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Disjunctive Graph
In the mathematical modeling of job shop scheduling problems, disjunctive graphs are a way of modeling a system of tasks to be scheduled and timing constraints that must be respected by the schedule. They are mixed graphs, in which vertices (representing tasks to be performed) may be connected by both directed and undirected edges (representing timing constraints between tasks). The two types of edges represent constraints of two different types: *If one task ''x'' must be performed earlier than a second task ''y'', this constraint is represented by a directed edge from ''x'' to ''y''. *If, on the other hand, two tasks ''x'' and ''y'' can be performed in either order, but not simultaneously (perhaps because they both demand the use of the same equipment or other resource), this non-simultaneity constraint is represented by an undirected edge connecting ''x'' and ''y''. Pairs of tasks that have no constraint on their ordering – they can be performed in either order or even simultane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Modeling
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science). The use of mathematical models to solve problems in business or military operations is a large part of the field of operations research. Mathematical models are also used in music, linguistics, and philosophy (for example, intensively in analytic philosophy). A model may help to explain a system and to study the effects of different components, and to make predictions about behavior. Elements of a mathematical model Mathematical models can take many forms, including dynamical systems, statistical m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Job Shop Scheduling
Job-shop scheduling, the job-shop problem (JSP) or job-shop scheduling problem (JSSP) is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. In a general job scheduling problem, we are given ''n'' jobs ''J''1, ''J''2, ..., ''Jn'' of varying processing times, which need to be scheduled on ''m'' machines with varying processing power, while trying to minimize the makespan – the total length of the schedule (that is, when all the jobs have finished processing). In the specific variant known as ''job-shop scheduling'', each job consists of a set of ''operations'' ''O''1, ''O''2, ..., ''On'' which need to be processed in a specific order (known as ''precedence constraints''). Each operation has a ''specific machine'' that it needs to be processed on and only one operation in a job can be processed at a given time. A common relaxation is the flexible job shop, where each operation can be processed on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Graph
In graph theory, a mixed graph is a graph consisting of a set of vertices , a set of (undirected) edges , and a set of directed edges (or arcs) . Definitions and notation Consider adjacent vertices u,v \in V. A directed edge, called an arc, is an edge with an orientation and can be denoted as \overrightarrow or (u,v) (note that u is the tail and v is the head of the arc). Also, an undirected edge, or edge, is an edge with no orientation and can be denoted as uv or ,v/math>. For the purpose of our application example we will not be considering loops or multiple edges of mixed graphs. A walk in a mixed graph is a sequence v_0,c_1,v_1,c_2,v_2,\dots,c_k,v_k of vertices and edges/arcs such that for all indices i, either c_i=v_v_ is an edge of the graph or c_i=\overrightarrow is an arc of the graph. This walk is a path if it does not repeat any edges, arcs, or vertices, except possibly the first and last vertices. A path is closed if its first and last vertices are the same, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (graph Theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex ''w'' is said to be ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
European Chapter On Combinatorial Optimization
The European Chapter on Combinatorial Optimization (also, EURO Working Group on Combinatorial Optimization, or EWG ECCO) is a working group whose objective is to promote original research in the field of combinatorial optimization at the European level. History ECCO is one of the working groups of EURO, the Association of European Operational Research Societies. The Group was founded in 1987 by Catherine Roucairol, Alexander Rinnooy Kan, and Dominique de Werra. Governance The group is managed by an Advisory Board of 4 members and a Coordinator. The Advisory Board is currently composed of Jacek Błażewicz, Van Dat Cung, Alain Hertz, and Paolo Toth. The current coordinator is Silvano Martello Silvano Martello (born in Bologna, Italy) is an Italian scientist and engineer, and an Emeritus Professor of Operations Research at the University of Bologna. He is known for his research in Operations Research and Mathematical Programming. In p .... Membership T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acyclic Orientation
In graph theory, an acyclic orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that does not form any directed cycle and therefore makes it into a directed acyclic graph. Every graph has an acyclic orientation. The chromatic number of any graph equals one more than the length of the longest path in an acyclic orientation chosen to minimize this path length. Acyclic orientations are also related to colorings through the chromatic polynomial, which counts both acyclic orientations and colorings. The planar dual of an acyclic orientation is a totally cyclic orientation, and vice versa. The family of all acyclic orientations can be given the structure of a partial cube by making two orientations adjacent when they differ in the direction of a single edge. Orientations of trees are always acyclic, and give rise to polytrees. Acyclic orientations of complete graphs are called transitive tournaments. The bipolar orientations are a special c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circular Dependency
In software engineering, a circular dependency is a relation between two or more modules which either directly or indirectly depend on each other to function properly. Such modules are also known as mutually recursive. Overview Circular dependencies are natural in many domain models where certain objects of the same domain depend on each other. However, in software design, circular dependencies between larger software modules are considered an anti-pattern because of their negative effects. Despite this, such circular (or cyclic) dependencies have been found to be widespread among the source files of real-world software. Mutually recursive modules are, however, somewhat common in functional programming, where inductive and recursive definitions are often encouraged. Problems Circular dependencies can cause many unwanted effects in software programs. Most problematic from a software design point of view is the ''tight coupling'' of the mutually dependent modules which reduces or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Acyclic Graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling). Directed acyclic graphs are sometimes instead called acyclic directed graphs or acyclic digraphs. Definitions A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Longest Path
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called ''simple'' if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means that the decision problem cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate. However, it has a linear time solution for directed acyclic graphs, which has important applications in finding the critical path in scheduling problems. NP-hardne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem. A more precise specification is: a problem ''H'' is NP-hard when every problem ''L'' in NP can be reduced in polynomial time to ''H''; that is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve any NP-hard problem would give polynomial time algorithms for all the problems in NP. As it is suspected that P≠NP, it is unlikely that such an algorithm exists. It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven. Moreover, the class P, in which all problems can be solved in polynomial time, is contained in the NP class. Defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gallai–Hasse–Roy–Vitaver Theorem
In graph theory, the Gallai–Hasse–Roy–Vitaver theorem is a form of duality between the colorings of the vertices of a given undirected graph and the orientations of its edges. It states that the minimum number of colors needed to properly color any graph G equals one plus the length of a longest path in an orientation of G chosen to minimize this path's The orientations for which the longest path has minimum length always include at least one This theorem implies that every orientation of a graph with contains a simple directed path with this path can be constrained to begin at any vertex that can reach all other vertices of the oriented Examples A bipartite graph may be oriented from one side of the bipartition to the other. The longest path in this orientation has length two, with only two vertices. Conversely, if a graph is oriented without any three-vertex paths, then every vertex must either be a source (with no incoming edges) or a sink (with no outgoing edges) an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
