1. Lecture 18 Forecasting (Continued) Books Introduction to Materials Management, Sixth Edition, J. R. Tony Arnold, P.E., CFPIM, CIRM, Fleming College, Emeritus, Stephen N. Chapman, Ph.D., CFPIM, North Carolina State University, Lloyd M. Clive, P.E., CFPIM, Fleming College Operations Management for Competitive Advantage, 11th Edition, by Chase, Jacobs, and Aquilano, 2005, N.Y.: McGraw-Hill/Irwin. Operations Management, 11/E, Jay Heizer, Texas Lutheran University, Barry Render, Graduate School of Business, Rollins College, Prentice Hall

2. Objectives When you complete this chapter you should be able to :  Compute three measures of forecast accuracy  Develop seasonal indexes  Conduct a regression and correlation analysis  Use a tracking signal

3. Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑ |Actual - Forecast| n Mean Squared Error (MSE) MSE = ∑ (Forecast Errors) 2 n

4. Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = ∑ 100|Actual i - Forecast i |/Actual i n n i = 1

5. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62

6. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD = ∑ |deviations| n = 82.45/8 = 10.31 For  =.10 = 98.62/8 = 12.33 For  =.50

7. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD10.3112.33 = 1,526.54/8 = 190.82 For  =.10 = 1,561.91/8 = 195.24 For  =.50 MSE = ∑ (forecast errors) 2 n

8. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD10.3112.33 MSE190.82195.24 = 44.75/8 = 5.59% For  =.10 = 54.05/8 = 6.76% For  =.50 MAPE = ∑ 100|deviation i |/actual i n i = 1

9. Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD10.3112.33 MSE190.82195.24 MAPE5.59%6.76%

10. Exponential Smoothing with Trend Adjustment When a trend is present, exponential smoothing must be modified Forecast including (FIT t ) = trend ExponentiallyExponentially smoothed (F t ) +(T t )smoothed forecasttrend

11. 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 Step 1: Compute F t Step 2: Compute T t Step 3: Calculate the forecast FIT t = F t + T t

12. Exponential Smoothing with Trend Adjustment ExampleForecast 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

13. Exponential Smoothing with Trend Adjustment ExampleForecast 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 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

14. Exponential Smoothing with Trend Adjustment ExampleForecast 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 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

15. Exponential Smoothing with Trend Adjustment ExampleForecast 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 FIT 2 = F 2 + T 1 FIT 2 = 12.8 + 1.92 = 14.72 units Step 3: Calculate FIT for Month 2

16. Exponential Smoothing with Trend Adjustment ExampleForecast 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 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

17. Exponential Smoothing with Trend Adjustment Example |||||||||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 ) with  =.2 and  =.4

18. 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 ^

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

20. Least Squares Method Time period Values of Dependent Variable 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)

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

22. 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 2001174174 20022794158 20033809240 200449016360 2005510525525 2005614236852 2007712249854 ∑x = 28∑y = 692∑x 2 = 140∑xy = 3,063 x = 4y = 98.86

23. 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 ^

24. Least Squares Example ||||||||| 200120022003200420052006200720082009 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 ^

25. 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

26. Seasonal Variations In Data The multiplicative seasonal model can adjust trend data for seasonal variations in demand

27. Seasonal Variations In Data 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 Steps in the process:

28. Seasonal Index Example Jan80851059094 Feb7085858094 Mar8093828594 Apr909511510094 May11312513112394 Jun11011512011594 Jul10010211310594 Aug8810211010094 Sept8590959094 Oct7778858094 Nov7572838094 Dec8278808094 DemandAverageAverage Seasonal Month2005200620072005-2007MonthlyIndex

29. Seasonal Index Example Jan80851059094 Feb7085858094 Mar8093828594 Apr909511510094 May11312513112394 Jun11011512011594 Jul10010211310594 Aug8810211010094 Sept8590959094 Oct7778858094 Nov7572838094 Dec8278808094 DemandAverageAverage Seasonal Month2005200620072005-2007MonthlyIndex 0.957 Seasonal index = average 2005-2007 monthly demand average monthly demand = 90/94 =.957

30. 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 Month2005200620072005-2007MonthlyIndex

31. 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 Month2005200620072005-2007MonthlyIndex Expected annual demand = 1,200 Janx.957 = 96 1,200 12 Febx.851 = 85 1,200 12 Forecast for 2008

32. Seasonal Index Example 140 140 – 130 130 – 120 120 – 110 110 – 100 100 – 90 90 – 80 80 – 70 70 – ||||||||||||JFMAMJJASONDJFMAMJJASOND||||||||||||JFMAMJJASONDJFMAMJJASOND Time Demand 2008 Forecast 2007 Demand 2006 Demand 2005 Demand

33. 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 972497459766 Trend Data

34. 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 Seasonal Indices

35. 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 9911 9265 9764 9520 9691 9411 9949 9724 9542 9355100689572 Combined Trend and Seasonal Forecast

36. 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

37. 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 ^

38. Associative Forecasting Example SalesLocal Payroll ($ millions), y($ billions), 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

39. 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

40. 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 $6 billion, then: Sales = 1.75 +.25(6) Sales = $3,250,000 3.25

41. 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 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Sales Area payroll 3.25

42. 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

43. 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

44. 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 $306,000 in sales

45.  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

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

47. 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

48.  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 For the Nodel Construction example: r =.901 r 2 =.81

49. 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

50. Multiple Regression Analysis y = 1.80 +.30x 1 - 5.0x 2 ^ In the Nodel example, 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 = $3,000,000

51.  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

52. Monitoring and Controlling Forecasts Tracking signal RSFEMAD= = ∑(Actual demand in period i - Forecast demand in period i)  ∑|Actual - Forecast|/n)

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

54. Tracking Signal ExampleCumulative 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

55. 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

56. 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

57. Focus Forecasting Developed at American Hardware Supply, focus forecasting is based on two principles: 1.Sophisticated forecasting models are not always better than simple ones 2.There is no single technique 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

58. 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

59. 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

60. FedEx Call Center Forecast 12% 12% – 10% 10% – 8% 8% – 6% 6% – 4% 4% – 2% 2% – 0% 0% – Hour of day A.M.P.M. 2468101224681012

61. Measuring Forecast Errors Mean Absolute Deviation (MAD) The sum of the absolute value of the individual forecast errors divided by the number of periods

62. Bias Bias exists when the cumulative actual demand varies from the cumulative forecast.

63. Seasonal Index Seasonal Index = period average demand avg. demand for all periods

64. End of Lecture 18