Total weighted completion time scheduling under resource allocation and time-dependent learning effects

This paper investigates a single-machine scheduling problem that integrates cumulative time-dependent learning effects with non-renewable resource allocation, where job processing times are controllable through resource investment. The actual processing time of each job is modeled as a nonlinear function of the allocated resource amount and the cumulative normal processing times of previously processed jobs, thereby reflecting proficiency gains in modern manufacturing environments. The objective is to minimize a weighted aggregate cost consisting of total weighted completion time and total resource consumption. By establishing convexity with respect to the resource variables, we derive an analytical optimal resource allocation policy for any fixed job sequence. This result transforms the original joint continuous-discrete optimization problem into an equivalent combinatorial sequencing problem. To solve the resulting problem, we develop an exact branch-and-bound (B&B) algorithm with a lower-bound scheme based on the monotonicity of the learning term. For larger instances, simulated annealing, tabu search, and genetic algorithms are developed to obtain high-quality feasible solutions within limited computational time. The computational results evaluate the pruning efficiency of the B&B algorithm and the stability of the proposed metaheuristics. The analysis also shows that, under strong learning effects, the classical WSPT rule is no longer universally dominant; in some cases, SPT-type sequences may be preferable. These findings provide useful insights for scheduling decisions in labor-intensive and high-precision manufacturing environments.