Complexity results on planar multifacility location problems with forbidden regions

This paper considers two planar facility location problems while employing the Manhattan travel metric. We first consider the p-median problem in the presence of arbitrarily shaped barriers and convex forbidden regions. For this problem we establish that the search for an optimal solution can be restricted to a finite set of easily identifiable points. Next, we consider the stochastic queue median problem in the presence of arbitrarily shaped barriers. A procedure to obtain a global optimum solution for this problem is established. The results of the paper are illustrated via numerical examples. Finally, we comment on a connection between network location problems and planar location problems which use the Manhattan travel metric.

Planar multifacility location problems with tree structure and finite dominating sets

This paper considers planar location problems with rectilinear distance and barriers where the objective function is any convex, nondecreasing function of distance. Such problems have a non-convex feasible region and a nonconvex objective function. Based on an equivalent problem with modified barriers, derived in a companion paper [3], the non convex feasible set is partitioned into a network and rectangular cells. The rectangular cells are further partitioned into a polynomial number of convex subcells, called convex domains, on which the distance function, and hence the objective function, is convex. Then the problem is solved over the network and convex domains for an optimal solution. Bounds are given that reduce the number of convex domains to be examined. The number of convex domains is bounded above by a polynomial in the size of the problem.

Facility location selection is a vital decision for companies that affects both cost and delivery time over the years. However, determination of the facility location is a NP-hard problem. A hybrid algorithm that combines revised weighted fuzzy c-means with Nelder Mead (RWFCM-NM) performs well when compared with well-known algorithms for the facility location problem (FLP) with deterministic customer demands and positions. The motivation of the study is both analyzing performance of the RWFCM-NM algorithm with probabilistic customer demands and positions and proposing a new approach for this problem. This paper develops two new algorithms for FLP when customer demands and positions are probabilistic. The proposed algorithms are a probabilistic fuzzy c-means algorithm and Nelder-Mead (Probabilistic FCM-NM), a probabilistic revised weighted fuzzy c-means algorithm and Nelder Mead (Probabilistic RWFCM-NM) for the un-capacitated planar multi-facility location problem when customer positions and customer demands are probabilistic with predetermined service level. Proposed algorithms performances were tested with 13 data sets and comparisons were made with four well known algorithms. According to the experimental results, probabilistic RWFCM-NM algorithm demonstrates superiority on compared algorithms in terms of total transportation costs.

Equitable location of facilities in a region with probabilistic barriers to travel

Given the demand between each origin-destination pair on a network, the planar hub location problem is to locate the multiple hubs anywhere on the plane and to assign the traffic to them so as to minimize the total travelling cost. The trips between any two points can be nonstop (no hubs used) or started by visiting any of the hubs. The travel cost between hubs is discounted with a factor. It is assumed that each point can be served by multiple hubs.

A globally convergent algorithm for facility location on a sphere

Inspired by the classical fractional cascading technique, we introduce new techniques to speed up the following type of iterated search in 3D: The input is a graph $\mathbf{G}$ with bounded degree together with a set $H_v$ of 3D hyperplanes associated with every vertex of $v$ of $\mathbf{G}$. The goal is to store the input such that given a query point $q\in \mathbb{R}^3$ and a connected subgraph $\mathbf{H}\subset \mathbf{G}$, we can decide if $q$ is below or above the lower envelope of $H_v$ for every $v\in \mathbf{H}$. We show that using linear space, it is possible to answer queries in roughly $O(\log n + |\mathbf{H}|\sqrt{\log n})$ time which improves trivial bound of $O(|\mathbf{H}|\log n)$ obtained by using planar point location data structures. Our data structure can in fact answer more general queries (it combines with shallow cuttings) and it even works when $\mathbf{H}$ is given one vertex at a time. We show that this has a number of new applications and in particular, we give improved solutions to a set of natural data structure problems that up to our knowledge had not seen any improvements. We believe this is a very surprising result because obtaining similar results for the planar point location problem was known to be impossible.

A classical problem in computational geometry is the planar point location problem. This problem calls for preprocessing a polygonal subdivision of the plane defined by n line segments so that, given a sequence of points, the polygon containing each point can be determined quickly on-line. Several ways of solving this problem in O(log n) query time and O(n) space are known, but they are all rather complicated. We propose a simple O(log n)-query-time, O(n)-space solution, using persistent search trees. A persistent search tree differs from an ordinary search tree in that after an insertion or deletion, the old version of the tree can still be accessed. We develop a persistent form of binary search tree that supports insertions and deletions in the present and queries in the past. The time per query or update is O(log m), where m is the total number of updates, and the space needed is O(1) per update. Our planar point location algorithm is an immediate application of this data structure. The structure also provides an alternative to Chazelle's hive graph structure, which has a variety of applications in geometric retrieval.

We study the planar maximum coverage location problem (MCLP) with rectilinear distance and rectangular demand zones in the case where “partial coverage” is allowed in its true sense, i.e., when covering part of a demand zone is allowed and the coverage accrued as a result of this is proportional to the demand of the covered part only. We pose the problem in a slightly more general form by allowing service zones to be rectangular instead of squares, thereby addressing applications in camera view-frame selection as well. More specifically, our problem, referred to as PMCLP-PCR (planar MCLP with partial coverage and rectangular demand and service zones), is to position a given number of rectangular service zones (SZs) on the two-dimensional plane to (partially) cover a set of existing (possibly overlapping) rectangular demand zones (DZs) such that the total covered demand is maximized. Previous studies on (planar) MCLP have assumed binary coverage, even when nonpoint objects such as lines or polygons have been used to represent demand. Under the binary coverage assumption, the problem can be readily formulated and solved as a binary linear program; whereas, partial coverage, although much more realistic, cannot be efficiently handled by binary linear programming, making PMCLP-PCR much more challenging to solve. In this paper, we first prove that PMCLP-PCR is NP-hard if the number of SZs is part of the input. We then present an improved algorithm for the single-SZ PMCLP-PCR, which is at least two times faster than the existing exact plateau vertex traversal algorithm. Next, we study multi-SZ PMCLP-PCR for the first time and prove theoretical properties that significantly reduce the search space for solving this problem, and we present a customized branch-and-bound exact algorithm to solve it. Our computational experiments show that this algorithm can solve relatively large instances of multi-SZ PMCLP-PCR in a short time. We also propose a fast polynomial time heuristic algorithm. Having optimal solutions from our exact algorithm, we benchmark the quality of solutions obtained from our heuristic algorithm. Our results show that for all the random instances solved to optimality by our exact algorithm, our heuristic algorithm finds a solution in a fraction of a second, where its objective value is at least 91% of the optimal objective in 90% of the instances (and at least 69% of the optimal objective in all the instances).

We describe algorithms for optimal single facility location problems with forbidden regions and barriers to travel. The former are those where location is not permitted, but one can travel through them, e.g., a lake. The latter are the regions where neither location nor travel is permitted, e.g., large parks in a city. Using the convexity properties of the objective function, in the first case, we develop an algorithm for finding the optimal solution. The objective function in the barrier case is shown to be non-convex. We use the concept of visibility to create a network with the location point as the source and use Dijkstra's algorithm to compute the shortest distance to all the other demand points. Using simulated annealing we find an approximate optimal solution. Numerical examples illustrate the implementation of the algorithms.

Restricted center problems under polyhedral gauges

Secure communication with low computational resources is a critical issue in the Internet-of-Things (IoT) implementations. It is more challenging in the presence of hybrid adversary enabled with full-duplex (FD) capability to perform eavesdropping and jamming simultaneously. In this work we aim to address this issue through optimal subcarrier allocation towards combating the FD hybrid attacker. We begin with secrecy performance analysis in a multi-user IoT system considering statistical channel state information only of all attacker links. Novel analytical expression for the exact intercept probability is derived and a closed-form approximation is also provided. We further propose an optimisation framework for fair subcarrier allocation with a novel objective of minimising maximum intercept probability among multiple users. Considering the proposed optimisation framework as a non-convex combinatorial, we propose a low-complexity sub-optimal solution by leveraging the integer linear program (ILP) structure of the problem. To reduce the complexity further, the original problem is mapped to the assignment model and solved by exploiting its special structure with graph theory tools providing an optimal solution in polynomial time. Comprehensive investigations, conducted to verify the analysis and quantify the secrecy performance, demonstrate that proposed optimal solutions yield significant enhancement in secrecy performance over relevant schemes.

A fuzzy max-T location problem is considered. The fuzzy max-T location problem is a generalization of a fuzzy maximin location problem by allowing in the objective function arbitrary triangular norms instead of the triangular norm defined by the minimum operation. Then we give conditions for the existence of its optimal solutions, and derive a relationship between its optimal solutions and efficient solutions of a fuzzy multicriteria location problem. Furthermore, we give some properties of triangular norms, and for triangular norms, we investigate the stability of optimal solutions of the fuzzy max-T location problem. 1 Introduction and preliminaries In a general continuous location model, finitely many points called demand points in R n , modeling existing facilities or customers, are given. Let di ∈ R n , i =1 , 2, ··· , � (≥ 2) be demand points. We put I ≡{ 1, 2, ··· , � }. Then a problem to locate a new facility in R n is called a single facility location problem. If one prefers the location of the facility near demand points, then the problem is formulated as follows: (1) min x∈Rn f (γ1(x − d1) ,γ 2(x − d2), ··· ,γ � (x − d� ))

We present an implementable feasible direction subgradient algorithm for minimizing the maximum of a finite collection of functions subject to constraints. It is assumed that each function involved in defining the objective function is the sum of a finite collection of basic convex functions and that the number of different subgradient sets associated with nondifferentiable points of each basic function is finite on any bounded set. Problems involving functions of $l_p$-norms, such as location and approximation problems, can be put in this form. Conditions are given which guarantee that the algorithm generates a feasible sequence converging to an optimal solution. The results of computational tests on some location problems are included. In these tests we explore the sensitivity of the algorithm to its parameters.