STOCHASTIC MODEL OF MANUFACTURING EQUIPMENT S MAINTENANCE PROCESSES

Transcription

1

2

3 STOHSTI MODEL OF MNUFTURING EQUIPMENT S MINTENNE PROESSES Pokorádi, L.; pokoradi@eng.unideb.hu bstract: From mathematical point of view, the manufacturing equipment s maintenance is a discrete state space stochastic process without after-effects so it can be approximated with a Markov-chain. fter setting up of transition probability matrix analysis of the process with system approach can be investigated with matrixalgebraic operations. The paper will be aimed to show the possibilities of use of Markov matrix in case of stationary maintenance processes. well-algorithmizable method for mathematical modeling of stationary stochastic industrial process will be shown. The shown modeling method can be used to estimate maintenance cost of investigated manufacturing equipment. Keywords: maintenance; maintenance management, stochastic modeling. INTRODUTION In case of production line operation by modification of maintenance technology, equipment s repairing time could be decreased, but it would procure the increasing of repairing cost. The task was to forecast the total repair cost if the new technology would be installed. The aims were to solve the task mentioned above and, in addition to develop a wellalgorithmizable method for this and similar problems. The solution and this paper has been inspired mainly by (Rohács & Simon, 989) and (Pokorádi, 8). The manufacturing system operation is a stochastic process based upon the equipment, their maintenance, their preparation, and also based upon the personnel carrying out repair, and upon the regulations. This process can be considered as a mathematically continuous time, discrete state space Markov-process, that can be approximated with a Markov-chain. The paper of El-Dancese dealt with a system of n-independent repairable units that can be described by a homogeneous continuous-time discrete-state Markov process (El-Dancese, 995). Transition probabilities matrix of the system had the form of a modified Kronecker sum of transition-rate matrices of its units was investigated. The mathematical basis of Markovprocesses can be read from books (Karlin & Taylor, 975) and (Korn, & Korn, 975). The paper will be aimed to show the possibilities of use of Markov matrix in case of stationary maintenance processes. wellalgorithmizable method for mathematical modeling of stationary stochastic industrial process will be shown. The shown modeling method is usable to estimate maintenance cost and available time of investigated manufacturing equipment. The paper will be organized as follows: Section shows the applied literatures and the main goals of investigation. Section words the Markov-processes shortly. Section 3 presents the maintenance processes. Section shows a case study for maintenance management decision. summation, conclusion and future work provide in the section 5.. MRKOV PROESSES stochastic processes η(t) whose development in the future is influenced by its development in the past only through its development in the present, that is stochastic process without aftereffects, is called Markov-process. If process η(t) during the investigated period can have an X value at any moment, it is called a continuous-time process. If η can only have some value at certain moments, the process is called a discrete-time one. stochastic process is considered to be of discrete state space, if the possible values of variable η constitute a finite set or a countable non-finite set. Finite or count non-finite stochastic process, that is the discrete state space one with no aftereffects, is called Markov-chain. In this case the hypothetical probability n, n ij = Pη n [ ( t ) = X η( t ) X ] P = j n i ()

4 is called transition probability which expresses the probability that n(t n ) = X j, supposing that n(t n ) = X i (Karlin & Taylor, 975). The limiting value n, n Pij β ij = lim () tn tn t t n is called transition probability (in this study: λ - failure and µ - repair) rate. 3. MINTENNE PROESSES The manufacturing system operation is a stochastic process based upon the equipment, their maintenance, their preparation, and also based upon the personnel carrying out repair, and upon the regulations. The operational process of technical system (which is the complex of events that happen to the system from its manufacturing to its discarding) is a random in time and frequency succession of socalled states of operation. This process can be described with the so called operational chain, which is, from mathematical point of view a Markov-chain (see Fig. ). n vector. The staying of the object of the operation in different states can also be characterized by the vector of costs of the staying in state of the operation. Knowing the characteristics above we are able to determine the total operational cost.. THE SE STUDY During a production line equipment s operation three main types of failures can be experienced. When the -type failures where repaired, the servicemen detect frequently that -type failure will occur shortly. Then, the -type failure repair is carried out. The statistical analysis of data has showed that the failure and repair rates have exponential featured distributions and independent on the operating time of the production line. The table. shows the main failure and repair data of investigated production line equipment. λ μ λ μ μ λ λ Figure. n operational chain (see Figure.) 3 When analyzing operational processes with the system approach, the actual succession of single states for equipment is no concern of ours. It is rather complicated to describe the whole operational process with an operational chain. In order to achieve a clearer survey it is advisable to describe the operational process as a directed graph. Within the graph of operation states are represented by the vertexes of the graph, and transitions from one state to another are represented by the directed edges of the graph (for example Fig. ). nalyzing the operational chain or the type graph, we assume that states are clearly marked, and transitions occur during zero- time. For characterization of transitions from one state to another we use their transition probability (failure or repair) rates. The probabilities of staying in states of operation can be arranged in a probability Figure. Graph of the maintenance Process Using of equipment; -type failure s repair; 3 -type failure s repair; -type failure s repair Table. Main Data of Statistical nalysis Failures -type -type -type MTF τ [hour] 5,6 39,35 96,5 Failure rate λ [hour - ] 6, ,79-3, - MTR τ [hour],, 6, Repair rate μ [hour - ] 9,8 -,5 -,63 - Mean Repairing ost [ ] λ v [hour - ],6

5 y modification of manufacturing technology, repairing time of -type failure can be decreased approximately till hours, but its cost is increased to around 5 Euros. The task was to forecast the total repair cost of the application of the new technology. The data analysis suggests that the process is stationary, and its Markov model can be written by following equations: = ( λ λ λ ) P μ P = λ P ( μ λ = λ P λ P ) P μ P = λ P μ P () when: P i is the probability of staying in the i-th state of the operation (see Figure.). During the solution of the system of equations () there is a problem, that the numerical algorithms provide (or can provide) the trivial solution. It is obvious that the concern of investigation is the solution different from the trivial one. ecause aim of the author was to develop a well-algorithmizable method for this problem, the system of equations with unknowns () was transformed into an system equations with 5 unknowns by equation 3 Σ = P i = i= μ P μ P P () of probability of total event space. Thus the system of equations has changed to ( λ λ λ ) λ λ λ μ ( μ λ ) λ μ μ μ μ 3. P P P = 3 P P Σ (3) The solution of the system of linear equations reached by any numerical algorithm will result in a solution of the system of equations () different from the trivial one. The solution of equation (3) by data of Table is following probabilities of the staying in states: P =,883 ; P =,86 ; P 3 =,38 ; P =,5696. Knowing the cost of different repairs, the total repairing cost: K Σ = T i= k i P i τ where: T investigated time-interval; k i cost of the i-th repair. i, () The total repairing cost of the present maintenance system for hours is: KΣ = = 77.88EURO.. 6. (5) For forecasting of the total repair cost if new technology the above calculation was completed by the following modification t = hours ; k = 5. Its results probabilities of the staying in states are P =,97 ; P =,9 ; P 3 =, ; P =,58, and total cost of modified maintenance system is: KΣ = =. 6EURO,, 6, (6) From the results mentioned above, the following conclusions can be drawn: the probability of staying in -type failure s repair state will decrease measurably (by.7 %); the probabilities of staying in other failures repair will increase slightly (by.38 and.8 %); These conclusions should be taken account during the organizing of the new maintenance system management. hours-related total repair cost will be reduced by Euros; the availability of the equipment will be increasing by.83 %, that generates more profit by improvement of the reliability and the productivity of the production line.

6 These positive economic consequences should be compared with installation cost of the new repairing technology. 5. LOSING REMRKS, FUTURE WORK The paper showed a well-algorithmizable method developed by the author for mathematical modeling of stationary stochastic industrial processes. The shown modeling method can be used to estimate maintenance cost of manufacturing systems or their equipment. The general conclusion of this paper is that the stationary Markov model of operational processes can be used to investigate maintenance systems. During prospective scientific research related to this field of mathematics and technical management science, the author will prepare other methods to investigate maintenance processes of technical systems. 6. REFERENES El-Damcese, M..(995), Markovian-Model for Systems with Independent Units, Microelectron. Reliab, Vol 35, No 7. pp Karlin, S. & Taylor H.M. (985), First ourse in Stochastic Processes, cademic Press, pp Korn, G.. & Korn, T.M. (975), Matematical Handbook for Scientists and Engineers, Műszaki Könyvkiadó, udapest, pp Pokorádi, L. (8), Systems and Processes Modeling, ampus Kiadó, Debrecen, pp.. (in Hungarian) Rohács, J. & Simon, I. (989), Handbook of irplanes and Helicopters Maintenance, Műszaki Könyvkiadó, udapest, pp. 53. (in Hungarian)