# MEC E 514 Reliability for Design - PowerPoint PPT Presentation

1 / 36

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 ...

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

MEC E 514 Reliability for Design

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

### 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:

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)