Abstract
Many location problems can be expressed as ordered median objective. In this paper, we investigate the ordered median objective when the demand points are generated in a circle. We find the mean and variance of the kth distance from the centre of the circle and the correlation matrix between all pairs of ordered distances. By applying these values, we calculate the mean and variance of any ordered median objective and the correlation coefficient between two ordered median objectives. The usefulness of the results is demonstrated by calculating various probabilities such as: What is the probability that the mean distance is greater than the truncated mean distance? What is the probability that the maximum distance is greater than 0.9? What is the probability that the range of distances is greater than 0.8? An analysis of an illustrative example also demonstrates the usefulness of the analysis.
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Appendix
Appendix
Proofs
Proof of Theorem 1
Substituting m=j−1 in the first sum and recalling that leads to:
All the terms, except the first one in the first sum, cancel out proving the Theorem.
Proof of Lemma 3
For simplicity of presentation, let . By Formula (5.2.3) in Arnold et al (1992).
Equation (15) is therefore by isolating all the terms that include p and applying Lemma 2:
Similarly to the derivation of Lemma 2, . Hence,
Yielding,
In obtaining Equation (16), we assume that j>i. However, the formula is correct also for j=i: μ ii =i/(n+1), which is the same as E i 2 by the proof of Theorem 4.
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Drezner, Z., Nickel, S. & Ziegler, HP. Stochastic analysis of ordered median problems. J Oper Res Soc 63, 1578–1588 (2012). https://doi.org/10.1057/jors.2012.2
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DOI: https://doi.org/10.1057/jors.2012.2