FUZZY MODELING OF MATERIAL HANDLING
SYSTEM AVAILABILITY
MODELOWANIE GOTOWOŚCI SYSTEMÓW
TRANSPORTU BLISKIEGO Z ZASTOSOWANIEM
LOGIKI ROZMYTEJ
Jarosław Smoczek
1, Janusz Szpytko
2 (1, 2) AGH University of Science and TechnologyFaculty of Mechanical Engineering and Robotics al. Mickiewicza 30, 30-059 Kraków
E-mail: (1) smoczek@agh.edu.pl, (2) szpytko@agh.edu.pl
Abstract: The paper presents comparison of classical method of system’s
availability determining with method based on heuristic estimation of capability of individual elements of considered system to realize a set of tasks with expected quality specified by assumed indicators. The proposed method of availability estimation was based on the fuzzy inference system and so called Mamdani implication. The attention in the paper is focused on an automated crane system which realize transportation tasks in material handling system.
Keywords: availability, material handling system, fuzzy logic
Streszczenie: W artykule przedstawione zostało porównanie klasycznej metody
oceny gotowości systemów o strukturze szeregowej i równoległej z proponowaną metodą opartą na heurystycznej ocenie zdolności elementów systemu do realizacji zbioru zadań i spełnienia wymagań określonych poprzez założone wskaźniki jakości. Przedstawiona metoda wyznaczania współczynnika gotowości systemu została zilustrowana przykładem zautomatyzowanego systemu transportu bliskiego realizowanego przez suwnice pomostową. Model gotowości rozważanego systemu zbudowany został z zastosowaniem rozmytego systemu wnioskowania oraz rozmytych implikacji typu Mamdani.
Słowa kluczowe: gotowość systemu, transport bliski, logika rozmyta
The safety, effectiveness, efficiency and dependability are important indicators of exploitation quality of material handling systems and devices. From transportation devices is required high availability to raise efficiency and productivity of manufacturing process (Szpytko, 2004; Szpytko and Kocerba, 2008a and 2008b), which is a major element of system effectiveness E . The availability A indicatesd the percent of the time that system is expected to operate satisfactorily and expresses probability that a system is capable to perform the tasks under stated conditions (Jaźwiński and Grabski, 2003; Grabski, 2008). The availability is usually estimated based on operating time, active repair time, idle time and logistic time. Frequently used terms referred to uptime and downtime of considered system are given by following definitions: Mean Time Between Failure (MTBF), Mean Time To Repair (MTTR), Mean Time Between Maintenance (MTBM), Mean Down-Time (MDT) and active maintenance time (M) (Blanchard, 2008; Ireson et all., 1996; Kececiouglu, 1995; Wang and Kececioglu, 2000). The inherent availability A is determined based on portion of downtime (meani corrective maintenance time) not including preventive maintenance (1).
MTTR MTBF MTBF i A (1)
However the achieved availability A is determined based on active maintenancea time (M) excepting logistic and administrative delay time (LDT and ADT).
M MTBM MTBM a A (2)
Consequently the operational availability A , which relates to total uptimeo (MTBM) and all downtime considerations (MDT), seems to be more relevant than either A or a A factors.i MDT MTBM MTBM o A (3)
Classical method of determining availability coefficient for series system of n elements (Fig. 1) is based on assumption that all elements are operational (4).
n i i s A A 1 (4) For redundant system with parallel structure (Fig. 2) the availability is derived from the equation (5).Fig. 2. The redundant availability system
n i i s A A 1 1 1 (5)The classical approach to determine the availability refers only to the uptime and downtime of considered system. The confidence of calculated availability is limited by uncertainty of collected by user data during monitoring of exploitation process. More over it is assumed that all elements of the system have the same influence to successfully meet operational demands, without taking into consideration the relationship between the level of requirements and expected quality of realized by system/elements tasks and risk degree of down-time occurring (repairs or necessary adjustments to meet increasing demands).
For this reason presented in the paper method of availability estimation is based also on heuristic method and fuzzy logic. The availability factor is calculated with using weighting coefficients of system’s elements determined based on heuristic knowledge of process operator/engineer and selected indicators of quality and effectiveness of the considered system.
2. FUZZY LOGIC APPROACH TO ESTIMATE AVAILABILITY
The proposed method of system’s availability determining is based on the assumption that each of system’s elements is capable to successfully perform a set of tasks under formulated set of conditions C with specified by user (process engineer or operator) weighting coefficients, which take values from the range1 , 0
i
h . Consequently the h coefficient is the heuristic factor which relatesi to the expected operational quality of i element indispensable to meet successfully operational demands. The heuristic coefficient is a function of conditions specified
for a system hi f
c1,c2,...,cm
. For a series system the proposed formula can be expressed as equation (6), while for a parallel system as (7).1 , 0 1
i n i h i s A h A i (6) 1 , 0 1 1 1
i n i h i s A h A iThe factor h can be estimated based on heuristic knowledge of user formulatedi by using if-then implication which express relationship between defined requirements and value of weighting coefficient. The fuzzy logic based algorithm can be useful to approximate the values of weighting coefficients specified for n elements and m conditions (Fig. 3).
Fig. 3. The fuzzy model used to estimate values of weighting coefficients of system’s elements
The fuzzy implication for m conditions specified for a system can be formulated as: IF 1 c is LM(c1) and/or c2 is LM(c2) and/or … cm isLM(cm) THEN k h1 is LM(h1k) and h2k is LM(h2k) and … hnk is LM(hnk) (8) where:
k = 1, 2, …, N - where N is the number of fuzzy rules in knowledge base defined by user,
) (cj
LM , LM(hi) - fuzzy sets (membership functions) formulated for input and output of fuzzy model.
The input and output of fuzzy model are fuzzified using membership functions, e.g. triangular functions presented in the figure 4.
Fig. 4. The example of triangular membership functions used for input and output fuzzifying
The crisp values of output H
h1,h2,...,hn
T are calculated based on the assumed methods of fuzzy implications aggregation and defuzzification.3. THE EXAMPLE OF ESTIMATING THE CRANE SYSTEM
AVAILABILITY
The proposed method of availability determining was addressed to overhead traveling crane with hoisting capacity Q = 15 000 [kg]. The crane automated system was assumed as a series structure (Fig. 5) with known availabilities of elements, respectively A1 0,98 for acting, supporting and safety subsystems and A2 0,97 for control subsystem consisting of anti-sway and anti-skew control systems. The anti-sway control system realizes positioning of a load shifted by crane’s movement mechanisms with expected tolerance, whereas the anti-skew of crane’s girders system reduces overloads that arise during transient states of crane’s power transmission systems working caused by non-uniform loading (this disadvantageous phenomenon affects unfavorable on exploitation of a wheel-rail system of the overhead traveling cranes).
Fig. 5. Simplified series structure of automated crane system
For transportation tasks realized by crane were specified two conditions, defined as expected accuracy of a load positioning c1(e 0,02;0,12 [m]) (positioning error calculated as a difference between desired and actual position of crane’s movement mechanisms), and risk of crane’s skew occurrence connected with mass
of a load value c2
m2 500;15000 [kg]
. The availability of the system to meet requirements specified by conditions c and 1 c can be formulated as:22 1 2 1h h s A A A (9) where: ) , ( , 1 2 1 2 1 h f c c
h - the weighting coefficients of subsystems.
The heuristic coefficient h takes the value 1 for acting, supporting and safety1
subsystems owing to the same high level of efficiency required from those elements of the considered system to meet specified by indicators c and 1 c2
conditions. The availability of control subsystem depends on heuristic weighting coefficient h , which relates to the level of expected effectiveness and efficiency,2
and which is the function of conditions c and 1 c , that describe required control2
quality.
The value of coefficient h can be determined by using fuzzy inference system2 type of Mamdani (Fig. 6), where the conditions c and 1 c , as well as the output2
of the fuzzy model h are fuzzified using triangular membership functions,2 described by linguistic terms Very Low (VL), Low (L), Medium (M), High (H) and Very High (VH) (Fig. 7).
Fig. 6. The fuzzy inference system used to estimate the heuristic coefficient h2
Fig. 7. The membership functions specified for input (c1 and c2)
The low requirements for control subsystem can be understood as the lack of necessity to use the automated control system to realize by crane the transportation tasks when the expected quality of control is low (control operations can be realized by operator). On the other hand, the high demands for transportation tasks performing require high effectiveness of control subsystem (intensity of subsystem exploitation is connected with risk of down-time occurring).
The knowledge base of the fuzzy inference system consists of if-then rules formulated by experienced user (engineer/operator of the system) which describe heuristic relationship between coefficient h and requirements 2 c and 1 c :2
IF c1 is {Low, Medium, High} and c2 is {Low, Medium, High}
THEN h2k is {Very Low, Low, Medium, High, Very High} (10) The knowledge base of the fuzzy system was assumed as nine implications presented in the table 1.
Table 1. The knowledge base of fuzzy model
1
c
Low Medium High
2
c
Low Very Low Low Medium
Mediu
m Low Medium High
High Medium High Very High
The crisp output h of the fuzzy model is calculated based on the assumed2
method used to realize fuzzy implications, aggregate the fuzzy set obtained from each implication, and finally method used to defizzify a fuzzy set. In the presented example the implications are realized using minimum method (11), which leads to obtain the fuzzy set LMk(h2) formulated as the equation (12).
( ), ( ), ( )
min ) (h2 LMk c1 LMk c2 LMk h2 k (11) 2 2 2 2) ( ) (h h h dh LMk
k 2 h (12)The combination of output of N implications results in the fuzzy set LM(h2), which is obtained as a sum of N fuzzy sets. The aggregation method can be expressed as follows: 2 2 1 2 2) ( ) (h h h dh LM N k k
2 h (13) The real value of the system’s output is calculated based on the final fuzzy set) (h2
LM by using largest of maximum (LOM) defuzzification method. The values of weighting coefficient h for examples of values of indicators 2 c and 1 c , as2
well as availability factors A calculated according to the equation (9) ares presented in the table 2.
Table 2. The weighting coefficient h and availability indicator values for2 examples of c and 1 c values2
c1 [m] 0,04 0,06 0,08 0,10 0,12 c2 [kg] 500 h2 = 0,5 As = 0,9652 h2 = 0,375 As = 0,9689 h2 = 0,25 As = 0,9726 h2 = 0,125 As = 0,9763 h2 = 0 As = 0,98 4 125 h2 = 0,625 As = 0,9615 h2 = 0,625 As = 0,9615 h2 = 0,375 As = 0,9689 h2 = 0,375 As = 0,9689 h2 = 0,125 As = 0,9763 7 750 h2 = 0,75 As = 0,9579 h2 = 0,625 As = 0,9615 h2 = 0,5 As = 0,9652 h2 = 0,375 As = 0,9689 h2 = 0,25 As = 0,9726 11 375 h2 = 0,875 As = 0,9542 h2 = 0,875 As = 0,9542 h2 = 0,625 As = 0,9615 h2 = 0,625 As = 0,9615 h2 = 0,375 As = 0,9689 15 000 h2 = 1 As = 0,9506 h2 = 0,875 As = 0,9542 h2 = 0,75 As = 0,9579 h2 = 0,625 As = 0,9615 h2 = 0,5 As = 0,9652
Owing to the classical method of series system’s availability calculating (4) the value of availability of presented in the example system (Fig. 5) equals
9506 , 0 s
A , whereas the proposed heuristic method of availability estimation results in considering the availability of system in the range of values
98 , 0 ; 9506 , 0 s
A . The availability interval relates to the range of probability that a system is capable to perform the tasks under stated conditions.
4. CONCLUSIONS
The presented in the paper method of system or technical object availability estimation indicates the probability that a system is capable to successfully perform scheduled operations as an interval given by individual abilities of system’s elements to meet specified by user requirements. Consequently the heuristic weighting coefficients of subsystems is the measure of efficiency and effectiveness required from individual subsystems to realize set of tasks with expected quality. The heuristic approach to determine availability of the system allows to use fuzzy inference method to model the relationship between user’s defined conditions and availability factor. The proposed Mamdani fuzzy model of availability estimating was applied to the example of automated crane system. In the considered fuzzy model were assumed triangular membership functions in fuzzification and LOM method used for defuzzifying.
The research project is financed from the Polish Science budget for the years 2008-2011.
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Prof. dr hab. inż. Janusz Szpytko (Full Prof., D.Sc., Ph.D., M.Sc., B.Sc. C.Eng.), AGH University of Science and Technology, Faculty of Mechanical Engineering and Robo-tics. Specialist in designing and exploitation of transport systems and devices, automatics, safety and reliability, monitoring and diagnostics, decision making systems, telematics. Author or co-author of more then 350 publications, both in Polish and English. Member of: STST KT PAN, TC IFAC, SEFI, ISPE, PTD, PTB, PSRA, ISA, SITPH and others. Visiting professor at the universities in: UK, France, Canada, Italy, Greece, Canada, Laos. Coordinator and member of several R&D projects both national and international. Organizer and member of several scientific and programme committees of international and national conferences and symposiums.
Dr inż. Jarosław SMOCZEK, AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics. Specialist in designing and exploitation of transport systems and devices, automatics, monitoring and diagnostics. Author or co-author of more then 50 publications, both in Polish and English.
