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ABSTRACT: We analyze an economic order quantity cost model with unit outofpocket holding costs, unit opportunity costs of holding, fixed ordering costs, and general purchasetransportation costs. We identify the set of purchasetransportation cost functions for which this model is easy to solve and related to solving a onedimensional convex minimization problem. For the remaining purchasetransportation cost functions, when this problem becomes a global optimization problem, we propose a Lipschitz optimization procedure. In particular, we give an easy procedure which determines an upper bound on the optimal cycle length. Then, using this bound, we apply a wellknown technique from global optimization. Also for the class of transportation functions related to full truckload (FTL) and lessthantruckload (LTL) shipments and the wellknown carload discount schedule, we specialize these results and give fast and easy algorithms to calculate the optimal lot size and the corresponding optimal orderuptolevel.01/2015; 
Article: Median graphs and Helly hypergraphs
10/2014; 
Dataset: The Interval Function of a Graph
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ABSTRACT: This is a paper in the 1988 Kalamazoo Proceedings. Note that the authors did not get the opportunity to proof read the paper. So there are still some disturbing typos.07/2013; 
Article: Quorum colorings of graphs
AKCE International Journal of Graphs and Combinatorics. 01/2013; 10(1).  [Show abstract] [Hide abstract]
ABSTRACT: A profile π=(x1,…,xk)π=(x1,…,xk), of length kk, in a finite connected graph GG is a sequence of vertices of GG, with repetitions allowed. A median xx of ππ is a vertex for which the sum of the distances from xx to the vertices in the profile is minimum. The median function finds the set of all medians of a profile. Medians are important in location theory and consensus theory. A median graph is a graph for which every profile of length 3 has a unique median. Median graphs have been well studied, possess a beautiful structure and arise in many arenas, including ternary algebras, ordered sets and discrete distributed lattices. They have found many applications, for instance in location theory, consensus theory and mathematical biology. Trees and hypercubes are key examples of median graphs.We establish a succinct axiomatic characterization of the median procedure on median graphs, settling a question posed implicitly by McMorris, Mulder and Roberts in 1998 [19]. We show that the median procedure can be characterized on the class of all median graphs with only three simple and intuitively appealing axioms, namely anonymity, betweenness and consistency. Our axiomatization is tight in the sense that each of these three axioms is necessary. We also extend a key result of the same paper, characterizing the median function for profiles of even length on median graphs.Discrete Applied Mathematics 01/2013; 161:838  846. · 0.72 Impact Factor 
Article: Pathneighborhood graphs
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ABSTRACT: A pathneighborhood graph is a connected graph in which every neighborhood induces a path. In the main results the 3sunfree pathneighborhood graphs are characterized. The 3sun is obtained from a 6cycle by adding three chords between the three pairs of vertices at distance 2. A P k graph is a pathneighborhood graph in which every neighborhood is a P k , where P k is the path on k vertices. The P k graphs are characterized for k≤4.Discussiones Mathematicae. Graph Theory. 01/2013; 4(4). 
Article: The ℓpfunction on trees
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ABSTRACT: A pvalue of a sequence π = (x1, x2,…, xk) of elements of a finite metric space (X, d) is an element x for which is minimum. The function ℓp with domain the set of all finite sequences defined by ℓp(π) = {x: x is a pvalue of π} is called the ℓpfunction on X. The ℓpfunctions with p = 1 and p = 2 are the wellstudied median and mean functions respectively. In this article, the ℓpfunction on finite trees is characterized axiomatically. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012Networks 09/2012; 60(2). · 0.65 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The median problem is a classical problem in Location Theory: one searches for a location that minimizes the average distance to the sites of the clients. This is for desired facilities as a distribution center for a set of warehouses. More recently, for obnoxious facilities, the antimedian was studied. Here one maximizes the average distance to the clients. In this paper the mixed case is studied. Clients are represented by a profile, which is a sequence of vertices with repetitions allowed. In a signed profile each element is provided with a sign from {+,}. Thus one can take into account whether the client prefers the facility (with a + sign) or rejects it (with a  sign). The graphs for which all median sets, or all antimedian sets, are connected are characterized. Various consensus strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity. Hypercubes are the only graphs on which Majority produces the median set for all signed profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming graphs, Johnson graphs and halfcubes.01/2012;  [Show abstract] [Hide abstract]
ABSTRACT: An antimedian of a profile $\\pi = (x_1, x_2, \\ldots , x_k)$ of vertices of a graph $G$ is a vertex maximizing the sum of the distances to the elements of the profile. The antimedian function is defined on the set of all profiles on $G$ and has as output the set of antimedians of a profile. It is a typical location function for finding a location for an obnoxious facility. The `converse' of the antimedian function is the median function, where the distance sum is minimized. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper such a characterization is obtained for the two classes of graphs on which the antimedian iswellbehaved: paths and hypercubes.Discrete Mathematics Algorithms and Applications 01/2012; 04(04). 
Article: The Lpfunction on trees
01/2011;  [Show abstract] [Hide abstract]
ABSTRACT: The general problem in location theory deals with functions that find sites to minimize some cost, or maximize some benefit, to a given set of clients. In the discrete case sites and clients are represented by vertices of a graph, in the continuous case by points of a network. The axiomatic approach seeks to uniquely distinguish certain specific location functions among all the arbitrary functions that address this problem by using a list of intuitively pleasing axioms. The median function minimizes the sum of the distances to the client locations. This function satisfies three simple and natural axioms: anonymity, betweenness, and consistency. They suffice on tree networks (continuous case) as shown by Vohra (1996) [19], and on cubefree median graphs (discrete case) as shown by McMorris et al. (1998) [9]. In the latter paper, in the case of arbitrary median graphs, a fourth axiom was added to characterize the median function. In this note we show that the above three natural axioms still suffice for the hypercubes, a special instance of arbitrary median graphs.Discrete Applied Mathematics 01/2011; 159:939944. · 0.72 Impact Factor  01/2011: pages 225  241; , ISBN: 9789814299145

Article: Guides and Shortcuts in Graphs
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ABSTRACT: The geodesic structure of a graphs appears to be a very rich structure. There are many ways to describe this structure, each of which captures only some aspects. Ternary algebras are for this purpose very useful and have a long tradition. We study two instances: signpost systems, and a special case of which, step systems. Signpost systems were already used to characterize graph classes. Here we use these for the study of the geodesic structure of a spanning subgraph F with respect to its host graph G. Such a signpost system is called a guide to (F,G). Our main results are: the characterization of the step system of a cycle, the characterization of guides for spanning trees and hamiltonian cycles.Discrete Mathematics 01/2011; 313(19). · 0.58 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This book is available at the Publisher: World Scientific Publishing Co, or via boostores or online bookstores.01/2011; World Scientific Publishing Co., ISBN: ISBN: 9789814299145 
Article: Maximal outerplanar graphs as chordal graphs, pathneighborhood graphs, and triangle graphs
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ABSTRACT: Maximal outerplanar graphs are characterized using three different classes of graphs. A pathneighborhood graph is a connected graph in which every neighborhood induces a path. The triangle graph $T(G)$ has the triangles of the graph $G$ as its vertices, two of these being adjacent whenever as triangles in $G$ they share an edge. A graph is edgetriangular if every edge is in at least one triangle. The main results can be summarized as follows: the class of maximal outerplanar graphs is precisely the intersection of any of the two following classes: the chordal graphs, the pathneighborhood graphs, the edgetriangular graphs having a tree as triangle graph.The Australasian Journal of Combinatorics [electronic only]. 01/2011; 52.  01/2011: pages 71  91; , ISBN: 9789814299145

Chapter: Median graphs. A structure theory
01/2011; , ISBN: ISBN: 9789814299145  [Show abstract] [Hide abstract]
ABSTRACT: A mean of a sequence π=(x 1 ,x 2 ,⋯,x k ) of elements of a finite metric space (X,d) is an element x for which ∑ i=1 k d 2 (x,x i ) is minimum. The function Mean whose domain is the set of all finite sequences on X and is defined by Mean(π)={xx is a mean of π} is called the mean function on X. In this note, the mean function on finite trees is characterized axiomatically.Discrete Mathematics Algorithms and Applications 01/2010; 2(3):313  329.  [Show abstract] [Hide abstract]
ABSTRACT: A mean of a sequence Ï€ = (x1, x2, . . . , xk) of elements of a finite metric space (X, d) is an element x for which is minimum. The function Mean whose domain is the set of all finite sequences on X and is defined by Mean(Ï€) = { x  x is a mean of Ï€ } is called the mean function on X. In this paper the mean function on finite trees is characterized axiomatically.Erasmus University Rotterdam, Econometric Institute, Econometric Institute Report. 01/2010;
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