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4-1 Operations Management Forecasting Chapter 4 - Part 2
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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
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4-3 Plot Data Period Actual 4 5 32 1 6 8 4 12 16 20
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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
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4-5 Trend Graph – Moving Average Period Actual MA Forecast 4 5 32 1 6 8 4 12 16 20
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4-6 ES = Exponential Smoothing with =0.5 ( F 2 =11) Exponential Smoothing for a Trend ? Period ES 1 8 211 313 11 415 12 3.0 519 13.5 5.5 ES Error 6 11 16.25 Sales
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4-7 Trend Graph – Exponential Smoothing and Moving Average Period Actual MA Forecast 4 5 32 1 6 8 4 12 16 20 ES Forecast
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4-8 Moving Averages and (simple) Exponential Smoothing are always poor. For a linear trend can use: Exponential Smoothing with Trend Adjustment (pp. 115-117). Linear Trend Projection (linear regression). For non-linear trend can use: Non-linear regression techniques. Forecasting a Trend
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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
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4-10 Plot of X,Y Data Time (x) Values of Dependent Variable (Y) Actual observation
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4-11 Least Squares Deviation Time (x) Values of Dependent Variable (Y) Actual observation Point on regression line
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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.
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4-13 Least Squares Equations Equation: y = a + bx Slope (p. 119): Y-Intercept:
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4-14 Linear Trend Projection Example 5 Perio d (x) 1 8 211 313 415 19 Sales (y) Given the sales for last 5 periods, forecast future sales using trend projection.
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4-15 Linear Trend Projection Example 5 Perio d (x) 1 8 211 313 415 19 Sales (y) xy 60 95 8 22 39 xy=224 x2x2 9 16 25 4 1 x 2 =55 x=3y=13.2
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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!
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4-17 Trend Graph MA Forecast ES Forecast Period Actual 4 5 32 1 6 8 4 12 16 20 TP Forecast
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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 120-124.
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4-19 Identify Independent and Dependent variables. 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
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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! Independent variable (number of ads). YX ii = + ab Dependent variable (sales). Linear Regression
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4-21 Least Squares Deviation Values of Independent Variable (x) Values of Dependent Variable (Y) Actual observation Point on regression line
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4-22 Linear Regression Equations (same as before) Equation: Slope: Y-Intercept:
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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
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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.
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4-25 Monthly Sales vs. Number of Ads Number of TV ads per month Sales 0
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4-26 Least Squares Line Number of TV ads per month Sales 0 What is sales forecast for small number of ads?
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4-27 Forecasting Outside Range of Observed Values is Unreliable Number of TV ads per month Sales 0 Forecast is for negative sales!
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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 Correlation
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4-29 Sample Coefficient of Correlation
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4-30 r = +1r = -1 r =.89r = 0 Y X Y X Y X X Coefficient of Correlation Y
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4-31 A good forecast has: No pattern or direction in forecast error. Error = Actual - Forecast A small forecast error. Mean square error (MSE). Mean absolute deviation (MAD). Mean absolute percentage error (MAPE). Guidelines for Selecting Forecasting Model
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4-32 Time Error 0 Desired Pattern Time Error 0 Trend Not Fully Accounted for Pattern of Forecast Error
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4-33 Suppose you have 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
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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 Actual 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|
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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|
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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%
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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. Signal should be within upper and lower control limits based on MAD. Tracking Signal
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4-38 Plot of a Tracking Signal Time Lower control limit Upper control limit Signal exceeded limit Tracking signal Acceptable range MAD + 0 -
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4-39 Tracking Signal Equation
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4-40 Based on Normal Distribution of forecast errors: 1 MAD = approximately 0.8 standard deviations. Limits at ±3 MAD (±2.4 std. dev.) mean that 98% of values should be within limits. Limits at ±4 MAD (±3.2 std. dev.) mean that 99.9% of values should be within limits. Use smaller limits to better control important items. (For example: ±2 MAD) Patterns, even if within limits, indicate better forecasts can be made. Tracking Signal Limits
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4-41 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090Cum|Error|
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4-42 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090 -10 -10Cum|Error| RSFE = Errors = -10 Error = Actual - Forecast = 90 - 100 = -10
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4-43 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090 -10 -1010Cum|Error| Cum |Error| = |Errors| = 10
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4-44 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090 -10 -1010 10.0Cum|Error| MAD = |Errors|/n = 10/1 = 10
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4-45 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS 110090 -10 -1010 10.0Cum|Error| TS = RSFE/MAD = -10/10 = -1
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4-46 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 -10 -1010 10.0Cum|Error|
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4-47 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 -10 -1010 10.0 -5Cum|Error| Error = Actual - Forecast = 94 - 99 = -5
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4-48 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS 110090 2 99 9994 -10 -1010 10.0 -5 -15Cum|Error| RSFE = Errors = (-10) + (-5) = -15
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4-49 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
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4-50 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
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4-51 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
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4-52 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|
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4-53 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|
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4-54 Demand and Forecast 70 80 90 100 110 120 130 140 01234567 Month Forecast Actual demand
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4-55 Tracking Signal 01234567 Time -3 -2 0 1 2 3 Tracking Signal 98% of points should be between these limits.
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4-56 Suppose you have 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
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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
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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
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4-59 Tracking Signals 501234 Year -3 -2 0 1 2 3 Tracking Signal Exponential Smoothing Linear Regression
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4-60 Forecasting in the Service Sector Examples: For staffing hospitals, fast-food restaurants, banking, etc. 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.
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4-61 Forecasting Summary Determine purpose of forecast first. Plot data. Use several appropriate methods. Continually monitor, evaluate and adjust methods to improve forecasts.
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