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Optimal Grouping in Binary Models: Sliding Window Systems (SWS)

Explore the hierarchical structure and reliability of sliding window systems (SWS) in binary models, including consecutive k-out-of-n and multi-state systems. Learn about the composition, distribution, and performance of elements in SWS for various applications like manufacturing, quality control, and service systems. Discover the algorithm for determining SWS reliability and optimal sequencing of elements for improved grouping solutions.

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Optimal Grouping in Binary Models: Sliding Window Systems (SWS)

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  1. r=n k=r Hierarchy of the Binary Models { k-out-of-r-from-n:F { r n { { k-out-of-n:F Consecutive k-out-of-n { k n n

  2. Pr{g>x} 1 r gnom 0 x Binary element Multi-state element Pr{g>x} 1 r gn 0 g1 g2 ... x

  3. r=n k=r Multi-state Models k-out-of-r-from-n Griffith (1986) Sliding Window Systems Levitin (2002) Consecutive k-out-of-n Chiang, Niu (1981), Bollinger (1982) Multi-state consecutive k-out-of-n Hwang, Yao (1989), Kossow, Preuss (1995) Weighted k-out-of-n Wu, Chen (1994) k-out-of-n Parallel Multi-state System

  4. Sliding window system definition k-out-of-r-from-n: Any function of r variables Any real value Acceptability function { r

  5. Sliding window systems { { { ... Total number of groups: n-r+1 { { ... Each element belongs to no more than r groups

  6. { SWS Applications:Manufacturing n { r

  7. { SWS Applications: Service System { n r

  8. { SWS Applications: Quality Control { n r Deviation Levels 3 2 1 0 1 2 3

  9. Representing Multi-state Elements and Groups Element State Distribution r-Group State Distribution gi+2,k gi+1,k ... gi,k S gi+r-1,k Cyclic Buffer ...

  10. Operator for Determining Group Unreliability Composition Operator gi+2,k gi+1,k gi+r,j ... gi,k S +gi+r,k-gi,j gi+r-1,k ...

  11. Like term collection in the the u-function gi+2 gi+1 gi+r,j ... gi gi+r-1 ... gi,1 gi,2 gi,3 gi,Ni gi+r-1 gi+r-1 ...

  12. Algorithm for SWS Reliability Determination

  13. Example of SWS reliability Determination 10 identical elements Element performance distribution P{G>x) x r:

  14. Ij= R/ rj No 1 2 3 4 5 6 7 8 9 10 r 0.87 0.90 0.83 0.95 0.92 0.89 0.80 0.85 0.82 0.95 g 200 200 400 300 100 400 100 200 300 200 Irrelevant element Reliability Importance of SWS Elements Most important element I w

  15. Optimal Sequencing of SWS Elements SWS Elements Performance distribution SWS Reliability R w 2,1,6,5,4,8,7,10,3,9 5,1,8,9,6,4,7,3,10,2 5,9,3,1,4,7,10,8,6,2

  16. A B Uneven allocation of SWS elements RA(3) =p4; RA(4)=0 RB(3) = p4+4(1-p)p3;RB(4) = p4 5—9—3—1—4—7—10—8—6—2 — —6,7,10— —2,5—1,4— —3,8,9——

  17. Optimal Grouping of SWS Elements in the Presence of Common Cause Failures

  18. Optimal Grouping Solutions for Different r and M r=3 r=5

  19. Ij= R/ sj Group Survivability Importance r=3 r=5

  20. r3=3, w3 r2=6, w2 r1=2, w1 g1g2 g3g4 Multiple sliding window systems { r2=5 { r1=3 …Gn G1 …

  21. { { >w3 { >w2 >w1 Example of SMWS

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