outline multiplicative seasonal influences additive seasonal influences

Outline

• Multiplicative Seasonal Influences• Additive Seasonal Influences• Seasonal influence with trend

LESSON 7: FORECASTING METHODS FOR SEASONAL SERIES

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.

Quarter Year 1 Year 2 Year 3 Year 4

1 45 70 100 1002 335 370 585 7253 520 590 830 11604 100 170 285 215

Total 1000 1200 1800 2200 Average 250 300 450 550

Seasonal Influences

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?

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.

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

= projected average quarterly demand

+ average seasonal index of that quarter.


1 45/250 = 0.18 70/300 = 0.23 100/450 = 0.22 100/550 = 0.182 335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.323 520/250 = 2.08 590/300 = 1.97 830/450 = 1.84 1160/550 = 2.114 100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39

Multiplicative Seasonal Influences

For example, seasonal Index, Year 1, Quarter 1=

Seasonal index = Actual Quarterly Demand

Average Quarterly Demand

Step 1: For each period, compute

Multiplicative influence:


1 45/250 = 0.18 70/300 = 0.23 100/450 = 0.22 100/550 = 0.182 335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.323 520/250 = 2.08 590/300 = 1.97 830/450 = 1.84 1160/550 = 2.114 100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39

Quarter Average Seasonal Index

1234


Step 2: For each quarter compute the average seasonal index


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 2600. 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.


Given projected average quarterly demand =650

The quarterly demand forecasts are obtained as follows:

Quarter Average Seasonal Index Forecast

1

2

3

4

Additive Seasonal Influences

For example, seasonal Index, Year 1, Quarter 1= 45-250 = -205

= Actual Quarterly Demand - Average Quarterly Demand

Step 1: For each period, compute seasonal index

Additive influence:


1 45-250 = -205 70-300 = -230 100-450 = -350 100-550 = -4502 335-250 = 85 370-300 = 70 585-450 = 135 725-550 = 1753 520-250 = 270 590-300 = 290 830-450 = 380 1160-550 = 6104 100-250 = -150 170-300 = -130 285-450 = -165 215-550 = -335

Quarter Average Seasonal Index

1 (-205-230-350-450)/4 = -308.752 (85+70+135+175)/4 = 116.253 (270+290+380+610)/4 = 387.504 (-150-130-165-335)/4 = -195.00


Step 2: For each quarter compute the average seasonal index


1 45-250 = -205 70-300 = -230 100-450 = -350 100-550 = -4502 335-250 = 85 370-300 = 70 585-450 = 135 725-550 = 1753 520-250 = 270 590-300 = 290 830-450 = 380 1160-550 = 6104 100-250 = -150 170-300 = -130 285-450 = -165 215-550 = -335


Given projected average quarterly demand =650

The quarterly demand forecasts are obtained as follows:

Quarter Average Seasonal Index Forecast

1 -308.75 650-308.75=341.25

2 116.25 650+116.25=766.25

3 387.50650+387.50=1037.50

4 -195.00 650-195.00=455.00

Difference between the highest and lowest demand increases

Multiplicative Influence

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Additive Influence

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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

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.


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.


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.


yxxxyyx and ,,,, 2

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.


yxxxyyx and ,,,, 2

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.


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.


Step 1 (Problem) Centered Moving Average

Centered (B/D)Period Demand MA(4) MA Ratio

A B C D E1 2052 1403 3754 5755 4756 2757 6858 965

Sample computation:

MA(4), Period 4: (205+140+375+575)/4=323.75

MA(4), Period 5: (140+375+575+475)/4=391.25

Observe that MA(4), period 4 is obtained from periods 1,2,3,4. So, MA(4), period 4 represents period (1+2+3+4)/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:(323.75+391.25)/2=357.5

Similalry, Centered MA is computed for periods 4, 5, and 6.

Step 1 (Sample Computation)Centered Moving Average

Sample computation:

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) = (357.5 +408.125)/2 = 382.813

For periods 7 and 8 centered MA = average of centered MA of periods 5 and 6 (again, a simplified approach) = (463.75 +551.25)/2 = 507.5

B/D ratio, period 1 = 205/382.813 = 0.54

Step 1 (Sample Computation)Centered Moving Average

FinalSeasonal Seasonal

Period Factors Factors1234

Total

Step 2 Seasonal Factors By CMA

Step 3 Deseasonalize

Seasonal Deseasonalized ReseasonalizedPeriod Demand Factors Demand Forecast

A B C D=B/C E1 205 0.76712 140 0.42523 375 1.17974 575 1.62805 475 0.76716 275 0.42527 685 1.17978 965 1.62809

10

Step 4a Forecast using Deseasonalized Data

DeseasonalizedDemand

x y xy x^21 267.24308752 329.2573623 317.88304754 353.18766245 619.22178816 646.75553247 580.66636688 592.7410333

SumAverage

Step 4b Forecast using Deseasonalized Data

)slope(Intercept

Slope 2

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A B C D E1 205 0.7671 267.24308752 140 0.4252 329.2573623 375 1.1797 317.88304754 575 1.6280 353.18766245 475 0.7671 619.22178816 275 0.4252 646.75553247 685 1.1797 580.66636688 965 1.6280 592.74103339 0.7671

10 0.4252

Step 4c Forecast using Deseasonalized Data


A B C D E1 205 0.7671 267.24308752 140 0.4252 329.2573623 375 1.1797 317.88304754 575 1.6280 353.18766245 475 0.7671 619.22178816 275 0.4252 646.75553247 685 1.1797 580.66636688 965 1.6280 592.74103339 0.7671 719.8792471

10 0.4252 776.8814164

Step 5Reseasonalize

Original Data

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Steps 4a, 4b: Regression

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Step 4c: Regression - Projection Forecast Deseasonalized Demand

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Step 5: Reseasonalize

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READING AND EXERCISES

Lesson 7

Reading:

Section 2.9, pp. 81-87 (4th Ed.), pp. 78-83 (4th Ed.)

Exercises:

33, 34, p. 87 (4th Ed.), pp. 82-83 (5th Ed.)

outline multiplicative seasonal influences additive seasonal influences

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