SRC Project 879 Progress report Task 879.1: Intelligent Demand Aggregation and Forecasting Task Leader: Argon Chen Co-Investigators: Ruey-Shan Guo Shi-Chung Chang Students: Jakey Blue, Felix Chang, Ken Chen, Ziv Hsia, B.W. Hsie, Peggy Lin
Outline � Dynamic demand disaggregation • Fundamental study of demand aggregation, forecast, and disaggregation
Problem Description Problem Description Aggregating demand for Total Forecast better forecast 1 P(1)=? P(n)=? P(2)=? P(3)….. Disaggregating for detailed planning USA Europe ….….. Africa How to disaggregate? Effect of Product Life Cycle 2 How to Consider PLC Effect in disaggregation? Stage Introduction Growth Maturity Decline
Proposed Methodology - - EWMA EWMA Proposed Methodology ● E Exponentially xponentially W Weighted eighted M Moving oving A Average statistic is introduced to catch the PLC verage statistic is introduced to catch the PLC ● ● Exponential weights Exponential weights ● Weights 0.5 0.45 α = 0.1 = 0.1 α − 0.4 = α − α n t w ( 1 ) 0.35 0.3 t 0.25 0.2 0.15 0.1 α : Exponential weight : Exponential weight 0.05 Time α 0 parameter parameter Weights 0.5 0.45 t : Exponential weight t : Exponential weight 0.4 α = 0.5 = 0.5 α 0.35 for time period “ “t t” ” for time period 0.3 0.25 n : Number of total n : Number of total 0.2 0.15 historical data historical data 0.1 0.05 Time 0 weight weight ● Different products have different “ α ” values for best α = 0.1 SSE performance. α = 0.5 (Demand is stable) (Demand is changing)
Use of EWMA in Disaggregation Sum of Weights = 1 Exponential Weight Time T=n-30 T=n-29 ………………….. T=n-2 T=n-1 T=n X History Data (Proportion or Demand) Time || T=n-30 T=n-29 ………………….. T=n-2 T=n-1 T=n+1 T=n P i,n+1
EWMA Disaggregation Disaggregation Formula Formula EWMA n ∑ Apply EWMA weights ⋅ w d i , t i , t to historical “demand” ˆ = = P t 1 + i , n 1 m n ∑∑ ⋅ w d Sum of all EWMA j , t j , t = = j 1 t 1 weighted demands and and α − α − n t n n ( 1 ) = ∑ Exponential weights ∑ = i i w 1 i , t − − α n 1 ( 1 ) = = t 1 t 1 i Time Week 1 Week 2 Week 3 Total Product w A ( α A =0.1) 0.3690 0.3321 0.2989 1 W B ( α B =0.5) 0.5714 0.2857 0.1429 1 Demand A 10 20 30 60 Demand B 40 80 20 140 EWMA d , = Demand of product = Demand of product “ “i i” ” at time at time “ “k k” ” i k 19.299 / 67.869 w , A x α A 3.690 6.642 8.967 19.299 = Weight of product “ “i i” ” at time at time “ “k k” ” = Weight of product i k = 0.284 n = Number of total historical data n = Number of total historical data m m = Number of total products = Number of total products 48.57 / 67.869 B x α B 22.856 22.856 2.858 48.57 α = Smoothing constant of product “ “i i” ” = Smoothing constant of product = 0.716 i
Characteristics of Industrial Demands Characteristics of Industrial Demands 1. Effect of PLC 2. “Standard deviation of demand is proportional to demand mean” (D. C. Heat & P. L. Jackson), (R. G. Brown) Product demand at different time period can be seen as different distributions with specific mean and standard deviation that is proportional to its mean ( μ t 3 ,C ×μ t3 ) ( μ t 4 ,C ×μ t4 ) ( μ t 2 ,C ×μ t2 ) ( μ t 1 ,C ×μ t1 ) PLC Introduction Growth Maturity Decline Stage 3. Product Substitution within the product family
The Simulated DRAM Demand Dataset The Simulated DRAM Demand Dataset Simulated demand 30000 30000 Data1 Data3 30000 Data2 25000 25000 25000 Total Total ● 3 products, 150-week Total 20000 20000 20000 demand data 15000 15000 15000 ● Product-1 is simulated 10000 10000 10000 as 256MB 5000 5000 5000 ● Product-2 is simulated 0 0 0 as 128MB 1 1 8 8 15 15 22 22 29 29 36 36 43 43 50 50 57 57 64 64 71 71 78 78 85 85 92 92 99 99 106 113 120 127 134 141 148 106 113 120 127 134 141 148 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 ● Product-3 is simulated as 512MB Resulting Proportion ● Each phase is simulated about 50 week length (1 year)
α ” “ α Determination of “ ” – – PLC Indicator PLC Indicator Determination of (S Sample ample A Auto uto- -C Correlation) orrelation) ( Sample Autocovari ance SAC = Sample Variance ● SAC is the correlation between the two consecutive data in the same data series ● SAC ↗ when the data trend is significant Stable ● SAC ↘ when data is without a trend (stable) Significant trend α trend SAC trend Proportion SAC trend α trend Significant trend α trend SAC trend PLC trend SAC trend α trend Time PLC Introduction Growth Maturity Decline Stage
SAC of Simulated Dataset SAC of Simulated Dataset Simulated Product-1 1.2 Product-1 Proportion Product-1 SAC 1 0.8 0.6 0.4 0.2 Time 0 0 20 40 60 80 100 120 140 160 -0.2 -0.4 Proportion α trend Time PLC Introduction Growth Maturity Decline Stage
Limitation of EWMA Method Limitation of EWMA Method Consider a “n-period” proportion data Proportion Historical Trend Future Trend double EWMA estimate Best EWMA estimate Limited Range of EWMA estimates Time n n+1 (Current Time) (Next Period) The EWMA statistic is not able to capture the future trend beyond the historical data range The Double EWMA smoothing constant β is introduced to estimate the “future trend”
PLC LC I Indicator ndicator D Dynamic ynamic D Double ouble E EWMA Method WMA Method P ˆ ˆ ˆ = + ∆ Estimate the proportion P M P + + + i , n 1 i , n 1 i , n 1 Mean level ( Σ M=1) n n m n ∑ ∑ ∑∑ ˆ ⋅ w d ⋅ ∆ − ⋅ ∆ ( v d ) M ( v d ) i , t i , t i , t i , t i , n j , t j , t ˆ = = t 1 M ˆ = = = ∆ = t 1 j 1 t 1 P + i , n 1 m n + i , n 1 ∑∑ m n Estimate the proportion ⋅ w d ∑∑ + ⋅ ∆ D v d j , t j , t n j , t j , t = = j 1 t 1 increase ( ΣΔ P=0) = = j 1 t 1 α − α − n t n = ∑ n ( 1 ) β − β − n t n n ( 1 ) ∑ = ∑ ∑ = i i w 1 = v i i 1 i , t − − α i , t n − − β 1 ( 1 ) n 1 ( 1 ) = = = = t 1 t 1 t 1 t 1 i i Exponential weights ˆ P , = Proportion estimates of product “i” at time “k” i k ˆ M , = Proportion’s mean estimates of product “i” at time “k” i k ˆ ∆ P , = Proportion difference estimates of product “i” at time “k” i k d , = Demand of product “i” at time “k” i k ∆ d , − d d = = Demand difference of product “i” at time “k” and “k-1” i k i , k i , k − 1 α i β , = Smoothing Constant of product “i” i n = Number of total historical data m = Number of total products
Indicator of β β – – S Sample ample A Auto uto- -C Correlation orrelation Indicator of n m n ∑ ∑∑ ˆ ⋅ ∆ − ⋅ ∆ ( v d ) M ( v d ) i , t i , t i , n j , t j , t β − β − n t ( 1 ) n = ∑ n ∑ ˆ = = = t 1 j 1 t 1 = ∆ = i i v 1 P + i , t i , n 1 − − β n m n 1 ( 1 ) ∑∑ = = + ⋅ ∆ t 1 t 1 i D v d n j , t j , t = = j 1 t 1 Since Δ Pi is estimated by β , SAC of proportion differences “ Δ pi” can be taken as indicator of β according to the same concept as α . ● β is expected to be higher when the slope of demand trend changes (as Δ Pi SAC)
Real Semiconductor Demand Real Semiconductor Demand
Semiconductor Product Proportions Semiconductor Product Proportions
Performance Comparison Performance Comparison Testing Data : Simulated Demand Data and Real Semiconductor Demand Data: 3 products, 121 weeks (60 � historical data, 61 � forecast) Testing Methods : 1. Conventional Method-A, 2. Conventional Method-B, 3. PIDE-SAC method : PIDE method indicated by SAC 4. PIDDE-SAC method : PIDE method with double EWMA statistics that indicated by SAC (SAC sample size 15, 25, 50 are tested as PIDDE-SAC-15, PIDDE-SAC-25, PIDDE-SAC-50 methods) Testing Results : Real Data Simulated Data Conventional Method Total PMSE Conventional Method Total PMSE Method-A 0.072740 Method-A 0.009766 Method-B 0.064664 Method-B 0.011467 PIDE Method (AC) Total PMSE PIDE Method (AC) Total PMSE PIDE-SAC-15 0.004540 PIDE-SAC-15 0.008208 PIDE-SAC-25 0.001962 PIDE-SAC-25 0.007813 PIDE-SAC-50 0.002375 PIDE-SAC-50 0.007875 PIDDE Method (SAC) Total PMSE PIDDE Method Total PMSE PIDDE-SAC-15 0.001036 PIDDE-SAC-15 0.008335 PIDDE-SAC-25 0.001017 PIDDE-SAC-25 0.008137 PIDDE-SAC-50 0.001109 PIDDE-SAC-50 0.010753
Outline • Dynamic demand disaggregation � Fundamental study of demand aggregation, forecast, and disaggregation
Concept of Aggregation, Forecasting and Disaggregation Bivariate VAR(1) demands Aggregating Forecasting based on AR(1) model Mean-proportional disaggregating
Critical Statistical Properties • Predictable Trend (PT): sum of autocorrelations over 30 lags • Correlation ( ρ ) • Coefficient of Variation (CV): degree of fluctuation
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