Uniform-machines scheduling

Uniform machine scheduling (also called uniformly-related machine scheduling or related machine scheduling) is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. We are given n jobs J₁, J₂, ..., J_n of varying processing times, which need to be scheduled on m different machines. The goal is to minimize the makespan - the total time required to execute the schedule. The time that machine i needs in order to process job j is denoted by p_i,j. In the general case, the times p_i,j are unrelated, and any matrix of positive processing times is possible. In the specific variant called uniform machine scheduling, some machines are uniformly faster than others. This means that, for each machine i, there is a speed factor s_i, and the run-time of job j on machine i is p_i,j = p_j / s_i.

In the standard three-field notation for optimal job scheduling problems, the uniform-machine variant is denoted by Q in the first field. For example, the problem denoted by " Q||${\displaystyle C_{\max }}$" is a uniform machine scheduling problem with no constraints, where the goal is to minimize the maximum completion time. A special case of uniform machine scheduling is identical-machines scheduling, in which all machines have the same speed. This variant is denoted by P in the first field.

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 Q||${\displaystyle \sum C_{i}}$. 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 Q||${\displaystyle \sum w_{i}C_{i}}$.