1. Operations Management Chapter 4 Forecasting

2. So What is Forecasting?  Process of predicting a future event  Forecasting is used for all business decisions  Production  Inventory  Personnel  Facilities ??

3.  Short-range forecast  1-3 months  Purchasing, job scheduling, workforce levels, job assignments, production levels  Medium-range forecast  12 to 18 months  Sales and production planning, budgeting  Long-range forecast  5 – 10 years  New product planning, facility location, research and development Forecasting Time Horizons

4. Distinguishing Differences  Medium/long range forecasts deal with more comprehensive issues and support management decisions regarding planning and products, plants and processes  Short-term forecasting usually employs different methodologies than longer-term forecasting  Short-term forecasts tend to be more accurate than longer-term forecasts

5. Influence of Product Life Cycle  Introduction and growth require longer forecasts than maturity and decline  As product passes through life cycle, forecasts are useful in projecting  Staffing levels  Inventory levels  Factory capacity Introduction – Growth – Maturity – Decline

6. Product Life Cycle Product design and development critical Frequent product and process design changes Short production runs High production costs Limited models Attention to quality IntroductionGrowthMaturityDecline OM Strategy/Issues Forecasting critical Product and process reliability Competitive product improvements and options Increase capacity Shift toward product focus Enhance distribution Standardization Less rapid product changes – more minor changes Optimum capacity Increasing stability of process Long production runs Product improvement and cost cutting Little product differentiation Cost minimization Overcapacity in the industry Prune line to eliminate items not returning good margin Reduce capacity Figure 2.5

7. Types of Forecasts  Economic forecasts  Address business cycle – inflation rate, money supply, housing starts, etc.  Technological forecasts  Predict rate of technological progress  Impacts development of new products  Demand forecasts  Predict sales of existing product

8. Strategic Importance of Forecasting  Human Resources – Hiring, training, laying off workers  Capacity – Capacity shortages can result in undependable delivery, loss of customers, loss of market share  Supply-Chain Management – Good supplier relations and price advance

9. Seven Steps in Forecasting  Determine the use of the forecast  Select the items to be forecasted  Determine the time horizon of the forecast (short, medium or long-term)  Select the forecasting model(s)  Gather the data  Make the forecast  Validate and implement results

10. The Realities!  Forecasts are seldom perfect  Most techniques assume an underlying stability in the system  Product family and aggregated forecasts are more accurate than individual product forecasts

11. ForecastingMethodsForecastingMethods QuantitativeQuantitativeQualitativeQualitative CausalCausal Time Series Forecasting Approaches

12.  Used when situation is vague and little data exist  New products  New technology  Involves intuition, experience  e.g., forecasting sales on Internet Qualitative Methods

13. Forecasting Approaches  Used when situation is ‘stable’ and historical data exist  Existing products  Current technology  Involves mathematical techniques  e.g., forecasting sales of color televisions Quantitative Methods

14. Overview of Qualitative Methods  Jury of executive opinion  Delphi method  Sales force composite  Consumer Market Survey

15.  Involves small group of high-level managers  Group estimates demand by working together  Combines managerial experience with statistical models  Relatively quick  ‘Group-think’ disadvantage Jury of Executive Opinion

16. Delphi Method  Iterative group process, continues until consensus is reached  3 types of participants  Decision makers  Staff  Respondents Staff (Administering survey) Decision Makers (Evaluate responses and make decisions) Respondents (People who can make valuable judgments)

17. Sales Force Composite  Each salesperson projects his or her sales  Combined at district and national levels  Sales reps know customers’ wants  Tends to be overly optimistic

18. Consumer Market Survey  Ask customers about purchasing plans  What consumers say, and what they actually do are often different  Sometimes difficult to answer

19. Overview of Quantitative Approaches 1.Naive approach 2.Moving averages 3.Exponential smoothing 4.Trend projection 5.Linear regression Time-Series Models Assume the future is a function of the past Associative Model Using other factors

20.  Set of evenly spaced numerical data  Obtained by observing response variable at regular time periods  The pattern of the data is an important factor in understanding how the time series has behaved in the past.  Forecast based only on past values  Assumes that factors influencing past and present will continue influence in future Time Series Forecasting

21. Trend Seasonal Cyclical Random Time Series Patterns

22. Patterns of Demand Demand for product or service |||| 1234 Year Average demand over four years Seasonal peaks Trend component Actual demand Random variation Figure 4.1 Cyclical

23.  Persistent, overall upward or downward pattern  Trends are usually the result of long term factors such as changes due to population, technology, age, culture, etc.  A trend pattern can be identified by analyzing multiyear movements in historical data. Trend Pattern

24.  Seasonal patterns are recognized by seeing the same repeating pattern of highs and lows over successive periods of time within a year.  Due to weather, customs, etc. Seasonal Pattern Number of PeriodLengthSeasons WeekDay7 MonthWeek4-4.5 MonthDay28-31 YearQuarter4 YearMonth12 YearWeek52

25.  Repeating up and down movements  Affected by business cycle, political, and economic factors  Multiple years duration  Often causal or associative relationships Cyclical Pattern 05101520

26.  Irregular, unsystematic, ‘residual’ fluctuations  Due to random variation or unforeseen events  Short duration and nonrepeating Random Pattern MTWTFMTWTFMTWTFMTWTF

27. Naive Approach  Assumes demand in next period is the same as demand in most recent period  e.g., If May sales were 48, then June sales will be 48  Sometimes cost effective and efficient

28.  MA is a series of arithmetic means  Used if little or no trend  Used often for smoothing  Provides overall impression of data over time Moving Average Method Moving average = ∑ demand in previous n periods n

29. January10 February12 March13 April16 May19 June23 July26 Actual3-Month MonthShed SalesMoving Average (12 + 13 + 16)/3 = 13 2 / 3 (13 + 16 + 19)/3 = 16 (16 + 19 + 23)/3 = 19 1 / 3 Moving Average Example 101213 (10 + 12 + 13)/3 = 11 2 / 3

30. Graph of Moving Average ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Shed Sales 30 30 – 28 28 – 26 26 – 24 24 – 22 22 – 20 20 – 18 18 – 16 16 – 14 14 – 12 12 – 10 10 – Actual Sales Moving Average Forecast

31.  Used when trend is present  Older data usually less important  Weights based on experience and intuition Weighted Moving Average Weighted moving average = ∑ (weight for period n) x (demand in period n) ∑ weights

32. January10 February12 March13 April16 May19 June23 July26 Actual3-Month Weighted MonthShed SalesMoving Average [(3 x 16) + (2 x 13) + (12)]/6 = 14 1 / 3 [(3 x 19) + (2 x 16) + (13)]/6 = 17 [(3 x 23) + (2 x 19) + (16)]/6 = 20 1 / 2 Weighted Moving Average 101213 [(3 x 13) + (2 x 12) + (10)]/6 = 12 1 / 6 Weights AppliedPeriod 3Last month 2Two months ago 1Three months ago 6Sum of weights Older data usually less important

33.  Increasing (n) smooth the forecast but makes it less sensitive to changes  Do not forecast trends well  Require extensive historical data Potential Problems With Moving Average

34. Moving Average And Weighted Moving Average 30 30 – 25 25 – 20 20 – 15 15 – 10 10 – 5 5 – Sales demand ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Actual sales Moving average Weighted moving average Figure 4.2

35.  Form of weighted moving average  Weights decline exponentially  Most recent data weighted most  Requires smoothing constant (  )  Ranges from 0 to 1  Subjectively chosen  Involves little record keeping of past data Exponential Smoothing

36. New forecast =last period’s forecast +  (last period’s actual demand – last period’s forecast) F t = F t – 1 +  (A t – 1 - F t – 1 ) whereF t =new forecast F t – 1 =previous forecast  =smoothing (or weighting) constant (0    1)

37. Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20

38. Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= 142 +.2(153 – 142)

39. Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= 142 +.2(153 – 142) = 142 + 2.2 = 144.2 ≈ 144 cars

40. Impact of Different  225 225 – 200 200 – 175 175 – 150 150 – |||||||||123456789123456789|||||||||123456789123456789 Quarter Demand  =.1 Actual demand  =.5

41. Choosing  The objective is to obtain the most accurate forecast no matter the technique We generally do this by selecting the model that gives us the lowest forecast error Forecast error= Actual demand - Forecast value = A t - F t

42. Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑ |actual - forecast| n Mean Squared Error (MSE) MSE = ∑ (forecast errors) 2 n

43. Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = 100 ∑ |actual i - forecast i |/actual i n n i = 1 The MAD and MSE values depend on the magnitude of the item being forecasted. If it is in Thousands the MAD and MSE will be very large. This problem can be avoided by using the MAPE

44. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 118017551755 2168176817810 31591751617314 417517321669 51901731717020 62051753018025 7180178219313 818217841864 84100

45. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 118017551755 2168176817810 31591751617314 417517321669 51901731717020 62051753018025 7180178219313 818217841864 84100 MAD = ∑ |deviations| n = 84/8 = 10.50 For  =.10 = 100/8 = 12.50 For  =.50

46. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 118017551755 2168176817810 31591751617314 417517321669 51901731717020 62051753018025 7180178219313 818217841864 84100 MAD10.5012.50 = 1,558/8 = 194.75 For  =.10 = 1,612/8 = 201.50 For  =.50 MSE = ∑ (forecast errors) 2 n

47. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 118017551755 2168176817810 31591751617314 417517321669 51901731717020 62051753018025 7180178219313 818217841864 84100 MAD10.5012.50 MSE194.75201.50 = 45.62/8 = 5.70% For  =.10 = 54.8/8 = 6.85% For  =.50 MAPE = 100 ∑ |deviation i |/actual i n i = 1

48. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 118017551755 2168176817810 31591751617314 417517321669 51901731717020 62051753018025 7180178219313 818217841864 84100 MAD10.5012.50 MSE194.75201.50 MAPE5.70%6.85%

49. Exponential Smoothing with Trend Adjustment Forecast including (FIT t ) = trend exponentiallyexponentially smoothed (F t ) +(T t )smoothed forecasttrend Exponential smoothing is a popular modeling approach in businessExponential smoothing is a popular modeling approach in business When a trend is present, exponential smoothing must be modified because of the lag in predictionWhen a trend is present, exponential smoothing must be modified because of the lag in prediction Taking trends into consideration the formula will beTaking trends into consideration the formula will be

50. Exponential Smoothing with Trend Adjustment F t =  (A t - 1 ) + (1 -  )(F t - 1 + T t - 1 ) T t =  (F t - F t - 1 ) + (1 -  )T t - 1 Where Ft = exponentially smoothed forecast for period t Tt = exponentially smoothed trend in period t Tt = exponentially smoothed trend in period t At = actual demand in period t At = actual demand in period t α = smoothing constant for the average (0≤α≤1) α = smoothing constant for the average (0≤α≤1) β = smoothing constant for the trend (0≤β≤1) β = smoothing constant for the trend (0≤β≤1)

51. Exponential Smoothing with Trend Adjustment Steps to compute a trend-adjusted forecast:Steps to compute a trend-adjusted forecast: –Step 1: Compute Ft –Step 2: Compute Tt –Step 3: Calculate the forecast FITt = Ft + Tt

52. Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 217 320 419 524 621 731 828 936 10 Table 4.1

53. Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 217 320 419 524 621 731 828 936 10 Table 4.1 F 2 =  A 1 + (1 -  )(F 1 + T 1 ) F 2 = (.2)(12) + (1 -.2)(11 + 2) = 2.4 + 10.4 = 12.8 units Step 1: Forecast for Month 2

54. Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 21712.80 320 419 524 621 731 828 936 10 Table 4.1 T 2 =  (F 2 - F 1 ) + (1 -  )T 1 T 2 = (.4)(12.8 - 11) + (1 -.4)(2) =.72 + 1.2 = 1.92 units Step 2: Trend for Month 2

55. Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 21712.801.92 320 419 524 621 731 828 936 10 Table 4.1 FIT 2 = F 2 + T 2 FIT 2 = 12.8 + 1.92 = 14.72 units Step 3: Calculate FIT for Month 2

56. Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 21712.801.9214.72 320 419 524 621 731 828 936 10 Table 4.1 15.182.1017.28 17.822.3220.14 19.912.2322.14 22.512.3824.89 24.112.0726.18 27.142.4529.59 29.282.3231.60 32.482.6835.16

57. Exponential Smoothing with Trend Adjustment Example Figure 4.3 |||||||||123456789123456789|||||||||123456789123456789 Time (month) Product demand 35 35 – 30 30 – 25 25 – 20 20 – 15 15 – 10 10 – 5 5 – 0 0 – Actual demand (A t ) Forecast including trend (FIT t )

58. Trend Projections Fitting a trend line to historical data points to project into the medium-to-long-range Linear trends can be found using the least squares technique y = a + bx ^ where y= computed value of the variable to be predicted (dependent variable) a= y-axis intercept b= slope of the regression line x= the independent variable ^

59. Least Squares Method Time period Values of Dependent Variable Figure 4.4 Deviation 1 Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y value) Trend line, y = a + bx ^

60. Least Squares Method Time period Values of Dependent Variable Figure 4.4 Deviation 1 Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y value) Trend line, y = a + bx ^ Least squares method minimizes the sum of the squared errors (deviations)

61. Least Squares Method Equations to calculate the regression variables b =  xy - nxy  x 2 - nx 2 y = a + bx ^ a = y - bx

62. Least Squares Example b = = = 10.54 ∑xy - nxy ∑x 2 - nx 2 3,063 - (7)(4)(98.86) 140 - (7)(4 2 ) a = y - bx = 98.86 - 10.54(4) = 56.70 TimeElectrical Power YearPeriod (x)Demandx 2 xy 1999174174 20002794158 20013809240 200249016360 2003510525525 2004614236852 2005712249854 ∑x = 28∑y = 692∑x 2 = 140∑xy = 3,063 x = 4y = 98.86

63. Least Squares Example b = = = 10.54  xy - nxy  x 2 - nx 2 3,063 - (7)(4)(98.86) 140 - (7)(4 2 ) a = y - bx = 98.86 - 10.54(4) = 56.70 TimeElectrical Power YearPeriod (x)Demandx 2 xy 1999174174 20002794158 20013809240 200249016360 2003510525525 2004614236852 2005712249854  x = 28  y = 692  x 2 = 140  xy = 3,063 x = 4y = 98.86 The trend line is y = 56.70 + 10.54x ^

64. Least Squares Example ||||||||| 199920002001200220032004200520062007 160 160 – 150 150 – 140 140 – 130 130 – 120 120 – 110 110 – 100 100 – 90 90 – 80 80 – 70 70 – 60 60 – 50 50 – Year Power demand Trend line, y = 56.70 + 10.54x ^

65. Least Squares Requirements 1.We always plot the data to insure a linear relationship 2.We do not predict time periods far beyond the database 3.Deviations around the least squares line are assumed to be random

66. Seasonal Variations In Data The multiplicative seasonal model can modify trend data to accommodate seasonal variations in demand 1.Find average historical demand for each season 2.Compute the average demand over all seasons 3.Compute a seasonal index for each season 4.Estimate next year’s total demand 5.Divide this estimate of total demand by the number of seasons, then multiply it by the seasonal index for that season

67. Seasonal Index Example Jan80851059094 Feb7085858094 Mar8093828594 Apr909511510094 May11312513112394 Jun11011512011594 Jul10010211310594 Aug8810211010094 Sept8590959094 Oct7778858094 Nov7572838094 Dec8278808094 DemandAverageAverage Seasonal Month2003200420052003-2005MonthlyIndex

68. Seasonal Index Example Jan80851059094 Feb7085858094 Mar8093828594 Apr909511510094 May11312513112394 Jun11011512011594 Jul10010211310594 Aug8810211010094 Sept8590959094 Oct7778858094 Nov7572838094 Dec8278808094 DemandAverageAverage Seasonal Month2003200420052003-2005MonthlyIndex 0.957 Seasonal index = average 2003-2005 monthly demand average monthly demand = 90/94 =.957

69. Seasonal Index Example Jan808510590940.957 Feb70858580940.851 Mar80938285940.904 Apr9095115100941.064 May113125131123941.309 Jun110115120115941.223 Jul100102113105941.117 Aug88102110100941.064 Sept85909590940.957 Oct77788580940.851 Nov75728380940.851 Dec82788080940.851 DemandAverageAverage Seasonal Month2003200420052003-2005MonthlyIndex

70. Seasonal Index Example Jan808510590940.957 Feb70858580940.851 Mar80938285940.904 Apr9095115100941.064 May113125131123941.309 Jun110115120115941.223 Jul100102113105941.117 Aug88102110100941.064 Sept85909590940.957 Oct77788580940.851 Nov75728380940.851 Dec82788080940.851 DemandAverageAverage Seasonal Month2003200420052003-2005MonthlyIndex Expected annual demand = 1,200 Janx.957 = 96 1,200 12 Febx.851 = 85 1,200 12 Forecast for 2006

71. Seasonal Index Example 140 140 – 130 130 – 120 120 – 110 110 – 100 100 – 90 90 – 80 80 – 70 70 – ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Time Demand 2006 Forecast 2005 Demand 2004 Demand 2003 Demand

72. San Diego Hospital 10,200 10,200 – 10,000 10,000 – 9,800 9,800 – 9,600 9,600 – 9,400 9,400 – 9,200 9,200 – 9,000 9,000 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec 676869707172737475767778 Month Inpatient Days 9530 9551 9573 9594 9616 9637 9659 9680 9702 972397459766 Figure 4.6 Trend Data

73. San Diego Hospital 1.06 1.06 – 1.04 1.04 – 1.02 1.02 – 1.00 1.00 – 0.98 0.98 – 0.96 0.96 – 0.94 0.94 – 0.92 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec 676869707172737475767778 Month Index for Inpatient Days 1.04 1.02 1.01 0.99 1.031.041.00 0.98 0.97 0.99 0.97 0.96 Figure 4.7 Seasonal Indices

74. San Diego Hospital 10,200 10,200 – 10,000 10,000 – 9,800 9,800 – 9,600 9,600 – 9,400 9,400 – 9,200 9,200 – 9,000 9,000 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec 676869707172737475767778 Month Inpatient Days Figure 4.8 9911 9265 9764 9520 9691 9411 9949 9724 9542 9355100689572 Combined Trend and Seasonal Forecast

75. Associative Forecasting Used when changes in one or more independent variables can be used to predict the changes in the dependent variable Most common technique is linear regression analysis We apply this technique just as we did in the time series example

76. Associative Forecasting Forecasting an outcome based on predictor variables using the least squares technique y = a + bx ^ where y= computed value of the variable to be predicted (dependent variable) a= y-axis intercept b= slope of the regression line x= the independent variable though to predict the value of the dependent variable ^

77. Associative Forecasting Example SalesLocal Payroll ($000,000), y($000,000,000), x 2.01 3.03 2.54 2.02 2.01 3.57 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Sales Area payroll

78. Associative Forecasting Example Sales, y Payroll, xx 2 xy 2.0112.0 3.0399.0 2.541610.0 2.0244.0 2.0112.0 3.574924.5 ∑y = 15.0∑x = 18∑x 2 = 80∑xy = 51.5 x = ∑x/6 = 18/6 = 3 y = ∑y/6 = 15/6 = 2.5 b = = =.25 ∑xy - nxy ∑x 2 - nx 2 51.5 - (6)(3)(2.5) 80 - (6)(3 2 ) a = y - bx = 2.5 - (.25)(3) = 1.75

79. Associative Forecasting Example 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Sales Area payroll y = 1.75 +.25x ^ Sales = 1.75 +.25(payroll) If payroll next year is estimated to be $600 million, then: Sales = 1.75 +.25(6) Sales = $325,000 3.25

80. Standard Error of the Estimate  A forecast is just a point estimate of a future value  This point is actually the mean of a probability distribution Figure 4.9 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Sales Area payroll 3.25

81. Standard Error of the Estimate wherey=y-value of each data point y c =computed value of the dependent variable, from the regression equation n=number of data points S y,x = ∑(y - y c ) 2 n - 2

82. Standard Error of the Estimate Computationally, this equation is considerably easier to use We use the standard error to set up prediction intervals around the point estimate S y,x = ∑y 2 - a∑y - b∑xy n - 2

83. Standard Error of the Estimate 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Sales Area payroll 3.25 S y,x = = ∑y 2 - a∑y - b∑xy n - 2 39.5 - 1.75(15) -.25(51.5) 6 - 2 S y,x =.306 The standard error of the estimate is $30,600 in sales

84.  How strong is the linear relationship between the variables?  Correlation does not necessarily imply causality!  Coefficient of correlation, r, measures degree of association  Values range from -1 to +1 Correlation

85. Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ]

86. Correlation Coefficient r = n∑xy - ∑x∑y [n∑x 2 - (∑x) 2 ][n∑y 2 - (∑y) 2 ] y x (a)Perfect positive correlation: r = +1 y x (b)Positive correlation: 0 < r < 1 y x (c)No correlation: r = 0 y x (d)Perfect negative correlation: r = -1

87.  Coefficient of Determination, r 2, measures the percent of change in y predicted by the change in x  Values range from 0 to 1  Easy to interpret Correlation

88. Multiple Regression Analysis If more than one independent variable is to be used in the model, linear regression can be extended to multiple regression to accommodate several independent variables y = a + b 1 x 1 + b 2 x 2 … ^ Computationally, this is quite complex and generally done on the computer

89. Multiple Regression Analysis y = 1.80 +.30x 1 - 5.0x 2 ^ Including interest rates in the model gives the new equation: An improved correlation coefficient of r =.96 means this model does a better job of predicting the change in construction sales Sales = 1.80 +.30(6) - 5.0(.12) = 3.00 Sales = $300,000

90.  Measures how well the forecast is predicting actual values  Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)  Good tracking signal has low values  If forecasts are continually high or low, the forecast has a bias error Monitoring and Controlling Forecasts Tracking Signal

91. Monitoring and Controlling Forecasts Tracking signal RSFEMAD= = ∑(actual demand in period i - forecast demand in period i)  ∑|actual - forecast|/n)

92. Tracking Signal Tracking signal + 0 MADs – Upper control limit Lower control limit Time Signal exceeding limit Acceptable range

93. Tracking Signal Example Cumulative AbsoluteAbsolute ActualForecastForecastForecast QtrDemandDemandErrorRSFEErrorErrorMAD 190100-10-10101010.0 295100-5-155157.5 3115100+150153010.0 4100110-10-10104010.0 5125110+15+5155511.0 6140110+30+35308514.2

94. Cumulative AbsoluteAbsolute ActualForecastForecastForecast QtrDemandDemandErrorRSFEErrorErrorMAD 190100-10-10101010.0 295100-5-155157.5 3115100+150153010.0 4100110-10-10104010.0 5125110+15+5155511.0 6140110+30+35308514.2 Tracking Signal Example Tracking Signal (RSFE/MAD) -10/10 = -1 -15/7.5 = -2 0/10 = 0 -10/10 = -1 +5/11 = +0.5 +35/14.2 = +2.5 The variation of the tracking signal between -2.0 and +2.5 is within acceptable limits

95. Adaptive Forecasting It’s possible to use the computer to continually monitor forecast error and adjust the values of the  and  coefficients used in exponential smoothing to continually minimize forecast error This technique is called adaptive smoothing

96. Focus Forecasting Developed at American Hardware Supply, focus forecasting is based on two principles: 1.Sophisticated forecasting models are not always better than simple models 2.There is no single techniques that should be used for all products or services This approach uses historical data to test multiple forecasting models for individual items The forecasting model with the lowest error is then used to forecast the next demand

97. Forecasting in the Service Sector  Presents unusual challenges  Special need for short term records  Needs differ greatly as function of industry and product  Holidays and other calendar events  Unusual events

98. Fast Food Restaurant Forecast 20% 20% – 15% 15% – 10% 10% – 5% 5% – 11-121-23-45-67-89-10 12-12-34-56-78-910-11 (Lunchtime)(Dinnertime) Hour of day Percentage of sales Figure 4.12