Report

Chapter 5 Forecasting To accompany Quantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna Power Point slides created by Jeff Heyl © 2008 Prentice-Hall, Inc. © 2009 Prentice-Hall, Inc. Introduction Managers are always trying to reduce uncertainty and make better estimates of what will happen in the future This is the main purpose of forecasting Some firms use subjective methods Seat-of-the pants methods, intuition, experience There are also several quantitative techniques Moving averages, exponential smoothing, trend projections, least squares regression analysis © 2009 Prentice-Hall, Inc. 5–2 Introduction Eight steps to forecasting : 1. Determine the use of the forecast—what objective are we trying to obtain? 2. Select the items or quantities that are to be forecasted 3. Determine the time horizon of the forecast 4. Select the forecasting model or models 5. Gather the data needed to make the forecast 6. Validate the forecasting model 7. Make the forecast 8. Implement the results © 2009 Prentice-Hall, Inc. 5–3 Introduction These steps are a systematic way of initiating, designing, and implementing a forecasting system When used regularly over time, data is collected routinely and calculations performed automatically There is seldom one superior forecasting system Different organizations may use different techniques Whatever tool works best for a firm is the one they should use © 2009 Prentice-Hall, Inc. 5–4 Forecasting Models Forecasting Techniques Qualitative Models Time-Series Methods Causal Methods Delphi Methods Moving Average Regression Analysis Jury of Executive Opinion Exponential Smoothing Multiple Regression Sales Force Composite Trend Projections Figure 5.1 Consumer Market Survey Decomposition © 2009 Prentice-Hall, Inc. 5–5 Time-Series Models Time-series models attempt to predict the future based on the past Common time-series models are Moving average Exponential smoothing Trend projections Decomposition Regression analysis is used in trend projections and one type of decomposition model © 2009 Prentice-Hall, Inc. 5–6 Causal Models Causal models use variables or factors that might influence the quantity being forecasted The objective is to build a model with the best statistical relationship between the variable being forecast and the independent variables Regression analysis is the most common technique used in causal modeling © 2009 Prentice-Hall, Inc. 5–7 Qualitative Models Qualitative models incorporate judgmental or subjective factors Useful when subjective factors are thought to be important or when accurate quantitative data is difficult to obtain Common qualitative techniques are Delphi method Jury of executive opinion Sales force composite Consumer market surveys © 2009 Prentice-Hall, Inc. 5–8 Qualitative Models Delphi Method – an iterative group process where (possibly geographically dispersed) respondents provide input to decision makers Jury of Executive Opinion – collects opinions of a small group of high-level managers, possibly using statistical models for analysis Sales Force Composite – individual salespersons estimate the sales in their region and the data is compiled at a district or national level Consumer Market Survey – input is solicited from customers or potential customers regarding their purchasing plans © 2009 Prentice-Hall, Inc. 5–9 Scatter Diagrams Annual Sales Scatter diagrams are helpful when forecasting time-series data because they depict the relationship between variables. 450 400 350 300 250 200 150 100 50 0 Televisions 0 2 4 6 8 10 12 Time (Years) © 2009 Prentice-Hall, Inc. 5 – 10 Scatter Diagrams Wacker Distributors wants to forecast sales for three different products YEAR TELEVISION SETS RADIOS COMPACT DISC PLAYERS 1 2 3 4 5 6 7 8 9 10 250 250 250 250 250 250 250 250 250 250 300 310 320 330 340 350 360 370 380 390 110 100 120 140 170 150 160 190 200 190 Table 5.1 © 2009 Prentice-Hall, Inc. 5 – 11 Scatter Diagrams Annual Sales of Televisions (a) Sales appear to be 330 – 250 – 200 – 150 – 100 – constant over time Sales = 250 A good estimate of sales in year 11 is 250 televisions 50 – | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 Time (Years) Figure 5.2 © 2009 Prentice-Hall, Inc. 5 – 12 Scatter Diagrams (b) Annual Sales of Radios 420 – Sales appear to be 400 – 380 – 360 – 340 – 320 – 300 – 280 – | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 increasing at a constant rate of 10 radios per year Sales = 290 + 10(Year) A reasonable estimate of sales in year 11 is 400 televisions Time (Years) Figure 5.2 © 2009 Prentice-Hall, Inc. 5 – 13 Scatter Diagrams (c) Annual Sales of CD Players This trend line may 200 – 180 – 160 – 140 – 120 – 100 – | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 not be perfectly accurate because of variation from year to year Sales appear to be increasing A forecast would probably be a larger figure each year Time (Years) Figure 5.2 © 2009 Prentice-Hall, Inc. 5 – 14 Measures of Forecast Accuracy We compare forecasted values with actual values to see how well one model works or to compare models Forecast error = Actual value – Forecast value One measure of accuracy is the mean absolute deviation (MAD) forecast error MAD n © 2009 Prentice-Hall, Inc. 5 – 15 Measures of Forecast Accuracy Using a naïve forecasting model YEAR ACTUAL SALES OF CD PLAYERS FORECAST SALES ABSOLUTE VALUE OF ERRORS (DEVIATION), (ACTUAL – FORECAST) 1 110 — — 2 100 110 |100 – 110| = 10 3 120 100 |120 – 110| = 20 4 140 120 |140 – 120| = 20 5 170 140 |170 – 140| = 30 6 150 170 |150 – 170| = 20 7 160 150 |160 – 150| = 10 8 190 160 |190 – 160| = 30 9 200 190 |200 – 190| = 10 10 190 200 |190 – 200| = 10 11 — 190 — Sum of |errors| = 160 MAD = 160/9 = 17.8 Table 5.2 © 2009 Prentice-Hall, Inc. 5 – 16 Measures of Forecast Accuracy Using a naïve forecasting model YEAR ACTUAL SALES OF CD PLAYERS FORECAST SALES ABSOLUTE VALUE OF ERRORS (DEVIATION), (ACTUAL – FORECAST) 1 110 — — 2 100 110 |100 – 110| = 10 3 120 100 |120 – 110| = 20 4 140 120 |140 – 120| = 20 5 170 140 |170 – 140| = 30 6 150 7 160 150 |160 – 150| = 10 8 190 160 |190 – 160| = 30 9 200 190 |200 – 190| = 10 10 190 200 |190 – 200| = 10 11 — 190 — forecast error 160 MAD 17.8 n 170 9 |150 – 170| = 20 Sum of |errors| = 160 MAD = 160/9 = 17.8 Table 5.2 © 2009 Prentice-Hall, Inc. 5 – 17 Measures of Forecast Accuracy There are other popular measures of forecast accuracy The mean squared error 2 ( error) MSE n The mean absolute percent error MAPE error actual 100% n And bias is the average error and tells whether the forecast tends to be too high or too low and by how much. Thus, it can be negative or positive. © 2009 Prentice-Hall, Inc. 5 – 18 Measures of Forecast Accuracy Year Actual CD Sales Forecast Sales |Actual -Forecast| 1 110 2 100 110 10 3 120 100 20 4 140 120 20 5 170 140 30 6 150 170 20 7 160 150 10 8 190 160 30 9 200 190 10 10 190 200 10 11 190 Sum of |errors| MAD 160 17.8 © 2009 Prentice-Hall, Inc. 5 – 19 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 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Forecast Actual 250 320 275 260 250 275 300 325 320 350 365 380 243 315 286 256 241 298 292 333 326 378 382 396 © 2009 Prentice-Hall, Inc. 5 – 20 Hospital Days Forecast Error Example Actual 243 |error| error2 |error/actual| JAN Forecast 250 7 49 0.03 FEB 320 315 5 25 0.02 MAR 275 286 11 121 0.04 APR 260 256 4 16 0.02 MAY 250 241 9 81 0.04 JUN 275 298 23 529 0.08 JUL 300 292 8 64 0.03 AUG 325 333 8 64 0.02 SEP 320 326 6 36 0.02 OCT 350 378 28 784 0.07 NOV 365 382 17 289 0.04 DEC 380 396 16 256 0.04 AVERAGE MAD= 11.83 MSE= 192.83 MAPE= .0381*100 = 3.81 © 2009 Prentice-Hall, Inc. 5 – 21 Time-Series Forecasting Models A time series is a sequence of evenly spaced events (weekly, monthly, quarterly, etc.) Time-series forecasts predict the future based solely of the past values of the variable Other variables, no matter how potentially valuable, are ignored © 2009 Prentice-Hall, Inc. 5 – 22 Decomposition of a Time-Series A time series typically has four components 1. Trend (T) is the gradual upward or downward movement of the data over time 2. Seasonality (S) is a pattern of demand fluctuations above or below trend line that repeats at regular intervals 3. Cycles (C) are patterns in annual data that occur every several years 4. Random variations (R) are “blips” in the data caused by chance and unusual situations © 2009 Prentice-Hall, Inc. 5 – 23 Demand for Product or Service Decomposition of a Time-Series Figure 5.3 Trend Component Seasonal Peaks Actual Demand Line Average Demand over 4 Years | | | | Year 1 Year 2 Year 3 Year 4 Time © 2009 Prentice-Hall, Inc. 5 – 24 Decomposition of a Time-Series There are two general forms of time-series models The multiplicative model Demand = T x S x C x R The additive model Demand = T + S + C + R Models may be combinations of these two forms Forecasters often assume errors are normally distributed with a mean of zero © 2009 Prentice-Hall, Inc. 5 – 25 Moving Averages Moving averages can be used when demand is relatively steady over time The next forecast is the average of the most recent n data values from the time series The most recent period of data is added and the oldest is dropped This methods tends to smooth out short-term irregularities in the data series Moving average forecast Sum of demands in previous n periods n © 2009 Prentice-Hall, Inc. 5 – 26 Moving Averages Mathematically Ft 1 Yt Yt 1 ... Yt n1 n where Ft 1 = forecast for time period t + 1 Yt = actual value in time period t n = number of periods to average © 2009 Prentice-Hall, Inc. 5 – 27 Wallace Garden Supply Example Wallace Garden Supply wants to forecast demand for its Storage Shed They have collected data for the past year They are using a three-month moving average to forecast demand (n = 3) © 2009 Prentice-Hall, Inc. 5 – 28 Wallace Garden Supply Example MONTH ACTUAL SHED SALES THREE-MONTH MOVING AVERAGE January 10 February 12 March 13 April 16 (10 + 12 + 13)/3 = 11.67 May 19 (12 + 13 + 16)/3 = 13.67 June 23 (13 + 16 + 19)/3 = 16.00 July 26 (16 + 19 + 23)/3 = 19.33 August 30 (19 + 23 + 26)/3 = 22.67 September 28 (23 + 26 + 30)/3 = 26.33 October 18 (26 + 30 + 28)/3 = 28.00 November 16 (30 + 28 + 18)/3 = 25.33 December 14 (28 + 18 + 16)/3 = 20.67 January — (18 + 16 + 14)/3 = 16.00 Table 5.3 © 2009 Prentice-Hall, Inc. 5 – 29 Weighted Moving Averages Weighted moving averages use weights to put more emphasis on recent periods Often used when a trend or other pattern is emerging Ft 1 ( Weight in period i )( Actual value in period) ( Weights) Mathematically w1Yt w2Yt 1 ... wnYt n1 Ft 1 w1 w2 ... wn where wi = weight for the ith observation © 2009 Prentice-Hall, Inc. 5 – 30 Weighted Moving Averages Both simple and weighted averages are effective in smoothing out fluctuations in the demand pattern in order to provide stable estimates Problems Increasing the size of n smoothes out fluctuations better, but makes the method less sensitive to real changes in the data Moving averages can not pick up trends very well – they will always stay within past levels and not predict a change to a higher or lower level © 2009 Prentice-Hall, Inc. 5 – 31 Wallace Garden Supply Example Wallace Garden Supply decides to try a weighted moving average model to forecast demand for its Storage Shed They decide on the following weighting scheme WEIGHTS APPLIED PERIOD 3 2 1 Last month Two months ago Three months ago 3 x Sales last month + 2 x Sales two months ago + 1 X Sales three months ago 6 Sum of the weights © 2009 Prentice-Hall, Inc. 5 – 32 Wallace Garden Supply Example THREE-MONTH WEIGHTED MOVING AVERAGE MONTH ACTUAL SHED SALES January 10 February 12 March 13 April 16 [(3 X 13) + (2 X 12) + (10)]/6 = 12.17 May 19 [(3 X 16) + (2 X 13) + (12)]/6 = 14.33 June 23 [(3 X 19) + (2 X 16) + (13)]/6 = 17.00 July 26 [(3 X 23) + (2 X 19) + (16)]/6 = 20.50 August 30 [(3 X 26) + (2 X 23) + (19)]/6 = 23.83 September 28 [(3 X 30) + (2 X 26) + (23)]/6 = 27.50 October 18 [(3 X 28) + (2 X 30) + (26)]/6 = 28.33 November 16 [(3 X 18) + (2 X 28) + (30)]/6 = 23.33 December 14 [(3 X 16) + (2 X 18) + (28)]/6 = 18.67 January — [(3 X 14) + (2 X 16) + (18)]/6 = 15.33 Table 5.4 © 2009 Prentice-Hall, Inc. 5 – 33 Wallace Garden Supply Example Program 5.1A © 2009 Prentice-Hall, Inc. 5 – 34 Wallace Garden Supply Example Program 5.1B © 2009 Prentice-Hall, Inc. 5 – 35 Exponential Smoothing Exponential smoothing is easy to use and requires little record keeping of data It is a type of moving average New forecast = Last period’s forecast + (Last period’s actual demand – Last period’s forecast) Where is a weight (or smoothing constant) with a value between 0 and 1 inclusive A larger gives more importance to recent data while a smaller value gives more importance to past data © 2009 Prentice-Hall, Inc. 5 – 36 Exponential Smoothing Mathematically Ft 1 Ft (Yt Ft ) where Ft+1 = new forecast (for time period t + 1) Ft = pervious forecast (for time period t) = smoothing constant (0 ≤ ≤ 1) Yt = pervious period’s actual demand The idea is simple – the new estimate is the old estimate plus some fraction of the error in the last period © 2009 Prentice-Hall, Inc. 5 – 37 Exponential Smoothing Example In January, February’s demand for a certain car model was predicted to be 142 Actual February demand was 153 autos Using a smoothing constant of = 0.20, what is the forecast for March? New forecast (for March demand) = 142 + 0.2(153 – 142) = 144.2 or 144 autos If actual demand in March was 136 autos, the April forecast would be New forecast (for April demand) = 144.2 + 0.2(136 – 144.2) = 142.6 or 143 autos © 2009 Prentice-Hall, Inc. 5 – 38 Selecting the Smoothing Constant Selecting the appropriate value for is key to obtaining a good forecast The objective is always to generate an accurate forecast The general approach is to develop trial forecasts with different values of and select the that results in the lowest MAD © 2009 Prentice-Hall, Inc. 5 – 39 Port of Baltimore Example Exponential smoothing forecast for two values of QUARTER ACTUAL TONNAGE UNLOADED 1 180 175 175 2 168 175.5 = 175.00 + 0.10(180 – 175) 177.5 3 159 174.75 = 175.50 + 0.10(168 – 175.50) 172.75 4 175 173.18 = 174.75 + 0.10(159 – 174.75) 165.88 5 190 173.36 = 173.18 + 0.10(175 – 173.18) 170.44 6 205 175.02 = 173.36 + 0.10(190 – 173.36) 180.22 7 180 178.02 = 175.02 + 0.10(205 – 175.02) 192.61 8 182 178.22 = 178.02 + 0.10(180 – 178.02) 186.30 9 ? 178.60 = 178.22 + 0.10(182 – 178.22) 184.15 FORECAST USING =0.10 FORECAST USING =0.50 Table 5.5 © 2009 Prentice-Hall, Inc. 5 – 40 Selecting the Best Value of QUARTER ACTUAL TONNAGE UNLOADED 1 180 175 5….. 175 5…. 2 168 175.5 7.5.. 177.5 9.5.. 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.3.. FORECAST WITH = 0.10 ABSOLUTE DEVIATIONS FOR = 0.10 Sum of absolute deviations MAD = Table 5.6 FORECAST WITH = 0.50 ABSOLUTE DEVIATIONS FOR = 0.50 82.45 Σ|deviations| n = 10.31 98.63 MAD = 12.33 Best choice © 2009 Prentice-Hall, Inc. 5 – 41 Port of Baltimore Example Program 5.2A © 2009 Prentice-Hall, Inc. 5 – 42 Port of Baltimore Example Program 5.2B © 2009 Prentice-Hall, Inc. 5 – 43 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. © 2009 Prentice-Hall, Inc. 5 – 44 PM Computers: Data 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, previous forecast of 40 Using MAD, what forecast is most accurate? © 2009 Prentice-Hall, Inc. 5 – 45 PM Computers: Moving Average Solution 2 month MA Abs. Dev 3 month WMA Abs. Dev Exp.Sm. Abs. Dev 37.00 37.00 3.00 39.10 1.90 38.50 2.50 40.50 3.50 40.14 3.14 40.43 3.43 39.00 6.00 38.57 6.43 38.03 6.97 41.00 9.00 42.14 7.86 42.91 7.09 47.50 4.50 46.71 3.71 47.87 4.87 46.50 0.50 45.29 1.71 44.46 2.54 45.00 11.00 46.29 9.71 46.24 9.76 51.50 MAD 51.57 5.29 53.07 5.43 4.95 Exponential smoothing resulted in the lowest MAD. © 2009 Prentice-Hall, Inc. 5 – 46 Exponential Smoothing with Trend Adjustment Like all averaging techniques, exponential smoothing does not respond to trends A more complex model can be used that adjusts for trends The basic approach is to develop an exponential smoothing forecast then adjust it for the trend Forecast including trend (FITt) = New forecast (Ft) + Trend correction (Tt) © 2009 Prentice-Hall, Inc. 5 – 47 Exponential Smoothing with Trend Adjustment The equation for the trend correction uses a new smoothing constant Tt is computed by Tt 1 (1 )Tt ( Ft 1 Ft ) where Tt+1 = Tt = = Ft+1 = smoothed trend for period t + 1 smoothed trend for preceding period trend smooth constant that we select simple exponential smoothed forecast for period t + 1 Ft = forecast for pervious period © 2009 Prentice-Hall, Inc. 5 – 48 Selecting a Smoothing Constant As with exponential smoothing, a high value of makes the forecast more responsive to changes in trend A low value of gives less weight to the recent trend and tends to smooth out the trend Values are generally selected using a trial-anderror approach based on the value of the MAD for different values of Simple exponential smoothing is often referred to as first-order smoothing Trend-adjusted smoothing is called second-order, double smoothing, or Holt’s method © 2009 Prentice-Hall, Inc. 5 – 49 Trend Projection Trend projection fits a trend line to a series of historical data points The line is projected into the future for medium- to long-range forecasts Several trend equations can be developed based on exponential or quadratic models The simplest is a linear model developed using regression analysis © 2009 Prentice-Hall, Inc. 5 – 50 Trend Projection Trend projections are used to forecast time- series data that exhibit a linear trend. A trend line is simply a linear regression equation in which the independent variable (X) is the time period 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. © 2009 Prentice-Hall, Inc. 5 – 51 Trend Projection The mathematical form is Yˆ b0 b1 X where Yˆ = predicted value b0 = intercept b1 = slope of the line X = time period (i.e., X = 1, 2, 3, …, n) © 2009 Prentice-Hall, Inc. 5 – 52 Trend Projection Value of Dependent Variable Dist7 Dist5 * * Dist3 * * Dist6 Dist4 Dist1 * * Dist2 Time * Figure 5.4 © 2009 Prentice-Hall, Inc. 5 – 53 Midwestern Manufacturing Company Example Midwestern Manufacturing Company has experienced the following demand for it’s electrical generators over the period of 2001 – 2007 YEAR ELECTRICAL GENERATORS SOLD 2001 2002 2003 2004 2005 2006 2007 74 79 80 90 105 142 122 Table 5.7 © 2009 Prentice-Hall, Inc. 5 – 54 Midwestern Manufacturing Company Example Notice code instead of actual years Program 5.3A © 2009 Prentice-Hall, Inc. 5 – 55 Midwestern Manufacturing Company Example r2 says model predicts about 80% of the variability in demand Significance level for F-test indicates a definite relationship Program 5.3B © 2009 Prentice-Hall, Inc. 5 – 56 Midwestern Manufacturing Company Example The forecast equation is Yˆ 56.71 10.54 X To project demand for 2008, we use the coding system to define X = 8 (sales in 2008) = 56.71 + 10.54(8) = 141.03, or 141 generators Likewise for X = 9 (sales in 2009) = 56.71 + 10.54(9) = 151.57, or 152 generators © 2009 Prentice-Hall, Inc. 5 – 57 Midwestern Manufacturing Company Example 160 – 150 – Generator Demand 140 – Trend Line Yˆ 56.71 10.54 X 130 – 120 – 110 – 100 – 90 – 80 – 70 – | | Actual Demand Line 60 – 50 – | Figure 5.5 | | | | | | 2001 2002 2003 2004 2005 2006 2007 2008 2009 Year © 2009 Prentice-Hall, Inc. 5 – 58 Midwestern Manufacturing Company Example Program 5.4A © 2009 Prentice-Hall, Inc. 5 – 59 Midwestern Manufacturing Company Example Program 5.4B © 2009 Prentice-Hall, Inc. 5 – 60 Seasonal Variations Recurring variations over time may indicate the need for seasonal adjustments in the trend line A seasonal index indicates how a particular season compares with an average season When no trend is present, the seasonal index can be found by dividing the average value for a particular season by the average of all the data © 2009 Prentice-Hall, Inc. 5 – 61 Seasonal Variations Eichler Supplies sells telephone answering machines Data has been collected for the past two years sales of one particular model They want to create a forecast that includes seasonality © 2009 Prentice-Hall, Inc. 5 – 62 Seasonal Variations SALES DEMAND MONTH YEAR 1 YEAR 2 AVERAGE TWOYEAR DEMAND MONTHLY DEMAND AVERAGE SEASONAL INDEX January 80 100 90 94 0.957 February 85 75 80 94 0.851 March 80 90 85 94 0.904 April 110 90 100 94 1.064 May 115 131 123 94 1.309 June 120 110 115 94 1.223 July 100 110 105 94 1.117 August 110 90 100 94 1.064 September 85 95 90 94 0.957 October 75 85 80 94 0.851 November 85 75 80 94 0.851 December 80 80 80 94 0.851 Total average demand = 1,128 Average monthly demand = Table 5.8 1,128 = 94 12 months Average two-year demand Seasonal index = Average monthly demand © 2009 Prentice-Hall, Inc. 5 – 63 Seasonal Variations The calculations for the seasonal indices are Jan. 1,200 0.957 96 12 July 1,200 1.117 112 12 Feb. 1,200 0.851 85 12 Aug. 1,200 1.064 106 12 Mar. 1,200 0.904 90 12 Sept. 1,200 0.957 96 12 Apr. 1,200 1.064 106 12 Oct. 1,200 0.851 85 12 May 1,200 1.309 131 12 Nov. 1,200 0.851 85 12 June 1,200 1.223 122 12 Dec. 1,200 0.851 85 12 © 2009 Prentice-Hall, Inc. 5 – 64 Regression with Trend and Seasonal Components Multiple regression can be used to forecast both trend and seasonal components in a time series One independent variable is time Dummy independent variables are used to represent the seasons The model is an additive decomposition model Yˆ a b1 X 1 b2 X 2 b3 X 3 b4 X 4 where X1 X2 X3 X4 = time period = 1 if quarter 2, 0 otherwise = 1 if quarter 3, 0 otherwise = 1 if quarter 4, 0 otherwise © 2009 Prentice-Hall, Inc. 5 – 65 Regression with Trend and Seasonal Components Program 5.6A © 2009 Prentice-Hall, Inc. 5 – 66 Regression with Trend and Seasonal Components Program 5.6B (partial) © 2009 Prentice-Hall, Inc. 5 – 67 Regression with Trend and Seasonal Components The resulting regression equation is Yˆ 104.1 2.3 X 1 15.7 X 2 38.7 X 3 30.1X 4 Using the model to forecast sales for the first two quarters of next year Yˆ 104.1 2.3(13) 15.7(0) 38.7(0) 30.1(0) 134 Yˆ 104.1 2.3(14) 15.7(1) 38.7(0) 30.1(0) 152 These are different from the results obtained using the multiplicative decomposition method Use MAD and MSE to determine the best model © 2009 Prentice-Hall, Inc. 5 – 68 Regression with Trend and Seasonal Components American Airlines original spare parts inventory system used only time-series methods to forecast the demand for spare parts This method was slow to responds to even moderate changes in aircraft utilization let alone major fleet expansions They developed a PC-based system named RAPS which uses linear regression to establish a relationship between monthly part removals and various functions of monthly flying hours The computation now takes only one hour instead of the days the old system needed Using RAPS provided a one time savings of $7 million and a recurring annual savings of nearly $1 million © 2009 Prentice-Hall, Inc. 5 – 69 Monitoring and Controlling Forecasts Tracking signals can be used to monitor the performance of a forecast Tacking signals are computed using the following equation RSFE Tracking signal MAD where forecast error MAD n © 2009 Prentice-Hall, Inc. 5 – 70 Monitoring and Controlling Forecasts Signal Tripped Upper Control Limit + Tracking Signal Acceptable Range 0 MADs – Lower Control Limit Time Figure 5.7 © 2009 Prentice-Hall, Inc. 5 – 71 Monitoring and Controlling Forecasts Positive tracking signals indicate demand is greater than forecast Negative tracking signals indicate demand is less than forecast Some variation is expected, but a good forecast will have about as much positive error as negative error Problems are indicated when the signal trips either the upper or lower predetermined limits This indicates there has been an unacceptable amount of variation Limits should be reasonable and may vary from item to item © 2009 Prentice-Hall, Inc. 5 – 72 Regression with Trend and Seasonal Components How do you decide on the upper and lower limits? Too small a value will trip the signal too often and too large will cause a bad forecast Plossl & Wight – use maximums of ±4 MADs for high volume stock items and ±8 MADs for lower volume items One MAD is equivalent to approximately 0.8 standard deviation so that ±4 MADs =3.2 s.d. For a forecast to be “in control”, 89% of the errors are expected to fall within ±2 MADs, 98% with ±3 MADs or 99.9% within ±4 MADs whenever the errors are approximately normally distributed © 2009 Prentice-Hall, Inc. 5 – 73 Kimball’s Bakery Example Tracking signal for quarterly sales of croissants TIME PERIOD FORECAST DEMAND ACTUAL DEMAND CUMULATIVE ERROR MAD 1 100 90 –10 –10 10 10 10.0 –1 2 100 95 –5 –15 5 15 7.5 –2 3 100 115 +15 0 15 30 10.0 0 4 110 100 –10 –10 10 40 10.0 –1 5 110 125 +15 +5 15 55 11.0 +0.5 6 110 140 +30 +35 30 85 14.2 +2.5 ERROR RSFE |FORECAST | | ERROR | TRACKING SIGNAL forecast error 85 MAD 14.2 n 6 RSFE 35 Tracking signal 2.5MAD s MAD 14.2 © 2009 Prentice-Hall, Inc. 5 – 74 Forecasting at Disney The Disney chairman receives a daily report from his main theme parks that contains only two numbers – the forecast of yesterday’s attendance at the parks and the actual attendance An error close to zero (using MAPE as the measure) is expected The annual forecast of total volume conducted in 1999 for the year 2000 resulted in a MAPE of 0 © 2009 Prentice-Hall, Inc. 5 – 75 Using The Computer to Forecast Spreadsheets can be used by small and medium-sized forecasting problems More advanced programs (SAS, SPSS, Minitab) handle time-series and causal models May automatically select best model parameters Dedicated forecasting packages may be fully automatic May be integrated with inventory planning and control © 2009 Prentice-Hall, Inc. 5 – 76