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| {{Expert-subject|Mathematics|date=November 2008}}
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| 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]].
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| 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.
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| == Machine environment ==
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| === Single stage problems ===
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| Each job comes with a given processing time.
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| ; 1
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| : there is a single machine
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| ; P
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| : there are <math>m</math> parallel identical machines
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| ; Q
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| : 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>
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| ; R
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| : 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>
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| 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.
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| === Multi-stage problem ===
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| ; O : [[Open Shop Scheduling|open shop problem]]
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| ; F : [[Flow Shop Scheduling Problem|flow shop problem]]
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| ; J : [[Job Shop Scheduling|job shop problem]]
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| == Job characteristics ==
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| 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.
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| ; <math>r_i</math>
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| : for each job a release time is given before which it cannot be scheduled, default is 0.
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| ; <math>d_i</math>
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| : 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.
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| ; pmtn
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| : the jobs may be preempted and execution resumed later, possibly on a different machine
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| ; <math>size_i</math>
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| : Each job comes with a number of machines on which it must be scheduled at the same time, default is 1.
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| 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.
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| ; prec
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| : an arbitrary precedence relation is given
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| ; sp-tree, tree, intree, outtree, chain
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| : specific partial orders
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| == Objective functions ==
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| 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>.
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| == Examples ==
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| Adapted from <ref name="initial specification">{{cite conference
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| | last1 = Graham | first1 = R. L.
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| | last2 = Lawler | first2 = E. L.
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| | last3 = Lenstra | first3 = J.K.
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| | last4 = Rinnooy Kan | first4 = A.H.G.
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| | title = Optimization and Approximation in Deterministic Suquencing and Scheduling: a Survey
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| | 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
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| | publisher = Elsevier
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| | year = 1979
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| | pages = (5) 287–326
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| }}</ref>
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| ; 1|prec|<math>L_\max</math>: a single machine, general precedence constraint, minimizing maximum lateness.
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| ; R|pnmt|<math>\sum C_i</math>: variable number of unrelated parallel machines, allowing preemption, minimizing total completion time.
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| ; J3|<math>p_{ij}</math>|<math>C_\max</math>: 3-machines job shop with unit processing times, minimizing maximum completion time.
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| == References ==
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| * 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)
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| *[http://www.informatik.uni-osnabrueck.de/knust/class/ Peter Brucker, Sigrid Knust. Complexity results for scheduling problems]
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| {{Reflist}}
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| [[Category:Theoretical computer science]]
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| [[Category:Scheduling (computing)]]
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