In the field of time-dependent scheduling, a job’s processing time is specified by a function of its start time. While monotonic processing time functions are well-known in the literature, this paper introduces non-monotonic functions with a convex, piecewise-linear V-shape similar to the absolute value function. They are minimum at an ideal start time, which is the same for all given jobs. Then, the processing time equals the job’s basic processing time. Earlier or later, it increases linearly with slopes that can be asymmetric and job-specific. The objective is to sequence the given jobs on a single machine and minimize the makespan. This is motivated by production planning of moving car assembly lines, in particular, to sequence a worker’s assembly operations such that the time-dependent walking times to gather materials from the line-side is minimized. This paper characterizes the problem’s computational complexity in several angles. NP-hardness is observed even if the two slopes are the same for all jobs. A fully polynomial time approximation scheme is devised for the more generic case of agreeable ratios of basic processing time and slopes. In the most generic case with job-specific slopes, several polynomial cases are identified.
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