1. Chapter 3 Lecture 4 Forecasting

2. Time Series is a sequence of measurements over time, usually obtained at equally spaced intervals – Daily – Monthly – Quarterly – Yearly Time Series Forecasts

3. Time ordered sequence of observations taken at regular observations taken at regular intervals. Statistical techniques that make use of historical data collected over a long period of time. Methods assume that what has occurred in the past will continue to occur in the future. Forecasts based on only one factor - time.

4. Time Series Patterns

5. Time Series Forecasts Naive Forecasts Techniques for Averaging Moving Average Weighted Moving Average Exponential Smoothing Techniques for Trend Trend Techniques for Seasonality Techniques for Cycles

6. Naive Forecasts Uh, give me a minute.... We sold 250 wheels last week.... Now, next week we should sell.... The forecast for any period equals the previous period’s actual value. The recent periods are the best predictors of the future

7. Simple to use Virtually no cost Quick and easy to prepare Data analysis is nonexistent Easily understandable Cannot provide high accuracy Naïve Forecasts

8. F(t) = A(t-1)F(t) = A(t-1) Stable time series data (stationer) F(t) = A(t-1) + (A(t-1) – A(t-2))F(t) = A(t-1) + (A(t-1) – A(t-2)) Data with trends trends F(t) = A(t-n)F(t) = A(t-n) Seasonal variations Uses for Naïve Forecasts at different data patterns

9. Naïve Forecasts DemandYear 3902006 4202007 3802009 4002010 4002011 Stationer F(t) = A(t-1) Ex.

10. Naïve Forecasts DemandYear 3902006 4202007 4402009 4802010 5202011 Trend F(t) = A(t-1) + (A(t-1) – A(t-2)) Ex.

11. Naïve Forecasts DemandYear 101/2008 907/2008 151/2009 1007/2009 121/2010 957/2010 121/2011 Seasonal F(t) = A(t-1) + (A(t-1) – A(t-2)) Ex.

12. Naïve Forecast Graph

13. A technique that averages a number of recent actual values, updated as new values become available. F t = forecast for time period t MA n = n period moving average A t-1 = actual value in period t-1 n = number of periods ( data points ) F t = MA n = n A t-n + … A t-2 + A t-1 Moving Averages, no pattern ( random variation )

14. Compute a 3-period moving average forecast given demand for shopping carts for the last five periods as shown: t=6 Compute a 3-period moving average forecast given demand for shopping carts for the last five periods as shown: t=6 F t = MA n = n A t-n + … A t-2 + A t-1 2006 20052004200320022001 period ???4140434042 demand F 6 = MA 3 = 3 A 3 + A 4 + A 5 = 41.33 Moving Averages Ex.

15. t=7 t=7 20072006 20052004200320022001 period ???41.334140434042 demand F 7 = MA 3 = 3 A 4 + A 5 + A 6 = 39.67 forecast 2007 200620052004200320022001 period ???384140434042 demand actual 2007 200620052004200320022001 period 39.67384140434042 demand forecast Moving Averages Ex. (cont.)

16. Moving Averages Ex.

17. Moving Averages Ex. (cont.)

18. Moving Averages Ex.

19. Moving Averages Ex. (cont.)

20. Moving Averages Ex.

21. Stability vs. Responsiveness Should I use a 2-period moving average or a 3- period moving average? The larger the “n” the more stable the forecast. A 2-period model will be more responsive to change. We must balance stability with responsiveness If responsiveness is required, average with few data points should be used,

22. As data points in an moving average technique increased, the sensitivity ( responsiveness ) of the average to new values decreased. If responsiveness is required, average with few data points should be used, Decreasing the number of data points in an moving average technique, increase the weight of more recent values Moving Averages

23. It is easy to compute It is easy to understand It is easy to compute It is easy to understand All values in the average are weighted equally, the oldest value has the same weight as the most recent value But Moving Averages most recent observations must be better indicators of the future than older observations Idea

24. Weighted Moving Averages Historical values of the time series are assigned different weights when performing the forecast

25. More recent values in a series are given more weight in computing the forecast. Trial and error used to find the suitable weighting scheme The sum of all weights must be = 1 Weighted average technique is more reflective of the most recent occurrences. Weighted Moving Averages

26. In a weighted moving average, weights are assigned to the most recent data. Formula: Weighted Moving Averages

27. Compute a 4-period weighted moving average forecast given demand for shopping carts for the last periods values and weights as shown: F6 = W5*A5 + W4*A4 + W3*A3 + W2*A2 F6 = 0.4(41) + 0.3(40) + 0.2(43) + 0.1(40) = 41 If the actual value of F6 is 39 then F7 = 0.4(39) + 0.3(41) + 0.2(40) + 0.1(43) = 40.2 6 54321 period ???4140434042 demand 0.40.30.20.10 weight Weighted Moving Averages Ex.

28. Market Mixer, Inc. sells can openers. Monthly sales for an eight-month period were as follows: MonthSalesMonthSales 1 450 5 460 2 425 6 455 3 445 7 430 4 435 8 420 Forecast next month’s sales using a 3-month weighted moving average, where the weight for the most recent data value is 0.60; the next most recent, 0.30; and the earliest, 0.10. Solution: PeriodSales Weighted Moving Average Forecast 1 450 2 425 3 445 4 435 (450*.10) + (425*.30) + (445*.60) = 440 5 460 (425*.10) + (445*.30) + (435*.60) = 437 6 455 (445*.10) + (435*.30) + (460*.60) = 451 7 430 (435*.10) + (460*.30) + (445*.60) = 455 8 420 (460*.10) + (455*.30) + (430*.60) = 441 9 (455*.10) + (430*.30) + (420*.60) = Comments: 1. Any forecasts beyond Period 9 will have the same value as the Period 9 forecast, i.e., 427. 3. WMA gives greater weight to more recent values in the moving average and is more responsive to recent changes in the data. 427 Weighted Moving Averages

29. The most recent observations might have the highest predictive value.The most recent observations might have the highest predictive value. Therefore, we should give more weight to the more recent time periods when forecasting. Therefore, we should give more weight to the more recent time periods when forecasting. Exponential Smoothing

30. F t = F t-1 +  ( A t-1 - F t-1 ) Weighted averaging method based on previous forecast plus a percentage of the forecast error Exponential Smoothing Determination of  is usually judgmental and subjective and often based on trial-and -error experimentation. The most commonly used values of  are between.10 and.50.

31. Exponential Smoothing Ex.

32. .1 .4 Actual Selecting a smoothing constant α is a matter of judgment or trial and error Picking a Smoothing Constant

33. Parabolic Exponential Growth Techniques for Trend Nonlinear Trends

34. Linear Trends Techniques for Trend

35. F t = Forecast for period t t = Specified number of time periods a = Value of F t at t = 0 b = Slope of the line F t = a + b t 0 1 2 3 4 5 t FtFt Techniques for Trend

36. b = n(ty)- ty nt 2 - (t) 2 a = y- bt n    Calculating a and b

37. Linear Trend Equation Ex.

38. y = 143.5 + 6.3t a= 812- 6.3(15) 5 = b= 5 (2499)- 15(812) 5(55)- 225 = 12495-12180 275-225 = 6.3 143.5 Linear Trend Equation

39. Moving average Weighted moving average Exponential smoothing Techniques for Averaging