M.B. Teitz, P. Bart, Heuristic methods for estimating generalized vertex median of a weighted graph, Operations Research 16 (5) (1968) 955-961.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....more generic and more robust. Still, it is not immediately clear that these methods can compete in computational eciency with k Means. 3 Interchange hill climbers The minimization of TWGD(P ) is typically solved approximately using interchange heuristics based on a hill climbing search strategy [13, 22, 26, 27, 38]. Hill climbers search the space of all partitions P = S 1 j : jS k of S by treating the space as if it were a graph: every node of the graph can be thought to correspond to a unique partition of the data. An edge exists between two nodes if the corresponding two partitions di er slightly. ....

....new node P is an improvement over the old (that is, if TWGD(P ) TWGD(P ) becomes the current node for time step t 1. Interchange hill climbers may de ne the neighborhood set N(P ) in varying ways. One general interchange heuristic, originally proposed in 1968 by Teitz and Bart [38], is a hill climber that is regarded as the best known benchmark [22] It has been remarkably successful in nding local optima of high quality in applications to facility location problems [27, 33] and with some improvements, very accurate for the clustering of large sets of low dimensional ....

M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955-961, 1968.

....open so as to minimize the sum of the distances from each user to the closest open facility. Being a well known NP complete problem [2] one is often compelled to resort to heuristics to deal with it in practice. Among the most widely used is the swapbased local search proposed by Teitz and Bart [10]. It has been applied on its own [8, 13] and as a key subroutine of more elaborate metaheuristics [3, 7, 9, 12] The e#ciency of the local search procedure is of utmost importance to the e#ectiveness of these methods. In this paper, we present a novel implementation of the local search procedure ....

....This is the case if there is a distance matrix, or if facilities and users are points on the plane, for instance. In this model, all values of # 1 and # 2 for a given solution S can be straighforwardly computed in O(pn) total time. 2 The Swap based Local Search Introduced by Teitz and Bart in [10], the standard local search procedure for the p median problem is based on swapping facilities. For each facility f i S, the procedure determines which facility f r S (if any) would improve the solution the most if f i and f r were interchanged (i.e. if f i were inserted and f r removed ....

M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16(5):955--961, 1968.

....the sum of the distances from each user to its closest open facility. Several algorithms for the p median problem have been proposed, including exact methods based on linear programming [2, 3, 8, 30] constructive algorithms [3, 17, 38] dual based algorithms [8, 21] and local search procedures [13, 16, 19, 24, 27, 35, 38]. More recently, metaheuristics capable of obtaining solutions of near optimal quality have been devised. Tabu search procedures have been proposed by Voss [37] and Rolland et al. 26] The latter method was compared in [29] with Rosing and ReVelle s heuristic concentration method [28] which ....

....solution quality. We could not use the randomization strategy normally used in GRASP, represented here by rgreedy; instead, we had to develop a faster alternative based on sampling. 4 Local Search The standard local search procedure for the p median problem, originally proposed by Teitz and Bart [35] and studied or used by several authors [10, 14, 15, 16, 24, 38] is based on swapping facilities. Given an initial solution S, the procedure determines, for each facility f ## S, which facility g S (if any) would improve the solution the most if f and g were interchanged (i.e. if f were ....

M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16(5):955--961, 1968.

....open so as to minimize the sum of the distances from each user to the closest open facility. Being a well known NP complete problem [2] one is often compelled to resort to heuristics to deal with it in practice. Among the most widely used is the swap based local search proposed by Teitz and Bart [10]. It is often applied on its own [8, 13] and also as a key subroutine of more elaborate metaheuristics [3, 7, 9, 12] The efficiency of the local search procedure is of utmost importance to the effectiveness of these methods. In this paper, we present a novel implementation of the local search ....

....This is the case if there is a distance matrix, or if facilities and users are points on the plane, for instance. In this model, all values of f 1 and f 2 for a given solution S can be straighforwardly computed in O(pn) total time. 2 The Swap based Local Search Introduced by Teitz and Bart in [10], the standard local search procedure for the p median problem is based on swapping facilities. For each facility f i S, the procedure determines which facility f r S (if any) would improve the solution the most if f i and f r were interchanged (i.e. if f i were inserted and f r removed ....

M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16(5):955--961, 1968.

....[16] However, Han and Ng proposed an algorithm CLARANS for this optimization that is a randomized interchange hill climber in order to obtain subquadratic algorithmic complexity. CLARANS can not guarantee local optimality. The best interchange hillclimber [11] is the Teitz and Bart heuristic [23] which requires quadratic time. Only in restricted cases, the time complexity of this type of hill climber has been reduced to subquadratic time (for example, D = 2 and Euclidean distance [9] Thus, we take the route of randomized variants of iterative algorithms [10] The iteration is the ....

M.B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955-961, 1968.

....[19] However, Han and Ng proposed an algorithm CLARANS for this optimization that is a randomized interchange hill climber in order to obtain subquadratic algorithmic complexity. CLARANS can not guarantee local optimality. The best interchange hill climber [11] is the Teitz and Bart heuristic [25] which requires quadratic time. Only in restricted cases, the time complexity of this type of hill climber has been reduced to subquadratic time (for example, D = 2 and Euclidean distance [9] Because medians are a more robust estimator of location than means, an algorithm optimizing the case a = ....

....The result is an even more robust algorithm, still with complexity O(n p n) similarity computations. The algorithm in this section is an interchange heuristics based on a hillclimbing search strategy. However, we require to adapt this to non crisp classification since all previous versions [8, 9, 11, 14, 17 19, 25] are for the crisp classification case. We will first present a quadratic time version of our algorithm, which we will name non crisp TaB in honor of Teitz and Bart [25] original heuristic. Later, we will use randomization to achieve subquadratic time complexity, as in the previous section. Our ....

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M.B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955--961, 1968.

....the objective function will include some measure of the distance or transportation cost (referred to hereafter simply as distance) between the demand nodes and the facilities to which their demand has been allocated. For example, in the p Median location allocation problem (refer to Hakimi [19641, Teitz and Bart [1968], and ReVelle and Swain [ 1970] the goal is to select p facilities and assign each demand node to its nearest facility in order to minimize the total weighted service distance (demand multiplied by distance to nearest facility) of all demand nodes. Because of the integral role of internodal ....

....with each iteration until no further beneficial swaps can be identified, at which point the algorithm terminates. GRIA was selected for this study because in test runs it has proven to be capable of finding consistently good p Median solutions in rapid time. In numerous test comparisons with the Teitz and Bart [1968] heuristic, well known for its robustness in solving p Median problems (see, for example, Rosing, Hillsman, and Rosing [19791) GRIA proved capable of generating solutions of comparable quality in substantially reduced time (Sorensen [1994] However, as with all heuristics, including that of ....

Teitz, M. and P. Bart, 1968. "Heuristic Methods for Estimating the Generalized Vertex Median of a Weighted Graph", Operations Research, 16(5), 955-961.

....p median problem. Keywords: p median problem, facility location, clustering, aggregation, point location, Voronoi diagram, Delauney triangulation. 1 Introduction The p median problem is a central problem that has remained in the focus of the facility location and operations research community [4, 9, 10, 12, 16, 17, 21, 23, 25, 28, 29, 30, 31, 32, 33]. The role of this fundamental problem is illustrated by Hillsman [13] where other important problems reduce to the p median problem. The NP hardness of the problem was discovered early, and researches have steadily progressed in obtaining faster approximation methods. Also, the advances in ....

....) space and time for formulating the linear program (for example, the objective function in Equation (2) has (n 2 ) terms and we have (n 2 ) constrains in Equation (3) The most common alternative is to solve the problem approximately with interchange heuristics. Typically, Teitz and Bart [33] (and its variants [7, 16] is the most successful. However, this approach still requires (n 2 ) time to obtain local optima. The other alternative is to use aggregation [18, 14] Here, a set A of a sites (with p a n) is chosen as aggregate units (A = f 1 ; a g S) Then, each ....

M.B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955-961, 1968.

....research literature uses p instead of k for the number of groups) However, the p median problem is known to be NP hard. It has a zero one integer programming formulation [36] with n 2 variables and n 2 1 constraints, and many heuristic techniques have been developed to for this problem [11, 37, 38, 39, 40, 44]. Lagrangian relaxation has been successfully used for small problems in facility location [10, 33, 46] While facility location problems may involve perhaps hundreds of points, the p median problem cannot be expected to be solved optimally for the large number of observations (several thousand or ....

....of points, the p median problem cannot be expected to be solved optimally for the large number of observations (several thousand or more) that knowledge discovery applications typically involve. For nding high quality approximate solutions, hill climbing variations of an interchange heuristic [11, 20, 31, 44] are considered very e ective [16, 32, 34] Other alternatives have been also explored: tabu search [37] genetic algorithms [24, 5, 6, 17, 18] and simulated annealing [31] The trade o of e ort versus quality tends to favor hill climbers over these methods. However, the alternatives ....

[Article contains additional citation context not shown here]

M.B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955-961, 1968.

....phase) a greedy adding procedure with random substitution (GRASP) is used to find the initial location of the facilities, where in each iteration the vertex with the best objective value is added to the set of locations. In the second phase, a one opt exchange heuristic, based on the well known Teitz and Bart (1968) procedure is used. For each one opt exchange of facilities, the objective is computed and, if its value is better than before the trade, the new set of locations is stored. Otherwise, the old solution is restored. The procedure is executed for all facilities and potential locations, until no ....

....and a novel procedure is developed for solving exactly such systems. A one opt meta heuristic has been used to obtain model solutions. This heuristic has two phases. The first one finds an initial solution to the problem and the second one tries to improve it, by using first the well known Teitz and Bart (1968) one opt heuristic, and then executing a tabu search. The heuristic was tested in 900 different 30 nodes networks with success. Further research on congested hub systems could include more exact models of the congestion, although any improvement on these could lead to intractable formulations. ....

Teitz, M, and P. Bart 1968: "Heuristic Methods for Estimating the Generalized Vertex Median on a Weighted Graph" Operations Research 16, 955-965.

....optimum of the optimisation criteria. They simply stop when the necessary conditions hold. This approach for obtaining iterative algorithms for a differentiable loss function is recurrent in many fields (for example, the iterative methods here are attributed in the facility location literature [12, 22] to the work of Copper [5, 6] 2.3 Maximum likelihood leads to Expectation Maximisation Statistical inference with finite mixtures offer a third representative based clustering approach. Inference using the principle of Maximum Likelihood selects a family of probabilistic models indexed by ....

M.B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955--961, 1968.

....research literature uses p instead of k for the number of groups) However, the p median problem is known to be NP hard. It has a zero one integer programming formulation [70] with n 2 variables and n 2 1 constraints, and many heuristic techniques have been developed to for this problem [18, 73, 74, 75, 76, 83]. Lagrangian relaxation has been successfully used for small problems in facility location [16, 61, 89] While facility location problems may involve perhaps hundreds of points, the p median problem cannot be expected to be solved optimally for the large number of observations (several thousand or ....

....of points, the p median problem cannot be expected to be solved optimally for the large number of observations (several thousand or more) that knowledge discovery applications typically involve. For finding high quality approximate solutions, hill climbing variations of an interchange heuristic [18, 38, 58, 83] are considered very effective [29, 59, 62] Other alternatives have also been explored: tabu search [73] genetic algorithms [7, 8, 31, 32, 45] and simulated annealing [58] The trade off of effort versus quality tends to favor hill climbers over these methods. However, the alternatives ....

[Article contains additional citation context not shown here]

M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955--961, 1968.

....the restriction to CH disjoint partitions can produce approximation algorithms that are at least as efficient as previous attempts in solving the unrestricted TWGD clustering problem. We illustrate this point now. We adapt well studied local search hill climbers known as interchange heuristics [8, 24, 34, 43] to TWGD restricted to CH disjoint partitions. These heuristics are typically used for the p medians problem (solving Equation (2) with the added restriction that the representative be data points) and recently they have been used for the general TWGD problem [33] Similar adaptations will carry ....

....t 1. When no better solution is found in the neighbourhood N(P t ) the search halts. The interchange hill climbers proposed to date define the neighbourhood set N(P t ) in varying ways. In finding a local optimum of high quality, an original heuristic proposed in 1968 by Teitz and Bart [43] has proven the most effective. We will refer to this heuristic as TaB. Its adaptation to solving TWGD works as follows [33] When searching for a profitable interchange, it considers the data points in turn, according to a fixed circular ordering ( x 1 ; x 2 ; x n) of the data. ....

M.B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955--961, 1968.

....restriction to CH disjoint partitions can produce approximation algorithms that are at least as efficient that previous attempts for solving the unrestricted TWGD clustering problem. We illustrate this point now. We adapt well studied local search hill climbers known as interchange heuristics [8, 24, 34, 43] to TWGD restricted to CH disjoint partitions. These heuristics are typically used for the p medians problem (solving Equation (2) with the added restriction that the representative be data points) Convex group clustering of large geo referenced data sets 5 and recently they have been used for ....

....step t 1. When no better solution is found in the neighbourhood N(P t ) the search halts. The interchange hill climbers proposed to date define the neighbourhood set N(P t ) in varying ways. In finding a local optimum of high quality, an original heuristic proposed in 1968 by Teitz and Bart [43] has proven the most effective. We will refer to this heuristic as TaB. Its adaptation to solving TWGD works as follows [33] When searching for a profitable interchange, it considers the data points in turn, according to a fixed circular ordering ( x 1 ; x 2 ; x n) of the data. ....

M.B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955--961, 1968.

....(Note that in the case of k median problems, only the third operation is needed since all feasible solutions have exactly k facilities. This heuristic was 2 proposed by Kuehn and Hamburger [7] and was subsequently shown to exhibit good practical performance in empirical studies (see, e.g. [2, 12]) For the k median problems we consider (facility location variants (i) iii) and (v) above) we define an (a; b) approximation algorithm as a polynomial time algorithm that computes a solution using at most bk facilities, whose cost is at most a times the cost of the optimal solution that uses ....

M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955--961, 1968.

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M.B. Teitz, P. Bart, Heuristic methods for estimating generalized vertex median of a weighted graph, Operations Research 16 (5) (1968) 955-961.

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M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16(5):955--961, 1968.

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M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955--961, 1968.

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M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955--961, 1968.

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M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16(5):955--961, 1968.

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M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955--961, 1968.

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M. B. Teitz and P. Bart. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16(5):955--961, 1968.

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M.B. Teitz and P. Bart, Heuristic methods for estimating the generalized vertex median of a weighted graph. Gpns. Res. 16, 955-961 (1968).

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Transportation. 5: 389-406. Teitz, M.B. and Bart, P., 1968. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research. 16: 955-961.

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Tietz, M. B. and Bart, P. #1968#. Heuristic methods for estimating the generalized vertex median of a weighted graph. Operations Research, 16:955#965.

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