1. Chapter 3 Lect 6 Forecasting

2. Seasonality – Repetition at Fixed Intervals Seasonal variations –Regularly repeating movements in series values that can be tied to recurring events. –It is applied to annually, monthly, weekly, daily and other regularly recurring patterns in data. Seasonality is –The amount that the actual values deviate from the average value of a series. Seasonal relative –Percentage of average or trend Centered moving average –A moving average positioned at the center of the data that were used to compute it. Techniques for Seasonality

3. THERE ARE TWO DIFFERENT MODELS OF SEASONALITY: ADDITIVE MODEL: SEASONALITY IS EXPRESSED AS QUANTITY (E.G., 20 UNITS), WHICH IS ADDED OR SUBTRACTED TO OR FROM THE SERIES AVERAGE (OR TREND). MULTIPLICATIVE MODEL: SEASONALITY IS EXPRESSED AS PERCENTAGE TO BE MULTIPLIED BY THE AVERAGE OF THE SERIES, IT IS ALWAYS CALLED SEASONAL RELATIVES OR SEASONAL INDEX. MODELS FOR SEASONALITY

4. Average Demand = average + seasonality Demand = average * seasonality relative SEASONALITY AND AVERAGE MODELS

5. Trend Demand = trend + seasonality Demand = trend * seasonality relative SEASONALITY AND TREND MODELS

6. Seasonal relatives Seasonal relatives are used in two different ways in forecasting to: Incorporate seasonality in a forecast Depersonalize the data

7. Incorporating seasonality into forecast Incorporating seasonality into forecast is useful when demand has both trend (or average) and seasonal components, which is conducted as following: Obtain trend estimates for the desired periods using trend equation. Add seasonality to the trend by multiplying (assuming multiplicative model is appropriate) these trend estimates by the corresponding seasonal relative.

8. Incorporating seasonality into forecast A furniture manufacturer wants to predict quarterly demand for certain seats for periods 15 and 16, which happen to be the second and the third quarters for a particular year. The series consists of both trend and seasonality. The trend portion of the demand is projected using the trend equation F t = 124 + 7.5 t. Quarter relatives are Q 1 = 1.2, Q 2 = 1.1, Q 3 = 0.75 and Q 4 = 0.95. use this information to predict demand for periods 15 and 16.  Solution The trend values at t = 15 and t = 16 are: F 15 = 124 + 7.5 (15) = 236.5 F 16 = 124 + 7.5 (16) = 244.0 Multiplying the trend value by the appropriate quarter relatives yield a forecast that includes both trend and seasonality. Given that t = 15 is a second quarter and t = 16 is a third quarter, the forecast will be: Period 15: 236.5 (1.1) = 260.15 Period 16: 244.0 (0.75) = 183.00 A furniture manufacturer wants to predict quarterly demand for certain seats for periods 15 and 16, which happen to be the second and the third quarters for a particular year. The series consists of both trend and seasonality. The trend portion of the demand is projected using the trend equation F t = 124 + 7.5 t. Quarter relatives are Q 1 = 1.2, Q 2 = 1.1, Q 3 = 0.75 and Q 4 = 0.95. use this information to predict demand for periods 15 and 16.  Solution The trend values at t = 15 and t = 16 are: F 15 = 124 + 7.5 (15) = 236.5 F 16 = 124 + 7.5 (16) = 244.0 Multiplying the trend value by the appropriate quarter relatives yield a forecast that includes both trend and seasonality. Given that t = 15 is a second quarter and t = 16 is a third quarter, the forecast will be: Period 15: 236.5 (1.1) = 260.15 Period 16: 244.0 (0.75) = 183.00 Ex.

9. Use the following information to deseasonalize sales for quarters 1 through 8 SalesQuarterPeriod 13211 14022 14633 15344 16015 16826 17637 18548 Q. R. 1.20 1.10 0.75 0.95 Des. Sal. 110 127.27 194.67 161.05 133.33 152.73 234.67 194.74 Incorporating seasonality into forecast Ex.

10. Seasonal Index – ratio of the average value of the item in a season to the overall average annual value. Example: average of year 1 January ratio to year 2 January ratio. (0.851 + 1.064)/2 = 0.957 If Year 3 average monthly demand is expected to be 100 units. Forecast demand Year 3 January: 100 X 0.957 = 96 units Forecast demand Year 3 May: 100 X 1.309 = 131 units Calculating seasonal relatives Ratio = demand / average demand

11. The manager of a parking lot has computed daily relatives for the number of cars per day in the lot. The computations are repeated here (about three weeks are shown for illustration). A seven period centered moving average is used because there are seven days (seasons) per week Calculating seasonal relatives Ex.

12. DayCarsCentered MA7Cars/MA7 Tues67 Wed75 Thur82 Fri9871.861.36 Sat9070.861.27 Sun3670.570.51 Mon55710.77 Tues6071.140.84 Wed7370.571.03 Thur8571..141.19 Fri9970.711.4 Sat8671.291.21 Sun4071.710.56 Mon52720.72 Tues6471.570.89 Wed7671.861.06 Thur8772.431.2 Fri9672.141.33 Sat88 Sun44 Mon50 Calculating seasonal relatives Ex.

13. The estimated relatives will be:  Monday: (0.77 + 0.72)/2 =.745  Tuesday: (0.84 + 0.89)/2 =.865  Wednesday: (1.03 + 1.06)/2 = 1.045  Thursday: (1.19 + 1.2)/2 = 1.195  Friday: (1.36 + 1.4 + 1.33)/3 = 1.36  Saturday: (1.27 + 1.21)/2 = 1.24  Sunday: (0.51 + 0.56)/2 = 0.535 Note The sum of the relatives must equal the number of periods (i.e., 7 in this example). If it is not, you have to multiply by a correction factor. In this example the sum is 6.985, therefore you have to multiply each factor by (7/6.985) Calculating seasonal relatives

14. The number of periods needed in a centered moving average is equal to the number of “seasons” involved  With monthly data, a 12-period moving average is needed.  With daily data, a 7-period moving average is needed.  With quarterly data, a 4-period moving average is needed Note: When the number of period is even, one additional step is needed, because the middle of an even period falls between two periods. The additional step requires taking two-period moving average of the even- numbered centered moving average. (See the following example) Calculating seasonal relatives

15.  Example (even period) Obtain estimates of quarter relatives for these data: Year1234 Quarter1234123412341 Demand14183546283660714554848858 Note: The number of seasons is 4 (an even number); therefore, a 4- centered moving average will be taken and then a 2-centered moving average is calculated to adjust the data. Calculating seasonal relatives

16. YearQuarterDemand (D)MA 4 MA 2 D/MA 2 1114 218 335301.17 446341.35 212839.380.71 23645.630.79 36050.881.18 47155.251.29 314560.500.74 25465.630.82 38469.381.21 488 4158 Calculating seasonal relatives

17. Average relatives Quarter 1 = (0.71+0.74)/2 = 0.725 Quarter 2 = (0.79+ 0.82)/2 = 0.805 Quarter 3 = (1.17 + 1.18 + 1.21)/3 = 1.187 Quarter 4 = (1.35 + 1.29)/2 = 1.32 The sum of these relatives is 4.037. Multiplying each by 4/4.037 will standardize relatives making their total equal 4. The resulting relatives are: Quarter 1 = 0.718, Quarter 2 = 0.798 Quarter 3 = 1.176, Quarter 4 = 1.308 Calculating seasonal relatives

18.  To deseasonalize the data divide the demand of each period by the corresponding seasonal relatives.  For a good forecast, use the deseasonalized data to fit the trend equation instead of the original data. Then multiply the trend by the associated relative of the period. Deseasonalize the data

19. 120 120 – 100 100 – 80 80 – 60 60 – 40 40 – 20 20 – Sales (millions of dollars) Actual data (y 1 ) Deseasonalized data (d t ) |11|111 |22|222 |33|333 |44|444 |11|111 |22|222 |33|333 |44|444 |11|111 |22|222 |33|333 |44|444 |11|111 |22|222 |33|333 |44|444 t 1998199920002001 Deseasonalize the data