Stochastic Models of Quality in Continuous Information Systems

Danielius Eidukas, Rimantas Kalnius Department of Electronics Engineering, Kaunas University of Technology

Studentu str. 50, LT-51368 Kaunas, Lithuania danielius.eidukas@ktu.lt

Abstract. Mathematical models of main stochastic characteristics of continuous multi-stage inter-operational control for the IS of mechatronics were created, when the flows of rejected IS are returned to the manufacture process for regeneration, and IS classification rules are analogous at the separate control stages, and when undeniable IS classification errors of the first and second kind are present. The efficiency of control operation is estimated in the way of modeling according to transformations of density functions of defectivity level at separate stages, by applying beta distribution and dynamics of defectivity level mean variation. It was shown that returning flows are determined by error probabilities and the level of defectivity before control, and the models of separate stages differ only in the magnitude of generalized error, which equals the product of generalized error probabilities of separate stages. Manufacturing peculiarities of IS and their components were employed during creation of models.

Keywords. Information systems, IS, efficiency, models, quality, beta-distribution.

1. 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].

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. When control is tightened (when new control stages are introduced, number of controlled parameters is increased, rules of acceptance are changed, etc.) losses due to reclamations decrease, but internal losses will grow due to increased control work rate and due to the fact, that more quality products will be falsely ranked as defective. On the other hand, when control is loose, losses due to reclamations will increase, but internal losses will decrease. Naturally a task emerges to select economically optimal variant of control. It is obtained by minimizing adequately-formulated function of losses.

Tasks solved during control are attributed to several areas of mathematical statistics, specifically – decision making theory, identification, classification, discriminational analysis, image recognition, etc. But in manufacture it is apparently more convenient (compare to other areas) to apply methods of mathematical statistics, for which aprioriinformation is required.

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.

Complex mechatronics products are defined in technical documentation as an entire series of parameters, the values of which determine the level of product quality. Parameters can be

703Proceedings of the ITI 2008 30th Int. Conf. on Information Technology Interfaces, June 23-26, 2008, Cavtat, Croatia

differentiated according to their importance regarding the implementation of purpose functions. International standard ISO-2859-0 [11] recommends to divide parameters into two – A, B – or three – A, B, C – classes (groups). Here A – most important or significant parameters, and B, C – secondary or less significant parameters. Such classification of parameters is convenient when analyzing problems of multiparametric product quality control [7–10]. When imitational modeling is applied, stochastic models of quality level are required for separate parameters, their groups and for entire product.

2. Control scheme and models

Inter-operational control fragment is presented in Fig. 1, which involves two stages of continuous control 1K and 2K (in both stages IS are classified according to analogical decision rules).

G K1 K21 p1 p2

ω θ τ

1, 1 2, 2

p12

q1 q2q12

Figure 1. The scheme of two-stage inter-operational continuous control

There is a probability ω that a defective IS after manufacture operations G will enter the control stage 1K which is characterized by classification error probabilities 1α and 1β ;probability ω is transformed to parameter θ after control operations. IS acceptance probability is p1 and rejection probability is q1.

Analogously in the second stage 2K with errors 2α and 2β parameter θ is transformed to τ, when probabilities p2, q2 are fixed. IS in such scheme is accepted with probability q12. Rejected IS are returned for reparation to manufacture process G .

3. Models with random values

The defective IS probability in the entirety of IS, as a random value [2], before control 1K is characterized by density ( )ωf (Fig. 2), before

2K – by density ( )θg and after control 2K – by density ( )τh .

G K1 K2( )ωf ( )θg ( )τh

ωµ θµp ,1p12

q1 q2q12

τµp ,2

Figure 2. Stochastic characteristics of two-stage control

The means of defectivity level respectively are τθω µµµ ,, :

( ) ( )

( )=

==

.,

,,,,

1

0

1

0

1

0

τττµ

θθθµωωωµ

τ

θω

dh

dgdf

(1)

We receive average sizes of accepted flows 1221 ,, ppp , and average flows of rejected

products 1221 ,, qqq :

( ) ( )[ ]

( ) ( )[ ] =−−−=

−−−=

;,~111

;~111

2112222

111

pppp

p

θ

ω

µβα

µβα (2)

where ,, 12122211 111 pqpqpq −=−=−= . (3)

Densities ( )ωf , ( )θg , ( )τh are presented in Fig. 3.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1

( ) ( ) ( )τωθ hfg ,,

θτω ,,ωµ

θµ

τµ

( )τh

( )θg

( )ωf

Figure 3. Densities of defectivity levels, when

2=a , 3=b , 25.0~ =β

704

4. Model with two – stage control

Let’s consider two-stage continuous control (modified), in which the returning flow is passed into its own repair (regeneration) operation Riafter each control stage Ki, i=1, 2 (Fig. 4).

G K1 K2g( ) h*( *) h*(τ*)µ *

1µ

1, 1 2, 2

1Q 2Q

*2µ

R1 R2*1β *

2β

Figure 4. The modified two-stage continuous control

Densities )(τh and )( 22 τh are shown in Fig. 5.

0 0,2 0,4 0,6 0,8 1

4

3

2

1

0

( ) .2,1),(),(, ** =ihhg iiii ττθ

)( *2

*2 τh 2

1~0 == ββ

)( 22 τh

)(τh

)( ** τh

)(θg

θ*, ii ττ

20β 0β

Figure 5. Densities of defectivity levels, when

a=b=1, 00 =v , 11 =v

5. Beta-distribution quality level

Assume, that IS is characterized by l–dimensional random vector of independent parameters ( )lXXXX ,,, 21= . Probability of defective IS iθ (in respect of the i-th parameter) is a random quantity (r.q.) with its density

( )iig θ , distribution function ( )iiG θ and main numerical characteristics: mean ii µθ =Ε and

dispersion 2iiV σθ = , where li ,,2,1= . For

practical application we are going to use additional dispersion characteristics: standard deviation ii Vθσ = and variation coefficient

i

iiv

µσ

= [2].

Probability of good product iη according to one parameter is

ii θη −=1 . (4)

R.q. iη is characterized by density ( )ii ηϕ ,distribution function ( )ii ηΦ , mean ii µη =Ε and dispersion 2

iii VV σθη == . The following relation formulas are valid:

( ) ( )−Φ−=−=

−=

iiii

iiii

ii

Gg

θθθϕθ

µµ

11),1()(

,1

. (5)

The probability of defective product for the entire product r.q. θ and probability of good product r.q. η are calculated as

−−=−=

=

∏

∏

=

=l

ii

l

ii

1

1

)1(11 θηθ

ηη

.

(6)

Since iη are inter-independent r.q., the following equation is valid

∏=

=l

iii

1

)()( ηϕηϕ . (7)

In general, the distribution function ( )ηΦ of the random multiplicative r.q. lηηη 1=according to [8,9] is defined by l-dimensional integral

∏=

=Φ l

l

ill

D

ddd ηηηηϕηϕηϕηη

...)()...()(...)( 211

2211 , (8)

here ηD – integration range. Then density ( )ηϕis expressed according to (9)

( )ηηηϕ

dd )(Φ= . (9)

We will provide the main formulas, required for the further analysis, when r.q. iθ and also iηare distributed according to the beta-distribution with shape parameters ia , ib and marking that:

( )iii baBe ,~θ , ( )iii abBe ,~η [8]:

−=

−=−−−

−−−

111

111

)1(),()(

)1(),()(ii

ii

ai

biiiiii

bi

aiiiiii

baB

baBg

ηηηϕ

θθθ, (10)

here ( ) ( ) ( )( )ii

iiiiii ba

baBbaB

+ΓΓΓ

=≡, – beta function,

( )izΓ – gamma function, ( ) ( ) ( )11 −Γ−=Γ iii zzz

705

or ( ) ( )!1−=Γ nn , when n is a whole number (h.n.), li 1= ;

++=

+++=

+=−=

+=

1)1()(

1,

22

ii

ii

iiii

iii

ii

iii

ii

ii

babababa

bab

baa

µµσ

µµµ ; (11)

)1( ++== iii

i

i

ii ba

ba

σµν . (12)

If 1=ia and iµ is sufficiently small ( 03.0<iµ ), then 1>>ib and 1≈iv , i.e. ii µσ ≈ .

Density ( )iig θ has maximum at the point iMθ (mode) [8]

21−+

−=ii

iiM ba

aθ . (13)

Respectively the maximum of density ( )ii ηϕis at the point iMiM θη −= 1 .

It is obvious, that (9)–(13) formulas may be applied for entire product, if r.q. ( )baBe ,~θ , or for separate groups A, B, C, if their defectivity levels Aθ , Bθ , Cθ are characterized by beta-distribution.

IS is characterized using two parameters 2,1=i , which are distributed according to beta-

distribution (4 – 14). Then according to (7), equations 21ηηη = , 2121 θθθθθ −+= are valid.

In order to avoid integration bands 0=iη we use dependency

( ) { } { }yYPyYPyF >−=<= 1 , (14)

here ( )yF is distribution function of r.q. Y ,{ }yYP > is probability, that yY > . In this way,

when yY > , integration range ηD (9) is defined by hyperbola 21ηηη = with upper variation interval limit 1=iη according to [1] (Fig. 6).

Figure 6. Two-dimensional integration space ηD , 2=l

We receive

( ) ( ) ( ) =−=Φ1 1

/222111

1

1η ηη

ηηϕηηϕη dd

⋅−= −−−1

/

12

11

11

21

1

21)(1ηηη

ηη bbBB (15)

211

21

121 )1()1( ηηηη ddaa −− −−⋅

( ) ( ) ==Φ=1

11

2111

1)(η

ηηηϕηϕ

ηηηηϕ d

dd

⋅−= −−−1

/

12

11

11

21

1

21)(1ηηη

ηη bbBB (16)

In order to obtain beta-densities in groups and for entire product, the condition

lbbb >> 21 according to (16) must be satisfied. Since 10== ilaa , when 5.0== µµwe have 10== lbb and ( )ibbi −+= 1010 ,

91÷=i . Then 119 =b , 128 =b , 137 =b , 146 =b ,155 =b , 164 =b , 173 =b , 182 =b , 191 =b .

Models of beta-densities according to (10) with their shape parameters: ( )ii b,1~θ , when

1010 =b , 191 =b . Average defectivity levels according to separate parameters:

05,01 =µ , 0526,02 =µ , 0556,03 =µ , 0588,04 =µ ,0625,05 =µ , 0667,06 =µ , 0714,07 =µ ,0769,08 =µ , 0833,09 =µ , 0909,010 =µ .

Densities ( )iig θ are monotonically decreasing functions from ( ) ii bg =0 to ( ) 01 =ig .

Densities and main numerical characteristics according to groups for both cases (Table 6) are slightly different, but for entire product densities

( )θg of defectivity level θ and numerical characteristics µ , 2σ practically coincides for both cases. It is obvious, that coincidence of all characteristics is the better the smaller µ value is.

6. Practical applications

Three groups A, B, C; %10=µ , 10=l ,2=r , 3=s , 2,1=∈ iA , jiB ≡=∈ 4,3 ,

kiC ≡÷=∈ 106 1=ia , 2=ja , 3=ka .Modeling results are presented in Table 1. In

columns AB, BC, AC modeling results are indicated, when one of parameter groups is eliminated and IS is evaluated according to

706

remaining two groups. For variant AC the condition of exact beta-distribution between A and C groups ( )222228 662 =+>= abb is not met any more, therefore density ( )ACACg θ is approximated using density

,)1()6,207(!16

)6,224()( 6,20616 θθθσ −Γ

Γ=ACg (18)

where 42 101,3 ⋅=ACθand remaining densities are described as in

,4,3,2,1,2,4%,6 =∈==∈==== iBiAslm σ2,11 == jaa with their parameters a, b (when

0>Mθ , densities are unimodal, and when 0=Mθ – monotonically decreasing curve).

Table 1. 10=l , %10=µ ; groups A, B, C, 2=r , 3=s

i a b µ ,%µ ,%σ υ ,%Mθ

1 1 229 229/230 0,435 0,432 ~1 02 1 228 228/229 0,437 0,435 ~1 03 2 226 226/228 0,877 0,616 0,7 0,442 4 2 224 224/226 0,885 0,621 0,7 0,446 5 2 222 222/224 0,893 0,627 0,7 0,450 6 3 219 219/222 1,35 0,773 0,57 0,910 7 3 216 216/219 1,37 0,784 0,57 0,922 8 3 213 213/216 1,39 0,794 0,57 0,935 9 3 210 210/213 1,41 0,805 0,57 0,948

10 3 207 207/210 1,43 0,817 0,57 0,962 A 2 228 228/230 0,87 0,611 0,7 0,44 B 6 222 222/228 2,63 1,06 0,4 2,21 C 15 207 207/222 6,76 1,68 0,25 6,36

23 207 207/230 10 1,97 0,2 9,65 AB 8 222 222/230 3,48 1,21 0,35 3,07 BC 21 207 207/228 9,21 1,91 0,21 8,85

AC*) 17 207,6 3933/ 4255 7,57 1,76 0,23 7,19

*) density ( )ACACg θ is approximated using beta-density ( )ACg θσ .

7. Conclusions

1. Received expressions give opportunity to model various desirable situations in the scheme of inter-operational control in visual manner, according to transformations of density of defectivity levels, by defining density parameters at desired point of scheme (density of beta distribution) and selecting real error probability values of separate control stages.

2. The control scheme with localized repair operations of rejected products (Fig. 3) is more efficient than the scheme in Fig. 2, in which the returning flows are returned back to the manufacture, but the maintenance of individual

repair workplaces is more expensive from economical point of view.

3. It is purposeful to use approximation of density ( )θg by beta-density ( )θσg when number of parameters 5<l , since when l value increases, the relative approximation error also increases.

4. Grouping of parameters according to their significance into classes (groups) A, B, C is recommended by ISO standards and it enables to highlight the influence of these groups onto the overall product defectivity level; it also enables modeling of situations when one group of parameters is eliminated or additional parameters are introduced.

7. References

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[3] Brogliato B., Lozano R., Maschke B., Egeland O. Dissipative Systems Analysis and Control Theory and Applications 2nd ed. ISBN-10 1-84628-516-X. Springer, 2006. – 576 p.

[4] Caramia M., Dell'Olmo P. Effective Resource Management in Manufacturing Systems Optimization Algorithms for Production Planning. ISBN-10: 1-84628-005-2. Springer, 2006. – 216 p.

[5] Conte G., Moog C. H., Perdon A. M. Algebraic Methods for Nonlinear Control Systems 2nd ed. ISBN-10 1-84628-594-1. Springer, 2006. – 178 p.

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[7] Eidukas D., Valinevi ius A., Kilius Š., Pl štys R., Vilutis G. Analysis of Communication channels of the Dwelling Control System // Proceedings of 28th

International Conference on

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[8] Murthy D.N.P., Blischke W. R. Warranty Management and Product Manufacture. ISBN-10: 1-85233-933-0. Springer, 2006. – 302 p.

[9] Kalnius R., Eidukas D. Applications of Generalized Beta-distribution in Quality Control Models // Electronics and Electrical Engineering. ISSN 1392-1215. – Kaunas: Technologija, 2007. – No. 1(73) – P. 5–12.

[10] Levitin G. The Universal Generating Function in Reliability Analysis and

Optimization. ISBN-10: 1-85233-927-6. Springer, 2005. – 442 p.

[11] International Standart ISO 2859-0: 1995. Sampling procedures for inspection by attributes – Part 0: Intoduction to the ISO 2859 attribute sampling system.

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

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