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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

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

- Series systems: System is failed if there is one component failed.
- Parallel systems: System is working if there is one component working.

- Series-parallel systems:
- Parallel-series systems:

- 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:

Decomposition Method

Bridge structure

Decomposition Method

How to deal with the following directed bridge structure ?

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

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)

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

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

Warm Standby System without Perfect Switching

- The general standby redundancy case with one active component
- Consider two components

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.

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.

Example 6.6: Minimal Path Sets

Find all path sets and all minimal path sets of the bridge network.

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.

Example 6.7: Minimal Cut Sets

Find all cut sets and all minimal cut sets of the bridge network.

Inclusion-Exclusion Method

- Used to evaluate system reliability based on minimal path (cut) sets.

Sum of Disjoint Products Method

- Sum of Disjoint Products (SDP) Method
- Another equation for the same purpose:

Example 6.8

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

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)

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.

Bounds on system reliability

- Not very good upper and lower bounds:
- Better upper and lower bounds
- An example

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

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.

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

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)

Redundancy Allocation

Problem:

Determine the optimal redundancy levels (number of components) of the subsystems (stages).

Redundancy Allocation

- Models 1:Minimizing cost subject to reliability requirement

- Design variables: number of components at each stage

- Objective: System cost

- Constraint: System reliability

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

Example: Redundancy Allocation

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

Example: Redundancy Allocation

- Coefficients used

- Optimization results:- X = (3, 2, 2, 3, 3), - R = 0.90447

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

Reliability Allocation

Problem:

Determine the optimal component reliabilities values for the subsystems (stages)

- Design variables: (r1, r2, …, rN)

Reliability-Redundancy Allocation

Problem:

Jointly determine the optimal redundancies and component reliabilities for the subsystems (stages)

- Design variables: (n1, n2, …, nN, r1, r2, …, rN)

Next Lecture

- Case Studies
- Reliability Software- Weibull++ (ReliaSoft)- BlockSim (ReliaSoft)

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