Chapter 5 Forecasting PDF

Chapter 5 Forecasting To accompany Quantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna © 2008 Prentice-Hall, Inc. Power Point slides created by Jeff Heyl © 2009 Prentice-Hall, Inc.  I...

Chapter 5 Forecasting To accompany Quantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna © 2008 Prentice-Hall, Inc. Power Point slides created by Jeff Heyl © 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  What is forecasting? A process of estimating or predicting future demand through past and present events is considered Forecasting. Information on potential future events and their effect on the business can be obtained from forecasting. As per Heizer and Render (2010), “Forecasting is considered art and science of estimating future events”. According to Louis Allen, forecasting is considered “a systematic attempt to probing the future through inference from facts that are already known”. © 2009 Prentice-Hall, Inc. 5–5 © 2009 Prentice-Hall, Inc. 5–6 © 2008 Prentice-Hall, Inc. © 2008 Prentice-Hall, Inc. © 2008 Prentice-Hall, Inc. © 2009 Prentice-Hall, Inc. 5 – 10  Forecasting Models Forecasting Techniques Qualitative Time-Series Causal Models Methods Methods Delphi Moving Regression Methods Average Analysis Jury of Executive Exponential Multiple Opinion Smoothing Regression Sales Force Trend Composite Projections Figure 5.1 Consumer Market Survey Decomposition © 2009 Prentice-Hall, Inc. 5 – 11  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 – 12  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 – 13  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 – 14  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 – 15  Scatter Diagrams Scatter diagrams are helpful when forecasting time-series data because they depict the relationship between variables. 450 400 Radios 350 Annual Sales 300 250 Televisions 200 150 pac t Di sc s 100 Com 50 0 0 2 4 6 8 10 12 Time (Years) © 2009 Prentice-Hall, Inc. 5 – 16  Scatter Diagrams Wacker Distributors wants to forecast sales for three different products YEAR TELEVISION SETS RADIOS COMPACT DISC PLAYERS 1 250 300 110 2 250 310 100 3 250 320 120 4 250 330 140 5 250 340 170 6 250 350 150 7 250 360 160 8 250 370 190 9 250 380 200 10 250 390 190 Table 5.1 © 2009 Prentice-Hall, Inc. 5 – 17  Scatter Diagrams (a) Sales appear to be Annual Sales of Televisions 330 – constant over time 250 –           Sales = 250 200 – A good estimate of 150 – sales in year 11 is 100 – 250 televisions 50 – | | | | | | | | | | 0 1 2 3 4 5 6 7 8 9 10 Time (Years) Figure 5.2 © 2009 Prentice-Hall, Inc. 5 – 18  Scatter Diagrams (b) 420 – Sales appear to be 400 – increasing at a Annual Sales of Radios 380 –   constant rate of 10 360 –   radios per year 340 –    Sales = 290 + 10(Year) 320 –   A reasonable 300 –  estimate of sales in 280 – year 11 is 400 | | | | | 0 1 2 3 4 5 6 7 8 9 10 | | | | | radios Time (Years) Figure 5.2 © 2009 Prentice-Hall, Inc. 5 – 19  Scatter Diagrams This trend line may (c) not be perfectly Annual Sales of CD Players 200 –  accurate because 180 –  of variation from 160 –  year to year   Sales appear to be 140 –  increasing 120 –  A forecast would  100 –  probably be a | | | | | | | | | | larger figure each 0 1 2 3 4 5 6 7 8 9 10 year Time (Years) Figure 5.2 © 2009 Prentice-Hall, Inc. 5 – 20  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) MAD MAD   forecast error n © 2009 Prentice-Hall, Inc. 5 – 21  Measures of Forecast Accuracy There are other popular measures of forecast accuracy The mean squared error MSE   ( error) 2 n The mean absolute percent error error  actual MAPE  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 – 22  Measures of Forecast Accuracy Using a naïve forecasting model ACTUAL ABSOLUTE VALUE OF SALES OF CD ERRORS (DEVIATION), YEAR PLAYERS FORECAST SALES (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 – 24  Measures of Forecast Accuracy Using a naïve forecasting model ACTUAL ABSOLUTE VALUE OF SALES OF CD ERRORS (DEVIATION), YEAR PLAYERS FORECAST SALES (ACTUAL – FORECAST) 1 110 — — 2 100 110 |100 – 110| = 10 3 120 100 |120 – 110| = 20 4 MAD  5  forecast error 160 140 170  17.8 120 140 |140 – 120| = 20 |170 – 140| = 30 6 150 n 170 9 |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 — This means that on the average, each forecast Sum of |errors| = 160 missed the actual value by 17.8 units. MAD = 160/9 = 17.8 Table 5.2 © 2009 Prentice-Hall, Inc. 5 – 25 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| 160 MAD 17.8 © 2009 Prentice-Hall, Inc. 5 – 26  Hospital Days Forecast Error Example Month Forecast Actual Ms. Smith forecasted JAN 250 243 total hospital inpatient days last year. Now FEB 320 315 that the actual data are MAR 275 286 known, she is APR 260 256 reevaluating her MAY 250 241 forecasting model. JUN 275 298 Compute the MAD, JUL 300 292 MSE, and MAPE for her AUG 325 333 forecast. SEP 320 326 OCT 350 378 NOV 365 382 DEC 380 396 © 2009 Prentice-Hall, Inc. 5 – 27  Hospital Days Forecast Error Example Forecast Actual |error| error2 |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 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 MAD= MSE= MAPE= AVERAGE 11.83 192.83.0381*100 = 3.81 © 2009 Prentice-Hall, Inc. 5 – 28  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 – 29  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 Sum of demands in previous n periods Moving average forecast  n © 2009 Prentice-Hall, Inc. 5 – 30  Moving Averages Mathematically Y  Yt  1 ...  Yt  n1 Ft 1  t n where Ft 1for time period t + 1 = forecast Yt = actual value in time period t n= number of periods to average © 2009 Prentice-Hall, Inc. 5 – 31 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 – 32  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 (16 + 19 + 23)/3 = 19.33 July 26 (19 + 23 + 26)/3 = 22.67 August 30 (23 + 26 + 30)/3 = 26.33 September 28 (26 + 30 + 28)/3 = 28.00 October 18 (30 + 28 + 18)/3 = 25.33 November 16 (28 + 18 + 16)/3 = 20.67 December 14 (18 + 16 + 14)/3 = 16.00 January — Table 5.3 © 2009 Prentice-Hall, Inc. 5 – 33  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 ...  w nYt  n1 Ft 1  w1  w2 ...  w n ere wi= weight for the ith observation © 2009 Prentice-Hall, Inc. 5 – 34  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 – 35  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 Last month 2 Two months ago 1 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 – 36  Wallace Garden Supply Example THREE-MONTH WEIGHTED MONTH ACTUAL SHED SALES MOVING AVERAGE 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 [(3 X 19) + (2 X 16) + (13)]/6 = 17.00 June 23 [(3 X 23) + (2 X 19) + (16)]/6 = 20.50 July 26 [(3 X 26) + (2 X 23) + (19)]/6 = 23.83 August 30 [(3 X 30) + (2 X 26) + (23)]/6 = 27.50 September 28 [(3 X 28) + (2 X 30) + (26)]/6 = 28.33 October 18 [(3 X 18) + (2 X 28) + (30)]/6 = 23.33 November 16 [(3 X 16) + (2 X 18) + (28)]/6 = 18.67 December 14 [(3 X 14) + (2 X 16) + (18)]/6 = 15.33 January Table 5.4 — © 2009 Prentice-Hall, Inc. 5 – 37  Wallace Garden Supply Example Program 5.1A © 2009 Prentice-Hall, Inc. 5 – 38  Wallace Garden Supply Example Program 5.1B © 2009 Prentice-Hall, Inc. 5 – 39  Exponential Smoothing Exponential smoothing is easy to use and requires little record keeping of data It is a type of moving average ecast = Last period’s forecast + (Last period’s actual demand – Last period’s forecast) Where  is a weight (or smoothing constant) 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 – 40  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 – 41 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 – 42 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 – 43  Port of Baltimore Example Exponential smoothing forecast for two values of  ACTUAL TONNAGE FORECAST FORECAST QUARTER UNLOADED USING  =0.10 USING  =0.50 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 Table 5.5 © 2009 Prentice-Hall, Inc. 5 – 44  Selecting the Best Value of  ACTUAL FORECAST ABSOLUTE ABSOLUTE TONNAGE WITH  = DEVIATIONS FORECAST DEVIATIONS QUARTER UNLOADED 0.10 FOR  = 0.10 WITH  = 0.50 FOR  = 0.50 1 180 175 175 5….. 5…. 2 168 175.5 177.5 7.5.. 9.5.. 3 159 174.75 172.75 15.75 13.75 4 175 173.18 165.88 1.82 9.12 5 190 173.36 170.44 16.64 19.56 6 205 175.02 180.22 29.98 24.78 7 180 178.02 192.61 1.98 12.61 Table 5.6 8 182 Best178.22 choice 3.78 186.30 4.3.. Sum of absolute deviations 82.45 98.63 Σ|deviations| © 2009 Prentice-Hall, Inc. 5 – 45 MAD = = 10.31 MAD = 12.33  Port of Baltimore Example Program 5.2A © 2009 Prentice-Hall, Inc. 5 – 46 Port of Baltimore Example Program 5.2B © 2009 Prentice-Hall, Inc. 5 – 47  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 – 48  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 ) = New forecast (Ft) t + Trend correction (Tt) © 2009 Prentice-Hall, Inc. 5 – 49  Exponential Smoothing with Trend Adjustment The equation for the trend correction uses a new smoothing constant  T is computed by t Tt 1 (1   )Tt   ( Ft 1  Ft ) where Tt+1 =smoothed trend for period t + 1 Tt =smoothed trend for preceding period  =trend smooth constant that we select Ft+1 =simple exponential smoothed forecast for period t + 1 Ft =forecast for pervious period © 2009 Prentice-Hall, Inc. 5 – 50  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-and- error 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, order double smoothing, smoothing or Holt’s method © 2009 Prentice-Hall, Inc. 5 – 51  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 – 52  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 – 53  Trend Projection The mathematical form is Yˆ b0  b1 X where = predicted Ŷ value b0= intercept b1= slope of the line X= time period (i.e., X = 1, 2, 3, …, n) © 2009 Prentice-Hall, Inc. 5 – 54  Trend Projection Dist7 * * Value of Dependent Variable Dist5 Dist6 * Dist3 * Dist4 Dist1 * Dist2 * * Time Figure 5.4 © 2009 Prentice-Hall, Inc. 5 – 55  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 74 2002 79 2003 80 2004 90 2005 105 2006 142 2007 122 Table 5.7 © 2009 Prentice-Hall, Inc. 5 – 56  Midwestern Manufacturing Company Example Notice code instead of actual years Program 5.3A © 2009 Prentice-Hall, Inc. 5 – 57  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 – 58  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 – 59  Midwestern Manufacturing Company Example 160 – 150 –  140 –  Trend Line 130 – Generator Demand Yˆ 56.71  10.54 X 120 –  110 –  100 – 90 –  80 –   70 –  Actual Demand Line 60 – 50 – | | | | | | | | | 2001 2002 2003 2004 2005 2006 2007 2008 2009 Figure 5.5 Year © 2009 Prentice-Hall, Inc. 5 – 60  Midwestern Manufacturing Company Example Program 5.4A © 2009 Prentice-Hall, Inc. 5 – 61  Midwestern Manufacturing Company Example Program 5.4B © 2009 Prentice-Hall, Inc. 5 – 62 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 – 63  Decomposition of a Time-Series Demand for Product or Service Trend Component Seasonal Peaks Actual Demand Line Average Demand over 4 Years | | | | Year Year Year Year 1 2 3 4 Time Figure 5.3 © 2009 Prentice-Hall, Inc. 5 – 64 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 – 65  If all variations in a time series are due to random variations, with no trend, seasonal, or cyclical component, some type of averaging or smoothing model would be appropriate. if there is a trend or seasonal pattern present in the data – techniques used are exponential smoothing with trend and trend projections there is a seasonal pattern present in the data, then a seasonal index may be developed and used with any of the averaging methods If both trend and seasonal components are present, then a method such as the decomposition method should be used. © 2009 Prentice-Hall, Inc. 5 – 66  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 – 67  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 – 68  Seasonal Variations SALES DEMAND AVERAGE AVERAGE TWO- MONTHLY SEASONAL MONTH YEAR 1 YEAR 2 YEAR DEMAND DEMAND INDEX January 80 100 94 0.957 90 February 85 75 94 0.851 80 March 80 90 94 0.904 85 April 110 90 94 1.064 100 May 115 131 94 1.309 123 June 120 110 94 1.223 115 July 100 110 94 1.117 105 August 110 90 94 1.064 1,128 100 Average two-year demand Average monthly demand = = 94 Seasonal index = 12 months Average monthly demand September 85 95 94 0.957 Table 5.8 90 © 2009 Prentice-Hall, Inc. 5 – 69  Seasonal Variations The calculations for the seasonal indices are 1,200 1,200 Jan. 0.957 96 July 1.117 112 12 12 1,200 1,200 Feb. 0.851 85 Aug. 1.064 106 12 12 1,200 1,200 Mar. 0.904 90 Sept. 0.957 96 12 12 1,200 1,200 Apr. 1.064 106 Oct. 0.851 85 12 12 1,200 1,200 May 1.309 131 Nov. 0.851 85 12 12 1,200 1,200 June 1.223 122 Dec. 0.851 85 12 12 © 2009 Prentice-Hall, Inc. 5 – 70  Seasonal Variation with Trend When both trend and seasonal components are present in a time series, a change from one month to the next could be due to a trend, to a seasonal variation, or simply to random fluctuations. To help with this problem, the seasonal indices should be computed using a Centered Moving Average (CMA) approach whenever trend is present. Using this approach prevents a variation due to trend from being incorrectly interpreted as a variation due to the season. © 2009 Prentice-Hall, Inc. 5 – 71  TABLE 5.10 Quarterly Sales ($1,000,000s) for Turner Industries © 2009 Prentice-Hall, Inc. 5 – 72  TABLE 5.11 Centered Moving Averages and Seasonal Ratios for Turner Industries © 2009 Prentice-Hall, Inc. 5 – 73  Centered Moving Average (CMA) To obtain the CMA, we take quarters 2, 3, and 4 of year 1, plus one-half of quarter 1 for year 1 and one-half of quarter 1 for year 2. The average will be We compare the actual sales in this quarter to the CMA and we have the following seasonal ratio: Thus, sales in quarter 3 of year 1 are about 13.6% higher than an average quarter at this time. © 2009 Prentice-Hall, Inc. 5 – 74  Since there are two seasonal ratios for each quarter, we average these to get the seasonal index. Thus, The sum of these indices should be the number of seasons (4) since an average season should have an index of 1. In this example, the sum is 4. If the sum were not 4, an adjustment would be made. We would multiply each index by 4 and divide this by the sum of the indices © 2009 Prentice-Hall, Inc. 5 – 75  Steps Used to Compute Seasonal Indices Based on CMAs 1. Compute a CMA for each observation (where possible). 2. Compute seasonal ratio = Observation/CMA for that observation. 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). © 2009 Prentice-Hall, Inc. 5 – 76  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 = time period X2 = 1 if quarter 2, 0 otherwise X3 = 1 if quarter 3, 0 otherwise X4 = 1 if quarter 4, 0 © 2009 Prentice-Hall, Inc. 5 – 77  Regression with Trend and Seasonal Components Program 5.6A © 2009 Prentice-Hall, Inc. 5 – 78  Regression with Trend and Seasonal Components Program 5.6B (partial) © 2009 Prentice-Hall, Inc. 5 – 79  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 Ŷ 104.1  2.3(13)  15.7(0)  38.7(0)  30.1(0) 134 Ŷ 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 – 80  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 – 81  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 MAD   forecast error n © 2009 Prentice-Hall, Inc. 5 – 82  Monitoring and Controlling Forecasts Signal Tripped Upper Control Limit Tracking Signal + Acceptable 0 MADs Range – Lower Control Limit Time Figure 5.7 © 2009 Prentice-Hall, Inc. 5 – 83  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 – 84  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 – 85  Kimball’s Bakery Example Tracking signal for quarterly sales of croissants TIME FORECAST ACTUAL |FORECAST | CUMULATIVE TRACKING PERIOD DEMAND DEMAND ERROR RSFE | ERROR | ERROR MAD SIGNAL 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 MAD   forecast error 85  14.2 n 6 RSFE 35 Tracking signal   2.5MADs MAD 14.2 The objective is to compute the tracking signal and determine whether forecasts are performing © 2009 Prentice-Hall, Inc. 5 – 86 adequately.  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 – 87  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 – 88