4-1 Operations Management Forecasting Chapter 4 - Part 2

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

4-3 Plot Data Period Actual

4-4 MA = 3 period Moving Average Moving Averages for a Trend Period MAMA MA Erro r ? Sales

4-5 Trend Graph – Moving Average Period Actual MA Forecast

4-6 ES = Exponential Smoothing with  =0.5 ( F 2 =11) Exponential Smoothing for a Trend ? Period ES ES Error Sales

4-7 Trend Graph – Exponential Smoothing and Moving Average Period Actual MA Forecast ES Forecast

4-8  Moving Averages and (simple) Exponential Smoothing are always poor.  For a linear trend can use:  Exponential Smoothing with Trend Adjustment (pp ).  Linear Trend Projection (linear regression).  For non-linear trend can use:  Non-linear regression techniques. Forecasting a Trend

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

4-10 Plot of X,Y Data Time (x) Values of Dependent Variable (Y) Actual observation

4-11 Least Squares Deviation Time (x) Values of Dependent Variable (Y) Actual observation Point on regression line

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.

4-13 Least Squares Equations Equation: y = a + bx Slope (p. 119): Y-Intercept:

4-14 Linear Trend Projection Example 5 Perio d (x) Sales (y) Given the sales for last 5 periods, forecast future sales using trend projection.

4-15 Linear Trend Projection Example 5 Perio d (x) Sales (y) xy  xy=224 x2x  x 2 =55 x=3y=13.2

4-16 TP = Trend Projection: Y = x Linear Trend Projection Example Period (x) MAMA ES MA Err Sales (y) ES Err. TP Err. TP Small errors!

4-17 Trend Graph MA Forecast ES Forecast Period Actual TP Forecast

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

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

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

4-21 Least Squares Deviation Values of Independent Variable (x) Values of Dependent Variable (Y) Actual observation Point on regression line

4-22 Linear Regression Equations (same as before) Equation: Slope: Y-Intercept:

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

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.

4-25 Monthly Sales vs. Number of Ads Number of TV ads per month Sales 0

4-26 Least Squares Line Number of TV ads per month Sales 0 What is sales forecast for small number of ads?

4-27 Forecasting Outside Range of Observed Values is Unreliable Number of TV ads per month Sales 0 Forecast is for negative sales!

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

4-29 Sample Coefficient of Correlation

4-30 r = +1r = -1 r =.89r = 0 Y X Y X Y X X Coefficient of Correlation Y

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

4-32 Time Error 0 Desired Pattern Time Error 0 Trend Not Fully Accounted for Pattern of Forecast Error

4-33 Suppose you have forecast sales with a linear regression model & exponential smoothing. Which model do you use? Linear RegressionExponential ActualModelSmoothing YearSalesForecastForecast (.9) Selecting Forecasting Model Example

4-34 MSE = Σ Error 2 / n = 1.10 / 5 = MAD = Σ |Error| / n = 2.0 / 5 = MAPE = Σ [| Error |/ Actual ]/ n = 1.2/5 = 0.24 = 24% Linear Regression Model 1.10 Year Actual F’cast Total ErrorError 2 |Error|

MSE = Σ Error 2 / n = 5.05 / 5 = 1.01 MAD = Σ |Error| / n = 3.11 / 5 = MAPE = Σ[ |Error|/Actual]/ n = /5 = = 21% Exponential Smoothing Model Year Y i F’cast Total ErrorError 2 |Error|

4-36 Which is Better??? Linear Regression Model: MSE = Σ Error 2 / n = 1.10 / 5 = MAD = Σ |Error| / n = 2.0 / 5 = 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 = MAPE = Σ[ |Error|/Actual]/ n = /5 = = 21%

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

4-38 Plot of a Tracking Signal Time Lower control limit Upper control limit Signal exceeded limit Tracking signal Acceptable range MAD + 0 -

4-39 Tracking Signal Equation

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

4-41 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS Cum|Error|

4-42 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS Cum|Error| RSFE =  Errors = -10 Error = Actual - Forecast = = -10

4-43 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS Cum|Error| Cum |Error| =  |Errors| = 10

4-44 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS Cum|Error| MAD =  |Errors|/n = 10/1 = 10

4-45 Tracking Signal - Month 1 MoF’cstAct Error RSFEMADTS Cum|Error| TS = RSFE/MAD = -10/10 = -1

4-46 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS Cum|Error|

4-47 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS Cum|Error| Error = Actual - Forecast = = -5

4-48 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS Cum|Error| RSFE =  Errors = (-10) + (-5) = -15

4-49 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS Cum|Error| Cum Error =  |Errors| = = 15

4-50 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS Cum|Error| MAD =  |Errors|/n = 15/2 = 7.5

4-51 Tracking Signal - Month 2 MoF’cstAct Error RSFEMADTS Cum|Error| TS = RSFE/MAD = -15/7.5 = -2

4-52 Tracking Signal - Month 3 MoF’cstAct Error RSFEMADTS Cum|Error|

4-53 Tracking Signal - Months 4-6 MoF’cstAct Error RSFEMADTS Cum|Error|

4-54 Demand and Forecast Month Forecast Actual demand

4-55 Tracking Signal Time Tracking Signal 98% of points should be between these limits.

4-56 Suppose you have forecast sales with a linear regression model & exponential smoothing. Which model do you use? Linear RegressionExponential ActualModelSmoothing YearSalesForecastForecast (.9) Selecting Forecasting Model Example - Revisited

4-57 Linear Regression Model Tracking Signal Year Y i F’cast ErrorMAD TS

4-58 Exponential Smoothing Model Tracking Signal 1.99 Year Y i F’cast Error MAD TS

4-59 Tracking Signals Year Tracking Signal Exponential Smoothing Linear Regression

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.

4-61 Forecasting Summary  Determine purpose of forecast first.  Plot data.  Use several appropriate methods.  Continually monitor, evaluate and adjust methods to improve forecasts.