Single Machine Scheduling with Nonmonotonic Piecewise Linear Time Dependent Processing Times

In this work, we introduce a single machine scheduling problem, in which the processing time of a job increases when its start time deviates from a given job dependent ideal time. This problem differs from similar problems found in literature by the fact that most time dependent problems only allow monotonic change in processing time. Its application lies in scheduling of a single worker who successively picks up components and takes them to a moving conveyor belt, for assembling them into a product. We solved the problem by developing a branch and bound procedure. Consider a conveyor line which moves a product with a constant speed along a straight line. The assembly consists of several independent jobs that need to be scheduled. We focus on a single worker only. In each of his or her jobs, he or she first picks up a component, then builds it into the product. Every part has its own fixed storage location along the conveyor line. Thus, the worker needs to walk from the product to the storage location and back. As this distance depends on the current product position, it is time dependent. Minimum distance occurs at the point in time when the product passes the storage location, to which we will refer as the ideal time. As the distance increases both before and after the ideal time, it is a nonlinear function of time. We assume a constant walking speed that is faster than the conveyor speed. Then, the walking time equals distance divided by speed. This walking time is added to the assembly time of a job, thus the processing time is nonlinearly time dependent. Scheduling with time dependent processing times has increasingly been studied in the last two decades. Often considered are learning or deterioration effects, sometimes both, e.g. Biskup (2008), Janiak, Krysiak & Trela (2011), as well as controllable processing times with influence on the cost (Shabtay & Steiner 2007). Monotonous functions of time, either increasing or decreasing, are most common (Gawiejnowicz 2008, p. 51), especially linear or piecewise linear cases (Alidaee & Womer 1999, Cheng, Ding & Lin 2004). What has been studied to a lesser extent are convex functions that are both decreasing and increasing over time. These could be used to model our application, because the walking time is minimum at the ideal time, and increases when deviating in both directions. One of the simplest objective functions in scheduling is to minimize the completion time of all jobs, namely Cmax. In our application, this leads to a maximization of the number of products assembled per time frame. Therefore, minimizing Cmax matches the application case.