sbco forecasting hb student
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D. Anthony CheversSBCO 6240 - Production and Operations Management
Lecture 6 – Forecasting |
Lecture #4 – Forecasting
Definition & purposeForecasting models
Simple moving averageWeighted moving averageExponential smoothingTrend projectionsRegression analysis
Forecast errorsExercises
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Lecture 6 – Forecasting | 3
My interest is in the future because I am going to spend the rest of my life there.
—Charles F. Kettering
Lecture 6 – Forecasting |
Some Reasons WhyForecasting is Essential
in OMNew Facility Planning – It can take 5 years to design
and build a new factory or design and implement a new production process.
Production Planning – Demand for products vary from month to month and it can take several months to change the capacities of production processes.
Workforce Scheduling – Demand for services (and the necessary staffing) can vary from hour to hour and employees weekly work schedules must be developed in advance.
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Lecture 6 – Forecasting |
Forecasting - Defined
Forecasting is the prediction/estimation of future activities. Types of forecast – Economic, Technological and Demand• The first step in planning is forecasting, or estimating the
future demand for products and services and the resources necessary to produce these outputs. E.g. Rising Star finals (2005); Craft Village –New Kgn for World Cup Cricket 2007 -closure & Diana Ross (Jazz Festival, Jan 2008); Girl you have Intimate potential – Gregory Isaacs (2008)
• Operations managers need long range forecasts to make strategic decisions about products, processes and facilities.
• Operations managers need short range forecasts to assist them in making decisions about production issues that span only the next few weeks.
Lecture 6 – Forecasting | 6
Factors to Consider
Costs associated with a modelRequired accuracyRelevance of past data & availabilityForecast horizonPattern of dataTime needed for analysis
Lecture 6 – Forecasting |
Introduction
Eight steps to forecasting:
• Determine the use of the forecast
• Select the items or quantities to be forecasted
• Determine the time horizon of the forecast
• Select the forecasting model or models
• Gather the data needed to make the forecast
• Validate the forecasting model
• Make the forecast
• Implement the results
Lecture 6 – Forecasting |
Forecasting Models
MovingAverage
Exponential Smoothing
Trend Projections
Time Series Methods
Forecasting Techniques
Delphi Methods
Jury of Executive Opinion
Sales ForceComposite
Consumer Market Survey
Qualitative Models
Causal Methods
Regression Analysis
Multiple Regression
Lecture 6 – Forecasting |
Lecture 6 – Forecasting |
Method Data Required Relative Cost
Forecast Horizon
Application
Subjective Models Delphi Method Cross-impact analysis Historical analogy
Survey Result Correlation between events Years of data for similar situation
High High High
Long Term Long Term Medium-Long
Technological Forecasting Technological Forecasting Life-Cycle Demand
Causal Models Regression Econometric
Past data for variables Past data for variables
Moderate Moderate-High
Medium-Term Medium-Long
Demand Forecasting Economic Condition
Time Series Models Moving Average Exponential Smoothing
N most recent observations Previous forecast and recent observation
Very Low Very Low
Short-Term Short-Term
Demand Forecasting Demand Forecasting
Lecture 6 – Forecasting |
Some Time Series Models
Simple Moving AverageK-Period MA =
∑(Actual Value in previous k periods) / k
Weighted moving averageK-Period WMA =
∑(weight for period i) (actual value in period i)
∑(weights)
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k
i=1 k
i=1
Lecture 6 – Forecasting |
Exponential Smoothing
Exponential Smoothing method: A sophisticated weighted moving average method that calculates the average of a time series by giving recent demands more weight than earlier demands. Exponential smoothing is the most frequent used formal forecasting method because of its simplicity and the small amount of date needed to support it.
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SES: Ft+1 = α + (1-α)
(Demand this period) (Demand this period) (Forecast calculated last period)(Forecast calculated last period)
= α Dt + (1-α)Ft
Where Where αα is a smoothing parameter (0 ≤ is a smoothing parameter (0 ≤ αα ≤ 1). ≤ 1).
Or an equivalent equation: Ft+1 = Ft +α(Dt-Ft)
Lecture 6 – Forecasting |
Equations - Simplified
1 SMA = Ʃ Demand# of periods
2 WMA = Ʃ (Demand x Weight) # of period
3 Expo.Smooth: Ft = Ft-1 + α (At-1 - Ft-1)
Lecture 6 – Forecasting |
Calculation of 3-Month Simple Moving Average
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MonthActual Shed
Sales3-Month Moving Average
January 10
February 12
March 13
April 16 (10+12+13)/3 = 11 2/3
May 19 (12+13+16)/3 = 13 2/3
June 23 (13+16+19)/3 = 16
July 26 (16+19+23)/3 = 19 1/3
Lecture 6 – Forecasting |
Limitations of Simple MA
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MonthActual Sales
3 Month MA
January 13
February 12
March 10
April 16 (13 + 12 + 10)/3 = 11.67
Simple MA does not capture trends nor seasonalities
Lecture 6 – Forecasting |
Calculating Weighted Moving Averages
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Weights Applied
Period
3 Last month
2 Two months ago
1 Three months ago3 * Sales last moth + 2 * Sales two months ago + 1 * Sales three months ago
6 Sum of weights
Lecture 6 – Forecasting |
Calculation of 3-Month Weighted-Moving Average
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MonthActual Shed Sales
3 Month MA
January 10
February 12
March 13
April 16 [(3*13) + (2*12) + (1*10)] 6 = 12
May 19 [(3*16) + (2*13) + (1*12)] = 14
June 23 [(3*19) + (2*16) + (1*14)] = 17
July 26 [(3*23) + (2*19) + (1*16)] = 20 ½
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Lecture 6 – Forecasting |
Exponential Smoothing
New forecast =
previous forecast + (previous actual - previous)
or:
where
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tttt FAFF
actual period previous
constant smoothing
forecast previous
forecast new
t
t-
t
A
F
F
Lecture 6 – Forecasting |
Equation [Ft = Ft-1 + α (At-1 – Ft-1)]
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Port of Baltimore Exponential Smoothing Forecasts for a = 0.10 and = 0.50
QUARTERACTUAL
TONNAGE UNLOADED
ROUNDED FORECAST USING =0.10*=0.10*
ROUNDED FORECAST
USING =0.50=0.50**
1 180 175 175
2 168 176 = 175.00 + 1.10(180 – 175) 178
3 159 175 =175.50 + 0.10(168-175.50) 173
4 175 173 =174.75 + 0.10(159-174.75) 166
5 190 173 =173.18 + 0.10(175-173.18) 170
6 205 175 =173.36 + 0.10(190-173.36) 180
7 180 178 =175.02 + 0.10(205-175.02) 193
8 182 178 =178.02 + 0.10(180-178.02) 186
9 ? 179= 178.22 + 0.10(182-178.22) 184
* Forecasts rounded to the nearest ton
TABLE 5.4
Lecture 6 – Forecasting |
Selecting the Smoothing Constant ()
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n||
MAD errorsforecast deviation absolute Mean
n)(
MSE errorsforecast
Error Square Mean
actual|errorforecast |
n1
MAPEError
Percent AbsoluteMean
errorsforecast Bias
Select to minimize:
Lecture 6 – Forecasting |
Table – Detailed Calculation
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[84/8 = 10.50 & 100/8 = 12.50; Select: α = 0.10]
Absolute Deviations and MADs for Port of Baltimore Example
QUARTERACTUAL
TONNAGE UNLOADED
ROUNDED FORECAST
USING =0.10*=0.10*
ABSOLUTE DEVIATIONS
FOR=0.10*=0.10*
ROUNDED FORECAST
USING=0.50=0.50**
ABSOLUTE FORECAST
USING =0.50=0.50**
1 180 175 5 175 5
2 168 176 8 178 10
3 159 175 16 173 14
4 175 173 2 166 9
5 190 173 17 170 20
6 205 175 30 180 25
7 180 178 2 193 13
8 182 178 4 186 4
Sum of absolute deviations 84 100
TABLE 5.5
MAD = 12.50 MAD = ●| deviations |
n= 10.50
Lecture 6 – Forecasting |
Forecast Error – MSE[n = 4 (# of periods)]
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MSE = forecast errors2 87.25 ? n
As a practice, find the MSE for = 0.50 and conclude findings.
Quarter
Actual Tonnage Unloaded
Rounded Forecast with 0.10 constant Error @ 0.10 (Error)2
Rounded Forecast with 0.50 constant
Error @ 0.50 (Error)2
1 180 175 5 25 ? ? ?
2 168 176 8 64 ? ? ?
3 159 175 16 256 ? ? ?
4 175 173 2 4 ? ? ?
Sum of errors squared = 349 ? ? ?
Lecture 6 – Forecasting |
Trend Projection
A time series forecasting method that fits a trend line to a series of historical data points and then projects the line into the future for medium-to-long range forecasts.
Look at linear (straight-line) trends only
Develop a linear trend line by a precise statistical method, we can apply the least squares method
A straight line that minimizes the sum of the squares of the vertical differences or deviations from the line to each of the actual observations.
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Lecture 6 – Forecasting |
The Least Squares Method
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for Finding the Best-Fitting Straight Line [Time on x-axis]
Time period
Val
ues
of d
epen
dent
var
iabl
e (v
alue
s)
Deviation Deviation
11(error(error))
Deviation Deviation
55
Deviation Deviation
77
Deviation Deviation
66
Deviation Deviation
44
Deviation Deviation
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Deviation Deviation
22
Actual observation (y value)
1 2 3 4 5 6 7
Trend Line, y=a + bx
Lecture 6 – Forecasting |
Least Squares Line
Described in terms of its y-intercept (the height at which it intercepts the y-axis) and its slope (the angle of the line)
If we compute the y-intercept and slope, we can express the line with the following equation: y = a + bx where y (called “y hat”) = computed value of the variable to
be predicted (dependent variable) a = y-axis intercept b = slope of the regression line (or the rate of change in y for
given change in x) [Slope = y/x]
x = the independent variable (which in this case is time)
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Lecture 6 – Forecasting |
Trend Projection – General Regression equation
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X nX
YXnXYb
Ya
Ywhere
bXaY
Intercept [ a = y – bx ] axis-variable) (dependent predicted be
to variable the of value computed
Lecture 6 – Forecasting |
Notations
ƅ = slope of the regression line
∑ = summation sign
x = known values of the independent variable
y = known values of the dependent variable
x = average of the value of the x’s
ӯ = average of the values of the y’s
n = number of data points or observations
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Lecture 6 – Forecasting |
Exercise - BurkeCampbellChevers
The demand for electrical power at BurkeCampbellChevers Inc. Over the period 1999 to 2005 is shown below, in megawatts. Let’s forecast 2006 demand by fitting a straight-line trend to these data:
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Year Electrical Power Demand
1999 74
2000 79
2001 80
2002 90
2003 105
2004 142
2005 122
We can designate 1999 as year 1, 2000 as year 2, etc.
Lecture 6 – Forecasting |
Solution: b = (∑XY – nXY) / (∑X2 – nX2)
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Year Time Period (x) Power Demand (y) x2 xy1999 1 74 1 742000 2 79 4 1582001 3 80 9 2402002 4 90 16 3602003 5 105 25 5252004 6 142 36 8522005 7 122 49 854
Sum 28 692 140 3063Average 4 98.86
x = 28/7 = 4 y = 692/7 = 98.86
b = 3,063 - (7)(4)(98.86) = 295 = 10.54
140 - (7)(42) 28
a = Y - b X
a = 98.86 - (10.54 x 4)
a = 98.86 - 42.16
a = 56.7
Lecture 6 – Forecasting |
Least Square Trend Equation
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Thus, the least square trend equation is: y = 56.70 + 10.54 x
To project demand in 2006, we first denote the year 2006 as x=8
Demand in 2006 = 56.70 + 10.54 (8)141.02 or 141 megawatts
Estimate for 2007 – Year=9 (x=9)Demand in 2006 = 56.70 + 10.54 (9)151.56 or 152 megawatts
Lecture 6 – Forecasting |
Electrical Power and the Computed Trend Line
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Pow
er d
eman
d
1997
160
150
140
130
120
110
100
90
80
70
60
50
1998 1999 2000 2001 2002 2003 2004 2005
Year
Trend Line, y= 56.70 + 10.54x
Lecture 6 – Forecasting |
Regression Analysis
Causal forecasting models usually consider several variables that are related to the quantity being predicted
This approach is more powerful than the time-series methods that use only the historical values for the forecasting variable
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Lecture 6 – Forecasting |
Regression Analysis For example, the sales of PC’s might be related to
advertising budget, the price charged, competitors’ prices, promotional strategies, the economy, disposable income, unemployment rates or time.
In this case, PC sales would be called the dependent variable (Y) and the other variables would be called independent variables (X).
We will use Y with one other variable (X)
The most common quantitative casual forecasting model is linear-regression analysis
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Lecture 6 – Forecasting |
Linear Regression Analysis
A straight-line mathematical model to describe the functional relationships between independent and dependent variables
Can use the same mathematical model employed in least squares method of trend projection to perform a linear regression analysis
The dependent variable that we want to forecast is still y-hat but the independent variable, x, need no longer be time.
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Lecture 6 – Forecasting |
Using Regression Analysis to Forecast
Question: If the Local Payroll is 6 in the 7th month, what is the Sales?
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YTriple A' Sales
($100,000's)
XLocal Payroll($100,000,000)
2.0 13.0 32.5 42.0 22.0 13.5 7
Y = 1.75 + 0.25X
Lecture 6 – Forecasting |
Solution b = (∑XY – nXY) / (∑X2 – nX2)
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Y X XY2 1 2 13 3 9 9
2.5 4 10 162 2 4 42 1 2 1
3.5 7 24.5 49Total 15 18 51.5 80
Average 2.50 3.00If X is 6
b = 51.5 - (6 x 3 x 2.5)80 - (6 x 3 x 3)51.5 - 4580 - 546.50 = 0.2526.0
a = 2.5 - (0.25 x 3)2.5 - 0.75 = 1.75
Y = = 1.75 + (0.25X) [Where X is 6]
= 1.75 + (0.25 x 6)
= 1.75 + 1.5 = 3.25
Y X XY X2
2 1 2 1
3 3 9 9
2.5 4 10 16
2 2 4 4
2 1 2 1
3.5 7 24.5 49
15 18 51.5 80
2.50 3.00
Total Average
If X is 6
Lecture 6 – Forecasting |
Solution - Projection
If the local Chamber of Commerce predicts that the payroll will be $600 million next period, we can estimate sales with the regression equation:
• 1.75 + 0.25(6)• 1.75 + 1.50 = 3.25
Thus sales = $325,000
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Lecture 6 – Forecasting |
Exercise 1Demand for heart transplant surgery at Washington General Hospital has increased steadily in the past few years, as seen in the table.
Year Outpatient Surgeries Performed
1 452 503 524 565 586 ?
_____________________________
The director of medical services predicted six years ago that the demand in year 1 would be for 41 surgeries
(a) Use exponential smoothing, first with a smoothing constant of 0.6 and then with one of 0.9, to develop forecasts for years 2 through 6.
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Lecture 6 – Forecasting |
Exercise 2
Tongren Construction Company renovates old homes in Balvenie, Mandeville. Over time, the company has found that its dollar volume of renovation work is dependent on the Balvenie area payroll. The following table lists Tongren’s revenues and the amount of money earned by wage earners in Balvenie during the following six years.
Tongren’s Sales Local Payroll ($000,000) ($000,000,000)
2 13 32.5 42 22 13.5 7
Forecast sales if the payroll in period #7 is 8.
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Lecture 6 – Forecasting |
Solution – Tutorial #1
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Exponential smoothing, α = 0.6:
Year Demand 3-yr Moving Average Abs. Dev.1 45 412 50 41.0 + 0.6(45-41) = 43.4 6.63 52 43.4 + 0.6(50-43.4) = 47.4 4.64 56 47.4 + 0.6(52-47.4) = 50.2 5.805 58 50.2 + 0.6(56-50.2) = 53.7 4.306 ? 53.7 + 0.6(58-53.7) = 56.3
∑ = 21.3
MAD = 5.3
Exponential smoothing, α = 0.9:
Year Demand 3-yr Moving Average Abs. Dev.1 45 412 50 41.0 + 0.9(45-41) = 44.6 5.43 52 44.6 + 0.9(50-44.6) = 49.5 2.54 56 49.5 + 0.9(52-49.5) = 51.8 4.205 58 51.8 + 0.9(56-51.8) = 55.6 2.406 ? 55.6 + 0.9(58-55.6) = 57.8
∑ = 14.5
MAD = 3.6
3-Year Moving AverageYear Demand 3-yr Moving Average Abs. Dev.
1 452 503 524 56 (45 + 50 + 52)3 = 49 7.005 58 (50 + 52 + 56)3 = 52.7 5.306 ? (52 + 56 + 58)3 = 55.3
∑ = 12.3
MAD = 6.15
Lecture 6 – Forecasting |
Solution – Tutorial #2
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Y X XY X2
2 1 2 13 3 9 9
2.5 4 10 162 2 4 42 1 2 1
3.5 7 24.5 49Total 15 18 51.5 80
Average 2.50 3.00If X is 8?
b = 6(51.5) - (18 x 15)6(80) - (18 x 18)309 - 270480 - 32439.00 = 0.25156.0
a = 2.50 - (0.25 x 3.0)2.50 - 0.75 = 1.75
Y = = 1.75 + (0.25 x 8)= 1.75 + 2.0 = 3.75
X Y XY X2
1 17 17 12 16 32 43 17 51 94 21 84 165 20 100 256 20 120 367 23 161 498 25 200 649 24 216 81
Total 45 183 981 285Average 5 20.33
b = 9(981) - (45 x 183) = (8829) - (8235)(9 x 285) - (45 x 45) (2565) - (2025)594 =540 1.1
a = 20.33 - (1.1 x 5) = 20.33 - (5.5)14.83
Y = 14.83 + (1.1 x 10) [Next year is 10]
14.83 + (11) = 25.83
Lecture 6 – Forecasting |
Normal Distribution
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.00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 .1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 .2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 .3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 .4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7828 .7852 .8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133. 9 .8159 .8186 .212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .86211.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .88301.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .90151.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .91771.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .94411.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .95451.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .96331.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .97061.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .98172.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .98572.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .98902.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .99162.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .99522.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .99642.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .99742.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .99812.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .99903.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .99933.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .99953.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .99973.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998
∞ 0 z +- ∞