MEC E 514 Reliability for Design

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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

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

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.

Review of Last Lecture

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

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

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
• 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)