Notation for theoretic scheduling problems

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 $m$ parallel identical machines
Q: there are $m$ parallel machines with different given speeds, length of job $i$ on machine $j$ is the processing time $p_{i}$ divided by speed $s_{j}$
R: there are $m$ parallel unrelated machines, there are given processing times $p_{ij}$ for job $i$ on machine $j$

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

Multi-stage problem

O: open shop problem
F: flow shop problem
J: job shop problem

Job characteristics

The processing time may be equal for all jobs ( $p_{i}=p$ , or $p_{ij}=p$ ) or even of unit length ( $p_{i}=1$ , or $p_{ij}=1$ ). This makes a difference because all release times, deadlines are assumed to be integer.

$r_{i}$: for each job a release time is given before which it cannot be scheduled, default is 0.
$d_{i}$: for each job a deadline is given after which it cannot be scheduled. If the objective is $\sum U_{i}$ for example, then this field is implicitly assumed.
pmtn: the jobs may be preempted and execution resumed later, possibly on a different machine
$size_{i}$: 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 $d_{i}$ and the completion time $C_{i}$ of job $i$ . We define lateness $L_{i}=C_{i}-d_{i}$ , earliness $E_{i}=\max\{0,d_{i}-C_{i}\}$ , tardiness $T_{i}=\max\{0,C_{i}-d_{i}\}$ , unit penalty $U_{i}=0$ if $C_{i}\leq d_{i}$ and $U_{i}=1$ otherwise. The common objective functions are $C_{\max },L_{\max },E_{\max },T_{\max },\sum C_{i},\sum L_{i},\sum E_{i},\sum T_{i}$ or weighted version of these sums, where every job comes with a priority $w_{i}$ .

Examples

Adapted from ^[1]

1|prec| $L_{\max }$: a single machine, general precedence constraint, minimizing maximum lateness.

R|pmtn| $\sum C_{i}$: variable number of unrelated parallel machines, allowing preemption, minimizing total completion time.

J3| $p_{ij}$ | $C_{\max }$: 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)
Peter Brucker, Sigrid Knust. Complexity results for scheduling problems

1 2 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.

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