Forecasting To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved.

Forecasting Predicting future events Predicting future events Usually demand behavior over a time frame Usually demand behavior over a time frame

Forecast: A statement about the future value of a variable of interest such as demand. Forecasts affect decisions and activities throughout an organization –Accounting, finance –Human resources –Marketing –MIS –Operations –Product / service design

AccountingTo estimate cost/profit values Finance To estimate cash flows and funding requirements, to develop budgets Human Resources To develop hiring/recruiting and training plans Marketing For Pricing, promotion, distribution decisions MIS To develop IT/IS systems and services Operations Schedules, MRP, workloads Product/service design New products and services Uses of Forecasts

Operations To assess LT capacity needs To develop production plans and schedules To plan orders for materials To plan work loads Product/service design New products and services Supply chain management To get agreement within firm and across supply chain partners. Uses of Forecasts

Uses of Forecasts: Summary  To help managers plan the system  To help managers plan the use of the system.

Assumes causal system past ==> future Forecasts are almost always wrong by some amount (they are rarely perfect) because of randomness Forecasts are more accurate for groups vs. individuals Forecasts are more accurate for shorter time periods. İe. Forecast accuracy decreases as time horizon increases Forecasts are not substitutes for calculated demand I see that you will get an A this semester. Features Common to All Forecasts

Elements of a Good Forecast Timely Accurate Reliable Meaningful Written Easy to use

Time Frame in Forecasting Short-range to medium-range Short-range to medium-range Daily, weekly monthly forecasts of sales data Daily, weekly monthly forecasts of sales data Up to 2 years into the future Up to 2 years into the future Long-range Long-range Strategic planning of goals, products, markets Strategic planning of goals, products, markets Planning beyond 2 years into the future Planning beyond 2 years into the future

Steps in the Forecasting Process Step 1 Determine purpose of forecast Step 2 Establish a time horizon Step 3 Select a forecasting technique Step 4 Gather and analyze data Step 5 Prepare the forecast Step 6 Monitor the forecast “The forecast”

Forecasting Process 6. Check forecast accuracy with one or more measures 4. Select a forecast model that seems appropriate for data 5. Develop/compute forecast for period of historical data 8a. Forecast over planning horizon 9. Adjust forecast based on additional qualitative information and insight 10. Monitor results and measure forecast accuracy 8b. Select new forecast model or adjust parameters of existing model 7. Is accuracy of forecast acceptable? 1. Identify the purpose of forecast 3. Plot data and identify patterns 2. Collect historical data

Approaches to Forecasting Qualitative methodsQualitative methods –Based on subjective methods Quantitative methodsQuantitative methods –Based on mathematical formulas

Quantitative Methods Used when situation is ‘stable’ and historical data exists –Existing products –Current technology Heavy use of mathematical techniques ******************************* E.g., forecasting sales of a mature product Qualitative Methods Used when situation is vague and little data exists –New products –New technology Involves intuition, experience ***************************** E.g., forecasting sales to a new market Forecasting Approaches

“Q2” Forecasting Quantitative, then qualitative factors to “filter” the answer

Approaches to Forecasting Judgmental (Qualitative)- uses subjective inputs Time series - uses historical data assuming the future will be like the past Associative models - uses explanatory variables to predict the future

Qualitative (Judgmental)Forecasting Executive opinions Sales force opinions Consumer surveys Outside opinion Delphi method –Opinions of managers and staff –Achieves a consensus forecast

Time Series A time series is a time-ordered sequence of observations taken at regular intervals (eg. Hourly, daily, weekly, monthly, quarterly, annually)

Time Series Models PeriodDemand What assumptions must we make to use this data to forecast?

Demand Behavior Trend Trend gradual, long-term up or down movement gradual, long-term up or down movement Cycle Cycle up & down movement repeating over long time frame; wavelike variations of more than one year’s duration up & down movement repeating over long time frame; wavelike variations of more than one year’s duration Seasonal pattern Seasonal pattern periodic oscillation in demand which repeats; short-term regular variations in data periodic oscillation in demand which repeats; short-term regular variations in data Irregular variations caused by unusual circumstances Irregular variations caused by unusual circumstances Random movements follow no pattern; caused by chance Random movements follow no pattern; caused by chance

Time Series Components of Demand... Time Demand... randomness

Time Series with... Time Demand... randomness and trend

Time series with... Demand... randomness, trend and seasonality May

Forms of Forecast Movement Time (a) Trend Time (d) Trend with seasonal pattern Time (c) Seasonal pattern Time (b) Cycle Demand Demand Demand Demand Random movement

Trend Irregular variatio n Seasonal variations Cycles Forms of Forecast Movement

Idea Behind Time Series Models Distinguish between random fluctuations and true changes in underlying demand patterns.

Time Series Methods Naive forecasts Naive forecasts Forecast = data from past period Forecast = data from past period Statistical methods using historical data Statistical methods using historical data Moving average Moving average Exponential smoothing Exponential smoothing Linear trend line Linear trend line Assume patterns will repeat Assume patterns will repeat Demand?

Naive Forecasts Uh, give me a minute.... We sold 250 wheels last week.... Now, next week we should sell.... The forecast for any period equals the previous period’s actual value.

Simple to use Virtually no cost Quick and easy to prepare Data analysis is nonexistent Easily understandable Cannot provide high accuracy Can be a standard for accuracy Naïve Forecasts

Stable time series data –F(t) = A(t-1) Seasonal variations –F(t) = A(t-n) Data with trends –F(t) = A(t-1) + (A(t-1) – A(t-2)) Uses for Naïve Forecasts

Techniques for Averaging Moving Average Weighted Moving Average Exponential Smoothing Averaging techniques smooth fluctuations in a time series.

Moving Average MA n = n i = 1  A t-i n where n =number of periods in the moving average A t- i =actual demand in period t- i Average several periods of data Average several periods of data Dampen, smooth out changes Dampen, smooth out changes Use when demand is stable with no trend or seasonal pattern Use when demand is stable with no trend or seasonal pattern

Moving Averages Moving average – A technique that averages a number of recent actual values, updated as new values become available. F t = MA n = n A t-n + … A t-2 + A t-1 Ft = Forecast for time period t MAn= n period moving average

Jan120 Feb90 Mar100 Apr75 May110 June50 July75 Aug130 Sept110 Oct90 ORDERS MONTHPER MONTH = = 110 orders for Nov Simple Moving Average F11 =MA 3 3-period moving average for period11

Jan120– Feb90 – Mar100 – Apr May June July Aug Sept Oct Nov –110.0 ORDERSTHREE-MONTH MONTHPER MONTHMOVING AVERAGE Simple Moving Average

Jan120– Feb90 – Mar100 – Apr May June July Aug Sept Oct Nov –110.0 ORDERSTHREE-MONTH MONTHPER MONTHMOVING AVERAGE = 91 orders for Nov Simple Moving Average F11 MA 5 =

Simple Moving Average Jan120– – Feb90 – – Mar100 – – Apr – May – June July Aug Sept Oct Nov – ORDERSTHREE-MONTHFIVE-MONTH MONTHPER MONTHMOVING AVERAGEMOVING AVERAGE

Smoothing Effects – – – – – – 0 0 – ||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNov Orders Month Actual

Smoothing Effects – – – – – – 0 0 – ||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNov 3-month Actual Orders Month

Smoothing Effects – – – – – – 0 0 – ||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNov 5-month 3-month Actual Orders Month

Weighted Moving Average WMA n = i = 1  W i A t-i where W i = the weight for period i, between 0 and 100 percent  W i = 1.00 Adjusts moving average method to more closely reflect data fluctuations Adjusts moving average method to more closely reflect data fluctuations n

Weighted Moving Averages Weighted moving average – More recent values in a series are given more weight in computing the forecast. F t = WMA n = n w n A t-n + … w n-1 A t-2 + w 1 A t-1

Weighted Moving Average Example MONTH WEIGHT DATA August 17%130 September 33%110 October 50%90 November forecast WMA 3 = 3 i = 1  Wi AiWi AiWi AiWi Ai = (0.50)(90) + (0.33)(110) + (0.17)(130) = orders

Table of Moving Averages and Weighted Moving Averages Period Actual Demand Three-Period Moving Average Forecast Three-Period Weighted Moving Average Forecast Weights = 0.5, 0.3,

Some Questions About Weighted Moving Averages What are the advantages? What do the weights add up to? Could we use different weights? Compare with a simple 3-period moving average.

Exponential Smoothing Weighted averaging method based on previous forecast plus a percentage of the forecast error A-F is the error term,  is the % feedback or a percentage of forecast error Sophisticated weight averaging model Needs only three numbers ( Ft, A, α,) F t+1 = F t + α(A t - F t )

Exponential Smoothing Premise--The most recent observations might have the highest predictive value. – Therefore, we should give more weight to the more recent time periods when forecasting. F t+1 = F t +  ( A t - F t ) Ft+1 =forecast for the next period At =actual demand for the present period Ft =previously determined forecast for the present period  = weighting factor, smoothing constant α  = weighting factor, smoothing constant

Smoothing in Action

Averaging method Averaging method Weights most recent data more strongly Weights most recent data more strongly Reacts more to recent changes Reacts more to recent changes Widely used, accurate method Widely used, accurate method Exponential Smoothing Where did the current forecast come from? What happens as α gets closer to 0 or 1? Where does the very first forecast come from?

Effect of Smoothing Constant 0.0  1.0 If  = 0.20, then F t +1 = 0.20  A t F t If  = 0, then F t +1 = 0  A t + 1 F t 0 = F t Forecast does not reflect recent data If  = 1, then F t +1 = 1  A t + 0 F t =  A t Forecast based only on most recent data

PERIODMONTHDEMAND 1Jan37 2Feb40 3Mar41 4Apr37 5May 45 6Jun50 7Jul 43 8Aug 47 9Sep 56 10Oct52 11Nov55 12Dec 54 Exponential Smoothing- Example 1

PERIODMONTHDEMAND 1Jan37 2Feb40 3Mar41 4Apr37 5May 45 6Jun50 7Jul 43 8Aug 47 9Sep 56 10Oct52 11Nov55 12Dec 54 F 2 =  1 + (1 -  )F 1 = (0.30)(37) + (0.70)(37) = 37 F 3 =  2 + (1 -  )F 2 = (0.30)(40) + (0.70)(37) = 37.9 F 13 =  12 + (1 -  )F 12 = (0.30)(54) + (0.70)(50.84) = Exponential Smoothing

FORECAST, F t + 1 PERIODMONTHDEMAND(  = 0.3) 1Jan37– 2Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan–51.79 Exponential Smoothing

FORECAST, F t + 1 PERIODMONTHDEMAND(  = 0.3)(  = 0.5) 1Jan37–– 2Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan– Exponential Smoothing

70 70 – – – – – – – 0 0 – ||||||||||||| Actual Orders Month Exponential Smoothing Forecasts

70 70 – – – – – – – 0 0 – ||||||||||||| Actual Orders Month  = 0.30 Exponential Smoothing Forecasts

70 70 – – – – – – – 0 0 – |||||||||||||  = 0.50 Actual Orders Month  = 0.30 Exponential Smoothing Forecasts

Exponential Smoothing-Example 2 Exponential Smoothing-Example 2

Picking a Smoothing Constant .1 .4 Actual

Trends in Time Series Data What do you think will happen to a moving average or exponential smoothing model when there is a trend in the data?

Same Exponential Smoothing Model as Before Since the model is based on historical demand, it always lags the obvious upward trend Period Actual Demand Exponential Smoothing Forecast

Trend-Adjusted Exponential Smoothing A variation of simple exponential smoothing can be used when a time series exhibits trend and it is called trend-adjusted exponential smoothing or double smoothing If a series exhibits trend, and simple smoothing is used on it the forecasts will all lag the trend: if the data are increasing, each forecast will be too low; if the data are decreasing, each forecast will be too high.

Trend-Adjusted Exponential Smoothing TAF t+1 = S t + T t Where S t = Smoothed forecast T t = Current trend estimate and S t =TAF t + α (A t – TAF t ) T t = T t-1 + β (TAF t – TAF t-1 – T t-1 ) α and β are smoothing constants

Simple Linear Regression Time series OR causal model Assumes a linear relationship: y = a + b(x) y x y: predicted variable x: predictor variable X” can be the time period or some other type of variable (examples?)

y = a + bt where a=intercept (at period 0) b=slope of the line t=the time period y=forecast for demand for period x Linear Trend Line

Calculating a and b: For trend line the t values will be used in place of x values

Calculating a and b (equations rearranged) b = n(ty) - ty nt 2 - ( t) 2 a = y - bt n   

y = a + bx where a = intercept (at period 0) b = slope of the line t= the time period y =forecast for demand for period x b = a = y - b t where n =number of periods t == mean of the t values y == mean of the y values  ty - nty  t 2 – nt 2  t n  y n Linear Trend Line Rearranged equations

t(PERIOD) y (DEMAND) Linear Trend Calculation Example

x (PERIOD) y (DEMAND) tyt Linear Trend Calculation Example

x (PERIOD) y (DEMAND) xyx t = = 6.5 y = = b = = = 1.72 a = y - bt = (1.72)(6.5) = (12)(6.5)(46.42) (6.5) 2  xy – nt y  x 2 – nt

Linear Trend Calculation Example x (PERIOD) y (DEMAND) xyx x = = 6.5 y = = b = = = 1.72 a = y - bx = (1.72)(6.5) = (12)(6.5)(46.42) (6.5) 2  xy - nxy  x 2 - nx Linear trend line y = t

Least Squares Example x (PERIOD) y (DEMAND) xyx x = = 6.5 y = = b = = = 1.72 a = y - bx = (1.72)(6.5) = (12)(6.5)(46.42) (6.5) 2  xy - nxy  x 2 - nx Linear trend line y = x Forecast for period 13 y = (13) y = units

Linear Trend Line – – – – – – – 0 0 – ||||||||||||| Demand Period

Linear Trend Line – – – – – – – 0 0 – ||||||||||||| Actual Demand Period

Linear Trend Line – – – – – – – 0 0 – ||||||||||||| Actual Demand Period Linear trend line

Example: Simplified Regression If we redefine the X values so that their sum adds up to zero, regression becomes much simpler –a now equals the average of the y values –b simplifies to the sum of the xy products divided by the sum of the x 2 values

Example: Simplified Regression Period (X) Perio d (X)' Demand (Y)X2XY

Dealing with Seasonality Quarter PeriodDemand Winter Spring2 240 Summer3 300 Fall4 440 Winter Spring6 720 Summer7 700 Fall8 880

What Do You Notice? Forecasted Demand = – x Period Period Actual Demand Regression Forecast Forecast Error Winter Spring Summer Fall Winter Spring Summer Fall

Regression picks up trend, but not seasonality effect

Seasonal Adjustments Repetitive increase/ decrease in demand Models of seasonality:  Additive (seasonality is expressed as a quantity that is added to or subtracted from the series average)  Multiplicative (seasonality is expressed as a percentage of the average (or trend)amount)

Seasonal Adjustments The seasonal percentages in the multiplicative model are referred to as seasonal relatives or seasonal indexes The seasonal percentages in the multiplicative model are referred to as seasonal relatives or seasonal indexes

Calculating Seasonal Index (1st Method): Winter Quarter (Actual / Forecast) for Winter Quarters: Winter ‘02:(80 / 90) = 0.89 Winter ‘03:(400 / 524.3) = 0.76 Average of these two = 0.83 Interpret!

Seasonally adjusted forecasts (1st method) For Winter Quarter [ – ×Period ] × 0.83 Or more generally: [ – × Period ] × Seasonal Index

Seasonally adjusted forecasts (first method) Forecasted Demand = – x Period Period Actual Demand Regression Forecast Demand/ Forecast Seasonal Index Seasonally Adjusted Forecast Forecast Error Winter Spring Summer Fall Winter Spring Summer Fall

Would You Expect the Forecast Model to Perform This Well With Future Data?

Seasonal Adjustments (2nd method) Use seasonal factor to adjust forecast Use seasonal factor to adjust forecast Seasonal factor = S i = DiDiDDDiDiDD

Seasonal Adjustment (2nd method) Total DEMAND (1000’S PER QUARTER) YEAR1234Total

Seasonal Adjustment (2nd Method) Total DEMAND (1000’S PER QUARTER) YEAR1234Total S 1 = = = 0.28 D1D1DDD1D1DD S 2 = = = 0.20 D2D2DDD2D2DD S 4 = = = 0.37 D4D4DDD4D4DD S 3 = = = 0.15 D3D3DDD3D3DD

Seasonal Adjustment (2nd Method) Total DEMAND (1000’S PER QUARTER) YEAR1234Total S i

Seasonal Adjustment (2nd Method) Total DEMAND (1000’S PER QUARTER) YEAR1234Total S i y = x = (4) = For 2002

Seasonal Adjustment (2nd Method) SF 1 = (S 1 ) (F 5 )SF 3 = (S 3 ) (F 5 ) = (0.28)(58.17) = 16.28= (0.15)(58.17) = 8.73 SF 2 = (S 2 ) (F 5 )SF 4 = (S 4 ) (F 5 ) = (0.20)(58.17) = 11.63= (0.37)(58.17) = Total DEMAND (1000’S PER QUARTER) YEAR1234Total S i y = t = (4) = For 2002

A Method for Computing Seasonal Relatives: Centered Moving Average  A commonly used method for representing the trend portion of a time series involves a centered moving average.  By virtue of its centered position it looks forward and looks backward, so it is able to closely follow data movements whether they involve trends, cycles, or random variability alone.

Computing Seasonal Relatives by Using Centered Moving Averages  The ratio of demand at period i to the centered average at period i is an estimate of the seasonal relative at that point.

Causal Models Time series assume that demand is a function of time. This is not always true. Sometimes it is possible to use the relationship between demand and some other factor to develop forecast 1. Pounds of BBQ eaten at party. 2. Dollars spent on drought relief. 3. Lumber sales. Linear regression can be used in these situations as well.

Associative Forecasting Concepts: Predictor variables - used to predict values of variable interest Regression - technique for fitting a line to a set of points Least squares line - minimizes sum of squared deviations around the line

Causal Modeling with Linear Regression Study relationship between two or more variables Study relationship between two or more variables Dependent variable y depends on independent variable x y = a + bx Dependent variable y depends on independent variable x y = a + bx

Linear Model Seems Reasonable A straight line is fitted to a set of sample points. Computed relationship

Linear Regression Formulas a = y - b x b = where a =intercept (at period 0) b =slope of the line x == mean of the x data y == mean of the y data  xy - nxy  x 2 - nx 2  x n  y n

Linear Regression Example xy (WINS)(ATTENDANCE) xyx

Linear Regression Example xy (WINS)(ATTENDANCE) xyx x = = y = = b = = = 4.06 a = y - bx = (4.06)(6.125) =  xy - nxy 2  x 2 - nx 2 (2,167.7) - (8)(6.125)(43.36) (311) - (8)(6.125) 2

Linear Regression Example xy (WINS)(ATTENDANCE) xyx x = = y = = b = = = 4.06 a = y - bx = (4.06)(6.125) =  xy - nxy 2  x 2 - nx 2 (2,167.7) - (8)(6.125)(43.36) (311) - (8)(6.125) 2 y = x y = (7) = 46.88, or 46,880 Regression equation Attendance forecast for 7 wins

Linear Regression Line 60,000 60,000 – 50,000 50,000 – 40,000 40,000 – 30,000 30,000 – 20,000 20,000 – 10,000 10,000 – ||||||||||| Wins, x Attendance, y

Linear Regression Line ||||||||||| ,000 60,000 – 50,000 50,000 – 40,000 40,000 – 30,000 30,000 – 20,000 20,000 – 10,000 10,000 – Linear regression line, y = x Wins, x Attendance, y

Correlation and Coefficient of Determination Correlation, r Correlation, r Measure of strength and direction of relationship between two variables Measure of strength and direction of relationship between two variables Varies between and Varies between and Coefficient of determination, r 2 Coefficient of determination, r 2 Percentage of variation in dependent variable resulting from changes in the independent variable. Percentage of variability in the values of the dependent variable that is explained by the independent variable. Percentage of variation in dependent variable resulting from changes in the independent variable. Percentage of variability in the values of the dependent variable that is explained by the independent variable.

Computing Correlation n  xy -  x  y [ n  x 2 - (  x ) 2 ] [ n  y 2 - (  y ) 2 ] r = Coefficient of determination r 2 = (0.947) 2 = r = (8)(2,167.7) - (49)(346.9) [(8)(311) - (49 )2 ] [(8)(15,224.7) - (346.9) 2 ] r = 0.947

More on Regression Models: Multiple Regression Study the relationship of demand to two or more independent variables y =  +  1 x 1 +  2 x 2 … +  k x k where  =the intercept  1, …,  k =parameters for the independent variables x 1, …, x k =independent variables

More on Regression Models: Multiple Regression Multiple regression –More than one independent variable y x z y = a + b1 × x + b2 × z

More on Regression Models: Nonlinear Regression Non-linear models –Example:y = a + b × ln(x)

Important Points in Using Regression Important Points in Using Regression  Always plot the data to verify that a linear relationship is appropriate  Check whether the data is time- dependent. If so use time series instead of regression  A small correlation may imply that other variables are important

Measuring Forecast Accuracy How do we know:  If a forecast model is “best”?  If a forecast model is still working?  What types of errors a particular forecasting model is prone to make? We need measures of forecast accuracy

Forecast Accuracy Error = Actual – Forecast ( Error = Actual – Forecast (Et = Dt – Ft) Find a method which minimizes error Mean Forecast Error Mean Forecast Error Mean Absolute Deviation (MAD) Mean Absolute Deviation (MAD) Mean Squared Error (MSE) Mean Squared Error (MSE) Mean Absolute Percent Deviation (MAPE) Mean Absolute Percent Deviation (MAPE)

Mean Absolute Deviation (MAD) where t = the period number t = the period number A t = actual demand in period t A t = actual demand in period t F t = the forecast for period t F t = the forecast for period t n = the total number of periods n = the total number of periods  = the absolute value  A t - F t  n MAD = What does this tell us that MFE doesn’t?

MFE and MAD: A Dartboard Analogy Low MFE and MAD: The forecast errors are small and unbiased

An Analogy Low MFE, but high MAD: On average, the arrows hit the bulls eye (so much for averages!)

An Analogy High MFE and MAD: The forecasts are inaccurate and biased

MAD Example PERIODDEMAND, A t F t (  =0.3)

MAD Example –– PERIODDEMAND, A t F t (  =0.3)(A t - F t ) |A t - F t |

MAD Example –– PERIODDEMAND, D t F t (  =0.3)(D t - F t ) |D t - F t |  A t - F t  n MAD= = =

MSE, and MAPE MSE = Actualforecast )   n ( MAPE = Actualforecas t  n / Actual*100) 

Example 10

Forecast Control Reasons for out-of-control forecasts Reasons for out-of-control forecasts (sources of forecast errors) (sources of forecast errors) Change in trend Change in trend Appearance of cycle Appearance of cycle Inadequate forecasting models Inadequate forecasting models Irregular variations Irregular variations Incorrect use of forecasting technique Incorrect use of forecasting technique

Controlling the Forecast A forecast is deemed to perform adequately when the errors exhibit only random variations Control chart –A visual tool for monitoring forecast errors –Used to detect non-randomness in errors Forecasting errors are in control if –All errors are within the control limits –No patterns, such as trends or cycles, are present

Tracking Signal Compute each period Compute each period Compare to control limits Compare to control limits Forecast is in control if within limits Forecast is in control if within limits Use control limits of +/- 2 to +/- 5 MAD Tracking signal = =  (A t - F t ) MADEMAD Bias: persistent tendency for forecasts to be greater or less than actual values

Tracking Signal Values ––– DEMANDFORECAST,ERROR  E = PERIODD t F t A t - F t  (A t - F t )MAD

Tracking Signal Values ––– DEMANDFORECAST,ERROR  E = PERIODA t F t A t - F t  (A t - F t )MAD TS 3 = = Tracking signal for period 3

Tracking Signal Values –––– DEMANDFORECAST,ERROR  E =TRACKING PERIODA t F t A t - F t  (A t - F t )MADSIGNAL

Tracking Signal Plot 3  3  – 2  2  – 1  1  – 0  0  – -1  -1  – -2  -2  – -3  -3  – ||||||||||||| Tracking signal (MAD) Period

Tracking Signal Plot 3  3  – 2  2  – 1  1  – 0  0  – -1  -1  – -2  -2  – -3  -3  – ||||||||||||| Tracking signal (MAD) Period Exponential smoothing (  = 0.30)

Tracking Signal Plot 3  3  – 2  2  – 1  1  – 0  0  – -1  -1  – -2  -2  – -3  -3  – ||||||||||||| Tracking signal (MAD) Period Exponential smoothing (  = 0.30) Linear trend line

Statistical Control Charts  = = = =  (A t - F t ) 2 n - 1 Using  we can calculate statistical control limits for the forecast error Using  we can calculate statistical control limits for the forecast error Control limits are typically set at  3  Control limits are typically set at  3 

Statistical Control Charts Errors – – – 0 0 – – – – ||||||||||||| Period

Statistical Control Charts Errors – – – 0 0 – – – – ||||||||||||| Period UCL = +3  LCL = -3 

Choosing a Forecasting Technique No single technique works in every situation Two most important factors –Cost –Accuracy Other factors include the availability of: –Historical data –Computers –Time needed to gather and analyze the data –Forecast horizon