Task 879.1: Intelligent Demand Aggregation and Forecasting Task - - PowerPoint PPT Presentation

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

SRC Project 879 Progress report

Outline

Dynamic demand disaggregation

• Fundamental study of demand

aggregation, forecast, and disaggregation

Stage Introduction

Growth Maturity Decline

Effect of Product Life Cycle Aggregating demand for better forecast

Total Forecast

Disaggregating for detailed planning How to disaggregate?

USA

P(1)=?

Europe ….….. Africa

P(n)=? P(2)=? P(3)…..

1

How to Consider PLC Effect in disaggregation?

2

Problem Description Problem Description

• E

Exponentially xponentially W Weighted eighted M Moving

• ving A

Average statistic is introduced to catch the PLC verage statistic is introduced to catch the PLC

t n t

w

−

− = ) 1 ( α α

α α: Exponential weight : Exponential weight parameter parameter t : Exponential weight t : Exponential weight for time period for time period “ “t t” ” n : Number of total n : Number of total historical data historical data

• Exponential weights

Exponential weights (Demand is stable)

α = 0.1

weight

(Demand is changing)

α = 0.5

weight

• Different products have

different “α” values for best SSE performance.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time Weights

α α = 0.1 = 0.1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time Weights

α α = 0.5 = 0.5

Proposed Methodology Proposed Methodology -

• EWMA

EWMA

T=n+1 History Data (Proportion or Demand) Time T=n-30 T=n-29 T=n T=n-1 T=n-2 ………………….. Exponential Weight Time T=n-30 T=n-29 T=n T=n-1 T=n-2 …………………..

Sum of Weights = 1

X || P i,n+1

Use of EWMA in Disaggregation

1 ) 1 ( 1 ) 1 (

1 1 ,

= − − − = ∑

∑

= − = n t n i t n i i n t t i

w α α α

∑∑ ∑

= = = +

⋅ ⋅ =

m j n t t j t j n t t i t i n i

d w d w P

1 1 , , 1 , , 1 ,

ˆ

and and

= = Demand of product Demand of product “ “i i” ” at time at time “ “k k” ” = Weight of product = Weight of product “ “i i” ” at time at time “ “k k” ” n n = Number of total historical data = Number of total historical data m m = Number of total products = Number of total products = Smoothing constant of product = Smoothing constant of product “ “i i” ”

k i

d ,

k i

w ,

i

α

Apply EWMA weights to historical “demand” Sum of all EWMA weighted demands Exponential weights

EWMA 140 20 80 40 Demand B 19.299 / 67.869 = 0.284 19.299 8.967 6.642 3.690 A x αA 60 30 20 10 Demand A 48.57 1 1 Total 48.57 / 67.869 = 0.716 2.858 22.856 22.856 B x αB 0.1429 0.2857 0.5714 WB(αB=0.5) 0.2989 0.3321 0.3690 wA(αA=0.1) Week 3 Week 2 Week 1

Time Product

EWMA EWMA Disaggregation Disaggregation Formula Formula

PLC Stage Introduction Growth Maturity Decline

(μt 2,C×μt2) (μt 3,C×μt3) (μt 4,C×μt4) (μt 1,C×μt1)

• 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

• 3. Product Substitution within the product family

Characteristics of Industrial Demands Characteristics of Industrial Demands

The Simulated DRAM Demand Dataset The Simulated DRAM Demand Dataset

Simulated demand Resulting Proportion

• 3 products, 150-week

demand data

• Product-1 is simulated

as 256MB

• Product-2 is simulated

as 128MB

• Product-3 is simulated

as 512MB

• Each phase is simulated

about 50 week length (1 year)

5000 10000 15000 20000 25000 30000

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148

Data1 Total

5000 10000 15000 20000 25000 30000

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148

Data2 Total

5000 10000 15000 20000 25000 30000

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148

Data3 Total

Variance Sample ance Autocovari Sample SAC =

• SAC is the correlation between the two consecutive data in the same data series
• SAC ↗ when the data trend is significant
• SAC ↘ when data is without a trend (stable)

PLC Stage Introduction Growth Maturity Decline SAC trend α trend Time Proportion PLC trend

Significant trend SAC trend α trend Stable α trend SAC trend Significant trend SAC trend α trend

Determination of Determination of “ “α α” ” – – PLC Indicator PLC Indicator ( (S Sample ample A Auto uto-

• C

Correlation)

• rrelation)

PLC Stage Introduction Growth Maturity Decline

α trend Time Proportion

Simulated Product-1

• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 20 40 60 80 100 120 140 160 Time

Product-1 Proportion Product-1 SAC

SAC of Simulated Dataset SAC of Simulated Dataset

Limitation of EWMA Method Limitation of EWMA Method

n (Current Time)

Time Proportion

Historical Trend Future Trend

Consider a “n-period” proportion data The EWMA statistic is not able to capture the future trend beyond the historical data range

Limited Range of EWMA estimates Best EWMA estimate

n+1 (Next Period)

The Double EWMA smoothing constant β is introduced to estimate the “future trend”

double EWMA estimate

∑∑ ∑∑ ∑

= = = = = +

∆ ⋅ + ∆ ⋅ − ∆ ⋅ = ∆

m j n t t j t j n m j n t t j t j n i n t t i t i n i

d v D d v M d v P

1 1 , , 1 1 , , , 1 , , 1 ,

) ( ˆ ) ( ˆ

= Proportion estimates of product “i” at time “k” = Proportion’s mean estimates of product “i” at time “k” = Proportion difference estimates of product “i” at time “k” = Demand of product “i” at time “k” = = Demand difference of product “i” at time “k” and “k-1” = Smoothing Constant of product “i” n = Number of total historical data m = Number of total products

k i

P, ˆ

k i

M , ˆ

k i

P, ˆ ∆

k i

d ,

k i

d , ∆

1 , , −

−

k i k i

d d

i i β

α ,

P PLC LC I Indicator ndicator D Dynamic ynamic D Double

• uble E

EWMA Method WMA Method

∑∑ ∑

= = = +

⋅ ⋅ =

m j n t t j t j n t t i t i n i

d w d w M

1 1 , , 1 , , 1 ,

ˆ

1 ) 1 ( 1 ) 1 (

1 1 ,

= − − − = ∑

∑

= − = n t n i t n i i n t t i

w α α α

1 ) 1 ( 1 ) 1 (

1 1 ,

= − − − = ∑

∑

= − = n t n i t n i i n t t i

v β β β

Estimate the proportion increase (ΣΔP=0) Estimate the proportion Mean level (ΣM=1)

1 , 1 , 1 ,

ˆ ˆ ˆ

+ + +

∆ + =

n i n i n i

P M P

Exponential weights

Indicator of Indicator of β β – – S Sample ample A Auto uto-

• C

Correlation

• rrelation

Since ΔPi is estimated by β, SAC of proportion differences “Δpi” can be taken as indicator of β according to the same concept as α.

∑∑ ∑∑ ∑

= = = = = +

∆ ⋅ + ∆ ⋅ − ∆ ⋅ = ∆

m j n t t j t j n m j n t t j t j n i n t t i t i n i

d v D d v M d v P

1 1 , , 1 1 , , , 1 , , 1 ,

) ( ˆ ) ( ˆ 1 ) 1 ( 1 ) 1 (

1 1 ,

= − − − = ∑

∑

= − = n t n i t n i i n t t i

v β β β

• β is expected to be

higher when the slope of demand trend changes (as ΔPi 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

(60historical data, 61forecast)

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 : Simulated Data Real Data

0.001109 PIDDE-SAC-50 0.001017 PIDDE-SAC-25 0.001036 PIDDE-SAC-15 Total PMSE PIDDE Method (SAC) 0.002375 PIDE-SAC-50 0.001962 PIDE-SAC-25 0.004540 PIDE-SAC-15 Total PMSE PIDE Method (AC) 0.064664 Method-B 0.072740 Method-A Total PMSE Conventional Method 0.010753 PIDDE-SAC-50 0.008137 PIDDE-SAC-25 0.008335 PIDDE-SAC-15 Total PMSE PIDDE Method 0.007875 PIDE-SAC-50 0.007813 PIDE-SAC-25 0.008208 PIDE-SAC-15 Total PMSE PIDE Method (AC) 0.011467 Method-B 0.009766 Method-A Total PMSE Conventional Method

Outline

• Dynamic demand disaggregation

Fundamental study of demand aggregation, forecast, and disaggregation

Concept of Aggregation, Forecasting and Disaggregation

Mean-proportional disaggregating

Aggregating

Forecasting based

• n AR(1) model

Bivariate VAR(1) demands

Critical Statistical Properties

• Predictable Trend (PT): sum of autocorrelations
• ver 30 lags
• Correlation (ρ)
• Coefficient of Variation (CV): degree of

fluctuation

Predictable Trend (PT)

0.2 0.4 0.6 0.8 1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

• 1
• 0.5

0.5 1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Predictable Trend Predictable Trend

• Aggregated time series:

PT’s of individual demands are close Forecast accuracy

• Autocorrelated time series: PT Statistical forecast accuracy

Demand Correlation ρ

Correlation (ρ):

• When ρ is strong and positive, the predictable trend

will be enhanced by aggregation and result in better forecast.

) ( ) ( ) (

2 2 1 1 2 1 x x x x x x

σ σ σ ρ =

t t x x

X X

2 1 2 1

and series demand

• f

covariance the is ) ( where σ

Coefficient of Variation: CV’s

CV: measuring the degree of fluctuation

Theorem 1: CV inheritance after mean-proportional disaggregation

Mean deviation Standard = CV

X1t and X2t : two interrelated time series Yt = X1t + X2t By mean-proportional disaggregation:

t t

Y X

1 2 1 1 ' 1

× + = µ µ µ

t t

Y X

1 2 1 2 ' 2

× + = µ µ µ

2 1 x x Y

V C V C CV ′ = ′ =

and Then,

2 1 x x

CV CV ≈

is preferable

1

1 1

< =

x Y Y

CV CV CV 1

2 2

< =

x Y Y

CV CV CV

& are preferable

Evaluation Scenarios

Demand Model:

• 14 Scenarios for evaluation

      +       ⋅       +       =      

− − t t t t t t

a a x x c c x x

2 1 1 2 1 1 22 21 12 11 2 1 2 1

ϕ ϕ ϕ ϕ

      − − − +       − − + +       − + − +       + − − +       − + + +       + − + +       + + − +       + + + + 4 . 3 . 3 . 4 . 4 . 3 . 3 . 4 . 4 . 3 . 3 . 4 . 4 . 3 . 3 . 4 . 4 . 3 . 3 . 4 . 4 . 3 . 3 . 4 . 4 . 3 . 3 . 4 . 4 . 3 . 3 . 4 .       − + +       + − +       − + +       + + + 4 . 3 . 4 . 4 . 3 . 4 . 4 . 3 . 4 . 4 . 3 . 4 .       − +       + + 4 . 4 . 4 . 4 .

Interrelated demands Unilaterally related Independent demands

Demand Planning Approaches & Evaluation

Approach 1: No aggregation; No statistical forecasting. Approach 3: No aggregation; Individual AR(1) Forecast Approach 5: No aggregation; VAR(1) Forecast Approach 2: Aggregation & disaggregation; No statistical Forecasting Approach 4: Aggregation; AR(1) Forecast; Disaggregation.

FSE ratio = FSE of Approach 5 FSE of other Approaches

• Approach 5
• Approach 4
• Approach 3
• Approach 2

Approach 1 Multivariate View Disaggregation Statistical Forecasting Aggregation

Evaluation Results

Aggregating-Forecasting-Disaggregating performs best when

• 1. Both PT’s > 0 (or < 0) AND
• 2. Correlation > 0.32 AND
• 3. 0.7 < (disaggregated CV / original CV) < 1.5

Aggregating-Forecasting-Disaggregating performs worst when

• 2. Both PT’s > 0 (or < 0) but Correlation < 0
• 1. One PT > 0 and the other < 0 OR

Otherwise, individual AR(1) forecast is the best.