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Transcript of Forecasting The Process of Predicting the Future presented by Your Local Engineering Management...
ForecastingThe Process of Predicting the Future
presented by
Your Local Engineering Management Office (LEMO)
My concern is the futuresince I plan to spend the restof my life there.
"Those who have knowledge, don't predict. Those who predict, don't have knowledge. " --Lao Tzu, 6th Century BC Chinese Poet Wise words from a long time ago.
Applications• sales• strategic planning• financial investments• inventory levels• production levels• work force sizing• energy requirements• economic planning
– unemployment– housing starts– inflation rates
Typical Forecasts
• Product Sales• Replacement part demands• Lead-times• Machine break rates• Expenditures• Market share• Unit costs• Labor rates
• Forecasts are used in manufacturing engineering in:– Inventory models– Machine loading– Production planning models– MRP systems – Manufacturing simulations
• A good forecast model is– accurate– computationally efficient– robust (to changes in patterns)
Laws of Forecasting• First Law: Forecasts are always wrong!• Second Law: Forecasts always change!• Third Law: The further into the future, the less
reliable the forecast!• Fourth Law: A good forecast includes a measure of
error!• Fifth Law: Aggregate forecasts are more accurate!• Sixth Law: Forecasts should not replace known
values!
Accuracy of forecasts depends on
• Accuracy of data
• sample size
• stability of the random process– variability– stationary vs non-stationary process
• length of forecasting period
• method used
• model selected
Forecasting Methods• Qualitative (subjective)
– historical analogy– market research
• customer surveys
– Expert opinion– Delphi technique– sales force composites
• Quantitative Models– regression analysis (causal models)– time-series models
• moving averages• exponential smoothing• Box-Jenkins• auto-regressive
Time Horizon
• Short Term– Sales, shift schedules, material and part requirements,
equipment failures– Days and weeks
• Intermediate– Product sales, labor requirements, resources– Weeks and months
• Long-term– Capacity requirements, long-term sales patterns, growth
trends, resource and labor costs– Months and years
Time SeriesTime Series: random variable indexed on timee.g. Dt = demands during month t
D1, D2 , …, Dt,… form a time series
Basic Premise: Can predict the future from thepast - the underlying process will continue as it has in the (recent) past.
Forecast:
where ai is the weight placed on the ith observation
1
n
t i t ii
F a D
Elements of Time Series Data• Trend (Gt)
– Constant (stationary) (Gt = b)
– linear (constant) (Gt = bt)
– quadratic (accelerated) (Gt = bt2)
– exponential (growth) (Gt = bt)
• Seasonal (St)
• Cyclical (Ct)
• Randomness (Noise) (et)
– no recognizable pattern
Trends
time
population
Seasonal Snow blower sales
spring summer fall winter spring summer fall winter
trend present
Cycles Unemployment Rate
1990 1991 1992 1993 1994 1995 1996
Cycles• long swings away from trend due to factors other than
seasonality– generally occurs over a number of years
• difficult to model– not as stable – rarely repeats at fixed intervals– amplitude varies– need several years of data (complete cycles) to distinguish from
trends
• causes of cycles include– psychological forces (fashions, music, food)– population demographics (college enrollments)– institutional (public policy, business practices, tax policies)– replacement cycles (technology changes, obsolescence)– education
The Road Ahead
• Stationary (constant) process– moving averages– exponential smoothing
• Trend only process– linear regression– Holt’s method (double exponential smoothing)
• Seasonal process– seasonal factors (stationary process)– Winter’s method (trend process)
but first a detour…
Evaluating ForecastsForecast Error
et = Ft - Dt
1
1
2
1
1
1
1
1100
n
t tt
n
tt
n
tt
nt
t t
E e
MAD en
MSE en
eMAPE x
n D
errors
Bad forecast
Wrong again
(measures bias)
Forecast Error Example
Time period (i)
Forecast (F) Actual (D) ei = F - A MSE MAD MAPE
1 120 122 -2.0 4 2 0.0163932 156 145 11.0 121 11 0.0758623 147 137 10.0 100 10 0.0729934 132 125 7.0 49 7 0.0565 122 119 3.0 9 3 0.025216 184 178 6.0 36 6 0.0337087 171 165 6.0 36 6 0.0363648 168 177 -9.0 81 9 0.0508479 145 153 -8.0 64 8 0.052288
10 136 148 -12.0 144 12 0.08108111 145 135 10.0 100 10 0.07407412 179 164 15.0 225 15 0.091463
sum 37.0 969.0 99.0 0.7average 3.1 80.8 8.3 5.6%
e = F - D
Forecast Model
Modelhistorical data forecast
error
Observedvalue
Multiplicative Model
Ft = Trend x seasonal x cyclical x irregular= aGt St Ct et
Additive Model
Ft = Trend + seasonal + cyclical + irregular = a + Gt + St + Ct + et
Notation
Given D1 , D2 , … , Dt demands have been observed and assuming an additive linear model:
Ft,t+ = forecast made at time tfor period t+ let Ft = Ft-1,t
,0
t t n t nn
F a D
D1, D2 , … , Dt are observed values of demand duringperiods 1, 2, …, t.
At time t, we have observed Dt , Dt-1 , …
2. Average forecast:
1
1
1
t
ii
t
DF
t
3. Moving averages:
Ft = (Dt-1+ Dt-2 + … + Dt-N ) / N and Ft-1,t+ = Ft
1. Last data point (LDP) Forecast: Ft = Dt-1
1
11
(1/ ) (1/ )
(1/ )
t t
t i t i t Ni t N i t N
t t t N
F N D N D D D
F N D D
Potential Forecasts
Time Series - Moving Averagesno trends/no cycles/no seasonal effects
Model: Dt = + t
Underlying constant of the process
E[t] = 0 and Var[t] = 2
Let’s hear it for the moving
average model!
Moving Averages Lag behind Trends
Period Demand MA(3) MA(6)1 22 43 64 8 45 10 66 12 87 14 10 78 16 12 99 18 14 11
10 20 16 1311 22 18 1512 24 20 17
(Simple) Exponential Smoothing
1 1
1 1 1 1 1
(1 ) ; 0 1t t t
t t t t t
F D F
F F D F
Why Ft is just the weighted sum of the current observation
and the previous estimate.
previousforecast
error frompreviousforecast
More (Simple) Exponential Smoothing
1 1
1 2 2
21 2 2
(1 )
(1 ) (1 )
(1 ) (1 )
t t t
t t t
t t t
F D F
D D F
D D F
10
(1 )it t i
i
F D
continuing:
0
(1 ) 11 (1 )
it
i
F
Note that the weights sum to one:
Moving Averages versus Exponential Smoothing
Average age of data:
moving average = (1/N) (1 + 2 + 3 + … + N) = (1/N) N (N+1)/2 = (N + 1)/2
exp smooth = 1
1
(1 ) 1/i
i
i
equating ages:1 1
22
N
N
Exampleif N = 10, then = .18182 if = .1, then N = 19
A Little Math Trick
1
1
1
1 1 1
20
(1 ) 1/
(1 )(1 ) (1 )
1 1 1(1 ) 1 1 1
1 (1 )
i
i
ii i
i i i
i
i
i
d di
d d
d d d
d d d
More Moving Averages versus Exponential Smoothing
age of data1 2 3 4 5 6
exponential smoothing with = .3
moving average with n = 6
weightplaced onith value
.3( ) .3 if i e
Considerations in the selection of the smoothing constant
If is small – response to change will be slowIf is large – response to change will be fast Normally .1 < < .3 Average age of the data:
Set: = 2/(n+1) to correspond to n-period moving average Minimize forecast error (MAD, MSE, RMSE, etc.)
1
1
11
. . .1, 10
.3, 3.33
i
k
i
e g A
A
Remember to go to Excel!
Moving Averages versus Exponential Smoothing – A Comparison
Similarities
• Assume stationary process (with adjustment of shifts in the mean)
• Single parameter model (N and α)
• Lags behind trend data
• Similar levels of accuracy
Differences
• Smoothing uses all past data, MA uses only the last N values
• Need to save N data points for MA
• MA weighs each observation by N while smoothing weights the Nth observation by (1-)N-1
Trend Based Methods
The Journey Continues…
"Wall Street indices predicted nine out of the last five recessions ! " --Paul A. Samuelson in Newsweek, Science and Stocks, 19 Sep. 1966.
Trend Data
x
xx
x xx
Model: Dt = + Bt + t
E[t] = 0 and Var[t] = 2
time
demands
Regression Analysis
yt = A + Bt + et with t = 1, 2, …, n
Ft+k = a +b (t + k)
where a & b are Least Square estimates
Least-Squares Estimates
1 1
2 2 2
1
( 1) / 2
( 1)
2
( 1)(2 1) ( 1)
6 4
where /
xy
xx
n n
xy i ii i
xx
n
ii
Sb
S
a D b n
n nS n iD D
n n n n nS
D D n
and Di is the demand fortime period i, i = 1,2, …, n
The Necessary ExampleQuarter Index Engine
Demands (failures)
1/2007 1 20
2/2007 2 25
3/2007 3 22
4/2007 4 28
1/2008 5 30
2/2008 6 32
3/2008 7 33
4/2008 8 31
GE's F110 engine family provides the most reliable power for the F-16C/D fighter aircraft. With a reputation for stall-free operation, the F110 continues to be the choice for F-16 operators and has been selected for twin-engine F-15 application.
Go to Excel…
A Second ExampleQuarter
Machine Failures
1 1982 2503 3214 4565 534
forecast6 615.27 7038 790.8
Machine Failuresy = 87.8x + 88.4
R2 = 0.9755
0
100
200
300
400
500
600
0 1 2 3 4 5 6
quarter
e.g. F6 = 88.4 + 87.8 (6) = 615.2
Trend Data - continued Double Exponential Smoothing (Holt’s method)
1 1
1 1
,
(1 )( )
( ) (1 )
0 1 ; 0 1
t t t t
t t t t
t t t t
S D S G
G S S G
F S G
Model: Dt = + Bt + t
Value of theseries (intercept)
Value of thetrend (slope)
A Holt’s Method Example
Set α = = .1; S0 = 200; G0 = 10
S1 =(.1)(200) + (.9)(200+10)=209.0
G1 = (.1)(209 - 200) + (.9)(10) = 9.9
S2 =(.1)(250) + (.9)(209+9.9)= 222.0
G2 = (.1)(222 - 209) + (.9)(9.9) = 10.2
S3 =(.1)(175) + (.9)(222+10.2)=226.5
G3 = (.1)(226.5 - 222) + (.9)(10.2) = 9.6
F3,4 = 226.5 + 9.6 = 236.1 and F3,5 = 226.5 +(2) 9.6 = 245.7
Dt
200250175186225285305190
Forecasting with Seasonal Effects
Experience the ups and downs of the four seasons
"I always avoid prophesying beforehandbecause it is much better to prophesy after the event has already taken place. "
--Winston Churchill
Seasonal Data with no trend
Model: Dt = ck + t , 1 k N
where = average (annual) demandck = seasonal factor for period k
t = random component
N = number of periods in a season
The road ahead is no longer straight
Seasonal Factors for a Series with no Trend
1. Compute the sample mean of the data
2. Divide each observation by the sample mean
3. Average the factors for like periods within each season.
4. Result are N seasonal factors
I can do that.
average for periodSeasonal Factor
overall average
Seasonal Indices – a real nice example
week 1 week 2 week 3 week 4Mon 16.2 17.3 14.6 16.1Tues 12.2 11.5 13.1 11.8Weds 14.2 15 13 12.9Thurs 17.3 17.6 16.9 16.6Fri 22.5 23.5 21.9 24.3
week 1 week 2 week 3 week 4Mon 0.986 1.053 0.889 0.980Tues 0.743 0.700 0.798 0.718Weds 0.865 0.913 0.791 0.785Thurs 1.053 1.072 1.029 1.011Fri 1.370 1.431 1.333 1.479
Divide by the Mean = 16.425
averageMon 0.977Tues 0.740Weds 0.839Thurs 1.041Fri 1.403
Example 2.6Cars on a toll bridgeData is in 1,000
forecasts16.0512.1513.78
17.123.05
x 16.425
Another example
Qtr 2001 2002 2003 avg index
1 124.5 157.4 144.1 142.0 0.691
2 181.0 192.3 178.4 183.9 0.896
3 287.1 281.8 251.5 273.4 1.332
4 240.1 217.1 208.6 221.9 1.081
205.3 1
Data are quarterly sales in 1,000 of gallons and are normally distributed with a mean of 200 and a std. dev. of 20
De-seasonalizing Our ExampleQtr Raw De-seasonnal
2001 1 124.5 180.0
2 181.0 202.1
3 287.1 215.5
4 240.1 222.1
2002 1 157.4 227.6
2 192.3 214.7
3 281.8 211.6
4 217.1 200.8
2003 1 144.1 208.4
2 178.4 199.1
3 251.5 188.8
4 208.6 193.0
Qtr index
1 0.691
2 0.896
3 1.332
4 1.081
for example:124.5/.691 = 180.0181.0/.896 = 202.1
Applying a MA Forecasting ModelMonth Sales 1-mo 2-mo 3-mo avg 4-mo 5-mo 6-mo avg 7-mo avg
2001/1 180
2 202.1 180.03 215.5 202.1 191.14 222.1 215.5 208.8 199.2
2002/1 227.6 222.1 218.8 213.2 204.92 214.7 227.6 224.9 221.7 216.8 209.53 211.6 214.7 221.2 221.5 220.0 216.4 210.34 200.8 211.6 213.2 218.0 219.0 218.3 215.6 210.5
2003/1 208.4 200.8 206.2 209.0 213.7 215.4 215.4 213.52 199.1 208.4 204.6 206.9 208.9 212.6 214.2 214.43 188.8 199.1 203.8 202.8 205.0 206.9 210.4 212.04 193 188.8 194.0 198.8 199.3 201.7 203.9 207.3
forecast 193.0 190.9 193.6 197.3 198.0 200.3 202.3MAD = 9.62 10.22 11.06 11.11 10.70 11.77 13.52RMSE = 10.9 12.1 12.7 13.0 11.9 13.4 14.8
Qtr sales 1-qtr 2-qtr 3-qtr 4-qtr 5-qtr 6-qtr 7-qtr
The Forecast for the next yearforecast 1-mo 2-mo 3-mo avg 4-mo 5-mo 6-mo avg 7-mo avg
2004 index 193.0 190.9 193.6 197.3 198.0 200.3 202.31 0.691 133.4 131.9 133.8 136.4 136.8 138.4 139.82 0.896 172.9 171.0 173.5 176.8 177.4 179.5 181.33 1.332 257.1 254.3 257.9 262.8 263.8 266.8 269.54 1.081 208.6 206.4 209.3 213.3 214.1 216.5 218.7
1-qtr 2-qtr 3-qtr 4-qtr 5-qtr 6-qtr 7-qtr
Seasonal Data with Trend
Model: Dt = ( + Gt)ct + t
where = the base constant at t = 0G = slope of trend componentct = seasonal factor for period tt = random component
The road ahead not only curves but also climbs.
Another better way when trend is present…
Could you review with us the 8 easy steps to applying the moving average method?
The 8 easy steps…
1. Obtain moving averages where N = length of season
2. Average and center adjacent values
3. Divide results of #2. into Dt
4. For each season, compute the “average” (i.e. the mean or median)
5. Adjust so sum is N by multiply each average by N / Total
6. Deseasonalize series by dividing each Dt by corresponding seasonal index
7. Forecast deseasonalized series using appropriate model
8. Apply corresponding seasonal indices to “reseasonalize” the series
He is right. There are 8 steps.
Example of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
Steps 1and 2 of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
More Steps 1 and 2 of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
Step 3 of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
26 / 18.50 = 1.405
17 / 19.13 = .889
Step 4 of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
(.600 + .516 + .478)/3 = .531
Another Step 4 of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
(1.082+ 1.143 + 1.143)/3 = 1.123
Step 5 of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
(4/ 3.995) .531 = .5321 1 1 1
if ; thenn n
i i
nx T x n
T
More Step 5 of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
(4/ 3.995) 1.123 = 1.124
Step 6 of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
12 / .532 = 22.6
More Step 6 of the first 6 easy steps…
year Period actuals - Dt #1- MA#2 -
Centered #3 - Index #4 Avg qtr #5 Adjust#6 - De-season
Y1 Qtr 1 1 10 0.531 0.532 18.8Y1 Qtr 2 2 20 1.123 1.124 17.8Y1 Qtr 3 3 26 18.25 18.50 1.405 1.372 1.374 18.9Y1 Qtr 4 4 17 18.75 19.13 0.889 0.969 0.970 17.5Y2 Qtr 1 5 12 19.50 20.00 0.600 22.6Y2 Qtr 2 6 23 20.50 21.25 1.082 20.5Y2 Qtr 3 7 30 22.00 22.00 1.364 21.8Y2 Qtr 4 8 23 22.00 22.50 1.022 23.7Y3 Qtr 1 9 12 23.00 23.25 0.516 22.6Y3 Qtr 2 10 27 23.50 23.63 1.143 24.0Y3 Qtr 3 11 32 23.75 23.75 1.347 23.3Y3 Qtr 4 12 24 23.75 24.13 0.995 24.7Y4 Qtr 1 13 12 24.50 25.13 0.478 22.6Y4 Qtr 2 14 30 25.75 26.25 1.143 26.7Y4 Qtr 3 15 37 26.75 Totals 3.995 4.000 26.9Y4 Qtr 4 16 28 28.9
23 / 1.124 = 20.5
Forecast period indices
#7 - Deseason Forecast
Y5 Qtr 1 17 0.532 28.0Y5 Qtr 2 18 1.124 28.7Y5 Qtr 3 19 1.374 29.3Y5 Qtr 4 20 0.970 29.9
Now easy step 7 …Forecast
year Period#6 - De-season
Y1 Qtr 1 1 18.8Y1 Qtr 2 2 17.8Y1 Qtr 3 3 18.9Y1 Qtr 4 4 17.5Y2 Qtr 1 5 22.6Y2 Qtr 2 6 20.5Y2 Qtr 3 7 21.8Y2 Qtr 4 8 23.7Y3 Qtr 1 9 22.6Y3 Qtr 2 10 24.0Y3 Qtr 3 11 23.3Y3 Qtr 4 12 24.7Y4 Qtr 1 13 22.6Y4 Qtr 2 14 26.7Y4 Qtr 3 15 26.9Y4 Qtr 4 16 28.9
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
0 5 10 15 20
y = 0.6402x + 17.137 R2 = 0.8444
7. Forecast using deseasonalized values
.6402 (18) + 17.137 = 28.7
Forecast period indices
#7 - Deseason Forecast
#8 - season adjusted
Y5 Qtr 1 17 0.532 28.0 14.9Y5 Qtr 2 18 1.124 28.7 32.2Y5 Qtr 3 19 1.374 29.3 40.3Y5 Qtr 4 20 0.970 29.9 29.0
Finally easy step 8…
y = 0.6402x + 17.137
8. Seasonalize forecast
.532 x 28.0 = 14.9
1.124 x 28.7 = 32.2
Now Begins Winter’s Model
Let Dt = demand in period tN = the number of periods (length of season)St = estimate of deseasonalized series in period tGt = estimate of trend term in period tct = estimate of seasonal component for period t
Yikes, this model has it all!
1 1
1 1
(1 ) ( )
( ) (1 )
(1 )
0 1 ; 0 1; 0 1
tt t t
t N
t t t t
tt t N
t
DS S G
c
G S S G
Dc c
S
The smoothing equations:
, ( )t t t t t NF S G c
Series
Trend
Seasonal
Forecast
Step 1: Calculate the average of each of the seasons:V1 = Davg1; V2 = Davg2 …, Vm = Davgm
Step 2: Set G0 = (Vm – V1 ) / [(m-1)N] (initial slope estimate)
Step 3: Calculate S0 = Davgm + G0 (N-1)/2 (value of series at t = 0)
a.
Step 4: Calculate the seasonal factors:
0
; 2 1 0[( 1) / 2 ]
tt
i
DC N t
V N j G
Vi = average of season i, j = period of season
Could you review with us the 6 easy steps to applying Winter’s Model?
Step 4 Explained
a.
Step 4: Calculate the seasonal factors:
0
; 2 1 0[( 1) / 2 ]
tt
i
DC N t
V N j G
Vi = average of season i, j = period of season
0[2.5 ]t iY V j G
mean of theith year
initial slopeestimate
for N = 4(quarters)
j=1,2,3,4
series+ trend
Step 4 b. Average the seasonal factors
2 1 1 01 0, ...,
2 2n N N
N
c c c cc c
Step 4 c. Normalize the seasonal factors
1
0
for 1 0jj N
ii
cc N N j
c
Step 5: Forecast for period t+
Ft+ = (St + Gt) ct+
Step 6: Next period, update model parameters with new data point.
1 1
1 1
(1 ) ( )
( ) (1 )
(1 )
0 1 ; 0 1; 0 1
tt t t
t N
t t t t
tt t N
t
DS S G
c
G S S G
Dc c
S
year quarterSales in 100's
Step 1 average j
Step 4a Initial Ct
Step 4b Avg each qtr
Step 4c normalize
Year 1 1 10 1.0 0.5904 0.5888 0.5932 20 2 1.1228 1.1010 1.1083 26 3 1.3913 1.3717 1.3814 17 18.25 4.0 0.8690 0.9115 0.918
Year 2 1 12 1 0.58722 23 2 1.0792 3.9730 4.003 30 3.0 1.35214 22 21.75 4 0.9539
Step 2. Initial Gt= 0.875Step 3. Initial ST= 23.0625
Example 2.8
0[( 1) / 2 ]t
ti
DC
V N j G
1
0
jj N
ii
cc N
c
G0 = (Vm – V1 ) / [(m-1)N]
S0 = Davg2 + G0 (N-1)/2
Sales in 100's alpha=.2
beta=.1
gamma=.1
before yr3 observed
Dt Ft St Gt Ct forecastyr 3 Qtr 1 16 14.19 24.58 0.940 0.595 14.190yr 3 Qtr 2 33 28.29 26.41 1.029 1.131 27.504yr 3 Qtr 3 34 37.90 26.91 0.976 1.350 35.476yr 3 Qtr 4 26 25.59 28.01 0.9883 0.915 24.376
More of Example 2.8
1 1
1 1
(1 ) ( )
( ) (1 )
(1 )
tt t t
t N
t t t t
tt t N
t
DS S G
c
G S S G
Dc c
S
Ft+ 1 = (St + Gt) c ; t = 0 Ft+ 1 = (St + Gt) ct+ ; t = 0, = 1,2,3,4
Year 4 forecast
Sales in 100's alpha=.2 beta=.1 gamma=.1 Normalize
Dt Ft St Gt Ct yr 3
yr 3 Qtr 1 16 14.19 24.58 0.940 0.595 0.5964yr 3 Qtr 2 33 28.29 26.41 1.029 1.131 1.1334yr 3 Qtr 3 34 37.90 26.91 0.976 1.350 1.3533yr 3 Qtr 4 26 25.59 28.015 0.9883 0.915 0.9170Yr 4 Qtr 1 1 17.30 3.990 4.000Yr 4 Qtr 2 2 33.99Yr 4 Qtr 3 3 41.92Yr 4 Qtr 4 4 29.31
Ft+ 1 = [28.015 + .9883] .5964 = 17.30Ft+ 2 = [28.015 + 2(.9883)] 1.1334 = 33.99Ft+ 2 = [28.015 + 3(.9883)] 1.3533 = 41.92Ft+ 2 = [28.015 + 4(.9883)] .9170 = 29.31
Bonus Topic
Casual Regression
Regression Model (Causal)
Model: Dt = a + bxt + et
where Dt = (demand) forecast for period txt = independent variable for period tet = random noisea,b slope, intercept to be estimated using least-squares
Two requirements are necessary to use this model:1. A causal relationship exists between x and D2. The value for x can be determined prior to
period t (a time lag exists)
Least-Squares Formulae
n n
tttt=1 t=1
n22
t
t=1
- xx D Db =
x xn
a = D - bx
Why not show the class how it
works in Excel?
Example – casual model
Month
Housing starts 3 mo. prior (in 1000's)
Sales of drywall (in 1000's)
House Starts Sales1 1.9 1022 2.1 1133 3.1 1634 4.2 2155 3.7 1926 5.5 2837 4.1 2128 2.9 1509 3.5 180
10 2.2 12011 1.4 8012 1.5 82
The Regression ModelDrywall Sales
y = 49.491x + 8.7823
R2 = 0.9993
0
50
100
150
200
250
300
0 1 2 3 4 5 6
Housing Starts
A Multiple Regression Example
0 1 2 1 2 2 2 3t t t t tY b b t b x b x b x e
Yt = drywall sales in month txt-1 = housing starts in month t-1xt-2 = housing starts in month t-2xt-3 = housing starts in month t-3
linear trend
Demand Sales
• Demands for items that are not in stock will result in backorders or lost sales
• Lost sales data is usually not available• Assume true monthly demand is normal
with a mean of 100 and a standard deviation of 30– If 110 items are stock each month– Then Pr{Demands > 110} .37
What about lost Sales?
Can we have some homework
problems?
Chapter 2: 12, 13 ,16-22, 24, 28- 30, 33 – 36.
"The future isn't what it used to be !" -- anonymous