1. Reliability-Redundancy Allocation for Multi-State Series-Parallel Systems Zhigang Tian, Ming J. Zuo, and Hongzhong Huang IEEE Transactions on Reliability, vol. 57, No. 2, June 2008 Presented by: Hui-Yu, Chung Advisor: Frank Yeong-Sung, Lin

2. Agenda Introduction Problem Formulation ◦ Design Variables ◦ System Utility Evaluation ◦ Formulation of System Cost ◦ Characteristics of the Optimization Problem ◦ Physical Programming-Based Optimization Problem Formulation ◦ Optimization Solution Method – Genetic Algorithm An Example ◦ The Joint Reliability-Redundancy Optimization Results ◦ The Redundancy Optimization Results ◦ Sensitivity Analysis for System Cost and System Utility Conclusions 2

3. Introduction Component – An “Entity” ◦ Can be connected in a certain configuration to form a subsystem, or system. Multi-State System ◦ Many systems can perform their intended functions at more than two different levels  From perfectly working to completely failed  Provide more flexibility for modeling Performance Measure – System Utility 3

4. Introduction State Distribution ◦ Used to describe the reliability of a MMS Two ways to improve the utility of a multi-state series-parallel system: ◦ To provide redundancy at each stage ◦ To improve the component state distribution  Make a component in states w.r.t. high utilities and probabilities 4

5. Introduction Previous studies on optimization of MMSs focused on only redundancies ◦ Only partial optimization The option of selecting different versions of components provides more flexibility 5

6. Notation & Acronym 6

7. Assumptions The states of the components in a subsystem is independent identically distributed (i.i.d.) The components, and the system may be in M + 1 possible states, namely, 0, 1, 2, …, M The multi-state series parallel systems under consideration are coherent systems 7

8. Agenda Introduction Problem Formulation ◦ Design Variables ◦ System Utility Evaluation ◦ Formulation of System Cost ◦ Characteristics of the Optimization Problem ◦ Physical Programming-Based Optimization Problem Formulation ◦ Optimization Solution Method – Genetic Algorithm An Example ◦ The Joint Reliability-Redundancy Optimization Results ◦ The Redundancy Optimization Results ◦ Sensitivity Analysis for System Cost and System Utility Conclusions 8

9. Problem Formulation The structure of a multi-state series- parallel system: N subsystems connected in series, each subsystem i has independent identically distributed components connected in parallel The prob. of component i in state j is 9

10. Design Variables State distributions ◦ i = 1, 2, …, N; j = 1, 2, …, M Redundancies ◦ i = 1, 2, …, N Reliability means the prob. of working 10

11. Reliability of a component Consider a three-state system ◦ Three States: { 0,1,2 } ◦ State Distributions: Statements: ◦ 1)The prob. that a component is in state 1 or 2 is the reliability of this component that its state is greater or equal to 1(“working” )  Reliability: ◦ 2) The prob. of component in state 2 is the reliability of it that its state is greater than or equal to 2  Reliability: 11

12. System Utility Evaluation System utility: The expected utility The prob. that the system is in state s or above: (s = 0, 1, …, M) The System Utility U: : Utility when the system is in state s 12

13. Formulation of System Cost The cost of subsystem i with parallel components: ◦ : cost-reliability relationship function for a component in subsystem i ◦ : cost of the components in subsystem i ◦ : interconnecting cost in parallel subsystem ◦, : characteristic constants 13

14. In a (M + 1) state MMS: ◦ Reliability of component i under treatment k: Assumption: ◦ There are M treatments that can influence the component’s state distribution, and treatment k will increase the prob. of the component in state k, but will not influence the prob. of the component in the states above k Formulation in System Cost 14

15. Formulation of System Cost The cost of the component: The system cost: 15

16. Characteristics of the Optimization Problem Objective to be optimized: ◦ System Utility, ◦ System Cost Determine and to maximize system utility and minimize cost Mixed integer optimization problem ◦ Continuous variables: state distributions ◦ Integer variables: redundancies 16

17. Characteristics of the Optimization Problem Formulated as a single-objective optimization problem: ◦ Either cost or utility can be a design objective, while the other can be a constraint 17

18. Characteristics of the Optimization Problem Formulated as a multi-objective optimization problem: ◦ Three approaches:  The surrogate worth trade-off method  The fuzzy optimization method  Physical programming method In this case, physical programming approach is used 18

19. Physical Programming-Based Optimization Problem Formulation Physical Programming Optimization ◦ The Decision Maker’s preference is considered in the optimization process ◦ Use of class functions Class Functions: ◦ The value reflects the preference of the designer on objective function value ◦ Four types of of “soft” class function:  Smaller is better, Larger is better, Value is better, and Range is better  Here, we use “Smaller is better” 19

20. Physical Programming-Based Optimization Problem Formulation Class-1S Class Function (for Cost) ◦ Monotonously increasing function ◦ Used to represent the objectives to be minimized Class-2S Class Function (for Utility) ◦ Monotonously decreasing function ◦ Used to represent the objectives to be maximized Design Objective Corresponding class function value 20

21. Physical Programming-Based Optimization Problem Formulation Transforming a physical programming problem to a single-objective optimization problem: f: aggregate objective function 21

22. Genetic Algorithm as the Optimization Solution Method Genetic Algorithm: ◦ Most effective algorithm to solve mixed integer optimization problems ◦ Chromosome: one solution in GA ◦ Population: a group of chromosome in each iteration Four stages in GA:  Initialization, selection, reproduction, termination 22

23. The procedure of GA Initialization ◦ Specify the GA operators ◦ Specify the GA parameters Evaluation ◦ Using fitness value to get P(k) and B(k) Construct new population ◦ Chromosome is replaced by the best fitness value. Terminate ◦ When reaching a maximal iteration 23

24. Agenda Introduction Problem Formulation ◦ Design Variables ◦ System Utility Evaluation ◦ Formulation of System Cost ◦ Characteristics of the Optimization Problem ◦ Physical Programming-Based Optimization Problem Formulation ◦ Optimization Solution Method – Genetic Algorithm An Example ◦ The Joint Reliability-Redundancy Optimization Results ◦ The Redundancy Optimization Results ◦ Sensitivity Analysis for System Cost and System Utility Conclusions 24

25. An Example 25

26. The Joint Reliability-Redundancy Optimization Results In Physical programming framework ◦ System utility: Class-2S objective function ◦ System cost: Class-1S objective function Mixed integer programming problem 9 design variables: GA parameters (run 30 times) Population Size Chromosome Length Selection Scheme Crossover rate Mutation Rate Maximum epoch 10015 Roulette- wheel 0.25 (One-point) 0.11000 26

27. The Joint Reliability-Redundancy Optimization Results The result of the optimization: 27

28. The Redundancy Optimization Results Consider different versions of components ◦ Otherwise, the results may not be optimal Integer programming problem The other conditions remain the same Component version for stage i 28

29. The Redundancy Optimization Results The Result of the optimization: 29

30. Sensitivity Analysis for System Cost and System Utility Sensitivity analysis of system cost: ◦ 9 design variables ◦ Model parameters,  Since they are affect the system costs Using the partial derivative to analyze ◦ While keeping all the others the same 30

31. Sensitivity Analysis for System Cost 31

32. Sensitivity Analysis for System Cost For any, is always positive ◦ System cost increases with the increase in The sensitivity of system cost decreases a bit with the increase of,( < 0.05 ) When > 0.05, the sensitivity always increases with the increase of System cost is more sensitive to stage 3 ◦ Since in stage 3 is larger 32

33. Sensitivity Analysis for System Cost Sensitivity w.r.t. the parameter, and ◦ Positive Constant Value  Cost increases with the increase of the parameter ◦ Positive and more sensitive 33

34. Sensitivity Analysis for System Utility 34

35. Sensitivity Analysis for System Utility For any, is always positive ◦ The system utility increases with the increases of System utility becomes less sensitive to with the increase of it. The utility is more sensitive to the distribution variables associated with state 2 35

36. Agenda Introduction Problem Formulation ◦ Design Variables ◦ System Utility Evaluation ◦ Formulation of System Cost ◦ Characteristics of the Optimization Problem ◦ Physical Programming-Based Optimization Problem Formulation ◦ Optimization Solution Method – Genetic Algorithm An Example ◦ The Joint Reliability-Redundancy Optimization Results ◦ The Redundancy Optimization Results ◦ Sensitivity Analysis for System Cost and System Utility Conclusions 36

37. Conclusions Two options too improve the system utility of a multi-state series-parallel system: ◦ Provide redundancy at each stage ◦ Improve the component state distributions Physical programming-based optimization is introduced and used in this problem Sensitivity Analysis ◦ Which can reflect the facts on the model 37

38. ~The End~ Thanks for Your Attention!!! 38