Understanding Forecasting Measures and Techniques
Explore common measures of error in forecasting including Mean Absolute Deviation (MAD), Mean Squared Error (MSE), and Mean Absolute Percent Error (MAPE). Learn how to compute forecast accuracy, develop seasonal indexes, conduct regression analysis, and utilize tracking signals. Compare forecast errors and study actual tonnage data to enhance your understanding of forecasting methods.
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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
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
Common Measures of Error Mean Absolute Deviation (MAD) |Actual - Forecast| n MAD = Mean Squared Error (MSE) (Forecast Errors)2 n MSE =
Common Measures of Error Mean Absolute Percent Error (MAPE) n 100|Actuali - Forecasti|/Actuali i = 1 MAPE = n
Comparison of Forecast Error Rounded Forecast with = .10 175 175.5 174.75 173.18 173.36 175.02 178.02 178.22 Absolute Deviation for = .10 5.00 7.50 15.75 1.82 16.64 29.98 1.98 3.78 82.45 Rounded Forecast with = .50 175 177.50 172.75 165.88 170.44 180.22 192.61 186.30 Absolute Deviation for = .50 5.00 9.50 13.75 9.12 19.56 24.78 12.61 4.30 98.62 Quarter 1 2 3 4 5 6 7 8 Actual Tonnage Unloaded 180 168 159 175 190 205 180 182
Comparison of Forecast Error |deviations| n Rounded Forecast with = .10 175 175.5 174.75 173.18 173.36 175.02 178.02 178.22 Absolute Deviation for = .10 5.00 7.50 15.75 1.82 16.64 29.98 1.98 3.78 82.45 Rounded Forecast with = .50 175 177.50 172.75 165.88 170.44 180.22 192.61 186.30 Absolute Deviation for = .50 5.00 9.50 13.75 9.12 19.56 24.78 12.61 4.30 98.62 MAD = Quarter 1 2 3 4 5 6 7 8 Actual Tonnage Unloaded 180 168 159 175 190 205 180 182 For = .10 = 82.45/8 = 10.31 For = .50 = 98.62/8 = 12.33
Comparison of Forecast Error (forecast errors)2 n Rounded Forecast with = .10 175 175.5 174.75 173.18 173.36 175.02 178.02 178.22 MAD Absolute Deviation for = .10 5.00 7.50 15.75 1.82 16.64 29.98 1.98 3.78 82.45 10.31 Rounded Forecast with = .50 175 177.50 172.75 165.88 170.44 180.22 192.61 186.30 Absolute Deviation for = .50 5.00 9.50 13.75 9.12 19.56 24.78 12.61 4.30 98.62 12.33 MSE = Quarter 1 2 3 4 5 6 7 8 Actual Tonnage Unloaded 180 168 159 175 190 205 180 182 For = .10 = 1,526.54/8 = 190.82 For = .50 = 1,561.91/8 = 195.24
Comparison of Forecast Error n 100|deviationi|/actuali i = 1 Rounded Forecast with = .10 175 175.5 174.75 173.18 173.36 175.02 178.02 178.22 MAD MSE Absolute Deviation for = .10 5.00 7.50 15.75 1.82 16.64 29.98 1.98 3.78 82.45 10.31 190.82 Rounded Forecast with = .50 175 177.50 172.75 165.88 170.44 180.22 192.61 186.30 Absolute Deviation for = .50 5.00 9.50 13.75 9.12 19.56 24.78 12.61 4.30 98.62 12.33 195.24 MAPE = Quarter 1 2 3 4 5 6 7 8 Actual Tonnage Unloaded 180 168 159 175 190 205 180 182 n For = .10 = 44.75/8 = 5.59% For = .50 = 54.05/8 = 6.76%
Comparison of Forecast Error Rounded Forecast with = .10 175 175.5 174.75 173.18 173.36 175.02 178.02 178.22 MAD MSE MAPE Absolute Deviation for = .10 5.00 7.50 15.75 1.82 16.64 29.98 1.98 3.78 82.45 10.31 190.82 5.59% Rounded Forecast with = .50 175 177.50 172.75 165.88 170.44 180.22 192.61 186.30 Absolute Deviation for = .50 5.00 9.50 13.75 9.12 19.56 24.78 12.61 4.30 98.62 12.33 195.24 6.76% Quarter 1 2 3 4 5 6 7 8 Actual Tonnage Unloaded 180 168 159 175 190 205 180 182
Exponential Smoothing with Trend Adjustment When a trend is present, exponential smoothing must be modified Exponentially smoothed (Ft) + forecast (Tt) smoothed trend Exponentially Forecast including (FITt) = trend
Exponential Smoothing with Trend Adjustment Ft = (At - 1) + (1 - )(Ft - 1 + Tt - 1) Tt = (Ft - Ft - 1) + (1 - )Tt - 1 Step 1: Compute Ft Step 2: Compute Tt Step 3: Calculate the forecast FITt = Ft + Tt
Exponential Smoothing with Trend Adjustment Example Forecast Including Trend, FITt 13.00 Month(t) 1 2 3 4 5 6 7 8 9 10 Actual Demand (At) 12 17 20 19 24 21 31 28 36 Smoothed Forecast, Ft 11 Smoothed Trend, Tt 2
Exponential Smoothing with Trend Adjustment Example Forecast Including Trend, FITt 13.00 Month(t) 1 2 3 4 5 6 7 8 9 10 Actual Demand (At) 12 17 20 19 24 21 31 28 36 Smoothed Forecast, Ft 11 Smoothed Trend, Tt 2 Step 1: Forecast for Month 2 F2 = A1 + (1 - )(F1 + T1) F2 = (.2)(12) + (1 - .2)(11 + 2) = 2.4 + 10.4 = 12.8 units
Exponential Smoothing with Trend Adjustment Example Forecast Including Trend, FITt 13.00 Month(t) 1 2 3 4 5 6 7 8 9 10 Actual Demand (At) 12 17 20 19 24 21 31 28 36 Smoothed Forecast, Ft 11 12.80 Smoothed Trend, Tt 2 Step 2: Trend for Month 2 T2 = (F2 - F1) + (1 - )T1 T2 = (.4)(12.8 - 11) + (1 - .4)(2) = .72 + 1.2 = 1.92 units
Exponential Smoothing with Trend Adjustment Example Forecast Including Trend, FITt 13.00 Month(t) 1 2 3 4 5 6 7 8 9 10 Actual Demand (At) 12 17 20 19 24 21 31 28 36 Smoothed Forecast, Ft 11 12.80 Smoothed Trend, Tt 2 1.92 Step 3: Calculate FIT for Month 2 FIT2 = F2 + T1 FIT2 = 12.8 + 1.92 = 14.72 units
Exponential Smoothing with Trend Adjustment Example Forecast Including Trend, FITt 13.00 14.72 17.28 20.14 22.14 24.89 26.18 29.59 31.60 35.16 Month(t) 1 2 3 4 5 6 7 8 9 10 Actual Demand (At) 12 17 20 19 24 21 31 28 36 Smoothed Forecast, Ft 11 12.80 15.18 17.82 19.91 22.51 24.11 27.14 29.28 32.48 Smoothed Trend, Tt 2 1.92 2.10 2.32 2.23 2.38 2.07 2.45 2.32 2.68
Exponential Smoothing with Trend Adjustment Example 35 Actual demand (At) 30 25 Product demand 20 15 Forecast including trend (FITt) with = .2 and = .4 10 5 0 | | 2 | 3 | 4 Time (month) | 5 | 6 | 7 | 8 | 1 9
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
Least Squares Method Actual observation (y value) Deviation7 Values of Dependent Variable Deviation5 Deviation6 Deviation3 Deviation4 Deviation1 (error) Deviation2 ^ Trend line, y = a + bx Time period
Least Squares Method Actual observation (y value) Values of Dependent Variable Deviation7 Deviation5 Deviation6 Least squares method minimizes the sum of the squared errors (deviations) Deviation3 Deviation4 Deviation1 Deviation2 ^ Trend line, y = a + bx Time period
Least Squares Method Equations to calculate the regression variables ^ y = a + bx xy - nxy x2 - nx2 b = a = y - bx
Least Squares Example Time Period (x) Electrical Power Demand Year x2 xy 2001 2002 2003 2004 2005 2005 2007 1 2 3 4 5 6 7 74 79 80 90 105 142 122 1 4 9 74 158 240 360 525 852 854 16 25 36 49 x = 28 x = 4 y = 692 y = 98.86 x2 = 140 xy = 3,063 3,063 - (7)(4)(98.86) 140 - (7)(42) xy - nxy b = = = 10.54 x2 - nx2 a = y - bx = 98.86 - 10.54(4) = 56.70
Least Squares Example Time Period (x) Electrical Power Demand Year x2 xy 1999 2000 2001 2002 2003 2004 2005 1 2 3 4 5 6 7 74 79 80 90 105 142 122 1 4 9 74 158 240 360 525 852 854 The trend line is 16 25 36 49 ^ y = 56.70 + 10.54x x = 28 x = 4 y = 692 y = 98.86 x2 = 140 xy = 3,063 xy - nxy 3,063 - (7)(4)(98.86) 140 - (7)(42) b = = = 10.54 x2 - nx2 a = y - bx = 98.86 - 10.54(4) = 56.70
Least Squares Example Trend line, y = 56.70 + 10.54x 160 150 140 130 120 110 100 90 80 70 60 50 ^ Power demand | | | | | | | | | 2001 2002 2003 2004 2005 2006 2007 2008 2009 Year
Seasonal Variations In Data The multiplicative seasonal model can adjust trend data for seasonal variations in demand
Seasonal Variations In Data Steps in the process: 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
Seasonal Index Example Demand 2005 2006 2007 Average 2005-2007 Average Monthly Seasonal Index Month Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec 80 70 80 90 113 110 100 88 85 77 75 82 85 85 93 95 125 115 102 102 90 78 72 78 105 85 82 115 131 120 113 110 95 85 83 80 90 80 85 100 123 115 105 100 90 80 80 80 94 94 94 94 94 94 94 94 94 94 94 94
Seasonal Index Example Demand 2005 2006 2007 Average 2005-2007 Average Monthly Seasonal Index 0.957 Month Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec 80 70 80 90 113 110 100 88 85 77 75 82 85 85 93 95 125 115 102 102 90 78 72 78 105 85 82 115 131 120 113 110 95 85 83 80 90 80 85 100 123 115 105 100 90 80 80 80 94 94 94 94 94 94 94 94 94 94 94 94 average 2005-2007 monthly demand average monthly demand Seasonal index = = 90/94 = .957
Seasonal Index Example Demand 2005 2006 2007 Average 2005-2007 Average Monthly Seasonal Index Month Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec 80 70 80 90 113 110 100 88 85 77 75 82 85 85 93 95 125 115 102 102 90 78 72 78 105 85 82 115 131 120 113 110 95 85 83 80 90 80 85 100 123 115 105 100 90 80 80 80 94 94 94 94 94 94 94 94 94 94 94 94 0.957 0.851 0.904 1.064 1.309 1.223 1.117 1.064 0.957 0.851 0.851 0.851
Seasonal Index Example Demand 2005 2006 2007 Average 2005-2007 Average Monthly Seasonal Index Month Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec 80 70 80 90 113 110 100 88 85 77 75 82 85 85 93 95 125 115 102 102 90 78 72 78 105 85 82 115 131 120 113 110 95 85 83 80 90 80 85 100 123 115 105 100 90 80 80 80 94 94 94 94 94 94 94 94 94 94 94 94 0.957 0.851 0.904 1.064 1.309 1.223 1.117 1.064 0.957 0.851 0.851 0.851 Forecast for 2008 Expected annual demand = 1,200 1,200 12 1,200 12 Jan x .957 = 96 Feb x .851 = 85
Seasonal Index Example 2008 Forecast 2007 Demand 2006 Demand 2005 Demand 140 130 120 Demand 110 100 90 80 70 | | F | | A | | J Time | J | A | S | O | N | J M M D
San Diego Hospital Trend Data 10,200 10,000 9745 9,800 Inpatient Days 9702 9659 9616 9573 9766 9724 9530 9,600 9680 9637 9594 9551 9,400 9,200 | | | | | | | | | | | | 9,000 Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec 67 68 69 70 71 72 Month 73 74 75 76 77 78
San Diego Hospital Seasonal Indices 1.06 1.04 1.04 1.04 1.03 Index for Inpatient Days 1.02 1.02 1.01 1.00 0.99 1.00 0.98 0.98 0.99 0.96 0.97 0.97 0.96 0.94 | | | | | | | | | | | | 0.92 Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec 67 68 69 70 71 72 Month 73 74 75 76 77 78
San Diego Hospital Combined Trend and Seasonal Forecast 10,200 10068 9949 10,000 9911 9764 9724 9,800 Inpatient Days 9691 9572 9,600 9542 9520 9,400 9411 9355 9265 9,200 | | | | | | | | | | | | 9,000 Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec 67 68 69 70 71 72 Month 73 74 75 76 77 78
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
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
Associative Forecasting Example Sales Local Payroll ($ billions), x 1 3 4 2 1 7 ($ millions), y 2.0 3.0 2.5 2.0 2.0 3.5 4.0 3.0 Sales 2.0 1.0 | 1 | 2 | 3 | 4 | 5 | 6 | 0 7 Area payroll
Associative Forecasting Example Sales, y 2.0 3.0 2.5 2.0 2.0 3.5 y = 15.0 Payroll, x x2 1 9 16 4 1 49 xy 2.0 9.0 10.0 4.0 2.0 24.5 1 3 4 2 1 7 x = 18 x2 = 80 xy = 51.5 51.5 - (6)(3)(2.5) 80 - (6)(32) xy - nxy b = = = .25 x2 - nx2 x = x/6 = 18/6 = 3 a = y - bx = 2.5 - (.25)(3) = 1.75 y = y/6 = 15/6 = 2.5
Associative Forecasting Example ^ y = 1.75 + .25x Sales = 1.75 + .25(payroll) If payroll next year is estimated to be $6 billion, then: 4.0 3.25 3.0 Sales 2.0 Sales = 1.75 + .25(6) Sales = $3,250,000 1.0 | 1 | 2 | 3 | 4 | 5 | 6 | 0 7 Area payroll
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 3.25 3.0 Sales 2.0 1.0 | 1 | 2 | 3 | 4 | 5 | 6 | 0 7 Area payroll
Standard Error of the Estimate (y - yc)2 n - 2 Sy,x = where y = y-value of each data point yc = computed value of the dependent variable, from the regression equation n = number of data points
Standard Error of the Estimate Computationally, this equation is considerably easier to use y2 - a y - b xy n - 2 Sy,x = We use the standard error to set up prediction intervals around the point estimate
Standard Error of the Estimate 39.5 - 1.75(15) - .25(51.5) 6 - 2 y2 - a y - b xy n - 2 Sy,x = = Sy,x = .306 4.0 3.25 3.0 Sales The standard error of the estimate is $306,000 in sales 2.0 1.0 | 1 | 2 | 3 | 4 | 5 | 6 | 0 7 Area payroll
Correlation 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 Coefficient n xy - x y r = [n x2 - ( x)2][n y2 - ( y)2]
y y Correlation Coefficient n xy - x y r = [n x2 - ( x)2][n y2 - ( y)2] x x (a) Perfect positive correlation: r = +1 (b) Positive correlation: 0 < r < 1 y y x x (d) Perfect negative correlation: r = -1 (c) No correlation: r = 0
Correlation Coefficient of Determination, r2, measures the percent of change in y predicted by the change in x Values range from 0 to 1 Easy to interpret For the Nodel Construction example: r = .901 r2 = .81
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 + b1x1 + b2x2 Computationally, this is quite complex and generally done on the computer
Multiple Regression Analysis In the Nodel example, including interest rates in the model gives the new equation: ^ y = 1.80 + .30x1 - 5.0x2 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
Monitoring and Controlling Forecasts Tracking Signal 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
