Introduction to

(Demand) Forecasting
 Module Outline
• The role of forecasting in contemporary production planning
frameworks
• Basic characterization of the (demand) forecasting problem
• Forecasting methods and some selection criteria
• A generic approach to quantitative forecasting
• Time series-based forecasting
• Building causal models through multiple linear regression
• Confidence Intervals and their application in forecasting
 Forecasting
• Def: The process of predicting the values of a certain
quantity, Q, over a certain time horizon, T, based on past
trends and/or a number of relevant factors.
• In the context of OM, the most typically forecasted
quantity is future demand(s), but the need of forecasting
arises also with respect to other issues, like:
– equipment and employee availability
– technological forecasts
– economic forecasts (e.g., inflation rates, exchange rates, housing
starts, etc.)
• The time horizon depends on
– the nature of the forecasted quantity
– the intended use of the forecast
 Forecasting future demand
• Product/Service demand: The pattern of order arrivals and
order quantities evolving over time.
• Demand forecasting is based on:
– extrapolating to the future past trends observed in the company
sales;
– understanding the impact of various factors on the company future
sales:
• market data
• strategic plans of the company
• technology trends
• social/economic/political factors
• environmental factors
• etc

• Rem: The longer the forecasting horizon, the more crucial

the impact of the factors listed above.
 Demand Patterns
• The observed demand is the cumulative result of:
– some systematic variation, resulting from the (previously)
identified factors, and
– a random component, incorporating all the remaining unaccounted
effects.
• (Demand) forecasting tries to:
– identify and characterize the expected systematic variation, as a set
of trends:
• seasonal: cyclical patterns related to the calendar (e.g., holidays,
weather)
• cyclical: patterns related to changes of the market size, due to, e.g.,
economics and politics
• business: patterns related to changes in the company market share,
due to e.g., marketing activity and competition
• product life cycle: patterns reflecting changes to the product life
– characterize the variability in the demand randomness
 Forecasting Methods
• Qualitative (Subjective): Incorporate factors like the
forecaster’s intuition, emotions, personal experience, and
value system; these methods include:
– Jury of executive opinion
– Sales force composites
– Delphi method
– Consumer market surveys
• Quantitative (Objective): Employ one or more
mathematical models that rely on historical data and/or
causal/indicator variables to forecast demand; major
methods include:
– time series methods: F(t+1) = f (D(t), D(t-1), …)
– causal models: F(t+1) = f(X1(t), X2(t), …)
 Selecting a Forecasting Method
• It should be based on the following considerations:
– Forecasting horizon (validity of extrapolating past data)
– Availability and quality of data
– Lead Times (time pressures)
– Cost of forecasting (understanding the value of
forecasting accuracy)
– Forecasting flexibility (amenability of the model to
revision; quite often, a trade-off between filtering out
noise and the ability of the model to respond to abrupt
and/or drastic changes)
 Applying a Quantitative Forecasting Method
Determine Method
•Time Series
•Causal Model

Collect data:
<Ind.Vars; Obs. Dem.>

Fit an analytical model - Determine

functional form
to the data: - Estimate parameters
F(t+1) = f(X1, X2,…) - Validate

Use the model for

forecasting future
demand
Update Model
Parameters
Monitor error:
e(t+1) = D(t+1)-F(t+1)

Yes Model No
Valid?
 Time Series-based Forecasting
Basic Model:

Dˆ (t   ),  1,2,...
Time Series
D(i ), i  1,..., t Model
Historical Forecasts
Data

Remark: The exact model to be used depends on the expected /

observed trends in the data.
Cases typically considered:
• Constant mean series
• Series with linear trend
• Series with seasonalities (and possibly a linear trend)
 A constant mean series
14.00

12.00

10.00

8.00
Series1
6.00

4.00

2.00

0.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The above data points have been sampled from a normal distribution with a
mean value equal to 10.0 and a variance equal to 4.0.
 Forecasting constant mean series:
The Moving Average model
The presumed model for the observed data:

D(t )  D  e(t )
where
D is the constant mean of the series and
e(t ) is normally 2distributed with zero mean and some unknown
variance 
Then, under a Moving Average of Order N model, denoted as MA(N),
the estimate of D returned at period t, is equal to:
N 1
ˆ 1
D (t ) 
N
 D(t  i)
i 0
 The Moving Average Model:
The selection of the model order, N, and
its impact on the model accuracy
Some “rules of thumb” for selecting an appropriate value for N:
• Smaller values of N give the model more flexibility since it focuses on the more
recent observations; this property is useful when the observed series experiences
frequent “jumps”.
• On the other hand, in case of a stationary series, larger values of N provide more
accuracy to the forecasts, since they reduce the variance of the forecasting error;
more specifically, defining the forecasting error as:
ˆ 1 N 1
 (t  1)  D (t )  D(t  1)  
N i 0
D(t  i )  D(t  1)
we obtain N 1
1 1
E[ (t  1)] 
N
 E[ D(t  i)]  E[ D(t  1)] 
i 0 N
( ND )  D  0
and
N 1
1 1 1 2
Var[ (t  1)]  2
N
Var[ D(t  i)]  Var[ D(t  1)] 
i 0 N 2
N 2
  2
 (1 
N
)
 Demonstrating the impact of order N on
the model performance
25.00

20.00

15.00
Series1
Series2
Series3
10.00

5.00

0.00
1

10

13

16

19

22

25

28

31

34

37

40
In the above plot, the blue series is the original data series, distributed according to N(10,4)
for the first 20 points, and N(20,4) for the last 20 points. The magenta series corresponds to
the predictions of a MA(5) forecasting model and the yellow series to the predictions of a
MA(10) forecasting model. As expected, the MA(5) model adjusts faster to the
experienced jump of the data mean value, but the mean estimates that it provides under
stationary operation are, in general, less accurate than those provided by the MA(10)
model.
 The Moving Average Model:
The selection of the model order N and
its impact on the model accuracy (cont.)
Remark 1: The definition of (t+1) as a linear combination of independent,
normally distributed random variables implies that it is also normally
distributed with the mean and variance computed in the previous slide.
Remark 2: Following a derivation similar to that in the previous slide, we can
ˆ (t )  D follows a normal distribution with zero
establish that the quantity D
mean and variance 2/N.
Remark 3: In practice, N is frequently selected through trial and error, by
applying different MA(N) models on the available data, and selecting the model
that minimizes one of the next criteria:
t
1
i) MAD(t )  
t  N i  N 1
|  (t ) |
t
1
ii) MSD (t )  
t  N i  N 1
( (t )) 2

iii) MAPE (t )  1   (t )
t

t  N i  N 1 D(t )
Remark 4: C.f. the attached spreadsheet for demonstrating examples.
 Forecasting constant mean series:
The Simple Exponential Smoothing model
The presumed model for the observed data series is the same as in
the case of the MA model, i.e.,
D(t )  D  e(t )
where D is an unknown constant and e(t ) is normally distributed
with zero mean and an unknown variance  2 .
The forecast Dˆ (t ) , at the end of period t, is computed through the
following recursion:
Dˆ (t )  aD(t )  (1  a) Dˆ (t  1)  Dˆ (t  1)  a[ D(t )  Dˆ (t  1)]
where (0,1) and it is known as the “smoothing constant”.
Remark: Notice that the updating equation can be considered as
a correction of the previous estimate in the direction suggested by the
forecasting error, D(t )  Dˆ (t  1) .
 The Simple Exponential Smoothing Model:
The role of the smoothing constant
We have:
Dˆ (t )  aD(t )  (1  a) Dˆ (t  1)
 aD(t )  a(1  a) D(t  1)  (1  a) 2 Dˆ (t  2)  ...
t 1
 a (1  a ) i D(t  i )  (1  a ) t Dˆ (0)
 i 0
Hence,
ˆ
1. The model considers all the past observations and the initializing value D (0)in
the determination of the estimate Dˆ (t ).
2. However, the weight / impact of the various data values decreases exponentially
with their age.
3. Furthermore, as 1, the model places more emphasis on the most recent
observations.
4. Finally, using the above formula it is easy to show that as t,
E[ Dˆ (t )]  D and Var[ D(t  1)  Dˆ (t )] 
2
2
2a
5. C.f. the attached spreadsheet for demonstrating examples.
 Demonstrating the impact of the smoothing
constant  and the initial estimate Dˆ (0)
on the model performance
25.00

20.00

15.00 Series1
Series2
Series3
10.00 Series4

5.00

0.00
1

10

13

16

19

22

25

28

31

34

37

40
In the above plot, the dark blue series is the original data series, distributed according to
N(10,4) for the first 20 points, and N(20,4) for the last 20 points. The magenta series is the
predictions of an ES(0.2) model initialized at the value of 10.0, the yellow series is the
predictions of an ES(0.2) model initialized as 0.0, and the light blue series is the predictions
of an ES(0.8) model initialized at 10.0. As expected, the ES(0.8) model adjusts faster to the
experienced jump of the data mean value, but the mean estimates that it provides under
stationary operation are, in general, less accurate than those provided by the ES(0.2) model.
Also, notice the (only) transient effect of the initial value on the model estimates.
 The inadequacy of SES and MA models
for data with linear trends
12

10

8
Dt
6 SES(0.5)
SES(1.0)
4

0
1 2 3 4 5 6 7 8 9 10

In the above plot, the blue series is a deterministic data series increasing linearly
with a slope of 1.0. The magenta and the yellow series are respectively the
predictions obtained from the application of a SES(0.5) and SES(1.0) model
initialized at the exact value of 1.0. It is clear that both of these models
systematically under-estimate the actual values, with the most inert model
SES(0.5) under-estimating the most. This should be expected since either of
these models (as well as any MA model) essentially averages the past
observations. Therefore, neither of the MA nor the SES model are appropriate for
forecasting a data series with a linear trend in it.
 Forecasting series with linear trend:
The Double Exponential Smoothing Model
The presumed model for the observed data:
D(t )  I  T  t  e(t )
where
I is the model intercept, i.e., the unknown mean value for t=0,
T is the model trend, i.e., the mean increase per unit of time, and
e(t ) is normally distributed with zero mean and some unknown
variance  2
The model forecasts at period t for periods t+, =1,2,…, are given by:
Dˆ (t   )  Iˆ(t )  Tˆ (t ) 
with the quantities Iˆ(t ) and Tˆ (t ) obtained through the following recursions:
Iˆ(t )  a  D(t )  (1  a)[ Iˆ(t  1)  Tˆ (t  1)]
Tˆ (t )   [ Iˆ(t )  Iˆ(t  1)]  (1   )  Tˆ (t  1)
The parameters a and  take values in the interval (0,1) and are the model
smoothing constants, while the values Iˆ(0)and Tˆ (0) are the initializing values.
 Forecasting series with linear trend:
The Double Exponential Smoothing Model
(cont.)
Remark 1: Similar to the Simple Exp. Smoothing model, the smoothing
constants are chosen empirically, by trial and error, using the MAD, MSD
and/or MAPE indices.
Remark 2: Also, it can be shown that for t, Iˆ(t )  I and Tˆ (t )  T
Remark 3: In principle, the variance of the forecasting error,  2 , can be
estimated as a function of the noise variance 2 through techniques similar to
those used in the case of the Simple Exp. Smoothing model, but in practice, it is
frequently approximated by ˆ 2  1.25MAD(t ) where

MAD(t )  g  (t )  (1  g )MAD(t 1)

for some appropriately selected smoothing constant g(0,1) or by ˆ   MSD (t )
2

Remark 4: Since, both, the MA and the Simple Exp. Smoothing models are
essentially averaging processes, their application on a series with a linear trend
will result in a systematic error known as lag.
Remark 5: The application of the Double Exp. Smoothing model, its convergent
properties, and the inadequacy of the MA and Simple Exp. Smoothing are
demonstrated in the attached spreadsheet.
 DES Example
12

10

8
Dt
6 DES(T0=1)
DES(T0=0)
4

0
1 2 3 4 5 6 7 8 9 10

The above plot demonstrates the application of the DES model on the data series
of slide 18. Both applied models have smoothing constants =0.5 and =0.2,
however, the magenta series corresponds to a model initialized so that the initial
prediction is exact (i.e., equal to 1.0) while the yellow series corresponds to an
initial estimate equal to 0.0. In the absence of variability in the original data, the
first model is completely accurate (the blue and the magenta series overlap
completely), while the second model overcomes the deficiency of the wrong
initial estimate and eventually converges to the correct values.
 Time Series-based Forecasting:
Accommodating seasonal behavior
In this case, the data demonstrate a periodic behavior (and maybe
some additional linear trend).

Example: Consider the following data, describing a quarterly

demand over the last 3 years, in 1000’s:

Year 1 Year 2 Year 3

Spring 90 115 120
Summer 180 230 290
Fall 70 85 105
Winter 60 70 100
Total 400 500 615
 Seasonal Indices
Plotting the demand data:

350

300

250

200
Series1
150

100

50

0
0 2 4 6 8 10 12 14

Remarks:
• At each cycle, the demand of a particular season is a fairly stable percentage of
the total demand over the cycle.
• Hence, the ratio of a seasonal demand to the average seasonal demand of the
corresponding cycle will be fairly constant.
• This ratio is characterized as the corresponding seasonal index.
 A forecasting methodology
Forecasts for the seasonal demand for subsequent years can be obtained by:
i. estimating the seasonal indices corresponding to the various seasons in the
cycle;
ii. estimating the average seasonal demand for the considered cycle (using, for
instance, a forecasting model for a series with constant mean or linear trend,
depending on the situation);
iii. adjusting the average seasonal demand by multiplying it with the
corresponding seasonal index.

Example (cont.):
Year 1 Year 2 Year 3 SI(1) SI(2) SI(3) SI
Spring 90 115 120 0.9 0.92 0.78 0.87
Summer 180 230 290 1.8 1.84 1.88 1.84
Fall 70 85 105 0.7 0.68 0.68 0.69
Winter 60 70 100 0.6 0.56 0.65 0.6
Total 400 500 615 4 4 4 4
Average 100 125 153.75
 Winter’s Method for Seasonal Forecasting
The presumed model for the observed data:
D(t )  ( I  T  t )  c(t 1) mod N 1  e(t )
where

• N denotes the number of seasons in a cycle;

• ci, i=1,2,…N, is the seasonal index for the i-th season in the cycle;

• I is the intercept for the de-seasonalized series obtained by dividing the

original demand series with the corresponding seasonal indices;

• T is the trend of the de-seasonalized series;

• e(t) is normally distributed with zero mean and some unknown variance  2
 Winter’s Method for Seasonal Forecasting
(cont.)
The model forecasts at period t for periods t+, 1,2,…, are given by:

Dˆ (t   )  [ Iˆ(t )  Tˆ (t )  ]  cˆ(t  1) mod N 1 (t )

Where the quantities Iˆ(t ) , Tˆ (t ) and cˆi (t ), i  1,..., N , are obtained from the
following recursions, performed in the indicated sequence:
D(t )
Iˆ(t ) : a  (1  a)[ Iˆ(t  1)  Tˆ (t  1)]
cˆ(t 1) mod N 1 (t  1)

Tˆ (t ) :   [ Iˆ(t )  Iˆ(t  1)]  (1   )  Tˆ (t  1)

D(t )
cˆ(t 1) mod N 1 (t ) : g  (1  g )  cˆ(t 1) mod N 1 (t  1)
ˆI (t )

cˆi (t ) : cˆi (t  1), i  (t  1) mod N  1

The parameters , , g take values in the interval (0,1) and are the model smoothing
constants, while Iˆ(0), Tˆ (0) and cˆi (0), i  1,..., N , are the initializing values.
 Causal Models:
An Introduction to Multiple Linear Regression
The basic model:
D  b0  b1  X 1  ...  bk X k  e
where
• Xi, i=1,…,k, are the model independent variables (otherwise known as the
explanatory variables);
• bi, i=0,…,k, are unknown model parameters;
• e is the a random variable following a normal distribution with zero mean and
some unknown variance 2.
Remark: It follows from the above that D follows a normal distribution N ( D ,  )
2

where D  b0  b1  X 1  ...  bk  X k

Our problem is to estimate <b0,b1,…,bk> and 2 from a set of n observations

{ D j ; X 1 j , X 2 j ,..., X kj , j  1,..., n}
 Estimating the parameters bi
According to the presumed model, the observed data satisfy the following equation:
 D1   1 X 11 ... X k1  b0   e1 
D   1 X ... X k 2   b1  e2 
 2   12

 ...  ... ... ... ...   ...   ... 
      
 Dn   1 X 1n ... X kn  bk  en 
or in a more concise form
d  X b  e
For any given value of the parameter vector b, the vector
e  d  X b
denotes the difference between the actual observations and the corresponding
mean values, and therefore, the estimate b̂ for the parameter vector b is selected
such that it minimizes the Euclidean norm of the resulting vector eˆ  d  X  bˆ .
It is easy to show through basic calculus that the minimizing value for b is equal to
bˆ  ( X T X ) 1 X T d
The necessary and sufficient condition for the existence of ( X T X ) 1 is that the
columns of matrix X are linearly independent.
 Characterizing the model variance
An unbiased estimate of 2 is given by
SSE
MSE  (Mean Squared Error)
n  k 1
where
SSE  eˆT  eˆ  (d  X  bˆ)T (d  X  bˆ) (Sum of Squared Errors)
Also, the quantity SSE/2 follows a Chi-square distribution with n-k-1 degrees of
freedom.
Given a point x0T=(1,x10,…,xk0), an unbiased estimator of D ( x0 ) is given by

Dˆ ( x0 )  bˆ0  bˆ1  x10  ...  bˆk  xk 0

This estimator is normally distributed with mean D ( x0 ) and variance  2 x0 ( X T X )1 x0
T

The random variable Dˆ ( x0 ) can function also as an estimator for any single
observation D(x0). Based on the above, it should be easy to see that the resulting
error Dˆ ( x0 )  D( x0 ) will have zero mean and variance  [1  x0 ( X X ) x0 ]
2 T T 1
 Assessing the goodness of fit
A rigorous characterization of the quality of the resulting approximation can be
obtained through Analysis of Variance, that can be traced in any introductory book
on statistics.

A more empirical test considers the coefficient of multiple determination

SSR
R 
2

SYY
where n
SSR  bˆT ( X T d )  nd 2   ( Dˆ j  d ) 2
n j 1
1
d   Dj
n j 1
and
SYY  SSE  SSR
A natural way to interpret R2 is as the fraction of the variability in the observed
data interpreted by the model over the total variability in this data.
 Multiple Linear Regression and Time
Series-based forecasting
Remark 1: For the previous analysis and results to carry on, the model needs to be
linear with respect to the parameters bi but not the explanatory variables Xi. Hence,
the factor multiplying the parameter bi can be any function fi of the underlying
explanatory variables.

Remark 2: A case of particular interest regarding Remark 1 above, is when the only
explanatory variable is just the time variable t. The resulting multiple linear
regression models essentially support time-series analysis.

Remark 3: Furthermore, it is worth-noticing that this approach enables the modeling

and analysis of more complex dependencies on time than those addressed by the
previously studied models of moving averages and exponential smoothing.

Remark 4: On the other hand, the model updating upon the obtaining of a new
observation is much more cumbersome for multiple linear regression-based models
than the updating performed by the models based on moving averages and
exponential smoothing (although there is an incremental linear regression model that
alleviates this problem).
 Confidence Intervals
Given a random variable X and p(0,1), a p100% confidence interval (CI) for
it is an interval [a,b] such that
P ( a  X  b)  p
In the case of forecasting applications, confidence intervals can be useful for the
following two reasons:
i. Monitoring the performance of the applied forecasting model, in particular,
the failure of an (series of) observation(s) to fall within the scope of a p-
confidence interval, for an appropriately selected p, can be perceived as a
signal for the model inadequacy.
ii. Adjusting an obtained forecast in order to achieve a certain performance
level, for instance, in the case of demand forecast, one might want to adopt
for planning purposes a demand value such that the actual demand will not
exceed this value with probability p.
In both of the above cases, the necessary confidence intervals can be obtained
by exploiting the statistics for the forecasting error, derived in the previous
slides.
Next we demonstrate this capability for the multiple linear regression model;
however, the presented methodology can be readily adjusted to the Moving
Average and Exponential Smoothing models.
Variance estimation and the t distribution
In all models presented in the previous slides, the variance of the forecasting
error is a function of the unknown variance, 2, of the model disturbance, e.
For instance, in the case of multiple linear regression, the variance of the
ˆ
forecasting error D ( x0 )  D( x0 ) is equal to  2
[1  x
T T 1
0 ( X X ) x0 ] .

Hence, one cannot take advantage directly of the normality of the forecasting
error in order to build the sought confidence intervals.
However, this problem can be circumvented by exploiting the additional fact
that the quantity SSE/2 follows a Chi-square distribution with n-k-1 degrees of
freedom.
Then, the quantity

[ Dˆ ( x0 )  D( x0 )]  1  x0 ( X T X ) 1 x0 Dˆ ( x0 )  D( x0 )
T

T 
SSE  2
MSE  [1  x0 ( X T X ) 1 x0 ]
T

n  k 1
follows a t distribution with n-k-1 degrees of freedom.
Remark: For large samples, T can also be approximated by a standardized
normal distribution.
 Adjusting the forecasted demand in order
to achieve a target service level p
Letting y denote the required adjustment, we essentially need to solve the following
equation:
ˆ ( x )  y)  p 
P( D( x0 )  D 0

D( x0 )  Dˆ ( x0 ) y
P(  ) p
MSE[1  x0 ( X T X ) 1 x0 ] MSE[1  x0 ( X T X ) 1 x0 ]
T T

y
 t p ,nk 1 
MSE[1  x0 ( X T X ) 1 x0 ]
T

y  t p ,nk 1 MSE[1  x0 ( X T X ) 1 x0 ]
T

The two-sided confidence interval that is necessary for the model performance
monitoring can be obtained through a straightforward modification of the above
reasoning.
 Suggested Readings
• For an introductory coverage, especially on time series models, any textbook
on Production Planning and/ or Operations Management, e.g., S. Nahmias,
Production and Operations Analysis, McGraw Hill.
• For a more in-depth coverage, cf. S. Makridakis, S. Wheelwright and R.
Hyndman, Forecasting: Methods and Applications, John Wiley & Sons.