1. 1 1 Slide © 2005 Thomson/South-Western EMGT 501 HW Solutions Chapter 15 - SELF TEST 3 Chapter 15 - SELF TEST 14

2. 2 2 Slide © 2005 Thomson/South-Western 15-3 a. Let x 1 = number of units of product 1 produced x 2 = number of units of product 2 produced,,,,,,,  0

3. 3 3 Slide © 2005 Thomson/South-Western b. In the graphical solution, point A provides the optimal solution. Note that with x 1 = 250 and x 2 = 100, this solution achieves goals 1 and 2, but underachieves goal 3 (profit) by $100 since 4(250) + 2(100) = $1200.

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5. 5 5 Slide © 2005 Thomson/South-Western c. The graphical solution indicates that there are four extreme points. The profit corresponding to each extreme point is as follows: Thus, the optimal product mix is x 1 = 350 and x 2 = 0 with a profit of $1400.

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7. 7 7 Slide © 2005 Thomson/South-Western d. The solution to part (a) achieves both labor goals, whereas the solution to part (b) results in using only 2(350) + 5(0) = 700 hours of labor in department B. Although (c) results in a $100 increase in profit, the problems associated with underachieving the original department labor goal by 300 hours may be more significant in terms of long-term considerations. e. Refer to the graphical solution in part (b). The solution to the revised problem is point B, with x 1 = 281.25 and x 2 = 87.5. Although this solution achieves the original department B labor goal and the profit goal, this solution uses 1(281.25) + 1(87.5) = 368.75 hours of labor in department A, which is 18.75 hours more than the original goal.

8. 8 8 Slide © 2005 Thomson/South-Western 15-14 a. b.

9. 9 9 Slide © 2005 Thomson/South-Western Home Work 16-9 and 16-35 Due Day: Nov 28, 2005 No Class on Nov 22, 2005

10. 10 Slide © 2005 Thomson/South-Western Chapter 16 Forecasting n Quantitative Approaches to Forecasting n The Components of a Time Series n Measures of Forecast Accuracy n Using Smoothing Methods in Forecasting n Using Trend Projection in Forecasting n Using Trend and Seasonal Components in Forecasting n Using Regression Analysis in Forecasting n Qualitative Approaches to Forecasting

11. 11 Slide © 2005 Thomson/South-Western Quantitative Approaches to Forecasting n Quantitative methods are based on an analysis of historical data concerning one or more time series. n A time series is a set of observations measured at successive points in time or over successive periods of time. n If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method. n If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method.

12. 12 Slide © 2005 Thomson/South-Western Time Series Methods n Three time series methods are: smoothing smoothing trend projection trend projection trend projection adjusted for seasonal influence trend projection adjusted for seasonal influence

13. 13 Slide © 2005 Thomson/South-Western Components of a Time Series n The trend component accounts for the gradual shifting of the time series over a long period of time. n Any regular pattern of sequences of values above and below the trend line is attributable to the cyclical component of the series.

14. 14 Slide © 2005 Thomson/South-Western Components of a Time Series n The seasonal component of the series accounts for regular patterns of variability within certain time periods, such as over a year. n The irregular component of the series is caused by short-term, unanticipated and non-recurring factors that affect the values of the time series. One cannot attempt to predict its impact on the time series in advance.

15. 15 Slide © 2005 Thomson/South-Western Measures of Forecast Accuracy n Mean Squared Error The average of the squared forecast errors for the historical data is calculated. The forecasting method or parameter(s) which minimize this mean squared error is then selected. n Mean Absolute Deviation The mean of the absolute values of all forecast errors is calculated, and the forecasting method or parameter(s) which minimize this measure is selected. The mean absolute deviation measure is less sensitive to individual large forecast errors than the mean squared error measure.

16. 16 Slide © 2005 Thomson/South-Western Smoothing Methods n In cases in which the time series is fairly stable and has no significant trend, seasonal, or cyclical effects, one can use smoothing methods to average out the irregular components of the time series. n Four common smoothing methods are: Moving averages Moving averages Centered moving averages Centered moving averages Weighted moving averages Weighted moving averages Exponential smoothing Exponential smoothing

17. 17 Slide © 2005 Thomson/South-Western Smoothing Methods n Moving Average Method The moving average method consists of computing an average of the most recent n data values for the series and using this average for forecasting the value of the time series for the next period.

18. 18 Slide © 2005 Thomson/South-Western Time Series A time series is a series of observations over time of some quantity of interest (a random variable). Thus, if is the random variable of interest at time i, and if observations are taken at times i = 1, 2, …., t, then the observed values are a time series.

19. 19 Slide © 2005 Thomson/South-Western Several typical time series patterns: Constant level Linear trend Seasonal effect

20. 20 Slide © 2005 Thomson/South-Western Constant level Example : the random variable observed at time I : the constant level of the model : the random error occurring at time i. forecast of the values of the time series at time t + 1, given the observed values,

21. 21 Slide © 2005 Thomson/South-Western Forecasting Methods for a Constant-Level Model (1) Last-Value Forecasting Method (2) Averaging Forecasting Method (3) Moving-Average Forecasting Method (4) Exponential Smoothing Forecasting Method

22. 22 Slide © 2005 Thomson/South-Western (1) Last-Value Forecasting Method By interpreting t as the current time, the last- value forecasting procedure uses the value of the time series observed at time, as the forecast at time t + 1. The last-value forecasting method sometimes is called the naive method, because statisticians consider it naïve to use just a sample size of one when additional relevant data are available.

23. 23 Slide © 2005 Thomson/South-Western (2) Averaging Forecasting Method This method uses all the data points in the time series and simply averages these points. This estimate is an excellent one if the process is entirely stable.

24. 24 Slide © 2005 Thomson/South-Western (3) Moving-Average Forecasting Method This method averages the data for only the last n periods as the forecast for the next period. The moving-average estimator combines the advantages of the last value and averaging estimators. A disadvantage of this method is that it places as much weight on as on.

25. 25 Slide © 2005 Thomson/South-Western (4) Exponential Smoothing Forecasting Method Where is called the smoothing constant. Thus, the forecast is just a weighted sum of the last observation and the preceding forecast for the period just ended.

26. 26 Slide © 2005 Thomson/South-Western Because of this recursive relationship between and, alternatively can be expressed as Another alternative form for the exponential smoothing technique is given by

27. 27 Slide © 2005 Thomson/South-Western Seasonal Factor It is fairly common for a time series to have a seasonal pattern with higher values at certain times of the year than others.

28. 28 Slide © 2005 Thomson/South-Western Quarter Three-Year Average Seasonal Factor Example

29. 29 Slide © 2005 Thomson/South-Western Quarter Actual Volume Seasonal Factor Seasonally Adjusted Volume Year

30. 30 Slide © 2005 Thomson/South-Western An Exponential Smoothing Method for a Linear Trend Model Linear trend Suppose that the generating process of the observed time series can be represented by a linear trend superimposed with random fluctuations.

31. 31 Slide © 2005 Thomson/South-Western The model is represented by Where is the random variable that is observed at time i, A is a constraint. B is the trend factor, and is the random error occurring at time i.

32. 32 Slide © 2005 Thomson/South-Western Adapting Exponential Smoothing to this Model Let Exponential smoothing estimate of the trend factor B at time t + 1, given the observed values, Given, the forecast of the value of the time series at time t + 1( ) is obtained simply by adding to the formula for.

33. 33 Slide © 2005 Thomson/South-Western The most recent observations are the most reliable ones for estimating the current parameters. latest trend at time t + 1 based on the last two values ( and ) and the last two forecasts ( and ). The exponential smoothing formula used for is

34. 34 Slide © 2005 Thomson/South-Western Then is calculated as where is the trend smoothing constant which must be between 0 and 1.

35. 35 Slide © 2005 Thomson/South-Western Getting started with this forecasting method requires making two initial estimates. initial estimate of the expected value of the time series initial estimate of the trend of the time series

36. 36 Slide © 2005 Thomson/South-Western The resulting forecasts for the first two periods are

37. 37 Slide © 2005 Thomson/South-Western Forecasting Errors The goal of several forecasting methods is to generate forecasts that are as accurate as possible, so it is natural to base a measure of performance on the forecasting errors.

38. 38 Slide © 2005 Thomson/South-Western The forecasting error for any period t is the absolute value of the deviation of the forecast for period t ( ) from what then turns out to be the observed value of the time series for period. Thus, letting denote this error,

39. 39 Slide © 2005 Thomson/South-Western Given the forecasting errors for n time periods (t =1, 2, …, n), two popular measures of performance are available. Mean Absolute Deviation (MAD) Mean Square Error (MSE)

40. 40 Slide © 2005 Thomson/South-Western The advantages of MAD (a) its ease of calculation (b) its straightforward interpretation The advantages of MSE (c) it imposes a relatively large penalty for a large forecasting error while almost ignoring inconsequentially small forecasting errors.

41. 41 Slide © 2005 Thomson/South-Western Causal Forecasting with Linear Regression In the preceding sections, we have focused on time series forecasting methods. We now turn to another type of approach to forecasting. Causal forecasting: Causal forecasting obtains a forecast of the quantity of interest by relating it directly to one ore more other quantities that drive the quantity of interest.

42. 42 Slide © 2005 Thomson/South-Western Linear Regression We will focus on the type of causal forecasting where the mathematical relationship between the dependent variable and the independent variable(s) is assumed to be a linear one. The analysis in this case is referred to as linear regression.

43. 43 Slide © 2005 Thomson/South-Western The number of variable A is denoted by X and the number of variable B is denoted by Y, then the random variables X and Y exhibit a degree of association. For any given number of variable A, there is a range of possible variable B, and vice versa. This relationship between X and Y is referred to as a degree of association model.

44. 44 Slide © 2005 Thomson/South-Western In some cases, there exists a functional relationship between two variables that may be linked linearly. The previous example is It follows that Both the degree of association model and the exact functional relationship model lead to the same linear relationship.

45. 45 Slide © 2005 Thomson/South-Western With t taking on integer values starting with 1, leads to certain simplified expressions. In the standard notation of regression analysis, X represents the independent variable and Y represents the dependent variable of interest. Consequently, the notational expression for this special time series model becomes

46. 46 Slide © 2005 Thomson/South-Western Method of Least Squares The usual method for identifying the “best” fitted line is the method of least squares. Regression Line

47. 47 Slide © 2005 Thomson/South-Western Suppose that an arbitrary line, given by the expression, is drawn through the data. A measure of how well this line fits the data can be obtained by computing the sum of squares of the vertical deviations of the actual points from the fitting line.

48. 48 Slide © 2005 Thomson/South-Western This method chooses that line a + bx that makes Q a minimum.

49. 49 Slide © 2005 Thomson/South-Western and where and

50. 50 Slide © 2005 Thomson/South-Western

51. 51 Slide © 2005 Thomson/South-Western Re-write (1)

52. 52 Slide © 2005 Thomson/South-Western Re-write (2) From (1)’ (1)’ in (2)

53. 53 Slide © 2005 Thomson/South-Western

54. 54 Slide © 2005 Thomson/South-Western Sales of Comfort brand headache medicine for the past ten weeks at Rosco Drugs are shown on the next slide. If Rosco Drugs uses a 3-period moving average to forecast sales, what is the forecast for Week 11? Example: Rosco Drugs

55. 55 Slide © 2005 Thomson/South-Western n Past Sales Week Sales Week Sales Week Sales Week Sales 1 110 6 120 1 110 6 120 2 115 7 130 2 115 7 130 3 125 8 115 3 125 8 115 4 120 9 110 4 120 9 110 5 125 10 130 5 125 10 130 Example: Rosco Drugs

56. 56 Slide © 2005 Thomson/South-Western Example: Rosco Drugs n Excel Spreadsheet Showing Input Data

57. 57 Slide © 2005 Thomson/South-Western Example: Rosco Drugs n Steps to Moving Average Using Excel Step 1: Select the Tools pull-down menu. Step 2: Select the Data Analysis option. Step 3: When the Data Analysis Tools dialog appears, choose M oving Average. Step 4: When the Moving Average dialog box appears: Enter B4:B13 in the Input Range box. Enter 3 in the Interval box. Enter C4 in the Output Range box. Select OK.

58. 58 Slide © 2005 Thomson/South-Western Example: Rosco Drugs n Spreadsheet Showing Results Using n = 3

59. 59 Slide © 2005 Thomson/South-Western Smoothing Methods n Centered Moving Average Method The centered moving average method consists of computing an average of n periods' data and associating it with the midpoint of the periods. For example, the average for periods 5, 6, and 7 is associated with period 6. This methodology is useful in the process of computing season indexes.

60. 60 Slide © 2005 Thomson/South-Western Smoothing Methods n Weighted Moving Average Method In the weighted moving average method for computing the average of the most recent n periods, the more recent observations are typically given more weight than older observations. For convenience, the weights usually sum to 1.

61. 61 Slide © 2005 Thomson/South-Western Smoothing Methods n Exponential Smoothing Using exponential smoothing, the forecast for the next period is equal to the forecast for the current period plus a proportion (  ) of the forecast error in the current period. Using exponential smoothing, the forecast for the next period is equal to the forecast for the current period plus a proportion (  ) of the forecast error in the current period. Using exponential smoothing, the forecast is calculated by: Using exponential smoothing, the forecast is calculated by:  [the actual value for the current period] +  [the actual value for the current period] + (1-  )[the forecasted value for the current period], (1-  )[the forecasted value for the current period], where the smoothing constant, , is a number between 0 and 1.

62. 62 Slide © 2005 Thomson/South-Western Trend Projection n If a time series exhibits a linear trend, the method of least squares may be used to determine a trend line (projection) for future forecasts. n Least squares, also used in regression analysis, determines the unique trend line forecast which minimizes the mean square error between the trend line forecasts and the actual observed values for the time series. n The independent variable is the time period and the dependent variable is the actual observed value in the time series.

63. 63 Slide © 2005 Thomson/South-Western Trend Projection n Using the method of least squares, the formula for the trend projection is: T t = b 0 + b 1 t. where: T t = trend forecast for time period t where: T t = trend forecast for time period t b 1 = slope of the trend line b 1 = slope of the trend line b 0 = trend line projection for time 0 b 0 = trend line projection for time 0 b 1 = n  tY t -  t  Y t b 1 = n  tY t -  t  Y t n  t 2 - (  t ) 2 n  t 2 - (  t ) 2 where: Y t = observed value of the time series at time period t where: Y t = observed value of the time series at time period t = average of the observed values for Y t = average of the observed values for Y t = average time period for the n observations = average time period for the n observations

64. 64 Slide © 2005 Thomson/South-Western If Rosco Drugs uses exponential smoothing to forecast sales, which value for the smoothing constant ,.1 or.8, gives better forecasts? Week Sales Week Sales Week Sales Week Sales 1 110 6 120 1 110 6 120 2 115 7 130 2 115 7 130 3 125 8 115 3 125 8 115 4 120 9 110 4 120 9 110 5 125 10 130 5 125 10 130 Example: Rosco Drugs (B)

65. 65 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B) n Exponential Smoothing To evaluate the two smoothing constants, determine how the forecasted values would compare with the actual historical values in each case. Let: Y t = actual sales in week t F t = forecasted sales in week t F 1 = Y 1 = 110 F 1 = Y 1 = 110 For other weeks, F t +1 =.1 Y t +.9 F t

66. 66 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B) Exponential Smoothing (  =.1, 1 -  =.9) Exponential Smoothing (  =.1, 1 -  =.9) F 1 = 110 F 2 =.1 Y 1 +.9 F 1 =.1(110) +.9(110) = 110 F 3 =.1 Y 2 +.9 F 2 =.1(115) +.9(110) = 110.5 F 4 =.1 Y 3 +.9 F 3 =.1(125) +.9(110.5) = 111.95 F 5 =.1 Y 4 +.9 F 4 =.1(120) +.9(111.95) = 112.76 F 6 =.1 Y 5 +.9 F 5 =.1(125) +.9(112.76) = 113.98 F 7 =.1 Y 6 +.9 F 6 =.1(120) +.9(113.98) = 114.58 F 8 =.1 Y 7 +.9 F 7 =.1(130) +.9(114.58) = 116.12 F 9 =.1 Y 8 +.9 F 8 =.1(115) +.9(116.12) = 116.01 F 10 =.1 Y 9 +.9 F 9 =.1(110) +.9(116.01) = 115.41

67. 67 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B) Exponential Smoothing (  =.8, 1 -  =.2) Exponential Smoothing (  =.8, 1 -  =.2) F 1 = 110 F 2 =.8(110) +.2(110) = 110 F 3 =.8(115) +.2(110) = 114 F 4 =.8(125) +.2(114) = 122.80 F 5 =.8(120) +.2(122.80) = 120.56 F 6 =.8(125) +.2(120.56) = 124.11 F 7 =.8(120) +.2(124.11) = 120.82 F 8 =.8(130) +.2(120.82) = 128.16 F 9 =.8(115) +.2(128.16) = 117.63 F 10 =.8(110) +.2(117.63) = 111.53

68. 68 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B) n Mean Squared Error In order to determine which smoothing constant gives the better performance, calculate, for each, the mean squared error for the nine weeks of forecasts, weeks 2 through 10 by: [( Y 2 - F 2 ) 2 + ( Y 3 - F 3 ) 2 + ( Y 4 - F 4 ) 2 +... + ( Y 10 - F 10 ) 2 ]/9

69. 69 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B)  =.1  =.8  =.1  =.8 Week Y t F t ( Y t - F t ) 2 F t ( Y t - F t ) 2 Week Y t F t ( Y t - F t ) 2 F t ( Y t - F t ) 2 1 110 1 110 2 115 110.00 25.00 110.00 25.00 2 115 110.00 25.00 110.00 25.00 3 125 110.50 210.25 114.00 121.00 3 125 110.50 210.25 114.00 121.00 4 120 111.95 64.80 122.80 7.84 4 120 111.95 64.80 122.80 7.84 5 125 112.76 149.94 120.56 19.71 5 125 112.76 149.94 120.56 19.71 6 120 113.98 36.25 124.11 16.91 6 120 113.98 36.25 124.11 16.91 7 130 114.58 237.73 120.82 84.23 7 130 114.58 237.73 120.82 84.23 8 115 116.12 1.26 128.16 173.30 8 115 116.12 1.26 128.16 173.30 9 110 116.01 36.12 117.63 58.26 9 110 116.01 36.12 117.63 58.26 10 130 115.41 212.87 111.53 341.27 10 130 115.41 212.87 111.53 341.27 Sum 974.22 Sum 847.52 Sum 974.22 Sum 847.52 MSE Sum/9 Sum/9 MSE Sum/9 Sum/9 108.25108.2594.1794.17

70. 70 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B) n Excel Spreadsheet Showing Input Data

71. 71 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B) n Steps to Exponential Smoothing Using Excel Step 1: Select the Tools pull-down menu. Step 2: Select the Data Analysis option. Step 3: When the Data Analysis Tools dialog appears, choose Exponential Smoothing. Step 4: When the Exponential Smoothing dialog box appears: Enter B4:B13 in the Input Range box. Enter 0.9 (for  = 0.1) in Damping Factor box. Enter C4 in the Output Range box. Select OK.

72. 72 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B) Spreadsheet Showing Results Using  = 0.1 Spreadsheet Showing Results Using  = 0.1

73. 73 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B) Repeating the Process for  = 0.8 Repeating the Process for  = 0.8 Step 4: When the Exponential Smoothing dialog box appears: Step 4: When the Exponential Smoothing dialog box appears: Enter B4:B13 in the Input Range box. Enter 0.2 (for  = 0.8) in Damping Factor box. Enter D4 in the Output Range box. Select OK.

74. 74 Slide © 2005 Thomson/South-Western Example: Rosco Drugs (B) Spreadsheet Results for  = 0.1 and  = 0.8 Spreadsheet Results for  = 0.1 and  = 0.8

75. 75 Slide © 2005 Thomson/South-Western The number of plumbing repair jobs performed by Auger's Plumbing Service in each of the last nine months is listed on the next slide. Forecast the number of repair jobs Auger's will perform in December using the least squares method. Example: Auger’s Plumbing Service

76. 76 Slide © 2005 Thomson/South-Western Month Jobs Month Jobs Month Jobs Month Jobs Month Jobs Month Jobs March 353 June 374 September 399 March 353 June 374 September 399 April 387 July 396 October 412 April 387 July 396 October 412 May 342 August 409 November 408 May 342 August 409 November 408 Example: Auger’s Plumbing Service

77. 77 Slide © 2005 Thomson/South-Western Example: Auger’s Plumbing Service n Trend Projection (month) t Y t tY t t 2 (month) t Y t tY t t 2 (Mar.) 1 353 353 1 (Apr.) 2 387 774 4 (Apr.) 2 387 774 4 (May) 3 342 1026 9 (May) 3 342 1026 9 (June) 4 374 1496 16 (June) 4 374 1496 16 (July) 5 396 1980 25 (July) 5 396 1980 25 (Aug.) 6 409 2454 36 (Aug.) 6 409 2454 36 (Sep.) 7 399 2793 49 (Sep.) 7 399 2793 49 (Oct.) 8 412 3296 64 (Oct.) 8 412 3296 64 (Nov.) 9 408 3672 81 (Nov.) 9 408 3672 81 Sum 45 3480 17844 285

78. 78 Slide © 2005 Thomson/South-Western Example: Auger’s Plumbing Service n Trend Projection (continued) = 45/9 = 5 = 3480/9 = 386.667 = 45/9 = 5 = 3480/9 = 386.667 n  tY t -  t  Y t (9)(17844) - (45)(3480) n  tY t -  t  Y t (9)(17844) - (45)(3480) b 1 = = = 7.4 b 1 = = = 7.4 n  t 2 - (  t ) 2 (9)(285) - (45) 2 n  t 2 - (  t ) 2 (9)(285) - (45) 2 = 386.667 - 7.4(5) = 349.667 = 386.667 - 7.4(5) = 349.667 T 10 = 349.667 + (7.4)(10) = T 10 = 349.667 + (7.4)(10) = 423.667423.667

79. 79 Slide © 2005 Thomson/South-Western Example: Auger’s Plumbing Service n Excel Spreadsheet Showing Input Data

80. 80 Slide © 2005 Thomson/South-Western Example: Auger’s Plumbing Service n Steps to Trend Projection Using Excel Step 1: Select an empty cell (B13) in the worksheet. Step 2: Select the Insert pull-down menu. Step 3: Choose the Function option. Step 4: When the Paste Function dialog box appears: Choose Statistical in Function Category box. Choose Forecast in the Function Name box. Select OK. more.......

81. 81 Slide © 2005 Thomson/South-Western Example: Auger’s Plumbing Service n Steps to Trend Projecting Using Excel (continued) Step 5: When the Forecast dialog box appears: Enter 10 in the x box (for month 10). Enter B4:B12 in the Known y’s box. Enter A4:A12 in the Known x’s box. Select OK.

82. 82 Slide © 2005 Thomson/South-Western Example: Auger’s Plumbing Service n Spreadsheet Showing Trend Projection for Month 10

83. 83 Slide © 2005 Thomson/South-Western Example: Auger’s Plumbing Service (B) Forecast for December (Month 10) using a three-period ( n = 3) weighted moving average with weights of.6,.3, and.1. Then, compare this Month 10 weighted moving average forecast with the Month 10 trend projection forecast.

84. 84 Slide © 2005 Thomson/South-Western Example: Auger’s Plumbing Service (B) n Three-Month Weighted Moving Average The forecast for December will be the weighted average of the preceding three months: September, October, and November. F 10 =.1 Y Sep. +.3 Y Oct. +.6 Y Nov. F 10 =.1 Y Sep. +.3 Y Oct. +.6 Y Nov. =.1(399) +.3(412) +.6(408) =.1(399) +.3(412) +.6(408) = = n Trend Projection F 10 = 423.7 (from earlier slide) F 10 = 423.7 (from earlier slide) 408.3408.3

85. 85 Slide © 2005 Thomson/South-Western Example: Auger’s Plumbing Service (B) n Conclusion Due to the positive trend component in the time series, the trend projection produced a forecast that is more in tune with the trend that exists. The weighted moving average, even with heavy (.6) placed on the current period, produced a forecast that is lagging behind the changing data.

86. 86 Slide © 2005 Thomson/South-Western Forecasting with Trend and Seasonal Components n Steps of Multiplicative Time Series Model 1. Calculate the centered moving averages (CMAs). 2. Center the CMAs on integer-valued periods. 3. Determine the seasonal and irregular factors ( S t I t ). 4. Determine the average seasonal factors. 5. Scale the seasonal factors ( S t ). 6. Determine the deseasonalized data. 7. Determine a trend line of the deseasonalized data. 8. Determine the deseasonalized predictions. 9. Take into account the seasonality.

87. 87 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop Business at Terry's Tie Shop can be viewed as falling into three distinct seasons: (1) Christmas (November-December); (2) Father's Day (late May - mid-June); and (3) all other times. Average weekly sales ($) during each of the three seasons during the past four years are shown on the next slide. Determine a forecast for the average weekly sales in year 5 for each of the three seasons.

88. 88 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop n Past Sales ($) Year Season 1 2 3 4 Season 1 2 3 4 1 1856 1995 2241 2280 1 1856 1995 2241 2280 2 2012 2168 2306 2408 2 2012 2168 2306 2408 3 985 1072 1105 1120 3 985 1072 1105 1120

89. 89 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop Dollar Moving Scaled Dollar Moving Scaled Year Season Sales ( Y t ) Average S t I t S t Y t / S t Year Season Sales ( Y t ) Average S t I t S t Y t / S t 1 1 1856 1.178 1576 1 1 1856 1.178 1576 2 2012 1617.67 1.244 1.236 1628 2 2012 1617.67 1.244 1.236 1628 3 985 1664.00.592.586 1681 3 985 1664.00.592.586 1681 2 1 1995 1716.00 1.163 1.178 1694 2 1 1995 1716.00 1.163 1.178 1694 2 2168 1745.00 1.242 1.236 1754 2 2168 1745.00 1.242 1.236 1754 3 1072 1827.00.587.586 1829 3 1072 1827.00.587.586 1829 3 1 2241 1873.00 1.196 1.178 1902 3 1 2241 1873.00 1.196 1.178 1902 2 2306 1884.00 1.224 1.236 1866 2 2306 1884.00 1.224 1.236 1866 3 1105 1897.00.582.586 1886 3 1105 1897.00.582.586 1886 4 1 2280 1931.00 1.181 1.178 1935 4 1 2280 1931.00 1.181 1.178 1935 2 2408 1936.00 1.244 1.236 1948 2 2408 1936.00 1.244 1.236 1948 3 1120.586 1911 3 1120.586 1911

90. 90 Slide © 2005 Thomson/South-Western n 1. Calculate the centered moving averages. There are three distinct seasons in each year. Hence, take a three-season moving average to eliminate seasonal and irregular factors. For example: 1 st MA = (1856 + 2012 + 985)/3 = 1617.67 1 st MA = (1856 + 2012 + 985)/3 = 1617.67 2 nd MA = (2012 + 985 + 1995)/3 = 1664.00 2 nd MA = (2012 + 985 + 1995)/3 = 1664.00etc. Example: Terry’s Tie Shop

91. 91 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop n 2. Center the CMAs on integer-valued periods. The first moving average computed in step 1 (1617.67) will be centered on season 2 of year 1. Note that the moving averages from step 1 center themselves on integer-valued periods because n is an odd number.

92. 92 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop n 3. Determine the seasonal & irregular factors ( S t I t ). Isolate the trend and cyclical components. For each period t, this is given by: S t I t = Y t /(Moving Average for period t ) S t I t = Y t /(Moving Average for period t )

93. 93 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop n 4. Determine the average seasonal factors. Averaging all S t I t values corresponding to that season: Season 1: (1.163 + 1.196 + 1.181) /3 = 1.180 Season 1: (1.163 + 1.196 + 1.181) /3 = 1.180 Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = 1.238 Season 2: (1.244 + 1.242 + 1.224 + 1.244) /4 = 1.238 Season 3: (.592 +.587 +.582) /3 =.587 Season 3: (.592 +.587 +.582) /3 =.587

94. 94 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop n 5. Scale the seasonal factors ( S t ). Average the seasonal factors = (1.180 + 1.238 +.587)/3 = 1.002. Then, divide each seasonal factor by the average of the seasonal factors. Average the seasonal factors = (1.180 + 1.238 +.587)/3 = 1.002. Then, divide each seasonal factor by the average of the seasonal factors. Season 1: 1.180/1.002 = 1.178 Season 1: 1.180/1.002 = 1.178 Season 2: 1.238/1.002 = 1.236 Season 2: 1.238/1.002 = 1.236 Season 3:.587/1.002 =.586 Season 3:.587/1.002 =.586 Total = 3.000 Total = 3.000

95. 95 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop n 6. Determine the deseasonalized data. Divide the data point values, Y t, by S t. n 7. Determine a trend line of the deseasonalized data. Using the least squares method for t = 1, 2,..., 12, gives: Using the least squares method for t = 1, 2,..., 12, gives: T t = 1580.11 + 33.96 t T t = 1580.11 + 33.96 t

96. 96 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop n 8. Determine the deseasonalized predictions. Substitute t = 13, 14, and 15 into the least squares equation: T 13 = 1580.11 + (33.96)(13) = 2022 T 13 = 1580.11 + (33.96)(13) = 2022 T 14 = 1580.11 + (33.96)(14) = 2056 T 14 = 1580.11 + (33.96)(14) = 2056 T 15 = 1580.11 + (33.96)(15) = 2090 T 15 = 1580.11 + (33.96)(15) = 2090

97. 97 Slide © 2005 Thomson/South-Western Example: Terry’s Tie Shop n 9. Take into account the seasonality. Multiply each deseasonalized prediction by its seasonal factor to give the following forecasts for year 5: Season 1: (1.178)(2022) = Season 1: (1.178)(2022) = Season 2: (1.236)(2056) = Season 2: (1.236)(2056) = Season 3: (.586)(2090) = Season 3: (.586)(2090) = 23822382 25412541 12251225

98. 98 Slide © 2005 Thomson/South-Western Qualitative Approaches to Forecasting n Delphi Approach A panel of experts, each of whom is physically separated from the others and is anonymous, is asked to respond to a sequential series of questionnaires. A panel of experts, each of whom is physically separated from the others and is anonymous, is asked to respond to a sequential series of questionnaires. After each questionnaire, the responses are tabulated and the information and opinions of the entire group are made known to each of the other panel members so that they may revise their previous forecast response. After each questionnaire, the responses are tabulated and the information and opinions of the entire group are made known to each of the other panel members so that they may revise their previous forecast response. The process continues until some degree of consensus is achieved. The process continues until some degree of consensus is achieved.

99. 99 Slide © 2005 Thomson/South-Western Qualitative Approaches to Forecasting n Scenario Writing Scenario writing consists of developing a conceptual scenario of the future based on a well defined set of assumptions. Scenario writing consists of developing a conceptual scenario of the future based on a well defined set of assumptions. After several different scenarios have been developed, the decision maker determines which is most likely to occur in the future and makes decisions accordingly. After several different scenarios have been developed, the decision maker determines which is most likely to occur in the future and makes decisions accordingly.

100. 100 Slide © 2005 Thomson/South-Western Qualitative Approaches to Forecasting n Subjective or Interactive Approaches These techniques are often used by committees or panels seeking to develop new ideas or solve complex problems. These techniques are often used by committees or panels seeking to develop new ideas or solve complex problems. They often involve "brainstorming sessions". They often involve "brainstorming sessions". It is important in such sessions that any ideas or opinions be permitted to be presented without regard to its relevancy and without fear of criticism. It is important in such sessions that any ideas or opinions be permitted to be presented without regard to its relevancy and without fear of criticism.

101. 101 Slide © 2005 Thomson/South-Western End of Chapter 16