Forecasting

What is Forecasting? Process of predicting a future event Underlying basis of all business decisions: Production Inventory Personnel Facilities

Short-range forecast Up to 1 year (usually less than 3 months) Job scheduling, worker assignments Medium-range forecast 3 months to 3 years Sales & production planning, budgeting Long-range forecast 3 years, or more New product planning, facility location Forecasts by Time Horizon

Long vs. Short Term Forecasting Long and Medium range forecasts deal with more comprehensive issues support management decisions regarding planning and products, plants and processes. Short-term forecasts usually employ different methodologies than longer-term forecasting tend to be more accurate than longer-term forecasts.

Influence of Product Life Cycle Stages of introduction and growth require longer forecasts than maturity and decline Forecasts useful in projecting Staffing levels, Inventory levels, and Factory capacity as product passes through life cycle stages Introduction, Growth, Maturity, Decline

Types of Forecasts Economic forecasts Address the business cycle (e.g., inflation rate, money supply, etc.) Technological forecasts Predict the rate of technological progress Predict acceptance of new products Demand forecasts Predict sales of existing products

Seven Steps in Forecasting Determine the use of the forecast Select the items to be forecasted Determine the time horizon of the forecast Select the forecasting model(s) Gather the data Make the forecast Validate and implement results

Realities of Forecasting Forecasts are seldom perfect Most forecasting methods assume that there is some underlying stability in the system Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches Used when situation is stable & historical data exist Existing products Current technology Involves mathematical techniques e.g., forecasting sales of color televisions Quantitative Methods Used when situation is vague & little data exist New products New technology Involves intuition, experience e.g., forecasting sales on Internet Qualitative Methods

Forecasting Approaches …The reality of all forecasting techniques is that they depend on both subjective and objective inputs… …That is to say that, regardless of the initial approach, all forecasting techniques are a blend of both art and science

Qualitative Methods Jury of executive opinion Pool opinions of high-level executives, sometimes augment by statistical models Delphi method Panel of experts, queried iteratively Sales force composite Estimates from individual salespersons are reviewed for reasonableness, then aggregated Consumer Market Survey Ask the customer

Involves small group of high-level managers Group estimates demand by working together Combines managerial experience with statistical models Relatively quick “Group-think” disadvantage © 1995 Corel Corp. Jury of Executive Opinion

Sales Force Composite Each salesperson projects his or her sales Combined at district & national levels Sales reps know customers’ wants Tends to be overly optimistic

Delphi Method Iterative group process 3 types of people Decision makers Staff Respondents Reduces ‘group-think’

Consumer Market Survey Ask customers about purchasing plans What consumers say, and what they actually do are often different Sometimes difficult to answer

Quantitative Approaches Naïve approach Moving average Weighted moving average Exponential smoothing Exponential smoothing with trend Trend projection Seasonally adjusted

Set of evenly spaced numerical data Obtained by observing response variable at regular time periods Forecast based only on past values Assumes that factors influencing past and present will continue influence in future Example Year: Sales: Time Series Models

Any observed value in a time series is the product (or sum) of time series components Multiplicative model: Y i = T i · S i · C i · R i Additive model: Y i = T i + S i + C i + R i Time Series Methods

Time Series Terms Stationary Data a time series variable exhibiting no significant upward or downward trend over time Nonstationary Data a time series variable exhibiting a significant upward or downward trend over time Seasonal Data a time series variable exhibiting a repeating patterns at regular intervals over time

Trend Seasonal Cycle Random Time Series Components

Persistent, overall upward or downward pattern Due to population, technology etc. Several years duration Trend Component

Regular pattern of up & down fluctuations Due to weather, customs, etc. Occurs within 1 year Seasonal Component

Repeating up & down movements Due to interactions of factors influencing economy Can be anywhere between years duration Cyclical Component

Erratic, unsystematic, ‘residual’ fluctuations Due to random variation or unforeseen events Union strike Tornado Short duration & non-repeating Random Component

Demand with Trend & Seasonality Year 1 Year 2 Year 3 Year 4 Seasonal peaksTrend component Actual demand line Average demand over four years Demand for product or service Random variation

Time Series Analysis There are many, many different time series techniques It is usually impossible to know which technique will be best for a particular data set It is customary to try out several different techniques and select the one that seems to work best To be an effective time series modeler, you need to keep several time series techniques in your “tool box”

Naive Approach Assumes demand in next period is the same as demand in most recent period e.g., If May sales were 48, then June sales will be 48 Sometimes cost effective & efficient

Naïve Example tANaïve

Naïve Forecast tANaïve

Naïve Forecast Chart

MA is a series of arithmetic means Used if little or no trend Used often for smoothing Provides overall impression of data over time Equation: MA n n  Demand in Previous Periods Periods Moving Average Method

3 period MA Example tA3MA

3 period MA Forecast tA3MA

3 period MA Forecast Chart

Older data may be considered less important as a predictor Weights based on intuition May be established as any numerical value Equation: WMA = Σ(Weight for period n) (Demand in period n) ΣWeights Weighted Moving Average Method

3 period WMA Example tA3WMA [7, 2, 1]

3 period WMA Forecast tA3WMA [7, 2, 1]

3 period WMA Forecast Chart

Increasing n makes forecast less sensitive to changes Do not forecast trend well Require a great amount of historical data Only account for random variation © T/Maker Co. Disadvantages of MA Methods

Form of weighted moving average Weights decline exponentially Most recent data weighted most Requires smoothing constant (  ) Ranges: 0 <  < 1 Subjectively chosen The larger the value of , the more responsive the model will be to historical data Exponential Smoothing Method

F t =  A t  (1-  )A t  (1-  ) 2 ·A t  (1-  ) 3 A t  (1-  ) t-1 ·A 0 F t = Forecast value A t = Actual value  = Smoothing constant F t = F t-1 +  (A t-1 - F t-1 ) Use for computing forecast If F 1 is unknown, then F 1 = A 1 Exponential Smoothing Equations

ES Example (  0.1  tAES α =

ES Forecast (  0.1  tAES α =

ES Forecast (  0.1) Chart

ES Example (  0.5  tAES α =

ES Forecast (  0.5  tAES α =

ES Forecast (  0.5) Chart

Which Model Is “Best” So Far? Naïve = 20 3MA = WMA =18.9 ES (a = 0.1) = ES (a = 0.5) =18.36

Exponential Smoothing with Trend Adjustment Forecast including trend (FIT t ) = exponentially smoothed forecast (F t ) + exponentially smoothed trend (T t ) FIT t = F t + T t

F t =  (Last period’s actual demand) + (1 -  )(Last period’s forecast + Last period’s trend) F t =  A t-1 + (1 -  )(F t-1 + T t-1 ) T t =  (Forecast this period - Forecast last period) + (1-  )(Trend estimate last period) T t =  (F t - F t-1 ) + (1-  )T t-1 Exponential Smoothing with Trend Adjustment - continued

F t = exponentially smoothed forecast of the data series in period If F 1 is unknown, then F 1 = A 1 T t = exponentially smoothed trend in period t If T 1 is unknown, then T 1 = 0 A t = actual demand in period t  = smoothing constant for the average Ranges: 0 <  < 1  = smoothing constant for the trend Ranges: 0 <  < 1 Exponential Smoothing with Trend Adjustment - continued

Used for forecasting linear trend line Assumes relationship between response variable, Y, and time, X, is a linear function Estimated by least squares method Minimizes sum of squared errors i YabX i  Linear Trend Projection

Least Squares Deviation Time Values of Dependent Variable Actual observation Point on regression line

Linear Trend Forecast Chart

Linear Trend Equations Equation: Slope: Y-Intercept:

Slope (b) Estimated Y changes by b for each 1 unit increase in X If b = 2, then sales (Y) is expected to increase by 2 for each 1 unit increase in advertising (X) Y-intercept (a) Average value of Y when X = 0 If a = 4, then average sales (Y) is expected to be 4 when advertising (X) is 0 Interpretation of Coefficients

Variation of actual Y from predicted Y Measured by standard error of estimate Sample standard deviation of errors Denoted S Y,X Affects several factors Parameter significance Prediction accuracy Random Error Variation

Least Squares Assumptions Relationship is assumed to be linear. Plot the data first - if curve appears to be present, use curvilinear analysis. Relationship is assumed to hold only within or slightly outside data range. Do not attempt to predict time periods far beyond the range of the data base. Deviations around least squares line are assumed to be random.

Answers: ‘how strong is the linear relationship between the variables?’ Coefficient of correlation Sample correlation coefficient denoted r Range: -1 < r < 1 Measures degree of association Used mainly for understanding Correlation

Coefficient of Correlation (r) R 2 = Coefficient of Determination = square of correlation coefficient (r), is the percent of the variation in y that is explained by the regression equation

Additive vs. Multiplicative Seasonality Additive Seasonal Effects Time Period Multiplicative Seasonal Effects Time Period

An Additive Seasonal Model where p represents the number of seasonal periods E t is the expected level at time period t. S t is the seasonal factor for time period t.

A Multiplicative Seasonal Model where p represents the number of seasonal periods E t is the expected level at time period t. S t is the seasonal factor for time period t.

Multiplicative Example (p. 124) Find average historical demand for each “season” by summing the demand for that season in each year, and dividing by the number of years for which you have data. Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons. Compute a seasonal index by dividing that season’s historical demand (from step 1) by the average demand over all seasons. Estimate next year’s total demand Divide this estimate of total demand by the number of seasons, then multiply it by the seasonal index for that season. This provides the seasonal forecast.

Seasonal Example (p. 124) t Demand Average Monthly DemandSeasonal OverallIndexForecast Jan ? Feb ? Mar ? Apr ? May ? Jun ? Jul ? Aug ? Sep ? Oct ? Nov ? Dec ?

You want to achieve: No pattern or direction in forecast error Error = (Y i - Y i ) = (Actual - Forecast) Seen in plots of errors over time Smallest forecast error Mean Absolute Deviation (MAD), or Mean Absolute Percentage Error (MAPE) Mean Squared Error (MSE) Selecting a Forecasting Model ^

Mean Square Error (MSE) Mean Absolute Deviation (MAD) Mean Absolute Percent Error (MAPE) Forecast Error Equations

Naïve Forecast Errors tANaïve|e|e2e MAD:1.91 MSE:4.45 Control Limit (+/-):4.22

3MA Forecast Errors tA3MA|e|e2e MAD:2.07 MSE:5.23 Control Limit (+/-):4.58

3 WMA Forecast Errors tA3WMA|e|e2e2 [7, 2, 1] MAD:2.00 MSE:4.70 Control Limit (+/-):4.34

ES (  0.1) Forecast Errors tAES|e|e2e2 α = MAD:3.48 MSE:14.72 Control Limit (+/-):7.67

ES (  0.5) Forecast Errors tAES|e|e2e2 α = MAD:2.05 MSE:4.79 Control Limit (+/-):4.38

Which Model Is “Best” So Far? The Naïve model has both the lowest MAD (1.91) and MSE (4.45) of the first five models tested Therefore, the Naïve model is the “best” However, it may be that one model has the lowest MAD or MAPE and another model has the lowest MSE…

So Which Model Do You Choose? If you only require the forecast with the smallest average deviation, choose the model with the smallest MAD or MAPE However, if you have a low tolerance for large deviations choose the model with the smallest MSE

Control Charts for Forecasting Once you have selected the “best” forecasting model… construct a control chart to monitor the continuing performance of the model’s forecasts: The center line is the average error = 0 The upper and lower control limits use a proxy of (+ or – 2 times the root mean square error) to approximate a 95% level of confidence.

Control Charts for Forecasting

Once you have constructed the chart plot each new forecast error and examine the trend for any patterns… If any patterns develop there is “cause for inspection” …making the existing model suspect and The parameters might need modification, or A new model must be developed

Patterns in Control Charts

Forecasting Quiz Suppose you had the following sales: Use the models: 4MA 3WMA [3, 2, 1] ES [alpha = 0.1] ES [alpha = 0.5] Forecast period 13 for each Find the MAD & MSE for each Answers… JanFebMar AprMayJun JulAugSep OctNovDec