# GROWTH CURVES

GROWTH CURVES FOR TECHNOLOGY FORECASTING

It has been observed that the growth pattern of many of the biological systems follow an `S’ shape curve. Initially the growth is slow, and then the growth rate increases and finally levels off into the natural limit. Fruits, vegetables, population of yeast cells etc. show an S shaped growth pattern in natural environment.

Scientists/researchers, in their continuous endeavour to evolve improved and realistic techniques of forecasting, found striking similarities between growths of biological systems and technological performances in some areas. For instance, such a behaviour was even apparent in a chemical reaction in a closed system with a predetermined quantity of reactants. Many of the engineering problems, like matching/achieving the thermodynamic efficiency of an engine to its natural limit of carnot cycle efficiency, manifest such a behaviour. This similarity between biological and technological systems has encouraged forecasters/scientists to use biological growth curves for technological forecasting.

Two methods namely Pearl curve and Gompertz curve are widely used in forecasting such growth patterns.

PEARL CURVE

Raymond Pearl (an American biologist) had shown that increase in population of organisms follows a growth pattern in an `S’ shaped curve (Figure -1). Pearl’s growth curve equation could be adapted for the growth of performance of technology with reference to functional capability which is given by

Where y = State of information or performance of technology or functional capability of technology at time t,

L upper limit of growth

t = time, a and b are constants.

‘y’ has an initial value of zero at ‘t’ = -∞

and reaches the maximum value of L at time t =∞ . The curve is symmetric about its inflection point y =1/2 L.

The constant `a’ determines where the curve will be on the time axis. The constant `b’ determines the steepness of the sharply rising portion. On the basis of some historical data points we determine the values of `a’ and `b’ which give a good fit to the data and then use the equation to forecast future progress as explained below.

The Pearl curve equation can be rewritten as

When Y is plotted against time we get a straight line which can be extrapolated into the future. This is the basis of graphical procedure for projections based on growth curve.

From the historical data we obtain a value of Y, corresponding to time t, and then carry out the minimization of

to obtain a regression fit of Y on t. From the future value of In

he expected functional capability at that time can be calculated. The initial slow rate of growth is caused usually by the resistance to the use or acceptance of the new technology. The upper flattening is due to approaching of a limit, which is physically the maximum possible attainable value. The performance of single technologies follows this type of S behaviour but the overall growth of technologies will be a net result of the growth of individual technologies.

Figure-1 : Pearl Curve

GOMPERTZ CURVE

This is another growth curve of the same class, named after its inventor-mathematician Benjamin Gompertz, that is frequently used in technological forecasting. It is represented as

Where, y= parameter of technological growth or a parameter representing technological capability

L = Upper limit of the parameter

b and K are constants

t = time

Like the Pearl curve the Gompertz curve (Figure 4.3) ranges from zero at t = -∞ , to the upper limit L at t =∞ . However, the curve is not symmetrical like Pearl curve.

The inflection point occurs at

The equation of Gompertz Curve can be rewritten as

When In (L/y) is plotted against time t, we get a straight line. Sum of least square method can be used for fitting the historical data for this curve also.

Figure-2 : Gompertz Curve

This curve can be used to predict the state of technology for which there is a limit, and when the growth in the initial stages is comparatively faster than that of the Pearl curve. Many technological trend capabilities have shown this type of growth. Data representing the functional capability of many technologies may follow one or more of the growth curves. Predictions into the future can then be made using a few initial points.

APPLICATIONS

Growth curves could be used for forecasting how and when a given technical approach will reach its upper limit. Analysis of most of the technologies shows that when a technical approach is new, growth is slow owing to initial problems. Once these are overcome, growth in performance is rapid. As the limit is approached, additional increments in performance are difficult. Growth curves can be used for forecasting parameters having a limit and they are useful for estimating demand for new technologies, performance characteristics of newer technological approaches etc. Conventional forecasts using linear extrapolation fail in case of systems which are growing but are bound by a limit. If extrapolation is done using initial data it leads to underestimation. Conversely, if extrapolation is based o n the rapid growing trend it leads to overestimation.