Measures of Dispersion and Variability: Range, QD, AD and SD
Lecture2 forecasting f06_604
1. CHAPTER 13
FORECASTING
Outline
• Forecasting and Choice of a Forecasting Methods
• Methods for Stationary Series:
– Simple and Weighted Moving Average
– Exponential smoothing
• Trend-Based Methods
– Regression
– Double Exponential Smoothing: Holt’s Method
• A Method for Seasonality and Trend
3. Decisions Based on Forecasts
• Production
– Aggregate planning,
inventory control,
scheduling
• Marketing
– New product
introduction, sales-
force allocation,
promotions
• Finance
– Plant/equipment
investment, budgetary
planning
• Personnel
– Workforce planning,
hiring, layoff
4. Characteristics of Forecasts
• Forecasts are always
wrong; so consider
both expected value
and a measure of
forecast error
• Long-term forecasts
are less accurate than
short-term forecasts
• Aggregate forecasts
are more accurate than
disaggregate forecasts
5. Forecasting
• Components of demand
• Evaluation of forecasts
• Time series: stationary series
• Time series: trend
– Linear regression
– Double exponential smoothing
• Time series: seasonality
6. Components of Demand
• Average demand
• Trend
– Gradual shift in average demand
• Seasonal pattern
– Periodic oscillation in demand which repeats
• Cycle
– Similar to seasonal patterns, length and
magnitude of the cycle may vary
• Random movements
• Auto-correlation
9. Components of Demand
Quantity
| | | | | | | | | | | |
J F M A M J J A S O N D
Months
(c) Seasonal influence: Data consistently show
peaks and valleys.
Year 1
10. Components of Demand
Quantity
| | | | | | | | | | | |
J F M A M J J A S O N D
Months
(c) Seasonal influence: Data consistently show
peaks and valleys.
Year 1
Year 2
16. Σ|Et |
nΣEt
2
n
RSFE = ΣEt
MAD =
MSE =
MAPE =
σ = MSE
Σ[|Et | (100)]/At
n
Measures of Forecast Error
Et = At - Ft
Choosing a MethodChoosing a Method
Forecast ErrorForecast Error
17. Absolute
Error Absolute Percent
Month, Demand, Forecast, Error, Squared, Error, Error,
t At Ft Et Et
2
|Et| (|Et|/At)(100)
1 200 225
2 240 220
3 300 285
4 270 290
5 230 250
6 260 240
7 210 250
8 275 240
-
Total
Choosing a MethodChoosing a Method
Forecast ErrorForecast Error
18. MSE = =
Measures of Error
MAD = =
MAPE = =
RSFE =
Choosing a MethodChoosing a Method
Forecast ErrorForecast Error
19. Choosing a MethodChoosing a Method
Forecast ErrorForecast Error
Running Sum Mean Absolute
of Forecast Errors Deviation
Method (RSFE - bias) (MAD)
Simple moving average
Three-week (n = 3) 23.1 17.1
Six-week (n = 6) 69.8 15.5
Weighted moving average
0.70, 0.20, 0.10 14.0 18.4
Exponential smoothing
α = 0.1 65.6 14.8
α = 0.2 41.0 15.3
20. Choosing a MethodChoosing a Method
Tracking SignalsTracking Signals Tracking signal =
RSFE
MAD
+2.0 —
+1.5 —
+1.0 —
+0.5 —
0 —
- 0.5 —
- 1.0 —
- 1.5 —
| | | | |
0 5 10 15 20 25
Observation number
Trackingsignal
Control limit
Control limit
21. Choosing a MethodChoosing a Method
Tracking SignalsTracking Signals Tracking signal =
RSFE
MAD
+2.0 —
+1.5 —
+1.0 —
+0.5 —
0 —
- 0.5 —
- 1.0 —
- 1.5 —
| | | | |
0 5 10 15 20 25
Observation number
Trackingsignal
Control limit
Control limit
Out of control
22. Choosing a MethodChoosing a Method
Tracking SignalsTracking Signals
C o n t r o l L im it
S p r e a d
( N u m b e r o f
M A D )
E q u iv a le n t
N u m b e r o f σ
( σ = 1 .2 5 M A D )
P e r c e n t a g e o f
A r e a w it h in
C o n t r o l L im it s
± 1 .0 ± 0 .8 0 5 7 .6 2
± 1 .5 ± 1 .2 0 7 6 .9 8
± 2 .0 ± 1 .6 0 8 9 .0 4
± 2 .5 ± 2 .0 0 9 5 .4 4
± 3 .0 ± 2 .4 0 9 8 .3 6
± 3 .5 ± 2 .8 0 9 9 .4 8
± 4 .0 ± 3 .2 0 9 9 .8 6
23. Problem 13-2: Historical demand for a product is:
Month Jan Feb Mar Apr May Jun
Demand 12 11 15 12 16 15
a. Using a weighted moving average with weights of 0.60,
0.30, and 0.10, find the July forecast.
b. Using a simple three-month moving average, find the July
forecast.
c. Using single exponential smoothing with α=0.20 and a June
forecast =13, find the July forecast.
d. Using simple regression analysis, calculate the regression
equation for the preceding demand data
e. Using regression equation in d, calculate the forecast in
July
24. Problem 13-15: In this problem, you are to test the validity of
your forecasting model. Here are the forecasts for a model
you have been using and the actual demands that
occurred:
Week 1 2 3 4 5 6
Forecast 800 850 950 950 1,000 975
Actual 900 1,000 1,050 900 900 1,100
Compute MAD and tracking signal. Then decide whether the
forecasting model you have been using is giving
reasonable results.
30. Time Series MethodsTime Series Methods
Simple Moving AveragesSimple Moving Averages
450 —
430 —
410 —
390 —
370 —
Patientarrivals
Week
| | | | | |
0 5 10 15 20 25 30
Actual patient
arrivals
3-week MA
forecast
31. Time Series MethodsTime Series Methods
Simple Moving AveragesSimple Moving Averages
Week
450 —
430 —
410 —
390 —
370 —
Patientarrivals
| | | | | |
0 5 10 15 20 25 30
Actual patient
arrivals
3-week MA
forecast
6-week MA
forecast
32. Taco Bell determined that
the demand for each 15-
minute interval
can be estimated from a 6-
week simple moving
average of sales.
The forecast was used to
determine the number of
employees needed.
33. Time Series MethodsTime Series Methods
Weighted Moving AverageWeighted Moving Average
450 —
430 —
410 —
390 —
370 —
Patientarrivals
Week
| | | | | |
0 5 10 15 20 25 30
Actual patient
arrivals
3-week MA
forecast Weighted Moving Average
Assigned weights
t-1 0.70
t-2 0.20
t-3 0.10
F4 =
34. Time Series MethodsTime Series Methods
Weighted Moving AverageWeighted Moving Average
450 —
430 —
410 —
390 —
370 —
Patientarrivals
Week
| | | | | |
0 5 10 15 20 25 30
Actual patient
arrivals
3-week MA
forecast Weighted Moving Average
Assigned weights
t-1 0.70
t-2 0.20
t-3 0.10
F5 =
40. Comparison of ExponentialComparison of Exponential
Smoothing and Simple MovingSmoothing and Simple Moving
AverageAverage
• Both Methods
– Are designed for stationary demand
– Require a single parameter
– Lag behind a trend, if one exists
– Have the same distribution of forecast error if
)1/(2 +=α N
41. Comparison of ExponentialComparison of Exponential
Smoothing and Simple MovingSmoothing and Simple Moving
AverageAverage
• Moving average uses only the last N periods
data, exponential smoothing uses all data
• Exponential smoothing uses less memory and
requires fewer steps of computation; store only
the most recent forecast!
42. Problem 13-20: Your manager is trying to determine what
forecasting method to use. Based upon the following historical
data, calculate the following forecast and specify what procedure
you would utilize:
Month 1 2 3 4 5 6 7 8 9 10 11 12
Actual demand 62 65 67 68 71 73 76 78 78 80 84 85
a. Calculate the three-month SMA forecast for periods 4-12
b. Calculate the weighted three-month MA using weights of 0.50,
0.30, and 0.20 for periods 4-12.
c. Calculate the single exponential smoothing forecast for periods 2-
12 using an initial forecast, F1=61 and α=0.30
d. Calculate the exponential smoothing with trend component
forecast for periods 2-12 using T1=1.8,F1=60,α=0.30,δ=0.30
e. Calculate MAD for the forecasts made by each technique in
periods 4-12. Which forecasting method do you prefer?
44. Turkeys have a long-term trend for increasing demand with a
seasonal pattern. Sales are highest during September to November
and sales are lowest during December and January.
57. Time Series MethodsTime Series Methods
Linear Regression AnalysisLinear Regression Analysis
| | | | | | | | | | | | | | |
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
80 —
70 —
60 —
50 —
40 —
30 —
Patientarrivals
Week
Yn = a + bXn
where
Xn = Weekn
58. Time Series MethodsTime Series Methods
Linear Regression AnalysisLinear Regression Analysis
| | | | | | | | | | | | | | |
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
80 —
70 —
60 —
50 —
40 —
30 —
Patientarrivals
Week
Yn = a + bXn
where
Xn = Weekn
59. Time Series MethodsTime Series Methods
Linear Regression AnalysisLinear Regression Analysis
• Standard error of estimate is computed as
follows:
2
)(
1
2
−
−
=
∑=
n
Yy
S
n
i
ii
yx
60. Time Series MethodsTime Series Methods
Linear Regression AnalysisLinear Regression Analysis
• An use of the standard error of estimate:
– Suppose that a manager forecasts that the demand
for a product is 500 units and Syx is 20. If the
manager wants to accept a stockout only 2% time,
how many additional units should be held in the
inventory?
61. • The method uses two smoothing constants α
and δ
Time Series MethodsTime Series Methods
Double Exponential SmoothingDouble Exponential Smoothing
ttt
tttt
ttt
TF
TFFT
AF
+=
−+−=
−+=
−−
−−
FIT
FIT
11
11
)1()(
)1(
δδ
αα
62. A Comparison of Methods
60
65
70
75
80
85
90
0 5 10 15
Months
Demand
Actual
3-Mo MA
3-Mo WMA
Exp Sm
Double Exp Sm
64. Quarter Year 1 Year 2 Year 3 Year 4
1 45 70 100 100
2 335 370 585 725
3 520 590 830 1160
4 100 170 285 215
Total 1000 1200 1800 2200
Average 250 300 450 550
Time Series MethodsTime Series Methods
Seasonal InfluencesSeasonal Influences
65. Quarter Year 1 Year 2 Year 3 Year 4
1 45 70 100 100
2 335 370 585 725
3 520 590 830 1160
4 100 170 285 215
Total 1000 1200 1800 2200
Average 250 300 450 550
Seasonal Index =
Actual Demand
Average Demand
Time Series MethodsTime Series Methods
Seasonal InfluencesSeasonal Influences
66. Quarter Year 1 Year 2 Year 3 Year 4
1 45 70 100 100
2 335 370 585 725
3 520 590 830 1160
4 100 170 285 215
Total 1000 1200 1800 2200
Average 250 300 450 550
Seasonal Index = =
Time Series MethodsTime Series Methods
Seasonal InfluencesSeasonal Influences
67. Quarter Year 1 Year 2 Year 3 Year 4
1 45/250 = 70 100 100
2 335 370 585 725
3 520 590 830 1160
4 100 170 285 215
Total 1000 1200 1800 2200
Average 250 300 450 550
Seasonal Index = =
Time Series MethodsTime Series Methods
Seasonal InfluencesSeasonal Influences
68. Quarter Year 1 Year 2 Year 3 Year 4
1 45/250 = 0.18 70/300 = 0.23 100/450 = 0.22 100/550 = 0.18
2 335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.32
3 520/250 = 2.08 590/300 = 1.97 830/450 = 1.84 1160/550 = 2.11
4 100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39
Time Series MethodsTime Series Methods
Seasonal InfluencesSeasonal Influences
69. Quarter Year 1 Year 2 Year 3 Year 4
1 45/250 = 0.18 70/300 = 0.23 100/450 = 0.22 100/550 = 0.18
2 335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.32
3 520/250 = 2.08 590/300 = 1.97 830/450 = 1.84 1160/550 = 2.11
4 100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39
Quarter Average Seasonal Index
1 (0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
2
3
4
Time Series MethodsTime Series Methods
Seasonal InfluencesSeasonal Influences
70. Quarter Average Seasonal Index Forecast
1 (0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
2
3
4
Projected Annual Demand = 2600
Average Quarterly Demand = 2600/4 = 650
Time Series MethodsTime Series Methods
Seasonal InfluencesSeasonal Influences
73. Time Series MethodsTime Series Methods
Seasonal Influences with TrendSeasonal Influences with Trend
Step 1: Determine seasonal factors
– Example: if the demands are quarterly, divide the average demand in
Quarter 1 by the average quarterly demand
Step 2: Deseasonalize the original data
– Divide the original data by the seasonal factors
Step 3: Develop a regression line on deaseasonalized data
– Find parameters a and b in Y=a+bX
– Where
– yi = deseasonalized data (not the original data)
– xi = time; 1, 2, 3, …, n
– n = Number of periods
74. Time Series MethodsTime Series Methods
Seasonal Influences with TrendSeasonal Influences with Trend
Step 4: Make projection using regression line
– For each i = n+1, n+2, …, compute yi by substituting a, b and xi in
the regression equation yi = a+bxi
Step 5: Reseasonalize projection using seasonal factors
– Multiply the projected values by the seasonal factors
75. Problem 13-21: Use regression analysis on deseasonalized
demand to forecast demand in summer 2006, given the
following historical demand data:
Year Season Actual Demand
2004 Spring 205
Summer 140
Fall 375
Winter 575
2005 Spring 475
Summer 275
Fall 685
Winter 965
