Forecasting

What is Forecasting?

Process of predicting a future event

Underlying basis of all business decisions: Production Inventory Personnel Facilities

Short-range forecastUp to 1 year (usually less than 3 months)Job scheduling, worker assignments

Medium-range forecast3 months to 3 yearsSales & production planning, budgeting

Long-range forecast3 years, or moreNew product planning, facility location

Forecasts by Time Horizon

Long vs. Short Term Forecasting

Long and Medium range forecastsdeal with more comprehensive issuessupport management decisions regarding

planning and products, plants and processes.

Short-term forecastsusually employ different methodologies

than longer-term forecastingtend 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 forecastsAddress the business cycle (e.g., inflation

rate, money supply, etc.)Technological forecasts

Predict the rate of technological progressPredict acceptance of new products

Demand forecastsPredict sales of existing products

Seven Steps in Forecasting

Determine the use of the forecastSelect the items to be forecastedDetermine the time horizon of the forecastSelect the forecasting model(s)Gather the dataMake the forecastValidate and implement results

Realities of Forecasting

Forecasts are seldom perfectMost 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 MethodsJury 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 managersGroup estimates demand by working

togetherCombines managerial experience with

statistical modelsRelatively 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 levelsSales reps know customers’ wantsTends to be overly optimistic

Delphi Method

Iterative group process3 types of people

Decision makersStaffRespondents

Reduces ‘group-think’

Consumer Market Survey

Ask customers about purchasing plansWhat consumers say, and what they

actually do are often differentSometimes difficult to answer

Quantitative Approaches

Naïve approachMoving averageWeighted moving averageExponential smoothingExponential smoothing with trendTrend projectionSeasonally 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

ExampleYear: 1998 1999 2000 2001 2002

Sales: 78.7 63.5 89.7 93.2 92.1

Time Series Models

Any observed value in a time series is the product (or sum) of time series components

Multiplicative model:Yi = Ti · Si · Ci · Ri

Additive model:Yi = Ti + Si + Ci + Ri

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

TrendTrend

SeasonalSeasonal

CycleCycle

RandomRandom

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 movementsDue to interactions of factors influencing

economyCan be anywhere between 2-30+ 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

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

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 48Sometimes cost effective & efficient

Naïve Example

t A Naïve     

1 11

2 13

3 14

4 15

5 13

6 15

7 17

8 18

9 19

10 15

11 17

12 20

13

Naïve Forecast

t A Naïve     

1 112 13 113 14 134 15 145 13 156 15 137 17 158 18 179 19 18

10 15 1911 17 1512 20 17

13 20

Naïve Forecast Chart

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12 13

Actual

Forecast

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:

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

3 period MA Example

t A 3MA     1 112 133 144 155 136 157 178 189 19

10 1511 1712 2013

3 period MA Forecast

t A 3MA     1 112 133 144 15 12.675 13 146 15 147 17 14.338 18 159 19 16.67

10 15 1811 17 17.3312 20 17

13 17.33

3 period MA Forecast Chart

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12 13

Actual

Forecast

Older data may be considered less important as a predictor

Weights based on intuitionMay be established as any numerical value

Equation:

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

3 period WMA Example

t A 3WMA    [7, 2, 1]1 112 133 144 155 136 157 178 189 19

10 1511 1712 2013

3 period WMA Forecast

t A 3WMA    [7, 2, 1]1 112 133 144 15 13.55 13 14.66 15 13.57 17 14.68 18 16.29 19 17.5

10 15 18.611 17 16.112 20 16.8

13 18.9

3 period WMA Forecast Chart

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12 13

Actual

Forecast

Increasing n makes forecast less sensitive to changes

Do not forecast trend wellRequire a great amount of historical

dataOnly account for random variation© 1984-1994 T/Maker Co.

Disadvantages of MA Methods

Form of weighted moving averageWeights decline exponentiallyMost recent data weighted most

Requires smoothing constant ()Ranges: 0 < < 1Subjectively chosen

The larger the value of , the more responsive the model will be to historical data

Exponential Smoothing Method

Ft = At - 1 + (1-)At - 2 + (1- )2·At - 3

+ (1- )3At - 4 + ... + (1- )t-1·A0

Ft = Forecast value

At = Actual value = Smoothing constant

Ft = Ft-1 + (At-1 - Ft-1)Use for computing forecast If F1 is unknown, then F1 = A1

Exponential Smoothing Equations

ES Example (0.1

t A ES    α = 0.11 112 133 144 155 136 157 178 189 19

10 1511 1712 2013

ES Forecast (0.1

t A ES    α = 0.11 11 112 13 113 14 11.24 15 11.485 13 11.8326 15 11.94887 17 12.253928 18 12.728539 19 13.25568

10 15 13.8301111 17 13.947112 20 14.25239

13 14.82715

ES Forecast (0.1) Chart

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12 13

Actual

Forecast

ES Example (0.5

t A ES    α = 0.51 112 133 144 155 136 157 178 189 19

10 1511 1712 2013

ES Forecast (0.5

t A ES    α = 0.51 11 112 13 113 14 124 15 135 13 146 15 13.57 17 14.258 18 15.6259 19 16.8125

10 15 17.9062511 17 16.4531312 20 16.72656

13 18.36328

ES Forecast (0.5) Chart

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12 13

Actual

Forecast

Which Model Is “Best” So Far?

Naïve = 20

3MA = 17.33

3WMA = 18.9

ES (a = 0.1) = 14.83

ES (a = 0.5) = 18.36

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

FITt = Ft + Tt

Ft = (Last period’s actual demand) + (1 - )(Last period’s forecast + Last period’s trend)

Ft = At-1 + (1 - )(Ft-1 + Tt-1)

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period)

Tt = (Ft - Ft-1) + (1- )Tt-1

Exponential Smoothing with Trend Adjustment - continued

Ft = exponentially smoothed forecast of the data series in period If F1 is unknown, then F1 = A1

Tt = exponentially smoothed trend in period t If T1 is unknown, then T1 = 0

At = 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 lineAssumes relationship between

response variable, Y, and time, X, is a linear function

Estimated by least squares methodMinimizes sum of squared errors

iY a bX i

Linear Trend Projection

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Linear Trend Forecast Chart

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12 13

Actual

Forecast

Linear Trend Equations

Equation: ii bxaY

Slope:22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept: xby a

Slope (b)Estimated Y changes by b for each 1 unit

increase in XIf 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 YMeasured by standard error of estimate

Sample standard deviation of errorsDenoted SY,X

Affects several factorsParameter significancePrediction 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 rRange: -1 < r < 1Measures degree of association

Used mainly for understanding

Correlation

r = 1 r = -1

r = .89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation (r)

R2 = 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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Time Period

Multiplicative Seasonal Effects

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Time Period

An Additive Seasonal Model

pnttnt SEYwhere

1)E-(1)S-Y(E tt-ptt

ptttt )S-(1)E-Y(S

10

10

p represents the number of seasonal periods

Et is the expected level at time period t. St is the seasonal factor for time period t.

A Multiplicative Seasonal Model

pnttnt SEYwhere

1)E-(1)/SY(E tt-ptt

ptttt )S-(1)/EY(S

10

10

p represents the number of seasonal periods

Et is the expected level at time period t. St 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 DemandAverage Monthly

Demand Seasonal 20062003 2004 2005 2003 - 2005 Overall Index Forecast

Jan 80 85 105 90 94 0.957 ?Feb 70 85 85 80 94 0.851 ?Mar 80 93 82 85 94 0.904 ?Apr 90 95 115 100 94 1.064 ?May 113 125 131 123 94 1.309 ?Jun 110 115 120 115 94 1.223 ?Jul 100 102 113 105 94 1.117 ?Aug 88 102 110 100 94 1.064 ?Sep 85 90 95 90 94 0.957 ?Oct 77 78 85 80 94 0.851 ?Nov 75 82 83 80 94 0.851 ?Dec 82 78 80 80 94 0.851 ?

You want to achieve:No pattern or direction in forecast error

Error = (Yi - Yi) = (Actual - Forecast)Seen in plots of errors over time

Smallest forecast errorMean 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

2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Naïve Forecast Errorst A Naïve |e| e2

         1 112 13 11.00 2.00 4.003 14 13.00 1.00 1.004 15 14.00 1.00 1.005 13 15.00 2.00 4.006 15 13.00 2.00 4.007 17 15.00 2.00 4.008 18 17.00 1.00 1.009 19 18.00 1.00 1.00

10 15 19.00 4.00 16.0011 17 15.00 2.00 4.0012 20 17.00 3.00 9.0013 20.00

MAD: 1.91MSE: 4.45

Control Limit (+/-): 4.22

3MA Forecast Errorst A 3MA |e| e2

         1 112 133 144 15 12.67 2.33 5.445 13 14.00 1.00 1.006 15 14.00 1.00 1.007 17 14.33 2.67 7.118 18 15.00 3.00 9.009 19 16.67 2.33 5.44

10 15 18.00 3.00 9.0011 17 17.33 0.33 0.1112 20 17.00 3.00 9.0013 17.33

MAD: 2.07MSE: 5.23

Control Limit (+/-): 4.58

3 WMA Forecast Errorst A 3WMA |e| e2

    [7, 2, 1]    1 112 133 144 15 13.50 1.50 2.255 13 14.60 1.60 2.566 15 13.50 1.50 2.257 17 14.60 2.40 5.768 18 16.20 1.80 3.249 19 17.50 1.50 2.25

10 15 18.60 3.60 12.9611 17 16.10 0.90 0.8112 20 16.80 3.20 10.2413 18.90

MAD: 2.00MSE: 4.70

Control Limit (+/-): 4.34

ES (0.1) Forecast Errorst A ES |e| e2

    α = 0.1    1 11 112 13 11 2.00 4.003 14 11.2 2.80 7.844 15 11.48 3.52 12.395 13 11.832 1.17 1.366 15 11.9488 3.05 9.317 17 12.25392 4.75 22.538 18 12.72853 5.27 27.799 19 13.25568 5.74 33.00

10 15 13.83011 1.17 1.3711 17 13.9471 3.05 9.3212 20 14.25239 5.75 33.0413 14.82715

MAD: 3.48MSE: 14.72

Control Limit (+/-): 7.67

ES (0.5) Forecast Errorst A ES |e| e2

    α = 0.5    1 11 112 13 11 2.00 4.003 14 12 2.00 4.004 15 13 2.00 4.005 13 14 1.00 1.006 15 13.5 1.50 2.257 17 14.25 2.75 7.568 18 15.625 2.38 5.649 19 16.8125 2.19 4.79

10 15 17.90625 2.91 8.4511 17 16.45313 0.55 0.3012 20 16.72656 3.27 10.7213 18.36328

MAD: 2.05MSE: 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.

MSE2

Control Charts for Forecasting

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 andThe parameters might need modification, orA 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…

Jan Feb Mar

15 25 20

     

Apr May Jun

35 30 25

     

Jul Aug Sep

20 30 35

     

Oct Nov Dec

40 30 35