Introduction to Management Science 8th Edition by Bernard W. Taylor III Chapter 5 Forecasting Chapter 5 - Forecasting

Chapter Topics Forecasting Components Time Series Methods Forecast Accuracy Time Series Forecasting Using Excel Time Series Forecasting Using QM for Windows Regression Methods Chapter 5 - Forecasting

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 pattern Chapter 5 - Forecasting

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. Chapter 5 - Forecasting

Forecasting Components Patterns (2 of 2) trend-line Figure 5.1 Forms of Forecast Movement: (a) Trend, (b) Cycle, (c) Seasonal Pattern, (d) Trend with Seasonal Pattern Chapter 5 - Forecasting

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. Chapter 5 - Forecasting

Forecasting Components Qualitative Methods Qualitative methods, the “jury of executive opinion,” is the most common type of forecasting method for long-term strategic planning. Performed by individuals or groups within an organization, sometimes assisted by consultants and other experts, whose judgments and opinion are considered valid for the forecasting issue. Usually includes specialty functions such as marketing, engineering, purchasing, etc. in which individuals have experience and knowledge of the forecasted item. Supporting techniques include the Delphi Method, market research, surveys, etc. Chapter 5 - Forecasting

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. Chapter 5 - Forecasting

Time Series Methods Moving Average (1 of 5) Moving average uses values from the recent past to develop forecasts. This dampens or smoothes out random increases and decreases. Useful for forecasting relatively stable items that do not display any trend or seasonal pattern. Formula for: Chapter 5 - Forecasting

Time Series Methods Moving Average (2 of 5) Example: Instant Paper Clip Supply Company forecast of orders for the next month. Three-month moving average: Five-month moving average: Chapter 5 - Forecasting

Three- and Five-Month Moving Averages Time Series Methods Moving Average (3 of 5) Figure 5.2 Three- and Five-Month Moving Averages Chapter 5 - Forecasting

Three- and Five-Month Moving Averages Time Series Methods Moving Average (4 of 5) Figure 5.2 Three- and Five-Month Moving Averages Chapter 5 - Forecasting

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. Moving average does not react well to changes (trends, seasonal effects, etc.) but is easy to use and inexpensive. Good for short-term forecasting. Chapter 5 - Forecasting

Weighted Moving Average (1 of 2) Time Series Methods Weighted Moving Average (1 of 2) In a weighted moving average, weights are assigned to the most recent data. Formula: Chapter 5 - Forecasting

Weighted Moving Average (2 of 2) Time Series Methods Weighted Moving Average (2 of 2) Determining precise weights and number of periods requires trial-and-error experimentation. Chapter 5 - Forecasting

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 where: Ft + 1 = the forecast for the next period Dt = actual demand in the present period Ft = the previously determined forecast for the present period  = a weighting factor (smoothing constant) use F1 = D1. Chapter 5 - Forecasting

Exponential Smoothing (2 of 11) Time Series Methods Exponential Smoothing (2 of 11) The most commonly used values of  are between .10 and .50. Determination of  is usually judgmental and subjective and often based on trial-and -error experimentation. Chapter 5 - Forecasting

Exponential Smoothing (3 of 11) Time Series Methods Exponential Smoothing (3 of 11) Example: PM Computer Services (see Table 5.4). 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 Chapter 5 - Forecasting

Exponential Smoothing Forecasts,  = .30 and  = .50 Time Series Methods Exponential Smoothing (4 of 11) Using F1 = D1 Table 5.4 Exponential Smoothing Forecasts,  = .30 and  = .50 Chapter 5 - Forecasting

Exponential Smoothing (5 of 11) Time Series Methods Exponential Smoothing (5 of 11) The forecast that uses the higher smoothing constant (.50) reacts more strongly to changes in demand than does the forecast with the lower constant (.30). 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 constants appropriate for data with trends. Chapter 5 - Forecasting

Exponential Smoothing Forecasts Time Series Methods Exponential Smoothing (6 of 11) Figure 5.3 Exponential Smoothing Forecasts Chapter 5 - Forecasting

Exponential Smoothing (7 of 11) Time Series Methods Exponential Smoothing (7 of 11) Adjusted exponential smoothing: exponential smoothing with a trend adjustment factor added. Formula: AFt + 1 = Ft + 1 + Tt+1 where: Tt = an exponentially smoothed trend factor: Tt + 1 = (Ft + 1 - Ft) + (1 - )Tt Tt = the last (previous) period’s trend factor  = smoothing constant for trend ( a value between zero and one). Reflects the weight given to the most recent trend data. Determined subjectively. Chapter 5 - Forecasting

Exponential Smoothing (8 of 11) Time Series Methods Exponential Smoothing (8 of 11) Example: PM Computer Services exponential smoothed forecasts with  = .50 and  = .30 (see Table 5.5). Start with T2 = 0.00 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 Chapter 5 - Forecasting

Exponential Smoothing (9 of 11) Time Series Methods Exponential Smoothing (9 of 11) Tt + 1 = (Ft + 1 - Ft) + (1 - )Tt Table 5.5 Adjusted Exponentially Smoothed Forecast Values Chapter 5 - Forecasting

Exponential Smoothing (10 of 11) Time Series Methods Exponential Smoothing (10 of 11) Adjusted forecast is consistently higher than the simple exponentially smoothed forecast. It is more reflective of the generally increasing trend of the data. Chapter 5 - Forecasting

Adjusted Exponentially Smoothed Forecast Time Series Methods Exponential Smoothing (11 of 11) Figure 5.4 Adjusted Exponentially Smoothed Forecast Chapter 5 - Forecasting

Time Series Methods Linear Trend Line (1 of 5) When demand displays an obvious trend over time, a least squares regression line , or linear trend line, can be used to forecast. Formula: Chapter 5 - Forecasting

Time Series Methods Linear Trend Line (2 of 5) Example: PM Computer Services (see Table 5.6) Chapter 5 - Forecasting

Time Series Methods Linear Trend Line (3 of 5) Table 5.6 Least Squares Calculations Chapter 5 - Forecasting

Time Series Methods Linear Trend Line (4 of 5) A trend line does not adjust to a change in the trend as does the exponential smoothing method. This limits its use to shorter time frames in which trend will not change. Chapter 5 - Forecasting

Time Series Methods Linear Trend Line (5 of 5) Figure 5.5 Chapter 5 - Forecasting

Seasonal Adjustments (1 of 4) Time Series Methods Seasonal Adjustments (1 of 4) A seasonal pattern is a repetitive up-and-down movement in demand. Seasonal patterns can occur on a monthly, weekly, or daily basis. A seasonally adjusted forecast can be developed by multiplying the normal forecast by a seasonal factor. A seasonal factor can be determined by dividing the actual demand for each seasonal period by total annual demand: Si =Di/D Chapter 5 - Forecasting

Seasonal Adjustments (2 of 4) Time Series Methods Seasonal Adjustments (2 of 4) Seasonal factors lie between zero and one and represent the portion of total annual demand assigned to each season. Seasonal factors are multiplied by annual demand to provide adjusted forecasts for each period. Chapter 5 - Forecasting

Demand for Turkeys at Wishbone Farms Time Series Methods Seasonal Adjustments (3 of 4) Example: Wishbone Farms Table 5.7 Demand for Turkeys at Wishbone Farms S1 = D1/ D = 42.0/148.7 = 0.28 S2 = D2/ D = 29.5/148.7 = 0.20 S3 = D3/ D = 21.9/148.7 = 0.15 S4 = D4/ D = 55.3/148.7 = 0.37 Chapter 5 - Forecasting

Seasonal Adjustments (4 of 4) Time Series Methods Seasonal Adjustments (4 of 4) Multiply forecasted demand for entire year by seasonal factors to determine quarterly demand. Forecast for entire year (trend line for data in Table 5.7): y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17 Seasonally adjusted forecasts: SF1 = (S1)(F5) = (.28)(58.17) = 16.28 SF2 = (S2)(F5) = (.20)(58.17) = 11.63 SF3 = (S3)(F5) = (.15)(58.17) = 8.73 SF4 = (S4)(F5) = (.37)(58.17) = 21.53 Note the potential for confusion: the Si are “seasonal factors” (fractions), whereas the SFi are “seasonally adjusted forecasts” (commodity values)! Chapter 5 - Forecasting

Forecast Accuracy Overview Forecasts will always deviate from actual values. Difference between forecasts and actual values referred to as forecast error. 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-bar) Average error, or bias (E) Chapter 5 - Forecasting

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: Chapter 5 - Forecasting

Mean Absolute Deviation (2 of 7) Forecast Accuracy Mean Absolute Deviation (2 of 7) Example: PM Computer Services (see Table 5.8). Compare accuracies of different forecasts using MAD: Chapter 5 - Forecasting

Computational Values for MAD Forecast Accuracy Mean Absolute Deviation (3 of 7) Table 5.8 Computational Values for MAD Chapter 5 - Forecasting

Mean Absolute Deviation (4 of 7) Forecast Accuracy Mean Absolute Deviation (4 of 7) The lower the value of MAD relative to the magnitude of the data, the more accurate the forecast. When viewed alone, MAD is difficult to assess. Must be considered in light of magnitude of the data. Chapter 5 - Forecasting

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 ( = .50): MAD = 4.04 Adjusted exponential smoothing ( = .50,  = .30): MAD = 3.81 Linear trend line: MAD = 2.29 Linear trend line has lowest MAD; increasing  from .30 to .50 improved smoothed forecast. Chapter 5 - Forecasting

Mean Absolute Deviation (6 of 7) Forecast Accuracy Mean Absolute 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: Chapter 5 - Forecasting

Mean Absolute Deviation (7 of 7) Forecast Accuracy Mean Absolute Deviation (7 of 7) MAPD for other three forecasts: Exponential smoothing ( = .50): MAPD = 8.5% Adjusted exponential smoothing ( = .50,  = .30): MAPD = 8.1% Linear trend: MAPD = 4.9% Chapter 5 - Forecasting

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 preponderance of errors are positive, forecast is consistently low; and vice versa. Cumulative error for trend line is always almost zero, and is therefore not a good measure for this method. Cumulative error for PM Computer Services can be read directly from Table 5.8. E =  et = 49.31 indicating forecasts are frequently below actual demand. Chapter 5 - Forecasting

Forecast Accuracy Cumulative Error (2 of 2) Cumulative error for other forecasts: Exponential smoothing ( = .50): E = 33.21 Adjusted exponential smoothing ( = .50,  =.30): E = 21.14 Average error (bias) is the per period average of cumulative error. Average error for exponential smoothing forecast: A large positive value of average error indicates a forecast is biased low; a large negative error indicates it is biased high. Chapter 5 - Forecasting

Comparison of Forecasts for PM Computer Services Forecast Accuracy Example Forecasts by Different Measures Table 5.9 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 forecasts. Linear trend is more accurate than all the others. Chapter 5 - Forecasting

Time Series Forecasting Using Excel (1 of 4) Exhibit 5.1 Chapter 5 - Forecasting

Time Series Forecasting Using Excel (2 of 4) Exhibit 5.2 Chapter 5 - Forecasting

Time Series Forecasting Using Excel (3 of 4) Exhibit 5.3 Chapter 5 - Forecasting

Time Series Forecasting Using Excel (4 of 4) Exhibit 5.4 Chapter 5 - Forecasting

Exponential Smoothing Forecast with Excel QM Exhibit 5.5 Chapter 5 - Forecasting

Time Series Forecasting Solution with QM for Windows (1 of 2) Exhibit 5.6 Chapter 5 - Forecasting

Time Series Forecasting Solution with QM for Windows (2 of 2) Exhibit 5.7 Chapter 5 - Forecasting

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. Chapter 5 - Forecasting

Regression Methods Linear Regression Linear regression relates demand (dependent variable ) to an independent variable. Linear function Chapter 5 - Forecasting

Linear Regression Example (1 of 3) Regression Methods Linear Regression Example (1 of 3) State University athletic department. Chapter 5 - Forecasting

Linear Regression Example (2 of 3) Regression Methods Linear Regression Example (2 of 3) Chapter 5 - Forecasting

Linear Regression Example (3 of 3) Regression Methods Linear Regression Example (3 of 3) Figure 5.6 Linear Regression Line Chapter 5 - Forecasting

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 strong linear relationship: correlation and anti-correlation. Chapter 5 - Forecasting

Regression Methods Correlation (2 of 2) Value for State University example: Chapter 5 - Forecasting

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 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. Chapter 5 - Forecasting

Regression Analysis with Excel (1 of 7) Exhibit 5.8 Chapter 5 - Forecasting

Regression Analysis with Excel (2 of 7) Exhibit 5.9 Chapter 5 - Forecasting

Regression Analysis with Excel (3 of 7) Exhibit 5.10 Chapter 5 - Forecasting

Regression Analysis with Excel (4 of 7) Exhibit 5.11 Chapter 5 - Forecasting

Regression Analysis with Excel (5 of 7) Exhibit 5.12 Chapter 5 - Forecasting

Regression Analysis with Excel (6 of 7) Exhibit 5.13 Chapter 5 - Forecasting

Regression Analysis with Excel (7 of 7) Exhibit 5.14 Chapter 5 - Forecasting

Regression Analysis with QM for Windows Exhibit 5.15 Chapter 5 - Forecasting

Multiple Regression with Excel (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 Chapter 5 - Forecasting

Multiple Regression with Excel (2 of 4) State University example: Chapter 5 - Forecasting

Multiple Regression with Excel (3 of 4) Exhibit 5.16 Chapter 5 - Forecasting

Multiple Regression with Excel (4 of 4) Exhibit 5.17 Chapter 5 - Forecasting

Example Problem Solution Computer Software Firm (1 of 4) Problem Statement: For 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. Chapter 5 - Forecasting

Example Problem Solution Computer Software Firm (2 of 4) Step 1: Compute the Exponential Smoothing Forecast. Ft+1 =  Dt + (1 - )Ft Step 2: Compute the Adjusted Exponential Smoothing Forecast AFt+1 = Ft +1 + Tt+1 Tt+1 = (Ft +1 - Ft) + (1 - )Tt Chapter 5 - Forecasting

Example Problem Solution Computer Software Firm (3 of 4) Chapter 5 - Forecasting

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 Chapter 5 - Forecasting

Example Problem Solution Building Products Store (1 of 5) Problem Statement: 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. Chapter 5 - Forecasting

Example Problem Solution Building Products Store (2 of 5) Chapter 5 - Forecasting

Example Problem Solution Building Products Store (3 of 5) Step 1: Compute the Components of the Linear Regression Equation. Chapter 5 - Forecasting

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. Chapter 5 - Forecasting

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 Chapter 5 - Forecasting

Chapter 5 - Forecasting