sim 119871 120579ℎ
119897is
the threshold for screening out weak signals by hidden-layernode 119897119908ℎ
119896119897is theweight of the connection between input node
119896 and hidden-layer node 119897 119896 = 1 sim 119901 119897 = 1 sim 119871 119909119894119895is
the 119896th input 119896 = 1 sim 119901 The remaining parameters aretransition variables (minus) and (times) denote fuzzy subtraction andmultiplication respectively
ℎ119897s are also transferred to the output layer with the same
procedure Finally the output of the FBPN is generated asfollows
119900119895= (1199001198951 1199001198952 1199001198953)
=1
1 + 119890minus119899119900
= (1
1 + 119890minus119899119900
1
1
1 + 119890minus119899119900
2
1
1 + 119890minus119899119900
3
)
(12)
where
119899119900
= (119899119900
1 119899119900
2 119899119900
3) = 119868119900
(minus) 120579119900
= (119868119900
1minus 120579119900
3 119868119900
2minus 120579119900
2 119868119900
3minus 120579119900
1)
(13)
119868119900
= (119868119900
1 119868119900
2 119868119900
3)
=sum
all 119897119908119900
119897(times) ℎ119897
cong (sum
all 119897min (119908
119900
1198971ℎ1198971 119908119900
1198973ℎ1198973)
sum
all 119897119908119900
1198972ℎ1198972 sum
all 119897max (119908119900
1198971ℎ1198971 119908119900
1198973ℎ1198973))
(14)
119900119895is the network output which is the normalized value of the
cycle time forecast of job 119895 120579119900 is the threshold for screeningout weak signals by the output node 119908119900
119897is the weight of
the connection between hidden-layer node 119897 and the outputnode 119897 = 1 sim 119871
Subsequently in the backward phase the training of theFBPN is decomposed into three subtasks determining thecenter value and upper and lower bounds of the parameters
First to determine the center of each parameter (such as119908ℎ
1198961198972 120579ℎ1198972 1199081199001198972 and 120579
119900
2) the FBPN is treated as a crisp network
Algorithms applicable for this purpose include the gradi-ent descent algorithms the conjugate gradient algorithmsthe Levenberg-Marquardt algorithm and others In thisstudy the Levenberg-Marquardt algorithm is applied TheLevenberg-Marquardt algorithm was designed for trainingwith second-order speed without having to compute theHes-sian matrix It uses approximation and updates the networkparameters in a Newton-like way as described below
The network parameters are placed in vector 120573 = [119908ℎ
11
119908ℎ
119901119871 120579ℎ
1 120579
ℎ
119871 119908119900
1 119908
119900
119871 120579119900
] The network output 119900119895
can be represented with 119891(x119895120573) The objective function of
the FBPN is to minimize RMSE or equivalently the sum ofsquared error (SSE)
SSE (120573) =
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573))2
(15)
The Levenberg-Marquardt algorithm is an iterative pro-cedure In the beginning the user should specify the initialvalues of the network parameter vector 120573 It is a commonpractice to set 120573T = (1 1 1) In each step the parametervector 120573 is replaced by a new estimate 120573 + 120575 where120575 = [Δ119908
ℎ
11 Δ119908
ℎ
119901119871 Δ120579ℎ
1 Δ120579
ℎ
119871 Δ119908119900
1 Δ119908
119900
119871 Δ120579119900
] Thenetwork output becomes 119891(x
119895120573 + 120575) it is approximated by
its linearization as
119891 (x119895120573 + 120575) asymp 119891 (x
119895120573) + J
119895120575 (16)
where
J119895=
120597119891 (x119895120573)
120597120573(17)
Mathematical Problems in Engineering 7
is the gradient vector of 119891 with respect to 120573 Substituting (16)into (15) gives
SSE (120573 + 120575) asymp
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573) minus J
119895120575)2
(18)
When the network reaches the optimal solution the gradientof SSE with respect to 120575 will be zero Taking the derivative ofSSE(120573+120575)with respect to 120575 and setting the result to zero gives
(JTJ) 120575 = JT (119873 (CT119895) minus 119891 (x
119895120573)) (19)
where J is the Jacobian matrix containing the first derivativeof network error with respect to the weights and biasesEquation (19) includes a set of linear equations that can besolved for 120575
24 Step 4 Determining the Upper Bounds of Parameters inthe FBPN In Chen andWang [33] and Chen and Lin [41] anNPmodel is constructed to adjust the connectionweights andthresholds in the FBPN such anNPmodel is not easy to solveThe proposedmethodology only makes iterative adjustmentsto the threshold on the output nodeThisway ismuch simplerand can also achieve good results
Substituting (13) into (12) gives
1199001198952
=1
1 + 119890minus119899119900
1198952
=1
1 + 119890minus(119868119900
1198952minus120579119900
2)
=1
1 + 119890120579119900
2minus119868119900
1198952
(20)
Therefore
ln(1
1199001198952
minus 1) = 120579119900
2minus 119868119900
1198952 (21)
So
119868119900
1198952= 120579119900
2minus ln(
1
1199001198952
minus 1) (22)
Assume that the adjustment made to the output nodethreshold is denoted as Δ120579
119900
= 120579119900
3minus 120579119900
2 After adjustment the
output from the new FBPN 1199001198953 determines the upper bound
of the cycle time
1199001198953
=1
1 + 119890minus119899119900
1198953
(23)
where119899119900
1198953= 119868119900
1198953minus 120579119900
3= 119868119900
1198953minus (120579119900
2+ Δ120579119900
) (24)
Substituting (24) into (23) gives
1199001198953
=1
1 + 119890minus(119868119900
1198953minus120579119900
2minusΔ120579119900)
(25)
and substituting (22) into (25) gives
1199001198953
=1
1 + 119890minus(120579119900
2minusln(11199001198952minus1)minus1205791199002minusΔ120579119900)
=1
1 + 119890ln(11199001198952minus1)+Δ120579119900
=1
1 + 119890Δ120579119900
(11199001198952
minus 1)
(26)
Actual cycle time
Upper bound 1Upper bound 2
Upper bound 3
Upper bound 5Upper bound 4
Upper bound 6
Final result
Figure 2 Iterative reduction of the upper bound
Obviously the maximum ofΔ120579119900 determines the lowest upper
bound since 1199001198953is the upper bound of the cycle time 119900
1198953ge
119873(CT119895)
1
1 + 119890ln(11199001198952minus1)+Δ120579119900
ge 119873(CT119895) (27)
Δ120579119900
le ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) (28)
Equation (28) holds for all jobs so
Δ120579119900
le min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(29)
According to (26) the optimal value of Δ120579119900 should be set to
the maximum possible value
Δ120579119900lowast
= min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(30)
Then the FBPNrsquos optimization results depend on theinitial conditions and therefore are different at every iterationAssume that the optimal value of 119900
1198953in the 119905th iteration is
denoted by 1199001198953(119905) then after some iterations
1199001198953
(all iterations) = min119905
1199001198953
(119905) (31)
In this way the upper bound of the cycle time is decreasedgradually (see Figure 2) Another merit of this approach isthat it does not rely on the parameters of the FBPN
Theorem 1 Theupper bound can be found by considering onlyjobs with119873(CT
119895) ge 1199001198952
8 Mathematical Problems in Engineering
Proof Divide jobs into two categories 1198621
(jobs with119873(CT
119895) lt 1199001198952) and 119862
2(jobs with 119873(CT
119895) ge 1199001198952) Equation
(30) becomes
Δ120579119900lowast
= min(min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)))
(32)
If119873(CT119895) lt 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) gt 0 (33)
Therefore
min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) gt 0 (34)
In contrast if119873(CT119895) ge 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) le 0
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) le 0
(35)
Finally
Δ120579119900lowast
= min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(36)
Theorem 1 is proved
25 Step 4 Determining the Lower Bounds of Parameters in theFBPN In a similar way the threshold on the output node canbe modified to determine the lower bound of the cycle timeforecast so that the actual value will be greater than the lowerbound of the network output The optimal value of Δ120579
119900 canbe obtained as
Δ120579119900lowast
= max119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(37)
Theorem2 The lower bound can be found by considering onlyjobs with 119873(CT
119895) lt 1199001198952
The proof of Theorem 2 resembles that of Theorem 1
Proof Assume that the optimal value of 1199001198951
in the 119905threplication is indicated with 119900
1198951(119905) then after some iterations
1199001198951
(all iterations) = max119905
1199001198951
(119905) (38)
In this way the upper bound of the cycle time is increasedgradually Δ120579
119900lowastdoes not rely on the parameters of the FBPN
3 Application and Analyses
Todemonstrate the application of the proposedmethodologyan illustrative example containing the data of 40 jobs (seeTable 6) was used
We standardize the data and obtain the correlationmatrix119877 The eigenvalues and eigenvectors of 119877 are calculated as
1205821= 266 120582
2= 115 120582
3= 094
1205824= 083 120582
5= 025 120582
6= 002
(39)
1199061=
[[[[[[[
[
020
052
016
056
007
059
]]]]]]]
]
1199062=
[[[[[[[
[
047
minus019
068
minus019
minus049
005
]]]]]]]
]
1199063=
[[[[[[[
[
minus027
031
minus032
003
minus085
minus002
]]]]]]]
]
1199064=
[[[[[[[
[
081
minus002
minus057
006
minus004
minus015
]]]]]]]
]
1199065=
[[[[[[[
[
minus012
minus072
minus007
065
minus018
009
]]]]]]]
]
1199066=
[[[[[[[
[
002
029
028
047
003
minus078
]]]]]]]
]
(40)
respectively The variance contribution rates are
1205781= 46 120578
2= 20 120578
3= 16
1205784= 14 120578
5= 4 120578
6= 0
(41)
Summing up 120578119902rsquos we obtain
120578Σ(1) = 46 120578
Σ(2) = 65 120578
Σ(3) = 81
120578Σ(4) = 95 120578
Σ(5) = 100 120578
Σ(6) = 100
(42)
A Pareto analysis chart is used to compare the percentageof variability explained by each principal component (seeFigure 3) There is a clear break in the amount of varianceaccounted for by each component between the first andsecond components However that component by itselfexplains less than 50 of the variance so more components
Mathematical Problems in Engineering 9
Table 6 The illustrative example
119895 1199091198951
1199091198952
1199091198953
1199091198954
1199091198955
1199091198956
CT119895
1 24 1261 181 781 112 092 9352 24 1263 181 762 127 090 9583 24 1220 176 761 127 089 10474 23 1282 178 802 127 094 10115 23 1303 180 780 175 093 10686 23 1281 183 782 175 093 11437 23 1242 184 741 163 089 11038 24 1262 182 681 139 086 12509 22 1260 182 701 98 086 118110 22 1260 179 700 257 087 119411 24 1301 163 722 99 084 126012 22 1221 184 641 131 082 124013 23 1323 159 740 247 087 118014 24 1362 181 782 191 095 122715 24 1261 181 762 219 091 123616 23 1321 177 801 219 096 121517 22 1343 180 822 219 097 122818 24 1321 177 762 54 093 126619 25 1343 179 781 54 096 128520 25 1300 180 740 54 092 127221 22 1320 181 721 54 091 131022 24 1321 182 742 49 092 126523 23 1262 165 680 201 080 130824 22 1240 161 722 103 082 133125 23 1183 183 661 53 082 129426 23 1282 184 701 53 088 131427 22 1202 177 680 248 084 132128 23 1202 178 681 248 085 135329 24 1202 185 701 82 086 122630 23 1202 158 721 98 081 130131 24 1343 181 760 67 094 128032 24 1381 185 801 67 097 128633 22 1362 156 780 67 091 125234 23 1282 179 782 223 092 121435 23 1320 180 782 176 093 125136 25 1340 176 801 462 097 122237 23 1320 182 781 168 095 118738 22 1361 181 781 141 094 120539 22 1381 179 781 95 097 112040 23 1363 178 802 179 097 1133
are considered To meet the requirement 120578Σ(119901) ge 85sim90
p is chosen as 3 We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so we reduce the issue to these threecomponents in order to visualize the data
The component scores are recalculated as the new inputsto the FCM-FBPN (see Table 7) These scores contain thecoordinates of the original data in the new coordinate systemdefined by the principal components
Subsequently jobs are classified using FCM based on thenew variables The results of the 119878 test are summarized inTable 8 In this case the optimal number of job categories is 5
Table 7 New inputs to the FCM-FBPN
1199111198951
1199111198952
1199111198953
minus056 091 minus019
minus013 087 minus034
051 057 minus037
minus097 minus010 020
minus087 minus020 minus026
minus075 014 minus051
057 056 minus066
130 118 minus055
155 031 047
137 minus087 minus104
111 minus059 091
304 063 minus020
051 minus244 minus002
minus194 012 minus043
minus030 035 minus129
minus162 minus084 minus048
minus204 minus124 minus017
minus087 077 089
minus192 134 064
minus058 170 034
022 023 129
minus062 131 073
254 minus126 minus016
239 minus164 120
302 157 014
089 121 066
256 minus074 minus119
219 minus013 minus154
161 190 minus042
272 minus123 087
minus127 099 071
minus256 107 078
minus037 minus244 247
minus060 minus051 minus082
minus106 minus027 minus017
minus254 minus136 minus341
minus131 minus002 minus018
minus132 minus063 067
minus177 minus058 132
minus213 minus066 012
Table 8 The results of the S test
Number of categories (119870) 119869119898
1198902
min 119878
2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043
However therewill be some categories with very few jobs Forthis reason the second best solution is used that is 4 cate-gories A common practice is to set a threshold of member-
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
is the gradient vector of 119891 with respect to 120573 Substituting (16)into (15) gives
SSE (120573 + 120575) asymp
119899
sum
119895=1
(119873 (CT119895) minus 119891 (x
119895120573) minus J
119895120575)2
(18)
When the network reaches the optimal solution the gradientof SSE with respect to 120575 will be zero Taking the derivative ofSSE(120573+120575)with respect to 120575 and setting the result to zero gives
(JTJ) 120575 = JT (119873 (CT119895) minus 119891 (x
119895120573)) (19)
where J is the Jacobian matrix containing the first derivativeof network error with respect to the weights and biasesEquation (19) includes a set of linear equations that can besolved for 120575
24 Step 4 Determining the Upper Bounds of Parameters inthe FBPN In Chen andWang [33] and Chen and Lin [41] anNPmodel is constructed to adjust the connectionweights andthresholds in the FBPN such anNPmodel is not easy to solveThe proposedmethodology only makes iterative adjustmentsto the threshold on the output nodeThisway ismuch simplerand can also achieve good results
Substituting (13) into (12) gives
1199001198952
=1
1 + 119890minus119899119900
1198952
=1
1 + 119890minus(119868119900
1198952minus120579119900
2)
=1
1 + 119890120579119900
2minus119868119900
1198952
(20)
Therefore
ln(1
1199001198952
minus 1) = 120579119900
2minus 119868119900
1198952 (21)
So
119868119900
1198952= 120579119900
2minus ln(
1
1199001198952
minus 1) (22)
Assume that the adjustment made to the output nodethreshold is denoted as Δ120579
119900
= 120579119900
3minus 120579119900
2 After adjustment the
output from the new FBPN 1199001198953 determines the upper bound
of the cycle time
1199001198953
=1
1 + 119890minus119899119900
1198953
(23)
where119899119900
1198953= 119868119900
1198953minus 120579119900
3= 119868119900
1198953minus (120579119900
2+ Δ120579119900
) (24)
Substituting (24) into (23) gives
1199001198953
=1
1 + 119890minus(119868119900
1198953minus120579119900
2minusΔ120579119900)
(25)
and substituting (22) into (25) gives
1199001198953
=1
1 + 119890minus(120579119900
2minusln(11199001198952minus1)minus1205791199002minusΔ120579119900)
=1
1 + 119890ln(11199001198952minus1)+Δ120579119900
=1
1 + 119890Δ120579119900
(11199001198952
minus 1)
(26)
Actual cycle time
Upper bound 1Upper bound 2
Upper bound 3
Upper bound 5Upper bound 4
Upper bound 6
Final result
Figure 2 Iterative reduction of the upper bound
Obviously the maximum ofΔ120579119900 determines the lowest upper
bound since 1199001198953is the upper bound of the cycle time 119900
1198953ge
119873(CT119895)
1
1 + 119890ln(11199001198952minus1)+Δ120579119900
ge 119873(CT119895) (27)
Δ120579119900
le ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) (28)
Equation (28) holds for all jobs so
Δ120579119900
le min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(29)
According to (26) the optimal value of Δ120579119900 should be set to
the maximum possible value
Δ120579119900lowast
= min119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(30)
Then the FBPNrsquos optimization results depend on theinitial conditions and therefore are different at every iterationAssume that the optimal value of 119900
1198953in the 119905th iteration is
denoted by 1199001198953(119905) then after some iterations
1199001198953
(all iterations) = min119905
1199001198953
(119905) (31)
In this way the upper bound of the cycle time is decreasedgradually (see Figure 2) Another merit of this approach isthat it does not rely on the parameters of the FBPN
Theorem 1 Theupper bound can be found by considering onlyjobs with119873(CT
119895) ge 1199001198952
8 Mathematical Problems in Engineering
Proof Divide jobs into two categories 1198621
(jobs with119873(CT
119895) lt 1199001198952) and 119862
2(jobs with 119873(CT
119895) ge 1199001198952) Equation
(30) becomes
Δ120579119900lowast
= min(min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)))
(32)
If119873(CT119895) lt 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) gt 0 (33)
Therefore
min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) gt 0 (34)
In contrast if119873(CT119895) ge 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) le 0
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) le 0
(35)
Finally
Δ120579119900lowast
= min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(36)
Theorem 1 is proved
25 Step 4 Determining the Lower Bounds of Parameters in theFBPN In a similar way the threshold on the output node canbe modified to determine the lower bound of the cycle timeforecast so that the actual value will be greater than the lowerbound of the network output The optimal value of Δ120579
119900 canbe obtained as
Δ120579119900lowast
= max119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(37)
Theorem2 The lower bound can be found by considering onlyjobs with 119873(CT
119895) lt 1199001198952
The proof of Theorem 2 resembles that of Theorem 1
Proof Assume that the optimal value of 1199001198951
in the 119905threplication is indicated with 119900
1198951(119905) then after some iterations
1199001198951
(all iterations) = max119905
1199001198951
(119905) (38)
In this way the upper bound of the cycle time is increasedgradually Δ120579
119900lowastdoes not rely on the parameters of the FBPN
3 Application and Analyses
Todemonstrate the application of the proposedmethodologyan illustrative example containing the data of 40 jobs (seeTable 6) was used
We standardize the data and obtain the correlationmatrix119877 The eigenvalues and eigenvectors of 119877 are calculated as
1205821= 266 120582
2= 115 120582
3= 094
1205824= 083 120582
5= 025 120582
6= 002
(39)
1199061=
[[[[[[[
[
020
052
016
056
007
059
]]]]]]]
]
1199062=
[[[[[[[
[
047
minus019
068
minus019
minus049
005
]]]]]]]
]
1199063=
[[[[[[[
[
minus027
031
minus032
003
minus085
minus002
]]]]]]]
]
1199064=
[[[[[[[
[
081
minus002
minus057
006
minus004
minus015
]]]]]]]
]
1199065=
[[[[[[[
[
minus012
minus072
minus007
065
minus018
009
]]]]]]]
]
1199066=
[[[[[[[
[
002
029
028
047
003
minus078
]]]]]]]
]
(40)
respectively The variance contribution rates are
1205781= 46 120578
2= 20 120578
3= 16
1205784= 14 120578
5= 4 120578
6= 0
(41)
Summing up 120578119902rsquos we obtain
120578Σ(1) = 46 120578
Σ(2) = 65 120578
Σ(3) = 81
120578Σ(4) = 95 120578
Σ(5) = 100 120578
Σ(6) = 100
(42)
A Pareto analysis chart is used to compare the percentageof variability explained by each principal component (seeFigure 3) There is a clear break in the amount of varianceaccounted for by each component between the first andsecond components However that component by itselfexplains less than 50 of the variance so more components
Mathematical Problems in Engineering 9
Table 6 The illustrative example
119895 1199091198951
1199091198952
1199091198953
1199091198954
1199091198955
1199091198956
CT119895
1 24 1261 181 781 112 092 9352 24 1263 181 762 127 090 9583 24 1220 176 761 127 089 10474 23 1282 178 802 127 094 10115 23 1303 180 780 175 093 10686 23 1281 183 782 175 093 11437 23 1242 184 741 163 089 11038 24 1262 182 681 139 086 12509 22 1260 182 701 98 086 118110 22 1260 179 700 257 087 119411 24 1301 163 722 99 084 126012 22 1221 184 641 131 082 124013 23 1323 159 740 247 087 118014 24 1362 181 782 191 095 122715 24 1261 181 762 219 091 123616 23 1321 177 801 219 096 121517 22 1343 180 822 219 097 122818 24 1321 177 762 54 093 126619 25 1343 179 781 54 096 128520 25 1300 180 740 54 092 127221 22 1320 181 721 54 091 131022 24 1321 182 742 49 092 126523 23 1262 165 680 201 080 130824 22 1240 161 722 103 082 133125 23 1183 183 661 53 082 129426 23 1282 184 701 53 088 131427 22 1202 177 680 248 084 132128 23 1202 178 681 248 085 135329 24 1202 185 701 82 086 122630 23 1202 158 721 98 081 130131 24 1343 181 760 67 094 128032 24 1381 185 801 67 097 128633 22 1362 156 780 67 091 125234 23 1282 179 782 223 092 121435 23 1320 180 782 176 093 125136 25 1340 176 801 462 097 122237 23 1320 182 781 168 095 118738 22 1361 181 781 141 094 120539 22 1381 179 781 95 097 112040 23 1363 178 802 179 097 1133
are considered To meet the requirement 120578Σ(119901) ge 85sim90
p is chosen as 3 We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so we reduce the issue to these threecomponents in order to visualize the data
The component scores are recalculated as the new inputsto the FCM-FBPN (see Table 7) These scores contain thecoordinates of the original data in the new coordinate systemdefined by the principal components
Subsequently jobs are classified using FCM based on thenew variables The results of the 119878 test are summarized inTable 8 In this case the optimal number of job categories is 5
Table 7 New inputs to the FCM-FBPN
1199111198951
1199111198952
1199111198953
minus056 091 minus019
minus013 087 minus034
051 057 minus037
minus097 minus010 020
minus087 minus020 minus026
minus075 014 minus051
057 056 minus066
130 118 minus055
155 031 047
137 minus087 minus104
111 minus059 091
304 063 minus020
051 minus244 minus002
minus194 012 minus043
minus030 035 minus129
minus162 minus084 minus048
minus204 minus124 minus017
minus087 077 089
minus192 134 064
minus058 170 034
022 023 129
minus062 131 073
254 minus126 minus016
239 minus164 120
302 157 014
089 121 066
256 minus074 minus119
219 minus013 minus154
161 190 minus042
272 minus123 087
minus127 099 071
minus256 107 078
minus037 minus244 247
minus060 minus051 minus082
minus106 minus027 minus017
minus254 minus136 minus341
minus131 minus002 minus018
minus132 minus063 067
minus177 minus058 132
minus213 minus066 012
Table 8 The results of the S test
Number of categories (119870) 119869119898
1198902
min 119878
2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043
However therewill be some categories with very few jobs Forthis reason the second best solution is used that is 4 cate-gories A common practice is to set a threshold of member-
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Proof Divide jobs into two categories 1198621
(jobs with119873(CT
119895) lt 1199001198952) and 119862
2(jobs with 119873(CT
119895) ge 1199001198952) Equation
(30) becomes
Δ120579119900lowast
= min(min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)))
(32)
If119873(CT119895) lt 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) gt 0 (33)
Therefore
min119895isin1198621
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) gt 0 (34)
In contrast if119873(CT119895) ge 1199001198952 then
ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1) le 0
min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1)) le 0
(35)
Finally
Δ120579119900lowast
= min119895isin1198622
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(36)
Theorem 1 is proved
25 Step 4 Determining the Lower Bounds of Parameters in theFBPN In a similar way the threshold on the output node canbe modified to determine the lower bound of the cycle timeforecast so that the actual value will be greater than the lowerbound of the network output The optimal value of Δ120579
119900 canbe obtained as
Δ120579119900lowast
= max119895
(ln(1
119873(CT119895)minus 1) minus ln(
1
1199001198952
minus 1))
(37)
Theorem2 The lower bound can be found by considering onlyjobs with 119873(CT
119895) lt 1199001198952
The proof of Theorem 2 resembles that of Theorem 1
Proof Assume that the optimal value of 1199001198951
in the 119905threplication is indicated with 119900
1198951(119905) then after some iterations
1199001198951
(all iterations) = max119905
1199001198951
(119905) (38)
In this way the upper bound of the cycle time is increasedgradually Δ120579
119900lowastdoes not rely on the parameters of the FBPN
3 Application and Analyses
Todemonstrate the application of the proposedmethodologyan illustrative example containing the data of 40 jobs (seeTable 6) was used
We standardize the data and obtain the correlationmatrix119877 The eigenvalues and eigenvectors of 119877 are calculated as
1205821= 266 120582
2= 115 120582
3= 094
1205824= 083 120582
5= 025 120582
6= 002
(39)
1199061=
[[[[[[[
[
020
052
016
056
007
059
]]]]]]]
]
1199062=
[[[[[[[
[
047
minus019
068
minus019
minus049
005
]]]]]]]
]
1199063=
[[[[[[[
[
minus027
031
minus032
003
minus085
minus002
]]]]]]]
]
1199064=
[[[[[[[
[
081
minus002
minus057
006
minus004
minus015
]]]]]]]
]
1199065=
[[[[[[[
[
minus012
minus072
minus007
065
minus018
009
]]]]]]]
]
1199066=
[[[[[[[
[
002
029
028
047
003
minus078
]]]]]]]
]
(40)
respectively The variance contribution rates are
1205781= 46 120578
2= 20 120578
3= 16
1205784= 14 120578
5= 4 120578
6= 0
(41)
Summing up 120578119902rsquos we obtain
120578Σ(1) = 46 120578
Σ(2) = 65 120578
Σ(3) = 81
120578Σ(4) = 95 120578
Σ(5) = 100 120578
Σ(6) = 100
(42)
A Pareto analysis chart is used to compare the percentageof variability explained by each principal component (seeFigure 3) There is a clear break in the amount of varianceaccounted for by each component between the first andsecond components However that component by itselfexplains less than 50 of the variance so more components
Mathematical Problems in Engineering 9
Table 6 The illustrative example
119895 1199091198951
1199091198952
1199091198953
1199091198954
1199091198955
1199091198956
CT119895
1 24 1261 181 781 112 092 9352 24 1263 181 762 127 090 9583 24 1220 176 761 127 089 10474 23 1282 178 802 127 094 10115 23 1303 180 780 175 093 10686 23 1281 183 782 175 093 11437 23 1242 184 741 163 089 11038 24 1262 182 681 139 086 12509 22 1260 182 701 98 086 118110 22 1260 179 700 257 087 119411 24 1301 163 722 99 084 126012 22 1221 184 641 131 082 124013 23 1323 159 740 247 087 118014 24 1362 181 782 191 095 122715 24 1261 181 762 219 091 123616 23 1321 177 801 219 096 121517 22 1343 180 822 219 097 122818 24 1321 177 762 54 093 126619 25 1343 179 781 54 096 128520 25 1300 180 740 54 092 127221 22 1320 181 721 54 091 131022 24 1321 182 742 49 092 126523 23 1262 165 680 201 080 130824 22 1240 161 722 103 082 133125 23 1183 183 661 53 082 129426 23 1282 184 701 53 088 131427 22 1202 177 680 248 084 132128 23 1202 178 681 248 085 135329 24 1202 185 701 82 086 122630 23 1202 158 721 98 081 130131 24 1343 181 760 67 094 128032 24 1381 185 801 67 097 128633 22 1362 156 780 67 091 125234 23 1282 179 782 223 092 121435 23 1320 180 782 176 093 125136 25 1340 176 801 462 097 122237 23 1320 182 781 168 095 118738 22 1361 181 781 141 094 120539 22 1381 179 781 95 097 112040 23 1363 178 802 179 097 1133
are considered To meet the requirement 120578Σ(119901) ge 85sim90
p is chosen as 3 We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so we reduce the issue to these threecomponents in order to visualize the data
The component scores are recalculated as the new inputsto the FCM-FBPN (see Table 7) These scores contain thecoordinates of the original data in the new coordinate systemdefined by the principal components
Subsequently jobs are classified using FCM based on thenew variables The results of the 119878 test are summarized inTable 8 In this case the optimal number of job categories is 5
Table 7 New inputs to the FCM-FBPN
1199111198951
1199111198952
1199111198953
minus056 091 minus019
minus013 087 minus034
051 057 minus037
minus097 minus010 020
minus087 minus020 minus026
minus075 014 minus051
057 056 minus066
130 118 minus055
155 031 047
137 minus087 minus104
111 minus059 091
304 063 minus020
051 minus244 minus002
minus194 012 minus043
minus030 035 minus129
minus162 minus084 minus048
minus204 minus124 minus017
minus087 077 089
minus192 134 064
minus058 170 034
022 023 129
minus062 131 073
254 minus126 minus016
239 minus164 120
302 157 014
089 121 066
256 minus074 minus119
219 minus013 minus154
161 190 minus042
272 minus123 087
minus127 099 071
minus256 107 078
minus037 minus244 247
minus060 minus051 minus082
minus106 minus027 minus017
minus254 minus136 minus341
minus131 minus002 minus018
minus132 minus063 067
minus177 minus058 132
minus213 minus066 012
Table 8 The results of the S test
Number of categories (119870) 119869119898
1198902
min 119878
2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043
However therewill be some categories with very few jobs Forthis reason the second best solution is used that is 4 cate-gories A common practice is to set a threshold of member-
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table 6 The illustrative example
119895 1199091198951
1199091198952
1199091198953
1199091198954
1199091198955
1199091198956
CT119895
1 24 1261 181 781 112 092 9352 24 1263 181 762 127 090 9583 24 1220 176 761 127 089 10474 23 1282 178 802 127 094 10115 23 1303 180 780 175 093 10686 23 1281 183 782 175 093 11437 23 1242 184 741 163 089 11038 24 1262 182 681 139 086 12509 22 1260 182 701 98 086 118110 22 1260 179 700 257 087 119411 24 1301 163 722 99 084 126012 22 1221 184 641 131 082 124013 23 1323 159 740 247 087 118014 24 1362 181 782 191 095 122715 24 1261 181 762 219 091 123616 23 1321 177 801 219 096 121517 22 1343 180 822 219 097 122818 24 1321 177 762 54 093 126619 25 1343 179 781 54 096 128520 25 1300 180 740 54 092 127221 22 1320 181 721 54 091 131022 24 1321 182 742 49 092 126523 23 1262 165 680 201 080 130824 22 1240 161 722 103 082 133125 23 1183 183 661 53 082 129426 23 1282 184 701 53 088 131427 22 1202 177 680 248 084 132128 23 1202 178 681 248 085 135329 24 1202 185 701 82 086 122630 23 1202 158 721 98 081 130131 24 1343 181 760 67 094 128032 24 1381 185 801 67 097 128633 22 1362 156 780 67 091 125234 23 1282 179 782 223 092 121435 23 1320 180 782 176 093 125136 25 1340 176 801 462 097 122237 23 1320 182 781 168 095 118738 22 1361 181 781 141 094 120539 22 1381 179 781 95 097 112040 23 1363 178 802 179 097 1133
are considered To meet the requirement 120578Σ(119901) ge 85sim90
p is chosen as 3 We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so we reduce the issue to these threecomponents in order to visualize the data
The component scores are recalculated as the new inputsto the FCM-FBPN (see Table 7) These scores contain thecoordinates of the original data in the new coordinate systemdefined by the principal components
Subsequently jobs are classified using FCM based on thenew variables The results of the 119878 test are summarized inTable 8 In this case the optimal number of job categories is 5
Table 7 New inputs to the FCM-FBPN
1199111198951
1199111198952
1199111198953
minus056 091 minus019
minus013 087 minus034
051 057 minus037
minus097 minus010 020
minus087 minus020 minus026
minus075 014 minus051
057 056 minus066
130 118 minus055
155 031 047
137 minus087 minus104
111 minus059 091
304 063 minus020
051 minus244 minus002
minus194 012 minus043
minus030 035 minus129
minus162 minus084 minus048
minus204 minus124 minus017
minus087 077 089
minus192 134 064
minus058 170 034
022 023 129
minus062 131 073
254 minus126 minus016
239 minus164 120
302 157 014
089 121 066
256 minus074 minus119
219 minus013 minus154
161 190 minus042
272 minus123 087
minus127 099 071
minus256 107 078
minus037 minus244 247
minus060 minus051 minus082
minus106 minus027 minus017
minus254 minus136 minus341
minus131 minus002 minus018
minus132 minus063 067
minus177 minus058 132
minus213 minus066 012
Table 8 The results of the S test
Number of categories (119870) 119869119898
1198902
min 119878
2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043
However therewill be some categories with very few jobs Forthis reason the second best solution is used that is 4 cate-gories A common practice is to set a threshold of member-
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
1 2 3 4Principal component
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
Varia
nce e
xpla
ined
()
()
Figure 3 The Pareto analysis chart
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Center valueActual value
Upper boundLower bound
Figure 4 The forecasting results from the FBPN
ship 120583119871to determine whether a job belongs to each category
With the decrease in the threshold each category containsmore jobs Because the overall forecasting performance isaffected by the outliers 120583
119871should not be set too high In this
case 120583119871is set to 03 The classifying results are shown in
Table 9After job classification 34 of the examples in each cate-
gory are used as the training example The remaining 14 isleft for testing A three-layer FBPN is then used to predict thecycle time of jobs in each category according to the new var-iables with the following settings
Single hidden layer
The number of neurons in the hidden layer 2 lowast 3 = 6
Convergence criterion SSE lt 10minus6 or 10000 epochs
have been run
Iterations 5
Table 9 The classifying results (120583119871= 03)
Category Jobs1 4ndash6 14ndash172 10-11 13 23-24 27 30 333 1 18ndash22 31-324 2-3 7ndash9 12 25-26 28-29 34ndash40
The forecasting results are shown in Figure 4 The perfor-mance level of the proposed methodology is compared withthose of statistical analysis (ie multiple linear regression)CART PCA-CART BPN FCM-BPN and PCA-BPN inTable 10 The forecasting precision is evaluated with theaverage range
average range =sum119899
119895=1
100381610038161003816100381610038161199001198953
minus 1199001198951
10038161003816100381610038161003816
119899 (43)
The average range of a nonbiased crisp approach can bemeasured with 6120590 (or 6 RMSE)
1205902
= RMSE2 119899
119899 minus 119896 minus 1 (44)
where
RMSE =radic
sum119899
119895=1(119886119895minus 119891119895)2
119899
(45)
119896 is the number of independent variables 119886119895and119891119895denote the
actual value and forecast of job j respectively 119899 is the totalnumber of data When 119899 rarr infin 1205902 asymp RMSE2 Theoreticallythe probability that such an interval contains the actual valueis about 997 under the assumption that residuals followa normal distribution The forecasting accuracy is measuredwith MAE MAPE and RMSE
MAE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816
119899
MAPE =sum119899
119895=1
10038161003816100381610038161003816119886119895minus 119891119895
10038161003816100381610038161003816119886119895
119899
(46)
The CART algorithm is based on that of Breiman et al [42]The results of CART are shown in Figure 5 In PCA-CARTthe new variables found by PCA are used as input variables todetermine the job cycle time The results of PCA-CART areshown in Figure 6 The adjustment of the lower and upperbounds for the proposed methodology is shown in Figure 7
According to the experimental results we have the fol-lowing
(1) The nonlinear nature of this problem is obvious sincethe performance of statistical analysis is poor andstatistical analysis is a linear approach
(2) The accuracy of the job cycle time forecasting by theproposed fuzzy data mining approach is better thanthose of the compared approaches because the pro-posed approach has achieved 25sim85 reductions inMAPE
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table 10 Comparison of the forecasting performance levels
Category Averagerange
MAE(hrs) MAPE RMSE
(hrs)Statistical analysis 594 73 61 99CART 438 50 42 73PCA-CART 294 36 31 49BPN 414 30 24 69FCM-BPN 228 15 12 38PCA-BPN 402 27 23 67The proposedmethodology 169 11 09 29
1301096 12407511 10792507
12721697 11985838
1199094lt 7405
1199092lt 13115
1199095lt 81
1199096lt 0855
Figure 5 Results of CART
(3) Moreover the precision level of the proposedmethodology measured in terms of the average rangeof forecasts is also superior to those of the comparedapproaches The proposed methodology can be usedto specify a very narrow range of the job cycle timeSince the internal due date is based on the cycle timeforecast a narrow range promotes the reliability ofthe internal due date
(4) The proposed method is unusually slow in its modifi-cation of the lower bound (see Figure 7) a moreeffective way to solve this problem is needed
(5) PCA seems to be useful for methods with job classifi-cation like CART The combination of PCA andCART has significantly better forecasting accuracythan either method in isolation
(6) The simple combination of PCA and BPN does nothave much effect The main effect of PCA is toimprove the correctness of the job classification
In order to test the performance of the proposedmethod-ology on real world situations it has been applied to schedulea problem of 457 jobs from awafer fabrication factory locatedin Taichung City Scientific Park Taiwan There are morethan ten products in the wafer fabrication factory The waferfabrication factory has a monthly capacity of 20000 wafersJobs are regularly released into the wafer fabrication factoryVarious types of priorities are assigned to these jobs Each
11941924 12380503 11996605 10329271
1221171 12809957
13069236
1199091lt 19
1199093lt 027
1199092lt 0541199093ltminus0395
1199091ltminus10151199092lt 087
Figure 6 Results of PCA-CART
product has 150sim200 steps and six to nine reentries to themost bottlenecked machineThe first-in first-out (FIFO) ruleis used to dispatch the jobs The production characteristicof ldquoreentryrdquo which is highly relevant to the semiconductorindustry is clearly reflected in this problem It also shows thedifficulties facing production planners and schedulers whoattempt to provide an accurate due date for a product witha very complicated routing For each job twelve parameterswere collected or retrieved from production managementinformation system (PROMIS) databases and reports After abackward elimination by regression analysis six parameters(the job size factory utilization the queue length on theroute the queue length before the bottleneck the work-in-process (WIP) and the average waiting time) were chosenas the inputs to the FBPN Here ldquothe average waiting timerdquorefers to the average of the waiting times of the three jobsthat were completed most recently not to the waiting timeof the job under consideration The proposed methodologyand several existing methods have been applied to this caseThe forecasting precision (measured in terms of the averagerange) and accuracy (measured in terms of RMSE) of theseapproaches are compared in Figures 8 and 9 respectivelyThe samples in this case exhibited notable variability whichlimited the performance of the forecasting methods Nev-ertheless it is obvious that the proposed methodology stilloutperformed the six existing methods in both regards It isalso noteworthy that the methods based on job classification(such as CART PCA-CART FCM-BPN and the proposedmethodology) achieved better forecasting accuracy thanmethods without job classification Such an advantage wasfurther strengthened by variable replacement techniques likePCA
4 Conclusions and Directions forFuture Research
Job cycle time forecasting is an important task in any waferfabrication factory In this field many data mining methods
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Job
Cycle
tim
e (hr
s)
Iteration 1
(a)
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
0 10 20 30 40Job
Iteration 2
(b)
Center valueActual value
Upper boundLower bound
0 10 20 30 40Job
Iteration 3
0
200
400
600
800
1000
1200
1400
1600
Cycle
tim
e (hr
s)
(c)
Figure 7 The adjustment of the lower bounds
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
Aver
age r
ange
0100200300400500600700800900
1000
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 8 Comparison of the forecasting precision
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
0 20 40 60 80
100 120 140 160 180
RMSE
CART
BPN
FCM
-BPN
The p
ropo
sed
met
hodo
logy
PCA-
CART
PCA-
BPN
Stat
istic
al
anal
ysis
Figure 9 Comparison of the forecasting precision
have been applied A fuzzy data mining approach for preciseand accurate job cycle time forecasting is presented in thisstudy The forecast of the job cycle time by the proposedmethodology is a fuzzy value to improve precision an effec-tive procedure tightens the upper and lower bounds of theforecast The forecasting accuracy is enhanced by a combina-tion of PCA FCM and FBPN
This study presents an illustrative example and a real casewith the data of 457 jobs from a wafer fabrication factoryAccording to the experimental results we have the following
(1) the forecasting accuracy (measured with MAEMAPE and RMSE) of the PCA-FCM-FBPN wassignificantly better than the accuracy levels of theother approaches
(2) Methods based on job classification have better fore-casting accuracy The incorporation of PCA makesthe relationships between variables clearer andstrengthens the advantage of job classification
(3) It is possible to specify a very narrow range of the jobcycle time using the proposed methodology
Theupper bound can be adjusted faster than the lower boundAn effectivemethod is needed to tackle this issue In additionthe FBPNapproach focuses on themodification of the thresh-old on the output node Similar concepts can be applied tomodify other parameters in the FBPN
Acknowledgment
This study was financially supported by the National ScienceCouncil of Taiwan
References
[1] P Backus M Janakiram S Mowzoon G C Runger and ABhargava ldquoFactory cycle-time prediction with a data-miningapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 19 no 2 pp 252-258 2006
[2] W L Pearn S H Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007
[3] T Chen A Jeang and Y-C Wang ldquoA hybrid neural networkand selective allowance approach for internal due date assign-ment in a wafer fabrication plantrdquo International Journal ofAdvanced Manufacturing Technology vol 36 no 5-6 pp 570ndash581 2008
[4] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intel-ligence to forecast and reduce semiconductor cycle timerdquo Jour-nal of Intelligent Manufacturing pp 1ndash14 2011
[5] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003
[6] Y Meidan B Lerner G Rabinowitz and M Hassoun ldquoCycle-time key factor identification and prediction in semiconductormanufacturing using machine learning and data miningrdquo IEEETransactions on SemiconductorManufacturing vol 24 no 2 pp237ndash248 2011
[7] L-XWang and J M Mendel ldquoGenerating fuzzy rules by learn-ing from examplesrdquo IEEE Transactions on Systems Man andCybernetics vol 22 no 6 pp 1414ndash1427 1992
[8] P-C Chang J-CHieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005
[9] P-C Chang and J C Hsieh ldquoA neural networks approach fordue-date assignment in a wafer fabrication factoryrdquo Interna-tional Journal of Industrial Engineering vol 10 no 1 pp 55ndash612003
[10] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal ofAdvanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004
[11] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007
[12] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopAdvanced Semiconductor Manufacturing (ASMC rsquo04) pp 339ndash343 May 2004
[13] T Chen Y C Wang and H R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009
[14] T Chen H C Wu and Y C Wang ldquoFuzzy-neural approacheswith example post-classification for estimating job cycle timein a wafer fabrdquo Applied Soft Computing Journal vol 9 no 4 pp1225ndash1231 2009
[15] I Tirkel ldquoCycle time prediction in wafer fabrication line byapplying data mining methodsrdquo in Proceedings of the 22ndAnnual IEEESEMI Advanced Semiconductor ManufacturingConference (ASMC rsquo11) p 5 Saratoga Springs NY USA May2011
[16] Y T Tai W L Pearn and J H Lee ldquoCycle time estimation forsemiconductor final testing processes with Weibull-distributedwaiting timerdquo International Journal of Production Research vol50 no 2 pp 581ndash592 2012
[17] A I Sivakumar and C S Chong ldquoA simulation based analysisof cycle time distribution and throughput in semiconductorbackend manufacturingrdquo Computers in Industry vol 45 no 1pp 59ndash78 2001
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
[18] H Okumura ldquoSimulation analysis of relation between tool var-iability and cycle timerdquo in Proceedings of the International Sym-posium on Semiconductor Manufacturing (ISSM) and e-Man-ufacturing andDesign Collaboration Symposium (eMDC rsquo11) pp1ndash10 Hsinchu Taiwan 2011
[19] C P L Veeger L F P Etman E Lefeber I J B F Adan J VanHerk and J E Rooda ldquoPredicting cycle time distributions forintegrated processing workstations an aggregate modelingapproachrdquo IEEE Transactions on SemiconductorManufacturingvol 24 no 2 pp 223ndash236 2011
[20] P C Chang J C Hsieh and TW Liao ldquoA case-based reasoningapproach for due date assignment in awafer fabrication factoryrdquoin Proceedings of the 4th International Conference on Case-BasedReasoning (ICCBR rsquo01) vol 2080 of Lecture Notes in ComputerScience pp 648ndash659 2001
[21] C Chiu P C Chang and N H Chiu ldquoA case-based expert sup-port system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003
[22] P-C Chang Y-WWang and C-H Liu ldquoCombining SOM andGA-CBR for flow time prediction in semiconductor manufac-turing factoryrdquo in Rough Sets and Current Trends in Computingvol 4259 of Lecture Notes in Computer Science pp 767ndash775Springer Berlin Germany 2006
[23] T Chen ldquoA hybrid look-ahead SOM-FBPN and FIR system forwafer-lot-output time prediction and achievability evaluationrdquoInternational Journal of Advanced Manufacturing Technologyvol 35 no 5-6 pp 575ndash586 2007
[24] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion time pre-dictionrdquo International Journal of AdvancedManufacturing Tech-nology vol 37 no 7-8 pp 782ndash792 2008
[25] TChen ldquoJob cycle time estimation in awafer fabrication factorywith a bi-directional classifying fuzzy-neural approachrdquo Inter-national Journal of Advanced Manufacturing Technology vol56 no 9 pp 1007ndash1018 2011
[26] P C Chang and T W Liao ldquoCombining SOM and fuzzy rulebase for flow time prediction in semiconductor manufacturingfactoryrdquo Applied Soft Computing Journal vol 6 no 2 pp 198ndash206 2006
[27] T Chen ldquoA hybrid SOM-BPN approach to lot output time pre-diction in a wafer fabrdquo Neural Processing Letters vol 24 no 3pp 271ndash288 2006
[28] T Chen ldquoPredicting wafer-lot output time with a hybrid FCM-FBPN approachrdquo IEEE Transactions on Systems Man andCybernetics B vol 37 no 4 pp 784ndash793 2007
[29] T Chen ldquoAn intelligent mechanism for lot output time predic-tion and achievability evaluation in a wafer fabrdquo Computers andIndustrial Engineering vol 54 no 1 pp 77ndash94 2008
[30] T Chen ldquoSemiconductor unit cost forecasting with fuzzy col-laborative forecastingmodelsrdquoAlgorithms vol 5 no 4 pp 449ndash468 2012
[31] Q Song and B S Chissom ldquoForecasting enrollments with fuzzytime seriesmdashpart IIrdquo Fuzzy Sets and Systems vol 62 no 1 pp1ndash8 1994
[32] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139
[33] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal due dateassignment in a wafer fabrication plantrdquo Robotics and Comput-er-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010
[34] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991
[35] H Nomura I Hayashi and N Wakami ldquoA learning method offuzzy inference rules by descent methodrdquo in Proceedings of theIEEE International Conference on Fuzzy Systems pp 203ndash210March 1992
[36] C-J Lin ldquoAGA-based neural fuzzy system for temperature con-trolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash333 2004
[37] J Watada H Tanaka and T Shimomura ldquoIdentification oflearning curve based on possibilistic conceptsrdquo in Applicationsof Fuzzy Set Theory in Human Factors Elsevier The Nether-lands 1986
[38] D Huang T Chen and M-JJ Wang ldquoA fuzzy set approach forevent tree analysisrdquo Fuzzy Sets and Systems vol 118 no 1 pp153ndash165 2001
[39] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol AC-19 pp 716ndash723 1974
[40] J Rissanen ldquoStochastic complexity and modelingrdquo The Annalsof Statistics vol 14 no 3 pp 1080ndash1100 1986
[41] T Chen and Y C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011
[42] L Breiman J Friedman C J Stone and R A Olshen Classifi-cation and Regression Trees Chapman amp Hall Boca Raton FlaUSA 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
