Summary: The modified Weiszfeld method [Y. Vardi
and C.H. Zhang
, Math. Program. 90, No. 3 (A), 90, 559–566 (2001; Zbl 0990.65064
)] is perhaps the most widely-used algorithm for the single-source Weber problem (SWP). In this paper, in order to accelerate the efficiency for solving SWP, a new numerical method, called Weiszfeld-Newton method, is developed by combining the modified Weiszfeld method with the well-known Newton method. Global convergence of the new Weiszfeld-Newton method is proved under mild assumptions. For the multi-source Weber problem (MWP), a new location-allocation heuristic, Cooper-Weiszfeld-Newton method, is presented in the spirit of Cooper algorithm [L. Cooper
, SIAM Rev. 6, 37–53 (1964; Zbl 0956.90014
)], using the new Weiszfeld-Newton method in the location phase to locate facilities and adopting the nearest center reclassification algorithm (NCRA) in the allocation phase to allocate the customers. Preliminary numerical results are reported to verify the evident effectiveness of Weiszfeld-Newton method for SWP and Cooper-Weiszfeld-Newton method for MWP.