The Components of the System.
(Continued )
Equation (6) from the preceeding page, namely :
N = B - C = b* n - c* n^ 2 (6 )
contains 2 adjustable parameters b and c , if for the sake of an example we choose
arbitrarely, b = 15 and c =1. We obtain values of N as given in Table C.1 below:
Table C.1
n P n^2 15n N
------ ---------------------------------------------------
1 90 1 15 14
2 45 4 30 26
3 30 9 45 36
4 22.5 16 60 44
5 18 25 75 50
6 15 36 90 54
no-> 7 13 49 105 56
no-> 8 11.3 64 120 56
9 10 81 135 54
10 9 100 150 50
12 7.5 144 180 36
14 6.4 196 210 14
---------------------------------------------------------------------------
In the above Table, n is the number of mowings for a 90 days season, P is the period or interval ( in days) bet. each mowing and N is the net benefit for the component. It is seen from the Table that N peaks at a value of 56 so that the optimized
interval is 12 days bet. the mowings.
Moving from an hypothetical situation to an actual one means confronting the
model against information data from an actual system. The process of validating the model in the case of gardening operations, like in any other operation, will yield
clearer results the more detailed and accurate is our data. (see Resouces,data aquisition)
The equation for the Net Benefit is of particular interest because it appears to
describe with reasonable approximation the behaviour of systems in a wide variety
of fields, from Economics to Ecology, Sociology, etc. The equation entails the interplay
of two fuctions with opposite signs, one parabolic, the other linear and as such reflects the optimization or minimization of variables so popular in systems analysis. A consideration of the graphical representation of the component and composite functions
as given in the Fig. below may facilitate the understanding of the behaviour. The fig.is
also from Lendrem (op.cit) for the time-sleeping behaviour of animals, so that the
coordinate values are not those of Table C-1.
.
Fig. C-1
In the upper graph of Fig. C-1 the behaviour of first and second order functions is contrasted. As in all the exponential functions, the slopes (derivatives) are initially low and increase with increasing values of the independent variable while in the linear case the slopes are, of course, the same in all the range. In our example, this behaviour leads to
increasing rate of change of the Cost with the number of mowings, whilst the rate of
change of the Benefit is the same for any number of mowings.
In the lower graph, the composite function, that is the sum of both, linear and quadratic is represented. The segmented line points to the value of the independent variable for which the dependent one reaches the maximum (in our example the maximum Net Benefit. Increasing the ner. of mowings over this point leads to decreasing values of N until eventually they become negative (horizontal full line in the Fig.)
meaning that Cost of the component is higher than the Benefits attained by performing
it. Although this may sound a bit obvious, it is surprising how often we fall into the
range of low or even negative net benefits from a component while distributing resources among the various system's components and it may be worthwhile to have the above
figure in a frame in front of our desks.
A Note of Caution: It is important to remember that a mathematical model like the
one of our example may hold for a given time interval which should be estimated and
allways specified. In our case the time interval was given as 90 days and any unqualified
extrapolation is unwarranted. A number of Components may result in a neglible increase
of the overall performance measure in the short term; that is they may be taken as irrelevant for the time period specified. A case in point may be the Component "Cost
Accountancy", if we neglect it for a certain short period no changes in the Garden may
be apparent to the eye. Nevertheless, before we close the office and give to the accountants a rake or a hoe, it is common sense to run the model for a longer period of
time; the brilliant idea of having more gardeners and less accountants may prove as a
dumb one in the long range.
A list of the possible Components that may be identified in a gardening system
may be long or short. If the aim of the exercise is just to make a list, the longer the better
because long lists are more impressive and prestigeous. If the aim is to build a workable
model, then the shorter the better. In a task oriented context a plaussible list may be the following one with 10 items: Lawn maintenance; Tree maint.; Bushes and Hedges; Weed
Control; Pest Control (IPCM); Flower beds and Containers; Inert elements maintenance;
Cost Accounting; Administration; Know How improvement.
Management is not included in the above list because it is in a higher hierarchical level: making decisions as to how to allocate the available Resources between the said
Components. This Topic is discussed in the next page.
