As metioed earlier, directly forecastig o idividual product demads usually result i a far-off forecast that ot oly impairs the quality of subsequet ma

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1 Semicoductor Product-mix Estimate with Dyamic Weightig Scheme Argo Che, Ziv Hsia ad Kyle Yag Graduate Istitute of Idustrial Egieerig, Natioal Taiwa Uiversity Roosevelt Rd. Sec. 4, Taipei, Taiwa, 6 ache@tu.edu.tw Abstract Semicoductor maufacturig techology is progressig rapidly. As show by the celebrated Moor s laws, ew geeratios of semicoductor products emerge i the pace of every 6 moths. As a result, techologies of differet geeratios ofte co-exist i the same fabricatio facility. For effective maufacturig resource plaig, accurate predictio of product mix is therefore crucial. Product-mix predictio is usually made by performig the demad disaggregatio fuctio i demad plaig. I this paper, weighted product-mix estimatio methodologies are first proposed. Dyamic weightig schemes are the developed to improve the accuracy of product mix predictio. The methodologies will be tested with simulated DRAM demads ad actual semicoductor demads of differet techology geeratios. INTRODUCTION Results of demad plaig serve as the basis of every plaig activity i the supply etwork ad ultimately determie the effectiveess of maufacturig/logistic operatios i the etwork. A commo demad plaig approach is to make forecast at a aggregated level first, ad the break dow (disaggregate) the forecast statistically ad/or judgmetally ito the idividual forecasts. This is kow to be a effective meas for better forecastig because the aggregated product-group demad is observed to fluctuate less ad easier to make forecast. This approach, however, requires a accurate product-mix predictio to break dow the aggregated demad forecast ito idividual product forecasts. Product-mix ca be expressed as demad proportios i a product group. For example, proportios of 64M, 8M ad 56M DRAM demads form the product-mix of the DRAM product group. Thus, product-mix estimatio is equivalet to demad disaggregatio. That is, to estimate the product-mix, we eed to estimate the proportios of idividual demads i the product group. A effective product mix estimate should take ito cosideratio the characteristics of semicoductor demads. There are three importat characteristics that will be cosidered i our proposed methodologies:. Product substitutability: most products withi the same product group are substitutable amog oe aother.. Product life cycle (PLC): Fast-paced techology developmet leads to fast PLC trasitio. 3. Variability proportioal to volume: the greater the demad volume, the more volatile the demad /3/$7. 3 IEEE. I Fig., we simulate the 8Mb, 56Mb ad 5Mb DRAM demads to resemble these three characteristics. Fig. Simulated DRAM demads It ca be see, substitutio of phasig out product demad by the emergig product leads to dramatic chage of productmix. It is difficult to estimate the product-mix without cosiderig the product life cycles ad substitutability. I this paper, we first propose weighted product-mix estimatio methodologies. Dyamic weightig schemes are the developed to more effectively capture the PLC for better product-mix predictio. Fially, both simulated DRAM demads ad actual semicoductor demads of differet techology geeratios are used to test the proposed methodologies. WEGHITED PRODUCT-MIX ESTIMATION Amog the may product-mix estimates (or demad disaggregatio methods) [], the followig two methods are most widely used: dit t Dt Method A: pˆ i = = Method B: dit Dt t t pˆ = = i = () where d it is the demad of product i at period t; is the k umber of historical demad periods; Dt = d it is the total i= demad at period t; k is the umber of products i the product group; ad P is the mea-proportioal estimate of product i. 84 ˆi

2 As metioed earlier, directly forecastig o idividual product demads usually result i a far-off forecast that ot oly impairs the quality of subsequet maufacturig plas but also sed the ripples to the dow-stream maufacturig activities via the supply chai. To capture the effect of PLC chage o the product-mix, we would give higher weights to the more recet data; that is, we like to have the product-mix estimate more iflueced by more recetly observed demads.thus, we apply expoetial weights to the productmix estimates i (), (7), ad (3), respectively, to obtai product-mix predictio for the ext time period. First, the mea-proportioal estimate i () ow becomes a expoetially weighted average of historical demads: wit dit p ˆ i, + = m () wjt d jt j= where pˆ i, + is the predictio of product i proportio for period +; ad wit is the weight applied to product i demad at time t ad satisfies: (3.7) time to compute all cadidates to search for the best { α, α, K, α k } to miimize (4) requires eormous computig power. The most commo method is Steepest Descet Search []. The search of the smoothig costats ca be treated as a k-variable (α ~α k ) steepest-descet search with (4) as the objective fuctio to miimize. Treat { α, α, K, α } as a k poit α i k-dimesio coordiate system. The steepestdescet search process here ca be see as a process to fid the poit i the k-dimesio coordiate system. Furthermore, the movemets of α from oe poit to aother are reached by adjustig each α i with fixed step size r. There are three possible movemets for each α i, addig oe step size to α i, (α i +r), subtractig oe step size from α i, (α I r), or keep the α i at the origial positio. The successive poits of α represetig the most SSE-reduced directio which is called the gradiet vector while all the other possible directios are called cadidate vectors. Termiatio of the search occurs at the poit where the gradiet vector becomes ull; i.e., the curret α poit has the smallest SSE value. A example of two-product steepest-descet smoothig costat search process is show i Fig. 3. w it t αi ( α i ) = ( α i ) = (3) (3.8) with the costat α i to cotrol the decliig rate over time as show i Fig.. Weights W eights α =. α =.5 Fig. Expoetial weights cotrolled by α values How to choose appropriate α i values i (3) becomes critical for accurate product-mix estimate. PLC LEADING INDICATOR To fid the best smoothig costats α i of forecast estimate at period c, that is to fid smoothig costats α i (i=~k), we miimize the mea squared forecast errors over s periods: k = + s k SSE( s) ( pˆ i, t pi, t ) (4) k+ i= Determiatio of good α i is a very time cosumig task. Suppose each α i has 99 possible values ( α i =. ~.99). The, there are 99 k possible { α, α, K, α k } cadidates. The Tim e Tim e Fig.3 Two-product steepest-descet search process The steepest descet method is still a computatio-itesive method. It would be much more efficiet if we kow the directio toward the miimum. Takig the effect of chagig demad variability ito accout, we iveted a PLC trasitio leadig idicator usig the sample oe-lag autocorrelatio (SAC) statistic: SAC t t ( s ) ( dτ d )( dτ + d ) s = = ˆ τ t ρ = (5) t ( s ) ( dτ d ) s τ = t where SAC t is the oe-lag sample autocorrelatio calculated at period t usig demad dataset { d t s+, K, dt }. Whe the product is at the growth or declie phase, the product-mix proportio sigificatly rises or falls ad the SAC becomes higher as well. If the product is mature i the market ad its 85

3 proportio is stable, SAC will be lower because oise dictates the chages of the product-mix proportio (Fig. 4). Fig. 4 Relatioship amog α, SAC ad PLC Sample size s i (5) determies how sesitive the SAC is to the PLC trasitio ad to the demad oise. Fig. 5 shows the SAC values with sample sizes s=5, 5 ad 5 for a simulated DRAM demad Time Produc t- Proportio Produc t- SAC(Sample size=5) Product- SAC(Sample size=5) Product- SAC(Sample size=5) Fig. 5 SAC calculated by differet sample sizes It ca be see that the SAC with a large sample size (s=5) is too slow to reflect the PLC trasitio while the SAC with small sample size (s=5), though resposive to the PLC trasitio, is too sesitive to the demad oise. The sample size of 5, approximately o half of oe PLC phase, gives the SAC a very good idicatio of PLC trasitio. Fig. 6 shows the smoothig costat estimated by steepest descet search (SDS), SAC calculated with s=5 ad the product-mix proportios of the simulated DRAM data (Fig. ). The smoothig costat estimate goes up whe the tred is risig or decliig but goes dow i the maturity phase. The tred of SAC ad the tred of SDS-estimated smoothig costats match each other pretty well. Therefore, SAC would be a good leadig idicator to determie the chagig tred of the smoothig costats. That is, the smoothig costat estimated at period t+ should be higher tha that at period t whe SAC t+ is higher tha SAC t ad vice versa. Usig the SAC as a leadig idicator for the search for the best smoothig costats has bee prove to cut the computig time to /55. Fig. 7 shows the procedure of the predictio scheme. Fig. 6 Smoothig costat estimates ad SAC t (DRAM) START Give the Iitial α i by Steepest-Descet Search Calculate SAC t New data available Calculate ew SAC t+ ad SAC tred (SAC t+ -SAC t ) Use the SAC tred to estimate the ew α at the ext period END Fig. 7 Dyamic product mix estimate scheme EVALUATION OF PROPOSED METHODOLOGY I order to evaluate the proposed product-mix predictio scheme, we use proportio mea squared error (PMSE): PMSE = k + m t = k + i= ( Pˆ, P, ) i t i t as the evaluatio measure ad compare the performace agaist two covetioal proportio estimate methods: methods A ad B. I additio to the simulated DRAM demads, actually semicoductor demad data as show i Fig. 6 is also tested. 86

4 The results (Table ) show sigificat improvemets by the proposed method (PLC Idicator Dyamic EWMA, PIDE) i both the simulated ad actual demad cases. Simulated Data Covetioal M ethod Total PMSE Method-A.774 Method-B PID E M ethod Total PMSE PID E.96 Actual Data Covetioal M ethod Total PMSE Method-A.9766 Method-B.467 PIDE Method Total PMSE PID E.783 Table Evaluatio Results ACKNOWLEDGEMENT This research is supported i part by ISMT ad SRC (879.) ad by NSC (NSC9-8-E--46). Fig. 8 Actual semicoductor demads It is diffcult to observe the relatioship amog the three idividul demads from Fig. 8. However, the product-mix proportios (Fig. 9) show clearly the relatioship amog these three products. REFERENCE [] Charles W. Gross ad Jeffrey E. Sohl Disaggregatio Methods to Expedite Product Lie Forecastig Joural of Forecastig, Vol. 9, 33-54, 99 [] Ward Cheey ad David Kicaid, Numerical Mathematics ad Computig, 4 th editio, Brooks/Cole Publishig Compay, 999 AUTHOR BIOGRAPHY Argo Che is a faculty member of the Graduate Istitute of Idustrial Egieerig at Natioal Taiwa Uiversity (NTU). He received his Ph.D. degree i idustrial egieerig ad eared his M.S. degrees i idustrial egieerig ad i statistics from State Uiversity of New Jersey, Rutgers. Dr. Che has bee workig closely with the semicoductor idustry i Taiwa as a priciple ivestigator of several research projects. I, Dr. Che was awarded a 3-year research project co-fuded by Semicoductor Research Corporatio (SRC) ad Iteratioal Sematech (ISMT). Dr. Che published his research work mostly i IEEE Trasactios o Semicoductor Maufacturig ad Techometrics. Fig. 9 Product-mix of actual semicoductor demads 87

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