MD 021 - Management and Operations
Forecasting
Outline
 Components of demand
 Judgment methods
 Linear regression
 Time series methods
 Forecast errors
1
Judgment Methods
Sales force estimates
Executive opinion
Market research
Delphi method
2
Linear Regression
Yi  a  bXi
where:
Y = dependent variable
X = independent variable
a = Y-intercept of the line
b = slope of the line
3
Measures of Forecast Accuracy in Linear
Regression
Coefficient of correlation
Coefficient of determination
Standard error of the estimate
4
Regression Analysis Example
The manager of Al’s Diner is interested in forecasting the number
of potato skin appetizers sold each week. He believes that the
number sold has a linear relationship to the price and uses linear
regression to determine if this is the case.
X
Y
Week (Price) (Appetizers)
1.
$2.70
760
2.
3.50
510
3.
2.00
980
4.
4.20
250
5.
3.10
320
6.
4.05
480
The Excel output is below:
Regression Statistics
Multiple R
0.843
R Square
0.711
Adjusted R Square
0.639
Standard Error
165.257
Observations
6
ANOVA
Regression
Residual
Total
Intercept
Price ($)
df
SS
1
4
5
269160
109239
378400
Coefficients Standard
Error
1454.604
295.939
-277.628
88.434
MS
F
Significance
F
269160 9.856
0.035
27309
t Stat
Pvalue
4.915 0.008
-3.139 0.035
5
Linear Regression Example
A professor is interested in determining whether average study
hours per week is a good predictor of test scores. The results of
her study are:
Hours
Score
3.0
90
2.1
95
5.8
65
3.8
80
4.2
95
3.2
60
5.3
85
4.6
70
A student says: "Professor, what can I do to get a B on the next
test. The professor asks, "On average, how many hours do you
spend studying for this course per week?" The student responds,
"About 2 hours." Use linear regression to forecast the student's test
score.
Regression Statistics
Multiple R
0.391
R Square
0.153
Adjusted R Square
0.0121
Standard Error
13.544
Observations
8
ANOVA
Regression
Residual
Total
Intercept
Study hours
df
SS
1
6
7
199.246
1100.753
1300
Coefficients Standard Error
97.325
17.301
-4.331
4.156
MS
F
Significance
F
199.246 1.0861
0.3375
183.458
t Stat
5.625
-1.042
P-value
0.0013
0.3375
6
Time Series Methods
Naive forecasts
Moving averages
Weighted moving averages
Exponential smoothing
Trend-adjusted exponential smoothing
Regression Method
Multiplicative seasonal method
7
Moving Average Method
n
Ft  MAn 
 At  i
i 1
n
Month
1
2
3
4
Customer
arrivals
800
740
810
790
 Use a 3-month moving average to forecast customer
arrivals for month 5.
F5 
 If the actual demand for month 5 is 805 customers, what
is the forecast for month 6?
F6 
8
Weighted Moving Average Method
Ft  wn At  n  wn 1 At  ( n 1)  ...  w1 At 1
Month
1
2
3
4
Customer
arrivals
800
740
810
790
 Let W1  0.50, W2  0.30, and W3  0.20.
Calculate the forecast for month 5.
F5 
 If the actual demand for month 5 is 805 customers, what
is the forecast for month 6?
F6 
9
Exponential Smoothing
Ft  (1   ) Ft 1  At 1
Month
1
2
3
4
Customer
arrivals
800
740
810
790
 Suppose F3  783 customers and   0.20.
What is the forecast for month 5?
F4 
F5 
 If D5  805, what is the forecast for month 6?
F6 
10
Trend-Adjusted Exponential Smoothing
St  TAFt   ( At  TAFt )
Tt  Tt 1   (TAFt  TAFt 1  Tt 1 )
TAFt 1  St  Tt
Month
1
2
3
4
5
Patients
48
52
50
54
55
 Using months 1-4, an initial estimate of the trend is 2
[(4-2+4)/3 = 2]. The starting forecast for month 5 is
54+2 = 56. Using   0.3 and   0.4 , forecast the
number of patients in month 6.
S5 
T5 
TAF6 
 If the actual number of patients in month 6 is 58, what is
the forecast for month 7?
S6 
T6 
TAF7 
11
Regression Method
Example: Garcia Garage
Month (t)
Number
of Number of Oil
time periods Changes (Y)
from t = 0
Jan.
1
41
Feb.
2
46
Mar.
3
57
Apr.
4
52
May
5
59
Jun.
6
51
Jul.
7
60
Aug.
8
62
1. Forecast the numbers of oil changes in September, October,
and November.
2. What is the average value of the trend?
Regression Statistics
Multiple R
0.817
R Square
0.668
Adjusted R
Square
0.613
Standard
Error
4.572
Observations
8.000
ANOVA
Regression
Residual
Total
Intercept
X Variable 1
df
1.000
6.000
7.000
Coefficients
42.464
2.452
SS
MS
F
252.595 252.595 12.085
125.405 20.901
378.000
Standard
Error
3.562
0.705
t Stat
11.921
3.476
Pvalue
0.000
0.013
Significance
F
0.013
Lower 95%
33.748
0.726
Upper
95%
51.181
4.179
12
Multiplicative Seasonal Method
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 4:
Calculate the trend line based on the
available data using regression.
Calculate the centered moving average,
with the number of periods equal to the
number of seasons.
Calculate the seasonal relative for a
period by dividing the actual demand for
the period by the corresponding centered
moving average.
Calculate the overall estimated seasonal
relative by averaging the seasonal
relatives from the same periods over the
cycle.
Calculate the trend values for each of the
periods to be forecast based on the trend
line determined in Step 1.
To get a forecast for a given period in a
future cycle, multiply the seasonal factor
by the trend values.
13
Multiplicative Seasonal Method Application
Quarter Demand CMA (4 seasons)
100
1
2
MA (2 periods)
Seasonal Relatives
Normalized S.R.
261.5
1.147227533
1.171002862
274
0.729927007
0.745054133
285.5
0.672504378
0.686441468
298
1.369127517
1.397501537
3.918786436
4
400
250
3
300
273
4
200
275
5
192
296
6
408
300
7
384
8
216
9
10
11
12
331
344
356
369
Total
(trend value*)
(trend value*)
(trend value*)
(trend value*)
227
480
417
275
(forecast)
(forecast)
(forecast)
(forecast)
* Using regression, the trend line is 218.86 + 12.48t.
14
Forecast Errors
 Bias--- systematic errors
 Random Errors --- variability
Example:
Day 1
100
Actual
Demand
Forecast 105
1
Forecast 50
2
Day 2
100
Day 3
100
Day 4
100
105
105
105
150
50
150
15
Forecast Error Measures
Bias:
n
 et
 t 1
n
Average error
Variability:
n
 et
2
Mean squared error
MSE
 t 1
n 1
Standard deviation
s
 MSE
n
 et
Mean absolute error
 t 1
n
MAD
n
[
Mean percent absolute error
MAPE 
t 1
et
(100)]
At
n
16
Control Chart for Forecast Errors
Upper Control Limit: UCL  0  z MSE
Lower Control Limit: LCL  0  z MSE
Z = the number of standard deviations from the mean
*Where to find “z” given the percentage of
the control chart, P0 ? Where to find “z”
given the probability for type I error,  ?
Normal Distribution Table (page 850, Table
B.2)
Look for “z” corresponds to the probability:
p

P{Z<= z} = 1  2 = 0.5+ 2 ,
0
P0 =1- 
e.g. A 95% control chart has  = 1-95% = 5%,
which means its probability for type I error is
5%. Thus probability in the table should be
0.975 (P = 1-0.025 or P = 0.5+ 0.475), which
corresponds to z = 1.96.
17
Summarizing Forecast Accuracy
Period
1
2
3
4
5
6
7
8
9
Actual (A)
113
85
96
86
121
100
142
92
72
Forecast (F) Error (E=A-F) Abs Error
95
18
18
80
5
5
103
-7
7
119
-33
33
117
4
4
125
-25
25
67
75
75
96
-4
4
116
-44
44
Total
MAD =
MSE =
s=
MAPE =
-11
215
Error Sq
324
25
49
1089
16
625
5625
16
1936
[(Abs E)/A] x
100
15.93
5.88
7.29
38.37
3.31
25.00
52.82
4.35
61.11
9705
214.06
23.9
1213.1
34.8
23.8%
18
Tracking and Analyzing Forecast Errors
Period
10
11
Actual (A)
102
107
Forecast (F)
130
102
Error (E=A-F)
-28
5
12
13
14
15
112
118
89
142
89
97
115
82
23
21
-26
60
16
17
18
100
94
111
130
137
89
-30
-43
22
Average error (periods 1-18)=
Standard deviation (periods 1-9) =
-0.39
34.8
2s control limits: 0 +/- 2(34.8) = 0 +/- 69.6
Total
4
80
UCL = 69.6
60
40
20
0
-20
10
11
12
13
14
15
16
17
18
-40
-60
LCL = -69.6
-80
19
Forecasting Summary Notes
Choosing a Forecasting Method
General considerations:
Method
Pros
Judgment
 Can be used in the absence of
historical data (e.g. new product)
 Helpful in identifying turning
points and preparing medium- and
long-term forecasts
Causal
 Most sophisticated method
 Very good for predicting turning
points and preparing medium- and
long-term forecasts
Time series
 Easy to implement
 Work well when demand
relatively stable
is
Cons
 Subjective estimates are subject to
the biases and motives of the
estimators
 Must
have historical data on
independent and dependent
variables
 Relationships can be difficult to
specify
 Rely exclusively on past demand
data
 Only useful for short-term
estimates
Specific considerations for time series methods:
Method
Pros
Naive forecast
 Easiest method, low cost
 Works well when random
errors are small
Simple moving average
 Easiest moving average
method
 To some extent, controls
for random error
Weighted moving average
 Weights
Cons
 Results in highly variable
forecasts if the random
errors are large
 Data must be retained for
n periods
 Forecast lags changes in
the underlying average of
demand
 Data must be retained for
n periods
 Forecast lags changes in
the underlying average of
demand
 Forecast lags changes in
the underlying average of
demand
can be varied to
be responsive to demand
pattern
 To some extent, controls
for random error
Exponential smoothing
 Requires little data
  can be varied to be
responsive to demand
pattern
 To some extent, controls
for random error
In general, emphasize recent demand (i.e. small n, large weights for recent observations,
large  ) for dynamic (i.e. uncertain) demand patterns. Emphasize historical experience
for stable demand patterns. If a trend is present, simple moving average, weighted
moving average, and exponential smoothing estimates will always lag actual demand.
20
Forecasting Notes
Choosing a Time Series Forecasting Method
Evaluating forecast performance: Forecast errors can be classified as either bias errors or
random errors. Bias errors are the results of systematic over- or underestimation.
Random errors are unpredictable. Ideally, a forecast should minimize both bias and
random errors.
Method
Purpose
Mean forecast
Measures bias
errors
Mean squared error (MSE) Measures the dispersion of forecast errors; large errors get
more weight than when using MAD
Mean absolute deviation
Measures the dispersion of forecast errors; method is
(MAD)
intuitive
Mean absolute percent
Measures the dispersion of forecast errors relative to the
error (MAPE)
level of demand
Forecast error control chart Determines whether the method of forecasting is
accurately predicting actual changes in demand
21