Abstract
For a scheduling problem to minimize the makespan on three uniform parallel machines we present a parametric analysis of the quality of a schedule with at most one preemption compared to the global optimal schedule with any number of preemptions. A tight bound is derived as a function of the relative speeds of the machines, provided that two of the machines have the same speed.
Highlights
In parallel machine scheduling, we are given the jobs of set N = {J1, J2, . . . , Jn} and m parallel machines M1, M2, . . . , Mm
There are three main types of scheduling systems with parallel machines: (i) identical parallel machines, for which the processing times are machine-independent, i.e., pi j = p j ; (ii) uniform parallel machines, which have different speeds, so that pi j = p j /si, where si denotes the speed of machine Mi ; and (iii) unrelated parallel machines, for which the processing time of a job depends on the machine assignment
For an instance of a scheduling problem on parallel machines, let S(∗q) and S∗p denote an optimal schedule with at most q preemptions, and an optimal preemptive schedule which uses an unlimited number of preemptions, respectively
Summary
If the machines are identical parallel, it is known that ρm(0) = 2 − 2/ (m + 1), as independently proved by Braun and Schmidt (2003) and Lee and Strusevich (2005) It is shown in Rustogi and Strusevich (2013) that the value of ρm(0) can be reduced for some instances that contain jobs with fairly large processing times. The single-parameter analysis performed in Soper and Strusevich (2014a) and in this paper for three uniform machines requires considerable technical efforts, which give an estimate of a possible difficulty of extending a similar analysis to a larger number of machines The latter can be seen as a goal for further research. A tight bound on the quality of a schedule with a single preemption is derived as a function of the relative speeds of the machines in Sects. 3 and 4
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