Exponential Smoothing with Trend. As we move toward mediumrange forecasts, trend becomes more important.


 Gavin Robertson
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1 Exponential Smoothing with Trend As we move toward mediumrange forecasts, trend becomes more important. Incorporating a trend component into exponentially smoothed forecasts is called double exponential smoothing. The estimate for the average and the estimate for the trend are both smoothed.
2 Exponential Smoothing with Trend Adjustment Forecast including trend (FIT t )OR Adjusted Forecast (AF t ) = exponentially smoothed forecast (F t ) + exponentially smoothed trend (T t ) That is, AF t = F t + T t We need to compute both Ft and Tt
3 or Exponential Smoothing with Trend Adjustment (contd.) F t = Last period s forecast + D (Last period s actual Last period s forecast) F t = F t1 + D (A t1 F t1 ) T t = E(This period s Forecast  last period s Forecast) + (1E) (Trend estimate last period) or T t = E(F t  F t1 ) + (1 E) T t1 for all t F t = exponentially smoothed forecast of the data series in period t T t = exponentially smoothed trend in period t A t = actual demand in period t α = smoothing constant for the average β = smoothing constant for the trend
4 Adjusted Exponential Smoothing Example PERIOD MONTH DEMAND 1 Jan 37 2 Feb 40 3 Mar 41 4 Apr 37 5 May 45 6 Jun 50 7 Jul 43 8 Aug 47 9 Sep Oct Nov Dec 54
5 Adjusted Exponential Smoothing Example Per Month Dem F t+1 T t+1 AF t+1 1 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan F 3 =F (A 2 F 2 ) = *3 = 38.5 T 3 = E(F 3  F 2 ) + (1  E) T 2 = (0.30)( ) + (0.70)(0) = 0.45 AF 3 = F 3 + T 3 = = T 13 = E(F 13  F 12 ) + (1  E) T 12 =(0.30)( ) + (0.70)(1.77) =1.36 AF 13 = F 13 + T 13 = = Forecast (D = 0.50)
6 Adjusted Exponential Smoothing Forecasts Actual Adjusted forecast (D = 0.50; E = 0.30)) Demand Forecast (D = 0.50) Period
7 Seasonal Adjustments Repetitive increase/decrease in demand Use seasonal factor to adjust forecast Seasonal factor = S = D i i D Where D i is the sum of demands of the period i in the time series data 6Dis net sum of demands of the entire period in the time series data
8 Example: Seasonal Adjustment [1] Demand (1000 s per quarter) Year Total Total S i Computed trend line for data y = X [Given to you] 2006 (year 4) forecast = (4) = Forecasted demand after seasonal adjustment for the year 2006 is Details x 0.28 = 16.28; x 0.20 = 11.63; x 0.15 = 8.73; x 0.37 = S D = = = D SF 1 = (S 1 ) (F 5 ) = (0.28)(58.17) = SF 2 = (S 2 ) (F 5 ) = (0.20)(58.17) = SF 3 = (S 3 ) (F 5 ) = (0.15)(58.17) = 8.73 SF 4 = (S 4 ) (F 5 ) = (0.37)(58.17) = 21.53
9 Example: Seasonal Adjustment [2] Quarter Year 1 Year 2 Year 3 Year Total Average
10 Example: Seasonal Adjustment [2] Quarter Year 1 Year 2 Year 3 Year Total Average Seasonal Index = Actual Demand Average Demand
11 Example: Seasonal Adjustment [2] Quarter Year 1 Year 2 Year 3 Year Total Average Seasonal Index = =
12 Example: Seasonal Adjustment [2] Contd. Quarter Year 1 Year 2 Year 3 Year /250 = Total Average Seasonal Index = =
13 Example: Seasonal Adjustment [2] Contd. Quarter Year 1 Year 2 Year 3 Year /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = 0.39
14 Example: Seasonal Adjustment [2] Contd. Quarter Year 1 Year 2 Year 3 Year /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = 0.39 Quarter Average Seasonal Index 1 ( )/4 =
15 Example: Seasonal Adjustment [2] Contd. Quarter Year 1 Year 2 Year 3 Year /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = 0.39 Quarter Average Seasonal Index 1 ( )/4 = ( )/4 = ( )/4 = ( )/4 = 0.50
16 Example: Seasonal Adjustment [2] Contd. Quarter Year 1 Year 2 Year 3 Year /250 = /300 = /450 = /550 = 0.18 Projected Annual Demand = 2600 [Given] 2 335/250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = 2.11 Average Quarterly Demand = 2600/4 = /250 = /300 = /450 = /550 = 0.39 Quarter Average Seasonal Index Forecast 1 ( )/4 = ( )/4 = ( )/4 = ( )/4 = 0.50
17 Example: Seasonal Adjustment [2] Contd. Quarter Year 1 Year 2 Year 3 Year /250 = /300 = /450 = /550 = 0.18 Projected Annual Demand = /250 = /300 = /450 = /550 = /250 Average = 2.08 Quarterly 590/300 = 1.97 Demand 830/450 = = 2600/ /550 = 650= /250 = /300 = /450 = /550 = 0.39 Quarter Average Seasonal Index Forecast 1 ( )/4 = (0.20) = ( )/4 = ( )/4 = ( )/4 = 0.50
18 Example: Seasonal Adjustment [2] Contd. Quarter Year 1 Year 2 Year 3 Year /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = /250 = /300 = /450 = /550 = 0.39 Quarter Average Seasonal Index Forecast 1 ( )/4 = (0.20) = ( )/4 = (1.30) = ( )/4 = (2.00) = ( )/4 = (0.50) = 325
19 Seasonalised Time Series Regression Analysis 1. Select a representative historical data set. 2. Develop a seasonal index for each season. 3. Use the seasonal indexes to DeSeasonalise the data. 4. Perform linear regression analysis on the DeSeasonalised data. 5. Use the regression equation to compute the forecasts. 6. Use the seasonal indexes to reapply the seasonal patterns to the forecasts.
20 Example: Computer Products Corp. Seasonalized Times Series Regression Analysis An analyst at CPC wants to develop next year s quarterly forecasts of sales revenue for CPC s line of Epsilon Computers. She believes that the most recent 8 quarters of sales (shown on the next slide) are representative of next year s sales.
21 Example: Computer Products Corp. Seasonalised Times Series Regression Analysis Representative Historical Data Set Year Qtr. ($mil.) Year Qtr. ($mil.)
22 Example: Computer Products Corp. Compute the Seasonal Indexes Quarterly Sales Year Q1 Q2 Q3 Q4 Total Totals Qtr. Avg Seas.Ind / 9.25
23 Example: Computer Products Corp. Time series data: Quarterly Sales Year Q1 Q2 Q3 Q Seasonal Index DeSeasonalised data for Q1 = { Actual Q1 sales / Seas. Index } DeSeasonalise the Data Quarterly Sales Year Q1 Q2 Q3 Q Y t = T t x S t x C t x R t Assum. There is no C t & R t Y t = T t x S t Y t (dese.)= (Y t /S t )
24 Example: Computer Products Corp. Perform Regression on Deseasonalized Data Yr. Qtr. x y x 2 xy Totals (74.01) 36( ) a = = (204) (36) 8(341.39) 36(74.01) b = = (204) (36) Y = X
25 Example: Computer Products Corp. Compute the DeSeasonalised Forecasts MODEL : Y = X Y 9 = (9) = Y 10 = (10) = Y 11 = (11) = Y 12 = (12) = Note: Average sales are expected to increase by.199 million (about $200,000) per quarter.
26 Example: Computer Products Corp. Seasonalised the Forecasts Seas. Deseas. Seas. Yr. Qtr. Index Forecast Forecast
27 Time Series Models & Classical Decomposition Decomposition time series models: Multiplicative: Additive: Y = T x C x S x e Y = T + C + S + e T = Trend component C = Cyclical component S = Seasonal component e = Error or random component
28 Time Series Models & Classical Decomposition Classical decomposition is used to isolate trend, seasonal, and other variability components from a time series model
29 Classical Decomposition Explained Basic Steps: 1. Determine seasonal indexes using the ratio to moving average method 2. Deseasonalize the data 3. Develop the trendcyclical regression equation using deseasonalized data 4. Multiply the forecasted trend values by their seasonal indexes to create a more accurate forecast
30 Start with multiplicative model Y = TCSe Then Se = (Y/TC)
31 Classical Decomposition: Illustration Gem Company s operations department has been asked to deseasonalize and forecast sales for the next four quarters of the coming year The Company has compiled its past sales data in Table 1 An illustration using classical decomposition will follow Table 1: Gem Company s Sales Data Original Forecasted Year Quarter Period Sales Sales t Y TS ? ? ? ?
32 Classical Decomposition Illustration: Step 1 (a) Compute the fourquarter simple moving average Ex: simple MA at end of Qtr 2 and beginning of Qtr 3 ( )/4 = Moving Year Quarter Period Sales Average t Y
33 Classical Decomposition Illustration: Step 1 (b) Compute the twoquarter centered moving average Ex: centered MA at middle of Qtr 3 ( )/2 = Table 2: FourQuarter Moving Average Simple Centered Moving Moving Year Quarter Period Sales Average Average t Y TC
34 Classical Decomposition Illustration: Step 1 (c) Compute the seasonalerror component (percent MA) Ex: percent MA at Qtr 3 (65/60.500) = Table 2: FourQuarter Moving Average Simple Centered Percent Moving Moving Moving Year Quarter Period Sales Average Average Average t Y TC Se=Y/(TC)
35 Classical Decomposition Illustration: Step 1 (d) Compute the unadjusted seasonal index using the seasonalerror components from Table 2 Ex (Qtr 1): [(Yr 2, Qtr 1) + (Yr 3, Qtr 1) + (Yr 4, Qtr 1)]/3 = [ ]/3 = Table 3: Seasonal Index Computation Unadjusted Adjusted Seasonal Adjusting Seasonal Quarter Average Index Factor Index 1 ( )/3 = x (4.000/4.004) = ( )/3 = x (4.000/4.004) = ( )/3 = x (4.000/4.004) = ( )/3 = x (4.000/4.004) =
36 Classical Decomposition Illustration: Step 1 (e) Compute the adjusting factor by dividing the number of quarters (4) by the sum of all calculated unadjusted seasonal indexes = 4.000/( ) = (4.000/4.004) Table 3: Seasonal Index Computation Unadjusted Adjusted Seasonal Adjusting Seasonal Quarter Average Index Factor Index 1 ( )/3 = x (4.000/4.004) = ( )/3 = x (4.000/4.004) = ( )/3 = x (4.000/4.004) = ( )/3 = x (4.000/4.004) =
37 Classical Decomposition Illustration: Step 1 (f) Compute the adjusted seasonal index by multiplying the unadjusted seasonal index by the adjusting factor Ex (Qtr 1): x (4.000/4.004) = Table 3: Seasonal Index Computation Unadjusted Adjusted Seasonal Adjusting Seasonal Quarter Average Index Factor Index 1 ( )/3 = x (4.000/4.004) = ( )/3 = x (4.000/4.004) = ( )/3 = x (4.000/4.004) = ( )/3 = x (4.000/4.004) =
38 Classical Decomposition Illustration: Step 2 Compute the deseasonalized sales by dividing original sales by the adjusted seasonal index Ex (Yr 1, Qtr 1): (55 / 0.942) = Table 4: Deseasonalizing Sales Adjusted Original Seasonal Deseasonalized Year Quarter Period Sales Index Sales t Y S TCe
39 Classical Decomposition Illustration: Step 3 Compute the trendcyclical regression equation using simple linear regression T t = a + bt tbar = 8.5 Tbar = 69.6 b = a = T t = t Table 5: Regression Equation Values Deseasonalized Year Quarter Period Sales t TCe = (Y/S) t(y/s) t
40 Classical Decomposition Illustration: Step 4 (a) Develop trend sales T t = t Ex (Yr 1, Qtr 1): T 1 = (1) = Table 6: Trend Sales Original Deseasonalized Trend Year Quarter Period Sales Sales Sales t Y TCe = (Y/S) T
41 Classical Decomposition Illustration: Step 4 (b) Forecast sales for each of the four quarters of the coming year Ex (Yr 5, Qtr 1): x = Table 7: Forecasted Sales Seasonal Trend Forecasted Year Quarter Period Index Sales Sales t S T TS
42 Classical Decomposition Illustration: Graphical Look 100 Graph 1: Comparison of Trend, Original, and Deseasonalized Sales 90 Sales ($) (Y/S) = TCe Deseasonalized T Trend Y Original Quarter
43 The Classical Decomposition Procedure Smooth the time series to remove random effects and seasonality. Calculate moving averages. Determine period factors to isolate the (seasonal) (error) factor. Determine the unadjusted seasonal factors to eliminate the random component from the period factors Calculate the ratio y t /MA t. Average all the y t /MA t that correspond to the same season.
44 The Classical Decomposition Procedure Contd. Determine the adjusted seasonal factors. Determine Deseasonalized data values. Calculate: [Unadjusted seasonal factor] [Average seasonal factor] Calculate: y t [Adjusted seasonal factors] t Determine a deseasonalized trend forecast. Use linear regression on the deseasonalized time series. Determine an adjusted seasonal forecast. Calculate: (y t /Ma t ) [Adjusted seasonal forecast].
45 Monitoring and Controlling Operations Forecasts Reasons for outofcontrol forecasts change in trend appearance of cycle politics weather changes promotions
46 Monitoring and Controlling a Forecasting Model Forecasts can be monitored using either Tracking Signal (TS) or Control Charts Why track the forecast? To plan around the error in the future To measure actual demand versus forecasts To improve our forecasting methods
47 Monitoring and Controlling a Forecasting Model Tracking Signal (TS) The TS measures the cumulative forecast error over n periods in terms of MAD TS = n i= 1 (Actual demand  Forecast demand ) If the forecasting model is performing well, the TS should be around zero TS indicates the direction of the forecasting error if the TS is positive  increase the forecasts, if the TS is negative  decrease the forecasts. i MAD i
48 Monitoring and Controlling a Forecasting Model Tracking Signal The value of the TS can be used to automatically trigger new parameter values of a model, thereby correcting model performance. If the limits are set too narrow, the parameter values will be changed too often. If the limits are set too wide, the parameter values will not be changed often enough and accuracy will suffer.
49 Tracking Signal Computation Mo Fcst Act Error RSFE Abs Error Cum MAD TS Error
50 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error Error = Actual  Forecast = = 10
51 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error RSFE = 6 Errors = NA + (10) = 10
52 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error Abs Error = Error = 10 = 10
53 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error Cum Error = 6 Errors = NA + 10 = 10
54 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum Error MAD MAD = 6 Errors /n = 10/1 = 10 TS
55 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error TS = RSFE/MAD = 10/10 = 1
56 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error Error = Actual  Forecast = = 5
57 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error RSFE = 6 Errors = (10) + (5) = 15
58 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error Abs Error = Error = 5 = 5
59 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error Cum Error = 6 Errors = = 15
60 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error MAD = 6 Errors /n = 15/2 = 7.5
61 Tracking Signal Computation Mo Forc Act Error RSFE Abs Error Cum MAD TS Error TS = RSFE/MAD = 15/7.5 = 2
62 Plot of a Tracking Signal Signal exceeded limit + Upper control limit Tracking signal MAD 0 Acceptable range  Lower control limit Time
63 Tracking Signals Actual Demand Forecast Actual demand Tracking Signal Time Tracking Singal
64 NOTE on TS ¾ The cumulative forecast error reflects the bias in forecasts, which is the persistent tendency for forecaststo be greater or less than the actual values of a time series. ¾ Tracking signal values are compared to predetermined limits based on judgment and experience. They often range from r3 to r8; for the most part, we shall use limits of ±4, which are roughly comparable to three standard deviation limits. ¾ Values within the limits suggest but do not guarantee that the forecast is performing adequately.
65 Statistical Control Charts The control chart approach involves setting upper and lower limits for individual forecast errors(instead of cumulative errors, as in the case with a tracking signal). The limits are multiples of the square root of MSE (The square root of MSE is used in practice as an estimate of the standard deviation, V, of the distribution of errors). V = (D t  F t ) 2 n This methods assumes (a) Forecast errors are randomly distributed around a mean of zero and (b) The distribution of errors is normal. 9 Using V we can calculate statistical control limits for the forecast error
66 Statistical Control Charts (Contd.) 9 Recall that for a ND, approximately 95% of the values (errors in this case) can be expected to fall within limits of 0 r 2V, and approximately 99.7% of the values can be expected to fall within r 3V of zero. 9 Hence, if the forecast is in control, 99.7% or 95% of the errors should fall within the limits, depending upon whether r 3V or r 2V limits are used. 9 Points that fall outside these limitsshould be regarded as evidence that corrective action is needed [that is the forecast is not performing adequately).
67 Statistical Control Charts Errors Period
68 Statistical Control Charts UCL = +3V 6.12 Errors LCL = 3V Period
69 Ranging Forecasts Forecasts for future periods are only estimates and are subject to error. One way to deal with uncertainty is to develop bestestimate forecasts and the ranges within which the actual data are likely to fall. The ranges of a forecast are defined by the upper and lower limits of a confidence interval.
70 Ranging Forecasts The ranges or limits of a forecast are estimated by: where: Upper limit = Y + t(s yx ) Lower limit = Y  t(s yx ) Y = bestestimate forecast t = number of standard deviations from the mean of the distribution to provide a given probability of exceeding the limits through chance s yx = standard error of the forecast
71 Ranging Forecasts The standard error (deviation) of the forecast is computed as: s = yx 2 y  a y  b xy n  2
72 Example: Railroad Products Co. Ranging Forecasts Recall that linear regression analysis provided a forecast of annual sales for RPC in year 8 equal to $20.55 million. Set the limits (ranges) of the forecast so that there is only a 5 percent probability of exceeding the limits by chance.
73 Example: Railroad Products Co. Ranging Forecasts Step 1: Compute the standard error of the forecasts, s yx. s yx (93).0801(15, 440) = = Step 2: Determine the appropriate value for t. n = 7, so degrees of freedom = n 2 = 5. Area in upper tail =.05/2 =.025 Statistical Table shows t =
74 Example: Railroad Products Co. Ranging Forecasts Step 3: Compute upper and lower limits. Upper limit = (.5748) = = Lower limit = (.5748) = = We are 95% confident that the actual sales for year 8 will be between $ and $ million.
75 Criteria/factor to be considered for Selecting a Forecasting Method Cost Accuracy Data available Time span Nature of products and services Impulse response and noise dampening
76 Criteria for Selecting a Forecasting Method Cost and Accuracy There is a tradeoff between cost and accuracy; generally, more forecast accuracy can be obtained at a cost. Highaccuracy approaches have disadvantages: Use more data Data are ordinarily more difficult to obtain The models are more costly to design, implement, and operate Take longer to use  Low/ModerateCost Approaches statistical models, historical analogies, executivecommittee consensus  HighCost Approaches complex econometric models, Delphi, and market research
77 Criteria for Selecting a Forecasting Method Availability of historical data Is the necessary data available or can it be economically obtained? If the need is to forecast sales of a new product, then a customer survey may not be practical; instead, historical analogy or market research may have to be used.
78 Criteria for Selecting a Forecasting Method Time Span What operations resource is being forecast and for what purpose? Shortterm staffing needs might best be forecast with moving average or exponential smoothing models. Longterm factory capacity needs might best be predicted with regression or executivecommittee consensus methods.
79 Criteria for Selecting a Forecasting Method Nature of Products and Services Is the product/service high cost or high volume? Where is the product/service in its life cycle? Does the product/service have seasonal demand fluctuations?
80 Criteria for Selecting a Forecasting Method Impulse Response and Noise Dampening An appropriate balance must be achieved between: How responsive we want the forecasting model to be to changes in the actual demand data Our desire to suppress undesirable chance variation or noise in the demand data
81 Reasons for Ineffective Forecasting Not involving a broad cross section of people Not recognizing that forecasting is integral to business planning Not forecasting the right things Not selecting an appropriate forecasting method Not tracking the accuracy of the forecasting models Not recognizing that forecasts will always be wrong
82 Forecasting in Small Businesses and StartUp Ventures Forecasting for these businesses can be difficult for the following reasons: Not enough personnel with the time to forecast Personnel lack the necessary skills to develop good forecasts These businesses are not datarich environments Forecasting for new products/services is always difficult, even for the experienced forecaster
83 Sources of Forecasting Data and Help Government agencies at the local, regional, state, and federal levels Industry associations Consulting companies
84 Some Specific Forecasting Data Consumer Confidence Index Consumer Price Index (CPI) Gross Domestic Product (GDP) Index of Leading Economic Indicators Personal Income and Consumption Producer Price Index (PPI) Purchasing Manager s Index Retail Sales
85 NOTE The wise decision maker does not limit forecasting decisions to a single technique but combines the subjective and objective methods. Furthermore, the approximate way of defining forecast could be Forecast = Projection r Judgment Good Forecasting has to be determined with the tool : DSS
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