Reliability Reliability in . . . Available Data and . . . How Parameters Are . . . Reducing Over-Conservative What Is the Accuracy . . . Expert Failure Rate New Approach: Main Idea Fuzzy Interpretation Estimates in the Presence of Analysis of the . . . Resulting Algorithm Limited Data: A New Home Page Probabilistic/Fuzzy Title Page Approach ◭◭ ◮◮ ◭ ◮ Carlos Ferregut, F. Joshua Campos, and Vladik Kreinovich Page 1 of 16 Future Aerospace Science & Technology (FAST) Center Go Back University of Texas at El Paso, El Paso, TX 79968, USA Full Screen ferregut@utep.edu, fjcampos@miners.utep.edu, vladik@utep.edu Close Quit
Reliability Reliability in . . . 1. Reliability Available Data and . . . • Failures are ubiquitous, so reliability analysis is an im- How Parameters Are . . . portant part of engineering design. What Is the Accuracy . . . New Approach: Main Idea • In reliability analysis of a complex system , it is impor- Fuzzy Interpretation tant to know the reliability of its components. Analysis of the . . . • Reliability of a component is usually described by an Resulting Algorithm exponential model : Home Page def P ( t ) = Prob(system is intact by time t ) = exp( − λ · t ) . Title Page ◭◭ ◮◮ • For this model, the average number of failures per unit time ( failure rate ) is equal to λ . ◭ ◮ • Another important characteristic – mean time between Page 2 of 16 failure (MTBF) θ – is, in this model, equal to 1 /λ . Go Back • Usually, the MTBF is estimated as the average of ob- Full Screen served times between failures. Close Quit
Reliability Reliability in . . . 2. Reliability in Aerospace Industry: A Challenge Available Data and . . . • In aerospace industry, reliability is extremely impor- How Parameters Are . . . tant, especially for manned flights. What Is the Accuracy . . . New Approach: Main Idea • Because of this importance, aerospace systems use unique, Fuzzy Interpretation highly reliable components; but: Analysis of the . . . – since each component is highly reliable, Resulting Algorithm – we have few ( ≤ 5) failure records, not enough to Home Page make statistically reliable estimates of λ . Title Page • So, we have to also use expert estimates. ◭◭ ◮◮ • Problem: experts are over-conservative, their estimates ◭ ◮ for λ are higher than the actual failure rate. Page 3 of 16 • In this paper, we propose an algorithm that reduces Go Back the effect of this over-conservativeness. Full Screen Close Quit
Reliability Reliability in . . . 3. Available Data and Main Assumptions Available Data and . . . • For each of n components i = 1 , . . . , n , we have n i How Parameters Are . . . observed times-between-failures t i 1 , . . . , t in i . What Is the Accuracy . . . New Approach: Main Idea • We also have expert estimates e 1 , . . . , e n for the failure Fuzzy Interpretation rate of each component. Analysis of the . . . • We usually assume the exponential distribution for the Resulting Algorithm failure times, i.e., the probability density λ i · exp( − λ i · t ). Home Page • Thus, the probability density corresponding to each Title Page observation t ij is equal to λ i · exp( − λ i · t ij ). ◭◭ ◮◮ • Different observations are assumed to be independent. ◭ ◮ • Different components are assumed to be independent. Page 4 of 16 • Thus, the probability density ρ corresponding to all Go Back observed failures is equal to the product: n i Full Screen n � � ρ = ( λ i · exp( − λ i · t ij )) . Close i =1 j =1 Quit
Reliability Reliability in . . . 4. How Parameters Are Determined Now Available Data and . . . n n i � � How Parameters Are . . . • Reminder: prob. is ρ = ( λ i · exp( − λ i · t ij )) . i =1 j =1 What Is the Accuracy . . . • Maximum Likelihood Approach: find λ i with the high- New Approach: Main Idea est probability ρ . Fuzzy Interpretation def Analysis of the . . . • Idea: ρ → max if and only if ψ = − ln( ρ ) → min: Resulting Algorithm n i n n � � � ψ ( λ i ) = − n i · ln( λ i ) + λ i · t ij . Home Page i =1 i =1 j =1 Title Page n n ◭◭ ◮◮ • So, ψ ( λ i ) = − � n i · ln( λ i ) + � n i · λ i · t i , where i =1 i =1 ◭ ◮ n i = 1 def � t i · t ij . Page 5 of 16 n i j =1 Go Back • Differentiating by λ i and equating the derivative to 0, Full Screen we get the traditional estimate λ i = 1 . t i Close Quit
Reliability Reliability in . . . 5. What Is the Accuracy of This Estimate? Available Data and . . . • Central Limit Theorem: when we have a large amount How Parameters Are . . . of data, the distribution of each parameter is ≈ normal: What Is the Accuracy . . . − ( λ i − µ i ) 2 � � New Approach: Main Idea ρ ( λ i ) = const · exp . 2 · σ 2 Fuzzy Interpretation i Analysis of the . . . • Thus, ψ ( λ i ) = const + ( λ i − µ i ) 2 , hence ∂ 2 ψ = 1 , Resulting Algorithm 2 · σ 2 ∂λ 2 σ 2 i i i Home Page � − 1 � ∂ 2 ψ and σ 2 i = . Title Page ∂λ 2 i ◭◭ ◮◮ n n • For ψ ( λ i ) = − � n i · ln( λ i ) + � n i · λ i · t i , we get ◭ ◮ i =1 i =1 ∂ 2 ψ = n i , so the standard deviation is σ i = λ i Page 6 of 16 . √ n i ∂λ 2 λ 2 i i Go Back • So, the relative accuracy of the estimate λ i is equal to Full Screen σ i 1 = . √ n i Close λ i Quit
Reliability Reliability in . . . 6. Confidence Interval Available Data and . . . • Based on λ i and σ i , we can form an interval that con- How Parameters Are . . . tains the actual failure rate with a given confidence: What Is the Accuracy . . . � � 1 − k 0 � � 1 + k 0 �� New Approach: Main Idea [ λ i − k 0 · σ i , λ i + k 0 · σ i ] = λ i · , λ i · . √ n i √ n i Fuzzy Interpretation Analysis of the . . . • We take k 0 = 2 if we want 90% confidence. Resulting Algorithm • We take k 0 = 3 if we want 99.9% confidence. Home Page • We take k 0 = 6 if we want 99.9999999% = 1 − 10 − 8 Title Page confidence. ◭◭ ◮◮ • Example: for n i = 5 and k 0 = 2, the confidence interval ◭ ◮ is approximately equal to [0 , 2 λ i ]. Page 7 of 16 • In other words, the actual failure rate can be 0 or it Go Back can be twice higher than what we estimated. Full Screen • Thus, if we only have 5 measurements, we cannot ex- tract much information about the actual failure rate. Close Quit
Reliability Reliability in . . . 7. New Approach: Main Idea Available Data and . . . • Experts provide estimates e i for the failure rates λ i . How Parameters Are . . . What Is the Accuracy . . . • Expert over-estimate , i.e., λ i = k i · e i for some k i < 1. New Approach: Main Idea • As usual, it is reasonable to assume that k i are nor- Fuzzy Interpretation mally distributed , with unknown k and σ 2 : Analysis of the . . . − ( k i − k ) 2 1 � � √ 2 · π · σ · exp . Resulting Algorithm 2 σ 2 Home Page • Approximation errors k i − k corresponding to different Title Page components are independent. ◭◭ ◮◮ • We then find λ i , k , and σ from the Maximum Likelihood ◭ ◮ Method ρ → max: � n n i � Page 8 of 16 � � · ρ ′ , where ρ = ( k i · e i · exp( − ( k i · e i · t ij )) Go Back i =1 j =1 Full Screen n − ( k i − k ) 2 � � 1 ρ ′ = � √ 2 · π · σ · exp . Close 2 σ 2 i =1 Quit
Reliability Reliability in . . . 8. Fuzzy Interpretation Available Data and . . . � � n i n How Parameters Are . . . · ρ ′ , � � • Reminder: ρ = ( k i · e i · exp( − ( k i · e i · t ij )) What Is the Accuracy . . . i =1 j =1 New Approach: Main Idea n − ( k i − k ) 2 1 � � where ρ ′ = Fuzzy Interpretation � √ 2 · π · σ · exp . 2 σ 2 Analysis of the . . . i =1 Resulting Algorithm • This formula is based on the assumptions of Home Page – Gaussian distribution, and Title Page – independence. ◭◭ ◮◮ • A similar formula can be obtained if we simply use: ◭ ◮ – Gaussian membership functions, and Page 9 of 16 – a product t-norm f & ( a, b ) = a · b to combine infor- Go Back mation about different components. Full Screen • In this case, we do not need independence assumptions . Close Quit
Reliability Reliability in . . . 9. Analysis of the Optimization Problem Available Data and . . . � � n i n How Parameters Are . . . · ρ ′ , � � • Reminder: ρ = ( k i · e i · exp( − ( k i · e i · t ij )) What Is the Accuracy . . . i =1 j =1 New Approach: Main Idea n − ( k i − k ) 2 1 � � where ρ ′ = � Fuzzy Interpretation √ 2 · π · σ · exp . 2 σ 2 Analysis of the . . . i =1 Resulting Algorithm def • We reduce ρ → max to ψ = − ln( ρ ) → min and use t i : Home Page n n n ( k i − k ) 2 Title Page � � � ψ = − n i · ln( k i )+ k i · n i · e i · t i + n · ln( σ )+ . 2 σ 2 ◭◭ ◮◮ i =1 i =1 i =1 • Equating derivatives w.r.t. σ , k , and k i to 0, we get: ◭ ◮ n n Page 10 of 16 σ 2 = 1 k = 1 � ( k i − k ) 2 ; � n · n · k i ; Go Back i =1 i =1 − n i + n i · e i · t i + k i − k Full Screen = 0 . σ 2 k i Close Quit
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