Introduction to Forecasting

COB 291

Spring 2000

Forecasting

A forecast is an estimate of future demand Forecasts contain error Forecasts can be created by subjective means

by estimates from informal sources OR forecasts can be determined

mathematically by using historical data OR forecasts can be based on both subjective

and mathematical techniques.

Qualitative Approaches

Executive committee consensus Delphi method Survey of sales force Survey of customers Historical analogy Market research

Quantitative Approaches

Based on the assumption that the “forces” that generated the past demand will generate the future demand (i.e., history will tend to repeat itself)

Analysis of the past demand pattern provides a good basis for forecasting future demand

Quantitative Approaches

Simple Linear Regression– Relationship between one independent variable,

x, and a dependent variable, y– Assumed to be linear– Form: Y=a+bX

• Y=dependent variable

• a=y-intercept

• X=independent variable

• b=slope of the regression line

Quantitative Methods - L.S. Regression ExamplePerfect Lawns, Inc., intends to use sales of

lawn fertilizer to predict lawn mower sales. The store manager feels that there is probably a six-week lag between fertilizer sales and mower sales. The pertinent data are shown below. =>

Quantitative Methods - L.S. Regression ExamplePeriod Fertilizer Sales Number of Mowers Sold

(Tons) (Six-Week Lag)

1 1.7 11

2 1.4 9

3 1.9 11

4 2.1 13

5 2.3 14

6 1.7 10

7 1.6 9

8 2 13

9 1.4 9

10 2.2 16

11 1.5 10

12 1.7 10

A) Use the least squares method to obtain a linear regression line for the data.

Quantitative Methods - L.S. Regression ExamplePeriod Fertilizer Sales Number of Mowers Sold (X) (Y) X2 Y2

(Tons) (X) (Six-Week Lag) (Y)

1 1.7 11 18.7 2.89 121

2 1.4 9 12.6 1.96 81

3 1.9 11 20.9 3.61 121

4 2.1 13 27.3 4.41 169

5 2.3 14 32.2 5.29 196

6 1.7 10 17.0 2.89 100

7 1.6 9 14.4 2.56 81

8 2 13 26.0 4.00 169

9 1.4 9 12.6 1.96 81

10 2.2 16 35.2 4.84 256

11 1.5 10 15.0 2.25 100

12 1.7 10 17.0 2.89 100

Quantitative Methods - L.S. Regression ExamplePeriod Fertilizer Sales Number of Mowers Sold (X) (Y) X2 Y2

(Tons) (X) (Six-Week Lag) (Y)

1 1.7 11 18.7 2.89 121

2 1.4 9 12.6 1.96 81

3 1.9 11 20.9 3.61 121

4 2.1 13 27.3 4.41 169

5 2.3 14 32.2 5.29 196

6 1.7 10 17.0 2.89 100

7 1.6 9 14.4 2.56 81

8 2 13 26.0 4.00 169

9 1.4 9 12.6 1.96 81

10 2.2 16 35.2 4.84 256

11 1.5 10 15.0 2.25 100

12 1.7 10 17.0 2.89 100

SUM 21.5 135 248.9 39.55 1575

Quantitative Methods - L.S. Regression Example

b

n X Y X Y

n X X

( )( )2 2

Quantitative Methods - L.S. Regression Example

b

( )( . ) ( . )( )

( )( . ) ( . )

.

..

12 248 9 215 135

12 39 55 215

84 3

12 356 826

2

Quantitative Methods - L.S. Regression Example

a

Y

n

b X

n

135

12

6 826 215

120 98

( . )( . ).

Quantitative Methods - L.S. Regression Example

Y X

Y

e

e

0 98 6 826

0 98 6 826 2 12 67

. . ( )

. . ( ) .

Predict lawn mower sales for the first week in August, given fertilier

sales six weeks earlier of two tons.

lawn mowers

Time Series Analysis

A time series is a set of numbers where the order or sequence of the numbers is important, e.g., historical demand

Analysis of the time series identifies patterns

Once the patterns are identified, they can be used to develop a forecast

Time Series Models

Simple moving average Weighted moving average Exponential smoothing (exponentially

weighted moving average)– Exponential smoothing with random fluctuations– Exponential smoothing with random and trend– Exponential smoothing with random and

seasonal component

Time Series Models Simple Moving Average

Sample Data (3-period moving average) t Dt Ft Dt-Ft | Dt-Ft |

Quarter Actual Demand Forecast Error Error

1 100

2 110

3 110

4 ? (100+110+110)/3=106.67

Time Series Models Simple Moving Average

Sample Data (3-period moving average) t Dt Ft Dt-Ft | Dt-Ft |

Quarter Actual Demand Forecast Error Error

1 100

2 110

3 110

4 80 (100+110+110)/3=106.67 80-106.67=-26.67 26.67

Time Series Models Simple Moving Average

Sample Data (3-period moving average) t Dt Ft Dt-Ft | Dt-Ft |

Quarter Actual Demand Forecast Error Error

1 100

2 110

3 110

4 80 (100+110+110)/3=106.67 80-106.67=-26.67 26.67

5 ? (110+110+80)/3 = 100.00

Time Series Models Simple Moving Average

Sample Data (3-period moving average) t Dt Ft Dt-Ft | Dt-Ft |

Quarter Actual Demand Forecast Error Error

1 100

2 110

3 110

4 80 (100+110+110)/3=106.67 80-106.67=-26.67 26.67

5 100 (110+110+80)/3 = 100.00 0 0

Time Series Models Exponential smoothing (exponentially weighted moving average)

S D S

F St t t

t t

( )1 1

1

Time Series Models Exponential smoothing (exponentially weighted moving average)

Where

t=time period

St=smoothed average at end of period t

Dt=actual demand in period t

=smoothing constant (0<<1)

Ft=forecast for period t

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha = 0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

1 ? 100

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

1 100 100

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

1 100 100 100-100=0

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

1 100 .2(100)+.8(100)=100 100 100-100=0

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

1 100 .2(100)+.8(100)=100 100 100-100=0

2 ? 100

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

1 100 .2(100)+.8(100)=100 100 100-100=0

2 110 100 110-100=10

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

1 100 .2(100)+.8(100)=100 100 100-100=0

2 110 .2(110)+.8(100)=102 100 110-100=10

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

1 100 .2(100)+.8(100)=100 100 100-100=0

2 110 .2(110)+.8(100)=102 100 110-100=10

3 ? 102

Time Series Models Exponential smoothing (exponentially weighted moving average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error

0 100

1 100 .2(100)+.8(100)=100 100 100-100=0

2 110 .2(110)+.8(100)=102 100 110-100=10

3 110 102 110-102=8

Make forecasts for periods 4-12.

Time Series Models Forecast Error

2 error measures:

Bias

tells direction (i.e., over or under forecast)

Mean Absolute Deviation

tells magnitude of forecast error

( )D F

n

t t

D F

n

t t

Characteristics of Good Forecasts

Stability Responsiveness Data Storage Requirements

BESM Example Cont’d

BESM - Expanded

The Basic Exponential Smoothing Model (BESM) is nothing more than a cumulative weighted average of all past demand (and the initial smoothed average).

Proof:

Demand Data with Trend

Time Series Models Exponential smoothing with trend enhancement

S D S T

T S S T

F S nT

t t t t

t t t t

t n t t

( )( )

( ) ( )

1

11 1

1 1

Demand Data with Trend and Seasonality

t Dt

0 1 402 703 604 885 706 1207 1008 130

Basic Model Applicationbase smoothing constant, alpha, =.20

This spreadsheet updates the ES model for Quarters 1 through 8 and makesforecasts for Quarters 9 through 12.

alpha= 0.20beta= 0.00gamma= 0.00

t Dt St Tt It Ft Dt-Ft ABS0 40.00 0.001 40.00 40.00 0.00 1.00 40.00 0.00 0.002 70.00 46.00 0.00 1.00 40.00 30.00 30.003 60.00 48.80 0.00 1.00 46.00 14.00 14.004 88.00 56.64 0.00 1.00 48.80 39.20 39.205 70.00 59.31 0.00 1.00 56.64 13.36 13.366 120.00 71.45 0.00 1.00 59.31 60.69 60.697 100.00 77.16 0.00 1.00 71.45 28.55 28.558 130.00 87.73 0.00 1.00 77.16 52.84 52.849 1.00 87.73

10 1.00 87.7311 1.00 87.7312 1.00 87.73

29.83 29.83BIAS MAD

FORECASTING DEMO

0.0020.0040.0060.0080.00

100.00120.00140.00

1 2 3 4 5 6 7 8

Period

Demand (D)

Forecast (F)

Forecast Error (D-F)

Abs. Forecast Error | D-F |

Trend-Enhanced Applicationbase smoothing constant, alpha, = .20 and trend smoothing constant, beta, = .30

This spreadsheet updates the ES model for Quarters 1 through 8 and makesforecasts for Quarters 9 through 12.

alpha= 0.20beta= 0.30gamma= 0.00

t Dt St Tt It Ft Dt-Ft ABS0 40.00 10.001 40.00 48.00 9.40 1.00 50.00 -10.00 10.002 70.00 59.92 10.16 1.00 57.40 12.60 12.603 60.00 68.06 9.55 1.00 70.08 -10.08 10.084 88.00 79.69 10.17 1.00 77.61 10.39 10.395 70.00 85.89 8.98 1.00 89.86 -19.86 19.866 120.00 99.90 10.49 1.00 94.87 25.13 25.137 100.00 108.31 9.87 1.00 110.39 -10.39 10.398 130.00 120.54 10.58 1.00 118.18 11.82 11.829 1.00 131.12

10 1.00 141.7011 1.00 152.2712 1.00 162.85

1.20 13.78BIAS MAD

FORECASTING DEMO

-50.00

0.00

50.00

100.00

150.00

1 2 3 4 5 6 7 8

Period

Demand (D)

Forecast (F)

Forecast Error (D-F)

Abs. Forecast Error | D-F |

Seasonal Indexes

seasonal index = actual demand / average demand

divide demand by its seasonal index to deseasonalize and

multiply demand by its seasonal index to seasonalize.

t Dt It Dt'1 8 0.8 102 24 1.2 203 27 0.9 304 44 1.1 405 40 0.8 506 72 1.2 607 63 0.9 708 88 1.1 80

Full Model for Exponential Smoothing NOTE: This model will allow you to

forecast with trend only, with trend and seasonality, with seasonality only, or with no trend and no seasonality.

Full Model for Exponential Smoothing (cont’d)

SD

IS T

T S S T

ID

SI

F S nT I

tt

tt t

t t t t

t mt

tt

t n t t t n

1

1

1

1 1

1 1

( )

t Dt St Tt It Ft

0 40 101 0.82 1.23 0.94 1.156789101112

t Dt St Tt It Ft

0 40 101 0.8 (40+10).8=402 1.23 0.94 1.156789101112

t Dt St Tt It Ft

0 40 101 40 0.8 (40+10).8=402 1.23 0.94 1.156789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 0.8 (40+10).8=402 1.23 0.94 1.156789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 1.23 0.94 1.156789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 1.23 0.94 1.15 .4(40/50)+.6(.8)=.806789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 1.2 (50+10)1.2=723 0.94 1.15 .4(40/50)+.6(.8)=.806789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 1.2 (50+10)1.2=723 0.94 1.15 .4(40/50)+.6(.8)=.806789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 1.2 (50+10)1.2=723 0.94 1.15 .4(40/50)+.6(.8)=.806789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 .3(59.67-50)+.7(10)=9.90 1.2 (50+10)1.2=723 0.94 1.15 .4(40/50)+.6(.8)=.806789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 .3(59.67-50)+.7(10)=9.90 1.2 (50+10)1.2=723 0.94 1.15 .4(40/50)+.6(.8)=.806 .4(70/59.67)+.6(1.2)=1.19789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 .3(59.67-50)+.7(10)=9.90 1.2 (50+10)1.2=723 0.9 (59.67+9.90)0.9=62.614 1.15 .4(40/50)+.6(.8)=.806 .4(70/59.67)+.6(1.2)=1.19789101112

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 .3(59.67-50)+.7(10)=9.90 1.2 (50+10)1.2=723 60 68.99 9.73 0.9 (59.67+9.90)0.9=62.614 88 78.97 9.8 1.1 86.585 70 88.52 9.73 .4(40/50)+.6(.8)=.80 71.026 120 98.78 9.89 .4(70/59.67)+.6(1.2)=1.19 116.847 100 109.46 10.12 0.89 96.488 130 119.18 10 1.11 132.229 0.8 10 1.2 11 0.9 12 1.1

We would like to forecast for quarters 9-12 (at end of qtr. 8)

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 .3(59.67-50)+.7(10)=9.90 1.2 (50+10)1.2=723 60 68.99 9.73 0.9 (59.67+9.90)0.9=62.614 88 78.97 9.8 1.1 86.585 70 88.52 9.73 .4(40/50)+.6(.8)=.80 71.026 120 98.78 9.89 .4(70/59.67)+.6(1.2)=1.19 116.847 100 109.46 10.12 0.89 96.488 130 119.18 10 1.11 132.229 0.8 10 1.2 11 0.9 12 1.1

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 .3(59.67-50)+.7(10)=9.90 1.2 (50+10)1.2=723 60 68.99 9.73 0.9 (59.67+9.90)0.9=62.614 88 78.97 9.8 1.1 86.585 70 88.52 9.73 .4(40/50)+.6(.8)=.80 71.026 120 98.78 9.89 .4(70/59.67)+.6(1.2)=1.19 116.847 100 109.46 10.12 0.89 96.488 130 119.18 10 1.11 132.229 0.8 (119.18+(1)10).8=102.8710 1.2 11 0.9 12 1.1

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 .3(59.67-50)+.7(10)=9.90 1.2 (50+10)1.2=723 60 68.99 9.73 0.9 (59.67+9.90)0.9=62.614 88 78.97 9.8 1.1 86.585 70 88.52 9.73 .4(40/50)+.6(.8)=.80 71.026 120 98.78 9.89 .4(70/59.67)+.6(1.2)=1.19 116.847 100 109.46 10.12 0.89 96.488 130 119.18 10 1.11 132.229 0.8 (119.18+(1)10).8=102.8710 1.2 (119.18+(2)10)1.2=166.9511 0.9 12 1.1

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 .3(59.67-50)+.7(10)=9.90 1.2 (50+10)1.2=723 60 68.99 9.73 0.9 (59.67+9.90)0.9=62.614 88 78.97 9.8 1.1 86.585 70 88.52 9.73 .4(40/50)+.6(.8)=.80 71.026 120 98.78 9.89 .4(70/59.67)+.6(1.2)=1.19 116.847 100 109.46 10.12 0.89 96.488 130 119.18 10 1.11 132.229 0.8 (119.18+(1)10).8=102.8710 1.2 (119.18+(2)10)1.2=166.9511 0.9 (119.18+(3)10)0.9=134.0012 1.1

t Dt St Tt It Ft

0 40 101 40 .2(40/.8)+.8(40+10)=50 .3(50-40)+.7(10)=10 0.8 (40+10).8=402 70 .2(70/1.2)+.8(50+10)=59.67 .3(59.67-50)+.7(10)=9.90 1.2 (50+10)1.2=723 60 68.99 9.73 0.9 (59.67+9.90)0.9=62.614 88 78.97 9.8 1.1 86.585 70 88.52 9.73 .4(40/50)+.6(.8)=.80 71.026 120 98.78 9.89 .4(70/59.67)+.6(1.2)=1.19 116.847 100 109.46 10.12 0.89 96.488 130 119.18 10 1.11 132.229 0.8 (119.18+(1)10).8=102.8710 1.2 (119.18+(2)10)1.2=166.9511 0.9 (119.18+(3)10)0.9=134.0012 1.1 (119.18+(4)10)1.1=175.07

E.S. Homework, Ex. 3

t Dt St Tt It Ft0 93 61 12 0.73 0.54 1.856789101112

t Dt St Tt It Ft0 93 61 1 (93+6)x1.00=99.002 0.73 0.54 1.856789101112

t Dt St Tt It Ft0 93 61 101 1 (93+6)x1.00=99.002 0.73 0.54 1.856789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 1 (93+6)x1.00=99.002 0.73 0.54 1.856789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 0.73 0.54 1.856789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 0.73 0.54 1.85 .6(101/99.8)+.4(1.00)=1.016789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 0.7 (99.8+6.24)x0.7=74.233 0.54 1.85 .6(101/99.8)+.4(1.00)=1.016789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 70 0.7 (99.8+6.24)x0.7=74.233 0.54 1.85 .6(101/99.8)+.4(1.00)=1.016789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 70 .4(70/0.7)+.6(99.8+6.24)=103.62 0.7 (99.8+6.24)x0.7=74.233 0.54 1.85 .6(101/99.8)+.4(1.00)=1.016789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 70 .4(70/0.7)+.6(99.8+6.24)=103.62 .3(103.62-99.8)+.7(6.24)=5.51 0.7 (99.8+6.24)x0.7=74.233 0.54 1.85 .6(101/99.8)+.4(1.00)=1.016789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 70 .4(70/0.7)+.6(99.8+6.24)=103.62 .3(103.62-99.8)+.7(6.24)=5.51 0.7 (99.8+6.24)x0.7=74.233 0.54 1.85 .6(101/99.8)+.4(1.00)=1.016 .6(70/103.62)+.4(0.7)=0.69789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 70 .4(70/0.7)+.6(99.8+6.24)=103.62 .3(103.62-99.8)+.7(6.24)=5.51 0.7 (99.8+6.24)x0.7=74.233 0.5 (103.62+5.51)x0.5=54.574 1.85 .6(101/99.8)+.4(1.00)=1.016 .6(70/103.62)+.4(0.7)=0.69789101112

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 70 .4(70/0.7)+.6(99.8+6.24)=103.62 .3(103.62-99.8)+.7(6.24)=5.51 0.7 (99.8+6.24)x0.7=74.233 58 111.88 6.34 0.5 (103.62+5.51)x0.5=54.574 205 116.49 5.82 1.8 212.85 120 120.91 5.4 .6(101/99.8)+.4(1.00)=1.01 123.536 94 130.28 6.59 .6(70/103.62)+.4(0.7)=0.69 87.157 70 137.02 6.64 0.51 69.88 250 142.38 6.26 1.78 255.719 110 0.7111 0.5112 1.77

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 70 .4(70/0.7)+.6(99.8+6.24)=103.62 .3(103.62-99.8)+.7(6.24)=5.51 0.7 (99.8+6.24)x0.7=74.233 58 111.88 6.34 0.5 (103.62+5.51)x0.5=54.574 205 116.49 5.82 1.8 212.85 120 120.91 5.4 .6(101/99.8)+.4(1.00)=1.01 123.536 94 130.28 6.59 .6(70/103.62)+.4(0.7)=0.69 87.157 70 137.02 6.64 0.51 69.88 250 142.38 6.26 1.78 255.719 1 (142.38+(1)6.26)1.00=148.6410 0.7111 0.5112 1.77

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 70 .4(70/0.7)+.6(99.8+6.24)=103.62 .3(103.62-99.8)+.7(6.24)=5.51 0.7 (99.8+6.24)x0.7=74.233 58 111.88 6.34 0.5 (103.62+5.51)x0.5=54.574 205 116.49 5.82 1.8 212.85 120 120.91 5.4 .6(101/99.8)+.4(1.00)=1.01 123.536 94 130.28 6.59 .6(70/103.62)+.4(0.7)=0.69 87.157 70 137.02 6.64 0.51 69.88 250 142.38 6.26 1.78 255.719 1 (142.38+(1)6.26)1.00=148.6410 0.71 (142.38+(2)6.26)0.71=109.9811 0.5112 1.77

t Dt St Tt It Ft0 93 61 101 .4(101/1.00)+.6(99)=99.80 .3(99.8-93)+.7(6)=6.24 1 (93+6)x1.00=99.002 70 .4(70/0.7)+.6(99.8+6.24)=103.62 .3(103.62-99.8)+.7(6.24)=5.51 0.7 (99.8+6.24)x0.7=74.233 58 111.88 6.34 0.5 (103.62+5.51)x0.5=54.574 205 116.49 5.82 1.8 212.85 120 120.91 5.4 .6(101/99.8)+.4(1.00)=1.01 123.536 94 130.28 6.59 .6(70/103.62)+.4(0.7)=0.69 87.157 70 137.02 6.64 0.51 69.88 250 142.38 6.26 1.78 255.719 1 (142.38+(1)6.26)1.00=148.6410 0.71 (142.38+(2)6.26)0.71=109.9811 0.51 (142.38+(3)6.26)0.51=82.1912 1.77