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Parallel systems: System is working if there is one component working. ... A general method for system reliability evaluation. A Path Set is a set of components ...

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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


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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


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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.


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Review of Last Lecture

  • Series-parallel systems:

  • Parallel-series systems:


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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:


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Decomposition Method

Bridge structure


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Decomposition Method

How to deal with the following directed bridge structure ?


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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


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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)


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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


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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


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Warm Standby System without Perfect Switching

  • The general standby redundancy case with one active component

  • Consider two components


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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.


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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.


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Example 6.6: Minimal Path Sets

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


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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.


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Example 6.7: Minimal Cut Sets

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


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Inclusion-Exclusion Method

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


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Sum of Disjoint Products Method

  • Sum of Disjoint Products (SDP) Method

  • Another equation for the same purpose:


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Example 6.8

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


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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)


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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.


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Bounds on system reliability

  • Not very good upper and lower bounds:

  • Better upper and lower bounds

  • An example


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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


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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.


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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


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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)


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Redundancy Allocation

Problem:

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


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Redundancy Allocation

  • Models 1:Minimizing cost subject to reliability requirement

- Design variables: number of components at each stage

- Objective: System cost

- Constraint: System reliability


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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


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Example: Redundancy Allocation

  • A five stage series-parallel system (Tillman et al, IEEE T. Rel., 1968)

  • Objective: Reliability

  • Constraints: cost, volume, weight


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Example: Redundancy Allocation

  • Coefficients used

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


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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


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Reliability Allocation

Problem:

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

  • Design variables: (r1, r2, …, rN)


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Reliability-Redundancy Allocation

Problem:

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

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


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Next Lecture

  • Case Studies

  • Reliability Software- Weibull++ (ReliaSoft)- BlockSim (ReliaSoft)


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