S.Sh. Kazhikenova

Karaganda University, Karaganda, Kazakstan

sauleshka555@mail.ru

About information estimation of technological

products quality

Entropy is a concept, which played a central role in a number of science fields, namely, in statistical mechanics and theory of information. Using the apparatus of probability theory permitted in the recent years to clear up the connections between various usages of entropy. The exact concepts of entropy and mathematics which appeared at the end of the 40-s and became stable first in the applied fields, namely, in the communication theory and cybernetics, were immediately subjected to a thorough mathematical processing and developed in some new fields of science. Fast, almost momentary introducing these concepts into different branches of mathematical, technical, social disciplines was conditioned by the fact that the corresponding mathematical apparatus for their analysis had already been prepared and, which is principal, there were problems which were waiting for the concept of entropy in one or another form, and solved with its help soon. In this connection on the basis of Shannon information entropy we have worked out a method of unifying still separate indices for extracting valuable components and their content in industrial products on processes and on the whole according to a technological scheme with the following using of this method for analyzing and comparative estimation of chemical-and-metallurgical industries. Algorithms of calculating the information capacity of the system suggested by Shannon permit to clear up the ratio of the determined information to the quantity of stochastic information and thus to give the possibility to determine the quality and quantity estimation of a certain technological scheme. In the general characteristic of the entropy-and-information analysis of any objects there is used Shannon statistic formula [1]:

, (1)

where i is a probability of detecting a system element; , .

We have considered this formula use for the quantity estimation of a product or a technical process quality uncertainty through the uncertainty of the major element of the system. As a probability of detecting the major element of a technological system there can be taken its content in the product expressed in a unit shares. For example, this is the content of the extracted chemical element in the products of a technological process. The same is true for the process of extracting an element in one or another product, because in such a case the extraction index is equal to the possibility of this element transition from one system state to another. To estimate the product quality or technological processes both of these two indices, content and extraction, can be in the equal degree be used.

Theorem 1. Let the own information of the technological system elements, consisting of N elements, be equal

. (2)

Then the entropy of the given discrete multitude is determined by:

, (3)

where i is a probability of detecting a system element; , .

Theorem 2. If a multitude of discrete probabilistic distributions has N elements, then the information entropy of the final discrete probabilistic distribution satisfies the condition:

.

Besides, then and only then, when the discrete multitude contains en element of the unit probabilistic element and then and only then, when the discrete multitude has a uniform distribution, i.e. ,

Theorem 3. If , are relative values of the information, entropy and on the basis of the law entropy and information sum preservation there is satisfied the condition:

, (4)

then is the solution of the equation:

,

where

The theorems proved show a continuous connection of the determination and stochastic components, of which the first is the dominating one and ensures stability, and the other determines the finest changes and the optimal information capacity of technological systems. The basis of the entropy-and-information analysis of technological processes is suggested by Shannon method of calculating the quantity of stochastic and determined information. Suggested by Shannon method of calculating the quantity of information and entropy turned out to be so universal, that its use is not limited by the narrow limits of purely technical applications. Before the publication of Shannon theory, R. Hartley suggested to determine the quantity of information by the formula which in relation to the levels has the form [2]:

, (5)

where n is a ordinary number of the level considered, ; k is the length of elements code at each of the levels of the hierarchy system; is the number of technological system level elements taken the starting point of counting, n=0.

Lets consider a technological scheme with , i.e. in such a case is the selection from a multitude of elements, an element and not element, containing in the product. Then (5) will take a form:

.

A principally important advantage of the information estimation of products quality or technological operations is that the suggested index, as any entropy-and-information value, can be summed for reflecting the whole system in this index. This property of additivity is immanently characteristic for entropy and information and is the base for expressing the law of their sum preservation [3]. Consequently, the technological uncertainty of different operations in the limits of a single technological scheme can be expressed by the system index of uncertainty:

bit/el., (6)

From the formulae for the determination and maximal information it follows that the determination and the system mponents of information are determined by the equalities:

bit/el., bit/el., (7)

bit/el., bit/el., (8)

The results of the calculations carried out for are presented in Table 1.

We have established the difference between the system and the level data, namely, that integral values of determination are less than those differentiated by levele due to taling into account the information of previous levels, characterized by larger stochastic properties. We see that when transiting to the higher structural level there comes into force the law or principle of the progressive increasing of diversity. As probabilities at these levels dont effect the production quality, in calculations its possible to take into account only inter-level correlations. The difference between harmonized, differentiated and integral models will be illustrated graphically in coordinates in accordance to Figure 1.

ble 1 Calculated entropy-and-information characteristics of technological processes in hierarchical system for ,

bit/el.

bit/el.

bit/el.

bit/el.

0

0

1,0

0

0

1,0

0

1

1,0000

2,0

0,5000

1,0000

3,0

0,3333

2

3,3333

4,0

0,8333

4,3333

7,0

0,6190

3

7,6667

8,0

0,9583

12,0000

15,0

0,8000

4

15,8667

16,0

0,9917

27,8667

31,0

0,8989

5

31,9556

32,0

0,9986

59,8222

63,0

0,9496

6

63,9873

64,0

0,9998

123,8095

127,0

0,9749

7

127,9968

128,0

1,0

251,8063

255,0

0,9875

8

255,9993

256,0

1,0

507,8056

511,0

0,9937

9

511,9999

512,0

1,0

1019,8055

1023,0

0,9969

10

1024,0000

1024,0

1,0

2043,8055

2047,0

0,9984

11

2048,0000

2048,0

1,0

4091,8055

4095,0

0,9992

12

4096,0000

4096,0

1,0

8187,8055

8191,0

0,9996

13

8192,0000

8192,0

1,0

16379,8055

16383,0

0,9998

14

16384,000

16384,0

1,0

32763,8055

32767,0

0,9999

15

32768,000

32768,0

1,0

65531,8055

65535,0

1,0

is level, is determination: 1 harmonized, 2 differentiated, 3 integral

Figure 1 Dependence of determination degree on level

 

Its obvious that the harmonized one is nearer to the integral one, for which the value of determination is less due to the contribution of the lower levels, characterized by larger stochastic determination. The integral one seems to depend significantly on the element code length. At the second level of an abstract technological scheme there is obtained a value practically coinciding with the golden section ratio. From here, in the arbitrary element base there must be especially widely spread three-level systems with the binary principle of organization.

LIST OF LITERATURE

 

1. Shannon K.E. Mathematical theory of connection // Works on the theory of the information and cybernetics. - M.:SILT,1963.With.243-332 .

2. Hartley R. Transfer of the information / the Theory of the information and its appendix. - M.: SILT, 1959. With. 5-35 .

3. lyshev V.P. Probabilistically determined image. - Almaty: Gylym, 1994. - 376 p.