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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
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