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Probability model quality of informations systems/Tikimybiniai daugiaparametriu gaminiu kokybes kontroles modeliai.

Introduction

In companies which utilize modern technologies databases of manufactured production and technological processes are constantly maintained and complemented. Data is often read and input automatically. By using computer networks and database control and analysis systems, information can be transmitted in real-time and can be used in decision making processes at each intermediate or final stage of manufacture. Thus there is a possibility to use information not only from the current but also from the previous stages of manufacture. That can increase the quality of information systems (IS) [1-4].

Modern IS control system involves control of raw materials, components, transitional control of elements or nodes, technological process and final product. In this way IS control system plays an important role by influencing cost and the final result.

Continuous inter-operational quality control is commonly applied in manufacture process of IS products. Selective control peculiarities of such IS are analyzed in publications [1-6] on the grounds of color picture tube manufacture specifics. In this paper we will describe the performance of multistage continuous inter-operational control with the help of stochastic models, when IS classification errors of the first and second kind are present. Main attention is paid to the transformation of production defectivity level probability distributions, which in turn allows to estimate the efficiency of inter-operational control in the way of modeling, and to select the required number of control stages and their characteristics [5-11].

[FIGURE 1 OMITTED]

Analyzing double-level control schematics shown in fig. 1, when IS is described as l-dimensional parameter vector [1 - 4], i = 1 - l.

Here [[omega].sub.i], [[theta].sub.i], [[tau].sub.i] - defective IS probabilities by i-th parameter before first level [K.sub.1], before and after second level [K.sub.2]. These are accidental values with densities [f.sub.i]([[omega].sub.i]), [g.sub.i]([[theta].sub.i]), [h.sub.i]([[tau].sub.1]) and distribution functions [F.sub.i]([[omega].sub.i]), [G.sub.i]([[theta].sub.i]), [H.sub.i]([[tau].sub.1]). Respectively [omega], [theta], [tau] - analogical characteristics for product by all controlled l parameters.

For further analysis good IS probabilities by i-th parameter are needed: [[xi].sub.i] = 1 - [[omega].sub.i], [[eta].sub.i] = 1 - [[eta].sub.i], [[zeta].sub.i] = 1 - [[tau].sub.i] accidental values with densities:

[[??].sub.i]([[xi].sub.i] = [f.sub.i](1 - [[xi].sub.i]), [[phi].sub.i]([[eta].sub.i]) = [g.sub.i](1 - [[eta].sub.i]), [[??].sub.i]([[zeta].sub.i]) = 1 - [h.sub.i](1 - [[zeta].sub.i]). (1)

And distribution functions

[[??].sub.i]([[xi].sub.i] = [F.sub.i](1 - [[xi].sub.i]), [[phi].sub.i]([[eta].sub.i]) = 1 - [G.sub.i](1 - [[eta].sub.i]), [[??].sub.i]([[zeta].sub.i]) = 1 - [H.sub.i](1 - [[zeta].sub.i]). (2)

Accidental values: [xi] = 1 - [omega], [eta] = 1 - [theta], [zeta] = 1 - [tau] - are good IS probabilities by all l parameters.

Analyzing situations, when all accidental values [[theta].sub.i], also accidental values [[eta].sub.i] densities [g.sub.i]([[theta].sub.i], [[phi].sub.i]([[eta].sub.i]) are known and written in beta law: [[theta].sub.i] ~ Be([a.sub.i], [b.sub.i]), [[eta].sub.i] ~ Be([b.sub.i], [a.sub.i]), here [a.sub.i], [b.sub.i] - beta law forms parameters [1 - 4]. Then [[tau].sub.i] is directly transformed (T), o [[omega].sub.i] - reverse transformed (A) accidental value [[theta].sub.i] with transformation parameter [[??].sub.i] :

[[??].sub.i] = [[beta].sub.i]/1 - [[alpha].sub.i], [[alpha].sub.i] + [[beta].sub.i] < 1 ; (3)

where [[alpha].sub.i]=const, [[beta].sub.i]=const - first and second kind errors probabilities [1] by i-th parameter, i = 1 - l.

Densities [h.sub.i]([[tau].sub.i]) ir [f.sub.i]([[omega].sub.i]) by analogy with [1] models are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (4)

where [B.sub.i] = B([a.sub.i], [b.sub.i]) = [GAMMA]([a.sub.i])[GAMMA]([b.sub.i])/[GAMMA]([a.sub.i] + [b.sub.i]) - beta function; [GAMMA](z) - gamma function; [??] = [[gamma].sub.i]/1 - [[alpha].sub.i] = 1 - [[??].sub.i], [[gamma].sub.i] = 1 - [[alpha].sub.i] - [[beta].sub.i],

[c.sub.i] = [[??].sub.i]/[[??].sub.i] [equivalent to] [[gamma].sub.i]/[[beta].sub.i], [GAMMA](1) = 0! = 1.

Accidental values [[theta].sub.i], [[eta].sub.i] densities and main numerical characteristics are shown in [1-2]. Distribution function [G.sub.i]([[theta].sub.i]) in common case is calculated by [6-9], for [a.sub.i], [b.sub.i] are whole numbers, we get [G.sub.i]([[theta].sub.i]) and [H.sub.i]([[tau].sub.i]) :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

when k=[a.sub.i]+[b.sub.i]-1, then in the sum [summation over (k)] setting in member

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

where [C.sup.M.sub.n] - combinations from n by m : n!/m!(n - m)!.

Accidental values [[tau].sub.i], [[omega].sub.i] averages [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and dispersions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] calculating in initial moment E[[omega].sup.k.sub.i], E[[tau].sup.k.sub.i], k = 1, 2 help [7 - 10]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

We get, that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[1] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

In common case, when [a.sub.i]>0, [b.sub.i]>0 applying Taylor series:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

If [a.sub.i], [b.sub.i] are whole numbers [greater than or equal to] 1, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Reversed A transformation models immediately got from T transformation models, to direct T transformation expressions instead of parameter [[??].sub.i] we set-in reversed value 1/[[??].sub.i] (analogically instead of -[[??].sub.i] we set-in [c.sub.i] or vice-versa):

j[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Then if [a.sub.i], [b.sub.i] are whole numbers [greater than or equal to] 1, we get (when k=1; 2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Double-parameter IS (l = 2), i=1; 2

Analogically with [1-4] models we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e. both parameters [K.sub.2] level (T transformation) not measured, we get [zeta] = [eta] and [PHI]([zeta]) = [PHI]([eta]), [??]([zeta] = [phi]([eta]).

Let IS describe as: [a.sub.i] = [b.sub.i] = 1, [g.sub.i]([[theta].sub.1])=[[phi].sub.i]([[eta].sub.i])=1, [phi]([eta]) = -ln [eta], then we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

where h(0) = f(0) = 0, h(1) = f(1) = [infinity].

When [[mu].sub.i] = [[bar.[mu]].sub.i] = 1/2, [[sigma].sup.2.sub.i] = 1/12, we get [bar.[mu]] = [[bar.[mu]].sub.1] [[bar.[mu]].sub.1] = 1/4, [mu] = 1 - [bar.[mu]] = 3/4 and when [[sigma].sup.2.sub.1] = [[sigma].sup.2.sub.2], [[bar.[mu].sup.2.sub.1] = [[bar.[mu].sup.2.sub.2], [[sigma].sup.2] = [[sigma].sup.2.sub.1] [[bar.[mu].sup.2.sub.2] + [[sigma].sup.2.sub.2] [[bar.[mu].sup.2.sub.1] + [[sigma].sup.2.sub.1] [[sigma].sup.2.sub.2] = [[sigma].sup.2.sub.1](2[[bar.[mu].sup.2.sub.1] + [[sigma].sup.2.sub.1]) = 7/144.

In partial case, when [[??].sub.2] =1 (unmeasured second IS parameter), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (24)

where 1 + [c.sub.1] = 1/[[??].sub.1].

It is obvious, that analogical expressions are when [[??].sub.1] = 1 (unmeasureable first parameter).

In partial case, when [[??].sub.2] = 1, we get: [[tau].sub.2] = [[omega].sub.2] = [[theta].sub.2] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Take that IS is characterized with parameters: [a.sub.i]=1, [b.sub.1]=2, [b.sub.2]=1; [[theta].sub.1]~Be(1,2), [[theta].sub.2]~Be(1,1), [theta]~Be(2,1), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

[??](1) = h(0)= 0, (26)

[??](0)= h(1)=[[??].sub.2](1+ [[??].sub.1]). (27)

In this case: [[mu].sub.1] = 1/3, [[mu].sub.2] = 1/2, [[bar.[mu].sub.1] = 2/3, [[bar.[mu].sub.2] = 1/2, [bar.[mu]] = 1/3, [mu] = 2/3, [[sigma].sup.2.sub.1] = 1/18, [[sigma].sup.2.sub.2] = 1/12, [[sigma].sup.2] = 1/18.

Partial cases ([[bar.[beta]].sub.2] = 1 or [[bar.[beta]].sub.1] = 1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Transformed densities approximations

It is obvious, if values [a.sub.i], [b.sub.i] grows, [??]([zeta]) (17) expression after integrating becomes more and more complicated, so it becomes disadvantageous for modeling, also integrating process becomes more complicated. In that case, for density g([theta]) [1-4] and for densities h([tau]) and f([omega]) we apply approximations.

In density g([theta]) regard there are three cases:

1. Known exact density g([theta]) model (not beta density).

Applying h([tau]) approximation to density [h.sub.[SIGMA]]([tau]):

[h.sub.[summation]]([tau])=[tau]/[??](1 + c[tau]), [theta]'([tau]) = [partial derivative][theta]([tau])/[partial deriative][tau] = 1/[??][(1 + c[tau]).sup.2]; (30)

where [theta]([tau]) = [tau]/[??][(1 + c[tau]).sup.2], [theta]'([tau]) = [partial derivative][theta]([tau])/[partial derivative][tau] = 1/[??][(1 + c[tau]).sup.2].

2. One-parameter IS (l = 1) either fixed [[theta].sub.i] value we have

[[??].sub.i] = [[beta].sub.i]/1 - [[alpha].sub.i] = [[eta].sub.i][[tau].sub.i]/[[theta].sub.i][[zeta].sub.i] = const. (31)

3. Double-parameter IS resultant normalized error probability [??] (direct transformation parameter for all IS) depends on parameters [[??].sub.1], [[??].sub.2], [[theta].sub.1], [[theta].sub.2].

Applying [7-10] models and get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (32)

where 1 - [alpha] = (1 - [[alpha].sub.1])1 - [[alpha].sub.2]); [beta] = p[tau]/[theta]; p = [p.sub.1][p.sub.2] - IS acceptance as good probability, [p.sub.i] = 1 - [[alpha].sub.i] - [[gamma].sub.i][[theta].sub.i] = (1 - [[alpha].sub.1]) x (1 - [[??].sub.i][[theta].sub.i]), i=1, 2 - IS acceptance as good by i-th parameter probability.

To limited [theta] values [theta]=0 and [theta]=1 we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Then parameter [??] is accidental value with its average value E[??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (34)

where [??]([theta]) - by (20).

To (30) model we set-in fixed parameter [??] value, which is equal to (34)

[[mu].sub.[tau]]([beta]) = E[tau], [[sigma].sup.2.sub.[tau]]([??]) = V[tau]. (35)

Then [??] is (35) or (36) equations solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (36)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

Analogically for A transformation we apply density f([omega]) approximation with density [f.sub.[summation]](u):

[f.sub.[summation]]([omega]) = [theta]'([omega] * g[[theta]([omega])] = [??] x g[[theta]([omega])]/[(1 - [??][omega]).sup.2]; (38)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Parameter [beta] is (25) equation solution.

For A transformation we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

Realizations

When [omega]=0 and [omega]=1, [??]([omega]) gets same values as (32).

[FIGURE 2 OMITTED]

Typical functions [[mu].sub.[tau]]([??]) and [[mu].sub.[omega]]([??]) dependencies curves are shown in fig. 2, if a1, At values are fixed. Numerical IS realizations are shown in fig. 3 - 6.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Conclusions

1. It is better to use beta densities for different parameters levels description for multiparameter measuring systems defect levels probabilities modeling, because it simplifies further modeling procedure.

2. Obtained expressions lets modeling of wanted situations in between operational-control schematics visually by defect level densities transformations (on computer display), giving densities parameters in desirable schematic place and choosing real separate parameters classification probabilities in addition with controlled parameters nomenclature.

3. Exact multiparameter expressions of IS defect levels expressions are very complicated because of multinomial integration procedure. Offered modeling variants should serve as useful instrument for control system design.

Received 2009 01 13

References

[1.] Eidukas D., Kalnius R. Stochastic Models of Quality Level of Mechatronic Products // Electronics and Electrical Engineering.--Kaunas: Technologija, 2008.--No. 3(83).--P. 43-48.

[2.] Eidukas D., Kalnius R. Stochastic Models of Quality in Continuous Information Systems // Proceedings of 30 International Conference on Information Technology Interfaces ITI 2008, June 23-26, Cavtat/Dubrovnik, Croatia.--P. 703-708.

[3.] Eidukas D. Stochastic Models Quality Electronics Systems // Electronics and Electrical Engineering.--Kaunas: Technologija, 2008.--No. 5(85).--P. 41-44.

[4.] Eidukas D., Kalnius R., Vaisvila A. Probability Distribution Transformation in Continuous Information Systems Control // ITI 2007: proceedings of the 29th international conference on Information Technology Interfaces, June 25-28, 2007, Dubrovnik, Croatia / University of Zagreb. University Computing Centre. Zagreb: University of Zagreb.--2007. P. 609-614.

[5.] Eidukas D., Baraisis P., Valinevicius A., Zilys M. Reliability and Efficiency of Electronics Systems.--Kaunas: Technologija, 2006.--316 p.

[6.] Kalnius R., Eidukas D. Applications of Generalized Beta-distribution in Quality Control Models // Electronics and Electrical Engineering.--Kaunas: Technologija.--2007. No. 1(73).--P. 5-12.

[7.] Kruopis J., Vaisvila A., Kalnius R. Mechatronikos gamini? kokybe. Vilnius: Vilniaus universiteto leidykla.--2005.--P. 518.

[8.] Caramia M., Dell'Olmo P. Effective Resource Management in Manufacturing Systems Optimization Algorithms for Production Planning. Springer.--2006.--P. 216.

[9.] Conte G., Moog C. H., Perdon A. M. Algebraic Methods for Nonlinear Control Systems 2nd ed. Springer.--2006.--P. 178.

[10.] Levitin G. The Universal Generating Function in Reliability Analysis and Optimization. Springer.--2005.--P. 442.

[11.] Kalnius R., Vaisvila A., Eidukas D. Probability Distribution Transformation in Continuous Production Control // Electronics and Electrical Engineering.--Kaunas: Technologija, 2006.--No. 4(68).--P. 29--34.

R. Kalnius, D. Eidukas

Department of Electronics Engineering, Kaunas University of Technology, Student? str. 50, LT-51368 Kaunas, Lithuania, phone:+370 3 7 351389, + 370 37 773529, e-mail: danielius.eidukas@ktu.lt
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Title Annotation:ELECTRONICS/ELEKTRONIKA
Author:Kalnius, R.; Eidukas, D.
Publication:Elektronika ir Elektrotechnika
Article Type:Report
Geographic Code:4EXLT
Date:May 1, 2009
Words:2371
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