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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
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
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
slide3

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.
slide4

Review of Last Lecture

  • Series-parallel systems:
  • Parallel-series systems:
slide5

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:
decomposition method
Decomposition Method

Bridge structure

decomposition method7
Decomposition Method

How to deal with the following directed bridge structure ?

consecutive k out of n f systems
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
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
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
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
Warm Standby System without Perfect Switching
  • The general standby redundancy case with one active component
  • Consider two components
load sharing systems
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
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
Example 6.6: Minimal Path Sets

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

path set and cut set method16
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
Example 6.7: Minimal Cut Sets

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

inclusion exclusion method
Inclusion-Exclusion Method
  • Used to evaluate system reliability based on minimal path (cut) sets.
sum of disjoint products method
Sum of Disjoint Products Method
  • Sum of Disjoint Products (SDP) Method
  • Another equation for the same purpose:
example 6 8
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
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
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 reliability23
Bounds on system reliability
  • Not very good upper and lower bounds:
  • Better upper and lower bounds
  • An example
today s agenda24
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
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
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
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
Redundancy Allocation

Problem:

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

redundancy allocation29
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 allocation30
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
Example: Redundancy Allocation
  • A five stage series-parallel system (Tillman et al, IEEE T. Rel., 1968)
  • Objective: Reliability
  • Constraints: cost, volume, weight
example redundancy allocation32
Example: Redundancy Allocation
  • Coefficients used
  • Optimization results:- X = (3, 2, 2, 3, 3), - R = 0.90447
another example of redundancy allocation
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
Reliability Allocation

Problem:

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

  • Design variables: (r1, r2, …, rN)
reliability redundancy allocation
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
Next Lecture
  • Case Studies
  • Reliability Software- Weibull++ (ReliaSoft)- BlockSim (ReliaSoft)
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