Notation for theoretic scheduling problems

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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.[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[edit]

Single stage problems[edit]

Each job comes with a given processing time.

there is a single machine
there are parallel identical machines
there are parallel machines with different given speeds, length of job on machine is the processing time divided by speed
there are parallel unrelated machines, there are given processing times for job on machine

The last three letters might be followed by the number of machines which is then fixed, here stands then for a fixed number.

Multi-stage problem[edit]

open shop problem
flow shop problem
job shop problem

Job characteristics[edit]

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.
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.

an arbitrary precedence relation is given
sp-tree, tree, intree, outtree, chain
specific partial orders

Objective functions[edit]

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 .


Adapted from [1]

a single machine, general precedence constraint, minimizing maximum lateness.
variable number of unrelated parallel machines, allowing preemption, minimizing total completion time.
3-machines job shop with unit processing times, minimizing maximum completion time.


  1. ^ a b Graham, R. L.; Lawler, E. L.; Lenstra, J.K.; Rinnooy Kan, A.H.G. (1979). "Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey". 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. Elsevier. pp. (5) 287–326.