1. Forecasting Y.-H. Chen, Ph.D. Production / Operations Management
International College
Ming-Chuan University

2. Forecasting Outline Introduction Forecasting Process
Forecasting Methods
Forecast Accuracy and Control
Forecast Method Selection and Usage

3. Introduction A forecast is a statement of future,
I see that you will get an A this semester.
A forecast
is a statement of future,
is a basis for planning,
is not for forecasting demand only,
requires a skillful blending of art and science,
assumes that the underlying system will continue to exist in the future, and
is rarely perfect.

4. Forecasting Process “The forecast” Step 6 Monitor the forecast
Step 1 Determine purpose of forecast
Step 2 Establish a time horizon
Step 3 Select a forecasting technique
Step 4 Gather and analyze data
Step 5 Prepare the forecast
Step 6 Monitor the forecast
“The forecast”

5. Elements of A Good Forecast
The forecast horizon must cover the time necessary to implement possible changes.
The degree of accuracy should be stated.
The forecast should be reliable; it should work consistently.
The forecast should be expressed in meaningful units.
The forecast should be in writing.
The forecast should be simply to understand and use, or consistent with historical data intuitively.

6. Additional Properties
Forecasts for groups of items tend to be more accurate than forecasts for individual items, because forecasting errors among items in a group usually have a canceling effect.
Forecast accuracy decreases as the time period covered by the forecast increases.
Timely
Accurate
Reliable
Meaningful
Written
Easy to use

7. Forecasting Methods Basic Methods Trend Seasonality Cycle Association
Judgmental Forecast
Statistical (Time Series) Forecast
Trend
Seasonality
Cycle
Association

8. Basic Forecasting Methods
Judgmental Forecast
Statistical (Time Series) Forecast
Averaging
Weighted Moving Average
Exponential Smoothing

9. Judgmental Forecast Executive opinions.
Mostly for long-range planning and introduction of new products. The view of one person may prevail.
Unable to distinguish between what customers would like to do and what they will actually do.
Could overly influenced by recent sales experiences. Low sales could lead to low estimates.
Conflict of interest. Low sales estimates lead to better sales performance.
Direct customer contact composites.
Consumer survey or point-of-sales (POS) data.
Expensive and time-consuming.
Possible existence of irrational patterns.
Low response rates.
Opinions of managers and staff.
Delphi method (Rand Corp., 1948): Managers and staff complete a series of questionnaires, each developed from the previous one, to achieve a consensus forecast.
Technological forecasting. Long-term single-time forecasting. Data are costly to obtain.

10. Statistical (Time Series) Forecast
It is extremely important to plot data and examine them before doing any analysis or forecast. A demand forecast should be based on a time series of past demand rather than sales or shipment.
Data patterns:
Trend
A long term upward or downward movement in data.
Seasonality
Short-term regular variations related to weather, holiday, or other factors.
Cycle
Wavelike variation lasting more than one year.
Irregular Variation
Caused by unusual circumstances, not reflective of typical behavior.
Random variation
Residual variation after all other behaviors are accounted for.

11. Data Patterns Trend Cycles Irregular variation 90 89 88
Seasonal variations

12. Simple Naive Forecast No cost. Quick and easy to prepare.
Easy to understand.
Can be applied to data with seasonality and trend

13. General Naive Forecast

14. Weighted Moving Average

15. Moving Average Example
Compute a 3-period moving average forecast given demand for shopping carts for the last five periods.
Period
Demand 1 42 2 40 3 43 4 5 41 6 ? 7
If the actual demand in period 6 turns out to be 39, what would be the moving average forecast for period 7?
41.33 39
40.00

16. Weighted Moving Average Example
Compute a weighted average forecast using a weight of .40 for the most recent period, .30 for the next most recent, .20 for the next, and .10 for the next.
Period
Demand 1 42 2 40 3 43 4 5 41 6 ? 7
If the actual demand in period 6 turns out to be 39, what would be the weighted moving average forecast for period 7?
41.0 39
40.2

17. Properties of Weighted Moving Average
Easy to compute and understand.
Moving average forecast lags and smoothens the actual forecast.
The number of data points in the average determines its sensitivity to each new data point: the fewer the data points in an average, the more responsive the average tends to be.
Weights can be added to values in the average to make the resulting average more responsive to some recent data points. However, weights involve the use of trial-and-error to find suitable weights.

18. Exponential Smoothing
Exponential smoothing is a weighted averaging method based on previous forecast plus a percentage of its forecast error.
Premise--The most recent observations might have the highest predictive value.
Therefore, we should give more weight to the more recent time periods when forecasting.

19. Properties of Exponential Smoothing
Commonly used values of alpha range from 0.05 to Low values are used when the underlying average tends to be stable; higher values are used when the underlying average is susceptible to change.
Moving average or naive forecast can be used to generate starting forecast for exponential smoothing.

20. Picking A Smooth Constant

21. Exponential Smoothing Example
Use exponential smoothing to develop a series of forecasts for the data, and compute (actual-forecast)=error for each period.
Use a smoothing factor of .10.
Use a smoothing factor of .40.
Plot the actual data and both sets of forecasts on a single graph.
Period (t)
Actual 1 42 2 40 3 43 4 5 41 6 39 7 46 8 44 9 45 10 38 11 12 ?

22. Exponential Smoothing Example
alpha=.10
alpha=.40
Period (t)
Actual
Forecast
Error 1
42.00 2
40.00
-2.00 3
43.00
41.80
1.20
41.20
1.80 4
41.92
-1.92 5
41.00
41.73
-0.73
41.15
-0.15 6
39.00
41.66
-2.66
41.09
-2.09 7
46.00
41.39
4.61
40.25
5.75 8
44.00
41.85
2.15
42.55
1.45 9
45.00
42.07
2.93
43.13
1.87 10
38.00
42.36
-4.36
43.88
-5.88 11
41.53
-1.53 12
40.92

23. Forecasting Method Extension
Trend
Linear Trend
Trend-Adjusted Exponential Smoothing
Seasonality
Cycle
Association

24. Linear Trend
A simple plot of data can often reveal the existence and nature of a trend.
Linear trends are fairly common and the easiest to work with.

25. Linear Trend Coefficients

26. Linear Trend Example
Calculate sales for a California-based firm over the last 10 weeks are shown in the table. Plot the data and visually check to see if a linear trend line would be appropriate. Then, determine the equation of the trend line, and predict sales for weeks 11 and 12.
Week
Unit Sales 1
700 2
724 3
720 4
728 5
740 6
742 7
758 8
750 9
770 10
775 11 ? 12

27. Linear Trend Example Solution
a. A plot suggests that a linear trend line would be appropriate.
b. For n=10, we have
Thus, the trend line is yt= t, where t=0 for period 0.
c. By letting t=11 and t=12, we have

28. Seasonality
Seasonality is regular movement in series values that can be tied to recurring events. It is also applied to daily, weekly, monthly, and other regular recurring patterns in data.
If a data series tends to vary around an average value, the seasonality is expressed in terms of that average; if trend is present, seasonality is expressed in terms of the trend value.
There are two models of seasonality: additive and multiplicative. In practice, businesses use the multiplicative model much more frequently than the additive model.

29. Cycle
Cycles are similar to seasonal variations but of longer duration, e.g., two to six years between peaks.
It is difficult to project cycles from past data, because turning points are difficult to identify.
A short moving average or a naive approach may be of some value.

30. Associative Forecasts
High correlation of a forecast with leading variables can be useful in computing the forecast.
The simple linear regression is the simplest and most widely used method.

31. Simple Linear Regression Coefficients

32. Simple Linear Regression Example
Sales, X
Profits, Y
(in millions of dollars) 7
0.15 2
0.10 6
0.13 4 14
0.25 15
0.27 16
0.24 12
0.20 20
0.44
0.34
0.17
Healthy Hamburgers has a chain of 12 stores in northern Illinois. Sales figures and profits for the stores are given in the following table. Obtain a regression line for the data and predict profit for a store assuming sales of $10 million.

33. Simple Linear Regression Example: Data Plot

34. Simple Linear Regression Example: Solutiony xy
x^2
y^2 7
0.15
1.05 49
0.0225 2
0.10
0.20 4
0.0100 6
0.13
0.78 36
0.0169
0.60 16 14
0.25
3.50
196
0.0625 15
0.27
4.05
225
0.0729
0.24
3.84
256
0.0576 12
2.40
144
0.0400
3.78 20
0.44
8.80
400
0.1936
0.34
5.10
0.1156
0.17
1.19
0.0289
132
2.71
35.29
1796
0.7159

35. An Important Measure of Simple Linear Regression
Correlation measures the strength and direction of the relationship between two variables.
+1, positive correlation.
-1, negative correlation.
0, zero correlation.
The square of the correlation coefficient provides a measure of how well a regression line “fits” the data. The values ranges from 0 to 1.00.
[0.80,1.00], good fit.
[0.25,0.80), moderate fit.
[0.00,0.25), poor fit.

36. Simple Linear Regression and Correlation Example
Sales of 19-inch color television sets and 3-month lagged unemployment are shown in the table below. Determine if unemployment levels can be used to predict demand for 19-inch color TVs and, if so, derive a predictive equation.
Period 1 2 3 4 5 6 7 8 9 10 11
Units sold 20 41 17 35 25 31 38 50 15 19 14
Unemployment %
7.2
4.0
7.3
5.5
6.8
6.0
5.4
3.6
8.4
7.0
9.0
(3-month lag)

37. Simple Linear Regression and Correlation Example: Data Plot

38. Simple Linear Regression and Correlation Example: Solutiony xy
x^2
y^2
7.2 20
144.0
51.8
400
4.0 41
164.0
16.0
1681
7.3 17
124.1
53.3
289
5.5 35
192.5
30.3
1225
6.8 25
170.0
46.2
625
6.0 31
186.0
36.0
961
5.4 38
205.2
29.2
1444
3.6 50
180.0
13.0
2500
8.4 15
126.0
70.6
225
7.0 19
133.0
49.0
361
9.0 14
81.0
196
70.2
305
1750.8
476.3
9907
Fairly high negative correlation.
Pertain to unemployment levels in the range 3.6 to 9.0.

39. Linear Regression Assumptions
No patterns such as cycles or trends should be apparent.
Deviations around the line should be normally distributed.
Predictions are best being made within the range of observed values.

40. Linear Regression Usage Guidelines
Always plot the data to verify that a linear relationship is appropriate.
The data may be time-dependent. If patterns appear, use analysis of time series or use time as an independent variable as part of a multiple regression analysis.
A small correlation may imply that other variables are important.

41. Linear Regression Summary
Simple linear regression applies only to linear relationship with one independent variable.
One needs a considerable amount of data to establish the relationship --- in practice, 20 or more observations.
All observations are weighted equally.

42. Forecast Accuracy
Error - difference between actual value and predicted value
Mean absolute deviation (MAD)
Average absolute error
Mean squared error (MSE)
Average of squared error

43. Forecast Accuracy: MAD & MSE=
Actual
forecast   n
MSE =
Actual
forecast) - 1 2   n (
The difference between these two measures is that MAD weights all errors evenly, and MSE weights errors according to their squared values.
For the usage of these measures, either MAD or MSE, a manager could compare the results of exponential smoothing with values of .1, .2, and .3, and select the one that yields the least MAD or MSE for a given set of data.
Example 10 (page 97).

44. Forecast Control
It is necessary to monitor forecast errors to ensure that the forecast is performing adequately over time. This is generally accomplished by comparing forecast errors to predefined values, or action limits.

45. Why Do We Need Forecast Control?
The omission of an important variable.
Appearance of a new variable.
A sudden or unexpected change in the variable (causing by severe weather or other nature phenomena, temporary shortage or breakdown, catastrophe, or similar events).
Being used incorrectly.
Data being misinterpreted.
Random variation.

46. Forecasting Control Methods
Tracking Signal
Control Chart

47. Forecast Control: Tracking Signal
A tracking signal focuses on the ratio of cumulative forecast error to the corresponding MAD.
The tracking signal often ranges from +/- 3 to +/- 8. For the most part, we shall use limits of +/- 4, which are roughly comparable to three standard deviation limits (99.7% CI). Values within the limits suggest --- but do not guarantee --- that the forecast is performing adequately.
MAD can be updated using the following exponential smoothing equation.

48. Forecast Control: Control Chart
The control chart sets the limits as multiples of the squared root of MSE.
Basic assumptions are
Forecast errors are randomly distributed around a mean of zero, and
The distribution of errors is normal.

49. Forecast Control: Control Chart
For a normal distribution, 95% of the errors fall within +/-2s, and approximately 99.7% of the errors fall within +/-3s. Errors fall outside these limits should be regarded as evidence that corrective action is needed.
The square root of MSE is used in practice as an estimate of the standard deviation of the distribution of errors.
For a normal distribution, 95% of the errors fall within +/-2s, and approximately 99.7% of the errors fall within +/-3s. Errors fall outside these limits should be regarded as evidence that corrective action is needed.
Plotting the errors with the help of a control chart can be very informative. A plot helps you to visualize the process and enables you to check for possible patterns, nonrandom errors, within the limit that suggest an improved forecast is possible.
The control chart approach is generally superior to the tracking signal approach. The major weakness of the tracking signal approach is its use of cumulative errors: individual errors can be obscured so that large positive and negative errors cancel each other.
Example 11 (Page 93).

50. Forecast Method Selection
Most important
Cost
Accuracy
Need to consider
Historical performance
Ability to respond to change
Two most important factors are cost and accuracy. Generally speaking, the higher the accuracy, the higher the cost, so it is important to weight cost-accuracy trade-offs carefully.
When deciding among forecasting alternatives, the operations manager need to consider
·        the historical performance of a forecast, and
·        the ability of a forecast to respond to changes.
Forecasts that cover short time frames tend to be more accurate than long-term forecasts. Management might choose to devote efforts to shortening the time horizon of forecast. This would require flexibility in operations to respond to such a forecast.