Intelligent Demand Aggregation and Forecasting

(1)

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

9Dynamic demand disaggregation

• Fundamental study of demand

(2)

StageIntroduction 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

● Method-B

● Method-A

( Average the Proportion of previous “n” periods to estimate the proportion next time period )

n

P

n t it n i

=

∑

= + 1 , 1 ,

n

D

n

d

P

n t t n t t i n i

∑

= = +

=

1 1 , 1 ,

d i,m: Demand of product i at

time bucket m

D m: Demand of the product family

at time bucket m

n : Total time period

P i,m: Proportion of product i at

time bucket m

n : Total time period

200 140 60 Total 50 100 50 Total 0.700 20 80 40 B 0.300 30 20 10 A Method-B Week 3 Week 2 Week 1 Time Product 0.667 0.333 Method-A 0.6 0.2 0.2 Proportion A 0.4 0.8 0.8 Proportion B 50 100 50 Total 20 80 40 B 30 20 10 A Week 3 Week 2 Week 1 Time Product

Conventional Disaggregation Methods

(3)

●

● E

E

xponentially

W

eighted

M

oving

A

verage statistic is introduced to catch the PLC

t

n

t

w

=

α

(

1 −

α

)

−

α

α: Exponential weight : Exponential weight parameter

parameter

t : Exponential weight t : Exponential weight

for time period for time period ““tt”” 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 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time Weights

α

= 0.1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time Weights

α

= 0.5

Proposed Methodology

-

EWMA

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

Sum of Weights =

1

X

||

P

i,n+1

(4)

1 )

1 (

1 )

1 (

1 1 ,

₋

=

−

=

∑

= − = n t i n t n i i n t t i

w

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

=

= Demand of product Demand of product ““ii””at time at time ““kk”” = Weight of product

= Weight of product ““ii””at time at time ““kk”” 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 ““ii””

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 Disaggregation Formula

1. Time horizon: 46-weeks semiconductor demand data.

2. Methods: conventional A, B; EWMA-A, EWMA-B

3. Historical data to determine proportions:30 weeks data

Total

Gen00

P1

P2

P3…

Gen01

….…..

Gen19

P14

Case Study

(5)

The result shows that :

EWMA-B has the smallest MSE (best performance)

Question: how to determine, dynamically if

possible, the value of α?

MSE Comparison

Total MSE

Method-B

4,407,671

Method-A

5,572,988

EWMA

1,567,397

Best Approach in Case Study: EWMA-B

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 αtrend Time Proportion PLC trend Significant trend SAC trend αtrend Stable

αtrend SAC trend

Significant trend

SAC trend

αtrend

Determination of

“

α

”

–

PLC Indicator

(

(6)

PLC

(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

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)

0 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 Product-1 Total 0 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 Product-2 Total 0 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 Product-3 Total

The Simulated DRAM Demand Dataset

(7)

PLC

αtrend Time Proportion

Simulated Product-1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 20 40 60 80 100 120 140 160 Time

Product-1 Proportion Product-1 SAC

SAC of Simulated Dataset

Real Semiconductor Demand

(8)

Semiconductor Product Proportions

Performance metric:

P

roportion

M

ean

S

quared

E

rror

Testing Results :

k

P

PMSE

k n n t m i t i t i

∑ ∑

++ + = =

−

=

1 1 1 2 , ,

)

ˆ

(

t i

P

, :Proportion of product “i” at time “t”

:Estimated proportion of product “i” at time “t” t

i

P

ˆ

_,

Simulated Data

Real Data

Conventional Method Total PMSE

Method-A 0.072740

Method-B 0.064664

PIDE Method Total PMSE

PIDE 0.001962

Conventional Method Total PMSE

Method-A 0.009766

Method-B 0.011467

PIDE Method Total PMSE

PIDE 0.007813

Performance Comparison

(9)

Outline

• Dynamic demand disaggregation

9Fundamental study of demand

planning approaches

Concept of Aggregation, Forecasting and

Disaggregation

Mean-proportional disaggregating Aggregating Forecasting based on AR(1) model

(10)

Critical Statistical Properties

• Predictable Trend (PT): sum of autocorrelations

over 30 lags

• Correlation (ρ)

• Coefficient of Variation (CV): degree of

fluctuation

Predictable Trend (PT)

0 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 0.5 1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Predictable Trend

(11)

Example

¾ Two AR(1) demands ( X

_1t

and X

_2t

).

,

4

2 1

=

x

=

x

PT

ρ

=

0

● MSE of X

1t

= 25.25

● MSE of X

2t

= 24.62

Forecasting Standard Error (FSE)

= =

MSE

of

X

1t

+

MSE

of

X

2t

9.99

Forecasts of X1t Forecasts of X2t 0 10 20 30 40 50 60 70 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 X1t X2t

Counter Example

¾ Two AR(1) demands (W

_1t

and W

_2t

).

,

25 .

0 ,

4

2 1

=

w

=

−

w

PT

ρ

=

0

● MSE of W

_1t

= 51.35

● MSE of W

_2t

= 11.93

FSE =

10.62

0 5 10 15 20 25 30 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 W1t W2t Forecasts of W 1t Forecasts of W 2t

(12)

Demand Correlation ρ

¾ Correlation (ρ):

• When

ρ

is strong and positive

, the predictable trend

will be enhanced by aggregation and result in better

forecast.

)

0 (

)

0 (

)

0 (

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

σ

ρ

=

t t x x

X

1 2 2 1

and

series

demand

of

covariance

the

is

)

0 (

where

σ

Example

¾ Two AR(1) demands (M

_1t

and M

_2t

).

,

4

2 1

=

m

=

m

PT

ρ

=

0 .

92

● MSE of M

1t

= 11.34

● MSE of M

2t

= 12.91

F o re c a s ts o f M 1 t F o re ca sts o f M 2 t 0 10 20 30 40 50 60 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 M1t M2t

(13)

Coefficient of Variation: CV’s

¾ CV: measuring the degree of fluctuation

Theorem 1: CV inheritance after mean-proportional disaggregation

Mean

deviation

Standard

=

CV

X

_1t

and X

_2t

: two interrelated time series

Y

_t

= X

_1t

+ X

_2t

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

C

V

C

V

CV

=

′

=

′

and

Then,

Individual CV’s Should be Close

508 .

0

097 .

0

₁ 2

=

<<

x

=

x

CV

0 20 40 60 80 100 120 1 2 3 4 5 6 7 8 9 10 X1 t X2 t Yt = X1 t + X2 t Mean-proportional disaggregation 0 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 X1 t 1 1

0 .

228

x x Y

C

V

CV

=

′

=

<<

0 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 X1 t X'1 t 0 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 X1 t X2 t X'1 t 0 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 X1 t X2 t X'1 t X'2 t 2 2

0 .

228

x x Y

C

V

CV

=

′

=

>>

2 1 x x

CV

≈

is preferable

1

1 2 21

=

≈

x x

CV

is preferable

(14)

The CV after Disaggregation Should be

Smaller than the Original CV

¾ Forecast is to predict trend, not the noise.

t t t

x

a

x

=

20 +

0 .

8 ⋅

₋₁

+

where

_a

~

_N

(

0 ,

5

2

)

t t t

x

ˆ

₊₁

=

20 +

0 .

8 ⋅

● The best forecast:

Trend

¾ The best forecast CV

<

Original CV

1 1 x x Y

C

V

CV

=

′

<

&

CV

Y

=

C

V

x

′

2

<

CV

x2

1

1 1

=

<

x Y Y

CV

1

2 2

=

<

x Y Y

CV

&

are preferable

Evaluation Scenarios

¾ Demand Model:

● 14 Scenarios for evaluation













+













⋅













+













=













− − t t t t t t

a

x

c

x

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

ϕ

      − − − +       − − + +       − + − +       + − − +       − + + +       + − + +       + + − +       + + + + 4 . 0 3 . 0 3 . 0 4 . 0 4 . 0 3 . 0 3 . 0 4 . 0 4 . 0 3 . 0 3 . 0 4 . 0 4 . 0 3 . 0 3 . 0 4 . 0 4 . 0 3 . 0 3 . 0 4 . 0 4 . 0 3 . 0 3 . 0 4 . 0 4 . 0 3 . 0 3 . 0 4 . 0 4 . 0 3 . 0 3 . 0 4 . 0       − + +       + − +       − + +       + + + 4 . 0 0 3 . 0 4 . 0 4 . 0 0 3 . 0 4 . 0 4 . 0 0 3 . 0 4 . 0 4 . 0 0 3 . 0 4 . 0