Demand Forecasting I
Time Series Analysis
Chris Caplice
ESD.260/15.770/1.260 Logistics Systems
Sept 2006
Agenda
“Predictions are usually difficult
– especially for the future”
Yogi Berra
Problem and Background
Four Fundamental Approaches
Time Series Methods
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Demand Processes
Demand Forecasting
„
„
„
Predict what will happen in the future
Typically involves statistical, causal or other model
Conducted on a routine basis (monthly, weekly, etc.)
Demand Planning
„
„
„
Develop plans for creating or affecting future demand
Results in marketing & sales plans – builds unconstrained
forecast
Conducted on a routine basis (monthly, quarterly, etc.)
Demand Management
„
„
„
„
Make decisions in order to balance supply and demand within
the forecasting/planning cycle
Includes forecasting and planning processes
Conducted on an on-going basis as supply and demand changes
Includes yield management, real-time demand shifting, forecast
consumption tracking, etc.
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Four Fundamental Approaches
Subjective
Objective
Judgmental
„
„
„
Time Series
Sales force surveys
Delphi techniques
Jury of experts
„
„
Experimental
„
„
„
„
“Black Box” Approach
Uses past to predict the
future
Causal / Relational
Customer surveys
Focus group sessions
Test Marketing
Simulation
„
„
„
Econometric Models
Leading Indicators
Input-Output Models
Often times, you will need to use a combination of approaches
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Cost of Forecasting vs Inaccuracy
Å Good Region Æ
Å Overly Naïve Models Æ
Å Excessive Causal Models Æ
Cost
Total Cost
Cost of Errors
In Forecast
Cost of Forecasting
Forecast Accuracy
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Time Series
The typical problem:
„
Generate the large number of short-term, SKU level, locally
disaggregated demand forecasts required for production, logistics,
and sales to operate successfully.
Predominant use is for:
„
„
„
„
„
Forecasting product demand of . . .
Mature products over a . . .
Short time horizon (weeks, months, quarters, year) . . .
Using models to assist in the forecast where . . .
Demand of items is independent
Special situations are treated differently
„
„
„
„
New product introduction
Old product retirement
Short life-cycle products
Erratic and sparse demand
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Time Series
„
Level (a)
Š Value where demand hovers around
„
Trend (b)
Š Persistent movement in one direction
Š Typically linear but can be exponential, quadratic, etc.
„
Seasonal Variations (F)
Š Movement that is periodic to the calendar
Š Hourly, daily, weekly, monthly, quarterly, etc.
„
Cyclical Movements (C)
a
time
Demand rate
Basic Components
Demand rate
Method of using past occurrences to model the future
Assumes some regular & recurring basis over time
b
Š Periodic movement not tied to calendar
Random Fluctuations (e or ε)
Š Irregular and unpredictable variations, noise
Combine components to model demand in period t
„
„
„
Multiplicative:
xt = (b)(F)(C)(e)
Additive:
xt = a + b(t) + Ft + Ct + et
Mixed: Combination xt = a + b(Ft)t + et
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Demand rate
„
time
S
time
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Time Series
Simple Procedure
1. Select an appropriate underlying model of
the demand pattern over time
2. Estimate and calibrate values for the model
parameters
3. Forecast future demand with the models
and parameters selected
4. Review model performance and adjust
parameters and model accordingly
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Time Series: Example
116
115
114
113
112
111
110
109
108
140
120
Volume
10
91
82
73
64
55
46
37
28
19
10
100
1
Demand
What is the forecast for period t+τ
made at the end of period t, ̂xt,t+τ?
80
60
40
Week
20
0
0
40
60
80
100
120
Week
So, what can I say?
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Time Series
How important is the history? Two
extreme assumptions . . . .
Naïve Forecast
Cumulative Forecast
„
„
All history matters equally
Pure stationary demand
„
„
Underlying Model:
x
t
Most recent dictates next
Random Walk, Last is Next
Underlying Model:
= a + e
x
t
where:
t
= xt-1 + e
t
where:
e t ~ iid (μ=0 , σ2=V[e])
Forecasting Model:
x t ,t +1 =
∑
e t ~ iid (μ=0 , σ2=V[e])
Forecasting Model:
t
x t ,t +1 = xt
x
i =1 i
t
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Time Series
Moving Average
„
„
Only include the last M observations
Compromise between cumulative and naïve
Š Cumulative model (M=n)
Š Naïve model (M=1)
„
Assumes that some step (S) occurred
Underlying Model:
x
t
= a + e
So, some questions
t
„
where:
„
e t ~ iid (μ=0 , σ2=V[e])
„
Forecasting Model:
x t ,t +1 =
∑
t
i =t +1− M
How do we find M?
What trade-offs are
involved?
How responsive are the
three models?
xi
M
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Moving Average Forecasts
116.00
115.00
114.00
113.00
112.00
111.00
110.00
109.00
ActDemand
MA10
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Naïve
MA20
12
91
10
0
82
73
64
55
46
37
28
19
1
10
108.00
MA3
Cumulative
© Chris Caplice, MIT
Time Series: Exponential Smoothing
Why should past observations all be weighted the same?
Value of observation degrades over time
Introduce smoothing constant (α)
Underlying Model:
x
t
= a + e
t
where:
e t ~ iid (μ=0 , σ2=V[e])
Forecasting Model:
Recall that
et = xt - ̂xt-1,t
̂xt,t+1 = ̂xt-1,t + αet (0<α<1)
or
̂xt,t+1=αxt + (1-α)̂xt-1,t
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Time Series: Exponential Smoothing
̂xt,t+1
= αxt + (1-α) ̂xt-1,t
but recalling that
̂xt-1,t = αxt-1 + (1-α) ̂xt-2,t-1
̂xt,t+1
= αxt + (1-α)(αxt-1 + (1-α) ̂xt-2,t-1)
̂xt,t+1
= αxt + α(1-α)xt-1 + (1-α)2 ̂xt-2,t-1
̂xt,t+1
= αxt + α(1-α)xt-1 + α(1-α)2xt-2 + (1-α)3 ̂xt-3,t-2
̂xt,t+1
= α(1-α)0xt + α(1-α)1xt-1 + α(1-α)2xt-2 + α(1-α)3xt-3...
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Time Series: Exponential Smoothing
Pattern of Decline in Weight
Effective Weight
1
0.8
alpha=.1
alpha=.5
alpha=.9
0.6
0.4
0.2
0
0
1
2
3
4
5
Age of Data Point
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Time Series: Non-Stationary Models
MA with Trend Data
30
Demand
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
10
11
12
Period
Demand
MA(3)
MA(6)
Note that MA and standard Exp Smoothing will just lag a trend
They only look at history to find the stationary level
Need to capture the ‘trend’ or ‘seasonality’ factors
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Time Series:
Level & Trended Data
Forecasting Model:
̂xt,t+τ = ̂at + τ ̂bt
Where:
̂at = αxt + (1-α)( ̂at-1 + ̂bt-1)
a
time
Demand rate
Underlying Model:
x t = a + bt + e t
where:
e t ~ iid (μ=0 , σ2=V[e])
Demand rate
Similar to exponential smoothing
Holt’s Method - smoothing constants for level (a) and
trend (b) terms
b
time
̂bt = β( ̂at - ̂at-1) + (1- β) ̂bt-1
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Time Series:
Level & Trended Data
This is a linear
weighted combination
of level at A and B
Forecast follows
on this line
Forecast
at t+1
Demand or Forecast
aˆt +1
xt
aˆt = α HW xt + (1 − α HW )(aˆt −1 + bˆt −1 )
A
B
A
bˆt = β HW (aˆt − aˆt −1 ) + (1 − β HW )bˆt −1
ât
B
aˆt −1
(aˆt − aˆt −1 )
C
bˆt −1
D
C
D
(aˆt −1 + bˆt −1 )
0
t-1
Source: Atul Agarwal MLOG’05
t
t+1
Time
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This is a linear weighted
combination of slopes
at C and D
Objective is to forecast t+1 & beyond
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Time Series:
Level & Seasonal Data
Underlying Model:
x t = aFt + e t
where:
e t ~ iid (μ=0 , σ2=V[e])
Demand rate
Multiplicative model using exponential smoothing
Introduces seasonal term, F, that covers P periods.
a
Forecasting Model:
̂xt,t+τ = ̂at ̂Ft+τ-P
Where :
̂at = α(xt/ ̂Ft-P) + (1- α) ̂at-1
̂Ft = γ(xt/ a
̂ t) + (1- γ) ̂Ft-P
Demand rate
time
S
time
An Example where P=4, t=12, τ=1:
̂x12,13 = ̂a12 ̂F9
̂a12 = α(x12/ ̂F8) + (1- α) ̂a12
̂F12 = γ (x12/ ̂a12) + (1- γ) ̂F8
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Time Series:
Level, Seasonal, & Trended Data
Where :
̂at = α(xt/ ̂Ft-P) + (1- α)( ̂at-1+ ̂bt-1)
̂bt = β( ̂at - ̂at-1) + (1- β) ̂bt-1
̂Ft = γ(xt/ ̂at) + (1- γ) ̂Ft-P
a
time
Demand rate
Forecasting Model:
̂xt,t+τ =( ̂at +τ ̂bt) ̂Ft+τ-P
b
time
Demand rate
Underlying Model:
x t = (a+bt) Ft + e t
where:
e t ~ iid (μ=0 , σ2=V[e])
Demand rate
Expand exponential model to include seasonality
Winter’s Method – Similar to Holt’s Method with added term
Seasonality is multiplicative
S
time
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Comments on Time Series Models
Most of the work is bookkeeping
„
„
Initialization procedures can be arbitrary
Adding seasonality greatly complicates calculations
Most of the value comes from sharing with users
„
„
Provide insights into explaining abnormalities
Assist in initial formulations and models
Picking appropriate smoothing factors
„
Level (α)
Š Stationary: ranges from 0.01 to 0.30 (0.1 reasonable)
Š Trend/Season: ranges from 0.02 to 0.51 (0.19 reasonable)
„
Trend (β)
Š Ranges from 0.005 to 0.176 (0.053 reasonable)
„
Seasonality (γ)
Š Ranges from 0.05 to 0.50 (0.10 reasonable)
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Questions, Comments,
Suggestions?