Extensive facility location problems on networks: an updated review

Location problems with extensive facilities represent a challenging field of research. According to the specialized literature, a facility is called extensive if, for purposes of location, it is too large in relation to its environment to be considered a point. There are many examples of this type of structures that appear in real-world applications both in the continuous space (straight lines, circles, strips) and in networks (paths, cycles, trees). There exists a recent literature review on the location of dimensional facilities on continuous space (Díaz-Báñez et al. in TOP 154:22–44, 2004; Schöbel in Location of dimensional facilities in a continuous space, 2015) that does not cover similar problems on networks. The goal of this paper is to review the location of dimensional facilities in networks. We mainly concentrate on the location of paths and trees considering the most common objective functions in the location literature, namely median and center. However, we also consider some other alternative criteria generalizing them, as the ordered median objective function, or related to equity, reliability, and robustness. We include the basic tools and techniques that are applicable to develop algorithms for this kind of problems. Moreover, we present the best known complexity results for each of the considered problems. Finally, some suggestions are also made for possible directions of future research.     

Path integrals, the ABL rule and the three-box paradox

The three-box problem is analysed in terms of virtual pathways, interference between which is destroyed by a number of intermediate measurements. The Aharonov-Bergmann-Lebowitz (ABL) rule is shown to be a particular case of Feynman's recipe for assigning probabilities to exclusive alternatives. The ‘paradoxical’ features of the three box case arise in an attempt to attribute, in contradiction to the uncertainty principle, properties pertaining to different ensembles produced by different intermediate measurements to the same particle. The effect can be mimicked by a classical system, provided an observation is made to perturb the system in a non-local manner.     

Locating tree-shaped facilities using the ordered median objective

In this paper we consider the location of a tree-shaped facility S on a tree network, using the ordered median function of the weighted distances to represent the total transportation cost objective. This function unifies and generalizes the most common criteria used in location modeling, e.g., median and center. If there are n demand points at the nodes of the tree T=(V,E), this function is characterized by a sequence of reals, =(₁, . . . ,_n), satisfying ₁₂..._n0. For a given subtree S let X(S)={x₁, . . . ,x_n} be the set of weighted distances of the n demand points from S. The value of the ordered median objective at S is obtained as follows: Sort the n elements in X(S) in nonincreasing order, and then compute the scalar product of the sorted list with the sequence . Two models are discussed. In the tactical model, there is an explicit bound L on the length of the subtree, and the goal is to select a subtree of size L, which minimizes the above transportation cost function. In the strategic model the length of the subtree is variable, and the objective is to minimize the sum of the transportation cost and the length of the subtree. We consider both discrete and continuous versions of the tactical and the strategic models. We note that the discrete tactical problem is NP-hard, and we solve the continuous tactical problem in polynomial time using a Linear Programming (LP) approach. We also prove submodularity properties for the strategic problem. These properties allow us to solve the discrete strategic version in strongly polynomial time. Moreover the continuous version is also solved via LP. For the special case of the k-centrum objective we obtain improved algorithmic results using a Dynamic Programming (DP) algorithm and discretization and nestedness results.Acknowledgement We would like to thank Noga Alon for the current version of the proof of Theorem 4.1, which simplifies our original proof significantly. J. Puerto also thanks the Spanish Ministerio de Ciencia y Tecnología through grant number BFM2001-2378 for partially supporting his research.     

Filtering Policies in Loss Queuing Network Location Problems

In this paper we deal with two stationary loss queuing network location models. We analyze the influence of filtering policies on the locational aspect of the problems. We assume that requests for service are placed at nodes of a transportation network and they arrive in time as independent homogeneous Poisson processes with different input rates. The considered policies only cover a given proportion of requests even if there are idle service units. This proportion is stationary and fixed in advance and only depends on the node where the request is originated. The objective is to find the location of the facilities together with the filtering policy to be applied that minimize the expected total cost per unit time with respect to a given cost structure. Properties and computational results are presented enabling the resolution of these problems efficiently and showing the good performance of filtering policies in terms of both the overall operating costs, and the demand that is served.     

Radical copolymerization of acrylic monomers. IV. Effect of substituents on the radical polymerization and copolymerization of phenyl acrylates

The kinetics of radical polymerization of phenyl, ortho-chlorophenyl, and para-chlorophenyl acrylates, as well as their copolymerization with methyl methacrylate, have been studied dilatometrically. The results obtained indicate that the overall rate of polymerization is affected by the flexibility of the growing radicals. However, the copolymerization of these monomers with methyl methacrylate gives overall rates rather similar for all three systems, being fundamentally regulated by the formation of reversible π complexes between the donor aromatic rings and the acceptor methacrylic double bonds. Dilatometric methods for the study of the copolymerization reactions have been tested and the corresponding binary bonding frequencies B_ij and conversion factors K_ij have been calculated for the copolymerization of ortho- and para-chlorophenyl acrylates with methyl methacrylate.     

One single-step synthesis of multifunctional methylene blue-coated magnetite nanoparticles

This work reports the preparation of hybrid nanoparticles with magnetic and fluorescent properties. The material is based on magnetite nanoparticles (NPs) coated with fluorophore methylene blue (MB). The synthesis of a multifunctional material with magnetic and fluorescent features is carried out in a single step by electrooxidation. The effect of the presence of methylene blue in the synthetic medium is discussed. The presence of MB polymer at the NP surface is demonstrated with visible UV, infrared and Raman spectroscopy. The NPs morphology, structure and size are determined by transmission electron microscopy (TEM) and X-ray diffraction. The magnetic properties are measured with a vibrating sample magnetometer (VMS). In overall, the results show that magnetite NPs generated electrochemically in the presence of MB present a core/shell structure, being the NP at the core surrounded by methylene blue polymer, leading to a nanocomposite or hybrid material.     

A maxmin location problem with nonconvex feasible region

The problem dealt with consists of locating a point in a given convex polyhedron which maximizes the minimum Euclidean distance from a given set of convex polyhedra representing protected areas around population points. The paper describes a finite dominating solution set for the optimal solution and develops a geometrical procedure for obtaining the optimal solution comparing a finite number of candidates.     

Optimal Positioning of a Mobile Service Unit on a Line

In this paper we address the problem of locating a mobile response unit when demand is distributed according to a random variable on a line. Properties are proven which reduce the problem to locating a non-mobile facility, transforming the original optimization problem into an one-dimensional convex program.In the special case of a discrete demand (a simple probability measure), an algorithm which runs in expected linear time is proposed.