Research Article An Iterative Procedure for Optimizing the ...downloads.hindawi.com/journals/mpe/2013/740478.pdf · Fuzzy-Neural Job Cycle Time Estimation Approach in a Wafer Fabrication

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 740478 15 pageshttpdxdoiorg1011552013740478

Research ArticleAn Iterative Procedure for Optimizing the Performance of theFuzzy-Neural Job Cycle Time Estimation Approach in a WaferFabrication Factory

Toly Chen and Yi-Chi Wang

Department of Industrial Engineering and Systems Management Feng Chia University 100 Wenhwa Road SeatwenTaichung 408 Taiwan

Correspondence should be addressed to Toly Chen tcchenfcuedutw

Received 28 October 2012 Accepted 27 December 2012

Academic Editor Peng Shi

Copyright copy 2013 T Chen and Y-C Wang This 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

Estimating the cycle time of each job in a wafer fabrication factory is a critical task to every wafer manufacturer In recent yearsa number of hybrid approaches based on job classification (either preclassification or postclassification) for cycle time estimationhave been proposed However the problemwith thesemethods is that the input variables are not independent In order to solve thisproblem principal component analysis (PCA) is considered useful In this study a classifying fuzzy-neural approach based on thecombination of PCA fuzzy c-means (FCM) and back propagation network (BPN) is proposed to estimate the cycle time of a jobin a wafer fabrication factory Since job classification is an important part of the proposed methodology a new index is proposedto assess the validity of the classification of jobs The empirical relationship between the S value and the estimation performance isalso found Finally an iterative process is employed to deal with the outliers and to optimize the overall estimation performanceA real case is used to evaluate the effectiveness of the proposed methodology Based on the experimental results the estimationaccuracy of the proposed methodology was significantly better than those of the existing approaches

1 Introduction

The competition in the semiconductor industry has beenvery intense How to obtain and maintain the competitiveedge is an important task for all manufacturers in thisindustry Quick response and on-time delivery are obviouslypressing needs for any modern enterprise To this endaccurate estimating and shortening the cycle time (flow timeor manufacturing lead time) of each job in the factory isa prerequisite [1ndash9] In a wafer fabrication factory a jobis usually composed of about 25 pieces of wafers and hashundreds of steps to be processed In addition a job mayvisit the same workstation more than once because the sameoperation may be needed multiple times A wafer fabricationfactory is therefore classified as a complicated reentrantproduction system

Estimating the cycle time of each job in awafer fabricationfactory is very important to the factory because it can

signal the manager if the orders are progressed as they wereexpected For example if the estimated cycle time of a jobis longer than as it was expected then this order may notbe completed to the customer before its due date Someproduction control actions should then immediately be takento accelerate the progress of the job [10]That iswhy this paperstudies the estimation of job cycle time in a wafer fabricationfactory

The existing approaches for the job cycle time estimationin a wafer fabrication factory can be classified into sixcategories statistical analysis production simulation (PS)back propagation network (BPN) case-based reasoning(CBR) fuzzy modeling methods and hybrid approaches [9]Among the six approaches statistical analysis is the easiestquickest and most prevalent in practical applications Mostof the statistical analyses used linear regression equationsto estimate the job cycle time (eg [11 12]) Pearn et al[13] fitted the distribution of the waiting time of a job with

2 Mathematical Problems in Engineering

a gamma distribution and then used a linear equation toestimate the job cycle time Recently Chien et al [14] usednonlinear regression equations instead and then found outthe relationship between the estimation error and somefactory conditions and job attributes with a BPN to furtherimprove the estimation accuracy The major disadvantageof statistical analysis is the lack of estimation accuracy [9]Conversely a huge amount of data and lengthy simulationtime required are two disadvantages of PS Neverthelesstheoretically PS is themost accurate job cycle time estimationapproach if the simulation model is completely valid and iscontinuously updated

Considering effectiveness (estimation accuracy) and effi-ciency (execution time) simultaneously Chang et al [8]Chang and Hsieh [15] and Sha and Hsu [16] estimated thecycle time of a job in a wafer fabrication factory using aBPN with a single hidden layer A BPN is an effective toolin modeling complex physical systems described by sets ofdifferent equations for prediction control and design pur-poses Compared with some statistical analysis approachesthe average estimation accuracy measured with root meansquared error (RMSE) was considerably improved with theBPNs For example an improvement of about 40 in RMSEwas achieved in the study of Chang et al [8] Chen [17]incorporated the job releasing plan of the wafer fabricationfactory into a BPN and constructed a ldquolook-aheadrdquo BPN forthe same purpose which led to an average reduction of 12in RMSE On the other hand much less time and fewer dataare required with a BPN than with PS Chen et al [18] andBeeg [19] estimated the cycle time of a job in a ramping upwafer fabrication factory In their studies Chen et al used aBPN-based method while Beeg tried to find out the impactof utilization for the cycle time

Chiu et al [20] established an expert system based onCBR for the job cycle time estimation To effectively considerthe uncertainty in the job cycle time fuzzy logic was used ina number of studies For example Chang et al [8] modifiedthe first step (ie partitioning the range of each input variableinto several fuzzy intervals) of the fuzzy modeling methodproposed by Wang and Mendel [21] called the WMmethodwith a simple genetic algorithm (GA) and proposed theevolving fuzzy rule (EFR) approach to estimate the cycle timeof a job in a wafer fabrication factory Their EFR approachoutperformed CBR and BPN in the estimation accuracyChen [9] constructed a fuzzy back propagation network(FBPN) that incorporated expert opinions to modify theinputs of the FBPN Chenrsquos FBPN surpassed the crisp BPNespecially with respect to efficiency

In recent years a number of hybrid approaches have beenproposed most of which classified jobs before estimatingthe cycle times For example Chen [7] combined self-organization map (SOM) and WM in which jobs wereclassified using a SOM before estimating the cycle timesof the jobs with WM Chen and Wang [22] constructed alook-ahead k-means- (kM-) FBPN for the same purpose anddiscussed in detail the effects of using different look-aheadfunctions More recently Chen [17] proposed the look-aheadSOM-FBPN approach for the job cycle time estimation in asemiconductor factory [23] Besides a set of fuzzy inference

rules were also developed to evaluate the achievability of acycle time forecast Subsequently Chen [24] added a selectiveallowance to the cycle time estimated using the look-aheadSOM-FBPN approach to determine the intermediate duedate Further Chen et al [23] showed that the suitabilityof combining the SOM and FBPN for the data could beimproved with the feedback of the estimation error by theFBPN to adjust the classification results of the SOM Chen etal [25] proposed a postclassification fuzzy-neural approachin which a job was not pre-classified but rather postclassifiedafter estimating the cycle time Experimental results showedthat the postclassification approach was better than thepreclassification approaches in certain cases In order tocombine the advantages of preclassifying and post-classifyingapproaches Chen [26] proposed a bi-directional classifyingapproach in which jobs are not only pre-classified but alsopostclassified Except few studies in which the historical dataof a real semiconductor factory were collected most studiesin this field used simulated data [27]

In short the followings have not done before

(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 How-ever this issue has rarely been addressed in previousstudies of this field

(2) Job classification has been shown to be conduciveto the estimation performance However most paststudies chose classifiers subjectively and did not eval-uate the performance of the classifier Needless to sayoptimizing the classifier for the subsequent estima-tion task

Principal component analysis (PCA) is a multivariate sta-tistical analysis method This method constructs a series oflinear combinations of the original variables to form a newvariable so that these new variables are unrelated to eachother as much as possible and the relationship among themcan be reflected in a better way In this study a fuzzy-neuralapproach based on the combination of PCA FCM and BPNis proposed to estimate the cycle time of a job in a waferfabrication factory The motivation of this study is explainedas follows

(1) While in the past some studies combined PCA andFCM the references on the combination of PCAFCM and BPN are still very limited Chen [28]applied PCA to modify the inputs to a BPN for thejob cycle time estimation The estimation accuracyof PCA-BPN was slightly better than that of BPNalone It seems that BPN can solve the dependenciesof the input variables for the job cycle time estimationproblems PCA seems to be more important forthe classification of jobs This provides us with amotivation to improve the existing job cycle timeestimation methods based on job classification

(2) FCM as a part of the preclassifying approach can-not be evaluated alone Its success depends on the

Mathematical Problems in Engineering 3

performance of the subsequent estimation task Thisprovides us with a motivation to assess the validity ofthe classification of jobs from this point of view

(3) The 119878 test is a commonly used method to determinethe best number of categories in FCM Howeverwhether this way directly favors the estimation per-formance has not been confirmed

The contribution compared with some previous works inthe literature includes the following

(1) With factors that are dependent on each other jobsmay bemisclassified if FCM is used aloneThismay beharmful to the estimation accuracy of BPN becauseincorrect examples are used to train the BPN Thefuzzy-neural approach replaces the original factorswith new independent factors and is expected tobe able to generate the correct classification resultsThe correctness of the classification results must bejudged from the estimation performance In order tomeasure that two new indexes are defined

(2) It is anticipated that the new factors found out byPCA have a more explicit relationship with the jobcycle time As a result the training of BPN may beaccelerated This also means that a more accuraterelationship between the factors and the cycle timecan also be generated with the same time

(3) A new index is proposed to assess the validity of theclassification of jobs

(4) The empirical relationship between the 119878 value andthe estimation performance is found

(5) Outliers that is jobs that cannot be classified def-initely have not been dealt with properly in thepast However the overall estimation performanceis often affected by the outliers For this reason aniterative process is established in this study which canoptimize the overall estimation performance

The differences between the proposed methodology and theprevious methods are summarized in Table 1

The remainder of this paper is organized as fol-lows Section 2 introduces the proposed PCA-FCM-BPNapproach An example is employed to illustrate the proposedmethodology A case with the real data from a wafer fabri-cation factory is investigated in Section 3 The performanceof the proposed methodology is compared with those of theexisting approaches for this real case Based on the resultssome points are made in analysis Finally the concludedremarks with a view to the future are given in Section 4

2 Methodology

Two characteristics of the proposed methodology are inputreplacement and job classification These features not aremathematical skills but also have implications for the oper-ations of a wafer fabrication factory First in the usefulinformation for the estimation of the job cycle time manyfactors are in fact mutually dependent For example it is

well known that the utilization of a factory increases whenthe work-in-process (WIP) level in the factory rises Bothutilization and theWIP level are important factors consideredin some job cycle time estimation approaches Whetherthe dependence of the factors will lead to problems in theclassification of jobs needs to be checked Therefore thereplacement of these factors with new independent variablesis worth a try

On the other hand a number of job cycle time estimationapproaches in this field classify jobs A well-known conceptis that the cycle time of a job is proportional to the WIPlevel of the factory according to Littlersquos law however thatonly holds when the factory utilization is 100 Thereforeit is reasonable to divide jobs into two categories jobs thatare released into the factory when the factory utilization is100 and jobs released when the factory utilization is lessthan 100

The architecture of the proposed methodology is shownin Figure 1

21 Variable Replacement Using PCA First PCA is used toreplace the inputs to the FCM-BPNThe combination of PCAand FCM has proven to be a more effective classifier thanFCM alone [29] PCA consists of the following steps

(1) Raw data standardization to eliminate the differencebetween the dimensions and the impact of largenumerical difference in the original variables Theoriginal variables are standardized as the following

119909lowast

119895119894=119909119895119894minus 119909119894

120590119894

119909119894=

sum119899

119895=1119909119895119894

119899

120590119894=radicsum119899

119895=1(119909119895119894minus 119909119894)2

119899 minus 1

(1)

where 119909119895119894is the 119894th attribute of job 119895 119895 = 1 sim 119899 119909

119894

and 120590119894indicate the mean and standard deviation of

variable 119894 respectively(2) Establishment of the correlation matrix 119877

119877 =1

119899 minus 1119883lowast119879

119883lowast

(2)

where 119883lowast is the standardized data matrix The

eigenvalues and eigenvectors of 119877 are calculated andrepresented as 120582

1sim 120582119898and 119906

1sim 119906119898 respectively

1205821ge 1205822ge 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)


Table 1 The differences between the proposed methodology and the previous methods

Method SOM-WM [7] SOM-FBPN [17]kM-FBPN [22ndash24] BPN-BPN [25] FCM-FBPN-RBF [26] The proposed

methodologyJob preclassification Yes No Yes YesJob postreclassification No Yes Yes YesParameter replacement No No No YesDealing with outliers No No No YesIteration No No No YeslowastRBF is radial basis function network

Job data PCA

FCM

Category 1examples

examples

1

2

1

2

1

1

2

1

2

1

Outliers

⋯

⋮⋮

⋮⋮

119873(119862119879119895)

119873(119862119879119895)

119890119895

119890119895

119900119895

119900119895

119901

119901

119911119901

119911119901

119911119901

1199111

1199111

1199111

1199112

1199112

1199112

119909119898

1199091

1199092

2119901

2119901

Category 119896

Figure 1 The architecture of the proposed methodology

where 119902 = 1 sim 119898 and the accumulated variancecontribution rate is

120578Σ(119901) =

119901

sum

119902=1

120578119902 (4)

where 119901 = 1 sim 119898 Choose the smallest 119901 value suchthat 120578Σ(119901) ge 85 sim 90 A Pareto analysis chart can

be used to compare the percent variability explainedby each principal component

(4) Formation of the following matrixes

119880119898times119901

= [1199061 1199062 119906

119901]

119885119899times119901

= 119883lowast

119899times119898119880119898times119901

(5)

119885119899times119901

= [119911119895119902] (119895 = 1 sim 119899 119902 = 1 sim 119901) is the

component scores which contain the coordinates ofthe original data in the new coordinate systemdefinedby the principal components and will be used as thenew inputs to the FFNN


Table 2 An example

119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956

1 24 1261 181 781 112 0922 24 1263 181 762 127 0903 24 1220 176 761 127 0894 23 1282 178 802 127 0945 23 1303 180 780 175 0936 23 1281 183 782 175 0937 23 1242 184 741 163 0898 24 1262 182 681 139 0869 22 1260 182 701 98 08610 22 1260 179 700 257 08711 24 1301 163 722 99 08412 22 1221 184 641 131 08213 23 1323 159 740 247 08714 24 1362 181 782 191 09515 24 1261 181 762 219 09116 23 1321 177 801 219 09617 22 1343 180 822 219 09718 24 1321 177 762 54 09319 25 1343 179 781 54 09620 25 1300 180 740 54 09221 22 1320 181 721 54 09122 24 1321 182 742 49 09223 23 1262 165 680 201 08024 22 1240 161 722 103 08225 23 1183 183 661 53 08226 23 1282 184 701 53 08827 22 1202 177 680 248 08428 23 1202 178 681 248 08529 24 1202 185 701 82 08630 23 1202 158 721 98 08131 24 1343 181 760 67 09432 24 1381 185 801 67 09733 22 1362 156 780 67 09134 23 1282 179 782 223 09235 23 1320 180 782 176 09336 25 1340 176 801 462 09737 23 1320 182 781 168 09538 22 1361 181 781 141 09439 22 1381 179 781 95 09740 23 1363 178 802 179 097

To illustrate the application of the proposedmethodologyan example is given in Table 2 To get a quick impression ofthe data a box plot is made in Figure 2 Note that there issubstantially more variability in 119909

1198952 1199091198954 and 119909

1198955than in the

remaining variablesSubsequently we standardize the data (see Table 3) and

obtain the correlation matrix as

119877 =

[[[[[[[

[

097 010 016 021 minus003 025

010 098 001 070 minus001 078

016 001 098 005 minus007 037

021 070 005 098 015 086

minus003 minus001 minus007 015 098 010

025 078 037 086 010 098

]]]]]]]

]

(6)

0 200 400 600 800 1000 1200 1400Values

1199091198956

1199091198955

1199091198954

1199091198953

1199091198952

1199091198951

Figure 2 The box plot

The eigenvalues and eigenvectors of 119877 are calculated asthe following

1205821= 266 120582

2= 115

1205823= 094 120582

4= 083

1205825= 025 120582

6= 002

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

]]]]]]]

]

(7)

respectively The variance contribution rates are

1205781= 46 120578

2= 20 120578

3= 16

1205784= 14 120578

5= 4 120578

6= 0

(8)


Table 3 The standardized data

119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956

1 088 minus053 040 074 minus040 0372 088 minus049 048 030 minus022 minus0053 088 minus127 minus017 029 minus022 minus0314 minus022 minus015 007 118 minus022 0655 minus022 024 037 071 035 0586 minus022 minus017 078 074 035 0457 minus022 minus087 090 minus014 021 minus0198 088 minus051 053 minus145 minus008 minus0879 minus132 minus055 060 minus101 minus056 minus08110 minus132 minus054 023 minus103 134 minus05511 088 020 minus187 minus057 minus056 minus11912 minus132 minus125 080 minus233 minus018 minus16413 minus022 060 minus240 minus016 122 minus06014 088 131 047 075 055 09415 088 minus053 040 031 088 01916 minus022 057 minus005 117 088 11317 minus132 097 033 162 088 13818 088 056 minus001 031 minus109 06219 197 096 015 074 minus109 10620 197 019 038 minus016 minus109 03621 minus132 055 051 minus057 minus109 01022 088 055 054 minus013 minus116 03223 minus022 minus052 minus159 minus147 067 minus20024 minus132 minus091 minus211 minus057 minus051 minus15425 minus022 minus195 072 minus189 minus111 minus16426 minus022 minus015 089 minus101 minus111 minus03627 minus132 minus159 minus007 minus147 123 minus12628 minus022 minus160 007 minus147 123 minus11129 088 minus160 093 minus103 minus076 minus08730 minus022 minus159 minus257 minus058 minus056 minus18631 088 097 041 027 minus094 07032 088 165 100 116 minus094 12733 minus132 130 minus273 071 minus094 01034 minus022 minus014 016 074 092 03335 minus022 055 034 075 036 05436 197 091 minus023 117 379 12837 minus022 055 061 073 027 09138 minus132 128 042 072 minus005 08139 minus132 166 014 072 minus060 13640 minus022 133 013 118 040 142

Summing up 120578119902rsquos we obtain the following

120578Σ(1) = 46 120578

Σ(2) = 65

120578Σ(3) = 81 120578

Σ(4) = 95

120578Σ(5) = 100 120578

Σ(6) = 100

(9)

A Pareto analysis chart is used to compare the percent vari-ability explained by each principal component (see Figure 3)There is a clear break in the amount of variance accountedfor by each component between the first and the second

1 2 3 40

102030405060708090

100

Principal component

0102030405060708090100

Varia

ncee

xpla

ined

()

()

Figure 3 The Pareto analysis chart

0 1 2 3 4

0

1

2

1st principal component

2nd

prin

cipa

l com

pone

nt

15

05

minus05

minus1

minus15

minus2

minus25

minus3 minus2 minus1

Figure 4 The component scores

components However that component by itself can onlyexplain less than 50 of the variance so more componentsmay be needed To meet the requirement 120578

Σ(119901) ge 85 sim

90 119901 is chosen as 3We can see that the first three principalcomponents explain roughly 80 of the total variability inthe standardized data so that might be a reasonable way toreduce the dimensions in order to visualize the data

Subsequently the component scores are computed (seeTable 4) which contain the coordinates of the original datain the new coordinate system defined by the principalcomponents and will be used as the new inputs to the FCM-BPN In Figure 4 the first two columns of the componentscores are plotted showing the data projected onto the firsttwo principal components

22 Classifying Jobs Using FCM After employing PCAexamples are then classified using FCM If a crisp clusteringmethod is applied instead then it is very likely that someclusters will have very few examples In contrast an examplebelongs to multiple clusters to different degrees in FCM


Table 4 New inputs to the FCM-BPN

1199111198951

1199111198952

1199111198953

minus056 091 minus019minus013 087 minus034051 057 minus037minus097 minus010 020minus087 minus020 minus026minus075 014 minus051057 056 minus066130 118 minus055155 031 047137 minus087 minus104111 minus059 091304 063 minus020051 minus244 minus002minus194 012 minus043minus030 035 minus129minus162 minus084 minus048minus204 minus124 minus017minus087 077 089minus192 134 064minus058 170 034022 023 129minus062 131 073254 minus126 minus016239 minus164 120302 157 014089 121 066256 minus074 minus119219 minus013 minus154161 190 minus042272 minus123 087minus127 099 071minus256 107 078minus037 minus244 247minus060 minus051 minus082minus106 minus027 minus017minus254 minus136 minus341minus131 minus002 minus018minus132 minus063 067minus177 minus058 132minus213 minus066 012

which provides a solution to this problem Similarly inprobability theory the naıve Bayes method provides theprobability that the item belongs to each class Howeverthe application of FCM can consider subjective factors inclassifying the jobs Algorithm 1

FCM classifies jobs byminimizing the following objectivefunction

Min119870

sum

119896=1

119899

sum

119895=1

120583119898

119895(119896)1198902

119895(119896) (10)

where119870 is the required number of categories 119899 is the numberof jobs 120583

119895(119896)indicates the membership that job 119895 belongs

to category 119896 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 isdescribed as follows

(1) Normalize the input data(2) Produce a preliminary clustering result(3) (Iterations) Calculate the centroid of each category as

the following

119911(119896)= 119911(119896)119902 119896 = 1 sim 119870

119911(119896)119902

=

sum119899

119895=1120583119898

119895(119896)119911119895119902

sum119899

119895=1120583119898

119895(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

(119911119895119901minus 119911(119896)119901)2

119895 = 1 sim 119899 119896 = 1 sim 119870

(11)

where 119911(119896)

is the centroid of category 119896 120583(119905)119895(119896)

is themembership that job 119895 belongs to category 119896 after the119905th 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 (12)

where 119889 is a real number representing the thresholdfor the convergence of membership

The performance of FCM is highly affected by the settings forthe initial values and therefore can be repeatedmultiple timesin order to find the optimal solution Finally the separatedistance test (119878 test) proposed by Xie and Beni [30] can beapplied to determine the optimal number of categories 119870 asfollows

Min 119878 (13)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)119902minus 119911(1198962)119902)2

)

119878 =119869119898

119899 times 1198902

min

119870 isin 119885+

(14)


A=[03857 7175 05381 01614 04281 05803]

c=2

[center U obj fun]=fcm(A c)

Jm=min (obj fun)

e2 min=9999

for i=1 c

for j=i+1 c

e2 sum=0

for k=1 3

e2 sum=e2 sum+(center (i k)minuscenter(j k))and 2

end

if e2 sumlte2 min

e2 min=e2 sum

end

end

end

e2 min

S=min (Jm)(40lowaste2 min)

Algorithm 1 The sample MATLAB code for the FCM approach

Table 5 The results of the 119878 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

Table 6 The classifying results (120583119871= 05)

Category Jobs1 1 2 18 19 20 22 31 322 3 7 8 9 12 25 26 293 4 5 6 14 16 17 34 35 37 38 39 404 10 11 23 24 27 30


Category Jobs1 1 2 18 19 20 21 22 31 322 2 3 7 8 9 12 25 26 28 293 4 5 6 14 15 16 17 33 34 35 36 37 38 39 404 10 11 13 23 24 27 28 30 33

The119870 value minimizing 119878 determines the optimal number ofcategories

The Fuzzy Logic Toolbox of MATLAB can be used toimplement the FCM approach A sample code is shown in

In the illustrative example the data have been standard-ized and therefore are not normalized again The results ofthe 119878 test are summarized in Table 5 In this case the optimalnumber of job categories was 5 However there will be somecategories with very few jobs For this reason the second bestsolution is used that is 4 categories A common practice is

to set a threshold of membership 120583119871to determine whether

a job belongs to each category For example if 120583119871= 05

then the classifying results are shown in Table 6 With thedecrease in the threshold each category will contain morejobs For example if 120583

119871= 03 then the classifying results are

shown in Table 7 Such a property can solve the problem ofan insufficient number of examples

We also note that the classification results are verydifferent according to the new variables compared with theresults based on the original variables In other words theresults of FCM and PCA-FCM are not the same

(1) The optimal number of categories in FCM is 6 whilethat in PCA-FCM is 5

(2) If jobs are divided into four categories in these twomethods then the results are compared in Figure 5Many jobs have been reclassified which means thatthe misclassification problem has been resolved aftervariable replacement

In Figure 5 there are also some outliers that cannot beclassified into any category

23 Estimating the Cycle Time Using BPN Finally the jobsexamples of a category are learned with the same BPN Arti-ficial neural networks have been proposed to solve a widevariety of problems usually characterized by sets of differentequations Although there have been some more advancedartificial neural networks such as compositional pattern-producing network cascading neural network and dynamicneural network a well-trained BPN with an optimized struc-ture can still produce very good results The configuration ofthe BPN is established as follows

(1) Inputs the new factors determined by PCAassociatedwith the 119895th examplejob These factors have to bepartially normalized so that their values fall within[01 09] [18]


0

1

2

3

4

5

0 10 20 30 40

Cate

gory

FCM

Job number

PCA-FCM

Figure 5 Comparison of the classification results by PCA-FCMandFCM

(2) Single hidden layer generally one or two hidden lay-ers are more beneficial for the convergence propertyof the BPN

(3) For simplicity the number of neurons in the hiddenlayer is twice that in the input layer An increase inthe number of hidden-layer nodes lessens the outputerrors for the training examples but increases theerrors for novel examples Such a phenomena is oftencalled ldquoover-fittingrdquo There has been some researchconsidering the relation among the complexity ofa BPN the performance for the training data andthe number of examples for example using Akaikersquosinformation criterion (AIC) or theminimumdescrip-tion length (MDL)

(4) Output the (normalized) cycle time estimate of theexample

The procedure for determining the parameter valuesis now described After preclassification a portion of theadopted examples in each category is fed as ldquotraining exam-plesrdquo into the BPN to determine the parameter values forthe category Two phases are involved at the training stageAt first in the forward phase inputs are multiplied withweights summated and transferred to the hidden layerThenactivated signals ℎ

119895119897are outputted from the hidden layer as

ℎ119895119897=

1

1 + 119890minus119899ℎ

119895119897

(15)

where

119899ℎ

119895119897= 119868ℎ

119895119897minus 120579ℎ

119897

119868ℎ

119895119897=

119901

sum

119902=1

119908ℎ

119902119897119911119895119902

(16)

ℎ119895119897rsquos are also transferred to the output layer with the same

procedure Finally the output of the BPN is generated as

119900119895=

1

1 + 119890minus119899119900

119895

(17)

where

119899119900

119895= 119868119900

119895minus 120579119900

119868119900

119895=

119871

sum

119897=1

119908119900

119897ℎ119895119897

(18)

The output 119900119895is comparedwith the normalized step flow time

119873(119862119879119895) for which RMSE is calculated as the following

RMSE = radicsum119899

119895=1(119900119895minus 119873(119862119879

119895))2

119899

(19)

Subsequently in the backward phase some algorithmsare applicable for training a BPN such as 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 the second-order speed without having to computethe Hessian matrix It uses approximation and updates thenetwork parameters in a Newton-like way as describedbelow

The network parameters are placed in vector 120573 =[119908ℎ

11 119908

ℎ

119901119871 120579ℎ

1 120579ℎ119871 1199081199001 119908

119900

119871 120579119900] The network output

119900119895can be represented with 119891(x

119895120573) The objective function

of the BPN is to minimize RMSE or equivalently the sum ofsquared error (SSE)

SSE (120573) =119899

sum

119895=1

(119873(119862119879119895) minus 119891 (x

119895120573))2

(20)

The Levenberg-Marquardt algorithm is an iterative pro-cedure In the beginning the user should specify the initialvalues of the network parameters 120573 Let 120573T = (1 1 1)

be a common practice In each step the parameter vector 120573is replaced by a new estimate 120573 + 120575 where 120575 = [Δ119908

ℎ

11

Δ119908ℎ

119901119871 Δ120579ℎ1 Δ120579

ℎ

119871 Δ1199081199001 Δ119908119900

119871 Δ120579119900]The network output

becomes 119891(x119895120573+120575) that is approximated by its linearization

as

119891 (x119895120573 + 120575) asymp 119891 (x

119895120573) + J

119895120575 (21)

where

J119895= 120597

119891 (x119895120573)

120597120573(22)

is the gradient vector of 119891 with respect to 120573 Substituting (21)into (20) leads to

SSE (120573+120575) asymp119899

sum

119895=1

(119873 (119862119879119895) minus 119891 (x

119895120573) minus J

119895120575)2

(23)


tn input=[0843 0831 sdot sdot sdot 0839 0859 sdot sdot sdot 09 09 sdot sdot sdot 0878 0889 sdot sdot sdot 0875 0858 sdot sdot sdot

0822 0827 sdot sdot sdot]

tn target=[0849 0849 sdot sdot sdot]

net=newff ([0 1 0 1 0 1 0 1 0 1 0 1] [12 1] logsig logsig trainlm)net=init (net)

net trainParam show=10

net trainParam lr=01

net trainParam epochs=1000

net trainParam goal=1eminus4

[net tr]=train (net tn input tn target)

tn output=sim (net tn input)

te input=[0825 0844sdot sdot sdot 0824 0835 sdot sdot sdot 09 09 sdot sdot sdot 0878 0889 sdot sdot sdot 0883 0875 sdot sdot sdot


te output=sim (net te input)

Algorithm 2 The sample MATLAB code for the BPN approach

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 givesthe following

(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)

where J is the Jacobian matrix containing the first derivativeof network error with respect to the weights and biasesEquation (24) includes a set of linear equations that can besolved for 120575

In the illustrative example 34 of the examples in eachcategory are used as the training example The remaining 14is left for testing A three-layer BPN is then used to estimatethe cycle time of jobs in each category according to the newvariables with the following setting

Single hidden layer

The number of neurons in the hidden layer 2lowast3 = 6

Convergence criterion SSE lt 10minus6 or 10000 epochs

have been run

For an outlier the BPNs of all categories are applied toestimate the cycle time The Neural Network Toolbox ofMATLAB is used to implement the BPN approach Thesample code is shown in Algorithm 2 The estimation accu-racy can be evaluated with mean absolute error (MAE)mean absolute percentage error (MAPE) and RMSE Theestimation performances are summarized in Table 8

Obviously the overall estimation performance is affectedby the outliers If the outliers can be dealt with properly theoverall estimation will be improved To this end an iterativefeedback control procedure is established in the next subsec-tion (see Figure 6) which can optimize the overall estimationperformance In the literature there have been a few controlmechanisms for various types of fuzzy systems [31ndash39] Onthe other hand we also compare the performances of thegradient descent algorithm and the Levenberg-Marquardtalgorithm as shown in Table 9

Table 8 The estimation performances

Category MAE (hrs) MAPE RMSE (hrs)1 18 16 422 1 01 33 6 04 124 23 18 53Outliers 149 119 152Total 42 35 85

Table 9 Comparing the performances of two training algorithms(group 4 convergence criterion SSE lt 10minus6)

Algorithm Number of epochs MAE MAPE RMSEGradient descent 10000 79 67 98Levenberg-Marquardt lt100 23 18 53

24 Iterative Optimization

241 The Effectiveness of the 119878 Test Job classification in theproposed methodology is based on the combination of FCM(or PCA-FCM) and the 119878 test according to which the bestnumber of categories is chosen This classification methodtakes into account only the similarity of the parameters ofjobs Whether it has a decisive impact for the subsequentcycle time estimation is not clear For this reason the cycletime estimation performances with different numbers ofcategories are compared to verify the results from the 119878 testThe results are shown in Figure 7 119884-axis is provided in alogarithmic scale to make the relationship clearer Clearlywhen the 119878 value becomes smaller the estimation error (interms of MAPE) is also reduced Therefore choosing theclustering results with the smallest 119878 value is helpful to theestimation accuracy

242The Correctness of Job Classification There are absoluterules for the classification of jobs in a wafer fabricationfactory It usually depends on the purpose of job classifica-tion apparently to enhance the estimation accuracy in the


Estimate the cycle time using the BPNs of all categories

Add the estimation errors to the input list

Improvement is negligible

Stop

Yes

No

Reclassify the jobs

Figure 6 The iterative process of dealing with outliers

001

01

10 2 4 6 8

Number of categories

MAPE119878

Figure 7 The relationship between the 119878 value and MAPE

proposedmethodologyTherefore a job is correctly classifiedif its cycle time is accurately estimated after classificationOtherwise the job is misclassified

Definition 1 (job misclassification) Assuming the cycle timeof job 119895 estimated by the BPN of category 119896 is indicated with119900119895(119896)The category of job 119895 determined by classifier119891 is119891(119895)

Then job 119895 is correctly classified if

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)

A strong requirement of inequality (25) is10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)

while a weak requirement of this inequality is10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)

Definition 2 (the correctness of classifying a job) The degreethat job 119895 is correctly classified by classifier 119891 is

120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)

Definition 3 (the correctnesscorrect percentage of the clas-sification results) The correctnesscorrect percentage of theclassification results by classifier 119891 is

120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)

In the illustrative example the correctness of job classificationis evaluated and the results are summarized in Table 10 Inthis example the correctness of the classification results is94

243 Feeding Back the Estimation Error and Reclassifi-cation Subsequently the estimation error is fed back tothe FCM classifier to adjust the classification results Thedifference with Chen and Wangrsquos method [40] is that inthe proposed methodology the BPNs of all categories areapplied to estimate the cycle time of a job [41] and thenthe estimation errors arising from these BPNs all becomeadditional inputs to the FCM and jobs are reclassified Thenew classification results are shown and compared with thatbefore error feedback in Figure 8 After job reclassificationsome outliers are assigned to the existing categories and theoverall estimation performance is improved in this way (seeTable 11) The correctness of job classification 120579(119891) is now97 Job reclassification continues until the improvement inthe overall estimation performance or in the correctness ofjob classification becomes negligible

3 Further Comparisons

To further evaluate the advantages andor disadvantagesof the proposed methodology eight existing approachesstatistical analysis CBR [20] BPN SOM-WM [7] EFR [21]SOM-FBPN [17] the postclassifying FBPN [25] and thebidirectional classifying BPN approach [26] were all applied


Table 10 The correctness of the classification results

119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100

to the collected data Three performance measures includingMAE MAPE and the minimal RMSE were evaluated

The proposed methodology was implemented on a PCwith an Intel Dual CPUE2200 22 GHz and 20GRAM FCM



Table 12 Comparisons of the performances of various approaches

MAE (hours) MAPE RMSE (hours)Statistical analysis 73 61 99CBR 81 65 104BPN 33 28 71SOM-WM 30 25 64EFR 30 26 65SOM-FBPN 22 20 38Postclassifying FBPN 40 27 88Bidirectional classifying BPN 19 19 37The proposed methodology 15 12 36

was implemented with the Fuzzy toolbox ofMATLAB 2006aIn addition BPN was implemented with the Neural NetworkToolbox under the following conditions

(1) Number of epochs per replication 10000

(2) Number of initial conditionsreplications 10

(3) Stop training ifMSElt 10minus6 is satisfied or 10000 epochshave been run

0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number

Before error feedbackAfter error feedback

Figure 8 Comparison of the classification results


Among the steps PCAandFCMcanbe done instantaneouslyThe training of BPN usually takes less than 1 minute perreplication

Theperformanceswith the nine approaches are comparedand summarized in Table 12

In statistical analysis a linear regression equation is usedto estimate the job cycle time In the CBR approach theweights of factors (the cycle times of the previous cases)are proportional to the similarities of the new job with theprevious cases The optimal value of parameter 119896 in the CBRapproach was equal to the value that minimized the RMSE[8] In the BPN approach there was one hidden layer with4sim8 nodes depending on the results of a preliminary analysisfor establishing the best configuration 34 of the collecteddata were used for training the BPN while the remainingdata were used for testing In SOM-FBPN and SOM-WMjobs were first classified with SOM Subsequently the exam-ples of different categories were then learned with differentFBPNs but with the same topology (or WM) In EFR jobsare classified using fuzzy partition In the post-classifyingFBPN approach a job was not pre-classified but ratherpost-classified after the estimation error has been generatedFor this purpose a BPN was used as the postclassificationalgorithm In the bidirectional classifying approach jobs arenot only preclassified but also postclassified The results ofpreclassification and postclassification are aggregated into asuitability index for each job Each job is then assigned to thecategory to which its suitability index is the highest

Statistical analysis was adopted as a comparison basisAccording to experimental results the following points aremade

(1) The combination of BPNandPCAcould reduce about50 of space for storing the input variables in themodeling of the wafer fabrication system

(2) From the effectiveness viewpoint the estimationaccuracy (measured with the MAPE) of the proposedmethodology was significantly better than those ofthe other approaches The average advantage overstatistical analysis is 80

(3) The standard deviation of the cycle time for this caseis 100 hours Compared with this the accuracy of theproposed methodology is good

(4) The estimation performance of the proposedmethod-ology was also better than the existing classifyingmethods such as SOM-WM SOM-FBN EFR SOM-FBPN the postclassifying FBPN and the bidirec-tional classifying BPN approach The advantage ofthe proposed methodology was reasonable due to thereplacement of the variables and the iterative processof dealing with the outliers

(5) In general the performances with the preclassify-ing approaches are better than that with the post-classifying approach

(6) The proposed methodology was also applied toother cases The results are summarized in Table 13

Table 13 Performances in other cases

RMSE Case I Case II Case III Case IVStatistical analysis 77 83 104 78CBR 74 78 96 72BPN 53 68 84 63SOM-WM 49 71 86 66EFR 50 50 62 47SOM-FBPN 38 53 66 50Postclassifying FBPN 62 93 113 86Bidirectional classifying BPN 24 31 38 28The proposed methodology 23 23 28 22

Wilcoxon signed-rank test [42] was then used tomake sure whether or not the differences between theperformance of the proposedmethodology and thoseof the eight existing approaches are significant1198670 When estimating the job cycle time the esti-mating performance of the proposedmethodol-ogy is the same as that of the existing approachbeing compared

1198671 When estimating the job cycle time the esti-mating performance of the proposedmethodol-ogy is better than that of the existing approachbeing compared

The results are summarized in Table 14 The null hypothesis1198670was rejected at 120572 = 005 showing that the proposed

methodology was superior to seven existing approaches inestimating the job cycle time

(7) To ascertain the effect of each treatment taken in theproposed methodology the performances of BPNFCM-BPN PCA-BPN and PCA-FCM-BPN (the pro-posed methodology) are compared in Table 15 Obvi-ously job classification (FCM) did contribute to theeffectiveness of the proposed methodology while theeffect of variable replacement (PCA) was not obviousThe simultaneous application of the two treatmentsfurther improved the estimation accuracy for thetesting data

4 Conclusions and Directions forFuture Research

Estimating the cycle time of each job in a wafer fabricationfactory is a critical task to the wafer fabrication factory andhas been widely studied in recent years In order to furtherenhance the accuracy of the job cycle time estimation PCAis applied to the FCM-BPN approach in this study which isan innovative treatment in this field Through replacing thevariables job classification can bemore accurate In additionthe relationship between the factors and the cycle time can beclearly specified

On the other hand since job classification is the core forthe proposedmethodology a new index is used to validate theclassification of jobs The empirical relationship between the119878 value and the estimation performance is also found Finally


Table 14 Results of the Wilcoxon sign-rank test

1198670

Statistical analysis 119885 = 183lowast

CBR 183lowast

BPN 183lowast

SOM-WM 183lowast

EFR 164SOM-FBPN 183lowast

Postclassifying FBPN 183lowast

Bidirectional classifying BPN 183lowastlowastP lt 005lowastlowastP lt 0025lowastlowastlowastP lt 001

Table 15 The effects of the treatments taken in the proposedmethodology

RMSE (hrs) BPN FCM-BPN PCA-BPN PCA-FCM-BPNTraining data 73 58 71 36

an iterative process is established to deal with the outliers tooptimize the overall estimation performance

An example is used to illustrate the proposed methodol-ogy According to the experimental results

(1) the estimation accuracy (measured with MAEMAPE and RMSE) using the proposed methodologywas significantly better than those with the existingapproaches

(2) the advantage of PCA is for improving the correctnessof job classification The simple combination of PCAand BPN does not show much advantage

(3) after combining with PCA the estimation accuracy ofFCM-BPN was significantly improved

(4) the overall estimation performance is often affectedby the outliersThe iterative procedure tries to removethe outliers and gradually improves the overall esti-mation performance

Some other issues for this topic can be further investi-gated Most of the existing methods are based on the jobclusteringThe aim of this study is to provide positive impactson certain measures for these methods However if thereare the other variable replacement techniques that can be aseffective is also worth exploring in future studies In additionthe iterative procedure used to optimize the results of job clas-sification is quite time consuming especially for a large-scaleproblem and therefore a more efficient way should be found

Acknowledgment

This work was supported by the National Science Council ofTaiwan

References

[1] 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

[2] T Chen Y C Wang and H C Wu ldquoA fuzzy-neural approachfor remaining cycle time estimation in a semiconductor man-ufacturing factory a simulation studyrdquo International Journal ofInnovative Computing Information and Control vol 5 no 8 pp2125ndash2139 2009

[3] T Chen and Y C Lin ldquoA fuzzy back propagation networkensemble with example classification for lot output time pre-diction in a wafer fabrdquo Applied Soft Computing Journal vol 9no 2 pp 658ndash666 2009

[4] T Chen ldquoA fuzzy-neural and multiple-bucket approach forestimating lot cycle time in a wafer fab with dynamic productmixrdquo Computers and Industrial Engineering vol 55 pp 423ndash438 2008

[5] T Chen ldquoA hybrid fuzzy-neural approach to job completiontime prediction in a semiconductor fabrication factoryrdquo Neu-rocomputing vol 71 no 16ndash18 pp 3193ndash3201 2008

[6] T Chen ldquoA SOM-FBPN-ensemble approach with error feed-back to adjust classification for wafer-lot completion timepredictionrdquo International Journal of Advanced ManufacturingTechnology vol 37 no 7-8 pp 782ndash792 2008

[7] T Chen ldquoA hybrid SOM-BPN approach to lot output timeprediction in a wafer fabrdquo Neural Processing Letters vol 24 no3 pp 271ndash288 2006

[8] P C Chang J C Hieh and TW Liao ldquoEvolving fuzzy rules fordue-date assignment problem in semiconductormanufacturingfactoryrdquo Journal of IntelligentManufacturing vol 16 no 4-5 pp549ndash557 2005

[9] T Chen ldquoA fuzzy back propagation network for output timeprediction in a wafer fabrdquo Applied Soft Computing Journal vol2 no 3 pp 211ndash222 2003

[10] T Chen ldquoA fuzzy-neural DBD approach for job scheduling ina wafer fabrication factoryrdquo International Journal of InnovativeComputing Information and Control vol 8 no 6 pp 4024ndash4044 2012

[11] S L Yang M Liu and L Li ldquoForecasting of productioncycle of engineer-to-order productsrdquo in Proceedings of the IEEE18th International Conference on Industrial Engineering andEngineering Management pp 510ndash513 2011

[12] T Chen ldquoA fuzzy-neural knowledge-based system for jobcompletion time prediction and internal due date assignmentin a wafer fabrication plantrdquo International Journal of SystemsScience vol 40 no 8 pp 889ndash902 2009

[13] W L Pearn S L Chung and C M Lai ldquoDue-date assignmentfor wafer fabrication under demand variate environmentrdquo IEEETransactions on SemiconductorManufacturing vol 20 no 2 pp165ndash175 2007

[14] C F Chien C Y Hsu and C W Hsiao ldquoManufacturing intelli-gence to forecast and reduce semiconductor cycle timerdquo Journalof Intelligent Manufacturing vol 23 no 6 pp 2281ndash2294 2011

[15] 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 TheoryApplications andPractice vol 10 no 1 pp 55ndash61 2003

[16] D Y Sha and S Y Hsu ldquoDue-date assignment in wafer fabri-cation using artificial neural networksrdquo International Journal of


Advanced Manufacturing Technology vol 23 no 9-10 pp 768ndash775 2004

[17] T Chen ldquoAn intelligent hybrid system for wafer lot output timepredictionrdquo Advanced Engineering Informatics vol 21 no 1 pp55ndash65 2007

[18] 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

[19] T Beeg ldquoWafer fab cycle time forecast under changing loadingsituationsrdquo in Proceedings of the IEEE Conference andWorkshopon Advanced Semiconductor Manufacturing pp 339ndash343 May2004

[20] C Chiu P C Chang and N H Chiu ldquoA case-based expertsupport system for due-date assignment in a wafer fabricationfactoryrdquo Journal of IntelligentManufacturing vol 14 no 3-4 pp287ndash296 2003

[21] L X Wang and J M Mendel ldquoGenerating fuzzy rules bylearning from examplesrdquo IEEE Transactions on Systems Manand Cybernetics vol 22 no 6 pp 1414ndash1427 1992

[22] T Chen and Y C Wang ldquoIncorporating the FCM-BPNapproach with nonlinear programming for internal duedate assignment in a wafer fabrication plantrdquo Robotics andComputer-Integrated Manufacturing vol 26 no 1 pp 83ndash912010

[23] TChenA Jeang andYCWang ldquoAhybrid neural network andselective allowance approach for internal due date assignmentin a wafer fabrication plantrdquo International Journal of AdvancedManufacturing Technology vol 36 no 5-6 pp 570ndash581 2008

[24] T Chen ldquoIncorporating fuzzy c-means and a back-propagationnetwork ensemble to job completion time prediction in asemiconductor fabrication factoryrdquo Fuzzy Sets and Systems vol158 no 19 pp 2153ndash2168 2007

[25] 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

[26] T Chen ldquoJob cycle time estimation in a wafer fabricationfactory with a bi-directional classifying fuzzy-neural approachrdquoInternational Journal of Advanced Manufacturing Technologyvol 56 no 9ndash12 pp 1007ndash1018 2011

[27] T Chen ldquoEvaluating themid-term competitiveness of a productin a semiconductor fabrication factory with a systematic proce-durerdquo Computers and Industrial Engineering vol 53 no 3 pp499ndash513 2007

[28] T Chen ldquoA PCA-FBPN approach for job cycle time estimationin a wafer fabrication factoryrdquo International Journal of FuzzySystem Applications vol 2 no 2 pp 50ndash67 2012

[29] X He and Q He ldquoApplication of PCA method and FCM clus-tering to the fault diagnosis of excavatorrsquos hydraulic systemrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo07) pp 1635ndash1639 August 2007

[30] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991

[31] H-C Chang G-S Liang C-W Chu and C-H Chou ldquoPri-oritizing service attributes for improvement using fuzzy zoneof tolerancerdquo International Journal of Innovative ComputingInformation and Control vol 8 no 1 pp 75ndash89 2012

[32] X Su P Shi L Wu and Y D Song ldquoA novel control design ondiscrete-time Takagi-Sugeno fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems In press

[33] X Su P Shi L Wu and Y-D Song ldquoA novel approach to filterdesign for T-S fuzzy discrete-time systems with time-varyingdelayrdquo IEEETransactions on Fuzzy Systems vol 20 no 6 ArticleID 6189779 pp 1114ndash1129 2012

[34] L Wu X Su P Shi and J Qiu ldquoModel approximation fordiscrete-time state-delay systems in the T-S fuzzy frameworkrdquoIEEE Transactions on Fuzzy Systems vol 19 no 2 pp 366ndash3782011

[35] L Wu and W X Zheng ldquoL2-Linfin control of nonlinear fuzzyito stochastic delay systems via dynamic output feedbackrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 39 no 5 pp 1308ndash1315 2009

[36] T Niknam H D Mojarrad and M Nayeripour ldquoA newhybrid fuzzy adaptive particle swarm optimization for non-convex economic dispatchrdquo International Journal of InnovativeComputing Information and Control vol 7 no 1 pp 189ndash2022011

[37] S Y Cho C W Ting and C Quek ldquoThermal facial patternrecognition for personal verification using fuzzy cmac modelrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 1 pp 203ndash222 2011

[38] R Yang Z Zhang and P Shi ldquoExponential stability on stochas-tic neural networks with discrete interval and distributeddelaysrdquo IEEE Transactions on Neural Networks vol 21 no 1 pp169ndash175 2010

[39] R Yang H Gao and P Shi ldquoNovel robust stability criteria forstochastic Hopfield neural networks with time delaysrdquo IEEETransactions on Systems Man and Cybernetics B vol 39 no 11pp 467ndash474 2009

[40] T Chen and Y C Wang ldquoA fuzzy-neural system with errorfeedback to adjust classification for forecasting wafer lot flowtime a simulation studyrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 221 no 5 pp 807ndash817 2007

[41] T Chen ldquoA collaborative fuzzy-neural system for global CO2

concentration forecastingrdquo International Journal of InnovativeComputing Information and Control vol 8 no 11 pp 7679ndash7696 2012

[42] F Wilcoxon ldquoIndividual comparisons by ranking methodsrdquoBiometrics Bulletin vol 1 no 6 pp 80ndash83 1945

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a gamma distribution and then used a linear equation toestimate the job cycle time Recently Chien et al [14] usednonlinear regression equations instead and then found outthe relationship between the estimation error and somefactory conditions and job attributes with a BPN to furtherimprove the estimation accuracy The major disadvantageof statistical analysis is the lack of estimation accuracy [9]Conversely a huge amount of data and lengthy simulationtime required are two disadvantages of PS Neverthelesstheoretically PS is themost accurate job cycle time estimationapproach if the simulation model is completely valid and iscontinuously updated

Considering effectiveness (estimation accuracy) and effi-ciency (execution time) simultaneously Chang et al [8]Chang and Hsieh [15] and Sha and Hsu [16] estimated thecycle time of a job in a wafer fabrication factory using aBPN with a single hidden layer A BPN is an effective toolin modeling complex physical systems described by sets ofdifferent equations for prediction control and design pur-poses Compared with some statistical analysis approachesthe average estimation accuracy measured with root meansquared error (RMSE) was considerably improved with theBPNs For example an improvement of about 40 in RMSEwas achieved in the study of Chang et al [8] Chen [17]incorporated the job releasing plan of the wafer fabricationfactory into a BPN and constructed a ldquolook-aheadrdquo BPN forthe same purpose which led to an average reduction of 12in RMSE On the other hand much less time and fewer dataare required with a BPN than with PS Chen et al [18] andBeeg [19] estimated the cycle time of a job in a ramping upwafer fabrication factory In their studies Chen et al used aBPN-based method while Beeg tried to find out the impactof utilization for the cycle time

Chiu et al [20] established an expert system based onCBR for the job cycle time estimation To effectively considerthe uncertainty in the job cycle time fuzzy logic was used ina number of studies For example Chang et al [8] modifiedthe first step (ie partitioning the range of each input variableinto several fuzzy intervals) of the fuzzy modeling methodproposed by Wang and Mendel [21] called the WMmethodwith a simple genetic algorithm (GA) and proposed theevolving fuzzy rule (EFR) approach to estimate the cycle timeof a job in a wafer fabrication factory Their EFR approachoutperformed CBR and BPN in the estimation accuracyChen [9] constructed a fuzzy back propagation network(FBPN) that incorporated expert opinions to modify theinputs of the FBPN Chenrsquos FBPN surpassed the crisp BPNespecially with respect to efficiency

In recent years a number of hybrid approaches have beenproposed most of which classified jobs before estimatingthe cycle times For example Chen [7] combined self-organization map (SOM) and WM in which jobs wereclassified using a SOM before estimating the cycle timesof the jobs with WM Chen and Wang [22] constructed alook-ahead k-means- (kM-) FBPN for the same purpose anddiscussed in detail the effects of using different look-aheadfunctions More recently Chen [17] proposed the look-aheadSOM-FBPN approach for the job cycle time estimation in asemiconductor factory [23] Besides a set of fuzzy inference

rules were also developed to evaluate the achievability of acycle time forecast Subsequently Chen [24] added a selectiveallowance to the cycle time estimated using the look-aheadSOM-FBPN approach to determine the intermediate duedate Further Chen et al [23] showed that the suitabilityof combining the SOM and FBPN for the data could beimproved with the feedback of the estimation error by theFBPN to adjust the classification results of the SOM Chen etal [25] proposed a postclassification fuzzy-neural approachin which a job was not pre-classified but rather postclassifiedafter estimating the cycle time Experimental results showedthat the postclassification approach was better than thepreclassification approaches in certain cases In order tocombine the advantages of preclassifying and post-classifyingapproaches Chen [26] proposed a bi-directional classifyingapproach in which jobs are not only pre-classified but alsopostclassified Except few studies in which the historical dataof a real semiconductor factory were collected most studiesin this field used simulated data [27]

In short the followings have not done before

(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 How-ever this issue has rarely been addressed in previousstudies of this field

(2) Job classification has been shown to be conduciveto the estimation performance However most paststudies chose classifiers subjectively and did not eval-uate the performance of the classifier Needless to sayoptimizing the classifier for the subsequent estima-tion task

Principal component analysis (PCA) is a multivariate sta-tistical analysis method This method constructs a series oflinear combinations of the original variables to form a newvariable so that these new variables are unrelated to eachother as much as possible and the relationship among themcan be reflected in a better way In this study a fuzzy-neuralapproach based on the combination of PCA FCM and BPNis proposed to estimate the cycle time of a job in a waferfabrication factory The motivation of this study is explainedas follows

(1) While in the past some studies combined PCA andFCM the references on the combination of PCAFCM and BPN are still very limited Chen [28]applied PCA to modify the inputs to a BPN for thejob cycle time estimation The estimation accuracyof PCA-BPN was slightly better than that of BPNalone It seems that BPN can solve the dependenciesof the input variables for the job cycle time estimationproblems PCA seems to be more important forthe classification of jobs This provides us with amotivation to improve the existing job cycle timeestimation methods based on job classification

(2) FCM as a part of the preclassifying approach can-not be evaluated alone Its success depends on the












2 Methodology







119909lowast

119895119894=119909119895119894minus 119909119894

120590119894

119909119894=

sum119899

119895=1119909119895119894

119899

120590119894=radicsum119899

119895=1(119909119895119894minus 119909119894)2

119899 minus 1

(1)


119894



119877 =1

119899 minus 1119883lowast119879

119883lowast

(2)



1sim 120582119898and 119906


1205821ge 1205822ge ge 120582

119898


120578119902=

120582119902

sum119898

119903=1120582119903

sdot 100 (3)





Job data PCA

FCM

Category 1examples

examples

1

2

1

2

1

1

2

1

2

1

Outliers

⋯

⋮⋮

⋮⋮

119873(119862119879119895)

119873(119862119879119895)

119890119895

119890119895

119900119895

119900119895

119901

119901

119911119901

119911119901

119911119901

1199111

1199111

1199111

1199112

1199112

1199112

119909119898

1199091

1199092

2119901

2119901

Category 119896



120578Σ(119901) =

119901

sum

119902=1

120578119902 (4)




119880119898times119901

= [1199061 1199062 119906

119901]

119885119899times119901

= 119883lowast

119899times119898119880119898times119901

(5)

119885119899times119901

= [119911119895119902] (119895 = 1 sim 119899 119902 = 1 sim 119901) is the



Table 2 An example

119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956

1 24 1261 181 781 112 0922 24 1263 181 762 127 0903 24 1220 176 761 127 0894 23 1282 178 802 127 0945 23 1303 180 780 175 0936 23 1281 183 782 175 0937 23 1242 184 741 163 0898 24 1262 182 681 139 0869 22 1260 182 701 98 08610 22 1260 179 700 257 08711 24 1301 163 722 99 08412 22 1221 184 641 131 08213 23 1323 159 740 247 08714 24 1362 181 782 191 09515 24 1261 181 762 219 09116 23 1321 177 801 219 09617 22 1343 180 822 219 09718 24 1321 177 762 54 09319 25 1343 179 781 54 09620 25 1300 180 740 54 09221 22 1320 181 721 54 09122 24 1321 182 742 49 09223 23 1262 165 680 201 08024 22 1240 161 722 103 08225 23 1183 183 661 53 08226 23 1282 184 701 53 08827 22 1202 177 680 248 08428 23 1202 178 681 248 08529 24 1202 185 701 82 08630 23 1202 158 721 98 08131 24 1343 181 760 67 09432 24 1381 185 801 67 09733 22 1362 156 780 67 09134 23 1282 179 782 223 09235 23 1320 180 782 176 09336 25 1340 176 801 462 09737 23 1320 182 781 168 09538 22 1361 181 781 141 09439 22 1381 179 781 95 09740 23 1363 178 802 179 097


1198952 1199091198954 and 119909

1198955than in the



119877 =

[[[[[[[

[

097 010 016 021 minus003 025

010 098 001 070 minus001 078

016 001 098 005 minus007 037

021 070 005 098 015 086


025 078 037 086 010 098

]]]]]]]

]

(6)

0 200 400 600 800 1000 1200 1400Values

1199091198956

1199091198955

1199091198954

1199091198953

1199091198952

1199091198951



1205821= 266 120582

2= 115

1205823= 094 120582

4= 083

1205825= 025 120582

6= 002

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

]]]]]]]

]

(7)


1205781= 46 120578

2= 20 120578

3= 16

1205784= 14 120578

5= 4 120578

6= 0

(8)



119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956



120578Σ(1) = 46 120578

Σ(2) = 65

120578Σ(3) = 81 120578

Σ(4) = 95

120578Σ(5) = 100 120578

Σ(6) = 100

(9)


1 2 3 40

102030405060708090

100

Principal component

0102030405060708090100

Varia

ncee

xpla

ined

()

()


0 1 2 3 4

0

1

2


2nd

prin

cipa

l com

pone

nt

15

05

minus05

minus1

minus15

minus2

minus25




Σ(119901) ge 85 sim






1199111198951

1199111198952

1199111198953




Min119870

sum

119896=1

119899

sum

119895=1

120583119898

119895(119896)1198902

119895(119896) (10)



to category 119896 119890119895(119896)



the following

119911(119896)= 119911(119896)119902 119896 = 1 sim 119870

119911(119896)119902

=

sum119899

119895=1120583119898

119895(119896)119911119895119902

sum119899

119895=1120583119898

119895(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

(119911119895119901minus 119911(119896)119901)2

119895 = 1 sim 119899 119896 = 1 sim 119870

(11)

where 119911(119896)





max119896

max119895

10038161003816100381610038161003816120583(119905)

119895(119896)minus 120583(119905minus1)

119895(119896)

10038161003816100381610038161003816lt 119889 (12)




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)119902minus 119911(1198962)119902)2

)

119878 =119869119898

119899 times 1198902

min

119870 isin 119885+

(14)


A=[03857 7175 05381 01614 04281 05803]

c=2


Jm=min (obj fun)

e2 min=9999

for i=1 c

for j=i+1 c

e2 sum=0

for k=1 3


end

if e2 sumlte2 min

e2 min=e2 sum

end

end

end

e2 min





1198902

min 119878

2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043


Category Jobs1 1 2 18 19 20 22 31 322 3 7 8 9 12 25 26 293 4 5 6 14 16 17 34 35 37 38 39 404 10 11 23 24 27 30


Category Jobs1 1 2 18 19 20 21 22 31 322 2 3 7 8 9 12 25 26 28 293 4 5 6 14 15 16 17 33 34 35 36 37 38 39 404 10 11 13 23 24 27 28 30 33
















0

1

2

3

4

5

0 10 20 30 40

Cate

gory

FCM

Job number

PCA-FCM







ℎ119895119897=

1

1 + 119890minus119899ℎ

119895119897

(15)

where

119899ℎ

119895119897= 119868ℎ

119895119897minus 120579ℎ

119897

119868ℎ

119895119897=

119901

sum

119902=1

119908ℎ

119902119897119911119895119902

(16)



119900119895=

1

1 + 119890minus119899119900

119895

(17)

where

119899119900

119895= 119868119900

119895minus 120579119900

119868119900

119895=

119871

sum

119897=1

119908119900

119897ℎ119895119897

(18)




119895=1(119900119895minus 119873(119862119879

119895))2

119899

(19)



11 119908

ℎ

119901119871 120579ℎ

1 120579ℎ119871 1199081199001 119908

119900





SSE (120573) =119899

sum

119895=1

(119873(119862119879119895) minus 119891 (x

119895120573))2

(20)



ℎ

11

Δ119908ℎ

119901119871 Δ120579ℎ1 Δ120579

ℎ

119871 Δ1199081199001 Δ119908119900



as

119891 (x119895120573 + 120575) asymp 119891 (x

119895120573) + J

119895120575 (21)

where

J119895= 120597

119891 (x119895120573)

120597120573(22)


SSE (120573+120575) asymp119899

sum

119895=1

(119873 (119862119879119895) minus 119891 (x

119895120573) minus J

119895120575)2

(23)

















(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)



Single hidden layer



have been run














Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















2 Methodology







119909lowast

119895119894=119909119895119894minus 119909119894

120590119894

119909119894=

sum119899

119895=1119909119895119894

119899

120590119894=radicsum119899

119895=1(119909119895119894minus 119909119894)2

119899 minus 1

(1)


119894



119877 =1

119899 minus 1119883lowast119879

119883lowast

(2)



1sim 120582119898and 119906


1205821ge 1205822ge ge 120582

119898


120578119902=

120582119902

sum119898

119903=1120582119903

sdot 100 (3)





Job data PCA

FCM

Category 1examples

examples

1

2

1

2

1

1

2

1

2

1

Outliers

⋯

⋮⋮

⋮⋮

119873(119862119879119895)

119873(119862119879119895)

119890119895

119890119895

119900119895

119900119895

119901

119901

119911119901

119911119901

119911119901

1199111

1199111

1199111

1199112

1199112

1199112

119909119898

1199091

1199092

2119901

2119901

Category 119896



120578Σ(119901) =

119901

sum

119902=1

120578119902 (4)




119880119898times119901

= [1199061 1199062 119906

119901]

119885119899times119901

= 119883lowast

119899times119898119880119898times119901

(5)

119885119899times119901

= [119911119895119902] (119895 = 1 sim 119899 119902 = 1 sim 119901) is the



Table 2 An example

119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956

1 24 1261 181 781 112 0922 24 1263 181 762 127 0903 24 1220 176 761 127 0894 23 1282 178 802 127 0945 23 1303 180 780 175 0936 23 1281 183 782 175 0937 23 1242 184 741 163 0898 24 1262 182 681 139 0869 22 1260 182 701 98 08610 22 1260 179 700 257 08711 24 1301 163 722 99 08412 22 1221 184 641 131 08213 23 1323 159 740 247 08714 24 1362 181 782 191 09515 24 1261 181 762 219 09116 23 1321 177 801 219 09617 22 1343 180 822 219 09718 24 1321 177 762 54 09319 25 1343 179 781 54 09620 25 1300 180 740 54 09221 22 1320 181 721 54 09122 24 1321 182 742 49 09223 23 1262 165 680 201 08024 22 1240 161 722 103 08225 23 1183 183 661 53 08226 23 1282 184 701 53 08827 22 1202 177 680 248 08428 23 1202 178 681 248 08529 24 1202 185 701 82 08630 23 1202 158 721 98 08131 24 1343 181 760 67 09432 24 1381 185 801 67 09733 22 1362 156 780 67 09134 23 1282 179 782 223 09235 23 1320 180 782 176 09336 25 1340 176 801 462 09737 23 1320 182 781 168 09538 22 1361 181 781 141 09439 22 1381 179 781 95 09740 23 1363 178 802 179 097


1198952 1199091198954 and 119909

1198955than in the



119877 =

[[[[[[[

[

097 010 016 021 minus003 025

010 098 001 070 minus001 078

016 001 098 005 minus007 037

021 070 005 098 015 086


025 078 037 086 010 098

]]]]]]]

]

(6)

0 200 400 600 800 1000 1200 1400Values

1199091198956

1199091198955

1199091198954

1199091198953

1199091198952

1199091198951



1205821= 266 120582

2= 115

1205823= 094 120582

4= 083

1205825= 025 120582

6= 002

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

]]]]]]]

]

(7)


1205781= 46 120578

2= 20 120578

3= 16

1205784= 14 120578

5= 4 120578

6= 0

(8)



119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956



120578Σ(1) = 46 120578

Σ(2) = 65

120578Σ(3) = 81 120578

Σ(4) = 95

120578Σ(5) = 100 120578

Σ(6) = 100

(9)


1 2 3 40

102030405060708090

100

Principal component

0102030405060708090100

Varia

ncee

xpla

ined

()

()


0 1 2 3 4

0

1

2


2nd

prin

cipa

l com

pone

nt

15

05

minus05

minus1

minus15

minus2

minus25




Σ(119901) ge 85 sim






1199111198951

1199111198952

1199111198953




Min119870

sum

119896=1

119899

sum

119895=1

120583119898

119895(119896)1198902

119895(119896) (10)



to category 119896 119890119895(119896)



the following

119911(119896)= 119911(119896)119902 119896 = 1 sim 119870

119911(119896)119902

=

sum119899

119895=1120583119898

119895(119896)119911119895119902

sum119899

119895=1120583119898

119895(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

(119911119895119901minus 119911(119896)119901)2

119895 = 1 sim 119899 119896 = 1 sim 119870

(11)

where 119911(119896)





max119896

max119895

10038161003816100381610038161003816120583(119905)

119895(119896)minus 120583(119905minus1)

119895(119896)

10038161003816100381610038161003816lt 119889 (12)




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)119902minus 119911(1198962)119902)2

)

119878 =119869119898

119899 times 1198902

min

119870 isin 119885+

(14)


A=[03857 7175 05381 01614 04281 05803]

c=2


Jm=min (obj fun)

e2 min=9999

for i=1 c

for j=i+1 c

e2 sum=0

for k=1 3


end

if e2 sumlte2 min

e2 min=e2 sum

end

end

end

e2 min





1198902

min 119878

2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043


Category Jobs1 1 2 18 19 20 22 31 322 3 7 8 9 12 25 26 293 4 5 6 14 16 17 34 35 37 38 39 404 10 11 23 24 27 30


Category Jobs1 1 2 18 19 20 21 22 31 322 2 3 7 8 9 12 25 26 28 293 4 5 6 14 15 16 17 33 34 35 36 37 38 39 404 10 11 13 23 24 27 28 30 33
















0

1

2

3

4

5

0 10 20 30 40

Cate

gory

FCM

Job number

PCA-FCM







ℎ119895119897=

1

1 + 119890minus119899ℎ

119895119897

(15)

where

119899ℎ

119895119897= 119868ℎ

119895119897minus 120579ℎ

119897

119868ℎ

119895119897=

119901

sum

119902=1

119908ℎ

119902119897119911119895119902

(16)



119900119895=

1

1 + 119890minus119899119900

119895

(17)

where

119899119900

119895= 119868119900

119895minus 120579119900

119868119900

119895=

119871

sum

119897=1

119908119900

119897ℎ119895119897

(18)




119895=1(119900119895minus 119873(119862119879

119895))2

119899

(19)



11 119908

ℎ

119901119871 120579ℎ

1 120579ℎ119871 1199081199001 119908

119900





SSE (120573) =119899

sum

119895=1

(119873(119862119879119895) minus 119891 (x

119895120573))2

(20)



ℎ

11

Δ119908ℎ

119901119871 Δ120579ℎ1 Δ120579

ℎ

119871 Δ1199081199001 Δ119908119900



as

119891 (x119895120573 + 120575) asymp 119891 (x

119895120573) + J

119895120575 (21)

where

J119895= 120597

119891 (x119895120573)

120597120573(22)


SSE (120573+120575) asymp119899

sum

119895=1

(119873 (119862119879119895) minus 119891 (x

119895120573) minus J

119895120575)2

(23)

















(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)



Single hidden layer



have been run














Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Job data PCA

FCM

Category 1examples

examples

1

2

1

2

1

1

2

1

2

1

Outliers

⋯

⋮⋮

⋮⋮

119873(119862119879119895)

119873(119862119879119895)

119890119895

119890119895

119900119895

119900119895

119901

119901

119911119901

119911119901

119911119901

1199111

1199111

1199111

1199112

1199112

1199112

119909119898

1199091

1199092

2119901

2119901

Category 119896



120578Σ(119901) =

119901

sum

119902=1

120578119902 (4)




119880119898times119901

= [1199061 1199062 119906

119901]

119885119899times119901

= 119883lowast

119899times119898119880119898times119901

(5)

119885119899times119901

= [119911119895119902] (119895 = 1 sim 119899 119902 = 1 sim 119901) is the



Table 2 An example

119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956

1 24 1261 181 781 112 0922 24 1263 181 762 127 0903 24 1220 176 761 127 0894 23 1282 178 802 127 0945 23 1303 180 780 175 0936 23 1281 183 782 175 0937 23 1242 184 741 163 0898 24 1262 182 681 139 0869 22 1260 182 701 98 08610 22 1260 179 700 257 08711 24 1301 163 722 99 08412 22 1221 184 641 131 08213 23 1323 159 740 247 08714 24 1362 181 782 191 09515 24 1261 181 762 219 09116 23 1321 177 801 219 09617 22 1343 180 822 219 09718 24 1321 177 762 54 09319 25 1343 179 781 54 09620 25 1300 180 740 54 09221 22 1320 181 721 54 09122 24 1321 182 742 49 09223 23 1262 165 680 201 08024 22 1240 161 722 103 08225 23 1183 183 661 53 08226 23 1282 184 701 53 08827 22 1202 177 680 248 08428 23 1202 178 681 248 08529 24 1202 185 701 82 08630 23 1202 158 721 98 08131 24 1343 181 760 67 09432 24 1381 185 801 67 09733 22 1362 156 780 67 09134 23 1282 179 782 223 09235 23 1320 180 782 176 09336 25 1340 176 801 462 09737 23 1320 182 781 168 09538 22 1361 181 781 141 09439 22 1381 179 781 95 09740 23 1363 178 802 179 097


1198952 1199091198954 and 119909

1198955than in the



119877 =

[[[[[[[

[

097 010 016 021 minus003 025

010 098 001 070 minus001 078

016 001 098 005 minus007 037

021 070 005 098 015 086


025 078 037 086 010 098

]]]]]]]

]

(6)

0 200 400 600 800 1000 1200 1400Values

1199091198956

1199091198955

1199091198954

1199091198953

1199091198952

1199091198951



1205821= 266 120582

2= 115

1205823= 094 120582

4= 083

1205825= 025 120582

6= 002

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

]]]]]]]

]

(7)


1205781= 46 120578

2= 20 120578

3= 16

1205784= 14 120578

5= 4 120578

6= 0

(8)



119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956



120578Σ(1) = 46 120578

Σ(2) = 65

120578Σ(3) = 81 120578

Σ(4) = 95

120578Σ(5) = 100 120578

Σ(6) = 100

(9)


1 2 3 40

102030405060708090

100

Principal component

0102030405060708090100

Varia

ncee

xpla

ined

()

()


0 1 2 3 4

0

1

2


2nd

prin

cipa

l com

pone

nt

15

05

minus05

minus1

minus15

minus2

minus25




Σ(119901) ge 85 sim






1199111198951

1199111198952

1199111198953




Min119870

sum

119896=1

119899

sum

119895=1

120583119898

119895(119896)1198902

119895(119896) (10)



to category 119896 119890119895(119896)



the following

119911(119896)= 119911(119896)119902 119896 = 1 sim 119870

119911(119896)119902

=

sum119899

119895=1120583119898

119895(119896)119911119895119902

sum119899

119895=1120583119898

119895(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

(119911119895119901minus 119911(119896)119901)2

119895 = 1 sim 119899 119896 = 1 sim 119870

(11)

where 119911(119896)





max119896

max119895

10038161003816100381610038161003816120583(119905)

119895(119896)minus 120583(119905minus1)

119895(119896)

10038161003816100381610038161003816lt 119889 (12)




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)119902minus 119911(1198962)119902)2

)

119878 =119869119898

119899 times 1198902

min

119870 isin 119885+

(14)


A=[03857 7175 05381 01614 04281 05803]

c=2


Jm=min (obj fun)

e2 min=9999

for i=1 c

for j=i+1 c

e2 sum=0

for k=1 3


end

if e2 sumlte2 min

e2 min=e2 sum

end

end

end

e2 min





1198902

min 119878

2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043


Category Jobs1 1 2 18 19 20 22 31 322 3 7 8 9 12 25 26 293 4 5 6 14 16 17 34 35 37 38 39 404 10 11 23 24 27 30


Category Jobs1 1 2 18 19 20 21 22 31 322 2 3 7 8 9 12 25 26 28 293 4 5 6 14 15 16 17 33 34 35 36 37 38 39 404 10 11 13 23 24 27 28 30 33
















0

1

2

3

4

5

0 10 20 30 40

Cate

gory

FCM

Job number

PCA-FCM







ℎ119895119897=

1

1 + 119890minus119899ℎ

119895119897

(15)

where

119899ℎ

119895119897= 119868ℎ

119895119897minus 120579ℎ

119897

119868ℎ

119895119897=

119901

sum

119902=1

119908ℎ

119902119897119911119895119902

(16)



119900119895=

1

1 + 119890minus119899119900

119895

(17)

where

119899119900

119895= 119868119900

119895minus 120579119900

119868119900

119895=

119871

sum

119897=1

119908119900

119897ℎ119895119897

(18)




119895=1(119900119895minus 119873(119862119879

119895))2

119899

(19)



11 119908

ℎ

119901119871 120579ℎ

1 120579ℎ119871 1199081199001 119908

119900





SSE (120573) =119899

sum

119895=1

(119873(119862119879119895) minus 119891 (x

119895120573))2

(20)



ℎ

11

Δ119908ℎ

119901119871 Δ120579ℎ1 Δ120579

ℎ

119871 Δ1199081199001 Δ119908119900



as

119891 (x119895120573 + 120575) asymp 119891 (x

119895120573) + J

119895120575 (21)

where

J119895= 120597

119891 (x119895120573)

120597120573(22)


SSE (120573+120575) asymp119899

sum

119895=1

(119873 (119862119879119895) minus 119891 (x

119895120573) minus J

119895120575)2

(23)

















(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)



Single hidden layer



have been run














Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2 An example

119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956

1 24 1261 181 781 112 0922 24 1263 181 762 127 0903 24 1220 176 761 127 0894 23 1282 178 802 127 0945 23 1303 180 780 175 0936 23 1281 183 782 175 0937 23 1242 184 741 163 0898 24 1262 182 681 139 0869 22 1260 182 701 98 08610 22 1260 179 700 257 08711 24 1301 163 722 99 08412 22 1221 184 641 131 08213 23 1323 159 740 247 08714 24 1362 181 782 191 09515 24 1261 181 762 219 09116 23 1321 177 801 219 09617 22 1343 180 822 219 09718 24 1321 177 762 54 09319 25 1343 179 781 54 09620 25 1300 180 740 54 09221 22 1320 181 721 54 09122 24 1321 182 742 49 09223 23 1262 165 680 201 08024 22 1240 161 722 103 08225 23 1183 183 661 53 08226 23 1282 184 701 53 08827 22 1202 177 680 248 08428 23 1202 178 681 248 08529 24 1202 185 701 82 08630 23 1202 158 721 98 08131 24 1343 181 760 67 09432 24 1381 185 801 67 09733 22 1362 156 780 67 09134 23 1282 179 782 223 09235 23 1320 180 782 176 09336 25 1340 176 801 462 09737 23 1320 182 781 168 09538 22 1361 181 781 141 09439 22 1381 179 781 95 09740 23 1363 178 802 179 097


1198952 1199091198954 and 119909

1198955than in the



119877 =

[[[[[[[

[

097 010 016 021 minus003 025

010 098 001 070 minus001 078

016 001 098 005 minus007 037

021 070 005 098 015 086


025 078 037 086 010 098

]]]]]]]

]

(6)

0 200 400 600 800 1000 1200 1400Values

1199091198956

1199091198955

1199091198954

1199091198953

1199091198952

1199091198951



1205821= 266 120582

2= 115

1205823= 094 120582

4= 083

1205825= 025 120582

6= 002

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

]]]]]]]

]

(7)


1205781= 46 120578

2= 20 120578

3= 16

1205784= 14 120578

5= 4 120578

6= 0

(8)



119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956



120578Σ(1) = 46 120578

Σ(2) = 65

120578Σ(3) = 81 120578

Σ(4) = 95

120578Σ(5) = 100 120578

Σ(6) = 100

(9)


1 2 3 40

102030405060708090

100

Principal component

0102030405060708090100

Varia

ncee

xpla

ined

()

()


0 1 2 3 4

0

1

2


2nd

prin

cipa

l com

pone

nt

15

05

minus05

minus1

minus15

minus2

minus25




Σ(119901) ge 85 sim






1199111198951

1199111198952

1199111198953




Min119870

sum

119896=1

119899

sum

119895=1

120583119898

119895(119896)1198902

119895(119896) (10)



to category 119896 119890119895(119896)



the following

119911(119896)= 119911(119896)119902 119896 = 1 sim 119870

119911(119896)119902

=

sum119899

119895=1120583119898

119895(119896)119911119895119902

sum119899

119895=1120583119898

119895(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

(119911119895119901minus 119911(119896)119901)2

119895 = 1 sim 119899 119896 = 1 sim 119870

(11)

where 119911(119896)





max119896

max119895

10038161003816100381610038161003816120583(119905)

119895(119896)minus 120583(119905minus1)

119895(119896)

10038161003816100381610038161003816lt 119889 (12)




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)119902minus 119911(1198962)119902)2

)

119878 =119869119898

119899 times 1198902

min

119870 isin 119885+

(14)


A=[03857 7175 05381 01614 04281 05803]

c=2


Jm=min (obj fun)

e2 min=9999

for i=1 c

for j=i+1 c

e2 sum=0

for k=1 3


end

if e2 sumlte2 min

e2 min=e2 sum

end

end

end

e2 min





1198902

min 119878

2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043


Category Jobs1 1 2 18 19 20 22 31 322 3 7 8 9 12 25 26 293 4 5 6 14 16 17 34 35 37 38 39 404 10 11 23 24 27 30


Category Jobs1 1 2 18 19 20 21 22 31 322 2 3 7 8 9 12 25 26 28 293 4 5 6 14 15 16 17 33 34 35 36 37 38 39 404 10 11 13 23 24 27 28 30 33
















0

1

2

3

4

5

0 10 20 30 40

Cate

gory

FCM

Job number

PCA-FCM







ℎ119895119897=

1

1 + 119890minus119899ℎ

119895119897

(15)

where

119899ℎ

119895119897= 119868ℎ

119895119897minus 120579ℎ

119897

119868ℎ

119895119897=

119901

sum

119902=1

119908ℎ

119902119897119911119895119902

(16)



119900119895=

1

1 + 119890minus119899119900

119895

(17)

where

119899119900

119895= 119868119900

119895minus 120579119900

119868119900

119895=

119871

sum

119897=1

119908119900

119897ℎ119895119897

(18)




119895=1(119900119895minus 119873(119862119879

119895))2

119899

(19)



11 119908

ℎ

119901119871 120579ℎ

1 120579ℎ119871 1199081199001 119908

119900





SSE (120573) =119899

sum

119895=1

(119873(119862119879119895) minus 119891 (x

119895120573))2

(20)



ℎ

11

Δ119908ℎ

119901119871 Δ120579ℎ1 Δ120579

ℎ

119871 Δ1199081199001 Δ119908119900



as

119891 (x119895120573 + 120575) asymp 119891 (x

119895120573) + J

119895120575 (21)

where

J119895= 120597

119891 (x119895120573)

120597120573(22)


SSE (120573+120575) asymp119899

sum

119895=1

(119873 (119862119879119895) minus 119891 (x

119895120573) minus J

119895120575)2

(23)

















(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)



Single hidden layer



have been run














Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119895 1199091198951

1199091198952

1199091198953

1199091198954

1199091198955

1199091198956



120578Σ(1) = 46 120578

Σ(2) = 65

120578Σ(3) = 81 120578

Σ(4) = 95

120578Σ(5) = 100 120578

Σ(6) = 100

(9)


1 2 3 40

102030405060708090

100

Principal component

0102030405060708090100

Varia

ncee

xpla

ined

()

()


0 1 2 3 4

0

1

2


2nd

prin

cipa

l com

pone

nt

15

05

minus05

minus1

minus15

minus2

minus25




Σ(119901) ge 85 sim






1199111198951

1199111198952

1199111198953




Min119870

sum

119896=1

119899

sum

119895=1

120583119898

119895(119896)1198902

119895(119896) (10)



to category 119896 119890119895(119896)



the following

119911(119896)= 119911(119896)119902 119896 = 1 sim 119870

119911(119896)119902

=

sum119899

119895=1120583119898

119895(119896)119911119895119902

sum119899

119895=1120583119898

119895(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

(119911119895119901minus 119911(119896)119901)2

119895 = 1 sim 119899 119896 = 1 sim 119870

(11)

where 119911(119896)





max119896

max119895

10038161003816100381610038161003816120583(119905)

119895(119896)minus 120583(119905minus1)

119895(119896)

10038161003816100381610038161003816lt 119889 (12)




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)119902minus 119911(1198962)119902)2

)

119878 =119869119898

119899 times 1198902

min

119870 isin 119885+

(14)


A=[03857 7175 05381 01614 04281 05803]

c=2


Jm=min (obj fun)

e2 min=9999

for i=1 c

for j=i+1 c

e2 sum=0

for k=1 3


end

if e2 sumlte2 min

e2 min=e2 sum

end

end

end

e2 min





1198902

min 119878

2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043


Category Jobs1 1 2 18 19 20 22 31 322 3 7 8 9 12 25 26 293 4 5 6 14 16 17 34 35 37 38 39 404 10 11 23 24 27 30


Category Jobs1 1 2 18 19 20 21 22 31 322 2 3 7 8 9 12 25 26 28 293 4 5 6 14 15 16 17 33 34 35 36 37 38 39 404 10 11 13 23 24 27 28 30 33
















0

1

2

3

4

5

0 10 20 30 40

Cate

gory

FCM

Job number

PCA-FCM







ℎ119895119897=

1

1 + 119890minus119899ℎ

119895119897

(15)

where

119899ℎ

119895119897= 119868ℎ

119895119897minus 120579ℎ

119897

119868ℎ

119895119897=

119901

sum

119902=1

119908ℎ

119902119897119911119895119902

(16)



119900119895=

1

1 + 119890minus119899119900

119895

(17)

where

119899119900

119895= 119868119900

119895minus 120579119900

119868119900

119895=

119871

sum

119897=1

119908119900

119897ℎ119895119897

(18)




119895=1(119900119895minus 119873(119862119879

119895))2

119899

(19)



11 119908

ℎ

119901119871 120579ℎ

1 120579ℎ119871 1199081199001 119908

119900





SSE (120573) =119899

sum

119895=1

(119873(119862119879119895) minus 119891 (x

119895120573))2

(20)



ℎ

11

Δ119908ℎ

119901119871 Δ120579ℎ1 Δ120579

ℎ

119871 Δ1199081199001 Δ119908119900



as

119891 (x119895120573 + 120575) asymp 119891 (x

119895120573) + J

119895120575 (21)

where

J119895= 120597

119891 (x119895120573)

120597120573(22)


SSE (120573+120575) asymp119899

sum

119895=1

(119873 (119862119879119895) minus 119891 (x

119895120573) minus J

119895120575)2

(23)

















(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)



Single hidden layer



have been run














Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199111198951

1199111198952

1199111198953




Min119870

sum

119896=1

119899

sum

119895=1

120583119898

119895(119896)1198902

119895(119896) (10)



to category 119896 119890119895(119896)



the following

119911(119896)= 119911(119896)119902 119896 = 1 sim 119870

119911(119896)119902

=

sum119899

119895=1120583119898

119895(119896)119911119895119902

sum119899

119895=1120583119898

119895(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

(119911119895119901minus 119911(119896)119901)2

119895 = 1 sim 119899 119896 = 1 sim 119870

(11)

where 119911(119896)





max119896

max119895

10038161003816100381610038161003816120583(119905)

119895(119896)minus 120583(119905minus1)

119895(119896)

10038161003816100381610038161003816lt 119889 (12)




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)119902minus 119911(1198962)119902)2

)

119878 =119869119898

119899 times 1198902

min

119870 isin 119885+

(14)


A=[03857 7175 05381 01614 04281 05803]

c=2


Jm=min (obj fun)

e2 min=9999

for i=1 c

for j=i+1 c

e2 sum=0

for k=1 3


end

if e2 sumlte2 min

e2 min=e2 sum

end

end

end

e2 min





1198902

min 119878

2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043


Category Jobs1 1 2 18 19 20 22 31 322 3 7 8 9 12 25 26 293 4 5 6 14 16 17 34 35 37 38 39 404 10 11 23 24 27 30


Category Jobs1 1 2 18 19 20 21 22 31 322 2 3 7 8 9 12 25 26 28 293 4 5 6 14 15 16 17 33 34 35 36 37 38 39 404 10 11 13 23 24 27 28 30 33
















0

1

2

3

4

5

0 10 20 30 40

Cate

gory

FCM

Job number

PCA-FCM







ℎ119895119897=

1

1 + 119890minus119899ℎ

119895119897

(15)

where

119899ℎ

119895119897= 119868ℎ

119895119897minus 120579ℎ

119897

119868ℎ

119895119897=

119901

sum

119902=1

119908ℎ

119902119897119911119895119902

(16)



119900119895=

1

1 + 119890minus119899119900

119895

(17)

where

119899119900

119895= 119868119900

119895minus 120579119900

119868119900

119895=

119871

sum

119897=1

119908119900

119897ℎ119895119897

(18)




119895=1(119900119895minus 119873(119862119879

119895))2

119899

(19)



11 119908

ℎ

119901119871 120579ℎ

1 120579ℎ119871 1199081199001 119908

119900





SSE (120573) =119899

sum

119895=1

(119873(119862119879119895) minus 119891 (x

119895120573))2

(20)



ℎ

11

Δ119908ℎ

119901119871 Δ120579ℎ1 Δ120579

ℎ

119871 Δ1199081199001 Δ119908119900



as

119891 (x119895120573 + 120575) asymp 119891 (x

119895120573) + J

119895120575 (21)

where

J119895= 120597

119891 (x119895120573)

120597120573(22)


SSE (120573+120575) asymp119899

sum

119895=1

(119873 (119862119879119895) minus 119891 (x

119895120573) minus J

119895120575)2

(23)

















(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)



Single hidden layer



have been run














Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A=[03857 7175 05381 01614 04281 05803]

c=2


Jm=min (obj fun)

e2 min=9999

for i=1 c

for j=i+1 c

e2 sum=0

for k=1 3


end

if e2 sumlte2 min

e2 min=e2 sum

end

end

end

e2 min





1198902

min 119878

2 196 014 0343 121 009 0344 086 007 0305 067 006 0266 053 003 043


Category Jobs1 1 2 18 19 20 22 31 322 3 7 8 9 12 25 26 293 4 5 6 14 16 17 34 35 37 38 39 404 10 11 23 24 27 30


Category Jobs1 1 2 18 19 20 21 22 31 322 2 3 7 8 9 12 25 26 28 293 4 5 6 14 15 16 17 33 34 35 36 37 38 39 404 10 11 13 23 24 27 28 30 33
















0

1

2

3

4

5

0 10 20 30 40

Cate

gory

FCM

Job number

PCA-FCM







ℎ119895119897=

1

1 + 119890minus119899ℎ

119895119897

(15)

where

119899ℎ

119895119897= 119868ℎ

119895119897minus 120579ℎ

119897

119868ℎ

119895119897=

119901

sum

119902=1

119908ℎ

119902119897119911119895119902

(16)



119900119895=

1

1 + 119890minus119899119900

119895

(17)

where

119899119900

119895= 119868119900

119895minus 120579119900

119868119900

119895=

119871

sum

119897=1

119908119900

119897ℎ119895119897

(18)




119895=1(119900119895minus 119873(119862119879

119895))2

119899

(19)



11 119908

ℎ

119901119871 120579ℎ

1 120579ℎ119871 1199081199001 119908

119900





SSE (120573) =119899

sum

119895=1

(119873(119862119879119895) minus 119891 (x

119895120573))2

(20)



ℎ

11

Δ119908ℎ

119901119871 Δ120579ℎ1 Δ120579

ℎ

119871 Δ1199081199001 Δ119908119900



as

119891 (x119895120573 + 120575) asymp 119891 (x

119895120573) + J

119895120575 (21)

where

J119895= 120597

119891 (x119895120573)

120597120573(22)


SSE (120573+120575) asymp119899

sum

119895=1

(119873 (119862119879119895) minus 119891 (x

119895120573) minus J

119895120575)2

(23)

















(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)



Single hidden layer



have been run














Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

2

3

4

5

0 10 20 30 40

Cate

gory

FCM

Job number

PCA-FCM







ℎ119895119897=

1

1 + 119890minus119899ℎ

119895119897

(15)

where

119899ℎ

119895119897= 119868ℎ

119895119897minus 120579ℎ

119897

119868ℎ

119895119897=

119901

sum

119902=1

119908ℎ

119902119897119911119895119902

(16)



119900119895=

1

1 + 119890minus119899119900

119895

(17)

where

119899119900

119895= 119868119900

119895minus 120579119900

119868119900

119895=

119871

sum

119897=1

119908119900

119897ℎ119895119897

(18)




119895=1(119900119895minus 119873(119862119879

119895))2

119899

(19)



11 119908

ℎ

119901119871 120579ℎ

1 120579ℎ119871 1199081199001 119908

119900





SSE (120573) =119899

sum

119895=1

(119873(119862119879119895) minus 119891 (x

119895120573))2

(20)



ℎ

11

Δ119908ℎ

119901119871 Δ120579ℎ1 Δ120579

ℎ

119871 Δ1199081199001 Δ119908119900



as

119891 (x119895120573 + 120575) asymp 119891 (x

119895120573) + J

119895120575 (21)

where

J119895= 120597

119891 (x119895120573)

120597120573(22)


SSE (120573+120575) asymp119899

sum

119895=1

(119873 (119862119879119895) minus 119891 (x

119895120573) minus J

119895120575)2

(23)

















(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)



Single hidden layer



have been run














Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























(JTJ) 120575 = JT (119873 (119862119879119895) minus 119891 (x

119895120573)) (24)



Single hidden layer



have been run














Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Stop

Yes

No

Reclassify the jobs


001

01

10 2 4 6 8


MAPE119878





10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816

le10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 for any 119896 = 119891 (119895)

(25)


119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (26)


119895)10038161003816100381610038161003816le max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816 (27)


120579 (119891 119895)

=

1 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816le min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

0 if 10038161003816100381610038161003816119900119895 (119891 (119895)) minus 119873 (119862119879119895)10038161003816100381610038161003816ge max119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

10038161003816100381610038161003816119900119895(119891 (119895)) minus 119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896) minus 119873 (119862119879

119895)10038161003816100381610038161003816

min119896 = 119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816minusmax

119896 =119891(119895)

10038161003816100381610038161003816119900119895(119896)minus119873 (119862119879

119895)10038161003816100381610038161003816

otherwise(28)


120579 (119891) =

sum119899

119895=1120579 (119891 119895)

119899sdot 100 (29)







119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119895 120579(119891 119895)

1 1002 1003 1004 1005 1006 1007 1008 1009 10010 10011 10012 10013 10014 10015 10016 10017 10018 10019 10020 10021 10022 10023 10024 10025 10026 06327 10028 05829 10030 10031 10032 10033 10034 10035 10036 10037 10038 05539 00040 100











0

1

2

3

4

5

0 10 20 30 40

Cate

gory

Job number


























1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198670


CBR 183lowast

BPN 183lowast

SOM-WM 183lowast













Acknowledgment


References





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article An Iterative Procedure for Optimizing the ...downloads.hindawi.com/journals/mpe/2013/740478.pdf · Fuzzy-Neural Job Cycle Time Estimation Approach in a Wafer Fabrication