1. www.izmirekonomi.edu.tr 1-1 Chapter 2 Forecasting

2. www.izmirekonomi.edu.tr 1-2 Introduction to Forecasting What is forecasting?  Primary Function is to Predict the Future Why are we interested?  Affects the decisions we make today Examples: who uses forecasting in their jobs?  forecast demand for products and services  forecast availability of manpower for manufacturing or services.  forecast inventory and materiel needs daily for mfg or services

3. www.izmirekonomi.edu.tr 1-3 Characteristics of Forecasts They are usually wrong! A good forecast is more than a single number  mean and standard deviation  range (high and low) Aggregate forecasts are usually more accurate Accuracy erodes as we go further into the future. Forecasts should not be used to the exclusion of known information

4. www.izmirekonomi.edu.tr 1-4 What Makes a Good Forecast It should be timely It should be as accurate as possible It should be reliable It should be in meaningful units It should be presented in writing The method should be easy to use and understand in most cases.

5. www.izmirekonomi.edu.tr 1-5 Forecast Horizons in Operation Planning Figure 2.1

6. www.izmirekonomi.edu.tr 1-6 Subjective Forecasting Methods Sales Force Composites  Aggregation of sales personnel estimates Customer Surveys Jury of Executive Opinion The Delphi Method  Individual opinions are compiled and reconsidered. Repeat until and overall group consensus is (hopefully) reached.

7. www.izmirekonomi.edu.tr 1-7 Judgmental Forecasts There may not be enough time to gather data and analyze quantitative data or no data at all. Expert Judgment – managers(marketing,operations,finance,etc.)  Be careful about who you call an “expert” Sales force composite  Recent experience may influence their perceptions Consumer surveys  Requires considerable amount of knowledge and skill Opinions of managers and staff  Delphi method: a series of questionnaire, responses are kept anonymous, new questionnaires are developed based on earlier results – Rand corporation (1948)

8. www.izmirekonomi.edu.tr 1-8 Objective Forecasting Methods Two primary methods: causal models and time series methods Causal Models Let Y be the quantity to be forecasted and (X 1, X 2,..., X n ) be n variables that have predictive power for Y. A causal model is Y = f (X 1, X 2,..., X n ). A typical relationship is a linear one. That is, Y = a 0 + a 1 X 1 +... + a n X n. What might be such variables for average income for Turkey for 2007?

9. www.izmirekonomi.edu.tr 1-9 Time Series Methods A time series is just collection of past values of the variable being predicted. Also known as naïve methods. Goal is to isolate patterns in past data. (See Figures on following pages) Trend Seasonality Cycles Randomness

10. www.izmirekonomi.edu.tr 1-10 Time Series Model Building A time-series is a time ordered sequence of observations taken at regular intervals over a period of time. The data may be demand, earnings, profit, accidents, consumer price index,etc. The assumption is future values of the series can be estimated from past values One need to identify the underlying behavior of the series - pattern of the data

11. www.izmirekonomi.edu.tr 1-11 Some Behaviors Typically Observed Trend  E.g., population shifts, change in income. Usually a long-term movement in data Seasonality  Fairly regular variations, e.g., Friday nights in restaurants, new year in shopping malls, rush hour traffic., etc. Cycles  Wavelike variations lasting more than a year, e.g. economic recessions, etc. Irregular variations  Caused by unusual circumstances, e.g., strikes, weather conditions, etc. Random variations  Residual variations after all other behaviors are accounted for. Caused by chance

12. www.izmirekonomi.edu.tr 1-12 Forecast Variations Trend Irregular variation Seasonal variations 90 89 88 Cycles Trend with seasonal pattern

13. www.izmirekonomi.edu.tr 1-13 Types of Time Series Models We will cover the following techniques in this section; Naïve Techniques for averaging  Moving average  Weighted moving average  Exponential smoothing Techniques for trend  Linear equations  Trend adjusted exponential smoothing Techniques for seasonality Techniques for Cycles

14. www.izmirekonomi.edu.tr 1-14

15. www.izmirekonomi.edu.tr 1-15 Notation Conventions Let D 1, D 2,... D n,... be the past values of the series to be predicted (demand). If we are making a forecast in period t, assume we have observed D t, D t-1 etc. Let F t, t +   forecast made in period t for the demand in period t +  where  = 1, 2, 3, … Then F t -1, t is the forecast made in t-1 for t and F t, t+1 is the forecast made in t for t+1. (one step ahead) Use shorthand notation F t = F t - 1, t.

16. www.izmirekonomi.edu.tr 1-16 Evaluation of Forecasts The forecast error in period t, e t, is the difference between the forecast for demand in period t and the actual value of demand in t. For a multiple step ahead forecast: e t = F t - , t - D t. For one step ahead forecast: e t = F t - D t.  e 1, e 2,.., e n forecast errors over n periods Mean Absolute Deviation  D = (1/n)  | e i | Mean Absolute Percentage Error MAPE = [(1/n)  | e i /Di| ]*100 Mean Square Error MSE = (1/n)  e i 2

17. www.izmirekonomi.edu.tr 1-17 Measures of Forecast Accuracy Error - difference between actual value and predicted value Mean absolute deviation (MAD)  Average absolute error Mean squared error (MSE)  Average of squared error Tracking signal  Ratio of cumulative error and MAD

18. www.izmirekonomi.edu.tr 1-18 MAD & MSE MAD = Actualforecast   n MSE = Actualforecast ) - 1 2   n (

19. www.izmirekonomi.edu.tr 1-19 Tracking Signal Tracking signal = ( Actual - forecast ) MAD  Tracking signal = ( Actual - forecast) Actual - forecast   n

20. www.izmirekonomi.edu.tr 1-20 Biases in Forecasts A bias occurs when the average value of a forecast error tends to be positive or negative (ie, deviation from truth). Mathematically an unbiased forecast is one in which E (e i ) = 0. See Figure 2.3 (next slide).  e i = 0

21. www.izmirekonomi.edu.tr 1-21 Forecast Errors Over Time Figure 2.3

22. www.izmirekonomi.edu.tr 1-22 Ex. 2.1 week F1D1| E1 | |E1/D1| F2D2| E2 | |E2/D2| 192884 0.0455 96915 0.0549 287881 0.0114 89 0 0.0000 395972 0.0206 92902 0.0222 490837 0.0843 93903 0.0333 588913 0.0330 90864 0.0465 693 0 0.0000 85894 0.0449 Which forecast is a better forecast? MAD, MAPE and MSE

23. www.izmirekonomi.edu.tr 1-23 MAD1 = 17/6 = 2.83 better MAD2 = 18/6 = 3.00 MSE1 = 79/6 = 13.17 MSE2 = 70/6 = 11.67 better MAPE1 = 0.0325 better MAPE2 = 0.0333

24. www.izmirekonomi.edu.tr 1-24 Forecasting for Stationary Series A stationary time series has the form: D t =  +  t where  is a constant (mean of the series) and  t is a random variable with mean 0 and var    Two common methods for forecasting stationary series are moving averages and exponential smoothing.

25. www.izmirekonomi.edu.tr 1-25 Moving Averages In words: the arithmetic average of the N most recent observations. For a one-step-ahead forecast: F t = (1/N) (D t - 1 + D t - 2 +... + D t - N )

26. www.izmirekonomi.edu.tr 1-26 Summary of Moving Averages Advantages of Moving Average Method  Easily understood  Easily computed  Provides stable forecasts Disadvantages of Moving Average Method  Requires saving all past N data points  Lags behind a trend  Ignores complex relationships in data

27. www.izmirekonomi.edu.tr 1-27 Moving Average Lags a Trend Figure 2.4

28. www.izmirekonomi.edu.tr 1-28 Exponential Smoothing Method A type of weighted moving average that applies declining weights to past data. Based on the idea: More recent data is more relevant 1. New Forecast =  (most recent observation) + (1 -  (last forecast) or 2. New Forecast = last forecast -  last forecast error) where 0 <  and generally is small for stability of forecasts ( around.1 to.2)

29. www.izmirekonomi.edu.tr 1-29 Exponential Smoothing (cont.) In symbols: F t+1 =  D t + (1 -  ) F t =  D t + (1 -  ) (  D t-1 + (1 -  ) F t-1 ) =  D t + (1 -  )(  )D t-1 + (1 -    (  )D t - 2 +... Hence the method applies a set of exponentially declining weights to past data. It is easy to show that the sum of the weights is exactly one. ( Or F t + 1 = F t -  F t - D t ) )

30. www.izmirekonomi.edu.tr 1-30 Weights in Exponential Smoothing Fig. 2-5

31. www.izmirekonomi.edu.tr 1-31 Comparison of ES and MA Similarities  Both methods are appropriate for stationary series  Both methods depend on a single parameter  Both methods lag behind a trend  One can achieve the same distribution of forecast error by setting  = 2/ ( N + 1). Differences  ES carries all past history. MA eliminates “bad” data after N periods  MA requires all N past data points while ES only requires last forecast and last observation.

32. www.izmirekonomi.edu.tr 1-32 Exponential Smoothing for different values of alpha So how does alpha effect forecast?

33. www.izmirekonomi.edu.tr 1-33 Example of Exponential Smoothing

34. www.izmirekonomi.edu.tr 1-34 Picking a Smoothing Constant .1 .4 Actual Lower values of  are preferred when the underlying trend is stable and higher values of  are preferred when it is susceptible to change. Note that if  is low your next forecast highly depends on your previous ones and feedback is less effective.

35. www.izmirekonomi.edu.tr 1-35 Using Regression for Times Series Forecasting n Regression Methods Can be Used When Trend is Present. – Model: D t = a + bt. n If t is scaled to 1, 2, 3,..., then the least squares estimates for a and b can be computed as follows: n Set S xx = n 2 (n+1)(2n+1)/6 - [n(n+1)/2] 2 Set S xy = n  i D i - [n(n + 1)/2]  D i Set S xy = n  i D i - [n(n + 1)/2]  D i _ – Let b = S xy / S xx and a = D - b (n+1)/2 These values of aandb provide the “best” fit of the data in a least squares sense. These values of a and b provide the “best” fit of the data in a least squares sense.

36. www.izmirekonomi.edu.tr 1-36 An Example of a Regression Line

37. www.izmirekonomi.edu.tr 1-37 Linear Trend Equation - Notation b is similar to the slope. However, since it is calculated with the variability of the data in mind, its formulation is not as straight-forward as our usual notion of slope. A linear trend equation has the form; Y t = a + bt 0 1 2 3 4 5 t Y y t =Forecast for period t, a= value of y t at t=0 and b is the slope of the line.

38. www.izmirekonomi.edu.tr 1-38 Insights For Calculating a and b b = n(ty) - ty nt 2 - ( t) 2 a = y - bt n    For further information refer to http://www.stat.psu.edu/~bart/0515.doc or any statistics book! Suppose that you think that there is a linear relation between the height (ft.) and weight (pounds) of humans. You collected data and want to fit a linear line to this data. Weight= a + b Height How do you estimate a and b?

39. www.izmirekonomi.edu.tr 1-39 More Insights For Calculating a and b Demand observed for the past 11 weeks are given. We want to fit a linear line (D=a+bT) and determine a and b that minimizes the sum of the squared deviations. (Why squared?) A little bit calculus, take the partial derivatives and set it equal to 0 and solve for a and b!

40. www.izmirekonomi.edu.tr 1-40 Linear Trend Equation Example

41. www.izmirekonomi.edu.tr 1-41 Linear Trend Calculation Question is forecasting the sales for the 6 th period. What do you think it will be? If we fit a line to the observed sales of the last five months,

42. www.izmirekonomi.edu.tr 1-42 Linear Trend Calculation 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 y = 143.5 + 6.3*6= 181.5

43. www.izmirekonomi.edu.tr 1-43 Other Methods When Trend is Present Double exponential smoothing, of which Holt’s method is only one example, can also be used to forecast when there is a linear trend present in the data. The method requires separate smoothing constants for slope and intercept.

44. www.izmirekonomi.edu.tr 1-44 Trends Adjusted Exponential Smoothing A variation of simple Exponential Smoothing can be used when trend is observed in historical data. It is also referred as double smoothing. Note that if a series has a trend and simple smoothing is used the forecasts will all lag the trend. If data are increasing each forecast will be low! When trend exists we may improve the model by adjusting for this trend. (C.C. Holt) Trend Adjusted Forecasts (TAF) is composed of two elements: a smoothed error and a trend factor; TAF t+1 = S t + T t where S t = smoothed forecast = TAF t +  (A t – TAF t ) T t = current trend estimate= T t-1 +  (TAF t – TAF t-1 – T t-1 )

45. www.izmirekonomi.edu.tr 1-45 Insights: TAES TAF t+1 = S t + T t where S t = smoothed forecast = TAF t +  (A t – TAF t ) T t = current trend estimate= T t-1 +  (TAF t – TAF t-1 – T t-1 )= (1-  T t-1 +  (TAF t – TAF t-1 ) Weighted average of last trend and last forecast error.  and  are smoothing constants to be selected by the modeler. S t is same with original ES – feedback for the forecast error is added to previous forecast with a percentage of  If there is trend ES will have a lag. We must also include this lag to our model. Hence T t is added where  T t is the trend and updated each period.

46. www.izmirekonomi.edu.tr 1-46 Forecasting For Seasonal Series Seasonality corresponds to a pattern in the data that repeats at regular intervals. (See figure next slide) Multiplicative seasonal factors: c 1, c 2,..., c N where i = 1 is first period of season, i = 2 is second period of the season, etc..  c i = N. c i = 1.25 implies 25% higher than the baseline on avg. c i = 0.75 implies 25% lower than the baseline on avg.

47. www.izmirekonomi.edu.tr 1-47 A Seasonal Demand Series

48. www.izmirekonomi.edu.tr 1-48 Quick and Dirty Method of Estimating Seasonal Factors Compute the sample mean of the entire data set (should be at least several seasons of data). Divide each observation by the sample mean. (This gives a factor for each observation.) Average the factors for like periods in a season. The resulting N numbers will exactly add to N and correspond to the N seasonal factors.

49. www.izmirekonomi.edu.tr 1-49 Deseasonalizing a Series To remove seasonality from a series, simply divide each observation in the series by the appropriate seasonal factor. The resulting series will have no seasonality and may then be predicted using an appropriate method. Once a forecast is made on the deseasonalized series, one then multiplies that forecast by the appropriate seasonal factor to obtain a forecast for the original series.

50. www.izmirekonomi.edu.tr 1-50 Seasonal series with increasing trend Fig 2-10

51. www.izmirekonomi.edu.tr 1-51 Initialization for Winters’s Method

52. www.izmirekonomi.edu.tr 1-52 Practical Considerations Overly sophisticated forecasting methods can be problematic, especially for long term forecasting. (Refer to Figure on the next slide.) Tracking signals may be useful for indicating forecast bias. Box-Jenkins methods require substantial data history, use the correlation structure of the data, and can provide significantly improved forecasts under some circumstances.

53. www.izmirekonomi.edu.tr 1-53 The Difficulty with Long-Term Forecasts

54. www.izmirekonomi.edu.tr 1-54 Tracking the Mean When Lost Sales are PresentFig. 2-13

55. www.izmirekonomi.edu.tr 1-55 Tracking the Standard Deviation When Lost Sales are Present Fig. 2-14

56. www.izmirekonomi.edu.tr 1-56 Case Study: Sport Obermeyer Saves Money Using Sophisticated Forecasting Methods Problem: Company had to commit at least half of production based on forecasts, which were often very wrong. Standard jury of executive opinion method of forecasting was replaced by a type of Delphi Method which could itself predict forecast accuracy by the dispersion in the forecasts received. Firm could commit early to items that had forecasts more likely to be accurate and hold off on items in which forecasts were probably off. Use of early information from retailers improved forecasting on difficult items. Consensus forecasting in this case was not the best method.