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On optimal assembly of systems

Author

Listed:
  • Cyrus Derman
  • Gerald J. Lieberman
  • Sheldon M. Ross

Abstract

We are concerned with the following reliability problem: A system has k different types of components. Associated with each component is a numerical value. Let {aj}, (j= 1,…,k), denote the set of numerical values of the k components. Let R(a1,…, ak) denote the probability that the system will perform satisfactorily (i. e., R(a1,…. ak) is the reliability of the system) and assume R(a1,…,ak) has the properties of a joint cumulative distribution function. Now suppose aj1 ≤… ≤ ajn are n components of type j (j= 1,…, k). Then n systems can be assembled from these components. Let N denote the number of systems that perform satisfactorily. N is a random variable whose distribution will depend on the way the n systems are assembled. Of all different ways in which the n systems can be assembled, the paper shows that EN is maximized if these n systems have reliability R(a1j,…, akj) (i = 1,…,n). The method used here is an extension of a well known result of Hardy, Littlewood, and Polya on sums of products. Furthermore, under certain conditions, the same assembly that maximizes EN minimizes the variance of N. Finally, for a similar problem in reliability, it is shown that for a series system a construction can be found that not only maximizes the expected number of functioning modules, but also possesses the stronger property of maximizing the probability that the number of functioning modules is at least r, for each 0 ≤ r ≤ n.

Suggested Citation

  • Cyrus Derman & Gerald J. Lieberman & Sheldon M. Ross, 1972. "On optimal assembly of systems," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 19(4), pages 569-574, December.
    • Handle: RePEc:wly:navlog:v:19:y:1972:i:4:p:569-574
    DOI: 10.1002/nav.3800190402
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    Citations

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    Cited by:

    1. Xiaoyan Zhu & Way Kuo, 2014. "Importance measures in reliability and mathematical programming," Annals of Operations Research, Springer, vol. 212(1), pages 241-267, January.
    2. Michael Katehakis & Ingram Olkin & Sheldon Ross & Jian Yang, 2013. "On the life and work of Cyrus Derman," Annals of Operations Research, Springer, vol. 208(1), pages 5-26, September.
    3. V.D. Dinopoulou & C. Melolidakis, 2001. "Asymptotically optimal component assembly plans in repairable systems and server allocation in parallel multiserver queues," Naval Research Logistics (NRL), John Wiley & Sons, vol. 48(8), pages 732-746, December.
    4. Wang, Dan & Si, Shubin & Cai, Zhiqiang & Zhao, Jiangbin, 2021. "Reliability optimization of linear consecutive-k-out-of-n: F systems driven by reconfigurable importance," Reliability Engineering and System Safety, Elsevier, vol. 216(C).
    5. Laurence A. Baxter & Farid Harche, 1992. "Note: On the greedy algorithm for optimal assembly," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(6), pages 833-837, October.
    6. Coit, David W. & Zio, Enrico, 2019. "The evolution of system reliability optimization," Reliability Engineering and System Safety, Elsevier, vol. 192(C).

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