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# MEC E 514 Reliability for Design

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1. MEC E 514 Reliability for Design Lecture 10:System Reliability EvaluationOptimal Reliability DesignNovember 8, 2006Instructor: Zhigang TianDepartment of Mechanical EngineeringUniversity of Albertahttp://www.ualberta.ca/~ztian/MECE514.htm

2. Today’s Agenda • System Reliability Evaluation • Optimal Reliability Design • Special structures • Cut set and path set method • Reliability bounds • General optimization model • Redundancy allocation • Reliability-redundancy allocation

3. Review of Last Lecture • Series systems: System is failed if there is one component failed. • Parallel systems: System is working if there is one component working.

4. Review of Last Lecture • Series-parallel systems: • Parallel-series systems:

5. Review of Last Lecture • k-out-of-n:G systems (k-out-of-n:F systems )Special cases: series, parallel • Consecutive k-out-of-n:F systems • Standby systems:

6. Decomposition Method Bridge structure

7. Decomposition Method How to deal with the following directed bridge structure ?

8. Consecutive-k-out-of-n:F Systems • R(k;n) is the probability that no k consecutive components failed in a n components system (independent components). Boundary Condition:R(k;n) = 1, if k> n

9. Standby system reliability • Parallel systems:. - R(t) = Pr (T=max(T1, T2, …, Tn))- Sometimes called “Hot standby” • Standby systems: standby components do not fail (Cold standby) or have lower failure rate (Warm standby).R(t) = Pr (T=T1+T2+ …+Tn) (cold standby system with perfect switching)

10. Cold Standby System with Perfect Switching • Consider the case with only one active component • R(t) = Pr (T=T1+T2+ …+Tn) • Special cases: Normal distribution; Exponential and Gamma distribution

11. Example 4.4 (Textbook) • Cold standby system with perfect sensing and switching • Two i.i.d. components following Exponential distribution with λ=0.01 hr-1. Mission time is t = 24 hrs. • (Result: R=0.9755) • MTTFs = N/ λ = 2 * MTTF

12. Warm Standby System without Perfect Switching • The general standby redundancy case with one active component • Consider two components

13. Load-sharing Systems • All the components are working and equally carrying the load • Examples: power generators, pumping systems. • Failure of one component results in that other component carry more loads, thus higher failure rate.

14. Path Set and Cut Set Method • A general method for system reliability evaluation • A Path Set is a set of components whose functioning will guarantee the system's functioning. • A Minimal Path Set is a path set in which the functioning of every component is absolutely necessary for the system to function. • At least one minimal path must contain all working components for the system to work.

15. Example 6.6: Minimal Path Sets Find all path sets and all minimal path sets of the bridge network.

16. Path Set and Cut Set Method • A Cut Set is a set of components whose failures will cause the system to fail. • A Minimal Cut Set is a cut set in which the failure of every component is absolutely necessary for the system to fail. • At least one minimal cut must contain all failed components for the system to fail.

17. Example 6.7: Minimal Cut Sets Find all cut sets and all minimal cut sets of the bridge network.

18. Inclusion-Exclusion Method • Used to evaluate system reliability based on minimal path (cut) sets.

19. Sum of Disjoint Products Method • Sum of Disjoint Products (SDP) Method • Another equation for the same purpose:

20. Example 6.8 Use the bridge network to illustrate using path set or cut set method for system reliability evaluation.

21. Comments: General System Reliability Evaluation Methods • Enumerating method (straight-forward, time-consuming) • Monte Carlo simulation method (efficiency-accuracy) • Decomposition method (human involvement -> not automated) • Path set and cut set methods (general, automated) • Event space method (similar to decomposition) • Path-tracing method (similar to path set method)

22. Bounds on system reliability • Reliability bounds give a range of system reliability; Not as accurate. • Easier to calculate: more efficient, or possible to calculate. • Here discuss reliability bounds based on minimal path (cut) sets • MPi denotes that the ith minimal path works, i.e., every component in the ith minimal path works. • MCi denotes that the ith minimal cut fails, i.e., every component in the ith minimal cut fails.

23. Bounds on system reliability • Not very good upper and lower bounds: • Better upper and lower bounds • An example

24. Today’s Agenda • System Reliability Evaluation • Optimal Reliability Design • Special structures • Cut set and path set method • Reliability bounds • General optimization model • Redundancy allocation • Reliability-redundancy allocation

25. General Optimal Reliability Design Make the right choices to optimize objectives • Objective: Reliability or Cost • Constraints: Cost or Reliability, volume, weight, etc. • Design variables: - Configuration (e.g. Redundancy) - Improve component reliability: components, processes, etc- Maintenance actions • Important things: identify design variables; evaluate objective and constraint functions.

26. Reliability Optimization Models • Redundancy allocation • Reliability allocation (continuous or discrete) • Reliability-redundancy allocation • Component assignment- A n-stage system with interchangeable components • Multi-objective optimization

27. Optimization Techniques • Mathematical programming methods- Software: Matlab Optimization Toolbox • Genetic Algorithms- Software: Matlab GA Toolbox • You can do OPTIMIZATION as long as you can: (1) identify design variables: what you can control (2) evaluate objectives (with respect to design variables)

28. Redundancy Allocation Problem: Determine the optimal redundancy levels (number of components) of the subsystems (stages).

29. Redundancy Allocation • Models 1:Minimizing cost subject to reliability requirement - Design variables: number of components at each stage - Objective: System cost - Constraint: System reliability

30. Redundancy Allocation • Models 2:Maximizing system reliability subject to budget requirement - Design variables: number of components at each stage - Objective: System reliability - Constraint: System cost

31. Example: Redundancy Allocation • A five stage series-parallel system (Tillman et al, IEEE T. Rel., 1968) • Objective: Reliability • Constraints: cost, volume, weight

32. Example: Redundancy Allocation • Coefficients used • Optimization results:- X = (3, 2, 2, 3, 3), - R = 0.90447

33. Another Example of Redundancy Allocation • A five-stage bridge system- where each block represents a stage (subsystem) that can have parallel redundancy. • Design variables: redundancy levels for the five stages

34. Reliability Allocation Problem: Determine the optimal component reliabilities values for the subsystems (stages) • Design variables: (r1, r2, …, rN)

35. Reliability-Redundancy Allocation Problem: Jointly determine the optimal redundancies and component reliabilities for the subsystems (stages) • Design variables: (n1, n2, …, nN, r1, r2, …, rN)

36. Next Lecture • Case Studies • Reliability Software- Weibull++ (ReliaSoft)- BlockSim (ReliaSoft)