This paper addresses the problem of identifying a planar subgraph of maximum weight of a given ed... more This paper addresses the problem of identifying a planar subgraph of maximum weight of a given edge weighted graph. The problem is NPComplete which justifies the use of heuristic (approximate) methods. It is shown how a new line of attack based on the graph theoretic concept of a wheel yields an efficient heuristic. An application to the layout of manufacturing
We describe a compact method to transform arc routing problem instances into node routing problem... more We describe a compact method to transform arc routing problem instances into node routing problem instances. Any node routing problem instance thus created must be solved by a branch-and-price process, such as the one described in this paper. The purpose is to make the number of nodes in the resulting transformed graphs greater by only one unit than the number r of required arcs (arcs having demand) in the original graph, that is, r+1 nodes. This low increase in the number of nodes represents an improvement compared to the methods previously presented by Pearn, Assad and Golden ( 3r+1 nodes) and by Longo, Poggi de Aragão and Uchoa and also by Baldacci and Maniezzo ( 2r+1 nodes). Using an adapted version of an existing branch-cut-and-price algorithm for a capacitated node routing problem on the transformed graph results in an effective approach for a capacitated arc routing problem. Computational experiments using this approach produced useful lower bounds in reasonable computational...
We have recently reported a method to identify the shortest possible phylogenetic tree for a set ... more We have recently reported a method to identify the shortest possible phylogenetic tree for a set of protein sequences [Foulds Hendy & Penny (1979) J. Mol. Evol. 13. 127--150; Foulds, Penny & Hendy (1979) J. Mol. Evol. 13, 151--166]. The present paper discusses issues that arise during the construction of minimal phylogenetic trees from protein-sequence data. The conversion of the data from amino acid sequences into nucleotide sequences is shown to be advantageous. A new variation of a method for constructing a minimal tree is presented. Our previous methods have involved first constructing a tree and then either proving that it is minimal or transforming it into a minimal tree. The approach presented in the present paper progressively builds up a tree, taxon by taxon. We illustrate this approach by using it to construct a minimal tree for ten mammalian haemoglobin alpha-chain sequences. Finally we define a measure of the complexity of the data and illustrate a method to derive a dir...
ABSTRACT This paper gives an introduction to analytical network techniques for dealing with proje... more ABSTRACT This paper gives an introduction to analytical network techniques for dealing with projects with stochastic evolution structure and feedback. Temporal analysis, cost minimization, and the scheduling of such projects under limited resources are discussed. The networks most commonly used to model these projects have several different node types, vector-valued arc weights, and cycles. They are called GERT networks. Specific properties of various classes of GERT networks are presented which can be exploited efficiently to solve certain scheduling problems. Examples and applications of these appropriate GERT networks are given.
Page 1. Facilities Layout Adjacency Determination: An Experimental Comparison of Three Graph Theo... more Page 1. Facilities Layout Adjacency Determination: An Experimental Comparison of Three Graph Theoretic Heuristics LR FOULDS University of Florida, Gainesville, Florida ... Choose vertex c so that the triangle la, b, c,} has highest weight. ...
ABSTRACT This article presents a survey of the current status of the research on structural and q... more ABSTRACT This article presents a survey of the current status of the research on structural and qualitative systems theory that has been conducted in the last few decades. This theory can be traced back to the work of Samuelson on comparative static analysis of a national economy. His problem was to find the direction of change in economic equilibrium induced by small changes in the exogenous variables of a system based solely on qualitative information such as the sign pattern of the parameters that specify the system. In the field of engineering, the work of Iri et al. on structural analysis of systems of equations and Lin's research on the structural controllability of dynamic systems were quite influential. Following these pioneering works, a variety of research has been carried out by mathematicians and engineers, as well as by economists. A few examples of problems are the structural solvability and decomposability of systems of equations, the structural characterization of Tinbergen's rule of economic policy, the structural controllability and decouplability of dynamic systems, and the sign-solvability of static systems. Using graph-theoretic terms with explanatory examples, this paper surveys the established results on the structural and qualitative analysis of systems in areas ranging from control theory to economics.
ABSTRACT The Steiner problem in graphs is concerned with finding a set of edges with minimum tota... more ABSTRACT The Steiner problem in graphs is concerned with finding a set of edges with minimum total weight which connects a given subset of points in a weighted graph. A branch and bound algorithm for solving this problem is presented together with an interesting application to a problem in molecular evolution. Computational experience gained in using the algorithm compares favorably, for certain classes of graphs, with that of existing methods.
ZOR Zeitschrift f�r Operations Research Methods and Models of Operations Research, 1989
ABSTRACT We consider the problem of finding maximal flows with respect to capacities which are li... more ABSTRACT We consider the problem of finding maximal flows with respect to capacities which are linear functions of a parametert [0,T]. Since this problem is a special case of a parametric linear program the classichorizontal approach can be applied in which optimal solutions are computed for successive subintervals of [0,T]. We discuss an alternative algorithm which approximates in each iteration the optimal solution for allt [0,T]. Thisvertical algorithm is a labeling type algorithm where the flow variables are piecewise linear functions. Flow augmentations are done alongconditional flow augmenting paths which can be found by modified path algorithms. The vertical algorithm can be used to solve the parametric flow problem optimally as well as to compute a good approximation for allt if the computation of the optimal solution turns out to be too time consuming.Wir betrachten maximale Fluprobleme, in denen die Kapazitten lineare Funktionen eines Parameterst [0,T] sind. Da dieses Problem ein Spezialfall eines parametrischen linearen Programms ist, kann man den klassischen horizontalen Ansatz anwenden, mit dem optimale Lsungen sukzessive auf Teilintervallen von [0,T] bestimmt werden. Wir stellen einen alternativen Algorithmus vor, der in jeder Iteration die optimale Lsung fr allet [0,T] approximiert. Dieser vertikalen Ansatz ist eine Art Markierungsalgorithmus, wobei die Fluvariablen stckweise lineare Funktionen sind. Fluvergrerungen werden aufbedingten fluvergrernden Wegen durchgefhrt, die mittels modifizierter krzester Wege Algorithmen gefunden werden knnen. Der vertikale Algorithmus kann sowohl zur Berechnung des optimalen parametrischen Flusses als auch zur Berechnung einer guten Approximation fr allet benutzt werden, falls sich herausstellt, da die Berechnung der optimalen Lsung zu zeitaufwendig ist.
LR Foulds Department of Management Systems University of Waikato Hamilton, New Zealand Editorial ... more LR Foulds Department of Management Systems University of Waikato Hamilton, New Zealand Editorial Board (North America): FW Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA KA Ribet Department of Mathematics University of ...
... complementing those given here. THEOREM .3 If e,, e2 are edges in T=(P, E) such that se&#... more ... complementing those given here. THEOREM .3 If e,, e2 are edges in T=(P, E) such that se'' Se". = ( Sex' Sex J ' then e, = ez . That is, f is onetoone. 138 DF ROBINSON AND LR FOULDS Proof. Assume e, =Le2. Then by Theorem 2 ...
One of the most important concepts in production planning is that of the establishment of an over... more One of the most important concepts in production planning is that of the establishment of an overall or aggregate production plan. In this paper, the problem of establishing an aggregate production plan for a manufacturing plant is considered. A new dynamic discrete-time model of capacity planning utilizing concepts arising in positive linear systems (PLS) theory is proposed and its controllability property is analyzed. Controllability is a fundamental property of the system with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control) but also in feedback control problems. Some new open problems regarding controllability of stationary and nonstationary PLS with linear constraints are posed in the paper. An optimal control problem for capacity planning is formulated and discussed.
This paper addresses the problem of identifying a planar subgraph of maximum weight of a given ed... more This paper addresses the problem of identifying a planar subgraph of maximum weight of a given edge weighted graph. The problem is NPComplete which justifies the use of heuristic (approximate) methods. It is shown how a new line of attack based on the graph theoretic concept of a wheel yields an efficient heuristic. An application to the layout of manufacturing
We describe a compact method to transform arc routing problem instances into node routing problem... more We describe a compact method to transform arc routing problem instances into node routing problem instances. Any node routing problem instance thus created must be solved by a branch-and-price process, such as the one described in this paper. The purpose is to make the number of nodes in the resulting transformed graphs greater by only one unit than the number r of required arcs (arcs having demand) in the original graph, that is, r+1 nodes. This low increase in the number of nodes represents an improvement compared to the methods previously presented by Pearn, Assad and Golden ( 3r+1 nodes) and by Longo, Poggi de Aragão and Uchoa and also by Baldacci and Maniezzo ( 2r+1 nodes). Using an adapted version of an existing branch-cut-and-price algorithm for a capacitated node routing problem on the transformed graph results in an effective approach for a capacitated arc routing problem. Computational experiments using this approach produced useful lower bounds in reasonable computational...
We have recently reported a method to identify the shortest possible phylogenetic tree for a set ... more We have recently reported a method to identify the shortest possible phylogenetic tree for a set of protein sequences [Foulds Hendy & Penny (1979) J. Mol. Evol. 13. 127--150; Foulds, Penny & Hendy (1979) J. Mol. Evol. 13, 151--166]. The present paper discusses issues that arise during the construction of minimal phylogenetic trees from protein-sequence data. The conversion of the data from amino acid sequences into nucleotide sequences is shown to be advantageous. A new variation of a method for constructing a minimal tree is presented. Our previous methods have involved first constructing a tree and then either proving that it is minimal or transforming it into a minimal tree. The approach presented in the present paper progressively builds up a tree, taxon by taxon. We illustrate this approach by using it to construct a minimal tree for ten mammalian haemoglobin alpha-chain sequences. Finally we define a measure of the complexity of the data and illustrate a method to derive a dir...
ABSTRACT This paper gives an introduction to analytical network techniques for dealing with proje... more ABSTRACT This paper gives an introduction to analytical network techniques for dealing with projects with stochastic evolution structure and feedback. Temporal analysis, cost minimization, and the scheduling of such projects under limited resources are discussed. The networks most commonly used to model these projects have several different node types, vector-valued arc weights, and cycles. They are called GERT networks. Specific properties of various classes of GERT networks are presented which can be exploited efficiently to solve certain scheduling problems. Examples and applications of these appropriate GERT networks are given.
Page 1. Facilities Layout Adjacency Determination: An Experimental Comparison of Three Graph Theo... more Page 1. Facilities Layout Adjacency Determination: An Experimental Comparison of Three Graph Theoretic Heuristics LR FOULDS University of Florida, Gainesville, Florida ... Choose vertex c so that the triangle la, b, c,} has highest weight. ...
ABSTRACT This article presents a survey of the current status of the research on structural and q... more ABSTRACT This article presents a survey of the current status of the research on structural and qualitative systems theory that has been conducted in the last few decades. This theory can be traced back to the work of Samuelson on comparative static analysis of a national economy. His problem was to find the direction of change in economic equilibrium induced by small changes in the exogenous variables of a system based solely on qualitative information such as the sign pattern of the parameters that specify the system. In the field of engineering, the work of Iri et al. on structural analysis of systems of equations and Lin's research on the structural controllability of dynamic systems were quite influential. Following these pioneering works, a variety of research has been carried out by mathematicians and engineers, as well as by economists. A few examples of problems are the structural solvability and decomposability of systems of equations, the structural characterization of Tinbergen's rule of economic policy, the structural controllability and decouplability of dynamic systems, and the sign-solvability of static systems. Using graph-theoretic terms with explanatory examples, this paper surveys the established results on the structural and qualitative analysis of systems in areas ranging from control theory to economics.
ABSTRACT The Steiner problem in graphs is concerned with finding a set of edges with minimum tota... more ABSTRACT The Steiner problem in graphs is concerned with finding a set of edges with minimum total weight which connects a given subset of points in a weighted graph. A branch and bound algorithm for solving this problem is presented together with an interesting application to a problem in molecular evolution. Computational experience gained in using the algorithm compares favorably, for certain classes of graphs, with that of existing methods.
ZOR Zeitschrift f�r Operations Research Methods and Models of Operations Research, 1989
ABSTRACT We consider the problem of finding maximal flows with respect to capacities which are li... more ABSTRACT We consider the problem of finding maximal flows with respect to capacities which are linear functions of a parametert [0,T]. Since this problem is a special case of a parametric linear program the classichorizontal approach can be applied in which optimal solutions are computed for successive subintervals of [0,T]. We discuss an alternative algorithm which approximates in each iteration the optimal solution for allt [0,T]. Thisvertical algorithm is a labeling type algorithm where the flow variables are piecewise linear functions. Flow augmentations are done alongconditional flow augmenting paths which can be found by modified path algorithms. The vertical algorithm can be used to solve the parametric flow problem optimally as well as to compute a good approximation for allt if the computation of the optimal solution turns out to be too time consuming.Wir betrachten maximale Fluprobleme, in denen die Kapazitten lineare Funktionen eines Parameterst [0,T] sind. Da dieses Problem ein Spezialfall eines parametrischen linearen Programms ist, kann man den klassischen horizontalen Ansatz anwenden, mit dem optimale Lsungen sukzessive auf Teilintervallen von [0,T] bestimmt werden. Wir stellen einen alternativen Algorithmus vor, der in jeder Iteration die optimale Lsung fr allet [0,T] approximiert. Dieser vertikalen Ansatz ist eine Art Markierungsalgorithmus, wobei die Fluvariablen stckweise lineare Funktionen sind. Fluvergrerungen werden aufbedingten fluvergrernden Wegen durchgefhrt, die mittels modifizierter krzester Wege Algorithmen gefunden werden knnen. Der vertikale Algorithmus kann sowohl zur Berechnung des optimalen parametrischen Flusses als auch zur Berechnung einer guten Approximation fr allet benutzt werden, falls sich herausstellt, da die Berechnung der optimalen Lsung zu zeitaufwendig ist.
LR Foulds Department of Management Systems University of Waikato Hamilton, New Zealand Editorial ... more LR Foulds Department of Management Systems University of Waikato Hamilton, New Zealand Editorial Board (North America): FW Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA KA Ribet Department of Mathematics University of ...
... complementing those given here. THEOREM .3 If e,, e2 are edges in T=(P, E) such that se&#... more ... complementing those given here. THEOREM .3 If e,, e2 are edges in T=(P, E) such that se'' Se". = ( Sex' Sex J ' then e, = ez . That is, f is onetoone. 138 DF ROBINSON AND LR FOULDS Proof. Assume e, =Le2. Then by Theorem 2 ...
One of the most important concepts in production planning is that of the establishment of an over... more One of the most important concepts in production planning is that of the establishment of an overall or aggregate production plan. In this paper, the problem of establishing an aggregate production plan for a manufacturing plant is considered. A new dynamic discrete-time model of capacity planning utilizing concepts arising in positive linear systems (PLS) theory is proposed and its controllability property is analyzed. Controllability is a fundamental property of the system with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control) but also in feedback control problems. Some new open problems regarding controllability of stationary and nonstationary PLS with linear constraints are posed in the paper. An optimal control problem for capacity planning is formulated and discussed.
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