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This article develops the mathematical implications of boundaries to the geometrical and analytical features of continuing place difficulties. The booklet will entice these operating in operations examine and administration technological know-how, and mathematicians attracted to optimization concept and its purposes.

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**Additional resources for Single Facility Location Problems with Barriers**

**Sample text**

Thus the objective is to minimize the total travel cost between the new facility X and Ex, modeling, for example, a warehouse, and a set of customers, respectively. Weber problems with diﬀerent distance functions and various extensions and variations have been extensively studied in the literature. For an overview of the Weber problem and related problems, see, for example, Wesolowsky (1993) and Drezner et al. (2002). t. 2), the weights wm , m ∈ M, are assumed to be nonnegative. A survey of the center problem and of related covering location problems is given, for example, in Plastria (2002).

Representation of a point p(t) on a permitted X-Y path in respect to u(t) and µ(t), t ∈ [0, 1]. 1. Let B be the unit sphere in Rn , centered at the origin, with 1 2 n x2i =1 . ∂(B) = X = (x1 , . . 5) by the functional G : Rn → R given by 1 2 n − 1. x2i G(X) = i=1 Let X, Y ∈ Rn , X, Y ∈ int(B), be two feasible points. t. 4, where for all u ¯ ∈ ∂(B) the normal vector N (¯ u) is given by N (¯ u) = ∇G(¯ u) = ∂ ∂ G(u), . . , G(u) ∂u1 ∂un = u¯. u=¯ u This can be easily veriﬁed, since the partial derivatives of G at a point u = (u1 , .

2000) introduce Weber problems with continuous demand over some given polyhedral set, possibly with holes acting as barriers to travel, and the Manhattan metric. While a polynomial-time algorithm is developed for the case that only one new facility is sought, NP-hardness is proven for the case of multifacility Weber problems if the number of new facilities is part of the input data. Lower and upper bounds as well as the relative accuracy of solutions for multifacility Weber problems with the Manhattan metric, with and without barriers, are also discussed in Batta and Leifer (1988).