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CHAPTER 13FORECASTING
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: Holts Method
A Method for Seasonality and Trend
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Forecasting
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Decisions Based on Forecasts
Production
Aggregate planning,inventory control,scheduling
MarketingNew product
introduction, sales-force allocation,
promotions Finance
Plant/equipmentinvestment, budgetary
planning
Personnel
Workforce planning,hiring, layoff
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Characteristics of Forecasts
Forecasts are always
wrong; so consider
both expected valueand a measure of
forecast error
Long-term forecasts
are less accurate thanshort-term forecasts
Aggregate forecasts
are more accurate than
disaggregate forecasts
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Forecasting
Components of demand
Evaluation of forecasts
Time series: stationary series Time series: trend
Linear regression
Double exponential smoothing Time series: seasonality
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Components of Demand
Average demand Trend
Gradual shift in average demand
Seasonal patternPeriodic oscillation in demand which repeats
Cycle
Similar to seasonal patterns, length andmagnitude of the cycle may vary
Random movements
Auto-correlation
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Qantity
Time
(a) Average: Data cluster about a horizontal line.
Components of Demand
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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
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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
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Components of Demand
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Components of Demand
Quantity
| | | | | |1 2 3 4 5 6
Years
(c) Cyclical movements: Gradual changes over
extended periods of time.
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Components of Demand
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Demand
Time
Trend
Random
movement
Dem
and
Time
Trend with
seasonal pattern
Components of Demand
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Snow Skiing
Seasonal
Long term growth trend
Demand for skiing products increased
sharply after the Nagano Olympics
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|Et|nEt
2
n
RSFE = EtMAD =
MSE =MAPE =
= MSE[|Et| (100)]/A t
n
Measures of Forecast Error
Et= A t- Ft
Choosing a Method
Forecast Er ror
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AbsoluteError Absolute Percent
Month, Demand, Forecast, Error, Squared, Error, Error,t A
t F
t E
tEt
2 |Et| (|E
t|/A
t)(100)
1 200 2252 240 2203 300 2854 270 2905 230 2506 260 240
7 210 2508 275 240
-
Total
Choosing a Method
Forecast Er ror
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MSE = =
Measures of Error
MAD = =
MAPE = =
RSFE =
Choosing a Method
Forecast Er ror
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Choosing a Method
Forecast Er ror
Running Sum Mean Absoluteof Forecast Errors Deviation
Method (RSFE - bias) (MAD)Simple mov ing averageThree-week (n= 3) 23.1 17.1Six-week (n= 6) 69.8 15.5Weighted mov ing average
0.70, 0.20, 0.10 14.0 18.4Exponent ial smooth ing = 0.1 65.6 14.8 = 0.2 41.0 15.3
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Choosing a Method
Tracking 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
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Choosing a Method
Tracking 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
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Choosing a Method
Tracking Signals
Control LimitSpread
(Number ofMAD)
Equivalent
Number of
(=1.25 MAD)
Percentage ofArea within
Control Limits
1.0 0.80 57.62
1.5 1.20 76.98
.0 1.60 89.04
2.5 2.00 95.44
3.0 2.40 98.36
3.5 2.80 99.48
4.0 3.20 99.86
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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 =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
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Problem 13-15: In this problem, you are to test the validity ofyour 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.
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Methods for Stationary Series
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Time Series Methods
Simple Moving Averages
Week
450
430
410
390
370P
atientarrivals
| | | | | |0 5 10 15 20 25 30
Actual patientarrivals
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Time Series Methods
Simple Moving Averages
Actual patientarrivals
450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
PatientWeek Arrivals
1 400
2 3803 411
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Time Series Methods
Simple Moving Averages
Actual patientarrivals
450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
PatientWeek Arrivals
1 400
2 3803 411
F4 =
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Time Series Methods
Simple Moving Averages
Actual patientarrivals
450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
PatientWeek Arrivals
2 380
3 4114 415
F5 =
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Time Series Methods
Simple Moving Averages450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
Actual patientarrivals
3-week MAforecast
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Time Series Methods
Simple Moving Averages
Week
450
430
410
390
370P
atientarrivals
| | | | | |0 5 10 15 20 25 30
Actual patientarrivals
3-week MAforecast
6-week MAforecast
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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.
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Time Series Methods
Weighted Moving Average450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
Actual patientarrivals
3-week MAforecast Weighted Moving Average
Assigned weights
t-1 0.70t-2 0.20t-3 0.10
F4 =
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Time Series Methods
Weighted Moving Average450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
Actual patientarrivals
3-week MAforecast Weighted Moving Average
Assigned weights
t-1 0.70t-2 0.20t-3 0.10
F5 =
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Time Series Methods
Exponential Smoothing450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
Exponential Smoothing = 0.10
Ft= A t -1+ (1 - )Ft - 1
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Time Series Methods
Exponential Smoothing450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
Exponential Smoothing = 0.10
Ft= A t -1+ (1 - )Ft - 1F3= (400 + 380)/2=390A 3 = 411
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Time Series Methods
Exponential Smoothing450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
F4 =
Exponential Smoothing = 0.10
Ft= A t -1+ (1 - )Ft - 1F3= (400 + 380)/2=390A 3 = 411
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Time Series Methods
Exponential Smoothing
Week
450
430
410
390
370P
atientarrivals
| | | | | |0 5 10 15 20 25 30
F4 =A 4 = 415
Exponential Smoothing = 0.10
Ft= A t+ (1 - )Ft - 1
F5 =
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Time Series Methods
Exponential Smoothing450
430
410
390
370P
atientarrivals
Week
| | | | | |0 5 10 15 20 25 30
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Compar ison of Exponential
Smoothing and Simple Moving
Average 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 N
C i f E ti l
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Compar ison of Exponential
Smoothing and Simple Moving
Average Moving average uses only the lastNperiods
data, exponential smoothing uses all data
Exponential smoothing uses less memory and
requires fewer steps of computation; store only
the most recent forecast!
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Problem 13-20: Your manager is trying to determine whatforecasting 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 =0.30
d. Calculate the exponential smoothing with trend component
forecast for periods 2-12 using T1=1.8,F1=60,=0.30,=0.30
e. Calculate MAD for the forecasts made by each technique in
periods 4-12. Which forecasting method do you prefer?
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Trend-Based Methods
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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.
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L inear Regression
Dep
endentvariable
Independent variable
X
Y
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L inear Regression
Dep
endentvariable
Independent variable
X
Y Regressionequation:Y= a+ bX
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L inear Regression
Dep
endentvariable
Independent variable
X
Y
ActualvalueofY
Value ofXusedto estimate Y
Regressionequation:Y= a+ bX
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L inear Regression
Dep
endentvariable
Independent variable
X
Y
ActualvalueofY
Estimate ofY fromregressionequation
Value ofXusedto estimate Y
Regressionequation:Y= a+ bX
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L inear Regression
Dep
endentvariable
Independent variable
X
Y
ActualvalueofY
Estimate ofY fromregressionequation
Value ofXusedto estimate Y
Deviation,or error
{
Regressionequation:Y= a+ bX
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L inear Regression
Sales AdvertisingMonth (000 units) (000 $)
1 264 2.52 116 1.3
3 165 1.44 101 1.05 209 2.0
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L inear Regression
Sales, y Advertising, xMonth (000 units) (000 $)
1 264 2.52 116 1.3
3 165 1.44 101 1.05 209 2.0
a= y- bx b=xy- nxyx2 - n(x)2
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a= y- bx b=xy - nxyx2 - nx2
Sales, y Advertising, xMonth (000 units) (000 $) xy x2
1 264 2.52 116 1.3
3 165 1.44 101 1.05 209 2.0
Totaly= x=
L inear Regression
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300
250
200
150
100
50
b= 109.229
Y=
Sales
(000s)
| | | |1.0 1.5 2.0 2.5
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L inear Regression
Sales, y Advertising, xMonth (000 units) (000 $) xy x2 y2
1 264 2.5 660.0 6.252 116 1.3 150.8 1.69
3 165 1.4 231.0 1.964 101 1.0 101.0 1.005 209 2.0 418.0 4.00
Total 855 8.2 1560.8 14.90y= 171 x= 1.64
nxy- xy[nx2 -(x) 2][ny2 - (y) 2]r=
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L inear Regression
Sales, y Advertising, xMonth (000 units) (000 $) xy x2 y2
1 264 2.5 660.0 6.25 69,6962 116 1.3 150.8 1.69 13,456
3 165 1.4 231.0 1.96 27,2254 101 1.0 101.0 1.00 10,2015 209 2.0 418.0 4.00 43,681
Total 855 8.2 1560.8 14.90 164,259y= 171 x= 1.64
r= 0.98 r2 = 0.96
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L inear Regression
Forecast for Month 6:
Advertising expenditure = $1750
Y=
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Time Series MethodsL inear 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
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Time Series MethodsL inear 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
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Time Series MethodsL inear Regression Analysis
Standard error of estimate is computed as
follows:
2
)(1
2
n
Yy
S
n
i
ii
yx
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Time Series MethodsL inear 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 themanager wants to accept a stockout only 2% time,
how many additional units should be held in the
inventory?
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The method uses two smoothing constants
and
Time Series MethodsDouble Exponential Smoothing
ttt
tttt
ttt
TF
TFFT
AF
FIT
FIT
11
11
)1()(
)1(
A Comparison of Methods
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A Comparison of Methods
6065
70
75
80
8590
0 5 10 15
Months
Dema
nd
Actual
3-Mo MA
3-Mo WMA
Exp Sm
Double Exp S
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Methods for Seasonal Series
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Quarter Year 1 Year 2 Year 3 Year 4
1 45 70 100 1002 335 370 585 725
3 520 590 830 11604 100 170 285 215
Total 1000 1200 1800 2200Average 250 300 450 550
Time Series MethodsSeasonal I nf luences
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Quarter Year 1 Year 2 Year 3 Year 4
1 45 70 100 1002 335 370 585 725
3 520 590 830 11604 100 170 285 215
Total 1000 1200 1800 2200Average 250 300 450 550
Seasonal Index =Actual Demand
Average Demand
Time Series MethodsSeasonal I nf luences
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Quarter Year 1 Year 2 Year 3 Year 4
1 45 70 100 1002 335 370 585 725
3 520 590 830 11604 100 170 285 215
Total 1000 1200 1800 2200Average 250 300 450 550
Seasonal Index = =
Time Series MethodsSeasonal I nf luences
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Quarter Year 1 Year 2 Year 3 Year 4
1 45/250 = 70 100 1002 335 370 585 725
3 520 590 830 11604 100 170 285 215
Total 1000 1200 1800 2200Average 250 300 450 550
Seasonal Index = =
Time Series MethodsSeasonal I nf luences
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Quarter Year 1 Year 2 Year 3 Year 4
1 45/250 = 0.18 70/300 = 0.23 100/450 = 0.22 100/550 = 0.182 335/250 = 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 = 2.114 100/250 = 0.40 170/300 = 0.57 285/450 = 0.63 215/550 = 0.39
Time Series MethodsSeasonal I nf luences
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Quarter Year 1 Year 2 Year 3 Year 4
1 45/250 = 0.18 70/300 = 0.23 100/450 = 0.22 100/550 = 0.182 335/250 = 1.34 370/300 = 1.23 585/450 = 1.30 725/550 = 1.323 520/250 = 2.08 590/300 = 1.97 830/450 = 1.84 1160/550 = 2.114 100/250 = 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
234
Time Series MethodsSeasonal I nf luences
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Quarter Average Seasonal Index Forecast
1 (0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
234
Projected Annual Demand = 2600Average Quarterly Demand = 2600/4 = 650
Time Series MethodsSeasonal I nf luences
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Seasonal I nf luences
Period
Demand
(a) Multiplicative influence
| | | | | | | | | | | | | | | |0 2 4 5 8 10 12 14 16
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Seasonal I nf luences
Period
| | | | | | | | | | | | | | | |0 2 4 5 8 10 12 14 16
Demand
(b) Additive influence
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Time Series MethodsSeasonal I nf luences 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
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Time Series MethodsSeasonal I nf luences with Trend
Step 4: Make projection using regression line
For each i = n+1, n+2, , computeyi by substituting a, b and xi in
the regression equationyi= a+bxi
Step 5: Reseasonalize projection using seasonal factors Multiply the projected values by the seasonal factors
Problem 13-21: Use regression analysis on deseasonalized
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Problem 13-21: Use regression analysis on deseasonalizeddemand to forecast demand in summer 2006, given the
following historical demand data:
Year Season Actual Demand2004 Spring 205
Summer 140
Fall 375
Winter 575
2005 Spring 475
Summer 275
Fall 685Winter 965
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Reading and Exercises
Chapter 13 pp. 518-539
Problems 1, 7, 13, 14,16