Summary
Quality control in a production company is an issue of great importance. Quali-ty process control is known as the methodology of monitoring of production process. Essential is to detect disorders and errors connected with production process which can cause manufactured details unfit for further processing. Fabricated details which do not match to specifications and requirements can become a reason of de-lay in handing over to a customer and can cause extra financial charges connected for example with additional production of a specific detail. Details with defects al-ready installed in a car can also expose participants of traffic to danger. Systems based on a fuzzy logic and knowledge of experts for many years have been used to support proceedings in science, also in quality control. Fuzzy systems are created and used, when one cope with problems using traditional methods or possessed in-formation is simply not enough. Because real data are given with support of tolerances, usage of interval and fuzzy analysis is closer to reality. There is a neces-sity of aggregation of various attributes with usage of unified system. In paper underneath fuzzy approach – model of detail evaluation in a production company of automotive industry is being presented.
Keywords: fuzzy logic, expert systems, production company, detail evaluation 1. Introduction
Quality control is an important element of a production company activity. Considering ex-penses and time, essential is to detect detail errors as quickly as it is possible, before a defective element will get to the next phase of the production process. That is why detail control should be often conducted by both: machines operators (visual control) and inspectors of quality control. First control should be conducted before initiating of production process and during it all, materi-als (used in detail production) delivered by suppliers should be checked. Next controls should be done in each phase of a production process till manufactured element is ready to be transferred to receiver. Quality inspectors due to perform appropriate analysis use quality control methods, which aim is to improve their work during process supervising. In paper, estimation of product quality will be conducted with reference to one of components manufactured in an enterprise of automotive sector.
1. Fuzzy process control Literature review
During last years supporting of quality control processes has been subject of many research, conceptions of fuzzy logic are being presented in many scientific elaborations. In the next part of
typical average value was accepted as a middle line of control chart characterizing process under control1.
Figure below presents conception of control chart with usage of linguistic values, proposed by Wang and Raz.
Figure 1. Linguistic values used in Wang and Raz control chart [6]
In 1997 Grzegorzewski proposed fuzzy control chart consisting two supplementary charts of observations, represented by fuzzy numbers. First chart contained middle line, which adopted major, middle fuzzy value of all samples to estimate level of the process. Each sample was changed into an interval symbolizing fuzzy set. Intervals acquired this way were placed on the chart. The situation, where the interval did not cut the middle line, was interpreted as an indication of situation under control. Each interval corresponded with value in the second chart, in which each value was interpreted as grade of certainty, indicating, that process is out of control2. El-Shal and Morris examined usage of fuzzy logic to modification of statistical quality control rules. They aimed to reduce false alarms and improve detection and quickness of real faults detecting3. Row-lands and Wang in 2000 released an article, where they presented evaluation and control of fuzzy
1
Raz T., Wang J.H., Probabilistic and membership approaches In the construction of control charts for linguistic data, Production Plann. Cont. 1 (1990) s. 147–157 [3].
2
Grzegorzewski P., Control charts for fuzzy data, Proc. Fifth European Congress on Intelligent Techniques and Soft Computing EUFIT’97, Aachen, 1997, s. 1326–1330 [2].
3
El-Shal S.M., Morris A.S., A fuzzy rule-based algorithm to improve the performance of statistical process control in quality systems, Journal of Intelligent and Fuzzy Systems 9 (2000) s. 207–223 [1].
logic in statistical process control. They were analyzing an integration of fuzzy logic and control charts while constructing fuzzy method of control and evaluation, based on usage of fuzzy logic to create rules connected with division into acceptance zones4. Taleb and Limam discussed various methods of control charts construction with linguistic values, based on fuzzy sets and probability theory. Comparison of fuzzy and probabilistic approaches, based on average periods length and samples under control was created with the real data5.
2. Modelling of details quality using inference method based on fuzzy sets analysis
The concept of inference bases on estimating with usage of entrance membership grades mi(x1), mi(x2), ... , mi(xn) so-called resultant membership function mi(y). Resultant function has a complicated shape, and its estimation can be done by realization of inference. Inference includes following elements:
- base of rules,
- inference mechanism,
- membership functions of the exit of the model.
Let us assume that detail called dumper, is being characterized by two parameters. Assign-ment of values of each parameter presents appropriately fig. 2 and fig. 3.
Figure 2. Assignment of values of parameter one Source: Own study.
4
Rowlands H., Wang L.R., An approach of fuzzy logic evaluation and control in SPC, Quality and Reliability Engineering Conference Proceedings, Orlando, FL, 1988, s 30–35 [4].
5
Taleb H., Limam M. On fuzzy and probabilistic control charts, International Journal of Production Research 40 (2002) s. 2849–2863 [5].
Figure 3. Assignment of values of parameter two Source: Own study.
In three-dimensional area we can compose two membership functions (fig. 4)
Figure 4. Illustration of conception of fuzzy inference with usage of Mamdani operator for rule: b) If x1 = x1o to x2 = x2o and entrance value x1 = x1’ and
c) for entrance value x1 = x1” Source: Own study.
Figure 4 presents way of integration of quality description with help of two membership func-tions relating to attributes of detail.
Membership functions refer to two completely different characteristics, parameters or quality values. The way of composing membership functions together is being called inference method and, in fact, it is conditional method, because according to initial value functions of different shapes are formed (compare fig. 2 and fig. 3). It brings about various effects of inference. When we have two or more shapes of resultant membership function, we get different combined evalua-tions of selected detail attributes (fig. 5).
Figure 1. Effect of diversity of combined evaluations as a consequence of variety of entrance data (x1’ and x1”) Source: Own study.
Figure 4 presents usage of Mamdani implication operator. One can also use other operators, for instance PROD implication. Effect of determination of resultant membership function is being presented on figure below (fig. 6.).
Figure 2. Inference with usage of implication operator PROD Source: Own study.
Each kind of inference one can relocate from three-dimensional or n-dimensional area to flat area (fig. 7).
Figure 7. Simplified way of realization of inference using Mamdani (a) and PROD (b) implication operators
Source: Own study.
In examples above there was only two-parametric descriptive structure of evaluated details and their parameters considered. In practice, one would like to consider more parameters while quality evaluation of a detail. Simplicity of proceeding while creating the inference model suggests conception of composing effects acquired from compliance of two parameters with further param-eters (represented by the membership function). Concept of composing together will be identified with inference operations based on selected operators. It can be shown more clearly – in the graphic form (fig. 8) by imitate sequent phases of parameters joining.
Figure 8. Multi-parametric inference – effect of composing of five membership functions Source: Own study.
After composing few membership functions one will get resultant function, which can be used in valuation of detail quality according to group of parameters. These parameters are represented by suitable integrant membership functions: mi(x(i)). In the example, Mamdani operator has been used, which is not a recommendation or preference, one can also use other operators, it depends on results of particular situation analysis and quality structure of examined set. Presented on figure 7 graphical algorithm of multi-parametric inference can be simplified by setting all membership functions in one horizontal line corresponding to axis of each argument (fig. 9).
Figure 9. Coaxial graphical algorithm of multi- dimensional inference Source: Own study.
One can create various configurations of parameters and settle them in different sequences. Reasoning in quality detail estimation, including its characteristics, will not depend on sequence of attaching parameters. For instance after changing parameters 4 and 5 from fig. 9 we can get different forms of resultant membership functions, but final estimation will be the same (fig. 10). That feature is characteristic only for some operators (for ex ample Mamdani operator).
Figure 10. Independence of final evaluation
from sequence of joining of parameters for Mamdani operator Source: Own study.
Independence of evaluation proceeds from nature of Mamdani implication operator, which for Mamdani can be described as follows:
mi’(xn) = mi’(x1=x1’, x2=x2’,..., xn=xn’)= min{mi(x1’), mi(x2’), ... , mi(xn’)} (1)
Result, which regards final evaluation can be presented as sharp, deterministic, or fuzzy, in-terval or approximate (for ex. ). It is connected with situation, when one (or all) parameter has uncertain, un-deterministic character. The easiest example is set of two parame-ters with one of them given in interval way (fig. 11). Result concerning evaluation will be received in fuzzy character type 0 (interval character). Low tolerance of the interval is nominated with usage of membership function
mi”(x2) = mi”(x1=x1”, x2=x2’) (2) and up with usage of function
mi’(x1) = mi’(x1=x1’, x2=x2’) (3)
Figure 11. Example of receiving final fuzzy evaluation of fuzzy parameter x1= =[x1’,x1”] and deterministic parameter x2 for Mamdani operator
Figure 12. Indication of dependency of final evaluation to the sequence of joining parameters for operator PROD
Source: Own study.
Result concerning fuzzy evaluation for operator PROD shows figure 10. Low tolerance of the interval is assigned with help of the membership function
mi”(x2) = mi”(x1=x1”, x2=x2’) (4) and up tolerance with help of function
mi’(x1) = mi’(x1=x1’, x2=x2’) (5)
Figure 3. Example of receiving final evaluation of fuzzy parameter x1= =[x1’,x1”] and deterministic parameter x2 for PROD operator
Source: Own study.
Examination of influence of fuzziness of parameters to a fuzziness of inference is an im-portant factor, which has to be included while choosing an operator to create fuzzy inference model concerning quality estimation of details, considering set of its parameters. Important issue is a positive correlation between changing of quality of one of parameters (mi(x1)), and correction of
inference level (mi’(x1,x2)). That result one can receive only with usage of Mamdani and PROD operators. According to tests, made to write this paper, the biggest changes in reasoning level are shown in case of Kleene-Dienes operator and Kleene-Dienes-Łukasiewicz operator.
In practice increase of quality because of random parameter will be adequate included in complex evaluation.
Experiment below contains two operators isolated as the most effective in aspect of quality es-timation of details or their sets (Mamdani and PROD operators). Plan of the experiment can be related with separate components and chosen sets of parameters. In the examined example each new membership function has its maximal value higher than maximal value of previous resultant function. That is why final results do not depend on the sequence of combining parameters. Results of the experiment also affirm it. As a tool, to the experiment one can use spreadsheet (for instance as help in data preparation – membership function normalization for each parameters) and a specially designed computer program.
To the experiment damper has been chosen. Maximal, optimal and minimal values of the de-tail are presented in table 1.
Table 1. Membership function characteristic of damper parameters
Source: Personal elaboration on base of real data.
Real and normalized values of damper parameters presents table 2. Table 2. Real and normalized values of the damper
0,00000001 – 0,000000001 > evaluation 2 0,000000001 – 0,0000000001 > evaluation 1
Exemplary changes of sequence of detail parameters are presented in table 3.
Table 3. Results of inference – resultant membership function for Mamdani and PROD operator and appropriate evaluations
Source: Own study.
Experiment proves, that change of sequence of joining parameters in case of use of Mamdani and PROD operators does not influence on final evaluation of examined parameters. Necessary calculations were made with help of the author’s program.
3. Conclusions
Production companies, not considering their activity, put pressure on manufacturing best qual-ity compounds. Thanks to unstoppable innovations during production, company can gain competitive predominance and satisfied customers will come back confident of quality and high reliability of offered products. Quality inspectors should use proper methods, thanks to which they could quickly and ably perform evaluations of manufactured elements. As we can see from exam-ples above, to create fuzzy model adequate to real estimations acquired from experts we should choose: detail or group of details, real values of each parameter, changes in sequence of joining parameters and evaluation of optimal sequence of parameter joining, which can ensure minimal and maximal values of detail evaluation.
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[1] El-Shal S.M., Morris A.S., A fuzzy rule-based algorithm to improve the performance of statistical process control in quality systems, Journal of Intelligent and Fuzzy Systems 9 (2000) s. 207–223.
[2] Grzegorzewski P., Control charts for fuzzy data, Proc. Fifth European Congress on Intelli-gent Techniques and Soft Computing EUFIT’97, Aachen, 1997, s. 1326–1330.
[3] Raz T., Wang J.H., Probabilistic and membership approaches In the construction of control charts for linguistic data, Production Plann. Cont. 1 (1990) s. 147–157.
[4] Rowlands H., Wang L.R., An approach of fuzzy logic evaluation and control in SPC, Quality and Reliability Engineering Conference Proceedings, Orlando, FL, 1988, s 30–35.
[5] Taleb H., Limam M., On fuzzy and probabilistic control charts, International Journal of Production Research 40 (2002) s. 2849–2863.
[6] Wang J.H., Raz T., On the construction of control charts using linguistic variables, Internat. J. Poduction Res. 28 (1990) s. 477–487.
i wymogami mog sta si powodem opó nienia w przekazaniu partii do klienta i spowodowa dodatkowe obcienia finansowe zwizane przykładowo z dodatkow produkcj okrelonego detalu. Wadliwe detale zamontowane w poje dzie mog na-razi uczestników ruchu drogowego na niebezpieczestwo. Systemy bazujce na logice rozmytej i wiedzy ekspertów przez wiele lat były wykorzystywane do wspierania procedur w nauce, równie w kontroli jakoci. Systemy rozmyte two-rzone i wykorzystywane s, kiedy napotyka si na problemy, korzystajc z tradycyjnych metod, lub posiadane dane s niewystarczajce. Poniewa dane rze-czywiste podawane s z uyciem tolerancji, uycie interwałowej i rozmytej analizy jest blisze rzeczywistoci. Istnieje potrzeba agregacji rónych atrybutów z uyciem zunifikowanego systemu. W artykule poniej zaprezentowano podejcie rozmyte – model oceny wyrobu w przedsibiorstwie produkcyjnym przemysłu motoryzacyjne-go.
Słowa kluczowe: logika rozmyta, systemy ekspertowe, przedsibiorstwo produkcyjne, ocena wyrobów
Aleksandra Ptak
Instytut Ekonometrii i Informatyki Wydział Zarz dzania
Politechnika Czstochowska ul. Armii Krajowej 19B e-mail: olaptak@zim.pcz.pl
