Fuzzy reliability analysis of washing unit in a paper plant using soft-computing based hybridized techniques

Transcription

1 Fuzzy reliability analysis of washing unit in a paper plant using soft-computing based hybridized techniques *Department of Mathematics University of Petroleum & Energy Studies (UPES) Dehradun , Uttarakhand, India

2 Outline Introduction Objective of the paper Assumptions Techniques Fuzzy Lambda-Tau(FLT) Genetic Algorithms based Lambda-Tau (GABLT) Neural Network and Genetic Algorithms based Lambda-Tau (NGABLT) Washing System Description Result and Discussion Conclusions

3 Introduction Researchers are paying more attention to the real life problems for improving the performance as well as profit margin of industrial systems. System performance is closely linked to the system reliability and when it is low, efforts are desired to improve it by reducing the failure rate or increasing the repair rate for each subsystem/component. Suitable maintenance strategies may be applied to improve the system reliability.

4 Objective In real life situation, it is difficult to achieve optimum performance of an industrial system for desired industrial goals using available resources and uncertain data due to system complexity and nonlinear behavior. The objective of the paper is to analyze the fuzzy reliability of washing system in a paper plant using available information and uncertain data through two soft-computing based hybridized techniques GABLT and NGABLT along with traditional FLT technique.

5 Assumptions Assumptions taken for the modeling methodology are: (i) component failures and repair rates are statistically independent, constant, very small and obey exponential distribution function, (ii) λ i < µ i, (iii) separate maintenance facility is available for each component. The repair process begins soon after a unit fails, (iv) after repairs, the repaired component is considered as good as new, (v) system structure is precisely known.

6 FLT Technique FLT technique is a traditional method for analyzing system fuzzy reliability. The methodology is based on qualitative modeling using PN and quantitative modeling using Lambda-Tau method (Table-1) of solution with basic events (AND-gates and OR-gates) represented by triangular fuzzy numbers. Various reliability indices(table-2) of the system are evaluated using fuzzy arithmetics.

7 Table-1:Basic Expressions of Lambda-Tau Methodology Gate λ AND τ AND λ OR τ OR [ ] n n n n n τ i n i=1 Expressions λ j τ j [ i=1 j=1 i=1 j=1 i j n j=1 ] n τ i i=1 i j λ i i=1 λ i τ i n λ i i=1

8 Table-2: Some Reliability Indices for Repairable System with Constant Repair Rate Model Reliability Indices Expressions Mean Time to Failure MTTF s = 1 λ s Mean Time to Repair MTTR s = 1 µ s = τ s Mean Time Between Failures MTBF s = MTTF s + MTTR s Expected Number of Failures ENOF s (0, t) = λsµst λ s+µ s + λ2 s (λ s+µ s) [1 e (λs+µs)t ] 2 Availability A s (t) = µs λ s+µ s + λs λ s+µ s e (λs+µs)t Reliability R s (t) = e λst

9 GABLT Technique In GABLT technique, two important tools, namely Lambda-Tau methodology and GA are hybridized. GABLT utilizes ordinary arithmetic and mathematical programming approach instead of fuzzy arithmetic. For finding system fuzzy reliability indices utilizing fuzzified failure and repair data, nonlinear optimization problems at each cut-level α is formulated.

10 Flow Chart of GABLT Technique - Design/ Maintenance Expert - Historical Records - Reliability Databases - Information Extraction - Failure rates and Repair times ( 's and 's) - System/ Reliability Analyst FUZZIFIER Obtain reliability indices function in the form of 's and 's usingfta and Table-1 and Table-2 results To construct fuzzy reliability indices membership function, formulate optimization problem at each cut level- and solve it using GA

11 Nonlinear Optimization Problem Minimize/Maximize: F (λ 1, λ 2,..., λ n, τ 1, τ 2,..., τ m ) or F (t/λ 1, λ 2,..., λ n, τ 1, τ 2,..., τ m ) Subject to : µ λi (x) α, µ τj (x) α, 0 α 1, i = 1, 2,..., n; j = 1, 2,..., m. Where F (λ 1, λ 2,..., λ n, τ 1, τ 2,..., τ m ) and F (t/λ 1, λ 2,..., λ n, τ 1, τ 2,..., τ m ) are time independent and dependent fuzzy reliability indices respectively. The obtained minimum and maximum value of F are denoted by F min and F max respectively. The membership function values of F at F min and F max are both α that is, µ F (F min) = µ F (F max) = α

12 Solution of Nonlinear Optimization Problem Using GA GA is used to solve the nonlinear optimization problems for each cut-level α. The flow chart of simple GA is given below. Begin Initialize population set generation = 0 Stopping criterion satisfied? Yes Stop No Evaluate Assign fitness Reproduction Crossover Mutation Generation = Generation + 1

13 Limitations of FLT and GABLT Techniques FLT is based on fuzzy arithmetic and required precise knowledge of components functional dependencies. GABLT technique requires precise knowledge of systems components functional dependencies and utilizes ordinary arithmetic. So these techniques cannot be applicable in future for those systems whose components functional dependencies are imprecisely known.

14 NGABLT Technique NGABLT is a softcomputing based technique in which fuzzy, artificial neural network(ann) and GA are hybridized. The major benefit of using ANN in NGABLT technique is that it can be used effectively in the situations where components functional dependency is precisely known (supervised learning) and where it is imprecisely known (unsupervised learning). In the present study, as system s components functional dependency is precisely known so supervised learning has been used.

15 Flow Chart of NGABLT Technique - Design/ Maintenance Expert - Historical Records - Reliability Databases - Information Extraction - Failure rates and Repair times ( 's and 's) Obtain reliability indices function in the form of 's and 's usingfta and Table-1 and Table-2 results - System/ Reliability Analyst FUZZIFIER Approximation of obtained reliability indices function using artificial neural network To construct fuzzy reliability indices membership function, formulate optimization problem at each cut level- and solve it using GA

16 Washing System Description The system consists of four main subsystems, defined as: Filter (A): It consists of single unit which is used to drain black liquor from the cooked pulp. Cleaners (B): In this subsystem three units of cleaners are arranged in parallel configuration. Each unit may be used to clean the pulp by centrifugal action. Failure of anyone will reduce the efficiency of the system as well as quality of paper. Screeners (C): Herein two units of screeners are arranged in series. These are used to remove oversized, uncooked and odd shaped fibers from pulp through straining action. Failure of any one will cause the complete failure of the system. Deckers (D): Two units of deckers are arranged in parallel configuration. The function of deckers is to reduce the blackness of pulp. Complete failure of decker occurs when both the components will fail.

17 Washing System Fault Tree Model WSF OR E1 E2 E3 E4 A AND OR AND B 1 B 2 B 3 C 1 C 2 D 1 D 2

18 Result and Discussion: Washing System Components Failure Rates and Repair Times Component Failure Rate(λ i ) Repair Time(τ i ) (failures/hrs) (hrs) A: Filter(i = 1) B: Cleaners(i = 2, 3, 4) C: Screeners(i = 5, 6) D: Deckers(i = 7, 8)

19 Washing System s Failure Rate and Repair Time Expressions The minimal cut sets are {A}, {B 1, B 2, B 3 }, {C i } i=1,2 and {D 1, D 2 }, obtained using matrix method. λ s = λ 1 + λ 5 + λ 6 + λ 2 λ 3 λ 4 (τ 2 τ 3 + τ 3 τ 4 + τ 2 τ 4 ) + λ 7 λ 8 (τ 7 + τ 8 ) τ s = (λ 1τ 1 + λ 5 τ 5 + λ 6 τ 6 + λ 2 λ 3 λ 4 τ 2 τ 3 τ 4 + λ 7 λ 8 τ 7 τ 8 ) λ s

20 GA and ANN Parameters Values to Apply GABLT and NGABLT Techniques for Washing System Reliability Analysis Reliability GABLT NGABLT Indices Parameters for Parameters for Parameters for GA ANN GA P s P c P m N i N d N h L r P s P c P m N i Failure Rate Repair Time MTBF ENOF Availability Reliability Notations: P s : Pop. size; P c : Prob. of crossover; P m : Prob. of mutation; N i : No. of iterations N d : No. of training data; N h : No. of hidden layers; L r : Learning rate

21 Fuzzy Reliability Indices Plots for Washing System Degree of Membership FLT GABLT NGABLT Degree of Membership FLT GABLT NGABLT Degree of Membership FLT GABLT NGABLT Failure Rate(hrs 1 ) (a) Failure Rate Repair Time(hrs) (b) Repair Time Mean Time Between Failures(hrs) (c) MTBF Degree of Membership FLT GABLT NGABLT Degree of Membership FLT GABLT NGABLT Degree of Membership FLT GABLT NGABLT Expected Number of Failures (d) ENOF Availability (e) Availability Reliability (f) Reliability

22 Approximation Error Plots for Washing System 5 x 10 3 Epoch :10000, Error : Epoch :10000, Error : Epoch :10000, Error : (a) Failure Rate (b) Repair Time (c) MTBF 0.02 Epoch :10000, Error : x 10 3 Epoch :10000, Error : Epoch :10000, Error : (d) ENOF (e) Availability (f) Reliability

23 Crisp and Defuzzified Values of Reliability Indices for Washing System Reliability Crisp Defuzzified values at (spread) Indices ±15% ±25% ±60% Failure Rate FLT: GABLT: NGABLT: Repair Time MTBF ENOF Availability Reliability

24 Conclusions In this paper various reliability indices of a washing system have been computed in the form of fuzzy membership functions by using FLT, GABLT and NGABLT techniques. Depending upon the confidence level α, the analyst can predict the behavior of the system. The defuzzified values of reliability indices for different level of uncertainties with their crisp values have been computed and tabulated.

25 Conclusions It is observed from the analysis that GABLT performs well in comparison of NGABLT and FLT techniques. If system analysts use GABLT results then they may predict the system behavior with more confidence. NGABLT technique provides the flexibility of extension of its present form in future for the systems whose components functional dependencies are imprecisely known. These hybridized techniques can be applied to a wide range of industrial systems for helping the reliability engineers to gain valuable information and also to evaluate and implement various maintenance strategies.

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