Unrelated-machines scheduling

Unrelated-machines scheduling is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. We need to schedule n jobs J₁, J₂, ..., J_n on m different machines, such that a certain objective function is optimized (usually, the makespan should be minimized). The time that machine i needs in order to process job j is denoted by p_i,j. The term unrelated emphasizes that there is no relation between values of p_i,j for different i and j. This is in contrast to two special cases of this problem: uniform-machines scheduling - in which p_i,j = p_i / s_j (where s_j is the speed of machine j), and identical-machines scheduling - in which p_i,j = p_i (the same run-time on all machines).

In the standard three-field notation for optimal job scheduling problems, the unrelated-machines variant is denoted by R in the first field. For example, the problem denoted by " R||${\displaystyle C_{\max }}$" is an unrelated-machines scheduling problem with no constraints, where the goal is to minimize the maximum completion time.

In some variants of the problem, instead of minimizing the maximum completion time, it is desired to minimize the average completion time (averaged over all n jobs); it is denoted by R||${\displaystyle \sum C_{i}}$. More generally, when some jobs are more important than others, it may be desired to minimize a weighted average of the completion time, where each job has a different weight. This is denoted by R||${\displaystyle \sum w_{i}C_{i}}$.

In a third variant, the goal is to maximize the minimum completion time, " R||${\displaystyle C_{\min }}$" . This variant corresponds to the problem of Egalitarian item allocation.