Parallel-machine scheduling with identical machine resource capacity limits and DeJong’s learning effect

We consider parallel-machine scheduling with identical machine resource capacity limits and DeJong’s learning effect. Each job has a resource consumption requirement and a normal processing time. The actual processing time of a job is a function of its normal processing time, subject to DeJong’s learning effect, while the resource consumption of a job is a function of its actual processing time. Each machine has the same resource capacity limit. The objective is to maximise the minimum machine load. Considering three resource consumption functions, namely, linear, concave, and convex, we show that all three scheduling models are NP-hard and propose two approximation algorithms for the models and analyse their worst-case ratios.