jafar fathali

In this paper, we investigate a case of the inverse 1-median problem on a tree by transferring the weights of vertices which has not been raised so far. This problem considers modifying the weights of vertices via transferring weights of the vertices with the minimum cost such that a given vertex of the tree becomes the 1-median with respect to the new weights. A linear programming model is proposed for this problem. The applicability and efficiency of the presented model are shown in numerical examples and a real-life problem dealing with transferring users in a social network.

Facility location problems are one of the most important issues for healthcare organizations and centers to achieve social welfare and respond to customer needs. Proper distribution of health and treatment facilities in cities is vital to minimize costs and improve the efficiency of health centers. The main contribution of the current article is dealing with the uncertainty issue in the p-median location-efficient problem. In this article, the p-median location problem along with network data envelopment analysis (Network DEA) is used in parallel mode to calculate the efficiency of health and treatment centers. In this issue, health centers are considered as parallel networks with two departments that operate independently. Due to the precision of the input and output values, triangular fuzzy numbers and the α-level fuzzy method have been used. The primary results that consider the uncertainty provide efficient solution and suggestions for the potential location of health centers in our case study.

Given a graph $G=(V,E)$,  a  function  $f:V\to \{0,1,2,3,4\}$ is a triple Roman  dominating function (TRDF)  of $G$, for each vertex $v\in V$,  (i) if $f (v ) = 0 $, then  $v$ must have either one neighbour in $V_4$, or either two neighbours in $V_2 \cup  V_3$ (one neighbour in $V_3$) or either three neighbours in $V_2 $, (ii) if $f (v ) = 1 $, then $v$ must have either one neighbour in  $V_3 \cup  V_4$  or either two neighbours in $V_2 $, and if $f (v ) = 2 $, then $v$ must have one neighbour in $V_2 \cup  V_3\cup  V_4$. The triple Roman  domination number of $G$ is the  minimum weight of an TRDF  $f$  of $G$, where the weight of $f$ is $\sum_{v\in V}f(v)$.  The triple  Roman  domination problem is to compute the  triple Roman  domination number of a given graph.  In this paper, we study the triple  Roman  domination problem. We show that   the problem is NP-complete for  the  star convex bipartite  and the   comb convex bipartite graphs and is APX-complete for graphs of degree at~most~4. We propose a linear-time algorithm for computing  the triple Roman  domination number of proper interval graphs.  We also   give an $( 2 H(\Delta(G)+1) -1  )$-approximation algorithm  for solving the problem  for any graph $G$,  where   $  \Delta(G)$ is the maximum degree of $G$ and $H(d)$ denotes the first $d$ terms of the harmonic  series. In addition, we prove  that  for any $\varepsilon>0$  there is no  $(1/4-\varepsilon)\ln|V|$-approximation  polynomial-time   algorithm for solving  the problem on bipartite and split  graphs, unless NP $\subseteq$ DTIME $(|V|^{O(\log\log|V |)})$.

In this paper we consider the inverse and reverse network facility location problems with considering the equity on servers. The inverse facility location with equality measure deals with modifying the weights of vertices with minimum cost, such that the difference between the maximum and minimum weights of clients allocated to the given facilities is minimized. On the other hand, the reverse case of facility location problem with equality measure considers modifying the weights of vertices with a given budget constraint, such that the difference between the maximum and minimum weights of vertices allocated to the given facilities is reduced as much as possible. Two algorithms with time complexity O(nlogn) are presented for the inverse and reverse 2-facility location problems with equality measures. Computational results show their superiority with respect to the linear programming models.

Traditionally, the minimax circle location problems concern finding a circle C in the plane such that the maximum distance from the given points to the circumference of the circle is minimized. The radius of the circle can be fixed or variable. In this paper we consider the inverse case, that is: a circle C with radius r0 is given and we want to modify the coordinate of existing points with the minimum cost such that the given circle becomes optimal. Mathematical models and some properties of the cases that circle C becomes optimal with comparing to all other circles, and circle C becomes the best circle with comparing to the circles with radius r0 are presented.

Location theory is one of the most important topics in optimization and operations research. In location problems, the goal is to find the location of one or more facilities in a way such that some criteria such as transportation costs, customer traveling distance, total service time, and cost of servicing are optimized. In this paper, we investigate the goal Weber location problem in which the location of a number of demand points on a plane is given, and the ideal is locating the facility in the distance R_i, from the i-th demand point. However, in most instances, the solution of this problem does not exist. Therefore, the minimizing sum of errors is considered. The goal Weber location problem with the l_p norm is solved using the stochastic version of the LBFGS method, which is a second-order limited memory method for minimizing large-scale problems. According to the obtained numerical results, this algorithm achieves a lower optimal value in less time with comparing to other common and popular stochastic optimization algorithms.

In this paper, we discuss the obnoxious and semi-obnoxious version of the backup 2-median problem on a tree. In the obnoxious case of the 2-median problem, all vertices have negative weights, whereas in the semi-obnoxious model the vertices may have either positive or negative weights. In these two problems, we should find the location of two facility servers on the tree so that the sum of minimum weighted distances from vertices in the tree to the set of functioning servers is minimized. In the backup model, each facility server may probably fail. If a facility server fails, the remaining server should serve the clients. Vertex optimality is an important property for the 2-median problem. This property indicates that the set of vertices involves an optimal solution of the 2-median problem. We verify that the vertex optimality holds for the semi-obnoxious backup 2-median problem on a tree network. In the obnoxious 2-median problem, the set of leaves contains an optimal solution, we show that this property does not hold for the obnoxious backup 2-median problem.

‎The objective of the classical version of the minisum circle location problem is finding a circle $C$ in the plane such that the sum of the weighted distances from the circumference of $C$ to a set of given points is minimized‎, ‎where every point has a positive weight‎. ‎In this paper‎, ‎we investigate the semi-obnoxious case‎, ‎where every existing facility has either a positive or negative weight‎. ‎The distances are measured by the Euclidean norm‎. ‎Therefore‎, ‎the problem has a nonlinear objective function and global nonlinear optimization methods are required to solve this problem‎. ‎Some properties of the semi-obnoxious minisum circle location problem with Euclidean norm are discussed‎. ‎Then a cuckoo optimization algorithm is presented for finding the solution of this problem‎.

In this paper, a fuzzy goal single facility location problem under to asymmetric Linex loss function is discussed. The aim of this paper is to determine the location of a facility center in the ideal radius to each of the demand points. In general, such a response is not always available. Therefore, minimizing the error function obtained from the distance of facility center to the ideal point is desirable. As, in many real-life situations, positive error and negative error of the same size often have different economic and material implications, a asymmetric Linex loss function is used for the first time where distinguishes between positive and negative errors with the same distance. In this paper, first, this problem is first investigated in a definite manner and by proving a theorem it is shown that the problem has a feasible solution and the optimal solution of the problem lies in the expanded rectangular shell of the demand points. In the following, To determine the optimal solution of the problem, a gradient quasi-Weisfield algorithm is presented, and by proposing some theorems, it is shown that this algorithm is convergent to the optimal solution of the problem. Also, to confirm the accuracy of the results obtained from this method, the obtained results are compared with the metaheuristic colonial competition algorithm. Finally, for the first time, the problem is modeled in fuzzy mathematical mode, and using a three-objective genetic algorithm its answers are compared and analyzed with a definite model.

This paper deals with the case of variable weights of the inverse model of the minimax circle location problem. The goal of the classic minimax circle location problem is finding a circle in the plane such that the maximum weighted distance from a given set of existing points to the circumference of the circle is minimized.  In the corresponding inverse model, a circle is given and we should modify the weights of existing points with minimum cost, such that the given circle becomes optimal. The radius of the given circle can be fixed or variable. In this paper, both of these cases are investigated and mathematical models are presented for solving them.

Location theory is an interstice field of optimization and operations research‎. ‎In the classic location models‎, ‎the goal is finding the location of one or more facilities such that some criteria such as transportation cost‎, ‎the sum of distances passed by clients‎, ‎total service time, and cost of servicing are minimized‎. ‎The goal Weber location problem is a special case of location models that have been considered recently by some researchers‎. ‎In this problem, the ideal is locating the facility in the distance $r_i$‎, ‎from the $i$-th client‎. ‎However‎, ‎in most instances‎, ‎the solution to this problem doesn't exist‎. ‎Therefore‎, ‎the minimizing sum of errors is considered‎. ‎In the previous versions of the goal location problem, the penalty functions have been considered by some symmetric functions such as square and absolute errors of distances between clients and ideal point‎. ‎In this paper‎, ‎we consider the asymmetric linex function as the error function‎. ‎We consider the case that the distances are measured by $L_p$ norm‎. ‎Some iterative methods are used to solve the problem and the results are compared with some previously examined methods.

In this paper we introduce a new facility location model, called backup multifacility location problem by considering the ideal radius for each customer. In this problem the location of clients are given in the plane. A radius is assigned to each client. We should find the location of new facilities, which some of them may fail with a given probability, such that the sum of weighted distances from new facilities to the radius distance of clients and sum of weighted distances between new facilities is minimized. Since in the most instance there dose not exist the location of a new facility such that its distance to each Customers be exactly equal to given radiuses, so we try to minimize the sum of the weighted square errors. We model the problem and propose an iterative method (weiszfeld like algorithm) for solving the presented problem. Then a discussion about convergence of presented method and some numerical examples are given. We show that the optimal solution lies in an extended rectangular hull of the existing points.

Location theory is an interstice field of optimization and operations research. In the classic location problem, the goal is finding the location of one or more facilities such that some criteria such as transportation cost, the sum of distances passed by clients, total service time and cost of servicing are minimized. In this paper, we consider the goal location problem. In the goal location problem, the ideal is locating the facility in the distances ri, from the i-th client. However, in the most instances, the solution of this problem doesn’t exist. Therefore, we consider the minimizing of distances between clients and ideal point. The minimizing sum of square errors and minimizing absolute errors under Lp norm are considered as the objective function. We use the Weiszfeld like, Gauss-Newton and imperialist competitive algorithms for solving the problem. Then we compare the results which obtained by these methods for some test problems.

In this paper we consider the inverse of backup 2-median problem. In this problem, a set of weighted points are given and we should change some parameters of the problem such as weights of vertices and edges and coordinates of points such that the two given points be the backup 2-median. We present mathematical models for inverse backup 2-median problems on graphs. In the case that the underlying network is a tree, linear models are presented for the problem with variable edges and weight of vertices. We also consider the continuous case of the problem with variable coordinates of vertices on the plane. In this case, we solve the model by PSO and a hybrid improved PSO methods. Computational results are compared for the varying amounts of parameters.   The inverse and backup location facility problems are two important branches of location theory that have been interested by many researchers in the recent decades.   Let n weighted points be given in the plane or on a graph. The inverse median models investigate to change some parameters of problem such as coordinates, edge lengths and vertex weights such that the given facilities be the median points. For more information about inverse location problems see Burkard et al. (2004). On the other hand, in the backup median problems supposed that some facilities may failed. Therefore the other facilities should serve the clients. The backup 2-median problem on trees has been considered by Wang et al. (2009). Fathali (2014) investigated the backup multi-facility location problem on the plane.
In this paper we consider the combination of inverse location and backup facility location problems. We want to change coordinates, weight of vertices or length of edges with minimum cost such that the given facilities be backup median facilities.  2. inverse Backup 2-Median On Trees: Let T= (V, E)  be a tree with n vertices. Each vertex  has a nonnegative weight. Let  be the distance between two points  and,  and  be the two given vertices in T which are assumed the location of facilities.   Each facility may fail with a probability. For any vertex, suppose that the cost of increasing and decreasing per unit of  is  and, respectively. Let  and  be the amounts by which the weight is increased and decreased, respectively. Then, the model of inverse backup 2-median problem can be written as follows. In this paper we investigated the backup 2-median problem with variable edge lengths and vertices weights on trees. The problem with variable coordinates on the plane is also considered. The models of mentioned problems and computational results which obtained by two PSO methods are presented.

One of the most reliable indicators of the evaluation of the same units is the use of mathematical programming based method called data envelopment analysis (DEA). DEA measures the efficiency score of a set of homogeneous decision making units (DMUs) based on observed input and output. The DEA method has been added to the literature by integrating Farrell's method in such a way that each evaluation unit has multiple inputs and multiple outputs. With the advancement and evolution of this approach, DEAis now one of the active areas of research in measuring performance and has been dramatically welcomed by world researchers. Charnes, Cooper, and Rhodes (CCR) [1] first proposed DEA method to evaluate the relative efficiency for not-for-profit organizations. So far, many studies and researches have been carried out in various associations and universities around the world about DEA and its applications. The simplicity of understanding and implementing the DEA method, along with its high precision and wide application in various political, cultural, social and economic fields has led many researchers to use this method to achieve their goals. So far, more than 50,000 articles, books, theses and more have been published on DEA theories and applications, calculations and issues.

Let a graph G = (V;E) be given. In the path center problem we want to find a path P in G such that the maximum weighted distance of P to every vertex in V is minimized. In this paper a genetic algorithm and a hybrid of genetic and ant colony algorithms are presented for the path center problem. Some test problems are examined to compare the algorithms. The results show that for almost all examples the hybrid method results better solutions than genetic algorithm.

In this paper, we consider a special case of Weber location problem which we call goal location problem. The Weber location problem asks to find location of a point in the plane such that the sum of weighted distances between this point and n existing points is minimized. In the goal location problem each existing point Pi has a relevant radius ri and it’s ideal for us to locate a new facility on the distance ri from Pi for i = 1, ..., n. Since in the most instances there does not exist the location of a new facility such that its distance to each point Pi be exactly equal to ri. So we try to minimize the sum of the weighted square errors. We consider the case that the distances in the plane are measured by the Euclidean norm. We propose a Weiszfeld like algorithm for solving the problem and also we use two modifications of particle swarm optimization method for solving this problem. Finally the results of these algorithms are compared with results of BSSS algorithm.

In this paper we study finding the (k,l)-core problem on a tree which the vertices have positive or negative weights. Let T=(V,E) be a tree. The (k,l)-core of T is a subtree with at most k leaves and with a diameter of at most l which the sum of the weighted distances from all vertices to this subtree is minimized. We show that, when the sum of the weights of vertices is negative, the (k,l)-core must be a single vertex. Then we propose an algorithm with time complexity of O(n2logn) for finding the (k,l) -core of a tree with pos/neg weight, which is in fact a modification of the one proposed by Becker et al. [Networks 40 (2002) 208].