Journal of Applied Mathematics

Volume 2019, Article ID 9520324, 16 pages

https://doi.org/10.1155/2019/9520324

## On the Location of a Constrained Tree Facility in a Tree Network with Unreliable Edges

^{1}Applied Mathematics and Information Science Department, Zewail City of Science and Technology, 6th of October City, Giza, Egypt^{2}Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt^{3}Department of Basic Sciences, Faculty of Engineering, Pharos University, Alexandria, Egypt

Correspondence should be addressed to Abdallah W. Aboutahoun; ge.ude.yticliawez@nuohata

Received 20 March 2019; Accepted 24 July 2019; Published 21 August 2019

Academic Editor: Wei-Chang Yeh

Copyright © 2019 Abdallah W. Aboutahoun and Eman Fares. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Given a tree network with vertices where each edge has an independent operational probability, we are interested in finding the optimal location of a reliable service provider facility in a shape of subtree with exactly leaves and with a diameter of at most which maximizes the expected number of nodes that are reachable from the selected subtree by operational paths. Demand requests for service originate at perfectly reliable nodes. So, the major concern of this paper is to find a location of a reliable tree-shaped facility on the network in order to provide a maximum access to network services by ensuring the highest level of network connectivity between the demand nodes and the facility. An efficient algorithm for finding a reliable – tree core of is developed. The time complexity of the proposed algorithm is Examples are provided to illustrate the performance of the proposed algorithm.

#### 1. Introduction

The classical location theory is concerned with the location of a service facility on a network. This facility can be a single point or a specified number of points, located at either a vertex or along an edge of the network. The location of the facility depends on the distances between the demand vertices (customers) and their respective server. The location of special types of subgraphs such as paths, trees, or forests is considered as an extensive facility location problem. These facilities formed of connected structures have many applications in the fields such as transportation, communication, and computer sciences. The three criteria for optimality which are extensively studied in the literature are the following:(1)The median criterion or the minisum criterion in which the sum of the distances from all the vertices of the network to the facility is minimized.(2)The center criterion or the minimax criterion in which the maximum distance from the demand vertex to the facility is minimized.(3)The centdian criterion in which the convex combination of the weighted average distance and the maximum weighted distance from the facility to the demand points is minimized.

In recent years, there has been a growing interest in studying the location of facilities formed of connected structures which are also called extensive facilities on a tree network. The location of a path-shaped and tree-shaped facilities under the above three criteria has been studied by a number of authors. In this paper, we are studying the location of a reliable tree-shaped facility on a tree network with unreliable edges. A core of a tree is defined to be a path that is optimal with respect to the property of minimizing the sum of the distances from each vertex in the tree to the path (criterion 1). The generalization of the core of the tree is a tree core which is a subtree containing exactly leaves that minimizes the sum of the distances from each vertex in the tree network to the selected tree core facility. The tree core is a subtree with diameter at most having leaves which minimizes the sum of the distances or the weighted distances from all vertices in the tree network to this subtree facility.

Finding the core of a tree network has been considered by a several authors. Becker et al. [1] presented two efficient algorithms for finding the core of a tree. For unweighted trees they developed an time algorithm, while for weighted trees they provided a procedure with time complexity of . Peng and Lo [2] presented a recursive time algorithm for finding a core of specified length, that is, a path with length equal to a specified value , in unweighted trees. Morgan and Slater [3] presented a linear time algorithm for finding the core of a tree network. Minieka and Patel [4] explored some properties of a core of a specified length of a tree network. A core of a tree is a set of mutually disjoint paths in that minimizes the sum of the distances of all vertices in from any of the two paths. Wang [5] developed an time algorithm for the core problem, where is the number of vertices in . Becker et al. [6] considered the problem of finding an optimal location of a path-shaped facility on a tree network using combinations of the center and the median criteria. They studied two problems: finding a path which minimizes the sum of the distances such that the maximum distance from the vertices of the tree to the path is bounded by a fixed constant and such that the length of the path is not greater than a fixed value; finding a path which minimizes the maximum distance with the sum of the distances being not greater than a fixed value and with bounded length. They gave divide-and-conquer algorithms.

Wang et al. [7] developed two algorithms for finding a tree core of a tree . The first algorithm has time complexity for the weighted tree (each edge has an arbitrary length). The second algorithm has time complexity for the unweighted tree (the lengths of all edges are 1). Also, Peng et al. [8] presented two algorithms to find a tree core of a tree with vertices. The time complexities of these two algorithms are and Becker et al. [9] provided two algorithms; the first one for unweighted trees has time complexity of , whereas the second one for weighted trees has time complexity of . Shioura and Uno [10] proposed a linear time algorithm for finding a tree core of a tree network. Minieka [11] described methods for finding an optimal location for a path-shaped or tree-shaped facility of a specified size in a tree network. Four optimization criteria were examined: minimizing distancesum, minimizing eccentricity, maximizing distancesum, and maximizing eccentricity. Solution algorithms were presented. Kim et al. [12] studied the problem of locating a subtree facility on a tree network. They considered the cost of establishing the subtree facility in addition to the transportation cost associated with the travel of customers to the subtree facility. The objective function is to select a tree-shaped facility that will minimize the sum of the setup cost and the total transportation cost. Tamir er al. [13] developed an algorithm of time complexity for finding the optimal location of a tree-shaped facility of a specified size in a tree network with nodes, using the centdian criterion: a convex combination of the weighted average distance and the maximum weighted distance from the facility to the demand points (nodes of the tree). Tamir et al. [14] studied the location of a subtree facility on a tree network, both discrete and continuous under the condition that existing facilities are already located. They used the center and the median criteria.

The problems of finding a single point, a path, or a subtree on tree network with distance edges have been studied extensively in literature, a single facility location problem on a network with unreliable edges has been also considered by a number of authors. Melachrinoudis and Helander [15] addressed the reliable 1-median (relisum) location problem of a single facility on an undirected tree with unreliable edges. The objective of the problem is to maximize the expected number of nodes reachable by operational paths. They developed two polynomial algorithms; an algorithm which is a modification of the Floyd-Warshall algorithm for finding all pairs of the shortest paths in a graph and an algorithm which set up a depth-first node traversal and the decomposition nature of an operational path. Xue [16] presented a linear time algorithm for the same problem which was displayed by Melachrinoudis and Helander [15]. Helander and Melachrinoudis [17] introduced path reliability measures for the expected number of accidents over a given planning horizon. Reliability refers to the probability of a hazmat transport vehicle completing a journey from an origin to a destination. They presented two different locations of the modeling frameworks: the reliable 1-median and a general framework for considering multiple routes. Santiváñez et al. [18] considered the problem of location a single facility on an undirected network with unreliable edges under the center criterion. The objective function is formally stated as either minimizing the maximum expected number of unsuccessful responses to demand requests over all nodes, called the reli-minmax problem, or maximizing the minimum expected number of successful responses to demand requests over all nodes, called the reli-maxmin problem. The problem is termed the reliable 1-center problem and finds applications in telecommunication and computer networks. As for subproblems of the most general reliable 1-center problem, Eiselt et al. [19] presented the problem of locating facilities on a network so that the total expected demand disconnected from the facilities is minimized. It is supposed that every edge has a probability of failure and that failures can never occur on two edges simultaneously. Eiselt et al. [20] showed how to optimally locate facilities on a network with unreliable node or edge in order to minimize the expected demand disconnected from the facilities. Ding and Xue [21] studied the problem of locating a node which maximizes the expected number of nodes that are reachable from it. Such a node is called a most reliable source (MRS) of the network. They presented a linear time algorithm for computing a most reliable source on an unreliable tree network where all links are immune to failures and each node has an independent transmitting probability and an independent receiving probability. Puerto et al. [22] considered the problem of locating two path-shaped facilities minimizing the expected service cost in the long run, assuming that paths may become unavailable and their failure probabilities are known in advance. They provided a polynomial time algorithm that solves the unreliable path location problem on tree networks in time, where is the number of vertices. Ding et al. [23] studied the problem of finding the 2-most reliable source (2-MRS) in an unreliable tree network. The 2-MRS problem aims to find a node pair from which the expected number of reachable nodes or the minimum reachability is maximized (i.e., Sum-Max 2-MRS and Min-Max 2-MRS).

The reliable tree core problem is stated as finding the optimal location of a tree-shaped facility (service provider) in an unreliable tree network with a diameter of at most and having leaves which maximizes the expected number of nodes that are reachable from it. This problem considers only edge failures while the nodes and the service provider facility are considered perfectly reliable. So, the model we present in this paper is stochastic in the sense that edges fail randomly which may cause a disconnection between demand nodes and service provider facility. The major concern of this work is to find a location of a service provider facility in order to provide a maximum access to network services, but in the context of connectivity, i.e., the probability that an operational path exists between two points of the network, and therefore the connectivity of the network is measured using path reliability.

For any two vertices , the vertex is called reachable from (or is called reachable from ) if there exists an operational path between them, i.e., every edge in this path is in an operational state. The objective is to identify a subtree that has exactly leaves, diameter of at most and the cumulative reliability from to , is maximized. This subtree is the reliable tree core facility of . The reliable tree core problem is motivated from a distributed database application in computer science; see Wang et al. [7] and Peng et al. [8].

The remainder of this paper is organized into five sections. In Section 2, we present notation, definitions, and some preliminary results. Section 3 is devoted to the formulation and a full description for the proposed problem. In Section 4, we present an efficient algorithm for finding a reliable tree core of . In Section 5, a numerical example is provided to illustrate the efficiency of the proposed algorithm. Finally, in Section 6, we give a concluding remarks and future research.

#### 2. Definitions and Preliminaries

Let be a tree network, where is the node set and is the edge set. Let be a unique path in tree from node to node and let be its length which is defined in this study to be the number of edges in the path. Let and denote the set of nodes and edges on the path , respectively. Let denote the probability that edge in operational state such that and is the probability that the edge is in a failed state. All vertices are assumed to be perfectly reliable and any vertex is called a leaf vertex if the number of edges incident to it is equal to . The number of edges that can be in a failed state has no limitations, we assume that failures occur independently. Let be the tree obtained by making rooted at a vertex The diameter of is while the height of is , where is called the root of tree . Let be an a vertex in , and we denote by a subtree of rooted at a vertex Let denote the set of children of and denote the parent of . If the vertex is adjacent to , is called a subtree of

For each edge we call the endpoint of closest to the root , and we call the endpoint of farthest to the root. Each edge divides into two disjoint subtrees denoted by and . The subtree is a subtree of rooted at a vertex as defined above and the subtree is a subtree of rooted also at but induced by the set of vertices , where is the set of vertices of the subtree . If and are two vertices in an unreliable communication network, we use to denote the reliability that a message can be transmitted correctly from to Let be a subtree of and be the set of nodes of The expected number of nodes reachable by operational paths from the nodes of subtree of is called the reachability or connectivity of the subtree

A reliable , −tree core is a subtree with leaves and with a diameter of at most which maximizes the expected number of nodes that are reachable by operational paths from the nodes of . In other words, we seek for a subtree of with diameter of at most , having leaves, which maximizes the expected number of successful responses to demand requests over all nodes. A demand request is originating at a node which requires that some entity moves from the service provider located at a subtree to node along some path in the tree network.

*Definition 1. *Under the assumption that edges fail independently, the reliability of a path is defined as the product of the reliability of arcs in the path , i.e.,and the probability that at least one edge in the path has failed deeming the path unusable is

Note that when consists of one edge and when Also, note that and as long as all vertices have the same weight. The path is operational if and only if all edges of are operational simultaneously. Since for any there exists a unique path in connecting and , then it always holds that in

*Definition 2. *For a rooted tree , the total connectivity of any vertex or the expected number of nodes reachable from is defined bywhere

For a vertex and a path of , the reliability from to is

Also, for a vertex and a subtree of , the reliability from to is

*Definition 3. *For a rooted tree , the total connectivity of any path is the sum of the reliabilities from all the vertices of not in to the path where is the maximum reliability from to a vertex in

*Definition 4. *For a rooted tree , the total connectivity of any subtree is the sum of the reliabilities from all the vertices of not in to the subtree where is the maximum reliability from to a vertex in

Following the same notation developed by Ding and Xue [21], let be a rooted tree at any vertex. For any vertex in , let be the subtree of rooted at vertex . Let be the set of vertices of and let be the subtree induced by the set of vertices such that and If is a leaf, then and If , then and The quantities corresponding to are often referred as below quantities, while the ones corresponding to are referred as upper quantities. The two subtrees and are connected by the edge , see Figure 1.