4. 4
Applications
We focus on the forecasting of demand
Forecasts are also used to predict
Profits/Revenues
Costs
Prices
Interest rates
Movements of key economic indicators
And more . . .
Concepts and techniques are the same
5. 5
Common Features
Assumes same causal system: past ==> future
Be alert to unplanned occurrences
Forecasts rarely perfect because of randomness
Forecasts more accurate for
groups vs. individuals
Forecast accuracy decreases
as time horizon increases
I see that you will
get an A this semester.
7. 7
Forecasting Approaches
Time series forecasts
& Associative models
Involves mathematical
techniques
e.g., forecasting sales of
color televisions
Used when situation is
‘stable’ & historical data
exist
Existing products
Current technology
Quantitative Methods
Judgmental forecasts
Involves intuition,
experience
e.g., forecasting sales on
Internet
Used when situation is
vague & little data exist
New products
New technology
Qualitative Methods
8. 8
Judgmental Forecasts
Executive opinions
A small group of upper-level managers
(marketing, operations and finance) may meet
and collectively develop a forecast.
Sales force opinions
The sales staff or the customer service staff is
often a good source of information. Sales staff
may under-report to reduce quota.
9. 9
Judgmental Forecasts (Con’t)
Consumer surveys
Consumers ultimately determine demand.
Expensive.
Outside opinion
Industrial reports, press.
Delphi method
• Opinions of managers and staff
• Achieves a consensus forecast
10. 10
Quantitative Forecasting Methods
Naïve Averaging Trend
Quantitative
Forecasting
Time Series
Forecasting
Associative
Models
Seasonality
Moving Average
Weighted Moving
Average
Exponential
smoothing
Linear trend
– linear trend equation
– trend-adjusted
exponential
smoothing (no req.)
Nonlinear trend
(not req.)
Additive
Multiplicative
Linear regression
Curvilinear
regression (not req.)
Multiple regression
(not req.)
12. 12
Time Series Forecasts
Time series : A time series is a time-ordered
sequence of observations taken at regular
intervals over a period of time (hourly, daily,
weekly, monthly, quarterly, annually).
Data : measurements of demand (sales,
earnings, profits, ……)
Example
Year: 1998 1999 2000 2001 2002
Sales: 78.7 63.5 89.7 93.2 92.1
Assumption : future will be like the past
14. 14
Naïve Forecast
The forecast for any period
=
The previous period’s actual value.
Uh, give me a minute....
We sold 250 wheels last
week.... Now, next week we
should sell....
15. 15
Naïve Forecasts
Simple to use
Virtually no cost
Quick and easy to prepare
Data analysis is nonexistent
Easily understandable
Cannot provide high accuracy
Can be a standard for accuracy
16. 16
Uses for Naïve Forecasts
Stable time series data
• Ft = At-1
Seasonal variations
• Forecast for this season = Actual value from last season
Data with trends
• Ft = At-1 + (At-1 – At-2)
Ft = Forecast for period t.
At-1 = Actual demand or sales for period t-1.
where
17. 17
Example 1: Naïve Forecasts
Forecast for period i is the actual value for
period i-1: Fi = Ai-1.
Period i Actual Demand Forecast
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
18. 18
Solution to Example 1
Period i Actual Demand Forecast
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
-
42
40
43
40
41
39
46
44
45
38
40
21. 21
Moving Average
Moving average – A technique that averages
a number of recent actual values, updated as
new values become available.
Ft = MAn =
n
At-i
i = 1
n
i = an index that corresponds to periods.
n = Number of periods (data points) in the moving
average period.
Ai = Actual value in period i.
MAn = Forecast based on most-recent n periods.
Ft = Forecast for time period t.
Where
22. 22
Example 2: Moving Average
Find moving average with n = 5.
Period i Actual Demand Forecast
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
23. 23
Solution to Example 2
Start from F6 (forecast for period 6).
Period
i
Actual
Demand
Forecast
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
-
-
-
-
-
41.2
40.6
41.8
42.0
43.0
42.4
42.6
43+40+41+39+46
5
=
46+44+45+38+40
5
=
26. 26
Weighted Moving Average
Assigns more weight to recent observed
values
More responsive to changes
Selection of weights is arbitrary, but weights
must add to one. The values for the weights
are always given.
27. 27
Example 3: Weighted Moving Average
Find weighted moving average using
Fi =0.4Ai-1 + 0.3Ai-2 + 0.2Ai-3 + 0.1Ai-4.
Period i Actual Demand Forecast
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
28. 28
Solution to Example 3
Start from F5 (forecast for period 5).
Period
i
Actual
Demand
Forecas
t
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
-
-
-
-
41.1
41.0
40.2
42.3
43.3
44.3
42.1
40.8
0.1(42)+.2(40)+.3(43)+.4(40)
=
=0.1(39)+.2(46)+.3(44)+.4(45)
29. 29
Shown solutions of Example 2 and 3
30
32
34
36
38
40
42
44
46
48
1 2 3 4 5 6 7 8 9 10 11 12
Observed
MA
WMA
30. 30
Exponential Smoothing
Current forecast = Previous forecast + α(Actual -
Previous forecast)
Ft = Ft-1 + (At-1 - Ft-1)
Ft = Forecast for period t
Ft-1 = Forecast for period t-1
α = Smoothing constant
At-1 =Actual demand or sales for period t-1
where
31. 31
Exponential Smoothing (Cont.)
Premise: The most recent observations might
have the highest predictive value.
• Therefore, we should give more weight to the more
recent time periods when forecasting.
Weighted averaging method based on previous
forecast plus a percentage of the forecast error
A-F is the error term, is the % feedback
32. 32
Example 4: Exponential Smoothing
Period (t) Actual (At) Ft (α = 0.1) Error (A-F) Ft ( α = 0.4)
1 42
2 40
3 43
4 40
5 41
6 39
7 46
8 44
9 45
10 38
11 40
12
Error (A-F)
36. 36
Picking a Smoothing Constant α
Using judgment or trial and error
Balancing smoothness and responsiveness
Usually range from 0.05 to 0.5
low α when stable
high α when susceptible to change
38. 38
Techniques for Trend
Linear trend
linear trend equation
Trend-adjusted exponential smoothing (Double
exponential smoothing), not required in exams
Nonlinear trend (not required in exams)
39. 39
Linear Trend Equation
Ft = Forecast for period t
t = Specified number of time periods from t = 0
a = Value of Ft at t = 0
b = Slope of the line
Ft = a + b t
0 1 2 3 4 5 t
Ft
40. 40
Calculating a and b
n = Number of periods
y = Value of the time series
t = Specified number of time periods from t = 0
t
n -
b =
n (ty) - y
t
2
( t)
2
a =
y - b t
n
41. Example 5:
Calculator sales for a California-based firm over the
last 10 weeks are shown in the following table.
Week (t) y yt t2
1
2
3
4
5
6
7
8
9
10
700
724
720
728
740
742
758
750
770
775
700
1448
2160
2912
3700
4452
5306
6000
6930
7750
1
4
9
16
25
36
49
64
81
100
55 7407 41358 385
42. 1. Plot the data, and visually check to see if a
linear trend line would be appropriate.
2. n = 10, t = 55, y =7407, ty = 41358, t2 = 385
42
Solution to Example 5
b =
10(41358) - 55(7407)
10(385) - 55(55)
=
413580- 407385
3850 - 3025
≈ 7.51
y = 699.40 + 7.51t
a =
7407 - 7.51(55)
10
≈ 699.40
t
n -
b =
n (ty) - y
t
2
( t)
2
a =
y - b t
n
43. 43
Solution to Example 5 (Cont.)
3. Then determine the equation of the trend
line, and predict sales for weeks 11 and 12.
y11 =699.40 + 7.51(11) = 782.01
y12 =699.40 + 7.51(12) = 789.51
44. 44
Solution to Example 5 (Cont.)
660
680
700
720
740
760
780
800
1 2 3 4 5 6 7 8 9 10
Observed
Trend line
45. Problem : A pizza boy is trying to forecast the volume of
Pizza sold based on the amount of money it has put into
advertisement in the local news paper. It has been tracking
the relationship between pizza sales and advertisement over
the past 4 months.
The results are as follows :
Pizza Sales
Advert.
Cost in $
58 135
43 90
62 145
68 145
Causal Method – Linear trend equation
46. What would happen to pizza sales,if pizza boy invested
$150 in advertizing for next month
In this problem, pizza sales are the dependent variable (Y)
And advertising cost is independent variable
Causal Method – Linear trend equation
47. Y X XY X2
Y2
58 135 7830 18225 3364
43 90 3870 8100 1849
62 145 8990 21025 3844
68 145 9860 21025 4624
Total 231 515 30550 68375 13681
X
___
= 4
515
___
Y
___
= 4
___
231
=
=
128.75
57.75
Causal Method – Linear trend equation
48. Step 1. Compute parameter b:
b = -------------- =
nΣ XY- Σ X Σ Y
__
nΣ X2- (ΣX)2
0.391
Step 2. Compute parameter a
a = =
Y-b X 7.41
Step 3. Substitute the value of a and b in the equation
Y =a+bX
Y = 7.41+ 0.391 X
__ __
__
Step 4. Generate a forecast for the dependent variable (Y)
Substitute the value for the independent variable (X)
Y = 7.41 + 0.391*150= 66 pizzas
Causal Method – Linear trend equation
50. 50
Associative Forecasting
Associative techniques rely on identification of related
variables that can be used to predict the variable of
interest (dependent variable)
Example 1: Crop yields are related to soil conditions and the
amounts and timing of water and fertilizer applications.
Example 2: Sales of beef may be related to the price per pound
(of beef) and the price of substitutes such as chicken, pork and
lamb.
Predictor (independent) variables - used to predict
values of variable of interest
Regression - technique for fitting a line to a set of points
52. 52
Linear Regression
The object of linear regression is to obtain an
equation of a straight line (Least squares line) that
minimizes the sum of squared vertical deviations of
the data points from the line.
yc = a + b x
yc = Predicated (dependent) variable.
x = Predictor (independent) variable.
a = Value of yc when x = 0.
b = Slope of the line.
Where
53. 53
Calculating Coefficients a and b
x
n -
b =
n (xy) - y
x
2
( x)
2
a =
y - b x
n
= y – b x
n = Number of paired observations
where
54. 54
Example 8 – Linear Model Seems Reasonable
593
.
1
132
1796
12
271
132
3529
12
2
b
x y xy x2
y2
7 15 105 49 225
2 10 20 4 100
6 13 78 36 169
4 15 60 16 225
14 25 350 196 625
15 27 405 225 729
16 24 384 256 576
12 20 240 144 400
14 27 378 196 729
20 44 880 400 1936
15 34 510 225 1156
7 17 119 49 289
132 271 3529 1796 7159
06
.
5
12
132
593
.
1
271
a
n = 12
x
n -
b =
n (xy) - y
x
2
( x)
2
a =
y - b x
n
= y – b x
55. 55
0
10
20
30
40
50
0 5 10 15 20 25
Computed
relationship
Example 8 – Linear Model Seems Reasonable
593
.
1
132
1796
12
271
132
3529
12
2
b
A straight line is fitted to a set
of sample points.
x y xy x2
y2
7 15 105 49 225
2 10 20 4 100
6 13 78 36 169
4 15 60 16 225
14 25 350 196 625
15 27 405 225 729
16 24 384 256 576
12 20 240 144 400
14 27 378 196 729
20 44 880 400 1936
15 34 510 225 1156
7 17 119 49 289
132 271 3529 1796 7159
06
.
5
12
132
593
.
1
271
a
n = 12
yc = 5.06 +1.593 x
57. 57
Accuracy and Control of Forecasts
Error: Difference between the actual value and
the value that was predicted for a given period
et = At – Ft
Forecast errors influence decisions in two
somewhat different ways
1. A choice between various forecasting alternatives
2. Evaluate the success or failure of a technique in use
58. 58
Measures of Forecast Accuracy
Mean Absolute Deviation (MAD)
MAD =
Actualt Forecastt
n
MAPE =
Actualt Forecastt
n
Actualt
× 100
