Abstract
The problem of assembling components into series modules to maximize the system reliability has been intensively studied in the literature. Invariably, the methods employed exploit special properties of the reliability function through standard analytical optimization techniques. We propose a geometric approach by exploiting the assembly polytope – a polytope generated by the potential assembly configurations. The new approach yields simpler proofs of known results, as well as new results about systems where the number of components in a module is not fixed, but subject to lower and upper bounds.
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References
E.R. Barnes A.J. Hoffman U.G. Rothblum (1992) ArticleTitleOn optimal partitions having disjoint convex and conic hulls Mathematical Programming 54 69–86 Occurrence Handle10.1007/BF01586042
L.A. Baxter F. Harche (1992) ArticleTitleNote: on the greedy algorithm for optimal assembly Naval Research Logistics 39 833–837
C. Derman G.J. Lieberman S.M. Ross (1972) ArticleTitleOn optimal assembly of systems Naval Research Logistics Quarterly 19 564–574
Du, D.Z. (1987), When is a monotonic grouping optimal? In: Osaki, S. and Cao, J. (eds.), Reliability Theory and Application, World Scientific, New Jersey, pp. 66–76.
D.Z. Du F.K. Hwang (1990) ArticleTitleOptimal assembly of an s-stage k-out-of-n system SIAM Journal on Discrete Mathematics 3 349–354 Occurrence Handle10.1137/0403030
E. El-Neweihi F. Proschan J. Sethuraman (1986) ArticleTitleOptimal allocation of components in parallel-series and series-parallel systems Journal of Applied Probability 23 770–777 Occurrence Handle10.2307/3214014
E. El-Neweihi F. Proschan J. Sethuraman (1987) ArticleTitleOptimal allocation assembly of systems using Schur functions and majorization Naval Research Logistics Quarterly 34 705–712
Fujishige, S. (1991), Submodular functions and optimization, Annals of Discrete Mathematics 47, North Holland, Amsterdam.
F.K. Hwang J.S. Lee U.G. Rothblum (2004) ArticleTitlePermutation polytopes corresponding to strongly supermodular functions Discrete Applied Mathematics 142 87–97 Occurrence Handle10.1016/j.dam.2002.11.010
F.K. Hwang U.G. Rothblum (1994) ArticleTitleOptimality of monotone assemblies for coherent systems composed of series modules Operations Research 42 709–720
F.K. Hwang U.G. Rothblum (1995) ArticleTitleSome comments on the optimal assembly problem Naval Research Logistics 42 757–771
F.K. Hwang U.G. Rothblum (1996) ArticleTitleDirectional-quasi-convexity, asymmetric Schur-convexity and optimality of consecutive partitions Mathematics of Operations Research 21 540–554 Occurrence Handle10.1287/moor.21.3.540
Hwang, F.K. and Rothblum, U.G. (2004), Partitions: Optimality and Clustering, World Scientific, forthcoming.
F.K. Hwang J. Sun E.Y. Yao (1985) ArticleTitleOptimal set-partitioning SIAM Journal on Algebraic and Discrete Methods 6 163–170
F.K. Hwang Y.M. Wang J.S. Lee (2002) ArticleTitleSortability of multi-partitions Journal of Global Optimization 24 463–472 Occurrence Handle10.1023/A:1021216024210
D.M. Malon (1990) ArticleTitleWhen is greedy module assembly optimal Naval Research Logistics Quarterly 37 847–854
S. Onn U.G. Rothblum (2004) ArticleTitleConvex combinatiorial optimization Discrete and Computational Geometry 32 549–566 Occurrence Handle10.1007/s00454-004-1138-y
L.S. Shapley (1971) ArticleTitleCores of convex games International Journal of Game Theory 1 11–26 Occurrence Handle10.1007/BF01753431
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Hwang, F.K., Rothblum, U.G. A Polytope Approach to the Optimal Assembly Problem. J Glob Optim 35, 387–403 (2006). https://doi.org/10.1007/s10898-005-3844-2
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DOI: https://doi.org/10.1007/s10898-005-3844-2