forecasting strategy for Management Science

MNG221 - Management Science
Forecasting

Lecture Outline
• Forecasting basics
• Moving average
• Exponential smoothing
• Linear trend line
• Forecast accuracy

Forecasting Basics
Forecasting

Forecasting Basics
• A Forecast – is a prediction of something
that is likely to occur in the future.
• A variety of forecasting methods exist,
and their applicability is dependent on
the:
–time frame of the forecast (i.e., how
far in the future we are forecasting),

Forecasting Basics
• A variety of forecasting methods exist,
and their applicability is dependent on
the:
–the existence of patterns in the
forecast (i.e., seasonal trends, peak
periods), and
–the number of variables to which the
forecast is related.

Forecasting Components
Forecasting

• Time Frames of Forecast:
–Short Range - encompass the
immediate future and are concerned
with the daily operations rarely goes
beyond a couple months into the
future.

–Medium Range - encompasses
anywhere from 1 or 2 months to 1
year.
–More closely related to a yearly
production plan and will reflect such
items as peaks and valleys in demand

–Long Range - encompasses a period
longer than 1 or 2 years.
–It is Related to management's attempt
to plan new products for changing
markets, build new facilities, or secure
long-term financing.

• Forecasts can exhibit patterns or trend:
–A trend is a long-term movement of
the item being forecast
–Random variations are movements
that are not predictable and follow no
pattern (and thus are virtually
unpredictable).

Forecasts can exhibit patterns or trend:
A cycle is an undulating movement in
demand, up and down, that repeats itself
over a lengthy time span (i.e., more than 1 year).
A seasonal pattern is an oscillating
movement in demand that occurs periodically
(in the short run) and is repetitive.
Seasonality is often weather related.

Forecasting Components: Forecast Patterns
Forms of forecast movement: (a) trend, (b) cycle, (c)
seasonal pattern, and (d) trend with seasonal pattern

Forecasting Methods
Forecasting

Forecasting Methods
The forecasting component determines to
a certain extent the type of forecasting
method that can or should be used.
• Time Series - is a category of statistical
techniques that uses historical data to
predict future behavior.

Forecasting Methods
• Regression (or causal) methods -
attempt to develop a mathematical
relationship (in the form of a regression
model) between the item being forecast
and factors that cause it to behave the
way it does.

Forecasting Methods
• Qualitative methods - use management
judgment, expertise, and opinion to
make forecasts.
• Often called "the jury of executive
opinion,"
• They are the most common type of
forecasting method for the long-term
strategic planning process.

Time Series Analysis
Forecasting Methods

Time Series Methods
• Time series methods tend to be
most useful for short-range
forecasting, (although they can
be used for longer-range
forecasting) and relate to only
one factor time.

Time Series Methods
• Two types of time series methods:
1.The Moving Average
a)Simple Moving Average
b)Weighted Moving Average
2.Exponential Smoothing.

Time Series – Moving Average
Moving Averages
• The moving average method uses
several values during the recent past to
develop a forecast.
• The moving average method is good for
stable demand with no pronounced
behavioral patterns.

Moving Averages
• Moving averages are computed for
specific periods, such as 3 months or 5
months, depending on how much the
forecaster desires to smooth the data.

Simple Moving Averages
• Moving average forecast may be computed for
specified time period as follows:
where
n = number of periods in the moving average
D = data in period i
n
D
MA
n
i
i
t
i


 1
,

Simple Moving Averages - Delivery Orders for 10-month period
Month Orders Delivered per Month
January 120
February 90
March 100
April 75
May 110
June 50
July 75
August 130
September 110
October 90

Simple Moving Averages Example
• The moving average from the demand
for orders for the last 3 months in the
sequence:

Simple Moving Averages Example
• The 5-month moving average is
computed from the last 5 months of
demand data, as follows:

Simple 3- and 5- month Moving Average

Simple 3- and 5- month Moving Average
Longer-period moving averages react more slowly to recent
demand changes than do shorter-period moving averages.

Weighted Moving Average
• The major disadvantage of the Simple
Moving Average method is that it does
not react well to variations that occur
for a reason, such as trends and seasonal
effects (although this method does
reflect trends to a moderate extent).

• The Simple Moving Average method can
be adjusted to reflect more closely more
recent fluctuations in the data and
seasonal effects.
• This adjusted method is referred to as a
Weighted Moving Average method.

• Weighted Moving Average - is a time
series forecasting method in which the
most recent data are weighted.
• It may be computed for specified time
period using the following:

• Weighted Moving Average -
Where:
Wi = the weight for period i, is
between 0% and 100%
∑Wi =1.00
Di = data in period i

For example, if the Instant Paper Clip Supply
Company wants to compute a 3-month
weighted moving average with a weight of
50% for the October data, a weight of 33% for
the September data, and a weight of 17% for
August, it is computed as.

Weighted Moving Average - Table

Time Series – Exponential Smoothing
• The Exponential Smoothing forecast
method is an averaging method that
weights the most recent past data more
strongly than more distant past data.
• There are two forms of exponential
smoothing:
1. Simple Exponential Smoothing
2. Adjusted Exponential Smoothing
(adjusted for trends, seasonal patterns, etc.)

Simple Exponential Smoothing
• The simple exponential smoothing
forecast is computed by using the
formula:
t
t
t F
D
F )
1
(
1 
 




where
Ft+1 = the forecast for the next period
Dt = the actual demand for the present period
Ft = the previously determined forecast for the
present periods
α = a weighting factor referred to as the
smoothing constant
t
t
t F
D
F )
1
(
1 
 




• The smoothing constant, α, is betw. 0 & 1.
• It reflects the weight given to the most
recent demand data.
»For example, if α = .20,
»Ft+1 = .20Dt + .80Ft
• This means that our forecast for the next
period is based on 20% of recent demand (Dt)
and 80% of past demand.

• The higher α is (the closer α is to one),
the more sensitive the forecast will be
to changes in recent demand.
• Alternatively, the closer α is to zero, the
greater will be the dampening or
smoothing effect.

• The most commonly used values of α
are in the range from .01 to .50.
• However, the determination of α is
usually judgmental and subjective and
will often be based on trial-and-error
experimentation.

Simple Exponential Smoothing Example
Period Month Demand
1 January 37
2 February 40
3 March 41
4 April 37
5 May 45
6 June 50
7 July 43
8 August 47
9 September 56
10 October 52
11 November 55
12 December 54
•A company - PM
Computer Services
has accumulated
demand data in table
for its computers for
the past 12 months.
•It wants to compute
exponential
smoothing forecasts,
using smoothing
constants (α) equal to
0.30 and 0.50.

• To develop the series of forecasts for the data
i, start with period 1 (January) and compute
the forecast for period 2 (February) by using α
= 0.30.
• The formula for exponential smoothing also
requires a forecast for period 1, which we do
not have, so we will use the demand for
period 1 as both demand and the forecast for
period 1.

• Thus the forecast for February is:
–F2 = αD1 + (1 - α)F1
–= (.30)(37) + (.70)(37) = 37 units

• The forecast for period 3 is computed
similarly: F3 = α D2 + (1 - α)F2
= (.30)(40) + (.70)(37) = 37.9 units
• The final forecast is for period 13, January,
and is the forecast of interest to PM
Computer Services: F13 = α D12 + (1 - α)F12
= (.30)(54) + (.70)(50.84) = 51.79 units

Period Month Demand
Forecast, Ft + 1
a = 0.30 a = 0.50
1 January 37
2 February 40 37.00 37.00
3 March 41 37.90 38.50
4 April 37 38.83 39.75
5 May 45 38.28 38.37
6 June 50 40.29 41.68
7 July 43 43.20 45.84
8 August 47 43.14 44.42
9 September 56 44.30 45.71
10 October 52 47.81 50.85
11 November 55 49.06 51.42
12 December 54 50.84 53.21
13 January 51.79 53.61

• In general, when demand is relatively stable, without any trend, using a
small value for α is more appropriate to simply smooth out the
forecast.
• Alternatively, when actual demand displays an increasing (or
decreasing) trend, as is the case, a larger value of α is generally better.

Adjusted Exponential Smoothing
• The adjusted exponential smoothing
forecast consists of the exponential
smoothing forecast with a trend
adjustment factor added to it.
• The formula for the adjusted forecast is:
AFt+1 = Ft+1 + Tt+1
where
T = an exponentially smoothed trend factor

• The trend factor is computed much the same
as the exponentially smoothed forecast.
• It is, in effect, a forecast model for trend:
Tt+1 = β(Ft+1 - Ft) + (1 - β)Tt
where
Tt = the last period trend factor
β = a smoothing constant for trend

• Like α, β is a value between 0 and 1.
• It reflects the weight given to the most
recent trend data.
• Also like α, β is often determined
subjectively, based on the judgment of
the forecaster.

• A high β reflects trend changes more
than a low β.
• It is not uncommon for β to equal α in
this method.
• The closer β is to one, the stronger a
trend is reflected.

Adjusted Exponential Smoothing Example
• PM Computer Services now wants to develop
an adjusted exponentially smoothed forecast,
using the same 12 months of demand.
• The adjusted forecast for February, AF2, is the
same as the exponentially smoothed forecast
because the trend computing factor will be
zero (i.e., F1 and F2 are the same and T2 = 0).

Adjusted Exponential Smoothing Example
• Thus, we will compute the adjusted forecast
for March, AF3, as follows, starting with the
determination of the trend factor, T3:
–T3 = β (F3 - F2) + (1 β)T2 = (.30)(38.5 - 37.0) +
(.70)(0) = 0.45, and
–AF3 = F3 + T3 = 38.5 + 0.45 = 38.95

• Period 13 is computed as follows:
–T13 = β(F13 - F12) + (1 β)T12
–= (.30)(53.61 - 53.21) + (.70)(1.77) =
1.36
and
• AF13 = F13 + T13 = 53.61 + 1.36 = 54.96
units

Period Month Demand
Forecast
(Ft +1)
Trend
(Tt +1)
Adjusted
Forecast
(AFt +1)
1 January 37 37.00
2 February 40 37.00 0.00 37.00
3 March 41 38.50 0.45 38.95
4 April 37 39.75 0.69 40.44
5 May 45 38.37 0.07 38.44
6 June 50 41.68 1.04 42.73
7 July 43 45.84 1.97 47.82
8 August 47 44.42 0.95 45.37
9 September 56 45.71 1.05 46.76
10 October 52 50.85 2.28 53.13
11 November 55 51.42 1.76 53.19
12 December 54 53.21 1.77 54.98
13 January 53.61 1.36 54.96

Time Series – Linear Trend Line
Linear Trend Line
• Linear regression is most often thought
of as a causal method of forecasting in
which a mathematical relationship is
developed between demand and some
other factor that causes demand
behavior.

Linear Trend Line
• However, when demand displays an
obvious trend over time, a least squares
regression line, or linear trend line, can
be used to forecast demand.
• A linear trend line is a linear regression
model that relates demand to time.

Linear Trend Line
• The linear regression takes form of a
linear equation as follows:
where
a = intercept
b = slope of the line
x = the time period
y = forecast for demand for period x
bx
a
y 


Linear Trend Line
• The parameters of the trend line may be
calculated as follows:
and
where
and




 2
2
x
n
x
y
x
n
xy
b
n
x
x


x
b
y
a 

n
y
y



Linear Trend Line Example
x (period) y (demand) xy x2
1 37 37 1
2 40 80 4
3 41 123 9
4 37 148 16
5 45 225 25
6 50 300 36
7 43 301 49
8 47 376 64
9 56 504 81
10 52 520 100
11 55 605 121
12 54 648 144
78 557 3,867 650

• Using these values for ẋ and ӯ the values, the
parameters for the linear trend line are
computed as follows:

Therefore, the linear trend line is
y = 35.2 + 1.72x
•To calculate a forecast for period 13, x = 13
would be substituted in the linear trend line:
y = 35.2 + 1.72(13) = 57.56
• A linear trend line will not adjust to a change
in trend as will exponential smoothing.

Time Series – Seasonal Adjustments
Seasonal Adjustments
• Many demand items exhibit seasonal behavior
or pattern, that is, a repetitive up-and-down
movement in demand.
• It is possible to adjust the seasonality of a
normal forecast by multiplying it by a seasonal
factor.
• A seasonal factor, which is a numerical value
is multiplied by the normal forecast to get a
seasonally adjusted forecast.

Seasonal Adjustments
• One method for developing a demand for seasonal factors
is dividing the actual demand for each seasonal period by
the total annual demand, according to the following
formula:
• The resulting seasonal factors are between 0 and 1
• These seasonal factors are thus multiplied by the annual
forecasted demand to yield seasonally adjusted forecasts
for each period.

Seasonal Adjustments Example
Demand (1,000s)
Year QUARTER 1 QUARTER 2 QUARTER 3 QUARTER 4 TOTAL
2003 12.6 8.6 6.3 17.5 45.0
2004 14.1 10.3 7.5 18.2 50.1
2005 15.3 10.6 8.1 19.6 53.6
Total 42.0 29.5 21.9 55.3 148.7
Next, multiply the forecasted demand
for the next year, 2006, by each of the
seasonal factors to get the forecasted
demand for each quarter.

Demand (1,000s)
2003 12.6 8.6 6.3 17.5 45.0
2004 14.1 10.3 7.5 18.2 50.1
2005 15.3 10.6 8.1 19.6 53.6
Total 42.0 29.5 21.9 55.3 148.7
• However, to accomplish this, we need a demand forecast for 2006.
• In this case, because the demand data in the table seem to exhibit a
generally increasing trend, we compute a linear trend line for the 3
years of data in the table to use as a rough forecast estimate:
y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17 or 58,170 turkeys.

Demand (1,000s)
2003 12.6 8.6 6.3 17.5 45.0
2004 14.1 10.3 7.5 18.2 50.1
2005 15.3 10.6 8.1 19.6 53.6
Total 42.0 29.5 21.9 55.3 148.7
Using this annual
forecast of demand, the
seasonally adjusted
forecasts, SFi, for 2006
are as follows:

Forecast Accuracy
• It is not probable that a forecast will be completely
accurate.
• Forecasts will always deviate from the actual demand
resulting in a Forecast error
• A Forecast Error is the difference between the forecast
and actual demand.
• There are different measures of forecast error:
– Mean Absolute Deviation (MAD),
– Mean Absolute Percent Deviation (MAPD),
– Cumulative Error (E),
– Average Error or Bias (Ē),
– Mean Squared Error (MSE).

Forecast Accuracy
Mean Absolute Deviation (MAD) – average absolute
difference between the forecast and actual values.
where:
Mean Absolute Percent Deviation (MAPD) – absolute
error between forecast and actual values.

Forecast Accuracy
Cumulative error – sum of the forecast error.
Average error – is the per-period average of cumulative
error.
Mean Squared Error (MSE)

 t
e
E

Forecast accuracy – Worked Example

Forecast Accuracy
Mean Absolute Deviation
• MAD is the average, absolute difference between the
forecast and the demand and is computed by the
following formula:

Forecast Accuracy
Mean Absolute Deviation Example
Period Demand, Dt
Forecast,
Ft (a = .30)
Error
(Dt-Ft) |Dt-Ft|
Error2
(Dt-Ft)2
1 37 37.00
2 40 37.00 3.00 3.00 9.00
3 41 37.90 3.10 3.10 9.61
4 37 38.83 1.83 1.83 3.35
5 45 38.28 6.72 6.72 45.15
6 50 40.29 9.71 9.71 94.28
7 43 43.20 0.20 0.20 0.04
8 47 43.14 3.86 3.86 14.90
9 56 44.30 11.70 11.70 136.89
10 52 47.81 4.19 4.19 17.56
11 55 49.06 5.94 5.94 35.28
12 54 50.84 3.16 3.16 9.98
520[*]
49.31 53.41 376.04
PM Computer Services,
forecasts were developed
using exponential smoothing
(with a = 0.30 and with a =
0.50), adjusted exponential
smoothing (a = 0.50, b =
0.30), and a linear trend line
for the demand data. The
company wants to compare
the accuracy of these
different forecasts by using
MAD.

Regression Methods
Forecasting

Regression Methods
• In contrast to times series techniques, regression is a
forecasting technique that measures the relationship of
one variable to one or more other variables.
• The simplest form of regression is linear regression.
• Simple Linear Regression relates one dependent variable
to one independent variable in the form of a linear
equation:

Regression Methods
Simple Linear Regression
• To develop the linear equation, the slope, b, and the
intercept, a, must first be computed by using the following
least squares formulas:
• Where

Regression Methods
Correlation
• Correlation in a linear regression equation is a measure of
the strength of the relationship between the independent
and dependent variables. The formula for the correlation
coefficient is:
• The value of r varies between -1.00 and +1.00, with a
value of ±1.00 indicating a strong linear relationship
between the variables.

Regression Methods
Correlation Example
We can determine the
correlation coefficient for the
linear regression equation
determined in our State
University example by
substituting most of the terms
calculated for the least squares
formula (except for Sy2
) into the
formula for r:

Regression Methods
Coefficient of Determination
• Another measure of the strength of the relationship
between the variables in a linear regression equation is the
coefficient of determination.
• The coefficient of determination is the percentage of the
variation in the dependent variable that results from the
independent variable.
• It is computed by simply squaring the value of r.
• For our example, r = .948; thus, the coefficient of
determination is:

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