IBM401 - Lecture 6

Forecasting
Techniques

Qualitative Time-Series Methods Causal
Models Methods

Delphi Moving Regression Analysis
Methods Average

Jury of Executive Exponential Multiple
Opinion Smoothing Regression

Sales Force Trend
Composite Projections

Consumer
Decomposition
Market Survey

 Time-series models attempt to predict the
future based on the past
 Common time-series models are
 Moving average
 Exponential smoothing
 Trend projections
 Decomposition
 Regressionanalysis is used in trend
projections and one type of decomposition
model

 Causal models use variables or factors that
might influence the quantity being
forecasted
 The objective is to build a model with the
best statistical relationship between the
variable being forecast and the
independent variables
 Regression analysis is the most common
technique used in causal modeling

 Wacker Distributors wants to forecast sales
for three different products
YEAR TELEVISION SETS RADIOS COMPACT DISC PLAYERS
1 250 300 110
2 250 310 100
3 250 320 120
4 250 330 140
5 250 340 170
6 250 350 150
7 250 360 160
8 250 370 190
9 250 380 200
10 250 390 190

(a)
330 –  Sales appear to be
constant over time
Annual Sales of Televisions

250 –          
Sales = 250
200 –
 A good estimate of
150 – sales in year 11 is
100 – 250 televisions
50 –
| | | | | | | | | |

0 1 2 3 4 5 6 7 8 9 10
Time (Years)

(b)
420 –  Sales appear to be
400 – increasing at a
380 –  constant rate of 10
Annual Sales of Radios



360 –
 radios per year
340 – 

 Sales = 290 + 10(Year)
320 – 
300 –    A reasonable
280 –
estimate of sales in
year 11 is 400
| | | | |
0 1 2 3 4 5 6 7 8 9 10
| | | | |
televisions
Time (Years)

 This trend line may
(c) not be perfectly
200 –
  accurate because of
Annual Sales of CD Players


180 – variation from year
160 –  to year

140 –   Sales appear to be

increasing
120 –

  A forecast would
100 –  probably be a larger
| | | | | | | | | |
figure each year
0 1 2 3 4 5 6 7 8 9 10
Time (Years)

 We compare forecasted values with actual values to
see how well one model works or to compare models

Forecast error = Actual value – Forecast value

 One measure of accuracy is the
 mean absolute deviation (MAD)

MAD 
 forecast error
n

 Using a naïve forecasting model
ACTUAL ABSOLUTE VALUE OF
SALES OF CD FORECAST ERRORS (DEVIATION),
YEAR PLAYERS SALES (ACTUAL – FORECAST)
1 110 — —
2 100 110 |100 – 110| = 10
3 120 100 |120 – 110| = 20
4 140 120 |140 – 120| = 20
5 170 140 |170 – 140| = 30
6 150 170 |150 – 170| = 20
7 160 150 |160 – 150| = 10
8 190 160 |190 – 160| = 30
9 200 190 |200 – 190| = 10
10 190 200 |190 – 200| = 10
11 — 190 —
Sum of |errors| = 160
MAD = 160/9 = 17.8

 Using a naïve forecasting model
ACTUAL ABSOLUTE VALUE OF
SALES OF CD FORECAST ERRORS (DEVIATION),
YEAR PLAYERS SALES (ACTUAL – FORECAST)
1 110 — —
2 100 110 |100 – 110| = 10
3 120 100 |120 – 110| = 20
4
MAD 
5
 forecast error  160  17.8
140
170
120
140
|140 – 120| = 20
|170 – 140| = 30
6 150 n 170 9 |150 – 170| = 20
7 160 150 |160 – 150| = 10
8 190 160 |190 – 160| = 30
9 200 190 |200 – 190| = 10
10 190 200 |190 – 200| = 10
11 — 190 —
Sum of |errors| = 160
MAD = 160/9 = 17.8

 There are other popular measures of forecast
accuracy
 The mean squared error

MSE 
 (error)2
n
 The mean absolute percent error
error
 actual
MAPE  100%
n
 And bias is the average error

A time series is a sequence of evenly
spaced events
 Time-series forecasts predict the future
based solely of the past values of the
variable
 Other variables are ignored

 A time series typically has four components
1. Trend (T) is the gradual upward or downward
movement of the data over time
2. Seasonality (S) is a pattern of demand
fluctuations above or below trend line that
repeats at regular intervals
3. Cycles (C) are patterns in annual data that
occur every several years
4. Random variations (R) are “blips” in the
data caused by chance and unusual
situations

Trend
Demand for Product or Service

Component

Seasonal Peaks

Actual
Demand
Line
Average Demand
over 4 Years

| | | |
Year Year Year Year
1 2 3 4
Time

 There are two general forms of time-series
models
 The multiplicative model

Demand = T x S x C x R

 The additive model
Demand = T + S + C + R

 Models may be combinations of these two
forms
 Forecasters often assume errors are normally
distributed with a mean of zero

 Moving averages can be used when demand is
relatively steady over time
 The next forecast is the average of the most
recent n data values from the time series
 This methods tends to smooth out short-term
irregularities in the data series

Sum of demands in previous n periods
Moving average forecast 
n

 Mathematically
Yt  Yt 1  ...  Yt  n1
Ft 1 
n

where
Ft 1 = forecast for time period t + 1
Yt
= actual value in time period t
n = number of periods to average

 Wallace Garden Supply wants to forecast
demand for its Storage Shed
 They have collected data for the past year
 They are using a three-month moving
average to forecast demand (n = 3)

MONTH ACTUAL SHED SALES THREE-MONTH MOVING AVERAGE
January 10
February 12
March 13
April 16 (10 + 12 + 13)/3 = 11.67
May 19 (12 + 13 + 16)/3 = 13.67
June 23 (13 + 16 + 19)/3 = 16.00
July 26 (16 + 19 + 23)/3 = 19.33
August 30 (19 + 23 + 26)/3 = 22.67
September 28 (23 + 26 + 30)/3 = 26.33
October 18 (26 + 30 + 28)/3 = 28.00
November 16 (30 + 28 + 18)/3 = 25.33
December 14 (28 + 18 + 16)/3 = 20.67
January — (18 + 16 + 14)/3 = 16.00

 Weighted moving averages use weights to put more
emphasis on recent periods
 Often used when a trend or other pattern is
emerging

Ft 1 
 ( Weight in period i )( Actual value in period)
 ( Weights)
 Mathematically
w1Yt  w2Yt 1  ...  wnYt  n1
Ft 1 
w1  w2  ...  wn
where
wi = weight for the ith observation

 Wallace Garden Supply decides to try a
weighted moving average model to
forecast demand for its Storage Shed
 They decide on the following weighting
scheme
WEIGHTS APPLIED PERIOD
3 Last month
2 Two months ago
1 Three months ago
3 x Sales last month + 2 x Sales two months ago + 1 X Sales three months ago
6
Sum of the weights

THREE-MONTH WEIGHTED
MONTH ACTUAL SHED SALES MOVING AVERAGE
January 10
February 12
March 13
April 16 [(3 X 13) + (2 X 12) + (10)]/6 = 12.17
May 19 [(3 X 16) + (2 X 13) + (12)]/6 = 14.33
June 23 [(3 X 19) + (2 X 16) + (13)]/6 = 17.00
July 26 [(3 X 23) + (2 X 19) + (16)]/6 = 20.50
August 30 [(3 X 26) + (2 X 23) + (19)]/6 = 23.83
September 28 [(3 X 30) + (2 X 26) + (23)]/6 = 27.50
October 18 [(3 X 28) + (2 X 30) + (26)]/6 = 28.33
November 16 [(3 X 18) + (2 X 28) + (30)]/6 = 23.33
December 14 [(3 X 16) + (2 X 18) + (28)]/6 = 18.67
January — [(3 X 14) + (2 X 16) + (18)]/6 = 15.33

 Exponential smoothing is easy to use and
requires little record keeping of data
 It is a type of moving average

New forecast = Last period’s forecast
+ (Last period’s actual demand - Last period’s
forecast)

Where  is a weight (or smoothing
constant) with a value between 0 and 1
inclusive

 Mathematically

Ft 1  Ft   (Yt  Ft )

where
Ft+1 = new forecast (for time period t + 1)
Ft = previous forecast (for time period t)
 = smoothing constant (0 ≤  ≤ 1)
Yt = pervious period’s actual demand

 The idea is simple – the new estimate is
the old estimate plus some fraction of
the error in the last period

 In January, February’s demand for a certain car
model was predicted to be 142
 Actual February demand was 153 autos
 Using a smoothing constant of  = 0.20, what is
the forecast for March?

New forecast (for March demand) = 142 + 0.2(153 – 142)
= 144.2 or 144 autos

 If actual demand in March was 136 autos,
the April forecast would be
New forecast (for April demand) = 144.2 + 0.2(136 – 144.2)
= 142.6 or 143 autos

 Selecting the appropriate value for  is key
to obtaining a good forecast
 The objective is always to generate an
accurate forecast
 The general approach is to develop trial
forecasts with different values of  and
select the  that results in the lowest MAD

 Exponential smoothing forecast for two values of 
ACTUAL
TONNAGE FORECAST FORECAST
QUARTER UNLOADED USING  =0.10 USING  =0.50
1 180 175 175
2 168 175.5 = 175.00 + 0.10(180 – 175) 177.5
3 159 174.75 = 175.50 + 0.10(168 – 175.50) 172.75
4 175 173.18 = 174.75 + 0.10(159 – 174.75) 165.88
5 190 173.36 = 173.18 + 0.10(175 – 173.18) 170.44
6 205 175.02 = 173.36 + 0.10(190 – 173.36) 180.22
7 180 178.02 = 175.02 + 0.10(205 – 175.02) 192.61
8 182 178.22 = 178.02 + 0.10(180 – 178.02) 186.30
9 ? 178.60 = 178.22 + 0.10(182 – 178.22) 184.15

ACTUAL ABSOLUTE ABSOLUTE
TONNAGE FORECAST DEVIATIONS FORECAST DEVIATIONS
QUARTER UNLOADED WITH  = 0.10 FOR  = 0.10 WITH  = 0.50 FOR  = 0.50
1 180 175 5….. 175 5….
2 168 175.5 7.5.. 177.5 9.5..
3 159 174.75 15.75 172.75 13.75
4 175 173.18 1.82 165.88 9.12
5 190 173.36 16.64 170.44 19.56
6 205 175.02 29.98 180.22 24.78
7 180 178.02 1.98 192.61 12.61
8 182 178.22 3.78 186.30 4.3..
Sum of absolute deviations 82.45 98.63
Σ|deviations|
MAD = = 10.31 MAD = 12.33
n

Best choice

 Like all averaging techniques, exponential
smoothing does not respond to trends
 A more complex model can be used that adjusts
for trends
 The basic approach is to develop an exponential
smoothing forecast then adjust it for the trend

Forecast including trend (FITt) = New forecast (Ft)
+ Trend correction (Tt)

 The equation for the trend correction uses a
new smoothing constant 
 Tt is computed by

Tt 1  (1  )T1   ( Ft 1  Ft )

where
Tt+1 = smoothed trend for period t + 1
Tt = smoothed trend for preceding period
= trend smooth constant that we select
Ft+1 = simple exponential smoothed forecast for
period t + 1
Ft = forecast for pervious period

 As with exponential smoothing, a high value of 
makes the forecast more responsive to changes in
trend
 A low value of  gives less weight to the recent
trend and tends to smooth out the trend
 Values are generally selected using a trial-and-error
approach based on the value of the MAD for
different values of 
 Simple exponential smoothing is often referred to as
first-order smoothing
 Trend-adjusted smoothing is called second-order,
double smoothing, or Holt’s method

Trend Projection
 Trend projection fits a trend line to a series
of historical data points
 The line is projected into the future for
medium- to long-range forecasts
 Several trend equations can be developed
based on exponential or quadratic models
 The simplest is a linear model developed
using regression analysis

Trend Projection
 The mathematical form is

ˆ
Y  b0  b1 X

where
ˆ
Y = predicted value
b0 = intercept
b1 = slope of the line
X = time period (i.e., X = 1, 2, 3, …, n)

Trend Projection

Dist7 *
*
Value of Dependent Variable

Dist5 Dist6

* Dist3 *
Dist4

Dist1 * Dist2
*
*

Time

 Midwestern Manufacturing Company has experienced
the following demand for it’s electrical generators
over the period of 2001 – 2007

YEAR ELECTRICAL GENERATORS SOLD
2001 74
2002 79
2003 80
2004 90
2005 105
2006 142
2007 122

r2 says model predicts
about 80% of the
variability in demand

Significance level for
F-test indicates a
definite relationship

 The forecast equation is
ˆ
Y  56.71 10.54 X

 To project demand for 2008, we use the coding
system to define X = 8
(sales in 2008) = 56.71 + 10.54(8)
= 141.03, or 141 generators

 Likewise for X = 9

(sales in 2009) = 56.71 + 10.54(9)
= 151.57, or 152 generators

160 –
150 – 
140 – 
Trend Line
130 – ˆ
Y  56.71 10.54 X
Generator Demand

120 – 
110 –

100 –
90 – 
80 –  
70 –  Actual Demand Line
60 –
50 –
| | | | | | | | |

2001 2002 2003 2004 2005 2006 2007 2008 2009
Year

 Recurring variations over time may
indicate the need for seasonal adjustments
in the trend line
 A seasonal index indicates how a particular
season compares with an average season
 When no trend is present, the seasonal
index can be found by dividing the average
value for a particular season by the
average of all the data

 Eichler Supplies sells telephone answering
machines
 Data has been collected for the past two
years sales of one particular model
 They want to create a forecast this
includes seasonality

AVERAGE
SALES DEMAND
AVERAGE TWO- MONTHLY SEASONAL
MONTH YEAR 1 YEAR 2 YEAR DEMAND DEMAND INDEX
January 80 100 90 94 0.957
February 85 75 80 94 0.851
March 80 90 85 94 0.904
April 110 90 100 94 1.064
May 115 131 123 94 1.309
June 120 110 115 94 1.223
July 100 110 105 94 1.117
August 110 90 100 94 1.064
September 85 95 90 94 0.957
October 75 85 80 94 0.851
November 85 75 80 94 0.851
December 80 80 80 94 0.851
Total average demand = 1,128
1,128 Average two-year demand
Average monthly demand = = 94 Seasonal index =
12 months Average monthly demand

 The calculations for the seasonal indices are
1,200 1,200
Jan.  0.957  96 July  1.117  112
12 12
1,200 1,200
Feb.  0.851  85 Aug.  1.064  106
12 12
1,200 1,200
Mar.  0.904  90 Sept.  0.957  96
12 12
1,200 1,200
Apr.  1.064  106 Oct.  0.851  85
12 12
1,200 1,200
May  1.309  131 Nov.  0.851  85
12 12
1,200 1,200
June  1.223  122 Dec.  0.851  85
12 12

Seasonal Variations with Trend
 When both trend and seasonal components are
present, the forecasting task is more complex
 Seasonal indices should be computed using a
centered moving average (CMA) approach
 There are four steps in computing CMAs
1. Compute the CMA for each observation (where
possible)
2. Compute the seasonal ratio = Observation/CMA
for that observation
3. Average seasonal ratios to get seasonal indices
4. If seasonal indices do not add to the number of
seasons, multiply each index by (Number of
seasons)/(Sum of indices)

Turner Industries Example
 The following are Turner Industries’ sales figures for
the past three years

QUARTER YEAR 1 YEAR 2 YEAR 3 AVERAGE
1 108 116 123 115.67
2 125 134 142 133.67
3 150 159 168 159.00
4 141 152 165 152.67
Average 131.00 140.25 149.50 140.25

Seasonal
Definite trend pattern

 To calculate the CMA for quarter 3 of year 1 we
compare the actual sales with an average quarter
centered on that time period
 We will use 1.5 quarters before quarter 3 and 1.5
quarters after quarter 3 – that is we take quarters 2,
3, and 4 and one half of quarters 1, year 1 and
quarter 1, year 2

0.5(108) + 125 + 150 + 141 + 0.5(116)
CMA(q3, y1) = = 132.00
4


 We compare the actual sales in quarter 3 to
the CMA to find the seasonal ratio

Sales in quarter 3 150
Seasonal ratio    1.136
CMA 132

YEAR QUARTER SALES CMA SEASONAL RATIO
1 1 108
2 125
3 150 132.000 1.136
4 141 134.125 1.051
2 1 116 136.375 0.851
2 134 138.875 0.965
3 159 141.125 1.127
4 152 143.000 1.063
3 1 123 145.125 0.848
2 142 147.875 0.960
3 168
4 165

 There are two seasonal ratios for each quarter so
these are averaged to get the seasonal index

Index for quarter 1 = I1 = (0.851 + 0.848)/2 = 0.85


 Scatter plot of Turner Industries data and
CMAs
200 – CMA

150 –    
  
  
100 –  
Sales

50 –
Original Sales Figures
0– | | | | | | | | | | | |

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

 Decomposition is the process of isolating linear
trend and seasonal factors to develop more accurate
forecasts
 There are five steps to decomposition
1. Compute seasonal indices using CMAs
2. Deseasonalize the data by dividing each number
by its seasonal index
3. Find the equation of a trend line using the
deseasonalized data
4. Forecast for future periods using the trend line
5. Multiply the trend line forecast by the
appropriate seasonal index

SALES SEASONAL DESEASONALIZED
($1,000,000s) INDEX SALES ($1,000,000s)
108 0.85 127.059
125 0.96 130.208
150 1.13 132.743
141 1.06 133.019
116 0.85 136.471
134 0.96 139.583
159 1.13 140.708
152 1.06 143.396
123 0.85 144.706
142 0.96 147.917
168 1.13 148.673
165 1.06 155.660

 Find a trend line using the deseasonalized data

b1 = 2.34 b0 = 124.78
 Develop a forecast using this trend a multiply the
forecast by the appropriate seasonal index
ˆ
Y = 124.78 + 2.34X
= 124.78 + 2.34(13)
= 155.2 (forecast before adjustment for
seasonality)
ˆ
Y x I1 = 155.2 x 0.85 = 131.92

 A San Diego hospital used 66 months of adult
inpatient days to develop the following seasonal
indices

MONTH SEASONALITY INDEX MONTH SEASONALITY INDEX
January 1.0436 July 1.0302
February 0.9669 August 1.0405
March 1.0203 September 0.9653
April 1.0087 October 1.0048
May 0.9935 November 0.9598
June 0.9906 December 0.9805

 Using this data they developed the following
equation
ˆ
Y = 8,091 + 21.5X
where
ˆ
Y = forecast patient days
X = time in months

 Based on this model, the forecast for patient days
for the next period (67) is

Patient days = 8,091 + (21.5)(67) = 9,532 (trend only)
Patient days = (9,532)(1.0436)
= 9,948 (trend and seasonal)

 Multiple regression can be used to forecast both
trend and seasonal components in a time series
 One independent variable is time
 Dummy independent variables are used to represent the
seasons
 The model is an additive decomposition model
ˆ
Y  a  b1 X 1  b2 X 2  b3 X 3  b4 X 4
where
X1 = time period
X2 = 1 if quarter 2, 0 otherwise

 The resulting regression equation is

ˆ
Y  104.1  2.3 X 1  15.7 X 2  38.7 X 3  30.1X 4

 Using the model to forecast sales for the first two
quarters of next year
ˆ
Y  104.1  2.3(13)  15.7(0)  38.7(0)  30.1(0)  134
ˆ
Y  104.1  2.3(14 )  15.7(1)  38.7(0)  30.1(0)  152

 These are different from the results obtained using
the multiplicative decomposition method
 Use MAD and MSE to determine the best model

 Tracking signals can be used to monitor the
performance of a forecast
 Tacking signals are computed using the following
equation

RSFE
Tracking signal 
MAD

where

MAD 
 forecast error
n
RSFE = Ratio of running sum of forecast errors
= ∑(actual demand in period i - forecast demand in period i)

Signal Tripped
Upper Control Limit Tracking Signal
+

Acceptable
0 MADs Range

–
Lower Control Limit

Time

 Positive tracking signals indicate demand is greater
than forecast
 Negative tracking signals indicate demand is less
than forecast
 Some variation is expected, but a good forecast will
have about as much positive error as negative error
 Problems are indicated when the signal trips either
the upper or lower predetermined limits
 This indicates there has been an unacceptable
amount of variation
 Limits should be reasonable and may vary from item
to item

 Tracking signal for quarterly sales of croissants

TIME FORECAST ACTUAL |FORECAST | CUMULATIVE TRACKING
PERIOD DEMAND DEMAND ERROR RSFE | ERROR | ERROR MAD SIGNAL
1 100 90 –10 –10 10 10 10.0 –1
2 100 95 –5 –15 5 15 7.5 –2
3 100 115 +15 0 15 30 10.0 0
4 110 100 –10 –10 10 40 10.0 –1
5 110 125 +15 +5 15 55 11.0 +0.5
6 110 140 +30 +35 35 85 14.2 +2.5

MAD 
 forecast error  85  14.2
n 6
RSFE 35
Tracking signal    2.5MAD s
MAD 14.2

 Adaptive smoothing is the computer monitoring
of tracking signals and self-adjustment if a limit
is tripped
 In exponential smoothing, the values of  and 
are adjusted when the computer detects an
excessive amount of variation

 Spreadsheets can be used by small and medium-
sized forecasting problems
 More advanced programs (SAS, SPSS, Minitab)
handle time-series and causal models
 May automatically select best model parameters
 Dedicated forecasting packages may be fully
automatic
 May be integrated with inventory planning and
control

 39 students
 Average:
 3.43
 High:
 14.50 100% class grade
 Grade
 A >= 85%
 B >= 70%, < 85%
 C >= 60%, < 70%
 D >= 50%, < 60%
 F < 50%

IBM401 - Lecture 6

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