Journal of Heuristics

, Volume 11, Issue 3, pp 201–217

# Multi-Path Approach for Reliability-Redundancy Allocation Using a Scaling Method

Article

## Abstract

The reliability-redundancy allocation problem is an optimization problem that achieves better system reliability by determining levels of component redundancies and reliabilities simultaneously. The problem is classified with the hardest problems in the reliability optimization field because the decision variables are mixed-integer and the system reliability function is nonlinear, non-separable, and non-convex. Thus, iterative heuristics are highly recommended for solving the problem due to their reasonable solution quality and relatively short computation time. At present, most iterative heuristics use sensitivity factors to select an appropriate variable which significantly improves the system reliability. The sensitivity factor represents the impact amount of each variable to the system reliability at a designated iteration. However, these heuristics are inefficient in terms of solution quality and computation time because the sensitivity factor calculations are performed only at integer variables. It results in degradation of the exploration and growth in the number of subsequent continuous nonlinear programming (NLP) subproblems. To overcome the drawbacks of existing iterative heuristics, we propose a new scaling method based on the multi-path iterative heuristics introduced by Ha (2004). The scaling method is able to compute sensitivity factors for all decision variables and results in a decreased number of NLP subproblems. In addition, the approximation heuristic for NLP subproblems helps to avoid redundant computation of NLP subproblems caused by outlined solution candidates. Numerical experimental results show that the proposed heuristic is superior to the best existing heuristic in terms of solution quality and computation time.

### Keywords

reliability optimization reliability-redundancy allocation scaling method iterative heuristic nonconvex mixed-integer nonlinear programming

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### References

1. Barlow, R.E. and F. Proschan. (1996). Mathematical Theory of Reliability. Philadelphia, PA: SIAM.Google Scholar
2. Bazaraa, M.S., H.D. Sherali, and C.M. Shetty. (1993). Nonlinear Programming: Theory and Algorithms 2nd edn. New York: John Wiley & Sons.Google Scholar
3. Bertsekas, D.P. (1999). Nonlinear Programming. 2nd edition, Belmont: Athena Scientific.Google Scholar
4. Gopal, K., K.K. Aggarwal, and J.S. Gupta. (1978). “An Improved Algorithm for Reliability Optimization.” IEEE Transactions on Reliability. 29, 325–328.Google Scholar
5. Gopal, K., K.K. Aggarwal, and J.S. Gupta. (1980). “A New Method for Solving Reliability Optimization Problem.” IEEE Transactions on Reliability. 29, 36–38.Google Scholar
6. Ha, C. (2004). “Reliability-Yield Allocation for Semiconductor Integrated Circuits: Modeling and Optimization.” College Station: Texas A&M University, PhD Theis.Google Scholar
7. Ha, C. and Way Kuo. (2004). “Reliability Redundancy Allocation: An Improved Realization for Nonconvex Nonlinear Programming Problems.” accepted by European Journal of Operational Research.Google Scholar
8. Hikita, M., Y. Nakagawa, K. Nakashima, and H. Narihisa. (1992). “Reliability Optimization of Systems by a Surrogate-Constraints Algorithm.” IEEE Transactions on Reliability. R-41(3), 473–480.
9. Hwang, C.L., F.A. Tillman, and W. Kuo. (1979). “Reliability Optimization by Generalized Lagrangian-Function and Reduced-Gradient Methods.” IEEE Transactions on Reliability. R-28(4), 316–319.Google Scholar
10. Kim, J.H. and B.J. Yum. (1993). “A Heuristic Method for Solving Redundancy Optimization Problems in Complex Systems.” IEEE Transactions on Reliability. 42(4), 572–578.
11. Kuo, W., C.L. Hwang, and F.A. Tillman. (1978). “A Note on Heuristic Methods in Optimal System Reliability.” IEEE Transactions on Reliability. R-27, 320–324.Google Scholar
12. Kuo, W., H. Lin, Z. Xu, and W. Zang. (1987). “Reliability Optimization with the Lagrange Multiplier and Branch-and-Bound Technique.” IEEE Transactions on Reliability. R-36, 624–630.Google Scholar
13. Kuo, W. and V.R. Prasad. (2000). “An Annotated Overview of System Reliability Optimization.” IEEE Transactions on Reliability. 49(2), 487–493.
14. Kuo, W., V.R. Prasad, F.A. Tillman, and C.L. Hwang. (2001). Optimal Reliability Design: Fundamentals and Applications. Cambridge: Cambridge University Press.Google Scholar
15. Nakagawa, Y. and K. Nakashima. (1977). “A Heuristic Method for Determining Optimal Reliability Allocation.” IEEE Transactions on Reliability. R-26(3), 156–161.Google Scholar
16. Powell, M.J.D. (1977). “Restart Procedures for the Conjugate Gradient Method.” Mathematical Programming. 12, 241–254.
17. Rockafellar, R.T. (1993). “Lagrange Multipliers, and Optimality.” SIAM Review. 35(2), 183–238.
18. Sun, X.L., K.I.M. Mckinnon, and D. Li. (2001). “A Convexification Method for a Class of Global Optimization Problems with Applications to Reliability Optimization.” Journal of Global Optimization. 21, 185–199.
19. Tillman, F.A., C.L. Hwang, and W. Kuo. (1977). “Determining Component Reliability and Redundancy for Optimum System Reliability.” IEEE Transactions on Reliability. 18, 162–165.Google Scholar
20. Xu, Z., W. Kuo, and H. Lin. (1990). “Optimization Limits in Improving System Reliability.” IEEE Transactions on Reliability. R-39(1), 51–60.