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**Prepared by Lee Revere and John Large**

Chapter 5 Forecasting Prepared by Lee Revere and John Large To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-1

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**Learning Objectives Students will be able to:**

Understand and know when to use various families of forecasting models. Compare moving averages, exponential smoothing, and trend time-series models. Seasonally adjust data. Understand Delphi and other qualitative decision-making approaches. Compute a variety of error measures. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-2

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**Chapter Outline 5.1 Introduction 5.2 Types of Forecasts**

5.3 Scatter Diagrams and Time Series 5.4 Measures of Forecast Accuracy 5.5 Time-Series Forecasting Models 5.6 Monitoring and Controlling Forecasts 5.7 Using the Computer to Forecast To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-3

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**Introduction Eight steps to forecasting:**

Determine the use of the forecast. Select the items or quantities to be forecasted. Determine the time horizon of the forecast. Select the forecasting model or models. Gather the data needed to make the forecast. Validate the forecasting model. Make the forecast. Implement the results. These steps provide a systematic way of initiating, designing, and implementing a forecasting system. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-4

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**Exponential Smoothing**

Types of Forecasts Forecasting Techniques No single method is superior Qualitative Models: attempt to include subjective factors Time-Series Methods: include historical data over a time interval Causal Methods: include a variety of factors Delphi Methods Moving Average Regression Analysis Jury of Executive Opinion Exponential Smoothing Multiple Regression Trend Projections Sales Force Composite Decomposition Consumer Market Survey To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-5

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Qualitative Methods Delphi Method interactive group process consisting of obtaining information from a group of respondents through questionnaires and surveys Jury of Executive Opinion obtains opinions of a small group of high-level managers in combination with statistical models Sales Force Composite allows each sales person to estimate the sales for his/her region and then compiles the data at a district or national level Consumer Market Survey solicits input from customers or potential customers regarding their future purchasing plans To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-6

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Scatter Diagrams Scatter diagrams are helpful when forecasting time-series data because they depict the relationship between variables. Radios Televisions Compact Discs To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-7

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**Measures of Forecast Accuracy**

Forecast errors allow one to see how well the forecast model works and compare that model with other forecast models. Forecast error = actual value – forecast value To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-8

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**Measures of Forecast Accuracy (continued)**

Measures of forecast accuracy include: Mean Absolute Deviation (MAD) Mean Squared Error (MSE) Mean Absolute Percent Error (MAPE) = å |forecast errors| n 2 = å (errors) n = å actual n error 100% To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-9

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**Hospital Days – Forecast Error Example**

Ms. Smith forecasted total hospital inpatient days last year. Now that the actual data are known, she is reevaluating her forecasting model. Compute the MAD, MSE, and MAPE for her forecast. Month Forecast Actual JAN 250 243 FEB 320 315 MAR 275 286 APR 260 256 MAY 241 JUN 298 JUL 300 292 AUG 325 333 SEP 326 OCT 350 378 NOV 365 382 DEC 380 396 To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-10

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**Hospital Days – Forecast Error Example**

Actual |error| error^2 |error/actual| JAN 250 243 7 49 0.03 FEB 320 315 5 25 0.02 MAR 275 286 11 121 0.04 APR 260 256 4 16 MAY 241 9 81 JUN 298 23 529 0.08 JUL 300 292 8 64 AUG 325 333 SEP 326 6 36 OCT 350 378 28 784 0.07 NOV 365 382 17 289 DEC 380 396 AVERAGE 11.83 192.83 3.68 MAD = MSE = MAPE = .0368*100 = To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-11

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**Decomposition of a Time-Series**

Time series can be decomposed into: Trend (T): gradual up or down movement over time Seasonality (S): pattern of fluctuations above or below trend line that occurs every year Cycles(C): patterns in data that occur every several years Random variations (R): “blips”in the data caused by chance and unusual situations To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-12

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**Components of Decomposition**

Actual Data Trend Cyclic Random To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-13

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**Decomposition of Time-Series: Two Models**

Multiplicative model assumes demand is the product of the four components. demand = T * S * C * R Additive model assumes demand is the summation of the four components. demand = T + S + C + R To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-14

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**Moving Averages Simple moving average = å demand in previous n periods**

Moving average methods consist of computing an average of the most recent n data values for the time series and using this average for the forecast of the next period. Simple moving average = å demand in previous n periods n To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-15

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**Wallace Garden Supply’s Three-Month Moving Average**

Actual Shed Sales Three-Month Moving Average January 10 February 12 March 13 April 16 May 19 June 23 July 26 ( )/3 = 11 2/3 ( )/3 = 13 2/3 ( )/3 = 16 ( )/3 = 19 1/3 To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-16

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**Weighted Moving Averages**

Weighted moving averages use weights to put more emphasis on recent periods. Weighted moving average = (weight for period n) (demand in period n) ∑ weights To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-17

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**Calculating Weighted Moving Averages**

Weights Applied Period Last month 3 Two months ago 2 Three months ago 1 3*Sales last month + 2*Sales two months ago + 1*Sales three months ago 6 Sum of weights To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-18

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**Wallace Garden’s Weighted Three-Month Moving Average**

Actual Shed Sales Three-Month Weighted Moving Average 10 12 13 16 19 23 January February March April May June July 26 [3*13+2*12+1*10]/6 = 12 1/6 [3*16+2*13+1*12]/6 =14 1/3 [3*19+2*16+1*13]/6 = 17 [3*23+2*19+1*16]/6 = 20 1/2 To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-19

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**Exponential Smoothing**

Exponential smoothing is a type of moving average technique that involves little record keeping of past data. New forecast = previous forecast + (previous actual –previous forecast) Mathematically this is expressed as: Ft = Ft-1 + (Yt-1 - Ft-1) Ft = new forecast Ft-1 = previous forecast = smoothing constant Yt-1 = previous period actual To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-20

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**Port of Baltimore Exponential Smoothing Example**

Qtr Actual Tonnage Unloaded Rounded Forecast using =0.10 1 180 175 2 168 176= ( ) 3 159 175 = ( ) 4 173 = ( ) 5 190 173 = ( ) 6 205 175 = ( ) 7 178 = ( ) 8 182 178 = ( ) 9 ? 179= ( ) To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-21

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**Port of Baltimore Exponential Smoothing Example**

Qtr Actual Tonnage Unloaded Rounded Forecast using =0.50 1 180 175 2 168 178 = ( ) 3 159 173 = ( ) 4 166 = ( ) 5 190 170 = ( ) 6 205 180 = ( ) 7 193 = ( ) 8 182 186 = ( ) 9 ? 184 = ( ) To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-22

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**Selecting a Smoothing Constant**

To select the best smoothing constant, evaluate the accuracy of each forecasting model. Actual Forecast with a = 0.10 Absolute Deviations Forecast with a = 0.50 180 175 5 168 176 8 178 10 159 16 173 14 2 166 9 190 17 170 20 205 30 25 193 13 182 4 186 MAD 10.0 12 The lowest MAD results from = 0.10 To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-23

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**PM Computer: Moving Average Example**

PM Computer assembles customized personal computers from generic parts. The owners purchase generic computer parts in volume at a discount from a variety of sources whenever they see a good deal. It is important that they develop a good forecast of demand for their computers so they can purchase component parts efficiently. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-24

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**PM Computers: Data Compute a 2-month moving average**

Period month actual demand 1 Jan 37 2 Feb 40 3 Mar 41 4 Apr 37 5 May 45 6 June 50 7 July 43 8 Aug 47 9 Sept 56 Compute a 2-month moving average Compute a 3-month weighted average using weights of 4,2,1 for the past three months of data Compute an exponential smoothing forecast using = 0.7 Using MAD, what forecast is most accurate? To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-25

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**PM Computers: Moving Average Solution**

MAD Exponential smoothing resulted in the lowest MAD. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-26

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**Exponential Smoothing with Trend Adjustment**

Simple exponential smoothing fails to respond to trends, so a more complex model is necessary with trend adjustment. Simple exponential smoothing - first-order smoothing Trend adjusted smoothing - second-order smoothing Low gives less weight to more recent trends, while high gives higher weight to more recent trends. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-27

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**Exponential Smoothing with Trend Adjustment**

Forecast including trend (FITt+1) = new forecast (Ft) + trend correction(Tt) Tt = (1 - )Tt-1 + (Ft – Ft-1) where Ti = smoothed trend for period 1 Ti-1 = smoothed trend for the preceding period = trend smoothing constant Ft = simple exponential smoothed forecast for period t Ft-1 = forecast for period t-1 To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-28

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**Trend Projection Trend projections are used to forecast**

time-series data that exhibit a linear trend. Least squares may be used to determine a trend projection for future forecasts. Least squares determines the trend line forecast by minimizing the mean squared error between the trend line forecasts and the actual observed values. The independent variable is the time period and the dependent variable is the actual observed value in the time series. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-29

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**Trend Projection (continued)**

The formula for the trend projection is: Y = b + b X where: Y = predicted value b1 = slope of the trend line b0 = intercept X = time period (1,2,3…n) To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-30

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**Midwestern Manufacturing Trend Projection Example**

Midwestern Manufacturing Company’s demand for electrical generators over the period of 1996 – 2000 is given below. Year Time Sales 1996 1 74 1997 2 79 1998 3 80 1999 4 90 2000 5 105 2001 6 142 2002 7 122 To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-31

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**Midwestern Manufacturing Company Trend Solution**

Sales = (time) To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-32

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**Midwestern Manufacturing’s Trend**

Trend Line Forecast points Actual demand line To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-33

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Seasonal Variations Seasonal indices can be used to make adjustments in the forecast for seasonality. A seasonal index indicates how a particular season compares with an average season. The seasonal index can be found by dividing the average value for a particular season by the average of all the data. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-34

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**Eichler Supplies: Seasonal Index Example**

Month Sales Demand Average Two-Year Monthly Seasonal Index Year 1 2 80 100 90 94 0.957 75 85 0.851 0.904 110 1.064 Jan Feb Mar Apr May 115 131 123 1.309 … Total Average Demand 1,128 Seasonal Index: = Average 2 -year demand/Average monthly demand To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-35

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**Seasonal Variations with Trend**

Centered Moving Average (CMA) is an approach that prevents a variation due to trend from being incorrectly interpreted as a variation due to the season. Steps of Multiplicative Time-Series Model 1. Compute the CMA for each observation. Compute seasonal ratio (observation/CMA). 3. Average seasonal ratios to get seasonal indices. 4. If seasonal indices do not add to the number of seasons, multiply each index by (number of seasons)/(sum of the indices). To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-36

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**Turner Industries Seasonal Variations with Trend**

Turner Industries’ sales figures are shown below with the CMA and seasonal ratio. CMA (qtr 3 / yr 1 ) = .5(108) (116) 4 Seasonal Ratio = Sales Qtr 3 = 150 CMA To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-37

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**Decomposition Method with Trend and Seasonal Components**

Decomposition is the process of isolating linear trend and seasonal factors to develop more accurate forecasts. There are five steps to decomposition: 1. Compute the seasonal index for each season. 2. Deseasonalize the data by dividing each number by its seasonal index. 3. Compute a trend line with the deseasonalized data. 4. Use the trend line to forecast. 5. Multiply the forecasts by the seasonal index. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-38

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**Turner Industries: Decomposition Method**

Turner Industries has noticed a trend in quarterly sales figures. There is also a seasonal component. Below is the seasonal index and deseasonalized sales data. * 108 0.85 = Seasonal Index for Qtr 1 = = 0.85 2 This value is derived by averaging the season rations for each quarter. Refer to slide 5-37. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-39

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**Turner Industries: Decomposition Method**

Using the deseasonalized data, the following trend line was computed: Sales = X To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-40

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**Turner Industries: Decomposition Method**

Using the trend line, the following forecast was computed: Sales = X For period 13 (quarter 1/ year 4): Sales = (13) = (before seasonality adjustment) After seasonality adjustment: Sales = (0.85) = Seasonal index for quarter 1 To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-41

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**Multiple Regression with Trend and Seasonal Components**

Multiple regression can be used to develop an additive decomposition model. One independent variable is time. Seasons are represented by dummy independent variables. Y = a + b X + b X + b X + b X Where X = time period X = 1 if quarter 2 = 0 otherwise X = 1 if quarter 3 X = 1 if quarter 4 1 2 3 4 To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-42

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**Monitoring and Controlling Forecasts**

Tracking signals measure how well predictions fit actual data. To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-43

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**Monitoring and Controlling Forecasts**

To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 5-44

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