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Activity Selection Problem
The activity selection problem is a combinatorial optimization problem concerning the selection of non-conflicting task (project management), activities to perform within a given time frame, given a set of activities each marked by a start time (si) and finish time (fi). The problem is to select the maximum number of activities that can be performed by a single person or machine, assuming that a person can only work on a single activity at a time. The activity selection problem is also known as the Interval scheduling, Interval scheduling maximization problem (ISMP), which is a special type of the more general Interval scheduling, Interval Scheduling problem. A classic application of this problem is in scheduling a room for multiple competing events, each having its own time requirements (start and end time), and many more arise within the framework of operations research. Formal definition Assume there exist ''n'' activities with each of them being represented by a start time '' ... [...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] |
Task (project Management)
In project management, a task is an activity that needs to be accomplished within a defined period of time or by a deadline to work towards work-related goals. It is a small essential piece of a job that serves as a means to differentiate various components of a project. A task can be broken down into assignments which should also have a defined start and end date or a deadline for completion. One or more assignments on a task puts the task under execution. Completion of all assignments on a specific task normally renders the task completed. Tasks can be linked together to create dependencies. Tasks completion generally requires the coordination of others. Coordinated human interaction takes on the role of combining the integration of time, energy, effort, ability, and resources of multiple individuals to meet a common goal. Coordination can also be thought of as the critical mechanism that links or ties together the efforts on the singular level to that of the larger task being ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Frame
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in the conscious experience. Time is often referred to as a fourth dimension, along with three spatial dimensions. Time has long been an important subject of study in religion, philosophy, and science, but defining it in a manner applicable to all fields without circularity has consistently eluded scholars. Nevertheless, diverse fields such as business, industry, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems. 108 pages. Time in physics is operationally defined as "what a clock reads". The physical nature of time is address ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine
A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecules, such as molecular machines. Machines can be driven by Animal power, animals and Human power, people, by natural forces such as Wind power, wind and Water power, water, and by Chemical energy, chemical, Thermal energy, thermal, or electricity, electrical power, and include a system of mechanism (engineering), mechanisms that shape the actuator input to achieve a specific application of output forces and movement. They can also include computers and sensors that monitor performance and plan movement, often called mechanical systems. Renaissance natural philosophers identified six simple machines which were the elementary devices that put a load into motion, and calculated the ratio of output force to input fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Scheduling
Interval scheduling is a class of problems in computer science, particularly in the area of algorithm design. The problems consider a set of tasks. Each task is represented by an ''interval'' describing the time in which it needs to be processed by some machine (or, equivalently, scheduled on some resource). For instance, task A might run from 2:00 to 5:00, task B might run from 4:00 to 10:00 and task C might run from 9:00 to 11:00. A subset of intervals is ''compatible'' if no two intervals overlap on the machine/resource. For example, the subset is compatible, as is the subset ; but neither nor are compatible subsets, because the corresponding intervals within each subset overlap. The ''interval scheduling maximization problem'' (ISMP) is to find a largest compatible set, i.e., a set of non-overlapping intervals of maximum size. The goal here is to execute as many tasks as possible, that is, to maximize the throughput. It is equivalent to finding a maximum independent set in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Competing
Competition is a rivalry where two or more parties strive for a common goal which cannot be shared: where one's gain is the other's loss (an example of which is a zero-sum game). Competition can arise between entities such as organisms, individuals, economic and social groups, etc. The rivalry can be over attainment of any exclusive goal, including recognition: Competition occurs in nature, between living organisms which co-exist in the same environment. Animals compete over water supplies, food, mates, and other biological resources. Humans usually compete for food and mates, though when these needs are met deep rivalries often arise over the pursuit of wealth, power, prestige, and fame when in a static, repetitive, or unchanging environment. Competition is a major tenet of market economies and business, often associated with business competition as companies are in competition with at least one other firm over the same group of customers. Competition inside a company is usu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operations Research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve decision-making. It is considered to be a subfield of mathematical sciences. The term management science is occasionally used as a synonym. Employing techniques from other mathematical sciences, such as modeling, statistics, and optimization, operations research arrives at optimal or near-optimal solutions to decision-making problems. Because of its emphasis on practical applications, operations research has overlap with many other disciplines, notably industrial engineering. Operations research is often concerned with determining the extreme values of some real-world objective: the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost). Originating in military efforts before World War II, its techniques have grown to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solution Set
In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities. For example, for a set of polynomials over a ring , the solution set is the subset of on which the polynomials all vanish (evaluate to 0), formally : The feasible region of a constrained optimization problem is the solution set of the constraints. Examples # The solution set of the single equation x=0 is the set . # For any non-zero polynomial f over the complex numbers in one variable, the solution set is made up of finitely many points. # However, for a complex polynomial in more than one variable the solution set has no isolated points. Remarks In algebraic geometry, solution sets are called algebraic sets if there are no inequalities. Over the reals, and with inequalities, there are called semialgebraic sets. Other meanings More generally, the solution set to an arbitrary collection ''E'' of relations (''Ei'') (''i'' varying in some index set ''I'') for a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greedy Algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time. For example, a greedy strategy for the travelling salesman problem (which is of high computational complexity) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids and give constant-factor approximations to optimization problems with the submodular structure. Specifics Greedy algorith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Solution
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] |
Pseudocode
In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine reading. It typically omits details that are essential for machine understanding of the algorithm, such as variable declarations and language-specific code. The programming language is augmented with natural language description details, where convenient, or with compact mathematical notation. The purpose of using pseudocode is that it is easier for people to understand than conventional programming language code, and that it is an efficient and environment-independent description of the key principles of an algorithm. It is commonly used in textbooks and scientific publications to document algorithms and in planning of software and other algorithms. No broad standard for pseudocode syntax exists, as a program in pseudocode is not an executa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Sets
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. Generalizations This definition of disjoint sets can be extended to a family of sets \left(A_i\right)_: the family is pairwise disjoint, or mutually disjoint if A_i \cap A_j = \varnothing whenever i \neq j. Alternatively, some authors use the term disjoint to refer to this notion as well. For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if A_i \cap A_j = \varnothing whenever A_i \neq A_j (every two ''distinct'' sets in the family are disjoint).. For example, the collection of sets is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
