demand forecasting i time series analysis
TRANSCRIPT
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8/8/2019 Demand Forecasting I Time Series Analysis
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Demand Forecasting I
Time Series Analysis
Chris CapliceESD.260/15.770/1.260 Logistics Systems
Sept 2006
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Chris Caplice, MIT2MIT Center for Transportation & Logistics ESD.260
Agenda
Problem and Background
Four Fundamental Approaches
Time Series Methods
Predictions are usually difficult
especially for the futureYogi Berra
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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 withinthe 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
SubjectiveJudgmental Sales force surveys
Delphi techniques
Jury of experts
Experimental Customer surveys
Focus group sessions Test Marketing
Simulation
Objective
Time Series Black Box Approach
Uses past to predict the
futureCausal / Relational
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
Cost of Forecasting
Forecast Accuracy
C
ost
Cost of ErrorsIn Forecast
Total Cost
Overly Nave Models Excessive Causal Models Good Region
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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
Basic Components
Level (a) Value where demand hovers around
Trend (b) Persistent movement in one direction Typically linear but can be exponential, quadratic, etc.
Seasonal Variations (F) Movement that is periodic to the calendar Hourly, daily, weekly, monthly, quarterly, etc.
Cyclical Movements (C) Periodic movement not tied to calendar
Random Fluctuations (e or ) Irregular and unpredictable variations, noise
time
Dema
ndrate
a
time
Demand
rate
b
time
Demandrate S
Method of using past occurrences to model the futureAssumes some regular & recurring basis over time
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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Time Series
Simple Procedure
1. Select an appropriate underlying model ofthe demand pattern over time
2. Estimate and calibrate values for the model
parameters3. Forecast future demand with the models
and parameters selected
4. Review model performance and adjustparameters and model accordingly
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Time Series: Example
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120
Week
Volume
What is the forecast for period t+made at the end of period t,xt,t+?
So, what can I say?
108
109
110
111
112113
114
115
116
110
19
28
37
46
55
64
73
82
91
10
Week
Demand
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Time Series
How important is the history? Twoextreme assumptions . . . .
Cumulative Forecast All history matters equally
Pure stationary demand
Underlying Model:
x t = a + e t
where:
e t ~ iid (=0 , 2=V[e])
Forecasting Model:
1, 1 =+ =
t
iit t
x
xt
Underlying Model:
x t = xt-1 + e t
where:
e t ~ iid (=0 , 2=V[e])
Forecasting Model:
Nave Forecast
Most recent dictates next
Random Walk, Last is Next
, 1+ =t t tx x
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Time Series
Moving Average Only include the last M observations Compromise between cumulative and nave
Cumulative model (M=n) Nave model (M=1)
Assumes that some step (S) occurred
Underlying Model:
x t = a + e t
where:
e t ~ iid (=0 , 2=V[e])
Forecasting Model:
1, 1
= + + =
t
ii t Mt t
xx
M
So, some questions
How do we find M?
What trade-offs areinvolved?
How responsive are thethree models?
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108.00
109.00
110.00
111.00
112.00
113.00
114.00
115.00
116.00
1 10 19 28 37 46 55 64 73 82 91 100
ActDemand Nave MA3
MA10 MA20 Cumulative
Moving Average Forecasts
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Time Series: Exponential Smoothing
Why should past observations all be weighted the same?
Value of observation degrades over time
Introduce smoothing constant ()
Underlying Model:
x t = a + e t
where:
e t ~ iid (=0 , 2=V[e])
Forecasting Model:
xt,t+1 =xt-1,t + et (0
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xt,t+1 = xt + (1-) xt-1,t
but recalling that xt-1,t = xt-1 + (1-)xt-2,t-1
xt,t+1 = xt + (1-)(xt-1 + (1-)xt-2,t-1)
xt,t+1 = xt + (1-)xt-1 + (1-)2 xt-2,t-1
xt,t+1 = xt + (1-)xt-1 + (1-)2xt-2 + (1-)3 xt-3,t-2
xt,t+1 = (1-)0xt + (1-)
1xt-1 + (1-)2xt-2 + (1-)
3xt-3...
Time Series: Exponential Smoothing
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Pattern of Decline in Weight
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
Age of Data Point
E
ffectiveW
eight
alpha=.1alpha=.5
alpha=.9
Time Series: Exponential Smoothing
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Time Series: Non-Stationary Models
Note that MA and standard Exp Smoothing will just lag a trendThey only look at history to find the stationary levelNeed to capture the trend or seasonality factors
MA with Trend Data
0
5
10
15
20
25
30
1 2 3 4 5 6 7 8 9 10 11 12
Period
Demand
Demand MA(3) MA(6)
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Time Series: Level & Trended Data
Similar to exponential smoothing
Holts Method - smoothing constants for level (a) andtrend (b) terms
Underlying Model:x t = a + bt + e t
where: et~ iid (=0 , 2=V[e])
Forecasting Model:xt,t+ =at + bt
Where: at = xt + (1-)(at-1 +bt-1)
bt = (at - at-1) + (1- )bt-1
time
D
emandrate
a
time
D
emandrate
b
time
D
emandrate
a
time
D
emandrate
a
time
D
emandrate
b
time
D
emandrate
b
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t-1 t t+1
Objective is to forecast t+1 & beyond
Forecastat t+1
)(11
+tt
ba
1
ta
ta
tx A
)(1 tt aa
0
1
tb
B
D
C
1 +ta
Forecast followson this line
DemandorForecast
Time
))(1( 11 ++= ttHWtHWt baxa
11)1()( += tHWttHWt baab
C D
A B
Source: Atul Agarwal MLOG05
This is a linearweighted combinationof level at and BA
This is a linear weightedcombination of slopesat andC D
Time Series: Level & Trended Data
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Chris Caplice, MIT19MIT Center for Transportation & Logistics ESD.260
Time Series: Level & Seasonal Data
Multiplicative model using exponential smoothingIntroduces seasonal term, F, that covers P periods.
Underlying Model:x t = aFt + e t
where: e t ~ iid (=0 , 2=V[e])
Forecasting Model:xt,t+ =at Ft+-P
Where :at = (xt/
Ft-P) + (1- )
at-1
Ft = (xt/at) + (1- )Ft-P
An Example where P=4, t=12, =1:x12,13 =a12 F9a12 = (x12/F8) + (1- )a12F12 = (x12/a12) + (1- )F8
time
Demandrate
a
time
Demandrate S
time
Demandrate
a
time
Demandrate
a
time
Demandrate S
time
Demandrate
time
Demandrate S
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Chris Caplice, MIT20MIT Center for Transportation & Logistics ESD.260
Expand exponential model to include seasonalityWinters Method Similar to Holts Method with added termSeasonality is multiplicative
Underlying Model:x t = (a+bt) Ft + e t
where: e t ~ iid (=0 , 2=V[e])
Forecasting Model:xt,t+ =(at + bt)Ft+-P
Where :at = (xt/
Ft-P) + (1- )(
at-1+
bt-1)bt = (at - at-1) + (1- )bt-1
Ft = (xt/at) + (1- )Ft-P
Time Series: Level, Seasonal, & Trended Data
time
Demandrate
a
time
Demandrate
b
time
Demandrate S
time
Demandrate
a
time
Demandrate
a
time
Demandrate
b
time
Demandrate
b
time
Demandrate S
time
Demandrate
time
Demandrate S
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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 ()
Stationary: ranges from 0.01 to 0.30 (0.1 reasonable)
Trend/Season: ranges from 0.02 to 0.51 (0.19 reasonable)
Trend ()
Ranges from 0.005 to 0.176 (0.053 reasonable)
Seasonality ()
Ranges from 0.05 to 0.50 (0.10 reasonable)
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Questions, Comments,
Suggestions?