1. Forecasting Operations Analysis and Improvement 2017 Spring
Dr. Tai-Yue Wang
Industrial and Information Management Department
National Cheng Kung University

2. Outline Introduction to Forecasting Subjective Forecasting Methods
Objective Forecasting Methods
Evaluating Forecasts
Methods for Forecasting Stationary Series
Trend-Based Methods
Methods for Seasonal Series
Box-Jenkins Models

3. Overview of Operations Planning Activities

4. Introduction to Forecasting
What is forecasting?
Primary Function is to Predict the Future
Why are we interested?
Affects the decisions we make today
Examples: who uses forecasting in their jobs?
forecast demand for products and services
forecast availability of manpower
forecast inventory and materiel needs daily

5. Introduction to Forecasting -- What Makes a Good Forecast
It should be timely
It should be as accurate as possible
It should be reliable
It should be in meaningful units
It should be presented in writing
The method should be easy to use and understand in most cases.

6. Introduction to Forecasting – The Time Horizon

7. Introduction to Forecasting – Characteristics of Forecasts
They are usually wrong!
A good forecast is more than a single number
mean and standard deviation
range (high and low)
Aggregate forecasts are usually more accurate
Accuracy erodes as we go further into the future.
Forecasts should not be used to the exclusion of known information

8. Subjective Forecasting Methods
Sales Force Composites
Aggregation of sales personnel estimates
Customer Surveys
Jury of Executive Opinion
The Delphi Method
Individual opinions are compiled and reconsidered. Repeat until and overall group consensus is (hopefully) reached.

9. Objective Forecasting Methods
Two primary methods: causal models and time series methods
Causal Models
Let Y be the quantity to be forecasted and (X1,
X2, , Xn) be n variables that have predictive power for Y. A causal model is Y = f (X1, X2, , Xn).
A typical relationship is a linear one. That is,
Y = a0 + a1X an Xn.

10. Time Series Methods Trend Seasonality Cycles Randomness
A time series is just collection of past values of the variable being predicted. Also known as naïve methods. Goal is to isolate patterns in past data. (See Figures on following pages)
Trend
Seasonality
Cycles
Randomness

12. Time Series Methods -- Notation Conventions
Let D1, D2, Dn, be the past values of the series to be predicted (demand). If we are making a forecast in period t, assume we have observed Dt , Dt-1 etc.
Let Ft, t + t = forecast made in period t for the demand in period t + t where t = 1, 2, 3, …
Ft -1, t is the forecast made in t-1 for t and Ft, t+1 is the forecast made in t for t+1. (one step ahead) Use shorthand notation Ft = Ft - 1, t .

13. Evaluation of Forecasts
The forecast error in period t, et, is the difference between the forecast for demand in period t and the actual value of demand in t.
For a multiple step ahead forecast:
et = Ft - t, t - Dt.
For one step ahead forecast: et = Ft - Dt.

14. Evaluation of Forecasts
MAD = (1/n) S | e i |
MSE = (1/n) S ei 2

15. Evaluation of Forecasts -- Biases in Forecasts
A bias occurs when the average value of a forecast error tends to be positive or negative.
Mathematically an unbiased forecast is one in which E (e i ) = 0. See Figure 2.3 on page 64 in text (next slide).

16. Forecast Errors Over Time Figure 2.3

17. Forecasting for Stationary Series
A stationary time series has the form:
Dt = m + e t where m is a constant and e t is a random variable with mean 0 and var s2 .
Two common methods for forecasting stationary series are moving averages and exponential smoothing.

18. Forecasting for Stationary Series -- Moving Averages
In words: the arithmetic average of the n most recent observations. For a one-step-ahead forecast:
Ft = (1/N) (Dt Dt Dt - n )

19. Forecasting for Stationary Series -- Moving Averages
Example:
Quarterly data:

20. Forecasting for Stationary Series -- Moving Averages --Summary
Advantages of Moving Average Method
Easily understood
Easily computed
Provides stable forecasts
Disadvantages of Moving Average Method
Requires saving all past N data points
Lags behind a trend
Ignores complex relationships in data

21. Forecasting for Stationary Series -- Moving Averages --Summary

22. Forecasting for Stationary Series -- Exponential Smoothing Method
A type of weighted moving average that applies declining weights to past data.
1. New Forecast = a (most recent observation)
+ (1 - a) (last forecast) or
2. New Forecast = last forecast a (last forecast error)
where 0 < a < 1 and generally is small for stability of forecasts ( around .1 to .2)

23. Forecasting for Stationary Series -- Exponential Smoothing Method
In symbols:
Ft+1 = a Dt + (1 - a ) Ft
= a Dt + (1 - a ) (a Dt-1 + (1 - a ) Ft-1)
= a Dt + (1 - a )(a )Dt-1 + (1 - a)2 (a )Dt
Hence the method applies a set of exponentially declining weights to past data. It is easy to show that the sum of the weights is exactly one.
(Or Ft = Ft a (Ft - Dt) )

24. Forecasting for Stationary Series -- Exponential Smoothing Method --Weights in Exponential Smoothing

25. Forecasting for Stationary Series -- Exponential Smoothing Method
Example
Quarterly data
What is F3 if a=0.1?

26. Forecasting for Stationary Series -- Exponential Smoothing Method

27. Forecasting for Stationary Series -- Comparison of ES and MA
Similarities
Both methods are appropriate for stationary series
Both methods depend on a single parameter
Both methods lag behind a trend
One can achieve the same distribution of forecast error by setting a = 2/ ( N + 1). .

28. Forecasting for Stationary Series -- Comparison of ES and MA
Differences
ES carries all past history. MA eliminates “bad” data after N periods
MA requires all N past data points while ES only requires last forecast and last observation.

29. Trend-Based Methods Regression Analysis
Regression Methods Can be Used When Trend is Present.
Model: Dt = a + bt.

30. Trend-Based Methods
If t is scaled to 1, 2, 3, , then the least squares estimates for a and b can be computed as follows:
Set Sxx = n2 (n+1)(2n+1)/6 - [n(n+1)/2] Set Sxy = n S iDi - [n(n + 1)/2] S Di
Let b = Sxy / Sxx and a = 𝐷 - b (n+1)/2
These values of a and b provide the “best” fit of the data in a least squares sense.

31. Trend-Based Methods -- Double exponential smoothing with Holt’s method
Double exponential smoothing, of which Holt’s method is only one example, can also be used to forecast when there is a linear trend present in the data. The method requires separate smoothing constants for slope and intercept.
where St=the intercept, Gt=slope, Dt=demand, a,b = constants
𝑆 𝑡 =𝛼 𝐷 𝑡 + 1−𝛼 𝑆 𝑡−1 + 𝐺 𝑡−1
𝐺 𝑡 = 𝛽 𝑆 𝑡 − 𝑆 𝑡−1 +(1−𝛽) 𝐺 𝑡−1

32. Trend-Based Methods -- Double exponential smoothing with Holt’s method
So the  step ahead forecast made in period t, denoted by Ft, t+,
𝐹 𝑡, 𝑡+𝜏 = 𝑆 𝑡 +𝜏 𝐺 𝑡

33. Trend-Based Methods -- Double exponential smoothing with Holt’s method
Example
Quarterly data
What is F3, 3+1 if a==0.1,
S0=200, and G0=10 ?

34. Methods For Seasonal Series --A Seasonal Demand Series

35. Methods For Seasonal Series – Seasonal Factors for Stationary Series
Seasonality corresponds to a pattern in the data that repeats at regular intervals.
Multiplicative seasonal factors: c1 , c2 , , cN where i = 1 is first period of season, i = 2 is second period of the season, etc..
S ci = N.
ci = implies 25% higher than the baseline on avg.
ci = implies 25% lower than the baseline on avg.

36. Methods For Seasonal Series – Seasonal Factors for Stationary Series
Procedure:
Compute the sample mean of all the data
Divide each observation by the sample mean  factors for each period of observed data
Average the factors for like periods within each season average all\factors corresponding to the same period of a season
Forecast your time series

37. Methods For Seasonal Series – Seasonal Factors for Stationary Series
Example:
Week 1
Week 2
Week 3
Week 4
Monday
16.2
17.3
14.6
16.1
Tuesday
12.2
11.5
13.1
11.8
Wednesday
14.2
15.0
13.0
12.9
Thursday
17.6
16.9
16.6
Friday
22.5
23.5
21.9
24.3

38. Methods For Seasonal Series – Seasonal Factors for Stationary Series
Solution:
Sample mean=16.425
Divide each observation by sample mean
Week 1
Week 2
Week 3
Week 4
Monday
0.986
1.053
0.889
0.980
Tuesday
0.743
0.700
0.798
0.718
Wednesday
0.865
0.913
0.791
0.785
Thursday
1.072
1.029
1.011
Friday
1.370
1.431
1.333
0.479

39. Methods For Seasonal Series – Seasonal Factors for Stationary Series
Solution:
Average all factors for Mondays thru Friday
Average Factors
Monday
0.98
Tuesday
0.74
Wednesday
0.84
Thursday
1.04
Friday
1.40

40. Methods For Seasonal Series – Seasonal Factors for Stationary Series
Solution:
Forecast the numbers by multiplying mean and each of average factor.
Average Factors
Monday
16.1
Tuesday
12.1
Wednesday
13.8
Thursday
17.1
Friday
23.0

41. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Steps of Seasonal decomposition Using Moving Average
Calculate the moving averages.
Center the moving average
Center the centered moving average on integer-valued periods.
Determine the seasonal and irregular factors.
Determine the average seasonal factors.

42. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Steps of Seasonal decomposition Using Moving Average
Scale the seasonal factors.
Determine the deseasonalized data.
Determine a trend line of the deseasonalized data.
Determine the deseasonalized predictions.
Take into account the seasonality.

43. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Example:
Demands for each periods
Period
Demand
MA(4) 1 10 2 20 3 26 4 17
18.25 5 12
18.75 6 23
19.50 7 30
20.50 8 22
21.75

44. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Example:
Demands for each periods

45. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Solution:
Moving average with k=3
Period
Demand
MA(4) 1 10 2 20 3 26 4 17
18.25 5 12
18.75 6 23
19.50 7 30
20.50 8 22
21.75

46. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Period
Demand
MA(4)
Centered MA 1 10 2 20
18.25 3 26
18.75 4 17
19.50 5 12
20.25 6 23
21.75 7 30
20.50 8 22
Solution:
Center the moving average

47. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Period
Demand
MA(4)
Centered MA
MA on Integer
periods 1 10 2 20
18.25 3 26
18.50
18.75 4 17
19.125
19.50 5 12
20.50 6 23
21.125
21.75 7 30 8 22
Center the centered moving average on integer-valued periods.

48. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Period
Demand (B)
MA(4)
Centered MA
MA on Integer
periods (C)
Seasonal and
irregular
factors (B/C) 1 10 2 20
18.25 3 26
18.50
1.405
18.75 4 17
19.125
0.888
19.50 5 12
0.600
20.50 6 23
21.125
1.089
21.75 7 30 8 22
Determine the seasonal and irregular factors by dividing (B) by (C)

49. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Period
Demand (B)
MA(4)
Centered MA
MA on Integer
periods (C)
Seasonal and
irregular
factors (B/C) 1 10 2 20
18.25 3 26
18.50
1.405
18.75 4 17
19.125
0.888
19.50 5 12
0.600
20.50 6 23
21.125
1.089
21.75 7 30 8 22
Determine the average seasonal factors. (not in this example)

50. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Period
Demand (B)
Centered
MA on Integer
periods (C)
Seasonal and
irregular
factors (B/C)
Scaled
seasonal factors
*(4/3.982) (D) 1 10 2 20 3 26
18.50
1.405
1.405*1.0045
=1.4113 4 17 ……
19.125
0.888
0.892 5 12
0.600
0.6027 6 23
21.125
1.089
1.094 7 30 8 22
Scale the seasonal factors.

51. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Determine the deseasonalized data.
Period
Demand (B)
Scaled
seasonal factors
*(4/3.982) (D)
Deseasonal Data (B/D) 1 10
10/0.6027=16.592 2 20
20/1.094=18.282 3 26
1.405*1.0045
=1.4113
26/1.4113=18.423 4 17 ……
0.892
17/0.892=19.058 5 12
19.91 6 23
1.094
21.024 7 30
21.257 8 22
24.664

52. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Determine a trend line of the deseasonalized data.
By regression,
Determine the deseasonalized predictions
𝐷 𝑡 = 𝑡
Period
Deseasonalized predictions 9
23.18 10
23.89 11
24.6 12
25.31

53. Methods For Seasonal Series – Seasonal decomposition Using Moving Average
Take into account the seasonality
Period
Deseasonalized
predictions
Seasonal Factors
Forecast 9
23.18
13.97 10
23.89
1.094
26.136 11
24.6
1.4113
34.72 12
25.31
0.892
22.577

54. Methods For Seasonal Series – Winter’s Method
Moving average with or without trend can be used to predict seasonal data, however, one needs to recalculate all factors if new data comes in.
Winter’s method is a triple exponential smoothing.
𝐷 𝑡 = 𝜇+ 𝐺 𝑡 𝑐 𝑡 + 𝜖 𝑡
Dt: demand at time t; 𝜇:intercept; Gt:Slope
ct: seasonal factor; 𝜖 𝑡 : error term

55. Methods For Seasonal Series – Winter’s Method

56. Methods For Seasonal Series – Winter’s Method
Assuming:
Length of the season, N
Seasonal factors are the same each season and 𝑐 𝑡 =𝑁
Three exponential smoothing equation are used:
The series
The trend
The seasonal factors

57. Methods For Seasonal Series – Winter’s Method
The series
the current level of the deseasonalized series, St
𝑆 𝑡 =𝛼 𝐷 𝑡 𝑐 𝑡−𝑁 +(1−𝛼)( 𝑆 𝑡−1 + 𝐺 𝑡−1 )
The trend
𝐺 𝑡 =𝛽[ 𝑆 𝑡 − 𝑆 𝑡−1 ]+(1−𝛽) 𝐺 𝑡−1
The seasonal factor
𝑐 𝑡 =𝛾( 𝐷 𝑡 𝑆 𝑡 )+(1−𝛾) 𝑐 𝑡−𝑁

58. Methods For Seasonal Series – Winter’s Method
The forecast made in period t for any future period t+ is
𝐹 𝑡, 𝑡+𝜏 = 𝑆 𝑡 + 𝜏𝐺 𝑡 𝑐 𝑡+𝜏−𝑁 if t  N
Note, appropriate seasonal factor would be:
𝑐 𝑡+𝜏−2𝑁 if N<𝜏  2N
𝑐 𝑡+𝜏−3𝑁 if 2N<𝜏  3N

59. Methods For Seasonal Series – Winter’s Method –Initialization procedure
In order to get the method started, one needs to obtain initial estimates of the series
Winters suggest minimum 2 seasons of data available
Assuming exactly two season are considered and the time now, t=0, so the past demands are labeled:
D-2N+1 , D-2N+2 , ….., D0

60. Methods For Seasonal Series – Winter’s Method –Initialization procedure
Calculate the sample means for the two separate seasons of data
𝑉 1 = 1 𝑁 𝑗=−2𝑁+1 −𝑁 𝐷 𝑗
𝑉 2 = 1 𝑁 𝑗=−𝑁+1 0 𝐷 𝑗

61. Methods For Seasonal Series – Winter’s Method –Initialization procedure
Define G0=(V2-V1)/N as the initial slope estimate
If m>2 (m:# of seasons), compute V1,,,,Vm as step 1 shows and define G0=(Vm-V1)/[(m-1)N]
Set S0=V2 + G0[(N-1)/2]

62. Methods For Seasonal Series – Winter’s Method –Initialization procedure
a. compute initial seasonal factor
𝑐 𝑡 = 𝐷 𝑡 𝑉 𝑖 −[ 𝑁+1 2 −𝑗] 𝐺 0 𝑓𝑜𝑟 −2𝑁+1≤𝑡≤0
where i=1 for the first season, i=2 for the second season
j: the period of the season,
j=1 for t=-2N+1and t=-N+1,
j=2 for t=-2N+2 and t=-N+2
b. average the seasonal factors by
𝑐 −𝑁+1 = 𝑐 −2𝑁+1 + 𝑐 −𝑁 , …, 𝑐 0 = 𝒄 −𝑵 + 𝒄 𝟎 𝟐 for exactly two seasons

63. Methods For Seasonal Series – Winter’s Method –Initialization procedure
c. Normalize the seasonal factor
𝑐 𝑗 = 𝑐 𝑗 𝑖=0 −𝑁+1 𝑐 𝑖
*Note: The above initialization method is only one of methods. Moving average, linear regression can be also used.

64. Methods For Seasonal Series – Winter’s Method --example
Initial data set:
𝑉 1 = 1 𝑁 𝑗=−2𝑁+1 −𝑁 𝐷 𝑗 =18.25
𝑉 2 = 1 𝑁 𝑗=−𝑁+1 0 𝐷 𝑗 =21.75
G0=(V2-V1)/N=0.875
S0=V2 + G0[(N-1)/2]=23.06
Initialization
𝑐 𝑡 = 𝐷 𝑡 𝑉 𝑖 −[ 𝑁+1 2 −𝑗] 𝐺 0 𝑓𝑜𝑟 −7≤𝑡≤0

65. Methods For Seasonal Series – Winter’s Method --example
Initial data set:
Initialization a.
For 𝑐 −7 , 𝑐 −6 , 𝑐 −5 , 𝑐 −4 , i=1; For 𝑐 −3 , 𝑐 −2 , 𝑐 −1 , 𝑐 0 , i=2
For 𝑐 −7 , 𝑐 −5 , 𝑐 −3 , 𝑐 −1 , j=1;
𝑐 −7 = 𝐷 −7 𝑉 1 −[ 5 2 −1](0.875) = −[1.5](0.875) =0.5904
For 𝑐 −6 , 𝑐 −4 , 𝑐 −2 , 𝑐 0 , j=2;
𝑐 −6 = 𝐷 −6 𝑉 1 −[ 5 2 −2](0.875) = −[1.5](0.875) =0.5904
And so on

66. Methods For Seasonal Series – Winter’s Method --example
Initial data set:
Initialization
b. Average seasonal factors:
Average 𝑐 −7 𝑎𝑛𝑑 𝑐 −3  𝑐 −3 =0.588
Average 𝑐 −6 𝑎𝑛𝑑 𝑐 −2  𝑐 −2 =1.010 …
c. Normalize seasonal factors
𝑐 −3 =0.5900, 𝑐 −2 =1.1100, …

67. Methods For Seasonal Series – Winter’s Method --example
Forecasting equations:
𝐹 𝑡, 𝑡+𝜏 = 𝑆 𝑡 + 𝜏𝐺 𝑡 𝑐 𝑡+𝜏−𝑁 if t  N
𝐹 0, 1 = 𝑆 0 + 𝐺 0 𝑐 −3 if t =0 and𝜏=1
𝐹 0, 2 = 𝑆 0 + 𝐺 0 𝑐 −2 if t =0 and 𝜏=2 …
For t=0, 𝐹 0, 1 =14.12, 𝐹 0, 2 =27.54, 𝐹 0, 3 =35.44, 𝐹 0, 3 =24.38

68. Methods For Seasonal Series – Winter’s Method --example
For t=1, D1 is available, and assume =0.2, =0.1, =0.1
𝑆 𝑡 =𝛼 𝐷 𝑡 𝑐 𝑡−𝑁 +(1−𝛼)( 𝑆 𝑡−1 + 𝐺 𝑡−1 )
𝑆 1 =0.2 𝐷 1 𝑐 1−𝑁 + 1−0.2 𝑆 0 + 𝐺 0
𝐺 𝑡 =𝛽[ 𝑆 𝑡 − 𝑆 𝑡−1 ]+(1−𝛽) 𝐺 𝑡−1
𝐺 1 =0.1 𝑆 1 − 𝑆 −0.1 𝐺 0
𝑐 𝑡 =𝛾( 𝐷 𝑡 𝑆 𝑡 )+(1−𝛾) 𝑐 𝑡−𝑁
𝑐 1 =0.1( 𝐷 1 𝑆 1 )+(1−0.1) 𝑐 1−𝑁
So 𝑆 1 =24.57, 𝐺 1 =0.9385, 𝑐 1 =0.5961

69. Methods For Seasonal Series – Winter’s Method --example
For t=1, D1 is available, and assume =0.2, =0.1, =0.1
Renorm: 𝑐 −2 , 𝑐 −1 , 𝑐 0 , 𝑎𝑛𝑑 𝑐 1 ,
Forecasting for t=1
𝐹 1, 2 = 𝑆 1 + 𝐺 1 𝑐 −2 if t =1 and𝜏=1
𝐹 1, 3 = 𝑆 1 + 𝐺 1 𝑐 −1 if t =1 and 𝜏=2 …
And so on repeat for the rest of data..

70. Practical Considerations
Overly sophisticated forecasting methods can be problematic, especially for long term forecasting. (Refer to Figure on the next slide.)
Tracking signals may be useful for indicating forecast bias.
TS= 𝑖=1 𝑛 𝑒 𝑖 𝑀𝐴𝐷
Box-Jenkins methods require substantial data history, use the correlation structure of the data, and can provide significantly improved forecasts under some circumstances.

71. The Difficulty with Long-Term Forecasts