Forecasting of Demand
Chapter 7 of Chopra
Read: Chap. 7.1-7.4; p207; p212-214 (upto/exclude
“Trend-corrected …”); 7.6; 7.7-upto p220 (exclude
“Trend-corrected …”); 7.10.
1
Global Sourcing
• components take different lead times before
reaching the destinations
wooden casing:
sea, Sweden
Cheap
peripheral
: van, HK
microprocessors:
air, Malaysia
destination:
USA
screws: train,
China (Sichuan)
LCD: truck,
China
microphone:
air, Japan
Resistors,
capacitors,
2
DVD Players
A CM/OED
3
About 10
pages
Murphy’s Law: If anything
can go wrong, it will.
4
When and In What Quantity to “Buy/Make” of
Each Parts/Finished Goods?
Push/Pull Processes (Chapter 1)
• With pull processes, execution is initiated
in response to a customer order -reactive Make-to-order, assemble-to-order
• With push processes, execution is initiated
in anticipation of customer orders -speculative Make-to-stock
Suppliers:
Parts, …
Procurement
process
Clock’s assembly
factory
Manufacturing/
Fulfillment
Processes
Orders
5
Learning Objectives
•
•
•
•
Describe types of forecasts
Describe time series
Use time series forecasting methods
Explain how to monitor & control forecasts
6
What Is Forecasting?
• Process of predicting a
future event
• “Forecasting is difficult
especially when it has to
deal with future” -- Mark
Twin
Sales will be
$200 Million!
• Underlying basis of
all business decisions
– Production
– Inventory
– Facilities, …...
7
Why forecast demand?
• We need to know how much to make
ahead of time, i.e. our production schedule
– How much raw material
– How many workers
– How much to ship to the warehouse in XXX
• We need to know how much production
capacity to build
8
Why Forecasting ?
• You’re managing merchandises for Park’n
Shop. Fruits take 3 wks to arrive from
Australia.
• You need to commit to a number of
containers NOW for the month of March in
order for a better price
• Coca-Cola Bottling: next quarter’s demand +
promotions -> production plan/ orders of
concentrates
9
Forecasting is Always Wrong
• “I think there is a world mkt for maybe 5 computers” Thomas Watson, Chairman of IBM, 1955
• “There is no reason anyone would want a computer in
their home.” - Ken Olson, CEO and Founder of Digital
Equipment Corp. , 1977
• “640K should be enough for anybody.” -- Bill Gates, 1981
• “Economists are good at explaining why their forecasts
always went wrong” -- Economist, xx, 1998
• “Fore. represents a constant pain for human
being” -- some one
10
Coping with Forecast
Errors
• Better forecasting methods (e.g., new SCM
concepts)
• Buffer mechanism (e.g., safety stock)
• Shorter lead time (i.e., reducing f horizon)
• Flexible ops (mass customisation approach)
11
Forecasting v.s. Planning
• Forecast:
– About what will happen in future
• Plan:
– About what should happen in future
– Forecasts as input
• All plans are based upon some fore.
explicitly or implicitly
12
Forecasting v.s. Planning
• When sales dept. shows sales
forecasts, be cautious. They may be
goals
• Both forecasting and planning are art
and science
– Quant f methods - educated guessing
• must be tempered by judgement bec’s
• quant f assumes future is a continuation of the
past
13
Types of Forecasts
by Time Horizon
• Short-range forecast
– Up to 1 year; usually < 3 months
– Procurement, worker assignments
• Medium-range forecast
– 3 months to 3 years
– Sales & production planning, budgeting
• Long-range forecast
– 3 + years
– Capacity planning, facility location
14
15
Types of Forecasts
by Item Forecast
• Key forecasts in business:
• Future demand for products, Sales
• Demand (sales = demand - lost sales)
• Future price of various commodities
• Lead times
• Processing times (learning curves) …
16
Forecasting Steps
•
•
•
•
•
•
•
•
•
Define objectives
Select items to be forecasted
Determine time horizon
Select forecasting model(s)
Gather data
Validate forecasting model
Make forecast
Implement results
Monitor forecast performance
17
Forecasting Approaches
Qualitative Methods
Quantitative Methods
• Used when situation is • Used when situation is
vague & little data exist ‘stable’ & historical data
– New products
exist
– New technology
3G
– Existing products
– Current technology
• Involves intuition,
experience
• Involves mathematical
techniques
• e.g., forecasting sales
on Internet
• e.g., forecasting sales
of milk, tissue papers,
…
18
Quantitative Forecasting
Methods
Qualitative
Simulation
Moving
Average
Quantitative
Forecasting
Time Series
Models
Exponential
Smoothing
時間序列
Trend
& Season
Causal
Models
因果關係
Regression
19
A future is continuation of the past (short run)
ERP: Enterprise Resource Planning
Black
Box
20
What’s a Time Series?
•
Set of evenly spaced numerical data
– Obtained by observing response variable at
regular time periods
•
Forecast based only on past values
– Assumes that factors influencing past,
present, & future will continue
21
1st & 2nd Law of Forecasting
1. In forecasting, we assume the future will
behave like the past
–
If behavior changes, our forecasts can be terrible
2. Even given 1, there is a limit to how
accurate forecasts can be (or nothing
can be predicted with complete accuracy)
–
The achievable accuracy depends on the
magnitude of the noise component
22
Monthly Demand for Sport-3506
Monthly Demand
160
140
De m and
120
100
80
60
40
20
0
0
5
10
15
20
25
30
35
40
Month
23
TS of a Raw Material’s Price
Raw Material Price
3.8
3.6
Price ($/Unit)
3.4
3.2
3
Price
2.8
2.6
2.4
2.2
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Period
24
Monthly Australian Red Wine Sales
Series
3000.
2500.
2000.
1500.
1000.
500.
0
20
40
60
80
100
120
140
25
Monthly new polio cases in the U.S.A.,
1970-1983
Series
14.
12.
10.
8.
6.
4.
2.
0.
0
20
40
60
80
100
120
140
160
26
Monthly Traffic Injuries (G.B)
beginning in January 1975
Series
2200.
2000.
1800.
1600.
1400.
1200.
1000.
0
20
40
60
80
100
120
27
Daily Dow Jones & HSI
Series 2
Series 1
14.00
14.00
13.50
13.50
13.00
13.00
12.50
12.50
12.00
12.00
11.50
11.50
11.00
11.00
10.50
10.50
10.00
10.00
9.50
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
30
Time Series Components
Sales
Original T.S.
Time
31
Time Series Components
Trend
Cyclical
Seasonal
Random
32
Trend Component
• Persistent, overall upward or downward
pattern
• Due to population, technology etc.
• Several years duration
Response
Mo., Qtr., Yr.
33
HK Regional Headquarters
34
Cyclical Component
• Repeating up & down movements
• Due to interactions of factors influencing
economy
• Usually 2-10 years duration
Cycle
Response
Mo., Qtr., Yr.

35
Electronics & Machinery Output
140.0
120.0
Index
100.0
80.0
Seri
60.0
40.0
20.0
0.0
0
1982
20
1985
40
60
1990 Yr
1995
80
2000 03
100
36
Port Unloading (Annual, 1993-2005)
160 000
140 000
120 000
100 000
80 000
60 000
40 000
20 000
0
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
37
Seasonal Component
• Regular pattern of up & down fluctuations
• Due to weather, customs etc.
• Occurs within 1 year
Spring Festives
Response
Mo., Qtr.
38
Quarterly
39
40
Random Component
• Erratic, unsystematic, ‘residual’
fluctuations
• Due to random variation or unforeseen
events
– Union strike
– Tornado
• Short duration &
nonrepeating
41
General
Time Series Models
• Any observed value in a time series is the
product (or sum) of time series components
• Multiplicative model
Yi = Ti · Si · Ci · Ri (if quarterly or mo. data)
• Additive model
Yi = Ti + Si + Ci + Ri (if quarterly or mo. data)
• Hybrids
42
Time Series Components
Sales
Original T.S.
Time
43
Time Series Components
Original T.S.
Cycle
Seasonal
Trend
Random
44
Purely Random Error No Recognizable Pattern
Demand
Demand
Sub-summary
Common Time Series Patterns
Increasing
Linear Trend
Time
Seasonal Pattern
Demand
Demand
Time
Time
Seasonal Pattern plus
Linear Growth
Time
45
Underlying model and definitions -Static Method
Systematic component = (level + trend) x seasonal
factor
L = estimate of level for period 0 (de-seasonalised
demand)
It assumes the estimates of level,
T= estimate
of trend
(increase/decrease
demand per
trend,
and seasonality
do notin vary
period)
as new demand is observed, at
St= Estimate of seasonal factor for period t
least for a fairly large number of
Dt= Actual demand observed for period t
periods.
Ft= Forecast of demand for period t
Ft+k = [ L+ (t+k)T ]St+k
Note: pp 207-211 on Static Forecast. – skip
46
HK Regional Headquarters
Static
You may use 1991-2002 to estimate the “trend line”; after 2003/05, you still use47
this line to project the future – not update it with 2003/05 new observation!
Monthly Demand for Sport-3506
Monthly Demand
160
140
De m and
120
100
80
60
40
20
0
0
5
10
15
20
25
30
35
40
Month
48
Adaptive forecasting
• The estimates of level, trend and seasonality
are updated after each demand observations
Lt  estimate of level at end of period t (de - seasonalis ed)
Tt  estimate of trend at end of period t
St  estimate of seasonal factor for period t
Dt  actual demand observed for period t
Ft  forecast of demand for period t (made in period t-1 or earlier)
Ft  k  (Lt  kTt )St  k
49
Moving Average
Lt  ( Dt  Dt 1    Dt ( N 1) ) / N
Lt 1  ( Dt 1  Dt  Dt 1    Dt ( N  2 ) ) / N
Ft  k  Lt
•
•
•
•
•
•
for all k
Assumes no trend and no seasonality =>
Level estimate is the average demand over most recent N periods
Update: add latest demand observation and and drop oldest
Forecast for all future periods is the same
Each period’s demand equally weighted in the forecast
How to choose the value of N?
–
–
N large =>
N small =>
50
Moving Average
Example
You’re manager of a museum store that
sells historical replicas. You want to
forecast sales (000) for 1998 using a 3period moving average.
1994
4
1995
6
1996
5
1997
3
1998
7
51
Moving Average
Solution
Time
All of what we need to know is
number: 5,Total
Demand thisMoving
is the forecast for all
Di which future
(N = 3)
periods!
1994 4
1995 6
1996 5
NA
Why do we need to calculate
the forecasts for the past
NA
periods?
NA
Forecasts
Moving
Avg. ( N = 3)
NA
NA
NA
1997 3
4 + 6 + 5 = 15 15/3 = 5.0
1998 7
6 + 5 + 3 = 14 14/3 = 4.7
1999 NA
5 + 3 + 7 = 15 15/3 = 5.0
Forecast for 1999
52
Moving Average Graph
Sales
8
6
4
2
0
94
Actual
Forecast
95
96 97
Year
98
99
53
Milk– weekly data / Pet products – monthly data
A pet supply product ( 6 varieties)
6000
5000
4000
3000
2000
1000
0
l
l
l
l
t v c
t v c
t v c
t v c
r
r
r
r
r y
r y
r y
r y
an eb a p a un Ju ug ep c o e an eb a p a un Ju ug ep c o e an eb a p a un Ju ug ep c o e an eb a p a un Ju ug ep c o e a
/2 J /F /M 2/A /M 2/J 2/ /A /S 2/O /N /D 3/J /F /M 3/A /M 3/J 3/ /A /S 3/O /N /D 4/J /F /M 4/A /M 4/J 4/ /A /S 4/O /N /D 5/J /F /M 5/A /M 5/J 5/ /A /S 5/O /N /D 6/J
0 02 02 0 02 0 00 02 02 0 02 02 0 03 03 0 03 0 00 03 03 0 03 03 0 04 04 0 04 0 00 04 04 0 04 04 0 05 05 0 05 0 00 05 05 0 05 05 0
20 20 20 20 20 20 2 20 20 20 20 20 20 20 20 20 20 20 2 20 20 20 20 20 20 20 20 20 20 20 2 20 20 20 20 20 20 20 20 20 20 20 2 20 20 20 20 20 20 2
54
Moving Average Method
•
Used if little or no trend
•
Used often for smoothing
– Provides overall impression of data over
time
•
Why “moving” not just overall mean?
55
Cereal Sales in HK
Quantity
(kg)
Year
56
Monthly Sales
Within a year
Month
57
Disadvantages of Moving
Averages
• Increasing N makes forecast
less sensitive to changes
• Do not forecast trend well
• Require much historical
data – N, while exponential only
last forecast!
58
Simple Exponential Smoothing
(No trend, no seasonality)
Lt 1  Dt 1  (1   ) Lt
Ft  n  Lt 1 for all n  1
• Rationale: recent past more indicative of future demand
observing
Dt+1 for
period t+1,
we of latest demand
• After
Update:
level estimate
is weighted
average
observation
andestimate
previous estimate
revise the
of the level
  is called
the smoothing
constant (0 <  < 1)
(as defined
in the textbook)
• Forecast for all future
Or periods is the same
• Assume
systematic
component
of demand
is the same for all
After
observing
D
for
period
t,
t
periods (L)
Lt = guess
Dt + (1-) Lt-1 
• Lt is the best
at period t of what the systematic demand
level is For all n1, Ft+n = Lt
59
Simple Exponential Smoothing
– Example 7-2
Data: 120, 127, 114, 122.
L0= 120.75
 = 0.1
F1 = L0 =120.75
Alternatively, if you are only
Note:interested
(120+127+114)/3
be estimated in a
D1= 120 in F5, then LL30 =can
=120.33 way! Here L
subjective
0
E1 = F1 – D1 = 120.75 – 120 = 0.75
=(120+127+114+122)/4
0.1- D4+0.9
L1 =  DL1 4+=(1
 ) L0 L3 = 12.2+108.2 =120.4
Especially when
there is
= (0.1)(120) + (0.9)(120.75)
=
120.68
=> F5 insufficient
= 120.4
data.
F2 = L1= 120.68, F3 = L2 = 121.31, …
F5 = L4= 120.72 => the forecast for period 5
60
Simple Exponential Smoothing –
Example: Tables 7-1 & 7-5
L0= 22083
F1 = L0
 = 0.1
D1=8000
E1 = F1 – D1 = 22083 – 8000 = 14083
L1 =  D1 + (1 -  ) L0
= (0.1)(8000) + (0.9)(22083) = 20675
F2 = L1= 20675,
F10 = L1 = 20675
Note: this example appears in pp 208-219
61
Simple Exponential Smoothing
Lt 1  Lt   ( L t  Dt 1 )


•
•
•
Et 1
Update: new level estimate is previous estimate adjusted by
weighted forecast error
How to choose the value of the smoothing constant ?
– Large   responsive to change, forecast subject to
random fluctuations
– Small   may lag behind demand if trend develops
Incorporates more information but keeps less data than
moving averages
–
–
Average age of data in exponential smoothing is 1/
Average age of data in moving average is (N+1)/2
If  is 0 then …
If  is 1 then ...
62
Understanding the exponential
smoothing formula
Lt 1  Dt 1  (1   ) Lt
 Dt 1  (1   )(Dt  (1   ) Lt 1 )
 Dt 1   (1   ) Dt  (1   ) 2 Lt 1

 Dt 1   (1   ) Dt   (1   ) 2 Dt 1     (1   ) k Dt  k  
• Demand of k-th previous period carry a weight of
hence the name exponential smoothing
• Demand of more recent periods carry more weight
63
Forecast Effect of
Smoothing Constant ()
The alpha parameter for exponential smoothing ...
Period .10
.30
.50
.70
1
.10
.30
.50
.70
Ft
= ·Dt - 1 +
2
.09
.21
.25
.21
·(1-)·Dt 3
.08
.15
.13
.06
+
4
.07
.10
.06
.02
·(1- )2·Dt - 3 +
5
.07
.07
.03
.01 ·(1- )3·D
t - 4 + ...
6
.06
.05
.02
.00
7
.05
.04
.01
8
.05
.02
.00
64
Exponential Smoothing
Example
You’re organising a international meeting. You
want to forecast attendance for 2000 using
exponential smoothing
( = .10). The 1994 forecast was 175.
1994
180
1995
168
1996
159
1997
175
1998
190
65
Exponential Smoothing
Solution
Lt = Lt-1 + · (Dt - Lt-1)
Time Actual
Forecast, Ft
( = .10)
1994
180
175.00 (Given)
1995
168
175.00 + .10(180 - 175.00) = 175.50
1996
159
175.50 + .10(168 - 175.50) = 174.75
1997
175
174.75 + .10(159 - 174.75) = 173.18
1998
190
173.18 + .10(175 - 173.18) = 173.36
1999
NA
173.36 + .10(190 - 173.36) = 175.02
66
Trend corrected exponential
smoothing (Holt’s model)
Update :
Lt  Dt  (1   )( Lt 1  Tt 1 )
Tt   ( Lt  Lt 1 )  (1   )Tt 1
Forecast :
Skipped
Ft  n  Lt  nTt
  is the smoothing constant for trend updating
• If  is large, there is a tendency for the trend
term to “flip-flop” in sign
• Typical  is 2
67
Holt’s model - Example
L0= 12015 T0=1549  = 0.1   0.2
F1 = L0 + T0 = 12015 + 1549 = 13564 , D1=8000
E1 = F1 – D1 = 13564 – 8000 = 5564
L1 =  D1 + (1 -  )(L0 + T0)
= (0.1)(8000) + (0.9)(13564) = 13008 Skipped
T1 =  (L1  L0) + (1 -  )T0
= (0.2)(13008  12015) + (0.8)(1549) = 1438
F2 = L1+T1= 13008+1438 = 14446,
F10 = L1 + 9 T1 = 13008 + 9(1438) = 25950
68
Trend and seasonality corrected
exponential smoothing (Winter’s model)
Update :
 Dt 1 
  (1   )( Lt  Tt )
Lt 1   
 St 1 
Tt 1   ( Lt 1  Lt )  (1   )Tt
Skipped
 Dt 1 
  (1   ) St 1
St 1 p   
 Lt 1 
Forecast :
Ft  n  ( Lt  nTt ) St  n
69
Trend- & Seasonality-corrected
Exp Smooth.
(level +trend ) x seasonal factor (Winter’s
Smoothed value
Method/Model)
for t+1
Need 3 revision eqns
,D, ==Actual
smooth. para.
St+1
==Seasonal
factor, p?
Tt+1
Trend
forecast
t+1
(1) Lt+1 =  (Dt+1 / St+1 )+ (1- ) (Lt + Tt )
(2) Tt+1 =  ( Lt+1 – Lt ) + (1- ) Tt
Skipped
(3) St+p+1 =  (Dt+1 / Lt+1 ) + (1-  ) St+1
Tt = Forecast for t+n
Forecast of t+n: Ft+n = (Lt + nTt ) St+n , n >1
70
Winter’s model - Example
L0= 18439 T0=524 S1 = 0.47, S2 =0.68, S3 =1.17, S4
=1.67,
 = 0.1,
  0.2,
 = 0.1,
F1 = (L0 + T0) S1 = (18439 + 524)(0.47) = 8913
D1=8000,
E1 = F1 – D1 = 8913 – 8000 = 913
L1 =  (D1/S1) + (1 -  )(L0 + T0)
=
T1 =  (L1  L0) + (1 -  )T0
= (0.2)(18769  18439) + (0.8)(524) = 485
S5 =  (D1/L1) + (1 - )S1
= (0.1)(8000/18769) + (0.9)(0.47) = 0.465
Skipped
F2 = (L1+T1)S2 =
F11 = (L1 + 10 T1)S11 =
71
Winter’s ES
• Why Dt+1 /St+1?
• How to initialize the forecast?
• How to choose alpha, beta and gamma
values?
Skipped
• Winter’s method is an extension of
Holt’s
72
Special Forecasting Difficulties for
Supply Chains
•
New products and service introductions
–
–
–
–
•
No past history
Use qualitative methods until sufficient data collected
Examine correlation with similar products
Use a large exponential smoothing constant
Lumpy derived demand
–
–
–
•
Not
required
Large but infrequent orders
Random variations “swamps” trend and seasonality
Identify reason for lumpiness and modify forecasts
Spatial variations in demand
–
Separate forecast vs. allocation of total forecasts
78
A Lumpy Demand Example
140
120
100
80
Skipped
Series1
60
40
20
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
79
Analysing Forecast Errors
•
•
Choose a forecast model
Monitor if current forecasting method/model accurate
–
–
•
Understand magnitude of forecast error
–
•
Consistently under-predicting? Over-predicting?
When should we adjust forecasting procedures?
In order to make appropriate contingency plans
Assume we have data for n historical periods
Et  Ft  Dt  forecast error in period t
At  Et  absolute deviation for period t
80
Measures of Forecast Error
•
Mean Square Error
(MSE)
1 n 2
MSEn   Et
– Estimate of variance (s )
of random component
n i 1
Mean Absolute Deviation
1 n
(MAD)
MAD

A

n
t
– If random component
n i 1
normally distributed,
s1.25 MAD
100  n Et
Mean Absolute Percent
 
MAPE n 
Error (MAPE)
n  i 1 Dt
2
•
•



81
Further Error Equations
• What does it mean when MFE  0 ?
• What does it mean when MFE =
MAD?
• What does it mean when MSE <
MAD?
• Why do we need MAPE?
82
Guidelines for Selecting
Forecasting Model
• No pattern or direction in forecast error
– Error = (Fore. -Actual )
– Seen in plots of errors over time
• Smallest forecast error
– Mean square error (MSE)
– Mean absolute deviation (MAD)
83
Pattern of
Forecast Error
Trend Not Fully
Accounted for
Desired Pattern
Error
Error
0
0
Time (Years)
Examples?
Time (Years)
84
Forecasting Steps
•
•
•
•
•
•
•
•
•
Define objectives
Select items to be forecasted
Determine time horizon
Select forecasting model(s)
Gather data
Validate forecasting model
Make forecast
Implement results
Monitor forecast performance
85
Tracking Errors
You have been using one!
•
Errors due to:
–
–
•
•
Random component
Bias (wrong trend, shifting
seasonality, etc.)
n
biasn   Et
i 1
Monitor quality of forecast with a
tracking signal
Alert if signal value exceeds
threshold
–
biast
TSt 
Indicates underlying environment
MADt
changed and model becomes
inappropriate
86
Monitoring:
Tracking Signal
• Tracking signal -- Checks for consistent
bias over many periods
• Measures how well forecast is
predicting actual values
• Ratio of running sum of forecast errors
(RSFE) to mean absolute deviation
(MAD)
– Good tracking signal has low values
87
TS = RSFE / MAD
RSFE(t)=RSFE(t-1)+E(t) = Bias
MAD = sum of | forecast errors| over time/ n
If TS is greater than some maximum value
then report a problem.
88
Tracking Signal Equation
RSFE
TS 
MAD
n
 (Fi  Di )
 i 1
MAD
 (Forecast Errors )

MAD
89
Tracking Signal Computation*
Mo Forc
Act Error RSFE Abs Cum MAD
Error Error
1
100
90
10
10
10
2
100
95
5
15
5
3
100 115
-15
0
4
100
90
10
5
100 115
6
100 130
TS
10 10.0
1
15
7.5
2
15
30 10.0
0
10
10
40 10.0
1
-15
-5
15
55 11.0
-.5
-30
-35
30
85 14.2 -2.5
90
Tracking Signal Plot
3
2
1
TS 0
-1
-2
-3
1
2
3
4
5
6
Time
91
Tracking Signal
• Limits used for tracking signal ratio
usually between (-3/6, 3/6)
6
0
Time
-6
• Used for monitoring
Re-evaluate the
model
92
Tracking Signal
• Cautious!
– Is it always good to have TS=0?
– TS: the smaller the better?
– Can TS be used for comparing models?
93
Summary so far
• Importance of forecasting in a supply chain
• Forecasting models and methods
• Exponential smoothing
– Stationary model
– Trend
– Seasonality
• Measures of forecast errors
• Tracking signals
94
A Remark
• Adaptive method
Observed Dt-1: Ft = f(Dt-1, …), observed Dt: Ft+1
= f(Dt , …), …
• Static method (Section 7.5) – it assumes the
estimates of level,. trend, and seasonality do not
vary as new demand is observed:
Observed Dt-1: Ft = f(Dt-1, …), observed Dt: Ft+1
= f(Dt-1 , …), …
95
• Forget all beyond this slide
96
Part 1 of As# 1
Chapter 7 in 3rd edition
• Discussion questions
All are posted as
downloadable
– Q4, Q9
• Exercises
– Q1, Q 2 & Q3.
•
The deadline: hand in the class
before ?. Part 2 will be released later.
97
Reading List (Chap. 7)
• Adaptive Forecasting, up to “Trend- and
Seasonality- … Winter’s Model)”.
• Section 7.6. Measures of Forecast Errors.
• Section 7.7, up-to “Trend- and
Seasonality- … Winter’s Model)”.
98
Moving Average Method
•
MA is a series of arithmetic means
•
Used if little or no trend
•
Used often for smoothing
– Provides overall impression of data over
time
•
Equation
Lt
Demand in Previous N Periods
N
99
Adaptive Forecasting
Moving Average Method
Systematic component of demand = Level
Chopra: p. 82
100
Trend-corrected Exp Smooth.
 Systematic component = level +trend
Dt = a t + b + Random
(Holt’s Model)
Need two revision eqns
(1) Level component
New forecast = Old forecast+ correction
= Lt-1 + Tt-1 + correction
Error = Dt – (Lt-1 + Tt-1 )
Lt = Lt-1 + Tt-1 +  (Dt – (Lt-1 + Tt-1 ) )
=  Dt + (1- ) (Lt-1 + Tt-1 )
101
Trend-corrected Exp Smooth.
(2) Trend component
Tt =  ( Lt -Lt-1) + (1- ) Tt
(3) Forecasting
Ft+1 = Lt + Tt ,
Ft+n = Lt + n Tt
Correction: Chopra, p84 under eqn 4.14: Should be
“After observing demand for period t+1”.
102
Trend- & Seasonality-corrected Exp Smooth.
Systematic component = (level +trend ) x
seasonal factor, with periodicity p.
Dt = (a t + b) s + Random
(Winter’s Model)
Need 3 revision eqns
(1) Lt =  (Dt / St )+ (1- ) (Lt-1 + Tt-1 )
(2) Tt =  ( Lt -Lt-1) + (1- ) Tt-1
(3) St+p =  (Dt/ Lt ) + (1-  ) St , t+p and t are the
same “season”, and St is the latest estimate of
seasonal factor which was made t-p periods ago
(for period t).
103
Forecast: Tt+n = (Lt + nTt ) St+n
Static Methods
•
It assumes that the estimates of level,
trend and seasonality do not vary as
new demand is observed – no need to
update
Systematic component = (Level+ trend)x
seasonal factor
Ft+n = [L+ (t+n ) T] St+n
104
Visual Inspection or Systematic
Diagnosing
105
Equations fh
106
Forecast Error Equations
• Mean Square Error (MSE)
• Mean Absolute Deviation (MAD)
107
Selecting Forecasting Model
Example
You’re an analyst for Hasbro Toys. You’ve
forecast sales with Holt’s model & expo.
smoothing. Which model do you use?
Actual Holt’s Model Expo Smooth
Year
Sales
Forecast
Forecast
1992
1
0.6
1.0
1993
1
1.3
1.0
1994
2
2.0
1.9
1995
2
2.7
2.0
1996
4
3.4
3.8
108
Holt’s Model
Evaluation
Year
1992
1993
1994
1995
1996
Total
MSE =
MAD =
Di
1
1
2
2
4
Fi
0.6
1.3
2.0
2.7
3.4
Error Error2 |Error|
0.4
0.16
0.4
-0.3
0.09
0.3
0.0
0.00
0.0
-0.7
0.49
0.7
0.6
0.36
0.6
0.0
1.10
2.0
Error2 / n = 1.10 / 5 = .220
|Error| / n = 2.0 / 5 = .400
109
Exponential Smoothing
Model Evaluation
Year
1992
1993
1994
1995
1996
Total
Di
1
1
2
2
4
Fi
1.0
1.0
1.9
2.0
3.8
Error Error2 |Error|
0.0
0.00
0.0
0.0
0.00
0.0
0.1
0.01
0.1
0.0
0.00
0.0
0.2
0.04
0.2
0.3
0.05
0.3
MSE =
Error2 / n = 0.05 / 5 = 0.01
MAD =
|Error| / n = 0.3 / 5 = 0.06
110
Further Error Equations
• Mean absolute percentage error
MAPE = i=1 | Ei / Di| x100/n
• Bias (Mean forecast error = MFE)
Bias = i=1 Ei
111
Further Error Equations
• What does it mean when MFE  0 ?
• What does it mean when MFE =
MAD?
• What does it mean when MSE <
MAD?
• Why do we need MAPE?
112
Tracking Signal Plot
113
Tracking Signal
• Limits used for tracking signal ratio
usually between (-6, 6)
6
0
Time
-6
• Used for monitoring
Re-evaluate the
model
114
Tracking Signal
• Cautious!
– Is it always good to have TS=0?
– TS: the smaller the better?
– Can TS be used for comparing models?
115
An Example
CLP Power has been collecting data on demand
for electric power in a recently developed
residential area for only the past 2 years.
1. What are weaknesses of the standard fore. methods
as applied to this set of data?
2. Propose your own approach to forecasting.
3. Forecast demand for each month of next year using
your model.
116