research article semiconductor yield forecasting using...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 672404 7 pageshttpdxdoiorg1011552013672404

Research ArticleSemiconductor Yield Forecasting UsingQuadratic-Programming-Based Fuzzy CollaborativeIntelligence Approach

Toly Chen and Yu-Cheng Wang

Department of Industrial Engineering and Systems Management Feng Chia University Taichung City 407 Taiwan

Correspondence should be addressed to Toly Chen tcchenfcuedutw

Received 26 March 2013 Accepted 17 May 2013

Academic Editor Yi-Chi Wang

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

Several recent studies have proposed fuzzy collaborative forecasting methods for semiconductor yield forecasting These methodsestablish nonlinear programming (NLP) models to consider the opinions of experts and generate fuzzy yield forecasts Such apractice cannot distinguish between the different expert opinions and can not easily find the global optimal solution In orderto solve some problems and to improve the performance of semiconductor yield forecasting this study proposes a quadratic-programming- (QP-) based fuzzy collaborative intelligence approach

1 Introduction

Yield is the proportion between goods and total productsGiven the same inputs the higher yield means more salableoutputs that increase the revenues Yield is a key performancemeasure in operations and has a dominant effect onmanufac-turing economics Yield improvement is also a crucial workfor a factoryrsquos competitiveness [1] In addition predictingthe future yield to estimate the possible gains is also a basisfor long-term production planning [2] The ramping of asemiconductor manufacturing factory should occur quicklyand then product yields should be maintained at a high levelto maximize profits This study is therefore committed topredicting the yield of a semiconductor product Semicon-ductor products are focused on because of the extensive usein modern life such as personal computer laptop cell phoneand others According to Chen and Wang [3] methods forpredicting the yield of a semiconductor product can bedivided into two categoriesmdashmacro yield modeling (MaYM)methods and micro yield modeling (MiYM) methods Thisstudy belongs to the MaYM category in which a learningmodel is used to predict the future yield of a semiconductorproduct After conversion some yield learning model canbe solved using linear regression [3] which is convenient

and explains why the related methods have been welcomedHowever there is much uncertainty in the yield learningprocess To deal with this issue several treatments have beentaken in the literature (Figure 1)

(1) ProbabilisticStochastic MethodsMethods of this sub-category assume that parameter distributions areknown in advance to a certain degree and thesedistributions can be modified in a Bayesian mannerafter actual values are observed The works of Spence[4]Majd andPindyck [5]Mazzola andMcCardle [6]and Anderson [7] all belong to this sub-category

(2) Fuzzy Methods Methods of this sub-category fit theyield learning process with a possibility regressionmodel and yield forecasts are given in fuzzy valuesto consider the uncertainty [1 3 8ndash12] In the workof Chen and Wang [3] the fuzzy yield learningcurve was fitted by solving a linear programming(LP) problem Wu et al [12] established a fuzzy backpropagation network (FBPN) to predict the yield ofsemiconductor The inputs of this network includethree categories the physical parameters electricaltest parameters and wafer defect parameters Chiangand Hsieh [13] applied a similar fuzzy neural network

2 Mathematical Problems in Engineering

Yield forecasting methods

MaYM methods MiYM methods

Stochastic methods

Fuzzy methods

Figure 1 Classification of yield forecasting methods

method in which grey relational analysis was usedto find the most influential parameters Lin [11] con-sidered the uncertainty of the defect clustering pat-tern and established a number of fuzzy rules to predictyield

On the other hand the concept of fuzzy collaborativeintelligence was proposed by Pedrycz in 2003 [14] and hasbeen applied to the forecasting of various properties such asyield [8 9] global CO

2concentration [15] job cycle time [16]

and others Among them yield forecasting received muchattention and is the focus of this study In [8] Chen andLin proposed a fuzzy collaborative intelligence approach forforecasting the yield of a semiconductor product In the fuzzycollaborative intelligence approach a group of experts predictthe future yield with a fuzzy number from different points ofviewThe forecasts by the experts are aggregated in such awaythat both the precision and accuracy can be improved at thesame time

(1) Accuracy the forecasted value should be as close aspossible to the actual value

(2) Precision a narrow interval containing the actualvalue is established

However the fuzzy collaborative intelligence approach req-uires the solving of two nonlinear programming (NLP) prob-lems which is not easy Furthermore the existing optimiza-tion packagesmay not be able to guarantee the global optima-lity of the solutions In addition the views of the experts arenot reflected properly on the objective function In order tosolve these two problems the following treatments have beentaken in this study

(1) A newpartitioningmethod has been proposed whichconsiders the effects on the objective function inpartitioning the range of parameters

(2) The nonlinear objective functions and constraintswere converted into quadratic ones using severalpolynomial fitting techniques Then a variety ofmethods such as the interior pointmethod the activeset method the augmented Lagrangian method theconjugate gradient method the gradient projectionmethod and the extension of the simplex algorithm

can be used to solve the quadratic programming (QP)problems

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

The rest of this paper is organized as follows Section 2reviews the related literature In Section 3 we proposed somecountermeasures to solve the problems in the work of Chenand Lin [8] in order to propose a QP-based fuzzy collabora-tive forecasting method Examples in the work of Chen andLin [8] were used to assess the effectiveness of the proposedmethodology and tomake a comparison with the NLP-basedmethod Finally the conclusions of this study are made inSection 4

2 Literature Review

Multiple analyses of a problem from diverse perspectivesraise the chance that no relevant aspects of the problemwill be ignored In addition as internet applications becomewidespread dealing with disparate data sources is becomingmore and more popular Technical constraints securityissues and privacy considerations often limit access to somesources Therefore the concepts of collaborative computingintelligence and collaborative fuzzy modeling have been pro-posed and the certain so-called fuzzy collaborative systemsare being established In a fuzzy collaborative system someexperts agents or systems with various backgrounds aretrying to achieve a common target Since they have differentknowledge and points of view they may use various methodsto model identify or control the common target The key ofsuch a system is that these experts agents or systems shareand exchange their observations settings experiences andknowledgewith each otherwhen achieving the commongoalThis features the fuzzy collaborative system distinct from theensemble of multiple fuzzy systems

In the limited literature fuzzy collaborative intelligenceand systems have been successfully applied to collaborativeclustering group forecasting agent negotiation assessmentand reputation information filtering robot control intrusiondetection object tracking and so forth Applications offuzzy collaborative intelligence in other fields remain to beinvestigated Several studies have argued that for certainproblems a fuzzy collaborative intelligence approach is moreprecise accurate efficient safe and private than typicalapproaches

Designing is a work in which the views may vary andtherefore needs coordination or cooperation Shai and Reich[17 18] defined the concept of infused design as an approachfor establishing effective collaboration between designersfrom different engineering fields Ostrosi et al [19] definedthe concept of consensus as the overlapping of design clustersof different perspectives

In demand forecasting Lo et al [20] believed that thedemand plan has to be compromised and achieved throughthe collaborative efforts of all the demand network stakehol-ders Buyukozkan and Vardaloglu [21 22] applied the fuzzycognitive map method to the collaborative planning fore-casting and replenishment of a supply chain The initial

Mathematical Problems in Engineering 3

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

Method Expertcollaboration

Partitioning the ranges ofparameters

Models tosolve Easiness to solve Optimality of

the solutionChen and Wang[9] No No LP Very easy Global

optimalChen and Lin [8] Yes Uniform partitioning NLP Very difficult Local optimalWu et al [12] No No FBPN Very difficult Local optimalThe proposedmethodology Yes Considering the effects on the

objective function QP Easy Globaloptimal

values of the concepts and the connection weights of thefuzzy cognitive map are dependent on the subjective beliefof the expert and can be modified after collaborationCheikhrouhou et al [23] thought that collaboration is neces-sary because of the unexpected events that may occur in thefuture demand

In the collaborative forecasting according to Poler et al[24] the comparison of collaboration methods and theproposing of software tools especially regard forecastingmethods for collaborative forecasting are still lacking Chen[16] put forward a fuzzy collaborative forecasting method topredict the job cycle time in a wafer fabrication factory inwhich each expert used a fuzzy multiple linear regressionequation to predict the cycle time of a job The cycle timeforecasts of different expertswere aggregated using the hybridfuzzy intersection (FI) and back propagation network (BPN)approachA similarmethodhas beenused byChen andWang[15] to predict the global CO

2concentrationThe traditional

method in this field is a simple time series analysis but it is notaccurate enough Nonlinearmethods such as artificial neuralnetworks [25 26] may be more effective

In all fuzzy collaborative intelligence methods the con-sensus of results is being sought Pedrycz and Rai [27]discussed the problem of collaborative data analysis by agroup of agents having access to different parts of dataand exchanging findings through their collaboration A two-phase optimization procedure was established so that theresults of communication can be embedded into the localoptimization results Sometimes however the consensusmaynot exist Therefore Chen [28] defined the concept of partialconsensus as the intersection of the views of some expertsIn the work of Chen and Wang [9] an agent considered thesettings of other agents to optimize its own setting

3 QP-Based Fuzzy CollaborativeIntelligence Approach

According to Gruber [29] the general yield learning modelof a semiconductor product is

119884119905= 1198840119890minus119887119905+119903(119905)

(1)

where 119887 and 1198840indicate the learning constant and the

asymptotic yield respectively 119887 ge 0 0 le 1198840 119884119905le 1 119905 is time

119905 = 1 sim 119879 119903(119905) is a homoscedastic serially noncorrelatederror term After converting to logarithms

ln119884119905= ln119884

0minus

119887

119905+ 119903 (119905) = 119886 minus

119887

119905minus 119903 (119905) (2)

where 119886 = ln1198840 In Chen and Linrsquos fuzzy collaborative

intelligence approach [8] to consider the uncertainty in yieldlearning the parameters in (2) are given in triangular fuzzynumbers as follows

119905= (1198841199051 1198841199052 1198841199053)

0= (11988401 11988402 11988403)

= (1198871 1198872 1198873)

119886 = (1198861 1198862 1198863)

(3)

and the following two NLP problems are to be solved

NLP problem I is

Min1198851=

119879

sum

119905=1

(ln1198841199053minus ln119884

1199051)119900119896 (4)

subject to

ln119910119905ge ln119884

1199051+ 119904119896(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119896(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(5)

NLP problem II is

Max 1198852=119898119896radic

sum119879

119905=1119904119905

119898119896

119879

(6)


subject to

119879

sum

119905=1

(ln1198841199053minus ln119884

1199051)119900119896

le 119879 sdot 119889119896

119900119896 (7)

ln119910119905ge ln119884

1199051+ 119904119905(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119905(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(8)

where = (1198871 1198872 1198873) and

0= (11988401 11988402 11988403) indicate the

learning constant and the asymptotic yield 119905 is time 119905 = 1 sim

119879 119900119896indicates the sensitivity of expert 119896 to the uncertainty

in the fuzzy forecast 119900119896ge 0 Chen and Wang [9] suggested

that 119900119896should be between 1 and 4 119904

119896is the satisfaction

level required by expert 119896 0 le 119904119896

le 1 119898119896reflects the

importance of the outlier to expert 119896 119898119896

ge 0 Chen andWang [9] suggested that 119898

119896should be between 1 and 4 119889

119896

is the required range by expert 119896 and 119889119896ge 0 Maximizing 119885

2

is equivalent to

Max 1198851015840

2=

119879

sum

119905=1

119904119905

119898119896 (9)

However there are two problems with Chen and Linrsquosmethod [8]

(1) The range of 119900119896was divided into six segments of

the same width and seven values of 119900119896 1 15 2

25 3 35 and 4 corresponding to seven linguis-tic terms (extremely insensitive very insensitivesomewhat insensitive moderate somewhat sensitivevery sensitive extremely sensitive) were provided foreach expert to choose from However the effects ofsuch a uniform partitioning were not proportionallyreflected on the objective function For example thedistance between 35 and 4 is the same as that between1 and 15 If ln119884

1199053minus ln1198841199051= 05 then (ln119884

1199053minus ln1198841199051)4minus

(ln1198841199053

minus ln1198841199051)35

= minus0026 but (ln1198841199053

minus ln1198841199051)15

minus

(ln1198841199053

minus ln1198841199051)1

= minus0146 Obviously minus0026 ≫

minus0146(2) The nonlinear objective functions and constraints are

difficult to handleFirst the range of 119900

119896should be partitioned in such a

way that the range of (ln1198841199053minus ln119884

1199051)119900119896 can be divided

into six equal parts According to (7) ln1198841199053minus ln119884

1199051is

to be smaller than119889119896Therefore the problembecomes

dividing the interval [1198891

1198961198894

119896] into six equal parts

indicated with [119900119896(119895) 119900119896(119895 + 1)] 119895 = 1 sim 6 119900

119896(1) =

1 119900119896(7) = 4 To this end the following equations are

to be solved individually

119889119900119896(2)

119896=

5119889119896+ 1198894

119896

6997888rarr 119900119896(2) = ln

((5119889119896+ 1198894

119896) 6)

ln (119889119896)

119889119900119896(3)

119896=4119889119896+ 21198894

119896

6997888rarr 119900119896(3)= ln

((4119889119896+ 21198894

119896) 6)

ln (119889119896)

119889119900119896(4)

119896=3119889119896+ 31198894

119896

6997888rarr 119900119896(4)= ln

((3119889119896+ 31198894

119896) 6)

ln (119889119896)

119889119900119896(5)

119896=4119889119896+ 21198894

119896

6997888rarr 119900119896(5)= ln

((2119889119896+ 41198894

119896) 6)

ln (119889119896)

119889119900119896(6)

119896=

119889119896+ 51198894

119896

6997888rarr 119900119896(6) = ln

((119889119896+ 51198894

119896) 6)

ln (119889119896)

(10)

For example if 119889119896= 05 then 119900

119896(1) = 1 119900

119896(2) = 122 119900

119896(3) =

15 119900119896(4) = 183 119900

119896(5) = 226 119900

119896(6) = 288 and 119900

119896(7) = 4

Subsequently the nonlinear objective function and con-straints can be converted into quadratic ones by minimizingthe mean absolute error of approximation when 0 le 119909 le 12

as

Min12

sum

119909=0001

10038161003816100381610038161003816119909ℎminus (120572119909

2+ 120573119909 + 120574)

10038161003816100381610038161003816

121 (11)

The range [0 12] is wide enough for most possible yieldvalues to fall into To this end a look-up table has beenestablished after extensive numerical simulation see Table 2The mean absolute percentage error (MAPE) was less than5 The MAPE will not be a serious problem if it is easier tofind the global optimal solution after approximation

For some frequently used values especially the fitted poly-nomials are shown below

11990915

cong 050271199092+ 05308119909 minus 00347

11990925

cong 143571199092minus 04847119909 + 00528

1199093cong 18000119909

2minus 08953119909 + 01051

11990935

cong 209971199092minus 12322119909 + 01508

1199094cong 23440119909

2minus 15053119909 + 01890

(12)

The application method of Table 2 is shown in Figure 2After approximation the following QP problems are

solved instead


Extremely SomewhatVery Somewhat Very Extremelyinsensitive insensitive insensitive Moderate sensitive sensitive sensitive

35 3252151NLP scale 4

QP scale 2099718143571050270 23441 005308

0 01890150801051005280

minus04847

minus00347

minus08953 minus12322minus15053

Figure 2 The application of the look-up table

Table 2 The look-up table

ℎ 120572 120573 120574

100 0 0 1101 00093 09927 minus00020102 00187 09852 minus00039

400 23440 minus15053 01890

QP problem I is

Min 1198853=

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)) )

(13)

subject to

ln119910119905ge ln119884

1199051+ 119904119896(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119896(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(14)

QP problem II is

Max 1198854=

119879

sum

119905=1

(120572 (119898 (119896)) 119904119905

2+ 120573 (119898 (119896)) 119904

119905+ 120574 (119898 (119896))) (15)

subject to

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)))

le 119879 sdot 119889119896

119900119896

ln119910119905ge ln119884

1199051+ 119904119905(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119905(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(16)

Take the case in Chen and Wang [9] as an example seeTable 3 In NLP models the four parameters 119900

119896 119904119896 119898119896 and

119889119896were set to 11 04 1 and 071 in their study The NLP

models were converted into the QP onesThen bothmethodswere applied to predict the yield After defuzzifying the yieldforecastswith the centroid-of-gravity formula the forecastingperformances by the twomethods were compared in Table 4

(1) Obviously the forecasting accuracy of the QPmodelswas better Considering MAPE the proposed QP-basedmethod reduced the prediction error up to 10

(2) On the other hand it is also possible to use the pro-posed QP-based method to improve the predictionprecision measured in terms of the average range offorecasts

(3) In theory as long as quadratic polynomials canvery precisely fit the original nonlinear objectivefunctions and constraints the prediction accuracy


Table 3 The case in Chen and Wang [9]

119905 1 2 3 4 5 6 7119884119905

3730 5850 5410 7410 6170 8000 7120

Table 4 The performances by the two methods

Model The average range(precision)

Mean absolute percentageerror (accuracy)

NLP I 034 145NLP II 037 172QP I 040 141QP II 028 143

of the proposed QP-based method must be betterthan the original NLP-based method Even with littleerror a similar effect still exists For this reasonit seems that it is not necessary to compare thesubsequent collaboration step Fuzzy collaborativeforecasting methods based on improved individualforecasts should perform better

(4) It is also easy to prove that the quadratic objectivefunction and constraints are convex because ln119884

1199053ge

ln1198841199051and 120572(119900(119896)) ge 0 That is very important to the

global optimality of the optimal solution

4 Discussion and Conclusion

Fuzzy collaborative forecasting methods have considerablepotential for the semiconductor yield forecasting In this fieldthe existing methods are mostly based on NLP models Sucha practice has some drawbacks To solve these problems weproposed a QP-based fuzzy collaborative forecasting methodand made a fair comparison with instances from the litera-ture

According to the experimental results the QP-basedapproach achieved a better prediction performance than theNLP-based method because it is easier to find the globaloptimal solution in the QP-based method than the methodbased on NLP In addition the disagreements between theviews of different experts are also better distinguished in theQP-based approach Furthermore theNLPmodels are in factformulated subjectively and certainly can be replaced by anyother formulation to achieve a better performanceThe errorof estimating the NLP model with the QP models becomesunimportant as well In other words the future focus shouldbe on the development of better QP models rather than theapproximation of the NLP models

Acknowledgment

This work was supported by the National Science Council ofTaiwan

References

[1] T Chen ldquoA FNP approach for evaluating and enhancingthe long-term competitiveness of a semiconductor fabricationfactory through yield learning modelingrdquo International Journalof Advanced Manufacturing Technology vol 40 no 9-10 pp993ndash1003 2009

[2] T Chen ldquoA fuzzy mid-term single-fab production planningmodelrdquo Journal of Intelligent Manufacturing vol 14 no 3-4 pp273ndash285 2003

[3] T Chen and M-J J Wang ldquoForecasting methods using fuzzyconceptsrdquo Fuzzy Sets and Systems vol 105 no 3 pp 339ndash3521999

[4] A M Spence ldquoThe learning curve and competitionrdquo BellJournal of Economics vol 12 pp 49ndash70 1981

[5] SMajd andR S Pindyck ldquoThe learning curve and optimal pro-duction under uncertaintyrdquo Rand Journal of Economics vol 20no 3 pp 331ndash343 1989

[6] J B Mazzola and K F McCardle ldquoA Bayesian approach tomanaging learning-curve uncertaintyrdquo Management Sciencevol 42 no 5 pp 680ndash692 1996

[7] K Anderson ldquoInnovative yield modeling using statisticsrdquo inProceedings of the 17th Annual SEMIIEEE Advanced Semicon-ductorManufacturing Conference (ASMC rsquo06) pp 219ndash221 May2006

[8] T Chen and Y C Lin ldquoA fuzzy-neural system incorporatingunequally important expert opinions for semiconductor yieldforecastingrdquo International Journal of Uncertainty Fuzziness andKnowlege-Based Systems vol 16 no 1 pp 35ndash58 2008

[9] T Chen and Y C Wang ldquoAn agent-based fuzzy collaborativeintelligence approach for precise and accurate semiconductoryield forecastingrdquo IEEE Transactions on Fuzzy Systems 2013

[10] JWatada H Tanaka and T Shimomura ldquoIdentification of lear-ning curve based on possibilistic conceptsrdquo in Applications ofFuzzy Set Theory in Human Factors Elsevier The Netherlands1986

[11] J S Lin ldquoConstructing a yield model for integrated circuitsbased on a novel fuzzy variable of clustered defect patternrdquoExpert Systems With Applications vol 39 no 3 pp 2856ndash28642012

[12] L Wu J Zhang and G Zhang ldquoA fuzzy neural networkapproach for die yield prediction of wafer fabrication linerdquo inProceedings of the 6th International Conference on Fuzzy SystemsandKnowledgeDiscovery (FSKD rsquo09) pp 198ndash202 August 2009

[13] Y-M Chiang and H-H Hsieh ldquoApplying grey relational anal-ysis and fuzzy neural network to improve the yield of thin-filmsputtering process in color filter manufacturingrdquo in Proceedingsof the 13th IFAC Symposium on Information Control Problems inManufacturing (INCOM rsquo09) pp 408ndash413 June 2009

[14] W Pedrycz ldquoCollaborative architectures of fuzzy modelingrdquoin Computational Intelligence Research Frontiers vol 5050 ofLecture Notes in Computer Science pp 117ndash139 Springer BerlinGermany 2008

[15] T Chen and Y C Wang ldquoA fuzzy-neural approach for globalCO2concentration forecastingrdquo Intelligent Data Analysis vol

15 no 5 pp 763ndash777 2011


[16] 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

[17] O Shai and Y Reich ldquoInfused design I Theoryrdquo Research inEngineering Design vol 15 no 2 pp 93ndash107 2004

[18] O Shai and Y Reich ldquoInfused design II Practicerdquo Research inEngineering Design vol 15 no 2 pp 108ndash121 2004

[19] E Ostrosi L Haxhiaj and S Fukuda ldquoFuzzy modelling ofconsensus during design conflict resolutionrdquo Research in Engi-neering Design pp 1ndash18 2011

[20] T S L Lo L H S Luong and RMMarian ldquoHolistic and colla-borative demand forecasting processrdquo inProceedings of the IEEEInternational Conference on Industrial Informatics (INDIN rsquo06)pp 782ndash787 August 2006

[21] G Buyukozkan and Z Vardaloglu ldquoAnalyzing of collaborativeplanning forecasting and replenishment approachusing fuzzycognitivemaprdquo in Proceedings of the International Conference onComputers and Industrial Engineering (CIE rsquo09) pp 1751ndash1756July 2009

[22] G B Buyukozkan O Feyzioglu and Z Vardaloglu ldquoAnalyzingCPFR supporting factors with fuzzy cognitive map approachrdquoWorld Academy of Science Engineering and Technology vol 31pp 412ndash417 2009

[23] N Cheikhrouhou FMarmier O Ayadi and PWieser ldquoA colla-borative demand forecasting process with event-based fuzzyjudgementsrdquo Computers and Industrial Engineering vol 61 no2 pp 409ndash421 2011

[24] R Poler J E Hernandez J Mula and F C Lario ldquoCollaborativeforecasting in networkedmanufacturing enterprisesrdquo Journal ofManufacturing Technology Management vol 19 no 4 pp 514ndash528 2008

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

[26] 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

[27] W Pedrycz and P Rai ldquoA multifaceted perspective at dataanalysis a study in collaborative intelligent agentsrdquo IEEE Trans-actions on Systems Man and Cybernetics B vol 38 no 4 pp1062ndash1072 2008

[28] T Chen ldquoA hybrid fuzzy and neural approach with virtualexperts and partial consensus for DRAM price forecastingrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 1B pp 583ndash598 2012

[29] H Gruber Learning and Strategic Product Innovation Theoryand Evidence for the Semiconductor Industry Elsevier TheNetherlands 1994

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Stochastic AnalysisInternational Journal of


Yield forecasting methods

MaYM methods MiYM methods

Stochastic methods

Fuzzy methods

Figure 1 Classification of yield forecasting methods

method in which grey relational analysis was usedto find the most influential parameters Lin [11] con-sidered the uncertainty of the defect clustering pat-tern and established a number of fuzzy rules to predictyield

On the other hand the concept of fuzzy collaborativeintelligence was proposed by Pedrycz in 2003 [14] and hasbeen applied to the forecasting of various properties such asyield [8 9] global CO

2concentration [15] job cycle time [16]

and others Among them yield forecasting received muchattention and is the focus of this study In [8] Chen andLin proposed a fuzzy collaborative intelligence approach forforecasting the yield of a semiconductor product In the fuzzycollaborative intelligence approach a group of experts predictthe future yield with a fuzzy number from different points ofviewThe forecasts by the experts are aggregated in such awaythat both the precision and accuracy can be improved at thesame time

(1) Accuracy the forecasted value should be as close aspossible to the actual value

(2) Precision a narrow interval containing the actualvalue is established

However the fuzzy collaborative intelligence approach req-uires the solving of two nonlinear programming (NLP) prob-lems which is not easy Furthermore the existing optimiza-tion packagesmay not be able to guarantee the global optima-lity of the solutions In addition the views of the experts arenot reflected properly on the objective function In order tosolve these two problems the following treatments have beentaken in this study

(1) A newpartitioningmethod has been proposed whichconsiders the effects on the objective function inpartitioning the range of parameters

(2) The nonlinear objective functions and constraintswere converted into quadratic ones using severalpolynomial fitting techniques Then a variety ofmethods such as the interior pointmethod the activeset method the augmented Lagrangian method theconjugate gradient method the gradient projectionmethod and the extension of the simplex algorithm

can be used to solve the quadratic programming (QP)problems

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

The rest of this paper is organized as follows Section 2reviews the related literature In Section 3 we proposed somecountermeasures to solve the problems in the work of Chenand Lin [8] in order to propose a QP-based fuzzy collabora-tive forecasting method Examples in the work of Chen andLin [8] were used to assess the effectiveness of the proposedmethodology and tomake a comparison with the NLP-basedmethod Finally the conclusions of this study are made inSection 4

2 Literature Review

Multiple analyses of a problem from diverse perspectivesraise the chance that no relevant aspects of the problemwill be ignored In addition as internet applications becomewidespread dealing with disparate data sources is becomingmore and more popular Technical constraints securityissues and privacy considerations often limit access to somesources Therefore the concepts of collaborative computingintelligence and collaborative fuzzy modeling have been pro-posed and the certain so-called fuzzy collaborative systemsare being established In a fuzzy collaborative system someexperts agents or systems with various backgrounds aretrying to achieve a common target Since they have differentknowledge and points of view they may use various methodsto model identify or control the common target The key ofsuch a system is that these experts agents or systems shareand exchange their observations settings experiences andknowledgewith each otherwhen achieving the commongoalThis features the fuzzy collaborative system distinct from theensemble of multiple fuzzy systems

In the limited literature fuzzy collaborative intelligenceand systems have been successfully applied to collaborativeclustering group forecasting agent negotiation assessmentand reputation information filtering robot control intrusiondetection object tracking and so forth Applications offuzzy collaborative intelligence in other fields remain to beinvestigated Several studies have argued that for certainproblems a fuzzy collaborative intelligence approach is moreprecise accurate efficient safe and private than typicalapproaches

Designing is a work in which the views may vary andtherefore needs coordination or cooperation Shai and Reich[17 18] defined the concept of infused design as an approachfor establishing effective collaboration between designersfrom different engineering fields Ostrosi et al [19] definedthe concept of consensus as the overlapping of design clustersof different perspectives

In demand forecasting Lo et al [20] believed that thedemand plan has to be compromised and achieved throughthe collaborative efforts of all the demand network stakehol-ders Buyukozkan and Vardaloglu [21 22] applied the fuzzycognitive map method to the collaborative planning fore-casting and replenishment of a supply chain The initial
















119884119905= 1198840119890minus119887119905+119903(119905)

(1)




ln119884119905= ln119884

0minus

119887

119905+ 119903 (119905) = 119886 minus

119887

119905minus 119903 (119905) (2)



119905= (1198841199051 1198841199052 1198841199053)

0= (11988401 11988402 11988403)

= (1198871 1198872 1198873)

119886 = (1198861 1198862 1198863)

(3)


NLP problem I is

Min1198851=

119879

sum

119905=1

(ln1198841199053minus ln119884

1199051)119900119896 (4)

subject to

ln119910119905ge ln119884

1199051+ 119904119896(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119896(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(5)

NLP problem II is

Max 1198852=119898119896radic

sum119879

119905=1119904119905

119898119896

119879

(6)


subject to

119879

sum

119905=1

(ln1198841199053minus ln119884

1199051)119900119896

le 119879 sdot 119889119896

119900119896 (7)

ln119910119905ge ln119884

1199051+ 119904119905(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119905(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(8)

where = (1198871 1198872 1198873) and

0= (11988401 11988402 11988403) indicate the











119896


2

is equivalent to

Max 1198851015840

2=

119879

sum

119905=1

119904119905

119898119896 (9)





1199053minus ln1198841199051= 05 then (ln119884

1199053minus ln1198841199051)4minus

(ln1198841199053

minus ln1198841199051)35

= minus0026 but (ln1198841199053

minus ln1198841199051)15

minus

(ln1198841199053

minus ln1198841199051)1






1199051)119900119896 can be divided


1199051is



1198961198894



119896(1) =



119889119900119896(2)

119896=

5119889119896+ 1198894

119896

6997888rarr 119900119896(2) = ln

((5119889119896+ 1198894

119896) 6)

ln (119889119896)

119889119900119896(3)

119896=4119889119896+ 21198894

119896

6997888rarr 119900119896(3)= ln

((4119889119896+ 21198894

119896) 6)

ln (119889119896)

119889119900119896(4)

119896=3119889119896+ 31198894

119896

6997888rarr 119900119896(4)= ln

((3119889119896+ 31198894

119896) 6)

ln (119889119896)

119889119900119896(5)

119896=4119889119896+ 21198894

119896

6997888rarr 119900119896(5)= ln

((2119889119896+ 41198894

119896) 6)

ln (119889119896)

119889119900119896(6)

119896=

119889119896+ 51198894

119896

6997888rarr 119900119896(6) = ln

((119889119896+ 51198894

119896) 6)

ln (119889119896)

(10)


119896(1) = 1 119900

119896(2) = 122 119900

119896(3) =

15 119900119896(4) = 183 119900

119896(5) = 226 119900

119896(6) = 288 and 119900

119896(7) = 4


as

Min12

sum

119909=0001

10038161003816100381610038161003816119909ℎminus (120572119909

2+ 120573119909 + 120574)

10038161003816100381610038161003816

121 (11)



11990915

cong 050271199092+ 05308119909 minus 00347

11990925

cong 143571199092minus 04847119909 + 00528

1199093cong 18000119909

2minus 08953119909 + 01051

11990935

cong 209971199092minus 12322119909 + 01508

1199094cong 23440119909

2minus 15053119909 + 01890

(12)


solved instead



35 3252151NLP scale 4

QP scale 2099718143571050270 23441 005308

0 01890150801051005280

minus04847

minus00347




ℎ 120572 120573 120574

100 0 0 1101 00093 09927 minus00020102 00187 09852 minus00039

400 23440 minus15053 01890

QP problem I is

Min 1198853=

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)) )

(13)

subject to

ln119910119905ge ln119884

1199051+ 119904119896(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119896(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(14)

QP problem II is

Max 1198854=

119879

sum

119905=1

(120572 (119898 (119896)) 119904119905

2+ 120573 (119898 (119896)) 119904

119905+ 120574 (119898 (119896))) (15)

subject to

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)))

le 119879 sdot 119889119896

119900119896

ln119910119905ge ln119884

1199051+ 119904119905(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119905(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(16)


119896 119904119896 119898119896 and








119905 1 2 3 4 5 6 7119884119905

3730 5850 5410 7410 6170 8000 7120







1199053ge






Acknowledgment


References
















15 no 5 pp 763ndash777 2011























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























119884119905= 1198840119890minus119887119905+119903(119905)

(1)




ln119884119905= ln119884

0minus

119887

119905+ 119903 (119905) = 119886 minus

119887

119905minus 119903 (119905) (2)



119905= (1198841199051 1198841199052 1198841199053)

0= (11988401 11988402 11988403)

= (1198871 1198872 1198873)

119886 = (1198861 1198862 1198863)

(3)


NLP problem I is

Min1198851=

119879

sum

119905=1

(ln1198841199053minus ln119884

1199051)119900119896 (4)

subject to

ln119910119905ge ln119884

1199051+ 119904119896(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119896(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(5)

NLP problem II is

Max 1198852=119898119896radic

sum119879

119905=1119904119905

119898119896

119879

(6)


subject to

119879

sum

119905=1

(ln1198841199053minus ln119884

1199051)119900119896

le 119879 sdot 119889119896

119900119896 (7)

ln119910119905ge ln119884

1199051+ 119904119905(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119905(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(8)

where = (1198871 1198872 1198873) and

0= (11988401 11988402 11988403) indicate the











119896


2

is equivalent to

Max 1198851015840

2=

119879

sum

119905=1

119904119905

119898119896 (9)





1199053minus ln1198841199051= 05 then (ln119884

1199053minus ln1198841199051)4minus

(ln1198841199053

minus ln1198841199051)35

= minus0026 but (ln1198841199053

minus ln1198841199051)15

minus

(ln1198841199053

minus ln1198841199051)1






1199051)119900119896 can be divided


1199051is



1198961198894



119896(1) =



119889119900119896(2)

119896=

5119889119896+ 1198894

119896

6997888rarr 119900119896(2) = ln

((5119889119896+ 1198894

119896) 6)

ln (119889119896)

119889119900119896(3)

119896=4119889119896+ 21198894

119896

6997888rarr 119900119896(3)= ln

((4119889119896+ 21198894

119896) 6)

ln (119889119896)

119889119900119896(4)

119896=3119889119896+ 31198894

119896

6997888rarr 119900119896(4)= ln

((3119889119896+ 31198894

119896) 6)

ln (119889119896)

119889119900119896(5)

119896=4119889119896+ 21198894

119896

6997888rarr 119900119896(5)= ln

((2119889119896+ 41198894

119896) 6)

ln (119889119896)

119889119900119896(6)

119896=

119889119896+ 51198894

119896

6997888rarr 119900119896(6) = ln

((119889119896+ 51198894

119896) 6)

ln (119889119896)

(10)


119896(1) = 1 119900

119896(2) = 122 119900

119896(3) =

15 119900119896(4) = 183 119900

119896(5) = 226 119900

119896(6) = 288 and 119900

119896(7) = 4


as

Min12

sum

119909=0001

10038161003816100381610038161003816119909ℎminus (120572119909

2+ 120573119909 + 120574)

10038161003816100381610038161003816

121 (11)



11990915

cong 050271199092+ 05308119909 minus 00347

11990925

cong 143571199092minus 04847119909 + 00528

1199093cong 18000119909

2minus 08953119909 + 01051

11990935

cong 209971199092minus 12322119909 + 01508

1199094cong 23440119909

2minus 15053119909 + 01890

(12)


solved instead



35 3252151NLP scale 4

QP scale 2099718143571050270 23441 005308

0 01890150801051005280

minus04847

minus00347




ℎ 120572 120573 120574

100 0 0 1101 00093 09927 minus00020102 00187 09852 minus00039

400 23440 minus15053 01890

QP problem I is

Min 1198853=

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)) )

(13)

subject to

ln119910119905ge ln119884

1199051+ 119904119896(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119896(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(14)

QP problem II is

Max 1198854=

119879

sum

119905=1

(120572 (119898 (119896)) 119904119905

2+ 120573 (119898 (119896)) 119904

119905+ 120574 (119898 (119896))) (15)

subject to

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)))

le 119879 sdot 119889119896

119900119896

ln119910119905ge ln119884

1199051+ 119904119905(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119905(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(16)


119896 119904119896 119898119896 and








119905 1 2 3 4 5 6 7119884119905

3730 5850 5410 7410 6170 8000 7120







1199053ge






Acknowledgment


References
















15 no 5 pp 763ndash777 2011























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










subject to

119879

sum

119905=1

(ln1198841199053minus ln119884

1199051)119900119896

le 119879 sdot 119889119896

119900119896 (7)

ln119910119905ge ln119884

1199051+ 119904119905(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119905(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(8)

where = (1198871 1198872 1198873) and

0= (11988401 11988402 11988403) indicate the











119896


2

is equivalent to

Max 1198851015840

2=

119879

sum

119905=1

119904119905

119898119896 (9)





1199053minus ln1198841199051= 05 then (ln119884

1199053minus ln1198841199051)4minus

(ln1198841199053

minus ln1198841199051)35

= minus0026 but (ln1198841199053

minus ln1198841199051)15

minus

(ln1198841199053

minus ln1198841199051)1






1199051)119900119896 can be divided


1199051is



1198961198894



119896(1) =



119889119900119896(2)

119896=

5119889119896+ 1198894

119896

6997888rarr 119900119896(2) = ln

((5119889119896+ 1198894

119896) 6)

ln (119889119896)

119889119900119896(3)

119896=4119889119896+ 21198894

119896

6997888rarr 119900119896(3)= ln

((4119889119896+ 21198894

119896) 6)

ln (119889119896)

119889119900119896(4)

119896=3119889119896+ 31198894

119896

6997888rarr 119900119896(4)= ln

((3119889119896+ 31198894

119896) 6)

ln (119889119896)

119889119900119896(5)

119896=4119889119896+ 21198894

119896

6997888rarr 119900119896(5)= ln

((2119889119896+ 41198894

119896) 6)

ln (119889119896)

119889119900119896(6)

119896=

119889119896+ 51198894

119896

6997888rarr 119900119896(6) = ln

((119889119896+ 51198894

119896) 6)

ln (119889119896)

(10)


119896(1) = 1 119900

119896(2) = 122 119900

119896(3) =

15 119900119896(4) = 183 119900

119896(5) = 226 119900

119896(6) = 288 and 119900

119896(7) = 4


as

Min12

sum

119909=0001

10038161003816100381610038161003816119909ℎminus (120572119909

2+ 120573119909 + 120574)

10038161003816100381610038161003816

121 (11)



11990915

cong 050271199092+ 05308119909 minus 00347

11990925

cong 143571199092minus 04847119909 + 00528

1199093cong 18000119909

2minus 08953119909 + 01051

11990935

cong 209971199092minus 12322119909 + 01508

1199094cong 23440119909

2minus 15053119909 + 01890

(12)


solved instead



35 3252151NLP scale 4

QP scale 2099718143571050270 23441 005308

0 01890150801051005280

minus04847

minus00347




ℎ 120572 120573 120574

100 0 0 1101 00093 09927 minus00020102 00187 09852 minus00039

400 23440 minus15053 01890

QP problem I is

Min 1198853=

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)) )

(13)

subject to

ln119910119905ge ln119884

1199051+ 119904119896(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119896(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(14)

QP problem II is

Max 1198854=

119879

sum

119905=1

(120572 (119898 (119896)) 119904119905

2+ 120573 (119898 (119896)) 119904

119905+ 120574 (119898 (119896))) (15)

subject to

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)))

le 119879 sdot 119889119896

119900119896

ln119910119905ge ln119884

1199051+ 119904119905(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119905(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(16)


119896 119904119896 119898119896 and








119905 1 2 3 4 5 6 7119884119905

3730 5850 5410 7410 6170 8000 7120







1199053ge






Acknowledgment


References
















15 no 5 pp 763ndash777 2011























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











35 3252151NLP scale 4

QP scale 2099718143571050270 23441 005308

0 01890150801051005280

minus04847

minus00347




ℎ 120572 120573 120574

100 0 0 1101 00093 09927 minus00020102 00187 09852 minus00039

400 23440 minus15053 01890

QP problem I is

Min 1198853=

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)) )

(13)

subject to

ln119910119905ge ln119884

1199051+ 119904119896(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119896(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(14)

QP problem II is

Max 1198854=

119879

sum

119905=1

(120572 (119898 (119896)) 119904119905

2+ 120573 (119898 (119896)) 119904

119905+ 120574 (119898 (119896))) (15)

subject to

119879

sum

119905=1

(120572 (119900 (119896)) (ln1198841199053minus ln119884

1199051)2

+120573 (119900 (119896)) (ln1198841199053minus ln119884

1199051) + 120574 (119900 (119896)))

le 119879 sdot 119889119896

119900119896

ln119910119905ge ln119884

1199051+ 119904119905(ln1198841199052minus ln119884

1199051)

ln119910119905le ln119884

1199053+ 119904119905(ln1198841199052minus ln119884

1199053)

ln1198841199051

= 1198861minus

1198873

119905

ln1198841199052

= 1198862minus

1198872

119905

ln1198841199053

= 1198863minus

1198871

119905

0 le 1198861le 1198862le 1198863

0 le 1198871le 1198872le 1198873

119905 = 1 sim 119879

(16)


119896 119904119896 119898119896 and








119905 1 2 3 4 5 6 7119884119905

3730 5850 5410 7410 6170 8000 7120







1199053ge






Acknowledgment


References
















15 no 5 pp 763ndash777 2011























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905 1 2 3 4 5 6 7119884119905

3730 5850 5410 7410 6170 8000 7120







1199053ge






Acknowledgment


References
















15 no 5 pp 763ndash777 2011























Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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