Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 5
Forecasting
Chapter 5 - Forecasting
1
Chapter Topics
Forecasting Components
Time Series Methods
Forecast Accuracy
Time Series Forecasting Using Excel
Time Series Forecasting Using QM for Windows
Regression Methods
Chapter 5 - Forecasting
2
Forecasting Components
A variety of forecasting methods are available for use
depending on the time frame of the forecast and the
existence of patterns.
Time Frames:
Short-range (one to two months)
Medium-range (two months to one or two years)
Long-range (more than one or two years)
Patterns:
Trend
Random variations
Cycles
Seasonal pattern
Chapter 5 - Forecasting
3
Forecasting Components
Patterns (1 of 2)
Trend - A long-term movement of the item being forecast.
Random variations - movements that are not predictable
and follow no pattern.
Cycle - A movement, up or down, that repeats itself over a
lengthy time span.
Seasonal pattern - Oscillating movement in demand that
occurs periodically in the short run and is repetitive.
Chapter 5 - Forecasting
4
Forecasting Components
Patterns (2 of 2)
trend-line
Figure 5.1
Forms of Forecast Movement: (a) Trend, (b) Cycle, (c) Seasonal
Pattern, (d) Trend with Seasonal Pattern
Chapter 5 - Forecasting
5
Forecasting Components
Forecasting Methods
Times Series - Statistical techniques that use historical data
to predict future behavior.
Regression Methods - Regression (or causal ) methods that
attempt to develop a mathematical relationship between the
item being forecast and factors that cause it to behave the
way it does.
Qualitative Methods - Methods using judgment, expertise
and opinion to make forecasts.
Chapter 5 - Forecasting
6
Forecasting Components
Qualitative Methods
Qualitative methods, the “jury of executive opinion,” is the
most common type of forecasting method for long-term
strategic planning.
Performed by individuals or groups within an organization,
sometimes assisted by consultants and other experts,
whose judgments and opinion are considered valid for the
forecasting issue.
Usually includes specialty functions such as marketing,
engineering, purchasing, etc. in which individuals have
experience and knowledge of the forecasted item.
Supporting techniques include the Delphi Method, market
research, surveys, etc.
Chapter 5 - Forecasting
7
Time Series Methods
Overview
Statistical techniques that make use of historical data
collected over a long period of time.
Methods assume that what has occurred in the past will
continue to occur in the future.
Forecasts based on only one factor - time.
Chapter 5 - Forecasting
8
Time Series Methods
Moving Average (1 of 5)
Moving average uses values from the recent past to
develop forecasts.
This dampens or smoothes out random increases and
decreases.
Useful for forecasting relatively stable items that do not
display any trend or seasonal pattern.
Formula for:
n
1
MAn  n  Di
i1
where:
n number of periods in the moving average
Di  data in period i
Chapter 5 
- Forecasting
9
Time Series Methods
Moving Average (2 of 5)
Example: Instant Paper Clip Supply Company forecast of
orders for the next month.
Three-month moving average:
3
1
MA3  3  D  90110130 110
3
i1 i
orders

Five-month moving average:
5
1
MA3  5  D  901101307550 91 orders
5
i1 i

Chapter 5 - Forecasting
10
Time Series Methods
Moving Average (3 of 5)
Figure 5.2
Three- and Five-Month Moving Averages
Chapter 5 - Forecasting
11
Time Series Methods
Moving Average (4 of 5)
Figure 5.2
Three- and Five-Month Moving Averages
Chapter 5 - Forecasting
12
Time Series Methods
Moving Average (5 of 5)
Longer-period moving averages react more slowly to
changes in demand than do shorter-period moving
averages.
The appropriate number of periods to use often requires
trial-and-error experimentation.
Moving average does not react well to changes (trends,
seasonal effects, etc.) but is easy to use and inexpensive.
Good for short-term forecasting.
Chapter 5 - Forecasting
13
Time Series Methods
Weighted Moving Average (1 of 2)
In a weighted moving average, weights are assigned to the
most recent data.
Formula:
n
WMAn   Wi Di
i1
where Wi  the weight for period i, between 0% and 100%
Wi  1.00
Example : Paper clip company we ights 50% for October, 33%
for September, 17% for August :
3
WMA   Wi Di  (.50)(90)  (.33)(110)  (.17)(130)  103.4 orders
3 i1
Chapter 5 - Forecasting
14
Time Series Methods
Weighted Moving Average (2 of 2)
Determining precise weights and number of periods
requires trial-and-error experimentation.
Chapter 5 - Forecasting
15
Time Series Methods
Exponential Smoothing (1 of 11)
Exponential smoothing weights recent past data more
strongly than more distant data.
Two forms: simple exponential smoothing and adjusted
exponential smoothing.
Simple exponential smoothing:
Ft + 1 = Dt + (1 - )Ft
where: Ft + 1 = the forecast for the next period
Dt = actual demand in the present period
Ft = the previously determined forecast for the
present period
 = a weighting factor (smoothing constant)
use F1 = D1.
Chapter 5 - Forecasting
16
Time Series Methods
Exponential Smoothing (2 of 11)
The most commonly used values of  are between
.10 and .50.
Determination of  is usually judgmental and subjective and
often based on trial-and -error experimentation.
Chapter 5 - Forecasting
17
Time Series Methods
Exponential Smoothing (3 of 11)
Example: PM Computer Services (see Table 5.4).
Exponential smoothing forecasts using smoothing constant
of .30.
Forecast for period 2 (February):
F2 =  D1 + (1- )F1 = (.30)(37) + (.70)(37) = 37 units
Forecast for period 3 (March):
F3 =  D2 + (1- )F2 = (.30)(40) + (.70)(37) = 37.9 units
Chapter 5 - Forecasting
18
Time Series Methods
Exponential Smoothing (4 of 11)
Using
F1 = D1
Table 5.4
Exponential Smoothing Forecasts,  = .30 and  = .50
Chapter 5 - Forecasting
19
Time Series Methods
Exponential Smoothing (5 of 11)
The forecast that uses the higher smoothing constant (.50)
reacts more strongly to changes in demand than does the
forecast with the lower constant (.30).
Both forecasts lag behind actual demand.
Both forecasts tend to be consistently lower than actual
demand.
Low smoothing constants are appropriate for stable data
without trend; higher constants appropriate for data with
trends.
Chapter 5 - Forecasting
20
Time Series Methods
Exponential Smoothing (6 of 11)
Figure 5.3
Exponential Smoothing Forecasts
Chapter 5 - Forecasting
21
Time Series Methods
Exponential Smoothing (7 of 11)
Adjusted exponential smoothing: exponential smoothing
with a trend adjustment factor added.
Formula:
AFt + 1 = Ft + 1 + Tt+1
where: Tt = an exponentially smoothed trend factor:
Tt + 1 = (Ft + 1 - Ft) + (1 - )Tt
Tt = the last (previous) period’s trend factor
 = smoothing constant for trend ( a value
between zero and one).
Reflects the weight given to the most recent trend data.
Determined subjectively.
Chapter 5 - Forecasting
22
Time Series Methods
Exponential Smoothing (8 of 11)
Example: PM Computer Services exponential smoothed
forecasts with  = .50 and  = .30 (see Table 5.5).
Start with T2 = 0.00
Adjusted forecast for period 3:
T3 = (F3 - F2) + (1 - )T2
= (.30)(38.5 - 37.0) + (.70)(0) = 0.45
AF3 = F3 + T3 = 38.5 + 0.45 = 38.95
Chapter 5 - Forecasting
23
Time Series Methods
Exponential Smoothing (9 of 11)
Tt + 1 = (Ft + 1 - Ft)
+ (1 - )Tt
Table 5.5
Adjusted Exponentially Smoothed Forecast Values
Chapter 5 - Forecasting
24
Time Series Methods
Exponential Smoothing (10 of 11)
Adjusted forecast is consistently higher than the simple
exponentially smoothed forecast.
It is more reflective of the generally increasing trend of the
data.
Chapter 5 - Forecasting
25
Time Series Methods
Exponential Smoothing (11 of 11)
Figure 5.4
Adjusted Exponentially Smoothed Forecast
Chapter 5 - Forecasting
26
Time Series Methods
Linear Trend Line (1 of 5)
When demand displays an obvious trend over time, a least
squares regression line , or linear trend line, can be used to
forecast.
Formula:
xy nxy
b  2
 x nx
a ybx
yabx
where:
a intercept (at period 0)
b slope of the line
x the time period
y forecast for demand
for period x
where:
n number of periods
x 1n  x
y 1n  y

Chapter 5 - Forecasting

27
Time Series Methods
Linear Trend Line (2 of 5)
Example: PM Computer Services (see Table 5.6)
x 78 6.5
12
y 557 46.42
12
xy nxy 3,867(12)(6.5)(46.42)

b

1.72
2
2
65012(6.5)
 x2 nx
a ybx46.42(1.72)(6.5)35.2
y35.21.72x linear trend line
for period 13, x  13, y35.21.72(13)57.56

Chapter 5 - Forecasting
28
Time Series Methods
Linear Trend Line (3 of 5)
Table 5.6
Least Squares Calculations
Chapter 5 - Forecasting
29
Time Series Methods
Linear Trend Line (4 of 5)
A trend line does not adjust to a change in the trend as
does the exponential smoothing method.
This limits its use to shorter time frames in which trend will
not change.
Chapter 5 - Forecasting
30
Time Series Methods
Linear Trend Line (5 of 5)
Figure 5.5
Linear Trend Line
Chapter 5 - Forecasting
31
Time Series Methods
Seasonal Adjustments (1 of 4)
A seasonal pattern is a repetitive up-and-down movement
in demand.
Seasonal patterns can occur on a monthly, weekly, or daily
basis.
A seasonally adjusted forecast can be developed by
multiplying the normal forecast by a seasonal factor.
A seasonal factor can be determined by dividing the actual
demand for each seasonal period by total annual demand:
Si =Di/D
Chapter 5 - Forecasting
32
Time Series Methods
Seasonal Adjustments (2 of 4)
Seasonal factors lie between zero and one and represent
the portion of total annual demand assigned to each
season.
Seasonal factors are multiplied by annual demand to
provide adjusted forecasts for each period.
Chapter 5 - Forecasting
33
Time Series Methods
Seasonal Adjustments (3 of 4)
Example: Wishbone Farms
Table 5.7
Demand for Turkeys at Wishbone Farms
S1 = D1/ D = 42.0/148.7 = 0.28
S2 = D2/ D = 29.5/148.7 = 0.20
S3 = D3/ D = 21.9/148.7 = 0.15
S4 = D4/ D = 55.3/148.7 = 0.37
Chapter 5 - Forecasting
34
Time Series Methods
Seasonal Adjustments (4 of 4)
Multiply forecasted demand for entire year by seasonal
factors to determine quarterly demand.
Forecast for entire year (trend line for data in Table 5.7):
y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17
Seasonally adjusted forecasts:
SF1 = (S1)(F5) = (.28)(58.17) = 16.28
SF2 = (S2)(F5) = (.20)(58.17) = 11.63
SF3 = (S3)(F5) = (.15)(58.17) = 8.73
SF4 = (S4)(F5) = (.37)(58.17) = 21.53
Note the potential for confusion: the Si are “seasonal factors” (fractions),
whereas the SFi are “seasonally adjusted forecasts” (commodity values)!
Chapter 5 - Forecasting
35
Forecast Accuracy
Overview
Forecasts will always deviate from actual values.
Difference between forecasts and actual values referred to
as forecast error.
Would like forecast error to be as small as possible.
If error is large, either technique being used is the wrong
one, or parameters need adjusting.
Measures of forecast errors:
Mean Absolute deviation (MAD)
Mean absolute percentage deviation (MAPD)
Cumulative error (E-bar)
Average error, or bias (E)
Chapter 5 - Forecasting
36
Forecast Accuracy
Mean Absolute Deviation (1 of 7)
MAD is the average absolute difference between the
forecast and actual demand.
Most popular and simplest-to-use measures of forecast
error.
Formula:
MAD 1n  Dt Ft
where:
t  the period number
Dt  demand in period t
Ft  the forecast for period t
n  the total number of periods
Chapter 5 - Forecasting

37
Forecast Accuracy
Mean Absolute Deviation (2 of 7)
Example: PM Computer Services (see Table 5.8).
Compare accuracies of different forecasts using MAD:
MAD 1n  Dt Ft  53.414.85
11

Chapter 5 - Forecasting
38
Forecast Accuracy
Mean Absolute Deviation (3 of 7)
Table 5.8
Computational Values for MAD
Chapter 5 - Forecasting
39
Forecast Accuracy
Mean Absolute Deviation (4 of 7)
The lower the value of MAD relative to the magnitude of the
data, the more accurate the forecast.
When viewed alone, MAD is difficult to assess.
Must be considered in light of magnitude of the data.
Chapter 5 - Forecasting
40
Forecast Accuracy
Mean Absolute Deviation (5 of 7)
Can be used to compare accuracy of different forecasting
techniques working on the same set of demand data (PM
Computer Services):
Exponential smoothing ( = .50): MAD = 4.04
Adjusted exponential smoothing ( = .50,  = .30):
MAD = 3.81
Linear trend line: MAD = 2.29
Linear trend line has lowest MAD; increasing  from .30 to
.50 improved smoothed forecast.
Chapter 5 - Forecasting
41
Forecast Accuracy
Mean Absolute Deviation (6 of 7)
A variation on MAD is the mean absolute percent deviation
(MAPD).
Measures absolute error as a percentage of demand rather
than per period.
Eliminates problem of interpreting the measure of accuracy
relative to the magnitude of the demand and forecast
values.
Formula:
 Dt  Ft 53.41
MAPD 

.103 or 10.3%
520
 Dt
Chapter 5 - Forecasting
42
Forecast Accuracy
Mean Absolute Deviation (7 of 7)
MAPD for other three forecasts:
Exponential smoothing ( = .50): MAPD = 8.5%
Adjusted exponential smoothing ( = .50,  = .30):
MAPD = 8.1%
Linear trend: MAPD = 4.9%
Chapter 5 - Forecasting
43
Forecast Accuracy
Cumulative Error (1 of 2)
Cumulative error is the sum of the forecast errors (E =et).
A relatively large positive value indicates forecast is biased
low, a large negative value indicates forecast is biased
high.
If preponderance of errors are positive, forecast is
consistently low; and vice versa.
Cumulative error for trend line is always almost zero, and is
therefore not a good measure for this method.
Cumulative error for PM Computer Services can be read
directly from Table 5.8.
E =  et = 49.31 indicating forecasts are frequently below
actual demand.
Chapter 5 - Forecasting
44
Forecast Accuracy
Cumulative Error (2 of 2)
Cumulative error for other forecasts:
Exponential smoothing ( = .50): E = 33.21
Adjusted exponential smoothing ( = .50,  =.30):
E = 21.14
Average error (bias) is the per period average of
cumulative error.
Average error for exponential smoothing forecast:
e
E  n t  49.31 4.48
11
A large positive value of average error indicates a forecast
is biased low; a large negative error indicates it is biased
high.
Chapter 5 - Forecasting
45
Forecast Accuracy
Example Forecasts by Different Measures
Table 5.9
Comparison of Forecasts for PM Computer Services
Results consistent for all forecasts:
Larger value of alpha is preferable.
Adjusted forecast is more accurate than exponential
smoothing forecasts.
Linear trend is more accurate than all the others.
Chapter 5 - Forecasting
46
Time Series Forecasting Using Excel (1 of 4)
Exhibit 5.1
Chapter 5 - Forecasting
47
Time Series Forecasting Using Excel (2 of 4)
Exhibit 5.2
Chapter 5 - Forecasting
48
Time Series Forecasting Using Excel (3 of 4)
Exhibit 5.3
Chapter 5 - Forecasting
49
Time Series Forecasting Using Excel (4 of 4)
Exhibit 5.4
Chapter 5 - Forecasting
50
Exponential Smoothing Forecast with Excel QM
Exhibit 5.5
Chapter 5 - Forecasting
51
Time Series Forecasting
Solution with QM for Windows (1 of 2)
Exhibit 5.6
Chapter 5 - Forecasting
52
Time Series Forecasting
Solution with QM for Windows (2 of 2)
Exhibit 5.7
Chapter 5 - Forecasting
53
Regression Methods
Overview
Time series techniques relate a single variable being
forecast to time.
Regression is a forecasting technique that measures the
relationship of one variable to one or more other variables.
Simplest form of regression is linear regression.
Chapter 5 - Forecasting
54
Regression Methods
Linear Regression
Linear regression relates demand (dependent variable ) to
an independent variable.
yabx Linear function
a ybx
xynxy
b 
2
 x2 nx
where:
x
x n  mean of the x data
y
y n  mean of the y data

Chapter 5 - Forecasting
55
Regression Methods
Linear Regression Example (1 of 3)
State University athletic department.
Wins
4
6
6
8
6
7
5
7
Attendance
36,300
40,100
41,200
53,000
44,000
45,600
39,000
47,500
Chapter 5 - Forecasting
x
(wins)
4
6
6
8
6
7
5
7
49
y
(attendance, 1,000s)
36.3
40.1
41.2
53.0
44.0
45.6
39.0
47.5
346.7
xy
145.2
240.6
247.2
424.0
264.0
319.2
195.0
332.5
2,167.7
x2
16
36
36
64
36
49
25
49
311
56
Regression Methods
Linear Regression Example (2 of 3)
x 49 6.125
8
y 346.9 43.34
8
xy nxy (2,167.70(8)(6.125)(43.34)
4.06

b 
2
2
(311)(8)(6.125)
 x2 nx
a ybx43.34(.406)(6.125)18.46
Therefore, y18.464.06x
Attendance forecast for x  7 wins is
y18.464.06(7)46.88 or 46,880
Chapter5 - Forecasting
57
Regression Methods
Linear Regression Example (3 of 3)
Figure 5.6
Linear Regression Line
Chapter 5 - Forecasting
58
Regression Methods
Correlation (1 of 2)
Correlation is a measure of the strength of the relationship
between independent and dependent variables.
Formula:
r
n xy   x  y
n x   x n y
2
2
2
  y 
2

Value lies between +1 and -1.
Value of zero indicates little or no relationship between

variables.
Values near 1.00 and -1.00 indicate strong linear
relationship: correlation and anti-correlation.
Chapter 5 - Forecasting
59
Regression Methods
Correlation (2 of 2)
Value for State University example:
r
(8)(2,167.7)(49)(346.7)
.948




2
(8)(311)(49)(49)(8)(15,224.7)(346.7) 




Chapter 5 - Forecasting
60
Regression Methods
Coefficient of Determination
The Coefficient of determination is the percentage of the
variation in the dependent variable that results from the
independent variable.
Computed by squaring the correlation coefficient, r.
For State University example:
r = .948, r2 = .899
This value indicates that 89.9% of the amount of variation in
attendance can be attributed to the number of wins by the
team, with the remaining 10.1% due to other, unexplained,
factors.
Chapter 5 - Forecasting
61
Regression Analysis with Excel (1 of 7)
Exhibit 5.8
Chapter 5 - Forecasting
62
Regression Analysis with Excel (2 of 7)
Exhibit 5.9
Chapter 5 - Forecasting
63
Regression Analysis with Excel (3 of 7)
Exhibit 5.10
Chapter 5 - Forecasting
64
Regression Analysis with Excel (4 of 7)
Exhibit 5.11
Chapter 5 - Forecasting
65
Regression Analysis with Excel (5 of 7)
Exhibit 5.12
Chapter 5 - Forecasting
66
Regression Analysis with Excel (6 of 7)
Exhibit 5.13
Chapter 5 - Forecasting
67
Regression Analysis with Excel (7 of 7)
Exhibit 5.14
Chapter 5 - Forecasting
68
Regression Analysis with QM for Windows
Exhibit 5.15
Chapter 5 - Forecasting
69
Multiple Regression with Excel (1 of 4)
Multiple regression relates demand to two or more
independent variables.
General form:
y = 0 +  1x1 +  2x2 + . . . +  kxk
where  0 = the intercept
 1 . . .  k = parameters representing
contributions of the independent
variables
x1 . . . xk = independent variables
Chapter 5 - Forecasting
70
Multiple Regression with Excel (2 of 4)
State University example:
Wins Promotion ($) Attendance
4
29,500
36,300
6
55,700
40,100
6
71,300
41,200
8
87,000
53,000
6
75,000
44,000
7
72,000
45.600
5
55,300
39,000
7
81,600
47,500
Chapter 5 - Forecasting
71
Multiple Regression with Excel (3 of 4)
Exhibit 5.16
Chapter 5 - Forecasting
72
Multiple Regression with Excel (4 of 4)
Exhibit 5.17
Chapter 5 - Forecasting
73
Example Problem Solution
Computer Software Firm (1 of 4)
Problem Statement:
• For data below, develop an exponential smoothing forecast
using  = .40, and an adjusted exponential smoothing
forecast using  = .40 and  = .20.
• Compare the accuracy of the forecasts using MAD and
cumulative error.
Period
1
2
3
4
5
6
7
8
Chapter 5 - Forecasting
Units
56
61
55
70
66
65
72
75
74
Example Problem Solution
Computer Software Firm (2 of 4)
Step 1: Compute the Exponential Smoothing Forecast.
Ft+1 =  Dt + (1 - )Ft
Step 2: Compute the Adjusted Exponential Smoothing
Forecast
AFt+1 = Ft +1 + Tt+1
Tt+1 = (Ft +1 - Ft) + (1 - )Tt
Chapter 5 - Forecasting
75
Example Problem Solution
Computer Software Firm (3 of 4)
Period
1
2
3
4
5
6
7
8
9
Chapter 5 - Forecasting
Dt
56
61
55
70
66
65
72
75

Ft

56.00
58.00
56.80
62.08
63.65
64.18
67.31
70.39
AFt

56.00
58.40
56.88
63.20
64.86
65.26
68.80
72.19
Dt - Ft
Dt - AFt


5.00
5.00
-3.00
-3.40
13.20
13.12
3.92
2.80
1.35
0.14
7.81
6.73
7.68
6.20


35.97
30.60
76
Example Problem Solution
Computer Software Firm (4 of 4)
Step 3: Compute the MAD Values
1
41.97
MAD(Ft )   Dt  Ft 
 5.99
n
7
1
37.39
MAD(AFt )   Dt  AFt 
 5.34
n
7
Step 4: Compute the Cumulative Error.

E(Ft) = 35.97
E(AFt) = 30.60
Chapter 5 - Forecasting
77
Example Problem Solution
Building Products Store (1 of 5)
Problem Statement:
For the following data,
Develop a linear regression model
Determine the strength of the linear relationship using
correlation.
Determine a forecast for lumber given 10 building
permits in the next quarter.
Chapter 5 - Forecasting
78
Example Problem Solution
Building Products Store (2 of 5)
Quarter
1
2
3
4
5
6
7
8
9
10
Chapter 5 - Forecasting
Building
Permits
x
8
12
7
9
15
6
5
8
10
12
Lumber Sales
(1,000s of bd ft)
y
12.6
16.3
9.3
11.5
18.1
7.6
6.2
14.2
15.0
17.8
79
Example Problem Solution
Building Products Store (3 of 5)
Step 1: Compute the Components of the Linear Regression
Equation.
x
92
 92
10
y
128.6
 12.86
10
 xy  n x y
(1,290.3)  (10)(9.2)(12.86)
b
 1.25
2 
2
2
(932)  (10)(9.2)
 x  nx
a  y  bx  12.86  (1.25)(9.2)  1.36
Chapter 5 - Forecasting
80
Example Problem Solution
Building Products Store (4 of 5)
Step 2: Develop the Linear regression equation.
y = a + bx, y = 1.36 + 1.25x
Step 3: Compute the Correlation Coefficient.
r
n xy   x  y
n x   x n y
2
r

2
2
  y 
2

(10)(1,170.3)  (92)(128.6)
(10)(932)  (92)(92)(10)(1,810.48)  (128.6)2 
Chapter 5 - Forecasting
 .925
81
Example Problem Solution
Building Products Store (5 of 5)
Step 4: Calculate the forecast for x = 10 permits.
Y = a + bx = 1.36 + 1.25(10) = 13.86 or 1,386 board ft
Chapter 5 - Forecasting
82
Chapter 5 - Forecasting
83