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Notation For Theoretic Scheduling Problems
Optimal job scheduling is a class of optimization problems related to scheduling. The inputs to such problems are a list of '' jobs'' (also called ''processes'' or ''tasks'') and a list of ''machines'' (also called ''processors'' or ''workers''). The required output is a ''schedule'' – an assignment of jobs to machines. The schedule should optimize a certain ''objective function''. In the literature, problems of optimal job scheduling are often called machine scheduling, processor scheduling, multiprocessor scheduling, or just scheduling. There are many different problems of optimal job scheduling, different in the nature of jobs, the nature of machines, the restrictions on the schedule, and the objective function. A convenient notation for optimal scheduling problems was introduced by Ronald Graham, Eugene Lawler, Jan Karel Lenstra and Alexander Rinnooy Kan. It consists of three fields: α, β and γ. Each field may be a comma separated list of words. The α field describes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Problem
In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ..., 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. Search space In the context of an optim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open-shop Scheduling
Open-shop scheduling or open-shop scheduling problem (OSSP) 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 ''open-shop scheduling'', each job consists of a set of ''operations'' ''O''1, ''O''2, ..., ''On'' which need to be processed in an ''arbitrary'' order. The problem was first studied by Teofilo F. Gonzalez and Sartaj Sahni in 1976. In the standard three-field notation for optimal job-scheduling problems, the open-shop variant is denoted by O in the first field. For example, the problem denoted by "O3, p_, C_\max" is a 3-machines jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Problem
In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset ''S'' of positive integers can be partition of a set, partitioned into two subsets ''S''1 and ''S''2 such that the sum of the numbers in ''S''1 equals the sum of the numbers in ''S''2. Although the partition problem is NP-complete, there is a pseudo-polynomial time dynamic programming solution, and there are Heuristic, heuristics that solve the problem in many instances, either optimally or approximately. For this reason, it has been called "the easiest hard problem". There is an optimization problem, optimization version of the partition problem, which is to partition the multiset ''S'' into two subsets ''S''1, ''S''2 such that the difference between the sum of elements in ''S''1 and the sum of elements in ''S''2 is minimized. The optimization version is NP-hard, but can be solved efficiently in practice. The partition problem is a special case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-objective Optimization
Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of MCDM, multiple-criteria decision making that is concerned with Mathematical optimization, mathematical optimization problems involving more than one Loss function, objective function to be optimized simultaneously. Multi-objective is a type of vector optimization that has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Earliness (scheduling)
In scheduling, tardiness is a measure of a delay in executing certain operations and earliness is a measure of finishing operations before due time. The operations may depend on each other and on the availability of equipment to perform them. Typical examples include job scheduling in manufacturing and data delivery scheduling in data processing networks. In manufacturing environment, inventory management considers both tardiness and earliness undesirable. Tardiness involves backlog issues such as customer compensation for delays and loss of goodwill. Earliness incurs expenses for storage of the manufactured items and ties up capital. Mathematical formulations In an environment with multiple jobs, let the deadline be d_i and the completion time be C_i of job i. Then for job i * lateness is L_i=C_i-d_i, * earliness is E_i = \max(0, d_i-C_i), * tardiness is T_i = \max(0, C_i-d_i). In scheduling common objective functions are C_\max, L_\max, E_\max, T_\max, \sum C_i, \sum L_i, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tardiness (scheduling)
In scheduling, tardiness is a measure of a delay in executing certain operations and earliness is a measure of finishing operations before due time. The operations may depend on each other and on the availability of equipment to perform them. Typical examples include job scheduling in manufacturing and data delivery scheduling in data processing networks. In manufacturing environment, inventory management considers both tardiness and earliness undesirable. Tardiness involves backlog issues such as customer compensation for delays and loss of goodwill. Earliness incurs expenses for storage of the manufactured items and ties up capital. Mathematical formulations In an environment with multiple jobs, let the deadline be d_i and the completion time be C_i of job i. Then for job i * lateness is L_i=C_i-d_i, * earliness is E_i = \max(0, d_i-C_i), * tardiness is T_i = \max(0, C_i-d_i). In scheduling common objective functions are C_\max, L_\max, E_\max, T_\max, \sum C_i, \sum L_i, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Throughput
Network throughput (or just throughput, when in context) refers to the rate of message delivery over a communication channel in a communication network, such as Ethernet or packet radio. The data that these messages contain may be delivered over physical or logical links, or through network nodes. Throughput is usually measured in bits per second (, sometimes abbreviated bps), and sometimes in packets per second ( or pps) or data packets per time slot. The system throughput or aggregate throughput is the sum of the data rates that are delivered over all channels in a network. Throughput represents digital bandwidth consumption. The throughput of a communication system may be affected by various factors, including the limitations of the underlying physical medium, available processing power of the system components, end-user behavior, etc. When taking various protocol overheads into account, the useful rate of the data transfer can be significantly lower than the maximum a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lateness (scheduling)
In scheduling, tardiness is a measure of a delay in executing certain operations and earliness is a measure of finishing operations before due time. The operations may depend on each other and on the availability of equipment to perform them. Typical examples include job scheduling in manufacturing and data delivery scheduling in data processing networks. In manufacturing environment, inventory management considers both tardiness and earliness undesirable. Tardiness involves backlog issues such as customer compensation for delays and loss of goodwill. Earliness incurs expenses for storage of the manufactured items and ties up capital. Mathematical formulations In an environment with multiple jobs, let the deadline be d_i and the completion time be C_i of job i. Then for job i * lateness is L_i=C_i-d_i, * earliness is E_i = \max(0, d_i-C_i), * tardiness is T_i = \max(0, C_i-d_i). In scheduling common objective functions are C_\max, L_\max, E_\max, T_\max, \sum C_i, \sum L_i, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Makespan
In operations research Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods to improve management and ..., the makespan of a project is the length of time that elapses from the start of work to the end. This type of multi-mode resource constrained project scheduling problem (MRCPSP) seeks to create the shortest logical project schedule, by efficiently using project resources, adding the lowest number of additional resources as possible to achieve the minimum makespan. The term commonly appears in the context of scheduling. Example There is a complex project that is composed of several sub-tasks. We would like to assign tasks to workers, such that the project finishes in the shortest possible time. As an example, suppose the "project" is to feed the goats. There are three goats to feed, one child can only feed on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series–parallel Graph
In graph theory, series–parallel graphs are graphs with two distinguished vertices called ''terminals'', formed recursively by two simple composition operations. They can be used to model series and parallel electric circuits. Definition and terminology In this context, the term graph means multigraph. There are several ways to define series–parallel graphs. First definition The following definition basically follows the one used by David Eppstein. A two-terminal graph (TTG) is a graph with two distinguished vertices, ''s'' and ''t'' called ''source'' and ''sink'', respectively. The parallel composition ''Pc = Pc(X,Y)'' of two TTGs ''X'' and ''Y'' is a TTG created from the disjoint union of graphs ''X'' and ''Y'' by merging the sources of ''X'' and ''Y'' to create the source of ''Pc'' and merging the sinks of ''X'' and ''Y'' to create the sink of ''Pc''. The series composition ''Sc = Sc(X,Y)'' of two TTGs ''X'' and ''Y'' is a TTG created from the disjoint union of gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Component (graph Theory)
In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components. The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component, one component that is significantly larger than the others; and of a percolation threshold, an edge probability above which a giant component exists and below which it does not. The components of a graph can be constructed in linear time, and a special case of the problem, connected-component labeling, is a basic technique in image analysis. Dynamic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Task Scheduling
Parallel task scheduling (also called parallel job scheduling or parallel processing scheduling) 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 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 ''parallel-task scheduling'', all machines are identical. Each job ''j'' has a ''length'' parameter ''pj'' and a ''size'' parameter ''q''j, and it must run for exactly ''pj'' time-steps on exactly ''q''j machines in ''parallel''. Veltman et al. and Drozdowski denote this problem by P, size_j, C_ in the three-field notation introduced by Graham et al. P means that there are several identical machines running in parallel; ''sizej'' means th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
