Research ArticleA Modified LotkandashVolterra Model forDiffusion and Substitution of MultigenerationDRAM Processing Technologies

Hui-Chih Hung Yu-Chih Chiu andMuh-CherngWu

Department of Industrial Engineering andManagement National Chiao Tung University 1001 University Rd Hsinchu 30010 Taiwan

Correspondence should be addressed to Hui-Chih Hung hhcnctuedutw

Received 10 December 2016 Accepted 2 May 2017 Published 21 May 2017

Academic Editor Fazal M Mahomed

Copyright copy 2017 Hui-Chih Hung et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We attempt to develop an effective forecastingmodel for the diffusion and substitution ofmultigenerationDynamic RandomAccessMemory (DRAM) processing technologiesWe considermarket share data and propose amodified LotkandashVolterramodel in whichan additional constraint on the summation of market share is introducedThemean absolute error is used to measure the accuracyof our market share predictions Market share data in DRAM industries from quarter one (Q1) of 2005 to 2013 Q4 is collectedto validate the prediction accuracy Our model significantly outperforms other benchmark forecasting models of both revenueand market share data including the Bass and LotkandashVolterra models Compared to prior studies on forecasting the diffusionand substitution of multigeneration technologies our model has two new perspectives (1) allowing undetermined number ofmultigeneration technologies and inconsecutive adoption of new technologies and (2) requiring less data for forecasting newborntechnologies

1 Introduction

Dynamic Random Access Memory (DRAM) a semiconduc-tor product that stores data bits in the form of capacitorsand transistors is a necessary and critical component widelyused in information technology (IT) products Growing withthe Personal Computer (PC) boom DRAM sector is oneof the lead products in the semiconductor industry for lastdecades [1] In todayrsquos modern IT industry the demandof DRAM is also driven by the data storage and big dataretrieval in social media and cloud computing [2] Beyondthatmobile DRAMalso drove record-high profits forDRAMvendors during the second half of 2014 [3] The total revenueof DRAM products reached 39 billion US dollars in 2010which accounted for 128 of the revenue of the semicon-ductor industry According to Taiwanrsquos Industrial TechnologyResearch Institute (ITRI) almost 70 of DRAM sales are inPCs (including notebooks) and servers About 10 of DRAMsales are in computer peripheral products and 20 are inmobile communications and consumer electronics productsTherefore the prices of DRAM products have an important

influence on the costs of other industries including com-puters mobile communications and consumer electronicsindustries

Because of price competition global DRAM manufac-turers continue to upgrade their technologies and pro-duction processes The development of new technologiesallows smaller chips which enables wafers to be cut intomore chips This greatly reduces the production cost perchip and enhances price competition The market sharesof different DRAM processing technologies are shown inFigure 1 [4ndash6] In quarter one (Q1) of 2005 processingtechnology based on 100ndash120 nanometers (nm) constitutedmainstream DRAM manufacturing and its market sharereached 85 Then processing technology based on 90nanometers began to develop By 2010 Q4 processing tech-nology based on 40 nanometers became mainstream andits market share reached 55 Currently processing tech-nology based on 30 nanometers is developing and has amarket share of 21 Figure 1 shows that the rapid substi-tution of processing technology is common in the DRAMindustry

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 3038203 12 pageshttpsdoiorg10115520173038203

2 Mathematical Problems in Engineering

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Time

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30

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50

60

70

80

90

Mar

ket s

hare

()

100ndash120 nm130ndash140 nm

2Xnm3Xnm4Xnm5Xnm6Xnm

7Xnm8Xnm9Xnm

Figure 1 Rapid substitution of DRAM processing technolo-gies Data sources DRAMeXchange IHS iSuppli andMorgan Stan-ley Research (2014) Data collation Industrial Technology ResearchInstitute of Taiwan and Taiwan Institute of Economic Research

The rapid substitution of processing technologies makesprediction important in the DRAM industry Accurate pre-diction of the substitution of processing technology helpsDRAMmanufacturers to invest appropriately in research anddevelopment and get the right timing for updating manu-facturing equipment With accurate prediction technologyleaders can plan and allocate resources for the next generationof technologies and maintain their leading positions Tech-nology followers can ignore obsolete technology upgradesto reduce their capital needs and operational risks Henceproducers need good forecasting models for predicting thesubstitution of DRAM processing technology

Many models such as the Bass model and the LotkandashVolterra model are widely used to forecast the diffusion andsubstitution of two technologies The Bass model mimicsthe diffusion process of an innovative product or technologyand shows how that new product or technology can beadopted as an interaction between users and potential usersThe LotkandashVolterra model is a mathematical model thatmimics the dynamics of two species competing for commonresources Both models are built based on the data ofadoptionshipment amounts andor product revenues whichmay be affected by industry long-term trends economiccycles and other confounding factors Given this sort ofinterference both models may yield inaccurate predictions

For multiple competing technologies Peterka extendedthe empirical model of Fisher and Pry to forecast the compe-tition of multiple technologies based onmarket share data [78] Norton and Bass generalized the Bassmodel to encompassthe diffusion and substitution of successive generations ofDRAM processing technologies [9] Though establishingsignificant milestones these forecasting models for multi-generation of technologies have the following three issues

First the number of multigeneration technologies must bepreset Second inconsecutive adoption of new technologiesis not allowedThird the number of fitting parameters growsquickly with the number of multigeneration technologiesand more data is required for prediction accuracy Howeverthe total number of multigeneration technologies may notalways be predictable in practice We also observe that newgenerations of DRAM processing technologies may come topremature end without adoption

Our study concentrates on the above three issuesWe firstconsider dividing lines to partition the relatively old and newDRAM processing technologies and their market share datais adopted Then a modified LotkandashVolterra model with anadditional constraint on the summation of market shares isproposed

To measure the prediction accuracy of market share weuse the mean absolute error (MAE) Market share data inDRAM industry from 2005Q1 to 2013Q4 are collected to val-idate the prediction accuracy Our modified LotkandashVolterramodel significantly outperforms other benchmark forecast-ingmodels which include the Bass and LotkandashVolterra mod-els based on either revenue or market share data Also ourmodified LotkandashVolterra model may help DRAM mangersdetermine the timing of capital investment decisions Moreimportantly this work sheds light on how to appropriatelyapply the market share data in forecasting technologicaldiffusion and substitution

The remainder of this paper is organized as followsRelated literature is reviewed in Section 2 and our modifiedLotkandashVolterra model is proposed in Section 3 In Section 4we present our data on the DRAM industry and empiricalstudies are implemented to verify the performance of ourmodel In Section 5 we summarize our work and discussdirections for future research

2 Literature Review

In this section we review the Bass and LotkandashVolterramodelsand other variations

21 Bass Model The Bass model was introduced by Bass in1969 [10] The model describes the diffusion process of newproducts or technologies being adopted and the interactionbetween users and potential users Bass relied on diffusiontheory to model the timing of adoption that leads to aparticular S-shaped growth pattern for new products ortechnologies [11] The diffusion rate of a new product in themarket is described by the following differential equation

119889119873 (119905)119889119905 = (119901 + 119902119873 (119905)) (119872 minus 119873 (119905)) (1)

where 119873(119905) is the cumulative number of adopters or usersat time 119905 and 119872 is the potential market size The parameter119901 is the coefficient of innovation and 119902 is the coefficient ofimitation

The Bass model and variations have been widely usedfor diffusion and sales forecasting models of new productsor technologies For example Park et al used the Bass

Mathematical Problems in Engineering 3

model to develop a market penetration forecasting modelof Hydrogen Fuel Cell Vehicles incorporating infrastructureand cost reduction effects [12] Cheng used the Bass modelto construct predictive models of technology and marketdiffusion for advanced ceramic powder materials He exam-ined the relationship over time between technology diffusionand market diffusion [13] Tsai et al considered the effectof price factors and modified the Bass model to study thediffusion of the new liquid crystal display TVs [14] Kong andBi studied the diffusion of electric vehicles in China with theBass model They examined two common business modelsof battery leasing and vehicle purchasing and analyzed theimpact and influence of social network to the innovationcoefficient and imitation coefficient [15]

22 LotkandashVolterra Model The LotkandashVolterra model wasfirst proposed by Lotka in 1925 [16] This model uses twononlinear differential equations to describe the dynamics andbalance of a biological system in which two species interactas predator and prey Volterra used Lotkarsquos model and hisown data to analyze the rise and fall of fish populations inthe Adriatic [17] Lotka and Volterra based their model onthe logistic equation including terms to account for speciesrsquointeractionsThe following twodifferential equations are usedto describe how two speciesrsquo population growth rates evolveover time

119889119909 (119905)119889119905 = (1198861 + 1198871119909 (119905) + 1198881119910 (119905)) 119909 (119905) (2)

119889119910 (119905)119889119905 = (1198862 + 1198872119910 (119905) + 1198882119909 (119905)) 119910 (119905) (3)

where 119909(119905) and 119910(119905) represent the populations of two com-peting species at time 119905 respectively The terms 1199092 and 1199102represent internal self-interaction of the same species Theterm 119909119910 represents the interaction between the two speciesThe set of differential equations contains basic parametersthat affect the growth rates of both species Parameter 119886119894 isthe logistic parameter of geometric growth for species 119894whenit is evolving Parameter 119887119894 is the limitation parameter of theniche capacity for species 119894 Parameter 119888119894 is the interactionparameter with the other species

The LotkandashVolterra model has been widely used to inves-tigate relationships between biological species For exampleSmitalova and Sujan proposed a competitive relationshipbetween two competing species [18] Geijzendorffer et alused the LotkandashVolterra model to predict the long-termcoexistence patterns of grassland species [19]

Outside the field of biology Goodwin introduced themodel into the field of economics by applying it to busi-ness cycles [20] Subsequently researchers used this modelto investigate competitive relationships between differentindustries products and technologies Kreng and Wangapplied the LotkandashVolterra model to analyze the competitionbetween LCD TVs and PDP TVs [21] Kim et al consideredthe dynamic competition of mobile phone subscription inKorea and applied the LotkandashVolterra model to show thecommensalism relationship [22] Watanabe et al studied thetransition process from monochrome TV to color TV in

the 1960s and used the LotkandashVolterra model to forecast thetransition from analog broadcasting to digital broadcastingin Japan [23]

23 Diffusion and Substitution of DRAM Processing Technolo-gies Forecasting market diffusion is important when a newproduct or technology is under developmentThe Bassmodelremains the most adopted and extends diffusion modelUnlike the Bass model the LotkandashVolterra model incor-porates interactions between competitors Although theydiffer crucially in their treatment of competitorsrsquo interactionsthe Bass and LotkandashVolterra models have both been usedto forecast the diffusion and substitution of new productstechnologies technologies and industries

For example Tsai and Li found that the developmentof the integrated circuit (IC) design industry had a positiveinfluence on the ICmanufacturing and IC packing industries[24] They used the Bass and LotkandashVolterra models toverify the commercial relationships between producers inthis Taiwanese IC cluster Chiang explored the innovationgrowth of 200mm and 300mm silicon wafers from Taiwanwith both the Bass and LotkandashVolterra models They foundthat the forecasting performances of the both models aresimilar [25] Chiang andWong used the two models to studythe competitive diffusion of desktop and notebook shipmentsin Taiwan and found evidence of predator-prey relationships[26]

Both models are built based on the data of adop-tionshipment amounts andor product revenues which maybe affected by industry long-term trends economic cyclesand other confounding factors Unfortunately this sort ofinterference is one of the most important characteristicsof DRAM industry and both models may yield inaccuratepredictions under this sort of interference It is thereforequestionable that the applicability of revenue data and theeffect of data characteristics are rarely analyzed in DRAMindustry

24Models forMultiple Competing Technologies Under rapidsubstitution multiple generations of DRAMprocessing tech-nologies coexist in the market Several models are proposedto encompass the diffusion and substitution of multiple com-peting technologies Peterka questioned the applicability ofrevenue data and adopted market share data [7] Competingtechnologies are considered pairwise to follow Fisher andPryrsquos empirical formula where the logarithmic function ofmarket share ratio is a linear function in time 119905 [8] ThePeterka model can be described by the following equations

ln119891119894 (119905)119891119895 (119905)

= 119896119894119895 minus 119888119894119895 (119905 minus 1199050) for any 119894 and 119895

sum119894

119891119894 (119905) = 1(4)

where 1199050 is the starting time of the model and 119891119894(119905) is themarket share of the 119894th competing technologies in time 119905 Notethat Peterka assumed that no technology can start from zeroin this model [4] A new technology must be initiated and

4 Mathematical Problems in Engineering

established by external investment before 1199050 Thus 119891119894(1199050) ispredetermined according to the external investment and 119896119894119895 =ln(119891119894(1199050)119891119895(1199050)) is the given ratio of initial market sharesAs a result this model may not be suitable for successivegenerations of DRAM processing technologies where futuregeneration may be zero at the starting time of the model

Norton andBass generalized the Bassmodel andmodeledthe demand growthdecline for successive generations ofmemory and logic circuits [9] They assume that the numberof successive generations is preset and each successive gener-ation has its own market potential and market penetrationA new generation of a product or technology always attractsusers from its potential market and the predecessor genera-tion The Norton-Bass model can be illustrated by a productof three successive generations

1198781 (119905) = 1198651 (119905)1198981 [1 minus 1198652 (119905 minus 1205912)]

1198782 (119905) = 1198652 (119905 minus 1205912) [1198982 + 1198651 (119905)1198981] [1 minus 1198653 (119905 minus 1205913)]

1198783 (119905) = 1198653 (119905 minus 1205912) 1198983 + 1198652 (119905 minus 1205912) [1198982 + 1198651 (119905)1198981]

(5)

where the 119894th generation 119894 is introduced at 120591119894 and 1205911 =0 Moreover 119898119894 is the potential sales of the 119894th generationand 119878119894(119905) is the sales of the 119894th generation in time period 119905Note that 119865119894(119905) is the cumulative distribution function of theadoption rate of the 119894th generation

The formula is

119865119894 (119905) =

1 minus exp [minus (119901119894 + 119902119894) 119905]1 + (119901119894119902119894) exp [minus (119901119894 + 119902119894) 119905]

for 119905 ge 1205911198940 otherwise

(6)

where the parameter 119901119894 is the coefficient of innovation and 119902119894is the coefficient of imitation of the 119894th generation

In these models there are three common issues Firstthe number of technologies must be preset Each technologyis assumed to be predictable and all newborn technologiescan go through the complete adoption cycle and competeswith earlier ones Unfortunately we observe that some newtechnologies may come to a premature end without adoptionin DRAM industry As a result the total number of newtechnologies may not be predetermined

Second the inconsecutive adoption of new technologiesis not allowed When adopting the new generation of DRAMprocessing technologies customers may intend to bypassthe immediate successor and adopt the later one due to theconsideration of market potential and transcendence As aresult some generations of DRAM processing technologiesmay not go through the complete adoption cycle and competewith earlier ones

Third the number of fitting parameters grows quicklywith the number of technologies andmore data is required forprediction accuracy For multiple competing technologiesthe existing forecasting models are designed to mimic thediffusion and substitution by equations among all technolo-gies Thus the number of fitting parameters grows quicklywith the number of technologies However it is natural thatmature technologies have plenty of data andnew technologies

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New process technology (less than Old process technology Total market revenues

Time

2000

6000

10000

Reve

nue (

mill

ion

USD

)

0

(9 nm and more than 90

Figure 2 DRAM revenue changes for new and old process tech-nologies Data sources DRAMeXchange IHS iSuppli and MorganStanley Research (2014) Data collation Industrial TechnologyResearch Institute of Taiwan and Taiwan Institute of EconomicResearch

have little data collectable in initial stages As a result maturetechnologies dominate the model and errors are cumulatedto new technologies In the worst case some models may notbe applied due to the insufficient data for parameters fitting

3 Model

Wenowpropose themodified LotkandashVolterramodel for fore-casting the diffusion and substitution of DRAM processingtechnologies

31 Market Share Data In this study we focus on the substi-tution between old and new DRAM processing technologiesAt a given point in time there usually coexist differentgenerations of processing technology on the DRAM marketFor example 60- 70- 80- 90- 100ndash120- and 130ndash140-nanometer processing technologies coexisted on the DRAMmarket in 2007 Q3 (see Figure 1) We first divide DRAMprocessing technologies into two groups old and new 90nanometers are used as the dividing line between the twoHence DRAM processing technologies of less than 90nanometers are classified as new technologies and those of90 nanometers and more than 90 nanometers are classifiedas old technologies

Data on shipments andor product revenues are typicallyused to forecast technological substitution However thesedatamay be affected bymarket fluctuations that have nothingto do with substitution Such factors include industry long-term trends economic cycles and other confounding factorsTo illustrate Figure 2 shows quarterly DRAM revenue datafrom 2005 Q1 to 2013 Q4 Unstable fluctuations in DRAMrevenue are apparent

Market shares measure the percentages of market salesaccounted for by competitive products or technologiesChanges in market shares reflect substitution not long-term trends economic cycles and other confounding factors

Mathematical Problems in Engineering 5

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New process technologies (less than Old process technologiesTotal market share

Time

0102030405060708090

100

Mar

ket s

hare

()

90 nm)(90 nm and more than 90 nm)

Figure 3 DRAM market share changes for new and old processtechnologies Data sources DRAMeXchange IHS iSuppli andMor-gan Stanley Research (2014) Data collation Industrial TechnologyResearch Institute of Taiwan and Taiwan Institute of EconomicResearch

Figure 3 shows that market shares fluctuate less than doesrevenue Hence our model is to use market share data forforecasting

32Model Formulation Usingmarket share data one naturalrestriction is that market shares must sum to 100 By setting119872 = 1 and 119910(119905) as the cumulative market share of newprocessing technologies at time 119905 the Bass model given by (1)can be rewritten as follows

119889119910 (119905)119889119905 = (119901 + 119902119910 (119905)) (1 minus 119910 (119905))

= 1198621 + 1198611119910 (119905) + 1198601119910 (119905)2(7)

where 1198621 = 119901 1198611 = 119902 minus 119901 and 1198601 = minus 119902When applying market share data to the LotkandashVolterra

model we let 119909(119905) and 119910(119905) be the market shares of old andnew DRAM processing technologies at time 119905 respectivelyGiven a total market share of 100 we have

119909 (119905) + 119910 (119905) = 1 (8)

From (8) (2) can be rewritten as

119889119909 (119905)119889119905 = (1198861 + 1198871119909 (119905) + 1198881119910 (119905)) 119909 (119905)

= ((1198861 + 1198881) + (1198871 minus 1198881) 119909 (119905)) 119909 (119905)

=1198612119909 (119905) + 1198602119909 (119905)2

(9)

where 1198612 = 1198861 + 1198881 and 1198602 = 1198871 minus 1198881Equations (7) and (9) show that the Bass and

LotkandashVolterra models of market shares degenerate toour model because total market shares must be 100 Thismodified LotkandashVolterra model is formulated as follows

119889119909 (119905)119889119905 = 119862 + 119861119909 (119905) + 119860119909 (119905)2

119910 (119905) = 1 minus 119909 (119905) (10)

33 Prediction Evaluation Traditionally both the meanabsolute percentage error (MAPE) and the mean squarederror (MSE) are used to evaluate prediction accuracy How-ever depending on the data and models used the MAPEand MSE are not necessarily the best measures of predictionaccuracyTherefore we use theMAE to evaluate the accuracyof our market share predictions

Given a number of forecast periods of 119899 and letting 119910(119905)and 119910(119905) denote the actual and forecast values for time 119905respectively the MAPE MSE and MAE are defined as

MAPE = 1119899119899

sum119905=1

10038161003816100381610038161003816100381610038161003816119910 (119905) minus 119910 (119905)

119910 (119905)10038161003816100381610038161003816100381610038161003816

MSE =119899

sum119905=1

(119910 (119905) minus 119910 (119905))2119899

MAE = 1119899119899

sum119905=1

1003816100381610038161003816(119910 (119905) minus 119910 (119905))1003816100381610038161003816

(11)

The MAPE conveys the percentage difference betweenthe actual values and forecast values Small 119910(119905) values cangenerate large MAPE values Actual values close to zero cangenerate infinitely large MAPEs Therefore the MAPE isnot suitable for prediction evaluation when actual values aresmall Given that market shares are by definition between 0and 1 the MAPE is not suited to our purposes

The MSE is the mean squared difference between theactual and forecast values Although it is not troubled by smallactual values because the MSE is dependent on the units ofmeasurement of119910(119905) it cannot be used for comparisons basedon different types of dataThus theMSE is not suitable for ourstudy because we want to compare the prediction accuracy ofboth revenue and market share forecasts

The MAE conveys the absolute difference between theactual and forecast values It is untroubled by small actualvalues Moreover when applied to market share data theMAE records the average absolute error as a percentageThusto ensure the comparability of our prediction evaluationswe compare the MAEs for market share predictions with theMAPEs for revenue predictions

4 Data Collection and Empirical Study

In this section our model is implemented to forecast thediffusion and substitution of DRAMprocessing technologiesOther benchmark forecasting models are used for com-parison These benchmark forecasting models include thestandard Bass and LotkandashVolterra models

Data on revenue and market share have different charac-teristics and influencesWe collect global revenue andmarketshare data on different DRAM processing technologies anduse them to evaluate prediction accuracy Our data sourcesinclude DRAMeXchange iSuppli and the industry analysisreports published by Taiwanrsquos ITRI In total 36 pairs ofrecords from 2005 Q1 to 2013 Q4 were collected

41 Numerical Study To distinguish between old and newprocessing technologies we use three dividing lines 90 80

6 Mathematical Problems in Engineering

Table 1 Revenues and market share for old and new processing technologies from 2005 Q1 to 2013 Q4

QuarterRevenue (millions USD) Market share

90 nm 80 nm 70 nm 90 nm 80 nm 70 nmOld New Old New Old New Old New Old New Old New

05Q1 6574 lt1 6574 lt1 6574 lt1 1 lt001 1 lt001 1 lt00105Q2 5715 lt1 5715 lt1 5715 lt1 1 lt001 1 lt001 1 lt00105Q3 6394 lt1 6394 lt1 6394 lt1 1 lt001 1 lt001 1 lt00105Q4 6372 lt1 6372 lt1 6372 lt1 1 lt001 1 lt001 1 lt00106Q1 6422 131 6553 lt1 6553 lt1 098 002 1 lt001 1 lt00106Q2 7158 457 7615 lt1 7615 lt1 094 006 1 lt001 1 lt00106Q3 8455 636 9091 lt1 9091 lt1 093 007 1 lt001 1 lt00106Q4 8897 1695 10592 lt1 10592 lt1 084 016 1 lt001 1 lt00107Q1 6961 2707 9475 193 9668 lt1 072 028 098 002 1 lt00107Q2 4380 2920 6716 584 7154 146 06 04 092 008 098 00207Q3 3584 4380 6929 1035 7645 319 045 055 087 013 096 00407Q4 2055 4173 4484 1744 5294 934 033 067 072 028 085 01508Q1 1408 4459 3462 2405 4457 1408 024 076 059 041 076 02408Q2 1042 5470 2800 3712 4298 2214 016 084 043 057 066 03408Q3 758 6134 2205 4687 3791 3101 011 089 032 068 055 04508Q4 286 3793 816 3263 1713 2366 007 093 02 08 042 05809Q1 95 3073 222 2946 919 2249 003 097 007 093 029 07109Q2 81 3984 203 3862 772 3293 002 098 005 095 019 08109Q3 122 5991 245 5868 856 5257 002 098 004 096 014 08609Q4 87 8595 174 8508 955 7721 001 099 002 098 011 08910Q1 93 9195 186 9102 836 8452 001 099 002 098 009 09110Q2 104 10317 208 10213 834 9587 001 099 002 098 008 09210Q3 108 10671 216 10563 647 10132 001 099 002 098 006 09410Q4 86 8558 173 8471 346 8298 001 099 002 098 004 09611Q1 83 8220 166 8137 332 7971 001 099 002 098 004 09611Q2 lt1 8147 lt1 8147 163 7984 lt001 1 lt001 1 002 09811Q3 lt1 6565 lt1 6565 66 6499 lt001 1 lt001 1 001 09911Q4 lt1 6453 lt1 6453 lt1 6453 lt001 1 lt001 1 lt001 112Q1 lt1 6271 lt1 6271 lt1 6271 lt001 1 lt001 1 lt001 112Q2 lt1 7023 lt1 7023 lt1 7023 lt001 1 lt001 1 lt001 112Q3 lt1 6429 lt1 6429 lt1 6429 lt001 1 lt001 1 lt001 112Q4 lt1 6864 lt1 6864 lt1 6864 lt001 1 lt001 1 lt001 113Q1 lt1 6870 lt1 6870 lt1 6870 lt001 1 lt001 1 lt001 113Q2 lt1 8531 lt1 8531 lt1 8531 lt001 1 lt001 1 lt001 113Q3 lt1 9300 lt1 9300 lt1 9300 lt001 1 lt001 1 lt001 113Q4 lt1 9748 lt1 9748 lt1 9748 lt001 1 lt001 1 lt001 1Data sources DRAMeXchange IHS iSuppli amp Morgan Stanley Research (2014)Data collation Industrial Technology Research Institute of Taiwan amp Taiwan Institute of Economic Research

and 70 nanometers as examples The raw data for each aregiven in Table 1

We are also interested in prediction accuracy at differentstages of diffusion The 99 95 90 80 70 60and 50 market shares for new processing technologies areearmarked as seven stages of diffusion In each stage recordspertaining to the time before the new processing technologiessurpassed the earmarkedmarket shares were used as trainingdata to fit the model parametersThe remaining records wereused as testing data to evaluate prediction accuracy

For example at the dividing line of 90 nanometers newprocessing technologies are those using 90 nanometers orless Table 1 shows that these new processing technologiesreached a 99 market share in 2009 Q4 Thus in Stage 119 pairs of revenue records for the old and new processingtechnologies from 2005 Q1 to 2009 Q3 were used as trainingdata to fit the Bass and LotkandashVolterra models by usingordinary least squares The remaining 17 pairs of revenuerecords were used as testing data to individually evaluate theMAPEs for the old and new processing technologies

Mathematical Problems in Engineering 7

Similarly 19 market share records from 2005 Q1 to 2009Q3 were used as training data to fit the Bass LotkandashVolterraand modified LotkandashVolterra models by using ordinary leastsquares in Stage 1 The remaining 17 market share recordswere used as testing data to individually evaluate the MAEsfor the old andnewprocessing technologiesThe results for allseven stages are reported in Table 2 Tables 3 and 4 report theresults of our numerical studies for the 80- and 70-nanometerdividing lines respectively

42 Analysis of Data Selection The results reported in theprevious subsection indicate that the lower the number ofrecords used as training data the higher the prediction errorsMoreover the market share predictions are significantlybetter than the revenue predictions from both the Bass andLotkandashVolterra models Consider the 90-nanometer dividingline as an example (see Table 2) When using revenue dataon the Bass model the prediction errors range from 907 toinfinity for the seven stages When using market share dataon the Bass model the prediction errors are much smallerbetween 056 and 1711 for the seven stages Similarlyfor the LotkandashVolterra model using revenue data generatesprediction errors of from 952 to infinity whereas usingmarket share data gives prediction errors of between 014and 9190 Results based on the other diving lines are similar(see Tables 3 and 4)

Arguably revenue data are not suitable for forecasting thediffusion and substitution of technologies This is becauserevenue data reflect market fluctuations caused by factorsother than substitution including industry long-term trendseconomic cycles and other confounding factors

For example global DRAM industry revenue has exhib-ited a prolonged growth trend particularly in terms ofdemand for computers and mobile communications prod-ucts In 2007 the DRAM industry invested huge amountsof capital to expand production capacity for upgrading toa new PC operating system Windows Vista However salesof Windows Vista fell far short of market expectations Thisled to an oversupply of DRAM products which caused spotprices and revenue to crash in 2007 In 2008 global DRAMrevenue continued to decline because of the global financialtsunami The consequent unexpected fall in global DRAMrevenue from 2006 Q4 to 2009 Q1 greatly affected predictionaccuracy

Market share data are more suitable than revenue datafor forecasting the diffusion and substitution of DRAMprocessing technologies This is because the market sharedata record percentages and because the total market shareis fixed at 100 Because market share data are unaffectedby seasonal factors and economic cycles they reflect mainlytechnological substitution Consequently using market sharedata generates more accurate predictions and is appropriatefor analyzing technological substitution

43 Analysis of Model Selection Having shown that usingmarket share data increases prediction accuracy for allforecasting models in this subsection we use market sharedata to compare the Bass LotkandashVolterra and modifiedLotkandashVolterra models

First note the symmetry of the modified LotkandashVolterramodel based on market share data In each stage theprediction errors for the old and new processing technologiesare the same This is because the market shares are forced tosum to 100 in (10) of the modified LotkandashVolterra modelAs a result the forecast values for the old and new processingtechnologies sum to 100 and must have the same MAEsSimilarly for the Bass model based on market share data theprediction errors for the old and new processing technologiesare highly symmetrical Again this arises because of thecharacteristics of the market share data

By contrast the LotkandashVolterra model based on marketshare data does not generate symmetric prediction errors forthe old and new processing technologies This is because themarket shares of old and new processing technologies aretreated as two species and there is no total population limitfor the two species in the LotkandashVolterra model The forecastmarket shares are not constrained to sum to 100 This iswhy the prediction errors from the LotkandashVolterra model arerelatively high

Let us compare the accuracy of the models based onmarket share data In Stages 1 to 7 of Table 2 (which reportsresults for the 90-nanometer dividing line) the Bass modelgenerates prediction errors of between 056 and 1711 theLotkandashVolterramodel generates prediction errors of between014 and 9190 and the modified LotkandashVolterra modelgenerates prediction errors of between 016 and 050(Tables 3 and 4 show that the other dividing lines generatesimilar results) Therefore when forecasting market sharesthemodified LotkandashVolterramodel outperforms the Bass andLotkandashVolterra models

5 Conclusions and Future Research

In this study we propose a modified LotkandashVolterra modelto forecast the diffusion and substitution of DRAM pro-cessing technologies Multiple generations of DRAM pro-cessing technologies are partitioned into old and new gen-erations and the quarterly data from 2005 Q1 to 2013 Q4is used for empirical studies In our model there are fourfeatures

First the type of data affects the prediction accuracy of thetechnological substitution trends Although shipment andorrevenue data have been widely used in existing research theymay be contaminated by confounding factors that reduceprediction accuracy This can be avoided by using marketshare data which are purged of such factors as long-termtrends and business cycles Consequently using market sharedata can generate more accurate forecasts of technologicalsubstitution

Second existing models for multiple competing tech-nologies require predetermined number of technologies andeach technology is assumed to go through the complete adop-tion cycle However both are not obvious in DRAM indus-try Focusing on these issues our modified LotkandashVolterramodel allows undetermined number of multigenerationtechnologies and inconsecutive adoption of new technolo-gies

8 Mathematical Problems in Engineering

Table2Predictio

nerrorsbasedon

different

datasets

mod

elsandstages90-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

sharefor

new

processin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out

119872=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

1917

98

Old

gt105

gt105

057

014

016

New

1113

1222

057

021

016

(2)

1620

95

Old

gt105

gt105

161

017

020

New

969

1113

056

018

020

(3)

1521

90

Old

gt105

gt105

245

024

024

New

940

952

244

032

024

(4)

1323

80

Old

gt105

gt105

505

029

029

New

907

2346

233

137

029

(5)

1224

70

Old

gt105

gt105

494

025

033

New

1620

6291

537

111

033

(6)

1125

60

Old

gt105

gt105

461

036

044

New

1105

9334

460

046

044

(7)

1026

50

Old

gt105

gt105

1712

4191

050

New

1450

9334

1711

9190

050

Mathematical Problems in Engineering 9

Table3Predictio

nerrorsbasedon

different

datasets

mod

elsandstages80-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

261

99

Old

lt001

lt001

lt001

lt001

lt001

New

lt001

lt001

lt001

lt001

lt001

(2)

243

95

Old

1950

2428

161

128

023

New

1236

1573

161

226

023

(3)

207

90

Old

14624

6670

113

073

086

New

1095

4937

113

072

086

(4)

1710

80

Old

71537

3235

10

3087

104

New

5394

5653

10

316

410

4

(5)

1611

70

Old

2474525

3608

780

087

110

New

4951

6009

780

137

110

(6)

1611

60

Old

2474525

3608

780

087

112

New

4951

6009

780

107

112

(7)

1413

50

Old

322327

3947

4026

141

114

New

2415

5919

40

27

152

114

10 Mathematical Problems in Engineering

Table4Predictio

nerrorsbasedon

different

datasets

mod

elsandstages70-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

ds

Num

bero

ftesting

perio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

2610

98

Old

gt105

gt105

141

007

006

New

871

1195

141

063

006

(2)

2313

95

Old

gt105

gt105

201

022

022

New

1396

1927

201

071

022

(3)

2016

90

Old

gt105

gt105

172

035

041

New

4106

4717

17

1071

041

(4)

1719

80

Old

gt105

gt105

089

051

058

New

5557

5906

089

073

058

(5)

1620

70

Old

gt105

gt105

951

077

062

New

5210

6519

95

0089

062

(6)

1620

60

Old

gt105

gt105

951

077

066

New

5210

6519

95

0089

066

(7)

1521

50

Old

gt105

gt105

2983

075

068

New

2127

9129

2981

098

068

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

Mathematical Problems in Engineering 3

model to develop a market penetration forecasting modelof Hydrogen Fuel Cell Vehicles incorporating infrastructureand cost reduction effects [12] Cheng used the Bass modelto construct predictive models of technology and marketdiffusion for advanced ceramic powder materials He exam-ined the relationship over time between technology diffusionand market diffusion [13] Tsai et al considered the effectof price factors and modified the Bass model to study thediffusion of the new liquid crystal display TVs [14] Kong andBi studied the diffusion of electric vehicles in China with theBass model They examined two common business modelsof battery leasing and vehicle purchasing and analyzed theimpact and influence of social network to the innovationcoefficient and imitation coefficient [15]

22 LotkandashVolterra Model The LotkandashVolterra model wasfirst proposed by Lotka in 1925 [16] This model uses twononlinear differential equations to describe the dynamics andbalance of a biological system in which two species interactas predator and prey Volterra used Lotkarsquos model and hisown data to analyze the rise and fall of fish populations inthe Adriatic [17] Lotka and Volterra based their model onthe logistic equation including terms to account for speciesrsquointeractionsThe following twodifferential equations are usedto describe how two speciesrsquo population growth rates evolveover time

119889119909 (119905)119889119905 = (1198861 + 1198871119909 (119905) + 1198881119910 (119905)) 119909 (119905) (2)

119889119910 (119905)119889119905 = (1198862 + 1198872119910 (119905) + 1198882119909 (119905)) 119910 (119905) (3)

where 119909(119905) and 119910(119905) represent the populations of two com-peting species at time 119905 respectively The terms 1199092 and 1199102represent internal self-interaction of the same species Theterm 119909119910 represents the interaction between the two speciesThe set of differential equations contains basic parametersthat affect the growth rates of both species Parameter 119886119894 isthe logistic parameter of geometric growth for species 119894whenit is evolving Parameter 119887119894 is the limitation parameter of theniche capacity for species 119894 Parameter 119888119894 is the interactionparameter with the other species

The LotkandashVolterra model has been widely used to inves-tigate relationships between biological species For exampleSmitalova and Sujan proposed a competitive relationshipbetween two competing species [18] Geijzendorffer et alused the LotkandashVolterra model to predict the long-termcoexistence patterns of grassland species [19]

Outside the field of biology Goodwin introduced themodel into the field of economics by applying it to busi-ness cycles [20] Subsequently researchers used this modelto investigate competitive relationships between differentindustries products and technologies Kreng and Wangapplied the LotkandashVolterra model to analyze the competitionbetween LCD TVs and PDP TVs [21] Kim et al consideredthe dynamic competition of mobile phone subscription inKorea and applied the LotkandashVolterra model to show thecommensalism relationship [22] Watanabe et al studied thetransition process from monochrome TV to color TV in

the 1960s and used the LotkandashVolterra model to forecast thetransition from analog broadcasting to digital broadcastingin Japan [23]

23 Diffusion and Substitution of DRAM Processing Technolo-gies Forecasting market diffusion is important when a newproduct or technology is under developmentThe Bassmodelremains the most adopted and extends diffusion modelUnlike the Bass model the LotkandashVolterra model incor-porates interactions between competitors Although theydiffer crucially in their treatment of competitorsrsquo interactionsthe Bass and LotkandashVolterra models have both been usedto forecast the diffusion and substitution of new productstechnologies technologies and industries

For example Tsai and Li found that the developmentof the integrated circuit (IC) design industry had a positiveinfluence on the ICmanufacturing and IC packing industries[24] They used the Bass and LotkandashVolterra models toverify the commercial relationships between producers inthis Taiwanese IC cluster Chiang explored the innovationgrowth of 200mm and 300mm silicon wafers from Taiwanwith both the Bass and LotkandashVolterra models They foundthat the forecasting performances of the both models aresimilar [25] Chiang andWong used the two models to studythe competitive diffusion of desktop and notebook shipmentsin Taiwan and found evidence of predator-prey relationships[26]

Both models are built based on the data of adop-tionshipment amounts andor product revenues which maybe affected by industry long-term trends economic cyclesand other confounding factors Unfortunately this sort ofinterference is one of the most important characteristicsof DRAM industry and both models may yield inaccuratepredictions under this sort of interference It is thereforequestionable that the applicability of revenue data and theeffect of data characteristics are rarely analyzed in DRAMindustry

24Models forMultiple Competing Technologies Under rapidsubstitution multiple generations of DRAMprocessing tech-nologies coexist in the market Several models are proposedto encompass the diffusion and substitution of multiple com-peting technologies Peterka questioned the applicability ofrevenue data and adopted market share data [7] Competingtechnologies are considered pairwise to follow Fisher andPryrsquos empirical formula where the logarithmic function ofmarket share ratio is a linear function in time 119905 [8] ThePeterka model can be described by the following equations

ln119891119894 (119905)119891119895 (119905)

= 119896119894119895 minus 119888119894119895 (119905 minus 1199050) for any 119894 and 119895

sum119894

119891119894 (119905) = 1(4)

where 1199050 is the starting time of the model and 119891119894(119905) is themarket share of the 119894th competing technologies in time 119905 Notethat Peterka assumed that no technology can start from zeroin this model [4] A new technology must be initiated and

4 Mathematical Problems in Engineering

established by external investment before 1199050 Thus 119891119894(1199050) ispredetermined according to the external investment and 119896119894119895 =ln(119891119894(1199050)119891119895(1199050)) is the given ratio of initial market sharesAs a result this model may not be suitable for successivegenerations of DRAM processing technologies where futuregeneration may be zero at the starting time of the model

Norton andBass generalized the Bassmodel andmodeledthe demand growthdecline for successive generations ofmemory and logic circuits [9] They assume that the numberof successive generations is preset and each successive gener-ation has its own market potential and market penetrationA new generation of a product or technology always attractsusers from its potential market and the predecessor genera-tion The Norton-Bass model can be illustrated by a productof three successive generations

1198781 (119905) = 1198651 (119905)1198981 [1 minus 1198652 (119905 minus 1205912)]

1198782 (119905) = 1198652 (119905 minus 1205912) [1198982 + 1198651 (119905)1198981] [1 minus 1198653 (119905 minus 1205913)]

1198783 (119905) = 1198653 (119905 minus 1205912) 1198983 + 1198652 (119905 minus 1205912) [1198982 + 1198651 (119905)1198981]

(5)

where the 119894th generation 119894 is introduced at 120591119894 and 1205911 =0 Moreover 119898119894 is the potential sales of the 119894th generationand 119878119894(119905) is the sales of the 119894th generation in time period 119905Note that 119865119894(119905) is the cumulative distribution function of theadoption rate of the 119894th generation

The formula is

119865119894 (119905) =

1 minus exp [minus (119901119894 + 119902119894) 119905]1 + (119901119894119902119894) exp [minus (119901119894 + 119902119894) 119905]

for 119905 ge 1205911198940 otherwise

(6)

where the parameter 119901119894 is the coefficient of innovation and 119902119894is the coefficient of imitation of the 119894th generation

In these models there are three common issues Firstthe number of technologies must be preset Each technologyis assumed to be predictable and all newborn technologiescan go through the complete adoption cycle and competeswith earlier ones Unfortunately we observe that some newtechnologies may come to a premature end without adoptionin DRAM industry As a result the total number of newtechnologies may not be predetermined

Second the inconsecutive adoption of new technologiesis not allowed When adopting the new generation of DRAMprocessing technologies customers may intend to bypassthe immediate successor and adopt the later one due to theconsideration of market potential and transcendence As aresult some generations of DRAM processing technologiesmay not go through the complete adoption cycle and competewith earlier ones

Third the number of fitting parameters grows quicklywith the number of technologies andmore data is required forprediction accuracy For multiple competing technologiesthe existing forecasting models are designed to mimic thediffusion and substitution by equations among all technolo-gies Thus the number of fitting parameters grows quicklywith the number of technologies However it is natural thatmature technologies have plenty of data andnew technologies

05Q

105

Q2

05Q

305

Q4

06Q

106

Q2

06Q

306

Q4

07Q

107

Q2

07Q

307

Q4

08Q

108

Q2

08Q

308

Q4

09Q

109

Q2

09Q

309

Q4

10Q

110

Q2

10Q

310

Q4

11Q

111

Q2

11Q

311

Q4

12Q

112

Q2

12Q

312

Q4

13Q

113

Q2

13Q

313

Q4

New process technology (less than Old process technology Total market revenues

Time

2000

6000

10000

Reve

nue (

mill

ion

USD

)

0

(9 nm and more than 90

Figure 2 DRAM revenue changes for new and old process tech-nologies Data sources DRAMeXchange IHS iSuppli and MorganStanley Research (2014) Data collation Industrial TechnologyResearch Institute of Taiwan and Taiwan Institute of EconomicResearch

have little data collectable in initial stages As a result maturetechnologies dominate the model and errors are cumulatedto new technologies In the worst case some models may notbe applied due to the insufficient data for parameters fitting

3 Model

Wenowpropose themodified LotkandashVolterramodel for fore-casting the diffusion and substitution of DRAM processingtechnologies

31 Market Share Data In this study we focus on the substi-tution between old and new DRAM processing technologiesAt a given point in time there usually coexist differentgenerations of processing technology on the DRAM marketFor example 60- 70- 80- 90- 100ndash120- and 130ndash140-nanometer processing technologies coexisted on the DRAMmarket in 2007 Q3 (see Figure 1) We first divide DRAMprocessing technologies into two groups old and new 90nanometers are used as the dividing line between the twoHence DRAM processing technologies of less than 90nanometers are classified as new technologies and those of90 nanometers and more than 90 nanometers are classifiedas old technologies

Data on shipments andor product revenues are typicallyused to forecast technological substitution However thesedatamay be affected bymarket fluctuations that have nothingto do with substitution Such factors include industry long-term trends economic cycles and other confounding factorsTo illustrate Figure 2 shows quarterly DRAM revenue datafrom 2005 Q1 to 2013 Q4 Unstable fluctuations in DRAMrevenue are apparent

Market shares measure the percentages of market salesaccounted for by competitive products or technologiesChanges in market shares reflect substitution not long-term trends economic cycles and other confounding factors

Mathematical Problems in Engineering 5

05Q

105

Q2

05Q

305

Q4

06Q

106

Q2

06Q

306

Q4

07Q

107

Q2

07Q

307

Q4

08Q

108

Q2

08Q

308

Q4

09Q

109

Q2

09Q

309

Q4

10Q

110

Q2

10Q

310

Q4

11Q

111

Q2

11Q

311

Q4

12Q

112

Q2

12Q

312

Q4

13Q

113

Q2

13Q

313

Q4

New process technologies (less than Old process technologiesTotal market share

Time

0102030405060708090

100

Mar

ket s

hare

()

90 nm)(90 nm and more than 90 nm)

Figure 3 DRAM market share changes for new and old processtechnologies Data sources DRAMeXchange IHS iSuppli andMor-gan Stanley Research (2014) Data collation Industrial TechnologyResearch Institute of Taiwan and Taiwan Institute of EconomicResearch

Figure 3 shows that market shares fluctuate less than doesrevenue Hence our model is to use market share data forforecasting

32Model Formulation Usingmarket share data one naturalrestriction is that market shares must sum to 100 By setting119872 = 1 and 119910(119905) as the cumulative market share of newprocessing technologies at time 119905 the Bass model given by (1)can be rewritten as follows

119889119910 (119905)119889119905 = (119901 + 119902119910 (119905)) (1 minus 119910 (119905))

= 1198621 + 1198611119910 (119905) + 1198601119910 (119905)2(7)

where 1198621 = 119901 1198611 = 119902 minus 119901 and 1198601 = minus 119902When applying market share data to the LotkandashVolterra

model we let 119909(119905) and 119910(119905) be the market shares of old andnew DRAM processing technologies at time 119905 respectivelyGiven a total market share of 100 we have

119909 (119905) + 119910 (119905) = 1 (8)

From (8) (2) can be rewritten as

119889119909 (119905)119889119905 = (1198861 + 1198871119909 (119905) + 1198881119910 (119905)) 119909 (119905)

= ((1198861 + 1198881) + (1198871 minus 1198881) 119909 (119905)) 119909 (119905)

=1198612119909 (119905) + 1198602119909 (119905)2

(9)

where 1198612 = 1198861 + 1198881 and 1198602 = 1198871 minus 1198881Equations (7) and (9) show that the Bass and

LotkandashVolterra models of market shares degenerate toour model because total market shares must be 100 Thismodified LotkandashVolterra model is formulated as follows

119889119909 (119905)119889119905 = 119862 + 119861119909 (119905) + 119860119909 (119905)2

119910 (119905) = 1 minus 119909 (119905) (10)

33 Prediction Evaluation Traditionally both the meanabsolute percentage error (MAPE) and the mean squarederror (MSE) are used to evaluate prediction accuracy How-ever depending on the data and models used the MAPEand MSE are not necessarily the best measures of predictionaccuracyTherefore we use theMAE to evaluate the accuracyof our market share predictions

Given a number of forecast periods of 119899 and letting 119910(119905)and 119910(119905) denote the actual and forecast values for time 119905respectively the MAPE MSE and MAE are defined as

MAPE = 1119899119899

sum119905=1

10038161003816100381610038161003816100381610038161003816119910 (119905) minus 119910 (119905)

119910 (119905)10038161003816100381610038161003816100381610038161003816

MSE =119899

sum119905=1

(119910 (119905) minus 119910 (119905))2119899

MAE = 1119899119899

sum119905=1

1003816100381610038161003816(119910 (119905) minus 119910 (119905))1003816100381610038161003816

(11)

The MAPE conveys the percentage difference betweenthe actual values and forecast values Small 119910(119905) values cangenerate large MAPE values Actual values close to zero cangenerate infinitely large MAPEs Therefore the MAPE isnot suitable for prediction evaluation when actual values aresmall Given that market shares are by definition between 0and 1 the MAPE is not suited to our purposes

The MSE is the mean squared difference between theactual and forecast values Although it is not troubled by smallactual values because the MSE is dependent on the units ofmeasurement of119910(119905) it cannot be used for comparisons basedon different types of dataThus theMSE is not suitable for ourstudy because we want to compare the prediction accuracy ofboth revenue and market share forecasts

The MAE conveys the absolute difference between theactual and forecast values It is untroubled by small actualvalues Moreover when applied to market share data theMAE records the average absolute error as a percentageThusto ensure the comparability of our prediction evaluationswe compare the MAEs for market share predictions with theMAPEs for revenue predictions

4 Data Collection and Empirical Study

In this section our model is implemented to forecast thediffusion and substitution of DRAMprocessing technologiesOther benchmark forecasting models are used for com-parison These benchmark forecasting models include thestandard Bass and LotkandashVolterra models

Data on revenue and market share have different charac-teristics and influencesWe collect global revenue andmarketshare data on different DRAM processing technologies anduse them to evaluate prediction accuracy Our data sourcesinclude DRAMeXchange iSuppli and the industry analysisreports published by Taiwanrsquos ITRI In total 36 pairs ofrecords from 2005 Q1 to 2013 Q4 were collected

41 Numerical Study To distinguish between old and newprocessing technologies we use three dividing lines 90 80

6 Mathematical Problems in Engineering

Table 1 Revenues and market share for old and new processing technologies from 2005 Q1 to 2013 Q4

QuarterRevenue (millions USD) Market share

90 nm 80 nm 70 nm 90 nm 80 nm 70 nmOld New Old New Old New Old New Old New Old New

05Q1 6574 lt1 6574 lt1 6574 lt1 1 lt001 1 lt001 1 lt00105Q2 5715 lt1 5715 lt1 5715 lt1 1 lt001 1 lt001 1 lt00105Q3 6394 lt1 6394 lt1 6394 lt1 1 lt001 1 lt001 1 lt00105Q4 6372 lt1 6372 lt1 6372 lt1 1 lt001 1 lt001 1 lt00106Q1 6422 131 6553 lt1 6553 lt1 098 002 1 lt001 1 lt00106Q2 7158 457 7615 lt1 7615 lt1 094 006 1 lt001 1 lt00106Q3 8455 636 9091 lt1 9091 lt1 093 007 1 lt001 1 lt00106Q4 8897 1695 10592 lt1 10592 lt1 084 016 1 lt001 1 lt00107Q1 6961 2707 9475 193 9668 lt1 072 028 098 002 1 lt00107Q2 4380 2920 6716 584 7154 146 06 04 092 008 098 00207Q3 3584 4380 6929 1035 7645 319 045 055 087 013 096 00407Q4 2055 4173 4484 1744 5294 934 033 067 072 028 085 01508Q1 1408 4459 3462 2405 4457 1408 024 076 059 041 076 02408Q2 1042 5470 2800 3712 4298 2214 016 084 043 057 066 03408Q3 758 6134 2205 4687 3791 3101 011 089 032 068 055 04508Q4 286 3793 816 3263 1713 2366 007 093 02 08 042 05809Q1 95 3073 222 2946 919 2249 003 097 007 093 029 07109Q2 81 3984 203 3862 772 3293 002 098 005 095 019 08109Q3 122 5991 245 5868 856 5257 002 098 004 096 014 08609Q4 87 8595 174 8508 955 7721 001 099 002 098 011 08910Q1 93 9195 186 9102 836 8452 001 099 002 098 009 09110Q2 104 10317 208 10213 834 9587 001 099 002 098 008 09210Q3 108 10671 216 10563 647 10132 001 099 002 098 006 09410Q4 86 8558 173 8471 346 8298 001 099 002 098 004 09611Q1 83 8220 166 8137 332 7971 001 099 002 098 004 09611Q2 lt1 8147 lt1 8147 163 7984 lt001 1 lt001 1 002 09811Q3 lt1 6565 lt1 6565 66 6499 lt001 1 lt001 1 001 09911Q4 lt1 6453 lt1 6453 lt1 6453 lt001 1 lt001 1 lt001 112Q1 lt1 6271 lt1 6271 lt1 6271 lt001 1 lt001 1 lt001 112Q2 lt1 7023 lt1 7023 lt1 7023 lt001 1 lt001 1 lt001 112Q3 lt1 6429 lt1 6429 lt1 6429 lt001 1 lt001 1 lt001 112Q4 lt1 6864 lt1 6864 lt1 6864 lt001 1 lt001 1 lt001 113Q1 lt1 6870 lt1 6870 lt1 6870 lt001 1 lt001 1 lt001 113Q2 lt1 8531 lt1 8531 lt1 8531 lt001 1 lt001 1 lt001 113Q3 lt1 9300 lt1 9300 lt1 9300 lt001 1 lt001 1 lt001 113Q4 lt1 9748 lt1 9748 lt1 9748 lt001 1 lt001 1 lt001 1Data sources DRAMeXchange IHS iSuppli amp Morgan Stanley Research (2014)Data collation Industrial Technology Research Institute of Taiwan amp Taiwan Institute of Economic Research

and 70 nanometers as examples The raw data for each aregiven in Table 1

We are also interested in prediction accuracy at differentstages of diffusion The 99 95 90 80 70 60and 50 market shares for new processing technologies areearmarked as seven stages of diffusion In each stage recordspertaining to the time before the new processing technologiessurpassed the earmarkedmarket shares were used as trainingdata to fit the model parametersThe remaining records wereused as testing data to evaluate prediction accuracy

For example at the dividing line of 90 nanometers newprocessing technologies are those using 90 nanometers orless Table 1 shows that these new processing technologiesreached a 99 market share in 2009 Q4 Thus in Stage 119 pairs of revenue records for the old and new processingtechnologies from 2005 Q1 to 2009 Q3 were used as trainingdata to fit the Bass and LotkandashVolterra models by usingordinary least squares The remaining 17 pairs of revenuerecords were used as testing data to individually evaluate theMAPEs for the old and new processing technologies

Mathematical Problems in Engineering 7

Similarly 19 market share records from 2005 Q1 to 2009Q3 were used as training data to fit the Bass LotkandashVolterraand modified LotkandashVolterra models by using ordinary leastsquares in Stage 1 The remaining 17 market share recordswere used as testing data to individually evaluate the MAEsfor the old andnewprocessing technologiesThe results for allseven stages are reported in Table 2 Tables 3 and 4 report theresults of our numerical studies for the 80- and 70-nanometerdividing lines respectively

42 Analysis of Data Selection The results reported in theprevious subsection indicate that the lower the number ofrecords used as training data the higher the prediction errorsMoreover the market share predictions are significantlybetter than the revenue predictions from both the Bass andLotkandashVolterra models Consider the 90-nanometer dividingline as an example (see Table 2) When using revenue dataon the Bass model the prediction errors range from 907 toinfinity for the seven stages When using market share dataon the Bass model the prediction errors are much smallerbetween 056 and 1711 for the seven stages Similarlyfor the LotkandashVolterra model using revenue data generatesprediction errors of from 952 to infinity whereas usingmarket share data gives prediction errors of between 014and 9190 Results based on the other diving lines are similar(see Tables 3 and 4)

Arguably revenue data are not suitable for forecasting thediffusion and substitution of technologies This is becauserevenue data reflect market fluctuations caused by factorsother than substitution including industry long-term trendseconomic cycles and other confounding factors

For example global DRAM industry revenue has exhib-ited a prolonged growth trend particularly in terms ofdemand for computers and mobile communications prod-ucts In 2007 the DRAM industry invested huge amountsof capital to expand production capacity for upgrading toa new PC operating system Windows Vista However salesof Windows Vista fell far short of market expectations Thisled to an oversupply of DRAM products which caused spotprices and revenue to crash in 2007 In 2008 global DRAMrevenue continued to decline because of the global financialtsunami The consequent unexpected fall in global DRAMrevenue from 2006 Q4 to 2009 Q1 greatly affected predictionaccuracy

Market share data are more suitable than revenue datafor forecasting the diffusion and substitution of DRAMprocessing technologies This is because the market sharedata record percentages and because the total market shareis fixed at 100 Because market share data are unaffectedby seasonal factors and economic cycles they reflect mainlytechnological substitution Consequently using market sharedata generates more accurate predictions and is appropriatefor analyzing technological substitution

43 Analysis of Model Selection Having shown that usingmarket share data increases prediction accuracy for allforecasting models in this subsection we use market sharedata to compare the Bass LotkandashVolterra and modifiedLotkandashVolterra models

First note the symmetry of the modified LotkandashVolterramodel based on market share data In each stage theprediction errors for the old and new processing technologiesare the same This is because the market shares are forced tosum to 100 in (10) of the modified LotkandashVolterra modelAs a result the forecast values for the old and new processingtechnologies sum to 100 and must have the same MAEsSimilarly for the Bass model based on market share data theprediction errors for the old and new processing technologiesare highly symmetrical Again this arises because of thecharacteristics of the market share data

By contrast the LotkandashVolterra model based on marketshare data does not generate symmetric prediction errors forthe old and new processing technologies This is because themarket shares of old and new processing technologies aretreated as two species and there is no total population limitfor the two species in the LotkandashVolterra model The forecastmarket shares are not constrained to sum to 100 This iswhy the prediction errors from the LotkandashVolterra model arerelatively high

Let us compare the accuracy of the models based onmarket share data In Stages 1 to 7 of Table 2 (which reportsresults for the 90-nanometer dividing line) the Bass modelgenerates prediction errors of between 056 and 1711 theLotkandashVolterramodel generates prediction errors of between014 and 9190 and the modified LotkandashVolterra modelgenerates prediction errors of between 016 and 050(Tables 3 and 4 show that the other dividing lines generatesimilar results) Therefore when forecasting market sharesthemodified LotkandashVolterramodel outperforms the Bass andLotkandashVolterra models

5 Conclusions and Future Research

In this study we propose a modified LotkandashVolterra modelto forecast the diffusion and substitution of DRAM pro-cessing technologies Multiple generations of DRAM pro-cessing technologies are partitioned into old and new gen-erations and the quarterly data from 2005 Q1 to 2013 Q4is used for empirical studies In our model there are fourfeatures

First the type of data affects the prediction accuracy of thetechnological substitution trends Although shipment andorrevenue data have been widely used in existing research theymay be contaminated by confounding factors that reduceprediction accuracy This can be avoided by using marketshare data which are purged of such factors as long-termtrends and business cycles Consequently using market sharedata can generate more accurate forecasts of technologicalsubstitution

Second existing models for multiple competing tech-nologies require predetermined number of technologies andeach technology is assumed to go through the complete adop-tion cycle However both are not obvious in DRAM indus-try Focusing on these issues our modified LotkandashVolterramodel allows undetermined number of multigenerationtechnologies and inconsecutive adoption of new technolo-gies

8 Mathematical Problems in Engineering

Table2Predictio

nerrorsbasedon

different

datasets

mod

elsandstages90-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

sharefor

new

processin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out

119872=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

1917

98

Old

gt105

gt105

057

014

016

New

1113

1222

057

021

016

(2)

1620

95

Old

gt105

gt105

161

017

020

New

969

1113

056

018

020

(3)

1521

90

Old

gt105

gt105

245

024

024

New

940

952

244

032

024

(4)

1323

80

Old

gt105

gt105

505

029

029

New

907

2346

233

137

029

(5)

1224

70

Old

gt105

gt105

494

025

033

New

1620

6291

537

111

033

(6)

1125

60

Old

gt105

gt105

461

036

044

New

1105

9334

460

046

044

(7)

1026

50

Old

gt105

gt105

1712

4191

050

New

1450

9334

1711

9190

050

Mathematical Problems in Engineering 9

Table3Predictio

nerrorsbasedon

different

datasets

mod

elsandstages80-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

261

99

Old

lt001

lt001

lt001

lt001

lt001

New

lt001

lt001

lt001

lt001

lt001

(2)

243

95

Old

1950

2428

161

128

023

New

1236

1573

161

226

023

(3)

207

90

Old

14624

6670

113

073

086

New

1095

4937

113

072

086

(4)

1710

80

Old

71537

3235

10

3087

104

New

5394

5653

10

316

410

4

(5)

1611

70

Old

2474525

3608

780

087

110

New

4951

6009

780

137

110

(6)

1611

60

Old

2474525

3608

780

087

112

New

4951

6009

780

107

112

(7)

1413

50

Old

322327

3947

4026

141

114

New

2415

5919

40

27

152

114

10 Mathematical Problems in Engineering

Table4Predictio

nerrorsbasedon

different

datasets

mod

elsandstages70-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

ds

Num

bero

ftesting

perio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

2610

98

Old

gt105

gt105

141

007

006

New

871

1195

141

063

006

(2)

2313

95

Old

gt105

gt105

201

022

022

New

1396

1927

201

071

022

(3)

2016

90

Old

gt105

gt105

172

035

041

New

4106

4717

17

1071

041

(4)

1719

80

Old

gt105

gt105

089

051

058

New

5557

5906

089

073

058

(5)

1620

70

Old

gt105

gt105

951

077

062

New

5210

6519

95

0089

062

(6)

1620

60

Old

gt105

gt105

951

077

066

New

5210

6519

95

0089

066

(7)

1521

50

Old

gt105

gt105

2983

075

068

New

2127

9129

2981

098

068

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

4 Mathematical Problems in Engineering

established by external investment before 1199050 Thus 119891119894(1199050) ispredetermined according to the external investment and 119896119894119895 =ln(119891119894(1199050)119891119895(1199050)) is the given ratio of initial market sharesAs a result this model may not be suitable for successivegenerations of DRAM processing technologies where futuregeneration may be zero at the starting time of the model

Norton andBass generalized the Bassmodel andmodeledthe demand growthdecline for successive generations ofmemory and logic circuits [9] They assume that the numberof successive generations is preset and each successive gener-ation has its own market potential and market penetrationA new generation of a product or technology always attractsusers from its potential market and the predecessor genera-tion The Norton-Bass model can be illustrated by a productof three successive generations

1198781 (119905) = 1198651 (119905)1198981 [1 minus 1198652 (119905 minus 1205912)]

1198782 (119905) = 1198652 (119905 minus 1205912) [1198982 + 1198651 (119905)1198981] [1 minus 1198653 (119905 minus 1205913)]

1198783 (119905) = 1198653 (119905 minus 1205912) 1198983 + 1198652 (119905 minus 1205912) [1198982 + 1198651 (119905)1198981]

(5)

where the 119894th generation 119894 is introduced at 120591119894 and 1205911 =0 Moreover 119898119894 is the potential sales of the 119894th generationand 119878119894(119905) is the sales of the 119894th generation in time period 119905Note that 119865119894(119905) is the cumulative distribution function of theadoption rate of the 119894th generation

The formula is

119865119894 (119905) =

1 minus exp [minus (119901119894 + 119902119894) 119905]1 + (119901119894119902119894) exp [minus (119901119894 + 119902119894) 119905]

for 119905 ge 1205911198940 otherwise

(6)

where the parameter 119901119894 is the coefficient of innovation and 119902119894is the coefficient of imitation of the 119894th generation

In these models there are three common issues Firstthe number of technologies must be preset Each technologyis assumed to be predictable and all newborn technologiescan go through the complete adoption cycle and competeswith earlier ones Unfortunately we observe that some newtechnologies may come to a premature end without adoptionin DRAM industry As a result the total number of newtechnologies may not be predetermined

Second the inconsecutive adoption of new technologiesis not allowed When adopting the new generation of DRAMprocessing technologies customers may intend to bypassthe immediate successor and adopt the later one due to theconsideration of market potential and transcendence As aresult some generations of DRAM processing technologiesmay not go through the complete adoption cycle and competewith earlier ones

Third the number of fitting parameters grows quicklywith the number of technologies andmore data is required forprediction accuracy For multiple competing technologiesthe existing forecasting models are designed to mimic thediffusion and substitution by equations among all technolo-gies Thus the number of fitting parameters grows quicklywith the number of technologies However it is natural thatmature technologies have plenty of data andnew technologies

05Q

105

Q2

05Q

305

Q4

06Q

106

Q2

06Q

306

Q4

07Q

107

Q2

07Q

307

Q4

08Q

108

Q2

08Q

308

Q4

09Q

109

Q2

09Q

309

Q4

10Q

110

Q2

10Q

310

Q4

11Q

111

Q2

11Q

311

Q4

12Q

112

Q2

12Q

312

Q4

13Q

113

Q2

13Q

313

Q4

New process technology (less than Old process technology Total market revenues

Time

2000

6000

10000

Reve

nue (

mill

ion

USD

)

0

(9 nm and more than 90

Figure 2 DRAM revenue changes for new and old process tech-nologies Data sources DRAMeXchange IHS iSuppli and MorganStanley Research (2014) Data collation Industrial TechnologyResearch Institute of Taiwan and Taiwan Institute of EconomicResearch

have little data collectable in initial stages As a result maturetechnologies dominate the model and errors are cumulatedto new technologies In the worst case some models may notbe applied due to the insufficient data for parameters fitting

3 Model

Wenowpropose themodified LotkandashVolterramodel for fore-casting the diffusion and substitution of DRAM processingtechnologies

31 Market Share Data In this study we focus on the substi-tution between old and new DRAM processing technologiesAt a given point in time there usually coexist differentgenerations of processing technology on the DRAM marketFor example 60- 70- 80- 90- 100ndash120- and 130ndash140-nanometer processing technologies coexisted on the DRAMmarket in 2007 Q3 (see Figure 1) We first divide DRAMprocessing technologies into two groups old and new 90nanometers are used as the dividing line between the twoHence DRAM processing technologies of less than 90nanometers are classified as new technologies and those of90 nanometers and more than 90 nanometers are classifiedas old technologies

Data on shipments andor product revenues are typicallyused to forecast technological substitution However thesedatamay be affected bymarket fluctuations that have nothingto do with substitution Such factors include industry long-term trends economic cycles and other confounding factorsTo illustrate Figure 2 shows quarterly DRAM revenue datafrom 2005 Q1 to 2013 Q4 Unstable fluctuations in DRAMrevenue are apparent

Market shares measure the percentages of market salesaccounted for by competitive products or technologiesChanges in market shares reflect substitution not long-term trends economic cycles and other confounding factors

Mathematical Problems in Engineering 5

05Q

105

Q2

05Q

305

Q4

06Q

106

Q2

06Q

306

Q4

07Q

107

Q2

07Q

307

Q4

08Q

108

Q2

08Q

308

Q4

09Q

109

Q2

09Q

309

Q4

10Q

110

Q2

10Q

310

Q4

11Q

111

Q2

11Q

311

Q4

12Q

112

Q2

12Q

312

Q4

13Q

113

Q2

13Q

313

Q4

New process technologies (less than Old process technologiesTotal market share

Time

0102030405060708090

100

Mar

ket s

hare

()

90 nm)(90 nm and more than 90 nm)

Figure 3 DRAM market share changes for new and old processtechnologies Data sources DRAMeXchange IHS iSuppli andMor-gan Stanley Research (2014) Data collation Industrial TechnologyResearch Institute of Taiwan and Taiwan Institute of EconomicResearch

Figure 3 shows that market shares fluctuate less than doesrevenue Hence our model is to use market share data forforecasting

32Model Formulation Usingmarket share data one naturalrestriction is that market shares must sum to 100 By setting119872 = 1 and 119910(119905) as the cumulative market share of newprocessing technologies at time 119905 the Bass model given by (1)can be rewritten as follows

119889119910 (119905)119889119905 = (119901 + 119902119910 (119905)) (1 minus 119910 (119905))

= 1198621 + 1198611119910 (119905) + 1198601119910 (119905)2(7)

where 1198621 = 119901 1198611 = 119902 minus 119901 and 1198601 = minus 119902When applying market share data to the LotkandashVolterra

model we let 119909(119905) and 119910(119905) be the market shares of old andnew DRAM processing technologies at time 119905 respectivelyGiven a total market share of 100 we have

119909 (119905) + 119910 (119905) = 1 (8)

From (8) (2) can be rewritten as

119889119909 (119905)119889119905 = (1198861 + 1198871119909 (119905) + 1198881119910 (119905)) 119909 (119905)

= ((1198861 + 1198881) + (1198871 minus 1198881) 119909 (119905)) 119909 (119905)

=1198612119909 (119905) + 1198602119909 (119905)2

(9)

where 1198612 = 1198861 + 1198881 and 1198602 = 1198871 minus 1198881Equations (7) and (9) show that the Bass and

LotkandashVolterra models of market shares degenerate toour model because total market shares must be 100 Thismodified LotkandashVolterra model is formulated as follows

119889119909 (119905)119889119905 = 119862 + 119861119909 (119905) + 119860119909 (119905)2

119910 (119905) = 1 minus 119909 (119905) (10)

33 Prediction Evaluation Traditionally both the meanabsolute percentage error (MAPE) and the mean squarederror (MSE) are used to evaluate prediction accuracy How-ever depending on the data and models used the MAPEand MSE are not necessarily the best measures of predictionaccuracyTherefore we use theMAE to evaluate the accuracyof our market share predictions

Given a number of forecast periods of 119899 and letting 119910(119905)and 119910(119905) denote the actual and forecast values for time 119905respectively the MAPE MSE and MAE are defined as

MAPE = 1119899119899

sum119905=1

10038161003816100381610038161003816100381610038161003816119910 (119905) minus 119910 (119905)

119910 (119905)10038161003816100381610038161003816100381610038161003816

MSE =119899

sum119905=1

(119910 (119905) minus 119910 (119905))2119899

MAE = 1119899119899

sum119905=1

1003816100381610038161003816(119910 (119905) minus 119910 (119905))1003816100381610038161003816

(11)

The MAPE conveys the percentage difference betweenthe actual values and forecast values Small 119910(119905) values cangenerate large MAPE values Actual values close to zero cangenerate infinitely large MAPEs Therefore the MAPE isnot suitable for prediction evaluation when actual values aresmall Given that market shares are by definition between 0and 1 the MAPE is not suited to our purposes

The MSE is the mean squared difference between theactual and forecast values Although it is not troubled by smallactual values because the MSE is dependent on the units ofmeasurement of119910(119905) it cannot be used for comparisons basedon different types of dataThus theMSE is not suitable for ourstudy because we want to compare the prediction accuracy ofboth revenue and market share forecasts

The MAE conveys the absolute difference between theactual and forecast values It is untroubled by small actualvalues Moreover when applied to market share data theMAE records the average absolute error as a percentageThusto ensure the comparability of our prediction evaluationswe compare the MAEs for market share predictions with theMAPEs for revenue predictions

4 Data Collection and Empirical Study

In this section our model is implemented to forecast thediffusion and substitution of DRAMprocessing technologiesOther benchmark forecasting models are used for com-parison These benchmark forecasting models include thestandard Bass and LotkandashVolterra models

Data on revenue and market share have different charac-teristics and influencesWe collect global revenue andmarketshare data on different DRAM processing technologies anduse them to evaluate prediction accuracy Our data sourcesinclude DRAMeXchange iSuppli and the industry analysisreports published by Taiwanrsquos ITRI In total 36 pairs ofrecords from 2005 Q1 to 2013 Q4 were collected

41 Numerical Study To distinguish between old and newprocessing technologies we use three dividing lines 90 80

6 Mathematical Problems in Engineering

Table 1 Revenues and market share for old and new processing technologies from 2005 Q1 to 2013 Q4

QuarterRevenue (millions USD) Market share

90 nm 80 nm 70 nm 90 nm 80 nm 70 nmOld New Old New Old New Old New Old New Old New

05Q1 6574 lt1 6574 lt1 6574 lt1 1 lt001 1 lt001 1 lt00105Q2 5715 lt1 5715 lt1 5715 lt1 1 lt001 1 lt001 1 lt00105Q3 6394 lt1 6394 lt1 6394 lt1 1 lt001 1 lt001 1 lt00105Q4 6372 lt1 6372 lt1 6372 lt1 1 lt001 1 lt001 1 lt00106Q1 6422 131 6553 lt1 6553 lt1 098 002 1 lt001 1 lt00106Q2 7158 457 7615 lt1 7615 lt1 094 006 1 lt001 1 lt00106Q3 8455 636 9091 lt1 9091 lt1 093 007 1 lt001 1 lt00106Q4 8897 1695 10592 lt1 10592 lt1 084 016 1 lt001 1 lt00107Q1 6961 2707 9475 193 9668 lt1 072 028 098 002 1 lt00107Q2 4380 2920 6716 584 7154 146 06 04 092 008 098 00207Q3 3584 4380 6929 1035 7645 319 045 055 087 013 096 00407Q4 2055 4173 4484 1744 5294 934 033 067 072 028 085 01508Q1 1408 4459 3462 2405 4457 1408 024 076 059 041 076 02408Q2 1042 5470 2800 3712 4298 2214 016 084 043 057 066 03408Q3 758 6134 2205 4687 3791 3101 011 089 032 068 055 04508Q4 286 3793 816 3263 1713 2366 007 093 02 08 042 05809Q1 95 3073 222 2946 919 2249 003 097 007 093 029 07109Q2 81 3984 203 3862 772 3293 002 098 005 095 019 08109Q3 122 5991 245 5868 856 5257 002 098 004 096 014 08609Q4 87 8595 174 8508 955 7721 001 099 002 098 011 08910Q1 93 9195 186 9102 836 8452 001 099 002 098 009 09110Q2 104 10317 208 10213 834 9587 001 099 002 098 008 09210Q3 108 10671 216 10563 647 10132 001 099 002 098 006 09410Q4 86 8558 173 8471 346 8298 001 099 002 098 004 09611Q1 83 8220 166 8137 332 7971 001 099 002 098 004 09611Q2 lt1 8147 lt1 8147 163 7984 lt001 1 lt001 1 002 09811Q3 lt1 6565 lt1 6565 66 6499 lt001 1 lt001 1 001 09911Q4 lt1 6453 lt1 6453 lt1 6453 lt001 1 lt001 1 lt001 112Q1 lt1 6271 lt1 6271 lt1 6271 lt001 1 lt001 1 lt001 112Q2 lt1 7023 lt1 7023 lt1 7023 lt001 1 lt001 1 lt001 112Q3 lt1 6429 lt1 6429 lt1 6429 lt001 1 lt001 1 lt001 112Q4 lt1 6864 lt1 6864 lt1 6864 lt001 1 lt001 1 lt001 113Q1 lt1 6870 lt1 6870 lt1 6870 lt001 1 lt001 1 lt001 113Q2 lt1 8531 lt1 8531 lt1 8531 lt001 1 lt001 1 lt001 113Q3 lt1 9300 lt1 9300 lt1 9300 lt001 1 lt001 1 lt001 113Q4 lt1 9748 lt1 9748 lt1 9748 lt001 1 lt001 1 lt001 1Data sources DRAMeXchange IHS iSuppli amp Morgan Stanley Research (2014)Data collation Industrial Technology Research Institute of Taiwan amp Taiwan Institute of Economic Research

and 70 nanometers as examples The raw data for each aregiven in Table 1

We are also interested in prediction accuracy at differentstages of diffusion The 99 95 90 80 70 60and 50 market shares for new processing technologies areearmarked as seven stages of diffusion In each stage recordspertaining to the time before the new processing technologiessurpassed the earmarkedmarket shares were used as trainingdata to fit the model parametersThe remaining records wereused as testing data to evaluate prediction accuracy

For example at the dividing line of 90 nanometers newprocessing technologies are those using 90 nanometers orless Table 1 shows that these new processing technologiesreached a 99 market share in 2009 Q4 Thus in Stage 119 pairs of revenue records for the old and new processingtechnologies from 2005 Q1 to 2009 Q3 were used as trainingdata to fit the Bass and LotkandashVolterra models by usingordinary least squares The remaining 17 pairs of revenuerecords were used as testing data to individually evaluate theMAPEs for the old and new processing technologies

Mathematical Problems in Engineering 7

Similarly 19 market share records from 2005 Q1 to 2009Q3 were used as training data to fit the Bass LotkandashVolterraand modified LotkandashVolterra models by using ordinary leastsquares in Stage 1 The remaining 17 market share recordswere used as testing data to individually evaluate the MAEsfor the old andnewprocessing technologiesThe results for allseven stages are reported in Table 2 Tables 3 and 4 report theresults of our numerical studies for the 80- and 70-nanometerdividing lines respectively

42 Analysis of Data Selection The results reported in theprevious subsection indicate that the lower the number ofrecords used as training data the higher the prediction errorsMoreover the market share predictions are significantlybetter than the revenue predictions from both the Bass andLotkandashVolterra models Consider the 90-nanometer dividingline as an example (see Table 2) When using revenue dataon the Bass model the prediction errors range from 907 toinfinity for the seven stages When using market share dataon the Bass model the prediction errors are much smallerbetween 056 and 1711 for the seven stages Similarlyfor the LotkandashVolterra model using revenue data generatesprediction errors of from 952 to infinity whereas usingmarket share data gives prediction errors of between 014and 9190 Results based on the other diving lines are similar(see Tables 3 and 4)

Arguably revenue data are not suitable for forecasting thediffusion and substitution of technologies This is becauserevenue data reflect market fluctuations caused by factorsother than substitution including industry long-term trendseconomic cycles and other confounding factors

For example global DRAM industry revenue has exhib-ited a prolonged growth trend particularly in terms ofdemand for computers and mobile communications prod-ucts In 2007 the DRAM industry invested huge amountsof capital to expand production capacity for upgrading toa new PC operating system Windows Vista However salesof Windows Vista fell far short of market expectations Thisled to an oversupply of DRAM products which caused spotprices and revenue to crash in 2007 In 2008 global DRAMrevenue continued to decline because of the global financialtsunami The consequent unexpected fall in global DRAMrevenue from 2006 Q4 to 2009 Q1 greatly affected predictionaccuracy

Market share data are more suitable than revenue datafor forecasting the diffusion and substitution of DRAMprocessing technologies This is because the market sharedata record percentages and because the total market shareis fixed at 100 Because market share data are unaffectedby seasonal factors and economic cycles they reflect mainlytechnological substitution Consequently using market sharedata generates more accurate predictions and is appropriatefor analyzing technological substitution

43 Analysis of Model Selection Having shown that usingmarket share data increases prediction accuracy for allforecasting models in this subsection we use market sharedata to compare the Bass LotkandashVolterra and modifiedLotkandashVolterra models

First note the symmetry of the modified LotkandashVolterramodel based on market share data In each stage theprediction errors for the old and new processing technologiesare the same This is because the market shares are forced tosum to 100 in (10) of the modified LotkandashVolterra modelAs a result the forecast values for the old and new processingtechnologies sum to 100 and must have the same MAEsSimilarly for the Bass model based on market share data theprediction errors for the old and new processing technologiesare highly symmetrical Again this arises because of thecharacteristics of the market share data

By contrast the LotkandashVolterra model based on marketshare data does not generate symmetric prediction errors forthe old and new processing technologies This is because themarket shares of old and new processing technologies aretreated as two species and there is no total population limitfor the two species in the LotkandashVolterra model The forecastmarket shares are not constrained to sum to 100 This iswhy the prediction errors from the LotkandashVolterra model arerelatively high

Let us compare the accuracy of the models based onmarket share data In Stages 1 to 7 of Table 2 (which reportsresults for the 90-nanometer dividing line) the Bass modelgenerates prediction errors of between 056 and 1711 theLotkandashVolterramodel generates prediction errors of between014 and 9190 and the modified LotkandashVolterra modelgenerates prediction errors of between 016 and 050(Tables 3 and 4 show that the other dividing lines generatesimilar results) Therefore when forecasting market sharesthemodified LotkandashVolterramodel outperforms the Bass andLotkandashVolterra models

5 Conclusions and Future Research

In this study we propose a modified LotkandashVolterra modelto forecast the diffusion and substitution of DRAM pro-cessing technologies Multiple generations of DRAM pro-cessing technologies are partitioned into old and new gen-erations and the quarterly data from 2005 Q1 to 2013 Q4is used for empirical studies In our model there are fourfeatures

First the type of data affects the prediction accuracy of thetechnological substitution trends Although shipment andorrevenue data have been widely used in existing research theymay be contaminated by confounding factors that reduceprediction accuracy This can be avoided by using marketshare data which are purged of such factors as long-termtrends and business cycles Consequently using market sharedata can generate more accurate forecasts of technologicalsubstitution

Second existing models for multiple competing tech-nologies require predetermined number of technologies andeach technology is assumed to go through the complete adop-tion cycle However both are not obvious in DRAM indus-try Focusing on these issues our modified LotkandashVolterramodel allows undetermined number of multigenerationtechnologies and inconsecutive adoption of new technolo-gies

8 Mathematical Problems in Engineering

Table2Predictio

nerrorsbasedon

different

datasets

mod

elsandstages90-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

sharefor

new

processin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out

119872=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

1917

98

Old

gt105

gt105

057

014

016

New

1113

1222

057

021

016

(2)

1620

95

Old

gt105

gt105

161

017

020

New

969

1113

056

018

020

(3)

1521

90

Old

gt105

gt105

245

024

024

New

940

952

244

032

024

(4)

1323

80

Old

gt105

gt105

505

029

029

New

907

2346

233

137

029

(5)

1224

70

Old

gt105

gt105

494

025

033

New

1620

6291

537

111

033

(6)

1125

60

Old

gt105

gt105

461

036

044

New

1105

9334

460

046

044

(7)

1026

50

Old

gt105

gt105

1712

4191

050

New

1450

9334

1711

9190

050

Mathematical Problems in Engineering 9

Table3Predictio

nerrorsbasedon

different

datasets

mod

elsandstages80-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

261

99

Old

lt001

lt001

lt001

lt001

lt001

New

lt001

lt001

lt001

lt001

lt001

(2)

243

95

Old

1950

2428

161

128

023

New

1236

1573

161

226

023

(3)

207

90

Old

14624

6670

113

073

086

New

1095

4937

113

072

086

(4)

1710

80

Old

71537

3235

10

3087

104

New

5394

5653

10

316

410

4

(5)

1611

70

Old

2474525

3608

780

087

110

New

4951

6009

780

137

110

(6)

1611

60

Old

2474525

3608

780

087

112

New

4951

6009

780

107

112

(7)

1413

50

Old

322327

3947

4026

141

114

New

2415

5919

40

27

152

114

10 Mathematical Problems in Engineering

Table4Predictio

nerrorsbasedon

different

datasets

mod

elsandstages70-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

ds

Num

bero

ftesting

perio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

2610

98

Old

gt105

gt105

141

007

006

New

871

1195

141

063

006

(2)

2313

95

Old

gt105

gt105

201

022

022

New

1396

1927

201

071

022

(3)

2016

90

Old

gt105

gt105

172

035

041

New

4106

4717

17

1071

041

(4)

1719

80

Old

gt105

gt105

089

051

058

New

5557

5906

089

073

058

(5)

1620

70

Old

gt105

gt105

951

077

062

New

5210

6519

95

0089

062

(6)

1620

60

Old

gt105

gt105

951

077

066

New

5210

6519

95

0089

066

(7)

1521

50

Old

gt105

gt105

2983

075

068

New

2127

9129

2981

098

068

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

Mathematical Problems in Engineering 5

05Q

105

Q2

05Q

305

Q4

06Q

106

Q2

06Q

306

Q4

07Q

107

Q2

07Q

307

Q4

08Q

108

Q2

08Q

308

Q4

09Q

109

Q2

09Q

309

Q4

10Q

110

Q2

10Q

310

Q4

11Q

111

Q2

11Q

311

Q4

12Q

112

Q2

12Q

312

Q4

13Q

113

Q2

13Q

313

Q4

New process technologies (less than Old process technologiesTotal market share

Time

0102030405060708090

100

Mar

ket s

hare

()

90 nm)(90 nm and more than 90 nm)

Figure 3 DRAM market share changes for new and old processtechnologies Data sources DRAMeXchange IHS iSuppli andMor-gan Stanley Research (2014) Data collation Industrial TechnologyResearch Institute of Taiwan and Taiwan Institute of EconomicResearch

Figure 3 shows that market shares fluctuate less than doesrevenue Hence our model is to use market share data forforecasting

32Model Formulation Usingmarket share data one naturalrestriction is that market shares must sum to 100 By setting119872 = 1 and 119910(119905) as the cumulative market share of newprocessing technologies at time 119905 the Bass model given by (1)can be rewritten as follows

119889119910 (119905)119889119905 = (119901 + 119902119910 (119905)) (1 minus 119910 (119905))

= 1198621 + 1198611119910 (119905) + 1198601119910 (119905)2(7)

where 1198621 = 119901 1198611 = 119902 minus 119901 and 1198601 = minus 119902When applying market share data to the LotkandashVolterra

model we let 119909(119905) and 119910(119905) be the market shares of old andnew DRAM processing technologies at time 119905 respectivelyGiven a total market share of 100 we have

119909 (119905) + 119910 (119905) = 1 (8)

From (8) (2) can be rewritten as

119889119909 (119905)119889119905 = (1198861 + 1198871119909 (119905) + 1198881119910 (119905)) 119909 (119905)

= ((1198861 + 1198881) + (1198871 minus 1198881) 119909 (119905)) 119909 (119905)

=1198612119909 (119905) + 1198602119909 (119905)2

(9)

where 1198612 = 1198861 + 1198881 and 1198602 = 1198871 minus 1198881Equations (7) and (9) show that the Bass and

LotkandashVolterra models of market shares degenerate toour model because total market shares must be 100 Thismodified LotkandashVolterra model is formulated as follows

119889119909 (119905)119889119905 = 119862 + 119861119909 (119905) + 119860119909 (119905)2

119910 (119905) = 1 minus 119909 (119905) (10)

33 Prediction Evaluation Traditionally both the meanabsolute percentage error (MAPE) and the mean squarederror (MSE) are used to evaluate prediction accuracy How-ever depending on the data and models used the MAPEand MSE are not necessarily the best measures of predictionaccuracyTherefore we use theMAE to evaluate the accuracyof our market share predictions

Given a number of forecast periods of 119899 and letting 119910(119905)and 119910(119905) denote the actual and forecast values for time 119905respectively the MAPE MSE and MAE are defined as

MAPE = 1119899119899

sum119905=1

10038161003816100381610038161003816100381610038161003816119910 (119905) minus 119910 (119905)

119910 (119905)10038161003816100381610038161003816100381610038161003816

MSE =119899

sum119905=1

(119910 (119905) minus 119910 (119905))2119899

MAE = 1119899119899

sum119905=1

1003816100381610038161003816(119910 (119905) minus 119910 (119905))1003816100381610038161003816

(11)

The MAPE conveys the percentage difference betweenthe actual values and forecast values Small 119910(119905) values cangenerate large MAPE values Actual values close to zero cangenerate infinitely large MAPEs Therefore the MAPE isnot suitable for prediction evaluation when actual values aresmall Given that market shares are by definition between 0and 1 the MAPE is not suited to our purposes

The MSE is the mean squared difference between theactual and forecast values Although it is not troubled by smallactual values because the MSE is dependent on the units ofmeasurement of119910(119905) it cannot be used for comparisons basedon different types of dataThus theMSE is not suitable for ourstudy because we want to compare the prediction accuracy ofboth revenue and market share forecasts

The MAE conveys the absolute difference between theactual and forecast values It is untroubled by small actualvalues Moreover when applied to market share data theMAE records the average absolute error as a percentageThusto ensure the comparability of our prediction evaluationswe compare the MAEs for market share predictions with theMAPEs for revenue predictions

4 Data Collection and Empirical Study

In this section our model is implemented to forecast thediffusion and substitution of DRAMprocessing technologiesOther benchmark forecasting models are used for com-parison These benchmark forecasting models include thestandard Bass and LotkandashVolterra models

Data on revenue and market share have different charac-teristics and influencesWe collect global revenue andmarketshare data on different DRAM processing technologies anduse them to evaluate prediction accuracy Our data sourcesinclude DRAMeXchange iSuppli and the industry analysisreports published by Taiwanrsquos ITRI In total 36 pairs ofrecords from 2005 Q1 to 2013 Q4 were collected

41 Numerical Study To distinguish between old and newprocessing technologies we use three dividing lines 90 80

6 Mathematical Problems in Engineering

Table 1 Revenues and market share for old and new processing technologies from 2005 Q1 to 2013 Q4

QuarterRevenue (millions USD) Market share

90 nm 80 nm 70 nm 90 nm 80 nm 70 nmOld New Old New Old New Old New Old New Old New

05Q1 6574 lt1 6574 lt1 6574 lt1 1 lt001 1 lt001 1 lt00105Q2 5715 lt1 5715 lt1 5715 lt1 1 lt001 1 lt001 1 lt00105Q3 6394 lt1 6394 lt1 6394 lt1 1 lt001 1 lt001 1 lt00105Q4 6372 lt1 6372 lt1 6372 lt1 1 lt001 1 lt001 1 lt00106Q1 6422 131 6553 lt1 6553 lt1 098 002 1 lt001 1 lt00106Q2 7158 457 7615 lt1 7615 lt1 094 006 1 lt001 1 lt00106Q3 8455 636 9091 lt1 9091 lt1 093 007 1 lt001 1 lt00106Q4 8897 1695 10592 lt1 10592 lt1 084 016 1 lt001 1 lt00107Q1 6961 2707 9475 193 9668 lt1 072 028 098 002 1 lt00107Q2 4380 2920 6716 584 7154 146 06 04 092 008 098 00207Q3 3584 4380 6929 1035 7645 319 045 055 087 013 096 00407Q4 2055 4173 4484 1744 5294 934 033 067 072 028 085 01508Q1 1408 4459 3462 2405 4457 1408 024 076 059 041 076 02408Q2 1042 5470 2800 3712 4298 2214 016 084 043 057 066 03408Q3 758 6134 2205 4687 3791 3101 011 089 032 068 055 04508Q4 286 3793 816 3263 1713 2366 007 093 02 08 042 05809Q1 95 3073 222 2946 919 2249 003 097 007 093 029 07109Q2 81 3984 203 3862 772 3293 002 098 005 095 019 08109Q3 122 5991 245 5868 856 5257 002 098 004 096 014 08609Q4 87 8595 174 8508 955 7721 001 099 002 098 011 08910Q1 93 9195 186 9102 836 8452 001 099 002 098 009 09110Q2 104 10317 208 10213 834 9587 001 099 002 098 008 09210Q3 108 10671 216 10563 647 10132 001 099 002 098 006 09410Q4 86 8558 173 8471 346 8298 001 099 002 098 004 09611Q1 83 8220 166 8137 332 7971 001 099 002 098 004 09611Q2 lt1 8147 lt1 8147 163 7984 lt001 1 lt001 1 002 09811Q3 lt1 6565 lt1 6565 66 6499 lt001 1 lt001 1 001 09911Q4 lt1 6453 lt1 6453 lt1 6453 lt001 1 lt001 1 lt001 112Q1 lt1 6271 lt1 6271 lt1 6271 lt001 1 lt001 1 lt001 112Q2 lt1 7023 lt1 7023 lt1 7023 lt001 1 lt001 1 lt001 112Q3 lt1 6429 lt1 6429 lt1 6429 lt001 1 lt001 1 lt001 112Q4 lt1 6864 lt1 6864 lt1 6864 lt001 1 lt001 1 lt001 113Q1 lt1 6870 lt1 6870 lt1 6870 lt001 1 lt001 1 lt001 113Q2 lt1 8531 lt1 8531 lt1 8531 lt001 1 lt001 1 lt001 113Q3 lt1 9300 lt1 9300 lt1 9300 lt001 1 lt001 1 lt001 113Q4 lt1 9748 lt1 9748 lt1 9748 lt001 1 lt001 1 lt001 1Data sources DRAMeXchange IHS iSuppli amp Morgan Stanley Research (2014)Data collation Industrial Technology Research Institute of Taiwan amp Taiwan Institute of Economic Research

and 70 nanometers as examples The raw data for each aregiven in Table 1

We are also interested in prediction accuracy at differentstages of diffusion The 99 95 90 80 70 60and 50 market shares for new processing technologies areearmarked as seven stages of diffusion In each stage recordspertaining to the time before the new processing technologiessurpassed the earmarkedmarket shares were used as trainingdata to fit the model parametersThe remaining records wereused as testing data to evaluate prediction accuracy

For example at the dividing line of 90 nanometers newprocessing technologies are those using 90 nanometers orless Table 1 shows that these new processing technologiesreached a 99 market share in 2009 Q4 Thus in Stage 119 pairs of revenue records for the old and new processingtechnologies from 2005 Q1 to 2009 Q3 were used as trainingdata to fit the Bass and LotkandashVolterra models by usingordinary least squares The remaining 17 pairs of revenuerecords were used as testing data to individually evaluate theMAPEs for the old and new processing technologies

Mathematical Problems in Engineering 7

Similarly 19 market share records from 2005 Q1 to 2009Q3 were used as training data to fit the Bass LotkandashVolterraand modified LotkandashVolterra models by using ordinary leastsquares in Stage 1 The remaining 17 market share recordswere used as testing data to individually evaluate the MAEsfor the old andnewprocessing technologiesThe results for allseven stages are reported in Table 2 Tables 3 and 4 report theresults of our numerical studies for the 80- and 70-nanometerdividing lines respectively

42 Analysis of Data Selection The results reported in theprevious subsection indicate that the lower the number ofrecords used as training data the higher the prediction errorsMoreover the market share predictions are significantlybetter than the revenue predictions from both the Bass andLotkandashVolterra models Consider the 90-nanometer dividingline as an example (see Table 2) When using revenue dataon the Bass model the prediction errors range from 907 toinfinity for the seven stages When using market share dataon the Bass model the prediction errors are much smallerbetween 056 and 1711 for the seven stages Similarlyfor the LotkandashVolterra model using revenue data generatesprediction errors of from 952 to infinity whereas usingmarket share data gives prediction errors of between 014and 9190 Results based on the other diving lines are similar(see Tables 3 and 4)

Arguably revenue data are not suitable for forecasting thediffusion and substitution of technologies This is becauserevenue data reflect market fluctuations caused by factorsother than substitution including industry long-term trendseconomic cycles and other confounding factors

For example global DRAM industry revenue has exhib-ited a prolonged growth trend particularly in terms ofdemand for computers and mobile communications prod-ucts In 2007 the DRAM industry invested huge amountsof capital to expand production capacity for upgrading toa new PC operating system Windows Vista However salesof Windows Vista fell far short of market expectations Thisled to an oversupply of DRAM products which caused spotprices and revenue to crash in 2007 In 2008 global DRAMrevenue continued to decline because of the global financialtsunami The consequent unexpected fall in global DRAMrevenue from 2006 Q4 to 2009 Q1 greatly affected predictionaccuracy

Market share data are more suitable than revenue datafor forecasting the diffusion and substitution of DRAMprocessing technologies This is because the market sharedata record percentages and because the total market shareis fixed at 100 Because market share data are unaffectedby seasonal factors and economic cycles they reflect mainlytechnological substitution Consequently using market sharedata generates more accurate predictions and is appropriatefor analyzing technological substitution

43 Analysis of Model Selection Having shown that usingmarket share data increases prediction accuracy for allforecasting models in this subsection we use market sharedata to compare the Bass LotkandashVolterra and modifiedLotkandashVolterra models

First note the symmetry of the modified LotkandashVolterramodel based on market share data In each stage theprediction errors for the old and new processing technologiesare the same This is because the market shares are forced tosum to 100 in (10) of the modified LotkandashVolterra modelAs a result the forecast values for the old and new processingtechnologies sum to 100 and must have the same MAEsSimilarly for the Bass model based on market share data theprediction errors for the old and new processing technologiesare highly symmetrical Again this arises because of thecharacteristics of the market share data

By contrast the LotkandashVolterra model based on marketshare data does not generate symmetric prediction errors forthe old and new processing technologies This is because themarket shares of old and new processing technologies aretreated as two species and there is no total population limitfor the two species in the LotkandashVolterra model The forecastmarket shares are not constrained to sum to 100 This iswhy the prediction errors from the LotkandashVolterra model arerelatively high

Let us compare the accuracy of the models based onmarket share data In Stages 1 to 7 of Table 2 (which reportsresults for the 90-nanometer dividing line) the Bass modelgenerates prediction errors of between 056 and 1711 theLotkandashVolterramodel generates prediction errors of between014 and 9190 and the modified LotkandashVolterra modelgenerates prediction errors of between 016 and 050(Tables 3 and 4 show that the other dividing lines generatesimilar results) Therefore when forecasting market sharesthemodified LotkandashVolterramodel outperforms the Bass andLotkandashVolterra models

5 Conclusions and Future Research

In this study we propose a modified LotkandashVolterra modelto forecast the diffusion and substitution of DRAM pro-cessing technologies Multiple generations of DRAM pro-cessing technologies are partitioned into old and new gen-erations and the quarterly data from 2005 Q1 to 2013 Q4is used for empirical studies In our model there are fourfeatures

First the type of data affects the prediction accuracy of thetechnological substitution trends Although shipment andorrevenue data have been widely used in existing research theymay be contaminated by confounding factors that reduceprediction accuracy This can be avoided by using marketshare data which are purged of such factors as long-termtrends and business cycles Consequently using market sharedata can generate more accurate forecasts of technologicalsubstitution

Second existing models for multiple competing tech-nologies require predetermined number of technologies andeach technology is assumed to go through the complete adop-tion cycle However both are not obvious in DRAM indus-try Focusing on these issues our modified LotkandashVolterramodel allows undetermined number of multigenerationtechnologies and inconsecutive adoption of new technolo-gies

8 Mathematical Problems in Engineering

Table2Predictio

nerrorsbasedon

different

datasets

mod

elsandstages90-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

sharefor

new

processin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out

119872=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

1917

98

Old

gt105

gt105

057

014

016

New

1113

1222

057

021

016

(2)

1620

95

Old

gt105

gt105

161

017

020

New

969

1113

056

018

020

(3)

1521

90

Old

gt105

gt105

245

024

024

New

940

952

244

032

024

(4)

1323

80

Old

gt105

gt105

505

029

029

New

907

2346

233

137

029

(5)

1224

70

Old

gt105

gt105

494

025

033

New

1620

6291

537

111

033

(6)

1125

60

Old

gt105

gt105

461

036

044

New

1105

9334

460

046

044

(7)

1026

50

Old

gt105

gt105

1712

4191

050

New

1450

9334

1711

9190

050

Mathematical Problems in Engineering 9

Table3Predictio

nerrorsbasedon

different

datasets

mod

elsandstages80-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

261

99

Old

lt001

lt001

lt001

lt001

lt001

New

lt001

lt001

lt001

lt001

lt001

(2)

243

95

Old

1950

2428

161

128

023

New

1236

1573

161

226

023

(3)

207

90

Old

14624

6670

113

073

086

New

1095

4937

113

072

086

(4)

1710

80

Old

71537

3235

10

3087

104

New

5394

5653

10

316

410

4

(5)

1611

70

Old

2474525

3608

780

087

110

New

4951

6009

780

137

110

(6)

1611

60

Old

2474525

3608

780

087

112

New

4951

6009

780

107

112

(7)

1413

50

Old

322327

3947

4026

141

114

New

2415

5919

40

27

152

114

10 Mathematical Problems in Engineering

Table4Predictio

nerrorsbasedon

different

datasets

mod

elsandstages70-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

ds

Num

bero

ftesting

perio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

2610

98

Old

gt105

gt105

141

007

006

New

871

1195

141

063

006

(2)

2313

95

Old

gt105

gt105

201

022

022

New

1396

1927

201

071

022

(3)

2016

90

Old

gt105

gt105

172

035

041

New

4106

4717

17

1071

041

(4)

1719

80

Old

gt105

gt105

089

051

058

New

5557

5906

089

073

058

(5)

1620

70

Old

gt105

gt105

951

077

062

New

5210

6519

95

0089

062

(6)

1620

60

Old

gt105

gt105

951

077

066

New

5210

6519

95

0089

066

(7)

1521

50

Old

gt105

gt105

2983

075

068

New

2127

9129

2981

098

068

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

6 Mathematical Problems in Engineering

Table 1 Revenues and market share for old and new processing technologies from 2005 Q1 to 2013 Q4

QuarterRevenue (millions USD) Market share

90 nm 80 nm 70 nm 90 nm 80 nm 70 nmOld New Old New Old New Old New Old New Old New

05Q1 6574 lt1 6574 lt1 6574 lt1 1 lt001 1 lt001 1 lt00105Q2 5715 lt1 5715 lt1 5715 lt1 1 lt001 1 lt001 1 lt00105Q3 6394 lt1 6394 lt1 6394 lt1 1 lt001 1 lt001 1 lt00105Q4 6372 lt1 6372 lt1 6372 lt1 1 lt001 1 lt001 1 lt00106Q1 6422 131 6553 lt1 6553 lt1 098 002 1 lt001 1 lt00106Q2 7158 457 7615 lt1 7615 lt1 094 006 1 lt001 1 lt00106Q3 8455 636 9091 lt1 9091 lt1 093 007 1 lt001 1 lt00106Q4 8897 1695 10592 lt1 10592 lt1 084 016 1 lt001 1 lt00107Q1 6961 2707 9475 193 9668 lt1 072 028 098 002 1 lt00107Q2 4380 2920 6716 584 7154 146 06 04 092 008 098 00207Q3 3584 4380 6929 1035 7645 319 045 055 087 013 096 00407Q4 2055 4173 4484 1744 5294 934 033 067 072 028 085 01508Q1 1408 4459 3462 2405 4457 1408 024 076 059 041 076 02408Q2 1042 5470 2800 3712 4298 2214 016 084 043 057 066 03408Q3 758 6134 2205 4687 3791 3101 011 089 032 068 055 04508Q4 286 3793 816 3263 1713 2366 007 093 02 08 042 05809Q1 95 3073 222 2946 919 2249 003 097 007 093 029 07109Q2 81 3984 203 3862 772 3293 002 098 005 095 019 08109Q3 122 5991 245 5868 856 5257 002 098 004 096 014 08609Q4 87 8595 174 8508 955 7721 001 099 002 098 011 08910Q1 93 9195 186 9102 836 8452 001 099 002 098 009 09110Q2 104 10317 208 10213 834 9587 001 099 002 098 008 09210Q3 108 10671 216 10563 647 10132 001 099 002 098 006 09410Q4 86 8558 173 8471 346 8298 001 099 002 098 004 09611Q1 83 8220 166 8137 332 7971 001 099 002 098 004 09611Q2 lt1 8147 lt1 8147 163 7984 lt001 1 lt001 1 002 09811Q3 lt1 6565 lt1 6565 66 6499 lt001 1 lt001 1 001 09911Q4 lt1 6453 lt1 6453 lt1 6453 lt001 1 lt001 1 lt001 112Q1 lt1 6271 lt1 6271 lt1 6271 lt001 1 lt001 1 lt001 112Q2 lt1 7023 lt1 7023 lt1 7023 lt001 1 lt001 1 lt001 112Q3 lt1 6429 lt1 6429 lt1 6429 lt001 1 lt001 1 lt001 112Q4 lt1 6864 lt1 6864 lt1 6864 lt001 1 lt001 1 lt001 113Q1 lt1 6870 lt1 6870 lt1 6870 lt001 1 lt001 1 lt001 113Q2 lt1 8531 lt1 8531 lt1 8531 lt001 1 lt001 1 lt001 113Q3 lt1 9300 lt1 9300 lt1 9300 lt001 1 lt001 1 lt001 113Q4 lt1 9748 lt1 9748 lt1 9748 lt001 1 lt001 1 lt001 1Data sources DRAMeXchange IHS iSuppli amp Morgan Stanley Research (2014)Data collation Industrial Technology Research Institute of Taiwan amp Taiwan Institute of Economic Research

and 70 nanometers as examples The raw data for each aregiven in Table 1

We are also interested in prediction accuracy at differentstages of diffusion The 99 95 90 80 70 60and 50 market shares for new processing technologies areearmarked as seven stages of diffusion In each stage recordspertaining to the time before the new processing technologiessurpassed the earmarkedmarket shares were used as trainingdata to fit the model parametersThe remaining records wereused as testing data to evaluate prediction accuracy

For example at the dividing line of 90 nanometers newprocessing technologies are those using 90 nanometers orless Table 1 shows that these new processing technologiesreached a 99 market share in 2009 Q4 Thus in Stage 119 pairs of revenue records for the old and new processingtechnologies from 2005 Q1 to 2009 Q3 were used as trainingdata to fit the Bass and LotkandashVolterra models by usingordinary least squares The remaining 17 pairs of revenuerecords were used as testing data to individually evaluate theMAPEs for the old and new processing technologies

Mathematical Problems in Engineering 7

Similarly 19 market share records from 2005 Q1 to 2009Q3 were used as training data to fit the Bass LotkandashVolterraand modified LotkandashVolterra models by using ordinary leastsquares in Stage 1 The remaining 17 market share recordswere used as testing data to individually evaluate the MAEsfor the old andnewprocessing technologiesThe results for allseven stages are reported in Table 2 Tables 3 and 4 report theresults of our numerical studies for the 80- and 70-nanometerdividing lines respectively

42 Analysis of Data Selection The results reported in theprevious subsection indicate that the lower the number ofrecords used as training data the higher the prediction errorsMoreover the market share predictions are significantlybetter than the revenue predictions from both the Bass andLotkandashVolterra models Consider the 90-nanometer dividingline as an example (see Table 2) When using revenue dataon the Bass model the prediction errors range from 907 toinfinity for the seven stages When using market share dataon the Bass model the prediction errors are much smallerbetween 056 and 1711 for the seven stages Similarlyfor the LotkandashVolterra model using revenue data generatesprediction errors of from 952 to infinity whereas usingmarket share data gives prediction errors of between 014and 9190 Results based on the other diving lines are similar(see Tables 3 and 4)

Arguably revenue data are not suitable for forecasting thediffusion and substitution of technologies This is becauserevenue data reflect market fluctuations caused by factorsother than substitution including industry long-term trendseconomic cycles and other confounding factors

For example global DRAM industry revenue has exhib-ited a prolonged growth trend particularly in terms ofdemand for computers and mobile communications prod-ucts In 2007 the DRAM industry invested huge amountsof capital to expand production capacity for upgrading toa new PC operating system Windows Vista However salesof Windows Vista fell far short of market expectations Thisled to an oversupply of DRAM products which caused spotprices and revenue to crash in 2007 In 2008 global DRAMrevenue continued to decline because of the global financialtsunami The consequent unexpected fall in global DRAMrevenue from 2006 Q4 to 2009 Q1 greatly affected predictionaccuracy

Market share data are more suitable than revenue datafor forecasting the diffusion and substitution of DRAMprocessing technologies This is because the market sharedata record percentages and because the total market shareis fixed at 100 Because market share data are unaffectedby seasonal factors and economic cycles they reflect mainlytechnological substitution Consequently using market sharedata generates more accurate predictions and is appropriatefor analyzing technological substitution

43 Analysis of Model Selection Having shown that usingmarket share data increases prediction accuracy for allforecasting models in this subsection we use market sharedata to compare the Bass LotkandashVolterra and modifiedLotkandashVolterra models

First note the symmetry of the modified LotkandashVolterramodel based on market share data In each stage theprediction errors for the old and new processing technologiesare the same This is because the market shares are forced tosum to 100 in (10) of the modified LotkandashVolterra modelAs a result the forecast values for the old and new processingtechnologies sum to 100 and must have the same MAEsSimilarly for the Bass model based on market share data theprediction errors for the old and new processing technologiesare highly symmetrical Again this arises because of thecharacteristics of the market share data

By contrast the LotkandashVolterra model based on marketshare data does not generate symmetric prediction errors forthe old and new processing technologies This is because themarket shares of old and new processing technologies aretreated as two species and there is no total population limitfor the two species in the LotkandashVolterra model The forecastmarket shares are not constrained to sum to 100 This iswhy the prediction errors from the LotkandashVolterra model arerelatively high

Let us compare the accuracy of the models based onmarket share data In Stages 1 to 7 of Table 2 (which reportsresults for the 90-nanometer dividing line) the Bass modelgenerates prediction errors of between 056 and 1711 theLotkandashVolterramodel generates prediction errors of between014 and 9190 and the modified LotkandashVolterra modelgenerates prediction errors of between 016 and 050(Tables 3 and 4 show that the other dividing lines generatesimilar results) Therefore when forecasting market sharesthemodified LotkandashVolterramodel outperforms the Bass andLotkandashVolterra models

5 Conclusions and Future Research

In this study we propose a modified LotkandashVolterra modelto forecast the diffusion and substitution of DRAM pro-cessing technologies Multiple generations of DRAM pro-cessing technologies are partitioned into old and new gen-erations and the quarterly data from 2005 Q1 to 2013 Q4is used for empirical studies In our model there are fourfeatures

First the type of data affects the prediction accuracy of thetechnological substitution trends Although shipment andorrevenue data have been widely used in existing research theymay be contaminated by confounding factors that reduceprediction accuracy This can be avoided by using marketshare data which are purged of such factors as long-termtrends and business cycles Consequently using market sharedata can generate more accurate forecasts of technologicalsubstitution

Second existing models for multiple competing tech-nologies require predetermined number of technologies andeach technology is assumed to go through the complete adop-tion cycle However both are not obvious in DRAM indus-try Focusing on these issues our modified LotkandashVolterramodel allows undetermined number of multigenerationtechnologies and inconsecutive adoption of new technolo-gies

8 Mathematical Problems in Engineering

Table2Predictio

nerrorsbasedon

different

datasets

mod

elsandstages90-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

sharefor

new

processin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out

119872=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

1917

98

Old

gt105

gt105

057

014

016

New

1113

1222

057

021

016

(2)

1620

95

Old

gt105

gt105

161

017

020

New

969

1113

056

018

020

(3)

1521

90

Old

gt105

gt105

245

024

024

New

940

952

244

032

024

(4)

1323

80

Old

gt105

gt105

505

029

029

New

907

2346

233

137

029

(5)

1224

70

Old

gt105

gt105

494

025

033

New

1620

6291

537

111

033

(6)

1125

60

Old

gt105

gt105

461

036

044

New

1105

9334

460

046

044

(7)

1026

50

Old

gt105

gt105

1712

4191

050

New

1450

9334

1711

9190

050

Mathematical Problems in Engineering 9

Table3Predictio

nerrorsbasedon

different

datasets

mod

elsandstages80-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

261

99

Old

lt001

lt001

lt001

lt001

lt001

New

lt001

lt001

lt001

lt001

lt001

(2)

243

95

Old

1950

2428

161

128

023

New

1236

1573

161

226

023

(3)

207

90

Old

14624

6670

113

073

086

New

1095

4937

113

072

086

(4)

1710

80

Old

71537

3235

10

3087

104

New

5394

5653

10

316

410

4

(5)

1611

70

Old

2474525

3608

780

087

110

New

4951

6009

780

137

110

(6)

1611

60

Old

2474525

3608

780

087

112

New

4951

6009

780

107

112

(7)

1413

50

Old

322327

3947

4026

141

114

New

2415

5919

40

27

152

114

10 Mathematical Problems in Engineering

Table4Predictio

nerrorsbasedon

different

datasets

mod

elsandstages70-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

ds

Num

bero

ftesting

perio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

2610

98

Old

gt105

gt105

141

007

006

New

871

1195

141

063

006

(2)

2313

95

Old

gt105

gt105

201

022

022

New

1396

1927

201

071

022

(3)

2016

90

Old

gt105

gt105

172

035

041

New

4106

4717

17

1071

041

(4)

1719

80

Old

gt105

gt105

089

051

058

New

5557

5906

089

073

058

(5)

1620

70

Old

gt105

gt105

951

077

062

New

5210

6519

95

0089

062

(6)

1620

60

Old

gt105

gt105

951

077

066

New

5210

6519

95

0089

066

(7)

1521

50

Old

gt105

gt105

2983

075

068

New

2127

9129

2981

098

068

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

Mathematical Problems in Engineering 7

Similarly 19 market share records from 2005 Q1 to 2009Q3 were used as training data to fit the Bass LotkandashVolterraand modified LotkandashVolterra models by using ordinary leastsquares in Stage 1 The remaining 17 market share recordswere used as testing data to individually evaluate the MAEsfor the old andnewprocessing technologiesThe results for allseven stages are reported in Table 2 Tables 3 and 4 report theresults of our numerical studies for the 80- and 70-nanometerdividing lines respectively

42 Analysis of Data Selection The results reported in theprevious subsection indicate that the lower the number ofrecords used as training data the higher the prediction errorsMoreover the market share predictions are significantlybetter than the revenue predictions from both the Bass andLotkandashVolterra models Consider the 90-nanometer dividingline as an example (see Table 2) When using revenue dataon the Bass model the prediction errors range from 907 toinfinity for the seven stages When using market share dataon the Bass model the prediction errors are much smallerbetween 056 and 1711 for the seven stages Similarlyfor the LotkandashVolterra model using revenue data generatesprediction errors of from 952 to infinity whereas usingmarket share data gives prediction errors of between 014and 9190 Results based on the other diving lines are similar(see Tables 3 and 4)

Arguably revenue data are not suitable for forecasting thediffusion and substitution of technologies This is becauserevenue data reflect market fluctuations caused by factorsother than substitution including industry long-term trendseconomic cycles and other confounding factors

For example global DRAM industry revenue has exhib-ited a prolonged growth trend particularly in terms ofdemand for computers and mobile communications prod-ucts In 2007 the DRAM industry invested huge amountsof capital to expand production capacity for upgrading toa new PC operating system Windows Vista However salesof Windows Vista fell far short of market expectations Thisled to an oversupply of DRAM products which caused spotprices and revenue to crash in 2007 In 2008 global DRAMrevenue continued to decline because of the global financialtsunami The consequent unexpected fall in global DRAMrevenue from 2006 Q4 to 2009 Q1 greatly affected predictionaccuracy

Market share data are more suitable than revenue datafor forecasting the diffusion and substitution of DRAMprocessing technologies This is because the market sharedata record percentages and because the total market shareis fixed at 100 Because market share data are unaffectedby seasonal factors and economic cycles they reflect mainlytechnological substitution Consequently using market sharedata generates more accurate predictions and is appropriatefor analyzing technological substitution

43 Analysis of Model Selection Having shown that usingmarket share data increases prediction accuracy for allforecasting models in this subsection we use market sharedata to compare the Bass LotkandashVolterra and modifiedLotkandashVolterra models

First note the symmetry of the modified LotkandashVolterramodel based on market share data In each stage theprediction errors for the old and new processing technologiesare the same This is because the market shares are forced tosum to 100 in (10) of the modified LotkandashVolterra modelAs a result the forecast values for the old and new processingtechnologies sum to 100 and must have the same MAEsSimilarly for the Bass model based on market share data theprediction errors for the old and new processing technologiesare highly symmetrical Again this arises because of thecharacteristics of the market share data

By contrast the LotkandashVolterra model based on marketshare data does not generate symmetric prediction errors forthe old and new processing technologies This is because themarket shares of old and new processing technologies aretreated as two species and there is no total population limitfor the two species in the LotkandashVolterra model The forecastmarket shares are not constrained to sum to 100 This iswhy the prediction errors from the LotkandashVolterra model arerelatively high

Let us compare the accuracy of the models based onmarket share data In Stages 1 to 7 of Table 2 (which reportsresults for the 90-nanometer dividing line) the Bass modelgenerates prediction errors of between 056 and 1711 theLotkandashVolterramodel generates prediction errors of between014 and 9190 and the modified LotkandashVolterra modelgenerates prediction errors of between 016 and 050(Tables 3 and 4 show that the other dividing lines generatesimilar results) Therefore when forecasting market sharesthemodified LotkandashVolterramodel outperforms the Bass andLotkandashVolterra models

5 Conclusions and Future Research

In this study we propose a modified LotkandashVolterra modelto forecast the diffusion and substitution of DRAM pro-cessing technologies Multiple generations of DRAM pro-cessing technologies are partitioned into old and new gen-erations and the quarterly data from 2005 Q1 to 2013 Q4is used for empirical studies In our model there are fourfeatures

First the type of data affects the prediction accuracy of thetechnological substitution trends Although shipment andorrevenue data have been widely used in existing research theymay be contaminated by confounding factors that reduceprediction accuracy This can be avoided by using marketshare data which are purged of such factors as long-termtrends and business cycles Consequently using market sharedata can generate more accurate forecasts of technologicalsubstitution

Second existing models for multiple competing tech-nologies require predetermined number of technologies andeach technology is assumed to go through the complete adop-tion cycle However both are not obvious in DRAM indus-try Focusing on these issues our modified LotkandashVolterramodel allows undetermined number of multigenerationtechnologies and inconsecutive adoption of new technolo-gies

8 Mathematical Problems in Engineering

Table2Predictio

nerrorsbasedon

different

datasets

mod

elsandstages90-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

sharefor

new

processin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out

119872=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

1917

98

Old

gt105

gt105

057

014

016

New

1113

1222

057

021

016

(2)

1620

95

Old

gt105

gt105

161

017

020

New

969

1113

056

018

020

(3)

1521

90

Old

gt105

gt105

245

024

024

New

940

952

244

032

024

(4)

1323

80

Old

gt105

gt105

505

029

029

New

907

2346

233

137

029

(5)

1224

70

Old

gt105

gt105

494

025

033

New

1620

6291

537

111

033

(6)

1125

60

Old

gt105

gt105

461

036

044

New

1105

9334

460

046

044

(7)

1026

50

Old

gt105

gt105

1712

4191

050

New

1450

9334

1711

9190

050

Mathematical Problems in Engineering 9

Table3Predictio

nerrorsbasedon

different

datasets

mod

elsandstages80-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

261

99

Old

lt001

lt001

lt001

lt001

lt001

New

lt001

lt001

lt001

lt001

lt001

(2)

243

95

Old

1950

2428

161

128

023

New

1236

1573

161

226

023

(3)

207

90

Old

14624

6670

113

073

086

New

1095

4937

113

072

086

(4)

1710

80

Old

71537

3235

10

3087

104

New

5394

5653

10

316

410

4

(5)

1611

70

Old

2474525

3608

780

087

110

New

4951

6009

780

137

110

(6)

1611

60

Old

2474525

3608

780

087

112

New

4951

6009

780

107

112

(7)

1413

50

Old

322327

3947

4026

141

114

New

2415

5919

40

27

152

114

10 Mathematical Problems in Engineering

Table4Predictio

nerrorsbasedon

different

datasets

mod

elsandstages70-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

ds

Num

bero

ftesting

perio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

2610

98

Old

gt105

gt105

141

007

006

New

871

1195

141

063

006

(2)

2313

95

Old

gt105

gt105

201

022

022

New

1396

1927

201

071

022

(3)

2016

90

Old

gt105

gt105

172

035

041

New

4106

4717

17

1071

041

(4)

1719

80

Old

gt105

gt105

089

051

058

New

5557

5906

089

073

058

(5)

1620

70

Old

gt105

gt105

951

077

062

New

5210

6519

95

0089

062

(6)

1620

60

Old

gt105

gt105

951

077

066

New

5210

6519

95

0089

066

(7)

1521

50

Old

gt105

gt105

2983

075

068

New

2127

9129

2981

098

068

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

8 Mathematical Problems in Engineering

Table2Predictio

nerrorsbasedon

different

datasets

mod

elsandstages90-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

sharefor

new

processin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out

119872=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

1917

98

Old

gt105

gt105

057

014

016

New

1113

1222

057

021

016

(2)

1620

95

Old

gt105

gt105

161

017

020

New

969

1113

056

018

020

(3)

1521

90

Old

gt105

gt105

245

024

024

New

940

952

244

032

024

(4)

1323

80

Old

gt105

gt105

505

029

029

New

907

2346

233

137

029

(5)

1224

70

Old

gt105

gt105

494

025

033

New

1620

6291

537

111

033

(6)

1125

60

Old

gt105

gt105

461

036

044

New

1105

9334

460

046

044

(7)

1026

50

Old

gt105

gt105

1712

4191

050

New

1450

9334

1711

9190

050

Mathematical Problems in Engineering 9

Table3Predictio

nerrorsbasedon

different

datasets

mod

elsandstages80-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

261

99

Old

lt001

lt001

lt001

lt001

lt001

New

lt001

lt001

lt001

lt001

lt001

(2)

243

95

Old

1950

2428

161

128

023

New

1236

1573

161

226

023

(3)

207

90

Old

14624

6670

113

073

086

New

1095

4937

113

072

086

(4)

1710

80

Old

71537

3235

10

3087

104

New

5394

5653

10

316

410

4

(5)

1611

70

Old

2474525

3608

780

087

110

New

4951

6009

780

137

110

(6)

1611

60

Old

2474525

3608

780

087

112

New

4951

6009

780

107

112

(7)

1413

50

Old

322327

3947

4026

141

114

New

2415

5919

40

27

152

114

10 Mathematical Problems in Engineering

Table4Predictio

nerrorsbasedon

different

datasets

mod

elsandstages70-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

ds

Num

bero

ftesting

perio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

2610

98

Old

gt105

gt105

141

007

006

New

871

1195

141

063

006

(2)

2313

95

Old

gt105

gt105

201

022

022

New

1396

1927

201

071

022

(3)

2016

90

Old

gt105

gt105

172

035

041

New

4106

4717

17

1071

041

(4)

1719

80

Old

gt105

gt105

089

051

058

New

5557

5906

089

073

058

(5)

1620

70

Old

gt105

gt105

951

077

062

New

5210

6519

95

0089

062

(6)

1620

60

Old

gt105

gt105

951

077

066

New

5210

6519

95

0089

066

(7)

1521

50

Old

gt105

gt105

2983

075

068

New

2127

9129

2981

098

068

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

Mathematical Problems in Engineering 9

Table3Predictio

nerrorsbasedon

different

datasets

mod

elsandstages80-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

dsNum

bero

ftestingperio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

261

99

Old

lt001

lt001

lt001

lt001

lt001

New

lt001

lt001

lt001

lt001

lt001

(2)

243

95

Old

1950

2428

161

128

023

New

1236

1573

161

226

023

(3)

207

90

Old

14624

6670

113

073

086

New

1095

4937

113

072

086

(4)

1710

80

Old

71537

3235

10

3087

104

New

5394

5653

10

316

410

4

(5)

1611

70

Old

2474525

3608

780

087

110

New

4951

6009

780

137

110

(6)

1611

60

Old

2474525

3608

780

087

112

New

4951

6009

780

107

112

(7)

1413

50

Old

322327

3947

4026

141

114

New

2415

5919

40

27

152

114

10 Mathematical Problems in Engineering

Table4Predictio

nerrorsbasedon

different

datasets

mod

elsandstages70-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

ds

Num

bero

ftesting

perio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

2610

98

Old

gt105

gt105

141

007

006

New

871

1195

141

063

006

(2)

2313

95

Old

gt105

gt105

201

022

022

New

1396

1927

201

071

022

(3)

2016

90

Old

gt105

gt105

172

035

041

New

4106

4717

17

1071

041

(4)

1719

80

Old

gt105

gt105

089

051

058

New

5557

5906

089

073

058

(5)

1620

70

Old

gt105

gt105

951

077

062

New

5210

6519

95

0089

062

(6)

1620

60

Old

gt105

gt105

951

077

066

New

5210

6519

95

0089

066

(7)

1521

50

Old

gt105

gt105

2983

075

068

New

2127

9129

2981

098

068

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

10 Mathematical Problems in Engineering

Table4Predictio

nerrorsbasedon

different

datasets

mod

elsandstages70-nano

meter

dividing

line

Stage

Num

bero

ftraining

perio

ds

Num

bero

ftesting

perio

ds

Earm

arkedmarket

share

forn

ewprocessin

gtechno

logies

Predictio

naccuracy

Processin

gtechno

logies

Revenu

e(MAPE

)Marketshare

(MAE)

Bassmod

elLV

mod

elBa

ssmod

elwith

out119872

=1

LVmod

elwith

out

119909+119910=1

Mod

ified

Lotkandash

Volterram

odel

(1)

2610

98

Old

gt105

gt105

141

007

006

New

871

1195

141

063

006

(2)

2313

95

Old

gt105

gt105

201

022

022

New

1396

1927

201

071

022

(3)

2016

90

Old

gt105

gt105

172

035

041

New

4106

4717

17

1071

041

(4)

1719

80

Old

gt105

gt105

089

051

058

New

5557

5906

089

073

058

(5)

1620

70

Old

gt105

gt105

951

077

062

New

5210

6519

95

0089

062

(6)

1620

60

Old

gt105

gt105

951

077

066

New

5210

6519

95

0089

066

(7)

1521

50

Old

gt105

gt105

2983

075

068

New

2127

9129

2981

098

068

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

Mathematical Problems in Engineering 11

Third the number of fitting parameters in our model isfixed and relatively small This increases not only the predic-tion accuracy but also the applicability of our model withlittle data Moreover the forecasting of a single generation oftechnology can be computed by the difference between twoour models with consecutive dividing lines

Fourth MAE is suitable for evaluating the predictionaccuracy of different forecasting models based on marketshare data There are limitations to adopt MAPE and MSEfor evaluating prediction accuracy The MAPE may generatemisleading measures of prediction accuracy for small actualvalues and the MSE is not suitable for evaluating forecastingmodels that use different types of data Therefore we usedthe MAE to evaluate the accuracy of our market sharepredictions

In the future we intend to apply our model to otherindustries and confirm our findings related to forecastingperformance and prediction accuracy This is importantbecause different industries have different characteristics Weare interested in studying these characteristics andmodifyingourmodel to fit individual industries For example integrateddevice manufacturers and IC industries are highly capitalintensive and their demands are highly influenced by seasonalfactors and business cycles Another example is the displayindustry in which old and new technologies are in fiercecompetition However alliances also exist among enterprisesfor technology supportThese interesting and special charac-teristics can be expected to reduce signal-to-noise ratios andto increase prediction accuracy We aim to investigate theseissues and develop new procedures for improving predictionaccuracy further

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N M Victor and J H Ausubel ldquoDRAMs as model organismsfor study of technological evolutionrdquo Technological Forecastingand Social Change vol 69 no 3 pp 243ndash262 2002

[2] A Grant ldquoMemory upgrade the demands of modern comput-ing call for a seismic shift in data storage and retrievalrdquo ScienceNews vol 184 pp 28ndash31 2013

[3] TrendForce ldquoTrendForce Reports mobile DRAM industry val-ued over US$36 billion in 4Q14 due to strong smartphoneshipmentsrdquo 2015 httppresstrendforcecomnodecreate_pdf1842

[4] IHS iSuppli ldquoDRAMRevenue inQ1Down 6 Percent to $83 Bil-lionrdquo 2011 httpstechnologyihscom394338dram-revenue-in-q1-down-6-percent-to-83-billion

[5] DRAMeXchange ldquoDRAMeXchange Little Change in 4Q11DRAM Revenue Ranking Mainly due to Mobile and ServerDRAM Production Increase by First-tier Manufacturersrdquo 2012httppresstrendforcecompress-516html

[6] DRAMeXchange ldquoTrendForce DRAM Industry Value Growsfor Fifth Consecutive Quarter in 4Q13rdquo 2014 httpwwwdram-exchangecomWeeklyResearchPost23655html

[7] V Peterka ldquoMacrodynamics of Technological Change MarketPenetration by New Technologies International Institute forApplied Systems Analysisrdquo Laxenburg Austria 1977

[8] J C Fisher and R H Pry ldquoA simple substitution modelof technological changerdquo Technological Forecasting and SocialChange vol 3 pp 75ndash88 1971

[9] J A Norton and F M Bass ldquoA diffusion theory model ofadoption and substitution for successive generations of high-technology productsrdquo Management Science vol 33 pp 1069ndash1086 1987

[10] F M Bass ldquoA new product growth model for consumerdurablesrdquoManagement Science vol 15 pp 215ndash227 1969

[11] V Mahajan E Muller and F M Bass ldquoDiffusion of new prod-ucts empirical generalizations andmanagerial usesrdquoMarketingScience vol 14 pp G79-G88 1995

[12] S Y Park J W Kim and D H Lee ldquoDevelopment of a marketpenetration forecasting model for Hydrogen Fuel Cell Vehiclesconsidering infrastructure and cost reduction effectsrdquo EnergyPolicy vol 39 no 6 pp 3307ndash3315 2011

[13] A-C Cheng ldquoExploring the relationship between technologydiffusion and new material diffusion the example of advancedceramic powdersrdquo Technovation vol 32 no 3-4 pp 163ndash1672012

[14] B-H Tsai Y Li and G-H Lee ldquoForecasting global adoptionof crystal display televisions with modified product diffusionmodelrdquo Computers amp Industrial Engineering vol 58 pp 553ndash562 2010

[15] D Y Kong and X H Bi ldquoImpact of social network and businessmodel on innovation diffusion of electric vehicles in ChinardquoMathematical Problems in Engineering vol 2014 Article ID230765 7 pages 2014

[16] A J Lotka Elements of physical biology Williams and WilkinsBaltimore USA 1925

[17] V Volterra ldquoFluctuations in the abundance of a species con-sidered mathematicallyrdquoNature vol 118 no 2972 pp 558ndash5601926

[18] K Smitalova and S Sujan A Mathematical Treatment ofDynamical Models in Biological Science Eills Horwood WestSussex UK 1991

[19] I R Geijzendorffer W van derWerf F J J A Bianchi and R PO Schulte ldquoSustained dynamic transience in a Lotka-Volterracompetition model system for grassland speciesrdquo EcologicalModelling vol 222 no 15 pp 2817ndash2824 2011

[20] R M Goodwin Socialism Capitalism and Economic GrowthCambridge University Press Cambridge UK 1967

[21] V B Kreng and H T Wang ldquoThe interaction of the marketcompetition between LCD TV and PDP TVrdquo Computers ampIndustrial Engineering vol 57 pp 1210ndash1217 2009

[22] J Kim D J Lee and J Ahn ldquoA dynamic competition analysison theKoreanmobile phonemarket using competitive diffusionmodelrdquoComputersamp Industrial Engineering vol 51 pp 174ndash1822006

[23] C Watanabe R Kondo and A Nagamatsu ldquoPolicy options forthe diffusion orbit of competitive innovationsmdashAn applicationof Lotka-Volterra equations to Japanrsquos transition from analog todigital TV broadcastingrdquo Technovation vol 23 no 5 pp 437ndash445 2003

[24] B-H Tsai and Y Li ldquoCluster evolution of IC industry fromTaiwan to Chinardquo Technological Forecasting amp Social Changevol 76 pp 1092ndash1104 2009

12 Mathematical Problems in Engineering

[25] S-Y Chiang ldquoAn application of LotkaVolterramodel to Taiwanstransition from 200 mm to 300 mm silicon wafersrdquo Technologi-cal Forecasting amp Social Change vol 79 pp 383ndash392 2012

[26] S-Y Chiang and G-G Wong ldquoCompetitive diffusion of per-sonal computer shipments in Taiwanrdquo Technological Forecastingamp Social Change vol 78 pp 526ndash535 2011

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of