1. LESSON 7: FORECASTING METHODS FOR SEASONAL SERIES
Outline
Multiplicative Seasonal Influences
Additive Seasonal Influences
Seasonal influence with trend

2. Turkeys have a long-term trend for increasing demand with a seasonal pattern. Sales are highest during September to November and sales are lowest during December and January.

3. Seasonal Influences
Quarter Year 1 Year 2 Year 3 Year 4
Total
Average
Consider the demand data shown above. Is the data seasonal? Are the demands in Quarter 1 consistently less than the average? What is the relationship of Quarter 1 demand with the average? How about the other quarters? 60

4. Seasonal Influences
Is it not true that Quarter 1 demand is less than the average? How much less do you expect the Quarter 1 demand from the average?
One can take different approaches to answer this question.
Answer 1: Quarter 1 demand is approximately 20% of the average.
Answer 2: Quarter 1 demand is approximately 200 units less than the average.
Are these answers true? We shall verify the correctness of the answers in this lesson. The first answer uses the concept of multiplicative seasonal influence and the second answer additive seasonal influence. We shall now define these two seasonal influences.

5. Seasonal Influences A seasonal influence is multiplicative if
the quarterly demand forecast of a quarter
= projected average quarterly demand
 average seasonal index of that quarter.
A seasonal influence is additive if
The quarterly demand forecast of a quarter
+ average seasonal index of that quarter.

6. Multiplicative Seasonal Influences
Quarter Year Year Year Year 4
1 45/250 = /300 = /450 = /550 = 0.18
2 335/250 = /300 = /450 = /550 = 1.32
3 520/250 = /300 = /450 = /550 = 2.11
4 100/250 = /300 = /450 = /550 = 0.39
Multiplicative influence:
Step 1: For each period, compute
Actual Quarterly Demand
Average Quarterly Demand
Seasonal index =
For example, seasonal Index, Year 1, Quarter 1 = 64

7. Multiplicative Seasonal Influences
Quarter Year Year Year Year 4
1 45/250 = /300 = /450 = /550 = 0.18
2 335/250 = /300 = /450 = /550 = 1.32
3 520/250 = /300 = /450 = /550 = 2.11
4 100/250 = /300 = /450 = /550 = 0.39
Quarter Average Seasonal Index 1 2 3 4
Step 2: For each quarter compute the average seasonal index

8. Multiplicative Seasonal Influences
The average seasonal indices can be used to get forecasts of the quarterly demands if the average quarterly demand is projected. The quarterly demand forecast of a quarter = projected average quarterly demand  average seasonal index of that quarter.
For example, suppose that the next year, in Year 5,
The projected annual demand is So, the projected average quarterly demand is 2600/4=650.
Then, the demand forecast in Quarter 1 = 650(0.20)=130. The next slide shows the demand forecast for the other quarters. 68

9. Multiplicative Seasonal Influences
Given projected average quarterly demand =650
The quarterly demand forecasts are obtained as follows:
Quarter Average Seasonal Index Forecast 1 2 3 4

10. Additive Seasonal Influences
Quarter Year Year Year Year 4
= = = = -450
= = = = 175
= = = = 610
= = = = -335
Additive influence:
Step 1: For each period, compute seasonal index
= Actual Quarterly Demand - Average Quarterly Demand
For example, seasonal Index, Year 1, Quarter 1
= = -205

11. Additive Seasonal Influences Quarter Average Seasonal Index
Quarter Year Year Year Year 4
= = = = -450
= = = = 175
= = = = 610
= = = = -335
Quarter Average Seasonal Index
1 ( )/4 =
2 ( )/4 =
3 ( )/4 =
4 ( )/4 =
Step 2: For each quarter compute the average seasonal index

12. Additive Seasonal Influences
Given projected average quarterly demand =650
The quarterly demand forecasts are obtained as follows:
Quarter Average Seasonal Index Forecast
=341.25
=766.25 =
=455.00

13. Difference between the highest and lowest demand increases71

14. Difference between the highest and lowest demand is constant72

15. Seasonal Influences with Trend
Step 1: Compute moving averages
Compute N-period moving averages
The value of N is set to the length of season. For example, if the data are quarterly demand, the length of the season is 4 (there are 4 quarters in a year), so N should be 4. If the data are monthly demand, N should be set to 12, etc.
Center the moving averages
Put the centered values back on periods
Compute the values for the periods in the beginning and end

16. Seasonal Influences with Trend
Step 2: Determine seasonal factors
For each period, compute a factor by dividing demands by moving average values
Average the factors that correspond to the same periods of each season
N seasonal factors will result
If the seasonal factors do not sum up to N, scale up or down each factor so that the factors sum up to N.
Note that the seasonal factors usually do not sum up to N. So, the seasonal factors should always be scaled up or down so that the factors sum up to N.

17. Seasonal Influences with Trend
Step 3: Deseasonalize the original data
Divide the original data by the seasonal factors.
This step removes the seasonality from the data.
Since we assume that the data contains seasonality and trend, there will only be trend after we remove the seasonality. Plus, recall that we already know how to deal with trend! For example, we can use regression to understand the trend.
After we remove the seasonality, the deseasonalized series is expected to be smooth. Then, we can use regression or other trend-based methods in Step 4.

18. Seasonal Influences with Trend
Step 4: Make a forecast using deseasonalized data
Use any trend-based method
Double exponential smoothing (not used in this lesson)
Linear regression (used in this lesson)
a. Compute
The series x denotes periods 1, 2, 3, … and y denotes the deseasonalized series. Note carefully that the series y does not denote the actual demand values but the deseasonalized demand that we get at the end of Step 3.

19. Seasonal Influences with Trend
Step 4: Make a forecast using deseasonalized data
a. Compute
Recall that the deseasonalized series has only the trend and no seasonality. In the entire Step 4, we consider the deseasonalized series and get a mathematical understanding of the trend. This mathematical understanding refers to slope and intercept that we find in Step 4b.
b. Compute slope and intercept
The slope and intercept approximately define the straight line that best fits the deseasonalized series.

20. Seasonal Influences with Trend
Step 4: Make a forecast using deseasonalized data
c. Forecast deseasonalized series
The deseasonalized series is projected to the required future period using slope and intercept.
For example, if we have data of 8 periods, we can project the deseasonalized series in the 9th, 10th or any other period in future using slope and intercept.
Note carefully that our job does not end with this step. Because, we have to put the effect of seasonality back. This is done in Step 5.

21. Seasonal Influences with Trend
Step 5: Reseasonalize forecast using seasonal factors
Multiply the forecasted values by the seasonal factors
This step is necessary to reverse the effect of deseasonalization that was done in Step 3.
Recall that the actual data is seasonal and we make a projection using the deseasonalized data in Step 4.
The forecast obtained in Step 4 does not contain seasonality.
Step 5 puts the effect of seasonality back into the forecast.

22. Step 1 (Problem) Centered Moving Average

23. Step 1 (Sample Computation) Centered Moving Average
MA(4), Period 4: ( )/4=323.75
MA(4), Period 5: ( )/4=391.25
Observe that MA(4), period 4 is obtained from periods 1,2,3,4. So, MA(4), period 4 represents period ( )/4 or period 2.5. Similarly, MA(4), period 5 represents period 3.5. Centered MA, period 3 is the average of these two values
Centered MA, Period 3:( )/2=357.5
Similalry, Centered MA is computed for periods 4, 5, and 6.

24. Step 1 (Sample Computation) Centered Moving Average
Centered MA cannot be obtained similarly for the first two and last two periods.
For periods 1 and 2 centered MA = average of centered MA of periods 3 and 4 (this is a simplified approach) = ( )/2 =
For periods 7 and 8 centered MA = average of centered MA of periods 5 and 6 (again, a simplified approach) = ( )/2 = 507.5
B/D ratio, period 1 = 205/ = 0.54

25. Step 2 Seasonal Factors By CMA

26. Step 3 Deseasonalize Seasonal Deseasonalized Reseasonalized Period
Demand
Factors
Demand
Forecast A B C
D=B/C E 1
205
0.7671 2
140
0.4252 3
375
1.1797 4
575
1.6280 5
475
0.7671 6
275
0.4252 7
685
1.1797 8
965
1.6280 9 10

27. Step 4a Forecast using Deseasonalized Data

28. Step 4b Forecast using Deseasonalized Data

29. Step 4c Forecast using Deseasonalized Data
Reseasonalized
Period
Demand
Factors
Demand
Forecast A B C D E 1
205
0.7671 2
140
0.4252 3
375
1.1797 4
575
1.6280 5
475
0.7671 6
275
0.4252 7
685
1.1797 8
965
1.6280 9
0.7671 10
0.4252

30. Step 5 Reseasonalize Seasonal Deseasonalized Reseasonalized Period
Demand
Factors
Demand
Forecast A B C D E 1
205
0.7671 2
140
0.4252 3
375
1.1797 4
575
1.6280 5
475
0.7671 6
275
0.4252 7
685
1.1797 8
965
1.6280 9
0.7671 10
0.4252

33. Steps 4a, 4b: Regression 1000 800 600 Demand 400 200 5 10 Period
Original
800
data
600
Demand
Desea-
sonalized
400
200
Reg-
ression 5 10
Period

34. Step 4c: Regression - Projection Forecast Deseasonalized Demand
1000
Original
800
data
600
Demand
Desea-
sonalized
400
200
Reg-
ression 5 10
Period

36. READING AND EXERCISES Lesson 7 Reading:
Section 2.9, pp (4th Ed.), pp (4th Ed.)
Exercises:
33, 34, p. 87 (4th Ed.), pp (5th Ed.)