Download - Forecasting Feb 2
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3-1
Linear Trend Equation
Ft = Forecast for period t t = Specified number of time periods a = Value of Ft at t = 0 b = Slope of the line
Ft = a + bt
0 1 2 3 4 5 t
Ft
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Calculating a and b
b = n (ty) - t y
n t2 - ( t)2
a = y - b tn
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Linear Trend Equation Example
t yW e e k t 2 S a l e s t y
1 1 1 5 0 1 5 02 4 1 5 7 3 1 43 9 1 6 2 4 8 64 1 6 1 6 6 6 6 45 2 5 1 7 7 8 8 5
t = 1 5 t 2 = 5 5 y = 8 1 2 t y = 2 4 9 9( t ) 2 = 2 2 5
Forecast week 6 & 7 sales?
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Linear Trend Calculation
y = 143.5 + 6.3t
a = 812 - 6.3(15)5 =
b = 5 (2499) - 15(812)5(55) - 225
= 12495-12180275 -225
= 6.3
143.5
Week (t) Sales (y) Sales (y)6 143.5 + 6.3 (6) 181.37 143.5 + 6.3 (7) 187.6
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Practice book problem
Calculate Sept forecast using linear trend method
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t Y tY From Table 31 with n = 7, t = 28, t2 = 140
50.)28(28)140(7)132(28)542(7
)t(tnYttYnb 22 =
=
=
86.167
)28(50.132n
tbYa ===
1 19 19 2 18 36
3 15 45 4 20 80 5 18 90 6 22 132 7 20 140 28 132 542
Book Problem # 2- Solution
Y= a + bt !For Sept., t = 8, and Yt = 16.86 + .50(8) = 20.86 (000)
1) Linear Trend
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Exponential Smoothing with Trend Adjustment Data
Figure 4.3
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Time (month)
Prod
uct d
eman
d
35 30 25 20 15 10
5 0
Actual demand (At)
There is an upward trend pattern
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Exponential Smoothing with Trend Adjustment
When a trend is present, exponential smoothing must be modified
Forecast including (FITt) = trend
Exponentially Exponentially smoothed (Ft) + smoothed (Tt) forecast trend
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Exponential Smoothing with Trend Adjustment
Tt = (Ft - Ft - 1) + (1 - )Tt - 1
Step 1: Compute Ft !!Step 2: Compute Tt !!Step 3: Calculate the forecast FITt = Ft + Tt
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Exponential Smoothing with Trend Adjustment Example
Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
Table 4.1
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Exponential Smoothing with Trend Adjustment Example
Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
Table 4.1
F2 = A1 + (1 - )(F1 + T1) F2 = (.2)(12) + (1 - .2)(11 + 2) = 2.4 + 10.4 = 12.8 units
Step 1: Forecast for Month 2
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Exponential Smoothing with Trend Adjustment Example
Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 12.80 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
Table 4.1
T2 = (F2 - F1) + (1 - )T1 T2 = (.4)(12.8 - 11) + (1 - .4)(2) = .72 + 1.2 = 1.92 units
Step 2: Trend for Month 2
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Exponential Smoothing with Trend Adjustment Example
Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 12.80 1.92 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
Table 4.1
FIT2 = F2 + T2 FIT2 = 12.8 + 1.92 = 14.72 units
Step 3: Calculate FIT for Month 2
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Exponential Smoothing with Trend Adjustment Example
Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 12.80 1.92 14.72 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
Table 4.1
! 15.18 2.10 17.28 17.82 2.32 20.14 19.91 2.23 22.14 22.51 2.38 24.89 24.11 2.07 26.18 27.14 2.45 29.59 29.28 2.32 31.60 32.48 2.68 35.16
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Exponential Smoothing with Trend Adjustment Example
Figure 4.3
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Time (month)
Prod
uct d
eman
d
35 30 25 20 15 10
5 0
Actual demand (At)
Forecast including trend (FITt) with = .2 and = .4
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Seasonal Variations In Data
The multiplicative seasonal model can adjust trend data for seasonal variations in demand
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Seasonal Index Example
140 130 120 110 100
90 80 70
| | | | | | | | | | | | J F M A M J J A S O N D
Time
Dem
and
2010 Forecast 2009 Demand 2008 Demand 2007 Demand
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Seasonal Variations In Data
1. Find average historical demand for each season 2. Compute the average demand over all seasons 3. Compute a seasonal index for each season 4. Estimate next years total demand 5. Divide this estimate of total demand by the
number of seasons, then multiply it by the seasonal index for that season
Steps in the process:
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Seasonal Index Example
Jan 80 85 105 90 94 Feb 70 85 85 80 94 Mar 80 93 82 85 94 Apr 90 95 115 100 94 May 113 125 131 123 94 Jun 110 115 120 115 94 Jul 100 102 113 105 94 Aug 88 102 110 100 94 Sept 85 90 95 90 94 Oct 77 78 85 80 94 Nov 75 72 83 80 94 Dec 82 78 80 80 94
Demand Average Average Seasonal Month 2007 2008 2009 2007-2009 Monthly Index
TOTAL 1050 1120 1204 AVE - 93.72
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Seasonal Index Example
Jan 80 85 105 90 94 Feb 70 85 85 80 94 Mar 80 93 82 85 94 Apr 90 95 115 100 94 May 113 125 131 123 94 Jun 110 115 120 115 94 Jul 100 102 113 105 94 Aug 88 102 110 100 94 Sept 85 90 95 90 94 Oct 77 78 85 80 94 Nov 75 72 83 80 94 Dec 82 78 80 80 94
Demand Average Average Seasonal Month 2007 2008 2009 2007-2009 Monthly Index
0.957
Seasonal index = Average 2007-2009 monthly demand Average monthly demand
= 90/94 = .957
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Seasonal Index Example
Jan 80 85 105 90 94 0.957 Feb 70 85 85 80 94 0.851 Mar 80 93 82 85 94 0.904 Apr 90 95 115 100 94 1.064 May 113 125 131 123 94 1.309 Jun 110 115 120 115 94 1.223 Jul 100 102 113 105 94 1.117 Aug 88 102 110 100 94 1.064 Sept 85 90 95 90 94 0.957 Oct 77 78 85 80 94 0.851 Nov 75 72 83 80 94 0.851 Dec 82 78 80 80 94 0.851
Demand Average Average Seasonal Month 2007 2008 2009 2007-2009 Monthly Index
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Seasonal Index Example
Jan 80 85 105 90 94 0.957 Feb 70 85 85 80 94 0.851 Mar 80 93 82 85 94 0.904 Apr 90 95 115 100 94 1.064 May 113 125 131 123 94 1.309 Jun 110 115 120 115 94 1.223 Jul 100 102 113 105 94 1.117 Aug 88 102 110 100 94 1.064 Sept 85 90 95 90 94 0.957 Oct 77 78 85 80 94 0.851 Nov 75 72 83 80 94 0.851 Dec 82 78 80 80 94 0.851
Demand Average Average Seasonal Month 2007 2008 2009 2007-2009 Monthly Index
Expected annual demand = 1,200
Jan x .957 = 96 1,200 12
Feb x .851 = 85 1,200 12
Forecast for 2010
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Seasonal Index Example
140 130 120 110 100
90 80 70
| | | | | | | | | | | | J F M A M J J A S O N D
Time
Dem
and
2010 Forecast 2009 Demand 2008 Demand 2007 Demand
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24 2011 Pearson Education, Inc.
Associative ForecastingUsed when changes in one or more independent variables can be used to predict the changes in the dependent variable. Some examples !-Sales of mountain bikes may be related to the percentage of the young population living in that area. -Ice cream sales can be related to temperature - Increase in fuel cost leads to price increases in
products and services !
!!!Most common technique is linear regression analysis same technique just as we did in the time series example
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Associative ForecastingForecasting an outcome based on predictor variables using the least squares technique
y = a + bx^
where y = computed value of the variable to be predicted (dependent variable)
a = y-axis intercept b = slope of the regression line x = the independent variable though to
predict the value of the dependent variable
^
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Associative Forecasting Example
Sales Area Payroll ($ millions), y ($ billions), x 2.0 1 3.0 3 2.5 4 2.0 2 2.0 1 3.5 7
4.0 3.0 2.0 1.0
| | | | | | | 0 1 2 3 4 5 6 7
Sale
s
Area payroll
Forecast sales amount for $ 6B???
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Associative Forecasting Example
Sales, y Payroll, x x2 xy 2.0 1 1 2.0 3.0 3 9 9.0 2.5 4 16 10.0 2.0 2 4 4.0 2.0 1 1 2.0 3.5 7 49 24.5 y = 15.0 x = 18 x2 = 80 xy = 51.5
x = x/6 = 18/6 = 3
y = y/6 = 15/6 = 2.5
b = = = .25xy - nxy x2 - nx2
51.5 - (6)(3)(2.5) 80 - (6)(32)
a = y - bx = 2.5 - (.25)(3) = 1.75
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Associative Forecasting Example
y = 1.75 + .25x^ Sales = 1.75 + .25(payroll)
If payroll next year is estimated to be $6 billion, then:
Sales = 1.75 + .25(6) Sales = $3,250,000
4.0 3.0 2.0 1.0
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Nod
els
sal
es
Area payroll
3.25
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Forecast Accuracy Error: difference between actual value and
predicted value Mean Absolute Deviation (MAD)
Average absolute error
Mean Squared Error (MSE) Average of squared error
Mean Absolute Percent Error (MAPE) Average absolute percent error
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MAD, MSE, and MAPE
MAD =Actual forecast
n
MSE = Actual forecast)
-1
2
n
(
MAPE = Actualforecast
n
/ Actual*100)(
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MAD, MSE, and MAPE
MAD Easy to compute Weights errors linearly
MSE Squares error More weight to large errors
MAPE Puts errors in perspective
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Example 10
Period Actual Forecast (A-F) |A-F| (A-F)^2 (|A-F|/Actual)*1001 217 215 2 2 4 0.922 213 216 -3 3 9 1.413 216 215 1 1 1 0.464 210 214 -4 4 16 1.905 213 211 2 2 4 0.946 219 214 5 5 25 2.287 216 217 -1 1 1 0.468 212 216 -4 4 16 1.89
-2 22 76 10.26
MAD= 2.75 (22 / 8 )MSE= 10.86 ( 76 / 7 )MAPE= 1.28 (10.26 / 8)
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Tracking Signal
Tracking signal = (Actual-forecast)MAD
Tracking signal Ratio of cumulative error to MAD
Bias: Persistent tendency for forecasts to be greater or less than actual values.
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Choosing a Forecasting Technique
No single technique works in every situation Two most important factors
Cost Accuracy
Other factors include the availability of: Historical data Computers Time needed to gather and analyze the data Forecast horizon