9/3/2023
Learning Objectives
Forecasting
Why forecasting is necessary?
• Planning and control for operations requires an
estimate of the demand for the product or the
service that the organization expects to provide
in the future
• Understand the importance of forecasting
• What are the different types of forecasting
methodologies
• Different types of models which we can build for
the forecasting
• Discuss and calculate various methods for
evaluating forecast accuracy
Defining Forecasting
• So, planning and control for operations requires
an estimation of the demand. So, whether it is a
manufacturing function or a service function you
need to have some kind of estimation that, what is
going to be the demand, and that estimation is
called forecasting
• This forecasting is equally responsible for product
organizations, the manufacturing organization or
the service organization
• Eg- Car Mfg OR Hospital Setup
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The Effect of Inaccurate Forecasting
Time span for forecasting
• Current needs/short term (1 day-30 days)
• Intermediate range (1 month- 12 months)
• Long range plans (more than a year)
Basic categories of Forecasting
• Extrapolative methods
• Causal methods
• Qualitative
Quantitative
Basic categories of Forecasting
• Extrapolative methods-Time Series
Method/analysis
Considers historical data or past data
The limitation of the Time Series analysis,
is that we are forecasting for the coming
period but the information, the data which
you are using for that purpose that is of the
previous period
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Basic categories of Forecasting
• Causal methods - cause effect method
Considers demand based on various factors
Eg- Demand of a product is directly related to the
advertisement expenses
Y = a + bX, where Y is demand and X is those
independent factors
Basic categories of Forecasting
• Qualitative
Historical data not available
Not knowing the factors which are affecting
the demand
Eg- Y = a + b1 X1 + b2 X2 + b3 X3
Basic categories of Forecasting
• Qualitative forecasting types
Delphi
Market survey
Brainstorming
Extrapolative Methods/Time series Methods
For our good Time Series analysis, we need to
identify that pattern in the historical data
1. Horizontal component
2. Trend
3. Seasonal
4. Cyclic
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Comparative analysis of components of
historical data
Horizontal
Trend
Seasonal
Cyclic
can very easily
forecast
requires some
amount of efforts
require more efforts
require
extraordinary efforts
some minor
fluctuations are
there but more or
less it is around a
straight line
will not be a smooth
curve. There will be
zigzag movements
but overall effect is
either increasing or
decreasing.
after a particular
when demand is
interval demand will going to increase,
increase to a high
that we do not know
level and for rest of
the period demand
remains to a low
level
Comparative analysis of components of
historical data
Product life cycle
Forms of Forecast Movement
Comparative analysis of components of
historical data
Failure rate of a product
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Time Series Forecasting
• We can pick models based on
Simple Moving Average method
1. Time horizon to forecast
2. Data availability
3. Accuracy required
4. Size of forecasting budget
5. Availability of qualified personnel
Simple Moving Average method
Time Series Forecasting
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Weighted moving average method
Weighted moving average method
• Weighted moving average method is a method
which is differentiating between different periods.
It gives different weight to the different periods’
demand
• While the simple moving average formula gives
equal weight, the weighted moving average
method permits an unequal weighting on prior
time periods
Weighted moving average method
Weighted Moving Average
• Adjusts moving average method to more closely
reflect data fluctuations
WMAn =
n
 W i Di
i=1
where
W i = the weight for period i,
between 0 and 100
percent
 Wi = 1.00
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Weighted Moving Average Example
MONTH
August
September
October
WEIGHT
DATA
17%
33%
50%
130
110
90
Weighted Moving Average Example
MONTH
August
September
October
WEIGHT
DATA
17%
33%
50%
130
110
90
3
November Forecast
WMA3 =

i=1
W i Di
November Forecast
WMA3 =
3
 W i Di
i=1
= (0.50)(90) + (0.33)(110) + (0.17)(130)
= 103.4 orders
Simple exponential smoothing
• In exponential smoothing method, we
assign weights, but these weights are
decreasing in the exponential order
from the present period to the past
periods
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Exponential Smoothing
•
•
•
•
•
Averaging method
Weights most recent data more strongly
Reacts more to recent changes
Widely used, accurate method
Smoothing constant, α
• applied to most recent data
Exponential Smoothing
Simple exponential smoothing
St = St-1 + α (D
(Dt- St-1)
OR
St = α Dt + (1(1-α) St-1
Here, we are updating the base value, updating the base value
means, we are trying to find out the current base value and this
current base value is nothing but the forecast for the next
period.
St = Ft+1
Ft +1 = Dt + (1 - )Ft
where:
Ft +1 = forecast for next period
Dt = actual demand for present period
Ft = previously determined forecast for
present period
=
weighting factor, smoothing constant
Therefore
Ft+1 = α Dt + (1(1-α) Ft
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Simple exponential smoothing
Effect of Smoothing Constant
0.0  1.0
If = 0.20, then Ft +1 = 0.20Dt + 0.80 Ft
If = 0, then Ft +1 = 0Dt + 1 Ft = Ft
Forecast does not reflect recent data
If = 1, then Ft +1 = 1Dt + 0 Ft =Dt
Forecast based only on most recent data
Simple exponential smoothing
• Use Simple moving average method with an
average period (AP) of 3 days to develop a
forecast of the call volume in the Day 13
F13= (168+198+159)/3=175 calls
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Use Exponential smoothing
• Use Weighted moving average method with an
average period (AP) of 3 days and weights of
0.1, 0.3 and 0.6 for oldest to recent datum to
develop a forecast of the call volume in the Day
13
• If a smoothing constant value of 0.25 is used
and the exponential smoothing forecast for day
11 was 180.76 calls, what is the exponential
smoothing forecast for Day 13?
Ft+1 = α Dt + (1(1-α) Ft
F13= α D12 + (1(1-α) F12
F12= α D11 + (1(1-α) F11
F13=0.1(168)+0.3(198)+0.6(159)=171.6 calls
= (0.25x198) + (0.75x180.76)
=185.07
Exponential Smoothing (α=0.30)
F13= α D12 + (1-α) F12
= (0.25x159) + (0.75x185.07)
=178.55
PERIOD
1
2
3
4
5
6
7
8
9
10
11
12
MONTH
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
DEMAND
37
40
41
37
45
50
43
47
56
52
55
54
F2 = D1 + (1 - )F1
F3 = D2 + (1 - )F2
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Exponential Smoothing
Exponential Smoothing (α=0.30)
PERIOD
1
2
3
4
5
6
7
8
9
10
11
12
MONTH
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
DEMAND
37
40
41
37
45
50
43
47
56
52
55
54
F2 = D1 + (1 - )F1
= (0.30)(37) + (0.70)(37)
= 37
F3 = D2 + (1 - )F2
= (0.30)(40) + (0.70)(37)
= 37.9
Exponential Smoothing
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
13
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
37
40
41
37
45
50
43
47
56
52
55
54
–
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
13
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
37
40
41
37
45
50
43
47
56
52
55
54
–
FORECAST, Ft + 1
( = 0.3)
( = 0.5)
–
–
Exponential Smoothing
FORECAST, Ft + 1
( = 0.3)
( = 0.5)
–
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
51.79
–
37.00
38.50
39.75
38.37
41.68
45.84
44.42
45.71
50.85
51.42
53.21
53.61
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Exponential Smoothing with Trend and
Seasonality
Exponential Smoothing with Trend and
Seasonality
• Simple Exponential Smoothing model was
having only one component that was the base
component and the fluctuations are around this
base value and we wanted to smoothen those
fluctuation
• Simple Exponential Smoothing doesn’t have any
trend and seasonality
• Winters and Pegels, over the period of 10 years,
from 1960s to 1970, developed exponential
smoothing models by incorporating the fact of
trend as well as seasonality in the historical data
Exponential Smoothing with Trend and
Seasonality
Exponential Smoothing with Trend and
Seasonality
• Hence, we have 9 different types of models and
we can develop our extrapolative or exponential
smoothing models for all these 9 types of
models
• When we have trend also in our data, we will
use one more smoothing constant that is beta,
beta will be used for smoothing the fluctuations
of your trend data
• When we have seasonality also in our demand
data, you will use one more smoothing constant
that is gamma
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Smoothing Constants
α= Smoothing constant for average
β = Smoothing constant for trend
γ = Smoothing constant for seasonality
Double exponential smoothing
• Incorporating a trend component into
exponential smoothing forecast is also known as
double exponential smoothing
• Two smoothing constants, alpha and beta are
used
• One is for smoothing the fluctuations of your
base value, and second the smoothing the
fluctuation of your trend value
Smoothing Constants
• We have possibility of using either alpha alone,
you can use alpha and beta, you can use alpha
and gamma and you can use alpha, beta,
gamma altogether
• Use of any smoothing constant depends upon
the type of demand data
• There is no restriction on the combination of
alpha, beta and gamma’s value, the only
limitation is the boundary condition of the values
will vary between 0 to 1
Adjusted Exponential Smoothing/Trend
corrected exponential smoothing/Holt’s model
• ADJUSTED FORECAST
AFt +1 = Ft +1+Tt +1
Where
Ft +1 = forecast of average for next period
Ft+1 = α Dt + (1-α) Ft
Tt +1 = forecast of trend for next period
Tt+1= (Ft+1 - Ft) + (1 - ) Tt
13
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Trend corrected exponential
smoothing(α=0.5 and β=0.30)
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
37
40
41
37
45
50
43
47
56
52
55
54
T3
= (F3 - F2) + (1 - ) T2
AF3 = F3 + T3
T13 = (F13 - F12) + (1 - ) T12
AF13 = F13 + T13 =
Trend corrected exponential smoothing/Holt’s
smoothing
model
(α=0.5 and β=0.30)
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
13
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
37
40
41
37
45
50
43
47
56
52
55
54
–
FORECAST
Ft +1
TREND
Tt +1
ADJUSTED
FORECAST AFt +1
Trend corrected exponential smoothing
(α=0.5 and β=0.30)
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
37
40
41
37
45
50
43
47
56
52
55
54
T3
= (F3 - F2) + (1 - ) T2
= (0.30)(38.5 - 37.0) + (0.70)(0)
= 0.45
AF3 = F3 + T3 = 38.5 + 0.45
= 38.95
T13 = (F13 - F12) + (1 - ) T12
= (0.30)(53.61 - 53.21) + (0.70)(1.77)
= 1.36
AF13 = F13 + T13 = 53.61 + 1.36 = 54.97
Trend corrected exponential smoothing
PERIOD
MONTH
DEMAND
FORECAST
Ft +1
TREND
Tt +1
ADJUSTED
FORECAST AFt +1
1
2
3
4
5
6
7
8
9
10
11
12
13
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
37
40
41
37
45
50
43
47
56
52
55
54
–
37.00
37.00
38.50
39.75
38.37
38.37
45.84
44.42
45.71
50.85
51.42
53.21
53.61
–
0.00
0.45
0.69
0.07
0.07
1.97
0.95
1.05
2.28
1.76
1.77
1.36
–
37.00
38.95
40.44
38.44
38.44
47.82
45.37
46.76
58.13
53.19
54.98
54.96
14
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Trend corrected exponential smoothing
Holt’s model
• Example 2. (Assignment-2)
An electronics manufacturer has seen demand
for its latest Smartphone increase over the past
6 months. Observed demand in thousands has
been D1=8415, D2=8732, D3=9014, D4=9808,
D5=10413 and D6=11961. Forecast demand for
period 7 using trend corrected exponential
smoothing with α=0.1 and β =0.2.
Time Series Decomposition
Seasonal variation (additive or multiplicative)
• Chronologically ordered data are referred to as a
time series
• A time series may contain one or many elements
• Trend, seasonal, cyclical, autocorrelation, and
random
• Identifying these elements and separating the
time series data into these components is known
as decomposition
• Seasonal variation may be either additive or
multiplicative
15
9/3/2023
Seasonal factor/index
• The seasonal factor is defined as the ratio of
amount sold during each season to the average
of all seasons
• It is the amount of correction needed in a time
series to adjust for the season of the year
Example: (seasonal factor/index)
• In past years, a firm sold an average of 1,000
units each year
•
•
•
•
200 in spring
350 in summer
300 in fall
150 in winter
• Find the seasonal factors
• Using those factors, if we expected demand for
next year to be 1,100 units, compute demand
per period
Example : Finding Seasonal Factors
Example 18.3: Forecast for Next Year
16
9/3/2023
Linear Regression Analysis
• Regression can be defined as the functional
relationship between two or more correlated
variables
• Linear regression refers to the special class of
regression where the relationship between
variables forms a straight line
• The major restriction in using linear regression
forecasting is that the past data and future
projections are assumed to fall in about a
straight line
Least Squares Example
x(PERIOD)
1
2
3
4
5
6
7
8
9
10
11
12
y(DEMAND)
37
40
41
37
45
50
43
47
56
52
55
54
xy
x2
Linear Regression Analysis
y = a + bx
xy - nxy
x2 - nx2
a = y-bx
b =
where
a = intercept
b = slope of the line
x = time period
y = forecast for
demand for period x
where
n = number of periods
x
x =
= mean of the x values
n
y
y = n = mean of the y values
Least Squares Example
xy
x2
1
2
3
4
5
6
7
8
9
10
11
12
37
40
41
37
45
50
43
47
56
52
55
54
37
80
123
148
225
300
301
376
504
520
605
648
1
4
9
16
25
36
49
64
81
100
121
144
78
557
3867
650
x(PERIOD)
y(DEMAND)
17
9/3/2023
Least Squares Example
x =
y =
b = xy - nxy =
x2 - nx2
a = y - bx
Linear trend line y = 35.2 + 1.72x
Forecast for period 13 y = 35.2 + 1.72(13) = 57.56 units
Least Squares Example
x = 78 = 6.5
12
y = 557 = 46.42
12
b = xy - nxy =
x2 - nx2
3867 - (12)(6.5)(46.42) =1.72
650 - 12(6.5)2
a = y - bx
= 46.42 - (1.72)(6.5) = 35.2
Trend and seasonality corrected exponential
smoothing (Winter’s model)
Example:
• A hospital wants to improve its forecasting by
applying both trend and seasonal indices to 66
months of data it has collected. It will then
forecast “patient days” over the coming year
18
9/3/2023
Trend and seasonality corrected exponential
smoothing (Winter’s model)
Trend and seasonality corrected exponential
smoothing (Winter’s model)
Determine the forecast till December i.e till 78th period
Using 66 months of inpatient days the following
equation was computed
y= 8090+21.5x
Where y= patient days
x=time in months
Based on this model the patients day forecast for
67th month should be ?
y67= 8090+(21.5X67)= 9530 days
Trend and seasonality corrected exponential
smoothing (Winter’s model)
Trend and seasonality corrected exponential
smoothing (Winter’s model)
The following table provides seasonal indices
based on the same 6 6months data.
19
9/3/2023
Trend and seasonality corrected exponential
smoothing (Winter’s model)
 Neither trend data nor the seasonal data alone
provide a reasonable forecast for the hospital.
 Only when the hospital multiplied the trend
adjusted data with respective seasonal index, it
obtain good forecasts
Thus for 67th month the patient days=
(Trend adjusted forecast)(Monthly seasonal index)
=(9530X1.04)
=9911
Advantage of combined Trend and Seasonal
Adjustments
 With trend only, the September forecast is
9702 but with both trend and seasonal
adjustments the forecast is 9411
 By combining trend and seasonal data the
hospital is better able to forecast inpatient
days and the related staffing and budgeting
vital to effective operations
Trend and seasonality corrected exponential
smoothing (Winter’s model)
Determine the combined trend and seasonal forecast
Trend and seasonality corrected exponential
smoothing (Winter’s model)
 Question:
If the slope of the trend line for patient days
is 22 and the index for December is 0.99,
what is the new forecast for December
inpatient days?
Answer: 9708 days
20
9/3/2023
Accuracy of Forecasts
Mean Absolute Deviation (MAD)
• Forecast Error (FE)
Difference between forecast and actual demand
FE= Ft –Dt
The popular methods of error measurement
•
MAD =
 Dt - Ft 
n
where
t = period number
Dt = demand in period t
Ft = forecast for period t
n = total number of periods
 = absolute value
Mean Absolute Deviation (MAD)
• Mean Absolute Percent Deviation (MAPD)
• Cumulative Error (E)
• Average error or bias
MAD Example
PERIOD
1
2
3
4
5
6
7
8
9
10
11
12
DEMAND, Dt
37
40
41
37
45
50
43
47
56
52
55
54
Ft ( =0.3)
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
MAD Calculation
(Dt - Ft)
–
|Dt - Ft|
–
MAD =
 Dt - Ft 
n
21
9/3/2023
MAD Example
PERIOD
1
2
3
4
5
6
7
8
9
10
11
12
DEMAND, Dt
37
40
41
37
45
50
43
47
56
52
55
54
Ft ( =0.3)
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
557
MAD Calculation
(Dt - Ft)
|Dt - Ft|
–
3.00
3.10
-1.83
6.72
9.69
-0.20
3.86
11.70
4.19
5.94
3.15
–
3.00
3.10
1.83
6.72
9.69
0.20
3.86
11.70
4.19
5.94
3.15
49.31
53.39
Other Accuracy Measures
Mean absolute percent deviation (MAPD)
|Dt - Ft|
MAPD =
Dt
Cumulative error
E = et
 Dt - Ft 
n
53.39
=
11
MAD =
= 4.85
Comparison of Forecasts
FORECAST
MAD
MAPD
E
(E)
Exponential smoothing (= 0.30)
Exponential smoothing (= 0.50)
Adjusted exponential smoothing
(= 0.50, = 0.30)
Linear trend line
4.85
4.04
3.81
9.6%
8.5%
7.5%
49.31
33.21
21.14
4.48
3.02
1.92
2.29
4.9%
–
–
Average error
(E) =
et
n
22
9/3/2023
Tracking Signal Values
Forecast Control
Tracking signal
• It helps to detect any drift in the forecasting system
Tracking signal =
(Dt - Ft)
E
MAD = MAD
DEMAND
Dt
FORECAST,
Ft
1
2
3
4
5
6
7
8
9
10
11
12
37
40
41
37
45
50
43
47
56
52
55
54
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
TS3 =
ERROR
D t - Ft
–
3.00
3.10
-1.83
6.72
9.69
-0.20
3.86
11.70
4.19
5.94
3.15
E =
(Dt - Ft)
MAD
–
3.00
6.10
4.27
10.99
20.68
20.48
24.34
36.04
40.23
46.17
49.32
–
3.00
3.05
2.64
3.66
4.87
4.09
4.06
5.01
4.92
5.02
4.85
DEMAND
Dt
FORECAST,
Ft
1
2
3
4
5
6
7
8
9
10
11
12
37
40
41
37
45
50
43
47
56
52
55
54
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
ERROR
D t - Ft
–
3.00
3.10
-1.83
6.72
9.69
-0.20
3.86
11.70
4.19
5.94
3.15
E =
(Dt - Ft)
MAD
–
3.00
6.10
4.27
10.99
20.68
20.48
24.34
36.04
40.23
46.17
49.32
–
3.00
3.05
2.64
3.66
4.87
4.09
4.06
5.01
4.92
5.02
4.85
Tracking Signal Plot
Tracking Signal Values
PERIOD
PERIOD
TRACKING
SIGNAL
–
1.00
2.00
1.62
3.00
4.25
5.01
6.00
7.19
8.18
9.20
10.17
6.10
= 2.00
3.05
23