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Combinatorial Optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead. Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science. Some research literature considers discr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Spanning Tree
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components. There are many use cases for minimum spanning trees. One example is a telecommunications company trying to lay cable in a new neighborhood. If it is constrained to bury the cable only along certain paths (e.g. roads), then there would be a graph containing the points (e.g. houses) connected by those paths. Some of the paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weight ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest-path Tree
In mathematics and computer science, a shortest-path tree rooted at a vertex ''v'' of a connected, undirected graph ''G'' is a spanning tree ''T'' of ''G'', such that the path distance from root ''v'' to any other vertex ''u'' in ''T'' is the shortest path distance from ''v'' to ''u'' in ''G''. In connected graphs where shortest paths are well-defined (i.e. where there are no negative-length cycles), we may construct a shortest-path tree using the following algorithm: # Compute dist(''u''), the shortest-path distance from root ''v'' to vertex ''u'' in ''G'' using Dijkstra's algorithm or Bellman–Ford algorithm. # For all non-root vertices ''u'', we can assign to ''u'' a parent vertex ''pu'' such that ''p''''u'' is connected to ''u'', and that dist(''p''''u'') + edge_dist(''p''''u'',''u'') = dist(''u''). In case multiple choices for ''pu'' exist, choose ''p''''u'' for which there exists a shortest path from ''v'' to ''p''''u'' with as few edges as possible; this tie-breaking rule is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest Path
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V such that v_i is adjacent to v_ for 1 \l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where this function has the smallest (or largest) value if such a point exists. Linear programs are problems that can be expressed in canonical form as : \begin & \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Algorithm
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expressed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reservoir
A reservoir (; from French ''réservoir'' ) is an enlarged lake behind a dam. Such a dam may be either artificial, built to store fresh water or it may be a natural formation. Reservoirs can be created in a number of ways, including controlling a watercourse that drains an existing body of water, interrupting a watercourse to form an embayment within it, through excavation, or building any number of retaining walls or levees. In other contexts, "reservoirs" may refer to storage spaces for various fluids; they may hold liquids or gasses, including hydrocarbons. ''Tank reservoirs'' store these in ground-level, elevated, or buried tanks. Tank reservoirs for water are also called cisterns. Most underground reservoirs are used to store liquids, principally either water or petroleum. Types Dammed valleys Dammed reservoirs are artificial lakes created and controlled by a dam constructed across a valley, and rely on the natural topography to provide most of the bas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Earth Science
Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres, namely biosphere, hydrosphere, atmosphere, and geosphere. Earth science can be considered to be a branch of planetary science, but with a much older history. Earth science encompasses four main branches of study, the lithosphere, the hydrosphere, the atmosphere, and the biosphere, each of which is further broken down into more specialized fields. There are both reductionist and holistic approaches to Earth sciences. It is also the study of Earth and its neighbors in space. Some Earth scientists use their knowledge of the planet to locate and develop energy and mineral resources. Others study the impact of human activity on Earth's environment, and design methods to protect the planet. Some use their knowledge about Earth proces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supply Chain Optimization
Supply-chain optimization (SCO) aims to ensure the optimal operation of a manufacturing and distribution of supply chain. This includes the optimal placement of inventory within the supply chain, minimizing operating costs including manufacturing costs, transportation costs, and distribution costs. Optimization often involves the application of mathematical modelling techniques using computer software. It is often considered to be part of supply chain engineering, although the latter is mainly focused on mathematical modelling approaches, whereas supply chain optimization can also be undertaken using qualitative, management based approaches. Applications Typically, supply-chain managers aim to maximize the profitable operation of their manufacturing and distribution supply chain. This could include measures like maximizing gross margin return on inventory invested ( GMROII) (balancing the cost of inventory at all points in the supply chain with availability to the customer), min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistics
Logistics is generally the detailed organization and implementation of a complex operation. In a general business sense, logistics manages the flow of goods between the point of origin and the point of consumption to meet the requirements of customers or corporations. The resources managed in logistics may include tangible goods such as materials, equipment, and supplies, as well as food and other consumable items. In military science, logistics is concerned with maintaining army supply lines while disrupting those of the enemy, since an armed force without resources and transportation is defenseless. Military logistics was already practiced in the ancient world and as the modern military has a significant need for logistics solutions, advanced implementations have been developed. In military logistics, logistics officers manage how and when to move resources to the places they are needed. Logistics management is the part of supply chain management and supply chain engine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: * An optimization problem with discrete variables is known as a '' discrete optimization'', in which an object such as an integer, permutation or graph must be found from a countable set. * A problem with continuous variables is known as a ''continuous optimization'', in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems. Continuous optimization problem The '' standard form'' of a continuous optimization problem is \begin &\underset& & f(x) \\ &\operatorname & &g_i(x) \leq 0, \quad i = 1,\dots,m \\ &&&h_j(x) = 0, \quad j = 1, \dots,p \end where * is the objective function to be minimized over the -variable vector , * are called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
