1. 4-1 Operations Management Forecasting Chapter 4 - Part 2

2. 4-2  Trend is increasing or decreasing pattern.  First, plot data to verify trend.  If trend exists, then moving averages and exponential smoothing will always lag. Forecasting a Trend

3. 4-3 Plot Data Period Actual 4 5 32 1 6 8 4 12 16 20

4. 4-4 MA = 3 period Moving Average Moving Averages for a Trend Period MAMA 1 8 211 313 41510.67 4.33 519 13.00 6.00 MA Erro r 6 15.67 ? Sales

5. 4-5 Trend Graph Period Actual MA Forecast 4 5 32 1 6 8 4 12 16 20

6. 4-6 MA = 3 period Moving Average ES = Exponential Smoothing with  =0.5 ( F 2 =11) Exponential Smoothing for a Trend ? Period MAMA ES 1 8 211 313 11 41510.67 12 4.33 3.0 519 13.00 13.5 6.00 5.5 MA Erro r ES Error 6 15.67 11 16.25 ? Sales

7. 4-7 Trend Graph Period Actual MA Forecast 4 5 32 1 6 8 4 12 16 20 ES Forecast

8. 4-8  Moving Averages and (simple) Exponential Smoothing are always poor.  For a linear trend can use:  Exponential Smoothing with Trend Adjustment (skip: pp. 90-92).  Linear Trend Projection (linear regression).  For non-linear trend can use:  Non-linear regression techniques. Forecasting a Trend

9. 4-9  Used for forecasting linear trend line.  PLOT TO VERIFY LINEAR RELATIONSHIP  Assumes linear relationship between response variable, Y, and time, X.  Y = a + bX  a = y-axis intercept; b = slope  Estimated by least squares method.  Minimizes sum of squared errors. Linear Trend Projection

10. 4-10 Plot of X,Y Data Time (x) Values of Dependent Variable (Y) Actual observation

11. 4-11 Least Squares Deviation Time (x) Values of Dependent Variable (Y) Actual observation Point on regression line

12. 4-12 Least Squares  Least squares line minimizes sum of squared deviations.  This reduces large errors.  Similar to MSE.  Deviations around least squares line are assumed to be random.

13. 4-13 b > 0 b < 0 a a Y X Linear Trend Projection Model

14. 4-14 Least Squares Equations Equation: Slope (p. 94): Y-Intercept:

15. 4-15 Linear Trend Projection Example Perio d (x) 1 8 211 313 415 519 Sales (y) xy 60 95 8 22 39  xy=224 x2x2 9 16 25 4 1  x 2 =55 x=3y=13.2

16. 4-16 TP = Trend Projection: Y = 5.4 + 2.6x Linear Trend Projection Example Period (x) MAMA ES 1 2 3 4 5 8 11 13 15 19 MA Err. 6 10.67 13.00 15.67 11 12 13.5 11 16.25 4.33 6.00 Sales (y) 3.0 5.5 ES Err. TP Err. TP 21.0 18.4 15.8 -0.8 0.6 Small errors!

17. 4-17 Trend Graph MA Forecast ES Forecast Period Actual 4 5 32 1 6 8 4 12 16 20 TP Forecast

18. 4-18 Models with Seasonality  Use if data exhibits seasonal patterns.  Daily, weekly, monthly, yearly.  Compute seasonal component.  Remove seasonality and forecast.  Factor in seasonal component.  See pages 96-100.

19. 4-19  Identify Independent and dependent variable.  Dependent variable (y): Entity to be forecast (demand).  Independent variable (x): Used to predict (or explain) dependent variable.  Determine relationship.  Plot data.  Consider time lags.  Calculate parameters.  Forecast.  Monitor. Associative Forecasting Methods

20. 4-20  Linear relationship between dependent & explanatory variables.  Example: Sales in month i ( Y i ) depends on advertising in month i ( X i ) (eg. number of ads)  Sales may also depend on advertising in previous months! YX ii = + ab Dependent variable (sales). Independent variable (number of ads). Linear Regression

21. 4-21 Least Squares Deviation Values of Independent Variable (x) Values of Dependent Variable (Y) Actual observation Point on regression line

22. 4-22 Linear Regression Equations (same as before) Equation: Slope: Y-Intercept:

23. 4-23  Slope ( b ):  Y changes by b units for each 1 unit increase in X.  If b = +2, then sales ( Y ) is forecast to increase by 2 for each 1 unit increase in advertising ( X ).  Y-intercept ( a ):  Average value of Y when X = 0.  If a = 4, then average sales ( Y ) is expected to be 4 when advertising ( X ) is 0. Interpretation of Coefficients

24. 4-24 Least Squares  Plot data to verify linearity!  If curve is present, use non-linear regression.  Forecast only in (or near) range of observed values!  May need future values of independent variable to make forecast.  Example: Summer hotel demand may depend on summer gasoline price.

25. 4-25 Monthly Sales vs. Number of Ads Number of TV ads per month Sales 0

26. 4-26 Least Squares Line Number of TV ads per month Sales 0 What is sales forecast for small number of ads?

27. 4-27 Forecasting Outside Range of Observed Values is Unreliable Number of TV ads per month Sales 0 Forecast is for negative sales!

28. 4-28  Answers: ‘ How strong is the linear relationship between the variables?’  Coefficient of correlation - r  Measures degree of association; ranges from -1 to +1  Coefficient of determination - r 2  Amount of variation explained by regression equation.  Used to evaluate quality of linear relationship. Correlation

29. 4-29 Sample Coefficient of Correlation

30. 4-30 r = +1r = -1 r =.89r = 0 Y X Y X Y X X Coefficient of Correlation Y

31. 4-31  You want to achieve:  No pattern or direction in forecast error.  Error = Actual - Forecast  Small forecast error.  Mean square error (MSE).  Mean absolute deviation (MAD).  Mean absolute percentage error (MAPE). Guidelines for Selecting Forecasting Model

32. 4-32 Time Error 0 Desired Pattern Time Error 0 Trend Not Fully Accounted for Pattern of Forecast Error

33. 4-33 You’re a marketing analyst for Hasbro Toys. You’ve forecast sales with a linear regression model & exponential smoothing. Which model do you use? Linear RegressionExponential ActualModelSmoothing YearSalesForecastForecast (.9) 110.61.00 211.31.00 322.01.00 422.71.90 543.41.99 Selecting Forecasting Model Example

34. 4-34 MSE = Σ Error 2 / n = 1.10 / 5 = 0.220 MAD = Σ |Error| / n = 2.0 / 5 = 0.400 MAPE = Σ [| Error |/ Actual ]/ n = 1.2/5 = 0.24 = 24% Linear Regression Model 1.10 Year Y i F’cast 110.6 0.40.160.4 211.3-0.30.090.3 322.0 0.00.000.0 422.7-0.70.490.7 543.4 0.60.360.6 Total0.02.0 ErrorError 2 |Error|

35. 4-35 1.99 MSE = Σ Error 2 / n = 5.05 / 5 = 1.01 MAD = Σ |Error| / n = 3.11 / 5 = 0.622 MAPE = Σ[ |Error|/Actual]/ n = 1.0525/5 = 0.2105 = 21% Exponential Smoothing Model Year Y i F’cast 111.000.00.000.0 211.000.00.000.0 321.001.01.001.0 421.900.10.010.1 542.014.042.01 Total0.35.053.11 ErrorError 2 |Error|

36. 4-36 Which is Better??? Linear Regression Model: MSE = Σ Error 2 / n = 1.10 / 5 = 0.220 MAD = Σ |Error| / n = 2.0 / 5 = 0.400 MAPE = Σ[ |Error|/Actual]/ n = 1.2/5 = 0.24 = 24% Exponential Smoothing Model: MSE = Σ Error 2 / n = 5.05 / 5 = 1.01 MAD = Σ |Error| / n = 3.11 / 5 = 0.622 MAPE = Σ[ |Error|/Actual]/ n = 1.0525/5 = 0.2105 = 21%

37. 4-37  Measures how well the forecast is predicting actual values.  To use:  Calculate tracking signal each time period.  Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD).  Plot tracking signal on graph.  Good tracking signal has low values.  Should be within upper and lower control limits (often based on MAD). Tracking Signal

38. 4-38 Plot of a Tracking Signal Time Lower control limit Upper control limit Signal exceeded limit Tracking signal Acceptable range MAD + 0 -

39. 4-39 Tracking Signal Equation

40. 4-40 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090Cum|Error|

41. 4-41 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090 -10 -10Cum|Error| RSFE =  Errors = -10 Error = Actual - Forecast = 90 - 100 = -10

42. 4-42 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090 -10 -1010Cum|Error| Cum |Error| =  |Errors| = 10

43. 4-43 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090 -10 -1010 10.0Cum|Error| MAD =  |Errors|/n = 10/1 = 10

44. 4-44 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090 -10 -1010 10.0Cum|Error| TS = RSFE/MAD = -10/10 = -1

45. 4-45 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 -10 -1010 10.0Cum|Error|

46. 4-46 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 -10 -1010 10.0 -5Cum|Error| Error = Actual - Forecast = 94 - 99 = -5

47. 4-47 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 -10 -1010 10.0 -5 -15Cum|Error| RSFE =  Errors = (-10) + (-5) = -15

48. 4-48 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 -10 -1010 10.0 -5 -1515Cum|Error| Cum Error =  |Errors| = 10 + 5 = 15

49. 4-49 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 -10 -1010 10.0 -5 -1515 7.5Cum|Error| MAD =  |Errors|/n = 15/2 = 7.5

50. 4-50 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 -10 -1010 10.0 -5 -1515 7.5-2Cum|Error| TS = RSFE/MAD = -15/7.5 = -2

51. 4-51 Tracking Signal - Month 3 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 3 98 98113 -10 -1010 10.0 -5 -1515 7.5-2 15 0 30303030 10101010 0Cum|Error|

52. 4-52 Tracking Signal - Months 4-6 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 3 98 98113 4105 95 95 5104119 6110140 -10 -1010 10.0 -5 -1515 7.5-2 15 0 30303030 10101010 0 -10 -10 40404040 10 15 5 55555555 11.45 30 35 85858585 14.2 2.47Cum|Error|

53. 4-53 Demand and Forecast 70 80 90 100 110 120 130 140 01234567 Month Forecast Actual demand

54. 4-54 Tracking Signal 01234567 Time -3 -2 0 1 2 3 Tracking Signal

55. 4-55  Upper and lower limits depend on the product being forecast.  98% of values should be within  3 MAD.  99.9% of values should be within  4 MAD.  Use smaller limits for high volume items.  For example: +2 MAD, -2 MAD  Patterns, even if within limits, indicate better forecasts can be made. Tracking Signal Limits

56. 4-56 You’re a marketing analyst for Hasbro Toys. You’ve forecast sales with a linear regression model & exponential smoothing. Which model do you use? Linear RegressionExponential ActualModelSmoothing YearSalesForecastForecast (.9) 110.61.00 211.31.00 322.01.00 422.71.90 543.41.99 Selecting Forecasting Model Example - Revisited

57. 4-57 Linear Regression Model Tracking Signal Year Y i F’cast 110.6 0.4 1.0 211.3-0.30.350.29 322.0 0.00.2330.43 422.7-0.70.35 -1.71 543.4 0.60.400.0 ErrorMAD TS

58. 4-58 Exponential Smoothing Model Tracking Signal 1.99 Year Y i F’cast 111.000.0 211.000.0 321.001.00.333.0 421.900.10.2754.0 542.010.6225.0 Error MAD TS

59. 4-59 Tracking Signals 501234 Year -3 -2 0 1 2 3 Tracking Signal Exponential Smoothing Linear Regression

60. 4-60 Forecasting in the Service Sector  Presents unusual challenges:  Large variability (during day, week, etc.).  Special need for short term forecasting.  Needs differ greatly as function of industry and product.  Issues of holidays and calendar.  Examples: Staffing for hospitals, fast-food restaurants, banking, etc.

61. 4-61 Forecasting Summary  Determine purpose of forecast first.  Plot data.  Use several appropriate methods.  Continually monitor, evaluate and adjust methods to improve forecasts.