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Uniform Machine Scheduling
Uniform machine scheduling (also called uniformly-related machine scheduling or related machine scheduling) is an optimization problem in computer science and Operations Research, operations research. It is a variant of optimal job scheduling. We are given ''n'' jobs ''J''1, ''J''2, ..., ''Jn'' of varying processing times, which need to be scheduled on ''m'' different machines. The goal is to minimize the makespan - the total time required to execute the schedule. The time that machine ''i'' needs in order to process job j is denoted by ''pi,j''. In the general case, the times ''pi,j'' are unrelated, and any matrix of positive processing times is possible. In the specific variant called ''uniform machine scheduling'', some machines are ''uniformly'' faster than others. This means that, for each machine ''i'', there is a speed factor ''si'', and the run-time of job ''j'' on machine ''i'' is ''pi,j'' = ''pj'' / ''si''. In the standard Optimal job scheduling, three-field notation for op ... [...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 ine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FPTAS
A fully polynomial-time approximation scheme (FPTAS) is an algorithm for finding approximate solutions to function problems, especially optimization problems. An FPTAS takes as input an instance of the problem and a parameter ε > 0. It returns as output a value is at least 1-\epsilon times the correct value, and at most 1 + \epsilon times the correct value. In the context of optimization problems, the correct value is understood to be the value of the optimal solution, and it is often implied that an FPTAS should produce a valid solution (and not just the value of the solution). Returning a value and finding a solution with that value are equivalent assuming that the problem possesses self reducibility. Importantly, the run-time of an FPTAS is polynomial in the problem size and in 1/ε. This is in contrast to a general polynomial-time approximation scheme (PTAS). The run-time of a general PTAS is polynomial in the problem size for each specific ε, but might be expon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow Shop Scheduling
Flow-shop 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 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 ''flow-shop scheduling'', each job contains exactly ''m'' operations. The ''i''-th operation of the job must be executed on the ''i''-th machine. No machine can perform more than one operation simultaneously. For each operation of each job, execution time is specified. Flow-shop scheduling is a special case of job-shop scheduling where there is strict order of all operations to be performed on all jobs. Flow-shop scheduling may apply as well to production facilities as to computing de ... [...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, 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 Optimal job scheduling, three-field notation for optimal job-scheduling problems, the open-shop variant is denoted by O in the first field. For example, the problem deno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These ''lattice-like'' structures all admi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Ordering
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''incompara ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Scheduling
List scheduling is a greedy algorithm for Identical-machines scheduling. The input to this algorithm is a list of jobs that should be executed on a set of ''m'' machines. The list is ordered in a fixed order, which can be determined e.g. by the priority of executing the jobs, or by their order of arrival. The algorithm repeatedly executes the following steps until a valid schedule is obtained: * Take the first job in the list (the one with the highest priority). * Find a machine that is available for executing this job. ** If a machine is found, schedule this job on that machine. **Otherwise (no suitable machine is available), select the next job in the list. Example Suppose there are five jobs with processing-times , and ''m''=2 processors. Then, the resulting schedule is , , and the makespan is max(18,12)=18; if ''m''=3, then the resulting schedule is , , , and the makespan is max(11,13,6)=13. Performance guarantee The algorithm runs in time O(n), where ''n'' is the number o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Longest Processing Time
Longest-processing-time-first (LPT) is a greedy algorithm for job scheduling. The input to the algorithm is a set of ''jobs'', each of which has a specific processing-time. There is also a number ''m'' specifying the number of ''machines'' that can process the jobs. The LPT algorithm works as follows: # Order the jobs by descending order of their processing-time, such that the job with the longest processing time is first. # Schedule each job in this sequence into a machine in which the current load (= total processing-time of scheduled jobs) is smallest. Step 2 of the algorithm is essentially the list-scheduling (LS) algorithm. The difference is that LS loops over the jobs in an arbitrary order, while LPT pre-orders them by descending processing time. LPT was first analyzed by Ronald Graham in the 1960s in the context of the identical-machines scheduling problem. Later, it was applied to many other variants of the problem. LPT can also be described in a more abstract way, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truthful Mechanism
In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information about what the others do, you fare best or at least not worse by being truthful. SP is also called truthful or dominant-strategy-incentive-compatible (DSIC), to distinguish it from other kinds of incentive compatibility. An SP game is not always immune to collusion, but its robust variants are; with group strategyproofness no group of people can collude to misreport their preferences in a way that makes every member better off, and with strong group strategyproofness no group of people can collude to misreport their preferences in a way that makes at least one member of the group better off without making any of the remaining members worse off. Examples Typical examples of SP mechanisms are majority voting between two alternatives, second- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identical-machines Scheduling
Identical-machines scheduling is an optimization problem in computer science and operations research. We are given ''n'' jobs ''J''1, ''J''2, ..., ''Jn'' of varying processing times, which need to be scheduled on ''m'' identical machines, such that a certain objective function is optimized, for example, the makespan is minimized. Identical machine scheduling is a special case of uniform machine scheduling, which is itself a special case of optimal job scheduling. In the general case, the processing time of each job may be different on different machines; in the case of identical machine scheduling, the processing time of each job is the same on each machine. Therefore, identical machine scheduling is equivalent to multiway number partitioning. A special case of identical machine scheduling is single-machine scheduling. In the standard three-field notation for optimal job scheduling problems, the identical-machines variant is denoted by P in the first field. For example, " P, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Longest-processing-time-first Scheduling
Longest-processing-time-first (LPT) is a greedy algorithm for job scheduling. The input to the algorithm is a set of ''jobs'', each of which has a specific processing-time. There is also a number ''m'' specifying the number of ''machines'' that can process the jobs. The LPT algorithm works as follows: # Order the jobs by descending order of their processing-time, such that the job with the longest processing time is first. # Schedule each job in this sequence into a machine in which the current load (= total processing-time of scheduled jobs) is smallest. Step 2 of the algorithm is essentially the list-scheduling (LS) algorithm. The difference is that LS loops over the jobs in an arbitrary order, while LPT pre-orders them by descending processing time. LPT was first analyzed by Ronald Graham in the 1960s in the context of the identical-machines scheduling problem. Later, it was applied to many other variants of the problem. LPT can also be described in a more abstract way, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
