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The | {{Expert-subject|Mathematics|date=November 2008}} | ||
A convenient '''notation for theoretic scheduling problems''' was introduced by [[Ronald Graham]], [[Eugene Lawler]], [[Jan Karel Lenstra]] and [[Alexander Rinnooy Kan]] in.<ref name="initial specification" /> It consists of three fields: [[Alpha (letter)|α]], [[Beta (letter)|β]] and [[Gamma|γ]]. | |||
Each field may be a comma separated list of words. The α field describes the machine environment, β the job characteristics, and γ the objective function. | |||
== Machine environment == | |||
=== Single stage problems === | |||
Each job comes with a given processing time. | |||
; 1 | |||
: there is a single machine | |||
; P | |||
: there are <math>m</math> parallel identical machines | |||
; Q | |||
: there are <math>m</math> parallel machines with different given speeds, length of job <math>i</math> on machine <math>j</math> is the processing time <math>p_{i}</math> divided by speed <math>s_j</math> | |||
; R | |||
: there are <math>m</math> parallel unrelated machines, there are given processing times <math>p_{ij}</math> for job <math>i</math> on machine <math>j</math> | |||
The last two letters might be followed by the number of machines which is then fixed, here <math>m</math> stands then for a fixed number. | |||
=== Multi-stage problem === | |||
; O : [[Open Shop Scheduling|open shop problem]] | |||
; F : [[Flow Shop Scheduling Problem|flow shop problem]] | |||
; J : [[Job Shop Scheduling|job shop problem]] | |||
== Job characteristics == | |||
The processing time may be equal for all jobs (<math>p_i=p</math>, or <math>p_{ij}=p</math>) or even of unit length (<math>p_i=1</math>, or <math>p_{ij}=1</math>). This makes a difference because all release times, deadlines are assumed to be integer. | |||
; <math>r_i</math> | |||
: for each job a release time is given before which it cannot be scheduled, default is 0. | |||
; <math>d_i</math> | |||
: for each job a deadline is given after which it cannot be scheduled. If the objective is <math>\sum U_i</math> for example, then this field is implicitly assumed. | |||
; pmtn | |||
: the jobs may be preempted and execution resumed later, possibly on a different machine | |||
; <math>size_i</math> | |||
: Each job comes with a number of machines on which it must be scheduled at the same time, default is 1. | |||
Precedence relations might be given for the jobs, in form of a partial order, meaning that if i is a predecessor of i' in that order, i' can start only when i is completed. | |||
; prec | |||
: an arbitrary precedence relation is given | |||
; sp-tree, tree, intree, outtree, chain | |||
: specific partial orders | |||
== Objective functions == | |||
Most objective functions depend on the deadline <math>d_i</math> and the completion time <math>C_i</math> of job <math>i</math>. We define lateness <math>L_i=C_i-d_i</math>, earliness <math>E_i = \max\{0, d_i-C_i\}</math>, tardiness <math>T_i = \max\{0, C_i-d_i\}</math>, unit penalty <math>U_i = 0</math> if <math>C_i\le d_i</math> and <math>U_i=1</math> otherwise. The common objective functions are <math>C_\max, L_\max, E_\max, T_\max, \sum C_i, \sum L_i, \sum E_i, \sum T_i</math> or weighted version of these sums, where every job comes with a priority <math>w_i</math>. | |||
== Examples == | |||
Adapted from <ref name="initial specification">{{cite conference | |||
| last1 = Graham | first1 = R. L. | |||
| last2 = Lawler | first2 = E. L. | |||
| last3 = Lenstra | first3 = J.K. | |||
| last4 = Rinnooy Kan | first4 = A.H.G. | |||
| title = Optimization and Approximation in Deterministic Suquencing and Scheduling: a Survey | |||
| booktitle = Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium | |||
| publisher = Elsevier | |||
| year = 1979 | |||
| pages = (5) 287–326 | |||
}}</ref> | |||
; 1|prec|<math>L_\max</math>: a single machine, general precedence constraint, minimizing maximum lateness. | |||
; R|pnmt|<math>\sum C_i</math>: variable number of unrelated parallel machines, allowing preemption, minimizing total completion time. | |||
; J3|<math>p_{ij}</math>|<math>C_\max</math>: 3-machines job shop with unit processing times, minimizing maximum completion time. | |||
== References == | |||
* B. Chen, C.N. Potts and G.J. Woeginger. "A review of machine scheduling: Complexity, algorithms and approximability". ''Handbook of Combinatorial Optimization'' (Volume 3) (Editors: D.-Z. Du and P. Pardalos), 1998, Kluwer Academic Publishers. 21-169. ISBN 0-7923-5285-8 (HB) 0-7923-5019-7 (Set) | |||
*[http://www.informatik.uni-osnabrueck.de/knust/class/ Peter Brucker, Sigrid Knust. Complexity results for scheduling problems] | |||
{{Reflist}} | |||
[[Category:Theoretical computer science]] | |||
[[Category:Scheduling (computing)]] |
Revision as of 21:06, 16 December 2013
Template:Expert-subject A convenient notation for theoretic scheduling problems was introduced by Ronald Graham, Eugene Lawler, Jan Karel Lenstra and Alexander Rinnooy Kan in.[1] It consists of three fields: α, β and γ.
Each field may be a comma separated list of words. The α field describes the machine environment, β the job characteristics, and γ the objective function.
Machine environment
Single stage problems
Each job comes with a given processing time.
- 1
- there is a single machine
- P
- there are parallel identical machines
- Q
- there are parallel machines with different given speeds, length of job on machine is the processing time divided by speed
- R
- there are parallel unrelated machines, there are given processing times for job on machine
The last two letters might be followed by the number of machines which is then fixed, here stands then for a fixed number.
Multi-stage problem
Job characteristics
The processing time may be equal for all jobs (, or ) or even of unit length (, or ). This makes a difference because all release times, deadlines are assumed to be integer.
- for each job a release time is given before which it cannot be scheduled, default is 0.
- for each job a deadline is given after which it cannot be scheduled. If the objective is for example, then this field is implicitly assumed.
- pmtn
- the jobs may be preempted and execution resumed later, possibly on a different machine
- Each job comes with a number of machines on which it must be scheduled at the same time, default is 1.
Precedence relations might be given for the jobs, in form of a partial order, meaning that if i is a predecessor of i' in that order, i' can start only when i is completed.
- prec
- an arbitrary precedence relation is given
- sp-tree, tree, intree, outtree, chain
- specific partial orders
Objective functions
Most objective functions depend on the deadline and the completion time of job . We define lateness , earliness , tardiness , unit penalty if and otherwise. The common objective functions are or weighted version of these sums, where every job comes with a priority .
Examples
Adapted from [1]
- R|pnmt|
- variable number of unrelated parallel machines, allowing preemption, minimizing total completion time.
References
- B. Chen, C.N. Potts and G.J. Woeginger. "A review of machine scheduling: Complexity, algorithms and approximability". Handbook of Combinatorial Optimization (Volume 3) (Editors: D.-Z. Du and P. Pardalos), 1998, Kluwer Academic Publishers. 21-169. ISBN 0-7923-5285-8 (HB) 0-7923-5019-7 (Set)
- Peter Brucker, Sigrid Knust. Complexity results for scheduling problems
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