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Weapon Target Assignment Problem
The weapon target assignment problem (WTA) is a class of combinatorial optimization problems present in the fields of optimization and operations research. It consists of finding an optimal assignment of a set of weapons of various types to a set of targets in order to maximize the total expected damage done to the opponent. The basic problem is as follows: :There are a number of weapons and a number of targets. The weapons are of type i = 1, \ldots, m . There are W_ available weapons of type i. Similarly, there are j = 1, \ldots, n targets, each with a value of V_ . Any of the weapons can be assigned to any target. Each weapon type has a certain probability of destroying each target, given by p_ . Notice that as opposed to the classic assignment problem or the generalized assignment problem, more than one agent (i.e., weapon) can be assigned to each ''task'' (i.e., target) and not all targets are required to have weapons assigned. Thus, we see that the WTA allows one to f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 discrete o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Time
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 expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Marriage Problem
In mathematics, economics, and computer science, the stable marriage problem (also stable matching problem or SMP) is the problem of finding a stable matching between two equally sized sets of elements given an ordering of preferences for each element. A matching is a bijection from the elements of one set to the elements of the other set. A matching is ''not'' stable if: In other words, a matching is stable when there does not exist any pair (''A'', ''B'') which both prefer each other to their current partner under the matching. The stable marriage problem has been stated as follows: The existence of two classes that need to be paired with each other (heterosexual men and women in this example) distinguishes this problem from the stable roommates problem. Applications Algorithms for finding solutions to the stable marriage problem have applications in a variety of real-world situations, perhaps the best known of these being in the assignment of graduating medical students t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Assignment Problem
The quadratic assignment problem (QAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics, from the category of the facilities location problems first introduced by Koopmans and Beckmann. The problem models the following real-life problem: :There are a set of ''n'' facilities and a set of ''n'' locations. For each pair of locations, a ''distance'' is specified and for each pair of facilities a ''weight'' or ''flow'' is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the sum of the distances multiplied by the corresponding flows. Intuitively, the cost function encourages facilities with high flows between each other to be placed close together. The problem statement resembles that of the assignment problem, except that the cost function is expressed in terms of quadratic inequal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Bottleneck Assignment Problem
In combinatorial optimization, a field within mathematics, the linear bottleneck assignment problem (LBAP) is similar to the linear assignment problem. In plain words the problem is stated as follows: :There are a number of ''agents'' and a number of ''tasks''. Any agent can be assigned to perform any task, incurring some ''cost'' that may vary depending on the agent-task assignment. It is required to perform all tasks by assigning exactly one agent to each task in such a way that the ''maximum cost'' among the individual assignments is minimized. The term "bottleneck" is explained by a common type of application of the problem, where the cost is the duration of the task performed by an agent. In this setting the "maximum cost" is "maximum duration", which is the bottleneck for the schedule of the overall job, to be minimized. Formal definition The formal definition of the bottleneck assignment problem is :Given two sets, ''A'' and ''T'', together with a weight function A weigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure Problem
In graph theory and combinatorial optimization, a closure of a directed graph is a set of vertices ''C'', such that no edges leave ''C''. The closure problem is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph... It may be solved in polynomial time using a reduction to the maximum flow problem. It may be used to model various application problems of choosing an optimal subset of tasks to perform, with dependencies between pairs of tasks, one example being in open pit mining. Algorithms Condensation The maximum-weight closure of a given graph ''G'' is the same as the complement of the minimum-weight closure on the transpose graph of ''G'', so the two problems are equivalent in computational complexity. If two vertices of the graph belong to the same strongly connected component, they must behave the same as each other with respect to all closures: it is not possible for a closure to contain one vertex without containing the other. F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Auction Algorithm
The term "auction algorithm"Dimitri P. Bertsekas. "A distributed algorithm for the assignment problem"original paper, 1979 applies to several variations of a Optimization (mathematics), combinatorial optimization algorithm which solves assignment problems, and network optimization problems with linear and convex/nonlinear cost. An ''auction algorithm'' has been used in a business setting to determine the best prices on a set of products offered to multiple buyers. It is an iterative procedure, so the name "auction algorithm" is related to a sales auction, where multiple bids are compared to determine the best offer, with the final sales going to the highest bidders. The original form of the auction algorithm is an iterative method to find the optimal prices and an assignment that maximizes the net benefit in a bipartite graph, the ''maximum weight matching problem'' (MWM).M.G. Resende, P.M. Pardalos. "Handbook of optimization in telecommunications"2006/ref>M. Bayati, D. Shah, M. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heuristic Algorithm
In mathematical optimization and computer science, heuristic (from Greek εὑρίσκω "I find, discover") is a technique designed for solving a problem more quickly when classic methods are too slow for finding an approximate solution, or when classic methods fail to find any exact solution. This is achieved by trading optimality, completeness, accuracy, or precision for speed. In a way, it can be considered a shortcut. A heuristic function, also simply called a heuristic, is a function that ranks alternatives in search algorithms at each branching step based on available information to decide which branch to follow. For example, it may approximate the exact solution. Definition and motivation The objective of a heuristic is to produce a solution in a reasonable time frame that is good enough for solving the problem at hand. This solution may not be the best of all the solutions to this problem, or it may simply approximate the exact solution. But it is still valuable b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization (mathematics)
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relaxation (approximation)
In mathematical optimization and related fields, relaxation is a modeling strategy. A relaxation is an approximation of a difficult problem by a nearby problem that is easier to solve. A solution of the relaxed problem provides information about the original problem. For example, a linear programming relaxation of an integer programming problem removes the integrality constraint and so allows non-integer rational solutions. A Lagrangian relaxation of a complicated problem in combinatorial optimization penalizes violations of some constraints, allowing an easier relaxed problem to be solved. Relaxation techniques complement or supplement branch and bound algorithms of combinatorial optimization; linear programming and Lagrangian relaxations are used to obtain bounds in branch-and-bound algorithms for integer programming. The modeling strategy of relaxation should not be confused with iterative methods of relaxation, such as successive over-relaxation (SOR); iterative methods of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch And Bound
Branch and bound (BB, B&B, or BnB) is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores ''branches'' of this tree, which represent subsets of the solution set. Before enumerating the candidate solutions of a branch, the branch is checked against upper and lower estimated ''bounds'' on the optimal solution, and is discarded if it cannot produce a better solution than the best one found so far by the algorithm. The algorithm depends on efficient estimation of the lower and upper bounds of regions/branches of the search space. If no bounds are available, the algorithm degenerates to an exhaustive search. The method was first proposed by Ailsa Land and Alison Doig whilst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
