Chapter 7 Forecasting n Quantitative Approaches to Forecasting n The Components of a Time Series n Measures of Forecast Accuracy n Using Smoothing Methods in Forecasting n Using Seasonal Components in Forecasting n Qualitative Approaches to Forecasting

Quantitative Approaches to Forecasting n Quantitative methods are based on an analysis of historical data concerning one or more time series. n A time series is a set of observations measured at successive points in time or over successive periods of time. n If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method. n If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method.

Time Series Demand for product or service |||| 1234 Year Average demand over four years Seasonal peaks Trend component Actual demand Random variation

Components of a Time Series n The trend component accounts for the gradual shifting of the time series over a long period of time. n Any regular pattern of sequences of values above and below the trend line is attributable to the cyclical component of the series

Components of a Time Series n The seasonal component of the series accounts for regular patterns of variability within certain time periods, such as over a year. n The irregular component of the series is caused by short-term, unanticipated and non-recurring factors that affect the values of the time series. One cannot attempt to predict its impact on the time series in advance. MTWTFMTWTFMTWTFMTWTF

Measures of Forecast Accuracy n Mean Squared Error The average of the squared forecast errors for the historical data is calculated. The forecasting method or parameter(s) which minimize this mean squared error is then selected. n Mean Absolute Deviation The mean of the absolute values of all forecast errors is calculated, and the forecasting method or parameter(s) which minimize this measure is selected. The mean absolute deviation measure is less sensitive to individual large forecast errors than the mean squared error measure.

Smoothing Methods n In cases in which the time series is fairly stable and has no significant trend, seasonal, or cyclical effects, one can use smoothing methods to average out the irregular components of the time series. n Four common smoothing methods are: Moving averages Moving averages Centered moving averages Centered moving averages Weighted moving averages Weighted moving averages

Smoothing Methods n Moving Average Method The moving average method consists of computing an average of the most recent n data values for the series and using this average for forecasting the value of the time series for the next period.

Sales of Comfort brand headache medicine for the past ten weeks at Rosco Drugs are shown on the next slide. If Rosco Drugs uses a 3-period moving average to forecast sales, what is the forecast for Week 11? Example: Rosco Drugs

n Past Sales Week Sales Week Sales Week Sales Week Sales Example: Rosco Drugs

n Excel Spreadsheet Showing Input Data

Example: Rosco Drugs n Steps to Moving Average Using Excel Step 1: Select the Tools pull-down menu. Step 2: Select the Data Analysis option. Step 3: When the Data Analysis Tools dialog appears, choose M oving Average. Step 4: When the Moving Average dialog box appears: Enter B4:B13 in the Input Range box. Enter 3 in the Interval box. Enter C4 in the Output Range box. Select OK.

Example: Rosco Drugs n Spreadsheet Showing Results Using n = 3

Smoothing Methods n Centered Moving Average Method The centered moving average method consists of computing an average of n periods' data and associating it with the midpoint of the periods. For example, the average for periods 5, 6, and 7 is associated with period 6. This methodology is useful in the process of computing season indexes.

Example: Rosco Drugs n Spreadsheet Showing Results Using n = 3

Trend Projection n Using the method of least squares, the formula for the trend projection is: T t = b 0 + b 1 t. where: T t = trend forecast for time period t where: T t = trend forecast for time period t b 1 = slope of the trend line b 1 = slope of the trend line b 0 = trend line projection for time 0 b 0 = trend line projection for time 0 b 1 = n  tY t -  t  Y t b 1 = n  tY t -  t  Y t n  t 2 - (  t ) 2 n  t 2 - (  t ) 2 where: Y t = observed value of the time series at time period t where: Y t = observed value of the time series at time period t = average of the observed values for Y t = average of the observed values for Y t = average time period for the n observations = average time period for the n observations

The number of plumbing repair jobs performed by Auger's Plumbing Service in each of the last nine months is listed on the next slide. Forecast the number of repair jobs Auger's will perform in December using the least squares method. Example: Auger’s Plumbing Service

Month Jobs Month Jobs Month Jobs Month Jobs Month Jobs Month Jobs March 353 June 374 September 399 March 353 June 374 September 399 April 387 July 396 October 412 April 387 July 396 October 412 May 342 August 409 November 408 May 342 August 409 November 408 Example: Auger’s Plumbing Service

n Trend Projection (month) t Y t tY t t 2 (month) t Y t tY t t 2 (Mar.) (Apr.) (Apr.) (May) (May) (June) (June) (July) (July) (Aug.) (Aug.) (Sep.) (Sep.) (Oct.) (Oct.) (Nov.) (Nov.) Sum

Example: Auger’s Plumbing Service n Trend Projection (continued) = 45/9 = 5 = 3480/9 = = 45/9 = 5 = 3480/9 = n  tY t -  t  Y t (9)(17844) - (45)(3480) n  tY t -  t  Y t (9)(17844) - (45)(3480) b 1 = = = 7.4 b 1 = = = 7.4 n  t 2 - (  t ) 2 (9)(285) - (45) 2 n  t 2 - (  t ) 2 (9)(285) - (45) 2 = (5) = = (5) = T 10 = (7.4)(10) = T 10 = (7.4)(10) =

Example: Auger’s Plumbing Service n Excel Spreadsheet Showing Input Data

Example: Auger’s Plumbing Service n Steps to Trend Projection Using Excel Step 1: Select an empty cell (B13) in the worksheet. Step 2: Select the Insert pull-down menu. Step 3: Choose the Function option. Step 4: When the Paste Function dialog box appears: Choose Statistical in Function Category box. Choose Forecast in the Function Name box. Select OK. more

Example: Auger’s Plumbing Service n Steps to Trend Projecting Using Excel (continued) Step 5: When the Forecast dialog box appears: Enter 10 in the x box (for month 10). Enter B4:B12 in the Known y’s box. Enter A4:A12 in the Known x’s box. Select OK.

Example: Auger’s Plumbing Service n Spreadsheet with Trend Projection for Month 10

Forecasting with Trend and Seasonal Components n Steps of Multiplicative Time Series Model 1. Calculate the centered moving averages (CMAs). 2. Center the CMAs on integer-valued periods. 3. Determine the seasonal and irregular factors ( S t I t ). 4. Determine the average seasonal factors. 5. Scale the seasonal factors ( S t ). 6. Determine the deseasonalized data. 7. Determine a trend line of the deseasonalized data. 8. Determine the deseasonalized predictions. 9. Take into account the seasonality.

Example: Terry’s Tie Shop Business at Terry's Tie Shop can be viewed as falling into three distinct seasons: (1) Christmas (November-December); (2) Father's Day (late May - mid-June); and (3) all other times. Average weekly sales ($) during each of the three seasons during the past four years are shown on the next slide. Determine a forecast for the average weekly sales in year 5 for each of the three seasons.

Example: Terry’s Tie Shop n Past Sales ($) Year Season Season

Example: Terry’s Tie Shop Dollar Moving Scaled Dollar Moving Scaled Year Season Sales ( Y t ) Average S t I t S t Y t / S t Year Season Sales ( Y t ) Average S t I t S t Y t / S t

n 1. Calculate the centered moving averages. There are three distinct seasons in each year. Hence, take a three-season moving average to eliminate seasonal and irregular factors. For example: 1 st MA = ( )/3 = st MA = ( )/3 = nd MA = ( )/3 = nd MA = ( )/3 = etc. Example: Terry’s Tie Shop

n 2. Center the CMAs on integer-valued periods. The first moving average computed in step 1 ( ) will be centered on season 2 of year 1. Note that the moving averages from step 1 center themselves on integer-valued periods because n is an odd number.

Example: Terry’s Tie Shop n 3. Determine the seasonal & irregular factors ( S t I t ). Isolate the trend and cyclical components. For each period t, this is given by: S t I t = Y t /(Moving Average for period t ) S t I t = Y t /(Moving Average for period t )

Example: Terry’s Tie Shop n 4. Determine the average seasonal factors. Averaging all S t I t values corresponding to that season: Season 1: ( ) /3 = Season 1: ( ) /3 = Season 2: ( ) /4 = Season 2: ( ) /4 = Season 3: ( ) /3 =.587 Season 3: ( ) /3 =.587

Example: Terry’s Tie Shop n 5. Scale the seasonal factors ( S t ). Average the seasonal factors = ( )/3 = Then, divide each seasonal factor by the average of the seasonal factors. Average the seasonal factors = ( )/3 = Then, divide each seasonal factor by the average of the seasonal factors. Season 1: 1.180/1.002 = Season 1: 1.180/1.002 = Season 2: 1.238/1.002 = Season 2: 1.238/1.002 = Season 3:.587/1.002 =.586 Season 3:.587/1.002 =.586 Total = Total = 3.000

Example: Terry’s Tie Shop n 6. Determine the deseasonalized data. Divide the data point values, Y t, by S t. n 7. Determine a trend line of the deseasonalized data. Using the least squares method for t = 1, 2,..., 12, gives: Using the least squares method for t = 1, 2,..., 12, gives: T t = t T t = t

Example: Terry’s Tie Shop n 8. Determine the deseasonalized predictions. Substitute t = 13, 14, and 15 into the least squares equation: T 13 = (33.96)(13) = 2022 T 13 = (33.96)(13) = 2022 T 14 = (33.96)(14) = 2056 T 14 = (33.96)(14) = 2056 T 15 = (33.96)(15) = 2090 T 15 = (33.96)(15) = 2090

Example: Terry’s Tie Shop n 9. Take into account the seasonality. Multiply each deseasonalized prediction by its seasonal factor to give the following forecasts for year 5: Season 1: (1.178)(2022) = Season 1: (1.178)(2022) = Season 2: (1.236)(2056) = Season 2: (1.236)(2056) = Season 3: (.586)(2090) = Season 3: (.586)(2090) =

Qualitative Approaches to Forecasting n Delphi Approach A panel of experts, each of whom is physically separated from the others and is anonymous, is asked to respond to a sequential series of questionnaires. A panel of experts, each of whom is physically separated from the others and is anonymous, is asked to respond to a sequential series of questionnaires. After each questionnaire, the responses are tabulated and the information and opinions of the entire group are made known to each of the other panel members so that they may revise their previous forecast response. After each questionnaire, the responses are tabulated and the information and opinions of the entire group are made known to each of the other panel members so that they may revise their previous forecast response. The process continues until some degree of consensus is achieved. The process continues until some degree of consensus is achieved.

Qualitative Approaches to Forecasting n Scenario Writing Scenario writing consists of developing a conceptual scenario of the future based on a well defined set of assumptions. Scenario writing consists of developing a conceptual scenario of the future based on a well defined set of assumptions. After several different scenarios have been developed, the decision maker determines which is most likely to occur in the future and makes decisions accordingly. After several different scenarios have been developed, the decision maker determines which is most likely to occur in the future and makes decisions accordingly.

Qualitative Approaches to Forecasting n Subjective or Interactive Approaches These techniques are often used by committees or panels seeking to develop new ideas or solve complex problems. These techniques are often used by committees or panels seeking to develop new ideas or solve complex problems. They often involve "brainstorming sessions". They often involve "brainstorming sessions". It is important in such sessions that any ideas or opinions be permitted to be presented without regard to its relevancy and without fear of criticism. It is important in such sessions that any ideas or opinions be permitted to be presented without regard to its relevancy and without fear of criticism.