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Outline Multiplicative Seasonal Influences Additive Seasonal Influences

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

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Outline Multiplicative Seasonal Influences Additive Seasonal Influences

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  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. Quarter Year 1 Year 2 Year 3 Year 4 1 45 70 100 100 2 335 370 585 725 3 520 590 830 1160 4 100 170 285 215 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?

  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 = projected average quarterly demand + average seasonal index of that quarter.

  6. Quarter Year 1 Year 2 Year 3 Year 4 1 45/250 = 0.18 70/300 = 0.23 100/450 = 0.22 100/550 = 0.18 2 335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.32 3 520/250 = 2.08 590/300 = 1.97 830/450 = 1.84 1160/550 = 2.11 4 100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39 Multiplicative Seasonal Influences Multiplicative influence: Step 1: For each period, compute Actual Quarterly Demand Average Quarterly Demand Seasonal index = For example, seasonal Index, Year 1, Quarter 1 =

  7. Quarter Year 1 Year 2 Year 3 Year 4 1 45/250 = 0.18 70/300 = 0.23 100/450 = 0.22 100/550 = 0.18 2 335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.32 3 520/250 = 2.08 590/300 = 1.97 830/450 = 1.84 1160/550 = 2.11 4 100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39 Quarter Average Seasonal Index 1 2 3 4 Multiplicative Seasonal Influences 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 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.

  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. Quarter Year 1 Year 2 Year 3 Year 4 1 45-250 = -205 70-300 = -230 100-450 = -350 100-550 = -450 2 335-250 = 85 370-300 = 70 585-450 = 135 725-550 = 175 3 520-250 = 270 590-300 = 290 830-450 = 380 1160-550 = 610 4 100-250 = -150 170-300 = -130 285-450 = -165 215-550 = -335 Additive Seasonal Influences Additive influence: Step 1: For each period, compute seasonal index = Actual Quarterly Demand - Average Quarterly Demand For example, seasonal Index, Year 1, Quarter 1 = 45-250 = -205

  11. Quarter Year 1 Year 2 Year 3 Year 4 1 45-250 = -205 70-300 = -230 100-450 = -350 100-550 = -450 2 335-250 = 85 370-300 = 70 585-450 = 135 725-550 = 175 3 520-250 = 270 590-300 = 290 830-450 = 380 1160-550 = 610 4 100-250 = -150 170-300 = -130 285-450 = -165 215-550 = -335 Additive Seasonal Influences Quarter Average Seasonal Index 1 (-205-230-350-450)/4 = -308.75 2 (85+70+135+175)/4 = 116.25 3 (270+290+380+610)/4 = 387.50 4 (-150-130-165-335)/4 = -195.00 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 1 -308.75 650-308.75=341.25 2 116.25 650+116.25=766.25 3 387.50 650+387.50=1037.50 4 -195.00 650-195.00=455.00

  13. Difference between the highest and lowest demand increases

  14. Difference between the highest and lowest demand is constant

  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 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.

  24. 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

  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 Seasonal Deseasonalized Reseasonalized Period Demand Factors Demand Forecast A B C D E 1 205 0.7671 267.2430875 2 140 0.4252 329.257362 3 375 1.1797 317.8830475 4 575 1.6280 353.1876624 5 475 0.7671 619.2217881 6 275 0.4252 646.7555324 7 685 1.1797 580.6663668 8 965 1.6280 592.7410333 9 0.7671 10 0.4252

  30. Step 5Reseasonalize Seasonal Deseasonalized Reseasonalized Period Demand Factors Demand Forecast A B C D E 1 205 0.7671 267.2430875 2 140 0.4252 329.257362 3 375 1.1797 317.8830475 4 575 1.6280 353.1876624 5 475 0.7671 619.2217881 6 275 0.4252 646.7555324 7 685 1.1797 580.6663668 8 965 1.6280 592.7410333 9 0.7671 719.8792471 10 0.4252 776.8814164

  31. Steps 4a, 4b: Regression 1000 Original 800 data 600 Desea- Demand sonalized 400 Reg- 200 ression 0 0 5 10 Period

  32. Step 4c: Regression - Projection Forecast Deseasonalized Demand 1000 Original 800 data 600 Desea- Demand sonalized 400 Reg- 200 ression 0 0 5 10 Period

  33. 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.)

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