1. Demand Forecasting Four Fundamental Approaches Time Series „„„
General Concepts
Evaluating Forecasts – How ‘good’ is it?
Forecasting Methods (Stationary) Š
Cumulative Mean Š
Naïve Forecast ŠŠ „ ŠŠŠŠ
Moving Average
Exponential Smoothing
Forecasting Methods (Trends & Seasonality)
OLS Regression
Holt’s Method
Exponential Method for Seasonal Data
Winter’s Model Other Models

2. Demand Forecasting
Forecasting is difficult – especially for the future
Forecasts are always wrong
The less aggregated, the lower the accuracy
The longer the time horizon, the lower the accuracy
The past is usually a pretty good place to start
Everything exhibits seasonality of some sort
A good forecast is not just a number – it should include a range, description of distribution, etc.
Any analytical method should be supplemented by external information
A forecast for one function in a company might not be useful to another function (Sales to Mkt to Mfg to Trans)

3. Four Fundamental Approaches
Subjective
Judgmental
Sales force surveys
Delphi techniques
Jury of experts
Experimental
Customer surveys
Focus group sessions
Test Marketing
Objective
Causal / Relational
Econometric Models
Leading Indicators
Input-Output Models
Time Series
“Black Box” Approach
Uses past to predict the future

4. Time Series Concepts
Time Series – Regular & recurring basis to forecast
Stationarity – Values hover around a mean
Trend- Persistent movement in one direction
Seasonality – Movement periodic to calendar
Cycle – Periodic movement not tied to calendar
Pattern + Noise – Predictable and random components of a Time Series forecast
Generating Process –Equation that creates TS
Accuracy and Bias – Closeness to actual vs Persistent tendency to over or under predict
Fit versus Forecast – Tradeoff between accuracy to past forecast to usefulness of predictability
Forecast Optimality – Error is equal to the random noise

5. Evaluating Forecasts Visual Review Errors Errors Measure MPE and MAPE
Tracking Signal

6. Demand Forecasting
Generate the large number of short-term, SKU level, locally dis-aggregated demand forecasts required for production, logistics, and sales to operate successfully. „
Focus on: ŠŠŠŠ Š
Forecasting product demand
Mature products (not new product releases)
Short time horizon (weeks, months, quarters, year)
Use of models to assist in the forecast
Cases where demand of items is independent

7. Forecasting Terminology
Historical Data 50
100
150
200
250
300
350
400 10 20 30 40
Historical Data
ExPost
Forecast
Initialization
Forecast

8. Forecasting Terminology
“We are now looking at a future from here, and the future we were looking at in February now includes some of our past, and we can incorporate the past into our forecast , the first half, which is now the past and was the future when we issued our first forecast, is now over”
Laura D’Andrea Tyson, Head of the President’s Council of Economic Advisors, quoted in November of 1993 in the Chicago Tribune, explaining why the Administration reduced its projections of economic growth to 2 percent from the 3.1percent it predicted in February.

9. Forecasting Problem
Suppose your fraternity/sorority house consumed the following number of cases of beer for the last 6 weekends: 8, 5, 7, 3, 6, 9
How many cases do you think your fraternity / sorority will consume this weekend?

10. Forecasting: Simple Moving Average Method
Using a three period moving average, we would get the following forecast: 1 2 3 4 5 6 7 8 9 10
Week
Cases

11. Forecasting: Simple Moving Average Method
What if we used a two period moving average? 1 2 3 4 5 6 7 8 9 10
Week
Cases

12. Forecasting: Simple Moving Average Method
The number of periods used in the moving average forecast affects the “responsiveness” of the forecasting method:
1 Period 1 2 3 4 5 6 7 8 9 10
Week
Cases
2 Periods
3 Periods

13. Forecasting Terminology
Applying this terminology to our problem using the Moving Average forecast:
Model
Evaluation
Initialization
ExPost
Forecast
Forecast

14. Forecasting: Weighted Moving Average Method
Rather than equal weights, it might make sense to use weights which favor more recent consumption values.
With the Weighted Moving Average, we have to select weights that are individually greater than zero and less than 1, and as a group sum to 1:
Valid Weights: (.5, .3, .2) , (.6,.3,.1), (1/2, 1/3, 1/6)
Invalid Weights: (.5, .2, .1), (.6, -.1, .5), (.5,.4,.3,.2)

15. Forecasting: Weighted Moving Average Method
A Weighted Moving Average forecast with weights of (1/6, 1/3, 1/2), is performed as follows:
How do you make the Weighted Moving Average forecast more responsive?

16. Forecasting: Exponential Smoothing
Exponential Smoothing is designed to give the benefits of the Weighted Moving Average forecast without the cumbersome problem of specifying weights. In Exponential Smoothing, there is only one parameter ():
= smoothing constant (between 0 and 1)

17. Forecasting: Exponential Smoothing
Initialization:

18. Forecasting: Exponential Smoothing
Using a = 0.4, t
A(t)
F(t) 1 8 2 5
6.5 3 7
5.9 4
6.34 6 9
5.4
6.84 10
Initialization
ExPost
Forecast
Forecast

19. Forecasting: Exponential Smoothing

20. Forecasting: Exponential Smoothing
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 2 3 4 5 6 7
Period
Weight
 = 0.1
 = 0.3
 = 0.5
 = 0.7
 = 0.9

21. Outliers (eloping point)

22. Data with Trends

23. Data with Trends 10 9 8 7 A(t) 6  = 0.3 5  = 0.5 4  = 0.7 3  = 0.91 2 3 4 5 6 7 8 9 10
A(t)
 = 0.3
 = 0.5
 = 0.7
 = 0.9

24. Forecasting: Simple Linear Regression Model
Simple linear regression can be used to forecast data with trends D I b a
D is the regressed forecast value or dependent variable in the model, a is the intercept value of the regression line, and b is the slope of the regression line. 35

25. Forecasting: Simple Linear Regression Model
Error
In linear regression, the
squared errors are minimized

26. Forecasting: Simple Linear Regression Model

27. Limitations in Linear Regression Model50
100
150
200
250 2 4 6 8 10 12 14 16
As with the simple moving average model, all data points
count equally with simple linear regression.

28. Forecasting: Holt’s Trend Model
To forecast data with trends, we can use an exponential smoothing model with trend, frequently known as Holt’s model:
L(t) = aA(t) + (1- a) F(t)
T(t) =  [L(t) - L(t-1) ] + (1- ) T(t-1)
F(t+1) = L(t) + T(t)
We could use linear regression to initialize the model

29. Holt’s Trend Model: Initialization
First, we’ll initialize the model:
L(4) = (9.9)=60.1
T(4) = 9.9

30. Holt’s Trend Model: Updating
= 0.3
b = 0.4 52
64.6
7.74
72.34 6
L(t) = aA(t) + (1- a) F(t)
L(5) = 0.3 (52) (70)=64.6
T(t) =  [L(t) - L(t-1) ] + (1- ) T(t-1)
T(5) = 0.4 [64.6 – 60.1] (9.9) = 7.74
F(t+1) = L(t) + T(t)
F(6) = = 72.34

31. Holt’s Trend Model: Updating
= 0.3
b = 0.4 63
69.54
6.62
76.16 7 72
L(6) = 0.3 (63) (72.34)=69.54
T(6) = 0.4 [69.54 – 64.60] (7.74) = 6.62
F(7) = = 76.16

32. Holt’s Model Results
Initialization
ExPost
Forecast
Forecast

33. Holt’s Model Results Initialization ExPost Forecast Forecast
Regression 50
100
150
200
250
300
350 5 10 15 20
Initialization
ExPost Forecast
Forecast

34. Forecasting: Seasonal Model (No Trend)

35. Seasonal Model Formulas
L(t) = aA(t) / S(t-p) + (1- a) L(t-1)
S(t) = g [A(t) / L(t)] + (1- g) S(t-p)
F(t+1) = L(t) * S(t+1-p)
p is the number of periods in a season
Quarterly data: p = 4
Monthly data: p = 12

36. Seasonal Model Initialization
Factor
S(t)
0.60
1.00
1.55
0.85
Quarter
Average
16.0
26.5
41.0
22.5
A(t)
2003
Spring 16
Summer 27
Fall 39
Winter 22
2004 26 43 23
S(5) = 0.60
S(6) = 1.00
S(7) = 1.55
S(8) = 0.85
L(8) = 26.5
Average Sales
per Quarter =
26.5

37. Seasonal Model Forecasting
A(t)
L(t)
Seasonal
Factor
S(t)
F(t)
2004
Spring 16
0.60
Summer 26
1.00
Fall 43
1.55
Winter 23
26.50
0.85
g = 0.3
= 0.4
2005
Spring 14
Summer 29
Fall 41
Winter 22
25.18
0.59
16.00
26.71
1.03
25.18
26.62
1.55
41.32
26.34
0.84
22.60
2006
Spring
Summer
Fall
Winter
15.53
27.02
40.69
22.25

38. Seasonal Model Forecasting5 10 15 20 25 30 35 40 45 50 2 4 6 8 12 14 16

39. Forecasting: Winter’s Model for Data with Trend and Seasonal Components
L(t) = aA(t) / S(t-p) + (1- a)[L(t-1)+T(t-1)]
T(t) = b [L(t) - L(t-1)] + (1- b) T(t-1)
S(t) = g [A(t) / L(t)] + (1- g) S(t-p)
F(t+1) = [L(t) + T(t)] S(t+1-p)

40. Seasonal-Trend Model Decomposition
To initialize Winter’s Model, we will use Decomposition Forecasting, which itself can be used to make forecasts.

41. Decomposition Forecasting
There are two ways to decompose forecast data with trend and seasonal components:
Use regression to get the trend, use the trend line to get seasonal factors
Use averaging to get seasonal factors, “de-seasonalize” the data, then use regression to get the trend.

42. Decomposition Forecasting
The following data contains trend and seasonal components:

43. Decomposition Forecasting
The seasonal factors are obtained by the same method used for the Seasonal Model forecast:
Seas.
Factor
0.80
1.35
1.05
0.79
1.00
Period
Quarter
Sales 1
Spring 90 2
Summer
157 3
Fall
123 4
Winter 93 5
128 6
211 7
163 8
122
Qtr. Ave.
109
184
143
107.5
Average to 1
Average =
135.9

44. Decomposition Forecasting
With the seasonal factors, the data can be de-seasonalized by dividing the data by the seasonal factors:
Regression on the De-seasonalized data will give the trend

45. Decomposition Forecasting Regression Results

46. Decomposition Forecast
Regression on the de-seasonalized data produces the following results:
Slope (m) = 7.71
Intercept (b) = 101.2
Forecasts can be performed using the following equation
[mx + b](seasonal factor)

47. Decomposition Forecasting50
100
150
200
250
300 1 2 3 4 5 6 7 8 9 10 11 12

48. Winter’s Model Initialization
We can use the decomposition forecast to define the following Winter’s Model parameters:
L(n) = b + m (n)
T(n) = m
S(j) = S(j-p)
So from our previous model, we have
L(8) = (7.71) =
T(8) = 7.71
S(5) = 0.80
S(6) = 1.35
S(7) = 1.05
S(8) = 0.79

49. Winter’s Model Example
= 0.3 b
= 0.4 g
= 0.2
Period
Quarter
Sales
L(t)
T(t)
S(t)
F(t) 1
Spring 90 2
Summer
157 3
Fall
123 4
Winter 93 5
128
0.8 6
211
1.35 7
162
1.05 8
122
162.88
7.71
0.79 9
Spring
152 10
Summer
303 11
Fall
232 12
Winter
171
176.41
10.04
0.81
136.47
197.85
14.60
1.39
251.71
215.00
15.62
1.06
223.07
226.37
13.92
0.78
182.19 13
Spring 14
Summer 15
Fall 16
Winter
195.19
352.41
283.09
220.87

50. Winter’s Model Example50
100
150
200
250
300
350
400 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

51. Evaluating Forecasts “Trust, but Verify”
Ronald W. Reagan
Computer software gives us the ability to mess up more data on a greater scale more efficiently
While software like SAP can automatically select models and model parameters for a set of data, and usually does so correctly, when the data is important, a human should review the model results
One of the best tools is the human eye

52. Visual Review How would you evaluate this forecast? 10 20 30 40 50 6010 20 30 40 50 60 1 2 3 4 5 6 7 8 9 11 12 13 14 15

53. Forecast Evaluation Where Forecast is Evaluated ExPost Initialization50
100
150
200
250
300
350
400 10 20 30 40
Do not include
initialization data
in evaluation
ExPost
Forecast
Initialization
Forecast

54. Errors 50
100
150
200
250
300
350
400 20 25 30 35 40
All error measures compare the forecast model to the actual data for the ExPost Forecast region

55. Errors Measure
All error measures are based on the comparison of forecast values to actual values in the ExPost Forecast region—do not include data from initialization.

56. Bias and MAD

57. Bias and MAD
Bias tells us whether we have a tendency to over- or under-forecast. If our forecasts are “in the middle” of the data, then the errors should be equally positive and negative, and should sum to 0.
MAD (Mean Absolute Deviation) is the average error, ignoring whether the error is positive or negative.
Errors are bad, and the closer to zero an error is, the better the forecast is likely to be.
Error measures tell how well the method worked in the ExPost forecast region. How well the forecast will work in the future is uncertain.

58. Absolute vs. Relative Measures
Forecasts were made for two sets of data. Which forecast was better?
Data Set 1
Bias = 18.72
MAD = 43.99
Data Set 2
Bias = 182
MAD = 912.5
Data Set 1
Data Set 2

59. MPE and MAPE
When the numbers in a data set are larger in magnitude, then the error measures are likely to be large as well, even though the fit might not be as “good”.
Mean Percentage Error (MPE) and Mean Absolute Percentage Error (MAPE) are relative forms of the Bias and MAD, respectively.
MPE and MAPE can be used to compare forecasts for different sets of data.

60. MPE and MAPE Mean Percentage Error (MPE)
Mean Absolute Percentage Error (MAPE)

61. MPE and MAPE
Data Set 1

62. MPE and MAPE
Data Set 2

63. MPE and MAPE
Data Set 1
Data Set 2

64. Tracking Signal
What’s happened in this situation? How could we detect this in an automatic forecasting environment? 10 20 30 40 50 60 1 2 3 4 5 6 7 8 9 11 12 13 14 15

65. Tracking Signal
The tracking signal can be calculated after each actual sales value is recorded. The tracking signal is calculated as:
The tracking signal is a relative measure, like MPE and MAPE, so it can be compared to a set value (typically 4 or 5) to identify when forecasting parameters and/or models need to be changed.

66. Tracking Signal t A(t) F(t) F(t) - A(t) RSFE | F(t) - A(t) | S MAD TS1
15.1 2
16.8
15.9 3
11.4
14.6
3.2
3.20
1.00 4
18.7
15.8
-2.9
0.3
2.9
6.1
3.05
0.10 5
11.8
2.8
3.1
8.9
2.97
1.04 6
17.2
15.4
-1.8
1.3
1.8
10.7
2.68
0.49 7
12.9
1.7
3.0
12.4
2.48
1.21 8
22.9
17.1
-5.8
-2.8
5.8
18.2
3.03
-0.92 9
24.0
19.2
-4.8
-7.6
4.8 23
3.29
-2.31 10
32.6
23.2
-9.4
-17.0
9.4
32.4
4.05
-4.20 11
38.5
27.8
-10.7
-27.7
43.1
4.79
-5.78 12
36.6
30.4
-6.2
-33.9
6.2
49.3
4.93
-6.88 13
40.6
33.5
-7.1
-41.0
7.1
56.4
5.13
-8.00 14
51.0
38.7
-12.3
-53.3
12.3
68.7
5.73
-9.31 15
51.9
42.7
-9.2
-62.5
9.2
77.9
5.99
-10.43

67. Tracking Signal 10 20 30 40 50 60 1 2 3 4 5 6 7 8 9 11 12 13 14 15
TS = -5.78