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Linear Programming Homework Illustrations and Solutions

Question: 1

From the following data, fit a free hand smooth curve and forecast the trend values thereby.

Year: Pardon of paddy (in 000’tonnes):

1992 65

1993 80

1994 100

1995 70

1996 80

1997 110

1998 115

1999 130

2000 90

2001 100

2002 150

2003 160

2004 120

Solution:

Graphic representation of the trend line relating to the production of Paddy by the method

of free hand curve.

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(b) Forecasting of the trend values by location on the trend line.

Year: Pardon of paddy (in 000’tonnes):

1992 64

1993 70

1994 77

1995 82

1996 85

1997 87

1998 100

1999 113

2000 107

2001 118

2002 121

2003 123

2004 132

From the above, it may be observed that there is a gradual rise in the trend values as

against zig zag yearly values.

The trend line thus drawn above can be extended to forecast the values for any future year.

But this method being purely of subjective nature, cannot always give dependable results.

(ii) Arbitrary Average Method

Under this method, the values of any two points of time preferably, those of the extreme

ones, or any other close to them are arbitrarily selected as the average of trend in the

series. Basing upon these two average, a trend line drawn on a graph paper in a free hand

smoothed manner to show the general tendency of the data. Referring to such a trend line,

the trend values of the different times are

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determine d by location. Alternatively, the trend values of the different times are computed

by adjusting the average change in value of the preceding point of time is represented by

𝑥 1, and that of the succeeding point of time by 𝑋 2, and the time gap between the two points

of time by N. The average change in the values between the two selected points of time is

computed by

𝑋 (𝑐) = 𝑋 2− 𝑋 1

𝑁

This average change in value is multiplied by the time gap ‘t’ between the selected point of

time and the observed time and the 4resultant figure is subtracted from the observed value

of the preceding times and added to the observed values of the succeeding times to find out

their respective trend values.

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Question: 2

From the following data, fit a trend line, and obtain the trend values for the different years

using the arbitrary average method:

Year : Sales (in ’000 tonnes) :

1996 20

1997 22

1998 24

1999 21

2000 23

2001 25

2002 25

2003 26

2004 24

Solution:

(a) Selecting arbitrarily the sales of 1997, and 2003 as the average points 𝑋 1 and 𝑋 2 we

obtain the trend line as follows:

Graphic representation of the Trend Line by the arbitrary average method.𝑋 1

(b) Computation of the trend values

(i) Locating on the above trend line we obtain the trend values for the different

years as follows:

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(ii) Alternatively

Taking the average change in the annual values we obtain the trend values for the different

years as follows:

Average change or 𝑋 𝑐 = 𝑋 2− 𝑋 1

𝑁

= 26−22

6 = 0.67

Adding this average change (.67) to the value of 𝑋 1 (22) successively for all the years

succeeding 1996 and subtracting this average change (.67) from the value of 𝑋 1 (22)

successively for all the years preceding 1996 we obtain the trend values as under:

Year : Trend Values

1996 21.33

1997 22

1998 22.67

1999 23.33

2000 24

2001 24.67

2002 25.33

2003 26

2004 24

(iii)Semi-average Method

Under this method, the trend line is fitted to a time series basing upon the average values

(usually arithmetic mean) of its two haves called semi-averages. For this, the entire series

is divided into two halves, leaving aside there value of the middle period, if there are odd

number of periods in the series. The average value of the first half portion of the series is

represented by 𝑋 1 and that of the second half portion by 𝑋 2. These two averages are placed

against the mid-point these two semi averages 𝑋 1 and 𝑋 2, in a straight linear manner. The

trend values for the different times are determined by locating the points on this line. The

trend line thus drawn can also be extended both the ways to estimate the trend values for

any earlier and later period.

Alternatively, the trend values for the different years can also be obtained by adjusting the

average change between the two semi average 𝑖. 𝑒.𝑋 2 = 𝑋 2−𝑋 1

𝑁 to any of the semi

average value in accordance with the times of deviations from the time of any of the semi

averages. Hence, the trend value for any year is computed by

T =𝑋 1 or 𝑋 2 +( 𝑋 2−𝑋 1

𝑁).x

Where, T = trend value to be computed for any year

𝑋 1= Semi average of the first half of the series.

𝑋 2 = Semi average of the second half of the series.

N = time difference between 𝑋 1 and 𝑋 2

And x = Time deviation from 𝑋 1 𝑎𝑛𝑑 𝑋 2 as the case may be.

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