Technical Note—Algorithms for Weber Facility Location in the Presence of Forbidden Regions and/or Barriers to Travel

Abstract

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A modelling framework for solving restricted planar location problems using phi-objects

“…To our knowledge, the first study on this problems is by Aneja and Parlar (1994), who describe algorithms for the single facility location problem with forbidden regions and barriers separately, with a metric defined on R n and with respect to a parameter 1 ⩽ p ⩽ ∞, namely, the l p metric, which has a general form represented as [|x − x i | p + |y − y i | p ] 1/p . For p = 1, the l p metric is known as the rectilinear distance.…”

“…Similarly, l 2 is known as the Euclidean distance and l ∞ is the Chebyshev distance. The algorithms proposed by Aneja and Parlar (1994) are applicable to cases where 1 < p ⩽ 2. For the forbidden region problem, they present algorithms for polygonal convex and non-convex forbidden regions.…”

Placing a finite size facility with a center objective on a rectangular plane with barriers

“…where a is the length of the edge V 1 V 2 , then an ETTL with respect to i between corners V 1 and V 2 is generated such that it is perpendicular to the edge V 1 V 2 at a distance 1 2 fjd i1 À d i2 j þ ag from the cell corner that is closest to user i. The cases between other adjacent cell corners can be dealt with similarly.…”

“…where b is the length of the edge V 2 V 3 , then an ETTL with respect to i between corners V 1 and V 3 is generated that touches the cell boundary at a distance 1 2 fjd i1 À d i3 j þ a þ bg from the cell corner that is closest to user i. It is pertinent to mention here that ETTLs generated due to diagonally opposite cell corners are inclined at 45°to the edges of a 4-cornered cell, e.g., the ETTL generated due to V 1 and V 3 makes an angle of 45°with the edges V 2 V 3 and V 3 V 4 .…”

“…Aneja and Parlar [1] and Butt and Cavalier [3] developed heuristics for the 1-median problem in the presence of polygonal barriers under the l p distance metric. Though the center problem in R 2 without barriers has been extensively studied in the literature (e.g., books of Drezner [5], Love et al [11] and Francis et al [6]), very few references can be obtained for the corresponding problem in the presence of barriers.…”