Minisum location problem with farthest Euclidean distances

A Barzilai-Borwein-based heuristic algorithm for locating multiple facilities with regional demand

“…The classical Weiszfeld procedure (2.3), however, may terminate unexpectedly when the singular case happens. Therefore, some alternatives capable of handling the singular case have been contributed, including the primal-dual algorithm [31], the variational inequality approach [25,26], and so on. Nevertheless, the Weiszfeld procedure and its variants are still significantly appreciated because of their attractive performance in practice.…”

“…The so-called farthest minisum location method (FMLM) developed in [25] is applicable for solving SMPs (1.2) arising in the location tasks. By dividing the entire plane into some fixed areas, each SMP in (1.2) is solvable via solving a polynomial number of CWPs.…”

“…, n} denote the set of regional demand. For convenience, we simply assume that {A j } are rectangles or circles throughout this paper; and the extension to the general case where {A j } are convex polygons can be referred to [25]. We denote the farthest point in A j to the facility x i by…”

“…Applying some strategies of decomposing the entire plane into a polynomial number of convex polygons (called fixed areas), Jiang and Xu [25] replaced f j (x) in (1.2) by some extreme point of A j (if A j is a rectangle), or the center of A j (if A j is a circle), within any of the decomposed fixed areas. Hence, (1.2) is solvable via solving a finite number of constrained Weber problems (CWPs) whose constraints are the reduced fixed areas.…”

“…Hence, (1.2) is solvable via solving a finite number of constrained Weber problems (CWPs) whose constraints are the reduced fixed areas. More specifically, for the ith SMP (1.2) at the kth iteration of the method in [25], let F i q (q ∈ {1, 2, . .…”