Location problems with costs being sums of powers of euclidean distances
Location problems with costs being sums of powers of euclidean distances
复制标题
成本为欧氏距离幂总和的位置问题
DOI：
10.1016/0305-0548(84)90017-0
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
Reuven Chen
中科院分区：
文献类型：
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作者：
Reuven Chen
进一步研究了极小化欧氏距离加权幂和的库珀问题。这个问题的实际意义在于，不是欧氏距离加权和最小化的韦伯问题，而是考虑了经济和规模不经济的韦伯问题，这导致了欧氏距离加权和最小化的问题。原始的库珀解是Weiszfeld方法的韦伯问题的修改；它可以被证明是一种最陡下降法，其步长由依赖于权重和距离的表达式确定，这在任何意义上都不是最优的。在本工作中，对最佳步长问题进行了研究。结果表明，将Cooper给出的步长乘以2n(其中n是欧几里得距离的幂)，大大改善了迭代过程。对于Cooper考虑的n≥L情形，即1≤n≤3，减少了达到某一终止准则所需的步数。对于较高的n值，原始库珀步长通常太大，不会收敛。用目前修正的方法，即使对于100次方或更多的次方，使用适当的归一化也可以解决问题。给出了这种步长的半直观证明，并给出了n=1，10,100解的数值例子。该方法可以很容易地扩展到三维(或更多)维的选址问题。
The Cooper problem of minimizing the sum of weighted powers of the Euclidean distances is further studied. The practical significance of this problem is that instead of the Weber problem where the weighted sum of the Euclidean distances is minimized, here economies and diseconomies of scale are allowed for, which results in a problem of minimizing the sum of weighted powers of the Euclidean distances. The original Cooper solution has been a modification of the Weiszfeld method for the Weber problem; it can be shown to be a steepest descent method with a step size determined by an expression depending on the weights and distances, which is not optimal in any sense. In the present work, the matter of the best step size to be taken is investigated. It is shown that multiplying the step size given by Cooper by 2 n where n is the power of the Euclidean distances, improves the iterative process substantially. For the cases with n≥ l considered by Cooper, namely 1≤ n≤ 3, the number of steps needed to reach a certain termination criterion is reduced. For higher values of n, the original Cooper step size is usually too large and there is no convergence. With the present amended method, the problem can be solved, using an appropriate normalization, even for powers of 100 and more. A semi-intuitive proof for this step size is given as well as numerical examples of solutions with n= 1, 10 and 100. The method can easily be extended to location problems in three (and more) dimensions.