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To select an appropriate curve the pattern of the representative curve of the analyzed project it was compared with several well known curves, all of them belonging to different electric systems; once chosen the curve, their equation was adopted. To verify the reasonability of the adopted equation, different points of the analyzed pattern were satisfied by the assignation of values to the equation coefficients (2). The representative equation of the Postulate of the MLO corresponds to an asymptotic exponential curve (3, 4, 5, 6) whose expression is:
Where I= value of the improvement; L= value of the MLO; e= constant value (2,7182818284..) (7), base of the natural logarithms; b = index of concavity; S = analyzed stage (in whole numbers).
As the curve representative of the analyzed project it is a succession of segments, this equation it doesn't satisfy it in all their points. However, it is completed there where it interests (integer values on abscissas, representing to each stage).
Determination of the value of the Maximum Limit of Optimization
Operating on the equation (a), it is determined the equation to calculate the value of the MLO:
The calculated values are probable: they depend on the balance that the Current Performing Conditions (CPC) maintains and of the tendencies contributed by the stage S.
Justification of the representative equation
It is justified the representative equation of the postulate analyzing the stationary state and the network theorems (8) of a serial Resistive Inductive circuit (9).
Applying the Kirchhoff voltages law to a Resistive Inductive circuit (RL), the differential equation is obtained:
where: L = inductance; R = resistance; E = electromotive force; i = current; t = time. In courses of Physics it is demonstrated that if the force E is applied to the circuit when the value of t is equal to zero, the differential equation solution is:
- i(t) = I (improvements).
- E/R = L (MLO value).
- R/L = b (concavity index).
- t = S (stage of unitary value).
The permanent state it is associates to the value of the MLO and the transitory state refers to the way through which that value is attained.
The exponential factor makes that the transitory state diminishes with the time. Operating on the index b it is modified the shape of the curve, determining better values of the MLO. This fact demonstrates what was said when treating the Conceptual and Transcendent Changes (CTC).
INDEX OF CONCAVITY
The shape of the representative curve of the electric circuit taken as example is known and constant, since it depends on the physical components of the circuit (Resistors, Inductors). But in the case of the curve that represents to a project it is not this way, because when fixing the variables on the coordinates axes they were transferred the consequences of the Current Performing Conditions (CPC) to the curve.
The Index of Concavity (index b) represents the curve shape.
When a Conceptual and Transcendent Change (CTC) modifies the slope and it elevates the trajectory of the curve, what modifying is been is really the value of the index b.
COMPUTER PROGRAMS
Several computer programs were developed to determine different analyzed project values. This computer programs make use of the representative equation of the Maximum Limit of Optimization Postulate, that is, the equation (a).
MLO FOR THE FIRST STAGE
The most probable MLO value it cannot be calculated in the first stage because it is not possible to know its index b without knowing the curve shape. To save this inconvenience it is supposed that the improvement of the second stage is placed inside the area B of the Figure 1. (If it were placed in the area A it would be a CTC. If it were placed in the area C, with negative slope, the project would worsen).
Figure 1
Then, it can be considered the tendency of the project calculating the values maximum, minimum and average of the Maximum Limit of Optimization, starting from appropriate index b. The values adopted next to equalize the denominator of the equation (b), they were chosen to facilitate the calculations.
Maximum value of the MLO
The denominator of the equation (b) it is the nearest possible to zero. Adopting 0.1, it is:
Operating, it is calculated the value for L maximum: index b = 0.1
This value determines a curve of minimum concavity: the improvement of the second stage will be steep and it will touch lightly the first stage prolongation, passing close to the straight line a.
Minimum value of the MLO
The denominator of the equation (b) it is the nearest possible to one. Adopting 0.95, it is:
Operating, it is calculated the value for L minimum: index b = 3
This value determines a maximum concavity curve: the improvement of the second stage will be flattened and it will pass near the straight z.
Average value of the MLO
It is considered an intermediate index of concavity: index b = 1.5
ANALYSIS OF THE IMPROVEMENTS OF THE FIRST STAGE
The TABLE 1, made by a computer program, considering the improvements of the first stage in intervals of 5 % and calculating the values of the MLO (L) minimum, average and maximum. When studying projects that can quantify their results, it could be more convenient to calculate the values of the MLO considering the improvements of the first stage in intervals of 1 %. Furthermore, to determine more adjusted values to the existent conditions, it could be used index b known or estimated.
It is also indicated in what stage (S) the desired result of 100 % would be reached (“0” indicates that with these improvement percentage and index b, never the 100 % could be reach). Some of the values of the MLO shown in the TABLE 1 overcome 100 %; these values like example of the calculations are shown; besides, it highlights a particularity that confirms the philosophy of the Conceptual and Transcendent Changes (CTC): the highest values in the MLO correspond to curves with high slopes (that is to say, with low index b).
For initial improvements until 25 %
In the TABLE 1 it is:
- If the improvement is smaller than 10 %, in any case one will be able to reach the desired result (the whole project should reconsider).
- If the improvement is between 10 % and 25 %, only under uncommon good conditions it could reach 100 %, but too many stages would be needed even this way (there is not coherence among good conditions and an improvement so low, for what the implementation should be revised or to restate the whole project).
For initial improvements higher to 25 %
It is necessary to know the index b of the representative curve of the project (this value will be able to be estimated, to be copied of similar previous projects, or to be calculated when knowing the improvement of the second stage):
- If the index b is high (b = 3), to have MLO values near to the desired result the improvement of the first stage it should be of 80 % or higher, and CTC should be produced even this way (if the improvement doesn't reach that value or if the necessary changes cannot take place, the CPC should be changed and the whole project must be restated).
- If the index b is an average value (b = 1.5) and the improvement is between 26 % and 55 %, CTC should be produced to obtain the desired result (if the changes are not enough the project should be restated).
- If the index b is an average value (b = 1.5) and the improvement is between 56 % and 75 %, the CPC facilitates the necessary CTC to reach the desired result (if the evaluator estimates that it is not this way, it will restate the project).
- If the index b is an average value (b = 1.5) and the improvement is higher than 75 %, the desired result will be reached naturally in 1, 2 or 3 stages, depending on the value of the improvement.
VALUE OF THE INDEX OF CONCAVITY (b)
The TABLE 1 presents values of the MLO for three different index b, but in the reality the form of the representative curve of a project can take any value between the values minimum and maximum proposed.
Not to forget that when the denominator of the equation (b) was equaled to a near value to zero was adopted 0.1 and that when equaling that denominator to a near value to one it was adopted 0.95. For the common cases the adopted values are enough, but higher precisions could be needed when analyzing scientific topics (to adopt near values to zero and smaller than 0.1 or near values to one and higher than 0.95).
In the TABLE 2, generated by a computer program, for an improvement of the first stage of 40 % the index b is shown in intervals of 0.1, the values of the MLO obtained according to each index b and the stage (S) in that the desired result of 100 % will be reached.
The data of the TABLE 2 are important for the evaluator: they allow him to predict the shape of the curve and to know the intensity that it will have the CTC to modify the tendency and to reach the desired result. Knowing if these changes are possible he will be able to accept their realization or he will decide to restate the whole project.
MLO VALUES FROM THE SECOND STAGE
All the stages, except the first one, know the history of the curve for their previous stages. In each stage, when combining previous data and current data, an index b it is calculated adjusted to the reality of that stage. With the indexes calculated in the successive stages they are determining MLO values each time more probable. This methodology allows a permanent control of the evolution of a project, since it actualized in each stage the forecast on its final result, noticing on the improvements or deteriorations produced.
In each stage (n), the only data are:
- The total improvement reached in the previous stage: (In-1)
- The total improvement reached in the current stage: (In)
- The improvement contributed by the current stage: (they decrease the estimate errors calculating this value for difference among the current and previous reached total improvements): (In - In-1).
SHAPE FACTOR (f)
To relate the existent data with the well known equation (a) and to determine the index b, an intermediate factor is calculated, the Shape Factor, that it is defined as the improvement contributed by the current stage divided by the total improvement reached in the current stage:
Determination of the index of concavity b of each stage
The Shape Factor relates the slope of the current stage with the resulting slope of all the previous stages, and it is associates to the transitory state. The exponential term of the equation (a) it is also associates to the transitory state. Equaling and operating:
REFERENCES
- B C Kuo, Automatic Control Systems (Cía. Editorial Continental S.A., México DF), 1965, Ed authorized by Prentice Hall, Inc, Englewood Cliffs, N.J.), (2.1).
- P F Smith and A Sullivan Gale, Elements of Analytic Geometry (Librería-Edit. Nigar, Buenos Aires, 1955, Edición autorizada por Ginn & Co, Boston, USA), II-V. Chapter III-32.
- F. Cernuschi y F. I. Greco, Teoría de Errores de Mediciones (Ed. Universitaria de Buenos Aires, Argentina, 1974), Capítulo V-6 (Ajuste de curvas).
- Staff Members of the Massachusetts Institute of Technology, Electric Circuits. Cía. Ed. Continental SA, México DF, 1963, autorizado por John Wiley & Sons, Inc, NY,3-6.
- F W Sears, Electricity and Magnetism (Ed. Aguilar, Madrid, 1959, authorized by Addison-Wesley Publishing Co, Inc, Reading, Massachusetts), Chapters 8-5 y 13-3.
- W. L. Everitt and G. E. Anner, Communication Engineering (Editorial Arbó S.A.C.I., Buenos Aires, 1961, Edition authorized by McGraw-Hill Book Co, Inc), Cap. II-14.
- Tablas Científicas, Ciba-Geigy S.A., Basilea, Switzerland, 6° Edición, 1973, (I).
- ibid 6, Chapter III.
- ibid 4, Chapter III-4.
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