## Abstract

This work is inspired by the problem of planning sequences of operations, as welding, in car manufacturing stations where multiple industrial robots cooperate. The goal is to minimize the station cycle time, i.e., the time it takes for the last robot to finish its cycle. This is done by dispatching the tasks among the robots, and by routing and scheduling the robots in a collision-free way, such that they perform all predefined tasks. We propose an iterative and decoupled approach in order to cope with the high complexity of the problem. First, collisions among robots are neglected, leading to a min-max Multiple Generalized Traveling Salesman Problem (MGTSP). Then, when the sets of robot loads have been obtained and fixed, we sequence and schedule their tasks, with the aim to avoid conflicts. The first problem (min-max MGTSP) is solved by an exact branch and bound (B&B) method, where different lower bounds are presented by combining the solutions of a min-max set partitioning problem and of a Generalized Traveling Salesman Problem (GTSP). The second problem is approached by assuming that robots move synchronously: a novel transformation of this synchronous problem into a GTSP is presented. Eventually, in order to provide complete robot solutions, we include path planning functionalities, allowing the robots to avoid collisions with the static environment and among themselves. These steps are iterated until a satisfying solution is obtained. Experimental results are shown for both problems and for their combination. We even show the results of the iterative method, applied to an industrial test case adapted from a stud welding station in a car manufacturing line. This paper is motivated by the problem of planning robot operations in welding applications in the automotive industry. Here, a number of welding tasks have been introduced along the car body: the goal is to let the robots perform such tasks while minimizing the cycle time (or makespan). The main diffic- lties, from the manufacturing engineer perspective, lie in assigning the tasks to the robots, deciding the order and the timing of the operations, avoiding collisions between the robots and the environment, and among the robots themselves. We present in this work an iterative approach, consisting of two steps: first, sequences for the robot operations are computed in order to minimize the cycle time, while neglecting collisions among robots; then, given the assignment of tasks to robots, the operations are reordered and scheduled while avoiding conflicts among robots. Robot motions are also automatically computed to avoid collisions with the static environment. We show an optimal algorithm, for the first part, based on implicit enumeration (B&B) and introduce a novel suboptimal algorithm, for the second part, to synchronize the robots. These algorithms are iterated while fetching information about the problem that are hard to compute, thus following a lazy approach. Tests on problems adapted from the literature and from the automotive industry, show clear improvements over more sequential approaches and good running times. The algorithms are especially suited for cases with up to 40 tasks and 4 robots, as for typical geometry stations. In future works, we will further investigate efficient heuristic optimization approaches in order to handle larger problems, consisting of more than 100 tasks, as for typical assembly stations.