Limit set

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. 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 two 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|prec|$L_{\max }$

a single machine, general precedence constraint, minimizing maximum lateness.

R|pnmt|$\sum C_{i}$

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

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