Forecasting Chapter 15

Forecasting Components Time Series Methods Forecast Accuracy Chapter Topics Forecasting Components Time Series Methods Forecast Accuracy Regression Methods Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Forecasting Components A variety of forecasting methods are available for use depending on the time frame of the forecast and the existence of patterns. Time Frames: Short-range (one to two months) Medium-range (two months to one or two years) Long-range (more than one or two years) Patterns: Trend Random variations Cycles Seasonal Patterns Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time Horizon in Forecasting Short-term forecasts (days or weeks) are required for inventory management, production plans and resource requirements planning. Medium-term forecasts (weeks and months) are required for estimating product family sales, resource requirements in aggregate production planning. Long-term forecasts (Months and years) are required for estimating capacity needs, growth trends in strategic planning.

Forecasting Components Patterns (1 of 2) Trend - A long-term movement of the item being forecast. Random variations - movements that are not predictable and follow no pattern. Cycle - A movement, up or down, that repeats itself over a lengthy time span. Seasonal pattern - Oscillating movement in demand that occurs periodically in the short run and is repetitive. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Forecasting Components Patterns (2 of 2) Figure 15.1 (a) Trend; (b) Cycle; (c) Seasonal; (d) Trend w/Season Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Forecasting Components Forecasting Methods Times Series - Statistical techniques that use historical data to predict future behavior. Regression Methods - Regression (or causal ) methods that attempt to develop a mathematical relationship between the item being forecast and factors that cause it to behave the way it does. Qualitative Methods - Methods using judgment, expertise and opinion to make forecasts. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time Series Methods Overview Statistical techniques that make use of historical data collected over a long period of time. Methods assume that what has occurred in the past will continue to occur in the future. Forecasts based on only one factor - time. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time-Series Forecasting Moving Averages Exponential Smoothing Time-Series Methods Time-Series Forecasting Moving Averages Exponential Smoothing Exponential Smoothing with Trend Adjustment Trend Projections using regression Trend Projections with seasonal pattern using regression (will not be covered) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time Series Methods Moving Average (1 of 5) Moving average uses values from the recent past to develop forecasts. This dampens or smoothes random increases and decreases. Useful for forecasting relatively stable items that do not display any trend or seasonal pattern. Formula for: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time Series Methods Moving Average (2 of 5) Example: Instant Paper Clip Supply Company forecast of orders for the month of November. Month June July August September October Order 50 75 130 110 90 Three-month moving average: Five-month moving average: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time Series Methods Moving Average (3 of 5) Table 15.2 Three- and Five-Month Moving Averages Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time Series Methods Moving Average (4 of 5) Figure 15.2 Three- and Five-Month Moving Averages Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time Series Methods Moving Average (5 of 5) Longer-period moving averages react more slowly to changes in demand than do shorter-period moving averages. The appropriate number of periods to use often requires trial-and-error experimentation. Disadvantage: Moving average does not react well to changes (trends, seasonal effects, etc.) Advantage: Easy to use and inexpensive. Good for short-term forecasting. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Weighted Moving Average Time Series Methods Weighted Moving Average In a weighted moving average, user defined weights are assigned to the most recent data. Determining precise weights and number of periods requires trial-and-error experimentation. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Exponential Smoothing (1 of 11) Time Series Methods Exponential Smoothing (1 of 11) Exponential smoothing weights recent past data more strongly than more distant data. Two forms: simple exponential smoothing and adjusted exponential smoothing. Simple exponential smoothing: Ft + 1 = Dt + (1 - )Ft or Ft + 1 = Ft + α (Dt – Ft) where: Ft + 1 = the forecast for the next period Dt = actual demand in the present period Ft = forecasted demand for the present period  = a weighting factor (smoothing constant). Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Exponential Smoothing (2 of 11) Time Series Methods Exponential Smoothing (2 of 11) Each new forecast is based on the previous forecast plus a percentage of the difference between that forecast and the actual value of the series at that point. The most commonly used values of  are between 0.10 and 0.50. Determination of  is usually judgmental and subjective and often based on trial-and -error experimentation. Exponential smoothing is a sophisticated weighted average. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Exponential Smoothing (3 of 11) Time Series Methods Exponential Smoothing (3 of 11) Example: PM Computer Services (see Table 15.4). Exponential smoothing forecasts using smoothing constant of .30. Forecast for period 2 (February): F2 =  D1 + (1- )F1 = (.30)(37) + (.70)(37) = 37 units Forecast for period 3 (March): F3 =  D2 + (1- )F2 = (.30)(40) + (.70)(37) = 37.9 units Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Exponential Smoothing (4 of 11) Time Series Methods Exponential Smoothing (4 of 11) Table 15.4 Exponential Smoothing Forecasts,  = .30 and  = .50 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Exponential Smoothing (5 of 11) Time Series Methods Exponential Smoothing (5 of 11) The forecasts that use the higher smoothing constant (.50) react more strongly to changes in demand than does the forecast with the lower constant (.30). But note that both forecasts lag behind actual demand. Both forecasts tend to be consistently lower than actual demand. Low smoothing constants are appropriate for stable data without trend; higher smoothing constants are appropriate for data with trends. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Exponential Smoothing (6 of 11) Time Series Methods Exponential Smoothing (6 of 11) Figure 15.3 Exponential Smoothing Forecasts Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Adjusted Exponential Smoothing (7 of 11) Time Series Methods Adjusted Exponential Smoothing (7 of 11) Adjusted exponential smoothing: exponential smoothing with a trend adjustment factor added. Formula AFt + 1 = Ft + 1 + Tt+1 where: T = an exponentially smoothed trend factor Tt + 1 = (Ft + 1 - Ft) + (1 - )Tt Tt = the last period trend factor  = smoothing constant for trend ( a value between zero and one) is a weight given to the most recent trend data. Determined subjectively. Notice that this formula for the trend factor reflects a weighted measure of the increase (or decrease) between the current forecast, Ft + 1, and the previous forecast, Ft.

Adjusted Exponential Smoothing The formula for the adjusted exponential smoothing forecast requires an initial value for Tt to start the computational process. This initial trend factor is most often an estimate determined subjectively or based on past data by the forecaster. In this case, we will start with the trend, Tt, equal to zero. By the time the forecast value of interest, F13, is computed, we should have a relatively good value for the trend factor. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Exponential Smoothing (8 of 11) Time Series Methods Exponential Smoothing (8 of 11) Example: PM Computer Services exponentially smoothed forecasts with  = .50 and  = .30 (see Table 15.5 next slide). Adjusted forecast for period 3: T3 = (F3 - F2) + (1 - )T2 = (.30)(38.5 - 37.0) + (.70)(0) = 0.45 AF3 = F3 + T3 = 38.5 + 0.45 = 38.95

Exponential Smoothing (9 of 11) Time Series Methods Exponential Smoothing (9 of 11) Table 15.5 Adjusted exponentially smoothed forecast values

Exponential Smoothing (10 of 11) Time Series Methods Exponential Smoothing (10 of 11) The adjusted forecast is consistently higher than the simple exponentially smoothed forecast. It is more reflective of the generally increasing trend of the data.

Exponential Smoothing (11 of 11) Time Series Methods Exponential Smoothing (11 of 11) Figure 15.4 Adjusted exponentially smoothed forecast

Time Series Methods Linear Trend Line (1 of 5) When demand displays an obvious trend over time, a least squares regression method, can be used to forecast the demand. The slope of the line (b) shows the amount by which y increases when x increases by 1 unit. Formula: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time Series Methods Linear Trend Line (2 of 5) Example: PM Computer Services (see Table 15.6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Least Squares Calculations Time Series Methods Linear Trend Line (3 of 5) Table 15.6 Least Squares Calculations Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Time Series Methods Linear Trend (5 of 5) Figure 15.5 Linear Trend Line Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Forecast Accuracy Overview Forecasts will always deviate from actual values. Difference between forecasts and actual values referred to as forecast error. We would like forecast error to be as small as possible. If error is large, either technique being used is the wrong one, or parameters need adjusting. Measures of forecast errors: Mean Absolute deviation (MAD) Mean absolute percentage deviation (MAPD) Cumulative error E=∑et Average error, or bias (E)/n Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Mean Absolute Deviation (1 of 7) Forecast Accuracy Mean Absolute Deviation (1 of 7) MAD is the average absolute difference between the forecast and actual demand. Most popular and simplest-to-use measures of forecast error. Formula: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Mean Absolute Deviation (2 of 7) Forecast Accuracy Mean Absolute Deviation (2 of 7) Example: PM Computer Services (see Table 15.8). Compare accuracies of different forecasts using MAD: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Mean Absolute Deviation (3 of 7) Forecast Accuracy Mean Absolute Deviation (3 of 7) Table 15.8 Computational Values for MAD and error Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Mean Absolute Deviation (5 of 7) Forecast Accuracy Mean Absolute Deviation (5 of 7) Can be used to compare accuracy of different forecasting techniques working on the same set of demand data (PM Computer Services): Exponential smoothing ( = .30): MAD = 53.41/11=4.85 Exponential smoothing ( = .50): MAD = 4.04 Linear trend line: MAD = 2.29 Linear trend line has lowest MAD; increasing  from .30 to .50 improved smoothed forecast. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Mean Absolute Percent Deviation (6 of 7) Forecast Accuracy Mean Absolute Percent Deviation (6 of 7) A variation on MAD is the mean absolute percent deviation (MAPD). Measures absolute error as a percentage of demand rather than per period. Eliminates problem of interpreting the measure of accuracy relative to the magnitude of the demand and forecast values. Formula: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Mean Absolute Deviation (7 of 7) Forecast Accuracy Mean Absolute Deviation (7 of 7) MAPD for other two forecasts: Exponential smoothing ( = .50): MAPD = 8.5% Linear trend: MAPD = 4.9% Again, Linear trend line has lowest MAPD; and increasing  from .30 to .50 improves smoothed forecast. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Forecast Accuracy Cumulative Error (1 of 2) Cumulative error is the sum of the forecast errors (E =et). A relatively large positive value indicates forecast is biased low, a large negative value indicates forecast is biased high. If majority of errors are positive, then forecasted values will be consistently low; and vice versa. The cumulative error for a trend line is always almost zero, and is therefore not a good measure for this method. Cumulative error for PM Computer Services for  = .30 can be read directly from Table 15.8: E =  et = 49.31 indicating that forecasts are frequently below the actual demand. Cumulative error for PM Computer Services for  = .50 is E = 33.21 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Forecast Accuracy Average Error Bias Average error (bias) is the per period average of cumulative error. A large positive value of average error indicates a forecast is biased low; a large negative error indicates it is biased high. Average error for exponential smoothing forecast for  = .30 : Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Table 15.9 Comparison of Forecasts for PM Computer Services Forecast Accuracy Example Forecasts by Different Measures Table 15.9 Comparison of Forecasts for PM Computer Services Results consistent for all forecasts: Larger value of alpha is preferable. Adjusted forecast is more accurate than exponential smoothing. Linear trend is more accurate than all the others. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Regression Methods Overview Time series techniques relate a single variable being forecast to time. Regression is a forecasting technique that measures the relationship of one variable to one or more other variables. Simplest form of regression is linear regression. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Regression Methods Linear Regression Linear regression relates demand (dependent variable ) to an independent variable. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Regression Methods Correlation (1 of 2) Correlation is a measure of the strength of the relationship between independent and dependent variables. Formula: Value lies between +1 and -1. Value of zero indicates little or no relationship between variables. Values near 1.00 and -1.00 indicate a strong linear relationship.

Coefficient of Determination Regression Methods Coefficient of Determination The coefficient of determination is the percentage of the variation in the dependent variable that results from the independent variable. Computed by squaring the correlation coefficient, r. For the State University example: r = .948, r2 = .899 This value indicates that 89.9% of the amount of variation in attendance can be attributed to the number of wins by the team, with the remaining 10.1% due to other, unexplained, factors.

Multiple Regression (1 of 4) Multiple regression relates demand to two or more independent variables. General form: y = 0 +  1x1 +  2x2 + . . . +  kxk where  0 = the intercept  1 . . .  k = parameters representing contributions of the independent variables x1 . . . xk = independent variables

EXAMPLE: The State University athletic department The State University athletic department wants to develop its budget for the coming year, using a forecast for football attendance. Football attendance accounts for the largest portion of its revenues, and the athletic director believes attendance is directly related to the number of wins by the team. The business manager has accumulated total annual attendance figures for the past 8 years: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Linear Regression Example (1 of 3) Regression Methods Linear Regression Example (1 of 3) State University Athletic Department. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Linear Regression Example (2 of 3) Regression Methods Linear Regression Example (2 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Linear Regression Example (3 of 3) Regression Methods Linear Regression Example (3 of 3) Figure 15.6 Linear Regression Line Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution Computer Software Firm (1 of 4) Problem Statement: For the data below, develop an exponential smoothing forecast using  = .40, and an adjusted exponential smoothing forecast using  = .40 and  = .20. Compare the accuracy of the forecasts using MAD and cumulative error.

Example Problem Solution Computer Software Firm (2 of 4) Step 1: Compute the Exponential Smoothing Forecast. Using Ft+1 =  Dt + (1 - )Ft or Ft + 1 = Ft + α(Dt – Ft ) Step 2: Compute the Adjusted Exponential Smoothing Forecast AFt+1 = Ft +1 + Tt+1 Tt+1 = (Ft +1 - Ft) + (1 - )Tt

Example Problem Solution Computer Software Firm (3 of 4)

Example Problem Solution Computer Software Firm (4 of 4) Step 3: Compute the MAD Values Step 4: Compute the Cumulative Error. E(Ft) = 35.97 E(AFt) = 30.60

Example Problem Solution Building Products Store (1 of 5) For the following data: Develop a linear regression model Determine the strength of the linear relationship using correlation. Determine a forecast for lumber given 10 building permits in the next quarter.

Example Problem Solution Building Products Store (2 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution Building Products Store (3 of 5) Step 1: Compute the Components of the Linear Regression Equation. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Example Problem Solution Building Products Store (4 of 5) Step 2: Develop the Linear regression equation. y = a + bx, y = 1.36 + 1.25x Step 3: Compute the Correlation Coefficient.

Y = a + bx = 1.36 + 1.25(10) = 13.86 or 1,386 board ft Example Problem Solution Building Products Store (5 of 5) Step 4: Calculate the forecast for x = 10 permits. Y = a + bx = 1.36 + 1.25(10) = 13.86 or 1,386 board ft