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Confidence in heuristic solutions?

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  • Kenneth Carling
  • Xiangli Meng

Abstract

Solutions to combinatorial optimization problems frequently rely on heuristics to minimize an intractable objective function. The optimum is sought iteratively and pre-setting the number of iterations dominates in operations research applications, which implies that the quality of the solution cannot be ascertained. Deterministic bounds offer a mean of ascertaining the quality, but such bounds are available for only a limited number of heuristics and the length of the corresponding interval may be difficult to control in an application. A small, almost dormant, branch of the literature suggests using statistical principles to derive statistical bounds for the optimum. We discuss alternative approaches to derive statistical bounds. We also assess their performance by testing them on 40 test $$p$$ p -median problems on facility location, taken from Beasley’s OR-library, for which the optimum is known. We consider three popular heuristics for solving such location problems; simulated annealing, vertex substitution, and Lagrangian relaxation where only the last offers deterministic bounds. Moreover, we illustrate statistical bounds in the location of 71 regional delivery points of the Swedish Post. We find statistical bounds reliable and much more efficient than deterministic bounds provided that the heuristic solutions are sampled close to the optimum. Statistical bounds are also found computationally affordable. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Kenneth Carling & Xiangli Meng, 2015. "Confidence in heuristic solutions?," Journal of Global Optimization, Springer, vol. 63(2), pages 381-399, October.
  • Handle: RePEc:spr:jglopt:v:63:y:2015:i:2:p:381-399
    DOI: 10.1007/s10898-015-0293-4
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    References listed on IDEAS

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    1. Kenneth Carling & Mengjie Han & Johan Håkansson, 2012. "Does Euclidean distance work well when the p-median model is applied in rural areas?," Annals of Operations Research, Springer, vol. 201(1), pages 83-97, December.
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    5. S. L. Hakimi, 1965. "Optimum Distribution of Switching Centers in a Communication Network and Some Related Graph Theoretic Problems," Operations Research, INFORMS, vol. 13(3), pages 462-475, June.
    6. Gonsalvez, David J. & Hall, Nicholas G. & Rhee, WanSoo T. & Siferd, Sue P., 1987. "Heuristic solutions and confidence intervals for the multicovering problem," European Journal of Operational Research, Elsevier, vol. 31(1), pages 94-101, July.
    7. Wilson, Amy D. & King, Russell E. & Wilson, James R., 2004. "Case study on statistically estimating minimum makespan for flow line scheduling problems," European Journal of Operational Research, Elsevier, vol. 155(2), pages 439-454, June.
    8. Ulrich Derigs, 1985. "Using Confidence Limits for the Global Optimum in Combinatorial Optimization," Operations Research, INFORMS, vol. 33(5), pages 1024-1049, October.
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    1. Carling, Kenneth & Han, Mengjie & Håkansson, Johan & Rebreyend, Pascal, 2015. "Testing the gravity p-median model empirically," Operations Research Perspectives, Elsevier, vol. 2(C), pages 124-132.

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